Decoherence of Bell states by local interactions with a dynamic spin environment

We study the evolution of a system of two qubits, each of which interacts locally with a spin chain with nontrivial internal Hamiltonian. We present an exact solution to this problem and analyze the dependence of decoherence on the distance between the interaction sites. In the strong coupling regime we find that decoherence increases with increasing distance. In the weak coupling regime the dependence of decoherence with distance is not generic (i.e., it varies according to the initial state). Decoherence becomes independent of distance when the latter is over a saturation length $l$. Numerical results for the Ising chain suggest that the saturation scale is related to the correlation length $\xi$. For strong coupling we display evidence of the existence of non-Markovian effects (such as environment-induced interactions between the qubits). As a consequence the system can undergo a quasiperiodic sequence of “sudden deaths and revivals” of entanglement, with a time scale related to the distance between qubits.

Figure 1

A system of two noninteracting qubits ($A$ and $B$), suffering decoherence from the local interaction with distant sites of an environment chain.

Figure 2

(Color online) Echo $L_{00,11}$ as a function of time for an Ising chain $\left(\gamma =1\right)$ of $N=100$ sites weakly coupled to a two-qubit system $\left(g=0.1\right)$. From top to bottom the transverse field is $\lambda =0.5$, 0.99, and 1.5. The distance between qubits is $d=0$ (full black), 1 (dashed green), 2 (dotted violet), and 3 (dashed-dotted magenta). In the almost critical case $\lambda =0.99$ we include also $d=4$ ($··–$, red), and 10 ($––·$, orange); these curves are not shown in the other plots because they are intertwined with the others. For comparison we display in full gray the limit of two independent environments.

Figure 3

(Color online) Echo $L_{01,10}$ as a function of time; all parameters are the same as in Fig. 2 (for $d=0$ this echo is always equal to 1).

Figure 4

(Color online) Saturation length as a function of the external field $\lambda$ (dots), for the Ising chain $\left(\gamma =1\right)$. The continuous red line shows the fit as a function of the correlation length $\xi$ of the chain $\left[l\simeq 1.1+0.21{\left(0.17+{\xi }^{-1}\right)}^{-2.2}\right]$. These results correspond to a chain of 500 spins, and the echoes have been smoothed to avoid spurious effects due to oscillations. Besides, the fact that $g>0$ is accounted for by a shift in the correlation length, given by ${\xi }_{\text{eff}}=\xi \left(\lambda +5.7\times {10}^{-3}\right)$.

Figure 5

(Color online) The echo for two qubits weakly interacting with a spin chain of $N=100$ sites has revivals or sharp decays due to finite-size effects. For a distance $d=N/2$ we show $L_{00,11}$ (full black) and $L_{01,10}$ (dashed red). When $d$ decreases the first peak and decay tends to disappear (the dotted blue line corresponds to $L_{00,11}$ for $d=30$). The result for two qubits coupled to independent environments is shown for comparison (full gray). The parameters are $\lambda =0.99$, $\gamma =1$, $g=0.1$.

Figure 6

(Color online) Envelope of the echo $L_{00,11}$ as a function of time for two qubits strongly interacting with two different sites of a chain with $N=100$, $\lambda =0.99$, $\gamma =1$, $g=50$. The plots correspond to the cases of distance $d=0$ (full black), 2 (dotted blue), and long distance limit (full gray). For the case $d=0$ a part of the fast oscillation is included. Here, in contrast to the weak coupling scenario, decoherence is stronger in the long distance limit, as the envelope for large $d$ corresponds to the square of the one for $d=0$.

Figure 7

(Color online) Envelope of the echo $L_{01,10}$ as a function of time for two qubits strongly interacting with two different sites of a chain with $N=100$, $\lambda =0.99$, $\gamma =1$, $g=50$. The plots correspond to the cases of distance $d=1$ (full black), 2 (dotted blue), and long distance limit (full gray). For the case $d=0$ there is no decoherence in this case, while for $d\ge 1$ decoherence turns out to be stronger as distance is increased.