Quantum state transfer along a ring with time-reversal asymmetry

Time-reversal symmetry breaking can enhance or suppress the probability of success for quantum state transfer (QST), and remarkably it can be used to implement the directional QST. In this paper we study the QST on a ring with time-reversal asymmetry. We show that the system will behave as a quantum state turnplate under some proper parameters, which may serve as time controlled quantum routers in complex quantum networks. We propose realizing the quantum state turnplate in the coupled resonator optical waveguide by controlling the coupling strength and the phase.

Figure 1

(a) The CROW ring. The circles mean the site resonators, and the link optical waveguides link the resonators as a ring. Every waveguide between two site resonators has two paths, the outer path and the inner path. The two paths have different lengths which induce the complex coupling strength $J$. (b) Single excitation graph. The circles correspond to the resonators in the CROW system. The arrow with labels $J_l$ represents the matrix element $J_l|l\rangle \langle l+1|$.

Figure 2

Numerical stimulation of the time evolution of the system consisting of three sites with the parameters $J_1=J_2=J_3=e^{i{\varphi }_i}$ and the initial state $|100\rangle$. (a) ${\varphi }_i=\frac{\pi }{6}$. (b) ${\varphi }_i=arctan(-\frac{2}{\sqrt{3}})$. (c) ${\varphi }_i=\frac{\pi }{7}$. $P\left(t\right)$ is the probability of the wave function. The solid line, dashed line, and dash-dotted line describe the probability at site 1, site 2, and site 3, respectively. At time $\tau$ the excitation transfers from site 1 to site 2 and it transfers to site 3 after the next time interval $\tau$. The system acts like a turnplate of the excitation.

Figure 3

The ring with nine nodes and $c_3$ symmetry. Under the bases $|l,i\rangle$ the Hamiltonian is block diagonalized consisting of three blocks which is implied by $c_3$. In every block the Hamiltonian represents a ring with length 3 and the coupling strength is the same as the original ring in the site bases with the total phase being $\varphi /3, \varphi /3+2\pi /3$, and $\varphi /3+4\pi /3$ where $\varphi$ is the total phase of the original ring.

Figure 4

Spectrum structure of the Hamiltonian $H_9$ with length $N=9$ and $C_3$ symmetry. The coupling strength is $J_1=J_3=e^{\frac{\pi }{18}}, J_2=4J_1$. So the total phase is $\varphi =\pi /2$. (a) Pictures of the characteristic polynomials of three equivalent Hamiltonians obtained from the Hamiltonian $H_9$. The dash-dotted line is the curve 0 corresponding to $l=-1$, and the other two lines can be obtained from curve 0 by translating along the vertical axis. (b) Spectrum of the Hamiltonian $H_9$. Every energy lever corresponds to one cross point of the curve and the horizontal axis in (a).

Figure 5

Numerically simulate the time evolution of the system with nine sites. $J_1=J_3=1, J_2=100$, and $\varphi =-\frac{\pi }{2}$. The initial state is $\left|\psi \right(0)\rangle =|100000000\rangle$ with $|1\rangle$ as the input state. The solid line presents the fidelity of the state transfer for the state of the first site at a different time. The dashed line and the dash-dotted line corresponding to the fidelity for the seventh site and fourth site, respectively.

Figure 6

Spectrum of the $H_0$ which is the main part of the Hamiltonian. Every zero energy level denotes the Hilbert space of the sites labeled by number $(1+l\times p)$, and every grouped $p-1$ level denotes the Hilbert space of the sites from $(2+l\times p)$ to $p+(l\times p)$, respectively, where $l=0,1,...,n-1$.

Figure 7

Numerically simulate the time evolution of the fidelity of the CROW system with nine resonators. $J_1=J_3=1, J_2=100$, and $\varphi =\frac{\pi }{2}$. The initial state is $\frac{1}{\sqrt{28}}\left(\right|0\rangle +2|1\rangle +3|2\rangle )\left(\right|1000000\rangle +|1100000\rangle )$ with $\frac{1}{\sqrt{14}}\left(\right|0\rangle +2|1\rangle +3|2\rangle )$ as the input state. The solid line represents the fidelity of the state transfer for the state of the first site at a different time. The dashed line and the dash-dotted line correspond to the fidelity for the fourth site and seventh site, respectively.