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SPIE Quantum Technologies

Nonclassicality and wave-particle duality in quantum interferometer

Chanseul Lee, Tai Hyun Yoon

Published Jun 10, 2024• 0 citations
Publisher

Abstract

We propose a novel quantum interferometer based on two stimulated Parametric Down-Conversion (PDC) crystals seeded by weak coherent idler fields to investigate the relationship between the nonclassicality of the signal photon state and the fidelity of the conjugate idler photon state. We show that the purity of the signal photon state is tightly bounded by the entanglement between the signal photon and its conjugate idler photon, i.e., the fidelity of the idler state in the bipartite system, resulting in a novel complementarity relation for wave-particle duality. Moreover, we find that the nonclassical measure of the signal photon state is identical to the fidelity of the conjugate idler photon states prepared in each crystal. In our novel interferometer, the input beam splitter of the conventional SU(2) interferometer is replaced with two seeded SU(1,1) PDC systems that emit only a single photon at once from two crystals, thus producing an entangled superposition state where the conjugate idler photon states act as path detectors with controllable fidelity. We demonstrate that the presented quantum interferometer consisting of two stimulated PDC crystals positioned in parallel is an ideal setup for investigating and quantitatively understanding the complementarity relation of wave-particle duality.

DOI
10.1117/12.3016880

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We propose a novel quantum interferometer based on two stimulated Parametric Down-Conversion (PDC) crystals seeded by weak coherent idler fields to investigate the relationship between the nonclassicality of the signal photon state and the fidelity of the conjugate idler photon state. We show that the purity of the signal photon state is tightly bounded by the entanglement between the signal photon and its conjugate idler photon, i.e., the fidelity of the idler state in the bipartite system, resulting in a novel complementarity relation for wave-particle duality. Moreover, we find that the nonclassical measure of the signal photon state is identical to the fidelity of the conjugate idler photon states prepared in each crystal. In our novel interferometer, the input beam splitter of the conventional SU(2) interferometer is replaced with two seeded SU(1,1) PDC systems that emit only a single photon at once from two crystals, thus producing an entangled superposition state where the conjugate idler photon states act as path detectors with controllable fidelity. We demonstrate that the presented quantum interferometer consisting of two stimulated PDC crystals positioned in parallel is an ideal setup for investigating and quantitatively understanding the complementarity relation of wave-particle duality.
To test the principle of complementarity and wave-particle duality quantitatively, we need a quantum composite system that can be controlled by experimental parameters. Here, we demonstrate that a double-path interferometer consisting of two parametric downconversion crystals seeded by coherent idler fields, where the generated coherent signal photons are used for quantum interference and the conjugate idler fields are used for which-path detectors with controllable fidelity, is useful for elucidating the quantitative complementarity. We show that the source purity μsμ_sμs​ is tightly bounded by the entanglement measure EEE by the relation μs=1−E2μ_s=\sqrt{1-E^2 }μs​=1−E2​ and the visibility VVV and detector fidelity FFF determine the coherence of the quantons, i.e., C=V∣F∣C = V|F|C=V∣F∣. The quantitative complementarity of the double-path interferometer we developed recently is explained in terms of the quanton-detector entanglement or quanton source purity that are expressed as functions of injected seed photon numbers. We further suggest that the experimental scheme utilizing two stimulated parametric downconversion processes is an ideal tool for investigating and understanding wave-particle duality and complementarity quantitatively.
We re-examine the properties of the states produced by nondegenerate coherently stimulated parametric down-conversion wherein the signal and idler modes are seeded with coherent states of light and where the nonlinear crystal is driven by a strong classical field as described by the parametric approximation. The states produced are the two-mode squeezed coherent states defined with a specific ordering of operators, namely, the displacement operators of the two modes acting on the double vacuum state followed by the action of the two-mode squeeze operator representing the down-conversion process. Though mathematically equivalent to the reverse ordering of operators, but with different displacement parameters, the ordering we consider is closely related to what could most easily be implemented in the laboratory. The statistical properties of the state are studied with an emphasis on how they, and its average photon number, are affected by the various controllable phases, namely, those of the classical pump field of the two input coherent states. We then consider the multiphoton interference effects that arise if the two beams are overlapped on a 50:50 beam splitter, investigating the role of the phases in controlling the statistical properties of the output states. Finally, we study the prospects for the application of the states to quantum-optical interferometry to obtain sensitivities in phase-shift measurements beyond the standard quantum limit.
Parametric down-conversion is a second-order nonlinear optical process annihilating a pump photon and creating a pair of photons in the signal and idler modes. Then, by using two parametric down-converters and introducing a path indistinguishability for the two generated idler modes, a quantum coherence between two conjugate signal beams can be induced. Such a double spontaneous or stimulated parametric down-conversion scheme has been used to demonstrate quantum spectroscopy and imaging with undetected idler photons via measuring one-photon interference between their correlated signal beams. Recently, we considered another quantum optical measurement scheme utilizing W-type tripartite entangled signal photons that can be generated by employing three spontaneous parametric down-conversion crystals and by inducing coherences or path-indistinguishabilities between their correlated idler beams and between quantum vacuum fields. Here, we consider an extended triple stimulated parametric down-conversion scheme for quantum optical measurement of sample properties with undetected idler and photons. Noting the real effect of vacuum field indistinguishability on the fringe visibility as well as the role of zero-point field energy in the interferometry, we show that this scheme is an ideal and efficient way to create a coherent state of W-type entangled signal photons. We anticipate that this scheme would be of critical use in further developing quantum optical measurements in spectroscopy and microscopy with undetected photons.
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 SSS 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 CCC, which is entangled with a third photon AAA. Therefore, the individual quantum state of photon CCC is undefined, which implements the undefined settings of the MZI for photon SSS. 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.
A single photon exhibits wave-particle duality in the Young's double-slit interferometer. The duality character-ized by an interference visibilityVVVas the wave-like property and a which-path informationDDDas the particle-like property has tradeoff relationD2+V2≤1D^{2}+V^{2}\leq 1D2+V2≤1known as complementarity [1]–[3]. There are a lot of studies about the complementary nature of photons from various viewpoints such as delayed-choice experiments, theoretical works related to the uncertainty principle, and so on. More recently, studies have expanded the concept of the wave-particle duality to higher-order interferometric scenarios, in which the input light includes two or more pho-tons [4], [5]. Despite having a long history, all previous studies have been related to phase correlation of optical states, which reason would be why the main focus in the first place is the investigation of the mysterious quantum nature of a photon, including the superposition principle.
A single photon exhibits wave-particle duality in the Young's double-slit interferometer. The duality characterized by an interference visibility and a which-path information has trade-off relation known as complementarity. These quantities are related to the first-order coherence, and the interference is based on the phase correlation between lights coming from two arms. However according to quantum optics theory, such a simple wave-particle picture is not enough to understand the nature because the theory showed an importance of higher-order coherence in the sense of both interference and statistical distribution of photons. Second-order intensity correlation is especially crucial to reveal distinctive quantum features of photons with no classical analogue. Here, in an intensity interferometric scenario as represented by the Hong-Ou-Mandel interferometer, we discuss a wave-particle duality of light based on a which-path information and a quantity characterizing a magnitude of the intensity interferometric effect. We show, for classical light, the two quantities obey the complementary principle similar to the case of the double-slit experiment, but do not for nonclassical light. The nonclassical light such as photons at two arms is allowed to show larger which-path information and intensity interference simultaneously beyond the complementary relation. Moreover, the violation reveals a new nonclassical nature of light although both of the above two quantities seem to be understandable classically, which is never found from a consideration of only one side of wave-particle duality.