Probing non-ergodicity and symmetry via coherent forward scattering in a shaken rotor

Abstract

The Coherent Backscattering (CBS) peak is a well-known interferential signature of weak localization in disordered or chaotic systems. More recently, a second interference feature -- the Coherent Forward Scattering (CFS) peak -- was predicted to emerge in the regime of strong localization. However, it has never been directly observed. Here we report the first direct observation of the CFS peak and demonstrate its dual role as a signature of non-ergodicity and as a probe of symmetries in quantum chaotic systems. Using a shaken rotor model realized with a Bose-Einstein condensate (BEC) of ultracold atoms in a modulated optical lattice, we investigate dynamical localization in momentum space. The CFS peak emerges in the position distribution as a consequence of non-ergodic dynamics, while its growth timescale reveals the underlying localization length. By finely tuning the modulation, we control time-reversal and parity symmetries and measure their distinct effects on both CBS and CFS peaks. Our results highlight the strong link of both the temporal growth and contrast of the CFS with symmetry and localization, making it a robust probe of these properties. This work opens new directions for characterizing non-ergodicity and symmetries in quantum chaotic or disordered systems, with possible applications in many-body localization and chaos.

Figures (12)

• Figure 1: Top: Sketch of the initial atomic density in a 1D lattice, which is subsequently periodically modulated in position ($\varphi(t)$) and amplitude ($F(t)$). Bottom: Experimentally measured Husimi quasi-distributions of (a) the initial squeezed Gaussian state peaked at $x=-\pi/2$, shown above the classical stroboscopic phase portrait, which highlights the classically chaotic region of size $L_p=L\hbar_\mathrm{eff}$. After undergoing periodic modulation for an evolution time $t>t_H$ sufficient for localization to set in, the coherent scattering peaks are measured: (b) In the absence of an appropriate time-reversal symmetry, only the CFS peak is observed. (c) When the dynamics possess the appropriate symmetry, both the CFS and the CBS peak, at $x=\pm\pi/2$ are observed. The Husimi distributions are obtained from a tomographic full quantum state reconstruction, combined for (b,c) with an average over realizations of chaotic dynamics (see text and Methods for details). Parameters are ($\mathbf{b}$) $\hbar_{\rm eff}=9.5$, $K=110\pm7$, $L=8.76\pm0.38$, $\xi_{\rm loc}=31.9\pm9.6$, $M=10$, ($\mathbf{c}$) $\hbar_{\rm eff}=9.55$, $K=113\pm7$, $L=8.99\pm0.20$, $\xi_{\rm loc}=38.9\pm 17.6$, $M=10$.
• Figure 2: Experimental protocol. A highly squeezed Gaussian state centered at $x=-x_0=-\pi/2$ is first prepared by a tailored lattice position (phase) modulation, using an optimal control (OC) algorithm for a fixed lattice depth $s_{\rm oc}$ and for a time $T_{\rm oc}$. The Husimi distribution of this initial state is represented in a. The lattice depth is subsequently shaken for $N$ time periods with different periodic modulation functions $f_m$, $1 \leq m\leq M$ (see frames b). The functions have a period $\nu^{-1}$ and contain $N_H$ harmonics with variable phases (see text). After modulation (see c), we perform a phase space rotation, holding the state for a quarter of the period in a static lattice of depth $s$. This transfers the probability density at the center of a lattice site onto the zero momentum component, whose population is measured by imaging the atoms after a time-of-flight (see d, e, f). This procedure is performed three times with three centers of rotation $x=x_r=-\pi/2,0,\pi/2$, set by a shift of the lattice phase, giving the evolutions and momentum populations $P_m(p,N|x_r)$ represented in panels d, e, f, and the bar diagrams on their right. This experiment is repeated $M$ times with different modulation function $f_m$, starting from the same initial state. By averaging all the contributions $P_1$,$...$,$P_M$ we obtain the signal encoding the CFS and CBS peaks and the background in the average amplitude of the zero momentum component. (g) Depending on the modulation regime and symmetries, the CBS or/and CFS peaks emerge from the background (blue, left: numerics, orange, right: experiment). The contrast is defined in g with respect to the background measured for $x_r=0$ (see text). Error bars indicate one standard deviation of the mean. Parameters: $s=24.2\pm 0.28$, $M=10$, $N_H=5$, $\nu=35.05$ kHz and $N=18$.
• Figure 3: Top row in each panel (a-d): Shape of periodic modulations of lattice depth (black) and position (red) corresponding to the symmetry regime (none, $\mathsf{T}$, $\mathsf{PT}$, $\mathsf{P+T}$). (a1–d1) Husimi representations of the maximum overlap Floquet eigenstate (see text) in each regime of symmetry. (a2,3–d2,3) Experimental (orange) and numerical (blue) contrasts obtained for different symmetries of the modulation, and in two different localization regimes: bounded $\xi_{\rm loc}/L\gtrsim 1$, and localized $\xi_{\rm loc}/L\lesssim 0.1$, where $\xi_{\rm loc}=\xi_p/\hbar_\mathrm{eff}$ is the localization length and $L=L_p/\hbar_\mathrm{eff}$ the extension of the chaotic sea, in units of $\hbar_\mathrm{eff}$. The signals are obtained using the method presented in Fig. \ref{['fig:Fig2_measurement']} with an additional average over 3 modulations periods ($N=[15,16,17]$ and $M=10$ if not otherwise stated). Parameters: ($\mathbf{a_2}$) $L=9.24\pm0.16$, $\xi_{\rm loc}=42\pm26$, $M=17$, ($\mathbf{a_3}$) $L=48.56\pm0.24$, $\xi_{\rm loc}=4.47\pm1.43$, $N=[6,7,8]$, ($\mathbf{b_2}$) $L=9.04\pm0.10$, $\xi_{\rm loc}=29.5\pm 16$, ($\mathbf{b_3}$) $L=23.96\pm0.38$, $\xi_{\rm loc}=1.52\pm 0.35$, ($\mathbf{c_2}$) $L=9.90\pm0.10$, $\xi_{\rm loc}=31\pm16$, ($\mathbf{c_3}$) $L=62.70\pm0.58$, $\xi_{\rm loc}=1.9\pm0.73$, ($\mathbf{d_2}$) $L=8.42\pm0.32$, $\xi_{\rm loc}=20.1\pm 8.5$, ($\mathbf{d_3}$) $L=25.78\pm0.42$, $\xi_{\rm loc}=0.77\pm 0.05$.
• Figure 4: CFS apparition timescale for various localization regimes and chaotic sea sizes $L$, and impact of $\mathsf{T}$-symmetry. Top panels show the averaged momentum distribution $\overline{P(p)}$ as a function of time, with the Husimi representation of the final state to the right. Bottom panels show the corresponding evolution of the averaged spatial density $\overline{P(x)}$. These numerical results are computed in a localized regime with large chaotic sea bounded by the dashed lines ($\mathbf{a_1}$), a classically bounded regime with a "small" chaotic sea ($\mathbf{b_1}$) and a bounded regime with a "large" chaotic sea ($\mathbf{c_1}-\mathbf{d_1}$). In $\mathbf{d_1}$, the modulation is chosen so that dynamics are $\mathsf{T}$-symmetry invariant, while all symmetries are broken for $\mathbf{a_1-c_1}$. Experimental and numerical signals (obtained as per Fig. \ref{['fig:Fig2_measurement']}) are shown at a short modulation time ($N=[4,5,6]$) for all cases ($\mathbf{a_2}-\mathbf{d_2}$), and at a longer time for cases $\textbf{c-d}$ ($N=[10,11,12]$, $\mathbf{c_3-d_3}$), showing the longer timescale of evolution of the CFS peak for larger classical bounds. The addition of $\mathsf{T}$-symmetry in $\textbf{d}$ strikingly impacts the CFS in the bounded regime shared by $\mathbf{c-d}$: the contrast is nonzero at short times, and saturates to an enhanced value (see text). Parameters: (a) $L=15.52\pm 0.34$, $\xi_{\rm loc}=5.97\pm2.8$, (b) $L=12.8\pm0.46$, $\xi_{\rm loc}=18.5\pm7.2$, (c) $L=15.9\pm0.24$, $\xi_{\rm loc}=52.5\pm26$, (d) $L=15.6\pm0.19$, $\xi_{\rm loc}=51\pm 31$.
• Figure A1: Estimation of the mean localization length $\xi_{\mathrm{loc}}$ and stochastic coefficient $K$. First row: $\mathbf{a_1}$ shows an example of modulation function and its stochastic coefficient $K_{1}$, and $\mathbf{b_1}$ the computed numerical evolution of the momentum density starting from the initial zero-momentum state (the condensate). $\mathbf{c_1}$ shows the time-averaged momentum density from period 10 to 20 obtained both numerically (purple) and experimentally (orange), showing good agreement. The slopes and sharp decrease indicate the localization length and classical box length $L$ respectively. Linear fits on the two slopes give left $\xi_{\mathrm{loc},1,l}$ and right $\xi_{\mathrm{loc},1,r}$ localization lengths. Second row: Similar computations can be performed for each modulation function $f_m$. Taking the mean values give the average $\xi_{\mathrm{loc}}$ and $K$ for this parameter regime. Parameter values are the same as for Figure \ref{['fig:Fig2_measurement']} of the main text (see table \ref{['tab:experimentalpara']}), in a broken symmetry case ($A_\varphi=2$). Parameters: $L=20.22 \pm 1.54$, $\xi_{\rm loc}=6.53\pm 2.87$.