The spectrum of spontaneous emission in the two-level system can be evalu-ated through the two-time correlations functions ⟨σ+(t1)σ−(t2)⟩. The spectrum of emission photons is the double Fourier transform of⟨σ+(t1)σ−(t2)⟩ [4] Substituting the exact expression of Eq.(3.27) and Eq.(3.28) into Eq.(5.3), we ob-tain respectively the spectrum of the Jaynes-Cummings model in the weak coupling region (λ > 2γ0) as In Fig.5.4(a), show the spontaneous emission spectra for different value of λ in the weak coupling region. The peaks of spectra are at the system frequency ω0. The widths of the spectra are determined by cut-off frequency λ. Where the λ increase, the width becomes a little bit narrower. When λ→ ∞, the spectra reach
the Markovian result S(ω)∝ [(ω −ω0)2+ (γ0/2)2]−1 [4]. The only difference between the non-Markovian and the Markovian spectra is the width.
However, the spontaneous emission spectra are more interesting in the strong coupling. Fig.5.4(b), shows the spontaneous emission spectra for different value of λ in the strong coupling region (λ < 2γ0). Note that the vertical axis in Fig.5.4(b) is in logarithmic scale. When the values of the cutoff frequence decrease, the spectrum from a single-peak structure centered at ω = ω0 to a double-peak structure centered at ω = ω0 ±d2, where d =√
2γ0λ− λ2 There exists a critical cutoff frequence λc at which the second derivative of S(ω) at ω = ω0 is zero, i.e., d2dωS(ω)2 |ω0,λc = 0. When the cutoff frequence is smaller then λc ≃ 1.2γ0, the two-peaks structure starts to develop.
The peak structure of spectrum may be understood from the two-time correlation function. Figure 5.4(c) shows a typical time-evolution of Re⟨σ+(t1)σ−(t2)⟩ oscillating with the frequency ω0 in the weak coupling region. The monotonically decay of the envelope of the two-time correlation function explains the spontaneous emission spectra in this region at ω = ω0. In contrast the envelope of, a typical time-evolution for λ < λc shown in Fig.5.4(d) is modulated by cos(dt2). As a result, the spectrum exhibits a double-peak structure centered at ω = ω0±d2. Another point is the height of the emission spectrum at ω = ω0 remains the same independent of the values of cutoff frequency.
−30 −2 −1 0 1 2 3
S(ω) in weak coupling region
λ= 2.001 γ0
S(ω) in strong coupling region λ= 0.050 γ0
Figure 5.4: spontaneous emission spectra in arbitrary unit for different value of cutoff frequency (a) in the weak coupling region(λ < 2γ0) (b) in the strong coupling region(λ > 2γ0), and time evolutions of the two-time correlation function (c) in the weak coupling with λ = 2.001γ0 (d) in the strong coupling with λ = 0.05γ0
Chapter 6 Conclusion
We have derived in Chapter 2 the perturbative non-Markovian time-convolutionless master equation for reduced effective density matrix ˜χs(t) through the cumulants expansion. The master equation can be directly applied to calculate the two-time correlation functions. The master equation is only based on a few requirements, (1) the effective density matrix χs(t) satisfies von Neumann equation, (2) initial system-bath is factorized in t = 0, (3) knowing the initial condition ˜χ(t2) is known and (4) the perturbative expansion series converges. We inserted the general interaction Hamiltonian up to fourth order, it is useful for any kind of problems.
We have calculated in Chapter 3 an exact two-time correlation function for a many-mode Jaynes-Cummings model with a Lorentz spectral density at zero-temperature. The exact two-time correlation function can be used to check the va-lidity and applicable region of the master equation approach developed in Chapter 2. We focus that the exact result of the two-time correlation function guide different from that obtaining the exact QRT method that neglects the non-Markovain bath correlation between t < t2 and t > t2. From the exact result of the two-time correla-tion funccorrela-tion, we were able to find an exact master for the reduced effective density matrix ˜χs. This allows us to make direct comparison between the exact two-time correlation with that obtained perturbatively.
We have calculate the two-time correlation function using the perturbative mas-ter equation up to fourth order. Here, we have used Aitken’s δ2 method to improve the perturbation master. To go beyond that higher order is a heavy and tedious
task. The perturbation master equation with Aitken’s delta-squared method can slightly improve the result of the two-time correlation function.
The perturbation result up to fourth order agrees with the exact result in the weak coupling region. In the strong coupling region, the perturbation method is valid only for t << tdiv. The contribution from the inhomogeneous terms depends strongly on the value of t2. The smaller the value of t2 the smaller contribution from the inhomogeneous terms.
Finally, we derived spontaneous emission spectrum analytically. The spectrum shows dramatically different structure in the weak and the strong coupling region.
In the weak coupling region, the spectrum has only one peak located at ω = ω0 and the spectrum width is determined by the cut-off frequency. In the strong coupling region, there exist a critical cut-off frequency λc below which the spectrum goes from a one-peak structure to a two-peak structure with peak centers located at ω = ω0± d/2.
In summary, the two-time correlation functions are important physical quantity.
They can provide additional information about the system, which the single-time ex-pectation values can not provide. We believe that we are the first group to calculate the exact two-time correlation function and the spontaneous emission spectrum for the many-mode JC model. The calculations provide significant insight into how the non-Markovian memory effect influences the behavior of the two-time correlation functions.
Although it is commendable to calculate the exact two-time correlation function, but not many problems can have the exact solutions. The perturbation master equation approach developed in this thesis can be applied to calculate the two-time correlation functions perturbatively for the non-Markovain open (disspative) quantum systems. We believe that this master equation approach that generalizes the QRT to the non-Markovain case will find broad applications in many different branches of physics.
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