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arXiv:2109.07998 · International Journal of Theoretical Physics

Strictly localized three-dimensional states close to single photons

Kai Ryen, Jan Gulla, Johannes Skaar

quant-phSubmitted Sep 16, 2021• Announced Sep 17, 2021• 5 citations
v1 · Sep 16, 2021v2 · Oct 1, 2021v3 · Dec 14, 2021
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Abstract

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 $n$-photons. We also provide an upper bound for the fidelity between a given single photon and any state strictly localized to a given volume. The bound is dependent on the size of the photon tail outside the volume.

Journal reference
Int. J. Theor. Phys. 61, 143 (2022)
DOI
10.1007/s10773-022-05133-7

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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 an upper bound for the fidelity between a given single photon and any state strictly localized to a given volume. The bound is dependent on the size of the photon tail outside the volume.
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.
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.
Exact single photons cannot be generated on demand due to their infinite tails. To quantify how close realizable optical states can be to some target single photon in one dimension, we argue that there are two natural but incompatible ways to specify the target state. Either it can be expressed as a photon with a chosen positive-frequency spectrum, or it can be described as an (unphysical) photon in a chosen positive-time pulse. The results show that for sufficiently short target pulses, the closest realizable states contain substantial multiphoton components. Upper and lower bounds for the maximum fidelity are derived and are expressed as functions of the size of the target state's tail, for negative time or negative frequency, respectively. We also generalize the bounds to arbitrary photon-number states.
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.
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.
Abstract The questions of whether a photon can be localized in an arbitrarily small volume and what is the allowable strength of that localization (the decrease with distance of the functional form) are questions of current interest. We propose a measure of localization for the single photon that is the expectation values of the electromagnetic field strength components in a coherent wavepacket state of mean photon number unity. As such, we deal with real quantities that have a physical meaning rather than complex amplitudes. It is seen that the real parts of complex amplitudes proposed previously as measures of localization are equal to our field expectations. With this measure, we examine two test states. The first has a well-resolved momentum. The field expectations show Gaussian (quadratic exponential) localization in all directions, although the localization length scale is much larger than the mean wavelength. For the other test state, with a spherically symmetric momentum distribution, we find almost exponential localization in all directions at t = 0. The profiles are scale invariant, so choosing the momentum width very large would make the localization length arbitrarily small. We conclude that there is no lower bound on the localization length scale of a photon as determined by this measure.