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arXiv:2501.03505 · American Journal of Physics

Interference of interference effects

Kevin Randles, S. J. van Enk

Published Aug 21, 2025• 0 citations
arXivHTMLPDF

Abstract

We analyze the interference of individual photons in two overlapping Mach–Zehnder interferometers joined via a common beam splitter. We show how two kinds of interference effects—single-photon Mach–Zehnder interference and two-photon Hong–Ou–Mandel interference—interfere with one another, partially canceling each other out. This configuration illustrates how quantum effects can combine nontrivially and, moreover, emphasizes that quantum interference happens at measurement. This work can serve as a bridge to more advanced quantum mechanical concepts, such as testing the predictions of quantum mechanics vs local realism.

DOI
10.1119/5.0256745

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We analyze the interference of individual photons in two overlapping Mach–Zehnder interferometers joined via a common beam splitter. We show how two kinds of interference effects—single-photon Mach–Zehnder interference and two-photon Hong–Ou–Mandel interference—interfere with one another, partially canceling each other out. This configuration illustrates how quantum effects can combine nontrivially and, moreover, emphasizes that quantum interference happens at measurement. This work can serve as a bridge to more advanced quantum mechanical concepts, such as testing the predictions of quantum mechanics vs local realism.
Two-photon interference effects arise because photons are indistinguishable particles. In the wellknown Hong-Ou-Mandel (HOM) effect, the transmission of two photons at a beam splitter interferes destructively with the reflection of both photons, requiring both photons to "bunch up" by leaving the beam splitter on the same side. Here, we show that the interference between locally propagating photons and photons exchanged by a mode swap can be implemented by post-selecting spatially separated photon outputs of a four-path interferometer. Even though the photons detected at spatially separated locations must have travelled along paths that never met up at the same beam splitter, the Hong-Ou-Mandel effect can be observed in correlations between the output ports that originate from the association of detection events with non-local output modes defined by the two single photon inputs. Local phase shifts can be used to map out non-classical correlations between the photons detected at different output locations, clarifying the role of linear optics in generating entanglement between spatially separated photons. Our work thus establishes a fundamental relation between multiphoton interference and entanglement, opening the door to new possibilities in optical quantum technologies.
This article reviews the historical and conceptual evolution of quantum physics, challenging P.A.M. Dirac’s famous dictum, “Each photon then interferes only with itself. Interference between two different photons never occurs.” We trace the progression from the early experiments by Forrester et al. and Magyar & Mandel, which first questioned this notion, to the advent of Spontaneous Parametric Down-Conversion (SPDC) following the invention of the laser and nonlinear optics. The article elucidates how D. N. Klyshko’s quantum description of SPDC laid the groundwork for the landmark Shih-Alley and Hong-Ou-Mandel (HOM) experiments, which demonstrated two-photon quantum interference and entanglement. Finally, we address the common misconception that photons must physically overlap at a beam splitter for two-photon interference to occur. We illustrate that quantum interference arises solely from the indistinguishability of probability amplitudes leading to a joint detection event, even in configurations where the photons never locally interact at a beam splitter.
We present experimental demonstrations of two-photon interference involving temporally separated photons within two types of interferometers: a Mach-Zehnder interferometer and a polarization-based Michelson interferometer. The two-photon states are probabilistically prepared in a symmetrically superposed state within the two interferometer arms by introducing a large time delay between two input photons; this state is composed of two temporally separated photons, which are in two different or the same spatial modes. We then observe two-photon interference fringes involving both the Hong-Ou-Mandel interference effect and the interference of path-entangled two-photon states simultaneously in a single interferometric setup. The observed two-photon interference fringes provide simultaneous observation of the interferometric properties of the single-photon and two-photon wavepackets. The observations can also facilitate a more comprehensive understanding of the origins of the interference phenomena arising from spatially bunched/anti-bunched two-photon states comprised of two temporally separated photons within the interferometer arms.
The photon in the interferometer This chapter introduces many of the ideas of quantum theory by exploring three specific “case studies” of quantum systems. Each is an example of a qubit , a generic name for the simplest type of quantum system. The concepts we develop will be incorporated into a rigorous mathematical framework in the next chapter. Our business here is to provide some intuition about why that mathematical framework is reasonable and appropriate for dealing with the quantum facts of life. Interferometers In Section 1.2 we discussed the two-slit interference experiment with a single photon. In that experiment, the partial waves of probability amplitude were spread throughout the entire region of space beyond the two slits. It is much easier to analyze the situation in an interferometer , an optical apparatus in which the light is restricted to a finite number of discrete beams . The beams may be guided from one point to another, split apart or recombined as needed, and when two beams are recombined into one, the result may show interference effects. At the end of the interferometer, one or more sensors can measure the intensity of various beams. (A beam is just a possible path for the light, so there is nothing paradoxical in talking about a beam of zero intensity.) Figure 2.1 shows the layout of a Mach–Zehnder interferometer , which is an example of this kind of apparatus.
Usually, the oscillations of interference effects are controlled by relative phases. We show that varying the amplitudes of quantum waves, for instance by changing the reflectivity of beam splitters, can also lead to quantum oscillations and even to Bell violations of local realism. We first study theoretically a generalization of the Hong-Ou-Mandel experiment to arbitrary source numbers and beam-splitter transmittivity. We then consider a Bell-type experiment with two independent sources and find strong violations of local realism for arbitrarily large source number NNN; for small NNN, one operator measures essentially the relative phase of the sources and the other their intensities. Since, experimentally, one can measure the parity of the number of atoms in an optical lattice more easily than the number itself, we assume that the detectors measure parity.