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arXiv:0907.0564 · Physical Review A

Vortex nucleation in a mesoscopic Bose superfluid and breaking of the parity symmetry

Daniel Dagnino, N. Barberán, Maciej Lewenstein

Published Nov 13, 2009• 22 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.80.053611

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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 \ensuremathΩc{\ensuremath{\Omega}}_{c}\ensuremathΩc​, the system undergoes an extra symmetry change or breaking. Well below \ensuremathΩc{\ensuremath{\Omega}}_{c}\ensuremathΩc​, the ground state is properly described by the mean-field theory with an even condensate wave function. Well above \ensuremathΩc{\ensuremath{\Omega}}_{c}\ensuremathΩc​, 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.
It is well known that a rotating Bose-Einstein condensate forms vortices to carry the angular momentum. For a first vortex to nucleate at the trap center, the rotational frequency must become larger than a certain critical value. The vortex nucleation process, however, is sensitive to the trap shape. It was shown earlier [Dagnino et al., Nat. Phys. 5, 431 (2009)] that, for a symmetry-breaking potential that preserves parity, at criticality the leading natural orbitals may become degenerate, giving rise to a ``maximally entangled'' quantum state, found from exact solutions for just a few bosons. Developing an effective two-state model, we show here that, in the limit of large particle numbers, the many-body ground state becomes either a so-called ``twin''-like or a ``Schr\"odinger cat''-like state. We corroborate this finding by a direct comparison to the exact numerical solution of the problem, feasible for moderate particle numbers $N\ensuremath{\lesssim}50$ within the lowest Landau level approximation. We show that the nature of the quantum state at criticality can be controlled by both the quadrupolar deformation and the flatness of the confining potential.
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.
A rapidly rotating Bose-Einstein condensate in a symmetric two-dimensional harmonic trap can be described with the lowest Landau-level set of single-particle states. The condensate wave function $\ensuremath{\psi}(x,y)$ is a Gaussian $\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}{r}^{2}/2)$, multiplied by an analytic function f(z)f(z)f(z) of the complex variable z=x+iyz=x+iyz=x+iy. The criterion for a quantum phase transition to a non-superfluid correlated many-body state is usually expressed in terms of the ratio of the number of particles to the number of vortices. Here a similar description applies to a rapidly rotating nonsymmetric two-dimensional trap with arbitrary quadratic anisotropy (${\ensuremath{\omega}}_{x}^{2}<{\ensuremath{\omega}}_{y}^{2}$). The corresponding condensate wave function $\ensuremath{\psi}(x,y)$ is a complex anisotropic Gaussian with a phase proportional to xy\mathit{xy}xy, multiplied by an analytic function f(z)f(z)f(z), where $z=x+i{\ensuremath{\beta}}_{\ensuremath{-}}y$ is a stretched complex variable and $0\ensuremath{\leqslant}{\ensuremath{\beta}}_{\ensuremath{-}}\ensuremath{\leqslant}1$ is a real parameter that depends on the trap anisotropy and the rotation frequency. Both in the mean-field Thomas-Fermi approximation and in the mean-field lowest Landau level approximation with many visible vortices, an anisotropic parabolic density profile minimizes the energy. An elongated condensate grows along the soft trap direction yet ultimately shrinks along the tight trap direction. The criterion for the quantum phase transition to a correlated state is generalized (111) in terms of N/LzN/{L}_{z}N/Lz​, which suggests that a nonsymmetric trap should make it easier to observe this transition, or (222) in terms of a ``fragmented'' correlated state, which suggests that a nonsymmetric trap should make it harder to observe this transition. An alternative scenario involves a crossover to a quasi one-dimensional condensate without visible vortices, as suggested by Aftalion et al., Phys. Rev. A 79, 011603(R) (2009).
We review some recent developments in the theory of rotating atomic gases. These studies have thrown light on the process of nucleation of vortices in regimes where mean-field methods are inadequate. In our review, we shall describe and compare quantum vortex nucleation of a dilute ultracold bosonic gas trapped in three different configurations: a one-dimensional ring lattice, a one-dimensional ring superlattice and a two-dimensional asymmetric harmonic trap. In all of them, there is a critical rotation frequency, at which the particles in the ground state exhibit strong quantum correlations. However, the entanglement properties vary significantly from case to case. We explain these differences by characterizing the intermediate states that participate in the vortex nucleation process. Finally, we show that noise correlations are sensitive to these differences. These new studies have, therefore, shown how novel quantum states may be produced and probed in future experiments with rotating neutral atom systems.
We review some recent developments in the theory of rotating atomic gases. These studies have thrown light on the process of nucleation of vortices in regimes where mean-field methods are inadequate. In our review we shall describe and compare quantum vortex nucleation of a dilute ultracold bosonic gas trapped in three dif-ferent configurations: a one-dimensional ring lattice, a one-dimensional ring super-lattice and a two-dimensional asymmetric harmonic trap. In all of them there is a critical rotation frequency, at which the particles in the ground state exhibit strong quantum correlations. However, the entanglement properties vary significantly from case to case. We explain these differences by characterizing the intermediate states that participate in the vortex nucleation process. Finally, we show that noise corre-lations are sensitive to these differences. These new studies have, therefore, shown how novel quantum states may be produced and probed in future experiments with rotating neutral atom systems.
We study the generation of vortices in rotating axially elongated magneto-optical traps, a situation which has been realized in a recent experiment (K. W. Madison, F. Chevy, W. Wohlleben, J. Dalibard, Phys. Rev. Lett. 84, 806 (2000)). We predict that at a critical frequency the condensate experiences a symmetry breaking and changes from a convex cloud to a state with one bended vortex. We also discuss several effects which enlarge the critical frequency with respect to other geometries of the trap: these are, (i) the failure of the Thomas-Fermi approximation on the transverse degrees of freedom of the condensate, (ii) the enhancement of the transverse asymmetry of the trap by means of rotation and (iii) the yet unobserved bending of the vortex lines.