A metasurface is a 2D array of subwavelength-sized elements engineered to control electromagnetic, acoustic, or matter waves [1]. By locally manipulating amplitude, phase, and polarization through resonant interactions and phase discontinuities, metasurfaces achieve unique wavefront shaping capabilities not possible with natural materials.

In matter wave manipulations, subwavelength slits act as quantum meta-atoms [1]. The effective potential energy of a quantum meta-atom plays a crucial role. It determines both the transmission amplitude and the phase of matter waves. These interactions occur as matter waves interact with a metasurface.

In this article, we will discuss how the effective potential energy of a quantum meta-atom controls the transmission and phase of de Broglie matter waves.

## Transmission Amplitude

**Effective Potential Energy (\( U_{\text{eff}} \)):**

– The effective potential energy \( U_{\text{eff}} \) is inversely proportional to the square of the slit width \( a \) in the dielectric film [1]:

\[ U_{\text{eff}} = \frac{\hbar^2 \pi^2}{2 m a^2} \]

where \( \hbar \) is the reduced Planck constant, \( m \) is the mass of the particle, and \( a \) is the slit width.

**Propagation of Matter Waves:**

– When the energy of the incident matter waves (\( E \)) is greater than \( U_{\text{eff}} \), the waves can propagate through the slits efficiently. This is because the effective potential energy is not high enough to fully impede the wave propagation.

– If \( E \) is less than \( U_{\text{eff}} \), the matter waves undergo evanescent decay within the slits, leading to negligible transmission due to tunneling effects.

**Transmission Efficiency:**

– At the point where \( E \) matches \( U_{\text{eff}} \), there is a transition region where the transmission efficiency can vary significantly. Properly designing the slit widths can maximize the transmission for specific wavelengths of matter waves.

## Transmission Phase

**Phase Shift:**

– The phase shift experienced by matter waves as they pass through the slits is also influenced by \( U_{\text{eff}} \). As the effective potential changes, it alters the propagation constant \( q_0 \) [1]:

\[ q_0 = \sqrt{\frac{2m (E – U_{\text{eff}})}{\hbar^2}} \]

– The transmission phase \( \phi_t \) can be expressed as:

\[ \phi_t = \arg(T_0) \]

where \( T_0 \) is the zero-order transmission coefficient, dependent on \( q_0 \).

**Phase Control:**

– By varying the slit width \( a \), the effective potential \( U_{\text{eff}} \) can be adjusted, thereby controlling the propagation constant \( q_0 \) and the resulting phase shift.

– This ability to tune the phase through changes in slit dimensions allows for precise control over the wavefront of the transmitted matter waves, enabling applications such as focusing, beam steering, and holography.

## Practical Example

**Waveguide Resonance and Rayleigh Anomaly:**

– At specific slit widths, the system can achieve waveguide resonance, where the transmission efficiency peaks due to constructive interference within the slits.

– At the Rayleigh anomaly (where the wavelength of the incident wave matches the periodicity of the slit array), the phase shift shows a predictable behavior, further enabling control over the wavefront.

## Summary

The effective potential energy \( U_{\text{eff}} \) of a quantum meta-atom influences:

**1. Transmission Amplitude:**

– Higher \( U_{\text{eff}} \) (narrower slits) reduces transmission for waves with energy \( E < U_{\text{eff}} \).

– Lower \( U_{\text{eff}} \) (wider slits) allows efficient propagation of matter waves.

**2. Transmission Phase:**

– Varying \( U_{\text{eff}} \) changes the propagation constant \( q_0 \), thereby controlling the phase shift.

– This tuning capability is crucial for wavefront manipulation, enabling advanced applications in quantum optics and related fields.

By manipulating \( U_{\text{eff}} \) through slit design, one can optimize the behavior of matter waves for various technological applications.

## References

[1] Xiao, Wan-yue, and Cheng-ping Huang. “Metasurfaces for de Broglie waves.” Physical Review B 104.24 (2021): 245429.