## The Berry phase and the Aharonov-Bohm effect on optical activity

Optics Express, Vol. 16, Issue 19, pp. 14675-14682 (2008)

http://dx.doi.org/10.1364/OE.16.014675

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### Abstract

The helical crystal structure in optically active media acts as the natural micro-solenoids for the electromagnetic waves passing through them, producing the longitudinal magnetic field in the direction of the axis of helices. Magnetic flux through the helical structure is quantized. The Berry phase is induced by rotation of the electrons around the helical structure. Optical rotation is related to the difference in the accumulative Berry phase between the right-, and the left-circularly polarized waves, which is proportional to the magnetic flux through the helical structure, according to the Aharonov-Bohm effect. The optical activity is the natural Faraday effect and the natural Aharonov-Bohm effect.

© 2008 Optical Society of America

## 1. Introduction

*r*)-, and the left (

*l*)-circularly polarized waves, the two components in optically active media are suggested to propagate at different speeds, and thus with circular birefringence [1]. Optically active media have the helical and dissymmetric crystal structure, which constrains the motions of the electrons to a helical path under the influence of the incident electric field [2

2. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. **9**, 432–457 (1937). [CrossRef]

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. **32**, 2936–2938 (2007). [CrossRef] [PubMed]

2. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. **9**, 432–457 (1937). [CrossRef]

*eυ*along the helices with the radius

*r*induces a magnetic field

*B*in the direction of the axis of helices [3

3. C. Z. Tan, “Quantum magnetic flux through helical molecules in optically active media,” Appl. Phys. B **82**, 633–636 (2006). [CrossRef]

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. **32**, 2936–2938 (2007). [CrossRef] [PubMed]

*e*and

*υ*are the charge and the velocity of an electron moving along the helical path. The radius is equal to the crystallographic radius of helical structure, or its multiplications [4

4. C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. **32**, 2936–2938 (2007). [CrossRef] [PubMed]

3. C. Z. Tan, “Quantum magnetic flux through helical molecules in optically active media,” Appl. Phys. B **82**, 633–636 (2006). [CrossRef]

**32**, 2936–2938 (2007). [CrossRef] [PubMed]

## 2. The Berry phase and the AB-effect on the motion of an electron around a closed contour

*B*, the motion of the electron around a contour

*C*is affected by the Lorentz force,

*eυB*, and the centrifugal force,

*mυ*

^{2}/

*r*, as that schematically shown in Fig. 1. Suppose the radius

*r*of the contour is perpendicular to the magnetic field vector

*B*. The direction of the charge flow can be found from Lenz’s law [3

3. C. Z. Tan, “Quantum magnetic flux through helical molecules in optically active media,” Appl. Phys. B **82**, 633–636 (2006). [CrossRef]

*p*(

*p*=

*mυ*) and the angular moment

*L*, with

*L*=

*r*×

*p*, where

*m*is the electron mass. In quantum mechanics, the angular momentum

*L*is quantized, that is [3

**82**, 633–636 (2006). [CrossRef]

**32**, 2936–2938 (2007). [CrossRef] [PubMed]

*j*is an integer, and

*ħ*is Planck’s constant. In the equilibrium state, the Lorentz and the centrifugal forces are balanced, namely

*eυB*=

*mυ*

^{2}/

*r*. Under these circumstances, the magnetic field is proportional to the angular momentum, and is hence quantized [3

**82**, 633–636 (2006). [CrossRef]

**32**, 2936–2938 (2007). [CrossRef] [PubMed]

*Φ*, through the closed contour is equal to

*πr*

^{2}

*B*, which is expressed by

*eυB*, and the centrifugal force,

*mυ*

^{2}/

*r*. Under this circumstance, the magnetic flux is the function of the angular momentum,

*L*, as depicted by Eq. (3). On the other hand, a quantal system in an eigenstate, slowly transported round the closed contour by varying parameters

*R*in its Hamitonian

*H*(

*R*), will acquire a geometrical phase factor in addition to the conventional dynamical phase factor [5

5. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A **392**, 45–57 (1984). [CrossRef]

5. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A **392**, 45–57 (1984). [CrossRef]

14. F. Plastina, G. Liberti, and A. Carollo, “Scaling of Berry’s phase close to the Dicke quantum phase transition,” Europhys. Lett. **76**, 182–188 (2006). [CrossRef]

*γ*. Particularly, when the electron is transported around the closed contour, the Berry phase

*γ*is equal to the phase shift of the AB-effect [6

6. Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. **115**, 485–491 (1959). [CrossRef]

15. M. P. Silverman, *More Than One Mystery, Explorations in Quantum Interference* (Springer-Verlag, Berlin, 1995). [CrossRef]

*π*.

## 3. The rotatory power and the Berry phase

*n*

_{l}and

*n*

_{r}for the

*l*-, and

*r*-waves. After passing through a distance

*d*, the rotation angle

*θ*of the plane of polarization is [1–4

**32**, 2936–2938 (2007). [CrossRef] [PubMed]

**32**, 2936–2938 (2007). [CrossRef] [PubMed]

*ρ*is the rotatory power. When the rotation is induced by a magnetic field

*B*, known as the Faraday effect, the rotation angle is expressed by [1–4

**32**, 2936–2938 (2007). [CrossRef] [PubMed]

*V*is referred to as the Verdet constant.

**32**, 2936–2938 (2007). [CrossRef] [PubMed]

**82**, 633–636 (2006). [CrossRef]

**32**, 2936–2938 (2007). [CrossRef] [PubMed]

*ρ*is proportional to the Verdet constant

*V*for optically active medium, with the ratio equal to the induced magnetic induction

*B*, namely

*ρ*=

*BV*. Using Eq. (2), we have [3

**82**, 633–636 (2006). [CrossRef]

**32**, 2936–2938 (2007). [CrossRef] [PubMed]

*ħV*/(

*er*

^{2}). The validity of Eq. (10) has been experimentally confirmed by the results of the rotatory power and the Verdet constant of α-quartz at different wavelengths of the incident light [3

**82**, 633–636 (2006). [CrossRef]

**32**, 2936–2938 (2007). [CrossRef] [PubMed]

16. P. Van Den Keybus and W. Grevendonk, “Comparison of optical activity and Faraday rotation in crystalline SiO_{2},” Phys. Status Solidi B **136**, 651–659 (1986). [CrossRef]

17. J. L. Dexter, J. Landry, D. G. Cooper, and J. Reintjes, “Ultraviolet optical isolators utilizing KDP-isomorphs,” Opt. Comm. **80**, 115–118 (1990). [CrossRef]

*B*is 82.98 Tesla. Recently, high number (

*j*=3) of quanta has been experimentally observed in α-quartz [4

**32**, 2936–2938 (2007). [CrossRef] [PubMed]

*r*of the helical structure in α-quartz is shown to be

*6a*

_{0}, where

*a*

_{0}is the cell parameter [4

**32**, 2936–2938 (2007). [CrossRef] [PubMed]

*r*-, and the

*l*-waves propagate in the medium with the wavelengths of

*λ*/

*n*

_{r}, and

*λ*/

*n*

_{l}, respectively. The

*r*-wave packet induces a charge flow in a sub-solenoid of the radius

*r*and the length

*λ*/

*n*

_{r}, whereas the

*l*-wave packet induces a charge flow in a sub-solenoid of the radius

*r*and the length

*λ*/

*n*

_{l}, as that schematically shown in Fig. 3. Note

*λ*is much larger than

*r*. The magnetic field

*B*

_{r}induced by the

*r*-waves is in the opposite direction of

*B*

_{l}induced by the

*l*-waves. Therefore, the total macroscopic magnetic field induced by the incident light waves is zero. In the equilibrium state, we have:

*eυ*

_{r}

*B*

_{l}=

*mυ*

^{2}

_{r}/

*r*, and

*eυ*

_{l}

*B*

_{r}=

*mυ*

^{2}

_{l}/

*r*, according to Lenz’s law, where

*υ*

_{r}and

*υ*

_{l}are the velocities of the electrons driven by the

*r*-, and

*l*-waves, respectively. Hence optical rotation vanishes when the plane-polarized light is reflected back in optically active medium. Conversely, optical rotation is doubled when the light beam is reflected back in the traditional Faraday effect.

*z*-axis). Slow transport of the electrons round a sub-solenoid induces a Berry phase

*γ*[5

5. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A **392**, 45–57 (1984). [CrossRef]

*d*, the numbers of sub-solenoids are

*n*

_{r}

*d*/

*λ*, and

*n*

_{l}

*d*/

*λ*for the

*r*-, and the

*l*-waves, respectively. Therefore, the difference in the accumulative Berry phase between the

*r*-, and the

*l*-waves is

*γ*(

*n*

_{r}-

*n*

_{l}) d/λ, which is equal to the rotation angle

*θ*, namely

*j*=

*1*, Eq. (13) leads to a classical result [1–4

**32**, 2936–2938 (2007). [CrossRef] [PubMed]

*ρ*=

*π*(

*n*

_{r}-

*n*

_{l})/

*λ*. This result indicates that optical rotation is associated with the difference in the accumulative Berry phase between the

*r*-, and the

*l*-waves. Comparing Eq. (10) with (13), the Verdet constant is given by

*A*=

*πr*

^{2}, is the area of the helix. Therefore, the Verdet constant depends on the dimensions of the helical structure.

## 4. Discussion

*z*-direction, the electromagnetic force on a moving electron is -

*e*(

*E*+

*υ*×

*B*), where

*E*(

*E*

_{x},

*E*

_{y}) is the electric field of the incident light [13]. In classical electrodynamics, the motion of the electrons is generally described by a differential equation of

*d*

^{2}

*s*/

*dt*

^{2}+

*ω*

_{0}

^{2}

*s*=-

*e*(

*E*+

*υ*×

*B*)/

*m*, where

*s*(

*s*

^{2}=

*x*

^{2}+

*y*

^{2}) is the displacement of the electron from its equilibrium position,

*ω*

_{0}is the resonance frequency, and

*υ*=

*ds*/

*dt*=[(

*dx*/

*dt*)

^{2}+(

*dy*/

*dt*)

^{2}]

^{1/2}denotes the velocity of the forced motion [13,19

19. C. Z. Tan, “Piezoelectric lattice vibrations at optical frequencies,” Solid State Commun. **131**, 405–408 (2004). [CrossRef]

20. C. Z. Tan, H. Li, and L. Chen, “Generation of mutual coherence of eigenvibrations in α-quartz at infrared frequencies by incidence of randomly polarized waves,” Appl. Phys. B **86**, 129–137 (2007). [CrossRef]

*E*(and hence to s):

*eυB*=

*mυ*

^{2}/

*r*. The equation of motion is then changed into the following forms:

*B*with respect to the

*z*-axis. If

*B*is in the direction of

*x*×

*y*, the sign in Eq. (15a) is “+”, and in (15b) is “-”. Conversely, if

*B*is in the direction of

*y*×

*x*, the sign in Eq. (15a) is “-”, and in (15b) is “+”. These are nonlinear differential equations. However, because

*r*is much larger than

*s*, (in α-quartz,

*r*=

*6a*

_{0}[4

**32**, 2936–2938 (2007). [CrossRef] [PubMed]

*ds*/

*dt*)

^{2}/

*r*.

## 5. Conclusions

## Acknowledgments

## References and links

1. | E. Hecht, |

2. | E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. |

3. | C. Z. Tan, “Quantum magnetic flux through helical molecules in optically active media,” Appl. Phys. B |

4. | C. Z. Tan and L. Chen, “Quantum effects in the optical activity of α-quartz,” Opt. Lett. |

5. | M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A |

6. | Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. |

7. | B. S. Deaver and W. M. Fairbank, “Experimental evidence for quantized flux in superconducting cylinders,” Phys. Rev. Lett. |

8. | R. Doll and M. Näbauer, “Experimental proof of magnetic flux quantization in a superconducting ring,” Phys. Rev. Lett. |

9. | W. L. Goodman and B. S. Deaver Jr., , “Detailed measurements of the quantized flux states of hollow superconducting cylinders,” Phys. Rev. Lett. |

10. | D. Yu. Sharvin and Yu. V. Sharvin, “Magnetic-flux quantization in cylindrical film of a normal metal,” JETP Lett. |

11. | R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, “Observation of |

12. | A. D. Stone and Y. Imry, “Periodicity of the Aharonov-Bohm effect in normal-metal rings,” Phys. Rev. Lett. |

13. | J. D. Jackson, |

14. | F. Plastina, G. Liberti, and A. Carollo, “Scaling of Berry’s phase close to the Dicke quantum phase transition,” Europhys. Lett. |

15. | M. P. Silverman, |

16. | P. Van Den Keybus and W. Grevendonk, “Comparison of optical activity and Faraday rotation in crystalline SiO |

17. | J. L. Dexter, J. Landry, D. G. Cooper, and J. Reintjes, “Ultraviolet optical isolators utilizing KDP-isomorphs,” Opt. Comm. |

18. | D. N. Nikogosyan, |

19. | C. Z. Tan, “Piezoelectric lattice vibrations at optical frequencies,” Solid State Commun. |

20. | C. Z. Tan, H. Li, and L. Chen, “Generation of mutual coherence of eigenvibrations in α-quartz at infrared frequencies by incidence of randomly polarized waves,” Appl. Phys. B |

**OCIS Codes**

(160.4760) Materials : Optical properties

(260.5430) Physical optics : Polarization

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: July 10, 2008

Revised Manuscript: August 13, 2008

Manuscript Accepted: August 13, 2008

Published: September 3, 2008

**Citation**

C. Z. Tan, "The Berry phase and the Aharonov-Bohm effect on optical activity," Opt. Express **16**, 14675-14682 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14675

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### References

- E. Hecht, Optics (Addison Wesley, New York, 2002).
- E. U. Condon, "Theories of optical rotatory power," Rev. Mod. Phys. 9, 432-457 (1937). [CrossRef]
- C. Z. Tan, "Quantum magnetic flux through helical molecules in optically active media," Appl. Phys. B 82, 633-636 (2006). [CrossRef]
- C. Z. Tan and L. Chen, "Quantum effects in the optical activity of ?-quartz," Opt. Lett. 32, 2936-2938 (2007). [CrossRef] [PubMed]
- M. V. Berry, "Quantal phase factors accompanying adiabatic changes," Proc. R. Soc. Lond. A 392, 45-57 (1984). [CrossRef]
- Y. Aharonov and D. Bohm, "Significance of electromagnetic potentials in the quantum theory," Phys. Rev. 115, 485-491 (1959). [CrossRef]
- B. S. Deaver, Jr., and W. M. Fairbank, "Experimental evidence for quantized flux in superconducting cylinders," Phys. Rev. Lett. 7, 43-46 (1961). [CrossRef]
- R. Doll and M. Näbauer, "Experimental proof of magnetic flux quantization in a superconducting ring," Phys. Rev. Lett. 7, 51-52 (1961). [CrossRef]
- W. L. Goodman and B. S. Deaver, Jr., "Detailed measurements of the quantized flux states of hollow superconducting cylinders," Phys. Rev. Lett. 24, 870-873 (1970). [CrossRef]
- D. Yu. Sharvin and Yu. V. Sharvin, "Magnetic-flux quantization in cylindrical film of a normal metal," JETP Lett. 34, 272-275 (1981).
- R. A. Webb, S. Washburn, C. P. Umbach, and R. B. Laibowitz, "Observation of h/e Aharonov-Bohm oscillations in normal-metal rings," Phys. Rev. Lett. 54, 2696-2699 (1985). [CrossRef] [PubMed]
- A. D. Stone and Y. Imry, "Periodicity of the Aharonov-Bohm effect in normal-metal rings," Phys. Rev. Lett. 56, 189-192 (1986). [CrossRef] [PubMed]
- J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1999).
- F. Plastina, G. Liberti, and A. Carollo, "Scaling of Berry???s phase close to the Dicke quantum phase transition," Europhys. Lett. 76, 182-188 (2006). [CrossRef]
- M. P. Silverman, More Than One Mystery, Explorations in Quantum Interference (Springer-Verlag, Berlin, 1995). [CrossRef]
- P. Van Den Keybus and W. Grevendonk, "Comparison of optical activity and Faraday rotation in crystalline SiO2," Phys. Status Solidi B 136, 651-659 (1986). [CrossRef]
- J. L. Dexter, J. Landry, D. G. Cooper, and J. Reintjes, "Ultraviolet optical isolators utilizing KDP-isomorphs," Opt. Comm. 80, 115-118 (1990). [CrossRef]
- D. N. Nikogosyan, Properties of Optical and Laser-Related Materials A Handbook (John Wiley & Sons, New York, 1997).
- C. Z. Tan, "Piezoelectric lattice vibrations at optical frequencies," Solid State Commun. 131, 405-408 (2004). [CrossRef]
- C. Z. Tan, H. Li, and L. Chen, "Generation of mutual coherence of eigenvibrations in ?-quartz at infrared frequencies by incidence of randomly polarized waves," Appl. Phys. B 86, 129-137 (2007). [CrossRef]

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