Guillermina Ramirez San JuanGuillermina Ramirez San Juan

Bloch wave in silicon Optical Lattice

PART I:

Brief description of Bloch Oscillation and Optical

Lattices

Background• 1rst formulated in the context of condensed matter physics. Pure quantum effect• It was predicted that a homogeneous static electric field induced oscillatory motion of electrons

in lattice• Every crystal structure has 2 lattices associated with it, the real lattice and the reciprocal lattice

Crystal lattices Reciprocal space unit cell (B-Z)

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e ir k ⋅R =1

Solve Schrodinger’s eq. for a particle in a periodic potential:

Solution proposed by Bloch: a wave function with a periodic squared modulus

The corresponding eigenvalues for this equation are:

Energy eigenvalues are periodic with periodicity k (reciprocal lattice vector)

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ˆ H Ψ = EΨ

ˆ H =ˆ p 2

2M+ V (x) V (x + d) = V (x)

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ψnq (x) =ψ nq (x + d) = e iqxunq (x)

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εn (q) = εn (q + k)

Bloch State

Particle in a periodic potential

• Periodicity of lattice leads to band structure of energy spectrum of the particle

Band structure for a particle in the periodic potential

and mean velocity : a) Free particle , b) €

U(z) = U0 sin2(π z /d)

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v0 q( )

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( U0 = 0 )

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U0 = E0 = h2π 2 /2md2

(a) (b)

• Particles in a periodic potentials subjected to an external force undergo oscillations instead of linear acceleration

Bloch Oscillations

• Eigenenergies and eigenstates are Bloch states

• Under the influence of a constant external force, evolves into the state according to

• The evolution is periodic and has a period of

• The mean velocity in is

• A wave packet with a well defined q in the nth band oscillates with an amplitude is the energy width of the nth band

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En (q)

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n,q⟩

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n,q(0)

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n,q(t)

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q(t) = q(0) + Ft /h

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TB =2πh

dF

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n,q(t)

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vn

q t( )( ) =1

h

dEn R q(t)( )( )

dq

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Δ n /2 F where Δn

•Electrons acted on by a static electric field oscillate

•Oscillations have never been observed in nature

•Studied in semiconductor super lattices, but oscillations are still dominated by relaxation process.

Solution: Optical lattices

• Studying Bloch oscillations, properties of condensed matter systems

• Study superfluid behaviour in the lattice

• Can be used for laser cooling atoms (lattice potential increases efficiency of some optical cooling methods)

• Study of many body quantum mechanics

• Atomic clocks

• Quantum computers?

Motivation to study optical lattices

Consider a 2 level atom in a standing plane wave. The temporal evolution if the system is given by:

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) H =

E e 0

0 E g

⎛

⎝ ⎜

⎞

⎠ ⎟+

p2

2M

1 0

0 1

⎛

⎝ ⎜

⎞

⎠ ⎟− 2Ωsin(ωL t)sin(kL x)

0 1

1 0

⎛

⎝ ⎜

⎞

⎠ ⎟M atomic massEg, Ee ground and excited electronic states Rabi frequency wL, kL frequency and wave vector of the standing wave

The wavefunction of this system is:

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Ψ(x, t) = exp(−iωL t)ψ e (x, t)e⟩+ψ g (x, t) g⟩

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ih∂ψ e (x, t)

∂t= hδψ e (x, t) +

ˆ p 2

2Mψ e (x, t) − hΩsin(kL x)ψ g (x, t)

ih∂ψ g (x, t)

∂t=

ˆ p 2

2Mψ g (x, t) − hΩsin(kL x)ψ e (x, t)

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where δ =Ee − Eg( )

h−ωL is the detuning

Optical Potential

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ih∂ Ψ

∂t= ˆ H Ψ(t)

The Hamiltonian of this system is:

We consider , then:

Then the problem reduces to solving a Schrodinger equation:

This eq. describes the motion of the atom along the standing wave. The potential is the optical lattice and has a spatial period of d=/2

The dept of the lattice is measured in units of recoil energy

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ψe (x, t) ≈ Ω /δ( )sin(kL x)ψ g (x, t)

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ih∂ψ g (x, t)

∂t=

ˆ p 2

2M+ V (x)

⎛

⎝ ⎜

⎞

⎠ ⎟ψ g (x, t)

V (x) = V0 sin2(kL x), V0 = −hΩ2 /δ

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ER = h2kL2 /2M

• Create a Bose-Einstein condensate or a cold gas of fermionic atoms with a well defined momentum spread

• Slowly ramp up lasers to create a lattice potential

• Put the lattice into the atoms and the atoms reorder to adapt to their new environment

Optical Lattices

Standing laser waves and cold neutral atoms play the role of the crystal lattices and electrons respectively

Description of the experiment performed by:

M.B Dahan,E.Peik, J.Richel, Y. Castin, C.Salomon. See [1]

PART II:

How to measure Bloch oscillations

Using laser cooling prepare a gas of free electrons with a momentum spread in the direction of the standing wave

• Precool Cs ( )using a MOT . Turn off magnetic field and 1D Raman cooling with horizontal beams

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6μK

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p = hk / 4

1. Cooling the atoms

Cloud of cold atoms

MOT with cloud of cold atoms visible in red

• Adiabatically switch on light potential, initial momentum distribution is turned into a mixture of Bloch states

• Laser is split in 2 beams with the same polarization and intensity. Beams are superimposed in counterpropagating directions

• Initially beams have the same frequency, their dipole coupling to the atom leads to the potential:

• Spontaneous emission can be neglected because interaction time is much shorter than the emission rate

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U(z ) = U0 sin2 kz

2. Setting the Potential

Atoms re-arrange and form optical lattice

Mimic external force by Introducing a tunable frequency difference between 2 counterpropagating laser waves. So atom feels a force:

This is done by applying a frequency ramp of duration

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ν( t)

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F = −ma = −mλd

dt

δν (t)

2

⎛

⎝ ⎜

⎞

⎠ ⎟

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ta

3. Applying external force

Schematic representation of Counterpropagating laser waves

• At a given acceleration time the standing wave is turned off fast

• Obtain atomic momentum distribution in the lab frame

• The distribution in the accelerated frame is obtained by a translation -ma

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ta

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ta

4. Measuring the oscillation

Source: See [1]

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Momentum distributions in the accelerated frame

for different values of ta between t a = 0 and t a = τ B = 8.2ms

Potential depth is U0 = 2.3ER and a = -0.85m/s2

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Mean atomic velocity v as a function of t a for values

of the potential depth : (a)U0 =1.4ER, (b)U0 = 2.3ER, (c)U0 = 4.4ER

Source: See [1]

• Initial momentum distribution is well defined and can be tailored at will

• Periodic potential can be turned on and off easily

• There is virtually� no dissipation or scattering from defects in the periodic potential

• We observe Bloch periods in the millisecond range, i.e. 10 orders of magnitude longer than in semiconductors.

Advantages of this Method

Source: See [2]

[1] M.B Dahan et al., Phys. Rev. Lett 76,24 (1996)

[2] M.Greiner & S.Folling, Nature 5, 736-738 (2008)

[3] D.Budker, D. Kimball & D.P DeMille, Atomic physics: An Exploration through Problems and Solutions (Oxford University Press, 2008)

[4] I.Bloch, Nature Phys. 1, 23-30 (2005)

[5] O.Morsch et al., Phys. Rev. Lett. 87,14 (2001)

References