Quantitative Acoustic Models for Superfluid Circuits

We experimentally realize a highly tunable superfluid oscillator circuit in a quantum gas of ultracold atoms and develop and verify a simple lumped-element description of this circuit. At low oscillator currents, we demonstrate that the circuit is accurately described as a Helmholtz resonator, a fundamental element of acoustic circuits. At larger currents, the breakdown of the Helmholtz regime is heralded by a turbulent shedding of vortices and density waves. Although a simple phase-slip model offers qualitative insights into the circuit’s resistive behavior, our results indicate deviations from the phase-slip model. A full understanding of the dissipation in superfluid circuits will thus require the development of empirical models of the turbulent dynamics in this system, as have been developed for classical acoustic systems.

Figure 1

Experimental system and oscillator frequency dependence in the nondissipative regime. (a) (Left) In situ experimental image for a reservoir radius $r=20\mu \mathrm{m}$, channel width $w=12.5\mu \mathrm{m}$, and channel length $\ell =35\mu \mathrm{m}$. (Right) Nondissipative dynamics of the population imbalance for this circuit geometry (red points), which are well fitted by a sinusoid (black curve, grey shading indicates 95% confidence interval). (b) As for (a) but with $\ell =1.5\mu \mathrm{m}$. (c) (Bottom) By measuring oscillations for different atom numbers [21], frequencies are extrapolated to $N=2\times {10}^6$ atoms (grey circles) for comparison with GPE simulations (orange diamonds) as a function of channel length for a fixed width $w=12.5\mu \mathrm{m}$. The black dash-dot line is a one-parameter fit of the acoustic model to the GPE data, where ${\mathcal{V}}_1={\mathcal{V}}_2=2\pi r^2{l}_z$ and $\mathcal{S}=2l_{z}w$, where $l_z$ is the Thomas-Fermi radius in the $z$ direction, resulting in an end correction of $\delta =2.18(1)\sqrt{\mathcal{S}}$. (Top) Shows the fit residuals. (d) Similar to (c), but for varying channel width with a fixed length $\ell =1.5\mu \mathrm{m}$, with the fit giving $\delta =2.14(3)\sqrt{\mathcal{S}}$. The speed of sound used in the fits was $c=\sqrt{\overline{n}g/m}$, where $\overline{n}$ is the spatially averaged atom number density of the GPE ground state. $c=2376\mu \mathrm{m}/\mathrm{s}$ and $c=2332\mu \mathrm{m}/\mathrm{s}$ in (c) and (d), respectively.

Figure 2

Acoustic Helmholtz oscillator and equivalent $LC$ circuit. (Left) A Helmholtz superfluid oscillator consists of two reservoirs of volume ${\mathcal{V}}_1$ and ${\mathcal{V}}_2$ coupled by a channel of cross sectional area $\mathcal{S}$ and length $\ell$. A pressure differential $\delta p$ leads to fluid exchange at velocity $\mathbf{u}$ between the reservoirs. The exchange of energy $E$ between the kinetic and potential terms leads to oscillations at frequency $\omega$, dependent on $\mathcal{S}$, $\mathcal{V}$, and the effective channel length ($\ell +\delta$), with the end correction $\delta$ accounting for fluid flow inside the reservoirs. (Right) Mapping $\delta p$ to voltage $\delta V$ and volume velocity $\mathcal{S}\mathbf{u}$ to current $\mathbf{I}$ results in an equivalent $LC$ circuit model for the system.

Figure 3

Superfluid atomtronic oscillation regimes, for channel width $w=12.5\mu \mathrm{m}$, and channel length $\ell =1.5\mu \mathrm{m}$. (Top row) Superfluid density dynamics, beginning from an initial $\eta =0.96(1)$ population imbalance. The Faraday images were taken following 3 ms time-of-flight expansion. In this regime, the large initial imbalance leads to phase-slip-induced dissipation; vortices can be clearly seen as dark holes in the density. (Bottom panel) Dynamics of the population imbalance for different initial imbalances. Red circles: $\eta =0.03(1)$; blue squares: $\eta =0.18(1)$; green diamonds: $\eta =0.60(5)$; black triangles: $\eta =0.96(1)$; continuous lines: GPE simulations. The data are offset for clarity, with horizontal lines denoting $\eta =0$. Uncertainties are 95% confidence intervals estimated from five datasets.

Figure 4

(a) Channel conductance as a function of programmed channel width, upon which $n_{1\text{D}}$ depends, for the GPE simulations (red diamonds) and experimental data (grey circles), for initial bias $\eta \sim 1$ and fixed channel length $\ell =10\mu \mathrm{m}$. The purple shaded area serves to guide the eye, highlighting a change in the conduction power law for small channel widths. For channel widths $w<3\mu \mathrm{m}$, the resolution of the DMD projection results in the channel floor rising as the width decreases, further reducing the density in the channel [21]. (b) Channel conductance as a function of the 1D channel density $n_{1\text{D}}$; the data fall on the same trend line. The power law fit (black dot-dashed line) demonstrates a deviation from the phase-slip model (see text). The light-grey region indicates 95% confidence intervals for the fit, and residuals are shown below. (c) Ratio of compressible and incompressible energies averaged over the resistive dissipation regime [exponential decay regime in Fig. 3], showing that compressible excitations are dominant for small channels.