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PHYSICAL REVIEW A

VOLUME 55, NUMBER 6

JUNE 1997

Causality in the excitation exchange between identical atoms P. R. Berman and B. Dubetsky Physics Department, University of Michigan, Ann Arbor, Michigan 48109-1120 ~Received 7 January 1997! The interaction between two identical atoms, separated by a distance r that is large compared to an optical wavelength, is considered using an amplitude approach in the Schro¨dinger picture. When atom 1 is excited by a radiation pulse whose duration is long compared to an optical period but short compared to the time it takes for light to travel between the atoms, the probability for atom 2 to be excited is found to be identically zero for times t,r/c, provided only r-dependent terms in the probability are retained. The r-dependent probability amplitude for atom 2 to be excited and atom 1 to be in its ground state is not identically equal to zero for t ,r/c. Rather it oscillates at the natural frequency of the optical transition. The Schro¨dinger picture approach allows one to clearly identify the various terms that give rise to the excitation probabilities. A diagrammatic technique is developed that facilitates the calculation. @S1050-2947~97!04405-3# PACS number~s!: 03.65.Ge, 32.80.Lg, 42.50.Fx

I. INTRODUCTION

A problem of fundamental importance in quantum electrodynamics relates to the excitation exchange between a pair of identical atoms. Suppose that both atoms are in their ground states and are separated by a distance r that is large compared to an optical wavelength. At t50, a pulse of radiation is incident on atom 1 that excites the atom to its lowest-lying electronic state. The pulse duration is much longer than an optical period, but much shorter than both the excited-state lifetime and the time t r 5r/c it takes for light to travel between the two atoms. The question to be addressed in this paper is, What is the earliest time for which atom 2 has a nonzero excitation probability? Based on the special theory of relativity, it might seem that this question has a trivial response: Atom 2 cannot be excited for times t ,r/c. It turns out, however, that, as is often the case, things are not necessarily that simple. Moreover, if one modifies the question slightly and asks, What is the earliest time at which the joint probability for atom 2 to be excited and atom 1 to be in its ground state is nonvanishing?, the problem takes on an added dimension. Recently, the question of causality in the resonant excitation exchange between two atoms was addressed by Hegerfeldt @1#. Using very general mathematical considerations, he concluded that atom 2 has a nonvanishing excitation probability for times t,r/c. Hegerfeldt’s conclusion was questioned by Milonni and co-workers @2#, who used a Heisenberg operator approach to prove that the excitation probability for atom 2 remains equal to zero for times t ,r/c, provided one considers only those terms that depend on the interatomic separation r. Milonni et al.’s result is consistent with that of Biswas et al. @3#, who also used a Heisenberg operator approach to establish the causal nature of the response. Thus it appears that Hegerfeldt’s result does not isolate the r-dependent contribution to the excitation of atom 2. To further investigate this topic, we calculate the excitation probability for atom 2 using the Schro¨dinger picture. Although the Heisenberg approach directly reveals the retardation effects, it conceals some of the interesting atomfield dynamics that can be seen using the Schro¨dinger ap1050-2947/97/55~6!/4060~10!/$10.00

55

proach. The price one pays for this clarity is a much more complicated calculation @4#. The Schro¨dinger picture also allows one to address the second question posed above regarding the joint probability P ge of finding atom 2 excited and atom 1 in its ground state. It has already been shown using both a Heisenberg operator approach @3,5# and a Schro¨dinger wave-function approach @6# that this joint probability is nonvanishing for times t ,r/c. This result appears paradoxical at first glance. The total probability for atom 2 to be excited is the sum of the joint probability P ge for atom 2 to be excited and atom 1 in its ground state plus the joint probability P ee for atom 2 to be excited and atom 1 in its excited state. How can this sum of positive-definite quantities ( P ge 1 P ee ) be zero for times t ,r/c when P ge itself is nonvanishing in this time interval? This apparent paradox is resolved in Secs. III and IV. The paper is organized as follows. In Sec. II the notation is established and the equations of motion are obtained. In Sec. III the probability for atom 2 to be excited is obtained. In order to carry out the calculation, which involves 96 terms, we develop a diagrammatic technique that greatly facilitates matters. This diagrammatic technique is applicable to a much wider range of problems involving time-dependent perturbation theory. Section IV is devoted to a calculation of the joint probability P ge . It is shown that, although P ge is not identically zero for t,r/c, it oscillates as sin v0t or cos v0t ~where v 0 is an optical frequency! in this time interval. One is led to conclude that P ge is effectively equal to zero for t,r/c, assuming that any measurement process is carried out on a time that is long compared to an optical period. The fact that P ge Þ0 for t,r/c is shown to be related to the nonvanishing of the vacuum field correlation function at different space-time points. This association was made by Biswas et al. @3#, without formal proof. Finally, we return to a discussion of Hegerfeldt’s paper @1# in Sec. V. II. EQUATIONS OF MOTION AND SOLUTION

The system under consideration is shown in Fig. 1. Two identical atoms are separated by a distance r that is large compared to the spatial extent of the electronic wave func4060

© 1997 The American Physical Society

55

CAUSALITY IN THE EXCITATION EXCHANGE . . .

FIG. 1. Schematic representation of the problem under discussion. Initially, atom 1 is in its excited state, atom 2 in its ground state, and there are no photons in the field. The atoms are separated by a distance r that is large compared to an optical wavelength.

tion of each of the atoms. Each atom has a ground state g and excited state e that are separated in frequency by v 0 . Initially, both atoms are in their ground states and there are no photons in the field. To avoid any confusion or spurious effects arising from the manner in which atom 1 is excited at t50, we assume that the excitation is produced by a ‘‘p’’ pulse of radiation whose duration t satisfies the inequalities r/c@ t @ v 21 0 .

~1!

These inequalities ensure that atom 2 cannot be excited during the excitation pulse. As a consequence, it is possible to take an initial condition in which atom 1 is excited and atom 2 in its ground state, provided time intervals are ‘‘course grained’’ on an interval that is large compared to v 21 0 . Specifically, it is assumed that about the initial time t50 and the retarded time t r 5r/c, we do not consider time intervals less than v 21 0 , that is,

v 0 u t u @1,

v 0 u t2r/c u @1.

~2!

From Eq. ~1!, it also follows that the calculation is restricted to interatomic separations larger than an optical wavelength @7#, k 0 r@1,

~3!

where k 0 5 v 0 /c. States other than g and e can be included in the calculation, but would not change the overall conclusions to be reached; as a consequence, such terms are neglected. The goal of the calculation is to map out the evolution of the atom-field system for times t.0. In the dipole approximation, it is possible to evaluate the field acting on atom i at its center-of-mass coordinate Ri (i 51,2). The Ri are taken to be fixed, classical variables in this calculation. In this limit, the Hamiltonian describing the two-atom system plus field is H5H 1 1H 2 1

(k \V k a †ka k1H 8 ,

~4!

where H 8 52 m1 •E~ R1 ! 2 m2 •E~ R2 ! , E~ R! 5

~5!

(k @ g k eka k exp~ ik•R! 1g k* eka †k exp~ 2ik•R!# , ~6!

4061

H i is the Hamiltonian for atom i, a k and a †k are the destruction and creation operators for a field mode having propagation vector k and frequency V k 5kc, mi is the dipole moment operator of atom i, g k 5i A\V k /2e 0 V, V is the quantization volume, and ek is the polarization vector for mode k ~there are two independent values of ek for each k!. The wave function for the system is expanded in an interaction representation as uC~ t !&5

(s

exp~ 2iE s t/\ ! b s ~ t ! u s & ,

~7!

where s is a label specifying states of the atoms and the field, E s is the energy associated with this state, and b is a probability amplitude. The expansion is written in terms of free states of the atoms and the field, implying that the ket us& can be expressed as u a 1 , a 2 ; $ k% & , where a 1 is the state of atom 1, a 2 is the state of atom 2, and $k% is a state of the field. To simplify matters, it is assumed that the ground and excited states of each of the atoms are nondegenerate. It follows from Schro¨dinger’s equation that the state amplitudes b s (t) evolve as b˙ s ~ t ! 5 ~ i\ ! 21

( s

8 b s 8~ t ! , exp~ iE ss 8 t/\ ! H ss 8

8

~8!

where E ss 8 5E s 2E s 8 . The initial condition is b e,g;0 ~ 0 ! 51,

~9!

with all other amplitudes equal to zero. To lowest nonvanishing order of perturbation theory, we wish to calculate the total probability P e~ 2 ! 5

u b a ,e; $ k% ~ t ! u 2 ( ( k $ %

a 5g,e

~10!

for atom 2 to be excited and the joint probability P ge 5

u b g,e; $ k% ~ t ! u 2 ( $ k%

~11!

for atom 1 to be in its ground state and atom 2 to be in its excited state. In calculating these probabilities, one finds terms that are r-independent terms ~representing the effect of vacuum fluctuations on atom 2 only! and terms that are functions of r. We retain only those terms that depend on the interatomic separation r. To lowest nonvanishing order in the atom-field interaction, the r-dependent contributions to P e (2) and P ge vary as u g k u 4 . It is relatively easy to map out the chains of atom-field interactions that give rise to the probabilities ~10! and ~11!. These chains are indicated schematically in Fig. 2. For example, Fig. 2~a! corresponds to two chains b e,g;0 →b g,g;k1 →b g,e;0 and b e,g;0 →b e,e;k1 →b g,e;0 , which give rise to the final-state amplitude b g,e;0 . When this amplitude is squared, it contributes to both P e (2) and P ge . As a second example, consider Fig. 2~b!, which corresponds to seven chains leading to the same final-state amplitude b e,e;k2 . One of these terms is first order in g k and the other

P. R. BERMAN AND B. DUBETSKY

4062

55

P b5

u b e,e;k u 2 , ( k

P c5

~14b!

2

2

(

u b g,e;k1 ,k2 u 2 ,

k1 ,k2

~14c!

and the P i (i5a,b,c) correspond to the absolute squares of the amplitudes represented schematically in Figs. 2~a!–2~c!. These amplitudes are obtained as a perturbative solution of Eq. ~8!. To guarantee the validity of the perturbative approach, it is assumed that Gt!1, where G is the excited-state decay rate. A diagrammatic technique is developed to evaluate the P i ’s. Readers not interested in the details of the calculation can proceed directly to Eq. ~42!. A. P a FIG. 2. Perturbation chains leading to the final-state amplitudes ~a! b g,e;0 , ~b! b e,e;k2 , and ~c! b g,e;k1 ,k2 . Each perturbation chain in ~a! and ~c! is second order in the atom-field coupling constant g k , while the top pathway in ~b! is of order g k and the remaining six pathways are of order g 3k .

Starting with the initial condition ~9!, one can solve Eq. ~8! for the amplitude b g,e;0 following the perturbative chains indicated schematically in Fig. 2~a!. Using Eqs. ~8!, ~9!, and ~12!, one finds 2ik1 •R1 b g,g;k1 ~ t ! 5 ~ i\ ! 21 g * k ~ 2 m* • e k1 ! e 1

six are third order in g k . When b e,e;k2 is squared, the term corresponding to the interference of the first- and third-order amplitudes is an r-dependent term that varies as u g k u 4 and contributes to P e (2) @8#. It turns out that such interference terms are critical for establishing the fact that P e (2) is identically zero for t,r/c. The remaining terms in Fig. 2 are considered in Sec. III. From Fig. 2 one sees that the energies of the states relevant to this calculation are E e,g;0 5\ v 0 , E g,e;0 5\ v 0 ,

E e,g;k 5\ ~ v 0 1V k ! , E g,e;k,k 8 5\ ~ v 0 1V k 1V k 8 ! ,

E

t

0

dt 8 exp@ i ~ V k 1 2 v 0 ! t 8 #

~15a!

2ik1 •R2 b e,e;k1 ~ t ! 5 ~ i\ ! 21 g * k ~ 2 m• ek1 ! e 1

3

E

t

0

dt 8 exp@ i ~ V k 1 1 v 0 ! t 8 # ,

~15b!

and

E g,g;k 5\V k ,

E e,e;k 5\ ~ 2 v 0 1V k ! ,

3

~12!

b g,e;0 ~ t ! 5 ~ i\ ! 21

gk ( k 1

1

E

t

0

dt 8 $ ~ 2 m• ek1 ! e ik1 •R2

E g,e;k 5\ ~ v 0 1V k ! ,

3e @ 2i ~ V k 1 2 v 0 ! t 8 # b g,g;k1 ~ t 8 !

E e,g;k,k 8 5\ ~ v 0 1V k 1V k 8 ! .

1 ~ 2 m* • ek1 ! e ik1 •R1 e @ 2i ~ V k 1 1 v 0 ! t 8 # b e,e;k1 ~ t 8 ! }

For these energies, one can determine from Fig. 2 that all contributions other than that in the top pathway of Fig. 2~a! correspond to ‘‘counterrotating’’ terms, in which one of the atoms is excited on the emission of a photon or one of the atoms is deexcited on the absorption of a photon. As is seen below, some of these counterrotating terms are critical to our discussion of causality, while others are not. III. CALCULATION OF P e „2…

52\ 22

1

1

1e 2ik1 •rI a ~ t;k1 ;2 !# ,

~16!

where r5R2 2R1

~17!

is the interatomic separation,

To order u g k u , the probability P e (2) for atom 2 to be excited is equal to

I a ~ t;k1 ;1 ! 5

E

t

0

dt 8 e 2i ~ V k 1 2 v 0 ! t 8

E

t8

0

dt 9 e i ~ V k 1 2 v 0 ! t 9 , ~18a!

~13!

where

I a ~ t;k1 ;2 ! 5 P a 5 u b g,e;0 u 2 ,

1

1

4

P e~ 2 ! 5 P a1 P b1 P c ,

u g k u 2 u m• ek u 2 @ e ik •rI a ~ t;k1 ;1 ! ( k

~14a!

E

t

0

dt 8 e 2i ~ V k 1 1 v 0 ! t 8

E

t8

0

dt 9 e i ~ V k 1 1 v 0 ! t 9 ~18b!

55


4063

refer to the two perturbative chains in Fig. 2~a!, and m 5 ^ e u m1 u g & 5 ^ e u m2 u g & is a matrix element. Although not indicated explicitly, each summation over k also includes a summation over two independent polarization vectors ek for each k. It follows that the probability P a 5 u b g,e;0 (t) u 2 is equal to P a5

(

k1 ,k2

C ~ k1 ,k2 !@ e ik1 •rI a ~ t;k1 ;1 ! 1e 2ik1 •rI a ~ t;k1 ;2 !#

3@ e ik2 •rI a ~ t;k2 ;1 ! 1e 2ik2 •rI a ~ t;k2 ;2 !# * ,

~19!

where C ~ k1 ,k2 ! 5\ 24 u g k 1 u 2 u g k 2 u 2 u m• ek1 u 2 u m• ek2 u 2 .

~20!

Rather than evaluating the integrals and summations appearing in Eqs. ~16!, ~18!, and ~19!, it is convenient at this point to introduce a diagrammatic technique that will greatly reduce the number of terms that need to be evaluated explicitly. Consider, for example, the product e ik1 •re 2ik2 •rI a (t;k1 ;1)I a* (t;k2 ;1) in Eq. ~19!. The product of the I’s can be written as I a ~ t;k1 ;1 ! I * a ~ t;k2 ;1 ! 5

E 8E 9E -E t

dt

0

t8

t

dt

0

dt

0

t-

dt i v

0

3A 1 ~ t i v ! A * 1 ~ t-!A* 2 ~ t 9 ! A 2~ t 8 ! ,

FIG. 3. Diagrammatic technique for obtaining all time-ordered contributions to the product t tiv iv * * t0 dt 8 * t08 dt 9 A * (t )A (t ) * dt * dt A (t )A (t ). The 9 8 2 1 2 0 0 1 first line is a shorthand notation for * * t0 dt 8 * t08 dt 9 * t09 dt - * t0- dt i v A 1 (t i v )A * (t )A (t )A (t ) and 9 8 2 1 2 each subsequent line is obtained by taking all possible permutations of the A’s, keeping the same relative position of the A’s that are connected by the square braces. These six diagrams correspond to the contribution to the probability P a 5 u b e,g;0 u 2 representing the absolute square of the top line of Fig. 2~a!.

~21! where

The remaining products A j ~ t ! 5e i ~ V k j 2 v 0 ! t .

~22!

We wish to replace expression ~21! by a sum of terms, each of which is a time-ordered expression. Note that all time orderings of t 8 ,t 9 ,t - ,t i v are allowed in Eq. ~21!, provided that t 8 .t 9 and t - .t i v . To keep track of the time-ordered terms that contribute, one can use the diagram shown as the top line of Fig. 3. This diagram is a shorthand notation for the expression

E 8E 9E -E t

dt

0

t8

dt

0

t9

dt

0

t-

0

dt A 1 ~ t ! A * 1 ~ t-!A* 2 ~ t 9 ! A 2~ t 8 ! . iv

iv

To get the other terms that contribute, one uses all other possible relative positions of the A’s in which A 1 appears to the left of A * 1 and A * 2 to the left of A 2 . The convention adopted is one in which the same relative position must be maintained for any two amplitudes connected by a square brace. The additional five terms are represented schematically in the remaining five lines of Fig. 3 and all six terms are listed in the first column of Table I. The exponential term at the top of the column multiples all terms in the column and the time ordering is such that the first term has argument t i v , the second has argument t - , etc. For example, the third term in the first column ~and the third line in Fig. 3! is a shorthand notation for

E 8E 9E -E t

dt

0

t8

0

dt

t9

0

dt

t-

0

dt i v A 1 ~ t i v ! A * 2 ~ t - ! A 2~ t 9 ! A * 1 ~ t8!.

e ik1 •re ik2 •rI a ~ t;k1 ;1 ! I a* ~ t;k2 ;2 ! , e 2ik1 •re 2ik2 •rI a ~ t;k1 ;2 ! I * a ~ t;k2 ;1 ! , and e 2ik1 •re ik2 •rI a ~ t;k1 ;2 ! I * a ~ t;k2 ;2 ! , appearing in Eq. ~19!, can be calculated using the same techniques. The fundamental diagrams for these terms are shown TABLE I. Terms that represent time-ordered contributions to the probability P a 5 u b g,e;0 u 2 . Each line represents a nested set of a time-ordered integrals with time increasing from left to right. For example, the first line in column 1 is a shorthand notation for * t0 dt 8 * t08 dt 9 * t09 dt - * t0- dt i v A 1 (t i v )A * 1 (t - )A * 2 (t 9 )A 2 (t 8 ). The exponential terms at the top of each column multiply each term in the column. The terms in columns 1–4 correspond to the time-ordered diagrams shown in Figs. 3 and 4. e ik1 •re 2ik2 •r

e ik1 •re ik2 •r

e 2ik1 •re 2ik2 •r

e 2ik1 •re ik2 •r

A 1 A 1* A 2* A 2 A 1 A 2* A 1* A 2 A 1A * 2 A 2A * 1 A* A 2 2A 1A * 1 A* 2 A 1A 2A * 1 A* 2 A 1A * 1 A2

2 A 1 A 1* A 2 2 *A 2 2 A 1 A 2 * A 1* A 2 2 2 * A 1A 2 A A 2 2 * 1 2 * A2 A A 2 2 1A * 1 2 A2 2 *A 1A 2 A * 1 2 A2 2 *A 1A * 1 A2

2 A2 1 A 1 * A 2* A 2 2 A 1 A 2* A 2 1 *A 2 2 * A2 A A 1 2 2A 1 * 2 2 A* 2 A 2A 1 A 1 * 2 2 A* 2 A 1 A 2A 1 * 2 2 A* 2 A 1 A 1 *A 2

2 2 2 A2 1 A 1 *A 2 *A 2 2 2 2 A 1 A 2 *A 1 *A 2 2 2 2 2 * A2 A A A 1 2 2 1 * 2 2 2 * A2 A A 2 2 1 A1 * 2 2 2 * A2 A 2 1 A2 A1 * 2 2 2 A 2 *A 2 1 A 1 *A 2


4064

55

FIG. 4. Diagrams from which the time-ordered contributions to the probability P a 5 u b e,g;0 u 2 can be obtained. These diagrams represent the remaining terms in the absolute square of the amplitude b e,g;0 for the perturbative chains of Fig. 2~a!.

in Fig. 4 and the contributions from these terms are listed in columns 2–4 Table I. The A 2 j ’s appearing in columns 2–4 are defined by

i~ Vk 1v0 !t . A2 j j ~ t ! 5e

~23!

The total contribution P a is obtained by summing all the terms in Table I, multiplying by C(k1 ,k2 ) @Eq. ~20!# and summing over k1 and k2 .

FIG. 5. Diagrams from which the time-ordered contributions to the probability P b 5 u b e,e;k2 u 2 can be obtained. These diagrams, in which c.c. stands for complex conjugate, represent the r-dependent contributions to P b that are fourth order in the atomfield coupling strength. They result from interference between the first- and third-order contributions to b e,e;k2 shown in Fig. 2~b!.

P b ~ 1,2! 5

b e,e;k2 ~ t ! 5 ~ i\ ! 3

FE

g* k ~ 2 m• ek2 ! 2

t

0

dt 8 e 2ik2 •R2 A 2 2 ~t8!

2\ 22

u g k u 2 u m• ek u 2 e ik •re 2ik •R ( k 1

1

1

3

E 9E -E t

dt

0

t9

0

2

1

1

dt

t-

0

G

2 dt i v A 1 ~ t i v ! A * 1 ~ t-!A2 ~ t9! .

~24! It follows that, to order u g k u 4 ,

UE

\ 22 u g k u 2 u m• ek u 2 ( k 2

2

Pb

In a similar manner one can calculate P b 5 ( k2 u b e,e;k2 u 2 . The perturbation chains leading to the amplitude b e,e;k2 are shown in Fig. 2~b!. For the moment, consider only the first two paths shown in Fig. 2~b!. It will be easy to generalize the result to include all paths. Carrying out perturbation theory using Eq. ~8! for the first two pathways of Fig. 2~b! using Eqs. ~9! and ~12!, one finds 21

2

2

5 B.

u b e,e;k u 2 ( k

2 3 3

F(

k1 ,k2

2

dt

0

t

0

0

iv dt i v A 2 2 ~t !

U

2

C ~ k1 ,k2 ! e ik1 •re ik2 •r

E 9E -E E 8 *8 G t

t

t9

0

dt

t-

0

2 dt i v A 1 ~ t i v ! A * 1 ~ t-!A2 ~ t9!

dt A 2 2 ~ t ! 1c.c. ,

~25!

where the argument ~1,2! is a reminder that this is the partial contribution to P b from the first two pathways of Fig. 2~b!. The second line of Eq. ~25!, proportional to u g k 2 u 2 , is independent of r, so is dropped from further consideration. The product of integrals appearing in the third line can be written as a time-ordered multiple integral using the diagram in line 1 of Fig. 5. In determining the various time orderings that are possible, one must require that A * 1 appears to the left of * . In this manner one arrives at A2 2 and A 1 to the left of A 1 the four terms in the first column of Table II. The second column corresponds to the complex conjugate of the first column. In writing this column, we have interchanged k1 and k2 , which is permissible since k1 and k2 appear as dummy indices in the sum for P b . The remaining ten columns in Table II correspond to contributions resulting from the interference of the five lower pathways with the top pathway

55


4065

TABLE II. Terms that represent time-ordered contributions to interference terms in the probability P b 5 u b e,e;k2 u 2 . The terms in columns 1–12 correspond to the time-ordered diagrams shown in Fig. 5. e ik1 •re ik2 •r 2 2 2A 1 A * 1 A2 A2 * 2 2 2A 1 A * 1 A 2 *A 2 2 2 2A 1 A 2 * A * 1 A2 2 2 2A 2 * A 1 A * 1 A2

e ik1 •re ik2 •r 2 2A 2 A 2 1 A 1* A 2 * 2 2 2A 2 A 1 A 2 * A 1* 2 2A 2 A 2 2 * A 1 A 1* 2 2A 2 * A 2 A 2 1 A* 1






2 2 2A * 2 A 2A 1 *A 1 2 2 2A * 2 A 2A 1 A 1 * 2 2 2A * 2 A 1 A 2A 1 * 2 2 2A 1 A * 2 A 2A 1 *

2 2 2 2A 2 1 A 1 *A 2 A 2 * 2 2 2 2A 1 A 1 * A 2 2 *A 2 2 2 2 2A 1 A 2 * A 1 * A 2 2 2 2 2 2A 2 2 *A 1 A 1 *A 2

2 2 2 2A 2 2 *A 2 A 1 *A 1 2 2 2 2 2A 2 * A 2 A 1 A 1 * 2 2 2 2A 2 2 *A 1 A 2 A 1 * 2 2 2 2A 1 A 2 * A 2 2 A1 *

2 2A 1 A 2 2 A* 1 A2 * 2 2 2A 1 A 2 A 2 * A * 1 2 2A 1 A 2 2 *A 2 A * 1 2 2A 2 2 *A 1A 2 A * 1

2 2 2A * 2 A 1 *A 2A 1 2 2 2A * 2 A 1 *A 1 A 2 2 2 2A * 2 A 1 A 1 *A 2 2 2 2A 1 A * 2 A 1 *A 2






2 2A 1* A 2 2 *A 2A 1 2 2 2A 1* A 2 * A 1 A 2 2 2A 1* A 2 1 A 2 *A 2 2 2 2A 1 A * 1 A 2 *A 2

2 2A 2 A 2 1 A 1* A 2 * 2 2 2A 2 A 1 A 2 * A 1* 2 2A 2 A 2 2 * A 1 A 1* 2 2A 2 * A 2 A 2 1 A* 1

2 2A 1* A 2 2 *A 2A 1 2 2 2A 1* A 2 * A 1 A 2 2 2A 1* A 2 1 A 2 *A 2 2 2 2A 1 A * 1 A 2 *A 2

2 2A 2 1 A 2 A 1* A 2 * 2 2 2A 1 A 2 A 2 * A 1* 2 2A 2 1 A 2 * A 2 A 1* 2 2A 2 * A 2 1 A 2A * 1

*A 2A 2 2A 2 2 *A 1 1 2 2A 2 * A 1* A 2 1 A2 2 *A 2 2A 2 2 *A 1 A 1 2 2 2A 1 A 2 * A * 1 A2

shown in Fig. 2~b!. The time-ordered integrals for these ten columns can be obtained from the diagrams that are shown in lines 2–6 of Fig. 5. The total contribution P b is obtained by summing all the terms in Table II, multiplying by C(k1 ,k2 ) @Eq. ~20!# and summing over k1 and k2 @9#. It should now be apparent how the paradox presented in the Introduction is resolved. Although the total probability P b is positive definite, the r-dependent part of P b represents interference terms that enter the calculation with a negative sign. In this manner they can cancel some of the ~acausal! terms in the joint probability P a . This is shown explicitly below. C.

Pc

Finally, we calculate P c 5 ( k1 ,k2 u b g,e;k1 ,k2 u 2 . To second order in g k , the amplitude b g,e;k1 ,k2 can be obtained by solving Eq. ~8! using perturbation theory for the chains shown in Fig. 2~c!. One finds b g,e;k1 ,k2 ~ t ! 52\ 22 g * k g k2 1

3 ~ m* • ek2 ! e

E E t

0

dt 8

t8

0

e

@A2 1 ~ t 9 ! A 2~ t 8 !

1A 2 ~ t 9 ! A 2 1 ~ t 8 !# 1 ~ m* • ek1 !~ m• ek2 ! 3e 2ik1 •R1 e 2ik2 •R2 @ A 2 2 ~ t 9 ! A 1~ t 8 ! 1A 1 ~ t 9 ! A 2 2 ~ t 8 !# % .

~26!

TABLE III. Terms that represent time-ordered contributions to the probability P c 5 u b g,e;k1 ,k2 u 2 . The terms in columns 1–4 correspond to the time-ordered diagrams shown in Fig. 6 e 2ik1 •re ik2 •r 2 A2 1 A 2A * 1 A2 * 2 2 * * A1 A1 A2 A2 2 A2 1 A 1* A 2 A 2 * 2 2 A 1* A 2 * A 1 A 2 2 A 1* A 2 1 A 2A 2 * 2 2 A 1* A 1 A 2 * A 2

D. P e „2…5P a 1P b 1P c

When all the terms in Tables I–III are added together to obtain P e (2), there is ~mercifully! significant cancellation. Thirty-two terms from Table II cancel 16 terms from Table I and 16 terms from Table III, leaving a total of 32 terms. If one sets

2 B *j B j 5 ~ A *j A j 2A 2 j A j *!, 2 2 B *j B 2 j 5~ A* j A j 2A j A * j !,

dt 9 $ ~ m1 • ek1 !

2ik2 •R1 2ik1 •R2

In taking the absolute square of this expression and summing over k1 and k2 , it appears that there are 16 terms; however, eight of these terms correspond to ‘‘double counting’’ since they are identical on the interchange of k1 and k2 . Of the remaining eight terms, four have no r dependence. As a consequence, one is left with the four r-dependent terms indicated schematically in Fig. 6. When all time orderings of these terms are taken, one arrives at the 24 contributions listed in Table III.




2 A2 2 *A * 1 A 2A 1 2 2 * A 2 A 2A 1 A * 1 2 A2 2 * A 2 A 1* A 1 2 A 2A 2 1 A 2 * A 1* 2 A 2 A 2 * A 1* A 2 1 2 A 2A 2 2 * A 1 A 1*

2 A2 1 A 2A 2 *A * 1 2 2 * A1 A2 A* 1 A2 2 A2 1 A 2 * A 2 A 1* 2 A 2 * A 1* A 2 1 A2 2 A2 2 * A 1 A 2 A 1* 2 A2 2 * A 1 A 1* A 2

2 A 2A 2 1 A* 1 A2 * 2 * * A 2A 1 A 2 A 2 1 2 A 2 A 1* A 2 1 A2 * 2 A 1* A 2 2 *A 2A 1 2 2 A 1* A 2 A 1 A 2 * 2 A 1* A 2 A 2 2 *A 1

~27!

then the remaining terms can be combined into the eight terms shown in Table IV plus the complex conjugate of these terms. In combining the terms appearing in Table IV, we took all the spatially dependent exponential factors to be the same, a procedure that is justified owing to the fact that the results are independent of the signs of k1 and k2 when the sums over k1 and k2 are carried out. We concentrate on the four terms in column 1 of Table IV since the remaining four terms can be treated in a similar manner. The first term in column 1 is TABLE IV. Terms that represent time-ordered contributions to the probability P e (2)5 P a 1 P b 1 P c . These terms, plus their complex conjugates, are obtained by summing all the terms in Tables I–III. e ik1 •re ik2 •r


2 A2 1 A 1 * B 2* B 2 A 1 A 1* B 2* B 2 A 1 B 2* B 2 A 1* A 1 B 2* A 1* B 2

2 A2 1 A 1* B 2 B 2 * 2 A 1* A 1 B 2 B 2 2 * 2 A 1* B 2 B 2 2 *A 1 2 2 A 1* B 2 A 1 B 2 *


4066

D ~ t,t 9 ! 5

55

E -E t

dt

0

t-

0

2 iv dt i v @ A 2 1 * ~ t - ! A 1 ~ t ! U ~ t 9 2t - !

iv iv 1A * 1 ~ t - ! A 1 ~ t ! U ~ t 9 2t !#

5

E -E E -E t-

t

dt

0

0

2 iv dt i v A 2 1 * ~ t - ! A 1 ~ t ! U ~ t 9 2t - !

t

1

dt

t

t-

0

dt i v A 1 ~ t - ! A 1* ~ t i v ! U ~ t 9 2t - ! , ~31!

FIG. 6. Diagrams from which the time-ordered contributions to the probability P c 5 u b g,e;k1 ,k2 u 2 can be obtained. These diagrams represent the r-dependent contributions to P c that result when one takes the absolute square of the amplitude b g,e;k1 ,k2 shown in Fig. 2~c!.

E 8E 9E -E t8

t

dt

0

t9

dt

0

5

dt

0

t-

t8

dt

dt

0

iv

0

t

dt

dt

0

S5S 1 1S 2 ,

0

2 iv A2 1 ~ t ! A 1 *~ t - ! B * 2 ~ t 9 ! B 2~ t 8 !

S 15 t-

dt i v

~28!

5

E 8E 9E -E dt

0

t9

dt

0

dt

0

t-

0

dt A 1 ~ t !@ A 1* ~ t - ! B 2* ~ t 9 ! B 2 ~ t 8 ! iv

5

E 8E 9E -E t8

dt

0

t

dt

0

dt

0

t-

iv

dt

dt

0

where

t8

0

0

t

t-

0

dt i v

E 8E 9E -E t8

t

dt

0

t

dt

dt

0

0

t-

0

dt i v B 2 ~ t 8 ! B 2* ~ t 9 ! B 1* ~ t - ! ~33a!

E 8E 9E -E t

dt

0

t8

t

dt

0

t

dt

0

dt i v

0

~33b!

~29!

Note that the term in square brackets is a sum of time-ordered terms equivalent to * t0 dt - A 1* (t - ) @ * t0 dt 8 * t08 dt 9 B 2* (t 9 )B 2 (t 8 ) # . The integration variable t i v must be less than both t - and t 9 , which is the reason the U function has been added. When Eqs. ~28! and ~29! are added together, one finds a sum t

dt

2 2 iv iv 3B 2 ~ t 8 ! B * 2 ~ t 9 ! A 1 * ~ t - ! A 1 ~ t ! U ~ t 9 2t ! .

0

E 8E

0

t

dt

iv

iv iv 3B 2 ~ t 8 ! B * 2 ~ t9!A* 1 ~ t - ! A 1 ~ t ! U ~ t 9 2t ! .

S5

dt

S 25

1B * 2 ~ t - ! B 2~ t 9 ! A * 1 ~ t 8 ! 1B * 2 ~ t-!A* 1 ~ t 9 ! B 2 ~ t 8 !# t

t8

t

3B 1 ~ t i v ! U ~ t 9 2t i v ! ,

where U(x)51 for x.0 and 0 for x,0. The remaining three terms in column 1 are equal to t8

E 8E 9E -E

iv 3B 2 ~ t 8 ! B * 2 ~ t 9 ! B 1~ t - ! B * 1 ~ t ! U ~ t 9 2t - !

0

2 2 iv 3B 2 ~ t 8 ! B * 2 ~ t 9 ! A 1 * ~ t - ! A 1 ~ t ! U ~ t 9 2t - ! ,

t

~32!

where

E 8E 9E -E t

where the second equality is obtained by interchanging the order of integration in the second term. Writing the integral t over t i v in the first term as * t0 2 * t - and using the definition ~27!, one can rewrite Eq. ~30! as

dt 9 B 2 ~ t 8 ! B * 2 ~ t 9 ! D ~ t,t 9 ! ,

For the moment, we neglect S 2 , which will turn out to be negligibly small. Multiplying Eq. ~33a! by exp@i(k1 exp[i(k11k2 )•r]C(k1 ,k2 ), where C(k1 ,k2 ) is given by Eq. ~20!, using the definitions ~27!, ~22!, and ~23!, changing from a sum over k to an integral using the prescription ( k→V/(2 p c) 3 * V 2k dV k dU k , and carrying out the sum over polarization vectors and angular integrations @6#, one finds that contribution to P e (2) from the S 1 term is

P e ~ 2;S 1 ! 5

~30!

E 8E 9E -E t

dt

0

t8

0

t

dt

dt

0

t-

dt i v

0

3F ~ t 8 ,t 9 ! F * ~ t - ,t i v ! U ~ t 9 2t i v ! , where

~34!

55


F ~ t,t 9 ! 5 p 21 v 23 0 5 p 21 v 23 0

E E

`

0 `

0

3 dV k 2 B 2 ~ t 8 ! B * 2 ~ t 9 ! V k G ~ r,k 2 ! 2

S

.

2

E

`

0

dV k $ exp@ i ~ V k 2 v 0 !~ t 8 2t 9 !#

E

`

2`

dV k exp@ i ~ V k 2 v 0 !~ t 8 2t 9 !#

3V 3k G ~ r,k ! ,

G ~ r,k ! 5 43 G

FS S

sin a m~ rˆ! 5

D

G

cos kr sin kr 1 cos2 a m~ rˆ! , ~ kr ! 2 ~ kr ! 3

u m xu 21 u m yu 2 , m2

cos a m~ rˆ! 5 2

E

u m zu 2 , m2

`

2`

~36!

~368!

dV k exp@ i ~ V k 2 v 0 !~ t 8 2t 9 !# sin kr ~37!

D

I 1 dI 1 dI 2 1 2 sin2 a m~ rˆ! r 3 r 2 dr r dr 2

S

12 2

D

I 1 dI 1 cos2 a m~ rˆ! . r 3 r 2 dr

~38!

If v 0 u t2r/c u @1, then d/dr @ e 2ik 0 r d (t 8 2t 9 2r/c) # . 2ik 0 e 2ik 0 r d (t 8 2t 9 2r/c) and F ~ t 8 ,t 9 ! .H ~ k 0 r ! d ~ t 8 2t 9 2r/c ! , where

3 iGe 2ix sin2 a m~ rˆ! . 4x

~40!

2 P e ~ 2;S 1 1S * 1 ! 52 u H ~ k 0 r ! u

3

E 8E 9E -E t

dt

0

t8

t

dt

0

dt

0

t-

0

dt i v d ~ t 8 2t 9

3

E

t8

0

E

t

0

dt 8 U ~ t 8 2r/c !

dt - U ~ t - 2r/c !

5 u H ~ k 0 r ! u 2 ~ t2r/c ! 2 U ~ t2r/c ! .

where k 0 5 v 0 /c, then

S

G

~39!

~41!

It is now an easy matter to show that the contribution from the term S 2 in Eq. ~32! is also proportional to U(t2r/c) @owing to the factors B 2 (t 8 )B * 2 (t 9 ) in Eq. ~33b!#, but is v 0 (t2r/c) times smaller that the S 1 contribution. In a similar manner one can show that the contribution from the sum of the terms in the second column of Table IV is proportional to U(t2r/c), but is smaller than the S 1 contribution by a factor v 0 (t2r/c). Thus, for v 0 (t2r/c)@1, one finds the probability for atom 2 to be excited is P e ~ 2 ! . u H ~ k 0 r ! u ~ t2r/c ! 2 U ~ t2r/c ! .

2ik 0 r 52 43 iGk 23 d ~ t 8 2t 9 2r/c ! , 0 e

F ~ t 8 ,t 9 ! 5

D

1 i 2 ˆ! 3 2 2 cos a m~ r x x

52 u H ~ k 0 r ! u 2

G5 43 @ m 2 /(4p « 0 \c) # ( v 30 /c 2 ) is the single-atom excited-state decay rate, and k5V k /c. The key point of the calculation is contained in Eq. ~35!. Inclusion of the counterrotating terms 2 of the type A 2 2 * (t 8 )A 2 (t 9 ) enables one to extend the V k integral to 2`, which, in turn, leads to a causal result, as we now show. If one defines I52 43 G p 21 k 23 0

D

i 1 1 1 2 sin2 a m~ rˆ! x3 x2 x

2r/c ! d ~ t - 2t i v 2r/c ! U ~ t 9 2t i v !

sin kr cos kr sin kr 1 2 sin2 a m~ rˆ! kr ~ kr ! 2 ~ kr ! 3

12 2

2

~35!

D

FS

Combining Eqs. ~35! and ~39! and adding in the complex conjugate leads to the contribution

2exp@ 2i ~ V k 1 v 0 !~ t 8 2t 9 !# % V 3k G ~ r,k ! 5 p 21 v 23 0

3 iGe 2ix 4

12 2

dV k 2 @ A 2 ~ t 8 ! A * 2 ~t9!

2 3 2A 2 2 * ~ t 8 ! A 2 ~ t 9 !# V k G ~ r,k 2 !

5 p 21 v 23 0

H ~ x ! 52

4067

~42!

This result is perfectly causal and agrees with the Heisenberg approach result @2#. It is interesting to note how the counterrotating terms enter the calculation. There are essentially two sets of counterrotating terms. The perturbation chains represented by the lower pathway of Fig. 2~a! and the four upper pathways of Fig. 2~b! contain counterrotating amplitudes that are essential for establishing causality. On the other hand, the perturbation chains represented by the lower three pathways of Fig. 2~b! and all the pathways in Fig. 2~c! contain counterrotating amplitudes that are smaller than the first set of terms by a factor v 0 (t2r/c). Little error is introduced if this second set of counterrotating terms is neglected. It is relatively easy to distinguish the two type of counterrotating terms in the diagrams of Figs. 3–6. The first set involves only products of A j ’s and their complex conjugates, while the second 2 involves only products of the form A j A 2 j * ~or A j A * j !. Had we included states other than e and g in the atoms, the additional states would give rise to ~causal! contributions having the same order of magnitude as the second set of counterrotating terms.

4068

P. R. BERMAN AND B. DUBETSKY IV. CALCULATION OF P ge

The joint probability P ge for atom 2 to be excited and atom 1 in its ground state is equal to P ge 5 u b g,e;0 u 1

(

2

u b g,e;k1 ,k2 u . u b g,e;0 u . 2

k1 ,k2

2

~43!

The probability amplitude b g,e;0 can be obtained from Eqs. ~16! and ~18!. Carrying out the sum over polarization vectors and k1 in Eq. ~18!, one can obtain @6# b g,e;0 ~ t ! 52

E 8E t

dt

0

t8

0

dt 9 p 21 v 23 0

E

`

0

dV k

3exp@ 2i ~ V k 2 v 0 !~ t 8 2t 9 !# V 3k G ~ r,k ! . ~44! Note that the critical difference between this equation and Eq. ~35! is that the V k integral is from 0 to ` rather than 2` to `. This integral has been evaluated in another paper @6#. One can replace the upper integration limit on the V k integral by some cutoff frequency, carry out the integrations, and then let the cutoff frequency go to infinity. In this manner, for ~45!

y[k 0 r@1 and v 0 u t2r/c u @1, one obtains 3 iG sin2 a m~ rˆ! M ~ s ! , b˙ g,e;0 5 4y

~46!

where M ~ s ! 5i p U ~ s 2 ! e iy 1cos y @ 2Ci~ s 1 ! 1Ci~ u s 2 u !# 2sin y @ Si~ s 1 ! 1Si~ s 2 ! 2 p U ~ s 2 !# , s 6 5 v 0 ~ t6r/c ! ,

~47! ~48!

Ci(x)52 * `x ds cos(s)/s, and Si(x)5 * x0 ds sin(s)/s. We are interested in determining whether b˙ g,e;0 is nonvanishing for times t,r/c. When t,r/c (s 2 ,0) and u s 6 u @1, one finds M ~ s ! ;cos y

F

sin~ s 2 ! sin~ s 1 ! 2 s2 s1

1sin y

F

G

cos~ s 2 ! cos~ s 1 ! 1 s2 s1

5sin~ v 0 t !

S

D

1 1 2 . s2 s1

G ~49!

This result is particularly satisfying. Although the joint probability is nonvanishing for times t,r/c, it oscillates at the optical frequency v 0 . If one course grains the result for times greater than v 21 0 , this contribution is essentially equal to zero. Attempts to measure this joint probability on a time would necessarily introduce additional scale less than v 21 0 contributions that would make it impossible to isolate the contribution being sought.

55

To gain some additional insight into this result, we return to Eq. ~16!, which can be written as b g,e;0 ~ t ! 52\ 22

E 8E t

dt

0

t8

0

dt 9 @ ^ m•E~ R2 ,t 8 !

3 m* •E~ R1 ,t 9 ! & e i v 0 ~ t 8 2t 9 ! 1 ^ m* •E~ R1 ,t 8 ! 3 m•E~ R2 ,t 9 ! & e 2i v 0 ~ t 8 2t 9 ! # ,

~50!

where the average is with respect to the vacuum field and E(R,t) is given by Eq. ~6! with a k replaced by a k e 2iV k t and a †k by a †k e iV k t . As postulated by Biswas et al. @3#, the nonvanishing of the joint probability can be related to the fact that the correlation function for the vacuum field at different space-time points differs from zero. V. DISCUSSION

The interaction between two identical atoms separated by a distance r has been considered. When atom 1 is excited by a radiation pulse whose duration is long compared to an optical period but short compared to the time it takes for light to travel between the atoms, the probability for atom 2 to be excited is identically zero for times t,r/c, provided only r-dependent terms in the probability are retained. This result remains valid for k 0 r,1 and for v 0 u t2r/c u ,1, although the functional form of the probability can change in these limits. The r-dependent joint probability for atom 2 to be excited and atom 1 to be in its ground state is not identically equal to zero for t,r/c; however, this joint probability is effectively equal to zero if measurements are carried out on a time scale greater than an inverse optical period. The calculations in the Schro¨dinger picture are particularly revealing. The vanishing of atom 2’s excitation probability for t,r/c occurs only when counterrotating terms are included in the calculation. The atom 2 excitation probability is given by P e (2)5 u b g,e;0 u 2 1 ( ku b e,e;ku 2 1 ( k1 ,k2 u b g,e;k1 ,k2 u 2 to fourth order in the atom-field coupling strength g k . The cancellation in the time region t ,r/c is critically dependent on terms in P b 5 ( ku b e,e;ku 2 that represent interference between first- and third-order contributions to the amplitude b e,e;k . Although the overall probability P b is positive definite, there are terms of order u g k u 4 that are negative and lead to the cancellation. The lead term in P b is of order u g k u 2 , but that term is r independent. We now return to Hegerfeldt’s paper @1#. Hegerfeldt argued that it is essential to use correct asymptotic states of the individual free atoms and the field. Such states, as discussed in the texts of Cohen-Tannoudji, Dupont-Roc, and Grynberg @10# and Compagno, Passante, and Persico @11#, are entangled atom-field states. Hegerfeldt proved that the expectation value of any positive operator is nonvanishing for t ,r/c if expectation values are taken with respect to the correct asymptotic states. The physical significance of this result is one we should like to address. As has been pointed out by Milonni and co-workers @2# and confirmed in this work, Hegerfeldt’s result is valid only for the r independent part of the expectation value. Consequently, it is the significance of the fact that r-independent contributions to transition probabilities are nonvanishing for t,r/c that must be questioned.

55


4069

Rather than talk about expectation values, it may prove more useful to consider experiments where some physical parameter, such as the excitation probability of atom 2, is measured. Even if atom 1 is absent, the excitation probability for atom 2 is nonvanishing at all times since the stable ground state of the system has some admixture of state 2. In some sense, this is related to the fact that the u g k u 2 contribution to P b is nonvanishing for t,r/c. To measure this nonvanishing probability, however, would require a probe pulse whose duration is less than an optical period. Such a pulse would dramatically affect the system and could not serve as a passive probe of the state 2 probability amplitude. It is also interesting to consider how the excitation of atom 1 affects the force between the two atoms. When both atoms are in their ground states, there is a ~retarded! van der Waals force of attraction between the atoms. To understand the physical consequences of the excitation of atom 1 at time

t50, one must in some way measure the change in the force between the two atoms for t.0. Although we have not carried out such calculations explicitly, we believe that the force ~or any other appropriate physical variable! will not change for times t,r/c provided the measurement process itself does not modify the force. In summary, it appears that life is what one would expect. Atom 2 cannot sense changes in atom 1 for times t,r/c.

@1# G. C. Hegerfeldt, Phys. Rev. Lett. 72, 596 ~1994!, and references therein. @2# P. W. Milonni, D. F. V. James, and H. Fearn, Phys. Rev. A 52, 1525 ~1995!; P. W. Milonni and P. L. Knight, ibid. 10, 1096 ~1974!, and references therein. @3# A. K. Biswas, G. Compagno, G. M. Palma, R. Passante, and F. Persico, Phys. Rev. A 42, 4291 ~1990!, and references therein. @4# Other authors have also used an amplitude approach in the Schro¨dinger picture rather than an operator approach in the Heisenberg picture to identify the physical mechanisms responsible for line-shape features in nonlinear spectroscopy. See, for example, B. Dubetsky and P. R. Berman, Phys. Rev. A 47, 1294 ~1993!; G. Grynberg and C. Cohen-Tannoudji, Opt. Commun. 96, 150 ~1993!; G. Grynberg, M. Pinard, and P. Mandel, Phys. Rev. A 54, 776 ~1996!; J. L. Cohen and P. R. Berman, ibid. 55, 3900 ~1997!. @5# In calculating the excitation probability for atom 2, P e (2), and the joint probability P ge , Biswas et al. @3# use an initial condition that differs from ours. They assume that each atom is in a superposition of ground and excited states. With this initial condition, the lowest-order contribution to both P e (2) and P ge is second order in the atom-field coupling strength g k rather than fourth order, as it is in our calculation. The calcu-

lations of Biswas et al. @3# are carried out only to order u g ku 2. P. R. Berman, Phys. Rev. A 55, 4466 ~1997!. Of course, to selectively excite only one of the atoms using optical radiation, it is necessary that the atoms be separated by more than an optical wavelength. If excitation occurred via a focused particle beam, smaller separations could be accommodated. There are additional pathways, such as b e,g;0 →b e,e;k2 →b g,e;k1 ,k2 →b e,e;k2 , which are not shown in Fig. 2~b!, since they lead to r-independent contributions to the probability P e (2). For cases in which the matrix element m is not purely real or purely imaginary, one should multiply the last six columns of Table II, all columns of Table III, and the second column of Table IV by C 8 (k1 ,k2 )5\ 24 u g k 1 u 2 u g k 2 u 2 ( m• ek1 ) 2 ( m* • ek2 ) 2 rather than C(k1 ,k2 )5\ 24 u g k 1 u 2 u g k 2 u 2 u m• ek1 u 2 u m• ek2 u 2 . C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, AtomField Interactions ~Wiley, New York, 1992!, pp. 229–238. G. Campagno, R. Passante, and F. Persico, Atom-Field Interactions and Dressed Atoms ~Cambridge University Press, Cambridge, England, 1995!, Chap. 6.

ACKNOWLEDGMENTS

We are pleased to acknowledge useful discussions of this problem with J. L. Cohen and P. Milonni. This research was supported by the National Science Foundation through Grant No. PHY-9414020 and by the U.S. Army Research Office under Grant No. DAAH04-93-G0503.

@6# @7#

@8#

@9#

@10# @11#