Signatures of topological quantum phase transitions in driven and dissipative qubit arrays

We study photonic signatures of symmetry broken and topological phases in a driven, dissipative circuit QED realization of spin-1/2 chains. Specifically, we consider the transverse-field XY model and a dual model with three-spin interactions. The former has a ferromagnetic and a paramagnetic phase, while the latter features, in addition, a symmetry protected topological phase. Using the method of third quantization, we calculate the nonequilibrium steady state of the open spin chains for arbitrary system sizes and temperatures. We find that the bilocal correlation function of the spins at both ends of the chain provides a sensitive measure for both symmetry-breaking and topological phase transitions of the systems, but no universal means to distinguish between the two types of transitions. Both models have equivalent representations in terms of free Majorana fermions, which host zero, one and two topological Majorana end modes in the paramagnetic, ferromagnetic, and symmetry protected topological phases, respectively. The correlation function we study retains its bilocal character in the fermionic representation, so that our results are equally applicable to the fermionic models in their own right. We propose a photonic realization of the dissipative transverse-field XY model in a tunable setup, where an array of superconducting transmon qubits is coupled at both ends to a photonic microwave circuit.

Figure 1

(a) Phase diagram of the TXY model in Eq. (2.1) with anisotropy parameter $\gamma$ in a transverse magnetic field $\overline{h}$. The red line marks the transition between PM and FM phases. The PM and FM phases correspond to phases with $n=0$ and $n=1$ Majorana end modes in the fermionic counterpart, respectively. The blue line corresponds to the transition from a ferromagnet in the x-direction to a ferromagnet in the y-direction. (b) Phase diagram of the 3SI model in Eq. (2.8) [see text below Eq. (2.8) for definition parameter ${\lambda }_{1,2}]$. The blue and red lines correspond to the equivalent transitions in (a), respectively. The 3SI model also has a phase with $n=2$ Majorana modes per end.

Figure 2

Scheme of a 1D array of transmon qubits. Each transmon, comprising a Josephson junction in parallel to a large capacitance, is grounded and connected to node $i$. Two neighboring transmons at node $i$ and $i+1$ are coupled via a Josephson junction and enclose a common loop, which is threaded by an external magnetic flux $\mathrm{\Phi }$ (we assume that the internal loop of the transmon remains unaffected by the external flux). The outer transmons at node $i=1,N$ are coupled capacitively to two external transmission lines, which act as a reservoir and an external drive.

Figure 3

Auxiliary qubit occupation $\langle {\sigma }_1^z\rangle$ (colour scale) as a function of bulk transverse field $\overline{h}$ and anisotropy parameter $\gamma =(J_x-J_y)/(J_x+J_y)$ for the NESS of the driven dissipative TXY model with $N=12$ (left) and $N=42$ (right) spins, i.e., $N-2$ bulk spins and two auxiliary spins. The auxiliary spins are coupled weakly to the ends of the chain with $J_{\mu ,1}=J_{\mu ,N-1}=0.02J_x$ ($\mu =x,y$) and even weaker to the external transmission lines (see Fig. 2) with dissipation rates ${\mathrm{\Gamma }}_{1,N}=\mathrm{\Gamma }=0.02J_x$. The number of thermal photons inside the transmission lines is assumed to be $n_{\mathrm{th}}=0.1$. Here, we have set $J_x=1$ such that a horizontal line with $\gamma =1$ corresponds to the special case of the transverse field Ising model with $J_y=0$.

Figure 4

Second-order cross correlation function $g_{1N}^{\left(2\right)}$ (color scale) as defined in Eq. (5.3) for $N=12$ (left panel) and $N=42$ (right panel). The same notation as in Fig. 3 applies.

Figure 5

Plot of ordering momentum $k_0$ and the set of discrete momenta $k_n$ for a $N=12$ TXY model. The values of $\overline{h}$, where $k_0$ and $k_n$ coincide correspond to the crests of the oscillations near the anisotropy line in Figs. 3 and 4.

Figure 6

Auxiliary qubit occupation $\langle {\sigma }_1^z\rangle$ (left) and second-order cross correlation function (right) in the NESS of the dual model (2.8) with dissipation from the boundaries. The system size is $N=42$.