Geometric scale invariance as a route to macroscopic degeneracy: Loading a toroidal trap with a Bose or Fermi gas

An easily scalable toroidal geometry presents an opportunity for creating large-scale persistent currents in Bose-Einstein condensates, for studies of the Kibble-Zurek mechanism, and for investigations of toroidally trapped degenerate Fermi gases. We consider in detail the process of isentropic loading of a Bose or Fermi gas from a harmonic trap into the scale-invariant toroidal regime that exhibits a high degree of system invariance when increasing the radius of the toroid. The heating involved in loading a Bose gas is evaluated analytically and numerically, both above and below the critical temperature. Our numerical calculations treat interactions within the Hartree-Fock-Bogoliubov-Popov theory. Minimal change in degeneracy is observed over a wide range of initial temperatures, and a regime of cooling is identified. The scale-invariant property is further investigated analytically by studying the density of states of the system, revealing the robust nature of scale invariance in this trap, for both bosons and fermions. We give analytical results for a Thomas-Fermi treatment. We calculate the heating due to loading a spin-polarized Fermi gas and compare with analytical results for high- and low-temperature regimes. The Fermi gas is subjected to irreducible heating during loading, caused by the loss of one degree of freedom for thermalization.

Figure 1

Harmonic Gaussian potential shown at $z=0$ for various values of $V_0$. For $V_0<0$, the potential is an attractive dimple trap; for $0<V_0<V_{\sigma }$, the potential forms a flat trap (i.e., flatter at the origin than the pure harmonic trap); for $V_0>V_{\sigma }$, a toroidal trap is realized with the minimum potential value ($V_m$) occurring at some nonzero radius ($r_m$). The harmonic approximation to the toroidal trap is shown for reference. Results for the case ${\sigma }_0=24.6\mu$m, $V_{\sigma }/K_B=50$ nK, and $V_0/K_B=-200$ nK, $50$ nK, $400$ nK.

Figure 2

(a) Condensate fraction versus temperature for fixed $V_{\sigma }/K_B=94$ nK and harmonic trap (dashed), $V_0/V_{\sigma }=1$ (dotted), 5, 10, and harmonic toroid (all solid and indistinguishable on this scale). The $\times$ indicates the results studied in subplot (b). The critical temperatures are 20, 13, and 10 nK, respectively. (b) Radial density at $z=0$ for the noncondensate (solid), condensate (dashed), and pure Thomas-Fermi (dotted) [from Eq. (16) with $\mu (N_0)$ from (17) evaluated for the same $N_0$ as the finite-temperature system]. Density profiles are shown for the case indicated with $\times$ in (a), at $T=0.8\hspace{0.5em}{T}_c$ with $V_{\sigma }/K_B=94$ nK, and $V_0/V_{\sigma }=5$. Calculations are for a system of ${10}^5$ ${}^{87}\mathrm{Rb}$ atoms ($a_s=5.77\hspace{0.5em}\mathrm{nm}$), with $[{\omega }_z,{\omega }_r]=2\pi [15.3,7.8]\mathrm{Hz}$ (so $V_{\sigma }=200\hspace{0.5em}\hslash \omega$).

Figure 3

Effect on the reduced temperature of adiabatically loading bosons into a toroid with (a) $V_{\sigma }=20\hspace{0.5em}\hslash \omega$ and (b) $V_{\sigma }=200\hspace{0.5em}\hslash \omega$. Other parameters are for $N={10}^5$ ${}^{87}\mathrm{Rb}$ atoms with $[{\omega }_z,{\omega }_r]=2\pi [15.3,7.8]\mathrm{Hz}$ and $V_0/V_{\sigma }=1$ [green (light gray) curve], 5 [red (dark gray) curve], 10 (black curve), and the harmonic toroid approximation (dashed curve). Also shown are the exact noninteracting results for the harmonic Gaussian trap (dashed-dotted curve), the harmonic toroid result Eq. (34) (dotted curve for $t_H<1$), and the limits Eq. (35) (dotted curve for $t_H>1$). For (b), $V_0/V_{\sigma }=5$ and $V_0/V_{\sigma }=10$ are indistinguishable on this scale.

Figure 4

Relative effect on the reduced temperature of adiabatic loading a spin polarized Fermi gas into a harmonic Gaussian toroid for (a) $V_{\sigma }=20\hspace{0.5em}\hslash \omega$ and (b) $V_{\sigma }=200\hspace{0.5em}\hslash \omega$. Numerical results are shown for $V_0/V_{\sigma }=1$ [green (light gray) curve], 5 [red (dark gray) curve], 10 (black curve), and the harmonic-toroid limit (dashed curve). Also shown are the Sommerfeld [horizontal dotted curve, from (45)] and Boltzmann [dotted curve, from (46)] approximations for the harmonic toroid. For (b) $V_0/V_{\sigma }=5$ and $10$ are almost indistinguishable on this scale.

Figure 5

Equipotential surfaces for the case of a (a) harmonic trap, (b) small toroidal trap, and (c) large toroidal trap. The correlation length is indicated (see text).

Figure 6

Density of states for the parameters of Fig. 3b and with $V_0=5V_{\sigma }$. Harmonic trap (dotted), series expansion with one (dashed), and two (solid) terms. With two terms, the series expansion and the exact numerical result (also solid) are indistinguishable on this scale.