<< . .

. 53
( : 97)



. . >>

is the positive-frequency part of the electric ¬eld (in the Schr¨dinger picture).
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

<< . .

. 53
( : 97)



. . >>