Simulating long-range hopping with periodically driven superconducting qubits

Quantum computers are a leading platform for the simulation of many-body physics. This task has been recently facilitated by the possibility to program directly the time-dependent pulses sent to the computer. Here, we use this feature to simulate quantum lattice models with long-range hopping. Our approach is based on an exact mapping between periodically driven quantum systems and one-dimensional lattices in the synthetic Floquet direction. By engineering a periodic drive with a power-law spectrum, we simulate a lattice with long-range hopping, whose decay exponent is freely tunable. We propose and realize experimentally two protocols to probe the long tails of the Floquet eigenfunctions and identify a scaling transition between long-range and short-range couplings. Our paper offers a useful benchmark of pulse engineering and opens the route towards quantum simulations of rich nonequilibrium effects.

Figure 1

Schematic plot of the Floquet Hamiltonian for a single qubit described by Eq. (3) with $h_x\left(t\right)=h_x(t+\tau )$. Each state is characterized by two quantum numbers: spin ($\uparrow$ or $\downarrow$) and Floquet index ($m$). The vertical lines show the energy scales: the driving frequency ${\omega }_p=2\pi /\tau$ and the two-level splitting, ${\omega }_z$. The first and second harmonics of the drive $h_x^{(\pm 1)}$ and $h_x^{(\pm 2)}$ correspond, respectively, to nearest neighbor and next-nearest neighbor couplings.

Figure 2

(a) Time-dependent drive with power-law spectrum, Eq. (4), with a cutoff at $M=5$, for different values of $\alpha$. (b) Time evolution of the qubit for $M=5$ and $\alpha =0$, theory (lines) and experiment (dots).

Figure 3

(a) Eigenstate of the Floquet Hamiltonian: numerical diagonalization of ${\widehat{H}}_F$ (continuous lines) and power-law asymptotics (dashed lines), Eq. (5). The normalization of the wave function is guaranteed by the value of $|{\varphi }_n{(m=0)|}^2$ (not shown). (b) Numerical diagonalization in the presence of a cutoff at $M=20$, giving rise to distinctive peak at $m=M$.

Figure 4

Amplitude of the discrete Fourier components $s_z\left(m\right)$: theory (lines) and experiment (dots). (a) For different values of $M$ at $\alpha =0$. (b) For different values of $\alpha$ at $M=10$. The pronounced peak at $m=M$ is an indicator of the long-range nature of the Floquet wave function (see text). For clarity, the curves with $\alpha >0$ in (b) are shifted vertically by ${10}^{2\alpha }$.

Figure 5

Time evolution of $d{s}_z/dt$ in the vicinity of $t=\tau$: theory (lines) and experiment (dots). The cutoff parameter $M$ allows us to identify the transition between long-range and short-range hopping at $\alpha =1$.

Figure 6

The quasienergies of Eqs. (2) and (5) for various values of the driving strength in the first Floquet-Brillouin zone. Each color represents a different quasienergy.

Figure 7

Time derivative $|d{s}_z/dt|$ at $t=\tau$ as a function of the cut-off parameter, $M$. This plot allows us to identify a transition between long-range and short-range hopping at $\alpha =1$. The dots are simulated data and the solid lines show the expected scaling behavior, Eq. (B11).