J. Phys. B: At. Mol. Phys. 20 (1987) 879-898. Printed in the UK

On the thermodynamics of quantum-electrodynamic frequency shifts G Barton School of Mathematical and Physical Sciences, University of Sussex, Brighton, BNl 9QH, UK

Received 14 May 1986, in final form 24 September 1986

Abstract. The spectra of atoms exposed to black-body radiation suffer temperature-dependent frequency shifts predicted earlier and now becoming measurable, whose thermodynamic status is clarified. To a good approximation, they are calculable as differences between the coupling-induced shifts in the Helmholtz free energies of the coupled system atom plus quantised electromagnetic field, in thermal equilibrium subject to the constraint that the atom is in a given state ii). Effectively, the state label i plays the same role as do macroscopically controlled variables in ordinary thermodynamics. With kT well below the electron rest-mass energy, such calculations require only the forward scattering amplitude of light from the atom; some previously overlooked but theoretically interesting relativistic features of the exact expressions are pointed out. For a harmonic oscillator, uniquely, the constrained free-energy shift is the same for all levels (and therefore also the same as the shift in unconstrained equilibrium); consequently there is no thermal frequency shift at all. Recent comments on black-body-induced Stark shifts by Ford et al are based on their calculation of the free energy of a harmonic oscillator in unconstrained equilibrium with the field. It is shown that as regards frequency shifts their comments are incompatible with, and misrepresent, previous correct work on such shifts.

1. Introduction

Ford et a1 (1985, FLOa for short) have calculated the free energy of a simple harmonic oscillator in thermal equilibrium with the quantised electromagnetic field. They treat the oscillator non-relativistically, and its coupling to the field in the electric-dipole approximation; but the equations that then result they solve exactly, i.e. to all orders in the coupling. In an expanded version of their paper (1986, FLOb for short), they attempt to apply its expressions to the quite different problem of spectroscopically measured frequency shifts under black-body irradiation, and claim that all previous work on this problem is wrong. In fact FLOb have failed to state intelligibly the nature of the physical problem that previous workers have addressed, and the nature of the quantities that one must calculate for comparison with such frequency shifts. From a thermodynamic point of view, the relevant problems relate to constrained rather than to overall equilibrium. This distinction is discussed in 0 2; roughly speaking, ‘overall’ or equivalently ‘unconstrained’ equilibrium specifies the canonical distribution for the field and the atom in mutual interaction (an oscillator may be regarded as an idealised atom), while the 0022-3700/87/050879

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C Barton constraints in question formally confine the atom to a prescribed quantum state. Frequency shifts measured spectroscopically under ideal conditions are given by differences between the Helmholtz free energies of such constrained interacting systems. FLOa calculate only the free energy in overall equilibrium. Though notable as a contribution to orthodox statistical physics, this is plainly insufficient to determine the transition frequency between specified atomic states. Nevertheless, FLOb try to extract such information from their expressions, but without any sustainable analysis to allow comparison with any system other than the oscillator. Indeed, once the real problem is solved in P 3, we shall see that the flaws in FLOb’s analysis may stem from their failure to appreciate that the strictly harmonic oscillator handled in the electric-dipole approximation is so special a case as to be totally unrepresentative, in this respect, of any atomic system. For instance, in the oscillator the unconstrained and all the constrained free-energy shifts happen to coincide and consequently it suffers no spectroscopically measurable thermal frequency shift at all. A further difficulty flows from the fact that FLO’s expression runs to all orders in the coupling: since they never isolate the leading-order term, which is relatively transparent, they miss its indications. In particular, they subdivide the shift in a manner that obscures its crucial low-temperature asymptotics, and then enshrine this subdivision in misleading nomenclature. (In fact, under most conditions the higher-order terms automatically retained in FLO’s mathematically exact solution are not warranted by the physics of their model, as shown in P 4.) FLO misrepresent earlier work. For instance, in the last paragraph on page L45 of FLOb, they claim that ‘. . . within the same framework as all previous investigators . . . we have assumed thermal equilibrium.’ By contrast, Knight (1972), Gallacher and Cooke (1979,1980) and Farley and Wing (1981), to cite only some, make it perfectly clear that they are not concerned with overall thermal equilibrium. Realistically, they consider instead the effects of incident radiation described by Planck’s formula, with the ‘temperature’ T treated simply as a parameter whose value is controlled by the experimenter. Farley and Wing give a particularly careful statement of the approximations involved. FLO’s criticisms of earlier work are not always consistent. For instance, FLOb’s second sentence (referring to Gallacher and Cooke ( 1979), though equally appropriate to the original work of Knight (1972)) does start by admitting that ‘These authors calculated the dynamic (AC) Stark shifts, induced by black-body radiation.. .’, which is not at all the same as FLOb’s other statement quoted in the preceding paragraph. It is of course perfectly natural to account for spectroscopic measurements like those of Hollberg and Hall (1984) by calculating the dynamic Stark shift, a phenomenon well understood quite irrespective of thermal equilibrium: see e.g. Townes and Schawlow (1959, or, specifically for thermal shifts, Farley and Wing (1981). The same expressions for the thermal shifts that emerge from this approach must be delivered by any correct theory if it is to be useful in understanding the experiment. Several misconceptions can be avoided by keeping this requirement in view. Admittedly, previous workers, being concerned explicitly with frequency shifts, approached these by calculating for individual levels the corrections denoted by A ( i ) below, without dwelling on the fact that from a thermodynamic point of view they are shifts in the (constrained) free energy rather than in the (constrained) total energy. In the approach through the dynamic Stark effect this would be quite superfluous, but the distinction is drawn in the original approach devised by the present writer (Barton 1972, B for short), though the constrained nature of these functions was taken for

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granted. In particular, B points out that the total energy shift is opposite in sign to the shift in free energy (which there is called the ‘adiabatic’ shift). In these circumstances, a careful restatement of the theory would seem to be appropriate, elucidating in particular its status with respect to statistical thermodynamics. The harmonic oscillator is too special a case to serve as a point of departure. Instead, we start from one (B) of the two original solutions, which is perhaps the more heuristic, and the one nearer to thermodynamics, merely comparing it at some crucial points with the other approach through the dynamic Stark shifts (Knight 1972). Once the theory is thus set up, its application to the harmonic oscillator is immediate and trivial. Section 2 constructs the general theory, and deals with thermodynamic questions. Section 3.2 applies the results to electrons and atoms, with only a few asides on the oscillator; the oscillator is then dealt with in § 3.3. The formulae given by the electricdipole approximation in § 3.2 are familiar and well established, and we eschew full detail. However we profit from the opportunity to make some other points that are new and interesting in principle, especially as regards the interplay between relativistic aspects and the high- and low-temperature asymptotics of the shifts; however, these points are unlikely to become relevant in practice at presently attainable experimental accuracies. Finally, § 4 uses the theory to show where FLOb is in error. Here one needs the leading-order approximation to their expression; to extract this calls for some quite careful analysis, which is relegated to the Appendix. 2. The general theory and its relation to thermodynamics We adapt to real photons an idea already applied to zero-point photons by Feynman (1961) and Power (1966) in order to calculate the ordinary (zero-temperature) Lamb shift. The argument proceeds by the following steps. (i) Start before the field is quantised, with the atom in state Ii), and determine the shifts Sw!) which its presence induces in the frequencies of the normal modes A of the field. What for brevity we call the atom could equally well be a free electron, a harmonic oscillator, or any ionic or molecular system. (ii) Quantise the field in the presence of the atom; write down the energy levels of the states I i, { nA}) of the coupled atom-field system (the nA are the photon numbers) and identify the level shifts due to the coupling. (iii) Spell out the conditions for the approximations implicit in step (ii) to apply, and verify that they are satisfied in the impending calculation of the thermal shifts. (iv) Determine the free-energy shifts ACi)(T) due to the coupling, subject to the constraint that the atom is in state Ii). (v) Note that the spectroscopically measurable frequency shifts are given by the differences between the A ( i ) , at least in the usual and probably excellent approximation where the measurement process is thermodynamically reversible. (This is why the function A ( i ) ( T )has traditionally, though perhaps by a slight abuse of language, been referred to as the ‘temperature-dependent level shift’.) (vi) Contrast A“) with the total energy shift A U ( i )subject to the same constraint to fixed Ii). Finally, (vii) link the A ( i )to the true free-energy shift AF( T ) of the joint atom-field system in overall (unconstrained) thermal equilibrium. Step (vii) has no direct bearing on spectroscopy, but prepares for the assessment of FLO’s approach. Throughout, we neglect the electron-gositron component of the black-body spectrum, whose effects are small, of order expi-pm), until kT = 1/p becomes comparable

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with the electron mass m. ( k is Boltzmann’s constant. Except in some landmark formulae we use natural units h = 1 = c. The fine-structure constant is a = e 2 / h c= 1/137.) (i) In real measurements, the Maxwell field satisfies appropriate boundary conditions on some surface bounding the experimental region, whose volume we call V. The normal modes depend on the shape of this region; moreover, the effects of the atom depend on its position relative to the boundaries. However, here we aim only at results that become exact in the limit V+CO,which is well defined. The limit is to be taken in such a way that the boundaries recede arbitrarily far from the atom. (Roughly speaking, thermal effects neglected in this approach would be proportional to some power of A T / where A T h c / k T is a typical thermal wavelength.) Then the normal modes of the Maxwell field are labelled by a two-valued polarisation index s and a wavenumber K, both subsumed into a single index A. The allowed values of K may be fixed conventionally by imposing fictitious periodic boundary conditions on the surfaces of a large (eventually infinite) volume V, and for sums over normal modes one has

-

C=CC=C A

s K

J

~ ( 2 ~ )d3K - ~

s

The second equality applies to any summand; the third applies if the radiation is unpolarised and isotropic, as in black bodies. In the absence of the atom the normal-mode frequencies are w , = cKA.The atom in state Ii) shifts these frequencies to w A+ S w t ’ and we determine the S w t ’ , following Feynman (1961). To this end, consider a sufficiently dilute medium consisting of N atoms in the volume V. The refractive index, differing little from unity, is p (w ) = 1

+ ( N / V)27rf‘”(w)/w2

where f “ ’ ( w ) is the real part of the coherent elastic forward scattering amplitude of light having frequency w by the atom in state ii) (see e.g. Ditchburn 1976, Newton 1982). The allowed values of K are the same as in vacuo, because they are determined by the boundary conditions independently of p ; but the phase velocity of light in the medium is l / p rather than 1, and the vacuum frequencies w A are therefore replaced by wA/~CL(WA\)‘wA[1-(N/V)2?Tf(i)(WA)/w2hl.

In the limits N / V-, 0 and V+ one atom as

CO

this identifies the frequency shift induced by just

S a t ’ = -2rf“’(wA)c2/wAV.

(2.2)

In the limit V + 00, the Sw t’ vanish, and the directly observable black-body spectrum remains Planckian; but we shall see presently that, after summation over A, observable level shifts survive even in the limit. (ii) Once the field is quantised, we adopt a manifestly inviting first approximation, to be discussed further in step (iii), with a Hamiltonian whose eigenstates and eigenvalues for the coupled system are /i, { n , } ) and

E ( i , { n , } ) = E‘”+C nA(wA +aut’). A

(2.3)

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In the absence of any interaction between the atom and the field, the atomic Hamiltonian is & , with eigenvalues E'": fiatom\i)= E("1i); and the Hamiltonian for the field is fifield =zAn*,+wA, where fifie[d\{nA})= z.hn,+@,+ I{ n A } ) . The circumflex identifies operators, and zero-point energies have been dropped because they are irrelevant to thermal effects. Then our Hamiltonian, incorporating the interaciion io the e5tent th!t it has the eigenvectors and eigenvalues specified above, is H = Hatom + Hfield + 6HeR;the effective coupling and the shift attributable to it in (2.3) are (2.4) tiE(i, { n A } )=(i, {nA}l&effli, { n A } )=C n , ~ w y ' . A

(2.5)

We shall treat 8fieffformally as a perturbation. Finally, and leaving the precise thermodynamic status of the outcome to be elucidated in step (iv), let us suppose for the moment that the atom is constrained to be and to remain in state Ii), while the state of the field is described by a canonical black-body density operator b B B with the unshifted frequencies: (2.6~) (2.6b) Here Z B B is the standard Planck black-body partition function, identified by the normalisation condition Tr $BB = 1. Then, using double angular brackets to denote canonical averages with given I i), we have as usual ((n*A))=Tr(bBBn*A)= l/[eXp(PwA)-1]= f i ( W ~ ) and, from (2.4), (2.5), the energy shift attributable to the coupling is

(2.7)

=z

A ( ~ ) ( T=((8fieff)) )

fiA(wA)awy).

A

Substitution from (2.1), (2.2) and (2.7) yields

which will prove to be the centrepiece of the theory. (iii) The true interaction between atom and field can, of course, create and destroy photons; unlike it is by no means diagonal in the n A . To upgrade our approximation into the true Hamiltonian, we would have to augment it by terms linear in the creation and annihilation operators for p h o t p (i.e. of the so-called dressed photons having frequencies CO,, h,).However, 6Heffhas the great virtue that through the agency of the forward scattering amplitude f it embodies the energy shifts induced by the interaction in those states that can be properly labelled by i and {nA}T. With f") to order a, this was established for the vacuum state, i.e. for zero-point photons, by Huang (1956), whose work underpins the application to the Lamb shift by Feynman (1961) and Power (1966); it has been commented on in a related context by Barton

+

t When referred to the conventional (completely non-interacting) basis, the wavefunctions of the dressed states, those we label as ti, {nA}), include perturbative (virtual) admixtures produced by the true underlying interaction. Our approximations and constraints exclude only the so-called on-shell admixtures, of states to which zero-order-energy-consewing transitions are possible. The Cauchy principal-value prescriptions encountered below reflect this exclusion.

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(1970). The same conclusions extend to thermal photons. (Related ideas have been elaborated by the Collbge de France group: Avan er a1 1976, Dalibard et a1 1982, 1984.) Feynman’s argument as applied in step (i) above is equally plausible withf‘” taken to any order in cy, but no explicit discussion like Huang’s or Power’s seems to be available beyond O ( a ) . Accordingly, though for purposes of discussion we shall sometimes assume that f‘” may be used to any accuracy, in actual applications we shall take f ‘ i ) only to order cy. Obviously, our method in its present formulation cannot deal with situations where off-diagonal matrix elements of the true Hamiltonian are significant, since H excludes these by construction. In particular, we cannot accommodate the effects of any coherence either between different normal modes, or even between different excitations (states with different n,) of a given normal mode. Where such coherence matters, it is probably much better to adopt the conventional description in terms of the dynamic Stark effect (cf the comments and references in 0 3). However, thermal excitation, like the effects of zero-point motion, lacks any coherence between states with different { n , } , as exemplified by the standard density operator PIBB which is diagonal in the { n , } . Thus, our method is tailor-made precisely for thermal shifts. One of its main virtues is heuristic: it concentrates attention on the forward scattering amplitudes as functions of frequency, which are simpler to pursue and perhaps more open to intuition than alternative attacks on the shifts (see e.g. the effects of atomic binding on thermal changes in the electron spin magnetic moment (Barton 1985b)). (iv) In order to elucidate the thermodynamic status of the constrained quantities A‘”( T), we start by stressing the familiar truth, that no spectroscopic information specific to the system can be obtained as long as the system remains in overall thermal equilibrium: all that one then sees is the universal black-body spectrum. (This is strictly true in the limit V + 00: cf the remark following equation (2.2) above.) Let us therefore consider, instead of overall equilibrium, the system in contact with a heat bath, but subject to the constraint that the atom is and remains in a prescribed state Ii) (see previous footnote). (For simplicity, think of the atom placed in an empty enclosure of volume V, whose walls are kept at temperature T. Though this is quite a far cry from the experimental arrangement of Hollberg and Hall (1984), the manifest differences do not in the end affect the argument.) For the moment we do not ask how such a constraint could be implemented in practice. Formally, the constrained variable, namely the state i of the atom, plays precisely the same role as the macroscopically controlled variables (like the volume of a fluid) in ordinary thermodynamics: work W is done on the system (atom plus field) when i changes through direct interaction between the atom and an external spectroscopic probe (say a laser beam). If the interaction is reversible, then W equals the increase in the constrained Helmholtz free energy F‘” introduced below. By contrast, the total energy U‘’’ of the system can also change through the exchange of heat with the heat bath, i.e. through the emission or absorption of photons by the walls. What we require is the change AF‘”( T ) induced in the constrained free energy by the level shifts SE( i, { n,}), equation (2.5), due in turn to the perturbation Seefl, equation (2.4). We shall presently establish this exactly (in the limit V + 0 0 ) ; but, for orientation, we start more modestly with the familiar thermodynamic perturbation formula to second order in the SE (see e.g. Landau and Lifshitz 1958, Peierls 1979): A F“ = (( SE )>- 4B [(( ( SE 1’)) - ((SE ))’I = A‘”

+ O( V - ’ ) .

(2.10)

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As in equations (2.6)-(2.8), the double angular brackets denote the expectation value calculated with the unperturbed constrained canonical distribution, i.e. with the density operator 1 i)( ilbBB. Hence the linear term ((SE))is indeed identically the same as A'i) in equations (2.8), (2.9); and it is easily seen from (2.5), (2.2) and (2.1) that the quadratic terms are of order 1/ V . The same result is readily derived from first principles. Since the energy E") of the atom is fixed by the constraint, the constrained partition function Z ( i )and the constrained free energy F") for atom plus field are given by the following equations, in a self-explanatory notation:

(2.11)

(2.13) The shift AF'" we require is defined by

F ( ' )= E ( ' ) + F,,+AF(')

(2.14)

where FBB

= -p-' In zBB = p-' In[ 1 -exp(-pw,)]

=-~p-~.rr~/45

is the unperturbed Planck free energy. By virtue of (2.1) and (2.2), equations (2.14), (2.13) and (2.12) entail FBB+AF")=p-1.rr-2

lorn

do~~Vln{l-exp[-p(w-2.rrf(~)(~))/wV]}.

(2.15)

When the logarithm is expanded by powers of 1/V, and then multiplied by V as in (2.15), the first term, of O( V), reproduces F B B ; the second term, of O(v"),reproduces AF'" = ((SE))= A''' as above; and, in the limit V+ CO, all higher terms vanish. In other words, (2.9) is exact under V+CO,in the sense that it is not then subject to further corrections of higher order in f'". (v) F ' f ' - F'" is the work that must be done on the system in any reversible process where the atom is promoted from state i to state J; e.g. by the induced absorption from a laser beam of a photon having this frequency, or of photons having frequencies that sum to this. If there is no black-body radiation (i.e. at zero temperature) this work reduces to E ( f ) - E'". Equations (2.14), (2.10) then show that the spectroscopic frequency shift induced by exposing the atom to black-body radiation is just A'f' - A'". In this operative sense we shall refer to A''' simply as the temperature-dependent level shift: that is how it is experienced by the spectroscopist, who need not encumber his native language with constantly repeated loan words from thermodynamics. Section 3 shows that A"), equation (2.9), tallies precisely with the expressions proposed before (except in papers with manifest errors, which it would be redundant to cite); hence no confusion needs to or should have arisen from comparisons with earlier work, and we now revert firmly to the shorter notation A") instead of A F " ) .

G Barton

To consolidate the argument one must understand the physical status of the formal constraint to a given atomic state Ii). Obviously it cannot be enforced directly, and certainly not by any macroscopic means. Instead, under the normal conditions of most spectroscopic experiments, it is satisfied (approximately) because the timescale of the measurement is too short for significant relaxation. A typical scenario would have kT 0) = 1. Substitution into (A.6) and recombination with J2 yields

The first term in (A.9) is just In[ 1 - exp( -pCl)], matching the first term on the right of (A.1). Changing the integration variable from y to x=pCly changes the second term of (A.9) into the second term on the right of (A.1). This establishes the desired equality in (A.1).

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References Avan P, Cohen-Tannoudji C, Dupont-Roc J and Fabre C 1976 J. Physique 37 993 Barton G 1970 Proc. R. Soc. A 320 251, appendix A -1972 Phys. Rev. A 5 468 -1985a The Spectrum ofAtomic Hydrogen 2nd edn, ed G W Series (Singapore: World Scientific) to be published - 1985b Phys. Let?. 162B 185 Bethe H A and Salpeter E E 1957 Quantum Mechanics o f o n e - and Two-Electron Atoms (Berlin: Springer) Dalibard J, Dupont-Roc J and Cohen-Tannoudji C 1982 J. Physique 43 1617 -1984 J, Physique 45 637 Ditchburn R W 1976 Light 3rd edn (London: Academic) D 15.45 Donoghue J F and Holstein B R 1983 Phys. Rev. D 28 340 Donoghue J F, Holstein B R and Robinett R W 1984 Phys. Rev. D 30 2561 Englert F 1959 Bull. Class. Sci. Acad. R. Belg. 45 782 Farley J W and Wing W H 1981 Phys. Rev. A 23 2397 Feynman R P 1961 La The‘orie Quantique des Champs (New York: Interscience) p 61 Ford G W, Lewis J T and O’Connell R F 1985 Phys. Rev. Lett. 55 2273 -1986 J, Phys. B: At. Mol. Phys. 19 L41 Friar J L and Fallieros S 1975 Phys. Rev. C 11 274, 277 Gallacher T F and Cooke W E 1979 Phys. Rev. Lett. 42 835 -1980 Phys. Rev. A 21 588 Goldberger M L and Low F E 1968 Phys. Rev. 176 1778 Hollberg L and Hall J L 1984 Phys. Rev. Lett. 53 230 Huang K 1956 Phys. Rev. 101 1173 Knight P L 1972 J. Phys. A : Math. Gen. Phys. 5 417 Landau L D and Lifshitz E M 1958 Statistical Physics (London: Pergamon) § 32 Newton R G 1982 Scattering Theory of Waves and Particles 2nd edn (New York: Springer) 5 1.5 Peierls R 1979 Surprises in Theoretical Physics (Princeton: Princeton University Press) 5 3.3 Power E A 1966 A m . J. Phys. 34 516 Townes C H and Schawlow A L 1955 Microwave Spectroscopy (New York: McGraw-Hill) 55 10.8, 10.9 Welton T A 1948 Phys. Rev. 74 1157