J . Phys. Chem. 1986, 90, 4931-4938 indicated above, the parameters chosen do reproduce the hole properties reported in ref 1. Additional information about the nature of the holes is obtained by extending the burn frequency to wavelengths remote from the peak of the SDF. Results of such burns are shown in Figure 3. From this figure the cause of the observed peak broadening and the shift of hole maximum become apparent. Each hole has two major components: one centered near the maximum of the S D F and the other peaked near the burn frequency. The component near the S D F maximum arises from the large number of absorbers around this wavelength, each absorbing through its broad phonon side band. Because of the large density of absorbers in the vicinity of the S D F maximum, the contributions of this term dominate the holes until wB is shifted several hundred wavenumbers from the S D F maximum. At large shifts from this frequency the other term begins to become significant and the hole begins to split. In these simulations the splitting is easily seen due to normalizing of the holes to the same value rather than scaling the hole maxima to the strength of the absorption. In an actual sample the splitting may not be detectable due to the weak absorption at the wB at which splitting will be seen. In summary, our theory for PHB together with linear electron-phonon coupling dataI2J3establish that the unusually large widths observed'q3 for P-870 of Rh. sphaeroides are the result of a multiphonon broadening process stemming from strong electron-phonon coupling. That is, the widths are not a manifestation of an ultrafast charge separation within the special Bchl pair
4931
following preparation (by excitation) of a neutral excitonic state.l,2 The same conclusion holds for P-960 of Rh. ~ i r i d i s Nevertheless, .~ the question of why P-870and P-960 exhibit Huang-Rhys factors comparable to those of charge-transfer states associated with a-molecular donoracceptor complexedg remains and is important. We note that theoretical calculations by Parson et al." indicate that both of the above dimer or excitonic states carry significant charge-transfer character (intradimer). A related question is why the primary electron donor state, P-700, of reaction centers of green algaeZoand spinachZ1yield photochemical holes which are more than 3 orders of magnitude sharper than those considered here. Sharp nonphotochemical holes have also been observed for the lowest excited state of self-aggregated dimers of Chl a.22
Acknowledgment. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. W-7405-Eng-82. This research was supported by the Director for Energy Research, Office of Basic Energy Science. (19) Haarer, D.; Philpott, M. R. In Spectroscopy and Excitation Dynamics of Condensed Molecular Systems, Agranovich, V. M., Hochstrasser, R. M., Eds, North-Holland: Amsterdam, 1983; Modern Problems in Condensed Matter Sciences Vol. 4, pp 27-82. (20) Maslov, V. G.; Chunaev, A. S.; Tugarinov, V. V. Mol. Bioi. 1981, 15, 788. (21) Golbeck, J.; Gillie, J. K.; Fearey, B. L.; Small, G. J., unpublished results. (22) Carter, T. P.; Small, G. J. J . Phys. Chem. 1986, 90, 1997.
FEATURE ARTICLE Pure Dephasing of a Two-Level System J. L. Skinner* and D. Hsu \
Department of Chemistry, Columbia University, New York, New York 10027 (Received: May 16, 1986)
The relevance of dephasing to many different branches of spectroscopy is discussed. Very general results for both quantum mechanical and stochastic formulations of the dephasing problem are presented and are shown to be intimately related. Our results are nonperturbative in nature, in contrast to the commonly accepted theories. Our results for specific quantum mechanical models are evaluated, and usefui expressions for the analysis of optical and vibrational dephasing experiments are presented.
I. Introduction In many branches of spectroscopy, the description of a system by only two quantum states plays a ubiquitous and fundamental role in our understanding of the absorption of radiation. For example, in magnetic resonance spectroscopy of electrons or spin nucleii, where the application of a static external magnetic field lifts the degeneracy of the two spin levels, many phenomena for isolated spins such as free induction decay or spin echoes can be understood simply by considering the dynamics of the two spin levels in applied time-dependent magnetic In vibrational relaxation or infrared spectroscopy experiments one can often (1) Abragam, A. Principles of Nuclear Magnetism; Oxford: London, 1961. ( 2 ) Slichter, C. P. Principles of Mugnetic Resonance; Springer-Verlag: Berlin, 1980; 2nd ed. ( 3 ) Farrar, T. C.; Becker, E. D.Pulse and Fourier Transform N M R , Academic: New York. 1971.
0022-3654/86/2090-4931$01.50/0
describe a molecule by only two vibrational levels4 This description is useful if the vibrational frequencies of the different modes are sufficiently different, and if either the temperature is low enough so that only the ground state is appreciably populated, or if the anharmonicity is great enough so that individual transitions for a single mode are well separated. In optical experiments one has traditionally analyzed both line shapes and quantum optics phenomena by considering only two electronic levels of atoms or molecule^.^ As long as the particular optical transition of interest is well separated from other transitions, this approximation is adequate. Other examples of two-level systems (TLS) that can be probed spectroscopically include tunnel-split levels in molecules (ammonia, for example), certain proton-transfer reactions, and configurational tunneling in glasses. (4) Oxtoby, D. Adu. Chem. Phys. 1979, 10, 1. 1981, 47 (part 2), 487. (5) Allen, L.; Eberly, J. H. Optical Resonance and Two-Level Atoms; Wiley: New York, 1975.
0 1986 American Chemical Society
4932 The Journal of Physical Chemistry, Vol. 90, No. 21, 1986 The absorption spectrum of a strictly isolated TLS consists of a single infinitely sharp line. The interaction of the TLS with other degrees of freedom (photons, phonons, other collective modes, molecular vibrations, rotations, translations, etc.) produces a broadening and shift of the absorption line, which in many cases becomes Lorentzian. In this case the fwhm line width in Hz, Av, is related to the total dephasing time T2 (defined below), by Au = l/aT2. The dephasing rate, l/T2, has contributions from both population and pure phase relaxation processes. (Population relaxation involves transitions between the two TLS levels, whereas pure phase relaxation conserves populations.) Thus, if the rate for the lower state to upper state transition is l/TL, and the rate for the upper state to lower state transition is l/Tu, then the total population relaxation rate, l / T , , is l/T1 = l / T L l/Tu. If the pure dephasing rate is l / T i (see below), then 1/T2 = l / T i l/2T,.l-5 As mentioned above, 1/T2 can be measured by absorption experiments. It can also be measured more directly by coherent transient techniques such as free induction decay or echo experiments. In either case 1/T2 is an important probe of the TLS-environment interaction and has therefore been the object of intense experimental scrutiny in all branches of spectroscopy. In the present article we focus only on the pure dephasing contribution to l/T2, which is in fact often the dominant contribution. Pure dephasing is best understood in the time domain. Suppose that the two quantum levels of the TLS have energy E, = hao and E , = ha,. Imagine creating a superposition state of the two levels at t = 0:
+
l+(O)) = a(0)lo)
+ b(0)11)
+
(1)
As time increases the state evolves according to I+(t)) =
a(t)lO)
+ b(t)ll)
(2)
Skinner and Hsu situations the off-diagonal element of the reduced density matrix for the TLS decays exponentially in time. This dephasing is usually due to the interactions of the TLS with either longwavelength phonons' or low-frequency local mode^.^^^ The latter process is sometimes called the exchange model. Most of the previous work on the subject of dephasing has been based on perturbation theory. For example, the calculation of 1/ T2 for N M R line widths is performed to second order in the fluctuating magnetic field.lJ For quantum mechanical problems, 1/ T2 and Aw are usually calculated to second order in the TLSbath (environment) interaction. Under some conditions (to be discussed below), the perturbative procedure is adequate. Often, however, it is not. In this article we discuss some nonperturbative (exact) results that we have obtained for the optical dephasing problem over the past several years. Our theoretical results have been published as a series of articles.1h12 In addition, we have compared the theory to several experiment~'~ and have very recently reviewed both theoretical and experimental aspects of the field.14 In fact, our results are substantially more general than we originally envisioned in that they apply to many different branches of spectroscopy; it is this generality that we will attempt to emphasize in the present Feature Article. In addition, we present several new results and derivations. The plan of the paper is as follows: in section 11, we formulate the problem quantum mechanically, and in section 111we provide a simplified derivation of 1 / T2 and Aw. In section IV we present results for specific model Hamiltonians. In section V we show how our results are easily extended to the case of stochastic fluctuations, and in section VI we conclude. 11. Quantum Mechanical Formulation of the Dephasing
Problem We assume that the full Hamiltonian can be written as the sum of TLS Hamiltonian, H$LS, a Hamiltonian for the surroundings (bath), H i , and an interaction Hamiltonian, H'i,,t:
with a(t) = a(O)e-'"O' b(t) = b(O)e+' which we can write as
+
I $ ( t ) ) = e-iYor[n(0)lO) e-iulorb(0)ll)]
(3)
with wlo E w , - wo, showing that there is a well-defined phase factor exp(-iwlot) between the amplitudes of the two states. Now imagine coupling the TLS to a thermal reservoir whose sole effect is to produce a time-dependent stochastic modulation of the energy difference E , - Eo. In this case the well-defined phase difference between the amplitudes gradually disappears and the system is said to be dephased. In the presence of these fluctuations, to describe the system properly one must use the density matrix.IZ2 In particular, the off-diagonal density matrix element is defined by ulo(t) (a(t)*b(t)), where the brackets denote an average over the stochastic process. If the time scale of the energy fluctuations is short compared to the TLS relaxation time, then, as we will see, the off-diagonal density matrix element decays exponentially in time:
where the pure dephasing rate is defined to be l / T i , and Aw is the frequency shift. (In what follows, since we do not discuss lifetime processes and therefore there is no confusion between the total dephasing rate and the pure dephasing rate, we will omit the prime on T2'.) One important example of the above is the broadening of NMR lines of molecules in solution. In this case the fluctuations in the TLS energy difference are due to fluctuating magnetic fields from the thermal motion of the liquid.Iq2 One usually treats the fluctuating field as a stochastic variable.6 In other cases, for example, the vibrational or optical dephasing of molecules in solids or on surfaces, one must treat the environment in a strictly quantum mechanical fashion. Nonetheless, one still finds that in many (6) Kubo, R. Adc. Chem. Phys. 1969, IS, 101.
where
and for the moment we leave Hrb unspecified. A, and A, are operators in the Hilbert space of the variables in Hb. H{ntis taken to be diagonal in the TLS eigenstates since this is a pure dephasing problem. (Any off-diagonal terms in H L t would lead to population relaxation.) Without loss of generality we can set Eo = 0, and defining Hb = H i A. and A = A, - A, and using the fact that l O ) ( O l + I l ) ( I1 = 1, we obtain
+
with
Hint
= All)(ll
where E , 3 h w , and Hb will remain unspecified. The density operator for the full Hamiltonian obeys the Liouville equation: ( 7 ) McCumber, D. E.; Sturge, M. D. J . Appl. Phys. 1963, 34, 1682. (8) Harris, C. B. J. Chem. Phys. 1977, 67, 5607. Harris, C. B.; Shelby, R. M.; Cornelius, P. A. Phys. Rev. Lett. 1977, 38, 1415. Shelby, R. M.; Harris, C. B.; Cornelius, P. A. J . Chem. Phys. 1979, 70, 34. (9) deBree, P.; Wiersma, D. A. J . Chem. Phys. 1979, 70, 790. (10) Hsu, D.; Skinner, J. L. J . Chem. Phys. 1984, 81, 1604. (11) " I , D.; Skinner, J. L. J . Chem. Phys. 1984, 81, 5471. (12) Hsu, D.; Skinner, J. L. J . Chem. Phys. 1985, 83, 2097. (13) Hsu, D.; Skinner, J. L. J . Chem. Phys. 1985, 83, 2107. (14) Skinner, J. L.; Hsu,D.Adu. Chem. Phys. 1986, 65, 1.
The Journal of Physical Chemistry, Vol. 90, No. 21, 1986 4933
Feature Article
(7)
A u = -1m (K)
(17)
1/ T2 = -Re (K)
(18)
with the solution
m
p ( t ) = e-iHf/hp(O)eiHf/h
As discussed in the Introduction, here we are interested in the decay of the off-diagonal element of the reduced density operator, defined by a(t)
= Trb [p(t)l
d o ) = 'd0)Pb
(10)
where e-oHb
=
[
ulo(t)= - i(uo
(1 1)
where
F ( t ) = dHd/he-i(Hb+A)t/h (...) =
Trb
eiHbr/hAe-iHbr/h
(14)
(TA(tl)A(fz) ...A(t,,) ) c
and (...), denotes a cumulant average." W e assume that the correlation function (TA(t)A(O)) decays to its infinite time value , we will call the bath of ( A ) 2 in a characteristic time T ~ which correlation time. W e showed p r e v i o ~ s l y 'that ~ for t >> T~
tK,,
(15)
where
K,, = (-i/h)"
+ t2 + ...t,l)...A(tl)A(0)),
(21)
(22)
where the sum is over all modes a = 1 to N , with frequencies w, and Boson creation and annihilation operators b,+ and b,. Exact analytical expressions for 1/ T2 and Au can be found for a quite general specification of the bath operator A.18,19 Here however, we will be content with the less general (but often adequate) assumption that
A=
W + -4' 2
U~#J
(23)
where
4 = Ch,(b,+
+ 6,)
(24)
Thus we have assumed that A can be expanded in terms of a collective coordinate, 6,that is a linear combination of the normal-mode coordinates (b,' + b,). We have also assumed that the expansion can be truncated after quadratic order. In what follows we will take W, the quadratic coupling constant, to be dimensionless. This implies that a, the linear coupling constant, ha, the expansion coefficients, and 4 all have units of the square root of energy. Our original derivationlo of A u and 1/ T2 was based on a diagrammatic expansion of eq 15. Since that time we have become aware of a simpler derivation20 that starts directly with eq 14? W e will present this simpler derivation here. Let us begin by writing the first few terms in eq 14. Defining
X
l m d t l ~ m d t 2 . . . ~ m d(A(tl twl
+ 1/2)
= Chw,(b,+b,
a
(F(t)) = ex~[CKn(t)I
Kn(t)
+ A w ) ) ~+ (1/Tz)2]-1
01
m
(-i/ h)"(1/n!) J'dtl x'dt2...J'dtn
[(u- (wo
111. Quantum Mechanical Derivation of A u and 1/T2 In order to provide exact expressions for the frequency shift, Aw, and dephasing rate, 1/T2, we assume that Hb is harmonic. Thus we write
(13)
and T is the chronological operator that positions the A(T)'s in order of increasing time from right to left. We next perform a cumulant expansion of ( F ( t J )to obtain" n- 1
a
Hb
In the above, A(T) is the Heisenberg expression for the operator A:
(20)
For completeness we also note that eq 21 implies a frequency shift and width (fwhm) in H z of 6v = A w / ~ Tand Av = l/?rT2, respectively.
[Ob..]
In what follows it will be convenient to write ( F ( t ) ) as a time-ordered exponential:l6
+ Au) + glo(t) Tl2 l
where Aw is the frequency shift. The statement that the off-diagonal element of the density matrix decays exponentially at long times is completely equivalent to the statement that the sharp feature of the absorption line shape is Lorentian if T2 >> T,, ~ i t h ' O , ' ~ Z(u)
alo(t) = uldO)e-'Oo'(F(t))
e
From the above we see that at long times ( t >> 7,) alo(t)indeed decays exponentially with a decay constant l/Tz. It also follows from eq 11 that for long times alo(t)obeys the Bloch equation:I5
Trb [e-oHb]
While it is not necessary to make this a s s ~ m p t i o n , it ' ~simplifies the following discussion. W e are particularly interested in the off-diagonal density matrix element ulo(t) = (llu(t)lO). From eq 8-10 we easily obtain:I5
A(7)
n-1
(9)
where Trb is a trace over all bath states. W e will assume that the initial density operator factorizes as a product of the reduced density operator and the equilibrium bath density operator:
Pb
K = CK,
(8)
A = a$
(25)
and
Therefore it follows that (F(2) )
a
e-'b'e-'/T2
(16)
where (15) Skinner, J. J . Chem. Phys. 1982, 77, 3398. (16) Fetter, A. L.; Walecka, J. D. Quantum Theory of Many-Particle Systems; McGraw-Hill: New York, 1971. (17) Kubo, R. J . Phys. SOC.Jpn. 1962, 27, 1100.
(18) " I , D.; Skinner, J. L., to be submitted for publication. (19) Osad'ko, I. S. Usp. Fiz. Nauk 1979, 128, 31 [Sou. Phys. Usp.1979, 22, 3111. (20) This derivation is a generalization (to the case of quantum mechanical dephasing and both linear and quadratic coupling) of one presented by Nitzan and Persson (see ref 21) for stochastic dephasing. (21) Nitzan, A.; Persson, B. J. N. J . Chem. Phys. 1985, 83, 5610. (22) This is also the approach taken in a paper by Abram, I. I . Chem. Phys. 1977, 25, 81.
4934
The Journal of Physical Chemistry, Vol. 90, No. 21, I986
B = (W/2)d2
Skinner and Hsu sin ( x t )
(26)
lim t--
we then have
=d(x) x
(34)
gives K l ( t ) = ( - i / h ) S ' d t L (B(tL))= (-i/h)(W/2)tC(O) 0
(27)
where the time-ordered correlation function, C(t),is defined by C(tL
- t2)
= (T+(tl)+(tA)
(28)
For future use we note that because of the time-ordering operator, C(t) is an even function of time. For K 2 ( t ) ,explicitly writing the cumulant average we have K2(t) = (-i/hI2(1 / 2 ) S r0 d t , J'dh
-
(35)
as t a. This shows explicitly that K,(t) 0: t for long times in agreement with eq 15; the values of Kn are defined accordingly. Now we can perform the summation in eq 19 to obtain K=
{(T-4(t1)4t2)) +
(TB(ti)B(tA) - (B)21 (36)
Using eq 26 we write (TB(t1)B(t2)) = ( w/a2(T+(tL)+(2l)+(t2)+(t2) ) With the finite-temperature version of Wick's theorem,I6 we can express the right-hand side as the sum of all possible contractions leaving (TB(ti)B(t,)) = (W/2)2{2C(t~- t J 2
We have used the fact that since C(t) is an even function of time, C(w) is an even function of w . From eq 24 and 28 we can explicitly write C(t)in terms of the coupling constants ha:
+ l)e-iw=lrl+ n(w,)eiw*lll]
C(t) = EhL12[(n(w,)
+ C(0)21 where
Thus the second cumulant is then
n(w,) = [exp(phw,) -
K2(t) =
(a2C(tL- t z ) + ( T P / 2 ) C ( t I+ t2I21
(-i/A)2(1/2)J'dtlJ'dt2
(29) Similarly for K 3 ( t ) we have K3(t) =
+
( - i / h ) ' ( l / 3 ! 1 S ' 0d r I S ' d0 t 2 ~ ' d t 3
Using Wick's theorem and performing the cumulant average, one obtains
c(t2- t 3 ) + (w3/3)C(tl - t 2 ) c(t2- t 3 ) c(t,- t l ) l
(30)
The procedure can be generalized for higher order terms to obtain
C(t2- t 3 ) ... C(t,_, - t,)
,...Sf 0
dt, (a2W"-2C(tL- t z ) X
+ ( W / n ) C ( t ,- t 2 ) C(tz- t 3 ) ... x C(tfl-1 - tfl)
C(t, - t l ) l (31)
Defining the Fourier transform of C(t) by 1 C(t ) = dw e-iwfc(w) 2* --
11-l
(38)
Defining the real and imaginary parts of C ( w ) by
+ ihQ(w)
(39)
n(w) = (2n(lwl) + 1)r(lwl)
(40)
C(w)
hn(w)
we find that where r ( w ) is the weighted density of states A
qw) = -ch,2qw
( TB(t1 )B(t2)B(t3)) c j
K,(t) = ( - i / h ) " ( l / 2 ) S r 0d t
(37)
LI
ha
Q(w)
= ?T S O -du
-
r(u)vP(
-> w2 - u2
(41) (42)
n(O),which is of course proportional to the real part of the total time integral of C(t), is given by n(0) = lim 2kW(w)/hw P O +
(43)
The weighted density of states evaluated at w = 0 must be zero, since all frequencies w, are positive. Requiring further that n(0) be noninfinite implies that hmW+ r ( w ) 0: w x with x L 1. For the rather special case of x = 1, the popular "Ohmic dissipation we see that (defining limw-o r ( w ) = A w )
1
n(0) = 2kTA/h, x = 1 (44) whereas in the more usual situation where x > 1, we find simply that
substituting this into the above, and performing the time integrations give
n(0) = 0 , x > 1 (45) In our previous derivationL0we implicitly assumed that x > 1 and hence that n(0) = 0. In this case eq 36 reduces to
K n ( t ) = (-i/h),(1/2)
x
-ia2Q(0) in [ l K = 2 h ( l - W Q ( 0 ) )- x - 2 *
+ iWc(w)/h]
(46)
which is identical with our earlier result,1° and also to the result of O ~ a d ' k o . ~ ~ (23) Chakravarty,S.; Leggett, A. J. Phys. Reu. Lett. 1984, 52, 5. Carrneli, B.; Chandler, D. J . Chem. Phys. 1985, 82, 3400. Harris, R. A.; Silbey, R. J . Chem. Phys. 1985, 83, 1069. Grabert, H.; Weiss, U. Phys. Reo. Lett. 1985, 54, 1605. Fisher, M. P. A.; Dorsey, A. T. Phys. Reo. Lett. 1985, 54, 1609. (24) Osad'ko, I. S. Fiz. Tuerd. Tela 1972,13, 1178. 1972, 14, 2927. 1975, 17, 3180 [Sou. Phys.-Solid State 1971, 13, 974. 1973, 14, 2522. 1976, 17, 20981; Zh. Eksp. Teor. Fiz. 1977, 72, 1575 [Sou. Phys.-JETP 1977,45, 8271. Osad'ko, I. S.; Zhdanov, S. A. Fiz. Tuerd. Tela 1976, 18, 766. 1977, 19, 1683 [Soc. Phys.-Solid State 1976, 18, 441. 1977, 19, 9821.
The Journal of Physical Chemistry, Vol. 90, No. 21, 1986 4935
Feature Article More generally (including the possibility that x = l ) , substituting eq 39 into eq 36 and using the definitions of 1/T2 and Aw from eq 17 and 18, we obtain a2 Q(O)(l - WQ(0))- W n ( O ) 2 Aw = 2 h I1 - WQ(O)l2 Wn(o)2
+
+
For linear bath coupling alone (W = 0), these expressions reduce simply to Aw = a 2 Q ( 0 ) / 2 h
(49)
1/T2 = a 2 n ( 0 ) / 2 h
(50)
+
Further substituting in n ( w ) = ( 2 4 ~ ) l ) r ( w ) from eq 40 (for w > 0) and defining the zero temperature and thermal contributions to Aw to be Awo and AUT, respectively, we find AwO =
To conclude this section we note, as in section 11, that the correlation function C(t)decays on a time scale T ~and , the results . is, the for 1/T2 and Aw are meaningful only if T2 >> T ~ That Bloch equations are valid only for t >> rC,and one is interested in times on the order of T2. Thus for the Bloch equations to be meaningful it follows that T2 >> 7,. In practice then, after one computes T2,one must verify that the separation of time scales indeed holds for the results to be meaningful.
IV. Results for Quantum Mechanical Model Problems In this section we consider models that relate to optical or vibrational dephasing of molecules or ions in crystals or on surfaces. In section A we discuss the Debye model for dephasing by acoustic phonons in bulk crystals, and in section B we discuss dephasing by pseudolocal phonons in bulk crystals or at surfaces. A. Debye Model. Here we consider dephasing by long wavelength acoustic phonons. In the Debye all three branches of acoustic phonons are described by the dispersion relation wqs = clql, where upsis is the phonon frequency, q is the wave vector, s is the phonon branch index, and c is the speed of sound. As a result of this dispersion relation, the phonon density of states is proportional to w 2 . The weighted density of states, I’(w), depends on the coupling constants, hqs, which determine the extent that each normal mode contributes to the collective coordinate 4. Consistent with the Debye model for the density of states is the long-wavelength approxiI1-ation for hqs,which is that h,, a (wqs)1/2.25 Thus the weighted density of states is proportional to w3. We choose the normalization’’ aJ
AUT = -
a2 WlI(O)2
+
which requires that -1 I W 5 density of states is 2 h ( l - r n ( O ) ) { ( l - WQ(0))2+ Wnn(O)2) 2 n ( w ) W T ( w ) ( l - WQ(w)) arctan ( 1 - W Q ( W ) ) ~(2n(w) + ~ ) W I I ( ~ ) ~
1-e (
I
+
(52)
n(o)
_1 -- -a2 T2
+
2 h ( I - WQ(0))2+ WrI(0)2 4n(w)(n(w)+ i)Wr(w)2 1 - 2 I n
[+ 1
+
]
(1 - W Q ( ~ ) )W ~r(w)2
=o
(57)
w
wwD
where h a D = ~ T Dand , T D is the Debye temperature of the crystal. With the above, our nonperturbative expression for AUT and 1/T2 from eq 52 and 5 3 are1’
(53)
In obtaining this expression for 1 / T 2we have used a result from Appendix A of our previous work.’O To summarize these results, we see that, for linear coupling alone ( a # 0, W = 0), Aw is temperature independent, and 1/T2 is zero, except in the case of Ohmic dissipation, when it is proportional to T. For both linear and quadratic coupling, we see that 1/T2= 0 for T = 0 K. For completeness, here we also give the perturbative expressions for 1/T2 and Aw, which are obtained by expanding eq 51-53 up to second order in a and W:
(54)
1
coth ( x T D / 2 T )W 9 a 2 x 6 / 4 ] - ’ (59)
T2 = O 4~D x l d x
In (1
+
where
WeI3 and Powell and c o - w ~ r k e r shave ~ ~ analyzed several optical experiments with eq 60 for 1/T2 and have found that the weak coupling assumption (IWl
