deviations from spontaneous lyman-α exponential decay revisited: a rigorous treatment within a...

September 10, 1999 15:2 WSPC/147-MPLB 0175

Modern Physics Letters B, Vol. 13, No. 16 (1999) 531–539c© World Scientific Publishing Company

DEVIATIONS FROM SPONTANEOUS LYMAN-α EXPONENTIAL

DECAY REVISITED: A RIGOROUS TREATMENT WITHIN A

TWO-LEVEL NONRELATIVISTIC MODEL

J. SEKE∗, A. V. SOLDATOV† and N. N. BOGOLUBOV, JR.‡

Institut fur Theoretische Physik, Technische Universitat Wien,Wiedner Hauptstrasse 8-10/136, A-1040 Wien, Austria

Received 15 July 1999

A recent paper by Facchi and Pascazio, in which the advantage of numerical evaluationsof deviations from exponential decay was pointed out, makes a revisitation of the subjectnecessary. All the more, since the above-mentioned authors have not estimated theerror bounds (being absolutely necessary in any calculation) for their results. On thecontrary, the accuracy of all results presented in this paper is proved by error estimates.Moreover, for the first time, the power of the simple iteration method in calculatingboth the exponential and non-exponential time evolution of an unstable atomic state isdemonstrated.

PACS Number(s): 32.90.+a, 42.50.p, 31.30.Jv

1. Introduction

Recently, Facchi and Pascazio1 published a paper in which the Seke–Herfort an-

alytical results for the Lyman-α spontaneous decay2 obtained a decade ago were

recalculated. The credit of this paper lies in focusing the attention to the new

computational facilities provided for numerical calculations of the non-exponential

decay in short- and intermediate-time regimes.1 The advantage of the numerical

calculations lies in the fact that the inverse of the Laplace-transformed equation

for the non-decay probability amplitude can be easily found.

Since in the mentioned paper,1 neither new analytical results for the deviations

from exponential decay were obtained, nor were any error estimations of the results

carried out, a revisitation of the analytical treatment seems to be urgently called

for.

Moreover, the purpose of the present paper is to demonstrate the power of the

simple iteration method in calculating both the exponential and non-exponential

time evolution of an unstable atomic state. It should be stressed that, in contrast

∗E-mail: [email protected]†,‡Permanent address: V. A. Steklov Mathematical Institute, Dept. of Statistical Mechanics, GSP-1, 117966, Gubkin Str. 8, Moscow, Russia. E-mail: [email protected], [email protected]

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532 J. Seke, A. V. Soldatov & N. N. Bogolubov, Jr.

to other authors, all analytical results presented here are provided with rigorous

error estimates.

2. Nonrelativistic Two-Level Hamiltonian and the

Non-Decay Probability Amplitude

The Hamiltonian for the nonrelativistic 2P → 1S decay in the interaction picture

reads as2:

HRT (t) =

∫ ∞0

dωH2P,1S(ω)ei(ω0−ω)t|2P 〉〈1S| ⊗ |v〉〈ω|+ H.c. (1)

where ω0 = E2P −E1S = (3/8)α2mec2/~ = 1.55×1016 s−1 is the energy separation

between 2P and 1S states,

H2P,1S(ω) = (λ/2π)1/2(−iω1/2)[1 + (ω/Ω)2]−2

is the transition matrix element, λ = γω0 (γ is the Einstein coefficient for spon-

taneous Lyman-α transition), Ω = 3/(2a0) = (3/2)αmec2/~ = 8.498 × 1018 s−1

(a0 is the Bohr radius) is the natural cutoff frequency, α is the fine-structure con-

stant and me is the electron mass. The state |v〉 is the photon vacuum state while

the state |ω〉 describes one-photon state with the frequency ω.

From the Schrodinger equation in the interaction picture, by using the Hamil-

tonian (1), the following integro-differential equation for the non-decay probability

amplitude b(t) (the probability amplitude of finding the atom in the initial state

2P and zero photons in the radiation field) can be derived:

b(t) = −∫ ∞

0

dω|H2P,1S(ω)|2∫ t

0

dτei(ω0−ω)τb(t− τ) . (2)

From now on, it will be more convenient to use scaled variables:

t = tΩ, ω0 = ω0/Ω, ω = ω/Ω .

The change of variables leads to

b(t ) = −χ∫ ∞

0

dωf(ω)

∫ t

0

dτei(ω0−ω)(t−τ)b(τ ) , (3)

where χ = λ/2π = 2/π(2/3)9α3 = 6.435× 10−9 and

f(ω) =ω

(1 + ω2)4. (4)

3. Iteration Method

The above Eq. (3) is a non-Markovian one. In the literature, this equation has

been formally treated by using the Laplace transform and its inverse. However,

this method is somewhat cumbersome and it has been overlooked that the iteration

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Deviations from Spontaneous Lyman-α Exponential Decay . . . 533

method combined with the method of complex function analysis make a more simple

treatment possible. The integration of Eq. (3) results in

b(t ) = b(0)− χ∫ t

0

dt1

∫ ∞0

dωf(ω)

∫ t1

0

dt2ei(ω0−ω)(t1−t2)b(t2) (5)

and the iterative solution of the nth order reads as

b(t ) = b(n)(t ) + brestn (t, n)

where

b(n)(t ) = b(0) +n∑

m=1

(−χ)m∫ t

0

dt1

∫ ∞0

dω1f(ω1)

∫ t1

0

dt2ei(ω0−ω1)(t1−t2)

×∫ tm

0

dtm+1

∫ ∞0

dωmf(ωm)

∫ tm+1

0

dtm+2ei(ω0−ωm)(tm+1−tm+2)b1(0) (6)

and the rest term is given by

brestn (t, n) = (−χ)n+1

∫ t

0

dt1

∫ ∞0

dω1f(ω1)

∫ t1

0

dt2ei(ω0−ω1)(t1−t2)

×∫ tn+2

0

dtn+3

∫ ∞0

dωn+1f(ωn+1)

×∫ tn+3

0

dtn+4ei(ω0−ωn+1)(tn+3−tn+4)b(tn+4) .

The error bound for each subsequent step of iteration follows immediately:

|brestn (t, n)| ≤ 1

(2n+ 2)!

(χ6

)n+1

t 2n+2. (7)

In deriving Eq. (7), we took into account that∫∞

0 dωf(ω) = 1/6. For any given

time interval, the uniform convergence of our iteration procedure is guaranteed by

the inequality (7). By applying the inverse transformation to the Laplace transform

of Eq. (6),

B(n)(u) =b1(0)

u− ω0

[1 +

n∑m=1

(−1)m

(u− ω0)m(χI(u))m

](8)

leads to the following formal solution:

b(n)(t ) =eiω0t

2πi

∫C

due−ituB(n)(u) (9)

where the contour of C runs from +∞+ iε to −∞+ iε (ε > 0). In deriving Eq. (8),

we used

I(u) =

∫ ∞0

dωf(ω)

(ω − u)= −C1(u) +

π

2C2(u) + f(u)[Log(u) + iπ] (10)

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and

C1(u) =f(u)

12(11 + 18u2 + 9u4 + 2u6), C2(u) =

f(u)

16(5− 15u2− 5u4−u6) (11)

which were first calculated in Ref. 2. Here and later on, we denote the multi-valued

natural logarithm of a complex variable by Log(u).

Let us now consider the inverse Laplace transform, given by Eq. (9), in the

first-order iteration only:

b(1)(t ) = b(0)(

1− χeiω0tR(1)(t ))

+D(t ) , (12)

where

R(1)(t ) = Res

[eitu

(u− ω0)2I(u);u = ω0

]= e−itω0

(dI(u)

du

∣∣∣∣u=ω0

− itI(ω0)

)

is the residue contribution stemming from the pole at ω0, and the term describing

the deviation from the exponential decay

D(t ) = b(0)χ(1− i)eiω0 t

∫ ∞0

dse−it((1−i)s)

((1− i)s− ω0)2f((1− i)s) (13)

arises as a consequence of the logarithmic singularity appearing in I(u) function.

In order to evaluate most effectively the branch cut contribution, stemming from

the complex Log(u), analogously as in Ref. 2, we deformed the path of integration

C in such a way that the dealing with the pole at u = −i is avoided.

It is obvious that the first iterative solution (cf. Eq. (12)) satisfies the initial

conditions

b(1)(0) = b(0),d

dtb(1)(t )

∣∣∣∣t=0

= 0 . (14)

Moreover, the deviation term D(t ) in Eq. (12), which stems from the branch

cut contribution, reveals correctly the long-time asymptotics t → ∞ (Abel’s

asymptotics):

D(t → +∞) = M(t) = −b(0)eiω0tχ

ω20 t

2. (15)

Thus, it has been shown, for the first time, that the first-order iteration already

leads to the correct asymptotics. Moreover, since the term D(t ) gives the correct

short-time deviation from the exponential decay (the largest deviation occurs at

t = 0 and is of order 10−7 (cf. Ref. 2) as well, it is evident that this term must

describe correctly the deviations for all times. To restore the correct complete time

evolution from the Laplace transform (8), no attention has to be paid to terms

describing branch cut contributions stemming from higher-order iterations. Thus,

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we can approximate the exact Laplace transform B(n)(u) as follows:

B(n)(u) ≈ b(0)

u− ω0

[1− χ

u− ω0I(u)

]

+b(0)

u− ω0

[n∑

m=2

(−1)mχm

(u− ω0)m

(I(ω0) +

dI(u)

du

∣∣∣∣u=ω0

(u− ω0)

)m]. (16)

The infinite resummation of the r.h.s. of Eq. (16) yields:

B(∞)(u) ≈ b(0)

u− ω0 + χ

(I(ω0) + dI(u)

du

∣∣∣u=ω0

(u− ω0)

)

− χb(0)

(u− ω0)2

(I(u)− I(ω0)− dI(u)

du

∣∣∣∣u=ω0

(u− ω0)

)(17)

where the inverse Laplace transform of the second term describes exactly the branch

cut contribution already obtained in the first-order iteration (12).

4. Solutions for the Non-Decay Probability Amplitude in Different

Time Regimes

The inverse Laplace transform of the r.h.s. of Eq. (17) yields an expression for the

probability amplitude:

b(t ) = R(t ) +D(t ) , (18)

R(t ) = b(0) exp

it χI(ω0)

1 + χdI(u)du

∣∣∣u=ω0

(1 + χdI(u)

du

∣∣∣∣u=ω0

)−1

= b(0)(1− χI ′(ω0) + ∆res)e−itω0(∆E2P−iλ2 ) ,

where Ref. 3 gives

I ′(ω0) ≈ −23

12− ln

(Ω

ω0

)+ iπ, |∆res| ≤ 6.8× 10−3λ ,

∆E2P =11λω0

24π− 5λΩ

64− λω0

2πln

Ω

ω0.

This expression is valid for the whole time evolution and may serve as a basis

for analytical approximations in different time regimes. Simple error estimations,

∆err, for any approximative solution of Eq. (3) follow immediately from the triangle

inequality, |a− b| ≤ |a− c|+ |c− b|, with a = b(t ), b = b(appr)(t ) and c = b(n)(t ):

|b(t )− b(appr)(t )| ≤ |b(t )− b(n)(t )|+ |b(appr)(t )− b(n)(t )| .

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It is easy to see that if the probability amplitude b(appr)(t ) lies within a given error

bound

|b(t )− b(appr)(t )| ≤ ∆(t ) (19)

then, the probability P (appr)(t ) = |b(appr)(t )|2 itself lies within the following error

bound:

|P (t )− P (appr)(t )| ≤ 2∆(t ) . (20)

Thus, expanding the r.h.s. of Eq. (18) in t up to the second order we get the

standard short-time approximation

b(s.t.)(t ) = 1− χ

12t 2 + · · · (21)

which leads to the corresponding probability:

P (s.t.)(t ) = 1 +χ

6t 2 + · · · , ∆err(t ) ≤ 4× 10−11, 0 ≤ t ≤ 1 (22)

where the coefficient√

6/χ is the so-called Zeno time.1

It can be easily seen that if the function f(ω) in Eq. (12) is approximated by ω,

a correct asymptotics, Eq. (15) follows. Therefore, this approximation can be used

in the intermediate- and long-time regime as well. This latter leads to:

b(t ) = R(t )− χb(0)eitω0

[1− (1− itω0)e−itω0Γ[0,−itω0]

], 10 ≤ t ≤ +∞ (23)

where Γ[0,−itω0] stands for the incomplete gamma function, defined for real as

well as complex argument z:

Γ[a, z] =

∫ +∞

z

ta−1e−tdt =

∫ +∞

0

(t+ z)a−1e−(t+z)dt. (24)

The error bound for the probability, P (t ), in the intermediate time interval 10 ≤t ≤ 1000 is: ∆err(t ) ≤ 6× 10−10.

As shown in Refs. 2 and 3, for times t > 105, it is possible to replace the D(t )

in Eq. (23) by its asymptotic solution. This leads to the following equation for the

non-decay probability3:

P (t ) = |b(0)|2

1− λ

π

[ln

(Ω

ω0

)− 23

12

]+ ∆1

exp(−λω0 t ) (25)

− λ

2π(ω0)2 t 2|b(0)|2 exp(−λω0)2 cos[ω0 + ∆E2P ] + ∆2(t) (26)

+

∣∣∣∣ λb(0)

2π(ω0)2t 2

∣∣∣∣2 (1 + ∆3(t)− |b(0)|2 + 1, t ≥ 8.5× 105 (27)

where the error estimates read: ∆1(t ) < 1.4× 10−2λ, ∆2(t ) < 34× 10−2, ∆3(t ) <

3.6× 10−1.

All regimes of the time evolution mentioned above are calculated numerically for

the same values of constants as in the numerical calculations of the inverse Laplace

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Fig. 1. Coincident numerical solutions (within indicated error bounds |∆|) between non-decayprobabilities P (t) − 1 obtained from exact (cf. Ref. 2) and approximated (cf. Eq. (17)) Laplacetransforms in different time regimes.

transform reported in Ref. 1. We prove numerically that the exact inverse Laplace

transform found analytically by Seke and Herfort2 and calculated later by Facchi

and Pascazio1 numerically matches with high precision our result (18) obtained by

the infinite resummation of the iteration series. In Fig. 1, where both results were

plotted simultaneously in various time regimes, no difference can be observed.

In Fig. 2, we plotted our approximated analytical results (21) (cf. Fig. 2(a)

and (23), (cf. Figs. 2(b)–(d)), which coincide (within the error bounds given in the

figures) with the exact numerical result in the corresponding time regimes.

5. Discussion

It was claimed in Ref. 1 that the so-called long-time asymptotic approximation (15)

already holds for t > 200 and this is in contradiction with the traditional point of

view on the subject expressed, for example, in Ref. 2. In the latter paper, it has

been proved that this approximation is valid at least for t ≥ 105. The discrepancy

between the two statements has its root in a fact somehow overlooked in Ref. 1.

Evaluating the relative error, the authors in Ref. 1 turned to a simple stand-by

criterion:

|bappr(t )| − |b(t )||bappr(t )| 1 (26)

and found that it already holds for t > 200 perfectly. However, they did not pay

attention to the mere fact that both, the exact as well as the approximate, solutions

are made up of two qualitatively different parts: the pole and the cut contribution.

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Fig. 2. Analytical approximations (21) and (23) (with indicated error bounds |∆|) versus exactnumerical solutions (dotted line) for the non-decay probability (with indicated error bounds |∆|)in different time regimes.

The pole contribution remains unchanged in the course of the approximation be-

cause it can be calculated easily as it is. It is precisely the cut contribution which

has to be approximated independently, thus, leading to the long-time asymptotics.

As a consequence, one should apply the criterion (26) to the cut contribution only

and not to the whole solution. In this case, Eq. (26) takes the form

|D(t )−M(t )||M(t )| 1 (27)

with D(t ) and M(t ) standing for the exact and asymptotic cut contributions,

respectively. In Ref. 2, only this approach was implemented and after a very involved

error estimation it has been proved that

|D(t )−M(t )||M(t )| < 16.6× 10−2 . (28)

Here, we were able to reconsider this classical result and managed to prove that

|D(t )−M(t )||M(t )| =

∣∣∣∣∣ω2

(1− 5πχ

32ω0

)2 ∫ ∞0

dse−it(1−i)s∂2

∂u2

f(u)

N0(u)N−1(u)

∣∣∣∣∣ (29)

where u = (1− i)s and

N0(u) = u− ω0 + χI0(u), N−1(u) = N0(u) + i2πχf(u) .

Equality (29) provides a basis for estimations of numerous effective error bounds

which can be easily calculated numerically, thus, avoiding the difficult analysis

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undertaken in Ref. 2. Finally, we put our choice on the inequality which follows

directly from (29):

|D(t )−M(t )||M(t )| ≤ ω2

(1− 5πχ

32ω0

)2 ∣∣∣∣∫ s0

0

dse−it(1−i)s∂2

∂u2

f(u)

N0(u)N−1(u)

∣∣∣∣+ ω2

(1− 5πχ

32ω0

)2

e−s0t∫ ∞s0

ds

∣∣∣∣ ∂2

∂u2

f(u)

N0(u)N−1(u)

∣∣∣∣ . (30)

Inserting for the arbitrary parameter s0 = 0.01, and for t = 17000, for example, the

equality

|D(t )−M(t )||M(t )| ≈ 0.0980574 ,

which does not contradict the result outlined in Ref. 2 (and has nothing to do with

the statement in Ref. 1), can be proved for sure. Thus, one can state that the long-

time asymptotics starts somewhere at t = 17000 and is valid for all later times if

one accepts the commonly adopted smallness criterion 0.1 as satisfactory.

Of course, this latter statement should not be misunderstood in the sense that

after t ≥ 25000 we have the long-time regime immediately. The long-time regime

is quite a different thing. It exists only when the asymptotic cut contribution term

M(t ) starts to dominate over the exact pole contribution term R(t ) and this takes

place for much larger times. We evaluated this time moment out of the following

criterion:

|R(t )||M(t )| ≤ 0.1 . (31)

It happened that the long-time regime in the above-mentioned sense starts at t =

3.91027× 1011 which corresponds roughly to the physical time t ≈ 10−8 sec.

Acknowledgments

J. Seke acknowledges the support by the Jubilaumsfonds der Oesterreichischen Na-

tionalbank zur Forderung der Forschungs- und Lehraufgaben der Wissenschaft un-

der Contract No. 7720. N. N. Bogolubov and A. V. Soldatov would like to acknowl-

edge the RF Support for Leading Scientific Schools, Grant No. 96-15-96033.

N. N. Bogolubov and A. V. Soldatov would like to express their gratitude to

the Osterreichische Akademie der Wissenschaften and personally to Professor Otto

Hittmair for inviting them to visit the Institut fur Theoretische Physik, Technische

Universitat Wien and thus, making this research possible.

References

1. P. Facchi and F. Pascazio, Phys. Lett. A241, 139 (1998).2. J. Seke and W. Herfort, Phys. Rev. A40, 1926 (1989).3. J. Seke and W. Herfort, Physica A178, 561 (1991).

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deviations from spontaneous lyman-α exponential decay revisited: a rigorous treatment within a...

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