VOLUME 17, NUMBER 9 PHYSICAL REVIEW LETTERS 29 AUGUsT 1966

and surface vibrations proposed by E. Boeker, %. M.De Muijnck, C. C. Jonker, Comptes Rendus du Con-gres International de Physique Nucleaire, Paris, 1964,edited by P. Gugenberger (Centre National de la Re-cherche Scientifique, Paris, 1964), Vol. II, p. 405.

2Recent electron scattering experiments give Bg 2)

=44 fm and a transition radius of about 3.3 fm; thus

Po is about 0.43. F. Gudden, private communication.~3The large vibrational amplitudes suggest that an-

harmonic terms might be of some importance. Infact, it is expected that they will improve the agree-ment between theory and experiment.

DOUBLE POLES AND NONEXPONENTIAL DECAYS IN ATOMIC PHYSICS*

K. E. La.ssilaInstitute for Theoretical Physics, Department of Physics, Stanford University, Stanford, California,

and Physics Department, Iowa State University, Ames, Iowa

and

Vesa RuuskanenDepartment of Nuclear Physics, University of Helsinki, Helsinki, Finland

(Received 25 July 1966)

Atomic resonance fluorescence is examined as a scattering problem, and Goldberger-Watson double poles are found to occur in situations producible in the laboratory. Sys-tematic study of this effect and associated nonexponential decay thus appears possible.

It was pointed out by Goldberger and Watson'that the evidence supporting purely exponentialdecay for every unstable particle is "far fromconvincing. " These authors showed, in partic-ular, that when the S matrix has multiple poles,the decay amplitude for the associated statebecomes a polynomial in time multiplied bythe usual exponential factor. Since then, sev-eral model situations have been discussed wheredouble poles can occur. ' In the present notephysical situations with double poles are pre-sented that can be produced and studied in thelaboratory.

The transition matrix, or T matrix, underdiscussion is that for resonance scattering oflight through more than one excited state. Thecalculational method and some results have

been discussed elsewhere' &; we now extendthese to the case where the T matrix has a dou-ble pole. This approach3 is similar to that ofthe Lee model' in that the calculation beginsfrom a second-quantized Hamiltonian in which

the unperturbed part gives the atomic energylevels and photon energies, and the interactionpart describes transitions between the levelswith photon emission and absorption. That suchan approach is appropriate had been noted ear-lier also by Kallen. '

Resonance flourescence through two excitedlevels is probably the case of most immediateexperimental interest. This process is describedby a. T matrix of the following form (subscripts1 and 2 refer to the two excited states; a sin-gle ground state is assumed, and the incidentphoton energy is ~):

where

T=g, (u u, , + i ,' I', )f—, +g, (&u e, +—i—,' I',—)f, +g, V„f, + g, V„f,

(u) —(u~)((u —(u )

&u~ = —,'(e, +&a,-i—,'I', -i—,'I', ) + 2[(v, -&u, -i—,'I", +i2I",)'+4V»V»]'",

is the energy (h = I) of the excited state lj),I"& is the corresponding width, and Vz& is thematrix element of an external (or possibly in-ternal) perturbation coupling the excited states.The fi and gi, in Franken's7 notation, are ab-

breviations for the absorption and emissionmatrix elements, respectively, connecting theexcited state ~i) with the ground state.

A double pole in T requires w+=~ . This

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VOI.UME 17, NUMBER 9 PHYSICAL RKVIKW LKTTKRS 29 AUGUsT 1966

condition, we note, can never be achieved when

the radiative widths of the two levels are equalor if no coupling interaction exists. Since 4)y

-(, can be varied for atomic levels by vary-ing an external magnetic field (1 and 2 thenrefer to Zeeman levels or perhaps hyperfinelevels as the case may be), w+ can be madeequal to w at crossing (~, = w, ) when the cou-pling interaction is chosen so that 4(I', —12)2

=4V»V», which can be taken real (4 l V» l').The case of S-P excited-state crossings (with

ground S state) is of particular interest becauseof its use in Lamb-shift measurements. '&' Thisis an especially simple example for our reso-nance-fluorescence discussion because not on-

ly is I; = I „but I', = g, =f, = 0, where 1 hasbeen chosen a,s S, and only state 2 (the P state)is excited in the absorption. With &~ labelingcrossing-point energy (or frequency) and dou-ble-pole conditions reached, —,'12:4V]2V2y,

by adjusting an external coupling perturbation(an electric field, for exa, mple), T becomesg2(w-u~)f2/(&u-&u~+ —,'il'2), which gives riseto a line shape of form (I', =- I')

I7'I'~((u —a )'/[(( —~ )'+,—', I'j'.

This function, multiplied by —,'I', is shown a.sthe solid curve in Fig. 1 compared with a Lo-rentz distribution with the same value for I",

A(t) ~ (1--,'rt) exp(--,'I' t). (5)

This same conclusion is reached by followingHeitler's arguments, "as Eqs. (3) and (4) canbe interpreted as the emitted-frequency distri-bution when one starts with two ensembles ofatoms in the respective excited configurations.

The addition of a, coupling perturbation is seento have a drastic effect on the line shape and

decay curve, although absorption and emissioncontinue to proceed only through the I' state.The double-pole condition —,'I"'=4

(V~' gives the

largest value of the coupling interaction for

indicated by the dashed line. This Lorentz shapeis the intensity curve given by Eq. (1) for theS--P case when the coupling perturbation is turnedoff. Following Ref. 1, from Eq. (4) we obtainexp(--,'I' t) as the time dependence of the prob-ability amplitude for finding the system in theexcited P state after initial resonance excita-tion when V=0. On the other hand, for the dou-ble-pole case,

L=LORENTZ CURVED = DOUBLE POLE I N & EN S I T Y

LLI

0.5

-o.5 r 0 o.5 r~-~„ IN uNlrs OF r

I.5r

which the (coupled) energy levels cross whenthe magnetic field is varied through the cross-ing point which still occurs at ~, =su, . Beforereaching the crossing point, however, the lev-els are considerably farther apart than theywould be for the same magnetic field valuewhen V=O. For larger values of V, anticross-ingx' conditions are reached.

It is not known whether the effect discussedwill lead to any new spectroscopic tools allow-ing more precise measurements (of the Lambshift, for example). We feel it of sufficientinterest to warrant study in its own right. "

There is, of course, no need to restrict one-self to S-P crossings in order to study doublepoles since many types of atoms are available.Nor is it necessary that the "double pole" beexcited by resonance radiation' should someother means be easier; the decay amplitudewill then be somewhat different from Eq. (5).Also, study of crossing situations in crystalsmay prove fruitful as long-lived states a,rerea.dily available.

In closing, we comment that the deriva, tionof Eq. (1) was done in matrix form so the cal-culation is easily generalized to, for example,three excited states where triple poles can oc-cur and might well be studied, "or to caseswith more than one ground state. These willbe included in a more complete publication.Also, we feel similar analyses can be donefrom Mower's'4 approach.

One of the authors (K.E.L.) would like to thank

Professor K. M. Watson, Professor G. %. Ser-ies, Professor L. I. Schiff, Professor A. L.Fetter, and Dr. R. M. Macfarlane for useful

I'IG. 1. Comparison of the double-pole intensity dis-tribution with that given by the corresponding Lorentzcurve. The solid line D for the double pole is given by41" (~—&u~) /[(u —&uz) + &~&I" I and the dashed line L = 41 2/

[((d—(d~) +gl ] .

49'

VOLUME 17, NUMBER 9 PHYSICAL REVIEW LETTERS 29 AUGUsT 1966

suggestions and discussions. He would alsolike to acknowledge a Fulbright award duringpart of his stay at the Research Institute forTheoretical Physics, Hensinki, Finland, wherethe present ideas originated.

*Work supported in part by the U. S. Air Forcethrough Air Force Office of Scientific Research Con-tract No. AF 49(638)-1389.

~M. L. Goldberger and K. M. Watson, Phys. Rev.136, B1472 (1964).

J. S. Bell and C. J. Goebel, Phys. Rev. 138, B1198(1965); H. Osborn, ibid. 145, 1272 (1966).

K. E. Lassila, Phys. Rev. 135, A1218 (1964).V. Ruuskanen, thesis, University of Helsinki, 1965

(unpublished) .T. D. Lee, Phys. Rev. 95, 1329 (1954).

6G. Kallen, Brandeis Summer Institute 1962 Lecturesin Theoretical Physics: Elementary Particle Physicsand Field Theory I, edited by K. W. Ford (W. A. Ben-jamin, Inc. , New York, 1962), p. 171.

P. A. Franken, Phys. Rev. 121, 508 (1961).R. T. Robiscoe and B. L. Cosins, Phys. Rev. Let-

ters 17, 69 (1966).G. W. Series, Phys. Rev. 136, A684 (1964).W. Heitler, The Quantum Theory of Radiation (Ox-

ford University Press, London, 1954), p. 200.~~T. G. Eck, L. L. Foldy, and H. Wieder, Phys. Rev.

Letters 10, 239 (1963).Measurements of the effect described should now be

possible in atomic physics. G. W. Series, private com-municationn.

~3R. M. Macfarlane, private communication.~4L. Mower, Phys. Rev. 142, 799 (1966).

SIGN OF THE K, -K2 MASS DIFFERENCE~

Gerald W. Meisner, Bevalyn B. Crawford, and Frank S. Crawford, Jr.Lawrence Radiation Laboratory, University of California, Berkeley, California

(Received 6 July 1966)

Evidence is presented that the long-lived neutral K is heavier than the short-lived.

+P —A+K' (4771 events)

and

w +P -Z'+Ko, Z'- A+ y (1269 events), (2)

where the A decays visibly via A-P+ m-. Thisis the same sample of K' we used in a previ-ous experiment to determine lm, -m, I by meansof secondary hyperon production, except thatin the present experiment we discard K' with

We have performed an experiment to mea-sure the sign of m, -m, using the method sug-gested by Camerini, Fry, and Gaidos. ' Wefind K, to be heavier than K,'. Our statisti-cal confidence level depends on the unresolvedFermi-Yang-type (F-Y) ambiguity that existsat present in the KN (strangeness S =+1) phaseshifts in isospin state I= 0. If the F solution(large positive p,„phase shift) is the correctsolution, we obtain Monte Carlo betting oddsof 45 to 1 for m, & m„assuming I m, -m, I

= 0.57w, '. If instead the Y solution (large pos-itive P~, pha. se shift) is correct, our bettingodds for m, & m, are 5 to 1.' We have not re-solved the F-Y ambiguity. '

The experiment uses 6040 K' mesons producedin the Alvarez 72-inch hydrogen bubble cham-ber via the reactions

momentum greater than 600 MeV/c, becauseof present lack of information on the I= 1 KX(S = —1) scattering amplitudes above 600 MeV/c.

The predicted K' direction from Reaction (1)is known to within about +0.5 deg; that fromReaction (2) is known to within about +20 deg.In the case of Reaction (1), we scan along thispredicted direction, within a cone +5 deg wide;for Reaction (2), we scan within the entire vol-ume downstream from the vertex. We look forelastic scatters

K +P-K +P,neutral

where the final K,' is detected by its visibledecay K,'- m++ m (double-vee events). Thereis no cutoff on the length of the recoil proton.We find 23 double-vee events with initial Kmomentum PK & 600 MeV/c. ' Our demand fora visible A decay gives us essentially 100%detection efficiency for finding double-vee events.There are no ambiguous events and no background.

For a K' produced at t = 0 with c.m. momen-tum hk, the probability P(x)dx that an elasticscatter of type (3) will occur at proper timet in lab distance interval dx and with c.m. scat-tering angle 0 (of the outgoing K with respectto the incident direction) in differential solid

492