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arXiv:2306.09793 · Physical Review A

Nonlocality of the energy density for all single-photon states

Maxime Federico, Hans Rudolf Jauslin

Published Oct 23, 2023• 2 citations
arXivHTMLPDF

Abstract

The nonlocality of single-photon states has been analyzed from several different but interrelated perspectives. In this article we propose a demonstration based on the electromagnetic energy density observable and on the antilocal property of the frequency operator $\mathrm{\ensuremath{\Omega}}=c{(\ensuremath{-}\mathrm{\ensuremath{\Delta}})}^{1/2}$. The present proof is based on the standard quantization of the electromagnetic field, which can be formulated equivalently in the momentum representations or in the position representations of Landau and Peierls [Z. Phys. 62, 188 (1930)] and of Bia\l{}ynicki-Birula [edited by E. Wolf (Elsevier, Amsterdam, 1996)]. Our proof extends to all single-photon states the results of Bia\l{}ynicki-Birula, which were formulated for two particular classes of states, those involving a uniform localization [Phys. Rev. Lett. 80, 5247 (1998)] or alternatively states that are electrically or magnetically localized [Phys. Rev. A 79, 032112 (2009)]. Our approach is formulated in terms of Knight's definition of strict localization [J. Math. Phys. 2, 459 (1961)], based on the comparison of expectation values of single-photon states of local observables with those of the vacuum.

DOI
10.1103/physreva.108.043720

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The nonlocality of single-photon states has been analyzed from several different but interrelated perspectives. In this article we propose a demonstration based on the electromagnetic energy density observable and on the antilocal property of the frequency operator \ensuremathΩ=c(\ensuremath−\ensuremathΔ)1/2\mathrm{\ensuremath{\Omega}}=c{(\ensuremath{-}\mathrm{\ensuremath{\Delta}})}^{1/2}\ensuremathΩ=c(\ensuremath−\ensuremathΔ)1/2. The present proof is based on the standard quantization of the electromagnetic field, which can be formulated equivalently in the momentum representations or in the position representations of Landau and Peierls [Z. Phys. 62, 188 (1930)] and of Bia\l{}ynicki-Birula [edited by E. Wolf (Elsevier, Amsterdam, 1996)]. Our proof extends to all single-photon states the results of Bia\l{}ynicki-Birula, which were formulated for two particular classes of states, those involving a uniform localization [Phys. Rev. Lett. 80, 5247 (1998)] or alternatively states that are electrically or magnetically localized [Phys. Rev. A 79, 032112 (2009)]. Our approach is formulated in terms of Knight's definition of strict localization [J. Math. Phys. 2, 459 (1961)], based on the comparison of expectation values of single-photon states of local observables with those of the vacuum.
States whose local expectation values are identical to those of vacuum outside of a given region of localization are known as strictly localized states. It is believed that states generated on-demand need to be strictly localized to satisfy causality requirements. Single particle states, however, cannot be strictly localized. In this case, to conform with observation, there should exist strictly localized states which are close enough to single particle states to be indistinguishable from them by standards of contemporary experimental precision. A class of photonic states has been constructed which is fully consistent with the above considerations in that they are strictly localized, causally evolving, and generally closer to single photons than experimental precision can distinguish. We call these eta-states. The author's work represents a versatile method of constructing this class of states in a fully physical, three dimensional electromagnetic field. This is to our best knowledge the first such state constructed. Furthermore, by repeated application of the operator which excites the state from vacuum, states close to n-photons follow readily. Several relevant properties of eta-states are derived analytically, such as their energy density expectation value, number expectation value, and their coefficients in a so-called pulse mode basis. Their fidelity, the coefficient corresponding to a single particle component, is studied analytically and numerically. General bounds on the fidelity of an arbitrary strictly localized state to a given single photon are also presented. The fidelity can be made arbitrarily close to 1, but is shown to decrease as the region of localization decreases relative to the carrier frequency of the pulse. This indicates the following: One may falsify the hypothesis that on-demand states must be strictly localized by observing fidelities exceeding this limit. On the other hand, the bounds may pose a real practical limit to the single-photon approximation in experiments and potential photonics applications in the future. The author's work also includes a potential candidate for a localized Dirac fermion state following a similar method to the photonic eta-state, with the notable feature that it is a superposition between a fermion and an antifermion.
Bell nonlocality of quantum states is an important resource in quantum information and then has various applications. It is usually detected by the violation of some Bell’s inequalities and the all-versus-nothing test. In the present paper, we aim to establish some mathematical methods for proving Bell nonlocality without inequalities, inspired by the work [Phys. Rev. Lett., 89, 080402 (2002)] regarding the GHZ paradox. For self-containedness, we recall the mathematical definition of Bell nonlocality proposed in [Sci. China-Phys. Mech. Astron. 62, 030311 (2019)] and then give some basic properties on it. Then we derive some necessary conditions for a multipartite state to be Bell local and obtain some sufficient conditions for a state to be Bell nonlocal in terms of “expectations” of local observables without invoking Bell inequalities. Unlike the standard approach to nonlocality detection based on violation of Bell inequalities, the obtained criteria are formulated in terms of certain relations for expectation values of local observables that are constructed from the well-known GHZ paradoxes.
Whilst quantum electrodynamics (QED) underwent an impressive development and reached its maturity in the middle of the last century, one of its basic concepts—the photon wave function in free space—was deprived of such fortune. Although the photon wave function in coordinate representation was introduced already in 1930 by Landau and Peierls, the concept was found to suffer from inherent difficulties that were not overcome during the century —see review (Bialynicki-Birula, 1996). The common explanation presented in textbooks, e.g., (Akhiezer & Berestetskii, 1965; Mandel & Wolf, 1995), 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.
We review what it means for a relativistic particle to be localized and argue that the most useful approach is through local measurement. This perspective ties localization directly to relativistic causality and measurable outcomes in spacetime. Strict localization in quantum field theory is formalized by states that give the same expectation value as vacuum for all local observables outside some spacetime region. With this formalism, single photons cannot be strictly localized, but coherent states can. The structure and interpretation of local observables and local operator algebras are discussed in connection with the Reeh-Schlieder theorem. Since probabilities of local measurement outcomes cannot vanish in vacuum, particle detection events must be phrased carefully. Key characteristics of strictly localized states are surveyed alongside concrete strategies for constructing them. The relationship between localization, causality, and on-demand sources is analyzed, finding that strictly localized states can represent localized, propagating information in quantum field theory. Explicit examples of localized photonic states are presented with an emphasis on quantum optics, including states that approximate single photons and (entangled) Bell states.
A first-quantized free photon is a complex massless vector field A=(Aμ) whose field strength satisfies Maxwell’s equations in vacuum. We construct the Hilbert space H of the photon by endowing the vector space of the fields A in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in H, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated symmetry axis and show that each choice of this axis specifies a particular position operator, a corresponding position basis, and a position representation of the quantum mechanics of a photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and give an explicit formula for the probability density of the spatial localization of the photon.