Gap Solitons in a Spin-Orbit-Coupled Bose-Einstein Condensate

Abstract

We report a diversity of stable gap solitons in a spin-orbit coupled Bose-Einstein condensate subject to a spatially periodic Zeeman field. It is shown that the solitons, can be classified by the main physical symmetries they obey, i.e. symmetries with respect to parity (P), time (T), and internal degree of freedom, i.e. spin, (C) inversions. The conventional gap and gap-stripe solitons are obtained in lattices with different parameters. It is shown that solitons of the same type but obeying different symmetries can exist in the same lattice at different spatial locations. PT and CPT symmetric solitons have anti-ferromagnetic structure and are characterized respectively by nonzero and zero total magnetizations. The progressively growing interest in the physics of mixtures of spinor Bose-Einstein condensates (BECs) and in particular in spin-orbit (SO) coupled BECs (SOBECs) [1,2], is motivated on the one hand by their fundamental importance for the atomic physics, and on the other hand by the rich possibilities they offer for emulating synthetic electric and magnetic fields in solids [3]. The last property allows one using neutral atoms to simulate numerous condensed matter phenomena [4] in a tunable way, what would be impossible in direct condensed matter experiments. In other words, a spinor BEC is a promising candidate for implementation of a quantum simulator, which due to the technologies available nowadays was already implemented using cold atoms [5], ions [6], or photonics [7]. In particular, we mention a recent experiment [8] with coupled (linear) arrays of optical waveguides simulating a discrete version of SO-coupling which in the present letter is considered for the matter waves.Meanwhile, the physics of BECs is characterized by two essential factors, which are not typical for the physics which is aimed to be simulated. First, a BEC is a nonlinear system, with the nonlinearity stemming from the twobody interactions [9]. Moreover, by changing the configuration of the system (i.e. of the laser beams) one can create effective nonlinear interactions of different types [10]. Respectively, such nonlinear objects as skyrmions [11], solitons [12,13], anti-ferromagnetic structures and symmetry breaking [14], have been recently reported for SOBECs. The second feature, is that a (quasi-) stationary state of the condensate requires the presence of the external potential. This, in particular, imposes constraints on the lower bound of the kinetic energy, what is particularly relevant for SO-BECs (see e.g. [14]). What concerns the external potentials, since the very first experiments with cold atoms loaded in optical lattices [15] it was widely recognized that lattices (i.e. periodic potentials) are particularly efficient in manipulating BECs [16]. Nowadays, the nonlinear properties of the atomic BECs held in optical lattices, described in the mean-field approximation are very well studied [17] (see also [18] for a brief review on BEC mixtures in optical lattices). It is therefore natural that studies of SO-BECs in pe...

Bose-Einstein condensates with localized spin-orbit coupling: Soliton complexes and spinor dynamics

“…(2) can be rewritten in the form in which stationary localized modes were thoroughly studied in optics [19]. For the case of constant SO coupling, bright solitons were found for constant [9] and periodic [11] Zeeman fields. Those solitons had two distinguishing features: in the limit of a small number of atoms they bifurcated from the linear spectrum, and the populations of the dark states were comparable and even equal.…”

“…Meantime, SO BECs feature physical factors which are usually absent in the emulated systems. This is, in particular, the intrinsic nonlinearity of BECs, originating from interatomic interactions and supporting solitons in homogeneous BECs [9,10] and in BECs with either Zeeman [11] or optical [12] lattices (both lattices are available experimentally [13,14]). …”

Bright solitons in a two-dimensional spin-orbit-coupled dipolar Bose-Einstein condensate

“…This achievement has ignited tremendous interest in this field because of the dramatic change in the single particle dispersion (induced by spin-orbit coupling) which in conjunction with the interaction leads to many exotic superfluids [35][36][37][38][39][40][41][42][43][44][45](also see [46][47][48][49][50][51][52][53] for review). Such change in dispersion also results in exotic solitons even when the interaction is contact (without dipole-dipole interactions), including 1D bright solitons [54][55][56][57][58][59][60] for a BEC with attractive contact interactions, 1D dark [61,62] and gap solitons [63][64][65] for a BEC with repulsive contact interactions, as well as 1D dark solitons for Fermi superfluids [66,67]. These solitons exhibit unique features that are absent without spin-orbit coupling, for instance, the plane wave profile with a spatially varying phase and the stripe profile with a spatially oscillating density for BECs, as well as the presence of Majorana fermions inside a soliton for Fermi superfluids.…”

Properties of spin–orbit-coupled Bose–Einstein condensates

“…The most famous example of this is the existence of a stripe phase that is an equal superposition of two minima in momentum space [40]. The ground state and collective excitations of SOC BECs have been investigated for a large number of different settings, such as homogeneous [40][41][42][43][44][45][46][47][48][49], harmonic [50][51][52][53][54], in the presence of a periodic optical lattice [55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72], a double-well [73][74][75], rotation [76][77][78][79][80][81], inside an optical cavity [82][83][84][85][86], and for particles with dipolar interaction …”