Abstract
We study the groundstate behavior of a BoseEinstein Condensate (BEC) in a Ramanlaserassisted onedimensional (1D) optical lattice potential forming a multilayer system. We find that, such system can be described by an effective model with spinorbit coupling (SOC) of pseudospin (N1)/2, where N is the number of layers. Due to the intricate interplay between atomic interactions, SOC and laserassisted tunnelings, the groundstate phase diagrams generally consist of three phases–a stripe, a plane wave and a normal phase with zeromomentum, touching at a quantum tricritical point. More important, even though the singleparticle states only minimize at zeromomentum for odd N, the manybody ground states may still develop finite momenta. The underlying mechanisms are elucidated. Our results provide an alternative way to realize an effective spinorbit coupling of Bose gas with the Ramanlaserassisted optical lattice, and would also be beneficial to the studies on SOC effects in spinor Bose systems with large spin.
Introduction
The realization of Ramaninduced artificial gauge fields in ultracold atomic gases^{1,2,3,4,5,6,7} provides a wellcontrollable way to investigate many fundamental phenomena induced by SOC^{8,9,10,11}. Among these studies, the spinorbit (SO) coupled Bose gases, which have no counterpart in conventional solid materials, are of particular interests in cold atom community. An important consequence brought by SOC is the degeneracy in the singleparticle ground states, which play a centre role in determining the manybody ground states of BECs. Many new phases as well as phase transitions are predicted to appear in diverse Bose systems with different types of SOC^{12,13,14,15,16,17,18,19,20,21,22,23,24}. For example, the stripe and plane wave phases^{12,13,14,15,16,17,18}, halfvortex (meron) ground states^{19,20,21,22,23}, and fractional skyrmion lattices^{24,25} may emerge in SO coupled BECs.
Despite of different proposals to generate SOC in ultracold atoms^{26,27,28,29,30,31,32,33,34,35,36}, so far for Bose gases, the artificial SOC has been realized only in one dimension^{1,2,5} or in 2D lattices^{7}. Recently, the technique of laserassisted tunneling^{37,38,39} is developed to produce strong magnetic fields in optical lattices^{40,41,42,43}. Such method provides a powerful and delicate way to manipulate atoms in lattice potential. Stimulated by these developments, some authors^{24,25} have proposed an alternative and realistic way to realize an effective 2D SOC in bilayer Bose systems based on the laserassisted tunneling. In such schemes, the prerequisite “internal” states to fabricate SOC are essentially replaced by the Ramanassisted “external” motional states in each layer, providing a new system to investigate the SO coupled BECs.
Motivated by the above advances, in this paper, we consider a gas of ultracold scalar bosons subjected to a Ramanassisted 1D optical lattice potential forming a multilayer system. Within the lowest band of the lattice, the system can be mapped to an effective model with SOC, where N different layers play a role of a pseudospin (N − 1)/2 coupled to the intralayer motion via the laserassisted tunneling of atoms between the layers. This scheme can avoid the using of near resonant light beams which cause heating in previous experiments^{1}, and can be applied to a wide range of atom species including fermions. Recently, a related scheme has been experimentally implemented to realize the effective SOC with double well potential formed by an optical superlattice^{44}. In such a scheme the double layer is in a direction of the atomic motion. On the other hand, we suggest to use a lattice with N sites in a direction perpendicular to the atomic motion. This resembles bosonic ladders^{43}, but the atoms now undergo a planar rather than a onedimensional motion.
We determine the ground states of the system in the presence of atomic interactions. Note that, the dynamics of a SO coupled BEC in a weakly tilted optical lattice has been studied^{45}, where the correlated Bloch oscillations with spin Hall effect are revealed. Here, the onedimensional optical lattice potential is sufficiently tilted and unlike the typical atomatom interactions in conventional spinor BEC^{46}, the special type of interactions from onsite repulsions in our system is quite different in the psuedospin representation, and can give rise to peculiar Ndependent phase diagrams with different behaviors: (1) For even N, by tuning the tunneling strength J, the singleparticle ground state may change from a single minimal with zeromomentum to double minima with finite momentum, with the corresponding manybody ground states evolving from a normal phase to a robust stripe phase. (2) For odd N, the singleparticle states only minimize at zeromomentum. However, when the interaction strength is increased, a stripe/plane wave phase with finite momentum can still emerge in the ground states. Such unique features reflect the competition and compromise between Ramanassisted tunneling and atomic interactions in this system.
Results
The model
We consider a threedimensional ultracold Bose gas (e.g. ^{87}Rb) loaded into a onedimensional optical lattice potential. Such potential are tight enough that the atoms only occupy the lowest energy band of the lattice potential (along zaxis), but move freely in the traverse xyplane, forming a stackeddisk configuration. Furthermore, we apply a linear gradient potential in zdirection to tilt the lattice, as depicted in Fig. (1). Such global tilt can be achieved by implementing a frequency shift between the lasers for the creating of the lattice potential^{47,48}, or by tilting the lattice along the direction of the gravitational field^{49,50}. The singleparticle Hamiltonian of this system reads:
with
where Ψ(r) annihilates a boson at position r. U_{0} and F are the strengths of optical and linear gradient potential respectively, and is a weak harmonic potential along zaxis with ω_{z} the trapping frequency. P and m are the momentum and mass of atom, and k_{o} is the wavevector of laser to generate the lattice potential.
When the tilting is not too large, the atoms can still move in the lowest state of each well, forming the energy band of the lattice potential. We can expand the field operator with the localized wannier function of the ith lattice, where N is the lattice number. The Hamiltonian. (1) then can be rewritten as:
Here, is the onsite energy and . When neglecting the small V(z), the energy differences between adjacent sites are mainly caused by the lineartilted part, which is given by Δ ≡ Δ_{ij} = ε_{i} − ε_{j}. Generally, there should also be a tunneling matrix element between adjacent sites <ij>. However, for a sufficiently tilted lattice potential, the intersite tunneling is much smaller than the energy mismatch between two sites, i.e. . For example, for Δ ≈ 8E_{R} with E_{R} the recoiling energy of the lattice^{39}. As a result, the direct tunneling is inhibited and hence can be neglected. To restore the atomic hopping between adjacent wells, we resort to the newly developed laserassisted process^{40,41,42,43}.
To this end, we implement two Raman lasers with wavevector k_{i} and frequency ω_{i} (i = 1, 2), which couple to the atomic internal state via a twophoton transition. This gives rise to an timedependent scalar potential V_{K} = 1V_{0}[e^{i(δk⋅r−δωt)} + 1e^{−i(δk⋅r−δωt)}], where V_{0} is controlled by the Raman beam intensities. δk = k_{1} − k_{2} and δω = ω_{1} − ω_{2} denotes the wave vector and frequency differences of the Raman lasers. Then along the zdirection, one has an additional overlap integral , i.e. , where we have assumed δω ~ Δ with the dominant contribution in K′ is from the overlap between adjacent sites <ij>, while other processes are far offresonance and can be neglected. Then by introducing Ncomponent spinor Φ = (ϕ_{1}ϕ_{2} … ϕ_{N})^{T}, and applying a unitary transformation , where
and δk_{⊥} ⋅ r_{⊥} = δk_{x}x + δk_{y}y, the Hamiltonian is transformed as (The definition of F_{z} is referred to Eq. 6). To write it explicitly, we have , and with being the laserassisted tunneling strength. Notice that, it is the factor due to the momentum transfer along the zaxis making the overlap integral J nonzero. Under the rotatingwave approximation, one can drop the counterrotating terms and arrive at the following effective singleparticle Hamiltonian
Here,
and
is the z component of angular momentum matrix with angular momentum , δ_{ε} = Δ − δω is the twophoton detuning. Furthermore, we have also included an additional harmonic trap V(z), which can be termed as an effective quadratic Zeeman energy (a is the lattice spacing) by adjusting the trapping center. Due to , Eq. (4) effectively describes a system with nontrivial spinorbit coupling, which reduces to a familiar form with equal Rashba and Dresselhaus contributions^{13,14,15} for N = 2 and N = 3. The major difference is that the internal spin states now are played by atoms in different wells and the total pseudospin can be varied via the lattice number N.
Before proceeding, we should mention that there are generally Bloch oscillations for atoms in a tilted optical lattice potential^{45,47,48,49,50}. For sufficiently tilted lattice case as discussed above, the energy difference Δ between adjacent sites is much larger than the suppressed intersite tunneling . In this regime, the Bloch oscillation becomes a rapid shivering motion with frequency Δ/h and amplitude . When the twophoton Raman transition is introduced, Δ reduces to a twophoton detuning δ_{ε}, and a considerable effective tunneling J is induced. In this case, it was shown that the coherent Bloch oscillations of frequency δ_{ε}/h would appear in the form of a periodic breathing dynamics^{51} or a periodic center of mass motion^{52}. In the following, we mainly concentrate on the groundstate behaviors for the resonant case with δ_{ε} = 0.
Without loss of generality, in the following we assume that the traverse momentum transfer δk_{⊥} is along the xaxis, i.e. with λ = δk_{⊥}, and set m = ħ = 1 and energy unit E_{λ} = λ^{2}/2 throughout the paper.
Single particle spectrum
We first discuss the single particle states of this system. In the absence of effective Zeeman fields, Hamiltonian. (4) bears the timereversal symmetry (TRS) with , resulting in a symmetrical singleparticle energy spectrum E(k) = E(−k). Due to the laserassisted tunneling J, the atomic states in different wells get mixed, and hence the degeneracy of psuedospin components is lifted with N energy branches. We are interested in the lowest branch, which is responsible for the determining of bosonic ground states.
In general, the singleparticle groundstate manifold can be classified into two categories: for even N, there may exist a twofold degeneracy; while for odd N, there is only one state in the ground subspace. As shown in Fig. (2a), we plot the lowest energy spectrum by diagonalizing Hamiltonian. (4) for different N (= 2, 3, 4) with J/E_{λ} = 0.1. We can see that for odd N (= 3), there is only one minimum state at k = 0. On the other hand, for even N (= 2, 4), double minima at ±k_{min} can be identified. Here, we have also included a weak harmonic trap. As the quadratic Zeeman term does not break the TRS, it would just modify the ground state energy for small δ_{z} without destroying the double degeneracy. On the other hand, a nonzero linear term δ_{ε} ≠ 0 would break the TRS and lead to asymmetric energy spectra. As a result, such possible degeneracy is lifted, with only one state left in the groundstate manifold.
In Fig. (2b), we plot the momentum evolution of minimal states as a function of J for even N (= 2, 4). One can find that, when J surpasses a critical J_{c}, k_{min} would converge to 0, indicating a tunneling induced transition would happen in the singleparticle ground state. As we will see below, above different behavior of the singleparticle states would have dramatic effects on the manybody ground states when the atomic interactions are included.
Ground state phase diagram
We now turn to investigate the manybody ground states of this system in the presence of atomatom interactions. Considering a shortranged case, the interactions for atoms situated in the same well are much stronger than that in different wells. Then, one can neglect the contribution from the latter and write the Hamiltonian for interacting atoms as
where denotes the atomic density in ith layer and with g_{0} the contact interaction strength. Notice that, the interactions here only happen in each pseudospin component, which keeps invariant under the unitary transformation U, and would play an important role in determining the groundstate configurations.
In the following we will discuss independently the “even” and “odd” N cases, which exhibit different behavior in the singleparticle spectra and the manybody ground states.
Even N
In this case, we find that for giving trapping potential δ_{z}, the phase diagrams in g_{2D} − J plane for different N have similar structures, and three different phases may appear: (I) “Stripe” phase, where the wavefunction is a superposition of two plane waves with opposite momenta ±k_{m} (k_{m} ≠ 0) and (II) “Plane Wave” phase, where only one plane wave component with finite momentum k_{m} contributes to the ground state; (III) a “Normal” phase with bosons condensed in the zeromomentum state of k = 0.
To be more specific and without loss of generality, we choose the simplest N = 2 for illustrations. In Fig. (3), we give the groundstate phase diagram in the g_{2D} − J plane for N = 2 by numerically minimizing the energy E_{G}. Generally, due to the interplay between atomic tunneling and atomatom interactions, above three phases may compete with each other and survive in three distinct regimes (labeled by colors), touching at a tricritical point.
In the dilute limit (g_{2D}/E_{λ} ≪ 1), above a critical tunneling strength, i.e. J > J_{c1} ≃ 0.5E_{λ}, the system is in the zeromomentum Normal phase. While for J < J_{c2} ≃ 0.41E_{λ}, a Stripe phase is favored. Between them (J_{c1} < J < J_{c2}), a Plane Wave phase is expected to have lower energy. The regime of such Plane Wave phase gets diminished with increasing of interaction g_{2D}, and finally disappears at a tricritical point around (J/E_{λ}, g_{2D}/E_{λ}) ≃ (0.38, 0.11), where three phases merge. Beyond the tricritical point, only Normal to Stripe phase transition survives (See Fig. (4b,d)). These features essentially reflect the competitions between kinetic and interaction energies of these states. In the weak interaction regime, the kinetic energy is dominant, and the system is always in a Normal phase when the single particle spectrum has only one minimum at k = 0. On the double minima side, the kinetic energies of Stripe and Plane Wave phases for the same k_{m} are degenerate, and would be further lifted by the atomic interactions.
In Fig. (4), we plot the groundstate momenta k_{m} and the interlayer polarization 〈F_{z}〉 as functions of tunneling J for two typical interaction strength. One can see that, the Plane Wave has homogeneous intralayer density but finite interlayer polarization 〈F_{z}〉 > 0, while the Stripe phase has inhomogeneous density n_{i}(r) with 〈F_{z}〉 = 0. On the other hand, since the atoms in the same layer repulse each other, the system tends to have both equal populations and homogeneous densities in each layer. Hence, close to the Normal phase, the Plane Wave phase with small 〈F_{z}〉 but homogeneous intralayer density is more favorable. While with decreasing of J, 〈F_{z}〉 becomes larger and larger, and the system transits into the Stripe phase. Moreover, the Plane Wave phase would be also suppressed by increasing of interactions and turn to a Normal phase continuously. Such two transitions finally meet at a quantum tricritical point.
Several remarks are on hand: first, we have taken δ_{z}/E_{λ} = 0.01 in numerical calculations, which gives no physical effects for N = 2 and would modify the phase boundaries slightly for N > 2. Second, compared to the effective model in ref. 1, here the external states in different layers play the role of spin rather than the internal states. The corresponding effective spinspin interaction takes the value c_{2}/c_{0} = (g_{↑} − g_{↑↓})/(g_{↑} + g_{↑↓}) = 1 with g_{↑↓} = 0, which is much larger than in the previous case, where c_{2} is very close to the degenerate point c_{2} = 0^{1,24}. This makes the Stripe phase in this system quite robust^{13}. Third, for N > 2, the phase diagrams are qualitatively unchanged around tricritical regime and similar analysis can be applied.
Odd N
When N is odd, the situation changes a lot and the phase diagrams may exhibit different behaviors. To be specifc, in the following we take N = 3 as an example to address this problem. Similar results can be found for N > 3.
In Fig. (5), we give the phase diagram for N = 3. It is interesting to see that even though the singleparticle spectrum is only minimized at the k = 0 state, the system can still be Plane Wave or Stripe phases carrying finite momenta in some regimes. In one hand, the zeromomentum Normal phase is predominant for small g_{2D}. In the other hand, the interaction energy would become significant with the increasing of interactions. As shown in Fig. (6e,f), the density of Normal phase in center layer is relative large. And for sufficient large g_{2D}, an instability to Plane Wave/Stripe phases with more delocalized atomic distribution and finite k_{m} (Fig. (6a,b)) would happen, where the increasing of kinetic energies is compensated by the decreasing of interaction energies. Furthermore, similar to the even N case, the Plane Wave phase with a finite 〈F_{z}〉 ≠ 0 (Fig. (6c,d)) only survives for moderate tunneling strength J, between the Normal and Stripe phases, and ends at a tricritical point.
It is worthy to stress that, the emerging of Plane Wave/Stripe phases for odd N is mainly driven by atomatom interactions, in a sharp contrast to the even N case, where the role is mainly played by atomic tunnelings. This may reflect the topological differences of singleparticle groundstate manifolds between these two cases.
Discussions and Conclusions
We now discuss some experimentrelated issues. First, our results are quite general and independent of specific atoms. Here, we take the ^{87}Rb as an example. The simplest N = 2 case can be achieved by a similar scheme as the bilayer configurations^{24,25}. For N > 2, one can resort to a superlattice potential with more than two nonequivalent sites^{39} or a linear tilt potential^{37}. For a standing wave with wavelength λ_{s} and depth U_{0} ~ 15E_{r}, where is the recoil energy, the trapping frequency in each well is about . If one choose Δ ~ E_{r}, the Ramanassisted tunneling with the bare tunneling J_{z} ≪ Δ, can be tuned up to J ~ 2π × 60 Hz by varying V_{0}. Note that, V_{0} ≪ Δ ≪ ω_{0} can be satisfied to ensure the validity of tightbinding and the lowest band approximations. To reach the scope of the phase diagram, one need E_{λ} ~ J. This can be done by arranging the opening angles of two Raman lasers with . For a typical harmonic trap with frequency ω_{z} ~ 2π × 10 Hz, is much smaller than E_{λ}. In the case of ^{87}Rb, g_{0} ~ 7.8 × 10^{−12} Hz cm^{3}, and the corresponding with , is limited to a weakly interacting regime.
Up to now, we have neglected the effects of effective Zeeman fields δ_{ε} and δ_{q}. For not too large δ_{ε} and/or δ_{q}, the phase boundaries would be modified quantitatively^{17} which are also confirmed in our case, while leaving the main results qualitatively unchanged. To detect these phases in experiments, one may implement the momentumresolved timeofflight measurements. The atom population in each well which characterizes 〈F_{z}〉 and , can be measured via insitu absorption imaging.
In conclusions, we have investigated the ground states and the associated phase diagrams of a BEC in a laserassisted 1D optical lattice potential forming a multilayer system. The unique Ndependence of the singleparticle spectra and the corresponding manybody groundstate configurations reflects the subtle competition between the effective SOC induced by laserassisted interlayer tunneling and atomatom interactions. Our results would have potential implications in searching new matter states in spinorbit coupled Bose systems with a large spin. In future studies, one may consider the effects of interlayer longrange interactions, and the extensions to multicomponent BECs and Fermi gases.
Methods
For weakly interacting Bose gases, the quantum fluctuations can be neglected safely. And one can adopt the variational method^{13,24} to investigate the ground states of the system. In meanfield level, the variational Ansatz of the groundstate wavefunction can be constructed as:
where Φ_{±k} e^{±ikx} are the eigenstates of the lowest energy branch with momentum ±k, determined by Eq. (4). a_{+} and a_{−} are complex amplitudes with normalization condition a_{+}^{2} + a_{−}^{2} = 1. In the dilute limit, one has k = k_{min} for double minima case and k = 0 for single minimum. While in general, k is dependent on the interactions^{13}. Minimizing the energy with respect to variational parameters a_{+}, a_{−} and k, one can obtain the groundstate phases as well as the phase diagrams.
Additional Information
How to cite this article: Sun, Q. et al. Ground states of a BoseEinstein Condensate in a onedimensional laserassisted optical lattice. Sci. Rep. 6, 37679; doi: 10.1038/srep37679 (2016).
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Acknowledgements
This work is supported by NSFC (Grants No. 11404225, No. 11474205, No. 21503138, No. 11504037 and No. 11434015), Foundation of Beijing/Chongqing Education Committees (Grants No. KM201510028005, KM201310028004, and No. KJ1500311), and CSTC under Grant No. cstc2015jcyjA50024. G. J. acknowledges a support by Lithuanian Research Council (Grant No. MIP 086/2015).
Author information
Affiliations
Department of Physics, Capital Normal University, Beijing, 100048, China
 Qing Sun
 , Jie Hu
 & AnChun Ji
College of Physics and Electronic Engineering, Chongqing Normal University, Chongqing, 401331, China
 Lin Wen
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China
 W.M. Liu
Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio Ave. 3, LT10222 Vilnius, Lithuania
 G. Juzeliūnas
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Contributions
Qing Sun performed the theoretical as well as the numerical calculations. Qing Sun, Jie Hu, Lin Wen, W.M. Liu, G. Juzeliūnas and AnChun Ji wrote and reviewed the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Qing Sun or AnChun Ji.
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Further reading

Absence of Landau damping in driven threecomponent Bose–Einstein condensate in optical lattices
Scientific Reports (2018)
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