Phase Operator in Quantum Mechanics : Phase and the Number-Phase Uncertainty of the Electromagnetic Field

A. V. Kozlovskiĭ

Abstract We examine various uncertainty relations: (electromagnetic-field phase (the so-called “Pegg–Barnett operator”))–(number of photons), (trigonometric functions of the phase)–(number of photons), and (trigonometric functions of the phase)–(field phase). An electromagnetic field in quantum states of phase superpositions of coherent states, which are general superpositions of two coherent states with identical moduli but arbitrary phases, is considered. The rigorous uncertainty relation (Cauchy inequality) and the soft uncertainty relation (Heisenberg inequality) are examined and compared.

Julio Alberto Gallardo

D. T. Smithey, M. Beck, J. Cooper +1 more

We have experimentally determined all of the quantities involved in the uncertainty relation for the phase and photon number of a mode of the electromagnetic field when the field mode is in a coherent state of small average photon number. This is accomplished by determining the quantum state of the field using optical homodyne tomography, which uses measured distributions of electric-field quadrature amplitude to determine the Wigner function and hence the density matrix. The measured state is then used to calculate the uncertainty product for the number and phase, as well as the expectation value of the commutator of the number and phase operators. The experimental results agree with the quantum-mechanical predictions. We also present measured phase- and photon-number distributions for these weak coherent states, as well as their measured complex wave functions.

V. M. Faǐn

V. M. Faǐn

M. Beck, D. T. Smithey, J. Cooper +1 more

The uncertainty principle applied to electromagnetic fields implies that the field phase and photon number cannot simultaneously have arbitrarily narrow distributions. Here we describe an experiment which determines, for the first time, the variances of phase and photon number for a field containing small numbers of photons. The measurements allow a test of the uncertainty relation for the phase and photon number of a mode of the electromagnetic field when the field mode is in a very weak coherent state. The phase of the field is described by the Pegg-Barnett phase operator formalism. [1] The measurements are accomplished by experimentally determining the quantum state of the field by using optical homodyne tomography. [2] This method of state characterization requires that an ensemble of many measurements be performed. [3] In this case, the relevant interpretation of the uncertainty relation is that the state of the field does not have well defined phase and photon number.