Decoherence induced by an interacting spin environment in the transition from integrability to chaos

We investigate the decoherence properties of a central system composed of two spins $1∕2$ in contact with a spin bath. The dynamical regime of the bath ranges from a fully integrable limit to complete chaoticity. We show that the dynamical regime of the bath determines the efficiency of the decoherence process. For perturbative regimes, the integrable limit provides stronger decoherence, while in the strong coupling regime the chaotic limit becomes more efficient. We also show that the decoherence time behaves in a similar way. On the contrary, the rate of decay of magnitudes like linear entropy or fidelity does not depend on the dynamical regime of the bath. We interpret the latter results as due to a comparable complexity of the Hamiltonian for both the integrable and the fully chaotic limits.

Figure 1

$P\left(s\right)$ distributions for regular $\alpha =0$ (upper panel) and chaotic $\alpha =\pi ∕2$ (lower panel) limits of $H_B$. In both cases there are $N=13$ spins and the histogram is built by collecting 50 different cases, characterized by different sets of parameters $\left\{{ϵ}_j\right\}$. The dashed line corresponds to the Poisson distribution, while the dotted line represents the Wigner distribution.

Figure 2

Parameter $\eta$ in function of $\alpha$ for $N=9$ (squares), $N=11$ (circles), and $N=13$ (triangles).

Figure 3

Power spectrum of ${\delta }_n$ statistic for $\alpha =0$ (upper left panel), $\alpha =\pi ∕10$ (lower left panel), $\alpha =\pi ∕5$ (upper right panel), and $\alpha =\pi ∕2$ (lower right panel). Squares correspond to $N=9$, circles correspond to $N=11$, and triangles correspond to $N=13$. The theoretical value for the integrable system is plotted with a dashed line and the theoretical value for chaotic systems with a solid line.

Figure 4

Density of states of $H_B$ with $\alpha =0$ (squares), $\alpha =\pi ∕10$ (circles), and $\alpha =\pi ∕2$ (triangles).

Figure 5

(Color online) Time evolution for different elements of the system density matrix: $⟨\uparrow \downarrow \mid \rho \mid \downarrow \uparrow ⟩$ (upper panel), $⟨\uparrow \downarrow \mid \rho \mid \uparrow \downarrow ⟩$ (middle panel), and $⟨\downarrow \downarrow \mid \rho \mid \downarrow \downarrow ⟩$ (lower panel), for regular $\alpha =0$ (black line; blue online) and chaotic $\alpha =\pi ∕2$ (gray line; red online) limits, as a function of the system-bath coupling strength $a$.

Figure 6

Matrix elements of pointer states of the system density matrix: $⟨\uparrow \downarrow \mid \rho \mid \downarrow \uparrow ⟩$ (upper panel), $⟨\uparrow \downarrow \mid \rho \mid \uparrow \downarrow ⟩$ (middle panel), and $⟨\downarrow \downarrow \mid \rho \mid \downarrow \downarrow ⟩$ (lower panel), for regular $\alpha =0$ (squares) and chaotic $\alpha =\pi ∕2$ (triangles) limits, and for an intermediate value $\alpha =\pi ∕10$ (circles), as a function of the system-bath coupling strength $a$.

Figure 7

Pointer states values of the system linear entropy, for regular $\alpha =0$ (squares) and chaotic $\alpha =\pi ∕2$ (triangles) limits, and for an intermediate value $\alpha =\pi ∕10$ (squares), as a function of the system-bath coupling strength $a$.

Figure 8

(Color online) Fidelity $F\left(t\right)$ for the central system perturbed by a bath composed by $N=15$ spins with $a_k=\sqrt{\frac{15}{11}}{10}^{-2}\forall k$, for different values of parameter $\alpha$: Solid black (blue online) corresponds to $\alpha =0$; dashed black (green online) corresponds to $\alpha =\pi ∕20$; dotted gray (cyan online) corresponds to $\alpha =\pi ∕10$; dotted black (black online) corresponds to $\alpha =\pi ∕5$; dashed gray (magenta online) corresponds to $\alpha =3\pi ∕10$; solid gray (orange online) corresponds to $\alpha =\pi ∕2$.

Figure 9

(Color online) Linear entropy $\Omega \left(t\right)$ for the central system perturbed by a bath composed by $N=15$ spins with $a_k=\sqrt{\frac{15}{11}}{10}^{-2}\forall k$, for different values of parameter $\alpha$: Solid black (blue online) corresponds to $\alpha =0$; dashed black (green online) corresponds to $\alpha =\pi ∕20$; dotted gray (cyan online) corresponds to $\alpha =\pi ∕10$; dotted black (black online) corresponds to $\alpha =\pi ∕5$; dashed gray (magenta online) corresponds to $\alpha =3\pi ∕10$; solid gray (orange online) corresponds to $\alpha =\pi ∕2$.