Hysteresis of noninteracting and spin-orbit-coupled atomic Fermi gases with relaxation

Hysteresis can be found in driven many-body systems such as magnets and superfluids. Rate-dependent hysteresis arises when a system is driven periodically while relaxing towards equilibrium. A two-state paramagnet driven by an oscillating magnetic field in the relaxation approximation clearly demonstrates rate-dependent hysteresis. A noninteracting atomic Fermi gas in an optical ring potential, when driven by a periodic artificial gauge field and subjected to dissipation, is shown to exhibit hysteresis loops of atomic current due to a competition of the driving time and the relaxation time. This is in contrast to electronic systems exhibiting equilibrium persistent current driven by magnetic flux due to rapid relaxation. Universal behavior of the dissipated energy in one hysteresis loop is observed in both magnetic and atomic systems, showing linear and inverse-linear dependence on the relaxation time in the strong and weak dissipation regimes. While interactions in general invalidate the framework for rate-dependent hysteresis, an atomic Fermi gas with artificial spin-orbit coupling can exhibit hysteresis loops of atomic currents. Cold atoms in ring-shape potentials are thus promising for demonstrating rate-dependent hysteresis and its associated phenomena.

Figure 1

Magnetization $m$ of a two-state paramagnetic system driven by an oscillating magnetic field $B$. (a) Equilibrium (gray curve) and dissipationless (red line) magnetization for the two-state paramagnet as a function of the magnetic field. Hysteresis loops of the magnetization emerge as the magnetic field is periodically modulated with dissipation characterized by a relaxation time of (b) $\tau /t_p=0.1$, (c) $\tau /t_p=1$, (d) and $\tau /t_p=10$. Here $t_p$ and $\tau$ are the period of the magnetic field and the relaxation time, respectively. We chose $\beta {\mu }_B{B}_0=1$, where $\beta =1/\left(k_{B}T\right),{\mu }_B$ is the magnetic moment, and $B_0$ is the magnitude of the magnetic field.

Figure 2

Total atomic current $J_{\mathrm{Tot}}$ over the flux $\varphi$ at $k_{B}T/E_u=10$ for noninteracting fermions in a ring. The left (right) column shows a system with $N_{\mathrm{tot}}=100 \left(N_{\mathrm{tot}}=101\right)$ particles. (a, b) Equilibrium (gray curves) and dissipationless (red curves) currents. (c–h) The current exhibits hysteresis loops as the flux is periodically modulated in the presence of dissipation with a relaxation time of $\tau =0.1t_0$ (c, d), $\tau =t_0$ (e, f), and $\tau =10t_0$ (g, h).

Figure 3

Maximal persistent current as a function of the normalized temperature $T/T_0$ for $N=101$ noninteracting fermions.

Figure 4

Hysteresis curve of nonequilibrium current assuming the loss of one atom per driving cycle. Initially there are $N=103$ fermions and $\tau =0.4t_0$, with a driving period of $2t_0$. We assume that atom loss occurs at every $5\tau$ increment. Here $T/T_0=10$. When atom loss occurs randomly, the curve will traverse both the even-number and the odd-number structures after a couple cycles. The curves will not be closed.

Figure 5

Energy dissipated in one hysteresis loop as a function of the relaxation time $\tau$ for (a) the magnetization of the two-state paramagnet and (b, c) the atomic currents from $N=100$ (b) and $N=101$ (c) noninteracting fermions in a ring. In the strongly dissipative regime, $E\propto \tau$ is apparent with small $\tau$. In the weakly dissipative regime, the inverse relationship $E\propto 1/\tau$ is observable with large $\tau$. Blue and red curves show the $\tau$ and $1/\tau$ dependence in the three cases. Energy units are $E_B={\mu }_B{B}_0$ (a) and $E_u={\left(2\pi \hslash \right)}^2/{\left(2m_{f}L\right)}^2$ (b, c). Temperatures follow the same values as in Figs. 1 and 2.

Figure 6

Hysteresis loops and dissipated energy of spin-orbit-coupled Fermi gases in a ring. (a–d) The total current, $J_{\mathrm{Tot}}$, as a function of the normalized flux, $\varphi$, at $k_{B}T/E_u=3$ with the spin-orbit coupling $\alpha /E_u=0.75$. (a) Equilibrium (gray line) and dissipationless (red line) currents for $N_{\uparrow }=N_{\downarrow }=100$ fermions. Currents exhibit hysteresis loops as the flux is periodically modulated with relaxation times $\tau /t_0=0.1$ (b), $\tau /t_0=1$ (c), and $\tau /t_0=10$ (d). (e) Energy FIG. 6. (Continued) dissipated in one hysteresis loop as a function of the relaxation time $\tau$. The linear (solid blue line) and inversely linear (dashed red line) dependences on $\tau$ in the strongly and weakly dissipative regimes similar to Fig. 5 are observable.