Angular momentum of rotating fermionic superfluids by Sagnac phonon interferometry

Abstract

Fermionic many-body systems provide an unrivaled arena to investigate how interactions drive the emergence of collective quantum behavior, such as macroscopic coherence and superfluidity. Central to these phenomena is the formation of Cooper pairs, correlated states of two fermions that behave as composite bosons and condense below a critical temperature. However, unlike elementary bosons, these pairs retain their internal structure set by underlying fermionic correlations, essential for understanding superfluid properties throughout the so-called Bose-Einstein condensate (BEC) to Bardeen-Cooper-Schrieffer (BCS) crossover -- a cornerstone of strongly correlated fermionic matter. Here, we harness a sonic analog of the optical Sagnac effect to disclose the composite nature of fermionic condensates across the BEC-BCS crossover. We realize an in-situ loop interferometer by coherently exciting two counter-propagating long-wavelength phonons of an annular fermionic superfluid with tuneable interparticle interactions. The frequency degeneracy between clock- and anticlock-wise sound modes is lifted upon controllably injecting a quantized supercurrent in the superfluid ring, resulting in a measurable Doppler shift that enables us to probe the elementary quantum of circulation and the angular momentum carried by each particle in the fermionic fluid. Our observations directly reveal that the superflow circulation is quantized in terms of $h/2m$, where $m$ is the mass of the constituents, in striking contrast to bosonic condensates where $h/m$ is the relevant circulation quantum. Further, by operating our interferometer at tunable temperature, we measure the thermal depletion of the superfluid in the unitary Fermi gas, demonstrating phonon interferometry as a powerful technique for probing fundamental properties of strongly-correlated quantum systems.

Figures (8)

• Figure 1: Sagnac-like matter-wave interferometer with annular Fermi superfluids. In analogy with optical Sagnac interferometers utilizing counter-propagating light waves in a loop, our interferometer is based on the excitation of two counter-propagating sound modes in an annular fermionic-pair condensate with tunable interactions. In the absence of rotation (a), both excitations follow the dispersion relation $\omega_0 = c_s k$. In a rotating system (b), they experience a relative Doppler shift, yielding $\omega_{\pm} = c_s k \pm \Omega$, where the frequency splitting $\Omega$ is proportional to the angular momentum per particle. Such Doppler splitting manifests as a precession of the standing-wave atomic density pattern, accruing a phase shift $\Delta \theta = \Omega t$ over a time $t$. (c), In our phonon loop-interferometer, the longest-wavelength phonon modes with $\textbf{k}\bar{R} = \pm \hat{\theta}$ are simultaneously excited by a weak potential perturbation $V_\text{pert} = -V_0 \cos{\theta}$ that is suddenly quenched off at $t=0$, defining the onset of the dynamics. (d), In-situ atomic density profiles recorded at different evolution times reveal the precession of the standing wave, with the angular shift $\Delta\theta = \Omega t$, from which we experimentally extract $\Omega$. Here, $\tau = 2\pi/\omega_0$ denotes the phonon oscillation period.
• Figure 1: Density profiles and chemical potential. (a), In situ image of our annular superfluid in the unitary regime in presence of the perturbation potential $V_{\text{pert}}$. (b), To obtain the interferograms of Figs. \ref{['fig2']}-\ref{['fig3']}, we unwrap the angular coordinate, and (c), integrate along the radial direction to obtain a one-dimensional signal. The green dashed line represent the average homogeneous density profile $n_{\theta}^0$, which is $\simeq1/2\pi$. (d), Fermi energy and chemical potential to trap frequency ratio as a function of the interaction parameter.
• Figure 2: Phonon interferometry across the BEC–BCS crossover. Temporal evolution of the azimuthal density modulation $\delta n(\theta,t)$ for a BEC at $1/k_Fa = 3.31(7)$ (a) and a unitary Fermi gas (b), both in the absence of circulation. Dashed lines in panels a(i) and b(i) are a guide to the eye of the nodal points of the standing wave. Panels a(ii) and b(ii) display the corresponding density-weighted angular moments $\langle \cos\theta \rangle$ (brown) and $\langle \sin\theta \rangle$ (black), as defined in the main text. For visual clarity, $\langle \cos\theta \rangle$ is vertically offset by $+0.1$ and $\langle \sin\theta \rangle$ is multiplied by 3. To extract the phonon frequency $\omega_0$, we fit the density modulations $\delta n(\theta,t)$ using Eq. \ref{['eqlinearresponse']}; the resulting fits are shown as continuous lines on the moments traces. (c), The phonon velocity, expressed as $c_s/v_F = \omega_0 \bar{R}/v_F$ with $v_F = \sqrt{2mE_F}$ the Fermi velocity. Insets show the corresponding power spectral densities of panels a(i) and b(i), confirming the monochromatic excitation with $\bar{R}k = \pm1$, and the absence of second-sound coupling. (d), Quality factor $Q = \omega_0/\Gamma$ across the BEC–BCS crossover. (e), Damping rate $\Gamma/k^2$ expressed in units of $\hbar/m$. Error bars in panels c-e stem from the standard fitting error of $\delta n$.
• Figure 2: Fitting procedures. (a), Temporal evolution of the normalized azimuthal density in a rotating superfluid of winding number $\textit{w}=1$, in the unitary regime. (b), 2D fit applied to the interferogram shown in a. (c), calculated values of $\langle \cos{\theta} \rangle$ and $\langle \sin{\theta} \rangle$, with their corresponding 1D sinusoidal fit (brown and black curves, respectively). (d), results for the phonon frequency (i), the decay rate (ii) and the precession frequency (iii) extracted with the two fitting method.
• Figure 3: Phonon Doppler shift and angular momentum per particle across the BEC–BCS crossover. Temporal evolution of the azimuthal density modulation $\delta n(\theta,t)$ in a rotating superfluid with a single quantum of circulation ($\textit{w} = 1$) for a BEC at $1/k_Fa = 3.31(7)$ (a) and a unitary Fermi gas (b). The color-scales are the same of Fig. \ref{['fig2']}a-b, respectively. Dashed lines in panels a(i) and b(i) are guides to the eye of the nodal points of the standing wave. Panels a(ii) and b(ii) display the corresponding density-weighted angular moments $\langle \cos\theta \rangle$ (brown) and $\langle \sin\theta \rangle$ (black); for clarity, $\langle \cos\theta \rangle$ is offset by $+0.1$ and $\langle \sin\theta \rangle$ is multiplied by 3. The oscillatory behavior of $\langle \sin\theta \rangle(t)$ associated to the precession of the density modulation provides an unambiguous signature of rotation. We extract the Doppler shift $\Delta_c$, fitting $\delta n(\theta,t)$ with Eq. \ref{['eqlinearresponse']}, as a function of the total imprinted phase employed to generate the persistent current for the BEC (c), and UFG (d) regimes. In addition, we show the mean winding number $\langle\textit{w}\rangle$ measured over repeated acquisitions after the imprinting protocol. (e), Measured angular momentum per particle $\ell_z=m\bar{R}\Delta_c$ as a function of winding number $\textit{w}$ for a BEC at $1/k_Fa = 3.31(7)$ (green squares), a unitary Fermi gas (blue circles), and a BCS superfluid at $1/k_Fa = -0.55(1)$ (orange triangles). (f), Angular momentum per particle across the BEC–BCS crossover. The black cross marks the critical interaction strength below which persistent currents are no longer observed. Colors of symbols identify measurements in panel e. The dashed line marks $\ell_z / \hbar\textit{w}= 1 / 2$. Error bars in panels c-d stem from the standard fitting error of $\delta n$, whereas in panels e-f they represent the standard error of the mean over at least 3 repetitions.