Opening-assisted coherent transport in the semiclassical regime

We study quantum enhancement of transport in open systems in the presence of disorder and dephasing. Quantum coherence effects may significantly enhance transport in open systems even in the semiclassical regime (where the decoherence rate is greater than the intersite hopping amplitude), as long as the disorder is sufficiently strong. When the strengths of disorder and dephasing are fixed, there is an optimal opening strength at which the coherent transport enhancement is optimized. Analytic results are obtained in two simple paradigmatic tight-binding models of large systems: the linear chain and the fully connected network. The physical behavior is also reflected in the Fenna-Matthews-Olson (FMO) photosynthetic complex, which may be viewed as intermediate between these paradigmatic models.

Figure 1

(a) The linear chain. One excitation can hop between $N$ sites, with on-site energies ${\omega }_i$ and with nearest neighbors connected by tunneling transition amplitude $\mathrm{\Omega }$. Site $N$ is connected to a decay channel, where the excitation can escape, with coupling strength ${\mathrm{\Gamma }}_{\mathrm{trap}}$. (b) The analogous fully connected model with $N=4$ sites.

Figure 2

(a) Schematic view of the two-site model in the presence of dephasing and coupling to the sink. (b) The imaginary part of the eigenvalues of the non-Hermitian Hamiltonian given in Eq. (20). Here $\mathrm{\Omega }=1,\mathrm{\Delta }=10,\gamma =0$. The vertical dashed line represents the superradiance transition (ST). (c) Transfer time as a function of decay width to the sink for the quantum model (solid curve) and for the Förster model (dashed curve). Here $\mathrm{\Omega }=0.1,\gamma =1,\mathrm{\Delta }=10$.

Figure 3

(a) Optimal coupling to the sink, ${\mathrm{\Gamma }}_{\mathrm{trap}}^{\mathrm{opt}}$, in a two-site system, as a function of the rescaled dephasing strength $\gamma /\mathrm{\Delta }$ in the regime $\mathrm{\Delta }\gg \mathrm{\Omega }$. Data refer to the case $\mathrm{\Delta }=100,\mathrm{\Omega }=1$. Symbols represent numerical simulations of the full quantum model, the dashed red curve shows the analytical result given by Eq. (30), and the blue arrow shows the asymptotic value given by Eq. (34). The solid horizontal line indicates the value at which the superradiance transition (ST) occurs for zero dephasing. (b) Minimal transfer times for the Förster model (solid curve) and for the full quantum calculation (symbols) are shown as functions of the rescaled dephasing strength $\gamma /\mathrm{\Delta }$.

Figure 4

The parameter regime of significant quantum enhancement of transport in the two-site model, Eq. (36), and the Förster regime, $\gamma <{\mathrm{\Gamma }}_{\text{trap}}/2$, where quantum effects are negligible. In the third regime, corresponding to very large opening ${\mathrm{\Gamma }}_{\text{trap}}$, quantum mechanics suppresses transport due to the quantum Zeno effect. Generalization to a disordered linear chain of arbitrary length is obtained by replacing the detuning $\mathrm{\Delta }$ in the two-site model with $W/\sqrt{6}$, where $W$ is the disorder strength.

Figure 5

Förster (solid), Leegwater (dotted), and quantum (symbols) average transfer times for the $N=3$ linear chain. Here we fix $\mathrm{\Omega }=1$. On the horizontal axis we normalize the trapping by ${\mathrm{\Gamma }}_{\text{trap}}^{\mathrm{opt}}$ as given by Eq. (42).

Figure 6

Comparison among Förster (solid cures), Leegwater (dotted and dashed curves), and quantum (symbols) average transfer times (rescaled by a factor $N^2$) for a linear chain with $N=10$ and $N=20$ sites. Here we fix $\mathrm{\Omega }=1,\gamma =10$, and $W=50$. On the horizontal axis we renormalize the trapping by ${\mathrm{\Gamma }}_{\text{trap}}^{\mathrm{opt}}$ [Eq. (42)].

Figure 7

The error in the Leegwater approximation, $\mathrm{\Delta }\tau ={〈\tau 〉}_W-{〈{\tau }_L〉}_W$, is shown as a function of system size $N$, dephasing strength $\gamma$, and disorder strength $W$, in the parameter regime of interest. Panels (a), (c), and (e) are for the fully connected network and (b), (d), and (f) are for the linear chain. The solid blue lines in panels (a) and (b) are fits to ${(a^{\prime }+b^{\prime }N)}^2/\sqrt{\gamma W}$ for the fully connected network (with $a^{\prime }=2.33,b^{\prime }=0.37$) and to $-{(a+bN)}^{2}W/{\gamma }^2$ for the chain (with $a=1.82,b=0.72$), which converge to Eqs. (56) and (43), respectively, in the large-$N$ limit. We fix $\mathrm{\Omega }=1$, and the other parameters are shown in the legend of each panel. In particular we fix ${\mathrm{\Gamma }}_{\mathrm{trap}}=0.8W$, which is close to the optimal opening ${\mathrm{\Gamma }}_{\mathrm{trap}}^{\mathrm{opt}}=\sqrt{6}W/3$ for both systems.

Figure 8

The function $C\left(N\right)$ in Eq. (60), which describes the $N$ dependence of the incoherent transfer time in the fully connected model, with a fit to $C\left(N\right)=2.34N+64.55$. Here $W=5000,{\mathrm{\Gamma }}_{\mathrm{trap}}=6000$, and $\gamma =\mathrm{\Omega }=1$.

Figure 9

Comparison among Förster (solid curves), Leegwater (dashed curves), and quantum (symbols) average transfer times for a fully connected network with $N=10$ sites and several values of the disorder strength $W$. Here we fix $\mathrm{\Omega }=1$ and $\gamma =5$.

Figure 10

Comparison among Förster (solid curves), Leegwater (dashed curves), and quantum (symbols) average transfer times for fully connected networks of several sizes $N$. Here we fix $W=500,\gamma =5$, and $\mathrm{\Omega }=1$. The analytic result (53) describes the behavior of the Leegwater and quantum ensemble-averaged transfer time in the region of strongest quantum transport enhancement.

Figure 11

A schematic illustration of the FMO Hamiltonian (62), with each bond indicating a coupling matrix element of magnitude at least $20{\mathrm{cm}}^{-1}$.

Figure 12

The average transfer time for the FMO system is shown as a function of coupling ${\mathrm{\Gamma }}_{\mathrm{trap}}$ between site 3 and the reaction center, at three different temperatures. The Förster, Leegwater, and quantum transfer times are represented by solid curves, dashed curves, and symbols, respectively.

Figure 13

(a) The minimal transfer time through the FMO complex (optimizing over the coupling ${\mathrm{\Gamma }}_{\mathrm{trap}}$ to the reaction center) is shown as a function of temperature, for the quantum, Leegwater, and incoherent (Förster) calculations. (b) The optimal coupling ${\mathrm{\Gamma }}_{\mathrm{trap}}^{\mathrm{opt}}$ is shown as a function of temperature, in the full quantum calculation and in the Leegwater approximation. In the incoherent model, the optimal coupling is always ${\mathrm{\Gamma }}_{\mathrm{trap}}^{\mathrm{opt}}=\infty$. The horizontal solid line indicates the location of the superradiance transition at zero temperature. In both panels, the dashed vertical line indicates room temperature, $T=300$ K.