Tunneling-induced Talbot effect

Abstract

We investigate the reforming of a plane wave into a periodic waveform in its propagation through a structural asymmetry four-level quantum dot molecule (QDM) system that is induced by an inter-dot tunneling process and present the resulting tunneling-induced Talbot effect. The tunneling process between two neighborhood dots is provided with the aid of a gate voltage. Using a periodic coupling field the response of the medium to the propagating plane probe beam becomes periodic. The needed periodic coupling field is generated with the interference of two coherent plane waves having a small angle and propagating almost parallel to the probe beam direction. In the presence of the tunneling effect of an electron between two adjacent QDs, for the probe beam propagating through the QDM system, the medium becomes transparent where the coupling fields interfere constructively. As a result, the spatial periodicity of the coupling field modulates the passing plane probe beam. We determine the minimum length of the QDM system to generate a periodic intensity profile with a visibility value equal to 1 for the probe field at the exit plane of the medium. It is also shown that by increasing the propagation length of the probe beam through the QDM medium, the profile of the maximum intensity areas becomes sharper. This feature is quantified by considering a sharpness factor for the intensity profile of the probe beam at the transverse plane. Finally, we investigate free space propagation of the induced periodic field and present the Talbot images of the tunneling-induced periodic patterns at different propagation distances for different values of the QDM medium lengths. The presented dynamically designing method of the periodic coherent intensity patterns might find applications in science and technology. For instance, in optical lithography, the need to use micro/nanofabricated physical transmission diffraction gratings, in which preparation of them is expensive and time-consuming, can be eliminated.

Figure 1

The energy levels of the QDM system before (a) and after (b) applying the external voltage. (c) The energy diagram of the four-level ladder type QDM system and two applied laser fields. The electron and hole are shown by $\ominus$ and $\oplus$, respectively.

Figure 2

A possible arrangement of the QDM system including the input probe field with the plane wavefront, standing coupling field, the output probe field profile and its image patterns.

Figure 3

Imaginary part of ${\rho }_{21}$ as a function of ${\mathrm{\Delta }}_p/\mathrm{\Gamma }$. Used parameters are ${\mathrm{\Gamma }}_2={\mathrm{\Gamma }}_4=\mathrm{\Gamma }$, ${\mathrm{\Omega }}_p=0.1\mathrm{\Gamma }$, ${\mathrm{\Omega }}_c=0.5\mathrm{\Gamma }$, and ${\mathrm{\Delta }}_c=0$.

Figure 4

The propagation of the probe field amplitude inside the QDM medium, $|E_p(x,z)/E_p(0)|$, and outside it, $|{\psi }_p(X,Z)/E_p(0)|$, in the absence of the tunneling effect, $T_e=0$. Used parameters are ${\mathrm{\Omega }}_{c0}=0.5\mathrm{\Gamma }$, $d=10\mathrm{\mu }m$, ${\lambda }_p=870$ nm, and $L=6.9\mathrm{\mu }$m under the multi-photon resonance condition, ${\mathrm{\Delta }}_p={\mathrm{\Delta }}_c=0$.

Figure 5

The propagation of the probe field amplitude inside the QDM medium, $|E_p(x,z)/E_p(0)|$, and outside it, $|{\psi }_p(X,Z)/E_p(0)|$, for the maximum value of the tunneling parameter, $T_e=\mathrm{\Gamma }$. Other used parameters are the same as in Fig. 4.

Figure 6

The visibility of the induced periodic waveform as a function of (a) medium length (for $T_e=\mathrm{\Gamma }$ and ${\mathrm{\Omega }}_{c0}=0.5\mathrm{\Gamma }$), (b) tunneling parameter (for $L=2.5\mathrm{\mu }$m and ${\mathrm{\Omega }}_{c0}=0.5\mathrm{\Gamma }$), and (c) strength of the coupling field (for $T_e=\mathrm{\Gamma }$ and $L=2.5\mathrm{\mu }$m).

Figure 7

The finesse of fringes of the induced periodic waveform versus (a) medium length (for $T_e=\mathrm{\Gamma }$ and ${\mathrm{\Omega }}_{c0}=0.5\mathrm{\Gamma }$), (b) tunneling parameter (for $L=2.5\mathrm{\mu }$m and ${\mathrm{\Omega }}_{c0}=0.5\mathrm{\Gamma }$), and (c) strength of the coupling field (for $T_e=\mathrm{\Gamma }$ and $L=2.5\mathrm{\mu }$m).

Figure 8

The visibility of the intensity profile of the quarter Talbot plane as a function of (a) medium length (for ${\mathrm{\Omega }}_{c0}=0.3\mathrm{\Gamma }$ and $T_e=\mathrm{\Gamma }$ (solid curve), ${\mathrm{\Omega }}_{c0}=0.5\mathrm{\Gamma }$ and $T_e=\mathrm{\Gamma }$ (dashed curve), and ${\mathrm{\Omega }}_{c0}=0.5\mathrm{\Gamma }$ and $T_e=0.5\mathrm{\Gamma }$ (dash-dotted curve)), (b) tunneling parameter (for ${\mathrm{\Omega }}_{c0}=0.3\mathrm{\Gamma }$ and $L=2.5\mathrm{\mu }$m (solid curve), ${\mathrm{\Omega }}_{c0}=0.5\mathrm{\Gamma }$ and $L=2.5\mathrm{\mu }$m (dashed curve), and ${\mathrm{\Omega }}_{c0}=0.5\mathrm{\Gamma }$ and $L=7.5\mathrm{\mu }$m (dash-dotted curve)), and (c) strength of the coupling field (for $L=2.5\mathrm{\mu }$m and $T_e=\mathrm{\Gamma }$ (solid curve), $L=7.5\mathrm{\mu }$m and $T_e=\mathrm{\Gamma }$ (dashed curve), and $L=7.5\mathrm{\mu }$m and $T_e=0.5\mathrm{\Gamma }$ (dash-dotted curve)). Other used parameters are the same as in Fig. 4.

Figure 9

The profiles of (a) the output probe field at $z=L$ and its diffracted field at (b) $Z=Z_T/5$, (c) $Z=Z_T/4$, (d) $Z=3Z_T/8$, (e) $Z=9Z_T/20$, and (f) $Z=Z_T/2$. Solid (dashed) curves with the left (right) vertical axis demonstrate the amplitude (intensity) of the field in all panels. The selected set of parameters are ${\mathrm{\Omega }}_{c0}=0.5\mathrm{\Gamma }$, $L=2.5\mathrm{\mu }$m, and $T_e=\mathrm{\Gamma }$. The other used parameters are the same as in Fig. 4.

Figure 10

The profiles of (a) the output probe field at $z=L$ and its diffracted field at (b) $Z=Z_T/5$, (c) $Z=Z_T/4$, (d) $Z=3Z_T/8$, (e) $Z=9Z_T/20$, and (f) $Z=Z_T/2$ under the same parameters of Fig. 9 except for $T_e=0.5\mathrm{\Gamma }$. Solid (dashed) curves with the left (right) vertical axis demonstrate the amplitude (intensity) of the field in all panels.

Figure 11

The propagation of the probe field amplitude inside the QDM medium, $|E_p(x,z)/E_p(0)|$, and outside it, $|{\psi }_p(X,Z)/E_p(0)|$, for different lengths of the QDM medium, i.e. (a) $L=2.5\mathrm{\mu }$m, (b) $L=5.25\mathrm{\mu }$m, and (c) $L=10.5\mathrm{\mu }$m. The tunneling parameter is fixed at $T_e=\mathrm{\Gamma }$, whereas other parameter values are the same as in Fig. 4.

Figure 12

The propagation of the probe field amplitude inside the QDM medium, $|E_p(x,z)/E_p(0)|$, and outside it, $|{\psi }_p(X,Z)/E_p(0)|$, for $T_e=0.5\mathrm{\Gamma }$ and different lengths of the QDM medium, i.e. (a) $L=1\mathrm{\mu }$m, (b) $L=2\mathrm{\mu }$m, and (c) $L=4.25\mathrm{\mu }$m. Other used parameters are the same as in Fig. 4.