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arXiv:2109.06472 · Physical Review A

Limits for realizing single photons

Jan Gulla, Kai Ryen, Johannes Skaar

Published Dec 13, 2023• 5 citations
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Abstract

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.

DOI
10.1103/physreva.108.063708

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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.
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.
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)).
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.
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.
The dynamics of a system interacting with an ultrashort pulse is known to depend on the phase content of said pulse. For linear absorption, phase control is possible over time-varying quantities, such as the population of metastable states, but not over time-independent quantities, such as the population of steady states. We derive here a strict upper bound for phase control that interpolates between these two cases --- the bound quantifies the approach to the steady state and resulting loss of one-photon phase control based on physical timescales. Significantly, this bound is violated by a number of numerical and experimental investigations. A careful analysis of the physical conditions underlying this result exposes multiphoton effects as a mechanism for these experiments.
Abstract Two complementary properties, such as frequency and duration of a pulse, cannot be precisely measured simultaneously. The lower bound on the time-bandwidth product, known also as uncertainty relation, plays a very important role in quantum theory. In this work, we consider single-photon pulses with arbitrary temporal and spectral profiles, and derive a lower bound for the single-photon time-bandwidth product. We investigate how this bound depends on the modal purity, the single-mode fidelity, and the effective number of modes. Our results can be used to solve the inverse problem, namely, to estimate characteristics of the single-photon pulse based on the time-bandwidth product obtained from temporal and spectral measurements of the pulse.