Generation of wave turbulence in dipolar gases driven across their phase transitions

TL;DR

The paper investigates wave turbulence emerging when a dipolar Bose-Einstein condensate is dynamically driven across supersolid and superfluid phase boundaries. Using an extended Gross-Pitaevskii framework that includes beyond-mean-field corrections, the authors show robust nonequilibrium quasi-steady states characterized by self-similar momentum distributions with algebraic tails $\tilde{n}(|\mathbf{k}|,t) \sim |\mathbf{k}|^{-\gamma}$, with $\gamma$ around $2.5$, and a cascade front indicating a direct energy cascade to large momenta. The roton minimum amplifies turbulence, and the turbulent state arises universally across initial phases, driving frequencies, and crossing directions, even in the presence of three-body losses. Dissipation can suppress high-momentum tails and modify the tail exponent, but the overall emergence of wave turbulence remains observable under experimentally relevant conditions, highlighting universal weak wave turbulence in anisotropically long-range interacting quantum gases. These findings advance the understanding of turbulence in complex quantum fluids and provide a route to experimentally access universal turbulent dynamics in dipolar systems.

Abstract

Ultracold quantum gases with long-range anisotropic interactions host novel exotic phases of matter, such as supersolids, exhibiting both rigid and superfluid characteristics. The impact of this interplay on the out-of-equilibrium dynamics of dipolar gases, and in particular its connection with universal turbulent behavior, remains highly unexplored. Here, upon considering a dipolar Bose-Einstein condensate of dysprosium atoms being dynamically driven across the supersolid-superfluid phase transition and vice versa, we unveil the emergence of a robust nonequilibrium quasi-steady state. This state displays self-similar momentum distributions exhibiting algebraic decay at large momenta, with scaling exponents supporting the existence of wave turbulence. We demonstrate that supersolidity sustaining higher-lying momenta, associated with the roton minimum, promotes the development of turbulence. Our results provide a stepping stone toward unraveling and exploiting turbulent and self-similar behavior in anisotropically long-range interacting quantum gases amenable in current experiments.

Figures (10)

• Figure 1: Emergence of wave turbulence in a SS upon dynamically crossing the SS-to-SF phase transition. (a)-(e) Three dimensional density isosurfaces [referring to $1\%$ (red), $10\%$ (yellow) and $25 \%$ (blue) of the 3D peak densities] at different time instants. The initial (final) 3D scattering length is $a_{\text{i}}=89 \, a_0$ ($a_{\text{f}}=98 \, a_0$) and the driving frequency $\omega_{\rm d}= 2\pi \times 127 \, \rm{Hz}$.
• Figure 2: Characterization of the self-similar turbulent response and its insensitivity on the driving frequency and initial state. (a) Compensated momentum distribution $\tilde{n}(\left| \textbf{k} \right|,t) (\left| \textbf{k} \right| l_{\rm s})^{\gamma}$ saturating to a plateau at large evolution times and momenta, obeying a power-law behavior with exponent $\gamma=2.5$. The dashed line determines the threshold for quantifying $\left| \textbf{k}_{\rm cf} \right|$, and the vertical dash-dotted and dotted lines mark $\left| \textbf{k}_{\rm{rot}} \right|$ and the forcing wavenumber, $\left| \textbf{k}_{\rm f} \right|$, respectively. The inset displays the ratio of integrated densities, $n_x/n_y$, signaling the isotropic dynamics at long evolution times. Panels (b) and (c) present the exponent dynamics $\gamma(t)$ with respect to the driving frequency, $\omega_{\rm d}$, in the case of the (b) SS-to-SF transition ($a_{\rm i}=89 \, a_0$, $a_{\rm f}=98 \, a_0$) and (c) vice versa ($a_{\rm i}=98 \, a_0$, $a_{\rm f}=91 \, a_0$). The dashed lines represent the mean values of the exponents over all $\omega_{\rm d}$. Other parameters are the same as in Fig. \ref{['Fig:Density_profiles']}.
• Figure 3: Dynamical crossing of a SS to the isolated droplets regime leading to wave turbulence. Time evolution of the scaling exponent under continuous driving from the SS ($a_{\rm i}=89~a_0$) to droplets ($a_{\rm f}=82~a_0$) phase for different driving frequencies (see legend). Saturation of the exponent around $2.41$ (dashed line) indicates the approach to a quasi-steady state. The latter encompassing a SF background is visualized in the inset depicting $\int dz~ \left| \Psi(\textbf{r},t) \right|^2$ pertaining to $\omega_{\rm d}=2\pi \times 126~\rm{Hz}$ at long evolution times.
• Figure 4: Wave turbulence develops faster in a SS as compared to a SF. Cascade front $\left| \textbf{k}_{\rm cf} \right| l_{\rm s}$ when transitioning from (a) SS-to-SF [$a_{\rm i}~(a_{\rm f})=89~(98)~a_0$], and (b) vice versa [$a_{\rm i}~(a_{\rm f})=98~(91)~a_0$] for different driving frequencies, $\omega_{\rm d}$. The black dashed lines represent the fits $\left| \textbf{k}_{\rm cf} \right| \sim t^{-\beta}$. The extended momentum distribution of the SS leads to a faster saturation of the cascade front compared to the SF.
• Figure 5: Onset of wave turbulence in a driven SS in the presence of three-body recombination. (a) Compensated spectrum with the inclusion of three-body losses for $a_{\rm i}=89~a_0$, $a_{\rm f}=98~a_0$ and $\omega_{\rm d}=2 \pi \times 127 ~{\rm Hz}$ as in Fig. \ref{['Fig:Momentum_exponents']}(a). Saturation of the spectrum at $t>500~{\rm ms}$ takes place manifesting the approach to turbulent behavior. The inset presents the $\left| \textbf{k} \right|^{-\gamma}$ power-law fitting of the momentum distribution in the range $\left| \textbf{k} \right| l_{\rm s} \simeq 1$. (b) Comparison between the exponent dynamics at large momenta with finite and zero three-body recombination (see legend). As it can be seen, there are no appreciable deviations imprinted in the exponent between the finite $L_3$ and $L_3=0$ scenario, while in both cases the exponent saturates close to $\approx 2.51$. The inset presents the particle number during the time evolution, normalized to the initial number $N_0=8 \times 10^4$. Even for the $\sim 40 \%$ atom losses occurring at long evolution times a SS structure persists.