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.
Ordinary intelligent states (OISs) hold equality in the Heisenberg uncertainty relation involving two noncommuting observables
A,B, whereas generalized intelligent states (GISs) do so in the more generalized uncertainty relation, the Schr\"odinger-Robertson inequality. In general, OISs form a subset of GISs. However, if there exists a unitary evolution
U that transforms the operators
A,B to a new pair of operators in a rotation form, it is shown that an arbitrary GIS can be generated by applying the rotation operator
U to a certain OIS. In this sense, the set of OISs is unitarily equivalent to the set of GISs. It is the case, for example, with the su(2) and the su(1,1) algebras which have been extensively studied, particularly in quantum optics. When these algebras are represented by two bosonic operators (nondegenerate case), or by a single bosonic operator (degenerate case), the rotation, or pseudorotation, operator
U corresponds to phase shift, beam splitting, or parametric amplification, depending on two observables
A,B.
The equality in the uncertainty principle for linear momentum and position is obtained for states which also minimize the uncertainty product. However, in the uncertainty relation for angular momentum and angular position both sides of the inequality are state dependent and therefore the intelligent states, which satisfy the equality, do not necessarily give a minimum for the uncertainty product. In this paper, we highlight the difference between intelligent states and minimum uncertainty states by investigating a class of intelligent states which obey the equality in the angular uncertainty relation while having an arbitrarily large uncertainty product. To develop an understanding for the uncertainties of angle and angular momentum for the large-uncertainty intelligent states we compare exact solutions with analytical approximations in two limiting cases.