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Physical Review A

Chiral superfluid in a one-dimensional bipartite optical lattice

A. H. Abbas

Published Feb 28, 2022• 0 citations
Publisher

Abstract

We study the formation of a chiral mode in a one-dimensional bipartite optical potential with a double-well dispersion. The geometry of the first excited state is tuned such that two degenerate energy minima are formed at the quasimomentum $q=\ifmmode\pm\else\textpm\fi{}\ensuremath{\hbar}{k}_{0}/2$ of the first Brillouin zone. Based on the coupled-mode theory, an unconventional superfluid order ${p}_{q}\ifmmode\pm\else\textpm\fi{}i{p}_{\ensuremath{-}q}$ is formed between the interacting condensate components within the double-well dispersion band under well-defined phase-locked steady states.

DOI
10.1103/physreva.105.023327

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We study the formation of a chiral mode in a one-dimensional bipartite optical potential with a double-well dispersion. The geometry of the first excited state is tuned such that two degenerate energy minima are formed at the quasimomentum q=\ifmmode±\else\textpm\fi\ensuremathℏk0/2q=\ifmmode\pm\else\textpm\fi{}\ensuremath{\hbar}{k}_{0}/2q=\ifmmode±\else\textpm\fi\ensuremathℏk0​/2 of the first Brillouin zone. Based on the coupled-mode theory, an unconventional superfluid order pq\ifmmode±\else\textpm\fiip\ensuremath−q{p}_{q}\ifmmode\pm\else\textpm\fi{}i{p}_{\ensuremath{-}q}pq​\ifmmode±\else\textpm\fiip\ensuremath−q​ is formed between the interacting condensate components within the double-well dispersion band under well-defined phase-locked steady states.
We present an ab initio study of the ground state of an ideal coupled two-component gas of ultracold atoms in a one-dimensional optical lattice, either bosons or fermions. Due to the internal two-level structure of the atoms, the Brillouin zone is twice as large as imposed by the periodicity of the lattice potential. This is reflected in the Bloch dispersion curves, where the energy bands regularly possess several local minima. As a consequence, when the system parameters are tuned across a resonance condition, a nonzero temperature phase transition occurs, which arises from an interplay between internal and kinetic atomic energies. For fermions, this phase transition is of topological character since the structure of the Fermi surface is changed across the critical value. It is shown that these phenomena are also expected to occur for two- and three-dimensional optical lattices.
We study the critical behavior of Bose–Einstein condensation in the second band of a bipartite optical square lattice in a renormalization group framework at one-loop order. Within our field theoretical representation of the system, we approximate the system as a two-component Bose gas in three dimensions. We demonstrate that the system is in a different universality class than the previously studied condensation in a frustrated triangular lattice due to an additional Umklapp scattering term, which stabilizes the chiral superfluid order at low temperatures. We derive the renormalization group flow of the system and show that this order persists in the low energy limit. Furthermore, the renormalization flow suggests that the phase transition from the thermal phase to the chiral superfluid state is first order.
Abstract This article provides a synopsis of our recent experimental work exploring Bose–Einstein condensation in metastable higher Bloch bands of optical lattices. Bipartite lattice geometries have allowed us to implement appropriate band structures, which meet three basic requirements: the existence of metastable excited states sufficiently protected from collisional band relaxation, a mechanism to excite the atoms initially prepared in the lowest band with moderate entropy increase, and the possibility of cross-dimensional tunneling dynamics, necessary to establish coherence along all lattice axes. A variety of bands can be selectively populated and a subsequent thermalization process leads to the formation of a condensate in the lowest energy state of the chosen band. As examples the 2nd, 4th and 7th bands in a bipartite square lattice are discussed. The geometry of the 2nd and 7th bands can be tuned such that two inequivalent energetically degenerate energy minima arise at the X ± -points at the edge of the 1st Brillouin zone. In this case even a small interaction energy is sufficient to lock the phase between the two condensation points such that a complex-valued chiral superfluid order parameter can emerge, which breaks time reversal symmetry. In the 4th band a condensate can be formed at the Γ-point in the center of the 1st Brillouin zone, which can be used to explore topologically protected band touching points. The new techniques to access orbital degrees of freedom in higher bands greatly extend the class of many-body scenarios that can be explored with bosons in optical lattices.
We study a spinor (two-component) Bose gas confined in a one-dimensional double-valley optical lattice which has a double-well structure in momentum space. Based on field theory analysis, it is found that spinor bosons in the double-valley band may form a spin-charge mixed chiral spin quasicondensate under certain conditions. Our numerical calculations in a concrete $\ensuremath{\pi}$-flux triangular ladder system confirm the robustness of the chiral spin order against interactions and quantum fluctuations. This exotic atomic Bose-Einstein condensate exhibits spatially staggered spin loop currents without any charge dynamics despite the complete absence of spin-orbit coupling in the system, creating an interesting approach to atom spintronics. The entanglement entropy scaling allows us to extract conformal-field-theory central charge and establish the low-energy effective field theory for the chiral spin condensate as a two-component Luttinger liquid. Our predictions should be detectable in atomic experiments through spin-resolved time-of-flight techniques.
We show theoretically that interaction effects in a double-well optical lattice can induce condensates in an excited band. For a symmetric double-well lattice, bosons condense into the bottom of the excited band at the edge of the Brillouin zone if the chemical potential is above a critical value. For an asymmetric lattice, a condensate with zero momentum is automatically induced in the excited band by the condensate in the lowest band. This is due to a combined effect of interaction and lattice potential, which reduces the band gap and breaks the inversion symmetry. Our work can be generalized to a superlattice composed of multiple-well potentials at each lattice site, where condensates can be induced in even higher bands.
It is shown that, in a reasonable approximation, the quantum state of ppp bosons in a bipartite square two-dimensional optical lattice is governed by the nonlinear boson model describing tunneling of boson pairs between two orthogonal degenerate quasimomenta on the edge of the first Brillouin zone. The interplay between the lattice anisotropy and the atomic interactions leads to the second-order phase transition between the number-squeezed and coherent phase states of the ppp bosons. In the isotropic case of the recent experiment [G. Wirth, M. \"Olschl\"ager, and A. Hemmerich, Nat. Phys. 7, 147 (2011)], the ppp bosons are in the coherent phase state, where the relative global phase between the two quasimomenta is defined only up to mod($\ensuremath{\pi}$): $\ensuremath{\varphi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$. The quantum phase diagram of the nonlinear boson model is given.