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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). References 1. 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Science 272, 1131 (1996); Myatt, C.J., King, B.E., Turchette, Q.A., Sackett, C.A., Kielpinski, D., Itano, W.M., Monroe, C.D. and Wineland, D.J., Nature 403, 269 (2000). 41. Lukin, M.D, Fleischhauer, M. and Cote, R. Phys.Rev.Lett. 87, 037901 (2001). 42. Davidovich, L., Zagury, N., Brune, M., Raimond, J.M. and Haroche, S. Phys.Rev.A. 50 R895 (1994). 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