Abstract
The fermionic Kitaev chain is a canonical model featuring topological Majorana zero modes1. We report the experimental realization of its bosonic analogue2 in a nano-optomechanical network, in which the parametric interactions induce beam-splitter coupling and two-mode squeezing among the nanomechanical modes, analogous to hopping and p-wave pairing in the fermionic case, respectively. This specific structure gives rise to a set of extraordinary phenomena in the bosonic dynamics and transport. We observe quadrature-dependent chiral amplification, exponential scaling of the gain with system size and strong sensitivity to boundary conditions. All these are linked to the unique non-Hermitian topological nature of the bosonic Kitaev chain. We probe the topological phase transition and uncover a rich dynamical phase diagram by controlling interaction phases and amplitudes. Finally, we present an experimental demonstration of an exponentially enhanced response to a small perturbation3,4. These results represent the demonstration of a new synthetic phase of matter whose bosonic dynamics do not have fermionic parallels, and we have established a powerful system for studying non-Hermitian topology and its applications for signal manipulation and sensing.
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Data availability
The data in this study are available from the Zenodo repository at https://doi.org/10.5281/zenodo.10279485.
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Acknowledgements
We acknowledge A. A. Clerk for discussions. M.B. acknowledges funding from the Swiss National Science Foundation (Grant No. PCEFP2_194268). J.d.P. acknowledges financial support from the ETH Fellowship programme (Grant No. 20-2 FEL-66). This work is part of the research programme of the Netherlands Organisation for Scientific Research. It is supported by the European Research Council (Starting Grant No. 759644-TOPP).
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J.J.S. and E.V. conceived the project. J.J.S. fabricated the sample, developed the experimental methods, performed the experiments and analysed the data. C.C.W., M.B., J.d.P. and A.N. developed the theory with input from J.J.S. and E.V. All authors contributed to the interpretation of the results and writing the manuscript. E.V. and A.N. supervised the work.
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Extended data figures and tables
Extended Data Fig. 1 Topological phase diagrams for varying chain lengths.
These panels show the calculated end-to-end gain \({\chi }_{{x}_{1}\to {x}_{N}}\) for finite open chains of length N = 4 (panel (a), corresponding to Fig. 3c), N = 10 (b), and N = 20 (c), as a function of phase φ and squeezing amplitude λ/J for J/γ = 5/16. As in Fig. 3c, the dashed lines depict the phase boundaries, determined by the structure (point gap and winding of origin) of the complex spectrum for an infinite chain under PBC. We recognize that the conditions for which the end-to-end gain exceeds unity align closely with the topological phase boundary that marks spectral winding of the origin. Slight deviations between those two are observed for smaller chains near φ = 0 and φ = π/2, and are thus associated with finite size effects. Still, the general structure of the phase diagram of the infinite chain is clearly recognized in the end-to-end gain for N = 4.
Extended Data Fig. 2 Experimental frequency accuracy.
(a) Thermomechanical reference spectra. Before each experimental run, a set of thermomechanical spectra is taken, with modulations off, to determine the value of the mechanical resonance frequencies ωj. A typical set is shown. Spectra are averaged over 50 runs and fitted by a Lorentzian function, with uncertainty (standard deviation) in the extracted ωj’s (grey dashed lines) between σω,fit/(2π) = 20 − 24 Hz, estimated from the fit residuals. (b) Frequency stability of the mechanical modes. Resonator frequencies ωj are extracted from thermomechanical resonance spectra, measured repeatedly over a duration of 90s. Error bars denote the ± 2σ fit uncertainty interval estimated from residuals. Over time, fluctuations in ωj are observed with standard deviation σω,fluct/(2π) = 43 − 64 Hz, depending on the mode. Correlated fluctuations across the modes indicate a common origin, that we attribute to fluctuations in the optical spring effect. This is compatible with the observation that mechanical modes with higher index and lower optomechanical couplings g0,j experience smaller frequency fluctuations.
Extended Data Fig. 3 Bar chart depiction of response matrices.
(a) Bar chart representation of the susceptibility matrices \({(\gamma {\chi }_{jk})}^{2}/4\) plotted in Fig. 1d, for G = 0.75, G = 1.0, and G = 1.25. (b) The same data as in (a), depicted in a basis of rotated quadratures chosen to numerically minimize the weight of the off-diagonal blocks. The close correspondence between (a) and (b) illustrates that the original (unrotated) calibration of quadratures was performed accurately, and represents the quadrature sensitivity of the response optimally.
Extended Data Fig. 4 Inferred experimental dynamical matrices.
The dynamical matrix \({{\mathcal{M}}}_{a}\) describing the evolution of mode operators \({a}_{j},{a}_{j}^{\dagger }\) is inferred from the measured susceptibility matrix χ through \({{\mathcal{M}}}_{a}={{\mathcal{T}}}^{-1}{({\rm{i}}\chi )}^{-1}{\mathcal{T}}\), where \({\mathcal{T}}\) is the matrix that transforms from mode basis \(\{{a}_{j},{a}_{j}^{\dagger }\}\) to the quadrature basis {xj, pj}. A perfect bosonic Kitaev chain has purely imaginary \({{\mathcal{M}}}_{a}\), that is, no on-site detunings and purely imaginary coupling rates. In our experiments, the real part of the diagonal elements of \({{\mathcal{M}}}_{a}\) indicates a small residual detuning (averaged across the four participating modes) of 3%, 5% and 7% of the mechanical linewidth γ for G = 0.75, 1, 1.25, respectively. The first two modes have higher optomechanical couplings g0 and are therefore more susceptible to fluctuations and nonlinearities in the optical spring effect.
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Slim, J.J., Wanjura, C.C., Brunelli, M. et al. Optomechanical realization of the bosonic Kitaev chain. Nature 627, 767–771 (2024). https://doi.org/10.1038/s41586-024-07174-w
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DOI: https://doi.org/10.1038/s41586-024-07174-w
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