Superradiant Phase Transition in a Superconducting Circuit in Thermal Equilibrium

We propose a superconducting circuit that shows a superradiant phase transition (SRPT) in thermal equilibrium. The existence of the SRPT is confirmed analytically in the limit of an infinite number of artificial atoms. We also perform a numerical diagonalization of the Hamiltonian with a finite number of atoms and observe an asymptotic behavior approaching the infinite limit as the number of atoms increases. The SRPT can also be interpreted intuitively in a classical analysis.

Figure 1

Superconducting circuit showing the SRPT in thermal equilibrium under a static external magnetic flux bias ${\mathrm{\Phi }}_{\text{ext}}={\mathrm{\Phi }}_0/2$.

Figure 2

Normalized inductive energy [Eq. (2)] versus $2\pi \varphi /{\mathrm{\Phi }}_0$ under the condition of ${\psi }_j=(1+L_g/L_{R0})\varphi$, which is obtained by $\partial U/\partial \varphi =0$. For $N{L}_R=L_{R0}>L_J-L_g$, the inductive energy shows two minima at $\varphi \ne 0$. This transition corresponds to the SRPT in the sense of the quantum phase transition. The parameter is $L_g=0.6L_J$.

Figure 3

(a) Normalized photonic amplitude ${\alpha }_{\text{eq}}/\sqrt{N}$ in the thermodynamic limit (infinite number of atoms $N\to \infty$) versus $L_{R0}=N{L}_R$ and temperature $k_{B}T/h$ in frequency units. The SRPT occurs at $L_{R0}^{\text{crit}}$ at $T=0$. (b) Atomic levels, potential, and wave functions. (c) Zero-point energy shift, (e) transition frequency, and (f) number of photons per atom versus $L_{R0}$. Dashed curves are obtained in the thermodynamic limit, and solid curves are calculated for $N=1,\dots ,5$. In panel (e), the dash-dotted curves represent the transition frequencies for exciting an odd number of bosons, while the solid curves are those for exciting an even number of bosons. (d) Photonic frequency ${\omega }_c$, renormalized atomic frequency ${\tilde{\omega }}_a$, and renormalized light-matter interaction strength $\tilde{g}$ versus $L_{R0}$. The SRPT occurs when $\tilde{g}$ reaches the critical interaction strength $\sqrt{{\tilde{\omega }}_a{\omega }_c}/2$ plotted by the dotted line. The parameters are $L_J=0.75\mathrm{nH}$, $L_g=0.6L_J=0.45\mathrm{nH}$, $C_J=24\mathrm{fF}$, and $C_{R0}=2\mathrm{fF}=C_R/N$.