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Rydberg to plasma evolution in a dense gas of very excited atoms

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1982 J. Phys. B: At. Mol. Phys. 15 L49

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J. Phys. B: A t. Mol. Phys. 15 1982) L49-L55. Printed in Grea t Britain

LETTER TO THE EDITOR

Rydberg to plasma evolution in a dense gas of

very excited atoms?

G V itrant, J M Raimond, M G ross and S Haroche

Labo ratoire d e Physique de I’Ecole Normale SupCrieure, 24 rue Lh omond, 75005 Paris,France

Received 5 October 19 81

Abstract. A Rydberg atom gas quickly evolves into a plasma ab ove a threshold density. Th e

process is attrib uted to Rydberg -Rydbe rg and Rydberg-electron collisions occurring fas terthan the me dium expansion. Observations are found in good agreem ent with a simple rateequ ation model. This effect is different from a M ott transition. Conditions for theoccurrence of this latter effect in a Rydberg atom gas are briefly discussed.

Th e developme nt of f ar infrared and m illimetre wave amplifiers an d detectors usingRydb erg atom s as their active medium naturally leads to th e study of th e behaviour of adense sam ple of very excited atoms. In those systems, th e Rydberg atom s are bound tointeract very strongly with each othe r via van de r Waals forces (Raimon d er a1 1981)and also to brea k at a very fast rate d ue to various ionising mechanisms. W orden et a1

(1978) have., for example, observed that strontium Rydberg states undergo strongionisation dom inate d by electron-Ryd berg collisions at densities in th e 10” cmV3range. In the course of ou r mase r and superradiance experim ents, we have observed arelated phe nom enon in a Cs high-density Ry dberg gas (atom s with principal qua ntu mnumber n = 30-45). T he ion-electron production presents a marke d thresholdbehaviour w ith a sh arp onset for a well defined atom density depending o n the principalquantum number U. Th e main characteristics of this effect are accou nted for by a sim plemodel based on rate equations describing an avalanche due to Rydberg-electroncollisions. This effect puts a n upper limit to th e density of achievable Rydberg ato msamples at room tem peratu re, the smallest average interatomic distance to Rydbergatom size ratio, x , being of th e ord er of 15 . Obviously, this sets an absolute upper limit

to th e inversion densities achievable in R ydberg m asers or de tectors (the actual limit isin fact smaller). M ore importantly, the avalanche mechanism reporte d here seem s alsoto preclude the observation of interesting phase transition phenomena expected forx - -5 (Mo tt transition in a densely packed R ydberg gas).

In this letter, we describe our experim ent an d give a simple interpretation. W e alsodiscuss th e possibility of eliminating this ava lanc he effect in an exp erim ent allowing usto observe the more fun dam ental Mott transition.

O ur expe rimental a pp ara tus is similar to th e on e described in previous publications(Fabre e t a1 1978 ). Th e Rydberg atoms a re prepare d at the intersection of a caesiumatomic beam with two collinear pulsed dye laser beams tuned on resonance with the

6 s + 6P3/2 and 6P312+ nD or 6P3/2+ nS transitions. T he sample has a length 1- 1cmi ork partly supported by DRET Contract No 80/187.

0022-3700/82/020049 + 07$02.00 @ 19 82 Th e Institute of Physics L49

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L50 Letter to th e Editor

and a diameter WO- 50* 50 pm defined by the laser beams focus. In this 2 x lo-'' m3volume, one can prepare, in the 10 ns laser duration, as many as 108-109 Rydberg

atoms. The initial Rydberg density (N- 1012-1013 mP3) decreases as the volume

expands at thermal velocity, and is reduced by a factor of ten after about 1ps.

The Rydberg medium is analysed by the selective field ionisation technique (Fabre

et a1 1978). A delay toafter the laser pulses, a ramp of electric field rising in about 1ps is

applied to the atoms. It reaches the thresholds for ionisation of successive levels at

different times. The resulting electrons are detected by a gain calibrated electron

multiplier: one thus gets a time-resolved spectrum of the Rydberg states populated in

the gas.

Figure 1 gives typical time-resolved signals for increasing Rydberg densities with

to-1~ s .n figure 1(a)ne observes a narrow peak corresponding to the level initially

prepared by the lasers. Figure l (6) shows broader structures characteristic of energy

transfers towards states less and more excited than the initial one (which ionise in

stronger and weaker fields, respectively, i.e. at later and earlier times). We attribute

these transfers to van der Waals interactions between Rydberg atoms (Raimond e t a11981). Above a well defined density, one observes suddenly, in addition to these

transfers, a sharp peak occurring near zero electric field and corresponding to free

electrons. Near the threshold density (figure l(c)), his peak is rather small and strongly

fluctuates from pulse to pulse, due to the intensity jitter of the laser. For slightly larger

densities, the signal from the free electrons becomes huge at the expense of the initial

level signal (figure l(d)). In figure2 the points represent the ratio 7 of the free-electron

to the total Rydberg atom signal, as a function of the initially prepared Rydberg density

No, or the case of the 41d level. The threshold effect clearly appears as a turn-on of the

h f i

I -- I - -

Timet o +,loo f,,o T ime

Figure 1. Time-resolved ionisation signals for various initial Rydberg densities. ( a )N = 10' ~ m - ~ ,b ) N = 2 .5 X 10" ~ m - ~ ,c ) N = 2.65 x 10 " cm-3 and ( d ) N =

4.5 x 10'' ~ m - ~ .

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Letter t o the Editor L51

Figure 2. Ionised fraction q after a delay to = 1~ s ,or n = 38.5 (41d level), as a function ofthe initial density (ranging from 4 x 107-5.7 lo1' ~ m - ~ ) .: xperimental results; fullcurve: theoretical prediction from equation (7) with U = 1.02 X mz (k= 0.25) (the

value u , / uR is 100).

ratio 7 above a given density NT. The value of NT depends on the principal quantum

number of the level. The points in figure 3 show the measured threshold variation as a

function of n on a logarithmic scale. NT decreases roughly as (n* is the effective

quantum number, equal to n -2.47 for D states, and to n -4.06 for S tates). Finally,

we have studied the influence of the delay to on this effect and observed that the

5 x 1 0 "

Lm

=

VIWO

+.-

10"

I I I I I

1 4 5 1 5 155 1 6 1 65 Lg(n*i *

I I I I I m

-0d 32d 34d 36d 38d 4 0d 42d 44d n

3 4 s 3 6 s 3 8 s 4 0 s 4 2 s 4 4 s 4 6 s n

Figure 3. Threshold density NT ( ~ m - ~ )s a function of n* on a logarithmic scale. 0 :experimental results; full line: predictions of equation (7 ) (k= 0.23), exhibiting a n*-4

dependence.

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L52 l e t t e r to the Editor

free-electron signal is not instantaneou s an d, near thre sho ld, requires a few tens ofnanoseconds to occur.

Th ese observations clearly indicate th at, abo ve a given density th e R ydberg gas isunstable and evo lves into a plasma, from which electrons ar e extracted by a very smallelectric field (we have checked tha t th e free-electron signal can be collected in a field as

small as 1 V cm-', ne ar threshold, as oppo sed to the 300 V cm-' field necessary tobrea k the initially prepare d R ydberg atoms for y1 - 0).

It is tempting to compare this Rydberg atom to plasma evolution with a metal-insulator phase transition of the Mott type (Mott 1974) . Let us recall that thesetransitions occur in a disordered system (atomic vapour, excitons in a dopedsemiconductor, . . .), when the average interparticle distance (particles = atom s, exci-tons, , , .) is of th e or de r of a few times th e particle size (M ott criterion); the electron sthen become delocalised, and th e M ott criterion can b e justified by electronic free-energy considerations. O ur Ry dbe rg to plasma transition does indeed exhibit som efeatur es reminiscent of such a Mott ph eno me non ; in particular the threshold behaviour

and the very s t rong dependence of the e lectron num ber on the initial density. T he M ottcriterion for such a R ydb erg gas can be estimated by comparing the negative bindingenergy g1 f a R ydb erg atom with the negative polarisation energy g 2 of the Rydb erggas by a free electron and a free ion, produced in the sample by the ionisation of aRydberg a tom. When 1g21s larger than lgll,which occurs for an in teratom ic distance toatomic size ratio x of the order of 3, the delocalised plasma state becomes the stableone , and the Ry dbe rg to plasma transition should occur.

Several feature s of the observed effect, however, do not sup port this interp retation :the existence of a long delay time com pared with the atom ic characteristic times

s), the order of magnitude of the threshold density, corresponding to x - 5

instead of 3, the NT versus y1 variation which should be y1? in a Mott transition.In fact, the relatively long time scale of the ionisation process indicates tha t the slow

motion of the nuclei plays an im porta nt part in the ph eno me non , whereas it is irrelevantin a Mo tt process, whose description is simple only at low temp eratur e. W e are thus ledto ta ke into account th e various Ry dbe rg atom collision processes.

Th e following mechanisms have to b e considered:

Cs*+ hv +Cs++ e -

Cs*+ C s+ Cs++Cs+ e-

Cs*+Cs+ Cs: +e-

CS"+CS*+ Cs*+Cs' + e-

Cs*+e- + Cs' +2e-

CS"+ CS++ CS++Cs' + e -

which respectively describe photoionisation, R ydberg -groun d-state ionising collisions,associative ionisation (Worden et a1 1978), Ryd berg-R ydberg ionising collisions(Olson 1976) , Rydberg-electron and Rydberg-ion collisions (Percival and Richards1975) . Photoionisation has an exceedingly small cross sec tion; furth erm ore it cann otoccur during m ost of the avalanche evo lution, since th e laser excitation lasts only 10 ns.

Th e second and third processes can also be neglected com pared with the fou rth on e in amedium w here the n um ber of Ry dbe rg atoms is of th e ord er of th e number ofgro und -state atom s (Rydberg-Rydberg cross sections are of the or der of t he geometric

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Letter t o the E ditor L53

cross sections, and th e Rydberg-ground-state cross sections are at least an ord er ofmagnitude sm aller for n - 0). In a neutral medium, t he last reaction is slower tha n thefifth, approximately in the ra tio of the ion to electron speeds (the cross sections ar e ofthe sam e orde r of m agnitud e). Finally reactions (4) and ( 5 ) are th e only processes o nenee d tak e into account in a simple model. Th e cross sections of the se two reactions ar e

of similar orders of magnitude (Olson 1976, Percival and R ichards 1975). Their ratesare in the ra tio of the electron to R ydberg densities times the Rydb erg to electronvelocities.

W e are thus led to describe the phen ome non in the following way: immediately afterthe laser excitation, Ryd berg atom s start to collide at a ra te inversely p roportiona l totheir geom etric ionisation cross section. This process provides t he first electrons in thesystem. Thes e electrons escape from th e sam ple with a n energy of the or der of lo-* eV ,i.e. the electron binding energy in th e Ry dberg states, which turns ou t to be of the o rderof th e thermal energy at the atomic beam tem perature. Af ter a few thousand electronshave escaped, th e remaining much slower ions build up a space charge which traps the

electrons produ ced subsequently. Thes e electrons travel back and forth in the medium,with a velocity 100 to 1000 times larger than that of the ions. Reaction ( 5 ) thenbecomes m ore and m ore impo rtant and rapidly domina tes the system evolution. Inoth er words, the initial Ryd berg-Rydberg collisions prod uce ‘seed’ blectrons w hichinduce an avalanche effect in the medium. T he exponential increase of t he electrondensity is limited only by the sample expansion. The threshold results from thecompetition between the avalanche and the expansion speeds. This interpretation linksthis effect to the on e observed in strontium (Worden e t a1 1978) and also to th e strongionisation observed in particular by McIlrath and Lucatorto (1977) in high-densityalkali gases irradiated on their principal resonance line by powerful laser beam s. W eshould n ote, how ever, the large difference of the ord ers of magnitude between this lasteffect and the Rydberg experiments. Whereas in McIlrath and Lucatorto (1977) thedensities are of the order of ~ m - ~ ,he R ydb erg ionisation effects are ob served fordensities as small as 1 O I 2 cmP3, reflecting the large ionisation cross sections in t heRyd berg gas.

Th e abov e explanation can be su pported by a qu antitative calculation based on arate equation. Let us call No the total initial population, NR the remaining Rydbergpopulat ion at t ime t after the excitation, U the total ionisation cross section for aRyd berg atom by another very excited a tom o r by an electron (that we assume to b e ofthe sam e orde r of magnitude), ue the mean electron velocity, ur the mean relativeRy dbe rg velocity; W the medium diameter at t ime t (W = WO u,t) (the length 1of the

medium is approximately constant during the experiment time);Rydberg-Rydberg collision rate:

1 NOCTU,

To l W i *-=-

T he evolution of N R is described by the following rate e quation

2dNR_- NRWO-NR)WO o e

d t M O T o (F)g’

1/To h e ini tial

(7)

T he first term in the right-hand side of this equa tion corresponds to R ydberg-Ry dbe rg collision effects. Th e (WO/W)2actor take s into account th e expansion of th emedium. Th e second term corresponds to th e Rydberg-electron collisions (the num ber

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L54 Letter to the Editor

of e lectron s at time t is N o - N ~ ) , hose collision rate is V J V R times larger tha n theRydberg-Rydberg one.

This equ atio n obviously describes an avalanche process in the Ry dberg gas. All theparam eters are known or measured, expect for (T which we assume to be r ( k a 0 n * 2 ) 2 ,

where k is a free pa ram eter. Solving (7) and defining th e threshold N r as the Novalue

giving

we can calculate NT ( n ) .W e can also calculate 7(No)t time to == I ps, We choose for k

th e value giving th e best fits with th e exp erim ental points of figure 2 and 3 : with k = 0.25one finds the full curves in these figures, which are in fair agreemeilt with theobservations. Using this k value, we have also calculated q ( t ) for increasing initialdensities No, and for n* = 38.5. Results are plotted in figure 4. The onset of theavalanche phenom enon is clearly appa rent for No= NT= 1 .4x 10'' cm - '~

0.0 T i m e a .L IO-'

Figure 4. Theoretical calculation of the ionised fraction q plotted against time for various

densities (curve A, 3 . 1 . ~o1 ' ~ m - ~ ;, 1.4 X 10" ~ m - ~ ;, 2.5 X 10" ~ m - ~ ;, 3.6 X

10" ~ m - ~ ;, 4.7 X 10 " with U = 1.02 x m2 and r;,/vR = 100.

This model thus explains fairly well th e main features of th e observed phe no m ena . It

assum es, how ever, a value of the Rydbe rg-Rydb erg and Rydberg-electron crosssection on e order of magnitude sm aller than the on e given by Olson (197 6) and Percivaland Richards (1975).

An explanation would be an overestimation of the Rydberg density (it is, inparticular, difficult to d ete rm ine the exact size of the m edium , which could be widerthan t he light focus du e to optical saturation). W e should also point out that th eRydberg ionisation occurs in a medium where the atom s are already strongly perturb edby van der Waals coupling between neutrals (Raimond et a1 19 81) . It is thus possibletha t th e two-body analysis developed here is too nai've.

L e t us conclude by a few remarks concerning the connection w ith various experi-men ts involving large num bers of Rydbe rg atom s.

It is clear tha t th e density in th e active medium of a maser o r an amplifier should beke pt lower than th e avalanche threshold. In fact the van der Waals dephasing, which

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Letter to the Ed i to r L55

already becomes important for x - 0, perturbs the electric dipoles and inhibits the

superradiance or maser effect well before the ionisation process described here occurs.

The avalanche threshold is thus certainly higher than the maximum possible density for

Rydberg masers or amplifiers.

As for the Mott transition effect, this experiment indicates that its observation in a

Rydberg system would be possible only if the effect of the atomic motion could be

suppressed. Rather than reducing the atom temperature to a fraction of a kelvin which

should be very difficult, it could be worth trying to perform the experiment in a time soshort that the atoms cannot move over distances larger than their own diameter. This

requires, in practice, picosecond excitation and detection times. Under these condi-

tions, it should be possible to ‘pack’ the atoms at much higher densities than in this

experiment and to observe phenomena related to Mott transitions.

Another recent subject of interest is the study of Rydberg-Rydberg ionising

collisions themselves. Our experiment shows that in a dense sample they are quickly

masked by the much faster electron-Rydberg processes. The only way to investigate

them would be to use a true collision apparatus involving two intersecting low-densityRydberg atomic beams.

References

Fabre C, Haroche S and Goy P 1978 Phys. Rev. A 18 22 9McIlrath T J and Lucatorto T B 1977 Phys. Rev. Lett. 38 1390Mott Sir N F 1974 Metal Insulator Transitions (Lond on: Taylor and Francis)Olson R E 1976 Phys. Rev. Lett. 43 126Percival I C and Richards D 1975 Advances in Atomic and Molecular Physics vol 11, ed D R Bates and

Raimond J M, Vitrant G and Haroche S 1981 J. Phys. B : At. Mol. Phys. 14 L655-60Worden E F, Paisner J A and Conway J G 1978 Opt. Lett. 3 156

B Bederson (New York: Academic) p 1