Characterization of a quasistatic environment with a qubit

We consider a qubit initialized in a superposition of its pointer states, exposed to pure dephasing due to coupling to a quasistatic environment, and subjected to a sequence of single-shot measurements projecting it on chosen superpositions. We show how with a few such measurements one can significantly diminish one's ignorance about the environmental state, and how this leads to an increase of coherence of the qubit interacting with a properly post-selected environmental state. We give theoretical results for the case of a quasistatic environment that is a source of an effective field of Gaussian statistics acting on a qubit, and for a nitrogen-vacancy center qubit coupled to a nuclear spin bath, for which the Gaussian model applies qualitatively provided one excludes from the environment nuclei that are strongly coupled to the qubit. We discuss the reason for which the most probable sequences of measurement results are the ones consisting of identical outcomes, and in this way we shed light on a recent experiment (D. D. Bhaktavatsala Rao et al., arXiv:1804.07111) on nitrogen-vacancy centers.

Figure 1

(Nonnormalized) probability density function of the value $P_h$ over the interval $\left(0,1\right)$.

Figure 2

Distributions of the effective frequency $\mathrm{\Delta }{\omega }_{\alpha },$ generated for $N=20$ nuclear spins located farther than 0.57 nm from the qubit, with “high field” results corresponding to $B_z=100\mathrm{mT}$. (a) Direct counting with resolution of $5\times {10}^{-5}\mu \mathrm{eV}$. (b) Coarse graining with 40 bins (symbols) and Gaussian fits (lines). These results are qualitatively representative for NV center not having nuclei closer than $\approx 0.5\mathrm{nm}$ from it, i.e., for the majority of spatial arrangements of the nuclei the distributions are well fit by Gaussians. An example of an outlier spatial arrangement that does not lead to such distribution is shown in Fig. 3.

Figure 3

(a) An example of spatial realization of environment consisting of $N=20$ nuclear spins located at least 0.57 nm from the qubit that leads to non-Gaussian character of distribution of $\mathrm{\Delta }\omega$. In (a) $R$ is the distance from the qubit, and $\theta$ is the angle between the vector connecting the nucleus to the qubit and the $z$ axis. The color scale corresponds to the qubit-nucleus coupling $v_k$ (in arbitrary units). The contour lines represent isolevels of magnitude of $v_k$ separated by steps equal to $10\%$ of the maximum value of $v_k$ visible in the figure. The presence of two nuclei located close to one another, and both about 0.6 nm from the qubit, marked by red circle in the lower left part of the panel, leads to a prominently non-Gaussian shape of the coarse-grained distributions of $\mathrm{\Delta }\omega$ shown in (b) (blue solid line for zero field, red dash-dotted line for $B_0=100\mathrm{mT}$ and bin width of $1\times {10}^{-4}\mu \mathrm{eV}$). Coarse-grained distributions with influence of these two spins removed are shown as dashed and dotted lines.

Figure 4

Probability of obtaining 16 measurement sequences $M_j$ for $n=4$ given by Eq. (51) in zero field and high field for $\tau =2.7\mu \mathrm{s}$ and $\tau =13.5\mu \mathrm{s}$, obtained using Gaussian approximation from Eq. (48), and parameters $\sigma$ of Gaussian distributions fit to data from Fig. 2.

Figure 5

Probabilities of obtaining all possible measurement sequences of four consecutive measurements for zero field (blue line, upper panels) and high field (red line, lower panels) cases for $N=20$ spins in the environment with $\tau =2.7\mu \mathrm{s}$ in the left panels, and $\tau =13.5\mu \mathrm{s}$ in the right panels. Different spatial arrangements of spins in the environment are labeled as sample $1\cdots 5$. For longer $\tau =13.5\mu \mathrm{s}$ the symmetry is more prominent and all the spatial arrangements of the nuclei give essentially the same result.

Figure 6

Probabilities of obtaining sequences $M_0$ of identical measurement results (all projections on $|+x\rangle$) versus number $n$ of measurements for zero field (blue) and high field (red) cases. The symbols correspond to an exact calculation for spatial arrangement of the nuclei used to generate Figs. 2 and 4, and the solid lines are the results of Gaussian approximation in large $n$ limit, Eq. (52). The dashed line shows the prediction of ${\mathcal{P}}_n\left(M_j\right)\propto 1/\sqrt{n}$ from Eq. (54) for the high field case.

Figure 7

Adaptive probability of getting an identical sequence $M_0$ in zero field (blue) and high field (red) regimes. The cross signs represent an exact calculation for spatial arrangement of the nuclei used to generate Figs. 2 and 4 and the solid lines are the results of Gaussian approximation in large $n$ limit, Eq. (55). In the inset, the probabilities over path length [the number of $|-x\rangle$ (failures) of measuring 100 with ${\widehat{P}}_x$] is displayed. The results are plotted for $\tau =13.5\mu \mathrm{s}$ corresponding to long time regime, in which both $M_0$ and $M_{2^n-1}$ sequences are the most probable ones [see Eq. (54)], as one can see in the inset.

Figure 8

Coarse-graining distributions of the effective frequency $\mathrm{\Delta }{\omega }_{\alpha }$ with post-selection Eq. (57) for our choice of $\tau =2.7\mu \mathrm{s}$. The similar distribution for longer $\tau =13.5\mu \mathrm{s}$ is exemplified in the lower panel. The number of measurements is $n=4$.

Figure 9

Signals correspond to the expectation value $\langle {\widehat{\sigma }}_x\left(t\right)\rangle$ with for the initial and the post-selected state of the environment from exact calculation and Gaussian approximation calculation at (a) zero field and (b) high field ($B=100\mathrm{mT}$).

Figure 10

Signals in Eq. (60), expected from all possible sequences $M_j,$ grouped according to their path length $k\left(j\right)=k$ with $n=4,\tau =2.7\mu \mathrm{s}$ in the zero field case.

Figure 11

Signals correspond to the expectation value $\langle {\widehat{\sigma }}_x\left(t\right)\rangle$ in exact calculation (top panel) and calculation within Gaussian approximation from Eq. (59) (bottom panel) of $\langle {\widehat{\sigma }}_x\left(t\right)\rangle$ with $N=20$ and $\tau =2.7\mu \mathrm{s}$, versus the number of measurements $n$ for zero field case. The analogous figures for the high field case are very similar. The solid lines are the exact isocoherence lines corresponding to $W=0.5$, while the dashed lines are obtained from Eq. (62).