Quantum Nondemolition Measurement Enables Macroscopic Leggett-Garg Tests

We show how a test of macroscopic realism based on Leggett-Garg inequalities (LGIs) can be performed in a macroscopic system. Using a continuous-variable approach, we consider quantum nondemolition (QND) measurements applied to atomic ensembles undergoing magnetically driven coherent oscillation. We identify measurement schemes requiring only Gaussian states as inputs and giving a significant LGI violation with realistic experimental parameters and imperfections. The predicted violation is shown to be due to true quantum effects rather than to a classical invasivity of the measurement. Using QND measurements to tighten the “clumsiness loophole” forces the stubborn macrorealist to recreate quantum backaction in his or her account of measurement.

Figure 1

Schematic representation of the Leggett-Garg test. An observable $Q$ is measured at different times $t_i$ (represented along the horizontal axis), giving results $Q_i$. Macrorealism assumes noninvasive measurements, with the consequence that correlations, e.g., $C_{13}=⟨Q_1{Q}_3⟩$, are equal, independently of which sequence was performed to obtain them. $S_1$ and $S_3$, which differ by the presence or absence of $Q_2$, give the same $C_{13}$ in macrorealism but not in quantum mechanics. LGIs can detect macrorealism violations using experimentally observed correlations.

Figure 2

Numerical evaluation of the reduced LGIs as a function of $\theta$. (a) $K_n^{\prime }$ for (from top to bottom) $n=3$, 5, 7, 9 in the presence of scattering. (b) Numerical evaluation of $K_9^{\prime }$. The two lower blue (upper green) curves are results with (without, i.e., ${\mathcal{B}}_{\mathrm{at}}=0$) backaction in Eq. (7). Solid (dashed) lines show results with (without) scattering. All plots are obtained using the same parameters, taken from Ref. [25] (see text).

Figure 3

Three-point LGI violations within longer measurement sequences. Upper and lower curves show $K_3$ vs $\theta$ for optimized seven- and nine-measurement sequences, respectively. Both plots are obtained for $N_L=5\times {10}^8$ including scattering and loss effects. Spirals illustrate optimal sequences in the seven-measurement protocol with delay $\theta =\pi /2$, for which $K_3=C_{35}+C_{37}+C_{57}+1$. Hollow orange circles indicate measurements used to compute correlators in $K_3$; filled green circles indicate measurements performed but discarded.