Topological edge/corner states and polaritons in dimerized/trimerized superconducting qubits in a cavity

We investigate a two-level topological system with an alternating XX coupling in a photon cavity. It is mapped to a free boson model equally coupled to a photon, whose interaction is highly nonlocal. We analyze the Su-Schrieffer-Heeger model and the breathing kagome model. Intriguing phenomena occur in the topological phase due to hybridization between the photon field and the topological edge or corner state. As the photon coupling increases, the symmetric edge or corner state is smoothly transformed into a polariton state coupled with the edge or corner state. Simultaneously, the photon state is smoothly transformed into the symmetric edge state or the ${\mathrm{C}}_{\text{3}}$-symmetric corner state. Furthermore, a state is detached from the bulk band and smoothly transformed into a polariton due to hybridization with the photon field.

Figure 1

(a) Illustration of the SSH model with each site equally coupled with a single photon. (b) That of the breathing kagome lattice.

Figure 2

Energy spectrum as a function of $\lambda$, describing bulk bands in cyan, the topological edges in cyan and the photon in red. (a) $\hslash g/J=0$, (b) $\hslash g/J=0.1$, and (c) $\hslash g/J=0.2$. The vertical axis is the energy in units of $J$. The color of each curve represents the photon number according to the color pallet given in Fig. 3. We have set the site number $N=80$. Note that we have used the abbreviation $g$ instead of $\hslash g/J$ in figures for brevity. This is the case for other figures.

Figure 3

Energy spectrum as a function of the coupling strength $g$ in the trivial phase with $\lambda =0.5$ for (a1) and (a2) and in the topological phase with $\lambda =-0.5$ for (b1)–(b3). The photon energy is taken so that $\hslash \mathrm{\Delta }\omega \equiv \hslash ({\omega }_0-{\omega }_{\text{s}})=-0.5J$. (b3) A detailed band structure at the anticrossing point in the topological phase, where there are three branches, the positive-energy branch, the zero-energy branch, and the negative-energy branch. The vertical axis is the energy in units of $J$. (b4) (f) The photon number $\langle {\widehat{a}}^{†}\widehat{a}\rangle$ as a function of the coupling strength $g$ in each state, where $\hslash g/J\le 0.7$ in (b4) and $\hslash g/J\le 3$ in (f). The photon number turns out to be $1/2$ both for the edge polariton and the detached polariton. (c) Analytical result of Eq. (33). (d) Color pallet showing the photon number. This color pallet is used for all figures. We have set the site number $N=80$.

Figure 4

(a1)–(a7) The wavefunction in the positive-energy branch starting from the edge state toward a polariton state for various $g$. (b1)–(b7) The wavefunction in the negative-energy branch starting from the photon state toward the edge state for various $g$. (c1)–(c7) The wavefunction of the detached state starting from a bulk state toward a polariton state for various $g$. The horizontal axis is the site index from 1 to 80, with the site number $N=80$. We have considered the topological phase ($\lambda =-0.5$). The anticrossing point is $\hslash g_c/J=0.158$. The color pallet is is given in Fig. 3, which indicates the photon number in each state.

Figure 5

(a) The critical coupling constant $\hslash g_{\text{c}}$ as a function of the photon energy $-\hslash \mathrm{\Delta }\omega$. The red curve indicates the numerical result, while blue curve indicates the analytical result. (b) Energy spectrum as a function of the coupling strength $g$. The photon energy $\hslash \mathrm{\Delta }\omega =-0.1J$. The anticrossing point is $\hslash g_c/J=0.07$.

Figure 6

Energy spectrum as a function of the coupling strength $g$ for various photon energy $\hslash \mathrm{\Delta }$. The photon energy is (a) $\hslash \mathrm{\Delta }\omega =-3J$, (b) $\hslash \mathrm{\Delta }\omega =-1.5J$, (c) $\hslash \mathrm{\Delta }\omega =0$, (d) $\hslash \mathrm{\Delta }\omega =0.5J$, (e) $\hslash \mathrm{\Delta }\omega =1.5J$, and (f) $\hslash \mathrm{\Delta }\omega =3J$.

Figure 7

Energy spectrum as a function of $\lambda$ in the breathing kagome model, describing three bulk bands in cyan, the topological corners in cyan and the photon in red. (a) $\hslash g/J=0$, (b) $\hslash g/J=0.1$, and (c) $\hslash g/J=0.2$. The vertical axis is the energy in units of $J$. A detached state from the bulk band is present in (b) and (c). We have set the site number $N=108$.

Figure 8

Energy spectrum of the breathing kagome model as a function of the coupling strength $g$ in the trivial phase with $\lambda =0.5$ for (a1) and (a2) and in the topological phase with $\lambda =-0.5$ for (b1) and (b2). The photon energy is taken so that $\hslash ({\omega }_0-{\omega }_{\text{s}})=-0.5J$. The color pallet is the same as in Fig. 3, which indicates the photon number in each state. A detailed band structure around the anticrossing point in the topological phase is almost identical to Fig. 3(b3). The main difference is that the zero-energy blanch contains two topological corner states except for the ${\mathrm{C}}_3$ symmetric one. The vertical axis is the energy in units of $J$. We have set the site number $N=108$.

Figure 9

Illustration of a Josephson circuit. A Josephson unit is composed of a Josephson junction (box with cross) and a shunted capacitance $C$. A Josephson circuit is a chain of Josephson junctions and alternating capacitances $C_A$ and $C_B$. An antenna is attached to each Josephson unit. The circuit is set in a cavity composed of two mirrors.