Quantifying non-Markovianity of continuous-variable Gaussian dynamical maps

We introduce a non-Markovianity measure for continuous-variable open quantum systems based on the idea put forward in H.-P. Breuer et al. [Phys. Rev. Lett. 103, 210401 (2009);], that is, by quantifying the flow of information from the environment back to the open system. Instead of the trace distance we use here the fidelity to assess distinguishability of quantum states. We employ our measure to evaluate non-Markovianity of two paradigmatic Gaussian channels: the purely damping channel and the quantum Brownian motion channel with Ohmic environment. We consider different classes of Gaussian states and look for pairs of states maximizing the backflow of information. For coherent states we find simple analytical solutions, whereas for squeezed states we provide both exact numerical and approximate analytical solutions in the weak coupling limit.

Figure 1

Non-Markovianity of the purely damping channel as a function of the coupling $\alpha$. The dotted (blue) line is the result obtained by maximizing over coherent states (the exact solution and the first-order approximation coincides). The solid lines are the exact non-Markovianities for squeezed states, whereas the dashed lines are the corresponding first-order approximations. The upper (black) curves correspond to $\phi =0.2$, while the lower (red) ones to $\phi =0.1$.

Figure 2

Diffusion coefficients as a function of time for an Ohmic environment with ${\omega }_c=1$ and (a) ${\omega }_0=4$ and (b) ${\omega }_0=6$. We plot the zero-temperature diffusion coefficient (solid blue), low temperature ($k_{B}T/\hslash {\omega }_c=1/5$, red dotted), intermediate temperature ($k_{B}T/\hslash {\omega }_c=1$, black dashed), and high temperature ($k_{B}T/\hslash {\omega }_c=4$, green dotted-dashed line). Time is measured in units of ${\omega }_c^{-1}$.

Figure 3

Non-Markovianity of QBM channel with Ohmic reservoir on coherent states as a function of the coupling $\alpha$ with ${\omega }_c=0.2$ and ${\omega }_0=1$, for different temperatures. Dashed (solid) lines represent the first-order approximate (numerical) solutions. Black curves correspond to $k_{B}T/\hslash {\omega }_0=0.2$ and blue curves to $k_{B}T/\hslash {\omega }_0=0.5$. The inset shows the temperature dependence for fixed coupling $\alpha =0.1$; ${\omega }_0=1$ (red solid) and ${\omega }_0=1.1$ (red dashed).

Figure 4

Non-Markovianity of QBM channel with Ohmic reservoir as a function of $\alpha$. We have set $k_{B}T/\hslash {\omega }_0=0.2$, ${\omega }_c=0.2$, ${\omega }_0=1$. Black and gray curves show the behavior of squeezed states. The dashed (solid) lines represent the first-order (numerical) solutions to the maximization problem performed with the constraint $r_1=r_2$ and for fixed phase $\phi =0.1$ (gray) or $\phi =0.05$ (black). The dotted (dotted dashed) blue line gives the first-order (numerical) measure for coherent states, with the maximization performed over all the characterizing parameters.

Figure 5

Non-Markovianity of QBM channel with Ohmic reservoir on coherent states. We have set ${\omega }_c=0.2$, ${\omega }_0=1$, and maximized at fixed angle $\phi =0.05$ over the squeezing parameters. Different lines correspond to different temperatures: $k_{B}T/\hslash {\omega }_0=0.3$ (black solid), $k_{B}T/\hslash {\omega }_0=0.9$ (blue dashed), $k_{B}T/\hslash {\omega }_0=4$ (red dotted), and $k_{B}T/\hslash {\omega }_0=8$ (green dotted-dashed).