Probing superfluidity of a mesoscopic Tonks-Girardeau gas

We study the dynamical response of a Tonks-Girardeau gas on a ring induced by a moving $\delta$-barrier potential. An exact solution based on the time-dependent Bose-Fermi mapping allows us to obtain the particle current, its fluctuations and the drag force acting on the barrier. The exact solution is analyzed numerically as well as analytically in the perturbative regime of weak barrier strength. In the weak barrier limit the stirring drives the system into a state with net zero current for velocities $v$ lower than $v_c=\pi \hslash /mL$, with $m$ the atomic mass and $L$ the ring circumference. At $v$ approaching $v_c$, angular momentum can be transferred to the fluid and a nonzero drag force arises. The existence of a velocity threshold for current generation indicates superfluid-like behavior of the mesoscopic Tonks-Girardeau gas, different from the nonsuperfluid behavior predicted for Tonks-Girardeau gas in an infinite tube.

Figure 1

Energy eigenvalues in units of ${\left(\pi \hslash \right)}^2/\left(m{L}^2\right)$ as a function of the stirring wave vector $q$ in units of $\pi /L$ for a barrier strength $m{U}_{0}L/{\hslash }^2=1$. Each parabola represents a state of different angular momentum. These states are coupled by the barrier that opens gaps at $q_n=n\pi /L$ and thus allows a change of the branch of angular momentum. Inset: The first avoided level crossing between states with zero and states with one quantum of angular momentum. Red parabolas correspond to occupied states induced by the out-of-equilibrium drive for the case $N=3$.

Figure 2

(a) Time-averaged, integrated particle current in units of $\left(\pi N\hslash \right)/\left(m{L}^2\right)$ and (b) current fluctuations in units of $\left(N\hslash k_F\right)/\left(mL\right)$, $k_F$ being the initial wave vector of the $N$th particle, for a TG gas [dotted (red) line] and a noninteracting Bose gas [dashed (cyan) line] subjected to the nonadiabatic stirring described in the text, as a function of the stirring wave vector $q$ in units of $\pi /L$, for a barrier strength $m{U}_{0}L/{\hslash }^2=1$ and $N=3$ particles. The behavior of the TG current following an adiabatic switching-on is also shown [solid (black) line]. Insets: Zoom-in on the second current (current fluctuation) peak. The numerical results (same notation and color codes as in the main figure) are compared with the results of the perturbative approach (thin solid lines).

Figure 3

Drag force exerted by the fluid on the moving barrier in units of $U_0{\rho }_0^2$, ${\rho }_0=N/L$ being the average density of the fluid, as a function of time in units of $t_0=m{L}^2/\pi \hslash$ for a TG gas [dotted (red) curves] and an ideal Bose gas [dashed (cyan) curves] for three values of stirring wave vector [(a) $q=0.25$ $q_c$, (b) $q=0.925$ $q_c$, and (c) $q=q_c$] for $N=3$ particles and a barrier strength $m{U}_{0}L/{\hslash }^2=1$. The numerical results are compared with the results of the perturbative approach (thin solid lines). In (c) for $q=q_c$ the analytical curves for ideal bosons and the TG gas are identical since $\delta q=0$ and therefore $\Delta E_q^{\left(l\right)}$ and $\{u_q^{\left(l\right)},v_q^{\left(l\right)}\}$ are independent of $l$.