**Key takeaways**: The article provides insights into Poisson’s spot for matter waves, highlighting its historical context, experimental observations, and mathematical framework. Key takeaways include the significance of wave-particle duality, the impact of van der Waals forces on diffraction patterns, and applications in nanotechnology and precision measurements using atom interferometers.

## What is Poisson’s spot?

**Poisson’s spot**, also known as the **Arago spot**, is a phenomenon where a bright point appears at the center of the shadow of a circular object due to wave diffraction. While originally observed with light, this effect is equally interesting when studied with matter waves. It offers profound insights into the wave-like behavior of particles.

## Historical Context

Poisson’s spot has an interesting historical context rooted in the early 19th-century debates on the nature of light. The phenomenon is named after the French physicist SimÃ©on Denis Poisson, who was a prominent advocate of the particle theory of light.

In 1818, Augustin-Jean Fresnel presented a comprehensive theory of wave optics to the **French Academy of Sciences**, including his light diffraction principles. SimÃ©on Denis Poisson, a member of the judging committee and a supporter of Isaac Newton’s particle theory of light, attempted to disprove Fresnel’s wave theory by deriving what he believed to be an absurd consequence.

Poisson argued that if light were a wave, then according to Fresnel’s theory, a bright spot should appear at the center of the shadow of a small circular obstacle due to constructive interference of the diffracted waves. He considered this prediction to be so counterintuitive that it would refute the wave theory.

However, another member of the committee, FranÃ§ois Arago, conducted an experiment to test this prediction. To the surprise of Poisson and the scientific community, Arago observed the predicted bright spot at the center of the shadow, thus providing strong evidence in favor of the wave theory of light. This experiment, often called the Poisson-Arago experiment, was a crucial turning point that helped establish the wave nature of light.

The discovery of Poisson’s spot validated Fresnel’s wave theory and paved the way for the acceptance of wave optics. Later, the phenomenon’s relevance extended beyond light to matter waves, further illustrating the wave-particle duality in quantum mechanics.

## Poisson’s Spot with Matter Waves

The advent of quantum mechanics introduced the concept of matter waves, first proposed by Louis de Broglie. According to de Broglie, particles such as electrons and atoms exhibit wave-like properties under certain conditions. This wave-particle duality is central to quantum mechanics and has been confirmed by numerous experiments.

Poisson’s spot for matter waves follows the same principles as for light. When a coherent beam of particles, such as electrons or atoms, passes by a circular obstacle, the wave nature of the particles causes them to interfere constructively at the center of the shadow, producing a bright spot.

## Differences Between Poisson’s Spot for Light and Matter Waves

Poisson’s spot manifests for both light and matter waves but with distinct differences. Light waves, with wavelengths in the nanometer range, form Poisson’s spot through interference and diffraction around obstacles, described by classical wave optics. Matter waves, with de Broglie wavelengths dependent on particle momentum, require quantum mechanical treatment and sophisticated setups for observation. The SchrÃ¶dinger equation governs their behavior, showing wavefunction interference.

Unlike light, matter waves are significantly affected by external electric and magnetic fields, altering diffraction patterns. Thus, while both phenomena highlight wave nature, the underlying physics, experimental requirements, and external field sensitivities differ substantially for light and matter waves.

## Significance of Studying Poissonâ€™s Spot for Matter Waves

**Wave-Particle Duality**: Demonstrates the fundamental principle of wave-particle duality in quantum mechanics, confirming the wave nature of particles like electrons, neutrons, and ions.**Quantum Interference**: Provides insights into quantum interference patterns, which are crucial for understanding and developing quantum technologies.**Coherence and Source Quality**: Assesses the coherence and quality of particle sources, essential for precision experiments in matter wave optics.**Field Effects**: Investigates the influence of external electric and magnetic fields on matter waves, aiding in the development of sensitive field detection techniques and quantum sensors.**Fundamental Physics**: Enhances understanding of fundamental physics concepts such as diffraction, superposition, and the behavior of wavefunctions in various potentials.**Technological Applications**: Supports the advancement of technologies like electron holography, matter-wave interferometry, and quantum computing.

## Mathematical Framework

The observation of Poisson’s spot for matter waves can be described mathematically using the principles of wave diffraction. The key equation governing this phenomenon is the Fresnel-Kirchhoff diffraction integral, which describes how waves propagate and interfere:

\[ I(\mathbf{r}) = \left| \frac{1}{i\lambda} \iint_S \frac{e^{ikr}}{r} \cos(\theta) U_0(\mathbf{r’}) dS \right|^2, \]

where:

\( I(\mathbf{r}) \) is the intensity at point \( \mathbf{r} \) on the observation screen.

\( \lambda \) is the wavelength of the matter wave.

\( r \) is the distance from a point on the aperture \( S \) to the observation point \( \mathbf{r} \).

\( \theta \) is the angle between the incident wave and the point on the aperture.

\( U_0(\mathbf{r’}) \) is the initial wave function at point \( \mathbf{r’} \) on the aperture.

In the context of matter waves, the de Broglie wavelength \( \lambda \) is given by:

\[ \lambda = \frac{h}{p}, \]

where \( h \) is Planck’s constant and \( p \) is the momentum of the particle.

## Developments So Far

**Particle-Wave Discrimination:**

In another study by Thomas Reisinger et al. (2011), the particle-wave discrimination in Poisson spot experiments was analyzed [1]. The study highlighted how matter-wave diffraction could be used to probe the fundamental nature of particle-surface interactions. It was shown that the phase shifts due to van der Waals potentials significantly impact the observed diffraction patterns, providing a sensitive method to study these interactions at the nanoscale.

**Gouy Phase and Partially Coherent Matter Waves:**

The study by I.G. da Paz and colleagues (2016) explored the role of the Gouy phase in the intensity of Poisson’s spot for partially coherent matter waves [2]. The Gouy phase is an additional phase shift experienced by a wave as it propagates through a focus, unique to optical and matter waves, causing a phase anomaly near the focal region. The research demonstrated that the Gouy phase significantly affects the central peak’s existence, corroborating the wavelike character of Poisson’s spot. The study provided an analytical model incorporating the Gouy phase and showed remarkable agreement with experimental data for deuterium molecules.

## Latest Advancement

In a recent study by Nicolas Gack and colleagues (2020) [3], indium atoms were used to demonstrate the Poisson’s spot with matter waves at submillimeter distances behind spherical silica particles.

The study investigates the effect of short-range van der Waals (vdW) forces on Poisson spot diffraction. The experiment is conducted in an ultrahigh-vacuum setup, where indium atoms are evaporated, and their diffraction patterns are recorded. The findings show that the vdW forces between the indium atoms and the particle surfaces cause a significant phase shift in the de Broglie waves, leading to an enhanced intensity of the Poisson spot. This enhancement is due to the effective widening of the Fresnel zones near the particles.

The observed diffraction intensities align well with theoretical predictions, confirming the sensitivity of Poisson spot diffraction to vdW forces. This study provides a sensitive method for probing atom-surface interactions, which has implications for nanoscale systems and the development of matter-wave diffractive optics.

## Future Prospect

The study of Poissonâ€™s spot for matter waves provides valuable insights into particle-wave interactions and surface interactions at the nanoscale. This research has significant implications for understanding forces like van der Waals interactions and their impact on particle behavior.

For instance, experiments with indium atoms have shown that van der Waals forces cause measurable phase shifts, enhancing the central spotâ€™s intensity. This sensitivity to small forces makes matter-wave diffraction a powerful tool for probing atomic and molecular interactions.

Understanding these interactions is crucial for developing advanced technologies such as atom interferometers. These devices can measure weak forces with high precision, finding applications in gravitational wave detection, inertial navigation, and fundamental physics tests.

Additionally, such studies contribute to the advancement of quantum sensors, nanotechnology, and material science, where precise control and measurement at the atomic scale are paramount.

## Summary

Poisson’s spot, or Arago spot, is a bright point at the center of a circular object’s shadow, caused by wave diffraction. Historically, it supported wave theory of light against particle theory. This phenomenon also applies to matter waves, showing wave-particle duality central to quantum mechanics.

Matter waves like electrons and atoms form Poisson’s spot through quantum interference, influenced by external fields. Recent studies highlight its importance in understanding quantum interference, field effects, and advancing technologies like electron holography and quantum sensors.

Mathematical models, such as the Fresnel-Kirchhoff diffraction integral, describe this phenomenon. Key advancements include analyzing particle-wave discrimination and phase shifts due to van der Waals forces, enhancing Poisson spot intensity. These studies offer insights into atomic interactions, essential for developing precise quantum devices and nanotechnology applications.

## References

[1] Reisinger, Thomas, Gianangelo Bracco, and Bodil Holst. “Particleâ€“wave discrimination in Poisson spot experiments.”Â *New Journal of Physics*Â 13.6 (2011): 065016.

[2] da Paz, I. G., et al. “Poisson’s spot and Gouy phase.”Â *Physical Review A*Â 94.6 (2016): 063609.

[3] Gack, Nicolas, et al. “Signature of short-range van der Waals forces observed in Poisson spot diffraction with indium atoms.”Â *Physical Review Letters*Â 125.5 (2020): 050401.