SUPERRADIANCE THEORY AND X-RAY LASERS : BASIC NOTIONS AND MODELS

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SUPERRADIANCE THEORY AND X-RAY LASERS : BASIC NOTIONS AND MODELS

A. Crubellier

To cite this version:

A. Crubellier. SUPERRADIANCE THEORY AND X-RAY LASERS : BASIC NOTIONS AND MOD- ELS. Journal de Physique Colloques, 1986, 47 (C6), pp.C6-211-C6-222. �10.1051/jphyscol:1986628�.

�jpa-00225871�

JOURNAL DE PHYSIQUE,

Colloque C6, suppl6ment au n o 10, Tome 47, octobre 1986

SUPERRADIANCE THEORY AND X-RAY LASERS : BASIC NOTIONS AND MODELS

A. CRUBELLIER

L a b o r a t o i r e Aimt? C o t t o n , CNRS 11, B S t . 5 0 5 , F-91405 O r s a y Cedex , F r a n c e

Rdsumd

-

La thdorie de l a superradiance e s t prdsent6e en i n s i s t a n t d'une part sur l e s propridtds e s s e n t i e l l e s du phdnomsne, qui apparaissent d&jb dans l e s modSles l e s p l u s simples, modSle de Dicke e t modsle semi-classique, e t , d'au- t r e p a r t , sur l a varidt6 des s i t u a t i o n s physiques dans lesquelles l e phdnoms- ne a dtd dtudig. Ceci permet de ddgager l e s p o s s i b i l i t d s e t l e s limites de l a

thdorie de l a superradiance come point de ddpart d'une analyse de 1'6mission cohzrente de rayons X dans l e s plasmas chauds.

Abstract

-

The e s s e n t i a l c h a r a c t e r i s t i c s of superradiance a r e described by using two simple models, the Dicke model and the semi-classical model. The main extensions t o r e a l i s t i c s i t u a t i o n s a r e a l s o presented and the p o s s i b i l i -

t y of superradiance theory being a s t a r t i n g point of a general approach of X-ray coherent emission i s discussed.

Coherent light production generally originates from stimulated emission in an inverted medium. A lot of different situations are however encountered, according to wether the pumping mechanism which creates the population inversion is continuous or instantane- ous, according to whether the active medium is or is not placed inside an optical cavity

...

Somehow controversed names are associated with the main typical cases of coherent light production. The word laser is generally restricted to devices including a resonator (11. In the absence of mirrors, two phenomena have been essentially studied: ampliGed spontaneous emission (ASE) and superradiance (SR) or superfluorescence 13-61. The term ASE usually refers to a steady-state regime whereas SR ideally deals with an infinitely short inversion rise time.

As compared to the other processes, superradiance exhibits quite peculiar features, such as, for instance, the n2 dependence (n being the atomic density) of the maximum of the emitted intensity, which is inseparable from a basically transient nature of the emission.

These properties are known to be due to the spontaneous-build-up of a collective dipole moment or, equivalently, to the occurence of an interference between the emitting dipoles.

In the laser regime or in ASE, dephasing effects continuously cancel the collective dipole moment and kill the interatomic interference. Transition from SR t o ASE, while decreasing the characteristic time of dephasing effects until it becomes smaller than the characteristic

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986628

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time of superradiance, has in particular already been observed 171 and analysed [8]. A general treatment can indeed be given which concerns SR and the transient regime preceding ASE as well. As long as it corresponds to the limiting case of negligible dephasing effects, SR provides in some sense the natural starting point of such a general theory of coherent light production, which would best apply to transient regimes, i.e., in particular, to all presently thought of X-ray laser schemes [9,10].

The present paper, together with the following one [ l l ] , intends to describe SR theory in order t o show how it could help in the understanding of the X-ray laser problem. A brief survey of this theory (sect. 2) immediatly suggests the interest and limitations of such an approach. The central role of the interatomic interference is then emphasised and the superradiance threshold condition is briefly recalled (sect. 3). Two basic models are then presented. The first one, the Dicke model 1121, is mostly idealised but it provides a right and deep understanding of the superradiance phenomenon (sect. 4). The semi-classical model (sect. 5) accounts for light propagation and is best adapted to numerical simulations.

Furthermore, as it will be shown in the following paper [ l l ] , this mode1 can be extended to realistic situations.

2.Survey of superradiance t h e o r y

Superradiance (or superfluorescence) is cooperative spontaneous emission. It occurs, like

"ordinary" non-cooperative spontaneous emission, for initially excited and then sponta- neously evolving atoms, but above a threshold of atomic density. Since the first observation of the phenomenon, in 1973 1131, many superradiance experiments have been performed 1141.

Most of them are schematically described by the figure.

filter de'tector

I ;

light atoms

01

(or

molecules,

ions ...)

The upper level of the relevant transition 1 -+ 2 is populated by a short laser pulse tuned at, A, and the light emitted at X is observed. When the atomic density n is higher than a threshold value, the properties of this light are quite different from those of non-cooperative spontaneous emission. First the emission is strongly directive; it is emitted in a small solid angle near the cylinder axis in both forward and backward directions [13]. The temporal shape of the radiated intensity is a pulse delayed with respect to the exciting pulse (figure);

the maximum intensity I, is proportional to n2, the delay time t d is proportional to l / n and so does the pulse width t , (since the pulse area is of course proportional t o n) [13].

The emitted light is essentially coherent; beats have been for instance observed between the superradiant light emitted by two different samples [5] or between different velocity classes of a same sample 1151; moreover, the polarisation of superradiant light has always been found to be complete [16-181. Finally, important shot to shot fluctuations of the different properties of the emitted light (delay time [19], maximum intensity [18], polarisation 116,171

...)

have been observed, which cannot be explained by experimental fluctuations.

A complete description of superradiance successively uses, in principle, a quantum-me- chanical and a classical description of the electromagnetic field. The collective decay begins indeed as the ordinary spontaneous emission and thus requires at first a fully quantum- mechanical description. However, atomic correlations spontaneously appear during the emission of the first few photons [20]. In other words the emitting dipoles spontaneously lock their relative phases. At the end of this very short period, the field has became quasi- classical [4,6,20] and a semi-classical treatment of the atom-field system is then valid. This qualitative description already explains the coherence properties of superradiant light and its macroscopic fluctuations as well. The latter are in some sense the "memory" of the initial quantum fluctuations 1211. One also notices that whereas spontaneous emission is responsible for the beginning of the emission, stimulated emission becomes essential as soon as the field behaves classically, i.e. during the main part of the emission.

Superradiance theory has been first concerned with an ideal case which corresponds, in particular, to two-level atoms and to an instantaneous pumping. Fully quantum-mechanical and semi-classical treatbents have been separately developed. Quantum-mechanical models are of course well suited to the description of the beginning of the emission. For extended samples, one is then led, however, to use the mean-field approximation [22], which cannot describe the details of the emission since it ignores the variation of the field envelope along the sample axis.

This variation is, on the contrary, easily accounted for in the semi-classical model [13,23].

The corresponding Maxwell-Bloch equations are indeed readily solved numerically, at least when they are written in the plane wave approximation, as it is suggested by the strong directivity of the superradiant Iight emitted by pencil-shaped samples. The validity of the semi-classical model, except for the first few emitted photons, has now been established by a careful study of the quantum initiation of superradiance [4,6,20,21,24]. In this paper I will briefly describe the simplest quantum-mechanical model, the Dicke model 1121, and the semi-classical model (13,231.

Many extensions of the original ideal case have now been studied. The influence on superradiance of the line broadening which is for instance due to Doppler effect or to atomic collisions has been early investigated [4,22,23,25]. The transition from SR to ASE, while increasing the line broadening, is now well understood [8]. In the particular case of atomic

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samples with linear dimensions much smaller than the atomic wavelength,the dephasing effect due to dipole-dipole interaction, which generally prevents the interatomic interference, has also been studied 126).

The so-called transverse effects, which appear since the plane wave approximation is never exactly valid, have now been quantitavely evaluated [27]. Pumping effects have been shown to occur as soon as the inversion rise time becomes of the order of the delay time of the superradiant pulse; in the limiting case of continuous pumping, the relaxation oscillations which have been predicted and observed (281 clearly confirm the essentially transient nature of superradiant emission.

A variety of specific properties are expected for superradiance in multilevel systems. In particular, the competition between transitions sharing a common level often leads to the inhibition of the emission on one of the two transitions [6,16,29,30]. Remarkable polarisation properties of superradiant light emitted on a transition between degenerate levels have also been observed and interpreted [1618,31].

The existence of a subradiance effect due to a destructive interatomic interference [32] has now been experimentally demonstrated (331. Other cooperative emission regimes, such as Raman SR [34] or collision-induced SR (351 have also been studied; it has been shown that one of the two processes always inhibits the other one (351. Studies of cooperative effects in resonance fluorescence [36] and in the absorption either of a blackbody radiation [37] or of a weak coherent field [38] must finally be mentioned.

In all cases, a satisfactory agreement between theory and experiment is obtained each time that experimental results are available. Whatever close the link between X-ray lasers and SR will be finally found to be, these theoretical studies should serve, because of their variety and completeness, as a guide when considering the specific situation encountered in X-ray laser experiments.

3. I n t e r a t o m i c interference and c o o p e r a t i v e emission

As superradiance is basically due to interatomic interference, the condition for the exis- tence of the interference obviously provides the cooperativity condition, i.e. the superra- diance threshold. Quite generally, emitting atoms are able to interfere one with the other when they are indistinguishable with respect t o the field they radiate. In this case, once a photon has been emitted, it is indeed impossible to know which atom has emitted it. Math- ematically this implies the invariance of the atom-field Hamiltonian with respect to atomic permutations. Different physical situations can be thought of.

Consider first the case of two two-level atoms. As it is easily guessed, they are indistin- guishable when the interatomic distance d is much smaller than the atomic wavelength

X

(391. For N two-level atoms, one would therefore think, at first sight, that the indistinguisha- bility condition is fulfilled in the so-called "small samples", which have linear dimensions much smaller than the atomic wavelength. It has however been shown that in this case the dipole-dipole interaction between the atoms cannot be neglected and that it generally pre- vents the permutation invariance of the problem 1261. The ideal case of N indistinguishable atoms has still been physically realised, by placing the atoms in a high-Q cavity [40]. It has indeed been shown that the atoms are then essentially coupled with a single mode of the

electromagnetic field: whatever the interatomic distance is, the permutation of two atoms only introduces a well-defined phase factor which can thus be eliminated.

In the general case of pencil-shaped samples, which is most commonly encountered in superradiance experiments, the indistinguishability of the atoms is only local and it is based on the plane wave approximation. The emitted superradiant field can indeed be approxi- mated by two plane wave packets traveling along the Oz cylinder axis in the forward and backward directions, and the amplitude of the field depends thus on the z-coordinate only.

The atoms which are in the neighbouring of a same wave plane are thus indistinguishable.

In other words, when a photon is emitted from a slice of the medium with a thickness much smaller than the atomic wavelength, it is basically impossible to find which particular atom has emitted it. The cooperativity condition can thus be written

where N is the total number of atoms and L is the sample length; this condition simply expresses that the number ,U of locally indistinguishable atoms, which are contained in a slice of thickness A, is very large. In fact, the cooperativity condition is most often written as (13,231

where TSR is the characteristic time of cooperative spontaneous emission, which is usually defined by T S R = 1 / N r ~ (where p is a form factor depending on the sample geometry (41]), and where Tsp = l / I '

,

which is simply the spontaneous lifetime of the upper level, is the characteristic time of ordinary spontaneous emission. Equivalence between conditions (1) and (2) is readily shown (321.

The interference bettveen the atoms can eventually be hindered by many physical pro- cesses, such as the above-mentioned dipole-dipole interaction, which are quite generally called dephasing effects. For instance, the Doppler effect due to the atomic motion or the dephasing of the atomic dipoles which is caused by the atomic collisions will generally prevent cooperative emission, unless a cooperativity condition is fulfilled, which is usually written as

where

T;

is the time which characterises the considered dephasing effect. This condition can also be viewed as an indistingishability condition. Consider, for instance, the case of Doppler effect. The corresponding characteristic time is

where 0 is the quadratic mean value of the z-component of the atomic velocity. The coop- erativit,~ condition can then be written as

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The right-hand member of this inequality is the time spent, in average, by an atom in a slice of thickness A; the condition therefore expresses that a same ensemble of atoms must remain indistinguishable for a long enough time interval.

The role of the interatomic interference in cooperative spontaneous emission most clearly appears in the elementary case of two two-level atoms. For the sake of simplicity we consider here the case of fully cooperative emission (d

<<

A) and we assume that one atom only is excited. It is then convenient to consider the following two-atom states,

with E = 3 ~ 1

(I+ ^>;

^(resp.

1-

>i)is the upper (resp. lower) state of the atom i). The decay rate of these two states is found to be [39]

For .s = + l , i.e. for the symmetric two-atom state, the decay rate is thus twice as large as in non-cooperative emission: this of course results from a constructive interference between the two atoms and it is the elementary process of superradiant emission. If, on the contrary,

E = -1, i.e. if the two-atom state is antisymmetrical, the decay rate is zero. The interference between the two atoms is then destructive and one photon is trapped by the atomic pair.

This is the elementary process of a phenomenon called subradiance. Although it was already mentioned by Dicke in 1954 1121, this phenomenon has only recently been analysed 1321 and observed 1331.

4. Dicke model

The Dicke model [12] was originally intended to apply to small samples, i.e. t o samples with linear dimensions much smaller than the -atomic wavelength. As mentioned above, it has been shown later [26] that the model generally fails in this case, because of dipole- dipole interaction. Nevertheless, the ideal case described by this model, i.e. the ideal case of N indistinguishable atoms, provides, despite its relative simplicity, a right and deep understanding of the superradiance phenomenon. As already mentioned in sect. 3, atoms in a high-Q cavity provide an actual realisation of this ideal case. Dicke model however only roughly applies to the ensembles of locally indistinguishable atoms of a pencil-shaped sample, since the interaction between these ensembles cannot be neglected, even in the mean-field approximation (221.

The Schrodinger point of view is used and a basis of collective states is thus first defined.

Each two-level atom is described as a particle of angular momentum 112. The corresponding pseudo-spin operators are

with the same notation as in sect. 3. Collective pseudo-spin operators are then defined by

and Dicke states lgrm

>

are finally defined as eigenstates of

RZ

and R2, with

with respective eigenvalues m and r ( r

+

1). The rules for the addition of angular momenta indicate that, for even (resp. odd) N , r takes integer (resp. half-integer) values between 0 (resp. 112) and N/2, whereas

rn

takes integer (resp. half-integer) values between -N/2 and N / 2 . The additional index g labels the different collective states corresponding t o the same set of r and m values. Physically speaking, m characterises the collective energy of the atoms whereas r , the Dicke's cooperation number, characterises the "cooperation ability" of the collective state. This quantum number has been shown t o be closely related to the symmetry properties with respect to the atomic permutations of the collective state [32] and it thus determines the either constructive or destructive nature of the interatomic interference. As, because of the indistinguishability of the atoms, the quantum number r is conserved during the whole evolution, SR finally appears, in the Dicke model, as a cascading emission among collective states of given r value.

The master equation describing cooperative emission can be formally derived from the equations of ordinary non-cooperative spontaneous emission of two-level atoms. These equa- tions are usually written as [42]

where p++, p-- and p+- are respectively the populations of the two levels and the coherence between them; they are equivalent to an operator equation involving the atomic density operator

,

The master equation describing cooperative spontaneous emission [43]

has

exactly the same form except that the individual pseudo-spin operators are replaced by the collective ones:

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A comparison of the two master equations clearly shows that cooperativity is due t o the interatomic interference. The decay rate of the system, which is found t o be

is also dominated by the interference terms. For a given Dicke state lgrm >,the decay rate is equal to

it thus depends on both the excitation of the system, i.e. on rn, and on the nature of the interatomic interference, i.e. on 7. During cooperative emission starting from a fully excited collective state, for instance, the decay rate, which is initially equal to the incoherent decay rate, first increases, as the energy decreases, up to N2/4 and then decreases down to zero.

These simple remarks already allow one t o catch a glimpse of the temporal shape of the emitted light. In fact the Dicke model qualitatively reproduces almost all observed features of superradiant emission. As in this model the atoms of each pair are simply assumed to constructively interfere, one therefore clearly understands the primordial role played in superradiance by the interatomic interference.

5.Semi-classical model

The basis of the semi-classical model of superradiance is readily described: each atom is considered as interacting with the classical electromagnetic field radiated by all atoms. The atoms, which are here assumed to be two-level atoms, are usually ccnsidered as a continuous medium, characterised by both a local macroscopic polarisation P and a local population inversion

N,

which are given by

where n(2) is the local atomic density, &is the dipole moment of the transition and the

~ ( 5 , t )

quantities are deduced from the corresponding atomic density matrix elements. As indicated in the Maxwell equation,

the local macroscopic polarisation induces an electromagnetic field, which in turn drives the evolution of the atoms. The latter evolution is ruled by the optical Bloch equations, which can be written as

where i is the density matrix of atom i and H is the atom-field Hamiltonian

H A and H I are respectively the atomic and interaction Hamiltonians, with

&

being the dipole operator of atom i. The important role of stimulated emission in su-

perradiant emission is, in this model, quite obvious. In fact spontaneous emission is merely ignored, as it is usual in the semi-classical theory. Difficulties must thus be expected a t the beginning of superradiant emission, which is ordinary spontaneous emission.

In view of an easier numerical resolution of the Maxwell-Bloch equations (eqs 17 and 18), two main approximations are then usually introduced. The first one is the above-mentioned plane wave approximation. It has been shown that this approximation best applies when the Fresnel number of the atomic sample is equal t o 1. The so-called transverse effects [27] which are then neglected may explain many former discrepancies between experiment and theory 151. The second approximation,the slowly varying envelope approximation (SVEA), simply states that the variation of the macroscopic polarisation and of the field during one optical period and over one wavelength is negligible; it thus holds as long as the atomic density remains much smaller than With these approximations and assuming in addition, for the sake of simplicity, that forward and backward emissions are not coupled so that forward emission can be separately considered, the Maxwell-Bloch equations can be written as 113,231

where

P

(5, t) and E (5, t) are the slowly varying envelopes of the macroscopic polarisation and of the field.

In the case of two-level atoms, a geometrical representation of the evolution is obtained by considering the local Bloch vector. This vector is defined by a a-component equal to N ( 5 , t ) / 2 and by a projection in the xy-plane equal to

IP (5,

t)l/d. As the evolution of the atoms is ruled by an Hamiltonian (eq 18), the statistical mixing of the atomic state is

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conserved by the evolution and so does the squared length of the local Bloch vector, which is simply related to the trace of the squared atomic density matrix. When starting from a complete population inversion, i.e. from a Bloch vector pointing upward, for instance, this vector will simply "fall" ,in a given plane, oscillate like a pendulum and finally stop when pointing downward. The radiated intensity roughly follows the variations of the squared local dipole moment, i.e. of the squared xy-projection of the Bloch vector. In particular, the delay time of the superradiant pulse is simply the time necessary for the dipole moment to reach its maximum value, which is equal to the initial population inversion. The "ringingn oscillations [23,44] which are observed in numerical simulations correspond to oscillations of the local Bloch vector; these oscillations are found to be almost suppressed when transverse effects are accounted for (271.

If both the electromagnetic field and the macroscopic polarisation are initially equal to zero, eqs (21) clearly show that the system is metastable and does not evolve at all. This is not surprising since ordinary spontaneous emission cannot be treated semi-classically. The problem of the initiation of superradiant emission in the semi-classical model will be evoked in the following paper [ll].

The solution of Maxwell-Bloch equations for two-level atoms in the plane wave approxi- mation (eqs 21) is rather easily computed. Furthermore, many additional physical processes, such as, for instance, Doppler effect or field losses 113,231, can be readily included in nu- merical simulations of more realistic situations. In fact the semi-classical model provides a treatment of all the effects listed in sect. 2: line broadening effects, dipole-dipole interac- tion, pumping effects, competition effects, polarisation effects, subradiance

...

The following paper [ll] will describe how the model can in particular be extended to the treatment of transverse effects, of inhomogeneous and homogeneous broadening and of a blackbody ra- diation triggering of the emission which can serve as a model for the actual initiation of superradiance.

6. Conclusion

Since Dicke's paper, in 1954, the number of papers in superradiance literature is rather impressive. Cooperative spontaneous emission is in fact an undoubtly attractive problem, readily formulated but still very complex, which has progressively revealed the many remark- able specific properties of superradiance. The phenomenon is presently well understood. As shown in the Dicke model, for instance, the essential role is played by an interatomic inter- ference, which is due to the (at least local) indistinguishability of the atoms with respect to the field they radiate. From a semi-classical point of view one can equivalently say that,at the beginning of the emission, the emitting dipoles spontaneously and definitively lock their relative phases. The interatomic interference can eventually be hindered by "dephasing"

processes; superradiant emission is then replaced by a transient regime essentially similar to ASE.

The reason why superradiance theory could help in the X-ray laser problem is mainly that X-ray coherent emission always occurs (or is expected to do so) in a more or less transient regime. As a first consequence, one is still allowed to wonder if, in some specific conditions, an interatomic interference could not play a role in coherent X-ray light production. Fur- thermore a general theory could readily be elaborated starting from SR theory which would apply to transient regimes of coherent emission whatever the dephasing effects be. In such a general theory, the large variety of realistic situations already studied in SR theory should

serve as a guide when considering the specific and complex situations encountered in X-ray laser experiments.

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