Effect of atomic beam alignment on photon correlation measurements in
  cavity QED
Effect of atomic beam alignment on photon correlation measurements
in cavity QED
L. Horvath and H. J. Carmichael
Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand
(Dated: November 3, 2018)
Quantum trajectory simulations of a cavity QED system comprising an atomic beam traversing a
standing-wave cavity are carried out. The delayed photon coincident rate for forwards scattering is
computed and compared with the measurements of Rempe et al. [Phys. Rev. Lett. 67, 1727 (1991)]
and Foster et al. [Phys. Rev. A 61, 053821 (2000)]. It is shown that a moderate atomic beam
misalignment can account for the degradation of the predicted correlation. Fits to the experimental
data are made in the weak-field limit with a single adjustable parameter—the atomic beam tilt from
perpendicular to the cavity axis. Departures of the measurement conditions from the weak-field limit
are discussed.
PACS numbers: 42.50.Pq, 42.50.Lc, 02.70.Uu
I. INTRODUCTION
Cavity quantum electrodynamics [1, 2, 3, 4, 5, 6] has
as its central objective the realization of strong dipole
coupling between a discrete transition in matter (e.g., an
atom or quantum dot) and a mode of an electromag-
netic cavity. Most often strong coupling is demonstrated
through the realization of vacuum Rabi splitting [7, 8].
First realized for Rydberg atoms in superconducting mi-
crowave cavities [9, 10] and for transitions at optical
wavelengths in high-finesse Fabry Perots [11, 12, 13, 14],
vacuum Rabi splitting was recently observed in mono-
lithic structures where the discrete transition is provided
by a semiconductor quantum dot [15, 16, 17], and in a
coupled system of qubit and resonant circuit engineered
from superconducting electronics [18].
More generally, vacuum Rabi spectra can be observed
for any pair of coupled harmonic oscillators [19] without
the need for strong coupling of the one-atom kind. Prior
to observations for single atoms and quantum dots, sim-
ilar spectra were observed in many-atom [20, 21, 22] and
-exciton [23, 24] systems where the radiative coupling is
collectively enhanced.
The definitive signature of single-atom strong coupling
is the large effect a single photon in the cavity has on
the reflection, side-scattering, or transmission of another
photon. Strong coupling has a dramatic effect, for exam-
ple, on the delayed photon coincidence rate in forwards
scattering when a cavity QED system is coherently driven
on axis [25, 26, 27, 28]. Photon antibunching is seen at
a level proportional to the parameter 2C1 = 2g
2/γκ [27],
where g is the atomic dipole coupling constant, γ is the
atomic spontaneous emission rate, and 2κ is the pho-
ton loss rate from the cavity; the collective parameter
2C = N2C1, with N the number of atoms, does not en-
ter into the magnitude of the effect when N ≫ 1. In the
one-atom case, and for 2κ ≫ γ, the size of the effect is
raised to (2C1)
2 [25, 26] [see Eq. (30)].
The first demonstration of photon antibunching was
made [29] for moderately strong coupling (2C1 ≈ 4.6)
and N = 18, 45, and 110 (effective) atoms. The mea-
surement has subsequently been repeated for somewhat
higher values of 2C1 and slightly fewer atoms [30, 31], and
a measurement for one trapped atom [32], in a slightly al-
tered configuration, has demonstrated the so-called pho-
ton blockade effect [33, 34, 35, 36, 37, 38]—i.e., the anti-
bunching of forwards-scattered photons for coherent driv-
ing of a vacuum-Rabi resonance, in which case a two-state
approximation may be made [39], assuming the coupling
is sufficiently strong.
The early experiments of Rempe et al. [29] and those
of Mielke et al. [30] and Foster et al. [31] employ systems
designed around a Fabry-Perot cavity mode traversed
by a thermal atomic beam. Their theoretical modeling
therefore presents a significant challenge, since for the
numbers of effective atoms used, the atomic beam car-
ries hundreds of atoms—typically an order of magnitude
larger than the effective number [40]—into the interac-
tion volume. The Hilbert space required for exact calcu-
lations is enormous (2100 ∼ 1030); it grows and shrinks
with the number of atoms, which inevitably fluctuates
over time; and the atoms move through a spatially vary-
ing cavity mode, so their coupling strengths are chang-
ing in time. Ideally, all of these features should be taken
into account, although certain approximations might be
made.
For weak excitation, as in the experiments, the lowest
permissible truncation of the Hilbert space—when cal-
culating two-photon correlations—is at the two-quanta
level. Within a two-quanta truncation, relatively simple
formulas can be derived so long as the atomic motion
is overlooked [27, 28]. It is even possible to account for
the unequal coupling strengths of different atoms, and,
through a Monte-Carlo average, fluctuations in their spa-
tial distribution [29]. A significant discrepancy between
theory and experiment nevertheless remains: Rempe et
al. [29] describe how the amplitude of the Rabi oscillation
(magnitude of the antibunching effect) was scaled down
by a factor of 4 and a slight shift of the theoretical curve
was made in order to bring their data into agreement with
this model; the discrepancy persists in the experiments
of Foster et al. [31], except that the required adjustment
http://arxiv.org/abs/0704.1686v2
is by a scale factor closer to 2 than to 4.
Attempts to account for these discrepancies have been
made but are unconvincing. Martini and Schenzle [41]
report good agreement with one of the data sets from
Ref. [29]; they numerically solve a many-atom master
equation, but under the unreasonable assumption of sta-
tionary atoms and equal coupling strengths. The unlikely
agreement results from using parameters that are very
far from those of the experiment—most importantly, the
dipole coupling constant is smaller by a factor of approx-
imately 3.
Foster et al. [31] report a rather good theoretical fit
to one of their data sets. It is obtained by using the
mentioned approximations and adding a detuning in the
calculation to account for the Doppler broadening of a
misaligned atomic beam. They state that “Imperfect
alignment . . . can lead to a tilt from perpendicular of
as much as 1◦”. They suggest that the mean Doppler
shift is offset in the experiment by adjusting the driving
laser frequency and account for the distribution about
the mean in the model. There does appear to be a dif-
ficulty with this procedure, however, since while such an
offset should work for a ring cavity, it is unlikely to do
so in the presence of the counter-propagating fields of a
Fabry-Perot. Indeed, we are able to successfully simulate
the procedure only for the ring-cavity case (Sec. IVC).
The likely candidates to explain the disagreement be-
tween theory and experiment have always been evident.
For example, Rempe et al. [29] state:
“Apparently the transient nature of the atomic mo-
tion through the cavity mode (which is not included
here or in Ref. [7]) has a profound effect in decorre-
lating the otherwise coherent response of the sam-
ple to the escape of a photon.”
and also:
“Empirically, we also know that |g(2)(0)− 1| is re-
duced somewhat because the weak-field limit is not
strictly satisfied in our measurements.”
To these two observations we should add—picking up on
the comment in [31]—that in a standing-wave cavity an
atomic beam misalignment would make the decorrelation
from atomic motion a great deal worse.
Thus, the required improvements in the modeling are:
(i) a serious accounting for atomic motion in a thermal
atomic beam, allowing for up to a few hundred inter-
acting atoms and a velocity component along the cavity
axis, and (ii) extension of the Hilbert space to include 3,
4, etc. quanta of excitation, thus extending the model
beyond the weak-field limit. The first requirement is
entirely achievable in a quantum trajectory simulation
[42, 43, 44, 45, 46], while the second, even with recent
improvements in computing power, remains a formidable
challenge.
In this paper we offer an explanation of the discrepan-
cies between theory and experiment in the measurements
Parameter Set 1 Set 2
cavity halfwidth
κ/2π 0.9MHz 7.9MHz
dipole coupling constant
gmax/κ 3.56 1.47
atomic linewidth
γ/κ 5.56 0.77
mode waist
w0 50µm 21.5µm
wavelength
852nm (Cs) 780nm (Rb)
effective atom number
N̄eff
18 13
oven temperature
473K 430K
mean speed in oven
voven
274.5m/s 326.4m/s
mean speed in beam
vbeam
323.4m/s 384.5m/s
TABLE I: Parameters used in the simulations. Set 1 is taken
from Ref. [29] and Set 2 from Ref. [31].
of Refs. [29] and [31]. We perform ab initio quantum tra-
jectory simulations in parallel with a Monte-Carlo sim-
ulation of a tilted atomic beam. The parameters used
are listed in Table I: Set 1 corresponds to the data dis-
played in Fig. 4(a) of Ref. [29], and Set 2 to the data dis-
played in Fig. 4 of Ref. [31]. All parameters are measured
quantities— or are inferred from measured quantities—
and the atomic beam tilt alone is varied to optimize the
data fit. Excellent agreement is demonstrated for atomic
beam misalignments of approximately 10mrad (a little
over 1/2◦). These simulations are performed using a two-
quanta truncation of the Hilbert space.
Simulations based upon a three-quanta truncation are
also carried out, which, although not adequate for the
experimental conditions, can begin to address physics be-
yond the weak-field limit. From these, an inconsistency
with the intracavity photon number reported by Foster
et al. [31] is found.
Our model is described in Sec. II, where we formu-
late the stochastic master equation used to describe the
atomic beam, its quantum trajectory unraveling, and the
two-quanta truncation of the Hilbert space. The previous
modeling on the basis of a stationary-atom approxima-
tion is reviewed in Sect. III and compared with the data
of Rempe et al. [29] and Foster et al. [31]. The effects
of atomic beam misalignment are discussed in Sec. IV;
here the results of simulations with a two-quanta trunca-
tion are presented. Results obtained with a three-quanta
truncation are presented in Sec. V, where the issue of
intracavity photon number is discussed. Our conclusions
are stated in Sec. VI.
II. CAVITY QED WITH ATOMIC BEAMS
A. Stochastic Master Equation: Atomic Beam
Simulation
Thermal atomic beams have been used extensively for
experiments in cavity QED [9, 10, 11, 12, 20, 21, 22, 29,
30, 31]. The experimental setups under consideration
are described in detail in Refs. [47] and [48]. As typi-
cally, the beam is formed from an atomic vapor created
inside an oven, from which atoms escape through a colli-
mated opening. We work from the standard theory of an
effusive source from a thin-walled oriface [49], for which
for an effective number N̄eff of intracavity atoms [11, 40]
and cavity mode waist ω0 (N̄eff is the average number
of atoms within a cylinder of radius w0/2), the average
escape rate is
R = 64N̄eff v̄beam/3π
2w0, (1)
with mean speed in the beam
v̄beam =
9πkBT/8M, (2)
where kB is Boltzmann’s constant, T is the oven tem-
perature, and M is the mass of an atom; the beam has
atomic density
̺ = 4N̄eff/πw
0l, (3)
where l is the beam width, and distribution of atomic
speeds
P (v)dv = 2u3(v)e−u
2(v)du(v), (4)
u(v) ≡ 2v/
π v̄oven, where
v̄oven =
8kBT/πM = (8/3π)v̄beam (5)
is the mean speed of an atom inside the oven, as cal-
culated from the Maxwell-Boltzmann distribution. Note
that v̄beam is larger than v̄oven because those atoms that
move faster inside the oven have a higher probability of
escape.
In an open-sided cavity, neither the interaction volume
nor the number of interacting atoms is well-defined; the
cavity mode function and atomic density are the well-
defined quantities. Clearly, though, as the atomic dipole
coupling strength decreases with the distance of the atom
from the cavity axis, those atoms located far away from
the axis may be neglected, introducing, in effect, a finite
interaction volume. How far from the cavity axis, how-
ever, is far enough? One possible criterion is to require
that the interaction volume taken be large enough to give
an accurate result for the collective coupling strength, or,
considering its dependence on atomic locations (at fixed
average density), the probability distribution over collec-
tive coupling strengths. According to this criterion, the
actual number of interacting atoms is typically an order
of magnitude larger than N̄eff [40]. If, for example, one
introduces a cut-off parameter F < 1, and defines the
interaction volume by [40, 50, 51]
VF ≡ {(x, y, z) : g(x, y, z) ≥ Fgmax}, (6)
g(x, y, z) = gmax cos(kz) exp
−(x2 + y2)/w20
the spatially varying coupling constant for a standing-
wave TEM00 cavity mode [52]—wavelength λ = 2π/k—
the computed collective coupling constant is [40]
N̄eff gmax →
N̄Feff gmax,
N̄Feff = (2N̄eff/π)
(1− 2F 2) cos−1 F + F
1− F 2
. (8)
For the choice F = 0.1, one obtains N̄Feff = 0.98N̄eff, a
reduction of the collective coupling strength by 1%, and
the interaction volume—radius r ≈ 3(w0/2)—contains
approximately 9N̄eff atoms on average. This is the choice
made for the simulations with a three-quanta trunca-
tion reported in Sec. V. When adopting a two-quanta
truncation, with its smaller Hilbert space for a given
number of atoms, we choose F = 0.01, which yields
N̄Feff = 0.9998N̄eff and r ≈ 4.3(w0/2), and approximately
18N̄eff atoms in the interaction volume on average.
In fact, the volume used in practice is a little larger
than VF . In the course of a Monte-Carlo simulation of
the atomic beam, atoms are created randomly at rate R
on the plane x = −w0
| lnF |. At the time, tj0, of its
creation, each atom is assigned a random position and
velocity (j labels a particular atom),
| lnF |
, vj = vj
cos θ
sin θ
, (9)
where yj(t
0) and zj(t
0) are random variables, uniformly
distributed on the intervals |yj(tj0)| ≤ w0
| lnF | and
|zj(tj0)| ≤ λ/4, respectively, and vj is sampled from the
distribution of atomic speeds [Eq. (4)]; θ is the tilt of
the atomic beam away from perpendicular to the cavity
axis. The atom moves freely across the cavity after its
creation, passing out of the interaction volume on the
plane x = w0
| lnF |. Thus the interaction volume has
a square rather than circular cross section and measures
| lnF |w0 on a side. It is larger than VF by approxi-
mately 30%.
Atoms are created in the ground state and returned
to the ground state when they leave the interaction vol-
ume. On leaving an atom is disentangled from the sys-
tem by comparing its probability of excitation with a
uniformly distributed random number r, 0 ≤ r ≤ 1, and
deciding whether or not it will—anytime in the future—
spontaneously emit; thus, the system state is projected
onto the excited state of the leaving atom (the atom will
emit) or its ground state (it will not emit) and propa-
gated forwards in time.
Note that the effects of light forces and radiative heat-
ing are neglected. At the thermal velocities considered,
typically the ratio of kinetic energy to recoil energy is of
order 108, while the maximum light shift h̄gmax (assum-
ing one photon in the cavity) is smaller than the kinetic
energy by a factor of 107; even if the axial component
of velocity only is considered, these ratios are as high as
104 and 103 with θ ∼ 10mrad, as in Figs. 10 and 11. In
fact, the mean intracavity photon number is considerably
less than one (Sec. V); thus, for example, the majority of
atoms traverse the cavity without making a single spon-
taneous emission.
Under the atomic beam simulation, the atom number,
N(t), and locations rj(t), j = 1, . . . , N(t), are chang-
ing in time; therefore, the atomic state basis is dynamic,
growing and shrinking with N(t). We assume all atoms
couple resonantly to the cavity mode, which is coher-
ently driven on resonance with driving field amplitude
E . Then, including spontaneous emission and cavity loss,
the system is described by the stochastic master equation
in the interaction picture
ρ̇ = E [â† − â, ρ] +
g(rj(t))[â
†σ̂j− − âσ̂j+, ρ]
(2σ̂j−ρσ̂j+ − σ̂j+σ̂j−ρ− ρσ̂j+σ̂j−)
2âρâ† − â†âρ− ρâ†â
, (10)
with dipole coupling constants
g(rj(t)) = gmax cos(kzj(t)) exp
x2j (t) + y
j (t)
, (11)
where â† and â are creation and annihilation operators
for the cavity mode, and σ̂j+ and σ̂j−, j = 1 . . .N(t), are
raising and lowering operators for two-state atoms.
B. Quantum Trajectory Unraveling
In principle, the stochastic master equation might be
simulated directly, but it is impossible to do so in prac-
tice. Table I lists effective numbers of atoms N̄eff = 18
and N̄eff = 13. For cut-off parameter F = 0.01 and an
interaction volume of approximately 1.3×VF [see the dis-
cussion below Eq. (8)], an estimate of the number of in-
teracting atoms gives N(t) ∼ 1.3×18N̄eff ≈ 420 and 300,
respectively, which means that even in a two-quanta trun-
cation the size of the atomic state basis (∼ 105 states)
is far too large to work with density matrix elements.
We therefore make a quantum trajectory unraveling of
Eq. (10) [42, 43, 44, 45, 46], where, given our interest
in delayed photon coincidence measurements, condition-
ing of the evolution upon direct photoelectron counting
records is appropriate: the (unnormalized) conditional
state satisfies the nonunitary Schrödinger equation
d|ψ̄REC〉
ĤB(t)|ψ̄REC〉, (12)
with non-Hermitian Hamiltonian
ĤB(t)/ih̄ = E(â† − â) +
g(rj(t))(â
†σ̂j− − âσ̂j+)
− κâ†â− γ
σ̂j+σ̂j−, (13)
and this continuous evolution is interrupted by quantum
jumps that account for photon scattering. There are
N(t)+1 scattering channels and correspondinglyN(t)+1
possible jumps:
|ψ̄REC〉 → â|ψ̄REC〉, (14a)
for forwards scattering—i.e., the transmission of a photon
by the cavity—and
|ψ̄REC〉 → σ̂j−|ψ̄REC〉, j = 1, . . . , N(t), (14b)
for scattering to the side (spontaneous emission). These
jumps occur, in time step ∆t, with probabilities
Pforwards = 2κ〈â†â〉REC∆t, (15a)
side = γ〈σ̂j+σ̂j−〉REC∆t, j = 1, . . . , N(t); (15b)
otherwise, with probability
1− Pforwards −
side,
the evolution under Eq. (12) continues.
For simplicity, and without loss of generality, we as-
sume a negligible loss rate at the cavity input mirror
compared with that at the output mirror. Under this
assumption, backwards scattering quantum jumps need
not be considered. Note that non-Hermitian Hamiltonian
(13) is explicitly time dependent and stochastic, due to
the Monte-Carlo simulation of the atomic beam, and the
normalized conditional state is
|ψREC〉 =
|ψ̄REC〉
〈ψ̄REC|ψ̄REC〉
. (16)
C. Two-Quanta Truncation
Even as a quantum trajectory simulation, a full im-
plementation of our model faces difficulties. The Hilbert
space is enormous if we are to consider a few hundred
two-state atoms, and a smaller collective-state basis is
inappropriate, due to spontaneous emission and the cou-
pling of atoms to the cavity mode at unequal strengths.
If, on the other hand, the coherent excitation is suffi-
ciently weak, the Hilbert space may be truncated at the
two-quanta level. The conditional state is expanded as
|ψREC(t)〉 = |00〉+ α(t)|10〉+
βj(t)|0j〉+ η(t)|20〉+
ζj(t)|1j〉+
j>k=1
ϑjk(t)|0jk〉, (17)
where the state |n0〉 has n = 0, 1, 2 photons inside the
cavity and no atoms excited, |0j〉 has no photon inside
the cavity and the j th atom excited, |1j〉 has one photon
inside the cavity and the j th atom excited, and |0jk〉 is
the two-quanta state with no photons inside the cavity
and the j th and kth atoms excited.
The truncation is carried out at the minimum level per-
mitted in a treatment of two-photon correlations. Since
each expansion coefficient need be calculated to domi-
nant order in E/κ only, the non-Hermitian Hamiltonian
(13) may be simplified as
ĤB(t)/ih̄ = E â† +
g(rj(t))(â
†σ̂j− − âσ̂j+)
− κâ†â− γ
σ̂j+σ̂j−, (18)
dropping the term −E â from the right-hand side. While
this self-consistent approximation is helpful in the ana-
lytical calculations reviewed in Sec. III, we do not bother
with it in the numerical simulations.
Truncation at the two-quanta level may be justified by
expanding the density operator, along with the master
equation, in powers of E/κ [25, 26, 53]. One finds that,
to dominant order, the density operator factorizes as a
pure state, thus motivating the simplification used in all
previous treatments of photon correlations in many-atom
cavity QED [27, 28]. The quantum trajectory formula-
tion provides a clear statement of the physical conditions
under which this approximation holds.
Consider first that there is a fixed number of atoms N
and their locations are also fixed. Under weak excitation,
the jump probabilities (15a) and (15b) are very small,
and quantum jumps are extremely rare. Then, in a time
of order 2(κ+γ/2)−1, the continuous evolution (12) takes
the conditional state to a stationary state, satisfying
ĤB |ψss〉 = 0, (19)
without being interrupted by quantum jumps. In view of
the overall rarity of these jumps, to a good approximation
the density operator is
ρss = |ψss〉〈ψss|, (20)
or, if we recognize now the role of the atomic beam,
the continuous evolution reaches a quasi-stationary state,
with density operator
ρss = |ψqs(t)〉〈ψqs(t)|, (21)
where |ψqs(t)〉 satisfies Eq. (12) (uninterrupted by quan-
tum jumps) and the overbar indicates an average over
the fluctuations of the atomic beam.
This picture of a quasi-stationary pure-state evolution
requires the time between quantum jumps to be much
larger than 2(κ+ γ/2)−1, the time to recover the quasi-
stationary state after a quantum jump has occurred. In
terms of photon scattering rates, we require
Rforwards +Rside ≪ 12 (κ+ γ/2), (22)
where
Rforwards = 2κ〈â†â〉REC, (23a)
Rside = γ
〈σ̂j+σ̂j−〉REC. (23b)
When considering delayed photon coincidences, after a
first forwards-scattered photon is detected, let us say at
time tk, the two-quanta truncation [Eq. (17)] is tem-
porarily reduced by the associated quantum jump to a
one-quanta truncation:
|ψREC(tk)〉 → |ψREC(t+k )〉,
where
|ψREC(t+k )〉 = |00〉+ α(t
k )|10〉+
N(tk)
k )|0j〉, (24)
α(t+k ) =
2η(tk)
|α(tk)|
, βj(t
k ) =
ζ(tk)
|α(tk)|
. (25)
Then the probability for a subsequent photon detection
at tk + τ is
Pforwards = 2κ|α(tk + τ)|2∆t. (26)
Clearly, if this probability is to be computed accurately
(to dominant order) no more quantum jumps of any kind
should occur before the full two-quanta truncation has
been recovered in its quasi-stationary form; in the ex-
periment a forwards-scattered “start” photon should be
followed by a “stop” photon without any other scatter-
ing events in between. We discuss how well this condi-
tion is met by Rempe et al. [29] and Foster et al. [31]
in Sec. V. Its presumed validity is the basis for com-
paring their measurements with formulas derived for the
weak-field limit.
III. DELAYED PHOTON COINCIDENCES FOR
STATIONARY ATOMS
Before we move on to full quantum trajectory simula-
tions, including the Monte-Carlo simulation of the atomic
beam, we review previous calculations of the delayed
photon coincidence rate for forwards scattering with the
atomic motion neglected. Beginning with the original
calculation of Carmichael et al. [27], which assumes a
fixed number of atoms, denoted here by N̄eff , all cou-
pled to the cavity mode at strength gmax, we then relax
the requirement for equal coupling strengths [29]; finally
a Monte-Carlo average over the spatial configuration of
atoms, at fixed density ̺, is taken. The inadequacy of
modeling at this level is shown by comparing the com-
puted correlation functions with the reported data sets.
A. Ideal Collective Coupling
For an ensemble of N̄eff atoms located on the cavity
axis and at antinodes of the standing wave, the non-
Hermitian Hamiltonian (18) is taken over in the form
ĤB/ih̄ = E â† + gmax(â†Ĵ− − âĴ+)
− κâ†â− γ
(Ĵz +Neff), (27)
where
Ĵ± ≡
σ̂j±, Ĵz ≡
σ̂jz (28)
are collective atomic operators, and we have written
2σ̂j+σ̂j− = σ̂jz + 1. The conditional state in the two-
quanta truncation is now written more simply as
|ψREC(t)〉 = |00〉+ α(t)|10〉+ β(t)|01〉+ η(t)|20〉+ ζ(t)|11〉+ ϑ(t)|02〉, (29)
where |nm〉 is the state with n photons in the cavity and
m atoms excited, the m-atom state being a collective
state. Note that, in principle, side-scattering denies the
possibility of using a collective atomic state basis. While
spontaneous emission from a particular atom results in
the transition |n1〉 → σ̂j−|n1〉 → |n0〉, which remains
within the collective atomic basis, the state σ̂j−|n2〉 lies
outside it; thus, side-scattering works to degrade the
atomic coherence induced by the interaction with the cav-
ity mode. Nevertheless, its rate is assumed negligible in
the weak-field limit [Eq. (22)], and therefore a calculation
carried out entirely within the collective atomic basis is
permitted.
The delayed photon coincidence rate obtained from
|ψREC(tk)〉 = |ψss〉 and Eqs. (24) and (26) yields the
second-order correlation function [27, 28, 54]
g(2)(τ) =
1− 2C1
1 + ξ
1 + 2C − 2C1ξ/(1 + ξ)
(κ+γ/2)τ
cos (Ωτ)+
(κ+ γ/2)
sin (Ωτ)
, (30)
with vacuum Rabi frequency
N̄effg2max − 14 (κ− γ/2)2, (31)
where
ξ ≡ 2κ/γ, (32)
C ≡ N̄effC1, C1 ≡ g2max/κγ. (33)
For N̄eff ≫ 1, as in Parameter Sets 1 and 2 (Table I), the
deviation from second-order coherence—i.e., g(2)(τ) =
630-3-6
210-1-2
FIG. 1: Second-order correlation function for ideal coupling
[Eq. (30)]: (a) Parameter Set 1, (b) Parameter Set 2.
1—is set by 2C1ξ/(1 + ξ) and provides a measure of the
single-atom coupling strength. For small time delays the
deviation is in the negative direction, signifying a photon
antibunching effect. It should be emphasized that while
second-order coherence serves as an unambiguous indica-
tor of strong coupling in the single-atom sense, vacuum
Rabi splitting—the frequency Ω—depends on the collec-
tive coupling strength alone.
Both experiments of interest are firmly within the
strong coupling regime, with 2C1ξ/(1+ ξ) = 1.2 for that
of Rempe et al. [29] (2C1 = 4.6), and 2C1ξ/(1+ ξ) = 4.0
for that of Foster et al. [31] (2C1 = 5.6). Figure 1 plots
the correlation function obtained from Eq. (30) for Pa-
rameter Sets 1 and 2. Note that since the expression is
a perfect square, the apparent photon bunching of curve
(b) is, in fact, an extrapolation of the antibunching ef-
fect of curve (a); the continued nonclassicality of the
correlation function is expressed through the first two
side peaks, which, being taller than the central peak, are
classically disallowed [26, 30]. A measurement of the in-
tracavity electric field perturbation following a photon
detection [the square root of Eq. (30)] presents a more
unified picture of the development of the quantum fluctu-
ations with increasing 2C1ξ/(1+ξ). Such a measurement
may be accomplished through conditional homodyne de-
tection [55, 56, 57].
In Fig. 1 the magnitude of the antibunching effect—
the amplitude of the vacuum Rabi oscillation— is larger
than observed in the experiments by approximately an
order of magnitude (see Fig. 3). Significant improvement
is obtained by taking into account the unequal coupling
strengths of atoms randomly distributed throughout the
cavity mode.
B. Fixed Atomic Configuration
Rempe et al. [29] extended the above treatment to the
case of unequal coupling strengths, adopting the non-
Hermitian Hamiltonian (18) while keeping the number
of atoms and the atom locations fixed. For N atoms in
a spatial configuration {rj}, the second-order correlation
function takes the same form as in Eq. (30)—still a per-
fect square—but with a modified amplitude of oscillation
[29, 58]:
(τ) =
[1 + ξ(1 + C{rj})]S{rj} − 2C{rj}
1 + (1 + ξ/2)S{rj}
(κ+γ/2)τ
cos (Ωτ) +
(κ+ γ/2)
sin (Ωτ)
, (34)
C{rj} ≡
C1j , C1j ≡ g2(rj)/κγ, (35)
S{rj} ≡
1 + ξ(1 + C{rj})− 2ξC1j
, (36)
where the vacuum Rabi frequency is given by Eq. (31)
with effective number of interacting atoms
N̄eff → N{rj}eff ≡
g2(rj)/g
max. (37)
C. Monte-Carlo Average and Comparison with
Experimental Results
In reality the number of atoms and their configuration
both fluctuate in time. These fluctuations are readily
taken into account if the typical atomic motion is suf-
ficiently slow; one takes a stationary-atom Monte-Carlo
average over configurations, adopting a finite interaction
volume VF and combining a Poisson average over the
number of atoms N with an average over their uniformly
distributed positions rj , j = 1, . . . , N . In particular, the
effective number of interacting atoms becomes
N̄eff = N
eff , (38)
where the overbar denotes the Monte-Carlo average.
Although it is not justified by the velocities listed in
Table I, a stationary-atom approximation was adopted
when modeling the experimental results in Refs. [29]
and [31]. The correlation function was computed as the
Monte-Carlo average
g(2)(τ) = g
(τ), (39)
with g
(τ) given by Eq. (34). In fact, taking a Monte-
Carlo average over normalized correlation functions in
this way is not, strictly, correct. In practice, first the
delayed photon coincidence rate is measured, as a sepa-
rate average, then subsequently normalized by the aver-
age photon counting rate. The more appropriate averag-
ing procedure is therefore
g(2)(τ) =
〈â†(0)â†(τ)â(τ)â(0)〉{rj}
〈â†â〉{rj}
, (40)
or, in a form revealing more directly the relationship to
Eq. (34), the average is to be weighted by the square of
the photon number:
g(2)(τ) =
〈â†â〉{rj}
〈â†â〉{rj}
, (41)
where
〈â†â〉{rj} =
1 + 2C{rj}
is the intracavity photon number expectation—in sta-
tionary state |ψss〉 [Eq. (19)]—for the configuration of
atoms {rj}.
Note that the statistical independence of forwards-
scattering events that are widely separated in time yields
the limit
(τ) → 1, (43)
which clearly holds for the average (39) as well. Equa-
tion (41), on the other hand, yields
g(2)(τ) →
〈â†â〉{rj}
〈â†â〉{rj}
≥ 1. (44)
A value greater than unity arises because while there are
fluctuations in N and {rj}, their correlation time is in-
finite under the stationary-atom approximation; the ex-
pected decay of the correlation function to unity is there-
fore not observed.
The two averaging schemes are compared in the plots
of Fig. 2, which suggest that atomic beam fluctuations
should have at least a small effect in the experiments;
although, just how important they turn out to be is not
captured at all by the figure. The actual disagreement
between the model and the data is displayed in Fig. 3.
The measured photon antibunching effect is significantly
630-3-6
210-1-2
FIG. 2: Second-order correlation function with Monte-Carlo
average over number of atoms N and configuration {rj}. The
average is taken according to Eq. (39) (thin line) and Eq. (41)
(thick line) for (a) Parameter Set 1, (b) Parameter Set 2.
smaller than predicted in both experiments: smaller by
a factor of 4 in Fig. 3(a), as the authors of Ref. [29]
explicitly state, and by a factor of a little more than 2 in
Fig. 3(b).
The rest of the paper is devoted to a resolution of this
disagreement. It certainly arises from a breakdown of the
stationary-atom approximation as suggested by Rempe
et al. [29]. Physics beyond the addition of a finite corre-
lation time for fluctuations of N(t) and {rj(t)} is needed,
however. We aim to show that the single most important
factor is the alignment of the atomic beam.
52.50-2.5-5
0.90.50-0.5-0.9
•••••
•••••••••••
FIG. 3: Second-order correlation function with Monte-Carlo
average, Eq. (41), over number of atoms N and configuration
{rj} compared with the experimental data from (a) Fig. 4(a)
of Ref. [29] (Parameter Set 1) and (b) Fig. 4 of Ref. [31]
(Parameter Set 2).
IV. DELAYED PHOTON COINCIDENCES FOR
AN ATOMIC BEAM
We return now to the full atomic beam simulation out-
lined in Sec. II. With the beam perpendicular to the
cavity axis, the rate of change of the dipole coupling con-
stants might be characterized by the cavity-mode transit
time, determined from the mean atomic speed and the
cavity-mode waist. Taking the values of these quanti-
ties from Table I, the experiment of Rempe et al. has
w0/v̄source = 182nsec, which should be compared with
a vacuum-Rabi-oscillation decay time 2(κ + γ/2)−1 =
94nsec, while Foster et al. have w0/v̄source = 66nsec and
a decay time 2(κ+ γ/2)−1 = 29nsec. In both cases, the
ratio between the transit time and decay time is ∼ 2;
thus, we might expect the internal state dynamics to fol-
low the atomic beam fluctuations adiabatically, to a good
approximation at least, thus providing a justifying for the
stationary-atom approximation. Figure 3 suggests that
this is not so. Our first task, then, is to see how well in
practice the adiabatic following assertion holds.
A. Monte-Carlo Simulation of the Atomic Beam:
Effect of Beam Misalignment
Atomic beam fluctuations induce fluctuations of the
intracavity photon number expectation, as illustrated
by the examples in Figs. 4 and 5. Consider the two
curves (a) in these figures first, where the atomic beam
is aligned perpendicular to the cavity axis. The ring-
ing at regular intervals along these curves is the tran-
sient response to enforced cavity-mode quantum jumps—
jumps enforced to sample the quantum fluctuations effi-
ciently (see Sec. IVB). Ignoring these perturbations for
the present, we see that with the atomic beam aligned
perpendicular to the cavity axis the fluctuations evolve
more slowly than the vacuum Rabi oscillation—at a simi-
lar rate, in fact, to the vacuum Rabi oscillation decay. As
anticipated, an approximate adiabatic following is plau-
sible.
Consider now the two curves (b); these introduce a
9.6mrad misalignment of the atomic beam, following up
on the comment of Foster et al. [31] that misalignments
as large as 1◦ (17.45mrad) might occur. The changes in
the fluctuations are dramatic. First, their size increases,
though by less on average than it might appear. The
altered distributions of intracavity photon numbers are
shown in Fig. 6. The means are not so greatly changed,
but the variances (measured relative to the square of the
mean) increase by a factor of 2.25 in Fig. 4 and 1.45
in Fig. 5. Notably, the distribution is asymmetric, so
the most probable photon number lies below the mean.
The asymmetry is accentuated by the tilt, especially for
Parameter Set 1 [Fig. 6(a)].
More important than the change in amplitude of the
fluctuations, though, is the increase in their frequency.
Again, the most significant effect occurs for Parameter
Set 1 (Fig. 4), where the frequency with a 9.6mrad tilt
approaches that of the vacuum Rabi oscillation itself;
clearly, there can be no adiabatic following under these
conditions. Indeed, the net result of the changes from
Fig. 4(a) to Fig. 4(b) is that the quantum fluctuations,
initiated in the simulation by quantum jumps, are com-
pletely lost in a background of classical noise generated
by the atomic beam. It is clear that an atomic beam
misalignment of sufficient size will drastically reduce the
photon antibunching effect observed.
For a more quantitative characterization of its effect,
we carried out quantum trajectory simulations in a one-
quantum truncation (without quantum jumps) and com-
puted the semiclassical photon number correlation func-
g(2)sc (τ) =
〈(â†â)(t)〉REC〈(â†â)(t+ τ)〉REC
〈(â†â)(t)〉REC
, (45)
where the overbar denotes a time average (in practice
an average over an ensemble of sampling times tk). The
photon number expectation was calculated in two ways:
740 760 780
FIG. 4: Typical trajectory of the intracavity photon number
expectation for Parameter Set 1: (a) atomic beam aligned
perpendicular to the cavity axis, (b) with a 9.6mrad tilt of the
atomic beam. The driving field strength is E/κ = 2.5× 10−2.
560 590 620
FIG. 5: As in Fig. 4 but for Parameter Set 2.
〈a†a〉REC/〈a
†a〉REC
31.50
FIG. 6: Distribution of intracavity photon number expecta-
tion with the atom beam perpendicular to the cavity axis
(thin line) and a 9.6mrad tilt of the atomic beam (thick line):
(a) Parameter Set 1, (b) Parameter Set 2.
first, by assuming that the conditional state adiabatically
follows the fluctuations of the atomic beam, in which
case, from Eq. (42), we may write
〈(â†â)(t)〉REC =
1 + 2C{rj(t)}
, (46)
and second, without the adiabatic assumption, in which
case the photon number expectation was calculated from
the state vector in the normal way.
Correlation functions computed for different atomic
beam tilts according to this scheme are plotted in Figs. 7
and 8. In each case the curves shown in the left column
assume adiabatic following while those in the right col-
umn do not. The upper-most curves [frames (a) and (e)]
hold for a beam aligned perpendicular to the cavity axis
and those below [frames (b)–(d) and (f)–(h)] show the
effects of increasing misalignment of the atomic beam.
A number of comments are in order. Consider first the
aligned atomic beam. Correlation times read from the
figures are in approximate agreement with the cavity-
mode transit times computed above: the numbers are
191nsec and 167nsec from frames (a) and (e), respec-
tively, of Fig. 7, compared with w0/v̄oven = 182nsec; and
68nsec and 53nsec from frames (a) and (e) of Fig. 8, re-
spectively, compared with w0/v̄oven = 66nsec. The num-
bers show a small decrease in the correlation time when
the adiabatic following assumption is lifted (by 10-20%)
but no dramatic change; and there is a corresponding
small increase in the fluctuation amplitude.
Consider now the effect of an atomic beam tilt. Here
the changes are significant. They are most evident in
frames (d) and (h) of each figure, but clear already in
frames (c) and (g) of Fig. 7, and frames (b) and (f) of
Fig. 8, where the tilts are close to the tilt used to generate
Figs. 4(b) and 5(b) (also to those used for the data fits in
Sec. IVB). There is first an increase in the magnitude of
the fluctuations—the factors 2.25 and 1.45 noted above—
but, more significant, a separation of the decay into two
40-4 40-4
FIG. 7: Semiclassical correlation function for Parameter Set 1,
with adiabatic following of the photon number (left column)
and without adiabatic following (right column); for atomic
beam tilts of (a,e) 0mrad, (b,f) 4mrad, (c,g) 9mrad, (d,h)
13mrad.
pieces: a central component, with short correlation time,
and a much broader component with correlation time
larger than w0/v̄oven. Thus, for a misaligned atomic
beam, the dynamics become notably nonadiabatic.
Our explanation of the nonadiabaticity begins with the
observation that any tilt introduces a velocity compo-
nent along the standing wave, with transit times through
a quarter wavelength of λ/4v̄oven sin θ = 86nsec in the
Rempe et al. [29] experiment and λ/4v̄oven sin θ = 60nsec
in the Foster et al. [31] experiment. Compared with
the transit time w0/v̄oven, these numbers have moved
closer to the decay times of the vacuum Rabi oscillation—
94nsec and 29nsec, respectively. Note that the distances
traveled through the standing wave during the cavity-
mode transit, in time w0/v̄oven, are w0 sin θ = 0.53λ (Pa-
rameter Set 1) and w0 sin θ = 0.28λ (Parameter Set 2).
It is difficult to explain the detailed shape of the correla-
tion function under these conditions. Speaking broadly,
though, fast atoms produce the central component, the
short correlation time associated with nonadiabatic dy-
namics, while slow atoms produce the background com-
ponent with its long correlation time, which follows from
an adiabatic response. Increased tilt brings greater sep-
aration between the responses to fast and slow atoms.
Simple functional fits to the curves in frame (g) of
Fig. 7 and frame (f) of Fig. 8 yield short correlation times
80-8 80-8
FIG. 8: As in Fig. 7 but for Parameter Set 2 and atomic
beam tilts of (a,e) 0mrad, (b,f) 10mrad, (c,g) 17mrad, (d,h)
34mrad.
of 40-50nsec and 20nsec, respectively. Consistent num-
bers are recovered by adding the decay rate of the vac-
uum Rabi oscillation to the inverse travel time through a
quarter wavelength; thus, (1/94+1/86)−1nsec = 45nsec
and (1/29+ 1/60)−1nsec = 20nsec, respectively, in good
agreement with the correlation times deduced from the
figures.
The last and possibly most important thing to note
is the oscillation in frames (g) and (h) of Fig. 7 and
frame (h) of Fig. 8. Its frequency is the vacuum Rabi
frequency, which shows unambiguously that the oscilla-
tion is caused by a nonadiabatic response of the intra-
cavity photon number to the fluctuations of the atomic
beam. For the tilt used in frame (g) of Fig. 7, the tran-
sit time through a quarter wavelength is approximately
equal to the vacuum-Rabi-oscillation decay time, while it
is twice that in frame (f) of Fig. 8. As the tilts used are
close to those giving the best data fits in Sec. IVB, this
would suggest that atomic beam misalignment places the
experiment of Rempe et al. [29] further into the nonadi-
abatic regime than that of Foster et al. [31], though the
tilt is similar in the two cases. The observation is consis-
tent with the greater contamination by classical noise in
Fig. 4(b) than in Fig. 5(b) and with the larger departure
of the Rempe et al. data from the stationary-atom model
in Fig. 3.
B. Simulation Results and Data Fits
The correlation functions in the right-hand column of
Figs. 7 and 8 account for atomic-beam-induced classi-
cal fluctuations of the intracavity photon number. While
some exhibit a vacuum Rabi oscillation, the signals are, of
course, photon bunched; a correlation function like that
of Fig. 7(g) provides evidence of collective strong cou-
pling, but not of strong coupling of the one-atom kind,
for which a photon antibunching effect is needed. We
now carry out full quantum trajectory simulations in a
two-quanta truncation to recover the photon antibunch-
ing effect—i.e., we bring back the quantum jumps.
In the weak-field limit the normalized photon corre-
lation function is independent of the amplitude of the
driving field E [Eqs. (30) and (34)]. The forwards photon
scattering rate itself is proportional to (E/κ)2 [Eq. (42)],
and must be set in the simulations to a value very much
smaller than the inverse vacuum-Rabi-oscillation decay
time [Eq. (22)]. Typical values of the intracavity pho-
ton number were ∼ 10−7 − 10−6. It is impractical, un-
der these conditions, to wait for the natural occurrence
of forwards-scattering quantum jumps. Instead, cavity-
mode quantum jumps are enforced at regular sample
times tk [see Figs. 4(a) and 5(a)]. Denoting the record
with enforced cavity-mode jumps by REC, the second-
order correlation function is then computed as the ratio
of ensemble averages
g(2)(τ) =
〈(â†â)(tk)〉REC〈(â†â)(tk + τ)〉REC
〈(â†â)(tl)〉REC
, (47)
where the sample times in the denominator, tl, are cho-
sen to avoid the intervals—of duration a few correlation
times—immediately after the jump times tk; this ensures
that both ensemble averages are taken in the steady state.
With the cut-off parameter [Eq. (6)] set to F = 0.01, the
number of atoms within the interaction volume typically
fluctuates around N(t) ∼ 400-450 atoms for Parameter
Set 1 and N(t) ∼ 280-320 atoms for Parameter Set 2;
in a two-quanta truncation, the corresponding numbers
of state amplitudes are ∼ 90, 000 (Parameter Set 1) and
∼ 45, 000 (Parameter Set 2).
Figure 9 shows the computed correlation functions for
various atomic beam tilts. We select from a series of such
results the one that fits the measured correlation function
most closely. Optimum tilts are found to be 9.7mrad
for the Rempe et al. [29] experiment and 9.55mrad for
the experiment of Foster et al. [31]. The best fits are
displayed in Fig. 10. In the case of the Foster et al. data
the fit is extremely good. The only obvious disagreement
is that the fitted frequency of the vacuum Rabi oscillation
is possibly a little low. This could be corrected by a small
increase in atomic beam density—the parameter N̄eff—
which is only known approximately from the experiment,
in fact by fitting the formula (31) to the data.
The fit to the data of Rempe et al. [29] is not quite
so good, but still convincing with some qualifications.
Note, in particular, that the tilt used for the fit might be
judged a little too large, since the three central minima
in Fig. 10(a) are almost flat, while the data suggest they
should more closely follow the curve of a damped oscil-
lation. As the thin line in the figure shows, increasing
the tilt raises the central minimum relative to the two
on the side; thus, although a better fit around κτ = 0
is obtained, the overall fit becomes worse. This trend
results from the sharp maximum in the semiclassical cor-
relation function of Fig. 7(g), which becomes more and
more prominent as the atomic beam tilt is increased.
The fit of Fig. 10(b) is extremely good, and, although it
is not perfect, the thick line in Fig. 10(a), with a 9.7mrad
tilt, agrees moderately well with the data once the un-
certainty set by shot noise is included, i.e., adding error
bars of a few percent (see Fig. 13). Thus, leaving aside
possible adjustments due to omitted noise sources, such
as spontaneous emission—to which we return in Sec. V—
and atomic and cavity detunings, the results of this and
the last section provide strong support for the proposal
that the disagreement between theory and experiment
presented in Fig. 3 arises from an atomic beam misalign-
ment of approximately 0.5◦.
One final observation should be made regarding the
fit to the Rempe et al. [29] data. Figure 11 replots
the comparison made in Fig. 10(a) for a larger range
of time delays. Frame (a) plots the result of our sim-
ulation for a perfectly aligned atomic beam, and frames
(b) and (c) shows the results, plotted in Fig. 10(a), cor-
responding to atomic beam tilts of θ = 9.7mrad and
10mrad, respectively. The latter two plots are overlayed
by the experimental data. Aside from the reduced am-
plitude of the vacuum Rabi oscillation, in the presence
of the tilt the correlation function exhibits a broad back-
ground arising from atomic beam fluctuations. Notably,
the background is entirely absent when the atomic beam
52.50-2.5-5
210-1-2
FIG. 9: Second-order correlation function from full quantum
trajectory simulations with a two-quanta truncation: (a) Pa-
rameter Set 1 and θ = 0mrad (thick line), 7mrad (medium
line), 12mrad (thin line); (b) Parameter Set 2 and θ = 0mrad
(thick line), 10mrad (medium line), 17mrad (thin line).
52.50-2.5-5
0.90.50-0.5-0.9
FIG. 10: Best fits to experimental results: (a) data from
Fig. 4(a) of Ref. [29] are fitted with Parameter Set 1 and
θ = 9.7mrad (thick line) and 10mrad (thin line); (b) data
from Fig. 4 of Ref. [31] are fitted with Parameter Set 2 and θ =
9.55mrad. Averages of (a) 200,000 and (b) 50,000 samples
were taken with a cavity-mode cut-off F = 0.01.
is aligned. The experimental data exhibit just such a
background (Fig. 3(a) of Ref. [29]); moreover, an esti-
mate, from Fig. 11, of the background correlation time
yields approximately 400nsec, consistent with the exper-
imental measurement. It is significant that this number
is more than twice the transit time, w0/v̄oven = 182nsec,
and therefore not explained by a perpendicular transit
across the cavity mode. In fact the background mimics
the feature noted for larger tilts in Figs. 7 and 8; as men-
tioned there, it appears to find its origin in the separa-
tion of an adiabatic (slowest atoms) from a nonadiabatic
(fastest atoms) response to the density fluctuations of the
atomic beam.
Note, however, that a correlation time of 400nsec ap-
pears to be consistent with a perpendicular transit across
the cavity when the cavity-mode transit time is defined
as 2w0/v̄oven = 364nsec, or, using the peak rather than
average velocity, as 4w0/
πv̄oven = 411nsec; the latter
definition was used to arrive at the 400nsec quoted in
Ref. [29]. There is, of course, some ambiguity in how a
transit time should be defined. We are assuming that the
time to replace an ensemble of interacting atoms with a
statistically independent one—which ultimately is what
determines the correlation time—is closer to w0/v̄oven
than 2w0/v̄oven. In support of the assumption we recall
that the number obtained in this way agrees with the
semiclassical correlation function for an aligned atomic
beam [Figs. 7 and 8, frame (a)].
C. Mean-Doppler-Shift Compensation
Foster et al. [31], in an attempt to account for the
disagreement of their measurements and the stationary-
atom model, extended the results of Sec. III B to include
52.50-2.5-5
FIG. 11: Second-order correlation function from full quantum
trajectory simulations with a two-quanta basis for Parameter
Set 1 and (a) θ = 0mrad, (b) θ = 9.7mrad, (c) θ = 10mrad.
Averages of (a) 15,000, and (b) and (c) 200,000 samples were
taken with a cavity-mode cut-off F = 0.01.
an atomic detuning. They then fitted the data using
the following procedure: (i) the component of atomic
velocity along the cavity axis is viewed as a Doppler shift
from the stationary-atom resonance, (ii) the mean shift is
assumed to be offset by an adjustment of the driving field
frequency (tuning to moving atoms) at the time the data
are taken, and (iii) an average over residual detunings—
deviations from the mean—is taken in the model, i.e.,
the detuning-dependent generalization of Eq. (34). The
approach yields a reasonable fit to the data (Fig. 6 of
Ref. [31]).
The principal difficulty with this approach is that a
standing-wave cavity presents an atom with two Doppler
shifts, not one. It seems unlikely, then, that adjusting
the driving field frequency to offset one shift and not the
other could compensate for even the average effect of the
atomic beam tilt. This difficulty is absent in a ring cavity,
though, so we first assess the performance of the outlined
prescription in the ring-cavity case.
In a ring cavity, the spatial dependence of the coupling
constant [Eq. (11)] is replaced by
g(rj(t)) =
gmax√
exp(ikzj(t)) exp
x2j (t) + y
j (t)
where the factor
2 ensures that the collective coupling
strength and vacuum Rabi frequency remain the same.
Figure 12(a) shows the result of a numerical implemen-
tation of the proposed mean-Doppler-shift compensation
for an atomic beam tilt of 17.3mrad, as used in Fig. 6 of
Ref. [31]. It works rather well. The compensated curve
(thick line) almost recovers the full photon antibunching
630-3-6
FIG. 12: Doppler-shift compensation for a misaligned atomic
beam in (a) ring and (b) standing-wave cavities (Parameter
Set 2). The second-order correlation function is computed
with the atomic beam perpendicular to the cavity axis (thin
line), a 17.3mrad tilt of the atomic beam (medium line), and
a 17.3mrad tilt plus compensating detuning of the cavity and
stationary atom resonances ∆ω/κ = kv̄oven sin θ/κ = 0.916
(thick line).
effect that would be seen with an aligned atomic beam
(thin line). The degradation that remains is due to the
uncompensated dispersion of velocities (Doppler shifts)
in the atomic beam.
For the case of a standing-wave cavity, on the other
hand, the outcome is entirely different. This is shown by
Fig. 12(b). There, offsetting one of the two Doppler shifts
only makes the degradation of the photon antibunching
effect worse. In fact, we find that any significant detuning
of the driving field from the stationary atom resonance
is highly detrimental to the photon antibunching effect
and inconsistent with the Foster et al. data.
V. INTRACAVITY PHOTON NUMBER
The best fits displayed in Fig. 10 were obtained from
simulations with a two-quanta truncation and premised
upon the measurements being made in the weak-field
limit. The strict requirement of the limit sets a severe
constraint on the intracavity photon number. We con-
sider now whether the requirement is met in the experi-
ments.
Working from Eqs. (23a) and (23b), and the solu-
tion to Eq. (19), a fixed configuration {rj} of N atoms
(Sec. III B) yields photon scattering rates [27, 28, 53]
Rforwards = 2κ〈â†â〉REC = 2κ
1 + 2C{rj}
, (49a)
Rside = γ
〈σ̂k+σ̂k−〉
g(rk)
1 + 2C{rj}
= 2C{rj}2κ〈â
†â〉REC, (49b)
with ratio
Rside
Rforwards
= 2C{rj} =
eff g
∼ 2N̄effg
. (50)
The weak-field limit [Eq. (22)] requires that the greater
of the two rates be much smaller than 1
(κ + γ/2); it is
not necessarily sufficient that the forwards scattering rate
be low. The side scattering (spontaneous emission) rate
is larger than the forwards scattering rate in both of the
experiments being considered—larger by a large factor of
70–80. Thus, from Eqs. (49a) and (50), the constraint on
intracavity photon number may be written as
〈â†â〉 ≪ 1 + γ/2κ
8N̄effg2max/κγ
, (51)
where, from Table I, the right-hand side evaluates as
1.2×10−2 for Parameter Set 1 and 4.7×10−3 for Param-
eter Set 2, while the intracavity photon numbers inferred
from the experimental count rates are 3.8×10−2 [29] and
7.6 × 10−3 [31]. It seems that neither experiment satis-
fies condition (51). As an important final step we should
therefore relax the weak-driving-field assumption (pho-
ton number ∼ 10−7–10−6 in the simulations) and assess
what effect this has on the data fits; can the simulations
fit the inferred intracavity photon numbers as well?
To address this question we extended our simulations
to a three-quanta truncation of the Hilbert space with
cavity-mode cut-off changed from F = 0.01 to F = 0.1.
With the changed cut-off the typical number of atoms
in the interaction volume is halved: N(t) ∼ 180–220
atoms for Parameter Set 1 and N(t) ∼ 150–170 atoms
for Parameter Set 2, from which the numbers of state
amplitudes (including three-quanta states) increase to
1, 300, 000 and 700, 000, respectively. The new cut-off
introduces a small error in N̄eff , hence in the vacuum
Rabi frequency, but the error is no larger than one or
two percent.
At this point an additional approximation must be
made. At the excitation levels of the experiments, even a
three-quanta truncation is not entirely adequate. Clumps
of three or more side-scattering quantum jumps can oc-
cur, and these are inaccurately described in a three-
quanta basis. In an attempt to minimize the error, we ar-
tificially restrict (through a veto) the number of quantum
jumps permitted within some prescribed interval of time.
The accepted number was set at two and the time inter-
val to 1κ−1 for Parameter Set 1 and 3κ−1 for Parameter
52.50-2.5-5
0.90.50-0.5-0.9
FIG. 13: Second-order correlation function from full quan-
tum trajectory simulations with a three-quanta truncation
and atomic beam tilts as in Fig. 10: (a) Parameter Set 1, mean
intracavity photon number 〈a†a〉 = 6.7×10−3; (b) Parameter
Set 2, mean intracavity photon numbers 〈a†a〉 = 2.2 × 10−4,
5.7×10−4, 1.1×10−3, and 1.7×10−3 (thickest curve to thinest
curve). Averages of 20,000 samples were taken with a cavity-
mode cut-off F = 0.1. Shot noise error bars are added to the
data taken from Ref. [29].
Set 2 (the correlation time measured in cavity lifetimes is
longer for Parameter Set 2). With these settings approx-
imately 10% of the side-scattering jumps were neglected
at the highest excitation levels considered.
The results of our three-quanta simulations appear in
Fig. 13; they use the optimal atomic beam tilts of Fig. 10.
Figure 13(a) compares the simulation with the data of
Rempe et al. [29] at an intracavity photon number that
is approximately six times smaller than what we estimate
for the experiment (a more realistic simulation requires a
higher level of truncation and is impossible for us to han-
dle numerically). The overall fit in Fig. 13 is as good as
that in Fig. 10, with a slight improvement in the relative
depths of the three central minima. A small systematic
disagreement does remain, however. We suspect that the
atomic beam tilt used is actually a little large, while the
contribution to the decoherence of the vacuum Rabi os-
cillation from spontaneous emission should be somewhat
more. We are satisfied, nevertheless, that the data of
Rempe et al. [29] are adequately explained by our model.
Results for the experiment of Foster et al [31] lead
in a rather different direction. They are displayed in
Fig. 13(b), where four different intracavity photon num-
bers are considered. The lowest, 〈â†â〉 = 2.2 × 10−4,
reproduces the weak-field result of Fig. 10(b). As the
photon number is increased, the fit becomes progressively
worse. Even at the very low value of 5.7× 10−4 intracav-
ity photons, spontaneous emission raises the correlation
function for zero delay by a noticeable amount. Then we
obtain g(2)(0) > 1 at the largest photon number consid-
ered. Somewhat surprisingly, even this photon number,
〈â†â〉 = 1.7×10−3, is smaller than that estimated for the
experiment—smaller by a factor of five. Our simulations
therefore disagree significantly with the measurements,
despite the near perfect fit of Fig. 10(b). The simplest
resolution would be for the estimated photon number to
be too high. A reduction by more than an order of mag-
nitude is needed, however, implying an unlikely error,
considering the relatively straightforward method of in-
ference from photon counting rates. This anomaly, for
the present, remains unresolved.
VI. CONCLUSIONS
Spatial variation of the dipole coupling strength has
for many years been a particular difficulty for cavity
QED at optical frequencies. The small spatial scale
set by the optical wavelength makes any approach to a
resolution a formidable challenge. There has neverthe-
less been progress made with cooled and trapped atoms
[13, 14, 32, 59, 60, 61], and in semiconductor systems
[15, 16, 17] where the participating ‘atoms’ are fixed.
The earliest demonstrations of strong coupling at opti-
cal frequencies employed standing-wave cavities and ther-
mal atomic beams, where control over spatial degrees of
freedom is limited to the alignment of the atomic beam.
Of particular note are the measurements of photon anti-
bunching in forwards scattering [29, 30, 31]. They pro-
vide a definitive demonstration of strong coupling at the
one-atom level; although many atoms might couple to
the cavity mode at any time, a significant photon anti-
bunching effect occurs only when individual atoms are
strongly coupled.
Spatial effects pose difficulties of a theoretical na-
ture as well. Models that ignore them can point the
direction for experiments, but fail, ultimately, to ac-
count for experimental results. In this paper we have
addressed a long-standing disagreement of this kind—
disagreement between the theory of photon antibunch-
ing in forwards scattering for stationary atoms in a cav-
ity [25, 26, 27, 28, 29] and the aforementioned experi-
ments [29, 30, 31]. Ab initio quantum trajectory sim-
ulations of the experiments have been carried out, in-
cluding a Monte-Carlo simulation of the atomic beam.
Importantly, we allow for a misalignment of the atomic
beam, since this was recognized as a critical issue in
Ref. [31]. We conclude that atomic beam misalignment
is, indeed, the most likely reason for the degradation
of the measured photon antibunching effect from pre-
dicted results. Working first with a two-quanta trunca-
tion, suitable for the weak-field limit, data sets measured
by Rempe et al. [29] and Foster et al. [31] were fitted
best by atomic beam tilts from perpendicular to the cav-
ity axis of 9.7mrad and 9.55mrad, respectively.
Atomic motion is recognized as a source of decorrela-
tion omitted from the model used to fit the measurements
in Ref. [29]. We found that the mechanism is more com-
plex than suggested there, however. An atomic beam
tilt of sufficient size results in a nonadiabatic response of
the intracavity photon number to the inevitable density
fluctuations of the beam. Thus classical noise is writ-
ten onto the forwards-scattered photon flux, obscuring
the antibunched quantum fluctuations. The parameters
of Ref. [29] are particularly unfortunate in this regard,
since the nonadiabatic response excites a bunched vac-
uum Rabi oscillation, which all but cancels out the anti-
bunched oscillation one aims to measure.
Although both of the experiments modeled operate at
relatively low forwards scattering rates, neither is strictly
in the weak-field limit. We have therefore extended our
simulations—subject to some numerical constraints—to
assess the effects of spontaneous emission. The fit to
the Rempe et al. data [29] was slightly improved. We
noted that the optimum fit might plausibly be obtained
by adopting a marginally smaller atomic beam tilt and
allowing for greater decorrelation from spontaneous emis-
sion, though a more efficient numerical method would be
required to verify this possibility. The fit to the Fos-
ter et al. data [31] was highly sensitive to spontaneous
emission. Even for an intracavity photon number five
times smaller than the estimate for the experiment, a
large disagreement with the measurement appeared. No
explanation of the anomaly has been found.
We have shown that cavity QED experiments can call
for elaborate and numerically intensive modeling before
a full understanding, at the quantitative level, is reached.
Using quantum trajectory methods, we have significantly
increased the scope for realistic modeling of cavity QED
with atomic beams. While we have shown that atomic
beam misalignment has significantly degraded the mea-
surements in an important set of experiments in the
field, this observation leads equally to a positive con-
clusion: potentially, nonclassical photon correlations in
cavity QED can be observed at a level at least ten times
higher than so far achieved.
Acknowledgements
This work was supported by the NSF under Grant No.
PHY-0099576 and by the Marsden Fund of the RSNZ.
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