Gastón García-Calderón

Resonance expansions in quantum mechanics

The goal of this contribution is to discuss various resonance expansions that have been proposed in the literature.

The effect of asymmetry on resonant tunneling in one dimension

Abstract This work considers the effect of asymmetric one-dimensional multibarrier potentials on resonant tunneling. It is shown, by using the properties of the propagator of the system, that this effect may lead to novel resonance phenomena which affect the decay properties of the quasistationary states of the system. As a consequence it is shown that in general the full width at half maximum of the transmission peak at resonance does not provide the corresponding lifetime as occurs in the symmetric case. The above considerations are illustrated by a simple analytical solvable model.

Transient effects and reconstruction of the energy spectra in the time evolution of transmitted Gaussian wave packets

We derive an analytical solution to the time-dependent Schr?dinger equation for the transmission of a Gaussian wave packet through an arbitrary potential of finite range. We consider the situation where the initial Gaussian wave packet is sufficiently broad in momentum space to include the full resonance structure of the system in the dynamical description. We find that the transmitted wave packet may be written as the free evolving Gaussian wave packet solution times a transient term. We demonstrate that both at very large distances and very long times the transient term tends to the transmission amplitude of the system and hence the time-evolving solution reproduces the resonance spectra of the system. We also prove that at a fixed distance and very long times the analytical solution is t?3/2 which extends previous analysis on this issue to arbitrary finite-range potentials. Our results are exemplified for a multibarrier system.

Chapter 7 - Theory of Resonant States: An Exact Analytical Approach for Open Quantum Systems

Abstract The theory of resonant states provides an exact analytical approach for the description of open quantum systems. Resonant states are defined by imposing purely outgoing boundary conditions to the solutions to the Schrodinger equation. This leads to complex energy eigenvalues and hence to a non-Hermitian quantum-mechanical formulation. This paper discusses some rigorous analytical results of this formalism for potentials of finite range with applications to scattering and the time evolution of decay.

Hermitian and non-Hermitian formulations of the time evolution of quantum decay

This work discusses Hermitian and non-Hermitian formulations for the time evolution of quantum decay, which involve, respectively, continuum wave functions and resonant states, to show that they lead to an identical description for a large class of well-behaved potentials. Our approach is based on the analytical properties of the outgoing Green function to the problem in the complex wave number plane.

Resonant tunneling transients and decay for a one-dimensional double barrier potential

Motivated by recent experimental work on quantum dots subjected to voltage pulses we consider a simple model to study the transition between off-resonance and on-resonance scattering states with the same incident energy in response to a sudden change in the well depth of a double barrier potential structure. The change displaces the real part of the resonance energy to coincide with the incident energy. The resonance buildup is not given by a pure exponential growth due to the interference between incident and resonance components represented by nearby poles, but the resonance lifetime is a relevant time scale. The reverse process (resonance depletion) that follows the opposite change in the well depth detunes the resonance level and the incident energy but, except for short and long time deviations, the decay is exponential with the lifetime of the displaced resonance. For a larger change in the well depth beyond a critical depth, trapping dominates rather than decay since the resonance becomes a bound s...

Transient tunneling effects of resonance doublets in triple barrier systems

Transient tunneling effects in triple barrier systems are investigated by considering a time-dependent solution to the Schrodinger equation with a cutoff wave initial condition. We derive a two-level formula for incidence energies E near the first resonance doublet of the system. Based on that expression we find that the probability density along the internal region of the potential is governed by three oscillation frequencies: one of them refers to the well known Bohr frequency, given in terms of the first and second resonance energies of the doublet, and the two others represent a coupling with the incidence energy E. This allows us to manipulate the above frequencies to control the tunneling transient behavior of the probability density in the short-time regime.

Transient time-domain resonances and the time scale for tunneling

Transient time-domain resonances found recently in time-dependent solutions to Schrodingers equation are used to investigate the issue of the tunneling time in rectangular potential barriers. In general, a time-frequencyanalysis shows that these transients have frequencies above the cutoff frequency associated with the barrier height, and hence correspond to nontunneling processes. We find, however, a regime characterized by the barrier opacity, where the peak maximum t m a x of the time-domain resonance corresponds to under-the-barrier tunneling. We argue that t m a x represents the relevant tunneling time scale through the classically forbidden region.

Survival probability of a single resonance

An exact single-level resonance formula for the survival probability S(t) in the full time interval, that depends only on the resonance energy r and the decay width ?r and fulfils time-reversal invariance, is used to discuss the non-exponential contributions to decay. At short times the formula behaves as S(t)?1-ct1/2 with c a constant, whereas at long times it behaves as S(t)?dt-3, d being a constant. With the time expressed in lifetime units, the onset of non-exponential decay is given at short times by ?S?4/[?(R2 + R + 1/4)] and at long times by ?L?5.41?ln (R) + 12.25, where R = r?/?r. The predictions of the formula are compared with numerical examples and some experimental results searching for non-exponential contributions to decay.

Decay widths for double‐barrier resonant tunneling

Calculations of the decay width and the partial decay widths for double‐barrier resonant tunneling structures are discussed using a description of resonances in terms of the complex energy poles of the transmission amplitude of the system and comparing it with the corresponding Wentzel–Kramers–Brillouin approximations. The effect of the asymmetry of the potential profile is emphasized.