Atom-modulated dynamic optical hysteresis in driven-dissipative systems

We introduce a two-level atom into a cavity as an ancilla to control the hysteresis of the system where the cavity is described by a driven-dissipative nonlinear Kerr model. We find that the dynamic optical hysteresis can be modulated by changing the atom-cavity coupling strength, and the nonlinearity induced by atom-cavity coupling weakens the Kerr nonlinearity in the weak coupling regime. This leads to the change in the critical point corresponding to the decrease in the hysteresis area. The physics behind the modulation is closely related to the effect of the atom-cavity coupling on the discontinuity of the first nonzero Liouvillian eigenvalue. A relative deviation of the metastable state and target steady state is defined to characterize the relation between the hysteresis and adiabaticity of the system. Taking only two Liouvillian eigenvectors into account in the composite system, we derive an expression for the dependence of the hysteresis area on the sweeping speed and discuss its feature in the slow sweeping limit.

Figure 1

(a) The cavity occupation numbers $\langle a^{†}a\rangle$ and (b) $\langle {\sigma }_z\rangle$ as a function of the sweeping parameter $\mathrm{\Delta }$ (in units of ${\kappa }_1$). In (a) and (b), solid lines stand for the hysteresis loop, whereas the dashed line denotes the steady state solution. The arrows indicate the sweeping direction. (c) and (d) are plotted for the hysteresis area $A$ and $A{}^{\prime }$ corresponding to the cavity occupation numbers $\langle a^{†}a\rangle$ and $\langle {\sigma }_z\rangle$, respectively, as a function of atom-cavity coupling strength $g$ (in units of ${\kappa }_1$) for different metastable residence times $\tau$. The other system parameters chosen are $F=8,U=0.5,{\kappa }_2=0.5$ in units of ${\kappa }_1$ and $N=121$.

Figure 2

(a) The hysteresis loop for the joint operator $\langle {\sigma }_z\times a^{†}a\rangle$ and (b) the witness $\epsilon$ averaged over the steady (dashed-dot line) and metastable (solid lines) state with $\tau =8/{\kappa }_1$ as a function of $\mathrm{\Delta }$ (in units of ${\kappa }_1$). The gray dashed line corresponds to $\epsilon =0$. The arrows indicate the sweeping direction. The other system parameters are the same as in Fig. 1.

Figure 3

The first nonzero eigenvalue $\ln (-{\gamma }_1/{\kappa }_1)$ of the Liouvillian superoperator as a function of driven-cavity detuning $\mathrm{\Delta }$ (in units of ${\kappa }_1$) and atom-cavity coupling constant $g$ (in units of ${\kappa }_1$), the discontinuity of the first nonzero Liouvillian eigenvalue vanishes in the function of $\mathrm{\Delta }$ when the coupling is strong enough. Other system parameters are the same as in Fig. 1.

Figure 4

$\ln (-{\tilde{\gamma }}_1/{\kappa }_1)$ and $\ln (-{\gamma }_1/{\kappa }_1)$ as a function of driven-cavity detuning $\mathrm{\Delta }$ (in units of ${\kappa }_1$) with $g/{\kappa }_1=1$. The other system parameters are the same as in Fig. 1.

Figure 5

(a) $\zeta$ as a function of driven-cavity detuning $\mathrm{\Delta }$ (in units of ${\kappa }_1$) for different $N$'s at the same metastable residence time $\tau =4/{\kappa }_1$. (b) The hysteresis area $A$ as a function of sweeping speed $v^{-1}$ (in units of ${\kappa }_1^{-2}$) with different atom-cavity coupling constant $g$. The calculation is performed with all the Liouvillian eigenvectors. Black dashed lines are the asymptotic lines, and the hexagrams are the numerical results of Eq. (26). The other parameters are the same as in Fig. 1.

Figure 6

Sarandy-Lidar geometric connection $A_{\mathrm{\Delta }}^{10}$ as a function of driven-cavity detuning $\mathrm{\Delta }$ (in units of ${\kappa }_1$) for different coupling strengths. The solid line corresponds to $A_{\mathrm{\Delta }}^{10}{\kappa }_1=0$. The other system parameters are the same as in Fig. 1.