PHYSICAL REVIEW A VOLUME 60, NUMBER 2 AUGUST 1999 Quantum optics of a Bose-Einstein conensate couple to a quantize light fiel M. G. Moore, O. Zobay, an P. Meystre Optical Sciences Center an Department of Physics, University of Arizona, Tucson, Arizona 85721 Receive 23 February 1999 We consier the interaction between a Bose-Einstein conensate an a single-moe quantize light fiel in the presence of a strong far-off-resonant pump laser. The ynamics is characterize by an exponential instability, hence the system acts as an atom-photon parametric amplifier. Triggere by a small injecte probe fiel, or simply by quantum noise, entangle atom-photon pairs are create which exhibit nonclassical correlations similar to those seen between photons in the optical parametric amplifier. In aition, the quantum statistics of the matter an light fiel epen strongly on the initial state which triggers the amplifier. Thus, by preparing ifferent initial states of the light fiel, one can generate matter waves in a variety of quantum states, emonstrating optical control over the quantum statistics of matter waves. S1050-2947 99 06708-6 PACS number s : 03.75. b, 42.50.Ct, 42.50.Dv I. INTRODUCTION In many ways, recently evelope Bose-Einstein conensates BEC s of trappe alkali-metal atomic vapors 1,2 are the atomic analog of the optical laser. In fact, with the aition of an output coupler, they are frequently referre to as atom lasers 3. Despite many interesting an important ifferences, the chief similarity behin the analogy is that both optical lasers an atomic BEC s involve large numbers of ientical bosons occupying a single quantum state. As a result, the physics of lasers an BEC s involves stimulate processes, which ue to Bose enhancement often completely ominate the spontaneous processes which play central roles in the nonegenerate regime. Just as the iscovery of the laser le to the evelopment of nonlinear optics, so too has the avent of BEC s le to remarkable experimental successes in the once theoretical fiel of nonlinear atom optics 4 8. Nonlinear optics typically involves the stuy of multiwave mixing, epitomize by phenomena such as parametric own-conversion an phase conjugation. Due to the presence of collisions, the evolution of the atomic fiel is also nonlinear, an multiwave mixing has been preicte 9 14 an observe in multicomponent conensates 15 as well as in scalar conensates 16. At the root of most optical phenomena is the ynamical interaction between optical an atomic fiels. Uner certain circumstances, one can formally eliminate the ynamics of the atomic fiel, resulting in effective interactions between light waves. Uner a ifferent set of conitions, one can eliminate the electromagnetic fiel ynamics, resulting in effective atom-atom interactions. These are the regimes of nonlinear optics an nonlinear atom optics, respectively. These regimes, therefore, represent limiting cases, where either the atomic or optical fiel is not ynamically inepenent, an instea follows the other fiel in some aiabatic manner which allows for its effective elimination. Outsie of these two regimes the atomic an optical fiels are ynamically inepenent, an neither fiel is reaily eliminate. In this paper we investigate the ynamics of couple quantum egenerate atomic an optical fiels in this intermeiate regime. In particular, we investigate a system which is analogous to the nonegenerate optical parametric amplifier OPA 17,18. However, whereas the OPA involves the creation of correlate photon pairs, this system involves the generation of correlate atom-photon pairs. The purpose of this paper is to evelop a etaile theory for the interaction of quantize atomic an optical fiels, with an emphasis on the manipulation an control of their quantum statistics an the generation of quantum correlations an entanglement between matter an light waves. The specific system we consier consists of a Bose- Einstein conensate riven by a strong far-off-resonant pump laser which interacts with a single moe of an optical ring cavity counterpropagating with respect to the pump. The strong pump laser is treate in the usual manner as a classical, uneplete light fiel, an furthermore it is assume to be etune far enough away from resonance that spontaneous emission may be safely neglecte. The cavity fiel, henceforth referre to as the probe, is assume to be weak relative to the pump, an is treate fully quantum mechanically as a ynamical variable. It is the ynamical interplay between this probe fiel an the atomic fiel which is the subject of interest. The pump serves as a sort of catalyst, inucing a strong atomic ipole moment, thus significantly enhancing the atom-probe interaction. Assuming that the probe fiel begins in or near the vacuum state, an the atomic fiel consists initially of a trappe BEC, the initial ynamics is ominate by a single process: the absorption of a pump photon by a conensate atom followe by the emission of a probe photon. We remark that in the far-off-resonant configuration, this is a twophoton virtual transition in which the excite atomic state population remains negligible. Due to atomic recoil, the absorption-emission process transfers the atom from the conensate groun state to a new state that is shifte in momentum space by the two-photon recoil. This new state constitutes a secon conensate component, which can be consiere as a momentum sie moe to the original 1050-2947/99/60 2 /1491 16 /$15.00 PRA 60 1491 1999 The American Physical Society
1492 M. G. MOORE, O. ZOBAY, AND P. MEYSTRE PRA 60 conensate. 1 As this sie moe is populate, it begins to interfere with the original conensate, resulting in fringes 21. These fringes are seen by the pump an probe fiels as a spatial ensity grating, which then enhances the photon scattering process. This interplay between interference fringes an scattering can act as a positive feeback mechanism, in which case the system is unstable, an is characterize by exponential growth. Any small signal, incluing quantum noise, will be sufficient to trigger the instability, resulting in the generation of exponentially growing sie moe an probe fiels. Of course this exponential growth is eventually reverse by high-intensity effects, so that the long-time ynamics is characterize by large-amplitue nonlinear oscillations. At the present time, we focus on the small-signal regime, characterize by exponentially growing fiels. In this regime, we emonstrate that the quantum state of the probe an sie moe fiels epens strongly on the initial conitions, so that, e.g., by injecting a small coherent light fiel into the probe, one can create an entirely ifferent quantum state than that generate from the amplification of quantum vacuum fluctuations. The ifferences are manifeste in both the quantum statistics of the iniviual fiel moes, as well as in nonclassical correlations an entanglement between them. The rest of this paper is organize as follows. Section II gives the backgroun an relates the current theory to previous works in a variety of fiels. Section III outlines the basic moel for a quantize many-boy atomic fiel interacting with a strong classical pump laser an a quantize optical cavity moe. In Sec. IV, couple-moe equations are evelope for the conensate an its momentum sie moes. These equations are then linearize in Sec. V, resulting in a three-moe moel which is exactly solvable. Section VI then iscusses the exponential instability, with an emphasis on the effects of collisions. In Sec. VII the quantum statistics of the atomic an electric fiels are investigate, an the extent to which they can be manipulate is etermine. In Sec. VIII atom-photon entanglement is iscusse, incluing an examination of two-moe squeezing between atomic an optical fiels. Finally, Sec. IX is a iscussion an conclusion, which inclues estimates of the important physical parameters, as well as potential experimental obstacles. II. BACKGROUND The system we escribe is in fact an extension into the ultracol regime of the theoretical work of Bonifacio an co-workers on the collective atomic recoil laser CARL 1 This new state may or may not be in the same internal groun state as the original conensate, epening on the polarizations of the pump an probe photons. If the magnetic sublevels are ifferent, then the transition woul be terme a Raman transition, if the sublevels are the same, then it may be thought of as Rayleigh scattering or two-photon Bragg scattering 19,20. As the states are alreay istinguishe by their center-of-mass momentum states, to istinguish them further by an aitional quantum number woul a nothing. Our moel eals specifically with the Bragg scheme; however, with only minimal moifications it coul be applie to the Raman scheme as well. 22,23. The original CARL theory treate the atomic centerof-mass motion classically, an approximation certainly vali for hot atoms, but not sufficient to escribe ultracol samples such as BEC s. Within this framework, the feeback mechanism which gives rise to the exponential instability in the CARL was outline using a slightly ifferent, but complementary physical picture, where the classical atomic centerof-mass motion in the optical potential of the couterpropagating pump an probe fiels is responsible for the grating formation. The theory was extene to the limit of zero temperature by assuming that all of the classical atoms begin from rest, leaing to the iscovery of the so-calle CARL cubic equation, which gives the exponential growth rate of the instability in terms of the relevant system parameters. Out of a esire to better unerstan the quantum statistics of the probe fiel, an attempt at a quantum T 0 theory was mae 24. However, this attempt explicitly assume that the wave functions of the iniviual atoms coul be localize in both momentum an position space to an extent which violates the Heisenberg uncertainty principle. Thus, rather than being a true quantum theory, it still treate the atoms as following classical trajectories, but now with small quantum fluctuations inclue. Both the original classical CARL moel, as well as the later quantum moel, fall within the ray-optics approximation for the atomic fiel. Clearly then, one woul expect such moels to break own as soon as the atomic ebroglie wavelength becomes comparable to the perio of the optical potential forme by the pump an probe fiels. As the wavelength of the optical potential is twice the optical wavelength, this breakown shoul occur near the atomic recoil temperature, which for typical alkali atoms is on the orer of K. As subrecoil temperature atomic vapors are achieve routinely through a variety of cooling techniques, a theory which properly treats the quantum motion of the atoms is require if one esires to investigate the behavior of the CARL in this regime. With the ultimate intent of extening the CARL theory into the BEC regime, so that the unique coherence properties of conensates might be further unerstoo an exploite by the interaction with ynamical light fiels, a quantum moel of the atomic motion was formulate 25, where it was confirme that the ray-optics versions i inee break own for temperatures of the orer of the recoil temperature or below. In fact, at T 0, a secon threshol for the existence of the exponential instability was iscovere, occurring when the bunching process is overcome by matter-wave iffraction. For T T R, however, it was shown that the previous theories make inistinguishable preictions from the quantum theory. We remark that while in this work the atomic center of mass motion was treate quantum mechanically, the light fiels were still treate classically; hence preictions concerning the quantum statistics of either the atomic or optical fiels coul not be mae. A full quantum moel of both the atomic an optical fiels was recently outline in Ref. 26, where the subjects of manipulating quantum statistics an atom-photon entanglement were first aresse. The present paper is a etaile elaboration an extension of that work, incluing to our knowlege, significant new physics. For example, utilizing the familiar s-wave scattering approach of BEC theory,
PRA 60 QUANTUM OPTICS OF A BOSE-EINSTEIN... 1493 the effects of atom-atom collisions are incorporate into the CARL theory. Also, in an extension of the OPA analogy, the existence of two-moe squeezing is shown to occur between a conensate sie moe an the probe optical fiel. The current approach also iffers from earlier work in that the familiar spontaneous symmetry breaking technique is no longer applie to the conensate. Instea it is assume that a conensate well below the critical temperature is better escribe by a number state than a coherent state, as recent work appears to emonstrate 27,28. The fully quantum moel is similar in many ways to a system first stuie by Zeng, Liu, an Zhang 29, an later extene by Kuang 30, in which the principle of manipulating the quantum statistics of a conensate by its interaction with a quantize light fiel was first propose. These papers, however, o not recognize the existence of unstable exponential solutions nor the fact that the system can be triggere from quantum noise. We note that the unstable exponential solutions, an the possibility to initiate them from quantum vacuum fluctuations, are both crucial components of this present work. Finally, we mention the connection to recent work on matter-wave amplification by Law an Bigelow 31, which also explores the interaction between conensates an quantize light fiels. In that work, however, the light fiel is assume to be heavily ampe, thus allowing for its ynamical elimination. As a result, only the properties of the atomic fiel are stuie in etail. CARL theory, incluing the present version, is also closely relate to the theory of recoil-inuce resonances RIR s 32, in which the effects of atomic recoil on the pump-probe spectroscopy of an atomic vapor is investigate. This theory treats the atomic center-of-mass motion quantum mechanically. The probe fiel, however, is not typically treate as a ynamical variable. Hence, it oes not inclue the effects of probe feeback, which are necessary for exponential behavior. A etaile comparison of the RIR an CARL theories is given in Ref. 33. III. BASIC MODEL In this section we erive a fully quantize moel of a gas of bosonic two-level atoms which interact with a strong, classical, uneplete pump laser an a weak, quantize optical ring cavity moe, both of which are assume to be tune far away from atomic resonances. As a result, singlephoton transitions between atomic internal groun an excite states are highly nonresonant, an the excite-state population remains negligible. In this case, one can safely neglect the effects of spontaneous emission as well as the two-boy ipole-ipole interaction. We must still, however, allow for two-photon virtual transitions in which the atomic internal state remains unchange, but ue to recoil may result in a change in the atom s centerof-mass motion. For example, an atom which absorbs a pump photon an emits a probe photon experiences a recoil kick equal to the ifference of the momenta of the two photons which for nearly counterpropagating pump an probe beams is of the orer of two optical momenta. These transitions, therefore, couple ifferent states of the atomic centerof-mass motion. Due to the quaratic ispersion relation of the atoms, these transitions will in general be nonresonant. For very col atoms, the resultant etunings are typically on the orer of the recoil frequency, i.e., much smaller than the natural linewith of the atomic transition a, whereas the one-photon transitions which we are neglecting have a etuning many orers of magnitue larger than a. Our theory begins with the secon-quantize Hamiltonian Ĥ Ĥ atom Ĥ probe Ĥ atom-probe Ĥ atom-pump Ĥ atom-atom, 1 where Ĥ atom an Ĥ probe give the free evolution of the atomic fiel an the probe moe respectively, Ĥ atom-probe an Ĥ atom-pump escribe the ipole coupling between the atomic fiel an the probe moe an pump laser, respectively, an Ĥ atom-atom contains the two-boy s-wave scattering collisions between groun-state atoms. The free atomic Hamiltonian is given by Ĥ atom r 3 ˆ r g 2 2m 2 V g r ˆ g r ˆ r e 2 ˆ e r, 2 2m 2 a V e r where m is the atomic mass, a is the atomic resonance frequency, ˆ e(r) an ˆ g(r) are the atomic fiel operators for excite- an groun-state atoms respectively, an V g (r) an V e (r) are their respective trap potentials. The atomic fiel operators obey the usual bosonic equal-time commutation relations ˆ j(r), ˆ j (r ) j, j 3 (r r ), an ˆ j(r), ˆ j (r ) ˆ j (r), ˆ j (r ) 0, where j, j e,g. The free evolution of the probe moe is governe by the Hamiltonian Ĥ probe ckâ Â, where c is the spee of light, k is the magnitue of the probe wave number k, an  an  are the probe photon annihilation an creation operators, satisfying the boson commutation relation Â, 1. The probe wave number k must satisfy the perioic bounary conition of the ring cavity, k 2 l/l, where the integer l is the longituinal moe inex, an L is the length of the cavity. The atomic an probe fiels interact in the ipole approximation via the Hamiltonian Ĥ atom-probe i gâ 3 r ˆ e r e ik r ˆ g r H.c., where g ck/(2 0 LS) 1/2 is the atom-probe coupling constant. Here is the magnitue of the atomic ipole moment, an S is the cross-sectional area of the probe moe in the vicinity of the atomic sample where it is assume to be approximately constant across the length of the atomic sample. In aition, the atoms are riven by a strong pump laser, which is treate classically an assume to remain uneplete. The atom-pump interaction Hamiltonian is given in the ipole approximation by 3 4
1494 M. G. MOORE, O. ZOBAY, AND P. MEYSTRE PRA 60 Ĥ atom-pump 0 2 e i 0 t 3 r ˆ e r e ik 0 r ˆ g r H.c., 5 where 0 is the Rabi frequency of the pump laser, relate to the pump intensity I 0 by 0 2 2 2 I 0 / 2 0 c, 0 is the pump frequency, an k 0 0 /c is the pump wave number. The approximation inicates that we are neglecting the inex of refraction insie the atomic gas, as we assume a very large etuning 0 a between the pump frequency an the atomic resonance frequency. Finally, the collision Hamiltonian is taken to be Ĥ atom-atom 2 2 3 r ˆ g r ˆ g r ˆ m g r ˆ g r, 6 where is the atomic s-wave scattering length. This correspons to the usual s-wave scattering approximation, an leas in the Hartree approximation to the stanar Gross- Pitaevskii equation for the groun-state wave function in the absence of the riving optical fiels. We limit ourselves to the case where the pump laser is etune far enough away from the atomic resonance that the excite-state population remains negligible, a conition which requires that a. In this regime the atomic polarization aiabatically follows the groun-state population, allowing the formal elimination of the excite-state atomic fiel operator. We procee by introucing the operators ˆ e (r) ˆ e(r)e i 0 t an  Âe i 0 t, which are slowly varying relative to the optical riving frequency. The new excite-state atomic fiel operator obeys then the Heisenberg equation of motion t ˆ e r i ˆ e r i 0 2 eik 0 r gâ e ik r ˆ g r, where we have roppe the kinetic energy an trap potential terms uner the assumption that the lifetime of the excite atom, which is of the orer 1/, is so small that atomic center-of-mass motion may be safely neglecte uring this perio. For the same reason, we are justifie in neglecting collisions between excite atoms, or between excite- an groun-state atoms in the collision Hamiltonian 6. We now aiabatically solve for ˆ e (r) by formally integrating Eq. 7 uner the assumption that ˆ g(r) varies on a time scale which is much longer than 1/. This yiels ˆ e r,t 1 0 2 eik 0 r igâ t e ik r ˆ g r,t 1 0 2 eik 0 r igâ 0 e ik r ˆ g r,0 e i t 7 The thir term on the right-han sie of Eq. 8 can be neglecte for most consierations if we assume that there are no excite atoms at t 0, so that this term acting on the initial state gives zero. The secon term may also be neglecte, as it is rapily oscillating at frequency, an thus its effect on the groun-state fiel operator is negligible when compare to that of the first term, which is nonrotating. 2 Dropping the unimportant terms, an then substituting Eq. 8 into the equation of motion for ˆ g(r), we arrive at the effective Heisenberg equation of motion for the groun-state fiel operator, t g r i ˆ 2m 2 V g r 4 m ˆ g r ˆ g r g 0 âe ik r â e ik r 2 0 2 g2 4 â â ˆ g r, where K k k 0 is the recoil momentum kick the atom acquires from the two-photon transition, an we have introuce the new slowly varying probe fiel operator â i( 0 * / 0 )Â, which still obeys the boson commutation relation â,â 1. Here the secon to last term is simply the optical potential forme from the counterpropagating pump an probe light fiels, an the last term gives the spatially inepenent light shift potential, which can be thought of as cross-phase moulation between the atomic an optical fiels. To complete our moel, in aition to Eq. 9, we also require the equation of motion for the slowly varying probe fiel operator. By again substituting Eq. 8, we fin that it obeys t â i â i g 0 2 3 r ˆ g r e ik r ˆ g r, 9 10 where 0, is the etuning between the pump an probe fiels. The probe frequency is given by ck, again assuming that the inex of refraction insie the conensate is negligible. IV. COUPLED-MODE EQUATIONS We assume that the atomic fiel is initially in a Bose- Einstein conensate with mean number of conense atoms N. Furthermore, we assume that N is very large, an that the conensate temperature is small compare to the critical temperature. These assumptions allow us to neglect the nonconense fraction of the atomic fiel. Thus our moel oes not inclue any effects of conensate number fluctuations. We now introuce the atomic fiel operator which annihilates an atom in the conensate groun state ĉ 0 3 r 0 * r ˆ g r, 11 ˆ e r,0 e i t. 8 2 We note that in much of the literature the secon an thir terms are simply ignore. We choose to keep them temporarily to emonstrate that the commutation relation for ˆ e(r) is preserve to orer 1/ by the proceure of aiabatic elimination.
PRA 60 QUANTUM OPTICS OF A BOSE-EINSTEIN... 1495 where 0 (r) r 0 satisfies the time-inepenent Gross- Pitaevskii equation 2m 2 V g r 4 m N 0 r 2 0 r 0, 12 being the chemical potential. By ifferentiating Eq. 11 with respect to time, an inserting Eqs. 9 an 12 we fin that the equation of motion for ĉ 0 is t 0 i ĉ 0 2 g2 4 â â ĉ 0 i 4 3 r m 0 * r N 0 r 2 ˆ g r ˆ g r ˆ g r i g 0 2 3 r 0 * r âe ik r â e ik r ˆ g r. 13 From this equation we see that the effect of the optical fiels is to couple the conensate moe to two sie moes, whose wave functions are given by r 0 r e ik r. 14 In principle, the collision term in Eq. 13 also couples the conensate moe to various neighboring moes. However, to be consistent with the assumption of a pure conensate (T 0), we assume that collisions alone o not populate any new atomic states. Defining the fiel operators for the first-orer conensate sie moes as For most conensate sizes an trap configurations, however, these integrals are many orers of magnitue smaller than unity. As a result, for typical conensates, the orthogonality approximation j j jj 19 yiels accurate results. The range of valiity of this approximation is iscusse in Appenix A, where we briefly examine how the theory shoul be moifie to properly take this nonorthogonality into account. In the following, however, we assume the valiity of Eq. 19, so that the states 0, an can be consiere as well efine an istinct moes of the atomic fiel. We now erive the Heisenberg equations for the momentum sie moe fiel operators, foun by ifferentiating Eq. 15 with respect to time an again inserting Eq. 9, yieling t, ĉ K2 2m 0 2 4 g2 â â ĉ, i 4 m 3 r, r N 0 r 2 ˆ g r ˆ g 0 g r ˆ g r i â ĉ 2 0, 2 âĉ 2,0 i Kk c m bˆ,, 20 where we have introuce four new fiel operators ĉ 2 an bˆ. The operators ĉ 2, which have the efinitions ĉ 3 r r ˆ g r 15 ĉ 2 3 r 2 r ˆ g r, 21 allows us to reexpress the equation of motion for the conensate moe fiel operator as t 0 i ĉ 0 2 4 g2 â â N 0 r 2 ˆ g r ˆ g r ˆ g r i g 0 âĉ 2 â ĉ, ĉ 0 i 4 3 r m 0 * r 16 where the operators ĉ j obey the bosonic commutation relations ĉ j,ĉ j j j, j, j,0,, 17 all other commutators being equal to zero. We note that the three states, 0, an are not mutually orthogonal, as their overlap integrals are given by 3 r 0 r 2 e i2k r, 0 3 r 0 r 2 e ik r. 18 are the annihilation operators for the secon-orer sie moes r 2 0 r e i2k r. 22 These moes will be optically couple to thir-orer sie moes, an so on so that a full theory of the nonlinear response of the system shoul inclue the entire manifol of sie moes. In this paper, however, we focus on the linear regime, where only the first-orer sie moes contribute significantly. The operators bˆ have the efinitions where bˆ 3 r r ˆ g r, r Kk c 1 e ik r ik 0 r. 23 24 Here k c is the momentum with of the conensate state along K, an is roughly given by k c 1/W c, where W c is the size of the conensate along K. The factor (Kk c ) 1 is simply a normalization coefficient. To unerstan the physical meaning of the bˆ term in Eq. 20, consier what happens to a single atom after it is trans-
1496 M. G. MOORE, O. ZOBAY, AND P. MEYSTRE PRA 60 ferre into the state (r) 0 (r)exp( ik r) at time t 0. Uner free evolution the wave packet of the atom, which initially has the shape of the conensate groun state, will move with group velocity K/m an sprea at the velocity k c /m. This evolution is escribe by the propagation equation r,t exp it /2m 2 r,0, which for short enough times becomes r,t 1 it K2 2m r t m K 0 r e ik r it 2m 2 0 r e ik r. 25 26 The first term on the right-han sie of Eq. 26 gives a phase shift ue to the kinetic energy of the atom, the secon term contributes a translational shift, an the thir an final term escribes spreaing. If we inclue the effects of the trap potential an collisions, this last term vanishes as all spreaing effects are balance by the trap potential for the groun state 0 (r). From Eqs. 26 an 24 we see that the state of the atom at time t can then be viewe as a coherent superposition of the state an the state. Thus the coupling to bˆ in Eq. 20 correspons physically to translational motion of the sie moe wave packet at the recoil velocity v r K/m. Since the probability at time t that the atom is still in the sie-moe state is simply the overlap between (r,t) an (r,0), it is clear that for times t W c /v r this probability will be essentially unity, an the coupling to bˆ can be ignore. V. LINEARIZED THREE-MODE MODEL From Eq. 20, we see that the first-orer sie moes are optically couple to both the conensate moe an to secon-orer sie moes. For times short enough that the conensate is not significantly eplete, the coupling back into the conensate is subject to Bose enhancement ue to the presence of N ientical bosons in this moe. The coupling to the secon-orer sie moe, in contrast, is not enhance. Hence for these time scales, the higher-orer sie moes are not expecte to play a significant role. In aition, we consier only times t W c /v r, so that the translational coupling can be neglecte. These arguments suggest eveloping an approach where the three atomic fiel operators ĉ 0, ĉ, an ĉ play a preominant role. Therefore, we expan the atomic fiel operator as ˆ g r r 0 ĉ 0 r ĉ r ĉ ˆ r, 27 where the fiel operator ˆ (r) acts only on the orthogonal complement to the subspace spanne by the state vectors 0,, an. As a result, ˆ (r) commutes with the creation operators for the three central moes. In the next step, we use Eq. 27 to expan the atomic polarization an collision terms in Eqs. 10, 16, an 20, with the eventual goal of eriving a close set of operator equations which fully escribes the system ynamics. At present, we are consiering four ominant moes, the conensate an first-orer sie moes, as well as the optical probe moe. In the linear regime, however, we will see that the conensate moe can be ynamically eliminate, resulting in an effective three-moe moel. In expaning the polarization an collision terms by means of Eq. 27, there are two principal consierations in etermining which are the ominant terms. The first is Bose enhancement, which, in the regime of negligible conensate epletion, strongly selects transitions involving the conensate moe. In orer to estimate this effect, we assign a weight of N for each occurrence of the operators ĉ 0 an ĉ 0 in a given term. The secon consieration is momentum conservation, which comes from the spatial integration in the polarization an collision terms. Integrals over slowly varying functions such as 0 (r) 2,or 0 (r) 4 are momentum selecte, an ominate over integrals of rapily oscillating functions such as 0 (r) 2 exp( ik r). With this approach we fin that the equation of motion for the probe fiel operator 10 becomes t â i â i g 0 ĉ 2 ĉ 0 ĉ 0 ĉ. 28 Thus we see that the probe annihilation operator is couple to the bilinear atomic fiel operators ĉ ĉ 0 an ĉ 0 ĉ. These operators correspon physically to interference fringes, i.e., a perioic moulation of the atomic ensity, which appear because the atoms are in a coherent superposition of the sie moe an conensate states. Gain in the probe can thus be interprete as Bragg scattering of the pump ue to the presence of interference fringes. By inserting Eq. 27 into Eq. 16, we further fin that the equation of motion for ĉ 0 ĉ 0 is given to leaing orer in the collision an optical terms by where t ĉ 0 ĉ 0 i 8 F 0 ĉ m ĉ 0 ĉ ĉ 0 i g 0 â ĉ 2 ĉ 0 ĉ 0 ĉ H.c., F 0 3 r 0 r 4. 29 30 Similarly we fin that the operators ĉ ĉ 0 an ĉ 0 ĉ obey the equations t ĉ ĉ 0 i K2 2m ĉ ĉ 0 an i 4 F 0 m ĉ ĉ 0 ĉ 0 ĉ ĉ 0 ĉ 0 i g 0 2 âĉ 2 0 ĉ 0 31
PRA 60 QUANTUM OPTICS OF A BOSE-EINSTEIN... 1497 t ĉ 0 ĉ i K2 2m ĉ 0 ĉ i 4 F 0 ĉ m ĉ 0 ĉ 0 ĉ ĉ 0 ĉ 0 i g 0 2 âĉ 2 0 ĉ 0. 32 an we have introuce the imensionless time r t, r K 2 /2m being the atomic recoil frequency, as well as the imensionless control parameters g 0 N 2 r, 41 / r, 42 We assume that all N atoms are initially in the conensate moe, so that t 0 1 N! ĉ 0 N 0, 33 0 being the vacuum state. We procee by linearizing the atomic fiel operators aroun their initial expectation values, which can be etermine from Eq. 33, together with the approximate commutation relations given by Eq. 19. This yiels an ĉ 0 ĉ 0 N 1 ˆ 0, ĉ ĉ 0 N ˆ, ĉ 0 ĉ N ˆ, 34 35 36 where ˆ 0, ˆ, an ˆ are therefore infinitesimal operators. In aition, we introuce a rescale probe fiel operator ˆ a â N, 37 which, provie that the mean number of photons in the probe moe is small compare to N, is also infinitesimal. This constraint is consistent with the assumption of negligible conensate epletion. Inserting these efinitions into the equation of motion 29 for ĉ 0 ĉ 0, an keeping only terms linear in the infinitesimal operators, gives t ˆ 0 0, 38 which has the trivial solution ˆ 0 0. As a result, this operator can be roppe from the linearize equations for ˆ an ˆ. This leas to a set of three couple infinitesimal operators whose linearize equations of motion can be expresse as ˆ im ˆ, where ˆ ( ˆ a, ˆ, ˆ ) T, the matrix M is given by M 1 1, 39 40 an 4 NF 0 m r. 43 Here is a imensionless atom-probe coupling constant, is the pump-probe etuning in units of the atomic recoil frequency, an gives the strength of collisions between the sie moes. In Appenix B we give the effective Hamiltonian from which Eq. 39 can be erive. The solution to Eq. 39 is then given by ˆ e im ˆ 0. 44 From this we see that the time epenence of the infinitesimal operators is etermine by the eigenvalue spectrum of the matrix M. For certain values of the control parameters,, an, one of the eigenvalues contains a negative imaginary part. When this occurs, the infinitesimal operators unergo exponential growth. This exponential instability is the focus of the next section, where we iscuss its properties in etail. VI. EXPONENTIAL INSTABILITY The eigenvalues of M are etermine by the characteristic equation 3 2 1 2 1 2 2 2 0, 45 which has either three real solutions, or one real an a pair of complex conjugate solutions. In the first case, the system is stable an exhibits only small oscillations aroun its initial state. In the secon case, the system is unstable an grows exponentially, even from noise. From Eq. 45 one fins that exponential instability occurs when 2 3 6 2 3/2 3 9 1 2 /27. 46 In Fig. 1 we plot the region of instability as a function of an 2. The shae region of Fig. 1 correspons to the instability region in the absence of collisions ( 0). As collisions are ae, the bounaries shift, illustrate by the ashe an otte curves, which show the bounaries for the cases 0.3 an 1.0. From Fig. 1, we see that for positive the bounary asymptotically reuces to 2 2 3 /27 1 2 /6, 47 i.e., the threshol value of 2 increases with the thir power of the etuning an is only weakly influence by the pres-
1498 M. G. MOORE, O. ZOBAY, AND P. MEYSTRE PRA 60 ominate, an the upper limit comes from the requirement that the sie moe populations remain a small fraction of the total atom number. This conition therefore formally efines the exponential growth regime. We note that for very short times 1, the behavior is of course not exponential. In this transient regime the instantaneous growth rate is not well approximate by. The rate of exponential growth has the explicit form ) 2 r q3 r 2 1/3 r q 3 r 2 1/3, 51 FIG. 1. Exponential instability region in the - 2 plane. The shae area gives the unstable omain in the absence of collisions ( 0). The ashe curves show how the bounaries change as collisions are inclue. They correspon to the cases 0.3 ashe line an 1.0 otte line. ence of collisions. On the other han, for negative the asymptotic behavior is given by where an r 1 3 1 2 2 3 27 q 1 9 3 6 2. 52 53 2 1 2 /2, 48 which only grows linearly with an is strongly affecte by interatomic collisions, which have the effect of reucing the unstable region. In earlier work 25, it was shown that this lower threshol occurs in the absence of collisions when atomic iffraction overcomes the bunching process. For positive scattering lengths, we note that formation of a ensity grating increases the mean-fiel energy. Collisions, therefore, shoul join iffraction in opposing the bunching process, resulting in a higher threshol for the instability. For negative scattering lengths, on the other han, bunching reuces the mean-fiel energy, hence collisions shoul enhance the bunching process an oppose iffraction, thus lowering the threshol. Finally, we note that the instability region for 2 1 is centere aroun 1 2 1. This is consistent with energy conservation in the scattering of a pump photon into the probe by an atom initially at rest. Once we have foun the eigenvectors an eigenvalues of M, we can reexpress solution 44 in the form ˆ Ue i U 1 ˆ 0, 49 where U is the matrix of eigenvectors of M, such that U ij is the ith component of the jth eigenvector, an is the iagonal matrix of eigenvalues, such that the ith iagonal element of is the ith eigenvalue of M. In the unstable regime, we have the eigenvalues 1, 2 i, an 3 i, where 1 an are real, an is real an positive. Thus 1 correspons to an oscillating solution, 2 an exponentially ecaying solution, an 3 correspons to an exponentially growing solution. Eventually, this exponentially growing solution will ominate, at which time we can neglect the other two terms, yieling the approximate solution ˆ j k jk ˆ k 0 e i, 50 where jk U j3 U 1 3k. The range of valiity for this approximation is roughly 1 ln( N), where the lower limit is set by the requirement that the exponentially growing terms As this equation is complicate an oes not provie much insight, in Fig. 2 we plot as a function of an 2 for three ifferent values of. Figure 2 a shows the limit of negligible collisions 0, an Figs. 2 b an 2 c show the cases 0.3 an 1, respectively. From these figures, we observe that the most significant effect of collisions is to shift the lower threshol. The values of in the vicinity of the maximum for fixe, on the other han, show less pronounce variations. In cases where 2 3, we have r 2 an q 3 r 2 2. In this case Eq. 51 reuces simply to ) /2 2/3. 54 Among other things, this shows that the gain scales as the number of atoms in the conensate to the 1 3 power. In Fig. 3, the growth rate is plotte versus 2 with 1, an roughly maximizes for fixe. The three ashe curves correspon to ifferent values of the collision parameter, while the soli line gives the approximate result 54. This shows that the approximation is a relatively accurate estimate of the maximum gain for all values of 2. VII. QUANTUM STATISTICS In this section we use solution 49 to compute some of the quantum-statistical properties of the system. This, however, first requires a more etaile iscussion of the physical meaning of the infinitesimal operators. The first, ˆ a â/ N, is clearly just a rescaling of the photon annihilation operator. From it one can compute all properties of the electric fiel an/or the photon statistics of the probe moe. The atomic sie moe operators ˆ ĉ ĉ 0 /N an ˆ ĉ 0 ĉ /N, however, are not simply rescalings of atom annihilation operators. Rather, they are irectly relate to the atomic ensity ˆ (r) ˆ g (r) ˆ g(r). To illustrate this point, we expan ˆ (r) accoring to Eq. 27 an linearize, yieling ˆ r N 0 r 2 1 2 e ik r ˆ ˆ H.c.. 55
PRA 60 QUANTUM OPTICS OF A BOSE-EINSTEIN... 1499 FIG. 3. The ashe lines show the growth rate as a function of 2 for the case 1, with the values of specifie in the figure. The soli line is the approximate expression given by Eq. 54. ĉ ĉ N ˆ ˆ. 57 Similarly the number operator for the sie moe is given by ĉ ĉ N ˆ ˆ. 58 FIG. 2. Exponential growth rate as a function of the scale pump-probe etuning an coupling parameter, for various values of the collision parameter. a shows the case 0, while b an c show the cases 0.3 an 1.0, respectively. From this expression we see that the sie moe operators ˆ inee escribe the appearance of a ensity moulation with wavelength 2 /K. In aition to the atomic ensity, one can also express the number operators for the sie moes in terms of ˆ. For example, we have, after linearization, N ˆ ˆ ĉ ĉ 0 ĉ 0 ĉ N ĉ N 1 ĉ. 56 N Hence with N 1 N the number operator for the sie moe can be expresse as From these number operators, one can therefore compute the number statistics of the sie moes in the linear regime. From the analytical solution 49 it is straightforwar to compute the properties of the atomic an optical fiels for an arbitrary initial conition. We focus on two conitions which appear reaily accessible experimentally. In the first one, the probe fiel an the atomic sie moes all begin in the vacuum state. In this case the exponential growth is triggere by vacuum fluctuations in both the probe fiel an the atomic ensity. A secon possible triggering mechanism involves injecting of a weak laser fiel into the probe moe. Both initial situations are investigate by assuming that the probe moe is initially in the coherent state, such that â, the vacuum case corresponing to 0. In aition, we assume throughout that the conensate sie moes begin in the vacuum state. Hence the initial state of the threemoe system can be expresse as,0,0, where the first inex refers to the probe moe, an the secon an thir inices give the states of the momentum sie moes. A. Electric fiel an atomic ensity The expectation value of the operator ˆ a is sufficient to compute the mean electric fiel, an likewise, the mean values of ˆ an ˆ are sufficient to compute the mean atomic ensity. We now give analytic solutions for these physical quantities an their quantum-mechanical uncertainties in the exponential growth regime, where all but the leaing exponential terms can be safely neglecte. The electric-fiel operator for the probe fiel is given by Ê r r E r, N ˆ a H.c., where r is the polarization unit vector, an 59
1500 M. G. MOORE, O. ZOBAY, AND P. MEYSTRE PRA 60 E r, 0 2 0 0 E r e i 0 / r 60 contains all constants of proportionality, the normalize spatial wave function of the probe moe E (r), an the oscillation at the pump frequency 0. The mean electric fiel is obtaine by inserting Eq. 50, an taking the quantum mechanical expectation value with respect to the initial state,0,0, which yiels r Ê r E r, aa e i c.c. 61 This correspons to an oscillating mean fiel with amplitue E 0 r 2 E r, aa e. 62 From this expression, we see that there is a nonzero mean fiel provie only that 0. We also see from Eq. 61 that the mean-fiel amplitue grows exponentially at the rate, an that its frequency is shifte by away from the pump frequency. Its phase, on the other han, has a somewhat complicate epenence on the system parameters. An analytic expression for this phase can be compute irectly from Eq. 61, but as we raw no specific conclusions from it, we o not give the explicit expression here. The variance in the electric fiel can also be compute in a straightforwar manner from Eq. 50, yieling E r & E r, a e. 63 This shows that the fluctuations also grow exponentially in time, irrespective of whether the mean fiel vanishes or not, an are in fact inepenent of. Hence these fluctuations can be attribute solely to the amplification of quantum noise, i.e., vacuum fluctuations in the probe electric fiel as well as atomic ensity fluctuations. While the mean fiel an the fluctuations both grow exponentially in time, the relative uncertainty, on the other han, is constant in time, given by E r E 0 r f,,. 64 & Here we have introuce the fluctuation function f,, U 3 1 U 1 3a, 65 which can be compute irectly from the eigenvectors of the matrix M, an therefore epens only on the parameters,, an. Figure 4 plots f (,, ) above the - 2 plane for various values of the collision parameter. Figure 4 a shows the limit of negligible collisions ( 0), where we see that f (,,0) is nearly flat in the vicinity of maximum gain ( 1), where it has a value somewhere between 1 an 2. It steaily increases from this value as the pump-probe etuning moves in the negative irection. Figures 4 b an 4 c show the cases 0.3 an 1.0, respectively. From these we see that the effect of increasing the collision parameter is to flatten f (,, ) as a function of an. In a similar manner, we next calculate the mean value an variance of the atomic ensity. By inserting Eq. 50 into Eq. FIG. 4. Comparison of the fluctuation function f (,, ) asa function of an 2 for various values of the collision parameter. a gives the limit of negligible collisions ( 0), where b an c show the cases 0.3 an 1.0, respectively. 55, an taking the expectation value with respect to the initial state, we fin the expectation value of the atomic ensity to be ˆ r N 0 r 2 0 r, cos K r, 66 where the amplitue 0 (r) of the ensity moulation is given by 0 r, N 0 r 2 a a e. 67 Thus we see that the mean atomic ensity is the sum of two contributions, the initial ensity of the conensate plus a ensity moulation which grows exponentially in time, provie of course that 0. Together with Eq. 61, this
PRA 60 QUANTUM OPTICS OF A BOSE-EINSTEIN... 1501 shows that the phase symmetry of the system is broken by the phase of the injecte fiel. Only in the case 0 oes the symmetry remain unbroken. We note that for the atomic sie moes, this is not symmetry breaking in the commonly use sense of nonzero mean fiels. Rather, it is the mean atomic ensity moulation which acquires a nonzero phase. Note also that the mean ensity moulation is not stationary, as its phase is given by, where, epens only on the system parameters,, an, an epens on both the system parameters an the phase of the injecte probe fiel arg. The variance in the atomic ensity can also be reaily compute in the exponential growth approximation, yieling r 2 N 0 r 2 e, 68 which shows that the ensity fluctuations grow exponentially in time, even in the case 0. The ratio between the ensity variance an the moulation amplitue is constant in time, an is given by the same expression as that for the probe, i.e., r 0 r f,,. 69 & From Eqs. 64 an 69, we see that in the case f (,, ), both the mean electric fiel an the mean atomic ensity moulation are quantum mechanically well efine, meaning that the quantum noise is small compare to their mean values. In this regime, both quantities coul be aequately treate as classical c-number fiels. Outsie of this regime, however, the quantum fluctuations play a significant role, an a classical escription no longer suffices. The main implication of these results is that by varying the system control parameters, an in particular the injecte fiel intensity an phase, one can vary the mean electric fiel an atomic ensity moulation continuously between two limits. For f (,, ) the fiels are ominate by quantum fluctuations, an we can expect to fin important nonclassical effects. In the limit of a strong injecte fiel, however, the fluctuations are not significant, an the atomic an optical fiels behave classically. B. Intensities We now turn to the number statistics of the three fiel moes, concentrating on the mean atom an photon numbers an their variances. It is convenient to reexpress the moe number operators, given by Eqs. 56 58, as Nˆ j N ˆ j ˆ j j, 70 where the inex j is again the moe label a,, or, an the function accounts for the fact that we have normally orere the infinitesimal fiel operators. It is then straightforwar to erive the full time-epenent solution for the mean occupation numbers N j Nˆ j as where N j a ja 2 2 a j 2 j, 71 FIG. 5. Logarithmic plot of the probe intensity N a as a function of time. The thick curves show the exact solution, given by Eq. 71, an the corresponing thin lines give the approximate solution of Eq. 75. The parameters chosen are 1, 2 1, an 0. Each pair of curves correspons to a ifferent value of the initial probe intensity 2, as specifie in the figure. 3 a ij k 1 U ik U 1 kj e i i. 72 The first term in Eq. 71 can be interprete as the stimulate contribution to the intensity, N j st a ja 2 2, 73 whereas the secon term gives the spontaneous contribution, N j sp a j 2 j, 74 present even in the case 0. In the exponential growth regime, Eq. 71 reuces to N j ja 2 2 j 2 e 2, 75 which shows that the moe occupation grows exponentially, even in the spontaneous case, where it was seen that the mean electric fiel an mean ensity moulation both vanish. The valiity of the exponential approximation 75 is emonstrate in Fig. 5, where we plot the logarithm of the probe intensity as a function of time. The parameters chosen for the plot are 1, 2 1, an 0. The thick lines give the full solution 71 for three ifferent values of the initial probe intensity 2 0, 1, an 10. The corresponing thin lines are the approximate solutions given by Eq. 75, which show goo agreement for 1. Turning now to quantum fluctuations in the occupation numbers, we fin that the relative uncertainties are given by N j N j 1 a ja 4 N j 2 1 N j. 76 From Eq. 71 it is clear that the secon term uner the raican in the above expression is 1, which means that as a function of time, the relative uncertainty is always between 1/N, characteristic of the fluctuations foun in a coherent state, an 1 1/N, which is the signature of thermal number fluctuations. While for very short times the relative uncertainty may fluctuate between the thermal an coherent limits, once the
1502 M. G. MOORE, O. ZOBAY, AND P. MEYSTRE PRA 60 7 the steay-state value for large is plotte as a function of the initial probe intensity 2. Thus we see that it is possible to vary the output continuously over the whole range between thermal an coherent limits, simply by varying the initial probe intensity. The parameters chosen for the figure are 1, 2 1, an 0. VIII. ATOM-PHOTON ENTANGLEMENT FIG. 6. The sie-moe number variance N /N is plotte vs time thick soli line. Also shown are the approximate solution given by Eq. 77 thick ashe line, as well as the variances for a thermal state thin soli line an a coherent state thin ashe line with the same mean value N. The parameters chosen are 1, 2 1, 0, an 1. exponentially growing terms ominate, the relative uncertainty eventually reaches a steay-state value given by N j N j 4 1 2 f 2,, 2. 77 Thus we see that when 2 f 2 (,, ), the relative uncertainty tens towars zero, while in the opposite case, it tens towar 1. These limits can be labele as the stimulate an spontaneous limits, respectively. In the intermeiate regime, the fluctuations can be varie continuously between the thermal an coherent limits, e.g., by varying the injecte laser intensity, thus achieving optical control over the quantum statistics of matter waves. The behavior of the particle number variances is illustrate by Figs. 6 an 7. In Fig. 6 the full time epenence of N /N is shown thick soli line. Also shown are the approximate solution given by Eq. 77 thick ashe line as well as the variances for a thermal state thin soli line an a coherent state thin ashe line with the same mean value N. The parameters chosen are 1, 2 1, 0, an 1. Thus we see that the variance always falls between those of thermal an coherent fiels. We also see that the long-time behavior is well approximate by Eq. 77. In Fig. We have previously iscusse the analogy between the present system an the nonegenerate optical parametric amplifier OPA. One of the most interesting applications of the OPA is the generation of entangle quantum optical states. We show that similar entanglements occur in the present system, but they are now between atomic an optical fiel moes. We first examine the two-moe intensity correlation functions, which give a measure of entanglement, an can be use to etermine whether or not nonclassical correlations exist between the three fiel moes. We then iscuss the issue of two-moe squeezing, an show how this phenomenon manifests itself in the present system. A. Two-moe intensity correlations The equal-time intensity correlation functions are efine in the usual manner as Nˆ inˆ j 2 ij Nˆ j. 78 Nˆ i Nˆ j g ij The two-moe correlation functions (i j) arise, e.g., if we consier a measurement of the intensity ifference between two moes, escribe by the operator whose variance is given by Nˆ ij Nˆ i Nˆ j, N ij 2 N i 2 2N i N j g 2 ij 1 N j 2. 79 80 For uncorrelate fiels, g (2) ij 1, an we have the usual rule for the aition of uncorrelate noise sources N ij 2 N i 2 N j 2. 81 If, however, there are correlations between the fluctuations in the intensities of the two moes, then we have g (2) ij 1, an the variance N ij will be less than that given by Eq. 81. For classical fiels positive P function, the two-moe (i j) correlations are constraine by the Cauchy-Schwartz inequality g 2 ij g 2 ii 1/2 g 2 jj 1/2. 82 Quantum-mechanical fiels, however, can violate this inequality, an are instea constraine by FIG. 7. The long-time limit of N j /N j is plotte against the initial probe intensity 2 thick soli line. It varies continuously between thermal upper ashe line an coherent lower ashe line limits. The parameters chosen are 1, 2 1, an 0. g 2 ij g ii 2 1 Nˆ i 1/2 g 2 jj 1 Nˆ j 1/2, 83 which reuces to the classical result in the limit of large intensities.