Higher-Order Topological Phase of Interacting Photon Pairs

Topological phases open a door to such intriguing phenomena as unidirectional propagation and disorder-resilient localization at a stable frequency. Recently discovered higher-order topological phases further extend the concept of topological protection enabling versatile control over localization in multiple dimensions. Motivated by the recent advances in quantum technologies such as large coherently operating qubit ensembles, we predict and investigate the higher-order topological phase of photon pairs emerging due to effective photon-photon interaction and described by the extended version of Bose-Hubbard model. Being feasible for state-of-the-art experimental capabilities, the designed model provides an interesting example of interaction-induced topological transitions in the few-particle two-dimensional system.

Figure 1

Artistic view of the system under study. The bulk site of the kagome lattice contains a nonlinear LC oscillator made of the grounded linear capacitance $C_g$ and Josephson junction $E_g$. The sites are connected to each other either via linear inductance $L$ or via Josephson junction $E_J$. To compensate the lack of neighbors for the edge and corner sites and avoid the formation of trivial defect states, grounding elements for them are modified as shown in the insets. White hexagon shows the unit cell choice. The location of the topological corner state of the bound photon pair is highlighted by red.

Figure 2

Single- and two-photon spectra. Density of states calculated for the qubit array for the following parameters: $L=10\mathrm{nH}$, $E_J=1\mathrm{GHz}$, $C=0.02$ fF, and $E_g=10\mathrm{GHz}$ for (a) the single-photon case and topologically trivial system of $n=120$ qubits, where $n$ is the number of unit cells along one side of the triangle, and (b) the two-photon case and the system size $n=8$. Bulk doublon states are accompanied by the doublon edge and corner states. The scales on the top and on the bottom of the plot correspond to the scattering states and doublon states, respectively. In both panels (a), (b) the grounding elements $E_g$, $C_g$ of the edge and corner sites are modified to avoid the formation of trivial Tamm-like states. (c) Phase diagram of topological regimes calculated for $C=0.02\mathrm{fF}$, $E_g=10\mathrm{GHz}$. Gray filling shows the regime when the single-photon model is trivial, while the two-photon system is topological. Black dot indicates the normalized parameters chosen for the calculations below: $L=10\mathrm{nH}$, $E_J=1\mathrm{GHz}$, $C=0.02$ fF, and $E_g=10\mathrm{GHz}$. (d) Topological doublon corner state for the system of size $n=6$. The color bar shows the value $S_n=\max [0,1+\ln (⟨n_i⟩/2)/\sigma ]$, where $\sigma =35$ is an auxiliary parameter chosen to highlight the exponential decay of the corner state.

Figure 3

Analysis of doublon bands topology. (a) The sketch of the unit cell containing three grounded qubits ($E_g$, $C_g$) connected through inductance $L$. The unit cells are connected via Josephson junctions $E_J$. Inset shows the examples of a doublon state and two-photon states with photons localized in the neighboring qubits. (b) The first Brillouin zone for the designed lattice. (c) Calculated doublon bands with symmetry indicators $p$ evaluated at high-symmetry points of the first Brillouin zone. Gray filling shows a complete photonic band gap.