Dissipatively stabilized superposition of squeezed coherent states via periodically breaking the qubit inversion symmetry

We propose a scheme to dissipatively stabilize superpositions of squeezed coherent states for single- and two-resonator modes in superconducting circuits, where a superconducting qubit is coupled to a single or two parametrically driven transmission-line resonators with different circuit designs. A modulated magnetic flux applied to the qubit can periodically break the inversion symmetry of the qubit and induce both the parametrically enhanced transverse and longitudinal couplings to the resonators, resulting in strong nonlinear two-photon interactions between the qubit and the resonators and enabling the scheme to be implemented in a relatively weak coupling regime. With an additional microwave drive applied to the qubit, the dissipation of the qubit used as a resource can help drive the resonators into the desired superpositions of squeezed coherent states at a steady state with high speed. Numerical simulations show that the target states with high fidelity and highly nonlinear properties can be created for a long time even in the presence of photon leakage out of the resonators. The scheme can be generalized to other quantum platforms and may have potential applications in the field of quantum information processing and the improvement of estimation precision in quantum metrology.

Figure 1

Schematic diagram of a TLR coupled to a superconducting qubit. The central line of the TLR is intersected by a SQUID loop, which can induce a parametric drive on the TLR when an external magnetic field is applied. The red segments in the superconducting lines denote Josephson junctions.

Figure 2

Time evolution of fidelity $F$ vs the dimensionless time $\gamma t$ by numerically solving the master equation (15) with two additional dissipator terms (a) for different $\kappa$ with ${\gamma }_{\varphi }=0$ and (b) for different ${\gamma }_{\varphi }$ with $\kappa =0$.

Figure 3

Wigner function of the created steady state in the absence of resonator dissipation with $r=0.5$.

Figure 4

Schematic diagram of a superconducting qubit coupled to two coupled TLR resonators.

Figure 5

Time evolution of fidelity $F$ vs the dimensionless time $\gamma t$ simulated using the Hamiltonian in Eq. (27) (a) for different $\kappa$ with ${\gamma }_{\varphi =0}$ and (b) for different ${\gamma }_{\varphi }$ with $\kappa =0$.

Figure 6

Wigner function of individual resonators in the absence of resonator dissipation of (a) resonator 1 and (b) resonator 2.

Figure 7

Two-dimensional cuts of the joint Wigner function of the generated state along (a) the $\mathrm{Im}({\beta }_1)-\mathrm{Im}({\beta }_2)$ plane and (b) the $\mathrm{Re}({\beta }_1)-\mathrm{Re}({\beta }_2)$ plane in the absence of resonator dissipation.

Figure 8

Schematic diagram of a superconducting qubit simultaneously coupled to two separated TLR resonators.

Figure 9

Time evolution of fidelity $F$ vs dimensionless time $\gamma t$ simulated using the Hamiltonian in Eq. (37) (a) for different $\kappa$ with ${\gamma }_{\varphi }=0$ and (b) for different ${\gamma }_{\varphi }$ with $\kappa =0$.

Figure 10

Two-dimensional cuts of the joint Wigner function of the generated state along (a) the $\mathrm{Im}({\beta }_1)-\mathrm{Im}({\beta }_2)$ plane and (b) the $\mathrm{Re}({\beta }_1)-\mathrm{Re}({\beta }_2)$ plane in the absence of resonator dissipation, which is simulated using the Hamiltonian in Eq. (37).