Cavity-induced spin-orbit coupling in an interacting bosonic wire

Cavity-induced spin-orbit coupling in an interacting bosonic wire

Catalin-Mihai Halati HISKP, University of Bonn, Nussallee 14-16, 53115 Bonn, Germany Physikalisches Institut, University of Bonn, Nussallee 12, 53115 Bonn, Germany    Ameneh Sheikhan Corresponding author: a_sheikhan@sbu.ac.ir. Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran HISKP, University of Bonn, Nussallee 14-16, 53115 Bonn, Germany    Corinna Kollath HISKP, University of Bonn, Nussallee 14-16, 53115 Bonn, Germany Physikalisches Institut, University of Bonn, Nussallee 12, 53115 Bonn, Germany
July 20, 2019
Abstract

We consider theoretically ultra-cold interacting bosonic atoms confined to a wire geometry and coupled to the field of an optical cavity. A spin-orbit coupling is induced via Raman transitions employing a cavity mode and a transverse running wave pump beam, the transition imprints a spatial dependent phase onto the atomic wavefunction. Adiabatic elimination of the cavity field leads to an effective Hamiltonian for the atomic degrees of freedom, with a self-consistency condition. We map the spin-orbit coupled bosonic wire to a bosonic ladder in a magnetic field, by discretizing the spatial dimension. Using the numerical density matrix renormalization group method, we show that in the continuum limit the dynamical stabilization of a Meissner superfluid is possible, for parameters achievable by nowadays experiments.

I Introduction

In the past years the experimental progress in the creation of ultracold atomic gases subjected to synthetic magnetic field or spin-orbit coupling has opened new and exciting possibilities to realize exotic quantum phases such as Meissner phases or topologically non-trivial phases in a well controlled way ZhangZhou2017 (); LinSpielman2009 (); LinSpielmanNature2009 (); StruckSengstock2011 (); MiyakeKetterle2013 (); AidelsburgerBloch2011 (); AtalaBloch2014 (); ManciniFallani2015 (); LiviFallani2016 (). For cold atoms loaded into an optical lattice a pair of Raman beams can be used to induce a tunneling process between neighboring lattice sites during which the wavefunction of atoms accumulates a position dependent phase. The imprint of such a phase is due to the running wave nature of one of the Raman beams and can be interpreted as the analog of an Aharanov-Bohm phase of charged particles in a magnetic field. The spin-orbit coupling of atoms in the continuum has been realized using two-photon Raman transitions which couple internal states of the atoms for bosons LinSpielmanNature2009 (); LinSpielman2011 () and fermions WangZhang2012 (); CheukZwierlein2012 (). The spin-orbit coupling couples the particle’s spin, here corresponding to the internal state, with its momentum, which induces chiral currents or topological effects GalitskiSpielman2013 (); ZhangZhang2016 (). One example is a Meissner state which corresponds to a helical liquid. In this state the spin and the momentum directions of the particles couple to each other, giving rise to the propagation of the two spins in opposite directions inducing a chiral current CornfeldSela2015 (); OregOppen2010 (); LutchynSarma2010 (); JaparidzeMalard2014 (); ColeSau2017 ().

The dynamic generation of gauge fields by a cavity-assisted tunneling has been proposed in recent years for cold atoms subjected to an optical lattice. The artificial magnetic field is induced dynamically via the feedback mechanism between the cavity field and the motion of atoms KollathBrennecke2016 (); BrenneckeKollath2016 (); WolffKollath2016 (); SheikhanKollath2016 (); ZhengCooper2016 (); BallantineKeeling2017 (); HalatiKollath2017 (). Phases for which the cavity mediated spin-orbit coupling plays an important role have been considered for standing-wave cavities DengYi2014 (); DongPu2014 (); PanGuo2015 (); PadhiGhosh2014 (), or ring cavities MivehvarFeder2014 (); MivehvarFeder2015 (); OstermannMivehar2018 (). Experimental steps took place in this direction, by realizing cavity mediated spin-dependent interactions LandiniEsslinger2018 () and spinor self-ordering of bosonic atoms coupled to a cavity KroezeLev2018 (). Theoretically, the steady state diagram has been determined for a ladder geometry in the case of noninteracting fermions BrenneckeKollath2016 (); KollathBrennecke2016 (); WolffKollath2016 () and interacting bosons HalatiKollath2017 (), where states with finite chiral currents have been found, or non-trivial topological properties in two dimensions SheikhanKollath2016 ().

In the present work we consider interacting bosons confined to a one-dimensional wire with a dynamically induced spin-orbit coupling via the coupling to a cavity mode. We consider a set of parameters close to the ones of existing experimental setups and characterize the self-organized states that arise. We show the dynamic stabilization of a state with a persistent chiral spin current, the Meissner superfluid, in the coupled atomic cavity system.

In Sec. II.1 we describe the setup of the interacting bosonic wire placed into the optical cavity and the theoretical model. In Sec. II.2 we derive an effective model for the atomic degrees of freedom and a stability condition by performing the adiabatic elimination of the cavity field. In Sec. II.3 we introduce the discretization which we use in order to simulate the system on a lattice. In Sec. II.4 we present the observables used for the characterization of the Meissner superfluid state. In Sec. III the typical parameters used within the numerical density matrix renormalization group method are given. The dynamical stabilization of the Meissner superfluid as a self-organized state with a finite cavity occupation is presented in Sec. IV.

Ii Model

ii.1 Description of the setup

Figure 1: (a) Sketch of the setup. The bosonic atoms in an optical cavity are confined in a one-dimensional wire. (b) Level scheme of the cavity-induced Raman coupling: and denote two internal states, and the excited internal electronic state. The energy offset between the two states is . The spin-orbit coupling is realized by two Raman processes each of which involve the cavity mode with vacuum Rabi frequency and one of the two transverse running wave pump beam with Rabi frequency . and are the frequencies of the excited state and the cavity mode in the rotating frame.

We study an ultracold bosonic gas placed in an optical cavity confined to a one-dimensional wire (Fig. 1). To create the spin-orbit coupling we use three internal states of an atom, as depicted in Fig. 1(b). The states and differ in energy by , and can be coupled using two balanced Raman transitions, each one involving a transverse running-wave pump laser and a standing-wave cavity mode (Fig. 1). The transverse pump laser beams have the frequencies , the Rabi frequencies and the wave-vectors , along the -direction, where the unit vectors along the three spatial directions are . The difference between the two pump beam frequencies is . The cavity mode has the frequency , vacuum Rabi frequency and the wave-vector . We assume that all the other cavity modes are far detuned from the possible transitions. The detuning between the cavity mode and the first pump beam is tuned such that it is close to the offset, , and the cavity and pump modes are considered to be far detuned from the internal atomic transition to the excited state, i.e. , thus, the excited state population is negligible and can be adiabatically eliminated. In the following we will use the rotating frame with the frequency .

The spin-orbit coupling is obtained via the cavity-induced Raman tunneling. During the Raman transition a spatially dependent phase factor is imprinted onto the atomic wavefunction, where and . This corresponds to a dynamically induced spin-orbit coupling. As the cavity mode does not give a contribution, the imprinted flux is , defining a corresponding length-scale . The strength of the spin-orbit coupling can be varied, in an experimental realization, by tilting the pump beams along the direction.

The effective description for the atoms after the adiabatic elimination of the excited state, derived similarly to Refs. RitschEsslinger2013 (); NagyDomokos2008 (), reads

(1)

describes the cavity mode in the rotating frame, where and and are the bosonic annihilation and creation operators for the cavity mode. The bosonic field operators and are the annihilation and creation operators of the atoms in the wire for the spin state . The length of the one-dimensional system is denoted by . The total number of bosonic atoms is , with the mean inter-particle spacing and the density of the wire . describes the kinetic term for the atoms along the direction of the wire, with the mass of the atoms. The repulsive contact interaction is given in , where the strength of the interaction of the atoms with the same spin is spin independent and denoted by and between atoms with different spin is , (). In nowadays experimental setups an external harmonic trapping potential is often present. We included this effect by adding the term , with . describes the coupling between the atoms and the cavity field. The creation or annihilation of a cavity photon is accompanied by a spin flip. By the spatial dependence of the imprinted phase during the spin flip a dynamically induced spin-orbit coupling for the atoms is realized. In order to prevent a privileged spin state we couple the spin flip in each direction to both the creation and the annihilation operators of the cavity field, by using two pump laser beams DimerCarmichael2007 (). The strength of this process is given by the amplitude , where is the Rabi frequency of the cavity mode. We fix the Rabi frequency of the second pump beam to , in order to realize the balanced Raman scheme.

We describe the dynamics of the coupled cavity atom system by a Lindblad equation. The Lindblad description is neccessary in order to take in addition to the unitary evolution, given by , also the cavity losses into account. The losses are due to the imperfections of the cavity mirrors. The evolution of an arbitrary operator is given by

(2)

with the dissipator , which describes the loss of cavity photons.

ii.2 Adiabatic elimination of the cavity field

In the following, employing the adiabatic elimination of the cavity field RitschEsslinger2013 (); KollathBrennecke2016 (); HalatiKollath2017 (), we derive an effective model for the bosonic atoms. Within this approximation, the cavity field is replaced with its steady state value, computed from the condition . Using Eq. (2) this condition becomes

(3)

This relates the expectation value of the cavity field to the expectation value of by

(4)

The model exhibits a symmetry, associated with the inversion of the sign of both the cavity field, , and of . We choose without loss of generality .

By performing a mean-field decoupling of the atomic and cavity degrees of freedom, we derive the following equations of motion for the atomic operators

(5)

with and the sign in the exponential is positive for the spin state and negative for .

The dynamics given by substituting the expectation value of the cavity field, Eq. (4), into the equations of motion of the bosonic operators, Eq. (II.2), can be effectively described by the following Hamiltonian

(6)

In the effective Hamiltonian, Eq. (6), gives the amplitude of the spin-orbit coupling. has to be computed self-consistently because it depends on the occupation of the cavity field and thus on the expectation value of . We have , with . We will call , loosely, the pump strength, since this is one of the experimental knobs used to tune . The non-trivial solutions of the self-consistency condition require , thus having a positive detuning .

The stability of the non-trivial steady states, obtained as the ground states of the effective model, needs to be investigated. A stability condition can be derived by considering perturbations around the steady state RitschEsslinger2013 (); HalatiKollath2017 (); Tian2016 (). Following an analogous stability analysis as performed in Ref. HalatiKollath2017 (), we get the following stability condition

(7)

ii.3 Discrete lattice model

In order to determine the self-consistent solutions we discretise the spatial dimension with a spacing . This maps the effective Hamiltonian, Eq. (6), to the corresponding model on a lattice. The derived lattice Hamiltonian reads

(8)

The bosonic operators and are the annihilation and creation operators of the atoms where labels the legs of the ladder and the rungs of the ladder. denotes the number of the rungs of the ladder and it is related to the physical size of the system by . The operator is the number operator and . corresponds to a Bose-Hubbard ladder in a magnetic field, where the two spins states represent the two legs of the ladder and spin flip processes are equivalent to the tunneling along the rungs of the ladder. Compared to our previous work from Ref. HalatiKollath2017 () on the bosonic ladder in a magnetic field, here we have additionally a non-local interaction along the rungs of the ladder, given by . This model has been studied previously in Refs. OrignacGiamarchi2001 (); PetrescuHur2015 (); Uchino2016 (); GreschnerVekua2016 (); StrinatiMazza2017 (). The parameters of the ladder model are renormalized such that in the continuum limit, , we would recover the Hamiltonian given by Eq. (6).

The procedure of determining the stable steady states of the model, Eq. (1), consists in four steps: First we perform ground-state searches for the effective discrete model, Eq. (8), using a matrix product state method (MPS), for a fixed discrete spacing and physical parameters of the continuum model: system size , particle number , magnetic flux , and interactions and , while varying . Next we compute the expectation value as a function of and solve the self-consistency equation. The self-consistency condition can be interpreted graphically, using the reformulation

(9)

The self-consistent solution correspond to the crossing of the two curves. The stability of the non-trivial solutions is inferred by comparing the slopes of the left-hand and right-hand sides of Eq. (9), we see from the stability condition, Eq. (7), that a solution is stable if the slope of is smaller than . The last step is to verify the convergence of the results in the continuum limit by considering different values of .

ii.4 Observables in the Meissner phase

In this section we will describe the observables that we use to characterize the non-trivial stable steady states of the system and how we can identify them with the corresponding observables computed on the ladder. We will focus on the properties of the Meissner superfluid, as this can be dynamically stabilized in the cavity, for the parameters considered in this work. The Meissner state is a ”zero momentum” state, as close to the non-interacting regime the bosonic quasiparticles are condensed in the single-particle dispersion minimum at momentum . It corresponds to a helical liquid, where the spin and the momentum directions of the particles lock to each other and with the two spins propagating in opposite directions CornfeldSela2015 (); OregOppen2010 (); LutchynSarma2010 (); JaparidzeMalard2014 (); ColeSau2017 (), giving rise to a chiral current.

We start with the local densities and currents, as their configurations can point towards the Meissner or vortex nature of the chiral phases. The local density , the local current , and the spin flip current are defined and computed as

(10)

with . From the local currents, one can compute global observables as the chiral current , which is a persistent spin current

(11)

The Meissner phase is characterized in an infinite homogeneous system by zero spin flip currents in the bulk and a finite value of the chiral current. The values of the chiral current and of the expectation value of can be computed by approximating the integrals with sums, .

In order to confirm the superfluid nature of the states we can use the single particle correlation and look for the algebraic decay, a characteristic feature of the standard Luttinger liquid paradigm Giamarchibook (). Here we use . In the Meissner superfluid the central charge , which can be interpreted as the number of gapless modes, is , as this phase has a gapless symmetric sector due to the superfluid nature of the state and a gapped antisymmetric sector. The central charge can be extracted from the scaling of the von Neumann entropy of an embedded subsystem of length in a system of length . We compute for a bipartition into two subsystems of length and . For open boundary conditions the von Neumann entropy for the ground state of gapless phases scales as VidalKitaev2003 (); CalabreseCardy2004 (); HolzeyWilczek1994 ()

(12)

where is a non-universal constant and we have neglected oscillatory terms LaflorencieAffleck2006 () due to the finite size of the system and logarithmic corrections AffleckLudwig1991 ().

In the numerical simulations, due to the lattice discretization and the open boundary conditions, we observe oscillations in the local density and currents, which are algebraically decaying away from the boundaries. In order to reduce the influence of these boundary effects on the computed observables, we calculate and from the average around which the oscillations occur, extracted by fitting the density and currents in the center of the system, with the function . We show that in the Meissner phase the amplitude of these oscillations decreases in the continuum limit and vanishes in the thermodynamic limit.

Iii Method

The numerical results presented in this work were obtained using a finite-size density matrix renormalization group (DMRG) algorithm in the matrix product state form White1992 (); Schollwock2005 (); Schollwock2011 (); Hallberg2006 (); Jeckelmann2008 (), using the ITensor Library itensor (). We simulate the discrete model Eq. (8) for ladders with a number of rungs between and , depending on the chosen discretisation , and with a bond dimension up to 1500. The convergence of the method to a sufficient accuracy was assured by comparing results with different truncation errors and bond dimensions. Since we are considering a bosonic model with finite interactions the local Hilbert space is infinite, thus, a cutoff for its dimension is needed. We use a maximal local dimension of three bosons per site, as we are dealing with a low density of particles on the ladder. The higher cutoff of four bosons per site gives consistent results.

Iv Results

iv.1 Identifying the stable stationary states in the homogeneous system

Figure 2: Graphical interpretation of the self-consistency condition for the parameters , , and . The expectation value is represented for (a) and multiple values of , together with an extrapolation to the limit , (b) for multiple system sizes in the continuum limit, and the extrapolation to the thermodynamic limit . The straight (red) lines represent the right-hand side of the self-consistency condition, Eq. (9), which is a linear function with slope . The intersections of the two curves give the solutions of the self-consistency condition. In the insets the extrapolations (a) and (b) of , for , are depicted.
Figure 3: The solutions of the self-consistency equation which are proportional to the cavity field versus the pump strength , for multiple , in the continuum and thermodynamic limits. In the grey area the stability of the solutions is not clear.

In this subsection we solve the self-consistency condition and identify the steady states which can be stabilized for different values of , for magnetic length , interaction and in a homogeneous system, . We show that the dynamic stabilization of a Meissner superfluid state is possible. We calculate the expectation value in the ground states of the effective model. The self-consistent solutions are given by intersections of this curve with the linear function , as we can see in Fig. 2(a) for and different values of . For the stability of the solutions we compare the slopes of the two curves, Eq. (9). If the derivative of is less than the slope of the linear function, the solution is stable.

For the considered parameters (Fig. 2), we can find stable solutions for , for all considered values of . In order to estimate the value of in the limit , we perform an extrapolation of our data for finite , as shown in the inset of Fig. 2(a) for . The value of in the continuum limit is labeled by in Fig. 2(a). Also the extrapolation leads to a stable solution, such that we are confident that the solutions remain stable in the continuum limit. In contrast for small values of , the stability is not clear, as has an almost linear behavior, as we observe from Fig. 2(a). This linear behaviour, together with the required extrapolation, makes our results inconclusive in this regime. It is also interesting to obtain some information about the stability of the system in the thermodynamic limit. Thus, we performed the above procedure for different system sizes and we performed an extrapolation to , as exemplified in the inset of Fig. 2(b) for . The values of in the continuum limit have been plotted in Fig. 2(b). We observe that for the self-consistent solutions remain stable also in the thermodynamic limit.

The non-trivial stable solutions obtained in the continuum limit are plotted in Fig. 3, for multiple system sizes. All the nontrivial stable solutions have a finite occupation of the cavity field, independent of the discretization or system size. In the following we will concentrate on the states which are stable both in the continuum and thermodynamic limit. Our results suggest a sudden onset of the occupation of the cavity mode above a certain threshold of the pump strength.

iv.2 Characterization of the Meissner superfluid steady state in the homogeneous system

Figure 4: (a) The local current for the parameters , , , , , and . The red curve represents the fit of the central part of the system, with the function . (b) The chiral current for the same parameters as a function of for different values of the discretization and the extrapolation to the continuum limit. In the inset the extrapolation is shown for . (c) The chiral current in the continuum limit for different system sizes and extrapolated to the thermodynamic limit. The extrapolation for is shown in the inset.
Figure 5: (a) The local current for the parameters , , , , , and . The red line represents the fit in the center of the system. (b) The amplitude of the oscillations in the spin flip current, , for the same parameters as a function of for different values of the discretization and the extrapolation to the continuum limit. In the inset the extrapolation is shown for . (c) in the continuum limit for different system sizes and the extrapolation to the thermodynamic limit. In the inset the extrapolation is shown for . In order to obtain an estimation of the spin flip current we enforce that it maintains a non-negative value in the limit.
Figure 6: The central charge for the parameters , , , , and multiple values of , as a function of . The central charge is extracted fitting the scaling of the entanglement entropy. The errorbars represent the fit error. The dashed horizontal line indicates the constant value .

In this section we show that the stable steady states, for , correspond to Meissner superfluid states. We focus on the behavior of the observables introduced in Sec. II.4. First we observe that all stable solutions, , have a finite value of the chiral current, shown in Fig. 4. In Fig. 4(a) we show the local current along the wire for one spin state, . In order to exemplify the extraction of the value of the chiral current we fit the oscillations due to the boundaries in the center part of the system for each leg, with the function . Extracting the chiral current as a function of for different values of (Fig. 4(b)) we obtain its behavior in the continuum . By going to a smaller discretization we obtain a larger value of the chiral current, thus, we expect that it has a finite value in the continuum. If we perform this for multiple sizes of the system, shown in Fig. 4(c), we obtain consistent results, indicating that the finite chiral current survives also in the thermodynamic limit. As we increase and go deeper into the Meissner phase, the chiral current seems to saturate.

The Meissner state is further characterized by balanced spin flip processes in the bulk of the system. In the discrete ladder representation this means that there are no currents on the rungs of the ladder. We show the behavior of this observable in Fig. 5. First the site-resolved spin flip current has been plotted in Fig. 5(a), together with the fit function , from which we extract the amplitude of the oscillations, . The results presented in Fig. 5 were computed by fitting the central part of the system. This differs with at most compared to the case if we would have considered the central of the system, . We observe that the amplitude of the spin flip current, , first increases with . It reaches a maximum for in the region were the stability is not clarified and then decreases for larger values of (Fig. 5(b)). The finite spin flip currents would indicate that a vortex state might be present for low values of . A phase transition may occur in the effective model between the Meissner state and a vortex state, close to the stability threshold. In Fig. 5(c), the amplitude of the spin flip current extrapolated to the continuum limit is represented as a function of for different system sizes. In the thermodynamic limit the spin flip current vanishes deep in the Meissner phase, as expected. However, in the intermediate regime for the amplitude of the spin flip current is larger than zero in the thermodynamic limit. This might be due to finite size effects or the extrapolation being too rough in this regime. The amplitude of the oscillations in the particle density also goes to zero deep for large values of , after taking both the continuum and thermodynamic limit (not shown).

Figure 7: The absolute value of the single particle correlations in a logarithmic plot for the parameters , , , , , for , in the Meissner superfluid phase. The correlations show an algebraic decay with distance, which corresponds to the superfluid phase. The straight (red) line are fits of the function , where .

Moreover, in the Meissner superfluid the central charge has the value, . This agrees with our numerical data within the uncertainties as presented in Fig. 6. Furthermore, we observe that for the central charge has a value , which is consistent with the assumption of a vortex state. It can be seen that close to the stability threshold both the error bars due to the fit error and the difference between the values of the central charge computed for different discretizations are increasing.

The superfluid nature of the stable stationary state solutions is in agreement with the algebraic decay of the single particle correlations, shown in Fig. 7 for . We mention that, in the weakly interacting limit, a Non-Luttinger liquid has been predicted PoZhou2014 () at the critical point of the transition between the vortex and Meissner superfluids, with an exponential decay of the single particle correlations.

To summarize, we found that the dynamical stabilization of a Meissner superfluid state is possible, both in the continuum and thermodynamic limit.

iv.3 The effect of the parabolic trap on the steady states

Figure 8: (a) Graphical interpretation of the self-consistency condition for the parameters , , , and . The expectation value is represented for multiple values of and an extrapolation to the limit . The straight (red) line represents the right-hand side of the self-consistency condition, which is a linear function with slope . The crossings of the two curves give the solutions of the self-consistency condition. (b) The local currents and for the same parameters and and .

In this section we will investigate the effect of the harmonic trapping on the dynamically organized steady states.

In the following, we will show that the Meissner state is stable also in the presence of a harmonic trap of strength , for , , . Due to the varying density, a coexistence of different states is possible across the trap. However in determining the stability of the dynamically organized states all regions are important, as , which enters the self-consistency condition, is a global observable. In the following, we will focus on the states that are present in the center of the trap, where the gradient of the trapping potential is small.

In Fig. 8(a), the expectation value as a function of is plotted in the trap. We obtain stable steady states for . In the following we focus on stationary state for , which is stable for all values of computed and in the extrapolation to the continuum limit. Due to the trapping potential, the large oscillations at the boundaries are partially suppressed, as we can observe from Fig. 8(b), compared to the homogeneous case. We identify the state realized for these parameters in the center of the trap as a Meissner superfluid, due to its finite chiral current, small values of the spin flip current and the algebraic decay of correlations (not shown). The local spin current has a maximum in the center of the trap (Fig. 8(b)), different from the homogeneous case where has a plateau-like behavior, see Fig. 4(a). As and are related via a continuity equation, at the edges of the trap the spin flip current has mostly negative values, for , or positive values, for . This implies that the amplitude of the oscillations of is small only in the central region of the trap. We note that the oscillations present in the spin flip currents are due to the finite size of the system and boundary effects.

Thus, we found that also in the presence of a harmonic trapping potential the dynamical stabilization of the Meissner superfluid is possible.

V Conclusions

In this work we find a dynamically stabilized Meissner superfluid of bosonic atoms confined in a one-dimensional wire coupled to an optical cavity. The bosonic atoms are prepared in two hyperfine states coupled to each other via Raman transitions, which involve the creation or annihilation of a cavity photon and a position dependent momentum transfer. By this coupling a cavity mediated spin-orbit coupling is induced if a finite cavity occupation forms. A dissipative dynamics takes place due to the leaking of photons out of the cavity mirrors. Above a certain pump strength a nontrivial chiral state is realized as a steady state of the dissipative attractor dynamics.

An experimental realization could use 87Rb atoms in an optical cavity BaumannEsslinger2010 (); WolkeHemmerich2012 (); LeonardDonner2017 () subjected to additional optical lattices which confine the atoms to one-dimensional structures. Previously, experiments of ultracold bosonic atoms in a cavity RitschEsslinger2013 () observed the Dicke phase transition BaumannEsslinger2010 (); KlinderHemmerich2015 (); DomokosRitsch2002 (); DimerCarmichael2007 (); NagyDomokos2008 (); PiazzaZwerger2013 () and an optical lattice has been added to investigate the influence of interaction KlinderHemmerichPRL2015 (); LandigEsslinger2016 (); HrubyEsslinger2018 (). Typical cavity parameters are and WolkeHemmerich2012 (). The cavity-induced spin-orbit coupling could be implemented using for example the two states from the 87Rb 5S\textsubscript1/2 manifold, and as used in Ref. LinSpielman2011 (). The scattering lengths of the two 87Rb states are and WideraBloch2006 (), with the Bohr radius.

Acknowledgments

We thank T. Donner and T. Esslinger for stimulating discussions. We acknowledge support from DFG (individual research grant, FOR 1807, TR 185 project B3 and SFB 1238 project C05) and the ERC (Grant Number 648166). A. S. thanks the research council of Shahid Beheshti University, G.C. (project number SAD/600/655).

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