Soliton turbulence of a strongly driven one-dimensional Bose gas

Abstract

We study the out-of-equilibrium dynamics of a weakly interacting one-dimensional Bose gas in a box trap, subjected to a drive realized by a periodically oscillating linear potential. After a transient regime, the gas reaches a quasi-steady state, characterized by the presence of several solitons. At weak driving amplitude, the solitons are only weakly perturbed by one another, while at strong driving amplitude a regime analogous to turbulence is reached, where the solitons are strongly intertwined with each other. We show that a hallmark of both regimes can be found in the momentum distribution, which displays a power-law decay $n(k) \sim k^{-2}$ at weak driving amplitude and $n(k) \sim k^{-α}$ with a power-law exponent $α\in [7,9]$ at large amplitude. We further characterize each of the two regimes by following the space-time maps and characterizing the solitons using the inverse scattering transform. The protocol analyzed in this study is amenable to experimental realization in current experimental setups.

Figures (10)

• Figure 1: Density profile maps $n(x,t)=N |\psi(x,t)|^2$ in units of the mean density $n_0=N/L$. Space coordinate $x$ is expressed in units of $L$ while time $t$ is in units of $T=2L /c_0$, corresponding to the time it takes to make a round trip in the box at the speed of sound. Interaction strength fulfills $\tilde{g}N=10^4$ with $\tilde{g}=gmL/\hbar^2$. The driving amplitude is set to (a)$U_0= 0.1\, \mu_0$ and (b)$U_0=\mu_0$.
• Figure 2: Density profile in units of $n_0$ at $t=8T$ for (a)$U_0=0.1\mu_0$ and (c)$U_0=1\mu_0$. The central part of their respective Lax spectrum is shown in (b) and (d), with the eigenvalue $\zeta_i$ as a function of the index $i$. The dashed black lines delimits the inner region of the spectrum where eigenvalues correspond to solitons. Details on the method to extract this information are given in the main text and in Appendix \ref{['app:counting']}.
• Figure 3: (a) Time averaged momentum distribution $\bar{n}(k)$ in units of $L$ for $U_0=\mu_0$ and various values of the particle number multiplied by the dimensionless interaction strength $\tilde{g}N$. (b) All curves collapse on a single one by a proper choice of units involving their respective healing length $\xi_0$. The dashed lines indicate the power law behaviors $k^{-2}$ and $k^{-7.4}$.
• Figure 4: Time averaged momentum distribution $\bar{n}(k)$ in units of $L$ as a function of the wave vector $k$ in units of $\xi_0^{-1}$ in double logarithmic scale, for weak driving amplitudes (a) and large driving amplitudes (c). For all curves, the power-law behavior is highlighted by the dashed black lines: $\bar{n}(k) \sim k^{-2}$ in (a), and $\bar{n}(k) \sim k^{-\alpha}$ in (c) with exponent $\alpha=$$7.37$ (yellow), $7.4$ (purple), $8.7$ (green) and $9$ (blue), from bottom to top. The vertical dotted lines in (c) indicate the corresponding value of $\xi_{\rm Lax}^{-1}$ for reference. The power-law behaviour is highlighted in panels (b) and (d) by displaying $k^2\bar{n}(k)$ (respectively $k^{\alpha} \bar{n}(k)$) in log-log scale for the same driving amplitude parameters $U_0$ as in (a) (respectively (c)). All curves were obtained for $\tilde{g}N=10^4$ and averaged over the last six oscillation periods.
• Figure 5: Wavevector $k_2^-$ corresponding to the onset of the $k^{-2}$ power law decay of the momentum distribution in the simulations as compared to the estimate $k_-=2n_s[1-\langle\nu^2\rangle]$, both in units of $\xi_0^{-1}$, as a function of the driving amplitude $U_0$ in units of $\mu_0$. See text for detail, and Appendix \ref{['app:counting']} for the estimate of the uncertainty.