Thermodynamics of solitonic matter waves in a toroidal trap

We investigate the thermodynamic properties of a Bose-Einstein condensate with negative scattering length confined in a toroidal trapping potential. By numerically solving the coupled Gross-Pitaevskii and Bogoliubov–de Gennes equations, we study the phase transition from the uniform state to the symmetry-breaking state characterized by a bright-soliton condensate and a localized thermal cloud. In the localized regime, three states with a finite condensate fraction are present: the thermodynamically stable localized state, a metastable localized state, and also a metastable uniform state. Remarkably, the presence of the stable localized state strongly increases the critical temperature of Bose-Einstein condensation.

Figure 1

Free energy $F$ as a function of the scaled interatomic strength $\gamma =2\mid a_s\mid ∕a_{\perp }$ for an attractive Bose gas of $N={10}^4$ atoms in a toroidal trap with $L∕a_{\perp }=100$. Dashed line: uniform solution. Solid line: localized solutions. The energies $F$ and $k_{B}T$ are in units of $\hslash {\omega }_{\perp }$.

Figure 2

Condensate fraction $N_0∕N$ as a function of the scaled interatomic strength $\gamma =2\mid a_s\mid ∕a_{\perp }$ for $N={10}^4$ atoms with $L∕a_{\perp }=100$. Dashed line: uniform solution. Solid line: localized solutions. The vertical dot-dashed line indicates the strength $N{\gamma }_c$ at which the uniform-to-localized phase transition takes place.

Figure 3

Azimuthal density profiles of the thermodynamically stable state for $N={10}^4$ atoms with $L∕a_{\perp }=100$ and $k_{B}T∕(\hslash {\omega }_{\perp })=2$. Length $z$ is in units of $a_{\perp }=[\hslash ∕(m{\omega }_{\perp }){]}^{1∕2}$.

Figure 4

(Color online) Phase diagram in the plane temperature $T$ vs interatomic strength $\gamma$ for $N={10}^4$ atoms in a toroidal trap, with $L=100$. The energy $k_{B}T$ is in units of $\hslash {\omega }_{\perp }$ and the length in units of $a_{\perp }$.