The recoil shift due to the finite mass of an absorbing atom is, when viewed in the right way, simply part of the total Doppler shift. This is true both in the low speed and relativistic limits and also for the reflection of light from a low-mass mirror

Barnett, Stephen M.

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On the recoil and Doppler shifts

Barnett, S.M.

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Strathprints Institutional RepositoryBarnett, Stephen M. (2010) On the recoil and Doppler shifts. Journal of Modern Optics, 57 (14-15).pp. 1445-1447. ISSN 0950-0340Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/Journal of Modern OpticsVol. 00, No. 00, DD Month 200x, 1–5On the recoil and Doppler shiftsStephen M. Barnett ∗Department of Physics, SUPA, University of Strathclyde, Glasgow G4 0NG, UKand Department of Physics, FNSPE, Czech Technical University in Prague, Brˇehova´ 7, 115 19 Praha 1,Czech Republic(October 2009)The recoil shift due to the finite mass of an absorbing atom is, when viewed in the right way, simply part of the total Doppler shift. Thisis true both in the low speed and relativistic limits and also for the reflection of light from a low-mass mirror.1 IntroductionThe absorption of a photon by an atom changes its internal state, but also modifies its kinetic energy andmomentum. This phenomenon, of course, underlies the field of laser cooling. The recoil shift is an additionto the familiar Doppler shift, which accounts for this effect [1]. This shift is typically in the kHz regionand, although small, has been measured [2].Our purpose, in this paper, is to suggest that the recoil shift is, in fact, part of the Doppler shift, butthat the correct velocity is not that of the atom before the absorption, but rather the arithmetical averageof this initial velocity and its value after the absorption.Our analysis relies only on the conservation of energy and momentum, and is readily extended tothe relativistic domain. We show that the required photon frequency for resonant absorption is thenarithmetical average of the Doppler shifts associated with the velocity of the atom before and after theabsorption.We conclude with a derivation of the Doppler shift relevant to the reflection of a photon from a low-massmoving mirror. I this case we find, surprisingly, that it is the geometric average of the Doppler shifts forthe initial and final mirror velocities that matters.2 Absorption of a photon by a moving atomWe consider the absorption of a photon by an atom moving towards the light source. As is well known,the frequency of the light, as it appears to the atom, is increased due to the Doppler shift. If the atomis moving with velocity v then the frequency in the atom’s frame ωa is related to that in the laboratoryframe, ω byωa = ω(1 +vc). (1)If the atom is moving very rapidly, then we need the to use the formulaωa = ωD(v) , (2)∗Corresponding author. Email: steve@phys.strath.ac.ukJournal of Modern OpticsISSN 0950-0340 print/ISSN 1362-3044 online c© 200x Taylor & Francishttp://www.tandf.co.uk/journalsDOI: 10.1080/0950034YYxxxxxxxx2where D(v) is the relativistic Doppler factorD(v) =√1 + vc1− vc(3)Clearly, for small velocities, this reduces to the non-relativistic expression (2).Fermi showed that we can derive the non-relativistic Doppler formula simply by imposing the conserva-tion of energy and momentum on the emission of a photon [3]. It is instructive to repeat Fermi’s calculationand we do so, but consider absorption rather than emission and specialize to a single spatial dimension.Let an atom of mass m be moving with velocity v towards a light source. Consider an optical transition atangular frequency ω0, and let the light have angular frequency ω in the laboratory frame. The abosorptionevent will reduce the velocity of the atom and we let the velocity after absorption be v′. Let us impose theconservation of energy and momentum:~ω0 +12mv′2 = ~ω +12mv2 (4)mv′ = mv − ~ωc. (5)Fermi’s approach to these equations was to square the second one and neglect terms of order c−2 to givemv′2 = mv2 − 2v~ωc. (6)Substituting this into (4) then gives the Doppler formula for resonanceω0 = ω(1 +vc). (7)It is not necessary, of course, to neglect terms of order c−2 and we can restore the term omitted inFermi’s approximation by replacing (6) withmv′2 = mv2 − 2v~ωc+1m(~ωc)2. (8)If we insert this into our energy conservation equation (4) then we obtain a resonant condition in whichthe Doppler shift is supplemented by an additional small shift, which is the recoil shift [1]ω0 = ω(1 +vc− ~ω2mc2). (9)We can see that the recoil shift is small, being the frequency multiplied by the ratio of the energy of thephoton to twice the rest-mass energy of the atom.Let us consider the effect of eliminating the mass rather than the final velocity from our energy andmomentum equations (4) and (5). To achieve this, we first rewrite the equations in the form12mv′2 − 12mv2 = ~ω − ~ω0 (10)mv′ −mv = −~ωc, (11)3and then divide the first equation by the second to obtainv′ + v2c=ω − ω0−ω⇒ ω0 = ω(1 +v′ + v2c). (12)The recoil shift has been absorbed into the Doppler shift, but the relevant velocity is (v′ + v)/2, that isthe mean, or arithmetical average of the velocities before and after the absorption.3 Relativistic treatment for a rapidly moving atomWe might ask whether the (total) Doppler shift in (12) is the average of the Doppler shifts at velocitiesv and v′ or, perhaps, the Doppler shift at the average velocity (v′ + v)/2. At low velocities, in the non-relativistic limit, there is no difference between these. To address the question, therefore, we need toperform a relativistic analysis. We let m be the rest mass and introduce the familiar Lorentz factorγ =1√1− v2c2, (13)with a similar expression for the factor, γ′, after the absorption. Our expressions for the conservation ofenergy and momentum (4) and (5) becomeγ′(mc2 + ~ω0) = γmc2 + ~ω (14)γ′(m+~ω0c)v′ = γmv − ~ωc. (15)Proceeding as before, we can eliminate the mass to obtain the effective form of the Doppler shift:ω0ω=γ(c+ v)− γ′(c+ v′)γγ′(v − v′)⇒ ω0 = ω 2D(−v) +D(−v′) . (16)In the relativistic regime, the required frequency of the field for resonant excitation of the atom is thearithmetic mean of the Doppler-shifted transition frequencies before and after the absorption:ω = ω0D(−v) +D(−v′)2. (17)For velocities in the non-relativistic regime this reduces to the simple formω = ω0(1− v′ + v2c)(18)as it should.44 Reflection from a low-mass mirrorIt is a well-known consequence of the Doppler effect that the frequency of light is changed if it is reflectedfrom a moving mirror. Let us consider a single photon, with angular frequency ω, and a mirror travellingtowards the source of the photon will velocity v. The frequency of the reflected photon, ω′, will then begreater than that of the incident photon, with the frequency up-shifted by the factor [4]ω′ω=1 + vc1− vc= D2(v) . (19)Let us suppose that the mirror is of sufficiently low mass, m, that the reflection of the photon canchange its velocity from v to v′ and impose on the mirror and the photon the conservation of energy andmomentum. We find thatγ′mc2 + ~ω′ = γmc2 + ~ω (20)γ′mv′ +~ω′c= γmv − ~ωc. (21)Eliminating, once more, the mass we find that the ratio of the frequencies isω′ω=γ(c+ v)− γ′(c+ v′)−γ(c− v) + γ′(c− v′)= D(v)D(v′) . (22)If we compare this with the high-mass mirror expression (19) then we see that the relevant Doppler factorhere is the geometric mean of those for the initial and final velocities of the mirror.We should investigate the limiting form of this Doppler shift in the non-relativistic regime. For smallvelocities, the ratio of frequencies (22) becomesω′ω≈(1 +vc)(1 +v′c)≈ 1 + v + v′c, (23)which is the Doppler shift for the sum of the mirror velocities before and after the reflection. Remarkably,we can interpret the relativistic expression in the same way if we use the relativistic velocity-additionformula so that the sum of the two velocities isV =v + v′1 + vv′c2. (24)The ratio of the frequencies of the reflected and incident photons is thenω′ω= D(v)D(v′) = D(V ) , (25)which is the relativistic Doppler shift at the (relativistic) sum of the velocities.5 ConclusionsFermi showed that the Doppler shift can be determined simply by applying the conservation of energy andmomentum to a radiative process [3]. His analysis provides, also, the recoil shift which supplements theDoppler shift. We have shown that the total shift is perhaps more correctly viewed as simply a Doppler5shift, but that the relevant velocity is the average of the absorbing body before and after the absorptiontakes place.We have shown that Fermi’s analysis can easily be extended into the relativistic regime in which atom ismoving at high velocity. In this case, the total frequency shift is again a Doppler shift: the required opticalfrequency for resonant absorption is given by the arithmetic mean of the Doppler shifts for the velocitiesbefore and after the absorption.Finally, we have applied the same reasoning to the problem of reflection of a photon from a movingmirror. We have considered the mirror to be of sufficiently low mass that the reflection of a single photonmakes a change to its velocity. In this case we found that the total frequency shift is again a Doppler shift,given by D2, with the relevant shift being the geometric mean of those for the initial and final velocitiesof the mirror:D =√D(v)D(v′) . (26)In this case, moreover, the nett shift can be interpreted simply as the familiar Doppler shift, given by D,with the relevant velocity being the relativistic sum of the initial and final velocities.In each case the mass of the atom or mirror makes no appearance and it is clear that, by identifying therelevant velocity, the recoil shift should be thought of as nothing more than part of the Doppler shift.AcknowledgementsI thank Paul Lett, Peter Milonni and Erling Riis for helpful comments and suggestions. I also thank IgorJex for his hospitality during my visit to Prague during which I had the time to complete this project.This work was supported by the projects MSM 684077039 and LC06002 of the Czech Republic, by theRoyal Society and by the Wolfson Foundation.References[1] S. Stenholm, Rev. Mod. Phys. 58, 699-739 (1986).[2] J. L. Hall, C. J. Borde´ and K. Uehara, Phys. Rev. Lett. 37, 1339-1342 (1976).[3] E. Fermi, Rev. Mod. Phys. 4, 87-132. (1932).[4] W. H. McCrea Relativity Physics Fouth Edition (Methuen and Co., London, 1954).

McCrea Relativity Physics Fouth Edition

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On the recoil and Doppler shifts

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