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arXiv:0712.2256 · Physical Review A

Spin-orbit coupled Bose-Einstein condensates

Tudor D. Stanescu, Brandon Anderson, Victor M. Galitski

Published Aug 11, 2008• 292 citations
arXivHTMLPDF

Abstract

We consider a many-body system of pseudo-spin-$1∕2$ bosons with spin-orbit interactions, which couple the momentum and the internal pseudo-spin degree of freedom created by spatially varying laser fields. The corresponding single-particle spectrum is generally anisotropic and contains two degenerate minima at finite momenta. At low temperatures, the many-body system condenses into these minima generating a type of entangled Bose-Einstein condensate. We show that in the presence of weak density-density interactions the many-body ground state is characterized by a twofold degeneracy. The corresponding many-body wave function describes a condensate of ``left-'' and ``right-moving'' bosons. By fine-tuning the parameters of the laser field, one can obtain a bosonic version of the spin-orbit coupled Rashba model. In this symmetric case, the degeneracy of the ground state is very large, which may lead to phases with nontrivial topological properties. We argue that the predicted type of Bose-Einstein condensates can be observed experimentally via time-of-flight imaging, which will show characteristic multipeak structures in momentum distribution.

DOI
10.1103/physreva.78.023616

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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.

•Jan 1, 2014•0 citations
We consider a many-body system of pseudo-spin-1∕21∕21∕2 bosons with spin-orbit interactions, which couple the momentum and the internal pseudo-spin degree of freedom created by spatially varying laser fields. The corresponding single-particle spectrum is generally anisotropic and contains two degenerate minima at finite momenta. At low temperatures, the many-body system condenses into these minima generating a type of entangled Bose-Einstein condensate. We show that in the presence of weak density-density interactions the many-body ground state is characterized by a twofold degeneracy. The corresponding many-body wave function describes a condensate of left-'' and right-moving'' bosons. By fine-tuning the parameters of the laser field, one can obtain a bosonic version of the spin-orbit coupled Rashba model. In this symmetric case, the degeneracy of the ground state is very large, which may lead to phases with nontrivial topological properties. We argue that the predicted type of Bose-Einstein condensates can be observed experimentally via time-of-flight imaging, which will show characteristic multipeak structures in momentum distribution.
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.
According to the “no-node" theorem, the many-body ground state wavefunctions of conventional Bose—Einstein condensations (BEC) are positive-definite, thus time-reversal symmetry cannot be spontaneously broken. We find that multi-component bosons with spin-orbit coupling provide an unconventional type of BECs beyond this paradigm. We focus on a subtle case of isotropic Rashba spin-orbit coupling and the spin-independent interaction. In the limit of the weak confining potential, the condensate wavefunctions are frustrated at the Hartree—Fock level due to the degeneracy of the Rashba ring. Quantum zero-point energy selects the spin-spiral type condensate through the “order-from-disorder" mechanism. In a strong harmonic confining trap, the condensate spontaneously generates a half-quantum vortex combined with the skyrmion type of spin texture. In both cases, time-reversal symmetry is spontaneously broken. These phenomena can be realized in both cold atom systems with artificial spin-orbit couplings generated from atom-laser interactions and exciton condensates in semi-conductor systems.
We study the system of spin-orbit (SO) coupled Bose–Einstein condensates (BEC) with Rabi coupling in quasi-two dimensions characterized by unequal Rashba and Dresselhaus couplings. The ground state properties and the phase diagram of the system are studied within the mean-field approximation at T = 0. The energy-momentum dispersion relation corresponding to the single-particle ground state exhibits an infinitely degenerate Rashba ring, whose degeneracy is destroyed by the Rabi coupling, leading to a single-point lowest energy. The effect of Rabi coupling and interaction in the system show up three different phases, namely stripe, plane wave and zero momentum phase. At a particular parametric condition, the system admits a critical point separating all three different phases, known as the tri-critical point. For further insight, the momentum distribution, the energy and the longitudinal/transverse spin polarization corresponding to the quantum phases are comprehensively discussed.
Realistic methods to create vortices in spin-orbit-coupled Bose-Einstein condensates are discussed. It is shown that, contrary to common intuition, rotation of the trap containing a spin-orbit condensate does not lead to an equilibrium state with static vortex structures but gives rise instead to nonequilibrium behavior described by an intrinsically time-dependent Hamiltonian. We propose here the following alternative methods to induce thermodynamically stable static vortex configurations: (i) to rotate both the lasers and the anisotropic trap and (ii) to impose a synthetic Abelian field on top of synthetic spin-orbit interactions. Effective Hamiltonians for spin-orbit condensates under such perturbations are derived for most currently known realistic laser schemes that induce synthetic spin-orbit couplings. The Gross-Pitaevskii equation is solved for several experimentally relevant regimes. The new interesting effects include spatial separation of left- and right-moving spin-orbit condensates, the appearance of unusual vortex arrangements, and parity effects in vortex nucleation where the topological excitations are predicted to appear in pairs. All these phenomena are shown to be highly nonuniversal and depend strongly on a specific laser scheme and system parameters.
An effective spin-orbit coupling can be generated in a cold atom system by engineering atom-light interactions. In this Letter we study spin-1/2 and spin-1 Bose-Einstein condensates with Rashba spin-orbit coupling, and find that the condensate wave function will develop nontrivial structures. From numerical simulation we have identified two different phases. In one phase the ground state is a single plane wave, and often we find the system splits into domains and an array of vortices plays the role of a domain wall. In this phase, time-reversal symmetry is broken. In the other phase the condensate wave function is a standing wave, and it forms a spin stripe. The transition between them is driven by interactions between bosons. We also provide an analytical understanding of these results and determine the transition point between the two phases.
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.