Loophole-Free Interferometric Test of Macrorealism Using Heralded Single Photons

We show unambiguous violations of the different macrorealist inequalities, like the Leggett-Garg inequality (LGI) and its variant called Wigner’s form of the Leggett-Garg inequality (WLGI) using a heralded, single-photon-based experimental setup comprising a Mach-Zehnder interferometer followed by a displaced Sagnac interferometer. In our experiment, negative result measurements are implemented as control experiments, in order to validate the presumption of noninvasive measurability used in defining the notion of macrorealism. Among the experiments to date testing macrorealism, the present experiment stands out in comprehensively addressing the relevant loopholes. The clumsiness loophole is addressed through the precision testing of any classical or macrorealist invasiveness involved in the implementation of negative result measurements. This is done by suitably choosing the experimental parameters so that the quantum mechanically predicted validity of the relevant two-time no-signaling in time (NSIT) conditions is maintained in the three pairwise experiments performed to show the violation of LGI or WLGI. Furthermore, importantly, the detection efficiency loophole is addressed in our experimental scheme by adopting suitable modifications in the measurement strategy, enabling the demonstration of the violation of LGI or WLGI for any nonzero detection efficiency. We also show how other relevant loopholes like the multiphoton emission loophole, coincidence loophole, and the preparation state loophole are all closed in the present experiment. We report a LGI violation of $1.32\pm 0.04$ and a WLGI violation of $0.10\pm 0.02$ in our setup, where the magnitudes of violation are respectively 8 times and 5 times the corresponding error values, while agreeing perfectly with the ranges of quantum mechanically predicted values of the LGI and WLGI expressions that we estimate by taking into account the nonidealities of the actual experiment. At the same time, consistent with quantum mechanical predictions, the experimentally observed probabilities satisfy the two-time NSIT conditions up to the order of ${10}^{-2}$. Thus, the noninvasiveness in our implemented negative result measurement is convincingly upper bounded to ${10}^{-2}$.

Popular Summary

Einstein famously asked: do you really believe that the moon is not there when nobody looks? This profoundly debated question concerns the notion of realism, namely that a system with well-defined properties is in a definite state at any instant, even when not measured. Remarkably, this notion has been catapulted from the domain of speculation to experimental physics through the testable algebraic inequalities derived by Bell and Leggett-Garg, respectively. The former has enabled testing realism when combined with the idea of locality (i.e., no action at a distance), a concept known as local realism. A fully conclusive experimental falsification of local realism using photons has only recently been accomplished.

On the other hand, the Leggett-Garg inequality enables testing realism in conjunction with the idea that a measurement can be done, ensuring that the effect on the system state is arbitrarily small. It is this notion of classical realism that has been decisively refuted in this experiment through the violations of two different forms of such an inequality, alongside perfect matching with the quantum mechanical predictions incorporating experimental nonidealities. Importantly, this experiment using single photons closes for the first time all the relevant loopholes through the various ingeniously developed strategies, thereby complementing the recent loophole-free testing of local realism using photons.

Furthermore, the present interferometric experiment exhibiting the most general unambiguous signature of nonclassicality of the single-photon states can provide a powerful platform for harnessing this nonclassicality toward various applications, such as in quantum communication and quantum metrology, for which the single photon is a ubiquitous workhorse.

Figure 1

Basic schematic of the experimental setup. Here BS1, BS2, BS3 are beam splitters; ${\theta }_1$, ${\theta }_2$ are phase modulators.

Figure 2

Schematic of the modified experimental setup. Here HWP1, HWP2, and HWP3 are the half-wave plates; PBS1 and PBS2 are the polarizing beam splitters; $L1$, $L4$, $L5$, and $L6$ are the focusing lens; $F1$ is the long-pass filter; $M$ is the dielectric mirror; $L2$ and $L3$ are the collimating lens; $F2$, $F3$, and $F4$ are the band-pass filters; $B1$, $B2$, $B3$, and $B4$ are the blockers; NPBS is the nonpolarizing beam splitter; and SPAD1, SPAD2$+$, and SPAD2$-$ are the single-photon avalanche detectors. Two arms of the AMZI are marked as 1 and 2, which represent the $+1$ and $-1$ arms, respectively. Similarly, two arms of the DSI are marked as 3 and 4, representing $-1$ and $+1$. SPAD2+ and $\mathrm{SPAD2}-$ are placed in the $+1$ and $-1$ arms, respectively.

Figure 3

The overall “hidden-variable” state space is $\mathrm{\Lambda }$. Its subspaces ${\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2,{\mathrm{\Lambda }}_3$ for which the photon is detected at $t_1,t_2,t_3$ are respectively represented by blue, green, and red circles. All the detectors are assumed to have the same detection efficiency. Different subspaces within ${\mathrm{\Lambda }}_1\cup {\mathrm{\Lambda }}_2\cup {\mathrm{\Lambda }}_3$ are denoted by the symbols $q,p,s,a,b,c,d$.

Figure 4

Splitting ratios of the NPBS are denoted as $T_i:R_i$ for $i=1,2,3,4$, depending on the input port of the NPBS. Here, $T_i+R_i=1$ for all $i$.

Figure 5

Plot for $\mathrm{SD}/M$ versus the number of iterations. The blue curve represents the interference case, whereas the red curve represents the noninterference case.