Equilibration of Non-interacting Photons and Quantum Signatures of Chaos

TL;DR

The intimate relation between out-of-time-order correlators and boson sampling is demonstrated and the unitary dynamics of a Floquet system may be exploited to perform sampling tasks with identical particles using single-mode phase shifters and multiport beamsplitters.

Abstract

Equilibration plays a fundamental role in our understanding of statistical mechanics and the long-time dynamics of many-body systems. In quantum systems, the route to equilibration is intimately related to level repulsion and quantum signatures of chaos that are encoded in their unitary evolution. Chaotic quantum systems exhibit the level statistics characteristic of ensembles of random matrices. In this work, we demonstrate that single-particle chaos leads to equilibration of many non-interacting photons. We show that the underlying mechanisms for equilibration are operator spreading and quantum interference. More specifically, we demonstrate that the unitary dynamics of a general Floquet system implemented using single-mode phase shifters and multiport beamsplitters leads to equilibration of photons. We propose a realistic photonic implementation of the multiparticle kicked rotor, which is a Floquet system that we use as a concrete example of our general approach.

Figures (5)

• Figure 1: Schematic illustration of regular vs chaotic Floquet dynamics dynamics of $N$ photons in $M$ modes. The dynamics of the modes are generated by an $M\times M$ unitary matrix $\boldsymbol{U}_{S}(mT)$, where $M$ is the number of modes and $T$ is the period. a) When the dynamics are regular, the photons remain localized with restricted operator spreading (shaded areas) and not all of them are able to interfere. b) In the chaotic regime, the operators spread with a typical linear light cone (shaded areas). This allows all the photons to effectively interfere after a characteristic time (dashed line). Due to causality, identical photons can only interfere when their lightcones overlap. This interefence is the underlying mechanism that allows equilibration of local observables at long times.
• Figure 2: Schematic of a photonic chip that implements the dynamics of the kicked rotor (see \ref{['eq:SingleParticle']}), for demonstrating equilibration of multiple photons. The yellow boxes represent the local phase-shifter $\hat{U}_1$ (see \ref{['eq:m-splitter']}). The multiport beam splitter is achieved by bringing the waveguides together, which implements the unitary $\hat{U}_2$ (see \ref{['eq:m-splitter']}. The black box is one cycle of the drive, whose dynamics is given by the Floquet operator $\hat{\mathcal{F}}$ (see \ref{['eq:UnitarySingleParticle']}).
• Figure 3: There is a crossover from Poissonian to Gaussian Orthogonal Ensemble (GOE) statistics in the consecutive level spacing ratio, $r$. (a) The average $\langle r \rangle$, where $\theta$ is the rotation angle of the $M$-port beam splitter, and $\Phi$ is the strength of the harmonic trapping potential. The contour line delineates $\langle r \rangle = 0.53590$ for the GOE. The probability distribution of consecutive level spacing ratios, $P(r)$, is depicted in (b) $W = 7/(16\Phi)$ and $\theta = 7.4/(16\Phi)$ (upward triangle); (c) $W = 3.5/(16\Phi)$ and $\theta = 7.4/(16\Phi)$ (star); (d) $W = 2/(16\Phi)$ and $\theta = 7.4/(16\Phi)$ (circle); (e) $W = 3/(16\Phi)$ and $\theta = 18/(16\Phi)$ (downward triangle). Calculated for $|\mathcal{E}_U| = 100$ disorder realizations, $M=300$ modes, $\Phi = \pi/4$, and $T=1$.
• Figure 4: Stroboscopic dynamics of the two-point spectral form factors, $\mathcal{R}(mT)$ [see \ref{['eq:2SpectralFormFactor']}], shows the characteristic dip, ramp, and plateau of chaotic systems. When the spectral statistics are (a) Poissonian the SFF is close to the long-time asymptote $\bar{\mathcal{R}}_2 = M$ (blue horizontal line). Instead, in (c)-(d) the SFF more closely resemebles that expected in a chaotic system [see \ref{['eq:sff-goe']}], with a dip, ramp, and plateau. In the crossover regime (b) the kicked rotor exhibits weak QSOC. We set the Heisenberg time as (a) $\tau_H \approx 535T$, (b) $585T$, (c) $605T$, (d) $300T$. The upwards triangle, star, circle, and downwards triangle correspond to the subfigures in \ref{['Fig2']}. Calculated for $|\mathcal{E}_U|=1000$ disorder realizations and $M=300$ modes.
• Figure 5: Dynamics over $m=12$ cycles for $N=2$ photons in $M=12$ modes for a single realization of disorder. $a)$ ($W=7/16\Phi$) and $b)$ ($W=1/8\Phi$) depict the dynamics of the mean number of photons $\langle\hat{n}_l(mT)\rangle$ for regular and chaotic unitaries, respectively. $d)$, $e)$ show $P_{\text{F}}(mT)$ for regular and chaotic unitaries, respectively. Clearly, in the regular regime, the system only explores a small portion of the available configurations. We benchmark our results using the a unitary evolution drawn from the Haar measure in $c)$ and f). The dynamics in $a)$ and $b)$ resemble the light cone structure illustrated in \ref{['Fig0']}. For the simulation we set $\Phi=\pi/4$ and $\theta = 7.4/16 \Theta$.