Collapse and revival of quantum coherence for a harmonic oscillator interacting with a classical fluctuating environment

We address the dynamics of nonclassicality for a quantum system interacting with a noisy fluctuating environment described by a classical stochastic field. As a paradigmatic example, we consider a harmonic oscillator initially prepared in a maximally nonclassical state, e.g., a Fock number state or a Schrödinger-cat-like state, and then coupled to either a resonant or a nonresonant external field. Stochastic modeling allows us to describe the decoherence dynamics without resorting to approximated quantum master equations and to introduce non-Markovian effects in a controlled way. A detailed comparison among different nonclassicality criteria and a thorough analysis of the decoherence time reveal a rich phenomenology whose main features may be summarized as follows: (i) Classical memory effects increase the survival time of quantum coherence and (ii) a detuning between the natural frequency of the system and the central frequency of the classical field induces revivals of quantum coherence.

Figure 1

Dynamics of quantumness according to the nonclassical depth criterion. Shown on the left is the dimensionless decoherence time $t_Q$ for a resonant interaction as a function of the memory parameter $\gamma$, for different values of the coupling $\lambda =1$ (solid brown curve), $\lambda =2$ (dashed black curve), and $\lambda =3$ (dot-dashed blue curve). For $\gamma \to \infty ,t_Q$ approaches the Markovian limit $t_Q^{\left(M\right)}$ independently of $\lambda$. On the right is a contour plot of $\sigma \left(t_Q\right)=1$, in the off-resonance case, as a function of $\gamma$ for a fixed value of the coupling $\lambda =1$ and different values of the detuning $\delta =0.3$ (solid red curve), $\delta =0.4$ (dotted green curve), and $\delta =0.5$ (dot-dashed purple curve). The dashed blue curve is chosen as a reference for the resonant case $\delta =0$. In the regions lying to the left of the curves we have $\sigma \left(t\right)<1$, i.e., nonclassicality. The vertical (dashed black) line denotes points at fixed $\gamma =0.05$ and the black circles indicate the three solutions of $\sigma \left(t_Q\right)=1$ for $\delta =0.3$. Correspondingly, the regions of non-classicality (NCL) and classicality (CL) are highlighted in the inset.

Figure 2

Dynamics of quantumness according to the Wigner negativity criterion. Shown on the top left is the Wigner decoherence time $t_W$ for a resonant interaction as a function of the memory parameter $\gamma$, for different values of coupling $\lambda =1$ (solid brown curve), $\lambda =2$ (dashed black curve), and $\lambda =3$ (dot-dashed blue curve). For $\gamma \to \infty ,t_W$ approaches the Markovian limit $t_W^{\left(M\right)}$ independently of $\lambda$. The top right shows a contour plot of $\sigma \left(t_W\right)=\frac{1}{2}$, in the off-resonance case, as a function of $\gamma$ for a fixed value of the coupling $\lambda =1$ and different values of the detuning $\delta =0.3$ (solid red curve), $\delta =0.4$ (dotted green curve), and $\delta =0.5$ (dot-dashed purple curve). The dashed blue curve is chosen as a reference for the resonant case $\delta =0$. In the regions lying to the left of the curves we have $\sigma \left(t\right)<\frac{1}{2}$, i.e., nonclassicality. The vertical (dashed black) line denotes points at fixed $\gamma =0.05$ and the black circle indicates the solutions of $\sigma \left(t_W\right)=\frac{1}{2}$ for $\delta =0.3$. Correspondingly, the regions of nonclassicality (NCL) and classicality (CL) are highlighted in the inset. The bottom plots show the ratio $t_W/t_Q$ as a function of $\gamma$ with the same values of $\lambda$ as in the top left panel. For $\gamma \gg 1$ the ratio approaches the Markovian value $\frac{1}{2}$, whereas for $\gamma \ll 1$ it approaches $\frac{1}{\sqrt{2}}$, due to the quadratic dependence on time of $\sigma \left(t\right)$.

Figure 3

Shown on the left is the Schrödinger-cat-state Vogel time $t_V$ as a function of $u$, with $\left|\alpha \right|=\sqrt{2}$. On the right is the Vogel time $t_V$ as a function of $u$ for the Fock state $|2\rangle$. In both panels, $\gamma =0.05$ and filled regions correspond to $|{\chi }_1\left[\rho \left(t_V\right)\right](u,0)|>1$. From bottom to top, the blue region represents the resonant interaction $\left(\delta =0\right)$, whereas the red $\left(\delta =0.3\right)$, green $\left(\delta =0.4\right)$, and purple $\left(\delta =0.5\right)$ regions correspond to the off-resonance case. The spots for the green and the purple regions indicate the presence of revivals of nonclassicality.

Figure 4

Shown on the left is the decoherence time $t_K$ for the Klyshko criterion as a function of $\gamma$ for the Schrödinger-cat state with $\alpha =\sqrt{2}$. On the right is the decoherence time $t_K$ for the Klyshko criterion as a function of $\gamma$ for the Fock state $|2\rangle$. In both panels, the dashed blue curve represents the resonant interaction $\left(\delta =0\right)$, whereas solid red $\left(\delta =0.3\right)$, dashed green $\left(\delta =0.4\right)$, and dot-dashed purple $\left(\delta =0.5\right)$ curves refer to the off-resonance case. In the regions lying to the left of the curves we have $B\left(1\right)<0$ (left) or $B\left(0\right)<0$ (right), i.e., nonclassicality.

Figure 5

Input-output fidelity as a function of the interaction time for a Schrödinger-cat state with $\alpha =\sqrt{2}$ (top left panel) and a Fock state with $n=2$ (top right panel) for different values of the memory parameter $\gamma$ and the detuning $\delta$ and for fixed coupling $\lambda =1$. Shown for comparison is the behavior of the variance $\sigma$ (bottom panel) in the same conditions. In all the panels the orange curves are for $\gamma =0.005$, green is for $\gamma =0.05$, and blue refers to $\gamma =1$. The solid curves correspond to zero detuning (resonant case) and the dashed ones are for $\delta =0.3$.

Figure 6

Dynamics of quantumness according to the nonclassical depth criterion for a Schrödinger-cat state evolving in classical environment with a power-law autocorrelation function. In both panels, the dashed blue curve represents the resonant case $\delta =0$, whereas the solid red $\left(\delta =0.3\right)$, dotted green $\left(\delta =0.4\right)$, dot-dashed purple $\left(\delta =0.5\right)$ curves refer to the off-resonance case. Shown on the left is the nonclassical depth time $t_Q$ as a function of $\gamma$ in the case of a Gaussian power-law process, for fixed $\beta =3$ and $\lambda =1$. For $\gamma \gg 1$ the nonclassical depth time $t_Q$ approaches the Markovian limit independently of $\delta$. Collapse and revival of quantumness are highlighted by the circles along the dashed black line at $\gamma =0.023$ and, correspondingly, in the inset. On the right is the nonclassical depth time $t_Q$ as a function of $\beta$ and fixed $\gamma =0.023$.