Entangled photon–electron states and the number-phase minimum uncertainty states of the photon field

Abstract

The exact analytic solutions of the energy eigenvalue equation of the system consisting of a free electron and one mode of the quantized radiation field are used for studying the physical meaning of a class of number-phase minimum uncertainty states. The states of the mode which minimize the uncertainty product of the photon number and the Suskind and Glogower (1964 Physics 1 49–61) cosine operator have been obtained by Jackiw (1968 J. Math. Phys. 9 339–46). However, these states have so far remained mere mathematical constructions without any physical significance. It is proved that the most fundamental interaction in quantum electrodynamics—namely the interaction of a free electron with a mode of the quantized radiation field—leads quite naturally to the generation of the mentioned minimum uncertainty states. It is shown that from the entangled photon–electron states developing from a highly excited number state, due to the interaction with a Gaussian electronic wave packet, the minimum-uncertainty states of Jackiw's type can be constructed. In the electron's coordinate representation, the physical meaning of the expansion coefficients of these states is the joint probability amplitudes of simultaneous detection of an electron and of a definite number of photons. The photon occupation probabilities in these states preserve their functional form as time elapses, but they depend on the location in space-time of the detected electron. An analysis of the entanglement entropies derived from the photon number distribution and from the electron's density operator is given.