In this letter we present an experimental study of the collective dipole oscillation of a spin-orbit coupled Bose-Einstein condensate in a harmonic trap. Dynamics of the center-of-mass dipole oscillation is studied in a broad parameter region, as a function of spin-orbit coupling parameters as well as oscillation amplitude. Anharmonic properties beyond effective-mass approximation are revealed, such as amplitude-dependent frequency and finite oscillation frequency at place with divergent effective mass. These anharmonic behaviors agree quantitatively with variational wave-function calculations. Moreover, we experimentally demonstrate a unique feature of spin-orbit coupled system predicted by a sum-rule approach, stating that spin polarization susceptibility-a static physical quantity-can be measured via dynamics of dipole oscillation. The divergence of polarization susceptibility is observed at the quantum phase transition that separates magnetic nonzero-momentum condensate from nonmagnetic zero-momentum phase. The good agreement between the experimental and theoretical results provides a bench mark for recently developed theoretical approaches.Many interesting quantum phases can emerge in solid state materials when electrons are placed in a strong magnetic field or possess strong spin-orbit (SO) coupling, such as the fractional quantum Hall effect [1] and the topological insulator [2]. In cold atom systems, albeit neutral atoms have neither charges nor SO coupling, the recent exciting experimental progress demonstrates that artificial gauge potentials can be synthesized in laboratory by laser control technique [3][4][5][6][7][8][9][10]. Synthetic gauge potential is becoming a powerful tool for simulating real materials with cold atoms. Moreover, the system of SO coupled bosons does not have an analogy in conventional condensed matter systems, and can exhibit many novel phases [11] such as striped superfluid phase [12,13] and half vortex phase [14][15][16][17].Collective excitations play an important role in studying physical properties of trapped atomic Bose-Einstein condensates (BEC) and degenerate Fermi gases. Collective dipole oscillation is a center-of-mass motion of all atoms. For a conventional condensate, the dipole oscillation is trivial: the frequency is just the harmonictrap frequency, independent of oscillation amplitude and interatomic interaction. This is known as Kohn theorem [18,19]. For a SO coupled condensate, however, it was found [4] that the dipole-oscillation frequency deviates from the trap frequency and the experimental data thereby can be explained by effective-mass approximation. Recently, much theoretical effort has been taken to understand dynamics of a SO coupled BEC [20][21][22][23][24][25], and many predicted unconventional properties remain to be experimentally explored. In particular, the so-called sum-rule approach predicts [25] a unique feature of SO coupled condensate: spin polarization susceptibility-a static physical quantity-can be measured via dynamics of dipole oscillatio...

Cold atoms with laser-induced spin-orbit (SO) interactions provide intriguing new platforms to explore novel quantum physics beyond natural conditions of solids. Recent experiments demonstrated the one-dimensional (1D) SO coupling for boson and fermion gases. However, realization of 2D SO interaction, a much more important task, remains very challenging.Here we propose and experimentally realize, for the first time, 2D SO coupling and topological band with 87 Rb degenerate gas through a minimal optical Raman lattice scheme, without relying on phase locking or fine tuning of optical potentials. A controllable crossover between 2D and 1D SO couplings is studied, and the SO effects and nontrivial band topology are observed by measuring the atomic cloud distribution and spin texture in the momentum 1 arXiv:1511.08170v1 [cond-mat.quant-gas] 24 Nov 2015space. Our realization of 2D SO coupling with advantages of small heating and topological stability opens a broad avenue in cold atoms to study exotic quantum phases, including the highly-sought-after topological superfluid phases.Spin-orbit (SO) interaction of an electron is a relativistic quantum mechanic effect, which characterizes the coupling between motion and spin of the electron when moving in an electric field. In the rest frame the electron experiences a magnetic field which is proportional to the electron velocity and couples to its spin by the magnetic dipole interaction, rendering the SO coupling. The SO interaction plays essential roles in many novel quantum states of solids. The recent outstanding examples include the topological insulators, which have been predicted and experimentally discovered in two-dimensional (2D) and 3D materials 1, 2 , and the topological superconductors 3, 4 , which host exotic zero-energy states called Majorana fermions 5,6 and still necessitate rigorous experimental verification. For topological insulators, the strong SO interaction leads to the so-called band inversion mechanism which drives a topological phase transition in such systems 7,8 . In superconductors, a triplet p-wave pairing is generically resulted when SO coupling is present, for which the superconductivity can be topologically nontrivial under proper conditions 9 .Recently, considerable interests have been drawn in emulating SO effects and topological phases with cold atoms, mostly driven by the fact that cold atoms can offer extremely clean platforms with full controllability to explore such exotic physics. In cold atoms the synthetic SO interaction can be generated by Raman coupling schemes which flip atom spins and transfer momentum where 1 is the 2 × 2 unit matrix, σ x,y,z are Pauli matrices acting on the spins, m is mass of an atom, V latt denotes the lattice potential in the x-z plane, M x,y are periodic Raman coupling potentials, and m z represents a tunable Zeeman field. Atoms can hop between nearest-neighboring sites due to lattice potential as well as the Raman coupling terms. Note that V latt is spin-independent and can induce hopping which conserves ...

Spin-orbit (SO) coupling has led to numerously exciting phenomena in electron systems.Whereas the synthesized SO coupling with ultracold neutral atoms gives us an opportunity to study SO coupling in bosonic systems, which exhibit many new phenomena of superfluidity and various symmetry breaking condensate phases. A richer structure of symmetry breaking always results in a nontrivial finite-temperature phase diagram, however, the thermodynamics of the SO coupled Bose gas at finite temperature is still unknown so far either in theory or experiment. Here we experimentally determine a novel finite temperature phase transition that is consistent with a transition between the stripe ordered phase and the magnetized phase. We also observe that the magnetic phase transition and the Bose condensate transition occur simultaneously as temperature decreases.Our work determines the entire finite-temperature phase diagram of SO coupled Bose gas and demonstrates the power of quantum simulation.Superfluidity is a phenomenon known for century in physics but the study of superfluidity still keeps producing novel physics. Recently SO coupling, which has played an important role in recently discovered topological insulator 1, 2 , has also been realized in ultracold degenerate gases [3][4][5][6][7][8][9] . The SO coupled Bose gases are predicted to exhibit a host of new phenomena of superfluidity. For instance, SO coupling leads to degenerate single-particle ground states, which can result in a new type of stripe superfluid with spatial density order [10][11][12][13] . SO coupling can significantly enhance low-energy density-of-state that dramatically increases quantum and thermal fluctuation effects and also magnifies interaction effects [14][15][16][17][18][19] . The absence of Galilean 2 invariance due to SO coupling yields unconventional behavior of superfluid critical velocity 18,20,21 .In this work we generate SO coupling in 87 Rb Bose gases by two contour-propagating laser beams as described in previous works 3, 6 . In this setup only the motion along the spatial direction of Raman laser (denoted byx) is coupled to spin, and the single-particle Hamiltonian alongx is given byWe focus on the case with δ = 0 where the system has an additional Z 2 symmetry (k x → −k x and σ z → −σ z simultaneously). The single-particle dispersion is shown in Fig. 1(a). To motivate our study of finite-temperature physics, we shall first summarize what are known at zero temperature.For Ω < Ω 2 4E r (E r = k 2 r /(2m)), there are two degenerate single-particle minima denoted by ±k min and their wave functions are represented by ψ L and ψ R , respectively, and these two degenerate states have opposite magnetization. Due to this degeneracy, wave function of Bose condensation should be determined by interactions in this regime. Theoretical results 11,12 have shown that for interaction parameters of 87 Rb atoms, the condensate wave function is in a superposition state (ψ L + ψ R )/ √ 2 for Ω < Ω 1 0.2E r and bosons condensate either into ψ L or into ψ R ...

Roton-type excitations usually emerge from strong correlations or long-range interactions, as in superfluid helium or dipolar ultracold atoms. However, in a weakly short-range interacting quantum gas, the recently synthesized spin-orbit (SO) coupling can lead to various unconventional phases of superfluidity and give rise to an excitation spectrum of roton-maxon character. Using Bragg spectroscopy, we study a SO-coupled Bose-Einstein condensate of ^{87}Rb atoms and show that the excitation spectrum in a "magnetized" phase clearly possesses a two-branch and roton-maxon structure. As Raman coupling strength Ω is decreased, a roton-mode softening is observed, as a precursor of the phase transition to a stripe phase that spontaneously breaks spatially translational symmetry. The measured roton gaps agree well with theoretical calculations. Furthermore, we determine sound velocities both in the magnetized and in the nonmagnetized phases, and a phonon-mode softening is observed around the phase transition in between. The validity of the f-sum rule is examined.

We use laser light shaped by a digital micro-mirror device to realize arbitrary optical dipole potentials for one-dimensional (1D) degenerate Bose gases of 87 Rb trapped on an atom chip. Superposing optical and magnetic potentials combines the high flexibility of optical dipole traps with the advantages of magnetic trapping, such as effective evaporative cooling and the application of radio-frequency dressed state potentials. As applications, we present a 160 µm long box-like potential with a central tuneable barrier, a box-like potential with a sinusoidally modulated bottom and a linear confining potential. These potentials provide new tools to investigate the dynamics of 1D quantum systems and will allow us to address exciting questions in quantum thermodynamics and quantum simulations.

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We study the decay behaviors of ultracold atoms in metastable states with spin-orbit coupling (SOC), and demonstrate that there are two SOC-induced decay mechanisms. One arises from the trapping potential and the other is due to interatomic collision. We present general schemes for calculating decay rates from these two mechanisms, and illustrate how the decay rates can be controlled by experimental parameters. We experimentally measure the decay rates over a broad parameter region, and the results agree well with theoretical calculations. This work provides an insight for both quantum simulation involving metastable dressed states and studies on few-body problems with SO coupling.

We experimentally study the dynamics and relaxation of a single density mode in a weakly interacting Bose gas trapped in a highly elongated potential. Although both the chemical potential and thermal energy of the gas exceed the level spacing of the transverse potential, we find that the observed dynamics are accurately described by the one-dimensional, integrable theory of generalized hydrodynamics. We attribute the absence of thermalization to the suppression of three-dimensional excitations, as outgoing states of the associated scattering processes obey fermionic statitistics, thus being susceptible to the Pauli exclusion principle. The mechanisms effectively preserves onedimensionality, and hence integrability, far beyond conventional limits.