[Paper Review] On Quantum - Classical Correspondence for Baker's Map

Kei Inoue, Makoto Ohya+1 more|ArXiv.org|Aug 23, 2001

Quantum Mechanics and Applications27 references11 citations

TL;DR

This paper provides a rigorous analytical derivation of the quantum-classical correspondence breakdown in the quantum baker's map, using Schack and Caves' symbolic formalism. It establishes that the discrepancy between quantum and classical trajectories emerges on a logarithmic timescale, $ t_h = \log_2(1/h) $, with the difference bounded by $ h2^{m-1} $, proving exact correspondence only for $ m \ll \log_2(1/h) $.

ABSTRACT

Quantum baker`s map is a model of chaotic system. We study quantum dynamics for the quantum baker's map. We use the Schack and Caves symbolic description of the quantum baker`s map. We find an exact expression for the expectation value of the time dependent position operator. A relation between quantum and classical trajectories is investigated. Breakdown of the quantum-classical correspondence at the logarithmic timescale is rigorously established.

Motivation & Objective

• To rigorously analyze the breakdown of quantum-classical correspondence in a chaotic quantum system.
• To determine the timescale at which quantum and classical trajectories diverge for the quantum baker's map.
• To establish an exact analytical expression for the time-dependent expectation value of the position operator in the quantum baker's map.
• To investigate the role of the Planck constant $ h $ in determining the onset of quantum-classical divergence.
• To provide a quantitative framework for understanding decoherence and information loss in quantum chaotic systems.

Proposed method

• Utilizes the symbolic dynamics formalism of Schack and Caves to describe the quantum baker's map in a finite-dimensional Hilbert space.
• Employs a discrete $ N $-qubit Hilbert space with $ D = 2^N $, where the Planck constant is $ h = 2^{-N} $, enabling exact computation.
• Derives an exact closed-form expression for the quantum expectation value of the position operator $ \langle \hat{q}_m \rangle $ at time $ m $, using matrix elements of the time-evolved state.
• Compares the quantum expectation value $ r_m^{(N)} $ with the classical trajectory $ q_m $, derived from the symbolic shift map.
• Applies bounds on the difference $ |r_m^{(N)} - q_m| $ using geometric series estimates on tail terms of the binary expansion.
• Establishes the logarithmic timescale $ t_h = \log_2(1/h) $ as the critical point where the quantum-classical divergence becomes significant.

Experimental results

Research questions

• RQ1 At what timescale does the quantum expectation value of the position operator begin to deviate significantly from the classical trajectory in the quantum baker's map?
• RQ2 How does the Planck constant $ h $ influence the onset of quantum-classical correspondence breakdown in chaotic systems?
• RQ3 Can the quantum-classical correspondence be quantitatively bounded, and what is the functional form of the deviation for finite $ h $?
• RQ4 Is the logarithmic timescale $ t_h = \log_2(1/h) $ a universal feature of quantum chaos in the baker's map model?
• RQ5 What is the exact analytical expression for the time evolution of the quantum position operator in this exactly solvable model?

Key findings

• The exact expression for the quantum expectation value of the position operator at time $ m $ is derived as $ r_m^{(N)} = \sum_{k=1}^{N-m} \xi_{m+k} 2^{-k} + \frac{1}{2^{N-m+1}} $, valid for $ 0 \leq m \leq N $.
• The difference between quantum and classical positions is bounded by $ |r_m^{(N)} - q_m| \leq \frac{1}{2^{N-m+1}} = h 2^{m-1} $, with equality achievable for certain initial states.
• The quantum-classical correspondence breaks down at the logarithmic timescale $ t_h = \log_2(1/h) $, where $ h = 2^{-N} $, as the bound $ h 2^{m-1} $ reaches $ \frac{1}{4} $ at $ m = N $.
• For $ m < t_h $, the difference remains small (bounded by $ \frac{1}{4} $), indicating a regime of good correspondence.
• The result confirms the conjecture that quantum-classical correspondence fails on a logarithmic timescale in chaotic systems, with exact analytical proof in this model.
• The analysis shows that the divergence is not gradual but emerges sharply at $ m \approx \log_2(1/h) $, marking a transition from quantum to classical-like behavior.