\[ \sigma ( \hat { \rho } _ { c l } ) = \left( \begin{array} { c c c c } { { 4 \alpha _ { c l } ^ { 2 } + 1 } } & { { 0 } } & { { - 4 \alpha _ { c l } ^ { 2 } } } & { { 0 } } \\ { { 0 } } & { { 1 } } & { { 0 } } & { { 0 } } \\ { { - 4 \alpha _ { c l } ^ { 2 } } } & { { 0 } } & { { 4 \alpha _ { c l } ^ { 2 } + 1 } } & { { 0 } } \\ { { 0 } } & { { 0 } } & { { 0 } } & { { 1 } } \end{array} \right) , \eqno ( 4 9 ) \]

where \( \alpha _ { c l } \) is the real and positive amplitude of the coherent states in the DOPOs. We have found that the Gaussian dis-cord calculated with Eq. (49) verges on \( D ^ { \leftarrow } \sim 0 . 0 2 3 5 6 \) for \( \alpha _ { c l } \gtrsim 5 0 \), which is in a good agreement with the values in the simulation. Eq. (48) clearly represents a mixture of Gaussian states, thus the result indicates a genuine quantum correlation between coherent states with a mutual injection path without excess noise.

# E. Distribution functions for quadrature amplitudes

Here, we focus on a low-loss case, where the distribution functions for the squeezed amplitude p around the oscillation threshold are o from Gaussian curves. Fig. 7 and 8 displays instantaneous distributions for x and p at some time points for \( \gamma _ { s } \, = \, 0 . 1 \), \( \gamma _ { c } \, = \, 0 . 2 \). In Fig. 7, the distribution for x gets broadened as the pumping rate increases. The dashed lines are Gaussian fitting curves for each time point. We see that it has some deviation from the fitting curve at \( \tau \) = 33 and 35. This indicates that the system is at the onset of the macroscopic bifurcation in x.

As shown in Fig. 8, both \( { \cal P } ( p _ { 1 } ) \) and \( \overline { { P ( p _ { 2 } ) } } \) at pumping rates around the oscillation threshold come to have small fringes at the sides of their central peaks. The fringes survive until the clear bifurcation in \( P ( x _ { 1 } ) \) and \( P ( x _ { 2 } ) \). On the other hand, they vanish when \( \gamma _ { s } \) and \( \gamma _ { c } \) are comparable to or larger than \( \zeta \) = 1. Therefore, \( P ( p _ { 1 } ) \) and \( P ( p _ { 2 } ) \) suggests the existence of the macroscopic superposition of the zero-phase state and \( \pi \)-phase state in a suciently closed two-DOPO system. For-mation of the superposition in such a slow pumping schedule with only a small \( \gamma _ { c } \) means that the quantum noise stored in the injection path is essential in the formation of superposition components here. The injection path contains a squeezed field for the two oscillators, which protects a macroscopic superpo-sition state from decoherence [47, 48]. It is worth noting that the theoretical model considered here is dierent from that in the previous studies. The side peaks in \( P ( p _ { 1 } ) \) and \( P ( p _ { 2 } ) \) are as high as those in an even cat state \| − αi + \|αi with \( \alpha \sim \) 0.9 although the state has a larger photon number than\|α\|2 = 0.81. This is because such a DOPO state does not cor-respond to a pure cat state. Note that the fringe signal will be a bit weaker than the flying optical cat states made with judicious techniques [64, 65]. Also, a larger g and a faster pumping schedule will give a clearer fringe due to the tran-sient eect, as the single DOPO case [62].

Here, we add the extra squeezing of the intracavity DOPO fields which supports the eect by the mutual injections. Fig. 9 displays the variances of \( p _ { 1 } \) and \( p _ { 2 } \) versus time (i.e. the pumping rate). When the system is below the threshold, they decrease with the rise in the pumping rate. Following the os-cillation of the DOPOs, they get back to the value for a co- herent state and the vacuum state (0.25). The minimum value∼ 0.043 is smaller than that for a single intracavity DOPO field [5] (0.125, meaning -3 dB squeezing). It suggests that the mutual injections enhance the squeezing in the DOPOs.

FIG. 7. (Color online) Distribution functions at dierent time points for (a) \( x _ { 1 } \) and (b) \( x _ { 2 } \). The dashed lines are Gaussian fitting curves with \( \sigma \) = (6.6, 9.1, 13.5, 20.0) for τ = (29, 31, 33, 35). 200,000 trajectories are used. \( \gamma _ { s } = 0 . 1 \), \( \gamma _ { c } = 0 . 2 \) and \( _ { g } \) = 0.01.

Compared to expectation values of observables, the con-vergence of the distribution functions (Fig. 8) to the num-ber of sampling is slower, because the sampled points have to cover the whole space where the distribution can have a non-negligible value. Thus, we have taken 200,000 runs to draw the curves here. Simultaneous formation of the side peaks in both \( P ( p _ { 1 } ) \) and \( P ( p _ { 2 } ) \) is a good indicator that the accuracy is not bad, because the two DOPOs obey the SDEs of the same form. However, numerical errors still lead to obvious negative values in some curves. Also, one of the p distribution func-tions is fluctuated a lot at some time points, leading to a larger fringe visibility and negative values.

# V. DISCUSSION

In this section, we discuss other theoretical schemes to sim-ulate the system considered here, the validity of the simulation in this study, and the possible contribution of the quantum ef-fects in the system to the performance of the coherent Ising machines.

FIG. 8. (Color online) Distribution functions at dierent time points for (a) \( p _ { 1 } \) and (b)\( p _ { 2 } \). Zoomed curves around the side peaks are added for both. 200,000 trajectories are used. \( \gamma _ { s } \, = \, 0 . 1 \), \( \gamma _ { c } \, = \, 0 . 2 \) and \( g \) = 0.01.

FIG. 9. (Color online) Variances of \( p _ { 1 } \) (blue curve) and \( p _ { 2 } \) (red curve). The two curves are almost identical due to the same form of the SDEs for each DOPO. 200,000 trajectories are used. \( \gamma _ { s } = 0 . 1 \),\( \gamma _ { c } = 0 . 2 \) and \( _ { g } \) = 0.01.

# A. Other theoretical schemes

First, we refer to the diculty in the simulation in this study with other theoretical schemes. Regarding a numerical anal-ysis for an open quantum system, direct integration on the master equation with the Fock state basis is the most stan-dard method as investigated in the previous relevant studies[47, 48]. It treats a series of ordinary dierential equations for the components of the density matrix for the system. Single-shot numerical integration for them gives all the information of the solution, thus we do not have to repeat stochastic sim-ulations or take ensemble averages over a number of samples. Also, it is relatively easy to get a good accuracy in numerical integration of an ordinary dierential equation. However, the basis has an infinite number of eigenstates hence we have to truncate some of them. Here, the more photons possible in the system, the more eigenstates needed. In addition, the number of modes crucially aects the complexity of the simulation. When we consider two DOPOs and the injection path with m eigenstates for each, the number of components of the den-sity matrix is \( m ^ { 6 } \). This amounts to unrealistic numbers such as 10 \( \mathsf { n n t } \) and 1000 \( 1 0 ^ { 6 } \) thus the simulation with the parameters here will be too costly.

Solving the Fokker-Planck equation will be useful if we can find a potential solution. However, it supposes a system at