Single-fluid model for rotating annular supersolids and its experimental implications

TL;DR

This work addresses how to describe and detect rotation in annular supersolids with partial angular-momentum quantization. It introduces a single-fluid hydrodynamic framework where the spatial phase of the global wavefunction sets both classical and superfluid flows, yielding an angular momentum $L = I_c(1-f_s)\Omega + N w f_s \hbar$ and a phase field $\phi_w(x,\Omega)$ that governs the current. Through numerical simulations of the extended Gross-Pitaevskii equation for dipolar atoms in a ring and phase-imprinting protocols, it demonstrates controlled excitation of persistent currents and metastable states across $0\le f_s\le 1$, and provides a concrete protocol using a phase imprinting potential $V_{PI}(x)=\hbar \phi_w(x;\Omega)/\tau$. It also proposes a time-of-flight readout that converts the superfluid angular momentum into a measurable classical rotation by ramping to a droplet crystal, enabling extraction of the initial angular momentum, with the momentum distribution revealing a $R_k$ hole that tracks $w$ while being insensitive to partial quantization. Together, the results reconcile the single-fluid picture with the two-fluid intuition in the appropriate limit and extend applicability to other density-modulated superfluids, including optical lattices and annular Josephson systems.

Abstract

The famous two-fluid model of finite-temperature superfluids has been recently extended to describe the mixed classical-superfluid dynamics of the newly discovered supersolid phase of matter. We show that for rigidly rotating supersolids one can derive a more appropriate single-fluid model, in which the seemingly classical and superfluid contributions to the motion emerge from a spatially varying phase of the global wavefunction. That allows to design experimental protocols to excite and detect the peculiar rotation dynamics of annular supersolids, including partially quantized supercurrents, in which each atom brings less than $\hbar$ unit of angular momentum. Our results are valid for a more general class of density-modulated superfluids.

Figures (6)

• Figure 1: Single fluid model. Microscopic velocity field (blue) and corresponding phase profile (magenta) for a rotating supersolid with $f_s \approx$ 50% in the classical manifold with zero circulation quanta (a), $w=0$, and with one circulation quantum (b), $w=1$. The supersolid density is plotted in gray. The horizontal axis is in units of the lattice period $d$. In this plot $\Omega/\Omega_c\approx 1.3$. c) Energy in the lab frame $E_0$ (dashed) and $E_1$ (dashed-dotted) versus the angular momentum $L$, for three different values of $f_s$. Thick lines indicate minimum energy states for a given $L$. $E_V=(N\hbar)^2/2I_c$ is the vortex energy ($\approx3\;µK \times k_B$ with our parameters). The stars indicate the local energy minima, corresponding to persistent currents.
• Figure 2: Phase imprinting of angular momentum to the supersolid. Supersolid ground state with $f_s \approx 0.5$ (a) and corresponding 2D potential (b) used to phase imprint a current. c) Supersolid density $\rho(x,t)$ as a function of time, with $\Omega/\Omega_c\approx 0.7$ and $f_s\approx 0.5$, showcasing the rigid body motion of the density. d) Angular momentum $L$ vs angular velocity $\Omega$ for a rotating supersolid. The points are the results of numerical simulations, while the straight lines represent the prediction of the model, Eq. \ref{['eq:L']}. The corresponding superfluid fractions are indicated. Full points represent ground states in the corotating frame, while open points are metastable states, see inset. The points for $w=1$ and $\Omega = 0$, surrounded by the box, are persistent current states corresponding to the energy minima in Fig. \ref{['fig:PhaseFields']}(c).
• Figure 3: Measure of the angular momentum of the supersolid. Superfluid ($L_s$, gray) and classical ($L_c$, black) angular momentum as a function of time during a quench to the droplet crystal regime, $f_s\rightarrow 0$. The initial state is a persistent current with fully superfluid angular momentum, $L/N\hbar = f_s$. After the ramp, the angular momentum is transferred into the classical component, measurable through the classical rotational velocity of the droplets (inset).
• Figure 4: Superfluid in a rotating optical lattice. a) Angular momentum $L$ vs angular velocity $\Omega$ for a superfluid in a rotating lattice for the $w=0$ and $w=1$ manifolds. Different colors correspond to different superfluid fractions, as displayed in the picture. b) Preparation of a classical rotational state ($w=0$) through a lattice rotation with angular velocity $\Omega(t) = \Omega(1-e^{-t/\tau})$, with $\Omega/\Omega_c\approx 0.6$, $\tau=50\;$ms. The points are the results of the simulation, while the continuous line is $L_c(t) = (1-f_s)I_c\Omega(t)$. The inset shows the density $\rho(x,t)$. c) Phase profile in the steady state of panel b) (lighter purple, thick) with the theoretical prediction based on the 1D model (purple, dashed). The density is plotted in gray for reference.
• Figure 5: Superfluid fraction and pinning lattice a) Superfluid fraction of the ground state of the supersolid as a function of the $s$-wave scattering length $a_s$, in units of the Bohr radius $a_0$. The full blue circles are results of simulations in which a pinning lattice is added in the ground state while the hollow purple points are without the lattice. The inset shows two degenerate ground states differing by a rotation of the clusters. The two wheels are references for the eyes. b) Superfluid fraction $f_s$ of a superfluid in an external optical lattice, as a function of the lattice depth $U$.