Space-time propagation of photon pulses in dielectric media, illustrations with beam splitters

Abstract

Photons are the elementary quantum excitations of the electromagnetic field. Quantization is usually constructed on the basis of an expansion in eigenmodes, in the form of plane waves. Since they form a basis, other electromagnetic configurations can be constructed by linear combinations. In this presentation we discuss a formulation constructed in the general formalism of bosonic Fock space, in which the quantum excitation can be constructed directly on localized pulses of arbitrary shape. Although the two formulations are essentially equivalent, the direct formulation in terms of pulses has some conceptual and practical advantages, which we illustrate with some examples. The first one is the passage of a single photon pulse through a beam splitter. The analysis of this formulation in terms of pulses in Fock space shows that there is no need to introduce “vacuum fluctuations entering through the unused port”, as is often done in the literature. Another example is the Hong-Ou-Mandel effect. It is described as a time dependent process in the Schrödinger representation in Fock space. The analysis shows explicitly how the two essential ingredients of the Hong-Ou-Mandel effect are the same shape of the pulses and the bosonic nature of photons. This formulation shows that all the phenomena involving linear quantum optical devices can be described and calculated on the basis of the time dependent solution of the corresponding classical Maxwell’s equations for pulses, from which the quantum dynamics in Fock space can be immediately constructed.

Isomorphism between the Białynicki-Birula and the Landau-Peierls Fock space quantization of the electromagnetic field in position representation

“…where η j are elements in H, the notation ¡ ¡ η j indicates that this term is missing and Ŝl are symmetrization operators. The bosonic creation-annihilation operators satisfy the following commutation relations [19, equation (3.192)], [30,32]…”

“…Remark. We point out that the interpretation of the field operators ⃗ Ψ † and ⃗ Ψ is not to be confused with that of the creation-annihilation operators B † Maxwell's equations [30], while each component Ψ † j is an operator-valued distribution which has to be integrated over the whole space acting on a test function to properly create a photon.…”

Non-local property of single-photon states: Illustration with spontaneously emitted photon from a Hydrogen atom

“…The ability to produce pulse-shaped single-photons [1] has brought back the importance of their description in position space [2,3]. Our recent works show that instead of defining photons through the decomposition of the field into plane waves, one can write the theory with bosonic creation-annihilation operators directly on pulses of arbitrary shape…”

“…where 𝜓 ⃗ is a classical complex representation of the pulse in position space [2,3]. The dynamics associated to this single-photon pulse is then given directly by 𝐵 ̂𝜓 ⃗⃗⃗ (𝑡) † |∅ >…”