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Shortcuts to adiabaticity, unexciting backgrounds, and reflectionless potentials

Fernando C. Lombardo, Francisco D. Mazzitelli

TL;DR

This work investigates shortcuts to adiabaticity (STA) for the time-dependent quantum harmonic oscillator (QHO) and for quantum field theory in non-stationary backgrounds, establishing a unifying view via a QHO–scattering correspondence. By leveraging the Lewis–Riesenfeld invariant framework and the Ermakov function, the authors design finite-time protocols that suppress excitations, and interpret particle creation as squeezing while STA completions correspond to anti-squeezing operations. They show that STA map to transmission resonances in the scattering picture and that unexciting backgrounds in QFT correspond to reflectionless potentials, with constructive methods such as the Kay–Moses approach and explicit Poschl–Teller examples. The results bridge quantum control, cosmology, and experimental platforms like circuit QED, suggesting practical schemes for excitation-free dynamics and potential applications in quantum thermodynamics and quantum technologies.

Abstract

We analyze shortcuts to adiabaticity (STA) and their completions for the quantum harmonic oscillator (QHO) with time-dependent frequency, as well as for quantum field theory (QFT) in non-stationary backgrounds. We exploit the analogy with one-dimensional quantum mechanics, and the well known correspondence between Bogoliubov coefficients in the QHO and transmission/reflection amplitudes in scattering theory. Within this framework, STA protocols for the QHO are equivalent to transmission resonances, while STA in QFT with homogeneous backgrounds correspond to reflectionless potentials. Moreover, using the connection between particle creation and squeezed states, we show how STA completions can be understood in terms of the anti-squeezing operator.

Shortcuts to adiabaticity, unexciting backgrounds, and reflectionless potentials

TL;DR

This work investigates shortcuts to adiabaticity (STA) for the time-dependent quantum harmonic oscillator (QHO) and for quantum field theory in non-stationary backgrounds, establishing a unifying view via a QHO–scattering correspondence. By leveraging the Lewis–Riesenfeld invariant framework and the Ermakov function, the authors design finite-time protocols that suppress excitations, and interpret particle creation as squeezing while STA completions correspond to anti-squeezing operations. They show that STA map to transmission resonances in the scattering picture and that unexciting backgrounds in QFT correspond to reflectionless potentials, with constructive methods such as the Kay–Moses approach and explicit Poschl–Teller examples. The results bridge quantum control, cosmology, and experimental platforms like circuit QED, suggesting practical schemes for excitation-free dynamics and potential applications in quantum thermodynamics and quantum technologies.

Abstract

We analyze shortcuts to adiabaticity (STA) and their completions for the quantum harmonic oscillator (QHO) with time-dependent frequency, as well as for quantum field theory (QFT) in non-stationary backgrounds. We exploit the analogy with one-dimensional quantum mechanics, and the well known correspondence between Bogoliubov coefficients in the QHO and transmission/reflection amplitudes in scattering theory. Within this framework, STA protocols for the QHO are equivalent to transmission resonances, while STA in QFT with homogeneous backgrounds correspond to reflectionless potentials. Moreover, using the connection between particle creation and squeezed states, we show how STA completions can be understood in terms of the anti-squeezing operator.
Paper Structure (8 sections, 38 equations, 1 figure)

This paper contains 8 sections, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. STA for the Harmonic Oscillator with Time-Dependent Frequency
  3. The Ermakov Equation
  4. STA Completions from the Ermakov Function
  5. STA Completions and Anti-Squeezing
  6. The Analogous Scattering Problem in Quantum Mechanics
  7. Unexciting Backgrounds in Quantum Field Theory and Reflectionless Potentials
  8. Conclusions

Figures (1)

  • Figure 1: Plot of the time dependent frequency $\omega(t)$ (blue curve) and the Ermakov function (orange curve). (a) The frequency follows a (soft) "step-like" evolution. The Ermakov function is initially constant and then oscillates, indicating an excited final state. (b) Similar behaviour for a "barrier-like" evolution. The dashed vertical line indicates the center of the barrier. (c) When the barrier is centered on an extremum of the Ermakov function, the Ermakov function becomes constant, indicating a STA completion.