Reconfigurable network for quantum transport simulations

We put forward a versatile, highly scalable, tunable electronic platform for the simulation of single-excitation quantum transport phenomena. Our system, comprising 10 state-of-the-art, fully reconfigurable electronic oscillators, is implemented by making use of functional blocks synthesized with operational amplifiers and passive linear electrical components. To test the robustness and precise control of our platform, we simulate different quantum transport protocols, such as the ballistic propagation of a single-excitation wave function in an ordered lattice, and its localization due to disorder. We implement the Su-Schrieffer-Heeger model to directly observe the emergence of topologically protected one-dimensional edge states. Furthermore, we present the realization of the so-called perfect transport protocol, a key milestone for the development of scalable quantum computing and communication. Finally, we show a simulation of the exciton dynamics in the B800 ring of the purple bacteria LH2 complex. The high fidelity of our simulations together with the low decoherence of our device make it a robust, versatile, and promising platform for the simulation of quantum transport phenomena.

Figure 1

Time evolution of an excitation (voltage signal) initialized in the central site of a one-dimensional network comprising nine nearest-neighbor-coupled oscillators. The rows (from top to bottom) depict the evolution for increasingly larger degree of disorder: (a), (b) $\mathrm{\Delta }=0$, (c), (d) $\mathrm{\Delta }=0.5$, and (e), (f) $\mathrm{\Delta }=0.9$. The results presented in all panels correspond to the average of 50 different disordered-array time evolutions.

Figure 2

Time evolution of a voltage signal (excitation) initialized in (a) the edge and (b) the bulk of a 10-oscillator Su-Schrieffer-Heeger (SSH) chain. (c) The dynamics of an excitation initialized at the edge of the system in the trivial phase, i.e., with the opposite bond dimerization pattern. The fast oscillating signals correspond to the experimentally measured squared voltages ${\left|V_n\right|}^2$, whereas the slowly varying envelope (solid lines) shows the theoretically predicted behavior of the quantum populations ${\left|c_n\right|}^2$. The insets show the lattice structure, as well as the initial excitation conditions in each case.

Figure 3

Time evolution of a voltage signal (excitation) initialized in the first site of the chain ($n=1$). The inset shows the lattice structure, as well as the initial excitation condition. Note that after the characteristic time $t_f=5.6\mathrm{s}$, the signal injected into the first site of the chain ($n=1$) is coherently transferred to the final site ($n=7$) with an efficiency of 0.61, that is, 61% of the total energy is recollected in the intended final site of the lattice.

Figure 4

Time evolution of a voltage signal (excitation) initialized in one of the sites of the B800 ring ($n=1$). The inset shows the ring structure, as well as the initial excitation condition. Note that the results are presented in a rescaled time window (3.2 s), which corresponds to a $\sim 0.6$ ps time evolution in the real photosynthetic complex.

Figure 5

Time evolution of a voltage signal (excitation) injected into site 1 of the modified B800 complex ($n=1$). The inset shows the ring structure with artificially introduced long-range interactions between sites 1, 4, and 7. The values of the couplings are given by the relations $J_{14}=J_{B800}/3, J_{47}=2J_{B800}/3$, and $J_{71}=J_{B800}$.

Figure 6

Schematics for the (a) oscillators, (b) couplings, and (c) parameter digital control. The input signals are indicated with red nodes, whereas the outputs are denoted with blue ones. $R_{fj}, C_{fj}, U_j$, and $M_j$ stand for resistors, capacitors, operational amplifiers, and analog multipliers, respectively. The voltage in the capacitor $V_n$ and the currents across the oscillator inductor $I_n$ are the electrical variables of interest, and the label $V_0$ refers to the interconnection of feedback signals. $V_{Rn}, V_{Ln}, V_{Cn}$, and $V_{IC}$ are voltage signals generated by the DACs, which allow one to configure the system parameters in an easy and accessible way via software. The communication between the microcontroller and the DACs is performed through the SPI protocol, which makes use of the digital signals LDAC, DATA, and CLK, and the configuration bits ($\mathrm{LE}j, A_0, A_1$, and $A_2$) sent to the demultiplexers. (d) Printed circuit board of 10 fully reconfigurable RLC oscillators.

Figure 7

Time evolution of the normalized energy in the electronic platform (solid line) and the quantum dissipative model (dashed line). The dash-dotted line shows an exponential curve fitting. The parameters used for obtaining the energy curves are those used in the experimental implementation of the SSH model.