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arXiv:1109.4945 · Physical Review A

Order by disorder in spin-orbit-coupled Bose-Einstein condensates

Ryan Barnett, Stephen Powell, Tobias Graß, Maciej Lewenstein, S. Das Sarma

Published Feb 13, 2012• 78 citations
arXivHTMLPDF

Abstract

Motivated by recent experiments, we investigate the system of isotropically interacting bosons with Rashba spin-orbit coupling. At the noninteracting level, there is a macroscopic ground-state degeneracy due to the many ways bosons can occupy the Rashba spectrum. Interactions treated at the mean-field level restrict the possible ground-state configurations, but there remains an accidental degeneracy not corresponding to any symmetry of the Hamiltonian, indicating the importance of fluctuations. By finding analytical expressions for the collective excitations in the long-wavelength limit and through numerical solution of the full Bogoliubov-de Gennes equations, we show that the system condenses into a single-momentum state of the Rashba spectrum via the mechanism of order by disorder. We show that in three dimensions the quantum depletion for this system is small, while the thermal depletion has an infrared logarithmic divergence, which is removed for finite-size systems. In two dimensions, on the other hand, thermal fluctuations destabilize the system.

DOI
10.1103/physreva.85.023615

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Motivated by recent experiments, we investigate the system of isotropically interacting bosons with Rashba spin-orbit coupling. At the noninteracting level, there is a macroscopic ground-state degeneracy due to the many ways bosons can occupy the Rashba spectrum. Interactions treated at the mean-field level restrict the possible ground-state configurations, but there remains an accidental degeneracy not corresponding to any symmetry of the Hamiltonian, indicating the importance of fluctuations. By finding analytical expressions for the collective excitations in the long-wavelength limit and through numerical solution of the full Bogoliubov-de Gennes equations, we show that the system condenses into a single-momentum state of the Rashba spectrum via the mechanism of order by disorder. We show that in three dimensions the quantum depletion for this system is small, while the thermal depletion has an infrared logarithmic divergence, which is removed for finite-size systems. In two dimensions, on the other hand, thermal fluctuations destabilize the system.
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.
Intensive theoretical studies have recently predicted that a Bose-Einstein condensate will exhibit a variety of novel properties if spin-orbit coupling is present. However, an unambiguous fact has also been pointed out: Rashba coupling destroys a condensate of noninteracting bosons even in high dimensions. Therefore, a conceptually important question arises as to whether or not a condensate exists in the presence of interaction and a general type of spin-orbit coupling. Here we show that interaction qualitatively changes the ground state of bosons under Rashba spin-orbit coupling. Any infinitesimal repulsion forces bosons either to condense at one or two momentum states or to form a superfragmented state that is a superposition of infinite numbers of fragmented condensates. The superfragmented state is unstable against the anisotropy of spin-orbit coupling in systems with large numbers of particles, leading to the revival of a condensate in current experiments.
We study a two-dimensional spin-12\frac{1}{2}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.
We present the collective excitation spectrum analysis of binary Bose-Einstein condensates (BECs) with spin-orbit (SO) and Rabi couplings in a quasi-two-dimensional system. In particular, we investigate the role of SO and Rabi coupling strengths in determining the dynamical stability of the coupled BECs using Bogoliubov--de Gennes (BdG) theory. Using the eigenenergy of the BdG spectrum, we confirm the existence of phonon, roton, and maxon modes with weak repulsive intra- and interspecies contact interactions. The depth of the minimum corresponding to the roton mode strongly depends on the coupling strength. We find that the increase of the SO coupling leads to instability, while the increase in the Rabi coupling stabilizes the system. Also, the eigenvectors of the BdG spectrum indicate the presence of a densitylike mode in the stable regime and spinlike modes in the unstable regimes. A phase diagram demonstrating the stability regime in the plane of SO and Rabi coupling strengths is obtained. Finally, we complement the observation of the excitation spectrum with the direct numerical simulation results of coupled Gross-Pitaevskii equations.
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.
We consider the spin-orbit-coupled Bose gas with repulsive mean-field interparticle interactions. We analyze the phase diagram of the system varying the temperature T>0T>0T>0, the chemical potentials, and interparticle and spin-orbit interaction couplings. Our results indicate that, for Rashba- and Weyl-type spin-orbit couplings, condensates featuring an ordering wave vector $\stackrel{P\vec}{Q}\ensuremath{\ne}\stackrel{P\vec}{0}$ are fragile with respect to thermal fluctuations and at T>0T>0T>0 the only stable thermodynamic phases involving the Bose-Einstein condensate (BEC) are those of uniform type with $\stackrel{P\vec}{Q}=\stackrel{P\vec}{0}$. On the other hand, the presence of the spin-orbit coupling stabilizes the $\stackrel{P\vec}{Q}=\stackrel{P\vec}{0}$ BEC state at any dimensionality d>1d>1d>1 and modifies either the order or the universality class of the corresponding phase transition. We emphasize the singular nature of the limit of vanishing spin-orbit interaction coupling vvv, sizable shifts of the phase boundaries upon varying vvv, and the role of the relative magnitudes of the interparticle interaction couplings for the character of the condensation transition.