[Paper Review] Quantum Ratchets on Maximally Uniform States in Phase Space: Semiclassical Full-Chaos Regime

Itzhack Dana|ArXiv.org|Oct 10, 2008

stochastic dynamics and bifurcation3 citations

TL;DR

This paper introduces a novel quantum ratchet mechanism in fully chaotic systems using maximally uniform quantum states—pure states uniformly distributed over a single Planck cell in phase space. Unlike classical systems, these states generate a nonzero directed current due to intrinsic quasicoordinate asymmetry within the Planck cell. The key result is that the variance of the current over all such states scales nearly proportionally to the chaotic diffusion coefficient and the square of the scaled Planck constant, demonstrating a fundamental quantum-classical divergence in ratchet effects.

ABSTRACT

A generic kind of quantum chaotic ratchet is introduced, based on initial states that are \emph{uniform} in phase space with the \emph{maximal possible} resolution of one Planck cell. Unlike a classical phase-space uniform density, such a state usually carries a \emph{nonzero} ratchet current, even in \emph{symmetric} systems. This quantum ratchet effect basically emerges from the generic asymmetry of the state quasicoordinates in the Planck cell. It is shown, on the basis of exact results, general arguments, and extensive numerical evidence, that in a semiclassical full-chaos regime the variance of the current over all the states is nearly proportional to the chaotic-diffusion rate and to the square of the scaled Planck constant. Experimental realizations are suggested.

Motivation & Objective

• To identify a generic quantum ratchet mechanism in fully chaotic systems that persists even in symmetric Hamiltonians.
• To resolve the quantum-classical paradox where classical ratchets vanish under uniform phase-space initial conditions, but quantum ratchets do not.
• To establish a semiclassical measure of quantum ratchet strength via statistical averaging over maximally uniform initial states.
• To demonstrate that system asymmetry plays only a minor role in the ratchet effect, primarily affecting the chaotic diffusion rate.

Proposed method

• Defining maximally uniform quantum states as pure states invariant under phase-space translation operators up to phase factors, with unit-cell area equal to Planck's constant h.
• Representing these states as delta-function comb wavefunctions in position and momentum space, parameterized by quasicoordinates (w1, w2) within a Planck cell.
• Deriving an exact expression for the quantum current as a function of quasicoordinates, showing it is generally nonzero even in symmetric systems due to quasicoordinate asymmetry.
• Computing the variance of the current over all quasicoordinates as a global measure of the ratchet effect, corresponding to a mixed state average.
• Using general semiclassical arguments and extensive numerical simulations to establish the scaling relation: variance ∝ Dql × ℏ², where Dql is the chaotic diffusion coefficient.
• Testing the scaling across different factorizations of the Planck cell (e.g., N1=N2 vs. N1=N, N2=1) and for symmetric and asymmetric kicked Harper models.

Experimental results

Research questions

• RQ1 Can a quantum ratchet effect exist in fully chaotic systems with symmetric Hamiltonians, despite the classical phase-space uniform density yielding zero current?
• RQ2 What is the origin of the quantum ratchet current in maximally uniform phase-space states, given the absence of classical asymmetry?
• RQ3 How does the strength of the quantum ratchet effect scale with the Planck constant and chaotic diffusion in the semiclassical regime?
• RQ4 To what extent does system asymmetry influence the ratchet effect, and can it be decoupled from the diffusion rate?
• RQ5 Is the ratchet variance independent of the shape of the Planck cell, as predicted by the scaling law?

Key findings

• The variance of the quantum current over all maximally uniform states scales nearly proportionally to the chaotic diffusion coefficient Dql and the square of the scaled Planck constant ℏ², as expressed in Eq. (9).
• The quantum ratchet effect arises from the intrinsic asymmetry of quasicoordinates within the Planck cell, even in completely symmetric systems, making it a fundamental quantum phenomenon.
• Numerical results confirm that the current variance decays as ℏ in the semiclassical regime, with a slope of −1.04 ± 0.06 for symmetric factorizations and −0.99 ± 0.01 for elongated ones, supporting the ℏ² scaling.
• The ratchet variance is nearly independent of the Planck-cell shape, validating the robustness of the scaling law across different lattice factorizations.
• The effect vanishes in the classical limit (ℏ→0), consistent with the classical result that uniform phase-space initial conditions yield zero current.
• Experimental realizations are feasible using atom-optics techniques, particularly via superpositions of plane waves to approximate the maximally uniform states.