Symmetry-Induced Logarithmic Relaxation in the Quantum Kicked Rotor

Abstract

We study the effect of discrete symmetries on coherent multiple scattering in the quantum kicked rotor. When the initial momentum is set to zero -- as in recent Bose-Einstein condensate experiments -- the effective pseudo-disorder becomes even under momentum inversion. The resulting discrete mirror symmetry of the dynamics profoundly alters spectral correlations: it generates quasi-degenerate Floquet doublets localised at opposite momenta, whose exponentially small splittings produce a hierarchy of exponentially large dynamical timescales. The coherent backscattering and forward-scattering peaks then exhibit a striking non-monotonic evolution and strongly asymmetric contrasts, followed by an exceptionally slow logarithmic relaxation toward a common asymptotic value -- a hallmark of glassy dynamics, here emerging in a fully coherent quantum system. That such archetypal glass-like behaviour arises from a single discrete symmetry constraint reveals an unexpected and deep connection between quantum coherence and slow relaxation phenomena.

Figures (9)

• Figure 1: Two hybridized localised eigenstates in momentum space (top panels) and in position space (bottom panels) obtained for $N = 129$, $K = 10$ and $\hbar_e = 2$ ($\xi_K = 6.25$). We represent the real part (in blue) and the imaginary part (in red) of an even eigenstate $\varphi_a^+$ (left panels) and of an odd eigenstate $\varphi^-_a$ (right panels) lying outside the central localisation box $|p| \leq \xi_K$ delimited by the vertical black dotted lines, but with the same localisation centres. The quasi-energy of the symmetric eigenstate $\varphi_a^+$ is $\epsilon_+ \approx -2.9548$ while that of the anti-symmetric eigenstate $\varphi_a^-$ is $\epsilon_- \approx -2.9568$, showing that these states form a doublet pair. As one can see, in position space both $\varphi_a^+$ and $\varphi_a^-$ are real and extended, while in momentum space $\varphi_a^+$ is real and $\varphi_a^-$ is purely imaginary. These properties are a signature of the parity-induced hybridization mechanism similar to a quantum double-well that we discuss in Section \ref{['sec:hybridization']}.
• Figure 2: Time evolution of the contrast $C(x,t)$ for the Quantum Kicked Rotor when the system starts in a position state $\ket{x_0}$. Top panels (a-c): the kick potential is even and symmetric random phases are used (parity and time reversal symmetry are preserved). Bottom panels (d-f): the kick potential is no longer even but symmetric random phases are still used (parity is broken and time-reversal symmetry is preserved). Each panel shows $C(x,t)$ at times $t = 1, 101, 1001, 50 \space 819$ and $1 \space 999 \space 998$ together with the time evolution of the CFS ($x=x_0$, dark colours) and CBS ($x=-x_0$, bright colours) peak values. For all plots, we have $M = 1024$ (lattice size $N=2M+1=2049$), $\hbar_e = 1$ and $x_0 = -\pi/2$. Panel (a): localised regime ($K = 6.32$, $\xi \sim \xi_K \approx 10 \ll M$). Panel (b): Intermediate regime ($K = 20$, $\xi \sim \xi_K \approx 100 < M$). Panel (c): Metallic regime ($K = 63.2$, $\xi \sim \xi_K \approx 1000 \approx M$). Panels (d,e,f): Same regimes when time-reversal is the sole symmetry of the system. Parameters are the same with the exception of (e) ($K = 11.2$). For readability, the CFS and CBS peaks have been coloured on the curves over a width equal to $2/\xi_K$, which is supposed to be the peak width at long times since $\xi \sim \xi_K$. The symmetric random phases $\alpha(p)$ are chosen randomly over a range $2\pi$ for $p \geq 0$ and duplicated for $p<0$. Data are then averaged over 1080 realizations of disorder. The contrast curves are smoothened for readability for $t>2$, with a moving average of width 80.
• Figure 3: Circles: Nearest-level spacing distribution $P(s)$ obtained for the HRQKR model \ref{['eq:qkr_halk_kick_evolution_operator']} with symmetric random phases in the even subspace $\mathcal{H}^+$ and with $\hbar_e=1$. The system size is $N=513$ and the data are averaged over $128$ disorder realizations. Since $\mathcal{H}^+$ contains $N/2$ states, quasi-energies have been rescaled by $2\Delta$ so that $\langle s\rangle=1$. The black circles have been obtained in the localised regime ($K=3$, $N_L=114$) while the red circles have been obtained in the metallic regime ($K=50$, $N_L=0.4$). Black solid line: Poisson distribution $P(s)=e^{-s}$. Red solid line: Wigner-Dyson distribution $P(s)=\frac{1}{2} \pi s\, e^{-\pi s^2/4}$.
• Figure 4: Main panel: Nearest-level spacing distribution $P(s)$ for the RQKR model \ref{['eq:qkr_halk_kick_evolution_operator']} with symmetric random phases in the full Hilbert space $\mathcal{H}=\mathcal{H}^+\oplus\mathcal{H}^-$ and with $\hbar_e=1$. Black data (averaged over $128$ disorder realizations): Metallic regime $K=50, N=513$ and $N_L = 0.4$. Blue data (averaged over $256$ disorder realizations): Localised regime $K=3, N=1025$ and $N_L=228$. Quasi-energies are rescaled by $\Delta=2\pi/N$ so that $\langle s\rangle=1$. Dashed red line: theoretical prediction $P_{\text{mix}}(s)=\frac{\pi s}{8}e^{-\frac{\pi s^2}{16}} \text{Erfc}(\frac{\sqrt{\pi}\,s}{4})+\frac{1}{2} e^{-\frac{\pi s^2}{8}}$ obtained for the mixture of two 2 independent GOE spectra Giraud2022. Inset: Numerically-extracted fraction of spacings in the Shnirelman peak for $0\leq s\leq 0.1$ and in the localised case. Parameters are $K=3$ and $N=127$, $257$, $513$ and $1025$. For $N=1025$, the estimated fraction is $\sigma_S=49.95\%$, compatible with the fraction $49.78\%$ computed with Eq. \ref{['eq:FracS']} and $\xi_K/N = 2.3 \, 10^{-3}$ ($N_L=228$). Red curve: Fit of the data with the theoretical prediction $\sigma_S= 1/2-\xi/N$, Eq. \ref{['eq:FracS']}. A reasonable agreement is obtained for $\xi = 14.34$.
• Figure 5: Integrated nearest-level spacing distribution $I(s)=\int_{0}^s P(s')ds'$ obtained for $\hbar_e=1$ as a function of $\ln s$ for $K=3$ (upper solid black curve) and $K=6$ (lower solid black curve). The system size is $N=1025$ and the data have been averaged over $256$ disorder realizations. Top panel: Separate fits for the linear and bell-shaped parts of the numerical data. The linear part for $K=3$ has been fitted over the interval $\ln s\in [-25,-5]$ with a straight line having a slope $\mu = 0.00358\simeq 3.67/N$ (upper red dashed line). The bell-shaped region around $\ln s =0$ has been fitted over the interval $\ln s \geq -2$ by the integrated Poisson distribution $I_P(s)=\int_0^s \sigma^2 e^{-\sigma s'} ds' = \sigma(1-e^{-\sigma s})$ plotted versus $\ln s$ (upper red dashed curve) where $\sigma$ is taken as a fit parameter. We have found $\sigma \approx 0.53$, giving $\sigma_S=0.47$ (see main text). The lower red dashed linear and curve fits for $K=6$ give $\mu=0.01097\simeq 11.24/N$, $\sigma\approx0.57$ and $\sigma_S = 0.43$. These $\sigma_S$ values should be compared with the fractions of spacings in the interval $[0,0.1]$, which are almost $0.5$ and $0.46$, respectively. This systematic but small deviation is probably due to the arbitrariness in the choice of the spacing $s$ delineating the Shnirelman peak and the Poisson tail. The vertical dashed line marks $s_{\max} = e^{-1}$. Bottom panel: Fit of the data obtained for $K=3$ and $K=6$ using Eq. \ref{['eq:IntP']} with the constant term $1/2$ replaced by $a$ and ($\sigma,a$) as free parameters. We find $\sigma\approx0.504$, $a\approx0.4855$ for $K=3$ and $\sigma \approx 0.511$, $a = 0.465$ for $K=6$. As one can see the agreement with Eq. \ref{['eq:IntP']} is quite good.