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International Convention on Information and Communication Technology, Electronics and Microelectronics

The electric field of a single photon

Andrea Aiello

Published May 23, 2022• 0 citations
Publisher

Abstract

It is well known that the expectation value of the electric field operator in a number (Fock) state vanishes. However, a measurement of the electric field must yield some nonzero value each time an experiment is performed. What values can be obtained, and with what probability, is perhaps less well known. In this work we determine the probability distribution for the electric-field amplitude of an optical beam prepared in a single-photon state, using some techniques of quantum field theory. For the purpose of illustration, we apply our theory to the analysis of an ideal experiment where two spatially separated detectors can measure either the amplitude of the electric field, or the intensity of a single-photon optical beam. We find that while the photon cannot be detected in two disjoint positions at the same time, a measurement of the amplitude of the electric field carried by ditto photon, would yield a nonzero value at both detectors. These results show that a quantum field is both local with respect to its discrete excitations, i.e., the particles, and nonlocal with respect to its continuous amplitude.

DOI
10.23919/mipro55190.2022.9803338

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It is well known that the expectation value of the electric field operator in a number (Fock) state vanishes. However, a measurement of the electric field must yield some nonzero value each time an experiment is performed. What values can be obtained, and with what probability, is perhaps less well known. In this work we determine the probability distribution for the electric-field amplitude of an optical beam prepared in a single-photon state, using some techniques of quantum field theory. For the purpose of illustration, we apply our theory to the analysis of an ideal experiment where two spatially separated detectors can measure either the amplitude of the electric field, or the intensity of a single-photon optical beam. We find that while the photon cannot be detected in two disjoint positions at the same time, a measurement of the amplitude of the electric field carried by ditto photon, would yield a nonzero value at both detectors. These results show that a quantum field is both local with respect to its discrete excitations, i.e., the particles, and nonlocal with respect to its continuous amplitude.
Abstract We present a detailed quantum field theory of single-photon states. The formalism we develop permits us to calculate the electric field of a single-photon light beam sent through a beam splitter (BS). This illustrates the nonlocal nature of quantum fields even when they are excited by discrete numbers of quanta. We find that the shape of the electromagnetic wave attached to a single photon entering the BS, determines the results of simultaneous measurements at two different locations behind the two ports of the BS. Therefore, our results confirm that a quantum field can be both local with respect to its discrete excitations, i.e. the photons, and nonlocal with respect to its continuous amplitude.
We present a time-dependent quantum calculations of the first order and second order photon correlation functions for the scattering of a single-photon pulse on a two-level atom (qubit) embedded in a one-dimensional open waveguide. Within Markov approximation we find the analytic expression for the quantum operator of positive frequency electric field. We restricted Hilbert space of initial states by the states with one and two excitations and show that the photon probability amplitudes are given by the off-diagonal matrix elements of the electric field operator between these states. For two-excitation initial state where the atom is excited and there exists a single photon in a waveguide we calculate the second order correlation function which describes the measurements by two detectors at two different space-time points. The second order correlation function exhibits the interference term showing that the measurements of two detectors are correlated. This interference is similar to that found in the Hanbury Brown and Twiss correlation experiment with two indistinguishable photons.
We have reconstructed the quantum state of optical pulses containing single photons using the method of phase-randomized pulsed optical homodyne tomography. The single-photon Fock state 1> was prepared using conditional measurements on photon pairs born in the process of parametric down-conversion. A probability distribution of the phase-averaged electric field amplitudes with a strongly non-Gaussian shape is obtained with the total detection efficiency of (55+/-1)%. The angle-averaged Wigner function reconstructed from this distribution shows a strong dip reaching classically impossible negative values around the origin of the phase space.
We calculate spatial localization and phase properties of spontaneously emitted photons. Our model is simple yet fully quantized: the emitting atom is a two-level atom located in a one-dimensional multimode optical cavity. Although the photon state vector does not have a position-space representation, the expectation value of the square of the electric field operator (intensity) is spatially localized and this pattern shifts at the speed of light. The emitted photon exhibits classical-like phase properties in the intensity expectation value when it “interferes with itself” after reflection. The phase properties of the emitted radiation are also evident at times sufficiently long for the radiation to have returned to the emitting atom before the decay of excited atomic state is complete.
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It had been a long standing problem that there is no consistent definition of photon position operator nor photon number density in the context of quantum theory. In this paper we derive the photon detection operator, which defines location of photon absorption, by applying the theory of indirect measurement to quantum electrodynamics. It is shown that the photon detection probability depends on electric properties of a photon-absorbing atom, in particular, on both electric and magnetic dipole moments of the atom. An experiment is proposed, in which the complementarity of wave-particle nature of light will be tested. It is also discussed that the complementarity is related to the non-commutativity of the electric and the magnetic fields.