Matter-wave bistability in coupled atom-molecule quantum gases
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J u l Matter-wave bistability in coupled atom-molecule quantum gases
Lei Jiang , Han Pu Andrew Robertson , , and Hong Y. Ling Department of Physics and Astronomy, and Rice Quantum Institute, Rice University, Houston, TX 77251, USA Department of Physics and Astronomy, Rowan University, Glassboro, New Jersey, 08028-1700, USA and Joint Quantum Institute and Condensed Matter Theory Center,Department of Physics, University of Maryland, College Park, MD 20742-4111, USA
We study the matter-wave bistability in coupled atom-molecule quantum gases, in which het-eronuclear molecules are created via an interspecies Feshbach resonance involving either two-speciesBose or two-species Fermi atoms at zero temperature. We show that the resonant two-channel Bosemodel is equivalent to the nondegenerate parametric down-conversion in quantum optics, while thecorresponding Fermi model can be mapped to a quantum optics model that describes a single-modelaser field interacting with an ensemble of inhomogeneously broadened two-level atoms. Using theseanalogy and the fact that both models are subject to the Kerr nonlinearity due to the two-bodys-wave collisions, we show that under proper conditions, the population in the molecular state inboth models can be made to change with the Feshbach detuning in a bistable fashion.
PACS numbers: 03.75.-b, 03.75.Ss, 05.30.Fk, 05.30.Jp
I. INTRODUCTION
We study the matter-wave bistability in coupledatom-molecule quantum gases, in which heteronuclearmolecules are created via an interspecies Feshbach res-onance involving either two-species Bose or two-speciesFermi atoms at zero temperature. We show that the res-onant two-channel Bose model is equivalent to the nonde-generate parametric down-conversion in quantum optics,while the corresponding Fermi model can be mapped to aquantum optics model that describes a single-mode laserfield interacting with an ensemble of inhomogeneouslybroadened two-level atoms. Using these analogies andthe fact that both models are subject to the Kerr nonlin-earity due to the two-body s-wave collisions, we show thatunder proper conditions, the population in the molecu-lar state in both models can be made to change with theFeshbach detuning in a bistable fashion.The ability to cool and trap neutral atoms down toquantum degenerate regime has created a host of new andexciting problems that are increasingly interdisciplinary,bridging in particular the atomic, molecular, and opti-cal physics and the condensed matter physics. The richknowledge and experience accumulated over the past sev-eral decades in these fields have dramatically acceleratedthe progress of ultracold atomic physics. An examplethat serves to illustrate how the interdisciplinary fieldslearn and benefit from each other is the phenomonon ofatomic pairing where a bosoinc molecule is coupled totwo bosonic or fermionic constituent atoms via Feshbachresonance or photoassociation. So far this is the only vi-able approach to create ultracold molecules. It is alsoan ideal test ground for studying coupled atom-moleculecondensates and the BCS-BEC crossover [2]. The lat-ter is thought to be underlying the mechanism of hightemperature superconductors and extensively studied inthe realm of condensed matter physics. In addition, thecoupled atom-molecule systems have deep quantum opti- cal analogies [4, 5]: bosoinc molecules coupled to bosonicatoms (which we will refer to as the bosonic model in thispaper) is the matter-wave analog of parametric couplingof photons which has important applications in generat-ing nonclassical light fields and, more recently, in quan-tum information science; while the system of bosonicmolecules coupled to fermionic atoms (which we will referto as the fermionic model) can be mapped to the Dickemodel where a light field interacts with an ensemble oftwo-level atoms, a model having fundamental importancein the field of quantum optics.In this work, we will further explore these quantumoptical analogies of the atom-molecule system and fo-cus on the important effects of binary collisional interac-tions between atoms which are largely ignored in previousstudies [4, 5]. We show that the atom-atom interactionintroduces extra nonlinear terms which, under certainconditions, give rise to matter-wave bistability in bothbosonic and fermionic models. Hence, we may establishthe connection between the coupled atom-molecule quan-tum gases and the nonlinear bistable systems [6] thathave been extensively studied in the 80’s in the contextof nonlinear optics, due both to its fundamental inter-est, and to its many practical applications in fast opticalswitches, optical memory, laser pulse shaping, etc.
II. BOSONIC MODEL
In what we call the bosonic model, a molecule as-sociated with annihilation operator ˆ a m is coupled totwo non-identical atoms labeled as | ↑i and | ↓i withcorresponding annihilation operators ˆ a ↑ and ˆ a ↓ , respec-tively. Here we consider two types of atoms in orderto make direct comparisons with the fermionic modelto be treated in the next section, for which only unlikefermionic atoms can pair with each other and form abosonic molecule. Futhermore, in this work we only con-sider zero-temperature homogeneous case so that all thebosons are condensed into zero center-of-mass momen-tum states.The second quantized Hamiltonian readsˆ H = δ ˆ a † m ˆ a m + g (cid:0) ˆ a † m ˆ a ↑ ˆ a ↓ + h.c. (cid:1) + X i,j χ ij ˆ a † i ˆ a † j ˆ a j ˆ a i , (1)where the detuning δ represents the energy differencebetween the molecular and atomic levels which can betuned by external field, g is the atom-molecule couplingstrength and χ ij = χ ji is the s -wave collisional strengthbetween modes i and j . This system has been studied inRef. [7]. For completeness and better comparison withthe fermionic model, we briefly state some of the mainresults relevant to the focus of this work — matter-wavebistability — and direct readers to Ref. [7] for more de-tails.For our purpose, we take the standard mean-field ap-proximation and replace operators ˆ a j with c -numbers a j = p N j e iϕ j . The mean-field Hamiltonian takes theform: H = 2Λ( y − y ) + 2 νy + (1 − y ) p y cos ϕ , (2)where y = 0 . − ( N ↑ + N ↓ ) /N ] = N m /N , ϕ = ϕ ↑ + ϕ ↓ − ϕ m , are a pair of conjugate variables, representing the molecu-lar population and phase mismatch, respectively. Otherquantities are defined as G = g √ N ,
Λ = N ( χ ↑↑ + χ ↓↓ + χ mm + 2 χ ↑↓ − χ m ↑ − χ m ↓ ) /G ,ν = [ δ + χ ↑↑ + χ ↓↓ + ( N − χ mm − N χ m ↑ − N χ m ↓ ] /G , with N ≡ N ↑ + N ↓ +2 N m a constant of motion represent-ing the total number of atoms, and we have assumed thatthe number of atoms in states | ↑i and | ↓i are equal, i.e., N ↑ = N ↓ . In addition, we will focus on the stationarystates with ϕ = π which has lower energies than the oneswith ϕ = 0. A. Quantum Optical Analogy
It is quite clear from the form of the second-quantizedHamiltonian in Eq. (1) that without the collisional termsour model will reduce to the trilinear Hamiltonian de-scribing the nondegenerate parametric down-conversionin quantum optics [8, 9]. In this analogy, the molecu-lar mode plays the role of the pump photon, where thetwo atomic modes are the signal and idler photons, re-spectively. The collisional terms would correspond to theKerr-type cubic nonlinearity which will be present in theoptical system if the light fields propagate in some non-linear medium [10].
B. Bistability
In the absence of the collisions or Kerr nonlinearity(i.e., Λ = 0), the system does not exhibit bistability.This can be seen by studying the properties of the mean-field Hamiltonian H in Eq. (2) which can be simplifiedas (taking ϕ = π ) H = 2 νy − (1 − y ) p y , (3)The stationary state correspond to the solution of ∂H∂y = 2 ν + 3 p y − / p y = 0 . (4)For a given detuning ν , the stationary state is unique: y ( ν ) = (cid:26) . , ν < − ( − ν + √ ν + 3) , ν ≥ − = 0, using Eq. (2), the stationary condition is givenby ∂H∂y = 2 ν ′ + 3 p y − / p y = 0 , (6)where we have defined ν ′ = ν + Λ(2 y − . (7)Note that Eqs. (4) and (6) have the same form. In otherwords,we can express the effect of collisions as a nonlinearphase shift for molecules that modifies the detuning ν .Consequently, the stationary solution for Λ = 0 shouldhave the same form as in Eq. (5) but with ν replaced by ν ′ , which makes y an implicit function of the detuning ν . To find the explicit dependence of y on ν , we canuse the graphic method as illustrated in Fig. 1. For theexample given, we obtain three stationary states. Furtheranalysis shows that the middle solution is dynamicallyunstable and the other two are stable solutions [7]. Sucha behavior is typical in bistable systems [6].The graphics of Fig. 1 also shows that, in order to havemultiple stationary solutions, the slope of the straightline (given by 1 / ν = −
1, andthis leads to the conditionΛ < − , (8)in order for the system to exhibit bistability. III. FERMIONIC MODEL
In the fermionic model, we denote ˆ a k ,σ as the anni-hilation operator for an atom with spin σ (= ↑ , ↓ ), mo-mentum ~ k , and energy ǫ k = ~ k / (2 m ), and as beforedenote ˆ a m as the annihilation operator for a molecule in y v ΄ FIG. 1: For given Λ and ν , the thick solid line represents y ( ν ′ ) and the thin dashed straight line represents Eq. (7).The intersects are the stationary solutions. Here we take1 /
2Λ = − . ν = 0 . state | m i with zero momentum. the second quantizedHamiltonian reads:ˆ H = X k ,σ ǫ k ˆ a † k σ ˆ a k σ + U X k , k ′ , q ˆ a † k ↑ ˆ a †− k + q ↓ ˆ a − k ′ + q ↓ ˆ a k ′ ↑ + ν ˆ a † m ˆ a m + g √ V X k (cid:0) ˆ a † m ˆ a − k ↓ ˆ a k ↑ + h.c. (cid:1) , (9)where V is the quantization volume. Hamiltonian (9) hasthe form of the two-channel model of BCS-BEC crossoverwhere only the condensed molecule part is considered[11]. Following the Hartree-Fock-Bogoliubov mean-fieldapproach [15] by dividing the two-body collision into apart related to the BCS gap potential ∆ = U p, where p = X k h ˆ a − k ↓ ˆ a k ↑ i /V , and a part related to the Hartree potential V h = U X k σ h ˆ a † k σ ˆ a k σ i / (2 V ) , (10)where we again assume equal population in | ↑i and | ↓i atomic states, i.e., h ˆ a † k ↑ ˆ a k ↑ i = h ˆ a † k ↓ ˆ a k ↓ i , we may expressthe Hamiltonian asˆ H = X k ,σ ( ǫ k + V h )ˆ a † k σ ˆ a k σ + ν ˆ a † m ˆ a m + X k h(cid:16) U p + g ˆ a m / √ V (cid:17) ˆ a † k ↑ ˆ a †− k ↓ + h.c i . (11)Defining ˆ N = 2ˆ a † m ˆ a m + P k ,σ ˆ a † k σ ˆ a k σ as the number op-erator which is a constant of motion, we may rewrite theterm proportional to V h in (11) as X k ,σ V h ˆ a † k σ ˆ a k σ = V h ( ˆ N − b † ˆ b )= V h ˆ N − (cid:16) U n − U h ˆ b † ˆ b i /V (cid:17) ˆ b † ˆ b , (12) where n = h ˆ N i /V is the constant total atom numberdensity. In our derivation, V h arises from the two-bodyatom-atom collision. In general, additional terms repre-senting atom-molecule and molecule-molecule collisionsare also present. These additional terms will modify thecoefficient U in the definition of V h [Eq. (10)], which is thecounterpart of Λ in the bosonic model, but the generalform of Eq. (12) will remain valid. In the following, wewill refer to this term as the collisional term. ThroughEq. (12), we have expressed the effect of the two-bodycollisions as a nonlinear energy shift of the molecules(along with a constant energy bias V h N ), in completeanalogy with the bosonic model. We remark that in theusual one-channel model of the mean-field BCS theoryvalid when the molecular population is negligible, thecollisional term just represents an unimportant constantenergy shift.As usual, ˆ a k σ ( t ) and ˆ a m ( t ) obey the Heisenberg equa-tions of motion based on Hamiltonian (11). By replacingBose operator ˆ a m with the related c-number c = h ˆ b i / √ V and Fermi operators ˆ a k σ ( t ) with the familiar u k ( t ) and v k ( t ) parameters through the Bogoliubov transformationˆ a k ↑ = u ∗ k ˆ α k ↑ + v k ˆ α †− k ↓ and ˆ a †− k ↓ = − v ∗ k ˆ α k ↑ + u k ˆ α †− k ↓ ,where ˆ α k σ are the Fermi quasiparticle operators, we ar-rive at the following set of mean-field equations of motion i ~ ˙ c = ν e c + gp , (13a) i ~ ˙ u k = − ǫ k u k + ∆ e v k , (13b) i ~ ˙ v k = ∆ e u k + ǫ k v k , (13c)where p = P k u ∗ k v k /V , ∆ e = gc + U p , and ν e = ν − U n + 2 U | c | , (14)is the effective detuning which contains a Kerr nonlinearterm 2 U | c | whose origin can be traced to the two-bodycollisional shift. This set of equations describes the dy-namics at zero temperature where the state of the systemcan be described as the quasiparticle vacuum. A. Quantum Optical Analog
In several previous studies where the collisional termis neglected, it has been pointed out that the fermionicmodel can be mapped to the Dicke model in quantumoptics [4, 12] as schematically shown in Fig. 2 (see belowfor details). In fact, this model was recently shown todisplay collective dynamics similar to photon echo andsoliton-like oscillations in transient collective coherentoptics [13]. Such a connection can be traced to the workof Anderson’s spin analogy [14] for the BCS problem.To show what is the quantum optical analogy of thecollisional term, let us rewrite Eqs. (13) in a form morefamiliar in cavity optics. To this end, we first introducea set of new variables P k = 2 u ∗ k v k , D k = | u k | − | v k | , E L = 2 i ∆ e , c n g m › | ¯ › | ¯ › | FIG. 2: (Color online) Mapping of the two-channel resonantFermi superfluid model to the Dicke model. The bosonicmolecules and the fermionic atoms in the former are mappedto the cavity laser field and an ensemble of two-level atoms inthe latter, respectively. See text for details. and recast Eqs. (13b), (13c) into ~ ˙ P k = − i ǫ k P k − E L D k , (15a) ~ ˙ D k = ( E ∗ L P k + E L P ∗ k ) / . (15b)Interpreting P k and D k as the microscopic polarizationand population inversion, respectively, Eqs. (15) then be-come the optical Bloch equation that describes the in-teraction between a local electromagnetic field E L anda fictitious two-level atom, characterized with a transi-tion energy 2 ǫ k [16]. This analogy is consistent with thefact that there exists a one-to-one mapping between pairsof fermion operators and Pauli matrices when the BCSpairing mechanism is taken into account [14].In this optical analogy, the local electric field E L = E + E i contains two contributions because of ∆ e = gc + U p .The first of these ( E = i gc ) is equivalent to an av-erage macroscopic field, whose dynamics is describedby Eq. (13a), which can now be interpreted as theMaxwell’s equation for the cavity field E with cavitydetuning ν e , driven by a macroscopic polarization den-sity p = P k P k / (2 V ) of an inhomogeneously-broadenedmedium [see Fig. 2]. The second part E i = i U p may beregarded as the internal field at the atom due to the col-lective dipole polarization of the nearby two-level atomsin the ensemble. As such, E L = E + E i here bears adirect analogy to the Lorentz-Lorenz relation in optics[17]. Note that had the collisional term been neglected(i.e., U = 0), there would have been no internal fieldcontribution, nor would there have been the Kerr nonlin-earity in the equation for the bosonic mode. For U = 0,both of these terms will be present. Under such a cir-cumstance, Eqs. (13a) and (15) represent the generalized optical-Bloch equations in which the Lorentz-Lorenz re-lation is explicitly incorporated [18], and hence can leadto interesting nonlinear phenomena just as they do inoptical systems. B. Bistability
Having established this analogy, we now look for thesteady state solution from Eqs. (13a) and (15). As iswell-known, the operation frequency of a laser field isnot known a priori ; but is established through the so-called mode pulling — the dynamical competition be-tween atomic and cavity resonances. A similar argumentholds for the molecular field c . For this reason, we adoptthe following steady-state ansatz c → c e − iµt/ ~ , P k → P k e − iµt/ ~ , D k → D k where the same symbols are used for both dynamicaland steady-state variables for notational simplicity. Themolecular chemical potential, 2 µ , is just the correspond-ing lasing frequency in the cavity optics model. From thesteady state equations obtained by inserting this station-ary ansatz into Eqs. (13a) and (15), we can easily findthat (a) there always exists a trivial solution or a “non-lasing” state with ∆ e = 0 or equivalently c = 0, whichcorresponds to the non-superfluid normal Fermi sea; and(b) a non-trivial solution with its µ , ∆ e and c determinedself-consistently from the gap equation1 U − g / ( ν e − µ ) = − V X k E k , (16)with E k = p ( ǫ k − µ ) + ∆ e , the number equation2 | c | + 1 V X k (cid:18) − ǫ k − µE k (cid:19) = n, (17)and an auxiliary relation | g ∆ e | = | c ( ν e − µ )[ U − g / ( ν e − µ )] | . (18)The integral in the gap equation (16) under the assump-tion of contact interaction is known to be ultraviolet di-vergent. To eliminate this problem, we renormalize theinteraction strength U and g , as well as the detuning ν in (16), while U in the collisional term is replaced by thebackground interaction strength U [19, 20].Note that there exists, in the single-mode inhomo-geneously broadened laser theory [21], a similar set ofsteady-state integral equations, which, due to lasers be-ing open systems, are obtained under different consider-ations. For example, the requirement that the cavity lossbalance the saturated gain leads to the “gap” equation,whose primary role is to limit the laser intensity; whilethe phase matching condition translates into the “num-ber” equation, whose main responsibility is to assign the e c (b) xx + c (a) x FIG. 3: (Color online) Free energy density f as a function of∆ e and | c | at ν = 0 .
02 (a) and ν = 0 . f ,∆ e , and ν are all in units of E F = (3 π n ) / / (2 m ), the Fermienergy of the non-interaction system. In all the examplesshown in this paper, the physical parameters correspondingto g and U are 1 . E F /k / F and − . E F /k F , respectively. (a) (b) FIG. 4: Molecular population | c | as a function of detuning.Vertical line in (a) indicate the critical point of a first-orderphase transition. In (a) the collisional term is included whileit is neglected in (b). amount of mode pulling of the laser field relative to thecavity resonance.An alternative way to derive Eqs. (16)-(18) is fromthe energy density. The zero-temperature energy density f (∆ e , c, µ ) ≡ h ˆ H i /V can be calculated using Hamilto-nian (11) and the Bogoliubov transformation as [20] f = X k ǫ k − µ − E k V − (∆ e − gc ) U + ( ν e − µ ) | c | + µn . (19)The extremum conditions ∂f /∂ ∆ e = ∂f /∂c = 0, leadto Eqs. (16) and (18), respectively, while the condition ∂f /∂µ = 0 results in the number equation (17).Figure 3 illustrates the energy density in the | c | -∆ e plane for different detuning ν . For any given pair of ( c ,∆ e ), µ is calculated self-consistently using the numberequation (17). Typically, f has only one extremum whichis a minimum point as shown in Fig. 3(a). However,in the regime ν ∈ ( − . , . E F , f possesses threeextrema: two of them are local minima and the thirda saddle point. An example with ν = 0 .
02 is shown inFig. 3(b).To gain more insights into the bistable behavior, wemay carry an analogous analysis as in Sec. II B. In the ab-sence of the collisional term, steady-state molecular pop-ulation | c | is a smooth monotonically decreasing func-tion of ν and the system does not exhibit bistability:As ν increases, molecules decompose into atoms. This isshown in Fig. 4(a). When collisional term is included, the relevant equations of motion maintain the same forms ifwe substitute ν by ν ′ = ν + 2 U | c | . (20)Hence the solution | c | as a function of ν ′ is representedby the same curve as in Fig. 4(a). To find | c | as a func-tion of ν , we need to find the intersects between this curveand the straight line representing Eq. (20). In direct anal-ogy to the graphic method in Fig. 1, for U sufficientlylarge and negative, these two curves have three inter-sects and the system exhibits bistability. One example isshown in Fig. 4(b). The vertical line in Fig. 4(b) indicatethe critical point of a first-order phase transition: acrossthis line, the ground state jumps from the upper branchto the lower one. For the parameters used, this occurs at ν c = − . E F .To check the stability of these steady states, we havesolved the dynamical equations (13) using the slightlyperturbed steady state solution as the initial condition.From the dynamical evolution of the system one cansee that, just like in the bosonic model, the states inthe upper and lower branches are dynamically stable:when slightly perturbed, they exhibit damped oscilla-tions around their equilibrium values. These oscillationscan be further understood from the excitation spectrumof the corresponding steady state. This can be done us-ing a linear stability analysis, which is also the standardtool for studying laser instabilities [21, 22]. The spectrumis found to contain a discrete part which determines theoscillation frequencies, and a continuous part which con-tributes to the damping of these oscillations at a muchlonger time scale [23]. By contrast, the states in the mid-dle branch are unstable as small perturbations will leadto large departures. C. Dynamics
The bistability has important ramifications in atom-molecule conversion dynamics. When the collisionalterm is unimportant and negligible, one can easily createbosonic molecules from fermionic atoms by adiabaticallysweeping the Feshbach detuning across the resonance.As long as the sweeping speed is sufficiently slow, themolecular population will follow the steady-state curve asshown in Fig. 4(a). By contrast, when bistability inducedby the collisional term occurs, the adiabaticity conditionwill necessarily break down. Fig. 5 displays the dynami-cal evolution of the bosonic population when the detun-ing is swept starting either from a large positive or a largenegative value. We can see that the steady-state curvecan be followed up to the point where the stable states ofthe upper and lower branches and the unstable states ofthe middle branch join each other (indicated by ν and ν in Fig. 5), where the population suddenly jumps betweenthe two stable branches. Note that the critical detuning ν c for the first-order phase transition as indicated by the ν (a) (b) |c| ν ν FIG. 5: (Color online) Dynamics of atom-molecule conversionas illustrated by the molecular population when the detuning ν is slowly swept. Curve (a) is obtained by sweeping ν frompositive to negative values, while curve (b) is obtained bysweeping ν in the opposite direction. The dotted line is thesteady-state molecular population, the same as in Fig. 4(b). vertical line in Fig. 4(b) lies between ν and ν . The dy-namical population curve thus exhibits hysteresis in thevicinity of the first-order phase transition. In this way,by tuning the detuning in the vicinity of ν or ν , anatom-molecule switch can be realized. Similar behavioris also found in the bosonic model. IV. CONCLUSION
In conclusion, we have studied the matter-wave bista-bility in coupled atom-molecule quantum gases in boththe bosonic and the fermionic models. These two casescan be mapped to two very different quantum opticalmodels: parametric downconversion in the former andgeneralized Dicke model in the latter. Nevertheless, oneimportant common feature for both cases is that bista-bility can be induced by collisional interactions whichgive rise to Kerr nonlinearity. We hope that our workwill motivate experimental efforts in demonstrating thematter-wave bistability we predicted here.
Acknowledgments
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