^{1}, J. G. Muga

^{2}, D. G. Austing

^{3}and G. García-Calderón

^{4}

### Abstract

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 state.

The authors are grateful for the support and encouragment of Dr. Rick Leavens. This work was supported by CERION-II (Canadian European Research Initiative on Nanostructures), “Ministerio de Ciencia y Tecnología-FEDER” (Grant No. BFM2003-01003), and UPV-EHU (Grant No. 00039.310-13507/2001). G.G.C. acknowledges partial financial support from DGAPA-UNAM under Grant No. IN 108003-2. D.G.A. was partly supported by the DARPA-QUIST program (Grant No. DAAD 19-01-1-0659).

I. INTRODUCTION

II. THEORY

A. Evolution of the initially external wave function

B. Evolution of the initially internal wave function

III. DOUBLE SQUARE BARRIER POTENTIAL

A. Resonance buildup

B. Resonance depletion/decay

C. Trapping

IV. CONCLUSION

### Key Topics

- Bound states
- 19.0
- Quantum wells
- 17.0
- Wave functions
- 13.0
- Quantum dots
- 11.0
- Boundary value problems
- 5.0

## Figures

Energy diagrams of the double barrier potential configurations I and II for the resonance buildup and decay processes. The resonance levels depicted are the real parts of the corresponding complex energies, see Table I. Note that the resonance state pole in I corresponds to a bound state in II, whereas the pole state is simply shifted from I to II. The bound (resonant) states are denoted by thick (thin) dashed lines. is the energy of the incident particles.

Energy diagrams of the double barrier potential configurations I and II for the resonance buildup and decay processes. The resonance levels depicted are the real parts of the corresponding complex energies, see Table I. Note that the resonance state pole in I corresponds to a bound state in II, whereas the pole state is simply shifted from I to II. The bound (resonant) states are denoted by thick (thin) dashed lines. is the energy of the incident particles.

Energy diagrams of the double barrier potential configurations II and III for the resonance tapping process . The resonance levels depicted are the real parts of the corresponding complex energies, see Table I. Note that the resonance state pole in II corresponds to a bound state in III, whereas the resonance pole is simply shifted on going from II to III. The bound (resonant) states are denoted by thick (thin) dashed lines. is the energy of the incident particles.

Energy diagrams of the double barrier potential configurations II and III for the resonance tapping process . The resonance levels depicted are the real parts of the corresponding complex energies, see Table I. Note that the resonance state pole in II corresponds to a bound state in III, whereas the resonance pole is simply shifted on going from II to III. The bound (resonant) states are denoted by thick (thin) dashed lines. is the energy of the incident particles.

Poles of the transmission amplitude for the three potential configurations of Figs. 1 and 2. For clarity we represent the poles in the “distorted” complex plane , . Solid (open) symbols correspond to even (odd) states. In the three configurations we use the same symbols for the pole pairs that can be connected by a continuous shift of the well depth: and (circles), and (squares), and (diamonds), and (triangles).

Poles of the transmission amplitude for the three potential configurations of Figs. 1 and 2. For clarity we represent the poles in the “distorted” complex plane , . Solid (open) symbols correspond to even (odd) states. In the three configurations we use the same symbols for the pole pairs that can be connected by a continuous shift of the well depth: and (circles), and (squares), and (diamonds), and (triangles).

Probability density (in arbitrary units) vs for different times in the buildup process (see Fig. 1): (dotted-dashed line), (long-dashed line), (dashed line), (dots) and the stationary state (solid line). We have chosen realistic parameters (Ref. 4), see Table I. The lifetime of for these parameters is .

Probability density (in arbitrary units) vs for different times in the buildup process (see Fig. 1): (dotted-dashed line), (long-dashed line), (dashed line), (dots) and the stationary state (solid line). We have chosen realistic parameters (Ref. 4), see Table I. The lifetime of for these parameters is .

vs time for the buildup process . The dots correspond to an approximation similar to Eq. (27) for the interior region of the double barrier potential. The dashed line marks the asymptotic level of the density. The double barrier parameters are in Table I. The triangle marks the value of the lifetime of the resonance .

vs time for the buildup process . The dots correspond to an approximation similar to Eq. (27) for the interior region of the double barrier potential. The dashed line marks the asymptotic level of the density. The double barrier parameters are in Table I. The triangle marks the value of the lifetime of the resonance .

vs time (solid line) at . The dashed and long-dashed lines correspond respectively to the initial and final values for the decay process , see also Figs. 1 and 2. The triangle marks the lifetime of the resonance , which is different from the lifetime relevant for the buildup process , compare with Fig. 5, as a consequence of the resonance displacement.

vs time (solid line) at . The dashed and long-dashed lines correspond respectively to the initial and final values for the decay process , see also Figs. 1 and 2. The triangle marks the lifetime of the resonance , which is different from the lifetime relevant for the buildup process , compare with Fig. 5, as a consequence of the resonance displacement.

vs time (solid line) at for the decay process : (a) very short times where expression (29) holds (dots) and (b) “long times” close to the transition time (diamond) where formula (30) (dashed line, almost indistinguishable from solid line) may be applied. The straight dashed line is the asymptotic value as .

vs time (solid line) at for the decay process : (a) very short times where expression (29) holds (dots) and (b) “long times” close to the transition time (diamond) where formula (30) (dashed line, almost indistinguishable from solid line) may be applied. The straight dashed line is the asymptotic value as .

Densities (normalized to 1 at the maximum) of the bound state (dashed line) and the on-resonant scattering state for (solid line).

Densities (normalized to 1 at the maximum) of the bound state (dashed line) and the on-resonant scattering state for (solid line).

Probability density in arbitrary units at vs time for the trapping process (solid line). Also plotted is the approximate solution using Eq. (33) (dotted line). The double barrier parameters are given in Table I.

Probability density in arbitrary units at vs time for the trapping process (solid line). Also plotted is the approximate solution using Eq. (33) (dotted line). The double barrier parameters are given in Table I.

## Tables

Incident energy , energies of the resonances, and/or bound states in the configurations I, II, and III of Figs. 4 and 5, and potential levels of the well . In all cases the barriers height is , , , and the effective mass is .

Incident energy , energies of the resonances, and/or bound states in the configurations I, II, and III of Figs. 4 and 5, and potential levels of the well . In all cases the barriers height is , , , and the effective mass is .

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