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Singularity spectrum of the resolvent in a nonmarkovian master equation
Volume 124, number 9
PHYSICS LETTERS A
19 October 1987
SINGULARITY SPECTRUM OF THE RESOLVENT IN A NONMARKOVIAN MASTER EQUATION E.S. H E R N A N D E Z ~ and H.M. C A T A L D O 2 Departamento de Fisica, Facultadde Ciencias Exactas y Naturales, Universidadde Buenos Aires, 1428 BuenosAires, Argentina
Received 18 March 1987; revised manuscript received 27 July 1987; accepted for publication 12 August 1987 Communicated by A.R. Bishop
The well-known standard nonmarkovian master equation is used to describe the damping of a harmonic macroscopic coordinate coupled to a thermally equilibrated Fermi gas. The time evolution of the phonon distribution is expressed in terms of a resolvent whose spectrum of singularities is computed and analyzed. It is seen that an apparent limitation of the model related to the appearance of undesired oscillations does not inhibit the expected behavior of the average phonon number and phonon number dispersion.
As a collective excitation sets in in a m a n y body system, the residual coupling to the intrinsic degrees o f freedom acts as a damping mechanism that drives the separate subsystems towards thermal equilibrium. The dynamics o f the coupled coordinates can, in principle, be established either on a heuristic or on a more fundamental basis. In general, one can, under suitable hypothesis and approximations, write down a set o f equations o f motion for the time-varying values o f some probabilities relevant to the problem and m a n y significant properties o f the solutions can be exposed if one examines the spectrum o f the generator o f the evolution. If the dynamics exhibits m e m o r y effects which cannot be disregarded, one should turn, if possible, to the investigation o f the singularities of the resolvent or Laplace transform of the time-dependent propagator. Our purpose in this work is to illustrate, by resort to a well-founded model [ 1-11 ], the kind o f analysis that can be done on the above mentioned basis. The model and the master equation. The system we aim at describing consists o f a quantal harmonic oscillator placed into a stationary Fermi gas to which it is coupled through a standard particle-phonon interaction o f the form [ 1-5,12] Him = ~ ( ) . ~ F + b~- b , + 2 * ~ r b ~+ b , ) .
(1)
otu
In this expression, F + and b + are, respectively, the creation operators for a phonon with energy kO and for a fermion state [A ), with F and bA the corresponding annihilators. The labels a, fl, ... (/z, v .... ) denote singleparticle (s.p.) states whose deexcitation (excitation) gives rise to the creation (destruction) o f an oscillator quantum. The coupling matrix elements are denoted as 2,~. The current model is a schematic approximation to a real Fermi system (i.e. a nucleus) where a harmonic collective mode can be excited and undergo subsequent decay due to the residual interaction with the intrinsic degrees o f freedom [ 1-5]. We assume that the heat reservoir is equilibrated at a temperature T, so that we can establish the distribution PA = [ 1 + exp(eA -- ~F)/T] -~ ,
(2)
with ¢~ the Fermi energy. The master equation is obtained employing a reduction procedure [ 1,6,7 ] on the Member of the Scientific Research Career, Consejo Nacional de Investigaciones Cientificas y Trcnieas (CONICET) of Argentina. 2 Fellow of CONICET of Argentina. 0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
489
Volume 124,number 9
PHYSICS LETTERSA
19 October 1987
SchriSdinger-von Neumann equation of motion for the density operator of the two coupled subsystems• As in prior works [ 1,11 ], we adopt the weak-coupling approximation in order to obtain an explicit expression for the collisional derivative in the reduced equation that describes the irreversible evolution of the oscillator density. However, opposite to these previous references, we do not pose a markovian hypothesis that might enable one to disregard memory effects. Considering that this latter density matrix #a is diagonal in a Fock basis, namely
ha= ~ p. ln>
(3)
n=0
with p, the occupation number of a state with n phonons, the master equation obtained in the current frame reads, /~n(t)=~g
~ 12~ul2
dze -~cos[(og.u-g2)z]
otp 0
×
{ [(n + 1 )p,,+, -np.l,_~p,,(l - p . ) + [np._, - ( n + 1 )p.],_~p.(1 -Pu) } •
(4)
In eq. (4), the symbol g indicates the degeneracy of momentum states kA due to spin and/or isospin, o9~u is a particle-hole-like frequency and 7 is the inelasticity spread or inverse lifetime rUo~r of correlations between fermions and phonons. In addition, and opposite to the view adopted in preceding papers [ 1-12 ], we have explicitly exposed the phonon number coefficients n and n + 1 that had been otherwise absorbed into the coupling parameters. As will be seen below, this statement permits one to unfold a spectrum of collisional frequencies whose eigenvalues can be related to the lifetimes of the moments of the phonon occupation probability. In order to examine the properties of the solutions of eq. (4) we perform a Laplace transformation, introducing the complex vector o~
/ ~ a ( s ) = ) dr e-ST/~a(r),
Re(s)>~0,
(5)
0
with components/~.(s). The transformed algebraic equation reads, in matrix form
[sI-M(s)]~a(s) = / ~ a ( t = 0 ) ,
(6)
where I is the identity in Fock space and the generator M(s) is " - W_ IV_ 0
M(s) =
IV+ -W+-2W_ 2W_
0 2W+ -2W+-3W_ •
•.
0 0 3W+ o
0 0 0 .
.•
nW_
... ... ...
-.
-nW+-(n+l)W_
(n+l)W+ ,°
(7) with the microscopic transition rate-like functions w+ = rv+_
490
I),,~ 12(7+s)
_a)
~p~ (1-p,~) }" tp (l-p.)
(8)
Volume 124, number 9
PHYSICS LETTERS A
19 October 1987
From eq. (6) one learns that the time evolution of the phonon density is described by the law
~a(t) = ~ Res[R(s), Sk] exp(skt)~a(O) ,
(9)
Sk
where it is clear that the matrix amplitude A(&) that accompanies each decaying exponential is the matrix residue of the resolvent R (s) = [ sI-M(s)] -~ at the pole Sk. The spectrum of singularities of the resolvent coincides with the spectrum of eigenvalues of the matrix M(s), which cannot be trivially computed, in general. It is however illustrative to analyze a limiting situation where this spectral problem can be exactly solved. This happens at zero temperature, where the Fermi distribution is a step function; as we examine expression (8), we may realize that the function W (s) acquires very small values, since it is entirely built out of "tails" of the Breit-Wigner-like kernel 7+s F,,,(7, s) = (7 +s) 2 + (09a~ -12) 2 '
(10)
that brings into account the fact that the interaction is not strictly energy-conserving. If we definitively neglect these contributions as we compare them with the W+ function, the secular problem to be solved concerns the matrix
M(s) = W÷ (s)
"0 1 0 -1 0 0
0
0
2 -2
0 3
...
...
...
(ll)
..
-n
n+l • ..
%
A straighforward algebraic calculation leads us to the eigenvalue spectrum, (12)
Sk=--kW÷(sk), k=0,1,2,.... and to corresponding eigenvectors ptk) with components p(k)
=(--1)n(~)p~k},
O~n<~k,
=0,
n>k.
(13) These eigenvectors are traceless vectors, i.e., Trp(k) =P~ .~okn/(-k) 1)"=0,
p~k) SO,
(14)
except for k=O, where the unitary trace equilibrium distribution corresponds to p(O) =6,0 •
(15)
It is now easy to verify that the form of the resolvent matrix is p(k)p+(k)
R(s)=k~o ~ s+kW+(s)'
(16)
where p+ (k) is the kth eigenvector of the adjoint secular problem with the spectrum given in (l 2),
p +(k)M(Sk) = S k P
+(k) .
(17) 491
Volume 124,number 9
PHYSICS LETTERSA
f o/~)
I
I
/t
lI
//
19 October 1987
/
!
/
/
~/~
Fig. 1. The singularityspectrum of the resolventof the integrodifferential equation (4) given as the set of intersections of the curves ~,- ~W+ (u/7) with the lines u/?k for positiveintegerk.
One obtains 0~
p+(k)=o, _{n'~_+,k, --\k)~,k ,
(18)
n>>.k,
with the orthonormalization condition T r p + ~k)p(k')=Okk' giving p~. (k) = ( _ 1 )k/p~k)
•
(19)
Consequently, from eqs. (9), ( 13 ), (16), (18) and (19) we obtain the final expression for the phonon occupation probabilities,
where the upper label i numbers the pole (i.e., i = 1,2,...) for a given k. This equation shows several properties of the decay process: first, we can realize that the actual decay rate of a population p, is the lowest nonvanishing pole of the function [s + n W+ (s)] -~, as indicated by eq. (12) and by the summation limits in eq. (20). Secondly, the evolution of each probability p, is independent of the initial occupation of the lowest-lying levels, however every population is fed from above. This is simply the statement that in this problem the probability flow is downwards. Let us first study the characteristics of the spectrum of singularitites Sk displayed in eq. (12). These poles can be computed by resort to the graphical method indicated in fig. 1, where we follow the scheme presented in ref. [ 12] and draw the curve 7-~ W+ (u/7) with u= - R e ( s ) for different choices of the strength parameters. The singularities Uk correspond to the intersections between a given W+ curve and the lines u/kT. In ref. [ 12 ], only the case k = 1 has been analyzed and it has been shown that three different regimes show up. They are illustrated by curves ( 1 ), (2), and ( 3 ) and respectively correspond to (i) strong interaction, two complex roots; (ii) average interaction, two real roots, the smallest being the macroscopic width inversely related to the lifetime of the oscillation, the highest being of microscopic origin; (iii) weak interaction, one real root related to the eigenvalue of the markovian problem, UM= I4~°) [ 1,5,6,12 ], by the approximate expression [ 12 ] W+(0) ul ~ 1 - W + ( O ) / 7 "
492
(21)
Volume 124,number 9
PHYSICSLETTERSA
19 October 1987
More details concerning properties of these roots and possible physical pictures of the three regimes can be found in ref. [ 12]. Fig. 1 exhibits samples of the singularity spectrum of the resolvent of the nonmarkovian problem (4). We can appreciate, as an example, that for the strength corresponding to curve (3), where only one k = 1 root appears, two roots contribute with k=2. Curve (4), which is associated to a rather weak interaction, gives rise to a spectrum with four poles, from k = 1 to k=4. We realize an important property: the spectrum truncates at a value k beyond which no real roots can exist. This limiting number can be found as follows: consider for example curve (3) and its two intersections with the straight line f ( v ) = v/2y. If we slowly increase the strength parameters - in other words, we raise the ordinate IV+ (0) - the two roots approach each other along the line v/27 until they merge into a double root. It can be seen that this happens at the intersection between the straight line v/2y and 1/2-v/2y, which means v/y= 1/2. These observations can be generalized in the following statement: for each positive integer k, there exists a maximum interaction strength - or equivalently, a maximum ordinate W~+k~(0) - for real roots to appear. For that amount of coupling, there is a double real root v~= 7/2 where the curve W+ (v/y) is tangent to the straight line v/k. A complementary statement can be posed: one could regard the curve W+ (v/y) as fixed and consider, accordingly, that for each function W+ (v/y) there exists a maximum positive integer kM beyond which no real roots appear. The appearance of complex poles brings in a delicate problem, since the oscillating functions Pn (t) could not be regarded as probabilities. Indeed, they could become negative or larger than unity. In ref. [ 12 ] it has been advanced that the regime associated to curve (1) in fig. 1 is related to a collapse of the weak coupling approximation and should not be regarded as the physical answer to the problem of collective damping in a situation where the interaction is strong enough to provide a large energy dispersion in the initial state. The difficulties increase as we consider the full spectrum for k > 1; we have learned from fig. 1 that a maximum k exists for even very weak coupling. This turns out to be specially cumbersome in the light of the following example. Let us consider an initial configuration PI(0)= 6/,o for a given positive integer no. The probabilities p, (t) are then (cf. eq. (20)) 1. P ° ( ' ) = ( - ) ,o
1)k(no)(~)Res[
k
1
s~] exp(s~t),
O<~n<~no n>no ,
=0,
(22)
and in particular, the initial population p,o(0)--1 evolves as 1
p,o(/) = ~
Res[s+no~V+(s),S~o] exp(S~o/).
(23)
It is clear then that if no> kM for the given function W+ (v/y), there are two complex - and conjugate [ 12] roots and P,o may become negative or larger than unity. This should mean, the master equation (4) does not provide a proper description of the damping of an excitation with no phonons, even in a situation where the weak coupling scheme holds. As in ref. [ 12 ], where no = 1, this failure of the dynamical picture can be traced to the presence of a large dispersion Tr[~a(nint--I~int) 2 ] at the initial time. One realizes that for sufficiently high no, this quantity can be large enough to exceed the squared inelasticity spread (by)2 even if the coupling strength 1212 remains close to a low figure. We can reach some deeper understanding of the extent to which the description (23) is unphysical, if we consider the moments of the phonon distribution, rather than the values of the probabilities themselves, in other words, let n p =Tr(/~a~O
(24) 493
Volume 124, number 9
PHYSICS LETTERS A
19 October 1987
be the pth m o m e n t , where h denotes the p h o n o n n u m b e r operator. We can derive a d y n a m i c a l system for these moments, from eq. (4), a n d the corresponding algebraic system in Laplace transform representation nP(s). We skip several steps and present here this final result in the form
snP(s) -nP(O) = - W+(s) pnP(s) ±k__L' ~ k - 1 ( - 1)P-knk(s)
,
l
where the second line should be d r o p p e d at zero temperature. The contents o f eq. (25) b e c o m e evident as we work out the first two m o m e n t s at T = 0 . They read
n(o) ~(s) - s + w+ ( s) ,
(26a)
n2(s) =
(26b)
n2(0) + W+ (s)t~(0) s+2W+(s) [ s + W+ (s)] [s+2W+(s)] '
and the general rule is illustrated, actually, that the p t h m o m e n t evolves with the frequencies sk for k<.p. This means that as the difficulty above discussed sets in, i.e., an oscillating " p r o b a b i l i t y " for n >/kM appears, the physical undesirability affects the m o m e n t s nP(t) for p~> kM. The lower m o m e n t s - average p h o n o n n u m b e r and dispersion - possess the expected, correct behavior: they decay to zero with real, macroscopic lifetimes associated to the smallest poles v~ and ~'2. This work was p e r f o r m e d u n d e r grant P I D 30529 from C O N I C E T , Argentina.
References [ 1] E.S. Hermindez and C.O. Dorso, Phys. Rev. C 29 (1984) 1510. [ 2 ] C.O. Dorso and E.S. Hern~mdez, Phys. Rev. C 29 (1984) 1523. [3] V. de la Mota, C.O. Dorso and E.S. Hermindez, Phys. Lett. B 143 (1984) 279. [4] E.S. Hermindez and C.O. Dorso, Phys. Rev. C 30 (1984) 1711. [5] E.S. Hermindez, Physica A 132 (1985) 28. [6] E.S. Hermindez and A. Kievsky, Phys. Rev. A 32 (1985) 1810. [7] H.M. Cataldo, E.S. Hern~indezand C.O. Dorso, Physica A 142 (1987) 498. [8] E.S. Hermindez and A. Kievsky, Phys. Rev. A 34 (1986). [9] A. Kievsky and E.S. Hermindez, Physica A 139 (1986) 149. [ 10] E.S. Hern~ndez and H.M. Cataldo, Physica A 142 (1987) 517. [ 11 ] A. Kievsky and E.S. Hermindez, to be published. [ 12] H.M. Cataldo and E.S. Hern~tndez, to be published.
494
PHYSICS LETTERS A
19 October 1987
SINGULARITY SPECTRUM OF THE RESOLVENT IN A NONMARKOVIAN MASTER EQUATION E.S. H E R N A N D E Z ~ and H.M. C A T A L D O 2 Departamento de Fisica, Facultadde Ciencias Exactas y Naturales, Universidadde Buenos Aires, 1428 BuenosAires, Argentina
Received 18 March 1987; revised manuscript received 27 July 1987; accepted for publication 12 August 1987 Communicated by A.R. Bishop
The well-known standard nonmarkovian master equation is used to describe the damping of a harmonic macroscopic coordinate coupled to a thermally equilibrated Fermi gas. The time evolution of the phonon distribution is expressed in terms of a resolvent whose spectrum of singularities is computed and analyzed. It is seen that an apparent limitation of the model related to the appearance of undesired oscillations does not inhibit the expected behavior of the average phonon number and phonon number dispersion.
As a collective excitation sets in in a m a n y body system, the residual coupling to the intrinsic degrees o f freedom acts as a damping mechanism that drives the separate subsystems towards thermal equilibrium. The dynamics o f the coupled coordinates can, in principle, be established either on a heuristic or on a more fundamental basis. In general, one can, under suitable hypothesis and approximations, write down a set o f equations o f motion for the time-varying values o f some probabilities relevant to the problem and m a n y significant properties o f the solutions can be exposed if one examines the spectrum o f the generator o f the evolution. If the dynamics exhibits m e m o r y effects which cannot be disregarded, one should turn, if possible, to the investigation o f the singularities of the resolvent or Laplace transform of the time-dependent propagator. Our purpose in this work is to illustrate, by resort to a well-founded model [ 1-11 ], the kind o f analysis that can be done on the above mentioned basis. The model and the master equation. The system we aim at describing consists o f a quantal harmonic oscillator placed into a stationary Fermi gas to which it is coupled through a standard particle-phonon interaction o f the form [ 1-5,12] Him = ~ ( ) . ~ F + b~- b , + 2 * ~ r b ~+ b , ) .
(1)
otu
In this expression, F + and b + are, respectively, the creation operators for a phonon with energy kO and for a fermion state [A ), with F and bA the corresponding annihilators. The labels a, fl, ... (/z, v .... ) denote singleparticle (s.p.) states whose deexcitation (excitation) gives rise to the creation (destruction) o f an oscillator quantum. The coupling matrix elements are denoted as 2,~. The current model is a schematic approximation to a real Fermi system (i.e. a nucleus) where a harmonic collective mode can be excited and undergo subsequent decay due to the residual interaction with the intrinsic degrees o f freedom [ 1-5]. We assume that the heat reservoir is equilibrated at a temperature T, so that we can establish the distribution PA = [ 1 + exp(eA -- ~F)/T] -~ ,
(2)
with ¢~ the Fermi energy. The master equation is obtained employing a reduction procedure [ 1,6,7 ] on the Member of the Scientific Research Career, Consejo Nacional de Investigaciones Cientificas y Trcnieas (CONICET) of Argentina. 2 Fellow of CONICET of Argentina. 0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
489
Volume 124,number 9
PHYSICS LETTERSA
19 October 1987
SchriSdinger-von Neumann equation of motion for the density operator of the two coupled subsystems• As in prior works [ 1,11 ], we adopt the weak-coupling approximation in order to obtain an explicit expression for the collisional derivative in the reduced equation that describes the irreversible evolution of the oscillator density. However, opposite to these previous references, we do not pose a markovian hypothesis that might enable one to disregard memory effects. Considering that this latter density matrix #a is diagonal in a Fock basis, namely
ha= ~ p. ln>
(3)
n=0
with p, the occupation number of a state with n phonons, the master equation obtained in the current frame reads, /~n(t)=~g
~ 12~ul2
dze -~cos[(og.u-g2)z]
otp 0
×
{ [(n + 1 )p,,+, -np.l,_~p,,(l - p . ) + [np._, - ( n + 1 )p.],_~p.(1 -Pu) } •
(4)
In eq. (4), the symbol g indicates the degeneracy of momentum states kA due to spin and/or isospin, o9~u is a particle-hole-like frequency and 7 is the inelasticity spread or inverse lifetime rUo~r of correlations between fermions and phonons. In addition, and opposite to the view adopted in preceding papers [ 1-12 ], we have explicitly exposed the phonon number coefficients n and n + 1 that had been otherwise absorbed into the coupling parameters. As will be seen below, this statement permits one to unfold a spectrum of collisional frequencies whose eigenvalues can be related to the lifetimes of the moments of the phonon occupation probability. In order to examine the properties of the solutions of eq. (4) we perform a Laplace transformation, introducing the complex vector o~
/ ~ a ( s ) = ) dr e-ST/~a(r),
Re(s)>~0,
(5)
0
with components/~.(s). The transformed algebraic equation reads, in matrix form
[sI-M(s)]~a(s) = / ~ a ( t = 0 ) ,
(6)
where I is the identity in Fock space and the generator M(s) is " - W_ IV_ 0
M(s) =
IV+ -W+-2W_ 2W_
0 2W+ -2W+-3W_ •
•.
0 0 3W+ o
0 0 0 .
.•
nW_
... ... ...
-.
-nW+-(n+l)W_
(n+l)W+ ,°
(7) with the microscopic transition rate-like functions w+ = rv+_
490
I),,~ 12(7+s)
_a)
~p~ (1-p,~) }" tp (l-p.)
(8)
Volume 124, number 9
PHYSICS LETTERS A
19 October 1987
From eq. (6) one learns that the time evolution of the phonon density is described by the law
~a(t) = ~ Res[R(s), Sk] exp(skt)~a(O) ,
(9)
Sk
where it is clear that the matrix amplitude A(&) that accompanies each decaying exponential is the matrix residue of the resolvent R (s) = [ sI-M(s)] -~ at the pole Sk. The spectrum of singularities of the resolvent coincides with the spectrum of eigenvalues of the matrix M(s), which cannot be trivially computed, in general. It is however illustrative to analyze a limiting situation where this spectral problem can be exactly solved. This happens at zero temperature, where the Fermi distribution is a step function; as we examine expression (8), we may realize that the function W (s) acquires very small values, since it is entirely built out of "tails" of the Breit-Wigner-like kernel 7+s F,,,(7, s) = (7 +s) 2 + (09a~ -12) 2 '
(10)
that brings into account the fact that the interaction is not strictly energy-conserving. If we definitively neglect these contributions as we compare them with the W+ function, the secular problem to be solved concerns the matrix
M(s) = W÷ (s)
"0 1 0 -1 0 0
0
0
2 -2
0 3
...
...
...
(ll)
..
-n
n+l • ..
%
A straighforward algebraic calculation leads us to the eigenvalue spectrum, (12)
Sk=--kW÷(sk), k=0,1,2,.... and to corresponding eigenvectors ptk) with components p(k)
=(--1)n(~)p~k},
O~n<~k,
=0,
n>k.
(13) These eigenvectors are traceless vectors, i.e., Trp(k) =P~ .~okn/(-k) 1)"=0,
p~k) SO,
(14)
except for k=O, where the unitary trace equilibrium distribution corresponds to p(O) =6,0 •
(15)
It is now easy to verify that the form of the resolvent matrix is p(k)p+(k)
R(s)=k~o ~ s+kW+(s)'
(16)
where p+ (k) is the kth eigenvector of the adjoint secular problem with the spectrum given in (l 2),
p +(k)M(Sk) = S k P
+(k) .
(17) 491
Volume 124,number 9
PHYSICS LETTERSA
f o/~)
I
I
/t
lI
//
19 October 1987
/
!
/
/
~/~
Fig. 1. The singularityspectrum of the resolventof the integrodifferential equation (4) given as the set of intersections of the curves ~,- ~W+ (u/7) with the lines u/?k for positiveintegerk.
One obtains 0~
p+(k)=o, _{n'~_+,k, --\k)~,k ,
(18)
n>>.k,
with the orthonormalization condition T r p + ~k)p(k')=Okk' giving p~. (k) = ( _ 1 )k/p~k)
•
(19)
Consequently, from eqs. (9), ( 13 ), (16), (18) and (19) we obtain the final expression for the phonon occupation probabilities,
where the upper label i numbers the pole (i.e., i = 1,2,...) for a given k. This equation shows several properties of the decay process: first, we can realize that the actual decay rate of a population p, is the lowest nonvanishing pole of the function [s + n W+ (s)] -~, as indicated by eq. (12) and by the summation limits in eq. (20). Secondly, the evolution of each probability p, is independent of the initial occupation of the lowest-lying levels, however every population is fed from above. This is simply the statement that in this problem the probability flow is downwards. Let us first study the characteristics of the spectrum of singularitites Sk displayed in eq. (12). These poles can be computed by resort to the graphical method indicated in fig. 1, where we follow the scheme presented in ref. [ 12] and draw the curve 7-~ W+ (u/7) with u= - R e ( s ) for different choices of the strength parameters. The singularities Uk correspond to the intersections between a given W+ curve and the lines u/kT. In ref. [ 12 ], only the case k = 1 has been analyzed and it has been shown that three different regimes show up. They are illustrated by curves ( 1 ), (2), and ( 3 ) and respectively correspond to (i) strong interaction, two complex roots; (ii) average interaction, two real roots, the smallest being the macroscopic width inversely related to the lifetime of the oscillation, the highest being of microscopic origin; (iii) weak interaction, one real root related to the eigenvalue of the markovian problem, UM= I4~°) [ 1,5,6,12 ], by the approximate expression [ 12 ] W+(0) ul ~ 1 - W + ( O ) / 7 "
492
(21)
Volume 124,number 9
PHYSICSLETTERSA
19 October 1987
More details concerning properties of these roots and possible physical pictures of the three regimes can be found in ref. [ 12]. Fig. 1 exhibits samples of the singularity spectrum of the resolvent of the nonmarkovian problem (4). We can appreciate, as an example, that for the strength corresponding to curve (3), where only one k = 1 root appears, two roots contribute with k=2. Curve (4), which is associated to a rather weak interaction, gives rise to a spectrum with four poles, from k = 1 to k=4. We realize an important property: the spectrum truncates at a value k beyond which no real roots can exist. This limiting number can be found as follows: consider for example curve (3) and its two intersections with the straight line f ( v ) = v/2y. If we slowly increase the strength parameters - in other words, we raise the ordinate IV+ (0) - the two roots approach each other along the line v/27 until they merge into a double root. It can be seen that this happens at the intersection between the straight line v/2y and 1/2-v/2y, which means v/y= 1/2. These observations can be generalized in the following statement: for each positive integer k, there exists a maximum interaction strength - or equivalently, a maximum ordinate W~+k~(0) - for real roots to appear. For that amount of coupling, there is a double real root v~= 7/2 where the curve W+ (v/y) is tangent to the straight line v/k. A complementary statement can be posed: one could regard the curve W+ (v/y) as fixed and consider, accordingly, that for each function W+ (v/y) there exists a maximum positive integer kM beyond which no real roots appear. The appearance of complex poles brings in a delicate problem, since the oscillating functions Pn (t) could not be regarded as probabilities. Indeed, they could become negative or larger than unity. In ref. [ 12 ] it has been advanced that the regime associated to curve (1) in fig. 1 is related to a collapse of the weak coupling approximation and should not be regarded as the physical answer to the problem of collective damping in a situation where the interaction is strong enough to provide a large energy dispersion in the initial state. The difficulties increase as we consider the full spectrum for k > 1; we have learned from fig. 1 that a maximum k exists for even very weak coupling. This turns out to be specially cumbersome in the light of the following example. Let us consider an initial configuration PI(0)= 6/,o for a given positive integer no. The probabilities p, (t) are then (cf. eq. (20)) 1. P ° ( ' ) = ( - ) ,o
1)k(no)(~)Res[
k
1
s~] exp(s~t),
O<~n<~no n>no ,
=0,
(22)
and in particular, the initial population p,o(0)--1 evolves as 1
p,o(/) = ~
Res[s+no~V+(s),S~o] exp(S~o/).
(23)
It is clear then that if no> kM for the given function W+ (v/y), there are two complex - and conjugate [ 12] roots and P,o may become negative or larger than unity. This should mean, the master equation (4) does not provide a proper description of the damping of an excitation with no phonons, even in a situation where the weak coupling scheme holds. As in ref. [ 12 ], where no = 1, this failure of the dynamical picture can be traced to the presence of a large dispersion Tr[~a(nint--I~int) 2 ] at the initial time. One realizes that for sufficiently high no, this quantity can be large enough to exceed the squared inelasticity spread (by)2 even if the coupling strength 1212 remains close to a low figure. We can reach some deeper understanding of the extent to which the description (23) is unphysical, if we consider the moments of the phonon distribution, rather than the values of the probabilities themselves, in other words, let n p =Tr(/~a~O
(24) 493
Volume 124, number 9
PHYSICS LETTERS A
19 October 1987
be the pth m o m e n t , where h denotes the p h o n o n n u m b e r operator. We can derive a d y n a m i c a l system for these moments, from eq. (4), a n d the corresponding algebraic system in Laplace transform representation nP(s). We skip several steps and present here this final result in the form
snP(s) -nP(O) = - W+(s) pnP(s) ±k__L' ~ k - 1 ( - 1)P-knk(s)
,
l
where the second line should be d r o p p e d at zero temperature. The contents o f eq. (25) b e c o m e evident as we work out the first two m o m e n t s at T = 0 . They read
n(o) ~(s) - s + w+ ( s) ,
(26a)
n2(s) =
(26b)
n2(0) + W+ (s)t~(0) s+2W+(s) [ s + W+ (s)] [s+2W+(s)] '
and the general rule is illustrated, actually, that the p t h m o m e n t evolves with the frequencies sk for k<.p. This means that as the difficulty above discussed sets in, i.e., an oscillating " p r o b a b i l i t y " for n >/kM appears, the physical undesirability affects the m o m e n t s nP(t) for p~> kM. The lower m o m e n t s - average p h o n o n n u m b e r and dispersion - possess the expected, correct behavior: they decay to zero with real, macroscopic lifetimes associated to the smallest poles v~ and ~'2. This work was p e r f o r m e d u n d e r grant P I D 30529 from C O N I C E T , Argentina.
References [ 1] E.S. Hermindez and C.O. Dorso, Phys. Rev. C 29 (1984) 1510. [ 2 ] C.O. Dorso and E.S. Hern~mdez, Phys. Rev. C 29 (1984) 1523. [3] V. de la Mota, C.O. Dorso and E.S. Hermindez, Phys. Lett. B 143 (1984) 279. [4] E.S. Hermindez and C.O. Dorso, Phys. Rev. C 30 (1984) 1711. [5] E.S. Hermindez, Physica A 132 (1985) 28. [6] E.S. Hermindez and A. Kievsky, Phys. Rev. A 32 (1985) 1810. [7] H.M. Cataldo, E.S. Hern~indezand C.O. Dorso, Physica A 142 (1987) 498. [8] E.S. Hermindez and A. Kievsky, Phys. Rev. A 34 (1986). [9] A. Kievsky and E.S. Hermindez, Physica A 139 (1986) 149. [ 10] E.S. Hern~ndez and H.M. Cataldo, Physica A 142 (1987) 517. [ 11 ] A. Kievsky and E.S. Hermindez, to be published. [ 12] H.M. Cataldo and E.S. Hern~tndez, to be published.
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