Macrorealism is a characteristic feature of many, but not all, classical systems. It is known, for example, that classical light can violate a Leggett-Garg inequality and, hence, reject a macrorealist interpretation. A recent experiment has used entangled light and negative measurements to demonstrate a loophole-free test of macrorealism [PRX Quantum $\mathbf{3}$, 010307 (2022)]. This paper shows that such an experiment, while soundly rejecting macrorealism, may nevertheless be open to a classical interpretation. This is done by offering an explicit classical model of heralded photon detection in an optical interferometer with beam blockers. A numerical analysis of the model shows good agreement with experimental observations and consistency with both local realism and a rejection of macrorealism.
We assess the most macroscopic matter-wave experiments to date as to the extent to which they probe the quantum-classical boundary by demonstrating interference of heavy molecules and cold atomic ensembles. To this end, we consider a rigorous Bayesian test protocol for a parametrized set of hypothetical modifications of quantum theory, including well-studied spontaneous collapse models, that destroy superpositions and reinstate macrorealism. The range of modification parameters ruled out by the measurement events quantifies the macroscopicity of a quantum experiment, while the shape of the pos…
We define a quantum witness for high-mass matter-wave interferometers that allows us to test fundamental assumptions of macroscopic realism. We propose an experimental realisation using absorptive laser gratings and show that such systems can strongly violate a macrorealistic quantum-witness equality. The measurement of the witness can therefore provide clear evidence of physics beyond macrorealism for macromolecules and nanoparticles.
In the tests for macrorealism proposed by Leggett and Garg, the temporal correlation functions of a dichotomic variable Q must be measured in a non-invasive way to rule out alternative classical explanations of Leggett-Garg inequality violations. Ideal negative measurements, in which a null result is argued to be a non-invasive determination of the system's state, are often used. From a quantum-mechanical perspective, such a measurement collapses the wave function and will therefore typically be found to be invasive under any experimental check. Here, a simple modified ideal negative measureme…
Tests of local realism and macrorealism have historically been discussed in very similar terms: Leggett-Garg inequalities follow Bell inequalities as necessary conditions for classical behavior. Here, we compare the probability polytopes spanned by all measurable probability distributions for both scenarios and show that their structure differs strongly between spatially and temporally separated measurements. We arrive at the conclusion that, in contrast to tests of local realism where Bell inequalities form a necessary and sufficient set of conditions, no set of inequalities can ever be neces…
We investigate quantum violation of macrorealism for multilevel spin systems under the condition of coarsening of measurement times -- i.e., when measurement times have experimental indeterminacy. This is studied together with the effect of coarsening of measurement outcomes for which individual outcomes cannot be unambiguously discriminated. In our treatment, along with different measurement outcomes being clubbed together into two groups in order to model the coarsening of measurement outcomes, importantly, varying degrees of coarsening of measurement time intervals have also been considered…
Macrorealism is a classical world view asserting that the properties of macro-objects exist independently and irrespective of observation. One practical approach to test this view in quantum theory is to observe the quantum coherence for macro-object in an interference experiment. An elegant and conceptually appealing approach for testing the notion of macrorealism in quantum theory is through the violation of Leggett-Garg inequality. However, a conclusive Leggett-Garg test hinges on how the non-invasive measurability criteria is guaranteed in an experiment and remains a debated issue till dat…
Fluctuations of the work performed on a driven quantum system can be characterized by the so-called fluctuation theorems. The Jarzynski relation and the Crooks theorem are famous examples of exact equalities characterizing non-equilibrium dynamics. Such statistical theorems are typically formulated in a similar manner in both classical and quantum physics. Leggett-Garg inequalities are inspired by the two assumptions referred to as the macroscopic realism and the non-invasive measurability. Together, these assumptions are known as the macrorealism in the broad sense. Quantum mechanics is prova…
Standard Leggett and Garg inequalities (SLGIs) were formulated for testing the incompatibility between the classical worldview of macrorealism and quantum mechanics. In recent times, various other formulations, such as Wigner form of LGIs (WLGIs), entropic LGIs (ELGIs) and the no-signaling in time (NSIT) condition have also been proposed. It is also recently argued that no set of SLGIs can provide the necessary and sufficient conditions for macrorealism but a suitable conjunction of NSIT conditions provides the same. In this paper, we first provide a comparative study of the various formulatio…
According to the world view of macrorealism, the properties of a given system exist prior to and independent of measurement, which is incompatible with quantum mechanics. Leggett and Garg put forward a practical criterion capable of identifying violations of macrorealism, and so far experiments performed on microscopic and mesoscopic systems have always ruled out in favor of quantum mechanics. However, a macrorealist can always assign the cause of such violations to the perturbation that measurements effect on such small systems, and hence a definitive test would require using non-invasive mea…
We consider two different types of conditions which were proposed to test macrorealism in the context of a system described by a single dichotomic variable Q. This is the view that a macroscopic system evolving in time possesses definite properties which can be determined without disturbing the future or past state. The Leggett-Garg (LG) inequalities, the most commonly-studied test, are only necessary conditions for macrorealism, but building on earlier work it is shown that when the four three-time LG inequalities are augmented with a certain set of two-time inequalities also of the LG form, …
For multilevel spin systems, robustness of the quantum mechanical (QM) violation of macrorealism (MR) with respect to coarse grained measurements is investigated using three different necessary conditions of MR, namely, the Leggett-Garg inequality (LGI), Wigner's form of the Leggett-Garg inequality (WLGI) and the condition of no-signalling in time (NSIT). It is shown that for dichotomic sharp measurements, in the asymptotic limit of spin, the algebraic maxima of the QM violations of all these three necessary conditions of MR are attained. Importantly, the QM violations of all these persist in …
The rise of quantum information theory has lent new relevance to experimental tests for non-classicality, particularly in controversial cases such as adiabatic quantum computing superconducting circuits. The Leggett-Garg inequality is a "Bell inequality in time" designed to indicate whether a single quantum system behaves in a macrorealistic fashion. Unfortunately, a violation of the inequality can only show that the system is either (i) non-macrorealistic or (ii) macrorealistic but subjected to a measurement technique that happens to disturb the system. The "clumsiness" loophole (ii) provides…
We show unambiguous violations of different macrorealist inequalities, like the LGI and the WLGI using a heralded, single-photon based experimental setup comprising one Mach-Zehnder interferometer followed by a displaced Sagnac one. The negative result measurements (NRM) are implemented in order to validate the presumption of non-invasive measurability used in defining macrorealism. Among all the experiments to date testing macrorealism, the present experiment stands out in comprehensively addressing the relevant loopholes. The clumsiness loophole is addressed through precision testing of any …
The Bell and Leggett-Garg tests offer operational ways to demonstrate that non-classical behavior manifests itself in quantum systems, and experimentalists have implemented these protocols to show that classical worldviews such as local realism and macrorealism are false, respectively. Previous theoretical research has exposed important connections between more general Bell inequalities and polyhedral combinatorics. We show here that general Leggett-Garg inequalities are closely related to the cut polytope of the complete graph, a geometric object well-studied in combinatorics. Building on tha…
The Leggett-Garg inequality attempts to classify experimental outcomes as arising from one of two possible classes of physical theories: those described by macrorealism (which obey our intuition about how the macroscopic classical world behaves), and those that are not (e.g., quantum theory). The development of cloud-based quantum computing devices enables us to explore the limits of macrorealism in new regimes. In particular, here we take advantage of the properties of the programmable nature of the IBM quantum experience to observe the violation of the Leggett-Garg inequality (in the form of…
Conceptually different from the decoherence program, we present a novel theoretical approach to macroscopic realism and classical physics within quantum theory. It focuses on the limits of observability of quantum effects of macroscopic objects, i.e., on the required precision of our measurement apparatuses such that quantum phenomena can still be observed. First, we demonstrate that for unrestricted measurement accuracy no classical description is possible for arbitrarily large systems. Then we show for a certain time evolution that under coarse-grained measurements not only macrorealism but …
Mounting experimental and theoretical evidence suggests that coherent quantum effects play a role in the efficient transfer of an excitation from a chlorosome antenna to a reaction center in the Fenna-Matthews-Olson protein complex. However, it is conceivable that a satisfying alternate interpretation of the results is possible in terms of a classical theory. To address this possibility, we consider a class of classical theories satisfying the minimal postulates of macrorealism and frame Leggett-Garg-type tests that could rule them out. Our numerical simulations indicate that even in the prese…
Several recent papers have shown that some forms of dispersion cancellation have classical analogs and that some aspects of nonlocal two-photon interferometry are consistent with local realistic models. It is noted here that the classical analogs only apply to local dispersion cancellation experiments [A.M. Steinberg et al., Phys. Rev. Lett. 68, 2421 (1992)] and that nonlocal dispersion cancellation [J.D. Franson, Phys. Rev. A 45, 3126 (1992)] is inconsistent with any classical field theory and has no classical analog. The local models that have been suggested for two-photon interferometry are…
We propose classical interferometry with low-intensity thermal radiation for the estimation of nonclassical independent Gaussian processes in material samples. We generally determine the mean square error of the phase-independent parameters of an unknown Gaussian process, considering a noisy source of radiation the phase of which is not locked to the pump of the process. We verify the sufficiency of passive optical elements in the interferometer, active optical elements do not improve the quality of the estimation. We also prove the robustness of the method against the noise and loss in both i…
The interaction of a trapped ion with a laser beam in the strong excitation regime is analyzed. In this regime, a variety of non--classical states of motion can be prepared either by using laser pulses of well defined area, or by an adiabatic passage scheme based on the variation of the laser frequency. We show how these states can be used to investigate fundamental properties of quantum mechanics. We also study possible applications of this system to build an ion interferometer.
We report on a novel experiment to generate non-classical atomic states via quantum non-demolition (QND) measurements on cold atomic samples prepared in a high finesse ring cavity. The heterodyne technique developed for the QND detection exhibits an optical shot-noise limited behavior for local oscillator optical power of a few hundred \muW, and a detection bandwidth of several GHz. This detection tool is used in single pass to follow non destructively the internal state evolution of an atomic sample when subjected to Rabi oscillations or a spin-echo interferometric sequence.
We generalize the operational quasiprobability involving sequential measurements proposed by Ryu {\em et al.} [Phys. Rev. A {\bf 88}, 052123] to a continuous-variable system. The quasiprobabilities in quantum optics are incommensurate, i.e., they represent a given physical observation in different mathematical forms from their classical counterparts, making it difficult to operationally interpret their negative values. Our operational quasiprobability is {\em commensurate}, enabling one to compare quantum and classical statistics on the same footing. We show that the operational quasiprobabili…
Induced Coherence in Quantum and Classical Interferometry: Origin and Control
The Institute of Mathematical Sciences（马蒂马拉科学院）；Homi Bhabha National Institute（霍米·巴巴国立学院）；Institute of Science, Banaras Hindu University（贝拿勒斯印度教大学理学院）；Centre for Quantum Science and Technology, Siksha ‘O’ Anusandhan（锡克沙'O'安苏丹量子科学与技术中心）；Indian Institute of Science Education and Research (IISER), Kolkata（加尔各答印度科学教育与研究学院）
Macro-realism is a fundamental feature of classical world that contradicts with the quantum theory. An elegant method of testing macrorealism is to apply the Leggett-Garg inequality (LGI), but the non-invasivity of measurement is challenging in practice. In this work, we report the LGI violation of path observable in a composite interference experiment with coherent light. Experiment results confirm the occurrence of destructive interference providing per se as evidence of macro-realism violation. And by using an exact weak measurement model in the present experiment, the advantage that the vi…
We study Fock state interferometry, consisting of a Mach-Zehnder Interferometer with two Fock state inputs and photon-number-resolved detection at the two outputs. We show that it allows discrimination of a discrete number of apriori-known optical phase shifts with an error probability lower than what is feasible with classical techniques under a mean photon number constraint. We compare its performance with the optimal quantum probe for M-ary phase discrimination, which unlike our probe, is difficult to prepare. Our technique further allows discriminating a null phase shift from an increasing…
We introduce and implement an interferometric technique based on chirped femtosecond laser pulses and nonlinear optics. The interference manifests as a high-visibility (> 85%) phase-insensitive dip in the intensity of an optical beam when the two interferometer arms are equal to within the coherence length of the light. This signature is unique in classical interferometry, but is a direct analogue to Hong-Ou-Mandel quantum interference. Our technique exhibits all the metrological advantages of the quantum interferometer, but with signals at least 10^7 times greater. In particular we demonstrat…
The usual formulation of Macrorealism is recast to make this notion fully concurrent with the basic ideas behind classical physics. The assumption of non-invasiveness of measurements is dropped. Instead, it is assumed that the current state of the system defines full initial conditions for its subsequent evolution. An example of a new family of temporal Bell inequalities is derived which can be applied to processes in which the state of the system undergoes arbitrarily many transformations (which was not the case in the original approach). An exponential (in terms of number of operations) viol…
We propose an interferometry technique, by using electromagnetically induced transparency phenomena, for measuring classical force. The classical force is estimated by measuring the phase at the output of the interferometer. The proposed measurement mechanism satisfies quantum non-demolition measurement conditions leading to back-action evasion. We further derive the sufficient condition under which the thermal noise in the interferometer is negligible. With no back-action noise and no thermal noise, the sensitivity of this technique is limited by shot noise only.
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…
Quantum pre- and post-selection (PPS) paradoxes occur when counterfactual inferences are made about different measurements that might have been performed, between two measurements that are actually performed. The 3 box paradox is the paradigm example of such a paradox, where a ball is placed in one of three boxes and it is inferred that it would have been found, with certainty, both in box 1 and in box 2 had either box been opened on their own. Precisely what is at stake in PPS paradoxes has been unclear, and classical models have been suggested which are supposed to mimic the essential featur…
Chirped-pulse interferometry (CPI) captures the metrological advantages of quantum Hong-Ou-Mandel (HOM) interferometry in a completely classical system. Modified HOM interferometers are the basis for a number of seminal quantum-interference effects. Here, the corresponding modifications to CPI allow for the first observation of classical analogues to the HOM peak and quantum beating. They also allow a new classical technique for generating phase super-resolution exhibiting a coherence length dramatically longer than that of the laser light, analogous to increased two-photon coherence lengths i…
A quantum system can be driven by either sinusoidal, rectangular, or noisy signals. In the literature, these regimes are referred to as Landau-Zener-Stuckelberg-Majorana (LZSM) interferometry, latching modulation, and motional averaging, respectively. We demonstrate that these pronounced and interesting effects are also inherent in the dynamics of classical two-state systems. We discuss how such classical systems are realized using either mechanical, electrical, or optical resonators. In addition to the fundamental interest of such dynamical phenomena linking classical and quantum physics, we …
A class of Adaptive Decoders (AD's) for coherent-state sequences is studied, including in particular the most common technology for optical-signal processing, e.g., interferometers, coherent displacements and photon-counting detectors. More generally we consider AD's comprising adaptive procedures based on passive multi-mode Gaussian unitaries and arbitrary single-mode destructive measurements. For classical communication on quantum phase-insensitive Gaussian channels with a coherent-state encoding, we show that the AD's optimal information transmission rate is not greater than that of a singl…
We demonstrate accurate phase measurement from low photon level interference data using a constrained optimization method that takes into account the expected redundancy in the unknown phase function. This approach is shown to have significant noise advantage over traditional methods such as balanced homodyning or phase shifting that treat individual pixels in the interference data as independent of each other. Our interference experiments comparing the optimization method with the traditional phase shifting method show that when the same photon resources are used, the optimization method prov…
We discuss advantages of using non-classical states of light for two aspects of optical imaging: creating of miniature images on photosensitive substrates, which constitutes the foundation for optical lithography, and imaging of micro objects. In both cases, the classical resolution limit given by the Rayleigh criterion is approximately a half of the optical wavelength. It has been shown, however, that by using multi-photon quantum states of the light field, and multi-photon sensitive material or detector, this limit can be surpassed. We give a rigorous quantum mechanical treatment of this pro…
Quantum coherence is one of the key features that fuels applications for which quantum mechanics exceeds the power of classical physics. This explains the considerable efforts that were undertaken to quantify coherence via quantum resource theories. An application of the resulting framework to concrete technological tasks is however largely missing. Here, we address this problem and connect the ability of an operation to detect or create coherence to the performance of interferometric experiments.
Non-classical states of light find applications in enhancing the performance of optical interferometric experiments, with notable example of gravitational wave-detectors. Still, the presence of decoherence hinders significantly the performance of quantum-enhanced protocols. In this review, we summarize the developments of quantum metrology with particular focus on optical interferometry and derive fundamental bounds on achievable quantum-enhanced precision in optical interferometry taking into account the most relevant decoherence processes including: phase diffusion, losses and imperfect inte…
In this paper, we investigate the phase sensitivities in two-path optical interferometry with asymmetric beam splitters. Here, we present the optimal conditions for the transmission ratio and the phase of the beam splitter to gain the highest sensitivities for a general class of non-classical states with parity symmetry. Additionally, we address the controversial question of whether the scheme with a combination of coherent state and photon-added or photon-subtracted squeezed vacuum state is better or worse than the most celebrated one using a combination of coherent state and squeezed vacuum …
Differential interferometry (DI) with two coupled sensors is a most powerful approach for precision measurements in presence of strong phase noise. However DI has been studied and implemented only with classical resources. Here we generalize the theory of differential interferometry to the case of entangled probe states. We demonstrate that, for perfectly correlated interferometers and in the presence of arbitrary large phase noise, sub-shot noise sensitivities -- up to the Heisenberg limit -- are still possible with a special class of entangled states in the ideal lossless scenario. These sta…
Even-order dispersion cancellation, an effect previously identified with frequency-entangled photons, is demonstrated experimentally for the first time with a linear, classical interferometer. A combination of a broad bandwidth laser and a high resolution spectrometer was used to measure the intensity correlations between anti-correlated optical frequencies. Only 14% broadening of the correlation signal is observed when significant material dispersion, enough to broaden the regular interferogram by 4250%, is introduced into one arm of the interferometer.
We demonstrate a two-pulse Ramsey-type interferometer for non-classical motional states of a Bose-Einstein condensate in an anharmonic trap. The control pulses used to manipulate the condensate wavefunction are obtained from Optimal Control Theory and directly optimised to maximise the interferometric contrast. They permit a fast manipulation of the atomic ensemble compared to the intrinsic decay and many-body dephasing effects, and thus to reach an interferometric contrast of 92% in the experimental implementation.
The phase resolution of interferometers is limited by the so-called Heisenberg limit, which states that the optimum phase sensitivity is inversely proportional to the number of interfering particles N, a 1/sqrt{N} improvement over the standard quantum limit. We have used simulated annealing, a global optimization strategy, to systematically search for quantum interferometer input states that approach the Heisenberg limited uncertainty in estimates of the interferometer phase shift. We compare the performance of these states to that of other non-classical states already known to yield Heisenber…
Quantum phenomena such as entanglement can improve fundamental limits on the sensitivity of a measurement probe. In optical interferometry, a probe consisting of $N$ entangled photons provides up to a $\sqrt{N}$ enhancement in phase sensitivity compared to a classical probe of the same energy. Here, we employ high-gain parametric down-conversion sources and photon-number-resolving detectors to perform interferometry with heralded quantum probes of sizes up to $N=8$ (i.e. measuring up to 16-photon coincidences). Our probes are created by injecting heralded photon-number states into an interfero…
