Generalized nonreciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering
Kejie Fang, Jie Luo, Anja Metelmann, Mathew H. Matheny, Florian Marquardt, Aashish A. Clerk, Oskar Painter
GGeneralized nonreciprocity in an optomechanical circuit via synthetic magnetism andreservoir engineering
Kejie Fang,
1, 2
Jie Luo,
1, 2
Anja Metelmann,
3, 4
Matthew H. Matheny,
1, 5
Florian Marquardt,
6, 7
Aashish A. Clerk, and Oskar Painter
1, 2 Kavli Nanoscience Institute, California Institute of Technology, Pasadena, California 91125, USA Institute for Quantum Information and Matter and Thomas J. Watson, Sr.,Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA Department of Physics, McGill University, 3600 rue University, Montr´eal, Quebec H3A 2T8, Canada Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA Department of Physics, California Institute of Technology, Pasadena, California 91125, USA Max Planck Institute for the Science of Light, G¨unther-Scharowsky-Straße 1/Bau 24, 91058 Erlangen, Germany Institute for Theoretical Physics, Department of Physics, Universit¨at Erlangen-N¨urnberg, 91058 Erlangen
Synthetic magnetism has been used to control charge neutral excitations for applications rangingfrom classical beam steering to quantum simulation. In optomechanics, radiation-pressure-inducedparametric coupling between optical (photon) and mechanical (phonon) excitations may be used tobreak time-reversal symmetry, providing the prerequisite for synthetic magnetism. Here we designand fabricate a silicon optomechanical circuit with both optical and mechanical connectivity betweentwo optomechanical cavities. Driving the two cavities with phase-correlated laser light results ina synthetic magnetic flux, which in combination with dissipative coupling to the mechanical bath,leads to nonreciprocal transport of photons with 35 dB of isolation. Additionally, optical pumpingwith blue-detuned light manifests as a particle non-conserving interaction between photons andphonons, resulting in directional optical amplification of 12 dB in the isolator through direction.These results indicate the feasibility of utilizing optomechanical circuits to create a more generalclass of nonreciprocal optical devices, and further, to enable novel topological phases for both lightand sound on a microchip.
Synthetic magnetism involving charge neutral elementssuch as atoms [2], polaritons [3–5], and photons [6–10] is an area of active theoretical and experimentalresearch, driven by the potential to simulate quantummany-body phenomena [11], reveal new topological waveeffects [12, 13], and create defect-immune devices for in-formation communication [7, 10]. Optomechanical sys-tems [14], involving the coupling of light intensity tomechanical motion via radiation pressure, are a partic-ularly promising venue for studying synthetic fields, asthey can be used to create the requisite large opticalnonlinearities [15]. By applying external optical drivingfields time-reversal symmetry may be explicitly brokenin these systems. It was predicted that this could enableoptically tunable nonreciprocal propagation in few-portdevices [16–19], or in the case of a lattice of optomechan-ical cavities, topological phases of light and sound [1, 20].Here we demonstrate a generalized form of optical non-reciprocity in a silicon optomechanical crystal circuit [21]that goes beyond simple directional propagation; this isachieved using a combination of synthetic magnetism,reservoir engineering, and parametric squeezing.Distinct from recent demonstrations of optomechan-ical nonreciprocity in degenerate whispering-gallery res-onators with inherent nontrivial topology [22–24], we em-ploy a scheme similar to that proposed in Refs. [1, 18] inwhich a synthetic magnetic field is generated via opticalpumping of the effective lattice formed by coupled op-tomechanical cavities. In such a scenario, the resultingsynthetic field amplitude is set by the spatial variationof the pump field phase and the field lines thread op- tomechanical plaquettes between the photon and phononlattices (see Fig. 1). To achieve nonreciprocal transmis-sion of intensity in the two-port device of this work –i.e., bonafide phonon or photon transport effects, not justnonreciprocal transmission phase – one can combine thissynthetic field with dissipation to implement the generalreservoir engineering strategy outlined in Ref. [25]. Thisapproach requires one to balance coherent and dissipa-tive couplings between optical cavities. In our system thecombination of the optical drives and mechanical dissipa-tion provide the “engineered reservoir” which is neededto mediate the required dissipative coupling.To highlight the flexibility of our approach, we use itto implement a novel kind of nonreciprocal device ex-hibiting gain [26, 27]. By using an optical pump which istuned to the upper motional sideband of the optical cav-ities, we realize a two-mode squeezing interaction whichcreates and destroys photon and phonon excitations inpairs. These particle non-conserving interactions can beused to break time-reversal symmetry in a manner that isdistinct from a standard synthetic gauge field. In a latticesystem, this can enable unusual topological phases andsurprising behavior such as protected chiral edge statesinvolving inelastic scattering [28] and amplification [29].Here, we use these interactions along with our reservoir-engineering approach to create a cavity-based optical di-rectional amplifier: backward propagating signals andnoise are extinguished by 35 dB relative to forward prop-agating waves which are amplified with an internal gainof 12 dB (1 dB port-to-port).The optomechanical system considered in this work is a r X i v : . [ phy s i c s . op ti c s ] A ug a J V O L O R M R M L b Forward Backward i i O L O R M R M L e i R e i L O L O R M R M L e i L e i R BL e i L R e i R Energy
FIG. 1.
Synthetic magnetic field in an optomechani-cal cavity system. a , In this scheme consisting of only twooptomechanical cavities, a two-dimensional plaquette can beformed from the synthetic dimension [1] created by radia-tion pressure coupling from the optical modes to the mechan-ical modes. Photon hopping at rate J and phonon hoppingat rate V occurs between the optical and mechanical cavi-ties, respectively, with J and V real for appropriate choiceof gauge. Pumping of the optomechancial cavities with phasecorrelated laser light ( | α L | e iφ L for the left cavity and ( | α R | e iφ R for the right cavity) results in a synthetic flux Φ B = φ L − φ R threading the 4-mode plaquette. b , Scheme for detecting thesynthetic flux through nonreciprocal power transmission ofan optical probe laser field. For forward (L → R) prop-agation, constructive interference set by the flux-dependentphase Φ B ≈ π/ → L) propagation direction resulting in destructiveinterference and reduced optical power transmission in theleft output waveguide. The power in this case is sunk intothe mechanical baths. shown schematically in Fig. 1a and consists of two in-teracting optomechanical cavities, labeled L (left) and R (right), with each cavity supporting one optical mode O L(R) and one mechanical mode M L(R) . Both the opticaland mechanical modes of each cavity are coupled togethervia a photon-phonon waveguide, resulting in optical andmechanical inter-cavity hopping rates of J and V , re-spectively (here we choose a local definition of the cavityamplitudes so both are real). The radiation pressure in-teraction between the co-localized optical and mechanicalmodes of a single cavity can be described by a Hamilto-nian ˆ H = (cid:126) g ˆ a † ˆ a (ˆ b + ˆ b † ), where ˆ a (ˆ b ) is the annihilationoperator of the optical (mechanical) mode and g is the vacuum optomechanical coupling rate [14] (here we haveomitted the cavity labeling).To enhance the effective photon-phonon interactionstrength each cavity is driven by an optical pump fieldwith frequency relatively detuned from the optical cavityresonance by the mechanical frequency (∆ ≡ ω p − ω c ≈± ω m ), with a resulting intra-cavity optical field ampli-tude | α | e iφ . In the good-cavity limit, where ω m (cid:29) κ ( κ being the optical cavity linewidth), spectral filter-ing by the optical cavity preferentially selects resonantphoton-phonon scattering, leading to a linearized Hamil-tonian with either a two-mode squeezing form ˆ H ent = (cid:126) G ( e iφ ˆ d † ˆ b † + e − iφ ˆ d ˆ b ) (blue detuned pumping) or a beam-splitter form ˆ H ex = (cid:126) G ( e iφ ˆ d † ˆ b + e − iφ ˆ d ˆ b † ) (red detunedpumping). Here G = g | α | is the parametrically en-hanced optomechanical coupling rate and ˆ d = ˆ a − α con-tains the small signal sidebands of the pump. For bothcases the phase of the resulting coupling coefficient isnonreciprocal in terms of the generation and annihilationof photon-phonon excitations. As has been pointed outbefore, such a nonreciprocal phase resembles the Peierlsphase that a charged particle accumulates in a mag-netic vector potential [30]. Crucially, the relative phaseΦ B = φ L − φ R is gauge independent (i.e. independent oflocal redefinitions of the ˆ a and ˆ b cavity amplitudes), im-plying it should have an observable effect. In the simplecase of ∆ = − ω m , Φ B is formally equivalent to having asynthetic magnetic flux threading the plaquette formedby the four coupled optomechanical modes (two opticaland two mechanical)[1, 8, 18]. For ∆ = + ω m , a non-zeroΦ B still results in the breaking of time-reversal symmetry,though the lack of particle number conservation meansthat it is not simply equivalent to a synthetic gauge field.Nonetheless, we will refer to it as a flux in what followsfor simplicity.To detect the presence of the effective flux Φ B , con-sider the transmission of an optical probe signal, on res-onance with the optical cavity resonances and coupled infrom either the left or the right side via external opticalcoupling waveguides as depicted in Fig. 1b. The probelight can propagate via two different paths simultane-ously: (i) direct photon hopping between cavities via theconnecting optical waveguide, and (ii) photon-phononconversion in conjunction with intervening phonon hop-ping via the mechanical waveguide between the cavities.As in the Aharonov-Bohm effect for electrons [32], thesynthetic magnetic flux set up by the phase-correlatedoptical pump beams in the two cavities causes a flux-dependent interference between the two paths. We de-fine the forward (backward) transmission amplitude as T R → L(L → R) ≡ d out,L(R) /d in,R(L) , where d out(in) is the am-plitude of the outgoing (incoming) electromagnetic signalfield in the corresponding coupling waveguide in units ofsquare root of photon flux. The optical transmission am-plitude in the forward direction has the general form T L → R [ ω ; ∆ = ± ω m ] = A ± [ ω ] (cid:0) J − Γ ± [ ω ] e − i Φ B (cid:1) , (1) Wavelength (nm)1534.4 1534.45 1534.55 1534.6 N o r m a l i z e d r e fl e c t i o n Frequency (GHz)5.75 5.77 5.79 5.81 5.83 P S D ( d B m / H z ) -132-131-130-129-128-127 e f cavity L cavity R M W M + M
10 μm O L M L optical coupler L photon+phonon waveguide optical coupler R waveguide modesoptical fiber-to-chip couplers L R a bc d
FIG. 2.
Silicon optomechanical crystal circuit. a , Scanning electron microscopy (SEM) image of the optomechanicalcrystal circuit studied in this work. The circuit is fabricated from a silicon-on-insulator microchip (see App. A). b , SEM of themain part of the circuit, which consists of a left and a right nanobeam optomechanical crystal cavity with a central unpatternednanobeam waveguide connecting the two cavities. A left and right optical coupler, which are each fed by an adiabatic fiber-to-chip coupler [31], are used to evanescently couple light into either of the two optical cavities. c , FEM simulated electrical field E y and magnitude of the displacement field for the localized optical and mechanical cavity modes, respectively, of the nanobeam. d , FEM simulated section of the corresponding optical and mechanical modes of the connecting waveguide. e, Optical reflectionspectrum of the left (blue) and right (orange) optical cavities. f , Optically transduced mechanical power spectral density (PSD)measured from the left (blue) and right (orange) optical cavities. M ± are the two hybridized mechanical cavity modes withfrequency ω M +( − ) / π = 5788 . .
1) MHz and M W is a mechanical waveguide mode with frequency ω M W / π = 5818 . where ω ≡ ω s − ω p and ω s is the frequency of the probelight. Γ ± is the amplitude of the effective mechanically-mediated coupling between the two optical cavities, andis given byΓ ± [ ω ] = V G L G R ( − i ( ω ± ω mL ) + γ iL )( − i ( ω ± ω mR ) + γ iR ) + V . (2)The prefactor A ± [ ω ] in Eq. (1) accounts for reflectionand loss at the optical cavity couplers, as well as themechanically-induced back-action on the optical cavities.This prefactor is independent of the transmission direc-tion, and for the reverse transmission amplitude T R → L ,only the sign in front of Φ B changes.The directional nature of the optical probe transmis-sion may be studied via the frequency-dependent ratio (cid:18) T L → R T R → L (cid:19) [ ω ; ∆ = ± ω m ] = J − Γ ± [ ω ] e − i Φ B J − Γ ± [ ω ] e + i Φ B . (3)Although the presence of the synthetic flux breaks time-reversal symmetry, it does not in and of itself resultin nonreciprocal photon transmission magnitudes uponswapping input and output ports [25, 33]. In our sys-tem, if one takes the limit of zero intrinsic mechanicaldamping (i.e. γ ik = 0), the mechanically-mediated cou-pling amplitude Γ ± [ ω ] is real at all frequencies. Thisimplies | T L → R | = | T R → L | , irrespective of the value of Φ B .We thus find that non-zero mechanical dissipation will be crucial in achieving any non-reciprocity in the magnitudeof the optical transmission amplitudes.The general reservoir-engineering approach to nonre-ciprocity introduced in Ref. [25] provides a frameworkfor both understanding and exploiting the above obser-vation. It demonstrates that nonreciprocity is genericallyachieved by balancing a direct (Hamiltonian) couplingbetween two cavities against a dissipative coupling of thecavities; such a dissipative coupling can arise when bothcavities couple to the same dissipative reservoir. Thebalancing requires both a tuning of the magnitude of thecoupling to the bath, as well as a relative phase whichplays a role akin to the flux Φ B . In our case, the dampedmechanical modes can play the role of the needed reser-voir, with the optical drives controlling how the opticalcavities couple to this effective reservoir. One finds thatat any given frequency ω , the mechanical modes induceboth an additional coherent coupling between the twocavities (equivalent to an additional coupling term in theHamiltonian) as well as a dissipative coupling (which isnot describable by a Hamiltonian). As is shown explic-itly in App, B, in the present setting these corresponddirectly to the real and imaginary parts of Γ ± [ ω ]. Hence,the requirement of having Im Γ[ ω ] (cid:54) = 0 is equivalent torequiring a non-zero mechanically-mediated dissipativecoupling between the cavities.Achieving directionality requires working at a fre-quency where the dissipative coupling has the correctmagnitude to balance the coherent coupling J , and a tun-ing of the flux Φ B . For | Γ ± [ ω ] | = J and arg(Γ ± ) = − Φ B ( (cid:54) = 0 , π ), one obtains purely uni-directional transportwhere the right optical cavity is driven by the left opticalcavity but not vice versa. One finds from Eq. (3) thatthe mechanically-mediated dissipative coupling betweenthe cavities is maximized at frequencies near the mechan-ical normal mode frequencies ω ≈ − ω m ± V ; to achievethe correct magnitude of coupling, the optical pumpingneeds to realize a many-photon optomechanical coupling G k ≈ ( Jγ ik ) / (see App. B for details). Note that ourdiscussion applies to both the choices of red-detuned andblue-detuned pumping. While the basic recipe for direc-tionality is the same, in the blue-detuned pump case theeffective reservoir seen by the cavity modes can give riseto negative damping, with the result that the forwardtransmission magnitude can be larger than one. We ex-plore this more in what follows.In order to realize the optomechanical circuit depictedin Fig. 1 we employ the device architecture of optome-chanical crystals [34–36], which allows for the realizationof integrated cavity-optomechanical circuits with versa-tile connectivity and cavity coupling rates [21, 37]. Fig-ure 2a shows the optomechanical crystal circuit fabri-cated on a silicon-on-insulator microchip. The main sec-tion of the circuit, shown zoomed-in in Fig. 2b, con-tains two optomechanical crystal nanobeam cavities,each of which has an optical resonance of wavelength λ ≈ ω m / π ≈ J/ π ≈
100 MHz and
V / π ≈ λ L(R) = 1534 .
502 (1534 . κ L(R) / π = 1 .
03 (0 .
75) GHz, and intrinsic cavitydamping rate κ iL(R) / π = 0 .
29 (0 .
31) GHz (c.f. Fig. 2e).Note that hybridization of the optical cavity resonancesis too weak to be observable in the measured left andright cavity spectra due to the fact that the optical cav-ity linewidths are much larger than the designed cavitycoupling J . The thermal mechanical spectra, as mea-sured from the two cavities using a blue-detuned pumplaser (see App. A), are shown in Fig. 2f where one cansee hybridized resonances M ± which are mixtures ofthe localized mechanical cavity modes M L and M R . Anearby phonon waveguide mode ( M W ) is also observablein both left and right cavity spectra. The optomechanicalcoupling rate and mechanical dissipation rate of M L(R) were measured before nano-oxidation tuning, yielding g , L(R) / π = 0 .
76 (0 .
84) MHz and γ iL(R) / π = 4 . .
9) MHz.The experimental apparatus used to drive and probethe optomechanical circuit is shown schematically inFig. 3a. As indicated, an optical pump field for theleft and right cavities is generated from a common diodelaser. The phase difference of the pump fields at the in-put to the cavities, and thus the synthetic magnetic flux,is tuned by a stretchable fiber phase shifter and stabilizedby locking the interference intensity of the reflected pumpsignals from the cavities. To highlight the unique kinds ofnonreciprocal transport possible in our setup, we presentresults for an experiment performed with blue-detunedpump fields with frequency ω p ≈ ω c + ω m ; as discussed,this will enable non-reciprocal transport with gain. Aninput optical probe signal is generated from either of theleft or right cavity pump beams by sending them throughan electro-optic modulator (EOM). A vector network an-alyzer (VNA) is used to drive the EOMs at modulationfrequency ω mod and detect the photocurrent generated bythe beating of the transmitted probe and reflected pumpsignals, thus providing amplitude and phase informationof the transmitted probe signal. Owing to the spectralfiltering of the cavities, only the generated lower sidebandof the blue-detuned pump at ω = − ω mod is transmittedthrough the circuit as a probe signal.Figure 3b shows the ratio of the forward and back-ward optical power transmission coefficients of the probelight ( | T L → R /T R → L | ) for several magnetic flux values be-tween Φ B = 0 and π . For these measurements the pumppowers at the input to the left and right cavity were setto P pL = − . P pR = − . n cL = 1000 and n cR = 1420. So as to remove differencesin the forward and reverse transmission paths externalto the optomechanical circuit, here the | T L → R /T R → L | ratio is normalized to 0 dB for a modulation frequency ω mod / π ≈ .
74 GHz, detuned far from mechanical res-onance in a frequency range where reciprocal transmis- λ-meterEDFA VOA EOMVNAEOM PDFPCFPC FPCFPC φ -shifterPD T r a n s m i ss i o n r a t i o ( d B ) a b d Frequency (GHz) c P p (dBm) A m p l i fi c a t i o n ( d B ) dB Frequency (GHz) e -30-20-100102030 pp L R cL cR | L | e i L | R | e i R B = B = B = B = B = B = B / T r a n s m i ss i o n r a t i o ( d B ) pp A tt e n u a t i o n ( d B ) FIG. 3.
Measurement of optical nonreciprocity. a , Experiment set-up. Red (blue) lines are optical (electronic) wiring.Blue-detuned pump light from a tunable diode laser is split into two paths and fed into the two cavities (red arrows). Part ofthe reflected pump laser light from the cavities (purple arrows) is collected by a photodetector (PD) and fed into a stretchablefiber phase shifter ( φ -shifter) to tune and lock the phase difference of the optical pumps. Each optical path can be modulatedby an electro-optic modulator (EOM) to generate an optical sideband which we use as the optical probe signal. The microwavemodulation signal with frequency ω mod is generated by port 1 of a vector network analyzer (VNA). After optical amplificationand photodetection, the transmitted optical probe signal through the optomechanical circuit is sent back to port 2 of the VNAto measure the phase and amplitude of the optical probe transmission coefficient. EDFA: Erbium doped fiber amplifier, FPC:fiber polarization controller, λ -meter: wavelength meter. b , The ratio of optical power transmission coefficients for right- andleft-propagation versus modulation frequency ( ω mod = − ω = ω p − ω s ), for three different synthetic flux values Φ B /π = 0 . .
26, and 0 .
34. The blue curves correspond to the fit of the theoretical model (c.f. Eq. 3) to the measured spectra. c , Thepower transmission coefficient ratio for Φ B with an additional π flux relative to those in b . d , Theoretical calculation of thepower transmission coefficient ratio for 0 ≤ Φ B ≤ π , where the six grey lines correspond to the six measured Φ B values in b and c . e , Peak forward signal amplification above background level (blue squares) and corresponding signal attenuation in thereverse direction (red circles) versus average optical pump power ( ¯ P p = (cid:112) P pL P pR ) for fixed flux value of Φ B = 0 . π . Thesolid curves are theoretical calculations based upon the theoretical model (c.f. Eq. 3 and SI) fit to the data in b and c . sion is expected. Closer to mechanical resonance, strongnonreciprocity in the optically transmitted power is ob-served, with a peak and a dip in | T L → R /T R → L | oc-curring roughly at the resonance frequencies of the hy-bridized mechanical modes M + and M − , respectively(c.f. Fig. 2c). The maximum contrast ratio betweenforward and backward probe transmission – the isola-tion level – is measured to be 35 dB for Φ B = 0 . π near the M + resonance. The forward transmission isalso amplified in this configuration (blue-detuned pump,∆ = + ω m ), with a measured peak probe signal amplifi-cation of 12 dB above the background level set by photonhopping alone ( J/ | Γ ± | (cid:29) J (cid:54) = κ/
2) and intrinsic optical cavity losses (see SI fordetails).From a two-parameter fit to the measured opticalpower transmission ratio spectra using Eq. 3 (see bluecurves in Figs. 3b and 3c), we obtain a waveguide-mediated optical and mechanical hopping rate of J/ π =110 MHz and V / π = 2 . | T L → R /T R → L | for a full 2 π range ofΦ B with the measured and fit optomechanical circuit pa-rameters. The pattern is seen to be odd symmetric withrespect to Φ B = π . Inserting an additional magnetic flux π into the measurements performed in Fig. 3b yields thespectra shown in Fig. 3c which displays a switch in theisolation direction as predicted by the model. The pumppower dependence of the peak (in frequency) forward sig-nal amplification and the corresponding backward signalattenuation relative to the background level far from me-chanical resonance are shown in Fig. 3e for a fixed mag-netic flux of Φ B = 0 . π . Good correspondence with thetheoretical power dependence (solid curves) is observed,with nonreciprocal amplification vanishing at low pumppower.One can also obtain nonreciprocal optical power trans-mission utilizing an even simpler system involving a sin-gle mechanical cavity. This is the situation we have forthe Fabry-Perot-like mechanical resonances that exist inthe central coupling waveguide (see M W resonance of d B d B a J bc G wR e i R G wL e i L O L O R M W i B B = B = B = B = B = B = FIG. 4.
Synthetic magnetic field with a single me-chanical cavity. a , Physical configuration for generation ofa synthetic magnetic field and optical nonreciprocity with twooptical modes parametrically coupled with a common dissi-pative mechanical waveguide mode. b,c
The ratio of opticalpower transmission coefficients for right and left propagationversus modulation frequency ω mod around the frequency ofthe waveguide mode M W for various Φ B . The blue curvescorrespond to a fit of the theoretical model (see App. B) tothe measured data. Fig. 2c). As depicted in Fig. 4a, the mode configura-tion in this case consists of two optical cavity modes ( O L and O R ) coupled together via the optical waveguide, onemechanical waveguide mode M W which is parametricallycoupled to each of the optical cavity modes, and the syn-thetic magnetic flux Φ B = φ L − φ R due to the relativephases of the optical pump fields threading the triangularmode space. In Fig. 4b and 4c we show the measurementof | T L → R /T R → L | for a series of different flux values Φ B with blue-detuned pumping (∆ ≈ + ω M W ) at levels of n cL = 770 and n cR = 1090. In this single mechanicalmode case the direction of the signal propagation is de-termined by the magnitude of the flux; Φ B ≤ π leadsto backward propagation and Φ B ≥ π to forward prop-agation. The lower contrast ratio observed is a result ofthe weaker coupling between the localized optical cavitymodes and the external waveguide mode, which for themodest pump power levels used here ( (cid:46) µ W) doesnot allow us to reach the parametric coupling requiredfor strong directional transmission.While our focus has been on the propagation of in-jected coherent signals through the optomechanical cir-cuit, it is also interesting to consider the flow of noise.As might be expected, the induced directionality of oursystem also applies to noise photons generated by the up-conversion of both thermal and quantum fluctuations ofthe mechanics; see App. C for detailed calculations. One finds that for the system of Fig. 2, the spectrally-resolvedphoton noise flux shows high directionality, but that thesign of this directionality changes as a function of fre-quency (analogous to what happens in the transmissionamplitudes). In contrast, in the single-mechanical modesetup of Fig. 4 the sign of the directionality is constantwith frequency, and thus the total (frequency-integrated)noise photon flux is directional depending upon the fluxmagnitude. The laser pump fields can thus effectivelyact as a heat pump, creating a temperature differencebetween the left and right waveguide output fields. Thecorresponding directional flow of quantum noise is espe-cially useful for quantum information applications, as itcan suppress noise-induced damage of a delicate signalsource like a qubit [25, 27].The device studied in this work highlights the poten-tial for optomechanics to realize synthetic gauge fieldsand novel forms of nonreciprocity enabled by harness-ing mechanical dissipation. Using just a few modes, itwas possible to go beyond simply mimicking the physicsof an isolator and realize a directional optical amplifier.By adding modes, an even greater variety of behaviourscould be achieved. For example, the simple addition ofa third optical cavity mode, tunnel-coupled to the firsttwo cavities but with no mechanical coupling, would real-ize a photon circulator similar to the phonon circulatorsconsidered in Ref. [18]. Scaling the synthetic gauge fieldmechanism realized here to a full lattice of optomechani-cal cavities would allow the study of topological phenom-ena in the propgation of both light and sound. Predictedeffects include the formation of back-scattering immunephotonic [1] and phononic [20] chiral edge states, topo-logically nontrivial phases of hybrid photon-phonon exci-tations [20], dynamical gauge fields [40], and, in the caseof non-particle-conserving interactions enabled by blue-detuned optical pumping, topologically protected inelas-tic scattering of photons [28] and even protected ampli-fying edge states [29].
ACKNOWLEDGMENTS
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Appendix A: Device Fabrication and Methods1. Device fabrication and atomic force microscope nano-oxidation tuning
The devices were fabricated from a silicon-on-insulator wafer with a silicon device layer thickness of 220 nm andburied-oxide layer thickness of 2 µ m. The device geometry was defined by electron-beam lithography followed byinductively coupled plasma reactive ion etching to transfer the pattern through the 220 nm silicon device layer. Thedevices were then undercut using an HF:H O solution to remove the buried oxide layer and cleaned using a piranhaetch.After device fabrication, we used an atomic force microscope to draw nanoscale oxide patterns on the silicon devicesurface. This process modifies the optical and mechanical cavity frequencies in a controllable and independent waywith the appropriate choice of oxide pattern. The nano-oxidation process was carried out using an Asylum MFP-3Datomic force microscope and conductive diamond tips (NaDiaProbes) in an environment with relative humidity of48%. The tip was biased at a voltage of − .
2. Optical transmission coefficient measurement
Frequency (GHz) T r an s m i ss i on c oe ff i c i en t ( d B ) -80-75-70-65-60-55-50-45-40-35 Frequency (GHz) ba T r an s m i ss i on c oe ff i c i en t ( d B ) B = B = FIG. 5. a Microwave signal power transmission through the optomechanical circuit for forward (right-propagation; blue) andbackward (left-propagation; blue curve) directions, with flux set to Φ B = 0 . π and cavity photon number n cL = 1000 and n cR = 1420. b Same as a but with Φ B = 1 . π . To measure the optical power transmission through the optomechanical circuit we used a vector network analyzer(VNA). The VNA outputs a microwave tone from port 1 with frequency ω mod to an electro-optic modulator whichmodulates the optical pump to generate an optical sideband corresponding to the optical probe. In the case of ablue-detuned pump from the optical cavity resonance, the probe field corresponds to the lower sideband (selected bythe filtering properties of the cavity itself). Both the optical probe and pump are launched into one optomechanicalcavity in the circuit. At the other cavity, the transmitted optical probe combines with a second pump and the beatingof the two is detected by a high-speed photodetector (both the first and second pump beams are from the samelaser source, and thus phase coherent). The photocurrent signal from the photodetector is sent into port 2 of theVNA to measure the microwave signal transmission coefficient T µ . Fig. 5 shows | T µ | for forward (right-propagating;blue curve) and backward (left-propagating; red curve) directions through the optomechanical circuit as a functionof the modulation frequency ω mod . In Fig. 5a the synthetic flux value is locked to Φ B = 0 . π whereas in Fig. 5bΦ B = 1 . π . In both flux settings the optical pumping levels were such that the left and right cavity photon numberswere n cL = 1000 and n cR = 1420, respectively. This is the raw transmission data corresponding to the normalizedtransmission ratio shown in Figs. 3b and 3c of the main text.While absolute optical transmission is not directly measured, the ratio of the optical transmission coefficients forforward and backward propagation can be obtained from the normalized microwave signal transmission coefficient ¯ T µ , | T L → R /T R → L | = | ¯ T µR / ¯ T µL | , (A1)where | ¯ T µ | is normalized using the value of | T µ | away from all mechanical resonances to remove all the externalasymmetry in the experimental setup for left and right propagation paths. These external asymmetries includemodulator efficiency, cable/fiber loss, etc. In our analysis the normalization level is the average value of | T µ | in thefrequency range of 5 . .
76 GHz. To be clear, the reason this calibration is necessary is because we don’t actuallyphysically swap the source and detector in our measurements. Rather, for the left-to-right transmission path we haveone modulator on the left side which generates the probe tone and one detector on the right side which measures thetransmission through to the right side. When we measure right-to-left transmission we have a different modulator onthe right side to generate the probe tone and a different detector on the left side to detect the transmitted probe. Ifthe modulator on the left side is different from the modulator on the right side, then for the same microwave drive thatexcites the modulators we would get different a different optical probe power in the sidebands of the pump. Similarlyif the left and right detectors have different efficiencies then they would produce a different photocurrent for thesame transmitted optical probe power. Since we measure in practice the ratio of the microwave drive to the detectedmicrowave photocurrent, this could cause an inherent asymmetry in the measured transmission for left-to-right andright-to-left transmission even if the optical transmission was perfectly symmetric.
3. Device characterization
To determine the components of optical cavity loss (intrinsic decay rate κ i , external waveguide-to-cavity coupling κ e , total cavity decay rate κ ) of both the left and right optical cavities we used a pump-probe scheme similar to thatused to measure the nonreciprocity of the optomechanical circuit. The pump beam in this case, however, is set to bevery weak so as to not resonantly excite the mechanics as the probe signal is swept across the optical cavity resonance.The cavity scans are plotted in Fig. 6a and 6b for the left and right cavities, respectively. We fit the phase responsecurves and get κ iL(R) / π = 0 .
29 (0 .
31) GHz, κ eL(R) / π = 0 .
74 (0 .
44) GHz, and κ L(R) / π = 1 .
03 (0 .
75) GHz. Theintrinsic and external optical cavity rates are used to determine the intra-cavity photon number for a given opticalpump power (specified at the input to the cavity).Thermal mechanical spectra of the two cavities are measured with a weak blue-detuned optical pump so as to avoidback-action; a single pump is used for each of the left and right cavity measurements. The reflected pump light fromthe cavity contains modulation sidebands from the thermal mechanical motion, which upon detection with a high-speed photodetector creates a photocurrent with the thermal motion of the mechanical cavity modes imprinted on it.Since the mechanical modes can be hybridized between left-cavity, right-cavity, and waveguide modes, a measurementwith the left-side pump produces a local measurement of the cavity modes as measured by the localized left opticalcavity mode, and similarly for the right-side pump and cavity. The intrinsic decay rate of the mechanical modes isinferred from the linewidth of the Lorentzian mechanical spectrum.Measurements of the mechanical mode spectra were performed both before and after the cavities were nano-oxidizedto tune their localized optical and mechanical modes into resonance. Measurements prior to nano-oxidation allowed usto determine the local (left and right) mechanical and optical cavity mode properties (i.e., the bare, uncoupled modeproperties). Knowing the left and right cavity mode properties from independent measurements allowed us to fit withfewer fitting parameters the measured forward and backward transmission curves of the hybridized cavities presentedin the main article text. Note that after nano-oxidation the left and right optical cavity modes were only very weaklyhybridized so as to maintain their left-cavity and right-cavity character. The mechanical modes were tuned to bestrongly hybdridized as evidenced in Fig. 2f of the main text. Figures 6c and 6d show the measured linewidth of themechanical cavity modes M L(R) versus optical pumping power. In Fig. 6c the left cavity was pumped with a bluedetuning ∆ = + ω mL ; in Fig. 6d the right cavigty was pumped with a blue detuning of ∆ = + ω mR . By fitting themeasured data with formula γ = γ i − g n c /κ ( n c corresponding to the intra-cavity photon number determined fromthe O L(R) measured cavity properties), we obtain g , L(R) / π = 0 .
76 (0 .
84) MHz and γ iL(R) / π = 4 . . J ) and mechanical ( V ) hopping rates between the two optomechanical cavities via the connectingwaveguide are determined from a global fitting using Eq. (1) of the main text for the group of measured transmissioncoefficient ratio curves in Figs. 3c and 3d with varying Φ B . The intra-cavity cavity photon number, optomechanicalcoupling rates and intrinsic mechanical decay rates are all taken as fixed and equal to the independently measured0 n cR
100 150 200 250 300 350 400 450 γ R / π ( M H z ) n cL
300 400 500 600 700 800 γ L / π ( M H z ) Wavelength (pm) -100 -50 0 50 100 N o r m a l i z e d r e fl e c t i o n dce T r a n s m i ss i o n ( d B ) Wavelength (pm) f Frequency (GHz) P ha s e ( deg r ee ) -20020406080100120140160180 Frequency (GHz) P ha s e ( deg r ee ) -180-160-140-120-100-80-60-40-200 a b FIG. 6. a , Left optical cavity phase response as measured by scanning the probe signal across the cavity resonance with weakblue-detuned pump. b , Same as in a for the right optical cavity. c , Measured back-action modified mechanical linewidth versusintra-cavity pump photon number for the left optical cavity. Here only left cavity pump beam is applied, and the pump is tunedto the upper motional sideband of the cavity (blue-detuned with ∆ = + ω mL ). d , Same as in c for the right-side cavity andright-side pump. Measurements in a - d were performed prior to nano-oxidation tuning. e Measured (circles) and theoretical(solid curves) optical reflection spectra using a left-cavity (blue) and right-cavity (red) optical pump. These measurements aretaken after nano-oxidation and the theoretical calculation includes the fit coupling ( J/ π = 110 MHz) between the left andright optical cavity modes and a splitting between the uncoupled modes. The wavelength origin is taken to correspond to theright optical cavity resonance. f Calculated optical transmission power from one optical port to the other of an optical probesignal near resonance of the coupled optical cavity modes. Here there is no pump beam, and so no coupling to phonons. Theparameters of the optical cavity modes are taken from the fit to the measured optical reflection spectra in e . values as described above.With the fit value of J from forward and reverse transmission measurements versus Φ B , and the measured cavitycoupling rates ( κ , κ i ) from the left and rigth optical cavity modes prior to nano-oxidation tuning, we fit the measuredoptical reflection spectra of the two weakly coupled optical cavity modes after nano-oxidation. This allows us todetermine the uncoupled left and right optical cavity mode frequencies. The measured and fit spectra as measuredfrom the left and right cavities are shown in Fig. 6e. As noted earlier, the measured spectra after nano-oxidationare still predominantly given by uncoupled left and right cavity modes. Based on the theoretical fit to the measuredoptical reflection spectra, we also calculate the transmission of an optical probe signal through the optomechanical1circuit in the absence of a pump beam (i.e., no coupling to phonons, just pure optical transmission) η = J √ κ eL κ eR | J + κ L κ R / − ( ω − ω cL )( ω − ω cR ) − iκ L ( ω − ω cL ) / − iκ R ( ω − ω cR ) / | . (A2)Fig. 6f shows the numerical result, and the minimum insertion loss for transmission from one port to the other portis found to be about 11 dB for a probe signal frequency in between the two cavity resonances. This is the estimatedport-to-port optical transmission effiency in absence of optomechanical amplification. Appendix B: Theory of optical nonreciprocity1. Input-output formula
We provide theoretical analysis of optical nonreciprocity in the coupled optomechanical cavity system. We firstconsider the case with two optical and two mechanical cavity modes. The Hamiltonian of this system can thus bewritten down as follows,ˆ H = (cid:88) k = L,R (cid:126) ω ck ˆ a † k ˆ a k + J (ˆ a † L ˆ a R + ˆ a L ˆ a † R ) + (cid:88) k = L,R (cid:126) ω mk ˆ b † k ˆ b k + V (ˆ b † L ˆ b R + ˆ b L ˆ b † R ) (B1)+ (cid:88) k = L,R (cid:126) g k (ˆ b † k + ˆ b k )ˆ a † k ˆ a k + (cid:88) k = L,R i (cid:126) √ κ ek α pk e − iω pk t − iφ k (ˆ a k − ˆ a † k ) , where J and V are the waveguide mediated optical and mechanical coupling strength (we gauged out the phase of J and V and take both of them to be real), and the last two terms are the optical driving fields (pumps) which havethe same frequency and correlated phases. We consider the situation where the optical cavities are nearly degenerate,i.e., ω cL (cid:39) ω cR ≡ ω c and both optomechanical systems are driven with a blue-detuned laser ( ω pk = ω c + ω mk ).We perform a displacement transformation ˆ a k = α k + ˆ d k , separating the classical steady state amplitude of the localoptical cavity field from its fluctuations. With this we can linearize the optomechanical interaction in the Hamiltonianof Eq. B1 in the usual manner. Assuming the good cavity limit (sideband resolved, ω mk (cid:29) κ k ), we apply a rotatingwave approximation and obtain for the equations of motions ( (cid:126) = 1) ddt ˆ d L = (cid:16) i ∆ L − κ L (cid:17) ˆ d L − √ κ eL ˆ d L, in − √ κ iL ˆ ξ L, in − iJ ˆ d R − iG L ˆ b † L e iφ L ,ddt ˆ d R = (cid:16) i ∆ R − κ R (cid:17) ˆ d R − √ κ eR ˆ d R, in − √ κ iR ˆ ξ R, in − iJ ˆ d L − iG R ˆ b † R e iφ R ,ddt ˆ b L = − (cid:16) iω mL + γ iL (cid:17) ˆ b L − √ γ iL ˆ b L, in − iV ˆ b R − iG L ˆ d † L e iφ L ,ddt ˆ b R = − (cid:16) iω mR + γ iR (cid:17) ˆ b R − √ γ iR ˆ b R, in − iV ˆ b L − iG R ˆ d † R e iφ R , (B2)with the total damping rates κ k = κ ek + κ ik , the detunings ∆ k = ω p − ω ck and the many-photon optomechanicalcouplings G k = g k α k . The latter contains the steady state amplitude of the local optical cavity field α k e iφ k , whichis related to the pump amplitudes through α L ( R ) e iφ L ( R ) = ( i ∆ R ( L ) − κ R ( L ) / √ κ eL ( R ) α pL ( R ) e − iϕ L ( R ) + iJ √ κ eR ( L ) α pR ( L ) e − iϕ R ( L ) ( i ∆ L − κ L / i ∆ R − κ R /
2) + J . (B3)We find the steady state amplitude is approximately √ κ ek α pk e − iϕ k /i ∆ k under the condition ∆ k ≈ ω mk (cid:29) κ k , J ,which means each cavity is effectively only driven by its own optical pump. Thus, each pump-enhanced optomechanicalcoupling and its phase can be independently controlled. The intrinsic noise operators ˆ ξ k, in and ˆ b k, in in the coupledmode equations B2 describe thermal and vacuum fluctuations impinging on the the cavities and the mechanical modesrespectively. The associated noise of a possible input signal is described via ˆ d k, in .2
2. Mechanically-mediated coupling
We perform a Fourier transform (ˆ b [ ω ] ≡ (cid:82) d t ˆ b ( t ) e + iωt ; ˆ b ( t ) ≡ (cid:82) d ω π ˆ b [ ω ] e − iωt ) of the coupled mode equations Eqs. B2and insert the resulting solution for ˆ b † L,R [ ω ] into the equations of the cavity operators. Ignoring the intrinsic noiseterms ˆ ξ in , k and ˆ b in , k for the moment, we obtain for the cavity operators in frequency space (Φ B = φ L − φ R ) (cid:101) χ − L, + [ ω ] ˆ d L [ ω ] = − √ κ eL ˆ d L, in [ ω ] − i (cid:0) J − Γ + [ ω ] e + i Φ B (cid:1) ˆ d R [ ω ] , (cid:101) χ − R, + [ ω ] ˆ d R [ ω ] = − √ κ eR ˆ d R, in [ ω ] − i (cid:0) J − Γ + [ ω ] e − i Φ B (cid:1) ˆ d L [ ω ] , (B4)with the modified susceptibility (cid:101) χ − k, + [ ω ] = (cid:0) − i ( ω + ∆ k ) + κ k + i Σ k, + [ ω ] (cid:1) . The frequency dependent coupling Γ + [ ω ]and the self-energy Σ k, + [ ω ] are defined asΓ + [ ω ] = V G R G L (cid:2) − i ( ω + ω mL ) + γ iL (cid:3) (cid:2) ( − i ( ω + ω mR ) + γ iR (cid:3) + V , Σ k, + [ ω ] = iG k V G ¯ k (cid:104) − i ( ω + ω m ¯ k ) + γ i ¯ k (cid:105) Γ + [ ω ] , (B5)here the coupling Γ + [ ω ] coincides with Eq. (2) of the main text. After eliminating the mechanical degrees of freedom,one finds both a ”local” modification of each cavity (described by the self energy Σ k, + [ ω ]) and an induced couplingbetween the cavities. The self-energies lead to damping (or anti-damping) of each cavity resonance as well as afrequency shift of the resonance. Here the subscript + indicates blue-detuning (∆ k = ω pk − ω c ≈ + ω mk ). The polesof the self energy read ω ± = − i γ iL + γ iR ) −
12 ( ω mL + ω mR ) ± (cid:115) V − (cid:20)
14 ( γ iL − γ iR ) − i ω mL − ω mR ) (cid:21) . (B6)The induced coupling has a coherent and a dissipative aspect. To illustrate this we separate the coupling into realand imaginary parts Γ + [ ω ] ≡ Γ Re [ ω ] + i Γ (cid:61) [ ω ]. The real and imaginary parts of this frequency-dependent couplinghave completely different physical interpretations. We see this, by considering again the coupling terms in Eq. (B4).We have ˆ d L [ ω ] ∼ (cid:2) − i (cid:0) J − Γ Re [ ω ] e + i Φ B (cid:1) − Γ (cid:61) [ ω ] e + i Φ B (cid:3) ˆ d R [ ω ] ≡ (cid:104) − i (cid:101) J [ ω ] − Γ (cid:61) [ ω ] e + i Φ B (cid:105) ˆ d R [ ω ] , ˆ d R [ ω ] ∼ (cid:2) − i (cid:0) J − Γ Re [ ω ] e − i Φ B (cid:1) − Γ (cid:61) [ ω ] e − i Φ B (cid:3) ˆ d L [ ω ] ≡ (cid:104) − i (cid:101) J ∗ [ ω ] − Γ (cid:61) [ ω ] e − i Φ B (cid:105) ˆ d L [ ω ] . (B7)For the given frequency of interest, we see that the real part of the induced coupling is completely equivalent to having aHamiltonian, coherent tunneling term between the cavities; we can absorb it into a redefinition of the coherent hoppingstrength J , i.e., J → (cid:101) J [ ω ]. In contrast, the coupling mediated by the imaginary part Γ (cid:61) [ ω ] is not equivalent to someeffective coherent tunneling interaction between the cavities, i.e., the Γ (cid:61) [ ω ] terms in ˆ d L and ˆ d R Eqs.(B7) cannot beincorporated into a definition of J . The terms involving Γ (cid:61) [ ω ] instead represent a dissipative coupling between thetwo cavities mediated by the mechanics. Such dissipative interactions (if we ignore their frequency dependence) canbe obtained in a master equation formalism via an effective Lindblad dissipator of the form 2Γ (cid:61) L (cid:104) d † L + e − i ∆ φ d † R (cid:105) ,where L [ˆ o ]ˆ ρ = ˆ o ˆ ρ ˆ o † − / o † ˆ o ˆ ρ − / ρ ˆ o † ˆ o is the standard Lindblad superoperator.
3. Directionality by balancing coherent and dissipative interactions
The dissipative coupling is crucial for directionality: by balancing the dissipative interaction against the coherentinteraction we obtain a nonreciprocal system (following the general recipe outlined in Ref.[25]). For example, if weaim for a directional transport from the left to the right cavity, we want to decouple the left cavity from the rightcavity (while still having the right cavity influenced by the left cavity). This is accomplished by balancing coherentand dissipative interactions, i.e., (cid:101) J [ ω ] ! = i Γ (cid:61) [ ω ] e i Φ B , (B8)in which case the coupling from the left to right cavity vanishes, cf. Eq. (B7), and we obtain a unidirectional couplingwhere the right cavity is driven by the left cavity but not vice versa. Crucially, this would not be possible without3the dissipative interaction, i.e., we need Γ (cid:61) [ ω ] (cid:54) = 0. Note, for the situation that Γ (cid:61) [ ω ] = 0, i.e., γ ik = 0, but finiteΦ B , we still obtain a directional dependent phase. However, to use this as the basic for nonreciprocal transmissionadditional interference processes have to implemented.The directionality condition Eq. (B8) can be reformulated in terms of the original J and the phase difference Φ B as used in Eq. (B4). This translates to the condition J = | Γ + [ ω ] | , Φ B = − arg(Γ + [ ω ]) , (B9)where we still aim for unidirectional behavior from left to right. For the case of a purely real coupling Γ + [ ω ] = Γ Re [ ω ]these conditions could still be satisfied, i.e., for Φ B = 0 and Γ Re [ ω ] = J . However, this means that there is effectivelyno coupling between the cavities, and thus no forward transport either. Note, that a sign change in arg(Γ + [ ω ]) wouldlead to the opposite situation, where the propagation direction would be from right to left.In general, the directionality balancing condition obtained here is frequency dependent, for the simple reason thatthe mechanically-mediated cavity-cavity coupling is frequency-dependent. If we could somehow fulfill the directionalitycondition in Eq. (B9) at every frequency, the cavity output field operators would be given by (using the standardinput-output relation ˆ d k, out = ˆ d k, out + √ κ ek ˆ d k )ˆ d L, out [ ω ] = [1 − κ eL (cid:101) χ L, + [ ω ]] ˆ d L, in [ ω ] , ˆ d R, out [ ω ] = [1 − κ eR (cid:101) χ R, + [ ω ]] ˆ d R, in [ ω ] − i √ κ eR κ eL (cid:101) χ R, + [ ω ] (cid:101) χ L, + [ ω ] | Γ + [ ω ] | (cid:16) e i + [ ω ]) − (cid:17) ˆ d L, in [ ω ] , (B10)where we neglected the noise contributions originating from the mechanical modes, i.e., the coupling to ˆ b n, in inEq. (B4), and the intrinsic cavity noise ξ in , k for simplicity. Here, we see again that the dissipative interaction iscrucial as we need arg(Γ + [ ω ]) (cid:54) = nπ, n ∈ Z , i.e., we need a finite imaginary part of Γ + [ ω ].The experimentally relevant situation is where dissipative and coherent interactions are only balanced at a singlefrequency (by appropriate tuning of phase and J ). Achieving this condition close to the normal modes resonancefrequencies is favorable given the resonantly-enhanced transmission. Enforcing directionality at ω = − ω m ± V forequal mechanical resonance frequencies, results in the directionality conditions ω mL = ω mR : Φ B = ∓ arctan 2 V ( γ iL + γ iR ) γ iL γ iR , J = V G R G L (cid:113) V ( γ iL + γ iR ) + γ γ , (B11)where the upper (lower) sign in the phase difference Φ B realizes directionality at ω = − ω m + V ( − ω m − V ). Directionalityhere means that an input signal injected on the left cavity is transmitted to the right cavity, whereas the backwardpropagation path, i.e., from right to left, is blocked.On the other side, if we assume identical bare mechanical damping of the mechanical modes ( γ iL = γ iR = γ i ), butunequal bare mechanical frequencies ( ω mL (cid:54) = ω mR ), then we find that at the frequencies of the hybridized mechanicalmodes Ω ± = − ( ω mL + ω mR ) ± (cid:113) V + ( ω mL − ω mR ) the directionality condition is modified to γ iL = γ iR : Φ B = ∓ arctan 4 (cid:113) V + ( ω mL − ω mR ) γ , J = V G L G R γ (cid:113) V + γ + ( ω mL − ω mR ) . (B12)where the upper (lower) sign in the phase difference Φ B realizes directionality at ω = Ω +( − ) . The directionalityconditions for a perfectly symmetric device, i.e., for equal mechanical resonance frequencies ( ω m ) and decay rates ( γ ),can simply be read off from either Eq. B12 or Eq. B11.
4. Nonreciprocal optical transmission: two blue-detuned pumps
From the equations for the cavity operators in Eqs. B4 we can calulate the transmission coefficients via input/outputtheory. Note, that although Eqs. B4 are formulated on the basis of noise operators, they as well describe thedynamics of the cavity field amplitudes d k around their steady state solution. The right transmission coefficient4 T L → R ≡ d R, out /d L, in and left transmission coefficient T R → L ≡ d L, out /d R, in are given by T R (cid:28) L [ ω ] = i √ κ eL κ eR (cid:2) J − Γ + [ ω ] e ∓ i Φ B (cid:3)(cid:101) χ − L [ ω ] (cid:101) χ − R [ ω ] + [Γ + [ ω ] + J − + [ ω ] J cos( φ )] ≡ A + [ ω ] (cid:2) J − Γ + [ ω ] e ∓ i Φ B (cid:3) , (B13)with the modified susceptibilities (cid:101) χ k [ ω ] as defined after Eq. (B4). The prefactor A + [ ω ] is the same for both transmissionamplitudes, it accounts for the mechanically-induced back-action on the optical cavities, cf. main text after Eq. (2).Note, that the corresponding prefactor for two red-detuned pumps is simply A − [ ω ] = − A ∗ + [ − ω ].We now assume a completely symmetric pair of mechanical cavities ( ω mL = ω mR = ω m and γ iL = γ iR = γ i ) andapply the corresponding directionality direction for symmetric parameters, cf. Eq. B12 or Eq. B11. The transmissioncoefficient for the through direction ( → ) under these conditions of perfect nonreciprocity is given by, T → [ − ω m ± V ] = (cid:114) κ eL κ eR κ R κ L (cid:115) ± i γ i V ∓ i γ i V i √C L C R (cid:104) C L (cid:0) ± i γ i V (cid:1) − (cid:16) ∓ i Vκ L (cid:17) (cid:0) ± i γ i V (cid:1)(cid:105) (cid:104) C R (cid:0) ± i γ i V (cid:1) − (cid:16) ∓ i Vκ R (cid:17) (cid:0) ± i γ i V (cid:1)(cid:105) , (B14)introducing the single cavity cooperativity C k ≡ G k /γ i κ k . Considering as well symmetric optical cavities ( κ eL = κ eR = κ e ; κ L = κ R = κ ) with symmetric optical pumping ( G L = G R = G ) the transmission coefficient simplifies to T → [ − ω m ± V ] V (cid:28) κ (cid:39) i C κ e κ (cid:2) − C ± i γ i V (1 − C ) (cid:3) , (B15)with C ≡ G /γ i κ and under the realistic assumption that the hopping rate V is much lower than the cavity decayrate κ . Here we work with blue-detuned pumping of both optical cavities (∆ ≈ + ω m ), which results in parametricamplification of each of the left and right mechanical modes and leads to amplification of the optical probe signal.This becomes apparent for the situation that the mechanical hopping rate is much faster than the intrinsic mechanicaldecay rate ( V /γ i (cid:29) C → √
2, hence the cooperativity by a factor of 2). Note, for the situation
V /γ i (cid:29) ω = − ω m ± V simplifies to J (cid:39) G L G R /γ i andΦ B → ∓ π/
5. Nonreciprocal optical transmission: two red-detuned pumps
The analysis for the the case of two red detuned pumps is similar to the blue-detuned case. The cavity operatorsin Eq.(B2) couple now to the mechanical lowering operators ˆ b k and vice versa, while the detuning between the cavityresonances and the external pump tones yields ∆ k = − ω mk . The ratio of transmission coefficients is found to be givenby the following expression T L → R T R → L = J − Γ − [ ω ] e − i Φ B J − Γ − [ ω ] e + i Φ B = J − V G L G R [ − i ( ω − ω mL )+ γ iL2 ][ − i ( ω − ω mR )+ γ iR2 ] + V e − i Φ B J − V G L G R [ − i ( ω − ω mL )+ γ iL2 ][ − i ( ω − ω mR )+ γ iR2 ] + V e + i Φ B (B16)where we have Γ − [ ω ] = Γ ∗ + [ − ω ], thus the ratio | T L → R /T R → L | is the same for blue and red detuned pumps evaluated atcorresponding frequencies. The reason for this is that the transmission is either amplified or suppressed simultaneouslyfor both directions and thus their ratio stay unchanged. Comparing to the blue detuned case, the perfect nonreciprocitycondition remains the same in the red detuned case, while the transmission coefficient for the through direction thehybridized mechanical modes Ω ± = ω m ± V is given by (assuming ω mL = ω mR , γ iL = γ iR = γ i and V (cid:28) κ k ) T → [ ω m ± V ] (cid:39) (cid:114) κ eL κ eR κ R κ L i √C L C R (cid:2) C L + 2 ± i γ i V ( C L + 1) (cid:3) (cid:2) C R + 2 ± i γ i V ( C R + 1) (cid:3) . (B17)From Eq. B17, we note in general an attenuated transmission for the red detuned case as T → ≤ (cid:112) κ eL κ eR / ( κ L κ R ) < V /γ i (cid:29) C k = 2 and/or κ k / G L G R γ i . Comparing thelatter to Eq. B11 we see the maximal through transmission efficiency is achieved when the optical cavity loss rate5 κ k / J for both cavities (impedance matching condition).
6. Nonreciprocity associated with a single mechanical waveguide mode
In our optomechanical circuits, we also observed optical nonreciprocity with a single mechanical waveguide mode. Inthis case, the Hamiltonian describing the interaction between two optical cavity modes and one mechanical waveguidemode is given by, ˆ H = (cid:88) k =L,R (cid:126) ω c ,k ˆ a † k ˆ a k + J (ˆ a † L ˆ a R + ˆ a L ˆ a † R ) + (cid:126) ω M W ˆ b † W ˆ b W (B18)+ (cid:88) k =L,R (cid:126) (cid:16) g , W k ˆ b W + g ∗ , W k ˆ b † W (cid:17) ˆ a † k ˆ a k + (cid:88) k =L,R i (cid:126) √ κ e k α p k e − iω p t − iφ k (ˆ a k − ˆ a † k ) . Going through a similar calculation using coupled mode equations, we find that the ratio of right and left opticaltransmission coefficients is T L → R T R → L = J ± i | G WL G WR |− i ( ω ± ω M W ) + γ iW2 e − i (Φ B ± Φ W ) J ± i | G WL G WR |− i ( ω ± ω M W ) + γ iW2 e + i (Φ B ± Φ W ) , (B19)where the upper (lower) sign corresponds to the blue (red) detuned case and Φ W = arg( G ∗ WL G WR ). The correspondingconditions for perfect directionality from left to right and at ω = ∓ ω M W are J = 2 | G WL G WR | γ iW , Φ B = ± π ∓ Φ W . (B20)This in turn leads to the transmission coefficients T → [ ∓ ω M W ] = (cid:114) κ eL κ eR κ L κ R i √C W L C W R ( C W L ∓ C W R ∓ . (B21)In the case of blue detuned tones an input signal is amplified and the corresponding gain increases for C W k → G ∗ WL G WR . This addition comes from the fact that wehave already chosen definitions for the local cavity mode amplitudes ( a L,R and b L,R ) such that the phase of theoptomechanical couplings of the localized cavity modes – G L ≡ | α L | g , L and G R ≡ | α R | g , R – are both zero. Withthese same definitions for amplitudes a L and a R we are not then free to set the phases of both G WL and G WL to bezero; not at least for the same set of pump phases φ L and φ R chosen for the localized cavity mode coupling. A simpleexample helps to illustrate this. The mode M W can be viewed as a hybridization between the localized left and rightcavity modes and a delocalized waveguide mode [21]. Using perturbation theory, we have for the mechanical modeamplitude of the hybridized mode M W , b W = b W (cid:48) + t L ω M (cid:48) W − ω mL b L + t R ω M (cid:48) W − ω mR b R , (B22)where b W (cid:48) is the unperturbed delocalized waveguide mode amplitude and ω M (cid:48) W is the unperturbed frequency of thedelocalized waveguide mode. t L(R) is the coupling coefficient between the delocalized waveguide mode and the localizedcavity mode M L(R) . The phases of t L and t R are determined by the field distribution of the hybridized mode M W in the left and right cavities, respectively, and cannot be (both) chosen arbitrarily . Using the mode decompositionof Eq. B22, we have that arg( g ∗ , WL g , WR ) = arg( t ∗ L t R ) as we have already chosen a local cavity mode amplitudebasis such that arg( g , L ) = arg( g , R ) = 0 and ω M (cid:48) W > ω mL , ω mR (this assumes of course that the left (right) opticalcavity mode only couples to the portion of b W which is due to b L ( b R ), which is a good approximation due to the factthat the optical cavities are in the far field of each other). Thus, by simultaneously measuring the flux-dependenttransmission near the resonance of the localized mechanical cavity modes and the hybridized mechanical waveguidemode we can determine the arg( g ∗ , WL g , WR ) in this mode basis (see Fig. 8 for example). For the M W mode in ourexperiment, we find arg( g ∗ , WL g , WR ) ≈ π , which means for this hybridized mode and chosen localized cavity modebasis the mechanical motion in the left cavity as seen by the left cavity optical mode is approximately 180 degreesout of phase with the motion in the right cavity as seen by the right cavity optical mode.6 Appendix C: Directional flow of quantum and thermal noise
Besides the nonreciprocal optical signal transmission, the flow of quantum and thermal noise in the optomechanicalcircuit is directional. This is a natural consequence of the system’s scattering matrix having a directional form; thescattering matrix determines both the transmission of coherent signals, as well as noise properties. To show this, wecalculate the symmetrized output noise spectral density via¯ S k, out [ ω ] = 12 (cid:90) d Ω2 π (cid:68)(cid:110) ˆ d k, out [ ω ] , ˆ d † k, out [Ω] (cid:111)(cid:69) , (C1)defined in the standard manner [41]. The mechanical and optical noise operators introduced in Eqs. B2 have zeromean and satisfy the canonical correlation relations: (cid:104) ˆ o k, in [ ω ]ˆ o † k (cid:48) , in [Ω] (cid:105) = (cid:104) o † k, in [ ω ] o k (cid:48) , in [Ω] (cid:105) + δ k,k (cid:48) δ ( ω + Ω) = (cid:0) n th o k + 1 (cid:1) δ k,k (cid:48) δ ( ω + Ω) , ˆ o k, in = ˆ d k, in , ˆ ξ k, in , ˆ b k, in . (C2)where n th o k is the thermal occupation of each bath. In what follows, we assume that we have no thermal occupationof the optical field. This is justified as we work with a very high optical frequency.Figure 7a-d depicts the output spectra for the situation that both pumps are blue detuned from the cavity by ω m .Here we assumed equal mechanical frequencies ω mL = ω mR = ω m and work in a rotating frame where the uncoupledmechanical resonance frequencies are shifted to zero. The remaining parameters are as used in the experiment,i.e., we take γ iL / π = 4 . γ iR / π = 5 . κ L / π = 1 .
03 GHz, κ R / π = 0 .
75 GHz, κ iL / π = 0 .
29 GHz, κ iR / π = 0 .
31 GHz,
V / π = 2 . J/ π = 110 MHz. The multiphoton couplings G L = G R used in the calculationare determined from Eq. B11.Figure 7a shows the result for zero temperature mechanical baths and a finite phase Φ B = 0 . π (determined fromEq. B11). As expected, the L and R output spectra are not identical: while each has a double-peaked structure(corresponding to the two normal mode resonances), the right output spectra ¯ S R, out [ ω ] has the upper-frequency peaklarger than the lower-frequency peak, while the situation is reversed for the left output spectra. This does not leadto any asymmetry in the total output photon number fluxes (i.e., intergrated over all frequencies). It does howeverlead to an asymmetry in the energy fluxes (i.e., as the higher energy peak is bigger for the right output spectrum,and the low energy peak is bigger for the left spectrum). Thus, the ”quantum heating” of zero-point fluctuationspreferentially cause an energy flow to the right (rather than to the left) for this choice of phase.It is also worth noting that if all dissipative rates are equal for the R and L cavities, then the L output spectrumis just the frequency-mirrored R output spectrum. The latter is visible in Fig. 7(c), where we plotted the outputspectra for symmetric parameters, i.e., we set γ iR / π = γ iL / π = 4 . κ R / π = κ L / π = 1 .
03 GHz, κ iR / π = κ iL / π = 0 .
31 GHz and Φ B = 0 . π (determined from Eq. B11 for the new γ iR ). However, having unequal decayrates, i.e., γ R (cid:54) = γ L and κ R (cid:54) = κ L , leads to a slight asymmetry even if the phase is set to zero, i.e., Φ B = 0, as visiblein Fig.7b. In Fig. 7g we plot the asymmetry ¯ S L, out [ ω ] − ¯ S R, out [ ω ] for all the four cases corresponding to Fig. 7a-d.For finite temperature, we find that the output spectrum has a roughly linear dependence on the mechanical bathtemperature: ¯ S k, out ( T ) = c k n th + ¯ S k, out (0) (assuming n th b L = n th b R ≡ n th ). This linear dependence is visible if wecompare Fig. 7c,d and Fig. 7e,f, where the latter show the output noise spectra for n th = 10 with symmetric cavityparameters. Additionally, we also calculate the added noise quanta to the transmitted signal¯ n k, add [ ω ] ≡ ¯ S k, out [ ω ] | T k [ ω ] | − , (C3)where is the half quanta noise of the vacuum optical fields injected from the coupler. Fig.7h shows the added noisefor left-right propagation with Φ B = 0 . π (and asymmetric experimental cavity parameters). The mechanical baths n th are varied as denoted in each graph. Even if the cavities and the mechanics are only driven by vacuum noise thestandard quantum limit (SQL) of half a quanta is not achieved. This is due to the limited amount of gain achievedin the experiment, i.e., the transmission coefficient is not high enough to suppress the noise contributions. Moreover,even in the large gain limit the added noise would be roughly one quanta due to the finite amount of intrinsic opticalcavity loss. Appendix D: Reciprocal device
Realizing optical nonreciprocity in the optomechanical circuits studied in this work is not simple or easy as justcreating a circuit with optical and mechanical coupling between two optomechanical cavities. One is limited by the7 a bc de fg h n th = n th = n th = FIG. 7. a-d
Output noise spectra at zero temperature for a set of parameters given in the text. e-f
Output noise spectra atfinite temperature with thermal phonon occupation of n th = 10. g Difference of the left and right output spectra for a-d . f Added noise for right-propagation signal. Frequency (GHz)5.77 5.78 5.79 5.8-10123456Frquency (GHz)5.77 5.78 5.79 5.8 P S D ( d B m / H z ) -132-131-130-129-128-127 a b N o r m a li z ed t r an s m i ss i on ( d B ) FIG. 8. Optical reciprocity in a circuit with large optical cavity coupling, J . a Mechanical spectra measured from the left(red) and right (blue) optical cavities. b Normalized optical signal power transmission coefficient for forward (red) and reverse(blue) optical signal propagation. practical realities of device power handling capability, finite optical and mechanical Q -factors, etc. As such, not all thecircuits that were tested exhibited nonreciprocal transmission and amplification; the effects were too weak to observein some circuits. This, however, was a useful test of our set-up as nonreciprocity could be effectively turned on andoff by looking at different circuits with only slightly different parameters.Eq. B12 sets the desired circuit parameters in order to achieve significant nonreciprocity, which for the optome-chanical coupling, optical and mechanical Q -factors, and the power handling capabilities of the nanobeam cavitiesrequires optical hopping rate between cavities to be less than J/ π ≈
500 MHz. Devices with larger coupling rates cansimply not be pumped hard enough to satisfy G k ≈ ( Jγ ik ) / . To confirm this, here we show another optomechanicalcrystal circuit with bare cavity wavelengths of λ L ( R ) = 1535 .
051 (1535 . J/ π = 1 . λ p = 1534 .
99 nm and synthetic flux of Φ B = π/