Bose-Einstein condensation from a rotating thermal cloud: Vortex nucleation and lattice formation

We develop a stochastic Gross-Pitaveskii theory suitable for the study of Bose-Einstein condensation in a rotating dilute Bose gas. The theory is used to model the dynamical and equilibrium properties of a rapidly rotating Bose gas quenched through the critical point for condensation, as in the experiment of Haljan et al. [Phys. Rev. Lett. 87, 210403 (2001)]. In contrast to stirring a vortex-free condensate, where topological constraints require that vortices enter from the edge of the condensate, we find that phase defects in the initial noncondensed cloud are trapped en masse in the emerging condensate. Bose-stimulated condensate growth proceeds into a disordered vortex configuration. At sufficiently low temperature the vortices then order into a regular Abrikosov lattice in thermal equilibrium with the rotating cloud. We calculate the effect of thermal fluctuations on vortex ordering in the final gas at different temperatures, and find that the BEC transition is accompanied by lattice melting associated with diminishing long-range correlations between vortices across the system.

Figure 1

Separation of the system into condensate and noncondensate bands. States beneath the energy $E_R$, defined in the rotating frame, form the condensate band, while states above $E_R$ form the noncondensate band consisting of high-energy, approximately thermalized atoms. In the $s$-wave scattering regime high-energy states $\left(E_S\le {ϵ}_{nlm}\right)$ are also eliminated from the theory [19].

Figure 2

Interactions between the condensate and noncondensate bands. In (a) two noncondensate band atoms collide. The collision energy is transferred to one of the atoms with the other passing into the condensate band. In (b) a condensate band atom collides with a noncondensate band atom with no change in condensate band population. The time-reversed processes also occur.

Figure 3

(Color online) (a) Variation of ideal gas BEC transition temperature with angular frequency of rotation $\Omega$, for $N=1.3\times {10}^5$ atoms. (b) Angular momentum per particle (solid line), for the same number of atoms at temperature $T=T_c$, with $⟨L_z⟩/\left(\hslash N\right)\pm \sigma \left(L_z\right)/\hslash$ (dashed lines) representing the size of fluctuations. Both the angular momentum per particle and its fluctuations diverge at the critical rotation frequency.

Figure 4

(Color online) The Wigner function for the rotating ideal Bose gas used as the initial state for our simulations of rotating Bose-Einstein condensation. Parameters are $T_0=12\text{nK},{\Omega }_0=0.979{\omega }_r,{\mu }_0=0.5{ϵ}_{000}$. (a) Mean density $⟨{\widehat{\psi }}^{†}\left(\mathbf{x}\right)\widehat{\psi }\left(\mathbf{x}\right)⟩$ computed by averaging over 5000 samples of the Wigner function. A single sample is shown in (b) with vortices ($+$ have positive angular momentum) shown for $r_v<3r_0$. (c) Condensate wave function according to the Penrose-Onsager criterion. The one-body density matrix used here is constructed from 500 samples. Due to the near degeneracy of angular momentum modes averaging must be taken over many samples to completely remove all vortices from the protocondensate. In this example the protocondensate contains $\sim 50$ atoms and many vortices are visible. In all plots the color map is scaled to represent the entire range of density in the data.

Figure 5

(Color online) Condensate number $N_0$ during a quench from ${\mu }_0=0.5{ϵ}_{000}$, $T_0=12\text{nK}$ to $\mu =3.5{ϵ}_{000}$, and a range of final temperatures $T$, at fixed rotation frequency ${\Omega }_0=0.979{\omega }_r$. The condensate number is found by diagonalizing the one-body density matrix found from short-time averaging single runs of the SGPE (see text). The final condensate number is suppressed at higher temperature.

Figure 6

(Color online) Condensate band density for a single trajectory of the SGPE. (a) Initial state for $1.3\times {10}^6$ ${}^{87}\text{R}\text{b}$ atoms at $T_0=12\text{nK}$, with ${\mu }_0=0.5{ϵ}_{000}$, and ${\Omega }_0=0.979{\omega }_r$. At time $t=0$ the noncondensate band is quenched to $T=1\text{nK}$, $\mu =3.5{ϵ}_{000}$, and $\Omega ={\Omega }_0$. (b)–(d) The condensate band undergoes rapid growth. (e) and (f) At this low temperature the vortices assemble into a regular Abrikosov lattice.

Figure 7

(Color online) Condensate mode calculated from the Penrose-Onsager criterion via short-time averaging, for the simulation data of Fig. 6. (a) Initial state. (b)–(d) After the quench the condensate undergoes a rapid growth into a highly disordered phase. (e) and (f) The system undergoes a prolonged stage of ordering into a periodic vortex lattice. At this temperature the condensate and condensate band are almost identical in the final equilibrium state.

Figure 8

(Color online) Condensate band density for a quenched rotating Bose gas. (a) Initial state for $1.3\times {10}^6$ ${}^{87}\text{R}\text{b}$ atoms at $T_0=12\text{nK}$, with ${\mu }_0=0.5{ϵ}_{000}$, and $\Omega =0.979{\omega }_r$. At time $t=0$ the noncondensate band is quenched to $T=11\text{nK}$, $\mu =3.5{ϵ}_{000}$, and $\Omega ={\Omega }_0$. (b)–(d) The condensate band undergoes rapid growth. (e) and (f) After growth, vortex order is suppressed by thermal fluctuations, frustrating the formation of a periodic vortex lattice seen at the much lower temperature of Fig. 6.

Figure 9

(Color online) Condensate mode calculated from the Penrose-Onsager criterion via short-time averaging, for the simulation data of Fig. 8. (a) Initial state. (b)–(d) After the quench the condensate growth into a highly disordered vortex configuration. (e) and (f) Suppression of vortex order is reflected in the disordered condensate wave function. At this higher temperature the condensate and condensate band are distinct, reflecting the disorder and the increased thermal fraction in the condensate band.

Figure 10

(Color online) Equilibrium condensate fraction versus temperature. The boxes show the result of SGPE simulations for the rotating Bose gas at constant rotation frequency $\Omega =0.979{\omega }_r$. The solid line is the ideal gas expression, which is invariant with respect to $\Omega$. For comparison the open circles are the experimental data for the nonrotating system reported in Ref. [41].

Figure 11

(Color online) Condensate density and vortex position histograms. Top, $T=1\text{nK}$; middle, $T=5\text{nK}$; bottom, $T=11\text{nK}$. The highly ordered Abrikosov lattice has its order degraded by thermal fluctuations for increasing temperature.

Figure 12

(Color online) Spatial variation of $G_1\left(Q\right)$ in a spherical trap $V\left(Q\right)=m{\omega }_r^2{Q}^2/2$. Growth rates are shown for $\mu =0.1k_{B}T$ (solid line) and $\mu =0.01k_{B}T$ (chain line). The dashed vertical line indicates the point $V\left(Q\right)=2E_R/3$ where the rates begin to vary. The condensate band terminates at $Q_R=\sqrt{2E_R/m{\omega }_r^2}$. The ratio $R_G$ (see text) is shown versus $\mu /k_{B}T$ (inset).