Microscopy of cavity-induced density-wave ordering in ultracold gases

Abstract

We demonstrate high-resolution in-situ imaging of density-wave ordering induced by cavity-mediated interactions in a unitary Fermi gas. We observe long-range spatial correlations throughout the formation of density waves, both for adiabatic preparation and following a quench, with a pattern controlled by the cavity mode structure. Our single-shot microscopic images together with the real-time readout of the cavity photons provide access to atom-photon correlations. We use this capability to investigate order fluctuations as a function of time following a quench and to directly confirm the correspondence between optical and atomic observables. Our system opens rich perspectives, from local patterning to correlation measurements in long-range interacting quantum gases.

Figures (10)

• Figure 1: Imaging of cavity-induced density waves. (a) Sketch of the experimental setup, with a unitary Fermi gas illuminated by a retroreflected side-pump beam with a wave vector $\textbf{k}_{\text{p}}$ (red). Light scattered by the atoms in the cavity mode (faint red) and leaking through the cavity mirrors is measured using heterodyne detection. A microscope objective oriented along the vertical direction collects absorption images of the atoms, using an imaging beam with wave vector $\textbf{k}$ (light orange). (b) Schematic representation of the phase transition. Above a critical pump strength $V_{0\text{c}}$, the system enters the ordered phase with a finite order parameter $\Theta$. (c)-(f) Single shot absorption images of the atomic cloud for $V_0=0.6V_{0\text{c}}$ (c) and $V_0=1.4V_{0\text{c}}$ (d). (e) and (f) present magnified views on the image centers. In white, the wave vectors of the pump beam ($\textbf{k}_{\text{p}}$), the cavity field ($\textbf{k}_{\text{c}}$) and the resulting difference of the two ($\mathbf{k}_{-}$) are illustrated.
• Figure 2: Analysis of absorption pictures in Fourier space and imaging across the phase transition. (a) Modulus squared of Fourier transforms of measured atomic densities at an applied pump potential $V_0=1.6(1)V_{0\text{c}}$ (average over 6 repetitions). (b) Line cuts of (a) through the peak at $\mathbf{k}_-$ ($\blacksquare$), indicated with a dashed square, along the $x-$direction (top) and the $y-$direction (bottom). In black ($\medbullet$) line cuts of (a) at $\mathbf{k}=0$ are shown for comparison. (c) Single-shot line cuts of $|n_{\text{m}, \mathbf{k}}|^2$ across the peak at $\mathbf{k}_-$ along $x$ for $V_0=0.6, 1.3, 1.4, 1.5, 1.7V_{0\text{c}}$ (bottom to top, curves vertically offset by 2 $\times$$10^8$ for clarity). (d) Zoom-in on the cloud centers of the corresponding absorption images in (c). (e) Fourier transform at $\mathbf{k}_{-}$ as a function of $V_0$ and $\tilde{\Delta}_c$. The black symbols represent the phase boundary extracted from the onset of the superradiant photon signal. (f) Correlation length of the density wave along the $x$ (top) and $y$-direction (bottom) as a function of $V_0$, above $V_{0\text{c}}$ for $\tilde{\Delta}_{\text{c}} =$$-2.6(2)\MHz$. The horizontal error bars represent the standard deviation of repeated measurements. The vertical error bars are obtained from a bootstrap resampling analysis.
• Figure 3: Density-wave ordering after a quench. (a) Mean density modulation amplitude as a function of time, following a quench to $V_0 = 14.4(4)V_{0\text{c}}$. Inset: mean cavity field amplitude for the same realizations. The vertical error bars represent the standard deviation of repeated measurements. (b) Variance of the density modulation amplitude. Inset: variance of the cavity field amplitudes. The vertical error bars represent the statistical uncertainty due to finite sample size. (c) Cavity field amplitude for three experimental realizations at time $t=16µs$, (, , ) (offset for clarity). The extracted field endpoints are shown as a square symbols. The pump field is represented by the orange line (). (d) Zoom-in on absorption images recorded at the end of the three experimental realizations shown in (c). (e) Correlation between density-wave modulation and cavity field amplitude, each data point representing an individual experimental realization. The solid line represents a linear fit to the data. The realizations from panels (c)-(d) are marked as square symbols. (f) Correlation between the phase of the density-wave and the phase of the cavity field referenced to the phase of the pump field.
• Figure 4: Density-wave ordering at the first transverse cavity mode TEM$_{01}$. (a) Sketch of the experiment geometry with the transverse profile of the electric field amplitude of the TEM$_{10}$ mode (red curve) and the full width at half maximum of the atomic density profile (dashed lines). (b) Zoom-in at the center and (c) wide field image of the atomic density for density-wave ordering close to the first transverse mode of the cavity. (d) Local map of the amplitude (brightness) and phase (color map) of the density wave pattern across the cloud, averaged over 100 repetitions.
• Figure S1: Imaging of density-waves as scattering by a thick grating. We image the atomic cloud using an imaging beam propagating along $\mathbf{k}$. The plane in blue illustrates the plane defined by the wave vectors of the density modulation. The modulation along $\mathbf{k}_{-}$ leads to diffraction of the imaging light into the orders labeled with $0, \pm 1$. For an imaging beam orthogonal to the plane (orange), along $\mathbf{k}_\perp$, the diffraction efficiency is suppressed due to the thickness of the atomic cloud along the imaging beam direction. For an imaging beam incoming at a finite angle $\alpha$ with respect to $\mathbf{k}_\perp$ (brown) the diffraction into one of the first orders is favored which leads to an overall increase of the scattering efficiency. Drawn is the experimental condition with $\alpha = 5.2(9)$°.