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Quantum Economics and Finance

Quantum Resonances Crossing: Quantitative Analysis in an Explicit Model

Andrea Sacchetti

Published Apr 15, 2025• 0 citations
Publisher

Abstract

Different kinds of crossings of quantum resonances can be observed even in simple models. When there is an exact crossing of the imaginary parts of the resonances then a damped beating phenomenon for the survival amplitude occurs, which is absent if there is no such crossing. This fact suggests the possibility of designing quantum sensors to determine in a very simple way whether the strength of an external DC electric field has an assigned value or not.

DOI
10.1177/29767032251332000

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Different kinds of crossings of quantum resonances can be observed even in simple models. When there is an exact crossing of the imaginary parts of the resonances then a damped beating phenomenon for the survival amplitude occurs, which is absent if there is no such crossing. This fact suggests the possibility of designing quantum sensors to determine in a very simple way whether the strength of an external DC electric field has an assigned value or not.
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.
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.
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.
A resonance, induced by a DC field in a two-photon transition of an atom between states with opposite parity, results in a number of new effects related to the polarization of photons. These effects are caused by the interference of the non-resonant second-order dipole-forbidden amplitude and of a resonant part of the DC field-induced third-order amplitude. The linear and circular dichroism for the cross section of the process is revealed and analysed in detail for a two-photon excitation of the state from the ground state with a resonance on an state, induced by a DC electric field. The numerical data for alkali atoms demonstrate that the effects have quantitative characteristics of sufficiently large magnitudes and may be observed in experiments.
We investigate finite-time field-induced ionization from a one-dimensional attractive delta-function well in a uniform dc electric field of arbitrary strength. Our aim is to determine the physical bound-state survival amplitude ab(t)a_b(t)ab​(t) at any observation time without constructing the complete time-dependent propagator or wavefunction. In the gauge-equivalent Kramers--Henneberger representation, the field appears as motion of the contact point, and the resulting dynamics reduces to a closed Volterra equation. Exact endpoint-phase factorization organizes the full chronological rescattering history into a relative-time convolution hierarchy and an exact resolvent. By introducing the accumulated bound-state amplitude and reorganizing its two-time domain in relative and complementary center times, the final contact contribution becomes an explicit boundary integral. All spatial integrations are carried out analytically. The field-driven contact-free term is obtained in closed form using the Faddeeva function, while the direct and repeated-rescattering terms are expressed as explicitly evaluable time integrals. The result applies at arbitrary dc-field strength and finite time, with no weak-field expansion, rescattering truncation, or asymptotic-time approximation. It is verified through the exact field-free limit and an independent numerical solution of the original physical Volterra equation. The formulation gives an exact nonperturbative solution of a fundamental ionization model and reveals otherwise hidden analytical structure in driven quantum dynamics.