IL NUOVO CIMENTO VoL 56A, N. 2 21 Matzo 1980

The Decay of Coherent Rotational States Subject to Random Quantum Measurements (*).

L. FO~DA

International Centre /or Theoretical Physics - Trieste, Italy Istituto di Fisiea Teoviea dell' Universitd~ - T~ieste, Italia

1~ -. MAI~KO~-BOIC~TNIK a n d M. ROSINA

Faculty o/ Natural Sciences and Technology, and J. Stefan Institute University E. Kardelj - f~jubljana, Yugoslavia

(rieevuto il 5 Febbraio 1980)

S u m m a r y . - - The influence of the measurement appara tus on the ,f-decay of coherent ro ta t ional states is discussed. I t is shown tha t the mathema- t ical procedure, which has been devised to describe the decay of an unstable part icle characterized by a unique quantum state, applies, with suitable changes, also to this case. For very short times, the decay probabi l i ty still shows the pulses discussed in a previous paper where the influence of the environment was not taken into account. For long times, the interact ion with the environment enforces a superposition of exponen- tials for the decay probabil i ty.

l . - I n t r o d u c t i o n .

R e c e n t l y , t h e d e s c r i p t i o n of t h e d e c a y of a n u n s t a b l e s y s t e m has been

r e f o r m u l a t e d r a t h e r r a d i c M l y (1) in o r d e r to m u k e p l ace for t h e e x p e r i m e n t a l

e v i d e n c e t h a t a n u n s t a b l e p a r t i c l e , wh ich l ives l ong e n o u g h to m e r i t an in-

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (~) See, for example, L. FOKDA, G. C. GHIRARDI and A. RI~II~I: ~ep. Prod. Phys., 41, 587 (1978), and references contained therein.

229

2~0 L. FONDA, N. ~¢IANKOC-BOI~STNIK and ~ . ROSINA

vest igat ion of its t ime development , unavo idab ly interacts repea ted ly wi th

its envi ronment before decaying. As thoroughly discussed in ref. (1), the en-

v i ronment (measuring apparatus) performs usually on the sys tem a typica l

quantum-mechanicM measuremen t of the yes-no type , in which the part icle

is found to be ei ther decayed or not. After each measuremen t of t ha t type , the wave funct ion jumps ab rup t ly onto the eigenmanifold corresponding to

the decaying products if the answer (( yes ~) is obtained, or to t h a t of the un- stable sys tem if the answer is (~ no ~).

A typica l example is t ha t of a charged unstable part icle t ravel l ing in a

bubble chamber. Each bubble of its t r ack corresponds to an in teract ion with

the environment . According to the location of the bubble, we say t ha t a measure-

men t has been per formed and the sys tem has been found undecayed or de-

cayed. I f the former s i tuat ion holds, the decay products have been localized

within a relat ive distance R f rom each other of the order of magni tude of

molecular distances. I f the sys tem is found decayed, then the decay products

have been localized outside this ve ry distance R. I t is easily recognized tha t ,

also for neut ra l unstable objects or for other types of decay, the same con-

siderations hold. When one t rea ts situations in which an isolated resonance is involved, i t

has been shown (3) t ha t appl icat ion of the project ion opera tor P~ (*) on a wave

packe t over lapping the isolated resonance leads essentially a unique q u a n t u m

state, if the t ime of flight T , of the decay products through the distance R is significantly smaller than the inverse of the width of the resonance T , ~<7-1 (a condition usually satisfied in practice) (**). Under such conditions, one gets an exponen t i a l decay law for the unstable system, if the inverse of the fre-

quency of the measurements is much smaller t han the l ifetime of the system. I n this paper we shall consider the effect of the measur ing appara tus on

the y-decay of the coherent ro ta t ional s tates formed in heavy- ion collisions. I n this ease one is not dealing with an isolated resonance, bu t ra ther wi th a superposit ion of resonances and therefore one is unable to associate a unique q u a n t u m sta te to the system. However , the general theory of ref. (1) can be

applied also to this ease, as we shall see.

(2) L. FONDA and G. C. GHIRARDI: N'UOVO Cimento A, 67, 257 (1970); 6, 553 (1971). (*) The operator Pa projects the states inside the sphere of radius R with centre the decaying nucleus. (**) The state is unique only up to a phase factor. If one neglects the part of the decay product wave function which is present inside the sphere of radius/% by defining the nondecay probability as A(t), one gets after the measurement at time t

A(t) [u(t)) ............ ~ ]A(t)-~ [u(O)),

where lu(O)} is the state obtained at the previous measurement at t = O.

T ~ E D E C A Y OF C O H E R E N T R O T A T I O N A L STATES S U B J E C T ETC. 231

2. - D e f i n i t i o n o f t h e s y s t e m .

We have shown in a previous paper (~) tha t under certain conditions, during heavy-ion collisions, the nucleus can be left in a coherent rotational state (~)

(2.z) I~f~'(t)} ---- ~ I a, I exp [-- iq~] exp [-- iE, t] Iq~}. I

Here I stands for both angular-momentum quantum numbers I and M. I~} is an eigenstate of the angular momentum and of the nuclear part of the t tamil tonian

(2.2) H o, I ~ } = E ~ I ~ } .

The Hamiltonian H e , describes a free nucleus having a well-pronounced rotat ional character for its excited states, i.e. the energies / ~ of these states follow the rule E±----(M(I Jr 1) with a good ~pproximation and the quadru- pole moments (%+~]Q[~s} and (~+~IQI?±} are dominant. The coherence of the state kp~(t)} is then provided by the fact tha t the amplitudes [axl are peaked around a mean value Io of the angular momentum, while the phases F~ are roughly equidistant (in the case of rcf. (4) they are exactly equidistant). Under these circumstances, the state [F~(t)} shows a periodicity in the time evolution if the energies E1 of the excited states obey the rule (oI(I-k 1) exactly. The period ~C2 is determined by the lowest energy of the rotational band: tO = z~/E~. I f this rule is obeyed in a good ~pproximation, the quasi- periodicity can be noticed (8).

As discussed in ref. (3), the system decays by 7 emission. Details of this decay have been discussed by using a nucleus-radiation interaction of the quadrupole type and velocity-independent nuclear forces. By means of first- order perturbation theory one can evaluate the probability P~(t) tha t the coherent state ]yJ~} be found undecayed at the time t (3):

(2.3)

(2.4)

P~(t) = 1 - ~: w ~_~( t ) ,

1 W~_~(t) -- 36~ ~ {A~,AzL, O(An,) + A~,AH, O(Azz,)}"

II'I~L'

t)~ t)~* sin (All,-- A~L,)t/2 exp [i(ALL"- Axe) t/2] , • ~ I ' 1 '°d,L'L

A I I ' - - A LL'

(3) N. MANKO~-BOR~TNIK, M. ROSINA and L. FOi',rDA: NUOVO Cimento A, 53, 440 (1979). (a) ~P. W. ATKII~S and J. C. DOBSO~q: P~'oc. t~. Soc. Londo~ Set. A, 32], 321 (1971).

232 n. FONDA, N. MANKOC-BOR~TNIK and ~. ROS~N/~

where Azz,-~ E,--El. , O(x) is 1 for x ~ 0 and zero otherwise, and

I f the rotat ional band is ideal, the t ime derivat ive of W~_~ shows a s t ructure in t ime which contains pulses whose width and shape depend on the width and shape of the amplitudes ai . One obtains narrow pulses if the absolute values of the amplitudes a1 are widely spread. I f the rota t ional band is not ideal, P~(t) is only quasi-periodic with pulses appearing quasi-periodically.

We have now to discuss how measurements can influence this decay pa t te rn . As pointed out in sect. 1, one is not dealing here with a clean isolated resonance, as the state (2.1) is a superposition of many resonant states. The main object of this paper is to see how to generalize the methods of ref. (1) to cover this case. We discuss this in the next section.

3 . - Decay of the coherent states in the presence of quantum random

measurements.

When the system lives long enough to meri t the detailed analysis of its t ime plot, as discussed in sect. 1, it cannot be considered as isolated, but , on the contrary, it must be considered as an open system in repeated interact ion with its environment. To write down an evolution equat ion for such a system, one must resort to the density operator formalism, since the wave funct ion is subjected to sudden quan tum jumps and therefore no causal equat ion can be obta ined for it.

Le t us summarize briefly the procedures of ref. (1). I f one is interested only in the history of the decaying nucleus and then disregards the destiny of the y- ray and its associated decayed nucleus, the effect of a measurement a t t ime t on the 0-operator can be visualized as the t ransformat ion

(3.1) o(t) ---> e:~(t) = P ~ o ( t ) P . ,

/~R being the projection operator within the sphere of radius R, as discussed in sect. 1.

On the other hand, evolution wi thout measurements through the t ime interval (0, t) is given by exp [-- iHt] 0(0) exp [iHt], where H is the total Hamil- tonian of the system. There follows tha t in general, when both evolutions with and wi thout measurements are taken into consideration, one can write the following equation for 0(t):

(3.2) do(t ) = ,~t dtoO(t) + (1 -- 2dt) exp[--iHdt]o(t ) exp[iHdt].

T H E D E C A Y O F C O H E R E N T R O T A T I O N A L S T A T E S S U B J E C T E T C . 2 S S

Here 2 is the mean frequency of the random measurements; t dt represents then the probability for the occurrence of a measurement in the interval dt of time. Equation (3.2) leads to the differential equation

dQ(t) (3.3) dt - - = - - i E H , e(t)] + t (Qv;( t)--e( t)) ,

which in turn is equivalent to the integral equation

(3.4) e(t) = exp [ - Xt] exp [ - i m ] Q(o) exp [ira] ÷ t

-~f~ d~ exp [-- 2~] exp [-- iH~] ~ ( t -- 6) exp [ i t t~] . (1

Since one assumes that the coherent state [~> is realized at t----Or one has the boundary condition

{3.5) q(o)= I~(o)><~(o)l.

The main question now is the determination of the operator /)R. If one disregards the presence of the photon within the sphere of radius R~ which in our opinion is an excellent approximation~ one can safely assume that P . is given by the projection operator on the photon vacuum state:

{3.6) 2R : ~ ]~x> <~vx[. I

The sum of course runs only over the states of the considered rotational band. By bracketing (3.4) on <~v~ I and ]~>, one gets

(3.7)

where

(3.8)

t

~L(t) = exp [--~t]N±~(t) -1- r'L, ~ !~ d(~ exp [--,~(SJP~,F~.,((~)~)~,~.,(t- ~),

N,L(t) = <q~[ exp [-- iHt] ~(0) exp [iHt] lq;L>,

We take the Laplace transform of (3.7). Defining

(3.9)

c o

M-~(s) =fexp r- st3:~(t) dt, 0

16 - I l Nuovo Cimento A .

234 L. FONDA, N. ~IANKO~-BORSTNIK and M. ROSIN&

we easily get

(3.10) e~(,) = 2v~(s + 2) + o o t

+ y. fat exp [ - st]fXd~ exp [ - 2~]~%,,~,(~)e,,,.,(t- ~). l ' I / 0 0

The application of the l~altung theorem yields

(3.11) -~ = Lv~(s 2) 2v~,,,~,(s eAs) + + ~ + 2) eF~,(s). I 'L."

The solution of (3.11) is tr ivial and can be be t te r visualized b y introducing the,

vectors Q, hi and the mat r ix P, whose elements are OIL, NzL and--P~LI'L', respec- t ively. I n this new notat ion eq. (3.11) can be wri t ten as

(3.12) OZ(s) = Nz(s + 2) + 2P-~(s_+ 2)Oz(s).

The solution of (3.12) is then

(3.13) O'S(s) = [1 -- 2P~(s + ,~.)]-lN'~(s + 2).

B y inverse Laplace transform, we obtain our original ~-matrix:

(3.1~) e+ico

if e(t) = ~ ds G--i c~

exp [st][1 - - 2P~e(s + 2)] -1NZ(s + 2),

where C must s tand to the r ight of all singularities of the integrand in t h e s-plane. The integrand exhibits poles at the zeroes s = - ~-1 of the deter-

m ~ n a n t I I 1 - ~V'*ll, ~.~. whe,

(3.15) co

det [[ ax.,, ¢),~L,- 2fd~ exp [-- 2a] exp [~/r,,]P.,L, vv(~)n = O, 0

which is of course an implicit equat ion for the unknowns z~.

In order to ex t rac t the contributions of the poles, one shifts the contour" in the s-plane so as to make i t run on the line s = -- 2. One eventual ly obta ins

(~.16) ~o,~(t) = z . ~ , ,~ e x p [ - - tiT.] +ico

+ .~x~[__-_. 2t] [as exp rst]{rl - zv-~(s)]-~ N~(s)},~ ~ J

--icO

T H E D E C A Y O F C O H E R E N T R O T A T I O N A L S T A T E S S U B J E C T ETC. ~

where the coefficients ~(~) are determined f rom the equations - - I L

(3.17)

co

• (n) •

0

The condition for the existence of a solution of (3.17) is of course the vanishing of the de te rminan t of the coefficients, i . e . eq. (3.15).

We see f rom the s t ructure of our solution (3.16) t h a t the interest ing t ime

region to explore mus t be of the order of 1/Z or smaller. I n fact , only in this way one can save the integral appear ing on the r.h.s, which contains the pulses which are present in the cas~ of no measurements . This t e r m behaves like exp [--At] and therefore dies out ra ther quickly in comparison with the slow

exponentials exp [--t/~,~]. I f one can then reach exper imenta l ly very-small-

t ime regions and if the f requency of the f luctuations is a t least an order of

magni tude higher t han 2, one would be able to see them, otherwise, for long t imes and/or high 2, one would see only a tr ivial superposit ion of exponentials

of the type

the periodic or quasi-periodic s t ructure present in the case of no measurements being washed out in this case b y the presence of the measur ing appara tus .

A thorough discussion of this point will be given in another paper.

$ $ $

The authors would like to acknowledge discussions with H. ]~KSTEIN~

G. FLEhlXNG~ G. C. GttIRARDI and R. E. PEIERLS.

@ R I A S S U N T 0

In questo lavoro si discute 1'influenza deli'apparato di misura sul decadimento y degli stati rotazionali coerenti. La procedura usata per la diseussione del deeadimento di una par~icella instabile c,%ratterizzata da uno state unico pub esser estesa, con oppor~uni cambiamenti, pure a coprire questo case. Per piceoli tempi, la probabilit~ di deeadi- mento presenta 1~ stesse fluttuazioni discusse in un altro lavoro dove l'iufluenz~ del- l 'ambiente non era stata presa in considcrazionc. Per tempi lunghi, l'interazione con l'ambiente produce come legge di decadimento una sovrapposizione di esponenziali.

Pe3IoMe He IIOay~eao.

16. - 11 Nuovo Cimento A .