Two-resonator circuit quantum electrodynamics: Dissipative theory

We present a theoretical treatment for the dissipative two-resonator circuit quantum electrodynamics setup referred to as quantum switch. There, switchable coupling between two superconducting resonators is mediated by a superconducting qubit operating in the dispersive regime, where the qubit transition frequency is far detuned from those of the resonators. We derive an effective Hamiltonian for the quantum switch beyond the rotating-wave approximation and provide a detailed study of the dissipative dynamics. As a central finding, we derive analytically how the qubit affects the quantum switch even if the qubit has no dynamics, and we estimate the strength of this influence. The analytical results are corroborated by numerical calculations, where coherent oscillations between the resonators, the decay of coherent and Fock states, and the decay of resonator-resonator entanglement are studied. Finally, we suggest an experimental protocol for extracting the damping constants of qubit and resonators by measuring the quadratures of the resonator fields.

Figure 1

(Color online) Sketch of the two-resonator circuit QED system under analysis, including schematically the interaction with bosonic heat baths (blue boxes with oscillator potentials). Either microstrip or coplanar waveguides could be employed as resonators (red lines). As in the text, the coupling qubit is exemplarily depicted as persistent-current flux qubit (green loop). The system is coupled to external circuits via coupling capacitors. The decay rates ${\kappa }_{\left\{A,B\right\}}$, $\gamma$ and ${\gamma }_{\varphi }$ are defined in Sec. 2C.

Figure 2

(Color online) Switch setting coefficient $g_{\text{SW}}$ (qubit in the ground state) for the RWA (blue dash-dotted line) and non-RWA case, Eq. (30) (red dashed line), compared to numerical data obtained from the diagonalization of Hamiltonian (1) (black solid line). All quantities are plotted against the qubit flux bias ${\Phi }_x^{\text{DC}}$. Parameters are $\Omega /2\pi =3.5\text{GHz}$, ${\delta }_Q/2\pi =4\text{GHz}$, $G/2\pi =0.0022\text{GHz}$, and $g/2\pi =0.24\text{GHz}$.

Figure 3

(Color online) (a) Rabi oscillations between the resonators at the effective interaction strength $g_{\text{GSW}}/2\pi =0.00315\text{GHz}$ (numerical value, analytically we find $g_{\text{GSW}}/2\pi =0.00328\text{GHz}$): Occupation numbers of resonator $A$ (solid red lines) and $B$ (dashed blue lines). The initial state is ${|g⟩}_{\text{qb}}{|1⟩}_A{|0⟩}_B$. (b) Time evolution of the qubit state in terms of $⟨{\sigma }_z⟩$ (solid black lines). The deviation from the ground state always lies below 0.5%. Numerical parameters: $\Omega /2\pi =3.5\text{GHz}$, ${\delta }_Q/2\pi =4\text{GHz}$, $\epsilon /2\pi =-6.37\text{GHz}$, $g/2\pi =0.24\text{GHz}$, $G/2\pi =0.0022\text{GHz}$, $T=20\text{mK}$. Decay rates: $J_A\left(\Omega \right)/2\pi =J_B\left(\Omega \right)/2\pi =0.00035\text{GHz}$, $J_x\left(\Omega \right)/2\pi =J_z\left(\Omega \right)/2\pi =0.035\text{GHz}$.

Figure 4

(Color online) Decay of the quantum switch out of the initial state ${|g⟩}_{\text{qb}}{|1⟩}_A{|0⟩}_B$. Numerical data obtained with the Bloch-Redfield QME (9) and $N=3$ states in each resonator: Decay of the expectation values of the number operators corresponding to the “center of mass coordinate”, $⟨A_{+}^{†}{A}_{+}⟩$ (red solid lines, thicker curve) and “relative coordinate”, $⟨A_{-}^{†}{A}_{-}⟩$ (blue solid lines, thinner curve), compared to analytical estimates, Eqs. (38, 39) (black dashed and dash-dotted lines, respectively). Decay rates are according to Eq. (34). Parameters: see Fig. 3.

Figure 5

(Color online) Dependence of the parameter ${\lambda }_{\Delta }$ (a) and the effective damping rate $\kappa +2{\kappa }_{\text{qb}}$ (b) (black solid lines), on the qubit energy bias $\epsilon$. The rate for qubit-enhanced decay ${\kappa }_{\text{qb}}$ is given by Eq. (35). The numerically extracted damping rates [red crosses in (b)] are related to the decay of the expectation value of the number operator corresponding to the “center of mass coordinate”, $⟨A_{+}^{†}{A}_{+}⟩$, out of the initial state ${|g⟩}_{\text{qb}}{|1⟩}_A{|0⟩}_B$ at different $\epsilon$. The dotted lines mark the limit of validity of the dispersive theory (see text). Parameters are chosen as in Fig. 3.

Figure 6

(Color online) Decay of the quantum switch out of the initial coherent state $|\alpha ⟩$ with $\alpha =1$. Simulations were run with $N=6$ states in each resonator. Numerical data: occupations of the “center of mass coordinate”, $⟨A_{+}^{†}{A}_{+}⟩$ (red solid lines, oscillating curve) and “relative coordinate”, $⟨A_{-}^{†}{A}_{-}⟩$ (blue solid smooth line), compared to analytical estimates, Eqs. (38, 39) (black dashed and dash-dotted lines, respectively). Initial transient effects are not depicted in the figure. Parameters: see Fig. 3.

Figure 7

(Color online) Decay of concurrence $C_{+}$ and $C_{-}$ for the entangled initial states ${|{\psi }_{+}⟩}_{\text{cav}}$ and ${|{\psi }_{-}⟩}_{\text{cav}}$ (solid red lines, thick curve, and solid blue lines, thin curve, respectively). The switch-setting off condition $g_{\text{SW}}=0$ is fulfilled by $\epsilon /2\pi =-8.916\text{GHz}/2\pi$. Here we use $G=0.00478\text{GHz}$, all other parameters are as in Fig. 3. The exponential decay corresponds to Eqs. (44, 45) (black dashed and dash-dotted lines, respectively).

Figure 8

(Color online) Decay of concurrence $C_{+}^i$ for the entangled initial state ${|{\psi }_{+}^i⟩}_{\text{cav}}$ (red solid lines, thick curve). The switch-setting off condition $g_{\text{SW}}=0$ is fulfilled by $\epsilon /2\pi =-8.916\text{GHz}$. Here we use $G=0.00478\text{GHz}$, all other parameters as in Fig. 3. The two-mode exponential (solid black line) corresponds to Eq. (47). The additional exponentials are given by Eqs. (44, 45) and highlight the different decay regimes.

Figure 9

(Color online) Time evolution of the auto-correlations and cross-correlations of the initial state ${|g⟩}_{\text{qb}}{|\alpha =1⟩}_A{|\beta =0⟩}_B$: Decay of $⟨X_A^2⟩+⟨X_B^2⟩+2⟨X_A{X}_B⟩$ [panel (a), red solid lines, oscillating] and $⟨X_A^2⟩+⟨X_B^2⟩-2⟨X_A{X}_B⟩$ [panel (b), blue solid lines, oscillating]. The exponential decay is compared to the analytical estimates given by Eqs. (38, 39) (black solid lines). Initial transient effects are not depicted in the figure. For parameters and initial conditions, see Fig. 3, 6, respectively.

Figure 10

(Color online) Sketch of the relevant decay mechanisms affecting the quantum switch. The qubit induces extra decoherence in higher dispersive order to the effective resonator-resonator dynamics. The arrows mark the decoherence channels, labeled by the corresponding decay rates ${\kappa }_{\left\{A,B,\text{qb}\right\}}$. Decay processes of fourth order in the parameters ${\lambda }_{\left\{\Delta ,\Sigma ,\Omega \right\}}$ are not displayed individually.