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arXiv:1106.1121 · Physical Review A

Complex and real unconventional Bose-Einstein condensations in high orbital bands

Zi Cai, Congjun Wu

Published Sep 26, 2011• 36 citations
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Abstract

We perform a theoretical study on the recently observed unconventional Bose-Einstein condensations (UBEC) in the high bands of optical lattices. These exotic states are characterized by complex-valued condensate wave functions with nodal points or real-valued wave functions with nodal lines; thus, they are beyond the ``no-node'' theorem of conventional BECs. A quantum phase transition is driven by the competition between the single-particle band and interaction energies. The complex UBECs spontaneously break time-reversal symmetry, exhibiting a vortex-antivortex lattice structure.

DOI
10.1103/physreva.84.033635

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•Sep 20, 2008•2 citations
We perform a theoretical study on the recently observed unconventional Bose-Einstein condensations (UBEC) in the high bands of optical lattices. These exotic states are characterized by complex-valued condensate wave functions with nodal points or real-valued wave functions with nodal lines; thus, they are beyond the ``no-node'' theorem of conventional BECs. A quantum phase transition is driven by the competition between the single-particle band and interaction energies. The complex UBECs spontaneously break time-reversal symmetry, exhibiting a vortex-antivortex lattice structure.
Feynman's "no-node" theorem states that the conventional many-body ground state wavefunctions of bosons in the coordinate representation are positive definite. This implies that time-reversal symmetry cannot be spontaneously broken. In this article, we review our progress in studying a class of new states of unconventional Bose–Einstein condensations beyond this paradigm. These states can either be the long-lived metastable states of ultracold bosons in high orbital bands in optical lattices as a result of the "orbital Hund's rule" interaction, or the ground states of spinful bosons with spin-orbit coupling linearly dependent on momentum. In both cases, Feynman's argument does not apply. The resultant many-body wavefunctions are complex-valued and thus break time-reversal symmetry spontaneously. Exotic phenomena in these states include the Bose–Einstein condensation at nonzero momentum, the ordering of orbital angular momentum moments, the half-quantum vortex, and the spin texture of skyrmions.
In the context of Gross-Pitaevskii theory, we investigate the unconventional Bose-Einstein condensations in the two-species mixture with ppp-wave symmetry in the second band of a bipartite optical lattice. An imaginary-time propagation method is developed to numerically determine the ppp-orbital condensation. Different from the single-species case, the two-species boson mixture exhibits two nonequivalent complex condensates in the intraspecies-interaction-dominating regime, exhibiting the vortex-antivortex lattice configuration in the charge and spin channels, respectively. When the interspecies interaction is tuned across the SU(2) invariant point, the system undergoes a quantum phase transition toward a checkerboardlike spin-density wave state with a real-valued condensate wave function. The influence of lattice asymmetry on the quantum phase transition is addressed. Finally, we present a phase-sensitive measurement scheme for experimentally detecting the unconventional Bose-Einstein condensation in our model.
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 investigate the unconventional Bose-Einstein condensations with the orbital degree of freedom in the three-dimensional cubic optical lattice, which gives rise to various exotic features absent in conventional scalar and spinor Bose-Einstein condensations. Orbital angular momentum moments are formed on lattice sites breaking time-reversal symmetry spontaneously. Furthermore, they exhibit orbital frustrations and develop a chiral ordering selected by the ``order-from-disorder'' mechanism.
According to the no-node theorem, 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 the 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.