[Paper Review] Observation of anisotropic superfluid density in an artificial crystal

Junheng Tao, Mingshu Zhao+1 more|arXiv (Cornell University)|Jan 3, 2023

Cold Atom Physics and Bose-Einstein Condensates2 references4 citations

TL;DR

This study derives a hydrodynamics model for Bose-Einstein condensates in harmonic traps with a small-period optical lattice, linking superfluid density to anisotropic sound velocities via the Josephson sum-rule. The key finding is that anisotropic superfluid density can induce a negative moment of inertia in rotating frames, a purely quantum effect absent in classical systems.

ABSTRACT

We experimentally and theoretically investigate the anisotropic speed of sound of an atomic superfluid (SF) Bose-Einstein condensate in a 1D optical lattice. Because the speed of sound derives from the SF density, this implies that the SF density is itself anisotropic. We find that the speed of sound is decreased by the optical lattice, and the SF density is concomitantly reduced. This reduction is accompanied by the appearance of a normal fluid in the purely Bose condensed phase. The reduction in SF density -- first predicted [A. J. Leggett, Phys. Rev. Lett. 1543--1546 (1970)] in the context of supersolidity -- results from the coexistence of superfluidity and density modulations, but is agnostic about the origin of the modulations. We additionally measure the moment of inertia of the system in a scissors mode experiment, demonstrating the existence of rotational flow. As such we shed light on some supersolid properties using imposed, rather than spontaneously formed, density-order.

Motivation & Objective

• To establish a theoretical framework linking superfluid density to elementary excitation spectra in anisotropic systems.
• To analyze the hydrodynamics of rotating Bose-Einstein condensates in harmonic traps with weak optical lattices.
• To investigate how lattice-induced anisotropy in superfluid density alters collective modes and moment of inertia.
• To identify quantum signatures such as negative moment of inertia arising from superfluid flow anisotropy.

Proposed method

• Derives the Josephson sum-rule relating superfluid density to the spectral function of the one-body Green’s function.
• Applies linear response theory to connect perturbations in the condensate wavefunction and current operator to the spectral density A(k,ω).
• Uses the continuity equation to relate current and density response, enabling derivation of superfluid density in the k→0 limit.
• Solves Bogoliubov-de-Gennes equations on a mean-field ground state to compute modified phonon spectra under optical lattice potential.
• Constructs a hydrodynamic model for rotating condensates, incorporating anisotropic superfluid fractions f^sf_ij into the momentum and energy equations.
• Derives expressions for the moment of inertia from superfluid and normal fluid contributions, validated against numerical simulations.

Experimental results

Research questions

• RQ1 How does anisotropic superfluid density emerge in a weak optical lattice and what is its relation to sound velocity anisotropy?
• RQ2 What is the hydrodynamic response of a Bose-Einstein condensate in a harmonic trap with a small-period lattice under rotation?
• RQ3 Can the moment of inertia become negative due to anisotropic superfluid flow, and what does this imply about quantum vs. classical behavior?
• RQ4 How does the order of frame transformation and band projection affect the superfluid contribution to the moment of inertia?

Key findings

• The superfluid density ρ^sf_ĥk is derived from the k→0 limit of the spectral function A(k,ω), with anisotropy determined by direction-dependent sound velocities.
• Bogoliubov-de-Gennes calculations show that a shallow optical lattice suppresses sound velocities differently along x and y, opening a gap at the Brillouin zone edge.
• In the rotating frame without lattice, the moment of inertia can become negative when superfluid density along one axis is sufficiently reduced, a purely quantum effect.
• With a rotating lattice synchronized to the trap, the superfluid contribution to the moment of inertia is given by I^sf/I_c = (f^sf_xx ω_x² - f^sf_yy ω_y²)² / [(f^sf_xx ω_x² + f^sf_yy ω_y²)(ω_x² + ω_y²)], which remains positive and matches simulation data.
• The normal fluid contribution to the moment of inertia is I^n/I_c = (f^n_xx ω_x² + f^n_yy ω_y²)/(ω_x² + ω_y²), derived from the sum rule over excited states.
• The total moment of inertia is the sum of superfluid and normal fluid contributions, with the superfluid part uniquely sensitive to anisotropic f^sf_ij and frame transformation order.