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arXiv:1112.2799 · Physical Review A

Critical temperature of a Rashba spin-orbit-coupled Bose gas in a harmonic trap

Hui Hu, Xia-Ji Liu

Published Jan 12, 2012• 29 citations
arXivHTMLPDF

Abstract

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 ${N}^{2/5}$ and ${N}^{2/3}$ in three and two dimensions, respectively, contrasted to the predictions of ${N}^{1/3}$ and ${N}^{1/2}$ in the absence of spin-orbit coupling. Finite-size corrections in three dimensions are also discussed.

DOI
10.1103/physreva.85.013619

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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 NNN, the critical temperature scales as N2/5{N}^{2/5}N2/5 and N2/3{N}^{2/3}N2/3 in three and two dimensions, respectively, contrasted to the predictions of N1/3{N}^{1/3}N1/3 and N1/2{N}^{1/2}N1/2 in the absence of spin-orbit coupling. Finite-size corrections in three dimensions are also discussed.
We investigate condensation and finite-temperature magnetic phase transitions in a coherently (Rabi) coupled Bose gas confined in a three-dimensional harmonic trap. For the noninteracting system, we derive analytical expressions for the critical temperature and condensate fraction using a semiclassical description of the single-particle spectrum and density of states. At fixed particle number, coherent coupling enhances the critical temperature relative to the uncoupled system, and this enhancement decreases with increasing particle number NNN. The finite-size correction, in contrast, lowers the transition temperature, with its effect diminishing for larger NNN. We then incorporate repulsive interactions within the Hartree--Fock--Bogoliubov--Popov framework to investigate the finite-temperature phase diagram in the temperature--Rabi coupling plane. The system undergoes successive transitions from a ferromagnetic to a paramagnetic condensate and, at higher temperature, to a thermal gas. The transition boundaries obtained from the vanishing of the condensate fraction show excellent agreement with the analytical Hartree-Fock predictions. We find that the interactions substantially suppress the variation in the condensation temperature with an increase in the coherent coupling.
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 have investigated the thermodynamic behaviour of ideal Bose gases with an arbitrary number of particles confined in a harmonic potential. By taking into account the conservation of the total number N of particles and using a saddle-point approximation, we derive analytically the simple explicit expression of mean occupation number in any state of the finite system. The temperature dependence of the chemical potential, specific heat, and condensate fraction for the trapped finite-size Bose system is obtained numerically. We compare our results with the usual treatment which is based on the grand canonical ensemble. It is shown that there exists a considerable difference between them at sufficiently low temperatures, especially for the relative small numbers of Bose atoms. The finite-size scaling at the transition temperature for the harmonically trapped systems is also discussed. We find that the scaled condensate fractions for various system sizes and temperatures collapse onto a single scaled form.
Cold atomic Bose-Einstein systems in the presence of simulated Rashba- Dresselhaus spin-orbit coupling exhibit novel physical features. With pure in-plane Rashba coupling the system is predicted in Bogoliubov-Hartree-Fock to have a stable Bose condensate below a critical temperature, even though the effective density of states is two-dimensional. In addition the system has a normal state at all temperatures. We review here the new physics when the system has such spin-orbit coupling, and discuss the nature of the finite temperature condensation phase transition from the normal to condensed phases.
Spin-orbit coupling is predicted to have a dramatic effect on thermal properties of a two-component atomic Bose gas. We show that in three spatial dimensions it lowers the critical temperature of condensation and enhances thermal depletion of the condensate fraction. In two dimensions we show that spin-orbit coupling destroys superfluidity at any finite temperature, modifying dramatically the cerebrated Berezinskii-Kosterlitz-Thouless scenario. We explain this by the increase of the number of low-energy states induced by spin-orbit coupling, enhancing the role of quantum fluctuations.
We study the stability of Bose condensates with Rashba-Dresselhaus spin-orbit coupling in three dimensions against quantum and thermal fluctuations. The ground state depletion of the plane-wave condensate due to quantum fluctuations is, as we show, finite, and therefore the condensate is stable. We also calculate the corresponding shift of the ground state energy. Although the system cannot condense in the absence of interparticle interactions, by estimating the number of excited particles we show that interactions stabilize the condensate even at nonzero temperature. Unlike in the usual Bose gas, the normal phase is not kinematically forbidden at any temperature; calculating the free energy of the normal phase at finite temperature, and comparing with the free energy of the condensed state, we infer that generally the system is condensed at zero temperature, and undergoes a transition to normal at nonzero temperature.