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Scientific Reports

Which-path information of path-polarization hybrid state in polarization-based nested Mach-Zehnder interferometer

Danbi Kim, Min-Su Kim, Han Seb Moon

Published Apr 12, 2025• 3 citations
Publisher

Abstract

The two-state vector formalism (TSVF) describes a quantum system between two measurements, with a pre- and post-selected states evolving forward and backward in time, respectively. In this paper, we investigate the which-path information and the TSVF analysis considering the polarization and path states in a polarization-based nested Mach-Zehnder interferometer (NMZI). By adjusting photon polarization in the multipath via polarization optics, we can investigate the which-path information of the photons after the polarization beam splitter in the polarization-based NMZI under conditions of constructive or destructive interference conditions. Considering the incomplete polarization states of photons in the NMZI, the interpretation of our experimental results does not contradict the TSVF analysis.

DOI
10.1038/s41598-025-89877-2

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The two-state vector formalism (TSVF) describes a quantum system between two measurements, with a pre- and post-selected states evolving forward and backward in time, respectively. In this paper, we investigate the which-path information and the TSVF analysis considering the polarization and path states in a polarization-based nested Mach-Zehnder interferometer (NMZI). By adjusting photon polarization in the multipath via polarization optics, we can investigate the which-path information of the photons after the polarization beam splitter in the polarization-based NMZI under conditions of constructive or destructive interference conditions. Considering the incomplete polarization states of photons in the NMZI, the interpretation of our experimental results does not contradict the TSVF analysis.
Recently, A. Danan et al. [Phys. Rev. Lett. 111, 240402 (2013)] performed a much-discussed experiment in which which-way information was obtained from the light in a nested Mach-Zehnder interferometer by weak measurement. The presented analysis using the two-state vector formalism drew the conclusions that the photons followed disconnected paths. We analyze this experiment using standard quantum optical methods and arrive at analytical expressions that match the experimental results without the need for such disconnected photon paths. We also propose a simple amendment to the experiment capable of displaying new phenomena, highlighting the advantages of our description.
The chapter discusses the Mach–Zehnder interferometer (MZI), which uses beam splitters in combination with mirrors to place a photon into a superposition of paths. In a final beam splitter, the two paths may be recombined to interfere, showing wave behavior. If the final beam splitter is removed, the photon is definitely in one path or the other, showing particle behavior. The polarization of light allows additional applications—one can use a polarizing beam splitter to separate and reconstitute the wave functions of a photon polarized at an angle. In a delayed-choice experiment, one decides which kind of experiment one is to do, particle or wave, after the photon has already entered the arms of the MZI. The chapter discusses a version of this that puts the MZI and a second-detector photon in an entangled state of particle and wave results.
We present the results of a theoretical work discussing the propagation of an electromagnetic wave through nested Mach-Zehnder interferometers using classical optics and standard quantum theory. We show that some seemingly surprising effects at first sight, which are often explained in the literature using the two-state vector formalism (TSVF), are a direct consequence of destructive or constructive interference and thus there is no need for the unconventional TSVF formulation. We show that the probability of a photon detection derived from the weak value used in TSVF can be interpreted as the probability of an encounter of two opposing photon fluxes.
Abstract We show that two-state vector formalism (TSVF), applied to quantum systems that make use of delicate interference effects, can lead to paradoxes. We consider a few schemes of nested Mach–Zehnder interferometers that make use of destructive interference. A particular interpretation of TSVF applied to these schemes makes predictions that are contradictory to quantum theory and can not always be verified. Our results suggest that TSVF might not be a suitable tool to describe quantum systems that make use of delicate quantum interference effects.
Which-path information of a quantum particle in interferometers is the key to infer the past of the quantum particle. It has led to many extensive discussions, including discussions of quantum complementarity and the path-visibility relation. The basics of these discussions are the description, detection, and control of which-path information. In this paper, we focus on the investigation of multidimensional which-path information in a nested Mach-Zehnder interferometer. A general expression of which-path information is given and can be partially extracted by different detection methods. Further analysis shows that the which-path information can be controlled by the phase differences and beam-splitting ratios between the arms of the nested Mach-Zehnder interferometer. Moreover, an alternative which-path information elimination phenomenon has been predicted and demonstrated experimentally. Our work can help us to understand the physics of quantum particles, with potential applications for quantum information processing and quantum metrology.
Linear-optical interferometers play a key role in designing circuits for quantum information processing and quantum communications. Even though nested Mach-Zehnder interferometers appear easy to describe, there are occasions when they provide unintuitive results. This paper explains the results of a highly discussed experiment performed by Danan et al. [Phys. Rev. Lett. 111, 240402 (2013).] using a standard approach. We provide a simple and intuitive one-state vector formalism capable of interpreting their experiment. Additionally, we cross-checked our model with a classical-physics-based approach and found that both models are in complete agreement. We argue that the quantity used in the mentioned experiment is not a suitable which-path witness, producing seemingly contraintuitive results. To circumvent this issue, we establish a more reliable which-path witness and show that it yields well-expected outcomes of the experiment.