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Physical Review A

Exponential and nonexponential buildup in resonant tunneling

Roberto Romo, Alberto Hernández-Maldonado, Jorge Villavicencio

Published Feb 27, 2013• 5 citations
Publisher

Abstract

The exponential and nonexponential regimes of the buildup process in resonant tunneling structures are analyzed by considering an analytic solution of the time-dependent Schr\"odinger equation. It is found that the buildup exhibits a purely exponential behavior in a finite time interval followed by a clear transition to a nonexponential regime. The buildup of the probability amplitude in the nonexponential regime follows a ${t}^{\ensuremath{-}3/2}$ time dependence, in the same fashion as the survival amplitude in quantum decay. For incidence energies at higher isolated resonances, $E={\ensuremath{\varepsilon}}_{n}$ with $n>1$, it is found that the exponential regime is split into two exponential subregimes: the first one is governed by the width ${\ensuremath{\Gamma}}_{n}$ of the resonance chosen for the incidence energy, and the second one is dominated by the width ${\ensuremath{\Gamma}}_{1}$ of the lowest resonance. The transition occurs directly from $n$ to 1 without jumps to other intermediate resonant states. This dynamics is discussed in comparison with the (opposite) and well-studied process of quantum decay, with which we find that there are striking similarities in both the exponential and nonexponential regimes. We also analyze the buildup in systems with resonance doublets, where the interference effects produced by the interacting resonances play an important role. In the latter case we find that the buildup of the probability amplitude exhibits a complex oscillatory behavior that can be characterized by a mix of oscillating contributions with well-defined Rabi type frequencies of the form $|E\ensuremath{-}{\ensuremath{\varepsilon}}_{n}|/\ensuremath{\hbar}$, where ${\ensuremath{\varepsilon}}_{n}$ are the nearest resonances around the incidence energy $E$.

DOI
10.1103/physreva.87.022121

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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 time evolution of the buildup process inside a double-barrier system for off-resonance incidence energies is studied by considering the analytic solution of the time dependent Schr\\"{o}dinger equation with cutoff plane wave initial conditions. We show that the buildup process exhibits invariances under arbitrary changes on the system parameters, which can be successfully described by a simple and easy-to-use one-level formula. We find that the buildup of the off-resonant probability density is characterized by an oscillatory pattern modulated by the resonant case which governs the duration of the transient regime. This is evidence that off-resonant and resonant tunneling are two correlated processes, whose transient regime is characterized by the same transient time constant of two lifetimes.
We consider a unified analytical approach for scattering and decay processes in one-dimensional resonant tunneling systems using the formalism of resonant states to address the issue of the differences and similarities in the time evolution of decay between the decay of an arbitrary state prepared initially within a system and the formation and subsequent decay of a quasistationary state in the scattering of a Gaussian wavepacket on that system. We find three distinctive regimes. A first regime, which refers only to the quasistationary state, that is characterized by a buildup time of the probability density at a given position within the internal region of the potential. Here we find that the buildup time has a dependence on position. A second regime, dominated by the exponentially decaying terms, where the decay of the quasistationary state proceeds in an almost identical fashion as for the initially prepared decaying state. And finally, a third regime that involves the transition to nonexponential decay at long times and its ulterior behavior as an inverse power of time. Here we find that the time scale of the transition occurs at different times, which implies a dependence on the parameters of the initial state.
We examine an analytical expression for the survival probability for the time evolution of quantum decay to discuss a regime where quantum decay is nonexponential at all times. We find that the interference between the exponential and nonexponential terms of the survival amplitude modifies the usual exponential decay regime in systems where the ratio of the resonance energy to the decay width is less than 0.3. We suggest that such a regime could be observed in semiconductor double-barrier resonant quantum structures with appropriate parameters.
Abstract By using an exact analytical formalism for quantum tunneling decay we show the existence of a regime of purely nonexponential decay for the survival probability in the full time interval and a propagating probability density along the external interaction region that evolves as a spreading process having no wavefront. This decaying regime follows from a complex pole of the propagator which in the wave number plane possesses an imaginary part larger than the corresponding real part, which implies a negative resonance energy and a decaying width, together with a dominant large overlap of the initial decaying state with the corresponding resonant state of the problem.
We study resonances and time decay in a quantum waveguide with two semitransparent barriers. The model is first analyzed as a one-dimensional Schrödinger operator with two delta interactions and then extended to a two-dimensional strip, where transverse Dirichlet modes reduce the problem, channel by channel, to the one-dimensional setting. We derive explicit scattering amplitudes and characterize resonance poles by analytic continuation. In the strong-barrier regime ρ=α−1→0+, the real parts of the resonances approach the eigenvalues of the decoupled cavity, while the imaginary parts are negative and of order α−2, so the lifetimes grow like α2. We also study the time evolution on short, intermediate, and long time scales: the survival probability is quadratic at short times, in accordance with the quantum Zeno effect; the intermediate regime is resonance-dominated and approximately exponential; and the long-time behavior is governed by the low-energy threshold and becomes algebraic. Numerical simulations complement the analysis by confirming these regimes and by showing the breakdown of exponential decay. They also provide an empirical fit for the breakdown time tbr in terms of the barrier strength α and the distance d1 between the barriers.