Phase diagram of a rotating Bose-Einstein condensate with anharmonic confinement

We examine the phase diagram of an effectively repulsive Bose-Einstein condensate of atoms that rotates in a quadratic-plus-quartic potential. With use of a variational method we identify the three possible phases of the system as a function of the rotational frequency of the trap and of the coupling constant. The derived phase diagram is shown to be universal and partly exact in the limit of weak interactions and small anharmonicity. The variational results are found to be consistent with numerical solutions of the Gross-Pitaevskii equation.

Figure 1

The phase diagram in the $\Omega ∕\omega –\sigma a$ plane for $\lambda =0.05$. The straight lines show the phase boundary between multiply quantized vortices given by Eq. (9), which denote discontinuous transitions. The dotted lines give the phase boundary between $m_0=2$ and $(m_1,m_0,m_2)=(0,2,4)$ (left), and between $m_0=3$ and $(m_1,m_0,m_2)=(0,3,6)$ (right), which denote continuous transitions. The dotted-dashed lines show the phase boundary between $m_0=1$ and $(m_1,m_0,m_2)=(0,2,4)$ (left), and between $m_0=2$ and $(0,3,6)$ (right), which denote discontinuous transitions. The straight lines do not have any physical meaning for $\sigma a$ higher than the point where they are cut by the dashed lines, as the phase boundary is given by the dotted-dashed lines.

Figure 2

The phase diagram of vortex states in a quadratic-plus-quartic potential in the $\Omega ∕\omega –\sigma a$ plane for $\lambda =0.05$ with $m_0=0–15$. The solid lines denote discontinuous transitions, and the dashed lines continuous transitions. The first circle on the left indicates the triple point, and the other circles show triple points on the lines of continuous transitions for $m_0=8,10,13$.

Figure 3

The same graph as in Fig. 2 around the phase boundary for $m_0=8$. Solid lines denote discontinuous transitions and dashed lines continuous. The slope of the dashed curve changes at the center of the circle, which is a triple point. In the small triangular region the state is dominated by the $(m_1,m_0,m_2)=(2,8,14)$ components, while above the circle it is dominated by the $(1,8,15)$ components. On the left the states are $m_0=7$ (bottom), and $(1,7,13)$ (top). On the right the states are $m_0=9$ (bottom), and $(2,9,16)$ (top).

Figure 4

Contour plots of the density of the cloud. The width in both axes is from $-4a_0$ up to $4a_0$. The white color denotes high density and the black low density. Each figure shows the density of the cloud in the state(s) with (a) $m_0=6$, (d) $(m_1,m_0,m_2)=(0,6,12)$; (b) $(1,7,13)$, (e) $m_0=7$; (c) $m_0=8$, (f) $(2,8,14)$, and (g) $(1,8,15)$. For $m_0⩾7$ there is always a node in the density at the center of the cloud.

Figure 5

Same graph as in Fig. 4 showing the phase of the order parameter of the condensate.

Figure 6

The same graph as in Fig. 2 for $0⩽m_0⩽8$, with the single-particle eigenvalue problem solved exactly. The circle denotes the triple point.

Figure 7

Same graph as in Fig. 2, for $\lambda =0.005$. The lines are identical to those in Fig. 2, except that the axes have been scaled according to the universality of the phase diagram. The crosses represent points of the continuous transitions computed numerically from the Gross-Pitaevskii equation (see Sec. 8).