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Annals of Physics

Quantum resonances crossing and survival amplitude for double-well potentials

Andrea Sacchetti

Published Jun 14, 2025• 1 citation
Publisher

Abstract

In the case of crossings of isolated pairs of quantum resonances for double-well potentials it is possible to observe that, under appropriate conditions, the associated metastable states are equally localised on both wells. This property, together with the fact that the imaginary parts of the two resonances coincide, results in the establishment of an interference phenomenon between the two metastable states that characterises the time dynamics of certain physical observables. In this way, it is possible to lay the theoretical foundations for the design of a quantum device to assess whether or not the model’s parameters assume pre-assigned values.

DOI
10.1016/j.aop.2025.170122

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In the case of crossings of isolated pairs of quantum resonances for double-well potentials it is possible to observe that, under appropriate conditions, the associated metastable states are equally localised on both wells. This property, together with the fact that the imaginary parts of the two resonances coincide, results in the establishment of an interference phenomenon between the two metastable states that characterises the time dynamics of certain physical observables. In this way, it is possible to lay the theoretical foundations for the design of a quantum device to assess whether or not the model’s parameters assume pre-assigned values.
While the phenomenon of the exact crossing of energy levels is a rarely occurring event, in the case of quantum resonances associated with metastable states this phenomenon is much more frequent and various scenarios can occur. When there is an exact crossing of the imaginary parts of the resonances in a two-level quantum system subject to an external DC electric field, then a damped beating phenomenon occurs, which is absent if there is no such crossing. This fact, tested numerically on an explicit one-dimensional model, suggests the possibility of designing quantum sensors to determine in a very simple way whether the external field strength has an assigned value or not.
Abstract We propose a theoretical model for a quantum sensor that can determine in a very simple way whether the intensity of an electric field has an assigned value or not. It is based on the fact that when an exact crossing of the imaginary parts of the resonances occurs in a double-well quantum system subject to an external DC electric field, a damped beating phenomenon occurs, which is absent if there is no such a crossing. This result is then tested numerically on an explicit one-dimensional model.
Double-well potentials are widely used to model phenomena in physics and chemistry. The system is assumed to be formed in a metastable state, its dynamical evolution providing the clues for the interpretation of the experimental data. Quantum mechanics predicts coherent oscillations of probability between wells if the double-well potential is nearly symmetric and irreversible exponential decay if the final well has an infinite width. For very asymmetric double-well potentials, these two extreme behaviors are expected to coexist. The purpose of the present paper is to investigate this coexistence and its evolution as a function of the width of (or density of states in) the second well. In this sense, increasing the density of states can be regarded as a mechanism for coherence breakdown. The dynamical evolution of the metastable state can be simulated by solving numerically the time-dependent Schroedinger equation (TDSE). This approach is general, intuitive, and gives access to time scales and dynamical effects. It also allows the inclusion of phenomenological dissipation.
For a model 1d asymmetric double-well potential we calculated so-called survival probability (i.e. the probability for a particle initially localised in one well to remain there). We use a semiclassical (WKB) solution of Schroedinger equation. It is shown that behaviour essentially depends on transition probability, and on dimensionless parameter which is a ratio of characteristic frequencies for low energy non-linear in-well oscillations and inter wells tunnelling. For the potential describing a finite motion (double-well) one has always a regular behaviour. For the small value of the parameter there is well defined resonance pairs of levels and the survival probability has coherent oscillations related to resonance splitting. However for the large value of the parameter no oscillations at all for the survival probability, and there is almost an exponential decay with the characteristic time determined by Fermi golden rule. In this case one may not restrict oneself to only resonance pair levels. The number of perturbed by tunnelling levels grows proportionally to the value of this parameter (by other words instead of isolated pairs there appear the resonance regions containing the sets of strongly coupled levels). In the region of intermediate values of the parameter one has a crossover between both limiting cases, namely the exponential decay with subsequent long period recurrent behaviour.
Since the 1990s, experiments on double-well heterostructures have highlighted a key phenomenon: applying an external DC electric field induces pronounced wave packet oscillations at specific field strengths. \ To gain a deeper insight into this effect, a comprehensive quantitative analysis of the properties of quantum resonances near a crossing point is required. \ This approach lays the theoretical foundation for devising a quantum sensor capable of rapidly and efficiently determining whether the strength of an external DC electric field lies within a narrow target range.