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Strict Localization and Single-Photon Approximation

Kai Ryen

Published Jan 1, 2021• 0 citations
OpenAlex

Abstract

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.

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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.
A class of strictly localized states which can be made arbitrarily close to single photons is constructed, and expressions for central properties are provided. It is demonstrated that single photon states can be well approximated by these states down to localization scales on the order of a few pulse cycles. The results readily generalize to states close to nnn-photons. We also provide upper and lower bounds for the fidelity between a given single photon and any state strictly localized to a given volume. These results constitute the limit of photon localization, complementary to the weak-localization limit (I. Bialynicki-Birula, Phys. Rev. Lett. 80, 5247 (1998)).
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.
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.
Single-photon pulses cannot be generated on demand, due to incompatible requirements of positive frequencies and positive times. Resulting states therefore contain small probabilities for multiphotons. We derive upper and lower bounds for the maximum fidelity of realizable states that approximate single-photon pulses. The bounds have implications for ultrafast optics; the maximum fidelity is low for pulses with few cycles or close to the onset, but increases rapidly as the pulse envelope varies more slowly. We also demonstrate strictly localized states that are close to single photons.
Abstract We employ reduced density matrices to investigate the localization of relativistic particles within the framework of (1+1)(1+1)(1+1) ( 1 + 1 ) -dimensional quantum field theory. In the case of bosons, we prove that scalar particles cannot be localized within any compact region, although they can be made arbitrarily close to localized states. For massless fermions, by exploiting properties of the reduced density matrices of coherent states, we show that the fermionic single-particle state which, under constructive bosonization, corresponds to a bosonic coherent state is strictly localized at a single spatial point; however, the energy of this state is infinite. Moreover, since the notion of spatial localization based on reduced density matrices is consistent with relativistic causality, even a particle with infinite energy that is localized at a single spatial point will neither spread outside the light cone nor propagate superluminally. Indeed, the explicit expression for the localized fermionic state derived in our work shows that a massless fermion can preserve its sharply localized profile while propagating at the speed of light, in full agreement with causality.
Although the group delay of classical pulses through a barrier may suggest superluminality, the information transfer is limited by the precursor, which propagates at the vacuum light speed. Single photons, however, have infinite tails, and the question of causality becomes meaningless. We solve this problem by introducing strictly localized states close to single photons, which are examples of optical states produced by on-demand single-photon sources. These states can be arbitrarily close to single photons while demonstrating causality for their leading edge.