Dissipation and ultrastrong coupling in circuit QED

Cavity and circuit QED study light-matter interaction at its most fundamental level. Yet, this interaction is most often neglected when considering the coupling of this system with an environment. In this paper, we show how this simplification, which leads to the standard quantum optics master equation, is at the root of unphysical effects. Including qubit relaxation and dephasing, and cavity relaxation, we derive a master equation that takes into account the qubit-resonator coupling. Special attention is given to the ultrastrong coupling regime, where the failure of the quantum optical master equation is manifest. In this situation, our model predicts an asymmetry in the vacuum Rabi splitting that could be used to probe dephasing noise at unexplored frequencies. We also show how fluctuations in the qubit frequency can cause sideband transitions, squeezing, and Casimir-like photon generation.

Figure 1

Excess in the mean photon number due to relaxation in the steady state of the ultrastrong qubit-resonator system. Initially, the system is in its true ground state $\widetilde{|g0\rangle }$, but, under the standard master equation (14), relaxation unphysically excites the system even at $T=0$. The black line, which corresponds to the left axis, represents the number of additional photons introduced in steady state by dissipation. The red dots, associated to the right axis, designate one minus the fidelity of the Rabi ground state $\widetilde{|g0\rangle }$ to the vacuum state $|g0\rangle$. The parameters are ${\omega }_a/2\pi ={\omega }_r/2\pi =6$ GHz, $\kappa /2\pi ={\gamma }_1/2\pi =0.1$ MHz, and no pure dephasing. Inset: mean photon number as a function of time for the system starting in its ground state with $g/2\pi =2$ GHz. In both the main plot and the inset, the blue dashed line indicates results for the fidelity and the photon number as obtained with the master equation presented in Sec. 3b

Figure 2

Transitions driven by noise. $X$ and ${\sigma }_x$ baths can only generate transitions between states of different parity. The ${\sigma }_z$ bath can generate transitions between any pair of levels of same parity.

Figure 3

Matrix elements under the RWA in the subspace $\left\{\right|g0\rangle ,|1-\rangle ,|1+\rangle \}$. (a) Relaxation matrix elements. Eigenstates are a mixture of qubit and resonator states, with an angle ${\theta }_1$. The fraction of the relaxation rate that comes from the qubit or the resonator bath is determined by the projection of the eigenstate on the qubit $|e0\rangle$ or resonator $|g1\rangle$ axis. (b) Dephasing matrix elements. The dephasing Hamiltonian rotates state vectors around the $|e0\rangle$ axis. Resulting vectors have a projection on the orthogonal eigenstate in the same doublet. This generates transitions between $|1,+\rangle$ and $|1,-\rangle$ if ${\theta }_1>0$.

Figure 4

Vacuum Rabi splitting. (a) Transition rates involved in the perturbative calculation in the three-level approximation. (b) Schematic plot of $\mathrm{Im}{\langle a\rangle }_s$ as a function of ${\omega }_d$. In general, the result is not symmetric.

Figure 5

Photon generation due to dephasing with the Lindbladian Eq. (16). Full line: white noise. Dotted line: white noise with a cut-off frequency increasing from bottom to top. For the bottom dotted line the cutoff is such that only transitions up to $\widetilde{|i,\pm \rangle }$ for $i=2$ are driven. For the top curve, transitions from $\widetilde{|g0\rangle }$ through $i=8$ are driven. Inset: photon generation rate $\beta$ as a function of $g$ for white noise. Points: numerical results. Line: perturbation theory Eq. (46). The parameters are ${\omega }_a/2\pi ={\omega }_r/2\pi =6$ GHz, $g/2\pi =1$ GHz, and ${\gamma }_{\phi }/2\pi =1$ MHz.

Figure 6

(a) Types of transitions allowed in the Bloch-Siegert regime ($g\ll \Sigma$). Red: transitions driven by the (even) ${\sigma }_z$ bath. Black: transitions induced by the (odd) $X$ and ${\sigma }_x$ baths. (b) Exact energy levels of the Rabi Hamiltonian with increasing coupling strength, obtained numerically. Crossings between levels in the spectrum lead to pairs of transitions with equal frequency. However, since in each of these pairs, one transition is even and the other is odd, they belong to different baths and these overlaps are not relevant for the master equation. Parameters are ${\omega }_r/2\pi ={\omega }_a/2\pi =6$ GHz.