Three-Dimensional Theory of the Magneto-Optical Trap

Transcription

1 ISSN , Journal of Experimental and Theoretical Physics, 015, Vol. 10, No. 4, pp Pleiades Publishing, Inc., 015. Original Russian Text O.N. Prudnikov, A.V. Taichenachev, V.I. Yudin, 015, published in Zhurnal Eksperimental noi i Teoreticheskoi Fiziki, 015, Vol. 147, No. 4, pp Three-Dimensional Theory of the Magneto-Optical Trap O. N. Prudnikov a, *, A. V. Taichenachev a,b, and V. I. Yudin a,b,c,d a Novosibirsk State University, Novosibirsk, Russia b Institute of Laser Physics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia c Novosibirsk State Technical University, Novosibirsk, Russia d Russian Quantum Center, Skolkovo, Moscow oblast, Russia * llf@laser.nsc.ru Received July 9, 014 Abstract The kinetics of atoms in a three-dimensional magneto-optical trap (MOT) is considered. A threedimensional MOT model has been constructed for an atom with the optical transition J g = 0 J e = 1 (J g, e is the total angular momentum in the ground and excited states) in the semiclassical approximation by taking into account the influence of the relative phases of light fields on the kinetics of atoms. We show that the influence of the relative phases can be neglected only in the limit of low light field intensities. Generally, the choice of relative phases can have a strong influence on the kinetics of atoms in a MOT. DOI: /S ATOMS, MOLECULES, OPTICS 1. INTRODUCTION The laser cooling of atoms has been a rapidly developing direction at the junction of laser and atomic physics since the mid-1980s. At present, laser-cooled atoms are widely used in precision spectroscopy and quantum frequency standards, to achieve Bose Einstein condensation, to model quantum effects in condensed media, to study interatomic collisions, and in other studies. The main source of cold atoms is a magneto-optical trap (MOT) that allows a comparatively large number of atoms to be cooled to ultralow temperatures (μk mk). A large number of experimental and theoretical studies of MOTs of various types have been performed over the past time [1 6. As a rule, the analytical MOT models are one-dimensional [7 11. This is because the theoretical description of the kinetics of atoms in a three-dimensional (3D) MOT model encounters considerable difficulties associated with the complex spatial structure of the light field produced by three pairs of interfering laser beams. As far as we know, there exist only two papers devoted to the construction of a 3D analytical MOT model in the semiclassical approximation [1, 13. The authors of [1 restrict themselves to an approximation linear in field intensity. In this approximation, the main MOT characteristics averaged over the spatial period coincide with the predictions of the 1D model. The authors of [13 found these characteristics in the form of a series in field intensity (up to the fourth order). They detected certain deviations from the predictions of the 1D model. It should be noted that the approach in [13 is limited. First, the series in field intensity converge only at low transition saturation. Second, a field configuration with fixed (zero) relative phases of waves propagating in orthogonal directions was used in [13 when the calculations were performed, while in experiments these phases generally change in an uncontrolled fashion. The spatial field configuration (local amplitudes, phases, and polarizations) depends on the relative phases, which can be important when the MOT characteristics are calculated. In this paper, we construct a 3D MOT model free from the above limitations in the semiclassical approximation for the atomic transition J g = 0 J e = 1 (J g, e is the total angular momentum in the ground and excited states). We derive analytical expressions for the main MOT characteristics valid for any transition saturation and an arbitrary field configuration. The averaging over the spatial period is performed for arbitrary relative phases. In doing so, we have detected a significant dependence of the averaged MOT characteristics on these phases. The results obtained in this paper may turn out to be important for the optimization of MOT parameters for the Mg, Ca, Sr, and Yb atoms that are used in the development and investigation of new-generation optical frequency standards [ FORMULATION OF THE PROBLEM Consider the motion of atoms with the total angular momenta J g = 0 and J e = 1 in the ground and 587

2 588 PRUDNIKOV et al. excited states in a monochromatic field with an arbitrary spatial configuration, E( r, t) = E( r)e iωt + c.c., (1) where E(r) is the complex vector field amplitude. In addition, we will assume that a static nonuniform magnetic field with a quadrupole configuration, 1 B( r) = β e z z -( e x x + e y y), () which together with the field (1) forms the MOT potential, is imposed on the system. Here, the coefficient β has the meaning of the magnetic field gradient along the z axis at the trap center. In the semiclassical approximation, where the photon momentum is much lower than the atomic momentum dispersion ( k Δp the kinetics of atoms is described by a Fokker Planck equation: p ( r, p) = -----F t M i ( r, p) p i + ij ( r, p) p i p j (3) where M is the atomic mass, (r, p) is the Wigner distribution function of atoms in phase space, and F i (r, p) and (r, p) are the Cartesian components of the force and the diffusion tensor (in momentum space) at point (r, p). To describe the distribution of cold atoms near the trap center (r = 0 it will suffice to restrict oneself to the linear terms in the expansion of the force in terms of the magnetic field and atomic velocity (v = p/m): F i ( r, p) F 0 r (4) where F (0) (r) is the light pressure force acting on a stationary atom in zero magnetic field, F (1) (r) is the linear (in magnetic field) magneto-optical force acting on i ( r, p F ( r) ξ ij ( r)v j, + + i the stationary atom, and ξ ij (r) is the tensor that defines the linear (in atomic velocity) correction to the light pressure force in zero magnetic field. The symmetric ( sym) part of the tensor ξ ij = (ξ ij + ξ ji )/ is the dissipation tensor specifying the linear (in velocity) friction. As a rule, for the diffusion tensor it will suffice to restrict oneself to the zeroth-order contributions in both magnetic field and velocity: ( r, p) ( r). 3. KINETIC COEFFICIENTS IN A FIELD WITH AN ARBITRARY CONFIGURATION (5) Using the methods from [18, 19, we can derive explicit analytical expressions for the vectors F (0) (r) and F (1) (r) and the tensors ξ ij and in a field with an arbitrary configuration (1) and at arbitrary transition saturation. The light pressure force is linear in field gradient (1): ( 0) F j = γ γ /4 + δ + Ω U (6) where U(r) = E(r)/E 0 is the dimensionless vector field amplitude normalized to the coordinate-independent scalar amplitude E 0. For example, the amplitude of one of the waves producing the field (1) or the root-mean-square field amplitude can be chosen as E 0. The following notation is used in Eq. (6): δ = ω ω 0 is the detuning of the field frequency from the transition frequency ω 0, γ is the spontaneous decay rate of the excited state, and Ω = de 0 / is the Rabi frequency corresponding to the scalar amplitude E 0 (d is the reduced matrix element of the dipole moment operator). The magneto-optical force F (1) (r) is linear in both field gradient (1) and magnetic field (): Ω δ γ - j U + i - [ U* j U U j U*, F j ( Ω U γ iδγ)ω = μ 0 g ( ( γ /4 + δ + Ω U )[ δγ + i( γ + Ω U j U* [ B U ) μ 0 g ) ( γ + 8Ω U 6iδγ)Ω ( U* [ B U )( ( γ /4 + δ + Ω U ) j U* U) + c.c. [ δγ + i( γ + Ω U ) (7) Here, μ 0 is the Bohr magneton and g is the Lande factor of the excited state. The dissipation tensor can be represented as JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 10 No

3 THREE-DIMENSIONAL THEORY OF THE MAGNETO-OPTICAL TRAP δγ 3 Ω 1 = G ( γ /4 + δ + Ω U )[ 4δ γ + ( γ + Ω U ) ij ξ ij sym δγ( γ Ω U )Ω G ( γ /4 + δ + Ω U ) [ 4δ γ + ( γ + Ω U ) ij δ ( 5γ6 + 31γ 4 Ω U + 40γ Ω 4 U 4 + 3Ω 6 U 6 + 0δ γ 4 + 8δ γ Ω U )Ω G γ( γ /4 + δ + Ω U ) 3 [ 4δ γ + ( γ + Ω U ) ij + ( γ / + Ω U )Ω 4 ( 4) ( 4δ γ γ 4 + 4Ω 4 U 4 )Ω G ( γ /4 + δ + Ω U ) 3 ij A ( γ /4 + δ + Ω U )[ 4δ γ + ( γ + Ω U ) ij + ( 96Ω6 U 6 + ( 64δ + 48γ )Ω 4 U 4 + ( 104δ 6γ )Ω U + 48δ 4 γ 3γ 6 )Ω A 8( γ /4 + δ + Ω U ) 3 [ 4δ γ + ( γ + Ω U ) ij, (8) ( n) where are real symmetric tensors quadratic in field gradients (1): = i U* j U + j U* i U, = ( U i U* U* i U) ( U j U* U* j U ( 3) = ( U i U* + U* i U) ( U j U* + U* j U ( 4) = i( U i U* U* i U) ( U j U* + U* j U) + i( U j U* U* j U) ( U i U* + U* i U (9) the local values of the field (1) and does not depend on its gradients: γ( k) Ω = γ /4 + δ + Ω U sp δ ij U U ( i U j * + U j U i *), (11) where δ ij is the Kronecker symbol. The second term, the induced diffusion tensor, results from the z ( n) A ij and are real antisymmetric tensors quadratic in field gradients: A ij = i( i U* j U j U* i U σ σ σ A ij = i( U* i U U i U* ) ( U j U* + U* j U) i( U i U* + U* i U) ( U* j U U j U* ). (10) x σ + B σ + y The diffusion tensor contains two components: σ + ( r) = ( r) + ( r). The first term attributable to the fluctuations of the spontaneous photon escape direction depends on Fig. 1. Spatial configuration of the light fields and the MOT magnetic field. JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 10 No

4 590 PRUDNIKOV et al. fluctuations of the light pressure force. The tensor ( sym) ξ ij, just as, is quadratic in field gradients: = γ G Ω ij γ + 4δ + 8Ω U 4( 3γ 4δ )Ω G ( γ + 4δ + 8Ω U ) 3 ij (1) Formulas (6) (1) generalize the analytical expressions for the kinetic coefficients derived in various limiting cases. For example, in the lowest approximation in field intensity (1 i.e., restricting ourselves to the terms of ~Ω, from (6) (1) we will obtain the results of [1. For simple 1D field configurations (1 our formulas reproduce the well-known results of 1D models (which are also presented in [13). 4. STANDARD MOT CONFIGURATION Three pairs of counterpropagating, mutually orthogonal laser beams are commonly used to cool and capture atoms in MOTs. The counterpropagating beams have orthogonal circular polarizations (the socalled σ + σ configuration). The laser beams near the trap center can be approximated by plane waves. Given the phases of the interfering waves, the corresponding field configuration can then be represented as ( z U( r) e ) +1 e ikz ( z = ( + e ) 1 e ikz ) (13) ( x + e ) 1 e ikx ( x ( + e ) +1 e ikx )e iφ x ( y e ) 1 e iky ( y + ( + e ) +1 e iky )e iφ y, where are the cyclic unit vectors corresponding to the right- and left-hand circular polarizations with respect to the j = x, y, z axis (Fig. 1). Since the choice of zero time and coordinate origin is arbitrary, two phases (in our case, Φ x and Φ y ) are sufficient for an unambiguous parametrization of the field configuration (13). The field (13) is normalized in such a way that E 0 corresponds to the amplitude of one wave. Note that the choice of wave polarization in (13) is reconciled with the quadrupole magnetic field () so as to ensure a stable capture of atoms in the MOT in all directions. For the transition J g = 0 J e = 1, the temperature of the atoms in the MOT is restricted by the Doppler limit, while the depth of the potential wells is insufficient for strong localization of atoms. Therefore, to calculate the MOT parameters, it will suffice to know the kinetic coefficients averaged over the spatial period of the light wave near the trap center. Since the magnetic field B(r) on size scales of + 4 [ 1δ γ + ( γ + Ω U ) Ω G γ ( γ + 4δ + 8Ω U ) 3 ij () j e ± 1 + 3δ ( γ + Ω U )Ω G γγ ( + 4δ + 8Ω U ) 3 ij. the order of the wavelength changes only slightly, this change may be neglected when the averaged MOT characteristics are calculated. The light pressure force F (0) (r) generally contains both potential and vortex components. Both these components for the configuration (13) become zero after averaging at any phases Φ x and Φ y : F ( 0) = 0. The averaged Cartesian components of the magnetooptical force (7) are proportional to the deflection from the trap center in the corresponding direction: F j = χ j r j, (14) where χ j are the string coefficients, which are generally not equal to one another. The dissipation and diffusion tensors become diagonal after averaging: ( sym) ξ ij = α j δ ij, = d j δ ij, (15) where α j and d j are, respectively, the friction and diffusion coefficients along the jth coordinate axis. The kinetic coefficients χ j, α j, and d j completely define the stationary distribution of atoms in coordinates and momenta (the stationary solution of Eq. (3) which is Gaussian: ( r, p) = () j ( r j, p j (16) () j p ( r j, p j ) C j χ j j r = exp exp j, Mk B T j k B T j where C j are the normalization coefficients and T j = d j /k B α j is the effective temperature. All coefficients χ j, α j, and d j for the configuration (13) depend on the phases Φ x and Φ y. This can be easily verified by representing these coefficients as power series of the saturation parameter S = Ω (γ /4 + δ ). The first term in the expansion of the coefficient χ z in powers of S coincides with the results of the 1D model: δγ χ x = kμ 0 gβ S + cos( Φ x Φ y ) + cos( Φ x ) γ [ sin( Φ x Φ y ) + sin( Φ x ) S δ + χ y kμ 0 gβ δγ = S + cos( Φ x Φ y ) + cos( Φ y ) γ [ sin( Φ x Φ y ) sin( Φ y ) S δ + (17) JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 10 No

5 THREE-DIMENSIONAL THEORY OF THE MAGNETO-OPTICAL TRAP 591 χ z = kμ 0 gβ δγ S + cos( Φ x ) + cos( Φ y ) γ [ sin( Φ y ) + sin( Φ x ) S δ + In the friction coefficient α x = k δγ ). S 1 + cos( Φ x ) + cos( Φ x Φ y ) γ [ sin( Φ x ) + sin( Φ x Φ y ) S δ + α y = k δγ S 1 + cos( Φ x ) + cos( Φ x Φ y ) γ [ sin( Φ y ) sin( Φ x Φ y ) S δ + α z = k δγ S 1 + cos( Φ x ) + cos( Φ y ) γ [ sin( Φ x ) + sin( Φ y ) S δ + ) (18) the dependence on the relative field phases Φ x and Φ y also appears in the terms of the expansion in powers of S higher than the first order. Note that since there is a direct dependence between the friction and string coefficients [11, these coefficients have a similar dependence on the field phases. The spontaneous and induced diffusion coefficients averaged over the spatial period are, respectively, d x k S = γs1 ----[ 35+ cos( Φ x Φ y ) 5 and d y k S = γs1 ----[ 35+ cos( Φ x Φ y ) 5 -+ cos( Φ x ) + cos( Φ y ) + d z d x d y k S = γs1 ----[ 35+ cos( Φ x Φ y ) 5 -+ cos( Φ x ) + cos( Φ y ) + = k γs1 + γ / δ ( cos( Φ γ /4 + δ x Φ y ) -+ cos( Φ x ) 14 S + k = γs 1 + d z γ / δ cos( Φ γ /4 + δ x Φ y ) -+ cos( Φ y ) 14 S + k = γs 1 + γ / δ cos( γ /4 + δ Φ ) x -+ cos( Φ y ) 14 S + ). And the phase-averaged expressions are χ x χ χ z y --- kμ 0 gβ δγs = = = S δ S + O( S 4 4 α x = α y = α z = k δγS 1 1S 60δ S + O( S 4 γ γ (19) (0) (1) d x = dy = dz = γ k [ S 14S + 16S 3 + OS ( 4 d x d y d z = = = γ k [ S 14S + 46S 3 + O( S 4 ). -+ cos( Φ x ) + cos( Φ y ) + O S 3 ( JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 10 No

6 59 PRUDNIKOV et al. χ z / kμ 0 gβ S Fig.. String coefficient χ z versus saturation parameter S at various relative field phases: Φ x = 0, Φ y = 0 (dashed line); Φ x = π/, Φ y = 0 (dash dotted line). The thin solid line indicates the dependence for the phase-averaged string coefficient. The thick solid line corresponds to the results for the 1D σ + σ light field configuration. The detuning is δ = γ/. ξ zz / k S Fig. 3. Friction coefficient averaged over the spatial period ξ zz versus saturation parameter S at various relative field phases: Φ x = 0, Φ y = 0 (dashed line); Φ x = π/, Φ y = 0 (dash dotted line). The thin solid line indicates the dependence for the phase-averaged friction coefficient. The thick solid line corresponds to the results for the 1D σ + σ light field configuration. The detuning is δ = γ/. In particular, at zero phases, Φ x = Φ y = 0, we have χ χ x = χ z y = --- = kμ 0 gβ δγs 1 4S 7δ S + O( S 4 () S 9δ + 5γ / S 1 35δ + 11γ / S 3 + O( S 4 i.e., the spatial components of the friction and diffusion coefficients along the x, y, and z axes are equal. We will estimate the laser cooling temperature as the ratio of the diffusion and friction coefficients averaged over the spatial period: k B T = , D ξ where we consider ξ = ξ xx + ξ yy + ξ zz /3 and D = D xx + D yy + D zz /3 as the average friction and diffusion coefficients (see Fig. 4 below). As can be seen from (18 the kinetic coefficients in the lowest order in saturation parameter S do not depend on the phases Φ x and Φ y and coincide with the corresponding expressions for the 1D model. However, the dependence on Φ x and Φ y appears already in γ α x = α y = α z = k δγS 1 3S δ S + O( S 4 ( sp d ) ( sp x = d ) ( sp y = d ) z = γ k [ S 16S + 88S 3 + OS ( 4 d x d y γ d z = = = γ k the next order in S. In this case, the terms even in Φ x and Φ y have the same parity in detuning δ as the principal terms independent of Φ x and Φ y, while the terms odd in Φ x, Φ y have the opposite parity in detuning. This general property is characteristic of all terms in the expansion of the kinetic coefficients in powers of S. As regards the quantitative characteristics of the phase dependences, at small S 1, when the restriction to a finite number of terms in the series is a good approximation, these dependences turn out to be weak (18). We have analyzed the averaged kinetic coefficients to within S 10. The following trends are observed in this case: the string and friction coefficients are k B T/ γ S Fig. 4. Temperature versus saturation parameter S at various relative field phases: Φ x = 0, Φ y = 0 (dashed line); Φ x = π/, Φ y = 0 (dash-dotted line); Φ x = π/4, Φ y = 0 (thin solid line). The thick solid line indicates the dependence of the friction coefficient for the 1D σ + σ light field configuration. The detuning is δ = γ/. JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 10 No

7 THREE-DIMENSIONAL THEORY OF THE MAGNETO-OPTICAL TRAP 593 smaller than those predicted by the 1D model, while the diffusion coefficient is larger. Accordingly, the temperature and the localization radius in the 3D model will be slightly larger than those in the 1D one, in agreement with the results of Minogin et al. [13. Moreover, the following qualitative deviations from the 1D theory can be pointed out. For example, there are no spatial gradients of the local light field amplitude leading to a Sisyphus friction mechanism in the 1D σ + σ light field configuration [19, 0. Accordingly, the friction coefficient in the 1D σ + σ light field configuration is sign-constant at any values of the saturation parameter and, hence, must lead to stable cooling at any field intensities. In contrast to the 1D theory, in the 3D configuration (13) there is always a modulation of the light field intensity U (r) dependent on the choice of relative phases Φ x and Φ y. This leads to a Sisyphus friction mechanism changing the sign of the dissipative force at some moderate values of the saturation parameter. Depending on the choice of relative phases, the sign of the dissipative force changes with increasing field intensity necessarily along two axes, leading to the exit from the regime of cooling for a MOT. For example, for the 3D field configuration (13) U = 6+ 4sin( kz) cos( ky) cosφ y 4 cos( kz) sin( kx) cosφ x 4 cos( kx) sin( ky) cos( Φ x Φ y (3) it can be seen that the absence of an intensity gradient and, hence, the sign constancy of the friction coefficient in one of the directions are possible only for the following choice of relative phases: (1) at Φ x = Φ y = π/, i.e., in the absence of an intensity gradient along z; () at Φ x = 0 and Φ y = π/, i.e., in the absence of an intensity gradient along y; (3) at Φ x = π/ and Φ y = 0, i.e., in the absence of an intensity gradient along x. The greatest deviations from the results of the 1D theory correspond to zero relative phases. This is related to the contributions attributable to the gradients of the local field amplitude, which are maximal at Φ x = Φ y = 0. A strong dependence on the phases is observed in moderate and strong fields, when the expansions in powers of S are inapplicable. In this case, we used a numerical procedure of averaging over the spatial field period (Figs. 4). For example, at detunings δ = γ/ (under the conditions of Fig. 3 the exit from the cooling regime for a MOT occurs at a saturation parameter S 1, which, for example, for 88 Sr atoms (the optical 1 S 0 1 P 1 transition, (λ = 461 nm) corresponds to an intensity I 85 mw cm per wave. 5. CONCLUSIONS A theory of the three-dimensional magneto-optical trap was developed. Analytical expressions were derived for the magneto-optical force and the friction and diffusion coefficients. The results and important deviations from previously developed models and, in particular, from the 3D MOT theories [1, 13 were presented. (1) In contrast to the 1D theory, the 3D MOT is formed by three pairs of interfering laser beams and has a more complex spatial structure. There is always a spatial modulation of the intensity U (r) dependent on the choice of relative phases Φ x and Φ y (13) in such a field. The intensity modulation leads to Sisyphus friction mechanisms changing the sign of the dissipative force at some large values of the saturation parameter, which qualitatively distinguishes the results of the 3D theory from those of the 1D one in moderately intense fields. () The Sisyphus friction mechanisms are unimportant in the limit of low light field intensities, leading to the absence of a phase dependence in the MOT operation. Accordingly, the results coincide with those of the 1D model and with the results from [1 in the first order in saturation parameter S. (3) A significant dependence on the relative phases of the light fields forming the MOT field manifests itself in moderately intense light fields. In special cases of choosing the relative phases, Φ x = Φ y = π/, Φ x = 0 and Φ y = π/, and Φ x = π/ and Φ y = 0, the dissipative character of the friction force is retained at any intensities only along one of the axes. In most of the cases where the relative field phases are not controlled (or change on time scales corresponding to the kinetic 1 evolution time scales, τ ω R friction changes to heating with increasing intensity along all three axes, leading to the exit from the cooling regime for a MOT. At small saturation parameters, this leads to temperatures higher than those predicted by the 1D model. In particular, at Φ x = Φ y = 0 the results of the expansion of the kinetic coefficients in powers of the saturation parameter coincide with the results of [13. ACKNOWLEDGMENTS This work was supported by the Ministry of Education and Science of the Russian Federation in the framework of the basic part of the state task no. 014/139, project no. 85; the Presidium of the Siberian Branch of the Russian Academy of Sciences; and the Russian Foundation for Basic Research (project nos , , , and ). REFERENCES 1. E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 3, 631 (1987). JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 10 No

8 594 PRUDNIKOV et al.. F. Shimizu, K. Shimizu, and H. Takuma, Opt. Lett. 16, 339 (1991). 3. T. Walker, D. Hoffmann, P. Feng, and R. S. Williamson III, Phys. Lett. A 163, 309 (199). 4. C. Chesman, E. G. Lima, F. A. M. de Oliveira, S. S. Vianna, and J. W. R. Tabosa, Opt. Lett. 19, 137 (1994). 5. M. Vangeleyn, P. F. Griffin, E. Riis, and A. S. Arnold, Opt. Express 17, (009). 6. J. Lee, J. A. Grover, L. A. Orozco, and S. L. Rolston, J. Opt. Soc. Am. B 30, 869 (013). 7. M. Drewsen, Ph. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, Appl. Phys. B: Lasers Opt. 59, 83 (1994). 8. H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, New York, 1999). 9. V. I. Balykin, V. G. Minogin, and V. S. Letokhov, Rep. Prog. Phys. 63, 149 (000). 10. S. Chang, T. Y. Kwon, H. S. Lee, and V. G. Minogin, Phys. Rev. A: At., Mol., Opt. Phys. 66, (00). 11. O. N. Prudnikov, A. V. Taichenachev, A. M. Tumaikin, and V. I. Yudin, Phys. Rev. A: At., Mol., Opt. Phys. 77, (008). 1. M. Gajda and J. Mostowski, Phys. Rev. A: At., Mol., Opt. Phys. 49, 4864 (1994). 13. S.-K. Choi, S. E. Park, J. Chen, and V. G. Minogin, Phys. Rev. A: At., Mol., Opt. Phys. 77, (008). 14. T. Hong, C. Cramer, W. Nagourney, and E. N. Fortson, Phys. Rev. Lett. 94, (005). 15. R. Santra, E. Arimondo, T. Ido, C. Greene, and J. Ye, Phys. Rev. Lett. 94, (005). 16. A. V. Taichenachev, V. I. Yudin, C. W. Oates, C. W. Hoyt, Z. W. Barber, and L. Hollberg, Phys. Rev. Lett. 96, (006). 17. Z. W. Barber, C. W. Hoyt, C. W. Oates, L. Hollberg, A. V. Taichenachev, and V. I. Yudin, Phys. Rev. Lett. 96, (006). 18. A. V. Bezverbnyi, O. N. Prudnikov, A. V. Taichenachev, A. M. Tumaikin, and V. I. Yudin, J. Exp. Theor. Phys. 96 (3 383 (003). 19. O. N. Prudnikov, A. V. Taichenachev, A. M. Tumaikin, and V. I. Yudin, J. Exp. Theor. Phys. 98 (3 438 (004). 0. A. P. Kazantsev, G. I. Surdutovich, and V. P. Yakovlev, Mechanical Action of Light on Atoms (World Scientific, Singapore, 1988; Nauka, Moscow, 1991). Translated by V. Astakhov JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 10 No