Detecting non-Abelian statistics of topological states on a chain of superconducting circuits

In view of the fundamental importance and many promising potential applications, non-Abelian statistics of topologically protected states have attracted much attention recently. However, due to the operational difficulties in solid-state materials, experimental realization of non-Abelian statistics is lacking. The superconducting quantum circuit system is scalable and controllable, and thus is a promising platform for quantum simulation. Here we propose a scheme to demonstrate non-Abelian statistics of topologically protected zero-energy edge modes on a chain of superconducting circuits. Specifically, we can realize topological phase transition by varying the hopping strength and magnetic field in the chain, and the realized non-Abelian operation can be used in topological quantum computation. Considering the advantages of the superconducting quantum circuits, our protocol may shed light on quantum computation via topologically protected states.

Figure 1

Numerical calculations in one dimension, consisting of 16 unit-cell lattices. (a) Eigenenergies of the finite system with the parameters $t_z/t_0=1, {\mathrm{\Delta }}_0/t_0=0.99$, and $h_z/t_0=0.3$. An energy gap is opened in the bulk and there are two zero modes in the middle of the gap. It is obvious that those two zero modes are localized at different edges. (b) Phase diagram of the chain with arbitrary ${\mathrm{\Delta }}_0$. The yellow and dark blue regions show the topologically invariant $\nu =1$ and $\nu =0$, respectively. Four red circles A, B, C, and D represent the parameters for demonstrating the non-Abelian quantum transformations. The red arrow means that ${\widehat{O}}_1$ is executed first and then ${\widehat{O}}_2$ follows; the blue arrow means that ${\widehat{O}}_2$ is executed first and then ${\widehat{O}}_1$. (a), (c), and (d) With the same parameters, the left and right zero-energy edge states of the chain are numerically calculated, where the red or purple line plots the amplitude of the $|\uparrow \rangle$ components of the wave functions and the blue or black dashed line plots the amplitude of the $|\downarrow \rangle$ components of the wave functions divided by a phase factor $i$.

Figure 2

Proposed setup of the superconducting circuit to mimic a spin-1/2 lattice model. (a) Spin-1/2 polariton lattice with two types of unit cells, R-type (colored red) and B-type (colored blue), arranged alternately, which are of a unit cell and are of different qubit and photon eigenfrequencies and JC coupling strengths. Each unit cell has two pseudo-spin-1/2 states simulated by the two single-excitation eigenstates of the JC model. The neighboring unit cells are coupled by a combination of a SQUID in series, to induce the tunable intercell photon hopping. (b) Detuned couplings of intercell spin states. In order to achieve the simulated Hamiltonian, two sets of driving strength are assigned: The coupling strength between $|\overline{\uparrow }{\rangle }_l\leftrightarrow {|\overline{\uparrow }\rangle }_{l+1}$ and $|\overline{\downarrow }{\rangle }_l\leftrightarrow {|\overline{\downarrow }\rangle }_{l+1}$ is $t_z$ (red) and the coupling strength between $|\overline{\uparrow }{\rangle }_l\leftrightarrow {|\overline{\downarrow }\rangle }_{l+1}$ and $|\overline{\downarrow }{\rangle }_l\leftrightarrow {|\overline{\uparrow }\rangle }_{l+1}$ is ${\mathrm{\Delta }}_0$ (black). (c) Polariton lattice in a rotating frame, where all polariton lattices can be considered the same, so that the proposed circuit simulates a 1D spin-1/2 tight-binding lattice model.

Figure 3

Dynamical detection of polaritonic topological edge states. The time evolution of the polaritonic density distribution $\langle {\sigma }^{+}{\sigma }^{-}+{\widehat{a}}^{†}\widehat{a}\rangle$ is shown when the JC lattice is in (a) $|{\mathrm{\Psi }}_f\left(x\right)\rangle$, which is obtained by the ${\widehat{O}}_1$ and ${\widehat{O}}_2$ quantum operations, and (b) $|{\mathrm{\Psi }}_f^{\prime }\left(x\right)\rangle$, obtained by the ${\widehat{O}}_2$ and ${\widehat{O}}_1$ quantum operations. Also shown are the edge-site populations $P_1\left(t\right)$ and $P_2\left(t\right)$ at 1.5 $\mu \mathrm{s}$ and the oscillation center $\nu /2$ of edge states (c) $|{\mathrm{\Psi }}_f\left(x\right)\rangle$ and (d) $|{\mathrm{\Psi }}_f^{\prime }\left(x\right)\rangle$ for different decay rates $\gamma$.