Simulation of 1D Topological Phases in Driven Quantum Dot Arrays

We propose a driving protocol which allows us to use quantum dot arrays as quantum simulators for 1D topological phases. We show that by driving the system out of equilibrium, one can imprint bond order in the lattice (producing structures such as dimers, trimers, etc.) and selectively modify the hopping amplitudes at will. Our driving protocol also allows for the simultaneous suppression of all the undesired hopping processes and the enhancement of the necessary ones, enforcing certain key symmetries which provide topological protection. In addition, we have discussed its implementation in a 12-QD array with two interacting electrons and found correlation effects in their dynamics, when configurations with different number of edge states are considered.

Figure 1

Topological invariant $\mathcal{W}$ as a function of the driving amplitude $\alpha /\omega$ and decay length of hopping amplitudes $\lambda$, for $q=1$. The maximum range of the hoppings included is fixed by $\lambda$, such that the smallest bare hopping amplitude is a factor ${10}^{-8}$ smaller than the largest one.

Figure 2

Quasienergy levels of a driven 12-quantum dot array and band structure in the thermodynamical limit (red and blue filling for the valence and conduction band, respectively), as a function of $\alpha /\omega$, including first- and third-neighbor hoppings, in the high-frequency regime. Second-neighbor hoppings have been suppressed through the driving protocol. The parameters are $\lambda /\omega =1.5$, $q=1$. Inset: each pair of edge states in the $\mathcal{W}=2$ phase has a different energy splitting, which can be also varied by tuning $\alpha /\omega$.

Figure 3

Occupation of each site of a driven 12-QD array as a function of time when two electrons with opposite spin are loaded into the system, for different values of $U$. The array has first- to third-neighbor couplings, whose values initially decay exponentially with distance, choosing $\lambda =1.5$. Considering the high-frequency regime with $\omega =100$ and $q=1$, the values for $\alpha$ have been chosen such that the desired hopping renormalization is realized. The four left plots correspond to $\alpha =230$, resulting in a topological phase with $\mathcal{W}=1$ (two edge states). The four right plots correspond to $\alpha =410$, yielding a configuration with $\mathcal{W}=2$ (four edge states).