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A thermal bath induced Rabi splitting on the profile of Mollow spectrum in single molecule spectroscopy
12 June 1998
Chemical Physics Letters 289 Ž1998. 287–297
A thermal bath induced Rabi splitting on the profile of Mollow spectrum in single molecule spectroscopy Gautam Gangopadhyay, Sharmistha Ghoshal S.N. Bose National Centre for Basic Sciences, JD Block, Sec-III, Salt Lake City, Calcutta 700091, India Received 25 November 1997; in final form 3 April 1998
Abstract We consider the system heat bath model to construct the dynamical evolution of the reduced density of the system when the reservoir has a finite bandwidth at finite temperature. The effect of Lorentzian bandwidth of the bath is shown in the resonance fluorescence spectra from a driven two-level single molecule. Our analysis reveals that depending on the availability of the bath modes, the width of the spectral peaks can change dramatically. With the increase in temperature, the central peak disintegrates into twin peaks, i.e. a reservoir induced extra resonance is observed at finite temperature. q 1998 Published by Elsevier Science B.V. All rights reserved.
The study of dissipative dynamics of quantum systems within the framework of system-heat bath models has drawn wide attention for the last few decades w1–3x. Although the major works so far done in this direction are confined in the domain of weak coupling of the system and the bath within the Markov approximation, recent experiments involving ultrafast time scale w4–6x and correlated laser pulse with adjustable memory time w7x, have given tremendous impetus to the study of system-heat bath model beyond the Markov approximation. The issue and relevance of non-Markovian relaxation processes have been addressed at length by many researchers. For instance, a cumulant expansion scheme w8x was considered by Villaeys et al. w9x to explain non-Lorentzian nature of optical absorption for the femtosecond transient processes. Gangopadhyay and Ray w10x constructed a system-operator concerned non-Markovian master equation for linear and nonlinear quantum optical model system in the weak coupling regime. Lewenstein et al. w11x observed a strong narrowing of the resonance fluorescence in a cavity with increasing driving field strength which they interpreted this as a dynamical decoupling of the atom from the vacuum field. Gardiner et al. w12x also deal with the effect of radiation-matter interaction in going beyond Markov approximation, where the memory time t is small but of the order of the decay time, gy1 . Very recently Brinati et al. w13x have found a non-Markovian effect at finite temperature on the absorption lineshape function of a driven two-level atom in the steady state regime. They have observed that the steady state absorption lineshape splits into twin peaks with increase in temperature. Recently single molecule w14x as a guest on a host matrix environment in condensed phase reveals a very exciting system to test many fundamental models of molecular physics, solid state physics and quantum optics. 0009-2614r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 4 2 9 - 1
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
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The impurity molecule on the host matrix can be represented as an electronic two-level system where the effect of vibrational dynamics can be made negligibly small. This is supported by a number of experiments as for example, single molecule showed narrow homogeneous Lorentzian line shapes w15x, saturation and antibunching due to optical nutation w16x. The pump-probe experiments w17x showed light shift and Autler- Townes-like w18x structures when the pump is in resonant with the molecule. Very recently, a pump-probe experiment with a new host-guest system w19x, e.g., benzanthracene in a napthalene crystal, showed hyper-Raman and sub-harmonic resonances on the profile of fluorescence w20,21x which is explained by Bloch equation. However, there is a big difference of two-state system in vacuum and in the presence of solid matrix, as an environment. So the property of the matrix environment may modify the Bloch equation picture a lot. In the present work we are considering the situation that the host system is a material whose modal density w22x can be controlled and thus the rate of the spontaneous emission from the impurity molecule can be designed appropriately. By constructing the appropriate modified Bloch equation we have shown that if the modal density in the frequency of interest is less than that of the empty space, the decay of the excited state will be retarded and if it is greater it will be accelerated. With a particular nature of modal density of the bath we have studied the fate of the spontaneous emission from the molecule in presence of strong driving field. For this purpose we have studied the resonance fluorescence spectrum w20,21x. Let us consider a system described by a Hamiltonian Hs . We assume the reservoir Hamiltonian is R and the interaction between system and reservoir is V. So the total Hamiltonian can be written as HT s H s q R q V ' H 0 q V .
Ž 1.
Generally we write V as a sum of the products of the form V s " Ý Q i Fi ,
Ž 2.
i
where Q i and Fi are the system and reservoir operators, respectively. Using the standard prescription w10x of deriving the reduced density equation of motion by averaging over the bath modes we obtain dr
i sy "
dt
t
w Hs , r x y Ý H d tX Q i QXj r X Ž tX . y QXi r X Ž tX . Q i ² Fi FjX :R i, j
0
y Q i r X Ž tX . QXj y r Ž tX . QXj Q i ² FjX Fi :R 4 ,
Ž 3.
where X
Ž Q j r Ž tX . . s exp X
Ž Fj . s exp
ž
i y "
ž
i y "
Hs Ž t y tX . Q j r Ž tX . exp
/
HR Ž t y tX . Fj exp
/ ž
i "
ž
i "
Hs Ž t y t X . ,
HR Ž t y t X . .
/
/
Ž 4a . Ž 4b .
The only assumption we made here is that at any time t,
k Ž t . ssŽ t . f Ž t . ,
Ž 5.
with f Ž t . as the reservoir density operator and then f is substituted as equilibrium bath density. As we do not assume Markov approximation w3x we need to supply the bandshape function for the bath to calculate the integral in Eq. Ž3..
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
289
In what follows we consider a two level system coupled to a reservoir at some temperature T, and driven by a classical monochromatic light field. Thus the system Hamiltonian we have Hs s
" v0 2
sz
Ž 6.
and the laser field interaction in the rotating wave approximation ŽRWA. is given by Vext Ž t . s " sq E0 eyi v t q sy E0 e i v t .
Ž 7.
Thus using the reduced system density operator equation of motion Ž3. and the strong field interaction we can construct the modified Bloch equation which upon invoking a slowly varying envelope approximation, i.e. ² sq Ž t .: s Sq Ž t . e i v t , and ² sz Ž t .: s S z Ž t . one finds S˙q Ž t . s iD Sq Ž t . q
i 2
E0 S z Ž t . y
t
H0 S
q
Ž tX . QX12 Ž t y tX . d tX
and S˙z Ž t . s iE0 Sq Ž t . y Sy Ž t . y
t
H0
Q0 Ž t y tX . q S z Ž tX . Q z Ž t y tX . d tX ,
Ž 8.
where X
QX12 Ž t y tX . s Ý < g j < 2 Ž 1 q 2 n Ž v j . . e iŽ v y v j .Ž tyt . , j
Q z Ž t y tX . s 2 Ý < g j < 2 Ž 1 q 2 n Ž v j . . cos Ž Ž v 0 y v j . Ž t y tX . . , j
Q0 Ž t y tX . s 2 Ý < g j < 2 cos Ž Ž v 0 y v j . Ž t y tX . . .
Ž 9.
j
with D s v 0 y v . In the literature w8–12x most of the workers have introduced the memory effect through the kernels Qq,y, z Ž t . X as proportional to some time dependent functions, for example, ey< Ž tyt .< r t . But in this way of introducing time dependent memory kernels lacks the knowledge of temperature dependence of the non-Markovian decay rate. An equivalent way of choosing exponential time decay in the bath correlation function can be done through the use of a distribution function of P Ž v X . with a finite width. We use Lorentzian distribution function of P Ž v X . as P Ž vX . s
P0 X
2
1q Ž v yv0 . t 2
,
Ž 10 .
where t is the characteristic time of the reservoir or the inverse of the reservoir bandwidth. t is normally of the order of gy1 but gt is always less than 1. P0 is a dimensionless constant quantity. This function is a prototype distribution function of finite bandwidth of the bath which leads to the initially excited system to the appropriate thermal equilibrium asymptotically and thus one can safely employ regression hypothesis which is necessary for the calculation of spectra and can offer an exact analytical formulation.
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
290
Using the bath distribution function we can calculate the following quantities QX12 Ž t y tX . s Q z Ž t y tX . s
g 2t
Ž 1 q 2 n Ž v 0 . . exp
g
Ž 1 q 2 n Ž v 0 . . exp
t
ž
žž
1 yiD y
t t y tX
y
t
/Ž
t y tX . ,
/
/
and t
H0
Q0 Ž t y tX . d tX s g Ž 1 y eyt r t . .
Ž 11 .
Thus the modified Bloch equation reduces to S˙q Ž t . s iD Sq Ž t . q
i 2
g E0 S z Ž t . y
2t
t
Ž 1 q 2 n Ž v 0 . . H Sq Ž tX . exp 0
žž
1 yiD y
t
/Ž
t y tX .
/
Ž 12a.
and S˙z Ž t . s iE0 Sq Ž t . y Sy Ž t . y g Ž 1 y eyt r t . y
g t
t
Ž 1 q 2 n Ž v 0 . . H S z Ž tX . exp 0
ž
t y tX y
t
/
d tX .
Ž 12b.
In the Markovian limit, i.e. when t ™ 0, we recover the usual Bloch equation. For example, one can show that Lim t ™ 0
t
X
1
H0 S Ž t . t exp i
ž
t y tX y
t
/
d tX s Si Ž t . ,
Ž 13 .
where Si is any one of Sq, Sy or S z . Taking the Laplace transform of Eqs. Ž12. one arrives at, pSq Ž p . y S0 s iD Sq Ž p . q
i 2
E0 S z Ž p . y Sq Ž p . Qq ,
pS z Ž p . y S z s iE0 Ž Sq Ž p . y Sy Ž p . . y g
ž
1
1 y
p
p q 1rt
Ž 14a.
/
y Sz Ž p . Q z ,
Ž 14b.
where `
Qq Ž p . '
H0
eyp t QX12 Ž t . d t s
g 2
yi
Ž1 q 2 n Ž v 0 . .
Dt y i Ž 1 q pt .
Ž 15a.
and Q z Ž p . s 2 pg Ž 1 q 2 n Ž v 0 . .
1
Ž 1 q pt .
,
Ž 15b.
with g s 2p < g Ž v 0 .< 2 P0 and nŽ v 0 . s we " v 0 r kT y 1xy1. Here we have considered nŽ v X . s nŽ v 0 . in order to obtain the qualitative results of non-Markovian effects. A more accurate estimation can be given with a Taylor series expansion of nŽ v X . around nŽ v 0 .. But for the rest of the discussion we consider the usual approximation of nŽ v X . s nŽ v 0 . to make life simple and also considering the fact that only the actual experimental result can dictate which kind of expansion of nŽ v X . is valid.
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
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Thus the steady state solution of Bloch equation reduces to DŽ1 q V 2 . q S z Ž t ™ `. s y 1 2
E02 Ž 1 q 2 n . Ž 1 q V 2 . q Ž 1 q 2 n .
2
gV 2
½
g2
Ž 1 q 2 n. q DŽ1 q V 2 . q
4
gV 2
Ž 1 q 2 n.
2
2
Ž 1 q 2 n. q
g2 4
Ž 1 q 2 n.
2
5
Ž 16 . and iE0 2
Sq Ž t ™ ` . s
g yi D q
2
Ž 1 q 2 n.
1yiV
Sz Ž t ™ `.
Ž 17 .
1qV 2
where V s Dt . This is interesting that the non-Markovian feature, i.e. the reservoir characteristic time t appears in the steady state solution of the Bloch equations. For nonzero detuning the steady state values are different than the usual case and steady state observables w13x will depend on t . Now we are interested about the dynamical features due to the modified Bloch equation. At finite t , the decay becomes slower due to longer correlation of the heat bath noise. In order to calculate the strong field resonance fluorescence w20,21x spectra SŽ n y v 0 ., i.e. `
S Ž n y v 0 . s Re Lim t ™`
H0
X
eyiŽ ny v 0 .t ² sq Ž t q tX . sy Ž t . :d tX .
Ž 18 .
We follow the standard prescription w20x of using regression hypothesis. The validity of regression hypothesis and a brief comment on quantum Markov process for the finite bandwidth of the reservoir is given in Appendix A. Calculating the Laplace transform of the quantity G Ž tX . s Lim t ™`² sq Ž t q tX . sy Ž t . : , Ž 19 . we obtain f Ž p. G Ž p . s LT Ž G Ž tX . . s , Ž 20 . pF Ž p . where F Ž p. s
1
Ž 1 q pt .
2
ž
t p2 q p q
g 2
Ž 1 q 2 n . t p 3 q p 2 q p Ž E02t q 3g Ž 1 q 2 n . r2 . q E02 .
/
Ž 21 .
From this F Ž p . one can still find the poles of Mollow the spectrum w20,21x. But it can be approximately factorized as follows
t2 F Ž p. s
Ž 1 q pt .
2
Ž p y lq . Ž p y ly . Ž p y L0 . Ž p y Lq . Ž p y Ly . .
Ž 22 .
f Ž p . can be expressed as f Ž p. s
1 2
Ž S z Ž t ™ `. q 1. p
ž
y Sy Ž t ™ ` . p q
½
p q Qy Ž p .
g pt q 1
/
iE0 2
p q Q z Ž 0. q
Ž p q Qy Ž 0. . .
E02 2
5 Ž 23 .
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
292
We have made the approximation that the detuning D s 0. Now taking g < E0 , F Ž p . can be factorized approximately and the roots of p for F Ž p . s 0, i.e. l ", L0 and L " can be given as
l "s y
1
1 . 1 y 2 k Ž 1 q 2 n. 4 2t 1 3 L0 s y q E02t q g Ž 1 q 2 n . t 2
1r2
,
and
L "s a " ib
Ž 24 .
where a s y 12 Ž E02t q 32 g Ž 1 q 2 n . . ,
(
b s E02 y a 2 and k s gt . Here l " are exact roots but L0 and L " are approximate roots. For numerical calculation of the spectrum we have calculated the poles L0 and L " by using a polynomial root finding algorithm. So the incoherent part of the spectra can be written as S Ž n y v 0 . s yRe q
CL0 i Ž v 0 y n . q L0 Cly
i Ž v 0 y n . q ly
q
CL q i Ž v 0 y n . q Lq
q
CL y i Ž v 0 y n . q Ly
q
Clq i Ž v 0 y n . q lq
Ž 25 .
where the C-values are the residues of f Ž p .rpF Ž p . at five simple poles of F Ž p .. For the purpose of demonstrating the dependence of the effect of tailored reservoir on the fluorescence spectra, we take t and n as parameters. In Fig. 1 we plot the fluorescence spectra and the modal density of the reservoir with detuning v 0 y v . For t s 0, i.e. for infinitely broadband reservoir, the fluorescence spectrum is
Fig. 1. Fluorescence spectrum SŽ n y v 0 .g r Modal Density of the bath P Ž n y v 0 . vs. Frequency detuning Ž n y v 0 .rg is plotted, with E0 rg s10.0 and ns 0.0 for three different values of k sgt s 0.0,0.05 and 0.1. ŽBoth scales are arbitrary..
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
293
Fig. 2. Same as in Fig. 1 with k s 0.05 for three different values of ns 0.0, 1.0 and 2.0.
the well-known Mollow triplet w20x. But with decrease in bandwidth of the reservoir, i.e. with increase in t , the side peaks gets narrower but the central peak does not change much. Under the strong driving field, the two state dipole moment oscillates at three different dressed frequencies, v 0 , v 0 " V R , where V R is the strong field induced Rabi frequency. The width of these peaks are mainly depends on the spontaneous decay rates induced by ordinary bath. Now when the value of t increases, the density of vacuum modes decreases considerably near the side peak frequencies and thus the width of the side peaks decreases sharply. But as the mode density of the reservoir near the central peak is not changed much due to the particular nature of the bath distribution function, the width of the central peak is not affected much. In Fig. 2 we show the spectra with increase in temperature but for low t values. With increase in temperature, the peaks are all broadened almost as in the case of ordinary reservoir. But with increase in temperature the side peaks are shifted outwards very slightly that means the strong field induced Rabi frequencies are increased very slightly and this is due to finite correlation of the bath, i.e. for nonzero values of t. In Fig. 3 we show the spectra at fixed nonzero temperature but for different values of t . With moderate increase in t , i.e. when the density of bath modes become very low near the range of Rabi frequencies, the width of the side peaks decreases sharply. But in these parameter ranges of temperature and t , the width of the
Fig. 3. Same as in Fig. 1 with ns1.0 for three different values of k s 0.05, 0.2 and 0.4.
294
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
Fig. 4. Ža.. Same as in Fig. 1 with k s 0.4 for three different values of ns 0.5, 1.0 and 2.0. Žb. Same as in Ža. but only the central peak is plotted here to show the splitting with increase in n.
central peak increases and with more increase in t the central peak disintegrates into twin peaks. For higher fixed value of t and with increase in temperature the splitting in the central peak is shown in Figs. 4Ža. and 4Žb.. Fig. 4Žb. is shown to plot the central peak only. This splitting in central peak for narrow bandwidth of the vacuum is induced by thermal quanta of the bath. We find this feature for the central peak only because the vacuum bandshape we have chosen is centered around the molecular resonance and thus the vacuum modes are present very little near the frequency range of v 0 " V R to dispense energy to the bath and the strong field induced Rabi sidebands are decoupled from the bath. If we would assume the bandshape maximum of the reservoir near the side peaks, we could have found the reversible dynamical feature through the side peaks also. The splitting of the central peak at higher temperature Ž n. and for shorter bandwidth of the bath can be traced from the poles near the molecular resonance, i.e. p s l "s yŽ1r2t .w1 . 1 y 2 k Ž1 q 2 n.41r2 x. For low values of t and n, 1 y 2 k Ž1 q 2 n. is positive and the two Lorentzians are getting added up together. But when t and n become of moderate values, l " become complex and thus the central peak breaks up into twin peaks. One can calculate the separation between the peaks as S s Ž1r '2 k .w2 k Ž2 n q 1. y 1x1r2 . Thus the separation between the twin peaks S, can be described as a thermal field induced Rabi splitting, which increases with decrease in bandwidth of the bath and with increase in temperature. In effect, a colored bath with very short bandwidth with large thermal photon number acts as an another coherent driving field. This situation is reminiscent to the case of absorption spectra when in addition to the strong driving field a low frequency rf modulation may spilt the absorption profile of Autler-Townes w18x.
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
295
We consider the system-heat bath model for a finite bandwidth of the bath to investigate the decaying dynamics of the system where we have made only one approximation that the system-bath joint density is always factorizable. Due to the evolution of the reduced system with memory, a strong driving field leads to a qualitatively different dynamics. We have given the modified Bloch equation for the Lorentzian bath mode distribution and in the limit of infinitely broadband vacuum, this reduces to the usual Bloch equation. The decay dynamics is studied by calculating the strong field resonance fluorescence spectra. When the bath mode density decreases near the range of Rabi frequencies, the peak width of the Mollow sidebands decreases very sharply. For narrower vacuum band, at finite temperature, the central peak breaks up into twin peaks. Thus a thermal bath induced Rabi splitting emerges on the profile of Mollow triplet. An appendix is provided to be more careful about the naming of the dynamics as non-Markovian and also on the validity of regression hypothesis for arbitrary bandshape function of the bath. Now for the experimental preparation of the band shape function of the vacuum one needs to make an impurity molecule in a crystal with a natural or artificial cut off of the phonon modes and it can also be a viable candidate to observe the resonance fluorescence in this circumstance, where the effect of temperature can also be probed satisfactorily. Although the Bloch equation picture is very general to describe many phenomena from NMR to Quantum Optics in gas phase systems, however, a single impurity molecule in condensed phase with a realistic host system demands modification of Bloch equation to simulate the effect of environment. This is true not only just to fit with the experimental data but also to understand the effect of various noise processes induced by solid matrix from very short to very long correlation time. In this case we considered a model colored noise environment through a Lorentzian distribution of the bath modes, however, it is interesting to see the effect of bath with a very long tail in the bath correlation function which will be discussed elsewhere.
Acknowledgements Authors are thankful to Dr. Samir Pal for an useful discussion. One of us ŽGG. is grateful to Prof. D.S. Ray for many stimulating discussions in the related issues.
Appendix A. A few comments on regression hypothesis and Markov property in the density matrix theory of damping First of all we show here that the approximation we have made in Eq. Ž5., i.e.
k Ž t . s s Ž t . f Ž t . ' s Ž t . f Ž 0.
Ž A.1 .
satisfy regression theorem w23,24x that means the two-time correlation function evolves in the same way as the expectation value. At t s 0, we have assumed
k Ž 0. s s Ž 0. f Ž 0. .
Ž A.2 .
The reduced system and reservoir density operators are defined as s Ž t . s TrB k Ž t . 4 , f Ž t . s Tr s k Ž t . 4 ,
Ž A.3 .
respectively. In terms of k Ž t ., the average of a system operator B, is given by ² B Ž t . : s Tr s TrB B Ž 0 . k Ž t . 4 4 s Tr s B Ž 0 . TrB U Ž t . k Ž 0 . Uy1 Ž t . 4 4 s Tr s B Ž 0 . s Ž t . 4 ,
Ž A.4 .
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
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where U is the total system-bath evolution operator. The two-time correlation function ² AŽ t . B Ž tX .: is given by ² A Ž t . B Ž tX . : s Tr s TrB A Ž t . B Ž tX . k Ž 0 . 4 4 sTr s TrB Uy1 Ž t . A Ž 0 . U Ž t . Uy1 Ž tX . B Ž 0 . U Ž tX . k Ž 0 . 4 4 s Tr s TrB A Ž 0 . U Ž t y tX . B Ž 0 . U Ž tX . k Ž 0 . Uy1 Ž tX . Uy1 Ž t y tX . 4 4 s Tr s A Ž 0 . TrB U Ž t y tX . B Ž 0 . k Ž tX . Uy1 Ž t y tX . 4 4 ,
Ž A.5 .
where A is also a system operator. Here we should note that the Eq. ŽA.5. is an exact expression. Comparing Eqs. ŽA.4. and ŽA.5., we can find that both sŽ t . and the function
V Ž t ,tX . s TrB U Ž t y tX . B Ž 0 . k Ž tX . Uy1 Ž t y tX . 4
Ž A.6 .
appear to evolve in the similar way but with subtle difference. Note that the initial time for the evolution of sŽ t . is t s 0 when the system-bath correlation vanishes. Without loss of generality one is interested in evaluating the correlation function ² AŽ t . B Ž tX .: for the case t G tX . Thus one can assume V Ž t,tX . s 0, for t F tX . So the initial time for the evolution of V Ž t,tX . is t s tX and thus a system-bath correlation will exist at t s tX . This correlation at the initial time in V Ž t,tX ., gives some extra correction factor to the evolution of V Ž t,tX . thereby causing V Ž t,tX . to evolve differently from sŽ t .. This extra factor will be destroyed provided k Ž tX . factorizes for all time, i.e.
k Ž tX . s s Ž tX . f Ž 0 . .
Ž A.7 .
This extra factor is calculated by Swain w23x. Thus when ŽA.7. is satisfied, the two-time correlation function like ŽA.5. evolves in the same way as single time expectation value ŽA.4., i.e. quantum regression theorem holds. Here we would like to comment on the Markov property in the context of density matrix theory of damping. By using the Langevin description of Heisenberg’s equation of motion, Lax w24x has proved that the Markov property implies the regression theorem as well as the converse. And in that derivation Markov property means the Langevin force at time t be uncorrelated to any information at earlier times. In terms of density matrix theory he w24x has proved that the regression theorem can be taken as a definition of Markov property by which quantum dynamics corresponds to classical dynamics. Thus according to Lax, in the density matrix theory Markov property is a derived condition when the regression theorem holds good. This precisely means correlation between system and bath does not evolve in time. Actually the memory can come in the picture in two different ways: Ži. When the joint system-bath density evolves with memory, that means when the factorization like ŽA.7. is not valid and this should be called as non-Markovian in the sense of Lax. Žii. Following the factorization assumption, Eq. ŽA.7., the reduced system density can evolve with memory, that is Markovian in the sense of Lax because regression theorem holds. In our work we follow the second condition but referred as non-Markovian to keep track with the earlier literature w8–13x. In the text we have named the master equation of the reduced density as a system operator concerned non-Markovian equation which evolves in time with memory and clearly needs a shape function of the vacuum band. Thus one can conclude that the regression theorem or Markov property of Lax w24x has more general validity than the standard Born-Markov theory ŽLouisell w3x. and can cover the case when the reduced system evolves with time convolution. As the factorization assumption is independent of bandwidth or memory function, so the physical validity of regression theorem lies on the choice of the memory function or the bandwidth of the bath. One possible way is to choose the the memory function so that the system should reach the appropriate thermal equilibrium in the long time limit.
References w1x P. Ullersma, Physica 32 Ž1966. 27, 56, 74, 90. w2x G.S. Agarwal, Phys. Rev. A 2 Ž1970. 2038; 3 Ž1971. 1783.
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297 w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x
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W.H. Louisell, Quantum Statistical Properties of Radiation ŽWiley, New York, NY, 1973.. P.C. Brecker, H.L. Fragnito, J.Y. Bigot, C.H. Britocruz, C.V. Shank, Phys. Rev. Lett. 63 Ž1990. 505. W. Vogel, D.G. Welsch, W. Wilhelmi, Phys. Rev. A 37 Ž1988. 3825. E.T.J. Nibbering, K. Duppen, D.A. Wiersma, J. Chem. Phys. 93 Ž1990. 5477. P. Tchenio, A. Debarre, J.C. Keller, J.L. Le Gouet, Phys. Rev. Lett. 62 Ž1989. 415. S. Mukamel, Phys. Rep. 93 Ž1982. 1. A.A. Villaeys, J.C. Vallet, S.H. Lin, Phys. Rev. A 43 Ž1991. 5030. G. Gangopadhyay, D.S. Ray, Phys. Rev. A 46 Ž1992. 1507. M. Lewenstein, T.W. Mossberg, R.J. Glauber, Phys. Rev. Lett. 59 Ž1987. 775. C.W. Gardiner, A.S. Perkins, Phys. Rev. A 50 Ž1994. 1792. J.R. Brinati, S.S. Mizrahi, G.A. Prataviera, Phys. Rev. A 50 Ž1994. 3304; 52 Ž1995. 2804. Single Molecule Optical Detection, Imaging and Spectroscopy, edited by T. Basche, W.E. Moerner, M. Orrit, U.P. Wild ŽVCH, Weinheim, Germany.. M. Orrit, J. Bernard, Phys. Rev. Lett. 65 Ž1990. 2716. T. Basche, W.E. Moerner, M. Orrit, H. Talon, Phys. Rev. Lett. 69 Ž1992. 1516. Ph. Tamarat, B. Lounis, J. Bernard, M. Orrit, S. Kummer, R. Kettner, S. Mais, T. Basche, Phys. Rev. Lett. 75 Ž1995. 1514. S.H. Autler, C.H. Townes, Phys. Rev. 100 Ž1955. 703. B. Lounis, F. Jelezko, M. Orrit, Phys. Rev. Lett. 78 Ž1997. 3673. B.R. Mollow, Phys. Rev. 188 Ž1969. 1969. P.L. Knight, P.W. Milloni, Phys. Rep. 66 Ž1980. 21. C.H. Keitel, P.L. Knight, L.M. Narducci, M.O. Scully, Optics Commun. 118 Ž1995. 143, and references therein. S. Swain, J. Phys. A: Math. Gen. 14 Ž1981. 2577. M. Lax, Phys. Rev. 172 Ž1968. 350.
Chemical Physics Letters 289 Ž1998. 287–297
A thermal bath induced Rabi splitting on the profile of Mollow spectrum in single molecule spectroscopy Gautam Gangopadhyay, Sharmistha Ghoshal S.N. Bose National Centre for Basic Sciences, JD Block, Sec-III, Salt Lake City, Calcutta 700091, India Received 25 November 1997; in final form 3 April 1998
Abstract We consider the system heat bath model to construct the dynamical evolution of the reduced density of the system when the reservoir has a finite bandwidth at finite temperature. The effect of Lorentzian bandwidth of the bath is shown in the resonance fluorescence spectra from a driven two-level single molecule. Our analysis reveals that depending on the availability of the bath modes, the width of the spectral peaks can change dramatically. With the increase in temperature, the central peak disintegrates into twin peaks, i.e. a reservoir induced extra resonance is observed at finite temperature. q 1998 Published by Elsevier Science B.V. All rights reserved.
The study of dissipative dynamics of quantum systems within the framework of system-heat bath models has drawn wide attention for the last few decades w1–3x. Although the major works so far done in this direction are confined in the domain of weak coupling of the system and the bath within the Markov approximation, recent experiments involving ultrafast time scale w4–6x and correlated laser pulse with adjustable memory time w7x, have given tremendous impetus to the study of system-heat bath model beyond the Markov approximation. The issue and relevance of non-Markovian relaxation processes have been addressed at length by many researchers. For instance, a cumulant expansion scheme w8x was considered by Villaeys et al. w9x to explain non-Lorentzian nature of optical absorption for the femtosecond transient processes. Gangopadhyay and Ray w10x constructed a system-operator concerned non-Markovian master equation for linear and nonlinear quantum optical model system in the weak coupling regime. Lewenstein et al. w11x observed a strong narrowing of the resonance fluorescence in a cavity with increasing driving field strength which they interpreted this as a dynamical decoupling of the atom from the vacuum field. Gardiner et al. w12x also deal with the effect of radiation-matter interaction in going beyond Markov approximation, where the memory time t is small but of the order of the decay time, gy1 . Very recently Brinati et al. w13x have found a non-Markovian effect at finite temperature on the absorption lineshape function of a driven two-level atom in the steady state regime. They have observed that the steady state absorption lineshape splits into twin peaks with increase in temperature. Recently single molecule w14x as a guest on a host matrix environment in condensed phase reveals a very exciting system to test many fundamental models of molecular physics, solid state physics and quantum optics. 0009-2614r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 4 2 9 - 1
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
288
The impurity molecule on the host matrix can be represented as an electronic two-level system where the effect of vibrational dynamics can be made negligibly small. This is supported by a number of experiments as for example, single molecule showed narrow homogeneous Lorentzian line shapes w15x, saturation and antibunching due to optical nutation w16x. The pump-probe experiments w17x showed light shift and Autler- Townes-like w18x structures when the pump is in resonant with the molecule. Very recently, a pump-probe experiment with a new host-guest system w19x, e.g., benzanthracene in a napthalene crystal, showed hyper-Raman and sub-harmonic resonances on the profile of fluorescence w20,21x which is explained by Bloch equation. However, there is a big difference of two-state system in vacuum and in the presence of solid matrix, as an environment. So the property of the matrix environment may modify the Bloch equation picture a lot. In the present work we are considering the situation that the host system is a material whose modal density w22x can be controlled and thus the rate of the spontaneous emission from the impurity molecule can be designed appropriately. By constructing the appropriate modified Bloch equation we have shown that if the modal density in the frequency of interest is less than that of the empty space, the decay of the excited state will be retarded and if it is greater it will be accelerated. With a particular nature of modal density of the bath we have studied the fate of the spontaneous emission from the molecule in presence of strong driving field. For this purpose we have studied the resonance fluorescence spectrum w20,21x. Let us consider a system described by a Hamiltonian Hs . We assume the reservoir Hamiltonian is R and the interaction between system and reservoir is V. So the total Hamiltonian can be written as HT s H s q R q V ' H 0 q V .
Ž 1.
Generally we write V as a sum of the products of the form V s " Ý Q i Fi ,
Ž 2.
i
where Q i and Fi are the system and reservoir operators, respectively. Using the standard prescription w10x of deriving the reduced density equation of motion by averaging over the bath modes we obtain dr
i sy "
dt
t
w Hs , r x y Ý H d tX Q i QXj r X Ž tX . y QXi r X Ž tX . Q i ² Fi FjX :R i, j
0
y Q i r X Ž tX . QXj y r Ž tX . QXj Q i ² FjX Fi :R 4 ,
Ž 3.
where X
Ž Q j r Ž tX . . s exp X
Ž Fj . s exp
ž
i y "
ž
i y "
Hs Ž t y tX . Q j r Ž tX . exp
/
HR Ž t y tX . Fj exp
/ ž
i "
ž
i "
Hs Ž t y t X . ,
HR Ž t y t X . .
/
/
Ž 4a . Ž 4b .
The only assumption we made here is that at any time t,
k Ž t . ssŽ t . f Ž t . ,
Ž 5.
with f Ž t . as the reservoir density operator and then f is substituted as equilibrium bath density. As we do not assume Markov approximation w3x we need to supply the bandshape function for the bath to calculate the integral in Eq. Ž3..
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
289
In what follows we consider a two level system coupled to a reservoir at some temperature T, and driven by a classical monochromatic light field. Thus the system Hamiltonian we have Hs s
" v0 2
sz
Ž 6.
and the laser field interaction in the rotating wave approximation ŽRWA. is given by Vext Ž t . s " sq E0 eyi v t q sy E0 e i v t .
Ž 7.
Thus using the reduced system density operator equation of motion Ž3. and the strong field interaction we can construct the modified Bloch equation which upon invoking a slowly varying envelope approximation, i.e. ² sq Ž t .: s Sq Ž t . e i v t , and ² sz Ž t .: s S z Ž t . one finds S˙q Ž t . s iD Sq Ž t . q
i 2
E0 S z Ž t . y
t
H0 S
q
Ž tX . QX12 Ž t y tX . d tX
and S˙z Ž t . s iE0 Sq Ž t . y Sy Ž t . y
t
H0
Q0 Ž t y tX . q S z Ž tX . Q z Ž t y tX . d tX ,
Ž 8.
where X
QX12 Ž t y tX . s Ý < g j < 2 Ž 1 q 2 n Ž v j . . e iŽ v y v j .Ž tyt . , j
Q z Ž t y tX . s 2 Ý < g j < 2 Ž 1 q 2 n Ž v j . . cos Ž Ž v 0 y v j . Ž t y tX . . , j
Q0 Ž t y tX . s 2 Ý < g j < 2 cos Ž Ž v 0 y v j . Ž t y tX . . .
Ž 9.
j
with D s v 0 y v . In the literature w8–12x most of the workers have introduced the memory effect through the kernels Qq,y, z Ž t . X as proportional to some time dependent functions, for example, ey< Ž tyt .< r t . But in this way of introducing time dependent memory kernels lacks the knowledge of temperature dependence of the non-Markovian decay rate. An equivalent way of choosing exponential time decay in the bath correlation function can be done through the use of a distribution function of P Ž v X . with a finite width. We use Lorentzian distribution function of P Ž v X . as P Ž vX . s
P0 X
2
1q Ž v yv0 . t 2
,
Ž 10 .
where t is the characteristic time of the reservoir or the inverse of the reservoir bandwidth. t is normally of the order of gy1 but gt is always less than 1. P0 is a dimensionless constant quantity. This function is a prototype distribution function of finite bandwidth of the bath which leads to the initially excited system to the appropriate thermal equilibrium asymptotically and thus one can safely employ regression hypothesis which is necessary for the calculation of spectra and can offer an exact analytical formulation.
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
290
Using the bath distribution function we can calculate the following quantities QX12 Ž t y tX . s Q z Ž t y tX . s
g 2t
Ž 1 q 2 n Ž v 0 . . exp
g
Ž 1 q 2 n Ž v 0 . . exp
t
ž
žž
1 yiD y
t t y tX
y
t
/Ž
t y tX . ,
/
/
and t
H0
Q0 Ž t y tX . d tX s g Ž 1 y eyt r t . .
Ž 11 .
Thus the modified Bloch equation reduces to S˙q Ž t . s iD Sq Ž t . q
i 2
g E0 S z Ž t . y
2t
t
Ž 1 q 2 n Ž v 0 . . H Sq Ž tX . exp 0
žž
1 yiD y
t
/Ž
t y tX .
/
Ž 12a.
and S˙z Ž t . s iE0 Sq Ž t . y Sy Ž t . y g Ž 1 y eyt r t . y
g t
t
Ž 1 q 2 n Ž v 0 . . H S z Ž tX . exp 0
ž
t y tX y
t
/
d tX .
Ž 12b.
In the Markovian limit, i.e. when t ™ 0, we recover the usual Bloch equation. For example, one can show that Lim t ™ 0
t
X
1
H0 S Ž t . t exp i
ž
t y tX y
t
/
d tX s Si Ž t . ,
Ž 13 .
where Si is any one of Sq, Sy or S z . Taking the Laplace transform of Eqs. Ž12. one arrives at, pSq Ž p . y S0 s iD Sq Ž p . q
i 2
E0 S z Ž p . y Sq Ž p . Qq ,
pS z Ž p . y S z s iE0 Ž Sq Ž p . y Sy Ž p . . y g
ž
1
1 y
p
p q 1rt
Ž 14a.
/
y Sz Ž p . Q z ,
Ž 14b.
where `
Qq Ž p . '
H0
eyp t QX12 Ž t . d t s
g 2
yi
Ž1 q 2 n Ž v 0 . .
Dt y i Ž 1 q pt .
Ž 15a.
and Q z Ž p . s 2 pg Ž 1 q 2 n Ž v 0 . .
1
Ž 1 q pt .
,
Ž 15b.
with g s 2p < g Ž v 0 .< 2 P0 and nŽ v 0 . s we " v 0 r kT y 1xy1. Here we have considered nŽ v X . s nŽ v 0 . in order to obtain the qualitative results of non-Markovian effects. A more accurate estimation can be given with a Taylor series expansion of nŽ v X . around nŽ v 0 .. But for the rest of the discussion we consider the usual approximation of nŽ v X . s nŽ v 0 . to make life simple and also considering the fact that only the actual experimental result can dictate which kind of expansion of nŽ v X . is valid.
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
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Thus the steady state solution of Bloch equation reduces to DŽ1 q V 2 . q S z Ž t ™ `. s y 1 2
E02 Ž 1 q 2 n . Ž 1 q V 2 . q Ž 1 q 2 n .
2
gV 2
½
g2
Ž 1 q 2 n. q DŽ1 q V 2 . q
4
gV 2
Ž 1 q 2 n.
2
2
Ž 1 q 2 n. q
g2 4
Ž 1 q 2 n.
2
5
Ž 16 . and iE0 2
Sq Ž t ™ ` . s
g yi D q
2
Ž 1 q 2 n.
1yiV
Sz Ž t ™ `.
Ž 17 .
1qV 2
where V s Dt . This is interesting that the non-Markovian feature, i.e. the reservoir characteristic time t appears in the steady state solution of the Bloch equations. For nonzero detuning the steady state values are different than the usual case and steady state observables w13x will depend on t . Now we are interested about the dynamical features due to the modified Bloch equation. At finite t , the decay becomes slower due to longer correlation of the heat bath noise. In order to calculate the strong field resonance fluorescence w20,21x spectra SŽ n y v 0 ., i.e. `
S Ž n y v 0 . s Re Lim t ™`
H0
X
eyiŽ ny v 0 .t ² sq Ž t q tX . sy Ž t . :d tX .
Ž 18 .
We follow the standard prescription w20x of using regression hypothesis. The validity of regression hypothesis and a brief comment on quantum Markov process for the finite bandwidth of the reservoir is given in Appendix A. Calculating the Laplace transform of the quantity G Ž tX . s Lim t ™`² sq Ž t q tX . sy Ž t . : , Ž 19 . we obtain f Ž p. G Ž p . s LT Ž G Ž tX . . s , Ž 20 . pF Ž p . where F Ž p. s
1
Ž 1 q pt .
2
ž
t p2 q p q
g 2
Ž 1 q 2 n . t p 3 q p 2 q p Ž E02t q 3g Ž 1 q 2 n . r2 . q E02 .
/
Ž 21 .
From this F Ž p . one can still find the poles of Mollow the spectrum w20,21x. But it can be approximately factorized as follows
t2 F Ž p. s
Ž 1 q pt .
2
Ž p y lq . Ž p y ly . Ž p y L0 . Ž p y Lq . Ž p y Ly . .
Ž 22 .
f Ž p . can be expressed as f Ž p. s
1 2
Ž S z Ž t ™ `. q 1. p
ž
y Sy Ž t ™ ` . p q
½
p q Qy Ž p .
g pt q 1
/
iE0 2
p q Q z Ž 0. q
Ž p q Qy Ž 0. . .
E02 2
5 Ž 23 .
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
292
We have made the approximation that the detuning D s 0. Now taking g < E0 , F Ž p . can be factorized approximately and the roots of p for F Ž p . s 0, i.e. l ", L0 and L " can be given as
l "s y
1
1 . 1 y 2 k Ž 1 q 2 n. 4 2t 1 3 L0 s y q E02t q g Ž 1 q 2 n . t 2
1r2
,
and
L "s a " ib
Ž 24 .
where a s y 12 Ž E02t q 32 g Ž 1 q 2 n . . ,
(
b s E02 y a 2 and k s gt . Here l " are exact roots but L0 and L " are approximate roots. For numerical calculation of the spectrum we have calculated the poles L0 and L " by using a polynomial root finding algorithm. So the incoherent part of the spectra can be written as S Ž n y v 0 . s yRe q
CL0 i Ž v 0 y n . q L0 Cly
i Ž v 0 y n . q ly
q
CL q i Ž v 0 y n . q Lq
q
CL y i Ž v 0 y n . q Ly
q
Clq i Ž v 0 y n . q lq
Ž 25 .
where the C-values are the residues of f Ž p .rpF Ž p . at five simple poles of F Ž p .. For the purpose of demonstrating the dependence of the effect of tailored reservoir on the fluorescence spectra, we take t and n as parameters. In Fig. 1 we plot the fluorescence spectra and the modal density of the reservoir with detuning v 0 y v . For t s 0, i.e. for infinitely broadband reservoir, the fluorescence spectrum is
Fig. 1. Fluorescence spectrum SŽ n y v 0 .g r Modal Density of the bath P Ž n y v 0 . vs. Frequency detuning Ž n y v 0 .rg is plotted, with E0 rg s10.0 and ns 0.0 for three different values of k sgt s 0.0,0.05 and 0.1. ŽBoth scales are arbitrary..
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
293
Fig. 2. Same as in Fig. 1 with k s 0.05 for three different values of ns 0.0, 1.0 and 2.0.
the well-known Mollow triplet w20x. But with decrease in bandwidth of the reservoir, i.e. with increase in t , the side peaks gets narrower but the central peak does not change much. Under the strong driving field, the two state dipole moment oscillates at three different dressed frequencies, v 0 , v 0 " V R , where V R is the strong field induced Rabi frequency. The width of these peaks are mainly depends on the spontaneous decay rates induced by ordinary bath. Now when the value of t increases, the density of vacuum modes decreases considerably near the side peak frequencies and thus the width of the side peaks decreases sharply. But as the mode density of the reservoir near the central peak is not changed much due to the particular nature of the bath distribution function, the width of the central peak is not affected much. In Fig. 2 we show the spectra with increase in temperature but for low t values. With increase in temperature, the peaks are all broadened almost as in the case of ordinary reservoir. But with increase in temperature the side peaks are shifted outwards very slightly that means the strong field induced Rabi frequencies are increased very slightly and this is due to finite correlation of the bath, i.e. for nonzero values of t. In Fig. 3 we show the spectra at fixed nonzero temperature but for different values of t . With moderate increase in t , i.e. when the density of bath modes become very low near the range of Rabi frequencies, the width of the side peaks decreases sharply. But in these parameter ranges of temperature and t , the width of the
Fig. 3. Same as in Fig. 1 with ns1.0 for three different values of k s 0.05, 0.2 and 0.4.
294
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
Fig. 4. Ža.. Same as in Fig. 1 with k s 0.4 for three different values of ns 0.5, 1.0 and 2.0. Žb. Same as in Ža. but only the central peak is plotted here to show the splitting with increase in n.
central peak increases and with more increase in t the central peak disintegrates into twin peaks. For higher fixed value of t and with increase in temperature the splitting in the central peak is shown in Figs. 4Ža. and 4Žb.. Fig. 4Žb. is shown to plot the central peak only. This splitting in central peak for narrow bandwidth of the vacuum is induced by thermal quanta of the bath. We find this feature for the central peak only because the vacuum bandshape we have chosen is centered around the molecular resonance and thus the vacuum modes are present very little near the frequency range of v 0 " V R to dispense energy to the bath and the strong field induced Rabi sidebands are decoupled from the bath. If we would assume the bandshape maximum of the reservoir near the side peaks, we could have found the reversible dynamical feature through the side peaks also. The splitting of the central peak at higher temperature Ž n. and for shorter bandwidth of the bath can be traced from the poles near the molecular resonance, i.e. p s l "s yŽ1r2t .w1 . 1 y 2 k Ž1 q 2 n.41r2 x. For low values of t and n, 1 y 2 k Ž1 q 2 n. is positive and the two Lorentzians are getting added up together. But when t and n become of moderate values, l " become complex and thus the central peak breaks up into twin peaks. One can calculate the separation between the peaks as S s Ž1r '2 k .w2 k Ž2 n q 1. y 1x1r2 . Thus the separation between the twin peaks S, can be described as a thermal field induced Rabi splitting, which increases with decrease in bandwidth of the bath and with increase in temperature. In effect, a colored bath with very short bandwidth with large thermal photon number acts as an another coherent driving field. This situation is reminiscent to the case of absorption spectra when in addition to the strong driving field a low frequency rf modulation may spilt the absorption profile of Autler-Townes w18x.
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
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We consider the system-heat bath model for a finite bandwidth of the bath to investigate the decaying dynamics of the system where we have made only one approximation that the system-bath joint density is always factorizable. Due to the evolution of the reduced system with memory, a strong driving field leads to a qualitatively different dynamics. We have given the modified Bloch equation for the Lorentzian bath mode distribution and in the limit of infinitely broadband vacuum, this reduces to the usual Bloch equation. The decay dynamics is studied by calculating the strong field resonance fluorescence spectra. When the bath mode density decreases near the range of Rabi frequencies, the peak width of the Mollow sidebands decreases very sharply. For narrower vacuum band, at finite temperature, the central peak breaks up into twin peaks. Thus a thermal bath induced Rabi splitting emerges on the profile of Mollow triplet. An appendix is provided to be more careful about the naming of the dynamics as non-Markovian and also on the validity of regression hypothesis for arbitrary bandshape function of the bath. Now for the experimental preparation of the band shape function of the vacuum one needs to make an impurity molecule in a crystal with a natural or artificial cut off of the phonon modes and it can also be a viable candidate to observe the resonance fluorescence in this circumstance, where the effect of temperature can also be probed satisfactorily. Although the Bloch equation picture is very general to describe many phenomena from NMR to Quantum Optics in gas phase systems, however, a single impurity molecule in condensed phase with a realistic host system demands modification of Bloch equation to simulate the effect of environment. This is true not only just to fit with the experimental data but also to understand the effect of various noise processes induced by solid matrix from very short to very long correlation time. In this case we considered a model colored noise environment through a Lorentzian distribution of the bath modes, however, it is interesting to see the effect of bath with a very long tail in the bath correlation function which will be discussed elsewhere.
Acknowledgements Authors are thankful to Dr. Samir Pal for an useful discussion. One of us ŽGG. is grateful to Prof. D.S. Ray for many stimulating discussions in the related issues.
Appendix A. A few comments on regression hypothesis and Markov property in the density matrix theory of damping First of all we show here that the approximation we have made in Eq. Ž5., i.e.
k Ž t . s s Ž t . f Ž t . ' s Ž t . f Ž 0.
Ž A.1 .
satisfy regression theorem w23,24x that means the two-time correlation function evolves in the same way as the expectation value. At t s 0, we have assumed
k Ž 0. s s Ž 0. f Ž 0. .
Ž A.2 .
The reduced system and reservoir density operators are defined as s Ž t . s TrB k Ž t . 4 , f Ž t . s Tr s k Ž t . 4 ,
Ž A.3 .
respectively. In terms of k Ž t ., the average of a system operator B, is given by ² B Ž t . : s Tr s TrB B Ž 0 . k Ž t . 4 4 s Tr s B Ž 0 . TrB U Ž t . k Ž 0 . Uy1 Ž t . 4 4 s Tr s B Ž 0 . s Ž t . 4 ,
Ž A.4 .
G. Gangopadhyay, S. Ghoshalr Chemical Physics Letters 289 (1998) 287–297
296
where U is the total system-bath evolution operator. The two-time correlation function ² AŽ t . B Ž tX .: is given by ² A Ž t . B Ž tX . : s Tr s TrB A Ž t . B Ž tX . k Ž 0 . 4 4 sTr s TrB Uy1 Ž t . A Ž 0 . U Ž t . Uy1 Ž tX . B Ž 0 . U Ž tX . k Ž 0 . 4 4 s Tr s TrB A Ž 0 . U Ž t y tX . B Ž 0 . U Ž tX . k Ž 0 . Uy1 Ž tX . Uy1 Ž t y tX . 4 4 s Tr s A Ž 0 . TrB U Ž t y tX . B Ž 0 . k Ž tX . Uy1 Ž t y tX . 4 4 ,
Ž A.5 .
where A is also a system operator. Here we should note that the Eq. ŽA.5. is an exact expression. Comparing Eqs. ŽA.4. and ŽA.5., we can find that both sŽ t . and the function
V Ž t ,tX . s TrB U Ž t y tX . B Ž 0 . k Ž tX . Uy1 Ž t y tX . 4
Ž A.6 .
appear to evolve in the similar way but with subtle difference. Note that the initial time for the evolution of sŽ t . is t s 0 when the system-bath correlation vanishes. Without loss of generality one is interested in evaluating the correlation function ² AŽ t . B Ž tX .: for the case t G tX . Thus one can assume V Ž t,tX . s 0, for t F tX . So the initial time for the evolution of V Ž t,tX . is t s tX and thus a system-bath correlation will exist at t s tX . This correlation at the initial time in V Ž t,tX ., gives some extra correction factor to the evolution of V Ž t,tX . thereby causing V Ž t,tX . to evolve differently from sŽ t .. This extra factor will be destroyed provided k Ž tX . factorizes for all time, i.e.
k Ž tX . s s Ž tX . f Ž 0 . .
Ž A.7 .
This extra factor is calculated by Swain w23x. Thus when ŽA.7. is satisfied, the two-time correlation function like ŽA.5. evolves in the same way as single time expectation value ŽA.4., i.e. quantum regression theorem holds. Here we would like to comment on the Markov property in the context of density matrix theory of damping. By using the Langevin description of Heisenberg’s equation of motion, Lax w24x has proved that the Markov property implies the regression theorem as well as the converse. And in that derivation Markov property means the Langevin force at time t be uncorrelated to any information at earlier times. In terms of density matrix theory he w24x has proved that the regression theorem can be taken as a definition of Markov property by which quantum dynamics corresponds to classical dynamics. Thus according to Lax, in the density matrix theory Markov property is a derived condition when the regression theorem holds good. This precisely means correlation between system and bath does not evolve in time. Actually the memory can come in the picture in two different ways: Ži. When the joint system-bath density evolves with memory, that means when the factorization like ŽA.7. is not valid and this should be called as non-Markovian in the sense of Lax. Žii. Following the factorization assumption, Eq. ŽA.7., the reduced system density can evolve with memory, that is Markovian in the sense of Lax because regression theorem holds. In our work we follow the second condition but referred as non-Markovian to keep track with the earlier literature w8–13x. In the text we have named the master equation of the reduced density as a system operator concerned non-Markovian equation which evolves in time with memory and clearly needs a shape function of the vacuum band. Thus one can conclude that the regression theorem or Markov property of Lax w24x has more general validity than the standard Born-Markov theory ŽLouisell w3x. and can cover the case when the reduced system evolves with time convolution. As the factorization assumption is independent of bandwidth or memory function, so the physical validity of regression theorem lies on the choice of the memory function or the bandwidth of the bath. One possible way is to choose the the memory function so that the system should reach the appropriate thermal equilibrium in the long time limit.
References w1x P. Ullersma, Physica 32 Ž1966. 27, 56, 74, 90. w2x G.S. Agarwal, Phys. Rev. A 2 Ž1970. 2038; 3 Ž1971. 1783.
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