Effects of local periodic driving on transport and generation of bound states

We periodically kick a local region in a one-dimensional lattice and demonstrate, by studying wave packet dynamics, that the strength and the time period of the kicking can be used as tuning parameters to control the transmission probability across the region. Interestingly, we can tune the transmission to zero which is otherwise impossible to do in a time-independent system. We adapt the nonequilibrium Green's function method to take into account the effects of periodic driving; the results obtained by this method agree with those found by wave packet dynamics if the time period is small. We discover that Floquet bound states can exist in certain ranges of parameters; when the driving frequency is decreased, these states get delocalized and turn into resonances by mixing with the Floquet bulk states. We extend these results to incorporate the effects of local interactions at the driven site, and we find some interesting features in the transmission and the bound states.

Figure 1

Schematic figure of a one-dimensional lattice where a fermion can hop between nearest-neighbor sites with amplitude $-\gamma$. A periodic $\delta$-function kick is applied at the central site $L_c$ of a lattice of length $L$. The kicking strength is $\alpha$.

Figure 2

(Top) Reflection probability $\mathcal{R}$ (red circles) and different conductance $G=(e^2/h)\mathcal{T}$ (blue squares) vs $\alpha$ of a wave packet which is centered at the site $L_o=50$ with width $\sigma =5$ on a lattice with $L=401$ sites. The kicking is done at the $L_c=201\mathrm{th}$ site. The kicking time period is $T=0.5$, and the momentum is centered at $k_c=\pi /2$. The wave packet is evolved up to a time $(L-2L_o)/|2sin{k}_c|$. (Bottom) Differential conductance $G=(e^2/h)\mathcal{T}$ vs $T$ when $\alpha =1$ is kept fixed. Other parameters are the same as in the top panel.

Figure 3

Transmission and reflection of a wave packet which is centered at the site $L_o=50$ with width $\sigma =5$ on a lattice with $L=801$ sites. The kicking is done at the $L_c=401\mathrm{th}$ site (denoted by a vertical blue line). The kicking time period is $T=1$, and the momentum is centered at $k_c=\pi /2$. The wave packet is shown at various intervals of time. From top to bottom, the different cases correspond to kicking strengths $\alpha =0$, 1.4, and 3. In the first case, the wave packet moves across the kicked site unhindered. In the second case, the original wave packet splits into two, one which transmits across the barrier and the other which reflects. In the third case, the wave packet gets completely reflected from the central site.

Figure 4

Differential conductance $G\left(E_c\right)=(e^2/h)\mathcal{T}\left(E_c\right)$ vs $\alpha$. $G$ is computed from the dynamics of a wave packet which is centered at the site $L_o=50$ with width $\sigma =10$ on a lattice with $L=401$ sites. The kicking is done at the 201th site, and the time period takes the values 0.4, 1.2, 2.4, and 3.6 in the four figures from top left to bottom right. The momentum is centered at $k_c=\pi /2$, so that $E_c=0$. The transmissions obtained from the exact wave packet dynamics and the NEGF formalism are compared. We see that the NEGF formalism matches the exact results for small $T$.

Figure 5

(Top) Bound state energy and (bottom) IPR for $\alpha =0.5$ and different values of $T$. The numerically calculated spectrum matches well the analytical expression as shown in Eqs. (17) and (20), for $L=401$ and $L_c=201$.

Figure 6

The maximum IPR value of the eigenstates of the time evolution operator as a function of $T$ and $\alpha$ for $V=0$ (left panel) and $V=-4$ (right panel). The system has $L=401$ sites and the central site is kicked periodically. For $V=0$, the IPR increases as $\alpha$ increases, while increasing $T$ reduces the IPR. When $T$ crosses $\pi /2$ the bound state (which has the largest IPR) ceases to exist.

Figure 7

(Left) Eigenvalues of time evolution operator for a system with 401 sites in which $\delta$-function kicks are applied at the 201th site with $\alpha =0.4$ and $T=1$. There is a bound state with IPR equal to 0.0936 and Floquet eigenvalue equal to $-0.4473-0.8944i$ shown by a large red dot. (Middle) Probability ${\left|\psi \left(n\right)\right|}^2$ of the bound state. (Right) Square of the modulus of the Fourier transform $|\tilde{\psi }{\left(k\right)|}^2$ of the bound state. It has peaks at $k=\pm \pi$.

Figure 8

(Left) Eigenvalues of time evolution operator for a system with 401 sites in which $\delta$-function kicks are applied to the 201th site with strength $\alpha =0.4$ and time period $T=2$. (There are more eigenvalues on the left side than on the right side; hence the plot looks solid on the left and dotted on the right.) There is a resonance state with Floquet eigenvalue equal to $-0.6388+0.7694i$ shown by a large red dot. (Middle) Probability ${\left|\psi \left(n\right)\right|}^2$ of the resonance state. (Right) Square of the modulus of the Fourier transform $|\tilde{\psi }{\left(k\right)|}^2$ of the resonance state. It has peaks at both $k=\pm \pi$ and $k=\pm 0.967$, thus showing a substantial mixing with the bulk states.

Figure 9

The maximum IPR value of the eigenstates of the time evolution operator as a function of the system size $L$ for (I) $\alpha =0.4$ and $T=2$ and (II) $\alpha =1.0$ and $T=2.5$. Curve I corresponding to the lower value of $\alpha$ has much larger fluctuations than curve II. The dotted lines show a fit of the form $\mathrm{IPR}=a/L^b$ for the average IPR; for (I) the values $a=0.31$ and $b=0.41$ give the best fit; for (II) $a=2.65$ and $b=0.83$.

Figure 10

Differential conductance $G=(e^2/h)\mathcal{T}$ vs $\alpha$ for a system with $V=-4$. We see that $\alpha$ can tune the conductance all the way from zero to 1. Here $L=401, k_c=\pi /2, L_o=50, \sigma =5$, and $T=0.5$.

Figure 11

Reflection probability ${\mathcal{R}}_{\uparrow }$ of a spin-up electron for a wave packet in a system with an on-site interaction $U$ and kicking strengths $\alpha =0$ (lower curve) and 0.2 (upper curve). Even with $\alpha =0$ the wave packet is not completely transmitted in the presence of a finite $U$. This happens because there is a finite probability for the incoming wave packet to get trapped in a bound state which lives near the interacting site. The effect of $U$ acts asymmetrically in the presence of a finite $\alpha$. The effect of $U$ reduces with increasing $\alpha$ and at large values of $\alpha , {\mathcal{R}}_{\uparrow }$ is independent of $U$. We have taken $L=51, T=0.25, k_c=\pi /2, \sigma =4$, and $L_o=6$.

Figure 12

(Top) The value of the two-body IPR for the state with the maximum IPR as a function of the Hubbard interaction $U$, with and without periodic driving. We have taken $L=51, T=0.5, \alpha =0.4$ (black disks) and $\alpha =0$ (red squares). (Bottom) A two-particle bound state wave function for $L=51, T=0.5, \alpha =0.4$, and $U=1$ where a much more localized bound state is produced by the driving.