Generalizednon-reciprocityinanoptomechanical ......ex D~G.ei˚dO†bOCe i˚dOOb†/(red-detuned pumping). Here GDg 0 j jis the parametrically enhanced optomechanical coupling rate

ARTICLESPUBLISHED ONLINE: 16 JANUARY 2017 | DOI: 10.1038/NPHYS4009

Generalized non-reciprocity in an optomechanicalcircuit via synthetic magnetism and reservoirengineeringKejie Fang1,2, Jie Luo1,2, Anja Metelmann3,4, Matthew H. Matheny1,5, Florian Marquardt6,7,Aashish A. Clerk3 and Oskar Painter1,2*

Synthetic magnetism has been used to control charge neutral excitations for applications ranging from classical beam steeringto quantum simulation. In optomechanics, radiation-pressure-induced parametric coupling between optical (photon) andmechanical (phonon) excitations may be used to break time-reversal symmetry, providing the prerequisite for syntheticmagnetism. Here we design and fabricate a silicon optomechanical circuit with both optical and mechanical connectivitybetween two optomechanical cavities. Driving the two cavities with phase-correlated laser light results in a synthetic magneticflux, which, in combination with dissipative coupling to the mechanical bath, leads to non-reciprocal transport of photons with35 dB of isolation. Additionally, optical pumping with blue-detuned light manifests as a particle non-conserving interactionbetween photons and phonons, resulting in directional optical amplification of 12 dB in the isolator through-direction. Theseresults suggest the possibility of using optomechanical circuits to create amore general class of non-reciprocal optical devices,and further, to enable new topological phases for both light and sound on a microchip.

Synthetic magnetism involving charge neutral elements such asatoms1, polaritons2–4, and photons5–9 is an area of active the-oretical and experimental research, driven by the potential tosimulate quantummany-body phenomena10, reveal new topologicalwave effects11,12, and create defect-immune devices for informationcommunication6,9. Optomechanical systems13, involving the cou-pling of light intensity to mechanical motion via radiation pressure,are a particularly promising venue for studying synthetic fields, asthey can be used to create the requisite large optical nonlinearities14.By applying external optical driving fields, time-reversal symmetrymay be explicitly broken in these systems. It was predicted that thiscould enable optically tunable non-reciprocal propagation in few-port devices15–18, or in the case of a lattice of optomechanical cavities,topological phases of light and sound19,20. Here we demonstrate ageneralized formof optical non-reciprocity in a silicon optomechan-ical crystal circuit21 that goes beyond simple directional propaga-tion; this is achieved using a combination of synthetic magnetism,reservoir engineering, and parametric squeezing.

Distinct from recent demonstrations of optomechanical non-reciprocity in degenerate whispering-gallery resonators withinherent non-trivial topology22–24, we employ a scheme similar tothat proposed in refs 17,20 , in which a synthetic magnetic fieldis generated via optical pumping of the effective lattice formed bycoupled optomechanical cavities. In such a scenario, the resultingsynthetic field amplitude is set by the spatial variation of the pumpfield phase, and the field lines thread optomechanical plaquettesbetween the photon and phonon lattices (see Fig. 1). To achievenon-reciprocal transmission of intensity in the two-port device of

this work—that is, bona fide phonon or photon transport effects,not just non-reciprocal transmission phase—one can combinethis synthetic field with dissipation to implement the generalreservoir-engineering strategy outlined in ref. 25. This approachrequires one to balance coherent and dissipative couplings betweenoptical cavities. In our system the combination of the optical drivesand mechanical dissipation provide the ‘engineered reservoir’which is needed to mediate the required dissipative coupling.

To highlight the flexibility of our approach, we use it toimplement a novel kind of non-reciprocal device exhibiting gain26,27.By using an optical pump which is tuned to the upper motionalsideband of the optical cavities, we realize a two-mode squeezinginteraction which creates and destroys photon and phononexcitations in pairs. These particle non-conserving interactions canbe used to break time-reversal symmetry in amanner that is distinctfrom a standard synthetic gauge field. In a lattice system, thiscan enable unusual topological phases and surprising behavioursuch as protected chiral edge states involving inelastic scattering28and amplification29. Here, we use these interactions along withour reservoir-engineering approach to create a cavity-based opticaldirectional amplifier: backward propagating signals and noise areextinguished by 35 dB relative to forward propagating waves, whichare amplified with an internal gain of 12 dB (1 dB port to port).

The optomechanical system considered in this work isshown schematically in Fig. 1a and consists of two interactingoptomechanical cavities, labelled L (left) and R (right), with eachcavity supporting one optical mode OL(R) and one mechanicalmode ML(R). Both the optical and mechanical modes of each cavity

1Kavli Nanoscience Institute, California Institute of Technology, Pasadena, California 91125, USA. 2Institute for Quantum Information and Matter andThomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA. 3Department of Physics, McGillUniversity, 3600 rue University, Montréal, Quebec H3A 2T8, Canada. 4Department of Electrical Engineering, Princeton University, Princeton,New Jersey 08544, USA. 5Department of Physics, California Institute of Technology, Pasadena, California 91125, USA. 6Max Planck Institute for theScience of Light, Günther-Scharowsky-Straße 1/Bau 24, 91058 Erlangen, Germany. 7Institute for Theoretical Physics, Department of Physics, UniversitätErlangen-Nürnberg, 91058 Erlangen, Germany. *e-mail: [email protected]

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS4009

J

V

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a

bForward Backward

OR

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ML MR ML MR

ML MR

iγ iγ

| L|ei Lφα

e−i Lφ e−i Rφei Rφ ei Lφ

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OL

Figure 1 | Synthetic magnetic field in an optomechanical cavity system. a, In this scheme consisting of only two optomechanical cavities, atwo-dimensional plaquette can be formed from the synthetic dimension created by radiation pressure coupling from the optical modes to the mechanicalmodes. Photon hopping at rate J and phonon hopping at rate V occurs between the optical and mechanical cavities, respectively, with J and V real forappropriate choice of gauge. Pumping of the optomechanical cavities with phase-correlated laser light (|αL|eiφL for the left cavity and (|αR|eiφR for the rightcavity) results in a synthetic fluxΦB=φL−φR threading the four-mode plaquette. b, Scheme for detecting the synthetic flux through non-reciprocal powertransmission of an optical probe laser field. For forward (L→R) propagation, constructive interference set by the flux-dependent phaseΦB≈π/2 of thedissipative phonon coupling path results in e�cient optical power transmission. The accumulated phase in the phonon coupling path is reversed for thebackward (R→L) propagation direction, resulting in destructive interference and reduced optical power transmission in the left output waveguide. Thepower in this case is sunk into the mechanical baths.

are coupled together via a photon–phonon waveguide, resultingin optical and mechanical inter-cavity hopping rates of J andV , respectively (here we choose a local definition of the cavityamplitudes so both are real). The two optical and two mechanicalmodes form a plaquette (with the mechanical modes adding asynthetic dimension20,30–33), through which the synthetic magneticflux is threaded. The radiation pressure interaction between theco-localized optical and mechanical modes of a single cavity canbe described by a Hamiltonian Ĥ=~g0â†â(b̂+ b̂†), where ~ is thePlanck constant, â(b̂) is the annihilation operator of the optical(mechanical) mode and g0 is the vacuum optomechanical couplingrate13 (here we have omitted the cavity labelling).

To enhance the effective photon–phonon interaction strength,each cavity is driven by an optical pump field with frequency ωprelatively detuned from the optical cavity resonance ωc by the me-chanical frequency ωm,∆≡ωp−ωc≈±ωm, with a resulting intra-cavity optical field amplitude |α|eiφ . In the good-cavity limit, whereωm� κ (κ being the optical cavity linewidth), spectral filtering

by the optical cavity preferentially selects resonant photon–phononscattering, leading to a linearized Hamiltonian with either a two-mode squeezing form Ĥent=~G(eiφ d̂†b̂†+e−iφ d̂ b̂) (blue-detunedpumping) or a beamsplitter form Ĥex=~G(eiφ d̂†b̂+e−iφ d̂ b̂†) (red-detuned pumping). Here G= g0|α| is the parametrically enhancedoptomechanical coupling rate and d̂ = â− α contains the smallsignal sidebands of the pump. For both cases, the phase of theresulting coupling coefficient is non-reciprocal in terms of thegeneration and annihilation of photon–phonon excitations. As hasbeen pointed out before, such a non-reciprocal phase resemblesthe Peierls phase that a charged particle accumulates in a magneticvector potential34. Crucially, the relative phase ΦB=φL−φR has anobservable effect as it is independent of the local redefinition of theâ and b̂ cavity amplitudes. In the simple case of ∆=−ωm, ΦB isformally equivalent to having a synthetic magnetic flux threadingthe plaquette formed by the four coupled optomechanical modes(two optical and two mechanical)7,17,20. For ∆=+ωm, a non-zeroΦB still results in the breaking of time-reversal symmetry, although

2


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NATURE PHYSICS DOI: 10.1038/NPHYS4009 ARTICLESthe lack of particle number conservation means that it is not simplyequivalent to a synthetic gauge field. Nonetheless, in what follows,we will refer to it as a flux for simplicity.

To detect the presence of the effective flux ΦB, consider thetransmission of an optical probe signal, on resonance with theoptical cavity resonances and coupled in from either the left orthe right side via external optical coupling waveguides, as depictedin Fig. 1b. The probe light can propagate via two different pathssimultaneously: either direct photon hopping between cavities viathe connecting optical waveguide; or photon–phonon conversion inconjunction with intervening phonon hopping via the mechanicalwaveguide between the cavities. As in the Aharonov–Bohm effectfor electrons35, the synthetic magnetic flux set up by the phase-correlated optical pump beams in the two cavities causes aflux-dependent interference between the two paths. We definethe forward (backward) transmission amplitude as TR→L(L→R) ≡dout,L(R)/din,R(L), where dout(in) is the amplitude of the outgoing(incoming) electromagnetic signal field in the correspondingcoupling waveguide in units of square root of photon flux. Theoptical transmission amplitude in the forward direction has thegeneral form

TL→R[ω;∆=±ωm]=A±[ω](J−Γ±[ω]e−iΦB

)(1)

where ω≡ωs−ωp and ωs is the frequency of the probe light. Γ±is the amplitude of the effective mechanically mediated couplingbetween the two optical cavities, and is given by

Γ±[ω]=VGLGR

(−i(ω±ωmL)+ γiL2 )(−i(ω±ωmR)+γiR2)+V 2

(2)

The prefactor A±[ω] in equation (1) accounts for reflection andloss at the optical cavity couplers, as well as the mechanicallyinduced back-action on the optical cavities (see Equation 15 inSupplementary Information). This prefactor is independent of thetransmission direction, while for the reverse transmission amplitudeTR→L only the sign in front ofΦB changes.

The directional nature of the optical probe transmission may bestudied via the frequency-dependent ratio(TL→R

TR→L

)[ω;∆=±ωm]=

J−Γ±[ω]e−iΦB

J−Γ±[ω]e+iΦB(3)

Although the presence of the synthetic flux breaks time-reversalsymmetry, it does not in and of itself result in non-reciprocal photontransmissionmagnitudes upon swapping input and output ports25,36.In our system, if one takes the limit of zero intrinsic mechanicaldamping (that is, γik = 0), the mechanically mediated couplingamplitude Γ±[ω] is real at all frequencies. This implies |TL→R| =|TR→L|, irrespective of the value of ΦB. We thus find that non-zero mechanical dissipation will be crucial in achieving any non-reciprocity in the magnitude of the optical transmission amplitudes.

The general reservoir-engineering approach to non-reciprocityintroduced in ref. 25 provides a framework for both understandingand exploiting the above observation. It demonstrates thatnon-reciprocity is generically achieved by balancing a direct(Hamiltonian) coupling between two cavities against a dissipativecoupling of the cavities; such a dissipative coupling can arise whenboth cavities couple to the same dissipative reservoir. The balancingrequires both a tuning of the magnitude of the coupling to the bath,as well as a relative phase which plays a role akin to the flux ΦB. Inour case, the damped mechanical modes can play the role of theneeded reservoir, with the optical drives controlling how the opticalcavities couple to this effective reservoir. One finds that at any givenfrequency ω, the mechanical modes induce both an additional

coherent coupling between the two cavities (equivalent to anadditional coupling term in the Hamiltonian) as well as a dissipativecoupling (which is not describable by a Hamiltonian). As is shownexplicitly in Supplementary Section IIB, in the present setting thesecorrespond directly to the real and imaginary parts of Γ±[ω].Hence, the requirement of having Im(Γ [ω]) 6= 0 is equivalent torequiring a non-zero mechanically mediated dissipative couplingbetween the cavities.

Achieving directionality requires working at a frequency wherethe dissipative coupling has the correct magnitude to balance thecoherent coupling J , and a tuning of the flux ΦB. For |Γ±[ω]|= Jand arg(Γ±)=−ΦB ( 6= 0, π), one obtains purely uni-directionaltransport, where the right optical cavity is driven by the leftoptical cavity but not vice versa. One finds from equation (3)that the mechanically mediated dissipative coupling between thecavities is maximized at frequencies near the mechanical normalmode frequencies ω≈−ωm±V ; to achieve the correct magnitudeof coupling, the optical pumping needs to realize a many-photon optomechanical coupling Gk≈ (Jγik)1/2 (see SupplmentarySection II for details). Note that our discussion applies to boththe choices of red-detuned and blue-detuned pumping. Althoughthe basic recipe for directionality is the same, in the blue-detunedpump case the effective reservoir seen by the cavity modes cangive rise to negative damping, with the result that the forwardtransmission magnitude can be larger than one. We explore thismore in what follows.

To realize the optomechanical circuit depicted in Fig. 1 weemploy the device architecture of optomechanical crystals37–39,which allows for the realization of integrated cavity-optomechanicalcircuits with versatile connectivity and cavity coupling rates21,40.Fig. 2a shows the optomechanical crystal circuit fabricated on asilicon-on-insulator microchip. The main section of the circuit,shown zoomed-in in Fig. 2b, contains two optomechanical crystalnanobeam cavities, each of which has an optical resonance ofwavelength λ≈1,530 nm and a mechanical resonance of frequencyωm/2π≈6GHz. The two optical cavities can be excited through twoseparate optical coupling paths, one for coupling to the left cavityand one for the right cavity. Both the left and right optical couplingpaths consist of an adiabatic fibre-to-chip coupler which coupleslight from an optical fibre to a silicon waveguide, and a near-fieldwaveguide-to-cavity reflective coupler. This allows separate opticalpumping of each cavity and optical transmission measurements tobe carried out in either direction. The two nanobeam cavities arephysically connected together via a central silicon beam sectionwhich is designed to act as both an opticalwaveguide and an acousticwaveguide. The central beam thus mediates both photon hoppingand phonon hopping between the two cavities even though thecavities are separated by a distancemuch larger than the cavitymodesize21,41. The numerically simulated mode profiles for the localizedcavities and the connecting waveguide are shown in Fig. 2c andd, respectively. The hopping rate for photons and phonons can beengineered by adjusting the number and shape of the holes in themirror section of the optomechanical crystal cavity along with thefree-spectral range of the connecting waveguide section21. Here weaim for a design with J/2π≈100MHz and V/2π≈3MHz so thatnon-reciprocity can be realized at low optical pump power, yet stillwith high transmission efficiency.

The optical and mechanical frequencies of the optomechanicalcavities are independently trimmed into alignment post-fabricationusing an atomic force microscope to oxidize nanoscale regions ofthe cavity. After nano-oxidation tuning, the left (right) cavity hasoptical resonancewavelength λL(R)=1,534.502 (1,534.499) nm, totalloaded damping rate κL(R)/2π=1.03 (0.75)GHz, and intrinsic cavitydamping rate κiL(R)/2π= 0.29 (0.31) GHz (see Fig. 2e). Note thathybridization of the optical cavity resonances is too weak to beobservable in the measured left and right cavity spectra due to



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Figure 2 | Silicon optomechanical crystal circuit. a, Scanning electron microscopy (SEM) image of the optomechanical crystal circuit studied in this work.The circuit is fabricated from a silicon-on-insulator microchip (see Methods). b, SEM of the main part of the circuit, which consists of a left and a rightnanobeam optomechanical crystal cavity with a central unpatterned nanobeam waveguide connecting the two cavities. A left and right optical coupler,which are each fed by an adiabatic fibre-to-chip coupler42, are used to evanescently couple light into either of the two optical cavities. c, Finite-elementmethod (FEM) simulated electrical field Ey and magnitude of the displacement field for the localized optical and mechanical cavity modes, respectively, ofthe nanobeam. d, FEM simulated section of the corresponding optical and mechanical modes of the connecting waveguide. e, Optical reflection spectrumof the left (blue) and right (orange) optical cavities. f, Optically transduced mechanical power spectral density (PSD) measured from the left (blue) andright (orange) optical cavities. M± are the two hybridized mechanical cavity modes with frequency ωM+(−)/2π=5,788.4 (5,779.1) MHz and MW is amechanical waveguide mode with frequency ωMW/2π=5,818.3 MHz.

the fact that the optical cavity linewidths are much larger thanthe designed cavity coupling J . The thermal mechanical spectra,as measured from the two cavities using a blue-detuned pumplaser (see Methods), are shown in Fig. 2f, where one can seehybridized resonances M±, which are mixtures of the localizedmechanical cavity modesML andMR. A nearby phonon waveguidemode (MW) is also observable in both left and right cavity spectra.The optomechanical coupling rate and mechanical dissipation rateof ML(R) were measured before nano-oxidation tuning, yieldingg0,L(R)/2π=0.76 (0.84) MHz and γiL(R)/2π=4.3 (5.9) MHz.

The experimental apparatus used to drive and probe theoptomechanical circuit is shown schematically in Fig. 3a. Asindicated, an optical pump field for the left and right cavities isgenerated from a common diode laser. The phase difference ofthe pump fields at the input to the cavities, and thus the syntheticmagnetic flux, is tuned by a stretchable fibre phase shifter andstabilized by locking the interference intensity of the reflectedpump signals from the cavities. To highlight the unique kindof non-reciprocal transport possible in our set-up, we presentresults for an experiment performed with blue-detuned pump fieldswith frequency ωp≈ωc+ωm; as discussed, this will enable non-reciprocal transport with gain. An input optical probe signal isgenerated from either of the left or right cavity pump beamsby sending them through an electro-optic modulator (EOM). Avector network analyser (VNA) is used to drive the EOMs atmodulation frequency ωmod and detect the photocurrent generatedby the beating of the transmitted probe and reflected pump signals,thus providing amplitude and phase information of the transmittedprobe signal. Owing to the spectral filtering of the cavities, only thegenerated lower sideband of the blue-detuned pump at ω=−ωmodis transmitted through the circuit as a probe signal.

Figure 3b shows the ratio of the forward and backward opticalpower transmission coefficients of the probe light (|TL→R/TR→L|2)for several magnetic flux values between ΦB= 0 and π. For thesemeasurements the pump powers at the input to the left and rightcavity were set to PpL=−14.2 dBm and PpR=−10.8 dBm, respec-tively, corresponding to intra-cavity photon numbers of ncL=1,000and ncR= 1,420. So as to remove differences in the forward andreverse transmission paths external to the optomechanical circuit,

here the |TL→R/TR→L|2 ratio is normalized to 0 dB for a modula-tion frequency ωmod/2π≈ 5.74GHz, detuned far from mechanicalresonance in a frequency range where reciprocal transmission isexpected. Closer to mechanical resonance, strong non-reciprocityin the optically transmitted power is observed, with a peak and adip in |TL→R/TR→L|2 occurring roughly at the resonance frequen-cies of the hybridized mechanical modes M+ and M−, respectively(see Fig. 2c). The maximum contrast ratio between forward andbackward probe transmission—the isolation level—is measured tobe 35 dB for ΦB=0.34π near the M+ resonance. The forwardtransmission is also amplified in this configuration (blue-detunedpump,∆=+ωm), with a measured peak probe signal amplificationof 12 dB above the background level set by photon hopping alone(J/|Γ±|�1). The corresponding port-to-port net gain is only 1 dBdue to impedance mismatching (J 6= κ/2) and intrinsic opticalcavity losses (see Supplementary Section IC for details). With ratherstraightforward improvements in the optical cavity and couplerdesign42,43, one should be able to attain a port-to-port gain equal tothat of the internal gain.

From a two-parameter fit to the measured optical powertransmission ratio spectra using equation (3) (see blue curvesin Fig. 3b,c), we obtain a waveguide-mediated optical andmechanical hopping rate of J/2π=110MHz andV/2π=2.8MHz,respectively, consistent with our design parameters. Figure 3d showsthe theoretical calculation of |TL→R/TR→L|2 for a full 2π range ofΦBwith the measured and fit optomechanical circuit parameters. Thepattern is seen to be odd symmetricwith respect toΦB=π. Insertingan additional magnetic flux π into the measurements performedin Fig. 3b yields the spectra shown in Fig. 3c, which displays aswitch in the isolation direction as predicted by the model. Thepump power dependence of the peak (in frequency) forward signalamplification and the corresponding backward signal attenuationrelative to the background level far from mechanical resonanceare shown in Fig. 3e for a fixed magnetic flux of ΦB = 0.28π.Good correspondence with the theoretical power dependence (solidcurves) is observed, with non-reciprocal amplification vanishing atlow pump power.

One can also obtain non-reciprocal optical power transmissionutilizing an even simpler system involving a single-mechanical

4




NATURE PHYSICS DOI: 10.1038/NPHYS4009 ARTICLES

EDFA

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B = 1.18πΦ B = 1.26πΦ B = 1.34πΦ

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Φ

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-meterλ

Figure 3 | Measurement of optical non-reciprocity. a, Experiment set-up. Red (blue) lines are optical (electronic) wiring. Blue-detuned pump light from atunable diode laser is split into two paths and fed into the two cavities (red arrows). Part of the reflected pump laser light from the cavities (purple arrows)is collected by a photodetector (PD) and fed into a stretchable fibre phase shifter (φ-shifter) to tune and lock the phase di�erence of the optical pumps.Each optical path can be modulated by an electro-optic modulator (EOM) to generate an optical sideband which we use as the optical probe signal. Themicrowave modulation signal with frequency ωmod is generated by port 1 of a vector network analyser (VNA). After optical amplification andphotodetection, the transmitted optical probe signal through the optomechanical circuit is sent back to port 2 of the VNA to measure the phase andamplitude of the optical probe transmission coe�cient. EDFA, erbium-doped fibre amplifier; FPC, fibre polarization controller; λ-meter, wavelength meter;VOA, variable optical attenuator. b, The ratio of optical power transmission coe�cients for right and left propagation versus modulation frequency(ωmod=−ω=ωp−ωs), for three di�erent synthetic flux valuesΦB/π=0.18, 0.26, and 0.34. The blue curves correspond to the fit of the theoretical model(see equation (3)) to the measured spectra. The dashed lines indicate the location of M±. c, The power transmission coe�cient ratio forΦB with anadditional π flux relative to those in b. d, Theoretical calculation of the power transmission coe�cient ratio for 0≤ΦB≤2π, where the six grey linescorrespond to the six measuredΦB values in b and c. e, Peak forward signal amplification above background level (blue squares) and corresponding signalattenuation in the reverse direction (red circles) versus average optical pump power (P̄p=

√PpLPpR) for fixed flux value ofΦB=0.28π. The solid curves are

theoretical calculations based upon the theoretical model (see equation (3)) fit to the data in b and c.

mode. This is the situation we have for the Fabry–Perot-likemechanical resonances that exist in the central coupling waveguide(see MW resonance of Fig. 2c). As depicted in Fig. 4a, the modeconfiguration in this case consists of two optical cavity modes(OL and OR) coupled together via the optical waveguide, onemechanical waveguide mode MW which is parametrically coupledto each of the optical cavity modes, and the synthetic magneticflux ΦB=φL−φR due to the relative phases of the optical pumpfields threading the triangular mode space. In Fig. 4b,c we showthe measurement of |TL→R/TR→L|2 for a series of different fluxvalues ΦB with blue-detuned pumping (∆≈+ωMW ) at levels ofncL = 770 and ncR=1,090. In this single-mechanical mode casethe direction of the signal propagation is determined by themagnitude of the flux; ΦB≤π leads to backward propagation andΦB≥π to forward propagation. The lower contrast ratio observedis a result of the weaker coupling between the localized opticalcavity modes and the external waveguide mode, which for themodest pump power levels used here (.100 µW) does not allowus to reach the parametric coupling required for strong direc-tional transmission.

Although our focus has been on the propagation of injectedcoherent signals through the optomechanical circuit, it is alsointeresting to consider the flow of noise. As might be expected, theinduced directionality of our system also applies to noise photonsgenerated by the upconversion of both thermal and quantumfluctuations of the mechanics; see Supplementary Section III fordetailed calculations. One finds that for the system of Fig. 2, thespectrally resolved photon noise flux shows high directionality,but that the sign of this directionality changes as a functionof frequency (analogous to what happens in the transmissionamplitudes). In contrast, in the single-mechanical mode set-up of

Fig. 4 the sign of the directionality is constant with frequency,and thus the total (frequency-integrated) noise photon flux isdirectional depending upon the flux magnitude. The laser pumpfields can thus effectively act as a heat pump, creating a temperaturedifference between the left and right waveguide output fields.The corresponding directional flow of quantum noise is especiallyuseful for quantum information applications, as it can suppressnoise-induced damage of a delicate signal source like a qubit25,27.Our calculations show that the added noise of the current deviceoperated as a directional amplifier is 2.5 noise quanta in the absenceof thermal mechanical noise. Challenges of reaching the standardquantum limit (0.5 added noise quanta) are primarily related tooperating the device at millikelvin temperatures44 where opticalabsorption heating and reduced thermal conductivity can lead toexcess thermal mechanical noise.

The device studied in this work also highlights the potential foroptomechanics to realize synthetic gauge fields and novel formsof non-reciprocity enabled by harnessing mechanical dissipation.Using just a few modes, it was possible to go beyond simplymimicking the physics of an isolator and realize a directionaloptical amplifier. By adding modes, an even greater variety ofbehaviours could be achieved. For example, the simple additionof a third optical cavity mode, tunnel-coupled to the first twocavities but with no mechanical coupling, would realize a photoncirculator similar to the phonon circulators considered in ref. 17.Not just limited to optical input–output devices, one may alsorealize non-trivial acoustic or photon–phonon polaritonic signalpropagation. Scaling the synthetic gauge field mechanism realizedhere to a full lattice of optomechanical cavities would allowthe study of topological phenomena in the propagation of bothlight and sound. Predicted effects include the formation of



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Figure 4 | Synthetic magnetic field with a single-mechanical cavity. a, Physical configuration for generation of a synthetic magnetic field and opticalnon-reciprocity with two optical modes parametrically coupled with a common dissipative mechanical waveguide mode. b,c, The ratio of optical powertransmission coe�cients for right and left propagation versus modulation frequency ωmod around the frequency of the waveguide mode MW for variousΦB. The blue curves correspond to a fit of the theoretical model (see Supplementary Section IIF) to the measured data.

back-scattering immune photonic20 and phononic19 chiral edgestates, topologically non-trivial phases of hybrid photon–phononexcitations19, dynamical gauge fields45, and, in the case of non-particle-conserving interactions enabled by blue-detuned opticalpumping, topologically protected inelastic scattering of photons28and even protected amplifying edge states29.

MethodsMethods, including statements of data availability and anyassociated accession codes and references, are available in theonline version of this paper.

Received 18 August 2016; accepted 9 December 2016;published online 16 January 2017

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AcknowledgementsThe authors would like to thank M. Roukes for the use of his atomic force microscope inthe nano-oxidation tuning of the cavities. This work was supported by the AFOSR-MURIQuantum Photonic Matter, the ARO-MURI Quantum Opto-Mechanics with Atoms andNanostructured Diamond (grant N00014-15-1-2761), the University of ChicagoQuantum Engineering Program (A.A.C., A.M.), the ERC Starting Grant OPTOMECH(F.M.), the Institute for Quantum Information and Matter, an NSF Physics FrontiersCenter (grant PHY-1125565) with support of the Gordon and Betty Moore Foundation,and the Kavli Nanoscience Institute at Caltech.

Author contributionsK.F., F.M., A.M., A.A.C. and O.P. came up with the concept. K.F., O.P. and J.L. planned theexperiment. K.F., J.L. and M.H.M. performed the device design and fabrication. K.F. andJ.L. performed the measurements. K.F., J.L., A.M., A.A.C. and O.P. analysed the data. Allauthors contributed to the writing of the manuscript.

Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints.Correspondence and requests for materials should be addressed to O.P.

Competing financial interestsThe authors declare no competing financial interests.



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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS4009MethodsThe devices used in this work were fabricated from a silicon-on-insulator waferwith a silicon device layer thickness of 220 nm and buried-oxide layer thickness of2 µm. The device geometry was defined by electron-beam lithography followed byinductively coupled plasma reactive ion etching to transfer the pattern through the220 nm silicon device layer. The devices were then undercut using an HF:H2Osolution to remove the buried-oxide layer and cleaned using a piranha etch.

After device fabrication, we used an atomic force microscope to draw nanoscaleoxide patterns on the silicon device surface. This process modifies the optical andmechanical cavity frequencies in a controllable and independent way with the

appropriate choice of oxide pattern. The nano-oxidation process was carried outusing an AsylumMFP-3D atomic force microscope and conductive diamond tips(NaDiaProbes) in an environment with relative humidity of 48%. The tip wasbiased at a voltage of−11.5V, scanned with a velocity of 100 nm s−1, and run intapping mode with an amplitude of 10 nm. The unpassivated silicon device surfacewas grounded.

Data availability. The data that support the plots within this paper and otherfindings of this study are available from the corresponding author uponreasonable request.


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Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineeringMethodsFigure 1 Synthetic magnetic field in an optomechanical cavity system.Figure 2 Silicon optomechanical crystal circuit.Figure 3 Measurement of optical non-reciprocity.Figure 4 Synthetic magnetic field with a single-mechanical cavity.ReferencesAcknowledgementsAuthor contributionsAdditional informationCompeting financial interestsMethodsData availability.

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Generalizednon-reciprocityinanoptomechanical ......ex D~G.ei˚dO†bOCe i˚dOOb†/(red-detuned pumping). Here GDg 0 j jis the parametrically enhanced optomechanical coupling rate