Memory-keeping effects and forgetfulness in the dynamics of a qubit coupled to a spin chain

Using recently proposed measures for non-Markovianity [H.-P. Breuer, E. M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009)], we study the dynamics of a qubit coupled to a spin environment via an energy-exchange mechanism. We show the existence of a point, in the parameter space of the system, where the qubit dynamics is effectively Markovian and that such a point separates two regions with completely different dynamical behaviors. Indeed, our study demonstrates that the qubit evolution can in principle be tuned from a perfectly forgetful one to a deep non-Markovian regime where the qubit is strongly affected by the dynamical backaction of the environmental spins. By means of a theoretical quantum process tomography analysis, we provide a complete and intuitive characterization of the qubit channel.

Figure 1

$\mathcal{N}(\Phi )$ versus $h/J$ with $h_0/J=0$, $J_0/J=1$ and $N=100$. To avoid spurious recursion effects, the dynamics is evaluated up to a temporal cutoff of $~2N/3$. We checked that the precise cutoff is not relevant and that the plot remains unchanged if $N$ is varied, provided $N\gg 1$. All quantities are dimensionless.

Figure 2

$\mathcal{N}(\Phi )$ versus $h/J$ for $h_0/J=0$ and three different values of $J_0/J$ for a chain of $N=200$ spins. For $J_0/J>1$, the Markovianity point disappears. All quantities are dimensionless.

Figure 3

We call $E_{+}$ ($E_{-}$) a discrete level lying above (below) the continuous energy band in the (single-particle) spectrum of the total Hamiltonian ${\stackrel{\wedge }{\mathcal{H}}}_0+{\stackrel{\wedge }{\mathcal{H}}}_{\Gamma }$. The parabolas $h/J=\pm [1-(J_0/J){}^2/2]$ divide the $(h/J,J_0/J)$ parameter plane in three regions with one, two, and no localized state (light-colored, dark-colored, and uncolored region in the plane, respectively). All quantities are dimensionless.

Figure 4

Average number of fermion excitations in the initial state for a qubit-chain system with $N+1=50$, coupling strength $J_0/J=1$, and three values of the magnetic field around the Markovianity point $(h/J=0.5,h_0/J=0)$. The initial state is always taken to be the tensor product of an equatorial state of $Q$ and the ground state of $\Gamma$.

Figure 5

Divisibility condition $\mathcal{C}$ against $t$ and $t_1$ for qubit $Q$ initially prepared in $(|0\rangle {}_0+|1\rangle {}_0)/\sqrt{2}$. Divisibility is guaranteed for $0⩽\mathcal{C}⩽1$. In panel (a) we have taken $J_0/J=1$ with $h_0/J=h/J=1/2$, while in panel (b) it is $h/J=1.1$. All quantities are dimensionless.

Figure 6

Dynamics of $10$ random pure initial states of qubit $Q$ pictured in the Bloch sphere. Each dot shows the qubit state (evolved from the corresponding input state) at a given instant of time. Different colors stand for different initial states. Time increases as the dots move toward point $(0,0,0)$, thus showing that $Q$ converges toward a state close to a fully mixed one.

Figure 7

(a) Distribution of density matrices evolved according to the dynamics induced by a quantum channel corresponding to $h/J=1/2$. We consider a sample of 500 random initial states. Regardless of the initial density matrix, the final state is the same, thus resulting in a zero-volume state distribution. This is no longer the case for $h/J=0.6$ [panel (b)], where a finite-volume distribution of final states is found.

Figure 8

(a): Optimized quantum process fidelity between a generalized amplitude damping channel and the dynamical map here under scrutiny. The process fidelity has been maximized with respect to the parameters of the generalized amplitude damping channel at each instant of time. In panels (b) and (c) we show the time behavior of $\mu$ and $\Gamma$. An analogous behavior is found for $\gamma$. The time dependence of the last two parameters, and in particular their positivity, are fully in line with the Markovian nature of the qubit effective dynamics.