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arXiv:quant-ph/0210009 · Physical review. B, Condensed matter

Transient tunneling effects of resonance doublets in triple barrier systems

Roberto Romo, Jorge Villavicencio, Gastón García‐Calderón

Published Jul 16, 2002• 20 citations
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Abstract

Transient tunneling effects in triple barrier systems are investigated by considering a time-dependent solution to the Schr\"odinger 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.

DOI
10.1103/physrevb.66.033108

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Transient tunneling effects in triple barrier systems are investigated by considering a time-dependent solution to the Schr"odinger 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.
An exact analytic solution of the time-dependent Schrödinger equation with cutoff wave initial conditions, that goes into the stationary solution at asymptotically long times, is used to investigate the transient behavior in the time evolution of the probability density along the internal and transmitted regions of a potential barrier. The approach involves the complex poles and residues (resonant states) of the outgoing Green's function of the problem. The solution exhibits absence of propagation along the classically forbidden internal region of the potential, and along the transmitted region it splits into two propagating structures. One of them is a transient effect describing the earliest response of the system to the initial condition whereas the other structure evolves towards the stationary solution.
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An exact analytical solution to the time-dependent Schr\"odinger equation with cutoff wave initial conditions is used to investigate the fast tunneling response of a rectangular potential barrier. We find that just across the tunneling region, the probability density exhibits at short times a transient behavior that may be characterized by a peak tp{t}_{p}tp​ and a width $\ensuremath{\Delta}t.$ We show that tp{t}_{p}tp​ provides the earliest tunneling response of the system and that the top-barrier S-matrix poles play an important role in the process. As a function of the barrier width, tp{t}_{p}tp​ exhibits two regimes. Along the first regime, tp{t}_{p}tp​ remains almost a constant; as the barrier width increases, a second regime appears where tp{t}_{p}tp​ grows linearly with the barrier width.
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.
Transient time-domain resonances found recently in time-dependent solutions to Schr\"odinger's equation are used to investigate the issue of the tunneling time in rectangular potential barriers. In general, a time-frequency analysis 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 tmax{t}_{\mathrm{max}}tmax​ of the time-domain resonance corresponds to under-the-barrier tunneling. We argue that tmax{t}_{\mathrm{max}}tmax​ represents the relevant tunneling time scale through the classically forbidden region.
The energy dependence of the delay time associated with tunnelling through smooth (symmetric) double and triple barriers is investigated. Sharp resonances in the delay are found to mirror resonances in the transmission rate. In the strong-localization limit, particles with off-resonant energies do not perceive the inner potential relief and tunnel through the given double, or triple, barrier in much the same way as they would tunnel through a simple, unstructured barrier. Verification of the derived relations through a high-precision numerical solution of the Schrödinger equation confirms the quality of the obtained expressions in wide energy ranges up to the barrier top.