Observation of nonclassical measurement statistics induced by a coherent spin environment

We demonstrate the role of measurement back action of a coherent spin environment in the dynamics of a spin (qubit) coupled to it, by inducing nonclassical (quantum random walk–like) statistics on its measurement trajectory. We show how the long lifetime of the spin bath correlates with measurements of the qubit over many repetitions. We use nitrogen vacancy centers in diamond as a model system and projective single-shot readout of the electron spin at low temperatures to simulate these effects. The employed measurement control could be used for purification of spin ensembles to desired quantum states.

Figure 1

(a) Schematic of the open-system model, where a central spin (NV) is coupled to a bath of nuclear spins. (b) The experimental scheme to test the nonclassicality in measurements induced by a long-lived (coherent) spin environment is shown through a circuit diagram. The NV spin is initialized in the same state, while the bath state is updated from previous (projective) measurements performed on the central spin. The NV spin evolves by interacting with a spin environment for a contact time $\tau$, through $U$, leading to phase evolution of its spin states.

Figure 2

The probability of finding the random walker at a position $x$ is shown for both (a) classical (CRW) and (c) quantum (QRW) walks. The corresponding measurement statistics for (b) CRWs and (d) QRWs displaying the probability of obtaining a measurement sequence $M_n$ for $N=4$ measurements performed after 20 steps of the walk.

Figure 3

The simulated probability of obtaining a measurement string $M_n$ for four measurements (top) is compared with the achievable relative purity $\left(P_R=\mathrm{Tr}\left[{\left({\rho }_B^{\left(M\right)}\right)}^2\right]/\mathrm{Tr}\left[{\left({\rho }_B\left(0\right)\right)}^2\right]\right)$ of the bath state (bottom) for the corresponding measurement string $M_n$. We have considered a spin bath with $N=7$ spins and couplings as given in the Appendix.

Figure 4

(a) Schematics of the level diagram and the pulse sequence used for experiments. (b) The free induction decay of the coherence of the NV spin $C\left(t\right)$ is plotted as a function of the time $t$. (c) The coincidence of finding a given measurement index $i$, corresponding to a measurement outcome which is the binary representation of $i$, is shown for four consecutive measurements. Measurements were performed at intervals of $\tau$. The theoretical simulations are detailed in the Appendix. (d) Left: Raw data for a measurement sequence (string) consisting of 246 measurement results (with evolution time $\tau =1.2\mu \mathrm{s}$) repeated 600 times. Violet color corresponds to the detection of at least one photon ($|b\rangle$), and yellow color symbolizes the measurement of no photon ($|d\rangle$), where $\left|b\right(d)\rangle =\frac{1}{\sqrt{2}}[|+1\rangle \pm |-1\rangle ]$. Right: Two measurement trajectories. (e) From the raw data shown in (d) we have extracted the probability of various measurement trajectories for the case of 40 consecutive measurements dashed blue line). In the same plot we also show the expected Gaussian statistics (solid red line) when all the measurements are equally probable, i.e., in the absence of back action.

Figure 5

(a) Raw data for a measurement sequence (string) consisting of 246 measurement results (with evolution time $\tau =1.2\mu \mathrm{s}$) repeated 600 times. Violet color corresponds to the detection of at least one photon ($|b\rangle$), and yellow color symbolizes the measurement of no photons ($|d\rangle$). (b) From the raw data we evaluate the probability of finding a similar measurement result (corresponding to spin state $|b\rangle$) after $n$ identical measurement results in any given measurement string described above.