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Physica Scripta

Absorption dynamics and delay time in complex potentials

Jorge Villavicencio, Roberto Romo, Alberto Hernández-Maldonado

Published Apr 4, 2018• 5 citations
Publisher

Abstract

Abstract The dynamics of absorption is analyzed by using an exactly solvable model that deals with an analytical solution to Schrödinger’s equation for cutoff initial plane waves incident on a complex absorbing potential. A dynamical absorption coefficient which allows us to explore the dynamical loss of particles from the transient to the stationary regime is derived. We find that the absorption process is characterized by the emission of a series of damped periodic pulses in time domain, associated with damped Rabi-type oscillations with a characteristic frequency, ω = ( E + ε )/ℏ, where E is the energy of the incident waves and − ε is energy of the quasidiscrete state of the system induced by the absorptive part of the Hamiltonian; the width γ of this resonance governs the amplitude of the pulses. The resemblance of the time-dependent absorption coefficient with a real decay process is discussed, in particular the transition from exponential to nonexponential regimes, a well-known feature of quantum decay. We have also analyzed the effect of the absorptive part of the potential on the dynamical delay time, which behaves differently from the one observed in attractive real delta potentials, exhibiting two regimes: time advance and time delay.

DOI
10.1088/1402-4896/aab41c

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Canadian Journal of Physics•Mar 1, 1981•5 citations
Abstract The dynamics of absorption is analyzed by using an exactly solvable model that deals with an analytical solution to Schrödinger’s equation for cutoff initial plane waves incident on a complex absorbing potential. A dynamical absorption coefficient which allows us to explore the dynamical loss of particles from the transient to the stationary regime is derived. We find that the absorption process is characterized by the emission of a series of damped periodic pulses in time domain, associated with damped Rabi-type oscillations with a characteristic frequency, ω = ( E + ε )/ℏ, where E is the energy of the incident waves and − ε is energy of the quasidiscrete state of the system induced by the absorptive part of the Hamiltonian; the width γ of this resonance governs the amplitude of the pulses. The resemblance of the time-dependent absorption coefficient with a real decay process is discussed, in particular the transition from exponential to nonexponential regimes, a well-known feature of quantum decay. We have also analyzed the effect of the absorptive part of the potential on the dynamical delay time, which behaves differently from the one observed in attractive real delta potentials, exhibiting two regimes: time advance and time delay.
Abstract We derive an exact analytic solution of time-dependent Schrödinger’s equation for cutoff quantum waves with exponential modulation to explore the dynamics along the internal region of a one-dimensional potential of arbitrary shape. The modulation of the wave packet is controlled by a single parameter given by its momentum distribution width, Δ, which allows to explore on the same footing the transient features of both cutoff spatially localized wave packets and extended plane waves. From the formal solution, we derive a simple and reliable formula that accurately describes both the buildup and decay of the probability density inside a resonant structure. The buildup-decay formula involves two exponential terms governed respectively by the resonance width Γ 1 of the lowest resonance of the system and the width Γ ¯ of a ‘virtual’ state induced by the incident wave packet. During the buildup process, we demonstrate that the contributions of both Γ 1 and Γ ¯ become essential. In the decay process, on the other hand, we find that there are two regimes characterized by a critical value Δ c , such that for Δ > Δ c the exponential decay is governed by Γ 1 , while for Δ < Δ c the exponential decay is governed by Γ ¯ . We also derive an analytic formula for a relevant timescale for the buildup-decay process, which depends in a simple fashion on the decay rates Γ 1 and Γ ¯ . Deviations from the exponential decay law at long times are also discussed.
The analytical solution to the time-dependent Schr\"odinger equation for tunneling using cutoff plane-wave initial conditions is in general given by the sum of two types of terms that exhibit a transient behavior. The time evolution of the probability density for the $\ensuremath{\delta}$ potential is compared with the free case to investigate in this case the role of these transient terms for the delay time. We find, by a dynamical calculation, that the delay time arises from the interference between these transient terms and we show that at very long times it goes into the phase delay time, given by the energy derivative of the phase of the transmission amplitude.
Transient phenomena of phase-modulated cutoff wave packets are explored by deriving an exact general solution to Schr\"odinger's equation for finite-range potentials involving arbitrary initial quantum states. We show that the dynamical features of the probability density are governed by a virtual self-induced two-level system with energies E+{E}_{+}E+​ and ${E}_{\ensuremath{-}}$ due to the phase modulation of the initial state. The asymptotic probability density exhibits Rabi oscillations characterized by the frequency $\mathrm{\ensuremath{\Omega}}=({E}_{+}\ensuremath{-}{E}_{\ensuremath{-}})/\ensuremath{\hbar}$, which are independent of the potential profile. It is also found that for a system with a bound state, the interplay between the virtual levels with the latter causes a quantum beat effect with a beating frequency, $\mathrm{\ensuremath{\Omega}}$. We also find a regime characterized by a time-diffraction phenomenon that allows us to measure unambiguously the delay time, which can be described by an exact analytical formula. It is found that the delay time agrees with the phase time only for the case of strictly monochromatic waves.
Abstract We derive an exact analytical solution to the time-dependent Schrödinger equation based on a resonant state expansion, to explore the time-evolution of cutoff plane waves scattered by finite range potentials, within a shifted quantum shutter model. The latter allows to control both, the position of the shutter and an hypothetical detector, which are initially separated by a distance Δ X . The dynamical advance-time (negative delay-time), Δ T , is explored for the particular case of a delta potential well, by measuring the difference of the first maxima of the time-diffraction pattern of the probability density, corresponding to the free and delta potential cases. We show that in general, Δ T exhibits a monotonic behaviour as a function of Δ X , and derive a simple formula for the timescale by using the symmetry and rescaling properties of the dynamical solution. We demonstrate that below a critical value of the shutter-detector separation, Δ X c , the monotonic behaviour of Δ T is hindered due to an oscillatory phenomena of the probability density, known as persistent oscillations , that drastically distorts the maxima used in the measurement process. These persistent oscillations are periodic Rabi-type oscillations that arise from an interplay between the incidence energy of the initial quantum wave and the bound state of the system, and their frequency play an important role in the dynamics of the delay time.
Analytic solutions to the time-dependent Schr\"odinger equation for cutoff wave initial conditions are used to investigate the time evolution of the transmitted probability density for tunneling. For a broad range of values of the potential barrier opacity $\ensuremath{\alpha},$ we find that the probability density exhibits two evolving structures. One refers to the propagation of a forerunner related to a time domain resonance [Phys. Rev. A 64, 0121907 (2001)], while the other consists of a semiclassical propagating wave front. We find a regime where the forerunners are absent, corresponding to positive time delays, and show that this regime is characterized by opacities $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}.$ The critical opacity ${\ensuremath{\alpha}}_{c}$ is derived from the analytical expression for the delay time, which reflects a link between transient effects in tunneling and the delay time.
Two different formulations of the problem of time-delay caused by scattering from an oscillating center of force are studied. In the first one, appropriate for local interactions, a semi-classical time-dependent solution of the Schrödinger equation is obtained by writing the wave function as a solution of two coupled equations one of which is the Hamilton–Jacobi equation. These equations are then solved using the narrow wave packet approximation. The classical motion of the center of this wave packet is found for two specific cases, a square well and a Morse potential. In the second formulation a time-independent many-channel solution of the Schrödinger equation is obtained for an exactly solvable quantum-mechanical problem with a potential consisting of a separable attractive part which oscillates and a fixed rigid wall. The results show that the time-delay is extremely sensitive to the values of the amplitude and the frequency of oscillations, and also to the energy of the incident particle.