Phase-space theory for dispersive detectors of superconducting qubits

Motivated by recent experiments, we study the dynamics of a qubit quadratically coupled to its detector, a damped harmonic oscillator. We use a complex-environment approach, explicitly describing the dynamics of the qubit and the oscillator by means of their full Floquet state master equations in phase-space. We investigate the backaction of the environment on the measured qubit and explore several measurement protocols, which include a long-term full readout cycle as well as schemes based on short time transfer of information between qubit and oscillator. We also show that the pointer becomes measurable before all information in the qubit has been lost.

Figure 1

Simplified circuit consisting of a qubit with one Josephson junction (phase $\gamma$, capacitance $C_q$, and inductance $L_q$) inductively coupled to a SQUID with two identical junctions (phases ${\gamma }_{1,2}$ and capacitance $C_S$) and inductance $L_S$. The SQUID is driven by an ac bias current $I_B\left(t\right)$ and the voltage drop is measured by a voltmeter with internal resistance $R$. The total flux through the qubit loop is ${\Phi }_q$ and through the SQUID is ${\Phi }_S$.

Figure 2

Probability density of momentum $P(p_0,t)$ (a), snapshots of it at different times (b), and expectation value of momentum for the two different qubit states (c). Here $\hslash \Omega ∕\left(k_{B}T\right)=2$, $\Delta ∕\Omega =0.45$, $\kappa ∕\Omega =0.025$ and $\hslash \nu ∕\left(k_{B}T\right)=1.9$, and ${\overline{p}}_0$ is the dimensionless momentum $p_0∕\sqrt{k_{B}Tm}$.

Figure 3

Discrimination time as function of the coupling strength between qubit and oscillator. Here $\hslash \Omega ∕\left(k_{B}T\right)=2$, $\kappa ∕\Omega =0.025$, and $\hslash \nu ∕\left(k_{B}T\right)=1.95$.

Figure 4

Dephasing rate dependence on driving: dependence on $\Delta$ for different driving strengths $F_0$ ($\kappa ∕\Omega ={10}^{-4}$ and $\nu =2\Omega$). Top inset: dependence of the decoherence rate on $F_0$ for different values of $\kappa$ ($\Delta ∕\Omega =5\times {10}^{-2}$ and $\nu =2\Omega$). Bottom inset: dependence of the decoherence rate on driving frequency $\nu$ for different vales of $\kappa$ $(\Delta ∕\Omega =0.5)$. Here $\hslash \Omega ∕k_{B}T=2$ and ${\overline{F}}_0$ is the dimensionless force $F_0\hslash ∕\left(k_{B}T\sqrt{m{k}_{B}T}\right)$.

Figure 5

Comparison of dephasing and/or measurement rate. Open symbols: $1∕\left({\tau }_{\varphi }\Omega \right)$ and filled symbols: $1∕\left({\tau }_m\Omega \right)$. Here $\nu =2\Omega$, $\kappa ={10}^{-4}\Omega$, and $\hslash \Omega ∕\left(k_{B}T\right)=2$. Inset: dependence on $\kappa$, ${\overline{F}}_0=200$. The crossings, conflicting with the quantum limit (Ref. 14), are signaling the limits of the Born approximation, as described in text.

Figure 6

Comparison of dephasing and/or discrimination rates. Open symbols: $1∕\left({\tau }_{\varphi }\Omega \right)$ and filled symbols: $1∕\left({\tau }_{\mathrm{discr}}\Omega \right)$. Here $\nu =1.1\Omega$, ${\overline{F}}_0=200$, and $\hslash \Omega ∕\left(k_{B}T\right)=2$. Inset: dependence on $\kappa$ for $\Delta ∕\Omega =0.1$.