Decoherence of Schrödinger cat states in light of wave/particle duality

TL;DR

This work challenges the conventional Lindblad-only view of decoherence for macroscopic Schrödinger cat states by deploying wave-particle correlator unravelings that couple direct photon detection with conditional homodyne measurements. Using quantum-trajectories and Fokker–Planck formalisms, it reveals how conditioning on different quadratures produces distinct diffusion dynamics: rapid population imbalance for position-like measurements and late-emerging interference for momentum-like measurements, all while the ensemble average adheres to the standard ME. The study identifies two key timescales, including a phase-localization time $t_m$ set by $\kappa t_m = \tfrac{1}{2}\ln(2A^2)$, and shows photon emissions act as diffusion markers of amplitude and phase inside the cavity. These results illuminate the contextual, wave–particle nature of decoherence in open quantum systems and offer a practical path to conditioned-state tomography via Wigner-function marginals.

Abstract

We challenge the standard picture of decohering Schrödinger cat states as an ensemble average obeying a Lindblad master equation, brought about locally from an irreversible interaction with an environment. We generate self-consistent collections of pure system states correlated with specific environmental records, corresponding to the function of the wave-particle correlator first introduced in Carmichael et al. [Phys. Rev. Lett. 85, 1855 (2000)]. In the spirit of Carmichael et al. [Coherent States: Past, Present and Future, pp. 75-91, World Scientific (1994)], we find that the complementary unravelings evince a pronounced disparity when the ``position'' and ``momentum'' of the damped cavity mode - an explicitly open quantum system - are measured. Intensity-field correlations may largely deviate from a monotonic decay, while Wigner functions of the cavity state display contrasting manifestations of quantum interference when conditioned on photon counts sampling a continuous photocurrent. In turn, the conditional photodetection events mark the contextual diffusion of both the net charge generated at the homodyne detector, and the electromagnetic field amplitude in the resonator.

Figures (3)

• Figure 1: Schematic illustration of the wave-particle correlator (center) realizing conditional homodyne detection, with a macroscopic coherent superposition as an initial source state. The device operates with a fraction $r$ of the input light flux going to an avalanche photodiode (APD) in the "trigger" channel, and the remaining $1-r$ directed to a balanced mode-matched homodyne detector (BHD). The BHD samples the quadrature phase amplitude that is in phase with the local oscillator (LO) field; $\theta$ is the LO phase. The cavity mode $a$ is prepared in the superposition of coherent states \ref{['eq:incat']} at time $t=0$, from which it decays to produce the scattered field (signal). Inset (i) depicts a contour plot of the Wigner function $W(x+iy;t=0)$ of the initial state \ref{['eq:incat']}, with $A=4$ and $\phi_0=0$. Insets (ii)--(iv) depict histograms of the cumulative charge $Q_{\theta=\pi/2}$ deposited in the BHD after the light has left the cavity, when the correlator operates with $r=0$ and $\theta=\pi/2$, for an initial Schrödinger cat state with $A=4$ and: $\phi_0=0$(ii), $\pi/2$(iii) and $\pi$(iv). The vertical lines indicate the location of $Q_{\pi/2}=0$.
• Figure 2: Conditioned Monte Carlo averages and state representations in single realizations. (a)1: Conditioned intracavity photon number $\langle a^{\dagger}a(t) \rangle_{\rm REC}$ against three cavity lifetimes, for $\theta=0$ (dashed line) and $\pi/2$ (solid line). The strokes underneath ($\theta=0$) and above ($\theta=\pi/2$) the main plots indicate photon emissions triggering the homodyne current generation at the BHD. The two insets depict relative frequency [$f(N_s)$] histograms of APD photon "click" resets, collected for $\theta=0$ (left) and $\theta=\pi/2$ (right), over 4,000 realizations. The curves in red depict the Poisson probability density $p(N_S)=\lambda^{N_s}e^{-\lambda}/N_s!$ for $\lambda=rA^2=8$. The small pink-shaded rectangular frame underneath shows a close-up of a different sample trajectory $\langle a^{\dagger}a(t) \rangle_{\rm REC}$ generated with $\theta=\pi/2$, in the course of half a cavity lifetime. The four red strokes underneath denote APD "clicks". (a)2 -- 4: Individual realizations of the intensity-field correlation function over its zero-delay value, $\tilde{h}_{\theta}(\tau)\equiv h_{\theta}(0; \tau)/h_{\theta}(0; 0)$, for $\theta=0$ in (2), $\pi/2$ in (3) and $(\pi/2-\theta)=0.0078$ in (4). Frame (a5) depicts the conditioned average of the quadrature amplitude $A_{\theta}\equiv \langle A_{\theta}(t) \rangle_{\rm REC}$ corresponding to the trajectory of (a4). The dotted strokes in (a5) mark photon triggers, while the long vertical line marks the time $\kappa t_m=(1/2)\ln(2A^2)\approx 1.733$. (b) Contour plot of the conditioned Wigner function $W_{\rm REC}(x+iy;t_1)$, at the time $\kappa t_1=0.004$ of the first photon trigger, along the trajectory of (a) generated for $\theta=0$; (c) Similar to (b), with $W_{\rm REC}(x+iy;t_2)$ at the time $\kappa t_2=0.096$ of the third photon trigger. The lower inset depicts a numerical approximation of the conditioned marginal distribution $P_{\rm REC}(y;t_2)=\int_{-\infty}^{\infty} W_{\rm REC}(x+iy;t_2)\,dx$. (d) Contour plot of the conditioned Wigner function $W_{\rm REC}(x+iy;t_1^{\prime})$, at the time $\kappa t_1^{\prime}=0.018$ of the first photon trigger, along the trajectory of (a) generated for $\theta=\pi/2$; (e) Similar to (d), with $W_{\rm REC}(x+iy;t_2^{\prime})$, at the time $\kappa t_2^{\prime}=0.098$ of the second photon trigger; (f) Similar to (d), with $W_{\rm REC}(x+iy;t_3^{\prime})$, at the time $\kappa t_3^{\prime}=0.392$ of the eighth photon trigger along the trajectory generated for $\theta=\pi/2$. The lower insets in (e, f) depict numerical approximations of the conditioned marginal distributions $P_{\rm REC}(y;t_k^{\prime})=\int_{-\infty}^{\infty} W_{\rm REC}(x+iy;t_k^{\prime})\,dx$ for $k=2, 3$, respectively. In all realizations, the initial state \ref{['eq:incat']} has $A=4$ and $\phi_0=0$, while the correlator operates with $r=0.5$. The time step size is $\kappa \Delta t=0.002$, and the Fock-state basis is truncated at the $30$-photon level; see App. \ref{['App:B2']}.
• Figure 3: Normalized distributions (histograms) of the net charge $Q_{\theta}$ deposited in the BHD at the time $t_1$ of the first photon trigger "click", for three different settings of the LO phase, $\theta=0$, $\theta=\pi/4$ and $\pi/2$, indicated accordingly. The two insets either side of the histograms depict distributions of waiting times for the first (right) and second (left) photon emissions. $\Delta n_1$ ($\Delta n_2$) is the number of times photon 1 (photon 2) is emitted in the time interval $[\tau,\tau+\Delta\tau)$CarmichaelKim2000. The continuous lines plot an ensemble average of 150,000 trajectories with bin size $\kappa \Delta \tau=0.0025$, while the dashed lines correspond to Eq. \ref{['eq:wapprox']} with $A$ (right) and $A\to A e^{-\kappa \tau_{\rm av}}$ (left). The initial state has: $A=20$, $\phi_0=\pi$, while the correlator operates with $r=0.05$.