arXiv:1105.4120v1 [quant-ph] 20 May 2011

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Bloch equations for individual atoms, the ensemble averaged atomic ... ERE describe the interaction of a broadband isotropic light field with an atomic two–.
A semiclassical optics derivation of Einstein’s rate equations Robert Hoeppner,∗ Eugenio Rold´an,† and Germ´an J. de Valc´arcel‡ ´ Departament de Optica, Universitat de Val`encia, Dr. Moliner 50, 46100 Burjassot, Spain, EU

arXiv:1105.4120v1 [quant-ph] 20 May 2011

Abstract We provide a semiclassical derivation of Einstein’s rate equations (ERE) for a two-level system illuminated by a broadband light field, setting a limit for their validity that depends on the light spectral properties (namely on the height and width of its spectrum). Starting from the optical Bloch equations for individual atoms, the ensemble averaged atomic inversion is shown to follow ERE under two concurrent hypotheses: (i) the decorrelation of the inversion at a given time from the field at later times, and (ii) a Markov approximation owed to the short correlation time of the light field. The latter is then relaxed, leading to effective Bloch equations for the ensemble average in which the atomic polarization decay rate is increased by an amount equal to the width of the light spectrum, what allows its adiabatic elimination for large enough spectral width. Finally the use of a phase-diffusion model of light allows us to check all the results and hypotheses through numerical simulations of the corresponding stochastic differential equations.

1

I.

INTRODUCTION

Lorentz’s equation1 and Einstein’s optical rate equations2 are well known, very valuable, and widely used heuristic models for the description of light–matter interaction. Both of them occupy an important place in the early teaching of quantum optics topics too (refractive index, laser action), although each of them applies to very different situations: roughly speaking, the Lorentz equation applies when weak or highly detuned fields are involved, whilst Einstein’s model applies to resonant interaction with broadband radiation. Of course the two models can be fully justified within the framework of quantum optics theory in the appropriate limits.3–6 Generically speaking, these formal derivations of heuristic models from first principles are important not only for esthetical reasons and completeness arguments, but also for the clarification of their applicability domain. In this respect the situation of the Lorentz and Einstein models is quite different: derivations of Lorentz model from the optical Bloch equations are easily found in many textbooks3,4 , but this is not the case for Einstein’s rate equations (ERE for short in the following). This is most surprising given the paramount importance of Einstein’s model.7 This does not mean that the connection between the optical Bloch equations and ERE has not been considered, as several quantum optics textbooks discuss ERE and provide derivations of Einstein’s A and B coefficients for spontaneous and stimulated processes8 , see4–6 . More general treatments can also be found in some textbooks such as those of Refs.4,9 . But these treatments do not focus on the derivation of ERE and, from our viewpoint, do not put enough emphasis on didactic aspects. We try to close this gap with the present article. In the following we derive ERE from the semiclassical Bloch equations. In Section II we present ERE briefly, and in Section III we introduce the optical Bloch equations and reduce them to an integro-differential equation for the atomic inversion. In Section IV we derive ERE by assuming (i) statistical decorrelation between inversion and field correlation function, and (ii) the Markov approximation. In Section V we remove the Markov approximation above and derive an effective Bloch model that incorporates the field incoherence; after discussing the differences between the solutions of the latter and those of ERE we show that ERE derives from this model after adiabatic elimination of the effective atomic coherence. Then, in Section VI we analyze a particularly simple case (that of a field having only phase noise) that allows us to show the validity of the statistical decorrelation above

2

and discuss a number of questions. Finally, we give our main conclusions in Section VII.

II.

EINSTEIN’S RATE EQUATIONS

ERE describe the interaction of a broadband isotropic light field with an atomic two– state system. Einstein2 postulated three basic light–matter interaction processes (stimulated absorption and emission, and spontaneous emission), and established the rate equations governing the evolution of the populations of each of the atomic states. Denoting by Ni the population of the lower (i = 1) and upper (i = 2) atomic states, and assuming that N = N1 + N2 is constant and large enough for individual absorptions and emissions to produce smooth temporal changes in Ni , the time evolution of the populations is given by dN2 = −AN2 + BW21 (N1 − N2 ) , dt dN1 dN2 =− , dt dt

(1a) (1b)

where W21 is the spectral energy density of the light field at the atomic transition Bohr frequency ω21 , and A and B are Einstein’s coefficients for spontaneous emission and stimulated processes, respectively, which neither depend on the field strength nor on time and verify 3 /π 2 c3 . In terms of the normalized population inversion n ¯ ≡ (N2 − N1 ) /N , A/B = ~ω21

1≥n ¯ ≥ −1, Eqs. (1) have the simpler form d¯ n = −A (¯ n + 1) − 2BW21 n ¯. dt

(2)

We address the reader to5 for a particularly clear presentation and discussion of ERE.

III.

BLOCH EQUATIONS FOR AN ENSEMBLE OF TWO–LEVEL ATOMS

Consider a generic light field whose electric component E~ we write as E~ (r, t) = 12 [E (r, t) + E∗ (r, t)] ,

(3)

where E (r, t) is the field negative-frequency part (that containing terms oscillating as e−iωt , ω > 0) interacting with a collection of identical two–level atoms or molecules (|2i and |1i denote the atoms’ excited and fundamental states) with Bohr frequency ω21 and electric dipole matrix elements h2| µ ˆ |1i = h1| µ ˆ |2i ≡ µz, which have been taken to be real vectors, 3

aligned parallel to the Cartesian z-axis, without loss of generality. Working in the Dirac picture, and after performing the rotating–wave approximation, the semiclassical optical Bloch equations for an individual atom, labelled by α and located at rα , can be written as3–5

dnα = −A (nα + 1) − i (Ω∗α σα − Ωα σα∗ ) , dt dσα 1 i = − Aσα − Ωα nα , dt 2 2 (α)

(α)

(4a) (4b)

(α)

where nα = ρ22 − ρ11 and σα = ρ12 exp (iω21 t) are, respectively, the population inversion and the slowly varying atomic coherence of atom α described by its density matrix ρ(α) , and Ωα (t) =

µ Ez (rα , t) eiω21 t , ~

(5)

is the complex Rabi frequency of the light field at the position of atom α, with Ez = z·E. The effect of spontaneous emission has been phenomenologically included (through the damping term proportional to A), as standard semiclassical theory cannot describe this process.8 First Eq. (4b) is integrated formally, Z 1 i t 0 0 σα (t) = − dt Ωα (t0 ) nα (t0 ) e− 2 A(t−t ) , (6) 2 0  where a transient term, σα (0) exp − 21 At , has been dropped (alternatively one can take σα (0) = 0 without loss of generality assuming that the interaction is turned on at that instant). Plugging this into Eq. (4a) we get dnα = −A (n + 1) dt Z

(7)

t

A

0

dt0 Ωα (t) Ω∗α (t0 ) nα (t0 ) e− 2 (t−t ) .

− Re 0

Let us emphasize that Eq. (7) is the equation of evolution for the population inversion of a single atom. As we are interested in the average evolution of the whole system — the ensemble of atoms — which is the quantity described by ERE, we write down next the evolution equation for the ensemble averaged inversion n ¯ (t) ≡ hnα (t)i ,

(8)

N 1 X hf (α)i = f (α) , N α=1

(9)

where averages are defined as

4

for any function possibly depending on α. The equation of evolution of n ¯ (t) is obtained from Eq. (7) and reads d¯ n = −A (¯ n + 1) − dt

Z

t

A

0

dt0 K (t, t0 ) e− 2 (t−t ) ,

(10a)

0

K (t, t0 ) = Re hΩα (t) Ω∗α (t0 ) nα (t0 )i .

(10b)

Equations (10) now describes the average dynamics of the system in an exact way, in the sense that no approximation has been done on the original Bloch equations in order to arrive at it. Clearly it is necessary to evaluate the correlation function K (t, t0 ) in order to perform the time integral in Eq. (10a) and then arrive at a connection of Bloch equations and ERE.

IV.

A FIRST DERIVATION OF ERE

A direct comparison between Eqs. (2) and (10) reveals a number of differences, the main one being that Eq. (2) has no memory (the time derivative of n ¯ at time t just depends on the value of n ¯ at the same time t) while in Eq. (10a) memory effects are present as in the integral it appears n ¯ (t0 ) with t0 < t. One then concludes that in order for (10) to merge with ERE in some limit two conditions must be verified. The first condition is that it should be possible to decorrelate K (t, t0 ) in (10b) as K (t, t0 ) ≈ Re hΩα (t) Ω∗α (t0 )i hnα (t0 )i ≡ C (t, t0 ) n ¯ (t0 ) ,

(11)

C (t, t0 ) ≡ Re hΩ∗α (t) Ωα (t0 )i ,

(12)

with

the field correlation function. This is a very reasonable approximation as K (t, t0 ) contains correlations between n ¯ (t0 ) and Ωα (t) with t ≥ t0 and Ωα (t) is assumed to have a random character. As for the second condition, C (t, t0 ) should be effectively zero but for |t − t0 | . tc , where the coherence time tc should be short, in the sense that A, |d¯ n/dt|  t−1 c , allowing the substitutions n ¯ (t0 ) → n ¯ (t) and exp [−A (t − t0 ) /2] → 1 under the integral in (10a). This last condition is a Markov approximation indeed as with it one assumes that the lack of correlation in the field (which implies a large enough amount of randomness in its evolution) provokes the complete loss of memory of the inversion n ¯ (t). We shall refer to the above conditions as (i) the decorrelation assumption and (ii) Markov approximation, respectively. 5

It is clear that for broadband (random-like) fields condition (ii) is verified and this is enough, once (i) holds, to derive ERE as we pass to show. Assuming then that (i) and (ii) hold Eq. (10) becomes d¯ n = −A (¯ n + 1) − n ¯ dt

Z

t

dt0 C (t, t0 ) ,

(13)

0

which has the same form as (2), and will match it exactly if the integral equals 2BW21 . Let us show that this is indeed the case. First we note that C (t, t0 ) = C (t0 , t) as follows straightforwardly from its definition. Then, if t is a bit larger than the correlation time tc the integral lower limit can be extended to −∞ without hurt and the symmetry of C allows writing

t

Z

1 dt C (t, t ) ≈ Re 2 0

0

0

Z

+∞

dτ hΩα (t) Ω∗α (t0 )i .

(14)

−∞

Using the Rabi frequency definition (5) and recalling that we are assuming that radiation is isotropic and unpolarized, what allows writing hEz (r, t) Ez∗ (r, t0 )i = 31 hE (r, t) · E∗ (r, t0 )i ,

(15)

we get µ2 d¯ n = −A (¯ n + 1) − 2 n ¯ dt 6~ Z +∞

dτ hE (rα , t) · E∗ (rα , t + τ )i e−iω21 τ ,

× Re

(16)

−∞

where we changed the integration variable t0 to τ = t0 − t. This last expression is fine as the integral is proportional to the light spectral energy density at frequency ω21 , W21 , as follows from Wiener-Khintchine theorem.5,10 As shown in Appendix I, the theorem can be expressed in the form ε0 W (ω) = Re 4π

Z

+∞

dτ hE (r, t) · E∗ (r, t + τ )i e−iωτ ,

(17)

−∞

and thus we obtain d¯ n = −A (¯ n + 1) − 2BW21 n ¯, dt πµ2 B= 2 , 3~ ε0

(18a) (18b)

which coincides with ERE (2) and gives the correct result for B.5 Thus we are able to provide a semiclassical derivation of ERE, with the correct expression for the B coefficient, whenever hypotheses (i) and (ii) above are verified. Of course both 6

assumptions are closely related as both are consequences of the randomness of the incoherent radiation field. In the following section we derive ERE again but now assuming only the decorrelation hypothesis, i.e. without invoking the Markov approximation.

V.

A SECOND DERIVATION OF ERE. EFFECTIVE BLOCH EQUATIONS

Let us return to the general case represented by the exact Eq. (10a). As stated, we shall assume again that the decorrelation hypothesis (11) holds for whichever broadband incoherent light field, but will remove the Markov approximation. Taking into account (5), the depolarized light field assumption (15), and the Wiener-Khintchine theorem (17), Eq. (10a) can be rewritten as Z t 1 2µ2 d¯ n 0 = −A (¯ n + 1) − dt0 n ¯ (t0 ) e− 2 A(t−t ) 2 dt 3ε0 ~ 0 Z +∞ 0 dωW (ω) ei(ω21 −ω)(t−t ) . ×

(19)

−∞

In order to be able to evaluate the frequency integral, we need some knowledge concerning the form of W (ω): We shall assume that the light field has a bell–shaped spectrum, namely a Lorentzian spectrum. For the sake of simplicity we shall also assume that its central frequency is resonant with the atomic transition, although this is inessential and the derivations that follow are easily generalized to the nonresonant case. Then we write W (ω) = W21

(∆/2)2 , (∆/2)2 + (ω − ω21 )2

(20)

where ∆ represents the width (FWHM) of the spectrum. The integral in ω in (19) can now be done, yielding d¯ n ˜ 21 q¯, = −A (¯ n + 1) − 2B W dt

(21)

where Z

t

q¯ (t) ≡ γ

0

dt0 n ¯ (t0 ) e−γ(t−t ) ,

(22)

0

γ = 12 (A + ∆) ,

(23)

˜ 21 is the effective spectral energy density W ˜ 21 ≡ W

∆ W21 , ∆+A 7

(24)

and the B coefficient reads as in (18b). It is clear that if n ¯ varies slowly during a time interval γ −1 then n ¯ (t0 ) can be picked out from the integral in (22) at t0 = t (this is the Markov approximation we made in the previous section) and, after performing the remaining integral, q¯ (t) = n ¯ (t) once γt  1, leading to ERE (18) but with the modified spectral energy density (24). We thus see a first correction to standard ERE that manifests when the light spectrum is not too wide as compared with ˜ 21 = W21 . But this is not the the natural linewidth A. Only in the limit ∆  A one has W way we are going to derive ERE in this section. The integro-differential Eq. (21) can be easily transformed into a pair of coupled linear differential equations by taking the time derivative of (22). One gets d¯ n ˜ 21 q¯, = −A (¯ n + 1) − 2B W dt d¯ q = −γ q¯ + γ n ¯, dt

(25a) (25b)

which are effective Bloch equations with q¯ playing the role of a kind of normalized average medium polarization. Let us recall that this set of equations is exact but for the decorrelation assumption (11), which should hold under a wide variety of conditions. Furthermore notice that for ∆ → 0, i.e. for a fully coherent field hence characterized by a constant Rabi frequency Ω0 (real without loss of generality), Eqs. (25) are equivalent to Eqs. (4) (for Ωα (t) = Ω0 ∀α) upon identifying q¯ with iAσ/Ω0 and W21 with Ω20 /∆, as it must be.13 Up to this point we have shown that by assuming the decorrelation hypothesis and by taking a Lorentzian spectrum for the radiation field, the average Bloch equations for the atom gas can be reduced to a pair of effective Bloch equations in which the effective medium polarization decays with a rate γ = (A + ∆) /2, where ∆ is the FWHM of the field spectrum. That amounts to say within the range of validity of the above assumptions, that the incoherence of the radiation field is in some way transferred to the average medium polarization, manifesting as an increase in its decay rate similar to the effect of nonradiative collisions.3

8

A.

Adiabatic elimination of the effective coherence

We now derive ERE from the effective Bloch equations by adiabatically eliminating the effective coherence. For that we integrate formally the equation of q¯ in Eqs. (25), Z t q¯ (t) = dτ e−γ(t−τ ) γ n ¯ (τ ) .

(26)

0

(note that q¯ (0) = 0, see Eq. (22)). Integrating repeatedly by parts we get   2 −2 d −1 d +γ q¯ (t) = 1 − γ − ... n ¯ (t) . dt dt2

(27)

For large enough γ we can approximate q¯ (t) ' n ¯ (t). Notice that this is the same as taking d¯ q /dt = 0 in Eqs. (25), which is the usual adiabatic elimination procedure. Using this result in the equation for n ¯ (t) we recover ERE, thus completing our second derivation. We can now estimate how large must be γ for the adiabatic elimination approximation be precise enough by comparing the first two terms in (27). The necessary condition is |d¯ n/dt|  γ |¯ n|, which using (25b) and (24) leads to ∆A BW21 

(28a) ∆ 4

(28b)

for ERE be valid. It is interesting to note here that in many textbooks rate equations are derived from optical Bloch equations through the adiabatic elimination of the medium polarization. In this case a constant Rabi frequency is assumed and the coherence relaxation rate is assumed to be much larger than the population one because of the existence of frequent dephasing collisions. Of course these are not the conditions that lead to ERE. However we have demonstrated here that by ensemble averaging the Bloch equations and assuming the decorrelation assumption and a Lorentzian shape for the radiation spectrum leads to effective Bloch equations in which the effective coherence has a decay rate which is increased (with respect to that of the inversion) by an amount equal to the radiation linewidth, i.e., similarly to the effect of dephasing collisions. Then for large enough ∆ the effective coherence can be adiabatically eliminated leading to the correct ERE.

9

B.

Comparison of ERE and effective Bloch equations solutions

Compared to ERE (18) the system (25) has an extra equation that allows for richer dynamics. We shall now compare the solutions of both models in two different time regimes, namely the short time and the long time regimes. This analysis will provide us with a better estimate of the necessary conditions for ERE be valid than that of inequality (28).

1.

The short time limit

For short times after the illumination has been switched on, say at t = 0, the predictions of both models given by ERE (18) and the effective Bloch Eqs. (25) differ. In the following ˜ 21 in order to be more fair with the we consider ERE (18) with W21 substituted by W comparisons. Assuming n ¯ (0) = −1 and q¯ (0) = 0 we easily obtain ˜ 21 t + O t2 n ¯ ERE (t) ≈ −1 + 2B W



1 ˜ 2 n ¯ Bloch (t) ≈ −1 + B W 21 t 2

(29a) (29b)

where the subscript indicates to which equations the result belongs. This means that ERE are overlooking the dynamics at the initial times, see Fig.1(b), in a way similar to what happens with the application of Fermi’s golden rule to the photoionization problem.14 A way to cure the problem consists in taking n ¯ (t0 ) out of the integral in (22) when considering the strongly incoherent limit (i.e. large ∆), and retaining the exact value of the integral, i.e. approximate q¯ (t) by (1 − e−γt ) n ¯ (t). With this approximation we get from (21) the modified ERE,  d¯ n ˜ 21 n = −A (¯ n + 1) − 2B W ¯ (t) 1 − e−γt , dt

(30)

that predicts a short time evolution as in (29b), see Fig.1(b). Hence (30) can be considered as a simple correction to the standard ERE (18) able to capture the short time dynamics of the system. The conclusion is then clear: the linear time dependence predicted by ERE at short times is an artifact as we are using an approximate equation in a region (t . γ −1 ) where it is not supposed to be valid.

10

2.

The long time limit

At long times after the illumination has been switched on, both ERE (18) and the effective Bloch Eqs. (25) reach a steady state given by the same expression, n ¯ (∞) = −

A . ˜ 21 A + 2B W

(31)

˜ 21 , and consequently (Remember that in the standard ERE (18) it appears W21 , and not W the asymptotic inversion value would differ trivially by this fact.) A natural question we can ask is how this steady state is approached: Is the time evolution of n ¯ monotonous, as standard ERE predict, or does it exhibit some kind of relaxation oscillations3 before reaching its steady state? The answer to this is given by a linear stability analysis of the corresponding equations. The stability analysis of ERE is performed by considering a situation in which n ¯ (t) is close to n ¯ (∞) and seeing how the small increment δ¯ n (t) ≡ n ¯ (t) − n ¯ (∞) evolves according to (18). One trivially gets d δ¯ n = λERE δ¯ n, dt

i h ˜ 21 , λERE = − A + 2B W

(32)

that leads to a monotonous evolution δ¯ n ∝ exp (λERE t), decreasing in time. The situation is slightly more complicated with the effective Bloch equations. We proceed here along the previous lines but now we introduce the two increments δ¯ n (t) ≡ n ¯ (t) − n ¯ (∞) and δ q¯ (t) ≡ q¯ (t) − q¯ (∞). Notice that q¯ (∞) = n ¯ (∞). We get in this case      ˜ n −A −2B W21 δ¯ n d δ¯ =  , dt δ q¯ γ −γ δ q¯ that leads to an evolution of the form exp (λ± t) with √ − (γ + A) ± R (∆ − A)2 λ± = , R= − 4BW21 ∆. 2 4

(33)

(34)

˜ 21 , i.e. ∆  A, 16B W ˜ 21 , the In the strongly incoherent limit defined by γ  A, 8B W above eigenvalues take the simple form λ+ → λERE , and λ− → −γ: Without entering into too many details we just mention that (the large and negative) eigenvalue λ− is responsible for a fast evolution (relaxation) leading to the equalization of δ q¯ and δ¯ n, and from then on the system evolves as governed by ERE. 11

However when incoherence is not so large, λ± may become complex, thus indicating an oscillatory approach to the steady state, i.e. the existence of relaxation oscillations (which are nothing but the manifestation of the underlaying Rabi oscillations) with frequency √ ωro = −R/2. This requires R < 0, i.e., BW21

(∆ − A)2 ∆ > −→ , 16∆ ∆A 16

(35)

where we used (24). This expression implies that there are relaxation oscillations in the approach to steady state whenever the light spectral energy density is large enough, in the sense of Eq. (35), in marked contrast with the ERE prediction. The observability of the oscillations must be considered however because these are damped oscillations according to (34): in order that relaxation oscillations are present their frequency should be larger than, say half their damping rate, i.e. BW21 > (BW21 )crit. (BW21 )crit. ≡

3∆ 3∆2 − 14A∆ + 5A2 −→ . ∆A 64 64∆

(36a) (36b)

As (35) is more restrictive than (36) we conclude that relaxation oscillations will occur when (35) is fulfilled. Hence we conclude that there will be no qualitative differences between the predictions of ERE and effective Bloch equations, i.e., that ERE are correct, whenever

BW21
103 the uncertainty is almost negligible.

VII.

CONCLUSIONS

We have provided a didactic derivation of the Einstein’s rate equations from the semiclassical optical Bloch equation for an ensemble of two–level atoms or molecules. The derivation is done in two forms. In the first one we assume (i) the statistical decorrelation between inversion and field correlation function, and (ii) the Markov approximation for the averaged inversion, this way leading quickly to ERE by using the Wiener-Kintchine theorem. In the second derivation, the Markov approximation is replaced by the assumption of a Lorentzian spectrum in the radiation field. This second derivation leads to a set of effective Bloch equations that contain information about the radiation spectrum whose bandwidth appears as an increase in the effective coherence decay rate. Then, for large enough spectral width, ERE are derived by adiabatically eliminating the effective coherence. Through the analysis and comparison of the solutions of both the ERE and effective Bloch models we have derived the conditions under which the former are applicable. Finally, we have studied numerically the decorrelation hypothesis and checked the different analyzed predictions in the special case of a field having only phase noise. We think our derivations will help students in understanding more clearly how the ERE model can be justified and under which conditions 15

it can be applied. We gratefully acknowledge help from Mar´ıa Gracia Ochoa in some numerical simulations. This work has been supported by the Spanish Government and the European Union FEDER through Project FIS2008-06024-C03-01.

VIII.

APPENDIX. THE WIENER-KHINTCHINE THEOREM AND ITS APPLI-

CATION TO EM RADIATION

The Wiener–Khintchine theorem relates the spectral energy density with the field’s autocorrelation function (see, e.g.5 ). It can be put in either form Z +∞ ε0 Re dτ hE (t) · E∗ (t + τ )i e−iωτ , W (ω) = 4π −∞ Z 2 +∞ ∗ dωW (ω) eiωτ , Re hE (t) · E (t + τ )i = ε0 −∞

(45a) (45b)

and this Appendix is devoted to show it. The derivation bases on considering the e.m. field defined inside a fictitious volume (a cube of side L) with periodic boundary conditions allowing for the existence of traveling waves as in free space, and then letting L → ∞ what allows passing to the continuum. The electric and magnetic fields in such case can be written in their more general form as X~ (r, t) =

XX n

X~n,σ ei(kn ·r−ckn t) + c.c.,

(46a)

σ

~ B, ~ where X~ = E,

n ∈ Z3 , σ = 1, 2, kn =

2π n, L

E~n,σ = L−3/2 eσ (n) An,σ ,

(46b)

B~n,σ = (ckn )−1 kn × E~n,σ ,

(46c)

kn = |kn |, the polarization unit vectors verify kn (n) · eσ (n) =

δσ,σ0 , and we take eσ (−n) = eσ (n) by convention. First we compute the average e.m. energy density contained in the volume, Z E  ε0 D ~ 2 1 ε0  ~ 2 2 ~2 u= E +c B = 3 d3 r E + c2 B~ 2 . 2 L L3 2

16

(47)

Upon using (46a) and taking into account that Z d3 rei(kn +kn0 )·r = L3 δn,−n0 ,

(48a)

L3

[kn × eσ (n)] · [k−n × eσ0 (n)] = −kn2 δσ,σ0 ,

(48b)

one obtains, after little algebra, u = 2ε0

XX n

L−3 |An,σ |2 .

(49)

σ

Finally passing to the continuum [d3 k = (2π/L)3 ] and working in spherical coordinates [d3 k = k 2 dkd2 Ω] we get straightforwardly Z +∞ u= dωW (ω) , 0 Z ε0 ω 2 X d2 Ω |Aσ (k)|2k=ω/c , W (ω) = 3 3 4π c σ 4π

(50a) (50b)

where dω = dck, and |Aσ (k)|2 = |An,σ |2 such that n = k/ (2π/L). Now we compute the Fourier transform Z +∞ dτ hE (r, t) · E∗ (r, t + τ )i e−iωτ , f [E] (ω) = Re

(51)

−∞

Comparing (3) with (46a) and applying a similar calculation as before we get f [E] (ω) = 8π

XX n

L−3 |An,σ |2 δ (ω − ckn ) .

(52)

σ

Passing to the continuum and working again in spherical coordinates we get straightforwardly Z ω2 X f [E] (ω) = 2 3 d2 Ω |Aσ (k)|2k=ω/c . π c σ 4π

(53)

Finally comparing (50b) and (53) we get (45).

IX. A.

FIGURE CAPTIONS Figure 1

Evolution of the ensemble averaged population inversion n ¯ as a function of the adimensional time At. In (a) Ω0 = 4 and we used the three values of ∆ indicated in the figure. 17

These results have been obtained by numerically integrating Eqs. (10) and the results coincide exactly with those provided by the effective Bloch Eqs. (25). In (b) we used  √  ˜ 21 = 2 and ∆ = 5, and we have represented the predictions of the Bloch Ω0 = 11 B W model (Eqs. (10), blue line), of ERE (Eqs. (44), red line), and of the modified ERE (Eqs. (30), brown line). The inset shows the different predictions for short times (see text).

B.

Figure 2

Evolution of the averaged inversion n ¯ obtained with the Bloch model Eqs. (10) for (a) Ω0 = 2 and ∆ = 10, and (b) Ω0 = 6 and ∆ = 1, for several values of the number of atoms N . In (b) the trajectories for N = 100 and N = 10000 (not labeled for the sake of clarity) are so close each other that we plotted the former with dashed line in order to distinguish them.

C.

Figure 3

Average population inversion ± its standard deviation after N trajectories for the steady state using (a) ∆ = 10, Ω0 = 2 (no transitory oscillations) and (b) ∆ = 1, Ω0 = 6 (strong transitory oscillations)



Electronic address: [email protected]



Electronic address: [email protected]



Electronic address: [email protected]

1

H. A. Lorentz, The Theory of Electrons (Teubner, Leipzig, 1909), Chap. 4.

2

A. Einstein, Quantum Theory of Radiation, Phys. Z. 18, 121-128 (1917); an English translation appears in The World of the Atom, edited by H. A. Boorse and L. Motz (Basic Books, New York, 1966), Vol.2, pp. 888-901.

3

P. W. Milonni and J. H. Eberly, Lasers, (John Wiley & Sons, New York, 1988)

4

B. W. Shore, The Theory of Atomic Coherent Excitation (John Wiley & Sons, New York, 1990).

5

R. Loudon, The Quantum Theory of Light (Oxford University Press, 2000).

6

J. N. Dodd, Atoms and Light: Interactions (Plenum Press, New York, 1991).

18

7

In some quantum optics textbooks rate equations are derived from optical Bloch equations through adiabatic elimination of the medium polarization, without taking into account the incoherence of the radiation. This correctly describes a situation in which the atomic coherence has a large decay rate as compared to that of the population inversion (see e.g.3 ). Notice, however, that this is not a true derivation of ERE unless it were demonstrated that the radiation incoherence is correctly described by an appropriate increase of the atomic coherence decay rate. Let us emphasize that for properly deriving ERE, one must take into account the broadband nature of the light field without imposing any a priori condition on the atomic decay rates. See the discussion following Eq. (28) in the text.

8

It is important to remark that standard semiclassical theory does not explain spontaneous emission. This means that the derivation of the A coefficient requires quantization of both matter and electromagnetic field (see, e.g.5 ). Contrarily, the standard semiclassical theory allows to derive an expression for the B coefficient, yielding the same result as obtained with the fully quantized theory.5 We must remark, however, that there is at least one formulation of the semiclassical theory that accounts for spontaneous emission: Self-field Quantum Electrodynamics, the theory developed by A. O. Barut and collaborators during the 1980 decade. See A. O. Barut and J. P. Dowling, Self-field quantum electrodynamics: the two–level atom, Phys. Rev. A 41, 2284-2294 (1990) and references therein.

9

S.M. Barnet and P.M. Radmore, Methods in theoretical quantum optics (Oxford University Press, 2003).

10

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995).

11

G.E.P. Box and M.E. Muller, A Note on the Generation of Random Normal Deviates, Ann. Math. Statist. 29, 610-611 (1958).

12

P.E. Kloeden and E. Platen, The numerical solution of stochastic differential equations, (Springer-Verlag, 1995).

13

For Ω ∈ R, it is easy to show that Re σ21 = 0 in Eqs. (4). Hence, Im σ21 can be identified with q¯ in this case.

14

H. Fearn and W. E. Lamb, Corrections to the golden rule, Phys. Rev. A 43, 2124-2128 (1991).

19

15

A random process ξ (t) of zero mean is said Gaussian if its n0 th-order correlation function verifies hξ (t1 ) ξ (t2 ) . . . ξ (tn )i = X

hξ (t1 ) ξ (t2 )i hξ (t3 ) ξ (t4 )i . . . hξ (tn−1 ) ξ (tn )i ,

all (n−1)!! pairings

for n even, and zero for n odd. i.e., for a Gaussian noise of zero mean all moments are known if the second order moment is. The Gaussian noise is said to be white when its second order moment verifies (39). See reference10 .

20

0.5

(a) ∆=1

averaged inversion

0.0

5 10

-0.5

-1.0 -0.7

0

1

2

3

5

ERE

(b)

EREmod

Bloch

-0.8

4

-0.98

Bloch

-0.9

-0.99

ERE

-1.00 0.0

-1.0

0

0.1

1

2

At 21 FIG. 1:

0.2

3

-0.1

(a) =1 -0.4

=1000

averaged inversion

=10 =100

-0.7

-1.0 0.2

0

1

2

3

4

(b)

=1 0.0 =10 -0.2

-0.4

2

4

At -0.6 22 FIG. 2:

6

averaged inversion

-0.45

(a) -0.55 -0.65 -0.75 0.04

(b) 0.00 -0.04 -0.08

1

10

100



23 FIG. 3:

1000

10000