Probing supersolidity through excitations in a spin-orbit-coupled Bose-Einstein condensate

Abstract

Spin-orbit-coupled Bose-Einstein condensates are a flexible experimental platform to engineer synthetic quantum many-body systems. In particular, they host the so-called stripe phase, an instance of a supersolid state of matter. The peculiar excitation spectrum of the stripe phase, a definite footprint of its supersolidity, has been difficult to measure experimentally. Here, we perform in situ imaging of the stripes and directly observe both superfluid and crystal excitations. We investigate superfluid hydrodynamics and reveal a stripe compression mode, thus demonstrating that the system possesses a compressible crystalline structure. Through the frequency softening of this mode, we locate the supersolid transition point. Our results establish spin-orbit-coupled supersolids as ideal systems to investigate supersolidity and its rich dynamics.

Figures (9)

• Figure 1: A robust high-contrast spin-orbit-coupled supersolid. (a) Spin-orbit-coupled BECs at small coupling have a dispersion relation with two minima, leading to two possible many-body phases: the unmodulated plane-wave phase (one minimum occupied) and the modulated supersolid-stripe phase (both minima occupied). (b) Two-photon Raman coupling scheme with two beams of frequency difference $\Delta\omega_\mathrm{R}$, coupling strength $\Omega$, detuning $\delta$, and recoil momentum per beam $\hbar k_\mathrm{R}$ (tunable through the angle of the Raman beams $\theta$), used to engineer spin-orbit coupling. (c) Dispersion relation $E(k)$ of the lower optically-dressed state (solid line), compared to that of the bare atomic states (dashed lines). Colorscale: spin polarization $P$ of the dressed state. The system can be modeled as a mixture of condensates in the left $\ell$ and right $r$ wells, which consist of a coherent superposition of the bare spin states (red and blue circles) and have modified effective interaction properties given by the effective scattering lengths $a_{\ell\ell}$, $a_{\ell r}$ and $a_{rr}$. (d) Effective interaction parameters at $\delta=0$ vs. Raman coupling $\Omega$, calculated via the mixture model. Vertical black dashed line: supersolid phase transition point $\Omega_\mathrm{c}$ separating the supersolid-stripe phase and the plane-wave phase. This panel shows how tuning interaction strengths can stabilize the stripe phase vs. the plane-wave phase. (e) Phase diagram of a homogeneous spin-orbit-coupled BEC featuring a large and stable supersolid stripe phase, calculated with the variational ansatz of Li2012Martone2015. $C$ is the contrast of the density modulation. Dashed lines: phase boundaries calculated with the mixture model. Gray band: region experimentally explored in this work, with detuning $\hbar\delta=0.00\pm0.02E_\mathrm{R}$. Parameters in (d) and (e) correspond to $^{41}$K at a magnetic field $B=51.7$ G and with a density $1e14\text{atoms}/cm^{3}$, which realizes a robust and high-contrast supersolid stripe phase.
• Figure 2: In situ observation of supersolid stripes. (a) Single shot experimental image of the in situ density profile of the cloud at $\hbar\Omega=2.00\pm0.08\,E_\mathrm{R}$ measured with matter-wave optics. The image is rescaled to represent the expected cloud aspect ratio before matter-wave optics magnification along $x$. Top right: corresponding density profile integrated along the stripes. The black line denotes a modulated Gaussian fit with stripe spacing $d$. Bottom right: histogram showing the spatial phase of the modulation $\phi$ over 350 realizations. (b) Spacing $d$ of the stripes vs. Raman coupling $\Omega$ measured from in situ (blue circles) and time-of-flight (gray squares, see methods). Solid line: single-particle theory prediction, obtained from the inverse of the momentum difference of the minima of the dispersion relation (shown in the inset for different spin-orbit coupling strengths $\Omega$) methods. Errorbars correspond to one standard error of the mean (eom). The blue shaded area shows the systematic error of the in situ data from the magnification calibration.
• Figure 3: Crystal Goldstone mode and superfluid hydrodynamics. (a) Population imbalance between side stripes $\eta$ vs. displacement of the central stripe $D$ with respect to the center of mass of the cloud (see inset and methods). For a harmonically-trapped system with superfluid flow, where a sliding of the density modulation is compensated by particle exchange between the stripes, the two quantities are linearly correlated and provide evidence for the crystal Goldstone mode. (b) Top panel: breathing mode of the overall cloud, observed through its size $\sigma$ vs. time. Solid line: sinusoidal fit used to extract the breathing frequency $\omega_{\mathrm{B}}$, yielding $\omega_{\mathrm{B}}/\omega_{\mathrm{D}}=1.5\pm0.1\sim\sqrt{5/2}$ with $\omega_{\mathrm{D}}$ the dipole mode frequency, as expected from superfluid hydrodynamics. Bottom panel: number of stripes in the supersolid cloud. It dynamically changes during half of a breathing oscillation (see shaded area in top panel), supporting the existence of superfluid flow across the system. Errorbars denote one standard eom.
• Figure 4: Stripe compression mode and supersolid phase transition. (a) In situ observation of the stripe compression mode. After a rapid ramp of the coupling strength $\Omega$ ending at $t=0$, the spacing of the stripes $d$ oscillates in time, demonstrating that the crystal structure is not stiff. Errorbars denote one eom. Bottom: sketch of the stripe compression in real space (left) and in momentum space (right). (b) Observation of mode softening from momentum-space measurements. Top: collective frequencies of the dipole mode (red squares) and stripe compression mode (blue circles), in units of the trapping frequency $\omega_x$, vs. coupling strength $\Omega$. Solid lines: mixture model predictions without fitting parameters. Vertical black dashed line: thermodynamic limit prediction of the critical point $\Omega_\mathrm{c}$methods. Bottom: example oscillations of the relative momentum of the dressed states $\hbar\Delta k$ for $\hbar\Omega=0$, $1.50\pm0.06$ and $2.10\pm0.08\,E_\mathrm{R}$, with the exponentially decaying sinusoidal fits (solid lines) used to extract the corresponding frequencies. Errorbars: error on the fit (top panel) and standard deviation of up to three repetitions (bottom panel). (c) Top: frequency softening of the stripe compression mode for magnetic fields $B= 51.52\pm0.02$, $51.75\pm0.02$ and $51.85\pm0.02$ G (from dark to light blue, respectively). Bottom: phase diagram of the supersolid-to-plane-wave phase transition, calculated from the mixture model. Solid blue lines: magnetic field values corresponding to the top panel. Gray dashed line: phase boundary predicted by the mixture model.
• Figure S1: Experimental setup and scattering length. (a) Sketch of the optical setup. Top panel: $x$-$z$-plane (side view) with confining beam, guide beam and direction of the magnetic field $B$. Bottom panel: $x$-$y$-plane (top view) with guide beam and Raman beams. (b) Top panel: location of the $\uparrow\downarrow$ Feshbach resonance through loss spectroscopy. Blue circles (red squares) show the atom number of state $\uparrow$ ($\downarrow$). The shaded area indicates the field $B_0$ of the resonance. Bottom panel: dependence of the $a_{\uparrow\downarrow}$ (purple dashed line), $a_{\uparrow\uparrow}$ (blue solid line) and $a_{\downarrow\downarrow}$ (red solid line) scattering lengths on the magnetic field $B$ as resulting from the loss spectroscopy measurements and Tanzi2018. Vertical black dotted lines line: magnetic field values used in this study.