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

Journal name:
Nature Photonics
Year published:
Published online


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.

At a glance


  1. Dynamically modulated photonic resonator lattice exhibiting an effective magnetic field for photons.
    Figure 1: Dynamically modulated photonic resonator lattice exhibiting an effective magnetic field for photons.

    A lattice of photonic resonators, with two square sublattices of resonators with frequency ωA (red) and ωB (blue), respectively. There is only nearest-neighbour dynamic coupling. The phase of the dynamic coupling on the horizontal bonds is zero. The phase on the vertical bonds is proportional to the column index, and within the same column the sign is flipped between two neighbouring bonds.

  2. Photon motion in an effective magnetic field.
    Figure 2: Photon motion in an effective magnetic field.

    a, Structure (part), comprising the resonator lattice shown in Fig. 1, used to demonstrate the Lorentz force for photons. Resonators are indicated by dots. Lattice parameters: ωA = 30, ωB = 0,  = 0, V = 6 (all in units of 2πc/a). The left part of the lattice has no effective magnetic field (φ = 0). The right part of the lattice has an effective magnetic field with the modulation phase set in a pattern according to Fig. 1. A Gaussian wave packet is initiated in the left part of the structure. The packet is described by equation (5), with w = √10 a. b, Trajectory of the centre of mass of the wave packet, after the wave packet (with k = −1.283/a) has entered the right part, where an effective magnetic field is present. Different symbols correspond to different φ. The wave packet has a circular trajectory. c, Radius of the trajectory as a function of 1/φ for k = −1.283/a. d, Radius of the trajectory as a function of k, for φ = 0.3.

  3. Photonic one-way edge mode in a dynamically modulated resonator lattice.
    Figure 3: Photonic one-way edge mode in a dynamically modulated resonator lattice.

    a, Projected Floquet band structure for a strip of the resonator lattice shown in Fig. 1. The strip is infinite along the y-axis, and has a width of 20a along the x-axis. The projection direction is along the y-axis. Parameters of the lattice: φ = π/2, ωA = 100, ωB = 0,  = 100, V = 2 (all in units of 2πc/a). There are four separate groups of bulk bands (green curves, the centre two bands do not split). In each bandgap between the bulk bands there are two one-way edge modes, which are located on the two edges of the strip. b, Field amplitude of the two edge modes indicated by red and blue dots in a. The fields are located on the two edges and decay exponentially into the bulk. c, Propagation of one-way edge modes. We consider a 20 × 20 lattice of resonators as shown in Fig. 1 (resonators are indicated as yellow dots), and excite the edge mode by locating a point source with ωs = 2(2πc/a) at the position indicated by the red marker. The field profile at time t = 40(a/2πc) is plotted. The propagation of the edge mode is unidirectional. d, Propagation of the one-way edge mode in the presence of a defect. The defect is a 3 × 3 sublattice of resonators with frequency ωd = 50(2πc/a) for each resonator, as indicated by the larger yellow dots. All other parameters are as in c. Notice that the edge mode propagates around the defect, indicating one-way propagation robust against defect.

  4. Dynamic coupling between photonic-crystal resonators.
    Figure 4: Dynamic coupling between photonic-crystal resonators.

    a, A photonic crystal made from dielectric rods (black) with a permittivity of 8.9 and radius of 0.2a (a is the lattice constant). The resonator on the left (red) is a rod with permittivity of 7.4 and radius of 0.12a, and is shifted 0.04a to the left from the lattice point. Such a resonator supports a monopole mode. The resonator on the right (blue) is a rod with permittivity of 9.1 and radius of 0.56a. It supports a quadrupole-xy mode. The resonator in the middle (yellow and brown) is a rectangular rod with permittivity of 8.9 and side lengths of 0.66a and 0.62a along the x-axis and y-axis, respectively. It supports a pair of split dipole modes. With these chosen parameters, the frequency of the monopole mode and px mode have the same frequency of 0.346(2πc/a), and the quadrupole mode and py mode have the same frequency of 0.352(2πc/a). The static coupling strength within each two equal-frequency pair is also similar. Dynamic modulation with  = 0.006(2πc/a) and permittivity modulation Δε = 0.002 is applied on the middle resonator. The modulation has a quadrupolar profile, with the modulation in the first and third quadrants (brown) and the modulation in the second and fourth quadrant (yellow) having a π phase difference. b, Electric field (along the z-axis) of the four resonant states, and the scheme of generating dynamic coupling between the monopole and quadrupole. c, The amplitude envelope of the monopole mode as a function of time when the quadrupole mode is initially excited, for the structure in a. Circles are data from finite-difference time-domain simulations. Red curve is the solution of the coupled mode equations describing the effective coupling between two resonant states.

  5. Realization of dynamic coupling between resonators in the microwave regime.
    Figure 5: Realization of dynamic coupling between resonators in the microwave regime.

    a, Two microwave RLC resonators with frequencies ωA and ωB, respectively, coupled through a transmission line waveguide incorporating a frequency conversion device. The frequency conversion device introduces dynamic coupling between the resonators. b, Circuit implementation of the frequency conversion device in a. The device is composed of two mixers (circled cross) in parallel, which are biased with opposite d.c. voltages such that the current only flows in one direction for each mixer. The phase of the local oscillator in the mixer is φ, which is used to implement the phase of dynamic coupling between the two microwave resonators.


  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, 494497 (1980).
  2. Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 15591562 (1982).
  3. Laughlin, R. Quantized Hall conductivity in two dimensions. Phys. Rev. B 23, 56325633 (1981).
  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, 21852190 (1982).
  5. Thouless, D., Kohmoto, M., Nightingale, M. & Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405408 (1982).
  6. Hatsugai, Y. Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71, 36973700 (1993).
  7. Yablonovitch, E. Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 58, 20592062 (1987).
  8. John, S. Strong localization of photons in certain disordered dielectric superlattices. Phys. Rev. Lett. 58, 24862489 (1987).
  9. Joannopoulos, J. D., Villeneuve, P. R. & Fan, S. Photonic crystals: putting a new twist on light. Nature 386, 143149 (1997).
  10. Pendry, J. B. Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 39663969 (2000).
  11. Smith, D. R., Pendry, J. B. & Wiltshire, M. C. K. Metamaterials and negative refractive index. Science 305, 788792 (2004).
  12. Shalaev, V. M. Optical negative-index metamaterials. Nature Photon. 1, 4148 (2007).
  13. Onoda, M., Murakami, S. & Nagaosa, N. Hall effect of light. Phys. Rev. Lett. 93, 083901 (2004).
  14. Raghu, S. & Haldane, F. D. M. Analogs of quantum-Hall-effect edge states in photonic crystals. Phys. Rev. A 78, 033834 (2008).
  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).
  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).
  17. Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljačić, M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772775 (2009).
  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).
  19. Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nature Phys. 7, 907912 (2011).
  20. Umucallar, R. O. & Carusotto, I. Artificial gauge field for photons in coupled cavity arrays. Phys. Rev. A 84, 043804 (2011).
  21. Kane, C. L. Graphene and the quantum spin Hall effect. Int. J. Mod. Phys. B 21, 11551164 (2007).
  22. Chen, W.-J. et al. Observation of backscattering-immune chiral electromagnetic modes without time reversal breaking. Phys. Rev. Lett. 107, 023901 (2011).
  23. Fang, K., Yu, Z. & Fan, S. Photonic Aharonov–Bohm effect based on dynamic modulation. Phys. Rev. Lett. 108, 153901 (2012).
  24. Winn, J. N., Fan, S., Joannopoulos, J. D. & Ippen, E. P. Interband transitions in photonic crystals. Phys. Rev. B 59, 15511554 (1999).
  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).
  26. Yu, Z. & Fan, S. Complete optical isolation created by indirect interband photonic transitions. Nature Photon. 3, 9194 (2009).
  27. Luttinger, J. M. The effect of a magnetic field on electrons in a periodic potential. Phys. Rev. 84, 814817 (1951).
  28. Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 22392249 (1976).
  29. Sherley, J. H. Solution of the Schrödinger equation with a Hamiltonian periodic in time. Phys. Rev. 138, B979B987 (1965).
  30. Samba, H. Steady states and quasi energies of a quantum-mechanical system in an oscillating field. Phys. Rev. A 7, 22032213 (1973).
  31. Xu, Q., Schmidt, B., Pradhan, S. & Lipson, M. Micrometre-scale silicon electro-optic modulator. Nature 435, 325327 (2005).
  32. Kuo, Y.-H. et al. Strong quantum-confined Stark effect in germanium quantum-well structures on silicon. Nature 437, 13341336 (2005).
  33. Villeneuve, P. R., Fan, S. & Joannopoulos, J. D. Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency. Phys. Rev. B 54, 78377842 (1996).
  34. Takahashi, Y. et al. High-Q nanocavity with a 2-ns photon lifetime. Opt. Express 15, 1720617213 (2007).
  35. Notomi, M., Kuramochi, E. & Tanabe, T. Large-scale arrays of ultrahigh-Q coupled nanocavities. Nature Photon. 2, 741747 (2008).
  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).
  37. Ishizaki, K. & Noda, S. Manipulation of photons at the surface of three-dimensional photonic crystals. Nature 460, 367370 (2009).
  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).
  39. Pozar, D. M. in Microwave Engineering Ch. 12, 618 (Wiley, 2005).
  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).
  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).
  42. Houck, A. A., Tureci, H. E. & Koch, J. On-chip quantum simulation with superconducting circuits. Nature Phys. 8, 292299 (2012).
  43. Hafezi, M. & Rabi, P. Optomechanically induced non-reciprocity in microring resonators. Opt. Express 20, 76727684 (2012).

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  1. Department of Physics, Stanford University, Stanford, California 94305, USA

    • Kejie Fang
  2. Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA

    • Zongfu Yu &
    • Shanhui Fan


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.

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The authors declare no competing financial interests.

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