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,andH.Cao2 1 Department of Physics, Peking University, Beijing 100871, P.R. China 2 Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208-3112, USA Received 10 July 2002 / Received in final form 12 November 2002 Published online 4 February 2003 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 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 a linear increase of k0a = ω0a/c with Zeff.Thismakes the pointlike electric-dipole approximation (neglecting the ik·x The spontaneous emission of an excited atom is an old factor e in the integral) poorer for large Zeff.(2)Dur- question of quantum theory. The first important progress ing the emission process, the frequency of the emitted in this respect is the proposition and evaluation of the photon does not need to be ω0 (such photon is the so- Einstein A coefficient γA, which gives the spontaneous called virtual photon) and the finite size correction will emission rate. It is well known that γA is proportional become larger for large ω. As to the Markov approxi- to d2,whered denotes the electric-dipole transition mo- mation, in the usual estimation, the characteristic cor- ment. The next important progress is the formulation of relation time is of order 2π/ω0, while the decay time of Weisskopf-Wigner theory. It takes γA as the percentage the atomic variable is of order 1/γA. Hence, the valid- decay rate of the upper-level population N2 during the ity of the Markovian approximation requires 2π/ω0 much smaller than 1/γA,namely2πγA/ω0 1. It can be shown whole emission process, leading to an exponential decay 2 of N2 and consequently a Lorentzian line profile in the that γA/ω0 increases as Zeff, hence large Zeff disadvan- atomic spectrum. This treatment actually contains two tages the Markovian approximation. approximations. One is the neglect of the finite size effect Later on, some people considered the correction to the on the electric-dipole transition along with the omission of Weisskopf-Wigner theory [1–4]. But, as far as we know, the contribution from all possible higher multipole tran- all of them applied the Laplace transform to solve the re- sitions, caused by the replacement of the plane wave fac- sultant differential-integral equation, and various kinds of tor eik·x by 1 in the transition matrix element evaluation. approximations were used in the inverse transform. Some Another is the so-called Markovian approximation on the papers even take the radiating atom as a pointlike electric reduced equations of the atomic variables derived by elim- dipole, so that the corresponding correlation spectrum di- inating the photon variables. They are differential-integral verges linearly with ω, and in the subsequent treatment an equations with correlation functions to denote the memory artificial cutoff of frequency is needed. Besides, majority effect. The Markovian approximation neglects this mem- of them only considers the case of Z =1. ory effect and hence changes them to simple differential In a preceding paper [5], two of us with other cowork- equations. ers restudied this problem by a totally different approach. The Weisskopf-Wigner theory applies well for the emis- The stochastic quantum trajectory formulation [6] is ap- sion of outer-shell electron or inner-shell electron of light plied for this investigation, and the non-Markovian cor- nuclei. But in the case of inner-shell electron emission of rection to the decay of upper-level population is taken heavy nuclei, it may lead to obvious deviation, since (1) ω0 into account by introduction of additional fictitious oscil- 2 is proportional to Zeff (where Zeffe is the charge of the lators [7,8]. However, the counter-rotating wave interac- atomic core felt by the emitting electron), and the radius a tion is still not taken into account. Now we will release of the electron cloud is proportional to 1/Zeff, leading to this limitation to include the counter-rotating wave term in the interaction, since its effect also becomes larger for a e-mail: [email protected] stronger coupling [9]. In this case there are two correlation 280 The European Physical Journal D spectra, hence more fictitious oscillators are needed to sim- field as reservoir, the resultant differential-integral equa- ulate them. Moreover, in the case of the rotating-wave ap- tions are given by proximation, the total quanta of the fictitious oscillators is either one or zero, while the counter-rotating wave in- t d teraction puts no limitation on the quanta numbers. Both σˆ−(t)= [u1(t − t ) − u¯1(t − t )]ˆσ3(t)ˆσ−(t )dt of these will increase the amount of numerical calculation. dt 0 t In Section 2 the relevant differential-integral equation 2iω0t +e [u2(t − t ) − u¯2(t − t )]ˆσ3(t)ˆσ+(t )dt for the atom operators is given, with the two correlation 0 spectra explicitly derived. The same hydrogen-like exam- − ˆ ˆ† ple as that of reference [5] is numerically studied for com- σˆ3(t) Σ1(t)+Σ2(t) (3a) parison. In Section 3 the quantum stochastic trajectory t d analysis [6] of the enlarged system [7,8] is carried out σˆ3(t)=−2 [u1(t − t ) − u¯1(t − t )]ˆσ+(t)ˆσ−(t )dt to give the time decay of the upper-level population. We dt 0 t see by this example that the correction due to counter- 2iω0t − e [u2(t − t ) − u¯2(t − t )]ˆσ+(t)ˆσ+(t )dt rotating wave term is rather small (except the end stage of 0 decay) even for the largest atomic number Z of hydrogen- ˆ ˆ† like atom, as conjectured by the usual estimation. +2ˆσ+(t) Σ1(t)+Σ2(t) +h.c. (3b)
whereσ ˆ3 is the atom population-difference operator 2 The spectra of correlation functions and Σˆj(t)’s are fluctuation operators defined in equa- for spontaneous emission of a two-level atom tion (5). There are totally four correlation functions in the above differential-integral equations: The A · P type Hˆint for the spontaneous emission of a 2 −i(ω−ω0)(t−t ) two-level atom is known as u1(t − t )= |gkj| e kj ∞ ∗ † ˆ − −i(ω−ω0)(t−t ) Hint(t)=i gkjσˆ+(t)ˆakj (t) gkjσˆ−(t)ˆakj(t) ≡ R1(ω)e dω, (4a) k,j 0 2 i(ω+ω0)(t−t ) ˆ† ∗ u¯1(t − t )= |g¯kj| e +¯gkjσˆ+(t)a kj(t) − g¯kjσˆ−(t)ˆakj(t) (1) kj ∞ i(ω+ω0)(t−t ) with counter-rotating wave terms included, wherea ˆkj ≡ R1(ω)e dω, (4b) † 0 (ˆakj) is the photon annihilation (creation) operator of i(ω−ω0)(t−t ) mode (k,j),σ ˆ± are atom-level change operators (ˆσ+ cor- u2(t − t )= gkjg¯kje responds to upward change andσ ˆ−downward change), gkj kj ∞ andg ¯kj are coupling constants for rotating-wave term and i(ω−ω0)(t−t ) counter-rotating wave term respectively, ≡ R2(ω)e dω, (4c) 0 −i(ω+ω0)(t−t ) e 2π u¯2(t − t )= gkjg¯kje gkj = − εkj · Gk, (2a) m Vkc kj ∞ ik·x † 3 ≡ −i(ω+ω0)(t−t ) Gk = e Ψ2 (x)∇Ψ1(x)d x, R2(ω)e dω, (4d) 0 e 2π g¯kj = − εkj · G¯ k, (2b) | |2 | |2 m Vkc since gkj = g¯kj . The correlation spectrum R1(ω)is expressed by −ik·x † 3 G¯ k = e Ψ (x)∇Ψ1(x)d x 2 2 V ω 2 R1(ω)= dΩk |gkj| in which εkj is the polarization vector of photon 3 3 (2π) c j mode (k,j), Ψ2(x)andΨ1(x) are the upper level and 2 lower level wave function respectively. They are two- e ω 2 2 = dΩk |Gk| −|nk · Gk| . (4e) component spinor functions. It is seen from equation (2) 4π2m2c3 that when Ψ2(x)andΨ1(x) have the same parity, G¯ k = −Gk;whenΨ2(x)andΨ1(x) have the opposite parity, Similar expression for R2(ω) can be written out directly. G¯ k = Gk; and similar relations for gkj andg ¯kj. The nk in equation (4e) is the unit vector in the direction The dynamical equations for both atom and photon of k. We see that there are two spectral functions R1(ω) ˆ operators are readily deduced from Hint. Eliminate the op- and R2(ω), not just only one spectral function as in the † eratorsa ˆkj and aˆ kjby taking the electromagnetic (e.m.) references [1–5]. C.-Q. Cao et al.: Spontaneous emission of an excited two-level atom 281
The two quantum fluctuation forces are expressed by into equation (4e) and carrying out the integration over θk and ϕk, one will obtain the first spectral function R1(ω). −i(ω−ω0)t Σˆ1(t)= gkjaˆkj(0)e , (5a) The spectrum R2(ω) can be calculated similarly. kj To see explicitly these spectral functions, we con- † † i(ω+ω0)t sider the same example of hydrogen-like atom studied in Σˆ2(t)= g¯kjaˆkj(0)e . (5b) reference [5]: Ψ1(x)andΨ2(x) carry quantum numbers kj (n1 =1,l1 =0)and(n2 =2,l2 =1,m2 = 1) respectively with the expression As a first step we calculate the coupling constant G¯ k and Gk. For simplicity, we omit the spin of the electron. 1 4 Substituting the wave function of hydrogen-like atom √ −r/a1 Ψ1(x)= N1e ,N1 = 3 , (12) 4π a1 Ψ1(x)=f1(r)Yl1 m1 (θϕ),Ψ2(x)=f2(r)Yl2 m2 (θϕ)(6)
−r/a2 4 and also the expansion formula of plane wave by spherical Ψ2(x)=N2re Y11(θϕ),N2 = 5 · (13) waves into equation (2a) and making use of the formula 3a2 of ∇Ψ1(x), one gets Gk as in [5] In this simple case, the gradient of Ψ1(x)maybeevaluated l directly. Substituting it into equation (2a) and carrying m2 4π(2l2 +1) i Gk =(−1) √ Ylm(θkϕk) out the angular integration, one obtains [5] 2l1 +1 lmµ 2l +1 × − { − } 4π N1N2 [ (l1 + 1)(2l1 +3) Al(ω) l1Bl(ω) Gk = X0(ω)Y00(θkϕk)n−1 × − 3 a1 C(l1 +1, 1,l1; m1 µ, µ, m1) ×C(l1 +1,l2,l; m1 − µ, −m2,m) 6 + X2(ω) Y2,−2(θkϕk)n+1 ×C(l1 +1,l2,l;0, 0, 0) + l1(2l1 − 1){Al(ω) 5 } − − +(l1 +1)Bl(ω) C(l1 1, 1,l1; m1 µ, µ, m1) 3 − Y2,−1(θkϕk)n0 ×C(l1 − 1,l2,l; m1 − µ, −m2,m) 5 × − n C(l1 1,l2,l;0, 0, 0)] µ (7) 1 + Y2,0(θkϕk)n−1 (14) in which C(l1 +1, 1,l1; m1 − µ, µ, m1) etc are Clebasch- 5 Gordan coefficients, nµ(µ =+1, 0, −1) are spherical bases, where 1 n+1 = √ (n1 +in2), 2 2 ∞ 4 ω a 2 2a 3 − 2 3 ω −r/a c 0 0 n0 = n3, X (ω)= r j r e dr = 2 2 3 , 0 c ω a 1+ 2 1 c n−1 = √ (n1 − in2)(8) (15) 2 ∞ 4 ω2a2 3 ω −r/a 8a c2 2 2 and X (ω)= r j r e dr = 2 2 3 (16) 0 c ω a ∞ 1+ c2 2 ω df1(r) Al(ω)= r jl r f2(r) dr, (9a) 0 c dr with ∞ 1 1 1 ω = + · (17) Bl(ω)= rjl r f2(r)f1(r)dr. (9b) a a1 a2 0 c In this simple example, both Al(ω)andBl(ω) are propor- The summation in equation (7) actually contains only fi- tional to Xl(ω). nite terms because of the angular momentum addition In equation (14) the angular momentum l only takes rule. For the first term in the square brackets, l is re- the value 0 and 2, hence we know from equation (10) that | − − |≤ ≤ stricted in the range l2 l1 1 l l2 + l1 +1,andfor in this example G¯ k = Gk,consistentwithwhatwesay the second term, in the range |l2 − l1 +1|≤l ≤ l2 + l1 − 1. below equation (2b). This is in turn implies the equality of G¯ The k canbeobtainedfromequation(7)simplyby the counter-rotating wave coupling constantg ¯kj and the the substitution rotating-wave coupling constant gkj. Thus the correlation l spectrum R2(ω)isgivenby Ylm(θkϕk) → (−1) Ylm(θkϕk). (10) V ω2 Substituting Gk and the formula 2 R2(ω)= 3 3 dΩk gkj (2π) c j µ=1 4π ∗ 2 nk = Y1µ(θkϕk)nµ (11) e ω 2 2 3 = dΩk Gk − (nk · Gk) . (18) µ=−1 4π2m2c2 282 The European Physical Journal D 2 It is easy to show that dΩkGk = 0. By further utilizing Actually in our example only electric dipole emission is the coupling rule for spherical harmonics, nk · Gk may allowed by the selection rule of angular momentum, thus 3 −4 be expressed as l=1 clYl,−1(θkϕk), hence we also have the factor (1 + ω0a/c) in R1(ω) just expresses the finite 2 dΩk(nk · Gk) = 0. Consequently we obtain size effect on the radiative electric dipole transition. In this example, both u2(t − t )and¯u2(t − t )are R2(ω)=0. (19) equal to zero, only two correlation functions u1(t−t )and u¯1(t − t ) remain. The former comes from the rotating- This may be a quite general result [10]. wave interaction and the latter from the counter-rotating From equations (4e, 11, 14), one gets the result wave interaction. In case we make the Markovian approx- for R1(ω)asR(ω) used in reference [5] and also in ref- imation on them, namely approximate them by Cδ(t − t) erences [3,4] and Cδ¯ (t − t) respectively to neglect the memory effect, ¯ γA ω the constants C and C will be determined by integration R1(ω)= 2 2 2 4 , (20) of equations (4a, 4b) respect to t from t to ∞ (note that 2πω0 (1 + ω a /c ) u1 andu ¯1 are defined only in the region t>t). The results in which γA is the Einstein A coefficient, given by γA = so obtained is 3 2 3 4ω0|d21| /3c and in our example the transition dipole moment d21 takes its absolute value as ∞ √ 1 R1(ω) C = πR1(ω0) − i ℘ dω, (25a) 64 2 2 ω − ω0 |d21| = ea, (21) 0 81 ∞ 1 R1(ω) C¯ =i dω, (25b) with e denoting the magnitude of electron charge. 2 0 ω + ω0 Actually there is only one parameter in the R1(ω), since ω0 may be expressed by a as where ℘ denotes taking the principal value. The imaginary ¯ parts of C nd C contribute just an level-shift which will 1 − be canceled out in equation (3b) because of the h.c. term. ω0 = (E2 E1)= 2 , (22) 6ma Taking the expectation value of equation (3b) and not- and the two atomic radii a1 and a2 are related to a by ing the upper level population Nˆ2(t) =(1+σˆ3(t) )/2, multiplying constants 3/2 and 3 respectively. Since a is one gets immediately proportional to the atomic number 1/Z , we see from equa- 2 tion (22) that ω0 is proportional to Z . d Nˆ2(t) = γ Nˆ2(t) (26a) The relation between γA/ω0and a is obtained from dt equations (21, 22), and may be expressed by with 2 2 3 2 γA 8 32 e aB γA · = 2 , (23a) γ =2πR1(ω0)= 2 2 2 4 (26b) ω0 3 243 c a (1 + ω0a /c ) where aB is the Bohr radius. The dependence of γA/ω0 We see that γ is different from γA by the finite size cor- 2 2 2 −4 on the atomic number Z is given by rection factor (1 + ω0a /c ) , which in our example is about 0.729. 2 2 3 γA 1 64 e = Z2, (23b) We would like to point out that the counter-rotating d ˆ ω0 6 81 c wave interaction will have no effect on dt N(t) in the case
−3 Markovian approximation is made, even if R2(ω) =0. hence the value γA/ω0 =10 corresponds to Z = 157, In the next section we will study the non-Markovian somewhat larger than the upper limit of the real nuclei. approach with the counter-rotating term included. AprofileofR1(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 ik·x 3 Quantum stochastic trajectory approach factor e in the integrand of Gk is omitted, Gk is then independent of k. Hence, R1(ω) will become linear in ω to the spontaneous emission as can be seen from equation (4e), and the coefficient is just γA/2πω0. We will call this spectrum as pointlike We have shown by equation (20) that the two-level dipole spectrum: atom with large atomic number Z has an evident non- D γA Markovian reservoir in its spontaneous emission. Hence R1 (ω)= ω. (24) 2πω0 we will, as described in references [7,8], introduce N additional fictitious harmonic oscillators, which interact It also corresponds to R1(ω)withωa/c =0inthedenom- with the atom in the same form as photons, to form inator. an expanded system. This expanded system thus has the C.-Q. Cao et al.: Spontaneous emission of an excited two-level atom 283
Hamiltonian where Hˆ is expressed by equation (27). The collapse op- erators acting on |Ψ(t) are N N ˆ 1 † H = ω0σˆ3 + ωjαˆjαˆj +i gjσˆ+αˆj Cˆj = Γj∆tαˆj,j=1, 2, ..., N. (30) 2 j=1 j=1 To proceed with concrete analysis and compare with the − ∗ † † − ∗ gj σˆ−αˆj +¯gjσˆ+αˆj g¯j σˆ−αˆj (27) result of rotating-wave approximation, we return to the example with Ψ2(x)andΨ1(x) described by equation (12). in which the energies of the two atom-levels are taken In reference [5] only two fictitious oscillators are in- ± 1 † as 2 ω0 respectively,α ˆj andα ˆj are annihilation and troduced to simulate the spectrum R(ω) ≡ R1(ω)ofthis ± (s) creation operators of the jth oscillator,g ¯j = gj for the example, the corresponding simulation spectrum R (ω) 2 1 ∓ relative parity of Ψ and Ψ equal to 1 respectively. Each fits pretty well with R(ω) in a large region ω/ω0 ≥ 0.5, of these oscillators, on the other hand, is assumed to in- which includes the important region around ω/ω0 =1. teract with its own reservoir with no memory, hence the But the R(s)(ω) drops down less slowly than R(ω)when expanded system is Markovian. In terms of Langevin equa- (s) (s) ω/ω0 → 0. At the origin, R (ω)/R (ω0) is still about tions in Heisenberg picture the dynamics of the above sys- 0.36 while R(ω) drops to zero (see Fig. 1 of Ref. [5]). tem is expressed by Now there is an additional spectrum R2(ω), we have N d to use one set of N oscillators to simulate both of them, σˆ−(t)=−iω0σˆ−(t) − gjαˆj(t) namely to make dt j=1 N 2 † ∼ (s) 1 |gj| Γj j 3 R1(ω) R (ω) ≡ , (31a) +¯g αˆj (t) σˆ (t), (28a) = 1 − 2 1 2 2π j=1 (ω ωj) + 4 Γj N d † N σˆ3(t)=2 gjαˆj(t)+¯gjαˆj(t) σˆ+(t) ∼ (s) 1 gjg¯jΓj dt R2(ω) R (ω) ≡ · (31b) j=1 = 2 − 2 1 2 2π j=1 (ω ωj) + 4 Γj N ∗ † ∗ +2 gj αˆj (t)+¯gj αˆj(t) σˆ−(t), (28b) In general, the simulation is not easy to be carried out. But j=1 in the case R2(ω) = 0, we may simply take four fictitious
d ∗ oscillators with following parameters: αˆj (t)=−iωjαˆj(t) − g σˆ−(t)+¯gjσˆ+(t) dt j √1 1 g1 = 0.011ω0,Γ1 =1.3ω0,ω1 = ω0, (32a) − Γj αˆj(t) − Fˆj(t) (28c) 2 2 √1 with g2 = 0.018ω0,Γ2 =2.4ω0,ω1 =1.85ω0, (32b) 2 ˆ ˆ ˆ i Fj (t) =0, Fi(t)Fj (t ) =0, g3 = √ 0.011ω0,Γ3 =1.3ω0,ω3 = ω0, (32c) R R 2 † Fˆi(t)Fˆj (t ) = δij Γjδ(t − t ). (28d) i R g4 = √ 0.018ω0,Γ4 =2.4ω0,ω4 =1.85ω0. (32d) 2 The last two terms of equation (28c) are the usual Markovian dissipation term and fluctuation term con- It is easy to check that the above set of parameters (s) tributed by the reservoir of jth oscillator. makes R2 (ω) = 0 no matter the relative parity be- (s) When the formal solution of equation (28c) is used to tween Ψ2 and Ψ1 is positive or negative, while R1 (ω) † (s) eliminate the variablesα ˆj andα ˆj in equations (28a, 28b), remains unchanged from the R (ω) of reference [5]. the reduced equation forσ ˆ+(t)andˆσ3(t) will have non- Since the numerical evolution takes place over discrete Markovian damping terms and non-Markovian fluctuation times with a small time step ∆t, the wave function |Ψ(t) terms, since each fictitious oscillator contributes a term is represented by a sequence |Ψ(tn) with tn = n∆t.Given with Lorentzian type spectrum. The next step is to se- the value of |Ψ(tn) , the next one |Ψ(tn+1) is determined lect parameters gj,¯gj, ωj and Γj to simulate the origi- by the following algorithm: nal damping terms and fluctuation terms. Now there are two spectral functions, not just one spectral function as (1) evaluate the four kinds of collapse probabilities during in rotating-wave approximation, the simulation task will interval (tn,tn−1): become more complicate. ˆ† ˆ As soon as the simulation task is accomplished, one Pj(tn)= Ψ(tn) Cj Cj Ψ(tn) may go back to the master-equation formulation and treat † it by quantum stochastic trajectory approach. The non- = Γj Ψ(tn) αˆjαˆj Ψ(tn) ∆t, j =1, 2, 3, 4 Hermitian Hamiltonian is now taken as (33) i † Hˆnh = Hˆ − Γj αˆjαˆj, (29) in which ∆t should be small enough to make P1(tn) 2 j and P2(tn) much small than 1; 284 The European Physical Journal D
may neglect this term for stable nuclei without introduc- ing obvious error. We note that the large difference between curve II and curve III comes from two respects as mentioned in Sec- tion 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)fromR(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 Fig. 1. The spontaneous decay of the upper level population wave coupling, it is totally due to the non-Markovian con- −3 with parameters (γA/ω0 =1×10 ). The curve I represents the tribution since in the present case R2(ω)=0,γ remains result without both rotating-wave and Markovian approxima- unchanged as specified in the end of Section 2. tion, the curve II represents the result with rotating-wave ap- proximation but without Markov approximation as calculated This work is supported by National Natural Science Founda- in reference [5], the curve III represents the Weisskopf-Wigner result. tion of China (project number 19774004), and also by the In- ternational Program of National Science Foundation of USA (ECS-9800068).
(2) generate four random numbers rj (j =1, 2, 3, 4) which have uniform probability distribution among the in- terval (0, 1); References (3) compare Pj (tn)withrj and derive |Ψ(tn+1) accord- ing to the following rule: if Pj (tn)