fulltext

Infrared divergence and many-photon atomic transitions

M. COLA(1_{) and F. MIGLIETTA(}1₎₍2₎

(1_{) Dipartimento di Fisica Nucleare e Teorica dell’ Università di Pavia} via Bassi 6, 27100 Pavia, Italy

(2_{) INFN, Sezione di Pavia - Pavia, Italy}

(ricevuto il 23 Dicembre 1996; approvato il 22 Aprile 1997)

Summary. — The coherent-state formalism, which was developed originally in order to treat the infrared divergence of QED, is applied in this paper to the study of n-photon stimulated-emission (or absorption) transitions in atomic physics, in cases in which the involved atomic states have different magnetic moments. We show the existence of a threshold for the electromagnetic energy-density, above which the n-photon transition is more probable than the corresponding electric-dipole one. The possibility of observing this effect for the 2 s_{1 O2}K 2 p1 O2 transition in hydrogen (the energy separation is due to the Lamb shift) is analyzed.

PACS 32.80 – Photon interactions with atoms.

1. – Introduction

Infrared divergence is a well-known problem in relativistic QED. It originates from the impossibility of separating a charged particle (electron or positron) from the soft-photon cloud which surrounds it [1]. In this paper we apply the coherent-state formalism, which was developed originally in order to cure the infrared divergence in relativistic QED [2-4], to a non-relativistic problem of absorption and stimulated emission of radiation by an atomic system. More specifically we analyze the interaction of an atom with a single mode of the radiation field, in which a very large number of photons is present. In particular we will be concerned with the study of transitions between two atomic levels, induced by a radiation-mode, whose frequency satisfies a

n-photon resonance condition

v_{mm 84 nv}0 (1.1)

(with n D1), where v0is the frequency of the radiation and vmm 8is the Bohr frequency given by

ˇ_v

mm 84 Em2 Em 8, (1.2)

in terms of the atomic energy levels Em, Em 8. We will assume that the long-wavelength approximation holds for every n, n 41 included (n41 corresponds to the usual

one-photon transition in electric-dipole approximation). The Coulomb gauge will be assumed also for the electromagnetic potential.

The analysis is based on the following physical idea. The atom cannot be separated from the classical magnetic field which surrounds it, owing to the atomic magnetic moment. From the quantum-electrodynamical point of view, the magnetic field consists of a condensate of transverse (i.e. physical) photons, which form a bound state with the atom [5]. We will show that, if the initial and final atomic states have different magnetic moments, these photons can partecipate in an atomic transition. This will be analyzed in details in sect. 3, where the probability per unit-time for a transition corresponding to a stimulated n-photon emission is calculated. The analysis will show the existence of a threshold for the electromagnetic energy-density, above which the n-photon transition is more probable than the corresponding electric-dipole transition. It will show also that the n-photon transition probability per unit-time (n D1) increases for decreasing v_{mm 8}and diverges both in the limits v_{mm 8K 0 and n K Q , in agreement} with the infrared divergence of QED.

2. – Magnetic moments and coherent states

We assume an atom, localized in r 40, which interacts with a single mode (k, a) of the electromagnetic field, described in the Coulomb gauge.

The unperturbed Hamiltonian H0is given by

H04 Ha1 ˇv0a†a , (2.1)

where v04 ck. The eigenstates and eigenvalues of Ha are labelled in the following way:

HaNmb 4 Em0Nmb. (2.2)

The eigenstates of the field are labelled by the number of photons. The interaction Hamiltonian H8

intwill be split in the following way:

H8

int4 (v 1 V) 1H.c. (2.3)

In eq. (2.3) we have separated the part of the interaction which is diagonal in the eigenstates of Ha

v 4

!

_mNmb amNvNmb amN (2.4)

(we assume amNVNmb 40 for every m). In the long-wavelength limit one obtains amNvNmb C2mBgm

g

ˇ_v0 2 e0c2V

h

1 O2 aekaQ amNjNmb , (2.5)

where j is the total angular momentum operator, mBis the Bohr magneton and

gm4 3 2 1 S(S 11)2L(L11) 2 J(J 11) (2.6)

electric-dipole approximation we obtain, for the non-diagonal part of the interaction, am 8 NVNmb 4 am 8 NV† Nmb C 2ivmm 8

g

ˇ 2 e0v0V

h

1 O2 ekaQ am 8 NdNmb 4 (2.7) 4 2i

g

nˇvm 8 m 2 e0V

h

1 O2 ekaQ am 8 NdNmb ,

where the resonance condition (1.1) has been used (we assume real polarization-vectors eka).

It is known, from the study of the infrared divergence of QED, that a physical electron cannot be separated from the coherent field consisting of soft photons which surround it. This result holds for bound electrons also, as in our case. For this reason it is convenient to introduce the following unitary operator:

D4

!

mNmb D(zm)amN , (2.8)

where D(z) is the displacement operator

D(z) 4exp [2z * a1za†_{] ,} (2.9)

which transforms the vacuum state into a coherent state. The complex numbers zmare assumed independent of time and they will be determined in the sequel

(

see eq. (2.13)

)

. By means of the unitary operator D of eq. (2.8) we can transform the time-evolution operator U(t) in the following way:

U(t) 4 DU×(t) D†_. (2.10)

Owing to the time-independence of D, the operator U× is solution to the following Schrödinger equation: iˇ d dt U×(t) 4 H × TU×(t) , (2.11)

where the new Hamiltonian H×Tis given by

H × T4 D†HTD4ˇv0a†a 1

!

mNmb][E 0 m1 ˇv0NzmN21 amNvNmb(zm1 z *m) ] 1 (2.12)

1a†(ˇv0zm1 amNvNmb) 1 a(ˇv0z *m1 amNvNmb)(amN 1 D†(Va 1V†a†) D. Equation (2.12) suggests the following choice for zm:

zm4 2 1 ˇv0 amNvNmb 4mBgm

g

1 2 e0c2ˇv0V

h

1 O2 aekaQ amNjNmb . (2.13)

With this choice for zm, we obtain the following simple expression for H×T:

H × T4

!

mNmb EmamN1Hint, (2.14) where Hint4 D†(Va 1V†a†) D . (2.15)

In eq. (2.14) we have absorbed into Emthe contribution, due to the mode (k , a), to the Lamb shift of the level Nmb. This contribution is given by the expression in squared brackets of eq. (2.12).

3. – Atomic transitions induced by the electromagnetic field

In order to calculate transition probabilities, it is convenient to go to the interaction picture. Let us define

HI(t) 4e i ˇH0t Hinte 2i ˇH0t 4 DI†]V † I(t) a † eiv0t 1 VI(t) ae2iv0 t ( DI(t) . (3.1) In eq. (3.1) VIis given by VI(t) 4e i ˇHat Ve 2i ˇHat (3.2)

and for DI, by simple calculations, one obtains the following expression: DI(t) 4e i ˇH0t De 2i ˇH0t 4

!

_mNmb eiv0a†a_D(z m) e2iv0a †_a amN4 (3.3) 4

!

_mNmb D(zmeiv0t) amN . Up to the first order in perturbation theory the S-matrix is given by the limit T KQ of the following expression:

S( 1 ) B 2 i ˇ



2T/2 T/2 dt HI(t) 42 i ˇ

!

m

!

_{m 8}e ixm 8 m_{Nm 8 b am 8 NVNmb}



2T/2 T/2 dt eivm 8 mt_Q (3.4) Q](a†_eiv0t 1 zm 8* ) D

(

zmm 8(t)

)

1 D

(

zmm 8(t)

)

(aeiv0t1 zm)(amN , where, for the sake of simplicity, we have assumed am 8 NVNmb real. In eq. (3.4) we have introduced the following abbreviations:

ix_{mm 8(t) 4} 1

2(zm 8* zm2 zm 8zm* ) , (3.5)

z_{mm 8(t) 4 (z}m2 zm8 ) eiv0t (3.6)

and the following properties of the displacement operator D:

D†_{(z 8) aD(z) 4D}†_{(z 8) D(z)(a1z) 4e}

1

2(z 8* z2z 8 z *)

D(z 82z)(a1z)

(3.7)

have been used.

Let the initial state be Nmb7NNb and the final one Nm 8b7NN 8b, where NNb, NN 8b are states with N, N 8 photons in the (k, a) mode. Let us assume that both N and N 8 are very large and that n 4NN 82NNbN. The S-matrix element between these states

is given by aN 8 Nam 8 NS( 1 ) Nmb Nnb C 2 i ˇ e ix_{m 8 m} am 8 NVNmbQ (3.8) Q



2T/2 T/2 dt eivm 8 mt_]eiv0t_{N 8}1 O2_{aN 821ND(z} mm 8) NNb1 1e2iv0t_N1 O2_{aN 8 ND(z} mm 8) NN21b1 (zm 8* 1zm)aN 8 ND(zmm 8) NNb( , according to eq. (3.4).

We observe that, for NzNb1, retaining the lowest power only in NzN we have

aNND(z)NN2nb 4 1

k

(N 2n)!aNN(a † 2 z * )N 2nD(z) N0b B (3.9) B 1

k

(N 2n)! aNNa†(N 2n)_{D(z) N0b 4}

o

N! n!(N 2n)!anND(z)N0b B B

o

N! (N 2n)! zn n! B (N1 O2_z)n n!

and in a similar way

aN 2nND(z)NNb B (2N

1 O2_{z * )}n

n! .

(3.10)

According to eqs. (3.9), (3.10) we obtain from eq. (3.8) for the n-photon stimulated-emission transition aN 1nNam 8 NS( 1 ) Nmb NNb B (3.11) B 2 iT ˇ dvm 8 m, nv0e ixm 8 m_{am 8 NVNmb} N nO2_(z m2 zm 8)n 21 (n 21)! ,

where the lowest power in Nzmm 8N only has been retained

(

i.e. the first term only in eq. (3.8), which descended from the term with the creation operator a† _{of eq. (3.4)}

₎

_. In a similar way, for a n-photon absorption process, we obtain

aN 2nNam 8 NS( 1 ) Nmb NNb B (3.12) B (2)n iT ˇ dvm 8 m, nv0e ix_{m 8 m} am 8 NVNmb N nO2_(z m* 2zm 8* )n 21 (n 21)! .

In the sequel we will refer to the n-photon stimulated-emission process of eq. (3.11). The transition probability per unit-time is given by

dP(n) dt B 1 TNaN 1 nNam 8 NS ( 1 ) Nmb NNb N2B (3.13) B T ˇ2_{[ (n 21)! ]}2dvm 8 m, nv0NamNVNm 8 b N 2

g

n E ˇ_v m 8 m

h

n Nzm2 zm8 N2(n 21)4 4 dvm 8 m, nv0 TW 2 e0ˇ2[ (n 21)! ]2 n2 n_g m 8 m n 21 _,

where E is the total energy initially present on the e.m. mode, W 4 E/V is the energy density and the adimensional parameter g_{m 8 m}is given by

g_{m 8 m4} W

G_{m 8 m} , (3.14)

in terms of the parameter G_{m 8 m}defined in the following way:

G_{m 8 m4 2 e}0c2m22B ˇ2v2m 8 mNekaQ [gmamNjNmb2gm 8am 8 NjNm 8b]N22. (3.15)

In the case n 41 we have the electric-dipole transition, which corresponds to dP( 1 ) dt B dvm 8 m, nv0 TW 2 e0ˇ2 . (3.16)

By a comparison of eq. (3.16) with eq. (3.13), both in resonance condition, we obtain the following ratio: dP(n) dP( 1 ) B n 2 n [ (n 21)! ]22_g m 8 m n 21 _. (3.17)

Equation (3.17), together with eq. (3.14), show that G_{m 8 m} is a critical value for the electromagnetic energy-density W. In fact, for W DGm 8 m, the n-photon transition probability per unit-time dPn_{/dt increases for increasing n and diverges in the limit}

n KQ, as one can read in eq. (3.17). This is a direct manifestation of the infrared

divergence of QED, in a non-relativistic problem concerning atomic physics.

In order to observe this effect, a very small value for v_{mm 8}is required. However a lower limit to v_{mm 8}is imposed by the natural line-width of either the atomic levels. For a 2 p level in hydrogen the line-width is approximately dv A108_s21_{. Therefore it is} reasonable to assume v_{mm 8D 10}8_s21_.

Let us consider the 2 s_{1 O2K 2 p1 O2} transition in hydrogen, for which v_{mm 8A 10}9_s21 (the separation energy is due to the Lamb shift) [6]. It is well known that the 2 s state is metastable (with a life-time T A1021_{s). Assuming it as initial state Nmb and assuming} the 2 p_{1 O2 state as final one Nm 8b we obtain the value Gmm 8A 2 . 5 3 10}22_{J 3cm}23_. Assuming in eq. (3.13) T Advmm 821 A 1028s , we obtain the following expression for the life-time tn of the 2 s level, in the presence of the energy-density W on the

electromagnetic mode: t21n 4 dP(n) dt B 10 17_n2 n [ (n 21)! ]22_Ggn_, (3.18)

where G is measured in J 3cm23_{. The observability of the effect requires t}

nb1021s. This corresponds, for the two-photon transition, to g c 1027_{and finally to the condition}

W c 1029_{J 3cm}23 _{for the energy-density W. This shows that the effect should be} observable, in principle.

4. – Conclusion

In this paper we have shown that, in atomic physics, the existence of the magnetic field, due to the atomic magnetic moment, has non-trivial consequences concerning the

atomic transitions induced by the electromagnetic field. In fact, from the

quantum-electrodynamical point of view, the magnetic field consists of a condensate of transverse (i.e. physical) photons which can partecipate in an electromagnetic transition, when the initial and final states have different magnetic moments. The analysis of the 2 s_{1 O2K 2 p1 O2} transition in hydrogen, performed at the end of sec. 3,

indicates the possibility, at least in principle, of observing a n-photon

stimulated-emission transition between these levels. The observation of such an effect would be very interesting from a fundamental point of view, because the mechanism of transition, as shown in sect. 3 , deeply involves the infrared divergent structure of QED.

R E F E R E N C E S

[1] BLOCHF. and NORDSIECKA., Phys. Rev., 52 (1937) 54. [2] CHUNGV., Phys. Rev. B, 140 (1965) 1110.

[3] GRECOM. and ROSSIG., Nuovo Cimento, LA1 (1967) 168. [4] KULISHP. P. and FADDEEVL. D., Teor. Mat. Fiz., 4 (1970) 153. [5] MIGLIETTAF., Nuovo Cimento A, 108 (1995) 205.

[6] The energy-separation between the two levels could be reduced, in principle, by the application of a static magnetic field.