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arXiv:2110.12930

A Simple Field Theoretic Description of Single-Photon Nonlocality

Andrea Aiello

Published Oct 25, 2021• 2 citations
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Abstract

We present a simple yet rigorous field theoretic demonstration of the nonlocality of a single-photon field. The formalism used allows us to calculate the electric field of a single-photon light beam sent through a beam splitter, which directly demonstrates that it is the light field, rather than the photon itself regarded as a particle, that exhibits nonlocality. Our results are obtained without using either inequalities or specific measurement apparatuses, so that they have perfectly general validity.

DOI
10.48550/arxiv.2110.12930

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We present a simple yet rigorous field theoretic demonstration of the nonlocality of a single-photon field. The formalism used allows us to calculate the electric field of a single-photon light beam sent through a beam splitter, which directly demonstrates that it is the light field, rather than the photon itself regarded as a particle, that exhibits nonlocality. Our results are obtained without using either inequalities or specific measurement apparatuses, so that they have perfectly general validity.
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.
A single-photon incident on a beam splitter produces an entangled field state, and in principle could be used to violate a Bell inequality, but such an experiment (without postselection) is beyond the reach of current experiments. Here we consider the somewhat simpler task of demonstrating Einstein-Podolsky-Rosen (EPR) steering with a single photon (also without postselection). We demonstrate that Alice's choice of measurement on her portion of the entangled state can affect Bob's portion of the entangled state in his laboratory, in a sense rigorously defined by us and Doherty [Phys. Rev. Lett. 98, 140402 (2007)]. Previous work by Lvovsky and coworkers [Phys. Rev. Lett. 92, 047903 (2004)] has addressed this phenomenon (which they called remote preparation) experimentally using homodyne measurements on a single photon. Here we show that, unfortunately, their experimental parameters do not meet the bounds necessary for a rigorous demonstration of EPR steering with a single photon. However, we also show that modest improvements in the experimental parameters, and the addition of photon counting to the arsenal of Alice's measurements, would be sufficient to allow such a demonstration.
Quantum nonlocality of the single-photon entangled state emerging from a beam splitter with arbitrary controlling parameters is investigated by means of the parity measurement. The condition of the maximal violation of the Bell-CHSH inequality is obtained for the single-photon entangled state. Under the condition of the small coherent intensity J(J=0.1) and the phase difference γαβ of the two coherent displacements being opposite to the internal phase shift φ of the beam splitter (γαβ=-φ), we find that the quantum nonlocality depends only on the reflection coefficient R of the beam splitter, while φ plays an important role for selecting γαβ. The strongest quantum nonlocality can be observed for a 50:50 beam splitter. If R deviates from the intermediate value 0.5, the quantum nonlocality decreases gradually, even vanishes. Our results provide a method to observe and manipulate the quantum nonlocality of the single-photon entangled state by adjusting the parameters of the coherent displacements and the beam splitter.
Quantum nonlocality of the single-photon entangled state emerging from a beam splitter with arbitrary controlling parameters is investigated by means of the parity measurement. The condition of the maximal violation of the Bell-CHSH inequality is obtained for the single-photon entangled state. Under the condition of the small coherent intensity J ( J = 0.1) and the phase difference γ αβ of the two coherent displacements being opposite to the internal phase shift ϕ of the beam splitter (γ αβ = −ϕ), we find that the quantum nonlocality depends only on the reflection coefficient R of the beam splitter, while ϕ plays an important role for selecting γ αβ . The strongest quantum nonlocality can be observed for a 50:50 beam splitter. If R deviates from the intermediate value 0.5, the quantum nonlocality decreases gradually, even vanishes. Our results provide a method to observe and manipulate the quantum nonlocality of the single-photon entangled state by adjusting the parameters of the coherent displacements and the beam splitter.
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.
We propose an experimentally feasible scheme to generate various types of entangled states of light fields by using beam splitters and single-photon detectors. Two beams of light fields are incident on two beam splitters respectively with each beam being asymmetrically split into two parts in which one part is supposed to be so weak that it contains at most one photon. We let the two weak output modes interfere at a third beam splitter. A conditional joint measurement on both weak output modes may result in an entanglement between the other two output modes. The conditions for the maximal entanglement are discussed based on the concurrence. Several specific examples are also examined.