Decoherence as attenuation of mesoscopic echoes in a spin-chain channel

An initial local excitation in a confined quantum system evolves, exploring the whole system and returning to the initial position as a mesoscopic echo at the Heisenberg time. We consider two weakly coupled spin chains, a spin ladder, where one is a quantum channel while the other represents an environment. We quantify decoherence in the quantum channel through the attenuation of the mesoscopic echoes. We evaluate decoherence rates for different ratios between sources of amplitude fluctuation and dephasing in the interchain interaction Hamiltonian. The many-body dynamics is seen as a one-body evolution with a decoherence rate given by the Fermi golden rule.

Figure 1

(a) Schematic representation of the spin ladder system given by the Hamiltonian of Eq. (1). Each spin chain has $M=5$ spins. (b) A system like in (a) but with periodic boundary conditions between spins $1$ and $M$. Light colors represent the quantum channel and the marked spin within them (red online) represents the site where the initial condition is placed.

Figure 2

Polarization at site $1$, $P_{1,1}(t)$, of the spin ladder of Fig 1a. (a) Upper (black/solid filled) curve represents the dynamics of an isolated chain given by $P_{1,1}^{\mathrm{isolated}}(t)$. Pattern-filled curves (color-filled) correspond to different values of the coupling $J_{\mathrm{y}}$ for a transversal isotropic interaction between chains. (b) Comparison of the dynamics behavior between different transversal interaction natures ($\mathit{XY}$, isotropic, dipolar) by keeping fixed the ratio $J_{\mathrm{y}}/J_{\mathrm{x}}=0.1$.

Figure 3

Natural logarithm of the ratio $P_{11}(t)/P_{1,1}^{\mathrm{isolated}}(t)$ for different values of the transversal coupling $J_{\mathrm{y}}$ as a function of time. The interaction between chains has an isotropic nature. The dashed line is a guide to the eye.

Figure 4

Maximum values of Fig. 3 for two different ratios $J_{\mathrm{y}}/J_{\mathrm{x}}$ as a function of time. The slope of the linear fit (solid line) gives the value of ${\tau }_{\varphi }$ associated with the perturbation $J_{\mathrm{y}}$.

Figure 5

Decoherence rates $\frac{1}{{\tau }_{\varphi }}$ for different transversal Hamiltonians as a function of $\left|J_{\mathrm{y}}\right|{}^2/\hslash J_{\mathrm{x}}$. The lineal dependence indicates the good agreement with the FGR behavior. Different slopes indicate that the decoherence rate is composed of two contributions: One is due to the injection of polarization produced by the $\mathit{XY}$ term of ${\stackrel{\wedge }{\mathcal{H}}}_{\mathrm{T}}$ and the other is associated with the dephasing process given by the Ising term.

Figure 6

$y_{\mathit{XY}}$ vs $x_{\mathit{ZZ}}$ for different natures of the transversal interaction, where $x_{\mathit{ZZ}}=a^2{\tau }_{\varphi }\left|J_{\mathrm{y}}\right|{}^2/(\hslash J_{\mathrm{x}})$ and $y_{\mathit{XY}}=b^2{\tau }_{\varphi }\left|J_{\mathrm{y}}\right|{}^2/\hslash J_{\mathrm{x}}$. The solid line is given by a linear fitting to the points.

Figure 7

Schematic representation of the two-spin channel (upper spins) coupled to independent spin environments (bottom spins). The darker spin (red online) of the two-spin channel represents the site where the initial condition is placed.