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Dynamic band structure measurement in the synthetic space

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Science Advances  08 Jan 2021:
Vol. 7, no. 2, eabe4335
DOI: 10.1126/sciadv.abe4335
  • Fig. 1 Illustration of dynamic band structure.

    (A) Movement of electrons under an external force F in k-space within the first Brillouin zone in a 1D solid-state system. (B) Physical picture of band structure movements for system in (A), which periodically shifts over time in k-space. (C) Theoretically calculated trajectory of the dynamic band structures based on (B) with F = 0, F > 0, and F < 0 from Eq. 4. (D) A ring resonator dynamically modulated by an EOM. In the near-resonance modulation case, the modulation frequency ΩM is chosen to be slightly different from resonant frequency ΩR, while ∆ is the modulation detuning. (E) The system in (D) can be mapped into a 1D tight-binding model for photons under an effective force along the synthetic frequency dimension.

  • Fig. 2 Experimental setup.

    The laser’s frequency is finely scanned by applying an external linear voltage ramp signal to its frequency modulation input. PC, polarization controller; DWDM, dense wavelength division multiplexing; PD, photodiode.

  • Fig. 3 Trajectories of the dynamic band structure under near-resonance modulation with fixed modulation strength.

    (A to E) Experimentally measured band structures with modulation frequency ΩM = 21.0, 21.5, 21.8, 22.0, and 23.0 MHz, respectively, and a fixed applied RF voltage of V = 1.5 V. (F to J) Theoretically calculated trajectories of the band structure in 1D solid-state system under external force F = 0.285, 0.143, 0.057, 0, and −0.285, respectively, and a fixed interaction strength J = 0.2 based on Eq. 4. (K to O) Numerically simulated trajectories of the band structure with modulation detuning ∆ = 0.285, 0.143, 0.057, 0, and −0.285, respectively, and a fixed coupling strength g = 0.2 and γ = 0.2 based on Eqs. 11 to 13. The bottom x axis in (A) to (E) and (K) to (O) represents one roundtrip time with period of 2π/ΩR, while x axis in (F) to (J) presents the first Brillouin zone in solids with period of 2π/a. The y axis represents the frequency detuning of the input laser from the resonant frequency normalized to FSR. The sign of modulation detuning is mapped to the sign of the effective force and hence controls the band shapes.

  • Fig. 4 Trajectories of dynamic band structures and transmission spectra with fixed modulation detuning.

    (A to D) Experimentally observed trajectories of the dynamic band structure with fixed modulation frequency ΩM = 20 MHz and measured transmission spectra from drop port. (E to H) Numerically simulated trajectories of the dynamic band structure with fixed modulation detuning ∆ = 0.57 and calculated transmission spectra from Eq. 13. a.u., arbitrary units.

Supplementary Materials

  • Supplementary Materials

    Dynamic band structure measurement in the synthetic space

    Guangzhen Li, Yuanlin Zheng, Avik Dutt, Danying Yu, Qingrou Shan, Shijie Liu, Luqi Yuan, Shanhui Fan, Xianfeng Chen

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