In this thesis, we consider the Bose-Einstein condensation of ideal Bose gas in prsence of
pseudo spin-orbit coupling. We take equal Rashba and Dresselhaus spin orbit coupling
which give rise to degenerate minima at �nite momenta in the ground state eigenvalue
and hence form an unconvensional condensate. In this work, we �rst considered the
quantum mechanics of this system to calculate energy eigenvalues and the wave function
both of which posses a signature of spin orbit coupling. Then we consider the
statistical mechanical properties of these systems. In the case of homogeneous system,
we �rst calculate the density of states and then the critical temperature of the system.
The critical temperature shows a non-analytic point at a particular value of a certain
parameter the Raman Coupling which is also known as the point of quantum phase transition
in the case of interacting system. Critical temperature at point shows a cusp like
structure and drops down to a minimum value at the same point. The critical temperature
can be approximated analytically in two di�erent Raman coupling regime before
and beyond the critical point from the density of state description. This description
can be found by approximating the density of states with that of a free particle with
reduced mass in the direction in which the spin orbit coupling takes place. We then
consider the magnetization of the system an its variance with the change in the Raman
coupling. The magnetization of the condensate vanishes at the same critical point. We
next consider the spin orbit coupled Bose gas in a harmonic con�nement. The critical
temeperature for this case will show a dip in around the same value of the critical point
as in the previous case. But, in this case the drop in the critical temeperature is less
sharp and smoothened due to the presense of an harmonic trap. Next we consider the
natural extension of the spin bosons to a large spin and we show that it can be exactly
mapped to the Dicke model. We can also estimate the critical temperature from this
description in a similar fashion.
In this thesis, we consider the Bose-Einstein condensation of ideal Bose gas in prsence of
pseudo spin-orbit coupling. We take equal Rashba and Dresselhaus spin orbit coupling
which give rise to degenerate minima at �nite momenta in the ground state eigenvalue
and hence form an unconvensional condensate. In this work, we �rst considered the
quantum mechanics of this system to calculate energy eigenvalues and the wave function
both of which posses a signature of spin orbit coupling. Then we consider the
statistical mechanical properties of these systems. In the case of homogeneous system,
we �rst calculate the density of states and then the critical temperature of the system.
The critical temperature shows a non-analytic point at a particular value of a certain
parameter the Raman Coupling which is also known as the point of quantum phase transition
in the case of interacting system. Critical temperature at point shows a cusp like
structure and drops down to a minimum value at the same point. The critical temperature
can be approximated analytically in two di�erent Raman coupling regime before
and beyond the critical point from the density of state description. This description
can be found by approximating the density of states with that of a free particle with
reduced mass in the direction in which the spin orbit coupling takes place. We then
consider the magnetization of the system an its variance with the change in the Raman
coupling. The magnetization of the condensate vanishes at the same critical point. We
next consider the spin orbit coupled Bose gas in a harmonic con�nement. The critical
temeperature for this case will show a dip in around the same value of the critical point
as in the previous case. But, in this case the drop in the critical temeperature is less
sharp and smoothened due to the presense of an harmonic trap. Next we consider the
natural extension of the spin bosons to a large spin and we show that it can be exactly
mapped to the Dicke model. We can also estimate the critical temperature from this
description in a similar fashion.
We consider a spin-orbit coupled configuration of spin-1/2 interacting bosons with equal Rashba and Dresselhaus couplings. The phase diagram of the system at T=0 is discussed with special emphasis on the role of the interaction treated in the mean-field approximation. For a critical value of the density and of the Raman coupling we predict the occurrence of a characteristic tricritical point separating the spin mixed, the phase separated, and the zero momentum states of the Bose gas. The corresponding quantum phases are investigated analyzing the momentum distribution, the longitudinal and transverse spin polarization, and the emergence of density fringes. The effect of harmonic trapping as well as the role of the breaking of spin symmetry in the interaction Hamiltonian are also discussed.
Spin-orbit-coupled bosons can exhibit rich equilibrium phases at low temperature and in the presence of particle-particle interactions. In the case with a 1D synthetic spin-orbit interaction, it has been observed that the ground state of a Bose gas can be a normal phase, stripe phase, or magnetized phase in different parameter regimes. The magnetized states are doubly degenerate and consist of a many-particle two-state system. In this work, we investigate the nonequilibrium quantum dynamics by switching on a simple one-dimensional optical lattice potential as external perturbation to induce resonant couplings between the magnetized phases, and predict a quantum spin dynamics which cannot be obtained in the single-particle systems. In particular, due to particle-particle interactions, the transition of the Bose condensate from one magnetized phase to the other is forbidden when the external perturbation strength is less than a critical value, and a full transition can occur only when the perturbation exceeds such critical strength. This phenomenon manifests itself a dynamical phase transition, with the order parameter defined by the time-averaged magnetization over an oscillation period, and the critical point behavior being exactly solvable. The thermal fluctuations are also considered in detail. From numerical simulations and exact analytic studies we show that the predicted many-body effects can be well observed with the current experiments.
We investigate theoretically Bose-Einstein condensation of an ideal, trapped Bose gas in the presence of Rashba spin-orbit coupling. Analytic results for the critical temperature and condensate fraction are derived based on a semiclassical approximation to the single-particle-energy spectrum and density of states and are compared with exact results obtained by explicitly summing discrete energy levels for a small number of particles. We find a significant decrease of the critical temperature and of the condensate fraction due to finite spin-orbit coupling. For a large coupling strength and a finite number of particles N, the critical temperature scales as N2/5 and N2/3 in three and two dimensions, respectively, contrasted to the predictions of N1/3 and N1/2 in the absence of spin-orbit coupling. Finite-size corrections in three dimensions are also discussed.
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.
In this paper we develop a quantum field approach to reveal the thermodynamic properties of the Bose-Einstein condensate with equal Rashba and Dresselhaus spin-orbit couplings. In the experimentally feasible regime, the phase transition from the separate phase to the single-minimum phase can be driven well by the tunable temperature. Moreover, the critical temperature, which is independent of the trap frequency, can be derived exactly. At the critical point, the specific heat has a large jump and can be thus regarded as a promising candidate to detect this temperature-driven phase transition. In addition, we obtain the analytical expressions for the specific heat and the entropy in the different phases. In the single-minimum phase, the specific heat and the entropy are governed only by the Rabi frequency. However, in the separate phase with lower temperature, we find that they are determined only by the strength of the spin-orbit coupling. Finally, the effect of the effective atom interaction is also addressed. In the separate phase, this effective atom interaction affects dramatically the thermodynamic properties, including the critical temperature.
A weakly interacting, spin-orbit coupled, two-component, ultracold Bose gas bound to a Bravais lattice is studied. Motivated by recent experimental advances in the field of synthetically spin-orbit coupled, ultracold, neutral atomic gases showing Bose-Einstein condensation, an analytic framework with which to describe such systems in the superfluid regime is presented. This is applied to a Rashba spin-orbit-coupled Bose gas in a two-dimensional optical lattice. The exotic nature of Bose-Einstein condensation in the presence of spin-orbit coupling is an interesting study by itself. Additionally, when the optical lattice is introduced, the system provides a highly controllable experimental testing ground for numerous condensed matter physics phenomena. Five phases of the system are considered, and their excitation spectra, critical superfluid velocities and free energies are found. In obtaining the free energy, the effects of terms in the Hamiltonian that are linear in excitation operators are included, and such terms have not been studied previously in this context. Minimization of the free energy at zero temperature is used to confirm the phase diagrams reported in the literature, where it has usually been obtained by neglecting the effect of excitations. The plane and stripe wave phases in the phase diagram are bosonic analogues of Fulde-Ferrell-Larkin-Ovchinnikov states in superconductors involving nonzero condensate momenta.