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On the Berry phase
Volume 149, number 4
PHYSICS LETFERS A
24 September 1990
On the Berry phase S.C. Tiwari Department of Mathematics, Banaras Hindu University, Varanasi 221005, India Received 12 July 1990; accepted for publication 26 July 1990 Communicated by V.M. Agranovich
Local chiral invariance is considered to obtain a topological phase in the polarisation state of light.
Berry reexamined the adiabatic theorem in nonrelativistic quantum mechanics and obtained a circuit dependent phase factor besides the usual dynamical phase [1]. Tomito and Chiao used a helically wound optical fiber along which the direction of the wave vector k is slowly changed, and found that the angle of rotation of linearly polarised light is determined by the solid angle of the path in k space [2]. If the parameter space is the Poincaré sphere then the Panchratnam phase is identified to be the Berry phase having a value given by half the solid angle, as verified experimentally [3]. Numerous papers have been written on this topic since the appearance of Berry’s work. The experiments described in refs. [2] and [3] are essentially classical, see, however, ref. [41 for the Berry phase in squeezed states, i.e. a typical quantum optics situation, The main aim of the present note is to relate the local chiral invariance with the Berry phase, and argue its similitude with the Aharonov—Bohm effect. A detailed discussion on chiral invariance in classical electrodynamics is presented in ref. [5]. Source free Maxwell equations are known to be invariant under the chiral transformations in the electric and magnetic fields, i.e.
left circularly polarised photons [6]. Classically chiral invariance implies that the relative phase of left and right circularly polarised light, or the absolute plane of polarisation of linearly polarised light cannot be determined. There, is, however, a difficulty here in the meaning of a right (or left) circularly polarised photon. A sensible interpretation appears to have been given by Good [7] which is such that following neutrino convention, left circularly polarised (LCP) light is a photon (‘y) and right circularly polarised (RCP) light is an antiphoton (?). Plane polarised light are their own antiparticles: ‘y~= ~(‘y +?). The photon wave function v= E+ iBis directly observable as photons obey Bose—Einstein statistics and hence enough photons can exist in a single state. In the experiments on Berry phase a light beam containing large number ofphotons is used and therefore one should seek the explanation of topological phase in terms of E and B fields, or the wave function ~=E+ lB. For this purpose following Good [8] the Maxwell curl equations can be written in the form
E—~E’=E cos O+B sin e,
where the 3 x 3 matrices S are defined as (SI)fk= ‘~k with E11~,, being the Levi-Civita density. If V is replaced by —p/ih then eq. (2) is the Schrodinger form
=
—E SIfl O+B COS 0.
(1)
Noether’s theorem leads to a corresponding conserved quantity which has been found to be proportional to the difference in the number of right and 0375-9601/90/S 03.50 © 1990
—
-~ ~
(2)
C öt
of Maxwell’s equations with the Hamiltonian 11= ~ —
Elsevier Science Publishers B.V. (North-Holland)
(3)
223
Volume 149, number 4
PHYSICS LETTERS A
The angular momentum commutation rules are satisfied by the matrices S, ~
,,
,]
— —
— .
S
,,.
( )
The subsidiary condition
ow,
ensures the transverse nature ofthe wave. The chiral transformation (1) is now replaced by the phase, or rather gauge transformation, adopting the prevalent terminology, (6) Obviously a local gauge transformation such that 0 is a function of space (and time) does not leave invariant the Schrodinger equation (2). Considering only space dependent 0 a possible generalisation of His (7)
It can be easily verified that the gauge transformation (6) together with the transformation W—~w——WI
(8)
leaves invariant the equation H ~ih
224
~
The gauge potential W physically represents the characteristics of the medium in which the light beam travels to phase attain picked the change itslight polarisation state. The extra up byinthe is determined by the line integral of W, and for a closed path one can write
(5)
H’=—cS’(p+hgW).
24 September 1990
(9)
Ø=g~W’d1
(10)
just as one obtains in the Aharonov—Bohm effect as a topological phase factor. Here g is a coupling constant, and the Hamiltonian (7) represents the interaction oflight with the medium such that the LCP or RCP photons are exchanged, thereby altering the polarisation state. One can introduce a quantisation condition for the phase 0 using Brouwer’s theorem so that the periods ofthe forms are integral multiples of some smallest value [9]. .
-
References [1] M.V. Berry, Proc. R. Soc. A 392 ( 1984) 45. [2] A. Tomito and R.Y. Chiao, Phys. Rev. Lett. 57 (1986) 937. [31R. Bhandari and J. Samuel, Phys. Rev. Lett. 60 (1988)1211. [4] R.Y. Chiao and T.F. Jordan, Phys. Lett. A 132 (1988) 77. [5] S.C. Tiwari, Polarised light and chiral invariance, submitted to Phys. Essays. [6] M.G. Calkin, Am. J. Phys. 33 (1965) 958. [7] RH. Good Jr., Am. J. Phys. 28 (1960) 659. [8] RH. Goodir., Phys. Rev. 105 (1957) 1914; S.C. Tiwari, Phys. Lett.A 133 (1988) 279. [9] R.M. Kiehn, J. Math. Phys. 18 (1977) 614; EJ. Post, Phys. Rev. D25 (1982) 3223.
PHYSICS LETFERS A
24 September 1990
On the Berry phase S.C. Tiwari Department of Mathematics, Banaras Hindu University, Varanasi 221005, India Received 12 July 1990; accepted for publication 26 July 1990 Communicated by V.M. Agranovich
Local chiral invariance is considered to obtain a topological phase in the polarisation state of light.
Berry reexamined the adiabatic theorem in nonrelativistic quantum mechanics and obtained a circuit dependent phase factor besides the usual dynamical phase [1]. Tomito and Chiao used a helically wound optical fiber along which the direction of the wave vector k is slowly changed, and found that the angle of rotation of linearly polarised light is determined by the solid angle of the path in k space [2]. If the parameter space is the Poincaré sphere then the Panchratnam phase is identified to be the Berry phase having a value given by half the solid angle, as verified experimentally [3]. Numerous papers have been written on this topic since the appearance of Berry’s work. The experiments described in refs. [2] and [3] are essentially classical, see, however, ref. [41 for the Berry phase in squeezed states, i.e. a typical quantum optics situation, The main aim of the present note is to relate the local chiral invariance with the Berry phase, and argue its similitude with the Aharonov—Bohm effect. A detailed discussion on chiral invariance in classical electrodynamics is presented in ref. [5]. Source free Maxwell equations are known to be invariant under the chiral transformations in the electric and magnetic fields, i.e.
left circularly polarised photons [6]. Classically chiral invariance implies that the relative phase of left and right circularly polarised light, or the absolute plane of polarisation of linearly polarised light cannot be determined. There, is, however, a difficulty here in the meaning of a right (or left) circularly polarised photon. A sensible interpretation appears to have been given by Good [7] which is such that following neutrino convention, left circularly polarised (LCP) light is a photon (‘y) and right circularly polarised (RCP) light is an antiphoton (?). Plane polarised light are their own antiparticles: ‘y~= ~(‘y +?). The photon wave function v= E+ iBis directly observable as photons obey Bose—Einstein statistics and hence enough photons can exist in a single state. In the experiments on Berry phase a light beam containing large number ofphotons is used and therefore one should seek the explanation of topological phase in terms of E and B fields, or the wave function ~=E+ lB. For this purpose following Good [8] the Maxwell curl equations can be written in the form
E—~E’=E cos O+B sin e,
where the 3 x 3 matrices S are defined as (SI)fk= ‘~k with E11~,, being the Levi-Civita density. If V is replaced by —p/ih then eq. (2) is the Schrodinger form
=
—E SIfl O+B COS 0.
(1)
Noether’s theorem leads to a corresponding conserved quantity which has been found to be proportional to the difference in the number of right and 0375-9601/90/S 03.50 © 1990
—
-~ ~
(2)
C öt
of Maxwell’s equations with the Hamiltonian 11= ~ —
Elsevier Science Publishers B.V. (North-Holland)
(3)
223
Volume 149, number 4
PHYSICS LETTERS A
The angular momentum commutation rules are satisfied by the matrices S, ~
,,
,]
— —
— .
S
,,.
( )
The subsidiary condition
ow,
ensures the transverse nature ofthe wave. The chiral transformation (1) is now replaced by the phase, or rather gauge transformation, adopting the prevalent terminology, (6) Obviously a local gauge transformation such that 0 is a function of space (and time) does not leave invariant the Schrodinger equation (2). Considering only space dependent 0 a possible generalisation of His (7)
It can be easily verified that the gauge transformation (6) together with the transformation W—~w——WI
(8)
leaves invariant the equation H ~ih
224
~
The gauge potential W physically represents the characteristics of the medium in which the light beam travels to phase attain picked the change itslight polarisation state. The extra up byinthe is determined by the line integral of W, and for a closed path one can write
(5)
H’=—cS’(p+hgW).
24 September 1990
(9)
Ø=g~W’d1
(10)
just as one obtains in the Aharonov—Bohm effect as a topological phase factor. Here g is a coupling constant, and the Hamiltonian (7) represents the interaction oflight with the medium such that the LCP or RCP photons are exchanged, thereby altering the polarisation state. One can introduce a quantisation condition for the phase 0 using Brouwer’s theorem so that the periods ofthe forms are integral multiples of some smallest value [9]. .
-
References [1] M.V. Berry, Proc. R. Soc. A 392 ( 1984) 45. [2] A. Tomito and R.Y. Chiao, Phys. Rev. Lett. 57 (1986) 937. [31R. Bhandari and J. Samuel, Phys. Rev. Lett. 60 (1988)1211. [4] R.Y. Chiao and T.F. Jordan, Phys. Lett. A 132 (1988) 77. [5] S.C. Tiwari, Polarised light and chiral invariance, submitted to Phys. Essays. [6] M.G. Calkin, Am. J. Phys. 33 (1965) 958. [7] RH. Good Jr., Am. J. Phys. 28 (1960) 659. [8] RH. Goodir., Phys. Rev. 105 (1957) 1914; S.C. Tiwari, Phys. Lett.A 133 (1988) 279. [9] R.M. Kiehn, J. Math. Phys. 18 (1977) 614; EJ. Post, Phys. Rev. D25 (1982) 3223.