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arXiv:0806.2012 · Physical Review A

Uniformly frustrated bosonic Josephson-junction arrays

Kenichi Kasamatsu

Published Feb 10, 2009• 27 citations
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Abstract

We derive a uniformly frustrated $XY$ model that describes two-dimensional Josephson-junction arrays consisting of rotating Bose-Einstein condensates trapped by both a harmonic trap and a corotating deep optical lattice. The harmonic trap makes the coupling constant of the model have a nonuniform parabolic dependence. We study the ground state through Monte Carlo simulations in a wide range of the frustration parameter $f$, revealing a rich variety of vortex patterns.

DOI
10.1103/physreva.79.021604

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We derive a uniformly frustrated XYXYXY model that describes two-dimensional Josephson-junction arrays consisting of rotating Bose-Einstein condensates trapped by both a harmonic trap and a corotating deep optical lattice. The harmonic trap makes the coupling constant of the model have a nonuniform parabolic dependence. We study the ground state through Monte Carlo simulations in a wide range of the frustration parameter fff, revealing a rich variety of vortex patterns.
We investigate the ground state structure of two-dimensional Josephson junction arrays consisting of Bose-Einstein condensates trapped by both a rotating harmonic potential and a corotating deep optical lattice with square symmetry. Monte Carlo simulations of the uniformly frustrated XY model in a wide range of the frustration parameter f reveal that the stable configuration of vortices for a simple fractional number f is a staircase structure, consisting of diagonal chains of vortices with periodic lattice of unit cells.
We consider a rotating Bose-Einstein condensate in a square optical lattice in the regime in which the Hamiltonian of the system can be mapped onto a Josephson junction array. In an approximate scheme where the couplings are assumed uniform, the ground state energy is formulated in terms of the vortex configuration. The results are compared with experimental results and also previously reported results for frustrated XY model. We also show that vortex configuration is robust with respect to change of couplings and therefore the results remain valid when we consider more realistic model with non-uniform couplings.
Under suitable conditions, the Hamiltonian of a gas of cold bosonic atoms inside a rotating optical lattice can be mapped onto that of a frustrated Bose-Hubbard model. Here, the role of magnetic frustration is played by the angular frequency of rotation of the lattice. The system can be fully frustrated at experimentally attainable values of the frequency of rotation, and a superlattice of vortices appears in the ground state. This has several implications on observable quantities such as the superfluid density and the momentum distribution function. In the limit of large on-site occupation and in dimensionality D ≥ 2, the system can be mapped into an array of Josephson junctions in a magnetic field.
Motivated by experiments on Josephson junction arrays in a magnetic field and ultracold interacting atoms in an optical lattice in the presence of a ``synthetic'' orbital magnetic field, we study the ``fully frustrated'' Bose-Hubbard model and quantum XY\mathrm{XY}XY model with half a flux quantum per lattice plaquette. Using Monte Carlo simulations and the density matrix renormalization group method, we show that these kinetically frustrated boson models admit three phases at integer filling: a weakly interacting chiral superfluid phase with staggered loop currents which spontaneously break time-reversal symmetry, a conventional Mott insulator at strong coupling, and a remarkable ``chiral Mott insulator'' (CMI) with staggered loop currents sandwiched between them at intermediate correlation. We discuss how the CMI state may be viewed as an exciton condensate or a vortex supersolid, study a Jastrow variational wave function which captures its correlations, present results for the boson momentum distribution across the phase diagram, and consider various experimental implications of our phase diagram. Finally, we consider generalizations to a staggered flux Bose-Hubbard model and a two-dimensional (2D) version of the CMI in weakly coupled ladders.
Trapped Bose-Einstein condensates (BECs) differ considerably from the standard textbook example of a uniform Bose gas. In an isotropic harmonic potential V(r) = ½Mω2 r 2, the single-particle ground state introduces a new intrinsic scale of length [the ground-state size d = √ℏ/(Mω)] and energy [the ground-state energy E 0 = 32 \frac{3} {2} 23​ ℏω]. When the trap rotates at a low angular velocity, the behavior of a single vortex illustrates the crucial role of discrete quantized vorticity. For more rapid rotation, the condensate contains a vortex array. The resulting centrifugal forces expand the condensate radially and shrink it axially; thus, the condensate becomes effectively two dimensional. If the external rotation speed approaches the frequency of the radial harmonic confining potential, the condensate enters the “lowest-Landau-level” regime, and a simple description again becomes possible. Eventually, the system is predicted to make a quantum phase transition to a highly correlated state analogous to the fractional quantum Hall states of electrons in a strong magnetic field.