The recent demonstrations of cascaded PDC (CPDC) and the hopeful prospects of realizing third-order PDC (TOPDC) for the generation of three-photon entangled states are paving the way for experimental studies on genuine three-photon interference. In this article, we formulate three-photon interference in terms of ``each three-photon interfering only with itself.'' We show that although a generalized two-alternative three-photon interference setup based on CPDC or TOPDC involves eight different length parameters, the interference can be fully characterized in terms of only three independent parameters. The first parameter is the three-photon path-length difference, which has a direct analog in the one-photon and two-photon cases, and the other two parameters quantify the path-asymmetry length. Unlike two-photon interference, which requires only one parameter to quantify path-asymmetry, two independent parameters are needed in three-photon interference. This results in a broader class of nonclassical three-photon effects, including three-photon HOM-type effects. Our work provides the theoretical basis for existing and future three-photon interference experiments exploring the rich and complex quantum correlations associated with three-particle entanglement and potentially enabling the development of novel protocols for harnessing those correlations.
Feynman had famously remarked that interference contains the only mystery of quantum mechanics [1]. The study of single-photon interference dates back to Thomas Young's classic double-slit experiment [2,3], while the study of multi-photon interference effects commences with the experiments of Hanbury-Brown and Twiss [4,5]. In a single-photon interference experiment, a single detector is used for measuring the probability of detecting a photon as a function of time or space, while in a multi-photon interference experiment, multiple detectors are used for measuring the joint probability of detection [6,7]. Interference effects with single-photon states are best described in terms of Dirac's famous dictum [8] that "Each photon interferes only with itself. Interference between two different photons never occurs." In his Nobel Lecture [9], Glauber praised Dirac's dictum as being "ringingly clear" and went on to describe its full im-port by noting that "it is not the photons that interfere physically, it is their probability amplitudes that interfere-and probability amplitudes can be defined equally well for arbitrary numbers of photons." Although Dirac's description for one-photon interference has been extended to interference effects involving two-photon fields [10,11,12,13,14,15,16,17], a comprehensive description is still pending for fields with three or more photons.
Quantum entanglement, which refers to the inseparability of the global quantum state of a composite multiparticle system into local states of the individual constituent particles, is a key feature of quantum theory that accounts for curious fundamental phenomena such as nonlocality [18,19] and useful practical applications such as teleportation [20,21,22,23]. At present, one of the most widely used experimental platform for generating entangled two-photon quantum states is parametric down-conversion (PDC) -a second-order χ (2) nonlinear optical process in which a single photon from an incident field known as the pump gets annihilated to produce two entangled photons termed as signal and idler [24]. The strong nonclassical correlations between the signal and idler photons manifest themselves quite strikingly through two-photon interference effects, wherein high-visibility interference fringes are observed in the two-photon coincidence rate, even in the absence of any modulation in the individual one-photon intensities [25]. For instance, the nonclassical temporal correlations of the signal and idler photons manifest themselves through temporal two-photon interference effects such as the Hong-Ou-Mandel (HOM) effect [26],
Franson effect [11], induced coherence without induced emission [27,28], and frustrated twophoton creation [29]. Such two-photon interference experiments have historically played a crucial role in advancing our fundamental understanding of two-particle entanglement and how it can be harnessed for quantum applications.
The theoretical description of temporal twophoton interference effects has been gradually developed over many years [10,11,12,13,14,15,16]. It is now known that the temporal twophoton interference effects can be quantitatively explained within the ambit of a single theoretical formalism based on describing them in terms of “each two-photon interfering only with itself” [15,16]. Within this formalism, it has been shown that although a generic two-alternative two-photon interference setup has six different tunable length parameters, the temporal interference effects can be completely characterized in terms of only two independent length parameters. In other words, the two-photon temporal coherence function factorizes into two parts corresponding to each parameter. One of the length parameters is called the two-photon pathlength difference, which plays a role analogous to the path-length difference parameter in onephoton interference. Thus, as a function of this parameter, one observes two-photon interference fringes with periodicity fixed by the central wavelength of the pump field, where the corresponding temporal coherence function is determined by the frequency correlations of the pump field. In essence, the theory shows that the pump’s temporal coherence gets transferred to the generated two-photon field [16]. The second parameter is called the two-photon path-asymmetrylength difference, which quantifies the path asymmetry between the two photons. The coherence function corresponding to this parameter is determined by the PDC phase-matching and depends on the frequency correlations of the downconverted photons. This parameter has no analog in one-photon interference and is responsible for exclusive two-photon effects. The Hong-Ou-Mandel (HOM) effect [26], which is one of the most famous two-photon coherence effects, can be understood in terms of how two-photon coherence varies as a function of two-photon pathasymmetry-length difference in a setup that involves mixing of two down-converted photons at a beam split
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