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 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.
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.
We show that nonclassicality of phase-space quasi-probability distributions is tied to violations of principles of physical reality in device-dependent scenarios. In this context, the nonclassicality problem has its dual form expressed as a device-dependent analog of Bell inequalities. This approach is applicable even to systems with only one spatial party. The derived inequalities are employed for testing nonclassicality with emblematic optical measurements: photocounting including the case of realistic photon-number resolution; unbalanced, balanced, and eight-port homodyne detection.
We analyze nonclassical correlations between outcomes of measurements conducted on two spatial radiation modes. These correlations cannot be simulated with statistical mixtures of coherent states or, more generally, with non-negative phase-space functions of quantum states and measurements. We argue that nonclassical correlations are naturally related to Bell nonlocality, the former being a more general class of quantum correlations. Indeed, it is known that local realistic as well as noncontextual models correspond to non-negative solutions to a system of linear equations for the joint probability distributions of all observables. We demonstrate that nonclassical correlations correspond to a particular solution to this system, which may have negative values even if local realism is not violated. A modification of Bell inequalities enables us to test such correlations. At the same time, our approach leads to a formulation of Bell inequalities applicable also to continuous variables. The results are illustrated with two-mode squeezed vacuum states and with hybrid entangled states (Schr\"odinger-Cat states), one mode being analyzed by balanced and the other one by unbalanced homodyne detection.
Quantum correlations play a fundamental role in quantum information science. The variety of their manifestations has become increasingly apparent following the development of novel light sources, protocols and photodetectors. One broad classification identifies two instances of non-classical correlations: particle and mode entanglement. These categories mirror two coexisting descriptions of quantum systems in terms of discrete and continuous variables of the electromagnetic field. The past decades have generated a number of promising results based on schemes which encompass elements from both frameworks, rather than viewing the two descriptions as mutually exclusive. In this context, it is possible to conceive and realise experiments where either the quantum resource or the detection system is 'hybrid'. Optical weak-field homodyne detectors bring together phase sensitivity and photon counting; as such, they represent a detection scheme which works across continuous and discrete variables of the radiation field. In this thesis we present a two-mode weak-field homodyne detection layout with added photon-number resolution and apply it to the study of a split single-photon state and a squeezed vacuum state. As a first test of the capabilities of this system, we investigate the reconstruction of relevant features of a given quantum resource - such as its photon statistics - with our detection scheme. Further, we experimentally demonstrate the observation of an instance of non-classical optical coherence which combines the continuous- and discrete-variable descriptions explicitly. The ability to probe phenomena at the interface of wave and particle regimes opens the way to novel, improved schemes for quantum information processing. From a more fundamental perspective, such hybrid approaches may shed light on the very roots of quantum enhancement.
Wave-particle duality and entanglement are two fundamental characteristics of quantum mechanics. All previous works on experimental investigations in wave-particle properties of single photons (or single particles in general) showed that a well-defined interferometer setting determines a well-defined property of single photons. Here we take a conceptual step forward and control the wave-particle property of single photons with a Bell state. By doing so, we experimentally test the complementarity principle in a scenario in which the setting of the interferometer is not defined at any instance of the experiment, not even in principle. To achieve this goal, we establish the three-photon entangled state, i.e., the entanglement between a single photon and a two-photon Bell state, send the photon of interest S into a quantum Mach-Zehnder interferometer (MZI), in which the output beam splitter of the MZI is controlled by the quantum state of the second photon C, which is entangled with a third photon A. Therefore, the individual quantum state of photon C is undefined, which implements the undefined settings of the MZI for photon S. This is realized by using three cascaded phase-stable interferometers for three photons. There is typically no well-defined setting of the MZI and thus the very formulation of the wave-particle objectivity from local hidden variable models becomes internally inconsistent.