Localized Single-Photon Wave Functions in Free Space

Abstract

We solve the joint open problems of photon localization and single-photon wave functions in the context of spontaneous emission from an excited atom in free space. Our wave functions are well-defined members of a discrete orthonormal function set. Both the degree and shape of the localization are controlled by entanglement mapping onto the atom wave function, even though the atom is remote from the photon.

Alternative perspective on photonic tunneling

“…However, in spite of QED's great success as well as the traditional conclusion that single photon cannot be localized [4][5], there have been many attempts to develop photon wave mechanics which is based on the concept of photon wave function and contains the first-quantized theory of the photon [6][7][8][9][10][11][12][13], and some recent studies have shown that photons can be localized in space [14][15][16]. These efforts have both theoretical and practical interests.…”

Photon localization barrier can be overcome

“…(6), (9), and (10). However, at any spatial location the wave function is square integrable with respect to time, thus the condition of the Paley-Wiener theorem has been satisfied.…”

“…The common explanation presented in textbooks (e.g., [3], [4]) may be summed up as follows: (i) no position operator exists for the photon, (ii) while the position wave function may be localized near a space-time point, the measurable quantities like the electromagnetic field vectors, energy, and the photodetection probability remain spread out due to their non-local relation with the position wave function. However, just before the turn of the century both of these widely-espoused notions were disproved [5], [6] and in the new century a fresh interest in the photon localization problem seems to have been awakened (see, e.g., [7],[8], [9]), meeting the needs of developments in near-field optics, cavity QED, and quantum computing.I. Bialynicki-Birula writes [6] that the statement "even when the position wave function is strongly concentrated near the origin, the energy wave function is spread out over space asymptotically like r −7/2 " (citation from [4], p. 638) is incorrect and that both wave functions may be strongly concentrated near the origin.…”

Momentum entanglement and disentanglement between an atom and a photon

“…As in the preceding section, to study the generation of momentum entanglement [3,4,5,6], it is usually convenient to assume the entangled system to be a close pure state system. However, in a realistic environment with T ≫ 0K, the interaction with environment will make the entangled system into a mixed state, and, as a result, cause the disentanglement.…”

“…As a physical realization, momentum entanglement has been extensively studied both theoretically [3,4,5,6,7] and experimentally [8]. With momentum entanglement between atom and photon, it is possible to define the best localized single-photon wavepacket even in free space [3], and realize the highest degree of continuous entanglement [4] up to date.…”