Vacuum-induced symmetry breaking in a superconducting quantum circuit

The ultrastrong-coupling regime, where the atom-cavity coupling rate reaches a considerable fraction of the cavity or atom transition frequencies, has been reported in a flux qubit superconducting quantum circuit coupled to an on-chip coplanar resonator. In this regime situations may arise where the resonator field $\widehat{X}=\widehat{a}+{\widehat{a}}^{†}$ acquires a nonzero expectation value in the system ground state. We demonstrate that, in this case, the parity symmetry of an additional artificial atom with an even potential is broken by the interaction with the resonator. Such a mechanism is analogous to the Higgs mechanism where the gauge symmetry of the weak force's gauge bosons is broken by the nonzero vacuum expectation value of the Higgs field. The results presented here open the way to controllable experiments on symmetry-breaking mechanisms induced by nonzero vacuum expectation values. Moreover, the mechanism proposed here can be used as a probe of the ground-state macroscopic coherence emerging from quantum phase transitions with vacuum degeneracy.

Figure 1

Schematic representation of the system. The quantum circuit (dashed black box) is constituted by a coplanar waveguide transmission line resonator ultrastrongly interacting with embedded identical flux qubits (only one is depicted in the figure).

Figure 2

(a) Contour plot of the vacuum expectation value $v\equiv \langle G\left|\widehat{X}\right|G\rangle$ of a resonator–three-qubit system in the dispersive regime $\left(\delta ={\omega }_{\mathrm{q}}-{\omega }_{\mathrm{c}}=0.7{\omega }_{\mathrm{c}}\right)$ as a function of the coupling rate $g$ and the external magnetic flux (encoded in the angle $\theta )$ threading the qubit. (b) Extracavity emission rate ${\varphi }_{\mathrm{out}}/{\gamma }_{\mathrm{c}}$ as a function of the modulation frequency of the resonator-qubit interaction rate $g=g_0+g_{1}sin\omega t$ for $\theta =0$ (solid gray line) and $\theta =\pi /10$ (dot-dashed red line). Numerical values for the constant part $g_0$ of the coupling rate and the modulation amplitude $g_1$ are, respectively, $g_0=0.15{\omega }_{\mathrm{c}}$ and $g_1=9\times {10}^{-4}{\omega }_{\mathrm{c}}$. The loss rates for the cavity and the qubit are ${\gamma }_{\mathrm{c}}={\gamma }_{\mathrm{q}}={10}^{-3}{\omega }_{\mathrm{c}}$. The dashed blue line describes the coherent part $|\langle G|\widehat{a}{|G\rangle |}^2$ of the emission rate for $\theta =\pi /10$.

Figure 3

Absorption spectrum as a function of the driving frequency of the coherent drive $\mathcal{E}\left(t\right)={\mathcal{E}}_{0}sin\left(\omega t\right)$ probing the probe qubit. The resonator is ultrastrongly coupled with one qubit (at positive detuning $\delta ={\omega }_{\mathrm{q}}-{\omega }_{\mathrm{c}}=0.7{\omega }_{\mathrm{c}})$ and at the P qubit (at negative detuning ${\delta }^{\prime }={\omega }_{{\mathrm{q}}^{\prime }}-{\omega }_{\mathrm{c}}=-0.6{\omega }_{\mathrm{c}})$ with coupling rates $g=0.5{\omega }_{\mathrm{c}}$ and $g^{\prime }=0.2{\omega }_{\mathrm{c}}$, respectively. The flux offset applied to the qubit coupled with the resonator corresponds to $\theta =\pi /4$, while it is zero for the P qubit $\left(\theta =0\right)$. Loss rates for the cavity and the qubits are ${\gamma }_{\mathrm{c}}={\gamma }_{\mathrm{q}}={\gamma }_{{\mathrm{q}}^{\prime }}=5\times {10}^{-4}{\omega }_{\mathrm{c}}$. The lowest energy peak corresponds to the dressed resonance frequency ${\omega }_{\mathrm{q}}^{\prime }$ of the P qubit. The higher energy peak arises from the excitation of the quantum field (the resonator in the USC regime) interacting with the P qubit.

Figure 4

Sketch of the relevant dressed energy levels and electric-dipole transitions.