Detecting non-Markovianity from continuous monitoring

We study how non-Markovianity of an open two-level system can be detected when continuously monitoring a part of its bosonic environment. Considering a physical scenario of an atom in a lossy cavity, we demonstrate that the properties of the time-dependent flux of the photons from the cavity allows the detection of memory effects in the atomic dynamics. This framework overlaps with effective descriptions for the memory part of the environment using pseudomode methods. Our central results show how the Markovian measurement record on the environment of an enlarged open system allows us to draw conclusions on the non-Markovianity of the original system of interest.

Figure 1

Top left: Atom excited state population for different values of $V$ and for $\delta =0\Gamma$. Top right: Atom excited state population for different values of $V$ and for $\delta =\Gamma$. Bottom left: Photon flux for different values of $V$ and for $\delta =0\Gamma$. Bottom right: Photon flux for different values of $V$ and for $\delta =\Gamma$. The legend here gives the values of $V$ for all of the panels.

Figure 2

Top: Regions in $(t,\delta )$ plane where $C\left(t\right)={\partial }_t{\left|c\left(t\right)\right|}^2>0$ (dark red, black) and $B\left(t\right)={\partial }_t\Gamma {\left|b\left(t\right)\right|}^2>0$ (orange, gray) for $V=\Gamma$. Bottom: Regions in $(t,V)$ plane for $\delta =\Gamma$ where $C\left(t\right)>0$ or $B\left(t\right)>0$. Every revival of atomic population $\mathrm{[}C\left(t\right)>0]$ is followed by increase in the photon flux $\mathrm{[}B\left(t\right)>0]$. There are also areas where $B\left(t\right)>0$ without any previous atomic revivals. For all parameters photon flux increases initially.

Figure 3

Oscillation frequencies $\Omega$ as a function of $V$ and $\delta$ near the transition from Markovian to non-Markovian dynamics. Below boundary black solid boundary curve dynamics is Markovian. Above Markovian boundary the dynamics of the open quantum system is non-Markovian. Maximal Markovian frequency ${\Omega }_M\approx 1.8\Gamma$ is marked with a white star. With a dotted black line we plot the contour where $\Omega (V,\delta )={\Omega }_M$. Non-Markovianity cannot be detected in the region between the solid and dotted curves. In this figure $V\in [0.05\Gamma ,1.2\Gamma ]$.

Figure 4

Spectrum of the photon flux. ${\Omega }_M\approx 1.8\Gamma$ is the largest Markovian frequency. In this figure we show evidence that for $(\delta ,V)$ pair $(2\Gamma ,2\Gamma )$ (dotted red line) non-Markovianity is detected because of a pronounced peak in the flux occurring at $\omega \approx 4.47\Gamma$. Pair $(0.0\Gamma ,0.9\Gamma )$ (dotted gray line) and $(\Gamma ,0.7\Gamma )$ are just below the nondetectability border and they show some structure near $\omega \approx {\Omega }_M$. $(1.7\Gamma ,0.3\Gamma )$ (solid blue line) is also near the detectability border and in the Markovian region but there is no significant contribution to the spectrum because the amplitude of $R\left(t\right)$ is very small, but there are oscillations.