The realization of artificial gauge fields and spin-orbit coupling for ultra-cold quantum gases promises new insight into paradigm solid state systems. Here we experimentally probe the dispersion relation of a spin-orbit coupled Bose-Einstein condensate loaded into a translating optical lattice by observing its dynamical stability, and develop an effective band structure that provides a theoretical understanding of the locations of the band edges. This system presents exciting new opportunities for engineering condensed-matter analogs using the flexible toolbox of ultra-cold quantum gases.PACS numbers: 03.75. Kk, 03.75.Mn, 03.75.Lm Spin-orbit coupling -the interaction between a particle's spin and its mechanical motion -plays a prominent role in condensed matter physics [1]. Even though the spin-orbit interaction is usually relatively weak, it can be important for bands close to the Fermi level [2]. The combination of spin-orbit coupling with a periodic potential resulted in the prediction and discovery of topological insulators [3,4]. Such spin-orbit coupled lattice systems, with the addition of strongly correlated manybody effects, can exhibit novel phases [5]. These systems have transformed our understanding and classification of insulators and have become a significant focus of recent research [6,7]. They afford the possibility of studying new phase transitions and realizing exotic spin models [8].Simulating model Hamiltonians relevant to condensed matter physics has developed into a major area of research for experiments with dilute gas Bose-Einstein condensates and degenerate Fermi gases [9,10]. Quantum gases in optical lattices are nearly disorder free and often exhibit long coherence times [9,10]. Additionally, quantum gases allow the modification of the interparticle interactions, e.g. by tuning the two-body scattering length [11] or engineering long range dipolar interactions [12], creating great flexibility for implementing model Hamiltonians. While many electronic condensedmatter systems naturally exhibit a band structure due the periodicity of an underlying crystal lattice, in ultracold quantum gases band structures can be engineered by loading the system into an optical lattice. Both an optical lattice potential [13] and spin-orbit coupling [14] can strongly modify the single-particle dispersion relation of a quantum gas, resulting in novel band structures.In this letter we perform a detailed study of a BoseEinstein condensate (BEC) with spin-orbit coupling [14][15][16][17][18] loaded into a shallow, translating one-dimensional optical lattice. We find that the system exhibits a number of dynamical instabilities induced by the periodic dispersion relation of the lattice [19]. The instabilities are marked by an initial exponential growth of excitations in the BEC, and are most significant in the vicinity of a band gap. We characterize the strengths of the instabilities by the loss rate of condensate atoms and find that a dynamical instability is present for lattice velocities exceeding a critical...

Dilute-gas Bose-Einstein condensates are an exceptionally versatile testbed for the investigation of novel solitonic structures. While matter-wave solitons in one-and two-component systems have been the focus of intense research efforts, an extension to three components has never been attempted in experiments, to the best of our knowledge. Here, we experimentally demonstrate the existence of robust dark-bright-bright (DBB) and dark-dark-bright (DDB) solitons in a spinor F = 1 condensate. We observe lifetimes on the order of hundreds of milliseconds for these structures. Our theoretical analysis, based on a multiscale expansion method, shows that small-amplitude solitons of these types obey universal long-short wave resonant interaction models, namely Yajima-Oikawa systems. Our experimental and analytical findings are corroborated by direct numerical simulations highlighting the persistence of, e.g., the DBB states, as well as their robust oscillations in the trap.PACS numbers: 03.75. Mn, 03.75.Lm Solitons are localized waves propagating undistorted in nonlinear dispersive media. They play a key role in numerous physical contexts [1]. Among the various systems that support solitons, dilute-gas Bose-Einstein condensates (BECs) [2,3] provide a particularly versatile testbed for the investigation of solitonic structures [4][5][6]. In single-component BECs, solitons have been observed either as robust localized pulses (bright solitons) [7][8][9][10][11] or density dips in a background matter wave (dark solitons) [12][13][14][15][16][17][18][19][20][21], typically in BECs with attractive or repulsive interatomic interactions, respectively. Extending such studies to two-component BECs has led to rich additional dynamics. Solitons have been observed in binary mixtures of different spin states of the same atomic species, so-called pseudo-spinor BECs [22,23]. In particular, darkbright (DB) [24][25][26][27][28], and related SO(2) rotated states in the form of dark-dark solitons [29,30], have experimentally been created in binary 87 Rb BECs. Interestingly, although such BEC mixtures feature repulsive intra-and inter-component interactions, bright solitons do emerge due to an effective potential well created by the dark soliton through the intercomponent interaction [31]. Such mixed soliton states have been proposed for potential applications. Indeed, in the context of optics where these structures were pioneered [32,33], the dark soliton component was proposed to act as an adjustable waveguide for weak bright solitons [34]. In multicomponent BECs, compound solitons of the mixed type could also be used for all-matter-wave waveguiding, with the dark soliton building an effective conduit for the bright one, similar to all-optical waveguiding in optics [35]. Apart from pseudospinor BECs, such mixed soliton states have also been predicted to occur in genuinely spinorial BECs, composed of different Zeeman sub-levels of the same hyperfine state [36][37][38]. Indeed, pertinent works [39,40] have studied the existence and dynamic...

A negative effective mass can be realized in quantum systems by engineering the dispersion relation. A powerful method is provided by spin-orbit coupling, which is currently at the center of intense research efforts. Here we measure an expanding spin-orbit coupled Bose-Einstein condensate whose dispersion features a region of negative effective mass. We observe a range of dynamical phenomena, including the breaking of parity and of Galilean covariance, dynamical instabilities, and self-trapping. The experimental findings are reproduced by a single-band Gross-Pitaevskii simulation, demonstrating that the emerging features -shockwaves, soliton trains, self-trapping, etc.-originate from a modified dispersion. Our work also sheds new light on related phenomena in optical lattices, where the underlying periodic structure often complicates their interpretation.

We measure the collective excitation spectrum of a spin-orbit coupled Bose-Einstein condensate using Bragg spectroscopy. The spin-orbit coupling is generated by Raman dressing of atomic hyperfine states. When the Raman detuning is reduced, mode softening at a finite momentum is revealed, which provides insight towards a supersolid-like phase transition. We find that for the parameters of our system, this softening stops at a finite excitation gap and is symmetric under a sign change of the Raman detuning. Finally, using a moving barrier that is swept through the BEC, we also show the effect of the collective excitation on the fluid dynamics.PACS numbers: 03.75. Kk, 03.75.Mn, 32.80.Qk, 71.70.Ej Since the achievement of Bose-Einstein condensation (BEC) in dilute atomic gases, the investigation of collective excitations has been a key tool to gain insight into this unusual state of matter [1]. For most atomic species used in BEC experiments the interactions between the ultracold atoms can be described by isotropic, short-range s-wave scattering, which leads to the well-known linear phonon excitation spectrum at low momentum. However, if long-range interactions, such as dipolar interactions, are present, the collective excitation spectrum of a BEC can exhibit a more complex structure: in addition to the typical low energy phonon spectrum, a roton-like structure can appear. It is characterised by a shoulder in the spectrum, which for certain parameters can turn into a parabolic minimum at a finite momentum [2][3][4][5].Interestingly, a similar parabolic minimum at a finite momentum can also exist in spin-orbit coupled (SOC) systems. In cold atomic gases, spin-orbit coupling can be implemented by Raman dressing of two or more atomic hyperfine states, which play the role of different (pseudo-)spins. The Raman lasers are arranged in such a way that a Raman transition between the states is accompanied by a change of momentum [6][7][8][9][10][11]. Since the Raman coupling strength and the detuning from the Raman resonance can be independently adjusted in an experiment, this provides a very flexible platform to engineer interesting dispersion relations and test spin-orbit coupled physics (for a review, see, e.g., [12][13][14][15]). In the single-particle picture, the effect of the Raman dressing is to displace two copies of the parabolic dispersion originating from the kinetic energy of the particle in opposite directions in momentum space. The Raman coupling opens a gap at the crossing of these two parabolas so that the resulting single-particle dispersion has the form of a double well in momentum space. The double well can be biased towards either minimum by changing the Raman detuning. When the Raman detuning or the Raman coupling exceed a critical value, one of the minima disappears and a single well dispersion results. * engels@wsu.eduIn the presence of nonlinear effects stemming from the s-wave scattering between the atoms in a Raman dressed BEC, the double well structure continues to exist. In a biased double wel...

Multi-component Bose-Einstein condensates exhibit an intriguing variety of nonlinear structures.In recent theoretical work, the notion of magnetic solitons has been introduced. Here we generalize this concept to vector dark-antidark solitary waves in multi-component Bose-Einstein condensates.We first provide concrete experimental evidence for such states in an atomic BEC and subsequently illustrate the broader concept of these states, which are based on the interplay between miscibility and inter-component repulsion. Armed with this more general conceptual framework, we expand the notion of such states to higher dimensions presenting the possibility of both vortex-antidark states and ring-antidark-ring (dark soliton) states. We perform numerical continuation studies, investigate the existence of these states and examine their stability using the method of Bogolyubovde Gennes analysis. Dark-antidark and vortex-antidark states are found to be stable for broad parametric regimes. In the case of ring dark solitons, where the single-component ring state is known to be unstable, the vector entity appears to bear a progressively more and more stabilizing role as the inter-component coupling is increased.