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Journal of Optics

Experimental observation of the Leggett-Garg inequality violation in classical light

Xiong Zhang, Tao Li, Zhenwei Yang, Xiangdong Zhang

Published Nov 28, 2018• 9 citations
Publisher

Abstract

Abstract Leggett-Garg (LG) inequalities have been derived to clarify the validity of generalizing quantum mechanics to macroscopic systems for more than thirty years. They have been used as tools to investigate the existence of quantum coherence in macroscopic systems. Here, we show that such inequality violation can also be observed in classical optical systems. Using the linear polarization degree of freedom as the dichotomic observables, in a classical optical experiment, we have observed the violation of LG inequality by implementing the non-invasive measurement. Furthermore, the coherence witness, which is analogous to the quantum witness, has also been observed in the classical optical experiment. Our results not only provide new physical insights into the macroscopic quantum correlation and quantum witness, but also show their potential applications in correlated information processes.

DOI
10.1088/2040-8986/aaf4ae

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Abstract Leggett-Garg (LG) inequalities have been derived to clarify the validity of generalizing quantum mechanics to macroscopic systems for more than thirty years. They have been used as tools to investigate the existence of quantum coherence in macroscopic systems. Here, we show that such inequality violation can also be observed in classical optical systems. Using the linear polarization degree of freedom as the dichotomic observables, in a classical optical experiment, we have observed the violation of LG inequality by implementing the non-invasive measurement. Furthermore, the coherence witness, which is analogous to the quantum witness, has also been observed in the classical optical experiment. Our results not only provide new physical insights into the macroscopic quantum correlation and quantum witness, but also show their potential applications in correlated information processes.
The Leggett-Garg (LG) inequalities were introduced to test for the possible presence of macroscopic quantum coherence. Since such effects may be found in various types of macroscopic oscillators, we consider the application of the LG approach to the one-dimensional quantum harmonic oscillator and also to more general bound systems, using a single dichotomic variable QQQ given by the sign of the oscillator position. We present a simple method to calculate the temporal correlation functions appearing in the LG inequalities for any bound system for which the eigenspectrum is (exactly or numerically) known. We apply this result to the quantum harmonic oscillator for a variety of experimentally accessible states, namely, energy eigenstates, and superpositions thereof. For the subspace of states spanned by only the ground state and first excited state, we readily find substantial regions of parameter space in which the LG inequalities at two, three, and four times can each be independently violated or satisfied. We also find that the violations persist, although are reduced, when the sign function defining QQQ is smeared to reflect experimental imprecision. For higher-energy eigenstates, we find that LG violations diminish, showing the expected classicalization. With a QQQ defined using a more general type of position coarse graining, we find two-time LG violations even in the ground state. We also show that two-time LG violations in a Gaussian state are readily found if the dichotomic variable at one of the times is taken to be the parity operator. To demonstrate the versatility of the approach, we repeat much of the LG analysis for the Morse potential, finding qualitatively similar physical results.
Leggett-Garg (LG) tests for macrorealism were originally designed to explore quantum coherence on the macroscopic scale. Interference experiments and systems modeled by harmonic oscillators provide useful examples of situations in which macroscopicity has been approached experimentally and are readily turned into LG tests for a single dichotomic variable QQQ. Applying this approach to the double-slit experiment in which a noninvasive measurement at the slits is included, we exhibit LG violations. We find that these violations are always accompanied by destructive interference. The converse is not true in general and we find that there are nontrivial regimes in which there is destructive interference but the two-time LG inequalities are satisfied, which implies that it is in fact often possible to assign (indirectly determined) probabilities for the interferometer paths. Similar features have been observed in recent work involving a LG analysis of a Mach-Zehnder interferometer and we compare with those results. We extend the analysis to the triple-slit experiment again finding LG violations, and we also exhibit examples of some surprising relationships between LG inequalities and NSIT conditions that do not exist for dichotomic variables. For the simple harmonic oscillator, we find an analytically tractable example showing a two-time LG violation with a gaussian initial state, echoing in simpler form recent results of Bose et al. [Phys. Rev. Lett. 120, 210402 (2018)].
In contrast to the spatial Bell's inequalities which probe entanglement between spatially separated systems, the Leggett–Garg inequalities test the correlations of a single system measured at different times. Violation of a genuine Leggett–Garg test implies either the absence of a realistic description of the system or the impossibility of measuring the system without disturbing it. Quantum mechanics violates the inequalities on both accounts and the original motivation for these inequalities was as a test for quantum coherence in macroscopic systems. The last few years has seen a number of experimental tests and violations of these inequalities in a variety of microscopic systems such as superconducting qubits, nuclear spins, and photons. In this article, we provide an introduction to the Leggett–Garg inequalities and review these latest experimental developments. We discuss important topics such as the significance of the non-invasive measurability assumption, the clumsiness loophole, and the role of weak measurements. Also covered are some recent theoretical proposals for the application of Leggett–Garg inequalities in quantum transport, quantum biology and nano-mechanical systems.
Despite the great success of quantum mechanics, questions regarding its application still exist and the boundary between quantum and classical mechanics remains unclear. Based on the philosophical assumptions of macrorealism and noninvasive measurability, Leggett and Garg devised a series of inequalities (LG inequalities) involving a single system with a set of measurements at different times. Introduced as the Bell inequalities in time, the violation of LG inequalities excludes the hidden-variable description based on the above two assumptions. We experimentally investigated the single photon LG inequalities under decoherence simulated by birefringent media. These generalized LG inequalities test the evolution trajectory of the photon and are shown to be maximally violated in a coherent evolution process. The violation of LG inequalities becomes weaker with the increase of interaction time in the environment. The ability to violate the LG inequalities can be used to set a boundary of the classical realistic description.
The Leggett–Garg (LG) test of macroscopic realism involves a series of dichotomic non-invasive measurements that are used to calculate a function which has a fixed upper bound for a macrorealistic system and a larger upper bound for a quantum system. The quantum upper bound depends on both the details of the measurement and the dimension of the system. Here we present an LG experiment on a three-level quantum system, which produces a larger theoretical quantum upper bound than that of a two-level quantum system. The experiment is carried out in nuclear magnetic resonance and consists of the LG test as well as a test of the ideal assumptions associated with the experiment, such as measurement non-invasiveness. The non-invasive measurements are performed via the modified ideal negative result measurement scheme on a three-level system. Once these assumptions are tested, the violation becomes small, despite the fact that the LG value itself is large. Our results showcase the advantages of using the modified measurement scheme that can reach higher LG values, since these leave a larger margin for violating the inequality in the face of experimental imperfections.