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 .