Two-frequency Jahn-Teller systems in circuit QED

We investigate the simulation of Jahn-Teller models with two nondegenerate vibrational modes using a circuit QED architecture. Typical Jahn-Teller systems are anisotropic and require at least a two-frequency description. The proposed simulator consists of two superconducting lumped-element resonators interacting with a common flux qubit in the ultrastrong coupling regime. We translate the circuit QED model of the system to a two-frequency Jahn-Teller Hamiltonian and calculate its energy eigenvalues and the emission spectrum of the cavities. It is shown that the system can be systematically tuned to an effective single-mode Hamiltonian from the two-mode model by varying the coupling strength between the resonators. The flexibility in manipulating the parameters of the circuit QED simulator permits the isolation of the effective single-frequency and pure two-frequency effects in the spectral response of Jahn-Teller systems.

Figure 1

(a) Schematic of a flux qubit galvanically coupled to two $LC$ resonators through coupling inductances $L_{c_{1,2}}$. Each resonator has resonant frequency ${\omega }_{1,2}={\left[(L_{1,2}+L_{c_{1,2}})C_{1,2}\right]}^{-1/2}$. The resonators are coupled to each other through the coupling capacitor $C_c$. (b) Implementation of the circuit using interdigitated finger capacitors. Feed lines individually coupled to each resonator can be employed to probe the internal photon state of the resonators.

Figure 2

Dependence of the lowest five eigenenergies of the two-frequency Jahn-Teller model on the mode-coupling parameter $\Delta$. (a) For the qubit-cavity coupling scaling parameter $k=0.1/\sqrt{2}$, the splitting of the first energy level into three at $\Delta =0$ is a pure Rabi splitting. The privileged effective single-mode regime is valid over a narrow band $\left|\Delta \right|<0.1$. (b) At $k=1/\sqrt{2}$ the energy-level transitions become flat over a broader band $\left|\Delta \right|<1$, deeper into the ultrastrong coupling regime. (c) Energy bands of the two-frequency Jahn-Teller model depending on the mode-coupling parameter $\Delta$ and the qubit-cavity coupling scaling parameter $k$. All the quantities plotted are dimensionless as explained in the text.

Figure 3

Resonator-qubit coupling $k_{\mathrm{eff}}$ dependence of the cavity emission spectrum for the privileged effective mode in the two-frequency Jahn-Teller model for different values of the resonator-resonator coupling strength $J$. (a) $J=0$ corresponds to the pure single-mode regime. $k_{\mathrm{eff}}\ll 1$ is the single-mode regime in the JC model. For $k_{\mathrm{eff}}\gtrsim 1$ one enters the regime of single-mode JT model. (b) $J=1/2$ shows three regimes: $k_{\mathrm{eff}}\ll 1$ (two-mode JC model), $k_{\mathrm{eff}}\gtrsim 1$ (two-mode JT model), and $k_{\mathrm{eff}}\gg J$ (effective single-mode JT model). This case simulates the frequency ratio of 3:1 of the two phonon modes in Fe${}^{2+}$ in ZnS. (c) $J=1/3$ simulates the frequency ratio of 2:1 of the two phonon modes in C${}_6$H${}_6$${}^{\pm }$. In this case the effective single-mode regime is more clear for $k_{\mathrm{eff}}>1.5$.

Figure 4

Variation of cavity emission spectrum for the privileged effective mode in the two-frequency Jahn-Teller model with resonator-resonator coupling $J$ at (a) $k_{\mathrm{eff}}=0.1$ and (b) $k_{\mathrm{eff}}=1$.