Quantum Optics1

8/3/2019 Quantum Optics1

http://slidepdf.com/reader/full/quantum-optics1 1/66

Quantum Optics Theory

Dr. Michael J. Hartmann

2010/2011



Contents

1 3

1.1 Quantisation of the Electromagnetic Field . . . . . . . . . . . . . 41.2 Fock or Number States . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Squeezed states . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Single-mode squeezed states . . . . . . . . . . . . . . . . 8

1.4.2 Multi-mode squeezed states . . . . . . . . . . . . . . . . 9

1.5 Quantum correlations and the Einstein-Podolsky-Rosen paradox . 10

2 11

2.1 Coherence properties of the electromagnetic field . . . . . . . . . 12

2.1.1 Field correlations . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2 Optical coherence: Young’s double slit experiment . . . . 12

2.2 First order optical coherence . . . . . . . . . . . . . . . . . . . . 13

2.3 Second order optical coherence: Photon correlation measurements 13

2.4 Phase dependent correlations: Homodyne detection . . . . . . . . 14

3 15

3.1 Representations of the Electromagnetic Field . . . . . . . . . . . 16

3.1.1 Expansion in number states . . . . . . . . . . . . . . . . 16

3.1.2 Expansion in coherent states . . . . . . . . . . . . . . . . 16

3.1.3 The Wigner function . . . . . . . . . . . . . . . . . . . . 17

4 18

4.1 Open quantum systems . . . . . . . . . . . . . . . . . . . . . . . 19

4.1.1 Damped Harmonic Oscillator . . . . . . . . . . . . . . . 21

4.1.2 P-Representation and Fokker-Planck Equations . . . . . . 23

5 25

5.1 Interactions Between Radiation and Atoms . . . . . . . . . . . . 26

5.1.1 Long-Wavelength Approx. and Dipole Representation . . 26

1



5.1.2 Two-Level Atoms and the Jaynes-Cummings Model . . . 27

5.1.3 Spontaneous Emission of a Two Level Atom . . . . . . . 285.1.4 Resonance Fluorescence . . . . . . . . . . . . . . . . . . 28

5.1.5 Power Spectrum of the Emitted Light . . . . . . . . . . . 30

5.1.6 Equations for Correlation Functions and the Quantum Re-

gression Theorem . . . . . . . . . . . . . . . . . . . . . . 32

5.1.7 Raman Transitions and Electromagnetically Induced Trans-

parency . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6 36

6.1 Cavity Quantum Electrodynamics . . . . . . . . . . . . . . . . . 37

6.1.1 Strong driving regime . . . . . . . . . . . . . . . . . . . 386.1.2 Strong coupling regime . . . . . . . . . . . . . . . . . . . 39

6.1.3 Input-output formalism . . . . . . . . . . . . . . . . . . . 39

6.1.4 Circuit QED . . . . . . . . . . . . . . . . . . . . . . . . 41

7 45

7.1 Light Forces on Atoms . . . . . . . . . . . . . . . . . . . . . . . 46

7.1.1 Concept of Doppler cooling . . . . . . . . . . . . . . . . 46

7.1.2 Semiclassical theory of light forces . . . . . . . . . . . . 46

7.1.3 Standing wave Doppler cooling . . . . . . . . . . . . . . 48

7.1.4 Limit of Doppler cooling . . . . . . . . . . . . . . . . . . 49

7.1.5 Cooling beyond the Doppler limit: Sisyphus cooling . . . 50

7.1.6 Optical lattices . . . . . . . . . . . . . . . . . . . . . . . 51

7.1.7 Many-particle representation: 2nd quantisation . . . . . . 52

7.1.8 Interactions between ultra-cold atoms . . . . . . . . . . . 54

7.1.9 Mott insulator to superfluid quantum phase transition . . . 56

7.1.10 Measuring the Mott insulator and superfluid phases . . . . 58

8 59

8.1 Trapped Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

8.1.1 Trapping potential: Paul trap . . . . . . . . . . . . . . . . 60

8.1.2 Manipulations with lasers . . . . . . . . . . . . . . . . . 628.1.3 Sideband cooling . . . . . . . . . . . . . . . . . . . . . . 62

8.1.4 Trapping multiple ions . . . . . . . . . . . . . . . . . . . 63

8.1.5 Ion trap quantum computer . . . . . . . . . . . . . . . . . 64

2



Chapter 1

3





The Hamilton function of the field is,

H =1

2

d 3r

Π · Π

ε 0+

(∇× A) · (∇× A)

µ 0

, (1.10)

where the canonically conjugate momentum to A is Π = ε 0 A. This form of the

Hamiltonian can be justified by showing that the Euler-Lagrange equations for

the corresponding Lagrangian reproduce Maxwell’s equations, (1.1) - (1.4). Using

eqs. (1.7) and (1.9), H can be written as a sum of harmonic oscillators,

H =∑ k

p2 k

2+ω 2 k

2q2

k , (1.11)

with,

q k (t ) =

h

2ω k

a k

e−iω k t + c.c.

(1.12)

p k (t ) = −i

hω k

2

a k

e−iω k t − c.c.

. (1.13)

q k and p k

play the roles of positions and momenta since their equations of motion,

q k = p k

and ˙ p k =

−ω k

2q k (1.14)

also follow from the Hamilton equations for H as in (1.10). This also shows that

(1.10) is indeed the Hamiltonian for the field.

In analogy to the harmonic oscillator, the electromagnetic field is now quan-

tised by promoting q k and p k

to become operators q k and ˆ p k

and imposing,q k

, ˆ p k

= ihδ k , k . (1.15)

The field now reads,

A( r , t ) =∑ k

h

2ε 0ω k a k

u k ( r )e−iω k

t + c.c. , (1.16)

where a† k

and a k are creation and annihilation operators for the quanta in mode k ,

a k

, a k

= 0 and

a k , a

† k

= δ k , k . (1.17)

These are bosonic commutation relations and show that the quanta of the field,

the photons, are bosons. The operator A( r , t ) is here written in Heisenberg picture

and fulfils d dt

A = − ih

A, ˆ H

with H as in (1.10).

5



1.2 Fock or Number States

The quantised electromagnetic field reads,

A( r , t ) =∑ k

h

2ε 0ω k

a k

u k ( r )e−iω k

t + c.c.

, (1.18)

where a† k

and a k create and annihilate photons in mode k , i.e.

a k

, a k

= 0 anda k

, a† k

= δ k , k . The operator a

† k

a k is therefore a number operator and it’s eigen-

values n k denote the number of photons in a mode k .

a†

k a k

|n k = n k

|n k . (1.19)

The eigenstates |n k of a

† k

a k are called Fock or number states, where a

† k

a k |0 =

0 denotes the vacuum state without photons. Since H = ∑ k hω k

(a† k

a k + 1

2), the

vacuum energy diverges. This has however no measurable consequences since

only energy differences can be observed.

Since they are eigenstates of an Hermitian operator, number states form a com-

plete orthonormal basis,

n k |n p = δ k , p (1.20)

∑n k

|n k n k

| = 1 . (1.21)

Number states are useful for calculations but difficult to generate experimentally.

1.3 Coherent States

A coherent state is generated by applying the shift operator

D(α ) = eα a†−α a (1.22)

onto the vacuum,

|α = D(α )|0 (1.23)

Such a shift is for example generated when a classical oscillating dipole emits an

electromagnetic field. Here, the dipole-field interaction reads,

V = −e d (t ) · E (t ) ≈ −ih( χ a − χ a†), (1.24)

6



where we have omitted fastly oscillating terms in the approximation. The time

evolution of this interaction Hamiltonian generates a shift D(α ) with α = χ t . Theshift operator D(α ) has the properties

D†(α ) a D(α ) = a +α (1.25)

D†(α ) a† D(α ) = a† +α (1.26)

D†(α ) = D−1(α ) = D(−α ) (1.27)

D(α ) = e− |α |22 eα a

†

e−α a . (1.28)

Coherent states have the following properties:

•eigenstates of a

a|α = α |α (1.29)

• expansion in Fock states

|α = e− |α |22

∞

∑n=0

α n√n!

|n (1.30)

• contain an indefinite number of photons

• normalised

α |α = 1 (1.31)

•probability to measure n photons is Poisson distributed

P(n) =|α |2n

n!e−|α |2

(1.32)

• not orthogonal

α |β = e−|α −β |2

(1.33)

• overcomplete d 2α |α α | = π (1.34)

• the position distribution for a harmonic oscillator, H = hω (a†a + 12

), that is

initially in a coherent state |α = | |α |eiδ

is a Gaussian with constant vari-ance σ x =

h

2mω and a mean value x(t ) =√

2|α |cos(ω t −δ ) that oscillates

with frequency ω .

P( x) =1√

2πσ xexp

− [ x − x(t )]2

2σ 2 x

(1.35)

In this sense, a coherent state of a harmonic oscillator is the state which

comes closest to our classical (i.e. non-quantum) notion of a harmonic os-

cillator.

7



1.4 Squeezed states

The expectation value and variance of an operator A in some state |ψ are defined

as,

A = A = ψ | A|ψ (1.36)

V ( A) = (∆ A)2 = A − A2. (1.37)

The uncertainty relation for two operators A and B reads,

∆ A∆ B ≥ 1

2|[ A, B]| . (1.38)

For the two field quadratures X 1 = a + a† and X 2 − i(a − a†), the uncertainty rela-

tion reads,

∆ X 1∆ X 2 ≥ 1. (1.39)

States, for which ∆ X 1∆ X 2 = 1 are called minimum uncertainty states.

1.4.1 Single-mode squeezed states

States, for which either ∆ X 1 < 1 ( and consequently ∆ X 2 > 1) or ∆ X 2 < 1 ( and

consequently ∆ X 1 > 1) are called squeezed states.

Squeezed states can for example be generated in a “degenerate parametricamplifier” via processes where a drive of frequency 2ω generates two photons of

frequency ω at a time. The Hamiltonian for this process reads,

H I = ih

2

χ a2 − χ (a†)2

, (1.40)

in an interaction picture with respect to the photon energy H 0 = hω a†a. Writing

χ = | χ |e2iφ we find for the quadratures Y 1 = ae−iφ + a†eiφ and Y 2 = −i(ae−iφ −a†eiφ ),

V (Y

1(t )) =

e−

2

| χ

|t V

(Y

1(0)) andV

(Y

2(t )) =

e2

| χ

|t V

(Y

2(0)), (1.41)

i.e. the variance in Y 1 becomes squeezed and the variance in Y 2 becomes amplified.

We thus define the squeezing operator to be,

S(ε ) = exp

ε

2a2 − ε

2(a†)2

, (1.42)

and a squeezed state to be,

|α ,ε = D(α )S(ε )|0. (1.43)

8



Writing ε = re2iφ with r =

|ε

|, we find

S†(ε )aS(ε ) = a cosh(r ) − a†e2iφ sinh(r ) (1.44)

S†(ε )a†S(ε ) = a† cosh(r ) − ae−2iφ sinh(r ). (1.45)

For a harmonic oscillator, H = hω (a†a + 12

), that is initially in a squeezed state

with |α ,ε = ||α |, re2iφ , we find for the position operator q =

h2ω (a + a†) that

its mean value oscillates harmonically just as for a coherent state,

q(t ) =

2h

ω |α |cos(ω t ). (1.46)

In contrast to a coherent state the variance (∆q)2 is however not constant but

oscillates with frequency 2ω ,

(∆q)2(t ) =h

2ω

cosh2(r ) + sinh2(r ) − 2cosh(r ) sinh(r ) cos(2ω t − 2φ )

. (1.47)

1.4.2 Multi-mode squeezed states

Multi-mode squeezed states can for example be generated in a “non-degenerate

parametric amplifier” via processes where a drive of frequency ω 1 +ω 2 gener-

ates one photon of frequency ω 1 and one photon of frequency ω 2 at a time. The

Hamiltonian for this process reads,

H I = ih χ a1a2 − χ a†

1a†2

, (1.48)

in an interaction picture with respect to the photon energies. We thus define the

two-mode squeezing operator to be,

S(G) = exp

Ga1a2 − Ga†1a

†2

, (1.49)

and a two-mode squeezed state to be, |α 1,α 2 = D(α 1) D(α 2)S(G)|0. The ex-

pansion of two-mode squeezed vacuum in terms of number states reads,

|ϕ

= S(G)

|0, 0

=

1

cosh(r )

∞

∑n=0

tanhn(r )

|n, n

(1.50)

This state is entangled , that means there is no way of writing,

|ϕ = |ϕ 1⊗ |ϕ 2 (1.51)

As a consequence, if we measure the number of photons in mode 1 to be n for the

state |ϕ of eq. (1.50) we know with certainty that mode 2 has n photons as well

although the variance in photon number is not zero for |ϕ .

Entanglement is the key resource for the exponential speed-up of quantum

computers as compared to classical computers.

9



1.5 Quantum correlations and the Einstein-Podolsky-

Rosen paradox

For a two mode squeezed state consider the quadratures ( j = 1, 2),

X φ j = a jeiφ + a

† j e−iφ with

X

φ j , X

φ + π 2

j

= 2i , (1.52)

which are canonically conjugate. Correlations between both modes can be quan-

tified by

V (φ ,ψ ) =1

2

X φ 1 − X

ψ 2

2

(1.53)

For the two mode squeezing dynamics generated by the Hamiltonian (1.48) we

have V (φ ,ψ )(t ) → 0 for t →∞, so that X φ 1 and X

ψ 2 become perfectly correlated.

For such a situation, Einstein, Podolsky and Rosen argued as follows: Sup-

pose the photons of both modes are in sufficiently separated laboratories so that

no communication at the speed of light can take place within the runtime of the

measurements. Assume further that V (φ ,−φ ) = 0. If now X φ 1 is measured, X

−φ 2 is

known with certainty. Therefore mode 2 must have been in an eigenstate of X −φ 2 ,

X −φ 2 |λ = λ |λ before the measurement. Since modes 1 and 2 are so far apart

that they can not interact, mode 2 must have been prepared in state |λ before the

photons have been taken apart.

If however, the experimentator in the laboratory of mode 1, after the photons

have been separated, decides to measure X φ 1 with φ = φ , he or she would find that

mode 2 must have been in the eigenstate X −φ 2 |λ = λ |λ before the photons have

been separated.

Since there are however values for φ and φ , for which X

−φ 2 , X

−φ 2

= 0 and

consequently |λ = |λ , this reasoning would require that mode 2 has been pre-

pared in two non-equal state at the same time.

The resolution to the paradox is that there are states with non-local correla-

tions. Information can however still only propagate at the speed of light or slower,

since both experimentators can not find out that their measurement results areactually correlated without exchanging information.

The existence of such non-local correlations has been verified several times in

experiments. For example Ou and Mandel measured the variances V ( X −φ 1 ) and

V ( X −φ +π /22 ) for a two mode squeezed state. Provided V (φ ,ψ )(t ) → 0, one has

V ( X −φ 1 ) → V ( X

−φ 2 ). With this reasoning, Ou and Mandel found a product of the

variances of V ( X −φ 2 )V ( X

−φ +π /22 ) = 1.4 ± 0.02, whereas the uncertainty relation

would require V ( X −φ 2 )V ( X

−φ +π /22 ) ≥ 4 if one measured V ( X

−φ 2 ) directly.

10



Chapter 2

11



2.1 Coherence properties of the electromagnetic field

2.1.1 Field correlations

If a photon is detected and hence absorbed by a photon detector it disappears

from the field. A field in the state (density matrix) ρ thus changes according to

ρ → E (+)ρ E (−) in the detection process. Since the information about the final

state | f of the field after the detection is irrelevant, the measured photon intensity

reads,

I ( r , t ) =∑| f

f | E (+)ρ E (−)| f = Tr

E (−) E (+)ρ

(2.1)

More generally, one can define the field correlations,

Gn( x1, . . . , xn, xn+1, . . . , x2n) = Tr

E (−)( x1) · · · E (−)( xn) E (+)( xn+1) · · · E (+)( x2n)ρ

(2.2)

where xn = ( r n,t n). These correlations have the properties,

G(1)( x, x) ≥ 0 (2.3)

Gn( x1, . . . , xn, xn, . . . , x1) ≥ 0 (2.4)

G(1)( x1, x1)G(1)( x2, x2) ≥G(1)( x1, x2)

2

(2.5)

(a

†

1a

1)

2

(a

†

2a

2)

2

≥ a†

1a

1a

†

2a

22

(2.6)

2.1.2 Optical coherence: Young’s double slit experiment

The electric field impinging at r on a screen which emerges from two pin holes at

positions r 1 and r 2 is a superposition of the field E 1 from pin hole 1 and the field

E 2 from pin hole 2,

E (+)( r , t ) = E (+)1 ( r ,t ) + E

(+)2 ( r , t ) (2.7)

Pin holes with sizes smaller than the wavelength can be treated as point sources

that emit spherical waves. The field E (+)( r , t ) thus reads,

E (+)( r , t ) ≈ i

hω

8π Lε 0

e−iω t

R

a1eiks1 + a2eiks2

, (2.8)

where s1 = | r − r 1|, s2 = | r − r 2| and we have approximated s−11 ≈ s−1

2 ≈ R−1. The

intensity on the screen thus shows interference fringes with maxima for k (s1 −s2) +φ = 2π n (n integer), i.e.

I ( r ,t ) ∝a

†1a1+ a

†2a2+ 2|a

†1a2|cos[k (s1 − s2) +φ ]

. (2.9)

12



2.2 First order optical coherence

The g(1)-function, which is defined by,

g(1)( x1, x2) =G(1)( x1, x2)

G(1)( x1, x1)G(1)( x2, x2), (2.10)

obeys |g(1)( x1, x2)| ≤ 1 due to eq. (2.5). The visibility of the interference fringes,

defined by,

ν =I max − I min

I max − I min. (2.11)

can be expressed in terms of the g(1)-function via the relation,ν = |g(1)( x1, x2)| 2√ I 1 I 2 I 1+ I 2 ,

where I 1 = G(1)( x1, x1) and I 2 = G(1)( x2, x2).

2.3 Second order optical coherence: Photon corre-

lation measurements

The probability to detect a photon at time t and a second at time t + τ is given by

the 2nd order correlation function G(2)(τ ) = E (−)(t ) E (−)(t +τ ) E (+)(t +τ ) E (+)(t ),

which is in fact independent of t for a steady beam. In analogy with first order

optical coherence, we define the g(2)-function,

g(2)(τ ) =G(2)(τ )

G(1)(0)2

. (2.12)

For an infinitely large time difference τ , the two photon detection events are ex-

pected to become independent, τ → ∞ ⇒ g(2)(τ ) → 1. For τ → 0 on the other

hand, we distinguish two scenarios, g(2)(0) > 1 called bunching and g(2)(0) < 1

called anti-bunching. Before proceeding to discuss several states of the field, we

note that,

g(2)(0) = 1 +V (n)

−n

n2 , (2.13)

where n = a†a and V (n) = a†a2− n2.

Examples:

• coherent states, E (+)|α = E (+)|α :

G(2)(τ ) = E (−)(t )E (−)(t + τ )E (+)(t + τ )E (+)(t ) =

G(1)(0)

2

⇒

g(2)(τ ) = 1. (2.14)

13



•fluctuating classical field

⇒G(2)(τ ) = d 2E P(E )E (t )E (t +τ )E (t +τ )E (t ).

g(2)(0) = 1 +

d 2E P(E )

|E |2 −|E |2

2

|E |22. (2.15)

Hence, whenever P(E ) ≥ 0 we have bunching, g(2)(0) < 1. This means

that every field that can be generated by mixing classical (coherent) fields

will display bunching. Consequently, if a field displays anti-bunching, it is

quantum in the sense that it cannot be generated by mixing classical fields.

•a number state

|n

shows anti-bunching as follwos form eq. (2.13).

g(2)(0) = 1− 1

n(2.16)

• squeezed states, |α ,ε with ε = re2iΦ, can show bunching if they are phase

squeezed, Φ = π 2

and α real. Amplitude squeezed states, Φ = 0 and α

real, can show anti-bunching for |α |2 2sinh2(r ) cosh2(r ). The squeezed

vacuum, α = 0, shows bunching.

2.4 Phase dependent correlations: Homodyne de-tection

Consider a beam splitter with transmittivity η that combines fields with

annihilation operators a and b to an output field c via the relation,

c =√ηa + i

1 −ηb. (2.17)

If the field with annihilation operator b is in a coherent state with large

amplitude β , the measured number of output photons obeys,

c†c ≈ (1 −η)|β |2 + |β | η(1 −η) X Φ+ π

2, (2.18)

where X Φ+ π 2

= ae−iΦ + a†eiΦ and β = |β |eiΦ. In eq. (2.18) we have ne-

glected a term η2a†a since |β | 1. After subtracting (1 −η)|β |2, the

photon number c†c thus allows to measure the quadrature X Φ+ π 2

. Chang-

ing the phase of the reference field b by π /2 then allows to also measure the

canonically conjugate quadrature X Φ.

14



Chapter 3

15



3.1 Representations of the Electromagnetic Field

3.1.1 Expansion in number states

Any density state of the electromagnetic field can be expanded in number states,

ρ =∞

∑n,m=0

cn,m|nm|. (3.1)

This particularly convenient for states which are diagonal in this basis, such as

thermal states,

ρ =

1 − e−hω

k BT ∞

∑n=0 e−hω

k BT n

|nn| =

1

1 + n

∞

∑n=0 n

1 + nn

|nn| (3.2)

where n = Tr(a†aρ) is the expectation value for the photon number.

3.1.2 Expansion in coherent states

Since the coherent states form an over-complete set, any state can as well be ex-

panded in terms of coherent states. Observing that the matrix elements n|O |m,

are the expansion coefficients of the Taylor series

e−|α

|2

α

|O

|α

=

∞

∑n,m=0

√n!m!

n|O

|m

(α )n (α )m

n!m!(3.3)

for any Hermitian operator O , we see that all matrix elements n|O |m can be

determined from only the diagonal elements α |O |α . Therefore the P-function,

defined by

ρ =

d 2α P(α ) |α α | (3.4)

contains all information about the state ρ and is thus an alternative form to repre-

sent it. For a coherent state we obviously have P(α ) = δ (2)(α −α 0).

P(α ) is not always positive and can therefore in general not be interpreted as

a probability distribution. This is evident from

g(2)(0) = 1 +

d 2α P(α )

|α |2 −|α |2

2

|α |22, (3.5)

where |α |2 =

d 2α P(α ) |α |2. For anti-bunching, P(α ) must have negative or

highly singular parts. The same holds for squeezing with either V ( X 1) < 1 or

V ( X 2) < 1 ( X 1 = a + a†, X 2 = −i(a − a†) since,

V ( X 1,2) = 1 +

d 2α P(α ) [α ±α − (α ±α )]2(3.6)

16



Whenever P(α ) is not a probability distribution, the corresponding field cannot

be generated by mixing classical fields and is thus regarded as quantum. Anti-bunching and squeezing are thus signatures of quantum fields.

3.1.3 The Wigner function

The characteristic function is defined as

χ (η) = Trρeηa†−ηa

(3.7)

and the Wigner function is its Fourier transform,

W (α ) =1

π 2

d 2ηeη

α −ηα

χ (η) =1

π 2

d 2ηTrρeη(a

†

−α

)−η

(a−α )

. (3.8)

Splitting the variables into real and imaginary parts, η = µ + iν , a = ω q+ip√2ω h

and

α = ω q+ip√2ω h

we get,

W (q, p) =2h

(2π )2

d µ d ν Tr

ρeiν (q−q)−iµ ( p− p)

, (3.9)

where µ =

2hω µ and ν =

2ω

hν , and see that W (q, p) is a Fourier transform of

ρ . It therefore contains all the information about the state ρ .

The Wigner function is a Gaussian convolution of the P-function,

W (α ) =2

π

d 2β P(β ) e−2|α −β |2

(3.10)

For a coherent state |α 0, the Wigner function is Gaussian with variances 1, cen-

tred at α 0,

W ( x1, x2) =2

π e− 1

2 ( x21+ x2

2), (3.11)

where x1 = 2Re[α −α 0] and x2 = 2Im[α −α 0]. For a squeezed state, on the other

hand, the variance is squeezed in one direction and enhanced in the conjugate one,

W ( x1, x2) = 2π

exp− x21

2e−2r

exp

− x22

2e2r

, (3.12)

with real (φ = 0) and positive squeezing parameter ε = r > 0. Finally, a number

state has the Wigner function,

W ( x1, x2) =2

π (−1)n Ln(4r 2)e−2r , (3.13)

with r 2 = x21 + x2

2, which can become negative. This is another signature that

number states are non-classical.

17



Chapter 4

18



4.1 Open quantum systems

In no realistic situation, quantum systems of interest are perfectly isolated. Hence

an interaction between the system under study and its environment has to be taken

into account. Since the environment is typically extremely large, it is impossible

to solve its dynamics exactly and approximations are needed. These approxima-

tions make use of the fact that the interaction between system and environment

is typically weak. In the following we will derive an equation of motion for the

reduced density matrix of the system under study and then simplify it via two

approximations to obtain an ordinary first order differential equation.

The Hamiltonian of the total system can be written as a sum of the Hamiltonian

for the system ( H S), the environment ( H E ) and interaction ( H I ),

H = H S + H E + H I (4.1)

If we write the density matrix of the whole system R in a product basis of system

states |sα and environment states |eβ , R = ∑α ,α ,β ,β cα ,β ;α ,β |sα , eβ sα , eβ |,the reduced density matrix for the system reads,

ρS = Tr E ( R) = ∑α ,α ,β

cα ,β ;α ,β |sα sα | (4.2)

The dynamics of R is given by the Liouville equation,

˙ R = −i[ H , R] (4.3)

where we have set h = 1 and H is as in eq. (4.1). Since eq. (4.3) is linear in R,

we can write as a shorthand ˙ R = −iLR ≡ −i[ H , R] and R(t ) = exp(−iLt ) R(0) ≡exp(−iHt ) R(0) exp(iHt ). Analogously to the Hamiltonian, we may write L = LS + L E + L I .

We now split the density matrix R into a relevant part in which we are in-

terested, Rrel = P R, and an irrelevant part, Rirr = Q R. Here, P and Q are

projection operators that fulfil P2 =P, Q2 =Q and 1 =P+Q.

For our purposes we will use for P a projector whose action on any operator

O is given by,

PO = ρ E ⊗ Tr E (O ). (4.4)

Using these projection operators and the identity,

e−i( L1+ L2)(t −t 0) = e−iL1(t −t 0) − i

t

t 0

ds e−iL1(t −s) L2e−i( L1+ L2)(s−t 0), (4.5)

which is valid for any two operators L1 and L2, we get the equations,

P ˙ R(t ) = −iP LP R(t )− iP LQ R(t ) = −iP LP R(t )− iP LQe−il(t −t 0) R(t 0)(4.6)

19



and

P ˙ R(t ) = −iP LP R(t )−

t

t 0

dsP LQe−iLQ(t −s) LP R(s)− iP LQe−iLQ(t −t 0) R(t 0)

(4.7)

To continue, we now make three assumptions,

1. the environment state ρ E is invariant under the environment dynamics,

[ H E ,ρ E ] = 0

2. the initial state factorises, R(t 0) = ρS(t 0) ⊗ρ E

3. the system environment interaction fulfilsP L I P = 0.

Observing that,P L E = L E P = 0 and [P, LS] = 0, and using these three assump-tions, we get the Nakajima-Zwanzig equation for the relevant part of the density

matrix,

˙ Rrel(t ) = −iLS Rrel(t ) − t

t 0

dsP L I Qe−iLQ(t −s) L I Rrel(s) (4.8)

This is an exact equation for the degrees of freedom of the system only. It however

contains a time integral over the history Rrel(s < t ) and is thus not Markovian.

To obtain a Markovian, ordinary differential equation for ρS, we now make

the following two approximations,

1. weak coupling approximation: we only keep the leading order terms in the

coupling L I in eq. (4.8). Higher orders would approximately contain addi-tional powers of H I (t − t 0). This approximation should thus be good as long

as || H I ||(t − t 0) 1.

2. Markov approximation: It will turn out that the kernel of the integral decays

fastly. For the time range that significantly contributes to the integral in eq.

(4.8), we can thus approximate ρS(t ) ≈ e−i( H S+ H E )(t −s)ρS(s)ei( H S+ H E )(t −s).

This means that for the time range that significantly contributes to the inte-

gral, ρS(s) in the integral kernel at an earlier time s can be linked to ρS(t )without taking H I into account.

Using these approximations, we get the equation,

ρS(t ) = −i[ H S,ρS(t )] − t

t 0

ds Tr E [ H I , [ H I (s − t ),ρ E ⊗ρs(t )]] , (4.9)

where we have switched back to the Hamiltonian picture and

H I (s − t ) = e−i( H S+ H E )(t −s) H I ei( H S+ H E )(t −s). Eq. (4.9) is now a Markovian or-

dinary differential equation since the right hand side only depends on ρs(t ). This

equation is a good approximation if the assumptions 1–3 are met and if the integral

kernel decays much faster than ρs(t ) can change due to the system-environment

interaction. Fortunately this scenario is often true for very large environments.

20



4.1.1 Damped Harmonic Oscillator

We will now apply the master eq. (4.9) to a specific model. To this end we

assume a harmonic oscillator that couples to an environment formed by harmonic

oscillators, H = H S + H E + H I with,

H S = ω 0

a†a +

1

2

, H E =∑

k

ω k

b

† k

b k +

1

2

(4.10)

H I = ∑ k

g k (a† + a)(b

† k

+ b k ) and ρ E =

1

Z exp

− H E

k BT

(4.11)

For evaluating the master eq. (4.9), we need H I in the interaction picture, H I (s−t ).This can be simplified further via the so called Rotating Wave Approximation,

H I (s − t ) ≈ a†Γ (s− t )eiω 0(s−t ) + aΓ †(s − t )e−iω 0(s−t ), (4.12)

where Γ (s − t ) = ∑ k g k

b k e−iω k

(s−t ). This approximation is good since the terms

that are neglected in eq. (4.12) belong to transitions that would not conserve the

energy of the system. Therefore their effect on the evolution of the system is

strongly suppressed.

Expanding the double commutator in the rhs of eq. (4.9), we find four terms,

Tr

[ H I , [ H I (s

−t ),ρ E

⊗ρS(t )]]

= Tr

H I H I (s

−t )ρ E

⊗ρS(t )

−−Tr H I (s − t )ρ E ⊗ρS(t ) H I −Tr H I ρ E ⊗ρS(t ) H I (s − t )+Trρ E ⊗ρS(t ) H I (s − t ) H I .As an example we will explicitly calculate the first one,

Tr H I H I (s − t )ρ E ⊗ρS(t ) =Γ †Γ †(s − t )

aae−iω 0(s−t ) +

Γ †Γ (s − t )

aa†e−iω 0(s−t )+

ΓΓ †(s − t )

a†ae−iω 0(s−t ) + ΓΓ (s − t )a†a†e−iω 0(s−t )ρS(t )

For evaluating the rhs of eq. (4.9) we thus need to calculate the integrals,

I 1 = t

t 0

ds

Γ †Γ †(s− t )

e−iω 0(s−t ), I 2 =

t

t 0

ds

Γ †Γ (s − t )

e−iω 0(s−t )

I 3 =

t

t 0

dsΓΓ †(s − t )

e−iω 0(s−t ), I 4 =

t

t 0

ds ΓΓ (s − t )e−iω 0(s−t )

Due to the form of ρ E , see eq. (4.10) and the coupling, see eq. (4.12), the first and

fourth integral vanishes, i.e. I 1 = I 4 = 0. For I 2 we find,

I 2 = t

t 0

ds∑ k , p

g k

g pTr

b k b

† pρ E

e−i(ω 0−ω p)(s−t ) =

0

t −t 0

d τ ∑ k

|g k |2n k

e−i(ω 0−ω k )τ

21



Since the environment is large, we can introduce a density of states per frequency

interval, η(ω ), and write,

I 2 = 0

t −t 0

d τ

∞0

d ω

2π η(ω )|g(ω )|2n(ω )e−i(ω 0−ω )τ

If we now assume that η(ω )|g(ω )|2n(ω ) = cω α for some positive α , we find, 0

t −t 0

d τ

∞0

d ω

2π ω α e−i(ω 0−ω )τ = ω α +1

0 Γ (α + 1) 0

t −t 0

d τ eiω 0τ

(iω 0τ )α +1

We see that the kernel in the integral decays as (iω 0τ )−(α +1) and rapidly goes to

zero for τ ω −10 . This asymptotics emerges since the bath correlations decay

rapidly. We can thus extend the lower bound of the time integral to −∞ and find,

Re

0

−∞d τ Γ (α + 1)eiω 0τ

(iτ )α +1

= πη(ω 0)|g(ω 0)|2n(ω 0).

Here the imaginary part only leads to a redefinition of the frequency ω 0.

If one already knows that the bath correlations decay rapidly, a faster route of

calculation can be made by extending the time integration to the range (−∞, 0)right away and using the relation, 0

−∞dt ei(ω −ω 0)t = πδ (ω −ω 0) − iPV

1

ω −ω 0

, (4.13)

where PV denotes the Cauchy principal value. With this calculation one finds,

I 2 =γ

2n(ω 0) + i∆ and I 3 =

γ

2[n(ω 0) + 1] + i∆, (4.14)

where γ = η(ω 0)|g(ω 0)|2, ∆ = PV ∞−∞

d ω 2π

η(ω )|g(ω )|2

ω −ω 0[n(ω ) + 1] and

∆ = PV ∞−∞

d ω 2π

η(ω )|g(ω )|2

ω −ω 0n(ω ). ∆ and ∆ are merely frequency shifts and we will

assumed they have been absorbed into ω 0. With these results we can write down

the master equation,

ρS = −i[ω 0a†a,ρS] +γ

2n(ω 0)(2a†ρSa − aa†ρS −ρSaa†) (4.15)

+ γ 2

[n(ω 0) + 1](2aρSa† − a†aρS −ρSa†a)

From eq. (4.15) we find the amplitude of the oscillator is damped, a(t ) =

a(0)e− γ 2 t , and that a†a(t ) = a†a(0)e−γ t + n(ω 0)(1 − e−γ t ). One can also

show that the oscillator will, for t → ∞, be damped to a thermal state with the

same temperature as the environment,

limt →∞

ρS =1

Z exp

− H S

k BT

(4.16)

22



4.1.2 P-Representation and Fokker-Planck Equations

For solving the master equation (4.15), various representations for the density

matrix ρS can be used. One version is to represent ρS in the number basis, ρ =

∑n,mρn,m|nm| (I will skip the index S from now on).

Here we will use the representation in terms of the P-function, ρ =

d 2α P(α ) |α α |.For representing equation (4.15) in a basis of coherent states |α , we define,

||α = e12 |α |2 |α . (4.17)

Using equation (1.29) and the expansion in Fock states, see eq. (1.30), we find,

α ||a† = α α || a†||α = ∂ ∂α

||α

a†||α = α ||α α ||a =∂

∂α α || (4.18)

After an integration by parts, that uses P(α ) → 0 for |α | →∞, we get the following

correspondences for the translation of equation (4.15) into an equation for P(α ),

aρ ↔ α P a†ρ ↔α − ∂

∂α

P

ρa† ↔ α P ρa ↔ α −

∂

∂α

P (4.19)

a†aρ ↔α − ∂

∂α

α P ρa†a ↔

α − ∂

∂α

α P

We move to an interaction picture with respect to H 0 = ω 0

a†a + 12

and get with

the help of the above mappings an equation of motion for the P-function in inter-

action picture, P I ,

∂

∂ t P I (α , t ) =

γ

2 ∂

∂α α +

∂

∂α α + γ n(ω 0)

∂ 2

∂α∂α P I (α , t ). (4.20)

This form of equation is known as a Fokker-Planck equation and also occurs in

classical stochastic processes. The analogy to classical stochastic processes be-

comes even more apparent with the following notation in terms of the real ( x1)

and imaginary ( x2) parts of α , α = x1 + ix2.

x =

x1

x2

, ∂ =

∂ ∂ x1

∂ ∂ x2

, A = −γ

2 x, D = γ n(ω 0)

10

01

, (4.21)

23



for which the Fokker-Planck equation (4.20) takes the form,

∂

∂ t P I ( x,t ) =

− ∂ · A +

1

2

2

∑ j,l=1

∂ 2

∂ x j∂ xl

D j,l

P I ( x, t ). (4.22)

Here A is called “drift vector” and D “diffusion matrix” since the dynamics of

the mean values x j(t ) =

dx1dx2 x jP I ( x, t ) only depends on A via d dt

x j = A jand the dynamics of the variances V ( x j) = x2

j− x j2 depends on A and D, i.e.d dt

V ( x j) = −γ V ( x j) +γ n(ω 0). We thus see that mean values decay at a rate γ /2

and variances approach their assymptotic value V ( x j, t →∞) → n(ω 0) at a rate γ ,

x

(t ) =

x

(0)e− γ

2 t , V ( x j, t ) = V ( x j, 0)e−γ t + n(ω 0)1

−e−γ t . (4.23)

Therefore, any squeezing that might be present in the initial state is damped away

at a rate γ .

Steady state solutions Often it is sufficient to know the asymptotic steady state

of a master equation which fulfils ∂ ∂ t

P = 0. From eq. (4.22) we see that this

implies, ∂ ∂ x j

− A j + 1

2 ∑2l=1

∂ ∂ xl

D j,l

P = 0 and try the ansatz,

A jP =1

2

2

∑l=1

∂

∂ xl D j,lP

⇔

2

∑l=1

D j,l∂

∂ xl

ln(P) = 2 A j −2

∑l=1

∂ D j,l

∂ xl

. (4.24)

Defining a function φ via P( x) =N −1 exp(−φ ( x)), whereN is a normalisation,we see that φ plays the role of a potential for a generalised force F via the relation F = − ∂φ , where

F j = 22

∑l=1

D−1 j,l

Al −

2

∑k =1

∂ Dl,k

∂ xk

. (4.25)

Here we have assumed that the matrix D is invertible. The existence of the po-

tential φ is then guaranteed if the force F fulfils the conditions ∂ jF l = ∂ lF j. As a

consequence φ can be expressed via an integral of F along a path in the complex

plane which is independent of the path taken,

φ ( x) =

x

F · d ν (4.26)

where d ν is the line element along the path. For our example given in equation

(4.21), we find φ ( x) = x1

0s1ds1

n(ω 0)+ x2

0s2ds2

n(ω 0)=

x21+ x2

2

2n(ω 0) , and therefore,

P( x) =1

2π n(ω 0)exp

− x2

1 + x22

2n(ω 0)

(4.27)

This is a Gaussian distribution centred at x = 0 with variance n(ω 0) in all direc-

tions.

24



Chapter 5

25



5.1 Interactions Between Radiation and Atoms

Since electromagnetic fields couple to charges and atoms are composed of a pos-

itively charged nucleus and negatively charged electrons, electromagnetic fields

couple to atoms. As the nucleus is much heavier than the electrons, only the latter

will be moved by the coupling to the fields. For one electron with charge −e, the

Hamiltonian describing this interaction reads,

H =1

2m

p + e A( r )

2

− eV (| r |) + H rad , (5.1)

where m is the mass, p the momentum and r the position of the electron. A and

V are the vector potential of the electromagnetic field and the Coulomb potentialrespectively. H rad is the energy of the free field. Using the Hamilton equations

of motion for the classical Hamilton function corresponding to (5.1), one can see

that these reproduce the known form of the Lorenz force, m r = −e E − e r × B.

The form of the coupling in Hamitonian (5.1) can furthermore be motivated by

observing that the probability to find a charged particle at a location r , |ψ ( r ,t )|2

is invariant under a local change of the phase of the wave function ψ , ψ ( r , t ) →ψ ( r , t ) = ψ ( r , t )eiφ ( r ,t ). Of course this transformation would not leave the dynam-

ics of the charged particle invariant. Therefore this change in the dynamics should

be caused by a coupling to fields which transform accordingly. One can show that

the Schrodinger equation corresponding to the Hamiltonian (5.1), ih

∂

∂ t ψ = H ψ ,is invariant under the transformation,

ψ ( r , t ) → ψ ( r , t )eiφ ( r ,t ), A → A +∇ χ , V → V − ∂

∂ t χ , (5.2)

with χ = heφ ( r , t ), which leaves both invariant, |ψ ( r , t )|2 as well as the physical

fields E and B. This formally justifies the form of the coupling.

5.1.1 Long-Wavelength Approx. and Dipole Representation

We consider a system of charged particles centered at r = 0 with Hamiltonian,

H =∑ j

1

2m j

p j + e A( r j)

2

+∑ j,l

1

8πε 0

q jql

| r j − r l|+ H rad , (5.3)

where the second term describes the Coulomb interaction of the particles with

charges q j. For atoms we have | r j − r l| ∼ aBohr ∼ 10−4λ with the Bohr radius

aBohr. We thus assume that the particles are distributed over a volume much

smaller than the wavelength of the radiation, | r j − r l| λ , so that we can ap-

proximate A( r j) ≈ A(0) and neglect any coupling of spins to the magnetic field.

26



The electric properties of the particle are then described by their dipole mo-

ment d = ∑ j q j r j and we can write the Hamiltonian (5.3) in a new form by ap-

plying the unitary transformation T = exp(− ih

d · A(0)). We find T r jT † = r j,

T p jT † = p j + q j A(0) and Ta k ,ν

T † = a k ,ν + (i u k ,ν

· d )/

2hω k ε 0, where u k ,ν

is

the mode function of the mode with vector k and polarization ν . The transformed

Hamiltonian reads,

H = T HT † = H atom + H rad − d · E (0), (5.4)

where H atom =∑ j

p2 j

2m j+∑ k ,ν

| u k ,ν · d |2 +∑ j,l

18πε 0

q jql

| r j− r l | .

For the relevant case of alkali atoms, only the outer valence electron is moved

by the radiation fields. We thus represent the Hamiltonian (5.4) in its eigenstates, H atom|α = E α |α and find for the coupling term,

− d · E (0) = −ih ∑α ,α , k

gα ,α , k

a k − g

α ,α , k a

† k

|α α |, (5.5)

where gα ,α , k

=

hω k 2ε 0V

e · α | d |α

5.1.2 Two-Level Atoms and the Jaynes-Cummings Model

Since

| E

|∝ 1√

V , atom photon interactions are strongest for light that is confined to

a small volume around the atom. This is achieved in cavities, like pairs of mirrorshat face each other, and causes the spectrum of the light to be discrete (only wave-

length that vanish at the mirrors are possible). The description of the atom-light

interactions can thus be restricted to only one atomic transition |α β | = |eg|and one mode a k

. Since h|gα ,α , k

| E e − E g, we apply a rotating wave approxi-

mation and arrive at the Jaynes-Cummings Hamiltonian,

H JC =hω A

2σ z + hω C a

†a + h(gσ −a† + gσ +a), (5.6)

where σ z = |ee|− |gg|, σ − = |ge| and σ + = |eg|. Since H JC conserves the

number of excitations, [ H JC ,|e

e

|+a†a] = 0, we diagonalise it for each excitation

number n separately and find for ω A = ω C the eigenvalues and eingenstates,

E n,± = hω C

n− 1

2

± h

|g|2n and |n,± =

|n − 1, e± |n, g√2

(5.7)

For each excitation number n, we thus find a doublet of states |n, + and |n,−,

separated by an energy 2h

|g|2n. If the atom is initially excited and no photons

are present, the excitation oscillates back and forth between atom and radiation

mode at a frequency 2√

n|g|. These oscillations are called Rabi oscillations and

2√

n|g| the Rabi frequency.

27



5.1.3 Spontaneous Emission of a Two Level Atom

Any atom always interacts with its electromagnetic environment according to the

Hamiltonian (5.4), i.e.

H =hω 0

2σ z +∑

k ,ν

hω k a

† k ,ν

a k ,ν − ih∑

k ,ν

(g k ,ν σ −a† k ,ν

− g k ,ν

σ +a k ,ν ), (5.8)

where ν counts the polarisations and g k ,ν =

hω k 2ε 0V

eν · d . Since |g k ,ν | hω 0,

the modes a k ,ν can be treated as a quantum environment in a master equation

approach. For ω 0

∼1015Hz, the thermal photon number at room temperature

is nω 0 ∼ 10−13 and we can assume that all a k ,ν are in their vacuum states. Thedynamics of the atomic levels is thus described by the master equation,

ρ = −i[ω 02σ z,ρ] +

γ

2(σ −ρσ + −σ +σ −ρ −ρσ +σ −), (5.9)

The spontaneous emission rate γ =ω 30 | d |2

3π hε 0c3 can here be calculated with the same

techniques as the damping rates in equation (4.15). Using the commutation rela-

tions of the σ j and the cyclic property of the trace, we find the dynamics,

|e

e

|(t ) =

|e

e

|(0) e−γ t (5.10)

σ −(t ) = σ −(0) e−(iω 0+γ /2)t (5.11)

The occupation probability of the excited state thus decays at rate γ and coher-

ences at rate γ /2.

5.1.4 Resonance Fluorescence

We now consider an atom that is continuously driven by a laser field and emits

photons into the surrounding vacuum. This phenomenon is called resonance fluo-

rescence. The driving laser corresponds to a coherent state for one mode a L with

the free time evolution,

|α l(t ) = e−iω La†

La L t |α l, where α l (t ) = e−iω Lt α L(0) (5.12)

For solving the full problem, one would need to solve the dynamics generated

by the Hamiltonian (5.8) for the initial state |ψ , 0 = |α L⊗ |0 k = k L⊗ |ψ atom, 0.

One can show that this dynamics is equivalent to the dynamics generated by the

Hamiltonian,˜ H = H − ihΩ L(t )σ + + ihΩ

L(t )σ − (5.13)

28



for an initial state

|ψ , 0

=

|0 k

⊗ |ψ atom, 0

where all modes a k

are in vacuum. In

equation (5.13), H is given by eq. (5.8) and Ω L(t ) = g Lα L(t ).For calculating the dynamics of the atomic degrees of freedom, we switch

to an interaction picture with respect to H 0 = hω L2σ z and write down the master

equation corresponding to ˜ H , where we treat all radiation modes as a quantum

environment,

ρ = −i[∆

2σ z − iΩ Lσ + + iΩ

Lσ −,ρ ] +γ

2(σ −ρσ + −σ +σ −ρ −ρσ +σ −), (5.14)

where ∆ = ω 0 −ω L and Ω L = g Lα L. Note that now all radiation modes are in

vacuum, hence the simple form of eq. (5.14).

Assuming Ω L to be real, we can now calculate the equations of motion for theexpectation values σ +, σ − and σ z. These can be written in compact form,

v = A v + b, (5.15)

where v = (σ +, σ −,σ z)T , b = (0, 0,−γ )T and

A =

i∆− γ /2 0 −iΩ L

0 −i∆− γ /2 iΩ L

−2iΩ L 2iΩ L −γ

(5.16)

Noting that the solution to the differential equation ˙ f (t ) = α f (t ) +β is f (t ) =ceα t −β /α , where c is an integration constant, we see that limt →∞ f (t ) = −β /α ,provided that Re[α ] < 0. For equation (5.15) this means, that a well defined

asymptotic state exists provided that all eigenvalues of A have negative real parts.

This is the case for A and also implies that A−1 exists. The steady state solution

can therefore be found from vss = A−1 b and reads,

σ +ss = i2(γ + i∆)Ω L

γ 2 + 4∆2 + 8Ω2 L

(5.17)

σ −ss = σ +ss (5.18)

σ zss = − γ 2 + 4∆2

γ 2 + 4∆2 + 8Ω2 L

(5.19)

From these equations one sees that σ zss < 0, which means that a simple laser

drive can not generate a population inversion and the occupation probability of

|g is always higher than for |e.

29



A time dependent solution of eq. (5.15) can be found by observing d dt

vSS = 0

and A vSS = 0 fromd

dt ( v − vSS) = A ( v − vSS) . (5.20)

We find

σ z(t ) = −1 +8Ω2

γ 2 + 8Ω2

1 − e− 3γ

4 t

cosh(κ t ) +

3γ

4κ sinh(κ t )

(5.21)

σ +(t ) =2iΩγ

γ 2 + 8Ω2

1 − e− 3γ

4 t

cosh(κ t ) +

κ

γ +

3γ

16κ

sinh(κ t )

,

where κ = 12

γ 2

4 − 16Ω2. For |Ω| > γ /8, κ is imaginary and the solutions oscil-late while approaching vSS.

5.1.5 Power Spectrum of the Emitted Light

The correlation function

E (−)( r , t ) E (+)( r , t + τ ) = ∑ k , k

h2ω k

ω k

4ε 20u

k u k e

iω k t e−iω k

(t +τ )a† k

a k

→ ∞−∞d ω

d Ωω

2

2ε 0c3(2π )3 hω nω e−iωτ (5.22)

can be written as a Fourier transform of the emitted energy per frequency interval

as the second line of equation (5.22) shows. Here the d Ω-integration is over the

entire solid angle. For deriving (5.22) we consider the steady state regime where

E (−)( r , t ) E (+)( r , t +τ ) is independent of t and hence a†

k a k = δ k , k nω k

for ω k >

0 and zero otherwise.

The power spectrum, S( r ,ω ), can thus be written as the Fourier (back) trans-

form of E (−)( r , t ) E (+)( r , t + τ ),

S( r ,ω ) = limt →∞

12π

∞−∞

d τ eiωτ E (−)( r , t ) E (+)( r , t + τ ) (5.23)

Since the master equation (5.14) only predicts the dynamics of the atomic degrees

of freedom but equation (5.23) specifies the power spectrum in terms of the emit-

ted field, we need to find the relation between the emitted field and the atomic

degrees of freedom.

30



Relation between emitted field and state of atom

From the Hamiltonian (5.13) one can derive the Heisenberg picture equations of

motion,

a k = −iω k

a k − g k

hσ − (5.24)

σ − = −iω 0σ − +σ z∑ k

g k

ha k

+σ zΩ(t ) (5.25)

We can formally integrate equation (5.24) to obtain,

a k (t ) = e−iω k t a k (0) −g

k h t

0dsσ −(s)e−iω k (t −s). (5.26)

Plugging this into the expression for E (+), and integrating over the entire solid

angle of k yields,

E (+)( r , t ) = − d

8π 2ε 0c2r

∞0

d ωω 2

eiω r c − e−iω r

c

t

0dsσ −(s)e−iω (t −s). (5.27)

The integral over d ω can now be calculated as follows: ∞

0 d ωω 2eiω y →

∞

0 d ωω 2eiω y−εω = 2!(ε −iy)3 −→

ε →0

2iy3 , where ε > 0. We get,

E (+)( r , t ) = − d

8π 2ε 0c2r

t

0dsσ −(s)

1

s − t − r c

3− 1

s − t + r c

3

(5.28)

The ds-integration can be done via the residual theorem. To this end we note

that for t large enough,

s − t ± r c

−3 ≈ 0 for s < 0 and that

s− t ± r c

−3 → 0 for

|s| →∞. In the complex plane we can thus integrate along the whole real line and

close the contour along the half-circle |s| → ∞. Applying the residual theorem

yields,

E (+)( r , t ) =

−

ω 20 d

4πε 0c

2

r σ −(t +

r

c

)

−σ

−(t

−

r

c

) , (5.29)

where we have approximated d 2

ds2σ −(s)

s=t ±r /c≈ −ω 20σ −(t ± r /c) since ω 0

|Ω|, |g k |. The term with σ −(t + r

c) can be ignored as it represents an incoming

field. If one takes into account the polarisation of the radiation field, one obtains

the general result,

E (+)( r , t ) = − ω 20 d

4πε 0c2r

d × r

r

× r

r σ −(t − r

c), (5.30)

31



5.1.6 Equations for Correlation Functions and the Quantum

Regression Theorem

Quantum Regression Theorem

If a set of operators Y j fulfil the equations of motion

d

dt Y j(t ) =∑

l

G j,l(t )Y l(t ), (5.31)

then the following equations of motion for correlation functions hold,

d

d τ Y k (t )Y j(t +τ ) =∑lG j,l(τ )Y k (t )Y l(t + τ ). (5.32)

For proving the quantum regression theorem, one shows that Y k (t )Y j(t + τ ) can

be read as an expectation value, Y k (t )Y j(t + τ ) = Tr(Y jσ (τ )), where σ (τ ) =e−iH τ ρ(t )Y k e

iH τ and Y j(t ) = Tr(Y jρ(t )). Therefore equations (5.32) hold with

the corresponding initial conditions.

We now apply (5.32) to equations (5.20) to find for the vector

u = limt →∞ σ +(t )σ +(t + τ )−σ +(t )σ +(t )

σ +(t )σ −(t + τ ) −σ +(t )σ −(t )σ +(t )σ z(t + τ ) −σ +(t )σ z(t )

(5.33)

the equationsd

d τ u = A u. (5.34)

Using σ +(t )σ +(t ) = 0, σ +(t )σ −(t ) = 12

(1+σ z(t )) and σ +(t )σ z(t ) = −σ z(t )),

we find from these for G(τ ) = σ +(t )σ −(t + τ ),

G(τ ) =4Ω2

γ 2 + 8Ω2 γ 2e−iω 0τ

γ 2 + 8Ω2+

e−(iω 0+ γ 2 )τ

2− X +e−(iω 0+ 3γ

4 −κ )τ

2+

X −e−(iω 0+ 3γ 4 +κ )τ

2 ,

(5.35)

where X ± = γ 2

γ 2+8Ω2

3γ ±4κ 4κ − γ

2κ − γ ±4κ 8κ and κ = 1

2

γ 2

4− 16Ω2.

Power spectrum

Using equation (5.30), the power spectrum can now be calculated as

S( r ,ω ) =I 0( r )

2π

∞−∞

d τ eiωτ G(τ ) =I 0( r )

π Re

∞0

d τ eiωτ G(τ )

, (5.36)

32



with I 0(

r ) =

ω 20 d

4πε 0c2r d

× r

r ×

r

r 2

. The second equality in equation (5.36) fol-

lows from the observation that G(−τ ) = G(τ ) together with a substitution of

integration variables.

The second expression in equation (5.36) contains integrals of the form

L(ω ) = Re

∞0

d τ ei(ω −ω 0)τ −Θτ

=Θ

(ω −ω 0)2 −Θ2, (5.37)

with ω , ω 0 real and Θ> 0. The function L(ω ) is called a Lorenzian. It has a peak

at ω = ω 0, L(ω 0) = 1/Θ, and its width at half the maximum is Θ, L(ω 0 ±Θ) =1/(2Θ). We thus see that the rate of exponential decay of a correlation function is

equal to the with of a spectral line as given by the Lorenzian in equation (5.37).

After performing the integration in equation (5.36), we find for a weak laser

drive, |Ω| γ /8,

S( r ,ω ) = I 0( r )4Ω2

γ 2 + 8Ω2δ (ω −ω 0) (5.38)

and for a strong laser drive, |Ω| γ /8,

S( r ,ω ) =I 0( r )

2π

2π

4Ω2

γ 2 + 8Ω2δ (ω −ω 0) +

1

2

γ 2

(ω −ω 0)2 + ( γ 2

)2(5.39)

+1

4

3γ 4

[ω − (ω 0 + 2Ω)]2 + ( 3γ 4 )2

+1

4

3γ 4

(ω − (ω 0 − 2Ω))2 + ( 3γ 4 )2

In the weak driving limit, the spectrum shows one delta peak at ω = ω 0, whereas

in the strong driving limit it shows one delta peak at ω = ω 0 and three Lorenz

peaks at ω = ω 0 and ω = ω 0 ± 2Ω. This is the famous “Mollow” spectrum.

Photon statistics

Apart from the spectral properties of the emitted light, one can also look at its pho-

ton statistics. Of particular interest here is the probability to detect two photons at

a specific time separation at the detector. This is quantified by the g(2)-function as

defined in equation (2.12). Here we find,

g(2)(τ ) = 1 − e− 3γ 4 τ

cosh(κτ ) +

3γ

4κ sinh(κτ )

, (5.40)

with κ = 12

γ 2

4−16Ω2. Importantly, we have anti-bunching, g(2)(0) = 0, indi-

cating that the photons are emitted one at a time and never two or more together.

This is due to the fact that the atom can only take one excitation at a time and after

one photon emission the atom needs to be reexcited before a second photon can

be emitted.

33



5.1.7 Raman Transitions and Electromagnetically Induced Trans-

parency

Here we make use of the fact that only specific transitions between atomic levels

have a non-vanishing dipole moment, j| d | j. More specifically, j| d | j = 0 only

if l j = l j ± 1 and m j = m j or m j ± 1, where L2|ψ = hl(l + 1)|ψ and L z|ψ =hm|ψ . We thus choose an atom the couples to two classical laser fields according

to the Hamiltonian

H = ω 3σ 33 +ω 2σ 22 + (Ω1e−iω L1t σ 31 +Ω2e−iω L2t σ 32 + H.c.) (5.41)

where we have chosen the atomic levels such that the transition

|1

↔ |2

has

no dipole moment. For example the 87Rb D2-line levels 52S1/2(F = 1) for |1,

52S1/2(F = 2) for |2 and 52P3/2(F = 1) for |3. This implies that any coupling to

electromagnetic fields on that transition is weaker by a factor aBohr/λ as compared

to the dipole transitions. Therefore spontaneous emission losses from level |2 are

negligible. In an interaction picture with respect to H 0 =ω L1σ 33 + (ω L1 −ω L2)σ 22

the Hamiltonian becomes time independent and reads,

H = ∆σ 33 +δσ 22 + (Ω1σ 31 +Ω2σ 32 + H.c.) (5.42)

with ∆= ω 3 −ω L1 and δ = ω 2 − (ω L1 −ω L2).

Raman Transitions

We now choose the lasers to be in a two photon resonance, δ = 0 and go to another

interaction picture with respect to H 0 = ∆σ 33,

H I (t ) = Ω1ei∆t σ 31 +Ω2ei∆t σ 32 + H.c. (5.43)

and formally integrate the Schrodinger equation d dt

|ψ , t = −iH I (t )|ψ , t to get

|ψ , t + T − |ψ , t T

=−

i

T

t +T

t

dsH I (s)|ψ , t

−1

T

t +T

t

ds s

t

drH I (s) H I (r )|ψ , t

+. . .

(5.44)

in an iterative expansion. We now consider a scenario where,

|Ω1,2| ∆ and|Ω1,2|2

∆T 1 ∆T (5.45)

In this regime, we find that the dominant contributions to the right hand side of

equation (5.44) come from the second term and that all other contributions are

negligible. Due to|Ω1,2|2

∆T 1 we can write

|ψ ,t +T −|ψ ,t T

≈ d dt

|ψ ,t and can thus

34



approximate (5.44) by an effective Schrodinger equation d dt

|ψ , t

=

−iH I ,e f f

|ψ , t

with

H I ,e f f =|Ω1|2 + |Ω2|2

∆σ 33 − |Ω1|2

∆σ 11 − |Ω2|2

∆σ 22 −

Ω

1Ω2

∆σ 12 + H.c.

,

(5.46)

Level |3 thus decouples and two photon transitions |1 ↔ |2 appear.

Electromagnetically Induced Transparency

For two photon resonance, δ = 0, the Hamiltonian (5.42) has an eigenstate

|ψ 0 =

Ω2

|1

−Ω1

|2

|Ω1|2 + |Ω2|2 (5.47)

which has no component in level |3 and hence does not contribute to spontaneous

emission. Therefore it is called a “dark state”.

If the field Ω1 is much weaker than the field Ω2, the dark state approaches

|1. We now study the propagation of a weak field E 1 = E 1( z)e−iω L1t + c.c. where

Ω1 = d 31

hE 1 in dependence of the strong field Ω2. We consider a one-dimensional

setting, where the wave equation in a medium reads,1

c2

∂ 2

∂ t 2− ∂ 2

∂ z2

E = −µ 0

∂ 2

∂ t 2P = − χ

c2

∂ 2

∂ t 2 E (5.48)

where the susceptibility χ is defined by P = ε 0 χ E and the polarizability in ourcase reads,

P = natoms

d 13ρ31e−iω L1t + d 23ρ32e−iω L2t + c.c.

(5.49)

Here, natoms is the density of the atoms and ρ the density matrix of one atom in the

interaction picture with H 0. To calculate ρ we solve the maser equation for one

atom that is driven by the two lasers and decays via spontaneous emission from

level |3 at a rate γ and from level |2 at a much smaller rate κ . In doing so we

make a perturbative expansion in the weak probe field E 1. To linear order we find

for the susceptibility χ ,

χ = −natoms

|d 13

|2

hε 0

2(2δ −

iκ )

(2∆− iγ )(2δ − iκ ) − 8Ω22

. (5.50)

We see that χ → 0 for δ = 0 and κ → 0. In this case the field E 1 decouples

from the polarization of the medium as can be seen from equation (5.48) and the

medium becomes transparent, even for ∆ = 0. Furthermore the group velocity of

E 1 becomes vg = c(n−ω L1dnd δ

)−1 ≈ c

1 + natoms|d 13|2

2hε 0Ω22

−1

, where n =√

1 + χ , can

become much smaller than c fornatoms|d 13|2

2hε 0Ω22

1. This phenomenon is called “slow

light”.

35



Chapter 6

36



6.1 Cavity Quantum Electrodynamics

Since the interaction strength between light and atoms g scales as g∝ 1/√

V where

V is the quantisation volume, it can be enhanced by confining the light field in a

small volume, a cavity. In the classical example of a Fabry-Perot cavity formed by

two mirrors that face each other, the electric field has to vanish at the mirror sur-

faces due to their high reflectivity. Hence the possible wavelength for resonance

modes for a cavity of length L are

λ n =2 L

n, (6.1)

where n is a positive integer. As the spectrum is discrete, an atom typically onlycouples with one transition to one single cavity mode. The Hamiltonian that de-

scribes this system reads,

H = ω 0|ee|+ω ca†a + g(a|eg|+ H.c.) + (Ωe−iω Lt a† + H.c.), (6.2)

where ω 0 is the transition frequency of the atom, ω c the resonance frequency of

the cavity mode and Ω represents a drive of the cavity by a classical laser field.

The photon annihilation operator for the cavity mode is a and |e (|g) denote the

excited (ground) state of the atom.

Since the atom-cavity system is not perfectly isolated, interactions with the

environment (here the electromagnetic vacuum) need to be taken into account.

This can be done with a master equation that describes the two dominant loss

processes, spontaneous emission of the atom and the leakage of photons out of

the cavity. It reads,

ρ = −i[ H ,ρ]+γ

2(2σ −ρσ + −σ +σ −ρ −ρσ +σ −)

+κ

2(2aρa† − a†aρ −ρa†a), (6.3)

where ρ is the density matrix of cavity mode and atom, σ − =

|g

e

|, σ + =

|e

g

|and γ and κ are the rates of sponatneous emission respectively cavity decay.As the Hamitonian (6.2) contains an explicit time dependence of the laser field,

it is convenient to work in an interaction picture with respect to H 0 = ω L|ee| +ω La†a. In this frame, the Hamiltonian can be taken to read,

H =∆

2σ z +δ a†a + g(a|eg| + H.c.) + (Ωa† + H.c.), (6.4)

where σ z = |ee|− |gg|, ∆ = ω 0 −ω L and δ = ω c −ω L. The daming terms in

equation (6.3) in turn stay invariant.

37



6.1.1 Strong driving regime

From equation (6.3) with Hamiltonian (6.4) one can derive the following equa-

tions of motions for operator expectation values,

d

dt a = −

κ 2

− iδ

a− iΩ− igσ −d

dt σ − = −

γ 2

− i∆

a + igaσ z (6.5)

d

dt σ z = − γ (σ z + 1) − 2ig

aσ +−a†σ −

,

which are nonlinear and not closed. Whereas they are difficult to solve for ageneral case one can find an approximate solution for the regime of a strong input

drive, i.e. |Ω| |g|. To this end we write a = α + δ a, where α is the coherent

part of the field, α = a, and δ a the quantum fluctuations around it. For strong

input drive one can expect that |α |2 δ a†δ a and neglect the fluctuations δ a.

This turns the equations (6.5) into a closed set and one finds for the steady state

with d dt

a = d dt

σ − = d dt

σ z = 0,

σ zss = −

1 +n

n0(1 + ∆2)

−1

, (6.6)

where n = |α |2 is the number of photons in the coherent part of the cavity field,

n0 = γ 2

2g2 and ∆ = 2∆γ . One finds, σ zss → −1 for n n0 and σ zss → 0 for

n n0. In the former case the atom is in its ground state |g and in the latter

in the fully mixed state 12

(|ee| + |gg|). The quantity n0 sets the number of

photons needed to excited the atom and is thus called “critical photon number”.

Moreover σ zss < 0 so that no population inversion can be achieved.

For a laser resonant with the cavity, δ = 0, one furthermore finds for the photon

number n in the steady state,

n3

+ 2(1−8C + ∆2

)n0n2

+ (1 + ∆2

)[(1 + 8C )2

+ ∆2

]n20n = 4 |

Ω

|2

κ 2 [n + (1 + ∆2

)n0]2

,(6.7)

where C = g2

2κγ is called the “cooperativity”. Since equation (6.7) is a 3rd order

polynomial in n it has 3 solutions. These are not always physically relevant since

the photon number n can only be positive. There exists however the possibility

of having 3 steady states which in fact happens for ∆ = 0 and sufficiently strong

drive Ω. Which of the steady states will be reached then depends on the initial

conditions for the system.

38



6.1.2 Strong coupling regime

We now focus on a regime of strong coupling, that is g κ and g γ , and assume

a weak driving laser, i.e. |Ω| |g|. The dynamics of the system is described by

eq. (6.3). Ignoring the weak drive and dissipation for now, we can approximate the

Hamiltonian by the Jaynes-Cummings model, c.f. eq. (5.6). Since H JC conserves

the number of excitations, [ H JC , |ee| + a†a] = 0, it can be diagonalised for each

excitation number n separately. For ω C = ω A, the eigenvalues and eigenstates are

given in eq. (5.7).

Photon blockade A test whether a cavity indeed works in the strong coupling

regime can be done via an effect called “photon blockade”. This effect is due tothe nonlinearity of the spectrum, that is the fact that E 2,− > 2 E 1,−. We now focus

on ω C = ω A and find from (5.7) that E 2,− −2 E 1,− = (2−√2)|g| > 0. A laser that

is resonant with the transition |0 → |1,− is thus not resonant with the transition

|1,−→ |2,−. Provided |Ω| |g| it can thus only bring one excitation into the

cavity and occupy state |1,− but it can not bring a second excitation into the

cavity, that is the cavity blocks a second photon from entering it.

The effect can be verified by measuring the photon statistics for the light leav-

ing the cavity. In this output light, the photons only come one by one but never in

pairs or bunches. The output light is thus anti-bunched as can be quantified by its

g(2)-function, c.f. eq. (2.12). The output field outside the cavity is related to the

light field in the cavity via the input-output formalism which we discuss next.

6.1.3 Input-output formalism

We now consider a Fabry-Perot cavity for which light can only be emitted to one

side, say the positive x-direction. An electric field outside the cavity then reads

(ignoring the polarisation) E = E (+) + E (−) with

E (+) = i∑k

hω k

2ε 0V bk e

−iω k (t − x/c) (6.8)

where [bk , b†k ] = δ k ,k . For highly reflective cavity mirrors, photons inside the

cavity will only couple to modes outside with ω k ≈ ω C and we can approximate hω k

2ε 0V =

hω C

2ε 0V . Note that ω k in the exponential cannot be approximated since

this would cause substantial errors for large t . We can thus define a new field

outside the cavity,

b( x,t ) = e−iω C (t − x/c) 1√2π

∞−∞

d ν b(ν )e−iν (t − x/c) (6.9)

39



where ν = ω

−ω C . With this definition,

b†( x, t )b( x, t )

is the number of photons

that pass point x per time unit, i.e. the photon flux at x.The coupling of photons in the cavity to modes outside can be described by,

V = ∞−∞

d ω h(ω )

a†b(ω ) + ab†(ω )

(6.10)

where h(ω ) is the coupling strength. Hence the Heisenberg equations of motion

for a and b(ω ) read,

b(ω , t ) = − iω b(ω , t )− ih(ω )a(t ) (6.11)

a(t ) = − i[ H JC , a(t )] − i

∞−∞

d ω h(ω )b(ω , t ) (6.12)

By formally integrating eq. (6.11), b(ω , t ) can be specified in terms of its initialcondition at t 0 < t , b(ω , t ) = e−iω (t −t 0)b(ω ,t 0) − ih(ω )

t t 0

dse−iω (t −s)a(s), or its

final condition at t 1 > t , b(ω , t ) = e−iω (t −t 1)b(ω , t 1) + ih(ω ) t 1

t dse−iω (t −s)a(s).

These expressions can then be plugged into eq. (6.12). Since the coupling h(ω )is weak (|h(ω )| ω C ), the cavity field only couples to a narrow frequency range

of fields outside and we can approximately take h(ω ) to be constant in this range.

We thus define h2(ω ) = κ 2π , where the meaning of κ is the rate at which photons

leak out of the cavity. With this approximation we find,

a(t ) = − i[ H JC , a(t )] − κ

2a(t ) +

√κ ain(t ) (6.13)

a(t ) = − i[ H JC , a(t )] + κ 2

a(t ) −√κ aout (t ) (6.14)

where ain = − i√2π

d ω e−iω (t −t 0)b(ω , t 0) and aout = − i√

2π

d ω e−iω (t −t 1)b(ω , t 1)

are the input and output fields. From (6.13) and (6.14) we find the input-output

relation,

aout (t ) =√κ a(t ) + ain(t ), (6.15)

which is valid for any intra cavity Hamiltonian, not only for H JC . From eqs.

(6.13-6.15) and causality, it can be shown that for any intra cavity operator X ,

X (t ), ain(t )=κθ (t

−t ) X (t ), a(t ) , X (t ), aout (t )= κθ (t

−t ) X (t ), a(t )

(6.16)where θ (τ ) = 1 for τ > 0, θ (τ ) = 1/2 for τ = 0 and θ (τ ) = 0 for τ < 0. From

these commutation relations one can show for cases where ain(t )|ψ = 0 (Note

that the laser does not contribute here.) that,

a†out (t )a

†out (t )aout (t )aout (t ) =κ 2a†(t )a†(t )a(t )a(t ) (6.17)

a†out (t )aout (t ) =κ a†(t )a(t ) (6.18)

Hence anti-bunching of photons in the cavity results in photon anti-bunching in

the output field. The photon blockade effect is thus visible in the output field.

40



6.1.4 Circuit QED

Since charges and currents generate electromagnetic fields and interact with them,

electronic circuits can be used to store fields. To keep oscillating fields for as long

as possible, the resistance of the circuit should be minimal and superconducting

circuits are thus the best candidates for this task. Basic elements of such circuits

are a capacitances and inductances. The energy stored in a capacitance is

E cap =Q2

2C where C =

Q

U (6.19)

is the capacitance, Q the charge and U the voltage drop. The energy stored in an

inductance in turn is E ind =

Φ2

2 Lwhere L =

Φ

I (6.20)

is the inductance, Φ the magnetic flux and I the current. The magnetic flux is

linked to a corresponding voltage drop by Faraday’s law of induction,

U =d Φ

dt (6.21)

The two basic elements can be combined in a row to form an LC-cicuit, which is

described by the Hamiltonian,

H LC =Q2

2C +Φ2

2 L=

C

2

d Φ

dt

2

+Φ2

2 L, (6.22)

and is an electronic form of a harmonic oscillator with mass m LC = C and fre-

quency ω LC = 1/√

LC . Hence it is quantised by imposing [Q,Φ] = −ih.

Transmission line

We now discuss the quantisation of electromagnetic fields carried by a transmis-

sion line. A transmission line can be modelled by a sequence of LC-circuits with

the Hamiltonian,

H tl = N

∑ j=1

C

2

d Φ j

dt

2

+ N

∑ j=2

1

2 L(Φ j −Φ j−1)2, (6.23)

and is thus an electronic form of a harmonic chain. The continuum limit of this

model can be taken by defining a capacitance per unit length, c = C /d and an

inductance per unit length, l = L/d , where d is the distance between neighbouring

41



capacitances respectively inductances. Observing that, (Φ j

−Φ j

−1)/d

→∂ Φ/∂ z

and ∑ N j=1 d →

Ltl/2− Ltl/2

dz, where Ltl is the length of the transmission line, we find,

H tl = Φ20

Ltl/2

− Ltl/2dz

c

2

d φ ( z)

dt

2

+1

2l

∂φ ( z)

∂ z

2

, (6.24)

where φ = Φ/Φ0 and Φ0 = π he

is the flux quantum (e is the elementary charge).

The corresponding Lagrange function ˜ L = Φ20

Ltl/2

− Ltl/2dzL with Lagrange den-

sity L = c2 φ

2 − 12l

(∂ zφ )2 leads to the Euler-Lagrange equations, ∂ L

∂φ − d

dt ∂ L ∂ φ

−∂ ∂ z

∂ L ∂ (∂ zφ )

= 0, which in our case read,d 2

dt 2− v2 ∂ 2

∂ z2

φ = 0. (6.25)

This is a wave equation that describes electromagnetic waves propagating in z-

direction at velocity v = 1/√

lc.

Transmission line resonator

A resonator or a cavity for the light fields in a transmission line can be engineered

by implementing boundary conditions for the electromagnetic fields which make

their spectrum discrete. This is simply done by cutting the transmission line a two

points such that two capacitances emerge. The boundary conditions at these two

points, − Ltl/2 and Ltl/2 are then∂φ ∂ z

z=− Ltl/2

= ∂φ ∂ z

z= Ltl/2

= 0 since no currents

can flow across the capacitances. The solution of equation (6.25) then reads,

φ ( z, t ) =

2

Ltl

∞

∑n=0

φ n,e(t ) cos

2nπ

Ltl z

+φ n,o(t ) sin

(2n + 1)π

Ltl z

, (6.26)

and the Hamiltonian (6.24) decomposes into a sum of harmonic oscillators for

each mode, H tl = Φ20∑

∞n=0

c2 φ

2n,e + c

2ω 2n,eφ

2n,e + c

2 φ 2n,o + c

2ω 2n,oφ

2n,o

, where ω n,e =

2nπ v Ltl

, ω n,o = (2n+1)π v Ltl

and v = 1/√

lc. Hence, the field can be quantised by quantis-

ing each of the harmonic oscillators separately, just as for a free electromagneticfield. We choose the length of the resonator such that only the mode 2, e is of

interest to us. We thus find a flux field Φ( z, t ) an the corresponding voltage field

U ( z, t ),

Φ( z,t ) = Φ0

h

Ltlω c(a + a†) cos

2π z

Ltl

(6.27)

U ( z,t ) = i

hω

Ltlc(a − a†) cos

2π z

Ltl

(6.28)

42



An artificial atom: the charge qubit

In circuit QED system the role of the atom is taken by a superconducting circuit

that includes a Josephson junction. This Josephson junction is nonlinear and its

energy eigenvalues are therefore not equidistant, just like those of an atom. The

cicuit we will discuss here is called charge qubit and consists of two superconduct-

ing electrodes separated by a thin insulating layer. In the superconducting regime,

the electrons form Cooper pairs which are bosonic particles. Their Hamiltonian

reads,

H =2

∑ j=1

E jc

† j c j +

U

2c

† j c j(c

† j c j − 1)

− J (c

†1c2 + c1c

†2), (6.29)

where c j annihilates a Cooper pair in electrode j, [c j, c†l

] = δ jl , U describes the

Coulomb interaction between Cooper pairs averaged over their separations, J is

the rate at which Cooper pairs tunnel through the insulating layer and E j is energy

of Cooper pairs in electrode j.

Instead of c j, the Cooper pairs can also be described in number and phase

variables, c j = e−iϕ j

n j with e−iϕ j = ∑∞n j=0 |n jn j + 1|. As can be seen from

[e−iϕ j , (e−iϕ j )†] = |00|, the phase operator ϕ can not be Hermitian. Since devi-

ations from Hermitianity only appear for small particle numbers we can however

ignore this complication and take ϕ to be Hermitian. One can furthermore show

that particle number and phase behave like position and momentum, that is,

[n j, ϕ l] = −iδ jl (6.30)

In terms of these variables, the Hamiltonian (6.29) reads,

H =2

∑ j=1

E jn j +

U

2n j(n j − 1)

− J

e−iϕ 2

n1n2eiϕ 1 + e−iϕ 1

n1n2eiϕ 2

(6.31)

We now introduce the new variables ˆ N = 12

(n1 + n2) and n = 12

(n1 − n2). Since the

state of the Cooper pairs is close to a coherent state with high particle number, we

can approximate ˆ N by a complex number, ˆ N = N , and assume n N . Conse-quently, we have,

√n1n2 =

√ N 2 − n2 ≈ N , and the Hamiltonian (6.31) simplifies

to,

H = ( E 1 − E 2)n +U n2 − 2 NJ cos(ϕ 1 − ϕ 2) (6.32)

where we have dropped an irrelevant constant. Introducing the charging energy

E C = U /4, the Josephson energy E J = 2 JN , the phase difference ϕ = ϕ 1 − ϕ 2 and

ng = ( E 2 − E 1)/(8 E C ) as new variables, the Hamiltonian takes the standard form,

H = 4 E C (n− ng)2 − E J cos(ϕ ) (6.33)

43



Noting that 4 E C n2 = Q2

2(C

J +C

g)

, we find that E C =e2

2(C

J +C

g)

, where e is the elemen-

tary charge (Note that the gate capacitance C g and the Josephson capacitance C J

are in parallel for one of the electrodes.). Furthermore ng can be related to a volt-

age V g applied at the gate of the junction by V g = 2eC g

ng. This gate voltage can be

employed to tune the operating point of the junction.

In a basis formed by eigenstates of n (n|n = n|n), the Hamitonian (6.33)

reads

H = 4 E C (n − ng)2|nn|− E J

2(|nn + 1|+ |n + 1n|) (6.34)

and one sees that for E C E J the spectrum becomes deviates strongest form an

equidistant one for ng = 12

.

Charge qubit and circuit cavity

If the charge qubit discussed above is inserted into a circuit cavity, not only a

constant gate voltage will apply to the Josephson junction but also the oscillating

voltage U ( z, t ) of equation (6.28). Hence we have,

ng = n DC g + n AC

g with n AC g = i

C g

2eU 0(a− a†), (6.35)

where U 0 =

hω

Ltlcand we have assumed that the qubit is located at z = 0. The

Hamitonian (6.33) generalises to H = 4 E C [n−n DC g + iC g2e U 0(a−a†)]2 − E J cos(ϕ )

and includes a coupling to the photons. Expanding the quadratic term and apply-

ing a rotating wave approximation that is valid forC 2g

2(C g+C J )U 20 hω , we find, [n−

n DC g + i

C g2e

U 0(a − a†)]2 ≈ [n − n DC g ]2 − i

C g2e

U 0(n − n DC g )(a − a†) +

C 2g2(C g+C J )

U 20 a†a.

With these approximations the Hamiltonian of the charge qubit and the circuit

cavity reads,

H = H JJ + H phot + H I (6.36)

with H JJ = 4 E C (n − n DC g )2 − E J cos(ϕ ), H phot = (hω +

C 2g2(C g+C J )

U 20 )a†a and H I =

−i2eC g

C g+C J U 0(n − n DC g )(a − a

†

).

For n DC g = 1

2and E C |2e

C gC g+C J

U 0| the dynamics can be constraint to the sub-

space spanned by states |0 and |1 of the charge qubit. Defining the states | ↑ =(|1 − |0)/

√2 and | ↓ = (|1 + |0)/

√2 and the Pauli operators

σ z = |↑↑|−|↓↓| and σ − = |↓↑|, the Hamiltonian (6.36) takes the form of

a Jaynes-Cummings Hamiltonian, H = E J

2σ z + hω a†a + (gσ +a + g∗σ −a†), with

ω =ω +C 2g

2(C g+C J )U 20 and g = ie

C gC g+C J

U 0. Compared to atoms, the Josephson junc-

tion’s dipole moment is 104 times larger giving rise to much stronger couplings.

44



Chapter 7

45



7.1 Light Forces on Atoms

So far we have assumed that the emitters, Josephson junctions or atoms, are fixed

at a certain position in space while they interact with light fields. In practice how-

ever atoms can move. If a free atom is illuminated with a laser, it can absorb

photons from the laser and re-emitt them either into the laser mode or the electro-

magnetic vacuum. In each such absorption or emission process the velocity of the

atom changes by the recoil velocity vrec = h k L/m.

7.1.1 Concept of Doppler cooling

Photon absorption and re-emission processes together with the Doppler effect canbe used to cool atoms. A moving atom will “see” the laser light at a Doppler

shifted frequency ω L = ω L − k L · v L. Hence an atom that moves in the opposite

direction of the laser photons, k L · v L < 0, sees a higher frequency, ω L > ω L, than

an atom at rest an vice versa. For atoms that are illuminated with a red detuned

laser, ω L < ω 0, this implies that atoms moving towards the laser source see laser

light that is closer to resonance that atoms moving away from the source.

The subsequent spontaneous emission occurs randomly in all directions and

therefore has no effect when averaged of several absorption and emission events.

The absorption in turn occurs predominantly when the atoms move towards the

laser source where the atoms experiences a recoil of vrec = h

k L/m in each ab-sorption event. Since the recoil velocity is in the opposite direction to the atom

velocity, the atom is on average slowed down.

7.1.2 Semiclassical theory of light forces

The Hamitonian of a 2-level atom interacting with a laser field reads,

H = H A + H AL = P2

2m+

hω 02

σ z + hΩ( R)σ +e−i(ω Lt +Φ( R)) + H.c.

, (7.1)

where P is the momentum and m the mass of the atom. Here we treat the externaldegrees of freedom of the atom classically. That is we neglect its uncertainty in

position R and momentum P. This is justified provided that,

∆ R λ L and k L∆v γ , (7.2)

where ∆ R is the position uncertainty, m∆v the momentum uncertainty, γ the spon-

taneous emission rate of the atom and k L and λ L the wavevector and wavelength

of the laser. The first condition says that the extension of the atom’s wave function

is small compared to λ L, whereas the second condition says that the atom travels

46



a distance much less than λ L between two spontaneous emission events. Both

conditions are consistent with Heisenberg’s uncertainty relation if,

hγ E R, (7.3)

where the recoil energy reads E R = h2k 2 L/(2m). This is called the “broad line

condition” and says that the internal dynamics is much faster than the external

motion. That is compared to the atom motion, the internal relaxation takes place

“instantaneously”. Hence for studying the motion of atoms, one can assume that

the internal degrees of freedom are at all times in the steady state of the master

equation,

ρ = −i

h [ H ,ρ ] +γ

2 (2σ −ρσ + −σ +σ −ρ −ρσ +σ −) (7.4)

We identify the semiclassical force on an atom with the derivative of its potential

energy with respect to its position mean value, i.e.

F = − ∂

∂ r H AL , (7.5)

where r = R and p = P. From this definition we obtain,

F = −2hΩ( r )u(t ) α ( r ) + v(t ) β ( r ) (7.6)

where α ( r ) =Ω−1( r ) ∂ ∂ r Ω( r ) is an intensity gradient and β ( r ) = ∂

∂ r Φ( r ) is a phase

gradient. u(t ) and v(t ) are given by,

u(t ) = Reσ +(t )e−i[ω Lt +Φ( r )]

, v(t ) = Im

σ +(t )e−i[ω Lt +Φ( r )]

, (7.7)

and are determined by the steady state solution of the equations,

u = −γ

2u + (∆+ Φ)v (7.8)

v = −γ

2 v− (∆+ Φ)u− 2Ωw (7.9)

w = −γ w + 2Ωv− γ

2, (7.10)

which follow from equation (7.4). Here, ∆ = ω L −ω 0 and w = 12σ z. For Φ in

turn we find,

Φ=∂ Φ

∂ t +∂ Φ

∂ r · d r

dt = v · ∂ Φ

∂ r (7.11)

47



Atom at rest:

For an atom at rest, v = 0 ⇒ Φ= 0, we find the steady state solutions

uss =∆

2Ω

s

1 + s, vss =

γ

4Ω

s

1 + s, wss = − 1

2(1 + s)(7.12)

where s = 8Ω2

4∆2+γ 2is the saturation parameter. Plugging u(t ) = uss and v(t ) = vss

into equation (7.6), we obtain

F = F RP + F DP with F RP = −2hΩvss β and F DP = −2hΩuss α (7.13)

where F RP is called “radiation pressure” force and F DP “dipole” force.The radiation pressure force is dissipative since it can be expressed as F RP =

−hγ p(e) β , where p(e) is the occupation probability of the excited state and hence

γ p(e) the total rate of spontaneous emission. Since p(e) < 1/2, F RP is bounded.

As an example for a plane wave laser with electric field E L = E 0 cos(ω Lt − k L · r )

one gets α = 0, β = − k L and hence F RP = −h k Lγ 2

2Ω2

∆2+2Ω2+γ 2/4.

The dipole force in contrast does not vanish in the limit γ → 0. It can be

expressed via a potential, F DP = − ∂ ∂ r

V DP. For ∆ Ω, we get V DP ≈ h∆Ω2

∆2+γ 2/4.

The dipole force is unbounded as can be seen for ∆ Ω and Ω → ∞, where

V DP ≈

hΩ2/∆→∞. As an example for a standing wave with electric field E

L=

2 E 0 cos(ω Lt ) cos( k L · r ) one gets for k L e z, β = 0 and F DP =16hΩ2

0∆cos(k L z) sin(k L z)

∆2+8Ω20 cos2(k L z)+γ 2/4

,

where Ω0 = d · E 0.

Friction force for moving atom

We now consider a plane wave laser field and an atom moving at velocity v, r = vt .

There is no dipole force F DP = 0, but Φ = − k L · v. We thus obtain the same

steady state solutions as in equation (7.12) but with ω L replaced by ω L − k L · v. the resulting radiation pressure force F RP = −h k L

γ 2

2Ω2

(∆

− k L

· v)2+2Ω2+γ 2/4

becomes

maximal for ω L − k L · v = ω 0.

7.1.3 Standing wave Doppler cooling

We now consider Doppler cooling in a standing wave along the z-direction, E L =

2 E 0 cos(ω Lt ) cos(k L z). As we have seen before, β = 0 for this case and there is

no radiation pressure force for an atom at rest. A moving atom in contrast can

nonetheless experience a friction force that cools it. Since there are no analytical

48



solutions to eqs. (7.8) for this case we seek here an expansion in powers of,

χ =k Lv z

γ =

2π v z

γλ L 1. (7.14)

To this end we define,

u =

u

v

w

, s =

0

0

γ /2

, B =

−γ /2 ∆ 0

−∆ −γ /2 −2Ω

0 2Ω −γ

(7.15)

and write equations (7.8) in the more compact form, u = B u + s. Since the atoms

move we have u = ∂ ∂ t

u + v z∂ ∂ z

u ≈ v z∂ ∂ z

u, where we have taken into account that

the internal dynamics is much faster than the motion of the atoms. We thus seek a

solution to the equation,

v z∂

∂ z u = B u + s (7.16)

in powers of χ . To this end we write u = u0 + u1 + . . . , where u0 is zeroth order

in χ and u1 is linear order in χ . The contributions of zeroth order in χ to equation

(7.17) read 0 = B u0 + s and can be solved for u0. The contributions of linear order

in χ to equation (7.17) read v z∂ ∂ z

u0 = B u1 and can be solved for u1 provided u0

has already been obtained. This procedure can be iterated further but we stop here

at linear order. The resulting force on the atoms reads,

F =

h∆s

1 + s k L tan(k L z)

1 +

γ 2(1

−s)

−2s2(∆2 + γ 2/4)

γ (∆2 + γ 2/4)(1 + s)2 k L tan(k L z)v z

. (7.17)

Here the first term corresponds to a trapping force where as the second term that

is proportional to v z is a friction force for ∆< 0 that results in cooling.

7.1.4 Limit of Doppler cooling

With the Doppler cooling concept described above, it is not possible to reach ar-

bitrarily low temperatures. This is due to the fact that the spontaneously emitted

photons on avergae carry a finite momentum of h k L. These spontaneous emission

events thus give the atom a recoil of energy E R =

h2 k 2 L

2m . As these recoils go inrandom directions they correspond to a heating process that counteracts the cool-

ing process. Doppler cooling will thus reach its limit when the associated heating

rate E R p(e)γ becomes equal to the cooling rate | F · v|. For low saturation s 1,

one obtains from this condition that Doppler cooling can reach a minimal kinetic

energy and hence minimal temperature for the atoms of,

m

2 v2 ≥ hγ

4and T ≥ hγ

2k B, (7.18)

where k B is Boltzmann’s constant. This is the so called Doppler limit.

49



7.1.5 Cooling beyond the Doppler limit: Sisyphus cooling

The mechanism of Sisyphus cooling makes use of the fact that light of different

polarisation couples to different atomic transitions. The total angular momentumˆ J of the valenz electron is composed of a angular momentum ˆ L and a spin S, ˆ J =ˆ L + S. Each absorption and emission of a photon changes the angular momentum

eigenvalue l ( ˆ L|ψ = hl(l + 1)|ψ ) by 1, l = l ± 1. Hence for the simplest case,

one has as ground states two states with m J = ±12

and as excited states four state

with m J = −3

2,−1

2, 1

2, 3

2.

The atom-photon coupling is V I = −e r · E . By Calculating the matrix elements

of the commutators [ ˆ L z, ˆ z] = 0 and [ ˆ L z, ˆ x± i ˆ y] = ±1h

( ˆ x± i ˆ y) in the basis formed by

states |l, m, where ˆ L|l, m = hl(l + 1)|l, m and ˆ L z|l, m = hm|l, m, one finds theequations,

l, m|ˆ z|l, m(m − m) = 0 and l, m| ˆ x ± i ˆ y|l, m(m − m 1) = 0. (7.19)

These restrict the possible transitions. Hence for light with polarisation along e z

which couples to atoms with d e z, only transitions with m = m have a non-

vanishing dipole matrix element and hence a non-vanishing coupling. In turn for

light with polaristaion along e x ± ie y, which couples to atoms with d e x ie y,

only transitions with m = m 1 couple.

For the level structure introduced above, this means that σ −

-polarisation cou-

ples to the transitions |g,−12 ↔ |e,−3

2 and |g, 12 ↔ |e,−1

2, π -polarisation cou-

ples to the transitions |g,−12 ↔ |e,−1

2 and |g, 1

2 ↔ |e, 1

2 and σ +-polarisation

couples to the transitions |g,−12 ↔ |e, 1

2 and |g, 1

2 ↔ |e, 3

2. The couplings are

however not all equally strong but carry weights given by Clebsh-Gordan coef-

ficients. For the level structure at hand, the spontaneous emission rate on the

transitions |e, 32 → |g, 1

2 and |e,−3

2 → |g,−1

2 is for example 3 times larger than

on the transitions |e, 12 → |g,−1

2 and |e,−1

2 → |g, 1

2.

For making use of these properties for cooling atoms, we put these atoms in a

standing wave formed by two counter-propagating laser fields, one with momen-

tum k e z and polarisation along e x and one with momentum

−k e z and polarisation

along e y. The resulting standing wave then has a polarisation that alternates be-tween σ − and σ + with a periode of λ /4, where λ = |k |/2π . That is, if the polar-

isation is σ − at z0, it changes to σ + at z0 +λ /4, back to σ − at z0 +λ /2, again to

σ + at z0 + 3λ /4 and so on.

Let us now assume that the atom is in one of the ground states, |g,−12 or |g, 1

2.

Since the transitions for different polarisations have different coupling strength,

the two ground state experience a different dipole potential, too. For |Ω|, |∆| γ ,the diploe potential reads V DP ≈ hΩ2( z)/∆ and is thus proportional to the square

of the dipole moment of the respective transition. Hence for σ −-light, the state

50



|g,

−12

is lower in energy (deeper dipole potential) than the state

|g, 1

2

and vice

versa.An atom in state |g,−1

2 at a location with σ −-light can thus only do the cyclic

transition, |g,−12 σ −−→ |e,−3

2 σ −−→ |g,−1

2, and will therefore stay in state |g,−1

2.

If this atom travels to a location with σ +-light, it has to climb uphill. At the

location with σ +-light it can however also do the transition, |g,−12 σ +−→ |e, 1

2 π −→

|g, 12, and be transferred to the state |g, 1

2. In this process the emitted photon has a

higher frequency than the absorbed photon. Hence the atom needs to loose kinetic

energy. It is therefore cooled. This process is called Sisyphus cooling due to its

analogy with the sentence of Sisyphus in the Greek myth.

Limit of Sisyphus cooling

Let us denote by U 0 the absolute value of energy difference between states |g,−12

and |g, 12 in a location with σ − or σ + polarisation. On average the atom will thus

loose an energy of U 0/2 in each cooling cycle. However only 1/6 of all photon ab-

sorption and re-emission events contribute to the cooling. Yet each spontaneously

emitted photon will heat the atom as spontaneous emission goes in random direc-

tions. Each such event increases the atom’s kinetic energy by roughly 1 E R. Hence

there is a minimal value for U 0 that is needed in order for the cooling to work.

Doing more precise numbers, one finds min[U 0] > 825

E R. The cooling process

will then come to a halt once the kinetic energy of the atom becomes small com-pared to U 0 as the atom is then unable to climb the potential hills. At this point

the atom’s kinetic energy is about 1 E R which is well below the Doppler limit.

7.1.6 Optical lattices

We consider a 2-level atom in a standing wave laser field. The Hamiltonian for the

internal degrees of freedom reads, H = −h∆2σ z + hΩ( z)(σ + +σ −) Since this is a

2×2 matrix, it can be diagonalised. Its eigenvalues read, E ± = ± h2

∆2 + 4Ω2( z).

For

|∆

| |Ω

|and ∆ < 0, one finds E

± ≈ h( |∆|

2+ Ω2( z)

|∆

|). The ground state thus

experiences an energy shift ∝ Ω2( z). One can associate a trapping potential to

this energy shift. This is the dipole potential, which we have derived here in the

dressed state picture.

For a standing wave we obtain a periodic potential, that is called an optical lat-

tice, V = −hΩ2

0

∆cos2(k z z). If we apply standing waves in all three spatial directions

we generate an optical lattice in 3d. A particle in this lattice has the Hamiltonian,

H = − h2

2m∑

α = x, y, z

∂ 2

∂α 2+ h

Ω20

∆∑

α = x, y, z

cos2(k α α ), (7.20)

51



Single-particle eigenstates

We first turn to find the single particle eigenstates of the Hamiltonian (7.20). To

model an infinitely large system in the Thermodynamic Limit, we assume pe-

riodic boundary conditions. Since the Hamiltonian is a sum of terms for each

coordinate α = x, y, z we can make a product ansatz for the wavefunction, Ψn, p =Ψn, p x

Ψn, p yΨn, p z

, whereh2

2m

∂ 2

∂ x2+V 0 (1 + cos(k x x))

Ψn, p x

= ε n, p xΨn, p x

, (7.21)

V 0 =−

hΩ2

0

∆

and H Ψn, p = (∑α = x, y, z ε n, pα )Ψn, p. The differential equation (7.21)

is known as Mathieu’s equation and its solutions can be expressed in terms of

Mathieu-functions C and S,

Ψn, p x= C C C (a, b, p x x) +C SS(a, b, p x x) (7.22)

Here C C and C S are expansion coefficients, a = mV 0h2k 2 x

− 2m

h2k 2 xε n, p x

and b = mV 0h2k 2 x

.

Notice that k α are the components of the laser wave vectors, whereas n and pα label the possible solutions.

For periodic boundary conditions with Ψn, p x

x + 2π

k x

=Ψn, p x

( x) only certain

values for p x are possible. These form bands that are labelled by n. For ultra-cold

atoms and V 0 > 5 E R the width of the lowest band (n = 0) is smaller than the gapto the next higher band (n = 1). Hence a collision of two ultra-cold atoms in the

n = 0 band cannot scatter one atom into a higher band. Therefore the dynamics of

ultra-cold atoms is confined to the lowest band and we can write Ψ p = Ψ0, p.

7.1.7 Many-particle representation: 2nd quantisation

We now turn to find a representation for the many-particle version of the Hamil-

tonian (7.20). This reads H =∑ N j=1 H j, where each H j has the form as in equation

(7.20) and the index j labels the particles. The Hilbert space of the system is then

spanned by product states of the form, |Ψ = |α 1,α 2, . . . ,α N , meaning that par-ticle j is in state |α j. Here we focus on bosons only so that possible states of

the many-particle system must be fully symmetric with respect to permutations of

particles. We write such states as a sum over all possible permutations of particles,

S|α 1,α 2, . . . ,α N =1√ N !∑

P

P|α 1,α 2, . . . ,α N , (7.23)

where the prefactor 1/√

N ! has been introduced since the sum ∑P contains N !

terms. If several particles are in the same state (α j = α l = . . . ) these states are

52



however not normalised. Correctly normalised and fully symmetric states can be

written,

|nα , nβ , . . . =1

nα ! nβ ! . . .S|α 1,α 2, . . . ,α N (7.24)

This notation should be read as: nα particles are in the state |α , nβ particles are in

the state |β etc. That is, there are nα values of j with |α j = |α etc. Importantly,

the states are fully characterised by the values nα , nβ , . . . . We can now define the

action of an annihilation and a creation operator on the states |nα , nβ , . . .,

aα | . . . , nα , . . . =√

nα | . . . , nα − 1, . . . (7.25)

We now express all operators that act on single or multiple particles in terms of

aα and a†α . Single particle operators have the form,

T = N

∑ j=1

T j with T j = ∑α ,α

T α ,α |α jα j|, (7.26)

where T α ,α = α j|T j|α j is independent of j as all particles are identical. One has,

∑ N j=1 |α jα j|| . . . , nα , . . . , nα , . . . = nα

√nα + 1 1√

nα | . . . , nα −1, . . . , nα + 1, . . .

Here the prefactor nα appears because a state |α is converted to a state |α in nα -terms of | . . . , nα , . . . , nα , . . . as given by equation (7.24). The two other

prefactors√

nα + 1 and 1√nα

then restore the correct normalisation of the state.

Hence a single particle operator T can be expressed as

T = N

∑ j=1

T j = ∑α ,α

T α ,α a†α aα . (7.27)

In a similar way, each two particle operator V such as an interaction potential, can

be expressed as,

V = N

∑ j=l

V j,l = ∑α ,β ,α ,β

V α β ,αβ a†α a

†β aα aβ . (7.28)

For a different basis, |λ = ∑α α |λ |α , the operator a†λ

= ∑α α |λ a†α creates

particles in the state |λ . Choosing |λ = | r to be position eigenstates, we definethe so called field operator to be,

Ψ†( r ) = a† r

=∑α

α | r a†α =∑

α

φ ∗α ( r )a†α (7.29)

where φ α ( r ) is the wave-function of state |α . Hence the many-particle Hamilto-

nian H = ∑ N j=1 H j can be written,

H =

d 3r Ψ†( r )

− h2

2m

∂ 2

∂ r 2+V (r )

Ψ( r ) (7.30)

53



Bloch and Wannier basis

The field operator Ψ( r ) can be expanded in any set of basis states according to

equation (7.29). Two important sets are the Bloch states Ψ k as defined in equation

(7.22) and their Fourier transforms, Wannier states w( r − r j) = 1√ N ∑ k

e−i k · r jΨ k ( r ),

Ψ( r ) =∑ k

Ψ k ( r )a k

=∑ j

w( r − r j)a j, (7.31)

where a j = 1√ N ∑ k

ei k · r j a k . In the Bloch basis, the Hamiltonian (7.30) reads,

H =

∑ k

ε k

a†

k a

k , (7.32)

whereas in the Wannier basis it reads

H ≈ ε ∑ j

a† j a j − J ∑

< j,l>

(a† j al + H.c.). (7.33)

with J j,l =

d 3r w∗( r − r j)− h2

2m∂ 2

∂ r 2+V (r )

w( r − r l) and ε = J j, j and J = J < j,l>.

The mode energies of the Bloch modes are connected to ε and J by, ε k = ε −

2 J cos | k | and can therefore obtained by calculating the centre and width of the

n = 0 Bloch band. They can be estimated to read,

ε ≈ 3 E R

V 0

E Rand J ≈ 4√

π E R

V 0

E R

34

e−

V 0 E R (7.34)

The Hamiltonian (7.33) thus describes particles that move on a periodic lattice,

where we have only kept tunnelling terms between neighbouring lattice sites and

neglected the much smaller long range tunnelling.

For V 0 E R the Wannier functions decay strongly as | r − r j| increases. One

can thus also obtaim ε by approximating in the potential cos2(k α α ) ≈ 1− (k α α )2

and modelling each lattice site as a harmonic oscillator. For the tunnelling ma-

trix element this approximation if for obvious reasons not good. So far we haveassumed that the atoms do not interact. They however do interact via a van der

Waals potenital.

7.1.8 Interactions between ultra-cold atoms

Here we consider the collision between two atoms of the same mass m. The

Hailtonian reads,

H 2b = p2

1

2m+

p22

2m+V (| r 1 − r 2|) (7.35)

54



By defining centre of mass, R = 12

( r 1 + r 2), P = p1 + p2 and relative coordinates

r = r 1 − r 2), p = p1 − p2, one can decouple the relative motion from the centre of mass motion. For the collision properties, only the relative motion is of interest,

which is described by the Hamiltonian,

H = p2

2µ +V (| r |) (7.36)

The scattering of the atoms can thus be modeled as the scattering of an atom with

reduced mass µ = m/2 at a given potenial V . To find the scattering properties we

therefore seek solutions of the eigenvalue equation H Ψ= h2k 2

2µ Ψ, where E = h2k 2

2µ is the energy of the incoming atom that is scattered. Since the relative momentum

p can be decomposed into a radial and an angular part, p2

= p2r + r −

2

L2

, withangular momentum L, we can make the product ansatz, Ψk ,l,m( r ) =

χ k ,l

kr Y l,m(θ ,φ ).

Here, the Y l,m are spherical harmonics that fulfill L2Y l,m = hl(l + 1)Y l,m. We thus

need to solve,− h2

2µ

d 2

dr 2+

h2l(l + 1)

2µ r 2+V (r )

χ k ,l(r ) =

h2k 2

2µ χ k ,l(r ) (7.37)

The angular momentum enters here as a centrifugal potential ∝ r −2. The van der

Waals potential V (r ) is given by V (r ) = −C 6r −6 outside the atom and has a hard

core, V →∞ at the atom radius. For large r the repulsive centrifugal potential thus

dominates. The sum of centrifugal and van der Waals potentials builds a potential

barrier that is much higher than the kinetic energy of ultra-cold atoms whenever

l ≥ 1. Therefore ultra-cold atoms can only undergo s-wave scattering with l = 0.

We thus can ignore the centrifugal term in equation (7.37).

For r much larger than the range of V , r r 0, we have V (r ) h2k 2

2µ and can

approximate equation (7.37) by − h2

2µ d 2

dr 2 χ k ,0(r ) = h2k 2

2µ χ k ,0(r ). The solution reads

χ k ,0 ≈ A sin(kr +δ 0) for r r 0 (7.38)

For r k −1 on the other hand, we have − h2

2µ d 2

dr 2 χ k ,0(r ) h2k 2

2µ χ k ,0(r ). This can be

seen by expanding χ k ,0(r ) =∑∞n=3 cnr −n, where terms with n < 3 do not contribute

as they would not be normalisable. For ultra-cold atoms, typical k s are howeversmall enough such that r 0 r k −1 exist. In this range we can thus approximate

(7.37) by − h2

2µ d 2

dr 2 χ k ,0(r ) = 0 and find the solution

χ k ,0 ≈ C 0(1−κ r ) for r 0 r k −1 (7.39)

In the range both solutions, (7.39) and (7.38) apply. We thus find,

χ k ,0 ≈ C 0(1 −κ r ) = A sin(kr +δ 0) ≈ A(sinδ 0 + kr cosδ 0) (7.40)

for r 0 r k −1 where we have used kr 1.

55



The phase δ 0 is called the s-wave scattering phase as it determines the phase

relation between the incoming wave and the outgoing wave. For r r 0, c.f.

(7.38), Ψ(r ) = 1√4π

sin(kr +δ 0)kr

= e−iδ 0

2i√

4π

e2iδ 0 eikr

kr − e−ikr

kr

, where eikr

kr is an outgoing

and e−ikr

kr an incoming spherical wave. For s-wave scattering, the effect of the

scattering potential is thus fully characterised by δ 0. From (7.40) we find,

cotδ 0 = −κ /k (7.41)

Instead of δ 0 one typically uses the s-wave scattering length aS defined by,

aS

=−

limk →0

tanδ 0

k , (7.42)

to characterise the interaction potential. Here we have, aS = κ −1. If we derive the

s-wave scattering phase δ 0 for the contact-potential,

V (r ) =4π h2

2µ aSδ (r ), (7.43)

we find cotδ 0 = −1/(kaS) which is identical to (7.41). For ultra-cold atoms the

interactions can thus be accurately described by the δ -potential (7.43).

Feshbach resonances

The interactions between ultra-cold atoms can furthermore be modified by the

use of Feshbach resonances that occur because the interaction between ultra-

cold alkali atoms depends on the relative orientation of the spins of their va-

lence electron. Hence, at an energy, where there are unbound scattering states

( h2k 2

2µ > V (r )) for the ↑↑-configuration, there can be a bound state ( h2k 2

2µ < V (r ))

for the ↑↓-configuration. Since the Hyperfine interaction couples the ↑↑- and the

↑↓-configuration, the scattering length changes dramatically as h2k 2

2µ approaches

the energy of a bound state of the

↑↓-configuration and vice versa. This is a Fes-

hbach resonance. The relative energies of ↑↑- and ↑↓-configuration can be tunedby applying a magnetic field and atoms can be driven through the Feshbach reso-

nance. When crossing the resonance, the scattering length typically changes sign.

One can thus switch between repulsive, aS > 0, and attractive, aS < 0, interactions.

7.1.9 Mott insulator to superfluid quantum phase transition

We now consider ultra-cold bosonic atoms in an optical lattice that interact via

a repulsive contact-potential as in equation (7.43). This system can undergo a

56



quantum phase transition from a Mott insulator to a superfluid phase. The Hamil-

tonian describing this system is composed of the noninteracting part, see equa-tion (7.33) and the interaction term given by, H I = 1

2

d 3r Ψ†( r )Ψ†( r )V (| r −

r |)Ψ( r )Ψ( r ) = U 2 ∑ j a

† j a

† j a ja j, where U = 4π h2

maS

d 3r |w( r − r )|4 ≈

≈

8π kaS E R

V 0

E R

34

.The full Hamiltonian now reads,

H = ε ∑ j

a† j a j − J ∑

< j,l>

(a† j al + H.c.) +

U

2∑

j

a† j a

† j a ja j, (7.44)

and is called the Bose-Hubbard Hamiltonian. We have already encountered a

2-site version of it when discussing Cooper pairs in a Josephson junction, see

equation (6.29).

The Bose-Hubbard model describes bosonic particles in a lattices potential

with short range interactions. This system resembles electrons in a solid, where

the lattice potential is created by the ionised atom bodies and the electrons interact

via Coulomb interactions. In our case the particles are however not Fermions

like electrons but Bosons. Nonetheless, the Bose-Hubbard model has become a

paradigmatic many-particle Hamiltonian. In optical lattices, the lattice sites are

however spaced much further apart than in a solid and the lattice depth can be

very accurately controlled via the laser intensities. The implementation in optical

lattices thus offers unprecedented experimental control and measurement access.

For U J and an integer number of particles per lattice site, the system isin a Mott insulating phase. Here the Hamiltonian can be approximated by H ≈ε ∑ j a

† j a j + U

2 ∑ j a† j a

† j a ja j. We assume a system of N particles and N lattice sites,

that is on average one particle per lattice site. Then the ground state is |GS =

∏ j |1 j. Since adding a second particle to a lattice site would require an additional

energy U , hopping of particles between lattice sites is suppressed and adding a

further particle to the system requires and energy U . The second property is called

“incompressibility” and is the defining property of the Mott insulator.

For U J , the Hamiltonian can be approximated by (7.33) and the ground

state of a system with N particles and N lattice sites is given by,

|GS =1

√ N ! (a

†

k =0) N

|vac. This state is called a superfluid. Since a

†

k =0 =1

√ N ∑ j a j,all particles are delocalised across the lattice and there is a phase coherence be-

tween the lattice sites.

The system can be driven through the phase transition between Mott insulator

and superfluid by tuning the ratio

J

U =

√2

kaS

e−

V 0 E R (7.45)

that can be modified by tuning V 0 = −Ω2

∆via the laser intensity or detuning.

57



7.1.10 Measuring the Mott insulator and superfluid phases

To measure which phase the ultra-cold atoms are in, the lattice potential is switched

off and the atoms fall down due to gravity. While falling the atomic cloud expands

ballistically since collisions are very unlikely once there is no longer a lattice po-

tential. After falling a certain distance, an absorption image of the atomic cloud

is taken by illuminating it with another laser and recording the shadow it creates.

In ballistic expansion the location of an atom is given by r = vt = h k m

t . Since the

number of particles is conserved,

d 3r n( r ) =

d 3k n( k ), where n( r ) and n( k ) are

the particle densities in position and in momentum space, one obtains,

n( r ) =m

ht 3

n( k ). (7.46)

Hence the observed position distribution n( r ) of the particles is proportional to the

momentum distribution n( k ) when the particles are released from the lattice.

In free space (with the lattice off) one has,

n( k ) =

d 3r

d 3r ei k ·( r − r )Ψ†( r )Ψ( r ) =

d 3q ei k · qw( q)

2

∑ j,l

ei k ·( r j− r l )a† j al,

(7.47)

with Wannier function w( q). The difference between Mott insulator and superfluid

enters in the correlations a† j al. For a superfluid of N particles on M lattice sites,

a† j al = N

M and one oberves interference peaks for k · ( r j − r l) = 2π m with m

integer. For a Mott insulator in turn, one has a† j al = N

M δ j,l and no interference

peaks appear.

58



Chapter 8

59



8.1 Trapped Ions

Ions are charged atoms and can therefore be trapped by applying and electromag-

netic potential. As they couple directly via their charge to the electromagnetic

fields, they can be subject to much stronger electromagnetic forces than neutral

atoms which couple due to their polarisation.

8.1.1 Trapping potential: Paul trap

The ions are trapped in a minimum of an electromagnetic potential. Close to

this minimum, the potential can always be approximated by a harmonic one, i.e.

V ( x, y, z) =V 02 (k x x

2

+ k y y2

+ k z z2

). Since the potential V should fulfil Laplace’sequation∆V = 0, one finds that k x + k y + k z = 0 which implies that at least one k is

negative and no trapping occurs in that direction. Therefore only time-dependent

potentials are necessary to trap charged particles in all directions of space. A

prominent trapping configuration is a Paul trap, for which,

V ( x, y, z) =V 0

2(k x x

2 + k y y2 + k z z

2) +V 1

2cos(ω r f t ) ( p x x

2 + p y y2), (8.1)

with 0 < k z = −(k x + k y) and p x = − p y. Typical values for the voltages are V 0 ≈0 − 50V and V 1 ≈ 100 − 500V whereas the radio frequency is typically ω r f ≈100kHz−100MHz.

The motion of a classical particle in the potential V as given in (8.1) can be

found exactly. For the z-direction, it is a harmonic oscillation described by the

equation of motion,

¨ z = − Z |e|m

V 0k z z (8.2)

where Z |e| is the charge of the ion and m its mass. For the x- and y-direction, in

turn, the equations of motion are Mathieu equations, e.g.

¨ x = − Z |e|m V 0k x +V 1 cos(ω r f t ) p x x, (8.3)

for which the solutions are known. We define the parameters, a x = 4 Z |e|V 0k xmω 2r f

and

q x = 2 Z |e|V 1 p x

mω 2r f

and assume that the voltages V 0 and V 1 are chosen such that a x 1

and q x 1. In this regime, the solution of (8.3) can be approximated by,

x(t ) ≈ A cos

a x +

q2 x

2

ω r f

2t

1 − q x

2cos

ω r f

2t

, (8.4)

60



and similarly for the y direction. The motion in the xy-plane thus decomposes into

a so called secular motion at frequency

a x + q2 x2ω r f

2and a much faster micro-

motion at frequencyω r f

2. Since q x 1, the micro-motion’s amplitude is also a lot

smaller than for the secular motion. Therefore the micro-motion can be ignored

in most cases.

With these approximations, the potential for trapped ions can be approximated

by a harmonic potential and the motion of the ions can thus be quantised in the

standard way,

H = hν a† z a z + hν t (a†

x a x + a† y a y) (8.5)

where ν = Z |e|m

V 0

k z

and ν t

= a x

+ q2 x

2

ω r f

2. One can arrange for ν

t ν so that

the transverse motion is in the ground state for cold ions. We now focus on this

regime, where the transverse motion can be ignored and we arrive at an effective

1d model (a = a z),

H = hν a†a (8.6)

61



8.1.2 Manipulations with lasers

Trapped ions interact with laser light via the dipole coupling as in eq. (5.4). Thus

the location of the ion enters into the interaction. The Hamiltonian reads,

H = hν a†a + hω A2σ z + h

Ω

2

σ −ei(ω L−k Lq A) + H.c.

, (8.7)

where ν is the frequency of the ions oscillation in the trap, ω A the atomic transition

frequency of the ion, Ω the Rabi frequency of the laser and q A =

h2mν (a + a†)

the quantised position of the ion in the trap. The Lamb-Dicke parameter,

η = k L

¯h

2mν =2π

λ L

¯h

2mν (8.8)

quantifies the ration of the ion’s zero point motion amplitude

h

2mν and the laser

wavelength λ L and is typically η ∼ 0.01 1. The Hamiltonian (8.7) can thus be

expanded in powers of η . For lasers of low enough intensity, such that ν Ω, we

can apply a rotating wave approximation. For different laser detunings we thus

obtain the following interactions: For ω A −ω L = 0 we drive so called “carrier

excitations” (h = 1),

H = ν a†

a +

ω A

2 σ z +

Ω

2

1 −η2

2 −η

2

a

†

aσ x, (8.9)

for ω A −ω L = ν we drive the so called “1. red sideband”,

H = ν a†a +ω A2σ z − iη

Ω

2σ −a† + iη

Ω

2σ +a, (8.10)

and for ω A −ω L = −ν in turn we drive the so called “1. blue sideband”,

H = ν a†a +ω A2σ z − iη

Ω

2σ −a + iη

Ω

2σ +a†. (8.11)

For ω A−ω L =

±2ν the second red and blue sidebands can be driven as well.

By choosing suitable laser frequencies one can thus generate a Jaynes-Cummings

interaction in Hamiltonian (8.10), c.f. eq. (6.2), and an anti-Jaynes-Cummings in

Hamiltonian (8.11). Driving the 1. red sideband can be used for cooling as we

will describe in the next section.

8.1.3 Sideband cooling

A laser that drives the 1. red sideband annihilates one phonon of the ion’s os-

cillation to excite the ion, see term σ +a in the Hamiltonian (8.10). If the ion’s

62



spontaneous emission rate γ obeys γ

η Ω

2, the ion’s excited state subsequently

decays via spontaneous emission to its ground state before a phonon can again becreated. The cycle thus reduces the ion’s vibrational energy by one phonon and

can therefore be employed for cooling the ion. Provided the sidebands can be well

resolved, i.e. ν γ , the process can cool an ion to its motional ground state.

8.1.4 Trapping multiple ions

One can not only load one but also multiple, say N ions into a trap. Here the

trapping Hamiltonian generalises to,

H =

N

∑ j=1

p2 j

2m +V with V =

N

∑ j=1

m

2 ν 2

z2 j +∑ j=l

Z 2e2

8πε 0

1

| z j − zl| , (8.12)

where the first part of the potential is the harmonic trap with axes along the z-

direction and the second part is the Coulomb interaction. For cold ions their mo-

tion can be approximated by expanding the potential to leading order around the

equilibrium positions z(0) j of the ions. Since the equilibrium positions are deter-

mined by ∂ V ∂ z j

z j= z

(0) j

= 0, the potential is to leading order quadratic in the devia-

tions form the equilibrium positions, q j = z j − z(0) j . We find,

H = N

∑ j=1

p2

j2m

+ m2ν 2

N

∑ j,l=1

A j,lq jql, (8.13)

with A j,l = δ j,l +δ j,l Z 2e2

4πε 0mν 2 ∑ N k =1,k = j

2

| z(0) j − z

(0)k

|3− (1 −δ j,l) Z 2e2

4πε 0mν 22

| z(0) j − z

(0)l

|3. The

matrix A can be diagonalised. It has non-negative eigenvalues µ m and eigenvectors b(m), m = 1, . . . , N . The Hamiltonian (8.12) can therefore be written as a sum of

decoupled harmonic oscillators for the normal modes given by Qm =∑ N j=1 b

(m) j q j

and Pm = ∑ N j=1 b

(m) j p j. Hence in terms of creation and annihilation operators,

H = N

∑m=1

hν ma†

m

am with ν m =√µ mν , am = mν m

2hQm +

i

mν mPm .

(8.14)

The energetically lowest normalmode is always the centre of mass mode Q0 =1√ N ∑

N j=1 q j with frequency ν 0 = ν . When adding laser drives that are tuned to the

first red sideband with ω A −ω L = ν 0 = ν , we obtain to leading order in η ,

H = ν 0a†0a0 +

ω A2

N

∑ j=1

σ z j − iη N

∑ j=1

Ω j

2√

N

σ − j a

†0 −σ + j a0

. (8.15)

Hence one can create interactions between ions via the centre of mass mode Q0.

63



8.1.5 Ion trap quantum computer

Classical computers represent information as bits, 0 and 1, that correspond e.g.

two values of a voltage. Hence a classical N -bit processor can process a bit string,

010011011100010010. . . , of length N at a time. Quantum computers on the other

hand represent information in two possible states of a two-level system, |0 and

|1, so called “qubits”. A quantum processor of N qubits can thus process a state,

|Ψ =2

∑ j1, j2,..., j N =1

c j1, j2,..., j N | j1, j2, . . . , j N , (8.16)

where the sum extends over 2 N orthogonal states. This indicates that a quantum

processor is able to process all 2 N

possible classical bit strings in parallel.Equation (8.16) also shows that one needs 2 N coefficients c j1, j2,..., j N

, that is 2 N

complex numbers to fully specify the state |Ψ. It is therefore very demanding to

simulate the dynamics of a large quantum system, i.e. with large N on a classical

computer. Moreover if one adds 1 qubit to the quantum system, the classical com-

puter has to double its power to still simulate it since 2 N +1 = 2×2 N . If a classical

computer can simulate a N qubit quantum system, it can also simulate a N qubit

quantum computer and hence do all calculations the quantum computer could do.

If the quantum computer is however increased by 1 qubit, the classical computer

should double in power to still be able to solve all problems the quantum computer

can solve. One can thus expect that quantum computers are more efficient than

classical computers. In fact there are two quantum algorithms which are known

to require less computational steps than any classical algorithm.

Grover’s algorithm can search an unsorted database of N entries in on average√ N steps, where a classical algorithm needs N /2 steps on average. Shor’s algo-

rithm can factorise a number x into a product of prime numbers in a number of

steps that grows polynomially in the number of digits of x. Classical algorithms

need a number of steps that grows exponentially in the number of digits of x.

Since the factorisation into prime numbers is unique, it is used in many encryp-

tion schemes which are secure due to the huge time it takes to find the factorisation

for large enough numbers.

It is known that universal computation (roughly speaking everything one canexpect a computer to do) can be done if single qubit rotations

α |0 +β |1 → α |0+β |1 ∀α ,α ,β ,β (8.17)

and controlled NOT respectively controlled phase gates,

|0, 0|0, 1|1, 0|1, 1

→

|0, 0|0, 1|1, 1|1, 0

respectively

|0, 0|0, 1|1, 0|1, 1

→

|0, 0|0, 1|1, 0

−|1, 1(8.18)

64



can be implemented. In fact, as shown by Cirac & Zoller in 1995, single qubit

rotations and controlled phase gates can be implemented with trapped ions.To see this, we choose a chain of N trapped ions and let two of their internal

states, say |g and |e represent a qubit,

|0 j ≡ |g j and |1 j ≡ |e j ∀ j. (8.19)

Furthermore we consider an auxiliary level in each ion, |e. The transition |g ↔|e with transition frequency ω e is thereby driven by a laser with one polarisation,

say σ + and the transition |g ↔ |e with transition frequency ω e is driven by a

laser with a different polarisation, say σ −. We thus have N qubits, that can couple

to each other via the centre of mass mode as described in the Hamiltonian (8.15).Single qubit rotations can be implemented for these ions by driving a carrier

excitation as describe by the Hamiltonian (8.9). A controlled phase gate between

ions j and l can be implemented by applying the sequence of unitary transforma-

tions,

U j,l = U (1) j U

(2)l

U (1) j , (8.20)

where

U (1) j = exp[

π

2(|e jg j|a −|g je j|a†)]

U (2)

l

= exp[π (|el

gl

|a

−|gl

el

|a†)].

Hence to implement the unitary transformation U (1) j , one applies a σ +-laser with

Rabi frequency Ω j to the first red sideband of ion j for a time t = π √

N /(ηΩ j).

This is described by Hamiltonian (8.10) with ω L = ω e −ν . To in turn implement

the transformation U (2)l , one applies a σ −-laser with Rabi frequency Ωl to the

first red sideband of ion l for a time t = 2π √

N /(ηΩl). This is described by

Hamiltonian (8.10) with ω L = ω e −ν .The action of the unitary U j,l can be seen as follows:

|g

j, g

l, 0

|g j, el, 0|e j, gl, 0|e j, el, 0

→

U (1)

j

|g

j, g

l, 0

|g j, el, 0−|g j, gl, 1−|g j, el, 1

→

U (2)l

|g

j, g

l, 0

|g j, el, 0|g j, gl, 1

−|g j, el, 1

→

U (1)

j

|g

j, g

l, 0

|g j, el , 0|e j, gl , 0

−|e j, el , 0,

(8.21)

where e.g. |g j, gl, 0 denotes a state with both ions, j and l in their ground states

d h t | 0 0 W th th t q ti (8 21) d ib

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