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We study the interplay between dephasing, disorder, and openness on transportefficiency in a one-dimensional chain of finite length $N$, and in particularthe beneficial or detrimental effect of dephasing on transport. The excitationmoves along the chain by coherent nearest-neighbor hopping $\Omega$, under theaction of static disorder $W$ and dephasing $\gamma$. The system is open due tothe coupling of the last site with an external acceptor system (sink), wherethe excitation can be trapped with a rate $\Gamma_{\rm trap}$, which determinesthe opening strength. While it is known that dephasing can help transport inthe localized regime, here we show that dephasing can enhance energy transfereven in the ballistic regime. Specifically, in the localized regime we recoverprevious results, where the optimal dephasing is independent of the chainlength and proportional to $W$ or $W^2/\Omega$. In the ballistic regime, theoptimal dephasing decreases as $1/N$ or $1/\sqrt{N}$ respectively for weak andmoderate static disorder. When focusing on the excitation starting at thebeginning of the chain, dephasing can help excitation transfer only above acritical value of disorder $W^{\rm cr}$, which strongly depends on the openingstrength $\Gamma_{\rm trap}$. Analytic solutions are obtained for short chains.

Optimal Dephasing for Ballistic Energy Transfer in Disordered Linear Chains