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Quantized fractional Thouless pumping of solitons

Nature Physics volume 19, pages 420–426 (2023)Cite this article

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Abstract

In many contexts, interaction between particles gives rise to emergent phenomena. An example is the fractional quantum Hall effect, where the interaction between electrons leads to fractionally quantized Hall conductance. In photonic systems, the nonlinear response of an ambient medium mediates the interaction between photons, and, in the mean-field limit, these dynamics are described by the nonlinear Schrödinger (also called Gross–Pitaevskii) equation. It was recently shown that at weak nonlinearity, soliton motion in nonlinear Thouless pumps—a dimensionally reduced implementation of a Chern insulator—could be quantized to the Chern number, because solitons track the single-band Wannier function throughout the pumping cycle. Here using arrays of coupled optical waveguides, we show that a sufficiently strong nonlinearity fractionally quantizes the motion of solitons. Specifically, we find that the soliton follows maximally localized multi-band Wannier functions and therefore returns to itself only after multiple cycles of the Thouless pump—but displaced by an integer number of unit cells—leading to a rich fractional plateau structure describing soliton motion. Our results represent an example of emergent behaviour in topologically non-trivial systems in the presence of interactions.

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Fig. 1: Model for quantized fractional Thouless pumping.
Fig. 2: Theory of quantized fractional Thouless pumping.
Fig. 3: Experimental observation of an f = −1/2 fractionally pumped soliton.
Fig. 4: Average centre-of-mass displacement per period, 〈x〉, calculated for an off-diagonal AAH model with 13 bands.

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Data availability

Data that support the findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.

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Acknowledgements

We acknowledge fruitful discussions with S. Gopalakrishnan and S. Vaidya. We further acknowledge the support of the ONR YIP programme under award no. N00014-18-1-2595, ONR-MURI programme N00014-20-1-2325, the AFOSR-MURI programme FA9550-22-1-0339, as well as the Packard Foundation (fellowship no. 2017-66821). C.J. gratefully acknowledges funding from the Alexander von Humboldt Foundation within the Feodor-Lynen Fellowship programme. Numerical calculations were performed on the Pennsylvania State University’s Institute for Computational and Data Sciences’ Roar supercomputer.

Author information

Authors and Affiliations

  1. Department of Physics, The Pennsylvania State University, University Park, PA, USA

    Marius Jürgensen, Sebabrata Mukherjee, Christina Jörg & Mikael C. Rechtsman

  2. Department of Physics, Indian Institute of Science, Bangalore, India

    Sebabrata Mukherjee

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  1. Marius Jürgensen
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Contributions

M.J. designed, fabricated and cut the devices and carried out all measurements. M.J. characterized the devices with input from C.J. S.M. designed and built the experimental beam shaping and fabrication set-up. The theoretical investigation was carried out by M.J. M.J. and M.C.R. wrote the manuscript, with input from C.J. M.C.R. supervised the project.

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Correspondence to Marius Jürgensen or Mikael C. Rechtsman.

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Extended data

Extended Data Fig. 1 Properties of solitons.

a. Linear energy bands (grey) for the model described in the main text at Ωz/2π=0.3 together with the nonlinear eigenvalue of a soliton (red) for increasing nonlinearity. Dashed line marks a nonlinear bifurcation point, with a corresponding slope change. b. Wavefunction of the soliton from a. c. Projection of the soliton wavefunction onto the linear energy eigenstates. The eigenstates are sorted via their eigenvalues from bottom (lowest) to top (highest).

Extended Data Fig. 2 Adiabaticity of soliton propagation.

a. Overlap of a numerically propagated soliton with the instantaneous soliton after one pump cycle with period L. b. Same data as a, but plotted as deviation from perfect overlap on a log-log scale confirming the existence of an adiabatic limit. The blue (orange) line depicts to \(gP/{J}^{\max }\)=0.55 (\(gP/{J}^{\max }\)=1.65), which corresponds to the integer (fractionally) pumped soliton in the adiabatic limit.

Extended Data Fig. 3 Linear waveguide characterization.

a. Hopping constant J as a function of separation s between waveguides. Each J is calculated via fitting 15 couplers with varying coupling length. Dashed line shows the exponential fit for the coupling function. b. On-site detuning in propagation constant for two waveguides written next to each other as a function of separation s. c. Transmission measurements of a straight waveguide including in- and out-coupling losses. Y-axis is plotted in log scale. The transmission measured for a propagation of ≈ 55 mm is excluded from the exponential fit (dashed line) due to a crack in the sample creating additional losses. The resulting propagation loss is (0.33 ± 0.02)dB/cm.

Extended Data Fig. 4 Nonlinear waveguide characterization.

a. Setup for nonlinear characterization of the waveguides including a pair of gratings (G1 and G2) to stretch and down-chirp the excitation pulse. The output facet is imaged onto a camera and simultaneously focused into a fiber-coupled optical spectrum analyzer (OSA). b. Measured output power for three straight waveguides as a function of the input power showing a linear dependence and hence no nonlinear absorption for input powers less than 6 mW. The threshold indicates the maximum power used in the experiments to prevent damage to the sample. c. Relative intensity in the two waveguides (I1 and I2) of a directional coupler as a function of power. Black lines are a least square fit, resulting in g=0.068/mm per mW input power.

Extended Data Fig. 5 Spectral analysis.

a. Measured normalized spectrum after propagation through a 76.15 mm long, straight waveguide in the uncut sample for increasing input power. The white lines mark the width in which 76% of the intensity is found (corresponding to the full-width-half-maximum of a Gaussian). b,c. Similar to a but spatially resolved and for an array showing fractional pumping. The insets show the waveguide modes at the output facet for increasing input power (from bottom to top). The white rectangle marks the spatial 1/e2 width for which the spectrum is measured.

Extended Data Fig. 6 Triple coupler.

a. Intensity distribution at the output facet for a coupler consisting of three waveguides for low input power. The coupler extends over a length of 5 mm and the top waveguide is excited. b. Normalized output intensities of a triple coupler as a function of the input power. The inset defines the color-coding. Due to the high coupling constant between the waveguides over a short propagation distance of 5 mm, the effects of nonlinearity are negligible.

Extended Data Fig. 7 Verification of soliton’s pumping behavior in the experiment.

a. Amplitude overlap, \({\sum }_{n}| {\Psi }_{n}^{{{{\rm{(Exp.)}}}}}| | {\Psi }_{n}^{{{{\rm{(S)}}}}}|\), between the measured output wavefunction, \({\Psi }_{n}^{{{{\rm{(Exp.)}}}}}\), and the numerically-calculated instantaneous soliton wavefunction, \({\Psi }_{n}^{{{{\rm{(S)}}}}}\), as a function of propagation length for linear propagation (\(gP/{J}^{\max }\)=0.04; shown in blue) and soliton propagation (\(gP/{J}^{\max }\)=2.15; shown in red). b. Center of mass displacement of the experimentally observed soliton and linear propagation (calculated using a higher order norm to suppress linear background effects; see text). Gray line indicates the numerically-calculated displacement of an instantaneous fractionally pumped soliton. c. Experimentally observed center of mass displacement after two periods for increasing input power showing a plateau at high input power. Gray line indicates the expected theoretical displacement of five sites (one unit cell) after two periods for the fractionally pumped soliton. Solid lines with squares show mean values, and shaded areas show one standard deviation for independently measured soliton propagation in eight different unit cells of the same lattice.

Extended Data Fig. 8 Band structure of a 13 site AAH model.

a. Additionally to the 13 linear energy bands (black) the nonlinear eigenvalues of four pumped solitons with \(gP/{J}^{\max }\)=0.04, 0.10, 0.78 and 3.08, which are part of the plateaux in Fig. 4 in the main text, are shown in green, red, blue and yellow, respectively. b Zoom-in onto the central group of three energy bands. c Zoom-in onto the lowest group of five energy bands.

Extended Data Fig. 9 Wannier function positions of a 13 site AAH model.

a. Center of mass position of single-band Wannier functions calculated for each of the 13 bands individually over one period and projected into a single unit cell. The number of windings is equal to the Chern number C of the corresponding band. b-g similar to a but for multi-band Wannier functions in c-g. The position of pumped solitons from the plateaus of Fig. 4 in the main text at nonlinearities \(gP/{J}^{\max }\)=0.04, 0.10, 0.78 and 3.08, are shown in green, red, blue and yellow, respectively.

Supplementary information

Supplementary Information

Supplementary Discussion and Supplementary Fig. 1.

Supplementary Video 1

In Supplementary Animation 1, we compare the measured propagation of the fractionally pumped soliton with numerical tight-binding simulations. The animation complements Fig. 3 and shows the integrated intensity per waveguide for two periods, measured by cutting the sample and observing the intensity distribution at the output facet for increasing input power.

Supplementary Video 2

In Supplementary Animation 2, we show the robustness of the fractionally pumped soliton against bandgap closings (and corresponding Chern number changes) amongst the participating bands. We modify the hopping functions and tune through two topological phase transitions which close the bandgap between the two lowest bands, interchanging their individual Chern numbers from C = {2, −3} to C = {−3, 2} and back to C = {2, −3}. While this process drastically alters the single-band Wannier functions, the multi-band Wannier function of the two lowest bands and therefore also the fractionally pumped soliton do not change their pumping behaviour. In Supplementary Animation 2, we use a five site AAH model with hoppings given by J_n =\left(-0.33-|\cos(\Omega z /2+ \frac{2\pi n}{5})|^{\gamma}\right)/b, where b is an arbitrary length, and we tune γ. The soliton is calculated for gP/Jmax =1.9.

Source data

Source Data Fig. 2

Numerical source data

Source Data Fig. 4

Numerical source data

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Jürgensen, M., Mukherjee, S., Jörg, C. et al. Quantized fractional Thouless pumping of solitons. Nat. Phys. 19, 420–426 (2023). https://doi.org/10.1038/s41567-022-01871-x

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