Superfluid oscillator circuit with a quantum current regulator

We examine the properties of the atomic current in a superfluid oscillating circuit consisting of a mesoscopic channel that connects two reservoirs of a Bose-Einstein condensate. We investigate the presence of a critical current in the channel and examine how the amplitude of the oscillations in the number imbalance between the two reservoirs varies with the system parameters. In addition to highlighting that the dissipative resistance stems from the formation of vortex pairs, we also illustrate the role of these vortex pairs as a quantum current regulator. The dissipation strength is discrete based on the number imbalance, which corresponds to the emergence of vortex pairs in the system. Our findings indicate that the circuit demonstrates characteristics of both voltage-limiting and current-limiting mechanisms. To model the damping behavior of the atomic superfluid circuit, we develop an equivalent LC oscillator circuit with a quantum current regulator.

Figure 1

The initial density distribution of the BEC in the oscillator circuit consisting of two reservoirs of radius $R=4.5a_o$ and a narrow channel of length $l=3a_o$ and width $d=0.4a_o$. Here, the potential bias is $V=0.1V_c$.

Figure 2

(a) The time evolution of the shifted number imbalance, $\eta \left(t\right)-\eta \left(0\right)$, for 51 different $V$ uniformly located in the region $[0,0.1]V_c$. The variable $V$ is observed to gradually increase from the yellow lines to the blue lines. The inset is the enlargement of the region in the red box. (b) The density distribution (left subfigure) and the corresponding phase distribution (right subfigure) in the red box region at $t=2.5t_0$ with $V=0.08V_c$. The length and the width of the channel in panels (a) and (b) are $l=2a_o$ and $d=0.6a_o$, respectively. (c) The density and the phase distributions at $t=2.5t_0$ with $V=0.062V_c, l=3a_o$, and $d=0.4a_o$.

Figure 3

The dependence of ${\omega }^2$ (or ${\omega }^{-2}$) on two factors of the channel: (a) the width $d$ and (b) the length $l$. The triangles accompanied by error bars, represent the statistical average of $\omega$ across different $V$ in the range of $[0,0.1]V_c$. The error bars indicate the range of ${\omega }^2$ (or ${\omega }^{-2}$) resulting from all $V$. Additionally, solid lines depict the fitting results obtained from Eq. (3) using $\delta =2.3d$.

Figure 4

(a) The amplitude $A$ and (b) the maximum current $I_{\max }$ as functions of $V/V_c$ for $l=2a_o$ and $d=0.6a_o$. The black dashed line in panel (a) is the result of the TF approximation. The red solid lines in panels (a) and (b) are determined by the equivalent quantum circuit described by Eqs. (6, 7, 8). (c) The critical amplitude $A_c$ and (d) the critical current $I_c$ with respect to the geometry of the channel. The blue dashed line in panel (d) is the fitting result, and the red dashed line is the result given by the Landau critical velocity.

Figure 5

(a) The dissipation strength $D={\eta }_0-A$ as a function of the potential bias $V$ for $l=2a_o$ and $d=0.6a_o$. The red solid line is the result of the equivalent quantum circuit described by Eqs. (6, 7, 8). (b)–(e) Vortices created for the biases marked in panel (a). (f) The dissipation step $D_s$ as a function of $l$ and $d$. (g) The energy loss (the blue dashed line) and the energy of a single vortex-pair (triangles) as a function of $d$ for different $l$.

Figure 6

The time evolution of $\eta \left(t\right)-\eta \left(0\right)$ for the equivalent quantum circuit (inset), where $\eta \left(t\right)=2Q\left(t\right)$, with $Q\left(t\right)$ being the charges stored in the capacitance. This simulation corresponds to the superfluid circuit of $l=2a_o$ and $d=0.6a_o$ with $\eta \left(0\right)\in [0,0.1]$ as shown in Fig. 2. In the simulation, the parameters $\omega =0.8984{\omega }_0, D_s=0.0145$, and $I_c=0.0289{\omega }_0$ were used. These values correspond to approximately $95\%$ and $141\%$ of the theoretical estimation values for ${\omega }^{\mathrm{th}}=0.9457{\omega }_0$ [obtained from Eq. (1)] and $D_s^{\mathrm{th}}=0.0103$ (derived from $E_{\text{loss}}=E_{\mathrm{vp}}$), respectively. The results for $A, I_{\max }$, and $D$ are shown in Figs. 4, 4, and 5, respectively.