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Quadrature nonreciprocity in bosonic networks without breaking time-reversal symmetry

Nature Physics volume 19, pages 1429–1436 (2023)Cite this article

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Abstract

Nonreciprocity means that the transmission of a signal depends on its direction of propagation. Despite vastly different platforms and underlying working principles, the realizations of nonreciprocal transport in linear, time-independent systems rely on Aharonov–Bohm interference among several pathways and require breaking time-reversal symmetry. Here we extend the notion of nonreciprocity to unidirectional bosonic transport in systems with a time-reversal symmetric Hamiltonian by exploiting interference between beamsplitter (excitation-preserving) and two-mode-squeezing (excitation non-preserving) interactions. In contrast to standard nonreciprocity, this unidirectional transport manifests when the mode quadratures are resolved with respect to an external reference phase. Accordingly, we dub this phenomenon ‘quadrature nonreciprocity’. We experimentally demonstrate it in the minimal system of two coupled nanomechanical modes orchestrated by optomechanical interactions. Next, we develop a theoretical framework to characterize the class of networks exhibiting quadrature nonreciprocity based on features of their particle–hole graphs. In addition to unidirectionality, these networks can exhibit an even–odd pairing between collective quadratures, which we confirm experimentally in a four-mode system, and an exponential end-to-end gain in the case of arrays of cavities.

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Fig. 1: Quadrature nonreciprocity (qNR) versus standard nonreciprocity (sNR) versus reciprocal transport.
Fig. 2: TRS and qNR.
Fig. 3: qNR transmission in ring networks.
Fig. 4: Steady-state response of a qNR chain.

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Data availability

The data in this study are available from the Zenodo repository at https://doi.org/10.5281/zenodo.7927433. Source data are provided with this paper.

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Acknowledgements

We thank A. A. Clerk, D. Malz, A. Alù and S. A. Mann for useful discussions. C.C.W. acknowledges funding received from the Winton Programme for the Physics of Sustainability and EPSRC (EP/R513180/1). J.d.P. acknowledges financial support from the ETH Fellowship programme (grant no. 20-2 FEL-66). M.B. acknowledges funding from the Swiss National Science Foundation (PCEFP2_194268). This work is part of the research programme of the Netherlands Organisation for Scientific Research (NWO). It is supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 732894 (FET-Proactive HOT) and the European Research Council (ERC starting grant no. 759644-TOPP).

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Author notes

  1. These authors contributed equally: Clara C. Wanjura, Jesse J. Slim.

Authors and Affiliations

  1. Cavendish Laboratory, University of Cambridge, Cambridge, UK

    Clara C. Wanjura

  2. Max Planck Institute for the Science of Light, Erlangen, Germany

    Clara C. Wanjura

  3. Center for Nanophotonics, AMOLF, Amsterdam, The Netherlands

    Jesse J. Slim, Javier del Pino & Ewold Verhagen

  4. Institute for Theoretical Physics, ETH Zürich, Zürich, Switzerland

    Javier del Pino

  5. Department of Physics, University of Basel, Basel, Switzerland

    Matteo Brunelli

  6. Faculty of Physics, University of Vienna, Vienna, Austria

    Andreas Nunnenkamp

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Contributions

C.C.W. developed the theoretical framework with J.d.P. and M.B. J.J.S. fabricated the sample, performed the experiments and analysed the data. E.V. and A.N. supervised the project. All authors contributed to the interpretation of the results and writing of the manuscript.

Corresponding authors

Correspondence to Ewold Verhagen or Andreas Nunnenkamp.

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Extended data

Extended Data Fig. 1 Optomechanical device.

(a) Scanning electron micrograph of a device as used in our experiments. Three suspended beams are defined in the top silicon device layer (thickness 220 nm). The flexural motion of each pair of adjacent beams is coupled to an optical resonance, hosted by a point defect in the sliced photonic crystal defined by the teeth. In the presented experiments, we address only one of the cavities from free-space, at normal incidence, to drive and read out the mechanical motion of a single beam pair. The inset shows a top view, revealing the narrow slit separating the beam pair. More details can be found in ref. 44. (b) Spectrum of thermal fluctuations imprinted on the intensity of a read-out laser reflected off the optical cavity. In this frequency range, four mechanical resonances can be identified, with the corresponding simulated displacement profiles shown above. These mechanical modes serve as the resonators in the presented experiments.

Extended Data Fig. 2 Schematic of the experimental set-up.

A sample holding the sliced nanobeam device is placed in a room-temperature vacuum chamber rotated by 45∘ relative to the vertical polarization of the incoming light. Spectrally resolved drive and detection lasers are combined in-fibre, launched into free-space, and focused onto the sample, coupling into the nanocavity from normal incidence. In a cross-polarized detection scheme, the horizontal component of the light radiated out from the cavity is transmitted by a polarizing beamsplitter (PBS). Subsequently, a fibre-based tunable bandpass filter (BPF) rejects light at the drive laser frequency and transmits detection laser light onto a fast photodetector (PD2). A high-frequency lock-in amplifier (LIA) serves to analyze (In) the intensity modulations of the drive laser (for calibration) and detection laser, while also driving (Out) the drive laser intensity modulator (IM) through an amplification stage (not shown). In addition, the electronic displacement signal is routed through a digital signal processor (DSP) that optionally generates a feedback signal to modify resonator damping rates. LP, linear polarizer.

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Wanjura, C.C., Slim, J.J., del Pino, J. et al. Quadrature nonreciprocity in bosonic networks without breaking time-reversal symmetry. Nat. Phys. 19, 1429–1436 (2023). https://doi.org/10.1038/s41567-023-02128-x

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