Generating Photon Number States on Demand via Cavity
Quantum Electrodynamics
Article (Published Version)
http://sro.sussex.ac.uk
Brattke, Simon, Varcoe, Benjamin T H and Walther, Herbert (2001) Generating Photon Number
States on Demand via Cavity Quantum Electrodynamics. Physical Review Letters, 86 (16). pp.
3534-3537. ISSN 0031-9007
This version is available from Sussex Research Online: http://sro.sussex.ac.uk/id/eprint/24465/
This document is made available in accordance with publisher policies and may differ from the
published version or from the version of record. If you wish to cite this item you are advised to
consult the publisher’s version. Please see the URL above for details on accessing the published
version.
Copyright and reuse:
Sussex Research Online is a digital repository of the research output of the University.
Copyright and all moral rights to the version of the paper presented here belong to the individual
author(s) and/or other copyright owners. To the extent reasonable and practicable, the material
made available in SRO has been checked for eligibility before being made available.
Copies of full text items generally can be reproduced, displayed or performed and given to third
parties in any format or medium for personal research or study, educational, or not-for-profit
purposes without prior permission or charge, provided that the authors, title and full bibliographic
details are credited, a hyperlink and/or URL is given for the original metadata page and the
content is not changed in any way.
VOLUME
86, NUMBER 16 PHYSICAL REVIEW LETTERS 16A
PRIL
2001
Generation of Photon Number States on Demand via Cavity Quantum Electrodynamics
Simon Brattke,
1,2
Benjamin T. H. Varcoe,
1
and Herbert Walther
1,2
1
Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
2
Sektion Physik der Universität München, 85748 Garching, Germany
(
Received 18 August 2000
)
Many applications in quantum information or quantum computing require radiation with a fixed num-
ber of photons. This increased the demand for systems able to produce such fields. We discuss the
production of photon fields with a fixed photon number on demand. The first experimental demonstra-
tion of the device is described. This setup is based on a cavity quantum electrodynamics scheme using
the strong coupling between excited atoms and a single-mode cavity field.
DOI: 10.1103/PhysRevLett.86.3534 PACS numbers: 42.50.Dv, 03.67.–a
In recent years there has been increasing interest in
systems capable of generating photon fields containing a
preset number of photons. This has chiefly arisen from
applications for which single photons are a necessary re-
quirement, such as secure quantum communication [1 –3]
and quantum cryptography [4]. Fock states are also use-
ful for generating multiple atom entanglements in systems
such as the micromaser. The generated field and the pump-
ing atoms are in an entangled state, this entanglement can
be transferred by the field to subsequent atoms, leading to
applications such as basic quantum logic gates [5]. In the
current experiment the micromaser employed a cavity with
a Q value of 4 3 10
10
corresponding to a photon lifetime
of 0.3 s which is the largest ever achieved in this type of
experiment and more than 2 orders of magnitude greater
than in related setups [5]. In this cavity, Fock states can be
used to entangle a large number of subsequent atoms. A
source of single photons or, more generally, arbitrary Fock
states is also a useful tool for further fundamental inves-
tigations of the atom-field interaction. It can be used to
obtain the reconstruction of purely quantum states of the
radiation field as represented by the Fock states [6].
Many sources for single photons have been proposed.
These include single-atom fluorescence [7], single-
molecule fluorescence [8], two-photon down-conversion
[9] and Coulomb blockade of electrons [10], state re-
duction [11], and using cavity quantum electrodynamics
[3,12–14]. On the other hand, only one source presented
recently, involving the transfer of atoms between a
magneto-optical trap and dipole trap [15], is, in principle,
able to produce n atoms. However, a reliable and deter-
ministic source of Fock states (or even single photons)
has not yet been demonstrated.
Using the one-atom maser or micromaser we present the
first experimental evidence for the operation of a reliable
and robust source of photon Fock states, which by virtue of
its operation also produces a predefined number of atoms
in a particular state. These atoms are entangled with the
generated field and, as mentioned above, can be further
entangled with subsequent atoms.
A basic requirement for reliably preparing a field in a
pre-set quantum state is the ability to choose the field state
in a controllable manner. Trapping states provide this con-
trol. Under trapping-state conditions a quantum feedback
between the atoms and the field acts to control the cav-
ity photon number. Using trapping states, one is therefore
able to provide photons on demand. This provides the ad-
ditional benefit of eliminating the need to detect the atoms
leaving the cavity, thus making these atoms available as a
source for further experiments. The method we describe
here is, in principle, also applicable to optical cavities [16]
and is therefore of broad use.
Under ideal conditions the micromaser field in a trap-
ping state is a Fock state; however, when the micromaser
is operated in a continuous wave (cw) mode, the field state
is very fragile and highly sensitive to external influences
and experimental parameters [17,18]. However, contrary
to cw operation, under pulsed operation the trapping states
are more stable and more practical, and usable over a much
broader parameter range than for cw operation.
The cw operation of the micromaser has been studied
extensively both theoretically [19] and experimentally. It
has been used to demonstrate quantum phenomena such
as, for example, sub-Poissonian statistics [20], the collapse
and revival of Rabi oscillations [21], and entanglement
between the atoms and cavity field [22].
The micromaser setup used for the experiments has
been described previously [17]. Briefly, a beam of
85
Rb
atoms is excited to the 63P
3兾2
Rydberg level by single-
step laser excitation 共l 苷 297 nm兲. The excited atoms en-
ter a high-Q superconducting microwave cavity housed in
a
3
He-
4
He dilution refrigerator which cools the cavity to
300 mK, corresponding to a thermal photon number n
th
苷
0.03. The cavity is tuned to a 21.456 GHz transition from
the 63P
3兾2
upper state to the 61D
5兾2
lower state of the
maser transition.
The emission probability, P
g
,ofa63P
3兾2
upper level
atom entering the cavity is given by
P
g
苷 sin
2
共
p
n 1 1 gt
int
兲 , (1)
3534 0031-9007兾01兾86(16)兾3534(4)$15.00 © 2001 The American Physical Society
VOLUME
86, NUMBER 16 PHYSICAL REVIEW LETTERS 16A
PRIL
2001
where n is the number of photons in the cavity, g is the
effective atom-field coupling constant 共艐41 kHz兲, and t
int
is the interaction time. We note that P
g
苷 0 when
p
n 1 1 gt
int
苷 kp , (2)
where k is an integer number of Rabi cycles. This is the
trapping-state condition. When it is fulfilled, the emis-
sion probability is zero and the field has reached an up-
per bound, thus preventing atoms from emitting. Trapping
states are denoted by the number of photons n and an
integer number of Rabi cycles k for which the emission
probability is zero, they are labeled 共n, k兲 [18]. It is this
mechanism that controls the emission probability of atoms
entering the cavity when the interaction time is tuned to
a trapping state where the Fock state is produced and sta-
bilized by the trapping condition. For short pulse lengths
the lower-state atom number will be the same as the pho-
ton number. For simplicity we will concentrate here on
the preparation of a one-photon Fock state although the
method can also be used to generate Fock states with higher
photon numbers.
For useful comparisons between experiment and theory,
Monte Carlo simulations [23] are used to calculate the rate
of production of lower-state atoms rather than the produc-
tion of photons in the cavity. As pulse lengths are rather
short 共t
pulse
苷 0.02t
cav
兲, there is little dissipation and the
probability of finding a one-photon state in the cavity fol-
lowing the pulse is very close to the probability of finding
an atom in the lower state.
To demonstrate the principle, Fig. 1 shows a simulation
of a sequence of 20 pulses of the pumping atoms, in which
an average of seven excited atoms per pulse are present.
Two operating conditions are presented comparing condi-
tions outside trapping conditions (gt
int
苷 1.67) with the
共1, 1兲 trapping state 共gt
int
苷 2.2兲. Below the pulse se-
quences, two distributions show the probability of finding
0–5 atoms (and hence photons) per pulse in the cavity.
Under the trapping condition only a single emission event
occurs, producing a single lower-state atom which leaves
a single photon in the cavity. Since the atom-cavity sys-
tem is in the trapping condition, the emission probability is
reduced to zero and the photon number is stabilized. The
variation of the time when an emission event occurs during
the atom pulses in Fig. 1 is due to the Poissonian spacing
of upper-state atoms entering the cavity and the stochastic-
ity of the quantum process. In Fig. 1a the broader photon
number distribution is due to the absence of a feedback
stabilization.
Figures 2(a)–2(c) present three curves obtained from
the computer simulations, which illustrate the behavior of
the maser under pulsed excitation as a function of inter-
action time for the same parameters as Fig. 1. The simu-
lations show the probability of finding the following: no
lower-state atom per pulse, P
共0兲
, related to the n 苷 0 Fock
component; finding exactly one lower-state atom per pulse,
P
共1兲
, related to the n 苷 1 Fock component; a correlation
parameter, P
共.1;1兲
, given by the conditional probability of
012345
Photon number
0
0.2
0.4
0.6
0.8
1
Probability
0 τ
pulse
Time
0
5
10
15
20
Pulse number
012345
Photon number
0
0.2
0.4
0.6
0.8
1
Probability
0 τ
pulse
Time
0
5
10
15
20
Pulse number
(a) (b)
FIG. 1. A simulation of a subset of twenty subsequent pulses
of the excitation laser and the associated probability distribution
for photons or lower-state atom production (filled circles repre-
sent lower-state atoms and open circles represent excited-state
atoms). The start and finish of each pulse is indicated by the
vertical dotted lines marked 0 and t
pulse
, respectively. The two
operating conditions are (a) outside the trapping-state conditions
(gt
int
苷 1.67) with a broad field distribution, and (b) the 共1, 1兲
trapping state 共gt
int
苷 2.2兲 with a near-Fock-state distribution.
Both distributions are sub-Poissonian but they are readily dis-
tinguishable experimentally. The parameters are indicated as
crosses in Fig. 2. The size of the atoms in this figure is ex-
aggerated for clarity. With the real atomic separation there are
0.06 atoms in the cavity on average (i.e., well into the one-
atom regime). The other parameters are t
pulse
苷 0.02t
cav
, n
th
苷
10
24
, and N
a
苷 7 (see also Ref. [23]).
finding a second lower-state atom in a pulse already con-
taining one. The value of P
共.1;1兲
represents the sum of all
Fock components n . 1, reaching a maximum value of
1 when there is no remaining population in the n 苷 1 or
n 苷 0 states and a value of 0 when there is no population
above the n 苷 1 state. This is the reason for its strong sup-
pression at the n 苷 1 trapping state. The correlation pa-
rameter is not defined at the vacuum trapping state, where
the exact one-to-one correspondence between lower-state
atoms and the field is lost. In the vacuum trapping state,
lower-state atoms can be produced only in the presence of
thermal photons or other noise effects [17].
P
共.1;1兲
is insensitive to the absolute values of the atomic
detection efficiency and can therefore be measured unam-
biguously in an experiment. It is therefore treated as the
most stable and useful observable of the interaction. The
probability P
共1兲
can be evaluated by using the formula
P
共1兲
苷 N
g
共1 2 P
共.1;1兲
兲 , (3)
3535
VOLUME
86, NUMBER 16 PHYSICAL REVIEW LETTERS 16A
PRIL
2001
01234567
gt
int
0
0.2
0.4
0.6
0.8
1
P
(>1;1)
0
0.2
0.4
0.6
0.8
1
P
(0)
0
0.2
0.4
0.6
0.8
1
P
(1)
(1,1) (1,2)(0,1) (0,2) (1,3)
(a)
(b)
(c)
FIG. 2. The probability of finding (a) no lower-state atoms
per pulse P
共0兲
, (b) exactly one lower-state atom per pulse P
共1兲
,
and (c) a second lower-state atom, if one has already been de-
tected P
共.1;1兲
. The crosses mark the positions of simulations
in Fig. 1. The parameters are t
pulse
苷 0.02t
cav
, N
a
苷 7 atoms,
and n
th
苷 10
24
. The maximum value of P
共1兲
is 97% for the
共1, 1兲 trapping state.
where N
g
is the lower-state atom probability. As the
probabilities must sum to 1, the three Fock components
n 苷 0, n 苷 1, and n $ 2 can be determined uniquely.
It follows from Fig. 2 that with an interaction time cor-
responding to the 共1, 1兲 trapping state, both one photon in
the cavity and a single atom in the lower state are pro-
duced with nearly 97% probability. Note that at no time
in this process is a detector event required to project the
field; the field evolves to the target photon number state,
when a suitable interaction time has been chosen so that
the trapping condition is fulfilled.
To maintain the 97% probability of emission, a mini-
mum number of atoms is required in each pulse. In fact,
for a given average number of atoms per pulse, there is an
upper bound to the probability of finding a single lower-
state atom per pulse. This is given by
P
max
苷 1 2 e
2P
g
N
a
. (4)
where N
a
is the average number of atoms (of any type) per
pulse and is considered the most important parameter when
comparing different operating conditions, noting that the
atomic beam intensity must be chosen to avoid violation
of the one-atom-at-a-time condition.
The inherent stability of the single-photon –single-atom
source is quite remarkable. Simulations show that stable
operating conditions extend from those considered here to
thermal photon numbers as high as n
th
苷 0.1 or for t
int
fluctuations up to 10%. While both of these values are con-
siderably higher than the current experimental parame-
ters, Fock states can still be prepared with an 80%–90%
fidelity [23]. This is an astounding result as it shows that
Fock state production is much more stable than was previ-
ously suspected and the highly stable low thermal photon
conditions required for cw trapping states [17,18,24] are
not specific requirements. For this reason this source is al-
ready being considered for use in such fundamental appli-
cations as phase diffusion measurements [19] and quantum
state reconstruction [6].
Experimentally, the correlation parameter P
共.1;1兲
is ob-
tained via atom pair correlations [25],
P
共.1;1兲
苷
N
gg
N
gg
1 N
eg
1 N
ge
, (5)
where, for example, N
eg
is the probability of detecting a
pair of atoms containing first an upper-state atom 共e兲 and
then a lower-state atom 共g兲 within a pulse. The number
of three atom events detected is negligible and can be
effectively ignored as a contributing factor. Equation (5)
provides a value both appropriate to the existent correlation
and equal to the total probability of finding more than
one lower-state atom per pulse (and thus more than one
photon in the cavity). Although P
共.1;1兲
is insensitive to the
absolute detector efficiency it does depend on the relative
detector efficiencies and the probability that a given atomic
level is detected in the wrong detector (miscounts).
The present setup of the micromaser was specifically
designed for cw operation. Nevertheless, the current ap-
paratus does permit a comparison between theory and ex-
periment in a relatively small parameter range.
A cavity pump rate of 60 atoms per cavity decay time
(usually called N
ex
) was obtained for a short range of
interaction times around the maximum in the Maxwell-
Boltzmann velocity distribution. This happens to corre-
spond to the interaction time for the 共1, 1兲 trapping state.
A pulse length of t
pulse
苷 0.066t
cav
leading to an aver-
age of 4 atoms per pulse was chosen as a compromise
between considerations of dissipative losses, the effect of
external influences, and a reasonable average atom number
per pulse. Figure 3 shows the results of the comparison of
theory and experiment for an experimental and theoretical
evaluation of P
共.1;1兲
around the 共1, 1兲 trapping state. Also
presented is a theoretical curve representing the parameter
P
共1兲
. The theoretical plot of P
共.1;1兲
is calculated for the
experimental conditions with no fit parameters. Shown are
conditions of no detector miscounts and measured detec-
tor miscounts of 7% in the lower-state detector and 2% in
the excited-state detector. When the miscounts are incor-
porated into the data there is an excellent match between
experiment and theory.
For the current experimental conditions Eq. (4) gives
a maximum probability for emitting into the mode of
P
max
苷 92% and a one-photon Fock state probability of
83.2%. This is in agreement with the lower-state atom-
atom probability N
g
苷 0.90 6 0.1% per pulse, obtained
when absolute detector efficiencies 共艐30%兲, detector mis-
counts, detector dark counts, and the finite lifetime of the
3536
VOLUME
86, NUMBER 16 PHYSICAL REVIEW LETTERS 16A
PRIL
2001
40 50 60 70
Interaction time (µs)
0.0
0.2
0.4
0.6
0.8
Probabilities P
(1)
, P
(>1;1)
P
(>1;1)
, miscounts
P
(>1;1)
, no miscounts
P
(1)
, no miscounts
01≥2
0
0.5
1
FIG. 3. Comparison between theory and experiment. The ex-
perimental data are evaluated using Eq. (5). The theoretical
curves are P
共1兲
and P
共.1;1兲
. P
共.1;1兲
is presented for no detec-
tor miscounts and miscounts of 7% in the lower-state detector
and 2% in the excited-state detector. Inset: a comparison of the
experimental (grey) and theoretical (black) Fock components at
the 共1, 1兲 trapping state for the experimental conditions. The cal-
culation is described in the text. The experimental parameters
were t
cav
苷 300 ms, t
pulse
苷 0.066t
cav
, pulse spacing of 1 s,
n
th
苷 0.03, and N
a
苷 4.
atoms are taken into account. The error stems from uncer-
tainties in these parameters.
Using Eq. (3) the distribution of emission events within
a pulse was extracted from the experimental data at the
position of the trapping state and is shown graphically in
the inset of Fig. 3, along with the theoretical distribution.
Following the evaluation above we get a success rate of
Fock state production of 85%. The length of the pump
pulse permits some dissipative losses, allowing a second
emission event to occur. This accounts for a small pro-
portion of observed two-atom events.
In this paper we have used trapping states to prepare a
one-photon Fock state with a success rate of 85 6 10%,
to be compared with the theoretical value under the
present experimental conditions of 83.2%. By improving
the experimental parameters, we can expect to reach con-
ditions for which nearly 97% of the pulses prepare single-
photon Fock states and a single atom in the lower state.
Englert and Walther [26] showed recently that, by using
only micromaser trapping states, it is possible to create
Greenberger-Horne-Zeilinger (GHZ) states [27] between
the atoms and cavity field. A thermal atomic source would
require postselective measurements while the current ap-
paratus can supply a sequence of single atoms with a very
small error in arrival times to a second cavity, preparing
the GHZ entangled states on demand. As the charac-
teristic time scale of the apparatus is the cavity decay
time, an arrival time error of 0.02t
cav
is effectively a d
function. Cavities with the appropriate properties for this
measurement are already in use in the micromaser, making
such a measurement possible in the near future. Further
possible applications are the realization of quantum logic
gates and, as mentioned above, the reconstruction of the
wave function of a Fock state.
[1] H. Zbinden et al., J. Cryptol. 13, 207 (2000); H.-K. Lo and
H. F. Chau, Science 283
, 2050 (1999).
[2] K. M. Gheri et al., Phys. Rev. A 58
, R2627 (1998); S. J.
van Enk et al., Phys. Rev. Lett. 78
, 4293 (1997); S. J.
van Enk et al., Science 279
, 205 (1998).
[3] J. I. Cirac et al., Phys. Rev. Lett. 78
, 3221 (1997); A. S.
Parkins et al., Phys. Rev. Lett. 71
, 3095 (1993); S. Parkins
and H. J. Kimble, J. Opt. B: Quantum Semiclassical Opt.
1
,
496 (1999).
[4] T. Jennewein et al., Phys. Rev. Lett. 84
, 4729 (2000); D. S.
Naik et al., Phys. Rev. Lett. 84
, 4733 (2000); W. Tittel
et al., Phys. Rev. Lett. 84
, 4737 (2000).
[5] A. Rauschenbeutel et al., Science 288
, 2024 (2000);
A. Rauschenbeutel et al., Phys. Rev. Lett. 83
, 5166 (1999).
[6] See, for example, C. T. Bodendorf et al., Phys. Rev. A
57
, 1371 (1998); M. S. Kim et al., Phys. Rev. A 58, R65
(1998), and references therein.
[7] F. Diedrich and H. Walther, Phys. Rev. Lett. 58
, 203 (1987).
[8] C. Brunel et al., Phys. Rev. Lett. 83
, 2722 (1999).
[9] C. K. Hong and L. Mandel, Phys. Rev. Lett. 56
, 58 (1986).
[10] J. Kim et al., Nature (London) 397
, 500 (1999).
[11] B. T. H. Varcoe et al., Nature (London) 403
, 743 (2000).
[12] G. M. Meyer et al., Europhys. Lett. 37
, 317 (1997).
[13] C. K. Law and J. H. Eberly, Phys. Rev. Lett. 76
, 1055
(1996); C. K. Law and H. J. Kimble, J. Mod. Opt. 44
, 2067
(1997); P. Domokos et al., Eur. Phys. J. D 1
, 1 (1998).
[14] A. Kuhn et al., Appl. Phys. B 69
, 373 (1999); P. W. H.
Pinkse et al., Nature (London) 404
, 365 (2000); J. Ye et al.,
Phys. Rev. Lett. 83
, 4987 (2000); C. J. Hood et al., Science
287
, 1447 (2000).
[15] D. Frese et al., Phys. Rev. Lett. 85
, 3777 (2000).
[16] A. Imamoglu et al., Phys. Rev. Lett. 83
, 4204 (1999);
M. Sherwin et al., Phys. Rev. A 60
, 3508 (1999); K. An
et al., Phys. Rev. Lett. 73
, 3375 (1994).
[17] M. Weidinger et al., Phys. Rev. Lett. 82
, 3795 (1999).
[18] P. Meystre et al., Opt. Lett. 13
, 1078 (1988).
[19] See, for example, M. O. Scully and M. S. Zubairy, Quan-
tum Optics (Cambridge University Press, Cambridge, UK,
1997).
[20] G. Rempe et al., Phys. Rev. Lett. 64
, 2783 (1990).
[21] G. Rempe et al., Phys. Rev. Lett. 58
, 353 (1987).
[22] B. Englert et al., Fortschr. Phys. 46
, 897 (1998).
[23] S. Brattke et al., Opt. Express 8
, 131 (2001).
[24] B. T. H. Varcoe et al., J. Opt. B: Quantum Semiclassical
Opt. 2
, 154 (2000).
[25] The use of atom pair correlations in the evaluation of
entangled atom-field states was reviewed in Ref. [22].
[26] B.-G. Englert and H. Walther, Opt. Commun. 179
, 283
(2000).
[27] D. Greenberger et al., Am. J. Phys. 58
, 1131 (1990).
3537
![FIG. 1. A simulation of a subset of twenty subsequent pulses of the excitation laser and the associated probability distribution for photons or lower-state atom production (filled circles represent lower-state atoms and open circles represent excited-state atoms). The start and finish of each pulse is indicated by the vertical dotted lines marked 0 and tpulse, respectively. The two operating conditions are (a) outside the trapping-state conditions (gtint 1.67) with a broad field distribution, and (b) the 1, 1 trapping state gtint 2.2 with a near-Fock-state distribution. Both distributions are sub-Poissonian but they are readily distinguishable experimentally. The parameters are indicated as crosses in Fig. 2. The size of the atoms in this figure is exaggerated for clarity. With the real atomic separation there are 0.06 atoms in the cavity on average (i.e., well into the oneatom regime). The other parameters are tpulse 0.02tcav, nth 1024, and Na 7 (see also Ref. [23]).](/figures/fig-1-a-simulation-of-a-subset-of-twenty-subsequent-pulses-390maowf.webp)
