Minimum Uncertainty Measurements of Angle and Angular Momentum

Abstract

The uncertainty relations for angle and angular momentum are revisited. We use the exponential of the angle instead of the angle itself and adopt dispersion as a natural measure of resolution. We find states that minimize the uncertainty product under the constraint of a given uncertainty in angle or in angular momentum. These states are described in terms of Mathieu wave functions and may be approximated by a von Mises distribution, which is the closest analogous of the Gaussian on the unit circle. We report experimental results using beam optics that confirm our predictions.PACS numbers: 42.50.Dv, 42.50.Vk Light carries and transfers energy as well as linear and angular momentum. The angular momentum contains a spin contribution, associated with polarization, and an orbital component, linked with the spatial profile of the light intensity and phase [1].The seminal paper of Allen et al [2] firmly establishes that the Laguerre-Gauss modes, typical of cylindrical symmetry, carry a well-defined angular momentum per photon. In the paraxial limit, this orbital component is polarization independent and arises solely from the azimuthal phase dependence e imφ , which gives rise to spiral wave fronts. The index m takes only integer values and can be seen as the eigenvalue of the orbital angular momentum operator. In consequence, the Laguerre-Gauss modes constitute a complete set and can be used to represent quantum photon states [3,4,5].The possibility of exploiting these light fields for driving micromachines, and their applications as optical tweezers and traps, have attracted a good deal of attention [6,7,8]. Moreover, entangled photons prepared in a superposition of states bearing a well-defined orbital angular momentum provide access to multidimensional entanglement. This is of considerable importance in quantum information and cryptography because with these states more information can be stored and there is less sensitivity to decoherence [9,10,11,12,13].Recently, by precise measurements on a light beam, an experimental test of the uncertainty principle for angle and angular momentum has been demonstrated [14,15]. The idea is to pass the beam through an angular aperture and measure the resulting distribution of angularmomentum states [16]. Moreover, one can even identify the form of the aperture that corresponds to the minimum uncertainty states for these variables.In the following, we deal with cylindrical symmetry; we are concerned with the planar rotations by an angle φ in the plane x-y, generated by the angular momentum along the z axis, which for simplicity will be denoted henceforth asL. In this respect, we recall that the proper definition of angular variables in quantum mechanics is beset by well-known difficulties [17,18]. For the case of a harmonic oscillator, the problems essentially arise from two basic sources: the periodicity and the semiboundedness of the energy spectrum. The first prevents the existence of a phase operator in the infinite-dimensional Hilbert space, but not of its exponential. ...

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