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Eur. Phys. J. D 23, 279–284 (2003) DOI: 10.1140/epjd/e2003-00033-9 THE EUROPEAN PHYSICAL JOURNAL D Spontaneous emission of an excited two-level atom without both rotating-wave and Markovian approximation C.-Q. Cao1,a , C.-G. Yu1 , and H. Cao2 1 2 Department of Physics, Peking University, Beijing 100871, P.R. China Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208-3112, USA Received 10 July 2002 / Received in final form 12 November 2002 c EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2003 Published online 4 February 2003 – Abstract. The spontaneous emission of an excited atom is analyzed by quantum stochastic trajectory approach without both rotating-wave approximation and Markovian approximation. The atom finite size effect is also taken into account. We show by an example that the correction due to the counter-rotating wave term is rather small, even for the largest atomic number of real nuclei. PACS. 42.50.Ct Quantum description of interaction of light and matter; related experiments – 42.50.Lc Quantum fluctuations, quantum noise, and quantum jumps 1 Introduction The spontaneous emission of an excited atom is an old question of quantum theory. The first important progress in this respect is the proposition and evaluation of the Einstein A coefficient γA , which gives the spontaneous emission rate. It is well known that γA is proportional to d2 , where d denotes the electric-dipole transition moment. The next important progress is the formulation of Weisskopf-Wigner theory. It takes γA as the percentage decay rate of the upper-level population N2 during the whole emission process, leading to an exponential decay of N2 and consequently a Lorentzian line profile in the atomic spectrum. This treatment actually contains two approximations. One is the neglect of the finite size effect on the electric-dipole transition along with the omission of the contribution from all possible higher multipole transitions, caused by the replacement of the plane wave factor eik·x by 1 in the transition matrix element evaluation. Another is the so-called Markovian approximation on the reduced equations of the atomic variables derived by eliminating the photon variables. They are differential-integral equations with correlation functions to denote the memory effect. The Markovian approximation neglects this memory effect and hence changes them to simple differential equations. The Weisskopf-Wigner theory applies well for the emission of outer-shell electron or inner-shell electron of light nuclei. But in the case of inner-shell electron emission of heavy nuclei, it may lead to obvious deviation, since (1) ω0 2 is proportional to Zeff (where Zeff e is the charge of the atomic core felt by the emitting electron), and the radius a of the electron cloud is proportional to 1/Zeff , leading to a e-mail: [email protected] a linear increase of k0 a = ω0 a/c with Zeff . This makes the pointlike electric-dipole approximation (neglecting the factor eik·x in the integral) poorer for large Zeff . (2) During the emission process, the frequency of the emitted photon does not need to be ω0 (such photon is the socalled virtual photon) and the finite size correction will become larger for large ω. As to the Markov approximation, in the usual estimation, the characteristic correlation time is of order 2π/ω0 , while the decay time of the atomic variable is of order 1/γA . Hence, the validity of the Markovian approximation requires 2π/ω0 much smaller than 1/γA , namely 2πγA /ω0 1. It can be shown 2 , hence large Zeff disadvanthat γA /ω0 increases as Zeff tages the Markovian approximation. Later on, some people considered the correction to the Weisskopf-Wigner theory [1–4]. But, as far as we know, all of them applied the Laplace transform to solve the resultant differential-integral equation, and various kinds of approximations were used in the inverse transform. Some papers even take the radiating atom as a pointlike electric dipole, so that the corresponding correlation spectrum diverges linearly with ω, and in the subsequent treatment an artificial cutoff of frequency is needed. Besides, majority of them only considers the case of Z = 1. In a preceding paper [5], two of us with other coworkers restudied this problem by a totally different approach. The stochastic quantum trajectory formulation [6] is applied for this investigation, and the non-Markovian correction to the decay of upper-level population is taken into account by introduction of additional fictitious oscillators [7,8]. However, the counter-rotating wave interaction is still not taken into account. Now we will release this limitation to include the counter-rotating wave term in the interaction, since its effect also becomes larger for stronger coupling [9]. In this case there are two correlation 280 The European Physical Journal D spectra, hence more fictitious oscillators are needed to simulate them. Moreover, in the case of the rotating-wave approximation, the total quanta of the fictitious oscillators is either one or zero, while the counter-rotating wave interaction puts no limitation on the quanta numbers. Both of these will increase the amount of numerical calculation. In Section 2 the relevant differential-integral equation for the atom operators is given, with the two correlation spectra explicitly derived. The same hydrogen-like example as that of reference [5] is numerically studied for comparison. In Section 3 the quantum stochastic trajectory analysis [6] of the enlarged system [7,8] is carried out to give the time decay of the upper-level population. We see by this example that the correction due to counterrotating wave term is rather small (except the end stage of decay) even for the largest atomic number Z of hydrogenlike atom, as conjectured by the usual estimation. 2 The spectra of correlation functions for spontaneous emission of a two-level atom The A · P type Ĥint for the spontaneous emission of a two-level atom is known as Ĥint (t) = i k,j gkj σ̂+ (t)âkj (t) − field as reservoir, the resultant differential-integral equations are given by d σ̂− (t) = dt t [u1 (t − t ) − ū1 (t − t )]σ̂3 (t)σ̂− (t )dt 0 t + e2iω0 t [u2 (t − t ) − ū2 (t − t )]σ̂3 (t)σ̂+ (t )dt 0 − σ̂3 (t) Σ̂1 (t) + Σ̂2† (t) (3a) t d σ̂3 (t) = −2 [u1 (t − t ) − ū1 (t − t )]σ̂+ (t)σ̂− (t )dt dt 0 t − e2iω0 t [u2 (t − t ) − ū2 (t − t )]σ̂+ (t)σ̂+ (t )dt 0 + 2σ̂+ (t) Σ̂1 (t) + Σ̂2† (t) + h.c. (3b) where σ̂3 is the atom population-difference operator and Σ̂j (t)’s are fluctuation operators defined in equation (5). There are totally four correlation functions in the above differential-integral equations: u1 (t − t ) = |gkj |2 e−i(ω−ω0 )(t−t ) kj ∞ ∗ gkj σ̂− (t)â†kj (t) ∗ σ̂− (t)âkj (t) +ḡkj σ̂+ (t)aˆ† kj (t) − ḡkj ≡ R1 (ω)e−i(ω−ω0 )(t−t ) dω, (1) ū1 (t − t ) = |ḡkj |2 ei(ω+ω0 )(t−t ) kj ∞ with counter-rotating wave terms included, where âkj (â†kj ) is the photon annihilation (creation) operator of mode (k, j), σ̂± are atom-level change operators (σ̂+ corresponds to upward change and σ̂− downward change), gkj and ḡkj are coupling constants for rotating-wave term and counter-rotating wave term respectively, gkj e =− m 2π εkj · Gk , V kc eik·x Ψ2† (x)∇Ψ1 (x)d3 x, 2π e εkj · Ḡk , =− m V kc (2a) ≡ Ḡk = R1 (ω)ei(ω+ω0 )(t−t ) dω, (4b) 0 u2 (t − t ) = gkj ḡkj ei(ω−ω0 )(t−t ) kj ∞ ≡ ū2 (t − t ) = 0 ≡ R2 (ω)ei(ω−ω0 )(t−t ) dω, (4c) gkj ḡkj e−i(ω+ω0 )(t−t ) kj ∞ Gk = ḡkj (4a) 0 R2 (ω)e−i(ω+ω0 )(t−t ) dω, (4d) 0 (2b) e−ik·x Ψ2† (x)∇Ψ1 (x)d3 x in which εkj is the polarization vector of photon mode (k, j), Ψ2 (x) and Ψ1 (x) are the upper level and lower level wave function respectively. They are twocomponent spinor functions. It is seen from equation (2) that when Ψ2 (x) and Ψ1 (x) have the same parity, Ḡk = −Gk ; when Ψ2 (x) and Ψ1 (x) have the opposite parity, Ḡk = Gk ; and similar relations for gkj and ḡkj . The dynamical equations for both atom and photon operators are readily deduced from Ĥint . Eliminate the operators âkj and aˆ† kj by taking the electromagnetic (e.m.) since |gkj |2 = |ḡkj |2 . The correlation spectrum R1 (ω) is expressed by R1 (ω) = V ω2 (2π)3 c3 = e ω 4π 2 m2 c3 2 dΩk |gkj |2 j dΩk |Gk |2 − |nk · Gk |2 . (4e) Similar expression for R2 (ω) can be written out directly. The nk in equation (4e) is the unit vector in the direction of k. We see that there are two spectral functions R1 (ω) and R2 (ω), not just only one spectral function as in the references [1–5]. C.-Q. Cao et al.: Spontaneous emission of an excited two-level atom The two quantum fluctuation forces are expressed by Σ̂1 (t) = gkj âkj (0)e−i(ω−ω0 )t , (5a) kj Σ̂2† (t) = kj ḡkj â†kj (0)ei(ω+ω0 )t . (5b) As a first step we calculate the coupling constant Ḡk and Gk . For simplicity, we omit the spin of the electron. Substituting the wave function of hydrogen-like atom Ψ1 (x) = f1 (r)Yl1 m1 (θ ϕ), Ψ2 (x) = f2 (r)Yl2 m2 (θ ϕ) (6) and also the expansion formula of plane wave by spherical waves into equation (2a) and making use of the formula of ∇Ψ1 (x), one gets Gk as in [5] 4π(2l2 + 1) il m2 √ Gk = (−1) Ylm (θk ϕk ) 2l1 + 1 2l + 1 lmµ ×[− (l1 + 1)(2l1 + 3){Al (ω) − l1 Bl (ω)} ×C(l1 + 1, 1, l1 ; m1 − µ, µ, m1 ) ×C(l1 + 1, l2 , l; m1 − µ, −m2 , m) ×C(l1 + 1, l2 , l; 0, 0, 0) + l1 (2l1 − 1){Al (ω) +(l1 + 1)Bl (ω)}C(l1 − 1, 1, l1 ; m1 − µ, µ, m1 ) ×C(l1 − 1, l2 , l; m1 − µ, −m2 , m) ×C(l1 − 1, l2 , l; 0, 0, 0)]nµ (7) in which C(l1 + 1, 1, l1 ; m1 − µ, µ, m1 ) etc are ClebaschGordan coefficients, nµ (µ = +1, 0, −1) are spherical bases, 1 n+1 = √ (n1 + in2 ), 2 n0 = n3 , 1 n−1 = √ (n1 − in2 ) 2 and ω df1 (r) r f2 (r) dr, c dr 0 ∞ ω r f2 (r)f1 (r)dr. Bl (ω) = rjl c 0 Al (ω) = ∞ r2 jl (9a) (9b) The summation in equation (7) actually contains only finite terms because of the angular momentum addition rule. For the first term in the square brackets, l is restricted in the range |l2 − l1 − 1| ≤ l ≤ l2 + l1 + 1, and for the second term, in the range |l2 − l1 + 1| ≤ l ≤ l2 + l1 − 1. The Ḡk can be obtained from equation (7) simply by the substitution Ylm (θk ϕk ) → (−1)l Ylm (θk ϕk ). Substituting Gk and the formula µ=1 4π Y1µ (θk ϕk )n∗µ nk = 3 µ=−1 into equation (4e) and carrying out the integration over θk and ϕk , one will obtain the first spectral function R1 (ω). The spectrum R2 (ω) can be calculated similarly. To see explicitly these spectral functions, we consider the same example of hydrogen-like atom studied in reference [5]: Ψ1 (x) and Ψ2 (x) carry quantum numbers (n1 = 1, l1 = 0) and (n2 = 2, l2 = 1, m2 = 1) respectively with the expression 4 1 −r/a1 Ψ1 (x) = √ N1 e , N1 = , (12) a31 4π 4 −r/a2 Y11 (θ ϕ), N2 = · (13) Ψ2 (x) = N2 re 3a52 In this simple case, the gradient of Ψ1 (x) may be evaluated directly. Substituting it into equation (2a) and carrying out the angular integration, one obtains [5] 4π N1 N2 X0 (ω)Y00 (θk ϕk )n−1 Gk = 3 a1 6 Y2,−2 (θk ϕk )n+1 + X2 (ω) 5 3 Y2,−1 (θk ϕk )n0 − 5 1 Y2,0 (θk ϕk )n−1 + (14) 5 where 2 2 ω 2a4 3 − ωc2a r e−r/a dr = X0 (ω) = r3 j0 , 2 2 3 c 0 1 + ωc2a (15) ∞ ω 4 ω 2 a2 8a c2 r e−r/a dr = X2 (ω) = r3 j2 (16) 2 2 3 c 0 1 + ωc2a (8) (10) with ∞ 1 1 1 = + · a a1 a2 (17) In this simple example, both Al (ω) and Bl (ω) are proportional to Xl (ω). In equation (14) the angular momentum l only takes the value 0 and 2, hence we know from equation (10) that in this example Ḡk = Gk , consistent with what we say below equation (2b). This is in turn implies the equality of the counter-rotating wave coupling constant ḡkj and the rotating-wave coupling constant gkj . Thus the correlation spectrum R2 (ω) is given by V ω2 R2 (ω) = (2π)3 c3 (11) 281 e2 ω = 4π 2 m2 c2 dΩk g 2kj j dΩk G2k − (nk · Gk )2 . (18) 282 The European Physical Journal D It is easy to show that dΩk G2k = 0. By further utilizing the coupling rulefor spherical harmonics, nk · Gk may 3 be expressed as2 l=1 cl Yl,−1 (θk ϕk ), hence we also have dΩk (nk · Gk ) = 0. Consequently we obtain R2 (ω) = 0. (19) This may be a quite general result [10]. From equations (4e, 11, 14), one gets the result for R1 (ω) as R(ω) used in reference [5] and also in references [3,4] R1 (ω) = ω γA , 2πω0 (1 + ω 2 a2 /c2 )4 (20) in which γA is the Einstein A coefficient, given by γA = 4ω03 |d21 |2 /3c3 and in our example the transition dipole moment d21 takes its absolute value as √ 64 2 |d21 | = ea, (21) 81 with e denoting the magnitude of electron charge. Actually there is only one parameter in the R1 (ω), since ω0 may be expressed by a as ω0 = 1 (E2 − E1 ) = , 6ma2 (22) and the two atomic radii a1 and a2 are related to a by multiplying constants 3/2 and 3 respectively. Since a is proportional to the atomic number 1/Z, we see from equation (22) that ω0 is proportional to Z 2 . The relation between γA /ω0 and a is obtained from equations (21, 22), and may be expressed by γA 8 = ω0 3 32 243 2 e2 c 3 a2B , a2 64 81 2 e2 c (23a) 3 Z 2, (23b) hence the value γA /ω0 = 10−3 corresponds to Z = 157, somewhat larger than the upper limit of the real nuclei. A profile of R1 (ω) is shown in Figure 1 of reference [5], in which the dimensionless parameter γA /ω0 (instead of a) in R(ω) is taken as 0.001. In case the atom is taken as a pointlike particle, the factor eik·x in the integrand of Gk is omitted, Gk is then independent of k. Hence, R1 (ω) will become linear in ω as can be seen from equation (4e), and the coefficient is just γA /2πω0 . We will call this spectrum as pointlike dipole spectrum: R1D (ω) = γA ω. 2πω0 ∞ 1 R1 (ω) C = πR1 (ω0 ) − i ℘ dω , 2 ω − ω0 0 ∞ 1 R1 (ω) C̄ = i dω , 2 ω + ω0 0 (25a) (25b) where ℘ denotes taking the principal value. The imaginary parts of C nd C̄ contribute just an level-shift which will be canceled out in equation (3b) because of the h.c. term. Taking the expectation value of equation (3b) and noting the upper level population N̂2 (t) = (1 + σ̂3 (t))/2, one gets immediately d N̂2 (t) = γ N̂2 (t) dt (26a) with where aB is the Bohr radius. The dependence of γA /ω0 on the atomic number Z is given by γA 1 = ω0 6 Actually in our example only electric dipole emission is allowed by the selection rule of angular momentum, thus the factor (1 + ω0 a/c)−4 in R1 (ω) just expresses the finite size effect on the radiative electric dipole transition. In this example, both u2 (t − t ) and ū2 (t − t ) are equal to zero, only two correlation functions u1 (t − t ) and ū1 (t − t ) remain. The former comes from the rotatingwave interaction and the latter from the counter-rotating wave interaction. In case we make the Markovian approximation on them, namely approximate them by Cδ(t − t ) and C̄δ(t − t ) respectively to neglect the memory effect, the constants C and C̄ will be determined by integration of equations (4a, 4b) respect to t from t to ∞ (note that u1 and ū1 are defined only in the region t > t ). The results so obtained is (24) It also corresponds to R1 (ω) with ωa/c = 0 in the denominator. γ = 2πR1 (ω0 ) = γA · (1 + ω02 a2 /c2 )4 (26b) We see that γ is different from γA by the finite size correction factor (1 + ω02 a2 /c2 )−4 , which in our example is about 0.729. We would like to point out that the counter-rotating d wave interaction will have no effect on dt N̂ (t) in the case Markovian approximation is made, even if R2 (ω) = 0. In the next section we will study the non-Markovian approach with the counter-rotating term included. 3 Quantum stochastic trajectory approach to the spontaneous emission We have shown by equation (20) that the two-level atom with large atomic number Z has an evident nonMarkovian reservoir in its spontaneous emission. Hence we will, as described in references [7,8], introduce N additional fictitious harmonic oscillators, which interact with the atom in the same form as photons, to form an expanded system. This expanded system thus has the C.-Q. Cao et al.: Spontaneous emission of an excited two-level atom Hamiltonian Ĥ = N N 1 ωj α̂†j α̂j + i gj σˆ+ αˆj ω0 σˆ3 + 2 j=1 j=1 − gj∗ σˆ− αˆj † + ḡj σˆ+ αˆj † − ḡj∗ σˆ− αˆj where Ĥ is expressed by equation (27). The collapse operators acting on |Ψ (t) are j = 1, 2, ..., N. (30) Ĉj = Γj ∆tα̂j , (27) in which the energies of the two atom-levels are taken as ± 21 ω0 respectively, α̂j and α̂†j are annihilation and creation operators of the jth oscillator, ḡj = ±gj for the relative parity of Ψ2 and Ψ1 equal to ∓1 respectively. Each of these oscillators, on the other hand, is assumed to interact with its own reservoir with no memory, hence the expanded system is Markovian. In terms of Langevin equations in Heisenberg picture the dynamics of the above system is expressed by N d gj αˆj (t) σ̂− (t) = −iω0 σ̂− (t) − dt j=1 + ḡj α̂†j (t) σ̂3 (t), (28a) N d gj αˆj (t) + ḡj α̂†j (t) σ̂+ (t) σ̂3 (t) = 2 dt j=1 +2 N gj∗ αˆj † (t) + ḡj∗ α̂j (t) σ̂− (t), (28b) j=1 d α̂j (t) = −iωj α̂j (t) − gj∗ σ̂− (t) + ḡj σ̂+ (t) dt 1 − Γj α̂j (t) − F̂j (t) 2 with 283 (28c) F̂j (t) = 0, F̂i (t)F̂j (t ) = 0, R R F̂i (t)F̂j† (t ) = δij Γj δ(t − t ). (28d) R The last two terms of equation (28c) are the usual Markovian dissipation term and fluctuation term contributed by the reservoir of jth oscillator. When the formal solution of equation (28c) is used to eliminate the variables α̂j and α̂†j in equations (28a, 28b), the reduced equation for σ̂+ (t) and σ̂3 (t) will have nonMarkovian damping terms and non-Markovian fluctuation terms, since each fictitious oscillator contributes a term with Lorentzian type spectrum. The next step is to select parameters gj , ḡj , ωj and Γj to simulate the original damping terms and fluctuation terms. Now there are two spectral functions, not just one spectral function as in rotating-wave approximation, the simulation task will become more complicate. As soon as the simulation task is accomplished, one may go back to the master-equation formulation and treat it by quantum stochastic trajectory approach. The nonHermitian Hamiltonian is now taken as i Ĥnh = Ĥ − Γj α̂†j α̂j , (29) 2 j To proceed with concrete analysis and compare with the result of rotating-wave approximation, we return to the example with Ψ2 (x) and Ψ1 (x) described by equation (12). In reference [5] only two fictitious oscillators are introduced to simulate the spectrum R(ω) ≡ R1 (ω) of this example, the corresponding simulation spectrum R(s) (ω) fits pretty well with R(ω) in a large region ω/ω0 ≥ 0.5, which includes the important region around ω/ω0 = 1. But the R(s) (ω) drops down less slowly than R(ω) when ω/ω0 → 0. At the origin, R(s) (ω)/R(s) (ω0 ) is still about 0.36 while R(ω) drops to zero (see Fig. 1 of Ref. [5]). Now there is an additional spectrum R2 (ω), we have to use one set of N oscillators to simulate both of them, namely to make N |gj |2 Γj 1 (s) , (31a) R1 (ω) ∼ = R1 (ω) ≡ 2π j=1 (ω − ωj )2 + 14 Γj2 N gj g¯j Γj 1 (s) R2 (ω) ∼ · (31b) = R2 (ω) ≡ 2π j=1 (ω − ωj )2 + 14 Γj2 In general, the simulation is not easy to be carried out. But in the case R2 (ω) = 0, we may simply take four fictitious oscillators with following parameters: 1 g1 = √ 0.011ω0, 2 1 g2 = √ 0.018ω0, 2 i g3 = √ 0.011ω0, 2 i g4 = √ 0.018ω0, 2 Γ1 = 1.3ω0 , ω1 = ω0 , (32a) Γ2 = 2.4ω0 , ω1 = 1.85ω0, (32b) Γ3 = 1.3ω0 , ω3 = ω0 , Γ4 = 2.4ω0 , ω4 = 1.85ω0. (32d) (32c) It is easy to check that the above set of parameters (s) makes R2 (ω) = 0 no matter the relative parity be(s) tween Ψ2 and Ψ1 is positive or negative, while R1 (ω) (s) remains unchanged from the R (ω) of reference [5]. Since the numerical evolution takes place over discrete times with a small time step ∆t, the wave function |Ψ (t) is represented by a sequence |Ψ (tn ) with tn = n∆t. Given the value of |Ψ (tn ), the next one |Ψ (tn+1 ) is determined by the following algorithm: (1) evaluate the four kinds of collapse probabilities during interval (tn , tn−1 ): Pj (tn ) = Ψ (tn ) Ĉj† Ĉj Ψ (tn ) = Γj Ψ (tn ) α̂†j α̂j Ψ (tn ) ∆t, j = 1, 2, 3, 4 (33) in which ∆t should be small enough to make P1 (tn ) and P2 (tn ) much small than 1; 284 The European Physical Journal D Fig. 1. The spontaneous decay of the upper level population with parameters (γA /ω0 = 1×10−3 ). The curve I represents the result without both rotating-wave and Markovian approximation, the curve II represents the result with rotating-wave approximation but without Markov approximation as calculated in reference [5], the curve III represents the Weisskopf-Wigner result. (2) generate four random numbers rj (j = 1, 2, 3, 4) which have uniform probability distribution among the interval (0, 1); (3) compare Pj (tn ) with rj and derive |Ψ (tn+1 ) according to the following rule: if Pj (tn ) < rj and for all j, i e− Ĥnh ∆t |Ψ (tn ) |Ψ (tn+1 ) = ; † i Ψ (tn )|e (Ĥnh −Ĥnh )∆t |Ψ (tn ) (34) may neglect this term for stable nuclei without introducing obvious error. We note that the large difference between curve II and curve III comes from two respects as mentioned in Section 1: the first one concerns the finite size correction, corresponding to the replacement of γA by γ with γ = 0.729 γA. This correction is calculated analytically and should be reliable. The second correction is the memory effect corresponding to the non-uniform of R(ω), namely the difference of R(ω) from R(ω0 ). This part is evaluated numerically and has some errors from simulation and from the pseudo character of the random numbers generated by program. Both corrections make the decay slower. As shown in reference [5], the first correction is much larger than the second. As to the difference between curves I and II which comes from the effect of counter-rotating wave coupling, it is totally due to the non-Markovian contribution since in the present case R2 (ω) = 0, γ remains unchanged as specified in the end of Section 2. This work is supported by National Natural Science Foundation of China (project number 19774004), and also by the International Program of National Science Foundation of USA (ECS-9800068). References 1. 2. 3. 4. 5. if Pl (tn ) > rl for some one index l and all other Pm (tn ) ≤ rm , 6. Ĉl |Ψ (tn ) |Ψ (tn+1 ) = · Ψ (tn )|Ĉl† Ĉl |Ψ (tn ) 7. 8. (35) 9. In the practical calculation, ∆t should be taken sufficiently small so that the case with any two (or more) Pj (tn )’s larger than the corresponding rj in the same interval (tn , tn + ∆t) will not happen. By carrying out the over and over steps specified above 1 1 from the initial state |0, 0, 0, 0 where is the atom 0 0 state in the upper level and |0, 0, 0, 0 denotes the state of four oscillators with their number of quanta all being zero, we get a quantum stochastic trajectory of the Monte Carlo wave function |Ψ (t). The expectation value of a given operator respect to |Ψ (t) is then calculated. Finally an ensemble average of 400 trajectories is carried out to give the resultant curve. The population on the upper level N̂2 (t) is shown in Figure 1. We see that the correction of counter-rotating wave term to the non-Markovian result obtained in reference [5] is quite small even Z is as large as 157. Hence one 10. P.L. Knight, P.W. Milonni, Phys. Rev. Lett. 56, 275 (1976) P. Carrazana, G. Vetri, Nuovo Clim. B 55, 191 (1980) P.T. Robisco, Phys. Lett. A 100, 407 (1984) J. Seke, W. Herfort, Phys. Rev. A 38, 833 (1988); Phys. Rev. A 40, 1926 (1989) C.-Q. Cao, W. Long, J. Wei, H. Cao, Phys. Rev. A 64, 043810 (2001) See for example, H.J. Carmichael, An Open System Approach to Quantum Optics (Springer-Verlag, Berlin, 1993), Lecture 7 A. Imamoglu, Phys. Rev. A 50, 3650 (1994) P. Stenius, A. Imamoglu, Quant. Semiclass. Opt. 8, 283 (1996) W.H. Louisell, Coupled mode and parametric electronics (Wiley, New York, 1960) We may show thatR2 (ω) = 0, provided m1 = m2 . First, see the integral dΩk G2k . Substituting equation (7) into it and carrying out the integration over the angular variables, we get a series over (l, m, n), and for each (l, m, n) there are four terms proportional to [Cl1 +1,l2 ,l;m1 −µ,−m2 ,m Cl1 +1,l2 ,l;m1 +µ,−m2 ,−m ], [Cl1 +1,l2 ,l;m1 −µ,−m2 ,m Cl1 −1,l2 ,l;m1 +µ,−m2 ,−m ], [Cl1 −1,l2 ,l;m1 −µ,−m2 ,m Cl1 +1,l2 ,l;m1 +µ,−m2 ,−m ], [Cl1 −1,l2 ,l;m1 −µ,−m2 ,m Cl1 −1,l2 ,l;m1 +µ,−m2 ,−m ] respectively. One may easily check that all of them unless m1 = m2 . Similarly, the integral will be zero (nk · Gk )2 dΩk contains four terms proportional to [Cl1 +1,l2 ,l;m1 −µ,−m2 ,m Cl1 +1,l2 ,l ;m1 −µ −m2 ,−m−µ−µ ], [Cl1 +1,l2 ,l;m1 −µ,−m2 ,m Cl1 −1,l2 ,l ;m1 −µ ,−m2 ,−m−µ−µ ], [Cl1 −1,l2 ,l;m1 −µ,−m2 ,m Cl1 +1,l2 ,l ;m1 −µ ,−m2 ,−m−µ−µ ], [Cl1 −1,l2 ,l;m1 −µ,−m2 ,m Cl1 −1,l2 ,l ;m1 −µ ,−m2 ,−m−µ−µ ] for each index-set (l, m, µ, l , µ ), they also will be zero unless m1 = m2