Collective Atomic Recoil in Ultracold Atoms: Advances and Applications

Transcription

1 Università degli Studi di Milano Facoltà di Scienze Matematiche, Fisiche e Naturali Dottorato di Ricerca in Fisica, Astrofisica e Fisica Applicata Collective Atomic Recoil in Ultracold Atoms: Advances and Applications Coordinatore Prof. Rodolfo Bonifacio Tutore Prof. Rodolfo Bonifacio Tesi di Dottorato di Mary Manuela Cola Ciclo XVI Anno Accademico

2 November 14, 2003 c M M Cola 2003

3 In memory of Prof. Giuliano Preparata

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6 If you do boast, consider this: you do not support the root, but the root supports you. (Rm. 11,18) The figure upwards shows evidence for a Bose Einstein condensation of sodium atoms. It is taken from PRL 75, 3969 (1995). The figure on the left shows evidence for a collective atomic recoil in a BEC. It is a courtesy of M. Inguscio and coworkers.

7 Acknowledgments First of all I wish to thank my tutor, Rodolfo Bonifacio, who gave me the possibility to approach the interesting physics of collective phenomena. Then I should sincerely thank Nicola Piovella, for his support and teachings during these years, especially for what concerned the physics of CARL. I also learned a lot from Matteo G.A. Paris: a great acknowledgment for his careful aid. He introduced me to the physics of quantum optics and quantum information. This let me to investigate fruitful topics in atom optics. A sincere thank to my Referee Francesco S. Cataliotti for his punctual and concerned correction of this thesis, and to Chiara Fort, Leonardo Fallani and Massimo Inguscio for all the hours of collaboration we spent together. I want to remember also other physicists with whom I often had stimulating discussions about frontiers of science, Emilio Del Giudice, Enrico Giannetto and Marco Giliberti. Thanks to Stefano Olivares, Andrea R. Rossi, Alessandro Ferraro and Gabriele Marchi. They keep cheerful the atmosphere of our group. A special thought to Carlo, for his patience and his encouragement, and to my friends Anna, Elisa and Federica for sharing the everyday difficulties. Finally I remember Giuliano Preparata: in a difficult moment of my life his passion for physics reminded me my passion for physics.

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9 Contents Introduction iv 1 Classical CARL Equations of motion The FEL limit The undamped case The damping case: adiabatic limit Linear stability analysis Experimental realizations Concluding remarks Quantum CARL First quantization Linear regime Concluding remarks Quantum field theory The CARL-BEC model Coupled-modes equations Linearized three-mode model Concluding remarks Superradiant Rayleigh scattering and matter waves amplification Directional matter waves produced by spontaneous scattering Dicke superradiance and emerging coherence Evidence for decoherence i

10 ii Contents 4.4 Seeding the superradiance Concluding remarks Superradiant Rayleigh scattering from a moving BEC Theoretical analysis Experimental features Seeding the superradiance Concluding remarks Entanglement generation The Hamiltonian model Spontaneous emission and small-gain regime Solution of the linear quantum regime Three mode entanglement High-gain regime The quasi-classical recoil limit ρ The quantum recoil limit ρ Atom-atom and atom-photon entanglement Concluding remarks Radiation to atom quantum mapping The entangled state The Bell measurement The displacement operation The readout system Concluding remarks Effects of decoherence and losses on entanglement generation Dissipative Master Equation Solution of the Fokker-Plank equation Evolution from vacuum and expectation values Numerical analysis for the relevant working regimes Concluding remarks Conclusion and Outlook 111

11 Contents iii A General solution of the linear model 115 B Wigner functions 119 C Homodyne and multiport homodyne detection 123 C.1 Matrix notation C.2 Balanced homodyne detection C.3 Double homodyne detection D Continuous variable teleportation as conditional measurement 129 D.1 Conditional quantum state engineering D.2 Joint measurement of two-mode quadratures D.3 Fidelity E Solution of the Fokker-Planck equation 137

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13 Introduction At the basis of most phenomena in atomic, molecular, and optical physics is the dynamical interaction between optical and atomic fields. In many ways, recently developed Bose Einstein Condensates (BECs) of trapped alkali atomic vapors [1, 2] are the atomic analog of the optical laser. In fact, with the addition of an output coupler, they are frequently referred to as atom lasers [3]. Despite many interesting and important differences, the main similarity is that both optical lasers and atomic BECs involve large numbers of identical bosons occupying a single quantum state. As a result, the physics of lasers and BECs involves stimulated processes which, due to Bose enhancement, often completely dominate the spontaneous processes which play central roles in the non-degenerate regime. Just like the discovery of the laser led to the development of nonlinear optics, so too the advent of BEC has led to remarkable experimental successes in the field of nonlinear atom optics [4, 5, 6, 7, 8, 9]. The regimes of nonlinear optics and nonlinear atom optics, therefore, represent limiting cases, where either the atomic or optical field is not dynamically independent because it follows the other field in some adiabatic manner which allows for its effective elimination. Outside of these two regimes the atomic and optical fields are dynamically independent, and neither field can be eliminated. The dynamics of coupled quantum degenerate atomic and optical fields in this intermediate regime is the topic of quantum atom optics, namely that extension of atom optics where the quantum state of a many-particle matter-wave field is being controlled, characterized and used in novel applications. Some advances and applications in this field will be the object of this thesis. One of the more relevant system in quantum atomic optics is composed of a BEC driven by a far off-resonant pump laser and coupled to a single mode of an optical 1

14 2 Introduction ring cavity. The mechanism that lies below this kind of physics is the so-called Collective Atomic Recoil Lasing (CARL) in his fully quantized version. The CARL mechanism was originally proposed as a new mechanism for the generation of coherent light [10, 11, 12]. It consist of three main ingredients: (1) a gas of two-level atoms (the active medium) (2) a strong pump laser, which drives the two-level atomic transition, and (3) a ring cavity which supports an electromagnetic mode (the probe) counterpropagating with respect to the pump. Under suitable conditions, the operation of the CARL results in the generation of a coherent light field due to the following mechanism. First, a weak probe field is initiated by noise, either optical in the form of spontaneously emitted light, or atomic in the form of density fluctuations in the atomic gas which backscatters the pump. Once initiated, the probe combines with the pump field to form a weak standing wave which acts as a periodic optical potential. The center-of-mass motion of the atoms on this potential results in a bunching, i.e. a modulation of their density, very similar to the combined effects of the wiggler and the light field leads to electron bunching in a Free Electron Laser (FEL) [13]. This bunching process is then seen by the pump laser as the appearance of a polarization grating in the active medium, which results in stimulated backscattering into the probe field. The resulting gain in the probe strength further amplifies the magnitude of the standing wave field, generating more bunching followed by an increase in stimulated backscattering, and so on. This positive feedback mechanism give rise to an exponential growth of both the probe intensity and the atomic bunching which leads to the perhaps surprising result that the presence of the ring cavity turns the ordinarily stable system of an atomic gas driven by a strong pump laser into an unstable system. CARL effect was verified experimentally by Bigelow et al. in a hot atomic cell [14]. Related experiments by Courtois et al. [15], using cold cesium atoms, and by Lippi et al. [16], using hot sodium atoms, measured the recoil induced smallsignal probe gain, which was interpreted in terms of coherent scattering from an induced polarization grating. However, these experiments missed a probe feedback mechanism, which is necessary to see the long time scale instability which characterizes the CARL. The first unambiguous experimental proof of the CARL effect has been obtained only very recently [17] in a system of cold atoms in a collision-less environment.

15 Introduction 3 In chapter 1 we review the essential conceptual framework of the original CARL showing that self-bunching via an exponential instability can occur under very general conditions. The original CARL theory considers the atoms as classical point particles moving in the optical potential generated by the light fields. From an atom optics point of view, this correspond to a ray atom optics treatments of the atomic field, in analogy with the ordinary ray optics treatment of electromagnetic fields. This description is valid provided that the characteristic wavelength of the matter-wave field remains much smaller than the characteristic length scale of any atom-optical element in the system. Such length, for the atomic field, is its De Broglie wavelength, determined by the atomic mass and the temperature T of the gas. The central atom-optical element of the CARL is the periodic optical potential, which acts as a diffraction grating for the atoms, and has the characteristic length scale of half the optical wavelength. Hence the classical description is valid provided that the temperature is high enough so that the thermal De Broglie wavelength is much smaller than the optical wavelength. However, the spectacular recent advances in atomic cooling techniques makes it likely that CARL experiments using ultracold atomic samples can and will be performed in the next future. In particular, subrecoil temperatures can now be achieved almost routinely. So CARL theory has been extended to this wave atom optics regime [18]. In this regime matter-wave diffraction plays a dominant role in the CARL dynamics. The main drawback of the semiclassical model is that, as it considers the center-of-mass motion of the atoms as classical, it cannot describe the discreteness of the recoil velocity, as has been observed in the experiment of Ref.[19] for an atomic sample below the recoil temperature. So, to extend the model in the region of ultracold atoms, a quantum mechanical description of the center-of-mass motion of the atoms should be included. In chapter 2 we present a way to work out this program simply performing a first quantization of the external variables of the atoms, position and momentum. This simple model gives a description of all the features of the considered system and in particular allows to define the main different regimes. In the conservative regime (no radiation losses), the quantum model depends on a single collective parameter, ρ, that can be interpreted as the average number of photons scattered per atom in the classical limit. When ρ 1, the semiclassical CARL regime is recovered, with many momentum levels populated

16 4 Introduction at saturation. On the contrary, when ρ 1, the average momentum oscillates between zero and q, where q is the difference between the incident and the scattered wave vectors, and a periodic train of 2π hyperbolic secant pulses is emitted. In the dissipative regime (large radiation losses) and in a suitable quantum limit ρ 1, a sequential superradiant scattering occurs, in which after each process atoms emit a π hyperbolic secant pulse and populate a lower momentum state. These results describe the regular arrangement of the momentum pattern observed in experiments of superradiant Rayleigh scattering from a BEC [20]. In chapter 3 we derive a quantum field theory model of a gas of bosonic two-level atoms which interact with a strong, classical, undepleted pump laser and a weak, quantized optical ring cavity mode, both of which are as usual assumed to be tuned far away from atomic resonances. Starting from the second-quantized hamiltonian of the system, we will write an effective model for the time evolution of the ground state atomic field operator and for the probe field operator (the CARL-BEC model), adiabatically eliminating the excited state atomic field operator and including effects of atom-atom collisions. In chapter 4 we review the experimental situations, such as superradiant Rayleigh scattering and matter waves amplification, that can be interpreted with the full quantistic version of CARL model in the dissipative regime, where the radiation emission is superradiant.[21] In chapter 5 we analyze some experiment performed at European Laboratory for Non-linear Spectroscopy (LENS) in Florence about superradiant Rayleigh scattering from a moving BEC. This allows to investigate the influence of the initial velocity of the condensate on superradiant Rayleigh scattering. The experiment gives evidence of a damping of the matter-wave grating which depends on the initial velocity of the condensate. We describe this damping in terms of a phase-diffusion decoherence process, in good agreement with the experimental results. Moreover we analyze the effect of seeding superradiance by a weak signal directed in the opposite direction with respect to the pump laser. One important consideration is to determine to which extent the quantum state of a many-particle atomic field like a BEC can be optically manipulated. In the single-particle case, the answer to this problem is known to a large extent. This is the domain of atom optics [22], where a number of optical elements for matter waves

17 Introduction 5 have now been developed, including gratings, mirrors, interferometers, resonators, etc. But these optical elements manipulate just the atomic field density, or at most first-order coherence properties. However, Schrödinger fields possess a wealth of further properties past their first-order coherence, including atom statistics, density correlation functions. In chapter 6 we investigate the properties, such as quantum fluctuations and entanglement, of the quantum system BEC-radiation in the linear regime for a good cavity regime. We obtain new analytical results, calculating explicitly the statistical properties for atoms and photons and evaluating the state of the coupled BEC-light system evolved from vacuum. In the limit of undepleted atomic ground state and unsaturated probe field, the quantum CARL Hamiltonian reduces to that for three coupled modes. By calculating the exact evolution of the state from the vacuum of the three modes we demonstrate that the evolved state is a fully inseparable three mode Gaussian one. Moreover we show how this three mode Gaussian state can provide a valuable source of atom-atom and atom-photon entanglement. Entanglement is a crucial resource in the manipulation of quantum information, and quantum teleportation [23, 24, 25, 26, 27] is perhaps the most impressive example of quantum protocol based on entanglement. It realizes the transferral of (quantum) information between two distant parties that share entanglement. There is no physical move of the system from one player to the other, and indeed the two parties need not even know each other s locations. Only classical information is actually exchanged between the parties. However, due to entanglement, the quantum state of the system at the transmitter location (say Alice) is mapped onto a different physical system at the receiver location (say Bob). In chapter 7 we show a scheme to realize radiation to atom continuous variable quantum mapping, i.e. to teleport the quantum state of a single mode radiation field onto the collective state of atoms with a given momentum out of a BEC. The atoms-radiation entanglement needed for the teleportation protocol is established through the CARL three linear model studied in chapter 6, whereas the coherent atomic displacement is obtained by the same interaction with the probe radiation in a classical coherent field. In chapter 8 the results obtained in chapter 6 are extended to include the effects of losses either due to the optical cavity or to the depletion of atomic modes. The calculation are performed by means of the Master equation formalism and a system-

18 6 Introduction atic comparison with respect to the ideal case is given. The results of this chapter are promising and give indications for future experiments.

19 Chapter 1 Classical CARL The CARL, a kind of hybrid between the FEL [13] and the ordinary laser, with physical features common to both, was thought and presented like a source of tunable coherent radiation. Its essential conceptual framework was first outlined in Ref.[10]. The ordinary laser and the FEL share an important physical trait: they generate electromagnetic waves through a noise-initiated process of self-organization. In an ordinary laser, for example, the energy is stored initially as excitation of internal degrees of freedom of the active medium, while in a FEL it is brought into the interaction region as translational kinetic energy of the incident electron beam. The spectral character of laser light is constrained mainly by the gain profile of the active medium, while in a FEL the frequency of the emitted radiation is assigned by the speed of the incident electrons, and can be varied, in principle, over a very wide range; hence, the FEL is intrinsically a widely tunable source. Furthermore, the laser gain originates from the induced atomic polarization, under the constraint that a suitable population inversion exists in the active medium; in a FEL, instead, amplification of coherent radiation follows the spontaneous emergence of a sufficiently large electron bunching, i.e. the appearance of a periodic spatial structure in the form of a longitudinal grating on the scale of the electromagnetic wavelength. Hence, light amplification in a FEL is the result of a coherent scattering process from the grating structure created within the active medium, and it comes at the expense of a recoil in the momentum of the individual electrons. The physics of the FEL and of the atomic lasers are unified in the CARL. The active medium now is a collection of two level atoms initial in their lower state and 7

20 8 Chapter 1. Classical CARL exposed to a strong pump field. For appropriate values of the parameters the atoms, through an exponential instability, can amplify a weak probe field counterpropagating with respect to the pump. As in the laser the active medium is characterized by bound states which play a key role in the amplification process but do not posses a population inversion. Common to the FEL, instead, is the existence of a reservoir of momentum that can be transformed partly into radiation through a kind of cooperative scattering. Furthermore, optical gain is initiated by the growth of a bunching parameter. What happens is that the incident optical wave creates an atomic polarization wave in the medium. This polarization couples with the backscattered radiation, creating a longitudinal self-consistent pendulum potential which traps and than bunches the particles giving rise to a coherent scattering. 1.1 Equations of motion The CARL model is based on the Hamiltonian of a collection of two-level atoms interacting with a strong pump field and a weak optical probe counterpropagating with respect to the pump. In addition to the internal atomic degrees of freedom, which are typical of laser models, the CARL model take explicit account of the center-of-mass motion. The explicit form of the Hamiltonian is N Ĥ = ω 1 â 1â 1 + ω 2 â 2â 2 + ω 0 Ŝ zj + +i ( g 1 â 1 j=1 N Ŝ j e ik 1ẑ j + g 2 â 2 j=1 N j=1 ˆp 2 j 2m ) N Ŝ j eik 2ẑ j h.c. j=1 (1.1) where N is the number of atoms in the interaction volume V, ω 1,2 = ck 1,2 are the carrier frequency of the probe and pump fields, respectively, k 1 and k 2 are the corresponding wave numbers and ω 0 is the atomic transition frequency when the atoms are at rest relative to the observer. The couplings constants are defined as [ ] 1/2 ωi g i = µ i = 1, 2 (1.2) 2 ɛ 0 V where µ is the modulus of the atomic dipole moment. Ŝ zj and Ŝ± j are the standard effective angular-momentum operators (in units of ) describing the evolution of the internal degrees of freedom of the j-th atom so that Ŝzj measures one-half the

21 1.1. Equations of motion 9 difference between the excited and ground state populations of the j-th atom; ẑ j and ˆp j denote, respectively, the position and momentum operators of the center of mass of the j-th atom and â i (i = 1, 2) are the photon creation operators of the pump field (index 1) and of the probe field (index 2). The operators obey the usual commutation relations: [Ŝzi, Ŝ± j ] = ±δ ij Ŝ ± j, [Ŝ+ i, Ŝ j ] = 2δ ij Ŝ zi (1.3) [ẑ i, ˆp j [ ] = i δ ij (1.4) â i, â j = δ ij (1.5) The Hamiltonian (1.1) admits two constants of the motion, N ˆp j + k 1 â 1â 1 k 2 â 2â 2 = constant (1.6) j=1 N Ŝ zj + â 1â 1 + â 2â 2 = constant; (1.7) j=1 the first represents the conservation of the total momentum and the second the conservation of the number of excitations. If we combine Eqs.(1.6) and (1.7) and eliminate the number operator of the driving field, â 2â 2, we can also write N (ˆp j + k 2 Ŝ zj ) + (k 1 + k 2 )â 1â 1 = constant (1.8) j=1 whose obvious physical implication is that the expectation value of the number operator for the probe field, â 1â 1, can grow either as the result of a loss of internal atomic energy or a decrease of the center-of-mass kinetic energy. This setting represents a generalization of the basic mechanism by which energy is produced in the laser and in the FEL; in fact, the laser Hamiltonian does not involve the momentum and position operators ˆp j and ẑ j, while the FEL Hamiltonian does not include the angular-momentum operators, descriptive of the internal degrees of freedom of the active medium. In this chapter we analyze the dynamical evolution of the coherent atomic recoil lasing within the framework of the standard semiclassical approximation. First we construct the Heisenberg equations of motion for the relevant operators and map

22 10 Chapter 1. Classical CARL the operator equation into their c-number counterparts in the usual factorized form obtaining dz j dt dp j dt da 1 dt = p j m (1.9) = k 1 g 1 a 1S j e ik 1z j + k 2 g 2 a 2S j e ik 2z j + c.c. (1.10) N = iω 1 a 1 + g 1 S j e ik 1z j (1.11) j=1 ds j dt ds zj dt = iω 0 S j + 2g 1 a 1 S zj e ik 1z j + 2g 2 a 2 S zj e ik 2z j (1.12) = ( g 1 a 1S j e ik 1z j + g 2 a 2S j eik 2z j + c.c. ). (1.13) where we have assumed that a 2 is a constant real number. This program is accomplished by introducing the appropriate slowly varying variables a o 1, a o 2, and S j according to the definitions a 1 (t) = { [ a 0 1(t) exp i ω 2 + k ] } 1 + k 2 m p(0) t (1.14) a 2 (t) = a 0 2 exp { iω 2 t} (1.15) S j (t) = S j(t) exp( ik 2 (z j + ct)) (1.16) where p(0) m v(0) is the average initial momentum of the atoms. Furthermore, we define the new position and momentum variables θ j (t) and δp j (t) [ θ j (t) = (k 1 + k 2 ) z j p(0) ] m t (1.17) δp j (t) = p j (t) p(0), (1.18) and the population difference between the ground and excited states of the j-th atom, The required equations of motion take the form D j (t) = 2S zj (t) (1.19) dθ j = k 1 + k 2 dt m δp j (1.20) d dt δp j = k 1 g 1 a 0 1 S j e iθ j + k 2 g 2 a 0 2 S j + c.c. (1.21) da 0 N 1 = iδ 1,2 a g 1 S j e iθ j (1.22) dt j=1

23 1.1. Equations of motion 11 ds j dt dd j dt [ ] ω2 δp j = i c m + δ 2,0 S j g 1 a 0 1D j e iθ j g 2 a 0 2D j γ S j, (1.23) = ( 2g 1 a 0 1 S j e iθ j + 2g 2 a 0 2 S j + c.c. ) γ (D j D eq j ) (1.24) where we have introduced the detuning parameters δ 2,1 = (k 1 + k 2 )[ v(0) v r,1 ], (1.25) δ 2,0 = k 2 [ v(0) v r,2 ], (1.26) with v r,1 = ω 1 ω 2 ω 1 + ω 2 c, v r,2 = ω 0 ω 2 ω 2 c (1.27) and added phenomenological decay terms to the polarization and population equations (D eq j = 1 because each atom is assumed to be in the ground state as it enters the interaction region). Note that the two resonance conditions δ 2,0 = δ 2,1 = 0, taken together, imply ω 1 = ω 0 1 β 0, ω 2 = ω β 0, (1.28) with β 0 = v(0)/c, i.e. they imply a resonance between the atomic transition frequency ω 0 and the Doppler-shifted frequencies of the probe and the pump beams. As our final step we introduce the so-called universal scaling and cast the working equations in dimensionless form. For simplicity we let k 1 k 2 k = ω/c, g 1 g 2 g and, furthermore, define the dimensionless parameter ρ = ( g ) 2/3 N ω R ( ) 1/3 N (1.29) V where ω R = 2 k 2 /m is the single-photon recoil frequency shift, the scaled time τ, and the new dependent variables P j and A 1,2 according to the definitions τ = ω R ρ t P j = δp j kρ A 1,2 = a0 1,2 Nρ (1.30) where A 2 is real for definiteness. The parameter ρ is a measure of the collective effects; in fact g N is the collective spontaneous Rabi frequency of an ensemble of N two-level atoms [28, 29].

24 12 Chapter 1. Classical CARL With these definitions the final form of the CARL equations of motion is dθ j dτ dp j dτ da 1 dτ = P j (1.31) = A 1e iθ j S j A 1 e iθ j Sj + 2A 2 ReS j, (1.32) = iδa N S j e iθ j N (1.33) j=1 ds j dτ dd j dτ = i 2 (P j + 2 )S j ρd j (A 1 e iθ j + A 2 ) Γ S j, (1.34) = [2ρ(A 1 e iθ j + A 2 )S j + h.c.] Γ (D j D eq ), (1.35) where the remaining parameters are defined as follow: δ = δ 2,1 = δ 2,0 ω R ρ ω R ρ (1.36) Γ = γ Γ ω R ρ = γ. ω R ρ (1.37) Eqs. ( ) form a closed, self-consistent set of equations for the internal and translational atomic degrees of freedom, coupled to the pump field A 2 and the probe field A 1, whose amplification was the main objective of the original works on CARL. In arriving at this result, as already mentioned, we have assumed k 1 k 2 k. If k 1 k 2, Eqs. ( ) are still valid in the so-called Bambini-Ranieri frame [30, 31] moving with a velocity v r,1 [see Eq.(1.27)], where the transformed frequencies coincide. We note that for a nearly resonant interaction, i.e. δ 2,1 0, it follows that v(0) v r,1. We can distinguish two cases: (a) Non relativistic particles [ v(0) c]; in this case we have ω 1 ω 2 and our equations are valid in the laboratory frame. (b) Relativistic particles [ v(0) c]; in this case it follows that ω 1 = c + v r,1 c v r,1 ω 2 = 1 + β 0 1 β 0 ω 2 4γ 2 ω 2, (1.38) where γ = (1 β 2 0) 1/2 ; hence ω 1 can be considerably larger than ω 2. Thus, this formulation can also account for the dynamics of relativistic particles; of course, in this case one needs an additional Lorentz transformation of Eqs.( ) back to the laboratory frame. We will never deal here with this case because our purpose is to show how the CARL model can be extended to the ultracold atoms regime. j

25 1.1. Equations of motion 13 Eqs. (1.34) and (1.35) are the optical Bloch equations, generalized for the inclusion of the atomic translational motion. In addition to the familiar detuning term S j in the polarization equation (1.34), one may note the appearance of the time dependent detuning contribution P j S j /2 resulting from the recoil suffered by atoms under the action of the pump and probe fields. If we ignore the probe field (A 1 = 0), Eqs. ( ) describe the usual cooling process for time long compared to Γ 1 and Γ 1. If, instead, we set A 2 = 0 and S j = 1, for all j, the modified Eqs. ( ) reduce to the traditional FEL equations. The structure of Eq.(1.33) indicates that the probe field A 1 can be amplified in the presence of an atomic polarization (but without the need of an initial population inversion) if the phase of the polarization has the appropriate value and if the atomic positions are properly bunched. If the scaled position variables are uniformly distributed between 0 and 2π, just as one has at the beginning of the evolution, no macroscopic field source exists even if the atomic polarization variables S j are maximized for all values of j because N e iθ j = 0 (1.39) j=1 Eqs. ( ) for a wide range of the parameters predict the development of an exponential instability for the probe field and for the bunching parameter 1 B = N N j=1 e iθ j. (1.40) The result of this instability is the growth of a macroscopic field and the spontaneous creation of a longitudinal spatial structure in the initially uniform atomic beam with a periodicity that matches the wavelength of the reflected field. This type of behavior can be easily demonstrated by numerical integration of Eqs. ( ) We now want to show that, under certain approximations, the atomic degrees of freedom can be eliminated and the CARL equations made formally identical to the FEL-model equations with universal scaling [13, 32]. This allows the simple description of a Hamiltonian instability leading to exponential growth of the probe field and of the bunching parameter. It will be demonstrated both without and in presence of atomic damping.

26 14 Chapter 1. Classical CARL 1.2 The FEL limit The undamped case Consider first the case in which Γ = 0 in the Bloch equations (this, in practice, means that we take Γ << 1 and τ << 1/Γ). Furthermore, let us assume that in Eqs. (1.34) and (1.35) we can neglect A 1 e iθ with respect to A 2, and P j with respect to 2. The first assumption implies that the atomic transition is dominated by the pump field, the second that the frequency shift due to recoil is always smaller than the initial detuning 2. In this approximation, the atomic variables become independent of j and the solution of the Bloch equations becomes: S 1 = 1 Ω 2 Ω2 + sin( Ω )τ (1.41) 2 S 2 = Ω Ω2 + Ω sin2 ( 2 )τ (1.42) 2 D = 1 2 Ω2 Ω2 + Ω sin2 ( 2 )τ (1.43) 2 where S 1 and S 2 are the real and imaginary part of the polarization S and Ω = 2ρ A 2 is the Rabi frequency of the pump field normalized to ω R ρ. In this way, we can eliminate the atomic variables from Eqs. differential equations for the recoil p j ( ) and obtain a closed set of and the self-consistent field A 1 with timedependent coefficients. A very simple description of this process follows if we further assume that the Rabi frequency is much larger than the collective recoil growth rate ω R ρ, i.e. Ω >> 1. In this case, we can take the time average of the atomic variables S 1 = 0 and S 2 = S 0 where S 0 = 1 Ω 2 Ω 2 +. (1.44) 2 Note that S 0 is maximized for = Ω, where S 0 = 1/4. Upon substituting in Eqs. ( ) we obtain dθ j dτ dp j dτ da dτ = P j (1.45) = S 0 (A e iθ j + Ae iθ j ) (1.46) = iδa + S 0 N N e iθ j (1.47) j=1

27 1.2. The FEL limit 15 where we have defined A = ia 1. Note that the time average has washed out the absorptive part S 1 of the polarization, leaving only the dispersive part S 2, which is antisymmetric in the detuning. In particular, in Eq. (1.32), the radiation pressure term 2A 2 ReS j does not play any role. Only the collective part survives and contributes to the process. Eqs. ( ) are formally identical to the FEL model equations in the Compton regime and the dispersive part of the polarization S 0 for CARL plays the role of the FEL wiggler field. To retrieve the FEL equations with no free parameters (universal scaling), we could redefine the CARL parameter as [10] ( ) gn 1/2 2/3 S ρ 0 = = ρs 2/3 0 (1.48) ω R and redefine all the scaled variables τ, P j, A,, δ and Ω now with respect to ρ. Performing these substitutions we obtain finally dθ j dτ dp j dτ da dτ = P j (1.49) = (A e iθ j + Ae iθ j ) (1.50) = iδa + 1 N e iθ j N (1.51) j= The damping case: adiabatic limit With strong damping we have Γ 1 and τ >> 1/Γ so that we can perform the adiabatic elimination of the polarization and population variables [11]. With simple calculations, one can show that we obtain again the FEL equations ( ) with A substituted by 2A and S 0 = 1 Ω (1.52) 2 Γ Ω 2 This result can be obtained by again assuming A 2 2 >> A 1 2 and P j << 2, as before. Furthermore, we assume γγ << ρ and Γ << in order to ignore the radiation-pressure term, and to minimize the effect of the real part of the polarization, respectively. These approximations are consistent with the fact that S 0 is maximized for = (Ω 2 + Γ 2 ) 1/2 >> Γ, provided Γ << Ω so that the maximum value of S 0 becomes S 0 = 1/2 2.

28 16 Chapter 1. Classical CARL 1.3 Linear stability analysis Equations ( ) admit two constants of motion: p + A 2 (1.53) and p is [ 0 A e iθ c.c ] = H (1.54) The first represents momentum conservation, the second defines the Hamiltonian from which ( ) can be derived. This Hamiltonian system is unstable. A linear analysis of ( ) leads to the identification of the conditions under which the process of self-amplification of the spontaneous emission takes place. Let us consider the initial conditions (where the system has an equilibrium solution) with zero field, no spatial modulation of the beam of particles in which θ j are randomly distributed. In this initial situations j e imθ j = 0 with m = 1, 2. Equations ( ) are generally valid also if the initial momentum has an arbitrary distribution f(p 0 ). In this case, we can label the particles by the initial position θ 0 and initial momentum p 0 so that 1 N N j=1 e iθ j is replaced by e iθ(θ 0,p 0,τ) = 1 2π f(p 0 )e iθ(θ 0,p 0,τ) dθ 0 dp 0 (1.55) 2π 0 where we have assumed a uniform distribution for θ 0. One can show that, by linearizing ( ) around the initial situation above and using a Laplace transformation, we find solutions of the form 3 c j e iλ jτ j=1 (1.56) where c j depends on the initial conditions and λ j are the roots of the cubic equation f(p 0 ) λ + (λ + p 0 ) dp 2 0 = 0 (1.57) In particular, for a cold case, i.e., if f(p 0 ) is a Dirac delta function centered at a value p 0 = δ, the cubic equation takes the form Note that for v 0 = 0, we have δ = (ω 2 ω 1 )/ω R ρ. λ + (λ + δ) = 0. (1.58)

29 1.3. Linear stability analysis 17 Exponential growth, and thus, unstable behavior, results if the cubic Eq.(1.58) has one real and two complex-conjugate roots. In this case, one of the imaginary parts of the eigenvalues measures the exponential growth rate of the unstable solution. For δ > δ c = (27/4) 1/3, all the roots are real and the system is stable, for δ < δ c the system is unstable and it shows a collective instability that leads to an exponential growth of the probe-field intensity. In particular, for δ = 0 the gain is maximum and the intensity grows as A 2 = e 3τ = e Gt (1.59) where G = 3 ω R ρ. (1.60) This shows that the growth rate of the collective instability described by this Hamiltonian is governed by ω R ρ. In general, the exponential gain depends on the detuning parameters δ and. The collective recoil process of the system produces an atomic longitudinal grating on the scale of the wavelength. This can be measured by the behavior of the bunching parameter. In fact, the bunching parameter is the source for the field in (1.51) and its modulus can range from 0, when the particles are randomly distributed in phase, to 1, when the particles are confined periodically to regions smaller than the wavelength. When the collective instability develops, the emitted radiation creates a correlation between the particles, and the beam becomes strongly modulated (self-bunching takes place). The time dependence of the bunching is also exponential and produces values that are almost unity. The strong bunching is due to the fact that in the high-gain exponential regime, the particles are trapped in the closed orbits of a pendulum phase space due the beat of the pump and of the self-consistent scatter field which appears explicitly in (1.32). We stress that this behavior takes place only for long times τ 1. In general, one must perform a careful analysis of the linear solution which has the form 3 A(δ, τ) = C j e iλ j(δ)τ (1.61) j=1 where λ j (δ) are the three roots of the cubic, and C j are fixed by the initial conditions. Here, we simply describe the results.

30 18 Chapter 1. Classical CARL Figure 1.1: Gain as a function of δ for different interaction times τ. (a) τ = 1: Small-gain Madey regime where G is an antisymmetric function of δ. (b) τ = 10: Characteristic symmetric shape of the gain curve in the high-gain exponential regime. Note that the gain curve is orders of magnitude larger than for τ = 1. For τ < 1, the time behavior is not exponential and one has the well known small-gain Madey regime [33, 34] (see Fig. 1.1a). The gain G(δ, τ) = [ A(δ, τ) 2 ] A 0 2 A 0 2 (1.62) is an oscillating function of τ, and for fixed τ is an antisymmetric function of δ, which is positive (gain) for δ > 0, zero for δ = 0 and negative for δ < 0 showing an absorptive behaviour. This gain is associated with the small bunching resulting from the fact that the particles are not trapped, i.e. move on open orbits of the pendulum phase space. However, when τ >> 1 and δ < δ c, the particles becomes trapped. As a consequence, the exponential behavior takes over and the gain G as a function of δ changes shape as τ increases, and acquires a symmetric dependence on δ, as shown in Fig. 1.1b. Its maximum at δ = 0 increases exponentially, as stated before. Hence, the exponential regime is observable only if the interaction time is sufficiently large so that τ >> 1. The behavior of the probe field as a function of the normalized time τ is shown in Fig. 1.2a. This is a numerical simulation of the exact set ( ) under

31 1.4. Experimental realizations 19 Figure 1.2: (a): Probe field as a function of τ for the exact CARL equations with Γ = 0, = 600, δ = 0; (b): Bunching parameter as a function of τ for the exact CARL equations, using the parameters of a. conditions of zero decay rate (Γ = 0). Fig. 1.2b shows the behavior of the bunching parameter as a function of τ. Note the very high value of saturation of the modulus of the bunching ( 0.8). 1.4 Experimental realizations The signature of CARL is an exponential growth of a seeded probe field oriented reversely to a strong pump interacting with an active medium. On the other hand, atomic bunching and probe gain can also arise spontaneously from fluctuations with no seed field applied. The underlying runaway amplification mechanism is particularly strong, if the reverse probe field is recycled by a ring cavity. After the theoretical proposal of CARL, experiments have been performed in order to observe its peculiar features. As we saw the most striking effect due to the CARL dynamics is the exponential growth of a seeded probe field oriented reversely to the pump field. On the other hand, atomic bunching and probe gain can also arise spontaneously from fluctuations with no seed field applied, particularly if the runaway amplification mechanism is enforced recycling the reverse probe field by a ring cavity. The first attempts to observe CARL action have been undertaken

32 20 Chapter 1. Classical CARL Figure 1.3: Scheme of the experimental setup in Tbingen. A Ti-sapphire laser is locked to one of the two counterpropagating modes (α + ) of a ring cavity. The beam α in can be switched off by means of a mechanical shutter (S). The atomic cloud is located in the free space waist of the cavity mode. The evolution of the interference signal between the two light fields leaking through one of the cavity mirrors and the spatial evolution of the atoms via absorption imaging are observed. Figure taken from Ref. [17]. only in hot atomic vapors [14, 16]. In particular P.R. Hemmer, N.P. Bigelow and coworkers [14] performed the first experiment in a strongly pumped atomic sodium vapor without the introduction of a counterpropagating probe. These experiments led to the identification of a reverse field with some of the expected characteristics. However, the gain observed in the reverse field can have other sources [35], which are not necessarily related to atomic recoil. The first unambiguous experimental proof of the CARL effect has been obtained only very recently [17] in a system of cold atoms in a collision-less environment. In this experiment (see Fig.1.3) a high-q ring cavity is pumped by a Ti-Sapphire laser locked to one mode (α + ) of the cavity. The 85 Rb atomic cloud is located in the free space waist of the cavity mode with a magneto-optical trap, working at a temperature of several 100µK. The reverse field α has been monitored as the beat signal between the field α itself and the pump α +. In this experiment, in contrast with the usual CARL model, the atoms are prepared already in a bunched state. In Fig.1.4(a) we can see that oscillations appear on the beat signal, showing the arising of the reverse field due to recoil effect even in the absence of a seed field. Notice that the amplitude of the oscillations is rapidly dumped, however they are still discernible after more than 1ms. Moreover as the interaction time between the

33 1.4. Experimental realizations 21 P beat (µw) (a) (b) ω/2π (khz) t (µs) 1 mm 800 (c) 600 (d) t (ms) (e) (f) Figure 1.4: (a) Recorded time evolution of the observed beat signal between the two cavity modes with N = 10 6 and P cav ± = 2W. At time t = 0 the pumping of the probe α has been interrupted. (b) Numerical simulation with the temperature adjusted to 200µK. (c) The symbols (X) trace the evolution of the beat frequency after switch-off. The dotted line is based on a numerical simulation. The solid line is obtained from numerical simulation with the assumption that the fraction of atoms participating in the coherent dynamics is 1/10 to account for imperfect bunching. (d) Absorption images of a cloud of atoms recorded for high cavity finesse at 0ms and (e) 6ms after switching off the probe beam pumping. All images are taken after a 1ms free expansion time. (f) This image is obtained by subtracting from image (e) an absorption image taken with low cavity finesse 6ms after switch-off. The intracavity power has been adjusted to the same value as in the high finesse case. Figure taken from Ref. [17]. pump and the atoms increases the detuning between probe and pump increases too (see Fig. 1.4(c)). As a consequence, the collective recoil gives rise to a detectable displacement of the atoms which has been indeed observed taking time-of-flight absorption images of atomic cloud at various times (see Fig. 1.4(d),(e),(f)). Besides

34 22 Chapter 1. Classical CARL these results, in experiment [17] a second set-up that differs from the original CARL proposal has been used. The usual CARL dynamics never reaches a steady state and the power of the reverse field decreases in time. Hence, in order to reach a stationary regime, a friction force has been introduced through an optical molassa so that a steady-state velocity of the atoms is reached when the velocity dependent dumping force balances the CARL acceleration. As a consequence the reverse field too reaches fixed detuning and amplitude, so that the lasing process becomes stationary. The work on this subject to explain the experiment is still in progress [37]. 1.5 Concluding remarks By taking into account the translational degrees of freedom of the active medium, we have described a mechanism that can lead to the exponential amplification of a weak probe. Roughly speaking, we can interpret the process of amplification as evolving in two steps: first, the external field creates a weak gain profile in the frequency response of a collection of independent driven atoms and begins the buildup of a spatial structure with the help of the atomic recoil; next the probe, whose carrier frequencies lies within a selected gain region of the active medium, undergoes exponential amplification. The role of the atomic recoil is essential to this process: not only it is the cause of the emergence of the spatial grating pattern, but it also reinforces the coherent growth of the signal to be amplified as energy is transferred from the atoms to the probe field. An alternative way of interpreting the probe amplification is to view it as the reflection of the pump field from the moving grating pattern or as a kind of coherent scattering from the bound states of the atoms. We stress that even though we have demonstrated the amplification of a probe signal, since the saturation value of the intensity and of the bunching is independent of the initial value of the probe, the process can be initiated from spontaneous emission noise.

35 Chapter 2 Quantum CARL The realization of Bose Einstein condensation in dilute alkali gases [1, 2] opened the possibility to study the coherent interaction between light and an ensemble of atoms prepared in a single quantum state. For example, Bragg diffraction [38] of a BEC by a moving optical standing wave can be used to diffract any fraction of the condensate into a selectable momentum state, realizing an atomic beam splitter. In particular, collective light scattering and matter-wave amplification caused by coherent centerof mass motion of atoms in a condensate illuminated by a far off-resonant laser [19, 39, 40] have been interpreted as superradiant Rayleigh scattering and can be investigated using a quantum theory based on a quantum multi-mode extention of the CARL model [21, 41, 42, 43]. The main drawback of the semiclassical model is that, as it considers the center-of-mass motion of the atoms as classical, it cannot describe the discreteness of the recoil velocity, as has been observed in the experiment of Ref.[19]. The original CARL theory, which treats the atomic center-of-mass motion classically, fails when the temperature of the atomic sample is below the recoil temperature T R = ω R /k B, M is the atomic mass and k B is the Boltzmann constant. So, to extend the model in the region of ultracold atoms, a quantum mechanical description of the center-of-mass motion of the atoms must be included. In this chapter we present a way to work out this program simply performing a first quantization of the external variables θ and P of atoms [20]. Even if not complete this model gives a simple description of all the features of the considered system and in particular allows to define the main different regimes. In the conservative regime (no radiation losses), the quantum model depends on 23

36 24 Chapter 2. Quantum CARL a single collective parameter, ρ, that can be interpreted as the average number of photons scattered per atom in the classical limit. When ρ 1, the semiclassical CARL regime is recovered, with many momentum levels populated at saturation. On the contrary, when ρ 1, the average momentum oscillates between zero and q, and a periodic train of 2π hyperbolic secant pulses is emitted. In the dissipative regime (large radiation losses) and in a suitable quantum limit (ρ < 2κ), a sequential superfluorescence scattering occurs, in which after each process atoms emit a π hyperbolic secant pulse and populate a lower momentum state. These results describe the regular arrangement of the momentum pattern observed in the aforementioned experiments of superradiant Rayleigh scattering from a BEC. 2.1 First quantization The atomic motion is quantized when the average recoil momentum is comparable to q where q = k 2 k 1 is the difference between the incident and the scattered wave vectors, i.e. the recoil momentum gained by the atom trading a photon via absorbtion and stimulated emission between the incident and scattered waves. The starting point of the following model is the classical model of equations ( ) derived in chapter 1 dθ j dτ dp j dτ da dτ = P j (2.1) = [ Ae iθ j + A e iθ j ] (2.2) = iδa + 1 N N e iθ j (2.3) for N two-level atoms exposed to an off-resonant pump laser, whose electric field has a frequency ω 2 = ck 2 with a detuning from the atomic resonance, 20 = ω 2 ω 0, much larger than the natural linewidth of the atomic transition, γ. j=1 The probe field has frequency ω 1 = ω 2 21 and electric field with the same polarization of the pump field. In the absence of an injected probe field, the emission starts from fluctuations and the propagation direction of the scattered field is determined either by the geometry of the condensate (as in the case of the MIT experiment [19], where the condensate has a cigar shape) or by the presence of an optical resonator tuned on a selected longitudinal mode.

37 2.1. First quantization 25 In order to quantize both the radiation field and the center-of-mass motion of the atoms, we consider θ j, p j = (ρ/2)p j = Mv zj / q and a = (Nρ/2) 1/2 A as quantum operators satisfying the canonical commutation relations [â, [ˆθj, ˆp j ] = iδ ] jj â = 1. (2.4) With these definitions, Eqs.(2.1)-(2.3) are transformed into the Heisenberg equations of motion dˆθ j dτ dˆp j dτ dâ dτ = 2 ρ ˆp j (2.5) ρ ] = [âe iˆθ j + â e iˆθ j (2.6) 2N ρ N = iδâ + e iˆθ j (2.7) 2N j=1 associated with the Hamiltonian: Ĥ = 1 ( N N ) ρ ˆp 2 j + i â e iˆθ j h.c. δâ â = ρ 2N j=1 j=1 N H j (ˆθ j, ˆp j ), (2.8) j=1 where Ĥ j (ˆθ j, ˆp j ) = 1 ρ ρ ˆp2 j + i 2N (â e iˆθ j ae iˆθ j ) δ N â â (2.9) We note that [Ĥ, ˆQ] = 0, where ˆQ = â â + j ˆp j is the total momentum in units of q. In order to obtain a simplified description of a BEC as a system of N noninteracting atoms in the ground state, we use the Schrödinger picture for the atoms (instead of the usual Heisenberg picture [44]), i.e. ψ(θ 1,..., θ N ) = ψ(θ 1 )... ψ(θ N ), (2.10) where ψ(θ j ) obeys the single-particle Schrödinger equation, i τ ψ(θ j) = H j (θ j, p j ) ψ(θ j ). (2.11) In this model we describe the scattered radiation field classically. Hence, considering the corresponding c-number a of the field operator â (i.e. its expectation value), eq.(2.7) yields: N da dτ = iδa + g ψ(θ j ) e iθ j ψ(θ j ). (2.12) j=1

38 26 Chapter 2. Quantum CARL Let now expand the single-atom wavefunction on the momentum basis, ψ(θ j ) = n c j(n) n j, where ˆp j n j = n n j, n =,... and c j (n) is the probability amplitude of the j-th atom having momentum n q. Remembering that we obtain [e ±iˆθ i, ˆp j ] = δ i,j e ±iˆθ i and e ±iˆθ j n j = n ± 1 j (2.13) N da dτ = iδa + g j=1 c j(n + 1)c j (n). (2.14) Introducing the collective density ˆϱ with matrix elements on the base { n } ϱ m,n = 1 N N c j (m) c j (n)e i(m n)δτ, (2.15) j=1 a straightforward calculation yields, from Eqs.(2.12) and (2.15), the following closed set of equations: dϱ m,n dτ da dτ = i(m n)δ m,n ϱ m,n + ρ 2 [A (ϱ m+1,n ϱ m,n 1 ) + A (ϱ m,n+1 ϱ m 1,n )] (2.16) = ϱ n,n+1 κa, (2.17) n= where δ m,n = δ + (m + n)/ρ and we have redefined the field as A = 2/ρNae iδτ. We have also introduced a damping term κa in the field equation, where κ = κ c /ω R ρ, κ c = c/2l and L is the sample length along the probe propagation, which provides an approximated model describing the escape of photons from the atomic medium. In the presence of a ring cavity of length L cav and reflectivity R, κ c = (c/l cav )lnr, as shown in the usual mean-field approximation [44]. Eqs.(2.16) and (2.17) determine the temporal evolution of the density matrix elements for the momentum levels. In particular, P n = ϱ n,n is the probability of finding the atom in momentum level n, ˆp = n nϱ n,n is the average momentum and B = ϱ n,n+1 (2.18) n is the bunching parameter. Eqs.(2.16) and (2.17), as we will show in the next chapter, are the equations for expectation values correspondent to those derived

39 2.1. First quantization 27 Figure 2.1: Classical limit of CARL for ρ 1 in the case κ = 0. (a): A 2 vs. τ as obtained from the classical eqs.(1)-(3) (dashed line) and from the quantum eqs.(7) and (8) for ρ = 10 (solid line); (b): population level p n vs. n at the occurring of the first maximum of A 2, at τ = The other parameters are δ = 0 and A(0) = Figure taken from Ref. [20]. in a complete second quantized treatment which introduces bosonic creation and annihilation operators of a given center-of-mass momentum [18]. For a constant field A, Eq.(2.16) describes a Bragg scattering process, in which m n photons are absorbed from the pump and scattered into the probe, changing the initial and final momentum states of the atom from m to n. Conservation of energy and momentum require that during this process ω 1 ω 2 = (m + n) ω R, i.e. δ m,n = 0. Eqs.(2.16) and (2.17) conserve the norm, i.e. m ϱ m,m = 1, and, when κ = 0, also the total momentum ˆQ = (ρ/2) A 2 + ˆp. Fig. 2.1a shows A 2 vs. τ, for κ = 0, δ = 0 and A(0) = 10 4, comparing the semiclassical solution with the quantum solution in the semiclassical limit that corresponds to ρ 1: the dashed line is the numerical solution of Eqs.(2.1)-(2.3), for a classical system of N = 200 cold atoms, with initial momentum p j (0) = 0 (where j = 1,..., N) and phase θ j (0) uniformly distributed over 2π, i.e. unbunched; the continuous line is the numerical solution of Eqs.(2.16) and (2.17) for ρ = 10 and a quantum system of atoms initially in the ground state n = 0, i.e. with ϱ n,m = δ n0 δ m0. We can notice that the quantum system behaves, with good approximation, classically. Because the maximum dimensionless intensity is A 2 1.4, the constant of motion ˆQ gives ˆp 0.7ρ and the maximum average number of emitted

40 28 Chapter 2. Quantum CARL photons is about â â Nρ. Hence in this limit the CARL parameter ρ can be interpreted as the maximum average number of photons emitted per atom (or equivalently, as the maximum average momentum recoil, in units of q, acquired by the atom) in the classical limit. Fig.2.1b shows the distribution of the population level P n at the first peak of the intensity of Fig. 2.1a, for τ = We observe that, at saturation, twenty-five momentum levels are occupied, with an induced momentum spread comparable to the average momentum. 2.2 Linear regime Let us now consider the equilibrium state with no probe field, A = 0, and all the atoms in the same momentum state n, i.e. with ϱ n,n = 1 and the other matrix elements zero. This is equivalent to assume the temperature of the system equal to zero and all the atoms moving with the same velocity n q, without spread. This equilibrium state is unstable for certain values of the detuning. In fact, by linearizing Eqs.(2.16) and (2.17) around the equilibrium state, the only matrix elements giving linear contributions are ϱ n 1,n and ϱ n,n+1, showing that in the linear regime the only transitions allowed from the state n are those towards the levels n 1 and n + 1. Introducing the new variables B n = ϱ n,n+1 + ϱ n 1,n and D n = ϱ n,n+1 ϱ n 1,n, Eqs.(2.16) and (2.17) reduce to the linearized equations: db n dτ dd n dτ da dτ = iδ n B n i ρ D n (2.19) = iδ n D n i ρ B n ρa (2.20) = D n κa, (2.21) where δ n = δ + 2n/ρ. Seeking solutions proportional to e i(λ δ n)τ, we obtain the following cubic dispersion relation: (λ δ n iκ) ( λ 2 1/ρ 2) + 1 = 0. (2.22) In the exponential regime, when the unstable (complex) root λ dominates, B(τ) e i(λ δ n)τ and, from Eq.(2.19), D n = ρλb n. The classical limit is recovered for ρ 1 when κ = 0 or ρ κ when κ > 1 and δ n δ, i.e. neglecting the shift due to the recoil frequency ω R. In this limit, maximum gain occurs for δ = 0,

41 2.2. Linear regime 29 with λ = (1 i 3)/2 when κ = 0 or λ = (1 + i)/ 2κ when κ > 1. Furthermore, ϱ n,n+1 ϱ n 1,n, so that the atoms may experience both emission and absorbtion. This result can be interpreted in terms of single-photon emission and absorption by an atom with initial momentum n q. In fact, energy and momentum conservation impose ω 1 ω 2 = (2n 1)ω R (i.e. δ n = ±1/ρ) when a probe photon is emitted or absorbed, respectively. Because in the semiclassical limit the gain bandwidth is ω ω R ρ ω R when κ = 0 (or ω κ c ω R when κ > 1) the atom can both emit or absorbe a probe photon. On the contrary, in the quantum limit the recoil energy ω R can not be neglected, and there is emission without absorbtion if ϱ n,n+1 ϱ n 1,n, i.e. This is true for ρ < 1 when κ = 0 with the unstable root B n D n, λ 1 ρ. (2.23) λ 1 ρ + δ n 2 1 (δ 2 n) 2 2ρ (2.24) (where δ n = δ n 1/ρ), and for ρ < 2κ when κ > 1 with Reλ 1 ρ + ρδ n 2[(δ n) 2 + κ 2 ], Imλ ρκ 2[(δ n) 2 + κ 2 ]. (2.25) In both cases, maximum gain occurs for δ n = 1/ρ (i.e. 21 = (1 2n)ω R ) within a bandwidth ω ω R ρ 3/2 and ω ω R ρ 2 /κ (respectively for κ = 0 and κ > 1), which are both less than the frequency difference 2ω R between the emission and absorbtion lines. Hence, in the quantum limit the optical gain is due exclusively to emission of photons, whereas in the semiclassical limit gain results from a positive difference between the average emission and absorbtion rates. When κ = 0, the resonant gain in the limit ρ < 1 is ρ G S = ω R ρ 2 = 3 8π Ω γ N eff, (2.26) where γ = µ 2 k 3 /3π ɛ 0 is the natural decay rate of the atomic transition, Ω 0 is the Rabi frequency of the pump and N eff = (λ 2 /A)(c/γL)N is the effective atomic number in the volume V = ΣL, where Σ and L are the cross section and the length of the sample. When κ > 1, the resonant superfluorence gain in the limit ρ < 2κ is G SF = ω Rρ 2 2κ = 3 ( ) 2 4π γ Ω0 λ 2 N. (2.27) 2 20 A

42 30 Chapter 2. Quantum CARL Figure 2.2: Quantum limit of CARL for ρ < 1 in the case κ = 0. (a) A 2 and (b) p vs. τ, for ρ = 0.2, δ = 5, A(0) = 10 5 and the atoms initially in the state n = 0. We note that p = (ρ/2)( A 2 A(0) 2 ). Figure taken from Ref. [20]. The above results show that the combined effect of the probe and pump fields on a collection of cold atoms in a pure momentum state n is responsible of a collective instability that leads the atoms to populate the adjacent momentum levels n 1 and n + 1. However, in the quantum limit ρ < 1 when κ = 0 (or ρ < 2κ when κ > 1) conservation of energy and momentum of the photon constrains the atoms to populate only the lower momentum level n 1. This holds also in the nonlinear regime, as we have verified solving numerically Eqs.(2.16) and (2.17). In the quantum limit above, the exact equations reduce to those for only three matrix elements, ϱ n,n, ϱ n 1,n 1 and ϱ n 1,n, with ϱ n 1,n 1 + ϱ n,n = 1. Introducing the new variables S n = S n 1,n and W n = ϱ n,n ϱ n 1,n 1, Eqs.(2.16) and (2.17) reduce to the well-known Maxwell-Bloch equations [45]: ds n dτ dw n dτ da dτ = iδ ns n + ρ 2 AW n (2.28) = ρ(a S n + h.c.) (2.29) = S n κa. (2.30) When κ = 0 and δ n = 0, if the system starts radiating incoherently by pure quantummechanical spontaneous emission, the solution of Eqs.(2.28)-(2.30) is a periodic train of 2π hyperbolic secant pulses [46] with [ ] ρ A 2 = (2/ρ) Sech 2 2 (τ τ n), (2.31)

43 2.2. Linear regime 31 Figure 2.3: Sequential superfluorescent (SF) regime of CARL. (a) A 2 and (b) p vs. τ, for ρ = 2, δ = 0.5, κ = 10, and the same initial conditions of fig.2.2. Figure taken from Ref. [20]. where τ n = (2n + 1)ln(ρ/2)/ ρ/2. Furthermore, the average momentum [ ] ρ ˆp = n + Th 2 2 (τ τ n) 1 (2.32) oscillates between n and n 1 with period τ n. We observe that the maximum number of photons emitted is â â peak = (ρn/2) A 2 peak = N, as expected. Fig. 2.2 shows the results of a numerical integration of Eqs.(2.16) and (2.17), for κ = 0, ρ = 0.2 and δ = 5, with the atoms initially in the momentum level n = 0 and the field starting from the seed value A 0 = The intensity A 2 and the average momentum ˆp vs. τ are in agreement with the predictions of the reduced Eqs.(2.28)-(2.30). In the superradiant regime, κ > 1, Eqs.(2.28)-(2.30) describe a single SF scattering process in which the atoms, initially in the momentum state n, decay to the lower level n 1 emitting a π hyperbolic secant pulse, with intensity and average momentum [ ] A 2 1 (τ τd ) = 4[κ 2 + (δ n) 2 ] Sech2, ˆp = n 1 2 τ SF { 1 + Th [ ]} (τ τd ) τ SF (2.33) where τ SF = 2(κ 2 + δ n 2 )/ρκ is the superfluorescence time [28], the delay time is τ D = τ SF Arcsech(2 S n (0) ) τ SF ln 2 S n (0) and S n (0) 1 is the initial polarization. Figures 2.3a and b shows A 2 and ˆp vs. τ calculated solving Eqs.(2.16) and (2.17) numerically with κ = 10, ρ = 2, δ = 0.5 and the same initial conditions of Fig.

44 32 Chapter 2. Quantum CARL 2.2. We observe a sequential SF scattering, in which the atoms, initially in the level n = 0, change their momentum by discrete steps of q and emit a SF pulse during each scattering process. We observe that for δ = 1/ρ the field is resonant only with the first transition, from n = 0 to n = 1; for a generic initial state n, resonance occurs when δ = (1 2n)/ρ, so that in the case of Fig. 2.3a the peak intensity of the successive SF pulses is reduced (by the factor 1/[κ 2 + (2n/ρ) 2 ]) whereas the duration and the delay of the pulse are increased. However, the pulse retains the characteristic Sech 2 shape and the area remains equal to π, inducing the atoms to decrease their momentum by a finite value q. We note that, although the SF time in the quantum limit (τ SF = 2κ/ρ at resonance) can be considerable longer than the characteristic superradiant time obtained in the classical limit, τ SR = 2κ, the peak intensity of the pulse in the quantum limit is always approximately half of the value obtained in the semiclassical limit (see Ref.[47] for details). 2.3 Concluding remarks We have shown that the CARL model describing a system of atoms in their momentum ground state (as those obtained in a BEC) and properly extended to include a quantum-mechanical description of the center-of-mass motion, allows for a quantum limit in which the average atomic momentum changes in discrete units of the photon recoil momentum q and reduce to the Maxwell-Bloch equations for two momentum levels. The behavior of the system is different for conservative and dissipative regimes. The regular arrangement of momentum pattern observed in the superradiant Rayleigh scattering experiments with BECs (see also chapter 4 for details) can be interpreted as being due to the sequential superfluorescence scattering.

45 Chapter 3 Quantum field theory In this chapter we derive a fully quantized model of a gas of bosonic two-level atoms which interact with a strong, classical, undepleted pump laser and a weak, quantized optical ring cavity mode, both of which are as usual assumed to be tuned far away from atomic resonances. Starting from the second-quantized hamiltonian of the system, we will write an effective model for the time evolution of the ground state atomic field operator and of the probe field operator, adiabatically eliminating the excited state atomic field operator and including effects of atom-atom collisions [48]. 3.1 The CARL-BEC model The second-quantized Hamiltonian of the system is Ĥ = Ĥatom + Ĥprobe + Ĥatom probe + Ĥatom pump + Ĥatom atom, (3.1) where Ĥatom and Ĥprobe give the free evolution of the atomic field and the probe mode respectively, Ĥ atom probe and Ĥatom pump describe the dipole coupling between the atomic field and the probe mode and pump laser, respectively, and Ĥatom atom contains the two-body s-wave scattering collisions between ground state atoms. The free atomic Hamiltonian is given by [ ) Ĥ atom = d 3 z ˆΨ g (z) ( 2 2m 2 + V g (z) ˆΨ g (z) ) ] + ˆΨ e (z) ( 2 2m 2 + ω 0 + V e (z) ˆΨ e (z), (3.2) 33

46 34 Chapter 3. Quantum field theory where m is the atomic mass, ω a is the atomic resonance frequency, ˆΨ e (z) and ˆΨ g (z) are the atomic field operators for excited and ground state atoms respectively, and V g (z) and V e (z) are their respective trap potentials. The atomic field operators obey the usual bosonic equal time commutation relations [ ˆΨj (z), ˆΨ ] j (z ) = δ j,j δ 3 (z z ) (3.3) [ ˆΨj (z), ˆΨ j (z )] = [ ˆΨ j (z), ˆΨ j (z )] = 0, (3.4) where j, j = {e, g}. The free evolution of the probe mode is governed by the Hamiltonian Ĥ probe = ck 1  Â, (3.5) where c is the speed of light, k 1 is the magnitude of the probe wave number k 1, and  and  are the probe photon annihilation and creation operators, satisfying the boson commutation relation [Â,  ] = 1. The probe wavenumber k 1 must satisfy the periodic boundary condition of the ring cavity, k 1 = 2πl/L, where the integer l is the longitudinal mode index, and L is the length of the cavity. The atomic and probe fields interact in the dipole approximation via the Hamiltonian Ĥ atom probe = i g 1  d 3 z ˆΨ e (z)e ik s z ˆΨg (z) + H.c., (3.6) where g 1 = µ[ck 1 /(2 ɛ 0 LS)] 1/2 is the atom-probe coupling constant. Here µ is the magnitude of the atomic dipole moment, and S is the cross-sectional area of the probe mode in the vicinity of the atomic sample (where it is assumed to be approximately constant across the length of the atomic sample). In addition, the atoms are driven by a strong pump laser, which is treated classically and assumed to remain undepleted. The atom-pump interaction Hamiltonian is given in the dipole approximation by Ĥ atom pump = Ω 2 e iω 2t d 3 z ˆΨ e (z)e ik 2 z ˆΨg (z) + H.c., (3.7) where Ω is the Rabi frequency of the pump laser, related to the pump intensity I by Ω 2 = 2µ 2 I/ 2 ɛ 0 c, ω 2 is the pump frequency, and k 2 ω 2 /c is the pump wavenumber. The approximation indicates that we are neglecting the index of refraction inside the atomic gas, as we assume a very large detuning 20 = ω 2 ω 0 between the pump frequency and the atomic resonance frequency.

47 3.1. The CARL-BEC model 35 Finally, the collision Hamiltonian is taken to be Ĥ atom atom = 2π 2 σ d 3 z m ˆΨ g (z) ˆΨ g (z) ˆΨ g (z) ˆΨ g (z), (3.8) where σ is the atomic s-wave scattering length. This corresponds to the usual s-wave scattering approximation, and leads in the Hartree approximation to the standard Gross-Pitaevskii equation for the ground state wavefunction (in the absence of the driving optical fields). We limit ourselves to the case where the pump laser is detuned far enough away from the atomic resonance that the excited state population remains negligible, a condition which requires that γ a. In this regime the atomic polarization adiabatically follows the ground state population, allowing the formal elimination of the excited state atomic field operator. First we write the Heisenberg equation of motion for the field operators. The commutation relation with the Hamiltonian are [ Ĥprobe] [ Ĥprobe] [ Ĥatom atom] ˆΨg (z), = ˆΨe (z), = ˆΨe (z), = 0 (3.9) ] ] ] [Â, Ĥ atom = [Â, Ĥ atom pump = [Â, Ĥ atom atom = 0 (3.10) [ Ĥatom] ) ˆΨg (z), = ( 2 2m 2 + V g (z) ˆΨ g (z) (3.11) [ Ĥatom probe] ˆΨg (z), = i g 1 Â ˆΨe (z)e ik 1 z (3.12) [ Ĥatom pump] ˆΨg (z), = Ω 2 eiω 2t ˆΨe (z)e ik 2 z (3.13) [ Ĥatom atom] ˆΨg (z), = 4π 2 σ m ˆΨ g (z) ˆΨ g (z) ˆΨ g (z) (3.14) [ Ĥatom] ) ˆΨe (z), = ( 2 2m 2 + ω 0 + V e (z) ˆΨ e (z) (3.15) [ Ĥatom probe] ˆΨe (z), = i g 1 Âe ik 1 z ˆΨg (z) (3.16) [ Ĥatom pump] ˆΨe (z), = Ω 2 e iω 2t e ik 2 z ˆΨg (z) (3.17) ] [Â, Ĥ probe = ck 1 Â (3.18) ] [Â, Ĥ atom probe = i g 1 d 3 z ˆΨ g (z)e ik 1 z ˆΨe (z) (3.19) so the equation of motions read i d ˆΨ ) g (z) = ( 2 dt 2m 2 + V g (z) + 4π 2 σ m ˆΨ g (z) ˆΨ g (z) ˆΨ g (z)

48 36 Chapter 3. Quantum field theory i d ˆΨ e (z) dt i dâ dt + (i g s  e iks z + Ω2 ) eiωt e ik z ˆΨ e (z) = ω 0 ˆΨe (z) + = ck 1  + i g 1 ( i g 1 Âe ik1 z + Ω ) 2 e iω 2t e ik 2 z ˆΨ g (z) (3.20) d 3 r ˆΨ e (z)e ik 1 z ˆΨ g (z) (3.21) where we have dropped the kinetic energy and trap potential terms under the assumption that the lifetime of the excited atom, which is of the order 1/, is so small that the atomic center-of-mass motion may be safely neglected during this period. For the same reason, we are justified in neglecting collisions between excited atoms, or between excited and ground state atoms in the collision Hamiltonian (3.8). We proceed by introducing the operators ˆΨ e(z) = ˆΨ e (z)e iω 2t and â = Âeiω 2t, which are slowly varying relative to the optical driving frequency. The new excited state atomic field operator obeys then the Heisenberg equation of motion i d [ ] dt ˆΨ e(z) Ω = 20 ˆΨ e (z) + 2 eik 2 z i g 1 âe ik 1 z ˆΨ g (z), (3.22) We now adiabatically solve for ˆΨ e(z) by formally integrating Eq. (3.22) under the assumption that ˆΨ g (z) varies on a time scale which is much longer than 1/ 20. This yields ˆΨ e(z, t) 1 [ Ω ] 2 eik 2 z ig 1 â(t)e ik 1 z ˆΨ g (z, t) [ ] Ω 2 eik 2 z ig 1 â(0)e ik 1 z ˆΨ g (z, 0)e i t + ˆΨ e(z, 0)e i 20t. (3.23) The third term on the r.h.s. of Eq. (3.23) can be neglected for most considerations if we assume that there are no excited atoms at t = 0, so that this term acting on the initial state gives zero. The second term may also be neglected, as it is rapidly oscillating at frequency 20, and thus its effect on the ground state field operator is negligible when compared to that of the first term, which is non-rotating. It is useful to keep them temporarily to demonstrate that the commutation relation for ˆΨ e (z) is preserved (to order 1/ 20 ) by the procedure of adiabatic elimination. Dropping the unimportant terms, and then substituting Eq. (3.23) into the equations of motion for ˆΨ g (z) and for Â, we arrive at the effective Heisenberg equations of motion for the ground state field operator and for the probe field operator

49 3.2. Coupled-modes equations 37 (CARL-BEC model) i d [ dt ˆΨ 2 g (z) = 2m 2 + V g (z) + 4π 2 σ m ˆΨ g (z) ˆΨ g (z) +i g ( â e iq z âe iq z) ( Ω )] 20g 2 â â ˆΨ 4 20 Ω 2 g (z), (3.24) i d dtâ = δâ + i g d 3 z ˆΨ g (z)e iq z ˆΨg (z), (3.25) where we have introduced the new coupling constant g = g 1 Ω/ 20 that contains the parameters of the pump field. The recoil momentum kick the atom acquires from the two-photon transition is q = k 1 k 2, the detuning between the pump and probe fields is δ 21 = ω 2 ω 1 and the probe frequency is given by ω 1 ck 1, again assuming that the index of refraction inside the condensate is negligible. The second term in Eq. (3.24) is simply the optical potential formed from the counterpropagating pump and probe light fields, and the last term gives the spatially independent light shift potential, which can be thought of as cross-phase modulation between the atomic and optical fields. 3.2 Coupled-modes equations We assume that the atomic field is initially in a BEC with mean number of condensed atoms N. Furthermore, we assume that N is very large and that the condensate temperature is small compared to the critical temperature. These assumptions allow us to neglect the non-condensed fraction of the atomic field. Thus this model does not include any effect of condensate number fluctuations. We introduce creation and annihilation operators for the atoms of a definite momentum p = n q. So we suppose we can write ˆΨ(z) = + n=0 ĉ n Φ n (z) (3.26) where Φ 0 (z) is the condensate ground state that satisfies the time independent Gross-Pitaevskii equation ( ) 2 2m 2 V g (z) 4π 2 σ m N Φ 0(z) 2 Φ 0 (z) = 0. (3.27)

50 38 Chapter 3. Quantum field theory Φ n (z) for n 0 are the n-th side modes with momentum n q Φ n (z) = Φ 0 (z)e inq z. (3.28) and ĉ m are bosonic operators obeying the commutation relations [ĉ n, ĉ n ] = d 3 zφ n(z)φ n (z) = δ nn [ĉ n, ĉ n ] = 0 (3.29) We are assuming that the states Φ n (z) form a complete orthonormal system. In general this is non true, as the overlap integrals Φ n Φ m = d 3 zφ n(z)φ m (z) = d 3 z Φ 0 (z) 2 e i(n m)q z (3.30) are not zero for n m and are not 1 for n = m. For most condensate sizes and trap configurations, however, these integrals are many orders of magnitude smaller than unity. As a result, for typical condensate, the orthogonality approximation yields accurate results. By properly taking into account the non-orthogonality of the atomic field modes, it can be shown that the only surviving effect in the linearized theory (see next section) is the modification of the atomic polarization term in the equation of motion for the probe field (3.25) to include a second scattering mechanism in which a condensate scatters a photon without changing its center of mass state. As a consequence of momentum conservation, this process is suppressed by a factor Φ n0 Φ n0 1 relative to the process which transfers the atom from the condensate in the state n 0 to the side mode state n 0 1. Bose enhancement, on the other hand, is stronger for this transition by a factor N, because we now have N identical bosons in both the initial and final states. Thus it is the product N Φ n0 Φ n0 1 which must be negligible if we have to make the orthogonality approximation. From Eq. (3.26) the atomic field operator which annihilates an atom in the n-th condensate side mode is defined ĉ n = d 3 zφ n(z) ˆΨ(z) (3.31) Taking the derivative with respect to time and substituting Eq.(3.24) we obtain i d ( ) = n2 ( q) 2 Ω 2 dtĉn 2m ĉn + + g2 1 â â ĉ n + i g ( ) â ĉ n+1 âĉ n ( ) ĉ m d 3 zφ n(z)e inq z 2 2 2m + V (z) Φ 0 (z) m +β ĉ mĉ k ĉ l d 3 zφ nφ m(z)φ k (z)φ l (z) (3.32) m,k,l

51 3.2. Coupled-modes equations 39 and inserting the Gross-Pitaevskii Eq. (3.27) finally i d ( ) = n2 ( q) 2 Ω 2 dtĉn 2m ĉn + + g2 1 â â ĉ n + i g ( ) â ĉ n+1 âĉ n βn ĉ m d 3 zφ n(z) Φ 0 (z) 2 Φ m (z) m +β ĉ mĉ k ĉ l d 3 zφ n+m Φ 0 (z) 2 Φ l+k (z) (3.33) m,k,l Substituting Eq. (3.26) in Eqs. (3.25) we obtain for the probe field operator dâ dt = i 21â + g d 3 z ˆΨ g (z)e iq z ˆΨg (z). (3.34) The source of the field equation (3.34) is the bunching operator ˆB = d 3 z ˆΨ g (z)e iq z ˆΨg (z) (3.35) If we consider an ideal condensate with a constant atomic density the ground state is independent from position variables Φ 0 (z) = Φ 0 = 1/ V and eq.(3.33) takes the simpler form i d dtĉn = ( ) n2 ( q) 2 Ω 2 2m ĉn + + g2 1 â â ĉ n + i g ( ) â ĉ n+1 âĉ n βn V ĉn + β ĉ V mĉ k ĉ n+m k (3.36) m,k and now the bunching operator is given by ˆB = n= ĉ nĉ n+1 (3.37) Generally in experiments that involves the CARL mechanism, like for example superradiant Rayleigh scattering, the laser pulse is applied when the trap is completely switched off and the condensate is in expansion, so it can be interesting to study the model when the effects of trap potential and of collisions are negligible. In this regime we get to the following model for the coupled modes d = iω R n 2 ĉ n + g ( ) â ĉ n+1 âĉ n 1 (3.38) dtĉn dâ = i 21 â + g c dτ nc n+1. (3.39) n=

52 40 Chapter 3. Quantum field theory We note that Eqs.(3.38) and (3.39) conserve the number of atoms, i.e. n ĉ nĉ n = N, and the total momentum, ˆQ = â â + n nĉ nĉ n. Defining the operators ˆϱ m,n = ĉ mĉ n from Eq.(3.38) we derive d dt ˆϱ mn = iω R (m 2 n 2 )ˆϱ mn +g { â (ˆϱ m+1,n ˆϱ m,n 1 ) + â (ˆϱ m,n+1 ˆϱ m 1,n ) } (3.40) Taking the expectation values for the operators, taking scaled variables and with the substitution A = 2/ρN â e iδτ, Eqs. (3.40) and (3.39) are equivalent to Eqs. (2.16) and (2.17) introduced with first quantization in chapter 2. A more realistic and complete model should take into account even effects of atomic decoherence and cavity losses. We will see possible approaches to this problem in some of the next chapters (see chapters 5 and 8), modifying the model in the proper way for the considered situation. 3.3 Linearized three-mode model From Eq. (3.40), we see that the first-order side modes are optically coupled to both the condensate mode and to second-order side modes. For times short enough that the condensate is not significantly depleted, the coupling back into the condensate is subject to Bose enhancement due to the presence of N identical bosons in this mode. The coupling to the second-order side mode, in contrast, is not enhanced. Hence for these time scales, the higher-order side modes are not expected to play a significant role. These arguments suggest developing an approach where, assuming that all N atoms are initially in the condensate mode n 0 with momentum n 0 q, the three atomic field operators ĉ n0, ĉ n0 1, and ĉ n0 +1 play a predominant role. Therefore, we expand the atomic field operator as ˆΨ g (z) = z Φ n0 ĉ n0 + z Φ n0 1 ĉ n0 1 + z Φ n0 +1 ĉ n ˆψ(z), (3.41) where the field operator ˆψ(z) acts only on the orthogonal complement to the subspace spanned by the state vectors Φ n0, Φ n0 1, and Φ n0 +1. As a result, ˆψ(z) commutes with the creation operators for the three central modes. With the assumption of negligible condensate depletion we can simply drop the operator ĉ n0 substituting it with its mean value ĉ n0 Ne in2τ/ρ. The system is

53 3.4. Concluding remarks 41 unstable for certain values of the detuning. In fact, by linearizing Eqs.(3.38) and (3.39) around the equilibrium state, the only equations depending linearly on the radiation field are those for ĉ n0 1 and ĉ n0 +1. Hence, in the linear regime, the only transitions involved are those from the state n 0 towards the levels n 0 1 and n With respect to scaled variables and introducing the operators â 1 = ĉ n0 1e i(n2 0 τ/ρ+ τ) (3.42) â 2 = ĉ n0 +1e i(n2 0 τ/ρ τ) (3.43) â 3 = âe i τ, (3.44) Eqs.(3.38) and (3.39) reduce to the linear equations for three coupled harmonic oscillator operators: dâ 1 dτ dâ 2 dτ dâ 3 dτ = iδ â 1 + ρ/2â 3 (3.45) = iδ + â 2 ρ/2â 3 (3.46) = ρ/2(â 1 + â 2 ), (3.47) with Hamiltonian ρ Ĥ = δ + â 2â 2 δ â 1â 1 + i 2 [(â 1 + â 2 )â 3 (â 1 + â 2)â 3 ], (3.48) where δ ± = δ ± 1/ρ and δ = + 2n 0 /ρ = (ω 2 ω 1 + 2n 0 ω R )/ρω R. Hence, the dynamics of the system is that of three parametrically coupled harmonic oscillators â 1, â 2 and â 3 [49], which obey the commutation rules [â i, â j ] = 0 and [â i, â j ] = δ ij for i, j = 1, 2, 3. Note that the Hamiltonian (3.48) commutates with the constant of motion We will solve exactly this model in chapter 6. C = â 2â 2 â 1â 1 + â 3â 3. (3.49) 3.4 Concluding remarks We have deduced an appropriate quantum field theory that extends into the ultracold regime of BEC the CARL model, so that the unique coherence properties of the condensates might be further understood and exploited by the interaction with

54 42 Chapter 3. Quantum field theory dynamical light fields. In the limit of no collisions and considering expectations values of bosonic operators this model reduces to the quantum one introduced in chapter 2 by first quantization. Furthermore we have linearized the fundamental equations reducing the theory to a three mode model. This will allow us to study extensively the quantum statistical features of the system and in particular to show its entanglement properties.

55 Chapter 4 Superradiant Rayleigh scattering and matter waves amplification The extensions of CARL into the regime of BEC focuses, as we shall see in the next chapters, mainly on exploiting the instability of the light-matter interaction in the good-cavity regime to parametrically amplify atomic and optical waves as well as to optically manipulate matter-wave coherence properties and generate entanglement between atomic and optical fields. Till now, experiments in a good-cavity regime has been performed only for the classical CARL [17]. However experiments by Ketterle and coworkers at MIT [19, 39] and by Kozuma and coworkers in Tokio [40] have demonstrated that this instability can play an important role also in the case in which laser light is scattered into the vacuum modes of the electromagnetic field in absence of the cavity. These experiments, that we review in the following, have demonstrated the formation of atomic matter waves in a cigar-shaped BEC pumped by an off-resonant laser beam, together with highly directional scattering of light along the major axis of the condensate. This emission has been interpreted as superradiant Rayleigh scattering, and successively investigated [21, 41, 42, 43] using the quantum extension of CARL. A variation of Dicke superradiance was observed in which the role of electronic coherence, which stores the memory of previous scattering events, is replaced by coherence between center-of-mass momentum states, i.e. interference fringes in the atomic density. Referring to the simple quantum model presented in chapter 2 these kind of experiments lay in the superradiant regime of CARL: the absence of the 43

56 44 Chapter 4. Superradiant Rayleigh scattering and matter waves amplification cavity can be described considering an appropriate high value of radiation losses κ. 4.1 Directional matter waves produced by spontaneous scattering First we describe the superradiant Rayleigh scattering experiment performed at M.I.T. in 1999 [19]. The geometry of the setup is shown in Fig. 4.1A. An elongated BEC is illuminated with a single off-resonant laser beam, travelling perpendicular to the long axis of the condensate. Atoms in the condensate absorb a photon from the laser beam and spontaneously emit a photon, receiving recoil momentum and energy. The direction of the spontaneous emission was expected to be random, leading to momentum diffusion and heating of the atomic cloud. This was confirmed by observing the momentum distribution of scattered atoms when the polarization of the incident beam was parallel to the long axis of the condensate. The momentum distribution was isotropic, modified by the dipolar emission pattern (Fig. 4.1B-D). However, when the polarization of the laser beam was perpendicular to the long axis of the condensate, highly directional beams of atoms were observed in time-offlight images (Fig. 4.1E-G). The observation of directional atomic beams strongly suggests the build-up of a high contrast matter wave grating in the condensate when these scattered atoms still overlapped with the condensate at rest. This build-up of the matter wave grating can be easily understood by following each step of the off-resonant Rayleigh scattering (Fig. 4.2). When a condensate is exposed to a laser beam with wave vector k, it absorbs a photon from the laser beam and spontaneously emits a photon with wave vector k s, generating an atom with recoil momentum q j = (k k s ) (Fig. 4.2a,b). Because light propagates at a velocity about 10 orders of magnitude faster than the atomic recoil velocity (3cm/s for sodium), the recoiling atoms remain within the volume of the condensate long after the photons have left and affect subsequent scattering events. They interfere with the condensate at rest to form a moving matter wave grating of wave vector q j, which diffracts the laser beam into the phase-matching direction, i.e. the same direction k s again (Fig.4.2c,d). This diffraction is a self-amplifying process because every diffracted photon creates another recoiling atom that increases the amplitude of the matter wave grating (Fig. 4.2e). Because each scattered photon creates

57 4.1. Directional matter waves produced by spontaneous scattering 45 Figure 4.1: Observation of superradiant Rayleigh scattering. (A) An elongated condensate is illuminated with a single off-resonant laser beam. Collective scattering leads to photons scattered predominantly along the axial direction and atoms at 45. (B to G) Absorption images after 20ms time of flight show the atomic momentum distribution after their exposure to a laser pulse of variable duration. When the polarization was parallel to the long axis, superradiance was suppressed, and normal Rayleigh scattering was observed (B to D). For perpendicular polarization, directional superradiant scattering of atoms was observed (E to G) and evolved to repeated scattering for longer laser pulses (F and G). The pulse durations were 25 (B), 100 (C and D), 35 (E), 75 (F), and 100 (G) µs. The field of view of each image is 2.8mm by 3.3mm. The scattering angle appears larger than 45 because of the angle of observation. All images use the same color scale except for (D), which enhances the small signal of Rayleigh scattered atoms in (C). Figure taken from Ref. [19]

58 46 Chapter 4. Superradiant Rayleigh scattering and matter waves amplification Figure 4.2: Build up of matter wave grating inside a condensate due to light scattering. A condensate illuminated by a single laser beam (a) scatters a photon spontaneously (b). The scattered atom interferes with the condensate at rest and forms a matter wave grating (c). Diffraction of light by the matter wave grating (d) transfers more atoms into the recoil mode, leading to higher dffration efficiency (e). As a result, the number of atoms in the recoil mode shows exponential growth. Figure taken from Ref. [50]. a recoiling atom, the diffraction efficiency of the grating gives the growth rate of each matter wave mode. The diffraction efficiency is proportional to the square of the depth of the density modulation, and therefore to the number of atoms in the recoil mode N j. This implies an exponential growth of N j, as long as one can neglect the depletion of the condensate at rest. gain can be understood as follows. The angular dependence of the Given the same contrast of the grating, one can show that the intensity of the scattered light in the phase-matching direction does not depend on the shape of the condensate. However, the total intensity of the scattered light, or the rate of scattering, is largest when the phase-matching direction of photon emission is along the long axis of the condensate, since the condition is fulfilled over a wider solid angle due to the finite size of the condensate. The gain equation for the number of atoms in recoil mode j can be either obtained from a semi-classical treatment or from a fully quantum mechanical treatment based on second-quantization and Fermi s Golden Rule Ṅ j = RN 0 sin 2 θ j 8π/3 Ω j(n j + 1) (4.1) Here, R is the rate for single-atom Rayleigh scattering, which is proportional to the laser intensity. The angular term reflects the dipolar emission pattern with θ j being the angle between the polarization of the incident light and the direction of emission.

59 4.2. Dicke superradiance and emerging coherence 47 The solid angle factor Ω j λ 2 /A j depends on the angle of photon emission through A j, which is the cross-sectional area of the condensate perpendicular to the direction of photon emission. Eq. (4.1) describes both the normal Rayleigh scattering at a constant rate Ṅj = RN 0 when N j 1 and exponential gain of the j-th recoil mode once N j becomes non negligible. Initially, the angular distribution of the scattered light follows the single-atom spontaneous (dipolar) emission pattern but can become highly anisotropic when stimulation by the atomic field becomes important. Eq. (4.1) is valid in absence of decoherence. 4.2 Dicke superradiance and emerging coherence This build-up of coherence in an atomic ensemble through spontaneous photon emission is analogous to the superradiance discussed by Dicke [51, 52, 53]. He first considered spontaneous emission from two excited atoms. If the spacing between these two atoms is smaller than the wavelength, the two dipoles are in phase after the first photon emission, resulting in a factor of two larger spontaneous emission rate for the second photon emission compared to the single-atom rate. If the spacing between the two atoms is larger than the wavelength, the phase relationship between the dipoles does not have to be symmetric any more, but it introduces a strong correlation between the direction of the first and the second photon emission. Consider angular distribution of spontaneously emitted photons from two excited state atoms separated by 2.5λ (see Fig.4.3). If the first emitted photon was found in one of the nodes of the radiation pattern from dipoles oscillating in phase, the probability distribution of the direction of the second photon emission is given by the solid line in Fig. 4.3, since the wave function after the first photon emission is in an anti-symmetric state. These correlations between the successive photon emission is further pronounced in superradiance from extended samples. For an anisotropic sample, the strength of the correlation is largest when photons are emitted along its longest axis ( endfire mode ). As a result, highly directional light beams come out from incoherently excited, elongated samples of atoms. The light beam has the same property as a laser beam and was called coherence-brightened laser, but its coherence is maintained by the phase relationship between the dipoles, not by recycling photons in a cavity.

60 48 Chapter 4. Superradiant Rayleigh scattering and matter waves amplification Figure 4.3: Dicke Superradiance from two atoms in excited states, separated by distances smaller or larger than the wavelenght of radiation. If the two atoms are spaced closer than the wavelength (a), the two dipoles oscillates in phase after the first photon emission and thus couples strongly to the radiation field. If two atoms are separated by more than the wavelength (b), the dipoles can acquire either in phase (probability f) or out of phase (probability 1 f) oscillation as a result of the first photon emission, and this phase difference affects the direction of the second photon emission. In case of two atoms separated by d = 2.5λ, if the first photon is emitted into one of the nodes of the dotted line (the radiation pattern from two dipoles in phase), it indicates out of phase oscillation of the dipoles, and the probability distribution of the second photon emission is given by the solid line (the radiation pattern from two dipoles oscillating out of phase). Figure taken from Ref. [50] The full analogy between the Dicke superradiance and the superradiant Rayleigh scattering can be summarized as follows. The condensate at rest dressed by the off-resonant laser beam corresponds to the electronically excited state in the Dicke state. It is important to notice that we are not referring to the fraction of atoms in electronically excited state in the dressed condensate. The dressed condensate can decay by spontaneous emission to a state with photon recoil, corresponding to the ground state. The rate of superradiant emission in Dicke s treatment is proportional to the square of an oscillating macroscopic dipole moment. In the case presented here, the radiated intensity is proportional to the matter wave interference pattern between the condensate and recoiling atoms.

61 4.3. Evidence for decoherence 49 Figure 4.4: Observation of directional emission of light. (A) The angular pattern of the emitted light along the axial direction showed a few bright spots with an angular width θ D (1/e 2 diameter) of 107 ± 20mrad, corresponding to the diffraction-limited angle of an object of 14mµ in diameter. The images were integrated over the entire duration of the light pulse. (B) The temporal evolution of the light intensity showed a strong initial increase characteristic of a stimulated process. For higher laser power, the pulse was shorter and more intense. The laser intensities were 3.8 (solid line), 2.4 (dashed line), and 1.4mW/cm 2 (dotted line), and the duration was 550ms. The inset shows a double peak in the temporal signal when the laser intensity was about 15mW/cm 2, which was above the threshold for sequential superradiant scattering. The photomultiplier recorded the light over an angle of 200mrad around the axial direction. (C) The dependence of the inverse initial rise time on the Rayleigh scattering rate shows a threshold for the stimulated process. The solid curve is a straight-line fit. Figure taken from Ref.[19]. 4.3 Evidence for decoherence The directional scattering of light was verified by directing the light onto a CCD camera that was positioned out of focus of the imaging system, and observing the angular distribution of photons emitted around the axial direction (Fig. 4.4A). The

62 50 Chapter 4. Superradiant Rayleigh scattering and matter waves amplification images consisted of bright spots with angular widths equal to the diffraction limit for a source with a diameter of 14µm. Typical images showed more than one such spot, and their pattern changed randomly under the same experimental condition. This large run to run variation arises from the amplification of initial quantum fluctuations. By replacing the camera with a photomultiplier, a time-resolved measurement of the scattered light intensity was obtained (Fig. 4.4b). Simple Rayleigh scattering would give a constant signal during the square-shaped incident pulse. Instead, a fast rise and a subsequent decay consistent with a stimulated process was observed. Measurement at variable laser intensities showed a threshold for the onset of superradiance and a shorter rise time for higher laser intensities. This behavior can be accounted for by introducing a loss term γ 0 : Ṅ j = (G j γ 0 )N j (4.2) where G j = RN 0 sin 2 θ j Ω j/(8π/3) is the gain coefficient. The exponential rate (G j γ 0 ) was determined by putting the initial rise in the light intensity. The inverse rise time Ṅj/N j versus the Rayleigh scattering rate R is shown in Fig. 4.4C. The slope gives G j /R, and the offset determines the loss γ 0. The offset in Fig. 4.4C determines the threshold for superradiance and yields 1/γ 0 = 35 µs. This decoherence rate of the matter wave grating was measured as a Bragg resonance width [54]. The observed full width at half-maximum of about 5kHz yields a decoherence rate of 32 µs, in good agreement with the value shown above. For higher laser powers, a distinct change in both the momentum patter of the atoms (Fig. 4.1F and G) and the photomultiplier traces (Fig. 4.4B) was observed. The atomic pattern showed additional momentum peaks that can be explained as a sequential scattering process in which atoms in the initial momentum peak undergo further superradiant scattering. These processes are time-delated with respect to the primary process and showed up as a second peak in the time-resolved photomultiplier traces (Fig. 4.4B). This cascade of superradiant scattering processes is exactly what predicted by the quantum CARL in the superradiant regime. Superradiance is based on the coherence of the emitting system, but it does not require quantum degeneracy. The condition for superradiance is that the gain exceed the losses or that the superradiant decay time be shorter than any decoherence time.

63 4.4. Seeding the superradiance 51 Above the BEC transition temperature T c, thermal Doppler broadening results in a 30 times shorter decoherence time than for a condensate. Therefore the threshold for superradiance in a thermal cloud is several orders of magnitude higher than for a condensate. 4.4 Seeding the superradiance In the field of atom optics it has long been speculated, in direct analogy with optical amplifiers, that it should be possible to coherently amplify a matter wave by using an appropriate gain medium. Matter wave amplification differs from light amplification in one important aspect. Since the total number of atoms is conserved (in contrast to photons), the active medium of a matter wave amplifier has to include a reservoir of atoms. One also needs a coupling mechanism which transfers atoms from the reservoir to the input mode while conserving energy and momentum. Amplification is realized if the transfer mechanism is accelerated by the build-up of atoms in the final state and irreversibility of the process is ensured by some form of dissipation. Spontaneous Rayleigh scattering from a condensate leads to the buildup of directional matter waves and this buildup can be regarded as matter wave amplification seeded by quantum fluctuations. This mechanism can be thought of as a stimulated matter-wave generator, in analogy with parametric generation in optical laser systems. However, because the process starts from spontaneously generated noise, the phase of the atomic grating is not controllable. Furthermore, there has been no experimental evidence as to whether this superradiant element maintains the long-range coherence properties of the condensate. The groups of Kozuma and Ketterle reported the first demonstration of an active, phase-coherent, matter-wave amplifier [39, 40]. Instead of relying on spontaneously generated noise, these experiments are performed establishing a small, coherent atomic seed in the BEC in a high momentum state using optical Bragg diffraction [38]. The superradiance effect was used solely as a gain medium. In particular we know describe the experiment performed in Tokio. Clear evidence of the increase in the population of the seed matter wave was observed. The phase of the amplified matter wave was locked to that of the injected seed, using a new type of Mach-Zehnder interferometer [55].

64 52 Chapter 4. Superradiant Rayleigh scattering and matter waves amplification Figure 4.5: (A) Direction along which the elongated condensate is illuminated by the superradiance (pulse A) and Bragg (pulses B and C) beams. (B through E) Absorption images showing the atomic momentum distribution 20ms after the condensate was exposed to various laser pulses. The 20ms time of flight (TOF) allows the momentum states enough time to separate completely in space. For these images, the width of the field of view is 940mm by 350mm, and the duration of pulse A is 520ms. Figure taken from Ref. [40] The experiment was performed with a condensate of Rb atoms formed in a dc magnetic trap with a standard evaporation technique. About 10 9 atoms was first trapped in an ultrahigh vacuum cell using a double magneto-optical trap. The atoms were then transferred into a dc magnetic trap and were further cooled by radio frequencyinduced evaporation. A BEC was created in the 5S 1/2, F = 1, m = 1 state. The condensate is highly elongated. After forming the BEC, the magnetic trap was turned off. After waiting long enough that the magnetic fields decayed completely away superradiance or Bragg laser pulses or both (Fig. 4.5A) was then applied. Both the superradiance pulse (pulse A in Fig. 4.5A) and the Bragg pair pulse (pulses B and C in Fig. 4.5A) were derived from an optically amplified, grating-stabilized diode laser. The laser beams were detuned from the 5S 1/2 F = 1 5P 3/2 F = 2 transition by /2π = 2GHz, had an intensity of 3mW/cm 2, and were applied along the axial (long) direction of the cigar-shaped

65 4.4. Seeding the superradiance 53 condensate. When only beam A was applied, the recoiled-atom growth rate from superradiance was given by Eq.(4.2). The dominant phase-matched grating causes photons to backscatter, and these atoms thereby acquire a momentum of 2 k where q k k s. The choice of applying the laser along the axial direction only supports this single superradiance mode. Applying a single pulse A 500µs after turning off the magnetic trap, when the BEC was still well elongated (Fig. 4.5B), resulted in the coherent buildup of the matter-wave grating because of spontaneous superradiance, and caused 62% of the atoms to be scattered into this q momentum state. In order to make a matter-wave amplifier, this spontaneous superradiance had to be suppress. That is, only an injected seed matter wave should be amplified. To do this, the time before applying pulse A after extinguishing the trap was increased in order to change the aspect ratio of the condensate. The condensate profile became more spherical because of the mean-field-driven expansion, and thus there was less of a preferred grating direction. When the pulse A was applied 1.8ms after extinguishing the magnetic trap (Fig. 4.5C), the coherent buildup of atoms in the q state was greatly suppressed (the condensate aspect ratio was reduced by a factor of 2, decreasing the gain term in Eq.(4.2) by a factor of 4). Coherent Bragg diffraction created the small matter-wave seed that was used to test the amplification process. The Bragg pulse was formed by a pair of linearly polarized, counterpropagating laser beams of slightly different frequencies [38]. The frequency difference between the two laser beams was chosen to fulfill a first-order Bragg diffraction condition that changes the momentum state of the atoms without changing their internal state. The atoms acquired a kinetic energy of 4 E r (E r h 3.8kHz is the 87 Rb single-photon recoil energy), and the necessary relative detuning was /2π = 15kHz. Figure 4.5D shows the result of diffracting 6.5% of the atoms into the 2 k state using a 15 µs Bragg diffraction pulse. The amplification was accomplished by successive application of the seed Bragg pulse and of the superradiance pulse A, before the diffracted atoms left the region of the condensate. Applying the superradiance pulse A after creating the seed matter wave (Fig. 4.5E) produced a striking increase of atoms in the 2 k state to 66% of the original condensate number. The population of the injected coherent atomic matter wave was thereby amplified by more than a factor of 10. This increase can

66 54 Chapter 4. Superradiant Rayleigh scattering and matter waves amplification Figure 4.6: (A) Population growth in the 2 k state is plotted as a function of the duration of pulse A both with (circles) and without (squares) the matter-wave seed. The solid curves are one-parameter fits (L j ) to Eq. (4.2). Here V j is approximated to be λ 2 /S, where S is the cross-sectional area of the condensate perpendicular to the direction of light emission, and λ is the optical wavelength. The best fit yielded L j = 3.8ms 1 for both curves. (B) Absorption image taken after 15ms TOF for a 50% split ratio. Although there are equal numbers of atoms in the two momentum states, their density distributions are not the same. The width of the field of view is 400mm by 260mm. Figure taken from Ref. [40] be seen in Fig. 4.6A, where population in the 2 k state is plotted as a function of the duration of the superradiance pulse A. At each pulse duration, no superradiance growth occurred without the injected seed (only normal Rayleigh scattering occurred). However, with the seed wave injected, the 2 k population increased as the duration of pulse A increased, allowing to control the gain of the matter-wave amplifier. The case where the duration of pulse A was chosen so that at the end of the pulse pair there were equal numbers of atoms (50%) in each momentum state is shown in Fig. 4.6B. It is important to examine the coherence properties of this amplified wave. The phase coherence of the seed wave may have been completely lost by the amplification process, a phenomenon analogous to strongly amplified spontaneous emission (ASE) [56], frequently seen in optical pulsed-dye laser amplifiers that are not well aligned. In order to determine whether the amplified matter wave was phase-coherent, its phase was studied with a novel Mach-Zehnder interferometer (Fig. 4.7) [55]. The long-range order was maintained and the phase was locked to that of the seed matter

67 4.4. Seeding the superradiance 55 Figure 4.7: Schematic of the Mach-Zehnder interferometer used to test the phase coherence of the amplified matter-wave seed. Figure taken from Ref. [40] wave. The phase-coherent matter-wave amplifier presented is capable of amplifying a matter wave whose momentum p is in the range 0 < p < 2 k, provided the phasematching condition is respected. If φ is the angle between the direction of propagation of the superradiance beam and that of the seed matter wave, the phase-matching condition becomes p = 2 k cos φ. This condition is relaxed by the momentum spread of the source condensate, which, for a particular choice of φ, can be thought of as the matter wave amplifier bandwidth. In this experiment, both the Bragg and the superradiance pulses were illuminated along the same direction. If a small fraction of atoms along a different direction was injected, it would be possible to study mode competition in the matter-wave amplification process. One could use phase-coherent matter-wave amplification to enhance the number of atoms in atom lithography or holography [57] experiments in order to reduce signal accumulation time. Furthermore, it should be possible to make a ring

68 56 Chapter 4. Superradiant Rayleigh scattering and matter waves amplification cavity for matter waves using multiple Bragg diffractions as mirrors. By combining such a matter-wave cavity and the phase-coherent amplification mechanism showed here, it should be possible to construct a new type of highbrightness atom laser. 4.5 Concluding remarks The experiments presented in this chapter were extensively analyzed through quantum extensions of CARL model in Ref. [21, 41, 42, 43] taking into account the particular 2D geometry of the system and inserting proper description of the atomic decoherence term. In general the rate of matter-wave decoherence in superradiance or CARL-type experiments is given by the ratio between the recoil velocity and the matter-wave coherence length. As the coherence length of a BEC is significatively larger than that of a non condensate atomic cloud, the threshold for superradiance above T C is much larger than below T C. This fact explains why superradiance was never observed above T C in the MIT experiment. We remark that the presence of an optical cavity would provide additional feedback, which could compensate for the lack of atomic coherence.

69 Chapter 5 Superradiant Rayleigh scattering from a moving BEC In this chapter we investigate the influence of the initial velocity of the condensate on superradiant Rayleigh scattering and the effect of seedindg superradiance by a weak signal directed in the opposite direction with respect to the pump laser. Our theoretical analysis is based on some experiments performed at European Laboratory for Non-linear Spectroscopy (LENS) in Florence by Francesco Saverio Cataliotti, Leonardo Fallani, Chiara Fort and Massimo Inguscio. An elongated BEC of rubidium atoms has been produced and exposed to a single off-resonant laser pulse directed along the condensate symmetry axis. The laser is far detuned from any atomic resonance so the only scattering mechanism present is Rayleigh scattering. Due to the geometry of these experiments, photons are back-scattered with k 1 k 2, where k 2 is the wave-vector of the laser photon, and the atoms move away from the original condensate with a relative momentum q 2 k 1 in the direction of the laser beam. The efficiency of the process is limited by the decoherence between the original and the recoiled atomic wavepackets causing the damping of the matter-wave grating. From data analysis we can identify two different mechanisms for decoherence, one resulting from Doppler and mean field broadening of the matter wave field, already descibed in chapter 4, and the other due to phase diffusion. The latter mechanism, dependent on the energy separation between the initial and final states of the system [44, 58], can be controlled by initially setting the condensate into motion. In particular, experimental results 57

70 58 Chapter 5. Superradiant Rayleigh scattering from a moving BEC (A) p 0 BEC trap (B) laser beam, k g (C) absorption imaging p 0 p + 2 k 0 Figure 5.1: Schematics of the experimental procedure. The condensate is set in motion by a sudden displacement of the magnetic trap center (A). When the condensate reaches the desired momentum p 0 we switch off the magnetic trap and flash the atoms with a far off resonance laser pulse directed along the condensate symmetry axis (B). After an expansion time allowing a complete separation of the momentum components (28 ms) we take an absorption image of the atoms (C). show that phase diffusion decoherence vanishes when the initial condensate momentum is such that after the interaction with the laser beam the scattered atomic wavepacket has the same kinetic energy of the original condensate in the laboratory frame. On the other hand, when counterpropagating laser beams are used, light forms a periodic structure on which the matter wave can perform Bragg scattering. Indeed this process too is coherent and the condensate undergoes Rabi oscillations between different momentum states. We study the transition between the superradiant regime and Rabi oscillations regime by introducing a small amount of laser light (seed beam) counterpropagating with respect to the pump beam. 5.1 Theoretical analysis The evolution of the system in this regime is described by the CARL-BEC model described in chapter 3 properly reduced to one dimension. With the assumption of constant atomic density we can consider the model for the density operators, Eq.

71 5.1. Theoretical analysis 59 (3.40), and taking expectation values ϱ m,n = ˆϱ m,n and a = â we write dϱ m,n dt da dt = i(ω m ω n )ϱ m,n +g{a(ϱ m,n 1 ϱ m+1,n ) + a (ϱ m 1,n ϱ m,n+1 )} (5.1) = gn ϱ n,n+1 κ c a. n (5.2) where ω n = 4ω r n 2 21 n, (5.3) the pump probr detuning is 21 = ω 2 ω 1 and the recoil frequency has been scaled so that ω R = 4ω r where ω r = k 2 1/2M. This model has been written in the mean field limit, so the propagation effects can be modelled with a damping term added in Eq. (5.2) where κ c c/2l and L is the condensate length. The nonlinear term has been neglected since the experiment has been performed after expansion From the analysis of experimental data we can deduce that this model fails in taking into account a decoherence mechanism depending on the initial momentum of the condensate. In literature the standard approach to the problem of decoherence is to write the proper master equation [59] for the system under investigation. In our case we write where dˆϱ dt = i [Ĥ, ˆϱ] τ 2 [Ĥ0, [Ĥ0, ˆϱ]], (5.4) ˆϱ = m,n ϱ m,n m n (5.5) Ĥ = Ĥ0 + ˆV (5.6) Ĥ 0 = 4 ω R ˆp 2 z 21ˆp z (5.7) ˆV = i g(â e 2ikz h.c.). (5.8) where ˆp z = ˆp/2 k is the normalized momentum operator with, in a Fock representation, eigenstates n and eigenvalues n. The phase destroying term with the double commutator in the right-hand side of Eq. (5.4) has appeared in many models of decoherence and induces diffusion in variables that do not commute with the Hamiltonian, preserving the number of atoms in the condensate. In it we have neglected the interaction ˆV in the weak-coupling limit g 2 N/κ c ω r. From Eqs.(5.4)

72 60 Chapter 5. Superradiant Rayleigh scattering from a moving BEC and (5.2) we now obtain: dϱ m,n dt da dt = i(ω m ω n )ϱ m,n + g{a(ϱ m,n 1 ϱ m+1,n ) + a (ϱ m 1,n ϱ m,n+1 )} τ 2 (ω m ω n ) 2 ϱ m,n (5.9) = gn ϱ n,n+1 κ c a. n (5.10) These equations differ from Eqs. ( ) for the damping term in Eq. (5.9) generated by the double commutator in Eq. (5.4). This term is fundamental to describe the experimental results and describes a phase-diffusion decoherence process, whose amplitude is characterized by a constant τ. It arises from a δ-correlated gaussian noise on the eigenenergies of the system and causes the decay of the off-diagonal matrix elements, so that the density matrix becomes diagonal in the basis of the recoil momentum states. This decoherence in the superposition state causes the decay of the matter wave gratings resulting from the quantum interference of the two momentum components. Therefore the self amplifying process for the backscattered radiation is stopped. In the experimental conditions at L.E.N.S the superradiant Rayleigh scattering involves only neighboring momentum states, i.e. transitions from the initial momentum state p 0 = n(2 k) to the final momentum state (n + 1)2 k. In this limit, the system is equivalent to a two-level system and Eqs.(5.9) and (5.2) reduce to the Maxwell-Bloch system [60, 20]: ds dt dw dt da dt = gaw γ n S (5.11) = 2g(AS + h.c.) (5.12) = gns (κ c i n )A, (5.13) where S = ϱ n,n+1 e i nt, A = ae i nt, W = P n P n+1 is the population fraction difference between the two states (where P n = ϱ n,n and P n + P n+1 = 1), n = ω 2 ω 1 4ω R (2n + 1) (5.14) is the detuning from the Bragg resonance with the scattered field (i.e. the condition for enegy conservation) and the decoherence rate γ n is given by: γ n = γ 0 + τ 2 2 n = γ 0 + τ [ ( p0 )] 2 ω 2 ω 1 4ω R 2 k + 1. (5.15)

73 5.1. Theoretical analysis 61 Population fraction Population fraction Population fraction Pulse length (ms) Pulse length (ms) Pulse length (ms) p 0= - k p = 0 0 p 0= + k Figure 5.2: Left) Time evolution of population in the original condensate (empty circles) and in the recoiled wavepacket (filled circles) for different pulse durations. The solid line is a fit with the hyperbolic tangent (5.19) predicted by the theoretical model, the dotted line is just one minus the fit curve. The momentum of the original condensate is set to k (top), 0 (center) and + k (bottom). Right) Plot of the atomic density profile after interaction with a 250µs pulse for the three cases of original momentum as on the left. The laser detuning and intensity are 13 GHz and 1.35 W/cm 2 respectively. To the decoherence rate γ n we have added an extra term γ 0 taking into account other decoherence decay mechanisms, as for instance Doppler and inhomogeneous broadenings of the two-photon Bragg resonance discussed in chapter 4. We note that in Eq. (5.13) S represents half of the amplitude of the matter-wave grating. In fact, if Ψ c n u n (z) + c n+1 u n+1 (z), (5.16) the longitudinal density is Ψ 2 2 λ {1 + 2Re[S exp(2ikz)]}, (5.17)

74 62 Chapter 5. Superradiant Rayleigh scattering from a moving BEC which describes a matter wave grating with a periodicity of half the laser wavelength. The main result is that the second term of Eq. (5.15), arising from a phase diffusion decoherence mechanism, depends on the frequency detuning between the incident and scattered radiation beams and on the initial momentum of the condensate, p 0 = n(2 k). We observe that the velocity-dependent term of the decoherence rate is invariant under Galilean transformation. In fact, in a frame moving with respect to the laboratory frame with a velocity v, the shift of p 0 compensates the Doppler shift of the frequency difference ω 2 ω 1. The parameters used in the experiment match those for the superfluorescent regime, in which the field loss rate κ c is much larger than the coupling rate g N. In this regime for t κ 1 c da/dt = 0 so that we can perform an adiabatic elimination putting A gns (κ c i n ). (5.18) The analytical solution for the fraction of atoms with initial momentum p 0 = n(2 k) is where P n = ( 1 2γ ) { n } G 2 Th [(G 2γ n) (t t 0 )], (5.19) G = 2g2 Nκ c (κ 2 c + 2 n) (5.20) is the superradiant gain and t 0 is a delay time. Furthermore, in the analyzed experiment κ c n, so that G 2g 2 N/κ c hence independent from the atomic velocity. Eq. (5.19) assumes the threshold condition G > 2γ n i.e. the gain must be larger than the decoherence rate. 5.2 Experimental features The experiment is performed with a cigar-shaped condensate of 87 Rb produced in a Ioffe-Pritchard magnetic trap by means of RF-induced evaporative cooling. The axial and radial frequencies of the trap are ω z /2π = 8.70(7) Hz and ω y /2π = 90.1(4) Hz respectively, with the z-axis oriented horizontally. After the evaporative ramp a collective dipole motion of the condensate inside the harmonic potential is induced along the z-axis, allowing to tune the atomic velocity. The dipole oscillation is

75 5.2. Experimental features 63 8 Decoherence rate (ms -1 ) Initial momentump 0 ( k units) Figure 5.3: Decoherence rate as a function of the initial momentum of the condensate. The solid line is a fit of the experimental data with a parabola centered in p 0 = k, as expected from the theoretical model. excited by non-adiabatically displacing the center of the magnetic trap. When the condensate has reached the maximum velocity in the magnetic potential, the trap is suddenly switched off and the cloud expand with a horizontal velocity proportional to the displacement of the trap (see Fig.5.1). A square pulse of light along the z-axis is applied, 2 ms after the release of the condensate, when the magnetic field of the trap is completely switched off, and the atomic cloud has still an elongated shape. After 2 ms of expansion the radial and axial sizes of the condensates are typically 10 and 70 µm, respectively. The pulse length is controlled with an acousto-optic modulator. The light comes from a diode laser red-detuned 13 GHz away from the rubidium D2 line at λ = 780 nm and has an intensity of 1.35 W/cm 2 corresponding to a Rayleigh scattering rate of roughly s 1. The linearly polarized laser beam is collimated and aligned along the z-axis of the condensate. In this geometry the superradiant light is backscattered and the self-amplified matter-wave propagates in the same direction of the incident light. After an expansion of 28 ms, when the two momentum components are spatially separated, an absorption image of the

76 64 Chapter 5. Superradiant Rayleigh scattering from a moving BEC cloud along the horizontal radial direction is taken.in order to minimize spurious reflections the laser beam has been aligned at a nonzero angle with respect to the normal to the vacuum cell windows. In section 5.3 we will discuss in detail the possible effect of some counterpropagating light seeding the superradiant process. In Fig. 5.1C we show a typical absorption image in which the left peak is the condensate in its original momentum state p 0 and the right peak is formed by atoms recoiling after the superradiant scattering at p k. The spherical halo centered between the two density peaks is due to non-enhanced spontaneous processes, i.e. random isotropic emission following the absorption of one laser photon. The number of atoms in both the original and the recoiled peaks is extracted from a 2D-fit of the pictures assuming a Thomas-Fermi density distribution.the population in the two momentum peaks has been studied as a function of the duration of the laser pulse for different initial velocities of the condensate. In Fig. 5.2 we report the population fractions of the initial wavepacket, P n, and of the scattered wavepacket, P n+1, as functions of the laser pulse duration for three different initial momenta p 0. The continuous line represent the fit with the theoretical function of Eq. (5.19). from the fit we extract the values of G and γ n for different p 0. The measured value of G = 19(3) ms 1 does not appreciably depend on p 0, as expected from the theoretical treatment. On the contrary we observe a strong dependence of the decoherence rate γ n on the initial momentum p 0. In Fig. 5.3 we plot the experimental points for the decoherence rate γ n as a function of the initial momentum of the atoms. The data show a parabolic behavior in good agreement with the prediction of Eq.(5.15) if one assumes ω 2 = ω 1 in the laboratory frame. Fitting the data with the theoretical curve we obtain the values γ 0 =4.2(2) ms 1 and τ=2.4(2) 10 7 s. The expected linewidth of the Bragg resonance [54] for our experimental parameters is γ 0 3 ms 1, close to the value obtained from the fit. Notice that the decoherence rate is minimized for p 0 = k. Indeed, if the initial momentum is k, after scattering the atoms in the laboratory frame have the same kinetic energy and, with the above assumption for the scattered light frequency ω 1, the phase destroying decoherence term in (5.4) is zero. This identifies a subspace which is decoherence free with respect to the phase destroying process [61].

77 5.3. Seeding the superradiance Seeding the superradiance In this section we discuss experiments performed adding a counterpropagating laser beam (seed beam) stimulating the superradiant scattering. This allowed us to investigate the crossover from the superradiant amplification of light to the Rabi oscillations regime due to the Bragg scattering induced by the presence oe the two laser beams. The scheme of the experiment is the same as the one presented above, with the only difference that now the condensate is illuminated by two beams coming from opposite directions. In these measurements the condensate was at rest in the laboratory frame (p 0 = 0) and the seed frequency ω 1 was detuned from the pump frequency ω 2 by 21 ω 2 ω 1 = 4ω R in order to satisfy the Bragg resonance condition 0 = 0. In order to provide the right detuning, the two laser beams were modulated by two indipendent acousto-optic modulators (AOM) driven by two different phase-locked carrier frequencies. In these conditions eqs. ( ) become ds dt dw dt da dt = gaw γ n S (5.21) = 2g(AS + h.c.) (5.22) = gns + i n A κ c (A A in ). (5.23) where A in is the scaled amplitude of the seed. In presence of a seed the dynamic of the system shows a competition between the CARL evolution when A in is small enough, and Rabi oscillations when A in become dominant. We can obtain an analytical solution of this system in the two limiting cases of A in = 0 and A in A. The first case has been previously illustrated and its analytic solution is reported in Eq.(5.19). In the second case the solution for the population fraction becomes, for n = 0 and γ n = γ 0, P n = 1 2 { [ 1 + e γ 0(t t 0 )/2 cos Ω(t t 0 ) + γ ]} 0 2Ω sin Ω(t t 0) (5.24) where Ω = Ω 0 γ 2 0/4 and Ω 0 = 2gA in is the Rabi frequency of the input signal. In Eq.(5.24) we have added a phenomenological delay time t 0 to account for shifts in the experimental timing. In the intermediate regime we have to resort to a numerical integration of Eqs.(5.21)-(5.23).

78 66 Chapter 5. Superradiant Rayleigh scattering from a moving BEC Figure 5.4: Time evolution of population of the condensate with p 0 = 0 as a function of the superradiant pulse length. The experimental data are fitted with the analytical curve of Eq (dotted line). The continuous line is the result of the numerical integration of Eqs.(5.21)-(5.23). In order to test the results of numerical integraion, we first present in Fig. 5.4 data referring to an experiment performed without seed, when the superradiance starts from noise. The effect of the noise is introduced in the model as an injected signal with frequency ω 1 = ω 2. The comparison with the exoerimental results allows us to determine the amplitude of the noise which triggers the onset of the superradiant process. The data in figure refers to an experiment performed with a pump beam intensity I 0 = 0.9W/cm 2 and detuning 15GHz. The dotted line is a fit of the experimental data with the curve of Eq. (5.19) giving G = 31ms 1, γ n = 6.44ms 1 and t 0 = 0.26ms as best parameters. The continuous line is instead the result of the numerical integration of Eqs.(5.21)-(5.23) where we assume an injected signal with a Rabi frequency Ω 0 = 9.7ms 1 corresponding to an equivalent intensity of I N = 1.1µW/cm 2. This value is chosen in such a way that the simulation has the best agreement with the experimental data. We define this value of the intensity as the equivalent input noise for this experimental set-up. We explain the oscillating

79 5.3. Seeding the superradiance 67 Figure 5.5: Time evolution of the population of the condensate at p 0 = 0 interacting with the superradiant pump beam and a counterpropagating seed beam (resonant with the Bragg transition 21 = 4ω r ) for different intensities. As the intensity increases the system goes from a superradiant regime to a Rabi oscillations regime. behaviour of the continuous line as an effect of the finite 0 when ω 2 = ω 1. We discuss now the results of the experiment performed varying the seed beam intensity I s. In Fig. 5.5 we show data obtained ranging the seeding intensity from I s = W/cm 2 to I s = W/cm 2. In this configuration the dynamics of the system goes from pure superradiant amplification of light (showing the typical tanh-like behaviour) to almost pure damped Rabi oscillations of population. Note that in the intermediate regime we observe some damped asymmetric oscillations of population, in which the building up of the scattered peak s faster than its decreasing, as a result of the interplay between the two processes. In Fig. 5.5D the experimental data are fitted with the analytic curve of Eq.(5.24) describing the Rabi oscillations regime, while in Fig. 5.5A,B,C the experimental results in the intermediate regime are compared with the numerical solution of Eqs.(5.21)-(5.23). The

80 68 Chapter 5. Superradiant Rayleigh scattering from a moving BEC parameters used in the numerical solution are the ones obtained from the fits of the data in Fig. 5.4 and Fig. 5.5D corresponding to the two limiting cases (pure superradiance and pure Rabi oscillations) for which analytical solutions exist. We now go back to discuss the role played by backdiffused light in the experiment described in the previous section. In an experimental apparatus it is very difficult to avoid the presence of light backreflected by the vacuum cell windows. In particular, considering the 1D geometry of this experiment, if some counterpropagating light exists, in the case p 0 = k this could cause stimulated Raman Bragg scattering of atoms in the same direction and with the same transfer of momentum 2 k, thus masking the effect of a pure superradiant scattering. Indeed, in this experimental setup the backdiffusion of a small amount of light was detected, caused by the poor quality of the cell windows. The magnitude of this light was first estimated directly measuring with a power-meter the intensity backscattered collinearly to the pump light. This intensity is W/cm 2, corresponding to of the pump intensity. The effect of this small amount of backdiffused light has also been evidenced performing the measurements without seed in the far-detuned regime. In this situation the spontaneous process triggering the superradiant amplification is suppressed (its rate being proportional to 1/ 2 21), while the stimulated Raman Bragg scattering can be predominant (its rate being proportional to 1/ 21, as can be seen in Fig.5.6), provided that some counterpropagating light exists. Indeed, in this regime (for GHz, I 3 W/cm 2 and t 0.5 ms), for an initial momentum p 0 = + k, the signature of a small Bragg scattering at p = k (i.e. in the direction opposite to the superradiant scattering) was observed, and this can be explained only assuming a backreflected light of W/cm 2. These two independent observations confirm that, in all the experiments described above, we should take into account the presence of some counterpropagating light at the same frequency ω 2 of the pump and a relative intensity of Coming back to the experiment on pure superradiant scattering, the spurious light backdiffused by the cell windows (approximately a fraction 10 5 of the pump light) is small enough to safely state that all the measurements discussed in the previous section have been made in a regime in which the dynamics of the system is completely dominated by superradiance. This amount of light is actually of the same order of magnitude of the equivalent input noise triggering the superradiant

81 5.3. Seeding the superradiance 69 Figure 5.6: Comparison between superradiant scattering and Bragg scattering. In graph we show the population of the scattered peak as a function of the pump beam detuning from the atomic resonance. The filled circles refer to superradiant scattering, while the open circles refer to stimulated Bragg scattering. The curves are fit to the experimental data with a 1/ α 21 function. The exponents obtained from the fits are consistent with the ones expected from the theory: α = 2 for superradiance and α = 1 for Bragg transitions. The two curves should not be directly compared, because obtained with different laser parameters (intensity and pulse duration). Note that the efficiency of the superradiant process approaches zero faster than the Bragg scattering efficiency. process. This can justify the assumption ω 1 = ω 2 used to fit the experimental data of Fig We remark that the presence of the backdiffused light cannot explain the results in terms of Bragg scattering, since the width of the Bragg resonance is one order of magnitude smaller than the range of momenta explored in our experiment and shown in Fig. 5.3 (so that only the experimental point at p 0 = k would be affected). Furthermore, the hyperbolic tangent dependence of the atomic population in Fig. 5.2 can only be explained by the self consistent amplification of the matter wave grating and of the backscattered light as described in the CARL-BEC model.

82 70 Chapter 5. Superradiant Rayleigh scattering from a moving BEC 5.4 Concluding remarks In this chapter we have studied superradiant Rayleigh scattering from a moving BEC. Using the CARL-BEC model in the mean-field limit we have shown that the efficiency of the overall process is fundamentally limited by the decoherence between the two atomic momentum states. Analyzing a first experiment we have studied the dependence of the decoherence rate on the initial momentum of the condensate. We identified a velocity dependent contribution to the decoherence rate, which can be minimized when the energy conservation condition is satisfied (i.e. the scattered and the unscattered atomic wavepacket have the same kinetic energy in the laboratory frame). In a second experiment, performed adding a counterpropagating beam, we have explored the transition from the pure superradiant regime to the Rabi oscillations regime induced by stimulated Bragg scattering. The theoretical model is in good agreement with the experimental results for the intermediate regime. As we will see in the next chapter, the fully quantized version of the CARL-BEC model offers the possibility of investigating the realization of macroscopic atom-atom or atom-photon entanglement [48, 62]. The control of decoherence showed in this chapter represents a significant step in this direction.

83 Chapter 6 Entanglement generation Entanglement is one of the most intriguing features of quantum mechanics. Of late, it has been recognized as a wonderful resource for quantum information processing. Creation of various entangled states is the first step towards development of quantum communication. Among the applications to quantum information there are quantum computation [63], quantum teleportation [23], quantum dense coding [64], and quantum cryptography [65]. In recent years, much progress has been made on creating quantum entanglement between macroscopic atomic samples [66, 67, 68, 69, 70, 71, 72] and to explore its applications to quantum communication [65, 73, 74] and quantum computation [75]. In particular, quantum entanglement between two separate macroscopic atomic samples [66] has been demonstrated experimentally. Regarding atomic BECs it has been shown that substantial manyparticle entanglement can be generated directly in a two-component weakly interacting BEC using the inherent inter-atomic interactions [67, 76] and in a spinor BEC using spin-exchange collision interactions [68, 70, 71]. Based on an effective interaction between two atoms from coherent Raman processes, a coherent coupling scheme to create massive entanglement of BEC atoms [72] was proposed. An entanglement swapping scheme between trapped BECs [77] has also been presented. Indeed, nowadays manipulation and control of quantum entanglement between BEC atoms has become one of important goals for experimental studies with BECs. As well known, one of the key problems in the experimental explorations of quantum entanglement is to coherently control interaction between the relevant particles. The strength of the inter-atomic interactions in atomic BECs can vary over a wide range 71

84 72 Chapter 6. Entanglement generation of values through changing external fields. This kind of control and manipulation of inter-atomic interactions has been experimentally realized through magnetical-fieldinduced Feshbach resonances in atomic BECs [78]. Therefore, atomic BECs provide us with an ideal experimental system for studying quantum entanglement. On the other hand, recently much attention has been paid to continuous variable quantum information processing in which continuous-variable-type entangled pure states play a key role. For instance, two-state entangled coherent states are used to realize efficient quantum computation [79] and quantum teleportation [80]. Twomode squeezed vacuum states have been applied to quantum dense coding [81]. In particular, following the theoretical proposal of Ref. [82], continuous variable teleportation has been experimentally demonstrated for coherent states of a light field [83] by using entangled two-mode squeezed vacuum states produced by parametric down-conversion in a subthreshold optical parametric oscillator. It is also has been shown that a two-state entangled squeezed vacuum state can be optically created and used to realize quantum teleportation of an arbitrary coherent superposition state of two equal-amplitude and opposite phase squeezed vacuum states [84, 85]. Therefore, it is an interesting topic to create entangled squeezed states in atomic BECs. In this chapter we will show that CARL provides an interesting mechanism to generate entanglement in BECs. In particular in a good-cavity regime the state evolved from vacuum through CARL dynamics is a fully inseparable three mode Gaussian state and, in appropriate regimes, can reduce to atom-atom and atomphoton entangled states. 6.1 The Hamiltonian model We now want to extensively study the three mode linearized model obtained in chapter 3. This approximation neglects both the depletion that occurs as atoms are trasferred into the side modes and the cross-phase modulation between the condensate and the scattered field. This is the matter-wave-optics analogous of the familiar classical undepleted pump approximation of nonlinear optic. Hence this means that we treat all strongly populated modes classically and all weakly populated modes quantum-mechanically.

85 6.1. The Hamiltonian model 73 As we have seen the system is unstable for certain values of the detuning. In fact, by linearizing Eqs. (3.38) and (3.39) around the equilibrium state, the only equations depending linearly on the radiation field are these for ĉ n0 1 and ĉ n0 +1. Hence, in the linear regime, the only transitions involved are those from the state n 0 towards the levels n 0 1 and n Introducing the operators the atomic field operator reduces to â 1 = ĉ n0 1e i(n2 0 τ/ρ+ τ) (6.1) â 2 = ĉ n0 +1e i(n2 0 τ/ρ τ) (6.2) â 3 = âe i τ, (6.3) ˆΨ(θ) 1 2π { N + â1 (τ)e i(θ+ τ) + â 2 (τ)e i(θ+ τ) } e i(n 0θ in 2 0 τ/ρ) (6.4) and Eqs. (3.38) and (3.39) reduce to the linear equations for three coupled harmonic oscillator operators [48]: dâ 1 dτ dâ 2 dτ dâ 3 dτ with Hamiltonian ρ Ĥ = δ + â 2â 2 δ â 1â 1 + i 2 where = iδ â 1 + ρ/2â 3 (6.5) = iδ + â 2 ρ/2â 3 (6.6) = ρ/2(â 1 + â 2 ), (6.7) [(â 1 + â 2 )â 3 (â 1 + â 2)â 3 ], (6.8) δ ± = δ ± 1 ρ (6.9) δ = ω 2 ω 1 + 2n 0 ω R ρω R. (6.10) Hence, the dynamics of the system is that of three parametrically coupled harmonic oscillators â 1, â 2 and â 3 [49], which obey the commutation rules [â i, â j ] = 0 and [â i, â j ] = δ ij for i, j = 1, 2, 3. Note that the Hamiltonian (6.8) commutates with the constant of motion Ĉ = â 2â 2 â 1â 1 + â 3â 3. (6.11)

86 74 Chapter 6. Entanglement generation It is worth to introduce also the atomic density operator, ˆn(θ) ˆΨ(θ) ˆΨ(θ), which, using Eq.(6.4), has the following linearized form: ˆn(θ) N ( 1 + ˆBe iθ + ˆB ) e iθ, (6.12) 2π where ˆB = 1 N e iδτ (â 1 + â 2 ) (6.13) is the linearized bunching operator. Furthermore, it is easy to show that the variance of the atomic density is ( ) 2 N ( [ (ˆn)] 2 = ˆn 2 ˆn 2 = 2 2π ˆB ˆB ˆB ˆB ). (6.14) Then, the expectation values ˆB and ˆB ˆB of the bunching operator are related to the average and variance of the atomic density, respectively. 6.2 Spontaneous emission and small-gain regime Before solving exactly the linear equations (6.5)-(6.7), we calculate the perturbative solution valid for short times τ. Solving Eqs. (6.5) and (6.6) keeping â 3 (τ) â 3 (0) constant, we obtain: â 1(τ) e iδ τ {â 1(0) + â 3 (0) ρ/2τ sinc(δ τ/2)} (6.15) â 2 (τ) e iδ +τ {â 2 (0) + â 3 (0) ρ/2τ sinc(δ + τ/2)}. (6.16) If the radiation field is initially in a coherent state with amplitude α and the atoms are in the vacuum for the state n 0 1 and n 0 + 1, i.e. if ψ(0) = 0, 0, α, then ˆn 1 â 1â 1 = (1 + α 2 )S (6.17) ˆn 2 â 2â 2 = α 2 S + (6.18) ˆn 3 â 3â 3 = S + α 2 [1 + G], (6.19) where S ± = (ρτ 2 /2)sinc 2 (δ ± τ/2) is the dimensionless spontaneous emission lineshape and G = S S + is the gain. In Eq. (6.19) we used the constant of motion (6.11) to obtain ˆn 3 = α 2 + ˆn 1 ˆn 2. The first term S in Eq. (6.19) is the average number of photons emitted spontaneously along the z-axis, with the usual

87 6.2. Spontaneous emission and small-gain regime G AIN Figure 6.1: Small-gain regime: G vs. δ for 1/ρ = 0 and τ = 1 (continuous line) and for 1/ρ = 10 and τ = 2 (dashed line). δ line shape centered at δ = 1/ρ, i.e. at ω 1 = ω 2 +(2n 0 1)ω R. The second term of Eq. (6.19) is the stimulated contribution, with the familiar antisymmetric dependence on the detuning [12]. In the limit ρ 1 we obtain G (τ 3 /2) d dx [ sinc 2 (x) ] x=δτ/2 = 4τ 3 [1 cos(δτ) (δτ/2) sin(δτ)]. (6.20) δ3 Expression (6.20) is known in the FEL literature as the Madey gain [91] and was originally obtained by Madey (for an electron beam travelling along a wiggler magnetic field) as the difference between the emission and absorption rates, shifted by the recoil frequency. Analog result can be obtained in the classical CARL, as we showed in chapter 1 (see Fig. 1.1). In the limit of small ρ, the solution (6.17)-(6.19) is valid for ρτ 2 1, whereas, in the limit ρ 1, the Madey gain (6.20) is valid for τ 1. Fig. 6.1 shows G as a function of δ for 1/ρ = 0 and τ = 1 (continuous line) and for 1/ρ = 10 and τ = 2 (dashed line). We observe that the Madey gain (6.20) does not depend explicitely on ρ and has the maximum value G max 0.27τ 3 at δτ = 2.6. Instead, in the limit ρ 1, the maximum gain is G max = ρτ 2 /2, occurring at the center of the

88 76 Chapter 6. Entanglement generation spontaneous emission line δ = 1/ρ. 6.3 Solution of the linear quantum regime The exact solution of Eqs. (6.5)-(6.7), can be obtained using the Laplace transform. Let s defined the Laplace transform of â j (t) ã j (µ) 0 dt e µt â j (t). (6.21) Applying this transformation to Eqs. (6.5)-(6.7) we obtain the following algebraic system ρ µã 1(µ) + iδ ã 1(µ) 2ã3(µ) = â 1(0) ρ µã 2 (µ) + iδ + ã 2 (µ) + 2ã3(µ) = â 2 (0) (6.22) ρ ρ µã 3 (µ) 2ã 1(µ) 2ã2(µ) = â 3 (0), The determinant of the system (6.22) is ( µ = µ µ + iδ n + i ) ( µ + iδ n i ) i, (6.23) ρ ρ hence ] â 1(0) + ρ ρ 2 â2(0) + ã 1(µ) = 1 { [ µ(µ + iδ + ) + ρ µ 2 ã 2 (µ) = 1 { ρ [ ρ ] µ 2 â 1(0) + 2 µ(µ + iδ ) ã 3 (µ) = 1 µ { ρ 2 (µ + iδ +) â 1(0) + â 2 (0) + } 2 (µ + iδ +) â 3 (0) } ρ 2 (µ + iδ ) â 3 (0) (6.24) } ρ 2 (µ + iδ )â 2 (0) + (µ + iδ + )(µ + iδ ) â 3 (0) Inverting the Laplace transform the exact solution of Eqs. (6.5)-(6.7) is the following [44, 49]: â 1(τ) = e iδτ [g 1 (τ)â 1(0) + g 2 (τ)â 2 (0) + g 3 (τ)â 3 (0)] (6.25) â 2 (τ) = e iδτ [h 1 (τ)â 1(0) + h 2 (τ)â 2 (0) + h 3 (τ)â 3 (0)] (6.26) â 3 (τ) = e iδτ [f 1 (τ)â 1(0) + f 2 (τ)â 2 (0) + f 3 (τ)â 3 (0)], (6.27)

89 6.3. Solution of the linear quantum regime Im λ (a) (b) (c) (d) (e) (f) Figure 6.2: Immaginary part of the unstable root of the cubic equation (29) vs. δ for 1/ρ = 0, (a), 0.5, (b), 3, (c), 5, (d), 7, (e) and 10, (f). δ where the explicit expressions of f i, g i and h i are given in appendix A, while the initial values f i (0) = δ i3, g i (0) = δ i1 and h i (0) = δ i2 verify the initial conditions for â i. The functions f i, g i and h i are the sum of three terms proportional to e iλjτ, where λ j are the complex roots of the cubic equations: (λ δ)(λ 2 1/ρ 2 ) + 1 = 0. (6.28) The characteristic equation (6.28) has either three real solutions, or one real and a pair of complex conjugate solutions. In the first case, the system is stable and exhibits only small oscillations around its initial state. In the second case, the system is unstable and grows exponentially, even from noise. In Fig. 6.2 we plot the immaginary part of λ as a function of δ for different values of ρ. The classical limit is obtained for ρ 1 (see Fig. 6.2a). In this case, the system is unstable for δ < 3/2 1/3 with maximum instability rate at δ = 0 and unstable root λ 3 = (1 i 3)/2. When ρ is smaller than one (Fig. 6.2c-f), the instability rate decreases and the peak of Im(λ) moves at δ = 1/ρ. This can be seen explicitely observing that the characteristic equation (6.28) has two resonances, one at δ = 1/ρ and the other

90 78 Chapter 6. Entanglement generation at δ = 1/ρ, corresponding to a mismatch between the probe and the pump field frequencies equal to ω 1 ω 2 = (2n 0 1)ω R. In the first case, the photon is absorbed from the probe and emitted into the pump, rising the atomic energy from n 2 0 ω R to (n 0 + 1) 2 ω R. In the second case, the photon is absorbed from the pump and emitted into the probe, decreasing the atom energy to (n 0 1) 2 ω R. The study of the solutions of Eq. (6.28) shows that the only unstable process is the second one. When ρ 1, the gain bandwidth (δω 1 ) GAIN = ρω R σ δ (where σ δ is defined as the interval of δ for which Im(λ) is at half of its maximum value) is of the order of ρω R ω R and the shift due to the energy recoil ω R is negligible, also if the system is initially at zero temperature, i.e. without thermal broadening. Because for ρ 1 the gain bandwidth is larger than the separation between the emission and absorption lineshape centers, each atom may absorb the photon either from the pump or from the probe. In this case, the probe field is amplified because, as we will see later, the average number of photons scattered from the pump into the probe is larger than that absorbed from the probe and emitted into the pump. On the other hand, the quantum recoil effect becomes visible when ρ < 1. In this case, the unstable root of the characteristic equation (6.28) is approximately λ 1/ρ + δ /2 (1/2) δ 2 2ρ. The imaginary part of λ reaches the maximum value ρ/2 for δ = 0 (i.e. δ = 1/ρ), with a gain bandwidth (δω 1 ) GAIN = (2ρ) 3/2 ω R less than ω R. In this case, the absorption lineshape center, δ = 1/ρ, is outside of the gain bandwidth, and the atom can only absorb the photon from the pump and emit it into the probe, whereas the inverse process is not allowed due to the energy conservation. This can be seen explicitly assuming that in the exponential regime â 1 and â 2 are proportional to exp[i(λ 3 δ)τ], where λ 3 is the unstable root of Eq. (6.28) with negative imaginary part. Then, from Eqs. (6.5) and (6.6), it follows that ( ) 1 ρλ3 â 2 â 1 + ρλ 1. (6.29) 3 In the case ρ 1, â 2 â 1, and the atoms have almost the same probability of transition from the momentum level n 0 to n or n 0 1, absorbing or emitting a photon, respectively. On the contrary, in the case ρ 1, then λ 3 1/ρ and Eq. (6.29) shows that â 2 â 1: the atoms can only emit a photon into the probe, due to the energy conservation. This makes the CARL in the quantum recoil limit ρ 1 an interesting example of two-level system coupled to a radiation mode, initially

91 6.3. Solution of the linear quantum regime 79 inverted and without incoherent spontaneous decay. We assume that the initial state is 0, 0, α, i.e. the vacuum state for â 1 and â 2 and a coherent state with amplitude α for â 3. The average occupation numbers ˆn i and the number variance σ 2 (ˆn i ) = ˆn 2 i ˆn i 2 can be calculated from Eqs. (6.25)-(6.27) yielding: for i = 1, 2, 3, where ˆn i = ˆn i sp + ˆn i st, (6.30) σ 2 (ˆn i ) = ˆn i (1 + ˆn i ) ɛ i ˆn i 2 st (6.31) ˆn 1 sp = g g 3 2, ˆn 1 st = αg 3 2, (6.32) ˆn 2 sp = h 1 2, ˆn 2 st = αh 3 2, (6.33) ˆn 3 sp = f 1 2, ˆn 3 st = αf 3 2 (6.34) and ɛ i = 1 + δ i1 / α 2. The average photon number ˆn 3 and the average number of atoms with momentum p = (n 0 1) q, i.e. ˆn 1 and ˆn 2, are the sum of two terms representing the spontaneous and the stimulated emission contributions. The first term originates from the fluctuations of the vacuum state and it is the only contribution in the absence of the initial probe field. We observe that the statistics of the spontaneous emission (for α = 0) is that of a chaotic (i.e. thermal) state, with number variance σi 2 ˆn i sp (1 + ˆn i sp ). Instead, if the stimulated emission dominates the spontaneous emission ( ˆn i st ˆn i sp ), then σi 2 ˆn i st [1 + 2 ˆn i sp + (1 ɛ i ) ˆn i st ]. We note that we obtain the Poisson statistics for a coherent field, i.e. σ 2 i ˆn i st, only for ˆn i sp 1. We calculate also the expectation value ˆB ˆB for the linearized bunching operator ˆB, ˆB 1 [ ˆB = g1 + h α 2 g 3 + h 3 2], (6.35) N from which, using Eq. (6.14), we obtain the atomic density fluctuations [18]: (ˆn) = 1 N π 2 g 1 + h 1. (6.36) We calculate now the equal-time intensity correlation and cross-correlation functions[18, 48], defined respectively as: g (2) i = â i (τ)â i (τ)â i(τ)â i (τ) ˆn i (τ) 2 (6.37)

92 80 Chapter 6. Entanglement generation g (2) i,j = ˆn i(τ)ˆn j (τ) ˆn i (τ) ˆn j (τ), (6.38) with i = 1, 2, 3 and i j. For a classical field there is an upper limit to the secondorder equal-time cross correlation function given by the Cauchy-Schwartz inequality g (2) i,j (τ) [g i(τ)] 1/2 [g j (τ)] 1/2. Quantum-mechanical fields, however, can violate this inequality and are instead constrained by [ g (2) i,j (τ) g (2) i (τ) + 1 ] 1/2 [ g (2) j (τ) + 1 ] 1/2, (6.39) ˆn i ˆn j which reduces to the classical results in the limit of large occupation numbers. We obtain the following expressions: g (2) i = 2 ɛ i ˆn i 2 st ˆn i 2 (6.40) g (2) 1,2 = 2 + ˆn 2 sp + ˆn 1 st (1 ɛ 1 ˆn 2 st ) α 2 ˆn 3 sp ˆn 1 ˆn 2 g (2) 1,3 = 2 + ˆn 3 sp + ˆn 1 st (1 ɛ 1 ˆn 3 st ) α 2 ˆn 2 sp ˆn 1 ˆn 3 (6.41) (6.42) g (2) 2,3 = 2 + ˆn 1 st ˆn 2 st ˆn 3 st α 2 ˆn 1 sp. (6.43) ˆn 2 ˆn 3 When the system builds up from noise (α = 0), g (2) i = 2, as expected for a thermal or chaotic field. More interesting information are obtained from the cross-correlation functions. In the spontaneous case α = 0, g (2) 1,2 = g (2) 1,3 = 2 + 1/ ˆn 1 and g (2) 2,3 = 2. Hence, both g (2) 1,2 and g (2) 1,3 violate the Cauchy-Schwartz inequality, while g (2) 2,3 is consistent with it. Furthermore, because ˆn 1 = ˆn 2 + ˆn 3, g (2) 1,2 and g (2) 1,3 are consistent with the quantum inequality (6.39), and close to their upper value for ˆn 3 = 0 and ˆn 2 = 0, respectively [18, 48]. Hence, we expect the existence of non classical correlations, as for instance two-mode entanglement, between the modes 1 and 2 or between the modes 1 and 3 when the average occupation numbers of the two modes are equal. 6.4 Three mode entanglement Entanglement between two particles is quite common, for example, EPR states, polarization states of twin-photons, down converted two-photon states in optical

93 6.4. Three mode entanglement 81 parametric oscillator and so on. In contrast, three particle entanglement is not so common, though recently three-photon GHZ entangled states [94] have been experimentally realized. Because of two dominant momentum side-modes involved in the CARL dynamics, a condensate exposed to a stronf far-off resonance pump laser seems to be a suitable candidate for exploring tripartite entanglement among these two condensate side-modes and the probe photon mode. We calculate the exact state vector ψ(τ) = Û(τ) 0, 0, 0 at the time τ when the system evolves from vacuum. At this aim, we disentangle the evolution operator Û(τ) = exp( iĥτ), where Ĥ is given by Eq.(6.8), writing it as the product of individual operators. The calculation, reported in detail in appendix A, yields 1 (m + n)! ψ(τ) = α1 m α2 n m + n, n, m, (6.44) 1 + ˆn1 m!n! where n,m=0 α 1 = f 1 g1 1 + ˆn 1 (6.45) α 2 = h 1 g ˆn 1. (6.46) We observe that α 1,2 2 = ˆn 3,2 /(1 + ˆn 1 ). The state in Eq. (6.44) is Gaussian, as it can be easily demonstrated by evaluating the characteristic function [ χ(λ 1, λ 2, λ 3 ) = Tr ψ(τ) ψ(τ) ˆD 1 (λ 1 ) ˆD 2 (λ 2 ) ˆD ] 3 (λ 3 ) = 0 Û (τ) ˆD 1 (λ 1 ) ˆD 2 (λ 2 ) ˆD 3 (λ [ 3 )Û(τ) 0 = exp 1 ( λ λ λ 3 2)], (6.47) where λ j are complex numbers, ˆD j (λ j ) = exp(λ j â j λ jâ j ) is a displacement operator for the j-th mode, and the primed quantities are given by λ 1 = f 1 λ 1 g 1 λ 2 h 1 λ 3 (6.48) λ 2 = f 2 λ 1 + g 2 λ 2 + h 2λ 3 (6.49) λ 3 = f 3 λ 1 + g 3λ 2 + h 3 λ 3. (6.50) Following Ref. [95], the characteristic function can be rewritten as [ χ(λ 1, λ 2, λ 3 ) = exp 1 ] 4 xt C x, (6.51)

94 82 Chapter 6. Entanglement generation where x T = (x 1, x 2, x 3, p 1, p 2, p 3 ),, ( ) T denotes transposition, λ j = 2 1/2 (p j ix j ), j = 1, 2, 3, and C denotes the covariance matrix of the Gaussian state, whose explicit expression can be easily reconstructed from Eqs. ( ). The covariance matrix determines the entanglement properties of ψ(τ), in fact, since ψ(τ) is Gaussian the positivity of the partial transpose is a necessary and sufficient condition for separability [95], which, in turn, is determined by the positivity of the matrices Γ j = Λ j CΛ j ij where Λ 1 = Diag(1, 1, 1, 1, 1, 1), Λ 2 = Diag(1, 1, 1, 1, 1, 1), Λ 3 = Diag(1, 1, 1, 1, 1, 1) and J is the symplectic block matrix ( ) 0 I J =, I 0 I being the 3 3 identity matrix. A numerical evaluation of the eigenvalues of Γ j shows that they are nonpositive matrices j. Correspondingly, the state (6.44) is fully inseparable i.e. not separable for any grouping of the modes. Ideally squeezed states between the modes 1 and 2 or the modes 1 and 3 can be obtained when ˆn 3 = 0 or ˆn 2 = 0, respectively, giving: ψ 1,2 = ψ 1,3 = ˆn ˆn1 α2 n, n n, 0. (6.52) n=0 α1 n, n 0, n. (6.53) n=0 The states (6.52) and (6.53) are pure bipartite states It is known that the von Neumann entropy is a good measure of entanglement for bipartite pure states [96]. So, for these states we can give a measure of the degree of entanglement. If we consider the reduced density operators ˆϱ i = Tr 1 [ˆϱ 1i ], where ˆϱ 1i = ψ 1i ψ 1i and i = 2, 3, we obtain the thermal state ˆϱ i = ˆn i m ( ) m ˆni m m, (6.54) 1 + ˆn i for which the entropy S i = Tr[ˆϱ i ln ˆϱ i ] is maximum, so that the states (6.52) and (6.53) are maximally entangled. The presence of the third mode in general reduces the entanglement between the other two modes [92]. We observe also that no twomode entanglement is possible between the states 2 and 3.

95 6.5. High-gain regime 83 In order to study the non classical correlations of atoms and photons, it can be useful to calculate also the two-mode relative number squeezing parameter [92], ξ i,j = σ2 (ˆn i ˆn j ) ˆn i + ˆn j, (6.55) although, as pointed correctly in [92], squeezing in the relative number of particles between states is not equivalent to entanglement. independent and coherent, then ξ i,j In fact, if the two states are = 1, whereas if they are squeezed, σ 2 (ˆn i ˆn j ) = 0, which imply ξ i,j = 0 [93]. Hence, it is expected that when ξ i,j decreases, the entanglement between the two states could improve. However, the condition σ 2 (ˆn i ˆn j ) = 0 is not a sufficient one for a signature for entanglement, showing only that the state is two-mode squeezed or, more generally, non-classically correlated. Because in the spontaneous case α = 0 we have ξ 1,2 = ˆn 3 ( ˆn 3 + 1) ˆn 1 + ˆn 2 (6.56) ξ 1,3 = ˆn 2 ( ˆn 2 + 1) ˆn 1 + ˆn 3, (6.57) maximum entanglement between the modes 1 and 2 or the modes 2 and 3 should occur when ˆn 3 = 0 or ˆn 2 = 0, respectively. 6.5 High-gain regime We now discuss the above results in the high-gain regime, i.e. for Imλ 3 τ 1, where λ 3 is the root of Eq. (6.28) with negative immaginary part. In this limit, the term proportional to exp(iλ 3 τ) will be dominant in the sum of the expressions (A.5)-(A.10) The quasi-classical recoil limit ρ 1 For ρ 1 and δ = 0, λ 3 (1 i 3)/2 and ˆn [ρ2 /2 + ρ + α 2 (ρ + 1)]e 3τ, (6.58) ˆn [ρ2 /2 + α 2 (ρ 1)]e 3τ, (6.59) ˆn (ρ/2 + α 2 )e 3τ. (6.60)

96 84 Chapter 6. Entanglement generation 10 2 <n 3 >/N <n 1,2 >/N <n i >/N Figure 6.3: Non linear evolution of the population fractions ˆn i for i = 1, 2, 3 (continuous lines) and of (ρ/2) ˆB + ˆB (dashed line), for ρ = 20 and δ = 0. We observe that ˆn 1 ˆn 2 = ˆn 3, in agreement with Eq.(6.11). τ The stimulated emission dominates the spontaneous emission when α 2 ρ/2. Furthermore, ˆn 1 ˆn 2 and ˆn 3 (2/ρ) ˆn 1, so that the number of emitted photons is much smaller than the occupation number of the motional states. The expectation value of the bunching parameter is: ˆB ˆB 1 9N [1 + (2/ρ) α 2 ]e 3τ 2 ρn ˆn 3. (6.61) Assuming that ˆB ˆB approaches a maximum value of the order of one, then the maximum average number of emitted photons is about ρn/2. Hence, ρ/2 can be interpreted as the average number of photons emitted per atom. In order to check the validity of the asymptotic expressions (6.58)-(6.61), we have solved numerically the non linear Eqs. (3.38) and (3.39), treating ĉ n and â as c-numbers. Fig. 6.3 shows the average population fractions ˆn i /N with i = 1, 2, 3 (continuous lines) and (ρ/2) ˆB + ˆB (dashed line), as they result from the simulation with ρ = 20, δ = 0 and a small initial seed simulating a small bunching at τ = 0. We observe that Eq. (6.61) is in good agreement with the simulation until τ 12, up to

97 6.5. High-gain regime 85 a maximum value of 0.77 of the expectation value of the bunching operator ˆB + ˆB. Instead, Eqs.(6.58) and (6.59) fit well the numerical result only until τ 8, up to the maximum value ˆn 1,2 0.34N, in agreement with the constraint ˆn 1,2 N/2 required by the conservation of the atomic number. When the exponentially growing terms dominate, the relative uncertainty of the occupation numbers reaches a steady-state value given by: σ(ˆn i ) ˆn i ρ(ρ + 4 α 2 ) ρ + 2 α 2 (6.62) for i = 1, 2, 3. We see that for α 2 ρ, σ(ˆn i )/ ˆn i 1, as should be for a thermal field when ˆn i 1. Instead, for α 2 ρ, Eq. (6.62) yields σ(ˆn i )/ ˆn i ρ/ α 1. Hence, the presence of an initial coherent field decreases the relative uncertainty of the occupation number [44, 18]. Finally we note that, in the high gain regime and in the quasi-classical limit ρ 1, the intensity correlation function for the three modes, Eq. (6.40), yields which tends to 1 for α 2 ρ [18]. ( ) g (2) α 2 2 i = 2 ɛ i, (6.63) ρ/2 + α The quantum recoil limit ρ 1 For ρ 1, the maximum rate of instability occurs at δ = 1/ρ, with λ 3 1/ρ i ρ/2 and: ˆn [1 + (ρ/2)3 + α 2 ]e 2ρτ, (6.64) ˆn 2 1 ( ρ ) 3 (1 + α 2 )e 2ρτ, (6.65) 4 2 ˆn (1 + α 2 )e 2ρτ. (6.66) In this case, the stimulated emission dominates the spontaneous emission when α 1. Furthermore, ˆn 1 ˆn 3 and ˆn 2 (ρ/2) 3 ˆn 1, so that the number of atoms ˆn 2 which absorb a photon from the probe gaining a recoil momentum q is much smaller than the number of atoms ˆn 1 which emit a photon into the probe loosing a recoil momentum. Hence, in the quantum recoil limit, emission dominates absorbtion.

98 86 Chapter 6. Entanglement generation The expectation value of the bunching parameter is: ˆB ˆB 1 4N [1 + α 2 ]e 2ρτ = 1 N ˆn 3. (6.67) When ˆB ˆB reaches a maximum value of the order of one, then the maximum average number of emitted photons is about N, i.e. all the atoms are tranferred from the ground motional state n 0 to the side-mode state n 0 1. In the asymptotic limit 2ρτ 1, the relative uncertainty of the occupation numbers are: σ(ˆn 1 ) ˆn 1 σ(ˆn 2 ) ˆn α 2 σ(ˆn 3) ˆn α 2 (6.68) 1 + α 2, (6.69) whereas ( ) g (2) α 2 2 i = 2 ɛ i, (6.70) 1 + α 2 tending again to 1 for α 2 1 [18]. 6.6 Atom-atom and atom-photon entanglement Now, we show the existence of two different regimes of CARL in which the initial vacuum state evolves into an entangled state. In particular, atom-atom entanglement can be obtained only in the limit ρ 1 and in a detuned, not fully exponential regime. On the contrary, in the limit ρ < 1, atom-photon entanglement can be obtained in the exponential regime if the average occupation number ˆn 2 remains smaller than one. As an example of atom-atom entanglement, we show in Fig. 6.4 the average occupation numbers ˆn i (i = 1, 2, 3), (a), and the two-mode squeezing parameter ξ 1,2, (b), as a function of δ for ρ = 100, α = 0, τ = 2 (we assume for an easier notation n 0 = 0). Because ˆn 3 ˆn 1 ˆn 2, it exists a region where ξ 1,2 is less than one, and the state has a form similar to that of Eq. (6.52). In general, this kind of entanglement is not very efficient, because the number of atoms is each quantum state does not grow exponentially and it remains of the order of ρ.

99 6.6. Atom-atom and atom-photon entanglement (a) <n 1 >, <n 2 > <n i > <n 3 > δ 2.0 (b) 1.5 ξ δ Figure 6.4: Atom-atom entanglement: occupation numbers ˆn i (i = 1, 2, 3), (a), and two-mode squeezing parameter ξ 1,2, (b), as a function of δ for ρ = 100, τ = 2 and α = 0.

100 88 Chapter 6. Entanglement generation 10 6 (a) 10 4 <n 1 >, <n 3 > <n i > <n 2 > τ 10 0 (b) 10-1 ξ τ Figure 6.5: Atom-photon entanglement: occupation numbers ˆn i (i = 1, 2, 3), (a), and two-mode squeezing parameter ξ 1,3, (b), as a function of τ for 1/ρ = δ = 10 and α = 0.

101 6.7. Concluding remarks 89 Atom-photon entanglement is more easily obtained in the limit ρ < 1, because in this case ˆn 2 ˆn 1 ˆn 3 also in the exponential regime. In fact, in this case ξ 1,3 = σ2 (ˆn 2 ) ˆn 1 + ˆn 3 = ˆn 2 (1 + ˆn 2 ) ˆn 1 + ˆn 3, (6.71) where ˆn i are given by Eqs. (6.64)-(6.66). Because ˆn i, for i = 1, 2, 3, grow exponentially, ξ 1,3 remains smaller than one only for ˆn 2 < 1. Hence, from Eq. (6.65) it follows that atom-photon entanglement occurs for τ < 1 2ρ (5ln2 3lnρ). (6.72) In Fig. 6.5 we show ˆn i (i = 1, 2, 3), (a), and the two-mode squeezing parameter ξ 1,3, (b), between atoms in the momentum state n 0 1 and photons, as functions of τ for 1/ρ = δ = 10 and α = 0. We observe that the average number of atoms in the momentum state n 0 + 1, ˆn 2, is more than three decades smaller than the average number of atoms in the momentum state n 0 1. Hence, the state of the system reduces from the three-mode entangled state (6.44) to the two-mode entangled state described by (6.53). As a confirm of this, ξ 1,3 in Fig. 6.5b remains much smaller than one, also in the exponential regime, until ˆn 2 becomes larger than one. 6.7 Concluding remarks In this chapter we have investigated the properties, such as quantum fluctuations and entanglement, of the linear model for the quantum system BEC-radiation in a good cavity regime. We obtained relevant new analytical results, calculating explicitly the statistical properties for atoms and photons and evaluating the state of the coupled BEC-light system evolved from vacuum. By calculating the exact evolution of the state from the vacuum of the three modes we have demonstrated that the evolved state is a fully inseparable three mode Gaussian one. Moreover we have shown how this three mode Gaussian state can provide a valuable source of atom-atom and atom-photon entanglement. The theory presented is valid in the linear regime, when the atomic ground state depletion and saturation of the radiation mode are neglected. Furthermore, we have neglected inhomogeneous effects due to the finite spatial extension of the atomic cloud and two-atom collisions. The effect of collisions has been considered [97],

102 90 Chapter 6. Entanglement generation showing that it can seriously limit the coherence of the scattering process and the entanglement. So it is of considerable interest to extend the analytical description of the linear regime of the quantum CARL to include this and other possible sources of decoherence. Moreover, the CARL Hamiltonian model may be extended to include a dissipative mechanism for the radiation mode. We shall se in chapter 8 how our results are modified taking into account effects of atomic decoherence and cavity radiation losses. From an experimental point of view the possibility to have BECs inside an optical cavity would allow a verification of quantum CARL in the good-cavity regime and an exploration of his entanglement properties. In this direction the set up of Ref. [17] described in chapter 1 is very promising.

103 Chapter 7 Radiation to atom quantum mapping Entanglement is a crucial resource in the manipulation of quantum information, and quantum teleportation [23, 24] is perhaps the most impressive example of quantum protocol based on entanglement. Teleportation is the transferral of quantum information between two distant parties that share entanglement. There is no physical move of the system from one player to the other, and indeed the two parties need not even know each other s locations. Only classical information is actually exchanged between the parties. However, due to entanglement, the quantum state of the system at the transmitter location (say Alice) is mapped onto a different physical system at the receiver location (say Bob). The information transferral is blind, i.e. the protocol should work also when the state to be teleported is completely unknown to both the sender and the receiver. Several teleportation protocols have been suggested either for qubits and continuous variable systems [83, 98, 99, 100, 101, 102, 103, 104, 105]. Moreover, interspecies teleportation schemes have been suggested either of atomic spin onto polarization states of light [74, 106, 107] or of motional state of a trapped ion and a light field [108]. In this chapter we show a scheme to realize radiation to atom continuous variable quantum mapping, i.e. to teleport the quantum state of a single mode radiation field onto the collective state of atoms with a given momentum out of a BEC. The atomsradiation entanglement needed for the teleportation protocol is established through the CARL interaction of a single radiation mode with the condensate studied in the 91

104 92 Chapter 7. Radiation to atom quantum mapping Figure 7.1: Schematic diagram of the proposed setup to realize radiation to atom continuous variable quantum mapping, i.e. teleportation of the quantum state of a single mode radiation field onto the collective state of atoms with a given momentum out of a BEC. The protocol proceeds as follows: the atomic mode a 1 and the radiation mode a 3 are entangled through the CARL dynamics. The outgoing radiation mode a 3 is then mixed (in a balanced beam splitter) at the sender s location (Alice) with another radiation mode a 4, excited in the state σ, which we want to teleport, and the joint measurement of a couple of two-mode quadratures is performed. The result of the measurement is sent to the receiver s location (Bob), where the corresponding coherent atomic displacement is performed. The latter is obtained through the same CARL interaction, by injecting a suitably modulated coherent pulse (M denotes a modulator). The overall dynamics is such that the ensemble of recoiling atoms in the mode a 1 is described by the density matrix τ, which approaches σ in the limit of high entanglement i.e. high gain of the CARL interactions. previous chapter, whereas the coherent atomic displacement is obtained by the same interaction with the radiation in a classical coherent field. The three basic ingredients of a quantum teleportation experiment are the following: i) an entangled state shared between two parties; ii) a joint Bell measurement performed on the system whose state is to be teleported and on one subsystems of the entangled state; iii) a device able to perform a given class of unitary transformation, conditioned to the results of the joint measurement. To verify teleportation,

105 7.1. The entangled state 93 it would be necessary moreover a readout system. In the following we describe the above points for our teleportation protocol, and discuss the feasibility conditions of our proposal. The setup is schematically illustrated in Fig The entangled state The entangled state supporting the teleportation protocol is the twin-beam-like state of a radiation mode and a collective mode of atoms with a given momentum out of a BEC studied in chapter 6. We have seen that this is obtained by the CARL interaction of a BEC with a single-mode quantized radiation field in the presence of a strong far off-resonant pump laser, in the good-cavity regime. The atom-radiation entangled state (6.53) ψ 1,3 = n1 α1 n, n 0, n. (7.1) is what supports our teleportation scheme. Incidentally, it has the same form of the twin-beam state of radiation used to realize continuous variable optical teleportation [83], and this will allows us to employ the same kind of Bell measurement scheme (see Appendix D for the definition of TWB and a simple description of the continuous variable optical teleportation). In the quantum limit ρ 1 and for 2ρτ 1 the population of the three oscillators are given by Eqs. ( ) for α = 0 (remember that the initial state from which ψ 1,3 is derived is vacuum) n=0 N 1 (τ) = ˆn 1 1 [ ( ] ρ e 4 2) 2ρτ, (7.2) N 2 (τ) = ˆn 2 1 ( ρ ) 3 e 2ρτ, (7.3) 4 2 N 3 (τ) = ˆn e 2ρτ. (7.4) so that N 1 N 3 N 2. Furthermore, maximal entanglement between modes 1 and 3 requires N 2 1, so that the interaction time must satisfy the following limits 1 2ρ τ int 1 2ρ (5ln2 3lnρ). (7.5)

106 94 Chapter 7. Radiation to atom quantum mapping 7.2 The Bell measurement The state ˆσ we want to teleport onto the atomic mode a 1 pertains to an additional radiation mode a 4. The Bell measurement is jointly performed on a 3 and a 4, and consists in the measurement of the real and the imaginary part of the complex operator The measurement of Ẑ R = Re[Ẑ] and ẐI = Ẑ = â 3 + â 4. (7.6) Im[Ẑ] corresponds to measuring the sum and difference quadratures ˆx 3 + ˆx 4 and ŷ 3 ŷ 4 of the two modes, where the quadrature ˆx of a mode b is the operator ˆx = 1 2 and the quadrature ŷ is the operator ŷ = 1 2i (ˆb + ˆb ), (7.7) (ˆb ˆb ). (7.8) Such a measurement can be experimentally implemented by multiport homodyne detection, i.e by mixing the two modes in a balanced beam splitter and then measuring two conjugated quadratures on the outgoing modes, if the two modes have the same frequencies [109, 110], or by heterodyne detection otherwise [111]. In appendix C we report a detailed description of homodyne and multiport homodyne detection. The measurement is described by the following probability operator-valued measurement (POVM) [112], acting on the Hilbert space of mode a 3 ˆΠ α = 1 π ˆD(α) ˆσ T ˆD (α) (7.9) where α is a complex number, ˆD(α) is the displacement operator ˆD(α) = exp{αâ 3 ᾱâ 3 } (7.10) and ( ) T denotes the transposition operation. The result of the homodyne measurement is classically transmitted to the receiver s location (Bob), where a displacement operation ˆD(α) is performed on the conditional state ˆϱ α (see below on how to implement coherent atomic displacement). In appendix D we review the dynamics

107 7.3. The displacement operation 95 of the conditional measurement [112, 113]. It is described by [ p α = Tr 13 ψ 13 ψ 13 Î1 ˆΠ ] α ˆϱ α = 1 p α Tr 3 [ ψ 13 ψ 13 Î1 ˆΠ α ] ˆτ α = ˆD(α)ˆϱ α ˆD (α), (7.11) where p α is the probability for the result α in the Bell measurement, Î the identity operator, ˆϱ α is the conditioned state of the atomic beam after the measurement, and ˆτ α is is the conditioned state after the displacement operation. The teleported state is the average over all the possible outcomes, i.e. ˆτ = d 2 α p α ˆτ α = d 2 α ˆD(α) Tr 3 [ ψ 13 ψ 13 Î1 ˆΠ ] ˆD α (α). (7.12) C C After performing the partial trace and with some algebra [112], one has where ˆτ = d 2 α πk exp{ α 2 K } ˆD(α)σ ˆD (α), (7.13) K = 1 + N 1 + N 3 (N 1 + N 3 )(N 1 + N 3 + 2). (7.14) Eq. (7.13) shows that the overall dynamics of our scheme is equivalent to that of a Gaussian noisy channel with parameter K [112, 114, 115]. The density matrix ˆτ, describing the final state of the atomic mode a 1 coincides with ˆσ in the limit N 1 + N 3 i.e. for high gain in the CARL dynamics. Notice that N, and in turn N 1 and N 3, may vary in the repeated preparations of the condensate, thus introducing additional noise in the teleported state. 7.3 The displacement operation The displacement operation ˆD(α) that should be performed on the conditional atomic state ˆϱ α can be obtained using the same CARL Hamiltonian (6.8) in the condensate, Ĥ = δ + â 2â 2 δ â 1â 1 + i ρ/2 [(â 1 + â 2 )â 3 (â 1 + â 2)â 3 ], (7.15) by injecting a suitably modulated pulse, i.e. by exciting the mode a 3 in a classical coherent state. In this case, assuming a short pulse, the terms proportional to â jâj

108 96 Chapter 7. Radiation to atom quantum mapping (j = 1, 2) in (7.15) can be discarded and the effective Hamiltonian may be written as Ĥ eff = ig N(γ â 2 γâ 1 ) + h.c (7.16) where γ is the amplitude of the modulated pulse. The evolution operator Û = exp(iĥeffτ) = ˆD 1 (α) ˆD 2(α) (7.17) coincides with the product of two displacement operators, one for each of the atomic modes, with amplitude given by α = gγ Nτ, where τ is an effective interaction time. The amplitude γ of the pulse can suitably be tuned to obtain the desired value of the amplitude α, matching the results of the Bell measurements. Remembering that the action of the displacement operator on a coherent state, apart from a phase factor, produces another coherent state ˆD 1 (α) β = e Im(α β) α + β (7.18) we note that the displacement operation has the effect to give a coherent kick to the collective atomic state on which we want to teleport the state ˆσ. The above dynamics displaces both the atomic modes, however without introducing quantum correlations. Therefore in this scheme we can ignore the effect on the atomic mode a 2, which does not participate to the teleportation protocol. The time duration of the pulse should be small when compared to the time scale of the CARL dynamics and the decoherence time of ψ 13 under free evolution. This is in order for two reasons: on one hand we have that the CARL dynamics should be switched off after producing the desired entangled state ψ 13, and therefore the whole protocol should be completed within the decoherence time. On the other hand, the displacement should be performed on a time scale comparable to that of the Bell measurement, i.e before the reset of the dynamics and the generation of the subsequent copy of the atom-radiation entangled state in the new condensate by CARL. Overall, our protocol may be described as a feed forward control scheme, randomized according to the statistics of the Bell measurement. 7.4 The readout system In order to discuss the readout part of our scheme, we should go back to the initial entangled scheme produced by the CARL dynamics. This should be more properly

109 7.5. Concluding remarks 97 written as ψ 13 = 1 (1 + N1 ) N β n n, 0, n, N n > (7.19) n=0 where the fourth entry in the ket describes the number of atoms in the condensate. Since N is a large number (of the order of ) writing the state as in Eq. (7.1) is perfectly admissible as far as we are concerned with its dynamics. However, this should be taken into account if we want to reconstruct the state of the output atomic beam. Let us consider, for the sake of simplicity, the initial light signal state as a pure state ˆσ = ϕ ϕ (7.20) ϕ = n ϕ n n. (7.21) In the limit of high CARL gain the teleported state on the atomic beam is given by ϕ = n ϕ n n, N n. (7.22) This indicates that any proper verification of the teleportation should involve a measurement also on the condensate, e.g. a two mode tomographic method involving the measurement of both momentum-mode and condensate quadratures [116, 117]. Such kind of measurements are at present experimentally challenging and therefore, in order to obtain an accessible readout system, we propose to check only the statistics of the population ϕ n 2, i.e. the diagonal part of the teleported state, which can be achieved by counting atoms. By choosing the initial radiation state ϕ as a squeezed vacuum or a Fock number state, we obtain an atomic teleported state with an even-odd or sub-poissonian atomic number distribution. Although this kind of measurements would be only a partial verification of teleportation it would show the transferral of a nonclassical such as sub-poissonian statistics. In turn, this implies that nonlocal correlations between the input radiation mode and the output atomic mode has been established and exploited. 7.5 Concluding remarks Quantum CARL provide an interesting mechanism for the generation of entanglement and no classical correlations so it is suitable for quantum information applications. In this chapter we have proposed a novel scheme to realize the interspecies

110 98 Chapter 7. Radiation to atom quantum mapping teleportation of the quantum state of a single mode radiation field onto the collective state of atoms with a given momentum out of a BEC. The entangled resource needed for the teleportation protocol is established through the CARL interaction of a single radiation mode with the condensate in presence of a strong far off-resonant pump laser, whereas the coherent atomic displacement is obtained through the same interaction by injecting a suitably modulated short coherent pulse.

111 Chapter 8 Effects of decoherence and losses on entanglement generation In chapter 6 we have considered the entanglement generation by the CARL mechanism in the linear regime, and in chapter 7 we have suggested a possible application to interspecies teleportation. The physical system was considered in ideal conditions: we have described it with a lossless cavity and no kind of decoherence for atoms. In order to have a more feasible description of the entanglement generation we now want to take into account effects of atomic decoherence and cavity radiation losses. 8.1 Dissipative Master Equation For the three modes model deduced in chapter 3 and deeply studied in chapter 6 with effective Hamiltonian ρ Ĥ = δ + â 2â 2 δ â 1â 1 + i 2 where [(â 1 + â 2 )â 3 (â 1 + â 2)â 3 ], (8.1) δ ± = δ ± 1 ρ and δ = ω 2 ω 1 ρω R, (8.2) the dynamics of the system in presence of atomic decoherence and cavity losses is described by the following Master equation dˆϱ ] [Ĥ, dτ = i ˆϱ + 2γ 1 L[â 1 ]ˆϱ + 2γ 2 L[â 2 ]ˆϱ + 2κL[â 3 ]ˆϱ, (8.3) 99

112 100 Chapter 8. Effects of decoherence and losses on entanglement generation where γ 1, γ 2 and κ are the damping rates and L[â i ] is the Lindblad superoperator L[â i ]ˆϱ = â i ˆϱâ i 1 2â iâi ˆϱ 1 2 ˆϱâ iâi. (8.4) The Master equation can be transformed into a Fokker-Planck (FP) equation for the Wigner function of the state ˆϱ W (α 1, α 2, α 3, τ) = 3 i=1 d 2 ξ i π 2 e ξ i α i α i ξ i χ(ξ 1, ξ 2, ξ 3, τ). (8.5) where α i and ξ j are complex numbers, χ is the characteristic function defined as [ χ(ξ 1, ξ 2, ξ 3 ) = Tr ˆϱ ˆD 1 (ξ 1 ) ˆD 2 (ξ 2 ) ˆD ] 3 (ξ 3 ), (8.6) and ˆD j (ξ j ) = exp(ξ j â j ξ j â j ) is a displacement operator for the j-th mode. Using the differential representation of the Lindblad superoperator, the FP equation is W { } τ = u T Au + u T A u W + u T Du W (8.7) where ( u T = (α1, α 2, α 3 ) u T = α 2,, α1 α 3 and A and D are respectively the following drift and diffusion matrixes A = γ 1 + iδ 0 ρ/2 0 γ 2 + iδ + ρ/2 ρ/2 ρ/2 κ D = ) γ γ κ 8.2 Solution of the Fokker-Plank equation (8.8). (8.9) The solution of the FP equation (8.7) can be written as a convolution W (u, τ) = d 2 u 0 W (u 0, 0) G(u, τ; u 0, 0) (8.10) where W (u 0, 0) is the Wigner function for the initial state and the Green function G(u, t; u 0, 0) is the solution of Eq.(8.7) for the initial condition G(u, 0; u 0, 0) = δ 3 (u u 0 ). The calculation of the Green function is reported in Appendix E and we have: G(u, τ; u 0, 0) = 1 { } π 3 det Q(τ) exp [u M(τ)u 0 ] Q 1 (τ) [u M(τ)u 0 ], (8.11)

113 8.3. Evolution from vacuum and expectation values 101 where and M(τ) e Aτ = Q(τ) = τ 0 f 11 (τ) f 12 (τ) f 13 (τ) f 12 (τ) f 22 (τ) f 23 (τ) f 13 (τ) f 23 (τ) f 33 (τ) (8.12) dτ M(τ )DM (τ ). (8.13) In Eq.(8.12) f ij (τ) are the sum of three terms proportional to e iω kτ, where ω k, with k = 1, 2, 3, are the three roots of the cubic equation: [ ( ) ] 2 1 [ω δ i (κ γ + )] ω 2 ρ + iγ iργ = 0 (8.14) and γ ± = (γ 1 ± γ 2 )/2 (see appendix E for details). 8.3 Evolution from vacuum and expectation values Let now assume a pure initial state ˆϱ = φ φ where φ = 0 1, 0 2, 0 3. The Wigner function for this initial state is W (u 0, 0) = namely it is a Gaussian with covariance matrix ( ) 3 2 e 2u 0 u 0, (8.15) π C(0) = 1 2 I (8.16) where I is the 3 3 identity matrix. The Green function (8.11) is also a Gaussian so the convolution (8.10) makes the Wigner function Gaussian at any time τ. After some algebra we get that the covariance matrix for the Wigner function is C(τ) = Q(τ) M(τ)M (τ), (8.17) Since the state of the system is Gaussian the covariance matrix completely characterize his properties and from (8.17) it is possible to derive all the significative expectation values of the three modes. Remind that C ij = (u i u i )(u j u j ). (8.18)

114 102 Chapter 8. Effects of decoherence and losses on entanglement generation In particular (remember ˆn i = â iâi and i = 1, 2, 3) C ii = ˆn i (8.19) and C ij = â iâj for i j. (8.20) The numbers variance and equal-time correlation functions for the mode numbers are calculated from the forth-order covariance matrix G ijkl = (u i u i )(u j u j )(u k u k ) (u l u l ), (8.21) using the following relations: G iiii = ˆn 2 i + ˆn i G ijij = ˆn iˆn j ˆn i ˆn j (8.22) (i j) (8.23) It is possible to demonstrate that the elements G ijkl are generated from the covariance matrix C as: From Eqs.(8.22)-(8.24) it follows that: G ijkl = C ki C lj + C li C kj. (8.24) σ 2 (n i ) = ˆn i (1 + ˆn i ) (8.25) g (2) i,i = â iâ iâiâ i ˆn i 2 = 2 (8.26) g (2) i,j = ˆn iˆn j ˆn i ˆn j = 1 + C ij 2 ˆn i ˆn j, (8.27) where σ 2 (n i ) = (ˆn 2 i ˆn i ) 2, with i = 1, 2, 3 and i j. Finally, we calculate the two-mode relative number squeezing parameter: ξ i,j = σ2 (ˆn i ˆn j ) ˆn i + ˆn j = σ2 (n i ) + σ 2 (n j ) 2 C ij 2. (8.28) ˆn i + ˆn j We observe, from Eqs.(8.25) and (8.26), that the statistics is that of a chaotic (i.e. thermal) state, as in the lossless case of chapter 6. If the two states are squeezed, σ 2 (n i n j ) = 0, which implies ξ i,j = 0. Hence, it is expected that when ξ i,j decreases,

115 8.4. Numerical analysis for the relevant working regimes 103 Figure 8.1: Growth rate G = Imλ γ vs. δ for the unstable root of the cubic equation (8.14) in the semi-classical limit, ρ = 100. In (a) κ = 0 and γ = 0.5, 1, 2; in (b) γ = 0 and κ = 1, 5, 10. The dashed lines represent the case κ = γ = 0. the entanglement between the two states could improve. However, as we said in chapter 6, the condition σ 2 (n i n j ) = 0 is not a sufficient one for a signature of entanglement, showing only that the state is two-mode squeezed or, more generally, non-classically correlated. 8.4 Numerical analysis for the relevant working regimes We now investigate the different regimes of operation of CARL. For sake of simplicity, we will discuss only the case with γ 1 = γ 2 = γ, so that γ + = γ and γ = 0. In this case the cubic equation (8.14) becomes: [ω δ i (κ γ)] (ω 2 1ρ ) + 1 = 0 (8.29) 2 We will discuss the four different regimes of CARL, as defined in chapter 2, i.e.: i) semi-classical good-cavity regime (ρ 1 and κ 1); ii) quantum goodcavity (κ 2 ρ < 1); iii) semi-classical superradiant regime (ρ 2κ > 1); iv) quantum superradiant regime (κ 2 2κ > ρ). We note also that the case γ = κ should deserve a special attention. In fact, in this case Eq.(8.29) is the same of

116 104 Chapter 8. Effects of decoherence and losses on entanglement generation Figure 8.2: Growth rate G = Imλ γ vs. δ for the unstable root of the cubic equation (8.14) in the quantum limit. In (a), ρ = 0.2, κ = 0 and γ = 0.2, 0.5, 1; in (b), ρ = 1, γ = 0 and κ = 0.5, 1, 5. The dashed lines represent the case κ = γ = 0. the case without losses. The effect of decoherence is just a overall factor exp( γτ) multiplying the functions f ij in the drift and diffusion matrixes M and Q. It is expected that this particular case should share statistical properties similar to the case free of decoherence, as will be discussed later. Before discussing the statistical properties of the system, we investigate the effect of decoherence and cavity losses on the CARL instability in the different regimes. In Fig.8.1 we show the gain G = Imλ γ for the unstable root of Eq. (8.29) vs. δ in the semiclassical limit (ρ = 100) for the good-cavity regime, κ = 0 and γ = 0.5, 1, 2 (Fig. 8.1a) and for the transition to the superradiant regime with γ = 0, i.e. for κ = 1, 5, 10 (Fig. 8.1b). The dashed lines in Fig. 8.1 and 8.2 represent the lossless case with γ = 0 and κ = 0. We observe that the gain is more negatively affected increasing γ instead of increasing κ. The same is true in the quantum limit of Fig. 8.2, where G is drawn vs. δ for ρ = 0.2, κ = 0 and γ = 0.2, 0.5, 1 (Fig. 8.2a, good-cavity regime) and for ρ = 1, γ = 0 and κ = 0.5, 1, 5 (Fig. 8.2b, superradiant regime). In the case γ = κ, G = G (0) γ, where G (0) = Imλ (0) is the solution for the case without losses, shown by the dashed lines of Fig. 8.1 and 8.2. We note that, increasing κ or γ, G goes to zero remaining positive for some value of δ. On the contrary the case γ = κ shows a gain threshold for G (0) = γ. Fig. 8.3 shows the populations and the atom-atom squeezing parameter versus

117 8.4. Numerical analysis for the relevant working regimes 105 Figure 8.3: Populations and the atom-atom squeezing parameter vs δ in the semiclassical regime for τ = 2, γ = 0 and different values of κ. δ in the semi-classical regime for τ = 2, γ = 0 and different values of κ. With respect to the ideal case the correlations measured by ξ 1,2 in the good-cavity regime remain nonclassical for about the same values of δ, but they are strongly affected in the transition to the superradiant regime. In Fig. 8.4 we show again populations and squeezing parameter in the semi-classical regime versus τ for δ = 5, γ = 0 and different values of κ. We note that correlations remains non classical even for long times only for small values of κ, than ξ 12 increases very fast. In Fig. 8.5 Figure 8.4: Populations and squeezing parameter in the semi-classical regime vs τ for δ = 5, γ = 0 and different values of κ

118 106 Chapter 8. Effects of decoherence and losses on entanglement generation Figure 8.5: Atom-atom squeezing parameter in the semi-classical regime vs δ for τ = 2 and vs τ for δ = 5, with κ = 1 and for different values of γ. we show for a fixed value of κ how correlations are affected by considering γ 0. The oscillations in time of ξ 12 are strongly damped by the presence of a small decoherence γ giving the remarkable results that for enough small values of γ κ we can have a stationary nonclassically correlated atom-atom state. This result could be very interesting for experimental applications. In Fig. 8.6 we show populations and atom-atom squeezing in this stationary regime for different values of κ = γ. In the good-cavity regime the number of nonclassically correlated atoms remains significative. Figure 8.6: Populations and atom-atom squeezing versus τ for different values of κ = γ.

119 8.4. Numerical analysis for the relevant working regimes 107 Figure 8.7: Populations and atom-photon squeezing parameter in the quantum regime vs τ for δ = 5, γ = 0 and different values of κ. In Fig. 8.7 we show the populations and the atom-photon squeezing parameter versus δ in the quantum regime for τ = 2, γ = 0 and different values of κ. Atomphoton correlations are very sensible to the presence of cavity losses and the time duration of non classicality decreases very fast increasing κ. In Fig. 8.8 we show for a fixed value of κ how correlations are affected by considering γ 0. As in the semiclassical regime the case κ = γ is particular. The corresponding state is not stationary in the proper sense but however it remains nonclassically correlated for Figure 8.8: Populations and atom-photon squeezing parameter in the quantum regime vs τ for δ = 5, κ = 0.1 and different values of γ.

120 108 Chapter 8. Effects of decoherence and losses on entanglement generation <n i > Figure 8.9: Populations and atom-photon squeezing in the quantum regime versus τ for different values of κ = γ. rather long times. In Fig. 8.9 we show populations and atom-photon squeezing in this particular regime for different values of κ = γ. In the good-cavity regime the number of nonclassically correlated atoms and photons remains significative only for small values of κ = γ. Increasing dissipation the modes remain not populated. The existence of the stationary states results from the particular form of the functions f ij, where, in the exponential regime, is present a competition between the exponential behaviour due to the instable root of the cubic equation and the damping due to losses. In order to quantify the differences between the three-mode entanglement ψ τ written in chapter 6 and the one described in this chapter we employ the fidelity (see appendix D for details about the definition) F = ψ τ ϱ ψ τ between the ideal pure state (6.44), obtained with an ideal cavity and no atomic decoherence, and the state ϱ corresponding to the evolved Wigner function W (u, t). In terms of the characteristic functions we have F = d 2 u π 3 χ ψ(u)χ( u) = where C ψ is the covariance matrix for ψ τ 1 det(c + C ψ ) (8.30) C ψ (τ) = 1 2 M ψ(τ)m ψ (τ), (8.31) where the matrix M ψ is the same that in Eq. (8.12) calculated for κ = γ = 0 so that f 1i = f i,f 2i = g i and f 3i = h i where f i, g i, h i are the functions introduced in

121 8.5. Concluding remarks 109 chapter 6 and explicitly reported in appendix A. A systematic numerical study of Eq. (8.30) is in progress. Here we give a first approximate result calculating the fidelity F to the first order in time τ for γ = 0. For the involved matrixes we have Therefore, for small τ we have M(τ) = exp(aτ) I + Aτ (8.32) M ψ (τ) = exp(a ψ τ) I + A ψ τ (8.33) C(τ) + C ψ (τ) I + Q(τ) Dτ. (8.34) [D + 12 (A + A + A ψ + A ψ ) ] τ (8.35) so that and C(τ) + C ψ (τ) 1 0 2τ ρ/ τ ρ/2 0 3κτ + 1 (8.36) det [C(τ) + C ψ (τ)] 1 + 3κτ + 2ρτ 2. (8.37) At the first order in τ for the fidelity we obtain F (τ) 1 3κτ. (8.38) and the result do not depend on atomic parameters. Notice that C C ψ for κ 0 and that det(c ψ ) = 1 at any time Concluding remarks In this chapter we have studied the linear regime for the CARL-BEC model taking into account effects of atomic decoherence and cavity radiation losses. In the superradiant regime the nonclassical correlations are destroyed by the presence of atomic decoherence and radiation losses. However for the superradiant regime it is possible to study a model that gives indication about the entanglement between the initial condensate and the momentum side modes generated via the CARL mechanism. In such a model it is necessary to take into account the depletion of the initial condensate, as we have seen in chapter 5, and the work is in progress.

122 110 Chapter 8. Effects of decoherence and losses on entanglement generation In the good-cavity regime the nonclassical correlations are preserved for enough small values of the parameters γ and κ. Moreover we have shown that the case κ = γ is special and can give rise to stationary nonclassical atom-atom and atomphoton states. These result are very promising for experimental applications. We have indeed discussed in chapter 7 how is important for applications to have long life entangled states. In such a way the whole protocol, as for example interspecies teleportation, could be completed within the decoherence time.

123 Conclusion and Outlook In this dissertation several aspects of the interaction among quantum degenerate atomic gases and the optical field have been analyzed and discussed, with the main focus on the appearence of quantum coherent effects. The system considered was a BEC driven by a far off-resonant pump laser and coupled to a single mode of an optical ring cavity. After a short introduction, in chapter 1 we have described the mechanism that lies below this kind of physics, the CARL in his original version. In particular we showed the positive feedback mechanism that give rise to an exponential growth of both the probe intensity and the atomic bunching. By taking into account the translational degrees of freedom of the active medium, CARL can lead to the exponential amplification of a weak probe. We interpreted the process of amplification as evolving in two steps: first, the external field creates a weak gain profile in the frequency response of a collection of independent atoms and begins the buildup of a spatial structure with the help of the atomic recoil; next the probe, whose carrier frequencies lies within a selected gain region of the active medium, undergoes exponential amplification. The role of the atomic recoil is essential to this process: not only it is the cause of the emergence of the spatial grating pattern, but it also reinforces the coherent growth of the signal to be amplified as energy is transferred from the atoms to the probe field. It is worth noticing that, even though we have demonstrated the amplification of a probe signal, since the saturation value of the intensity and of the bunching is independent of the initial value of the probe, the process can be initiated from spontaneous emission noise. In order to extend the model in the region of ultracold atoms, a quantum mechanical description of the center-of-mass motion of the atoms have been included. Thus, in chapter 2 we presented a way to work out this program simply performing 111

124 112 Conclusion and Outlook a first quantization of the external variables of the atoms, position and momentum. This simple model allows for a quantum limit in which the average atomic momentum changes in discrete units of the photon recoil momentum q and therefore gives a description of all the main features of the considered system. We observed that the behavior of the system is different for conservative and dissipative regimes. In the conservative regime (no radiation losses), the quantum model depends on a single collective parameter, ρ, that can be interpreted as the average number of photons scattered per atom in the classical limit. When ρ 1, the semiclassical CARL regime is recovered, with many momentum levels populated at saturation. On the contrary, when ρ 1, the average momentum oscillates between zero and q, and a periodic train of 2π hyperbolic secant pulses is emitted. In the dissipative regime (large radiation losses) and in a suitable quantum limit (ρ 2κ), a sequential superradiant scattering occurs, in which after each process atoms emit a π hyperbolic secant pulse and populate a lower momentum state. These results describe the regular arrangement of the momentum pattern observed in experiments of superradiant Rayleigh scattering from a BEC. In chapter 3 we have deduced a quantum field theory for the CARL in a BEC introducing bosonic creation and annihilation operators of a given center-of-mass momentum mode. In the limit of no collisions and taking expectation values for operators, this model reduces to the quantum model introduced in chapter 2 by first quantization. Moreover, in the limit of undepleted atomic ground state and unsaturated probe field, we have deduced the corresponding linearized three mode model that has been later extensively investigated in chapter 6 and 8. So far, experiments in a good-cavity regime has been performed only for the classical CARL. However recent experiments have demonstrated that this instability can play an important role also in the case when laser light is scattered into the vacuum modes of the electromagnetic field in absence of the cavity. In chapter 4 we reviewed some of the experimental realizations, such as superradiant Rayleigh scattering and matter waves amplification, that can be interpreted with the full quantum version of CARL model in the dissipative regime, where the radiation emission is superradiant. These experiments have demonstrated the formation of atomic matter waves in a cigar-shaped BEC pumped by an off-resonant laser beam, together with highly directional scattering of light along the major axis of the condensate.

125 Conclusion and Outlook 113 A variation of Dicke superradiance was observed in which the role of electronic coherence, which stores the memory of previous scattering events, is replaced by coherence between center-of-mass momentum states, i.e. interference fringes in the atomic density. Referring to the simple quantum model presented in chapter 2 these kind of experiments lay in the superradiant regime of CARL: the absence of the cavity can be described considering a high value of radiation losses. In chapter 5 we analyzed some experiment performed at LENS in Florence about superradiant Rayleigh scattering from a moving BEC. This allowed to investigate the influence of the initial velocity of the condensate on superradiant Rayleigh scattering. Using the CARL-BEC model in the mean-field limit we have shown that the efficiency of the overall process is fundamentally limited by the decoherence between the two atomic momentum states. Analyzing a first experiment we have studied the dependence of the decoherence rate on the initial momentum of the condensate. We identified a velocity dependent contribution to the decoherence rate, which can be minimized when the energy conservation condition is satisfied (i.e. the scattered and the unscattered atomic wavepacket have the same kinetic energy in the laboratory frame). In a second experiment, performed adding a counterpropagating beam, we have explored the transition from the pure superradiant regime to the Rabi oscillations regime induced by stimulated Bragg scattering. Unlike experimental realizations, the theoretical extensions of CARL into the regime of BEC focused mainly on exploiting the instability of the light-matter interaction in the good cavity regime to parametrically amplify atomic and optical waves as well as to optically manipulate matter-wave coherence properties and generate entanglement between atomic and optical fields. In chapter 6 we have deeply investigated the properties of the linear model for the quantum system BEC-radiation, such as quantum fluctuations and entanglement, in a good cavity regime. We have shown that the evolved state is a fully inseparable three mode Gaussian one. Moreover we have shown that, for different regimes, the general state may reduce to a thermal atom-atom or atom-photon entangled state. These two-modes states have the same form of the twin-beam state of radiation used to realize continuous variable optical teleportation, and this allowed us to employ the same kind of Bell measurement scheme. Indeed in chapter 7 we have proposed a novel scheme to realize the interspecies teleportation of the quantum state of a single mode radiation field onto

126 114 Conclusion and Outlook the collective state of atoms with a given momentum out of a BEC. The twinbeam-like atom-photon entangled state established through the CARL interaction provides the entangled resource needed for the teleportation protocol, whereas the coherent atomic displacement is obtained through the same interaction by injecting a suitably modulated short coherent pulse in the probe mode. Finally in chapter 8 the results obtained in chapter 6 have been extended to include the effects due to atomic decoherence and cavity losses. The calculation are performed by means of the Master equation formalism and a systematic numerical analysis of the effects of decoherence and losses on non classical properties of the system has been given. In the good-cavity regime the nonclassical correlation are preserved for enough small values of the parameters γ and κ. Moreover we have shown that the case κ = γ is special and can give rise to stationary nonclassical atom-atom and atom-photon states. These results are very promising for experimental applications. We have indeed discussed in chapter 7 how is important for applications to have long life entangled states. In such a way the whole protocol, as for example interspecies teleportation, could be completed within the decoherence time. Future extensions of the work presented in these thesis concerns either theoretical and experimental features. From a theoretical point of view it will be very interesting to study a model that gives indication about the possible entanglement between the momentum side modes generated via the CARL mechanism and the initial condensate. In such a model it would be necessary to take into account the depletion of the initial condensate that, as we have seen in chapter 5, is not negligible in the superradiant regime. Regards to the good-cavity regime, the experimental set up of Ref. [17] described in chapter 1 is very promising for experimental investigation of the features of quantum CARL. The cell for cold atoms used by the group of Prof. Zimmermann is inside the optical cavity and, with appropriate variations to the apparatus, conditions for BEC inside an optical cavity could be reached. This would open new interesting scenarios either for the study of fundamental physics or for applications of BEC to quantum information implementations.

127 Appendix A General solution of the linear model The general solution of the linear problem associated with the three mode Hamiltonian ρ Ĥ = δ + â 2â 2 δ â 1â 1 + i 2 was given in Eqs.(6.25)-(6.27) and reads: [(â 1 + â 2 )â 3 (â 1 + â 2)â 3 ], (A.1) â 1(τ) = e iδτ [g 1 (τ)â 1(0) + g 2 (τ)â 2 (0) + g 3 (τ)â 3 (0)] (A.2) â 2 (τ) = e iδτ [h 1 (τ)â 1(0) + h 2 (τ)â 2 (0) + h 3 (τ)â 3 (0)] (A.3) â 3 (τ) = e iδτ [f 1 (τ)â 1(0) + f 2 (τ)â 2 (0) + f 3 (τ)â 3 (0)], (A.4) where the expressions of the quantities f i, g i and h i (i = 1, 2, 3) are ρ 3 f 1 (τ) = i (λ j + 1/ρ) eiλ jτ = g 3 (τ) (A.5) 2 j=1 j ρ 3 f 2 (τ) = i (λ j 1/ρ) eiλ jτ = h 3 (τ) (A.6) 2 j f 3 (τ) = g 1 (τ) = j=1 3 (λ 2 j 1/ρ 2 ) eiλ jτ j=1 j 3 [(λ j δ)(λ j + 1/ρ) ρ/2] eiλ jτ j=1 g 2 (τ) = ρ 2 3 j=1 j (A.7) (A.8) e iλ jτ j = h 1 (τ) (A.9) 115

128 116 Appendix A. General solution of the linear model h 2 (τ) = 3 [(λ j δ)(λ j 1/ρ) + ρ/2] eiλ jτ j=1 j (A.10) where j = λ j (3λ j 2δ) 1/ρ 2 and λ 1, λ 2 and λ 3 are the roots of the cubic Eq.(6.28), (λ δ)(λ 2 1/ρ 2 ) + 1 = 0. (A.11) Because the commutation rules for the operators a i and a i are preserved at the time τ, they imply the following relations between the functions f i, g i and h i : We now show that the evolution operator 1 + f 1 2 = f f 3 2 (A.12) g 1 2 = 1 + g g 3 2 (A.13) 1 + h 1 2 = h h 3 2 (A.14) g 1 f 1 = g 2 f 2 + g 3 f 3 (A.15) g 1 h 1 = g 2 h 2 + g 3 h 3 (A.16) h 1 f 1 = h 2 f 2 + h 3 f 3. (A.17) Û(τ) = exp( iĥτ), where Ĥ is given by Eq.(A.1), can be disentangled into those of individual operators. This allows to calculate how the state ψ(τ) evolves from the vacuum state 0, 0, 0. Introducing the five operators Ĵ1 = â 1 â 1 + â 3â 3, Ĵ 2 = â 3â 3 â 2â 2, Ĵ = â 2 â 3, ˆK = â1 â 3 and ˆM = â 1 â 2, the Hamiltonian (A.1) can be written in the following form: Ĥ = Ĉ + δ 3 (Ĵ2 Ĵ1) + 1 ρ (Ĵ1 + Ĵ2) + i ρ/2(ĵ ˆK Ĵ + ˆK ), (A.18) where Ĉ = (δ/3)(1 + 2Ĉ) 1/ρ and Ĉ is the constant of motion (3.49). The operators Ĵ1, Ĵ 2, Ĵ, ˆK and ˆM satisfy the following commutation relations: [Ĵ, Ĵ ] = Ĵ2 (A.19) [ ˆK, ˆK ] = Ĵ1 (A.20) [ ˆM, ˆM ] = Ĵ1 + Ĵ2 (A.21) [Ĵ1, Ĵ] = Ĵ (A.22) [Ĵ2, Ĵ] = 2Ĵ (A.23) [ ˆK, Ĵ1] = 2 ˆK (A.24) [ ˆK, Ĵ2] = ˆK (A.25)

129 117 [ ˆM, Ĵ1] = [ ˆM, Ĵ2] = ˆM (A.26) [ ˆK, Ĵ] = ˆM (A.27) [ ˆM, Ĵ ] = ˆK (A.28) [ ˆM, ˆK ] = Ĵ (A.29) and [Ĵ1, Ĵ2] = [Ĵ, ˆK] = [Ĵ, ˆM] = [ ˆK, ˆM] = 0. Hence, they form a closed algebra. We observe that the operators Ĵx = i( ˆM ˆM ), Ĵy = i( ˆK ˆK ) and Ĵz = i(ĵ Ĵ ) are the generators of SU(1,1) Lie algebra, whose statistical properties have been extensively discussed in ref. [118, 119]. Because Ĉ commutates with Ĥ, we can write Ĥ = Ĉ + Ĥ and neglect the inessential phase factor e iĉ τ in the evolution operator Û(τ), that can be written in the form of a Baker-Hausdorff equation: Û(τ) = e iĥ τ = e α 1 ˆK e α 2 ˆM e α 3Ĵ e α 4Ĵ1 e α 5Ĵ2 e α 6Ĵe α 7 ˆKe α 8 ˆM, (A.30) where α i (i = 1,..8) are complex functions depending on τ. Applying Û(τ) to the vacuum state we obtain ψ = Û(τ) 0, 0, 0 = eα 4 e α 1 ˆK e α 2 ˆM 0, 0, 0 = e α 4 m,n=0 m,n=0 (m + n)! α1 m α2 n m+n, n, m. m!n! (A.31) The constant α 4 can be obtained from the normalization condition: ψ ψ = 1 = e 2Reα (m + n)! 4 α 1 2m α 2 2n e 2Reα 4 = m!n! 1 α 1 2 α 2, (A.32) 2 where we used the formula: Γ(a + n) zn n! = Γ(a)(1 z) a (A.33) n=0 and Γ(a) is the Gamma function. Hence, neglecting the immaginary part of α 4, which introduces an inessential phase factor, e α 4 = 1 α 1 2 α 2 2. (A.34) The calculation of the average occupation numbers for the modes 2 and 3, using the state (A.31), yields: ˆn 2 = ˆn 3 = α α 1 2 α 2 2 (A.35) α α 1 2 α 2, 2 (A.36)

130 118 Appendix A. General solution of the linear model which, once inverted and because ˆn 1 = ˆn 2 + ˆn 3, gives: e α 4 = ˆn1. (A.37) The two functions α 1 and α 2 can be obtained calculating the expectation values of â 1 â 3 and a 2â 3 and using the Heisenberg picture of the operators, Eqs.(A.2)-(A.4): â 1 â 3 = f 2 g2 + f 3 g3 = f 1 g1 (A.38) a 2â 3 = f 1 h 1, (A.39) where we used Eq.(A.15). Conversely, evaluating these expectation values using the state (A.31) we obtain: â 1 â 3 = e 2α 4 α 1 (A.40) â 2â 3 = e 2α 4 α 1 α2. (A.41) Finally, combining the two results, we obtain, after some algebra: α 1 = f 1 g1 1 + ˆn 1 (A.42) α 2 = h 1 g ˆn 1. (A.43)

131 Appendix B Wigner functions In this appendix we review some simple formulas that connect the generalized Wigner functions [120, 121] of s-ordering with the density matrix, and vice-versa. These formulas prove very useful for quantum mechanical applications as, for example, for connecting master equations with Fokker-Planck equations, or for evaluating the quantum state from Monte Carlo simulations of Fokker-Planck equations, and finally for studying positivity of the generalized Wigner functions in the complex plane. As a method to express the density operator in terms of c-number functions, the Wigner functions often lead to considerable simplification of the quantum equations of motion as, for example, the transformation of Master equations in operator form into more treatable Fokker-Planck differential equations (see, for example [122]). Using the Wigner function one can express quantum-mechanical expectation values in form of averages over the complex plane (the classical phase-space), the Wigner function playing the role of a c-number quasi-probability distribution, which generally can also have negative values. More precisely, the original Wigner function allows to easily evaluate expectations of symmetrically ordered products of the field operators, corresponding to the Weyl s quantization procedure [123]. Generalized s-ordered Wigner function W s (α, α ) are defined as follows [124] W s (α, α ) = C d 2 λ π 2 eαλ α 1 λ+ 2 s λ 2 Tr{ ˆD(λ)ˆϱ}, (B.1) where the integral is performed on the complex plane with measure d 2 λ = d Re[λ] d Im[λ], ˆϱ being the density operator, ˆD(α) exp{αâ α â} denotes the displacement op- 119

132 120 Appendix B. Wigner functions erator, â and â ([a, a ] = 1) are the annihilation and creation operators of the field mode of interest and s is a real number. The Wigner functions in equation (B.1) allow to evaluate s-ordered expectation values of the field operators through the following relation Tr{:(â ) n â m : s ˆϱ} = d 2 α W s (α, α ) α n α m. (B.2) C The particular cases s = 1, 0, 1 correspond to anti-normal, symmetrical, and normal ordering, respectively. In these cases the generalized Wigner function W s (α, α ) are usually denoted by the following symbols and names 1 Q(α, π α ) W (α, α ) P (α, α ) for s = 1 (Q-function); for s = 0 (usual Wigner function); for s = 1 (P -function). (B.3) For the normal (s = 1) and anti-normal (s = 1) orderings, the following simple relations with the density matrix are well known Q(α, α ) α ˆϱ α, (B.4) ˆϱ = d 2 α P (α, α ) α α, (B.5) C α being a coherent state. Among the three particular representations (B.3), the Q- function is positively definite and infinitely differentiable (it actually represents the probability distribution for ideal joint measurements of position and momentum of the harmonic oscillator). On the other hand, the P -function is known to be possibly highly singular and the only pure states for which it is positive are the coherent states [125]. Finally, the usual Wigner function has the remarkable property of providing the probability distribution of the quadratures of the field in form of marginal distribution, namely d Im[α] W (αe iϕ, α e iϕ ) = ϕ Re[α] ˆϱ Re[α] ϕ, (B.6) where x ϕ denotes the (unnormalizable) eigenstate of the field quadrature ˆX ϕ = 1 2 (â e iϕ + âe iϕ) (B.7) with real eigenvalue x. Notice that any couple of quadratures ˆX ϕ, ˆXϕ+π/2 is canonically conjugate, namely [ ˆX ϕ, ˆX ϕ+π/2 ] = i/2, and it is equivalent to position and

133 121 momentum of a harmonic oscillator. Usually, negative values of the Wigner function are viewed as signature of a non-classical state, the most eloquent example being the Schrödinger-cat state [126], whose Wigner function is characterized by rapid oscillations around the origin of the complex plane. From equation (B.1) one can notice that all s-ordered Wigner functions are related to each other through Gaussian convolution W s (α, α ) = d 2 β W s (β, β 2 ) { C π(s s) exp { s s 2 = exp 2 α α } 2 α β 2 s s (B.8) } W s (α, α ), (s > s). (B.9) Equation (B.8) shows the positivity of the generalized Wigner function for s < 1, as a consequence of the positivity of the Q function. From a qualitative point of view, the maximum value of s, keeping the generalized Wigner functions as positive, can be considered as an indication of the classical nature of the physical state [127, 128]. An equivalent expression for W s (α, α ) can be derived as follows [129]. We rewrite equation (B.1) as W s (α, α ) = Tr{ˆϱ ˆD(α) Ŵs ˆD (α)}, (B.10) where Ŵ s = C d 2 λ π 2 e 1 2 s λ 2 ˆD(λ). (B.11) Through the customary Baker-Campbell-Hausdorff (BCH) formula } eâ e ˆB = exp {Â + ˆB + 1[Â, ˆB] 2, (B.12) which holds when [Â, [Â, ˆB]] = [ ˆB, [Â, ˆB]] = 0, one writes the displacement in normal order, and, integrating with respect to arg[λ] and λ, one obtains Ŵ s = 2 π(1 s) n=0 ( ) n 1 2 (â ) n â n = n! s 1 where we used the normal-ordered forms 2 π(1 s) ( )â â s + 1 s 1, (B.13) :(â â) n : = (â ) n â n = â â(â â 1)... (â â n + 1), (B.14)

134 122 Appendix B. Wigner functions and the identity :e xâ â: = ( x) l (â ) l â l = (1 l! x)â â. (B.15) l=0 The density matrix can be recovered from the generalized Wigner functions using the following expression ˆϱ = 2 ( d 2 α W s (α, α )e 2 1+s α 2 e 2α 1+s s 1 â 1 + s C s + 1 )â â e 2α 1+s â. (B.16) For the proof of equation (B.16) the reader is referred to reference [129]. In particular, for s = 0 one has the inverse of the Glauber formula ˆϱ = 2 d 2 α W (α, α ) ˆD(2α)( )â â, C whereas for s = 1 one recovers equation (B.5) that defines the P -function. (B.17)

135 Appendix C Homodyne and multiport homodyne detection C.1 Matrix notation For pure states in the bipartite Hilbert space H 1 H 2 we will use the notation C. = ij C ij i 1 j 2 (C.1) C. = ij C ij1 i 2 j (C.2) where C ij = [C] ij are the elements of the matrix C and i n is the standard basis of H n, n = 1, 2 [130]. A given matrix C individuates a linear operator from H 1 to H 2, given by Ĉ = ij C ij i 12 j. (C.3) In the following we will consider H 1 and H 2 to be both the Hilbert space of a harmonic oscillator. Thus we will refer only to square matrices and omit the indices for bras and kets. We have the following identities  ˆB C = ACB T where (..) ] T denotes transposition with respect to the standard basis, and A B = Tr [ ˆB. Normalization of state C thus implies Tr[Ĉ Ĉ] = 1. Also useful are the following relations about partial traces: Tr 2 [ A B ] =  ˆB and Tr 1 [ A B ] =  T ˆB where (..) denotes complex conjugation, and about partial transposition ( C C ) θ = (Ĉ Î)Ê(Ĉ Î), where Ê = ij i j j i is the swap operator. 123

136 124 Appendix C. Homodyne and multiport homodyne detection Notice that  ˆB and ÂT ˆB in the above formulas should be meant as operators acting on H 1 and H 2 respectively. C.2 Balanced homodyne detection Balanced homodyne detection [131, 132, 133] is a scheme to measure any given quadrature of a radiation field, that is the observables ˆX ϕ = 1 2 (â e iϕ + âe iϕ) (C.4) where ϕ [0, 2π). The scheme of a balanced homodyne detector is showed in Fig. c b Figure C.1: Scheme of the balanced homodyne detector. C.1: the signal mode â is mixed with a strong laser beam mode ˆb in a balanced 50/50 beam splitter. The mode ˆb is the so-called local oscillator (LO) mode of the detector. It operates at the same frequency of â and is excited by a laser in a strong coherent state z ( z 1). Since in all experiments using homodyne detectors the signal and the LO beams are generated by a common source, we assume that they have a fixed phase relation. In this case the LO phase gives the reference ϕ for the quadrature measurement and then we identify the phase of the LO with the phase difference between the two modes. As we will see, by tuning ϕ = arg[z] we can measure the quadrature ˆX ϕ at different phases. After the beam splitter, the two modes are detected by two identical photodetectors (usually linear photodiodes) and finally the difference of photocurrents at zero frequency is electronically processed and rescaled by 2 z. The modes at the

137 C.2. Balanced homodyne detection 125 output of the 50/50 beam splitter are given by ĉ = â ˆb 2 and ˆd = â + ˆb 2, (C.5) and the output of the dector, i.e. the difference of photocurrents, is given by the following operator Î = ˆd ˆd ĉ ĉ 2 z = â ˆb + ˆb â 2 z. (C.6) In order to evaluate the probability distribution of the output photocurrent Î when the signal mode â is described by the generic state ˆϱ [134, 135] consider the moments generating function of the photocurrent Î { } χ(λ) = Tr ˆϱ z z e iλî, (C.7) and get the probability distribution of Î as the Fourier transform of χ(λ) P (Î) = + dλ 2π e iλîχ(λ). (C.8) Evaluating (C.7) by standard disentangling formulas for exponential operator [134, 136, 137] one can recast χ(λ) in normal-order with respect to â. This form is suitable to take the strong-lo limit z, where only the lowest order terms in λ/ z are retained, and â â is neglected with respect to z 2. One has lim χ(λ) = exp i λ } z 2 eiϕ â exp { λ2 exp i λ 2 e iϕ â = exp{iλ 8 ˆX ϕ } a a, (C.9) where ϕ = arg[z]. The generating function in equation (C.9) thus correspond to the POVM ˆΠ(x) = + dλ 2π exp{iλ( ˆX ϕ x)} = δ( ˆX ϕ x) x ϕϕ x, i.e. the projector on the eigenstate x φ of the quadrature ˆX ϕ with eigenvalue x. (C.10) In conclusion, the balanced homodyne detector achieves the ideal measurement of the quadrature ˆX ϕ in the strong LO limit. In this limit, the probability distribution of the output photocurrent Î approaches exactly the probability distribution p(x, ϕ) = ϕ x ˆϱ x ϕ of the quadrature ˆX ϕ, and this for any state ˆϱ of the signal mode â.

138 126 Appendix C. Homodyne and multiport homodyne detection In order to take into account the non-unit detector quantum efficiency, we perform the following substitution ĉ = η ĉ 1 η û, (C.11) ˆd = η ˆd 1 η ˆv, (C.12) û and ˆv being vacuum modes; now, the output current is scaled by 2 z η, i.e. Î η 1 {[ ] } 1 η â + (û + ˆv) 2 z 2η ˆb + h.c., (C.13) where only terms containing the strong LO mode ˆb are retained. The POVM is then obtained by replacing ˆX ϕ ˆX ϕ + 1 η 2η (û ϕ + ˆv ϕ ) (C.14) in equation (C.10), with ŵ ϕ = (ŵ e iϕ + ŵe iϕ )/2, ŵ = û, ˆv, and tracing the vacuum modes û and ˆv. One then obtains where ˆΠ η (x) = = = + dλ 2π eiλ( ˆX ϕ x) 0 e iλ 1 η 2η ûϕ 0 2 = 1 + 2π 2 η 1 2π 2 η exp { dx e (x x ) 2 η x ϕϕ x (x ˆX ϕ ) η } + dλ 2π eiλ( ˆX ϕ x) e λ2 1 η 8η, (C.15) 2 η = 1 η 4η. (C.16) Thus in the nonideal case the POVM is just the Gaussian convolution of the ideal one with a Gaussian conditional probability. C.3 Double homodyne detection In order to achieve the estimation of the complex amplitude the following multiport homodyne scheme [138, 110, 139, 140, 141] can be employed: the mode under investigation is mixed with another reference mode in a balanced beam splitter and

139 C.3. Double homodyne detection 127 then, on the two outgoing modes, two conjugated quadratures, say ˆx ˆX 0 = 1 2 (ĉ + ĉ ) (C.17) ŷ ˆX π 2 i 2 ( ˆd ˆd), (C.18) are measured by balanced homodyne detection with LO phase-shifted by π/2. The two output photocurrents are a couple of commuting selfadjoint operators, corresponding to the real and the imaginary part of the complex operator Ẑ = ReẐ + iimẑ, with [Ẑ, Ẑ ] = [ReẐ, ImẐ] = 0, where Ẑ is given by Ẑ = â ˆb, (C.19) and the arbitrary phases of modes have been suitably chosen. The overall photocurrent Ẑ is a normal operator, and therefore the probability measure of the detector is given by the projector over the orthogonal eigenvectors of Ẑ. Using the matrix notation introduced above we write the eigenvectors of In fact one has Ẑ D(z) = (â ˆb )( ˆD a (z) Îb) I = ( ˆD a (z) Îb)(â ˆb + z) Ẑ with eigenvalue z as 1 π D(z). n n n=0 = z( ˆD a (z) Îb) Î = z D(z). (C.20) The orthogonality of such eigenvectors can be verified through the relation D(z) D(z ) = Tr[ ˆD (z) ˆD(z )] = πδ (2) (z z ), (C.21) where δ (2) (α) denotes the Dirac delta function over the complex plane. The (orthogonal) POVM of the detector is given by Êz = D(z) D(z). If ˆR is the joint density matrix describing the quantum state of modes â and ˆb the statistics of the measurement is described by the probability density K z = Tr ab [ ˆR Ê z ]. Let us now consider the situation in which ˆR = ˆϱ ˆσ is factorized. Let us say that ˆϱ is the state under investigation and ˆσ a known reference state usually referred to as the probe of the detector. In this case we have that the statistics of the outcomes may be also described as follows [ K z = Tr ab ˆϱ ˆσ ˆΠ ] z = 1 ]] π Tr a [ˆϱ Tr b [Î ˆσ D(z) D(z) = 1 π Tr [ a [ˆϱTrb D(z)σ T D(z) ]] = 1 π Tr a [ˆϱ ˆΠ ] z (C.22)

140 128 Appendix C. Homodyne and multiport homodyne detection with ˆΠ z = 1 π ˆD(z)ˆσ T ˆD (z), which is the POVM of the detector referring only to the mode â. If the probe is in the vacuum ˆσ = 0 0 the setup measures the Q-function Q z = z ˆϱ z of the state ˆϱ, i.e. the distribution in the complex plane of the amplitude of the signal. Notice that the role of signal and probe may be exchanged, as the statistics can be also written as K z = 1 π Tr b [ ˆσ ˆΠ ] z where the POVM acting on the mode ˆb is given by (C.23) ˆΠ z = 1 π ˆD T (z)ˆϱ T ˆD (z). (C.24) The effect of non-unit quantum efficiency can be taken into account in analogous way as for homodyne detection. The Ẑ current is scaled by an additional factor η1/2, and vacuum modes û and ˆv are introduced, thus giving Ẑ η = â ˆb 1 η 1 η + û ˆv. η η Upon tracing over modes û and ˆv, one obtain the POVM ) d 2 β z β 2 Ê zη = exp ( D(β) D(β) π 2 η 2 η (C.25) (C.26) where 2 η = (1 η)/η.

141 Appendix D Continuous variable teleportation as conditional measurement The original teleportation protocol by Bennett et al. [23] concerned quantum states in a two-dimensional Hilbert space and the correspondent experiments have been mostly performed in the optical domain for polarization qubit [99, 98]; continuous variable quantum teleportation (CVQT) has been also realized by optical means and successful teleportation of coherent states has been reported [82]. In optical CV teleportation entanglement is provided by the twin beam state of radiation (TWB). In quantum mechanics, the reduction postulate provides an intrinsic mechanism to achieve effective nonlinear dynamics. In fact, if a measurement is performed on a portion of a composite entangled system, e.g. the bipartite entangled systems made of two modes of radiation, the other component is conditionally reduced according to the outcome of the measurement [142]. The resulting dynamics is highly nonlinear, and may produce quantum states that cannot be generated by currently achievable nonlinear processes. The efficiency of the process, i.e. the rate of success in getting a certain state, is equal to the probability of obtaining a certain outcome from the measurement. The nonlinear dynamics induced by conditional measurements has been analyzed for a large variety of tasks [143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159], among which we mention photon adding and subtracting schemes [144, 145], optical state truncation of coherent states [146], generation of cat-like states [147, 148, 149], state filtering by active cavities [150, 152], synthesis of arbitrary unitaries [153], and generation of optical qubit by conditional 129

142 130 Appendix D. Continuous variable teleportation as conditional measurement interferometry [154]. Particularly interestign is the use of conditional measurements on entangled twinbeam of radiation (TWB) to engineer quantum states, i.e. to produce, manipulate, and transmit nonclassical light. The reason to choose TWB as entangled resource for conditional measurements is twofold. On one hand, TWBs are the natural generalization to continuous variable (CV) systems of Bell states, i.e. maximally entangled states for qubit systems. On the other hand, and more importantly, TWBs are the only CV entangled states that can be reliably produced with current technology, either by parametric downconversion of the vacuum in a nondegenerate parametric amplifier [160], or by mixing two squeezed vacua from a couple of degenerate parametric amplifiers in a balanced beam splitter [83, 161]. In particular the action of homodyne detection on TWB represents a tunable source of squeezed light, with high conditional probability and robustness to experimental imperfections. The joint measurement of the sum- and difference-quadratures of two modes, corresponding to the measurement of the real and the imaginary parts of the complex photocurrent Ẑ = â+ˆb, a and b being two modes of the field is realized by generalized heterodyne detection if the two modes have different frequencies, and by multiport homodyne detection if they have the same frequency. If one of the two modes is a beam of the TWB, whereas the second mode is excited in a given reference state, usually referred to as the probe of the measurement, this conditional measure corrispond to a CV quantum teleportation. D.1 Conditional quantum state engineering The general measurement scheme we are going to consider is schematically depicted in Fig. D.1. The first stage consists of a non-degenerate optical parametric amplifier (NOPA) obtained by a χ (2) nonlinear optical crystal cut either for type I or type II phasematching. In the parametric approximation (i.e. pump remaining Poissonian during the evolution [162]) the crystal couples two modes of the radiation field according to the effective Hamiltonian Ĥ κ = κ(â ˆb + âˆb), (D.1)

143 D.1. Conditional quantum state engineering 131 ρ x U x σ x = U x ρ x U x + NOPA a cc b E x V + Π x = V E x V Figure D.1: Scheme for quantum state engineering assisted by entanglement. At first, a twin-beam of the modes â and ˆb is produced by spontaneous downconversion in a nondegenerate parametric optical amplifier. Then, mode b is (possibly) subjected to the unitary transformation V and then revealed by a measurement apparatus described by the probability operator-valued measure (POVM) Êx. Overall, the quantum operation on the mode ˆb is described by the POVM ˆΠ x = ˆV Ê x ˆV. The conditional state of mode â is given by ˆϱ x, and this state may be further modified by a unitary transformation Ûx depending on the outcome of the measurement, whose value may be sent to the receiver location by classical communication. The overall conditional state is thus ˆσ x = Ûx ˆϱ x Û x. Figure taken from Ref. [112] where κ represents the effective nonlinear coupling, sometimes referred to as as the gain of the amplifier, and â and ˆb denote modes with wave vectors satisfying the phase-matching condition k a + k b = k p, k p being the wave vector of the pump. For vacuum input we have parametric downconversion, with the output given by the so-called twin-beam state of radiation λ = 1 λ 2 p=0 λ p pp pp = p a p b (D.2) where λ = tanh κ τ and τ represents an effective interaction time. The TWB λ is an entangled state in the bipartite Hilbert space H a H b, where H j, j = a, b, are the Fock space of the two modes respectively.

144 132 Appendix D. Continuous variable teleportation as conditional measurement TWBs are pure states and therefore their degree of entanglement can be quantified by the excess Von-Neumann entropy S = 1(S[ˆϱ 2 a] + S[ˆϱ b ] S[ˆϱ]) [96, 163, 164, 165]. The entropy of a two-mode state ˆϱ is defined as S[ˆϱ] = Tr {ˆϱ log ˆϱ} whereas the entropies of the two modes â and ˆb are given by S[ˆϱ j ] = Tr j {ˆϱ j log ˆϱ j }, j = a, b, with ˆϱ a = Tr b {ˆϱ} and ˆϱ b = Tr a {ˆϱ} denoting partial traces. The degree of entanglement of the state λ, in terms of the average number of photons of the TWB N = 2λ 2 /(1 λ 2 ), is given by S = log(1 + N 2 ) + N 2 log(1 + 2 N ). (D.3) Notice that for pure states S represents the unique measure of entanglement [166]. TWBs are the maximally entangled states for a given average number of photons, and the degree of entanglement is a monotonically increasing function of N. A measurement performed on one of the two modes reduces the other one according to the projection postulate. Each possible outcome x occurs with probability P x, and corresponds to a conditional state ˆϱ x on the other subsystem. We have [ P x = Tr ab λ λ Îa ˆΠ ] x = (1 λ 2 ) q λ 2q q ˆΠ x q = (1 λ 2 )Tr b [λ 2ˆb ˆb ˆΠx ] (D.4) ˆϱ x = 1 [ Tr b λ λ P Îa ˆΠ ] x x = 1 λ2 P x pq = λâ â ˆΠx λâ â Tr b [λ 2ˆb ˆb ˆΠx ] λ p+q p ˆΠ x q p q (D.5) where ˆΠ x is the probability operator-valued measure (POVM) describing the measurement, and Îa denotes the identity operator on H a. In the last equalities of both Eqs. (D.4) and (D.5) we have already performed the trace over the Hilbert space H a. Also notice that in the last expression for ˆϱ x in Eq. (D.5), ˆΠ x should be meant as an operator acting on H a. Our scheme is general enough to include the possibility of performing any unitary operation on the beam subjected to the measurement. In fact, if Ê x is the original POVM and ˆV the unitary, the overall measurement process is described by ˆΠ x = ˆV Ê x ˆV, which is again a POVM. In

145 D.2. Joint measurement of two-mode quadratures 133 a b S c Figure D.2: Scheme for continuous variable teleportation as conditional measurement. the following we always consider ˆV = Î, i.e. no transformation before the measurement. A further generalization consists in sending the result of the measurement (by classical communication) to the reduced state location and then performing a conditional unitary operation Ûx on the conditional state, eventually leading to the state ˆσ x = Ûx ˆϱ x Û x. This degree of freedom will be used in the next section, where we analyze CV quantum teleportation as a conditional measurement. D.2 Joint measurement of two-mode quadratures We now assume that mode ˆb is subjected to the measurement of the the real and the imaginary part of the complex operator Ẑ = ˆb + ĉ, where ĉ is an additional mode excited in a reference state Ŝ. The measurement of Re[Ẑ] and Im[Ẑ] corresponds to