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Particle Trajectories for Quantum Maps

Yonah Borns-Weil, Izak Oltman

Abstract

We study the trajectories of a semiclassical quantum particle under repeated indirect measurement by Kraus operators, in the setting of the quantized torus. In between measurements, the system evolves via either Hamiltonian propagators or metaplectic operators. We show in both cases the convergence in total variation of the quantum trajectory to its corresponding classical trajectory, as defined by propagation of a semiclassical defect measure. This convergence holds up to the Ehrenfest time of the classical system, which is larger when the system is less chaotic. In addition, we present numerical simulations of these effects. In proving this result, we provide a characterization of a type of semi-classical defect measure we call uniform defect measures. We also prove derivative estimates of a function composed with a flow on the torus.

Particle Trajectories for Quantum Maps

Abstract

We study the trajectories of a semiclassical quantum particle under repeated indirect measurement by Kraus operators, in the setting of the quantized torus. In between measurements, the system evolves via either Hamiltonian propagators or metaplectic operators. We show in both cases the convergence in total variation of the quantum trajectory to its corresponding classical trajectory, as defined by propagation of a semiclassical defect measure. This convergence holds up to the Ehrenfest time of the classical system, which is larger when the system is less chaotic. In addition, we present numerical simulations of these effects. In proving this result, we provide a characterization of a type of semi-classical defect measure we call uniform defect measures. We also prove derivative estimates of a function composed with a flow on the torus.
Paper Structure (18 sections, 13 theorems, 161 equations, 7 figures, 1 algorithm)

This paper contains 18 sections, 13 theorems, 161 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model
  3. Previous Work
  4. Background
  5. Semiclassical Analysis on the Real Line
  6. Fourier Integral Operators
  7. Semiclassical Defect Measures
  8. The Quantized Torus
  9. Measurement Procedure
  10. Evolution Procedure
  11. Main Result and Proof
  12. Numerical Illustrations
  13. Acknowledgements
  14. Uniformity of Defect Measures
  15. Coherent States
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose $\rho=\rho_N:H_N\to H_N$ is a density operator with semiclassical defect measure $\mu$. Fix $n\in \mathbb{N}$, and let $(q_{N,0},q_{N,1}, \dots , q_{N,n})$ be random variables with law $\mathbb{P}_{N,\rho}^{(n)}$ given by eq:quantummeasure and let $(q_0, q_1,\dots , q_n)$ be random variables

Figures (7)

  • Figure 1: A numerical simulation demonstrating this paper's main result. Each grey curve represents the trajectory of a quantum particle (under the evolution procedure discussed in this paper) on a torus with identical initial conditions and a fixed Hamiltonian. A single trajectory is plotted in red and the corresponding classical trajectory is plotted as a blue dotted line. Up until time $0.4$, most of the positions of the quantum particles are within $0.1$ of the classical trajectory, but shortly after become decoherent. More numerics are provided in § \ref{['s:numerics']}. This simulation is discussed at the end of § \ref{['ss:model']} and § \ref{['s:numerics']}.
  • Figure 2: Evolution Procedure Here we simulate the trajectory of a quantum particle for three time steps under the discussed evolution procedure. The specifics of this simulation are discussed in § \ref{['s:numerics']}.
  • Figure 3: Marginal distributions of $\bm{\mathbb{P}_{N, \rho}^{(n)}}$ vs $\bm{P_\mu^{(n)}}$: We perform Algorithm \ref{['qmap']} (observation then evolution by the quantum cat map). We plot the relative frequencies of the observed positions (approximating the marginal distribution of ${\mathbb{P}_{N, \rho}^{(n)}}$ against the marginal distributions of ${P_\mu^{(n)}}$). See § \ref{['s:numerics']} for more details.
  • Figure 4: Convergence of marginal distributions of $\bm{\mathbb{P}_{N, \rho}^{(n)} \to P_\mu^{(n)}}$: We run the same simulation as in Figure \ref{['fig:fig3']}, but vary $N$ from $100$ to $4000$ and compare $\mathbb{P}_{N, \rho}^{(n)}$ to $P_\mu^{(n)}$. See § \ref{['s:numerics']} for more details.
  • Figure 5: Changing the Lyapunov exponent: We present the same numerics as in Figure \ref{['fig:fig3']} but evolve by quantizations of three different matrices given by \ref{['eq:matrices']} with 3 different Lyapunov exponents. See § \ref{['s:numerics']} for more details.
  • ...and 2 more figures

Theorems & Definitions (30)

  • Theorem : Main result
  • Definition 2.1: symbol class
  • Definition 2.2: Weyl quantization
  • Definition 2.3: Lyapunov exponent
  • Definition 2.4: quantized flow
  • Definition 2.5: defect measure
  • Proposition 2.6
  • Proposition 2.7
  • Definition 2.8: uniform defect measure
  • Proposition 2.9
  • ...and 20 more