Robust generation of entangled fields against inhomogeneous parameters of coupled superconducting resonators and qubits

We propose an efficient approach for the generation of a stable entangled state of microwave fields in circuit quantum electrodynamics. The system under consideration consists of two linearly coupled superconducting resonators, each of which is coupled to a superconducting flux qubit. By individually driving the qubits to engineer the desired interactions with the resonators, we show that the energy relaxation of qubits can be exploited to steer the two resonators into the two-mode entangled state via a dissipative dynamical process. Since our protocol is based on quantum reservoir engineering, neither specific initialization nor unitary dynamics is required. Moreover, the distinct advantage of our scheme is that the quantum state preparation is robust in relation to inhomogeneous parameters, i.e., there is no need for identical qubit-field couplings, as well as the same resonance frequencies of the resonators. These features make the present scheme pretty feasible for experimental implementation.

Figure 1

Schematic of two directly coupled superconducting transmission line resonators, each coupled to a gap-tunable superconducting flux qubit. The flux qubit is made up of four Josephson junctions.

Figure 2

Logarithmic negativity $E_N$ vs the dimensionless time $\mathrm{\Gamma }t$ by numerically solving the master equation. The parameters are ${\delta }_1/2\pi =10$ GHz, ${\delta }_2/2\pi =9.5$ GHz, ${\omega }_1/2\pi =6$ GHz, ${\omega }_2/2\pi ={\omega }_1/2\pi -\mathrm{\Delta }$, $J/2\pi =0.3$ GHz, $g_1/2\pi =50$ MHz, $g_2/2\pi =60$ MHz, $\mathrm{\Gamma }/2\pi ={\mathrm{\Gamma }}_1/2\pi ={\mathrm{\Gamma }}_2/2\pi =10$ MHz, ${\xi }_1^1={\xi }_1^2=0.14$, and ${\xi }_2^1={\xi }_2^2=0.2$ for (a) different quality factors $Q$ and (b) different $\mathrm{\Delta }$.

Figure 3

Time evolution of purity $\mu$ by numerically solving the master equation for (a) different quality factors $Q$ and (b) different $\mathrm{\Delta }$. The parameters are the same as in Fig. 2.