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 atomsinteracting with microwave photons in a superconducting cavity areused to perform quantum information manipulations. We describequantum phase gates using atoms and photons as « qubits », variousengineered entanglement experiments involving up to three particlesat 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 andentanglement. We finally recall previous experiments performed withthe same set-up, in which we had generated and studied quantumsuperpositions of coherent fields with different phases, the so called« Schrödinger cat states » of the field. The decoherence of these stateswas experimentally studied. We conclude by discussing some of theperspectives 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 statesuperpositions and entanglement in order to perform communication and computationoperations which are impossible according to classical logic. Information is coded intosimple quantum objects, usually two-level systems called “quantum bits” or qubits. Quantuminformation procedures involve the manipulation of several qubits at a time, preparing them instate superposition and entangling them together by using quantum gates. Ingenious schemesinvolving 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 : haroche@physique.ens.fr

quantum algorithms of various kinds, including one to factorize large numbers much morerapidly than by classical methods.

The practical implementation of these operations is made very difficult by the phenomenon ofdecoherence which very rapidly destroys the quantum coherences and the entanglement inmany bit systems[2]. While some processes in quantum communication (cryptography[3],teleportation[4]) have been successfully implemented, the development of quantumcomputation requires the simultaneous manipulation of many bits and presents formidablechallenges to theorists – who must develop error correcting codes fighting the effects ofdecoherence[5] - and to experimentalists who must find the best qubit candidates. Thesequbits should be easily prepared and manipulated, and at the same time should remain wellisolated 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]. Inthese experiments, which operate with huge number of systems in a macroscopic ensemble,the existence of entanglement is however questionable[7]. The manipulation of individualqubits and their controlled coupling in quantum gates has been so far achieved only inquantum optics, with two kinds of systems. In ion traps[8], the bits are single ions codinginformation in internal degrees of freedom. The bits are coupled by the combined effect oflaser light and Coulomb interaction between the ions. In Cavity QED experiments, the bits areatoms or single photons stored in a cavity. In optical cavity QED work, photons can becoupled 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 radiationprovides the dynamical coupling for the gate operations. With both ion traps and microwavecavity QED systems, the basic operations of quantum information processing have beendemonstrated and some limitations and difficulties linked to decoherence have beeninvestigated.

Other systems, based on the quantum properties of mesoscopic objects such as Josephsonjunctions[11] or quantum dots[12] are also investigated, with recent very promisingresults[13,14]. Entanglement of qubits remain however to be demonstrated with thesesystems. The development of this field of research thus follows two complementary paths: inquantum 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 becomestrong enough to permit coherent operations and quantum control. The mesoscopic approachseems to be better suited for practical applications, but decoherence is there much less knownthan in quantum optics. The progress in quantum information processing thus clearly requiresa combination of efforts in both fields, quantum optics exploring simple systems and

developing basic tools which mesoscopic physicists should then try to implement in morepractically scalable systems.

The Göteborg Nobel Symposium on quantum coherence has provided a very interestingopportunity to bring together atomic and solid state physicists and to discuss the respectivemerits 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-cavityQED experiments performed at Ecole Normale Supérieure in Paris. These experiments haveallowed us to demonstrate the operation of quantum gates and to engineer entanglementbetween two and three qubits. They have provided also text book demonstrations of thequantum concept of complementarity and have lead to a quantitative study of the phenomenonof decoherence. I give here only the essential results of this work, up to the end of 2001 andbriefly discuss the perspectives they open in the near future. Details about our set up andexperimental 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 twospherical superconducting mirrors facing each other, in a Fabry-Perot configuration. It storesmicrowave 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, atmid - distance between the mirrors. The cavity is resonant or nearly resonant with thetransition between two Rydberg levels e and g (with principal quantum numbers 51 and 50respectively). Each atom behaves essentially as a two-state e-g system. The set - up is cooledto about 1 K to minimize thermal radiation noise. The atoms are detected after they havecrossed the cavity by field ionization and a binary information (atom found in e or g) isacquired.

Some experiments are performed by directly detecting the atomic final energy state andgaining in this way information on the atom -cavity field energy exchange process. In otherexperiments, 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. Thisis performed with the help of two auxiliary classical microwave field pulses sandwiching thecavity (see Figure 1). The successive application of these pulses constitutes a Ramseyinterferometer [26]. Interference fringes are obtained in the probability for finding the atom ina given final state, when the frequency of the Ramsey pulses is tuned across a transitionbetween 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 interactionprocess.

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 theposition of each atom at any time in the apparatus and we can apply different sets ofoperations 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 knownJaynes-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 theupper or in the lower state is a periodic function of the atom-cavity interaction time, withfrequency 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 haveexperimentally reconstructed the Rabi oscillation versus time (Figure 2). We can interrupt thecoherent oscillation on demand, by simply applying an electric field across the cavity mirrorswhich 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 whichsurvives 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 theRabi 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 eor 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 atomsare 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 andstudying their correlations [16]. This massive EPR pair of particles could be used for Bell’sinequality tests. Note that pairs of entangled atoms have also been realized with ions intraps[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 atomsinteracting at successive times with the same field which plays the role of a catalyst forentanglement. 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 toavoid this problem and to entangle in a more robust way the two atoms together, withoutrelying on the transient production of a real photon in the cavity[29]. To achieve this, wemake two atoms “collide” together inside the cavity. The two atoms have slightly differentvelocities and the “second” one overtakes the “first” at cavity center. The cavity is slightlyoff-resonant with the e-g transition, to avoid real photon emission processes. Virtual exchangeof photons between the atom and the cavity do however occur, which again brings the twoatoms 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 enhancedby the presence of the cavity mirrors around the atoms. The collision impact parameter ishuge, the two atoms interacting typically at distances of the order of a millimeter. Byadjusting the atom’s velocities and the atom-cavity detuning, we can fix all the parameters ofthis collision and realize a coherent control of the collision, which is of great interest forquantum 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 onephoton 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 signof the atomic state is unaltered. If we consider that the atom on one hand and the field in thecavity 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 isthe control qubit and the field the target bit). We have studied the operation of this gate andshown that it works in a fully coherent and reversible way [18]. Similar gates have beenoperated 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 itundergoes a 2π 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 settingthe 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 non-

destructive scheme and shown that the same photon can be measured repeatedly by successiveatoms, without being absorbed. This is quite different from ordinary photon countingprocedures, which absorb light quanta. Previous quantum non-demolition procedures basedon non-linear optical processes were restricted to the detection of fields containing largephoton numbers [31].

6. Multi-particle entanglement

By combining quantum Rabi pulses of various duration and auxiliary Ramsey pulses onsuccessive atoms crossing the cavity, one can generate and analyse entangled states involvingmore 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 three-particle entanglement involving the two atoms and the photon field is generated. The fieldstate is finally copied on a third atom, initially in the lower level of the atomic transitionresonant 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-particleentangled state are analysed by performing various measurements on the three atoms. Thesemeasurements involve the application of auxiliary Ramsey pulses after the atoms have

interacted with the cavity field [20]. The procedure can be generalized to situations ofincreasing complexity, with larger numbers of atoms. Entanglement involving more than twoparticles 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 foundingfathers of quantum theory[34]. Interferences, exhibiting the “wave” nature of quantumsystems are observed under some experimental conditions, while they disappear under others,

giving to the systems a “particle” aspect. Fundamentally, the notions of information andentanglement play an essential role here: fringes are observed in an interferometer if nothingcan 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 theappearance of an entanglement between the system and the “which-path” detector and thevisibility of the fringes can be directly related to the degree of this entanglement. The linkbetween 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 aYoung interferometer is at the same time a beam splitter in an interferometer and (provided itis 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 andone looses the interferences. In modern language, the state of the slit gets in this caseentangled with the particle. Following previous suggestions[36, 37], we have performed aversion of this experiment in which the Young apparatus is replaced by a Ramseyinterferometer[21]. The beam splitters are then microwave fields which transform atomicstates into superpositions and interferences fringes can be observed, as discussed above, whenthe 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 physicalchange 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 whichmany photons are continuously recycled (equivalent to “heavy classical slits in Young’sexperiment)[38]. In our Cavity QED experiments, we usually employ two classical pulsessandwiching the cavity to realize the Ramsey pulses. In a modified version, we have realizedone of the pulses by using a coherent field stored in the high Q cavity itself, the other pulsebeing 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 thisfield 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 theorder 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 changeof one photon becomes more and more conspicuous and fringes vanish. In other words, as thefield in C gets more and more entangled to the atom, we loose progressively the fringecontrast. Figure 3b shows the fringe contrast versus the average photon number n and

compares the experimental signal (squares) with the theoretical predictions (solid line). Theagreement is very good.

We could also perform a quantum eraser variant of this experiment. In the case when the fieldin C is the vacuum, the which-path information corresponds to the presence of exactly 1photon in C after the passage of the atom. We can suppress this information by sending asecond 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 aπ/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 factstrictly equivalent to the entanglement experiment between the two atoms describedabove[16].

8. Schrödinger cats and decoherence

Entanglement can also be realized when the atoms and the cavity field are non-resonant. Inthis 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 toa real index of refraction shifting the field’ s frequency in the cavity. The Rydberg atom tocavity coupling is so strong that a single atom, crossing the cavity can displace its frequencyby several KHz, a quite noticeable effect. Moreover, this shift has a sign depending upon theatom’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 simpleentanglement 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, containingon average 3 to 10 photons in the cavity, by coupling it to an external source. We have thenlet this field interact with a single atom in a state superposition. The field frequency shiftresulted in the appearance of two field components with different phases, correlated to the twoatom’s energy states. We have studied in details this situation, strongly reminiscent of thefamous Schrödinger’s cat, suspended coherently between life and death [22]. By sending aftera delay a second probe atom across the cavity, we have monitored the disappearance of thecoherence between the two field components and studied how it occurred faster and faster asthe separation between these components was increased. This experiment provided a

quantitative test of decoherence theories. Similar Schrödinger cat and decoherence studieshave been performed in ion trap systems [40].

9. Perspectives

We have performed other experiments with this set-up, which will not be described here. Letme mention the entanglement of two different field modes in the cavity[23], the preparationand 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 clearsignature 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 suffersindeed some experimental limitations we must overcome to increase the fidelity of each qubitoperation and to be able to manipulate coherently larger numbers of qubits. One limitationcomes from the cavity whose Q-factor should be increased, while keeping its open geometrywhich is very convenient for atom manipulation. We are presently developing new methods toprepare spherical mirrors with a superconducting coating which look promising to achievethis goal. A small thermal field due to field leakage inside the cavity was present in ourrecent experiments. We solved this problem by absorbing this field with the help of a streamof 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 requiredto completely suppress this leakage effect.

Another limitation comes from the atom preparation. In order to ensure that we do not preparemore than one atom per pulse, we rely on a weak Poissonian process, preparing on averagemuch less than one atom. This is not efficient when we perform many atom correlationexperiments. We then need to prepare, by sequences of pulses, several atoms crossing theapparatus one after the other. If each atom is prepared with a probability much smaller thanone, the overall efficiency of preparation becomes very low. Methods to preparedeterministically 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 atomictrap and to take advantage of the strong interaction between Rydberg atoms which shifts theirenergy levels. This would prevent the excitation of more than one atom (so called dipole-dipole blockade effect [41]). An alternative method would be to monitor the flux of atoms inthe atomic beam by fluorescence techniques and to apply an adiabatic excitation process intothe Rydberg state exactly at the time when one atom has been observed. In this way, we hopeto be able to manipulate sequences of more than three atoms in succession, opening the wayto many interesting applications: teleportation of qubits carried by material particles [42],realization of field superpositions involving many photons in distinct cavities (non localSchrödinger cat states) [43], operations of simple quantum algorithms etc…

Acknowledgements

This work is supported by the Commission of the European Community, the Centre Nationalde la Recherche Scientifique (CNRS) and the Japan Science and Technology Corporation(JST- International cooperative project, Quantum Entanglement Project).

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Figure captions:

Figure 1. Scheme of the Rydberg atom-Cavity QED experimental set-up. Atoms from anatomic beam are velocity selected by laser induced atomic pumping before being prepared bya pulsed process into a circular Rydberg state. They cross a Fabry-Perot cavity C withsuperconducting mirrors (see picture of mirrors in inset). They are detected downstream bystate selective field ionization in detector D. Optionally, the atoms can be subjected in R1 andR2 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 initiallyin vacuum. Repeating the experiment for various atom-cavity interaction times, wereconstruct the probability Pe to detect finally the atom in e. The damping of the oscillation isdue to field damping, combined with various experimental imperfections. The points showthree typical times for which the coherent Rabi oscillation can be interrupted by applying anelectric field across the mirrors, resulting in three important Rabi pulses ( π/2, π and 2πpulses respectively). [from ref [14]]

Figure 3. (a) Fringes observed with a Ramsey interferometer in which one of the pulses isclassical and the other produced by a coherent field in the cavity C, for increasing averagephoton 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 areexperimental and the curve corresponds to a theoretical model accounting for theentanglement between the atom and the field in C (including a correction for the finite fringecontrast of the interferometer)(by permission of Nature (reference [20]).

Detector (D)Cavity (C)

R1 R2

Velocit

y selec

tion

(optic

al pumpin

g)

Pulsed circularstate preparation