Center-of-mass rotation and vortices in an attractive Bose gas

The rotational properties of an attractively interacting Bose gas are studied using analytical and numerical methods. We study perturbatively the ground-state phase space for weak interactions, and find that in an anharmonic trap the rotational ground states are vortex or center-of-mass rotational states; the crossover line separating these two phases is calculated. We further show that the Gross-Pitaevskii equation is a valid description of such a gas in the rotating frame and calculate numerically the phase-space structure using this equation. It is found that the transition between vortex and center-of-mass rotation is gradual; furthermore, the perturbative approach is valid only in an exceedingly small portion of phase space. We also present an intuitive picture of the physics involved in terms of correlated successive measurements for the center-of-mass state.

Figure 1

The ground-state phase-space diagram of a condensate in a rotating trap with $\lambda =0.15$ as a function of the rotating frequency and the scattering length. Different ground states are marked by the following symbols: $엯$, nonrotating state; $▿$, vortex state (the number is the circulation of a multiply quantized vortex); $*$, c.m. rotational state. The solid lines give the same phase-space diagram obtained from the equations of motion where the c.m. and internal motion are decoupled, using a Gaussian ansatz for the wave functions.

Figure 2

Density plots for different ground-state configurations. Bright shades indicate high density. On the left, we have a doubly quantized vortex for $a=-1.0\mathrm{a.u.}$, in the center c.m. rotational state for $a=-4.0\mathrm{a.u.}$ and on the right c.m. state for $a=-10.0\mathrm{a.u.}$ Anharmonicity is $\lambda =0.15$ and the rotation frequency is fixed to $\Omega ∕\omega =1.35$. The unit of length for $x$ and $y$ is ${\left(\hslash ∕m\omega \right)}^{1∕2}$.

Figure 3

The angular momentum (in units of $\hslash$) per atom as a function of the dimensionless anharmonicity parameter $\lambda$ for the c.m. state. The parameter $\Omega ∕\omega$ is chosen such that in perturbation theory $L∕N=1$. Because we compare this to the results from GP simulations (marked by the symbol $*$) we have to fix the dimensionless interaction parameter $gN$ also. The parameter set for the lowest $\lambda$ is $\left(\lambda =0.01;\Omega =1.03;gN=0.015\right)$ and the others are $\left(0.02;1.06;0.03\right)$, $\left(0.05;1.15;0.075\right)$, and $\left(0.1;1.3;0.15\right)$, respectively. The dashed line is variationally calculated for Gaussian trial wave functions in the decoupling approximation.

Figure 4

Density plot of a c.m. rotational state obtained by solving the 3D GP equation. On the left $z=0$ plane and on the right $y=0$ plane. The relevant parameters are $\Omega ∕\omega =1.40$, $a=-10.0\mathrm{a.u.}$, and $\lambda =0.15$. The unit of length for $x$, $y$, and $z$ is ${\left(\hslash ∕m\omega \right)}^{1∕2}$.

Figure 5

Probability (unnormalized) of observing an atom at $\theta$ when another atom was detected at $\theta =0$ for $t=0$. The solid line is the time correlation Eq. (22) at $t=0$ and the dashed line represents $t=1.0$. The unit of time is $1∕\omega$.