We analyze vortex nucleation in mesoscopic two-dimensional Bose superfluid in a rotating trap. We explicitly include a weakly anisotropic stirring potential, breaking thus explicitly the axial symmetry. As the rotation frequency passes the critical value ${\ensuremath{\Omega}}_{c}$, the system undergoes an extra symmetry change or breaking. Well below ${\ensuremath{\Omega}}_{c}$, the ground state is properly described by the mean-field theory with an even condensate wave function. Well above ${\ensuremath{\Omega}}_{c}$, the mean-field solution works also well, but the order parameter becomes odd. This phenomenon involves therefore a discrete parity symmetry breaking. In the critical region, the mean-field solutions exhibit dynamical instability. The true many-body state is a strongly correlated entangled state involving two macroscopically occupied modes (eigenstates of the single-particle density operator). We characterize this state in various aspects: (i) the eligibility for adiabatic evolution, (ii) its analytical approximation given by the maximally entangled combination of two single modes, and finally (iii) its appearance in particle detection measurements.
Abstract Using exact diagonalization, we investigate the many-body ground state for regular vortex patterns in a rotating Bose-condensed gas of N spinless particles, confined in a quasi-two-dimensional harmonic trap and interacting repulsively via finite-range Gaussian potential. The N -body Hamiltonian matrix is diagonalized in given subspaces of quantized total angular momentum L z , to obtain the lowest-energy eigenstate. Further, the internal structure of these eigenstates is analyzed by calculating the corresponding conditional probability distribution. Specifically, the quantum mechanically stable as well as unstable states in a co-rotating frame are examined in the moderately rotating regime corresponding to angular momenta 4 N ≤ L z < 5 N for N = 16 bosons. In response to externally impressed rotation, the patterns of singly quantized vortices are formed, shaping into canonical polygons with a central vortex at the trap center. The internal structure of unstable states reveals the mechanism of entry, nucleation and pattern formation of vortices with structural phase transition, as the condensate goes from one stable vortical state to the other. The stable polygonal vortex patterns having discrete p -fold rotational symmetry with p = 5 and p = 6 are observed. The hexagonal vortex pattern with p = 6 symmetry is a precursor to the triangular vortex lattice of singly quantized vortices in the thermodynamic limit. For unstable states, quantum melting of vortex patterns due to uncertainty in positions of individual vortices, is also briefly discussed.
I review free and trapped Bose‐Einstein condensates (BECs) [1, 2, 3, 4, 5]. In an isotropic harmonic potential V(r) = 12Mω2r2, 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 E0 = 32ℏω]. When the trap rotates [6, 7, 8], there is a critical angular velocity for the appearance of one or a few quantized vortices. For more rapid rotation [9, 10, 11, 12, 13] 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 [14, 15]. 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 [16, 17, 18, 19, 20].
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 =
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.
We consider a gas of
N(< 15) Bose particles with hard-core repulsion, contained in a quasi-two-dimensional harmonic trap and subjected to an overall angular velocity $\ensuremath{\Omega}$ about the z axis. Exact diagonalization of the $n\ifmmode\times\else\texttimes\fi{}n$ many-body Hamiltonian matrix in given subspaces of the total (quantized) angular momentum
Lz, with $n\ensuremath{\sim}{10}^{5}$ (e.g., for
Lz=N=15,n=240782) was carried out using Davidson's algorithm. The many-body variational ground-state wave function, as also the corresponding energy and the reduced one-particle density-matrix $\ensuremath{\rho}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})={\ensuremath{\sum}}_{\ensuremath{\mu}}{\ensuremath{\lambda}}_{\ensuremath{\mu}}{\ensuremath{\chi}}_{\ensuremath{\mu}}^{*}(\mathbf{r}){\ensuremath{\chi}}_{\ensuremath{\mu}}({\mathbf{r}}^{\ensuremath{'}})$ were determined. With the usual identification of $\ensuremath{\Omega}$ as the Lagrange multiplier associated with
Lz for a rotating system, the ${L}_{z}\ensuremath{-}\ensuremath{\Omega}$ phase diagram (or the stability line) was determined that gave a number of critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}}, i=1,2,3,\dots{},$ at which the ground-state angular momentum and the associated condensate fraction, given by the largest eigenvalue of the reduced one-particle density matrix, undergo abrupt jumps. For a given N, a number of (total) angular momentum states were found to be stable at successively higher critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}}, i=1,2,3,\dots{}.$ All the states in the regime
N>Lz>0 are metastable. For
Lz>N, the
Lz values for the stable ground states generally increased with increasing critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}},$ and the condensate was strongly depleted. The critical ${\ensuremath{\Omega}}_{\mathrm{ci}}$ values, however, decreased with increasing interaction strength as well as the particle number, and were systematically greater than the nonvariational yrast-state values for the
Lz=N single vortex state. We have also observed that the condensate fraction for the single vortex state (as also for the higher vortex states) did not change significantly even as the two-body interaction strength was varied over several $(\ensuremath{\sim}4)$ orders of magnitude in the moderately to the weakly interacting regime.
With the imminent advent of mesoscopic rotating Bose-Einstein condensates in the lowest Landau level regime, we explore lowest Landau level vortex nucleation. An exact many-body analysis is presented in a weakly elliptical trap for up to 400 particles. Striking non-mean-field features are exposed at filling factors >>1. For example, near the critical rotation frequency pairs of energy levels approach each other with exponential accuracy. A physical interpretation is provided by requantizing a mean-field theory, where 1/N plays the role of Planck's constant, revealing two vortices cooperatively tunneling between classically degenerate energy minima. The tunnel splitting variation is described in terms of frequency, particle number, and ellipticity.