Abstract
Imposing chirality on a physical system engenders unconventional energy flow and responses, such as the Aharonov–Bohm effect^{1} and the topological quantum Hall phase for electrons in a symmetrybreaking magnetic field. Recently, great interest has arisen in combining that principle with broken Hermiticity to explore novel topological phases and applications^{2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}. Here we report phononic states with unique symmetries and dynamics that are formed when combining the controlled breaking of timereversal symmetry with nonHermitian dynamics. Both of these are induced through timemodulated radiation pressure forces in small nanooptomechanical networks. We observe chiral energy flow among mechanical resonators in a synthetic dimension and Aharonov–Bohm tuning of their eigenmodes. Introducing particlenonconserving squeezing interactions, we observe a nonHermitian Aharonov–Bohm effect in ringshaped networks in which mechanical quasiparticles experience parametric gain. The resulting complex mode spectra indicate fluxtuning of squeezing, exceptional points, instabilities and unidirectional phononic amplification. This rich phenomenology points the way to exploring new nonHermitian topological bosonic phases and applications in sensing and transport that exploit spatiotemporal symmetry breaking.
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Data availability
The data in this study are available from the Zenodo repository at https://doi.org/10.5281/zenodo.6320519.
Change history
31 October 2022
A Correction to this paper has been published: https://doi.org/10.1038/s41586022054649
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Acknowledgements
We thank C. Wanjura, A. Nunnenkamp and M. Brunelli for discussions, and M. SerraGarcia, S. Rodriguez, F. Koenderink and O. Zilberberg for critical reading of the manuscript. This work is part of the research programme of the Netherlands Organisation for Scientific Research (NWO). We acknowledge support from the European Research Council (ERC starting grant no. 759644TOPP) and the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 732894 (FET Proactive HOT). J.d.P. acknowledges financial support from the ETH Fellowship programme (grant no. 202 FEL66).
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J.d.P. developed the theoretical framework. J.J.S. fabricated the sample, performed the experiments, analysed the data and aided in constructing the theoretical approach. E.V. conceived and supervised the project. All authors contributed to the interpretation of results and writing of the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Estimation of beamsplitter interaction strengths.
a, Mode splitting induced by a beamsplitter interaction observed in thermomechanical spectra. Each column corresponds to a beamsplitter interaction induced between a pair of resonators i ↔ j (left: 1 ↔ 2, middle: 2 ↔ 3, right: 1 ↔ 3) by a single drive laser modulation at frequency Δω_{ij} = ω_{i} − ω_{j}, where ω_{i,j} is the frequency of resonator i, j. Thermomechanical spectra (top row: resonator i, bottom row: resonator j) are recorded for increasing modulation depth c_{m}. The linear relationship \({J}_{{\rm{est}}}={c}_{{\rm{m}}}\sqrt{{\rm{\delta }}{\omega }_{i}{\rm{\delta }}{\omega }_{j}}/2\) is used to estimate the coupling strength J_{est} (top axis) from c_{m}, where δω_{i,j} is the optical spring shift of mode i, j. The estimated mode splitting (dashed) is slightly larger than observed, presumably due to frequencydependent transduction (at d.c. and frequency Δω_{ij}) in the measurement of c_{m}. The difference is quantified by extracting Lorentzian peak frequencies from the spectra and subsequently fitting those linearly against modulation depth, and results in an observed mode splitting slope that is 78%, 90% and 90% of the estimated slope, respectively. The average estimation offset of 86% is applied to all (beamsplitter and squeezing) interaction strength calculations in our experiments. b, Time evolution of the coherent amplitude (in units of their zeropoint fluctuations x_{zpf}) of a pair of resonators (1, blue and 2, red) coupled through a beamsplitter interaction (strength J/(2π) = 5 kHz). Resonator 1 is initially (time t < 0) driven to a highamplitude steady state by a coherent drive laser modulation. At t = 0, the drive is switched off and the interaction is switched on. Rabi oscillations induced by the coupling interaction are observed, where energy is transferred back and forth between the resonators until the coherent energy in the resonators is dissipated. These dynamics illustrate the possibility for a transfer scheme in the strongcoupling regime where couplings are interrupted after a Rabi semicycle, that is, a time t_{π} = π/(2J). The energy transfer efficiency for this process can be calculated^{67} to be ~64% for corresponding parameters and 70% for the coupling rates presented in Fig. 1.
Extended Data Fig. 2 Frequency and linewidth modulation in the squeezing dimer.
a, Experimental resonance frequencies (top) and linewidths (bottom) obtained by fitting a superposition of Lorentzian lineshapes to the thermomechanical spectra in Fig. 3c, d. Grey curves indicate theoretical values of Re(ϵ) (top) and Im(ϵ) (bottom). Two peaks were fitted to the spectra for Φ = 0 (left), as for that flux both eigenvalues are expected to be doubly degenerate for all J. The observed branching of frequencies and linewidths is characteristic of an exceptional point. Four peaks were instead fitted for Φ = π (middle), where the exceptionalpoint behaviour completely vanishes, and spectra are fitted well with a combination of broad and narrow peaks at two frequencies. When varying flux in the rightmost panel, the grey shaded areas depict the regions near Φ = 0, π where a fit of two peaks provided better results than a fit of four. Note that near the exceptional point, the nonLorentzian nature of the spectrum causes the fitted values of the Lorentzian linewidths to deviate from the theoretical Im(ϵ). This origin of the deviation is confirmed by applying the same fit procedure to theoretically predicted spectra (inset, bottom left), which shows the same deviation. Error estimation is described in Methods section ‘Error estimation’. b, Thermomechanical spectra for several values of J/η, for Φ = 0 (blue) and Φ = π (red). Solid lines show Lorentzian fits. c, As in b, but for different values of Φ at J = η.
Extended Data Fig. 3 Network graph representation of general quadratic Hamiltonians.
Schematic of an arbitrary Hamiltonian matrix \( {\mathcal H} \), acting on a Nambulike vector \({\boldsymbol{\alpha }}=({a}_{1},{a}_{2},\cdots ,{a}_{N},{a}_{1}^{\dagger },{a}_{2}^{\dagger },\cdots ,{a}_{N}^{\dagger })\). Particle annihilation (hole creation) operators, a_{i}, are represented by blue nodes, and hole annihilation (particle creation) operators are represented by orange nodes. \( {\mathcal H} \) includes excitationconserving interactions (matrix \({\mathscr{A}}\)), which link particle operators (for example, terms \({{\mathscr{A}}}_{ij}{a}_{i}^{\dagger }{a}_{j}\)) and hole operators (for example, terms \({{\mathscr{A}}}_{ji}^{\ast }{a}_{j}{a}_{i}^{\dagger }\)). Squeezing interactions (with complex amplitude matrix \( {\mathcal B} \)) contain pairs \({ {\mathcal B} }_{ij}{a}_{i}^{\dagger }{a}_{j}^{\dagger }\) that can be visualized to either annihilate two particles i, j or to annihilate a particle in i and create hole in j, hence the connection between particle and hole networks (green). Mutatis mutandis, terms \({ {\mathcal B} }_{ij}^{\ast }{a}_{i}{a}_{j}\) can be similarly visualized.
Extended Data Fig. 4 Calculated eigenstates of the \({{\boldsymbol{a}}}_{{\bf{1}}},{{\boldsymbol{a}}}_{{\bf{2}}},{{\boldsymbol{a}}}_{{\bf{3}}}^{\dagger }\) loop in the singly conjugated trimer studied in Fig. 4.
a, Phase diagram for the imaginary part of the eigenfrequencies, showing the stabilitytoinstability boundary in ξ–Φ space, where \(\xi =J/(2\sqrt{2}\eta )\) and γ_{i} = 0. Such boundary is associated with a secondorder exceptional contour. b, Cuts of the eigenfrequency Riemann surfaces along Φ = 0, shown as a red dashed trajectory in the phase diagram, as a function of the ratio \(\xi =J/(2\sqrt{2}\eta )\). The squared weights of the J = 0 eigenstates in the full eigenvectors are shown in the colour scale. The weights are calculated from the symplectic projections (Σ_{z} product) on the gainy/lossy combinations a_{g}, a_{l} and the passive mode a_{−}. A secondorder exceptional point (denoted EP2), found for \(J=2\sqrt{2}\eta \), is highlighted. As \(J < 2\sqrt{2}\eta \), \({{\mathscr{P}}}_{{\rm{gl}}}{\mathscr{T}}\) symmetry is spontaneously broken, inducing eigenstate localization. The antisymmetric 1–2 mode a_{−} is detached from this mechanism and remains uncoupled. Real and imaginary parts are rescaled by η. c, As in b, but along the cut Φ = π/2 (corresponding to the blue dashed line in a, which shows the thirdorder exceptional point (EP3, at \(J=\sqrt{2}\eta \)). The \({{\mathscr{P}}}_{{\rm{gl}}}{\mathscr{T}}\)symmetry broken states are now hybrid combinations of a_{g}/a_{−} and a_{l}/a_{−} modes. Such combinations break \({{\mathscr{P}}}_{12}{\mathscr{T}}\) symmetry as well, as explained in the text.
Extended Data Fig. 5 Experimental setup.
a, Electron micrograph (left, tilt 45°; inset, top view) showing a device as used in our experiments. In the top silicon device layer (thickness 220 nm), three suspended beams are defined with teeth separated by a narrow slit (~50 nm). Between each outer beam and the central beam, a photonic crystal cavity is defined that hosts an optical mode (right, simulated electric field y component, E_{y}). The mode’s energy is strongly confined to the narrow slits, inducing large parametric interaction with flexural mechanical resonances of the two beams. The cavity’s offcentre position ensures coupling to both even and odd resonances. In the presented experiments, we only use one of the two cavities. The widths of the outer beams’ straight sections are intentionally made unequal, such that the mechanical resonances of all beams are detuned. The top layer is supported by pedestals etched out in the buried silicon oxide layer. b, Schematic of the experimental setup. The ultrahighfrequency lockin amplifier (LIA) ports serve to (Out) drive the intensity modulator through an amplification stage (not shown) and to (In) analyse intensity modulations of the drive laser (for calibration) and detection laser. For timeresolved measurements, the signal generator (SG) is programmed to (Out) actuate the drive signal switches and trigger the lockin amplifier acquisition. The digital signal processor (DSP) optionally generates a feedback signal to modify resonator damping rates. BPF, optical bandpass filter; IM, intensity modulator; LP, linear polarizer; PBS, polarizing beamsplitter; PD1, PD2, photodiodes; SWs, microwave switches.
Extended Data Fig. 6 Optical spring shift and optothermal backaction.
a, Thermomechanical noise spectra of the first few mechanical modes imprinted on an unmodulated single drive/detection laser, as the laser’s frequency (ω_{L}) is swept across the cavity resonance. The four most intense peaks around frequencies ω_{i}/(2π) ≈ {3.7, 5.3, 12.8, 17.6} MHz correspond to flexural modes (labelled i) of the individual beam halves and show frequencytuning characteristic to the optical spring effect, and the other peaks represent nonlinearly transduced harmonics of those modes.^{65}. PSD, power spectral density.b, Magnification of the PSD of the first four resonators. c, From the spectra in b, resonance frequencies ω_{i} (blue circles) and linewidths γ_{i} (red circles) are extracted. The resonance frequencies are fitted using the standard optical spring model (solid blue). Across all resonators, we find agreement in the fitted cavity resonance ω_{c}/(2π) = 195.62 THz and linewidth κ/(2π) = 320 GHz (Q factor, Q ≈ 600). The small sideband resolution ω_{i}/κ ≈ 10^{−5} suggests very little change in linewidth due to dynamical cavity backaction (dashed red). The linewidth modulations we observe suggest the presence of an optothermal retardation effect^{66}. Displayed errors correspond to fit uncertainty, smaller than plot markers on the fitted frequencies (Methods section ‘Error estimation’). On each panel, blue/red colourcoded arrows indicate the y scale for each plotted quantity. d, Drive laser frequency sweep, now using a separate, fixedfrequency, fardetuned detection laser. The fixed transduction of mechanical motion onto this detection laser allows a comparison of resonance peak area A_{i}(ω_{L}) versus linewidth γ_{i}(ω_{L}) as the drive laser frequency ω_{L} is varied (a.u., arbitrary units). The resonance peak area of mode i is proportional to the variance \(\langle {X}_{i}^{2}\rangle \) of its displacement x_{i}, which is proportional to its temperature T_{i}. Dynamical backaction modifies the effective mode temperature through \({T}_{i}={T}_{0}({\tilde{\gamma }}_{i}/{\gamma }_{i})\) (ref. ^{29}), where T_{0} is the initial temperature and \({\tilde{\gamma }}_{i}\) is the mode’s intrinsic linewidth, determined by switching off the drive laser. Our data are well explained by linear fits of A_{i}(ω_{L}) versus \({\tilde{\gamma }}_{i}/{\gamma }_{i}({\omega }_{{\rm{L}}})\) (dashed), confirming the effective temperature model.
Extended Data Fig. 7 Singlemode squeezing and linewidth modulation by parametric driving.
a, Parametric gain induced by a singlemode squeezing interaction observed in thermomechanical spectra. Each row corresponds to a separate experiment where resonator i (1 through 4) is subjected to a singlemode squeezing interaction of strength η. As η is increased, the resonance transitions from the broad intrinsic linewidth to a narrow parametric resonance. b, The phasespace distribution of the thermal fluctuations of resonator i (left: 3; right: 4) subject to a singlemode squeezing interaction of strength η/(2π) = 1 kHz with squeezing angle θ = π/2 reveals a squeezed thermal state. The squeezed (antisqueezed) quadrature X (Y), measured in units of the thermal equilibrium amplitude \(\sqrt{{\bar{n}}_{i}}\), are referenced using the propagation delay (Methods). c, Fitted Lorentzian fullwidth at halfmaximum linewidths of the resonances shown in a. Even though a superposition of two degenerate resonances is expected—a broadened resonance of the antisqueezed quadrature and a narrowed resonance of the squeezed quadrature—only a single one can be successfully fitted in each spectrum. This reflects the fact that the highly populated narrowed resonance dominates the broadened resonance. As the parametric gain η is increased, each resonator’s squeezed quadrature linewidth is expected to decrease by Δγ = −2η (dashed lines), until parametric threshold is reached at η = γ_{i}/2, where γ_{i} is the intrinsic linewidth of resonator i. The fitted linewidths follow the expected trend quite closely for intermediate η, whereas for lower η the narrow resonance is presumably not yet fully dominant and for larger η highamplitude nonlinear effects are prominent. Error bars correspond to the fit uncertainty, and are smaller than the symbol size in most points (Methods section ‘Error estimation’).
Extended Data Fig. 8 Damping rate adjustment by feedback.
Resonator thermomechanical spectra (top row) and fitted fullwidth halfmaximum linewidths (bottom row) adjusted by feeding back electronically filtered and phaseshifted resonator displacement signals onto the drive laser modulation (left two columns, resonator 1; right two columns, resonator 2). The resonator linewidth (circles) and frequency shift (crosses) vary sinusoidally with the feedback phase ϕ_{fb} (odd columns). By fitting the linewidth variation (solid black), the optimal phase shift to increase the damping rate is selected. The frequency variation (dashed grey) expected from the fitted linewidth modulation, relative to the resonator frequency with feedback off (dashed red), lags by π/2 radians. For the optimal feedback phase shift, an increase in linewidth is observed for increasing gain G, whereas the resonator frequency remains unaffected (even columns). The slope of the linear fit (solid black) can be used when setting a resonator’s linewidth to a desired value. Error bars reflect the fit uncertainty and control parameter stability, and are typically smaller than the plot marker size (Methods section ‘Error estimation’).
Extended Data Fig. 9 Resonator coherent response.
a, Amplitude a_{i} (blue, left axis) and phase ϕ_{i} (red, right axis) of the complex response \({a}_{i}({\Delta })={{\rm{e}}}^{{\rm{i}}({\phi }_{i}({\Delta })+{\alpha }_{i})}{a}_{i}({\Delta })\) of resonators 1 through 4 (resonance frequencies ω_{i}) to a drive laser modulation at a frequency ω_{d} close to resonance (drive detuning Δ = ω_{d} − ω_{i}). α_{i} is the phase offset due to signal delay through the setup. A Lorentzian response \({a}_{i}={{\rm{e}}}^{{\rm{i}}{\alpha }_{i}}{A}_{i}\frac{{\gamma }_{i}/2}{{\rm{i}}{\gamma }_{i}/2{\Delta }}\) is fitted to the data (dashed). b, Phase offset α_{i} versus resonance frequency ω_{i}/(2π). A linear fit (dashed) of α_{i} = −ω_{i}τ implies a signal delay τ = 106.7 ns. Error bars in a reflect the spread in repeated measurement, whereas the error bars in b correspond to fit uncertainty and are smaller than the symbol size (Methods section ‘Error estimation’). a.u., arbitrary units.
Extended Data Fig. 10 Tunable singlemode and effective twomode squeezing in the squeezing dimer.
a, Intraresonator squeezing as a function of the beamsplitter coupling J. Two values Φ = 0, π of the flux are shown for equal singlemode squeezing strengths η_{1} = η_{2} = 0.5 kHz. The level of singlemode squeezing is expressed by the ratio of the smallest (\(\Delta {R}_{{\rm{sq}}}^{2}\)) and largest (\(\Delta {R}_{{\rm{a}}}^{2}\)) eigenvalues of the covariance matrix of the quadrature amplitudes recorded for each resonator. These eigenvalues indicate the amplitude variance along the squeezed and antisqueezed principal quadrature components, respectively. For Φ = π, where the squeezed (antisqueezed) quadratures X_{i} (Y_{i}) of both resonators are coupled (see Fig. 2d), the slight initial imbalance in variance ratio is reduced as J increases while the value of the variance ratio remains low. By contrast, for Φ = 0—when the squeezed quadrature X_{i} in one resonator is coupled to the antisqueezed quadrature Y_{j} in the other—we observe cancellation of singlemode squeezing as the variance ratio tends to 1 with increasing J. This agrees well with theory (dashed line), where for simplicity we have assumed equal dissipation rates \(\bar{\gamma }=2.2\,{\rm{kHz}}\) equal to the average of the experimental losses γ_{i} = {2.6, 1.9} kHz, as well as equal bath occupations. Owing to dynamical (optothermal) backaction, for this particular experiment the effective bath occupations \({\bar{n}}_{1}\approx {\bar{n}}_{2}\) only differed by a few per cent. b, Twomode squeezing observed in the crossresonator amplitude distribution of quadratures X_{1} and Y_{2} for Φ = 0, J = 3.5 kHz and η_{1} = η_{2} = 0.5 kHz. The dashed ellipse depicts the standard deviation of the principal components of the quadrature covariance matrix and shows positive correlations between X_{1} and Y_{2} (covariance σ(X_{1}, Y_{2}) = 0.08). c, Covariance of the coupled quadrature pairs X_{1}Y_{2} and Y_{1}X_{2} as a function of J, with η_{1} = η_{2} = 0.5 kHz. No correlations are found for flux Φ = π, when singlemode squeezing is strongest and independent of J (compare with a). However, for Φ = 0, positive correlations σ(X_{1}, Y_{2}), σ(Y_{1}, X_{2}) > 0 are found when J is increased, as predicted by theory (dashed line). A tradeoff between the squeezing axes rotation towards the standard twomode squeezing limit and the decrease in the overall squeezing level as J is increased leads to a maximum covariance (although not optimal squeezing level for the rotated quadratures) at a coupling J_{opt}. For the simple theory model with equal dissipation and bath occupation that we use it is given by \({J}_{{\rm{opt}}}^{2}=({\gamma }^{2}4{\eta }^{2})/4\). Error bars in a and c reflect statistical uncertainty and control parameter stability (Methods section ‘Error estimation’).
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del Pino, J., Slim, J.J. & Verhagen, E. NonHermitian chiral phononics through optomechanically induced squeezing. Nature 606, 82–87 (2022). https://doi.org/10.1038/s41586022046090
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DOI: https://doi.org/10.1038/s41586022046090
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