Quantum scattering in driven single- and double-barrier systems

We investigate the quantum transmission through laterally driven single- and double-barrier systems in the nonlinear regime of strong driving. A broad parameter range is explored, distinguishing in particular between different frequency regimes. The applicability of an effective, time independent, potential description in the high-frequency regime is explored. Moreover, we analyze in detail the inelastic processes and their dependence on parameters, resonant tunneling, and photon assisted tunneling. For the single-barrier problem we address driving laws that differ from the purely sinusoidal one. In this context, we encounter reduced spatial and temporal symmetries and demonstrate the corresponding effects on quantum pumping. For the double barrier, the focus of our studies lies on the impact of the variation in the relative phase of the barriers on the transmission.

Figure 1

Setup of the double-barrier system at initial time $t=t_0$.

Figure 2

(Color online) Dependence of the transmission on the initial phase of the barrier oscillation at different barrier frequencies. $\sigma =2$, $E=0.8V_0$, $a=2$, $V_0=5$.

Figure 3

Shapes of the effective potentials for $V_0=5$ and $a=2$ (left), $a=0.3$ (right). Dashed line: Corresponding harmonic approximation around the minimum.

Figure 4

(Color online) Transmission for high frequencies ($\omega =4,7,10$, effective potential). $V=5$, $a=2$, broad wave packets: $\sigma =30$.

Figure 5

(Color online) Parameters: $V_0=5$, $a=2$. (a) Transmission for intermediate frequencies with $\varphi =0$, $\sigma =10$. (b) Reflection coefficients for several Floquet channels for $\varphi =0$, $\sigma =10$. (c) and (d) Comparison between transmission curves for different phases with a broad $\left(\sigma =10\right)$ wave packet (c) and a narrow $\left(\sigma =2\right)$ wave packet (d).

Figure 6

(Color online) Examples of different driving laws for $V_0=5$. $x_m\left(t\right)$ is the position of the barrier center. (a) $x_m\left(t\right)=2\left[\text{sin}\left(\omega t\right)+0.2\text{cos}\left(2\omega t\right)\right]$, (b) $x_m\left(t\right)=2\left[\text{sin}\left(\omega t\right)-0.3\text{sin}\left(2\omega t\right)\right]$, and (c) $x_m\left(t\right)=\text{sin}\left(2\omega t\right)+\phantom{[}\text{sin}\left(3\omega t\right)]$. First column: Position of the barrier center as a function of time for $\omega =1$. Second column: Effective potential, i.e., time average over one period $T=2\pi /\omega$. Third column: Transmission curves for $\omega =4$ employing left and right incident wave packets.

Figure 7

(Color online) Rabi oscillations between the two wells of the effective potential displayed in Fig. 6b. Parameters: $V_0=40$, driving law: $x_m\left(t\right)=2\left[\text{sin}\left(\omega t\right)-0.3\text{sin}\left(2\omega t\right)\right]$. (a) Scattering off the effective potential. (b) Scattering off the oscillating potential, $\omega =10$.

Figure 8

(Color online) Transmission of a double barrier as a function of the initial energy of the wave packet for high frequencies. The curves converge to the transmission of the effective potential. Note that the curve for $\omega =40$ almost exactly coincides with the transmission through the effective potential, only at about $E=43$ there is a small dip. Parameters: $V=5$, $a_0=2$, $a_1=a_2=0.5$, $\varphi =0$, and $\sigma =30$.

Figure 9

(Color online) (a) Transmission and (b) dwelltime of a double barrier as a function of initial energy of the wave packet for two different relative phases of the barrier oscillation. Parameters: $V=5$, $a_0=2$, $a_1=a_2=0.5$, $\omega =10$, and $\sigma =30$.

Figure 10

Value of the transmission function and dwelltime at $E=\hslash \omega +0.5$ (lowest first-order resonance) as a function of $\omega$. Top row: $\varphi =0$, bottom row: $\varphi =\pi$. Parameters: $V=5$, $a_0=2$, $a_1=a_2=0.5$, and $\sigma =30$. The dashed lines indicate the frequency $\omega =10$ that was considered in the above discussion.

Figure 11

(Color online) Transmission and reflection coefficients decomposed into the Floquet channels as a function of the initial energy. The three channels with the largest occupation are displayed $\left(n=0,\pm 1\right)$. Parameters: $V=5$, $a_0=2$, $a_1=a_2=0.5$, $\omega =10$, and $\sigma =30$, different relative phases $\varphi$ of the barrier oscillation. Top row: $\varphi =0$, bottom row: $\varphi =\pi$. The arrows indicate the energies of maximum discrepancies between the two relative phases (top and bottom). Note that at low energies the wave packet is entirely reflected without undergoing inelastic interactions, therefore $R_1\left(E\right)\to 1$ for $E\to 0$.

Figure 12

(Color online) Time evolution of the probability density ${\left|\Psi \left(x,t\right)\right|}^2$ over one oscillation period in the midst of the scattering process. Parameters: $V=5$, $a_0=2$, $a_1=a_2=0.5$, $\omega =10$, $\sigma =30$, and $E_0=16$, (a) relative barrier phase $\varphi =0$, (b) $\varphi =\pi$.

Figure 13

(Color online) Transmission and dwelltime curves of a system where only one of the two barriers is oscillating, the other one static. Parameters: $V=5$, $a_0=2$, and $a_1=0.5$ or $a_2=0.5$, $\omega =10$, and $\sigma =30$.

Figure 14

(Color online) Transmission and dwelltime curves of a system of two barriers oscillating with different frequencies. Parameters: $V=5$, $a_0=2$, $a_1=a_2=0.5$, $\varphi =0$, and $\sigma =30$.