Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://qilab.phys.msu.ru/papers/JETP-1999-89(4)-en.pdf Äàòà èçìåíåíèÿ: Mon Feb 4 18:39:20 2008 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:50:42 2012 Êîäèðîâêà:

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

VOLUME 89, NUMBER 4

OCTOBER 1999

Photoinduced chirality of hydrogen peroxide molecules
B. A. Grishanin*) and V. N. Zadkov
Institute of Physics and International Laser Center, M. V. Lomonosov Moscow State University, 119899 Moscow, Russia

Submitted 20 April 1999 ´ Zh. Eksp. Teor. Fiz. 116, 1250 ­ 1263 October 1999 The feasibility of using nonlinear optical techniques to control the chiral states of molecules is examined with the hydrogen peroxide molecule as an example. Raman excitation of optical activity owing to a transition among states with different chiral symmetries is proposed, along with an experimental scheme for detecting the corresponding photoinduced optical rotation in hydrogen peroxide vapor. © 1999 American Institute of Physics. S1063-7761 99 00910-5

1. INTRODUCTION

One of the most intriguing mysteries of nature is the chiral purity of the biological world. For example, with enviable consistency, nature chooses levorotatory helical DNA molecules and dextrorotatory sugar molecules, although the levo- and dextro-configurations are energetically equivalent.1 Research on the chiral purity of the biological world and, as a whole, on the nature of the chiral symmetry of molecules, the construction of chiral properties and their experimental study, and attempts at controlling chirality are currently among the most pressing areas of modern physics.2­4 In particular, the physical consequences of the existence of a universal mechanism for breaking of chiral symmetry5,6 owing to the failure of reflection symmetry as the result of a weak interaction through neutral currents7,8 are the subject of ongoing discussion. Besides the spontaneous breakdown of chiral symmetry, there is practical interest in research on the feasibility of deliberately controlling changes in the chiral symmetry, induced, for example, by light. If this is possible, then in optics we have at our disposal a powerful arsenal of experimental methods, both for exciting and for probing molecular systems,9 which may be used in setting up an experiment on photoinduced chirality. In this paper we study the feasibility of exciting chirally asymmetric states of the hydrogen peroxide molecule (H2O2 ) by means of a laser pulse with specially selected parameters. The hydrogen peroxide molecule Fig. 1a is the simplest chiral molecule whose geometry is not invariant under the spatial inversion ( x , y , z ) ( x , y , z ) . After this transformation, the initial position of the atoms of the molecule cannot be recovered by a rotational transition, since the torsional angles 1,2 are inequivalent. The choice of direction of the vector nO from one oxygen atom of the molecule to the other makes it possible to assign a definite sign to the torsion angle HOOH after choosing between the right- and left-handed coordinate systems. According to theoretical calculations and experiment,10­13 the equilibrium torsion angle 120° , where a positive sign in the gaseous phase is corresponds to the so-called d-state ``dextro'' or right1063-7761/99/89(10)/8/$15.00 669

handed indicated in Fig. 1a and a negative sign, to the socalled l-state ``levo'' or left-handed . Let us examine qualitatively what happens when a chiral H2O2 molecule interacts with an electromagnetic field. The existence of reflection symmetry between the minima of the torsional potential means that the eigenstates of the torsional Hamiltonian are split as a result of tunneling through the lower barrier and are described by even and odd wave functions S , A tunneling through the upper barrier and the additional splitting owing to it are negligible13 . The relatively large splitting E 0 11.4 cm 1 corresponds to a lack of stationary chirally asymmetric states, which, prior to the establishment of equilibrium, oscillate at the tunneling frequency relative to the stationary states S and A . Thus, in the H2O2 molecule, in principle it is possible only to obtain an oscillatory optical rotation effect, as opposed to heavy molecules, for which the period of these oscillations can correspond to arbitrarily long times and the initial chirally asymmetric state is stable. The Hamiltonians for the interaction of the molecule with an electromagnetic field ^ H
I

dk E rk

in the dipole ( H D ) and quadrupole ( H Q ) approximations have a qualitative difference owing to the fact that in the dipole approximation for E( rk ) E( r0 ) ] the contributions of the protons add, while in the quadrupole approximation E( rk ) ( rk r0 ) E( r0 ) they subtract, so that H D is an even function, while H Q is odd. For the corresponding off12 12 diagonal matrix elements H D and H Q of the S A transition for the dipole and quadrupole Hamiltonians, we obtain H
12 D

0,

H

12 Q

0.

1

This implies that the quadrupole interaction excites precession between states of a chiral molecule that are split owing to the d l tunneling transition between the right- and lefthanded chiral configurations, while the dipole interaction only produces a modulation in the energy of the eigenstates, which therefore oscillates at the frequency of the exciting field. The existence of the quadrupole contribution offers the
© 1999 American Institute of Physics


670

JETP 89 (4), October 1999

B. A. Grishanin and V. N. Zadkov

FIG. 1. a Geometry of an H2O2 molecule in the dconfiguration. The wave vector k of the laser field E is directed along the z axis, nO is a vector in the direction of the O ­ O bond, 1,2 are the torsion angles relative to the x nO plane, the equilibrium valence angles are H 100° , and a O 1.461 å and a H 0.964 å are the lengths of the O ­ O and O ­ H bonds, respectively. b Model function for the torsional potential V as a function of the torsion angle and the position of the lowest energy levels for the symmetric dots and antisymmetric continuous line states.

opportunity, in principle, to selectively excite the d- or lenantiomers using a fundamental quantum optical effect-- the coherent precession of a two-level system driven by a coherent electromagnetic field pulse. In this paper we discuss the conditions under which this photoinduced chirality effect optical activity can be detected experimentally. The excitation mechanism is first discussed using a single-photon model, and then a more realistic scheme of two-photon excitation using two lasers with orthogonal polarizations is proposed. As the following analysis shows, if a molecule is initially oriented along the z axis, then in principle a laser pulse of some duration can be used to bring it from an initial S- or A-state uniformly distributed across the d- and l-configurations, and which has no optical activity, into a state with a definite configuration and a sign corresponding to rotation of the plane of polarization of the incident field. This photoinduced chirality effect can be observed experimentally in its purest form in hydrogen peroxide vapor, since the interpretation of data for the liquid phase is made more complicated by the strong intermolecular interaction. Let us estimate the order of magnitude of the effect in a single-photon excitation scheme. The condition for complete orientation of the dipole moment of the molecules along the z axis under the influence of a field of strength E is Eea H kT , which corresponds to E 106 V/cm at room temperature. If we do not require 100% orientation of the molecules, then much lower fields can be used to obtain a substantially lower degree of orientation, Eea H / kT , which can nevertheless be used to establish experimentally the presence of one or the other chiral configuration of the molecules by measuring the optical activity of the medium. In the frequency range of the electronic susceptibility, it can be estimated to lowest order by assuming that describes the fraction of the molecules strictly oriented in a given direction, while the remainder of the molecules is not oriented along the field. Then, for an upper bound estimate, on multiplying by the characteristic magnitude of the this small factor 102 deg·cm3/g·dm for materials with specific rotation the most distinct optical activity, we obtain 10 1

deg·cm3/g·dm for resonant excitation of optical activity at E 103 V/cm. A 1-mm thick layer of the vapor at standard temperature and pressure yields a rotation angle of the order of 10 1 sec. This value is at the limit of sensitivity for linear polarization spectroscopic techniques. Thus, even neglecting the other complicating factors in a single-photon excitation scheme with initial orientation of the molecules by a constant electric field, observing the laser induced optical rotation will be difficult. Another important point is the finite lifetime r of the chiral state owing to intermolecular collisions. It should at least be longer than the time the field propagates in the active region, c L / c 10 10 s 1 , and a lower bound estimate is 1 , where N is the concentration of the molr (N v ) ecules, v is the thermal speed, and is the collision cross section. For atmospheric pressure and room temperature we obtain r 10 9 s 1 c . The biharmonic excitation scheme using two laser pulses proposed here satisfies this restriction, and furthermore can be used to efficiently induce a given chiral state in the excited molecules.
2. THEORETICAL MODEL FOR PHOTOINDUCED CHIRALITY OF H2 O2 MOLECULES 2.1. Reduced model for the dynamics of a free molecule

For an approximate description of the dynamics of molecules in a laser radiation field, it is appropriate to simplify the complete Hamiltonian by taking advantage of the small ratio of the masses of the protons and oxygen atoms. Then the dynamics of the protons can be treated in the adiabatic approximation with respect to the coordinates of the oxygen atom, and the dynamics of the latter reduces, in the simplest case, to averaging over the direction of the unit vector nO ( sin cos , sin sin , cos ), which specifies the orientation of the O ­ O-bond. Here the average over the z-direction Fig. 1a can be taken considering only the rotational dynamics of the protons relative to the O ­ O bond, given that the potential of the proton bond depends only on the torsion angle 2 1.


JETP 89 (4), October 1999

B. A. Grishanin and V. N. Zadkov

671

Given these considerations, the characteristic Hamiltonian in the reduced model, which accounts for the rotation of a molecule relative to the O ­ O bond as a whole and the torsional vibrations, has the form ^^ HH where
2 2 2 H H

^ H,

2

along the z axis, i.e., 0, and the field E1 is oriented along the y axis, the rotational symmetry reduces to a uniform distribution over the angle ~ ; the quadrupole moment has a dependence of the form sin ~ . Here the orientation shows up in Eq. 5 , which takes the form ~ E 1d 2J
2 1

u 1 t cos ~ cos

2

cos

1

t

1

,

6

^ H

H

4 m Hr
2

~
2

2

,

3

^ H

m Hr

2 H

2

V

,

4

with cos ~ having the same angular dependence as the derivative of the quadrupole Hamiltonian for the field E2 x , so that ^ H
Q

where ~ ( 1 2 ) /2, m H is the proton mass, and V ( ) is the torsional potential. The vibrations of the valence angles HOO are neglected here for simplicity. Besides the uncertainty in the direction of nO mentioned above, the initial state of the molecule with respect to the average angle ~ includes an uncertainty associated with the rotation of the molecule relative to that direction, which corresponds to the third rotational degree of freedom of the molecule and supplements the two angles that specify the direction of nO . Here the corresponding frequencies of the transitions between the levels for free rotation about the O ­ O axis are
nn 1

^ HQ / ~

~ cos2 ~ .

An order of magnitude estimate in the visible range is ~ 10
10

I 1,

7

7.84· n 1/2 cm

1

,

and are a factor of four less than the corresponding frequencies of free internal rotation in view of Eqs. 3 and 4 , which largely determines the form of the torsional potential U ( ).
2.2. Photoexcitation of rotational degrees of freedom of the molecule

In a field Et E1 u 1 t cos
1

t

1

with envelope u 1 ( t ) and frequency 1 n n 1 , the quantum features of the resulting excitation are unimportant and the response is described by the classical equation J d 2~ dt
2

where I 1 is the intensity of the field E1 in W/cm2 and the angle is in radians. This estimate reaches order unity only in fields stronger than the intra-atomic fields. The dynamics of the torsional vibrations corresponding ^ to the Hamiltonian H are largely quantum mechanical owing to the relatively small mass of the proton compared to the heavier atoms. This shows up in the tunneling between the dand l-states of the local potential minima Fig. 1b , which leads to the formation of superposition eigenstates S and A that are split in energy and have equally represented d- and l-configurations, as well as to a nonrigidity of the molecular configuration owing to the quantum mechanical indeterminacy of the wave functions with respect to the torsion angle. The form of the torsional potential of the H2 O2 molecule and the eigenenergies of the torsional potential have been studied both by ab initio computational techniques and by analyzing experimental spectroscopic data. Figure 1b shows the potential function and structure of the eigenlevels.14 An estimate of the uncertainty in the local states with respect to the torsion angle using the formula for position fluctuations in the ground state of a harmonic oscillator yields / m Hr
2 H 0 1/2

20° .

~

E t d,

^ The Hamiltonian H I for interaction with a laser field in the dipole approximation, allowing only for the displacement of the proton charges, is ^ H
D

where J is the moment of inertia of the molecule. To zeroth order in the deviation, in the expression for the force on the right-hand side of the equation and for pulses that are not too short, with durations 1 1/ 1 , the response in the form of the change in the angle ~ at frequency 1 is given by ~ E1 t d ~ J
2 1

E L ea H Re e e1 e2 ,

8

.

5

This change leads to a modulation at frequency 1 in the quadrupole Hamiltonian acting on the internal rotation, and therefore to partial orientation of the molecule, which shows up as a loss of symmetry in the quadrupole Hamiltonian. In particular, when the O ­ O axis of a molecule is oriented

where a H is the H ­ O distance, e1,2 are the corresponding unit vectors for the directions of the proton bonds, e is the polarization vector of the laser field, and e is the proton charge. The quadrupole component of the interaction Hamiltonian depends on the choice of the coordinate center. The displacement of the center leads to an additional dipole term which, however, is small and can be discarded, since it describes the same qualitative features of the interaction as the main dipole contribution. Thus, the coordinate center must be chosen as in Fig. 1 in order to obtain the simplest form of the Hamiltonian. The corresponding expression is


672

JETP 89 (4), October 1999

B. A. Grishanin and V. N. Zadkov

^ H

Q

EL d

Q

knOa O E L ea H Re i e e2 e1 , 2

9

where a O is the O ­ O distance. This choice of the coordinate center on the O ­ O line for light propagating along that line i.e., k nO) means that the dependence of the quadrupole Hamiltonian H Q on the torsion angle is determined solely by the odd function sin( /2) . To calculate Eqs. 8 and 9 , one must represent the polarization vector of the field in the form e ( e x , e y ,0) , where the coefficients e x and e y are, in general, complex numbers, and calculate the corresponding coefficients, C
Hk

of the hydrogen bonds serve as coordinate operators. The projection coefficients of the proton dipole moments in the x or y directions of the laser polarization vector e are calculated according to the formulas C
Hk

nx O nOOH ,
1

H

O nO ,

k

O nn

x nO

, /2

nx nO nx , S
Hk

ny O nOOH ,
2

H

O nO ,

k

On

nx nO

, /2

nx nO nx as the scalar product of the vectors nx , y and the vectors obtained a by rotating the vector nx initially by an angle /2 nx nO in the plane of nx nO until reaching a perpendicular to the nO axis, which is used as a basis axis for reading the torsion angles, b by rotating nx around the O--O axis by the torsion angle k , and c by a subsequent rotation of nx in the OOHk plane by an angle H H /2 until the unit vector along nHk is obtained. Given the nonzero value of H , the analytic expressions for these coefficients end up being very cumbersome, so here we give them only for the approximation H 0, i.e., for directions of the hydrogen bonds orthogonal to the O ­ O axis.1 The resulting relative error is less than 10% because H 10° is small. The three-dimensional rotation matrix for a rotation about the n axis is

nx n

Hk

,

S

Hk

ny nHk ,

where nx and ny are the unit vectors along the x and y axes and nHk are the unit vectors along the O ­ Hk bonds ( k 1,2) . In this notation, the Hamiltonians 8 and 9 take the form ^ H
D

E L ea H Re e ey S
H1

iL

ex C

H1

C

H2

S

H2

,
iL

^ H

Q

k La O E L ea H Re ie 2 ey S
H2

ex C

H2

C

H1

S

H1

,

10

where k L is the modulus of the wave vector of the laser field, L is the phase of the laser field, and the rotation angles 1,2

n O n,

2 x y z

n 2 cos y n x n y cos n x n z cos

2 n z cos

n xn n

n xn n xn

n z sin n y sin

y 2 y z

n x n y cos n cos n y n z cos
2 x 2 z

n z sin n cos n x sin

n xn n yn n

z

n x n z cos n y n z cos n cos
2 x 2 y

n y sin n x sin n cos .

n yn

z 2 z

Hence, the coefficients can finally be written in the form C C S
H1 H2

C

H1

C

H2

C cos /2 , cos2

S

H1

S

H2

S cos /2 , 12

1 sin

2

cos

2

cos cos sin2
1

1 2

, ,
1

1 sin2 16 cos2 sin 2 2 8 sin
1

cos2 sin2 2

C 2 sin 2
1

2 1 sin2 2 16 cos2 2 sin 2 2

cos ~ , sin2 ~ 8 sin ~ 2
1

S
1

2 sin2 ~ sin 2

2 sin 2 sin 2 sin 2

~ ~ ~

H1

8 sin
1

sin 2 2 ,
2 1

2 sin 2
1

sin 2 2
1 1

8 sin ~ sin 2 ~ 2

sin 2 sin2
2

2

,

S

H2

16 cos2 sin 2 2 8 sin
2

sin2 2

2 sin 2
2

8 sin
2

sin 2 2 .
2

2 sin 2
2

sin 2 2
2

sin 2

11

which depend on the torsional angle via the even function cos( /2) . Here ~ ( 1 2 ) /2 is the average rotation angle of the hydrogen bonds. For the differences of the coefficients determining the quadrupole potential, we obtain an expression that depends on the torsion angle in terms of the odd function sin( /2) : C
H2

For the sum of the coefficients that determine the dipole potential we obtain

C

H1

C sin

2

,

S

H2

S

H1

S sin

2

,


JETP 89 (4), October 1999

B. A. Grishanin and V. N. Zadkov

673

C S

2 1 sin2 16 cos2 2 cos 2 2 2

cos2 2 sin2 ~

sin ~ , sin2 ~
1

2 cos 2 cos 2 cos 2

~ ~ ~

where V 12 V 21 because the eigenfunctions k are real. After averaging over the oscillations in the field and the atomic polarization of the components at frequencies 2 L in the rotating wave approximation,15 we obtain 0 ^ H
L

cos 2

2

8 cos ~ cos 2 ~ 2

8 cos ~ .

1 QS AS e 2
i~ L

i~ L

13

For a simplified analysis of the interaction of the rotational motion of a molecule about the O ­ O axis and the torsional vibrations, the numerical parameters of the torsional potential U ( ) are important. The corresponding values of the frequency splitting of the levels,14 E 0 11.44 cm 1 , E 1 116.34 cm 1 , and E 2 206.57 cm 1 , exceed the characteristic free rotation frequencies of the molecule as a whole for n 1 ; this means that it is possible to study transitions between the eigenstates of the torsional Hamil^ tonian H directly in terms of the classical rotational coordi~. nate
3. PHOTOEXCITATION DYNAMICS IN A TWO-LEVEL MODEL

1 QS AS e 2

, 0

14

where, with Eqs. 10 , 12 , and 13 , we obtain Q k L a OE L ea
H

1 sin2

cos2

sin ~ ,

15

where ~ L is the initial phase of the laser field, which also includes a phase contribution, determined by Eq. 10 , to the polarization of the field and the orientation of the molecule, and S
AS A

sin

2

S

d

16

If it is assumed that the frequencies of the transition ^ between the eigenstates of the torsional Hamiltonian H greatly exceed the rotation frequency of the proton bonds, then laser excitation is possible for a negligibly small deviation of the orientation angles and of the molecule and a small deviation of the rotational angle ~ . Here it is possible to excite the symmetric/antisymmetric states S and A into a coherent superposition CA A CS S . For S A -transitions with high enough transition energy E n , which is nonzero because of tunneling between the d- and l-states, it can be estimated as E exp 2 m Ha
2 H

is a dimensionless matrix element describing the tunneling effect. As a result, the Hamiltonian 14 is described by the matrix 2 Q Se 2 AS
i~ L

^ H

Q Se 2 AS 2

i~ L

,

17

where L 12 is the detuning of the laser field. This operator can be expressed in terms of the Pauli matrices: ^ H 2 ^, , QS
AS

cos ~ L , sin ~

L

.

18

V

/

,

The temporal evolution operator corresponding to the operator 17 , Ut T exp i/ ^ H dt
L

where V and are the characteristic height and width of the potential barrier. When the excitation frequency is choE n / , only sen to be the resonant transition frequency 0 the resonant matrix elements will be significant in the Hamiltonians 10 and the torsional dynamics of the molecule can be examined in a two-level approximation. Given the form 10 for the Hamiltonian, we have ^ H
D

can be calculated analytically for a a rectangular pulse E const, or b zero offset 0, using the equations a) Ut cos 2 t ^ i sin I ^ 1 cos ~ ^ i sin I
2

cos ,

^ H

Q

sin .

2
L

t

^ ^ 2 sin ~

3

We show the form of the 2 2 matrix for the single-photon ^ ^ interaction Hamiltonian H L for the total Hamiltonian H I ^ ^ H D H Q for the case of linear polarization e x 1, e y 0. Here there is no advantage in using circular polarization in the quadrupole approximation, because the polarization does not appear in the dependence of the quadrupole moment on the coordinate for this transition. Given the representation of the interaction Hamiltonian in terms of the Pauli matrices for this transition, we obtain ^ H I V 12 ^ V
12

QS

AS

L

, ^ 2 sin ~

19a

b)

Ut

cos
2 0

2

2

^ 1 cos ~

L

L

, 19b

V 21 ^
L

V
L

12

t^

1

t
L

where is the total Rabi frequency, 0 QS AS is the Rabi frequency, and ( t ) dt is the angle of the laser pulse. 0, in accordance with Eq. 19b , the initial incoFor herent states, represented in the form ^
0

0 cos

t~

cos

t^

1

sin

L

t^

2

,

^ /2 w ^ 3 /2 I

1 w 1,


674

JETP 89 (4), October 1999

B. A. Grishanin and V. N. Zadkov

transform into the states ^
t

1 ^ w ^ 3 cos I 2 ^ 2 sin cos
L

^ 1 sin t~
L

sin ,

L

t~

L

20

where, besides the transformation 19 , we have included free precession at the laser frequency, which is applicable in the standard form for the theory of the interaction16 and the rotating wave approximation to the operators for the physical quantities. The components with ^ 1 and ^ 2 describe the contribution of the coherent superposition of the states S and A . In particular, for the lower initial state, which corre/2, a /2-pulse, for which sponds to w 1 for L t ~ L /2 and cos 1 in Eq. 20 and /2 /4 and cos( /2) 1/ 2 in Eq. 19b , produces a jump from the initial S state into the chiral states 1,2 ( S A )/ 2, ^ ^ 1 ) /2. Thus, which correspond to the density matrices ( I for fixed angles , ,~ it is possible to bring a molecule into a d- or l-state with 100% probability by an appropriate choice of parameters. In general, however, the resulting transformation of the state must be averaged over the angles . This averaging can be accomplished using a standard superoperator technique.17 For the result 20 of the excitation of an incoherent initial state, upon averaging only over the sign of the parameter Q, which depends on the orientation angle ~ of the hydrogen bonds along the x axis, which enters into the factor sin in Eq. 20 , we obtain the following structure for the density matrix: ^
t

describe, respectively, left- and right-handed chiral states, 1 for ^ which correspond to degrees of chirality ^ l,d l , d . For a state ( t ) excited by a rectangular laser pulse of duration p with phase p , the corresponding dependence of the degree of chirality of the excited state on the detuning and phases , ~ L can be calculated analytically. For zero frequency offset, the analytic dependence of the degree of chirality on the angle and phase of the laser pulse has the form sin sin ~ L .

The qualitative dependence for nonzero offset is shown in Fig. 2a. The important point is the dependence of the effect on the phase of the laser field.
4. EXPERIMENTAL SCHEME FOR OBSERVING PHOTOINDUCED CHIRALITY IN H2O2

^ w ^ 3 cos I

/2.

This means that when there is no preferred orientation of the molecules in ~ , the density matrix transforms incoherently, i.e., diagonal density matrices transform into diagonal matrices. As a result of this transformation, the symmetry of the state, S or A, does not change, but the squared modulus of the wave function preserves reflection symmetry under the transformation . Therefore, in order for reflection symmetry to be disrupted, the molecules must be oriented beforehand.
3.1. Excitation of initially oriented molecules

If there is an inhomogeneity in the distribution of the angles ~ owing to the existence of a preferred orientation of the molecules, then the excitation of incoherent states can contain a coherent component associated with the excitation of chiral states that differ from A and S . The scalar characteristic of the degree of chirality is the average 2 where
l,d l

Given the features of the hydrogen peroxide molecule analyzed above, we propose the following experiment for observing photoexcited optical activity as an indicator of the breakdown of reflection symmetry. As a working two-level system, the most appropriate is the S A -transition with n 1, which corresponds to a wavelength 86 m and frequency 12 116.34 cm 1 , which is substantially greater than the corresponding frequency for n 0 11.44 cm 1 ) . To create a population in this transition it is easiest to use Raman excitation by two lasers tuned to the frequency of the S S -transition n 0 n 1. Since only dipole-active transitions are used here, we can obtain an essentially complete saturation regime in the active volume of the medium, so that in this stage there are no important limitations and we can start with n S 1 for estimating the initial population on this transition. In view of the oscillations in the degree of chirality excited in this scheme, it seems appropriate to use a laser with frequency 1 for probing and another frequency 2 , which differs by the frequency 1 2 12 of the precession in the degree of chirality, for detection. This can be done efficiently using two crossed polarizers. Detection of nonzero polarization of the output field perpendicular to the probe field corresponds to detection of the rotation effect. Here the probe field can be simultaneously employed to create an effect analogous to the orientation by an electrostatic field described in Sec. 1. According to Eq. 6 , it leads to an angular displacement by this frequency. Therefore, the quadrupole Hamiltonian corresponding to excitation by the other laser field at frequency 2 and with a polarization direction along the x axis receives a correction at the resonant frequency 12 , Q E 1E 2e 2a 8J
2 1 2 H

^

l

1 2

2

d

^

d

1 , 2

21

k La

O

1 sin2

cos2 22

cos2 ~ u 1 t u 2 t . 1 2
S A

Here an additional dependence on the torsional angle of the form cos( /2) should be included in the matrix element 16 ,


JETP 89 (4), October 1999

B. A. Grishanin and V. N. Zadkov

675

FIG. 2. a Degree of chirality of the excited state as a function of the relative detuning / 0 and phase ~ L of the laser field. b Layout of an experiment with twofrequency excitation. Radiation from one of the lasers is also used as the probe radiation.

which reduces to the substitution sin( /2) ( sin )/2; this matrix element is also nonzero, while the additional time dependence of the form cos( 1t 1) reduces to replacing the ~ L 1 laser frequency, L 1 2 , and the phase, . Equation 22 yields the order of magnitude estimate 2 Q
p

frequency 2 not to interfere with detection of the effect at the same frequency, the beams must be slightly skewed so that this does not have a significant effect on the field distribution in the active region.
5. CONCLUSION

/

10

4

I 1I

2p

,

where the pulse duration p is in seconds and the intensities I 1 and I 2 are in W/cm2 . Thus, in order to obtain the effect of a /2-pulse for pulse lengths 1 ns, the average geometric intensities of the pulses employed must be of order I 0 104 / p 1013 W/cm2 . This quantity only yields an upper bound on the laser power to be used, which may actually be bounded by the substantially lower intensities corresponding to small Q p / . The minimum measured rotation angle min , in conjunc/ L , detertion with the expected angular rotation mines the length L of the active region. For , we can pro2 k L a O( 1 )