Oscillations and Decay of Superfluid Currents in a One-Dimensional Bose Gas on a Ring

We study the time evolution of a supercurrent imprinted on a one-dimensional ring of interacting bosons in the presence of a defect created by a localized barrier. Depending on interaction strength and temperature, we identify various dynamical regimes where the current oscillates, is self-trapped, or decays with time. We show that the dynamics is captured by a dual Josephson model and involve phase slips of thermal or quantum nature.

Figure 1

(a) Sketch of the quench protocol: a one-dimensional (1D) Bose gas on a ring in the presence of a localized barrier; e.g., a tightly focused repulsive optical potential (red) creating a dip in the density (blue) is quenched out of equilibrium by phase imprinting. (b) Energy landscape of the homogeneous 1D Bose gas on a ring: states with integer values of the current per particle correspond to local minima of the energy. Quench (black arrow) transfers the system from the initial zero-current state (light blue circle) to the state with one unit of current (light red circle). Depending on the parameters, the barrier can resonantly couple the $+1$ and -1 states (light gray arrow) or induce an adiabatic transition between the $+1$ and 0 states (dashed blue arrow).

Figure 2

Classical field simulations of the quench dynamics in the mean-field regime for $g=20\times {\hslash }^2/(mL)$ and $N=1000$ (corresponding to $\gamma =0.02$). (a) Average current per particle (black solid lines, in units of $\mathrm{\Pi }=\hslash /(Nm)$) as a function of time (in units of $\tau =m{L}^2/\hslash$), at $T=0$. The horizontal black dotted (dashed) lines indicate the values $J=0$ ($\pm 1$). From top to bottom, ${\lambda }_{\mathrm{GP}}=\{0.8,1,1.05,2\}$. (b) Current at $T={\mu }_0/k_B$, averaged over 100 realizations of the classical field, for barrier strengths ${\lambda }_{\mathrm{GP}}=\{0.6,0.9,1.5,2\}$, where black solid lines indicate simulations and red dashed curves indicate fits from the model function $J(t)=A{e}^{-{\mathrm{\Gamma }}_{A}t}+B\cos [\omega t+\varphi ]e^{-{\mathrm{\Gamma }}_{B}t}$. (c) Current for $\gamma =2\times {10}^{-5}$ at $T=0$ simulations (solid lines) and two-mode model (dashed lines) for ${\lambda }_{\mathrm{GP}}=\{0.05,0.1,0.15,0.2\}$ (blue, red, magenta, and black, respectively). (d) Damping rate $\mathrm{\Gamma }$ (in units of $1/\tau$) (extracted from the fit, with a maximum among ${\mathrm{\Gamma }}_A$ and ${\mathrm{\Gamma }}_B$) as a function of ${\lambda }_{\mathrm{GP}}$ for $T=\{0.5,1,1.5,1.75\}\times {\mu }_0/k_B$ (solid blue, dashed red, solid yellow, and dashed violet, respectively). (e) Zoomed-in view of a single classical field trajectory, at $T={\mu }_0/k_B$ and ${\lambda }_{\mathrm{GP}}=0.6$, evidencing a phase slip: a jump in the current (top panel) corresponds to the reflection of a slow soliton at the barrier, which is visible in the density deviation map [58] (middle panel) and to a singularity in the phase profile (bottom panel).

Figure 3

Exact solutions in the Tonks-Girardeau regime. (a) Average current per particle (in units of $\mathrm{\Pi }=\hslash /Nm$) vs time (in units of $\tau =m{L}^2/\hslash$) after the quench for $N=23$, at $T=0$, for barrier strength ${\lambda }_{\mathrm{TG}}=\{0.1,0.5,1,4\}$. The horizontal black dotted (dashed) lines indicate the values $J=0$ ($\pm 1$). (b) Current at $T=E_F/k_B$ (black solid) for ${\lambda }_{\mathrm{TG}}=\{0.1,0.5,1,4\}$ from top to bottom and fits (red dashes, with same fitting function as in Fig. 2). (c) Frequency $\omega /N$, and (d) damping rate $\mathrm{\Gamma }/N$ obtained from the fit vs ${\lambda }_{\mathrm{TG}}$ for $N=11$ (solid blue) and $N=23$ (dashed red). Other curves in Fig. 3: frequency for universal Rabi oscillations ${\omega }_R={\pi }^{2}N{\lambda }_{\mathrm{TG}}$ (black solid) and first excitation frequency at the Fermi sphere (black dashed) [48]. (e) Frequency of the excitations produced in the quench (relative amplitude in color map) vs ${\lambda }_{\mathrm{TG}}$ for $N=23$ at $T=0$.