Quantifying non-Markovianity due to driving and a finite-size environment in an open quantum system

We study non-Markovian effects present in a driven qubit coupled to a finite environment using a recently proposed model developed in the context of calorimetric measurements of open quantum systems. To quantify the degree of non-Markovianity we use the Breuer-Laine-Piilo (BLP) measure [H.-P. Breuer, Phys. Rev. Lett. 103, 210401 (2009)]. We show that information backflow only occurs in the case of driving, in which case we investigate the dependence of memory effects on the environment size, driving amplitude, and coupling to the environment. We show that the degree of non-Markovianity strongly depends on the ratio between the driving amplitude and the coupling strength. We also show that the degree of non-Markovianity does not decrease monotonically as a function of the environment size.

Figure 1

Schematic of a qubit interacting with the calorimeter. The calorimeter is composed of a finite number of two-level systems, resonant with the qubit. Arrows illustrate the direction of the flow of information. ${\mathcal{I}}_{\text{int}}$ represents the information in the qubit and ${\mathcal{I}}_{\text{ext}}$ represents the information in the calorimeter (see text for details). (a) Markovian case, where there is constant loss of information by the qubit and by the whole system. (b) Non-Markovian case, where there is backflow of information into the qubit but the system, as a whole, still behaves as Markovian.

Figure 2

Rate of change of the information in the qubit, ${\dot{\mathcal{I}}}_{\text{int}}$, the information in the calorimeter, ${\dot{\mathcal{I}}}_{\text{ext}}$, and the total information, ${\dot{\mathcal{I}}}_{\text{int}}+{\dot{\mathcal{I}}}_{\text{ext}}$, as a function of time. Inset: Region of the evolution that contributes the BLP measure in Eq. (6). Parameters used are $\beta \hslash {\omega }_0=2,{\lambda }_0/\hslash {\omega }_0=0.08,g/\hslash {\omega }_0=0.066,N=20,\theta =1.69$, and $\varphi =0$.

Figure 3

BLP measure $[{10}^2\mathcal{N}\left(\mathrm{\Phi }\right)$, color scale] as a function of the driving strength ${\lambda }_0$ and the coupling constant $g$ in a grid of $40\times 40$ points for 5 (top), 50 (middle), and 100 (bottom) two-level systems in the calorimeter. The dashed line indicates the line of constant maximum ${\mathcal{N}}_{\max }\left(\mathit{N}\right)$, given by ${\lambda }_0/g=a_N$.

Figure 4

Log-linear plot of the ratio $a_N$ and log-log plot of ${\mathcal{N}}_{\max }\left(\mathit{N}\right)$ as a function of the calorimeter size $N$.

Figure 5

Log-log plot of ${\omega }_0{t}_R$ along the line of maximum $\mathcal{N}\left(\mathrm{\Phi }\right)$ (${\lambda }_0=a_{N}g$) for the three calorimeter sizes $N=5$, 50, and 100.