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arXiv:2502.04010 · Physical Review Research

Optical interference by amplitude measurement

Yunxiao Zhang, Xuan Tang, Xueshi Guo, Liang Cui, Xiaoying Li, Z. Y. Ou

Published Mar 7, 2025• 4 citations
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Abstract

Interference effects are usually observed by intensity measurement. Path indistinguishability by the quantum complementarity principle requires projection of the interfering fields into a common indistinguishable mode before detection. On the other hand, the essence of wave interference is the addition of amplitudes of the interfering fields. Therefore, if amplitudes can be directly measured and added, interference can occur even though the interfering fields are in well-distinguishable modes. Here, we make a comprehensive study in both theory and experiment of a technique by homodyne measurement of field amplitudes to reveal interference. This works for both classical and quantum fields even though there exists distinguishability in the interfering paths of light. This directly challenges the complementarity principle. We present a resolution of this issue from the viewpoint of measurement that emphasizes either particle or wave. This technique is particularly useful for recovering interference in unbalanced interferometers with path imbalance beyond a coherence length of the input field and can be applied to remote sensing to extend the applicable range. Since the amplitude-based interference phenomena studied here are fundamentally different from the traditional intensity-based interference phenomena, our approach leads to a new paradigm to study coherence between optical fields.

DOI
10.1103/physrevresearch.7.013255

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Interference effects are usually observed by intensity measurement. Path indistinguishability by the quantum complementarity principle requires projection of the interfering fields into a common indistinguishable mode before detection. On the other hand, the essence of wave interference is the addition of amplitudes of the interfering fields. Therefore, if amplitudes can be directly measured and added, interference can occur even though the interfering fields are in well-distinguishable modes. Here, we make a comprehensive study in both theory and experiment of a technique by homodyne measurement of field amplitudes to reveal interference. This works for both classical and quantum fields even though there exists distinguishability in the interfering paths of light. This directly challenges the complementarity principle. We present a resolution of this issue from the viewpoint of measurement that emphasizes either particle or wave. This technique is particularly useful for recovering interference in unbalanced interferometers with path imbalance beyond a coherence length of the input field and can be applied to remote sensing to extend the applicable range. Since the amplitude-based interference phenomena studied here are fundamentally different from the traditional intensity-based interference phenomena, our approach leads to a new paradigm to study coherence between optical fields.
Optical interference usually requires two interfering paths to be balanced within the coherence length. For the case of unbalanced paths, interference effects can be recovered in the frequency-domain analysis of the superposed resultant optical field because optical frequency filtering in the spectral analysis of the optical field lengthens the coherence length to overcome the path imbalance. However, since optical detection usually is intensity measurement that loses the phase information of the detected optical field, the output photocurrent of intensity measurement cannot recover the lost interference effect in the postprocessing of the data such as electronic spectral analysis. On the other hand, if amplitude measurement can be used to measure the resultant optical field from the path-unbalanced superposition of two fields, the output photocurrent carries complete information of two fields and can restore interference in postdetection processing, even though no interference shows up in direct intensity measurement. In this paper, we use homodyne detection method for amplitude measurement to analyze the output of a path-unbalanced interferometer in the spectral domain. Since complete information about the superposed optical field is contained in the photocurrent, spectral analysis shows interference fringes in frequency that reveal information about the path imbalance. It is found that shot noise is the limiting factor in the visibility of interference but can be reduced with a squeezed state input in the unused port of the unbalanced interferometer. This approach is a different paradigm from the traditional one for studying the coherence between optical fields and should find wide application in remote sensing and precision measurement.
It is known that quantum interference can disappear with the mere possibility of distinguishability without actually performing the act. We create such distinguishability in an unbalanced SU(1,1) interferometer and indeed observe no interference in the direct photodetection of the outputs. On the other hand, such distinguishability can be erased with a projective measurement. Here, we report a method of homodyne detection that can also recover interference effect. We find that it is the indistinguishability in amplitude measurement that leads to the recovery of interference, and the quantum nature of homodyne detection and the detector's slow response time both play an essential role. This is different from the quantum eraser schemes mentioned above. It demonstrates that quantum interference occurs in the measurement processes. With no need for path compensation, the unbalanced interferometers studied here should have practical applications in quantum metrology and sensing.
Traditionally, observation of optical interference in intensity requires the balance of the paths of the two interfering fields within the coherence length of the optical fields. With intensity correlation fourth-order measurement between two detectors, such a requirement can be relaxed. Here, we report the observation of interference in a path-unbalanced Mach-Zehnder interferometer by mixing the output field with a weak coherence reference field before making a fourth-order intensity correlation measurement. Detailed theoretical analysis is made and is consistent with the experimental observation. As a side product of the study, when the coherent reference field is not present, the path-unbalanced interferometer realizes a Hong-Ou-Mandel interference effect that can be used to measure the coherence time of the input field with g(2)=1{g}^{(2)}=1g(2)=1. Such a technique of recovering interference, which is otherwise absent with the traditional observation method, should find wide applications in optical sensing by extending the applicable range.
Over the past several decades, single photon-based path interference in double-slit experiment has been well demonstrated for the particle nature of photons satisfying complementarity theory, where the quantum mechanical interpretation of the single photon interference fringe is given by Born rule for complex amplitudes in measurements. Unlike most conventional methods using entangled photon pairs, here, a classical coherent light source is directly applied for the proof of the same self-interference phenomenon. Instead of double slits, we use a Mach-Zehnder interferometer as usual, where the resulting self-interference fringe of coherent single photons is exactly the same as that of the coherent photon ensemble of the laser, demonstrating that the classical coherence based on the wave nature is rooted in the single photon self-interference. This unexpected result seems to contradict our common understanding of decoherence phenomena caused by multi-wave interference among bandwidth distributed photons in coherence optics.