Figure 1 : From short-term quantum squeezing to long-term phase coherence and number entanglement.

From: Classical synchronization indicates persistent entanglement in isolated quantum systems

Figure 1

Quantum dynamics of two coupled oscillators or modes; the initial state being a two-mode coherent state |z, Δφ〉=|0, π/2〉. (a) For sufficiently small coupling strengths (here K=0.1ω) the two modes simply dephase. The Husimi density Q(z, Δφ; t)≡|〈z, Δφ|Ψ(t)〉|2 (ref. 43) spreads out along the phase direction. (b) Squeezing and number entanglement emerge for a sufficiently large coupling strength (here K=0.8ω). At intermediate times, the Husimi density is Q(z, Δφ; t) is compressed in the phase direction. For long times, the quantum state is trapped in the right half of phase space implying preservation of phase coherence. The grey lines in (a,b) show trajectories of the classical mean-field system (2) bearing the Kuramoto model. (ce) Evolution of (c), the squeezing parameter ξ2 (refs 13, 19), (d) the phase coherence α12 and (e) the number entanglement W12 for K=0.8ω (blue line) and K=0.1ω (red line). Short-term quantum squeezing, long-term phase coherence, and number entanglement are observed for a strong coupling K=0.8ω. Parameters for all panels are N=40, U=0.4/N, T=2π/ω and . The initial state is a two-mode coherent state |z, Δφ〉=|0, π/2〉.