aa r X i v : . [ c ond - m a t . qu a n t - g a s ] M a r Bistability in Feshbach Resonance
Hong Y. Ling
Department of Physics and Astronomy, Rowan University, Glassboro, New Jersey, 08028-1700, USA (Dated: November 20, 2018)A coupled atom-molecule condensate with an intraspecies Feshbach resonance is employed toexplore matter wave bistability both in the presence and in the absence of a unidirectional opticalring cavity. In particular, a set of conditions are derived that allow the threshold for bistability, dueboth to two-body s-wave scatterings and to cavity-mediated two-body interactions, to be determinedanalytically. The latter bistability is found to support, not only transitions between a mixed (atom-molecule) state and a pure molecular state as in the former bistability, but also transitions betweentwo distinct mixed states.
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I. INTRODUCTION
The subject of optical bistability [1], to which LorenzoNarducci contributed greatly during his prime years oflife, was brought to spotlight again by recent experi-mental demonstration of optical bistability in a micro-cavity with a cavity field at the level of a single photon[2, 3]. Instead of thermal gases, as typically employedby Lorenzo’s generation, where the thermal de Broglie’swavelength of the particles is far smaller than the inter-particle spacing, more recent trend in cavity quantumelectrodynamics (QED) focuses on cavity systems withultracold quantum gases [2, 3], opening the door to stud-ies such as cavity-mediated bistable Mott-insulator to su-perfluid phase transition [4, 5], where the particle statis-tical nature becomes an essential feature of the cavityproblem. In contrast to 1980s, when the surge of in-terest in optical bistability was inspired by the prospectof its use as a switch in an all-optical computer [1], thecurrent surge of interest in the same topic was, however,motivated largely by the equivalence of the cavity con-densate system [3] to the cavity opto-mechanical system[6]. The study of the latter falls into the realm of cav-ity optomechanics, a rapidly emerging field which aimsto use the cavity-assisted radiation force [7] to cool me-chanical device, ranging from nano- or micro-mechanicalcantilevers to macroscopic mirrors in LIGO project, downultimately to their quantum mechanical ground state [8].In this paper, we focus on a cavity system containing acoupled homonuclear atom-molecule condensate as illus-trated in Fig. 1. In the absence of the cavity, molecularstate | i is coupled only to free atomic state | i by anintraspecies Feshbach resonance [9] characterized with astrength α ′ and a detuning ǫ ′ [the shaded part in Fig.1(a)]. In the cavity QED setting, state | i is coupled,besides to state | i , also to excited molecular state | i by a unidirectional ring cavity of a total length L . Thecavity is driven by an external laser of wavenumber k ,polarization ˆ ε , and frequency ω , which is tuned far awayfrom the | i ↔ | i molecular transition. In addition, thecavity is assumed to possess a sufficiently large intermodespacing L /c (with c being the light speed in vacuum) sothat only the cavity mode with a longitudinal mode fre- FIG. 1: (Color online) (a) The schematic energy diagram(with the shaded part representing the intraspecies Fesh-bach process) for (b) a unidirectional ring cavity + coupledhomonuclear atom-molecule condensate system. quency ω c closest to ω is relevant to our study, where ω c L /c equals integer multiple of 2 π . Further, the cav-ity mode is assumed to overlap with the condensate in aspatial region, characterized with a length L and a cross-sectional area A , large enough so that the condensate canbe treated as a uniform system with an effective volume V a = LA and a total atom number N a (counting thosein molecules).A similar cavity + condensate system has recently beenstudied [10], in which atoms are converted into moleculesby photoassociation as opposed to magnetoassociation(Feshbach resonance) in our model. There, absorptionof a cavity photon will convert two atoms into a molecule,and emission of a cavity photon will dissociate two atomsinto a molecule. This is reminiscent of the absorptivebistability in a driven optical cavity containing an en-semble of two-level atoms. In contrast, the cavity field inour model is to introduce a phase shift to the molecularfield, not the population exchange between atoms andmolecules. As a result, the bistability in our model isof dispersive nature. Further, interaction between elec-tronic dipoles and a cavity field in Ref. [10] gives rise toatom-molecule coupling, which, since the cavity field isitself a dynamical variable, changes with time. In con-trast, in our model, it is the hyperfine interaction - thecoupling between electron spins and nuclear spins of twocolliding atoms that result in atom-molecule coupling,which is therefore fixed by the atomic internal structure,independent of cavity field. In this respect, our model isanalogous more closely to a cavity system with a spin-1condensate (also recently proposed [11]) than to a cav-ity system with photoassociation [10]; atom exchangesamong different internal states in the spin-1 condensateare accomplished by spin-exchange interaction, which isalso independent of cavity field.The interest in Feshbach resonance stems primarilyfrom its use as an effective tool to coherently createmolecular BECs from the existing atomic BECs. Ina typical experiment, conversion of atomic BECs intomolecular BECs is carried out by ramping the Feshbachdetuning across the resonance. This method relies onthe existence of a mixed (or dressed) atom-molecule state[12], and the ability of this state to change its composi-tion from predominantly atomic to predominantly molec-ular species when the Feshbach detuning ǫ ′ is tuned fromabove to bellow the resonance. The question that wewant to pursue, in this paper, is how the molecular pop-ulation in state | i can be made to vary with the Feshbachdetuning in a bistable fashion, instead of monotonouslyas in typical situations. A bistable crossover adds newmeaning to the Feshbach resonance: whether the systemis in the atomic or molecular extreme is determined notonly by the Feshbach detuning but also by the history ofthe system.To this end, we first formulate, in Sec. II, a semi-classical mean-field description of our model in the limitwhere both the excited molecular field and the opticalcavity field can be adiabatically eliminated. We thenexplore the matter-wave bistability due to the two-bodys-wave collisions in Sec. III A and that due to thecavity-mediated effective interaction between two Fes-hbach molecules in Sec. III B. Finally, we provide asummary in Sec. IV. II. THE BASIC EQUATIONS
In this section, we take a semiclassical approach, inwhich optical fields are treated classically while matterfields are treated quantum mechanically, to formulate atheoretical description of the proposed cavity + conden-sate system. This is the same approach that Lorenzoembraced in many of his works [13], except that quan-tization is now performed at a level for a many-bodysystem, instead of a single-body system as in a typicalsemiclassical laser theory. To begin with, we expand op-tical field E ( r , t ) in terms of the slowly varying amplitude F in space r and time t according to E ( r , t ) = 12 F ˆ εe i ( kz − ωt ) + c.c, (1) and matter field ˆ ψ ( r , t ) in terms of ˆ ψ i = ˆ c i / √ N a accord-ing toˆ ψ ( r , t ) = √ n a h ˆ ψ | i + ˆ ψ | i + ˆ ψ e i ( kz − ωt ) | i i , (2)where ˆ c i denotes the operator in momentum space for an-nihilating a bosonic particle in state | i i and n a = N a /V a is the total atom number density. The expansion inEq. (2) is carried out in a frame rotating at the laserfrequency ω . In arriving at Eq. (2), we have assumedthat the particles in states | i and | i are all condensedto their respective zero momentum modes, and those instate | i to the mode with ~ k momentum in accordancewith momentum conservation during photon-atom inter-action.The coupled atom-molecule system, within these ap-proximations, is then described by the following Hamil-tonian ˆ H/ ~ N a = ǫ ′ ˆ ψ † ˆ ψ + √ n a (cid:18) α ′ ψ † ˆ ψ + h.c (cid:19) + n a χ ′ ij ψ † i ˆ ψ † j ˆ ψ j ˆ ψ i ( i, j = 1 or 2)+ ( ǫ ′ − ∆ a ) ˆ ψ † ˆ ψ − (cid:16) Ω ˆ ψ † ˆ ψ + h.c (cid:17) , (3)where repeated indices are to be summed from 1 to 2. InEq. (3), the first line describes the Feshbach resonanceof strength α ′ , the second line the s-wave collisions ofstrength χ ′ ij (= χ ′ ji ) between states | i i and | j i , and thelast line the part of Hamiltonian involving excited state | i . In the last line, the first term denotes the energy ofstate | i in the rotating frame, where ∆ a is the laser de-tuning, and the second term stands for the laser-inducedelectric dipole interaction, where Ω = µ F/ ~ is theRabi frequency, µ = h | ˆ µ · ˆ ε | i the matrix element,and ˆ µ the electric dipole moment operator. Finally, col-lisions involving the final excited state | i are ignoredsince state | i remains virtually empty in our model.An important concept in the semiclassical approachis the macroscopic polarization defined as P ( r , t ) = D ˆ ψ † ( r , t ) ˆ µ ˆ ψ ( r , t ) E . This polarization is found, when theuse of both Eq. (2) and the selection rule are made, topossess the same mathematical form as the optical fieldin Eq. (1), P ( r , t ) = 12 P ˆ εe i ( kz − ωt ) + c.c., (4)where P = 2 n a µ D ˆ ψ ˆ ψ ∗ E represents the slowly varyingpart of the polarization. The evolution of the opticalfield is then governed by the Maxwell’s equation, which,under the slowly varying envelope approximation, takesthe form c ∂ Ω ∂z + ∂ Ω ∂t = i µ ωc ~ µ P , (5)where µ ( ǫ ) is the magnetic (electric) permeability invacuum. Equation (5) clearly shows that polarizationplays the role of a bridge between the classical opticalfield in Eq. (1) and the quantum matter fields in Eq.(2).The final component in the semiclassical approach per-taining to any QED problems is the boundary condition,which, in our case and under the assumption that bothinput and output mirrors have the same reflectivity R (transmissivity T = 1 − R ), can be cast into the form[14] Ω (0 , t ) = T Y + R Ω (
L, t − ∆ t ) e i ∆ c L /c , (6)where Y is the scaled amplitude of the incident field,∆ t = ( L − L ) /c the transit free propagation time insidethe cavity, and ∆ c = ω − ω c the cavity mode frequencydetuning. Equation (6) links the field entering ( z = 0)to that leaving the condensate ( z = L ), and hence imple-ments the concept of feedback, which is the most impor-tant feature of any optical cavities. The school led byBonifacio, Lugiato, and Narducci distinguishes itself byits insistence on a rigorous (first-principle) treatment ofthe boundary condition - a treatment made up of botha transformation mapping two non-isochronous events inEq. (6) into two isochronous events [15] and a set ofconditions embodying the notion of “mean-field limit”[16]. Following this treatment, we explicitly eliminatethe spatial derivative in Eq. (5), rendering Eq. (5) into d Ω dt = ( i ∆ c − κ ) Ω + κY + ig N a D ˆ ψ ˆ ψ † E , (7)where κ = c | ln R | / L is the cavity damping rate, g = µ p ω/ ~ ǫ V c the “Rabi frequency” per photon, and V c = L A the total cavity volume.In this paper, we restrict our study to the parameterregime in which both the time scale for the optical field(on the order of 1 /κ ) and that for the molecular fieldof state | i (on the order of 1 / ∆ a ) are far shorter thanthose for the Feshbach degrees of freedom, namely, thematter fields corresponding to states | i and | i . Thisrestriction allows us to adiabatically eliminate the formerfast variables in favor of the latter slow ones, reducing,from the Heisenberg’s equations for ˆ ψ i and the Maxwell’sequation for Ω, a set of equations involving only the slowdegrees of freedom ψ and ψ : i dψ dt = χ i n i ψ + αψ ψ ∗ , (8a) i dψ dt = [ ǫ ′ + χ i n i + F ( n )] ψ + α ψ . (8b)where we have adopted the standard mean-field theory,treating ˆ ψ i as c -numbers ψ i . In Eqs. (8), n = | ψ | , n = | ψ | , α = α ′ √ n a , and χ ij = χ ′ ij n a are the renor-malized quantities, and F ( n ) = κ ηN c κ + (∆ c − ηN a n ) , (9) represents the phase shift to the molecular field inducedby the cavity optical field, where N c = Y /g repre-sents the number of injected intracavity photons and η = g / ∆ a measures the effective atom-cavity couplingstrength. (In this paper,without loss of generality, welimit ∆ a > η is always positive.) III. DISCUSSIONS
In our system, stationary solutions can be divided intomixed atom-molecule states in which each species has a finite density n i , and pure molecular or atomic phase inwhich one of species has a zero density. In this section,we study bistability in the Feshbach process by focusingon the mixed atom-molecule state where each species canbe described by the field ψ i = √ n i e iθ i characterized witha well defined phase θ i . (Note that such a descriptioncannot be applied to pure molecular or atomic phases asthe phase associated with an empty component is notdefined.) For such a state, we can apply the total atomnumber conservation n + 2 n = 1 and simplify Eqs. (8)into a set of equations involving only two variables - amolecular density n ≡ n and a phase mismatch θ = θ − θ . This set of equations, in a unit system inwhich ǫ = ( ǫ ′ − χ − χ ) /α is defined as the effectiveFeshbach detuning, χ = [ χ + 4 ( χ − χ )] /α as theeffective Kerr nonlinear coefficient for the molecular field, δ = ∆ c /κ as the cavity detuning, and τ = αt as the time,take the form dndτ = − (1 − n ) √ n sin θ, (10a) dθdτ = ǫ + χn + 12 1 − n √ n cos θ + f ( n ) , (10b)where f ( n ) = 2 B/C δ − Cn ) . (11)In arriving at Eqs. (10), we have replaced N a and N c inEq. (9) in favor of two unitless parameters, C = ηN a /κ = g N a / ∆ a κ, (12)and B = η N c N a / (2 κα ) . (13)In contrast to C , which is bose enhanced by atom num-ber only, B is bose enhanced by both photon and atomnumbers. Note that when ∆ a is replaced with the decayrate of the excited state | i , C becomes the so-calledatomic “cooperative parameter” [17]. As we will seeshortly, cavity-mediated bistability depends crucially onthe values of C and B .As in other studies [18, 19], Eqs. (10) of the typeincluding Feshbach resonance support two branches ofsteady-states: one with θ = 0 and the other with θ = π .This feature is expected since at zero temperature the in-traspecies Feshbach resonance represents a matter waveanalog of the second harmonic generation in nonlinearoptics where the phase-matching condition plays an im-portant role. In this paper, without the loss of generality,we take α >
0. Under such a circumstance, the branchwith θ = π not only always has a lower energy than thebranch with θ = 0, but also has the property of consist-ing primarily atom species in the limit of a large positiveFeshbach detuning. For these reasons, we will focus onthe branch with θ = π , determined at steady state by ǫ = − χn + 12 1 − n √ n − f ( n ) , (14)where for notational simplicity, same symbols are usedto stand for the steady state variables.To carry out the stability analysis, we apply the stan-dard linearization procedure that Lorenzo used exten-sively in the context of his interest in laser instabilities[20, 21], and derive from Eqs. (10) a set of linearly cou-pled equations ddτ (cid:18) δnδθ (cid:19) = (cid:20) − n ) √ n − dǫdn (cid:21) (cid:18) δnδθ (cid:19) , (15)where δn and δθ are small departures from the corre-sponding steady state variables, and ǫ is given by Eq.(14). The eigenvalues of Eqs. (15) are then found totake two values: ± p − (1 − n ) √ ndǫ/dn , which, since n < .
5, clearly indicate that only when dǫ/dn > dn/dǫ > n as a function of ǫ is positive is unstable. Note thatsuch a conclusion may not hold, if we do not impose, inthe previous section, the conditions that allow ψ and Ωto be adiabatically eliminated. In the case of absorptiveoptical bistability, it is well known that the upper branchwhich is stable according to the slope criterion may be-come unstable; instability there is manifested in the formof self-pulsings [22].Thus, we see that the stability analysis here amountsto analyzing the points at which dǫ/dn = 0, which, bydefinition, simply corresponds to the critical transitionpoints in a bistable (or multi-stable) system. As aresult, in what follows, we carry out bistability study byfocusing on the equation d ( n ) = χ − h ( n ) , (16)derived from Eq. (14) under the condition that dǫ/dn =0, where d ( n ) = − B ( δ − Cn ) h δ − Cn ) i , (17) h ( n ) = − n − / − n − / . (18) A. Collision-Induced Bistability
In our model, the molecular field (at state | i ) is sub-ject to two types of self phase modulations: one, de-scribed by a Kerr type of nonlinear term χn, originatesfrom short-range s-wave scatterings, and another, de-scribed by f ( n ) in Eq. (11), stems from cavity-mediatedlong-range two-body collisions. In order to differentiatetheir roles in the formation of matter wave bistability, wefirst remove the cavity component and study the bista-bility due solely to the Kerr nonlinearity by solving χ = h ( n ) , (19)obtained from Eq. (16) by setting d ( n ) = 0. As onemay easily verify, h ( n ) in Eq. (18) is a monotonouslyincreasing function of n . As a result, in order for Eq.(19)to have real roots within n < . χ must be less than h ( n = 0 .
5) or χ ≤ χ th ≡ − √ . (20)This condition is similar to that of Ref. [23] for a het-eronuclear atom-molecule system with an interspeciesFeshbach resonance [18]. It simply reflects the fact thatfor bistability to take place, there must be a sufficientlystrong positive feedback between the molecular popula-tion and the effective Feshbach detuning ǫ + χn . A nega-tive χ fulfills this positive feedback; as can be seen, with anegative χ , an increase in the molecular density decreasesthe effective detuning, which, in turn, further increasesthe molecular density, or vice versa. Such a chain ofpositive reaction under condition (20) can lead to theformation of the critical transition points around whichthe molecular population changes in a runaway fashion.Indeed, under condition (20), Eq. (16) is found to sup-port a critical point with a critical molecular density n (1) cri given by, q n (1) cri = (cid:18) | χ | (cid:19) (cid:18)r χ + 1 (cid:19) + (cid:18) | χ | (cid:19) (cid:18)r χ − (cid:19) − χ , (21)and a critical Feshbach detuning ǫ (1) cri determined by Eq.(14) when n is replaced with n (1) cri in Eq. (21). Figure 2(a)shows a typical example of bistability (with χ = 2 χ th ).In addition to the mixed state, there is a pure molecularstate with n = 0 . n (2) cri = 0 . , ǫ (2) cri = − χ − √ . (22) ε n ε χ (a) ε (1)cri ε (2)cri (b)( ε (2)cri , n (2)cri )( ε (1)cri , n (1)cri ) FIG. 2: (a) Molecular population n as a function of ǫ when χ = 2 χ th = − √
2. (b) ǫ (1) cri and ǫ (2) cri at different values of χ . The threshold for bistability is reached when the two crit-ical points become degenerate. Clearly, this happens at n = 0 . χ = χ th . Figure 2(b) shows that when χ is bellow χ th , the size of the hysteresis loop (measuredby ǫ (2) cri − ǫ (1) cri ) increases with | χ | .To see the implication of condition (20) to a realisticsystem, consider, for example, the Feshbach resonancelocated at the magnetic field 85 . Na [26, 27].This resonance has a width of 0 . µ T or equivalently aFeshbach coupling strength of α ′ = 4 . × − m / s − .Taking the total atom number density to be 10 m − and using 3.4 nm as the s-wave scattering length forsodium atoms [28], we find that χ = 1.18 × s − and α =4.22 × s − = 3 . χ . If we further as-sume χ = χ , we see from Eq. (20) that this requires χ > (cid:0) χ + 4 χ + 2 √ α (cid:1) > . χ . B. Cavity-Mediated Bistability
The above example means to illustrate that Eq. (20)can be fulfilled only when both the total atom densityand the two-body interspecies collisional strength are suf-ficiently large, a condition which is difficult to meet undertypical systems. In this subsection, we turn our atten-tion to the cavity model in Fig. 1, and pursue, from Eq.(16), the question of under what cavity parameters canbistability occur even when condition (20) breaks down.Note that unlike Eq. (19), which only contains one crit-ical point, Eq. (16) can typically support two criticalpoints. Thus, the threshold for bistability in this cavitymodel can, in principle, occur at any value of n , insteadof always at n = 0 . √ y ( √ y ) − γ √ y + 1 = 0 , (23)where y = Cn − δ and γ = 2 p B/ [ χ − h ( n )]. Thischange of variable is motivated by the realization thatwhen condition (20) breaks down or equivalently χ +2 √ >
0, the quantity χ − h ( n ) is always positive so B t h ( n ) −2 −1 0 1 200.10.20.30.40.5 ε n ≈ FIG. 3: (Color online) (a) B th ( n ) as a function of n . (b)Molecular population n as a function of ǫ for B = B th (0 .
2) =4 .
73 (solid line), B = 2 B th (0 .
2) (dashed line), and B =4 B th (0 .
2) (dot-dashed line) for a cavity system with C = 20, χ = 0 , and δ = δ th (0 .
2) = 3 . that only when Cn − δ > n , we regard Eq.(23) as a transcendental equation for δ , and require Eq.(23) to support a real root of multiplicity 2 (another tworoots are a complex conjugate pair). This requirementallows us to conclude that bistability develops when B ≥ B th ( n ) ≡ √ χ − h ( n )] , (24)and the bistability threshold at a given n (and C ) isreached when B = B th ( n ) and δ = δ th ( n ) ≡ Cn − / √ B th ( n ) as a function of n is displayed in Fig. 3(a).This simple threshold relation seems to hold quite wellas long as C is sufficiently large. Consider, for example,a cavity system with C = 20. Figure 3(b) shows how n changes with ǫ under different values of B . The solidline, produced with B = B th (0 .
2) = 4 .
73 [point A in Fig3(a)] and δ = δ th (0 .
2) = 3 .
42, shows that the bistabilitythreshold indeed occurs at the theoretically predicted lo-cation with n = 0 . ǫ = − .
58 determined from Eq.(14)]. The dashed and dot-dashed lines in Fig. 3(b),produced with B = 2 B th (0 .
2) and B = 4 B th (0 . B is increased beyond its threshold value B th (0 . L = 200 µm and finesse F = 3 . × . The cav-ity is driven by an external laser tuned ∆ a = 2 π × D line, characterized with a wave-length 780 nm and a linewidth 2 π × L = 30 µm and A = (10 µm ) . In such a cavity-condensate sys-tem, we have η = 2 π × . κ = 2 π × . . × . This figure is three orders of magnitude smallerthan the threshold photon number in a typical laser [17],and can be further reduced with an appropriate choice ofthe system parameters.Matter wave bistability at small photon numbers canbe understood as follows. In contrast to the phase shiftdue to the s-wave scattering, which is linearly propor-tional to the molecular density, the phase shift, arisingfrom the feedback between optical and matter fields, isnonlinearly proportional to the molecular density by wayof Eq. (11) in a resonant fashion. On one hand, the sensi-tivity (the change of this phase shift versus the change ofthe molecular density) is bose-enhanced by the collectivenature of the condensate system. On the other hand, thepeak of this phase shift for a given cavity photon numbercan be significantly amplified in a microcavity environ-ment where photons are confined into a tiny volume. Asa result, molecular bistability is possible under a weakcavity field. IV. CONCLUSION
In this paper, we have studied the matter wave bista-bility in an intraspecies Feshbach resonance model with and without the assistance of an optical cavity. In par-ticular, we have arrived at a set of conditions that allowthe bistability thresholds under the two different settingsto be estimated analytically. In the absence of a cavity,bistability is possible only when the effective Kerr nonlin-earity stemming from s-wave scatterings [Eq. (20)] is suf-ficiently negative. In the presence of a cavity, even whencondition (20) breaks down, bistability can still occur,provided that B in Eq. (13), a key parameter describingthe cavity-mediated two-body interaction, is sufficientlylarge [Eq. 24]. An important difference between systemswith and without a cavity is that the former supports onestable mixed state while the latter can support two dif-ferent stable mixed states. (Both systems contain a puremolecular state.) Thus, a bistable transition in the lat-ter system can take place not only between a mixed stateand a pure molecular state as in the former model, butalso between two different mixed states. V. ACKNOWLEDGEMENT
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