Quantum mechanics, developed a century ago, has long challenged conventional views of nature. At its core lies the principle of wave-particle duality, which shows that quantum objects can behave like both waves and particles, depending on how they are observed. A new study from the Stevens Institute of Technology, published in Physical Review Research, introduces a formula that precisely defines the relationship between these two behaviors.
Wave-particle duality is the concept that quantum entities, like photons and electrons, behave as both waves and particles. Experiments such as Young’s double-slit show interference patterns (wave behavior), while the photoelectric effect demonstrates discrete energy packets (particle behavior). Louis de Broglie extended this idea to matter, proposing that particles have associated wavelengths. Quantum mechanics, through the Schrödinger equation, formalizes this duality, showing that the nature observed depends on measurement. This duality challenges classical intuition and underpins technologies like electron microscopy and quantum computing.
“Wave-particle duality is the cornerstone of quantum mechanics,” said Xiaofeng Qian, lead author of the paper and Assistant Professor of Physics at Stevens. “Researchers have been working to quantify wave-particle duality for half a century, but this is the first complete framework to fully quantify wave-like and particle-like behaviors with optimum quantitative measures that are relevant at the quantum level.”
Earlier models expressed wave-ness and particle-ness as an inequality, where the sum of the two behaviors was equal to or less than one. This suggested that if an object displayed fully wave-like properties, it would show no particle-like properties, and vice versa. However, these models allowed for scenarios in which both behaviors could increase simultaneously, contradicting the exclusive nature of the relationship.
To address this, Qian and his colleagues introduced coherence as a new variable. Coherence describes the hidden potential for wave-like interference. “Coherence is a tricky concept, but it’s essentially a hidden description of the potential for wave-like interference,” Qian explained. “And the conventional measure visibility represents the amount of wave-ness can be extracted. When we quantify and compensate for coherence, alongside the standard metrics for wave-ness and particle-ness, we find they add up to exactly one.”
The team’s analysis revealed what they call a duality ellipse (DE) equality, a closed-form mathematical relationship that unifies visibility (wave-ness), predictability (particle-ness), and coherence. This framework provides a complete embodiment of Bohr’s complementarity principle, showing that the interplay of these quantities can be represented as an ellipse. In perfectly coherent systems, the relationship forms a quarter-circle; as coherence declines, the ellipse flattens.
The researchers extended their framework to quantum imaging with undetected photons (QIUP), a technique where one of a pair of entangled photons scans an aperture. If the photon passes through freely, coherence remains high; if it collides with the aperture walls, coherence decreases. By measuring the wave-ness and particle-ness of the entangled partner photon, the team could deduce coherence and reconstruct the aperture’s shape. The study introduces the concept of an imaging duality ellipse (IDE), which directly connects wave-particle duality to the transmittance profile of the object being scanned. This relation enables characterization of objects through duality measurements alone and remains robust against experimental imperfections such as decoherence and misalignment.
“This shows that the wave-ness and particle-ness of a quantum object can be used as a resource in quantum imaging, and potentially many other quantum information or computational tasks,” Qian said. Importantly, imaging remained possible even when external factors like temperature or vibrations reduced overall coherence. Because these factors affect both high and low coherence scenarios equally, the difference between them can still be detected. “The ellipse gets squeezed, but we’re still able to extract the information of the object we need,” Qian noted.
The findings advance both the theoretical and practical understanding of quantum duality. By providing a systematic framework for quantifying coherence alongside wave-ness and particle-ness, the research offers a toolkit for optimizing coherence-driven quantum technologies, including imaging and sensing. Still, Qian emphasized that further research is needed, especially in more complex multipath quantum systems. “The mathematics make it look simple, but we’re a long way from exhausting the weirdness of quantum mechanics,” he said. “There are still plenty of frontiers left for us to explore.”
Source: Stevens University of Technology, APS
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