o

Adapting the general result (11.27) to the cavity problem gives the RWA interaction

Hamiltonian

Hrwa = ’d · E(+) σ+ ’ d— · E(’) σ’

gκ a† σ’ ;

—

= ’i gκ aκ σ+ + i (12.4)

κ

κ κ

where d = d is the dipole matrix element; the coupling frequencies are

1 2

ωκ d · E κ (R)

gκ = K (ω0 ’ ωκ ) ; (12.5)

20

K (ω0 ’ ωκ ) is the RWA cut-o¬ function; and R is the position of the atom.

We will now drastically simplify this model in two ways. The ¬rst is to assume

that the center-of-mass motion of the atom can be treated classically. This means

that ω0 should be interpreted as the Doppler-shifted resonance frequency. In many

cases the Doppler e¬ect is not important; for example, for microwave transitions in

Rydberg atoms passing through a resonant cavity, or single atoms con¬ned in a trap.

The second simpli¬cation is enforced by choosing the cavity parameters so that the

lowest (fundamental) mode frequency is nearly resonant with the atomic transition,

while all higher frequency modes are well out of resonance. This guarantees that only

the lowest mode contributes to the resonant Hamiltonian; consequently, the family of

annihilation operators aκ can be reduced to the single operator a for the fundamental

mode. From now on, we will call the fundamental frequency the cavity frequency

ωC and the corresponding mode function E C (R) the cavity mode.

The total Hamiltonian for the Jaynes“Cummings model is therefore HJC = H0 +

Hint , where

H0 = ωC a† a + ( ω0 /2) σz , (12.6)

Hint = ’i gaσ+ + i ga† σ’ , (12.7)

and

ωC d · E C (R)

g= . (12.8)

20

By appropriate choice of the phases in the atomic eigenstates | and | , we can

1 2

always arrange that g is real.

12.1.2 Dressed states

The interaction Hamiltonian in eqn (12.7) has the same general form as the interac-

tion Hamiltonian (11.25) for the Weisskopf“Wigner model of Section 11.2.2, but it is

greatly simpli¬ed by the fact that only one mode of the radiation ¬eld is active. In

¿¿

The Jaynes“Cummings model

the Weisskopf“Wigner case, the in¬nite-dimensional subspaces Hse are left invariant

(mapped into themselves) under the action of the Hamiltonian. Since the Hamiltonians

have the same structure, a similar behavior is expected in the present case.

The product states,

| j , n (0) = | |n (n = 0, 1, . . .) , (12.9)

j

where the | j s (j = 1, 2) are the atomic eigenstates and the |n s are number states

for the cavity mode, provide a natural basis for the Hilbert space HJC of the Jaynes“

Cummings model. The | j , n (0) s are called bare states, since they are eigenstates of

the non-interacting Hamiltonian H0 :

H0 | j , n (0) = ( + n ωC ) | j , n (0) . (12.10)

j

Turning next to Hint , a straightforward calculation shows that

Hint | 1 , 0 (0) = 0 , (12.11)

which means that spontaneous absorption from the bare vacuum is forbidden in the

resonant wave approximation. Consequently, the ground-state energy and state vector

for the atom“¬eld system are, respectively,

ω0

=’ and |G = | 1 , 0 (0) .

µG = (12.12)

1

2

Furthermore, for each photon number n the pairs of bare states | 2 , n (0) and

| 1 , n + 1 (0) satisfy

√

Hint | 2 , n (0) = i g n + 1 | 1 , n + 1 (0) ,

√ (12.13)

Hint | 1 , n + 1 (0) = ’i g n + 1 | 2 , n (0) .

Consequently, each two-dimensional subspace

Hn = span | 2 , n (0) , | 1 , n + 1 (0) (n = 0, 1, . . .) (12.14)

is left invariant by the total Hamiltonian. This leads to the natural decomposition of

HJC as

HJC = HG • H0 • H1 • · · · , (12.15)

where HG = span | 1 , 0 (0) is the one-dimensional space spanned by the ground state.

In the subspace Hn the Hamiltonian is represented by a 2 — 2 matrix

√

’2ig n + 1

1 10 δ

√

+

HJC,n = n + ωC , (12.16)

’δ

01 2 2ig n + 1

2

where δ = ω0 ’ ωC is the detuning. This construction allows us to reduce the solution

of the full Schr¨dinger equation, HJC |¦ = µ |¦ , to the diagonalization of the 2 — 2-

o

matrix HJC,n for each n. The details are worked out in Exercise 12.1. For each subspace

¿ Cavity quantum electrodynamics

Hn , the exact eigenvalues and eigenvectors, which will be denoted by µj,n and |j, n

(j = 1, 2), respectively, are

1 „¦n

µ1,n = n+ ωC + , (12.17)

2 2

|1, n = sin θn | 2 , n (0) + cos θn | 1 , n + 1 (0) , (12.18)

1 „¦n

ωC ’

µ2,n = n+ , (12.19)

2 2

|2, n = cos θn | 2 , n (0) ’ sin θn | 1 , n + 1 (0) , (12.20)

where

δ 2 + 4g 2 (n + 1)

„¦n = (12.21)

is the Rabi frequency for oscillations between the two bare states in Hn . The probability

amplitudes for the bare states are given by

„¦n ’ δ

cos θn = ,

2

(„¦n ’ δ) + 4g 2 (n + 1)

√ (12.22)

2g n + 1

sin θn = .

2

(„¦n ’ δ) + 4g 2 (n + 1)

The bare (g = 0) eigenvalues

(0)

µ1,n = (n + 1/2) ωC + δ/2 ,

(12.23)

(0)

µ2,n = (n + 1/2) ωC ’ δ/2

are degenerate at resonance (δ = 0), but the exact eigenvalues satisfy

√

µ1,n ’ µ2,n = „¦n 2 g n + 1 . (12.24)

This is an example of the ubiquitous phenomenon of avoided crossing (or level

repulsion) which occurs whenever two states are coupled by a perturbation.

The eigenstates |1, n and |2, n of the full Jaynes“Cummings Hamiltonian HJC are

called dressed states, since the interaction between the atom and the ¬eld is treated

exactly. By virtue of this interaction, the dressed states are entangled states of the

atom and the ¬eld.

12.2 Collapses and revivals

With the dressed eigenstates of HJC in hand, we can write the general solution of the

time-dependent Schr¨dinger equation as

o

∞ 2

’iµG t/

Cj,n e’iµj,n t/ |j, n ,

|Ψ (t) = e CG |G + (12.25)

n=0 j=1

where the expansion coe¬cients are determined by the initial state vector according

to CG = G |Ψ (0) and Cj,n = j, n |Ψ (0) (j = 1, 2) (n = 0, 1, . . .). If the atom

¿

Collapses and revivals

is initially in the excited state | 2 and exactly m cavity photons are present, i.e.

|Ψ (0) = | 2 , m (0) , the general solution (12.25) specializes to |Ψ (t) = | 2 , m; t , where

„¦n t „¦n t

| 2 , n; t ≡ e’i(n+1/2)ωC t cos | 2 , n (0)

+ i cos (2θn ) sin

2 2

„¦n t

’ ie’i(n+1/2)ωC t sin (2θn ) sin | 1 , n + 1 (0) . (12.26)

2

At resonance, the probabilities for the states | 2 , m (0) and | 1 , m + 1 (0) are

√

2

2 , m |Ψ (t)

(0)

= cos2 g m + 1t ,

P2,m (t) =

(12.27)

√

2

+ 1 |Ψ (t)

(0) 2

P1,m+1 (t) = 1, m = sin g m + 1t ,

so”as expected”the system oscillates between the two atomic states by emission and

absorption of a single photon. The exact periodicity displayed here is a consequence

of the special choice of an initial state with a de¬nite number of photons. For m > 0,

this is analogous to the semiclassical problem of Rabi ¬‚opping driven by a ¬eld with

de¬nite amplitude and phase. The analogy to the classical case fails for m = 0, i.e. an

excited atom with no photons present. The classical analogue of this case would be

a vanishing ¬eld, so that no Rabi ¬‚opping would occur. The occupation probabilities

P2,0 (t) = cos2 (gt) and P1,1 (t) = sin2 (gt) describe vacuum Rabi ¬‚opping, which is

a consequence of the purely quantum phenomenon of spontaneous emission, followed

by absorption, etc.

For initial states that are superpositions of several photon number states, exact

periodicity is replaced by more complex behavior which we will now study. A super-

position,

∞

|Ψ (0) = Kn | 2 , n (0) , (12.28)

n=0

of the initial states | 2 , n (0) that individually lead to Rabi ¬‚opping evolves into

∞

|Ψ (t) = Kn | 2 , n; t , (12.29)

n=0

so the probability to ¬nd the atom in the upper state, without regard to the number

of photons, is

∞ ∞

2 2

2

2 , n |Ψ (t) |Kn | 2 , n | 2 , n; t

(0) (0)

P2 (t) = = . (12.30)

n=0 n=0

At resonance, eqn (12.27) allows this to be written as

∞

√

11 2

|Kn | cos 2 n + 1gt .

P2 (t) = + (12.31)

2 2 n=0

If more than one of the coe¬cients Kn is nonvanishing, this function is a sum of

oscillatory terms with incommensurate frequencies. Thus true periodicity is only found

¿ Cavity quantum electrodynamics

for the special case |Kn | = δnm for some ¬xed value of m. For any choice of the Kn s

the time average of the upper-level population is P2 (t) = 1/2.

In order to study the behavior of P2 (t), we need to make an explicit choice for the

Kn s. Let us suppose, for example, that the initial state is |Ψ (0) = | 2 |± , where |± is

a coherent state for the cavity mode. The coe¬cients are then |Kn | = e’|±| |±| /n!,

2

2 2n

and 2∞

√

1 e’|±|

2n

|±|

P2 (t) = + cos 2 n + 1gt . (12.32)

2 2 n=0 n!

Photon numbers for the coherent state follow a Poisson distribution, so the main

contribution to the sum over n will come from the range (n ’ ∆n, n + ∆n), where

2

n = |±| is the mean photon number and ∆n = |±| is the variance. For large n,

the corresponding spread in Rabi frequencies is ∆„¦ ∼ 2g. At very early times, t

1/g, the arguments of the cosines are essentially in phase, and P2 (t) will execute an

almost coherent oscillation. At later times, the variation of the Rabi frequencies with

photon number will lead to an e¬ectively random distribution of phases and destructive

interference. This e¬ect can be estimated analytically by replacing the sum over n

with an integral and evaluating the integral in the stationary-phase approximation.

The result,

1 e’|gt|

2

cos (2 |±| gt) for gt 1 ,

P2 (t) = + (12.33)

2 2

describes the collapse of the upper-level population to the time-averaged value of

1/2. This decay in the oscillations is neither surprising nor particularly quantal in

character. A superposition of Rabi oscillations due to classical ¬elds with random ¬eld

strengths would produce a similar decay.

What is surprising is the behavior of the upper-level population at still later times.

A numerical evaluation of eqn (12.32) reveals that the oscillations reappear after a

rephasing time trp ∼ 4π |±| /g. This revival”with P2 (t) = O (1)”is a speci¬cally

quantum e¬ect, explained by photon indivisibility. The revival is in turn followed by

another collapse. The ¬rst collapse and revival are shown in Fig. 12.1.

The classical nature of the collapse is illustrated by the dashed curve in the same

¬gure, which is calculated by replacing the discrete sum in eqn (12.32) by an integral.

The two curves are indistinguishable in the initial collapse phase, but the classical

(dashed) curve remains ¬‚at at the value 1/2 during the quantum revival. Thus the

experimental observation of a revival provides further evidence for the indivisibility

of photons. After a few collapse“revival cycles, the revivals begin to overlap and”as

shown in Exercise 12.2”P2 (t) becomes irregular.

¿

The micromaser

22

1

0.8

0.6

0.4

0.2

CJ

5 10 15 20 25 30

Fig. 12.1 The solid curve shows the probability P2 (t) versus gt, where the upper-level

population P2 (t) is given by eqn (12.32), and the average photon number is n = |±|2 = 10.

The dashed curve is the corresponding classical result obtained by replacing the discrete sum

over photon number by an integral.

12.3 The micromaser

The interaction of a Rydberg atom with the fundamental mode of a microwave cavity

provides an excellent realization of the Jaynes“Cummings model. The con¬guration

sketched in Fig. 12.2 is called a micromaser (Walther, 2003). It is designed so that”

with high probability”at most one atom is present in the cavity at any given time. A

velocity-selected beam of alkali atoms from an oven is sent into a laser excitation region,

where the atoms are promoted to highly excited Rydberg states. The size of a Rydberg

atom is characterized by the radius, aRyd = n2 2 /me2 , of its Bohr orbit, where np is

p

Atomic beam oven

Maser cavity

Field ionization

Velocity selector

Laser excitation

Channeltron

detectors Atomic beam

Fig. 12.2 Rubidium Rydberg atoms from an oven pass successively through a velocity selec-

tor, a laser excitation region, and a superconducting microwave cavity. After emerging from

the cavity, they are detected”in a state-selective manner”by ¬eld ionization, followed by

channeltron detectors. (Reproduced from Rempe et al. (1990).)

¿ Cavity quantum electrodynamics

the principal quantum number, and 2 /me2 is the Bohr radius for the ground state of

the hydrogen atom. These atoms are truly macroscopic in size; for example, the radius

of a Rydberg atom with np 100 is on the order of microns, instead of nanometers.

The dipole matrix element d = np |er| np + 1 for a transition between two adjacent

Rydberg states np + 1 ’ np is proportional to the diameter of the atom, so it scales

as n2 . On the other hand, for transitions between high angular momentum (circular)

p

states the frequency scales as ω ∝ 1/n3 , which is in the microwave range. According to

p

2

eqn (4.162) the Einstein A coe¬cient scales like A ∝ |d| ω 3 ∝ 1/n5 . Thus the lifetime

p

„ = 1/A ∝ n5 of the upper level is very long, and the neglect of spontaneous emission

p

is a very good approximation.

The opposite conclusion follows for absorption and stimulated emission, since the

relation (4.166) between the A and B coe¬cients shows that B ∝ n4 . For the same

p

applied ¬eld, the absorption rate for a Rydberg atom with np 100 is typically 108

times larger than the absorption rate at the Lyman transition between the 2p and

1s states of the hydrogen atom. Since stimulated emission is also described by the

Einstein B coe¬cient, stimulated emission from the Rydberg atom can occur when

there are only a few photons inside a microwave cavity.

As indicated in Fig. 12.2, a single Rydberg atom enters and leaves a supercon-

ducting microwave cavity through small holes drilled on opposite sides. During the

transit time of the atom across the cavity the photons already present can stimulate

emission of a single photon into the fundamental cavity mode; conversely, the atom

can sometimes reabsorb a single photon. The interaction of the atom with a single

mode of the cavity is described by the Jaynes“Cummings Hamiltonian in eqn (12.7).

By monitoring whether or not the Rydberg atom has made a transition, np + 1 ’ np ,

between the adjacent Rydberg states, one can infer indirectly whether or not a single

microwave photon has been deposited in the cavity. This is possible because of the

entangled nature of the dressed states in eqns (12.18) and (12.20). A measurement of

the state of the atom, with the outcome | 2 , forces a reduction of the total state vector

of the atom“radiation system, with the result that the radiation ¬eld is de¬nitely in

the state |n . In other words, the number of photons in the cavity has not changed.

Conversely, a measurement with the outcome | 1 guarantees that the ¬eld is in the

state |n + 1 , i.e. a photon has been added to the cavity.

The discrimination between the two Rydberg states is easily accomplished, since

the ionization of the Rydberg atom by a DC electric ¬eld depends very sensitively

on its principal quantum number np . The higher number np + 1 corresponds to a

larger, more easily ionized atom, and the lower number np corresponds to a smaller,

less easily ionized atom. The electric ¬eld in the ¬rst ionization region”shown in Fig.

12.2”is strong enough to ionize all (np + 1)-atoms, but too weak to ionize any np -

atoms. Thus an atom that remains in the excited state is detected in the ¬rst region.

If the atom has made a transition to the lower state, then it will be ionized by the

stronger ¬eld in the second region. In this way, it is possible sensitively to identify the

state of the Rydberg atom. If the atom is in the appropriate state, it will be ionized and

release a single electron into the corresponding ionizing ¬eld region. The free electron

is accelerated by the ionizing ¬eld and enters into an electron-multiplication region

of a channeltron detector. As explained in Section 9.2.1, the channeltron detector

¿

The micromaser

can enormously multiply the single electron released by the Rydberg atom, and this

provides an indirect method for continuously monitoring the photon-number state of

the cavity.

A frequency-doubled dye laser (» = 297 nm) is used to excite rubidium (85 Rb)

atoms to the np = 63, P3/2 state from the np = 5, S1/2 (F = 3) state. The cavity is

tuned to the 21.456 GHz transition from the upper maser level in the np = 63, P3/2

state to the np = 61, D5/2 lower maser state. For this experiment a superconducting

cavity with a Q-value of 3—108 was used, corresponding to a photon lifetime inside the

cavity of 2 ms. The transit time of the Rydberg atom through the cavity is controlled

by changing the atomic velocity with the velocity selector. On the average, only a

small fraction of an atom is inside the cavity at any given time. In order to reduce

the number of thermally excited photons in the cavity, a liquid helium environment

reduces the temperature of the superconducting niobium microwave cavity to 2.5 K,

corresponding to the average photon number n ≈ 2.

If the transit time of the atom is larger than the collapse time but smaller than

the time of the ¬rst revival, then the solution (12.32) tells us that the atom will come

into equilibrium with the cavity ¬eld, as seen in Fig. 12.1. In this situation the atom

leaving the chamber is found in the upper or lower state with equal probability, i.e.

P2 = 1/2. When the transit time is increased to a value comparable to the ¬rst revival

time, the probability for the excited state becomes larger than 1/2. The data in Fig.

12.3 show a quantum revival of the population of atoms in the upper maser state that