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Quantum information with atoms and photons in a cavity:
entanglement, complementarity and decoherence studies
S. Haroche(*)*
Laboratoire Kastler Brossel,
Département de Physique de l’ Ecole Normale Supérieure,
24 rue Lhomond, 75231, Paris, France
and
Collège de France, 11 Place Marcelin-Berthelot, 75005, Paris
Abstract: We review recent experiments in which Rydberg atoms
interacting with microwave photons in a superconducting cavity are
used to perform quantum information manipulations. We describe
quantum phase gates using atoms and photons as « qubits », various
engineered entanglement experiments involving up to three particles
at a time and ideal non-destructive measurements of single photons.
We also analyse an atomic interferometry experiment which
illustrates the link between the notions of complementarity and
entanglement. We finally recall previous experiments performed with
the same set-up, in which we had generated and studied quantum
superpositions of coherent fields with different phases, the so called
« Schrödinger cat states » of the field. The decoherence of these states
was experimentally studied. We conclude by discussing some of the
perspectives opened by this line of research.
PACS numbers: 03.65.Ud ; 03.65 Yz ; 03.67-a.
1. Introduction
The physics of quantum information[1] seeks to utilize the quantum concepts of state
superpositions and entanglement in order to perform communication and computation
operations which are impossible according to classical logic. Information is coded into
simple quantum objects, usually two-level systems called “quantum bits” or qubits. Quantum
information procedures involve the manipulation of several qubits at a time, preparing them in
state superposition and entangling them together by using quantum gates. Ingenious schemes
involving the manipulation of qubits have been proposed by theorists of quantum information:
quantum key distribution in cryptography, teleportation of quantum states, dense coding,
*
e-mail address : [email protected]
quantum algorithms of various kinds, including one to factorize large numbers much more
rapidly than by classical methods.
The practical implementation of these operations is made very difficult by the phenomenon of
decoherence which very rapidly destroys the quantum coherences and the entanglement in
many bit systems[2]. While some processes in quantum communication (cryptography[3],
teleportation[4]) have been successfully implemented, the development of quantum
computation requires the simultaneous manipulation of many bits and presents formidable
challenges to theorists – who must develop error correcting codes fighting the effects of
decoherence[5] - and to experimentalists who must find the best qubit candidates. These
qubits should be easily prepared and manipulated, and at the same time should remain well
isolated from their decoherence inducing environment.
Nuclear spins of organic molecules in liquid phase at room temperature have been used as
qubits in various experiments which have illustrated some simple quantum algorithms[6]. In
these experiments, which operate with huge number of systems in a macroscopic ensemble,
the existence of entanglement is however questionable[7]. The manipulation of individual
qubits and their controlled coupling in quantum gates has been so far achieved only in
quantum optics, with two kinds of systems. In ion traps[8], the bits are single ions coding
information in internal degrees of freedom. The bits are coupled by the combined effect of
laser light and Coulomb interaction between the ions. In Cavity QED experiments, the bits are
atoms or single photons stored in a cavity. In optical cavity QED work, photons can be
coupled together via strong dispersive non-linear interactions induced by single atoms,
leading to conditional dynamics[9]. In microwave cavity QED[10], the bits are Rydberg
atoms or single millimeter wave photons. The strong interaction between atoms and radiation
provides the dynamical coupling for the gate operations. With both ion traps and microwave
cavity QED systems, the basic operations of quantum information processing have been
demonstrated and some limitations and difficulties linked to decoherence have been
investigated.
Other systems, based on the quantum properties of mesoscopic objects such as Josephson
junctions[11] or quantum dots[12] are also investigated, with recent very promising
results[13,14]. Entanglement of qubits remain however to be demonstrated with these
systems. The development of this field of research thus follows two complementary paths: in
quantum optics, one tries to increase progressively the complexity of systems, adding one
atom or one photon at a time while preserving the quantum features. In mesoscopic physics,
one follows the opposite way, miniaturizing systems until the quantum properties become
strong enough to permit coherent operations and quantum control. The mesoscopic approach
seems to be better suited for practical applications, but decoherence is there much less known
than in quantum optics. The progress in quantum information processing thus clearly requires
a combination of efforts in both fields, quantum optics exploring simple systems and
developing basic tools which mesoscopic physicists should then try to implement in more
practically scalable systems.
The Göteborg Nobel Symposium on quantum coherence has provided a very interesting
opportunity to bring together atomic and solid state physicists and to discuss the respective
merits of both their approaches for the implementation of quantum information. In this article,
which summarizes my contribution to the Symposium, I review the Rydberg atom-cavity
QED experiments performed at Ecole Normale Supérieure in Paris. These experiments have
allowed us to demonstrate the operation of quantum gates and to engineer entanglement
between two and three qubits. They have provided also text book demonstrations of the
quantum concept of complementarity and have lead to a quantitative study of the phenomenon
of decoherence. I give here only the essential results of this work, up to the end of 2001 and
briefly discuss the perspectives they open in the near future. Details about our set up and
experimental procedures can be found in a recent paper in Reviews of Modern Physics[9] as
well as in other publications of our group[15-25].
2. The Cavity QED set-up
The core of our set-up (sketched in Figure 1) is an open geometry cavity made of two
spherical superconducting mirrors facing each other, in a Fabry-Perot configuration. It stores
microwave photons (51.1 GHz frequency, 6 mm wavelength) for up to 1 millisecond and let
them interact with velocity selected circular Rydberg atoms crossing the cavity one by one, at
mid - distance between the mirrors. The cavity is resonant or nearly resonant with the
transition between two Rydberg levels e and g (with principal quantum numbers 51 and 50
respectively). Each atom behaves essentially as a two-state e-g system. The set - up is cooled
to about 1 K to minimize thermal radiation noise. The atoms are detected after they have
crossed the cavity by field ionization and a binary information (atom found in e or g) is
acquired.
Some experiments are performed by directly detecting the atomic final energy state and
gaining in this way information on the atom -cavity field energy exchange process. In other
experiments, atomic state superpositions are used. Atoms are sent across the cavity in a
superposition of energy levels and the states are mixed again after the cavity interaction. This
is performed with the help of two auxiliary classical microwave field pulses sandwiching the
cavity (see Figure 1). The successive application of these pulses constitutes a Ramsey
interferometer [26]. Interference fringes are obtained in the probability for finding the atom in
a given final state, when the frequency of the Ramsey pulses is tuned across a transition
between two Rydberg levels. By studying how the fringes phase and amplitude is affected by
the presence of photons in the cavity, one gains information on the atom-photon interaction
process.
The velocity of each atom crossing the apparatus is selected by laser induced optical pumping,
before the atoms are prepared in the initial Rydberg state[7]. In this way, we know the
position of each atom at any time in the apparatus and we can apply different sets of
operations on successive atoms, allowing us to engineer complex multi-atom quantum states.
3. Quantum Rabi oscillation
Consider an experiment performed with a cavity exactly resonant with the e-g transition.
When the atom is inside C, the atom-cavity system evolves according to the well known
Jaynes-Cummings Hamiltonian[ 27] which can be written as:
H = - (hΩ/2) [ a†e > < g  + a g > < e]
†
where Ω is the so called vacuum Rabi frequency, and a , a are the creation and
annihilation operators of one photon in the mode of C. Under the action of this Hamiltonian,
an atom sent in e in an initially empty cavity emits and reabsorbs reversibly a photon in it
and undergoes a quantum Rabi oscillation [15]. The probability for finding the atom in the
upper or in the lower state is a periodic function of the atom-cavity interaction time, with
frequency Ω/2π. In our experiment, this frequency is 50 KHz, a very large value due to the
exceptionally strong coupling of circular Rydberg atoms to microwaves. By sending atoms
one by one across the cavity and varying the atom-field interaction time, we have
experimentally reconstructed the Rabi oscillation versus time (Figure 2). We can interrupt the
coherent oscillation on demand, by simply applying an electric field across the cavity mirrors
which detunes by Stark effect the atomic transition from the cavity frequency . In this way,
we can freeze the Rabi oscillation after a π/2 Rabi pulse, realizing a coherent superposition
 e , 0> +  g , 1> of an atom in level e correlated to a cavity in vacuum and an atom in g
combined with a single photon in the cavity. This is an entangled atom-field state which
survives after the atom has left the cavity. We can also interrupt the coupling after a π Rabi
pulse, which amounts to swapping the atom and the field states. Finally, we can adjust the
Rabi oscillation to perform a 2π pulse, which amounts to a change of sign of the atom-field
state (analogous to a 2π rotation of a spin 1/2 like system). These operations provide us all the
tools required for operating quantum gates. We can indeed consider the two level atom (in e
or g) or the field (in 0 or 1 photon state) as qubits interacting together.
4. Atom-atom entanglement
4.1 Two-atom entanglement via real photon exchange
The quantum Rabi oscillation provides a simple and flexible tool to entangle atoms together.
After a first atom has been entangled to the cavity field by a π/2 pulse, we send a second
atom, initially in g which undergoes a π Rabi pulse, reabsorbing with unit probability the
photon emitted by the first atom. In this way, the cavity ends up in vacuum and the two atoms
are entangled together in a state of the form  e , g > +  g , e >. We have checked this
entanglement by performing various measurements on the final states of the two atoms and
studying their correlations [16]. This massive EPR pair of particles could be used for Bell’s
inequality tests. Note that pairs of entangled atoms have also been realized with ions in
traps[28].
4.2 Two-atom entanglement induced by a controlled collision in the cavity
In the above experiment, the entanglement results from an indirect process, the atoms
interacting at successive times with the same field which plays the role of a catalyst for
entanglement. The quality of the entangled pair depends crucially on the cavity damping time.
Any photon loss between the two atoms will destroy the quantum coherence. It is possible to
avoid this problem and to entangle in a more robust way the two atoms together, without
relying on the transient production of a real photon in the cavity[29]. To achieve this, we
make two atoms “collide” together inside the cavity. The two atoms have slightly different
velocities and the “second” one overtakes the “first” at cavity center. The cavity is slightly
off-resonant with the e-g transition, to avoid real photon emission processes. Virtual exchange
of photons between the atom and the cavity do however occur, which again brings the two
atoms in a linear entangled superposition of the  e , g > +  g , e > type. This process
amounts to a van der Waals collision between two Rydberg atoms, whose effect is enhanced
by the presence of the cavity mirrors around the atoms. The collision impact parameter is
huge, the two atoms interacting typically at distances of the order of a millimeter. By
adjusting the atom’s velocities and the atom-cavity detuning, we can fix all the parameters of
this collision and realize a coherent control of the collision, which is of great interest for
quantum information processing. We have checked the properties of the entangled pair by
performing correlation measurements on the two atoms emerging from the cavity[17].
5. Atom-photon phase gate and quantum non demolition measurement of photons
After a full cycle of quantum Rabi oscillation (2π Rabi pulse), the combined atom-field
system comes back to the initial state, but the phase of its wave function has undergone a π-
phase shift. For example, if the atom is initially in the lower state of the transition with one
photon in the cavity, it emerges in the same state, leaving the photon in the cavity, but with
the sign of its wave function changed. On the contrary, if the cavity is initially empty, the sign
of the atomic state is unaltered. If we consider that the atom on one hand and the field in the
cavity on the other hand are quantum bits carrying binary information, the 2π Rabi pulse
couples them according to the conditional dynamics of a quantum phase gate (the photon is
the control qubit and the field the target bit). We have studied the operation of this gate and
shown that it works in a fully coherent and reversible way [18]. Similar gates have been
operated in ion trap experiments [30].
We have also applied our quantum phase gate to perform a non-destructive measurement of a
single photon in the cavity [19]. The phase change of the atom’s wave function when it
undergoes a 2 π Rabi pulse can be translated into an inversion of the phase of the fringe
pattern of the Ramsey interferometer sandwiching the cavity (see section 2 above). By setting
the interferometer at a fringe extremum, we correlate in this way the photon number (0 or 1)
to the final state of the atom. The atom appears as a “meter” measuring, without destroying it,
a single photon in the cavity. We have demonstrated the operation of this quantum nondestructive scheme and shown that the same photon can be measured repeatedly by successive
atoms, without being absorbed. This is quite different from ordinary photon counting
procedures, which absorb light quanta. Previous quantum non-demolition procedures based
on non-linear optical processes were restricted to the detection of fields containing large
photon numbers [31].
6. Multi-particle entanglement
By combining quantum Rabi pulses of various duration and auxiliary Ramsey pulses on
successive atoms crossing the cavity, one can generate and analyse entangled states involving
more than two particles[20]. For example, by applying a π/2 Rabi pulse on a first atom, one
entangles it to a 0/1 photon field. A second atom then undergoes a 2π Rabi pulse combined to
Ramsey interferometry, in order to measure this field. Before this atom is detected, a threeparticle entanglement involving the two atoms and the photon field is generated. The field
state is finally copied on a third atom, initially in the lower level of the atomic transition
resonant with the cavity mode. This atom undergoes a π Rabi pulse, absorbing the photon and
getting in this way correlated to the first two atoms. The characteristics of this three-particle
entangled state are analysed by performing various measurements on the three atoms. These
measurements involve the application of auxiliary Ramsey pulses after the atoms have
interacted with the cavity field [20]. The procedure can be generalized to situations of
increasing complexity, with larger numbers of atoms. Entanglement involving more than two
particles has also been studied by other quantum optics techniques [32,33].
7. Complementarity and entanglement in a Cavity QED interference experiment
The concept of complementarity has been at the heart of the discussions between the founding
fathers of quantum theory[34]. Interferences, exhibiting the “wave” nature of quantum
systems are observed under some experimental conditions, while they disappear under others,
giving to the systems a “particle” aspect. Fundamentally, the notions of information and
entanglement play an essential role here: fringes are observed in an interferometer if nothing
can reveal the path followed by the system. If an information about this path leaks into a
“which-path” detector, the fringes vanish. The leakage of information is related to the
appearance of an entanglement between the system and the “which-path” detector and the
visibility of the fringes can be directly related to the degree of this entanglement. The link
between fringe visibility and “which-path” information has been discussed in [35].
In a famous thought experiment [33] , Bohr has discussed the situation where the slit in a
Young interferometer is at the same time a beam splitter in an interferometer and (provided it
is light enough), plays also the role of a “which-path” detector. By detecting the recoil of a
very light slit, one could indeed get informed about the particle path through the system and
one looses the interferences. In modern language, the state of the slit gets in this case
entangled with the particle. Following previous suggestions[36, 37], we have performed a
version of this experiment in which the Young apparatus is replaced by a Ramsey
interferometer[21]. The beam splitters are then microwave fields which transform atomic
states into superpositions and interferences fringes can be observed, as discussed above, when
the relative phase between the two atomic states is varied.
The visibility of the fringes in an usual Ramsey experiment relies on the fact that no physical
change in the pulses themselves can inform us about the path followed by the atom’s internal
state. This is clearly the case when these pulses are produced by classical fields in which
many photons are continuously recycled (equivalent to “heavy classical slits in Young’s
experiment)[38]. In our Cavity QED experiments, we usually employ two classical pulses
sandwiching the cavity to realize the Ramsey pulses. In a modified version, we have realized
one of the pulses by using a coherent field stored in the high Q cavity itself, the other pulse
being a classical pulse applied afterwards. The cavity field could be either very weak,
containing a very small photon number (vacuum field in the limiting case), or “strong”
containing a relatively large photon number. In all cases, we set the interaction time with this
field to perform a π/2 pulse. The corresponding Ramsey fringes are shown in Figure 3a. When
the field in C is “strong” we see fringes. They vanish when we decrease the field intensity.
The explanation is simple. When the field is strong, the photon number fluctuation, of the
order of √N, is too large to allow us to be able to detect the photon number change
∆N = 1 produced by the atom undergoing the e → g transition. The path in the Ramsey
interferometer is unknown and fringes are visible. When the field becomes small, the change
of one photon becomes more and more conspicuous and fringes vanish. In other words, as the
field in C gets more and more entangled to the atom, we loose progressively the fringe
contrast. Figure 3b shows the fringe contrast versus the average photon number n and
compares the experimental signal (squares) with the theoretical predictions (solid line). The
agreement is very good.
We could also perform a quantum eraser variant of this experiment. In the case when the field
in C is the vacuum, the which-path information corresponds to the presence of exactly 1
photon in C after the passage of the atom. We can suppress this information by sending a
second atom across C to absorb this photon (π Rabi pulse). The path information is then
encoded into this atom and must be erased by mixing the two level of this second atom with a
π/2 classical pulse after the cavity. Fringes are then restored in a signal obtained by
correlating the results of the state measurements on the two atoms. This experiment is in fact
strictly equivalent to the entanglement experiment between the two atoms described
above[16].
8. Schrödinger cats and decoherence
Entanglement can also be realized when the atoms and the cavity field are non-resonant. In
this case, matter and radiation cannot exchange energy, but dispersive light-shift type effects
[39] can be used to entangle the two systems. A non-resonant atomic medium corresponds to
a real index of refraction shifting the field’ s frequency in the cavity. The Rydberg atom to
cavity coupling is so strong that a single atom, crossing the cavity can displace its frequency
by several KHz, a quite noticeable effect. Moreover, this shift has a sign depending upon the
atom’s energy state. If the atom is sent across the cavity in a superposition of states, the cavity
ends up with two different frequencies at the same time. This leads to very simple
entanglement situations. These experiments have been performed about five years ago[22]
and we only recall here their main result. We have prepared a small coherent field, containing
on average 3 to 10 photons in the cavity, by coupling it to an external source. We have then
let this field interact with a single atom in a state superposition. The field frequency shift
resulted in the appearance of two field components with different phases, correlated to the two
atom’s energy states. We have studied in details this situation, strongly reminiscent of the
famous Schrödinger’s cat, suspended coherently between life and death [22]. By sending after
a delay a second probe atom across the cavity, we have monitored the disappearance of the
coherence between the two field components and studied how it occurred faster and faster as
the separation between these components was increased. This experiment provided a
quantitative test of decoherence theories. Similar Schrödinger cat and decoherence studies
have been performed in ion trap systems [40].
9. Perspectives
We have performed other experiments with this set-up, which will not be described here. Let
me mention the entanglement of two different field modes in the cavity[23], the preparation
and detection of two-photon Fock states[24] and the measurement at the origin of phase space
of the Wigner function of a one photon state[25], exhibiting a negative value which is a clear
signature of the quantum nature of this field. Many other interesting experiments are possible.
Before performing them, we need however to improve various aspects of our set-up. It suffers
indeed some experimental limitations we must overcome to increase the fidelity of each qubit
operation and to be able to manipulate coherently larger numbers of qubits. One limitation
comes from the cavity whose Q-factor should be increased, while keeping its open geometry
which is very convenient for atom manipulation. We are presently developing new methods to
prepare spherical mirrors with a superconducting coating which look promising to achieve
this goal. A small thermal field due to field leakage inside the cavity was present in our
recent experiments. We solved this problem by absorbing this field with the help of a stream
of atoms crossing the cavity prior to each experimental sequence. This procedure was rather
cumbersome to implement and worked only partially. We have now taken the steps required
to completely suppress this leakage effect.
Another limitation comes from the atom preparation. In order to ensure that we do not prepare
more than one atom per pulse, we rely on a weak Poissonian process, preparing on average
much less than one atom. This is not efficient when we perform many atom correlation
experiments. We then need to prepare, by sequences of pulses, several atoms crossing the
apparatus one after the other. If each atom is prepared with a probability much smaller than
one, the overall efficiency of preparation becomes very low. Methods to prepare
deterministically one and exactly one atom per pulse are being investigated. A promising one
would consist in preparing a small sample of ground state atoms in a tightly confined atomic
trap and to take advantage of the strong interaction between Rydberg atoms which shifts their
energy levels. This would prevent the excitation of more than one atom (so called dipoledipole blockade effect [41]). An alternative method would be to monitor the flux of atoms in
the atomic beam by fluorescence techniques and to apply an adiabatic excitation process into
the Rydberg state exactly at the time when one atom has been observed. In this way, we hope
to be able to manipulate sequences of more than three atoms in succession, opening the way
to many interesting applications: teleportation of qubits carried by material particles [42],
realization of field superpositions involving many photons in distinct cavities (non local
Schrödinger cat states) [43], operations of simple quantum algorithms etc…
Acknowledgements
This work is supported by the Commission of the European Community, the Centre National
de la Recherche Scientifique (CNRS) and the Japan Science and Technology Corporation
(JST- International cooperative project, Quantum Entanglement Project).
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43. Davidovich, L., Maali, A.,
Phys.Rev.Lett. 71, 2360 (1993).
Brune, M.,
Raimond, J.M. and Haroche, S.
Figure captions:
Figure 1. Scheme of the Rydberg atom-Cavity QED experimental set-up. Atoms from an
atomic beam are velocity selected by laser induced atomic pumping before being prepared by
a pulsed process into a circular Rydberg state. They cross a Fabry-Perot cavity C with
superconducting mirrors (see picture of mirrors in inset). They are detected downstream by
state selective field ionization in detector D. Optionally, the atoms can be subjected in R1 and
R2 before and after they cross C to classical radiation performing π /2 pulses (Ramsey
interferometer).
Figure 2. Quantum Rabi oscillation signal: an atom in level e is sent across the cavity initially
in vacuum. Repeating the experiment for various atom-cavity interaction times, we
reconstruct the probability Pe to detect finally the atom in e. The damping of the oscillation is
due to field damping, combined with various experimental imperfections. The points show
three typical times for which the coherent Rabi oscillation can be interrupted by applying an
electric field across the mirrors, resulting in three important Rabi pulses ( π /2, π and 2π
pulses respectively). [from ref [14]]
Figure 3. (a) Fringes observed with a Ramsey interferometer in which one of the pulses is
classical and the other produced by a coherent field in the cavity C, for increasing average
photon number N in this field. We see that fringe visibility increases with photon number.
(b) Fringe contrast plotted as a function of the average photon number in C. The points are
experimental and the curve corresponds to a theoretical model accounting for the
entanglement between the atom and the field in C (including a correction for the finite fringe
contrast of the interferometer)(by permission of Nature (reference [20]).
V
(o elo
pt ci
ic ty
al se
pu le
m cti
pi on
ng
)
Detector (D)
Cavity (C)
R1
Pulsed circular
state preparation
R2
e
1 .0
2
0 .8
0 .6
p
p /2
P
0 .4
0 .2
0 .0
0
p
2 0
4 0
ti (m s )
6 0
8 0
1 0 0
P
=
g
0 .0
0 .2
0 .4
0 .6
0 .8
0 .2
0 .4
0 .6
0 .8
0 .2
0 .4
0 .6
0 .8
0 .2
0 .4
0 .6
0 .8
1 .0
-1
0
1
B / F 2
N = 1 2 .8
N = 2
N = 0 .3 1
N = 0
F r in g e s c o n tr a s t
0 .0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0
2
4
6
N
8
1 0
1 2
1 4
>
1 6