Vacuum Degeneracy of a Circuit QED System in the Ultrastrong Coupling Regime

We investigate theoretically the quantum vacuum properties of a chain of $N$ superconducting Josephson atoms inductively coupled to a transmission line resonator. We derive the quantum field Hamiltonian for such a circuit QED system, showing that, due to the type and strength of the interaction, a quantum phase transition can occur with a twice degenerate quantum vacuum above a critical coupling. In the finite-size case, the degeneracy is lifted, with an energy splitting decreasing exponentially with increasing values of $g^2{N}^2$, where $g$ is the dimensionless vacuum Rabi coupling per artificial atom. We determine analytically the ultrastrong coupling asymptotic expression of the two degenerate vacua for an arbitrary number of artificial atoms and of resonator modes. In the ultrastrong coupling regime the degeneracy is protected with respect to random fluctuations of the transition energies of the Josephson elements.

Figure 1

A chain of $N$ Josephson atoms ($F$ stands for fluxonium [4]) are inductively coupled to a transmission line resonator. By tuning the external magnetic flux, the flux-dependent potential for each fluxonium has a symmetric double well structure with two states $|0⟩$ and $|1⟩$ (with energy difference $\hslash {\omega }_F$) forming the two-level system (parameters used for the inset, $E_J/E_{C_J}=3$, $E_J/E_{L_J}=20$).

Figure 2

First 30 energy eigenvalues versus dimensionless vacuum Rabi coupling for $N=5$ fluxonium atoms, $N_m=3$ resonator modes (${\omega }_F={\omega }_{k=1}$). Upper inset: The difference between the energy eigenvalues and the ground state energy is plotted. Lower inset: Normalized energy difference (log scale) between the first two quasidegenerate levels versus the dimensionless coupling. In the ultrastrong coupling limit, the two ground states are excellently approximated by the analytical formula in Eq. (7) (98% overlap for the largest coupling here considered).

Figure 3

Normalized energy splitting (log scale) between the two quasidegenerate vacua versus $g^2$ for different values of $N$ ($2,3,4,\dots ,11$), where $g$ is the dimensionless vacuum Rabi coupling frequency per artificial atom. (a) Results for the case of identical fluxonium atoms with same transition frequency ${\omega }_F$. $\delta /{\omega }_F$ decreases as $\exp \mathbf{(}-\beta (N)g^2\mathbf{)}$. Inset: $\beta (N)$ vs $N^2$. (b) Averaged splitting $⟨\delta ⟩$ in presence of a random distribution: ${\omega }_{j,F}={\omega }_F+{\Delta }_j={\omega }_F(1+0.5{\xi }_j)$, where $j$ labels the site and ${\xi }_j$ is a Gaussian variable with variance equal to 1. The results have been averaged over 100 disorder configurations. The standard deviation $\sigma =\sqrt{⟨{\delta }^2⟩-⟨\delta {⟩}^2}$ has the same exponential dependence (see inset).