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arXiv:2601.11256 · Physics

Shortcuts to adiabaticity, unexciting backgrounds, and reflectionless potentials

Fernando C. Lombardo, Francisco D. Mazzitelli

quant-phSubmitted Jan 16, 2026• Announced Jan 19, 2026• 0 citations
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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.

Journal reference
Physics 2026, 8(1) (Special Issue: A Themed Issue in Honor of Professor Viktor Dodonov, 55 Years in Quantum Physics)
DOI
10.3390/physics8010010
Comments
10 pages, 1 figure

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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.
We investigate the shortcuts to adiabaticity (STA) of a quantum harmonic oscillator under time-dependent frictional force, using an invariant-based inverse-engineering method with a class of invariants characterized by a time-dependent frictional coefficient. We discuss the implementation of a shortcut protocol in a generalized framework for any arbitrary time-dependent frictional interaction and study a class of such interaction from which the STA for the quantum harmonic oscillator can be obtained. We also discuss the application of the above protocol for the harmonic oscillator with time-varying mass. For an illustration, we consider the coupled photonic lattice as a harmonic oscillator with time-varying mass and frequency and discuss implementing the above protocol.
Shortcuts to adiabaticity (STA) are relevant in the context of quantum systems, particularly regarding their control when they are subjected to time-dependent external conditions. In this paper, we investigate the completion of a nonadiabatic evolution into a shortcut to adiabaticity for a quantum field confined within a one-dimensional cavity containing two movable mirrors. Expanding upon our prior research, we characterize the field's state using two Moore functions that enables us to apply reverse engineering techniques in constructing the STA. Regardless of the initial evolution, we achieve a smooth extension of the Moore functions that implements the STA. This extension facilitates the computation of the mirrors' trajectories based on the aforementioned functions. Additionally, we draw attention to the existence of a comparable problem within nonrelativistic quantum mechanics.
The quantum harmonic oscillator with time-dependent frequency is a paradigmatic model of driven quantum dynamics and one of the few nontrivial systems that admits an exact analytical solution. In this review paper, we present a unified treatment of the time-dependent oscillator based on the Lewis-Riesenfeld invariant method, Bogoliubov transformations and the Ermakov-Pinney equation. We show how these approaches naturally connect to squeezing for the description of excitations production, and to the breakdown of adiabaticity under generic frequency protocols. Exact results for sudden quenches and smooth ramps are discussed in detail. By explicitly bridging invariant methods and squeezing formalism, this review is meant to provide a comprehensive framework for understanding nonequilibrium dynamics in quadratic potentials, with applications ranging from thermodynamics and condensed matter to quantum control theory.
The expansion of a harmonic potential that holds a quantum particle may be realized without any final particle excitation but much faster than adiabatically via "shortcuts to adiabaticity" (STA). While ideally the process time can be reduced to zero, practical limitations and constraints impose minimal finite times for the externally controlled time-dependent frequency protocols. We examine the role of different time-averaged energies (total, kinetic, potential, nonadiabatic) and of the instantaneous power in characterizing or selecting different protocols. Specifically, we prove a virial theorem for STA processes, set minimal energies (or times) for given times (or energies), and discuss their realizability by means of Dirac impulses or otherwise.
We present a technique for efficiently transitioning a quantum system from an initial to a final stationary state in less time than is required by an adiabatic (quasistatic) process. Our approach makes use of Nelson's stochastic quantization, which represents the quantum system as a classical Brownian process. Thanks to this mathematical analogy, known protocols for classical overdamped systems can be translated into quantum protocols. In particular, one can use classical methods to find optimal quantum protocols that minimize both the time duration and some other cost function to be freely specified. We have applied this method to the time-dependent harmonic oscillator and tested it on two different cost functions: (i) the cumulative energy of the system over time and (ii) the dynamical phase of the wave function. In the latter case, it is possible to construct protocols that are "adiabatically optimal," i.e., they minimize their distance from an adiabatic process for a given duration.
Shortcuts to adiabaticity (STA) provide control protocols to guide the dynamics of a quantum system through an adiabatic reference trajectory in an arbitrary prescheduled time.Designing STA proves challenging in complex quantum systems when the dynamics of the degrees of freedom span different time scales.We introduce counterdiabatic Born-Oppenheimer dynamics (CBOD) as a framework to design STA in systems with a large separation of energy scales.CBOD exploits the Born-Oppenheimer approximation to separate the Hamiltonian into effective fast and slow degrees of freedom and calculate the corresponding counterdiabatic drivings for each sub-system.We show the validity of the CBOD technique via an example of coupled harmonic oscillators, which can be solved exactly for comparison, and further apply it to a system of two-charged particles.