Solitary waves in a phononic integrated circuit

Abstract

Solitons are universal nonlinear excitations that appear in settings as varied as optics, water waves, and quantum gases [1-5]. While reduced models of soliton dynamics are well established, their validity and dynamical behaviour in strongly nonlinear regimes with frequent interactions remain largely unexplored experimentally. Progress has been constrained by the difficulty of simultaneously achieving precise control of dispersion and nonlinearity, together with the temporal and spatial resolution required for dynamical observations. Here we overcome these difficulties by producing acoustic solitons in integrated phononic waveguides. We exploit the interplay between waveguide dispersion and mechanical Kerr nonlinearity to generate 'dark' solitons that persist over metre-scale propagation distances. The slow phonon velocity allows direct imaging of hundreds of dark soliton collisions -- two orders of magnitude more than have previously been accessible [6, 7] -- as well as soliton fission and the melting of a soliton Wigner crystal. Furthermore, the unprecedented dynamical resolution allows us to verify two long-predicted aspects of dark soliton behaviour: the existence of a collisional phase shift and two depth-dependent collision regimes [8, 9]. These results not only illuminate fundamental nonlinear energy transport processes, but also show a path towards acoustic versions of soliton-enabled technologies such as frequency combs and mode-locked lasers [1, 2, 10].

Figures (20)

• Figure 1: Dispersion and mechanical Kerr nonlinearity in an on-chip acoustic waveguide. a, Schematic illustration of out-of plane guided acoustic waves in a membrane waveguide. Note the section view is for illustrative purposes; in the physical device the extremity of the waveguide is also clamped (see Fig. \ref{['fig:Fig2']}a). b, Waveguide dispersion relation for the fundamental (n=1; solid blue), and second order (n=2; solid orange) acoustic modes (left axis). Grey shading: below cut-off; blue shading: single mode region romeroPropagationImagingMechanical2019mauranyapinTunnelingTransverseAcoustic2021. Group velocity $v_g$ (green, dash-dot --- right axis). c, Pulse chirp arising from GVD and self-phase modulation (SPM) arising from the mechanical Kerr nonlinearity. In our system, GVD and SPM combine to broaden bright pulses ($i \,\&\, ii$), and counterbalance in dark pulses leading to solitary wave solutions ($iii \,\&\, iv$) agrawalNonlinearFiberOptics2019.
• Figure 2: Experimental setup and actuation protocol.a, Bottom: color optical photograph of the fabricated device, showing four groups of four acoustic waveguides of length $L=1$ cm and width $W=25$$\mu$m. The iridescence is caused by the periodic array of sub-wavelength release holes mauranyapinTunnelingTransverseAcoustic2021romeroAcousticallyDrivenSinglefrequency2024. Top: zoom-ins show scanning electron microscope images of the top waveguide. (i): waveguide termination; (ii): mid-section of waveguide; (iii): waveguide extremity with electrostatic actuation electrode. b, Optical and electronic drive and readout scheme. AOM: acousto-optic modulator; VBS: variable beam-splitter; SA: spectrum analyzer; AWG: arbitrary waveform generator; FPC: fiber polarization controller. See Supplementary Information for more details. c, Illustration of the actuation pulse that produces a dark solitary wave. d, Diagram of the soliton's progression through the waveguide and resulting measured power spectrum: (i) before, (ii) during, and (iii) after passing the optical fiber probe.
• Figure 3: Acoustic dark solitary waves.a, Sketch of predicted dark pulse behaviour in the low amplitude (linear) and high amplitude (nonlinear) regimes. Note the soliton compression highlighted by the red and blue arrows, and the shedding of grey solitary waves on either side of the pulse. b, Experimental pulse evolution (left panel) and NLSE simulation (right panel) in the low amplitude regime ($A_0=1.3$ nm). c, Experimental pulse evolution (left panel) and NLSE simulation (right panel) in the high amplitude regime ($A_0=26.3$ nm). d, Cross-sections of the measured RMS displacement at the coordinates shown in (c) (blue), along with a black and grey soliton model fit (dashed orange) - see Supplementary Information.
• Figure 4: Soliton fission and solitary wave repulsion.a, Schematic illustration of the break-up of a broad ($T\gg T_s$) initial dark pulse into its solitary wave components. b Experiment (left) and NLSE simulation (right). Here $A_0=59.5\,\mathrm{nm}$. c, Zoom-in of the wave crossings (white boxes in b). d, Sketch of wave crossings in the case of no interactions (left) and repulsive interactions (right). The collisional phase shift is apparent, as each soliton emerges ahead of its pre-collision trajectory huangDarkSolitonsTheir2001thurstonCollisionsDarkSolitons1991.
• Figure 5: Programmable soliton-soliton collider. a, Sketch of the waveguide initialisation to perform head-on collisions. b, Experimental data (top) and NLSE simulation (bottom). c, Sketch of the waveguide initialisation to perform overtaking collisions. d, Experimental data (top) and NLSE simulation (bottom). e, Role of the parity of the injected pulse: an appropriately-sized tanh pulse (odd) forms a single solitary wave, while an even $\tanh^2$ pulse forms a pair of dark solitary waves. f, Experimental data (top) and NLSE simulation (bottom). All NLSE simulations are performed with no free parameters (see Supplementary Information).