Enhanced stripe phases in spin-orbit-coupled Bose-Einstein condensates in ring cavities
EEnhanced stripe phases in spin-orbit-coupled Bose-Einstein condensates in ringcavities
Farokh Mivehvar and David L. Feder ∗ Institute for Quantum Science and Technology, Department of Physics and Astronomy,University of Calgary, Calgary, Alberta, Canada T2N 1N4 (Dated: August 12, 2018)The coupled dynamics of the atom and photon fields in optical ring cavities with two counter-propagating modes give rise to both spin-orbit interactions as well as long-ranged interactions be-tween atoms of a many-body system. At zero temperature, the interplay between the two-body andcavity-mediated interactions determines the ground state of a Bose-Einstein condensate. In thiswork, we find that cavity quantum electrodynamics in the weak-coupling regime favors a stripe-phasestate over a plane-wave phase as the strength of cavity-mediated interactions increases. Indeed, thestripe phase is energetically stabilized even for condensates with attractive intra- and inter-speciesinteractions for sufficiently large cavity interactions. The elementary excitation spectra in bothphases correspond to linear dispersion relation at long wavelengths, indicating that both phasesexhibit superfluidity, though the plane-wave phase also displays a characteristic roton-type feature.The results suggest that even in the weak coupling regime cavities can yield interesting new physicsin ultracold quantum gases.
I. INTRODUCTION
The experimental realization of Bose-Einstein conden-sation (BEC) has opened many opportunities for real-izing new many-body phases [1–3]. Ultracold atomstrapped in laser-generated optical lattice potentials ex-perience crystalline environments and exhibit a variety ofintriguing phenomena [4], most notably the superfluid–Mott-insulator phase transition [5]. There are numerousproposals for inducing gauge fields in quantum gases bymeans of laser light [6], and recently abelian [7] and non-abelian [8] gauge fields have been realized. In the latterwork an equal combination of Rashba and Dresselhausspin-orbit (SO) couplings were induced via two-photonRaman transitions. These developments have set thestage for realizing topological states in these systems [9].The single-particle energy dispersion of a SO-coupledatom is a momentum-space double well, which is two-folddegenerate in the symmetric case [7]. In a Bose-Einsteincondensate (BEC) of atoms, the two-body interactionslift this degeneracy and drive the BEC into either a planewave phase (PWP) or a stripe phase (SP), depending onthe strength and sign of the intra- and inter-species two-body interactions [10–13]. In the PWP, all atoms con-dense into one of the two single-particle energy minima,while the SP is a superposition state of the minima andthe total BEC density exhibits faint fringes [14]. Addi-tional phases are found for fully three-dimensional SOinteractions [15]. When a SO-coupled quantum gas isconfined in an optical lattice, the ground state of thesystem exhibits a variety of magnetic orderings in theMott-insulator regime, such as ferromagnetic, antiferro-magnetic, spin spiral, vortex and antivortex crystals, andskyrmion crystal phases [16–18]. The superfluid to Mott- ∗ Corresponding author: [email protected] insulator phase transition of SO-coupled quantum gaseshas also been investigated [16, 19].In laser-based approaches to generating SO couplings,the radiation field is treated classically and one ignoresthe back-action of the atoms on it. Confining the radia-tion field to within an optical cavity leads to a coherentexchange of energy and momentum between atoms andphotons [20]. The back-action of the atoms on the photonfields is no longer negligible, leading to complex coupleddynamics of the matter and radiation fields in which bothentities are affected by one another and must be treatedon the same footing [21]. As a consequence, cavity-mediated long-range interactions are induced betweenatoms, yielding novel collective phenomena in atomicsystems [22]. A few schemes have been recently pro-posed to induce SO coupling in ultracold atoms via cavityquantum electrodynamics [23–26] and to couple a laser-induced SO-coupled BEC to the cavity field [27]. Theseschemes exhibit a wealth of physics, including strong syn-thetic magnetic fields, a cavity-mediated Hofstadter spec-trum, and a variety of magnetic orders.In this work we investigate the ground state and theelementary excitations of a spinor BEC at zero tempera-ture subject to ring-cavity-induced SO interactions [23].Here we consider lossy cavities where a steady-state pho-ton population is maintained by the application of ex-ternal pump lasers. The cavity photons mediate infinite-range interactions between atoms, whose strengths canbe tuned experimentally by adjusting the amplitudes ofthe pump lasers. The sign of these interactions can bemade positive or negative depending on the cavity detun-ing, the frequency difference between the applied pumplasers and the cavity. These cavity-mediated interac-tions compete with the inherent two-body interactionsbetween atoms to determine the ground state of the SO-coupled BEC. In particular, stripe phases are always fa-vored when positive cavity-mediated interactions domi-nate the two-body-interactions, even in the case where a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y the intrinsic atomic interactions (both intra- and inter-species) are attractive. Asymmetry in the strength ofcavity-mediated interactions for different spin compo-nents yields stripe-phase states with an arbitrary num-ber of atoms in the left or right minimum of the single-particle dispersion relation, so that the magnetizationvaries continuously from zero in the stripe phase to unityin the plane-wave phase. This behavior allows us to iden-tify a novel stripe-phase order parameter, and to identifyits associated mean-field critical exponent.Consideration of the quantum fluctuations around themean-field ground states reveals that the particle-hole el-ementary excitation spectra in both PWP and SP havethe usual linear sound-like dispersion relation at longwavelengths, an indication of superfluidity. In the PWP,the dispersion relation also exhibits a roton-type featureat the same wave vector that charactizes the fringe peri-odicity in the SP, which could be used experimentally asa distinguishing feature. The critical transition betweenthe PWP and SP occurs when the energy of this mini-mum falls below zero. Unlike for the PWP, in the SP thespeed of sound depends strongly on the cavity-mediatedinteractions. The speed of sound is found fall below zeroat a critical value of the cavity interactions and inter-species interactions strength, but this appears to signala phase transition to a phase-separated state. Overall,the ring-cavity environment provides an experimentallyconvenient framework for exploring exotic ground statesof SO-coupled BECs.The manuscript is organized as follows. In Section II,we start from the full atom-photon Hamiltonian den-sity for a lossy but pumped cavity, to derive an effectiveatomic Hamiltonian with the photon fields eliminated.The ground state of this effective Hamiltonian is exploredin Section III using both a variational method and bysolving the generalized Gross-Pitaevskii equations. Theremainder of this Section is devoted to an analysis of theelementary excitations. A discussion of the results andconclusions are found in Sec. IV. Appendices A and Bprovide details of the adiabatic elimination of the atomicexcited state and cavity fields, respectively. II. MODEL AND HAMILTONIAN
Consider spin-1 bosonic atoms inside a ring cavity withtwo driven counter-propagating running modes ˆ A e ik z and ˆ A e − ik z , where ˆ A j is the annihilation operator forthe photon in j th mode with wave vector k j = ω j /c and z is the direction along the cavity axis. Withoutloss of generality, one can assume that the wave vec-tors k and k of the two modes are approximately equalto each other, k R ≡ k ≈ k [28]. The mode ˆ A e ik R z ( ˆ A e − ik R z ) propagates to the right (left) and solely in-duces the atomic transition | a (cid:105) → | e (cid:105) ( | b (cid:105) → | e (cid:105) ), where {| a (cid:105) , | b (cid:105)} are non-degenerate pseudospin states of interestand | e (cid:105) is an excited state. The two cavity modes ˆ A j areassumed to be sufficiently populated to justify omitting associated degenerate modes ˆ A (cid:48) j . In principle, a state-independent external potential V ext ( r ) would need to beimposed to confine atoms inside the cavity. The single-particle Hamiltonian density in the dipole and rotating-wave approximations is H (1) = H (1)at + H cav + H (1)ac , (1)with H (1)at = (cid:20) − (cid:126) m ∇ + V ext ( r ) (cid:21) I × + (cid:88) τ ∈{ a,b,e } ε τ σ ττ ,H cav = (cid:126) (cid:88) j =1 , ω j ˆ A † j ˆ A j + i (cid:126) (cid:88) j =1 , (cid:16) η j ˆ A † j e − iω p j t − H.c. (cid:17) , H (1)ac = (cid:126) (cid:104)(cid:16) G ae e ik R z ˆ A σ ea + G be e − ik R z ˆ A σ eb (cid:17) + H.c. (cid:105) , where ε τ are the internal atomic-state energies, σ ττ (cid:48) = | τ (cid:105) (cid:104) τ (cid:48) | , and I × is the identity matrix in the internalatomic-state space. The atom-photon coupling for thetransition τ ↔ τ (cid:48) is denoted G ττ (cid:48) , and H.c. stands forthe Hermitian conjugate. The cavity mode ˆ A † j is drivenby a pump laser with frequency ω p j and amplitude η j ,indicated by the second sum in H cav . In this work, in or-der to simplify the analytical calculations, V ext ( r ) is setto zero. In reality, one might imagine a very weak (al-most unbound) confining potential along the cavity axis z but a standard harmonic trap in the radial direction.The details of the transverse confining potential are notimportant for the analysis presented in this work.After expressing Hamiltonian (1) in the rotating frameof the pump lasers [29] and assuming that the atomicdetunings ∆ = ω − ε ea / (cid:126) and ∆ = ω − ε eb / (cid:126) arelarge compared to ε ba / (cid:126) = ( ε b − ε a ) / (cid:126) , one can adi-abatically eliminate the atomic excited state to obtainan effective Hamiltonian H (cid:48) (1)SO for the ground pseudospinstates {| (cid:105) , | (cid:105)} ≡ {| b (cid:105) , | a (cid:105)} . The details are presentedin Appendix A. In the limit of a very weak confiningpotential along the cavity axis ˆ z , one can assume thatthe momentum p z = (cid:126) k z is a good quantum number.Alternatively one could consider approximately uniformquantum gases in a box potential where V ext ( r ) = 0 ex-cept at the boundaries; such a potential has recently beenrealized experimentally [30]. One can then transformto the co-moving frame of the cavity modes by apply-ing the unitary transformation U = e − ik R zσ z (where σ z = σ − σ is the third Pauli matrix, see also Ap-pendix A). The kinetic-energy part of the Hamiltoniandensity H (cid:48)(cid:48) (1)SO ≡ U H (cid:48) (1)SO U † associated with the momen-tum p z , Eq. (A5), then takes the familiar form of an equalRashba-Dresselhaus SO coupling: m ( p z I × + (cid:126) k R σ z ) ,which is characterized by a double-well energy disper-sion [8].In the presence of dissipation, such as when the decayrate κ of both cavity modes is non-zero, one should inprinciple numerically solve the associated master equa-tion [31]. That said, in the weak-coupling regime when κ is the dominant energy scale, κ (cid:29) ( G ae , G be ), the mas-ter equation approach is equivalent to including dissipa-tion in the Heisenberg equations of motion for the cavityfields: ∂ t ˆ A j = − i [ ˆ A j , H (cid:48)(cid:48) (1)SO ] / (cid:126) − κ ˆ A j [21]. The cavityfields quickly reach steady states, allowing them to beadiabatically eliminated. Setting ∂ t ˆ A j = 0 one obtainssteady-state expressions for ˆ A j that can be substitutedinto H (cid:48)(cid:48) (1)SO to yield an effective atomic Hamiltonian; thedetails are relegated to Appendix B.The resulting effective many-body Hamiltonian reads H eff = (cid:90) d r (cid:18) ˆ Ψ † H (1)SO ˆ Ψ + 12 g ˆ n + 12 g ˆ n + g ˆ n ˆ n (cid:19) + (cid:88) τ =1 , U τ ˆ N τ + U ± ˆ S + ˆ S − + U ∓ ˆ S − ˆ S + + 2 U ds ˆ N ˆ S x , (2)where ˆ Ψ ( r ) = ( ˆ ψ ( r ) , ˆ ψ ( r )) T are the bosonic field opera-tors obeying the commutation relation [ ˆ ψ τ ( r ) , ˆ ψ † τ (cid:48) ( r (cid:48) )] = δ τ,τ (cid:48) δ ( r − r (cid:48) ), ˆ N τ = (cid:82) ˆ n τ ( r ) d r = (cid:82) ˆ ψ † τ ( r ) ˆ ψ τ ( r ) d r is thetotal atomic number operator for pseudospin τ ∈ { , } ,ˆ N = ˆ N + ˆ N is the total atomic number operator, andthe x -component of the total spin operator is definedin a usual way ˆ S x = ( ˆ S + + ˆ S − ) using the collectivepseudospin raising and lowering operators ˆ S + = ˆ S †− = (cid:82) ˆ ψ † ( r ) ˆ ψ ( r ) d r . The atoms in this system experiencetwo kinds of interactions, reflected in the effective Hamil-tonian H eff : the standard two-body contact interactionsand the cavity-mediated long-ranged interactions. Here g τ ≡ g ττ denotes the two-body intra-species interactionstrength and g the two-body inter-species interactionstrength. The strength of the cavity-mediated interac-tions are found in Appendix B: U = 4 (cid:126) G ∆ c (∆ − κ )∆ (∆ + κ ) η ,U ± ( ∓ ) = 4 (cid:126) G ∆ c ∆ (∆ + κ ) (cid:34) ∆ − (cid:32) η η (cid:33) κ (cid:35) η ,U ds = 4 (cid:126) G ∆ c (cid:0) ∆ − κ (cid:1) ∆ (∆ + κ ) η η , (3)where G ≡ G ae = G be , ∆ ≡ ∆ = ∆ , and ∆ c ≡ ω p j − ω j . The single-particle part of the effective Hamil-tonian density has the familiar form of the equal Rashba-Dresselhaus SO coupling: H (1)SO = − (cid:126) m (cid:104) ∇ ⊥ − ( − i∂ z + k R σ z ) (cid:105) + V ext ( r )+ 12 (cid:126) δσ z + (cid:126) Ω R σ x , (4)with the effective two-photon detuning and Raman cou- pling given by (see Appendix B) δ = 2 G (∆ − κ )∆(∆ + κ ) ( η − η ) , Ω R = 2 G (∆ − κ )∆(∆ + κ ) (cid:20) − G ∆ c ∆(∆ − κ ) (cid:21) η η . (5)Before proceeding further, consider briefly some realis-tic order-of-magnitude estimates for various parametersused in the theory based on current experiments in ul-tracold atomic gases and cavity QED. The first exper-imental realization of a synthetic SO coupling was car-ried out on Rb atoms using two counter-propagatingRaman laser beams with wavelength λ R = 804 . E R = 2 . × − J) [8]; the two-body interactionstrengths for the desired pseudospin states of Rb atomsare reported to be g = 5 . × − Jm and g = g =4 . × − Jm . With typical average BEC densities¯ n of order 10 − m − [2], one obtains g τ ¯ n/E R ∼ V U τ /g τ ∼
1. Most experimentalwork is focused on the strong-cavity limit, where G (cid:29) κ ;typical atom-cavity coupling and cavity decay rates for Rb are G ae ∼ G be ∼ κ ∼ π ×
10 MHz [32, 33].One can attain
V U τ /g τ ∼ ∼
26 THz, η = η = − ∆ c = 10 MHz (for example, ∆ c ≈ κ and η ≈ . κ in Ref. 32), and a volume V = 10 − mm ; forthese parameters one also obtains (cid:126) Ω R /E R ∼ × − .The weak coupling regime relevant to the present workcan be attained by increasing the value of κ , for ex-ample by decreasing the reflectivity of the cavity mir-rors. Choosing κ ∼ π ×
100 MHz one can never-theless ensure
V U τ /g ∼ V = 10 − mm as well as stronger pump fields and cav-ity detuning η = η = − c = 3 GHz; these choicesyield (cid:126) Ω R /E R ∼ × − . Further increasing the driv-ing field intensities up to η = η = 15 GHz at the fixed∆ c = − V U τ /g ∼
30 while (cid:126) Ω R /E R ∼ G / [∆(∆ c + iκ )]and G N τ / [∆(∆ c + iκ )] are assumed to be small. Usingthe weak-coupling values considered above and assuminga typical average BEC particle number N τ ∼ , it isstraightforward to verify that both G / | ∆(∆ c + iκ ) | (cid:28) G N τ / | ∆(∆ c + iκ ) | ∼ − (cid:28)
1. Making use of κ (cid:28) ∆ c and defining ξ ≡ G / ∆∆ c (cid:28)
1, one can writeΩ R ≈ ξη η ∆ c ; δ ≈ ξ ∆ c (cid:0) η − η (cid:1) ; U ds ≈ (cid:126) ξ ∆ c η η ; U = U ∓ ( ± ) ≈ (cid:126) ξ ∆ c η . (6)If η = η then (cid:126) δ = 0 and U ds = U = U ∓ ( ± ) with U / (cid:126) Ω R = ξ (cid:28)
1. Alternatively, if both pump fields arenon-zero ( η , η (cid:54) = 0), then defining δU ≡ U − U oneobtains δU/ (cid:126) δ = U ds / (cid:126) Ω R = ξ (cid:28)
1. These relations willbe important below when choosing parameters for thetheoretical calculations.
III. GROUND STATE AND EXCITATIONS:ANALYTICS
The above analysis indicates that as long as η and η are not too different from one another then δ (cid:28) Ω R ; in the following we therefore restrict calculationsto δ (cid:39)
0. The effective single-particle Hamiltoniancan be diagonalized, and expressed in the form H (1)SO = (cid:80) k ,λ = ± (cid:15) λ ( k ) ˆ ϕ † λ ( k ) ˆ ϕ λ ( k ) with single-particle energy dis-persion relation˜ (cid:15) ± (˜ k ) ≡ (cid:15) ± ( k ) E R = ˜ k + 1 ± (cid:113) k z + ˜Ω R , (7a)and spinor eigenstates φ − ( k ) = (cid:18) sin θ k − cos θ k (cid:19) ; φ + ( k ) = (cid:18) cos θ k sin θ k (cid:19) , (7b)where ‘+’ and ‘ − ’ designate the upper and lower band,respectively, and sin 2 θ k = ˜Ω R / (cid:113) k z + ˜Ω R . The unitlessparameters ˜ k = k /k R and ˜Ω R = (cid:126) Ω R /E R are defined forconvenience, where E R = (cid:126) k R / m is the recoil energy.Recall that using experimentally motivated parametersas discussed toward the end of Sec. II, one can choose˜Ω R ∼ O (1). Note that in deriving this result we haveassumed that the condensate is confined in a box poten-tial with negligible occupation of transverse momentumstates, i.e. ˜ k = (0 , , ˜ k z ). In fact, the nature of the trans-verse confinement is not important in the current work;for example, instead assuming a strong radial oscillatorpotential V ( ρ ) = mω ρ ρ / k by ˜ k z + (cid:126) ω ρ /E R under the assumption that the conden-sate occupied the ground state of the radial oscillator.The energy dispersion with respect to ˜ k z consists oftwo bands with a band gap of 2 ˜Ω R at the origin ˜ k = 0.The lower energy band ˜ (cid:15) − (˜ k ) is a symmetric double wellalong the ˜ k z direction with the two minima located at˜ k z = ± ˜ k ≡ ± (cid:113) − ˜Ω R / , (8)for ˜Ω R <
2, and it has a single minimum at ˜ k z = 0when ˜Ω R > k ⊥ = 0). The operators ˆ Φ ( k ) =( ˆ ϕ + ( k ) , ˆ ϕ − ( k )) T annihilate a boson at momentum k inthe upper and lower bands and are related to the field op-erators through ˆ Ψ ( r ) = (cid:80) k ,λ = ± e i k · r φ λ ( k ) ˆ ϕ λ ( k ). Note that the laboratory-frame bosonic field operators ˜ Ψ ( r )(which gives the observable atomic density distribution)are related to ˆ Ψ ( r ) by the unity transformation U , i.e.˜ Ψ ( r ) = U † ˆ Ψ ( r ).The single-particle ground state of the symmetric dou-ble well (i.e. when ˜Ω R <
2) is two-fold degenerate; theatom is either in the left minimum at ˜ k = − ˜ k =(0 , , − ˜ k ) or the right minimum at ˜ k = ˜ k = (0 , , ˜ k ).The non-interacting N -particle ground state, when thecavity-mediated interactions are also absent, is therefore( N + 1)-fold degenerate (any number of pseudospin-upatoms, up to N , can reside in the left well). Nonetheless,the two-body and cavity-mediated interactions competewith each other to lift this degeneracy. A. Variational Approach
In order to determine the nature of the ground state,we assume the following ansatz for the BEC condensatewavefunction, (cid:20) ψ ψ (cid:21) = √ ¯ n (cid:26) c e − ik z (cid:20) cos θ k − sin θ k (cid:21) + c e ik z (cid:20) sin θ k − cos θ k (cid:21)(cid:27) (9)where k = k R ˜ k and ¯ n = N/V is the average particledensity, with N and V being the total particle numberand volume, respectively. The variational parameters are c and c with the normalization constraint | c | + | c | =1. Once they are determined, one can find the relevantground-state quantities such as the total density n ( r ) = | ψ ( r ) | + | ψ ( r ) | , and the magnetization per particle s z ( r ) = [ | ψ ( r ) | − | ψ ( r ) | ] / ¯ n : n ( r ) = ¯ n [1 + 2 | c c | cos(2 k z + γ ) sin 2 θ k ] , (10) s z ( r ) = (cid:0) | c | − | c | (cid:1) cos 2 θ k , (11)where γ is the relative phase between c and c . Notethat the magnetization s z is homogeneous while the to-tal density n ( r ) exhibits fringes in the z direction pro-vided that c c (cid:54) = 0. Constraining ˜Ω R <
2, one can writesin 2 θ k = ˜Ω R / θ k = ˜ k ; then these take thesimpler form n ( z ) = ¯ n (cid:104) R | c c | cos(2 k z + γ ) (cid:105) and s z = ˜ k (cid:0) | c | − (cid:1) . The energy functional E [ c , c ] = E + E int is obtained from Eq. (2) by replacing the fieldoperators ˆ ψ τ with the corresponding condensate wave-functions ψ τ . This yields E = − N E R ˜Ω R / E int = N | g | V (cid:40) sgn( g ) + ˜ g + 4 ˜ U + 2 δ ˜ U − U ds ˜Ω R + (cid:104) g − sgn( g ) − ˜ g + 4 (cid:16) ˜ U ss − ˜ U (cid:17) − δ ˜ U (cid:105) ˜Ω R
8+ 12 (cid:0) | c | − | c | (cid:1) (cid:16) − ˜Ω R (cid:17) / (cid:2) sgn( g ) − ˜ g − δU (cid:3) − | c c | (cid:34) sgn( g ) + ˜ g + 4 ˜ U + 2 δ ˜ U − g − (cid:16) g ) + 3˜ g + 8 ˜ U + 4 δ ˜ U − g (cid:17) ˜Ω R (cid:35)(cid:41) , (12)where the two-body interaction strengths are rescaled by | g | (for example ˜ g = g / | g | ) and the cavity-mediatedinteraction strengths are rescaled by | g | /V (for exam-ple ˜ U = V U / | g | ). In the above equations we havedefined 2 ˜ U ss ≡ ˜ U ± + ˜ U ∓ and δ ˜ U ≡ ˜ U − ˜ U , andsgn( g ) = g / | g | = ± g . Again,recall that using experimentally motivated parameters asdiscussed toward the end of Sec. II, ˜ U ∼ ˜Ω R ∼ O (1). E is the single-particle contribution to the energy andis independent of c i , as expected. Minimizing E int withrespect to c i determines the ground state of the system.The parameters ˜ U and δ ˜ U (or ˜ U ) are the only cavity-mediated interaction parameters having an effect on theground state.Consider first the simplest case where ˜ g = sgn( g )and δ ˜ U = 0, so that only that last line of Eq. (12) con-tributes to the interaction energy. Then the energy isminimized either with ( c , c ) = (1 ,
0) or (0 , c = c = 1 / √ − k or k ), la-beled the plane wave phase (PWP). In the PWP the to-tal density is uniform. The magnetization takes the value s z = ± ˜ k = ± (1 − ˜Ω R / / , with the upper (lower) signcorresponding to c = 1 ( c = 0). For small ˜Ω R themagnetization approaches unity. Note that the PWP istwofold degenerate; that is, all atoms can condense inthe left ( c = 1) or right minimum ( c = 1). The sec-ond solution set corresponds to atoms condensing intoa superposition state of plane waves. It is characterizedby the broken translational symmetry and the resultingdensity n ( z ) = n [1 + ˜Ω R cos(2 k z + γ )] exhibits spatialvariations in the z (i.e. SO-coupling) direction, so this isreferred to as the stripe phase (SP). In this phase the den-sity oscillations have greatest contrast for large ˜Ω R → s z is zero.The SP solution yields a lower energy than the PWPsolution when term in square brackets in the last line ofEq. (12) is positive. (Recall ˜ g = sgn( g ) and δ ˜ U = 0 sothat the middle line vanishes identically.) The cavity in-teraction strength that favors the SP solution is therefore ˜ U > ˜ U , where˜ U ≡ g − sgn( g )] − [˜ g − g )] ˜Ω R − ˜Ω R ) , (13)is the critical cavity interaction for the SP-PWP tran-sition. In the limit of small ˜Ω R , this becomes ˜ U (cid:39) [˜ g − sgn( g )] + [˜ g + sgn( g )] ˜Ω R . If ˜ g = sgn( g )the SP is favored for any non-zero, positive cavity inter-action in the limit ˜Ω R →
0. In the other hand when˜Ω R → g (cid:54) = − sgn( g ), the critical cavity interac-tion ˜ U diverges and SP is only favored for very largepositive cavity interaction.It is important to verify that the total interaction en-ergy, Eq. (12), remains positive; the system is stableonly if ∂ E int /∂N >
0. Let us examine this first inthe SP where c = c = 1 / √
2, for a special case where˜ U ds = ˜ U ss = ˜ U (and ˜ g = sgn( g ) and δ ˜ U = 0 as before).One obtains E int = N | g | V (cid:26)
18 [˜ g + sgn( g )] (cid:16) R (cid:17) + 12 ˜ U (cid:16) − ˜Ω R (cid:17) (cid:27) . (14)Surprisingly, the SP is energetically stable for two-component attractive BECs in the presence of spin-orbit interactions as long as the inter-species interactionstrength is sufficiently large and positive. Substitutingthe critical cavity interaction ˜ U into Eq. (14) yields theconstraint ˜ g (cid:62) sgn( g ) ˜Ω R (cid:104) (2 − ˜Ω R ) − (cid:105) ˜Ω R + 16 . (15)In the limit of ˜Ω R →
0, for the lowest possible val-ues of the cavity interaction favoring the SP phase ˜ U (cid:38) ˜ U = [˜ g − sgn( g )], the SP is energetically stableas long as ˜ g (cid:62)
0, with no constraint on the sign ofthe intra-species interaction strength. Thus, the infinite-range cavity-mediated atom-atom interactions stabilizeattractive two-component BECs against collapse, even inthe absence of a confining potential. For larger values of˜ U even the inter-species interactions can be attractive.The coefficient of ˜ U in Eq. (14) is strictly positive.Therefore, for a given parameter set { sgn( g ) , ˜ g , ˜Ω R } one can choose arbitrary large positive values of the cav-ity interaction strength to strongly favor SP without com-promising stability (i.e. to satisfy ˜ U > ˜ U while ensur-ing that E int (cid:62) U which favors a stable SP satisfies˜ U > max − [˜ g + sgn( g )] (cid:16) R (cid:17) (cid:16) − ˜Ω R (cid:17) , ˜ U . (16)The stability of PWP can be investigated in a similarmanner. The plane wave phase is favored when ˜ U < ˜ U .The positivity constraint of the interaction energy in thePWP E int = N | g | V (cid:26) sgn( g ) + 18 [˜ g − sgn( g )] ˜Ω R + ˜ U (cid:16) − ˜Ω R (cid:17) (cid:27) > , (17)imposes a lower bound in the cavity interaction − g ) + [˜ g − sgn( g )] ˜Ω R (cid:16) − ˜Ω R (cid:17) < ˜ U < ˜ U , (18)beyond which PWP is unstable. Thus, even the PWPbecomes energetically stable for attractive spin-orbit cou-pled two-component BECs if the cavity-mediated inter-actions are judiciously chosen.Figure 1 depicts the phase diagrams in the { ˜ U , ˜Ω R } and { ˜ U , ˜ g } parameter planes. The phase diagrams arecomprised of two physical regions: the SP and PWP, de-noted by black and white in Fig. 1, respectively. Thedark (light) grey indicates the regions where the SP(PWP) is energetically unstable. Figure 1(a) showsthe phase diagram in the { ˜ U , ˜Ω R } parameter space forsgn( g ) = ˜ g = 1 and different values of ˜ g . The stripephase is favored over an ever-larger parameter space as˜ U increases as long as | ˜Ω R | < { ˜ U , ˜ g } parameter plane for sgn( g ) = ˜ g = − R = 0 .
1, where Eq. (13) reveals that thephase boundary is linear in ˜ g for fixed ˜Ω R .Relaxing the constraint considered above that δ ˜ U = 0in Eq. (12), one can prepare any arbitrary superpositionstate, i.e. arbitrary c and c subject to | c | + | c | = 1.The plane-wave phase is no longer degenerate; rather, theminimum favored depends on the sign of δ ˜ U . Figure 2shows the dependence of | c | in the { ˜ U , ˜Ω R } plane forsgn( g ) = ˜ g = δ ˜ U = 1, and ˜ g = 2. Under these condi-tons the SP with | c | = | c | is found only for very large˜ U (cid:29) ˜ U , i.e. far from the SP-PWP phase boundary˜ U . Whereas for ˜ U → ˜ U +1c , | c | increases monotonicallyuntil the PWP with | c | = 1 is attained for ˜ U < ˜ U (note that the critical value ˜ U (cid:39) ˜ U and is weaklydependent on δ ˜ U , as discussed below). The plane-wave g (cid:142) (cid:61) (cid:142) (cid:61) (cid:142) (cid:61) PWPSP (cid:45) (cid:87)(cid:142) R U (cid:142) (a) PWPSP (cid:45) g (cid:142) U (cid:142) (b) FIG. 1: Phase diagrams in the (a) { ˜ U , ˜Ω R } and (b) { ˜ U , ˜ g } parameter planes. The stripe and plane-wave phases are de-noted by back and white, respectively; dark (light) grey in-dicates the regions where the SP (PWP) is unstable. (a)Phase diagram for sgn( g ) = ˜ g = 1 and different values of˜ g = 0 .
1, 1, and 2. (b) Phase diagram for sgn( g ) = ˜ g = − R = 0 . phase begins to be unstable in the left bottom corner ofthis figure.The magnetization s z = ˜ k (cid:0) | c | − (cid:1) as a functionof ˜ U is illustrated with the black solid curve in Fig. 3for sgn( g ) = ˜ g = δ ˜ U = 1, ˜ g = 2, and ˜Ω R = 0 .
1. Forcontrast, the magnetization when δ ˜ U = 0 is also shown(blue dashed curve). Note that while the sign of themagnetization in the PWP is arbitrary for the δ ˜ U = 0case (a spontaneously broken symmetry in the groundstate), in the present case the sign of s z always followsthat of δ ˜ U .
On the PWP side, the magnetization is fixedat its maximal value s z = ˜ k ; for ˜ U (cid:38) ˜ U on the SPside, the magnetization decreases sharply before reachingan asymptotic value deep within the SP phase.For small δ ˜ U and ˜Ω R , the SP-PWP phase transitionoccurs at almost the same value of the critical cavityinteraction ˜ U = 0 . δ ˜ U = 0. Near the phase transition point on the SPside, one can write c = 1 − x and c = √ x , where x (cid:28) c + c (cid:39) O ( x ). Setting the term proportionalto x in E int [ c = 1 , c = 0] − E int [ c = 1 − x , c = √ x ]equal to zero yields a modified critical cavity interaction˜ U = ˜ U − (cid:34) − (4 − ˜Ω R ) / − ˜Ω R − ˜Ω R (cid:35) δ ˜ U . (19)In the small ˜Ω R limit this may be simplified to ˜ U (cid:39) [˜ g − sgn( g )] + [˜ g + sgn( g ) + δ ˜ U ] ˜Ω R , which is thesame critical cavity interaction ˜ U obtained above in thesmall ˜Ω R limit, save for the δ ˜ U -dependent correction.The behavior of the magnetization for ˜ U > ˜ U sug-gests that one can define the order parameter for thestripe phase to be P = 1 − s z / ˜ k = 2(1 − c ). As desired,this vanishes in the PWP (here we only consider a PWPwith momentum − k ) and takes a nonzero value in SP.The order parameter is shown in the inset of Fig. 3. The FIG. 2: (Color online) Density plot of | c | in the { ˜ U , ˜Ω R } parameter plane for sgn( g ) = ˜ g = δ ˜ U = 1, and ˜ g = 2.The plane-wave phase begins to be unstable in the left bottomcorner. discontinuity in the derivative of P with ˜ U suggests thatthe SP-PWP quantum (zero-temperature) phase transi-tion is second order. It is therefore of interest to deter-mine the (mean-field) exponent β for the order parame-ter P in the vicinity of the transition point. Substituting˜ U = ˜ U + χ into the energy functional E int and mini-mizing it with respect to c yields c = (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) δ ˜ U (cid:16) − ˜Ω R (cid:17) / + χ (cid:16) − ˜Ω R (cid:17) δ ˜ U (cid:16) − ˜Ω R (cid:17) / + 2 χ (cid:16) − ˜Ω R (cid:17) . (20)The order parameter P = 2(1 − c ) computed using thisexpression for c is illustrated as the green dashed curvein the the inset of Fig. 3, and is in excellent agreementwith the numerical results of the variational approach,shown as the black solid curve. Taylor expanding c inEq. (20) for small χ and ˜Ω R up to first and second order,respectively, one obtains c MF1 (cid:39) − χ/ δ ˜ U (the term pro-portional to χ ˜Ω R is also omitted). This yields the mean-field order parameter P MF = 2 χ/δ ˜ U = 2( ˜ U − ˜ U ) β /δ ˜ U and a critical exponent β = 1. The behavior of the orderparameter near the transition point fits well to P , as isshown by the orange dashed curve in the inset of Fig. 3.In principle, it is not valid to consider δ ˜ U (cid:54) = 0 while atthe same time assuming that ˜ δ ≡ (cid:126) δ/E R = 0. Rather,if η (cid:54) = η (cid:54) = 0 but η ∼ η , then Eqs. (6) state that˜ δ ∼ δ ˜ U whenever ˜ U ∼ ˜Ω R . That said, in Fig. 3 theparameters are chosen so that ˜Ω R = 0 . (cid:28) δ ˜ U = 1. Onecan therefore expect ˜ δ (cid:28) δ ˜ U by a similar ratio, whichagain justifies neglecting it.Consider briefly the effect of keeping a non-zero butsmall value of ˜ δ . The single-particle dispersions of the U (cid:142) s z U (cid:142) P FIG. 3: (Color online) The magnetization s z as a function of˜ U shown as the black solid curve for sgn( g ) = ˜ g = δ ˜ U = 1,˜ g = 2, and ˜Ω R = 0 .
1. The blue dashed curve represents themagnetization when δ ˜ U = 0. The red dotted curves are themagnetization computed from solutions of the coupled Gross-Pitaevskii equations in the SP and PWP assuming ˜ U ss =˜ U ds = ˜ U for the same parameters as the solid black curve,and | g | ¯ n/E R = 1. Inset: the SP order parameter P is shownas a function of ˜ U (black curve); an analytical approximation(dashed green curve) and the behavior near the critical point(orange dashed curve) are shown for comparison. spin-orbit Hamiltonian (4) become˜ (cid:15) ± (˜ k ) = ˜ k z + 1 ± (cid:114) (cid:16) k z + ˜ δ (cid:17) + ˜Ω R (21)rather than the expressions given in Eq. (7a). The asso-ciated (orthogonal) eigenvectors have the same form asEqs. (7b) but now sin 2 θ k = ˜Ω R / (cid:114) (cid:16) k z + ˜ δ (cid:17) + ˜Ω R .For ˜ δ (cid:54) = 0, the lower double-well dispersion curve ˜ (cid:15) − is nolonger symmetric; rather, the right well is lower (higher)when ˜ δ > δ < δ precludesa simple form like Eq. (8) for the location of the energyminima, but in the limit when both ˜Ω R (cid:28) δ (cid:28) k ≈ − ˜Ω R (cid:32) − ˜ δ (cid:33) . (22)The lowest-order contribution of ˜ δ is a correction to thecoefficient of the already small ˜Ω R -dependent term, andtherefore the value of ˜ k is well-approximated by assum-ing ˜ δ = 0. Likewise, the BEC approximation consistsof ˜ k z with ˜ k ; because 4˜ k z → k ≈ (cid:29) ˜ δ in the ex-pressions for the single-particle energies and eigenvectorsabove, ˜ δ can be similarly neglected in the calculations. B. Coupled Gross-Pitaevskii Equations
While the variational calculation discussed in the pre-vious section has revealed that a ring cavity can stabi-lize stripe phases in interacting spin-orbit coupled Bose-Einstein condensates, it is important to verify the re-sults using a more rigorous approach. In this section, thecoupled Gross-Pitaevskii (GP) equations are derived forboth PWP and SP ans¨atze and the ground state proper-ties are obtained from their solutions.
1. Plane wave phase
The GP equations can be obtained directly from themany-particle Hamiltonian (2): (cid:20) (cid:126) m ˆ (cid:52) + g | ψ | + g | ψ | + 2 U N + 2 U ds S x (cid:21) ψ + (cid:2) (cid:126) Ω R + 2 U ss S − + U ds N (cid:3) ψ = µψ , (cid:20) (cid:126) m ˆ (cid:52) + g | ψ | + g | ψ | + 2 U N + 2 U ds S x (cid:21) ψ + (cid:2) (cid:126) Ω R + 2 U ss S + + U ds N (cid:3) ψ = µψ , (23)where ˆ (cid:52) = −∇ ⊥ + ( − i∂ z + k R ) and ˆ (cid:52) = −∇ ⊥ +( − i∂ z − k R ) and the BEC wavefunctions for the two spincomponents are denoted by ψ rather than ψ ( r )to save space. These equations can be simplified in theplane-wave phase by assuming homogeneous wavefunc-tions ψ τ ( r ) = e ± ik z ¯ ψ τ , where the upper (lower) signcorresponds to a condensate in the right (left) minimum.The GP equations are then recast as (cid:104) ˜ µ − (˜ k ± (cid:105) ¯ ψ − ˜Ω R ¯ ψ = | g | E R (cid:26) (cid:104)(cid:16) sgn( g ) + 2 ˜ U (cid:17) | ¯ ψ | + (cid:16) ˜ g + 2 ˜ U ss (cid:17) | ¯ ψ | (cid:105) ¯ ψ + ˜ U ds (cid:0) | ¯ ψ | + | ¯ ψ | (cid:1) ¯ ψ + ˜ U ds ¯ ψ ¯ ψ ∗ (cid:27) ; (cid:104) ˜ µ − (˜ k ∓ (cid:105) ¯ ψ − ˜Ω R ¯ ψ = | g | E R (cid:26) (cid:104)(cid:16) ˜ g + 2 ˜ U (cid:17) | ¯ ψ | + (cid:16) ˜ g + 2 ˜ U ss (cid:17) | ¯ ψ | (cid:105) ¯ ψ + ˜ U ds (cid:0) | ¯ ψ | + 2 | ¯ ψ | (cid:1) ¯ ψ + ˜ U ds ¯ ψ ¯ ψ ∗ (cid:27) , (24)where again the upper (lower) sign in each equation cor-responds to a condensate in the right (left) minimum,and the chemical potential is expressed in recoil energyunits, ˜ µ ≡ µ/E R .The chemical potential can be obtained from the firstof Eqs. (24) and then substituted into the second. Underthe assumption that both condensate wavefunctions are real, ˜ U = ˜ U ss = ˜ U ds , and sgn( g ) = ˜ g , one obtains | g | E R (cid:104)(cid:0) ˜ g − sgn( g ) (cid:1) (cid:0) ¯ ψ − ¯ ψ (cid:1) ¯ ψ ¯ ψ + ˜ U (cid:0) ¯ ψ − ¯ ψ (cid:1)(cid:105) ± k ¯ ψ ¯ ψ + ˜Ω R (cid:0) ¯ ψ − ¯ ψ (cid:1) = 0 . (25)For the plane-wave phase, both ¯ ψ and ¯ ψ are assumedto be constant, so that ¯ ψ + ¯ ψ = ¯ n and ¯ ψ − ¯ ψ = ¯ ns z .Inserting these into Eq. (25) gives (cid:112) − s z (cid:20) ∓ k + s z | g | ¯ nE R (cid:16) ˜ g − sgn( g ) (cid:17)(cid:21) + 2 s z (cid:18) ˜Ω R + ˜ U | g | ¯ nE R (cid:19) = 0 . (26)When ˜ U = 0 and ˜Ω R ≈
0, this expression is approx-imately correct when s z ≈
1, consistent with the vari-ational results in this regime. Recall that in the varia-tional approach, the magnetization s z = ˜ k is constant[c.f. Eq. (11)], solely determined by ˜Ω R . Unlike the vari-ational result, however, it is immediately apparent fromthe second term in Eq. (26) that the magnetization mustdecrease monotonically as ˜ U is increased.The magnetization s z obtained via numerical solutionof Eq. (26) is shown as the red dotted curve in Fig. 3 for acondensate in the left well (i.e. choosing the lower sign) ofthe PWP for ˜ U ≤ ˜ U . Parameters are ˜ U = ˜ U ss = ˜ U ds ,sgn( g ) = ˜ g = | g | ¯ n/E R = δ ˜ U = 1, ˜ g = 2, and˜Ω R = 0 .
1. As expected, the magnetization decreasesmonotonically with ˜ U from its maximum at ˜ U = 0.The difference between the results of the two methodshas its origins in the fact that the variational ansatz,Eq. (9), is a single-particle wavefunction which satisfiesthe GP equations in PWP only when all the two-bodyand cavity-mediated interactions are zero. In principle,the variational ansatz could be remedied by allowing both k and θ k to be variational parameters [12]. The depen-dence of the solution of GP equations on the two-bodyand cavity-mediated interactions will be investigated fur-ther in Sec. III C 1 in the calculation of the elementaryexcitations in the PWP.
2. Stripe phase
The momentum dependence of the condensate in theSP is not as readily apparent as it is for the PWP. Itis therefore convenient to instead construct an effectivelow energy Hamiltonian by first mapping the completeHamiltonian (2) into the lower band and then derivingthe low energy coupled GP equations [14, 34]. This isreasonable because the occupation of the upper band (cid:15) + ( k ) can be assumed to be small at low temperatures k B T (cid:28) (cid:126) Ω R . Furthermore, only states in the vicinity ofthe two minima ± ˜ k will be occupied.The field operators ˆ Ψ ( r ) can then be expanded in thelower band basis around the two minima (recall that φ − ( k ) is the two-component spinor in the lower band):ˆ Ψ ( r ) (cid:39) (cid:88) q < q c (cid:104) e i ( − k + q ) · r φ − ( − k + q ) ˆ ϕ − ( − k + q )+ e i ( k + q ) · r φ − ( k + q ) ˆ ϕ − ( k + q ) (cid:105) , (27)where the sum over q need only be taken up to somemaximum q c . Approximating the spinor φ − ( ± k + q ) (cid:39) φ − ( ± k ) in the limit ˜Ω R (cid:28) ϕ (cid:48) ( q ) ≡ ˆ ϕ − ( − k + q ) and ˆ ϕ (cid:48) ( q ) ≡ ˆ ϕ − ( k + q ) [14],the field operators readˆ Ψ ( r ) = e − i k · r φ − ( − k ) ˆ ψ (cid:48) ( r ) + e i k · r φ − ( k ) ˆ ψ (cid:48) ( r ) , (28)where ˆ ψ τ (cid:48) ( r ) = (cid:80) q e i q · r ˆ ϕ τ (cid:48) ( q ). In the small ˜Ω R limitand keeping terms only up to second order in ˜Ω R andnoting that k (cid:39) (1 − ˜Ω R / k R , the field operators canbe further simplified to (cid:20) ˆ ψ ( r )ˆ ψ ( r ) (cid:21) (cid:39) (cid:34) (1 − ˜Ω R ) e − ik z ˜Ω R e ik z − ˜Ω R e − ik z − (1 − ˜Ω R ) e ik z (cid:35) (cid:20) ˆ ψ (cid:48) ( r )ˆ ψ (cid:48) ( r ) (cid:21) . (29)Note that the lab-frame pseudospin field operator ˆ˜ ψ τ maps correctly to the corresponding dressed pseudospinfield operator ˆ ψ τ (cid:48) in the ˜Ω R → Ψ ( r ) = U † ˆ Ψ ( r ). Substituting Eq. (29) back into theoriginal Hamiltonian (2) and only keeping terms to sec-ond order in ˜Ω R yields the effective low-energy Hamilto-nian: H e = (cid:90) d r (cid:18) ˆ Ψ (cid:48)† H (1)e ˆ Ψ (cid:48) + 12 g (cid:48) ˆ n (cid:48) + 12 g (cid:48) ˆ n (cid:48) + g (cid:48) ˆ n (cid:48) ˆ n (cid:48) (cid:19) + 12 U (cid:48) ˆ N (cid:48) + 12 U (cid:48) ˆ N (cid:48) + U (cid:48) ˆ N (cid:48) ˆ N (cid:48) , (30)where ˆ Ψ (cid:48) ( r ) = ( ˆ ψ (cid:48) ( r ) , ˆ ψ (cid:48) ( r )) T , as before ˆ N τ (cid:48) = (cid:82) ˆ n τ (cid:48) ( r ) d r = (cid:82) ˆ ψ † τ (cid:48) ( r ) ˆ ψ τ (cid:48) ( r ) d r is the total atomic num-ber operator for the dressed pseudospin τ (cid:48) ∈ { (cid:48) , (cid:48) } , andwe have introduced the dressed interaction parameters g (cid:48) τ ≡ g τ (cid:48) τ (cid:48) = g τ −
18 ( g τ − g ) ˜Ω R ,g (cid:48) ≡ g (cid:48) (cid:48) = g + 18 ( g + g ) ˜Ω R ,U (cid:48) τ ≡ U τ (cid:48) τ (cid:48) = 2 U τ − U ds ˜Ω R −
14 ( U τ − U ss ) ˜Ω R ,U (cid:48) ≡ U (cid:48) (cid:48) = − U ds ˜Ω R + 18 ( U + U + 2 U ss ) ˜Ω R , (31)with τ ∈ { , } and τ (cid:48) ∈ { (cid:48) , (cid:48) } .The single-particle part of the effective low energyHamiltonian H (1)e = ( − (cid:126) / m )[ ∇ ⊥ +(1 − ˜Ω R / ∂ z ] can beeasily diagonalized [14], yielding the effective low energy dispersion (cid:15) e ( k ) /E R = ˜ k ⊥ + (1 − ˜Ω R / k z . It is impor-tant to note that the lowest single-particle energy statefor both dressed pseudospins is the k = 0 momentumstate, not k = ± k as it was for the actual pseudospins.Then the effective low energy GP equations for the SPcan be obtained from H e , Eq. (30): (cid:104)(cid:16) ˜ g (cid:48) + ˜ U (cid:48) (cid:17) | ψ (cid:48) | + (cid:16) ˜ g (cid:48) + ˜ U (cid:48) (cid:17) | ψ (cid:48) | (cid:105) ψ (cid:48) = ¯ µψ (cid:48) , (cid:104)(cid:16) ˜ g (cid:48) + ˜ U (cid:48) (cid:17) | ψ (cid:48) | + (cid:16) ˜ g (cid:48) + ˜ U (cid:48) (cid:17) | ψ (cid:48) | (cid:105) ψ (cid:48) = ¯ µψ (cid:48) , (32)where the dressed pseudospin wavefunctions ψ τ (cid:48) are as-sumed to be homogeneous and unitless parameters havebeen introduced for convenience: ˜ g (cid:48) τ = g (cid:48) τ / | g | , ˜ g (cid:48) = g (cid:48) / | g | , ˜ U (cid:48) τ = V U (cid:48) τ / | g | , and ˜ U (cid:48) = V U (cid:48) / | g | . Here¯ µ = µ/ | g | which has units of inverse volume. Thesealgebraic equations have the solution n (cid:48) = 2 ˜ U + ˜ g (cid:48) − ˜ g (cid:48) − (cid:16) ˜ U + 3 ˜ U (cid:17) ˜Ω R ˜ g (cid:48) + ˜ g (cid:48) − g (cid:48) + 2 (cid:16) ˜ U + ˜ U (cid:17) (cid:16) − ˜Ω R (cid:17) ¯ n,n (cid:48) = 2 ˜ U + ˜ g (cid:48) − ˜ g (cid:48) − (cid:16) U + ˜ U (cid:17) ˜Ω R ˜ g (cid:48) + ˜ g (cid:48) − g (cid:48) + 2 (cid:16) ˜ U + ˜ U (cid:17) (cid:16) − ˜Ω R (cid:17) ¯ n, (33)where n (cid:48) + n (cid:48) = ¯ n . Note that although the GP equa-tions for the SP, Eq. (32), depend on the cavity parame-ters ˜ U ss and ˜ U ds , these solutions do not; rather, ˜ U and ˜ U are the only cavity interaction parameters that affect ψ τ (cid:48) ,consistent with the variational approach of Sec. III A.The dressed magnetization s (cid:48) z = ( n (cid:48) − n (cid:48) ) / ¯ n can eas-ily be obtained from Eq. (33), and the actual magneti-zation s z = s (cid:48) z (1 − ˜Ω R /
8) up to O ( ˜Ω R ) is found usingEq. (29): s z = (cid:104) ˜ g (cid:48) − ˜ g (cid:48) + 2 δ ˜ U (cid:16) − ˜Ω R (cid:17)(cid:105) (cid:16) − ˜Ω R (cid:17) ˜ g (cid:48) + ˜ g (cid:48) − g (cid:48) + 2 (cid:16) ˜ U + ˜ U (cid:17) (cid:16) − ˜Ω R (cid:17) . (34)The SP magnetization s z is displayed as a functionof ˜ U ( (cid:62) ˜ U ) in Fig. 3 with the red dotted curve forsgn( g ) = ˜ g = δ ˜ U = 1, ˜ g = 2, and ˜Ω R = 0 .
1. Thebehavior is indistinguishable from the magnetization ob-tained from the variational approach, Eq. (11). The crit-ical cavity interaction for the SP-PWP phase transitioncan be obtained from Eq. (33) by setting n (cid:48) = ¯ n (orsetting s (cid:48) z = 1):˜ U L1c = 14(4 − ˜Ω R ) (cid:26) − (cid:104) ˜ g − sgn( g ) − g − δ ˜ U (cid:105) ˜Ω R + 8 [˜ g − sgn( g )] (cid:27) , (35)for a phase transition from SP to a PWP at the left min-imum. Instead setting n (cid:48) = 0 (or s (cid:48) z = −
1) for a phase0transition from SP to a PWP at the right minimum, oneobtains˜ U R1c = 14(4 − ˜Ω R ) (cid:26) − (cid:104) ˜ g − sgn( g ) − g − δ ˜ U (cid:105) ˜Ω R + 8 (cid:104) ˜ g − sgn( g ) − δ ˜ U (cid:105) (cid:27) . (36)Note that when sgn( g ) = ˜ g and δ ˜ U = 0, the two criticalcavity interactions ˜ U L1c and ˜ U R1c become equal to the value˜ U found using the variational approach, Eq. (13). C. Elementary Excitations: Bogoliubov theory
Thus far we have treated the bosons as classical fields,having replaced the field operators with their expecta-tion values ˆ ψ τ → ψ τ ≡ (cid:104) ˆ ψ τ (cid:105) . In this section, we considerthe quantum fluctuations of the fields and obtain theelementary excitation spectrum using Bogoliubov the-ory. This is accomplished by writing the field operatorsas ˆ ψ τ = ψ τ + δ ˆ ψ τ , where δ ˆ ψ τ is the quantum fluctu-ation operator. These expressions are substituted intothe time-dependent GP equations and the resulting equa-tions are linearized, i.e. terms are retained only up to firstorder in the fluctuations. One then obtains a set of time-dependent coupled equations for δ ˆ ψ τ which yields, afterdiagonalization, the elementary excitation spectrum.
1. Plane wave phase
Following the approach taken in Sec. III B 1 for thePWP, it is reasonable to define the bosonic field operatorˆ ψ τ ( r , t ) ≡ e ± ik z (cid:104) ¯ ψ τ + δ ˆ ψ τ ( r , t ) (cid:105) , (37)where ¯ ψ τ are the time-independent, homogeneous solu-tions of the coupled GP equations (24) in the PWP. Toconsider time-dependent fluctuations around the equi-librium solutions it is convenient to replace the chem-ical potential (which is the eigenvalue of the time-independent GP equations) by a time-dependent oper-ator, µ → µ + i (cid:126) ∂ t . The time-dependent fluctuations canthen be expressed using the usual Bogoliubov approachin terms of particle and hole excitations with amplitudes¯ u τ, q e i ( q · r − ωt ) and ¯ v ∗ τ, q e − i ( q · r − ωt ) , respectively.Consider the specific case of a condensate in the leftminimum − ˜k of the double-well single-particle disper-sion relation; for condensation in the right well one needonly replace ˜ k in what follows with − ˜ k . Substitut-ing Eq. (37) into the time-dependent GP equations andkeeping only linear terms in the fluctuations, one obtainsthe following non-Hermitian eigenvalue equation for eachvalue of q : M g ¯ ψ g ¯ ψ ¯ ψ ∗ + (cid:126) Ω eff g ¯ ψ ¯ ψ − g ¯ ψ ∗ − M − g ¯ ψ ∗ ¯ ψ ∗ − g ¯ ψ ∗ ¯ ψ − (cid:126) Ω ∗ eff g ¯ ψ ∗ ¯ ψ + (cid:126) Ω ∗ eff g ¯ ψ ¯ ψ M g ¯ ψ − g ¯ ψ ∗ ¯ ψ ∗ − g ¯ ψ ¯ ψ ∗ − (cid:126) Ω eff − g ¯ ψ ∗ − M ¯ u , q ¯ v , q ¯ u , q ¯ v , q = (cid:126) ω ( q ) ¯ u , q ¯ v , q ¯ u , q ¯ v , q , (38)where M / = E R (cid:104) ˜ q ∓ k − q z (cid:105) + g | ¯ ψ | − (cid:126) Ω eff ¯ ψ ¯ ψ ,M / = E R (cid:104) ˜ q ∓ k + 1)˜ q z (cid:105) + g | ¯ ψ | − (cid:126) Ω ∗ eff ¯ ψ ¯ ψ , (cid:126) Ω eff = (cid:126) Ω R + | g | ˜ U ds ¯ n + 2 | g | ˜ U ss ¯ ψ ¯ ψ ∗ . (39)In deriving the Bogoliubov Hamiltonian (38), we madeuse of the fact that ˆ N τ = (cid:82) ˆ ψ † τ ( r , t ) ˆ ψ τ ( r , t ) d r = (cid:82) | ¯ ψ τ | d r = V | ¯ ψ τ | = N τ , because ¯ ψ τ is homogeneousby assumption and (cid:82) δ ˆ ψ τ ( r , t ) d r = 0 because the spatialintegral of either Bogoliubov amplitude ¯ u τ, q e i ( q · r − ωt ) or¯ v ∗ τ, q e − i ( q · r − ωt ) is zero for any q (cid:54) = 0. A similar argumentensures that ˆ S + = S + and ˆ S − = S − as well. Note alsothat the chemical potential in Eq. (38) has been elimi-nated using the coupled GP equations (24). Diagonalizing Eq. (38) yields the spectrum ω PW ± ( q ) ofcollective excitations. The results are shown in Fig. 4(a)for the parameters sgn( g ) = ˜ g = | g | ¯ n/E R = 1,˜ g = 2, and ˜Ω R = 0 .
1, when all the cavity-mediatedinteraction terms are zero ( ˜ U = ˜ U = ˜ U ss = ˜ U ds = 0),i.e. the system is deep in the PWP. The lower curve ex-hibits the usual superfluid sound-like linear dispersionaround the origin ˜ q z ≡ q z /k R = 0 (around the left min-imum of the single-particle energy dispersion where allthe atoms are condensed) and a roton-type minimumaround ˜ q z (cid:39)
2. As the cavity interactions are increased,the energy of the roton minimum lowers. For parame-ters ˜ U = 0 . δ ˜ U = 1 .
5, ˜ U ss = ˜ U ds = 0, and the otherparameters same as in Fig. 4(a), this minimum coincideswith zero energy (i.e. the excitation energy at the ori-gin ˜ q z = 0); see the black solid curve in Fig. 4(b). Thered dashed-dotted curve represents the elementary exci-tation spectrum for the same values of ˜ U and δ ˜ U but1 (cid:45) q (cid:142) z (cid:209) Ω (cid:177) PW (cid:72) q (cid:76) (cid:144) E R (a) (cid:45) q (cid:142) z (cid:209) Ω (cid:177) PW (cid:72) q (cid:76) (cid:144) E R (b) FIG. 4: (Color online) Elementary excitation spectrum in thePWP for sgn( g ) = ˜ g = | g | ¯ n/E R = 1, ˜ g = 2, and ˜Ω R =0 .
1. ( ˜ U , ˜ U , ˜ U ss , ˜ U ds ) = (0 , , ,
0) in (a), and (0 . , . , , . , . , . , .
5) in (b) for the black solid and red dashed-dotted curves, respectively. for ˜ U ss = ˜ U ds = 0 .
5. In this case, (cid:126) Ω eff /E R [cf. Eq. (39)]is somewhat bigger than the bare ˜Ω R = 0 . q z (cid:39) q z (cid:39) k R canbe reduced below zero by further increasing the cavityinteraction strength ˜ U . This signals a dynamic instabil-ity toward the formation of the SP; recall from Eq. (10)that the density modulation in the SP has wave vector2 k (cid:39) k R for ˜Ω R →
0. The critical cavity interactionsfor the black solid and the red dashed-dotted excitationspectra in Fig. 4(b) are ˜ U (cid:39) . .
53, respectively, and these are in good agreement with that of the varia-tional approach, where Eq. (19) predicts a phase transi-tion between the PWP and the SP at the critical value˜ U (cid:39) . g ) = ˜ g = δ ˜ U = 1,˜ g = 2, and ˜Ω R = 0 . U ss = ˜ U ds = 0 in the PWP,then the critical cavity interaction ˜ U obtained from theanalysis of the elementary excitations and the variationalmethod would match exactly with each other for anyrange of parameters. Nevertheless, they begin to devi-ate from one another as ˜ U ss and ˜ U ds become larger andlarger, because Eq. (19) is independent of these cavityinteraction parameters while both the coupled GP equa-tions and the Bogoliubov Hamiltonian depend explicitlyon them (the latter through (cid:126) Ω eff ). That said, we havecompared the critical phase transition point ˜ U obtainedfrom both the variational approach and the elementaryexcitation spectrum in the PWP and have found thatwhen ˜ U = ˜ U ss = ˜ U ds they agree with one another withina ∼
8% error for ˜ g in the range of ∼ −
8, assumingsgn( g ) = ˜ g = | g | ¯ n/E R = δ ˜ U = 1 and ˜Ω R = 0 .
2. Stripe phase
The derivation of the Bogoliubov excitation spectrumbegins with the corresponding time-dependent, effectivelow energy GP equations in the SP [c.f. Eq. (32)]: i (cid:126) ∂∂t ˆ ψ (cid:48) = (cid:16) H (1)e + g (cid:48) | ˆ ψ (cid:48) | + g (cid:48) | ˆ ψ (cid:48) | + U (cid:48) ˆ N (cid:48) + U (cid:48) ˆ N (cid:48) − µ (cid:17) ˆ ψ (cid:48) ,i (cid:126) ∂∂t ˆ ψ (cid:48) = (cid:16) H (1)e + g (cid:48) | ˆ ψ (cid:48) | + g (cid:48) | ˆ ψ (cid:48) | + U (cid:48) ˆ N (cid:48) + U (cid:48) ˆ N (cid:48) − µ (cid:17) ˆ ψ (cid:48) . (40)As in the PWP case, the low energy field operators are replaced with ˆ ψ τ (cid:48) ( r , t ) = ψ τ (cid:48) + δ ˆ ψ τ (cid:48) ( r , t ) in these equations.Here ψ τ (cid:48) are the time-independent, homogeneous solutions of the effective low energy GP equations in the SP, Eq. (33),and δ ˆ ψ τ (cid:48) ( r , t ) are the quantum fluctuations. Linearizing Eq. (40) yields the Bogoliubov Hamiltonian: (cid:15) e ( q ) + g (cid:48) | ψ (cid:48) | g (cid:48) ψ (cid:48) g (cid:48) ψ (cid:48) ψ ∗ (cid:48) g (cid:48) ψ (cid:48) ψ (cid:48) − g (cid:48) ψ ∗ (cid:48) − (cid:15) e ( q ) − g (cid:48) | ψ (cid:48) | − g (cid:48) ψ ∗ (cid:48) ψ ∗ (cid:48) − g (cid:48) ψ ∗ (cid:48) ψ (cid:48) g (cid:48) ψ ∗ (cid:48) ψ (cid:48) g (cid:48) ψ (cid:48) ψ (cid:48) (cid:15) e ( q ) + g (cid:48) | ψ (cid:48) | g (cid:48) ψ (cid:48) − g (cid:48) ψ ∗ (cid:48) ψ ∗ (cid:48) − g (cid:48) ψ (cid:48) ψ ∗ (cid:48) − g (cid:48) ψ ∗ (cid:48) − (cid:15) e ( q ) − g (cid:48) | ψ (cid:48) | u (cid:48) , q v (cid:48) , q u (cid:48) , q v (cid:48) , q = (cid:126) ω ( q ) u (cid:48) , q v (cid:48) , q u (cid:48) , q v (cid:48) , q , (41)which can be diagonalized to give the spectrum of theelementary excitations: (cid:126) ω SP ± ( q ) = (cid:115) (cid:15) ( q ) + (cid:15) e ( q ) (cid:18) D ± (cid:113) D − D (cid:19) , (42) with D = g (cid:48) n (cid:48) + g (cid:48) n (cid:48) ,D = ( g (cid:48) g (cid:48) − g (cid:48) ) n (cid:48) n (cid:48) . (43)We have again used the fact that ˆ N τ (cid:48) = N τ (cid:48) .Surprisingly, the Bogoliubov Hamiltonian in the SPdoes not depend explicitly on the cavity parameters andthe form of the excitation spectrum coincides with the2 U (cid:142) v (cid:166) (cid:72) (cid:177) (cid:76) (cid:72) mm (cid:144) s (cid:76) U (cid:142) v (cid:166) (cid:72) (cid:45) (cid:76) U (cid:142) v (cid:166) (cid:72) (cid:43) (cid:76) FIG. 5: (Color online) The speed of sound in the transversedirection v ( ± ) ⊥ is shown as a function of ˜ U for ˜ U = 1 / U = 5 / R = 0 .
4, sgn( g ) = ˜ g = 1, ˜ g = 0 . g ¯ n/E R = 1, and m isthe mass of Rb atom. The insets show the results closer tothe origin. quasiparticle spectrum of a Raman-induced stripe phaseBEC [14]. That said, the excitation spectrum implicitlydepends on the cavity parameters ˜ U (cid:48) τ through n τ (cid:48) , as canbe seen in Eq. (33). Both ω SP ± ( q ) are gapless and exhibitlinear dispersion at long wavelengths, the characteristicof superfluidity in this phase; the slope of the dispersionrelation at long wavelength corresponds to the speed ofsound in the medium. In the transverse direction, oneobtains v ( ± ) ⊥ = dω SP ± ( q ) dq ⊥ (cid:12)(cid:12)(cid:12) q → = 1 √ m (cid:114) D ± (cid:113) D − D , (44)and the speed of sound in the z (SO-coupling) directionis nearly the same for small ˜Ω, v ( ± ) z = v ( ± ) ⊥ (cid:113) − ˜Ω R / v ( ± ) ⊥ as a function of ˜ U for ˜ U = 1 / U = 5 / g ) = ˜ g = 1, ˜ g = 0 . R = 0 .
4, and g ¯ n/E R = 1. The mass is assumed to bethat of Rb. As ˜ U is increased above zero, the speedof sound in the positive branch v (+) ⊥ (the blue curves)first decreases quickly and reaches a minimum around δ ˜ U = ˜ U − ˜ U ∼ v ( − ) ⊥ (black curves) has the opposite be-havior, first increasing sharply to a maximum again near δ ˜ U ∼ v ( ± ) ⊥ closeto the origin. The asymptotic behaviour of the speed ofsound can be understood by noting that for large positive˜ U (cid:29) ˜ U , n (cid:48) approaches ¯ n and n (cid:48) approaches zero [c.f.Eqs. (33)]. As a consequence D → v ( − ) ⊥ → v (+) ⊥ → (cid:112) g (cid:48) ¯ n/m . For the solid curves (where ˜ U = 1 / v ( − ) ⊥ becomes zero at ˜ U (cid:39)
37, consistent with the value at which thedressed magnetization s (cid:48) z becomes unity for this choiceof parameters. This signifies an instability toward theformation of a different phase.The condition that the speed of sound must be non-negative imposes the constraint D (cid:62)
0. This conditionmarks the onset of a phase transition at the critical point˜ g (cid:48) (c)12 = (cid:112) ˜ g (cid:48) ˜ g (cid:48) , which does not depend on any cavity-mediated interaction parameters and is solely determinedby the two-body interactions and ˜Ω R . This critical pointis not consistent with the previous results obtained fromthe variational approach, the effective low-energy GPequations in the SP, or the elementary excitations inthe PW which all consistently predict a critical pointfor the PWP-SP phase transition that depends on thecavity-mediated interaction parameters. To verify thatthere was not an error in the calculations, the elementaryexcitations were computed directly in momentum spaceby Fourier transferring the effective low-energy Hamilto-nian (30), and treating the fluctuations around the con-densate ϕ τ (cid:48) ( q = 0) to second order in ˆ ϕ τ (cid:48) ( q ) for smallmomenta q . The results were identical with the real-space analysis, Eq. (42). Interestingly, the critical inter-species interaction ˜ g (cid:48) (c)12 above defines a phase boundarybetween the stripe phase and a phase-separated statein Raman-induced spin-orbit coupled BECs [14]. It istherefore conceivable that there is another phase betweenthe SP and the PWP induced by the cavity interactions,whose signature is the observed inconsistency in the crit-ical point. IV. DISCUSSION AND CONCLUSIONS
In this work we have shown that cavity-mediated long-ranged interactions between atoms can profoundly al-ter the nature of the ground state and the elemen-tary excitations of a cavity-induced spin-orbit-coupledtwo-component BEC, for ring-type cavities in the weak-coupling regime. Specifically, experimentally tunablecavity-mediated interactions compete with the standardtwo-body interactions to yield both plane-wave andstripe phase ground states. Indeed, positive long-rangecavity interactions can stabilize fully attractive BECs(condensates where intra-species collisional interactionsare negative, independent of the sign of the inter-speciesinteraction) against collapse in the stripe phase. The col-lective excitations of the plane-wave phase ground statesare found to have a distinctive roton-type excitation spec-trum reminiscent of that of superfluid He, which can beused as a signature of the phase. The stripe phase has astandard linear dispersion relation; the associated speedof sound is found to go negative at a critical value ofthe cavity interaction strength, signalling an instabilitytoward another (likely phase-separated) phase. The re-sults suggest that cavity QED, even in the weak-couplingregime, can yield interesting new physics for spin-orbitcoupled BECs.3The results raise interesting avenues for future inves-tigations. This work assumed a fictional experimentalconfiguration where the momentum is a good quantumnumber in the direction of the applied spin-orbit inter-actions. In reality the condensate would be confined inthis direction, and even a weak harmonic potential couldchange the physics. While the stripe phase would likelyremain robust, as it is essentially a weak standing wavesuperimposed on the background condensate density pro-file, the plane-wave phase has no analog in a confinedgeometry. Another loose end is the nature of the phasehinted at in the limit of a large difference δ ˜ U betweenthe cavity-mediated interactions between the two kindsof spin components ˜ U and ˜ U . For large δ ˜ U , the soundvelocity in the stripe phase was found to go negative, asignature of the dynamic instability of the phase. Whileother work suggests that this signals aHowever, a few intriguing issues and questions remainunclear and deserve further investigations. These in-clude the inconsistency in the critical phase transitionpoint, how the combined SO coupling effect, the two-body interactions, and the cavity-mediated long-ranged interactions change the superfluid–Mott-insulator phasetransition as well as the nature of magnetic orders inthe Mott-insulating regime when an optical lattice im-posed inside the cavity. Furthermore, whether it is pos-sible to have a superfluid–Mott-insulator-like phase tran-sition with solely the cavity-mediated long-range interac-tions, whether there is more interesting physics in strong-coupling regime, and how the cavity fields are affected bythe atoms. Some of these questions are the subject of ourcurrent works with some promising preliminary resultsand will be published elsewhere. Acknowledgments
The authors are grateful to Paul Barclay andChristoph Simon for constructive criticisms. We alsothank Han Pu and Lin Dong for stimulating correspon-dence. This work was supported by the Natural Sciencesand Engineering Research Council of Canada and Al-berta Innovates-Technology Futures.
Appendix A: Adiabatic Elimination of the Atomic Excited State
We first express the single-particle Hamiltonian density H (1) , Eq. (1), in the rotating frame of pump lasers [29] byapplying the unitary transformation U = exp (cid:110) i (cid:104)(cid:16) ˆ A † ˆ A − σ aa (cid:17) ω p1 + (cid:16) ˆ A † ˆ A − σ bb (cid:17) ω p2 (cid:105) t (cid:111) , to obtain H (cid:48) (1) = (cid:20) p m + V ext ( r ) (cid:21) I × + (cid:126) δ (cid:48) σ aa − σ bb ) − (cid:126) a1 + ∆ a2 ) σ ee + (cid:126) (cid:104)(cid:16) G ae e ik R z ˆ A σ ea + G be e − ik R z ˆ A σ eb (cid:17) + H.c. (cid:105) − (cid:126) (cid:0) ∆ c1 ˆ A † ˆ A + ∆ c2 ˆ A † ˆ A (cid:1) + i (cid:126) (cid:104)(cid:0) η ˆ A † + η ˆ A † (cid:1) − H.c. (cid:105) , (A1)where we have defined the atomic and the two-photon (or relative-atomic) detunings∆ a1 = ω p1 − (cid:126) ( ε e − ε a ) , ∆ a2 = ω p2 − (cid:126) ( ε e − ε b ) , δ (cid:48) = ( ω p1 − ω p2 ) − (cid:126) ( ε b − ε a ) = ∆ a1 − ∆ a2 , (A2a)and cavity detunings ∆ c j = ω p j − ω j , j = 1 , , (A2b)with respect to the pump lasers. Let us now assume that the detunings ∆ = ω − ε ea / (cid:126) = − ∆ c1 + ∆ a1 and∆ = ω − ε eb / (cid:126) = − ∆ c2 + ∆ a2 are large compared to ε ba / (cid:126) = ( ε b − ε a ) / (cid:126) so that we can adiabatically eliminatethe dynamic of the atomic excited state | e (cid:105) from the Hamiltonian (A1) and obtain an effective Hamiltonian for theground pseudospins { , } ≡ { b, a } . Following the standard adiabatic elimination procedure [23, 35], we first find theHeisenberg equations of motion i (cid:126) ˙ σ eτ = [ σ eτ , H (cid:48) (1) ] for ˙ σ ea and ˙ σ eb , and then (after transferring to slowly rotatingvariables) set them equal to zero to find the steady-state solutions σ (ss) ea and σ (ss) eb . After substituting these steady-statesolutions back in H (cid:48) (1) (A1) and dropping terms diagonal in σ ee , we arrive at the single-particle Hamiltonian densityfor pseudospins H (cid:48) (1)SO = (cid:20) p m + V ext ( r ) (cid:21) I + ˆ ε σ + ˆ ε σ + (cid:126) Ω (cid:48) R (cid:16) e ik R z ˆ A † ˆ A σ + e − ik R z ˆ A † ˆ A σ (cid:17) + H (cid:48) cav , (A3)4where H (cid:48) cav = − (cid:126) (cid:0) ∆ c1 ˆ A † ˆ A + ∆ c2 ˆ A † ˆ A (cid:1) + i (cid:126) (cid:104)(cid:16) η ˆ A † + η ˆ A † (cid:17) − H.c. (cid:105) , and ˆ ε = − (cid:126) δ (cid:48) (cid:126) G be ∆ ( ˆ A † ˆ A + 12 ) , ˆ ε = (cid:126) δ (cid:48) (cid:126) G ae ∆ ( ˆ A † ˆ A + 12 ) . (A4)Here, Ω (cid:48) R = ∆ +∆ ∆ ∆ G ae G be is the two-photon Rabi frequency and I ≡ I × is the identity matrix in the pseudospinspace. Note the hat on ˆ ε τ , implying that it depends on the cavity field operators. After transferring to the co-movingframe of the cavity modes by applying the unitary transformation U = e − ik R ( σ − σ ) z to the Hamiltonian density(A3), we obtain the SO-coupled single-particle Hamiltonian density H (cid:48)(cid:48) (1)SO = 12 m (cid:110) p ⊥ I + (cid:2) p z I − (cid:126) k R ( σ − σ ) (cid:3) (cid:111) + V ext ( r ) I + (cid:88) τ =1 , ˆ ε τ σ ττ + (cid:126) Ω (cid:48) R (cid:16) ˆ A † ˆ A σ + ˆ A † ˆ A σ (cid:17) + H (cid:48) cav . (A5)One can identify (cid:126) k R ( σ − σ ) with eA ∗ z /c as in the minimal coupling Hamiltonian, that is, eA ∗ z /c ≡ (cid:126) k R ( σ − σ ) = − (cid:126) k R σ z , where σ z = σ − σ is the third Pauli matrix. Nonetheless, we emphasis that here A ∗ z is a matrix acting inthe internal pseudospin states, in contrast to the ordinary vector potential whose components are scaler fields. Thenthe single-particle Hamiltonian reads H (cid:48)(cid:48) (1)SO = 12 m (cid:90) ˆ Ψ † (cid:2) p ⊥ I + ( p z I + (cid:126) k R σ z ) + V ext ( r ) I (cid:3) ˆ Ψ d r + (cid:88) τ =1 , ˆ ε τ ˆ N τ + (cid:126) Ω (cid:48) R (cid:16) ˆ A † ˆ A ˆ S + + ˆ A † ˆ A ˆ S − (cid:17) + H (cid:48) cav , (A6)where ˆ Ψ ( r ) = ( ˆ ψ ( r ) , ˆ ψ ( r )) T are the bosonic field operators, ˆ N τ = (cid:82) ˆ ψ † τ ( r ) ˆ ψ τ ( r ) d r is the total atomic numberoperator for pseudospin τ , ˆ N = ˆ N + ˆ N is the total atomic number operator, and ˆ S + = ˆ S †− = (cid:82) ˆ ψ † ( r ) ˆ ψ ( r ) d r arethe collective pseudospin raising and lowering operators. Appendix B: Adiabatic Elimination of the Cavity Fields
By noting that the cavity field operator commutes with the atomic interaction Hamiltonian [ ˆ
A, H int ] = 0, then theHeisenberg equations of motion of the cavity field operators are determined by the single-particle Hamiltonian H (cid:48)(cid:48) (1)SO ,Eq. (A6): ∂ t ˆ A j = − i [ ˆ A j , H (cid:48)(cid:48) (1)SO ] / (cid:126) − κ ˆ A j , where the cavity-mode decay − κ ˆ A j is included phenomenologically. Theycan be recast in the matrix form, ddt (cid:18) ˆ A ˆ A (cid:19) = i (cid:18) ˆ α − ˆ α − ˆ α ˆ α (cid:19) (cid:18) ˆ A ˆ A (cid:19) + (cid:18) η η (cid:19) , (B1)where the elements of the ”operator” matrix ˆ α are given byˆ α = (∆ c1 + iκ ) − G ae ∆ ˆ N , ˆ α = (∆ c2 + iκ ) − G be ∆ ˆ N , ˆ α = ˆ α † = Ω (cid:48) R ˆ S − . (B2)If the cavity decay rate κ is large, then the cavity fields reach steady states very quickly. By setting ∂ t ˆ A = ∂ t ˆ A = 0in Eq. (B1), one can simultaneously solve the two equations of motion to obtain formal expressions for the steady-statefield amplitudes ˆ A ss j . However, one should take special care in solving these equations since the cavity fields andatomic fields commute with one another and this can give rise to ambiguities in solving these equations. In order toavoid such ambiguities, we symmetrize the equations of motion and exercise symmetrization procedure in all resultsfollowing from the equations of motion. Thus, after setting ∂ t ˆ A = ∂ t ˆ A = 0 in Eq. (B1), we re-express equations ofmotion as i (cid:16) ˆ α ˆ A ss1 + ˆ A ss1 ˆ α (cid:17) − i (cid:16) ˆ α ˆ A ss2 + ˆ A ss2 ˆ α (cid:17) + η = i (cid:16) ˆ α ˆ A ss2 + ˆ A ss2 ˆ α (cid:17) − i (cid:16) ˆ α ˆ A ss1 + ˆ A ss1 ˆ α (cid:17) + η = 0 . (B3)5Equation (B3) can then be rearrangedˆ A ss1 = 14 (cid:104)(cid:0) ˆ α − ˆ α + ˆ α ˆ α − (cid:1) ˆ A ss2 + ˆ A ss2 (cid:0) ˆ α − ˆ α + ˆ α ˆ α − (cid:1)(cid:105) + i ˆ α − η , (B4a)ˆ A ss2 = 14 (cid:104)(cid:0) ˆ α − ˆ α + ˆ α ˆ α − (cid:1) ˆ A ss1 + ˆ A ss1 (cid:0) ˆ α − ˆ α + ˆ α ˆ α − (cid:1)(cid:105) + i ˆ α − η , (B4b)where ˆ α − and ˆ α − are the inverse operators of ˆ α and ˆ α , respectively, such that ˆ α ˆ α − = ˆ α − ˆ α = ˆ1 andˆ α ˆ α − = ˆ α − ˆ α = ˆ1. In order to make the subsequent analyses somewhat easier and trackable, we assume that alldual variables (except η j at this moment) are equal, namely, ∆ = ∆ ≡ ∆, ∆ c1 = ∆ c2 ≡ ∆ c , and G ae = G be ≡ G .We also introduce ˜∆ c ≡ ∆ c + iκ for a shorthand. We expand the inverse operators to the second order in a smallunitless parameter ξ ≡ G / ∆ ˜∆ c (cid:28) (cid:104) ˆ N τ (cid:105) ∼ one still has ξ (cid:104) ˆ N τ (cid:105) ∼ − (cid:28)
1, see Sec. II for moredetails), ˆ α − = (cid:18) ˜∆ c − G ∆ ˆ N (cid:19) − (cid:39) ˜∆ − (cid:18) G ∆ ˜∆ c ˆ N + 4 G ∆ ˜∆ ˆ N (cid:19) , ˆ α − = (cid:18) ˜∆ c − G ∆ ˆ N (cid:19) − (cid:39) ˜∆ − (cid:18) G ∆ ˜∆ c ˆ N + 4 G ∆ ˜∆ ˆ N (cid:19) , (B5)such that ˆ α ˆ α − = ˆ α − ˆ α = ˆ α ˆ α − = ˆ α − ˆ α = ˆ1 + O ( ξ ). Note that the error in symmetrizing Eq. (B4) is alsoof order O ( ξ ). This can be easily checked by substituting, say, Eq. (B4a) in the first equation of (B3). Equations(B4a) and (B4b) can now be simultaneously solved, yieldingˆ A ss1 = i ˆΓ − (cid:104) η ˆ α − + η (cid:0) ˆ α − ˆ α ˆ α − + ˆ α ˆ α − ˆ α − + ˆ α − ˆ α − ˆ α + ˆ α − ˆ α ˆ α − (cid:1)(cid:105) , ˆ A ss2 = i ˆΓ − (cid:104) η ˆ α − + η (cid:0) ˆ α − ˆ α ˆ α − + ˆ α ˆ α − ˆ α − + ˆ α − ˆ α − ˆ α + ˆ α − ˆ α ˆ α − (cid:1)(cid:105) , (B6)where ˆΓ = (cid:104) −
12 ˜∆ (ˆ α ˆ α + ˆ α ˆ α ) (cid:105) up to ξ , by noting ˆ α = ˆ α † ∝ Ω (cid:48) R = 2 G / ∆ and (B5). We then haveˆΓ − (cid:39) (ˆ α ˆ α + ˆ α ˆ α ) = 1 + 2 G ∆ ˜∆ (cid:16) ˆ S + ˆ S − + ˆ S − ˆ S + (cid:17) , (B7)up to O ( ξ ). Using Eqs. (B2), (B5)-(B7), and retaining terms up to ξ , we obtainˆ A ss1 = i ˜∆ c (cid:26) η + 2 G ∆ ˜∆ c (cid:16) η ˆ N + η ˆ S − (cid:17) + 4 G ∆ ˜∆ (cid:104) η ˆ N + η (cid:16) ˆ S + ˆ S − + ˆ S − ˆ S + (cid:17) + η ˆ N ˆ S − (cid:105)(cid:27) , ˆ A ss2 = i ˜∆ c (cid:26) η + 2 G ∆ ˜∆ c (cid:16) η ˆ N + η ˆ S + (cid:17) + 4 G ∆ ˜∆ (cid:104) η ˆ N + η (cid:16) ˆ S + ˆ S − + ˆ S − ˆ S + (cid:17) + η ˆ N ˆ S + (cid:105)(cid:27) . (B8)By substituting steady-state solutions (B8) and their Hermitian conjugates in the Hamiltonian H (cid:48)(cid:48) (1)SO , Eq. (A6),exercising symmetrization procedure again and retaining terms up to ξ , we can find an effective Hamiltonian whichdepends solely on the atomic operators. After some tedious though straightforward algebra, we obtain the cavity-field-eliminated effective many-body Hamiltonian H eff = (cid:90) d r (cid:18) ˆ Ψ † H (1)SO ˆ Ψ + 12 g ˆ n + 12 g ˆ n + g ˆ n ˆ n (cid:19) + (cid:88) τ =1 , U τ ˆ N τ + (cid:16) U ± ˆ S + ˆ S − + U ∓ ˆ S − ˆ S + (cid:17) + 2 U ds ˆ N ˆ S x , (B9)where the cavity-field-eliminated, effective single-particle Hamiltonian density takes the familiar form H (1)SO = − (cid:126) m [ ∇ ⊥ − ( − i∂ z + k R σ z ) ] + V ext ( r ) + 12 (cid:126) δσ z + (cid:126) Ω R σ x , (B10)with effective two-photon detuning and Raman coupling given by δ ≡ G (∆ − κ )∆(∆ + κ ) ( η − η ) , Ω R = 2 G ∆(∆ + κ ) (cid:18) ∆ − κ − G ∆ c ∆ (cid:19) η η = Ω (cid:48) R (∆ + κ ) (cid:18) ∆ − κ − G ∆ c ∆ (cid:19) η η . (B11)6(Note that δ (cid:48) = 0, since we have assumed ∆ a1 = ∆ a2 ≡ ∆ a ; cf. Eqs. (A2) and (A4).) The coefficients of thecavity-mediated long-range interactions are found to be U = 4 (cid:126) G ∆ c (∆ − κ )∆ (∆ + κ ) η , U = 4 (cid:126) G ∆ c (∆ − κ )∆ (∆ + κ ) η , U ds = 4 (cid:126) G ∆ c (cid:0) ∆ − κ (cid:1) ∆ (∆ + κ ) η η ,U ± = 4 (cid:126) G ∆ c ∆ (∆ + κ ) (cid:2) ∆ η − ( η + 2 η ) κ (cid:3) , U ∓ = 4 (cid:126) G ∆ c ∆ (∆ + κ ) (cid:2) ∆ η − ( η + 2 η ) κ (cid:3) . (B12)The terms with coefficients U / , U ± / ∓ , and U ds in the effective Hamiltonian (B9) are the cavity-mediated long-rangeinteractions. Note that in the special case of η = η ≡ η , one has δ = 0 and U = U = U ± = U ∓ = U ds ≡ U . [1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E.Wieman, and E. A. Cornell, Science , 198 (1995).[2] K. Davis, M.-O. Mewes, M. R. Andrews, N. J. vanDruten, D. S. Durfee, D. M. Kurn, and W. Ketterle,Phys. Rev. Lett. , 3969 (1995).[3] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. , 885 (2008).[4] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski,A. Sen(De), and U. Sen, Advances in Physics , 243(2007).[5] M. Greiner, O. Mandel, T. Esslinger, T. W. H¨ansch, andI. Bloch, Nature , 39 (2002).[6] J. Dalibard, F. Gerbier, G. Juzeli¯unas, and P. ¨Ohberg,Rev. Mod. Phys. , 1523 (2011).[7] Y.-J. Lin, R. L. Compton, K. Jim´enez-Garca, J. V. Porto,and I. B. Spielman, Nature , 628 (2009).[8] Y.-J. Lin, K. Jim´enez-Garcia, and I. B. Spielman, Nature , 83 (2011).[9] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011).[10] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev.Lett. , 160403 (2010).[11] T.-L. Ho and S. Zhang, Phys. Rev. Lett. , 150403(2011).[12] Y. Li, L. P. Pitaevskii, and S. Stringari, Phys. Rev. Lett. , 225301 (2012).[13] W. Zheng and Z. Li, Phys. Rev. A , 053607 (2012).[14] Q.-Q. L¨u and D. E. Sheehy, Phys. Rev. A , 043645(2013).[15] R. Liao, O. Fialko, J. Brand, and U. Z¨ulicke, preprintarXiv:1504.07370 (2015).[16] W. S. Cole, S. Zhang, A. Paramekanti, and N. Trivedi,Phys. Rev. Lett. , 085302 (2012).[17] J. Radi´c, A. D. Ciolo, K. Sun, and V. Galitski, Phys.Rev. Lett. , 085303 (2012).[18] Z. Cai, X. Zhou, and C. Wu, Phys. Rev. A , 061605(R)(2012).[19] T. Graß, K. Saha, K. Sengupta, and M. Lewenstein, Phys. Rev. A , 053632 (2011).[20] H. J. Kimble, Physica Scripta T76 , 127 (1998).[21] H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger,Rev. Mod. Phys. , 553 (2013).[22] K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger,Nature , 1301 (2010).[23] F. Mivehvar and D. L. Feder, Phys. Rev. A , 013803(2014).[24] L. Dong, L. Zhou, B. Wu, B. Ramachandhran, and H. Pu,Phys. Rev. A , 011602(R) (2014).[25] Y. Deng, J. Cheng, H. Jing, , and S. Yi, Phys. Rev. Lett. , 143007 (2014).[26] L. Dong, C. Zhu, and H. Pu, e-print: arXiv:1504.01729(2015).[27] B. Padhi and S. Ghosh, Phys. Rev. A , 023627 (2014).[28] The applied frequencies are assumed to be ω = ω + ∆ ω with | ∆ ω | /ω j (cid:28)
1. In the general case when k (cid:54) = k ,one can define k c ≡ ( k + k ) and ∆ k c ≡ k − k .All the results still hold, except that p z is replaced by p z − (cid:126) ∆ k c after transferring to the co-moving frame ofthe cavity modes, which is just a Galilean transformationof the momentum.[29] C. Maschler, I. Mekhov, and H. Ritsch, Eur. Phys. J. D , 545 (2008).[30] A. L. Gaunt, T. F. Schmidutz, I. Gotlibovych, R. P.Smith, and Z. Hadzibabic, Phys. Rev. Lett. , 200406(2013).[31] P. Meystre and M. Sargent, Elements of Quantum Op-tics, 3rd. ed. (Springer, Berlin, 1999).[32] P. M¨unstermann, T. Fischer, P. Maunz, P. W. H. Pinkse,and G. Rempe, Phys. Rev. Lett. , 4068 (2000).[33] F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. K¨ohl,and T. Esslinger, Nature , 268 (2007).[34] T. D. Stanescu, B. Anderson, and V. Galitski, Phys. Rev.A , 023616 (2008).[35] C. C. Gerry and J. H. Eberly, Phys. Rev. A42