[Paper Review] Quantum Imprint of the Anharmonic Oscillator

P. Lo Chiatto, S. Schenk+1 more|arXiv (Cornell University)|Aug 2, 2023

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper investigates the semiclassical limit of the anharmonic oscillator using exact WKB methods in a 't Hooft-like double scaling limit, showing that the tunneling action and the transition amplitude to highly excited states coincide—except for an irreducible instanton contribution. This discrepancy reveals a fundamental quantum imprint, establishing a 'quantum imprint rule' that quantum theories are intrinsically gapped from classical behavior, even in the semiclassical regime.

ABSTRACT

We study the anharmonic double well in quantum mechanics using exact Wentzel-Kramers-Brillouin (WKB) methods in a 't Hooft-like double scaling limit where classical behavior is expected to dominate. We compute the tunneling action in this double scaling limit, and compare it to the transition amplitude from the vacuum to a highly excited state. Our results, exact in the semiclassical limit, show that the two expressions coincide, apart from an irreducible and surprising instanton contribution. Thus, the semiclassical limit of the anharmonic oscillator betrays its quantum origin as a rule, which we dub the "quantum imprint rule," showing that the quantum theory is intrinsically gapped from classical behavior. Besides an example of the failure of reductionism and an example of a resurgent connection between perturbative and nonperturbative physics, this work provides a possible classification of theories according to their quantum imprints.

Motivation & Objective

• To investigate the emergence of classical physics from quantum mechanics in the semiclassical limit of the anharmonic oscillator.
• To resolve the apparent contradiction between classically equivalent double-well potentials with different quantum vacuum structures.
• To explore how nonperturbative effects, particularly instantons, manifest in highly excited quantum states.
• To establish a classification of quantum theories based on their 'quantum imprint'—the residual quantum signature in the classical limit.
• To demonstrate that perturbative and nonperturbative physics are connected via resurgence, even in the high-energy semiclassical regime.

Proposed method

• Employing exact Wentzel-Kramers-Brillouin (WKB) methods in a 't Hooft-like double scaling limit to access the semiclassical regime.
• Computing the tunneling action in the double scaling limit to analyze quantum corrections to classical behavior.
• Comparing the tunneling action to the transition amplitude from the vacuum to a highly excited state using resurgent asymptotics.
• Applying Borel-Laplace resummation to handle divergent perturbative series and extract nonperturbative information.
• Using resurgence theory to connect perturbative expansions with nonperturbative instanton effects, particularly poles in the Borel plane.
• Identifying the failure of Borel-Laplace summability as a signal of instanton contributions, which encode the quantum imprint.

Experimental results

Research questions

• RQ1 How does the quantum behavior of the anharmonic oscillator manifest in the semiclassical limit, particularly in highly excited states?
• RQ2 Why do classically equivalent double-well potentials (with different vacuum structures) yield distinct quantum behaviors in the semiclassical regime?
• RQ3 To what extent can perturbative expansions in quantum mechanics be resummed to capture nonperturbative effects like instantons?
• RQ4 What is the role of resurgence in connecting perturbative and nonperturbative physics in the high-energy limit?
• RQ5 Can the quantum imprint—defined as the irreducible quantum deviation from classicality—be used to classify quantum field theories?

Key findings

• The tunneling action and the transition amplitude to a highly excited state coincide in the double scaling limit, confirming consistency between semiclassical and quantum descriptions.
• An irreducible instanton contribution persists even in the semiclassical limit, breaking full classical recovery and revealing a quantum imprint.
• The failure of Borel-Laplace summability is directly linked to the presence of instantons, with poles in the Borel plane encoding their action and relative importance.
• The quantum theory is fundamentally gapped from classical behavior, as the instanton contribution cannot be removed, establishing a 'quantum imprint rule'.
• The results demonstrate a resurgent connection between perturbative and nonperturbative physics, even in the high-energy regime.
• The work provides a framework for classifying quantum theories by their quantum imprint, based on the nature of their nonperturbative corrections in the classical limit.