The excitation spectrum of a spin-orbit coupled Bose-Einstein condensate is investigated by the use of the Bose-Hubbard model and a mean-field approach. The undiagonalized Hamiltonian matrix for a general inversion-symmetric Bravais lattice is found. Analytical expressions for the excitation spectrum is then found in the limiting case of on-site interactions and nearest-neighbor hopping on a one-dimensional chain and a two-dimensional quadratic lattice. The densities of particles in the ground state acting as mean-field parameters are found self-consistently by minimization of the free energy, and they are seen to be in agreement with published literature.
The excitation spectrum of a spin-orbit coupled Bose-Einstein condensate is investigated by the use of the Bose-Hubbard model and a mean-field approach. The undiagonalized Hamiltonian matrix for a general inversion-symmetric Bravais lattice is found. Analytical expressions for the excitation spectrum is then found in the limiting case of on-site interactions and nearest-neighbor hopping on a one-dimensional chain and a two-dimensional quadratic lattice. The densities of particles in the ground state acting as mean-field parameters are found self-consistently by minimization of the free energy, and they are seen to be in agreement with published literature.
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.
In this Master's thesis, the behavior of ultracold, synthetically spin orbit coupled, weakly interacting, two-component Bose gases residing on Bravais lattices, is explored analytically. In particular, utilizing a mean field approach, general expressions for the chemical potentials are deduced for the superfluid phase, and non-diagonal expressions for the Hamiltonian are subsequently provided. An application of these expressions is presented for the case of a pure condensate residing on a square lattice, yielding an analytically derived phase diagram that coincides with numerical results found in the literature. Finally, future continuations of this work are suggested.
Binary mixtures of Bose-Einstein condensates (BECs) trapped in deep optical lattices and subjected to equal contributions of Rashba and Dresselhaus spin-orbit coupling (SOC) are investigated in the presence of a periodic time modulation of the Zeeman field. SOC tunability is explicitly demonstrated by adopting a mean-field tight-binding model for the BEC mixture and by performing an averaging approach in the strong modulation limit. In this case, the system can be reduced to an unmodulated vector discrete nonlinear Schr\"odinger equation with a rescaled SOC tuning parameter $\ensuremath{\alpha}$, which depends only on the ratio between amplitude and frequency of the applied Zeeman field. We consider the attractive interaction case and focus on the effect of the SOC tuning on the localized ground states. The dependence of the spectrum of the linear system on $\ensuremath{\alpha}$ has been analytically characterized. In particular, we show that extremal curves (ground and highest excited states) of the linear spectrum are continuous piecewise functions (together with their derivatives) of $\ensuremath{\alpha}$, which consist of a finite number of decreasing band lobes joined by constant lines. This structure also remains in the presence of inter- and intra-species interactions, the nonlinearity mainly introducing a number of localized states in the band gaps. The stability of ground states in the presence of the modulating field has been demonstrated by real-time evolutions of the original (unaveraged) system. Localization properties of the ground state induced by the SOC tuning, and a parameter design for possible experimental observation, have also been discussed.
We study a two-dimensional spin-21 interacting Bose-Einstein condensate with a Rashba-Dresselhaus spin-orbit coupling. The analytical expression of the superfluid density is derived from the linear response theory and Bogoliubov approximation. We find that the superfluid density can be divided into three terms, which originate from the condensate, the interaction between the condensate and the Bogoliubov excitations, and only the excitations. We show that the condensate-excitation interaction changes the bare mass in the superfluid density into the effective mass. In the isotropic spin-orbit coupling limit, the effective mass diverges, and the superfluid density is predicted to be zero within previous studies; however, our work shows that the excitation-induced superfluid density remains nonzero, which means that quantum fluctuations rescue the superfluidity. Our results show the importance of the quantum fluctuations in understanding the superfluid properties of spin-orbit-coupled Bose-Einstein condensates.
Abstract Employing the variational principle and the two-mode approximation, we have carried out a study on the nonlinear energy spectrum and current density of spin-1 spin-orbit-coupled (SOC) Bose-Einstein condensates (BECs) in a one-dimensional shallow optical lattice. The energy spectrum structure diagrams in the momentum space are given. When certain conditions for the system parameters are satisfied, we find that the loop structure appears at the boundary of the Brillouin zone. Furthermore, we analyze the influence of the parameters in the system on the critical condition and width of the loop structure. It is revealed that the optical lattice and Raman coupling can suppress the emergence of the loop. On the contrary, the quadratic Zeeman strength, spin-orbit-coupling and atomic interactions are favorable for the generation of the loop, which leads to the complication of the energy band structure. Intriguingly, the change of the band structure is closely related to the current density distribution of the condensates. The spin-orbit-coupling leads to a strong asymmetry in the current density distribution of different spin states in the momentum space, so as to cause a separation phenomenon. However, the optical lattice and Raman coupling weaken this asymmetry and promote the current density distribution of different spin states to re-coincidence. Our results provide a theoretical basis for further understanding the nonlinear energy spectrum characteristics of this system.
Dynamical characteristics of internal and external states of a Bose-Einstein condensate are generally different and independent, thus requiring different experimental manipulation techniques. The spin-orbit coupling recently achieved in Bose-Einstein condensates essentially connects spin and motion degree of freedom, endowing spin states with the ability to respond to orbital manipulation, and vice versa. In this work, a dynamical response effect, induced by simultaneously manipulating the internal and external states of a spin-orbit-coupled Bose-Einstein condensate, is predicted. Here, the “simultaneously manipulating the internal and external states” means that the driving field combines the Zeeman field applied to the internal state of the atom and the orbital potential affecting the external states of the atom. Specifically, the Bose-Einstein condensate is assumed to be activated by an abruptly applied Zeeman field and a sudden shake of the trapping potential. After some reasonable simplification and approximation of the model (i.e. neglecting the inter-atomic interactions and modelling the shake of the trapping potential by a short time-dependent pulse), an analytical relationship connecting spin frequency spectrum and the parameters of the driving fields is derived. The numerical calculations based on directly integrating the Gross-Pitaevskii equation are in good agreement with the results from the analytical relationship. The physical origin of the predicted spin dynamical response can be traced back to the quantum interference among different spin-orbit states. Due to the fact that a series of characteristic parameters of the condensate can be manifested in the spin frequency spectrum, the dynamical response effect predicted here provides a candidate method for determining and calibrating various system parameters by measuring the spin frequency spectrum.