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Cited by 179Quantum Mechanics and Non-Hermitian PhysicsTopological Materials and PhenomenaAcoustic Wave Phenomena Research

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Experimental band structure spectroscopy along a synthetic dimension

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Dutt, Avik, Minkov, Momchil, Lin, Qian, Yuan, Luqi, Miller, David A. B., Fan, Shanhui·2019·Nature Communications
DOI 10.1038/s41467-019-11117-9PMC PMC6635488
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Abstract

There has been significant recent interest in synthetic dimensions, where internal degrees of freedom of a particle are coupled to form higher-dimensional lattices in lower-dimensional physical structures. For these systems, the concept of band structure along the synthetic dimension plays a central role in their theoretical description. Here we provide a direct experimental measurement of the band structure along the synthetic dimension. By dynamically modulating a resonator at frequencies commensurate with its mode spacing, we create a periodically driven lattice of coupled modes in the frequency dimension. The strength and range of couplings can be dynamically reconfigured by changing the modulation amplitude and frequency. We show theoretically and demonstrate experimentally that time-resolved transmission measurements of this system provide a direct readout of its band structure. We also realize long-range coupling, gauge potentials and nonreciprocal bands by simply incorporating additional frequency drives, enabling great flexibility in band structure engineering.

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Fig. 1 Dynamically modulated ring cavity and its band structure in the synthetic frequency dimension. a A sinusoidal signal VM modulates the index of a part of the ring (left) at the mode spacing ΩR, creating coupling J between the different frequency modes (right). EOM electro-optic phase modulator. b Band structure of the nearest-neighbor coupled 1D tight-binding model, εk = 2J cos kΩ. c Theoretical time-resolved steady-state response of the ring (Eq. 16) for a cavity loss rate 2γ = J/5. The time t plays the role of the Bloch quasimomentum k
Fig. 1 Dynamically modulated ring cavity and its band structure in the synthetic frequency dimension. a A sinusoidal signal VM modulates the index of a part of the ring (left) at the mode spacing ΩR, creating coupling J between the different frequency modes (right). EOM electro-optic phase modulator. b Band structure of the nearest-neighbor coupled 1D tight-binding model, εk = 2J cos kΩ. c Theoretical time-resolved steady-state response of the ring (Eq. 16) for a cavity loss rate 2γ = J/5. The time t plays the role of the Bloch quasimomentum kCC BY · PubMed Central
Fig. 2 Experimental setup. a Schematic. b Detailed setup. Cavity length ~13.5 m, Cavity mode spacing or free-spectral range ΩR = 2π ⋅ 15.04 MHz. EOM electro-optic phase modulator, PD photodiode, SOA semiconductor optical amplifier, amp RF amplifier
Fig. 2 Experimental setup. a Schematic. b Detailed setup. Cavity length ~13.5 m, Cavity mode spacing or free-spectral range ΩR = 2π ⋅ 15.04 MHz. EOM electro-optic phase modulator, PD photodiode, SOA semiconductor optical amplifier, amp RF amplifierCC BY · PubMed Central
Fig. 3 Band structure for on-resonance modulation ΩM = ΩR = 2π ⋅ 15.04 MHz. a–c Experimentally measured time-resolved transmission. d–f Theoretical transmission based on Eq. (15). The bottom x-axis represents time (the quasimomentum) in units of π/ΩR, while the top x-axis of (a–c) presents the time in real units. The y-axis represents the input laser’s detuning from the ring’s resonance frequency, allowing us access to different quasienergies εk (left: real units, right: normalized to FSR). On increasing the coupling J (theory) or the amplitude of modulation Vpp (experiment), cosinusoidal bands with increasing extent in quasienergy are observed, in agreement with εk = 2J cos kΩ. See also Supplementary Movie 1. g–i Corresponding time-averaged transmission through the modulated cavity. In (g), typical Lorentzian resonances of the unmodulated cavity are seen. In (h), the photonic density of states (DOS) in synthetic space is observed. The experimental data (blue) matches the expected density of states (red dashed) of a 1D tight-binding model with nearest neighbor coupling. In the bottom panels (c), (f, i), the modulation-induced coupling is strong enough to cause overlap of Floquet bands
Fig. 3 Band structure for on-resonance modulation ΩM = ΩR = 2π ⋅ 15.04 MHz. a–c Experimentally measured time-resolved transmission. d–f Theoretical transmission based on Eq. (15). The bottom x-axis represents time (the quasimomentum) in units of π/ΩR, while the top x-axis of (a–c) presents the time in real units. The y-axis represents the input laser’s detuning from the ring’s resonance frequency, allowing us access to different quasienergies εk (left: real units, right: normalized to FSR). On increasing the coupling J (theory) or the amplitude of modulation Vpp (experiment), cosinusoidal bands with increasing extent in quasienergy are observed, in agreement with εk = 2J cos kΩ. See also Supplementary Movie 1. g–i Corresponding time-averaged transmission through the modulated cavity. In (g), typical Lorentzian resonances of the unmodulated cavity are seen. In (h), the photonic density of states (DOS) in synthetic space is observed. The experimental data (blue) matches the expected density of states (red dashed) of a 1D tight-binding model with nearest neighbor coupling. In the bottom panels (c), (f, i), the modulation-induced coupling is strong enough to cause overlap of Floquet bandsCC BY · PubMed Central
Fig. 4 Band-structure engineering with long-range hopping and synthetic gauge potentials. a Time-resolved transmission for a modulation frequency ΩM = 2ΩR, coupling exclusively next-nearest neighbors. While the band structure is similar to Fig. 3, the first Brillouin zone is halved, and two periods of the cosine band structure are seen in ΩRt∈[−π, π]. b Schematic of lattice for VM = V1 cos ΩRt + V2 cos (nΩRt + ϕ), where V1 (solid lines) determines the strength of nearest-neighbor coupling and V2 (dashed lines) determines the long-range hopping, here shown for n = 2. The phase of the V2 links is +ϕ when going up in frequency and −ϕ when going down in frequency, as imposed by the modulation53,54. Bottom: Equivalent lattice representation by a triangular chain threaded by a phase ϕ per plaquette. c Measured nonreciprocal band structure for V2/V1 = 0.40, ϕ = π/2, n = 2, which shows strong asymmetry about k = 0. See also Supplementary Movie 2. d Same as c, but for ϕ = 0, n = 3. Black overlays in (c) and (d) indicate expected band structures
Fig. 4 Band-structure engineering with long-range hopping and synthetic gauge potentials. a Time-resolved transmission for a modulation frequency ΩM = 2ΩR, coupling exclusively next-nearest neighbors. While the band structure is similar to Fig. 3, the first Brillouin zone is halved, and two periods of the cosine band structure are seen in ΩRt∈[−π, π]. b Schematic of lattice for VM = V1 cos ΩRt + V2 cos (nΩRt + ϕ), where V1 (solid lines) determines the strength of nearest-neighbor coupling and V2 (dashed lines) determines the long-range hopping, here shown for n = 2. The phase of the V2 links is +ϕ when going up in frequency and −ϕ when going down in frequency, as imposed by the modulation53,54. Bottom: Equivalent lattice representation by a triangular chain threaded by a phase ϕ per plaquette. c Measured nonreciprocal band structure for V2/V1 = 0.40, ϕ = π/2, n = 2, which shows strong asymmetry about k = 0. See also Supplementary Movie 2. d Same as c, but for ϕ = 0, n = 3. Black overlays in (c) and (d) indicate expected band structuresCC BY · PubMed Central

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Citations

Cites

Research works printed but not linked: 3

  1. Ashcroft, N. W. & Mermin, N. D. Solid State Physics. (Holt, Rinehart and Winston, New York, 1976).
  2. Martin, I., Refael, G. & Halperin, B. Topological frequency conversion in strongly driven quantum systems. Phys. Rev. X 7, 041008 (2017).
  3. Goldman, N. & Dalibard, J. Periodically driven quantum systems: effective Hamiltonians and engineered gauge fields. Phys. Rev. X 4, 031027 (2014).

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Metadata

Authors
Dutt, A., Minkov, M., Lin, Q., Yuan, L., Miller, D. A. B., & Fan, S.
Affiliations
  • Stanford University (US)
  • Shanghai Jiao Tong University (CN)
Corresponding author
  • Yuan, Luqi
  • Fan, Shanhui
ORCID
Venue
Nature Communications
Publisher
Springer Science and Business Media LLC
Published
2019-07-16
Volume
10
Issue
1
Pages
3122-3122
ISSN
2041-1723 (ISSN-L)
DOI
10.1038/s41467-019-11117-9
PMC
PMC6635488
OpenAlex
W2951236500
Cited by
178
References
75
Citations by year
2026: 22 · 2025: 37 · 2024: 23 · 2023: 21 · 2022: 17 · 2021: 30 · 2020: 22 · 2019: 5
FWCI
8.13
Type
journal article
Language
en
Open access
Licence
Locations
  • Nature Communications (publishedVersion, cc-by)
  • Europe PMC (PubMed Central) (publishedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion, cc-by-nc-sa)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
  • Figshare (submittedVersion, cc-by)
  • arXiv (Cornell University)
Funders
  • United States Department of Defense | United States Navy | Office of Naval Research (N00014-17-1-3030)
  • United States Department of Defense | United States Air Force | AFMC | Air Force Office of Scientific Research (FA9550-17-1-0002)
  • Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (P300P2_177721)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Non-Hermitian Physics
  • Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Topological Materials and Phenomena
  • Physical Sciences › Physics and Astronomy › Acoustics and Ultrasonics › Acoustic Wave Phenomena Research
Keywords
synthetic dimensions, electronic band structure, long-range coupling, gauge potential, band structure engineering

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