在光子帶隙中原子的自發(fā)衰減

黃江　謝欽 邱彩虹

摘要 研究了一個二能級原子在具有雙洛倫茲譜密度的光子帶隙中激發(fā)態(tài)布居的自發(fā)衰減動力學(xué)行為.通過對比自發(fā)衰減動力學(xué)行為在馬爾科夫近似條件下和非馬爾科夫條件下的演化特性，發(fā)現(xiàn)了在這兩種條件下演化動力學(xué)有著明顯區(qū)別.非馬爾科夫記憶效應(yīng)使得流入環(huán)境中的部分信息回復(fù)至系統(tǒng)中來，從而導(dǎo)致了原子布居的震蕩.另外，研究結(jié)果表明耦合強度與衰減率有著密切聯(lián)系.

關(guān)鍵詞 光子帶隙；雙洛倫茲譜；非馬爾科夫

Quantum communication and information has attracted much attention of many physicists working in the area of quantum mechanics for a few decades[16]. Due to the development of experiments， the corresponding theoretical studies are also developed rapidly. As we know that every realistic quantum systems would interact with its environment， which causes the dissipation and decoherence[712]. Dissipation and decoherence in open quantum systems can be described by the standard techniques of the theory of quantum Markovian process with Lindbald structure quantum master equation. One believe that the Markovian process happens when the systemreservoir coupling is weak. However， with the rapid development of experimental technology， the strong coupling is considered in some works. The process with strong coupling is called as nonMarkovian process. Recently， the nonMarkovian dynamics of the open quantum system were investigated within exact master equation[1320]. In this paper， we consider an atom spontaneous decay regulation in the photonic band gap. We find that the population regulation of the excited state is much different under Markovian approximation and nonMarkovian condition， the corresponding physical explanation is given， too. At last we conclude the paper briefly.

This work takes our attention on the spontaneous decay of a qubit under a photonic band gap， the twolevel atom is captured in a photonic band gap which forms with multimodes. In order to build the model and solve the evolution problem， we much take the population of the atom and decay rate into consideration. The total Hamiltonian is

H=HS+HE+VI=ω0σ+σ-+∑kωkakak+∑k（gkakσ-g*kakσ+），

where ω0 is the transition frequency， σ+ and σ- are the raising and lowering operators， ωk respects the environment multimodes， ak and ak are the creation and annihilation operators， gk is the coupling constants， respectively. In the interaction picture， the Hamiltonina is

湖南師范大學(xué)自然科學(xué)學(xué)報第38卷

第1期

黃江等：在光子帶隙中原子的自發(fā)衰減

HI（t）=σ+（t）A（t）+σ-（t）A+（t），

with σ±（t）=σ±e±iω0t， A（t）=∑kgkake-iωkt.

The initial state of the atom can be written as：

|ψ0〉=c0|〉A(chǔ)|0〉E+c1|1〉A(chǔ)|0〉E+∑kck|0〉A(chǔ)|k〉E，

here |0〉A(chǔ) is the state of atom and |0〉E is for environment. Evolves after time t into state：

|ψt〉=c0|0〉A(chǔ)|0〉E+c1（t）|1〉A(chǔ)|0〉E+∑kck（t）|0〉A(chǔ)|k〉E，

The state |0〉E denotes the vacuum state of the environment and |k〉E=bk|0〉E the state with one particle in mode k. In the interacting picture， the state |ψt〉 of the total system obeys the Schrdinger equation：

ddt|ψt〉=-iHI|ψt〉，（1）

The amplitude c0 is constant in time since HI（T）|0〉A(chǔ)|0〉E=0， with the amplitudes c1 and c2 are time dependent， the time evolution of these amplitudes are easily derived from the differential equation （1）：

1（t）=-i∑kgkei（ω0-ωk）tck（t），（2）

k（t）=-ig*ke-i（ω0-ωk）tc1（t），（3）

Assuming that ck（0）=0， i.e. there are no photons in the initial state. We solve equation （3） and insert the solution into equation （2） to get a closed equation for the coefficient c1（t）：

1（t）=-∫t0f（t-t1）c1（t）dt1，（4）

where the kernel f（t-t1）=∫dωJ（ω）ei（ω0-ω）（t-t1） is related to the spectral density J（ω） of the reservoir. We now can solve the equation （4） just by a Laplace transformation and a specific

form of the spectral density J（ω）.

For solving the probability amplitudes c0， c0 and ck， we can get the timeevolution of the reduced density matrix of the qubit by constructing an exact dynamical map：

ρA=trE[|ψt〉〈ψt|]=|c1（t）|2c*0c1（t）

c0c*1（t）|c1（t）|2+∑k|ck（t）|2=

|c1（t）|2c*0c1（t）

c0c*1（t）1-|c1（t）|2.

Differentiating this expression with respect to time and recalling that ck（0）=0 means there are in the vacuum state， we get：

ddtρA（t）=ddt|c1（t）|2c*01（t）

c0*1（t）-ddt|c1（t）|2，（5）

After computation， equation （5） can be written as：

ddtρA（t）=iS（t）[σ+σ-，ρA（t）]+γ（t）[σ-ρA（t）σ+-12σ+σ-ρA（t）-12ρA（t）σ+σ-]，（6）

where

S（t）=-2 1（t）c1（t），

γ（t）=-2 1（t）c1（t），

The exact master equation （6） is of the reduced system dynamics. Here， S（t） acts as a timedependent Lamb shift and γ（t） is a timedependent decay rate. In this paper， we just care about the decay rate γ（t） here.

For that the environment is the photonic band gap， we require the spectral density is a doubleLorentzian form

J（ω）=Ω202π[W1Γ1（ω-ω0）2+（Γ1/2）2-W2Γ2（ω-ω0）2+（Γ2/2）2]，

where Ω20 is the overall coupling strength， Γ1 is the bandwidth of the background， Γ2 is the width of the gap， W1 and W2 are the relative strength of the background and the gap， respectively. This two parameters have a relation as W1-W2=1.

Substitute J（ω） in f（t- t1） and use the residue theorem， we can solve the corresponding function f（t- t1） as