Orbital Angular Momentum Exchange in the Interaction of Twisted Light with Molecules

VOLUME 89, N UMBER 14 PHYSICA L R EVIEW LET T ERS 30 SEPTEMBER 2002 Orbital Angular Momentum Exchange in the Interaction of Twisted Light with Molecules M. Babiker,1 C. R. Bennett,2 D. L. Andrews,3 and L. C. Dávila Romero3 1 Department of Physics, University of York, Heslington, York, YO10 5DD, United Kingdom 2 RD114, QinetiQ, St. Andrews Road, Malvern, Warcs WR14 3RS, United Kingdom 3 School of Chemistry, University of East Anglia, Norwich, NR4 7TJ, United Kingdom (Received 29 April 2002; published 11 September 2002) In the interaction of molecules with light endowed with orbital angular momentum, an exchange of orbital angular momentum in an electric dipole transition occurs only between the light and the center of mass motion; i.e., internal ‘‘electronic-type’’ motion does not participate in any exchange of orbital angular momentum in a dipole transition. A quadrupole transition is the lowest electric multipolar process in which an exchange of orbital angular momentum can occur between the light, the internal motion, and the center of mass motion. This rules out experiments seeking to observe exchange of orbital angular momentum between light beams and the internal motion in electric dipole transitions. DOI: 10.1103/PhysRevLett.89.143601 During the past decade or so, the orbital angular momentum (OAM) associated with certain types of laser light has been the focus of much attention in both theoretical and experimental contexts. Interest evolved from work by Allen et al. [1], who showed that LaguerreGaussian light carries OAM in discrete units of h
associated with the azimuthal phase dependence of the field distribution. Since then, a number of experiments have demonstrated the influence which the OAM of light imparts on polarizable matter, leading to novel features, such as the optical spanner effect [2 – 4]. The manifestation of OAM in interactions of Laguerre-Gaussian light with atoms has been explored theoretically, leading to predictions of a light-induced torque which can be used to control the rotational motion of atoms and ions [5]. Berry [6] showed theoretically that the OAM is an intrinsic property of all types of azimuthal phase-bearing light, independent of the choice of axis about which it is defined. A review of the work carried out until 1999 is given in Ref. [7]. More recently, O’Neil et al. [8] investigated the classification of OAM in terms of intrinsic and extrinsic types in the context of Laguerre-Gaussian light, and Muthukrishnan and Stroud [9] explored the entanglement of internal and external angular momenta in a single atom. There is also a very recent report on the measurement of optical OAM [10]. It can be argued that, if orbital angular momentum is indeed an intrinsic property of light, then in its interaction with a bound system of charges such as an atom or a molecule, an exchange should arise between light and matter, especially in a transition between the energy levels, just as the photon spin angular momentum manifests itself in the interaction of circularly polarized light in a radiative transition. The purpose of this article is to explore the validity of this argument using a prototypical model of a molecule interacting with a simple form of light carrying OAM. We focus on an electromagnetic light mode of frequency ! and orbital angular momentum lh,
with an 143601-1 0031-9007=02=89(14)=143601(4)$20.00 PACS numbers: 42.50.Vk, 32.80.Lg, 42.50.Ct electric field vector distribution expressible in cylindrical polar coordinates r
rk ; z  rk ; 0 ; z as follows [11]: 0 Ekl r; t

^ Frk eikz!t eil ; (1) where
^ is a wave polarization vector and Frk  is a scalar distribution function depending only on the radial coordinate. For the molecule we consider a bound system of charges in simplest form, namely, a hydrogenic twoparticle system consisting of a spinless electron (referred to as e  ) of mass m1 and charge e and a spinless nucleus (referred to as e  ) of mass m2 and charge e, with e the magnitude of electron charge. It is straightforward to derive the Power-Zienau-Woolley (PZW) Hamiltonian of this system in interaction with the light field [12,13], expressible as a sum of four parts: 0 0  H0 H
HM  H fields  Hint : 0 HM (2) P2 =2M
is the center of mass Hamiltonian, which is essentially its kinetic energy operator, with P the center of mass momentum and M
m1  m2 its total mass. The center of mass momentum is conjugate to the center of mass coordinate R, defined in terms of the particle position vectors qi ; i
1; 2, by R
m1 q1  m2 q2 =M. Figure 1 schematically shows the position vectors q1 and q2 and that of the center of mass R. We are, however, interested in the possibility of the center of mass rotating about a beam axis in which case we should write L2z P2  z ; (3) 2I 2M where Lz is the angular momentum operator, I is the moment of inertia of the atomic center of mass about the z axis, and Pz is the center of mass momentum axial vector component. The second term in Eq. (2) pertains to the internal ‘‘electronic-type’’ motion, p2 e2 0 H  ; (4)
2 40 q 0 HM
where 
m1 m2 =M is the reduced mass and p is the  2002 The American Physical Society 143601-1 VOLUME 89, N UMBER 14 z PHYSICA L R EVIEW LET T ERS e+ ~ kl r; t is the second quantized form of the elecwhere E tric field in Eq. (1); P r is the electric polarization defined in closed integral form by q CM e- q2 30 SEPTEMBER 2002 P r
X 
1;2 R e Z1 0 dq  R r
R  q  R : (7) q1 For simplicity, we have ignored all magnetic interactions. Note that, although the electric polarization field defined in Eq. (7) appears to be a function of q1 and q2 , it can be written entirely in terms of the relative coordinate q using y q1|| R|| q2|| q1;2  R
m2;1 q=M: x q|| FIG. 1. The particle position vectors and that of the center of mass for the two-particle model of the molecule. The projections of these vectors in the x-y plane are also shown. momentum conjugate to the internal coordinate q
q1  q2 . The second term in Eq. (4) is the Coulomb potential binding the two-particle system, with q
jqj. The third term in the total Hamiltonian is defined by Hfield
h!a
ykl akl ; (5) which is the field Hamiltonian in quantized form with akl the annihilation operator of the light mode of frequency !, orbital angular momentum lh,
and axial wave vector k
kz^ . The validity of such quantization for beams with OAM has recently been vindicated in work by Dávila Romero et al. [11]. Finally, the last term in (2) is the Hamiltonian describing the coupling between the light and matter. In the PZW scheme, this can be written as [12,13] Z ~ kl r; t; Hint
 d3 r P r E (6) Hint
e
Z d3 r e q M Z1 0 Z1 0 We start by specifying zero-order states of the overall motion, comprising the center of mass motion (rotational and translational), the internal ‘‘electronic-type’’ motion, and the field state. The appropriate states are product 0 states of the three-subsystem Hamiltonian H 0
HM  0 0 H  Hfield expressible as jPz ; Lz ; j; fNkl gi. The unperturbed center of mass motion in this state is represented by an axial translational state with linear momentum Pz , together with a rotational eigenstate of the angular momentum operator Lz with eigenvalues hL
z . The internal motion enters in terms of the hydrogenic excited discrete states jji  jei of energy Ee and a ground state jji  jgi of energy Eg . The notation jei and jgi stand for jne ; le ; me i and jng ; lg ; mg i, respectively, where nj ; lj ; mj with j  e; g are hydrogenic state quantum numbers. Finally, the ketjfNkl gi is the number state of the light field. The coupling between matter and field invokes the interaction matrix element Mif , where jii  0 jPz ; Lz ; e; fNkl gi and jfi  jP0z ; L0z ; g; fNkl gi. Specifically, Mif
hPz ; Lz ; e; fNkl gj Z d3 rP r ~ kl r; tjP0z ; L0z ; g; fN 0 gi: E kl (9) To proceed, we express Eq. (6) as follows:      m2 m2 m1 m1 ~ kl r; t d q r
R   q  q r
R   q E M M M M ~ kl R  m2 q=M; t  m1 E ~ kl R  m1 q=M; tg
H1  H 2 ; dfm2 E int int where we have used Eqs. (7) and (8) and the electric field is that given by Eq. (1), evaluated at r
R  m2 q=M in 1 2 Hint and at r
R  m1 q=M in Hint . The most useful form of the interaction Hamiltonian in Eq. (10) is a multipolar. It should, however, be emphasized that, while we can treat the magnitude of the internal coordinate as small relative to a wavelength, we cannot say that the internal azimuthal angle is small, or set it equal to that of 143601-2 (8) (10) the center of mass. We need to incorporate the full azimuthal angular dependence which must be split into internal and center of mass dependences. To establish the azimuthal angular dependence, we consider projections of relevant vectors in the x-y plane. Figure 2 shows the vectors Rk , m2 qk =M and their sum Rk  m2 qk =M in the con1 text of Hint . A similar figure can be constructed for the 143601-2 VOLUME 89, N UMBER 14 PHYSICA L R EVIEW LET T ERS 30 SEPTEMBER 2002 vectors Rk ; m1 qk =M and their sum Rk  m1 qk =M in M sin  R  sin   2 the context of Hint .
; (11) 1 m2 qk Rk Concentrating on Hint first, the azimuthal angles associated with the relevant vector projections as shown in which, after expanding the sines, yields Fig. 2 are as follows:  is the azimuthal angle of the vector Rk  m2 qk =M, while R is that for the center of m2 qk sin  MRk sinR  tan
: (12) mass coordinate Rk and  is the internal azimuthal m2 qk cos  MRk cosR  angle, which is also the azimuthal angle of the vector ~ kl and making use of Eq. (1), we have m2 qk =M. The sine rule immediately gives Substituting for E Z1 em2 1 Hint dFjRk  m2 qk =MjeikRz m2 qz =M eil ei!t akl  H:c:; (13)

^ q M 0 where H.c. is the Hermitian conjugate. The azimuthal exponential factor in Eq. (13) is 1l 0 m q cos  MR cos   i m q sin  MR sin  2 k k R 2 k k R A: eil
@q 2 2 m2 qk cos  MRk cosR   m2 qk sin  MRk sinR  (14) This can be simplified to give 0 eil
@q m2 qk ei  MRk eiR m2 qk 2  M2 R2k  2m2 Mqk Rk cos  R  1l A: (15) We now make use of the ‘‘multipolar’’ approximation, m2 qk =M  Rk ; m2 qz =M  Rz : (16) These facilitate the next steps starting from Eq. (13) involving the expansion around Rk of the function FjRk  m2 qk =Mj, together with the expansion of the exponential term containing z components around Rz . A similar treatment is needed to expand the azimuthal factor in Eq. (15). These steps are followed by integration over . We obtain ^ k
cos  R , where carets denote unit vectors, from Eq. (13) using q^ k R     qm em m kq 1 il1R i il1R i  Hint  2
^ qeikRz 1  i 2 z ei!t akl  FRk eilR  k 2 G R e e R e e  G H:c:; k k l l M M 2M (17) il R  arising from differentiating FR  with respect to R , and expanding e , the functions G k k k l  1 dF lF  : (18) G l Rk 
4 dRk Rk 2 to the analogue of Eq. (17): Following similar steps, we can reduce the expression for Hint    qk m1  em1 m1 kqz i!t 2 il1 i il1 i  il ikR R R R z Gl Rk e Hint  e H:c: e  Gl Rk e akl  FRk e  1i e
^ qe M M 2M (19) The total interaction Hamiltonian is the sum of the expressions in Eqs. (17) and (19). The interaction Hamiltonian in the electric dipole approximation emerges from the sum of the terms linear in the vector components of the internal coordinate q. We have Hint dipole
e
^ qeikRz FRk eilR ei!t akl  H:c:; (20) and we see that, besides the internal operator e
^ q, the dipole approximation involves only the center of mass cylindrical coordinates Rk ; R ; Rz . Substitution of this in Eq. (9), writing the explicit forms of the translational and rotational eigenstates of the center of mass motion, 143601-3 and performing the space integrals, we obtain 1=2 i!t Mif
22 hej
^ djgiNkl e Lz L0z ;l
 Pz  P0z  hkM; k (21) where d
eq is the electricRdipole moment vector and Mk is the integral Mk
1 0 dRk Rk FRk . The Dirac delta function in Eq. (21) exhibits conservation of the center of mass axial linear momentum with conventional linear momentum transfer between the light and the center of mass. The Kronecker delta expresses conservation of orbital angular momentum; there is clearly OAM transfer of magnitude lh
between the light and the center of mass rotational motion. In this electric dipole 143601-3 VOLUME 89, N UMBER 14 PHYSICA L R EVIEW LET T ERS y Φ Φ R + (φ − Φ) q e- R x CM φ FIG. 2. The vector projections in the x-y plane and the corresponding azimuthal angles for the vectors Rk , 1 . m2 qk =M and their sum Rk  m2 qk =M in the context of Hint approximation, the internal motion does not participate in any exchange of momentum with the light beam, neither linear momentum nor orbital angular momentum. Consider next terms quadratic in vector components of q . These correspond to quadrupole interaction and are classifiable into three kinds. The first is of the form 1 Hint qq

^ qqz eilR FRk eikRz ei!t akl  H:c:; (22) 30 SEPTEMBER 2002 we conclude that a transfer of OAM occurs between the internal motion and the light beam, enhancing the beam by one unit to l  1h
which is transferred to the center of mass rotation. It is easy to check that the integrals over the internal azimuthal angle  lead to the usual quadupole selection rule jme  mg j
0; 1; 2, where me and mg are the azimuthal quantum numbers of the respective internal states jei and jgi involved in the transition. In conclusion, we have demonstrated by explicit analysis that, in the interaction of light possessing orbital angular momentum with atoms or molecules, the major mechanism of exchange occurs in the electric dipole approximation and involves only the center of mass motion and the light beam. The internal ‘‘electronic-type’’ motion does not participate in any OAM exchange with the light beam to this leading order. It is only in the weaker electric quadrupole interaction that an exchange involving all three subsystems (the light, the atomic center of mass, and the internal motion) can take place. This involves one unit of OAM exchanged between the light beam and the internal motion resulting in the light beam possessing l  1h,
which are then transferred to the center of mass motion. These conclusions rule out any experiments which seek to observe orbital angular momentum exchange involving light beams and the internal states of molecular systems via electric dipole transitions. The authors are grateful to Professor Les Allen for useful discussions and for providing a copy of Ref. [8] prior to publication. L. C. Dávila Romero wishes to thank the EPSRC for financial support. where  is a constant. The second type is of the form 2 ikRz i!t e akl qq
)
^ qqk ei eil1R G Hint l Rk e  H:c:; (23) where ) is a constant. The third is of the form 3 ikRz ei!t a Hint qq
)
^ qqk ei eil1R G kl l Rk e  H:c: (24) 1 qq, Eq. (22) is inserted It is easy to show that once Hint in the matrix element in Eq. (9), this term cannot mediate any transfer of OAM between the light and the internal motion. However, transfer of OAM does occur between the light and the center of mass motion, as in the electric dipole case. By contrast, we see in the expression for 2 Hint qq in Eq. (23) that a factor ei now appears in the matrix element between internal states jei and jgi, and the center of mass azimuthal phase factor is now eil1R . This is indicative of transfer of OAM from the light beam to the internal motion, leaving only l  1h
which are transferred to the center of mass rotation. Similarly when 3 Hint qq, Eq. (24), is substituted in the matrix element, 143601-4 [1] L. Allen, M.W. Beijesbergen, R. J. C. Spreeuw, and J. P. O. Woerdman, Phys. Rev. A 45, 8185 (1992). [2] J. Courtial et al., Opt. Commun. 144, 210 (1997). [3] M. E. J. Frise et al., Nature (London) 394, 348 (1998). [4] H. He et al., Phys. Rev. Lett. 75, 826 (1995). [5] M. Babiker, W. L. Power, and L. Allen, Phys. Rev. Lett. 73, 1239 (1994). [6] M.V. Berry, in Proceedings of the Meeting in Singular Optics, Frunzenskoe, Crimea, 1998, SPIE Proceedings Vol. 3487 (SPIE, Bellingham, WA, 1998), p. 6. [7] L. Allen, M. J. Padgett, and M. Babiker, Prog. Opt. XXXIX, 291 (1999). [8] A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002). [9] A. Muthukrishnan and C. R. Stroud, Jr., J. Opt. B 4, S73 (2002). [10] J. Leach et al., Phys. Rev. Lett. 88, 257901 (2002). [11] L. C. Dávila Romero, D. L. Andrews, and M. Babiker, J. Opt. B 4, S66 (2002). [12] V. E. Lembessis, M. Babiker, C. Baxter, and R. Loudon, Phys. Rev. A 48, 1594 (1993). [13] J. C. Guillot and J. Robert, J. Phys. A 35, 5023 (2002). 143601-4