Feedback-induced nonlinearity and superconducting on-chip quantum optics

Quantum coherent feedback has been proven to be an efficient way to tune the dynamics of quantum optical systems and, recently, those of solid-state quantum circuits. Here, inspired by the recent progress of quantum feedback experiments, especially those in mesoscopic circuits, we prove that superconducting circuit QED systems, shunted with a coherent feedback loop, can change the dynamics of a superconducting transmission line resonator, i.e., a linear quantum cavity, and lead to strong on-chip nonlinear optical phenomena. We find that bistability can occur under the semiclassical approximation and photon antibunching can be shown in the quantum regime. Our study presents alternate perspectives for engineering nonlinear quantum dynamics on a chip.

Figure 1

Schematic diagram of the quantum input-output system with the input field $b_{\mathrm{in}}$ and the output field $b_{\mathrm{out}}$. Here, $S$, $L$, and $H$ are, respectively, the scattering matrices, the Lindblad operator, and the Hamiltonian of the input-output system.

Figure 2

Schematic diagram of the cascade system with two cascaded-connected components in (a) and the feedback-control systems in (b), which can be seen as the controlled system cascaded-connected with itself.

Figure 3

Schematic diagram of the coherent feedback network using superconducting quantum circuits. The controlled system is a TLR which can be considered as a linear cavity. The controller in the feedback loop is another TLR coupled with a dc-SQUID-based superconducting charge qubit which acts as a nonlinear component.

Figure 4

Magnitude of the steady field $|A_0{|}^2$ in the controlled cavity vs the intensity of the driving field $\left|\varepsilon \right|$, with system parameters ${\Delta }_s/2\pi =100$ MHz, $\Delta /2\pi =4.9$ MHz, $\chi /2\pi =10$ MHz, $\gamma /2\pi =6$ MHz, ${\gamma }_f/2\pi =8$ MHz, and $\kappa /2\pi =3$ MHz. In the parameter regime $p_2>\sqrt{3}{p}_1$, we observe a hysteresis curve, indicating bistability. In the parameter regime $p_2\le \sqrt{3}{p}_1$, bistability disappears.

Figure 5

The numerical and analytical solutions for the second-order correlation function $g^{\left(2\right)}\left(0\right)$ in Eq. (25) vs the tunable parameter $K$. The system parameters for this simulation are $\chi =10$ MHz, ${\Delta }_s=50$ MHz, $\gamma /2\pi =2$ MHz, ${\gamma }_f/2\pi =2.5$ MHz, $\kappa =1$ MHz, and $\varepsilon =0.1\kappa$.

Figure 6

(a) $g^{\left(2\right)}\left(0\right)$ for the controlled TLR and (b) $g^{\left(2\right)}\left(0\right)$ for the controller vs the parameter $K=\Delta /\chi +1$. The red dotted curve and the blue curve correspond to different detuning frequencies ${\Delta }_s=50$ and 10 MHz. The other system parameters are $\chi =10$ MHz $\gamma /2\pi =2$ MHz, ${\gamma }_f/2\pi =2.5$ MHz, $\kappa =1$ MHz, and $\varepsilon =0.1\kappa$.

Figure 7

Second-order correlation function vs $K$ and ${\Delta }_s$, with $\chi =10$ MHz, $\gamma /2\pi =2$ MHz, ${\gamma }_f/2\pi =2.5$ MHz, $\kappa =1$ MHz, and $\varepsilon =0.1\kappa$. Note that $g^{\left(2\right)}\left(0\right)<1$ occurs when ${\Delta }_s-(K-1)=0$, namely, $\omega ={\omega }_s$ and $K=1$ with any ${\Delta }_s$.

Figure 8

Second-order correlation function vs the normalized driving strength $\varepsilon /\kappa$, for different values of the detuning frequency ${\Delta }_s$. Here the system parameters are $\chi =10$ MHz, $\gamma /2\pi =2$ MHz, ${\gamma }_f/2\pi =2.5$ MHz, and $\kappa =1$ MHz. (a) $K=1$. (b) $K=2$.

Figure 9

Second-order correlation function vs the nonlinear strength $\chi$ of the controller, where the parameters of the system are $K=1$, $\gamma /2\pi =2\mathrm{MHz}$, ${\gamma }_f/2\pi =2.5$ MHz, $\kappa =1$ MHz, and $\varepsilon =0.1\kappa$.