Robust Negativity in the Quantum-to-Classical Transition of Kerr Dynamics

TL;DR

This work quantifies how a single-mode Kerr oscillator subject to loss undergoes a quantum-to-classical transition, revealing three dynamical regimes: a short-time Gaussian regime where unitary Kerr dynamics align with classical Gaussian flow, a non-Gaussian mean-field regime where robust macroscopic Wigner negativity emerges (characterized by Airy-type Wigner functions), and a long-time kitten regime where distinguishable cat-like superpositions form but are highly fragile to losses. By combining analytic mean-field reductions, Wigner-function analyses, and numerical simulations, the authors show that loss can suppress intricate kitten-state interference yet leave a window of robust nonclassicality in the intermediate regime, with negativity persisting in the macroscopic limit when the loss rate does not scale too quickly with system size. They also derive scaling laws for negativity under loss and provide a simplified model that captures the essential competition between nonlinearity and diffusion. The results broaden the understanding of quantum-to-classical transitions and point toward loss-tolerant non-Gaussian resources for continuous-variable quantum information processing.

Abstract

We quantify the quantum-to-classical transition of the single-mode Kerr nonlinear dynamics in the presence of loss. We establish three time scales that govern the dynamics, each with distinct characteristics. For times short compared to the Ehrenfest time, the evolution is classical, characterized by Gaussian dynamics. For sufficiently long times, as we increase the initial photon number, unitary Kerr evolution would generate macroscopic superpositions of coherent states (so-called kitten states), but this is severely restricted in the presence of small photon loss so that expectation values of observables coincide with their classical values. The intermediate time scale, however, shows resilient quantum behavior in the macroscopic limit. We show that in the mean-field non-Gaussian regime, the Kerr Hamiltonian (with small photon loss) generates a significant amount of Wigner-negativity, and classical flow is recovered only if the loss rate grows with system size. Our results broaden the usual understanding of quantum-to-classical transitions and demonstrate the potential for creating robust nonclassical resources for continuous-variable quantum information processing in the presence of loss.

Figures (15)

• Figure 1: Evolution of the Wigner function according to the Kerr Hamiltonian with an initial coherent state ($\alpha_{0}=4$). From left to right, as time evolves, the coherent state squeezes and rotates in the Gaussian regime (the first two frames). Around time $\kappa t \approx 1/\alpha_{0}^{3/2}$, the non-Gaussian evolution, characterized by a cubic Hamiltonian, begins (third frame). This cubic Hamiltonian generates an Airy Wigner function (Sec. \ref{['sec:MFC']}). At time $\kappa t \approx 1/\alpha_{0}$, the first distinguishable Kitten states form. These states are characterized by extensive sub-Planck phase-space structure. We will quantify the quantum-to-classical transition in these three separate regimes.
• Figure 2: Wigner functions of three cat states, $|\psi\rangle_{\rm cat} \propto |\alpha_c\rangle + \ket{-\alpha_c}$, with (a) $\alpha_{c} = 1$, (b) $\alpha_{c} = 1.5$, and (c) $\alpha_{c} = 2$. We identify the two coherent states as distinguishable in the second frame, based on the criteria discussed in the text. The separation between the coherent state Gaussian has to be at least $6\Delta \alpha$ for kitten states to be sufficiently separable.
• Figure 3: Wigner functions of $\ket{\psi(t_{M,N})}$ (Eq. \ref{['eq:kittenstatedefinition']}) at three different times with $\alpha_{0}=6$. From top to bottom: $\ket{\psi(t_{1,2})}$, $\ket{\psi(t_{1,9})}$ and $\ket{\psi(t_{1,12})}$. For increasing $N$, the kitten states become less distinguishable, consistent with the distinguishability criteria of Eq. \ref{['eq:nmaxdef']}.
• Figure 4: (a) Normalized deviation in moments and (b) quadratures between the open quantum and classical systems at the distinguishable kitten-state time scale. (a) $|\langle\hat{a}^5\rangle-\langle\alpha^5\rangle|/{\alpha_0^5}$ as a function of time with $\gamma/\kappa = 10^{-4}$ for $\alpha_{0} \in \{10,50,100 \}$. (b) $|\langle\hat{X}^5\rangle-\langle X^5\rangle|$ as a function of time with the same parameters, normalized by the initial non-central Gaussian moment for $X^5$. In both plots, as $\alpha_{0}$ increases, the deviation is exponentially suppressed, revealing that a quantum-to-classical transition is occurring as we scale the system.
• Figure 5: The surviving fraction of the first distinguishable kitten state ($n_{s}$ in Eq. \ref{['eq:fraction_alive']}) as a function of $\gamma/\kappa$ for four different values of $\alpha_0^{2}$. As $\gamma/\kappa$ increases, the fraction of the kitten state that survives ($n_{s}$) decreases exponentially in $\alpha_{0}$, signifying the fragility of the kitten states to loss.