Role of relaxation in the quantum measurement of a superconducting qubit using a nonlinear oscillator

We analyze the relaxation of a superconducting flux qubit during measurement. The qubit state is measured with a nonlinear oscillator driven across the threshold of bifurcation, acting as a switching dispersive detector. This readout scheme is of quantum nondemolition type. Two successive readouts are used to analyze the evolution of the qubit and the detector during the measurement. We introduce a simple transition rate model for characterizing the qubit relaxation and the detector switching process. Corrected for qubit relaxation the readout fidelity is at least 95%. Qubit relaxation strongly depends on the driving strength and the state of the oscillator.

Figure 1

(a) Qubit and readout circuit. (b) Bistability diagram of the oscillator for the two qubit states $|g⟩$ and $|e⟩$. (c) Oscillator driving amplitude for qubit readout. (d) Oscillator switching probability $P\left(H\right)$ as a function of the switching plateau driving amplitude $I_{\text{sw}}$ for the qubit prepared initially with a Rabi pulse, $0\left(g\right)$, $\pi \left(e\right)$, and $3\pi /2\left(s\right)$. (e) First derivative of $P\left(H\right)\left(I_{\text{sw}}\right)$.

Figure 2

(a) Oscillator driving sequence used to analyze the qubit relaxation. In the first driving pulse the oscillator is set in either $L$ or $H$ (see text). (b) Decay of the probability for the measurement outcome $L$ (due to the qubit relaxation). (c) Qubit relaxation rates versus the oscillator driving amplitude $I_{\text{drive}}$ for the oscillator in the state $L$ or $H$.

Figure 3

(a) Oscillator driving amplitude. (b) Switching probability as a function of the duration of the switching plateau (solid line). (c) Effective potential barrier between $L$ and $H$, for the qubit state $|g⟩$ or $|e⟩$. (d) Schematic of the model used to describe the evolution of the qubit and the oscillator during the readout pulse, including qubit relaxation rates ${\Gamma }_{\downarrow }^L$ and ${\Gamma }_{\downarrow }^H$ and oscillator switching rates ${\Gamma }_{\text{sw}}^e$ and ${\Gamma }_{\text{sw}}^g$. $P_0^g\left(H\right)$, $P_0^e\left(H\right)$, $P_0^e\left(e,H\right)$, and $P_0^e\left(e,L\right)$ describe the regime I (see text).

Figure 4

(a) Sequence of two successive readouts used to analyze the qubit state after the first measurement. (b) Conditional probability for the measurement outcome $L$ after a switching event in the first readout. (c) $P_0^g\left(H\right)$, $P_0^e\left(H\right)$, and $P_0^e\left(e,H\right)$. (d) Oscillator switching rates ${\Gamma }_{\text{sw}}^e$ and ${\Gamma }_{\text{sw}}^g$, qubit relaxation rates ${\Gamma }_{\downarrow }^L$ and ${\Gamma }_{\downarrow }^H$ as a function of the switching plateau driving amplitude $I_{\text{sw}}$.