Minimum uncertainty measurements of angle and angular momentum

Area of Science:

• Quantum mechanics
• Quantum optics
• Wave mechanics

Background:

• The angle-angular momentum conjugate pair is fundamental in quantum mechanics.
• Existing descriptions face challenges in defining resolution and optimal states.
• Understanding these properties is crucial for quantum information processing.

Purpose of the Study:

• To develop an accurate description of the angle-angular momentum conjugate pair.
• To identify optimal quantum states that minimize uncertainty under specific constraints.
• To experimentally validate the theoretical framework using photonic systems.

Main Methods:

• Describing the angle-angular momentum using the exponential of the angle.
• Deriving optimal states (intelligent states) using Mathieu wave functions.
• Employing Gaussian approximations with von Mises distributions.
• Applying the theory to the spatial degrees of freedom of a photon.
• Conducting an experiment with computer-controlled spatial light modulators.

Main Results:

• Dispersion is established as a natural measure of resolution.
• Mathieu wave functions are identified as optimal states for angle-angular momentum.
• Gaussian approximations to these states are provided using von Mises distributions.
• The theory is successfully applied to photon spatial modes.
• Experimental verification confirms the theoretical predictions.

Conclusions:

• The proposed exponential description offers a more accurate framework for angle-angular momentum.
• Mathieu wave functions represent a significant finding for optimal quantum states in this context.
• The experimental validation demonstrates the practical applicability of the theory in quantum optics.