Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://qilab.phys.msu.ru/papers/Laser-Physics-20(1)107-118(2010).pdf Äàòà èçìåíåíèÿ: Mon Dec 21 12:45:25 2009 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 20:04:54 2012 Êîäèðîâêà:

ISSN 1054 660X, Laser Physics, 2010, Vol. 20, No. 1, pp. 107­118.
© Pleiades Publishing, Ltd., 2010. Original Russian Text © Astro, Ltd., 2010.

PAPERS

Coherent Control of Chirality in Ensemble of Randomly Oriented Molecules Using a Sequence of Short Laser Pulses1
D. V. Zhdanov and V. N. Zadkov
International Laser Center and Faculty of Physics, Moscow State University, Moscow, 119991 Russia
e mail: zhdanov@phys.msu.ru, zadkov@phys.msu.ru Received December 29, 2008; published online July 2, 2009

Abstract--The universal algorithm of choosing the train of short laser pulses, which could selectively excite the specific enantiomers in a racemic mixture of rotationally cold molecules, is proposed. Its application is illustrated on example of laser control of content of different chiral conformations in the mixture of hydrogen peroxide molecules. Methods of detection of racemity breaking in case of dynamical chirality are also dis cussed. DOI: 10.1134/S1054660X0917023X
1

1. INTRODUCTION

The problem of absolute asymmetric synthesis (AAS) is one of enigmas of the modern molecular sci ence, which still has no general solution despite it is constantly at focus of attention of scientists of differ ent branches of physics, chemistry, biology, and life sciences since 1849 when Louis Pasteur produced a pure sample of levotartaric acid which was the first demonstration of enantiomers of chiral molecule (a pair of isomers which are mirror reflections of each over). The AAS is targeted at formation of an excess of given enantiomer (called sometimes also as "enrich ment", "purification" or "distillation") in a racemic (i.e., equally weighted) mixture of enantiomers. The importance of the problem originates from amazing fact of chiral purity of molecules constituting cells of all living beings. Thus, understanding the principles of asymmetric synthesis both casts light on fundamental question of the origin of life and also removes acute shortage of reagents for many practical applications in medicine, chemistry, pharmaceutical and food indus tries etc. The most vivid examples of a successful solu tion of the AAS problem are suggested by chemists and are based on a sophisticated choice of chiral catalysts [1]. Although remedying the situation for a few partic ular cases they however help a little to shine a light on how to solve the problem as a whole. Therefore, the nowadays hopes are pinned on pos sible physical ways of realizing the AAS [2]. In partic ular, recent progress in laser technologies raises an increasing interest to searching the solution among the optical methods, which are incidentally the only phys ical methods that allow reliable detecting and distin guishing the enantiomers. Nevertheless, among a large number of different theoretical proposals [3, 4] only a few of them have been verified experimentally. Among
1

The article is published in the original.

them are scenarios of the AAS based on utilization of magneto chiral effect [5] and electroquadrupole interaction with circularly polarized light [6­9]. Unfortunately, similarly to the case of asymmetric catalysis the necessity in rather specific molecular properties cause these scenarios to have very narrow applicability (in particular, they are deliberately inap plicable to small molecules). To overcome this shortcoming, it is natural to employ more conventional and widely used mecha nisms of laser coherent control of molecular dynam ics. That is why many theoreticians look towards elab oration of the AAS driven by pure electrodipole inter actions with laser radiation [10­37]. However, none of these scenarios occurs suitable for practical imple mentation. The most common reasons for that are impracticable assumptions of preliminary molecular orientation (for scenarios developed in [11­27]) or precise spatial localization (for scenarios developed in [31­36] and zero initial temperature of the molecules. Moreover, analysis of the results of the AAS is also a nontrivial problem due to the nonstable (dynamical) character of chirality of almost all sample molecules. In most of the cited above papers the scope of consid eration does not go beyond the bounds of particular scheme of excitation, which is often closely locked on a specific target molecule via peculiar limitations on its properties (such as existence of achiral excited elec tronic states for the scenarios given in [32, 33]). In our recent paper [38], we made an attempt to systemize and generalize the ideas of laser control of molecular chirality suggested in these works, and derive a number of universal obligatory conditions on the parameters of field molecule interaction in the AAS scenarios. We also successfully employed these conditions to elaborate a new scheme of the AAS based on the laser assisted orientation dependent selection of molecules. In this scheme, we take advan

107


108

ZHDANOV, ZADKOV

tages of combined adiabatic and nonadiabatic regimes of interaction with strong multicolor laser pulses. Numerical tests prove that well known property of sta bility of adiabatic couplings with respect to variations of the laser pulses parameters allows this scheme to be effective for distillation in a macroscopic part of iso tropic gaseous racemate even at room temperature. However, the adiabatic couplings compared to nona diabatic ones are shown to demand orders of magni tude higher laser intensities and pulse durations [16, 20] and, therefore, in many cases may be unrealizable within sub breakdown fields strengths or ineffective due to the strong parasitic effects such as photodisso ciation [26]. In this paper, we will analyze in detail one of the alternative approaches based on a nonadiabatic reso nant impact of the sequence of laser pulses of simple form. Various schemes of the AAS of such kind were already elaborated for the case of preliminary oriented molecules [13­26]. However, the case of isotropic racemate is scantily explored, though there exist the particular proposal for the specific case of molecules with achiral excited state [32, 33]. The aim of this work is to generalize the results of the above works and our previous investigations [36, 37] and to elaborate an algorithm of development of the AAS scenarios via action of a train of short laser pulses for the case of randomly oriented molecules bearing on obligatory conditions derived in [38]. Our results also complement the material of [38] with the methodological example of application of these con ditions in case of essentially quantum mechanical character of rapid rotations of small molecules and also with detailed consideration of the case of dynam ical chirality. Also the problem of heterogeneous spa tial distribution of enantiomeric excess, which is native for all suggested schemes, will be considered in the context of possible ways of detection of induced chiral symmetry breaking. As a computational exam ple of the AAS, the hydrogen peroxide molecule (H2O2) will be considered. It is worth to stress, that for many reasons (hardly achievable necessary laser pulses, difficult detection of the signal) this molecule is a poor candidate for the real AAS experiment. But due to its simplicity and availability of detailed data about its structure, this molecule is ideal for both qual itative and quantitative illustrations of the general solution in principle, suggested in this work. The paper is organized as follows. Section 2 gives a brief insight in theoretical description of molecular chirality and overview of obligatory conditions derived in [38]. Section 3 presents the abstract description of conceptual basics and limitations of proposed general algorithm for construction of the laser driven AAS scenarios based on illumination of the molecules by trains of short laser pulses. Next two sections illustrate an application of this algorithm on example of hydro gen peroxide molecules. In the auxiliary Section 4, general molecular parameters and peculiarities of

mathematical description of rovibrational dynamics are introduced. In Section 5 the scenario of laser driven enantiomeric enrichment of racemic H2O2 vapor is proposed and analyzed in detail. Possible method of detection of the media symmetry breaking is also discussed. In conclusion, Section 6 summarizes the results of the paper. 2. MATHEMATICAL MODEL OF MOLECULAR CHIRALITY 2.1. Mathematical Background The mirror mutual correspondence of different enantiomeric forms of a molecule (often called L and D forms) means that if we introduce a set of quantum states |Lk (k = 1, 2...), corresponding to L form of the molecule, then one can simply obtain the correspond ing set for the D form applying spatial inversion oper ^ ator E * : ^ |D k = E * |L k , (1)

and vise versa. However, it is well known that the eigen ^ molecular hamiltonian H 0 (up to negligibly small cor rections due to the weak interactions) is invariant with ^ respect to E * : ^ ­1 ^ ^ ^ E * H0 E * = H0 . (2)

Comparing Eqs. (1) and (2) one can deduce that |Lk can be in fact only approximate eigenstates, while the ^ true eigenstates |k have definite parity: E * |k = ^ |k, E * |­k = ­|­k. Here each pair of indices of opposite sign enumerates so called chiral doublet states with identical quantum numbers except parity. It is easy to see that the states |Lk and |Dk, which cor respond to Eq. (1), can be expressed as coherent superposition of states of k th chiral doublet: |L k = 1 ( | k + | ­k ) , 2 1 |D k = ( | k ­ | ­k ) 2 (k > 0).

(3)

Thus, even without any external influence, the L and D molecular conformations should self interconvert at the time tun = / |E­k ­ Ek |, where Ek and E­k are the energies corresponding to the states |k and |­k, respectively. Note that time tun varies from a few pico seconds up to few millions of years, depending on the parameters of the potential barrier separating localized chiral states in a particular molecule. In the case of small tun (compared with typical time scale of an experiment) it is said about dynamical molecular chirality, while other limiting case is designated as sta ble chirality.
LASER PHYSICS Vol. 20 No. 1 2010


COHERENT CONTROL OF CHIRALITY

109 | 2 1

To quantify the enantiomeric constitution of a molecular mixture we will use the value of relative excess of the L forms of molecule over the D forms ( 1 1, = ±1 correspond to the pure L/D enan tiomers). This quantity is often called the degree of chirality and corresponds to following quantum oper ator: ^ =

| 1 P |


| q q N |

N+1





q+1

q+1



q+1 |

q+2

| N

k=1





|L k L k| ­ |D k D k| =

k = ­





| k ­k| .

(4)

Fig. 1. The laser induced chain of dipole transitions for the selective excitation of enantiomers.

2.2. Obligatory Conditions of Symmetry Breaking In this work, we will focus on enantiomeric enrich ment of an initially racemic mixture of chiral mole cules (i.e., ways of controllable breaking the equality = 0) by couplings of molecular dipole moment d with the complex electric field produced by a series of laser pulses j: =

(ii) Configurational structure of the transition dipole moments of the rovibronic transitions consti tuting the chain must be noncoplanar. ^ ^ (iii) Each of the three components ^ d , ^ d ,
x' x' y' y'


j

j

=2


j

j A j ( t ) cos ( j t + j ) ,

(5)

where Aj(t), j, and j are the envelopes, frequencies, and unit vectors along the polarization of the respec tive pulses. Mathematically, these couplings can be ^ ^ described via semiclassical additive H I ( ) = ­ d to the eigen Hamiltonian. In our previous work [38], we develop an universal theoretical background for elaborating the scenarios of such kind, revealing the set of six symmetry based conditions on the parameters of field molecule inter action, which are obligatory for selective excitation of different chiral forms of molecule. In the following we will briefly outline these basic results. First of all, we have shown that the necessary con dition of radiational selectivity is the coherent linkage of the states of the chiral doublets (see Fig. 1): (i) The selective excitation of enantiomers in the isotropic racemic mixture can be achieved only as a result of N photon processes where N is an odd num ber that link coherently the rovibronic states | 1 and | N + 1 = ­1 of the chiral doublets by the chain of dipole transitions through the even number of inter mediate rovibronic states | p (p = 2, 3, ..., N) (see Fig. 1).2 The next two conditions refine the above result for the case of chiral molecules, which rotational dynam ics is well described by the model of rigid rotor (i.e., the object with fixed principal momenta of inertia) with principal axes of inertia directed along unit vec tors ^ x' , ^ y' , and ^ z' :
2

^ and ^ z' d z' of the dipole moment must be responsible for the odd number of transitions in the chain. The forth condition clarifies the polarization struc ture of the incident laser field (5): (iv) Polarization configuration of the incident radi ation should be noncoplanar. Specifically, let us assume that ' is a set of indices of the K laser field components with directions j ' of the polarization vec tors orthogonal to the plane and N is the number of rovibronic transitions in the chain of transitions induced by these K components. Then, N has to be odd if the directions of polarization vectors of remain ing components lie in . The following condition deals with the case of heavy molecules and transitory resonant laser excita tion when one can neglect both quantum features in rotational dynamics and any classical rotational dislo cations of molecules during irradiation. For this case it is possible to introduce the following obligatory crite rion for the choice of both polarization of the field components and intermediate vibronic levels in the above chain: (v) Let us define as k, l and k, l the directions of the transition dipole moments of vibronic transitions between k th and l th vibronic states in the molecular frame of L and D molecular conformations corre sponding to above chain. Denote as j ( k, l ) the polar izations of the field components, driving the corre sponding transitions. Then, the following inequality is obligatory for the selective excitation of enantiomers
l d

p=1


N

N



j ( p,

p+1

l ) p,

p+1

()


(6)

­

The origin of this result is also discussed in [10]. LASER PHYSICS Vol. 20 No. 1 2010

p=1





j ( p,

p+1

d ) p,

p+1

()


0.


110

ZHDANOV, ZADKOV

Here, ... denote orientational statistical averaging across all the Euler angles which should be considered as parameters. The last aspects considered in [38] are the role of phase matching between the components j and anal ysis of spatial distribution of induced degree of chiral ity. It is found that the leading term in the dependence of the degree of chirality ' of the excited part of the molecules on their spatial location r has the form: ' Re e
0 i
0

(3) Then, laser action should be assumed as a train of nonoverlapping pulses, where each pulse is respon sible for the corresponding transition in the chain. The order of pulses should satisfy to enumeration of transi tions. For the case shown in Fig. 1 the pulse j ( 1, 2 ) should come first, the pulse
j ( 2, 3 )

should be the

p=1



N

i sgn (

e

, p p+1

)

j( , ) p p+1

,

(7)

where j = j ­ kj r,
0

p,

p+1

1 = ( E p ­ E



p+1

) , and 0,

j are some position independent quantities (here the case of stable chirality is assumed). Thus, the typi cal region of space where one can expect selective excitation of a given form of the molecule has dimen sions of typical wavelength of the incident radiation. However, from Eq. (7) the following condition of spa tially homogeneous excitation of enantiomers of the same configuration in a macroscopic region can be exactly deduced: (vi) The incident laser field must consist of a set of two or more subsets (u, v, ...) of the components with different frequencies, such as the wavevectors of all the components in the u subset are collinear to the direc tion u (|u | = 1), and u w (u w). Also, for the components of each subset the following phase matching condition must be fulfilled:

next one etc., so that the pulse j ( N, N + 1 ) will be the final pulse in the sequence. (4) Thereafter, one should define the polarization of each pulse guiding by conditions (iv). (5) By perforce of special spatial distribution (homogeneous one inclusive!) of induced degree of chirality one further has to perform the choice of directions of propagation of the pulses guiding by Eq. (7) or condition (vi). (6) Finally, the amplitudes and durations of the pulses ought to be assigned to tune the areas S j ( m, n ) = 2
j ( m, n ) d m,
n

to be equal to /2, and the areas of the rest of the pulses to be equal to . Let us clarify the functional concept of the result ing control scheme. For clarity and simplicity of fur ther considerations only sublevel | 1 in Fig. 1 will be assumed to be initially populated. As result of action of the first pulse, half of the population of the racemic state | 1 is transferred into the state | 2 resulting in coherent superposition of the form
i*(t)



A

j ( m, n )

( t ) dt of the first and last pulses

1 ( | 1 + 2

j ( p,


p+1

N

sgn (
)u

p,

p+1

)

j ( p,

p+1

)

= 0.

(8)

3. ENERGETICAL SEPARATION OF ENANTIOMERS: THE GENERAL ALGORITHM Now we are ready to introduce the universal algo rithm for constructing scenarios of energetical separa tion of enantiomers. The concept of the suggested algorithm directly bears on the above conditions and was prompted by Fig. 1. The algorithm is outlined below: (1) First, one should choose the chain of electodi pole transitions (see Fig. 1) between rovibronic molec ular states linking the initially populated sublevels of the chiral doublets (in Fig. 1 these are the sublevels | 1 and | N + 1 ) so that conditions (i)­(iii) are sat isfied. (2) Next, the transitions of the chain are enumer ated in series according to arbitrary chosen traversal direction as shown in Fig. 1.

e | 2 ) (*(t) is a time varying phase). The latter laser pulses will transport coherently the molecules excited by the first pulse further along the chain of transitions so that the resulting state of the molecules just before the last pulse will be 1 ( | 1 + 2 e | N ) (**(t) is also a time varying phase). The application of the final /2 pulse will raise simul taneously the transitions | 1 | ­N and . Taking into account the infinitesi ^ mality of the matrix elements L m| d |D n (the latter is simply the existence condition for the localized chiral conformations) and symmetry properties of the states | m one can easily prove the approximate equality of the dipole momenta of both these transitions and absence of the essential differences in the transition frequencies. Thus, the effect of the last pulse is two fold: first, half of the population of the state | N (i.e., a quarter of the total number of the molecules) will be transferred into the state | ­1 , and, second,
­
1

i ** ( t )

| N

|

LASER PHYSICS

Vol. 20

No. 1

2010


COHERENT CONTROL OF CHIRALITY (a) H a x' H
0

111

x'

(b)

U

g

z' D form

y'

a1 L form

O

z'

170 cm-1 386 cm-1

2460 cm-
0

0

O



1.5 ps
1

11.4 cm-

1

y'

1

2

0



0

-

Fig. 2. (a) Geometry of the enantiomers of hydrogen peroxide molecule and molecule fixed coordinate frame {x', y', z'} (axis x' is parallel to the bisector of dihedral angle HOOH ). The ground vibronic state parameters are as follows: = 115.16° and 244.84°, 1 = 99°, a1 = 1.451 å, a0 = 0.962 å [41, 42]. (b) Potential energy of the ground state of H2O2 as a function of the con torsional coordinate (see [41]).

half of the molecules in the state | 1 (also, a quarter of their total number) will be transferred into the state | ­N . Thus, the molecules in 1th and Nth vibronic levels come to be in superposition states |
­

1 ( | 1 ± 2

1 - ( | N + | ­N ) 2 sgn(N) |D ( L ) N (the choice of sign is tuned by the phase matching between the laser pulses). Thereby, the partial racemity of the molecular state in each of these levels becomes broken, i.e., energetical separation of the enantiomers is achieved.3 It is important to note two native peculiarities of the proposed scheme, which are the direct conse quences of a strict linkage of the chain of transitions, both to preselected rovibronic state and to the appointed intermediate states of a definite symmetry. If the initial state is an incoherent superposition of a large number of different states | m the laser can excite a wide majority of different chains of transitions satisfying the conditions of energetical separation but with the opposite contributions to the partial degrees of chirality of each vibronic level. Therefore, first, the proposed method is expected to be effective only at the low enough temperatures compared with characteris tic values of the molecular rotational constants. Sec ond, the necessity of excitation of the transitions between strongly defined rovibronic states excludes
1

the possibility of nonresonant excitation of the transi tions and also limits the maximum frequency detun ings, peak amplitudes, and minimum durations of each pulse in the sequence. Now let us make the dipper insight into suggested method, familiarizing with it on example of the AAS for hydrogen peroxide molecules. 4. HYDROGEN PEROXIDE--THE SIMPLEST CHIRAL MOLECULE The hydrogen peroxide molecule (H2O2) is a con venient model object for our studies due to its simplic ity and availability of both, reach data on its molecular structure [41, 42] and previous investigations of the laser driven chiral dynamics [36, 37, 43, 44]. Geometry of the enantiomers of H2O2 in its ground vibronic state 0 is shown in Fig. 2a. Like most sim plest chiral molecules, these configurations are not stable because of large energetic splitting in the chiral doublets h
0 tun

) sgn(1) |L ( D ) 1 and

11.4 cm

­1

[41]. Thus, the L and

D forms mutually interconvert during tun 1.5 ps. The substantial feature of the H2O2 molecule is existence of a single dominant transformation channel between the chiral isomers of the molecule: the con formations are changed as a result of enlarging the dihedral angle HOOH from 115.16° to 244.84° with out essential alteration of other valent angles and bond lengths (Fig. 2a). The latter gives a possibility to describe the molecular chirality using one dimen sional Hund model [45]. Recall that this model assumes the potential energy surface to have two sym metrically spaced minima along some contorsional coordinate. The wavepackets localized in each of these wells correspond to the distinct chiral conformations whereas the properties of the potential barrier between the wells define the self interconversion time.

3

Note that the laser impact will lead to the same result also in the case of initial state being the arbitrary noncoherent superposi tion of the states | and | . In the latter case, the paral
1 ­1

lel excitation of one more chain of transitions will occur. This chain starts from the state | and consists of all the levels
­1

| with indices k coinciding with the level indices of the con
k

sidered above chain taken with reversed signs. LASER PHYSICS Vol. 20 No. 1 2010


112

ZHDANOV, ZADKOV

Table 1. The lowest vibrational contorsional levels of the ground electronic state of H2O2 molecule [42] Mode index
0 1 2

Mode description Ground vibrational state Symmetric valent vibrations of hydrogen atoms Symmetric torsion vibrations (changing the angles OOH) Antisymmetric valent vibrations of hydrogen atoms

Energy of the lowest symmetric contorsional state |...(s) (cm­1) 0 3609.8 1395.9 3610.7

Energy of the lowest antisym metric contorsional state |...(a) (cm­1) 11.4 3618.0 1398.3 3618.8

5

Such a contorsional coordinate in the case of hydrogen peroxide is designated as = HOOH , and the corresponding two well potential is sketched in Fig. 2b. From Fig. 2b it is evident that the molecular eigenstates can be splitted into pairs of closely spaced levels |(s) and |(a) (here index indicates other quantum numbers describing vibronic state), which are symmetric and antisymmetric on , correspond ingly. Thereby, the quasistable chiral states localizing in "left" and "right" wells are described by the follow ing superpositions:
(l) (s) (a) | = 1 ( | + | ) , 2 (d) (s) (a) 1 | = ( | ­ | ) . 2

(9)

Despite close connection between Eqs. (3) and (9) it must be emphasized that the vibronic states |(s) and |(a) should not be confused with rovibronic states |k and |­k. In particular, contrary to the latter ones, the states |(s) and |(a) have not definite symmetry with ^ respect to the inversion operation E * unless the sym metry of rotational state is exactly specified. Neverthe less, the states |(s) and |(a) are sufficiently more use ful for description of chiral dynamics, especially in the case of dynamic chirality, because their structure is well adopted for description of the tunneling effects in
Table 2. Properties of several vibrational dipole transitions for L enantiomer of H2O2 molecule calculated using results of [42] Vibrational transition | |
(l) 0 (l) 0

Approximate values (±25%) of x' , y' , and z' components of transition dipole momenta [D] {0.51, 0, 0} {0, 0.51, 0.13} {0.14, 0, 0} {0.094, 0, 0}


l)

| |

(l) 1 (l) 5 (l) 2 (l) 2

|0 ( |1 (

| |

l)

specific molecules and for splitting of vibrational, rotational, and contorsional variables. That is why we will rest upon these states through the rest of the paper. We will elaborate the method of coherent control of molecular chirality based on the subsequent excitation of the different lowest vibrational modes of the lowest contorsional doublet and the ground electronic state of H2O2 molecule. Brief description of the properties of several lowest vibrational modes and dipole transi tions, which will be used in our control scenario, are given in Tables 1 and 2 for reference. In particular, from Table 1 one can see that doublet splittings between the lowest contorsional states and thereby the times of self interconversion between the chiral con formations are strongly dependent on vibrational state of the molecule. The important advantage of hydrogen peroxide as a model molecule for theoretical analysis is the feasibil ity of several substantial simplifications in the rota tional dynamics description. First, note that due to the substantive differences in masses of O and H atoms the directions of principal inertia axes are close to direc tions of the frame axes x', y', z' shown in Fig. 2. More over, the corresponding approximate principal momenta of inertia Ix', Iy', and Iz' satisfy relations Ix' Iy' Iz' and negligibly depend on contorsional and vibrational state. Thus, the rotational dynamics of the molecule, despite its essential nonrigidity, can be rather accurately described within the rigid symmetric top model with z' taken as its axis (e.g., corresponding errors in the rotational energies for the angular momentum J 3 do not exceed 0.1%). We should stress, however, that in this simple approximation we lose the possibility of correct account of selection rules caused by indistinguishability of identical particles (for such an account the choice of axis x', which is the symmetry axis of the second order, as a quantization axis would be preferable). Thus, the H2O2 molecule can be treated as a rigid rotor and thus falls within the scope of the obligatory conditions of Subsection 2.2. The latter proves the feasibility of applying the algo rithm proposed in previous section to this molecule. Keeping in mind that our consideration is aimed only at demonstration the proposed solution in princi ple, we will make one more severe assumption arising
LASER PHYSICS Vol. 20 No. 1 2010


COHERENT CONTROL OF CHIRALITY

113

0

from large difference (about an order of magnitude) between the characteristic classical frequency of molecular rotation around the z' axis and analogous frequencies for the two another axes. Note that for the rotational temperature lower than 10­15 K the typical classical period of rotations around the latter axes does not fall below the few picoseconds. Thus, with the use of the pulse sequences of picosecond duration we can additionally assume the direction of the axis z' to be rotationally "frozen" during the laser illumination. As a result, we come to our main reduced one dimen sional model of molecular rotations, in which the only dynamical variable is the Euler angle describing the rotations around axis z' (corresponds to the quantum number k of the angular momentum projection on this axis) whereas remaining rotational degrees of freedom (i.e., angles and ), characterizing the direction of z', are treated as parameters. Thus, the consideration of laser impact on chaotically oriented molecules in the frame of this model among quantum mechanical computations includes subsequent statistical averag ing over two angular parameters. Let us turn now to elaboration of scenario of the laser driven AAS, using the above data and approxi mations. 5. LASER COHERENT CONTROL OF DYNAMIC CHIRALITY IN H2O MOLECULES

partial contributions and of the states 0 and 2 into the local degree of chirality becomes nonzero: 0 2 0 1 (t) = 0 cos( 2 tun t ) and = 2 2 1 ­ 0 cos( 2 tun t ). The value of 0 here is propor 2 tional to the number of excited molecules and oscillat ing terms describing the tunneling between the differ ent enantiomeric forms. Due to the distinct oscillation frequencies, the overall local degree of chirality takes the form:
2



(t) = (t) + (t)
0 2





= 0 cos ( 2 (

0 tun

­ tun ) t /2 ) cos ( 2 (
2



0 tun

+

2 tun

(10) ) t /2 ) .

2

First of all, let us flesh out our problem settling. Suppose that initially we have a racemic mixture of free noninteracting H2O2 molecules at zero rotational temperature. As the initial vibrational contorsional state we will take the incoherent superposition of (s) (a) equally populated states | 0 and | 0 . Our main task will be searching for the laser action, which could raise the excess of predefined enantiomer (at a given time instant) in a specific point of space. It must be emphasized that we will not demand the carrying out of condition (vi) of achieving the enantiomeric excess of the same enantiomer simultaneously in a macro scopic region of space because such additional complication seems to be inexpedient in the case of molecules possessing the fast oscillating dynamical chirality. As it happens, the solution of the task can be obtained simply via energetical separation of different enantiomeric forms between two vibrational levels with the distinct doublet splitting tun between its chiral sublevels. Consider, for example, the levels 0 and 2 ( tun = 342 GHz, tun = 72 GHz). Suppose that in a certain point of space at the time instant t = 0 the chiral conformations were energetically separated via selective population of vibrational state 2 in a number of molecules in D conformation. Thus, the
0 2





Thus, the value of is still equal to zero at t = 0, but becomes nonzero at the latter time moments varying accordingly with the biharmonic law with the peak amplitude 0. To elaborate the scenario of energetical separation, we used the algorithm from Section 3 and the data listed in Tables 1 and 2 with additional request of the low number of "links" in the chain of transitions and large enough values of all the employed transition dipole momenta. The resulted proposal is shown in Fig. 3. The laser action is composed of three laser pulses 1, 2, and 3 with the different carrier fre quencies, propagating along two mutually orthogonal directions 1 and 2, as shown in Fig. 3b. From Fig. 3c, however, one can see that the corresponding chain of transitions consists of not three but five links because the first pulse 1 excites the first three transi tions simultaneously. Using Fig. 3b one can easily ver ify the fulfilment of the conditions of oddness of tran sitions in the chain, noncoplanarity of polarization configuration of the laser field, and condition (iv) for each of the planes xz, yz, and xy chosen as plane . The configuration of the matrix elements of the dipole momenta for employed rovibrational transi tions is shown in Fig. 3a. After the first look on the fig ure, a thorough reader could wonder whether the shown configuration really satisfies the noncoplanar ity condition (ii). Indeed, the directions of the dipole momenta of all the vibrational transitions belong to the same plane defined by vectors x' and d 0, 5 . Thus, be molecules "glued" and immobilized, the proposed scheme would be unable to work due to evident viola tion of the condition (v). However, in the previous sec tion, we have seen that at the laser interaction dura tions of the picosecond time scale one have to take into account at least the fast rotations of molecules around the axis z'. It is now evident that we are beyond the scope of applicability of the "frozen" approxima tion and condition (v) and obliged to perform exact consideration of transitions between the distinct rota tional vibrational sublevels. Note that condition (ii)

LASER PHYSICS

Vol. 20

No. 1

2010


114 (a) y' d
l 0, 5

ZHDANOV, ZADKOV (b) y y'
2 3 1 2

(c)
2 1 1



5 1

(d

l 0, 5)

z'
(d
5 l 0,

l l 0, 1 d l 0, 2 1 , 2

d

x'

z

x
1 1


3

d

2

k=1 k=0

0

Fig. 3. The laser control scenario of the AAS for randomly oriented hydrogen peroxide molecules. (a) Configuration of the employed transition dipole momenta (see also Tables 1 and 2). (b) Polarizations and directions of propagation (1 and 2) of the laser pulses. The constituents 1 and 2 of the first pulse are shown by solid arrows, while the single component 3 of the second pulse along with its direction of propagation are shown by dashed arrows. (c) The laser driven chain of transitions. Signs "­" and "+" indicate whether or not particular transition results in change of the state symmetry with respect to the contorsional coordi nate.

was derived exactly for such detailed level of consider ation. It can be shown that the dipole momenta of rov ibrational transitions in the case of rigid rotor like molecules are always collinear to its principal axes of inertia [38]. Thus, even in a reduced one dimensional model of rotations the z' and y' projections of the 5 dipole moment of vibrational transition 0 (shown in Fig. 3b by grey arrows) produce the distinct "links" for the chain of transitions: in the first case there is no changes of the quantum number k occur whereas k = ±1 in the second one. Suppose that the first two links in the chain shown in Fig. 3c are excited by different projections of d 0, 5 . Then the effect of the corresponding two photon transition is that despite the molecules return back into its initial vibrational state, it still appear in the new rotational state with |k | = 1. Using Fig. 3a one can easily prove now that such transition chain satisfy both conditions (ii) and (iii). Thus, we are assured that proposed scenario satisfy all the necessary conditions of energetical separation.

0.4 /2 0 /2 0.2 0 2 3/2





Fig. 4. Dependence of the local amplitude 0 photo induced chirality on the Euler angles and for the opti mal parameters of the laser impact (see the text).

( , )



(, ) 0

' )z

0.8 0.6

To quantify the efficacy of the proposed scenario let us estimate the achievable amplitude 0 of local pho toinduced oscillations of the degree of chirality [Eq. (10)]. To do this, we will first calculate in frame of one dimensional model of rotations the peak degrees ( , ) of chirality 0 produced by the laser pulses in each local subensemble of molecules having the same ori entation of axis z'(, ) and then will perform the aver ( , ) aging over such subensembles: 0 = 0 , . It is turned out that it is possible to derive an ana ( , ) lytical expression for 0 under several extra assumptions: · suppose that laser pulses do not overlap; · relaxation and intermode interaction can be neglected; · equilibrium values of contorsional coordinate for l d all the employed states | m and | m are roughly esti mated as = 90° and 270°, correspondingly; · laser pulses frequencies are resonant to the cor responding transitions shown in Fig. 3c; · calculations can be done within the bounds of rotation wave approximation; · all the nonresonant transitions can be neglected. In frame of the last assumption we account for only the dipole transitions detuned less than 30 cm­1 from the resonance (neglecting the corresponding detun ings itself). Practically, it means that the levels with k < 2 only are to be included in calculations. This assump tion is rather correct unless the laser pulses durations are below the picosecond scale. Simple but cumbersome computation shows that ( , ) within above assumptions the value of 0 does not explicitly depend on the envelope shapes and dura tions of the pulses (see Eq. (5)), but can be expressed in terms of their integral characteristics sj = A j dt ( j = 1, 2, 3):
LASER PHYSICS Vol. 20 No. 1



2010


COHERENT CONTROL OF CHIRALITY

115

â sin s s
1

( , ) 0

=2
1

( 1, x' ) 0, 1

( 1, y' ) 0, 5

( 1, z' ) 0, 5

2

­

2
1 1

1

+

2



1

­ 1( 2

( 1, x' ) 2 0, 1 )



+ sin

+

2 2

­
2 2

2 â[ 2 â sin 1 s 2 s â cos
1 2 2 1

1

1( 2

( 1, x' ) 2 0, 1 )

+

1

(11)

­

2 ­1 1] ( 1, x' ) 0, 2

( 2, x' ) 1, 2



sin ( 2 s s

3

)
2

2

­

1

­ cos

1

1

+

2 â sin ( 1 ­ 2 ­ 3 ) .

2



In this expression we used decomposition of the dipole moment operator introduced while formulating the condition (iii), and also the following notations:
( j, ' ) m, n

= 1( 4

1

(

j '

(l) ^ (l) ) m | d ' | n , ( 1 , y' ) 2 0, 5 ) ( 1, z' ) 2 0, 5 )

1

( , ) =

( 1, x' ) 4 0, 1 )

+ 8(

(

, .
( , )

2

( , ) = 1 ( 2

( 1, x' ) 2 0, 1 )

+(

( 1 , y' ) 2 0, 5 )

+ 2(

( 1, z' ) 2 0, 5 )

The results of numerical averaging of the 0 over all possible directions of molecular axis z' for dif ferent values of s1, s2, and s3 confirm that locally aver aged amplitude 0 is, in general, also nonzero. Its peak value 0 ~ 0.23 corresponds to the following param eters: s1 = 6 â 10­4 V s/m, s2 = 1.4 â 10­2 V s/m, s3 = 4.5 â 10­4 V s/m, 1 ­ 2 ­ 3 = /2. It is interesting to take a look at the corresponding ( , ) dependence 0 shown in Fig. 4. One can see that the orientation of the axis z' affects only the amplitude ( , ) of 0 , not its sign. Thus, the asymmetric synthesis proceeds with essentially different efficacy, but in the same direction independently of orientation of the molecules. Note also that several particular orienta tions of molecular axis z' correspond to very large ( , ) amplitudes of 0 up to 0.8. Thus, the efficacy of the proposed scenario of AAS in partially oriented mole cules is comparable with the efficacy of the previously suggested scenarios of AAS specifically designed for the entirely preoriented molecules [11­24]. The simple and visual dynamical model described above is however rather questionable due to the large number of arguable suppositions. So, to verify the reli
LASER PHYSICS Vol. 20 No. 1 2010

ability of the results we complement the above calcula tions with ones made in fully numerical manner using more accurate totally quantum mechanical three dimensional description of rotations in the frame of a symmetric top model with both extended calculation basis (all the levels with J < 4 were included) and explicit account of all detunings from resonance. Note that such an account of detunings raises the depen dence of results of laser impact on shape and duration of the laser pulses and requires to be attended by their exact specification. For our calculations we choose the simplest model of rectangular nonoverlapping pulses coursed each other with zero time delays. Results of our calculations confirm that our previ ous simple model allows rather satisfactory estimates of optimal parameters sj which however are somewhat understated (by a factor of 1.35 for s1 and 1.25 for s2) over predictions given by more precise calculations. As expected, the optimal pulse durations j are found to be in the picosecond range: 1 = 1.2 ps, 2 = 1.65 ps, 3 = 1 ps (corresponding pulse intensities are 8.7 â 1010, 1.5 â 1013, and 1.9 â 1010 W/cm2, respectively). At the same time, the simple model substantially overes timates the extremal value of 0 (0.23 instead of 0.12). Nevertheless, one can conclude that the mutual agree ment between the results of two calculations is quite satisfactory. Let us conclude the discussion with several remarks on possible ways of experimental revealing of the effect of the AAS in the case of dynamical chirality. The sim plest method is possibly the detection of constituents of nonlinear response of molecular ensemble to laser impact caused by chiral symmetry breaking. The idea of the method is the following. Recall that our controlling laser field consists of two groups of pulses propagating in the mutually orthogonal direc tions 1 and 2, as shown in Fig. 3b. Now let us con sider the macroscopic molecular ensemble localized in the finite volume. It is evident that the time delay t between arrival of pulses from distinct groups (specifi cally, between the pulses 2 and 3) has to be different for the molecules located in different parts of the vol ume. This, certainly, raises the spatial dependence of the results on laser impact. For example, the depen dence of partial chirality

0

of molecules in the

2

ground vibrational state 0 (the expression for is identical) on spatial point r of their localization in the frame of assumptions of the simple model reads as ( r, t ) = 0 cos [ 2 tun t ­ ( k 1 ­ k 2 ­ k 3 ) r + ] . (12)
0 0 0







Here, kj is the wavevector of j th pulse and is a phase shift, which within the constraints of a simple model is independent on r and t. Note that described above more precise calculations revealed the slight (as com pared to (k1 ­ k2 ­ k3)r term) dependance of and,


116 00 cos ( ) 0.002 0.001 0 -0.001 t, 10 5 10
v0 0 v

ZHDANOV, ZADKOV

-12

,s

15

20

25

sin ( )

Fig. 5. The dependence of amplitude and phase of the laser induced oscillations of the averaged local degree of chirality on time delay t between the pulses 2 and 3 (this estimate was made for the initial temperature T = 300).


thus, 0 on t (see Fig. 5). Figure 5 shows that assumption of the constant is satisfactory fulfilled for the delays t up to 30 ps, i.e., in space areas with char acteristic dimensions less than ~1 cm. It follows from Eq. (12) that the spatial distribution of local degree of chirality at any time instant has the configuration of running spatial grid. Let us show that this grid like laser induced chiral asymmetry can be revealed via the effects accompanying the propagation of the tail of the last laser pulse 3 through the mixture of enantiomers which certainly are already energeti cally separated to that moment. It can be shown that such an interaction affects the component Qyz of the molecular quadrupole moment Q which begins to oscillate according to following law:
0

Q yz cos ( 3 t ­ k 3 r + 3 ) â cos ( 2 tun t ­ ( k 1 ­ k 2 ­ k 3 ) r + )
0



(13)

= cos ( ( 3 + 2 ) t ­ ( k 1 ­ k 2 ) r + ( + 3 ) ) + ... . In view of the fact that |k1 ­ k2 | |k3 |, it is easy to see from (13) that oscillations of Qxy up to small detuning 2 tun 4 are spatially phase matched to amplify the signal response wave with the frequency signal = 3 + ~ 2 0 polarized parallel to the axis y and running along the axis z. Origination of this signal wave can be treated as indication of the racemity breaking. Phe nomenologically, the appearance of the signal wave can be interpreted as a result of media symmetry ^ breaking with respect to the inversion operation E * . First, the signal wave can be treated as a result of emerging the even order susceptibilities, which lead to generation of the wave with the difference frequency
0

0 tun



~(1 ­ 2) = 3 of the pulses 1 and 2. Another explanation of the signal wave is that it is a result of scattering of the last pulse 3 by running a grid of the spatial molecular chirality distribution. This latter explanation seems to be physically more adequate. Unfortunately, application of the described scheme to the case of hydrogen peroxide molecules seems to be extremely difficult. One of the reasons is mentioned in the final part of Section 3 and raised by strong decrease of effectiveness of the proposed scenario at nonzero initial temperature of molecules. In particu lar, our estimates show that the amplitude of the local degree of chirality induced in the sample at room tem perature is expected to be less than 10­3. On the other hand, preliminary deep cooling of the molecules, e.g., in a supersonic jet, could drastically affect the number of exposed molecules. In any case, the peak intensity of the response signal will be by many orders of mag nitude lower than the intensity of the incident laser field. For example, let us consider the case of H2O2 molecules at normal conditions. From Fig. 5 one can see that the spatial dependence (5) of partial chirality distribution holds on only in the volume with the dimensions about 1 cm. However, our estimates show that even illumination of this maximal volume cannot result in the peak intensity of the response signal exceeding 10­8 W/cm2. Another problem arises from strong sensitivity of the results of the laser impact on the laser pulses parameters. For instance, 20% variation of the inten sity of the first pulse 1 leads both to half as much decrease of amplitude 0 and to shift by of the phase of oscillations of the degree of chirality. Thus, the effective generation of the response signal is possible only if the given field approximation is valid for inci dent laser field and also if the cross section intensity distribution of the pulses on the scales of the reaction volume can be neglected. Finally, it is necessary to stress that during the above consideration we neglected the effects of tunnel ion ization, which can be unwarrantable for the obtained optimal intensities and durations of the laser pulses. 6. CONCLUSIONS In conclusion, we proposed universal algorithm for construction of the schemes of selective laser excita tion of given enantiomers for arbitrary randomly ori ented chiral molecules which was further successfully applied to the problem of absolute asymmetric synthe sis on example of rotationally cold hydrogen peroxide molecules. Specifically, we considered the method of producing the local enantiomeric excess in a racemic mixture of dynamically chiral molecules via energeti cal separation of enantiomers. We showed that the sequence of three strong (with peak intensities of about ~1010­1013 W/cm2) ps laser pulses with frequen cies in the mid IR range can stimulate the oscillations
LASER PHYSICS Vol. 20 No. 1 2010

4

This detuning can be compensated via the slight declination (on the order of 0.05 rad) between directions of propagation of the pulses 1 and 2 or by replacing the pulse 1 with the pair of pulses of the same frequency with slightly distinct wavevectors directions.


COHERENT CONTROL OF CHIRALITY

117

of the degree of chirality with an amplitude of up to 0.12. These results, thus, confirm usefulness and effectiveness of the proposed algorithm as the base for developing the control schemes. Also, we proposed a method for detection of produced nonuniform distri bution of chiral content based on the registration of nonlinear response of the media. Analysis of the hydrogen peroxide molecule case shows that the correct description of the laser driven chiral dynamics needs an explicit quantum mechani cal consideration of almost all vibrations as well as the molecular rotations because the degrees of freedom of both types hold important information about symme try properties of interaction. Nevertheless, we found that all the key features of the laser driven dynamics can be satisfactory reproduced within a simple semi classical model of rotations. Moreover, in the frame of this model we have been able to rather accurately determine the optimal properties of the laser impact, although obtained twice overestimated values for the peak local degree of chirality compared to the direct quantum mechanical computations using symmetric top approximation. We also revealed that our scenario allows achieving very high enantiomeric excess 0.8 in the case of partly oriented molecules (with only one fixed axis). At the same time, it is necessary to stress the weak nesses of the proposed scenario. First, the specific sce nario for hydrogen peroxide molecules employs the laser pulses with both experimentally hardly achiev able frequencies and also extremely high amplitudes, which can raise strong parasitic tunnel ionization. However, these shortcomings are not of fundamental nature and could be eliminated by replacing the rovi brational transitions with rovibronic ones. Unfortu nately, there exist a more principal weaknesses, which is native for all the schemes of considered type. Namely, the result is qualitatively changed by rather slight (about 10­20%) variations of amplitudes of constituents of the laser impact and effectiveness of the method drastically breaks up at nonzero initial temperatures of molecules. These weaknesses turn the registration of the effect of asymmetric synthesis in H2O2 vapor to a very challenging task due to the weak ness of the response signal (below 10­8 W/cm2 at nor mal temperature and pressure of the sample). ACKNOWLEDGMENTS This paper has been prepared for the special issue of Laser Physics dedicated to the memory of a great Russian scientist, Prof. Vladilen Letokhov, with whom one of the authors (VNZ) has a privilege to collaborate on many various occasions for more than 25 years starting from the time when VNZ used to work with Prof. Letokhov's department at the Institute of Spec troscopy on a part of his PhD thesis. This specific work has been also deeply discussed at the scientific seminar within the same department and we greatly appreciate
LASER PHYSICS Vol. 20 No. 1 2010

all the comments and suggestions by Vladilen and many members of his labs. We are grateful also to Prof. Boris Grishanin, Dr. Julia Vladimirova, Prof. Oleg Sarkisov, and Prof. Nikolay Zefirov for valuable discussions of the results of this paper. REFERENCES
1. Nobel Prize in Chemistry for 2001 was awarded for the development of catalytic asymmetric synthesis, with one half jointly to William S. Knowles and Ryoji Noyori "for their work on chirally catalysed hydrogenation reactions" and the other half to K. Barry Sharpless "for his work on chirally catalysed oxidation reactions." 2. P. Compain, V. Desvergnes, C. Ollivier, et al., New J. Chem. 30, 823 (2006). 3. L. D. Barron, Chem. Soc. Rev. 15, 189 (1986). 4. M. Avalos, R. Babiano, P. Cintas, et al., Chem. Rev. 98, 2392 (1998). 5. G. L. J. A. Rikken and E. Raupach, Nature 405, 932 (2000). 6. N. P. M. Huck, W. F. Jager, B. Lange, et al., Science 273, 1686 (1996). 7. Y. Shimizu and S. Kawanishi, Chem. Commun. 1333 (1996). 8. Y. Shimizu, J. Chem. Soc., Perkin Trans. 1, 1275 (1997). 9. H. Nishino, A. Nakamura, and Y. Inoue, J. Chem. Soc., Perkin Trans. 2, 1693 (2001); A. Nakamura, H. Nishino, and Y. Inoue, J. Chem. Soc., Perkin Trans. 2, 1701 (2001); H. Nishino, A. Nakamura, H. Shitomi, et al., J. Chem. Soc., Perkin Trans. 2, 1706 (2001). 10. M. Shapiro and P. Brumer, Rep. Prog. Phys. 66, 859 (2003). 11. Y. Fujimura, L. GonzÞlez, K. Hoki, et al., Chem. Phys. Lett. 306, 1 (1999). 12. K. Hoki, Y. Ohtsuki, and Y. Fujimura, J. Chem. Phys. 114, 1575 (2001). 13. L. GonzÞlez, K. Hoki, D. KrÆner, et al., J. Chem. Phys. 113, 11134 (2000). 14. M. Shapiro, E. Frishman, and P. Brumer, Phys. Rev. Lett. 84, 1669 (2000). 15. D. Gerbasi, M. Shapiro, and P. Brumer, J. Chem. Phys. 115, 5349 (2001). 16. L. GonzÞlez, D. KrÆner, and I. R. Sola, J. Chem. Phys. 115, 2519 (2001). 17. A. S. Leal, D. KrÆner, and L. GonzÞlez, Eur. Phys. J. D 14, 185 (2001). 18. K. Hoki, L. GonzÞlez, and Y. Fujimura, J. Chem. Phys. 116, 2433 (2002). 19. K. Hoki, L. GonzÞlez, and Y. Fujimura, J. Chem. Phys. 116, 8799 (2002). 20. Y. Ohta, K. Hoki, and Y. Fujimura, J. Chem. Phys. 116, 7509 (2002). 21. E. Frishman, M. Shapiro, and P. Brumer, J. Phys. B: At. Mol. Opt. Phys. 37, 2811 (2004). 22. K. Hoki, D. KrÆner, and J. Manz, Chem. Phys. 267, 59 (2001).


118

ZHDANOV, ZADKOV 35. I. Thanopulos, P. KrÀl, and M. Shapiro, J. Chem. Phys. 119, 5105 (2003). 36. Yu. V. Vladimirova, B. A. Grishanin, D. V. Zhdanov, and V. N. Zadkov, Moscow Univ. Phys. Bull., No. 6, 37 (2005). 37. B. A. Grishanin, H. Takahashi, Yu. V. Vladimirova, D. V. Zhdanov, and V. N. Zadkov, Las. Phys. 15, 1247 (2005). 38. D. V. Zhdanov and V. N. Zadkov, J. Chem. Phys. 127, 244312 (2007). 39. D. V. Zhdanov and V. N. Zadkov, Phys. Rev. A 77, R011401 (2008). 40. D. V. Zhdanov and V. N. Zadkov, Phys. Rev. A 78, 033407 (2008). 41. P. A. L. Bachi Tom, V. I. Tyulin, and V. K. Matveev, Rus. J. Phys. Chem. 73, 1988 (1999). 42. J. Koput, S. Carter, and N. C. Handy, J. Chem. Phys. 115, 8345 (2001). 43. B. A. Grishanin and V. N. Zadkov, JETPH 89, 669 (1999). 44. S. S. Bychkov, B. A. Grishanin, V. N. Zadkov, and H. Takahashi, J. Raman Spectr. 33, 962 (2002). 45. F. Hund, Z. Phys. 43, 805 (1927).

23. D. KrÆner, M. F. Shibl, and L. GonzÞlez, Chem. Phys. Lett. 372, 242 (2003). 24. D. KrÆner and L. GonzÞlez, Chem. Phys. 298, 55 (2004). 25. D. KrÆner and L. GonzÞlez, Chem. Phys. 5, 3933 (2003). 26. K. Hoki, L. GonzÞlez, M. F. Shibl, et al., J. Phys. Chem. A 108, 6455 (2004). 27. L. GonzÞlez, J. Manz, B. Schmidt, and M. F. Shibl, Chem. Phys. 7, 4096 (2005). 28. P. Brumer, E. Frishman, and M. Shapiro, Phys. Rev. A 65, 015401 (2001). 29. S. S. Bychkov, B. A. Grishanin, and V. N. Zadkov, JETPH 116, 31 (2001). 30. S. S. Bychkov, B. A. Grishanin, and V. N. Zadkov, Las. Phys. 11, 1088 (2001). 31. P. KrÀl and M. Shapiro, Phys. Rev. Let. 87, 183002 (2001). 32. E. Frishman, M. Shapiro, D. Gerbasi, et al., J. Chem. Phys. 119, 7238 (2003). 33. D. Gerbasi, M. Shapiro, and P. Brumer, J. Chem. Phys. 124, 074315 (2006). 34. P. KrÀl, I. Thanopulos, M. Shapiro, et al., Phys. Rev. Lett. 90, 033001 (2003).

LASER PHYSICS

Vol. 20

No. 1

2010