By minimizing the coupled mean-field energy functionals, we investigate the ground-state properties of a rotating atomic boson-fermion mixture in a two-dimensional parabolic trap. At high angular frequencies in the mean-field lowest-Landau-level regime, quantized vortices enter the bosonic condensate, and a finite number of degenerate fermions form the maximum-density-droplet state. As the boson-fermion coupling constant increases, the maximum density droplet develops into a lower-density state associated with the phase separation, revealing characteristics of a Landau-level structure.
We investigate single‐particle quantum states of degenerate fermions, which interact repulsively with a Bose‐Einstein condensate in a rotating harmonic trap. The boson‐fermion coupling partially lifts the degeneracies of the fermionic energy spectrum in a slow rotating regime, and causes level discontinuities of occupied states upon the formation of quantized vortices in the condensate. In a fast rotating regime, the Fermi energy is located in a highly quasidegenerate single band, and hence the lowest‐Landau‐level description is applicable. The probability of finding fermions on the vortex cores is significantly enhanced when the numbers of vortices and fermions are comparable. However, increasing the number of fermions expands the fermionic cloud outward, and phase separation occurs due to the Pauli pressure while the shape of the bosonic condensate remains unchanged.
We study the global density profile of a rapidly rotating Bose-Einstein condensate in a harmonic trap with transverse frequency
ω⊥. By introducing an additional variational degree of freedom to the lowest Landau level wave function, we demonstrate that with increasing strength of the interparticle interaction, the global density profile changes from a Gaussian to the inverted parabolic one characteristic of Thomas-Fermi theory. The criterion for the lowest Landau level wave function to be a good approximation for the global structure is that the mean field energy be small compared with
ℏω⊥/Nv, where
Nv is the number of vortices in the cloud. This condition is more stringent than the requirement that the mean field energy be small compared with
ℏω⊥ which is necessary for the lowest Landau level wave function to be a good approximation to the local structure. Our results show that the lowest Landau level wave function is inappropriate for the global structure of the system realized in recent experiments even though this wave function can describe the local structure well.
A rapidly rotating Bose-Einstein condensate in a symmetric two-dimensional trap can be described with the lowest Landau-level set of states. In this case, the condensate wave function $\ensuremath{\psi}(x,y)$ is a Gaussian function of
r2=x2+y2, multiplied by an analytic function
P(z) of the single complex variable
z=x+iy; the zeros of
P(z) denote the positions of the vortices. Here, a similar description is used for a rapidly rotating anisotropic two-dimensional trap with arbitrary anisotropy $({\ensuremath{\omega}}_{x}∕{\ensuremath{\omega}}_{y}\ensuremath{\leqslant}1)$. The corresponding condensate wave function $\ensuremath{\psi}(x,y)$ has the form of a complex anisotropic Gaussian with a phase proportional to
xy, multiplied by an analytic function $P(\ensuremath{\zeta})$, where $\ensuremath{\zeta}\ensuremath{\propto}x+i{\ensuremath{\beta}}_{\ensuremath{-}}y$ and $0\ensuremath{\leqslant}{\ensuremath{\beta}}_{\ensuremath{-}}\ensuremath{\leqslant}1$ is a real parameter that depends on the trap anisotropy and the rotation frequency. The zeros of $P(\ensuremath{\zeta})$ again fix the locations of the vortices. Within the set of lowest Landau-level states at zero temperature, an anisotropic parabolic density profile provides an absolute minimum for the energy, with the vortex density decreasing slowly and anisotropically away from the trap center.
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) of the complex variable
z=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, multiplied by an analytic function
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 (
1) in terms of
N/Lz, which suggests that a nonsymmetric trap should make it easier to observe this transition, or (
2) 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 present results for the ground states of a system of spin-1 bosons in a rotating trap. We focus on the dilute, weakly interacting regime, and restrict the bosons to the quantum states in the lowest Landau level (LLL) in the plane (disk), sphere, or torus geometries. We map out parts of the zero-temperature phase diagram, using both exact quantum ground states and LLL mean-field configurations. For the case of a spin-independent interaction we present exact quantum ground states at angular momentum
L< N. For general values of the interaction parameters, we present mean-field studies of general ground states at slow rotation and of lattices of vortices and skyrmions at higher rotation rates. Finally, we discuss quantum Hall liquid states at ultrahigh rotation.
We show that, within mean-field theory, the density profile of a rapidly rotating harmonically trapped Bose-Einstein condensate is of the Thomas-Fermi form as long as the number of vortices is much larger than unity. Two forms of the condensate wave function are explored: (i) the lowest Landau level (LLL) wave function with a regular lattice of vortices multiplied by a slowly varying envelope function, which gives rise to components in higher Landau levels; (ii) the LLL wave function with a nonuniform vortex lattice. From variational calculations, we find it most favorable energetically to retain the LLL form of the wave function but to allow the vortices to deviate slightly from a regular lattice. The predicted distortions of the lattice are small, but in accord with recent measurements at lower rates of rotation.