Finite-time quantum-to-classical transition for a Schrödinger-cat state

The transition from quantum to classical, in the case of a quantum harmonic oscillator, is typically identified with the transition from a quantum superposition of macroscopically distinguishable states, such as the Schrödinger-cat state, into the corresponding statistical mixture. This transition is commonly characterized by the asymptotic loss of the interference term in the Wigner representation of the cat state. In this paper we show that the quantum-to-classical transition has different dynamical features depending on the measure for nonclassicality used. Measures based on an operatorial definition have well-defined physical meaning and allow a deeper understanding of the quantum-to-classical transition. Our analysis shows that, for most nonclassicality measures, the Schrödinger-cat state becomes classical after a finite time. Moreover, our results challenge the prevailing idea that more macroscopic states are more susceptible to decoherence in the sense that the transition from quantum to classical occurs faster. Since nonclassicality is a prerequisite for entanglement generation our results also bridge the gap between decoherence, which is lost only asymptotically, and entanglement, which may show a “sudden death.” In fact, whereas the loss of coherences still remains asymptotic, we emphasize that the transition from quantum to classical can indeed occur at a finite time.

Figure 1

Vogel nonclassicality condition as a function of $\tau =\gamma t$ and $u$ for $v=0$, $n=100$, and $\alpha =2$. The solid line corresponds to ${\Phi }_t(u,0)=1$. The state is nonclassical in the area under the curve. The time ${\tau }_V$ is the time at which the state becomes classical.

Figure 2

Vogel nonclassicality condition ${\Phi }_t(u,0)=1$ as a function of $\tau =\gamma t$ and $u$ for $v=0$, $n=100$ and initial separations $\alpha$ ranging from 1 to 10 (lines from left to right, respectively). The dashed line marks the saturation time ${\tau }_V(\alpha \to \infty )$ after which the state is classical.

Figure 3

The Klyshko quantity $B\left(1\right)$ as a function of the rescaled time $\tau =\gamma t$ for different values of the thermal noise and different separation amplitude (left, $\alpha =2$; right, $\alpha =3$). Nonclassical states turn into classical ones at a threshold time ${\tau }_K$ depending on both the amplitude and the thermal noise. In both plots solid lines are for $n=1$, dashed for $n=10$, and dotted for $n=100$.

Figure 4

The nonclassicality condition $B\left(1\right)<0$ is satisfied in the gray areas of the plot showing the transition time from quantum to classical as a function of the initial wave-packet separation. Here $\tau =\gamma t$, and (from larger to smaller areas) $n=1,10,100$. The border of each gray area individuates the function ${\tau }_K\left(\alpha \right)$.