In previous chapter, we had seen the perturbation of higher order need calculate many terms. We can see from Eqs.(2.73)-(2.78) that the second order expansion of the master equation for effective density matrix ˜χs contains near 30 terms, and the fourth order expansion of master equation contains near 600 terms. We can expect that the sixth order contains at least thousands of terms, and the calculation would become very tedious and complicated. Therefore, we provide an easier method, which can improve the accuracy of the perturbation time-dependent decay rates without really going to calculate the higher order contributes of Kn and In.
The methos is Aitken’s delta-squared method. It is a numerical method used for accelerate the rate of convergence of the sum of a series. Aitken’s delta-squared method can be described as follows. Suppose Sn =∑n
i=0Xi is a partial sum of Xi to the nth term of a slowly convergent sequence where exact result is achieved when n → ∞ . The new sequence Sn′ transformed by Aitken’s δ2 method will converges faster or closer to the exact result then Sndoes. The expression of the new sequence is form from Sn and previous two sequence Sn−1 and Sn−2 as
Sn′ = Sn− (Sn− Sn−1)2
Sn− 2Sn−1+ Sn−2. (4.17) In one case, we have calculate the decay rate up to fourth order to obtain γ4(t) =
∆γ2(t) + ∆γ4(t). If we set the 0th and 2nd order decay rates to be γ0(t) = 0 and γ2(t) = ∆γ2(t), we may apply Aitken’s δ2 method to find a new decay rate as
γ4′(t) = γ4(t)− (γ4(t)− γ2(t))2
γ4(t)− 2γ2(t) + 0. (4.18) Similarly, the effective 0th, 2nd and 4th order decay rates i0(t1, t2) = 0, i2(t1, t2) =
∆i2(t1, t2) and i4(t1, t2) = i2(t1, t2) + ∆i4(t1, t2). One can also apply Aitken’s δ2 method to find a new effective decay rate
i′4(t1, t2) = i4(t1, t2)− (i4(t1, t2)− i2(t1, t2))2
i4(t1, t2)− 2i2(t1, t2) + 0. (4.19)
In Fig. 4.1(a),we show the decay rates calculated by different methods for λ = 2.001γ0(in the weak-coupling region). The black sold line is exact decay obtained from Eq. (4.15), the blue dotted lien is the 2nd-order decay rate γ2(t), the green dot-dashed line is the 4th order decay rate γ4(t), and the red dashed line is decay rate γ4′(t) obtained by Aitken’s δ2 method . It is obvious that the 4th order perturbation result is better than the 2nd order one, and the Aitken’s δ2 method can improve the decay rate as the result obtained from it is closer to the exact result than the 4th order perturbation. In section 5.3, we apply Aitken’s δ2 method to perturbative master equation up to 4th order and then to obtain the two-time correlation function.
Figures4.1(b) - 4.1(d) show the effective decay rates with different value of t2. One can see that the decay time of the effective decay rate is about τB∼ λ−1 which is the bath correlation time. The strength of the effective decay rate dependent strongly on t2. When t2 is small, the strength of effective decay rate is also small.
When t2 increases, the strength of the effective decay rate also increase, but it would reach a steady state and will not increase any more at largest t2.
0 2 4 6 8 10
Figure 4.1: Time-dependent decay rates and effective rates obtained by different methods.
Chapter 5
Comparison between the exact
result and the perturbation results
With the various master equation for the reduced density matrix ˜Xs obtained, we can find the solution of ˜χs then trace the product of ˜σ+χ˜s(t1, t2) over the system state to obtain the two-time correlation function⟨σ+(t1)σ−(t2). In this chapter, we will show the time evolution of two-time correlation function⟨σ+(t1)σ−(t2)⟩ obtained using different methods.
To eliminate the oscillating factor of eiω0(t1−t2)and to make the time evolution be-haviors clearly the absolute value of two-time correlation function (|⟨σ+(t1)σ−(t2)⟩|) illustrated, we plot in all the figures shown in this Chapter.
The different methods and corresponding time evolution were shown in the fig-ures are summarized below. The first method is the perturbative mater equation approach for the reduced effective density matrix.The time-evolutions calculated us-ing Eq.(2.64) with different perturbaion order are presented. We denote K2withI2 in black dashed line as calculation using Eq.(2.64) with homogeneous and inhomo-geneous terms up to 2nd order, K4withI2 in green dotted line as with homogeneous terms up to 4th order and inhomogeneous terms up to 2nd order, K4withI2 in pur-ple solid line as with homogeneous and inhomogeneous terms up 4th order. We also plot the Markovain time evolution to 2nd order in blue dot-solid line as Markovian.
The second method is the exact direct evaluation by operator technique. The time evolution obtained by the exact result Eq.(3.11) in red solid line denoted as Exact.
Another result obtain by Eq.(3.13) that neglects the bath correlation between t < t2
and t > t2 but treats the reduce time evolution from t2 to t1 exactly is plotted in pink solid line and demoted as Exact QRT
The initial states of the environments is in the zero-temperature vacuum state,
∑
k|0⟩k, and the initial system state is set to be|ϕ(0)⟩ = √12(|0⟩s+|1⟩s)
5.1 Numerical result in the weak coupling region of λ > 2γ
0Figure 5.1: Two-time correlation functions of|⟨σ+(t1)σ−(t2)⟩| obtained by different methods with λ = 2.001γ0 for different value of (a) t2 = 4γ0 ,(b)0.1γ0 respectively.
In Fig. (a), when we consider perturbation of homogeneous and inhomogeneous term up to 4th order, the result is better then exact QRT case, even 2nd order perturbation it is also better then exact QRT in short time region. In Fig. (b), if the t2 is not large enough do not have enough memory about the time before t < t2 (i.e. t2 < λ−1) , QRT is applicable . The initial condition of ˜Xs(t2) was obtained by exact operator method, it make the contributing of inhomogeneous terms to be clear.
In this section, we consider the region with λ > 2γ0 (referred to as the weak coupling region ). Specifically, we choose the cutoff frequency λ = 2.001γ0. In this region, the two-time correlation functions will decrease monotonically.
We can see from Fig.5.1(a) that the difference between the exact result and the result by the exact QRT method is obvious. The reasons is that the QRT that neglects the bath correlation between t < t2 and t > t2 does not consider the non-Markovain memory effect of the bath comes from t < t2 that may affect the system
dynamics in t > t2 .
Next, we compare the perturbation results in Fig.5.1(a). The two-time correla-tion funccorrela-tion obtained by perturbacorrela-tion method with homogeneous and inhomoge-neous term up to 4th order is closer to the result by the exact operator evaluation then the exact QRT, which demonstrates clearly the validity of the evolution equa-tion Eq.(2.64).
As expected, the result of K4withI4 is more accurate than the result of K2withI2. One can also observe that even the second order perturbation result with inhomo-geneous contribution is better than the exact QRT in the short time region. After γ0t > 3 the inhomogeneous contribution dies out a shown in Fig.4.1(c), the exact QRT result is then close to the exact result. The Markovain result also seem better than the exact QRT in the short time region. This is because Markovain result result assume a time-dependent decay rate γM ∼ γ2(t→ ∞), so it has a large decay rate then all other cases in the short time region.
The more high-order terms are considered, the more accuracy the results are.
However, to include the higher-order perturbation contribution require much more tedious calculations. An alternative scheme of Aitken’s δ2 method to improve accu-racy introduced in Sec.4.2, will be discussed in Sec.5.3.
The difference between K4withI4 and K4withI2 is that K4withI4 containing the 4th order contribution of the inhomogeneous terms. We can see from Fig.4.1(c) that the contribution of ∆i4(t1, t2) is very small. Furthermore,the bath correlation time or memory time is about τB ∼ λ−1, so the contribution of the inhomogeneous terms becomes less affect t = t1− t2 > λ−1
The difference between the Exact result and the Exact QRT result is not sig-nificant. For small t2, the inhomogeneous contribution from t < t2 is expected to be small. This can be seen from Fig.4.1(b) and Fig4.1(c) that the magnitude of the effecttive decay rate i(t) coming from inhomogeneous contribution for γ0t = 0.2 is about 5 times smaller than that for γ0t = 4 case. Thus we conclude that for t2 << λ−1 and for τB ∼ λ−1 << γ0−1, the memory effect of the bath coming from
the t < t2 to affect the system dynamics in t > t2 will be small and thus the QRT is valid in this case.