Non–Linear Coupling

hω_ξ(g_I(ξ)Z_ξ+g_II(ξ)Z_ξ²). (2.45) The studies reported below are based on Eq. (2.44) for the system reservoir coupling and extend the investigations on non–Markovian nuclear dynamics in previous sections to the case where two–quantum processes govern the dissipa-tive nuclear dynamics. In particular, we will demonstrate that the term propor-tional toZ_ξ²results in a correlation function which contains a time–independent part. The reasonableness of such a function which mainly determines the structure of the memory kernel entering the non–Markovian Quantum Master Equation (QME) is demonstrated and the breakdown of a Markovian Redfield–

theory is shown. As the basic technique to solve the non–Markovian QME we use our Laguerre polynomial expansion method explained in Appendix F.

2.6.1 Non–Linear Coupling

In contrast to an expression where Φ depends linearly on the reservoir coordi-nates Eq. (2.45) results in a thermal averaged expectation value of Φ which

does not vanish but gives

<Φ>R=^X

ξ

¯

hωξgII(ξ)(1 + 2n(ωξ)) (2.46) Here,

< ... >R= trR{Rˆeq...} (2.47) denotes the thermal average with respect to the reservoir DOF. ( ˆReq is the respective equilibrium statistical operator as in previous sections.) The non-vanishing <Φ>_R leads to the appearance of the so–called mean–field term in the QME proportional to < HS−R >R. To allow for excitation energy dissi-pation via a coupling to the reservoir of passive coordinates, we provide that initially the active coordinate has been prepared in a non–equilibrium state by a photo-excitation process into an excited electronic state. The complete QME governing the dynamics of the excited state the density operator ˆσ (compare 2.27) (≡∆ ˆρ_ee) reads

∂

∂tσ(t) =ˆ −i

¯

h[He+< H_S−R^(e) >R,∆ˆσ(t)]₋

− D(tˆ −t_field; ∆ˆσ) + ˆF(t, t_field;E),

(2.48) where the time argument t_field indicates the time region just before the action of the light–pulse. This time argument can be considered as the initial time of the density matrix propagation since we have ˆσ= 0 for t < t_field (note that we will set t_field = 0 in the following). The light field enters Eq. (2.48) again via the source term ˆF introduced in Appendix D.

A straightforward calculation gives for the correlation function (compare also [OF89])

C(t) =C_I(t) +C_II(t). (2.49) The first term on the right–hand side corresponds to the linear part of expan-sion (2.45) whereas the second term is originated by the quadratic contribution in Eq. (2.45). There is no mixing between both since expectation values of an odd number of reservoir coordinates vanishes. C_I(t) can be written in the standard form [Wei93, MK99]

C_I(t) =

Z

dω ω² e^−iωt(1 +n(ω))J_I(ω)−J_I(−ω)

(2.50)

with the Bose–Einstein distribution n(ω) and the spectral density J_I(ω) =

P

ξg_ξ^(I)2δ(ω−ω_ξ). For the second contribution to C(t) we get

C_II(t) =C_II⁽¹⁾(t) +C_II⁽²⁾ . (2.51) The first time–dependent contribution is obtained as

C_II⁽¹⁾(t) = 1 argument 2ωξ indicates that the considered type of system–reservoir coupling results in relaxation processes where transitions within the spectrum of the active system are accompanied by the emission or absorption of two reservoir quanta. For the second, time–independent part of the correlation function, Eq.

(2.51) one obtains The absence of any time–dependence shows thatC⁽²⁾, if inserted into the dissi-pative part of Eq. (2.48) (see Eq. (D.5)), results in a memory extending up to the beginning of the time–evolution. Consequently, such a retarded dissipation has to be considered within the non–Markovian version of the QME. Chang-ing to the Fourier–transformed correlation functionC_II(ω) the partC_II⁽²⁾ would become proportional to δ(ω). This singular frequency dependence, again, in-dicates that it cannot be described within a standard Markov approximation of the QME (Redfield theory).

However, the presence of a time–independent part in C(t) does not mean that long–living correlations appear in the reservoir. This becomes obvious if one perturbs the reservoir with a coupling expression like that given in Eq.

(2.44), where K(Q), however, has to be understood as an external field and not as a quantum-mechanical operator. Asking for the linear response of the reservoir if the external fieldK has been applied one ends up with a generalized linear susceptibility (see, for example [MK99]). It is given as −i¯hΘ(t)(C(t)− C(−t)), where the unit step function Θ(t) takes notice of causality. Since the anti-symmetrized version of the correlation function enters the susceptibility any contribution of the time–independent part C_II⁽²⁾, Eq. (2.53) is canceled, i.e.

the susceptibility goes to zero for t→ ∞.

0 200 400 600 800 1000 0.0

0.2 0.4 0.6 0.8 1.0

C2/C1(t=0)

T [K]

a b

c

d

e

Figure 2.11: Ratio between the time-independent part C_II⁽²⁾ of the correlation function and the time-dependent part C_II⁽¹⁾(t) at time t=0 versus temperature and for different values of the correlation time t_c = 1/ω_c: (a) t_c = 50f s, (b) t_c = 30f s, (c) t_c = 20f s, (d) t_c = 10f s and (e) t_c = 5f s. Pairs of the correlation time and temperature taken to compute curves in further figures are indicated by a dot.

In the calculations, both spectral densities JI and JII are used in the com-mon form J(ω) = Θ(ω)J₀j(ω). The normalized quantity j(ω) is taken as 1/ω_c² ×exp(−ω/ω_c), and Θ(ω) denotes the unit step function. The cut–off frequency ωc mainly determines the memory time of the non–Markovian dy-namics. We again denote the inverse of ω_c byt_c and call it correlation time.

Before presenting numerical results we will concentrate on the effect the time–independent part C_II⁽²⁾ causes on the dissipative dynamics. Such a con-sideration would be most useful if C_II⁽²⁾ dominates on C_II⁽¹⁾. According to Eqs.

(2.52) and (2.53) this should be the case for higher temperatures. Fig. 2.11 shows the ratio C_II⁽²⁾/C_II⁽¹⁾(t = 0) versus temperature for different tc = 1/ωc. Indeed, for k_BT ¯h/t_c the dissipative dynamics are governed by the time–

independent part of C(t).

2.6.2 Numerical Results

In this section we calculate the non-Markovian dynamics of the system used also in previous sections i.e. the system consisting of two electronic states (see

Fig. 2.1). For the case where the time-independent part of the correlation function is predominant one can recall the analytical solution we have derived in section 2.4. For the analysis below we will orient on the results obtained on non–Markovian effects in the earlier chapters. There, the interplay has been discussed of optical state preparation, vibrational motion in the excited elec-tronic state, and decay of reservoir coordinate correlations. It could be shown that the most pronounced deviation from non–retarded dynamics is obtained for the case of impulsive excitation. For such an excitation where the length of the light–pulse is short compared to the two other characteristic times the vibrational state populations P_M show characteristic oscillations modulating the redistribution of the probability among different vibrational levels. The doubling of the period characterizing the modulations of P_M just after the excitation process could be identified as a clear signature of retarded, i.e. non–

Markovian dynamics. If the pulse length becomes larger or comparable to the vibrational period these modulations vanish but the overall time–dependence of the level populations show just a deviation from that of the Markovian case.

In Fig.2.2 in section 2.4 we have compared the full non–Markovian results for the case of a bilinear coupling and a long correlation time t_c = 50f s with the corresponding Markovian results as well as the analytical solution for the case of an infinitely long correlation time. The system was excited by an infinitely short laser pulse at time t = 50f s and the coupling with the bath in Fig. 2.2 part (a) is chosen so, that we cannot notice any considerable relaxation in our 300f slong time window. The non–Markovian results with finite memory show clearly both the features of the analytical solution with infinite memory and those of Markovian theory. Since the relaxation is negligible the differences are reduced to irregularities in the oscillation pattern of the non–Markovian population probabilities.

As already stated, the presence of a time–independent partC_II⁽²⁾of the corre-lation function leads to a breakdown of the Markov–approximation. Therefore, it is impossible to compare retarded with non–retarded dynamics. Instead, we use the irregular modulation of the level populations as an indication for the presence of retardation effects. If for a given parameter set temperature is in-creased we expect the dominance of the irregular modulation of the P_M since C_II⁽²⁾ governs the whole dissipative dynamics.

Thus, in Fig.2.12 we present the time–dependence of the vibrational level population. To meet conditions near an impulsive excitation we describe the excitation by a laser–pulse of 5f s duration, and the correlation time t_c is set equal to 20f s. The magnitude J₀ of the spectral density (system reservoir coupling strength) has been chosen to observe non–Markovian effects in the dynamics even without the presence of the constant term C_II⁽²⁾. Accordingly

0 100 200 300

Figure 2.12: Vibrational population dynamics for the parameters: C_I+C_II⁽¹⁾= 2.2 × 10⁻⁴1/f s (Fourier-transformed correlation function), τp = 5f s, and t_c = 10f s. Part (a): T = 900K and Part (b): T = 10K. For the chosen pa-rameters the linear coupling to the bath would result in population dynamics not showing any non-Markovian effects similarly to part (b). The modulation of the populations by a frequency ω ≈ ω_vib/2 is clearly due to the constant contribution C_II⁽²⁾.

the curves in Fig. 2.12 part (a) valid for the case k_BT h/t¯ _c show a typical non–Markovian behavior (irregular modulation of the populations) surviving up to large times. In contrast, at low temperature the irregular modulation of P_M is only present at an early part of the dynamics. After a sufficiently long time (say from t ≈ 150f s) the low temperature dynamics in Fig. 2.12 part (b) exhibits the modulation with the frequency typical for Markovian dynamics. Consequently, this offers a clear distinction between Markovian and non–Markovian dynamics.

Another aspect of the given description is the fact that we can observe a non–Markovian behavior even for parameter sets for which the bilinear cou-pling, Eq. (1.14) or a description whereC_II⁽²⁾ has been removed would not show characteristic retardation effects. This indicates that the time–independent term C_II⁽²⁾ is the only source of the non–Markovian effects in this case. In Fig. 2.13 we present respective results for a duration of the laser pulse of 5f s,

0 100 200 300

Figure 2.13: The vibrational population dynamics for the parameters C_I + C_II⁽²⁾ = 2.2⁻⁴1/f s (Fourier-transformed correlation function), τp = 5f s, and t_c = 20f s. Part (a): T = 200K and Part (b): T = 10K. For the sake of clarity we only show the second and the first excited vibrational levels. Non-Markovian effects can also be observed for low temperature, but in the initial part of the dynamics only. The presence of the time-independent part of the correlation function prolongs these effects essentially.

and a correlation time tc = 10f s. In part (a) of this figure, where the tem-perature equals 900K one can observe an irregular modulation of the level populations over the whole time. In the low temperature case (T = 10K) as displayed in part (b), however, any irregular modulation is absent and the time–dependence can be well reproduced by means of corresponding Markov–

theory. The results obtained at T = 900K while dropping C_II⁽²⁾ (not shown) exhibit the same time–dependence as in the low temperature case. From what was written above we can draw the following general conclusion. Whenever an anharmonic coupling between active and passive coordinates becomes no-ticeably strong non–Markovian effects may become predominant. And, this would be even the case when a bilinear coupling with a comparable spectral density does not lead to any retardation effect in the course of the dissipative time–evolution.