Survival of quantum effects for observables after decoherence

When a quantum nonlinear system is linearly coupled to an infinite bath of harmonic oscillators, quantum coherence of the system is lost on a decoherence time scale ${\tau }_D$. Nevertheless, quantum effects for observables may still survive environment-induced decoherence and be observed for times much larger than the decoherence time scale. In particular, we show that the Ehrenfest time, which characterizes a departure of quantum dynamics for observables from the corresponding classical dynamics, can be observed for a quasiclassical nonlinear oscillator for times $\tau ⪢{\tau }_D$. We discuss this observation in relation to recent experiments on quantum nonlinear systems in the quasiclassical region of parameters.

Figure 1

Comparison between the numerically obtained decoherence time ${\tau }_D$ and the approximate relation given by Eq. (19), obtained retaining only one term in the master equation. The dashed line corresponds to ${\tau }_D\left(\text{theory}\right)={\tau }_D\left(\text{numerics}\right)$. Data are obtained by varying parameters in the following regions: $20⩽I_0⩽100$, ${10}^{-3}⩽\overline{\mu }⩽4$, ${10}^{-2}⩽\overline{\beta }⩽1$, and ${10}^{-5}⩽\gamma ⩽{10}^{-2}$.

Figure 2

The average position as a function of the dimensionless time $\tau$. In all cases $I_0=50$, $\overline{\beta }=1$. (a) Parameters are $\overline{\mu }=0.1$ and $\gamma ={10}^{-4}$. The dashed curve corresponds to $\exp \left(-\tau ∕{\tau }_D\right)$, where ${\tau }_D=18>{\tau }_E\approx 0.7$. (b) is an enlargement of the first bump of (a). Two more curves have been added: a dashed-dotted line corresponds to the average position for $\gamma ={10}^{-2}$, so that ${\tau }_D=0.18<{\tau }_E$, and a dotted line that corresponds to the envelope $\exp (-{\tau }^2∕2{\tau }_E^2)$. (c) is an enlargement of the third bump of (a). The dotted line corresponds to the envelope $\exp (-{\tau }_R∕{\tau }_D)\exp \left[-{(\tau -{\tau }_R)}^2∕2{\tau }_E^2\right]$, where ${\tau }_R=10\pi$. (d) Parameters are $\overline{\mu }=\gamma ={10}^{-2}$, so that ${\tau }_D⪡{\tau }_E⪡{\tau }_{\gamma }<{\tau }_R$.

Figure 3

The average position $x\left(\tau \right)$ in the “classical” limit: ${\tau }_D⪡{\tau }_{\mathrm{cl}}⪡{\tau }_{\gamma }⪡{\tau }_E⪡{\tau }_R$. Parameters are $\overline{\mu }={10}^{-4}$, $\overline{\beta }=1$, $\gamma =0.01$, and $I_0=50$, so that ${\tau }_D=0.92$, ${\tau }_{\mathrm{cl}}=2\pi$, ${\tau }_{\gamma }=200$, ${\tau }_E=707$, and ${\tau }_R=\pi \times {10}^4$. The correct decay (dotted line), as given by the relaxation time ${\tau }_{\gamma }$, is compared with the “wrong” decay (dashed line) given by the Ehrenfest time ${\tau }_E$.

Figure 4

The Fourier spectrum of the average position versus the rescaled frequency, where ${\omega }_{\mathrm{cl}}=1+2\overline{\mu }{I}_0$ is the classical frequency. At the dashed black line we show the theoretical expression, Eq. (25), for the closed system. Parameters $\overline{\beta },\overline{\mu },I_0$ are the same as in Fig. 2, while (a) $\gamma ={10}^{-5}$, ${\tau }_D=180⪢{\tau }_E=0.7$, (b) $\gamma ={10}^{-4}$, ${\tau }_D=18>{\tau }_E=0.7$ (c) $\gamma ={10}^{-3}$, ${\tau }_D=1.8>{\tau }_E=0.7$, and (d) $\gamma ={10}^{-2}$, ${\tau }_D=0.18<{\tau }_E=0.7$. As one can see, decreasing of the decoherence time has no effect on the width of the Fourier spectrum.