Cavity-Based Reservoir Engineering for Floquet-Engineered Superconducting Circuits

Considering the example of superconducting circuits, we show how Floquet engineering can be combined with reservoir engineering for the controlled preparation of target states. Floquet engineering refers to the control of a quantum system by means of time-periodic forcing, typically in the high-frequency regime, so that the system is governed effectively by a time-independent Floquet Hamiltonian with novel interesting properties. Reservoir engineering, on the other hand, can be achieved in superconducting circuits by coupling a system of artificial atoms (or qubits) dispersively to pumped leaky cavities, so that the induced dissipation guides the system into a desired target state. It is not obvious that the two approaches can be combined, since reaching the dispersive regime, in which system and cavities exchange excitations only virtually, can be spoiled by driving-induced resonant transitions. However, working in the extended Floquet space and treating both system-cavity coupling as well as driving-induced excitation processes on the same footing perturbatively, we identify regimes, where reservoir engineering of targeted Floquet states is possible and accurately described by an effective time-independent master equation. We successfully benchmark our approach for the preparation of the ground state in a system of interacting bosons subjected to Floquet-engineered magnetic fields in different lattice geometries.

Figure 1

$M$ artificial atoms are coupled to $L$ pumped and leaky cavities. The atoms are periodically driven to Floquet engineer effective Hamiltonians with desired properties, while the cavities induce dissipation in a controlled way.

Figure 2

(a) and (b) Examples of unwanted excitation-photon conversion assisted by the Floquet drive. In (a), an excitation tunnels from the array to a cavity releasing an energy $m\hslash \omega$ to the drive, and leaks to the environment being lost. In (b), a photon tunnels from the cavity to the array. (c) Fundamental energy scales in the system.

Figure 3

(a) Ladder geometry and coupling to the cavities. (b) Single-excitation (left) and two-excitation (right) level structure and transition energies addressed by the cavities (arrows). (c) Stroboscopic evolution of the populations $p_{\eta }(t)=⟨\eta |\widehat{\rho }(t)|\eta ⟩$ of the eigenstates $|\eta ⟩$ of ${\widehat{H}}_S^{\mathrm{eff}}$, for the full driven master equation (solid) and the effective master equation (dashed). The gray shaded area indicates the fraction of discarded states in postselection (black dashed line for the effective model). The parameters are $\mathrm{\Phi }=\pi /2$, $\hslash \omega =20J$, $\mathit{\delta }/\hslash \omega =(1.76,1.7)$, $U=8J$, ${\kappa }_1={\kappa }_2=0.1J$, $\mathcal{E}/J=(1.2,0.5)$, $g_1=g_2=J$. Inset: The size of the arrows and circles reproduces the current pattern and excitation density in the final state. (d) As (c) but for two excitations in the hard-core bosons subspace. The parameters are $\mathrm{\Phi }=\pi /2$, $\hslash \omega =20J$, $\mathit{\delta }/\hslash \omega =(1.69,1.68,1.7)$, $U=8J$, $\mathit{\kappa }/J=(0.05,0.05,0.05)$, $\mathcal{E}/J=(0.6,1.6,0.4)$, $\mathit{g}/J=(1,1,1)$.

Figure 4

(a) Bands in a 100 sites ladder and (b) interband cooling of one excitation using two cavities. Parameters $\mathrm{\Phi }=0.8\pi$, $\hslash \omega =20J$, $\lambda \simeq 1.16\hslash \omega$, $d_1=d_2=J$, ${\delta }_1={\delta }_2=34.4J$, $g_1=g_2=J$, ${\mathcal{E}}_1={\mathcal{E}}_2=1.4J$. The shaded region indicates the fraction of discarded records in postselection.