Disorder-Enhanced and Disorder-Independent Transport with Long-Range Hopping: Application to Molecular Chains in Optical Cavities

Overcoming the detrimental effect of disorder at the nanoscale is very hard since disorder induces localization and an exponential suppression of transport efficiency. Here we unveil novel and robust quantum transport regimes achievable in nanosystems by exploiting long-range hopping. We demonstrate that in a 1D disordered nanostructure in the presence of long-range hopping, transport efficiency, after decreasing exponentially with disorder at first, is then enhanced by disorder [disorder-enhanced transport (DET) regime] until, counterintuitively, it reaches a disorder-independent transport (DIT) regime, persisting over several orders of disorder magnitude in realistic systems. To enlighten the relevance of our results, we demonstrate that an ensemble of emitters in a cavity can be described by an effective long-range Hamiltonian. The specific case of a disordered molecular wire placed in an optical cavity is discussed, showing that the DIT and DET regimes can be reached with state-of-the-art experimental setups.

Figure 1

(a),(b) Two different setups for a disordered chain with excitation pumping ${\gamma }_p$ at one edge of the chain and draining ${\gamma }_d$ at the opposite edge. Here, $\mathrm{\Omega }$ is the hopping between nearest-neighbor sites. The arrows indicate the hopping paths available for an excitation (gray circle) present at the center of the chain. The energy of the sites is disordered. (a) A long-range coupling $-\gamma /2$ is present between each pair of sites. (b) The chain is placed inside an optical cavity, where $g$ is the coupling of each site to the cavity mode.

Figure 2

(a) Normalized typical current $\hslash I^{\mathrm{typ}}/\mathrm{\Omega }$ versus the normalized static disorder $W/\mathrm{\Omega }$. (b) Average variance $⟨{\sigma }^2⟩$ versus the normalized static disorder $W/\mathrm{\Omega }$ for the ground state (triangles) and the excited states (all other sets). The blue curves show the case $\gamma =0$, the orange squares the perturbative approach (details are given in the SM [39]). Dashed vertical lines indicate the different critical disorders given by Eqs. (3), (10), and (11). (c),(d) Average shape of the eigenfunctions (details concerning the process of averaging are given in SM [39]), $⟨|\mathrm{\Psi }{|}^2⟩$ versus the site basis $k$. Different disorder regimes are shown. (c) (DET) $W_1\le W\le W_2$, $W/\mathrm{\Omega }=1$ (black) and $W/\mathrm{\Omega }=44.02$ (red); (d) (DIT) $W_2<W<W_{\mathrm{gap}}$, $W/\mathrm{\Omega }={10}^2$ (black) and $W/\mathrm{\Omega }={10}^3$ (red). Here, $N={10}^4$, ${\gamma }_p={\gamma }_d=\gamma =\mathrm{\Omega }$, and $N_r=100$ disorder configurations. In (c),(d) symbols are compared with blue curves indicating the case $\gamma =0$.

Figure 3

(a) Typical current $I^{\mathrm{typ}}$, Eq. (9), versus the static disorder $W$. The results for a linear chain in an optical cavity Eq. (12) (crosses) are compared with a long-range hopping model Eq. (1) (circles). Parameters for the linear chain in an optical cavity are $N={10}^4$, $\mathrm{\Omega }=0.0124\mathrm{eV}$, $\hslash {\omega }_c=2\mathrm{eV}$, $\mu \approx 36$ D, $g_c=3.188\mathrm{eV}$, ${\gamma }_p={\gamma }_d=0.0124\mathrm{eV}$. The long-range hopping model has been obtained using the same $\mathrm{\Omega }$ value and setting $\gamma =2g_c/N$ in Eq. (1). The number of disorder configurations $N_r$ is such that $N_r\times N={10}^6$. (b) Normalized typical current $I^{\mathrm{typ}}{N}^2$ versus the static disorder $W$ for a linear chain in an optical cavity for different $N$ values, as indicated in the legend. Vertical dashed lines represent the values of $W_1$ for different system sizes. Other parameters are the same as in (a).