Experimental investigation of the uncertainty principle for radial degrees of freedom

While the uncertainty principle for linear position and linear momentum, and more recently for angular position and angular momentum, is well established, its radial equivalent has so far eluded researchers. Here we exploit the logarithmic radial position, ln   r , and hyperbolic momentum, P H , to formulate a rigorous uncertainty principle for the radial degree of freedom of transverse light modes. We show that the product of their uncertainties is bounded by Planck’s constant, Δ ln   r · Δ P H ≥ ℏ / 2 , and identify a set of radial intelligent states that satisfy the equality. We illustrate the radial uncertainty principle for a variety of intelligent states, by preparing transverse light modes with suitable radial profiles. We use eigenmode projection to measure the corresponding hyperbolic momenta, confirming the minimum uncertainty bound. Optical systems are most naturally described in terms of cylindrical coordinates, and our radial uncertainty relation provides the missing piece in characterizing optical quantum measurements, providing a new platform for the fundamental tests and applications of quantum optics.