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The optical Faraday effect and optical MCD
of Moiecdar Liquids, 65 (1993) 127-136 Elsevicr Scienx Publishers B.V., Arm&dam
JournaC
T’HZ QPTICAL
FARUIAY
EFFECT
127
AND
OPTICAL
MCD
M.W. EVANS 72.2Theory Center, Cornell University, 2 i 5orutory, Penmylvania
Ithaca, .YY 14853, and Materials Research state University, University Park, PA 76802, USA
(Received 25 September 199x) ABSTFLACT Based on the recent theoretical deduction of the photon’s static magnetic field operator &I, the opticnl Faraday effect and optical MCD are proposed. It is shown that &r rotates the plane of a linearly poirtized probe, an effect which is fkquency dependent and which gives rise to optical magnetic circular dichrrjism (optical MCD), in which the conventional magnet is replaced by a &I gewrtltxw, a circularly polarized loser generating a magnetostatic field, each phokx of which carries the quantum of flux &r.
INTRODUCTION
This paper continues a series of articles in which the consequences are developed of the recent deduction [l-6] that the photon generates a magnetic
flux quantum,
an operator
Here BO is the scalar magnetic flux density amplitude of a beam of N photons (a circularly polarized generator laser), 3 is the photon’s angular momentum operatcr [71 whose eigenvalues are &i&E, where MJ, is plus or minus one, and where A is the reduced Plan& constant. The operator 41 ch anges sign with the circular polarity of the generator laser, is unlocalized in space, and has eigenvalues &A&.23,where 23, is the laser’s scalar flux density amplitude. Its classical equivalent is the axial vector I+, a novel magnetostatic flux density generated by circularly polarized electromagnetic plane waves [l-6]. The theoretical existence_ of &, is supported by the experimexdal evidence for light induced magnetic ef0167-7322/93/$06.00
0 1993 -
Elsevier Science Publishers B.V. All rights reserved.
fects. The first to be described (in the sixties) was the inverse Faraday eTTect[S-141, and recently it has been shown 115,161 that NMR resonances arti shifted in new and useful ways by the magnetizing effect of circularly polarized argon ion radiation at frequencies far from optical resonance (i.e. where the sample is transparent $0 the argon ion radiation and dozs noL absorb it). Both these effects can be described in terms of the operator It has also been proposed &V or its classical equivaient, BJI [l-6,17-23]. [4-61 that there exists an optical Z?eman effect, in which &,, splits singlet electric dipole transition frequencies in atoms; an anomalous optical Zeeman effect for atomic triplet states; and an optical Paschen-Back effect. It has also been proposed theoretically [2,3] that & can be detected by examining spectrally a circularly polarized laser at visible frequencies reflected from an electron beam. The existence of A,, also implies that of an optical Faraday effect, in which it rotates the plane of a linearly polarized probe, an effect which is frequency dependent, and which gives rise therefore to optical ,magnetic circular dichroism (optical MCD spectroscopy). These effects are developed theoretically in this paper for atoms. Mason [24] has given an interesting discussion of cause and effect in chirality, a discussion which includes some pertinent historical analysis of interest here. In 1846, Faraday showed experimentally [25] that a magnetostatic field induces optical activity in flint glass and other isotropic transparent media. In 1884, Pasteur 1261 proposed on the basis of this result that the magnetic field represented a source of chirality. It is now known [27j that the magnetically induced optical activity in tfie Faraday effect has a fundamentally different symmetry [2S-31] frdlrnthat of natural optical activity. Nonetheless, it is interesting for our purpose in this paper to nota, following Mason 1241, that le Be1 in 1874 [32] had independently proposed that circularly polarized radiation also provides a “chiral force” of the type envisioned by Pasteur emanating from the magnetostatic field used by Faraday. Enantio-differentiating photoreactions were indeed reported by Kuhn and Braun in 1929 [33], and it has been proposed repeatedly (e-g- Bonner E3431that circularly polarized solar irradiation may be responsible for the well known preponderance in nature of one enantiomer over another. However, Mason [24] favours the universal and parity violating mechanism of the electroweak force as the &gin of this dissymmetry because the electroweak force does not depend on time and location on the earth’s surface. (Natural solar radiation is only 0.1% circularly polarized, and equally and oppositely so at dawn and dusk 1241.)
129
It is therefore interesting for our purpose to note that both circular polarity in light and the magnetostatic field have been proposed independently (by !e Be! and Pasteur respectively) as sources of chiraiity. It had therefore been sensed more than one hundred years ago that these two concepts have something in common in their effect on material. Per&an et al. [2i, in their first paper on the experimental tiemonstration of the magnetizing effect of circularly polarized giant ruby laser radiation, edged towards the concept of &,, by describing the effect of the laser as being due to an effective magnetostatic field. In view of these indications, those by Kielich and co-workers 112,131, and by Atkins and Miller [14], the present author appears to have demonstrated conclusively [l-6] the fact that circularly polarized radiation generates the classical magnetostatic field BII whose equivalent in quantum field theory is the photon’s flux quantum A,,, a nave; fundamental property of quantum field theory. As noted elsewhere, the concept of A,, must be clearly and carefully distinguished from the well known (“standard IUPAC”) oscillating 3 field of the electromagnetic plane wave; the two are quite different [l-6] in nature. We have thereforo resolved [l-6] the conjectures of le Be1 (1874) and Pasteur (1884) insofar as to show that circularly polarized light can indeed act in the same way as a magnetostatic field, a finding which implies the existence of several novel types of spectroscopy in circumstances where a conventionally applied magnet would be replaced by a circularly polarized laser. In Section 1, the contemporary quantum theory of Faraday’s effect of 1846 is developed succinctly for atoms, whereby it becomes relatively straightforward to show that the flux quantum & must also generate Faradafs observation of optical activity in all material, inherently (structurally) chiral or otherwise. In our ease, the magnet used by Faraday is replaced by a circularly polarized laser which generates 811, and is therefore referred to as the “generator laser? Section 2 develops the frequency dependence of the optics1 Faraday effect in atoms through the properties of the magnetically (i.e. An> perturbed antisymmetric polarizability of conventional contemporary Faraday effect theory [35], and therefore arrives at expressions in atoms for optical Finally a discussion is given of possible experimental configurations and order of magnitudes of the expected optical Faraday effect in terms of the intensity in watts per unit area of the generator laser.
nm3.
130
1. OPTICALLY
INDUCED
FARADAY
ROTATION
IN ATOMS
The contemporary quantum theory of the Faraday effect is based on the well known work by Serber 1361which was the precursor for the well known A, B and C terms 371. This section develops the Serber theory for use with a flux quantum B 11from the generator laser, and shows that the analogy between the optical and conventional Faraday effects is easily forged by using & in place of the conventional magnetic field BO from a magnet. The starting point for the theory of the conventional Faraday effect in quantum mechanics is an nr?quationfor the rotation of the plane of linearly po’ltized radizl+yisn. This is derived [3?] by a consideraticn ofthe effect of a magnetostatic field on the antisymmetric part of an atomic or molecular property tensor called the antisymmetric polarizability, q-“:
where qjh” is a perturbation tensor of order three. Before embarking on detailed theoretical development it is instructive to consider the fundamental symmetries of these atomic property tensors (we restrict consideration in this paper to atoms) and to recall that the complex electronic imaginary electric polarizability, of which CY+” is the antisymmetric, component, is derived from time dependent perturbation theory within whose framework 1371 qj” is a product of two transition electric dipole moment operators. The perturbation tensor qjk” in this context involves -P moment matrix element. two of these and one transition magnetic c: This structure introduces frequency depenacixx into the conventional Faraday effect, as is well known, leading to MCD. Similarly, frequency dependence occurs in the optical Faraday effect, leading to optical MCD. The fundamental symmetries considered are parity inversion (represented by the operator $1 and motion reversdi (by the operator fi. In this context q f’ is negative to P and positive to P, while qjk” is positive to both p and ?‘.‘Therefore CQ” is finite only in the presence of a p negative influence, such as IBOor &, and this influence is mediated by qjk”, which is finite for all atoms and molecules and is described by the ubiq.uitous B term 1371. The contemporary theory of Faraday’s effect depends on a perturbation of a quantity qj” which is itself the result of semi-classical time dependent second order perturbation theory. The basic reason for t-his is that the observable in the Faraday effect is an angle of rotation (A9) (or alternatively a change in ellipticity Aq) which must be calculated from Maxwell’s equations or Rayleigh refringent scattering theory [37].
131
Although the A terns is closely related to the Zeeman effect 1383 the observables of the two effects are quite different in nature, being traditionally an angle of rotation (Faraday effect) and a frequency shiR in the visible frequency region (Zeeman effect). In consequence the latter can be described by an energy perturbation, while Faraday’s effect needs perturbation of the antisymmetric polar&ability, because energy does not appear directly in Mm*ell’s equations, which are needed to calculate refractive i.ndices and therefrom A9 and Av. With these considerations, the starting point for our development of the optical Faraday effect and optical MCD is the equation I371 for angle of rotation in the conventicnal quantum theory of the Faraday effect:
(3)
Here, A8 is the angle of rotation of plane polarized probe radiation of angular frequency w parallel to the coaventicnally generated magnetostatic flux density Bz;- The quantity ATis the total number of molecules per unit volume in a set of degenerate quantum states of the atom, individually designated [37] yG ah= &Zis the degeneracy and the sum is over all componsnts of the degenerate set, with Nd = Net,. In eqn. (3), b is the vacuum permeability, e the speed of light, 2 the sample length, mnz the 2 component of the atomic magnetic dipole moment in state n; and li7’ ia the thermal energy Der atom. Recall that the atomic property tensors qj” and qjk’* are derived from semi-classical, time dependent, petiurbation theory and are frequency dependent in general, so that A6 mapped over a frequency range has the appearance of a spectrum - the well known, conventional MCI) spectrum. Our task here is to incorporate the novel flux quantum fl, [l-6] into eqn. (31, and thus generate the optical Faraday elect and optical MCIX In order to proced, we note that (oI;i”’ and qjk’* as used in eqn. (3) F expectation values of the respective quantum mechanical operators aij” and &+N in the same way that rn~ is y expectation value of the magneiic electronic dipole moment operator, m,. The quantity BOz is a classical magnetostatic vector component, and AB is an expectation value of the operator A& It is convenient to transform the appropriate cartesian components of the operators &” and &Yk”into spherical form 137,391, using the CondonlShortley phase convention:
132
(4)
Both kYt8and GYk”are purely imaginary in.lhe appppriate spherical representation, indicating that the operators r4 and ig are anti-hermitian, with purely imaginary eigenvalues 1371. (Kate preftlly, however, that the p symmetry of& is negative, while that of ic$ is positive. Both have positive @ symmetry.) The next step in our development for atoms is to replace thz vector component BOz of the conventional quantum theory bv “tile quantum mechanical operator defined by eqn. (I), which has aE ~nxted with it the anguiar momentum quantum numbor J. An immediate consequence of this replacement is the necessity to consider the magnetic field flux quantum using irreducible tensorial methods of angular momentum coupling theory in quantum mechanics [39+&l]. In other words we are now ccnsidering a quantized photon angular momentum interacting with an atom, w*hichin general contains, as usual, quantized orbita and spin electronic angular momentum. Without loss of generality we restrict consideration to atomic singlet states, in which there is a net orbital electronic angular momentum &, but no net spin angular momentum &. With these considerations, eqn. (3) becomes, for the optical Faraday effect:
133
(6)
where we have used an unco~&d representation (39411 to describe the net angular momentum generated during the interaction of photon and atom. This is ju*stified through the fact that the axim-uthal components of the operators m, and &, are both Nell defined in the uncoupled representation [3941]_ whereas in the coupled representation of the same problem the to%1 angular momentum is defined but the individual azimuthal components are not. With these considerations, the expectation value of the angle ofro’kkion in the optical Faraday effect is (7) (3flL - UL + 1)) ml&a
+ (L(L + 1) (2L + i) (U, + 3j (U, - I))*
1
where we have used the fact that the expectation value of the photon’s B n operator is kMJBO, positive for left circularly polarized ra&ation and negative for right circularly polarized radiation from the generator laser. This is an expression for the angle of rotation induced in plane polarized probe radiation in a sam 3e of atoms by a circularly poltized laser generating the flux quantum 1 II_The observation of such a rotation would provide a test for the existence of &. Equation (7) is written out in terms of reduced matrix elements of dipole moment and atomic polarizability operators, matrix elemenk which are products of eleckic and magnetic dipole transition dipole moment matrix elements from time dependent perturbation theory [S71. These introduce frequency dependence into the angle of rotation of the optical Faraday effect. The selection rules governing the various atomic property tensors are as foilows: &AL=o;AML=@; (9) ~:AL=O;f2;AM~=O; theAL=k
I part is parity forbidden,
as in magnetic dipole transitions.
134 2. OPTICAL
MCD,
FREQUENCY DEPENDENCE OF AI3
The origin of frequency dependence in the optical Faraday effect can be traced to semi-classical time dependent perturbation theory, which produces expressions for the polarizability components as given in the conventional theory of magnetic electronic optical activity [371. These can be further developed as usual in terms of reduced matrix elements of electric and magnetic transition dipole moment operators. For a given generator laser intensity and frequency therefore the optical MCD spectrum is a plot of the &Ii induced angle of rotation A0 against the frequency ef the linearly polarized pro,&!. Experimentally, this is built up in princziple by replacing the conventional magnet of MCD apparatus by the circularly polarized generator laser. DISCUSSION it is interesting to note that using the concept of &I1 the theory of the optical Faraday effect can also be developed and understood simply by replacing the magnetic flux density vector component Boz wherever it occurs in the convectional theory of MCD by the quantity kB&‘& where B. is the *magneticflux drmsity ampiitude of the generator laser and &WJ the two possible azimuthal quantum numbers of the photon. Thus, the optical Faraday effect can be’developed along the lines of the conventional counterpart in Serbeis A, B, and C terms, a convenient description of which is given by Barron 1371 in his equations (6.2.2) and (6.2.3). It follows that the optical MCD spectrum, which would test for the existence of&,, would have the same characteristics as the conventional spectixun. If this were to be confirmed experimentily, it would 3.~ strong evidence for the photon’s fkdamental flux quantum A11 in%roduced in refs. [l-6]. If the optical MCI) spectrum were found to differ from the conventional MCD spectrum, it would indicate the presence of other mechanisms of magnetization by the generator laser, such as the induction of a magnetic dipole moment through E42-451
(9) conjuwhere mgk is a hyperpo larizability, and I$$) is the antis~etric gate product of the generator laser 142451. Finally, for an order of magnitude estimation of the expected angle of rotation in a linearly polarized probe due to a gexaeratorlaser of intensity I* = lo6 watts per square metre, we use antisy,nmetric polarizabilities
135
computed ab initio by Manakov, 0vsiannikov and Kielich 1131in atomic S = l/2 ground states as a guide to orders of magnitude. For example, in Cs at 9440 cm-r we have 2 = 3.4~10‘~’ G%z2Z1. Focussing attention on the term in 2 in eqn. (7), and usingN= 6~10~” molecules per metre cubed, and the Bohr magneton for A1o, we obtain for a generator laser delivering at 300 K: Bo - lo-’ 12 = IO4 tesla
(10)
an angle of rotation of 0.8 radians per metre, measurable easily with a spectropolarimeterThis result should be proportional to the square root of the intensity, Ic of the generator laser and inversely proportional to temperature. There is also a contibution from $he rank three perturbating tensor of eqn. (7). These features would add to the evidence for the existence of& already available f&m the inverse Faraday effect I&14] and light induced N1LLR shif%s 115, 161. ACKNOWLEDGMENTS The Leverticllme Trust, Cornell Theory Center and the MRL zt Penn State are thanked for research support. REFERENCES 1
2 3 4 r z 7 8 9 10 11
12 13 14
15 16 17
M.W. Evans, Physica 8, in press (1992). M.W. Evans, Phys. Rev. Lett., submitted for publication. M-W. Evans, Physica Es,submitted for publication. S. Box-man,Chem. Eng. News, April 6th, 1992. M.W. Evans, Phys. Rev. I&t., submitted for publication. M-W. Evans, submita& for publication_ P.W. Atkins, Molecukr Qx~tun~ Mechanics. oxford Univ. Press, 1983,2nd ed. P.S. Per&an, Phys. Rrzv. 130 (1963) 919. Y-X Shen, PhBx. Rev., 134 11964)661. J.P. van der Ziel, P.S. Per&an md LD. Malmstrom, Phys. Rev., 143 (1965) 199. P-S- Pershan, J-P.van der Ziel, and L.D_ Malmstrom, Phys. Rev., 143 (19661574. S. Kidich, Proc. Phys. Sac., 66 (1965) 709. N.L. Manakov, V.D. 0vsianGk6v_ and S. Kielich, Acta Whys. Pclon., A53 (19781561, 595. P.W. A&ins and M-H. Miller, Mol. Phys., 15 (1968) 503. M.W. Evans, J.Phys. Chem., 95 (1991) 2256. W.S. Warren, S. Mayr, D. Goswami and AP. W&i., Jr., Sc+z~ce, 255 (1992j 1683. M-W. Evans, Int. J. Mod. Phys., B, 5 (1991) 1963 keview).
136 1s 19 20 21 22 23 24 25 26 27 2s 29 30 31 32 33 34 35 36 37 3s 39 46 41 42 43 44 45
M.W. Evnns. Chem. Phys., 157 (1991) I. M.W. Evans, S. WoBnink and G. WagniL?rc, Physica B. 173 (1991) 357; 175 (1991) 416. S. WoBniak. M.;rir_Evans and G. Wagnihrc. Mol. Phys.. 75 (1992) 81. S. Wofniak, M.W. E;vane and G. Wagnii?rc, Mol. Phys-, 75 (1992) 99. M.W. Evans, Physica B, 168 (1991) 9. M.W. Evans. .J. Mol. Spect.. 146 (1991) 351. S.F. Mason. Bio Systems, 20 (1987) 27. M. Faraday. Phil. &lag., 28 (IS461 294. L. Pasteur, Rev. Scientifiquc, 7 (1884) 2. L.D. Bar-ran. Chcm. Sot. Rev., 15 (1986) 189. M.W. Evans, in 1. Prigoginc and S-A. Rice, Advances in Chemical Physics. Wi!cy Intcrscience. New York, 1992, Vol. 81. SF. Mason, Molecular Optical Activity s?d the Chiral Discriminations. Cambridge Univ. Press, Cambridge, 1982. M.W. Evans, Phys. Rev. L&t., 50 (1983) 371. P.G. de Gennes, C.R. H&d. Scanccs Gcnd. Sci., Ser. B, 270 (197C) 891. J.A. le Eel, Buli. Sot. Chim. France. 22 (18741337_ W, Kuhn onci F. Braun: Natum-issenschaften, 17 (1929) 227. W-A. Bonncr, Origins Life, 14 (1984) 383. S.B. Piep:ro and P.N. Schatz. Group Theory in Spectmscopy with Applications to Magnetic Circular Dichmism. WiIcy, New York, 1983. R. Scrbzr, Phys. Rev., 41(1932) 489. L.D. Barron, Molecular Light Scattering and Optical Activity. Cambridge Univ. Press, Cambridge, 1982. B.W. Shore, The Theory of Coherent Atomic Exiitation. Wiley, New York. 1990. %Jol.2. A-R. Edmonds. Angular Momentum in Quan.tum Mechanics. Princeton Univ. Press. Princeton, 1960,2nd. ed. B.L. Silver, Irreducib!e Tensor \Iet.hods. d\cademic. New York, 1976. R. Zare, Angular Momentum, Wiley, New York, 1988. C. Kielicb, Acta Phys. Polon., 32 (1966) 385. S. Kielich, Acta Phys. Polon., 29 (196d) 675. S. Kielich ;ind A. Pie’kzra, Acta Phys. Polon., 18 (1958) 1297. A. Pickara and S. Kielich. J. Chern. Phys., 29 (1958) 1297.
JournaC
T’HZ QPTICAL
FARUIAY
EFFECT
127
AND
OPTICAL
MCD
M.W. EVANS 72.2Theory Center, Cornell University, 2 i 5orutory, Penmylvania
Ithaca, .YY 14853, and Materials Research state University, University Park, PA 76802, USA
(Received 25 September 199x) ABSTFLACT Based on the recent theoretical deduction of the photon’s static magnetic field operator &I, the opticnl Faraday effect and optical MCD are proposed. It is shown that &r rotates the plane of a linearly poirtized probe, an effect which is fkquency dependent and which gives rise to optical magnetic circular dichrrjism (optical MCD), in which the conventional magnet is replaced by a &I gewrtltxw, a circularly polarized loser generating a magnetostatic field, each phokx of which carries the quantum of flux &r.
INTRODUCTION
This paper continues a series of articles in which the consequences are developed of the recent deduction [l-6] that the photon generates a magnetic
flux quantum,
an operator
Here BO is the scalar magnetic flux density amplitude of a beam of N photons (a circularly polarized generator laser), 3 is the photon’s angular momentum operatcr [71 whose eigenvalues are &i&E, where MJ, is plus or minus one, and where A is the reduced Plan& constant. The operator 41 ch anges sign with the circular polarity of the generator laser, is unlocalized in space, and has eigenvalues &A&.23,where 23, is the laser’s scalar flux density amplitude. Its classical equivalent is the axial vector I+, a novel magnetostatic flux density generated by circularly polarized electromagnetic plane waves [l-6]. The theoretical existence_ of &, is supported by the experimexdal evidence for light induced magnetic ef0167-7322/93/$06.00
0 1993 -
Elsevier Science Publishers B.V. All rights reserved.
fects. The first to be described (in the sixties) was the inverse Faraday eTTect[S-141, and recently it has been shown 115,161 that NMR resonances arti shifted in new and useful ways by the magnetizing effect of circularly polarized argon ion radiation at frequencies far from optical resonance (i.e. where the sample is transparent $0 the argon ion radiation and dozs noL absorb it). Both these effects can be described in terms of the operator It has also been proposed &V or its classical equivaient, BJI [l-6,17-23]. [4-61 that there exists an optical Z?eman effect, in which &,, splits singlet electric dipole transition frequencies in atoms; an anomalous optical Zeeman effect for atomic triplet states; and an optical Paschen-Back effect. It has also been proposed theoretically [2,3] that & can be detected by examining spectrally a circularly polarized laser at visible frequencies reflected from an electron beam. The existence of A,, also implies that of an optical Faraday effect, in which it rotates the plane of a linearly polarized probe, an effect which is frequency dependent, and which gives rise therefore to optical ,magnetic circular dichroism (optical MCD spectroscopy). These effects are developed theoretically in this paper for atoms. Mason [24] has given an interesting discussion of cause and effect in chirality, a discussion which includes some pertinent historical analysis of interest here. In 1846, Faraday showed experimentally [25] that a magnetostatic field induces optical activity in flint glass and other isotropic transparent media. In 1884, Pasteur 1261 proposed on the basis of this result that the magnetic field represented a source of chirality. It is now known [27j that the magnetically induced optical activity in tfie Faraday effect has a fundamentally different symmetry [2S-31] frdlrnthat of natural optical activity. Nonetheless, it is interesting for our purpose in this paper to nota, following Mason 1241, that le Be1 in 1874 [32] had independently proposed that circularly polarized radiation also provides a “chiral force” of the type envisioned by Pasteur emanating from the magnetostatic field used by Faraday. Enantio-differentiating photoreactions were indeed reported by Kuhn and Braun in 1929 [33], and it has been proposed repeatedly (e-g- Bonner E3431that circularly polarized solar irradiation may be responsible for the well known preponderance in nature of one enantiomer over another. However, Mason [24] favours the universal and parity violating mechanism of the electroweak force as the &gin of this dissymmetry because the electroweak force does not depend on time and location on the earth’s surface. (Natural solar radiation is only 0.1% circularly polarized, and equally and oppositely so at dawn and dusk 1241.)
129
It is therefore interesting for our purpose to note that both circular polarity in light and the magnetostatic field have been proposed independently (by !e Be! and Pasteur respectively) as sources of chiraiity. It had therefore been sensed more than one hundred years ago that these two concepts have something in common in their effect on material. Per&an et al. [2i, in their first paper on the experimental tiemonstration of the magnetizing effect of circularly polarized giant ruby laser radiation, edged towards the concept of &,, by describing the effect of the laser as being due to an effective magnetostatic field. In view of these indications, those by Kielich and co-workers 112,131, and by Atkins and Miller [14], the present author appears to have demonstrated conclusively [l-6] the fact that circularly polarized radiation generates the classical magnetostatic field BII whose equivalent in quantum field theory is the photon’s flux quantum A,,, a nave; fundamental property of quantum field theory. As noted elsewhere, the concept of A,, must be clearly and carefully distinguished from the well known (“standard IUPAC”) oscillating 3 field of the electromagnetic plane wave; the two are quite different [l-6] in nature. We have thereforo resolved [l-6] the conjectures of le Be1 (1874) and Pasteur (1884) insofar as to show that circularly polarized light can indeed act in the same way as a magnetostatic field, a finding which implies the existence of several novel types of spectroscopy in circumstances where a conventionally applied magnet would be replaced by a circularly polarized laser. In Section 1, the contemporary quantum theory of Faraday’s effect of 1846 is developed succinctly for atoms, whereby it becomes relatively straightforward to show that the flux quantum & must also generate Faradafs observation of optical activity in all material, inherently (structurally) chiral or otherwise. In our ease, the magnet used by Faraday is replaced by a circularly polarized laser which generates 811, and is therefore referred to as the “generator laser? Section 2 develops the frequency dependence of the optics1 Faraday effect in atoms through the properties of the magnetically (i.e. An> perturbed antisymmetric polarizability of conventional contemporary Faraday effect theory [35], and therefore arrives at expressions in atoms for optical Finally a discussion is given of possible experimental configurations and order of magnitudes of the expected optical Faraday effect in terms of the intensity in watts per unit area of the generator laser.
nm3.
130
1. OPTICALLY
INDUCED
FARADAY
ROTATION
IN ATOMS
The contemporary quantum theory of the Faraday effect is based on the well known work by Serber 1361which was the precursor for the well known A, B and C terms 371. This section develops the Serber theory for use with a flux quantum B 11from the generator laser, and shows that the analogy between the optical and conventional Faraday effects is easily forged by using & in place of the conventional magnetic field BO from a magnet. The starting point for the theory of the conventional Faraday effect in quantum mechanics is an nr?quationfor the rotation of the plane of linearly po’ltized radizl+yisn. This is derived [3?] by a consideraticn ofthe effect of a magnetostatic field on the antisymmetric part of an atomic or molecular property tensor called the antisymmetric polarizability, q-“:
where qjh” is a perturbation tensor of order three. Before embarking on detailed theoretical development it is instructive to consider the fundamental symmetries of these atomic property tensors (we restrict consideration in this paper to atoms) and to recall that the complex electronic imaginary electric polarizability, of which CY+” is the antisymmetric, component, is derived from time dependent perturbation theory within whose framework 1371 qj” is a product of two transition electric dipole moment operators. The perturbation tensor qjk” in this context involves -P moment matrix element. two of these and one transition magnetic c: This structure introduces frequency depenacixx into the conventional Faraday effect, as is well known, leading to MCD. Similarly, frequency dependence occurs in the optical Faraday effect, leading to optical MCD. The fundamental symmetries considered are parity inversion (represented by the operator $1 and motion reversdi (by the operator fi. In this context q f’ is negative to P and positive to P, while qjk” is positive to both p and ?‘.‘Therefore CQ” is finite only in the presence of a p negative influence, such as IBOor &, and this influence is mediated by qjk”, which is finite for all atoms and molecules and is described by the ubiq.uitous B term 1371. The contemporary theory of Faraday’s effect depends on a perturbation of a quantity qj” which is itself the result of semi-classical time dependent second order perturbation theory. The basic reason for t-his is that the observable in the Faraday effect is an angle of rotation (A9) (or alternatively a change in ellipticity Aq) which must be calculated from Maxwell’s equations or Rayleigh refringent scattering theory [37].
131
Although the A terns is closely related to the Zeeman effect 1383 the observables of the two effects are quite different in nature, being traditionally an angle of rotation (Faraday effect) and a frequency shiR in the visible frequency region (Zeeman effect). In consequence the latter can be described by an energy perturbation, while Faraday’s effect needs perturbation of the antisymmetric polar&ability, because energy does not appear directly in Mm*ell’s equations, which are needed to calculate refractive i.ndices and therefrom A9 and Av. With these considerations, the starting point for our development of the optical Faraday effect and optical MCD is the equation I371 for angle of rotation in the conventicnal quantum theory of the Faraday effect:
(3)
Here, A8 is the angle of rotation of plane polarized probe radiation of angular frequency w parallel to the coaventicnally generated magnetostatic flux density Bz;- The quantity ATis the total number of molecules per unit volume in a set of degenerate quantum states of the atom, individually designated [37] yG ah= &Zis the degeneracy and the sum is over all componsnts of the degenerate set, with Nd = Net,. In eqn. (3), b is the vacuum permeability, e the speed of light, 2 the sample length, mnz the 2 component of the atomic magnetic dipole moment in state n; and li7’ ia the thermal energy Der atom. Recall that the atomic property tensors qj” and qjk’* are derived from semi-classical, time dependent, petiurbation theory and are frequency dependent in general, so that A6 mapped over a frequency range has the appearance of a spectrum - the well known, conventional MCI) spectrum. Our task here is to incorporate the novel flux quantum fl, [l-6] into eqn. (31, and thus generate the optical Faraday elect and optical MCIX In order to proced, we note that (oI;i”’ and qjk’* as used in eqn. (3) F expectation values of the respective quantum mechanical operators aij” and &+N in the same way that rn~ is y expectation value of the magneiic electronic dipole moment operator, m,. The quantity BOz is a classical magnetostatic vector component, and AB is an expectation value of the operator A& It is convenient to transform the appropriate cartesian components of the operators &” and &Yk”into spherical form 137,391, using the CondonlShortley phase convention:
132
(4)
Both kYt8and GYk”are purely imaginary in.lhe appppriate spherical representation, indicating that the operators r4 and ig are anti-hermitian, with purely imaginary eigenvalues 1371. (Kate preftlly, however, that the p symmetry of& is negative, while that of ic$ is positive. Both have positive @ symmetry.) The next step in our development for atoms is to replace thz vector component BOz of the conventional quantum theory bv “tile quantum mechanical operator defined by eqn. (I), which has aE ~nxted with it the anguiar momentum quantum numbor J. An immediate consequence of this replacement is the necessity to consider the magnetic field flux quantum using irreducible tensorial methods of angular momentum coupling theory in quantum mechanics [39+&l]. In other words we are now ccnsidering a quantized photon angular momentum interacting with an atom, w*hichin general contains, as usual, quantized orbita and spin electronic angular momentum. Without loss of generality we restrict consideration to atomic singlet states, in which there is a net orbital electronic angular momentum &, but no net spin angular momentum &. With these considerations, eqn. (3) becomes, for the optical Faraday effect:
133
(6)
where we have used an unco~&d representation (39411 to describe the net angular momentum generated during the interaction of photon and atom. This is ju*stified through the fact that the axim-uthal components of the operators m, and &, are both Nell defined in the uncoupled representation [3941]_ whereas in the coupled representation of the same problem the to%1 angular momentum is defined but the individual azimuthal components are not. With these considerations, the expectation value of the angle ofro’kkion in the optical Faraday effect is (7) (3flL - UL + 1)) ml&a
+ (L(L + 1) (2L + i) (U, + 3j (U, - I))*
1
where we have used the fact that the expectation value of the photon’s B n operator is kMJBO, positive for left circularly polarized ra&ation and negative for right circularly polarized radiation from the generator laser. This is an expression for the angle of rotation induced in plane polarized probe radiation in a sam 3e of atoms by a circularly poltized laser generating the flux quantum 1 II_The observation of such a rotation would provide a test for the existence of &. Equation (7) is written out in terms of reduced matrix elements of dipole moment and atomic polarizability operators, matrix elemenk which are products of eleckic and magnetic dipole transition dipole moment matrix elements from time dependent perturbation theory [S71. These introduce frequency dependence into the angle of rotation of the optical Faraday effect. The selection rules governing the various atomic property tensors are as foilows: &AL=o;AML=@; (9) ~:AL=O;f2;AM~=O; theAL=k
I part is parity forbidden,
as in magnetic dipole transitions.
134 2. OPTICAL
MCD,
FREQUENCY DEPENDENCE OF AI3
The origin of frequency dependence in the optical Faraday effect can be traced to semi-classical time dependent perturbation theory, which produces expressions for the polarizability components as given in the conventional theory of magnetic electronic optical activity [371. These can be further developed as usual in terms of reduced matrix elements of electric and magnetic transition dipole moment operators. For a given generator laser intensity and frequency therefore the optical MCD spectrum is a plot of the &Ii induced angle of rotation A0 against the frequency ef the linearly polarized pro,&!. Experimentally, this is built up in princziple by replacing the conventional magnet of MCD apparatus by the circularly polarized generator laser. DISCUSSION it is interesting to note that using the concept of &I1 the theory of the optical Faraday effect can also be developed and understood simply by replacing the magnetic flux density vector component Boz wherever it occurs in the convectional theory of MCD by the quantity kB&‘& where B. is the *magneticflux drmsity ampiitude of the generator laser and &WJ the two possible azimuthal quantum numbers of the photon. Thus, the optical Faraday effect can be’developed along the lines of the conventional counterpart in Serbeis A, B, and C terms, a convenient description of which is given by Barron 1371 in his equations (6.2.2) and (6.2.3). It follows that the optical MCD spectrum, which would test for the existence of&,, would have the same characteristics as the conventional spectixun. If this were to be confirmed experimentily, it would 3.~ strong evidence for the photon’s fkdamental flux quantum A11 in%roduced in refs. [l-6]. If the optical MCI) spectrum were found to differ from the conventional MCD spectrum, it would indicate the presence of other mechanisms of magnetization by the generator laser, such as the induction of a magnetic dipole moment through E42-451
(9) conjuwhere mgk is a hyperpo larizability, and I$$) is the antis~etric gate product of the generator laser 142451. Finally, for an order of magnitude estimation of the expected angle of rotation in a linearly polarized probe due to a gexaeratorlaser of intensity I* = lo6 watts per square metre, we use antisy,nmetric polarizabilities
135
computed ab initio by Manakov, 0vsiannikov and Kielich 1131in atomic S = l/2 ground states as a guide to orders of magnitude. For example, in Cs at 9440 cm-r we have 2 = 3.4~10‘~’ G%z2Z1. Focussing attention on the term in 2 in eqn. (7), and usingN= 6~10~” molecules per metre cubed, and the Bohr magneton for A1o, we obtain for a generator laser delivering at 300 K: Bo - lo-’ 12 = IO4 tesla
(10)
an angle of rotation of 0.8 radians per metre, measurable easily with a spectropolarimeterThis result should be proportional to the square root of the intensity, Ic of the generator laser and inversely proportional to temperature. There is also a contibution from $he rank three perturbating tensor of eqn. (7). These features would add to the evidence for the existence of& already available f&m the inverse Faraday effect I&14] and light induced N1LLR shif%s 115, 161. ACKNOWLEDGMENTS The Leverticllme Trust, Cornell Theory Center and the MRL zt Penn State are thanked for research support. REFERENCES 1
2 3 4 r z 7 8 9 10 11
12 13 14
15 16 17
M.W. Evans, Physica 8, in press (1992). M.W. Evans, Phys. Rev. Lett., submitted for publication. M-W. Evans, Physica Es,submitted for publication. S. Box-man,Chem. Eng. News, April 6th, 1992. M.W. Evans, Phys. Rev. I&t., submitted for publication. M-W. Evans, submita& for publication_ P.W. Atkins, Molecukr Qx~tun~ Mechanics. oxford Univ. Press, 1983,2nd ed. P.S. Per&an, Phys. Rrzv. 130 (1963) 919. Y-X Shen, PhBx. Rev., 134 11964)661. J.P. van der Ziel, P.S. Per&an md LD. Malmstrom, Phys. Rev., 143 (1965) 199. P-S- Pershan, J-P.van der Ziel, and L.D_ Malmstrom, Phys. Rev., 143 (19661574. S. Kidich, Proc. Phys. Sac., 66 (1965) 709. N.L. Manakov, V.D. 0vsianGk6v_ and S. Kielich, Acta Whys. Pclon., A53 (19781561, 595. P.W. A&ins and M-H. Miller, Mol. Phys., 15 (1968) 503. M.W. Evans, J.Phys. Chem., 95 (1991) 2256. W.S. Warren, S. Mayr, D. Goswami and AP. W&i., Jr., Sc+z~ce, 255 (1992j 1683. M-W. Evans, Int. J. Mod. Phys., B, 5 (1991) 1963 keview).
136 1s 19 20 21 22 23 24 25 26 27 2s 29 30 31 32 33 34 35 36 37 3s 39 46 41 42 43 44 45
M.W. Evnns. Chem. Phys., 157 (1991) I. M.W. Evans, S. WoBnink and G. WagniL?rc, Physica B. 173 (1991) 357; 175 (1991) 416. S. WoBniak. M.;rir_Evans and G. Wagnihrc. Mol. Phys.. 75 (1992) 81. S. Wofniak, M.W. E;vane and G. Wagnii?rc, Mol. Phys-, 75 (1992) 99. M.W. Evans, Physica B, 168 (1991) 9. M.W. Evans. .J. Mol. Spect.. 146 (1991) 351. S.F. Mason. Bio Systems, 20 (1987) 27. M. Faraday. Phil. &lag., 28 (IS461 294. L. Pasteur, Rev. Scientifiquc, 7 (1884) 2. L.D. Bar-ran. Chcm. Sot. Rev., 15 (1986) 189. M.W. Evans, in 1. Prigoginc and S-A. Rice, Advances in Chemical Physics. Wi!cy Intcrscience. New York, 1992, Vol. 81. SF. Mason, Molecular Optical Activity s?d the Chiral Discriminations. Cambridge Univ. Press, Cambridge, 1982. M.W. Evans, Phys. Rev. L&t., 50 (1983) 371. P.G. de Gennes, C.R. H&d. Scanccs Gcnd. Sci., Ser. B, 270 (197C) 891. J.A. le Eel, Buli. Sot. Chim. France. 22 (18741337_ W, Kuhn onci F. Braun: Natum-issenschaften, 17 (1929) 227. W-A. Bonncr, Origins Life, 14 (1984) 383. S.B. Piep:ro and P.N. Schatz. Group Theory in Spectmscopy with Applications to Magnetic Circular Dichmism. WiIcy, New York, 1983. R. Scrbzr, Phys. Rev., 41(1932) 489. L.D. Barron, Molecular Light Scattering and Optical Activity. Cambridge Univ. Press, Cambridge, 1982. B.W. Shore, The Theory of Coherent Atomic Exiitation. Wiley, New York. 1990. %Jol.2. A-R. Edmonds. Angular Momentum in Quan.tum Mechanics. Princeton Univ. Press. Princeton, 1960,2nd. ed. B.L. Silver, Irreducib!e Tensor \Iet.hods. d\cademic. New York, 1976. R. Zare, Angular Momentum, Wiley, New York, 1988. C. Kielicb, Acta Phys. Polon., 32 (1966) 385. S. Kielich, Acta Phys. Polon., 29 (196d) 675. S. Kielich ;ind A. Pie’kzra, Acta Phys. Polon., 18 (1958) 1297. A. Pickara and S. Kielich. J. Chern. Phys., 29 (1958) 1297.