Volker Schweikhard- Ultracold Atoms in a Far Detuned Optical Lattice

Ultracold Atoms in a Far Detuned

Optical Lattice

by

Volker Schweikhard

Diplomarbeit

November 2001

Carried out at 5. Physikalisches Institut

Universitat Stuttgart

Under the Supervision of Prof. Dr. Tilman Pfau

2

Contents

1 Introduction and Motivation 7

2 Action of Light on Matter 13

2.1 The Spontaneous Force: Optical Molasses . . . . . . . . . . . . . . . . . . 13

2.2 Magneto-Optical Trapping of Atoms . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Doppler Theory of Forces in the MOT . . . . . . . . . . . . . . . . 15

2.2.2 An Elongated Three Dimensional Trap Configuration . . . . . . . . 16

2.2.3 MOT Temperature and Density . . . . . . . . . . . . . . . . . . . . 18

2.2.4 Loading and Decay Dynamics of the 3D-MOT . . . . . . . . . . . . 22

2.3 A Cold Atomic Beam from a 2D-MOT . . . . . . . . . . . . . . . . . . . . 23

2.3.1 Operation Principle of the 2D-MOT . . . . . . . . . . . . . . . . . . 23

2.3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Optical Dipole Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Optical Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5.1 Time Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5.2 Parametric Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 The Rubidium BEC Experiment 33

3.1 Laser-Cooling the Element Rubidium . . . . . . . . . . . . . . . . . . . . . 34

3.2 Description of the Experimental Setup . . . . . . . . . . . . . . . . . . . . 35

3.2.1 Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.2 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.3 2D-MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.4 3D-MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Preparation of a Large Number of Ultracold Atoms 41

4.1 Two-Dimensional MOT as a Bright Source of Cold Atoms . . . . . . . . . 41

4.1.1 Time of Flight Measurements . . . . . . . . . . . . . . . . . . . . . 42

4.1.2 Transverse Velocity Distribution in the 2D-MOT . . . . . . . . . . 45

4.1.3 Measured Beam Divergence . . . . . . . . . . . . . . . . . . . . . . 46

3

4 CONTENTS

4.1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Characterization of a Large Elongated 3D-MOT . . . . . . . . . . . . . . . 48

4.2.1 Loading of Atoms: Interplay between 2D- and 3D-MOT . . . . . . 49

4.2.2 Analysis of the Decay of the Trapped Atom Number . . . . . . . . 52

4.2.3 Trapped Atom Number and Density . . . . . . . . . . . . . . . . . 54

4.2.4 Influence of Trapping Laser Detuning . . . . . . . . . . . . . . . . . 56

4.2.5 Velocimetry of Cold Atoms . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.6 Temperature Measurements . . . . . . . . . . . . . . . . . . . . . . 60

4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5 The Mott Insulator Phase Transition in an Optical Lattice 67

5.1 Tunnelling of Interacting Atoms in an Optical Lattice . . . . . . . . . . . . 70

5.2 Bloch Oscillations of a Phase Fluctuating BEC . . . . . . . . . . . . . . . . 75

6 Design of an Optical Lattice and Experimental Preparations 79

6.1 Numerical Values for Time and Energy Scales . . . . . . . . . . . . . . . . 79

6.2 Accelerating the Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.3 A 3-Dimensional Far Detuned Optical Lattice . . . . . . . . . . . . . . . . 84

6.4 Characterization of the Verdi V10 Laser . . . . . . . . . . . . . . . . . . . 89

6.4.1 Gaussian Beam Parameters . . . . . . . . . . . . . . . . . . . . . . 89

6.4.2 Spectral Measurement of Amplitude Noise . . . . . . . . . . . . . . 90

6.4.3 Frequency Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.4.4 Proposal of a Stabilization Scheme for the Verdi Laser . . . . . . . 95

7 Conclusion and Outlook 97

A Capture Velocity of the 3D-MOT 99

B Polarization Spectroscopy for Laser Stabilization 101

C Time of Flight Temperature Measurement 105

D Spectral Noise Measurements 107

E Bloch Oscillations 109

F Latest 2D-MOT Model Results 111

G List of Developed Mathematica Programs 113

CONTENTS 5

Bibliography 114

6 CONTENTS

Chapter 1

Introduction and Motivation

This work was carried out at the Rubidium Bose-Einstein condensation (BEC) experi-

ment at the University of Stuttgart, which has the objective to prepare BEC with a large

number of atoms at a high repetition rate in an apparatus with good optical access. This

will enable new types of experiments, including the study of atom-atom interactions in

optical lattices, of magnetic transport of Bose-Einstein condensates, and the exploration

of novel regimes of nonlinear optics and atom optics.

This work consists of two parts: The first is the preparation of a large sample of

ultracold 87Rb atoms in a magneto-optical trap (MOT), which will serve as a starting

point for evaporatively cooling these atoms to the Bose-Einstein condensation (BEC)

transition at a phase space density of ρ = nΛ3 = 2.612, where n is the number density of

the atomic sample and Λ =√

2π2

mkBTis the thermal deBroglie wavelength of an atom of

mass m at temperature T . The characterization of this source of trapped cold atoms is

described in chapter 4, after an introduction to light forces in chapter 2 and a description

of the experimental setup in chapter 3.

The starting point of the cooling cycle is a dilute 87Rb vapor in a glass cell, slightly above

room temperature, at a vapor pressure of 1 · 10−6mbar, with a phase space density of

10−16. This vapor is transversely cooled in a 9 cm long two-dimensional MOT [1, 2],

which efficiently produces a highly collimated beam of cold atoms with a flux of above

6 · 1010 atoms/s, directed into an ultrahigh vacuum (UHV) chamber.

As many as 8 · 1010 atoms from this beam are collected in a large volume elongated

three-dimensional MOT in an UHV chamber, which represents an 80fold increase

compared to a state-of-the-art Rubidium MOT [3]. The loading rate is above 2 · 1010

atoms per second, more than 100 times higher than in [3]. The density in the MOT is

approximately 2 · 1010/cm3, a typical value for large MOT’s. The MOT temperature of

a few millikelvin can be reduced within 5 ms, without significant loss of atoms, to 300

µK by switching the magnetic fields off to create an optical molasses. At this point the

7

8 CH. 1 Introduction and Motivation

atomic sample has an estimated phase space density of ρ = 1 · 10−7, a factor of 109 higher

than the dilute vapor in the glass cell, and only a factor of 3 · 107 below the Bose-Einstein

condensation threshold of ρ = 2.612.

The final cooling stage to BEC, forced evaporative cooling in a magnetic trap (MT),

continuously removes the hottest atoms from the MT, while elastic collisions re-thermalize

the trapped sample to a lower temperature and higher density. Typically a factor 103 to

104 in atom number is lost before reaching the BEC transition.

Therefore the large number of ultracold atoms prepared in this diploma thesis represents

an ideal starting point for this technique: Condensates of 107 atoms may be produced,

a factor of 10 more than in [3], providing significantly better signal-to-noise ratio in

measurements. On the other hand it may be possible to increase the speed of evaporative

cooling such that the production cycle of one BEC takes substantially less than a minute,

which would allow for experiments which require a large number of data points.

The second focus of this work is a three-dimensional far-detuned optical lattice

with large potential depth1, which is suitable to study atom transport under the influence

of an external force. An external force can be the inertial force on the atoms when the

lattice is accelerated, by frequency-chirping two acousto-optical modulators ( AOM’s ).

Numerical calculations were performed to find experimental signatures of a theoretically

predicted quantum phase transition of Bosons in an optical lattice, from a superfluid

phase at low lattice potential depths to a Mott insulator phase in deep periodic potentials.

These calculations will be described in chapter 5.

In chapter 6, accelerateable lattice schemes are examined, which make most efficient use

of the available laser power. The beam shape and noise properties of the proposed lattice

laser, a 10W Nd : Y V O4 VERDI laser at a wavelength of 532nm, are characterized, as

well as other components for the lattice, such as two AOM’s with high efficiency over a

large bandwidth, employed to accelerate the lattice.

Light Forces and Magneto-Optical Trapping

It was understood in the 1970s that light forces could accelerate and trap [4], as well as

cool [5] neutral atoms. The realization of these ideas, however, began only in the 1980s,

with the laser deceleration of an atomic beam in 1982 [6], and a remarkable series of exper-

iments between 1985 and 1987, where atoms were laser cooled and and viscously confined

( but not trapped ) in a light field ( optical molasses, 1985 ) [7], purely magnetically

trapped ( 1985 ) [8], confined in an optical dipole trap ( 1986 ) [9], and magneto-optically

trapped in 1987 [10], following a general concept proposed in 1986 [11].

1Our interest in optical lattices will be motivated in the following sections.

9

The magneto-optical trap (MOT) offers a much deeper trapping potential than the mag-

netic or dipole trap, and has been the subject of extensive theoretical and experimental

study, especially in the early 1990’s [12, 13, 14, 15], where fundamental limitations to

reachable temperatures and densities were explored, and found to prevent further progress

toward the goal of achieving Bose-Einstein condensation (BEC).

Improvements in the trapping technique, such as the dark spot MOT [16, 17], the com-

pressed MOT [18], and a large volume MOT that could trap 3 ·1010 Cs atoms [19], allowed

to obtain larger numbers of trapped atoms which were still dense and cold. This made

the MOT a good starting point for evaporative cooling in a purely magnetic trap [20],

leading to the achievement of BEC in 1995 [21], which has since then become one of the

driving fields of atomic physics [22, 23].

The accumulation of a high number of atoms under ultrahigh vacuum (UHV) conditions,

as required for the achievement of BEC, also stimulated the development of efficient

sources for cold atomic beams, which collect atoms from a low vacuum source and trans-

fer them into an UHV environment. The most commonly used sources are a Zeeman

slower [6] and a double MOT system [24], but recently alternative concepts, such as a

two-dimensional MOT [25], or a low velocity intense source (LVIS), [26] have been devel-

oped.

Optical Lattices

Not long after the basic work on laser cooling and trapping was done, optical lattices,

periodic light field configurations made from interfering laser beams, started to attract

interest. In the years from 1992 to 1994, lattices were studied, in which the laser

frequency was set close to an atomic resonance. In these near-resonant lattices, trapping

due to the dipole force and cooling due to additional spontaneous processes, take place

simultaneously. For an extensive review, see [27]. In order to observe effects of a coherent

time evolution of atoms in an optical lattice, the detuning δ of laser frequency ω from

atomic resonance at ωat and the intensity I have to be increased. This lowers the photon

scattering rate Γsc ∝ Iδ2, whereas the dipole potential depth Udip ∝ I

δshows a weaker

detuning dependence. If powerful, tightly focused far detuned lasers are used, deep

potentials with negligible scattering result.

The interest in far-detuned optical lattices is manifold. The first association of

the word ”lattice” in physics is the ordered structure of atoms in a solid state crystal. An

analogy between neutral atoms moving in an array of light-induced potential wells with

electrons in the periodic potential of a solid can be found, and has been exploited to view

fundamental concepts of solid state physics in ”a new light”. An example is the study

of quantum transport, namely the observations of Bloch oscillations [28], Wannier-Stark

10 CH. 1 Introduction and Motivation

resonances and Landau-Zener tunnelling ( for a review of these discoveries, see [29] ).

Optical lattices offer some attractive features not found in solid state physics. Above all,

they are free of defects, allowing for ”clean” measurements. The shape of the potential

is exactly known and is tunable in strength ( by laser intensity ), periodicity ( by

choosing appropriate laser wavelengths ), and structure ( by laser beam geometry and

polarizations ) [30]. This allows for continuous tuning of oscillation frequencies of atoms

in individual potential wells, and tunnelling rates between neighboring wells. Optical

lattices can be loaded with Bosonic or Fermionic atoms, and the population of energy

bands can be tailored ”at will”. As an example, the selective population of only a small

fraction of the first Brillouin zone has been achieved, allowing a clean observation of

Bloch oscillations [28]. Further improvement in experimental techniques has recently

allowed the observation of interesting phenomena as the quantum Zeno and Anti-Zeno

effect [31], and aspects of quantum chaos [32, 33].

In the field of atom optics - the manipulation of atomic motion with light - opti-

cal lattices or standing waves were utilized to diffract atoms - both for fundamental

studies ( see for example [34] ) and for applications with technological significance as

atom lithography [35]. Furthermore, the quantization of atomic motion in an optical

lattice can be exploited for cooling, making optical lattices a tool for the creation of high

brightness atomic sources [36, 37].

Quantum computing is another issue that immediately comes to mind when one

thinks of an array of isolated atoms under perfect control by purely optical means. An

optical lattice loaded with atoms could present a highly parallel assembly of qubits, and

detailed suggestions have already been made how to implement quantum logic operations

in an optical lattice [38, 39]. The problem of addressing single lattice sites remains to be

solved, however.

The extremely good control over atomic motion, and the good isolation from their

environment ( at low densities including also isolation from other atoms of the same

species ), combined with long storage and coherence times, make optical lattices an ideal

system for precision measurements. One recent example is the proposal by Steve Chu et.

al. [40] to measure a possible electron dipole moment a hundred times more accurately

than in present atomic beam experiments.

Bose-Einstein Condensates in Optical Lattices

Bose-Einstein condensates represent a macroscopic phase coherent quantum state of

matter. The exploration of the interaction of coherent matter with coherent light

11

fields has only begun, but already produced a variety of interesting observations,

such as the coherent amplification of matter waves [41]. In optical lattice light fields,

number-squeezed matter states [42] and an atomic analog to the Josephson effect [43]

have been demonstrated.

If a BEC at a typical density of 1014 /cm3 to 1015 /cm3 is loaded into an optical

lattice with potential well volumes of the order of(λ2

)3 ≈ 10−14 cm3, an occupation of

about one atom per lattice site results, and strong short-range interatomic interactions

are possible in a flexible and well-controlled environment. In typical non-condensed

atomic samples, on the other hand, densities do not exceed 1011 /cm3. At the resulting

low occupation of lattice wells, atomic interactions can be completely ignored.

The behavior of interacting ”Cold Bosonic Atoms in Optical Lattices” was theoretically

explored in a publication in 1998 [44]. The authors predict a zero temperature quantum

phase transition of the free superfluid Boson gas, to a Mott insulator phase, with

increasing lattice potential depth, whereby the site-to-site tunnelling rate is decreased,

but short-range interatomic interactions are enhanced. This behavior will be described

in Chapter 5.

The first experimental transfer of a BEC into a one dimensional optical lattice

[45] in the same year, shortly after the first demonstration of BEC confinement in an

optical dipole trap made of a single focused laser beam [46], created a new boost for

experiments with optical lattices. Bose-Einstein condensates in one- and two-dimensional

optical lattices have been achieved by several other groups during the last months

[43, 47, 48], but so far no transfer into a three-dimensional optical lattice has been

reported2. Reference [48] shows beautiful measurements of the momentum distribution

of a BEC in an optical lattice, which corresponds to the reciprocal lattice points, as long

as the condensate undergoes a coherent evolution, but fills out the first Brillouin zone

homogeneously when the coherence is willingly destroyed.

Optical lattices are a quickly developing research field, and interesting new devel-

opments are to be expected.

2While this thesis is completed, a paper is being submitted for publication by the Munichgroup, reporting on the observation of the superfluid - Mott insulator phase transition ina three-dimensional optical lattice [49]. The measurements clearly demonstrate the lossof phase-coherence between different lattice sites in the insulating phase. Coherence isshown to reappear reversibly after returning to the superfluid phase, in contrast to a statein which the phases between lattice sites have been willingly dephased. The existence ofan energy gap and resonances in the excitation spectrum of the insulating phase is shown,in contrast to the gapless excitation spectrum of the superfluid phase.

Chapter 2

Action of Light on Matter

The first part of this chapter deals with the forces asserted on atoms by light fields tuned

close to an atomic resonance ( δ = ωl − ωat ≈ Γ, where ωl is the laser frequency, ωat is

an atomic resonance frequency, and Γ is the natural linewidth of the atomic transition ).

It is shown how these dissipative forces can cool and trap atoms, and produce atomic

beams. Here, these techniques will be described separately. In chapters 3 and 4, it will

be shown how they are combined to produce a large sample of ultracold atoms, in a

system consisting of a 2D-MOT source of an atomic beam and a 3D-MOT for trapping

and cooling these atoms.

The second part of this chapter deals with the trapping of atoms by the conservative

forces produced by intense light fields detuned far away from an atomic resonance (δ Γ).

A light field is characterized by three quantities, which can be position-dependent:

its amplitude, phase, and polarization1. Gradients of all three quantities can cause forces

on atoms moving in the light field. Phase gradients are the cause of the spontaneous

force or scattering force described below, intensity and polarization gradients give rise

to the (conservative) dipole force described in section 2.4, which is employed in optical

lattices to trap atoms. Together with spontaneous processes, polarization and intensity

gradients can provide a type of laser cooling not described here [50].

2.1 The Spontaneous Force: Optical Molasses

Consider a laser beam of intensity I, wavevector k = 2πλ

and frequency ωl = ck incident

on an atom at rest, with a resonance transition at a frequency ωat of natural linewidth Γ

and a saturation intensity Isat,

1The frequency of the light field is ignored, as it is not position-dependent in the experi-ments described here

13

14 CH. 2 Action of Light on Matter

Isat =πhcΓ

3λ3(2.1)

It is convenient to define a saturation parameter s, as the ratio of incident intensity to

saturation intensity:

s =I

Isat(2.2)

The scattering of photons from the laser beam exerts a force on the atom, whose strength

is given by the product of the momentum k, which an atom receives when absorbing a

photon, and the scattering rate Γ2· s

1+s+( 2δΓ

)2, resulting in the spontaneous, or scattering

force

Fsp =kΓ

2· s

1 + s+ (2δΓ)2

(2.3)

Again, δ = ωl − ωat denotes the detuning between laser frequency and atomic resonance.

In the following, it will be shown how moving atoms can be cooled by making use of

radiation pressure imbalance brought about by the Doppler effect. We consider an atom

moving with velocity v in the light field of two counterpropagating beams ( wave vectorsk± ) of equal frequency, each causing a force

F± =k±Γ2

· s

1 + s+ (2(δ∓k·v)Γ

)2(2.4)

If the frequency ωl is red-detuned with respect to the atomic resonance transition (i.e.

δ < 0), the Doppler effect will shift the laser beam closer to resonance towards which

the atom moves, and will shift the opposite beam further out of resonance. In this way

a radiation pressure imbalance is produced which tends to decelerate the atom to zero

velocity. The total force F = F+ + F− can be approximated for |k · v| Γ, as

F =8k2δ · s · v

Γ(1 + s+ (2δΓ)2)2

= −βv (2.5)

This viscous force with friction coefficient β dampens the atomic motion and cools the

atoms. This ”optical molasses” can easily be extended to three dimensions. The range of

initial atomic velocities that can be cooled (capture range) is determined by |k · v| < δ.

Only for these velocities the laser light is red-detuned and atoms lose kinetic energy in

the absorption and re-emission cycle. The force is of spontaneous nature and therefore

exhibits saturation when s exceeds one. As a function of detuning, the force strength first

increases to a maximum at δ ≈ −Γ, and then decreases ∝ δ−3.

If momentum could be transferred from the light field to the atom and back in infinites-

imally small amounts, cooling could approach the absolute zero temperature. As the

2.2 Magneto-Optical Trapping of Atoms 15

photon momentum is discrete, we have to take in account the re-emission of photons by

the excited atoms, which has so far been left out of the discussion. This process leads

to a momentum transfer to the atom into a random direction, and sets a limit to cool-

ing. Equating the heating rate due to this ”momentum diffusion” with the cooling rate,

leads to an equilibrium temperature, which depends on the trapping laser detuning. The

minimum equilibrium temperature, the Doppler temperature TD,

kBTD =Γ

2(2.6)

is reached at a detuning of δ = −Γ2, where Γ again denotes the natural linewidth of

the transition employed for cooling. For larger detunings, the equilibrium temperature

increases slightly, but stays on the order of TD for the full range of detunings used in

practice. For 87Rb, TD = 143.6µK.

2.2 Magneto-Optical Trapping of Atoms

2.2.1 Doppler Theory of Forces in the MOT

The force described so far cools the atoms in velocity space, but does not provide spatial

confinement. A spatial dependence of the force in the magneto-optical trap (MOT) is

achieved by adding an inhomogeneous magnetic field to the optical molasses described

above. Consider as a one-dimensional example the Zeeman effect of an atom with a

J = 0 ground state and a J = 1 excited state in a linearly increasing magnetic field

B(x) = ∇B · x, as shown in figure 2.1. The transition used to trap 87Rb is a F = 2 −→F = 3 hyperfine transition (F : total electronic plus nuclear angular momentum), but the

principles of the model described here apply in general to transitions F −→ F ′ = F + 1

[51].

If an atom moves to the right, out of the center of the trap, the Zeeman effect shifts the

∆mJ = −1 transition closer to resonance with the counterpropagating red-detuned laser

beam, which can drive this transition if it is left circularly polarized (σ−). An analogous

argument applies to motion to the left. The resulting force F = F+ + F− is

F± =k±Γ2

· s

1 + s+ (2(δ∓k·v±µ′B(x)

)

Γ)2

(2.7)

Provided |k · v| |δ| and |µ′B(x)

| |δ|, one obtains the approximation

F = −βv − κx (2.8)

with a friction constant β given by equation (2.5), and the spring constant κ = µ′∇Bk

β.

This is the force of a damped harmonic oscillator, and accordingly the motion of captured


x

m = 0J

m ' = 0J

m ' = -1J

m ' = 1J

L L

rC-rC

Figure 2.1: Toy model for magneto-optical trapping, based on a J=0 −→ J=1 transition.

The excited state exhibits Zeeman splitting in a linear magnetic field B(x) = ∇B · x.The capture volume is bounded by rc, where the Zeeman splitting exceeds the detuning

|δ| > |µ′B(x)

|.

atoms is a damped oscillation towards the trap center. The relevant timescale for this

motion is the ratio [10] βκ≈ 2.5ms for the parameters of our 3D-MOT.

To form a MOT, usually six circularly polarized laser beams of a few millimeters to cen-

timeters diameter are overlapped with the center of a spherical quadrupole field produced

by two circular coils in anti-Helmholtz configuration. This results in a linearly increasing

magnetic field strength in all directions from the trap center. The field gradients obey∇ · B = 0, or ∂zBz = −2∂rBr, which results in a Gaussian-shaped trapped atom cloud

with a near-spherical aspect ratio of√2 between the radial and z-axis.

2.2.2 An Elongated Three Dimensional Trap Configuration

The trap configuration used here ( figure 2.2 ), differs from this standard configuration:

A radial quadrupole field is produced by four long straight bars with alternating current

direction (Ioffe bars). The boundaries of the trapping volume are calculated in Appendix

A: the diameter is between 1 and 1.5 cm, depending on the radial gradient ∇B.

Axial confinement is produced by two ”pinch” coils with currents flowing in anti-Helmholtz

sense, but the distance between the coils is much larger than their radius. In this way

an axially elongated trap is produced. The magnetic field along the z-axis has a nearly

vanishing axial gradient in the center and increases non-linearly from the center, limiting

the trapping volume at about +/-4 cm. In figure 2.3, the values of the magnetic field

components along radial (left) and axial (right) lines through the trap center are shown.

Two orthogonal pairs of counterpropagating elliptical laser beams are shone in perpendic-

ularly to the axis of the trap, with sizes of 10 cm length by 2 cm height. On the axis a pair


Pinch Coils

Ioffe Bars

x

z

y

Bias Coils

Figure 2.2: Magnetic coil configuration of an elongated 3D-MOT, and polarizations of the

respective laser beams. The black arrows give the current directions. The bias coils are

not used in the MOT, but are necessary for a tightly confining magnetic trap.

-100 -80 -60 -40 -20 0 20 40 60 80 100-30

-20

-10

0

10

20

30

R[mm]

B[G

]

B

B

B

B[G

]

-100 -80 -60 -40 -20 0 20 40 60 80 100-100

-50

0

50

100

150

Z[mm]

B

B

x

y,z

z

x,y

Figure 2.3: Computed values of the magnetic field components along lines through the trap

center in radial (left) and axial (right) direction for the experimentally applied currents

of 24A in the pinch coils, and 40A in the Ioffe bars. z denotes the trap symmetry axis.

The radial gradient has a value of 14G/cm.

of 12mm diameter counterpropagating beams is used. The polarizations are all circular,

as shown in figure 2.2. Together with the magnetic field a cylindrical trap volume of

above 1 cm diameter and 8 cm length results. This elongated configuration with a large

trapping volume offers two distinct advantages over a conventional setup:

First, the density in a MOT is limited to about 1011 atoms per cubic centimeter by pro-

cesses described in section 2.2.3, which requires a large trapping volume if large numbers


of atoms are to be trapped.

Second, loading with an atomic beam is facilitated by the elongated trap. We use an

atomic beam that enters the trap under an angle of 27 (determined by technical consid-

erations) to its long axis, resulting in a larger stopping distance for atoms, compared to a

standard MOT. Thus atoms with a larger velocity ( about 65m/s, compared to a value

of 35m/s for a typical MOT ) can be captured, allowing for a fast loading of large atom

numbers. A calculation of the capture velocity is given in appendix A.

2.2.3 MOT Temperature and Density

The simple Doppler-theory description of the MOT given above, is not sufficient for a

quantitative understanding of the MOT parameters temperature, density, and trapped

atom number. Atom-atom interactions arising from multiple scattering of photons play

an important role, as well as the attenuation of the laser beams when the cloud becomes

optically thick. A complete understanding of these complicated many-body effects has

not been reached, but empirical scaling laws for the defining parameters of the trap could

be found. Different regimes of trapping can be distinguished, as described below [12].

Furthermore, in the MOT as well as in optical molasses, cooling beyond the Doppler limit

(2.6) can take place [52]. This will not be described here, as no evidence for sub-Doppler

effects was found in our MOT, for reasons described below.

The Temperature Limited Regime

At low atom numbers ( below about 104 ) the trap is in the so-called temperature-limited

regime, where atom-atom interactions can be neglected. The temperature of the trapped

atoms is only dependent on intensity I and detuning δ of the trapping laser [12]:

T = C0 + C1TD · I

Isat· Γδ

(2.9)

The value C0 is a few times the recoil temperature Trec ( cf. equation (2.36) ), and

C1 ≈ 0.25 typically. The shape of the atom cloud in this regime is Gaussian, the radii

determined by temperature and the spring constants κii of the trap, following

kBT =1

2κiir

2i (2.10)

As the volume in this regime is independent of atom number, the density n increases

linearly with atom number N.


Multiple Scattering

For larger atom numbers ( on the order of 105 to 108 ) the trapped atom cloud expands

and heats up due to an outward pointing radiation pressure. This effective repulsive force

between atoms comes about when a photon from the trap laser is absorbed by a first atom,

and after re-emission is subsequently absorbed by a second atom. This process results in

a relative momentum of 2k between the atoms. As the probability of reabsorption falls

off as r−2, an inverse square law for the force results. This repulsive force increases with

density and must be balanced by the trapping forces. This limits the atomic density to

n ≈ 1011 atoms/cm3. Upon further addition of atoms the cloud expands, resulting in a

number-independent homogenous density distribution across the cloud.

The same process also results in heating: The random momentum transfer to the atoms

by multiple scattering is called momentum diffusion and leads to an enhanced kinetic

energy. The MOT temperature is empirically found to scale with atom number as

T ∝ N13 (2.11)

Optically Thick MOTs

When the column density n · l of the MOT, where l is a characteristic size of the trapped

atom cloud, becomes too large, the near-resonant trapping light is attenuated:

I = I0 · e−nσabs(δ)·l (2.12)

with the detuning-dependent absorption cross section

σabs(δ) =σ0

1 +(

2δΓ

)2 (2.13)

For the 87Rb cooling transition, σ0 = 3λ2

2πis the resonant absorption cross section2. If the

beams are only partially attenuated, the resulting radiation pressure imbalance between

each pair of counterpropagating beams causes an inward pointing force, in addition to the

force described in section 2.2.1.

If, on the other hand, for still higher atom numbers the cloud becomes optically thick for

the trapping light, both counterpropagating beams are fully attenuated in the trap center,

severely decreasing the trapping potential for the inner parts of the cloud and causing it

to expand further: Following [16] one demands the column density not to exceed

2This result is exact whenever the atom can be approximated as a two-level system [51].The 87Rb D2 line used for cooling ( cf. section 3.1 ) contains multiple Zeeman levels butcan, due to optical pumping in the circularly polarized MOT light, be approximated as atwo-level transition.


n · l ≤ b0, b0 = 5 · 109cm−2 (2.14)

which corresponds to a resonant attenuation of e−20 in (2.12), on the order of magnitude

found experimentally in our MOT 3. Assuming a spherical MOT one has n = N/V ∼ N/l3,

and to reach the upper limit of (2.14), it has to hold that l ∼ √N . Therefore in this

regime the density decreases with increasing atom number according to

n ∼ N−1/2 (2.15)

In summary, the density in the different regimes of trapping is plotted in figure 2.4, where

the values of the proportionality constants in equation 2.15, and between density and

number in the temperature-limited regime are taken from [16].

104

105

106

107

108

109

1010

1011

108

109

1010

1011

Experiment

Temperature Limited RegimeMultiple Photon ScatteringOptically Thick Spherical MOTOptically Thick Elongated MOT

Densi

ty[c

m-3]

Trapped Atom Number

Figure 2.4: A rough estimate of density limits for a MOT. In the temperature limited

regime, n grows linearly with atom number. In the multiple scattering regime, n is con-

stant. When the atom cloud becomes optically thick to the trapping light, n decreases,

following (2.15). In this regime, the density limit can be increased by an elongated trap-

ping volume, according to (2.16). Here, an aspect ratio a = 6 is chosen, as observed in

our MOT. The experimental result will be discussed in section 4.2.3.

For trapping a large number of atoms an ever larger trapping volume is necessary unless

further improvements in the trapping technique, e.g. the dark spot MOT [16, 17], are

3The MOT light is detuned by δ ≈ −2Γ, which leads, according to equation (2.13), toσabs ≈ σ0

17 , i.e. to a weaker attenuation of only ≈ e−1.


utilized to increase the density.

Cold Collisions

A further density limiting factor in the MOT are cold collisions, which can eject atoms

from the trap by converting internal into kinetic energy [53],[54]. They can be divided

into light-assisted collisions, in which at least one atom in an excited state is involved,

and collisions between atoms in different hyperfine ground-states ( see section 3.1 ). Loss

rates depend on density and the amount of excitation by light, as well as on the trap

depth. Light-assisted collisions involve the large fraction of atoms in the MOT in the

excited state, and are subdivided in two processes: If an atom absorbs a photon while

being far away from other atoms, and re-emits it after moving into the region of attractive

interaction with another atom, the energy of the re-emitted photon can be substantially

smaller than the energy absorbed. In another process, the fine structure state of an

excited atom can change in a collision. In both cases the energy difference is converted

into kinetic energy.

For shallow traps ( operated at low trapping light intensities, large detunings, cf. equation

2.8 ) also the second category, collisions between ground state atoms, provides enough

energy to eject atoms from the trap, if the hyperfine ground state of one atom is changed

(in a Rb MOT this always corresponds to an energy release as the cycling transition

involves the F = 2 ground state, which lies 6.8 GHz above the F = 1 state ).

Wether the density is limited by radiation pressure or by collisions, can be seen in the

decay curve of the trapped atom number after stopping the loading process. Only in the

second case a nonexponential decay will be observed, as will be explained in section 2.2.4.

Influence of an Elongated MOT Volume on Density

In addition to facilitating loading from an atomic beam, and providing space for a large

number of atoms, an elongated trapping volume with aspect ratio a between axial and

radial cloud dimension tends to increase the density limit in the optically thick regime,

as the attenuation of the radial beams is less severe in an elongated cloud. Instead of

n ∼ N/l3 one has n ∼ N/(al · l2), and the density limit (2.14) leads instead of (2.15) to

n ∼ a12 ·N−1/2 (2.16)

The difference in the density limit between a spherical and an elongated trap is just a

factor a12 , but for an aspect ratio of about 6 this allows to trap a 2.5 times higher number

of atoms in the same volume.


2.2.4 Loading and Decay Dynamics of the 3D-MOT

A large amount of information about a trap can be gained by observing its loading and

decay. The rate equation governing the time dependence of the atom number of a trap

loaded at a rate RL, while simultaneously suffering one-body losses at a rate ΓL and

two-body losses described by a coefficient β, is given by

dN

dt= −ΓLN +RL − β

∫n2(r)dr (2.17)

Assuming a roughly constant density over the volume V of the trapped cloud, this gives

dN

dt= −ΓLN +RL − β′N2 (2.18)

with β′ = βV. The term βN2 describes cold collisions ( cf. section 2.2.3 ), and β depends on

the intensity and detuning of the trap lasers. Its value for 87Rb under our experimental

parameters lies around β = 10−12cm3/s [54]. The loss rate ΓL includes losses due to

collisions of trapped atoms with atoms from the 2D-MOT beam ( see Chapter 3 for a

description of the experimental setup ), from the thermal beam entering through the

differential pumping tube, and collisions with the background gas in the UHV chamber.

Neglecting the term βN2 for the moment, equation 2.17 has a solution ( initial condition

N(t = 0) = 0 ) corresponding to an exponential loading curve:

N(t) =RL

ΓL(1− Exp(−ΓLt)) (2.19)

The decay of the trapped atom number after terminating the loading process (RL = 0)

follows a similar differential equation

dN

dt= −ΓDN − β′N2 (2.20)

but this time the decay constant ΓD includes only collisions with the background gas, as

usually a mechanical shutter in the differential pumping tube is closed when observing

decay curves. If the shutter stays open during the decay, and only the 2D-MOT B-field

is extinguished to terminate loading, we denote the respective loss rate by Γ′D. If the

two-body term is again neglected, the solution to equation (2.20) is a purely exponential

decay, with the steady state value N(t = ∞) = RL

ΓLof equation (2.19) as initial condition:

N(t) =RL

ΓLExp(−ΓDt) (2.21)

Note that the steady state number at t = 0, from which the decay starts, is determined by

the loading rate and the decay constant under loading conditions, ΓL, whereas the decay

rate of the trapped atom number is determined by ΓD.

In summary, the decay rates can be split up into

2.3 A Cold Atomic Beam from a 2D-MOT 23

ΓD = Γbg (2.22)

Γ′D = Γbg + Γth (2.23)

ΓL = Γbg + Γth + Γ2D−MOT (2.24)

with Γbg the loss rate due to the background gas in the UHV chamber, Γth due to the

thermal flux through the differential pumping tube, and Γ2D−MOT losses due to col-

lisions with the atoms in the untrapped high velocity tail of the 2D-MOT flux distribution.

A contribution of two-body losses to the decay is seen as a nonexponential decay

curve, usually consisting of a fast drop of the atom number at short timescales, followed

by a slower exponential decay when the density has decreased sufficiently. The solution

to equation (2.18) including two-body decay is [55]

N(t) =RL

ΓL

Exp(−ΓDt)

1 + β′RL

ΓDΓL(1− Exp(−ΓDt))

(2.25)

which will be used in section 4.2.2 to determine the two-body loss coefficient β of our

trap.

2.3 A Cold Atomic Beam from a 2D-MOT

A two-dimensional magneto-optical trap (2d-MOT) can produce a well-collimated atomic

beam with a high flux of cold atoms [1, 25]. The setup consists of a radial magnetic

quadrupole field with a zero value on the z axis, located in a vapor cell. Perpendicular

to the axis, four laser beams are shone in, providing radial cooling and trapping. In the

axial direction the atoms are free to move and no cooling takes place, which results in

two atomic beams propagating in both axial directions. One of these beams enters the

trapping region of our 3D-MOT through a differential pumping tube, which suppresses

thermal background flux to the UHV chamber.

2.3.1 Operation Principle of the 2D-MOT

Cooling atoms into the atomic beam requires a sufficient interaction time. For atoms with

low longitudinal velocities the interaction time is limited by the transverse velocity, and

the capture velocity can be determined by a one-dimensional calculation. Atoms below

this capture velocity experience a damped oscillatory motion to the axis of zero magnetic

field, are cooled and contribute to the atomic beam.

For large longitudinal velocities the interaction time is limited by the longitudinal, not


the transverse motion: these atoms traverse the length of the 2D-MOT too fast to be

sufficiently cooled in the radial dimension. Some do not hit the differential pumping tube

due to a too large position offset, others enter the tube with a too large radial velocity

to pass through it. In this limit the interaction time, and hence also the transverse

capture velocity vc, decreases with longitudinal velocity vz, as vc ∝ 1vz. This suppresses

the contribution of large longitudinal velocity classes to the atomic beam.

Only in the limit of an infinite 2D-MOT length one would expect an atomic beam with

thermal longitudinal velocity distribution, and a transverse velocity distribution close to

the Doppler limit. For finite length the longitudinal velocity distribution in the atomic

beam is non-thermal and is shifted to much lower values than a thermal distribution at

room temperature (√〈v2

z,th〉 =√

kBTm

≈ 169m/s), although no longitudinal laser cooling

is present. The effect of the finite cooling time on the transverse velocity distribution is

a substantial broadening beyond the Doppler limit.

2.3.2 Model

In order to describe the flux, the flux distribution and the mean atomic velocity of the

2D-MOT beam quantitatively, a model proposed in [26] was extended. To obtain the flux

distribution in the atomic beam, one has to take in account the following:

• The flux of atoms into the 2D-MOT beam is integrated over the cylindrical surface

of the trapping volume A = 2πrc · L (L: MOT length; rc, see figure 2.1). In this

integral, atom losses from the beam due to light assisted collisions [14] are taken

into account by an exponential factor Exp (−Γcoll(n)τ). Γcoll(n) ∝ n (n is the Rb

density in the vapor cell) is the light-assisted collision rate, and τ = lvz

is the time

an atom spends in the beam before passing the distance l from its starting point to

the UHV side.

• The Maxwell velocity distribution is integrated over all trappable radial velocities,vc(vz ,l)∫

0

2πvrdvrvr

π3/2u3Exp(

−v2

u2

), where the factor 2πvr arises from the transforma-

tion from cartesian to spherical velocity coordinates, and atoms must be additionally

weighted proportional to their radial velocity vr, to take in account the larger spa-

tial volume from which faster atoms can enter the trapping volume within a given

time. Furthermore, u =√

2kBTm

is the most probable velocity in a thermal velocity

distribution at temperature T , and v =√

2 · v2r + v2

z is the modulus of the velocity.

Following the arguments in the preceding section, the capture velocity vc(vz, l) is

modelled by vc(vz, l) =vc,0

1+ vzvz,0

. Here vc,0 ≈ 35m/s is the maximum radial capture

velocity, as determined in [2]. In our model we replace the path length l of an atom

in the 2D-MOT by its mean value L/2 (L is the full 2D-MOT length). The capture

2.3 A Cold Atomic Beam from a 2D-MOT 25

velocity falls off on a scale given by vz,0 = vc,0L

2·2rc : Above this longitudinal velocity,

the interaction time of atoms with the light field is no longer determined by the

radial motion tint,r =2 rcvc,0

, but by the longitudinal motion, tint,z =L

2 vzwhere again

a mean path length L2in the 2D-MOT has been assumed. Equating tint,r = tint,z

yields vz,0.

• The loading rate into the beam increases with 87Rb density n in the vapor cell.

At high density however, losses occur even before atoms can be captured into the

beam. This is taken in account phenomenologically by a factor 1

1+Γtrap(n)

Γout

. The rate

Γtrap ∝ n describes the loss rate in the capture process, and Γout determines the

outcoupling rate from the 2D-MOT trapping region into the atomic beam.

Thereby the flux distribution is given by (C = 16√πcollects all constants)

Φ(vz, n) =n

1 + Γtrap(n)

Γout

CA

L

0∫−L

dl

Exp

(−Γcoll

l

vz

) vc(vz ,l)∫0

dvrv2r

u3Exp

(−v2

u2

) (2.26)

If now the capture velocity vc(vz, l) is substituted by our model vc(vz, L), the length and

velocity integrals can be separated, which results in

Φ(vz, n) =n

1 + Γtrap(n)

Γout

· C · 2πrc · vzΓcoll(n)

Exp

(−v2z

u2

)· (2.27)

(1− Exp

(−Γcoll(n)

Lvz

)) vc(vz ,L)∫0

dvrv2r

u3Exp(

−v2r

u2

)

For small lengths L, the flux distribution grows more than linearly with L: the term1

Γcoll(n)

(1− Exp

(−Γcoll(n)

Lvz

))) ≈ L

vz, and furthermore with increasing length, more fast

atoms can be captured, increasing the value of the radial velocity integral. The flux

distribution shows saturation, when L ≈ 〈vz〉Γcoll(n)

. As a function of pressure, the termn

1+Γtrap(n)

Γout

alone would lead to a linear increase at low densities, and to saturation at

higher values. In addition, the term 1Γcoll(n)

(1− Exp

(−Γcoll(n)

Lvz

))is constant at low

densities, and falls off as 1Γcoll(n)

∝ 1nat high densities. Therefore, an optimum pressure

must be found to maximize the flux, and the length should be adjusted close to 〈vz〉Γcoll(n)

.

The total flux of atoms is given by the integral over all longitudinal velocity components

Φ(n) =

∫ ∞

0

Φ(vz, n)dvz (2.28)

and the mean longitudinal velocity is


〈vz〉 = 1

Φ(n)

∫ ∞

0

vz Φ(vz, n)dvz (2.29)

Results of this model are compared to experimental data in Appendix F.

2.4 Optical Dipole Potentials 27

2.4 Optical Dipole Potentials

As described above, intensity gradients of a light field cause a conservative force on atoms.

The atom, as a polarizable object in an inhomogeneous time-dependent electric field (

light field ), develops an induced oscillating dipole moment. If the detuning is negative

(δ < 0, red-detuned dipole trap), the dipole oscillates in phase with the field, and the

atoms are pulled towards regions of high intensity. In the opposite case of a blue-detuned

trap, the dipoles oscillate 180 out of phase and the atoms are expelled from the high-

intensity regions. This makes trapping geometries slightly more complicated, but offers

the advantage that atoms are trapped in dark regions where unwanted light scattering is

suppressed.

The potential energy of an atom in a light field I(r) can be calculated to be [56]:

Udip(r) = − 1

2ε0cRe(α)I(r) (2.30)

where α is the dynamic atomic polarizability, and I(r) is the intensity of the light field.

Any position dependence of I(r) gives rise to a dipole force Fdip = −∇Udip. The scatteringrate of photons by the atom is given by

Γsc(r) =1

ε0cIm(α)I(r) (2.31)

α can be calculated semiclassically [56], resulting for a two-level atom in

Udip(r) =3πc2

2ω3at

(Γ

δ− Γ

δ+

)I(r) (2.32)

Γsc(r) =3πc2

2ω3at

(ω

ωat

)3 (Γ

δ− Γ

δ+

)2

I(r) (2.33)

As usual, δ = ω − ωat is the detuning of the light field at frequency ω from the atomic

resonance at ωat, and δ+ = ω + ωat represents the so-called counter-rotating term. In

usual configurations, this term is a minor correction to the rotating wave approximation,

in which the potential depth scales as Udip ∝ Iδ, whereas the spontaneous scattering rate

scales as Γsc ∝ Iδ2. It can be seen that one can obtain a required potential depth at an

arbitrarily low scattering rate by increasing both intensity and detuning, the practical

limits being set by the available laser power.

For multilevel atoms, however, the result (2.32) is not complete. The full dipole

potential contains further contributions from all higher lying excited states, which have

to be summed up.

If one looks closer at the dipole potential due to one line, the transition strength is

distributed over all fine structure and hyperfine structure components. Furthermore,


the contribution of every fine structure component, in addition to the scalar potential

considered above, contains a vector term which acts differently on the ground state

magnetic sublevels mF , therefore called a fictitious magnetic field [57]. For the example

of the dominant 5S1/2 −→ 5PJ transition of 87Rb, which is split into D1 line (J = 1/2)

at a wavelength of λ = 794nm and D2 line (J = 3/2, λ = 780nm), one obtains

Udip(r) =∑

J=1/2,J=3/2

πc2Γ

2ω3at,J

(1

δJ− 1

δ+,J

) [c2J I + iσJ(E∗(r)× E(r)) · F

F

]I(r) (2.34)

Here, E(r) is the local polarization unit vector of the total, three-dimensional field config-

uration, and ∗ denotes the complex conjugate. F and F denote the total atomic angular

momentum operator and its modulus. I is the unity operator, cJ is a coefficient for the

transition 5S1/2 −→ 5PJ , and σJ =

+1−1 for the transition to 5PJ , with J =

3/21/2 .

The vector term does not vanish if the polarization has some degree of ellipticity. As in

our proposed lattice configuration ( see section 6.3 ), without control of the relative phases

between the standing waves, there is no control over the ellipticity of the polarization [58],

and the vector term seems to be relevant. For a the case of a trap laser of wavelength

532nm however, where the detuning is much larger than the excited state fine structure

splitting (δ1/2 ≈ δ3/2), the contributions to the dipole potential due to the D1 and D2

lines cancel in the fictitious magnetic field term, while they add up in the scalar term,

and we are left with

Udip(r) =∑

J=1/2,J=3/2

πc2Γ

2ω3at,J

c2J

(1

δJ− 1

δ+,J

)I(r) (2.35)

For our case, c23/2 = 2, and c21/2 = 1, whereby the two-level result for the dipole potential,

equation (2.32), is recovered.

If instead a Ti:Sa laser ( emission range from 700 to 1000 nm ) should be used as the trap

laser, equation (2.34) must be used.

2.5 Optical Lattices

An array of identical microscopic ( sizes on the order of the trapping laser wavelength

) dipole traps is called an optical lattice. These microscopic potential wells are usually

created by interference of a number of spatially overlapping laser beams. The force on

atoms in such a lattice can be thought of as arising from redistribution of photons between

the different overlapping laser modes. Therefore a relevant energy scale is the recoil energy

Erec, which an atom of mass m experiences when absorbing a photon of wavevector k:

2.5 Optical Lattices 29

Erec =

2k2

2m(2.36)

For a 87Rb atom absorbing a lattice photon of λ = 532nm, Erec = 5.35 · 10−30 J , or

in temperature units, Trec = Erec

kB= 387nK. A number of lattice field configurations

have been described in [30]. Here, I will concentrate on the configuration envisaged in

our experiment, which consists of three mutually orthogonal pairs of standing waves with

mutually orthogonal polarizations and equal frequency ω [59, 60]. The resulting electric

field is

E(r, t) =6∑i=1

Ei(t)uiExp(i(ki · r − ωt+ φi(t))) (2.37)

where the wave vectors ki and polarization vectors ui satisfy

k1 = −k4 = ke1, u1 = u4 = e2; (2.38)

k2 = −k5 = ke2, u2 = u5 = e3; (2.39)

k3 = −k6 = ke3, u3 = u6 = e1; (2.40)

where k = 2πλ, and the φi(t) describe a slow variation of the phases of the lattice beams

due to noise in the laser or mechanical noise, and can be written as

φ1(t) = −φ4(t); φ2(t) = −φ5(t); φ3(t) = −φ6(t); (2.41)

The amplitudes of the counterpropagating waves are chosen equal, but for technical rea-

sons the amplitudes of the different standing waves are not necessarily equal, such that

E1(t) = E4(t); E2(t) = E5(t); E3(t) = E6(t); (2.42)

The intensity of this configuration is given by the sum of intensities in all three standing

waves. As these have orthogonal polarizations, they do not interfere with each other:

I(r, t) =1

2ε0c

∣∣∣ E(r, t)∣∣∣2 =

3∑j=1

4Ij(t)cos2(ki · r − ωt+ φj(t)) (2.43)

Here, Ij(t) = 12ε0c |Ej(t)|2 is the single beam intensity ( assumed constant throughout

the lattice region ). The dipole potential Udip(r) of this laser configuration is given by

inserting (2.43) into equation (2.32) ( for simplicity, here we assume the Ij equal ):

Udip(r) = U0

(1

6(1− cos(2kx)) +

1

6(1− cos(2ky)) +

1

6(1− cos(2kz))

)(2.44)


Here, U0 is the full peak-to-peak potential depth. According to equation (2.32),

U0 =3πc2

2ω3at

(Γ

δ− Γ

δ+

)· 3I0 (2.45)

The peak intensity in each of the three standing waves is I0 = 4 · 2Pπw2 , where the factor 4

results from constructive interference of the two counterpropagating beams’ E fields, and2Pπw2 is the peak intensity of a Gaussian beam of waist w and a total power P .

2.5.1 Time Scales

Let us now consider some energy- and timescales relevant for the motion of atoms in

optical lattices. Numerical values of these quantities will be given in chapter 6. In these

microscopic potential wells the atomic wavepackets experience a quantum mechanical

motion described by a band structure. In the case of very high potential wells, tunnelling

between neighboring potential wells is negligible, and the atomic motion can be described

by an oscillation frequency in a harmonic oscillator potential

ωosc =2

√U0

3· Erec (2.46)

The tunnelling rate of an atom in a state Ψ between neighboring potential wells separated

by a lattice vector d, is given by

Γtun(r) =1

〈Ψ(r) |

(p2

2m+ Udip(r)

)| Ψ(r + d)〉 (2.47)

The scattering rate Γsc of lattice photons sets the limit for coherent time evolution of the

atomic wavepackets, and causes heating of a trapped atomic sample at a rate Pheat =

2ErecΓsc. The scattering rate Γsc can be expressed as

Γsc(r) =1

(ω

ωat

)3 (Γ

δ− Γ

δ+

)〈Ψ(r) | Udip(r) | Ψ(r)〉 (2.48)

Here 〈Ψ(r) | Udip | Ψ(r)〉 denotes the expectation value of the potential energy of an atom

in a state Ψ in the optical potential.

2.5.2 Parametric Heating

Apart from scattering of lattice photons there is another source of heating in optical

dipole traps and lattices. If the amplitude or the frequency of the lattice laser is noisy,

the motion of the atoms in the potential wells is not a harmonic oscillation. This heats

2.5 Optical Lattices 31

the atoms up over time with a rate Γp, that we will now examine. The heating rates from

amplitude and frequency noise add up, and should be substantially lower than the inverse

duration of envisaged experiments.

The following Hamiltonian [61, 62] describes the harmonic oscillator approximation of a

lattice potential well or a dipole trap, disturbed by a small time-dependent amplitude

variation ε(t):

H =p2

2m+

1

2mω2[1 + ε(t)]x2 (2.49)

Because the perturbation has the same spatial symmetry as the unperturbed Hamiltonian,

transitions to the adjacent vibrational level are forbidden. Therefore only noise at twice

the oscillation frequency of atoms in the trap can drive transitions to levels separated

by two vibrational quantum numbers from the initial state. The heating rate Γp,1 of an

ensemble of atoms with initial energy 〈E〉 in a trap with trap frequency νtrap =ωosc2π

is

Γp,1(νtrap) =〈E〉〈E〉 = π2ν2

trap · S(2νtrap) (2.50)

where S(ν) is the relative spectral amplitude noise density in units [ 1Hz

]

S(ν) =2

π

∫ ∞

0

dτcos(2πντ)〈ε(t)ε(t+ τ)〉 = ∆P 2(ν)

P 2(2.51)

and the last equality follows from the Wiener-Khinchine theorem. ∆P 2 is the square of

the RMS spectral noise power in units [W2

Hz], and P is the laser power. Thus, the final

result is

Γp,1(νtrap) = π2ν2trap ·

∆P 2(2νtrap)

P 2(2.52)

Also position shifts of the potential minimum, as can be caused by frequency noise in the

lattice laser, can lead to parametric heating. The appropriate Hamiltonian to describe

this is

H =p2

2m+

1

2mω2[x− εx(t)]

2 (2.53)

leading to a heating rate of

Γp,2 =〈E〉〈E〉 = π2ν2 · S(ν)〈x2〉 (2.54)

〈x2〉 is the mean square position of an atom at t = 0. Note that S(ν) is given by the same

autocorrelator expression as in (2.51), but this time εx has units [m], therefore S(ν) has

the unit [m2

Hz]. For this kind of perturbation, transitions between adjacent levels occur. In

this case S(ν) is related to measurable quantities by


S(ν) =

(∆x · ∆ωl(ν)

ωl

)2

(2.55)

where ∆ωl(ν) is the Fourier component of laser frequency jitter at frequency ν, and

∆x = N λ2is the distance from the retroreflector mirror to the lattice center, which should

be kept as small as possible. This relation can be understood as follows: A frequency (or

wavelength) jitter causes a jitter in lattice periodicity. The node of the electric field at the

retroreflecting mirror stays fixed, but every subsequent potential minimum is shifted. For

the N’th minimum this shift amounts to εx(t) = N λ2

∆ωLωL

. Measurements of the quantities

(2.51) and (2.55), and the related heating rates will be presented in section 6.4.

Chapter 3

The Rubidium BEC Experiment

The production of a BEC places stringent requirements on the quality of the vacuum

surrounding the magnetic trap in which evaporative cooling is performed. On the other

hand one wants to collect, cool and trap a high number of atoms in a short time, and

load them efficiently into this magnetic trap. The problem is solved up to now most

commonly by two alternative techniques, both of which employ a MOT in the UHV

chamber as the final step before loading the atoms into the magnetic trap:

1.) The MOT can be loaded from an atomic beam produced by a Zeeman slower.

The flux can be up to 1011 atoms/s, but only a fraction of around 40% can be trapped by

a MOT. The reasons for this are a large divergence because only the longitudinal motion

is cooled, and furthermore a large fraction of the beam is not cooled at all, and the large

flux of thermal atoms can perturb the MOT.

2.) A double MOT system [24] is widely used: a low vacuum MOT (possibly in a

vapor cell) allows fast accumulation of a high atom number, which is then transferred in

a pulsed operation to the UHV part of the apparatus. In this system, 1.5 · 1010 Rb atoms

were trapped, with a loading time constant of 95 s , much too slow for the purpose of

fast creation of Bose-Einstein condensates.

Two other atomic beam production techniques have recently been developed:

3.) An atomic beam from a low velocity intense source (LVIS) [26]), can transfer

5 · 109 atoms/s from a vapor cell MOT to a UHV chamber in continuous operation, and

an order of magnitude higher flux when operated in a pulsed mode.

4.) A two-dimensional-MOT is a relatively simple setup that has previously been

demonstrated to produce a flux of 9 · 109 atoms/s with low velocities [25].

33

34 CH. 3 The Rubidium BEC Experiment

Here, an improved 2D-MOT, 3D-MOT system is described. It employs large elon-

gated trapping volumes for both the 2D-MOT and the 3D-MOT. The 2D-MOT produces

a well-collimated beam of atoms with a longitudinal velocity of around 50m/s. By

allowing these high longitudinal velocities, atoms from a larger fraction of velocity space

are collected into the beam and produce a higher atom flux than the double MOT system

or the LVIS.

The elongated setup of the 3D-MOT allows to catch and cool substantially higher beam

velocities than a standard MOT and completes an efficient system for fast collection of a

high number of ultracold atoms.

3.1 Laser-Cooling the Element Rubidium

The element Rubidium is an alkali metal, located in the first column of the periodic

table. It occurs in two isotopes, 85Rb with 72% relative abundance and 87Rb with 28%

abundance, the latter being used in our experiment. As Rubidium possesses a single

electron outside a closed shell, the electronic level scheme and the optical spectrum is

relatively simple. The transitions from the ground state to the first excited state ( a fine

structure doublet ) are located at 780 nm (D2 line, 5S1/2 −→ 5P3/2) and 794 nm (D1 line,

5S1/2 −→ 5P1/2) respectively. Diode lasers for this wavelength are commercially available,

and if high power is required, the Ti:Sa laser is ideally suited, as its gain maximum is

located at 780 nm. The coupling of the total electronic angular momentum to the nuclear

spin of I = 3/2 leads to a splitting of both ground state and the excited states, as shown

for the D2 line in figure 3.1. For laser cooling, the 5S1/2,F = 2 −→ 5P3/2,F′ = 3 transition

at 780, 25 nm in the D2 multiplet is used. The laser is red-detuned by around −2 Γ to

the 5S1/2,F = 2 −→ 5P3/2,F′ = 3 transition. Circularly polarized light forms a closed

transition (”cycling transition”) between the mF = F and mF ′ = F ′ Zeeman sublevels.

The presence of the other hyperfine levels, however, complicates the situation. Impurity

of the circular polarization causes unwanted off resonant excitation of the 5P3/2,F′ = 2

level, located at a 267MHz lower frequency than the cycling transition. From there,

spontaneous emission can transfer the atoms into the 5S1/2,F=1 ground state. There

they cannot participate in cooling, because of the 6.835GHz frequency difference to the

F = 2 ground state of the cycling transition. For efficient cooling and trapping, a laser

has to be used on the transition 5S1/2,F = 1 −→ 5P3/2,F′ = 2 at 780, 23 nm, to repump

atoms back into the cycling transition.

The saturation intensity Isat ( equation (2.1) ) for the 87Rb cycling transition with a

natural linewidth Γ = 2π · 5.98MHz has a value of 1.67mW/cm2, which means that

with 1W available laser power, an area of 120 cm2 could be homogeneously illuminated

3.2 Description of the Experimental Setup 35

RepumperMOT-Laser

Figure 3.1: Hyperfine splitting of the levels involved in laser cooling the isotope 87Rb. The

cooling laser operates slightly red-detuned to the 5S1/2,F = 2 −→ 5P3/2,F′ = 3 transition,

and repumping on the 5S1/2,F = 1 −→ 5P3/2,F′ = 2 transition is necessary to obtain a

closed transition.

at a saturation parameter (2.2) of s = 5. This allows to operate the 2D-MOT with an

illuminated area of about 60 cm2 in parallel with the 3D-MOT ( A ≈ 70 cm2 ) by a single

Ti:Sa laser.

The room temperature Rb-vapor-pressure is 3 · 10−7 mbar. This turns out to be

well suited for operation of a 2D-MOT in a vapor cell to obtain a high flux in an atomic

beam. On the other hand it makes the achievement of ultrahigh vacuum conditions for

the 3D-MOT harder than e.g. for Na ( room temperature vapor pressure of 3 ·10−11mbar

), but does not pose a principle problem.

3.2 Description of the Experimental Setup

The basic experimental setup consists a vacuum apparatus divided in two parts. An

expansion of the central parts of the setup is shown in figure 3.2. In a Rubidium vapor

cell (135 × 35 × 35mm) a cold atomic beam is produced by two-dimensional magneto-

optical cooling. The glass cell, whose long axis (z-axis) is horizontally aligned, is separated


from a large ultrahigh vacuum (UHV) chamber by a differential pumping tube, through

which the transversely cooled atoms enter the UHV part of the apparatus. As described

in section 2.3.1, the finite cooling time for longitudinally fast atoms results in an atomic

beam with a longitudinal velocity substantially lowered compared to a thermal source.

This beam enters the trapping volume of a large elongated 3D-MOT, at an angle of 27

to the long trap axis. This design allows for a capture velocity from the atomic beam of

around 65m/s ( see Appendix A ), much larger than in any standard MOT.

4

To Rb Reservoirand Pumps

To Pumps

3D-MOT

Differential Pumping Tube

Vapor Cell2D-MOT

2D-MOT Coils

Elliptical 2D-MOT Beams Elliptical 3D-MOT Beams

Ioffe BarsPinch Coils

Figure 3.2: Top view of the central parts of the apparatus: The Rb reservoir supplies a

dilute room temperature vapor to the 2D-MOT, from where a cold atomic beam emerges

to the 3D-MOT in the UHV chamber, through a differential pumping tube. Compare the

magnetic coil configuration for the 3D-MOT to figure 2.2.

3.2.1 Laser System

The laser setup for both 2D and 3D-MOT consists of a Ti:Sapphire laser pumped at

532nm by a 10W Verdi laser, providing 1.3W of cooling light at 780nm. The locking

scheme, as well as the beam splitting optics are depicted and explained in figure 3.3.

The laser power to the experiment is directed through an AOM, for fast switching of

intensity and detuning, and is split by a λ2plate and a polarizing beamsplittter cube

between 2D-MOT and 3D-MOT. After that it is further split into the individual beams.

Combinations of spherical and cylindrical telescopes are used to create the elongated


elliptical beam profiles for 2D- and 3D-MOT.

MOT-

AOM

AOM

Axial-Radial2D-3D

+200 MHz

”= +95 MHz

Laser(MBR 110)

Pump-Laser(Verdi V10)

Stabilization

Cylindrical Telescopef=-60 mm (h=w=25 mm);f=400 mm (h=30mm; w=150 mm)

Spherical Telescopesf=30 mm; f=200 mm

Spherical Telescopesf=30 mm; f=150 mm

Repumper

(Dl100)

Vertical3D-MOTBeams

Axial 3D-MOTBeams

Vertical2D-MOTBeams

Horizontal3D-MOTBeams

Horizontal2D-MOTBeams

Combination of /2Plate andPolarizingBeamsplitter Cube

Figure 3.3: Schematic of the laser setup for 2D-MOT and 3D-MOT. The locking scheme

of the Ti:Sa laser is as follows: A small fraction of the power is split up and passes

through an AOM, and is directed into a polarization spectroscopy setup. The frequency

ω′′ = ω + 95MHz is stabilized to ω2,3 − 133.5MHz + x, where x = +(2 − 3)Γ. To the

experiment, the frequency ω′ = ω + 200MHz has a detuning δ = ω2,3 − ω′ ≈ −2Γ

A grating stabilized diode laser is employed as the repump laser, whose frequency is

locked to the 5S1/2,F = 1 −→ 5P3/2,F′ = 2 transition using polarization spectroscopy,

analogous to the setup described in appendix B. The repumping light is overlapped with

the full area of both horizontal 2D-MOT beams. In the 3D-MOT, it is overlapped with

the vertical beam pair. Repump light in the horizontal beam pair would be inefficient,

as the atomic beam has a velocity component in the direction of these beams, and atoms

are Doppler shifted out of resonance with the repump frequency.

For the analysis of the atomic beam a second grating-stabilized diode-laser is used. This

probe-laser is locked to the 5S1/2,F = 2 −→ 5P3/2,F′ = 3 transition. A third diode laser,

whose setup will be described in section 4.2.5, is used for various analysis techniques of

the 3D-MOT.


3.2.2 Vacuum system

The full vacuum setup is shown in figure 3.4. The vacuum system for the 2D-MOT

( laboratory jargon: the low vacuum side ) has been described accurately in [2]. A Rb

reservoir is separated by a valve from the 2D-MOT, so that the reservoir can be exchanged

without breaking the vacuum. This part of the apparatus is pumped by a turbo pump,

and an ion pump of 20 l/s pump power. The 130mm long differential pumping tube has

a constant diameter of 6mm in the first half of its length, widening in the second half

towards the end to 9.6mm. It can maintain a pressure difference between the low-vacuum

side and the ultrahigh vacuum side of more than three orders of magnitude. An additional

feature for this purpose is a motorized mechanical shutter inside the tube, that is only

opened while loading the 3D-MOT. The UHV chamber is pumped by a 200 l/s ion pump,

and optionally by an additional Titanium sublimator. Pressures of 2 · 10−11 mbar are

reached, allowing trap lifetimes of tens of seconds.

2-3/4” DN40

2-3/4” DN40

4

Ion Pump 20 l/s

Turbo PumpIon Pump 200 l/s

Rb Reservoir

Ti Sublimator

Vapor Cell2D-MOT

3D-MOT

5 Way Cross andDifferential Pumping Tubewith Shutter

Figure 3.4: Top view of the full vacuum system. To the left, tilted by 27, the low vacuum

side is shown with the rubidium reservoir and the 2D-MOT vapor cell. The right part is

the UHV chamber, pumped by a 200 l/s ion pump and an optional Ti-sublimator.


3.2.3 2D-MOT

A set of four racetrack-shaped coils ( see figure 4.1 ) produces a radial magnetic quadrupole

field with field gradients of typically 17G/cm. Two pairs of circularly polarized coun-

terpropagating laser beams with large elliptical beam profiles ( the illuminated area per

beam has 95mm width, and 15mm height ) are shone in perpendicularly to the line of

zero magnetic field. Following the model described in section 2.2, atoms are transversely

cooled to the axis and form an atomic beam, whose parameters will be described in chap-

ter 4. If the 2D-MOT is operated alone, the average intensity across the beam profile of

the cooling laser is I = Pπwzwρ

= 38mW/cm2 at a power of P = 180mW per beam, for a

horizontal waist wz ≈ 25mm, and a vertical waist wρ ≈ 6mm. Comparison to the satu-

ration intensity (2.1), Isat = 1.67mW/cm2 of the cycling transition, gives the saturation

parameter s ≈ 20. At such high intensities also the wings of the Gaussian laser intensity

distribution contribute fully to cooling and trapping in the MOT. Therefore not the 1/e2

diameters of the beam profiles are relevant, but the full illuminated area of 95× 15mm.

Electric heating rods around the glass cell provide homogeneous and stable heating, al-

lowing to work at Rubidium partial pressures up to 4 · 10−6mbar, substantially higher

than the room temperature Rb-vapor-pressure (3 ·10−7mbar). A liquid nitrogen reservoir

allows cooling the differential pumping tube, thereby reducing the Rubidium pressure

down to about 10−7mbar.

3.2.4 3D-MOT

Magnetic Fields

The magnetic field configuration described in section 2.2 ( figure 2.2 ) is produced by the

following setup:

The radial quadrupole field is produced by four parallel straight current-carrying bars

(Ioffe bars), aligned in a square at distances of 25 mm to each other, with 100mm length

in the UHV chamber, and a total 500mm length, extending further to the back side of

the UHV chamber. This extension is provided to allow for magnetic transport of Bose-

Einstein condensates into an optional second glass cell that can be attached to the UHV

chamber along the trap axis. Each Ioffe bar is made of six square profile copper bars of

8mm side, with a round hole along the center for high pressure ( 16 bar ) water cooling

( needed only for operation of a magnetic trap with currents of up to 1000A ). The

currents through the six copper ”wires” in each bar run in parallel, and the current flows

in alternating directions in the four bars. Typically, in the MOT a radial magnetic field

gradient of 14G/cm is produced by a current of 40A in each copper ”wire”.

Two round coils (pinch coils) of 18mm inner diameter, oriented with their symmetry axis

along the zero field line of the radial configuration, at a distance of 110mm to each other,


create the axial field. They consist of 10 windings of square-shaped copper wire of 4mm

side length, with a central round hole for water cooling. Usually, in the MOT a current

of 24A flows through each winding.

The earth magnetic field and possible stray fields from the ion pumps are compensated by

three separately adjustable orthogonal pairs of coils wrapped around the vacuum chamber.

Optics

The optical setup for the 3D-MOT ( cf. figure 3.2 ) consists of four large radial beams

with elliptical beam profiles identical to the four 2D-MOT beams, shone in through large

vacuum windows ( CF 160, 160 mm diameter ). 2 counterpropagating axial beams with

a spherical beam profile of 12 mm illuminated diameter are introduced into the chamber

through two CF 16 windows with approximately 12 mm diameter of good optical quality.

All polarizations are circular, and are shown in figure 2.2.

The single beam power in the 3D-MOT, operated in connection with the 2D-MOT is is

about 100mW per radial beam, giving a saturation parameter of s ≈ 12. This is still

enough that, as in the 2D-MOT, the full beam profile takes part in cooling and trapping.

Chapter 4

Preparation of a Large Number of

Ultracold Atoms

4.1 Two-Dimensional MOT as a Bright Source of

Cold Atoms

In this section the measurements will be described which characterize the source of cold

atoms from which the three dimensional MOT is loaded. The beam parameters of rele-

vance are:

• the flux Φ (2.28), i.e. the number of atoms per unit time that enter the UHV

chamber in the atomic beam (units [atoms/s] = [at/s]), and provides a measure of

how fast the 3D-MOT will be loaded with atoms.

• the flux distribution (2.27) ( flux per velocity class, in units[at/sm/s

] = [at/m]) shows,

which fraction of the total flux lies below the estimated capture velocity of the 3D-

MOT, and can be trapped. The mean velocity 〈v〉 should be less than the capture

velocity.

• the beam diameter and transverse velocity distribution in the beam determine the

divergence, which has to be small enough for the atoms to enter the trapping volume

of the 3D-MOT.

• the brightness ( total flux divided by solid angle into which the beam spreads, units

[atomss·sr ] ) is the quantity that summarizes loading a 3D-MOT from an atomic beam:

a high total flux into a small enough solid angle to be trapped, is required.

The results described in the following were obtained in the course of this diploma thesis,

but the measurements have already been described in detail in another diploma thesis

[2], and are to be published [1]. Here only the information necessary to understand the

41

42 CH. 4 Preparation of a Large Number of Ultracold Atoms

results relevant to the further process of loading the 3D-MOT from the 2D-MOT beam

will be stated.

4.1.1 Time of Flight Measurements

The atomic flux distribution and the total flux of the atomic beam, as well as the mean

atomic velocity were measured as a function of varying laser intensity, with varying Ru-

bidium pressure, and by varying the length of the 2D-MOT. The results are summarized

in the following.

CharacterizationSetup

CCD Camera orPhotodetector

Fluorescence

Elliptical MOT Beams

Probe Beam

Plug Beam

Glass cell

MOT-Coils

Differential Pumping

Cold Rb-Beam

TurboPumpTurboPump

Ion GaugeIon Gauge

Rb - ReservoirRb - Reservoir Valve

Valve Valve

Vapor CellVapor Cell

5-Way Cross5-Way Cross

Detection Section(6-Way-Cross)

Detection Section(6-Way-Cross)

IonPump

IonPump

Figure 4.1: Schematic view of the preliminary vacuum system ( left ) and setup for charac-

terization of the 2D-MOT atomic beam. The beam is analyzed by a time-of-flight-method

behind the differential pumping tube. A plug beam shuts off the atomic beam, and a probe-

laser is shone in perpendicularly, to excite fluorescence in the fading-out atomic beam.

The fluorescence is detected by a CCD-camera or a photodetector.

The detection of the atomic beam is done in a preliminary UHV system ( see figure

4.1 ), consisting of two six-way-crosses, both mounted on the axis of the atomic beam.

One is used for observation, and another one is needed to fix the ion pump ( pump

flux 20 l/s ). All free ports are closed with vacuum windows. A resonant light sheet is

shone in perpendicularly to the beam through a window of the first six-way-cross, and

the fluorescence is measured on a calibrated photo diode or CCD camera [2], installed

perpendicularly to both atomic beam and probe laser, whose polarization is chosen to

maximize the fluorescence signal.

The characterization of the atomic beam employs a time of flight method: The 2D-MOT

4.1 Two-Dimensional MOT as a Bright Source of Cold Atoms 43

beam is abruptly turned off by shining an intense near-resonant beam ( plug beam )

from the side into the observation cross. The photon pressure deflects all atoms below a

longitudinal velocity of about 130 m/s [2]. The temporal dependence of the fluorescence

while the beam fades out is a measure for the flux distribution:

Φ(v) =η

dprobe

l

v

dS

dt(4.1)

S is the signal votage from the photodiode, dprobe the width of the probe-beam light

sheet along the atomic beam axis, l the distance between plug-beam and probe-beam.

The factor η contains all calibration parameters of the detection system, and v is the

longitudinal velocity of the atoms.

The plug beam blocks the thermal background flux up to a velocity of 130m/s which

should therefore also be visible in the TOF signal. From later measurements of the

loading time constant of the three-dimensional MOT, with and without the 2D-MOT

turned on, however, one can deduct that the thermal background flux up to velocities

around 60m/s is at least a factor of 10 lower than the flux from the 2D-MOT beam.

The optimum trap laser detuning and magnetic field gradient were not varied dur-

ing the experiments described here: The detuning was set to -2Γ, and the magnetic field

gradient stayed fixed at 15 G/cm. The optimum detuning (field gradient) results from

a compromise between the force strength and a large capture velocity (capture volume),

cf. equations (2.5) and (2.8).

Beam Parameters as a Function of Laser Power

The dependence of the atomic flux on incident laser power was measured at a Rb pressure

of 1.5 · 10−6mbar, and the results are shown in figure 4.2.

The flux increases linearly at powers below 50mW per beam and shows saturation above

this value. Only the amplitude, not the shape of the flux distribution changes with power.

An optimum total flux of above 6·1010 atoms/s is obtained at a power of 160mW per laser

beam . If operated in connection with the 3D-MOT, only a fraction of the full 1.3W laser

power will be available for the 2D-MOT. One estimates that a little less than half the

power will be used for the 2D-MOT, as the 3D-MOT employs equal telescopes for the

radial beams and only small axial beams. Under these conditions a flux of 5·1010 atoms/s

should be obtained.

The mean velocity is constant at about 57m/s over the whole power range covered. As

shown in appendix A, the expected capture velocity of the 3D-MOT is about 65m/s,

which means that a substantial amount of the flux should be trappable, although clearly

not the full flux.


0 20 40 60 80 100 120 140 160 180 200-1

0

1

2

3

4

5

6

7

8170mW135mW115mW95mW70mW50mW30mW

velocity [m/s]

Ato

mic

Flu

x[1

0a

tom

s/m

]8

0 20 40 60 80 100 120 140 160 180

1x1010

2x1010

3x1010

4x1010

5x1010

6x1010

7x1010

Ato

mFlu

x[1

/s]

Laser Power per Beam [mW]

Figure 4.2: Left: atomic flux distribution measured for different incident laser powers per

beam; Right: the integrated total flux shows saturation at about 100 mW.

Dependence on Rb Pressure

To measure the dependence of the beam parameters on the Rb pressure in the vapor cell,

the Rb reservoir was either heated or cooled as described in section 3.2.3. The pressure

is measured by passing a probe laser through the glass cell, and recording the absorption

strength of the 87Rb D2-line [2].

0 50 100 150 2000

5

10

15

20

25

30

35

40

45

502.94e-74.97e-78.45e-71.59e-62.34e-63.01e-63.5e-6

7

6

5

4

3

2

11 2 30.5 1.5 2.5 3.5

Pressure [10 mbar]-6Velocity [ m/s]

70

60

50

40

Me

an

Ve

locity

[m/s

]

Ato

mic

Flu

x[1

0a

t/s]

10

Flu

xD

istr

ibu

tio

n[1

0a

t/m

]8

Figure 4.3: Pressure dependence of flux distribution, total flux and mean velocity. at high

pressures the flux saturates and even decreases due to collisions of atoms in the beam with

the background Rb gas in the vapor cell.

As shown in figure 4.3, the mean atomic velocity in the beam increases linearly with Rb

pressure, from 42m/s to 67m/s. Therefore the pressure determines which fraction of the


flux can be trapped in the 3D-MOT. At the same time also the total flux is strongly depen-

dent on pressure, increasing sharply at low pressures, saturating at 2·10−6mbar and even

decreasing slightly above this value. Therefore, for loading the 3D-MOT, a compromise

has to be found between a higher total flux at high pressure, and an increased trappable

fraction of the flux due to a lower mean velocity. The optimum pressure is anticipated to

lie in the controllable interval, between 1·10−6mbar and 1.5·10−6mbar.

The saturation and decrease of the flux is due to an enhanced collision rate of atoms

in the beam with the background Rubidium, when the mean free path of a beam atom

becomes comparable to the length of the MOT.

Collisions can also explain the increase of the mean velocity with pressure, as slow atoms

spend more time in the vapor cell and are kicked out with higher probability than fast

atoms. A quantitative understanding of the linear increase over the full pressure range,

however has not been reached. A comparison between the experimental results and pre-

dictions from the model (section 2.3.2 ) is given in Appendix F.

Dependence on 2D-MOT Length

As explained in section 2.3.1, the mean velocity should increase with length, reaching

the thermal velocity in the limit of infinite length. This also means that the flux should

increase more than linearly with length, as not only the trapping volume increases but

also the trapped fraction of the thermal velocity distribution. The length dependence

was measured at a pressure of 1.5 · 10−6mbar. To vary the 2D-MOT length, a cover was

moved over part of the illuminated area in each beam, while keeping the power of 21mW

per beam in the illuminated area constant. Both the increase in velocity and in flux can

be observed in figure 4.4. At very short lengths the thermal background dominates the

flux, increasing the mean velocity.

As the model predicts, no saturation of the flux is found as the length is increased, even

though saturation as a function of pressure has already taken place, cf. Appendix F.

4.1.2 Transverse Velocity Distribution in the 2D-MOT

Transverse Doppler spectroscopy was carried out in the working 2D-MOT by passing

a small collimated probe beam transversely through the 2D-MOT approximately 1 cm

before the entrance hole to the tube. The frequency was scanned around resonance and

the absorption profile was recorded on a photodetector. A typical result is shown in figure

4.5. Two important features can be seen on top of the Doppler profile of the surrounding

thermal vapor in the glass cell. First, two dips centered symmetrically around resonance

correspond to atoms lost from the thermal vapor by cooling to lower velocities into the

atomic beam. Second, the central peak corresponds to the transverse velocity distribution


4 5 6 7 8 95,0x10

9

1,0x1010

1,5x1010

2,0x1010

2,5x1010

3,0x1010

3,5x1010

Ato

mFlu

x[1

/s]

MOT-Length [cm]

4 5 6 7 8 9

56

57

58

59

60

61

62

Mean

Velo

city

[m/s

]

Length [cm]

Figure 4.4: Dependence of flux on length of the 2D-MOT. For comparability of the data

points, in each measurement the MOT was illuminated with the same total power of 21mW

per beam.

in the 2D-MOT. Division of the data by a Gaussian fit to the ( unperturbed ) high velocity

wings of the thermal vapor reveals these features more clearly and allows for the extraction

of the transverse capture velocity of the 2D-MOT ( half the velocity difference between

the two dips ) and of the mean transverse velocity in the 2D-MOT ( half width at half

maximum of the central peak ).

These two quantities are plotted in figure 4.6 against total laser power in the 2D-MOT.

The capture velocity is between 29m/s and 38m/s, increasing with power and showing

saturation. The mean transverse velocity in the 2D-MOT is around 5 to 7m/s, increasing

slightly with laser intensity. This value is much larger than the Doppler velocity of

vD ≈ 12 cm/s [51], as has been anticipated from the finite cooling time ( cf. section

2.3.1).

4.1.3 Measured Beam Divergence

The beam divergence was measured by recording with a CCD camera the spatial fluo-

rescence profile of a resonant light sheet, shone in perpendicularly to the beam. This

was done at two distances from the end of the differential pumping tube. From the half

widths at half maximum intensity of the fluorescence signal and the distance between

the two observation points the divergence angle is deducted. The measured divergence

is α = 32mrad. This is to be compared to the geometrically allowed divergence by the

differential pumping tube of 59mrad [2]. From these numbers it can be seen that the

tube does not geometrically filter out a large fraction of the beam. Therefore in contrast

to [25, 26], where the flux distribution is shifted to very low velocities, determined by


2vc

0.24

0.2

0.16

0.12

0.002

0.002

0.004

0.004

0.006

0.006

0.008

0.008

1.08

1.00

0.92

Frequency [a.u.]

Figure 4.5: Doppler spectroscopy absorption signal of the vapor cell with running 2D-

MOT. The broad peak is the Doppler broadened profile of the thermal gas, whereas the

small central feature is due to the transverse velocity distribution in the 2D-MOT. The

dips correspond to atoms lost out of the thermal distribution due to cooling.

0 200 400 600 800 1000

26

28

30

32

34

36

38

40

Captu

reV

elo

city

[m/s

]

total MOT power [mW]

0 200 400 600 800 1000

5,5

6,0

6,5

7,0

7,5

mean

transv

ers

eve

loci

tyin

the

MO

T[m

/s]

total MOT power [mW]

Figure 4.6: Capture velocity and mean transverse velocity of the 2D-MOT as a function

of laser power.

geometrical filtering by a small aperture, here the unperturbed velocity distribution of

our 2D-MOT source is seen in the atomic beam.

With the solid angle Ω = π tan2(α2), the brightness of our 2D-MOT source is Φ

Ω=

7.5·1013at/sr/s [2], more than an order of magnitude larger than the brightness of [25, 26].


4.1.4 Conclusions

The above measurements have shown that the 2D-MOT provides a well-collimated atomic

beam with a high flux of up to 6 · 1010 atoms per second, a large fraction of which has

velocities below 65m/s and should therefore be trappable. By varying the Rb vapor

pressure, the mean velocity and the total flux of atoms are changed drastically. This will

allow to optimize the loading of the 3D-MOT with the atomic beam. Saturation as a

function of laser power takes place already when only 50% of the available laser power are

directed to the 2D-MOT. Therefore 2D- and 3D-MOT can be operated jointly without

dramatically reducing the atomic flux.

The following questions will be answered in the next section: What is the loading rate

of the 3D-MOT? Does a high flux of atoms with the relatively high velocity of 50m/s

cause perturbations of the 3D-MOT? Will there be considerable trap loss or heating by

collisions with thermal atoms transmitted through the tube?

4.2 Characterization of a Large Elongated 3D-MOT

The purpose of the following measurements is twofold: The first focus is on the

combination of 2D-MOT and 3D-MOT as an efficient tool for the preparation of a large

sample of trapped ultracold atoms. Therefore, loading curves under different conditions

are examined, as well as the possible influence of the atomic beam on trap losses. Second,

the trapped atom number, density and temperature are characterized, to optimize the

conditions for loading the trapped atoms into a magnetic trap and evaporative cooling.

A high initial number of atoms is necessary for evaporation, as a large fraction of atoms

is lost in this process. A high density and a low temperature provide a high initial phase

space density, which reduces fraction of atoms that are lost.

In most experiments, the number of atoms is explored as a function of time, or

when varying experimental parameters. For this purpose either the fluorescence of the

atoms in the trapping light field, integrated over the whole cloud, is imaged onto a

photodiode, or images of the fluorescing cloud are recorded on a CCD camera. Both

devices are mounted above the top vacuum window and are calibrated so that, if

necessary, the absolute number of atoms can be extracted from the signals. For accurate

number determination a dark background image is always subtracted from the MOT

image. The CCD camera is calibrated according to

Natoms =τsctac

1

ηΩ

1

ηfilter

1

η bitsphoton

Nbits (4.2)

The acquisition time for a single picture is tac = 20ms, ηΩ = 4.67 ·10−5 is the efficiency of

4.2 Characterization of a Large Elongated 3D-MOT 49

the camera due to the finite solid angle it presents to the MOT, ηfilter = 8.4 · 10−3 is the

transmission of a filter to prevent the CCD camera from saturating, η bitsphoton

= 2.0 · 10−3

is the quantum efficiency of the camera at 780 nm, and Nbits is the number of recorded

counts. Finally, τsc = 129+50−30 ns is the mean scattering time of atoms in the MOT,

1

τsc= Γsc =

Γ

2· s

1 + s+ 4( δΓ)2

(4.3)

with a saturation parameter s = 12 ( cf. section 3.2.4 ), is calculated from the single beam

intensity in the 3D-MOT of about 100mW of an elliptical Gaussian beam of 25mm and

6mm beam waists. The detuning is δ ≈ −2Γ, where Γ = 2π 5.98 · 106Hz is the natural

linewidth of the cooling transition. The error in the scattering time is estimated from the

uncertainties in the saturation parameter of about 25% and in the detuning of ±12Γ. The

resulting calibration factor is

Natoms = 8360+3240−1944 ·Nbits (4.4)

which shows that the atom number is determined with a relative uncertainty of +38%

and −23%.

The photodiode is calibrated by the proportionality of photo voltage to CCD count num-

ber, the proportionality factor depending on the transmission of filters used before the

photodiode.

4.2.1 Loading of Atoms: Interplay between 2D- and 3D-MOT

Loading and decay curves are examined as a function of two parameters, the ratio R =P2D−MOT

P3D−MOTof laser power directed to 2D- and 3D-MOT, and the low vacuum Rb pressure.

Power Splitting between 2D- and 3D-MOT

Figure 4.7 shows loading curves of the 3D-MOT for different power splitting ratios R.

At small ratios, the 3D-MOT is essentially loaded from the thermal background gas that

passes through the differential pumping tube. Loading is very slow, with time constants

up to 20 s, and the steady state atom number is only about 15of the number at optimum

splitting ratio.

With increasing power in the 2D-MOT, the flux of the cold atomic beam increases,

resulting in faster loading and a higher steady state atom number in the 3D-MOT. The

optimum splitting ratio is between R = 0.35 and R = 0.39, where the trap suddenly

becomes unstable, which one can see in the large fluctuations in the steady state atom

number.

This instability can be seen by eye as a struggle between loading and a sudden collapse


0 10 20 30 40

0

2

4

6

8

10

R = 0,35

R = 0,54

R = 0,71

R = 0,85

R = 0,39R = 0,35

R = 0,24

R = 0,17

R = 0,07

Flu

ores

cenc

e[a

.u.]

Time [s]

0 10 20 30 40

0

2

4

6

8

10

0 10 20 30 40

0

2

4

6

8

10

0 10 20 30 40

0

2

4

6

8

10

0 10 20 30 40

0

2

4

6

8

10

0 10 20 30 40

0

2

4

6

8

10

0 10 20 30 40

0

2

4

6

8

10

0 10 20 30 40

0

2

4

6

8

10

0 10 20 30 40

0

2

4

6

8

10

Figure 4.7: Loading curves of the 3D-MOT as a function of power splitting ratio R between

2D-MOT and 3D-MOT. For low splitting ratios (low power in the 2D-MOT), loading is

slow. The loading rate increases monotonically with R, but for values of R ≥ 0.35 the

3D-MOT becomes unstable and losses prevent the accumulation of a higher steady-state

number.

and loss of atoms when the cloud reaches a certain size. It can happen as a more or less

regular temporal pattern, but also irregularly. The ”frequency” of growth and collapse,

and the critical MOT size depend crucially on laser alignment, the intensity balances

between counterpropagating beams as well as between different pairs of beams, and the

laser detuning.

Further increasing the splitting ratio still increases the loading rate, but instability occurs

already at lower atom numbers, determined by either the weaker trapping potential of

the 3D-MOT with decreasing power, or the increasing flux of atoms from the beam, or a

combination of both.

Rubidium Pressure in the Vapor Cell

Now we consider loading as a function of Rb pressure in the 2D-MOT. For any pressure the

loading process is optimized by choosing the power splitting ratio that provides maximum

steady state atom number. This procedure always results in stable, purely exponential

loading curves without signs of instability.

Figure 4.8 shows a) the optimum steady state atom number, as determined from the

photodiode signal, b) the loss constant ΓL ( equation 2.19 ), c) the loading rate RL, which

equals the trapped atomic flux from the 2D-MOT. It is determined by dividing steady


state atom number by loading time constant ( this is only valid for exponential loading

curves, not for loading curves that show instability and a cutoff of the loading curve

). Figure c) shows an example of such a loading curve, and indeed all loading curves

obtained when optimizing the steady-state atom number are strictly exponential.

0,5 1,0 1,5 2,0 2,5

3,6

3,8

4,0

4,2

4,4

4,6

4,8

5,0

Tra

pped

Ato

mN

um

ber[1

010]

Rubidium Vapor Pressure[10-6

mbar]

6 8 10 12 14 16 18 201

2

3

4

5

6

Flu

ore

scence

[a.u

.]

Time [s]

a b

c d

0,5 1,0 1,5 2,0 2,5

1,8

2,0

2,2

2,4

2,6

2,8

3,0

3,2

3,4

Loss

Tim

eC

onst

ant1/

L[s

]

Rubidium Vapor Pressure [10-6

mbar]

0,5 1,0 1,5 2,0 2,51,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4

2,6

2,8

Loadin

gR

ate

RL[A

tom

s/s]

Rubidium Vapor Pressure [10-6mbar]

Figure 4.8: a) optimum trapped atom number Ntrap in the 3D-MOT as a function of Rb

pressure in the vapor cell. For each data point the power splitting ratio between 2D-MOT

and 3D-MOT was adjusted to optimize Ntrap. The lines are a guide-to-the-eye fit. b) loss

time constant (ΓL)−1, under the same conditions. c) Trapped atom flux Φtrap. d) typical

3D-MOT loading curve, fitted by (2.19), from which Ntrap and ΓL are determined.

There are several important pieces of information to be drawn out of these measurements.

First, the steady state atom number reaches its optimum of (here) 5 · 1010 atoms at

pressures around 1 · 10−6mbar. This pressure is far lower than the 2.5 · 10−6mbar at

which the 2D-MOT reaches its optimum flux ( see section 4.1.1 ). This is due to a

decreasing optical power with pressure in the 2D-MOT: The pressure dependence of

the 2D-MOT flux in section 4.1.1 was measured at constant laser power. Here instead

the power in the 2D-MOT was varied for every pressure to optimize the trapped atom

number. For the lowest pressures, about 42% of the power are directed into the 2D-MOT


- much cooling light is needed to generate enough flux to load the 3D-MOT fast. This

fraction continuously decreases to about 30% at high pressure, where the increased mean

velocity of beam atoms ( cf. figure 4.3 ) demands a higher intensity in the 3D-MOT to

trap a large fraction of atoms from the beam.

Second, the loss constant ΓL shows a complementary trend to the atom number,

reaching a minimum at the same pressure 1 · 10−6mbar at which the number is optimum.

This means that the trapped atom flux reaches its maximum Φtrap ≈ 2.4 · 1010 at/s at a

Rb pressure of 1 · 10−6mbar. Comparing the trapped flux to the flux one expects from

the 2D-MOT under these circumstances ( section 4.1.1 ), one can deduct the loading

efficiency and hence a measure for the capture velocity of the 3D-MOT:

The expected flux of the 2D-MOT at a pressure of 1 · 10−6mbar is about

Φ(p=1·10−6 mbar) = 4.6 · 1010 at/s, about 77% of the optimum flux of Φmax = 6 · 1010 at/s.

Furthermore, here we shine in only 40% of the maximum power of 180mW per beam.

The flux is thereby reduced to 75% of the optimum flux. Taken together one expects a

total flux of Φtot = 0.75 · 0.77 · Φmax = 3.5 · 1010 atoms/s. The fraction of the flux below

a certain cutoff velocity has been calculated in [2]. Equating the result to the trapped

flux Φtrap ≈ 2.2 · 1010 at/s, one obtains a cutoff velocity of about 80m/s, even higher

than the 65m/s expected from the calculation in appendix A. The trapping efficiency of

atoms out of the beam is ηtrap =Φtrap

Φtot= 0.62.

A further information from the complementary behavior of trapped atom number and

loading time 1ΓL

is that the loss rate ΓL depends substantially on the flux from the

2D-MOT.

4.2.2 Analysis of the Decay of the Trapped Atom Number

Loss Mechanisms in the 3D-MOT

Decay curves of the trapped atom number were recorded at UHV conditions of 10−10mbar,

and a low vacuum side Rb pressure of 2 · 10−6mbar, to determine the decay constants

ΓD and Γ′D ( equation 2.22 ): If loading is terminated by turning off the 2D-MOT

magnetic field and closing the mechanical shutter simultaneously, the decay constant

is ΓD = (4, 71±0.34 s)−1. If the shutter is left open and the loading process is terminated

by turning off the 2D-MOT B-field only, a higher decay rate Γ′D = (2, 16 ± 0.07 s)−1 is

obtained. In the same experimental run, the loss rate under loading conditions was deter-

mined from a loading curve, yielding a value of ΓL = (1, 40 ± 0.01 s)−1. Given the three

values ΓD, Γ′D and ΓL one can single out the contributions of background gas, thermal

beam and cold 2D-MOT beam according to equation 2.22 to the 3D-MOT losses, with

the following result:


Γbg = (4, 71± 0.34 s)−1 (4.5)

Γth = (3, 99± 0.46 s)−1 (4.6)

Γ2D−MOT = (3, 98± 0.16 s)−1 (4.7)

These numbers show that the 2D-MOT beam and the thermal flux from the low vacuum

side have a significant influence on the loss rate and therefore the equilibrium properties

of the MOT.

Even when the mechanical shutter is closed during decay, the trap lifetime depends on the

pressure at the low vacuum side of the setup: Lifetimes were found to vary between 11.3

s at a Rb pressure of 4.2 · 10−7mbar and 7.2 s at a Rb pressure of 2.8 · 10−6mbar. This

could be explained by a continuous atomic beam of Rubidium that was initially deposited

on the inner walls of the tube.

From the measurements in this section the following conclusions were drawn:

• Trap lifetimes of about 15 s were achieved by improving the UHV pressure to

2 · 10−11mbar, whereby Γbg is reduced. Further improvements are necessary for

evaporative cooling to Bose-Einstein condensation.

• The loss constant Γth is reduced by working at a lower Rb pressure than 2·10−6mbar,

which is possible without lowering the loading rate ( section 4.2.1). If necessary,

the differential pumping tube of 6mm diameter can be replaced by a smaller one.

This will lower the flux of the 2D-MOT beam, but might be overcompensated by a

reduction in losses due to the thermal background flux, and due to a lower atomic

beam velocity.

• The differential pumping tube should be continuously cooled in experiments, and

heated from time to time in between experiments to clean it from adsorbed 87Rb.

Nonexponential Decay

In general, the decays are dominated by one-body losses due to collisions with background

gas atoms. Non-exponential decays are rarely seen, indicating a low atom density. One

counter-example is shown on a semi-logarithmic scale in figure 4.9.

By fitting equation (2.25) to the data, the two-body loss constant β is determined from

the fit parameter B via β = B·V ΓDN0

, with ΓD = 1τ. The initial atom number N0 is

determined separately by a CCD camera, as well as a volume of V = 3.14 cm3, with an

uncertainty of δVV

= 30%. The error in β is given by δββ

= δBB

+ δVV

+ δN0

N0+ δτ

τ, with

δB and δτ given from the fit, and an estimated uncertainty in N0 of 30%. The result

is β = (4.5 ± 3.1) · 10−12 cm3/s, which is comparable to β = 1 · 10−12 cm3/s expected


0 5 10 15 20 25 30 35 40

6,7x108

1,0x109

3,3x109

6,7x109

1,0x1010

3,3x1010

6,7x1010

N(t)=N

0*Exp(-(t-t

0)/ )/(1+B*(1-Exp(-(t-t

0)/ )))

Chi^2 = 0.00337R^2 = 0.99528

t0

0.12208 ±0

N0

(6.6781±0.02141) 1010

6.84894 ±0.10872

B 0.95606 ±0.04976

Tra

pped

Ato

mN

um

ber

Time [s]

Figure 4.9: Non-exponential decay of the atom number in the trap after termination of the

loading process. The fit function is equation (2.25). From the parameter B the two-body

loss constant β is determined.

from [54]. This confirms the statement that the elongated trap shape does not have a

significant influence on two-body collisions.

4.2.3 Trapped Atom Number and Density

Atom Number

The trapped atom number is determined by the detection of fluorescence on a calibrated

CCD camera or photodiode, as described in section 4.2. The highest number of trapped

atoms determined in this way from equation (4.4) is (8.5+3.3−2.0)·1010 atoms. Typical numbers

obtained in daily operation are about 30% lower.

As an independent check of the atom number, the following measurement is performed:

A probe laser beam is transmitted through the short axis of the cloud, and its absorption

signal is recorded while the laser frequency is swept around the F = 2−→F ′ = 3 resonance.

The laser intensity is kept below saturation in order to avoid power broadening. In this

way a trace like displayed in figure 4.10 results.

The absorption line is broadened due to optical thickness of the cloud, according to

equations (2.12) and (2.13). A fit of (2.12) to the absorption trace determines the optical

density


400 500 600 700 800 900

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

Abso

rptio

nS

ignal[

a.u

.]

Frequency [a.u.]

0,0

0,5

1,0

1,5

2,0

2,5

36 MHz

Satu

ratio

nS

pect

rosc

opy

Sig

nal[

a.u

.]

78,5 MHz

Figure 4.10: Absorption of a probe laser beam scanned around the MOT cycling transition

is broadened due to the optical thickness of the atom cloud. The upper trace shows two

doppler-free resonances of a saturation spectroscopy signal, as a frequency reference.

OD = σ0

∫ b

−bn(x)dx ≈ n · l · σ0 (4.8)

of the atom cloud. The resonant photon absorption cross section is σ0 = 3λ2

2π=

2.91 · 10−13m2, n(x) is the ( approximately constant ) number density of trapped atoms

along the laser axis, −b and b are the boundaries of the cloud, and the absorption length

l of the laser beam is taken to be the length between the points where the cloud density

falls off to one half of its constant central value.

The broadest absorption curves found in this way are 41MHz wide, or 6.9 natural

linewidths. The fit gives an optical density of n · l · σ0 = 35. The atom number is

then determined by

N = n · l · A = OD · Aσ0

(4.9)

The area A is perpendicular to the laser beam axis, and is determined by a CCD camera

image with the CCD viewing the MOT on the axis of the laser beam. A is taken to be

the area inside the curve on which the density falls off to one half of the central value.

The atom numbers determined in this manner lie in the range of 4 · 1010 atoms, approxi-

mately a factor 2 lower than the numbers measured by fluorescence.


Density

In a very similar way, also the cloud density can be determined: The optical density is

obtained as above. Now, the length l is determined by taking a CCD picture, with the

camera mounted perpendicularly to the laser beam axis. The position of the laser beam in

the cloud is determined separately by increasing the laser power, thus blowing a ”tunnel”

into the cloud by radiation pressure, and taking an additional picture of the disturbed

MOT. The density is obtained by n = ODσ0l

.

The determined average densities lie in a range between 1 and 2 · 1010 atoms/cm3. This

value is by approximately a factor of 10 higher than expected from Ketterle’s semi-

empirical model [16], equation (2.15). The reasons for this could be our elongated MOT

shape ( aspect ratio a = 6 ), which leads to a factor of 2.5 higher density, given by equa-

tion (2.16), compared to a spherical MOT. Furthermore, there could be a difference in

typical densities between Sodium ( for which the empirical data in [16] are taken ) and

Rubidium.

4.2.4 Influence of Trapping Laser Detuning

It is suspected that the 3D-MOT works best at a larger detuning than the 2D-MOT:

The size of the trapping volume is terminated at the position where the Zeeman shift

equals the detuning ( Appendix A ). The large capture velocity needed for the 3D-MOT

to trap a sufficient fraction of atoms from the beam requires a large trapping volume

and therefore a large detuning, whereas for the 2D-MOT with a capture velocity of only

35m/s, a smaller detuning is optimum. In figure 4.11 the fluorescence intensity of the

3D-MOT is shown as a function of time, while the trapping laser detuning was scanned at

a rate of 400 kHz/sec. The polarization spectroscopy signal from the stabilization of the

trapping laser is shown as a frequency reference. This spectroscopy signal is offset with

respect to the fluorescence by an amount of +(200−95)MHz, corresponding to the drive

frequencies of the AOM’s used in the spectroscopy beam and towards the experiment,

respectively ( cf. figure 3.3 ).

The 3D-MOT fluorescence reaches its maximum at a detuning δ3D = −(2.3± 0.3) Γ from

the F = 2 −→ F ′ = 3 transition. This is to be compared to the optimum detuning of the

2D-MOT, δ2D = −1.9 Γ [2]. The optimum 3D-MOT detuning is only slightly higher than

that of the 2D-MOT. This can be understood, as the 3D-MOT is only optimally loaded

when the detuning is optimum for 2D-MOT operation.

The FWHM width of the fluorescence is 2.5 Γ. This does mean, that only in this region

efficient collection of atoms and trapping takes place. However, for short times it should

be possible to increase the detuning significantly further without deteriorating the trap

function - only the atom collection becomes inefficient. This is important for loading the


400 500 600 700

1,2

1,4

1,6

1,8

2,0

2,2

2,4

2,6

2,8

3,0

2D

+ (200-95) MHz

133.5MHz

MOT Fluorescence

Spectroscopy Signal

Flu

ore

scenc

[a.u

.]

Frequency [a.u.]

4,8

5,0

5,2

5,4

5,6

5,8

6,0

5

6

Figure 4.11: Fluorescence intensity of the 3D-MOT as a function of trapping laser detun-

ing. As a frequency reference, the polarization spectroscopy signal is shown.

magneto-optically trapped atoms into a purely magnetic trap, as will be discussed below.

4.2.5 Velocimetry of Cold Atoms

The last important property of the trapped sample of atoms is its temperature. Usually,

MOT temperatures are measured by releasing the cloud of atoms. By temporal expan-

sion due to thermal motion, information about velocity space is transferred into position

space and provides a measure for the atoms’ temperature. This information is easily in-

terpretable if the source of expanding atoms is point-like. In the case of a large MOT the

time of flight (TOF) data are hard to interpret, as finite-size effects are still considerable

after long expansion times. Therefore a velocimetry method was applied, that does not

rely on expansion, but provides local information about temperature in the cloud. Here,

the setup, the physical principle and an attempt to implement this method are described.

Two AOM Double-Pass Configurations for a Diode Laser

A diode laser (TUI Optics DL 100) is stabilized by doppler-free polarization spectroscopy

( see appendix B ) close to the 5S1/2,F=2 −→ 5P3/2,F′=3 cycling transition.

Its output power of about 25mW is split into two beams with a variable ratio of powers,

who are directed into a pair of AOM double pass configurations, as shown in figure


toexperiment

Amplifier2W

Amplifier2W

f=100

f=100

f=500

f=500

f=300

Driver:

FrequencyAmplitude A1TTL

Driver:

FrequencyAmplitude A2TTL

/4

/4

from spectroscopy

AOM80 MHz

AOM

80 MHz+

Figure 4.12: Two AOM double pass configurations allow to introduce a controlled frequency

difference between two laser beams. The laser beam enters from the spectroscopy setup for

its stabilization, is split into two beams with variable intensities, which are frequency offset

by 160MHz and 160MHz + 2δAOM , respectively, and directed towards the experiment.

4.12. The beams enter the double pass by reflection at a polarizing beamsplitter (PBS)

cube, and become circularly polarized after the λ4plate. After passing the AOM, the

first diffraction order beams acquire frequency shifts of 80MHz and 80MHz + δAOM .

They are retroreflected into themselves, and are further diffracted and frequency shifted

by the same amount, resulting in total detunings of 160MHz and 160MHz + 2δAOM ,

where 2δAOM < 1MHz is required for constant diffraction efficiency. The total efficiency

of each double pass configuration ( ratio of optical output power after the final PBS

cube to incident power before entering the first PBS cube ) is about 60%, allowing for

a frequency-shifted output of about 14mW if one beam is operated separately. The two

beams of variable relative frequency detuning are overlapped in the center of the MOT

for velocimetry measurements, as shown in diagram 4.14.


Physical Principle

The underlying mechanism for the velocimetry method [63] is stimulated optical Compton

scattering (SOCS): A photon from one of the laser beams can be coherently scattered into

the other beam by purely stimulated processes. If the polarizations of the two beams are

equal, the internal state of the atom is never changed in this two-photon process, via

a virtual intermediate state. Scattering is sensitive to the velocity vat of the scattering

atom. Energy and momenetum conservation demand

q · vat = 2δAOM +|q|22m

(4.10)

where |q| = 2|k| sin(Θ2) is the scattering vector of the photon ( wavevector k ), and Θ is

the angle between the two laser beams.

Scattering takes place from both beams, and the net number of scattered photons from one

beam into the other is proportional to the population difference of the atoms in resonance

with either direction of the two-photon process. If the relative laser detuning 2δAOM is

scanned around zero, the resulting transmitted intensity through the MOT is proportional

to the derivative of the atomic velocity distribution. For a sample of atoms in thermal

equilibrium, the velocity distribution is Gaussian, characterized by a temperature, and

the Raman signal SR is the derivative of this Gaussian [64]

SR ∝ 2δAOMkΘ| vat|Exp

(−(2π · 2δAOM)2

2(kΘ| vat|)2); (4.11)

which is plotted for an expected T = 300µK and Θ = 10 in figure 4.13

-150 -100 -50 50 100 150

-0.6

-0.4

-0.2

0.2

0.4

0.6

[kHz]

S R[a.u.]

Figure 4.13: Calculated Raman signal, equation (4.11), due to stimulated optical Compton

scattering of a sample of atoms at a temperature of 300µK. The two Raman beams are

overlapped at an angle of Θ = 10. Peak separation and width are proportional to the

temperature.


Photon redistribution constitutes a gain for one of the beams, and a loss for the other.

Therefore the transmission signals of both beams are complementary to each other, which

means that subtraction of both transmission curves results in an addition of both Raman

signals, while fluctuations ( e.g. of the MOT fluorescence ), are suppressed.

Experimental Implementation

In the experimental setup for this method, two beams with intensities below saturation

are crossed in the MOT, shone in through the lower vacuum window, enclosing an angle

of about 10 ( figure 4.14 ).

While the relative detuning between the beams is scanned between 2δAOM = ±400MHz

and ±1MHz, the transmission signals from both beams were recorded on identical pho-

todiodes, and the difference signal was observed. The absolute detuning of both beams

with respect to the F = 2−→F ′ = 3 transition is chosen to lie just outside a wing of

the absorption curve, about 30MHz or 5 natural linewidths from resonance, a compro-

mise between not too high absorption and decreasing signal amplitude with increasing

absolute detuning. The beam overlap in the MOT was secured by adjusting the beams

with diaphragms closed to 1mm diameter. To ensure that the beams crossed in the atom

cloud, the MOT was reduced in size by closing diaphragms in both the axial and radial

MOT beams. Thus only the central part of the MOT is present, and is blown away by

radiation pressure when hit by a resonant laser beam. Subsequently the intensities of the

beams are reduced, and their detuning from resonance is increased. Thereby a small hole

is blown into the MOT, which is observed for both beams to ensure overlap.

It was not possible to obtain an interpretable signal. The reason for this failure was that

the AOM’s were not phase-locked, and their frequencies showed broader widths than the

expected Raman signal, which was thereby smeared out. It was therefore decided to apply

instead time-of flight methods for a rough temperature measurement.

4.2.6 Temperature Measurements

Two different methods were applied for temperature characterization, both time of flight

methods. Therefore they are not ideally suited for characterizing a large MOT, but precise

enough for our purposes. These methods will be described in the following sections.

Decay of Atomic Density

In this method, the cloud is released from the 3D-MOT in the following sequence: First,

the magnetic field of the 2D-MOT is turned off, followed by a 10ms waiting period for

the atomic beam to fade out. Then the 3D-MOT magnetic field is turned off, followed

1ms later by the cooling light. A probe laser beam is shone in vertically, along the axis


DensityMeasurement

Velocimetry

Time of Flight

d(t) l(t)~~

d0

l =2r0 0

Figure 4.14: Left: setup for three temperature measurement methods for cold atoms -

velocimetry is a nonlinear spectroscopy method in which two beams are overlapped in

the MOT, and at least one of the transmitted intensities is recorded. In the time of

flight (TOF) method the atoms are released from the MOT and pass through a resonant

laser beam shined in 7 cm below the trap. From the time-dependent absorption signal the

temperature is deduced. The third method employs a probe beam directed along the axis of

the falling and expanding cloud, which is scanned around resonance to obtain an absorption

spectrum. The time dependence of this signal allows the temperature extraction. Right:

definitions of MOT size parameters.

of the falling cloud ( to be precise, under an angle of 3.7 to the vertical, to avoid clipping

the MOT beams ). The frequency of this laser is scanned around resonance, and after

a variable time delay, the absorption spectrum is recorded on a photodiode above the

vacuum chamber, resulting in absorption profiles as shown in figure 4.10.

The column density is extracted, by fitting equations (2.12) and (2.13) to this absorption

signal. As the cloud expands, the column density decreases with time ( see figure 4.14 ),

according to n(t) · l(t) = NV (t)

· l(t) with


V (t) =4π

3r(t)2d(t) (4.12)

l(t) = 2r(t) = 2√r20 + v2

tht2 (4.13)

d(t) =√d2

0 + v2tht

2 (4.14)

so that the optical density n ·l ·σ0, with σ0 from (2.13), has the following time dependence:

(n · l · σ0) (t) =Nσ0

4π3

√r20 + v2

tht2√d2

0 + v2tht

2(4.15)

This temporal dependence results first only in a decreasing width of the absorption

profile, until the cloud becomes transparent to the laser light after about 20 ms. From

then on the width of the signal stays fixed at about the natural linewidth, only the depth

of the signal decreases until it finally vanishes after ≈ 250ms, when the last atoms have

hit the lower vacuum window. Absorption traces are taken from 2 to 200ms after the

release of atoms from the MOT. Each trace is fitted by (2.12) to obtain the quantity

n · l · σ0, and gives one data point in figure 4.15. The resulting curve is fitted by the

function (4.15), and from the fit parameter vth =√

3kBTm

the temperature is calculated.

10-3

10-2

10-1

0,1

1

10

Opt

ical

Thi

ckne

ssn

l

time [s]

"far detuned" MOT

10-3

10-2

10-1

0,01

0,1

1

10

Optic

alT

hic

kness

nl

time [s]

MOT near resonance

Figure 4.15: Decrease of the column density of the expanding atom cloud with time. Left:

MOT under normal operating conditions, i.e. at a detuning of δ ≈ −2Γ right: MOT

further detuned from resonance. Note the different scales on the y axis. From these decay

curves the temperatures of the atom clouds can be determined.

In figure 4.15 two curves are shown, for a MOT under usual operation conditions (left)

and for a larger detuning (right). The magnitude of the detuning was not measured, but

is estimated to be −(4 − 5)Γ from section 4.2.4. The temperature of the MOT under

usual conditions is T = 3.1 ± 0.5mK, and for larger detuning T = 550 ± 100µK. The


temperature under normal conditions is surprisingly high compared to other MOTs, with

temperatures typically a few 100 µK, close to the Doppler limit.

There are two reasons for this result:First, the number of atoms in our MOT is about two

orders of magnitude larger than in typical 87Rb MOT’s. From the number dependence of

the MOT temperature, equation(2.11), two orders of magnitude in number can account

for a factor of 5 increased temperature. Second, due to the elongated trapping volume,

the axial laser beams are attenuated severely. Thereby the central part of the cloud

experiences heating due to multiple scattering, which is uncooled in the axial direction.

Time of Flight

To check the temperature results of the decay of atomic density the time of flight method

(TOF) was used. Atoms were released from the MOT by first turning off the magnetic

fields, leaving the atoms in a σ+ σ− optical molasses. The light fields were turned off after

a variable cooling time in the molasses, and TOF traces are shown for several molasses

durations.

0,05 0,10 0,15 0,20 0,25 0,30

1,06

1,08

1,10

1,12

1,14

1,16

1,18

1,20

1,22

1,24

T: Molasses Duration

MOT shutoff

T=25 msT= 3 msT= 1 msT=0.6 msT=0.2 msT=0 ms

dete

ctor

sign

al[a

.u.]

time [s]

Figure 4.16: Time of Flight absorption traces for different durations of molasses post-

cooling. For t ≥ 0.6ms, the signal is clearly seen to decrease in width and increase in

height, indicating the progressive cooling. For shorter cooling times, the signal is double-

peaked, which we attribute to cuts into the transverse velocity distribution at high temper-

atures due to the Ioffe bars.

In figure 4.16, temporal absorption profiles recorded by the time of flight method are


shown. For the shortest molasses times and without molasses, the signal consists of a

sharp spike immediately after turning off the magnetic fields, and a long flat tail. With

increasing molasses duration, a second peak rises and becomes sharper, while for durations

longer than 0.6 ms the initial peak vanishes.

For a duration longer than 0.2 ms the second peak can be fitted by the function N(t),

equation (C.3), see appendix C. From this fit, the temperature is determined and shown

in figure 4.17, as a function of molasses duration:

0 5 10 15 20 25

0

200

400

600

800

1000

1200

1400

Tem

pera

ture

[ K

]

Molasses Duration [ms]

Figure 4.17: Temperature of atoms for different durations of molasses cooling. The tem-

perature of the relatively hot steady-state MOT falls off within a few milliseconds to a new

equilibrium temperature of 300µK.

The temperature drops from values above 1 mK in the MOT to a steady state value

of 300µK in the molasses, roughly consistent with (4.17), which predicts ≈ 200µK for

our parameters. The time scale of this decay ( about 2 to 3ms ) is somewhat surprising.

From simple Doppler cooling one would expect a cooling time of atoms with initial thermal

velocity vi, to a velocity vf ≈ 17 cm/s ( corresponding to T = 300µK ):

tcool ≈ m(vi − vf )τsck

≈ 20µs (4.16)

where τsc = 130 ns has been taken ( cf. equation 4.3 ), and vi = 1m/s, corresponding to

a temperature of 10 mK. Obviously, the real cooling takes two orders of magnitude longer

than expected. One might attribute this to an inhibition of cooling by multiple scattering,

which becomes less pronounced as the density in the molasses decreases due to diffusion

4.3 Conclusions 65

out of the original MOT trapping volume. One expects a significant reduction in multiple

scattering on timescales when the density decreases to half its initial value, which should

happen, according to the results of section 4.2.6, after a time t ≈ r0vi

= 10ms. Here,

r0 ≈ 1 cm is the initial MOT size, and again vi is taken to be 1m/s. Cooling is faster

than expected from this argument, as after the a time of 2 ms no substantial change in

density can have occurred for any reasonable initial temperature.

From the measurements in this section is is clear that even after molasses post-cooling

of the MOT, no sub-Doppler temperatures could be reached ( TD = 150µK for 87Rb ).

This is not surprising, given that the molasses was still operated at high intensity and a

detuning close to resonance ( δ ≈ −2Γ ). According to equation (4.17) lower (possibly

sub-Doppler) temperatures can be reached by decreasing the intensity and increasing the

detuning of the molasses light.

4.3 Conclusions

The 2D- 3D-MOT system described in the previous chapters is ideally suited to collect

high numbers of atoms (8 · 1010) on short time scales of a few seconds, but it has two

severe drawbacks when loading a magnetic trap is considered.

The first is the relatively low density (2 · 1010 /cm3), and corresponding large volume, as

well as a slightly irregular shape caused by inhomogeneities of the laser beam profiles.

The second is the high temperature caused by increased momentum diffusion due to multi-

ple scattering of trap laser photons by the large number of atoms. For loading a magnetic

trap from this MOT the following scheme is proposed, based on the results of [18]:

First, change the detuning from −2Γ to −(5 − 10)Γ and wait about 1 to 5ms for equi-

libration of the atoms. This leads to a reduced photon scattering without affecting the

function of the trap. A decrease in temperature by a factor of 5 to 10 should result.

Second, ramp up the radial magnetic field gradients from 15G/cm to around 50G/cm,

and also increase the axial confinement. In this way a compressed MOT results, and a

density increase by a factor of 10 should be possible. The density equilibrates on such

short time scales that the loss of atoms is negligible. Furthermore the irregular shape

vanishes and a Gaussian shaped cloud results.

After a short equilibration phase of the compressed MOT, the magnetic fields should be

turned off, creating an optical molasses. Ramping down the light intensity over a few

ms should result in sub-Doppler temperatures, according to (4.17), without a large loss

of atoms. It is important to ramp down the intensity at a large detuning because (4.17)

breaks down at δ < 5Γ, resulting in higher final temperatures than expected.

The cold and dense sample of a large number atoms produced in this way is ideally suited

for loading into a magnetic trap and evaporative cooling to Bose-Einstein condensation.

Chapter 5

The Mott Insulator Phase Transition

in an Optical Lattice

Quantum mechanical particles in periodic potentials form delocalized Bloch states.

In the case of a Bose-Condensate in an optical lattice, these delocalized states with

long-range phase coherence share most of the properties of the superfluid free Boson gas.

With increasing lattice potential depth, repulsive short-range atom-atom interactions1

lead to a Mott quantum phase transition to ”insulating” number states on each lattice

site. In contrast to temperature-driven Mott transitions in solid state physics, the phase

transition takes place at T = 0, as the values of two competing energy terms in the

Hamiltonian are varied across a critical value.

If an external potential with linear position dependence is applied, the particles in

the superfluid phase undergo Bloch oscillations, as described in Appendix E. In the

insulating state, the condensate phase is abruptly destroyed, which suppresses the Bloch

oscillations, and may allow the observation of the phase transition.

In the following sections, numerical work on the Mott insulator transition, pre-

dicted by Jaksch et.al. in 1998 [44], is presented. In this publication, the behavior of

Bosons cooled to the lowest energy band of an optical lattice is considered, governed by

the Hamiltonian

H = −J∑i,j

b†i bj +∑i

εi · ni + 1

2U

∑i

ni(ni − 1) (5.1)

Here, J is the single particle tunnelling matrix element between neighboring lattice sites i

and j, εi is the value of an optional external potential at site i, e.g. from a magnetic field.

1Only repulsive interactions are considered, as only in this case large stable Bose con-densates can be formed. The interaction energy is denoted by U , which should not beconfused with the dipole potential UDip(r) and the lattice depth U0.

67

68 CH. 5 The Mott Insulator Phase Transition in an Optical Lattice

U is a short-range interaction energy between two particles caused by cold ( ”s-wave” )

collisions. b†i and bi are Bosonic creation and annihilation operators of particles at lattice

site i, and ni = b†i bi is the particle number operator at site i. The double sum in the first

term over all sites i, and all nearest neighbor sites j, describes the hopping of a Boson

from site j (where it is ”destroyed” by bj), to site i (where it is ”recreated” by b†i ).

The third term in (5.1) describes the interaction of ni particles with their ni−1 partners at

site i, and the 12is there to count the two-particle interactions only once. One variationally

minimizes the grand canonical expectation value (µ is the chemical potential)

〈Ψ | H | Ψ〉 − µ〈Ψ | N | Ψ〉 (5.2)

with respect to parameters f(i)n of an ansatz for the ground state wave function | Ψ〉 =∏

i | Φi〉, where i denotes the lattice site, and | Φi〉 =∑∞

n=0 f(i)n | n〉i is a superposition

of Fock number states at lattice site i. This procedure leads to different types of ground

state wavefunctions, as a function of the parameter

U

zJ(5.3)

where z is the number of nearest neighbor lattice sites. For low values of UzJ, the ground

state corresponds to a superfluid phase of delocalized states with non-vanishing number

fluctuations at each single lattice site (f(i)n = 0 for many n), whereas for high values of

UzJ

the ground state is insulating, characterized by Fock states with fixed occupation

number at each lattice site and vanishing number fluctuations (f(i)n = 1 for one fixed n).

The physics underlying the phase transition is a competition between the parame-

ters U and J . Tunnelling particles will experience an energy gain by their delocalization,

given by the width of the energy band induced by the tunnel coupling J in (5.1). For a

Gaussian wavefunction

Ψ(r) =

(1

πβ2

) 34

Exp

(−

(r

2β

)2)

(5.4)

of width β =(

mωosc

) 12at each lattice site, with ωosc given by (2.46), the tunnelling matrix

element is given by [65]

J =ωosc8

[1−

(2

π

)2] (

d

β

)2

Exp

(−1

4

(d

β

)2)

(5.5)

Here, d is the lattice constant, d = λ/2 in the case of a standing wave. The reason for

the decrease of J with increasing potential depth is the compression of the width β ∝

69

U− 1

40 of the wavepackets, and correspondingly the decrease of the overlap of neighboring

wavepackets. Numerically J is given by

J = 1.800 [Erec] ·(

P [W ]

π(w [mm])2

)· Exp

(−2.742

(P [W ]

π(w [mm])2

)(1/2))

(5.6)

The competing interaction energy term U suppresses tunnelling: If a Boson tunnels to

a site already occupied by another one, they experience the on site repulsion described

by the third term in the Hamiltonian (5.1). This repulsion is increased at high potential

depths, when the tighter compression of β leads to a higher density:

U =4πas

2

mβ3π32

(5.7)

Here, as is the s-wave scattering length2, as = 5.3nm for 87Rb, and β was defined below

(5.4). U can be numerically expressed by

U = 0.328 [Erec] ·(

P [W ]

π(w [mm])2

) 34

= · 16.77 [kHz] ·(

P [W ]

π(w [mm])2

) 34

(5.8)

If, with increasing potential depth, the repulsion U becomes comparable in strength to

the tunnel coupling J, more precisely if the parameter (5.3) reaches the value UzJ

= 5.8,

tunnelling becomes unfavorable, and the transition from the superfluid to the insulating

phase takes place. This parameter is plotted in figure 5.1 as a function of the power of a

single lattice beam, focused to a waist of w = 0.5mm. It can be seen that 10W of laser

power, if available to every standing wave, are more than enough to observe the phase

transition in a lattice of 1mm dimensions.

The phase diagram of the Boson fluid at T = 0 is shown in figure 5.2. Both the chemical

potential µ and the tunnelling matrix element J are normalized by the on-site interaction

U . The phase diagram consists of a superfluid phase, and lobes of insulating phases at

low values zJU. Each of these lobes corresponds to an integer number of Bosons at each

lattice site, this number increases with increasing chemical potential µU. The critical value

for the transition to the lobe with N = 1 particle per site is UzJ

= 5.8.

The formalism of the Hubbard model was also applied to spinor condensates in an optical

lattice, yielding a more structured phase diagram with exotic phases [67]. Numerical

calculations concerning the Mott insulator phase transition can be found in [65].

2As the range of interaction, as λ2 , the interaction takes place only if two particles meet

a the same lattice site, and is hence called on-site interaction.


U6J

P [W]

P [W]P [W]0 2 4 6 8 10

0.1

0.150.2

0.3

0.5

0.7

1

1.52

0 2 4 6 8 100.1

1

10

100

0 2 4 6 8 100.001

0.002

0.005

0.01

0.02

0.05

0.1

0.2

5.83

JERec

UERec

Figure 5.1: Parameters of the Mott-Hubbard model: The on-site repulsion U, the tun-

nelling matrix element J (both in units of the recoil energy Erec), and the unitless ratio

U/6J , all plotted logarithmically against laser intensity in each lattice beam. The ratio

U/6J can be easily tuned experimentally over many orders of magnitude around the critical

ratio for the phase transition, U/6J = 5.83, by variation of the lattice laser intensity.

5.1 Tunnelling of Interacting Atoms in an Optical

Lattice

The Model

In this section a simple model is developed and numerically explored to describe the

tunnelling probability of a single particle out of an accelerated optical lattice. We consider

a one-dimensional model for the quantum mechanical transmission coefficient of a particle

with mass m and energy E through an arbitrary potential barrier V (x), from a starting

point b1 to a final position b2:

T = Exp

(−2

∫ b2

b1

dx

√2m(V (x)− E)

2

)(5.9)

We apply this model to a particle trapped in a lattice potential with an additional external

potential, and calculate the probability of tunnelling out of the lattice, i.e. past the last

potential hill with an energy higher than the tunnelling particle’s total energy, as depicted

5.1 Tunnelling of Interacting Atoms in an Optical Lattice 71

MIN=3

MIN=1

MIN=2

<N>=3

<N>=2

<N>=1

J/U

SF

SF

/U

J /Uc

Figure 5.2: Phase diagram of Bosons in an optical lattice at T = 0. The y-axis is the

chemical potential µ, in units of the interaction energy U, µU. In the superfluid phase µ is

proportional to the atomic density, whereas in the insulating phase the occupation at each

lattice site is a step function of µ. The x-axis is the ratio JU[66], which can in a simplified

way be thought of as the inverse potential depth. The critical value for the transition to

the N = 1 insulating phase is JcU

= (z × 5.83)−1.

in the left part of figure 5.3.

E E

V V

z z

Figure 5.3: Tunnelling out of a lattice with external linear potential. Left: The lattice

is empty except for the tunnelling particle. Right, the lattice is occupied with one (red)

particle per site, leading to an additional effective barrier for the tunnelling (grey) particle

at each site whose chemical potential exceeds the tunnelling particle’s energy E.

In the tunnelling probability through an occupied lattice, we take in account the repul-

sive interaction between particles at doubly occupied sites by replacing the pure dipole


potential by an effective potential, as shown in the right part of figure 5.3. This poten-

tial is given at any point by either the dipole potential or the chemical potential at the

respective lattice site, whatever value is larger. The chemical potential can be considered

as the energy needed to add a particle to a system (here: to a lattice site). Adding a

particle to an already occupied site requires the oscillator ground state energy ωosc2

(

measured from the bottom of the respective potential well ) plus the on-site repulsion U.

If this potential energy is higher than the energy of the tunnelling particle, it constitutes

an effective barrier that decreases the tunnelling probability through the lattice. This

effective potential is depicted in the right part of figure 5.3.

We consider three cases: First, the transmission probability Tn=0 of a particle in an other-

wise empty lattice. The starting energy E in equation (5.9) is the oscillator ground state

energy of the particle’s starting lattice site, and the potential V (x) is the pure dipole

potential. This case we associate with a low-density superfluid phase in figure 5.2.

Second, we assume every lattice site to be occupied with one particle, resulting in a trans-

mission probability Tn=1, where now V (x) is the effective potential described above. This

case represents the N = 1 insulating phase.

Third, we assume the particle under consideration to start from a doubly occupied lattice

site, with all other sites being singly occupied, with a transmission probability Tn=1+ε.

Here, the starting energy is ωosc2

+ U , and again V(x) is the effective potential.

For the dipole potential, equation (2.44) was taken, with a peak intensity I0 determined

by the beam waist w = 0.75mm and the laser power P = 10W in one beam. The external

potential Vext(x) = −F · x is simulated by an acceleration of the lattice, leading to an

inertial force F = m · a on the atoms, see section 6.2.

Results

The resulting ratios of transmission rates are shown in figures 5.4 and 5.5, as a function

of the lattice acceleration a. The acceleration is scaled such that at acceleration a = 1,

the potential is tilted enough that the initial energy of the tunnelling particle is equal to

the chemical potential of a singly occupied neighboring potential well (right inset of figure

5.4 ). Then the tunnelling barriers for an empty lattice and a lattice with one particle per

site become equal, and so do the transmission coefficients Tn=1 and Tn=0. For the lattice

parameters chosen, a = 1 corresponds to a real acceleration of 360m/s.

Figure 5.4 shows the ratio of transmission through the lattice occupied with one parti-

cle per site, to the transmission through an empty lattice. The resulting transmission

coefficients Tn=1 and Tn=0 are different for low accelerations, when many potential wells

have to be passed, and in the occupied lattice the repulsion from many other particles

contributes to the potential barrier. When the acceleration is increased, more potential

wells are energy-shifted below the particle’s energy E and do no longer contribute to the

5.1 Tunnelling of Interacting Atoms in an Optical Lattice 73

0,0 0,2 0,4 0,6 0,8 1,0

1E-6

1E-5

1E-4

1E-3

0,01

0,1

1R

atio

Tn=

1/Tn=

0

Acceleration [skaled units]

Laser Parameters : P=10 WLattice Beam Waist 0.75 mm

z

z

V

V

E

EV E

Figure 5.4: Ratio of transmission coefficients Tn=1 to Tn=0, as a function of scaled ac-

celeration (described in the main text). For a = 0.5, exactly one effective ”interaction

barrier” (left inset) has to be overcome in the occupied lattice in addition to the dipole

potential, leading to a lower transmission coefficient. If a > 1, the tunnel barriers (and

hence the transmission coefficients) for a particle in an empty lattice and one in a lattice

with unity occupation become equal, as shown in the right inset.

barrier. Each kink in the curve results when one more well is shifted such that its chemical

potential is lower than the tunnelling particle’s energy.

In figure 5.5, the ratio between transmission Tn=1+ε starting from a doubly occupied

lattice site, to the transmission Tn=1 starting from a singly occupied site is shown. In

both cases every other site is singly occupied. Starting from a doubly occupied site,

the particle’s energy E contains the on-site repulsion, making the potential barrier

lower, compared to a particle starting from a singly occupied site. Also here, a marked

difference is seen between the two tunnelling probabilities, showing the same features as

in figure 5.4. Because of the different energies E, the transmission coefficients do not

become equal at any acceleration.

So far, only ratios between transmission coefficients have been considered. The

experimentally measured quantity is a tunnelling probability P per particle during the

lattice acceleration phase. To obtain this quantity, the transmission coefficient T has to


0,0 0,2 0,4 0,6 0,8 1,0

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

Ratio

Tn=1/T

n=1+

Acceleration [skaled units]

Laser Parameters : P=10 WBeam Waist 0.75 mm

Figure 5.5: The ratio of the transmission coefficients Tn=1+ε to Tn=1, shows that particles

in the superfluid phase Tn=1+ε have a higher probability of tunnelling out of the lattice

than particles in the insulating state Tn=1. This marked difference can be used to ”clean”

the lattice from the superfluid phase, leaving behind a pure insulating phase.

be multiplied by the ratio of acceleration time to Bloch period. This ratio is the number

of tunnelling ”attempts” per particle out of the lattice during the experiment. This

quantity is shown in figure 5.6 for the n = 0 and n = 1 case. The beam waist is chosen to

be 0.9mm, which yields a measurable tunnelling probability per atom for the observation

of the first kink at large accelerations.

The transmission probability is only high for large accelerations. Therefore probably only

the first kink and the difference between the n = 0 superfluid phase and the n = 1 insu-

lating phase will be experimentally observable. The experimental procedure will be, to

load condensates of different density into the lattice, and measure the tunnelling output

for each density at different accelerations.

As the condensate is approximately Gaussian shaped, its density is not constant, which

makes it experimentally difficult to observe density-dependent effects. Also here, tun-

nelling can help: Sites initially multiply occupied can be depleted until occupied by only

one particle, due to the higher tunnelling rate compared to singly occupied sites. With

this tunnelling pre-selection uniform occupations should be created.

5.2 Bloch Oscillations of a Phase Fluctuating BEC 75

0,0 0,2 0,4 0,6 0,8 1,0

1E-4

1E-3

0,01

0,1

1

corresponds to 360m/s2

laser power 10W @ 532 nmbeam waist 0.9 mm

tunnelin

gpro

babili

typerato

m

acceleration [scaled units]

empty latticeone atom per site

Figure 5.6: Here, the tunnelling probabilities per atom during an acceleration time t =

10ms, Pn=0 and Pn=1, are shown. The rightmost kink at a tunnelling probability of 20%

should be visible, as well as the difference between the two curves for lower accelerations.

5.2 Bloch Oscillations of a Phase Fluctuating BEC

We now consider an array of Bose-Einstein condensates in the potential wells of a one-

dimensional optical lattice (lattice constant d). If an external force F is applied on the

atoms, and if there is perfect phase coherence between the different condensates, the

tunnelling current will form a pulsed output [45] with a periodicity

TBloch =h

F · d (5.10)

which is called the Bloch period. In the following, a model is described to quantitatively

understand this pulsed current. The influence of phase fluctuations of condensates in the

individual potential wells is considered. Such fluctuations can be of technical nature, in-

duced by noise in the lattice, but they can also be forced by the number-phase uncertainty

relation

∆N ·∆Φ ≥ 2π (5.11)

If tunnelling between neighboring potential wells is suppressed by increasing the potential

barriers between them, the number fluctuations of atoms in individual sites are squeezed

below Poissonian statistics, therefore increasing the phase uncertainty [42].

The Mott insulator phase transition described in the last section is marked by an abrupt


formation of Fock states with a fixed number of atoms on each single lattice site, whereby

the phases are abruptly randomized.

For the study of both phenomena, it is important to understand the influence of phase

fluctuations on the pulsed tunnelling output due to Bloch oscillations of the BEC.

An example of Bloch pulses is displayed in the left diagram in figure 5.7. It is computed

by phase-coherently summing over the matter-wave tunnelling outputs of each single

potential well. In this way one arrives at a wavefunction Ψ(z, t) and a probability density

n(z, t) = Ψ∗Ψ for the outcoupled atoms to be found at a position z at a time t [45]

Ψ(z, t) =

jmax∑j=1

AjExp

(i

(m

2

3

√2g(|z − zj|) 3

2 − ωjt− Φj

))(5.12)

Here, Aj is an amplitude factor proportional to the occupation of the j′th potential well

( located at position zj = j · λ/2 ), ωj =mgzj

is the frequency at which the phase of a

wavepacket in the j′th well evolves, and jmax = 30 is the number of occupied potential

wells. For the acceleration, the earth′s gravity is taken, so that the model describes the

output of a vertical standing wave under the influence of gravity. If the lattice is accel-

erated to simulate the external potential, g has to be replaced by a general acceleration

a, and the output has to be transformed into the laboratory frame, as equation (5.12)

describes the outcoupled wavefunction in the rest frame of the lattice.

For the results in figure 5.7 a Gaussian occupation with FWHM 2/5 of the 30 lattice

wells was chosen, but the exact shape of the occupation is uncritical for the output.

The chosen observation point z lies 30.000 potential wells below the center of the lattice,

corresponding to a distance of 8 mm.

For the left diagram in figure 5.7, all phases Φj were set to zero (perfect phase coherence

between all potential wells). In this situation, very similar to a mode-locked laser, the

output is sharply pulsed, with a width inversely proportional to the number of contribut-

ing potential wells, and a repetition period corresponding to the oscillation frequency of

the atomic wavepackets (given by equation (5.10)). The role of the equidistant mode

frequencies of a mode-locked laser is taken over by the frequencies ωj by which the phases

of the wave packets in the j′th wells evolve.

The right diagram shows the output under the same circumstances, except that now the

phases Φj are allowed to fluctuate. More precisely, the phase Φ0 of wavepacket in the first

potential well is set to zero, and the phases Φj+1 are subsequently computed from Φj by

adding a random number with a Gaussian probability distribution of width σΦ. To obtain

reliable results, the calculation is repeated ten times with new sets of random numbers,

and the output functions n(z,t) are averaged. For the right diagram in figure 5.7, σΦ = 4π10

is chosen.

If the strength σΦ of the phase fluctuations is increased, the Bloch pulses get smeared

5.2 Bloch Oscillations of a Phase Fluctuating BEC 77

-0.00806 -0.00804 -0.00802 -0.00798

20

40

60

80

100

120

140

-0.00806 -0.00804 -0.00802 -0.00798

20

40

60

80

100

120

140

Without Phase Fluctuations

Position below lattice [m] Position below lattice [m]

With Phase Fluctuations:

Probability Density [a.u.] Probability Density [a.u.]

Figure 5.7: Pulsed tunnelling output n(z,t) of a BEC in an optical lattice due to Bloch

oscillations. Left: perfect phase coherence among condensates across the lattice is as-

sumed. Right: the visibility of Bloch oscillations deteriorates under the influence of phase

fluctuations.

out. A measure of this broadening is the ratio of the width w of a pulse to the separation

TBloch between two subsequent pulses, which is plotted over σΦ in figure 5.8.

0,0 0,2 0,4 0,6 0,8

0,0

0,1

0,2

0,3

0,4

0,5

0,6Model: y = Sqrt ( a^2 + ( / ( b ) )^4 )

Chi 2 = 0.00036R^2 = 0.9876

a 0.09497 ±0.00971b 1.108616 ±0.015209

Ratio

:P

eak

Wid

th/P

eak

Dis

tance

Phase Fluctuation [ ]

Figure 5.8: The ratio of width to spatial separation of Bloch pulses increases under the

influence of phase fluctuations. The data points are results of a numerical simulation of

equation (5.12). The fit based on the analytical result (5.14) shows excellent agreement.


The resulting curve can be qualitatively (and, as it turns out, also quantitatively) under-

stood from an analogy to diffraction from a multislit grating. The ratio of the diffraction

peak width ( corresponding to the temporal width w of one Bloch pulse ) to their angular

separation ( corresponding to the Bloch period ) is the inverse of the number of slits within

the coherence length of the wavepacket ( here the number of potential wells within which

the phase fluctuations of the condensates are less than π ). The result of this analogy is

that the ratio of pulse width w to period TBloch is given by

w

TBloch=

√(w0

TBloch

)2

+(σΦ

π

)4

(5.13)

where w0 is the (temporal) width of the Bloch-pulse without phase fluctuations ( de-

termined solely by the number of potential wells that are populated ). The calculation

leading to equation (5.13) is described in appendix E. In figure 5.8, the function

w

TBloch=

√a2 +

(σΦ

bπ

)4

(5.14)

is fitted to the numerical results based on equation (5.12), where a =(

w0

TBloch

)and b

is a parameter that determines the ”stiffness” of the coherence, i.e. which amount of

phase fluctuations σΦ ( in units of π ) is necessary to broaden a Bloch pulse, that is

infinitesimally wide without phase fluctuations, such that its temporal width exactly

equals the Bloch period. In the model (5.13), b = 1, and the fit in figure 5.8 yields

b = (1.11± 0.02), remarkably close to the analytical model.

In summary, we have developed a numerical model that determines the amount of

Bloch pulse broadening by a given strength of random phase fluctuations, and compared

it to a simple analytical calculation. The results are in excellent agreement. No

assumptions are made about the cause of the phase fluctuations.

If one is interested in number squeezing by suppressed site-to-site tunnelling in an optical

lattice, one has to find a model for number fluctuations caused by tunnelling of atoms

to neighboring lattice sites, as a function of potential depth. By the number-phase

uncertainty relation (5.11) and our model (5.13) one can relate these fluctuations to the

broadening of Bloch pulses, which is the quantity to be measured experimentally.

The Mott insulator phase transition could then be detected in the following scheme:

The Bloch pulse width is measured with increasing potential depth. When the potential

depth reaches the critical value for the Mott phase transition, one expects a sudden

decrease of the number-fluctuations, with respect to those predicted by a model based

on single-particle tunnelling. The corresponding increase in phase fluctuations should be

measurable as an abrupt increase of the Bloch pulse width.

Chapter 6

Design of an Optical Lattice and

Experimental Preparations

In this chapter the design and experimental issues of a 3D far detuned optical lattice are

discussed. The lattice is designed specifically to study the Mott insulator phase transition.

An extremely low scattering rate of lattice photons is therefore required not to destroy

the coherent evolution of condensate wavepackets. At the same time a sufficiently strong

trapping potential has to be maintained. These requirements are met by far-detuned

optical lattices, as discussed in section 2.5. A Verdi V10 laser, producing a single mode

output of 10W at λ = 532nm, was chosen as the lattice laser. In this way a blue-detuned

optical lattice results in which atoms are localized in dark regions. In practice, a CO2

laser also fulfills the requirements, but its wavelength of 10.6µm and hence a factor of

20 larger lattice periodicity, is not well suited for the desired tight confinement of single

atoms at individual lattice sites.

6.1 Numerical Values for Time and Energy Scales

In this section, numerical values for the characteristic energies and time scales of optical

lattices are given, and it is discussed if experiments in a lattice of ≈ 10W laser power per

beam, focused to a diameter of 1mm are feasible. These values ensure a ratio UzJ

5.83 (

equation (5.3) ), as required to observe the insulating state of Bosons in an optical lattice.

To get a feeling for the lattice potential depth U0, we express it as

U0 = 3.655 [Erec] · P [W ]

π(w [mm])2= kB · 1.42 [µK] · P [W ]

π(w [mm])2(6.1)

with the recoil energy (2.36). For a power of P = 10W per lattice beam and a waist

radius of w = 0.5mm, one obtains U0 = 46.6Erec = kB · 18.1µK.

In figure 6.1, the oscillation frequency (2.46), the tunnelling rate of a Gaussian wave

79

80 CH. 6 Design of an Optical Lattice and Experimental Preparations

packet ( expressed by the frequency Γtun = Jcorresponding to equation 5.5 ), and the

photon scattering rate for atoms in the ground state of an individual lattice site, equation

(2.48), are shown as a function of laser power. The lattice laser has a wavelength of

532nm, and is focused to 1mm diameter.

100

200

300

0.1

0.2

0.5

1

2

5

2 4 6 8 10

0.002

0.004

0.006

0.008

P [ W ]

P [ W ]

P [ W ] sc

[Hz]

tun

[kHz]

osc

[kHz]

Figure 6.1: Time scales in an optical lattice as a function of laser power, focused to

a diameter of 1mm. The oscillation frequency gives the time scale of atomic motion

in individual lattice sites, whereas the tunnelling rate between neighboring lattice sites

describes the delocalization of wavepackets. The scattering rate of lattice photons sets a

limit to coherent time evolution.

Oscillation frequencies are of the order of a few hundred kilohertz. This means that the

lattice is particularly sensitive to noise in the region below 1MHz. Measurements of

amplitude and frequency noise in this range will be presented in section 6.4.

Tunnelling rates are a factor 50 ( at P ≈ 2W ) to 5000 ( at P ≈ 10W ) lower than

oscillation frequencies, which shows that the description of atomic motion by an oscillation

in individual wells is an excellent approximation for lattice intensities above 2W . The

oscillation frequency for a 87Rb atom can be expressed numerically by

6.2 Accelerating the Lattice 81

ωosc = 112.57 [kHz] ·√

P [W ]

π(w [mm])2(6.2)

which corresponds to a three-dimensional harmonic oscillator ground state energy, ex-

pressed in units of the recoil energy (2.36):

3

2ωosc = 3.31 [Erec] ·

√P [W ]

π(w [mm])2(6.3)

The expectation value (2.48) for the photon scattering rate is calculated for the Gaussian

wavefunction (5.4), resulting in

Γsc =1

(ω

ωat

)3 (Γ

δ− Γ

δ+

)U0

1

232

Exp

(−(βk)2

2

) (Exp

((βk)2

2

)− 1

)(6.4)

In figure 6.1 one sees that the scattering rate even for the highest lattice intensities is

below 0.01Hz, which means a photon is scattered every 2πΓsc

≈ 600 s on average. Clearly

this does not pose an experimental limitation. For a tightly localized wavepacket ( βk 1

) in a deep potential, the scattering rate can be approximated by

Γsc = 2.41 · 10−3 [1/s] ·√

P [W ]

π(w [mm])2(6.5)

6.2 Accelerating the Lattice

For the simulation of transport measurements of atoms in an optical lattice described in

the last chapter, a linear potential was applied by imposing a time-dependent frequency

difference between two counterpropagating lattice beams, by use of one or two frequency-

tuned acousto-optical modulators (AOM’s). This method offers more precise control and

faster switching compared to the use of homogeneous electric or magnetic fields.

The acceleration can be understood from the following argument: The beams are detuned

with respect to each other in the laboratory frame. By a transformation to a moving frame

with suitable velocity, the Doppler effect shifts the beams to equal frequencies. In this

frame one obtains again a standing wave. The velocity of this frame is obtained from the

Doppler shift

∆ω = k · v (6.6)

With a linear frequency ramp ∆ω(t) = R · t one obtains v(t) = a · t, a linearly accelerated

lattice with an acceleration


a =R

k(6.7)

The transformation from the laboratory frame to the linearly accelerated lattice frame

creates an inertial force F = −m · a on atoms accelerated with the lattice, which can be

interpreted as arising from a linear potential V = m · a · x.

Limitations to Acceleration

There are two limitations on the possible acceleration time and reachable final velocity:

First, the size of the central lattice region, in which the potential depth does not vary

substantially, is of the order of x = 0.2mm, if one takes a fifth of the 1/e2 diameter of the

lattice beams. For well-defined experimental conditions, the BEC should not leave this

region during the experiment. This leads to an upper limit for the acceleration time of

t =

√2x

a(6.8)

Second, the diffraction efficiency of an AOM is high over a finite frequency plateau,

∆ωAOM ≈ 10MHz. If only one of the AOM’s is frequency chirped, this leads to

t =∆ωAOMa · k (6.9)

If both AOM’s are chirped, the limiting time is doubled for a fixed acceleration. Both

limits are plotted as a function of acceleration in figure 6.2.

0 2000 4000 6000 8000 10000Acceleration

0.0002

0.0005

0.001

0.002

0.005

Accele

ration

Tim

e

[m/s^2]

[s]

2

0 2000 4000 6000 8000 10000Acceleration

0.25

0.5

0.75

1

1.25

1.5

1.75

Fin

al

Velo

city

[m/s

]

[m/s^2]

Figure 6.2: Limits on final velocity and acceleration time of an optical lattice. Yellow and

violet curves: restriction by equation (6.9). Violet: detuning of one AOM by 10MHz,

yellow: detuning of both AOMs by 10MHz in the opposite sense. Blue, green, red curves:

limited by equation (6.8), for different lattice sizes xL (here determined by 1/5 of the 1/e2

diameter of the lattice beams, such that the intensity can safely be assumed constant).

Red: xL = 0.25mm, green: xL = 0.2mm, blue: xL = 0.16mm.

6.2 Accelerating the Lattice 83

Reachable final velocities are on the order of 150 recoil velocities for 87Rb, limited by

the lattice size for low accelerations, and by the AOM efficiency at high accelerations.

The Acousto-Optical Modulators

Two acousto-optical modulators ( type: NEOS N3085-XQ) for the lattice acceleration

double pass configuration ( cf. figure 6.8 ) were characterized. Two criteria had to be

met: First, a high diffraction efficiency η is required, as the lattice beam passes four times

through an AOM before forming a standing wave with the incident beam, whereby the

power is reduced by a factor η4. Second, the efficiency as a function of frequency should

be as constant as possible over a range of 10MHz to allow for large lattice accelerations

without loss of power.

8 10 12 14 16

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1 mm

6 mm

2 mm

Inte

nsi

tyin

1.D

iffra

ctio

nO

rder[a

.u.]

x Position [mm]0 5 10 15 20

70

75

80

85

90

Just below crystal top (glued to AOM case)

1mm above transducer

Diff

ract

ion

Effic

iency

[%]

z Position [mm]

Figure 6.3: Peak diffraction efficiency as a function of position: left, along the phonon

standing wave (z-direction), right: perpendicular to the phonon column. The three curves

were measured at different z-positions, measured with respect to the entrance aperture in

the direction of the transducer.

To obtain optimum efficiency, the beam position on the crystal’s input face was varied

along two perpendicular axes. The results are shown in figure 6.3. Along the phonon

standing wave (z-direction), the efficiency shows a smooth decrease as the beam is moved

away from the transducer. Perpendicular to this axis, the diffraction efficiency shows a

sharp peak of 2.5mm FWHM, which broadens slightly when the beam is positioned at

larger distance to the transducer. The peak efficiency is around 90%, as required.

In figure 6.4, the frequency dependence of the peak diffraction efficiency is shown. It

reveals a broad peak of 35MHz FWHM width, centered around 80MHz, with a slight

double peak structure, consisting of two peaks of η = 90.7% efficiency at 72MHz and

92MHz. The efficiency is nearly constant in a range from 70MHz to 95MHz, providing


an useable width of about 20MHz for acceleration, which meets the requirements.

60 70 80 90 100 110

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,090,7% 90,7%

86,4%E

ffiz

iency

infir

stD

iffra

ctio

nO

rder

Frequency [MHz]

RF-Power=5.5W

Figure 6.4: Frequency dependence of the AOM efficiency

The AOM efficiency further shows a marked polarization dependence, with a peak ef-

ficiency of 90% for s-polarization ( defined here to be perpendicular to the transducer

emission direction ), and a minimum efficiency of 45% for p-polarization.

6.3 A 3-Dimensional Far Detuned Optical Lattice

Lattice Schemes

In this section a three-dimensional lattice configuration made out of three mutually or-

thogonal standing waves is considered. The polarizations of the three standing waves are

chosen orthogonal to each other to prevent any interference between them. One of the

three standing waves can be accelerated by use of acousto-optical modulators (AOMs).

Several possible lattice schemes were examined. All of them are designed to recycle the

power of at least one of the lattice beams and use it for a second lattice beam. In this

way the intensity can be larger than in a scheme where the power is simply split up into

three standing waves.

In Figure 6.5, a scheme is shown, in which the laser power is recycled most efficiently. The

Verdi laser beam first passes through a stabilization setup, then through an optical diode

with a transmission of 88%, into the first lattice beam. It is retroreflected to form the first

standing wave, and directed to the second optical diode. On its way the polarization is

adjusted to be orthogonal to the polarization of the first standing wave. The light passes

through a further optical diode into the second lattice beam and is again retroreflected.

6.3 A 3-Dimensional Far Detuned Optical Lattice 85

/2

/2

AOM

Servo

/2

Verdi

M M

M

M

M

AO

M

AO

M

0.9 0.88

0.88

0.9

0.9

10W 0.98 7.76W

7.76W

6.01W

5.29W

3.47W

Problem: retroreflectedback to the lattice

beampropagates

Figure 6.5: Schematic of a three-dimensional optical lattice made out of a single, multiply

recycled laser beam. The unitless numbers are the transmissions of the optical elements,

furthermore the powers in each beam are given.

In the third lattice beam, it double-passes through a combination of two AOM’s. The

first AOM diffracts the light into first order, the second one into minus first order. After

retroreflection the beam passes back on the same path. If the AOM frequencies are equal,

there is no net frequency shift of the laser beam. If instead there is a frequency difference

δ between the AOM’s, the beam frequency will be shifted by 2δ with respect to the

incoming beam, thus forming a ”moving standing wave” as described in section 6.2. The

different powers in the three standing waves can be transformed into equal intensities by

choosing different beam diameters.

This setup is optimum as far as the recycling of laser power is concerned, but will be

hard to adjust experimentally, as all lattice beam positions depend on the settings of the

previous ones. Furthermore it is not possible to switch to a 2D or 1D lattice configuration

that still can be accelerated. Another drawback is that the last retroreflected lattice beam

propagates back through the lattice, leading to an uncontrolled perturbation of the light

field. This problem could be solved by installing a third optical diode on the path between

the second and third standing wave.

A solution to all three problems is is shown in figure 6.6. The power for the accelerated

standing wave is split off before the first optical diode, leaving the setup otherwise un-

changed. It offers the opportunity of installing an one-dimensional lattice that can be

accelerated, by simply turning the λ/2 plate before the first optical diode, and a two-

dimensional lattice by blocking the reflected output of the first optical diode. The second

optical diode in this case prevents the backreflection of the second lattice beam from


/2

AOM

Servo

/2

Verdi

M M

AO

M

AO

M

0.9 0.88

0.88

0.9

0.9

10W 0.98 3.38 W

3.88 W

3.00 W4.41 W

2.89 W

/2

/2

0.5

0.5

Figure 6.6: In this scheme it is possible to change from one- to two- and three-dimensional

lattice geometries, and it is easier to adjust than the one in figure 6.5.

re-entering the first standing wave.

A third configuration was considered, shown in 6.7, avoiding the power imbalance in the

accelerated arm of the lattice. The accelerated arm is split into two beams of equal

power. Both beams are sent through a standard AOM double pass configuration before

being overlapped in the lattice. This configuration yields a 20% lower power in the weakest

beam than in the first two configurations, resulting in a slightly smaller lattice volume at

fixed required intensity.

The setup in figure 6.6 is considered to be the optimum compromise between simplicity,

flexibility and power recycling.

Introducing the Lattice into the Experiment

In our experimental setup, there are two possible locations for an optical lattice: First,

overlapped with the center of the magnetic trap, where Bose-Einstein condensation takes

place. In the last months magnetic transport of ultracold atoms over tens of centimeters

[68] and of a BEC over millimeters in a microstructure trap [69] has been demonstrated

to work well. Therefore also another option seems possible: The installment of an optical

lattice in a further glass cell attached to the back side of the vacuum chamber, centered

between the extensions of the Ioffe bars ( cf. figure 3.2 ).

6.3 A 3-Dimensional Far Detuned Optical Lattice 87

AOM

Servo

/2

Verdi

M M

0.9 0.88

0.9f

f f

f

10W 0.98 3.11W

3.11W

2.41W2.14W

2.14W

/2

0.4

0.6

/2

M

M

0.50.5

/4

0.9

AOM

AOM

A O MA O M

/4

0.88

Double Pass

Double Pass

/2

Figure 6.7: This lattice scheme circumvents the need to pass a single beam four times

through an AOM for acceleration of one lattice beam.

In this thesis, however, only the first alternative is considered and shown in figure

6.8. The six lattice beams are to be shone in through the four large vacuum windows

and the two small windows on the long axis of the 3D-MOT ( all anti reflex coated for

532, 780, and 1064 nm), and all six beams will be overlapped at the center of the UHV

chamber. To avoid conflicts with the MOT laser beams, the four lattice beams through

the large windows will be tilted by an angle of about 6. The axial lattice beams must

pass the same windows as the MOT beams. This is solved by using flipper mirrors which

either pass the MOT beams or the lattice beams into the vacuum chamber. This is no

disadvantage as far as the loading of a condensate into the lattice is concerned: there

will be a phase of at least a few seconds of evaporative cooling in between collection of

atoms in the MOT and possible experiments with a BEC in an optical lattice, during

which the flippers can be switched.

For one of the axial beams a double mirror must be used. This mirror is high reflection

coated for 780nm and 532nm, and optimized not to distort the phase of 780nm light,

in order to preserve the circular polarization needed for the MOT. The focusing optics

for the lattice beams has to be mounted outside the vacuum chamber, which places a

minimum limit of 500mm on the useable focal lengths. As it is planned to focus the

beams to waist diameters of about one millimeter, corresponding to Rayleigh ranges of

1.5m, this does not pose a problem. Focused waist radii down to 100µm are possible.

The AOM double pass configuration will be set up on a separate platform, for acceleration


2-3

/4”D

N40

2-3

/4”D

N40

4

AOM AOM

Additional Optics Platform

Double Mirror780nm, 532nmDown Mirror

Down Mirror

Mirror532nm

Flipper Mirror780 nm

Flipper Mirror780 nm

Cylindrical Lenses Rails

Angle: 6°

MOT Beams 780 nm

Lattice Beams 532 nm

Figure 6.8: Side view and top view of the setup for the three-dimensional optical lattice

(green laser beams) and the 3D-MOT (red laser beams).

of the lattice along the symmetry axis of the magnetic trap.

All retroreflector mirrors have to be mounted stably, because they define the position of

the standing wave nodes. Therefore it might be useful, in contrast to figure 6.8, to shine

the vertical beam in from the upside and mount the retroreflector mirror on the optical

table, not on the upper platform, which might be susceptible to mechanical noise.

A lattice position in an attached glass cell would have considerable advantages to

the proposed setup, as the path lengths to possible positions of retroreflector mirrors are

smaller, and conflicts between MOT beams and lattice beams can be avoided. However,

6.4 Characterization of the Verdi V10 Laser 89

direct condensation by evaporation into the lattice, which could allow for interesting

experiments, is not possible in this setup. In the case of transfer of a pre-formed

condensate into an optical lattice, the phases of sub-condensates in individual potential

wells are initially locked, whereas by condensing directly into a deep lattice potential,

there is no reason why this should be so. Lowering the potential wells subsequently

might cause the phases of the fractionized condensates to lock at a later instant. To my

knowledge, there are no experiments to date which study the details of the formation of

the macroscopically defined condensate phase.

6.4 Characterization of the Verdi V10 Laser

In this section, the Coherent Verdi-V10 laser is characterized as the lattice laser. Noise

and beam shape properties are measured. The Verdi is a diode-pumped Nd : Y V O4 laser,

frequency doubled by a lithium triborate (LBO) crystal to 532nm. The cavity is in a bow-

tie configuration, made of a single piece of INVAR and placed on a water-cooled baseplate

for good thermal stability. The Nd : Y V O4 crystal is actively temperature-stabilized to

30C and pumped via optical fibers by the output of two diode arrays located in the power

supply. The pump power is constantly monitored, and a decrease due to ageing of the

pump diodes, is compensated by increasing the pump current. A temperature stabilized

etalon and an optical diode provide single-frequency operation. A photodiode monitors

the output power of up to 10.5W at 532nm and provides feedback to the pump diodes

for active amplitude stabilization.

The output coupler mirror is dichroic with a high reflectivity at 1064 nm and high trans-

mission at 532nm. The polarization of the laser light is perpendicular to the baseplate.

6.4.1 Gaussian Beam Parameters

To obtain an optical lattice with well-defined and equal potential wells all over the lattice

volume, a beam profile with well-defined phase fronts is required. The spatial profile of

the Verdi beam is a nearly circular, Gaussian shaped 00-mode, of high quality, compared

to diode lasers. This means that not much power will be lost in beam shaping.

The beam profile was measured by moving a razor-blade mounted on a translation

stage into the beam, and recording the power as a function of blade position. From the

positions at which the power falls off to 97.7% percent and to 2.3%, the 1/e2 diameter of

the beam is found. This measurement is repeated at several distances up to 3m from the

output coupler to find the longitudinal variation of the beam width. From the curve in

figure 6.9 one obtains a beam waist (1/e2 radius) w0 = (1, 37± 0.01)mm, and a Rayleigh

parameter ( the longitudinal distance from the position of the waist, where the beam has


0 500 1000 1500 2000 2500 3000

2,62

2,77

2,92

3,08

3,23

3,38

3,54

3,69

Model: y=2 w0(1+(x/z

0)2)1/2

Chi^2 = 3.09877R^2 = 0.99231

2w0

(2.738 ±0.012) mm

z0

(3361 ±74) mmB

eam

Dia

mete

r[m

m]

Distance from Verdi Output Coupler [mm]

Figure 6.9: Beam waist of the Verdi laser as a function of distance from the output coupler

mirror. From these data, the beam divergence and the Rayleigh range are determined.

expanded by a factor of√2 ) z0 = (3361± 74)mm, yielding a beam divergence angle of

θ0 = w0

z0= 4.08 · 10−4 rad. The specified beam waist is 1, 13± 0, 11mm, and the specified

divergence is less than 5 · 10−4 rad, close to the measured values.

6.4.2 Spectral Measurement of Amplitude Noise

In this section, measurements will be described which lead to a prediction of the para-

metric heating rate due to amplitude fluctuations of the lattice laser light.

The amplitude noise of the Verdi laser was measured using a calibrated fast photodiode

with a DC and an AC output channel. The DC output of the photodiode has a 1/e

bandwidth of about 5kHz. The AC output, which was used in all amplitude noise mea-

surements, has a transfer function with a broad peak centered at 600 kHz, which falls

off to the 1/e level at 100 kHz and 1.4 MHz. It is therefore suitable for measurements

in the few 100 kHz regime, which is important for trapping atoms in a lattice, because

these noise frquencies are comparable with oscillation frequencies of atoms in the lattice

potential wells. Frequencies substantially lower than 100 kHz are not important in typ-

ical experiments, which last only a few milliseconds. Frequencies above a few MHz are

averaged out during the much slower atomic motion.

The amplitude noise was measured by recording the AC output of the photodiode, which

was illuminated with 3.25mW of Verdi light. The signal was amplified and recorded on

a spectrum analyzer ( Anritsu MS 2601B, used with a preamplifier SRS-SR560 at an


60 80 100 200 400 600 8001000

1E-13

1E-12

1E-11

1E-10

1E-9

1E-8

rela

tive

spect

raln

ois

epow

erP

opt2/P

opt2

[1/H

z]

Frequency [kHz]

Figure 6.10: Relative spectral amplitude noise density of the Verdi laser. The peaks are

caused by a pulsed transformer in the power supply and are insufficiently suppressed by

an internal noise reduction board.

amplification of 50, without filters ). As will be described in appendix D, the data were

calibrated by the detector transfer function, resulting in the relative spectral noise density∆P 2

opt

P 2opt

displayed in figure 6.10.

From equation (2.52) the parametric heating rate was computed, and is shown in figure

6.11. Up to 400 kHz, the heating rate shows a base value of 0.1s−1, which means the the

energy of a trapped sample of atoms will be doubled only after 10 seconds, much longer

than typical experiment durations. The peaks at trap frequencies of 50 kHz, 100 kHz,

315 kHz, and 600 kHz, could pose a problem. They are caused by a pulsed transformer

in the power supply, whose noise is insufficiently suppressed by a possibly defect noise

reduction unit within the power supply. The relevant parts will be replaced by Coherent,

hopefully removing these peaks.

6.4.3 Frequency Noise

A Confocal Resonator for Frequency Noise Measurements

For the measurement of frequency noise, and possibly its stabilization, a confocal resonator

was built. It was known from another group using the Verdi, that its linewidth, measured

on a timescale of 0.1ms, is 0.02MHz, and that this line drifts by ≈ 0.5MHz within

1ms. To explore this drift in more detail, a resonator of 10MHz linewidth was designed.

If the laser frequency is set to a slope of the transmission curve, the frequency drift is


20 40 60 80 100 200 400 600 800

0,01

0,1

1

10

100

Heatin

gra

te=

<dE

/dt>

/<E

>[1

/s]

Trap frequency [kHz]

Figure 6.11: Parametric heating rate, equation (2.50), due to the measured amplitude

noise, as a function of oscillation frequency (2.46).

converted into an amplitude signal by monitoring the transmission as a function of time

on a photodiode.

A confocal resonator was chosen, in which the distance between the mirrors equals the radii

of curvature, because all transverse modes are degenerate in frequency, which makes it

easier to obtain a high spectral resolution. The distance L between the mirrors was chosen

to be 100mm, resulting in a free spectral range of ∆ν = c2L

= 1.5GHz. Therefore, a

finesse F = ∆νδν

= 150 is needed to obtain a FWHM transmission width of δν = 10MHz.

From F = πr12

1−r one computes a mirror reflectivity of r = 97, 9% for F = 150. The

mirrors that were produced on request have a reflectivity of 98, 4%, corresponding to a

theoretically reachable finesse of 194. Experimentally, a finesse of 100 ( corresponding to

a FWHM δν = 15MHz) was reached when the input beam was mode-matched to the

transverse resonator ground mode by focusing it with a f = 350mm lens to a radius of

≈ 100µm at the center of the resonator.

One of the mirrors is mounted on a piezo-ceramic tube, which allows to continuously tune

the length of the cavity by an amount corresponding to about four free spectral ranges.

The lowest resonance frequency of the piezo is ωres ≈ 4 kHz, therefore it can be scanned

at frequencies up to 3 kHz.

The resonator cavity consists of a tube made of a rod of INVAR, which has a low thermal

expansion coefficient. To ensure high thermal frequency stability, a wall thickness of

10mm was chosen. One of the mirrors is glued to a 10mm thick disk of INVAR, with a 8

mm diameter hole in the center for beam transmission. This disk has a fine thread on its


outside, fitting to a thread on one side of the cavity tube. The other mirror is mounted

on the piezo ring, which itself is glued on top of a 55 mm long INVAR cylinder, which

can be slid into the cavity tube, and glued once the confocal geometry has been found.

The cavity is mounted on a stable fixed aluminum holder.

Frequency Noise Measurement

For the measurement of frequency noise, the resonator transmission was measured on the

DC output of the photodiode described above, and displayed on the spectrum analyzer.

The cavity length was adjusted such that the laser line lies on a transmission flank,

by tuning the piezo voltage. The frequency jitter ∆ωl(ν)ωl

is extracted in the following

procedure:

The dBm noise signal x is converted into a power by P [mW ] = 10−310x[dBm]

10 . This noise

power corresponds to a certain intensity modulation at the photodetector behind the

resonator, which can on the slope of the known resonator transmission curve be converted

into a frequency modulation. The calibration is explained in appendix D.

The result of this procedure, the relative spectral frequency noise, is shown in figure 6.12.

The main features are a broadband low frequency 1/ν noise below 10 kHz, bounded by

a plateau with several spikes between 10 and 50 kHz. Above 100 kHz, the noise level

rises to a broad peak centered around 300 kHz. The photodiode transfer function did not

allow for measurements at frequencies beyond 600 kHz, therefore it cannot be seen from

figure 6.12 that the high frequency rise really corresponds to a peak, but the noise was

also monitored on an oscilloscope, where clearly a dominant period of ≈ 300 kHz could

be seen in the noise. It should be stressed here, that it can not be determined wether the

source of noise is the confocal cavity, the laser, or both. The measurement was carried

out, when the INVAR cylinder with the piezo crystal and one mirror was not yet glued

to the INVAR tube, which in hindsight renders the measurement doubtful.

From the relative spectral frequency noise ∆ωl(ν)ωl

, the heating rate (2.55) is calculated.

For the distance ∆x between retroreflector mirror and lattice center, 1m is taken, and for

the mean position square 〈x2〉 the value β2 = (

mωosc) of the harmonic oscillator ground

state in a single potential well is taken, where ωosc = 2πνtrap is the oscillation frequency.

The result is shown in figure 6.13.

This high heating rate, if really caused by the laser and not by the cavity, would make it

inevitable to stabilize the laser frequency, to suppress noise by a factor 100 to 1000, and

lower the heating rate due to frequency noise to a level comparable to the amplitude noise

heating rate ( due to the squared dependence of the heating rate on ∆ωl(ν) this would

reduce heating by a factor 104 to 106 ).

Not shown in figure 6.12 is a low frequency drift between resonator and laser, on a

timescale of seconds. The amplitude of this drift is larger than the width of the resonator


10 100

1E-11

1E-10

1E-9

Rela

tive

Fre

quency

Nois

e

L(

)/ L

[Hz-1

/2]

Frequency [kHz]

Figure 6.12: Relative spectral frequency noise density of the Verdi laser. In this case it

cannot be distinguished wether the noise originates from the laser or the confocal resonator,

with which it was measured.

10 100

10-1

100

101

102

103

104

105

106

107

Heatin

gra

te=

<dE

/dt>

/<E

>[1

/s]

Trap Frequency [kHz]

Figure 6.13: Parametric heating rate caused by frequency noise of the lattice laser, as

given by equation (2.55), as a function of oscillation frequency.

transmission resonance, shifting laser and resonator out of resonance within seconds.


6.4.4 Proposal of a Stabilization Scheme for the Verdi Laser

Both amplitude and frequency of the Verdi laser can be stabilized in the following scheme,

depicted in figure 6.14, with an AOM as the active regulator. The first diffraction order

of the AOM is directed to the experiment, and a part of the power is split off for analysis

of amplitude and frequency.

Wedge

Wedge

ConfocalResonator

Piezo

Photodetector

Photodetector

To Experiment

AmplitudeFrequency

PI-Regulator

PI-Regulator

SlowIntegrator

AmplitudeInput

FrequencyInput

Driver70-100 MHz8 W

Coherent Verdi V10First Diffraction Order

AOM85 MHz

Figure 6.14: Proposed scheme for stabilization of both amplitude and frequency of the

Verdi laser, which seems feasible based on the previously described noise measurement

data.

Amplitude stabilization is achieved by recording a small fraction of the intensity on a fast

photo diode, and sending a feedback signal through a PI regulator to the amplitude input

of the AOM.

Frequency noise stabilization requires two steps: The low frequency drift, which shifts the

cavity and the laser out of resonance within seconds, is not relevant to experiments, as

they take place within milliseconds. Therefore the cavity can be allowed to follow this

drift, by providing feedback to the piezo voltage. This is done by feeding the resonator

transmission signal to a slow integrator, and an error signal from there to the piezo driver.

In this way, the laser line can be held constantly on the slope of the transmission curve.

The transmitted signal, recorded on a photo detector can then generate an error signal

from a PI regulator to the frequency input of the AOM driver. In this way, frequency

noise in the Hz to MHz range should be compensated.

Chapter 7

Conclusion and Outlook

In this diploma thesis, one bottleneck in the production of BEC’s was removed: the

fast accumulation of a high number of ultracold trapped atoms in an ultrahigh vacuum

environment. We use an efficient 2D-MOT / 3D-MOT system to trap around 8 · 1010

atoms within 2 to 3 seconds in an UHV chamber with a background pressure of

2 · 10−11mbar. The 2D-MOT is unique due to its large length of 9 cm, illuminated

with a high laser intensity I ≈ 20 Isat so that also the wings of the Gaussian intensity

distribution contribute to cooling and trapping. The 3D-MOT has a large elongated

trapping volume in a geometry ideally suited to trap atoms from the 2D-MOT atomic

beam.

With the high number of atoms captured in this way it will likely be possible to produce

very large BEC’s or to further improve the speed of evaporative cooling, which is the

second major bottleneck in BEC production.

For the creation of a BEC, the atoms captured in the MOT are loaded into a magnetic

trap (MT). The field configuration for this magnetic trap differs from the MOT in

three points: First, the axial anti-Helmholtz configuration is turned into a Helmholtz

configuration by reversing the current in one coil. Second, the currents both in the pinch

coils and the Ioffe bars are increased dramatically to 500A and 1000A respectively.

Third, to obtain a tight radial confinement, the offset field on the axis produced by the

pinch coils has to be compensated to near zero. Therefore two bias coils, approximately

in a Helmholtz configuration, at a current of 500 amperes, produce a homogeneous offset

field antiparallel to the pinch coils field.

The next steps towards achieving BEC will be the optimization of the magnetic trap and

the loading process from the MOT to the MT, as well as the radio frequency technology

to perform forced evaporative cooling.

After achieving BEC, it is planned to install a far-detuned optical lattice and load

it with a BEC. An optimum scheme for the three dimensional optical lattice was designed

97

98 CH. 7 Conclusion and Outlook

in this thesis. It allows for high laser power in each lattice arm by recycling retroreflected

light for successive lattice beams. The lattice can be accelerated by frequency-chirping

one lattice beam by use of two acousto-optical modulators (AOM’s). The acceleration

exerts an inertial force on atoms trapped in the lattice and thus allows transport

measurements of ultracold atoms in a periodic potential. The AOM’s were characterized

and found to have high efficiency over a frequency range of more than 10MHz, as

required for lattice acceleration.

The lattice laser, a 10W single mode laser at 532nm, was characterized in amplitude

and frequency noise. Parametric heating rates due to both types of noise were calculated.

Amplitude noise was found to be sufficiently low for an optical lattice, with the exception

of two peaks at 100 and 200 kHz, which will be removed by replacing defect components.

The frequency drift between the laser and a confocal cavity built in this thesis was

measured, yielding an upper limit on the laser frequency noise, but no unambiguous

result for the parametric heating rate. For the suppression of both frequency and

amplitude noise, a stabilization scheme using an AOM, with a confocal cavity as a

frequency reference was suggested.

Our novel BEC setup has good optical access, which allows a wide range of new

experiments. Although the Mott insulator phase transition of Bosons in an optical lattice

has now been observed [49], the properties of the insulating phase remain an interesting

subject of study. Numerical calculations performed in this thesis identify signatures of

the phase transition and will help interpret the results of transport measurements of

atoms in the superfluid and insulating phases.

The list of planned experiments further includes atom-atom interactions in confined

space, especially dipole-dipole interactions between neutral atoms. Coherences and

entanglement in optical lattices represent a second field of study. Magnetic transport of

the BEC is possible in our setup and could be combined with a second ongoing project,

in which microstructured atom traps are examined.

Appendix A

Capture Velocity of the 3D-MOT

In this appendix, the capture velocity of an elongated MOT loaded from an atomic beam

entering at an angle α, in our case α = 27, is calculated.

Boundaries of the Capture Volume

The boundaries of the trapping volume are calculated as shown in figure 2.1: Atoms can

only be captured where the detuning is larger than the Zeeman splitting, as only then

energy is lost in a radiative cycle.

Radially, the field increases linearly with a gradient ∇B, and equating the Zeeman shift

with the detuning

µ′∇Bxi

= δ (A.1)

and inserting the experimental values δ = −2.3Γ, µ′ = 5/6µB [2] (where µB is the

Bohr magneton), and ∇B = 14G/cm, leads to xi = 8.4mm, i.e. a trap diameter of

L1 = 16.8mm. On the long trap axis, the magnetic field increases nonlinearly, terminating

the trapping volume at a position where

µ′B(x3)

= δ (A.2)

This value B = 11.8G is reached at an axial position x3 = ±40mm, whereby the MOT

length is L3 = 80mm.

Equation of Motion and Capture Velocity

If the photon scattering rate Γsc (4.3) in the light field of the 3D-MOT is assumed to be

saturated over the full trapping volume, Γsc =Γ2, with the natural linewidth of the cycling

transition Γ = 2π · 5.98MHz, atoms trapped and cooled from the atomic beam follow a

simplified equation of motion

99

100 APPENDIX A Capture Velocity of the 3D-MOT

mxi = −kiΓ

2(A.3)

where i = 1, 2 denote the radial directions, and i = 3 is the axial direction. ki is the

wavevector of the laser beam counterpropagating to the respective velocity component.

The atomic motion is taken to be in the x1, x3 plane, with initial conditions x1(t = 0) =

v0sin(α), x3(t = 0) = v0cos(α), x1(t = 0) = x0, and x3(t = 0) = z0. Integrating the

equation of motion twice leads to

xi =−kΓ

2mt+ xi(t = 0) xi =

−kΓ

4mt2 + xi(t = 0)t+ xi(t = 0) (A.4)

By demanding a particle with initial velocity vmax just to be brought to a halt ( xi(t =

tF ) = 0 ) after traversing the full MOT volume ( xi(t = tF ) − xi(t = 0) = Li ) at a

time tF , one obtains two conditions for the maximum allowed initial velocity vmax, from

motion x1 and x3 direction. The more stringent of both limits gives the capture velocity

vc:

vmax,1 =

√L1kΓ

msin(α)= 65m/s vmax,3 =

√L3kΓ

mcos(α)= 140m/s (A.5)

One might object that the diameter of the axial laser beams (6mm) is smaller than the

calculated diameter of the trapping region (L1

2= 8.4mm). But for the slowing of the

radial velocity components, the radial laser beams are responsible, which illuminate a

volume of about 20× 20× 80mm, larger than the 16.4mm required for equation A.3.

Therefore, the estimated capture velocity of the elongated 3D-MOT, loaded from an

atomic beam, is vc = 65m/s, whereas for usual MOT’s the capture velocity is around

35m/s.

Appendix B

Polarization Spectroscopy for Laser

Stabilization

The following scheme is employed to stabilize diode lasers to an atomic hyperfine transition

[70]: The laser is protected from optical feedback by an optical isolator, and the transverse

mode profile is converted into a spherical shape (≈ 1mm diameter) by an anamorphic

prism pair. Then a part of the beam is split up for stabilization and diagnostics, while

most power is directed to the experiment. A Fabry Perot interferometer monitors single

mode operation, and a wavemeter is used to coarse set the operating wavelength (up to

≈ 0.05nm). The spectroscopy setup consists of a Rb vapor cell in which a strong (800µW)

circularly polarized pump beam and a counterpropagating weak linearly polarized probe

beam (10µW) are overlapped.

After passage through the cell, the probe beam is split into two orthogonally polarized

components whose intensities are recorded on two equal photodiodes. Away from a hy-

perfine transition the intensities are adjusted to be equal, but in the vicinity of a hyperfine

transition, a frequency dependent polarization rotation results in a dispersive intensity

imbalance on the photodiodes.

The difference signal of the two photodiodes serves as negative feedback for the frequency

control of the laser diode, which is done by moving the grating mounted on a piezo crystal.

A typical polarization spectroscopy signal is shown in figure B.2, revealing a dispersive

signal for every allowed hyperfine transition from the 5S1/2,F=2 ground level, and the

associated ”crossover” signals [70]. For comparison, a single detector signal is displayed,

showing the saturation spectroscopy resonance structures on top of the doppler broadened

profile.

The existence of resonances of widths on the order of Γ is explained as follows: If both

pump and probe beam interact with the same velocity class of the Doppler broadened

absorption profile, the linearly polarized probe light experiences a frequency dependent

polarization rotation: It can be thought of as split into two opposite circularly polarized

101

102 APPENDIX B Polarization Spectroscopy for Laser Stabilization

/4

Photodetector B

Rb - Spektroscopy Cell

To AOM Doublepassand Experiment

Optical Isolator

Laser DiodeTUI DL100

AnamorphicPrism Pair

Photodetector A

PIDRegulator

polarisierenderStrahlteilerwürfel

f=50 mm

To Wavemeter andFabry Perot Resonator

500 W

27,5 mW

10 WProbe Beam

800 WPump Beam

Figure B.1: Setup for polarization spectroscopy laser stabilization. The spectroscopy con-

sists of a circularly polarized pump beam and a counterpropagating linearly polarized probe

beam, which are overlapped in the spectroscopy cell. After passage through the cell, the

probe beam is divided into equal intensity components of perpendicular linear polarization,

monitored on two photodiodes. For each beam, the optical power is given.

components, which experience slightly different frequency dependent refractive indices

due to the different light shifts of the Zeeman sublevels induced by optical pumping by

the circularly polarized pump beam.

The resonances 2 → F ′ occur if pump and probe laser interact with zero velocity atoms on

the same hyperfine transition. The crossover resonances (F ′1, F

′2) appear when the pump

APPENDIX B Polarization Spectroscopy for Laser Stabilization 103

laser excites a distinct velocity class v = 0 on one hyperfine transition to a sublevel F ′1,

and the probe laser experiences polarization rotation by the same velocity class on the

transition to a different sublevel F ′2.

0,002 0,004 0,006 0,008

-0,6

-0,5

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

Pola

riza

tion

Spect

rosc

opy

[a.u

.]

Ferquency [a.u.]

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

2->1(1,2)

2->2

(1,3) (2,3)

2->3 Satu

ratio

nS

pect

rosc

opy

[a.u

.]

Saturation Spectroscopy

Polarization Spectroscopy

Figure B.2: Polarization spectroscopy signal of the Rb 5S1/2,F = 2 −→ 5P3/2,F′ transition

at 780.25nm (each hyperfine transition is denoted by 2 → F ′), compared to the saturation

spectroscopy signal under the same conditions. 2 → 3 is the MOT cycling transition, and

(F ′1, F

′2) denotes crossover resonances explained in the text.

Each of the slopes of the polarization spectroscopy signal can be converted into a feedback

signal for the laser frequency control unit.

104 APPENDIX B Polarization Spectroscopy for Laser Stabilization

Appendix C

Time of Flight Temperature

Measurement

Here, the calculation of the fit function for the time-of-flight absorption signal, equation

C.3, is described:

The starting point is the description of the trapped sample by a function n1(x,y,z,vx,vy ,vz)

in phase space, which takes the distribution of atoms in velocity and space into account:

n1(x, y, z, vx, vy, vz) =N

8π3σxσyσzσ3v

· (C.1)

Exp(−

(x2

2σ2x+ y2

2σ2y+ z2

2σ2z

))· Exp

(−

(v2x

2σ2v+

v2y

2σ2v+ v2

z

2σ2v

))

Here, σx, σy, σz denote the 1/e2 widths of the trapped atom cloud, and σv =√

kBTm

is

the velocity spread. The spatial distribution at a point (x, y, z) and a time t after release

from the trap, is obtained by integrating equation C.1 over all initial (t = t0) velocities

(vx0, vy0, vz0) of atoms which reach the point (x, y, z) at time t.

n2(x, y, z, t) = (C.2)∫ ∞∫−∞

∫dv0 n1(x− vx0(t− t0), y − vy0(t− t0) +

g2(t− t0)

2, z − vz0(t− t0), vx0, vy0, vz0, t0)

To obtain the measured detector signal, the spatial distribution is integrated along the

x-direction of the laser beam, resulting in

N(t) = aExp(− (9.81(t−t0)2+2b)2

8(σ2+(t−t0)2σ2v))

( (t−t0)2σ2 + 1

σ2v)√σ4σ6

vσ2z

√1σ2v+ (t−t0)2

σ2z

√(σ2 + (t− t0)2σ2

v + c (C.3)

105

106 APPENDIX C Time of Flight Temperature Measurement

Here, σ ≈ 8mm and σz ≈ 60mm are the 1/e2 radial and axial sizes of the atom cloud,

σv = vth =√

kBTm

is the thermal velocity of the atoms, from which their temperature is

determined. t0 is the time at which the atoms are released out of the molasses, b ≈ −10 cm

is the vertical distance between the MOT center and the detection point, and a and c are

an amplitude and offset.

The use of seven fit parameters (a, t0, b, c, σ, σv, σz) makes the fitting procedure more

complicated but still solvable. The program Origin is used for fitting. One starts by

making physically reasonable assumptions for the trap dimensions σ and σz and the

distance b, setting these parameters fixed. The release time t0 can be initialized with the

respective molasses duration time, but the best fits are obtained by setting t0 to zero.

The offset c can be determined from the absorption trace at large times and is also fixed

initially, leaving only the amplitude a and the velocity spread σv floating. After several

iterations the function N(t) fits the data reasonably well, and the fit parameters have

reached a fixed value. Now one can subsequently allow the other parameters to float. It

turns out that the results do not depend on the order in which the parameters are allowed

to float.

The restriction of atomic motion by the Ioffe bars close to the trap volume is not taken into

account in equation C.3, and can lead to a systematic error in temperature determination.

Appendix D

Spectral Noise Measurements

On the spectrum analyzer, a signal x in [dBm] is given. This corresponds to a power

y [mW ] = 10−310x [dBm]

10 (D.1)

This electrical power corresponds to an optical power Popt incident on a photodiode

y =U2PD

R∝ P 2

opt (D.2)

where UPD is the photodiode voltage, and R = 50Ω is the input resistance of the spectrum

analyzer. To obtain the proportionality constant, a calibration signal has to be measured

on the photodiode.

Amplitude Noise

In this case, the calibration was done by recording the output of the first diffraction order

of an acousto-optical modulator (AOM) on a spectrum analyzer ( Anritsu MS 2601B,

used with a preamplifier SRS-SR560 at an amplification of 50, without filters ). The

RF input to the AOM was set in the linear regime of its characteristic, and slightly AC-

modulated with a SRS DS345 function generator at different frequencies, to record the

detector transfer function. The mean diffracted light intensity was kept at the same level

as in the actual noise measurement, P = 3.25mW . The AOM modulation depth was

carefully chosen not to saturate the diode at any frequency, i.e. the modulated light am-

plitude A was decreased to a level only slightly above the noise level, A = 0.3µW . In this

way the diode is calibrated under conditions as close to the measurement as possible.

The calibration signal on the spectrum analyzer is ( cf. D.2 ): ycal(ν) ∝ P 2cal. The

proportionality constant involves the frequency dependent detector transfer function.

With Pcal = A · sin(ωAOM t), and ωAOM = 85MHz, one finds that P 2cal is given by

P 2cal =

1T

∫ T

0A2 · sin2(ωAOM t)dt =

A2

2.

The measured noise signal is ynoise(ν) ∝ ∆P 2opt(ν), and calibration results in

107

108 APPENDIX D Spectral Noise Measurements

∆P 2opt(ν) =

P 2cal

ycal(ν)· ynoise(ν) (D.3)

From this, the ratio∆P 2

opt(ν)

P 2opt

and the heating rate, equation 2.52, is extracted.

Frequency Noise

In the measurement of frequency noise, one has to calibrate the power measured at the

spectrum analyzer to an intensity modulation on the photodetector behind the confocal

resonator. This intensity modulation is related to the original laser frequency modulation

by the known resonator transfer function.

This intensity modulation is calibrated in a separate measurement: The photodetector is

moved away from the resonator and monitors the modulated power in the first diffraction

order of an AOM, yielding the signal ycal on the spectrum analyzer (already converted

in watts). The AOM is modulated at a low frequency of about 1 Hz, and the intensity

modulation amplitude is equivalent to the amount of intensity modulation in the frequency

noise measurement, that occurs when the resonator transmission changes from 90% to

10%.This change in transmission occurs when the frequency shifts by roughly one FWHM

on the slope of the transmission function. In this way one obtains a relation for the

measured noise signal ynoise when the frequency jitter equals ∆ωl(ν) ≈ 1 · FWHM =

15MHz (the numerical value for FWHM is computed from the measured resonator finesse

of 100, and the free spectral range c2L

is determined by the cavity length L = 100 mm (

cf. section 6.4.3 )).

The calibration is then given by

∆ωl(ν) =FWHM

ycal· ynoise (D.4)

From this, the relative spectral frequency noise, equation (2.55), is inferred.

Appendix E

Bloch Oscillations

Physical Background

The energy spectrum of a particle in a periodic potential consists of a series of energy

bands, enumerated by an index n. The eigenstates are delocalized Bloch states charac-

terized by a wavevector k and the band index n. If now an additional weak force F is

applied, the Bloch states undergo a temporal evolution. It can be shown that the equation

of motion of the wavevector k is

∂k

∂t= F ⇒ ∆k =

1

F ·∆t (E.1)

whence k(t) changes linearly in time. As momentum space is periodic for a periodic

potential with lattice constant d, it holds for the i’th component ki of k

ki ≡ ki +mi · 2πd

(E.2)

it follows that the evolution of k can be viewed as periodic in time, with a period

TBloch =h

|F | · d (E.3)

This oscillation in k-space also corresponds to an oscillation in real space, as the velocity

associated with a wavepacket formed from Bloch states in the n’th energy band En(<k) is

v =1

grad<kEn(<k) (E.4)

therefore, as the energy bands (and their gradients) are periodic in k, and k is periodic

in time, a velocity periodic in time results.

In optical lattices, Bloch oscillations are observed by a periodically pulsed tunnel

current out of the lattice. The tunnelling mechanism is called Landau-Zener tunnelling.

109

110 APPENDIX E Bloch Oscillations

During its evolution in time k periodically crosses the first Brillouin zone Boundary,

where the energy gap between the first and second energy band is smallest, and tunnelling

to the second band is most probable. This leads to a tunnelling current out of the lattice,

that is periodic with TBloch. Tunnelling probabilities across the much smaller energy gaps

to higher bands and into the continuum of unbound states are much larger than from

the first to the second band. Therefore a particle in the second energy band will nearly

instantaneously leave the lattice.

A Model For the Width of Bloch Pulses

For a condensate in an optical lattice with relative phase fluctuations σφ(1) between

neighboring wells, the relative phase fluctuations between the N’th and the first lattice

well, given by the quadratic addition of Gaussian noise between subsequent potential

wells, amount to

σφ(N) =√N · σφ(1) (E.5)

One can assume, that Nmax potential wells contribute to a coherent tunnel output, where

σφ(Nmax) = π, or

Nmax =

(π

σφ(1)

)2

(E.6)

( thus the coherence length in the optical lattice is Nmax · λ2 = λ2

(π

σφ(1)

)2

).

We now use an analogy to diffraction from a multislit grating, where the ratio of the width

w of a diffraction peak to the separation ∆ between neighboring peaks is

w

∆=

1

Nmax

(E.7)

where Nmax is the number of slits in the coherence range of the incident waves.

We attribute the same ratio w∆

to the temporal width and separation of subsequent Bloch

pulses out of an optical lattice due to phase fluctuations. By accounting for the finite

width w0 of the Bloch pulses in the absence of phase fluctuations, which is caused by the

finite extension of the occupied lattice volume, we arrive at

w

∆=

√w2

0 +

(1

Nmax

)2

=

√w2

0 +

(σφ(1)

π

)4

(E.8)

which corresponds to equation (5.13).

Appendix F

Latest 2D-MOT Model Results

Here, results of the model described in section 2.3.2 are compared with experimental data.

50 100 150 200

1 109

2 109

3 109

4 109

5 109

45

50

55

60

65

5 10 15 20 25 30 35

3 1010

4 1010

5 1010

6 1010

0 50 100 150 200-1

0

1

2

3

4

5

6

7

8

v[m/s] v[m/s]

<v>[m/s]Flux[ atoms / s ]

Pressure [10 mbar] Pressure [10 mbar]-7 -7

[a.u.]

[a.u.]

^

Figure F.1: Pressure dependence of the flux distribution (first row, left: theory; right:

experiment), the total flux (second row, left) and the mean velocity (right). Theoretical

results are colored green, and experimental data are marked red.

Figure F.1 shows the pressure dependence. The calculated total flux is multiplied by a

factor of 1/12 to match the experimentally measured one. This factor can be accounted

111

112 APPENDIX F Latest 2D-MOT Model Results

0 50 100 150

0

0.5

1

1.5

2

2.5 72 mm

67 mm

62 mm

57 mm

52 mm

47 mm

42 mm

92 mm

82 mm

77 mm

Flux[ atoms / s ]

Length [mm]

v[m/s] v[m/s]

[a.u.]^

50 100 150 200

2 108

4 108

6 108

8 108

1 109

[a.u.]

^

60 70 80 90

1.5 1010

2 1010

2.5 1010

3 1010

60 70 80 90

50

55

60

Length [mm]

<v>[m/s]

Figure F.2: Length dependence of the flux distribution (first row, left: theory; right: exper-

iment), the total flux (second row, left) and the mean velocity (right). Theoretical results

are colored green, and experimental data are marked red.

for by the overall efficiency of the capture process. The linear dependence of mean velocity

on pressure is not well reproduced by the model.

In figure F.2 the length dependence is shown. This time the calculated flux is multiplied

by 1/24 to match the data, consistent with a lower capture efficiency at the low laser

power of 21mW per beam, which was used in this measurement. The experimental mean

velocity at short MOT lengths shows a higher value than expected, probably due to the

thermal background flux that becomes comparable to the very low 2D-MOT flux at low

lengths.

Both figures were obtained with the same set of fit parameters, and show good qualitative

agreement of the flux distribution and the total flux on experimental parameters. The

variable parameters were vc,0 = 22m/s, Γcoll = 170 [1/s] nn0, and Γtrap

Γout= 0.07 n

n0, where

n0 = 6.5 · 109/cm3 is the density of room temperature Rb, corresponding to a vapor

pressure of 3 · 10−7mbar. The MOT diameter was fixed at a value of d = 2 rc = 1 cm.

From Γcoll = vth nσeff one can extract the effective cross section for collisions to remove

atoms from the beam. With vth = 270m/s one obtains σeff = 10−12 cm2, in good

agreement with a model for light assisted collisions [14, 25].

Appendix G

List of Developed Mathematica

Programs

LatticeParametersFinal.nb: Calculation of tunnelling matrix element J, on-site interac-

tion U, and ratios UzJ

in 1d, 2d and 3d, as a function of laser power, for a given focused

laser waist.

MottTunnelling.nb: Calculates the tunnelling probabilities of a particle out of the

effective potential of a possibly occupied optical lattice, as a function of acceleration.

The three cases described in section 5.1 are considered: Tn=0, Tn=1, Tn=1+ε. The program

also shows the effective potential for selected accelerations.

Blochoscillation06b.nb: Calculates the Bloch oscillation tunnelling output of a condensate

in an optical lattice, under the influence of gravity. It includes random Gaussian phase

fluctuations of variable strength, and averages the tunnelling output over ten distinct

runs with different random phase fluctuations.

Velocimetry04.nb: Calculates the expected Raman signal as a function of relative

detuning between the two laser beams for different temperatures of trapped atoms, at a

given intersection angle of both Raman beams.

2DMotModel.nb calculates the dependence of the atomic beam flux distribution,

the total flux and the mean velocity on 2D-MOT length, and Rb pressure.

113

114 APPENDIX G List of Developed Mathematica Programs

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Thanks

I would like to thank all who made my diploma thesis such an interesting and fulfilling

time. First of all, I would like to thank Prof. Pfau, who always fully supported me, who

gave me the freedom to do a rather independent project and planned with me when it

turned out that it could not be finished. He always took the time for fruitful discussions

when necessary.

I can rarely imagine a more interesting place for a diploma thesis, because of the

great spirit in the group, as well as the stimulating competition to the ”Chromium

guys” ( not to forget the ”microstructure guy” and the ”Ytterbium guys” recently added

to the list ). The build up of so many new experiments creates a really unique atmosphere.

I would like to thank Juergen Schoser, who has worked very close with me and

showed me a great lot about experimental physics. It was real fun working together

with you! I will especially remember the many long measuring nights on the 2D-MOT.

Thanks also for your long distance advice on my diploma thesis, and the many emails

concerning the 2D-MOT model.

I thank Robert Loew for his work with me on optical lattices and on the 3D-MOT, for

always taking time to answer questions, and for carefully reading my manuscript.

I thank Dr. Yuri Ovchinnikov for giving his invaluable experimental advice, and for

working together with me on the 3D-MOT characterization.

Thanks to Alex Bataer for introducing me into the details of the experiment, and his

great work on the 2D-MOT. Thanks to Rolf Heidemann for his support on the 3D-MOT

characterization. A big thanks also to Axel Grabowski. Without you the setup and

debugging of the UHV system would have driven me crazy. Piet Schmitt - it was always

good to hear your laughter in the adjacent lab. Thanks for many little ”problem solvers”

in electronics and optics ( the ”mysterious” beam splitter etc... ). I thank Joerg Werner

for his ”software assistance” to the Rubidium project, and many answers whenever

computers were concerned. Thanks to Sven Hensler for his tips concerning the diode

laser, and to Dr. Axel Goerlitz for discussions on the optical lattice setup. Thanks to my

fellow diploma students - Thomas Binhammer and Axel Griesmaier .

I would especially like to thank Karin Otter for doing all the things physicists tend to

121

122 THANKS

forget, but especially for always having the right motivating words - both in the lab or

the office and ”long distance”.

I want to take this opportunity to tell my parents how thankful I am for their

support through all the years. You have enabled me to do what I like best, provided the

basis for the completion of my studies, and supported with great care every step I took

on my way.

To my friend and partner Isabell: I enjoy every moment of being together with

you. Thank you for giving your support and sharing your ideas and energy with me over

the last four years.

Documents

Volker Schweikhard- Ultracold Atoms in a Far Detuned Optical Lattice