Tuning non-Markovianity by spin-dynamics control

We study the interplay between forgetful and memory-keeping evolution enforced on a two-level system by a multi-spin environment whose elements are coupled to local bosonic baths. Contrarily to the expectation that any non-Markovian effect would be buried by the forgetful mechanism induced by the spin-bath coupling, one can actually induce a full Markovian–to–non-Markovian transition of the two-level system's dynamics, controllable by parameters such as the mismatch between the energy of the two-level system and of the spin environment. For a symmetric coupling, the amount of non-Markovianity surprisingly grows with the number of decoherence channels.

Figure 1

(a) Scheme of the system: The central spin 0 interacts with $N$ peripheral spins, each affected by its own local environment. (b) Evolution of states ${\left|+\right.〉}_x$ (red trajectory) and ${\left|-\right.〉}_x$ (blue one) for a star with $N=4$ peripheral sites. The Bloch spheres in the left (right) column correspond to the isotropic (anisotropic) spin-spin coupling. The top (bottom) row is for the resonant (off-resonant at $\Delta =1/2$) case with $\gamma /J=1$ and $T=0$. In the isotropic cases, the final state of spin 0 is pure, while for $\lambda =\pm 1$ it is mixed. $\Delta \ne 0$ prevents the intersections of the trajectories, which are the dynamical points at which the trace distance is strictly null.

Figure 2

(a) Trace distance between the optimal states ${\rho }_{0,j}\left(t\right)(j=1,2)$ for $N=6$ peripheral spins with $\lambda =0$ and $\gamma /J=0.5$ (dot-dashed line), $\gamma /J=1$ (dashed line), and $\gamma /J=1.5$ (solid line). As the relaxation time becomes shorter, the revivals of $D({\rho }_{0,1}\left(t\right),{\rho }_{0,2}\left(t\right))$ are suppressed as a result of a reduction of information back-flow from the baths. (b) $\mathcal{N}$ against $\Delta$ for $\gamma /J=\sqrt{N}$. The two lines correspond to ${\theta }_{1,2}=\pi /2,({\varphi }_1,{\varphi }_2)=(0,\pi )$ (solid blue curve) and $({\theta }_1,{\theta }_2)=(0,\pi )$ (dashed red curve), which are the optimal states in different detuning regions: $\mathcal{N}$ is the topmost curve in each region. There is a finite window of detunings (light-shadowed region marked as M) where $\mathcal{N}=0$ (NM marks regions where $\mathcal{N}\ne 0$). Inset: $\mathcal{N}$ against $\Delta$ for $\gamma /J=0.5,0.75,1,1.25$, and 1.5 (from top to bottom curve).

Figure 3

(a) $\mathcal{N}$ against $N$ and $\Delta$ for $\gamma /J=0.1$ and $\overline{n}=3$. Differently from $T=0$, except for a small range of values, the detuning has no effect on the character of the dynamics of spin 0. Strikingly, $\mathcal{N}$ grows with $N$ (almost linearly for $N\gg 1$). (b) Analytic behavior of $\mathcal{N}$ versus $N$ for $\lambda =0$, $\Delta =0$, $\gamma =J$ at $T=0$ ($\overline{n}=0$). Inset: we present the case corresponding to $\lambda =-1$ [other parameters as in (b)].

Figure 4

Ratio $\mathcal{R}\left(t\right)={\mathcal{N}}^{+}\left(t\right)/{\mathcal{N}}^{-}\left(t\right)$ versus $\lambda$ for a star of $N=8$ sites at $T=0$, with $\gamma /J=\sqrt{8}$ (left plots) and $\gamma /J=1$ (right plots) for three different values of the detuning: $\Delta /J=0$ for (a) and (d), $\Delta /J=0.7$ for (b) and (e), while $\Delta /J=3$ for (c) and (f).