Fast and robust quantum state transfer assisted by zero-energy interface states in a splicing Su-Schrieffer-Heeger chain

We propose a fast, robust, and long-distance quantum state transfer (QST) protocol via a splicing Su-Schrieffer-Heeger (SSH) chain, where the interchain couplings vary with the change in the phase parameter and the single or splicing SSH chain can be designed by adjusting it. It is found that the existence of a zero-energy interface state (IFS) not only can improve the speed of QST but also can realize long-distance QST. Furthermore, we give the phase diagram in the parameter space of the transfer time $T$ and the system's size $N$, where the different regions that can successfully implement QST via a single- or splicing-SSH-chain protocol are given. Therefore, we can choose the optimal QST protocol by adjusting only the phase parameter for different transfer times and system sizes. By numerically investigating the resilience of each protocol to static disorder, we reveal that the splicing-SSH-chain protocol is quite robust to both diagonal and off-diagonal disorders and clearly outperforms the single-SSH-chain protocol. By considering the environmental influence, rendering the Hamiltonian non-Hermitian by allowing energy to radiate away, our work shows that the QST protocol assisted by zero-energy IFS is more robust than previously expected and also outperforms the single-SSH-chain protocol.

Figure 1

Schematic diagram of different SSH chains. (a) Splicing SSH chain, where the coupling strengths of the four subchains are set as $J_n=J_{n,0}$ for $n\in [1,N_1-1], J_n=J_{n,\frac{\pi }{2}}$ for $n\in [N_1,N_1+N_2-1], J_n=J_{n,\pi }$ for $n\in [N_1+N_2,N-N_4-1]$, and $J_n=J_{n,\frac{3\pi }{2}}$ for $n\in [N-N_4,N-1]$. (b) Single SSH chain, where the coupling strength equably is set as $J_n=J_{n,0}$ for $n\in [1,N-1]$.

Figure 2

Energy spectrum under the open boundary condition for an odd-sized splicing SSH chain. (a) Energy $E$ vs the phase parameter $\theta$. (b) Enlargement of the rectangular region in (a). (c)–(f) Zero-energy states corresponding to $\theta =0,0.5\pi ,1.25\pi$ and $2.5\pi$, respectively. The other parameters are chosen to be $g_0=g_1=1, N=41$, and $(N_1,N_2,N_3,N_4)=(10,10,10,11)$.

Figure 3

The distribution of the zero-energy IFS. (a)–(d) The chain length of the splicing SSH chain is designed as $(N_1,N_2,N_2,N_4)=(11,10,10,10)$, (10,11,10,10), (10,10,11,10), and (10,10,10,11), where the red, black, and blue zero-energy IFSs can be obtained at $\theta =0.75$, 1.25, and 1.75, respectively. The other parameters are chosen to be $g_0=g_1=1, N=41$.

Figure 4

(a) and (b) QST via splicing-SSH-chain protocols for the interface at even and odd sites. (c) The fidelity as a function of the transfer time for different protocols, where the black dashed line and red solid line correspond to the splicing-SSH-chain protocols of the interface at even and odd sites and the blue solid line corresponds to the single-SSH-chain protocol. The other parameters are chosen to be $g_0=g_1=1, N=41$.

Figure 5

The fidelity as a function of the total chain length $N$ for a fixed transfer time (a) $T=200$ and (b) $T=500$, where blue and black dash-dotted lines correspond to the results of the single and splicing SSH chains, respectively. The other parameters are chosen to be $g_0=g_1=1$.

Figure 6

Phase diagram of the quantum state transfer in the parameter space $(T,N)$. (a) The result of the splicing SSH chain. (b) The result of the single SSH chain. (c) The total phase diagram derived from (a) and (b), where the parameter space can be divided into four regions. The other parameters are chosen to be $g_0=g_1=1$.

Figure 7

The average fidelity as a function of the disorder strength $w_s$ for different protocols and forms of disorder. The black dotted lines and red solid lines correspond to the results of the splicing SSH chain in which the splicing interfaces are at even and odd sites, and the blue solid lines are the results obtained via the single SSH chain. The other parameters are chosen to be $g_0=g_1=1, N=25$, and $T=800$.

Figure 8

Complex energies of the odd-sized splicing-SSH-chain as a function of the phase parameter $\theta$. The other parameters are chosen to be $g_0=g_1=1, N=41, \gamma =0.02$, and $(N_1,N_2,N_3,N_4)=(10,10,10,11)$.

Figure 9

The fidelity as a function of the parameter $\gamma$ for a fixed total chain length and transfer time, (a) $N=41, T=500$ [region III in Fig. 6] and (b) $N=25, T=800$ [region II in Fig. 6], where blue and black dash-dotted lines correspond to the results of the single and splicing SSH chains, respectively. The other parameters are chosen to be $g_0=g_1=1$.

Figure 10

The distribution of the zero-energy interface state and corresponding quantum state transfer for three different splicing paths with a fixed total chain length $N=17$ and transfer time $T=500$: (a) and (b) $(N_1,N_2,N_2,N_4)=(4,4,4,5)$, (c) and (d) $(N_1,N_2,N_2,N_4)=(4,6,2,5)$, and (e) and (f) $(N_1,N_2,N_2,N_4)=(4,2,2,9)$.

Figure 11

The quantum state transfer for two different splicing paths with a fixed total chain length $N=101$ and transfer time $T=500$: (a) $(N_1,N_2,N_2,N_4)=(26,25,25,25)$ and (b) $(N_1,N_2,N_2,N_4,\cdots ,N_{20})=(6,5,5,5,\cdots ,5)$.