Nondestructive Rydberg Atom Counting with Mesoscopic Fields in a Cavity
P. Maioli,1T. Meunier,1S. Gleyzes,1A. Auffeves,1G. Nogues,1M. Brune,1J.M. Raimond,1and S. Haroche1,2
1Laboratoire Kastler Brossel, De ´partement de Physique de l’Ecole Normale Supe ´rieure,
24 rue Lhomond, F-75231 Paris CEDEX 05, France
2Colle `ge de France, 11 place Marcelin Berthelot, F-75231 Paris CEDEX 05, France
(Received 11 October 2004; published 24 March 2005)
We present an efficient, state-selective, nondemolition atom-counting procedure based on the dispersive
interaction of a sample of circular Rydberg atoms with a mesoscopic field contained in a high-quality
superconducting cavity. The state-dependent atomic index of refraction, proportional to the atom number,
shifts the classical field phase. A homodyne procedure translates the information from the phase to the
intensity. The final field intensity is readout by a mesoscopic atomic sample. This method opens promising
routes for quantum information processing and nonclassical state generation with Rydberg atoms.
DOI: 10.1103/PhysRevLett.94.113601 PACS numbers: 32.80.Rm, 03.67.–a, 42.50.Dv, 42.50.Pq
In a quantum measurement, a microscopic system leaves
its imprint on a meter. When the meter is a mesoscopic
object, it is cast by the interaction into a quantum super-
position state. This leads to particularly interesting situ-
ations on the border between quantum and classical
worlds, where the decoherence processes  can be un-
veiled and studied. In this context, giant circular Rydberg
atoms present many advantages. They are so strongly
coupled to millimeter-wave fields that a single atom may
appreciably modify a mesoscopic field containing tens of
photons stored in a high-quality cavity . In the disper-
sive regime, the state-dependent atomic index of refraction
shifts the field phase. In the resonant case, the Rabi oscil-
lation also leads to an entanglement between the atom and
the field. Many experiments in cavity quantum electro-
dynamics (CQED) have used atomic control of the field
phase: the study of the decoherence of a mesoscopic super-
position of states , the observation of the Rabi oscilla-
tion collapse and revival , and the exploration of the
underlying atom-field entanglement [5,6].
In this Letter, we show that the information exchange
between circular atoms and a mesoscopic field acting as a
meter can be used,in the dispersiveregime, to detect and to
count the atoms in a nondestructive manner. The atomic
sample acts as a piece of transparent dielectrics, with a
state-dependent index of refraction, shifting the field phase
by an amount proportional to the atom number Na. A
displacement in the Fresnel plane transforms this phase
information onto the intensity . Different atom numbers
or different atomic states correspond, after this amplifica-
tion process, to distinguishable photon numbers, read out
by a mesoscopic atomic ensemble.
This experiment mirrors a quantum nondemolition de-
tection of photons , in which a single photon leaves its
imprint on the phase of an atomic state superposition. This
scheme is also in the same league as quantum information
experiments , in which individual quantum systems are
measured via an amplification effect. The ion trap ap-
proach uses a state-selective detection, with nearly 100%
efficiency, based on the ion’s fluorescence in resonant laser
light . Individual neutral atoms in optical traps can be
detected in a similar way [10,11]. Atoms in a high-quality
optical cavity can be counted through the induced cavity
transmission modification [12–14]. Recently, a cavity-shift
effect has also been used to monitor the quantum state of a
single superconducting circuit . All these detection
schemes rely on an amplification process, encoding the
measured state in a field of many photons. Note that, in our
case, this encoding process involves all photons at once,
generating a mesoscopic superposition which, as shown
below, could be used, besides the atomic measurement
presented here, for other interesting purposes.
An efficient nondestructive Rydberg atom counter
opens a realm of possibilities for CQED experiments.
Inserted between atomic sample preparation and use, it
would make it possible to preselect samples with a definite
atom number. Single atoms could be prepared more effi-
ciently than with the present method , which uses
samples with much less than one atom on average at the
expense of long data acquisition times. Efficient single-
atom generation is essential to quantum feedback proce-
dures . Multiatom samples lead to interesting collec-
tive behaviors .
We describe here the main features of the setup,
sketched in Fig. 1(a) (see  for more details). Rubidium
atoms, effusing from oven O, are velocity selected by
Doppler-resolved and time-of-flight techniques. They are
excited in box B, in a pulsed process, to the circular state
jei (jgi), with principal quantum number 51 (50). The atom
number per sample follows a Poisson distribution. Its
average value, Na, can be adjusted from 0 to 15. The
samples cross the superconducting microwave Fabry-
Perot cavity C (field energy damping time Tc? 860 ?s),
tuned close to resonance with the jei ! jgi transition at
51.099 GHz. The cavity contains initially a small residual
thermal field, with an average photon number nt? 1. An
electric field applied across the cavity mirrors tunes the
atoms, via the Stark effect, in or out of resonance with C.
PRL 94, 113601 (2005)
25 MARCH 2005
2005 The American Physical Society
The atom-cavity coupling is determined by the ‘‘vacuum
Rabi oscillation’’ frequency, ?0=2? ? 49 kHz. The atoms
are finally destructively counted by the field-ionization
A source S, coupled to C, realizes a displacement D1
preparing a coherent field j?i (we neglect, for the time
being, the thermal field). Its amplitude ? is assumed to be
real and positive (zero phase), without loss of generality. In
an independent experiment, we calibrate the average pho-
ton number, n ? j?j2, with a 10% accuracy using the light
shift  experienced in C by the transition from jgi to the
circular state jii (principal quantum number 49) at
54.3 GHz. This shift is measured by a Ramsey interfer-
We then send into C an atomic sample As, containing
Naatoms, all prepared in the same state. In the dispersive
regime, ? ? ?0(? being the atom-field detuning), the
atoms remain in their initial state but atoms in jei, at cavity
center, shift the mode frequency by Na?2
Atoms in jgi produce an opposite shift. After Ashas
crossed the cavity, this transient frequency displacement
results in a field phase shift ?Na?, with ? ? ?2
where tiis the effective interaction time, accounting for the
Gaussian mode geometry. The initial field state j?i is thus
transformed, within irrelevant quantum phases, into
j?e?iNa?i. Figure 1(b) presents the three possible fields
after Ashas crossed C, for Na? 0 or 1 atom prepared in
jei or jgi. The information about atomic state and atom
number is encoded in the field phase.
The source S then performs a second displacement, D2,
with an amplitude ??ei’. The phase ’, controlled to
within a degree, can be tuned between 0 and 2?. This
‘‘homodyne’’ procedure transfers the phase imprint left by
Asonto the intensity. The phase ’ is chosen to optimize
the difference between photon numbers corresponding to
different atomic samples. For instance, in the simple case
depicted in Fig. 1(b), an optimal choice is ’ ? ?, leading
to the coherent states represented as gray circles. The final
photon number is then zero for an atom in jei, large when
Ascontains no atom, much larger yet for an atom in jgi.
Reading out the photon number determines the atomic
Before implementing the complete measurement proto-
col, we have measured, in a first experiment, the dispersive
phase shift produced by a single atom. The average atom
number in As(initial state jei, velocity v1? 200:0 ?
0:8 m=s) is slightly less than 1. The atom-cavity detuning,
?=2? ? 50 kHz, does not fulfill the dispersive coupling
condition. Numerical simulations, however, predict that
the atom-cavity system adiabatically follows the smooth
variation of the dressed states during the atom’s transit
across the mode. The atom exits C in its initial state. The
expected phase shift ? is obtained through the simulations
and is shown to decrease slightly with increasing n due to
After interaction with As, C contains a phase shifted
coherent field. Its phase ? (ideally equal to 0, ?, and 2?
for 0, 1, and 2 atoms in As) is determined by sending into
C after D2a single resonant probe atom Ap, initially in jgi,
with a velocity v2? 335:0 ? 1:2 m=s. For ’ ? ?, C is
empty after D2, and Apends up in jgi. When ’ is notice-
ably different from ?, C is left containing a large field,
which saturates the jgi ! jei transition, and Apis detected
in jei with 50% probability. We determine the field phase
distribution  by averaging many realizations of the
experiment and recording the probability Pg?’? for finding
the probe atom Apin jgi as a function of ’. Note that, due
to cavity losses, the field amplitude decays during the time
interval between D1and D2. In the experiment, the am-
plitude setting for D2takes into account this predictable
Figure 2 presents the experimental results. The open
circles (and the dashed Gaussian fit) correspond to the
phase distribution of the initial thermal field displaced by
the amplitude ? ?
centered at ’ ? 0 is 30:5?? 1:7?, in good agreement with
the value 2
into account the initial residual thermal field.
The phase distribution when Ascrosses C is plotted as
solid squares. We now observe three peaks. One, centered
at ’ ? 0, corresponds to the events where no atom is
present in As. The others correspond to one or two de-
phasing atoms. From a Gaussian fit on the single-atom
peak (solid line in Fig. 2), we get ? ? 39:6?? 1?, in
good agreement with the numerical prediction ? ? 39?.
The two-atom peak is centered at ’ ? 78:5?? 2?, close to
2?. Even with the small ? value used here, the index of
refraction is proportional to Na.
The dephasing atoms in Asshould ideally finally be
found still in their initial state, jei. We observe instead a
spurious 15% transfer rate to jgi. We have checked that
. The 1=e width of the narrow peak
? 30:1?expected when taking
FIG. 1 (color online).
tus. (b) Pictorial representation of the cavity field evolution when
a single atom in jei or jgi crosses the cavity. The displacement
D1is represented by the solid straight arrow, the atom-induced
phase shift by the curly arrows, and D2by dashed arrows.
Coherent states are depicted as circles, symbolizing the quantum
fluctuations around the classical amplitude. The open circles
represent the fields after the atom-cavity interaction, the gray
circles the fields after D2.
(a) Scheme of the experimental appara-
PRL 94, 113601 (2005)
25 MARCH 2005
these atoms prepared in jei and finally detected in jgi
produce the same dephasing as the ones finally detected
in jei. We infer that the transfer occurs at the exit of C. It
might be due to nonadiabatic transitions between dressed
levels resulting from stray fields in this region. This trans-
fer does not affect the field phase shift.
The efficiency ? of D, a quantity not easily measured by
other methods, can be deduced from this experiment.
Neglecting the width difference (10%) between the three
peaks, the Pg?’? curve exhibits directly the histogram of
the actual atom number in the dephasing sample. The real
mean atom number Na? 0:68 ? 0:07 can then be deduced
from the ratio of the areas of the ‘‘zero’’ and ‘‘one-atom’’
peaks, centered at ’ ? 0 and ’ ? ?. Comparing Nawith
the average number of atoms detected by D (0:60 ? 0:01),
we get 80 < ? < 100%. The efficiency is higher than
previously reported ones , thanks to an improved design
of the electrostatic lenses routing the electrons from the
ionization condenser to the dynode multipliers.
The cavity field readout technique used above retrieves
only a small part of the information stored in C. Using as a
probe a mesoscopic atomic ensemble, Ap, we can now
count the dephasing atom number in a single shot. The
atoms in As(average number Na? 0:09) are prepared in
state jgi. The cavity field, immediately before D2, now
contains n ? 57:7 photons on average. The atomic veloc-
ity is v3? 154 ? 0:8 m=s, and the atom-cavity detuning is
set at ?=2? ? 50 kHz. The phase shift per atom is then
measured to be ? ? 35:8?, in good agreement with nu-
The phase information is transferred onto the intensity
by the displacement D2, with amplitude ??ei’. The phase
’ ? 14:7?results from an optimization of the detection
efficiency described below. The photon number after D2is
then 3.7, 42, and 108 for Na? 0, 1, and 2 atoms, respec-
tively. We send through C many atoms in level jgi. They
absorbthe cavity field.The total numberofatoms exiting C
in jei thus measures the photon number. Since the number
of atoms per sample is limited by the available laser
excitation power, Apis made up of five samples with a
velocity 335 ? 5:3 m=s, separated by a 120 ?s time inter-
val and containing on average 11.5 atoms each. The first
absorbing sample crosses the cavity 398 ?s after Asand
296 ?s after D2. Because of cavity losses during the
readout operation, the maximum number of absorbed pho-
tons is about 50. Note that absorbing samples are also used
before the beginning of each experimental sequence to
remove leftover fields .
We record the conditional probability distribution
Pe?NjNa? for detecting N atoms in jei in Ap, provided
the field-ionization counter has detected Naatoms in As.
Figure 3(a) presents the histograms corresponding to Na?
0;1, and 2 (inset: field configuration before and after D2).
These histograms are deduced from 400000 realizations of
the experiment. Because of the small average atom number
Na, the statistics for Pe?N;2? are poor (1500 events). This
low Navalue, however, ensures that the cross contamina-
ditional probability distribution Pe?NjNa? for detecting N atoms
in jei in the readout samples, provided the field-ionization coun-
ter has detected Naatoms in the dephasing sample (atoms pre-
pared in jgi), for Na? 0 (open circles), Na? 1 (solid circles),
and Na? 2 (solid diamonds). The detection thresholds N1and
N2are indicated by vertical bars. (b) Histograms of Pe?Nje?
(solid squares), Pe?Nj0? (open circles) and Pe?Njg? (solid
circles). Vertical bars indicate the thresholds Neand Ng. The
insets in (a) and (b) show the cavity fields in phase space before
(open circles) and after (gray circles) the displacement D2with
the same conventions as in Fig. 1(b).
Nondemolition detection of an atomic sample. (a) Con-
prepared in level jei). Probability Pg?’? for finding the probe
atom in jgi versus the displacement phase ’. Open circles,
reference field; solid squares, phase distribution after interaction
with the dephasing atom sample. The error bars reflect statistical
fluctuations. The peaks corresponding to zero, one, and two
atoms are clearly separated. The lines are Gaussian fits.
Cavity field phase distributions (dephasing sample
PRL 94, 113601 (2005)
25 MARCH 2005
tion of the histograms due to the finite efficiency of D is Download full-text
The average number of atoms detected in jei in the
readout sample are 4.3, 13.8, and 25.4 for 0, 1, and 2
dephasing atoms, respectively,lowerthan theinitial photon
number due to cavity relaxation. For atom counting, we set
two thresholds N1? 8 and N2? 21 [indicated by vertical
bars in Fig. 3(a)]. The number Naof atoms in the dephas-
ing sample is then measured to be 0, 1, and 2 when N is
found, in a single run of the experiment, to satisfy N ? N1,
N1< N ? N2, or N2< N, respectively.
Let us estimate the error rates. The probability for fail-
ing to detect a single atom corresponds to the area of the
Pe?Nj1? histogram below N1. Accordingly, the proba-
bility for a ‘‘dark count’’ is the area of Pe?Nj0? above
N1. We thus get the probabilities P?njq? for detecting a q
atoms sample as an n one: P?0j1? ? 18%, P?1j0? ? 7%,
P?1j2? ? 30%, P?2j1? ? 9%. The chosen ’ value mini-
mizes the first two error rates.
Note that the Pe?N;1? histogram is contaminated by a
small contribution having the shapeofPe?N;0?.This is due
to an imperfection in the dephasing sample preparation for
the low atomic velocity v3. About 8% of the atoms are in
high angular momentum ‘‘elliptical’’ states, detected to-
gether with jgi in D. These states are far off resonancewith
C and do not appreciably interact with it. The dispersive
detection scheme presented here can distinguish between
circular and elliptical states, a feature not achieved by
field-ionization detection. Afairevaluation ofthe detection
error rate should not take this sample preparation imper-
fection into account. We then get P?0j1? ? 13%, P?1j0? ?
6%, corresponding to a 87% detection efficiency, and to a
6% dark count rate.
We have performed a similar experiment with a dephas-
ing sample containing either no atom or an atom in jei or
jgi. The phase ’ is now 53:5?. The numbers of photons in
C after D2are 5.5, 46.8, and 114 for jei, no atom, and jgi,
respectively. The corresponding histograms Pe?Njx?,
where x stands for ‘‘e,’’ ‘‘g,’’ or ‘‘0,’’ i.e., ‘‘no atom,’’
are plotted in Fig. 3(b) (the fields are represented in the
inset). We set two thresholds Ne? 8 and Ng? 22, any
detection with Ne< N ? Ngcorresponding to a no atom
event. The crossed error rates are then P?0je? ? 21%,
P?0jg? ? 21%, P?ej0? ? 7%, P?gj0? ? 9%, P?ejg? ?
1:3%, P?gje? ? 1:9%
These imperfections are mainly due to cavity relaxation,
making most of the photons available in C immediately
after D2escape absorption by the readout samples. This is
confirmed by a simple numerical model of the detection,
taking into account all known sources of imperfection,
which is in fair agreement with observation (the average
number of atoms in Apdetected in jei is reproduced to
This limitation could be circumvented by shortening the
experimental sequence and getting rid of the residual ther-
mal field, which broadens the atom number histograms.
Thermal field leaking from room temperature can be com-
bated with improved screening and with cold microwave
absorbers. A single absorbing sample with 100 atoms or
more could cross C shortly after D2and efficiently absorb
most of the information-carrying photons. A new Rydberg
state excitation scheme, under development, has already
provided a significant increase in the atom number, open-
ing the way to an efficient readout with a single sample.
Here, our numerical model shows that the discrimination
between 0, 1, and 2 atoms could be excellent, with error
rates well below a percent.
This method opens new perspectives for experiments
with circular Rydberg atoms and superconducting cavities,
making it possible to operate easily with well-defined atom
numbers. Let us note finally that this scheme ultimately
entangles the internal state of an atom to the excitation
level of a many atoms sample. Provided the whole process
is coherent, it is thus possible to prepare atomic meso-
scopic superposition states, which open fascinating per-
spectives for decoherence tests.
Laboratoire Kastler Brossel is a laboratory of Universite ´
Pierre et Marie Curie and ENS, associated with CNRS
(UMR 8552). We acknowledge support by the European
Community and by the Japan Science and Technology
corporation (International Cooperative Research Project:
 W.H. Zurek, Rev. Mod. Phys. 75, 715 (2003).
 J.-M. Raimond et al., Rev. Mod. Phys. 73, 565 (2001).
 M. Brune et al., Phys. Rev. Lett. 77, 4887 (1996).
 M. Brune et al., Phys. Rev. Lett. 76, 1800 (1996).
 A. Auffeves et al., Phys. Rev. Lett. 91, 230405 (2003).
 T. Meunier et al., Phys. Rev. Lett. 94, 010401 (2005).
 G. Nogues et al., Nature (London) 400, 239 (1999).
 D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics
of Quantum Information (Springer, Berlin, 2000).
 D. Leibfried, R. Blatt, C. Monroe, and D.J. Wineland,
Rev. Mod. Phys. 75, 281 (2003).
 N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier,
Nature (London) 411, 1024 (2001).
 D. Schrader et al., Phys. Rev. Lett. 93, 150501 (2004).
 J. McKeever et al., Phys. Rev. Lett. 93, 143601 (2004).
 P. Maunz et al., Nature (London) 428, 50 (2004).
 Y. Shimizu et al., Phys. Rev. Lett. 89, 233001 (2002).
 A. Wallraff et al., Nature (London) 431, 162 (2004).
 S. Zippilli, D. Vitali, P. Tombesi, and J.-M. Raimond,
Phys. Rev. A 67, 052101 (2003).
 S. Haroche and J.-M. Raimond, in Advances in Atomic and
Molecular Physics, edited by D.R. Bates and B. Bederson
(Academic Press, New York, 1985), Vol. XX, p. 347.
 M. Brune et al., Phys. Rev. Lett. 72, 3339 (1994).
PRL 94, 113601 (2005)
25 MARCH 2005