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Quantum variance and fluctuations for Walsh-quantized baker's maps

Laura Shou

TL;DR

The paper analyzes Walsh-quantized baker maps as models of quantum chaos on the torus, establishing a detailed link between quantum matrix-element fluctuations and classical baker-map correlations. It proves that for $D eq4$, the scaled diagonal matrix-element fluctuations are asymptotically Gaussian with variance $V(a)$ determined by classical time-correlation sums, and the empirical distribution converges to $ ext{N}(0,V(a))$; for $D=4$ a fractal subset induces a subtle, observable-dependent shift, producing a Gaussian mixture in the limit. It also proves a quantum-ETH-type statement for off-diagonal matrix elements: after appropriate normalization by a classical-correlation factor $f_a(oldsymbol{ alpha},oldsymbol{ beta})$, off-diagonal entries converge to standard complex Gaussian distributions, both within and across eigenspaces. The analysis hinges on phase-cancellation mechanisms near the Ehrenfest time, precise Weingarten calculus for random quadratic forms, and a careful treatment of classical correlations encoded by the baker map. Overall, the results provide a precise rate of convergence in quantum ergodicity for these highly degenerate quantum maps and reveal how microscopic eigenstate structure, via classical correlations, shapes fluctuations and ETH-like behavior in a discrete-time quantum chaotic system.

Abstract

The Walsh-quantized baker's maps are models for quantum chaos on the torus. We show that for all baker's map scaling factors $D\ge2$ except for $D=4$, typically (in the sense of Haar measure on the eigenspaces, which are degenerate) the empirical distribution of the scaled matrix element fluctuations $\sqrt{N}\{\langle \varphi^{(j)}|\operatorname{Op}_{k,\ell}(a)|\varphi^{(j)}\rangle-\int_{\mathbb{T}^2}a\}_{j=1}^{N}$ for a random eigenbasis $\{\varphi^{(j)}\}_{j=1}^{N}$ is asymptotically Gaussian in the semiclassical limit $N\to\infty$, with variance given in terms of classical baker's map correlations. This determines the precise rate of convergence in the quantum ergodic theorem for these eigenbases. We obtain a version of the Eigenstate Thermalization Hypothesis (ETH) for these eigenstates, including a limiting complex Gaussian distribution for the off-diagonal matrix elements, with variances also given in terms of classical correlations. The presence of the classical correlations highlights that these eigenstates, while random, have microscopic correlations that differentiate them from Haar random vectors. For the single value $D=4$, the Gaussianity of the matrix element fluctuations depends on the values of the classical observable on a fractal subset of the torus.

Quantum variance and fluctuations for Walsh-quantized baker's maps

TL;DR

The paper analyzes Walsh-quantized baker maps as models of quantum chaos on the torus, establishing a detailed link between quantum matrix-element fluctuations and classical baker-map correlations. It proves that for , the scaled diagonal matrix-element fluctuations are asymptotically Gaussian with variance determined by classical time-correlation sums, and the empirical distribution converges to ; for a fractal subset induces a subtle, observable-dependent shift, producing a Gaussian mixture in the limit. It also proves a quantum-ETH-type statement for off-diagonal matrix elements: after appropriate normalization by a classical-correlation factor , off-diagonal entries converge to standard complex Gaussian distributions, both within and across eigenspaces. The analysis hinges on phase-cancellation mechanisms near the Ehrenfest time, precise Weingarten calculus for random quadratic forms, and a careful treatment of classical correlations encoded by the baker map. Overall, the results provide a precise rate of convergence in quantum ergodicity for these highly degenerate quantum maps and reveal how microscopic eigenstate structure, via classical correlations, shapes fluctuations and ETH-like behavior in a discrete-time quantum chaotic system.

Abstract

The Walsh-quantized baker's maps are models for quantum chaos on the torus. We show that for all baker's map scaling factors except for , typically (in the sense of Haar measure on the eigenspaces, which are degenerate) the empirical distribution of the scaled matrix element fluctuations for a random eigenbasis is asymptotically Gaussian in the semiclassical limit , with variance given in terms of classical baker's map correlations. This determines the precise rate of convergence in the quantum ergodic theorem for these eigenbases. We obtain a version of the Eigenstate Thermalization Hypothesis (ETH) for these eigenstates, including a limiting complex Gaussian distribution for the off-diagonal matrix elements, with variances also given in terms of classical correlations. The presence of the classical correlations highlights that these eigenstates, while random, have microscopic correlations that differentiate them from Haar random vectors. For the single value , the Gaussianity of the matrix element fluctuations depends on the values of the classical observable on a fractal subset of the torus.

Paper Structure

This paper contains 28 sections, 19 theorems, 203 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline and notation
  4. Background and preliminaries
  5. Classical baker's map
  6. Quantum baker's map
  7. Walsh-quantized baker's maps
  8. Coherent state bases
  9. Walsh quantization of observables
  10. Useful properties
  11. Random quadratic forms
  12. Proof outline
  13. Time evolution and average expectation values
  14. Quantum vs classical time evolution
  15. Averaged expected variance
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $D\ge2$ and consider the $N\times N$ Walsh-quantized $D$-baker's map $\hat{B}_k$ (defined in eqn:walsh-action or eqn:wbaker) for $N=D^k$, $k\in\mathbb{N}$. Consider random orthonormal eigenbases $(\varphi^{(k,m)})_{m=1}^{D^k}$ of $\hat{B}_k$ chosen according to Haar measure within each eigenspac where $\ell=\ell(k)\in\llbracket0:k\rrbracket$ is a quantization parameter as described in eqn:opkl

Figures (4)

  • Figure 1: Graphical plot of the matrix entries of $\hat{B}_k^t$ for $D=3$ and $k=4$ in the position basis according to \ref{['eqn:wbaker']}, for $t=1,\ldots,8$ from left to right, top to bottom. The phases are plotted in color, with matrix entries that are zero shown in white. Noting that the $(0,0)$ entry of the matrix $\hat{B}_k^t$ is the top left, we see for early $t$ that the nonzero matrix entries of $\hat{B}_k^t$ resemble the graph of the classical map $q\mapsto D^tq\;\mathrm{mod}\;1$, which is the $D$-baker's map action on the position coordinate. At $t$ approaches the Ehrenfest time $k=4$ (top right), the correspondence breaks down as the matrix $\hat{B}_k^t$ becomes fully dense. Afterwards, however, $\hat{B}_k^t$ "reverses" form and becomes sparse again.
  • Figure 2: Phases of $\langle\varepsilon'\!\cdot \varepsilon|\hat{B}_k^t|\varepsilon'\!\cdot \varepsilon\rangle$ for $D=3$, $k=8$, $\ell=4$, and $t=7,8$, plotted as a function of the quantum rectangle $[\varepsilon'\!\cdot \varepsilon]\subset\mathbb{T}^2$ in coordinates $q=\varepsilon_\ell\cdots\varepsilon_1$ and $p=\varepsilon_k\cdots\varepsilon_{\ell+1}$. Entries where the value is zero are shown in white. There are no large groupings of a single phase, suggesting that the phases vary enough over nearby rectangles to cause cancellations.
  • Figure 3: The graph $G_{k,\mathbf{t},p}$ describing \ref{['eqn:trace-partial-p']} for $p>1$. The hanging edges in the bottom row connect to the corresponding variable in the top row.
  • Figure 4: Three regions of times $t$: the set $\mathcal{T}_Q$ when $\hat{B}_k^t$ is sparse (and $t\ne -2k$), the set $\mathcal{T}_\mathrm{dense}$ where $\hat{B}_k^t$ is dense, and the time $t=-2k$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Theorem 2.1: CollinsSniady2006
  • Proposition 3.1: average expectation values
  • proof : Proof of Proposition \ref{['prop:averages']}(i)
  • Proposition 3.2: wbaker
  • ...and 29 more