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arXiv:2109.10170 · Physical Review Applied

Optimal Interferometry for Bell Nonclassicality Induced by a Vacuum–One-Photon Qubit

Tamoghna Das, Marcin Karczewski, Antonio Mandarino, Marcin Markiewicz, Marek Żukowski

Published Sep 27, 2022• 12 citations
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Abstract

Bell nonclassicality of a single-photon superposition in two modes, often referred to as ``nonlocality of a single photon,'' is one of the most striking nonclassical phenomena discussed in the context of foundations of quantum physics. Here we show how to robustly violate local realism within the weak-field homodyne measurement scheme for any superposition of one photon with vacuum. Our modification of the previously proposed setups involves tunable beam splitters at the measurement stations, and the local oscillator fields significantly varying between the settings, optimally being on or off. As photon-number-resolving measurements are now feasible, we advocate for the use of the Clauser-Horne Bell inequalities for detection events using precisely defined numbers of photons. We find a condition for optimal measurement settings for the maximal violation of the Clauser-Horne inequality with weak-field homodyne detection, which states that the reflectivity of the local beam splitter must be equal to the strength of the local oscillator field. We show that this condition holds not only for the vacuum--one-photon qubit input state, but also for the superposition of a photon pair with vacuum, which suggests its generality as a property of weak-field homodyne detection with photon-number resolution. Our findings suggest a possible path to employ such scenarios in device-independent quantum protocols.

DOI
10.1103/physrevapplied.18.034074

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Bell nonclassicality of a single-photon superposition in two modes, often referred to as ``nonlocality of a single photon,'' is one of the most striking nonclassical phenomena discussed in the context of foundations of quantum physics. Here we show how to robustly violate local realism within the weak-field homodyne measurement scheme for any superposition of one photon with vacuum. Our modification of the previously proposed setups involves tunable beam splitters at the measurement stations, and the local oscillator fields significantly varying between the settings, optimally being on or off. As photon-number-resolving measurements are now feasible, we advocate for the use of the Clauser-Horne Bell inequalities for detection events using precisely defined numbers of photons. We find a condition for optimal measurement settings for the maximal violation of the Clauser-Horne inequality with weak-field homodyne detection, which states that the reflectivity of the local beam splitter must be equal to the strength of the local oscillator field. We show that this condition holds not only for the vacuum--one-photon qubit input state, but also for the superposition of a photon pair with vacuum, which suggests its generality as a property of weak-field homodyne detection with photon-number resolution. Our findings suggest a possible path to employ such scenarios in device-independent quantum protocols.
Abstract Nowadays photon-number resolving weak-field homodyne measurements allow realization of emblematic gedankenexperiments revealing correlations of optical fields. This covers experiments on (a) excitation of a pair of spatial modes by a single photon, and (b) two spatial modes in a weakly squeezed vacuum state, involving constant local oscillator strengths. Proving Bell nonclassicality of such correlations demands measurements of complementary observables. We show that typical arrangement of weak-field homodyne detection with measurement settings defined by phases of local oscillators of constant strength does not provide enough complementarity for confirming Bell nonclassicality. In the case of experiment (a) we provide an exact local hidden variable model restoring all quantum probabilities, whereas in the case of experiment (b) we show that the claims of its nonclassicality are unfounded. A full complementarity of wave aspects vs particle (number) can be achieved in a measurement situation in which respectively the local oscillators are on or off. This is shown to lead to an operational scenario, in the case of both experiments (a) and (b), which reveals indisputable violations of local realism. Such schemes may find possible applications in device-independent quantum protocols.
Abstract We reconsider the all-optical weak homodyne-measurement based experimental schemes aimed at revealing Bell nonclassicality (‘nonlocality’) of a single photon. We focus on the schemes put forward by Tan et al (TWC, 1991) and Hardy (1994). In our previous work we show that the TWC experiment can be described by a local hidden variable model, hence the claimed nonclassicality is apparent. The nonclassicality proof proposed by Hardy remains impeccable. We investigate which feature of the Hardy’s approach is crucial to disclose the nonclassicality. There are consequential differences between TWC and Hardy setups: (i) the initial state of Hardy is a superposition of a single photon excitation with vacuum in one of the input modes of a 50–50 beamsplitter. In the TWC case there is no vacuum component. (ii) In the final measurements of Hardy’s proposal the local settings are specified by the presence or absence of a local oscillator field (on/off). In the TWC case the auxiliary fields are constant, only phases are varied. We show that in Hardy’s setup the violation of local realism occurs due to the varying strength of the local oscillators. Still, one does not need to operate in the fully on/off detection scheme. Thus, the nonclassicality in a Hardy-like setup cannot be attributed to the single-photon state alone. It is a consequence of its interference with the photons from auxiliary local fields. Neither can it be attributed to the joint state of the single photon excitation and the local oscillator modes, as this state is measurement setting dependent. Despite giving spurious violations of local realism, the TWC scheme can serve as an entanglement indicator, for the TWC state. Nevertheless an analogue indicator based on intensity rates rather than just intensities overperforms it.
Entanglement potentials are popular measures of the nonclassicality of single-mode optical fields. These potentials are defined by the amount of entanglement (measured by, e.g., the negativity or concurrence) of the two-mode field generated by mixing a given single-mode field with the vacuum on a balanced beam splitter. We generalize this concept to define the potentials for Bell nonlocality and quantum steering in specific measurement scenarios, in order to quantify single-mode nonclassicality in a more refined way. Thus, we can study the hierarchy of three types of potentials in close analogy to the well-known hierarchy of the corresponding two-mode quantum correlations. For clarity of our presentation, we focus on the analysis of the nonclassicality potentials for arbitrary vacuum-one-photon superpositions (VOPSs), corresponding to a photon-number qubit. We discuss experimentally feasible implementations for the generation of single-mode VOPS states, their mixing with the vacuum on a balanced beam splitter, and their two-mode Wigner-function reconstruction using homodyne tomography to determine the potentials. We analyze the effects of imperfections, including phase damping and unbalanced beam splitting on the quality of the reconstructed two-mode states and nonclassicality potentials. Although we focus on the analysis of VOPS states, single-mode potentials can also be applied to study the nonclassicality of qudits or continuous-variable systems.
We investigate quantum nonlocality of a single-photon entangled state under feasible measurement techniques consisting of on-off and homodyne detections along with unitary operations of displacement and squeezing. We test for a potential violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality, in which each of the bipartite party has a freedom to choose between two measurement settings, with each measurement yielding a binary outcome. We find that single-photon quantum nonlocality can be detected when two or less of the four total measurements are carried out by homodyne detection. The largest violation of the CHSH inequality is obtained when all four measurements are squeezed-and-displaced on-off detections. We test the robustness of violations against imperfections in on-off detectors and single-photon sources, finding that the squeezed-and-displaced measurement schemes perform better than the displacement-only measurement schemes.
The question of “non-locality of a single photon'', which started with a paper by Tan, Walls and Collett (TWC, 1991) stirred a thirty years long debate. This hampered attempts to use the TWC interferometric scheme in quantum cryptography. The scheme involves a single photon 50-50 beam-split into two modes propagating to two spatially separated observation stations at which weak homodyne measurements are made. The physics and non-classicality of such an arrangement has been understood only recently, and points out that an unquestionable Bell non-classicality, as was suggested by Hardy (1994), can be observed when the local measurement settings differ by the weak local oscillator being on or off, and additionally the homodyning for the on case is not balanced. Based on that, we present a single-photon based device-independent quantum key distribution scheme secure even against no-signaling eavesdropping. In our protocol the random bits of the cryptographic key are obtained by measurements on the single photon, that is for off settings at both Alice and Bob sides, while the security is positively tested if for eavesdropping testing runs one observes a violation of a specific Bell inequality involving the on and off weak homodyne measurements as alternative local settings. The security analysis presented here is based on a decomposition of the correlations into extreme points of a no-signaling polytope, which allows for identification of the optimal strategy for any eavesdropping constrained only by the no-signaling principle. For this strategy, the key rate is calculated, which is then connected with the violation of a specific Clauser-Horne inequality. We also adapt this analysis to propose a self-testing quantum random number generator based on the old idea that employs the randomness of reflection and transmission events of a quantum light impinged on a 50-50 beamsplitter.
Entanglement does not always require one particle per party. It was predicted some thirty years ago that a single photon traversing a beam splitter could violate a Bell inequality. Although initially debated, single-photon nonlocality was eventually demonstrated via homodyne measurements. Here, we present an alternate realisation that avoids the complexity of homodyne measurements and potential loopholes in their implementation. We violate a Bell inequality by performing joint measurements on two copies of the same single-photon entangled state, where one photon acts as a phase reference for the other, making it self-referential. We observe CHSH parameters of 2.71±0.092.71\pm 0.092.71±0.09 and 2.23±0.072.23\pm 0.072.23±0.07, depending on the joint measurements implemented. This offers a new perspective on single-photon nonlocality and a more accessible experimental route, potentially applicable to general mode-entangled states in diverse platforms.