A SIMPLIFIED MODEL FOR COMPUTING TEMPERATURE OF A DILUTE POPULATION OF LASER COOLED STORED IONS

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TEMPERATURE OF A DILUTE POPULATION OF LASER COOLED STORED IONS

J. André, A. Teboul, F. Vedel, M. Vedel

To cite this version:

J. André, A. Teboul, F. Vedel, M. Vedel. A SIMPLIFIED MODEL FOR COMPUTING TEMPERA-

TURE OF A DILUTE POPULATION OF LASER COOLED STORED IONS. Journal de Physique

Colloques, 1981, 42 (C8), pp.C8-315-C8-325. �10.1051/jphyscol:1981839�. �jpa-00221735�

JOURNAL DE P H Y S I Q U E

CoZZoque C8, suppZ6ment au n012, Tome 42, dgcembre 1981 page C8-315

A SIMPLIFIED MODEL FOR COMPUTING TEMPERATURE OF A DILUTE POPULATION OF LASER COOLED STORED IONS

J. Andr6, A. Teboul, F. Vedel and M. Vedel

Physique des Interactions Ioniques e t MoZ6cuZaires*, Universitg de Provence, 13397 Mars& ZZe, France

A b s t r a c t - Energetic and t e m ~ o r a l aspects o f t h e l a s e r c o o l i n o o f a s t o r e d i o n i c

~ o p u l a t i o n a r e n u m e r i c a l l y i n v e s t i q a t e d . The process which causes t h e c o o l i n g i s described as a Raman s c a t t e r i n q . Adantod formalisms and a l a o r i t h m s are develop- ped. The c o o l i n g l i m i t i s computed and t h e optimal c o n d i t i o n s t o decrease t h e temperature are discussed.

I - I n t r o d u c t i o n - T h e p o s s i b i l i t y t h a t an atomic system be cooled by l i c l h t was consi- dered as e a r l y as 1950('). I n 1975, a new method ( l a s e r c o o l i n g ) was suqqested

,n,

by H.G. Dehmelt and 3.5. \&!ineland(") i n o r d e r t o decrease t h e temperature o f a s t o r e d i o n ( o r of a s t o r e d i o n i c p o p u l a t i o n ) . P r e s e n t l y t h e ~ r ~ ( ~ " ) a n d y 7 ) show t h a t i t i s an e f f i c i e n t method and t h e t ions a r e confined w i t h a very weak t r a n s l a t i o n a l energy. However, a l l t h e problems a r e n o t y e t solved and i t remains i m - p o r t a n t t o know t h e i n f l u e n c e o f parameters such as i o n i c weiqht, a b s o r p t i o n frequen- cy, n a t u r a l l i n e w i d t h , storaqe frequency, e t c . . .

For several months we have been i n t e r e s t e d i n those problems, f o r a very d i l u t e i o n i c p o p u l a t i o n (so t h a t Coulomb i n t e r a c t i o n s can be neglected) s t o r e d i n a 9.F.

t r a p . Me must f i r s t e s t a b l i s h equations and s e l e c t numerical algorithms. Then we w i l l use them i n order t o compute e n e r q e t i c o r temporal parameters ( f o r example i o n i c temperature).

Me present here o u r f i r s t r e s u l t s . They a r e obtained from an onedimensional mo- d e l : we assume t h a t ions a r e always l o c a t e d on t h e s-mmetry a x i s Oz and t h a t t h e i n - c i d e n t electromagnetic wave i s plane and perpendicular t o t h i s a x i s . I n t h e f o l l o w i n g we discuss e s s e n t i a l l y t h e e n e r g e t i c aspects o f t h i s problem and w i l l n o t t r e a t t h e equations ornumei-ical a l g o r i t h m s i n a d e t a i l e d manner. General aspects o f i o n storage i n a R.F. t r a p a r e supposed t o be known.

11- E N E R G E T I C D I S T R I B U T I O N O F I O N S - 11-I-Transition probabilities - I n o r d e r t o compute t r a n s i t i o n p r o b a b i l i t i e s , i t i s necessary t o choose a model f o r t h e i n t e r - a c t i o n between i o n s and t h e electromagnetic wave. Such a problem was discussed i n a fundamental a r t i c l e by D.J. Wineland and W.M. ~ t a n o ( ~ ) which we f o l l o w here.

The i o n t r a p i s assumed t o be a harmonic p o t e n t i a l w e l l . We b e l i e v e t h i s as- sumption t o be r e a l i s t i c , p a r t i c u l a r l y i f experimental c o n d i t i o n s correspond t o a

0, q -1. The t r a n s l a t i o n a l s t a t e o f an i o n ( t h e mass o f which i s denoted by M) i s described by t h e i n t e g e r n o r by t h e enerqy En :

En = (n + l / 2 ) h w . ^(11-1)

* ERA 898

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981839

w i s t h e fundamental frequency o f t h e s t o r e d ions. Me suppose t h a t t h e i o n s possess a r e - sonant e l e c t r i c d i p o l e t r a n s i t i o n between t h e ground s t a t e and t h e f i r s t e x c i t e d s t a t e The energy d i f f e r e n c e between these two l e - v e l s i s denoted by five .Woreover, we assume t h a t t h e h a l f n a t u r a l l i n e w i d t h y (expres- sed i n r d / s ) i s l a r g e , so t h a t o t h e r therma- l i s i n g parameters can be neglected. The fundamental frequency o f a R.F. quadrupole t r a p i s n o t very l a r g e (a few :lrd/s) and i t s a t i s f i e s t h e c o n d i t i o n :

w < Y (11-2)

which i s always assumed t o be r e a l i z e d . We consider t h a t t h e electromagnetic wave i s monochromatic and t h a t i t s i n t e n s i - t y i s low enough t o n e g l e c t s a t u r a t i o n e f - f e c t s . We denote by h t h e wavelength and by

v t h e frequency o f t h i s wave.

The motion o f every i o n i s approxima- t e l y p e r i o d i c so t h a t i t s Doppler spectrum i s composed o f many l i n e s separated from each o t h e r by t h e frequency w/2n . ^The

c e n t r a l l i n e corresponds t o an i o n a t r e s t . Because o f (11-2), t h e l i f e - t i m e o f t h e ex-

Figure 1 - Spectral parameters (a) natural linewidth,

(b) Doppler structure,

(c) difference between the absor- ption frequency (v0 ) and the laser frequency ( v ) .

In this example, B = 4, m = - 3 .

c i t e d i e v e l i s s m a l l e r than 2 n /a, and we 1 1

consider t h e i n t e r a c t i o n between t h e i o n s and t h e electromagnetic wave as a sequence o f i n e l a s t i c s c a t t e r i n g processes. During every process, t h e t r a n s l a t i o n a l s t a t e o f an i o n can change and t r a n s i t from s t a t e n t o a random s t a t e Nf.

The f i r s t q u a n t i t y we have t o compute i s t h e t r a n s i t i o n p r o b a b i l i t y t h a t an i o n w i l l be i n t h e s t a t e nf a t i n s t a n t t t 6 t when i t was i n a s t a t e n a t i n s t a n t t.

This q u a n t i t y (denoted by p ( n + n f ) 6 t ) s a t i s f i e s an equation which i s adapted t o t h e onedimensional model from a more general expression :

.. ^{< nf 1 s x j >< j leikxl n >}

P (n'nf)= C ; I ^{O : j}

-m t ( j - n ) - i B l 2

k s 2"k , An = In-nlj,n, = I n f (n,nl).

It i s n o t necessary here t o g i v e e x p l i c i t l y t h e constant C and we p u t

p ( n + n l ) = C $ ( n - t n 1 ) (11-4)

TABLE 1

= 27~ ?lrd/s A = 0.224

LU

= 10 Mrdls A = 0.141 w = 271 Mrd/s A = 0.105

w = 10 Mrd/s A = 0.066

By contrast, parameters A, B and m govern t h e nature o f t h e s t a t i s t i c a l energy d i s t r i b u t i o n and e x p l i c i t expressions w i l l be given f o r them.

The constant A i s d e f i n e d by A = Rlhw , where R i s t h e c l a s s i c a l " r e c o i l energy":

7 2

R = h - k /2M = 2 n 2 h 2 / ( ~ A 2 ) . I n t h i s work, A i s always l e s s than one. Several exam"1es a r e i n d i c a t e d i n t a b l e I.

The constant B i s equal t o Y/U . For a given value o f

, t h e l a r g e r i s t h e na- t u r a l l i n e w i d t h , t h e l a r g e r i s B (see f i g . I ) . Due t o (11-2) B i s always g r e a t e r than one i n t h e present work.

The t h i r d important parameter i s : m = 2n(v-vo)/w where

i s t h e d i f f e - rence between t h e absorption frequency o f t h e i o n a t r e s t and t h e frequency o f t h e i n c i d e n t electromagnetic wave (see f i g . I). Cooling can o n l y occur i f m i s negative.

I t may be remarked t h a t t h i s d i f f e r e n c e i s equal t o Y when m = -B.

11-2 - Computation of the equilibrium distribution of ions - When t h e i o n i c po- p u l a t i o n begins t o i n t e r a c t w i t h t h e electromagnetic wave, t h e s t a t e s o f t h e i o n s a r e s t a t i s t i c a l l y d i s t r i b u t e d around a l a r g e number (10 t o 4 lo8, accordinn t o expe- r i m e n t a l c o n d i t i o n s ) . For a given ion, whose s t a t e i s denoted by n, a s c a t t e r i n o pro- cess changes n i n t o a random s t a t e Nf . I f m i s negative, on average t h i s i o n has t o decrease i t s e x t e r n a l energy i n o r d e r t o conserve o v e r a l l energy. Thus, the mean value nf (n) i s l e s s than n a t t h e beginning o f t h e c o o l i n g process. Nevertheless, f i g u r e 2 i n d i c a t e s t h a t t h i s behaviour cancels f o r a value which i s denoted by no and i t reverses f o r s t a t e s l e s s t h a n no. Hence i t f o l l o w s t h a t n decreases u n t i l i t reaches no, then i t stays. However, t h i s model i s much t o o rough, because we o n l y took i n t o account mean values and disregarded t h e i n f l u e n c e o f t h e standard devia- t i o n o ( n ) . I n a more r e a l i s t i c model, we have t o consider t h a t a f t e r each s c a t t e r i n g process, Nf i s d i s t r i b u t e d w i t h i n t h e range ( i i f ( n ) - 2o(n), iif(n) + 2o(n) )(see f i g . 2). Hence, a t e q u i l i b r i u m , t h e i o n s t a t e Nf l i e s between 0 and an upper value which can be much g r e a t e r than no.

The above considerations a r e t h e same f o r any i o n o f t h e p o p u l a t i o n w i t h which

we associate a s t a t i s t i c d i s t r i b u t i o n which i s independant o f t i m e (because o f

equi 1 i brium) and which i s c h a r a c t e r i z e d by a sequence o f probabi 1 it i e s :

{ ~ ( n ) = Pr [N(e)=n] } . The c l a s s i c a l balance equation can be w r i t t e n :

Figure 2 - Inrluence of t h e scattering process. The dashed straight l i n e c o r r e s p n d s to nf = n. I n the figure are drawn

fif (n) (middle curve) , ^Ef +

dn)

(upper curve) and iif (n) - ^2a (n) (lower curve).

n

5 ,a'

, f

<

<.--

_ ^<.-

./

This equation reduces t o

and i t may be denoted s y m b o l i c a l l y by

z, n ( n l ) Q(nl,n) = 0 n

Expression (11-7) i s an i n f i n i t e system o f coupled a l g e b r i c equations. I n order t o i n t e g r a t e them, i t may be noted t h a t p ( n o + n) converges r a p i d l y t o 0 as n ' v a r i e s from n. Therefore, t h e behaviour o f Q(n:n) i s t h e same. Yoreover ~ ( n ) i s a r e g u l a r f u n c t i o n . Hence, we can assume t h a t w i t h i n t h e range o f values f o r which Q i s non-0,

a ( n l ) can be approximated by t h e f i r s t t h r e e terms o f i t s T a y l o r ' s s e r i e s . Due t o t h i s remark, expression (11-7) i s g r e a t l y simp1 i f i e d t o t h e recurrence r e 1 a t i o n

: [aZ (n) + al(n))l a(n+tl) = n(n) [Za2(n) - a 0 ( n ) l + a(n-1) Icrl(n) -ap(") 1 .

(11-8) aO(n) = E , Q ( n 4 ,n), al(n) = Z , (n'-n)Q(n1,n)/2, a2(n) = C , ( n ' - n ) )(n19nY2. 2

n n n

a ( n ) must be known f o r two a r b i t r a r y values o f n t o s o l v e t h i s equation. ble choose n(0) = 1 and &ii ^{-a(n) =} 0. Note t h a t sequence { n l does n o t meet t h e c o n d i t i o n

Z a ( n ) = l ,

n=O (11-9)

b u t t h a t i t i s n o t d i f f i c u l t t o normalize n i f necessary.

11-3 - Numerical r e s u l t s - I n f i g u r e 3 a r e p l o t t e d f u n c t i o n s

a g a i n s t n f o r

several values o f A, B and m. The curves a r e very r e g u l a r . Except f o r v e r y small

values o f n and f o r l a r g e absolute values. of m, t h e p o i n t s on t h e curve a r e described by expressions which are r o u g h l y geometric progressions. Such a behaviour seems t o i n d i c a t e t h a t t h e energy d i s t r i b u t i o n i s v e r y near t h e Boltzmann d i s t r i b u t i o n . This assumption i s o f t e n made a p r i o r i .

I n f a c t , an accurate study o f

shows d i f f e r e n c e s w i t h t h e above d i s t r i b u t i o n . This i s obvious f o r t h e l a r g e magnitudes o f m, b u t i t a l s o can be t r u e f o r o t h e r values. For example, curves l o g ( ~ ( n ) ) a r e drawn i n f i g u r e 4 and i t may be seen t h a t these curves o f t e n d i f f e r from a s t r a i g h t l i n e .

111 - ^{I O N} TEMPERATURE - Temperature seems t o be a n a t u r a l parameter i n order t o des- c r i b e t h e e n e r g e t i c p r o p e r t i e s o f i o n s . However, due t o t h e p r e v i o u s remarks, we have t o choose a s p e c i f i c d e f i n i t i o n adapted t o t h e problem.

111-1 - D e f i n i t i o n o f t h e i o n temperature - For a Boltzmann d i s t r i b u t i o n , t h e energy p r o b a b i l i t y law i s given by

vo ( E k ) = K exp ( - Ek/ K T ) . ^(111-1)

Since i t i s e a s i e r t o u s e s t a t e n o t a t i o n n, t h e previous expression w i l l be w r i t t e n v O ( n ) = x exp(-& / K T ) = (lqlqn , ^{q = exp} (-kw/

T) . ^(111-2)

fi

a (n) A

4

(n) A =

B = 5 B = 5

1 .

-

0.9 -

- _{m= -1}

0.8

- ⁿ 0.6 n

0 5 0 ^I 5

1

" ^s (n)

(a) Figure 3 - Curves a (n) f o r several va- lues of A, B and m.

A =

Figure 3a s h m s the general behaviour of B - 5 v (n) .

rn = -1 I n f i g u r e 3b and 3c, curves a (n) are drawn for the first values of n.

n

I I

+

0 5 0

The temperature T i s t h e o n l y parameter of t h e d i s t r i b u t i o n . For any o t h e r d i s t r i b u - t i o n , t h e temperature i s d e f i n e d b y u s i n g a p a r t i c u l a r p r o p e r t y of t h e Boltzmann d i s t r i b u t i o n . For example, i t i s usual t o d e f i n e T i n terms o f t h e mean value of N

ti w

T = - [ l n ( l + ~ / r n ~ ( ~ ) ) J , (111-3)

where

m

m, (N) = n : o n a ( n )

Thus, f o r any d i s t r i b u t i o n , i f mean values e x i s t , temperatures can be defined. How- ever, i t i s e b v i o u s l y necessary t o know t h e second member o f (111-4), a s e r i e s which sometimes converges t o o slow1 y t o be e a s i l y computed.Therefore, temperature wi 11 be defined here i n a d i f f e r e n t way. Denote by n * . the s m a l l e s t s t a t e which s a t i s f i e s the equation ( f o r t h e Boltzmann d i s t r i b u t i o n ) :

Figure 4 - V a r i i d t i a n of log (n) versus n for several values of A, B and m.

J

--..----

The s t a t e n* i s such t h a t t h e s t a t e o f an i o n i s w i t h i n t h e range (0,n

) 95 p e r c e n t o f t h e time. For t h e Boltzmann d i s t r i b u t i o n , (111-6) i s reduced t o :

q n* 3 0.05 , (111-6)

Hence

For a d i f f e r e n t d i s t r i b u t i o n , t h e above equation a l s o d e f i n e s a temperature. Note t h a t expression (111-6) i s e a s i e r t o compute than equation (111-5). Moreover T i s

* *

p r o p o r t i o n a l t o n , which i s an i n t e r e s t i n g parameter i t s e l f , thus n can be used

t o c h a r a c t e r i z e temperature. The e r r o r o f T i s always g r e a t e r than kw/3~,however t h i s

lower l i m i t i s small enough f o r t h e present study.

n* or T(a.u.1

= 0.025 (1)

80 - _A

a

0.050 (2) A = 0.100 (3)

A = 0.200 (4)

4

2 0

- - ^{3 & l}

-=- - -

- - -

(9)

0 -

O l b 3 4 5 6 7 8 9 1 0 *

n*orT (a.u.4

200 -

A = 0.050

CI

(-m)

0 .,

0 2 4 6 8 10 12 14 16 18 20

Figure 5 - Variation of n* ^versus (-in) for several

of A and B.

Note ^that the lowest value of n* : (s) i s reached when (-m) is appmxi-

mately equal t o B.

111-2-Influence o f t h e magnitude of m upon T-In f i n u r e 5, a r e p l o t t e d n* ( o r T expressed i n a r b i t r a r y u n i t ) a g a i n s t (-m), f o r given values o f A and B. I f t h e sca- l e s a r e n o t considered, the curves o f f i q u r e s 5a and 5b a r e very s i m i l a r . As (-m) increases from 0 t h e s t a t e n ( o r t h e temflerature T) decreases r a o i d l y and reaches a minimum, denoted by nm(or T, ) and then increases s l o w l y . Such a behaviour can be explained by two a n t a g o n i s t i c e f f e c t s . On t h e one hand, i f a s c a t t e r i n q process changes t h e i n i t i a l s t a t e n i n t o Pif, t h e mean value o f t h i s s t a t e ml(n) decreases w i t h (-m) and t h e r e f o r e , t h e decrease i n T tends t o be more e f f i c i e n t with(-m).

On t h e o t h e r hand, an accurate study o f t r a n s i t i o n p r o b a b i l i t i e s shows t h a t t h e stan- dard d e v i a t i o n o ( n ) increases w i t h (-m) and hence a t e q u i l i b r i u m T increases

(because n* increases).

I t may be seen t h a t t h e lowesttemperature (T,) i s reached f o r m approximately equal t o -B. This value corresponds t o an electromagnetic wave which d i f f e r s from the a b s o r p t i o n frequency by h a l f a n a t u r a l l i n e w i d t h . T h i s r e s u l t was p r e v i o u s l y obtained by Wineland and ~ t a n o ( ~ ) .

111-3 - Study o f minimum temperature - n, ,, ( o r Tm) v a r i e s as a f u n c t i o n o f A and B. Curves o f f i g u r e 5a represent n* (A) f o r t 3 = 5. They have a minimum f o r nm approximately equal t o 15. The curve drawn i n f i q u r e 5b represents n*(A) f o r S = 10 and a minimum occurs f o r nm near 30. These r e s u l t s i n d i c a t e t h a t nm i s aporoximately equal t o 3B. Therefore Tm can be w r i t t e n :

hw fi

T m z T B = 2 . (111-8)

This v a l u e was proposed i n previous papers. However, s i n c e 3 8 i s n o t e x a c t l y nm, Tm depends a l s o upon B. I n f i g u r e 6, n, i s drawn as a f u n c t i o n o f 6 i n t h e ranse 2 < B < 4 0 f o r d i f f e r e n t values o f A choosen w i t h i n t h e domain 0.025< A< 0.400.

Numerical values o f nm a r e always g r e a t e r than 3B, and t h e d i f f e r e n c e

an = nm - 3B ( 1 1 1 4 )

increases w i t h A. It may be noted t h a t An i s a l i n e a r f u n c t i o n o f A f o r B g r e a t e r than 5.

For 8 l e s s than 5, An i s o f t h e o r d e r o f n, . I t r e s u l t s t h a t A i s as e f f i c i e n t as B f o r l i m i t i n g c o o l i n g . Hence, expression (111-8) i s perhaps n o t r e a l i s t i c , c h i e f - f l y f o r l i g h t ions, f o r a weak n a t u r a l l i n e w i d t h , and f o r l a r a e frequency electroma- g n e t i c waves. By c o n t r a s t , t h e c o n t r i b u t i o n o f A f o r l i m i t i n g nm decreases as D

increases from 5. For these values, expression (111-8) seems t o be r e a l i s t i c and nm may be w r i t t e n :

nm = =A + 38, (111-10)

t h e r e f o r e , t h e temperature i s o f t h e form :

T = " R + - li ^Y (111-11)

m

Note t h a t t h e above expression i s independent o f w .

By c o n t r a s t , w i s one o f t h e most important parameters when t h e purpose i s t o

o b t a i n t h e s m a l l e s t magnitude f o r nm. I n f i g u r e 6c, i s drawn t h e path associated

Figure 6 - Influence of the parameters upon T

, and q , , . In figure 6a are drawn curves

%(A) . ^{Each curve} corresponds to a fixed value of A.

In figure 6b is plotted , T against y .

The solid line correspmds t o the lowest temperature. The black circle corresponds t c an ion .Mg+ { see table 1) . Suppse that the temperature of an ion (mass M I wavelergth 1 ) is represented by the white star, the tem- perature of an ion whose mass is 2M is represented by the black star, and an ion whose wave length is 2X is represented by

a white circle. When w increases, the state % o f an ion follows the path drawn in the figure 6c.

0

1 0 2

w i t h nm as wincreases, a l l t h e o t h e r parameters remaining constant. \le see t h a t

...

u s i n g a h i g h fundamental frequency i s a very e f f i c i e n t method t o decrease nm. Per- haps, t h i s suggests t h a t s t a t i c t r a p s are b e t t e r i n o r d e r t o observe ions a t r e s t

I V - OPTIFlAL FREQUENCY OF THE ELECTRO~IAGNETIC WAVE - Figure 5 shows t h a t a magni- tude of m le-ss than B has t o be avoided, b u t t h a t g r e a t e r values a l l o w approximately t h e same e f f i c i e n t c o o l i n g . However, we have n o t y e t taken i n t o account temporal considerations. For e f f i c i e n c y , t h e c o o l i n g r a t e must a l s o be large. To i n t r o d u c e t h i s new concept, we must d e f i n e a c h a r a c t e r i s t i c parameter denoted by ^{T .}

For an ion, the mean w a i t i n g time ( v n ( n ) ) between t h e i n i t i a l s t a t e n and the

0

f i n a l s t a t e no s a t i s f i e s :

II ( n o ) p ( n + n ' ) d n t + ~ ( n ) p ( n + n ' ) dn' = 1 .

no i ^(Iv-l)

"0 0

This equation i s deduced from a more general equation explained i n reference (8) Here,

i s d e f i n e d by the f o l l o w i n g expression :

r = u n (nm+l) (IV-2)

m

I t may be mentionned t h a t s i n c e p i s known t o w i t h i n a m u l t i p l i c a t i v e constant

u and

are known t o w i t h i n 1 / C . Therefore,

(-m) can o n l y be expressed i n a r b i t r a - r y u n i t s , which i s s u f f i c i e n t here. I n f i g u r e 7 are superposed n and *

as f u n c t i o n s o f (-m). On t h e one hand, c o o l i n g i s l i m i t e d f o r m l e s s than -B because o f tempera- t u r e . On t h e o t h e r hand, c o o l i n g process i s r e s t r i c t e d f o r rn g r e a t e r than -B because o f

.

To sum up, the d i f f e r e n c e between the absorption frequency o f t h e ions ( a t r e s t ) and t h e electromagnetic wave frequency must n o t be t o o d i f f e r e n t from h a l f t h e na- t u r a l 1 i newidth.

rl,,, 9 (a.u.) 7

1 0

Figure 7 - ^{Variation of q-,,}

m i % p r e s s e d in arbi- trary units) versus L-m) .

whkte circles : rpn,

V- CONCLUSION - Our f i r s t numerical r e s u l t s have been presented. They a r e s t i l l i n - complete and they i n v o l v e an onedimensional model. However, previous works show t h a t t h i s model i s r e a l i s t i c , f o r energy c o n s i d e r a t i o n s and t h a t i t i s q u a l i t a t i v e l y c o r - r e c t f o r temporal s t u d i e s . Moreover, our computations c o n f i r m previous r e s u l t s g i - ven by o t h e r authors. This f a c t g i v e s new arguments f o r t h e e f f i c i e n c y o f t h i s sim- p l i f i e d model.

2

-.

0 ,

black circles :

.

(-m)

>

1 2 3 %

5 6 7

i

We have shown t h a t , f o r fundamental frequencies l e s s than t h e n a t u r a l l i n e - width, the l a t e r i s t h e main parameter which l i m i t s t h e temperatures. I n t h e opposi- t e c o n d i t i o n , c o o l i n g can i n v o l v e t h e r e c o i l energy. Computations a l s o show an o p t i - mal d i f f e r e n c e between i n c i d e n t frequency and a b s o r p t i o n frequency, o f t h e order o f h a l f the n a t u r a l l i n e w i d t h .

Several o t h e r problems were n o t examined y e t . For example, i t would be neces- sary t o know t h e decrease o f t h e i o n i c p o p u l a t i o n as the c o o l i n g process occurs.

I n t h i s model, i o n i c temperature i s always very low and i t i s l i k e l y t h a t o t h e r processes tend t o increase it. P a r t i c u l a r l y i t would be i n t e r e s t i n g t o know t h e i n f l u e n c e o f c o l l i s i o n s . A more fundamental problem i s t h e increase o f tempera- t u r e under t h e i n f l u e n c e o f space charge, b u t a general treatment needs a three- dimensional model and i t w i l l be necessary f u r t h e r t o improve algorithms.

REFERENCES

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( 5 ) D.J. WINELAND, R.E. DRULLINGER and F.L. WALLS, Phys. Rev. L e t t . 4J, 1639 (1978) ( 6 ) N. NEUHAUSEN, M. HOHENSTATT, P. TOSCHEK and H.G. DEHHELT, Phys. Rev. L e t t .

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-

(7) N. NEUHAUSEN, #. HOHENSTATT, P. TOSCHEK and H.G. DEHYELT, Appl. Phys. 2 ,

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