Coherent control of correlated nanodevices: A hybrid time-dependent numerical renormalization-group approach to periodic switching

The time-dependent numerical renormalization-group approach (TD-NRG), originally devised for tracking the real-time dynamics of quantum-impurity systems following a single quantum quench, is extended to multiple switching events. This generalization of the TD-NRG encompasses the possibility of periodic switching, allowing for coherent control of strongly correlated systems by an external time-dependent field. To this end, we have embedded the TD-NRG in a hybrid framework that combines the outstanding capabilities of the numerical renormalization group to systematically construct the effective low-energy Hamiltonian of the system with the prowess of complementary approaches for calculating the real-time dynamics derived from this Hamiltonian. We demonstrate the power of our approach by hybridizing the TD-NRG with the Chebyshev expansion technique in order to investigate periodic switching in the interacting resonant-level model. Although the interacting model shares the same low-energy fixed point as its noninteracting counterpart, we surprisingly find the gradual emergence of damped oscillations as the interaction strength is increased. Focusing on a single quantum quench and using a strong-coupling analysis, we reveal the origin of these interaction-induced oscillations and provide an analytical estimate for their frequency. The latter agrees well with the numerical results.

Figure 1

(a) Real-time dynamics of the impurity charge $n_d\left(t\right)$ in the resonant-level model following a sudden quench of the level energy from $E_d=-2{\Gamma }_0$ to $E_d=2{\Gamma }_0$ at time $t=0$. The red solid line shows the exact time evolution on the Wilson chain, obtained by exact diagonalization of the single-particle eigenmodes. The dashed blue line depicts the TD-NRG result, while the solid black line shows the exact analytical solution for a continuous band [given by Eq. (50) of Ref. 31]. (b) A two-dimensional contour plot of the exact time-dependent occupancies $n_i\left(t\right)$ of the first 26 sites along the Wilson chain, labeled $i=0,...,25$. The red arrows indicate instances in time when reflected currents reach the impurity site. The very same times are marked by the black arrows in panel (a). Parameters: $\Lambda =1.8$, ${\Gamma }_0/D={10}^{-2}$, the number of states kept in the TD-NRG is $N_s=800$, and the chain length is $N=40$.

Figure 2

The exact single-particle energy levels of the RLM on a Wilson chain with $\Lambda =1.8$, ${\Gamma }_0/D={10}^{-2}$, and $E_d=0$. The chain length is $N=40$. Panel (a) depicts positive energies on a logarithmic scale. Panel (b) shows the full spectrum on a linear scale.

Figure 3

The full Wilson chain of length $N$ is divided into a subchain of length $m$ and the “environment” $R_{m,N}$. The Hamiltonian ${\mathcal{H}}_m$ can be viewed either as acting only on the subchain of length $m$ or as acting on the full chain of length $N$, but with the hopping matrix elements $t_m,...,t_{N-1}$ all set to zero. The former picture is the traditional one. In the TD-NRG, one adopts the latter point of view.

Figure 4

A hybrid Wilson chain where the first $M+1$ hopping matrix elements decrease according to $t_n\propto {\Lambda }^{-n/2}$ with $\Lambda >1$, and all further hopping matrix elements are held constant and equal to $t_M$.

Figure 5

Comparison of the exact continuum-limit solution for $n_d\left(t\right)$ in the RLM (solid black line) and its exact numerical evaluation for various hybrid chains of the type depicted in Fig. 4. The full red line depicts the case of a pure Wilson chain of length $N=40$. (a) Keeping the total chain length fixed at $N=40$ and varying the partitioning parameter $M$. (b) For $M=12$ and two different chain lengths $N=40$ and 112. Model parameters are ${\Gamma }_i={\Gamma }_f={\Gamma }_0$, $E_d^f=-E_d^i=2{\Gamma }_0$, ${\Gamma }_0/D={10}^{-2}$, and $\Lambda =1.8$.

Figure 6

Comparison of the hybrid NRG-CET approach and exact diagonalization for the RLM on a Wilson chain with $\Lambda =1.4$ and $N=84$. The TD-NRG resummation is applied up to iteration $M$, beyond which the CET is used to track the time evolution of $|{\chi }_M\left(t\right)\rangle$. The number of states retained in the course of the NRG is equal to $N_s=1024$. The resulting occupancy $n_d\left(t\right)$ is averaged over $N_z=8$ equally distributed values of the twist parameter $z$, both for the hybrid-NRG and in exact diagonalization. Model parameters are ${\Gamma }_a={\Gamma }_b={\Gamma }_0$, $E_d^b=-E_d^a={\Gamma }_0$, and $\tau =5/{\Gamma }_0$, with ${\Gamma }_0/D={10}^{-5}$. Upon decreasing $M$, the hybrid NRG-CET approach gradually converges onto the exact result.

Figure 7

Same as Fig. 6 but for $\tau =1/{\Gamma }_0$ and $N_z=1$ (i.e., no $z$ averaging). Note the much stronger $M$ dependence as compared to $\tau =5/{\Gamma }_0$.

Figure 8

The time-dependent response to repeated switchings between ${\Gamma }_a={\Gamma }_0/2$ and ${\Gamma }_b={\Gamma }_0$ for fixed $E_d^a=E_d^b={\Gamma }_0/2$. All other parameters are the same as in Fig. 6.

Figure 9

Same as Fig. 6, for $U/D=1$ and $E_d^b=-E_d^a={\Gamma }_{\mathrm{eff}}$. Here, ${\Gamma }_a={\Gamma }_b={\Gamma }_0=6\times {10}^{-9}D$ was adjusted so as to maintain a fixed ${\Gamma }_{\mathrm{eff}}={10}^{-5}D$. The black line shows for comparison the exact time evolution on the Wilson chain for $U=0$ and ${\Gamma }_a={\Gamma }_b={\Gamma }_0={10}^{-5}D$ (taken from Fig. 6).

Figure 10

The real-time dynamics following a single quantum quench from $E_d^a=-{\Gamma }_{\mathrm{eff}}$ to $E_d^b={\Gamma }_{\mathrm{eff}}$, for the IRLM with ${\Gamma }_a={\Gamma }_b={\Gamma }_0$ and different values of the Coulomb repulsion $U$. Here, ${\Gamma }_0$ was adjusted separately for each value of $U$ so as to maintain a fixed ${\Gamma }_{\mathrm{eff}}={10}^{-5}D$. All remaining parameters are as in Fig. 6.