Theoretical foundations of laser spectroscopy

—

.4p>.

(3.25)

In this representation the kinetic energy becomes a pure c-number =

_~_-(J,~ —p~),

(3.26)

and the difficulties are transferred to the representation of the interaction operator. This is the representation used from the beginning in section 2.

Here we have reached the momentum representation via the Wigner function (3.22). It is,

5

however, possible to proceed in the opposite way and set

p(p, r)

3r e_~~~p(R, r) d

=

=

e~~~
—

.4r> d3R.

(3.27)

This representation preserves the spread of both the momentum and the position as variables whereas the classical position and momentum have been transformed away. This representation

was introduced in [126] and was shown to offer certain advantages in treating the problems of laser spectroscopy. We shall call this the Shirley representation and return to it in section 4.6. The equation of motion without interaction terms becomes ~-p(p,r) =

—

~

~-p(p,r).

(3.28)

168

S. Stenhoim, Theoretical foundations of laser spectroscopy

Summarizing the various possibilities we present the following table [126] over possible representations of the translational degrees of freedom Variables

r R

I

Fourier transform

p

Fourier transform

~P

Position representation

Wigner representation

Shirley representation

Momentum representation

It is, of course, possible to regain the momentum representation by transforming the Shirley representation with respect to r. In the classical limit we expect the spread in the one-particle wave packet to become negligible. In the Wigner representation the density matrix should then describe a particle following a classical trajectory. To see how this emerges we look at the solution of the free-particle equation of motion

p(t)

=

exp

[—

iHt/h]p(0) exp [iHt/h]

(3.29)

exp[_

(3.30)

giving

=

~

_P~)t].

Transforming into the position representation we find the diagonal elements —

R0 I p1 R

—

R0>

exp [i(p1 ~P2)~ (R

=

—

R0)/h] exp

P1 P2

—

(p~_p~)t]

~ ~exp [ip (R

—

R0)/h] exp [ jp Pt/Mh]p(p, P)

4~~ exp [ip (R

—

R0

.

=~

[

.

—

Pt/M)/h]p(p, P).

(3.31)

If we want to localize the wave packet in space, its spread r must become small; it is then natural to regard the spread in momentum space p to be large. We must, however, remember that the variables (r, p) do not form a Fourier transform pair and hence the statement above is not a strictly mathematical consequence of our definitions. Assuming p(p, P) to depend but little onp we obtain from (3.31) in the limit of a large volume —

R0I IR

—

R0>

=

5

3Pö(R d

—

R 0

—

Pt/M)p(p, P).

(3.32)

Here we can clearly see the emergence of the classical trajectory of a particle initially at R0 and propagating with velocity P/M. The function p(~,P) is equivalent with the initial momentum (velocity) distribution and the dependence on ~ is negligible. Apart from the use of momentum instead of velocity, the result obtained in (3.32) is equivalent with those given by physical arguments

169

S. Stenholm, Theoreticalfoundations oflaser spectroscopy

in eqs. (3.2) and (3.10). The particular form (3.32) corresponds to an ensemble formed by particles injected at R0 but with a spread in velocities (momenta). It is trivial to collect additional particles

into the ensemble by summing over the initial position or time ofinjection. Subsequently removing the momentum summation we regain eq. (3.2). In the case when thep dependence in (3.31) cannot be neglected we must write 3Ppwigner(R R R0I P I1~ R0> = ~jx d 0 Pt/M;P). (3.33) —

5

—

—

—

Hence the general case corresponds to a Wigner function with velocity P/M which propagates with free-particle kinematics R(t)

=

R0 + Pt/M,

(3.34)

but its shape is determined by the initial preparation and need not correspond to that of a classical particle with a well defined trajectory. 3.3. Quantized fields and multiphoton transitions In this section we shall look into the equation of motion for a two-level system acted upon by one quantized field mode. The Hamiltonian is taken from (2 .47) in the form H/h

=

w21a~a2+ Ilbtb

—

g12(a~a1+ a~a2)(b+ bt),

(3.35)

where the translational modes have been omitted. Denoting the density matrix elements by = p1~4n,m)with In>, rn> being the occupation number eigenstates of the operator btb and ij = 1,2, we find

i-ã_p22(n,m) = f~(n m)p22(n,m) —

2p + n” 12(n

—

—

1, m)

g12[(n + 1)”2 p12(n + 1,m) 2p

—

(rn + 1)’’ 21(n, rn + 1)

—

2p m” 21(n, m

—

1)]

(3.36)

—

1)]

(3.37)

12

i.~-p11(n,m)= f~(n m)p11(n,m) g12[(n + 1)’ p21(n + 1,m) 2p 2p 2p + n~ 21(n m) (rn + 1)~ 12(n,m + 1) m’’ 12(n, m —

—

—

i

t21(n~rn)

=

[f(n —

—

1,

—

—

m) + w21]p21(n, m) 2p 2p g12[(n + 1)” 11(n + 1,m) + n” 11(n 2p 12p (m + 1)~ 22(n,m + 1) m’ 22(n, m —

—

—

—

1,m) 1)].

(3.38)

In these equations no pumping or decay terms have been written out, but they can easily be introduced in the conventional way; see section 3.1. No energy conservation argument has yet been invoked to eliminate terms which are far off resonance (the Rotating Wave Approximation), and hence we have preserved the option to consider multi-photon processes which are at resonance near ~ kf~,

(3.39)

170

S. Stenhoim, Theoreticalfoundations of laser spectroscopy

where k is an integer. The strength of these resonances is measured by the parameter (g2 1/C~), which is usually very small in the optical regime but can be made large for radio frequency fields. The main resonance is displaced by the Bloch-Siegert shift [26] 1ii2~O/~

=

(3.40)

where n0 is the average photon number in the field. The progress in radio frequency experiments

has made the influence of the counter rotating terms the object of great interest over the last decades. The first experimental observation [102] was followed by the correct theoretical explanation [153], which was based on a quantized Hamiltonian like (3.35). The experimental work has been continued mainly in France, where the theoretical understanding has evolved within the frame of a perturbation treatment centered on “dressed atomic states”, which incorporate the main effects of the field exactly and the remainder is used to couple the levels. Detailed descriptions of these developments are available in refs. [41, 68, 69]. We go back to eqs. (3.36)—(3.38) and notice that the diagonal part of the Hamiltonian depends only on

v=n—m.

(3.41)

The dependence on n and m separately enters only in the square roots, characteristic of the transition matrix elements of a quantized field. If we assume an intense field this dependence is weak and we can assume that the combination (3.41) is the most important variable. To see this explicitly we introduce the other variable

p

n + m

=

—

2n0,

(3.42)

which describes deviations from the average photon number n0. For large values of n0 we have 12 = [n 2= \/~O + O(n~’’2), (n + 1)1/2 ~ n’ 0 + 4(/2 + v)]~ (rn + 1)1/2 ~ rn’12 = [n 112 = ~o + O(n~”2). (3.43) 0 + ~p — v)] When the correction terms of order n~112are neglected we find that the equation of motion for p(v, p) retains no explicit dependence on the variable p. We have i

p~ 2(v,p)

=

f~vp22(v,p) —

i ~p21(v, p)

=

p21(v

—

—

g12~0[p,2(v + 1, p + 1) + p12(v

p22(v

—

1, p

—

1)

l,p + 1)— p21(v + l,p —1)],

[~v + w21]p21(v, p) —

—

1,

p + 1)

—

g12~j~0[p11(v+ 1, p + 1) + p11(v

—

p22(v + 1, p

—

1)],

(3.44) —

1,

p

—

1)

(3.45)

and similar equations for Pu and P12~As there is no dependence on p, we can assume a solution p(v, p) independent of p. Then we can easily see that the equations (3.44)—(3.45) can be derived with a semiclassical Hamiltonian obtained from (2.63) in the form H/h

=

w21a~a2— p12Ecos1~t(a1a2+ a~a1).

The density matrix equation of motion gives for instance

(3.46)

S. Stenholm, Theoreticalfoundations of laser spectroscopy

i~-.p2~ =

C02,P2u

—

p,2Ecos1~t(,p1,

—

ill

(3.47)

P22).

Using a Fourier expansion of the elements of the density matrix =

~ e~’~p~(v),

(3.48)

we find that (3.47) is equivalent with (3.45) when we identify p1.,.(v)

(3.49)

p~,(v,p)

and 2g,2\/~0= p12E,

(3.50)

in agreement with (2.60). We have thus shown how to obtain the classical limit of a strong field from the equation of motion for the density matrix. The present derivation was given in [134], where also correction terms were obtained.

The equation of motion (3.47) has periodic coefficients and in accordance with Floquet theory an expansion of the type (3.48) is expected to be useful. For the state amplitudes it was introduced by Autler and Townes [10], and the full power of Floquet theory was applied to the problem by Shirley [124]. The expansion (3.48) for the density matrix was introduced in [131—132]where the

strengths and shifts of higher multiphoton resonances were computed by means of a continued fraction solution. Recently a large body ofwork has been devoted to this problem; a detailedreview ofthe various theories and the experimental situation can be found in [137]. Finally we make some points about the use of quantized fields in this type of problems. The occupation number states form a basis for the density matrix only. We need not assume that we are anywhere near a well-defined photon number. The calculations could as easily be carried out

in the coherent state representation, but its introduction is not imperative as was discussed in section 2.3. The photons are not a necessary requirement for the occurrence of multi-photon processes; they occur as well in the semiclassical theory of eq. (3.47). The notion of a k-photon process stems entirely from the energy conservation requirement (3.39), which matches the time dependent phase ofthe field to the phase of the matter. In this section we have shown that the density matrix of a quantized field corresponds to the Fourier expansion coefficient p(n m) in the semiclassical limit. It is interesting to notice

I I

—

that a similar relation holds true for matter in the WKB-approximation [92]. It is shown that here too the spectrum asymptotically approaches an evenly spaced ladder. It then follows [92, §48]

that .-+exp(iw~~t) .-+exp[iw(n

—

m)t]p(n

—

m,n + m)

(3.51)

depends only weakly on n + rn and describes the classical Fourier coefficient (n m) in an ordinary series expansion. This result has been seen to follow explicitly for the quantized field too. —

I 72

S. Stenhoim, Theoreticalfoundations of laser spectroscopy

4. Spectroscopic effects of atomic velocity 4.1. The theory of saturation spectroscopy After our preliminary investigations we are now going to consider the results of atomic motion on the interaction with light. The most interesting application is that of saturation spectroscopy, where one wave of amplitude E~traverses the sample in one direction and another of amplitude E_ in the opposite direction. The first one saturates the medium and the second one probes the modifications induced by the first field. When this takes place inside a laser medium the two amplitudes are equal, E + = E -, and we have a standing wave cavity mode. It was, indeed, in this configuration that the first saturation resonance, the Lamb dip, was found theoretically by Lamb [88]. The role of saturation in spectroscopic investigation has grown to a very large extent; an extensive review is presented in ref. [39]. Sample

E

Detector

Fig. 4.1.

The experimental situation is shown in fig. 4.1. We describe the two fields classically and neglect all spontaneous effects. From eq. (2.63) we then obtain the Hamiltonian for a two-level system (cf. [136]), H/h

~

=

(~+ w5)a~a,5

ks= 1,2 —

L U

.1. ‘c’ i 2P12 L~~ak+q2ak! k

~‘

+

e

jOt

—

...L

t

~‘

ak_~.,2ak1

—

e

— jOt

t e’ t E — elOt ak_~1a~1..~+ eiLlt1-~-ak+qlak2

Starting with an initial momentum hk, the Hamiltonian mixes only those states that have the momentum of the form h(k + nq) where n is an integer. The continuum of translational eigenstates is, thus, replaced by a denumerably infinite set ofcoupled states. Starting from another momentum we generate another set of states coupled to each other but not to the previous ones. This shows that we can replace the continuous labels of the density matrix by two integers n and m as follows

=

p5p(n,m).

(4.2)

From eq. (4.1) it follows that the equation of motion for the density matrix is i -~—p22(n,rn)

=

(Ck+flq —

.1.

—

%+mq)P22(fl’

IL’

(

—jOt

2p!2L~+~e

rn)

p12~n—

i

eiOt p21~n,m— eiOtp2i(n,m + 1))],

~,rn,—

+ E_(eiQtpi2(n + 1,m) —

(4.3)

S. Stenholm, Theoreticalfoundations of laser spectroscopy

i ~!-p21(n, m)

=

(co21

+

Ck+nq

—

173

4+mq)P21(~’m)

_ip,2e_~t[E÷(pii(n

—

+ E_(p1,(n + 1,m)

p22(n,m

—

1,rn)

—

p22(n,rn + 1))

—

1))],

(4.4)

and similar equations for Pu, and Pu2~As we can see the mathematical structure of these equations is very similar to that of eqs. (3.44)—(3.45). If we introduce the new variables v=n—m,

(4.5)

p=n+rn,

(4.6)

we find that only in the combination 2 hk

hq

rn2) = Vqv + evp (4.7) there is an explicit dependence on the variable p. This dependence is multiplied by the recoil energy 4+nq

=

—

8k+mq =

~ç~q(n— m) + ~-~(n2

—

hq2/2M, which is due to the fact that absorbing or emitting a photon the atom has to com-

pensate the change of momentum of the field. This will affect its kinetic energy by an amount being equal hc, which can, however, usually be neglected. The remaining term Vq is the nonrelativistic Doppler shift of the atom moving with velocity V = hk/M. We want to make the rotating wave approximation and hence we set p

21(n,m) = e~~°~j5(v,p) and denote the detuning by A

=

w2,

—

(4.8)

f~.

(4.9)

The rapidly oscillating terms exp [± i21tj are omitted from the equation of motion. If we introduce the variables v, p into (4.3)—(4.4) and also the adequate pumping and decay parameters (see section 3.1) we find the steady state equations [Y2

+ i(Vqv + ~vp)]p22(v,p)

=

~2~vO

+ ~pi2[E+(,~*(_v + l,p + E_(~*(_v

—

[y, + i(Vqv + evp)]p11(v, p)

—

1)—~(v+ l,p.— 1))

l,p + 1)— ~(v —

l,p

[712

—

l,p

1))],

(4.10)

= ‘~1~vO

+ ~p12[E~(~(v + l,p + 1)— ~*(v + E_(15(v

+

1)— ~*(_v

—

+ l,p + 1))

l,p

—

1))],

p22(v

—

l,p + 1))

—

(4.11)

+ i(A + Vqv + evp)]~(v,p) =

+ ~p12[E+(p11(v

—

l,p

+ E_(p1,(v + l,p + 1)

—

—

1)

—

p22(v + l,p

—

1))].

(4.12)

174

S. Stenhoim, Theoreticalfoundations of laser spectroscopy

These equations can form the basis for a large number of calculations in the theory of saturation spectroscopy. To show the main effects included in these equations we consider the situation with only one travelling wave present, E = 0 say. From eq. (4.12) we find that ~

ip1~E~ [‘/12 + i(A + Vq + ev(p

—

—

1))](P22(0~P)

— pij(O,p

—2)).

(4.13)

Inserting this into eqs. (4.10) and (4.11) we find 2-~-—I÷L(A + qV+ ~(p i))[P22(O,P) Pii(O,P —2)], 2712 — + I~L(A+ qV + e(p + l))[P22(O, p + 2) Pui(°,p)], ‘/u 2712 —

P22(O,P) Puu(°’p) =

—

(4.14)

‘/2

=~—~—

—

(4.15)

where the dimensionless intensity parameter is 1+ =p~ 2E~/y~y2

(4.16)

and the Lorentzian is

L(x)

Y~2/(Y~2+

x).

(4.17)

Combining eqs. (4.14) and (4.15) we can solve for the population difference 0,P)

—

Piu(0,p —2)

=

N[i

— ~

(A

+

qV + ep —e)~+ V 1~(1+ ~I±)]’

P22(

(4.18)

where N

=

22/72

—

(4.19)

2,/yr

is the population difference without field and the coherence factor is ‘7 =

(Yu +

1

72)/2712

(4.20)

according to (3.9). If coherence is destroyed e.g. by collisions, the factor ~ is decreased and the degree of saturation is decreased too. The travelling wave modifies the population at the position Vq=—A+e—ep.

For e

(4.21)

0 we have the usual Doppler shifted velocity V = A/q but eq. (4.21) shows how recoil corrections enter. From the left-hand side we can see that the two coupled populations occur at velocities differing by the amount =

qAV = ep

—

—

e(p

—

2)

=

2e

hq =

=

J’q,

(4.22)

where Vr is the velocity change enforced by recoil; the situation is illustrated in fig. 4.2. Only two different velocity groups are coupled by E~and hence the solution is simple. The field transfers population between the levels and creates a Bennett hole [21] on one level and an ear on the other.

S. Stenhoim, Theoreticalfoundations oflaser spectroscopy

2>

175

-v

ii> I

~ Vr ~ Fig. 4.2.

The width of these contains power broadening and is given by F

712[1 + ,7I~]1l2.

=

(4.23)

To be able to calculate the amount of power absorbed we consider the expression (4.24)

W= and introduce the dipole moment from (2.44) and the field from (2.58). We find

w

i(~h

=

+ E_

~ (E+ <(a~_q,ak2e <(a~+q1ak2 e

—

a;+q2akl

—

e’~)> (4.25)

ak~_~2ak~ e_~t)>).

The contributions from the two waves can directly be separated to give 10t— e~°~]= —hflp W_

=

-~!~p~E_ ~ [ e

12E_ ~ Im i~~— 1, 1), (4.26)

where (4.8) has been used. The sum over all k-values can be transformed into an integral over velocity v W_

= =

hk/M and we fmd _h1~Pi2E_j’dvImi~(_1~1)~

(4.27)

where a constant factor has been dropped. Similarily we find for the absorption from the other wave

$

W÷= —hQp12E~ dv Im j~~(1, —1).

(4.28)

We calculate the absorption from one travelling wave using (4.18) and (4.13). We find Impl(1,_1)=_2L(A+Vq_6)N~~~2~

2

____ 712

2~

Y~2

(429)

(A+qV—e)+F

Inserted into (4.28) this gives W÷=

—

h(’l ~

$

dv~—~~ ~

+

~

(4.30)

176

S. Stenholm, Theoreticalfoundations of laser spectroscopv

where the velocity profile N(v) is taken into account. If N(v) is Gaussian, (4.30) can be expressed in terms of the plasma dispersion function. If the width of N(v), qu, is much broader than the line width, qu ~ F, and the resonance velocity from (4.21) with p = 0 falls close to the center of the distribution N(v), we can take out N and obtain the power absorption W~=

~ I~(1+ ‘7J+Y’12.

_~h12~~

(4.31)

For large enough power the absorption grows like J1/2 only. For both field amplitudes E~,E. nonzero no closed solutions are available when recoil is retained. Usually we can, however, assume e = 0 and then the problem can be reduced to the situation normally encountered in nonlinear laser spectroscopy. 4.2. Recoil-free nonlinear spectroscopy We shall now neglect the influence of recoil on our equations (4.10)—(4.12) describing saturation spectroscopy and set e = 0. As we see the resulting equations lack an explicit dependence on the variable p exactly like the equations we encountered in section 3.3. We can hence look for a solution depending only on the variable v, namely p(v). To see the consequences of this situation we assume the standing wave configuration E~= E

=

(4.32)

E,

characteristic of the field inside a laser. We define the new variables s(v)

=

[~(v) ~*(_v)]

d(v)

=

[p

(4.33)

—

22(v) — p11(v)].

(4.34)

From eqs. (4.10)—(4.12) we obtain (‘/2

+ iVqv)p22(v)

=

22ö%,0

—

(Yi

+ iVqv)p11(v)

=

~

+ 4ip12E[s(v + 1) + s(v

+ i(A + Vqv)]p(v)

[‘/12

=

.4ip12E[s(v + 1) + s(v

—4ip12E[d(v + 1) + d(v

— 1)] —

—

1)]

1)].

(4.35)

Solving these for s and d we find the coupled recurrence relations s(v)

=

—p12ED1(v)[d(v + 1) + d(v

—

d(v)

=

—p,2ED2(v)[s(v + 1) + s(v

—

1)], 1)]

(4.36)

N550,

+

(4.37)

where N is given in (4.19) and D1(v)

=

-[

D2(v)

=

i[ 1 2[Vqv —

.

2[Vqv+A—1y12 l’/i

+

+

Vqv

.

1,

Vqv—A—iy12j

—172]

(4.38) (4.39)

177

S. Stenholm, Theoreticalfoundations oflaser spectroscopy

The coupled equations (4.36) and (4.37)were first given in ref. [139],eqs. (61a, b) with some obvious changes in notation. The most noticeable one is the change of sign due to the choice of cos-functions here in contrast to the sine-functions of ref. [139]. In [139] it was observed that the solution of the homogeneous set (4.38)—(4.39) can have nonvanishing coefficients d(v) for even v only and s(v) for odd v only. These two equations can be

combined into the one equation x(v)

p,2ED(v)[x(v + 1) + x(v

=

—

1)] + ~

when x(v)

(4.40)

= d(v)/N for v = even and x(v) = s(v)/N for v = odd and D(v) is the function (4.39) or (4.38) respectively. The infinite set of equations (4.40) can be solved by the continued fractions

x(1) x(0)

—

s(1) d(0)

—

—

x(—1)

—

x(0)

—

—

p,2ED(1) 2D(1)D(2)

—

1

—

s(—1) d(0)

—

( .41)

p~2E 1 p~ 2D(2)D(3) 2E 1—... —

p

—

1

—

—

12ED(—1) 2D(—1)D(—2) p~2E 1—...

(4.42)

Clearly from (4.38) and (4.39) follows that

D(—v) s(—1)

= =

_D(v)*

(4.43)

_s(1)*,

(4.44)

as can be seen directly from (4.33) too. Inserting (4.41) and (4.42) into (4.40) for v relation

[ d(0)

0 we find the

s(1)1_~

.

—

+ Pi

=

2ED(0)2iIm~~] N E2”l

2 —

=

L +

1\~ +

P12

m 1

—

(4.45) D(1) 2D(1)D(2) 1—... J p~2E

—

N.

The saturation parameter

p~

/1

2(—

i\ +

—

J

p~ ‘7

2 2E

~

(4.46)

‘/172 2E \Y1 ‘/2/ is seen to agree with the definition (4.16) and (4.20). From the results (4.41), (4.42) and (4.45) all coefficients can be calculated. This solution was used in [139] to evaluate the properties ofa strong

signal laser in single mode operation. Here we do not want to investigate the properties in greater detail but refer the reader to ref. [139] and also [129] and [117]. An independent derivation of the continued fraction solution was given in [50]. A detailed discussion of the numerical calculations is given in [156] and [22]. In this work we have derived the basic recurrence relations (4.36)—(4.37) from our general equations of motion (4. 10)—(4. 12). We want to establish the correspondence with the semiclassical

178

S. Stenholm, Theoreticalfoundations of laser spectroscopy

starting point of refs. [139] and [50]. To this end we remember that the density matrix element is given by (4.2) as p(v)

=

,

(4.47)

where hk0 is the arbitrary initial momentum and no dependence on m + n is assumed. Remembering that we only couple a discrete set of momenta we can replace the continuous Fourier transforms of section 3.2 by discrete ones. In the language of that section (4.47) is in the (p, P) representation. Ifwe go to the Wigner representation by transforming away the dependence on p we can take the discrete transform p(Z)

~ e’~p(v),

=

(4.48)

which at the same time is the generating function for the variables p(v). Introducing this transformation into the equations (4.35) we find (‘/2

+ V~)P22(Z)=

iA +

[‘/12+

22

—

ip12EcosqZ[~(Z)

—

v~]~~(z) —ip,2EcosqZ[p22(Z) =

,5*(Z)],

—

(4.49)

p,1(Z)]

(4.50)

and similar equations for Pit and ~ These are the rotating wave steady state equations for the atom in the classical standing wave field E(Z, t)

=

2E cos qZ cos Ot

=

E[cos (qZ

—

fit) + cos (qZ + ft)],

(4.51)

as we expect from the superposition of two fields like (2.58). The velocity dependence enters in the total time derivative d/dt

Cl/Clt + v~Cl/Clr

=

(4.52)

as in eq. (3.8). If we keep in mind the time dependence of the position Z of the moving atom we can replace VCl/~Zby d/dt; see section 3.1. We have now considered the case of a standing wave. It is, however, possible to utilize the continued fraction technique also for unequal amplitudes E~~ E. To see this we return to eqs. (4.10)—(4.12) with e = 0 and use the notation (4.34) to obtain =

d(v)

—4p,2D~(v)[E+d(v

—

1) + E.d(v + 1)]

No~0— p12D2(v)[E÷(~5(v+ 1)

=

—

(4.53)

,5*(_v + 1)) + E...(15(v

— 1) —

~*(_v

—

1))],

(4.54)

with D2 given in (4.39) and

D~(v)= 1/(A + Vqv

(4.55)

— i712).

The relation =

—.4pu2D~(v)[E+d(—v + 1) + Ed(—v

allows us to eliminate ~ from (4.54) to obtain

—

1)]

(4.56)

179

S. Stenholm, Theoreticalfoundations of laser spectroscopy

—

.4D2(v)p~2[E~.(D~*(v + 1)

—

D~(—v+ 1)) + E~(D~(v 1) —

N~5~0 + .4p~2D2(v)E+E_[(D~(v+ 1)

=

+ (D~(v 1) —

—

D*(_v + 1))d(v

—

—

D(—v

—

—

D~(—v 1))]}d(v) —

1))d(v + 2)

2)].

(4.57)

This is again a three term recursion relation for the even coefficients d(v) and it can be solved directly using the continued-fraction technique as above. Shirley [125] has developed a different

method to treat the case of unequal amplitudes, but the result is, ofcourse, the same. In order to see the implications of (4.57) we consider first the situation of one wave only, i.e. E_ = 0. Then only the term d(0) is significant and we obtain [1 + .4(1 + 1’~ p~2E~y122 ld(0) [ \y, ‘Y2J (A + Vq) + ‘/12]

N,

(4.58)

which for c

= 0 agrees with (4.18). Setting E~= 0 we obtain the corresponding result for a wave travelling in the opposite direction. From eq. (4.34) it follows that d(0) is the population difference averaged over the length of the cavity. eq. (4.57) for the approximation we 2~andWe E~can but try neglect all cross-saturation terms E +where E This include that the self-saturating implies we set d(v) = terms 0 for vE ~ 0 and neglects all variations of the population difference. From (4.57) we immediately obtain -.

d(0)

= [1

+

.4(~_+ I) 7,

~

Y~ Yi~

(E~.L(A + Vq) + E~L(A

—

Vq))]

N,

(4.59)

where L is defined in (4.17). This is the well-known result of the rate equation approximation [139, 129], and it shows two Bennett holes at V = ±A/qwithout any interference between them. For well separated holes the approximation is good even if it does not reproduce the detailed structure of the velocity distribution. The observable quantities are, however, integrals of the

velocity distribution and the rate equation approximation averages the correct result for rather high intensities. When A ~ 0 and the holes overlap it is no longer so good. The degree of saturation is exaggerated and the Lamb dip structure becomes too pronounced, see the discussion in [5]. The velocity distribution (4.59) is compared with the exact one calculated from the continued

fraction in fig. 4.3.

d 0

0

d0

10

20

qV/y

0 Fig. 4.3.

10

20

qV/7

180

S. Stenholm, Theoreticalfoundations of laser spectroscopy

4.3. Standing wave features In the laser experiments the standing wave is the natural configuration. As there are several interesting features connected with this system, we consider some of these in the present section. For a standing wave we set E~= E in the rate equation result (4.59). From the equations (4.36)—(4.37) we can easily verify that the result involves the coefficients d(0) and s( + 1), s( — 1). It is of some interest to see what physics is contained in the next approximation, which includes d(± 2). From (4.37) we obtain d( ±2)

=

—p12ED2( ± 2)s( ± 1).

(4.60)

Inserting this into (4.36) we obtain the relations p,2ED1(±1) s(±1)= — 1 + p~2E2D1(±l)D2(±2)~°). (4.61) To obtain d(0) these are inserted into (4.37) and D,(+ 1) from the rate equation approximation is replaced by (Vq + A

—

iy12)(Vq

— A — i712)

(Vq _

[A

=

— i)’,2)

2

2

2

(Vq—iy12)—A—p12E

where we have assumed 2 + p~

Vq ~

—

(Vq 4(2Vq 2

— i712) —

2{1/(2Vq—

iy12)p~2E

— i[ — —~

i 71)

1

1/(2Vq

+

1

1 .

2[Vq—f~—iy12

+

— i72)}

.

i,

Vq+(~—iy12j

2]”2 ~

(4.

)

(4.63)

>~

2E

when all the ys are equal (4.62) is exact. Similarly we find that D( — 1) is replaced by

1

—

D 1(—1) ,~ 1[ 2D p~2E 1(—1)D2(—2) — 2[Vq

—

1 1 f~+ iy~2+ Vq + ~ + iy~2

464

(

Inserting (4.61) with (4.62) and (4.64) into (4.37) we obtain d(0)

=

[1

+

I’7(L(Vq

— f~) +

L(Vq +

1))]_1

N,

(4.65)

where we used I for the dimensionless intensity (4.16) belonging to each travelling wave. The result (4.65) is similar to the rate equation result (4.59); the Bennett holes have only been shifted to ±0. Their position for large detunings follow from (4.63) in the form 2/2A+ (4.66) ...

0 = A + p~2E

Thus the holes always occur at larger velocities than in the rate approximation. The reason is the direct repulsion between the two holes due to their mutual interaction, as originally suggested by Bennett [21]. The shift in (4.66) is a Dynamic Stark shift or light shift (see the discussion in ref. [137]) and we can see that it greatly resembles the Bloch—Siegert shift (3.40) when the frequency 0 is replaced by the detuning A. This analogy has been used [85] to establish a correspondence between the multiphoton problem and that of a standing wave with large detuning A. Some experimental evidence exists for higher order resonances also [55, 115].

S. Stenholm, Theoreticalfoundations of laser spectroscopy

181

When we let A go to zero we find that the two holes do not meet but will remain separated by the distance qAV = 2p12E,

(4.67)

which means that there occurs a Stark splitting as in the radio frequency case [10]. The line shape obtained is of the type shown in fig. 4.4. It is as if the two holes were unable to overlap too much

due to their repulsion. Thus they can saturate atoms over a broader velocity range and achieve a larger laser intensity than that given by the rate equation result. Hence we can see that the Lamb dip will be more shallow than that predicted by the rate approximation. In [156] it is shown that the approximation retaining v = 3 is very close to the exact results.

qV Fig. 4.4.

The considerations above can, however, at best be qualitative as we know that the approximations leading to (4.65) can be valid only for Al ‘/12. For A = 0 all the D-functions of (4.38) and (4.39) have resonances at V = 0 and it is difficult to see what the result of their interplay leads to. It is possible to obtain an exact solution of the case A = 0 if one assumes that >~‘

1/,

=

72

=

7.

‘/12

(4.68)

This was first noted by Rautian [114] and independently in refs. [127] and [50]. Here we use the method of ref. [50] as modified in [133]. Under these assumptions we can write eq. (4.40) in the form x(v) where e

=

x(v)

(p,2E’\ =

~

v

1 —

.[x(v + 1) + x(v

— 1)]

+

ö50,

(4.69)

(y/Vq). This is the recurrence relation for Bessel’s functions and if we set =

(2p12E\

x(0)J~_1~Vq )/

/

(2p,2E\ Vq )‘

(4.70)

J_IEI\

we can determine x(0) from the relation (4.69) at v x(—1)

=

—

x(1)*,

= 0.

Noticing that (4.71)

we find x(0)

[ =

[

iz(J,_1~(z) Jl+IE(z)\1 1 — —~

2e

\J..je(Z)

—

j

J+1~(z)jj

,

(4.72)

182

S. Stenhoim, Theoreticalfound ations of laser spectroscopy

where z

2p,2E/Vq.

=

(4.73)

Using the properties of Bessel’s functions [149] we find Ju±~~(z) = ± ~J~1~(z) ~-J±1~(z)

(4.74)

—

and further JIE(z)Jl

...IE(z)

—

=

—

=

—

J1~(z)J1+1~(z) J1~(z)J- ~E(z)+

~IJjE(z)I2 +

(~

- IE(z)

Jjr(z)

—

JIE(z)

J - j~(z))

sinh ire.

(4.75)

Inserting the relation (4.75) into (4.72) we obtain 2 x(0)

d(0)

=

(4.76)

J(2Pu2E)~

:t~

When the line width y goes to zero we find the averaged population difference p

22(V)

—

p11(V)

=

d(0)

=

N~JO(2!~2E)~.

(4.77)

As a function of V this shows the oscillating structure of fig. 4.5. The minima are zeroes at qV

=

etc.

qV =

~

(4.78)

O5~7

0.5

1.0 ~

p,2E

Fig. 4.5.

For a finite value of e the expansion =

~0

+

~

=

y/q V the function

uv v=O

=

~0

+

2 y0

I~ 12 never becomes exactly zero as can be seen from (4.79)

S. Stenholm, Theoreticalfoundations of laser spectroscopy

and hence

d(0)

IJIEI2

N~

=

N

=

~::~

+ c2ir2

183

(4.80)

~

Because the zeroes of the Bessel function of second kind Y 0 do not coincide with those of J0, the expression (4.80) is never exactly zero. The considerations above are valid as long as Vq > y. When the velocity goes to zero, the argument of the Bessel function blows up, but at the same time the complex order e grows to infinity. This shows the complexity ofthe point V = 0. Its special role was seen in ref. [127] and is discussed in [22] too. In the next section we will obtain the exact result at V = 0 which was derived as the asymptotic limit of (4.72) in ref. [50, appendix C]. The structure near V ~ 0 was noticed in [139] but only as a bump like that in fig. 4.4. The full structure was derived from the continued fraction by Feldman and Feld [50] but it was earlier documented by Rautian [114]. There seems to be no experimental evidence for this type of behaviour in spite of its theoretical interest.

4.4. The solution for stationary atoms In this section we shall considerthe situation for those atoms, which do not move in the direction of the light beam. They experience no Doppler shift and hence resonate only with the field at resonance, A = 0. In the case of one relaxation rate only (4.68) it is shown in [127] that the population as a function of velocity is given by 2 (481) k=O d(0) ~ (—1Y’(2k)’(k’Y [1 + (qV/y)2][1 + 22(qV/y)2]. [1 + k2(qV/y)2] This result has the interesting property that for V ~ 0 we have a convergent series for all values — —

..

N

of I as the terms of (4.81) for large values of k asymptotically become 1)(2k)!(k!~4(Iy2/q2V2~’,

(—

which give a convergent series. For large velocities the expansion parameter is decreased by the factor (y/q V)2. At zero velocity, however, the series (4.81) sums to

d(0)

N[1 + 41]_u/2

=

(4.82)

This power series does, however, converge only for I

=

p~

2/y~ <~,

(4.83)

2E

and hence the special nature of the point V = 0 is obvious. We want to look at the stationary atoms directly from the exact continued fraction solution

(4.45) which converges well for all values of the velocity. When V = 0, the functions D of (4.38) and (4.39) lose their dependence on v and we have only D,

=

i

2 +

Y~2) =

iL(A)/y,

712/(A 1.1 — .k•~i/ / \ — L~2 — 21~.1/Y1 rt i‘f7~, —

2 / D771217172. .

(4.84)

184

5. Stenholm. Theoretical foundations of laser spectroscopi

The continued fraction becomes =

1

—

p2E2DD 1 — p~ 2D 2E 2C’

(4.86)

because the sequence repeats itself. The combination 2= i D2p~2E

(4.87)

712171

enters into the equation for C of the form 2 + C + L(A) =0. C 71217’ 171712 The solutions to this equation are C

=

—

[1

± ~1

+

(4.88)

4 171L(A)].

(4.89)

2712171

Inserting this into (4.45) we find the two solutions 2N. (4.90) d(0) = ± [1 + 4171L(A)] - “ At this point we notice that only one of the two solutions can be physical. The solution with the negative sign shows that d(0) = — N at zero field intensity, which contradicts the definition of N in (4.19). The ensuing result in (4.90) is seen to agree with the summation (4.82), when all relaxation rates are equal, viz.’7 1, and we tune to resonance, A = 0. Finally we want to comment on the fact that there are two solutions of the basic equation (4.88). The same holds true for the basic recurrence relation (4.40) because it is a second order difference equation. Thus for the equation (4.69) there is the additional solution ~ The latter, however, diverges as (v 1)! whereas the function J~ — (v ! ) — Thus only the latter can be the coefficient of a convergent Fourier series. The convergence of the series is discussed by Holt [73] and Kuroda and Ogura [84]. The situation with respect to the solution of the second order difference equation can be illustrated by the analogy with a second order differential equation ~.

—

d2y/ds2

—

ic2y

=

(4.91)

ö(s).

For s ~ 0 the solution is y

=

Ae’~+ BeKS.

(4.92)

If a bounded solution is needed there is no way to choose A and B so that (4.92) would be valid for all s. The delta-function in (4.91) allows us to choose the solution in the form y(s)

=

—e~5/2ic;

s

y(s)

=

—e~/2i~

s <0,

>

0 (4.93)

which is seen to reproduce the delta-function. The shape of the solution (4.93) is shown in fig. 4.6. A similar situation prevails with the difference equation (4.40). The homogeneous equation has got two solutions but one is regular for large positive values of v and the other one for large

S. Stenholm, Theoreticalfoundations of laser spectroscopy

185

a-ET1 w362 586 m367 586 lSBT S

Fig. 4.6.

negative values of v. The delta-function at v = 0 allows us to join these two at this point and gives the unique solution (4.45). The situation is like that of fig. 4.6 when s takes the integer values v. The two linearly independent solutions can bejoined at v = 0. By joining them at another arbitrary point we can develop a Green function for the difference equation and solve a general inhomogeneous difference equation, see ref. [5, appendix 3]. In this section we have considered some special features of the solution for the velocity distribution. In particular we have looked at the limiting point at V = 0, which cannot be computed from the series expansion (4.81) and is quite complicated in the analytical expression (4.76). It is shown that we can easily solve this problem directly and obtain the analytic result (4.90). The procedure calls forth some comments on the general solution of the second order difference equation and its generalizations. This discussion has taken us far into the mathematical formalism and next we want to return to a physically relevant problem. 4.5. Probe spectroscopy The first observations of narrow resonances within the Doppler width were those connected with the Lamb dip inside an operating laser [49]. There the saturation due to one wave is probed by the other. But the fact that the probe is strongly saturating means that the magnitude of the dip is decreased. Actually, for large values of the intensity the dip disappears because the powerbroadening exceeds the Doppler width. This was pointed out by Greenstein [59] within the rate

approximation. As we have seen this implies that also the less pronounced Lamb dip of the exact theory will be obliterated. The exact nature of the process is discussed in [156]. It was soon realized that it was advantageous to perform the experiment outside the laser cavity [12] and with one weak probe beam [94]. This way the full magnitude of the Bennett hole caused by the strong wave could be sampled by the unsaturating probe. This theory can be developed quite straightforwardly and displays many points of physical interest. We first assume the counterpropagating amplitude to vanish, E_ = 0 and use as the zeroth approximation the running wave solution (4.13) and (4.18) with recoil neglected, s = 0, —

~(O) =

~

= N[1

—

171÷(A + Vq)2

±Y12(l

+

?l1+)1’

(4.94a)

ip, 2E÷[‘/12

+ i(A + Vq)]h1~ — p~fl.

The strong field has created two modifications to the field-free situation. The population difference

S. Stenholm, Theoreticalfound ations oflaser spectroscopy

186

has been modified in (4.94a), but due to the coherent nature of the field there has arisen a polarization which follows the phase of the field, (4.94b). To calculate the response to the probe field E we write the perturbation approximation to (4.10)—(4.12) with recoil neglected. We must remember that only ~5(v= 1) and p22(v = 0), Pi 1(v = 0) are nonvanishing in the lowest We obtain 1~(— 1) approximation. = ~ip [‘/12 + i(A — Vq)]p~ 12E+[p~(—2) — p~’~(—2)]

+

4ip12E...[pç°~(0)—

p~(0)]

(4.95) 1~(— 1) — [Yi

— 2Vqi]pç’~(—2)

[‘/2

—

[‘/12

=

i(A +

1)— —

— —

~i)(_

2E...~5(°)*(1)

(4.96)

1)] + ~ip,2E...i5(O)*(1)

(4.97)

3Vq)]~3W*(3)= —~ip12E+[pç’~(—2)—

The absorption of the probe beam is related to ~1(

2[712

4ipi

—

~ip12E~[j5~

2Vqi]p~.’~(—2) = 1ipi2E÷[i3~)*(3)—

—

~(1)*(3)]

ip12E... + i(A Vq)] —

~

(4.98)

p~(—2)].

and we solve

~5(1)( — 1) by (4.27)

—

—

ip12E~ 2[y~ + i(A Vq)] —

~

—2)— pç’~(—2)]. (4.99)

The first term is induced by the population modifications caused by the strong beam and gives a rate contribution. The second term contains coherence effects and has to be solved from (4.96)— (4.98). As the inhomogeneous term here is ~3(0)* we see that the coherence created by the strong field will induce changes in the second term of (4.99). Thus we have both population induced and coherence induced contributions to the response. Combining eq. (4.96) with (4.97) and (4.95) with (4.98) we obtain the coupled pair [p~’~(—2)

— p~’~(—2)]

[~1)(1)

—

~1)*(3)]

x

=

=

— ~5(1)*(3)]

—p,2E+D2(—2)[j5”~(—1)

—

[pW(-2)

~iPi2E+[7 + i(A

-

p~’~(-2)]

-

—

Vq) +

[

p12E

‘/12

—

i + i(A

[‘/12

+ pi2ED2(_2),5~~*, (4.100)

i(A +

1 3Vq)]

—

PuiJ’

(4.101)

Vq)

where D2(v) is given in (4.39). Solving these we find

i(A

~Y12+

=

—

Vq) +

_____________

[p~’~(—2)— P~1~...2)][l—

ip~2E~ED2(—2), 7,2

+ i(A

—

Vq)

—

(0))

712 — i(A +

3Vq))]

______________

+ p12ED2(_2)~O)*.

(4.102)

~P22

Because of (4.95) we can write

~W(—2)—p~(—2) =~ip~2E~ED2( —2)H

1

[‘/12

+iA— Vq)+ 712 —i(A+ Vq)]~22P

11) (4.103)

187

S. Stenholnl, Theoreticalfoundations of laser spectroscopy

where H

=

1

~P~2E~D2(2)(712 + i~A Vq) + 712

—

—

—

i(A + 3Vq)}

(4.104)

We define f(Vy’

—2iD2(—2)7172(y, +

=

(4.105)

72)’

and find that we can write 2

(1)(2\ P22k .‘

—

(1)12\ Pii~

.‘

x [~ —

— —

—

~°fl ~

—

p12E~E_ 7172 71217

[2fV+nI+~i2(712

p~2E~E_ ‘/172

712

‘/12

[~

‘/12 —

i(A

—

1 + i(A

Vq) +

7,2

—

1 + Vq) i(A

+ i(~— Vq) +

‘/12

—

i(A + 3Vq))]

—

—

p~°flB(V, A, It).

(4.106)

Vq)

Inserting this into the equation (4.99) for the observable (4.27) we find 2B(V,A, 1+)]. (4.107) 13(l)(_1)=iPl~E_

~

y~2+(A—Vq)

The observable will be given by the integrand of (4.27) and we have —0p 12E_ Im ~(1)(

1)= ..o(P~E~.)(2+(A— Vq)2)~ —p~°fl[1—p1÷Re $(V~A, I)]. (4.108)

The absorption of the probe was written in the form (4.108) by Haroche and Hartman [70]; the same calculations were also carried out by Baklanov and Chebotaev [17]. They have also discussed the case of a probe parallel with the strong beam [18].

The result (4.108) is written in a form where the various terms have simple physical meaning. The first three is the absorption frequency 0 times the matrix element p12E_ squared times the Lorentzian for energy conservation. This is multiplied by the population difference, and if I + ~ 0 these factors correspond to the linear absorption of the wave E_. For 1÷~ 0 the population difference is modified to (4.94) and we see that the absorption consists of a product of two Lorentzians, one of which is power-broadened by 1+. This is the result ofthe rate approximation, coming from the first term of (4.99). In addition we have a coherence contribution in (4.108) which contains the function ReB, which was introduced in ref. [70]. The definition (4.106) shows that in addition to resonance behaviour at Vq = ± A we have a resonance at Vq

=

—iA.

(4.109)

This is the emergence of the first multiphoton resonance which is contained in the general solution (4.45); we also have resonant behaviour at V = 0 from the function f of (4.105). These features are discussed further in [85]. The macroscopic observable involves the integral of (4.108) over velocity. The special features of this are discussed in [17].

S. Stenhoim, Theoretical foundations of laser spectroscopy

188

4.6. Recoil effects In section 4.1 we considered the effects caused by atomic recoil due to the photon momentum. In the resonance condition for a travelling wave in (4.21) we have a recoil shift of the Bennett hole by AV = J’~.,see (4.22). In a standing wave the consequence is that we have different positions for the holes and “ears” caused by saturation, see fig. 4.7. Thus the two features in the velocity distribution overlap for two different values of the detuning at A = ±e= ±~V~q. The Lamb dip is, consequently, split up into two, and as they relate to different levels they have different strength if ‘/2 The recoil in fluorescence is discussed in ref. [86]. ~.

lv

v~

-~

Fig. 4.7.

The first calculation of recoil effects was done by Kol’chenko et al. [81] using a perturbation approximation for the Wigner distribution. We can easily derive their result again from (4.10)— (4.12) using the perturbation form p~’~(v,p) = ~ipj 2D2(v,p)[E+(j~~~)*(_v + l,p

p~(v+ l,p

— 1) —

1(v —

+ E(5~~)*(_v 1,p + 1) — j5~” ~ip 12D1(v,p)[E+(j~(v + 1, p + 1)

l,p

+

— ~(n)*(

—

— 1))

1))],

(4.110)

—

p~~ ‘~(v,p) 1~ 1i

=

+ E..(p~”~(v — ~~v,p)

=

l,p

—

1) —

~ip12D12(A,v,p)[E+(p~(v

,3(~~)*(_v l,p —

— l,p

—

1)

—

+ E..(p~(v+ l,p + 1)— p~(v+ l,p

—

—

v + 1, p + 1))

1))],

(4.111)

p~(v l,p + 1)) —

1))],

(4.112)

where Dk(v,p)

=

~

y, + iv(Vq + ep)

(k

=

1,2)

(4.113)

and D12(A,v,p)=

—~------~—————-----.

‘/12

+ i(A + Vqv + evp)

(4.114)

189

S. Stenholm, Theoreticalfoundations of laser spectroscopy

The initial value for an iterative solution of the set (4.110) to (4.112) is obtained from 2k/’/k (k = 1,2). (4.115) p~~T(O,p) = This generates the terms: ~5( 1 ~(1,p) and j3~1)( 1, p). In second order we generate the elements: p~(0,p), p~(±2, p) (k = 1, 2) and finally from (4.112) we obtain the third order result ~5~3ki, 1), which enters the expression for the observed absorption (4.28). We find —

—

—

1)

=

—

i(~p

3ND

12)

12(A,1,

—

1)[E~(7’ + 7172

72)

~

L(qV + A

—

712

+ E~E+(D12(A,1, —1)(D2(2, —2) + D1(2,0)) +

2

L(qV

—

A + e)

72712

+

2

L(qV—A—3s)+D1(2,0)D12(—A, 1, +1)+D2(2, —2)D(—A, 1,

71712

J (4.116)

where we have introduced the Lorentzians (4.17). It is possible to generate the fifth order terms in the same manner by further iteration of (4.1 10)—(4.112) but the result becomes unwieldy, and the range of additional validity is small. Baklanov [14, 16] has used this result to calculate the power dependent shift of the Lamb dip. The experimental result is obtained when we integrate (4.116) over the velocity distribution. In order to be able to observe the small recoil effects, one has to perform measurements on molecular transitions with very narrow lines. Then the Doppler width, qu, is usually much broader than the frequency region over which the resonance behaviour occurs. We can, consequently, integrate (4.116) over a flat velocity distribution. Then a considerable simplification occurs as we have from (4.113) and (4.114) Dk(v, p)

cc

[Vq + cii

—

1’Yk/VI

—‘

D, 2(A, v, p) cc [Vq + A/v + ep

—

iy, 2/v]

(4.117)

‘.

In (4.116) we see that we have only D-functions with positive values of v and hence they have their poles in the upper complex half-plane of the variable V. Hence the integration over velocity can be closed in the lower half-plane and only the Lorentzian terms will contribute. We are left with the terms 3~(1, —1)

=

—

(~p,

Im~

2)~—~—L(A + Vq

—

c)[E~(7’ + Y2)~L(A + qV— c)

712

+ E~E + 72712 2 L(qV

7172

—

A + e) +

YuYi2 2

7i2

L(q V

—

A

—

1. 3c)J

(4.118)

The integral is now trivially executed after substitution into (4.28) to give ~

=

const. [i~

+ 72 + ~I+I_(2~_L(A 2712 712

‘/1

—

c) + -~-~-L(A + e))].

(4.119)

‘/12

The first term is a self-saturation term and the term proportional to 1÷I_is the cross-saturation effect which is split into two Lamb dips with the relative weights (Y,/’/12) and (‘/2/7,2) respectively.

190

5. Stenholm, Theoretical foundations of laser spectroscopv

is given by the zero of its derivative The maximum of the signal W~3~ 2 + Y~2] = 0. 71(A — e)/[(A — e)2 + ‘/~2] + 72(A + e)/[(A + e) For a small splitting 2e ~ ‘/12 we obtain A(max)

=

(4.120)

e(y 1

—

72)/(Y1

+

(4.121)

‘/2),

first derived in [81]. For larger values of e the Lorentzians become split. The present derivation is obtained in [5]. Thus the conclusion is that, for well resolved recoil effects, the line becomes a doublet, in other cases it is shifted when the levels decay at different rates. Experimentally the only system where recoil effects are important is the methene CH4 absorber at the 3.39 jim He—Ne laser line. There e ~ 2kHz and this resolution has been achieved as a part of the work on new frequency standards. Two groups have been able to resolve the splitting: the Boulder group with J. Hall and collaborators [63, 64] and the Novosibirsk group with Chebotaev et al. [13] and [95, section 10.2.2]. Here the lines are of equal width and the spectrum is approximately symmetric. The slight asymmetry observed [63] may contain information on collision processes in the gas. In the experiments the fields are so large that the perturbation series is presumably no longer valid. The treatment of section 4.5 can be generalized to the case with recoil included [136], and one can calculate the response of a weak probe field with a strong counter running saturating wave [4], and also [31]. The rate approximation can be generalized [5]and shows the filling in of the splitting due to power broadening, see fig. 4.8. We see that for the assymmetric case the peak value remains very close to the peak of the stronger line.

w

/; \

REA -

E=3;

I,=I..= 30

w

REA = ~

=

1,=1= 3O~

Fig. 4.8.

2

S. Stenholm, Theoreticalfoundations of laser spectroscopy

191

As usual, we expect the rate approach to exaggerate the structure and the true curves will have less pronounced splitting than the rate results. In ref. [126] the Shirley representation (see section 3.2) is used to calculate the saturation spectroscopy Lamb dip including recoil. It is possible to

attack the problem numerically and obtain results that go beyond the rate equation calculations. The general conclusions given above are born out: The structure of the rate approximation resonances shows too much variation, the true result is smoother, but it is also found that the positions of the peaks are shifted only by an inperceptible amount. This method is presently utilized to obtain more detailed information about the strong signal behaviour of recoil-split Lamb dips. 4.7. Concluding remarks In this part of our article we have looked at velocity selective interaction between radiation and matter, which gives rise to saturation spectroscopy. This field of non-linear spectroscopy was born with the observation of the Lamb dip in the lasers [49]. Since then the theoretical understanding has progressed according to the following brief historical summary: For a standing wave Lamb published the third-order perturbation theory in 1964 [88]. Within the perturbation theoretic framework, the recoil effect was included in 1969 [81]. The same year the general strong signal solution was obtained [139]. The solution with one strong wave and a weak probe wave was obtained 197 1—72, [17, 18, 70]. The general case with two different amplitudes for the two counter propagating waves was solved by Shirley in 1973, [125]. The inclusion of recoil into the probe theory was carried out in 1973—74 [136,4] and the generalization to the rate approximation appeared in 1976 [5]. The recoil calculations [126] are still being carried out to generalize the strong signal theory. There are many aspects oflaser theory, which have not been discussed in this article. In particular we have omitted a discussion of the actual operation of the laser device, which can be based on a self-consistent determination of the field amplitude [88]. General discussions of these aspects are found in refs. [129] and [117]. For lasers the multimode operation is a very important aspect. Its theory has been developed within the frame of the semiclassical theory and a detailed discussion is found in [117]. One problem which arises in spectroscopy of very narrow atomic and molecular lines is the

limit imposedon the interaction time by the transverse motion. Manyparticles transit the diameter of the laser beam in a time shorter than their life time. Only the very slow particles interact long enough to give a very sharp line. Theoretical discussions of this problem are found in [20, 143]

and numerical evaluations of the spectroscopic implications are given in [32, 142]. In this work we have discussed only transitions between two levels. Spectroscopists can find many phenomena of interest connected with three levels. The semiclassical description of these phenomena was first given by Feld and Javan [51] and Hänsch, Toschek et al. [65,67]; for a recent review see [38]. Some recent experimental observations of the complex line shapes possible in a three level system are given in [119, 120]. A special branch of three level spectroscopy is the Doppler-free variant suggested by Vasilenko

et al. [145]. The advantage here is the utilization ofall atoms due to the lack of velocity selectivity. A recent review of this very successful experimental technique is found in ref. [27], and also in [95,chapter 4].

In short, non-linear laser spectroscopy has grown to a very large area of research which much

192

S. Stenholm, Theoreticalfoundations of laser spectroscopy

exceeds the scope of the present article. Here we concentrate on the basic formulation and its most immediate consequences. For additional details we must refer the reader to the many recent review articles.

5. Spontaneous emission 5.1. Rate equations for spontaneous emission A feature characteristic of quantum electrodynamics is the occurrence of spontaneous decay of excited atomic states. The energy captured in a bound state can be released in the form of radiation escaping into empty space where previously no electromagnetic energy was present. With spontaneous emission we understand the transformation of atomic energy into electromagnetic radiation emitted into an empty eigenmode of the field. There the energy is supposed to escape without being able to recollect itself into an absorption process. The vacuum of electromagnetism thus acts as an ideal heat bath in the sense of irreversible thermodynamics. The emitted energy can never retain enough coherence to possess any finite recurrence time. The similarity between the spontaneous emission and the reservoir theory of a heat bath has been developed in detail by Cohen-Tannoudji [42]. The general theory has been reviewed by Haake [62] and Agarwal [3]. In this approach the method of projection onto the subspace of interest is central; it was originally developed by Zwanzig [157, 158] and developed further by Fano [48]. The first work where an irreversible equation of motion was derived for the density matrix is the paper by Landau [91]. Nowadays such equations are usually called master equations after the quantum mechanical rate equation derived by Pauli [110]. What are the properties we expect an ideal reservoir to possess? Firstly it has to be able to relax rapidly enough to distribute any energy injected into a few of its degrees of freedom rapidly over a number of modes proportional to the size of the reservoir. This energy is then assumed to remain incoherent and unable to return to the emitting system for any finite Poincaré period. Finally the reservoir must be large enough to be able to absorb this energy without any discernible change of its state. The necessary conditions are clarified by Fano [47]. For the actual derivations of irreversible (master) equations powerful techniques are available [3, 62]. Under the assumption of relaxation into an ideal reservoir, all the essential physics is found in the derivation by Wangsness and Bloch [148] of the phenomenological Bloch equation for nuclear magnetic resonance. We will derive the rate equations for spontaneous emission directly from the approach of ref. [148]. We work in the space including only the atomic states lik> where i = 1, 2 and E2 — E1 = hw21 and k is the translational state with momentum hk. The field space is taken to be the vacuum 0> and the one photon states lq> with the electromagnetic energy hOq. The Hamiltonian is given by the quantum terms of (2.63) only h[>~(ek + w~)a~Sak~ + >~0qb~bq — ~g21(q)(a~+q2a,1bq + kci q kq The only matrix element coupling the basis states to each other is H

=

<0;2k~H~q;1k’> =

—hg21(q)ö,,+q

and the equation of motion for the density matrix gives

aLqia,2bq).

(5.1)

(5.2)

S. Stenholm, Theoreticalfoundations oflaser spectroscopy

i

<0;ik~~ 10;ik’>


(ek

—

Ck.

+ w~)<0;ikl ~ 10;ik’>

{g21(q)[ö~2
—~

i

=

I q ;jk’>

=

(ek

—

—

Cl’

193

—

q~ l0;ik’>

—

p q;1, k’

<0;ikl

—

q> o~2]},

(5.3)

I q;jk’>

+ w~~)
g2,(q)[ö11 <0;2,k + q~p~q;jk’> ö,1]. —

(5.4)

In (5.4) we have neglected terms leading to states with two photons because these correspond to correlation properties of the vacuum, which is outside the scope of the present treatment. In order to evaluate the interaction terms of (5.3) and (5.4) we need to calculate the matrix elements of type from the equation

I

i~-
l0;ik’>

=

(Oq — (D~~ + C1

—

Cl.)



—g21(q)ö31 <0;2,k + q~p~0;ik’> + ~g2u(q’)5~2
—

q’>.

(5.5)

In order to impose the properties of an ideal reservoir on the system we assume that the sum over q’ can be neglected. This omits states like and ; the former matrix elements represent coherence between the photon states Iq’> and and the latter the population on level q>. For a reservoir these must both be negligible. From (5.5) we, consequently, obtain

I



ig21(q)

=

dt’ exp [i(Oq

—

I~>

w~1+ c1

—

e1.)(t

—

t)]o~1<0;2, k +

q~~(t’)0; ik’> (5.6)

Inside the integral we must introduce further approximations. The main time dependence of the density matrix comes from the energy difference between the basis states. This can be exhibited explicitly in an interaction representation <0;2, k + q~p(t’) 10;ik’>

=

exp [—i(w2, + ~

—

s1~)t]<0; 2,

k + q~~5(t’)0; ik’>

(5.7)

The time variation ofj5 can now be assumed slow compared with that of the exponential. The main contribution to the integral comes from the upper limit and hence we can write <0;2,

k + q~~(t’)I0;ik’> ~ <0;2, k + ql j5(t) =

l0;ik’>

exp [i(w21+

—

c1~)t]<0;2,k + qI p(t) 10;ik’>.

(5.8)

We obtain the integral t J

0

t

dt’ ei~0(t’_t)=

$

dx ~

(5.9)

0

where A00qWii+CkCk~W2iCk+q+Sk~QqW21+CkCk+q

(5.10)

194

S. Stenholm, Theoreticalfoundations of laser spectroscopy

because w21 + w1 = w21. We have to carry out the integration in (5.9) in such a way that it has a well defined asymptotic limit for large times, in order to obtain the correct irreversible behaviour. This problem is solved in scattering theory (see e.g. [123, section 11.6]). We set

J

1

e’~’~dx

=

—

eL~0+~~)t

i(A0

—

~

=

i)

+ ilrö(A0))

(5.11)

in the limit of large times. Introducing (5.7)—(5.1 1) into (5.6) we obtain = ~j,g2t(q)( —

W21

6k+q

1

+

61

—

l?7J . ~<0;2,k+

(5.12)

qjp(t)~0;ik’>.

—

This is now to be introduced into the equations (5.3) and (5.4). Before this we define the reduced density matrix where we look at the marginal distribution with respect to the atomic coordinates only and trace away the unobserved photon degrees of freedom p,~.(k)


=

Tracefp

<0;ikl

~

Iik>

+ ~

I0;ik>

(5.13)

+....

As we have assumed the one photon states to suffice for our description we can obtain by summing the equations (5.3) (5.4) to obtain i


p~1k’>

=


1k’>

~

[w 11+

—

6k’]


p~1k’>

~g21(q){~12
—

—

Cl

~j2<0;tk~p~q;1,k

—

—

~I~

q>

I0;.ik’>

+

~ti

<0;2,k + qj p q;jk’>

5~1}.

—

(5.14)

We insert the approximate expression (5.12) and assume that inside the sum over q it is sufficient to include only the first term of (5.13). Then we obtain the general relaxation equation for the density matrix of matter. It is given in the form i



=

[w~~+

61

clq

+ ~ —

~1

—

11

81’]


1k’>

—

~

1g21(q);2[o.2

6k’ — 6k’+q + i 1 (021 + Cl — 6k+q — 117

—

—

Qq

—

61 8k’—q — 8k

—

j2

ç~ —

~21


P12

1k’>

+

117

—

117

(

5 15

To see the implications of this result we specify to the three density matrix elements of interest P22, Pit and P21~We find first
P22

1k>

=

—F


1k>,

(5.16)

195

S. Stenholm, Theoreticalfoundations of laser spectroscopy

where F

=

=

~

6k

Ig2iI2(0

—

g2

2iv ~

5(Oq

1I

+

~2t —

~k-q

+

~2i

—

‘17

—

—

~21

+

~k-q

—

Cl

+

—

6k—q

—

(5.17)

Cl).

t

This describes simple decay from the upper level. For the lower level we obtain ~—

1k>


=

~ G(k, q)
qI P22 1k + q>,

(5.18)

where G(k q)

=

1g2,I(

.

—

\Qq~W21+6k~6k+q~117

2irIg 25(Oq 2jj

=

—

(021

+

6k

—

__________________

0q(021+616k+q+1l1

Ck+q).

(5.19)

This shows that atoms appearing on the lower level with momentum hk may originate at any other momentum value h(k + q) satisfying the energy conservation condition = (~2,

+

61

—

Cic+q)

~

(5.20)

~

It is easily seen that when the photon momentum hq can be omitted we obtain the usual decay equations because >~G(k,q)= F.

(5.21)

For the off-diagonal elements we obtain the expression

i~

= (w21

+ A

—

i~F),

(5.22)

where 1 6k—q Cl (5.23) + is a level shift for the two-level system which is equivalent with the well-known Lamb shift for —

q

q

—

~21

multilevel systems. In this paper we have derived the equations (5. 16)—(5.23) directly in the form desired. This approach appears useful because it displays explicitly the various approximations utilized. Especially interesting is the fact that we need only the first term of the expansion (5.13) to obtain the equation of motion for the sum up to the second term. This casts an interesting light upon the situation leading to a general relaxation equation. The derivation presented here is a particularization of the general theory of relaxation as given by ref. [148]. A particularly detailed discussion is given in [42]. There the relaxation terms are derived without the recoil; c = 0. How to obtain the results of this section from the general theory is explained in [138]. The same type of approach can also be used to describe optical pumping

with laser light. This work is reviewed in ref. [42].

196

S. Stenhoim, Theoretical foundations of laser spectroscopv

5.2. Some simple consequences First we consider the radiative decay rate of the upper level (5.17). We find F

=

2ir~

g21l

t~(Oq

~21

—

+

6kq

—

2~

= ( 2

w

—

)2$d~dcoso~ dq2 Vh

(5.24)

21)

where the definition of g21, eq. (2.46) has been used and the polarization sum is reintroduced. We have F

=

2(27~)28O(C)~f~~c050h2 3 ~icp~

3, (5.25) 2= hp~2w~1/3irE0c which agrees with the usual result for radiative decay. The integral kernel occurring in (5.18) extends over a range of length 2q. Writing the integration explicitly over one direction only, we find

26

=

(h/2(27r)

0)(w21/c)

~ G(k, q)
1k

+ q>

= (2)~

=

(~)2

d~d

cos

0q2 dqö(cq

—

Wq)


1k

+ q>

~ Jdzjg 2il2

(5.26)

=FJdz,

where we have written z = q cos B and used the definition (5.25) of F. When we neglect the z-dependence of the density matrix we regain the result (5.21). Finally we consider the frequency shift A=

(2

)3Jd~dcosoPi2[2

]~d~[~(1

—

~-)

—

~

+

s.].

(5.27)

When we neglect the recoil ~ ~2t,

hk/M ~

C

(5.28)

we find that for the upper limit of integration A~Jq2dq

(5.29)

is badly divergent. This divergence is due to our use of the interaction (2.28); if we had used the vector potential form we would have had only a linear divergence J dq. Power and Zienau [112] have shown that we can decrease the divergence by one power of q when we take into account the

S. Stenholm, Theoreticalfoundations of laser spectroscopy

197

polarization energy (2.30). The last step of the renormalization procedure is to subtract the second order shift due to the free electron with the Hamiltonian (5.30) H(free) = —eA v + eA2/2m. Then one obtains the logarithmic divergence A

Jd~/~~

(5.31)

originally derived by Bethe [24]. When the calculation proceeds from the relativistic equation for the electron Kroll and Lamb [83] showed that the result becomes finite after renormalization. This suggests that part of our trouble comes from the nonrelativistic approach and, indeed, no problems are encountered if we accept that only photons with h0q

.~

Mc2

(5.32)

are important. As we have assumed that our bound neutrals are stable, we must neglect photons with energies much less than the upper limit in (5.32) because of the dissociation danger. Such a cut-off remedy cannot, of course, be the basis for a consistent theory. It is interesting to notice that when recoil is retained in (5.27) the asymptotic behaviour changes. For large values of q we obtain (5.33) with one power less of q. When we carry out the renormalization as above it turns out that even the nonrelativistic shift becomes finite. This was pointed out by Lamb, but the only publication is a footnote in the paper by Welton [150] (as far as is known to the present author). This compensation can, however happen only when =

hq2/2M

>

cq

hq

.~.

2Mc,

>

(5.34)

which is clearly beyond the region of validity assumed by our nonrelativistic approach. Recently the quantum mechanical calculations of the Lamb shift have been reconsidered in some detail [1, 2, 34, 104, 105, 116]. In particular it is shown that a two-level system is special in that, even without recoil, one can obtain a finite shift by including the counter-rotating term. The calculation gives the result A cc ~

I d~( ~q

J

—

q w

—

21/c

(5.35)

q

q + w21/c)

The first term is identical with (5.23) for c = 0, and the second term gives a totally negligible contribution near the singularity. For large values of q we have A

$dqq2[1

+~-~-~

—

(i

—

~-~-~)] = ~

(5.36)

which.is equivalent with the result (5.33). The result (5.35) is, however, more satisfying because the compensation starts to be effective as soon as

198

S. Stenholm, Theoreticalfoundations of laser spectroscopv ~

=

>~‘

(5.37)

~21,

which may well be valid in the nonrelativistic regime. Next we are going to look at the effect of the spontaneous emission terms derived in section 5.1, when we introduce a strong field. In the rest of this part we neglect the momentum changes induced by recoil and the integration in (5.18) disappears in accordance with (5.21); the momentum variable becomes a label of the atoms only. We assume that we can add the strong field effects directly to the equations derived for spontaneous emission without any interference phenomena. This can be justified by the extremely short correlation time possible for the spontaneously emitted field [42]. The exact validity conditions for this approximation still remain to be formulated. We consider the two-level system with spontaneous emission 12> —~ 1> as derived in section 5.1. The field is taken to be of the form

I

E

=

E0cos~i,

(5.38)

where we have the phase I/i

=

lit

—

kz.

(5.39)

We introduce the rotating wave approximation by setting P21

(5.40)

P21 ~

=

and we introduce A=w21—Q+k~.

(5.41)

If the atoms do not move in the propagation direction of the light beam, we simply set ~ = 0. We are interested in atoms that continue their interaction with the field for such a long time that they can achieve equilibrium. Hence we assume the probability to be conserved and set P22

+ Pu

=

1.

(5.42)

The equations of motion are then = iFp22

1~P22

=

+

—

(5.43)

P21)

—iFp 22

=

~(1~12

(A

—

c(P12

— i~F)~21

—

— ~(Pii

(5.44)

P21)

—

P22),

(5.45)

where =

4pE0.

(5.46)

In steady state the first two equations become identical and with (5.42) we have to solve the set iFp22

=

~(I~2u

—

P12) 2P22).

(A

—

i~F)~21 = cx(1

—

(5.47)

199

S. Stenhoim, Theoreticalfoundations of laser spectroscopy

The solution to this system is P22 = ‘~ 1 +

L(A) 2I0L(A)

~2 2 + ~F2 +

—

8 (5.4)

2~2’

A

—

where p2E~

4l~2

LA

‘

(F/2)2

—

549

)A2+(F/2)2

(.

)

and also

~

[

1

1

A + i~F

(A + i~F)

2 + ~F2 + 2~2 = P~~• (5.50) 0L(A)j = ~ A We see that we have a power-broadened line width [*F2 + ~p2Efl”2 and for very large intensities we obtain

P21 = (A

P22 ~ P11

—

~

i~F)[i + 21

(5.51)

~.

For fast enough flipping rates, the atom spends equal times on the two levels, which gives an intuitive feeling for the result (5.51). Our next step is to consider the effect of the strong field modifications of p on the spontaneously emitted light. This is possible if we continue the approximation scheme started in section 5.1, but first we will consider the relevant experimental situation in the next section. 5.3. Resonance fluorescence in a strong field We are now going to consider the experimental system in fig. 5.1. A laser beam crosses an atomic beam at right angles, and the travelling wave is assumed to strongly saturate the particles in the region of intersection. The detector is placed in the third orthogonal direction and records the spectrum of the spontaneously emitted side light. In this configuration both the strong field and

Laser

Particles

Spectral Analyser

1S1V’\ Fig. 5.1.

Fig. 5.2.

200

S. Stenholm. Theoretical foundations of laser spectroscopy

the spontaneous emission experience only a very small Doppler shift, and the interaction time is determined by the beam velocity. For strong enough fields the atom spends about half its time in the upper state and spontaneous decay processes occur at the rate IF. This means that during one interaction time many spontaneously emitted photons can emerge, because the strong field readily re-excites the atom after each decay process. The situation is depicted in fig. 5.2. In this situation it is not possible to use any perturbation approach, and the equations must be solved more exactly. The first detailed treatment of the ensuing problem was presented by Mollow [106], who used the Fourier transforms of correlation functions and a quantum regression philosophy. He continued his work in [107, 108], and his results have been corroborated by many workers e.g. [15, 35, 37,43, 71, 80, 141]. A discussion of the problem is given by Cohen-Tannoudji in [42]. Recently it has been suggested [36, 37] that the correlation between the different spontaneously emitted photons will bear witness of the quantum mechanical nature of the process. Such correlations are of interest because they purport to contain results not explainable in semiclassical terms but inherently dependent on the field theoretic aspects of the interaction. Experimentally the predicted effects have been observed [122, 146], and verified in some detail in [60]. Photon correlation experiments are in progress and experimental results are expected soon. To be able to calculate the result of a spectral measurement according to fig. 5.1 we must look at the rate of emission of photons of momentum hq. We consider only states with, at most, one photon and let the observable be =

p q;l >]spOnt

[
=

F<0;2I ~ 10;2>.

(5.52)

From the equation of motion for the density matrix we find in steady state

[~<~ii

p I~1>]

—i[ <0;21 p q;1>

=

—


p 0;2> <0;21 V q~l>].

spont

(5.53) Using the result (5.2) we find for the observable the expression 14’~ = 21m { <0;21 p q;1>}

=

In order to use this result we must know <0;21 mation for this in eq. (5.12) giving <0;21 ~

Iq;i>

=

g21(q)

1 q

with

P22

=

I4~=

—

~21

.

—2hg21(q)Im

<0;2I p q;1>.

(5.54)

q; 1>. We have already calculated one approxi-

P22

+ ~17

(5.55)

<0;21 i 10;2>. Inserting this into (5.54) we find ~~hg21~25(liq

—

W21)P22.

(5.56)

This is the result of the spontaneous decay at the atomic frequency liq = W21, and the strong field only serves to build up the population P22W Summing over the variable q we regain the decay equation (5.18) with (5.21). The processes of interest to us do not appear in the approximation

201

S. Stenholm, Theoreticalfoundations oflaser spectroscopy

(5.55). In order to consider those we must obtain a better estimate for

<0;2I t Iq; 1>.

To do this

we develop in detail an approach used by Baklanov [15] and find it to be the next approximation in a chain, where we carried out the first step in section 5.1. 5.4. The basic equations for fluorescence We consider now the full problem with one strong wave of amplitude E like in section 5.2 and spontaneous emission included. From eq. (2.63) we write the Hamiltonian H/h

=

w21a~a2+ >~Qqb~bq >~g21(q)[a~a,bq+ ala2bfl —

q

q

—

~pE(a~a, emnt + a~a2e~t). (5.57)

Calculating the equation of motion we obtain for the different elements the results i~_<0;2Ip~q;1>

(°~21~

—<0;2~p~q;2>]

+g21(q)<0;2IpIO;2>

~g21(q’),

—

(5.58)

i~<0;1Ip~q;1>= _liq<0;1~pIq;1>+ g21(q)<0;1~p~0;2> ~

—

emn1], t<0;1Ip q;2>

i~ <0;2~p q;2>

=

<0;21 p ~q;2>

~ —

—

IpE[e~°

e1’tt<0;2I p q;1>]

~g 21(q’)(

i~-<0;1~p~q;2> = —

—

(5.59)

(liq

—

<0;2~p~q’,q;1>),

(5.60)

+ w21)<0;1~plq;2>

~ (5.61) In the same way as we defined the reduced density matrix in eq. (5.13) we want to define now the matrices P21q

=

<0;2I p q;1> + ~
(5.62)

Pttq

=

<0;1~p~q;1> + ~.,

(5.63)

P22q

=

<0;2~p~q;2>+ >J1,

(5.64)

P12q

=

<0;1I p

(5.65)

q;2> + ~ .

S. Stenholm, Theoretical foundations of laser spectroscopy

202

The new density matrix elements appearing in (5.62—65) have the equations motion i~(~°21_liq) —~pEe~t[] + g 21(q) + g21(q’) —

—

~g21(q”) ,

(5.66)

i~= —Qq+g21(q)

i


Iq’, q;2>

=

+ g t_et] _IPE[~ liq21(q’)[ — <0;2~p~q’,q;1>], t — e~t] —~pE[e~ — ~ g~(q”)( — )~

—

(5.67)

(5.68)

i~=—(liq+w 2i)

1~t[ — ] —~pEe + ~g 21(q”) — g21(q’) <0;21 t q’, q;2>.

(5.69)

The equations (5.58~5.61)should be compared with eq. (5.3) and equations (5.66)—(5.69) with eq. (5.4). In order to carry out the same type of approximation as in section 5.1 we have to set ~ 0,

~ 0,

0,

~0.

(5.70)

These correspond to higher order photon correlations which we neglect. Adding the equations 1~P21q

=

(co21

—

liq)P21q

—

~pEe_iOt[piiq

—

P22q]

+g 21(q)p22 =

~liqpijq

—

—

~g21(q’)[

IPE[e~Qtp2iq —

=

~liqP22q

—

—

IPE[e_~tpi2q —

~g21(q’)(

],

(5.71)

—

<0;2~p~q’,q;1>],

(5.72)

e_~tpi2q]

+ q21(q)p,2 + ~g21(q’)[ i ~jJP22q

—

—

ep2lq]

<0;21 i

q’, q;1>),

(5.73)

203

S. Stenholm, Theoreticalfoundations of laser spectroscopy

1~Pi2q

=

—

(liq

+

~21)P12q

—

~pE eI~~t[p 22q

Piiq]

—

+ ~g21(q’)[<0;1~ ~ q’, q;1>

The matrix elements

P22

and

—

Iq’, q;2>].

<0;21 i

(5.74)

are identical with those defined in eq. (5.13). Introducing now

P12

the approximations leading to (5.12)—(5. 15) of section 5.1 we can derive the relations

~lq;j>

=

~ q’, q, i>

~i1~2t(~)(~

=

—

~i1~21(~)(~

(02,

—

~2,

—

i~)

(5.75)

+

(5.76)

j17)Pi2~.

Here we have also, as in section 5.1, assumed that inside the summations only the first terms of (5.62)—(5.65) need to be used. With (5.75)—(5.76) we find
—

Ig2i(q’)I~(~

~

—

~g21(q’)[

=

~

—

= ~

—

iIFP2iq + shift term,

j17)P21~ =

(5.77)

~21

+

~21

iiq~ —

—

j17>22~

=

iFP22q,

(5.78)

1~P22q,

(5.79)

<0;2Ip Iq’,q, 1)’)

Ig2i(q’)I~(~

j17 — <0;2~p~q’,q,2>) —

—

= ~

—

<0;2~p~q’,q;1>]

—

Ig2,(q’)l~(~

~g21(q’)(

~21

~21

Ig

2i(q’)I~(~

—

+

—

~

o~21+

—

=

j17)P22~ =

‘

—i~FP~+ shift term.

(5.80)

We thus find that the terms retained exactly reproduce the decay terms derived in eqs. (5.16), (5.18) and (5.22). If we insert (5.77)—(5.80) into eqs. (5.71)—(5.74), neglect the shift and introduce the ansatz Pllq P12q

—

—

e

— —

e

iczt-

—

PI1q’

2i1t—

P12q”

P22q

—

P21q

—

..

iflt-

e P22q’ e

—2int— P21q’

P12

—

—

—

P12

e

jOt

(5.81)

we find the equations

4Piiq

—

‘~P21q

=

(A + v

—

1IF)P2lq

—

~

P22q) + gp

22,

(5.82)

204

S. Stenholni, Theoreticalfoundations of laser spectroscopv

v~iiq+

~~~11q

=

~P22q

=

(v

—

=

—

(A

~

i

~F~

iF)~22q

v

—

+

22q

~12q

~12q

—

(5.83)

&12’

(5.84)

P2uq),

—

+ iIF)~i2q—

+

P21q)

—

~22q

—

(5.85)

~i ~)‘

where: g

=

g21(q),

A

= ~21

(5.86)

—

(5.87)

V=Q~liq

IiE.

=

(5.88)

The equations (5.82)—(5.85) have been given by Baklanov [15] but without a detailed derivation. We solve these coupled equations in steady state with P22 and P12 as the inhomogeneous terms, which are given by (5.48) and (5.50). The solution is written in the form 2]p22 + c~(v — A iIF)(v — iF)j~ v[(v — A — iIF)(v — iF) — 2~ 12 P21q = —g vD (5.89) —

,

where

D

=

[(v

—

A

—

iIF)(v + A

—

iIF)(v

2(v— ilfl].

—

iF)

—

(5.90)

4~

Eliminating P12 by (5.50) we finally find the relationship [(V

-

P21q

=

—

A

—

iIF)(v

—

iF)

—

(v

2~2

—

A

—

iIF)(v

—

~~P 22[

D

+

vD

iF)(A

—

iIF)1

j.

(5.91)

The observed quantity is the imaginary part of this expression according to (5.54). In the next section we will consider the physical implications of (5.91). Finally a note relating to the derivation by Mollow [106]. He uses a quantum regression theorem relating the time evolution of correlation functions to that of the expectation values. The same physical feature can be seen in our derivation in the fact that except the term in v and the inhomogeneous terms, the eqs. (5.82)—(5.85) have the same structure as eqs. (5.43)—(5.45). 5.5. Evaluation of the fluorescence spectrum The result (5.91) inserted into (5.54) gives the spectrum of the observed fluorescence. Firstly we notice the pole at v = 0, i.e. =

li.

(5.92)

In this case we observe fluorescence only at the frequency of the incident laser light. This is the elastic scattering component noticed already by Heitler [72, V §20]. In order to separate this we

S. Stenholm, Theoretical foundations of laser spectroscopy

add the small imaginary part

—

205

i17 to v and obtain

l4’~= 2hg ImP2lq~V,,O = 2hg2p22(.~~

Im

~lt2~~i~2))

~

2 + ~F2)ö(v), (5.93) 27thg2(~) (A where (5.48) has been used. If we introduce the last form of the relation (5.50) we find that =

(A + i~F)p

= 2 (A2 + ~F2)p~ 2 22/~t~ 2/c~ and hence the elastic scattering rate (5.93) can be written in the form =

II~2iI2

25(v), (5.94) 21I see ref. [15]. This is the spontaneous scattering rate of a dipole moment cc P21 driven by a coherent field. A fluorescence spectrum narrower than the natural width has been observed in [46, 57]. For small intensities of the driving field we obtain from (5.50) that ~

=

hgp

h A2±~F25(v),

=

(5.95)

which is the result derived in [72]. For strong fields we obtain ~

P22 ~ ~

and

irh 2+~F2 —~g2A 2

(5.96)

Thus we see that the elastic scattering intensity goes to zero for large intensities. We have now been able to interpret the second term of (5.91) in the limit v ~ 0. We want to separate this singularity from the expression (5.91) in order to have a more regular spectral density. To do this we define A(v)

=

(v

—

A

—

i~F)(v iF)(A —

— iIF)

(5.97)

and subtract the term (5.93). We consider M

1 A(v) vD(v)

A(0) D(0)’

598

.

(.

where A(0)

=

iF(A2 + ~F2),

D(0)

=

iF[A2 + ~F2 +

(5.99) (5.100)

2~c2].

We find A(v)D(0) —D(v)A(0)

=

—v(A

—

i~F)[(v — A

—

iIF)(v

— iF)(A

+

iIF)

—

2ot2(v + A

—

(5.10 1)

206

S. Stenholm, Theoreticalfoundations of laser spectroscopy

which leads to the expression M(V) If

— —

—

\~A2+A~F2 + — ilF

(

2~2)

\[(v

—

—ilF)(v

A

—

iF)(vD(v) + ill’)

—

. ) 2~2(v+A —ilF)] . 5102

we furthermore define N(v)

(v

=

A

—

ilF)(v— iF)

—

(5.103)

2~2

—

we write —2hg Im

=

Pl2qlinelastic

2hg2p

=

(5.104)

22 Im (N(v) + M(v)).

To see the shape of the spectrum we consider the particular situation when the strong field is exactly at resonance, A = 0. Then the function D becomes D(v)

=

(v

iIF)[v

—

ilF)(v

—

—

iF)

—

(5.105)

4~2]

The zeroes of this function are v ~ ilF

(5.106) (5.107)

~ —i~F±2ct~.

2

v = —i~F±~%/4c~2 — ~F We, consequently, have three resonances, one at the incident laser frequency v Stark shifted components at ±2c~. For (5.103) we find

N(v)

=

1 =

(v — ilF)(v — iF) ilF)[(v — ilF)(v —

—

v

—

1[ ilF + 2[

Inserting this into We

=

v

—

iF)

—

(v

4~2] =

11 ilF + 2 ~v

1 —

[

1 — ilF)[1

+

(v

0 and two

2~2

— 2~

—

i~F)(v+ 2~— i~F)

1/ 1 i~F)+ 2~v+ 2~— i~F)j~ ‘\

2c~ —

(5.108)

we find the result

(5.104)

hg2p

=

22~ 2 For A

1

2~2

—

=

+ ~ (~

~F2

—

2~+

~3fl2)

+

~ 2~F~(~F)2)].

(5.109)

0 we find from (5.102) the expression

M( V)

(

—

1 2~22 iF + 1F2))(v

=

17 iF 4(,,,(2cx2 + ~F2))

iF[ ~

\[7 [~

[iIF(v 2~ — i~F)(v+ — iF) — 2~ 2~2] — i~F)

—

+ iF 2

—

1 + 2~—

2\ / 1 F 32) ~ + 2x —

1

1 i~F

y

— 2~

—

i~Fj +

\ i~F)

iF

7

—

—

—

F2\

1

1

~)

~,,v — 2~

—

i~F

7/r’\2\

0y~~))~

(5.110)

in the limit ofa large driving field. For large enough fields P22 ~ land we can see that the expression (5.109) remains of order unity, whereas the expression (5.110) disappears as (F/ce). In addition, the

S. Stenholin, Theoreticalfoundations of laser spectroscopy

207

imaginary part of (5.110) is dispersive and only contributes an asymmetry to the Lorentzians of (5.109). We notice that the pole near v ~ 0 does not appear in M. In the limit c’ -÷ cc the behaviour of the spectrum is given by N(v) only. As also the signal is of order unity it will dominate the elastic scattering of (5.96), which is of order (F/~)2. The result (5.109) is thus the observed spectrum with the two sidebands 50% broader than the central resonance and their peaks only ~ of the central one, see fig. 5.3. This is the result first derived by Mollow [106] and verified experimentally in [60]. For A ~ 0 a numerical evaluation shows lines similar to fig. 5.3, but distorted in shape. The main part of the spectrum is again given by N, because of the factor cc ~2 in front of the expression (5.102).

~rI

Fig.

-

5.3.

6. Mechanical force exerted by light 6.1. Introduction We noticed already in section 4 that emission and absorption of a light quantum is accompanied

by a mechanical compensation of the change of momentum in the field. This radiation pressure is not found exclusively in quantum theory but is also an integral property of Maxwell’s classical theory. It is interesting to notice that the momentum transferred in radiative processes formed a central argument in Einstein’s early paper [45] on the quantum theory of radiation. He showed that the black-body distribution not only gives energy equilibrium with matter but also that it leads to a momentum equilibrium between the radiation and atoms assumed to be distributed according to a Maxwellian. The approach seems to attach a more concrete interpretation to the photon than we are prepared to admit today, but the reality of light pressure is beyond doubt. Earlier observations of light pressure had to utilize nonresonant light, and hence the result was due entirely to diffuse scattering or total absorption. Only the development of the laser provided an intense light source of high directionality, which makes it possible to observe the effect unambiguously [8]. Tuning the laser to a resonantly absorbing transition one can enhance the rate of the processes considerably [6]. Recently a large number of papers have discussed the mechanical manifestations of the interaction between laser light and matter. In section 6 of the present paper we shall review the basic theory for the mechanical effects and some of the physical phenomena suggested. The story is largely unfinished; experimental verification of the results has only recently

been initiated and the potential applications have still to materialize. On the side of fundamental research many of the suggestions are based on heuristic considerations and incomplete computations only. What novel phenomena and basic tests of our physical understanding will eventually

5. Stenholm, Theoreticalfoundations of laser spectroscopv

208

emerge remains to be seen. For the moment we are entangled in the mathematical difficulties encountered when we want to treat processes which are of such high order that the accumulated recoil momentum acquires macroscopic dimensions. The present exposition can, consequently, be nothing but tentative and suggestive of further developments. 6.2. The resonant force on a particle We consider a particle interacting with two counter-propagating light fields of the type displayed in eqs. (4.3—4.4). Here we neglect the direct recoil i.e. Cl

—

6k~ =

h(k ±k’)(k

—

k’) ~ V(k

—

k’),

(6.1)

where V is the atomic velocity (see also eq. (4.7)). Then the only variable necessary in p(v, p) is 1’ = n — m. To be able to let the field act on the atom during an arbitrary time interval we consider decay from the upper level to the lower one only. This implies that the lower level is the ground state (or a sufficiently metastable state), and that the density is so low that no collision processes need to be considered. The relaxation rates are then given by the same terms as in section 5.2, and we obtain the equations

[F + iVqv]p22(v) iVqvp11(v)

—

=

lipi2[E+(,~*(_v + 1)

Fp22(v)

=

lipi2[E+(~(v + 1)

—

j5(v + 1)) + E_(~*(_v—

1)

—

—

~*(_v + 1)) + E_(j3~v—

1)

—

j~(v—

1))],

—

(6.2)

1))],

(6.3) [IF+i(A+

If we set E_

Vqv)]~ö(v)=~ip12[E+(p11(v—1)—p22(v— 1))+E..(p11(v+ 1)— p22(v + 1))].

0 and v = 0 for Pu and P22 and v = I for valid for one travelling wave. The conservation of probability =

Pit + P22

~,

=

(6.4)

we re-obtain the eqs. (4.13)—(4.15)

1

(6.5)

gives with (4.2), (4.5) and (4.6) p11(v) + p22(v)

=



=

‘5n,m =

t~v,o.

(6.6)

The rate equation approximation is obtained, when we assume only v = 0, ±1 to be important; then we take the results of the two travelling waves to contribute independently. We find the equations Fp22(0)

lipi2[E+(,5*(1)

=

—

~(1))

[IF + i(A + Vq)]~(1)= Iipi2E+ IIIF + i(A

—

+

—

Vq)]~(—1)= ~ip12E_

E_(~*(_1)

—

~(—1))],

ip12E+p22(0), —

ip12E_p22(0).

(6.7)

(6.8) (6.9)

The first equation follows from both (6.2) and (6.3). Solving we find p(±l) -

1p12E±Vq)] [Pti(O) 2[IF+i(A±

—

P22(O)]

(6.10)

209

S. Stenholm, Theoretical foundations oflaser spectroscopy

and 1

—

=

Pi,(O)

P22(O)

=

I[1

—

1

+

2(J~L(A+ Vq) + I_L(A

—

Vq))]’

(6.11)

where (6.12) 1±= p~2E~/F2 and L(x) is given in (5.49). In order to evaluate the pressure exerted on the particle we have to calculate the average force acting on the induced dipole F

=

.

(6.13)

We write according to (2.44) and (2.58) the result F

=

—hp 12 ~.(iq)[E÷
—

a~÷q2akl emot>

—

E_
—

~

e_bot>], (6.14)

which gives directly (k gives the velocity dependence) t— F = —~ihp12q[E÷(p21(1) etQt and further with (4.8) F

=

—

P12(—

1) e_iOt)

P12(l) c_lQt)]

(6.15)

E_(p21(— 1) e’°

—

hqp,

2[E÷ Im~(1) E_ Imj~(—1)].

(6.16)

—

From (6.10) we obtain Im~(±1)= P12E±L(A±Vq)[p11(O)

—

(6.17)

P22(O)],

which gives for the force (6.16), when also (6.11) is used, F

+ Vq)Vq) LL(A hqF1 +I~L(A 2(I~L(A+ + I_L(A Vq) — Vq)) —

=

—

(6.18)

This is the force connected with absorption and spontaneous re-emission. Because each elementary process transfers the momentum hq to the atom and we have F events per second the order of magnitude F hqF is correct; when A > 0 we have resonance with the field I_ for positive velocities V and with 1~for negative V. In the former case the force is negative and in the latter case it is positive. All atoms are thus forced towards slower motion, i.e. the atoms are retarded, see fig. 6.1. For A <0 the roles of the two fields are changed and we have acceleration of the atoms. The force (6.18) was derived by Ashkin [6] who suggested its use for isotope separation [7], see also [77, 78]. The derivation given by Letokhov et al. [98,99] gives the force without the average over space implied by our quantized atomic states 1k>. Then they obtain the force (6.18) multiplied by the factor 2qz = 1 — cos 2qz (6.19) 2 sin and in addition they obtain the force Fjnduced

=

2hq[I÷(A + Vq)L(A + Vq) + I_(A

—

Vq)L(A

—

Vq)][p 22(O)

—

Pt ,(O)] sin 2qz. (6.20)

210

S. Stenholm, Theoreticalfoundations oflaser spectroscopy F

A >0

Fig. 6.1.

As the particle travels with velocity V it will take a time At ~ )L/2V

=

ir/qV

(6.21)

to traverse the period of the functions in (6.19) and (6.20). As the forces are of order hqF we find a change in velocity Av

hqF/MqV

hF/MV.

=

For atoms near resonance, qV ~ Al, we obtain a Doppler shift contribution qAv hq2F/MIA~—~ CF/A < F,

(6.22)

(6.23)

and consequently the spatially varying forces can be neglected. Looking at the equation of motion for the particles near resonance we find from (6.18) MJ~”= hqF 1 +21

=

F 0.

(6.24)

This gives the solution V(t)

=

V(O) +

t

(6.25)

and in order for the resonance condition to remain valid we have to require

A(t)

=

~21

—

li(t)

=

±qV(t)= ±q[v(o) + ~ ~].

(6.26)

By sweeping the laser frequency adiabatically and linearly we can thus “push” all the atoms to higher or lower velocities as we wish. The time it takes to cool the gas down to a Doppler width of F is given by =

~

(6.27)

because we need (u/T’) steps, where u

=

~.J2kT/M

(6.28)

and Vr is the recoil velocity change. Each step can, however, be executed in the time F’ and hence

211

S. Stenholm, Theoreticalfoundations of laser spectroscopy

we get (6.27). We return to the question of cooling in section 6.4. The present discussion is based on [98, 99]; we refer to these for additional details. 6.3. Radiation-induced mod~/icationsof the velocity We have already in sections 4 and 5 derived all the results necessary to describe the mechanical effects on spectroscopic phenomena. The equations (4.10)—(4.12) describe the behaviour of a twolevel system in two counter-propagating waves of arbitrary amplitudes. On the other hand, the

equations (5.16), (5.18) and (5.22) describe the influence of spontaneous emission on the two-level system. If we want to consider steady state situations where the atoms are pumped into one of the levels and decay out after the average time y 1 we can set Yi

=

72

=

Y~

712

=

(6.29)

)‘~,

where we have retained the possibility that collisions make y~> y. We then combine our equations to give /d

\

+ Y)P22(O~P)

=

llp,2[E+(.o*(1, ~z

—

1)

+ E...(1o*(_ 1, jA + 1) +

~

=

—

—

j3~l,j.i

~5(—1, JL + 1))]

~ + ~ip,2[E÷(fi(1,jA + 1)

—

1,~

+ E_(~(—1,jA—

1)—

1))

—

—

Fp22(O, j.t),

(6.30)

~*(1,jA+ 1)) —1))] +IF

JP

22o~~ + 2~)d~,

~*(

(6.31)

—1

where the integral is obtained directly from (5.26) and (4.6). Finally we have

[d =

.

1_.

+ y~+ IF + i(A + vVq + cV/.L)]P(v~~i) ~ip12[E+(p11(v—1,JL—1)—p22(v—1,Jz+1)) +E_(p11(v+1,/.4+1)—p22(v+1,/.z—1))]. (6.32)

We have explicitly written the equations of the diagonal elements for v = 0 only. If we set one travelling wave amplitude, E_ say, equal to zero we find that only the elements j~(l,~.t 1) are involved and from (6.32) we see that they couple back to the terms v = 0 only. This set of equations is closed except for the integral term in (6.31), and it has been treated in detail in [138]. In general the integral term (6.31) complicates the mathematical problem considerably because instead of —

having a denumerably infinite set ofcoupled equations we need to solve the system for a continuum ofvalues of p. or equivalently for a continuous mixing of all values of k. If we for a moment assume that the phase relaxation rate y~is large, we can solve the coherence equation (6.32) adiabatically Sand introduce this into (6.30)—(6.3 1) (see [89]).If we also neglect the terms p~(± 2, ji) we find =

~2

~

+ IF + i(A+ Vq + 6~~](P11(O~P 1) —

P22(O,jA

+ 1)),

(6.33)

S. Stenhoim, Theoretical foundations of laser spectroscopv

212

~(—i,jA)

~

~2

+ IF + i(A— Vq

—

+ 1)—

8jA)](P11LP

P22(O’~

1)).

—

(6.34)

Inserting these into (6.30) and (6.31) we obtain the equations

(~-+ Y + F)P2~(0~~)(y + F)i+L(A + Vq + =

+ (y + F)TJI(A

—

Vq

6jA

—

— 6)[pii(O,1L

—2)

— 8)[pii(O,1L

+

—

2)

p22(0,jA)]

—

P22(0,jA)]

(6.35)

and (~ + Y)Pi ~(0,jA)

+ (y + F)T_L(A

=

—

~ + (y + F)T~L(A+ Vq + e~+ 6)[p22(O, ~ + 2) Vq

—

c~+ 6)[p22(O,jA —2)

—

p11(O,jA)]

+

4F

— Pi

JP22(O~P —

1

~)]

~(O,

+

2~)d~,

(6.36)

with 2./4(y

1± L(x)

p12E

=

1 + lFXy + F), 2/[(y± + IF)2 +

(6.37) (6.38)

x2].

= (y~+ IF) The equations (6.35) and (6.36) are typical rate equations with rates-in and rates-out. If we

remember that p is to be regarded as a continuous variable we have to set Vq +

8jA

=

q(v +

qv,

(6.39)

where v is the velocity variable and =

hq/M.

(6.40)

We can see that for the rate-out terms only the value at v is important, whereas the rate-in terms v depend on the populations at v ±Vr (since /2 couples to p ±2) whereas the spontaneous decay smears the distribution continuously over the interval (v — Vr, v + I’), see fig. 6.2. at

//

\\\ V

Fig. 6.2.

S. Stenholm, Theoreticalfoundations of laser spectroscopy

213

If we set = 0 we find that only P22(O’ ji) and Pi ~(O, p 2) couple and we regain exactly the travelling wave result of [138]. If, on the other hand, we set F = 0, we obtain a set of coupled difference equations solved numerically for recoil in [5]; see section 4.6. In steady state we can set the time derivatives equal to zero, and eqs. (6.35) and (6.36) remain valid also when y~= y. This describes the situation where atoms are introduced at rate ,~and inter—

act for a time y’ before they are removed out ofreach ofthe interaction. This description is only an approximation to a solution of the real time dependent problem, but if y
p,2E~~ y,

(6.41)

we expect the treatment to be accurate enough. It reduces the calculation to a steady state problem which considerably simplifies its solution. It still remains a major problem to solve the equations (6.35) and (6.36) exactly. The computational problem is rather hard, but has been successfully treated for certain parameter ranges in our group [75]. For a standing wave results like fig. 6.3 are found.

A different approach is to expand the integral +1

fF

jC P22(O,

11

+ 2~)d~= ~

=

~-

J~

p22(k + x) dx

~-1p22(k)x’dx

=

fp22(k) + ~Fq2~p22(k)

+....

(6.42)

In this approximation we obtain an equation of the Fokker—Planck type. Such equations have been derived in [113] and solved for a particular case in [19]. The validity condition of the cut-off in the expansion (6.42) is q~p22 4

(6.43)

P22

and because the structure in the population is of width F we find that 2( 8 ~\ hq2 P22 q ~P22 = hq~~)P22 A~T~ 0 ‘~

(6.44)

and hence we require by (6.43) that

~ F ~ qu; a condition normally satisfied in gases confined to a cell, but not necessarily always for the transverse velocity spread in a particle beam. Even with the approximation (6.43) it is far from trivial to obtain solutions for physically interesting cases. 6.4. Cooling and heating with light Hänsch and Schawlow [66] have pointed out that isotropic broad-band radiation restricted mainly to the frequencies below resonance will act to cool the particles. Indeed, the atoms moving

5. Stenholm, Theoreticalfbundations of laser spectroscopv

214

towards the radiation will be Doppler shifted into resonance and absorb momentum from the oncoming radiation thus being retarded and eventually cooled. The limit of this process is given by the natural line width and the final velocity is reduced by the factor (F/qu) 10 1_b- 2~ Thus the temperature reduction may be p

‘final

— —

(r’’ 1

k

\2’7’

1qu,

,~

‘initial ~

ir~—3y ‘‘~

‘initial

In order to have saturated conditions for all velocities we need for each velocity group

p12E

F

(6.46)

and the number of frequency-intervals to be covered is (qu/F). The total power needed is 2—~ cC 2(qu/F) = cc 2. (6.47) W —~ c~0E 0(F/p12) 0Fqu/p~2—~ b0~W/cm Less power need be used if we can sweep the frequency from far below resonance up to F away from the center position of the line. This situation was discussed in section 6.2 according to the treatment of [98, 99]. We sweep the frequency linearly from ~2 1 qu to ~2 1 — F according to (6.26). The velocity has then been reduced to —

V 1

=

F/q

(6.48)

and we need the absorption and re-emission of

N1

=

qu/e

2U/Vr

=

—~

iO~

(6.49)

photons. The power needed is, however, much smaller than (6.47) by the factor (F/qu). When the adiabatic process is finished there is, however, a small force left. If we expand the expression (6.18) near V ~ 0 we find at A = F/2 F0

=

78F”~ ~-j~-)

V

21 ‘\ —hq(\1 + 21)qV 7

=

—f3V,

(6.50)

where we have assumed a standing wave 1~= 1_ and /1 is a constant. Thus the equation of motion

becomes M~= —/3V

(6.51)

and there appears a friction-like force tending to damp the velocity down. The ultimate limit for this cooling process is determined by the random velocity kicks that are delivered to the atom by the spontaneous emission of photons in a random direction. The number of such processes is given by N2

=

p22Ft

(6.52)

and the average velocity change is equal to

yr.

Because the processes are uncorrelated the average

change in 2 velocity = I’.2N AVis derivable from the random walk equation

AV

2

=

2Dt

(6.53)

giving the diffusion constant

D

2N 1 Vr ~ 2

=

2 b/hq\ p

~

22F,

(6.54)

215

S. Stenholm, Theoreticalfoundations of laser spectroscopy

see [99] and [113]. The Fokker—Planck equation corresponding to the problem (6.51) and a random perturbation with the diffusion coefficient (6.54) is [147]

1

817w

OW =

82

Wj + D

~

W.

(6.55)

For an ensemble of particles starting at the initial value V1 of (6.41) the solution to (6.55) is [144] 112 exp [—(V V 2/D(t)] (6.56) W(V, t) = [irD(t)] 1 exp (—flt/M)) —

where

D(t)

=

2DM [1

—

exp

(—

2f3t/M)].

(6.57)

For t 4 $/M the distribution (6.56) is a delta function at velocity V, and for large times t (fl/M) we obtain a Gaussian centered at V = 0 and independent of the initial velocity V 1. The most ~‘

probable velocity is given by 2 = [(h~)2 = [2DM]”

FM 1(1

+~)]

(~)

(6.58)

in the limit of large fields. The time needed for this cooling is approximately t, 0o1

‘~‘

The cooling T

=

M/2fl down

(6.59)

1/4Cq. to the velocity (6.58) corresponds to a temperature

iO~K.

~—MAV2 =

(6.60)

To solve the problem of cooling from the 5equations of section 6.3 becomes difficult; steps to achieve complete cooling.rather It is, however, according (6.49) we have to include N1 —~ i0 possible totoachieve a considerable cooling ofatoms within the velocity interval ~oo1

=

J’(F/y),

(6.61)

which are retarded to a line width much less than u; see fig. 6.3. The velocity distribution is, in

general, not Maxwellian and hence a temperature can be defined only approximately; see [75]. 922 922+

~

~

Laser frequency Fig. 6.3.

Laser frequency Fig. 6.4.

S. Stenholm, Theoretical /~undationsof laser spectroscopy

216

Some approximate calculations have been published also by Krasnov and Shaparev [82]; see in addition the discussion in [77, 78]. For ideal cooling the radiation field must consist of three orthogonal standing waves. Even in one wave it is possible to remove the energy at one velocity component and after some relaxation period the gas is cooled. For spectroscopic line narrowing it is often enough to cool one component. By tuning the laser to a frequency above the resonance it is, of course, also possible to heat the gas or accelerate the particles. The theoretical treatment of this case is similar to the case of cooling, but it results in a double-peaked distribution, which cannot be represented by a Maxwellian even approximately. Thus describing it as heating involves a major inaccuracy; see the general behaviour indicated in fig. 6.4. 6.5. Trapping of cooled atoms When the atomic system has been cooled, it would be important to be able to trap the particles in a confined volume. Then exceedingly high resolution spectroscopy becomes possible, and also low energy chemistry and scattering experiments can be performed. For a particle of polarizability ~ the dielectric force is given by F

=

(6.62)

l~~E2.

This type of force was considered by Askar’yan [9] and suggested for trapping by Letokhov [93]. Design of optical trapping devices has been suggested by Ashkin [6]. The spectroscopic implications have been considered by Letokhov and Pavlik [100] and the quantum mechanical problem of trapping into a volume of the size 2~is discussed by Letokhov and Minogin [96]. If we have a laser beam of shape E

=

E0(r) cos f~tcos qz,

(6.63)

we find from (6.62) the result 2

2

1

2~

1

F= r (E0) cos qz + ~kE0sin 2qz]. The longitudinal motion is governed by the equation ~.

.1

=

~

~—

E~sin 2qz,

(6.64)

(6.65)

which is the classical pendulum equation, see fig. 6.5. For low enough energy the particle becomes trapped and oscillates at the frequency of the linearized equation (6.65) 112qEo. (6.66) = (~l/2M) The solution of this problem is discussed in [100] and [77]. At the end of a resonant cooling cycle the final vibrational energy is given by (6.58) as hF. The potential barrier given in (6.65) has to be large enough to trap particles of this energy and hence we find E~> 2hF/lcsl,

(6.67)

217

S. Stenholm, Theoreticalfoundations of laser spectroscopy

l~IEg

2qz Fig. 6.5.

which gives a power requirement of iO~W/cm2. In order to achieve lower values of the power one would have to tune near resonance, which enhances c’~ considerably [6]. This case does, however, appear to fail to provide trapping inside a volume of order )~,[97], and only slowing down into a macroscopic volume can take place. In the case of a finite laser beam the radial force can be calculated. For the Gaussian beam ‘~

E~= A e~2I02,

(6.68)

we obtain the equation of motion MP

=

—

—i-

(6.69)

r,

2a which provides stability of the particle in the beam if (n

—

1)

>

0,

(6.70)

i.e. the particle has a higher refractive index than the vacuum; see [6]. This force is, however, rather small because the beam waist a cannot be made small over a long distance and a A.. When the particles can be trapped by the potential we introduce the field ~‘

E

E

=

0[cos(1+t

—

q÷z)+ cos(li...t + qz)]

(6.71)

instead of (6.63). The longitudinal force then becomes 2(lit

F

—

~Aqz) cos2(lAlit

—

2qz) cos2(lit

—

qz)}

2ccE~ {cos

=

—

2~xqE~ sin (Alit

—

~Aqz) ~

—

c~qE~ sin (Alit

—

2qz)

(6.72)

where we have assumed li÷~ fL. Thus, instead of the equation (6.65) we have the motion in a moving potential I

=

~ E~sin (2qz

—

Alit),

(6.73)

where the minima are translated with the speed Vph

=

Ali/2q.

(6.74)

If All is small enough but adiabatically increasing we can accelerate particles trapped in the potential well, see [77]. In [78] also the effect of pulses is discussed and in [79] the polarization dependence is found to give rise to birefringence.

S. Stenholm, Theoreticalfoundations of laser spedroscopv

218

The general problem of trapping and manipulating slow particles by light is not yet clarified, and except for trapping by nonresonant dielectric forces [6] no experimental verifications are available. 6.6. Beam deflection by light It has been suggested that the pressure of light can be used to deflect a particle beam [7]. As the tuning of the frequency can make the interaction selective with respect to different atomic species it provides a way to separate them, which is of potential use for separation of isotopes and possibly other particles. Cotector

r”~Beam~[cl

L.__J

1T1~ L.....J

Skimmer

Resonant Light Interaction Region

Fig. 6.6.

The basic physical situation is shown in fig. 6.6. The particle beam emerges from an oven into the interacting region of a travelling wave. The particles absorb resonantly from the beam and re-emit the energy isotropically. As a consequence they are on the average diverted by an angle equal to

e

VrFt/ Vii

VJ’L/V1~,

(6.75)

where J~is the particle velocity and L is the length of the interaction region. The beam is, however, spread out over a transverse velocity band V1

—~

2OV~

(6.76)

and consequently the separation by a final skimmer can never be very efficient. If, however, the yield of desired species is allowed to become very low, the purity of the separated atoms can be made very high. The beam deflection method is well established experimentally [25, 74, 111, 121]. It has been used for the detection of atomic species [74, 111] and for separation of the isotopes of Ba [23]. This case is advantageous due to the existence of a resonance transition in the dye laser range and the small mass giving a large value for V~.= hq/M

‘~

h/AM.

(6.77)

The Fokker—Planck treatment of ref. [19] gives some idea about the growth of the transverse velocity and its mean spread. In ref. [66] a pair of transverse standing waves are suggested as a means to cool the transverse components of the beam velocity. This would collimate the beam and reshape it to desired form.

S. Stenholm, Theoreticalfoundations of laser spectroscopy

219

A light wave along the beam could be used to accelerate its atoms but unfortunately the transverse velocity diffusion will defocus it at the same rate. It is at present unclear to what extent laser light

can be utilized to shape particle beams into experimentally desirable forms.

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