Photon momentum operator

Victor Nikolaevich Popov, Vladimir Sergeevich Yarunin

Toshio Inagaki

A quantum-mechanical description of a photon in refractive mediums is proposed. Here we assume that the medium is isotropic and linear, and the material constants (the dielectric constant and permeability) are both the functions of the position coordinates only and dispersionless. It is found that the momentum operator is generally ill defined as well as the position operator. The momentum operator, however, becomes approximately well defined when the extents of the appreciable changes of the material constants are very large compared to the photon wavelengths. Then, it is shown that the various expressions of the physical quantities in the classical electromagnetic theory, such as the energy and momentum, can be obtained from the expectation values of the corresponding operators. The Abraham-Minkowski controversy in the classical electromagnetic theory is glanced from the viewpoint of our quantum-mechanical approach.

Kyle E. Ballantine, Michaël Mazilu

When light scatters from an object, it can impart some physical quantity such as momentum or angular momentum. This can act as a measurement on the photon, which collapses on to an eigenstate of the measurement operator. However the corresponding operator is not the same as that describing the total linear or angular momentum in free space. Optical eigenmodes provide a powerful method to describe this interaction by expanding the field as a linear combination of some basis modes and examining the eigenvalues and eigenvectors of the quadratic measure in question. We extend this to the quantum case by writing the quantum operator corresponding to a given measurement such as energy, momentum or angular momentum as a superposition of creation and annihilation operators for each eigenmode. Upon measurement we find that the possible states of a single photon are simply the classical eigenmodes of the measurement. As an application, we examine the force and torque on a general, possibly anisotropic, material. By looking at eigenvalues of the measurement operator we show that the amount of a given quantity transferred in an interaction with matter is not in general the expected amount which a photon carries in free space, even at the single photon level. In particular the difference in linear or angular momentum from before and after is in general not equal to ~k or ~ which are the eigenvalues of these quantities in free space.

J. Bertrand

Michał Dobrski, Maciej Przanowski, Jaromir Tosiek +1 more

Vlasta Peřinová, A. Lukš, J. Křepelka

Quasiprobability distributions of the photon number and the momentum-like quadrature appropriate to partial symmetrizations of products of the photon-number and quadrature operators allowing these operators to alternate at most (2r-2) times are good approximations to the Wigner function for these simultaneously considered noncommuting quantities related to the fully symmetric ordering. It is shown that the r-parametrized quasiprobability distributions must take on values at multiples of 1/r. A scheme for simultaneous measurement of the photon number and the momentum-like quadrature is proposed. The r-parametrized quasidistributions can be compared with a joint distribution of the photon number and the momentum-like quadrature measured simultaneously.