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Realizing effective magnetic field for photons by controlling the phase of dynamic modulation



The goal to achieve arbitrary control of photon flow has motivated much of the recent research on photonic crystals and metamaterials. As a new mechanism for controlling photon flow, we introduce a scheme that generates an effective magnetic field for photons. We consider a resonator lattice in which the coupling constants between the resonators are harmonically modulated in time. With appropriate choice of the spatial distribution of the modulation phases, an effective magnetic field for photons can be created, leading to a Lorentz force for photons and the emergence of topologically protected one-way photon edge states that are robust against disorders—without the use of magneto-optical effects.

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Figure 1: Dynamically modulated photonic resonator lattice exhibiting an effective magnetic field for photons.
Figure 2: Photon motion in an effective magnetic field.
Figure 3: Photonic one-way edge mode in a dynamically modulated resonator lattice.
Figure 4: Dynamic coupling between photonic-crystal resonators.
Figure 5: Realization of dynamic coupling between resonators in the microwave regime.


  1. 1

    Klitzing, K. V., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494–497 (1980).

    ADS  Article  Google Scholar 

  2. 2

    Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562 (1982).

    ADS  Article  Google Scholar 

  3. 3

    Laughlin, R. Quantized Hall conductivity in two dimensions. Phys. Rev. B 23, 5632–5633 (1981).

    ADS  Article  Google Scholar 

  4. 4

    Halperin, B. I. Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B 25, 2185–2190 (1982).

    ADS  Article  Google Scholar 

  5. 5

    Thouless, D., Kohmoto, M., Nightingale, M. & Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).

    ADS  Article  Google Scholar 

  6. 6

    Hatsugai, Y. Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71, 3697–3700 (1993).

    ADS  MathSciNet  Article  Google Scholar 

  7. 7

    Yablonovitch, E. Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 58, 2059–2062 (1987).

    ADS  Article  Google Scholar 

  8. 8

    John, S. Strong localization of photons in certain disordered dielectric superlattices. Phys. Rev. Lett. 58, 2486–2489 (1987).

    ADS  Article  Google Scholar 

  9. 9

    Joannopoulos, J. D., Villeneuve, P. R. & Fan, S. Photonic crystals: putting a new twist on light. Nature 386, 143–149 (1997).

    ADS  Article  Google Scholar 

  10. 10

    Pendry, J. B. Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966–3969 (2000).

    ADS  Article  Google Scholar 

  11. 11

    Smith, D. R., Pendry, J. B. & Wiltshire, M. C. K. Metamaterials and negative refractive index. Science 305, 788–792 (2004).

    ADS  Article  Google Scholar 

  12. 12

    Shalaev, V. M. Optical negative-index metamaterials. Nature Photon. 1, 41–48 (2007).

    ADS  Article  Google Scholar 

  13. 13

    Onoda, M., Murakami, S. & Nagaosa, N. Hall effect of light. Phys. Rev. Lett. 93, 083901 (2004).

    ADS  Article  Google Scholar 

  14. 14

    Raghu, S. & Haldane, F. D. M. Analogs of quantum-Hall-effect edge states in photonic crystals. Phys. Rev. A 78, 033834 (2008).

    ADS  Article  Google Scholar 

  15. 15

    Haldane, F. D. M. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008).

    ADS  Article  Google Scholar 

  16. 16

    Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljačić, M. Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Phys. Rev. Lett. 100, 013905 (2008).

    ADS  Article  Google Scholar 

  17. 17

    Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljačić, M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772–775 (2009).

    ADS  Article  Google Scholar 

  18. 18

    Yu, Z., Veronis, G., Wang, Z. & Fan, S. One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal. Phys. Rev. Lett. 100, 023902 (2008).

    ADS  Article  Google Scholar 

  19. 19

    Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nature Phys. 7, 907–912 (2011).

    ADS  Article  Google Scholar 

  20. 20

    Umucallar, R. O. & Carusotto, I. Artificial gauge field for photons in coupled cavity arrays. Phys. Rev. A 84, 043804 (2011).

    ADS  Article  Google Scholar 

  21. 21

    Kane, C. L. Graphene and the quantum spin Hall effect. Int. J. Mod. Phys. B 21, 1155–1164 (2007).

    ADS  Article  Google Scholar 

  22. 22

    Chen, W.-J. et al. Observation of backscattering-immune chiral electromagnetic modes without time reversal breaking. Phys. Rev. Lett. 107, 023901 (2011).

    ADS  Article  Google Scholar 

  23. 23

    Fang, K., Yu, Z. & Fan, S. Photonic Aharonov–Bohm effect based on dynamic modulation. Phys. Rev. Lett. 108, 153901 (2012).

    ADS  Article  Google Scholar 

  24. 24

    Winn, J. N., Fan, S., Joannopoulos, J. D. & Ippen, E. P. Interband transitions in photonic crystals. Phys. Rev. B 59, 1551–1554 (1999).

    ADS  Article  Google Scholar 

  25. 25

    Dong, P., Preble, S. F., Robinson, J. T., Manipatruni, S. & Lipson, M. Inducing photonic transitions between discrete modes in a silicon optical microcavity. Phys. Rev. Lett. 100, 033904 (2008).

    ADS  Article  Google Scholar 

  26. 26

    Yu, Z. & Fan, S. Complete optical isolation created by indirect interband photonic transitions. Nature Photon. 3, 91–94 (2009).

    ADS  Article  Google Scholar 

  27. 27

    Luttinger, J. M. The effect of a magnetic field on electrons in a periodic potential. Phys. Rev. 84, 814–817 (1951).

    ADS  MathSciNet  Article  Google Scholar 

  28. 28

    Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976).

    ADS  Article  Google Scholar 

  29. 29

    Sherley, J. H. Solution of the Schrödinger equation with a Hamiltonian periodic in time. Phys. Rev. 138, B979–B987 (1965).

    ADS  Article  Google Scholar 

  30. 30

    Samba, H. Steady states and quasi energies of a quantum-mechanical system in an oscillating field. Phys. Rev. A 7, 2203–2213 (1973).

    ADS  Article  Google Scholar 

  31. 31

    Xu, Q., Schmidt, B., Pradhan, S. & Lipson, M. Micrometre-scale silicon electro-optic modulator. Nature 435, 325–327 (2005).

    ADS  Article  Google Scholar 

  32. 32

    Kuo, Y.-H. et al. Strong quantum-confined Stark effect in germanium quantum-well structures on silicon. Nature 437, 1334–1336 (2005).

    ADS  Article  Google Scholar 

  33. 33

    Villeneuve, P. R., Fan, S. & Joannopoulos, J. D. Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency. Phys. Rev. B 54, 7837–7842 (1996).

    ADS  Article  Google Scholar 

  34. 34

    Takahashi, Y. et al. High-Q nanocavity with a 2-ns photon lifetime. Opt. Express 15, 17206–17213 (2007).

    ADS  Article  Google Scholar 

  35. 35

    Notomi, M., Kuramochi, E. & Tanabe, T. Large-scale arrays of ultrahigh-Q coupled nanocavities. Nature Photon. 2, 741–747 (2008).

    ADS  Article  Google Scholar 

  36. 36

    Povinelli, M. L., Johnson, S. G., Fan, S. & Joannopoulos, J. D. Emulation of two-dimensional photonic crystal defect modes in a photonic crystal with a three-dimensional photonic band gap. Phys. Rev. B 64, 075313 (2001).

    ADS  Article  Google Scholar 

  37. 37

    Ishizaki, K. & Noda, S. Manipulation of photons at the surface of three-dimensional photonic crystals. Nature 460, 367–370 (2009).

    ADS  Article  Google Scholar 

  38. 38

    Lira, H., Yu, Z., Fan, S. & Lipson, M. Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip. Phys. Rev. Lett. 109, 033901 (2012).

    ADS  Article  Google Scholar 

  39. 39

    Pozar, D. M. in Microwave Engineering Ch. 12, 618 (Wiley, 2005).

    Google Scholar 

  40. 40

    Koch, J., Houck, A. A., Le Hur, K. & Girvin, S. M. Time-reversal-symmetry breaking in circuit-QED-based photon lattices. Phys. Rev. A 82, 043811 (2010).

    ADS  Article  Google Scholar 

  41. 41

    Underwood, D., Shanks, W. E., Koch, J. & Houck, A. A. Low-disorder microwave cavity lattices for quantum simulation with photons. Phys. Rev. A 86, 023837 (2012).

    ADS  Article  Google Scholar 

  42. 42

    Houck, A. A., Tureci, H. E. & Koch, J. On-chip quantum simulation with superconducting circuits. Nature Phys. 8, 292–299 (2012).

    ADS  Article  Google Scholar 

  43. 43

    Hafezi, M. & Rabi, P. Optomechanically induced non-reciprocity in microring resonators. Opt. Express 20, 7672–7684 (2012).

    ADS  Article  Google Scholar 

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This work was supported in part by the US Air Force Office of Scientific Research (grant no. FA9550-09-1-0704) and the US National Science Foundation (grant no. ECCS-1201914).

Author information




K.F. conceived the mechanism for achieving an effective magnetic field and performed the calculations. All authors contributed to the design of the study, discussion of the results and writing of the manuscript.

Corresponding author

Correspondence to Shanhui Fan.

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The authors declare no competing financial interests.

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Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nature Photon 6, 782–787 (2012).

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