Fig. 4
From: Coherent Ising machines—optical neural networks operating at the quantum limit
Quantum states of DOPO fields at the threshold. a The density matrix elements \(\left\langle {X\left| {\hat \rho } \right|X^\prime} \right\rangle\) of the two out-of-phase coupled DOPO pulses in a high-Q ring cavity at the threshold pump rate. The in-phase amplitude probability distribution \(\left\langle {X\left| {\hat \rho } \right|X} \right\rangle\) and the quantum coherence \(\left\langle {X\left| {\hat \rho } \right| - X} \right\rangle\) of the two DOPO pulses are given by a horizontal cut at X−X′ = 0 and a vertical cut at X+X′ = 0, respectively. The round trip loss is 0.1%. b The density matrix elements \(\left\langle {X\left| {\hat \rho } \right|X^\prime} \right\rangle\) for a superposition state, \(1{\mathrm{/}}\sqrt 2 \left( {\left| \alpha \right\rangle + \left| { - \alpha } \right\rangle } \right)\) and a mixed state, \(\hat \rho = 1{\mathrm{/}}2\left( {\left| \alpha \right\rangle \left\langle \alpha \right| + \left| { - \alpha } \right\rangle \left\langle { - \alpha } \right|} \right)\). (c) The corresponding Wigner functions to Fig. 4a40
