- Email: [email protected]
Force pattern on atoms in a monochromatic light field with arbitrary polarization
1 March 1998
Optics Communications 148 Ž1998. 151–158
Full length article
Force pattern on atoms in a monochromatic light field with arbitrary polarization A.V. Bezverbnyi a , G. Nienhuis b, A.M. Tumaikin
c
a
b
Far East State Maritime Academy, 690055 VladiÕostok, Russia Huygens Laboratorium, UniÕersity of Leiden, P.O. Box 9504, 2300 RA Leiden, The Netherlands c NoÕosibirsk State UniÕersity, 630090 NoÕosibirsk, Russia Received 17 September 1997; accepted 20 November 1997
Abstract The light force on slow atoms in a light field with an arbitrary polarization distribution is explicitly evaluated in the case of an atomic transition J ™ J with half-integer J values. The expression of the force is based on an exact analytical expression for the steady-state density matrix of the atoms. The force is naturally separated in contributions resulting from gradients of the field intensity, the polarization ellipticity, and the phase. In areas of spatial localization, where the total force vanishes, the atomic motion is governed by a potential, which is harmonic only for J s 1r2. The results are illustrated for specific simple models of a one- or two-dimensional optical lattice. q 1998 Elsevier Science B.V. PACS: 42.50.Vk; 32.80.Pj
1. Introduction Controlling the center-of-mass motion of atoms with light has been a field of growing interest in the last few years. The key feature is the radiative force that light exerts on the atoms by exchange of photon momentum, and which is determined by the gradient of the atom–field interaction. The force in general has a contribution due to the gradient of the phase, the intensity and the polarization, which specify the local properties of the light field. A detailed study of the radiative force has been made for many different specific cases. A few characteristic atomic models are Ži. atoms with two nondegenerate states in travelling or standing waves of weak or strong intensity w1,2x, or in a bichromatic field w3x, Žii. atoms with three nondegenerate states driven by traveling or standing waves w4–7x, and Žiii. atoms with two degenerate levels driven by a monochromatic field with polarization gradients or in an external magnetic field. Prototype cases of sub-Doppler cooling in one dimension are well-known examples w8–10x.
The light force also plays a crucial role in the formation of 2D or 3D optical lattices of cold atoms w11–14x. The polarization gradient force depends on the internal state the atom, which in general is anisotropic due to the interaction with the polarized field. When the radiation field drives the transition between two levels with Zeeman degeneracy, the steady-state density matrix for both levels can be easily evaluated in the simple case of linear or circular polarization w15x. The case of arbitrary elliptical polarization has also received attention w16x, but a general expression for the steady-state density matrix is lacking in general, except for small values of Je and J g w8,17,18x. Recently we have shown w19x that a general analytical expression can be given for the steady-state density matrix for a transition between two states with the same total angular momentum J g s Je s J Ž J half-integer., when it is driven by light with an arbitrary elliptical polarization. In this paper we analyze the radiative force of a weak monochromatic field with arbitrary polarization acting on slow atoms. This is a standard situation for ultradeep laser
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 6 9 7 - 4
A.V. BezÕerbnyi et al.r Optics Communications 148 (1998) 151–158
152
cooling w8x. The field drives the transition between two states with the same total angular momentum J g s Je s J Ž J half-integer.. We give compact explicit expressions for the force. As an example, this general result is then applied to specific one- and two-dimensional field patterns.
The evolution of the density matrix r for the internal state of the atom is described by the master equation w20x d dt
g r s yi w H , r x y
2
g Pe r y
2
r Pe q g Ý Qb† r Qb , b
Ž7. 2. Radiative force for arbitrary polarization The transition between two degenerate levels of an atom is driven by a monochromatic radiation field with arbitrary polarization. The electric field is parametrized as Re E Ž r . expŽyi v t . with E Ž r . s Re A Ž r . e Ž r . e i f Ž r . ,
Ž1.
where the amplitude A is real, and the polarization vector e is normalized. In order that the position-dependent phase f be uniquely defined Žmodulo 2p ., we assume that the component of the polarization vector e along the long axis of the polarization ellipse is real. This implies that the component along the short axis is imaginary. We parametrize the polarization vector as e s z1cos e q iz2 sin e ,
E P E s A2 e 2 i f cos2 e ,
Ž3.
from which all three field parameters can directly be extracted. The Hamiltonian for the atom coupled to the field is in the rotating-wave and dipole approximations Ž " s 1. H s y 12 D Ž Pe y Pg . y 12 V Ž V e i f q V †eyi f . ,
Ž4.
with Pe and Pg the projectors on the substates of the excited and the lower levels, and D s v y v o the detuning of the laser frequency from the atomic resonance. The real Rabi frequency V is determined by the product of the amplitude A and the reduced transition dipole. The operator V is the raising part of the reduced dipole moment along the polarization direction, which can be expressed as VsePQ .
Ž5.
The reduced dipole operator Q is defined by the requirement that its spherical components have matrix elements between the Zeeman substates of the ground state and the excited state that are equal to Clebsch–Gordan coefficients, so that w20x ² Je , M e < Qb < J g , M g : s ² Je , M e < J g , M g ;1, b : for b s y1, 0, 1.
F s Tr r 12 = Ž V V e i f . q c.c.
Ž6.
Ž8.
When the rate of optical pumping is low compared to g , the excited-state population is small, and one can eliminate the excited state from the evolution equation. Adiabatic elimination of the optical coherences gives rise to a closed evolution equation for the ground-state submatrix, which we denote as s . Introducing the saturation parameter
V 2r4 Gs
Ž2.
with z1 and z2 two orthogonal real unit vectors, which determine the direction of the axes of the polarization ellipse, respectively. The right-hand Cartesian coordinate frame spanned by these two vectors and their cross product z3 s z1 = z2 is termed the polarization frame. The ellipticity angle e can be assumed to attain values between ypr4 and pr4, representing the range between left and right circular polarization. The amplitude A, the ellipticity angle e and the phase f of the field are determined by E according to the equations E ) P E s A2 ,
with g the rate coefficient of spontaneous decay. The solution r of the master equation Ž7. determines the average force on the atom, which is given by
Ž9.
g 2r4 q D 2
leads to the ground-state master equation w21x 1
d
g G dt
ssy
ž
1 2
D qi
g
/
V † Vs y
q Ý Qb† Vs V †Qb .
ž
1 2
D yi
g
/
s V †V
Ž 10.
b
This equation describes the evolution of the ground state due to optical pumping. The first two terms on the r.h.s. describe the loss and the light shifts of the levels. The last term is the corresponding gain term. A remarkable feature of this low-intensity master equation is that the evolution rate is simply proportional to the optical-pumping rate g G, and thereby to the intensity. This implies that scaling down the intensity means that the evolution of the ground-state density matrix s is simply slowed down by the same factor. In this limit of low saturation rate, the average force on the atom is an expectation value with respect to the ground-state density matrix s , as given by w21x Fsy
ir4
gr2 q i D
Tr s Ž V V †eyi f . = Ž V V e i f . q c.c.
Ž 11. One notices from Ž11. that the average force contains the gradients of the Rabi frequency V , the phase f , and the operator V, which is determined by the polarization vector e. The contribution containing =V has a dispersive dependence on the detuning D, and arises from the Hermitian part of the polarizability of the atoms w21x. It is proportional to the intensity gradient, and it has the nature of the dipole force. The contribution containing =f has an absorptive dependence on the detuning. As usual, this phase-gradient term has the nature of radiation pressure,
A.V. BezÕerbnyi et al.r Optics Communications 148 (1998) 151–158
and it can be expressed in the dissipative anti-Hermitian part of the polarizability. Finally, the contribution to the force that contains the gradient of the operator V or V † results from polarization gradients. This contribution generally contains both a dissipative and a dispersive part w21x. Polarization gradients play an essential part in sub-Doppler cooling w8x. The strength of the optical-pumping operator V † V is independent of the polarization vector. From the definition Ž5. one directly finds Tr V † V s
2 Je q 1 3
,
Ž 12.
irrespective of the type of polarization.
3. Steady-state radiative force for transition J ™ J We consider the situation of cold atoms, which do not move appreciably compared with the variations in the polarization pattern during the characteristic time of optical pumping Žg G .y1. Then we may assume that the density matrix of the atoms reaches the internal steady state that corresponds to the local polarization. Since the polarization varies typically over a wavelength, this requires that the atomic velocity Õ obeys the inequality Õ < Ggrk .
Ž 13.
Normally grk is considerably larger than the velocity grm that corresponds to optimized Doppler cooling w8x Ž m is the atom mass.. Hence, with sub-Doppler cooling techniques and for moderately low values of the saturation parameter G, the inequality Ž13. can easily be realized, except maybe very close to the field nodes. Hence, in the immediate vicinity of a node, the steady-state forces will tend to be smeared out. However, since the forces are very small in this region anyhow, this does not affect the pattern of the forces that the atoms experience. This justifies to approximate the force pattern by the steady-state force at any position. Obviously, the steady-state solution of Eq. Ž10. depends only on the polarization of the field, but not on the intensity or the phase. In the polarization frame, the steady-state density matrix depends exclusively on the ellipticity angle e .
'
3.1. Steady-state density matrix Recently it has been demonstrated that in the special case of a transition between two degenerate atomic levels with equal half-integer angular momenta J, an exact operator expression can be given for the steady-state solution of the master equation Ž7. w19x. In the absence of saturation, the ground-state density matrix, which is the steadystate solution of Eq. Ž10. is given by the simple exact expression
ss ŽV †V .
y1
rNJ ,
Ž 14.
153
with NJ a normalization constant. This steady-state solution is obvious, if one recognizes that in these cases the excited-state operator V Ž V † V .y1 V † occurring in the last term in Ž10. is isotropic. This implies that for a transition J ™ J with half-integer J, the excited-state density matrix is isotropic in the steady state, independent of the polarization. Eq. Ž14. shows that the steady-state density matrix s is diagonal on the basis of eigenstates of V † V, and that the populations are proportional to the inverse of the corresponding eigenvalues. Hence, the population accumulates preferentially in the states that are weakly coupled to the field. These eigenvalues and the normalization constant N depend exclusively on the ellipticity angle e w19x. For circular polarization, the ground-state operator V † V has one eigenvalue zero, and cannot be inverted. In the polarization frame, the Zeeman state with maximal value of the magnetic quantum number M is then a dark state, that is not coupled to the light field. In the steady state, only this dark state is populated. This degenerate case can be formally treated by adding an infinitesimal number d to V † V in Ž14., and taking the limit d ™ 0. 3.2. Force and field parameters An expression for the radiative force is obtained by substituting Eq. Ž14. in Ž11.. The result is Fsy
ir4
1
gr2 q i D NJ
Ž 2 J q 1 . Ž V =V q i V 2 =f .
qV 2 Tr Vy1 = V q c.c.
Ž 15.
Note that for the transition J ™ J the operator V is represented by a square matrix, which can be inverted in general for half-integer J. Recall that for equal integer J values there is a dark state for all polarizations w22x. The first two terms in Ž15. represent the dipole force and radiation pressure, which are proportional to the gradients of the intensity and the phase, respectively. The factor Ž2 J q 1.rN measures the effective coupling strength. The last term in Ž15. depends only on the gradient of the ellipticity angle, and not on the orientation of the polarization frame. This is typical for the force in the steady state. In order to evaluate the ellipticity-gradient force, we apply the matrix identity detV s expwTr ŽlnV .x, which holds for square matrices. After differentiation this gives Tr Vy1 = V s
1 detV
=detV .
Ž 16 .
The determinant of V is most easily found when we express the polarization vector e as linear superposition of circular and linear polarization w22x. For this purpose, we choose a Cartesian reference frame, which follows from the frame z1, z2 , z3 by a rotation along z2 , over an angle arccosŽtan e .. The polarization Ž2. is expressed in the new frame after the substitution z1cos e s u1sin e q u 3'cos2 e ,
Ž 17 .
A.V. BezÕerbnyi et al.r Optics Communications 148 (1998) 151–158
154
field parameters A, f and e . An explicit alternative expression for the force is Fs
2 Jq1 NJ
ž
g G =f y D =G y GD
1 cos2 e
/
=cos2 e .
Ž 21.
Fig. 1. Sketch of the relation between the quantization axes z3 and u 3 . The direction of z3 is normal to the polarization ellipse, which is the intersection of a plane with the cylinder with axis u 3 . Hence the polarization is a linear combination of u 3 and circular polarization in the plane normal to u 3 .
while noting that z2 s u 2 . The polarization vector e is then e s Ž u1 q iu 2 . sin e q u 3'cos2 e ,
Ž 18.
which is a superposition of circular polarization in the u1 u 2 plane, and linear polarization along u 3 , which serves as the quantization axis of this frame. The relative strength of the circular polarization is 2 sin2e , and the linear polarization has the complementary strength cosŽ2 e .. Geometrically speaking, u 3 is oriented along the axis of the cylinder whose intersection with the polarization plane coincides with the polarization ellipse w22,19x. This is illustrated in Fig. 1. When we take this new axis u 3 as quantization axis, the operator V couples a Zeeman substate < M : of the ground state only to the two substates < M : and < M q 1: of the excited states, as indicated in the level diagram in Fig. 2. This shows that detV is just the product of the diagonal elements. Hence we find J
detV s detV † s Ž cos2 e .
Jq1r2
Ł
² J , M < J , M ;1,0: .
Ms J
Ž 19.
One notices that the expression in brackets is fully independent of the value of J. The three terms in Eq. Ž21. correspond respectively to the radiation pressure, which is proportional to the phase gradient, the dipole force, determined by the intensity gradient, and the polarizationgradient force. Remarkably, this latter term is simply proportional to the gradient of lnŽcos2 e ., for each value of J. Moreover, its dependence on the detuning D is a dispersion curve, which indicates that the polarization gradient force can be entirely expressed in terms of the Hermitian, non-absorptive part of the polarizability w21x. The polarization-gradient force depends only on the gradient of the ellipticity angle, and not on the orientation of the polarization frame z1, z2 , z3. In the presence of saturation, the expression for the steady-state density matrix is only slightly modified w19x. Also in that case, the excited state remains fully unpolarized. The resulting expression for the force is Fs
2 Jq1 NJ q Ž 2 J q 1 . G
ž
= g G =f y D =G y GD
1 cos2 e
/
=cos2 e .
Ž 22 .
Hence, the effect of saturation is a multiplicative factor, that depends on the position-dependent intensity and ellipticity angle. 3.3. Localization points In order to evaluate the force explicitly, the expressions for the normalization constant NJ are needed. These are found directly from the matrix representation of the operator V with z3 as quantization axis, when using the explicit
Eq. Ž15. for the force can now be expressed in terms of a single gradient Fsy
ir4
1 2 Jq1 V 2
gr2 q i D NJ
2
EPE
= Ž E P E . q c.c.
Ž 20. Eq. Ž20. constitutes the general result of this paper. It is exact in case of the mentioned assumptions Žno saturation, steady state.. It is remarkable that the force can be expressed as the gradient of the square Žnot the absolute square. of the complex field E. Notice that according to Eq. Ž3., this expression depends on the gradients of all
Fig. 2. Level diagram for J s 3r2, with the quantization axis along u 3 . Transitions corresponding to linear polarization and to right-hand circular polarization are driven.
A.V. BezÕerbnyi et al.r Optics Communications 148 (1998) 151–158
expressions for the Clebsch–Gordan coefficients. The result can be expressed as w19x NJ s
2J
J Ž J q 1.
Ý
cos2 e
Aa
as0
ž
1 y cos2 e cos2 e
a
/
,
Ž 23.
with J
A0 s
1
Ý nsyJ Jy a
Aa s
n2
, nq a
1
Ý n syJ
n
2
Ž J q b .Ž J y b q 1 .
Ł
2b
bs n q1
2
First we consider the field configuration consisting of two linearly polarized counterpropagating waves of equal intensity, with an angle u between the polarization directions. This scheme has been shown to be efficient for sub-Doppler w24x or sub-recoil cooling based on polarization-gradient forces and coherent-population trapping w25– 27x. It is equivalent to two circularly polarized standing waves oscillating in phase, with a spatial phase shift equal to u . The electric field is written as
Ž 24. 6
F Žr. s
,
N3r2 s
160
20
. Ž 25 . cos 2 e 3cos 2 e cos 2 2 e In general, NJ equals Žcos2 e .y2 Jy1 times a polynomial in cos 2 2 e of the order J y 1r2. This shows that the force Ž21. Žor Ž22.. disappears when cos2 e s 0, which is the case at points with circular polarization. Typically, in an optical lattice the detuning is large, so that radiation pressure can be neglected. In the neighborhood of a point of circular polarization, NJy1 is well approximated by its leading term. Then the force can be written as F s y=F , in terms of the effective potential 2
GD J Ž J q 1. A 2 J
4
y
Ž cos2 e . 2 Jq1 .
Ž 26.
This demonstrates that in general an atom is attracted to a point of circular polarization when the detuning D is positive. Normally one would expect that cos2 e varies linearly with position. In that case, the restoring force is harmonic only in the special case that J s 1r2. Moreover, the force disappears also at the nodes of the field. At positive detuning, this force will be attractive in the neighborhood of the equilibrium point. In summary, atoms will tend to be localized at positions of circular polarization, or at the nodes of the field. These points are conveniently characterized as the zeros of E P E.
'2
ž
u
žž ˆ
xcos
2
u q ysin ˆ
u
q xcos ˆ
For J s 1r2 and J s 3r2, we find N1r2 s
E
EŽ r. s
.
155
2
u y ysin ˆ
2
2
/
eik z
/ /
eyi k z ,
Ž 27 .
with E the real amplitudes of the two traveling waves. The field parameters Ž3. are then determined by the identities E ) P E s E 2 Ž 1 q cos2 kzcos u . , E P E s E 2 Ž cos2 kz q cos u . .
Ž 28 .
Obviously, the one-dimensional lattice has 2 k as its reciprocal lattice constant, so that the periodicity of the lattice is half an optical wavelength. The phase f is zero, whereas the ellipticity angle is given by cos2 e s
cos2 kz q cos u
. 1 q cos2 kzcos u For the total force Ž20. Žor Ž21.. we find F s zˆ
2 J q 1 G0 D NJ
cos2 e
2 ksin2 kz ,
Ž 29.
Ž 30.
where G 0 is the saturation parameter corresponding to the field amplitude E. The force vanishes at the points of circular polarization, where NJ approaches infinity. These localization points occur at the positions z where 2 kz s Ž2 n q 1.p " u . There are two such points in a lattice constant, which is half of a wavelength. For small values of u , these points form closely spaced pairs, corresponding to the nodes of the two circularly polarized standing waves that compose the light field. In between such a pair, the
4. Two linearly polarized traveling waves in one dimension In this and the subsequent section, we give some explicit examples of the force pattern in optical lattices, which arise when atoms are bound in the periodic optical potential that is created by a monochromatic light field composed of a few traveling waves w11,10,13x. In general, when the traveling waves have wave vectors k m , the reciprocal lattice is spanned by the reciprocal lattice vectors k m y k mX . For two beams in one dimension, or three beams in two dimensions, the polarization pattern and the resulting force distribution is independent of the phase of the beams w23x.
Fig. 3. Plot of cos2 e as a function of 2 kz for the one-dimensional lattice of Section 4. The selected values of u correspond to cos u s 0 Žsolid line., 0.7, 0.99 and 1 Ždotted line..
A.V. BezÕerbnyi et al.r Optics Communications 148 (1998) 151–158
156
polarization varies rapidly between two opposite circular polarizations, as expressed by a similar rapid variation of the function cos2 e . This behavior of the ellipticity angle is illustrated in Fig. 3. The force Ž30. also disappears at the points where 2 kz s np , so that sin2 kz vanishes. However, for non-zero values of u , these equilibrium points are not stable. In the standing-wave case u s 0, localization occurs at the nodes 2 kz s Ž2 n q 1.p of the field.
5. Three linearly polarized traveling waves in two dimensions Next we consider the symmetric two-dimensional configuration of three linearly polarized traveling waves, so that 3
EsE
Ý
e mexp Ž ik m P rm . .
Ž 31.
ms 1
The wave vectors make an angle 2pr3 with each other, so that k 1 s kxˆ ,
k2 s
k 2
Ž yxˆ q '3 yˆ . ,
k3 s
k 2
Ž yxˆ y '3 yˆ . . Ž 32.
The periodicity of the field pattern is spanned by two reciprocal lattice vectors, which we choose as b 1 s k 2 y k 1 and b 2 s k 3 y k 1. This means that the pair of lattice vectors of a unit cell, which are biorthogonal to b 1 and b 2 , is given by a1 s
2p
Ž yxˆ q '3 yˆ . , 3k
a2 s
2p 3k
Ž yxˆ y '3 yˆ . . Ž33.
The unit cell is a diamond with an angle pr3, and its sides have length 2 lr3. The three linear polarization vectors are taken to have the same angle u with the z-axis, so that e 1 s ysin ˆ u q zcos ˆ u, e 2 s y 12 Ž xˆ'3 q yˆ . sin u q zcos ˆ u, e 3 s 12 Ž xˆ'3 y yˆ . sin u q zcos ˆ u.
Ž 34 .
Hence, in addition to the translational invariance over the lattice vectors, the field pattern has an obvious threefold rotational symmetry. A full parametrization of the field and of the steady-state force is given by the functions Ž3.. In order to facilitate the discussion, we introduce the scalar function 3
Zs
Ý
exp Ž ik m P r . ,
Ž 35.
Since the three wave vectors k m add up to zero, one directly proves the identity 3
Z2s
exp Ž 2 ik m P r . q 2 Z ) .
Ý
Ž 37.
ms 1
According to Eq. Ž3., the field parameters A, f and e are determined by the expressions E ) P E s E 2 Ž 3 q c Ž Z ) Z y 3. . , E P E s E 2 Ž Z 2 y 2 Ž1 y c . Z ) . .
Ž 38.
When we parametrize the position vector as a linear combination r s j 1 a1 q j 2 a 2 of the lattice vectors, we find Z s exp w y2p i Ž j 1 q j 2 . r3 x = w 1 q exp Ž 2p i j 1 . q exp Ž 2p i j 2 . x ,
Ž 39 .
which is symmetric for an interchange of j 1 and j 2 . This expression shows immediately that Z ) Z s 1 when j i s 1r2 q n for i s 1 or 2, and also for j 1 s j 2 q n q 1r2, for all polarization angles u . The intensity E ) P E s E 2 Ž3 y 2 c . is uniform along these lines. From Eq. Ž37. one derives that Z has two zeros in each unit cell, which occur at j 1 s 1r3, j 2 s 2r3 and at j 1 s 2r3, j 2 s 1r3. The maximal value of < Z
Z s 2 Ž 1 y c . exp Ž 2p inr3 . ,
Ž 40.
with n s 0, 1, 2. These are the points of circular polarization or the nodes of the field. At the solution Z s 0, the intensity of the field is E ) P E s 3 E 2 Ž1 y c ., and at the other solutions in Ž37. we find E ) P E s E 2 Ž1 y c .Ž2 c q 1.Ž3 y 2 c .. The fact that the intensity is the same for these solutions with non-zero values of Z suggests that these solutions are physically equivalent. This is confirmed by the combined rotational and lattice symmetry of the configuration. The function Z obtains a phase factor at translation over the lattice vectors Ž33., as expressed by Z Ž r y a i . s exp Ž 2p ir3 . Z Ž r .
for
i s 1,2 .
Ž 41.
Hence, from a solution of Z s 2Ž1 y c . one arrives at solutions of the last identity Ž37. by a simple translation over a lattice vector. Combined with the invariance of Z under rotations over 2pr3, this allows us to find all solutions after finding only the points where Z s 2Ž1 y c .. We now illustrate these general considerations in a few explicit cases.
ms 1
and the scalar product c s e m P e mX between two different polarization vectors as c s cos 2u y 12 sin2u .
Ž 36.
5.1. Parallel polarization Õectors When all polarizations are parallel to the z-axis, so that c s 1, the polarization is linear, and the ellipticity angle e
A.V. BezÕerbnyi et al.r Optics Communications 148 (1998) 151–158
is zero everywhere, Hence, the polarization-gradient force vanishes. In the standard situation for optical lattices that the detuning D is large compared with g , the force Ž21. can be expressed as a purely conservative dipole force F s =F with
Fs
Ž2 J q 1. D NJ
G,
Ž 42.
with G s G 0 Z ) Z. The normalization factor NJ s J Ž J q 1. A 0 follows from Eq. Ž23. in the special case that cos2 e s 1. The solutions Ž37. of the localization points now all merge to the single condition Z s 0, which in the present case corresponds to the nodes of the field. These are the small black dots in Fig. 4, which form a hexagonal lattice with side 2 l'3 r9. 5.2. Normal polarization Õectors A second relatively simple case occurs when cos u s 1r c s 0. This implies that the three polarization
'3 , so that
157
vectors form an orthonormal set, and the intensity E ) E s 3 E 2 is uniform. This configuration has been considered by Mauri and Arimondo as an example of velocity-selective coherent population trapping w28x. For D 4 g , when radiation pressure is negligible, only the last term in Ž21. is nonzero, and the force can be derived from a potential function of cos2 e . For J s 1r2, this potential has the form
F Žr. s
GD 6
cos 2 2 e .
Ž 43.
The localization points, which are the zeros of E P E s E 2 Ž Z 2 y 2 Z ) . now represent the points of circular polarization only. From Ž37. one finds in addition to the zeros of Z the points Ž j 1, j 2 . s Ž1r6,1r3. where Z s 2, Ž2r3,5r6. where Z s 2 expŽ2p ir3. and Ž1r6,5r6. where Z s 2 expŽy2p ir3., and the corresponding points with j 1 and j 2 interchanged. These additional solutions are indicated by the open dots in Fig. 4. The localization points for this case form again a hexagonal lattice, but now with side l'3 r9.
5.3. Polarization Õectors in plane All polarization vectors fall in the xy plane when c takes its minimal value y1r2. In this case, both the polarization and the intensity E ) P E s E 2 Ž9 y Z ) Z .r2 vary with position. The localization points. which solve Eq. Ž37., contain points of circular polarization, determined by Z s 0, and field nodes, indicated by Z s 3 expŽ2p inr3.. The points of circular polarization are the small black dots in Fig. 4. The nodes are located at the lattice sites, corresponding to integer values of j i , and they are indicated by the large black dots in Fig. 4. These two types of localization points combine into a regular triangular lattice, with lattice side 2 l'3 r9. At the points of circular polarization, the intensity is maximal. Hence, localization points correspond to the points where the intensity takes extreme values in this particular configuration. The polarization is linear at all straight lines connecting neighboring lattice points. These are represented by the drawn lines in Fig. 4. Fig. 4. Two-dimensional lattice pattern as discussed in Section 5. The inset gives the directions of the three k vectors. The thick arrows denote the two lattice vectors a1 and a 2 , which have length 2 l r3. The unit cells are indicated by the thick lines, and at the thin lines the intensity is uniform, for all polarization angles u . The small black dots are the locations where Zs 0. For parallel polarizations Ž u s 0. these are the field nodes, which are the only localization points. For normal polarizations, the localization points are the points of circular polarization, which are represented by the small dots Žblack and open.. For polarizations in the plane Ž u sp r2., the small black dots are the points of circular polarization, whereas the field nodes fall at the lattice sites, denoted by the large black dots.
5.4. Polarization at small angles u Finally, we consider the configuration where the polarization vectors make a small angle with the z-axis. Then we can write c s 1 y h , with a small value of h. The localization points are now determined by the equations Z s 0 and Z s 2h expŽ2p inr3.. The intensity at these localization points is given by E ) P E s 3h E 2 , to first order in h. As h deviates from zero, each localization point corresponding to the small black dots in Fig. 4
158
A.V. BezÕerbnyi et al.r Optics Communications 148 (1998) 151–158
separates into four closely spaced points. The four localization points around Ž j 1, j 2 . s Ž1r3,2r3. are specified by
References
Ž 1 y h'3 rp ,2 q h'3 rp . Ž 1 y h'3 rp ,2 y 2h'3 rp . , Ž 1 q 2h'3 rp ,2 q h'3 rp . .
w1x V.G. Minogin, V.S. Letokhov, Laser light pressure on atoms, Gordon & Breach, New York, NY, 1987. w2x A.P. Kazantsev, Y.I. Surdutovich, V.P. Yakovlev, Mechanical action of light, World Scientific, Singapore, 1990. w3x A.P. Kazantsev, I.V. Krasnov, JETP 95 Ž1989. 104. w4x J. Javanainen, Phys. Rev. Lett. 64 Ž1990. 519. w5x S. Chang, B.M. Garraway, V.G. Minogin, Opt. Commun. 77 Ž1991. 19. w6x A.I. Sidorov, R. Grimm, V.S. Letokhov, J. Phys. B 24 Ž1991. 3733. w7x M.G. Prentiss, N.P. Bigelow, M.S. Shahriar, P.R. Hemmer, Opt. Lett. 16 Ž1991. 169. w8x J. Dalibard, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6 Ž1989. 2023. w9x B. Sheehy, S.-Q. Shang, R. Watts, S. Hatamian, H. Metcalf, J. Opt. Soc. Am. B 6 Ž1989. 2165. w10x B. Lounis, P. Verkerk, J.-Y. Courtois, C. Salomon, G. Grynberg, Europhys. Lett. 21 Ž1993. 13. w11x N.P. Bigelow, M.G. Prentiss, Phys. Rev. Lett. 65 Ž1990. 29. w12x P. Verkerk, B. Lounis, C. Salomon, C. Cohen-Tannoudji, J.-Y. Courtois, G. Grynberg, Phys. Rev. Lett. 68 Ž1992. 1861. w13x P. Jessen, C. Gerz, P.D. Lett, W.D. Phillips, S.L. Rolston, R.J.C. Spreeuw, C.I. Westbrook, Phys. Rev. Lett. 69 Ž1992. 49. w14x A. Hemmerich, T.W. Hansch, Phys. Rev. Lett. 70 Ž1993. ¨ 410. w15x G. Nienhuis, Phys. Rev. A 26 Ž1982. 3137. w16x G. Nienhuis, Opt. Commun. 59 Ž1986. 353. w17x A.V. Bezverbnyi, V.S. Smirnov, A.M. Tumaikin, Zh. Eksp. Teor. Fiz. 105 Ž1994. 62 wJETP 78 Ž1994. 33x. w18x D. Suter, Opt. Commun. 86 Ž1991. 381. w19x V. Taichenachev, A.M. Tumaikin, V.I. Yudin, G. Nienhuis, Zh. Eksp. Teor. Fiz. 108 Ž1995. 415 wJETP 81 Ž1995. 224x. w20x G. Nienhuis, Phys. Rep. 138 Ž1986. 151. w21x G. Nienhuis, P. van der Straten, S-Q. Shang, Phys. Rev. A 44 Ž1991. 462. w22x A.M. Tumaikin, V.I. Yudin, Zh. Eksp. Teor. Fiz. 98 Ž1990. 81 wJETP 71 Ž1990. 43x. w23x K.I. Petsas, A.B. Coates, G. Grynberg, Phys. Rev. A 50 Ž1994. 5173. w24x V. Finkelstein, P. Berman, J. Guo, Phys. Rev. A 45 Ž1992. 1829. ¨ P. Zoller, M.S. Shahriar, M. w25x P. Marte, R. Dum, R. Taieb, Prentiss, Phys. Rev. A 49 Ž1994. 4826. w26x M. Weidemuller, T. Esslinger, M.A. Ol’shanii, A. Hem¨ merich, T.W. Hansch, Europhys. Lett. 27 Ž1994. 109. ¨ w27x R. Dum, M.A. Ol’shanii, Phys. Rev. Lett. 76 Ž1996. 1788. w28x F. Mauri, E. Arimondo, Europhys. Lett. 8 Ž1991. 171.
3 Ž j 1 , j 2 . s Ž 1,2 . ,
Ž 44.
Around the point Ž2r3,1r3., the four solutions of Z s 0 and Z s 2h expŽ2p inr3. Ž n s 0,1,2. are
Ž 2 q h'3 rp ,1 y h'3 rp . Ž 2 y 2h'3 rp ,1 y h'3 rp . , Ž 2 q h'3 rp ,1 q 2h'3 rp . .
3 Ž j 1 , j 2 . s Ž 2,1 . ,
Ž 45.
For small values of u , the polarization varies rapidly in these regions, giving rise to large gradients of cos2 e .
6. Conclusions The effects of optical pumping on the internal state of atoms depend strongly on the polarization of the driving field. Characteristic for situations of polarization-gradient cooling or optical lattices is that the polarization varies continuously through all possible values of the ellipticity. We have applied a closed analytical expression for the steady-state density matrix of an atom driven by a radiation field with arbitrary polarization, to study the pattern of radiative forces on the atom. The treatment is valid for transitions J ™ J with arbitrary half-integer J-values. The expressions Ž20. or Ž21. for the force depend on the value of J only through the normalization factor NJ , that is a function of the ellipticity of the local polarization. The force contribution from the polarization gradient is purely redistributive, so that it is independent of the anti-Hermitian part of the polarizability w21x. The force disappears both at the nodes and at the points of circular polarization. For large positive values of the detuning, these equilibrium points are stable. As an illustration, we analyze the force pattern for a physically interesting class of one- and twodimensional optical lattices.
Acknowledgements This work was supported by a grant from the Netherlands Organization for Research ŽNWO..
Optics Communications 148 Ž1998. 151–158
Full length article
Force pattern on atoms in a monochromatic light field with arbitrary polarization A.V. Bezverbnyi a , G. Nienhuis b, A.M. Tumaikin
c
a
b
Far East State Maritime Academy, 690055 VladiÕostok, Russia Huygens Laboratorium, UniÕersity of Leiden, P.O. Box 9504, 2300 RA Leiden, The Netherlands c NoÕosibirsk State UniÕersity, 630090 NoÕosibirsk, Russia Received 17 September 1997; accepted 20 November 1997
Abstract The light force on slow atoms in a light field with an arbitrary polarization distribution is explicitly evaluated in the case of an atomic transition J ™ J with half-integer J values. The expression of the force is based on an exact analytical expression for the steady-state density matrix of the atoms. The force is naturally separated in contributions resulting from gradients of the field intensity, the polarization ellipticity, and the phase. In areas of spatial localization, where the total force vanishes, the atomic motion is governed by a potential, which is harmonic only for J s 1r2. The results are illustrated for specific simple models of a one- or two-dimensional optical lattice. q 1998 Elsevier Science B.V. PACS: 42.50.Vk; 32.80.Pj
1. Introduction Controlling the center-of-mass motion of atoms with light has been a field of growing interest in the last few years. The key feature is the radiative force that light exerts on the atoms by exchange of photon momentum, and which is determined by the gradient of the atom–field interaction. The force in general has a contribution due to the gradient of the phase, the intensity and the polarization, which specify the local properties of the light field. A detailed study of the radiative force has been made for many different specific cases. A few characteristic atomic models are Ži. atoms with two nondegenerate states in travelling or standing waves of weak or strong intensity w1,2x, or in a bichromatic field w3x, Žii. atoms with three nondegenerate states driven by traveling or standing waves w4–7x, and Žiii. atoms with two degenerate levels driven by a monochromatic field with polarization gradients or in an external magnetic field. Prototype cases of sub-Doppler cooling in one dimension are well-known examples w8–10x.
The light force also plays a crucial role in the formation of 2D or 3D optical lattices of cold atoms w11–14x. The polarization gradient force depends on the internal state the atom, which in general is anisotropic due to the interaction with the polarized field. When the radiation field drives the transition between two levels with Zeeman degeneracy, the steady-state density matrix for both levels can be easily evaluated in the simple case of linear or circular polarization w15x. The case of arbitrary elliptical polarization has also received attention w16x, but a general expression for the steady-state density matrix is lacking in general, except for small values of Je and J g w8,17,18x. Recently we have shown w19x that a general analytical expression can be given for the steady-state density matrix for a transition between two states with the same total angular momentum J g s Je s J Ž J half-integer., when it is driven by light with an arbitrary elliptical polarization. In this paper we analyze the radiative force of a weak monochromatic field with arbitrary polarization acting on slow atoms. This is a standard situation for ultradeep laser
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 6 9 7 - 4
A.V. BezÕerbnyi et al.r Optics Communications 148 (1998) 151–158
152
cooling w8x. The field drives the transition between two states with the same total angular momentum J g s Je s J Ž J half-integer.. We give compact explicit expressions for the force. As an example, this general result is then applied to specific one- and two-dimensional field patterns.
The evolution of the density matrix r for the internal state of the atom is described by the master equation w20x d dt
g r s yi w H , r x y
2
g Pe r y
2
r Pe q g Ý Qb† r Qb , b
Ž7. 2. Radiative force for arbitrary polarization The transition between two degenerate levels of an atom is driven by a monochromatic radiation field with arbitrary polarization. The electric field is parametrized as Re E Ž r . expŽyi v t . with E Ž r . s Re A Ž r . e Ž r . e i f Ž r . ,
Ž1.
where the amplitude A is real, and the polarization vector e is normalized. In order that the position-dependent phase f be uniquely defined Žmodulo 2p ., we assume that the component of the polarization vector e along the long axis of the polarization ellipse is real. This implies that the component along the short axis is imaginary. We parametrize the polarization vector as e s z1cos e q iz2 sin e ,
E P E s A2 e 2 i f cos2 e ,
Ž3.
from which all three field parameters can directly be extracted. The Hamiltonian for the atom coupled to the field is in the rotating-wave and dipole approximations Ž " s 1. H s y 12 D Ž Pe y Pg . y 12 V Ž V e i f q V †eyi f . ,
Ž4.
with Pe and Pg the projectors on the substates of the excited and the lower levels, and D s v y v o the detuning of the laser frequency from the atomic resonance. The real Rabi frequency V is determined by the product of the amplitude A and the reduced transition dipole. The operator V is the raising part of the reduced dipole moment along the polarization direction, which can be expressed as VsePQ .
Ž5.
The reduced dipole operator Q is defined by the requirement that its spherical components have matrix elements between the Zeeman substates of the ground state and the excited state that are equal to Clebsch–Gordan coefficients, so that w20x ² Je , M e < Qb < J g , M g : s ² Je , M e < J g , M g ;1, b : for b s y1, 0, 1.
F s Tr r 12 = Ž V V e i f . q c.c.
Ž6.
Ž8.
When the rate of optical pumping is low compared to g , the excited-state population is small, and one can eliminate the excited state from the evolution equation. Adiabatic elimination of the optical coherences gives rise to a closed evolution equation for the ground-state submatrix, which we denote as s . Introducing the saturation parameter
V 2r4 Gs
Ž2.
with z1 and z2 two orthogonal real unit vectors, which determine the direction of the axes of the polarization ellipse, respectively. The right-hand Cartesian coordinate frame spanned by these two vectors and their cross product z3 s z1 = z2 is termed the polarization frame. The ellipticity angle e can be assumed to attain values between ypr4 and pr4, representing the range between left and right circular polarization. The amplitude A, the ellipticity angle e and the phase f of the field are determined by E according to the equations E ) P E s A2 ,
with g the rate coefficient of spontaneous decay. The solution r of the master equation Ž7. determines the average force on the atom, which is given by
Ž9.
g 2r4 q D 2
leads to the ground-state master equation w21x 1
d
g G dt
ssy
ž
1 2
D qi
g
/
V † Vs y
q Ý Qb† Vs V †Qb .
ž
1 2
D yi
g
/
s V †V
Ž 10.
b
This equation describes the evolution of the ground state due to optical pumping. The first two terms on the r.h.s. describe the loss and the light shifts of the levels. The last term is the corresponding gain term. A remarkable feature of this low-intensity master equation is that the evolution rate is simply proportional to the optical-pumping rate g G, and thereby to the intensity. This implies that scaling down the intensity means that the evolution of the ground-state density matrix s is simply slowed down by the same factor. In this limit of low saturation rate, the average force on the atom is an expectation value with respect to the ground-state density matrix s , as given by w21x Fsy
ir4
gr2 q i D
Tr s Ž V V †eyi f . = Ž V V e i f . q c.c.
Ž 11. One notices from Ž11. that the average force contains the gradients of the Rabi frequency V , the phase f , and the operator V, which is determined by the polarization vector e. The contribution containing =V has a dispersive dependence on the detuning D, and arises from the Hermitian part of the polarizability of the atoms w21x. It is proportional to the intensity gradient, and it has the nature of the dipole force. The contribution containing =f has an absorptive dependence on the detuning. As usual, this phase-gradient term has the nature of radiation pressure,
A.V. BezÕerbnyi et al.r Optics Communications 148 (1998) 151–158
and it can be expressed in the dissipative anti-Hermitian part of the polarizability. Finally, the contribution to the force that contains the gradient of the operator V or V † results from polarization gradients. This contribution generally contains both a dissipative and a dispersive part w21x. Polarization gradients play an essential part in sub-Doppler cooling w8x. The strength of the optical-pumping operator V † V is independent of the polarization vector. From the definition Ž5. one directly finds Tr V † V s
2 Je q 1 3
,
Ž 12.
irrespective of the type of polarization.
3. Steady-state radiative force for transition J ™ J We consider the situation of cold atoms, which do not move appreciably compared with the variations in the polarization pattern during the characteristic time of optical pumping Žg G .y1. Then we may assume that the density matrix of the atoms reaches the internal steady state that corresponds to the local polarization. Since the polarization varies typically over a wavelength, this requires that the atomic velocity Õ obeys the inequality Õ < Ggrk .
Ž 13.
Normally grk is considerably larger than the velocity grm that corresponds to optimized Doppler cooling w8x Ž m is the atom mass.. Hence, with sub-Doppler cooling techniques and for moderately low values of the saturation parameter G, the inequality Ž13. can easily be realized, except maybe very close to the field nodes. Hence, in the immediate vicinity of a node, the steady-state forces will tend to be smeared out. However, since the forces are very small in this region anyhow, this does not affect the pattern of the forces that the atoms experience. This justifies to approximate the force pattern by the steady-state force at any position. Obviously, the steady-state solution of Eq. Ž10. depends only on the polarization of the field, but not on the intensity or the phase. In the polarization frame, the steady-state density matrix depends exclusively on the ellipticity angle e .
'
3.1. Steady-state density matrix Recently it has been demonstrated that in the special case of a transition between two degenerate atomic levels with equal half-integer angular momenta J, an exact operator expression can be given for the steady-state solution of the master equation Ž7. w19x. In the absence of saturation, the ground-state density matrix, which is the steadystate solution of Eq. Ž10. is given by the simple exact expression
ss ŽV †V .
y1
rNJ ,
Ž 14.
153
with NJ a normalization constant. This steady-state solution is obvious, if one recognizes that in these cases the excited-state operator V Ž V † V .y1 V † occurring in the last term in Ž10. is isotropic. This implies that for a transition J ™ J with half-integer J, the excited-state density matrix is isotropic in the steady state, independent of the polarization. Eq. Ž14. shows that the steady-state density matrix s is diagonal on the basis of eigenstates of V † V, and that the populations are proportional to the inverse of the corresponding eigenvalues. Hence, the population accumulates preferentially in the states that are weakly coupled to the field. These eigenvalues and the normalization constant N depend exclusively on the ellipticity angle e w19x. For circular polarization, the ground-state operator V † V has one eigenvalue zero, and cannot be inverted. In the polarization frame, the Zeeman state with maximal value of the magnetic quantum number M is then a dark state, that is not coupled to the light field. In the steady state, only this dark state is populated. This degenerate case can be formally treated by adding an infinitesimal number d to V † V in Ž14., and taking the limit d ™ 0. 3.2. Force and field parameters An expression for the radiative force is obtained by substituting Eq. Ž14. in Ž11.. The result is Fsy
ir4
1
gr2 q i D NJ
Ž 2 J q 1 . Ž V =V q i V 2 =f .
qV 2 Tr Vy1 = V q c.c.
Ž 15.
Note that for the transition J ™ J the operator V is represented by a square matrix, which can be inverted in general for half-integer J. Recall that for equal integer J values there is a dark state for all polarizations w22x. The first two terms in Ž15. represent the dipole force and radiation pressure, which are proportional to the gradients of the intensity and the phase, respectively. The factor Ž2 J q 1.rN measures the effective coupling strength. The last term in Ž15. depends only on the gradient of the ellipticity angle, and not on the orientation of the polarization frame. This is typical for the force in the steady state. In order to evaluate the ellipticity-gradient force, we apply the matrix identity detV s expwTr ŽlnV .x, which holds for square matrices. After differentiation this gives Tr Vy1 = V s
1 detV
=detV .
Ž 16 .
The determinant of V is most easily found when we express the polarization vector e as linear superposition of circular and linear polarization w22x. For this purpose, we choose a Cartesian reference frame, which follows from the frame z1, z2 , z3 by a rotation along z2 , over an angle arccosŽtan e .. The polarization Ž2. is expressed in the new frame after the substitution z1cos e s u1sin e q u 3'cos2 e ,
Ž 17 .
A.V. BezÕerbnyi et al.r Optics Communications 148 (1998) 151–158
154
field parameters A, f and e . An explicit alternative expression for the force is Fs
2 Jq1 NJ
ž
g G =f y D =G y GD
1 cos2 e
/
=cos2 e .
Ž 21.
Fig. 1. Sketch of the relation between the quantization axes z3 and u 3 . The direction of z3 is normal to the polarization ellipse, which is the intersection of a plane with the cylinder with axis u 3 . Hence the polarization is a linear combination of u 3 and circular polarization in the plane normal to u 3 .
while noting that z2 s u 2 . The polarization vector e is then e s Ž u1 q iu 2 . sin e q u 3'cos2 e ,
Ž 18.
which is a superposition of circular polarization in the u1 u 2 plane, and linear polarization along u 3 , which serves as the quantization axis of this frame. The relative strength of the circular polarization is 2 sin2e , and the linear polarization has the complementary strength cosŽ2 e .. Geometrically speaking, u 3 is oriented along the axis of the cylinder whose intersection with the polarization plane coincides with the polarization ellipse w22,19x. This is illustrated in Fig. 1. When we take this new axis u 3 as quantization axis, the operator V couples a Zeeman substate < M : of the ground state only to the two substates < M : and < M q 1: of the excited states, as indicated in the level diagram in Fig. 2. This shows that detV is just the product of the diagonal elements. Hence we find J
detV s detV † s Ž cos2 e .
Jq1r2
Ł
² J , M < J , M ;1,0: .
Ms J
Ž 19.
One notices that the expression in brackets is fully independent of the value of J. The three terms in Eq. Ž21. correspond respectively to the radiation pressure, which is proportional to the phase gradient, the dipole force, determined by the intensity gradient, and the polarizationgradient force. Remarkably, this latter term is simply proportional to the gradient of lnŽcos2 e ., for each value of J. Moreover, its dependence on the detuning D is a dispersion curve, which indicates that the polarization gradient force can be entirely expressed in terms of the Hermitian, non-absorptive part of the polarizability w21x. The polarization-gradient force depends only on the gradient of the ellipticity angle, and not on the orientation of the polarization frame z1, z2 , z3. In the presence of saturation, the expression for the steady-state density matrix is only slightly modified w19x. Also in that case, the excited state remains fully unpolarized. The resulting expression for the force is Fs
2 Jq1 NJ q Ž 2 J q 1 . G
ž
= g G =f y D =G y GD
1 cos2 e
/
=cos2 e .
Ž 22 .
Hence, the effect of saturation is a multiplicative factor, that depends on the position-dependent intensity and ellipticity angle. 3.3. Localization points In order to evaluate the force explicitly, the expressions for the normalization constant NJ are needed. These are found directly from the matrix representation of the operator V with z3 as quantization axis, when using the explicit
Eq. Ž15. for the force can now be expressed in terms of a single gradient Fsy
ir4
1 2 Jq1 V 2
gr2 q i D NJ
2
EPE
= Ž E P E . q c.c.
Ž 20. Eq. Ž20. constitutes the general result of this paper. It is exact in case of the mentioned assumptions Žno saturation, steady state.. It is remarkable that the force can be expressed as the gradient of the square Žnot the absolute square. of the complex field E. Notice that according to Eq. Ž3., this expression depends on the gradients of all
Fig. 2. Level diagram for J s 3r2, with the quantization axis along u 3 . Transitions corresponding to linear polarization and to right-hand circular polarization are driven.
A.V. BezÕerbnyi et al.r Optics Communications 148 (1998) 151–158
expressions for the Clebsch–Gordan coefficients. The result can be expressed as w19x NJ s
2J
J Ž J q 1.
Ý
cos2 e
Aa
as0
ž
1 y cos2 e cos2 e
a
/
,
Ž 23.
with J
A0 s
1
Ý nsyJ Jy a
Aa s
n2
, nq a
1
Ý n syJ
n
2
Ž J q b .Ž J y b q 1 .
Ł
2b
bs n q1
2
First we consider the field configuration consisting of two linearly polarized counterpropagating waves of equal intensity, with an angle u between the polarization directions. This scheme has been shown to be efficient for sub-Doppler w24x or sub-recoil cooling based on polarization-gradient forces and coherent-population trapping w25– 27x. It is equivalent to two circularly polarized standing waves oscillating in phase, with a spatial phase shift equal to u . The electric field is written as
Ž 24. 6
F Žr. s
,
N3r2 s
160
20
. Ž 25 . cos 2 e 3cos 2 e cos 2 2 e In general, NJ equals Žcos2 e .y2 Jy1 times a polynomial in cos 2 2 e of the order J y 1r2. This shows that the force Ž21. Žor Ž22.. disappears when cos2 e s 0, which is the case at points with circular polarization. Typically, in an optical lattice the detuning is large, so that radiation pressure can be neglected. In the neighborhood of a point of circular polarization, NJy1 is well approximated by its leading term. Then the force can be written as F s y=F , in terms of the effective potential 2
GD J Ž J q 1. A 2 J
4
y
Ž cos2 e . 2 Jq1 .
Ž 26.
This demonstrates that in general an atom is attracted to a point of circular polarization when the detuning D is positive. Normally one would expect that cos2 e varies linearly with position. In that case, the restoring force is harmonic only in the special case that J s 1r2. Moreover, the force disappears also at the nodes of the field. At positive detuning, this force will be attractive in the neighborhood of the equilibrium point. In summary, atoms will tend to be localized at positions of circular polarization, or at the nodes of the field. These points are conveniently characterized as the zeros of E P E.
'2
ž
u
žž ˆ
xcos
2
u q ysin ˆ
u
q xcos ˆ
For J s 1r2 and J s 3r2, we find N1r2 s
E
EŽ r. s
.
155
2
u y ysin ˆ
2
2
/
eik z
/ /
eyi k z ,
Ž 27 .
with E the real amplitudes of the two traveling waves. The field parameters Ž3. are then determined by the identities E ) P E s E 2 Ž 1 q cos2 kzcos u . , E P E s E 2 Ž cos2 kz q cos u . .
Ž 28 .
Obviously, the one-dimensional lattice has 2 k as its reciprocal lattice constant, so that the periodicity of the lattice is half an optical wavelength. The phase f is zero, whereas the ellipticity angle is given by cos2 e s
cos2 kz q cos u
. 1 q cos2 kzcos u For the total force Ž20. Žor Ž21.. we find F s zˆ
2 J q 1 G0 D NJ
cos2 e
2 ksin2 kz ,
Ž 29.
Ž 30.
where G 0 is the saturation parameter corresponding to the field amplitude E. The force vanishes at the points of circular polarization, where NJ approaches infinity. These localization points occur at the positions z where 2 kz s Ž2 n q 1.p " u . There are two such points in a lattice constant, which is half of a wavelength. For small values of u , these points form closely spaced pairs, corresponding to the nodes of the two circularly polarized standing waves that compose the light field. In between such a pair, the
4. Two linearly polarized traveling waves in one dimension In this and the subsequent section, we give some explicit examples of the force pattern in optical lattices, which arise when atoms are bound in the periodic optical potential that is created by a monochromatic light field composed of a few traveling waves w11,10,13x. In general, when the traveling waves have wave vectors k m , the reciprocal lattice is spanned by the reciprocal lattice vectors k m y k mX . For two beams in one dimension, or three beams in two dimensions, the polarization pattern and the resulting force distribution is independent of the phase of the beams w23x.
Fig. 3. Plot of cos2 e as a function of 2 kz for the one-dimensional lattice of Section 4. The selected values of u correspond to cos u s 0 Žsolid line., 0.7, 0.99 and 1 Ždotted line..
A.V. BezÕerbnyi et al.r Optics Communications 148 (1998) 151–158
156
polarization varies rapidly between two opposite circular polarizations, as expressed by a similar rapid variation of the function cos2 e . This behavior of the ellipticity angle is illustrated in Fig. 3. The force Ž30. also disappears at the points where 2 kz s np , so that sin2 kz vanishes. However, for non-zero values of u , these equilibrium points are not stable. In the standing-wave case u s 0, localization occurs at the nodes 2 kz s Ž2 n q 1.p of the field.
5. Three linearly polarized traveling waves in two dimensions Next we consider the symmetric two-dimensional configuration of three linearly polarized traveling waves, so that 3
EsE
Ý
e mexp Ž ik m P rm . .
Ž 31.
ms 1
The wave vectors make an angle 2pr3 with each other, so that k 1 s kxˆ ,
k2 s
k 2
Ž yxˆ q '3 yˆ . ,
k3 s
k 2
Ž yxˆ y '3 yˆ . . Ž 32.
The periodicity of the field pattern is spanned by two reciprocal lattice vectors, which we choose as b 1 s k 2 y k 1 and b 2 s k 3 y k 1. This means that the pair of lattice vectors of a unit cell, which are biorthogonal to b 1 and b 2 , is given by a1 s
2p
Ž yxˆ q '3 yˆ . , 3k
a2 s
2p 3k
Ž yxˆ y '3 yˆ . . Ž33.
The unit cell is a diamond with an angle pr3, and its sides have length 2 lr3. The three linear polarization vectors are taken to have the same angle u with the z-axis, so that e 1 s ysin ˆ u q zcos ˆ u, e 2 s y 12 Ž xˆ'3 q yˆ . sin u q zcos ˆ u, e 3 s 12 Ž xˆ'3 y yˆ . sin u q zcos ˆ u.
Ž 34 .
Hence, in addition to the translational invariance over the lattice vectors, the field pattern has an obvious threefold rotational symmetry. A full parametrization of the field and of the steady-state force is given by the functions Ž3.. In order to facilitate the discussion, we introduce the scalar function 3
Zs
Ý
exp Ž ik m P r . ,
Ž 35.
Since the three wave vectors k m add up to zero, one directly proves the identity 3
Z2s
exp Ž 2 ik m P r . q 2 Z ) .
Ý
Ž 37.
ms 1
According to Eq. Ž3., the field parameters A, f and e are determined by the expressions E ) P E s E 2 Ž 3 q c Ž Z ) Z y 3. . , E P E s E 2 Ž Z 2 y 2 Ž1 y c . Z ) . .
Ž 38.
When we parametrize the position vector as a linear combination r s j 1 a1 q j 2 a 2 of the lattice vectors, we find Z s exp w y2p i Ž j 1 q j 2 . r3 x = w 1 q exp Ž 2p i j 1 . q exp Ž 2p i j 2 . x ,
Ž 39 .
which is symmetric for an interchange of j 1 and j 2 . This expression shows immediately that Z ) Z s 1 when j i s 1r2 q n for i s 1 or 2, and also for j 1 s j 2 q n q 1r2, for all polarization angles u . The intensity E ) P E s E 2 Ž3 y 2 c . is uniform along these lines. From Eq. Ž37. one derives that Z has two zeros in each unit cell, which occur at j 1 s 1r3, j 2 s 2r3 and at j 1 s 2r3, j 2 s 1r3. The maximal value of < Z
Z s 2 Ž 1 y c . exp Ž 2p inr3 . ,
Ž 40.
with n s 0, 1, 2. These are the points of circular polarization or the nodes of the field. At the solution Z s 0, the intensity of the field is E ) P E s 3 E 2 Ž1 y c ., and at the other solutions in Ž37. we find E ) P E s E 2 Ž1 y c .Ž2 c q 1.Ž3 y 2 c .. The fact that the intensity is the same for these solutions with non-zero values of Z suggests that these solutions are physically equivalent. This is confirmed by the combined rotational and lattice symmetry of the configuration. The function Z obtains a phase factor at translation over the lattice vectors Ž33., as expressed by Z Ž r y a i . s exp Ž 2p ir3 . Z Ž r .
for
i s 1,2 .
Ž 41.
Hence, from a solution of Z s 2Ž1 y c . one arrives at solutions of the last identity Ž37. by a simple translation over a lattice vector. Combined with the invariance of Z under rotations over 2pr3, this allows us to find all solutions after finding only the points where Z s 2Ž1 y c .. We now illustrate these general considerations in a few explicit cases.
ms 1
and the scalar product c s e m P e mX between two different polarization vectors as c s cos 2u y 12 sin2u .
Ž 36.
5.1. Parallel polarization Õectors When all polarizations are parallel to the z-axis, so that c s 1, the polarization is linear, and the ellipticity angle e
A.V. BezÕerbnyi et al.r Optics Communications 148 (1998) 151–158
is zero everywhere, Hence, the polarization-gradient force vanishes. In the standard situation for optical lattices that the detuning D is large compared with g , the force Ž21. can be expressed as a purely conservative dipole force F s =F with
Fs
Ž2 J q 1. D NJ
G,
Ž 42.
with G s G 0 Z ) Z. The normalization factor NJ s J Ž J q 1. A 0 follows from Eq. Ž23. in the special case that cos2 e s 1. The solutions Ž37. of the localization points now all merge to the single condition Z s 0, which in the present case corresponds to the nodes of the field. These are the small black dots in Fig. 4, which form a hexagonal lattice with side 2 l'3 r9. 5.2. Normal polarization Õectors A second relatively simple case occurs when cos u s 1r c s 0. This implies that the three polarization
'3 , so that
157
vectors form an orthonormal set, and the intensity E ) E s 3 E 2 is uniform. This configuration has been considered by Mauri and Arimondo as an example of velocity-selective coherent population trapping w28x. For D 4 g , when radiation pressure is negligible, only the last term in Ž21. is nonzero, and the force can be derived from a potential function of cos2 e . For J s 1r2, this potential has the form
F Žr. s
GD 6
cos 2 2 e .
Ž 43.
The localization points, which are the zeros of E P E s E 2 Ž Z 2 y 2 Z ) . now represent the points of circular polarization only. From Ž37. one finds in addition to the zeros of Z the points Ž j 1, j 2 . s Ž1r6,1r3. where Z s 2, Ž2r3,5r6. where Z s 2 expŽ2p ir3. and Ž1r6,5r6. where Z s 2 expŽy2p ir3., and the corresponding points with j 1 and j 2 interchanged. These additional solutions are indicated by the open dots in Fig. 4. The localization points for this case form again a hexagonal lattice, but now with side l'3 r9.
5.3. Polarization Õectors in plane All polarization vectors fall in the xy plane when c takes its minimal value y1r2. In this case, both the polarization and the intensity E ) P E s E 2 Ž9 y Z ) Z .r2 vary with position. The localization points. which solve Eq. Ž37., contain points of circular polarization, determined by Z s 0, and field nodes, indicated by Z s 3 expŽ2p inr3.. The points of circular polarization are the small black dots in Fig. 4. The nodes are located at the lattice sites, corresponding to integer values of j i , and they are indicated by the large black dots in Fig. 4. These two types of localization points combine into a regular triangular lattice, with lattice side 2 l'3 r9. At the points of circular polarization, the intensity is maximal. Hence, localization points correspond to the points where the intensity takes extreme values in this particular configuration. The polarization is linear at all straight lines connecting neighboring lattice points. These are represented by the drawn lines in Fig. 4. Fig. 4. Two-dimensional lattice pattern as discussed in Section 5. The inset gives the directions of the three k vectors. The thick arrows denote the two lattice vectors a1 and a 2 , which have length 2 l r3. The unit cells are indicated by the thick lines, and at the thin lines the intensity is uniform, for all polarization angles u . The small black dots are the locations where Zs 0. For parallel polarizations Ž u s 0. these are the field nodes, which are the only localization points. For normal polarizations, the localization points are the points of circular polarization, which are represented by the small dots Žblack and open.. For polarizations in the plane Ž u sp r2., the small black dots are the points of circular polarization, whereas the field nodes fall at the lattice sites, denoted by the large black dots.
5.4. Polarization at small angles u Finally, we consider the configuration where the polarization vectors make a small angle with the z-axis. Then we can write c s 1 y h , with a small value of h. The localization points are now determined by the equations Z s 0 and Z s 2h expŽ2p inr3.. The intensity at these localization points is given by E ) P E s 3h E 2 , to first order in h. As h deviates from zero, each localization point corresponding to the small black dots in Fig. 4
158
A.V. BezÕerbnyi et al.r Optics Communications 148 (1998) 151–158
separates into four closely spaced points. The four localization points around Ž j 1, j 2 . s Ž1r3,2r3. are specified by
References
Ž 1 y h'3 rp ,2 q h'3 rp . Ž 1 y h'3 rp ,2 y 2h'3 rp . , Ž 1 q 2h'3 rp ,2 q h'3 rp . .
w1x V.G. Minogin, V.S. Letokhov, Laser light pressure on atoms, Gordon & Breach, New York, NY, 1987. w2x A.P. Kazantsev, Y.I. Surdutovich, V.P. Yakovlev, Mechanical action of light, World Scientific, Singapore, 1990. w3x A.P. Kazantsev, I.V. Krasnov, JETP 95 Ž1989. 104. w4x J. Javanainen, Phys. Rev. Lett. 64 Ž1990. 519. w5x S. Chang, B.M. Garraway, V.G. Minogin, Opt. Commun. 77 Ž1991. 19. w6x A.I. Sidorov, R. Grimm, V.S. Letokhov, J. Phys. B 24 Ž1991. 3733. w7x M.G. Prentiss, N.P. Bigelow, M.S. Shahriar, P.R. Hemmer, Opt. Lett. 16 Ž1991. 169. w8x J. Dalibard, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6 Ž1989. 2023. w9x B. Sheehy, S.-Q. Shang, R. Watts, S. Hatamian, H. Metcalf, J. Opt. Soc. Am. B 6 Ž1989. 2165. w10x B. Lounis, P. Verkerk, J.-Y. Courtois, C. Salomon, G. Grynberg, Europhys. Lett. 21 Ž1993. 13. w11x N.P. Bigelow, M.G. Prentiss, Phys. Rev. Lett. 65 Ž1990. 29. w12x P. Verkerk, B. Lounis, C. Salomon, C. Cohen-Tannoudji, J.-Y. Courtois, G. Grynberg, Phys. Rev. Lett. 68 Ž1992. 1861. w13x P. Jessen, C. Gerz, P.D. Lett, W.D. Phillips, S.L. Rolston, R.J.C. Spreeuw, C.I. Westbrook, Phys. Rev. Lett. 69 Ž1992. 49. w14x A. Hemmerich, T.W. Hansch, Phys. Rev. Lett. 70 Ž1993. ¨ 410. w15x G. Nienhuis, Phys. Rev. A 26 Ž1982. 3137. w16x G. Nienhuis, Opt. Commun. 59 Ž1986. 353. w17x A.V. Bezverbnyi, V.S. Smirnov, A.M. Tumaikin, Zh. Eksp. Teor. Fiz. 105 Ž1994. 62 wJETP 78 Ž1994. 33x. w18x D. Suter, Opt. Commun. 86 Ž1991. 381. w19x V. Taichenachev, A.M. Tumaikin, V.I. Yudin, G. Nienhuis, Zh. Eksp. Teor. Fiz. 108 Ž1995. 415 wJETP 81 Ž1995. 224x. w20x G. Nienhuis, Phys. Rep. 138 Ž1986. 151. w21x G. Nienhuis, P. van der Straten, S-Q. Shang, Phys. Rev. A 44 Ž1991. 462. w22x A.M. Tumaikin, V.I. Yudin, Zh. Eksp. Teor. Fiz. 98 Ž1990. 81 wJETP 71 Ž1990. 43x. w23x K.I. Petsas, A.B. Coates, G. Grynberg, Phys. Rev. A 50 Ž1994. 5173. w24x V. Finkelstein, P. Berman, J. Guo, Phys. Rev. A 45 Ž1992. 1829. ¨ P. Zoller, M.S. Shahriar, M. w25x P. Marte, R. Dum, R. Taieb, Prentiss, Phys. Rev. A 49 Ž1994. 4826. w26x M. Weidemuller, T. Esslinger, M.A. Ol’shanii, A. Hem¨ merich, T.W. Hansch, Europhys. Lett. 27 Ž1994. 109. ¨ w27x R. Dum, M.A. Ol’shanii, Phys. Rev. Lett. 76 Ž1996. 1788. w28x F. Mauri, E. Arimondo, Europhys. Lett. 8 Ž1991. 171.
3 Ž j 1 , j 2 . s Ž 1,2 . ,
Ž 44.
Around the point Ž2r3,1r3., the four solutions of Z s 0 and Z s 2h expŽ2p inr3. Ž n s 0,1,2. are
Ž 2 q h'3 rp ,1 y h'3 rp . Ž 2 y 2h'3 rp ,1 y h'3 rp . , Ž 2 q h'3 rp ,1 q 2h'3 rp . .
3 Ž j 1 , j 2 . s Ž 2,1 . ,
Ž 45.
For small values of u , the polarization varies rapidly in these regions, giving rise to large gradients of cos2 e .
6. Conclusions The effects of optical pumping on the internal state of atoms depend strongly on the polarization of the driving field. Characteristic for situations of polarization-gradient cooling or optical lattices is that the polarization varies continuously through all possible values of the ellipticity. We have applied a closed analytical expression for the steady-state density matrix of an atom driven by a radiation field with arbitrary polarization, to study the pattern of radiative forces on the atom. The treatment is valid for transitions J ™ J with arbitrary half-integer J-values. The expressions Ž20. or Ž21. for the force depend on the value of J only through the normalization factor NJ , that is a function of the ellipticity of the local polarization. The force contribution from the polarization gradient is purely redistributive, so that it is independent of the anti-Hermitian part of the polarizability w21x. The force disappears both at the nodes and at the points of circular polarization. For large positive values of the detuning, these equilibrium points are stable. As an illustration, we analyze the force pattern for a physically interesting class of one- and twodimensional optical lattices.
Acknowledgements This work was supported by a grant from the Netherlands Organization for Research ŽNWO..