Noise Spectroscopy Using Correlations of Single-Shot Qubit Readout

A better understanding of the noise causing qubit decoherence is crucial for improving qubit performance. The noise spectrum affecting the qubit may be extracted by measuring dephasing under the application of pulse sequences but requires accurate qubit control and sufficiently long relaxation times, which are not always available. Here, we describe an alternative method to extract the spectrum from correlations of single-shot measurement outcomes of successive free induction decays. This method only requires qubit initialization and readout with a moderate fidelity and also allows independent tuning of both the overall sensitivity and the frequency region over which it is sensitive. Thus, it is possible to maintain a good detection contrast over a very wide frequency range. We discuss using our method for measuring both $1/f$ noise and the fluctuation spectrum of the nuclear bath of GaAs spin qubits.

Figure 1

Measurement cycle: Qubit initialization (I), evolution (E) for time $\tau$, and measurement (M) of the outcome. The delay time between two evolutions is $\Delta t$. While the initialization and measurement time set a lower limit on $\Delta t$, the idle time can be varied to adjust the delay between successive evolutions.

Figure 2

(a) Autocorrelation of return probability $P$ for evolution times from 50 ns (black curve) to $5\mu \mathrm{s}$ (blue curve) and exponential cutoff ($\gamma =1$) for the spectrum in Eq. (5). Note that, for the dotted part of the blue curve, $\Delta t<\tau$ and thus the correlation function does not exist. (b) Exponent of the dominant term of Eq. (2). Note the three different regimes $\Delta t\ll 1/{\omega }_E$, $1/{\omega }_E\ll \Delta t\ll 1/{\omega }_L$, and $1/{\omega }_L\ll \Delta t$ corresponding to $\Delta t^2$, $\Delta t^1$, and $\Delta t^0$ behavior, respectively, as discussed in the text. Short evolution times $\tau {\omega }_E\ll 1$ result in a prefactor ${\tau }^2$ and show a qualitatively similar behavior.

Figure 3

(a) Autocorrelation function for the singlet-triplet qubit. Adjusting the evolution time allows for detection of the crossover and slope of curves with different cutoff behavior, thus providing well distinguishable characteristics on an experimentally measurable scale. The lowest (blue) curve shows the autocorrelation in the absence of a cutoff. For $\Delta t>600\mu \mathrm{s}$, the scale is semilogarithmic. Note that the plateau at large $\Delta t$ directly reflects the prefactor of the $1/{\omega }^2$ region of the spectrum. (b) Autocorrelation function for the $1/f^{\alpha }$ spectrum. For $\tau \ll \Delta t\ll 1/{\omega }_L$, exact $1/f$ noise leads to a constant result. Deviations from this behavior (curves with negative and positive slope) allow for identification of $\alpha$.