Prediction of Short Time Qubit Readout via Measurement of the Next Quantum Jump of a Coupled Damped Driven Harmonic Oscillator

The dynamics of the next quantum jump for a qubit [two level system] coupled to a readout resonator [damped driven harmonic oscillator] is calculated. A quantum mechanical treatment of readout resonator reveals nonexponential short time behavior which could facilitate detection of the state of the qubit faster than the resonator lifetime.

Figure 1

Graph of the logarithmic norm of the system vs $\tau =\kappa t$: $\log [W(\tau )]/\overline{n}=-\tau +2\alpha (\tau )/\sqrt{\overline{n}}+{\alpha }^2(\tau )/\overline{n}$. It shows that the time evolution changes from ${\tau }^3$ to $3-\tau$ at times $t=O(1/\kappa )$. A precise determination of the short-time behavior of the norm and the next quantum jump rate is essential for qubit readouts at times $t\lesssim 1/\kappa$.

Figure 2

Comparison of error for qubit state detection for dispersive homodyne readout ${\epsilon }_{\mathrm{DR}}$ and next jump readout $\epsilon$. The dimensionless time is $\tau =\kappa t$. The error for the next jump detection decreases with increasing dispersion and for this curve we have taken $\chi /\kappa =20$; for dispersive readout $\chi /\kappa =1/2$, which optimizes the SNR. In each case the resonator drive is set to $\overline{n}=100$. For dispersive homodyne readout the error becomes very small at times longer than the resonator lifetime. For coherent qubit detection via the next jump the error is minimized at short time and becomes 50% at long times (not shown). However, at very short time $\chi t<1$ the error $\epsilon$ is large as the dispersion prevents one from distinguishing whether the atom is in $|G⟩$ or $|B⟩$.

Figure 3

The scaled logarithmic decrement $Y$ of probability for the next quantum jump as a function of time.