Inference of dispersion measure from incoherent time-steady sources
aa r X i v : . [ a s t r o - ph . C O ] D ec Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 18 October 2018 (MN L A TEX style file v2.2)
Inference of dispersion measure from incoherent time-steadysources
Christopher M. Hirata & Matthew McQuinn Department of Physics and Department of Astronomy, The Ohio State University, 191 West Woodru ff Avenue, Columbus, Ohio 43210, USA Department of Astronomy, University of California, Berkeley, California 94720, USA; Hubble fellow
11 December 2013
ABSTRACT
Several recent papers have proposed schemes by which a dispersion measure, and hence elec-tron column, could be obtained from a time-steady, incoherent radio source at a cosmologicaldistance (such as an active galactic nucleus). If correct, this would open a new window on thedistribution of intergalactic baryons. These schemes are based on the statistical properties ofthe received radiation, such as the 2- or 4-point correlation function of the received electricfield, and in one case on the quantum nature of the electromagnetic field. We show, on thebasis of general principles, that these schemes are not sensitive to dispersion measure (or havean extremely small signal-to-noise ratio), because (i) the classical 2-point correlation functionis una ff ected by dispersion; (ii) for a source with a large number of incoherently emitting elec-trons, the central limit theorem obliterates additional information in higher-order functions;and (iii) such an emitter produces a radiation density matrix that is equivalent to a statisticaldistribution of coherent states, which contains no information that is not already in the statis-tics of the classical waveforms. Why the proposed observables do not depend on dispersionmeasure (or have extremely tiny dependences) is discussed in detail. Key words: methods: statistical — radio continuum: general — intergalactic medium.
Estimates of dispersion measure (DM) using the frequency-dependent arrival time of pulses from pulsars have provided someof the most useful constraints on ionized gas in the interstel-lar medium of the Milky Way Galaxy (Taylor & Cordes 1993;Cordes & Lazio 2002) and in the Galactic halo (e.g. Fang et al.2013). These estimates provide a direct measurement of the elec-tron column density to each source, R n e d ℓ . They are indepen-dent of clumping factors, gas temperatures, and (in some wave-bands) extinction, in contrast to emission-based probes of ionizedgas. They are also independent of the metallicity and ionizationstate, unlike absorption line tracers. It would be extremely useful tohave a similar electron column density probe for the intergalacticmedium (e.g. McQuinn 2013).Unfortunately, pulsars are intrinsically faint and the most dis-tant ones observed thus far are in the Magellanic Clouds; for theforeseeable future, pulsar-based DMs are unmeasureable at cosmo-logical distances. Two recent developments have renewed interestin dispersion measure as a cosmological probe. One has been thediscovery of radio transients with durations of order millisecondsor less with a frequency-dependent arrival time consistent withplasma dispersion (Thornton et al. 2013; see also Lorimer et al.2007). These transients’ implied DM is far greater than that ex-pected from the Milky Way and is consistent with an origin atcosmological distances ( z of order unity). However, a terrestrialorigin has not been excluded (see e.g. Burke-Spolaor et al. 2011; Loeb et al. 2013). The other major development – and the subjectof this paper – is the suggestion that DM could be measured in a continuous rather than pulsed source (Lovelace & Richards 2013;Lieu & Duan 2013). If correct, then dispersion measures could beobtained using radio-loud active galactic nuclei (AGNs) as back-lights, opening a completely new window on the study of inter-galactic gas.There are two recently proposed schemes for measuring theDM to a continuous synchrotron source. One uses the 2-point cor-relation function of the electric field between two neighbouringfrequency channels. The idea is that in the presence of disper-sion, these channels’ cross-correlation should peak at a nonzerolag, with the signal arriving first in the higher-frequency channel(Lovelace & Richards 2013). The other scheme is to use the 2-point correlation function of intensity fluctuations (which is a 4-point correlation function of the received electric field). It relieson the intuition that since dispersion within a frequency channelsmears out the received synchrotron pulse from each electron, thetimescale at which the intensity fluctuates should be increased rel-ative to the DM = ∼ ∆ f − (Lieu & Duan 2013).This paper shows that no scheme for measuring the DM to atime-steady incoherent radio source can work. The conceptual ba-sis for this result is that the 2-point correlation function of the elec-tric field contains no phase information and, hence, is insensitiveto the phase shift that is produced by dispersion. While the non- c (cid:13) Hirata & McQuinn
Gaussian (or connected) part of higher-order statistics, such as the3- and 4-point correlation functions, are indeed sensitive to disper-sion measure, we show that the non-Gaussian part is undetectablefor any potential incoherent source. This is a consequence of thethe central limit theorem: the number of electrons contributing toa source’s flux at any time is large and, hence, the signal is veryGaussian. There are exceptions to this argument: (1) the relativephase of both polarizations is observable in the 2-point correlationfunction (which is why rotation measure is measurable), and (2)the relative phase of the received wavefront is altered both spa-tially or temporally by density gradients along the sightline (yield-ing scintillations). However, neither of these loopholes is relevant tothe determination of the dispersion measure, which is polarization-independent, does not vary with observer position over accessiblebaselines, and does not produce amplitude fluctuations.This paper will also explain why the proposed observables inLovelace & Richards (2013) and Lieu & Duan (2013) do not de-pend on DM, despite it seeming plausible that they would. In thecase of the former observable (the 2-point correlation function be-tween adjacent bandpasses), in addition to the e ff ect discussed inLovelace & Richards (2013) where an electronic pulse arrives ear-lier in the higher band, dispersion also distorts the pulse shape.We show that this distortion correlates the beginning of the lower-frequency signal with the end of the higher-frequency signal, ex-actly canceling the impact of the delay on observed correlations. Inthe case of the latter observable (the temporal correlation of inten-sity fluctuations, or the 4-point correlation function of the electricfield), for dispersed electron pulses, some electrons contribute cor-related fluctuations in the intensity at unequal times – i.e. they con-tribute positively to h δ I ( t ) δ I ( t ) i – but some electrons contributeanti-correlated fluctuations. The net result is that the intensity fluc-tuations decorrelate on timescales longer than ∼ ∆ f − even if thepulse from each electron is dispersed into a train whose temporallength is many times longer.The only part of the intensity correlation function that doesnot decorrelate on timescales longer than ∆ f − is the connectedpart, where all 4 electric fields are contributed by the same elec-tron; we show that this connected part is suppressed by a factor of1 / N e , where N e is the number of electrons that contribute to the ob-served electric field at any given time. For any realistic AGN thisis negligible. Recently, Lieu et al. (2013) claimed that the suppres-sion is only by a factor of 1 / ¯ n γ , where ¯ n γ is the photon occupationnumber seen by the observer. This would correspond to a quan-tum correction to the correlation function since it is proportionalto Planck’s constant: 1 / ¯ n γ ∼ h f / kT sys . We show that quantum cor-rections do not enable one to measure the DM; the basic reason isthat for near-classical sources, quantum intensity fluctuations (pho-ton Poisson noise) are a feature of the uncertainty principle andits impact on the measurement process, and dispersion acts on theunderlying wave function before the measurement is made.The outline of this paper is as follows. In §
2, we developthe formalism for correlation functions of the electric field re-ceived at a detector after processing through an arbitrary linear fil-ter. We use this to prove in § §
4, the procedure suggested byLovelace & Richards (2013) is studied in detail as a special case.Intensity fluctuations are considered in §
5, where we show that thecomponent that depends on dispersion is completely undetectable.We further discuss the calculation in Lieu & Duan (2013). The pos-sibility of masers as a source for DM measurements is briefly con- sidered in §
6. We conclude in §
7. Appendix A discusses the role ofquantum mechanics in the emission process.
Consider an optically thin incoherent synchrotron source. The nota-tion here parallels that of Lovelace & Richards (2013). Index nota-tion will be used as follows: capital Roman alphabet
ABC ... denotesignal channels (i.e. filtered electric fields in the chosen polarizationstate and filtered to the frequency band); lower case Roman indices i jk ... denote pulses from relativistic electrons in the source (the ob-served signal is the sum of the electric field from each such pulse);and Greek indices αβγ... denote polarization states. All summationsare kept explicit. Fourier transforms here will use the normalizationconvention E ( t ) = Z ∞−∞ ˜ E ( f ) e − π i ft d f ↔ ˜ E ( f ) = Z ∞−∞ E ( t ) e π i ft d t , (1)and convolutions are denoted by ∗ , with[ E ∗ G ]( t ) ≡ Z ∞−∞ E ( t ′ ) G ( t − t ′ ) d t ′ ↔ [ ] E ∗ G ]( f ) = ˜ E ( f ) ˜ G ( f ) , (2)where tildes indicate the Fourier transform.The electric field from the source as emitted is given by E em α ( t ) = X j E j ψ j α ( t − t j ) , (3)where the summation is over the pulse from each relativistic elec-tron that contributes to the emission, E j is a normalization of thesource strength, ψ j α ( t ) is the pulse profile, and t j is the time of the j th pulse. The rate of pulses is Γ and has units of s − ; the pulsesare assumed independent so the number in any time interval δ t isPoisson-distributed with mean Γ δ t . Eq. (1) of Lovelace & Richards(2013) is similar except that the treatment here keeps the polariza-tion index and the finite width of the pulse (i.e. it is taken as somefunction ψ j rather than a Dirac δ -function). This makes the treat-ment here more general, but neither of these details is important tothe final result.In coming to the Earth, the signal passes through an ionizedcloud of some dispersion measure DM, which introduces a phasedelay given by φ ( f ). It follows that the received field is˜ E α ( f ) = e i φ ( f ) ˜ E em α ( f ) . (4)Written in the time domain, one has E α = E em α ∗ D , where D ( t ) = Z ∞−∞ e i[ φ ( f ) − π ft ] d f (5)is the delay kernel. (Plasma birefringence or Faraday rotation isignored here as it is not germane to the problem at hand.)An electric field observed with an actual detector A is sensitiveto some polarization state p A α with some linear filter χ A :¯ E A ( t ) = X α p A α [ χ A ∗ E α ]( t ) . (6)Here the bar is used to indicate an electric field processed throughthe polarization and spectral response of the telescope, feeds, andhardware and software filters, as opposed to the unbarred E α ( t )which denotes a free-space electric field incident on the telescope.It is assumed that the bandpass filter falls o ff fast enough that wecan approximate ˜ χ A ( f ) = f χ A (0) = c (cid:13) , 000–000 ispersion of incoherent sources While Eq. (6) represents the electric field for an arbitrarybandpass, it is typical of a frequency channel in a radio receiverto have some central frequency f A and a bandpass shape (deter-mined by some combination of hardware and software) with somewidth, B A . In this case, the output electric field is determined by thecentral frequency and a bandpass function ∆ A :¯ E A ( t ) = e − π i f A t X α p A α Z ∞−∞ E α ( t ′ )e π i f A t ′ ∆ A ( t − t ′ ) d t ′ . (7)The operation considered here is – again followingLovelace & Richards (2013) – a multiplication against a lo-cal oscillator at frequency f A (i.e. the complex exponential e π i f A t )followed by a convolution with the bandpass-limiting filter ∆ A , andfinally by the prefactor that mixes the signal back to the originalfrequency. The filter ∆ A is taken to have temporal width ∼ B − A sothat it passes a frequency width of order B A , and is normalized tounit transmission at the band center, ˜ ∆ A (0) =
1. A simple exampleis of course the top-hat in frequency ∆ A ( τ ) = sin( π B A τ ) / ( πτ ),though many other choices are possible. It is easily seen thatEq. (7) is equivalent to the general linear filter, Eq. (6), with χ A ( τ ) ≡ e − π i f A τ ∆ A ( τ ) ↔ ˜ χ A ( f ) = ˜ ∆ A ( f − f A ) . (8)Note that since the filter χ is positive frequency only, χ A ( τ ) andhence ¯ E A ( t ) are complex functions. The real and imaginary parts of¯ E A ( t ) correspond to the cosine-like and sine-like parts of E α ( t ) (seeEq. 7) in the same sense that a complex phasor is used to describea real oscillatory function.This paper is concerned with time-steady radio sources and,hence, we are interested in correlation functions of ¯ E A ( t ). The “1-point correlation function” we write as h ¯ E A ( t ) i , where h ... i denotesan ensemble average (which is equivalent for time-steady sourcesto a time average). It vanishes trivially so long as the filter A ex-cludes the DC component, i.e. so long as ˜ χ A (0) =
0. Thus, only2-point and higher correlation functions are of interest. The 2-pointcorrelation function is C EAB ( δ t ) ≡ h ¯ E A ( t ) ¯ E ∗ B ( t + δ t ) i , (9)where the average is taken over both the types of electron pulses( E j and ψ j α ) and the pulse times ( t j ) of each pulse. In principle onemay define the 2-point correlation function without the complexconjugate: C (2 , AB ( δ t ) ≡ h ¯ E A ( t ) ¯ E B ( t + δ t ) i , (10)but this is zero because ¯ E A ( t ) and ¯ E B ( t + δ t ) are both positive-frequency functions (their Fourier transforms are zero at f < ffi ciently long period of time is zero.Higher-order correlation functions may be defined analo-gously and some have a simple physical interpretation. For exam-ple, the correlation function of intensity fluctuations is a 4-pointcorrelation function of the electric field, since the intensity is pro-portional to the square of the electric field. We note that all the sta-tistical properties of the radiation are encoded in n -point functionsof the field.The formalism presented here, and used in the main text, isbased on classical signals and correlation functions rather thanquantum states and operator expectation values. The use of clas-sical electrodynamics in writing down Eq. (3) is justified in detailin § The last step was done implicitly in Lovelace & Richards (2013) in writ-ing their Eq. (2).
In Lovelace & Richards (2013), it was claimed that the 2-pointcorrelation function of overlapping frequency channels with finite | f A − f B | could be used to measure the dispersion measure to atime-steady synchrotron source. Using the tools of §
2, we will nowformally evaluate this correlation function and show that there isno DM information contained therein. Section 4 discuses at lengthwhy the intuition that dispersion creates a temporal lag between thehigher and lower-frequency channels fails.Substituting our expression for E A ( t ) (Eq. 6) into the expres-sion for the correlation function (Eq. 9) yields C EAB ( δ t ) = X αβ jk p A α p ∗ B β D E j E ∗ k [ χ A ∗ D ∗ ψ j α ]( t − t j ) × [ χ ∗ B ∗ D ∗ ∗ ψ ∗ k β ]( t + δ t − t k ) E . (11)One now splits the sum into two parts: one with j = k and onewith j , k . In each case we may replace the average over t j andsummation over j with Γ R ∞−∞ d t j , giving C EAB ( δ t ) = Γ X αβ Z ∞−∞ d t j p A α p ∗ B β D |E j | [ χ A ∗ D ∗ ψ j α ]( t − t j ) × [ χ ∗ B ∗ D ∗ ∗ ψ ∗ j β ]( t + δ t − t j ) E + Γ X αβ Z ∞−∞ d t j Z ∞−∞ d t k p A α p ∗ B β × D E j E ∗ k [ χ A ∗ D ∗ ψ j α ]( t − t j ) × [ χ ∗ B ∗ D ∗ ∗ ψ ∗ k β ]( t + δ t − t k ) E . (12)The second term (i.e. the term where the electric field comes fromtwo di ff erent pulses) should be zero because distinct pulses do notcorrelate: This term can be broken down into two separate integralscontaining pulses j and k , and each one individually vanishes uponintegration over t j (or t k ) since ˜ χ A (0) = ˜ χ B (0) =
0. Thus, only thefirst term of Eq. (12) survives. With the replacement τ = t − t j , theintegral over t j becomes an integral over τ such that C EAB ( δ t ) = Γ X αβ Z ∞−∞ d τ p A α p ∗ B β D |E j | [ χ A ∗ D ∗ ψ j α ]( τ ) × [ χ ∗ B ∗ D ∗ ∗ ψ ∗ j β ]( τ + δ t ) E . (13)Further simplification can be achieved by defining the time-reversaloperator R via[ R E ]( t ) = E ( − t ) ↔ [ g R E ]( f ) = ˜ E ( − f ) . (14)With the help of this operator and its trivial distributive propertyover convolution R ( E ∗ G ) = ( R E ) ∗ ( R G ), Eq. (13) reduces to C EAB ( δ t ) = Γ X αβ p A α p ∗ B β D |E j | [( R χ A ) ∗ ( RD ) ∗ ( R ψ j α ) ∗ χ ∗ B ∗ D ∗ ∗ ψ ∗ j β ]( δ t ) E . (15)The final simplification involves the dispersion kernel D . Onesees that[ ] ( RD ) ∗ D ∗ ]( f ) = [ ] ( RD )]( f )[ f D ∗ ]( f ) = ˜ D ( − f ) ˜ D ∗ ( − f ) = | ˜ D ( − f ) | . (16)However, the dispersion kernel has a Fourier transform ˜ D ( f ) = e i φ ( f ) , where φ ( f ) is a real function. It follows that | ˜ D ( − f ) | = RD and D ∗ in Eq. (15) cancel out: C EAB ( δ t ) = Γ X αβ p A α p ∗ B β D |E j | [( R χ A ) ∗ ( R ψ j α ) ∗ χ ∗ B ∗ ψ ∗ j β ]( δ t ) E . (17) c (cid:13) , 000–000 Hirata & McQuinn
The correlation function of E A ( t ) with E B ( t ) depends on the strengthand shape of the pulses as well as the polarization and bandpassresponse of the detectors. However, the dispersion dependence hasdropped out. Therefore, dispersion has no e ff ect on the correlationfunction of observables that are linear in the electric field . It is instructive to explicitly work through the electric field cor-relation function in a simple example to see how it does notdepend on DM. The example considered here follows that inLovelace & Richards (2013), taking ψ j α ( t ) to be a δ -function andconsidering a single polarization (so that the polarization indicesneed not be kept). The bandpass filter shapes ˜ ∆ A ( f ) will be taken tobe identical for all channels, and only their central frequencies willdi ff er. The bandpass function ∆ ( f ) will be kept arbitrary at first, andthen two explicit examples will be given: a Gaussian bandpass anda tophat bandpass.We approximate the phase delay (cf. Eq. 4) as quadratic infrequency and given by φ ( f ) = φ A + π T A ( f − f A ) + π D ( f − f A ) . (18)(We approximate φ to be a quadratic function of f across all chan-nels so that D does not require a subscript.) In accordance withEq. (5), the stationary-phase time delay as a function of frequencyis T ( f ) = (2 π ) − d φ ( f ) / d f , so that T A = T ( f A ) is the delay at thecenter of the frequency band and D = d T / d f encapsulates how thepulses is broadened by dispersion (units: s GHz − ). Note that forplasma dispersion T > D <
0. In §
5, we will need to usesome numerical estimates for D ; its relation to the usual dispersionmeasure is D = − DM10 pc cm − ! f − s GHz − , (19)where 10 pc cm − is roughly the expected DM to a source at z = The dispersion kernel is the Fourier transform of e i φ ( f ) or D ( t ) = e i φ A Z ∞−∞ e π i[ T A ( f − f A ) + D ( f − f A ) / − ft ] d f = e i φ A e − π i f A t Z ∞−∞ e π i[( T A − t )( f − f A ) + D ( f − f A ) / d f = e i φ A e − π i f A t e − π i( t − T A ) / D Z ∞−∞ e π i D ( f − f ) d f = √− π D e i( φ A − π/ e − π i f A t e − π i( t − T A ) / D , (20)where we have set f = f A + ( t − T A ) / D in the third line.With this D ( t ), ignoring polarization, and if the emitted fieldis a sequence of δ -functions, the received field is E ( t ) = X j E j √− π D e i( φ A − π/ e − π i f A ( t − t j ) e − π i( t − t j − T A ) / D . (21)The filtered field is then given by Eq. (7):¯ E A ( t ) = X j E j e i( φ A − π/ √− π D e − π i f A ( t − t j ) Z ∞−∞ e − π i( t ′ − t j − T A ) / D ∆ ( t − t ′ ) d t ′ = X j E j e i φ A e − π i f A ( t − t j ) ∆ ( D ) ( t − t j − T A ) , (22) where ∆ ( D ) ( τ ) ≡ e − i π/ √− π D Z ∞−∞ e − π i τ ′ / D ∆ ( τ − τ ′ ) d τ ′ (23)is the dispersed bandpass function. The dispersed bandpass is for-mally equivalent to taking the original bandpass and smearing itwith a Gaussian of complex width √ D / (2 π i); as a check, one mayverify that for D → ∆ (0) ( τ ) = ∆ ( τ ). One expects the e ff ect of dispersion on thesignal within band to be modest when the smearing width is lessthan the intrinsic width, | D | / < B − : this is the standard smearingcriterion, encapsulated in Lovelace & Richards (2013, Eq. 10). In-deed, the fractional e ff ect on the variance of ∆ ( D ) should be of order | D | B .Using the notation of Eq. (22), the correlation function of¯ E A ( t ) with ¯ E B ( t ) is C EAB ( δ t ) = Γ h|E j | i e i( φ A − φ B ) Z ∞−∞ e − π i f A ( t − t j ) ∆ ( D ) ( t − t j − T A ) × e π i f B ( t + δ t − t j ) ∆ ( D ) ∗ ( t + δ t − t j − T B ) d t j . (24)Setting T BA = T B − T A and f BA = f B − f A , and defining ¯ f = ( f A + f B ) / T = ( T A + T B ) /
2, we may make the substitution τ = t − t j − ¯ T + δ t /
2. Also we recognize that φ A − φ B = − π ¯ T f BA . Withthese simplifications, C EAB ( δ t ) = Γ h|E j | i e π i ¯ f δ t Z ∞−∞ e π i f BA τ ∆ ( D ) (cid:18) τ + T BA − δ t (cid:19) × ∆ ( D ) ∗ (cid:18) τ − T BA − δ t (cid:19) d τ. (25)In Eq. (25), the prefactor consists only of an overall normalizationand a phase (2 π ¯ f δ t ) that is a property only of the instrument andsoftware; thus only the integral is interesting, which is given by J ( s ) = Z ∞−∞ e π i f BA τ ∆ ( D ) ( τ + s ) ∆ ( D ) ∗ ( τ − s ) d τ, (26)so that C EAB ( δ t ) = Γ h|E j | i e π i ¯ f δ t J (cid:18) T BA − δ t (cid:19) . (27) One might expect the modulus of the correlation function to bemaximal when δ t ≈ T BA as this likely maximizes the overlapof the functions inside the integral in Eq. (26). Indeed, if ∆ ( D ) ( τ )were a time-symmetric function such as a Gaussian of real stan-dard deviation or a sinc-function, then |J ( s ) | would be symmetricaround s =
0, and the correlation function would peak exactly at δ t = T BA . In this way, it would be possible to measure T BA andhence D ≈ − T BA / f BA from the correlation of two adjacent fre-quency channels.However, since the time delay between the two frequencychannels is D f BA , and we must have f BA . B in order for the signalsat the two frequencies to be coherent over a correlation time ∼ B − ,the ratio of the delay to the correlation time is | D f BA | B . | D | B .Thus according to the preceding discussion (following Eq. 23), thetime delay between the frequencies f A and f B is of the same orderof magnitude in terms of fractional e ff ect on J ( s ) as the deviationof the dispersed response ∆ ( D ) from the instrumental response ∆ .One must determine whether |J ( s ) | is really peaked at, or symmet-ric around, s = c (cid:13) , 000–000 ispersion of incoherent sources symmetric under s ↔ − s , so our attention turns instead to the phasestructure. Of particular interest is the possibility that ∆ ( D ) ( t ) couldexhibit a “phase acceleration,” i.e. that α = d [arg ∆ ( D ) ( t )] / d t maybe nonzero. In this case, and taking for example f BA > α > s < s >
0, sincedd τ arg[e π i f BA τ ∆ ( D ) ( τ + s ) ∆ ( D ) ∗ ( τ − s )] = π f BA + d arg ∆ ( D ) ( τ + s )d τ − d arg ∆ ( D ) ( τ − s )d τ ∼ π f BA + α s . (28)(The last step is only schematic since – except in special cases suchas the Gaussian bandpass – α is not constant.) Thus if α > | I ( s ) | should be enhanced for s < s > α < C AB ( δ t ).This situation is clarified next for two explicit cases: a Gaus-sian and a tophat. In each case, ∆ ( D ) ( t ) will be evaluated, and it willbe shown that it exhibits a positive phase acceleration. The Gaussian bandpass is the simplest choice for the purposes ofanalytic calculation. It is defined by˜ ∆ ( f ) = e − f / σ f ↔ ∆ ( t ) = √ π σ t e − t / σ t , (29)where σ f = / (2 πσ t ). The power-equivalent bandwidth is B ≡ Z ∞−∞ | ˜ ∆ ( f ) | d f = √ π σ f = √ π σ t . (30)According to Eq. (23), the dispersed bandpass is obtained by con-volving the instrumental response ∆ ( t ) with a Gaussian of variance D / (2 π i). Thus one has ∆ ( D ) ( t ) = √ π σ ( D ) t e − t / σ ( D )2 t , (31)where σ ( D ) t ≡ r σ t + D π i . (32)Note that since D < D / (2 π i) is on the positive imaginary axis,so that σ ( D ) t lies in the first octant, i.e. 0 < arg σ ( D ) t < π/
4. Fur-thermore, σ ( D ) − t lies in the fourth quadrant, so that ∆ ( D ) ( t ) takesthe form of a Gaussian envelope with varying phase. Decomposing σ ( D ) − t = a − i b with a and b real gives a = σ t σ t + ( D / π ) and b = − D / πσ t + ( D / π ) . (33)The phase of ∆ ( D ) ( t ) varies as arg ∆ ( D ) ( t ) = constant + bt /
2. It is this“phase acceleration” of ∆ ( D ) ( t ) that will lead to a shift in the peakof |J ( s ) | . One notes that the phase shift at t = D , we haved [arg ∆ ( D ) ( t )] / d t | t = = − π DB . The dispersed bandpass, ∆ ( D ) ( t ),for this Gaussian case are plotted in the top panels of Fig. 1. Thetop-left panel shows D =
0, the top-middle shows D = − . B − , Other definitions of bandwidth are possible and have no e ff ect on thecalculation; this one is chosen for definiteness, and for consistency with thebandwidth B in Lovelace & Richards (2013). and the top-right shows D = − . B − . The phase acceleration isclearly visible for finite D : ∆ ( D ) ( t ) is positive-frequency at t < t > J ( s ) = π | σ ( D ) t | Z ∞−∞ e π i f BA τ e − ( τ + s ) / σ ( D )2 t e − ( τ − s ) / σ ( D ) ∗ t d τ = e − as π | σ ( D ) t | Z ∞−∞ e π f BA + bs ) τ e − a τ d τ = e − as π / a / | σ ( D ) t | e − ( π f BA + bs ) / a = √ π a / | σ ( D ) t | e − [( a + b ) s + π f BA bs + π f BA ] / a = √ π σ t e − π f AB σ t e − ( s − Df BA / /σ t . (34)Here the first equality is substitution into Eq. (26); the second is analgebraic conversion of σ ( D ) t into a and b ; the third is a Gaussianintegral; the fourth is an expansion of the exponent in terms of s ;and the fifth is a simplification using the identities a / | σ ( D ) t | = σ t (for the prefactor), and a / ( a + b ) = σ t and b / ( a + b ) = − D / (2 π )(for completing the square in the exponent).Equation (34) implies that the overlap integral between thetwo dispersed response functions is not at s = s = − D f BA /
2. This is a direct consequence of the phase acceler-ation term b ,
0. One then concludes that the observed correlationfunction C AB ( δ t ) peaks not at δ t = T BA but at δ t = T BA − s peak = T BA − D f BA = . (35)The last equality is the first step in this section ( §
4) where the factthat D = T BA / f BA has been explicitly used. Thus one concludes thateven though the signal at frequency f B is delayed relative to that at f A , the warping of the signal within each band due to dispersionproduces an equal and opposite shift of the peak of the correlationfunction, leading to no net observable e ff ect. The tophat bandpass is ∆ ( t ) = sin( π Bt ) π t ↔ ˜ ∆ ( f ) = Π fB ! , (36)where the unit tophat function is Π ( x ) = | x | < and 0 oth-erwise. It is used as an example in Lovelace & Richards (2013).The calculation for this bandpass is more involved than for theGaussian, but a similar result will be derived. The power-equivalentbandpass (see Eq. 30) is trivially shown to be B .The dispersed bandpass is given by ∆ ( D ) ( t ) = e − i π/ √− π D Z ∞−∞ e − π i t ′ / D sin[ π B ( t − t ′ )] π ( t − t ′ ) d t ′ . (37)The simplest form can be obtained by substituting the relationsin[ π B ( t − t ′ )] π ( t − t ′ ) = π Z π B − π B e i u ( t − t ′ ) d u (38)into Eq. (37), and then performing the Gaussian t ′ integral to get: ∆ ( D ) ( t ) = π Z π B − π B e i[ Du / (4 π ) + ut ] d u . (39) c (cid:13) , 000–000 Hirata & McQuinn -0.5 0 0.5 1 1.5-3 -2 -1 0 1 2 3 ∆ (D) (t)/B BtGaussian, D=0ReIm -0.5 0 0.5 1 1.5-3 -2 -1 0 1 2 3 ∆ (D) (t)/B BtGaussian, D=-0.5B -2 ReIm -0.5 0 0.5 1 1.5-3 -2 -1 0 1 2 3 ∆ (D) (t)/B BtGaussian, D=-1.5B -2 ReIm-0.4-0.2 0 0.2 0.4 0.6 0.8 1 -4 -3 -2 -1 0 1 2 3 4 ∆ (D) (t)/B BtTophat, D=0ReIm -0.4-0.2 0 0.2 0.4 0.6 0.8 1 -4 -3 -2 -1 0 1 2 3 4 ∆ (D) (t)/B BtTophat, D=-0.5B -2 ReIm -0.4-0.2 0 0.2 0.4 0.6 0.8 1 -4 -3 -2 -1 0 1 2 3 4 ∆ (D) (t)/B BtTophat, D=-1.5B -2 ReIm
Figure 1.
The real and imaginary parts of the dispersed bandpass function, ∆ ( D ) ( t ), for D = − . B − , and − . B − , for the Gaussian (upper row) and tophat(lower row) bandpasses. The scales are di ff erent for the two rows. Note the behaviour of the phase in the dispersed cases: ∆ ( D ) ( t ) is positive-frequency at t < t > J ( s ), Eq. (26), this phaseacceleration leads to a peak in the correlation integral at positive s (for f B > f A or f BA > ff setin the peak of the correlation integral between pulses in frequency channels f A and f B that cancels the dispersion-induced delay in arrival time at frequency f A relative to f B , and leads to zero observed delay in the peak of the cross-correlation function between the two channels. The substitution u ′ = u − π t / D turns this into a Fresnel integral: ∆ ( D ) ( t ) = π e − i π t / D Z π B − π t / D − π B − π t / D e i Du ′ / (4 π ) d u ′ = e i π t / (2 w ) w (cid:20) F ∗ (cid:18) wB + tw (cid:19) − F ∗ (cid:18) − wB + tw (cid:19)(cid:21) , (40)where w ≡ √− D / F is the Fresnel integral , F ( z ) = Z z e i πς / d ς. (41)The Fresnel integral as a function of real z traces out the familiar“Cornu spiral” in the complex plane, arcing from F ( −∞ ) = − − ito F ( ∞ ) = + i. According to Eq. (40), the bandpass function ∆ ( D ) ( t ) is the separation vector between two points on the spiralwith parameter z = t / w ± wB , with the instantaneous direction ofmotion at z = t / w removed by a phase rotation e i π t / (2 w ) , and withthe normalizing factor 2 w . The dispersed tophat bandpass functionis shown in the bottom panels of Fig. 1.Equation (40) shows that there is a phase acceleration of ∆ ( D ) ( t ) analogous to that which occurred for the Gaussian case.There is no analytic expression for the phase in this case, but itis possible to do a Taylor expansion of arg ∆ ( D ) ( t ) to order w t or Dt (in the two parameters w and t ) and thus obtain the phaseacceleration at the center of the pulse:arg ∆ ( D ) ( t ) = π DB − π DB t + ..., (42) This is given by F ( z ) = C ( z ) + i S ( z ) in the notation ofAbramowitz & Stegun (1972, § This is straightforward by brute force expansion of Eq. (39). i.e. there is a negative phase shift at t = − ( π / DB . This is just as for the Gaussiancase, albeit with a di ff erent prefactor.One is now interested in the integral J ( s ), which is obtainedby integrating two copies of ∆ ( D ) ( t ). Substituting Eq. (40) intoEq. (26) gives J ( s ) = Z ∞−∞ e π i( f BA + s / w ) τ w (cid:20) F ∗ (cid:18) wB + τ + sw (cid:19) − F ∗ (cid:18) − wB + τ + sw (cid:19)(cid:21) × (cid:20) F (cid:18) wB + τ − sw (cid:19) − F (cid:18) − wB + τ − sw (cid:19)(cid:21) d τ = Z ∞−∞ e π i( f BA + s / w ) τ w Z wB − wB Z wB − wB e − i π [( τ + s ) / w + x ] / × e i π [( τ − s ) / w + y ] / d x d y d τ = Z ∞−∞ e π i( f BA + s / w ) τ w Z wB − wB Z wB − wB e i( y − x ) / × e i π [ − τ s / w + τ ( y − x ) / w − s ( x + y ) / w ] d x d y d τ, (43)where in the second line the Fresnel integral has been re-expandedusing the fundamental theorem of calculus. The integral appearingin the last expression can be simplified by performing the τ integralfirst, which leads to a δ -function: J ( s ) = π w Z wB − wB Z wB − wB e i π [( y − x ) / − s ( x + y ) / w ] δ (cid:18) π f BA + π y − xw (cid:19) d x d y . (44)The δ -function enforces that y − x = w f BA , and hence that ( y − x ) / = w f BA ( y + x ). Switching coordinates to z = ( x + y ) / v = ( y − x ) / w , so that d x d y = w d z d v , and then trivially integratingthe δ -function, one finds J ( s ) = w Z z max − z max e π i z ( wf BA − s / w ) d z , (45) c (cid:13)000
The real and imaginary parts of the dispersed bandpass function, ∆ ( D ) ( t ), for D = − . B − , and − . B − , for the Gaussian (upper row) and tophat(lower row) bandpasses. The scales are di ff erent for the two rows. Note the behaviour of the phase in the dispersed cases: ∆ ( D ) ( t ) is positive-frequency at t < t > J ( s ), Eq. (26), this phaseacceleration leads to a peak in the correlation integral at positive s (for f B > f A or f BA > ff setin the peak of the correlation integral between pulses in frequency channels f A and f B that cancels the dispersion-induced delay in arrival time at frequency f A relative to f B , and leads to zero observed delay in the peak of the cross-correlation function between the two channels. The substitution u ′ = u − π t / D turns this into a Fresnel integral: ∆ ( D ) ( t ) = π e − i π t / D Z π B − π t / D − π B − π t / D e i Du ′ / (4 π ) d u ′ = e i π t / (2 w ) w (cid:20) F ∗ (cid:18) wB + tw (cid:19) − F ∗ (cid:18) − wB + tw (cid:19)(cid:21) , (40)where w ≡ √− D / F is the Fresnel integral , F ( z ) = Z z e i πς / d ς. (41)The Fresnel integral as a function of real z traces out the familiar“Cornu spiral” in the complex plane, arcing from F ( −∞ ) = − − ito F ( ∞ ) = + i. According to Eq. (40), the bandpass function ∆ ( D ) ( t ) is the separation vector between two points on the spiralwith parameter z = t / w ± wB , with the instantaneous direction ofmotion at z = t / w removed by a phase rotation e i π t / (2 w ) , and withthe normalizing factor 2 w . The dispersed tophat bandpass functionis shown in the bottom panels of Fig. 1.Equation (40) shows that there is a phase acceleration of ∆ ( D ) ( t ) analogous to that which occurred for the Gaussian case.There is no analytic expression for the phase in this case, but itis possible to do a Taylor expansion of arg ∆ ( D ) ( t ) to order w t or Dt (in the two parameters w and t ) and thus obtain the phaseacceleration at the center of the pulse:arg ∆ ( D ) ( t ) = π DB − π DB t + ..., (42) This is given by F ( z ) = C ( z ) + i S ( z ) in the notation ofAbramowitz & Stegun (1972, § This is straightforward by brute force expansion of Eq. (39). i.e. there is a negative phase shift at t = − ( π / DB . This is just as for the Gaussiancase, albeit with a di ff erent prefactor.One is now interested in the integral J ( s ), which is obtainedby integrating two copies of ∆ ( D ) ( t ). Substituting Eq. (40) intoEq. (26) gives J ( s ) = Z ∞−∞ e π i( f BA + s / w ) τ w (cid:20) F ∗ (cid:18) wB + τ + sw (cid:19) − F ∗ (cid:18) − wB + τ + sw (cid:19)(cid:21) × (cid:20) F (cid:18) wB + τ − sw (cid:19) − F (cid:18) − wB + τ − sw (cid:19)(cid:21) d τ = Z ∞−∞ e π i( f BA + s / w ) τ w Z wB − wB Z wB − wB e − i π [( τ + s ) / w + x ] / × e i π [( τ − s ) / w + y ] / d x d y d τ = Z ∞−∞ e π i( f BA + s / w ) τ w Z wB − wB Z wB − wB e i( y − x ) / × e i π [ − τ s / w + τ ( y − x ) / w − s ( x + y ) / w ] d x d y d τ, (43)where in the second line the Fresnel integral has been re-expandedusing the fundamental theorem of calculus. The integral appearingin the last expression can be simplified by performing the τ integralfirst, which leads to a δ -function: J ( s ) = π w Z wB − wB Z wB − wB e i π [( y − x ) / − s ( x + y ) / w ] δ (cid:18) π f BA + π y − xw (cid:19) d x d y . (44)The δ -function enforces that y − x = w f BA , and hence that ( y − x ) / = w f BA ( y + x ). Switching coordinates to z = ( x + y ) / v = ( y − x ) / w , so that d x d y = w d z d v , and then trivially integratingthe δ -function, one finds J ( s ) = w Z z max − z max e π i z ( wf BA − s / w ) d z , (45) c (cid:13)000 , 000–000 ispersion of incoherent sources where the range of integration is such that x = z − w f BA and y = z + w f BA are both between − wB and wB – i.e. we have z max = w ( B −| f BA | ) if | f BA | < B and 0 otherwise. The integral is then J ( s ) = sin[2 π ( z max / w )( w f BA − s )]2 π ( w f BA − s ) . (46)Substituting back in the expressions for w and z max : J ( s ) = Θ ( B − | f BA | ) sin[2 π ( B − | f BA | )( s + D f BA / π ( s + D f BA / . (47)From Eq. (47), one sees that the overlap function again de-pends only on the dispersion D through an overall o ff set of thehorizontal scale: J ( s ) is shifted to be centered at s = − D f BA / § ∆ ( D ) ( t ) introduces an o ff set in the correlation of the two chan-nels that exactly cancels the delay di ff erence T BA between the twocentral frequencies. The intensity fluctuations from a source are related to the 4-pointcorrelation function of the electric field. Defining the mean inten-sity in a channel ¯ I A = h| ¯ E A ( t ) | i and an intensity fluctuation δ I A ( t ) = | ¯ E A ( t ) | − ¯ I A , (48)one can find the intensity correlation function in two channels: C δ IAB ( δ t ) = h ¯ E A ( t ) ¯ E ∗ A ( t ) ¯ E B ( t + δ t ) ¯ E ∗ B ( t + δ t ) i − ¯ I A ¯ I B . (49)The next task is to evaluate the intensity correlation function anddetermine how it depends on dispersion measure. It will be shownthat there is indeed a dependence on DM: the argument leading toEq. (17) that showed that the 2-point correlation function was inde-pendent of DM does not apply to higher-order correlation func-tions, since only in the 2-point case can the integral over pulseepochs t j be converted to a convolution of RD and D ∗ . (For higher-point correlation functions, a more complicated set of integrals over D applies.) However only the connected part of the electric field4-point function can depend on DM, since the disconnected partconsists of 2-point functions and is thus independent of DM asshown in §
3. If the number of independent emitting electrons islarge, we will find that the disconnected part dominates. Section § § ff ers from the conclusion in Lieu & Duan (2013). The connected part of the 4-point function of any set of zero-mean vari-ables is defined as h wxyz i c ≡ h wxyz i − h wx ih yz i − h wy ih xz i − h wz ih xy i . Itvanishes for Gaussian fields. Substituting into Eq. (49) yields C δ IAB ( δ t ) = X jklm D E j E ∗ k E l E ∗ m [ D ∗ Ψ jA ]( t − t j ) [ D ∗ ∗ Ψ ∗ kA ]( t − t k ) × [ D ∗ Ψ lB ]( t + δ t − t l ) [ D ∗ ∗ Ψ ∗ mB ]( t + δ t − t m ) E − X jklm D ǫ j ǫ ∗ k [ D ∗ Ψ jA ]( t − t j ) [ D ∗ ∗ Ψ ∗ kA ]( t − t k ) E × D ǫ l ǫ ∗ m [ D ∗ Ψ lB ]( t − t l ) [ D ∗ ∗ Ψ ∗ mB ]( t − t m ) E . (50)To reduce clutter, we have introduced the notation Ψ jA ( τ ) ≡ X α p A α [ χ A ∗ ψ j α ]( τ ) , (51)which is the electron pulse ( ψ j α ) observed through the instrumentbandpass χ A and polarization state p A α . As always, it is assumedthat there is no DC response: ˜ χ A (0) =
0, or R ∞−∞ Ψ jA ( τ ) d τ = ˜ Ψ jA (0) = C δ IAB result from each index be-ing equal to at least one other index (on account of the “no DCresponse” condition). Breaking the first term of Eq. (50) into thesedi ff erent components yields several terms: (i) a term with j = k and l = m ; (ii) a term with j = l and k = m ; (iii) a term with j = m and k = l ; and (iv) a term with j = k = l = m . The term of the form(i) cancels the second term in Eq. (50). The term of the form (ii)reduces to X jk D |E j | [ D ∗ Ψ jA ]( t − t j ) [ D ∗ Ψ jB ]( t + δ t − t j ) E × D |E k | [ D ∗ ∗ Ψ ∗ kA ]( t − t k ) [ D ∗ ∗ Ψ ∗ kB ]( t + δ t − t k ) E . (52)The summations over j and k are seperable and so the term of type(ii) in Eq. (50) reduces to a product of 2-point correlation func-tions, C (2 , AB ( δ t ) C (2 , ∗ AB ( δ t ). These functions, defined by Eq. (10), areequal to zero since they are averages of the correlation of positive-frequency functions. Thus the term of the form (ii) vanishes.The term of the form (iii) has a similar expression: X jk D |E j | [ D ∗ Ψ jA ]( t − t j ) [ D ∗ ∗ Ψ ∗ jB ]( t + δ t − t j ) E × D |E k | [ D ∗ ∗ Ψ ∗ kA ]( t − t k ) [ D ∗ Ψ kB ]( t + δ t − t k ) E . (53)This reduces to C EAB ( δ t ) C E ∗ AB ( δ t ), which is generally nonzero (on ac-count of the location of the complex conjugates); this is equivalentto Eq. (8) of Lieu et al. (2013). Thus Eq. (50) reduces to C δ IAB ( δ t ) = | C EAB ( δ t ) | + C δ IAB ; j = k = l = m ( δ t ) . (54)Here the first term is that of form (iii) and the second is of form (iv).Further computations will focus on this last term that is of interestsince we already know that | C AB ( δ t ) | is independent of dispersionmeasure. The last term is easily seen to be the connected part of thecorrelation function, and we will use this nomenclature below.[Note that in the absence of the connected part C δ Ij = k = l = m ( δ t ),the variance of the intensity fluctuation in a channel C δ IAA (0) is equalto the square of the mean intensity ¯ I A = C EAA (0). This is appro-priate for a complex time series ¯ E A ( t ), since in the Gaussian limitits “intensity” follows a rescaled χ distribution with 2 degrees offreedom. For a real time series with only 1 degree of freedom thedisconnected contribution to the variance is twice the mean inten-sity squared. Mathematically this arises because Ψ j has only posi-tive frequencies. A real Ψ j would result in a factor of 2 in front ofthe | C EAB ( δ t ) | term because the terms of form (ii) would contributeequally to those of form (iii).] c (cid:13) , 000–000 Hirata & McQuinn
The connected term is given by C δ IAB ; j = k = l = m ( δ t ) = Γ Z ∞−∞ d τ D |E j | [ D ∗ Ψ jA ]( τ ) × [ D ∗ ∗ Ψ ∗ jA ]( τ ) × [ D ∗ Ψ jB ]( τ + δ t ) × [ D ∗ ∗ Ψ ∗ jB ]( τ + δ t ) E = Γ D |E j | (cid:8) [ R| ( D ∗ Ψ jA | ] ∗ |D ∗ Ψ jB | (cid:9) ( δ t ) E , (55)where we have used the same trick as in § τ in Eq. (55) a convolution by applying a time reversal operator, R .(Our formulae for C δ Ij = k = l = m and C E double count the case i = j = k = l in a manner that does not matter in the limit that a largenumber of pulses are contributing at any one time.)We consider the case of a single filter A , and drop its subscriptfor convenience. In Fourier space, C j = k = l = m ( δ t ) becomes˜ C δ Ij = k = l = m ( f ) = Γ (cid:28) |E j | Z ∞−∞ d f ′ ˜ D ( f ′ ) ˜ Ψ j ( f ′ ) f D ∗ ( f − f ′ ) f Ψ ∗ j ( f − f ′ ) × Z ∞−∞ d f ′′ g RD ( f ′′ ) g R Ψ j ( f ′′ ) g RD ∗ ( f − f ′′ ) g R Ψ ∗ j ( f − f ′′ ) (cid:29) = Γ (cid:28) |E j | Z ∞−∞ d f ′ ˜ D ( f ′ ) ˜ Ψ j ( f ′ ) ˜ D ∗ ( − f + f ′ ) ˜ Ψ ∗ j ( f − f ′ ) × Z ∞−∞ d f ′′ ˜ D ( − f ′′ ) g R Ψ j ( f ′′ ) ˜ D ∗ ( f − f ′′ ) g R Ψ ∗ j ( f − f ′′ ) (cid:29) = Γ (cid:28) |E j | Z R d f ′ d f ′′ e − i[ φ ( f ′ ) − φ ( − f + f ′ ) + φ ( − f ′′ ) − φ ( f − f ′′ )] × ˜ Ψ j ( f ′ ) ˜ Ψ ∗ j ( f ′ − f ) ˜ Ψ j ( − f ′′ ) ˜ Ψ ∗ j ( f − f ′′ ) (cid:29) , = Γ (cid:28) |E j | Z R d f ′ d f ′′ e − i[ φ ( f ′ ) − φ ( f ′ − f ) + φ ( f ′′ ) − φ ( f ′′ − f )] × ˜ Ψ j ( f ′ ) ˜ Ψ ∗ j ( f ′ − f ) ˜ Ψ j ( f ′′ ) ˜ Ψ ∗ j ( f + f ′′ ) (cid:29) , (56)where in the second equality we used that g RD ( f ) = ˜ D ( − f ), f D ∗ ( f ) = ˜ D ∗ ( − f ), and g RD ∗ ( f ) = ˜ D ∗ ( f ). The last equality involveda change of variables: f ′′ → − f ′′ .Let us evaluate ˜ C j = k = l = m for a broadband source viewedthrough a Gaussian filter of 1 σ width σ f and mean f A , with σ f ≪ f A . The total normalization is absorbed into E j :˜ Ψ j ( f ) = e − ( f − f A ) / σ f . (57)We take the phase shift φ to be quadratic for frequencies near f A , i.e. we use Eq. (18). With these assumptions, Eq. (56) be-comes a 2-dimensional Gaussian integral over frequency, peakednear ( f ′ , f ′′ ) ≈ ( f , f ). This evaluates to e C δ Ij = k = l = m ( f ) = π Γ h|E j | i σ f exp − f (1 + π D σ f )2 σ f . (58)The last factor depends on dispersion.The Fourier transform of Eqn (58) is C δ Ij = k = l = m ( δ t ) = π Γ σ f h|E j | i q (2 πσ f ) − + π D σ f exp − π σ f + π D σ f δ t . (59)Compare this to the disconnected part for the assumed waveform | C E ( δ t ) | = π Γ σ f h|E j | i e − π σ f δ t . (60)Our same reasoning holds as before that the disconnected part ismuch larger as δ t →
0. It is only at significant temporal lags thatthe connected part becomes larger than C E ( t ). Here we estimated the signal-to-noise ( S / N ) at which C δ Ij = k = l = m ( δ t )can be detected in the most optimistic limit that the synchrotronsource dominates the instrumental system temperature. At largetime lags the S / N with which C j = k = l = m ( δ t ) can be measured in agiven sample (i.e. in a time of order σ − f ) is SN = | C δ Ij = k = l = m ( δ t ) || C E (0) | ≈ √ π Γ | D | σ f e − δ t / (2 D σ f ) , (61)where the latter approximate equality used Eq.s (59) and (60). Thedisconnected part can be measured for ∼ D σ f temporal lags (thewidth of the Gaussian D σ f , divided by the sample time σ − f ) eachwith σ f t independent samples, meaning the cumulative signal tonoise is (cid:18) SN (cid:19) ∼ σ f t Γ | D | . (62)To evaluate Eq. (62), we need an estimate for the numberof electrons that contribute at any time. The number of electronswhose emission is beamed towards an observer from a cosmo-logical synchrotron source can be obtained from the formulaefor synchrotron radiation (e.g Rybicki & Lightman 1986, § P ν ∼ e B / ( m e c ) F ( ν/ν c ) (units: erg s − Hz − ), the function F ( x )is peaked near x ∼ F ∼ ν c ∼ γ eB / ( m e c ), and thatin any instant an electron illuminates Ω ∼ γ − . The power radi-ated per electron per unit frequency is then P ν ∼ e ν/ ( γ c ). Thenumber of electrons contributing to the radiation at any one time isthen N e = Ω d f ν / P ν , since Ω / (4 π ) is the fraction of electrons con-tributing, 4 π d f ν is the total power emitted per unit frequency (inall directions), and P ν is the contribution of any one electron. Thisevaluates to N e ∼ Ω d f ν P ν ∼ cd f ν e ν ∼ f ν ! (cid:18) ν (cid:19) − d L ! . (63)Thus, since Γ ∼ N e f , Eq. (62) evaluates to (cid:18) SN (cid:19) ∼ t ( σ f / f ) N e f | D | ∼ − t yr ( σ f / f )( N e / ) f GHz | D s / GHz | . (64)The factors of order unity need not be computed here; the signal-to-noise ratio is completely negligible.This S / N estimate was for the correlation function of δ I . Wecould imagine instead measuring h E ( t ) E ∗ ( t + δ t ) E ( t + δ t ) E ∗ ( t + δ t ) i – the general four point function of the electric field –, which willincrease the number of independent lags from | D | σ f to ( | D | σ f ) .However, this increase is not comparable to N e as detection wouldrequire. In addition, we could look at even higher order momentsand compare the connected to disconnected part, but it is clear thatsince S / N is proportional to ∼ N − ne with n > /
2, where 1 / , and other factors simply are insu ffi cientto o ff set this large number. This equation assumes that | D | σ f ≫
1: see Eq. (59). From Eq. (19), thisis trivially satisfied except for extraordinarily narrow filters. If such a narrowfilter were used, then we should replace | D | in the estimates below by σ − / f .Of course this would then give the detectability of the connected correlationfunction, and not of the dispersion. For synchrotron emission, a nonzero odd point function requires an asym-metry in the orientation of the source’s gyrating electrons, such as wouldoccur if the system has nonzero net magnetic flux.c (cid:13) , 000–000 ispersion of incoherent sources In the Lieu & Duan (2013) equation for the waveform (their Eq. 1),DM enters (correctly) as a pure phase, and so the argument that thedisconnected part must not depend on dispersion ( §
3) has to remainvalid. Lieu & Duan (2013) calculate the disconnected part of theintensity correlation function at zero lag (which does not dependon DM when temporally averaged; their Eq. 11). They claim thatthe timescale of intensity variations is what depends on DM, andtheir calculations are based on intensity variations involving sumsover 2 electrons – thus they are indeed calculating the disconnectedpart. Here we show explicitly that the disconnected part of the vari-ation in the Lieu & Duan (2013) calculation is indeed independentof DM if we follow their calculation through to its conclusion inthe case of D ,
0. (Lieu & Duan 2013 do not provide the completecalculation.) In particular, we show that the coherence time of in-tensity fluctuations is ∼ σ t ≡ (2 πσ f ) − regardless of D , and is notthe width of the measured pulse from a single electron, which is ∼ D σ f if | D | ≫ σ t .In the notation of Lieu & Duan (2013), the observed wave-form (equivalent to our Gaussian bandpass, but defining ∆ ω ≡ πσ f ) is Ψ LD j ( t ) = Ac s π + i ξ ∆ ω e − at + i a ξ t − i ω t + i φ j . (65)This is Lieu & Duan (2013, Eq. 6) for the waveform where DMenters via ξ , and we have used their parameter definitions a ≡ ∆ ω + ξ ) (66)and ξ = d ω d k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω ( ∆ ω ) t j − t e c = − d( v − )d ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ω ( ∆ ω ) v Lc = − π D ( ∆ ω ) . (67)The amplitude is contained in Ψ LD j in the notation of Lieu & Duan(2013), so we may take E j = j . Note also that the timeof the event t j appears in the function ψ in Lieu & Duan (2013),whereas here we include it by using t − t j as the argument.We can now recompute the correlation functions using thisnotation. The electric field 2-point correlation function C E ( δ t ) is C E ( δ t ) = Γ [ R Ψ LD j ∗ Ψ LD ∗ j ]( t ) = π / | A | c Γ∆ ω e − ∆ ω δ t / − i ω δ t , (68)and the mean intensity is¯ I = C E (0) = π / | A | c Γ∆ ω, (69)in agreement with Lieu & Duan (2013, Eq. 8).The disconnected contribution to the intensity fluctuationcan be obtained by taking the fluctuating part of the intensity(Lieu & Duan 2013, Eq. 7), I ( t ) = ℜ X j < k Ψ LD j ( t − t j ) Ψ LD ∗ k ( t − t k ) , (70)and finding the correlation function at lag δ t , where ℜ denotes the Lieu & Duan (2013, Eq. 6) are missing a factor of i in front of the phase. Here we use the notation of Lieu & Duan (2013), where t j − t e is thepropagation time, and take L to be the distance to the source, so that thepropagation time is L / v g . We work to lowest order in the intergalactic elec-tron density so that we may approximate v g ≈ c . The first equality us basedon v g = d ω/ d k , and the last equality used that D = (2 π ) − d( L / v g ) / d f . real part. The leading part in the limit where N e ≫ j , k ): C δ I ( δ t ) = h I ( t ) I ( t + δ t ) i = (cid:28) X j < k ℜ [ Ψ LD j ( t − t j ) Ψ LD ∗ k ( t − t k )] ×ℜ [ Ψ LD j ( t − t j + δ t ) Ψ LD ∗ k ( t − t k + δ t )] (cid:29) . (71)Substitution of Eq. (65) gives C δ I ( δ t ) = (cid:28) π | A | c (1 + ξ ) X j < k e − a [( t − t j / − t k / + δ t / + δ t + ( t k − t j ) ] × cos n − a ξ [( t − t j ) − ( t − t k ) ] + ω t jk + φ jk o × cos n − a ξ [( t − t j + δ t ) − ( t − t k + δ t ) ] + ω t jk + φ jk o(cid:29) , (72)where φ jk ≡ φ j − φ k and t jk = t j − t k , and we have completed thesquare in the Gaussian envelope.It is readily seen that the Gaussian envelope in Eq. (72) canhave a nonzero contribution only when | δ t | . a − / , and hence it istempting to conclude that the coherence time is now ∼ a − / , whichgrows with | D | . But this is not so: a more accurate conclusion fromEq. (72) is that the coherence time must be . a − / . To see how thisworks, one must actually perform the average in Eq. (72). Let usperform the average over phases φ jk first, using the rule that h cos( α + φ jk ) cos( β + φ jk ) i =
12 cos( α − β ) . (73)Then with some algebraic simplification, Eq. (72) reduces to C δ I ( δ t ) = π | A | c (1 + ξ ) (cid:28)X j < k e − a [( t − t j / − t k / + δ t / + δ t + t jk ] × cos(2 a ξ t jk δ t ) (cid:29) . (74)The key to the coherence time is the cosine factor. At zero lag( δ t = as in Lieu & Duan (2013, Eq. 10), and theiraverage value is simply . However, when we look at nonzerocoherence times, there is an oscillatory factor: if t jk is an integermultiple of π/ ( a ξ δ t ), then the argument of the cosine is an inte-ger multiple of 2 π and the two electrons j and k cause positivelycorrelated intensity fluctuations at t and at t + δ t . However, if t jk is a half-integer multiple of π/ ( a ξ δ t ), then that pair of electronscauses negatively correlated intensity fluctuations. This oscillatoryor “fringing” behaviour has a simple interpretation for highly dis-persed pulses ( ξ ≫ j and k simultane-ously, then due to dispersion, the pulses have propagated through atime that di ff ers by t jk and hence their frequencies di ff er by Dt jk . If Dt jk δ t is an integer then the relative phase of the pulses from j and k is identical at t and t + δ t , and so that pair of electrons either inter-feres constructively at both t and t + δ t or destructively at both t and t + δ t . But if Dt jk δ t is a half-integer then the relative phase changesby π , and so that pair of electrons interferes constructively at either t or t + δ t and destructively at the other. The intensity correlationfunction at lag δ t thus receives a positive or negative contribution.This means that the intensity fluctuations can become uncor-related either if | δ t | ≫ a − / or if the range ∼ a − / of t jk al-lowed by the Gaussian envelope contains many fringe periods, i.e. This follows trivially from the product-to-sum rule.c (cid:13) , 000–000 Hirata & McQuinn is ≫ / ( a ξ δ t ). The latter condition is a − / ≫ a ξ δ t → δ t ≫ a / ξ ∼ p + ξ − ∆ ω . (75)Thus we expect the coherence time to be the minimum of a − / ∼ p + ξ / ∆ ω and p + ξ − / ∆ ω , i.e. it should be of order ∆ ω − ,regardless of the dispersion parameter ξ .To complete our computation of the intensity fluctuation cor-relation function, we replace the summation in Eq. (74) with anintegral over t j and t k (and include a prefactor of the rate Γ ). Infact it is easiest to switch variables to τ ≡ t − ( t j + t k ) / + δ t / t jk ; the Jacobian is unity. Moreover, the region of integration is over R , with a factor of to avoid double-counting pairs. The result is C δ I ( δ t ) = π | A | c (1 + ξ ) Γ e − a δ t Z ∞−∞ d τ e − a τ × Z ∞−∞ d t jk e − a t jk cos(2 a ξ t jk δ t ) . (76)The Gaussian ( τ ) and Gaussian-oscillatory ( t jk ) integrals are easilyevaluated as √ π / a and ( √ π / a )e − a ξ δ t respectively. The result is C δ I ( δ t ) = π | A | c a (1 + ξ ) Γ e − a δ t e − a ξ δ t = ¯ I e − ∆ ω δ t / , (77)where in the second equality we have used Eq. (66) for a andEq. (69) for ¯ I , and finally note that since we were calculatingthe disconnected part C δ I ( δ t ) = | C E | ( δ t ) , where C E is given byEq. (68). The discussion thus far has been entirely classical, and it has beenshown that the connected part of the intensity correlation function,which is suppressed by 1 / N e relative the to the disconnected part, isthe only part that is sensitive to DM. On the other hand, Lieu et al.(2013) have argued that only the electrons whose emitted photonsreach the observer should be counted in this calculation. In theirpicture, in the limit where other noise sources are negligible (i.e.where the source is much brighter than the sky integrated over thebeam width and with no thermal emission in the instrument), the S / N at which the connected part of the intensity correlation func-tion can be detected is ∝ / ¯ n γ in place of 1 / N e in our expressions,where ¯ n γ is the source photon occupation number (see discussionin their § T b ∼
1K in the beam of a radio instrument, they have¯ n γ ∼ ν − T b K , which is much smaller than N e ∼ . This moti-vates the question of how our classical derivations are modified byquantum mechanics. The basic question is as follows: are the quan-tum (Poisson) intensity fluctuations a feature of the source that canbe dispersed by passage through a plasma like any classical inten-sity fluctuation, as Lieu et al. (2013) assumed? Or is the Poissonnoise somehow immune to dispersion due to its quantum nature?It is shown in Appendix A that – within the full context ofquantum field theory, and with appropriate approximations – thedensity matrix of the received radiation is equivalent to a statisticalsuperposition of coherent states. These states are displaced vacuumstates, i.e. states in which the electromagnetic field operators ˆ A ( x ),ˆ E ( x ), etc. are equal to a classical field solution, plus the operators The calculation in Lieu et al. (2013) is actually classical; while incorpo-rating the Poisson noise term by constructing their signal as a sum of pulses,their signal “ Φ ( t )” is a number, not an operator. corresponding to the quantum vacuum fields. Their properties aredescribed in detail in § A3. The statistical distribution that goes intothese coherent states is merely the classical waveform emitted bythe electrons, averaged over a randomly chosen phase in their or-bits. It is thus apparent that the full quantum state of the receivedradiation can contain no information that would not be present clas-sically. In particular, the intensity fluctuations measured by a pho-toelectric detector, which correspond to a normal-ordered 4-pointcorrelation function of the electric field (Mandel & Wolf 1995, Ch.12), are equal to the classical correlation function computed fromthe coherent state amplitudes according to the optical equivalencetheorem (Sudarshan 1963) – i.e. those computed in the precedingsections. This result formally settles the question in favour of theclassical analysis presented herein: the intensity correlation func-tion is independent of DM up to corrections of order 1 / N e .To understand intuitively why this is so, one must consider thenature of Poisson noise for identical bosonic particles. Considera single-mode coherent state of complex amplitude v . A coherentstate, or displaced vacuum state, is a right eigenstate of the annihi-lation operator ˆ a with eigenvalue v (and not a right eigenstate of ˆ a † with eigenvalue v ∗ ). It is for this reason that the optical equivalencetheorem for coherent states applies to normal-ordered correlationfunctions: e.g. ¯ n γ ≡ h ˆ N i = h ˆ a † ˆ a i = | v | and h ˆ a † ˆ a i = | v | . Thevariance of the photon occupation number is given by h ˆ N i − h ˆ N i = h ˆ a † ˆ a ˆ a † ˆ a i − | v | = h ˆ a † [ˆ a , ˆ a † ]ˆ a i = ¯ n γ ; (78)this is in fact the Poisson noise term, and it arose entirely fromoperator-ordering considerations (the square of the photon numberoperator ˆ N = ˆ a † ˆ a is not normal-ordered). Since the phase delaydue to dispersion applies to the coherent state amplitudes, and notdirectly to numbers of photons, this added Poisson noise is not dis-persed.To frame this discussion in the context of a full statistical dis-tribution, the number of photons in that mode is Poisson-distributedwith mean | v | ; in a statistical mixture of such states where v is com-plex Gaussian distributed with zero mean and variance ¯ n γ , then theoccupation number is Bose-Einstein distributed, with mean ¯ n γ andvariance ¯ n γ + ¯ n γ (here ¯ n γ is the classical variance, and ¯ n γ is theadditional contribution from Poisson noise). This is a good modelfor the photon density matrix in a single mode of the electromag-netic field in the limit of a large number of emitting electrons N e ,since then the central limit theorem will force the amplitudes v tohave a Gaussian distribution. It is to this underlying semiclassicalGaussian field that the machinery in the previous sections shouldbe applied. The previous sections showed that dispersion is undetectable for in-coherent sources of radiation because the radiation is highly Gaus-sian. A class of extragalactic sources exist where the emissionis stimulated and potentially less Gaussian: the 1.6 GHz OH and22 GHz H O “mega-masers” (e.g. Lo 2005). Astrophysical masersdo not have a well-defined cavity like a laboratory laser. Instead,maser radiation is broadband in nature, without the phase coher-ence of a laboratory laser.Still, maser radiation may exhibit correlations between non-equal frequencies and hence non-Gaussianity (Menegozzi & Lamb1978; Field & Richardson 1984; Dinh-v-Trung 2009). Saturationof the population inversion and as a result in the growth of the elec-tric field occurs first in modes that have higher than average ampli- c (cid:13) , 000–000 ispersion of incoherent sources tudes. It acts to reduce the variance of the intensity, C δ I ( δ t = C δ I ( f ) that is ∼ −
10% the disconnected partat line center (Dinh-v-Trung 2009), although there are many ef-fects that may reduce the connected term relative to these predic-tions. This level of non-Gaussianity is comparable to the observa-tional limit on the connected part of C δ I (0) in Galactic OH masers(Evans et al. 1972).However, the connected contribution to C δ I ( δ t ) in a maserowes its existence to correlations between modes separated by onthe order of the homogeneous line width of the masing molecules, ∆ f hom (Dinh-v-Trung 2009). In order for plasma dispersion to havea significant impact on the received C δ I ( δ t ), we require | D | ∆ f &
1. Taking parameters appropriate for OH masers at z ∼
1, we find | D | ∆ f ∼ − assuming DM = pc cm − , and even smallervalues result for H O masers. Therefore, the connected part ofmaser emission is not significantly altered by dispersion.
The measurement of a dispersion measure to a cosmological ra-dio source would open up a new window on the study of inter-galactic baryons. Since all confirmed cosmological radio sourcesare constant over timescales much larger than those a ff ected bydispersion, the temporal delay in the arrival of di ff erent frequen-cies cannot be used to measure dispersion unlike for Galactic pul-sars. However, recently several schemes have been proposed tomeasure the dispersion to such continuous sources using (1) the2-point correlation function of the electric field, i.e. the delay ofa lower-frequency channel relative to a higher-frequency channel(Lovelace & Richards 2013); (2) the 2-point correlation functionof intensity fluctuations, which should have a longer timescale dueto the spreading of arrival time within a given frequency channel(Lieu & Duan 2013); and (3) the quantum corrections to the inten-sity fluctuations, i.e. dispersion of the Poisson noise fluctuations inthe source intensity (Lieu et al. 2013). Our analysis has shown that,due to various subtleties, none of these methods work. Indeed, un-der very general assumptions, the observed signal from a continu-ous point source has no information beyond the 2-point correlationfunction of the electric field (which evaluated at zero lag yields theintensity and polarization Stokes parameters). This result is a con-sequence of the central limit theorem and the large number of inco-herently emitting electrons contributing to the observed waveformfor any astrophysical source.The possibility of measuring cosmological dispersion mea-sures remains enticing, but will only be possible using sources thatare time-variable or coherent (and hence potentially non-Gaussian).Masers are coherent, but for typical parameters the dispersionacross their very narrow line widths is too small. Thus, given thecalculations presented herein, the availability of DM measurementsas a probe of the intergalactic medium remains contingent on thecosmological interpretation of the fast radio bursts (Thornton et al.2013) or the existence of some similar class of fast radio transients. ACKNOWLEDGMENTS
We thank Eric Hu ff , Richard Lovelace, and David Weinberg foruseful comments.During the preparation of this paper, CH has been supported by the US Department of Energy under contract DE-FG03- 02-ER40701, the David and Lucile Packard Foundation, the SimonsFoundation, and the Alfred P. Sloan Foundation. MM acknowl-edges support by the National Aeronautics and Space Administra-tion through Hubble Postdoctoral Fellowship awarded by the SpaceTelescope Science Institute, which is operated by the Associationof Universities for Research in Astronomy, Inc., for NASA, undercontract NAS 5-26555. REFERENCES
Abramowitz M., Stegun I. 1972, Handbook of MathematicalFunctions, Dover, New YorkBurke-Spolaor S., Bailes M., Ekers R., Macquart J., Crawford F.2011, ApJ, 727, 18Cordes J., Lazio T. 2002, arXiv:astro-ph / APPENDIX A: QUANTUM MECHANICAL TREATMENTOF SYNCHROTRON PULSES
In the main text, we have treated the emission of synchrotron radia-tion classically. In particular, Lieu et al. (2013) argued that the rateof pulses Γ appearing in the correlation function formulae shouldbe the number of photons received per unit time ˙ N ph − rec , whereasin our classical treatment it is the number of electrons whose syn-chrotron beams sweep over the observer per unit time ˙ N el − beam . In practical situations, ˙ N ph − rec ≪ ˙ N el − beam . Since the dimensionless Lieu et al. (2013) use the symbol λ for this rate instead of Γ here and inLovelace & Richards (2013).c (cid:13) , 000–000 Hirata & McQuinn connected part of the intensity fluctuation C j = k = l = m ( δ t ) / ¯ I is pro-portional to 1 / Γ , and it is this part that is sensitive to DM, it isimportant that we resolve this issue. Is the classical calculation cor-rect, so that Γ is large and the connected intensity fluctuations aretiny, as found in the main text? Or should only the photons that arereceived by the observer contribute to Γ , as claimed by Lieu et al.(2013), resulting in much larger connected intensity fluctuations?The purpose of this appendix is to resolve this issue witha quantum mechanical calculation of an appropriately simplifiedproblem. Glauber (1963b) showed that a classical current sourcecoupled to a quantized radiation field initially in the vacuum stateproduces a “coherent” photon state, i.e. one obtained from thevacuum state by displacing the wave function with the displace-ment given by the classical field amplitude. We consider here afully quantized source, and show that with suitable approximationsthe outgoing photon state is a statistical superposition of coher-ent states, each corresponding to the classical field amplitude fromelectrons with a random distribution of phases in their orbits. Theconclusion is that the received radiation field is in fact statisticallyindistinguishable from the classical field with appropriate measure-ment noise (including the familiar h f per mode in the case of acoherent receiver) added in. Therefore, quantum corrections to thereceived electric field do not provide additional information thatwould be inaccessible classically; in particular they do not add anyinformation about the dispersion measure.Throughout we use the c.g.s. unit system and the Schr¨odingerpicture of quantum mechanics. A1 Assumptions
Consider the problem of a synchrotron-emitting cloud. For simplic-ity, we will take the cloud to be optically thin. We will furthermoreignore processes that create or destroy electrons (we are interestedonly in the synchrotron radiation), ignore the electron spin, and willassume the cloud to be su ffi ciently dilute that the identical natureof the electrons can be neglected (i.e. we can number them 1 ... N ,and treat their motion as independent degrees of freedom, ignoringwave function anti-symmetrization or state blocking). Of these as-sumptions, only the optically thin condition is likely to be violatedin a realistic AGN.We further assume here that the magnetic field configurationpermits separation of variables. This is not likely to be true in anactual source, but since our only use of this assumption is to con-struct wave packets in action-angle space (instead of in position-momentum space, which would lead to a much more extended for-malism) we do not think it is of critical importance. In particular,we follow each wave packet through only a portion of an orbit, sowe expect that exact integrability (or not) would not a ff ect the con-clusions. Also the semi-classical limit will be taken in which allquantum numbers are large, here meaning that the change in quan-tum number in emission of a single photon is small compared withthe quantum number itself. A2 Formalism and Hilbert space
The electromagnetic field is quantized as a wave, with a set ofdiscrete indices α for the various modes and a continuous index k ∈ R + = (0 , ∞ ) describing the wave number. There are manypossible choices of mode with one continuous index (e.g. sphericalwaves, where k is a continuous and the discrete quantum numbersare angular momentum jm and electric or magnetic type parity Eor M) but we do not specify these yet. The appropriately normal-ized transverse radiative magnetic vector potential operator is thengiven by A (rad) ( x ) = X α Z ∞ d k π r π ~ ck h Z α ( x ; k )ˆ a α ( k ) + Z ∗ α ( x ; k )ˆ a † α ( k ) i , (A1)where ˆ a α ( k ) is an annihilation operator, ˆ a † α ( k ) is a creation operator,and the mode functions Z α ( x ; k ) are complete over the space of di-vergenceless vector fields. They obey the orthonormality relation Z Z ∗ α ( x ; k ) · Z β ( x ; k ′ ) d x = πδ αβ δ ( k − k ′ ) (A2)and the eigenvalue equation ∇ Z α ( x ; k ) = − k Z α ( x ; k ). The annihi-lation operators mutually commute, but have a nontrivial commu-tation with the creation operators[ˆ a α ( k ) , ˆ a † β ( k ′ )] = πδ αβ δ ( k − k ′ ) . (A3)The Hamiltonian is – aside from an irrelevant additive constant –ˆ H rad = X α Z ∞ d k π ~ ck ˆ a † α ( k )ˆ a α ( k ) . (A4)The radiative part of the electric field is conjugate to the magneticvector potential: E (rad) ( x ) = X α Z ∞ d k π √ π c ~ k i h Z α ( x ; k )ˆ a α ( k ) − Z ∗ α ( x ; k )ˆ a † α ( k ) i . (A5)Our next interest is in the electrons. Since we are neglectingspin, we take these to be described by a complex scalar wave equa-tion in a time-independent background magnetic vector potential A (bg) ( x ). The Lagrangian isˆ L el = Z R d x n ~ | ˙ˆ ψ ( x ) | − | [ − i c ~ ∇ − e A (bg) ] ψ ( x ) | − m c | ˆ ψ ( x ) | o . (A6)As usual without an electric potential, the conjugate momentum isˆ π ( x ) = ~ ˆ˙ ψ ∗ ( x ) and the Hamiltonian ˆ H el is equal to ˆ L el but with a + instead of a − in the second two terms. The complex wave equationis separable as ψ ( x ) = φ n ( x )e − i ω n t , where the mode functions aregiven by ~ ω n φ n ( x ) = m c φ n ( x ) + [ − i c ~ ∇ − e A (bg) ( x )] φ n ( x ) . (A7)As this is an eigenvalue equation with eigenvalue ω n and a positive-definite Hermitian right-hand side, we may choose the φ n ( x ) tobe L -orthonormal: R R φ n ( x ) φ ∗ n ′ ( x ) d x = δ nn ′ , and complete: P n φ n ( x ) φ ∗ n ( y ) = δ (3) ( x − y ). The “electron” wave operator and itsconjugate momentum may then be written asˆ ψ ( x ) = X n √ ~ ω n (ˆ b n + ˆ d † n ) φ n ( x ) (A8) The formulae given here for the quantized electromagnetic field are stan-dard; we have used those of Mandel & Wolf (1995, § k -index. Only a 1-dimensional δ -function appears here since k is taken to be anumber rather than a vector; the directional dependence is captured in thediscrete indices, which may be e.g. angular momentum indices.c (cid:13)000
The electromagnetic field is quantized as a wave, with a set ofdiscrete indices α for the various modes and a continuous index k ∈ R + = (0 , ∞ ) describing the wave number. There are manypossible choices of mode with one continuous index (e.g. sphericalwaves, where k is a continuous and the discrete quantum numbersare angular momentum jm and electric or magnetic type parity Eor M) but we do not specify these yet. The appropriately normal-ized transverse radiative magnetic vector potential operator is thengiven by A (rad) ( x ) = X α Z ∞ d k π r π ~ ck h Z α ( x ; k )ˆ a α ( k ) + Z ∗ α ( x ; k )ˆ a † α ( k ) i , (A1)where ˆ a α ( k ) is an annihilation operator, ˆ a † α ( k ) is a creation operator,and the mode functions Z α ( x ; k ) are complete over the space of di-vergenceless vector fields. They obey the orthonormality relation Z Z ∗ α ( x ; k ) · Z β ( x ; k ′ ) d x = πδ αβ δ ( k − k ′ ) (A2)and the eigenvalue equation ∇ Z α ( x ; k ) = − k Z α ( x ; k ). The annihi-lation operators mutually commute, but have a nontrivial commu-tation with the creation operators[ˆ a α ( k ) , ˆ a † β ( k ′ )] = πδ αβ δ ( k − k ′ ) . (A3)The Hamiltonian is – aside from an irrelevant additive constant –ˆ H rad = X α Z ∞ d k π ~ ck ˆ a † α ( k )ˆ a α ( k ) . (A4)The radiative part of the electric field is conjugate to the magneticvector potential: E (rad) ( x ) = X α Z ∞ d k π √ π c ~ k i h Z α ( x ; k )ˆ a α ( k ) − Z ∗ α ( x ; k )ˆ a † α ( k ) i . (A5)Our next interest is in the electrons. Since we are neglectingspin, we take these to be described by a complex scalar wave equa-tion in a time-independent background magnetic vector potential A (bg) ( x ). The Lagrangian isˆ L el = Z R d x n ~ | ˙ˆ ψ ( x ) | − | [ − i c ~ ∇ − e A (bg) ] ψ ( x ) | − m c | ˆ ψ ( x ) | o . (A6)As usual without an electric potential, the conjugate momentum isˆ π ( x ) = ~ ˆ˙ ψ ∗ ( x ) and the Hamiltonian ˆ H el is equal to ˆ L el but with a + instead of a − in the second two terms. The complex wave equationis separable as ψ ( x ) = φ n ( x )e − i ω n t , where the mode functions aregiven by ~ ω n φ n ( x ) = m c φ n ( x ) + [ − i c ~ ∇ − e A (bg) ( x )] φ n ( x ) . (A7)As this is an eigenvalue equation with eigenvalue ω n and a positive-definite Hermitian right-hand side, we may choose the φ n ( x ) tobe L -orthonormal: R R φ n ( x ) φ ∗ n ′ ( x ) d x = δ nn ′ , and complete: P n φ n ( x ) φ ∗ n ( y ) = δ (3) ( x − y ). The “electron” wave operator and itsconjugate momentum may then be written asˆ ψ ( x ) = X n √ ~ ω n (ˆ b n + ˆ d † n ) φ n ( x ) (A8) The formulae given here for the quantized electromagnetic field are stan-dard; we have used those of Mandel & Wolf (1995, § k -index. Only a 1-dimensional δ -function appears here since k is taken to be anumber rather than a vector; the directional dependence is captured in thediscrete indices, which may be e.g. angular momentum indices.c (cid:13)000 , 000–000 ispersion of incoherent sources and ˆ π ( x ) = X n − i r ~ ω n b n − ˆ d † n ) φ n ( x ) , (A9)where the sum is over ω n >
0. It is readily verified that these oper-ators obey the proper commutation relations with ˆ b n and ˆ d n inter-preted as annihilation operators for independent quantum harmonicoscillators (for the particle and antiparticle), and with a Hamilto-nian ˆ H el = X n ~ ω n (ˆ b † n ˆ b n + ˆ d † n ˆ d n ) , (A10)again with an irrelevant constant subtracted o ff . While the antiparti-cle operators are necessary for the overall consistency of the theory,none of our operations will involve states with antiparticles and soin what follows we suppress terms involving ˆ d n .The interaction of matter and radiation to first order in theradiation amplitude is described byˆ H = e Z ˆ A (rad) · n − i ~ c ˆ ψ † ∇ ˆ ψ + i ~ ˆ ψ ∇ ˆ ψ † + e A (bg) ˆ ψ † ˆ ψ o d x ;(A11)the term responsible for emission of synchrotron radiation then hasthe formˆ H (I)int = e X nn ′ α Z d k π r π c ~ k ω n ω n ′ ˆ a † α ( k )ˆ b n ˆ b † n ′ Z d x Z ∗ α ( x ; k ) · n i ~ c φ ∗ n ′ ∇ φ n ( x ) − i ~ c φ n ∇ φ ∗ n ′ ( x ) + e A (bg) φ ∗ n ′ φ n ( x ) o . (A12)The Hermitian conjugate ˆ H (I) † int is also present. A term ˆ H (II)int contain-ing two factors of the radiation Hamiltonian is also present, but wedo not need its explicit form. The total interaction Hamiltonian isthus ˆ H int = ˆ H (I)int + ˆ H (I) † int + ˆ H (II)int .The relevant Hilbert space thus consists of the photon andelectron degrees of freedom. A3 Setup of the problem; initial conditions
In the synchrotron emission problem, the initial state of the elec-tromagnetic field is usually taken to be the vacuum, | vac i . We willbe slightly more general here in order to derive results that are use-ful later, and choose the coherent state (Glauber 1963a,b) | v α ( k ) i ,defined by | v α ( k ) i = exp nX α Z d k π [ v α ( k )ˆ a † α ( k ) − v ∗ α ( k )ˆ a α ( k )] o | vac i . (A13)It is important to note that a coherent state for a photon field islabeled by a set of complex functions v α ( k ) for each mode. The op-erator in brackets is an anti-Hermitian linear combination of thegeneralized coordinate operators ˆ A and generalized momentumoperators ˆ E . Thus the coherent state can be thought of as a dis-placed vacuum state: if v α ( k ) is real, then the complex exponentialin Eq. (A13) is a displacement operator in the coordinate-space rep-resentation of the wave function; if v α ( k ) is purely imaginary, thenit is a displacement operator in the momentum-space representationof the wave function.A general discussion of coherent states and their propertiescan be found in Mandel & Wolf (1995, § • An annihilation operator acting on a coherent state returns thestate’s value, ˆ a α ( k ) | v β ( k ′ ) i = v α ( k ) | v β ( k ′ ) i . • Any density matrix on the photon space can be represented for-mally as a statistical superposition of coherent states (though notnecessarily with positive weight). • In a coherent state, the expectation value of any normal-orderedoperator ˆ a † α ( k ) ... ˆ a α M ( k M ) is obtained by replacing ˆ a † α ( k ) → v ∗ α ( k )and ˆ a α ( k ) → v α ( k ) (the optical equivalence theorem). • Finally, the coherent state is not an eigenstate of the freeHamiltonian, but it does evolve simply as exp( − i ˆ H rad t / ~ ) | v α ( k ) i = | e − i ckt v α ( k ) i , where the (irrelevant) zero-point energy of the radia-tion Hamiltonian has been removed. A4 WKB approximation and correspondence principle
The absorption and emission of radiation by charged particles ina potential with a separable Hamiltonian has a long history butis not often covered in standard texts. Pioneering work, predat-ing quantum field theory, can be found in van Vleck (1924a,b). Afull derivation of the results is given here however, to be consistentwith the formalism of quantum field theory and the notation usedelsewhere in this appendix. We use the Wentzel-Kramers-Brillouin(WKB) or eikonal approximation to the solutions to the wave equa-tion with classical Hamiltonian H cl ( x , p ) = q m c + c [( p − e A (bg) ( x )] . (A14)The WKB approximation to the stationary states can be for-mulated in terms of action-angle variables. In cases where thequantum-mechanical wave equation is separable, the classicalHamilton-Jacobi equation is also separable and hence one can con-struct a set of conjugate action-angle variables, the angles { θ µ } µ = (periodic over the domain from 0 to 2 π , and here taken to be afunction of the actions and the spatial coordinates) and the actions { ~ n µ } µ = . The solution W ( x , n ) to the Hamilton-Jacobi equation, i.e. ~ ω n = H cl ( x a , ∂ W /∂ x a ) is Hamilton’s characteristic function andhas units of action. It is the generating function for the canonicaltransformation from ( x a , p a ) → ( ~ n µ , θ µ ). The relation is given ex-plicitly by p a = ∂ W ∂ x a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x , n and θ µ = ~ − ∂ W ∂ n µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x , n ; (A15)see e.g. Goldstein (1980, § ~ ω n = H cl . The Hamiltonian is asmooth function of the quantum numbers for a separable system;the three classical fundamental frequencies are given by ~ Ω µ = ∂ H cl /∂ n µ . Since there are three actions (and hence three quantumnumbers) it is convenient to index the states by the triplet of integers n ∈ Z ; thus the index for the electron mode functions φ n willhenceforth be written in boldface.In a one-dimensional quantum mechanical system (where x , p , θ , and n have only one component and so are written as scalars),the WKB solution may be written as φ . o . f . n ( x ) = C X streams | ∂ x /∂θ | / e i W ( x , n ) / ~ , (A16)where the summation is over the di ff erent streams, i.e. the di ff erentpossible values of momentum (or angle) at fixed action ~ n and po-sition x . (In textbook examples, there are usually two streams, onewith positive velocity and one with negative velocity.) The pref-actor C does not depend on position. The characteristic function W ( x , n ) is normally written in quantum mechanics texts as the “ac-tion” R p d x , but by Eq. (A15) the momentum is equal to ∂ W /∂ x and so these forms are equivalent. The denominator is normally c (cid:13) , 000–000 Hirata & McQuinn written as the classical velocity ˙ x cl in quantum mechanics texts, butit may also be written as ∂ x /∂θ since the conversion factor ˙ θ cl = Ω is independent of x . The requirement of the normalization of thewave function forces C = (2 π ) − / (up to an overall and irrelevantphase). We have ignored the phase shift at turning points, since inour calculations below only a small portion of the orbit is consid-ered.In a multiple degree of freedom system that separates in thethree coordinates ( x , x , x ), Hamilton’s characteristic function canbe written for a given set of actions as a sum of functions in eachseparated variable. Then the total wave function is a product ofthe wave functions in each of the three coordinates, and Eq. (A16)generalizes to φ n ( x ) = (2 π ) − / X streams (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ x ∂θ ∂ x ∂θ ∂ x ∂θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − / e i W ( x , n ) / ~ . (A17)The summation is over the di ff erent streams, i.e. the di ff erent solu-tions for θ at fixed n and x . In fact, Eq. (A17) is even more generalthan that: if the wave equation separated in some other coordinatesystem ( X , X , X ), then the transformation of a quantum mechan-ical wave function back to Cartesian coordinates ( x , x , x ) intro-duces an additional factor of | det( ∂ X /∂ x ) | / , leading to φ n ( x ) = (2 π ) − / X streams | det T | − / e i W ( x , n ) / ~ , (A18)where we have defined the 3 × T = ( ∂ x a /∂θ µ ) | n . There-fore we may use Eq. (A18), even if the usual Cartesian coordinatesystem is not the system in which the motion separates.For appropriate choice of photon modes, the basis functionscan be taken to be local plane waves in the emitting region (whichwe place at the origin). An example of such a basis is the basis ofspherical waves centered at a distant observer. Then Z α ( x ; k ) = Y α ( k )ˆ ǫ α ( k ) e i k ˆ s α ( k ) · x , (A19)where Y α, k is a scalar amplitude, ˆ ǫ α ( k ) is a unit vector polariza-tion, and ˆ s α ( k ) is a unit vector in the direction of the local wavevector. The mode equation guarantees that this wave vector hasnorm k , and since the modes are transverse (divergenceless) wehave ˆ ǫ α ( k ) ⊥ ˆ s α ( k ).The interaction Hamiltonian, Eq. (A12), is thenˆ H (I)int = e π X nn ′ α Z d k π r π c ~ k ω n ω n ′ ˆ a † α ( k )ˆ b n ˆ b † n ′ Y ∗ α ( k ) Z e − i k ˆ s α ( k ) · x × ˆ ǫ ∗ α ( k ) | det T | · h − c ∇ x W ( x , n ) + e A (bg) ( x ) i e i( n − n ′ ) · θ cl d x , (A20)where the di ff erence in quantum numbers ∆ n = n ′ − n is kept onlyin the relative phase of the di ff erent wave functions. Here θ cl ( x , n )is the classical angle at positon x and action ~ n (i.e. obtained fromthe canonical transformation) and is equal to θ µ = ~ − ∂ W /∂ n µ byEq. (A15). It appears in the complex exponential because the di ff er-ence in W between two di ff erent values of n has been replaced by apartial derivative times ∆ n . We have suppressed here the sum overstreams, as only one stream (the one beamed toward the observer)is significant at any one time.The integrals in Eq. (A20) can be simplified as follows. First,note that the classical velocity of the wave packet is u cl = ∇ p H cl = c c p cl − e A (bg) ( x ) H cl = c c ∇ x W ( x , n ) − e A (bg) ( x ) H cl ( n ) . (A21)We may thus writeˆ H (I)int = ~ X α, n , ∆ n Z d k π ˆ a † α ( k )ˆ b † n +∆ n ˆ b n F α, n , ∆ n ( k ) , (A22) where F α, n , ∆ n ( k ) = − r π c ~ k Y ∗ α ( k ) F α, n , ∆ n ( k ) (A23)and F α, n , ∆ n ( k ) = ec Z e − i k ˆ s α ( k ) · x ˆ ǫ ∗ α ( k ) · u cl ( x , n )e − i ∆ n · θ cl d θ (2 π ) . (A24)In this expression, we have used the Jacobian to convert the d x integral to a d θ integral, and used that the classical Hamiltoniancorresponding to a given orbit is equal to ~ ω n .The interpretation of F α, n , ∆ n ( k ) is straightforward. In classical electrodynamics, the current density J cl ( x , t ) is multi-periodic withfrequencies given by m · Ω , where m ∈ Z is a triplet of integers. Theclassical current density is given by e u cl δ (3) ( x − x cl ), where x cl is theparticle’s classical position. Its 4-dimensional Fourier transform is˜ J cl ( k , ω ) = Z J cl ( x , t ) e − i( k · x − ω t ) d x d t = Z d θ Z d t δ (3) [ θ − θ (0) − Ω t ] e u cl e − i k · x cl e i ω t = X m Z d θ (2 π ) Z d t e i m · [ θ − θ (0) − Ω t ] e u cl e − i k · x cl e i ω t = X m Z d θ (2 π ) e u cl e − i k · x cl e i m · [ θ − θ (0)] δ ( ω − m · Ω ) , (A25)where θ (0) is the angle at t =
0. The classical vector potential˜ A cl , obs ( ω ) radiated by the particle, measured at a large distance R from the system in direction ˆ s α , is then given byˆ ǫ ∗ α · ˜ A cl , obs ( ω ) = ecR X m Z d θ (2 π ) ǫ ∗ α · u cl e i k ω/ c e − i k ˆ s α · x cl e i m · [ θ − θ (0)] δ ( ω − m · Ω ) = e i ω R / c R X m π F α, n , m ( ω/ c )e − i m · θ (0) δ ( ω − m · Ω ) , (A26)where in the intermediate steps we have taken k = ω/ c and usedthe Green’s function expansion for electromagnetic fields (Jackson1998, Eq. 6.48) to equate the received frequency-domain mag-netic vector potential to the 4-dimensional Fourier transform ofthe source. The conclusion is that the coe ffi cient in the interac-tion Hamiltonian that contains the detailed dependence on the wavefunction properties is in fact equal to the classical radiated electro-magnetic field. This is a manifestation of the correspondence prin-ciple.The time-domain version of Eq. (A26) isˆ ǫ ∗ α · A cl , obs ( t ) = e i ω R / c R X m F α, n , m ( ω/ c )e − i m · θ (0) e − i m · Ω t . (A27)Note that F α, n , m ( k ) has units of magnetic flux. A5 Quantum emission – formalism
Now let us follow the quantum behavior of the photon state throughthe emission of a single pulse, to second order in the interactionstrength. In propagating from a starting time t s to an ending time t e ,the state varies as | Ψ ( t e ) i = ∞ X j = ( − i) j ~ j Z t e > t >... t j > t s d t ... d t j ˆ U ( t e , ) ˆ H int ˆ U ( t , ) ˆ H int ... × ˆ U ( t j − , j ) ˆ H int ˆ U ( t j , s ) | Ψ ( t s ) i , (A28) c (cid:13)000
Now let us follow the quantum behavior of the photon state throughthe emission of a single pulse, to second order in the interactionstrength. In propagating from a starting time t s to an ending time t e ,the state varies as | Ψ ( t e ) i = ∞ X j = ( − i) j ~ j Z t e > t >... t j > t s d t ... d t j ˆ U ( t e , ) ˆ H int ˆ U ( t , ) ˆ H int ... × ˆ U ( t j − , j ) ˆ H int ˆ U ( t j , s ) | Ψ ( t s ) i , (A28) c (cid:13)000 , 000–000 ispersion of incoherent sources where ˆ U ( δ t ) = e − i( ˆ H rad + ˆ H el ) δ t / ~ . The shorthand t a , b ≡ t a − t b has beenintroduced. One may define a “rotated” Hamiltonian byˆ H rot ( t ) = ˆ U ( t e − t ) ˆ H int ˆ U † ( t e − t ) , (A29)with which | Ψ ( t e ) i = ∞ X j = ( − i) j ~ j Z t e > t >... t j > t s d t ... d t j ˆ H rot ( t ) ˆ H rot ( t ) ... ˆ H rot ( t j ) × ˆ U ( t es ) | Ψ ( t s ) i . (A30)The ordering of time here is key, because while the interactionHamiltonian H always commutes with itself, it does not commutewith the unperturbed Hamiltonian or hence with ˆ U ( δ t ). It followsthat the unequal-time rotated Hamiltonians do not necessarily com-mute with each other. However, by taking the logarithm of the op-erator on the first line of Eq. (A30), we find | Ψ ( t e ) i = e − i ˆ O ˆ U ( t es ) | Ψ ( t s ) i , (A31)whereˆ O = ~ Z t e t s d t ˆ H rot ( t ) − i ~ Z t e t s d t Z t t s d t [ ˆ H rot ( t ) , ˆ H rot ( t )] + ... (A32)is a Hermitian operator.The terms in ˆ O can be understood most easily if H rot is brokendown into individual terms (each with some number of annihilationand creation operators of particular modes) such that H rot , a ∝ e − i ω a t .Then – taking the limit of large t es – the time integrals may beperformed to give ˆ O = t es ~ X a W ( ω a ) ˆ H int , a + t es ~ X a , b W ( ω a + ω b ) P ω a − ω b × [ ˆ H int , a , ˆ H int , b ] + ..., (A33)where P denotes a principal part , and W is a window function: W ( s ) = t es Z t e t s e i st ea d t a = e i st es / sin( st es / st es / W (0) = W ( − s ) = W ∗ ( s ), and R ∞−∞ W ( s ) d s = π/ t es . Tosimplify the second-order term we used the identity that for longtimes , Z t e t s d t a Z t a t s d t b e i ζ t ea e i η t eb ≈ t W ( ζ ) W ( η ) + t es W ( ζ + η ) P ζ − η . (A35) In the commonly used interaction picture formulation of quantum fieldtheory, the interaction Hamiltonian operator is ˆ H rot ( t ) and the field operatorsevolve from one time to another according to ˆ U ( δ t ). This operator is easily seen to be Hermitian since each term H rot , a willhave a conjugate term H † rot , a with the opposite frequency. In the second-order term in O , the Hermitian conjugate term appears with a − sign inaddition to the usual complex conjugates since 1 / ( ω a − ω b ) flips sign. This iswhy there is no factor of i in this term, even though for Hermitian operatorsˆ A and ˆ B it is i[ ˆ A , ˆ B ] rather than [ ˆ A , ˆ B ] is Hermitian. This is a principal part in the sense that one averages over the two pos-sible pole displacements, P ( z − ) = [( z + i ǫ ) − + ( z − i ǫ ) − ]. This wayfor an analytic function f , the conventional principal part of the integral R z − f ( z ) d z is equal to R P ( z − ) f ( z ) d z . This can be proven by splitting the integral into terms symmetric under ζ ↔ η and antisymmetric. The symmetric term becomes the product oftwo W -functions, while the antisymmetric term can be split into a doubleintegral over ( t a + t b ) / t ab . Approximating the range of integrationover t ab as 0 < t ab < ∞ gives the result. The exact antisymmetry of thisterm under ζ ↔ η implies that the inverse 1 / ( ζ − η ) should be taken to bethe principal part. The interpretation of Eq. (A33) is straightforward: the long-timeevolution is dominated by a series of interactions with 1 vertex,with 2 vertices, and higher-order terms (not shown here). Interac-tions with multiple vertices contain a propagator (inverse frequencydenominator). This is the familiar expansion of particle scatteringin quantum field theory. The exponential in Eq. (A31) allows mul-tiple interactions to take place; it is of minor importance for single-particle scattering but is critical for understanding the coherenceproperties of light.
A6 Quantum emission – application
It is now time to consider the density matrix evolution of the ra-diation field during the above process. Suppose that we start in acoherent photon state and a definite action for the electron, i.e. | Ψ ( t s ) i = | v α ( k ) i ⊗ | n i . (A36)Here | n i indicates an electron in the state with quantum numbers n ,i.e. | n i ≡ ˆ b † n | vac i . The unperturbed unitary evolution takes this to | Ψ (0)e i = e − i ω n t es | e − i ckt es v α ( k ) i ⊗ | n i . (A37)The final photon density matrix is ρ rad ( t e ) = Tr el h e − i ˆ O | e − i ckt es v α ( k ) i ⊗ | n ih e − i ckt es v α ( k ) | ⊗ h n | e i ˆ O i , (A38)where the trace is over the electron state. The trace may be sim-plified with a resolution of the identity operator into angle statesas I el = Z d ϑ (2 π ) | ϑ ih ϑ | , (A39)where | ϑ i ≡ P m e − i ϑ · m | m i . Then ρ rad ( t e ) = Z d ϑ (2 π ) h ϑ | e − i ˆ O | e − i ckt es v α ( k ) i⊗| n ih e − i ckt es v α ( k ) |⊗h n | e i ˆ O | ϑ i . (A40)Now consider the e ff ect of the terms in ˆ O that are first-orderin e , which correspond to the elementary emission and absorptionprocesses. We now make the approximation that, over a small rangein electron quantum numbers near n , the amplitude F β, n ′ , ∆ n ( k ′ ) isroughly constant. This assumption eliminates self-absorption, sinceself-absorption with the absorbing electron in a given level n is re-lated to the fact that transitions from n ↔ n + n ↔ n − Then ˆ O re-duces toˆ O ≈ t es X β, ∆ n Z d k ′ π W ( ck ′ − Ω · ∆ n ) F β, n , ∆ n ˆ Σ † ∆ n ˆ a † β ( k ′ ) + h . c ., (A41)where ˆ Σ ∆ n = P m ˆ b † m ˆ b m +∆ n is the state shift operator. Given our pre-vious approximations, this is now the only operator in ˆ O that actson the electron Hilbert space. But in Eq. (A40), it acts on an anglestate, which is an eigenstate :ˆ Σ ∆ n | ϑ i = e − i ϑ · ∆ n | ϑ i and ˆ Σ † ∆ n | ϑ i = e i ϑ · ∆ n | ϑ i . (A42) The treatment of the propagator poles is di ff erent because here we donot have the boundary conditions of a scattering problem. It can be seen that this situation will occur semi-classically fromEq. (A27). Considering only one of the electron degrees of freedom andassuming a harmonic oscillator, the squared amplitude of emitted radiationis proportional to the action, | F β, n ′ , ( k ′ ) | ∝ n ′ . Technically this is only true if the range of quantum numbers is overall integers, since otherwise the eigenstate formula fails for e.g. n < ∆ n .Since our analysis does not involve states with small quantum numbers, thisis not a problem.c (cid:13) , 000–000 Hirata & McQuinn
We may thus make the replacement in Eq. (A40):ˆ
O ≈ t es X β, ∆ n Z d k ′ π W ( ck ′ − Ω · ∆ n ) F β, n , ∆ n e i ϑ · ∆ n ˆ a † β ( k ′ ) + h . c ., (A43)and use h n | ϑ i = e − i ϑ · n . This leaves Eq. (A40) in the form ρ rad ( t e ) = Z d ϑ (2 π ) ˆ D| e − i ckt es v α ( k ) ih e − i ckt es v α ( k ) | ˆ D † , (A44)where ˆ D = exp nX β Z d k ′ π [ u β ( k ′ )ˆ a † β ( k ′ ) − u ∗ β ( k ′ )ˆ a β ( k ′ )] o (A45)and u β ( k ′ ) = − i t es X ∆ n W ( ck ′ − Ω · ∆ n ) F β, n , ∆ n e i ϑ · ∆ n . (A46)Note that ˆ D is a displacement operator; acting on a coherent state | v α ( k ) i , it gives another state e i χ | u α ( k ) + v α ( k ) i , whose amplitude isthe input amplitude plus the displacement u α ( k ). The phase χ is notneeded here, since it cancels out in the density matrix Eq. (A44).The result of Eq. (A44) is that the output state of the radiation,after interaction with a single electron in a quantum state | n i , isa statistical superposition of coherent states, where the statisticalaverage is taken over angles ϑ . The coherent state is displaced by u α ( k ), given by Eq. (A46).The above machinery is now well-suited to studying the quan-tum state of the radiation after interaction with many electrons start-ing from an initial vacuum state | vac i . The interaction with eachelectron adds another term to u α ( k ), thus placing the photon ulti-mately in a statistical superposition of coherent states with ampli-tude P N e i = u α ( k ), where the statistical superposition is taken over the3 N e angles ϑ ... ϑ N e : ρ rad ( t e ) = Z d ϑ ... d ϑ N e (2 π ) N e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N e X i = u α ( k ) + * N e X i = u α ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A47) A7 Relation to classical waveform
The interpretation of this result is easiest if we realize that to everycoherent state | u α ( k ) i of a quantum field there corresponds a classi-cal field configuration obtained via the substitution ˆ a α ( k ) → u α ( k )in Eq. (A1). The quantum state is the vacuum displaced by this clas-sical solution, and as such a measurement on a quantum system in anonnegative-weight statistical superposition of coherent states (e.g.Eq. A47) can contain no more information than the classical sys-tem in the corresponding statistical distribution. In particular, ac-cording to the optical equivalence theorem (Sudarshan 1963), thenormal-ordered quantum correlation functions of the field, such asthe intensity fluctuations that would be measured by a photoelectricdetector, are equal to the classically computed moments based onthe probability distribution of u α ( k ).We are therefore motivated to learn about the functions u α ( k )produced in single-electron interactions and their statistical proper-ties. We do this by finding the corresponding received field at theobserver. Substituting W ( s ) → ( π/ t es ) δ ( s ) into Eq. (A46), we find u α ( k ) = i π + X ∆ n δ ( ck − Ω · ∆ n ) r π c ~ k Y ∗ α ( k ) F α, n , ∆ n ( k )e i ϑ · ∆ n , (A48)where the + sign on the summation indicates that the sum is taken over states with Ω · ∆ n >
0. Substitution into Eq. (A1) with thenyields, with some simplification, the classical-equivalent field A coh ( x ) = π i + X α, ∆ n Y ∗ α ( k ) Z α ( x ; k ) F α, n , ∆ n ( k ) k e i ϑ · ∆ n + c . c ., (A49)where here k = ω/ c and ω = Ω · ∆ n . The final step is the evaluationof the mode functions. Let us take a set of modes propagating nearthe direction ˆ s , which will be taken to be toward the observer, andtake the polarization vectors to be either horizontal or vertical. Thenthe behaviour of the modes near the source (origin) will be that onemode is a top-hat with cross sectional area A (taken to be largecompared to the emitting region): then near the origin Z α ( x ; k ) = √ A ˆ ǫ α e i k ˆ s · x (A50)within the area A and 0 otherwise, so that Y α ( k ) = A − / . Aside fromthese 2 modes (2 since there are both polarizations), the remainingphoton modes do not contribute. The Kirchho ff di ff raction formula(Jackson 1998, Eq. 10.85) then gives Z α ( x ; k ) at the observer byintegrating over the area A , Z α ( x obs ; k ) = A / ˆ ǫ α k e i kR π i R . (A51)Thus: A coh ( x ) = + X α, ∆ n e i kR R ˆ ǫ α F α, n , ∆ n ( k )e i ϑ · ∆ n + c . c .. (A52)This is equivalent to Eq. (A27) with relabeled phase factors, show-ing that the quantum emission process from an optically thin elec-tron cloud leads to a statistical superposition of coherent radia-tion states, with amplitude given by the classical field configura-tion, and statistical weight uniformly distributed over the classicalangles (phases) of the electron trajectories .In the case where dispersion is present, each k -oscillator re-mains a harmonic oscillator but the frequency changes adiabati-cally as the wave enters and exits an ionized cloud with continuouselectron density. Since the time evolution of a coherent state in aharmonic oscillator (Hamiltonian proportional to ˆ a † ˆ a ) is that thecomplex amplitude v picks up a phase, the cloud changes a coher-ent state | v α ( k ) i to a re-phased coherent state | e − i φ ( k ) v α ( k ) i . Againthe e ff ect is exactly as in classical physics, except that it acts on acoherent state displacement rather than a classical complex number(Glauber 1966). c (cid:13)000