State transfer in static and dynamic spin chains with disorder

We examine the speed and fidelity of several protocols for state or single excitation transfer in finite spin chains subject to diagonal and off-diagonal disorder. We find that, for a given chain length and maximal achievable interspin exchange ($\mathit{XY}$) coupling strength, the optimal static spin-coupling protocol, implementing the fastest state transfer between the two ends of the chain, is more susceptible to off-diagonal ($\mathit{XY}$ coupling) disorder, as compared to a much slower but robust adiabatic transfer protocol with time-dependent coupling strengths.

Figure 1

Dynamics of single-excitation transfer in a spin chain of length $N=25$ for (a) sequential SWAP, (b) spin-coupling, and (c) adiabatic protocols. The top panels correspond to noiseless spin chains, ${\sigma }_h={\sigma }_J=0$, yielding complete excitation transfer $|A_N(t_{\mathrm{out}})|{}^2=1$ for all cases (a), (b), and (c). The graph above (c) shows the time dependence of couplings $J_{\mathrm{even}}$ and $J_{\mathrm{odd}}$ normalized to $J_{\max }$ [cf. Eq. (5)]. The bottom panels illustrate the results for noisy chains with ${\sigma }_h={\sigma }_J=0.15J_{\max }$ averaged over 1000 independent realizations, leading to $\langle |A_N(t_{\mathrm{out}})|{}^2\rangle \simeq 0.2$, $0.42,$ and $0.96$ for (a), (b), and (c), respectively. Time is measured in units of $J_{\max }^{-1}$ and the evolution terminates at the corresponding $t_{\mathrm{out}}$.

Figure 2

Averaged (over 1000 realizations) fidelity $\langle F\rangle$ in noisy spin chains for (a) sequential SWAP, (b) spin-coupling, and (c) adiabatic protocols. The top panel shows the dependence of $\langle F\rangle$ on the diagonal disorder ${\sigma }_h$ with ${\sigma }_J=0$ (upper plots) and on the off-diagonal disorder ${\sigma }_J$ with ${\sigma }_h=0$ (lower plots), for the chains of lengths $N=15$, 25, 51. The inset in each graph displays the corresponding transfer probabilities $\langle |A_N(t_{\mathrm{out}})|{}^2\rangle$. The bottom panel shows $\langle F\rangle$ versus both ${\sigma }_h$ with ${\sigma }_J$ in spin chains with $N=25$. ${\sigma }_{h,J}$ are measured in units of $J_{\max }$.