[Paper Review] Sub-Planck Structure, Decoherence, and Many-Body Environments

Andrew Jordanand, Mark Srednicki|ArXiv.org|Dec 21, 2001

Quantum many-body systems16 references3 citations

TL;DR

This paper challenges Zurek's claim that sub-Planck-scale phase-space structure in chaotic systems generically leads to orthogonal states under small displacements. Using the Berry-Voros ansatz and analysis of many-body systems, it shows that for few degrees of freedom, overlaps exhibit ringing with power-law decay, not exponential suppression; however, for large d, the overlap decays exponentially, restoring Zurek's conclusion. The key contribution is establishing that many-body environments are essential for effective quantum decoherence via this mechanism.

ABSTRACT

In a recent paper [Nature 412, 712 (2001)], Zurek has argued that (1) time evolution typically causes chaotic quantum systems to generate structure that varies on the scale of phase-space volume elements of size $(\hbar^2/A)^d$, where A is a classical action characteristic of the state and d is the number of degrees of freedom, and that (2) this structure implies that a small change in a phase-space coordinate X by an amount $δX \sim \hbar X/A$ generically results in an orthogonal state. While we agree with (1), we argue that (2) is not correct if the number of degrees of freedom is small. Our arguments are based on the Berry-Voros ansatz for the structure of energy eigenstates in chaotic systems. We find, however, that (2) becomes valid if the number of degrees of freedom is large. This implies that many-body environments may be crucial for the phenomenon of quantum decoherence.

Motivation & Objective

• To assess whether sub-Planck-scale phase-space structure in chaotic systems leads to orthogonal states under small displacements, as claimed by Zurek.
• To investigate the validity of Zurek's conclusion that small displacements δx ∼ ℏ/P or δp ∼ ℏ/L generically produce orthogonal states.
• To determine under what conditions—particularly the number of degrees of freedom—this orthogonality arises.
• To reconcile the behavior of the displacement matrix element ⟨D(δp,δx)⟩ with both the Wigner function's small-scale structure and its large-scale ergodic distribution.

Proposed method

• Uses the Berry-Voros ansatz to model the large-scale Wigner function as δ(H(x,p)−E), representing a uniform distribution over the energy surface.
• Applies the Fourier transform relation (eq. 5) between the Wigner function and the displacement matrix element ⟨D(δp,δx)⟩ to analyze how phase-space structure affects overlap.
• Analyzes a dilute gas of N hard spheres in a 3D box, modeling it as a single particle in a 3N-dimensional chaotic billiard to apply random wave conjecture.
• Derives the displacement matrix element using Bessel functions and sinc functions for position and momentum components, then takes the large-N limit to obtain Gaussian decay.
• Compares the behavior of ⟨D(δp,δx)⟩ for few-body (d ≪ 1) and many-body (d ≫ 1) systems, showing power-law vs. exponential decay of recurrence peaks.
• Uses the Wigner function's small-scale structure (from Zurek) and large-scale structure (from Berry-Voros) to reconcile apparent contradictions in the behavior of the overlap.

Experimental results

Research questions

• RQ1 Does sub-Planck-scale structure in the Wigner function of a chaotic system generically lead to orthogonal states under small displacements?
• RQ2 Is the conclusion that δx ∼ ℏ/P or δp ∼ ℏ/L causes orthogonal states valid for systems with few degrees of freedom?
• RQ3 How does the number of degrees of freedom d affect the decay of the overlap ⟨D(δp,δx)⟩ for small displacements?
• RQ4 Can the ringing behavior of the overlap in few-body systems be reconciled with the Wigner function's small-scale structure?
• RQ5 Under what conditions does the overlap ⟨D(δp,δx)⟩ decay exponentially rather than oscillate with power-law decay?

Key findings

• For systems with few degrees of freedom, the overlap ⟨D(δp,δx)⟩ exhibits ringing with recurrent zeroes at δx ∼ ℏ/P and δp ∼ ℏ/L, and peak heights decay as a power law.
• The overlap does not decay exponentially for small displacements in few-body systems; instead, it shows persistent oscillations with power-law suppression.
• For many-body systems (d ≫ 1), the overlap ⟨D(δp,δx)⟩ decays exponentially as exp(−P²|δx|²/6ℏ²) and exp(−L²|δp|²/24ℏ²), leading to rapid suppression.
• The exponential decay in many-body systems restores Zurek's conclusion that small displacements produce nearly orthogonal states.
• The transition from power-law to exponential decay is driven by the increase in degrees of freedom, which suppresses recurrence peaks.
• The analysis confirms that many-body environments are crucial for effective quantum decoherence via this mechanism.