Transient probing of the symmetry and the asymmetry of electron interference

The transient processes of electron transport in nano-scale devices exhibit special phenomena that exist only in the transient regime. Besides how fast the steady states are approached, one interesting aspect of transient transport arises from its strong dependence on the initial state of the system. Here we address the issue of how the symmetries embedded in the initial state interplay with those of the system structure in the course of transient transports. We explicitly explore the transient currents arising from various initial occupations in a double-quantum-dot Aharonov-Bohm interferometer. We find symmetry relations between the transient in-tunneling and out-tunneling dynamics for initially empty or full quantum dots when the energy levels in the electrodes are symmetrically distributed with respect to the energy levels in the quantum dots. This is true for whatever applied fluxes. We also find the flux-even components of the currents and the flux-odd components of the currents exhibit distinct cross-lead symmetric relations.

Figure 1

A schematic sketch of the DQD device under consideration. Two single-level QDs with on-site energies ${\varepsilon }_1$ and ${\varepsilon }_2$ are coupled to the two separate leads with the tunneling amplitudes $t_{i\alpha k}$. The device is threaded by the the Aharonov-Bohm flux $\varphi$.

Figure 2

Various transient components of the currents with up-down symmetry, ${\overline{\mathrm{\Gamma }}}_{11}^L={\overline{\mathrm{\Gamma }}}_{22}^L=0.1\mathrm{\Gamma }$ and ${\overline{\mathrm{\Gamma }}}_{11}^R={\overline{\mathrm{\Gamma }}}_{22}^R=0.9\mathrm{\Gamma }$, at degeneracy $\delta \varepsilon =0$. In plots (a1), (a2), (b1), and (b2) different line styles are for different fluxes as indicated on the right. In plot (c) we show the decay rates as a function of flux under the bonds and on-site energies specified above. We have set ${\mu }_L={\mu }_R={\mu }_0={\varepsilon }_0$ in this figure as well as in Figs. 6 and 7. The temperature is taken to be $k_{B}T=0.1\mathrm{\Gamma }$ for all figures, such that the thermal effects due to finite temperature become unimportant.

Figure 3

The even components of the initial-occupation-induced currents with up-down symmetry, ${\overline{\mathrm{\Gamma }}}_{11}^L={\overline{\mathrm{\Gamma }}}_{22}^L=0.1\mathrm{\Gamma }$ and ${\overline{\mathrm{\Gamma }}}_{11}^R={\overline{\mathrm{\Gamma }}}_{22}^R=0.9\mathrm{\Gamma }$, are shown in plots (a) and (b). Here we set $\delta \varepsilon =-0.5\mathrm{\Gamma }$ such that $|{\mathrm{\Gamma }}_{12}{\left(\varphi \right)|}^2>\delta {\varepsilon }^2$ and the corresponding decay rates are plotted in (c).

Figure 4

The even components of the initial-occupation-induced currents with nondegenerate QDs of on-site energy splitting, $\delta \varepsilon =-4\mathrm{\Gamma }$, and with up-down symmetry, ${\overline{\mathrm{\Gamma }}}_{11}^L={\overline{\mathrm{\Gamma }}}_{22}^L=0.1\mathrm{\Gamma }$ and ${\overline{\mathrm{\Gamma }}}_{11}^R={\overline{\mathrm{\Gamma }}}_{22}^R=0.9\mathrm{\Gamma }$. Different line styles are for different fluxes, as indicated on the right.

Figure 5

The odd components of the initial-occupation-induced currents. Plots (a) and (b) are with the same parameters used in Fig. 3 for $|{\mathrm{\Gamma }}_{12}{\left(\varphi \right)|}^2>\delta {\varepsilon }^2$ and in Fig. 4 for $|{\mathrm{\Gamma }}_{12}{\left(\varphi \right)|}^2<\delta {\varepsilon }^2$, respectively.

Figure 6

Various transient components of the currents with left-right symmetry, ${\overline{\mathrm{\Gamma }}}_{11}^L={\overline{\mathrm{\Gamma }}}_{11}^R=0.1\mathrm{\Gamma }$ and ${\overline{\mathrm{\Gamma }}}_{22}^L={\overline{\mathrm{\Gamma }}}_{22}^R=0.9\mathrm{\Gamma }$, at degeneracy $\delta \varepsilon =0$. Different line styles are for different fluxes as indicated on the right of plot (c).

Figure 7

Various components of the transient currents with left-right symmetry, ${\overline{\mathrm{\Gamma }}}_{11}^L={\overline{\mathrm{\Gamma }}}_{11}^R=0.1\mathrm{\Gamma }$ and ${\overline{\mathrm{\Gamma }}}_{22}^L={\overline{\mathrm{\Gamma }}}_{22}^R=0.9\mathrm{\Gamma }$, at nondegeneracy $\delta \varepsilon =-2\mathrm{\Gamma }$. Different line styles are for different fluxes, as indicated on the top. The parts that are even in the flux are shown in plots (a1), (a2), (a3) and the corresponding odd parts are shown in plots (b1), (b2), and (b3).

Figure 8

The difference between in-tunneling and out-tunneling currents, quantified by the even component of Eq. (49), $\mathrm{\Delta }{I}_{\alpha }^{+}(t,\varphi )$. Here we take $\varphi =\pi /2$. In plots (a1) and (b1), the bonds are ${\overline{\mathrm{\Gamma }}}_{11}^L=0.2, {\overline{\mathrm{\Gamma }}}_{11}^R=0.6, {\overline{\mathrm{\Gamma }}}_{22}^L=0.7$, and ${\overline{\mathrm{\Gamma }}}_{22}^R=0.5$. Different line styles there correspond to different $\delta \varepsilon$, as shown on the right of (b1). In plots (a2) and (b2), the bonds are ${\overline{\mathrm{\Gamma }}}_{11}^L={\overline{\mathrm{\Gamma }}}_{22}^L=0.1\mathrm{\Gamma }$ and ${\overline{\mathrm{\Gamma }}}_{11}^R={\overline{\mathrm{\Gamma }}}_{22}^R=0.9\mathrm{\Gamma }$ and and the on-site energies are $\delta \varepsilon =-2\mathrm{\Gamma }$. Different line styles there correspond to different positions of ${\mu }_0$, as indicated to the right of (b2). In plots (a3) and (b3), we set degenerate QDs with the setting of unequal bonds used in (a1) and (b1) for all the curves. Different line styles here are for different positions of ${\mu }_0$, as shown on the right of (b3).

Figure 9

The odd components of $\mathrm{\Delta }{I}_{\alpha }(t,\varphi )$ [plot (a)], and its composition currents, ${\overline{I}}_{\alpha }^{\text{em.}}(t,\varphi )$ [plot (b)] and $I_{\alpha }^{[1+2]}(t,\varphi )$ [plot (c)], at $\varphi =\pi /2$ with other parameters used in Figs. 8 and 8.

Figure 10

The even and the odd components of the directly observable currents that start at different initial occupations. The applied bias is $\text{eV}=0.5\mathrm{\Gamma }$ and the flux is taken at $\varphi =\pi /2$. The plots (a1) and (a2) are with ${\mu }_0={\varepsilon }_0$, while the plots (b1) and (b2) are with ${\mu }_0={\varepsilon }_0+\mathrm{\Gamma }$. In all the plots we have let ${\overline{\mathrm{\Gamma }}}_{11}^{\alpha }={\overline{\mathrm{\Gamma }}}_{22}^{\alpha }={\mathrm{\Gamma }}^{\alpha }$, with $\delta \varepsilon =-2\mathrm{\Gamma }$. The red solid lines in (a1) and (b1) are for the even component ${\left.I_R^{+}(t,\varphi )\right|}_{n_1\left(t_0\right)=n_2\left(t_0\right)=0}$, while those in (a2) and (b2) are for the odd component ${\left.I_R^{-}(t,\varphi )\right|}_{n_1\left(t_0\right)=n_2\left(t_0\right)=0}$ evaluated with $({\mathrm{\Gamma }}^L,{\mathrm{\Gamma }}^R)=(0.1,0.9)\mathrm{\Gamma }$. The black long-dashed lines in (a1) and (b1) are for the even components, $-{\left.I_L^{+}(t,\varphi )\right|}_{n_1\left(t_0\right)=n_2\left(t_0\right)=1}$, and those in (a2) and (b2) are for the odd components, ${\left.I_L^{-}(t,\varphi )\right|}_{n_1\left(t_0\right)=n_2\left(t_0\right)=1}$, evaluated with interchanged bonds as $({\mathrm{\Gamma }}^L,{\mathrm{\Gamma }}^R)=(0.9,0.1)\mathrm{\Gamma }$. The green dash-dotted lines are drawn to indicate the zero value. One sees that in the long-time limit, the even components (a1) and (b1) are away from zero while the odd components (a2) and (b2) overlap with zero.