The coherence of a quantum system is one of the key signatures of many-body physics. While a three-dimensional Bose-Einstein condensate (BEC) exhibits true long-range coherence, a Bose gas confined in a one-dimensional (1D) potential displays quasi-long-range order. Experimentally, these coherence properties are typically probed via the momentum distribution, which corresponds to the Fourier transform of the one-body correlation function. In two-dimensional (2D) systems, for example, momentum distribution enables the observation of the Berezinskii-Kosterlitz-Thouless (BKT) transition from a thermal cloud to a superfluid. With trimodal distribution fitting, an intermediate quasicondensate phase is also revealed. In 1D systems, the momentum distribution of a quenched Tonks-Girardeau (TG) Bose gas can be mapped to that of a non-interacting Fermi gas at a specific evolution time, achieving the rapidity distribution. On the other hand, spin-orbit coupling (SOC) has emerged as one of the central topics in ultracold atomic physics. In additional to creating an artificial electric or magnetic field, researchers also observe negative-mass dynamics in a spin-orbit-coupled BEC by engineering the curvature of the band dispersion. This phenomenon is directly observable in the density distribution of the system. In 1D systems, SOC also introduces rich physical phenomena; while low-energy 1D quantum systems are conventionally described by Tomonaga-Luttinger liquid (TLL) theory, SOC notably alters the energy-momentum dispersion. This modification gives rise to unconventional Luttinger liquids whose properties are governed by the spin-orbit coupling strength.In this work, we first investigate the qualitative properties of a 1D Bose gas and demonstrate the temperature dependence of the momentum distribution, revealing how the thermal fluctuations destroy the quasi-long-range order. In the second part, we probe the effects of 1D SOC on a 1D Bose gas, focusing on how SOC modifies the energy-momentum dispersion and the resulting momentum distribution. We demonstrate that within a specific parameter regime, the momentum distribution and its associated coherence properties undergo distinct qualitative changes. The occupied higher momentum states induce phase fluctuations and destroy the coherence of the system, showing signals of the non-Luttinger liquid physics. We probe the coherence properties of a 1D system along these two parameter axes.
The coherence of a quantum system is one of the key signatures of many-body physics. While a three-dimensional Bose-Einstein condensate (BEC) exhibits true long-range coherence, a Bose gas confined in a one-dimensional (1D) potential displays quasi-long-range order. Experimentally, these coherence properties are typically probed via the momentum distribution, which corresponds to the Fourier transform of the one-body correlation function. In two-dimensional (2D) systems, for example, momentum distribution enables the observation of the Berezinskii-Kosterlitz-Thouless (BKT) transition from a thermal cloud to a superfluid. With trimodal distribution fitting, an intermediate quasicondensate phase is also revealed. In 1D systems, the momentum distribution of a quenched Tonks-Girardeau (TG) Bose gas can be mapped to that of a non-interacting Fermi gas at a specific evolution time, achieving the rapidity distribution. On the other hand, spin-orbit coupling (SOC) has emerged as one of the central topics in ultracold atomic physics. In additional to creating an artificial electric or magnetic field, researchers also observe negative-mass dynamics in a spin-orbit-coupled BEC by engineering the curvature of the band dispersion. This phenomenon is directly observable in the density distribution of the system. In 1D systems, SOC also introduces rich physical phenomena; while low-energy 1D quantum systems are conventionally described by Tomonaga-Luttinger liquid (TLL) theory, SOC notably alters the energy-momentum dispersion. This modification gives rise to unconventional Luttinger liquids whose properties are governed by the spin-orbit coupling strength.In this work, we first investigate the qualitative properties of a 1D Bose gas and demonstrate the temperature dependence of the momentum distribution, revealing how the thermal fluctuations destroy the quasi-long-range order. In the second part, we probe the effects of 1D SOC on a 1D Bose gas, focusing on how SOC modifies the energy-momentum dispersion and the resulting momentum distribution. We demonstrate that within a specific parameter regime, the momentum distribution and its associated coherence properties undergo distinct qualitative changes. The occupied higher momentum states induce phase fluctuations and destroy the coherence of the system, showing signals of the non-Luttinger liquid physics. We probe the coherence properties of a 1D system along these two parameter axes.
The dissertation explores the effects of synthetic spin-orbit coupling on the behaviour of quantum gases in several different contexts. We first study realistic methods to create vortices in spin-orbit-coupled (SOC) Bose-Einstein condensates (BEC). We propose two different methods to induce thermodynamically stable static vortex configurations: (1) to rotate both the Raman lasers and the anisotropic trap; and (2) to impose a synthetic Abelian field on top of synthetic spin-orbit interactions. We solve the Gross-Pitaevskii equation for several experimentally relevant regimes and find new interesting effects such as spatial separation of left- and right-moving spin-orbit-coupled condensates, and the appearance of unusual vortex arrangements. Next we consider cold atoms in an optical lattice with synthetic SOC in the Mott-insulator regime. We calculate the parameters of the corresponding tight-binding model and derive the low-energy spin Hamiltonian which is a combination of Heisenberg model, quantum compass model and Dzyaloshinskii-Moriya interaction. We find that the Hamiltonian supports a rich classical phase diagram with collinear, spiral and vortex phases. Next we study the time evolution of the magnetization in a Rashba spin-orbit-coupled Fermi gas, starting from a fully-polarized initial state. We model the dynamics using a Boltzmann equation, which we solve in the Hartree-Fock approximation. The resulting non-linear system of equations gives rise to three distinct dynamical regimes controlled by the ratio of interaction and spin-orbit-coupling strength λ: for small λ, the magnetization decays to zero. For intermediate λ, it displays undamped oscillations about zero and for large λ, a partially magnetized state is dynamically stabilized. Motivated by an interesting stripe phase which appears in BEC with SOC [Li et al., Phys. Rev. Lett. 108, 225301 (2011)], we study the finite-temperature phase diagram of a pseudospin-1/2 Bose gas with contact interactions. We show that strong inter-spin interactions can lead to the appearance of magnetically ordered phases at temperatures above the superfluid transition. For the case of inter-spin attraction, we also discuss the possibility of a bosonic analogue of the Cooper-paired phase, however this state is not energetically favourable. We extend our calculations to a spin-orbit-coupled Bose gas to investigate the possibility of stripe ordering in the normal phase. However, within our approximations, we do not find an instability towards stripe formation. Finally, we consider a two-dimensional Bose gas at zero temperature with an underlying quartic single-particle dispersion in one spatial direction. This Hamiltonian can be realized using the NIST scheme of spin-orbit coupling [Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature 471, 83 (2011)], or using the shaken lattice scheme of Parker et al. [C. V. Parker, L.-C. Ha and C. Chin, Nature Physics 9, 769 (2013)]. By numerically comparing energies of various trial wave-functions, we show that, at low densities, the ground state is strongly correlated, in contrast to a typical mean-field BEC. The trial wave-function with the lowest energy is of Jastrow-type and it describes a state with finite, but strongly reduced, condensate fraction.
Spinor Bose gases may exhibit both phase and ferromagnetic order. Consequently, they may support superflow of both mass and spin currents. In this thesis we utilize a stochastic Gross-Pitaevskii model to investigate the superfluid properties of a two-dimensional spin-1 Bose gas with ferromagnetic interactions. Owing to the low dimensionality, the superfluidity of this system can not be attributed to the presence of a Bose-Einstein condensate, with thermal fluctuations precluding long-range order. Rather, mass and spin superfluidity emerge via respective Berezinskii-Kosterlitz-Thouless (BKT) transitions. Such transitions are characterized by the emergence of quasi-long-range order, driven by the binding of topological defect pairs. First, we consider an easy-plane ferromagnetic gas. We find the system superfluidity may be understood by direct application of existing BKT theory. In particular, mass and spin superfluidity are respectively associated with quasi-long-range order in the global phase and local magnetization. The former arises from the binding of mass vortex pairs, while the latter arises from the binding of polar-core spin vortex pairs. Following this, we introduce axial magnetization, and consider a gas in the broken-axisymmetric phase. We find axial magnetization acts to couple mass and spin superflows via a mechanism analogous to the superfluid drag exhibited by two-component superfluids. This coupling renders invalid the direct applicability of existing BKT theory: here mass and spin superfluidity may each be associated with both phase and magnetic order. We remedy this with a generalization making concrete the relation between superfluidity and quasi-long-range order in this system. In addition, through variations of the quadratic Zeeman energy and axial magnetization we identify three distinct superfluid phases, which serve as finite-temperature generalizations of the polar, broken-axisymmetric, and easy-axis ferromagnetic ground states.
We analyze the two-body momentum correlation function for a uniform, weakly interacting, one-dimensional Bose gas. We show that the strong positive correlation between opposite momenta, expected in a Bose-Einstein condensate with a true long-range order, almost vanishes in a phase-fluctuating quasicondensate where the long-range order is destroyed. Using the Luttinger liquid approach, we derive an analytic expression for the momentum correlation function in the quasicondensate regime, showing (i) the reduction and broadening of the opposite-momentum correlations (compared to the singular behavior in a true condensate) and (ii) an emergence of anticorrelations at small momenta. We also numerically investigate the momentum correlations in the crossover between the quasicondensate and the ideal Bose-gas regimes using a classical field approach and show how the anticorrelations gradually disappear in the ideal-gas limit.
We investigate spin-incoherent Luttinger liquid of a one-dimensional spin-1 Bose gas in a harmonic trap. In this regime highly degenerate spin configurations emerge since the energy splitting between different spin states is much less than the thermal energy of the system, while the temperature is low enough that the lowest energetic orbitals are occupied. As an example we numerically study the momentum distribution of a one-dimensional spin-1 Bose gas in Tonks-Girardeau gas limit and in the sector of zero magnetization. We find that the momentum distributions broaden as the number of atoms increase due to the averaging of spin function overlaps. Large momentum (p) asymptotic is analytically derived, showing the universal 1/p4 dependence. We demonstrate that the spin-incoherent Luttinger liquid has a momentum distribution also distinct from spinless bosons at finite temperature.
We present a systematic theoretical study of the BCS-BEC crossover in two-dimensional Fermi gases with Rashba spin-orbit coupling (SOC). By solving the exact two-body problem in the presence of an attractive short-range interaction we show that the SOC enhances the formation of the bound state: the binding energy E(B) and effective mass m(B) of the bound state grows along with the increase of the SOC. For the many-body problem, even at weak attraction, a dilute Fermi gas can evolve from a BCS superfluid state to a Bose condensation of molecules when the SOC becomes comparable to the Fermi momentum. The ground-state properties and the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature are studied, and analytical results are obtained in various limits. For large SOC, the BKT transition temperature recovers that for a Bose gas with an effective mass m(B). We find that the condensate and superfluid densities have distinct behaviors in the presence of SOC: the condensate density is generally enhanced by the SOC due to the increase of the molecule binding; the superfluid density is suppressed because of the nontrivial molecule effective mass m(B).
We study the contacts, large-momentum tail, radio-frequency spectroscopy, and some other universal relations for an ultracold one-dimensional (1D) two-component Fermi gas with spin-orbit coupling (SOC). Different from previous studies, we find that the ${q}^{\ensuremath{-}8}$ tail in the spin-mixing (off-diagonal) terms of the momentum distribution matrix is dependent on the two SOC parameters in the laboratory frame for 1D systems, where q is the relative momentum. This tail can be observed through time-of-flight measurement as a direct manifestation of the SOC effects on the many-body level. Besides the traditional 1D even-wave scattering length, we find that two new physical quantities must be introduced due to the SOC. Consequently, two new adiabatic energy relations with respect to the two SOC parameters are obtained. Furthermore, we derive the pressure relation and virial theorem at short distances for this system. To find how the SOC modifies the large-momentum behavior, we take the SOC parameters as perturbations since the strength of the SOC should be much smaller than the corresponding strength scale of the interatomic interactions. In addition, by using the operator product expansion method, we derive the asymptotic behavior of the large-momentum distribution matrix up to the ${q}^{\ensuremath{-}8}$ order and find that the diagonal terms of the distribution matrix include the contact of traditional 1D even-wave scattering length as the leading term and the SOC modified terms beyond the leading term, the off-diagonal term is beyond the subleading term and is corrected by the SOC parameters. We also find that the momentum distribution matrix shows spin-dependent and anisotropic features. Furthermore, we calculate the momentum distribution matrix in the laboratory frame for the experimental implication. In addition, we calculate the high-frequency tail of the radio-frequency spectroscopy and find that the presence of the contact related to the center-of-mass momentum in the radio-frequency spectral is due to the SOC effects. This paper paves the way for exploring the profound properties of many-body quantum systems with SOC in one dimension.