Self-consistent projection operator theory in nonlinear quantum optical systems: A case study on degenerate optical parametric oscillators

Nonlinear quantum optical systems are of paramount relevance for modern quantum technologies, as well as for the study of dissipative phase transitions. Their nonlinear nature makes their theoretical study very challenging and hence they have always served as great motivation to develop new techniques for the analysis of open quantum systems. We apply the recently developed self-consistent projection operator theory to the degenerate optical parametric oscillator to exemplify its general applicability to quantum optical systems. We show that this theory provides an efficient method to calculate the full quantum state of each mode with a high degree of accuracy, even at the critical point. It is equally successful in describing both the stationary limit and the dynamics, including regions of the parameter space where the numerical integration of the full problem is significantly less efficient. We further develop a Gaussian approach consistent with our theory, which yields sensibly better results than the previous Gaussian methods developed for this system, most notably standard linearization techniques.

Figure 1

Sketch of the self-consistent projection operator theory for the DOPO, which consists of an optical cavity containing a crystal with second-order optical nonlinearity, pumped by a laser at frequency ${\omega }_p=2{\omega }_s$ (pump mode), and capable of producing a field at the subharmonic frequency ${\omega }_s$ (signal mode) via down-conversion in the crystal. For the sake of illustration, we consider in the figure a doubly resonant semimonolithic configuration in which each face of the crystal acts as a mirror for one of the modes, but is transparent for the other, allowing the creation of independent cavities for the pump and signal modes via two additional partially transmitting mirrors [48]. In our approach, the full problem described by the state $\rho \left(t\right)$ and the Liouvillian $\mathcal{L}$ is mapped onto two coupled equations for the signal and pump modes. In one of the equations, the signal mode considered as the system is described by an effective master equation for its reduced state ${\dot{\rho }}_s\left(t\right)={\mathcal{L}}_s\left({\rho }_p\right){\rho }_s\left(t\right)$ with an effective Liouvillian depending on the state of the pump, which plays here the role of an environment. The other equation considers the reversed scenario with the pump taking the role of the system while the signal is interpreted as the environment, leading to the effective equation ${\dot{\rho }}_p\left(t\right)={\mathcal{L}}_p\left({\rho }_s\right){\rho }_p\left(t\right)$. In this way, the two equations form a closed set.

Figure 2

Accuracy tests of the c-MoP theory for steady-state expectation values as a function of the injection parameter $\sigma$. In all plots, we set ${\gamma }_p={\gamma }_s=1$. The cases $\chi =1$ and $\chi =0.1$ are considered in (a)–(c) and (d)–(f), respectively. The rescaled pump amplitude $\chi \langle a_p\rangle$ is shown in (a) and (d); (b) and (e) show the signal photon number $\langle a_s^{†}{a}_s\rangle$; finally, (c) and (f) show the $g^{\left(2\right)}$ function of the signal, which is equal to 1 for a coherent state (or a balanced mixture of coherent states differing only in phase). The gray thin solid lines show the classical prediction from Eq. (2), showing that the classical threshold where the signal field is switched on lies at $\sigma =1$. The blue solid curves represent the results obtained from the numerical solution of the c-MoP equations (16), (17), (19), and (20). The red stars show the result obtained from the full master equation (1) up to injection parameters $\sigma$ where the numerics are tractable for us. Finally, the black dashed curves represent the mean-field theory; see Eq. (10). Apart from the classical solution, all theories conserve the $Z_2$ symmetry, i.e., $\langle a_s\rangle =0$.

Figure 3

Wigner functions of the c-MoP density matrix for the signal mode (a) without and (b) with the Gaussian state approximation for ${\gamma }_p={\gamma }_s=1,\chi =0.1$, and for different values of $\sigma$. In the absence of injection, $\sigma =0$, the signal state is in vacuum, and one finds the characteristic uncertainty circle showing the same quantum noise in any direction of phase space. Upon approaching the threshold, the state becomes squeezed, with the highest squeezing levels obtained around $\sigma =1$, what can be appreciated by the clear gap present between the end of the uncertainty region and the frame along the $p$ axis, in contrast to the $\sigma =0$ case where the uncertainty region is almost touching the frame. Above threshold, two symmetric peaks appear and the squeezing reaches some asymptotic value as we move away from threshold. Note how above threshold the state can be approximated by a balanced mixture of two symmetry-breaking states. Indeed, let us remark that while for $\sigma <1$ we are plotting the unique solution that appears when applying the Gaussian state approximation onto the c-MoP equations (which we have called below-threshold solution in the text), for $\sigma >1$ we have chosen to plot the Wigner function corresponding to a balanced mixture of the two above-threshold symmetry-breaking solutions with opposite phase which coexist with the symmetry-preserving Gaussian solution.

Figure 4

Accuracy tests of the c-MoP theory with and without the Gaussian state approximation for steady-state observables as a function of the injection $\sigma$. In all of the plots, we set ${\gamma }_p={\gamma }_s=1$ and $\chi =0.01$. As in Fig. 2, we show (a) the rescaled pump field amplitude $\chi \langle a_p\rangle$, (b) the signal photon number $\langle a_s^{†}{a}_s\rangle$, and (c) the $g^{\left(2\right)}$ function of the signal. The red stars show the results obtained from the c-MoP equations (16), (17), (19), and (20), up to injection parameters $\sigma$ where the numerics are tractable. For comparison, the quasiexact method of Drummond and collaborators [36, 37] is shown as a dark yellow dot-dashed line. The blue solid and the green dashed curves represent the below- and above-threshold solutions, respectively, obtained from a Gaussian state approximation on the c-MoP equations. The black thin dotted curve displays the results of mean-field theory [see Eq. (10)], which in this context can be understood as the below-threshold solution of a Gaussian state approximation on the full master equation (1). Finally, the gray thin solid lines represent the prediction of the standard linearization theory in (a) and (b), and the coherent-state prediction $g^{\left(2\right)}=1$ of the classical equations (2) in (c).

Figure 5

Accuracy tests of the c-MoP theory for transient-time evolution. The initial state is chosen to be the vacuum. We set ${\gamma }_p={\gamma }_s=1$, investigate the classical threshold point $\sigma =1$, and choose (a) $\chi =0.1$ and (b) $\chi =0.05$. The plots display the signal photon number as a function of time in units of the dissipation rates, while the insets show the $g^{\left(2\right)}$ function of the signal mode. The red stars in (a) show the result obtained from the numerical simulation of the full master equation (1), while the red line in (b) indicates its steady-state values only. The blue solid curves represent the results obtained from the numerical integration of the c-MoP equations (16), (17), (19), and (20). Finally, the green dashed and black dotted lines represent the time evolution obtained from a Gaussian state approximation on the c-MoP equations and the full master equation (mean field), respectively.