Continuous-time quantum walk on an extended star graph: Trapping and superradiance transition

A tight-binding model is introduced for describing the dynamics of an exciton on an extended star graph whose central node is occupied by a trap. On this graph, the exciton dynamics is governed by two kinds of eigenstates: many eigenstates are associated with degenerate real eigenvalues insensitive to the trap, whereas three decaying eigenstates characterized by complex energies contribute to the trapping process. It is shown that the excitonic population absorbed by the trap depends on the size of the graph, only. By contrast, both the size parameters and the absorption rate control the dynamics of the trapping. When these parameters are judiciously chosen, the efficiency of the transfer is optimized resulting in the minimization of the absorption time. Analysis of the eigenstates reveals that such a feature arises around the superradiance transition. Moreover, depending on the size of the network, two situations are highlighted where the transport efficiency is either superoptimized or suboptimized.

Figure 1

(a) Representation of the extended star graph with $N=1+N_1(N_2+1)$ nodes $(\ell ,s)$. Graphical representation of the block Hamiltonian (b) $H^{\left(k\right)}$ with $k\ne N_1$, (c) $H^{\left(N_1\right)}$, and (d) $\mathcal{H}$ (see the text).

Figure 2

(a) $N_2$ dependence of the limiting probability for $N_1=3$ and (b) $N_1$ dependence of the limiting probability for $N_2=3$. Limiting probabilities to observe the exciton on the excited site (black full line), the excited star (green dotted line), the other stars (blue dashed line), and the central core (red short dashed line).

Figure 3

Time evolution of the absorbed population $P_A\left(t\right)$ for (a) $\mathrm{\Gamma }=3\mathrm{\Phi }$ and for (b) $\mathrm{\Gamma }=20\mathrm{\Phi }$. ($N_1=8,N_2=3)$ blue full line, ($N_1=3,N_2=8)$ blue dashed line, $(N_1=6,N_2=3)$ black full line, and $(N_1=3,N_2=6)$ black dashed line.

Figure 4

$\mathrm{\Gamma }$ dependence of the absorption time $\tau$. Configuration A: $(N_1=3,N_2=8)$ dashed line. Configuration B: $(N_1=8,N_2=3)$ full line.

Figure 5

$\mathrm{\Gamma }$ dependence of (a) the width and (b) the energy of the decaying states for configurations A (dashed lines) and B (full lines): ${\gamma }_0^{(N_1,N_2)}$ and ${\omega }_0^{(N_1,N_2)}$ (green curve); ${\gamma }_{\pm }^{(N_1,N_2)}\mathrm{and}{\omega }_{\pm }^{(N_1,N_2)}$ (red curve).

Figure 6

$\mathrm{\Gamma }$ dependence for configuration A of (a) the superradiant state and (b) the subradiant states; $\mathrm{\Gamma }$ dependence for configuration B of (c) the superradiant state and (d) the subradiant states. The figures show the weight ${\mathrm{\Pi }}_n$ of each decaying state on $|0,0\rangle$ ($n=1$, full line), $|{\chi }_{N_1},0\rangle$ ($n=2$, dashed line), and $|{\chi }_{N_1},{\mu }_{N_2}\rangle$ ($n=3$, dotted line).

Figure 7

$\mathrm{\Gamma }$ dependence of the IPR for configuration A (dashed line) and for configuration B (full line). Red curves refer to the subradiant states, and green curves refer to the superradiant state.

Figure 8

Time evolution of the population at the superradiance transition (a) for configuration A (i.e., for $\mathrm{\Gamma }={\mathrm{\Gamma }}_A^{\left(ST\right)}$) and (b) for configuration B (i.e., for $\mathrm{\Gamma }={\mathrm{\Gamma }}_B^{\left(ST\right)}$). The figure shows the absorbed population $P_A\left(t\right)$ (blue curves), the population $P_3\left(t\right)$ of the state $|{\chi }_{N_1},{\mu }_{N_2}\rangle$ (red curves), the population $P_2\left(t\right)$ of the state $|{\chi }_{N_1},0\rangle$ (green curves), and the population $P_1\left(t\right)$ of the state $|0,0\rangle$ (black curves) (see the text).

Figure 9

Size dependence of the minimum value of the absorption time $\tau$.