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Journal of Optics B Quantum and Semiclassical Optics

A quantum mechanical description of electromagnetic and atom Mach-Zehnder interferometers

Y. Ben-Aryeh, Doron ludwin, A. Mann

Published Apr 27, 2001• 7 citations
Publisher

Abstract

A quantum mechanical description of a non-absorbing electromagnetic (em) beam splitter (BS) transformation is given by the use of momentum creation operators, which also include momentum shift operators operating on the BS. This BS transformation is used for analysing photon-statistics effects in the em Mach-Zehnder interferometer. Also the relation between this BS transformation and Welcher-Weg measurements is discussed. The analogy between em and atom interferometers is described with an emphasis on the use of high-order correlation functions. The entanglement between the interfering beams in these interferometers and the BS is taken into account in our analysis for treating special effects.

DOI
10.1088/1464-4266/3/3/311

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A quantum mechanical description of a non-absorbing electromagnetic (em) beam splitter (BS) transformation is given by the use of momentum creation operators, which also include momentum shift operators operating on the BS. This BS transformation is used for analysing photon-statistics effects in the em Mach-Zehnder interferometer. Also the relation between this BS transformation and Welcher-Weg measurements is discussed. The analogy between em and atom interferometers is described with an emphasis on the use of high-order correlation functions. The entanglement between the interfering beams in these interferometers and the BS is taken into account in our analysis for treating special effects.
The 'collapse' of the wave function in a general measuring process is analyzed by a pure quantum mechanical (QM) approach. The problem of the delayed choice and Welcher-Weg (WW) experiments is analyzed for Mach-Zehnder (MZ) interferometer. The WW effect is related to complementarity principle and orthogonality of wavefunctions although it produces small momentum changes between the electromagnetic (EM) field and the beam-splitters (BS's). For QM states for which we have a superposition of states and interference effects there is no reality before measurement, but by excluding such effects the system gets a real meaning already before measurement.
It is generally believed that the “mystery” behind understanding Quantum Mechanics (QM) and the global drives to construct quantum computers using “Entanglement”, can be understood from how the two-beam superposition effect (SE) emerges out of a 2-slit, and a Mach-Zehnder Interferometer (MZI). This chapter rehabilitates the classical analyses of these two superposition effects, while exploiting functional asymmetries, either deliberately introduced or intrinsic in the two-beam apparatuses. We are using mathematical formalism already well-established in classical optics since 1800’s, but with the extra emphasis that any final data generation requires some real physical interaction between the detector and the two light signals simultaneously stimulating the detecting molecules. This critical step of physical interaction process is not explicitly underscored by either the classical or the quantum physics. However, the asymmetry, either in the propagation or in the interaction process, is utilized to bring out the contradictions with the QM interpretations. For the 2-slit system, we deliberately introduce asymmetry on one of the slits. For MZI, the asymmetry is already built into the classical reflection property of a typical beams splitter.
In this paper we describe within the formalism of Quantum Optics (QO) a double Mach-Zehnder interferometer (MZI). For single photon Fock states this experimental setup is shown to exhibit a counter-intuitive behavior: for certain values of the path length difference of the first MZI, the singles photon-count statistics at the output detectors does not change, whatever the difference in path length for the second MZI. For simultaneously impinging light quanta, we show that this setup is able to show the same HOM antibunching effect previously obtained with a beam splitter. However, by adding substantial delays in each MZI, we can obtain the same effect even if the "photon wave packets" do not overlap at the second beam splitter.
We discuss complementarity and uncertainty in a gedanken Which-Way (Welcher-Weg) experiment in a Mach-Zehnder interferometer. Although a Welcher-Weg measurement can be performed with only a negligible amount of momentum change in the detector itself, the change in the interference pattern must correspond to a change in the momentum distribution of the particle. Entanglement between the Mach-Zehnder interferometer and the Welcher-Weg measurement introduces orthogonality that disables the interference of the particle, but simultaneously, it also generates a momentum transfer between the particle and other parts of the whole entangled system (including macroscopic parts). This analysis can be related to the Delayed Choice interpretation. We show where the momentum change takes place and we conclude that entanglement is the key to understand how complementarity (i.e. Which Way versus interference) and the position-momentum uncertainty relation are interwoven.
The realization of the optical stochastic (or nonstationary) field interferometer (Sto-IF) is reported. By this device, whose properties are determined by those of the stochastic beam splitter (Sto-BS), relevant Bose-Einstein correlations are generated within the investigated particle field. A very general quantum theory of Sto-BS and Sto-IF is reported. Because it accounts for any distribution function of the Sto-BS driving field, this theory embodies the formal content and the results of previous interferometry theories. All theoretical results are substantiated by an experiment carried out with a Mach-Zehnder Sto-IF. Basic considerations on Bohr's complementarity in stochastic interferometry are expressed on the basis of Shannon's information theory applied to the particle welcher weg problem.
This study explores how optical beam splitters provide a conceptual and operational link between classical and quantum descriptions of light. In the classical picture, they are modeled through energy-conserving field amplitudes and a four-port scattering matrix, which successfully explain interference in systems such as Michelson and Mach–Zehnder interferometers. Moving to the quantum domain, the field amplitudes are replaced by bosonic creation and annihilation operators that obey the Weyl–Heisenberg algebra. Using the Glauber displacement operator, we define coherent states whose expectation values reproduce classical monochromatic fields while exhibiting Poissonian photon statistics. This unified approach shows that beam splitters diagnose the nature of incident light: coherent states yield classical interference, while indistinguishable single-photons reveal quantum signatures such as antibunching and Hong–Ou–Mandel interference. The analysis relies on operator formalism and correlation functions that capture both average-field behavior and nonclassical statistics. As beam-splitting technologies transition from bulk optics to integrated photonics and metasurfaces, this framework offers a consistent basis for modeling optical circuits across the classical-quantum divide. By highlighting the dual role of the beam splitter as a passive device and a quantum probe, we emphasize its importance for quantum optics, quantum information, and hybrid photonic platforms.