Mechanical On-Chip Microwave Circulator
S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. B. Dieterle, O. Painter, J. M. Fink
MMechanical On-Chip Microwave Circulator
S. Barzanjeh, ∗ M. Wulf, M. Peruzzo, M. Kalaee,
2, 3
P. B. Dieterle,
2, 3
O. Painter,
2, 3 and J. M. Fink † Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics,California Institute of Technology, Pasadena, CA 91125, USA Institute for Quantum Information and Matter,California Institute of Technology, Pasadena, CA 91125, USA (Dated: June 7, 2017)
Nonreciprocal circuit elements form an inte-gral part of modern measurement and commu-nication systems. Mathematically they requirebreaking of time-reversal symmetry, typicallyachieved using magnetic materials [1] and morerecently using the quantum Hall effect [2], para-metric permittivity modulation [3] or Josephsonnonlinearities [4, 5]. Here, we demonstrate anon-chip magnetic-free circulator based on reser-voir engineered optomechanical interactions [6,7]. Directional circulation is achieved with con-trolled phase-sensitive interference of six distinctelectro-mechanical signal conversion paths. Thepresented circulator is compact, its silicon-on-insulator platform is compatible with both su-perconducting qubits [8] and silicon photonics,and its noise performance is close to the quan-tum limit. With a high dynamic range, a tun-able bandwidth of up to 30 MHz and an in-situreconfigurability as beam splitter or wavelengthconverter [9, 10], it could pave the way for su-perconducting qubit processors with multiplexedon-chip signal processing and readout.
Nonreciprocal devices are quintessential tools to sup-press spurious modes, interferences and unwanted signalpaths. More generally, circulators can be used to realizechiral networks [11] in systems where directional matter-light coupling is not easily accessible. In circuit quantumelectrodynamics circulators are used for single port cou-pling or as isolators to protect the vulnerable cavity andqubit states from electromagnetic noise. State of the artpassive microwave circulators are based on magneto-opticeffects which require sizable magnetic fields incompati-ble with ultra-low loss superconducting circuits forminga major roadblock towards a fully integrated quantumprocessor based on superconducting qubits.Many recent theoretical and experimental efforts havebeen devoted to overcome these limitations both in theoptical [12–14] and microwave regimes [2–5, 15–18]. Inparallel, the rapidly growing field of optomechanical andelectromechanical systems has shown promising potentialfor applications in quantum information processing andcommunication, in particular for microwave to optical ∗ [email protected] † jfi[email protected] conversion [19, 20] and amplification [21]. Very recently,several theoretical proposals [6, 22, 23] have pointed outthat optomechanical systems can lead to nonreciprocityand first isolators have just been demonstrated in theoptical domain [24–26]. Here, we present an on-chip mi-crowave circulator using a new and tunable silicon elec-tromechanical system.The main elements of the microchip circulator deviceare shown in Fig. 1 a-b. The circuit is comprised ofthree high-impedance spiral inductors ( L i ) capacitivelycoupled to the in-plane vibrational modes of a dielectricnanostring mechanical resonator. The nanostring oscilla-tor consists of two thin silicon beams that are connectedby two symmetric tethers and fabricated from a high re-sistivity silicon-on-insulator device layer [27]. Four alu-minum electrodes are aligned and evaporated on top ofthe two nanostrings, forming one half of the vacuum gapcapacitors that are coupled to three microwave resonatorsand one DC voltage bias line as shown schematically inFig. 1c (see App. A for details).The voltage bias line can be used to generate an attrac-tive force which pulls the nanobeam and tunes the oper-ating point frequencies of the device [9]. Fig. 1 d showsthe measured resonance frequency change as a functionof the applied bias voltage V dc . As expected, resonators1 and 3 are tuned to higher frequency due to an increasedvacuum gap while resonator 2 is tuned to lower frequency.A large tunable bandwidth of up to 30 MHz as obtainedfor resonator 2, the ability to excite the motion directlyand to modulate the electromechanical coupling in-siturepresents an important step towards new optomechani-cal experiments and more practicable on-chip reciprocaland nonreciprocal devices.As a first step we carefully calibrate and character-ize the individual electromechanical couplings and noiseproperties. We then measure the bidirectional frequencyconversion between two microwave resonator modes asmediated by one mechanical mode [10]. The incomingsignal photons can also be distributed to two ports withvarying probability as a function of the parametric drivestrength and in direct analogy to a tunable beam split-ter. We present the experimental results, the relevantsample parameters and the theoretical analysis of thisbidirectional frequency conversion process in App. B.Directionality is achieved by engaging the second me-chanical mode, a method which was developed in paral-lel to this work [28, 29] for demonstrating nonreciproc- a r X i v : . [ qu a n t - ph ] J un Resonator 1Nanostring Port 1 Port 2 V dc (V)- 4 - 2 0 2 411.3811.379.589.599.609.779.789.799.80 F r e q u e n c y ( G H z ) C s,1 C m,1 C m,3 C m,2 C m L C s,3 L C s,2 L a a port 1 a a port 2a a V dc mode 2mode 1 a dc b V dc ¹ m100 ¹ mResonator 3Resonator 2 Resonator 1Resonator 2Resonator 3 FIG. 1.
Microchip circulator and tunability. a , Scan-ning electron micrograph of the electromechanical device in-cluding three microwave resonators, two physical ports, onevoltage bias input ( V dc ) and an inset of the spiral inductorcross-overs (green dashed boxed area). b , Enlarged view ofthe silicon nanostring mechanical oscillator with four vacuum-gap capacitors coupled to the three inductors and one voltagebias. Insets show details of the nanobeam as indicated by thedashed and dotted rectangles. c , Electrode design and elec-trical circuit diagram of the device. The input modes a i,in couple inductively to the microwave resonators with induc-tances L i , coil capacitances C i , additional stray capacitances C s,i , and the motional capacitances C m,i . The reflected tones a i,out pass through a separate chain of amplifiers each, and aremeasured at room temperature using a phase locked spectrumanalyzer (not shown). The simulated displacement of the low-est frequency in-plane flexural modes of the nanostring areshown in the two insets. d , Resonator reflection measurementof the three microwave resonators of an identical device, as afunction of the applied bias voltage and a fit (dashed lines)to ∆ ω = α V + α V with the tunabilties α / π = 0 . and α / π = 0 .
05 MHz/V with a total tunablebandwidth of 30 MHz for resonator 2 at 9.8 GHz. ity in single-port electromechanical systems. We beginwith the theoretical model describing two microwave cav-ities with resonance frequencies ω i and total linewidths κ i with i = 1 , ω m,j and damping rates γ m,j with j = 1 ,
2. To es-tablish the parametric coupling, we apply four microwavetones, with frequencies detuned by δ j from the lower mo-tional sidebands of the resonances, as shown in Fig. 2a.In a reference frame rotating at the frequencies ω i and ω m,j + δ j , the linearized Hamiltonian in the resolved side-band regime ( ω m,j (cid:29) κ , κ ) is given by (¯ h = 1) H = − (cid:88) j =1 , δ j b † j b j + (cid:88) i,j =1 , G ij (cid:16) e iφ ij a i b † j + e − iφ ij a † i b j (cid:17) + H off , (1)where a i ( b j ) is the annihilation operator for the cavity i (mechanics j ), G ij = g ,ij √ n ij and g ij are the ef-fective and vacuum electromechanical coupling rates be-tween the mechanical mode j and cavity i respectively,while n ij is the total number of photons inside the cavity i due to the drive with detuning ∆ ij , and φ ij is the rel-ative phase set by drives. Here, ∆ = ∆ = ω m, + δ and ∆ = ∆ = ω m, + δ are the detunings of thedrive tones with respect to the cavities and H off de-scribes the time dependent coupling of the mechanicalmodes to the cavity fields due to the off-resonant drivetones. These additional coupling terms create cross-damping [30] and renormalize the mechanical modes, andcan only be neglected in the weak coupling regime for G ij , κ j (cid:28) ω j , | ω m, − ω m, | .To see how the nonreciprocity arises we use the quan-tum Langevin equations of motion along with the input-output theorem to express the scattering matrix S ij ofthe system described by the Hamiltonian (1), and relat-ing the input photons a in ,i ( ω i ) at port i to the outputphotons a out ,j ( ω j ) at port j via a out ,i = (cid:80) j =1 , S ij a in ,i with i = 1 ,
2. The dynamics of the four-mode sys-tem described by Hamiltonian (1) is fully captured bya set of linear equations of motion as verified in App. C.Solving these equations in the frequency domain, us-ing the input-output relations, and setting φ = φ,φ = φ = φ = 0, the ratio of backward to forwardtransmission reads λ := S ( ω ) S ( ω ) = √ C C Σ m, ( ω ) + √ C C Σ m, ( ω ) e iφ √ C C Σ m, ( ω ) + √ C C Σ m, ( ω ) e − iφ . (2)Here, Σ m,j = 1 + 2 i (cid:2) ( − j δ − ω (cid:3) /γ m,j is the inverse ofthe mechanical susceptibility divided by the mechanicallinewidth γ m,j and C ij = 4 G ij / ( κ i γ m,j ) is the optome-chanical cooperativity. Note that, in Eq. (2) we assumethe device satisfies the impedance matching condition onresonance i.e. S ii ( ω = 0) = 0 which can be achieved inthe high cooperativity limit ( C ij (cid:29) φ and the de-tuning δ to obtain nonreciprocal transmission. When thecooperativities for all four optomechanical couplings areequal ( C ij = C ) then perfect isolation, i.e. λ = 0, occursfor tan[ φ ( ω )] = δ ( γ m, + γ m, ) + ω ( γ m, − γ m, ) γ m, γ m, / − δ − ω ) . (3)Equation 3 shows that on resonance ( ω = 0) tan[ φ ] ∝ δ ,highlighting the importance of the detuning δ to obtainnonreciprocity. Tuning all four drives to the exact redsideband frequencies ( δ = 0) results in bidirectional be-havior ( λ = 1). At the optimum phase φ given byEq. (3), ω = 0, and for two mechanical modes with iden-tical decay rates ( γ m, = γ m, = γ ) the transmission inforward direction is given by S = −√ η η i δ (1 − iδ/γ ) C γ (cid:16) δ /γ C (cid:17) (4) dB Phase = -102.6 C C C Phase = +102.6 C C C Data DataTheory TheoryC M M C abc G G a a a a G G P S D ± + ! m,1 ! d,11 ± + ! m,2 ! d,12 ± + ! m,1 ! d,21 ± + ! m,2 ! d,22 ! S ≠ S T r a n s m i ss i o n ( d B ) Probe detuning (Hz) P u m p ph a s e ( d e g ) -40-700-180-90090180-180-90090180 -350 350 7000 -700 -350 350 7000-30-20-100 j S j j S j j S j j S j j S j j S j j S j j S j FIG. 2.
Optomechanical isolator. a , Mode coupling dia-gram for optomechanically induced nonreciprocity. Two mi-crowave cavities ( C and C ) are coupled to two mechanicalmodes ( M and M ) with the optomechanical coupling rates G ij (where i, j = 1 , δ i from the lowermotional sidebands of the resonances. All four pumps arephase-locked while the signal tone is applied. Only one ofthe microwave source phases is varied to find the optimal in-terference condition for directional transmission between port1 and 2. b , Measured power transmission (dots) in forward | S | (cavity 1 → cavity 2) and backward directions | S | (cavity 2 → cavity 1) as a function of probe detuning for twodifferent phases φ = ± . c , Experimental data (top) and theoretical model (bot-tom) of measured transmission coefficients | S | and | S | asa function of signal detuning and pump phase φ . Dashed-linesindicate the line plot locations of panel b. where η = κ ext , /κ is the resonator coupling ra-tio and κ i = κ int ,i + κ ext ,i is the total damping rate.Here κ int ,i denotes the internal loss rate and κ ext ,i theloss rate due to the cavity to waveguide coupling. Equa-tion (4) shows that the maximum of the transmission inforward direction, | S | = η η [1 − (2 C ) − ], occurs when2 C = 1 + 4 δ /γ and for large cooperativities C (cid:29) ω , ω ) / π = (9 . , .
82) GHz coupled totwo different physical waveguide ports and measurementlines with ( η , η ) = (0 . , . ω m, , ω m, ) / π = (4 . , .
64) MHzwith intrinsic damping rates ( γ m, , γ m, ) / π = (4 ,
8) Hz.The vacuum optomechanical coupling strengths for thesemode combinations are ( g , , g , , g , , g , ) / π =(33 , , ,
31) Hz.Figure 2 b shows the measured transmission of thewavelength conversion in the forward | S | and back-ward directions | S | as a function of probe detuningfor two different phases as set by one out of the fourphase locked microwave drives. At φ = − . . . . φ = 102 . φ , which are symmetric andbidirectional around φ = 0. We find excellent agreementwith theory over the full range of measured phases withless than 10% deviation to independently calibrated drivephoton numbers and without any other free parameters.For bidirectional wavelength conversion, higher coop-erativity enhances the bandwidth. In contrast, the band-width of the nonreciprocal conversion is independent ofcooperativity and set only by the intrinsic mechanicallinewidths γ m,i , which can be seen in Eq (2). This high-lights the fact that the isolation appears when the en-tire signal energy is dissipated in the mechanical environ-ment, a lossy bath that can be engineered effectively [7].In the present case it is the off-resonant coupling be-tween the resonators and the mechanical oscillator whichmodifies this bath. The applied drives create an effec-tive interaction between the mechanical modes, whereone mode acts as a reservoir for the other and vice versa.This changes both the damping rates and the eigenfre-quencies of the mechanical modes. It therefore increasesthe instantaneous bandwidth of the conversion and auto-matically introduces the needed detuning, which is fullytaken into account in the theory.The described two-port isolator can be extended to aneffective three-port device by parametrically coupling thethird microwave resonator capacitively to the dielectricnanostring, as shown in Fig. 1 a. The third resonator ata resonance frequency of ω / π = 11 .
30 GHz is coupledto the waveguide with η = 0 .
52 and to the two in-planemechanical modes with ( g , , g , ) / π = (22 ,
45) Hz.Similar to the isolator, we establish a parametric cou- abc T r a n s m i ss i o n ( d B ) -40-30-20-100 P u m p ph a s e ( d e g ) -180-90090180-180-90090180 Probe detuning (Hz)-700 -350 350 7000 -700 -350 350 7000 -700 -350 350 7000Data Data DataTheory Theory Theory j S j j S j j S j j S j j S j j S j Phase = -54 Phase = -54 Phase = -54 C C C C C C C C C j S j j S j j S j j S j j S j j S j C M M C G G G G G G a a a a C a a dB FIG. 3.
Optomechanical circulator. a , Mode coupling di-agram describing the coupling between three microwave cav-ities ( C , C and C ) and two mechanical modes ( M and M ) with optomechanical coupling rates G ij (where i = 1 , , j = 1 , b , Measured power transmis-sion (dots) in forward ( | S | , | S | and | S | ) and backwarddirections ( | S | , | S | and | S | ) as a function of probe de-tuning for a pump phase φ = −
54 degrees. The solid linesshow the prediction of the coupled-mode theory model dis-cussed in the text. c , Measured S parameters (top) and the-oretical model (bottom) as a function of detuning and pumpphase. Dashed-lines indicate the line plot positions shown inpanel b . pling between cavity and mechanical modes using sixmicrowave pumps with frequencies slightly detuned fromthe lower motional sidebands of the resonances, which forcertain pump phase combinations can operate as a three-port circulator for microwave photons, see Fig 3 a. Usingan extra microwave source as probe signal, we measure the power transmission between all ports and directionsas shown in Fig. 3 b for a single fixed phase of φ = − S , , with an insertion loss of (3 . .
8, 4 .
4) dB and an isolation in the backward direction S , , of up to (18 .
5, 23, 23) dB. The full dependence ofthe circulator scattering parameters on the drive phaseis shown in Fig. 3 c where we see excellent agreementwith theory. The added noise photon number of the de-vice is found to be ( n add , , n add , , n add , ) = (4 , . , . n add , , n add , , n add , ) =(4 , , .
5) in the backward direction, limited by the ther-mal occupation of the mechanical modes and discussedin more detail in App. D.In conclusion, we realized a frequency tunable mi-crowave isolator / circulator that is highly directional andoperates with low loss and added noise. Improvementsof the circuit properties will help increase the instanta-neous bandwidth and further decrease the transmissionlosses of the device. The external voltage bias offers newways to achieve directional amplification and squeezingof microwave fields in the near future. Direct integrationwith superconducting qubits should allow for on-chip sin-gle photon routing as a starting point for more compactcircuit QED experiments.
Acknowledgements
We thank Nikolaj Kuntner forthe development of the Python virtual instrument paneland Georg Arnold for supplementary device simulations.This work was supported by IST Austria and the Eu-ropean Union’s Horizon 2020 research and innovationprogram under grant agreement No 732894 (FET Proac-tive HOT). SB acknowledges support from the EuropeanUnion’s Horizon 2020 research and innovation programunder the Marie Sklodowska Curie grant agreement No707438 (MSC-IF SUPEREOM).
Author contributions
SB and JMF conceived the ideasfor the experiment. SB developed the theoretical model,performed and analyzed the measurements. SB, MW,MP and JMF designed the microwave circuit and builtthe experimental setup. MK, PD, JMF and OP designedthe mechanical nanobeam oscillator. JMF and MK fab-ricated the sample. PD and OP contributed to samplefabrication. SB and JMF wrote the manuscript. JMFsupervised the research.
Additional information
Correspondence and requestsfor materials should be addressed to SB and JMF.
Competing financial interests
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I. SUPPLEMENTARY INFORMATIONAppendix A: Circuit properties
The electromechanical microwave circuit shown in Fig. 1 a, includes three high-impedance microwave spiral in-ductors ( L i ) capacitively coupled to the in-plane vibrational modes of a dielectric nanostring mechanical resonator,creating three LC resonators with frequencies ω i = 1 / √ L i C i with i = 1 , ,
3. The nanostring resonator fabricatedfrom a high resistivity smart-cut silicon-on-insulator wafer with 220 nm device layer thickness has a length of 9 . µ mand consists of two metalized beams that are connected with two tethers at their ends. The vacuum gap size for themechanically compliant capacitor fabricated with an inverse shadow technique [31] is approximately 60 nm.The electromechanical coupling between the nanostring mechanical resonator and each LC circuit is given by g i = x zpf ∂ω i ∂v = − x zpf ζ i ω i C m,i ∂C m,i ∂v , (A1)where v is the amplitude coordinate of the in-plane mode, ζ i = C m,i C Σ ,i is the participation ratio of the vacuum gapcapacitance C m,i to the total capacitance of the circuit C Σ ,i = C m,i + C s,i , where C s,i is the stray capacitance of thecircuit including the intrinsic self-capacitance of the inductor coils. Eq. (A1) indicates that large electromechanicalcoupling g i requires a large participation ratio. We can make the coil capacitance C L,i relatively small by using asuspended and tightly wound rectangular spiral inductor with a wire width of 500 nm and wire-to-wire pitch of 1 µ m[32]. Knowing the inductances L i of the fabricated inductors based on modified Wheeler, as well as the actuallymeasured resonance frequencies ω i along with vacuum-gap capacitance C m (from FEM simulations), we can findthe total stray capacitance including the intrinsic self-capacitance of the each inductor coil correspondingly. Carefulthermometry calibrated mechanical noise spectroscopy measurements similar to the ones in [32] yield the measuredelectromechanical coupling for each mode combination as outlined in the table below. ω π (GHz) κ int π (MHz) κ ex π (MHz) κ π (MHz) η = κ ex κ L (nH) C s (fF) C m (fF) g π (Hz) g π (Hz)Cavity 1 9.55 0.62 1.8 2.42 0.74 48.2 5.3 0.45 33 34Cavity 2 9.82 0.28 1.7 1.98 0.86 48.3 4.98 0.45 13 31Cavity 3 11.32 1.42 1.58 3 0.52 34.4 5.29 0.45 22 45We use finite-element method (FEM) numerical simulations to find the relevant in-plane mechanical modes of thestructure and optimize their zero point displacement amplitudes and mechanical quality factor. Our simulations areconsistent with the measured mechanical frequencies for a tensile stress of ∼
600 MP in a ∼
70 nm thick electron beamevaporated aluminum layer [33]. The associated effective mass and zero-point displacement amplitude along with themeasured linewidths and resonance frequencies of the first two in-plane modes of the nanostring are presented in thetable below. ω m π (MHz) γ m π (Hz) m eff (pg) x zpf (fm)first mechanical mode 4.34 4 4 22second mechanical mode 5.64 8 2.2 26 Appendix B: Bidirectional frequency conversion
To understand the optomechanical frequency conversion, we first theoretically model our system to see how fre-quency conversion arises. Figure 4 a shows an electromechanical system, in which two microwave cavities withresonance frequencies ω and ω and linewidths κ and κ are coupled to a mechanical oscillator with frequency ω m and damping rate γ . The electromechanical coupling is driven by two strong drive fields, E and E , near thered sideband of the respective microwave modes at ω d, = ω − ω m , see Fig. 4 b. In the resolved-sidebandlimit ( ω m (cid:29) κ , γ ) the linearized electromechanical Hamiltonian in the rotating frames with respect to the externaldriving fields is given by (¯ h = 1) H = (cid:88) i =1 , ∆ i a † i a i + ω m b † b + (cid:88) i =1 , G i (cid:16) a i b † + ba † i (cid:17) , (B1)where a is the annihilation operator for the microwave signal field 1 (microwave signal field 2), b is the annihilationoperator of the mechanical mode, ∆ = ω − ω d = ω m is the detuning between the external driving fieldand the relevant cavity resonance, and G i = g i √ n i is the effective electromechanical coupling rate between the C M C abc G a a a a G P S D ! m ! m ! m ! d,1 ! d,2 ! S » S C C = 95 j S j j S j j T j -1400.5 1 10 Signal (dBm) n
Signal -100 -80 j T j FIG. 4.
Bidirectional frequency conversion . a , The microwave-mechanical mode diagram for the frequency conversion.Two microwave cavities C and C are parametrically coupled to a mechanical mode with coupling rates G and G , whichgives rise to frequency conversion between the two microwave cavities. b , Power spectral densities (PSD) of the mechanicalmode and microwave cavities and the drive tone frequencies indicated with vertical arrows near the red sidebands of themicrowave modes at ω d, = ω − ω m . c , Experimental demonstration (dots) and theoretical prediction (solid lines) of thefrequency conversion between two microwave cavities at resonance frequencies ( ω , ω ) / π = (9 . , .
82) GHz as a function ofcooperativity C for C = 95. Here, | T | = | S | · | S | (yellow dots), | S | (red dots) and | S | (blue dots) demonstrate themagnitude of the transmission and reflection coefficients on resonance with the cavities, respectively. As predicted by Eq. (B2),the transmission between the two cavities is maximum for C = C ≈
95. The inset shows the dynamic range of the devicewhere the transmission coefficient is measured as function of the signal input power P signal or mean total number of signalphotons inside the microwave cavities n signal . mechanical resonator and cavity i with n i = E i κ i +4∆ i being the total number of photons inside the cavity. Notethat, the fast-oscillating counter-rotating terms at ± ω m are omitted from the Hamiltonian under the rotating waveapproximation.The first and second terms of Hamiltonian (B1) describe the free energy of the mechanical and cavity modes whilethe last term of the Hamiltonian indicates a beam splitter-like interaction between mechanical degree of freedomand microwave cavity modes. In fact this term allows both optomechanical cooling (with rate Γ i = 4 G i /κ i ) andbidirectional photon conversion between two distinct microwave frequencies. In the photon conversion process, firstan input microwave signal at frequency ω with amplitude a in , ( ω ) is down-converted into the mechanical mode atfrequency ω m , i.e. a ( ω ) H ∝ a b † −−−−−→ b ( ω m ). Next, during an up-conversion process the mechanical mode transfers itsenergy to the output of the other microwave cavity at frequency ω and amplitude a out , ( ω ), i.e. b ( ω m ) H ∝ ba † −−−−→ a ( ω ).Likewise, an input microwave signal at frequency ω can be converted to frequency ω by reversing the conversionprocess. In fact, the Hermitian aspect of the Hamiltonian (B1) makes the conversion process bidirectional and holdsthe time-reversal symmetry.The photon conversion efficiency, which is defined as the ratio of the output-signal photon flux over the input-signalphoton flux, is given by | S | = (cid:12)(cid:12)(cid:12) a out , ω a in , ω (cid:12)(cid:12)(cid:12) . Since the conversion process is bidirectional therefore | S | = | S | = | T | .In the steady state and in the weak coupling regime the conversion efficiency reduces to | T | = 4 η η C C (1 + C + C ) , (B2)where C = g , n κ γ m is the electromechanical cooperativity for cavity 1 (2) and η = κ ext , κ is the outputcoupling ratio in which κ i = κ int ,i + κ ext ,i is the total damping rate while κ int ,i and κ ext ,i show the intrinsic andextrinsic decay rate of the microwave cavities, respectively. Likewise, the reflection coefficients due to impedancemismatch are given by | S | = (cid:16) C + C − η (1 + C )1 + C + C (cid:17) , (B3) | S | = (cid:16) C + C − η (1 + C )1 + C + C (cid:17) . (B4)Note that for the lossless microwave cavities ( η i = 1), near unity photon conversion can be achieved in the limitthat C = C = C and C (cid:29)
1. The former condition balances the photon-phonon conversion rate for each cavitywhile the later condition guarantees the mechanical damping rate γ m is much weaker than the damping rates Γ i = γ m C i . Under these two conditions, the ideal photon conversion is achieved i.e. | T | = 1 (perfect transmission) and | S | = | S | = 0 (no reflection). The denominator of Eq. (B2) indicates that the bandwidth of the conversion isgiven by Γ T = γ m + Γ + Γ , which is the total back-action-damped linewidth of the mechanical resonator in thepresence of the two microwave drive fields.We perform coherent microwave frequency conversion using the intermediate nanostring resonator as a couplingelement between two superconducting coil resonators at ω / π = 9 .
55 GHz and ω / π = 9 .
82 GHz as shown inFig 1 a. The microwave cavities are accessible by ports”, i.e. semi-infinite transmission lines giving the modes finiteenergy decay rates leading to the cavity linewidths κ / π = 2 .
42 MHz and κ / π = 1 .
98 MHz with associatedoutput coupling ratios η = 0 .
74 and η = 0 .
86, indicating that both cavities are strongly overcoupled to the twodistinct physical ports 1 and 2. The fundamental mode of the mechanical oscillator has a resonance frequency of ω m / π = 4 . γ m / π = 4Hz. Measuring the mechanical resonator noisespectrum along with the off-resonant reflection coefficients of each cavity and measurement line, we calibrate the gainand attenuation in each input-output line and accurately back out the vacuum optomechanical coupling rate for eachcavity of g / π = 33Hz and g / π = 13Hz.Figure 4 c shows the measured scattering parameters | S | (red line), | S | (blue line), and | T | = | S | · | S | (yellow line) versus the electromechanical cooperativity C at C = 95. As predicted by Eq. (B2) at C = C (cid:39) | T | = 0 .
64, which is dominated by internal losses of the cavities limiting the maximumreachable conversion efficiency to | T | ≤ η η = 0 . −
80 dBm input signal power, corresponding to about 10 signalphotons inside the cavities. At even higher signal powers the transmission efficiency is degraded abruptly, becausethe probe tone acts as an additional strong drive invalidating the transducer model, and also because of an increaseof the resonance frequency shifts and resonator losses. Appendix C: General theory of a coupled electromechanical system1. Hamiltonian of a multi-mode electromechanical transducer
In this section we present a general theory to describe the nonreciprocal behavior of our on-chip electromechanicaltransducer, shown in Fig 1a of the main paper. We begin with an optomechanical system comprised of three microwavecavities with frequencies ω i and linewidths κ i where i = 1 , , ω m.i and damping rates γ m,i where i = 1 ,
2. To tune a desired coupling intoresonance, we assume the cavities are coherently driven with six microwave tones, with frequencies detuned from thelower motional sidebands of the resonances by δ ,i . The Hamiltonian of the system is (¯ h = 1)[34] H = (cid:88) i =1 ω i a † i a i + (cid:88) i =1 ω m,i b † i b i + (cid:88) i =1 2 (cid:88) j =1 g ,ij a † i a i ( b j + b † j ) + H d , (C1)where a i is the annihilation operator for the cavity i , b j is the annihilation operator of the mechanical mode j , and H d = (cid:88) i =1 2 (cid:88) j =1 E ij ( a i e i ( ω d,ij t + φ ij ) + a † i e − i ( ω d,ij t + φ ij ) ) , (C2)0describes the Hamiltonian of the pumps with amplitude E ij = E ∗ ij , frequency ω d,ij , and phase φ ij .We can linearize Hamiltonian (C1) by expanding the cavity modes around their steady-state field amplitudes, a i → a i − (cid:80) j =1 α ij e − iω d,ij t , where | α ij | = 4 |E ij e − iφ ij | / ( κ i + 4∆ ij ) is the mean number of photons inside the cavity i induced by the microwave pump due to driving mechanical mode j , the κ i = κ int ,i + κ ext ,i is the total damping rateof the cavity while κ int ,i and κ ext ,i show the intrinsic and extrinsic decay rate of the microwave cavities, respectively.Here, ∆ ij = ω i − ω d,ij is the detuning of the drive tone with respect to cavity i . In the rotating frame with respectto (cid:80) i =1 ω i a † i a i + (cid:80) i =1 ( ω m,i + δ ,i ) b † i b i , the linearized Hamiltonian becomes H = − (cid:88) i =1 δ ,i b † i b i + (cid:88) i =1 (cid:110)(cid:16) (cid:88) j =1 (cid:2) α ij e i ∆ ij t a † i + α ∗ ij e − i ∆ ij t a i (cid:3)(cid:17)(cid:16) (cid:88) j =1 g ,ij (cid:2) b j e − i ( ω m,j + δ ,j ) t + b † j e i ( ω m,j + δ ,j ) t (cid:3)(cid:17)(cid:111) . (C3)By setting the effective cavity detunings so that ∆ = ∆ = ∆ = ω m, + δ , and ∆ = ∆ = ∆ = ω m, + δ , and neglecting the terms rotating at ± ω m, and ω m, + ω m, , the above Hamiltonian reduces to H = − (cid:88) i =1 δ ,i b † i b i + (cid:88) i =1 2 (cid:88) j =1 (cid:16) G ij a † i b j + G ∗ ij a i b † j (cid:17) + H off . (C4)where G ij = g ,ij | α ij | e − iφ ij is the effective coupling rate between the mechanical mode j and cavity i and H off describes off-resonant/time dependent interaction between mechanical modes and the cavity fields, and it is given by H off = (cid:88) i =1 2 (cid:88) j =1 (cid:104) F ij a † i b j e ( − j − iδω m t + H.c. (cid:105) (C5)where δω m = ω m, − ω m, + δ , − δ , and we define following off-resonant optomechanical coupling parameters F = g , | α | e − iφ , F = g , | α | e − iφ ,F = g , | α | e − iφ , F = g , | α | e − iφ , (C6) F = g , | α | e − iφ , F = g , | α | e − iφ . The off-resonant Hamiltonian (C5) has an essential role in the nonreciprocity aspect of our device, therefore, itis important to discuss the physical roots of such off-resonant couplings [30, 35]. Inspection of Hamiltonians (C4)and (C5) reveals that each drive tone generates two different types of interactions: Resonant coupling in which thedrive tone couples a single mechanical mode to a single cavity mode, described by the time-independent part of theHamiltonian (C4). Each drive tone also generates an interaction which couples the other mechanical mode to thecavity off-resonantly. The Hamiltonians (C5) explain this off-resonant coupling between cavity fields and mechanicalmodes. As we will see, these off-resonant couplings alter the mechanical damping rate, which changes the isolationbandwidth and also cools the mechanical modes. In addition, the coupling also introduces mechanical frequency shiftsand introduces an effective detuning for the drive tones. Note that, within the rotating wave approximation (RWA)the non-resonant/time-dependent components of the effective linearized interactions can be neglected in the weakcoupling regime and when the cavity decay rates κ i are much smaller than the two mechanical frequencies ω m,i andtheir difference | F ij | , κ i (cid:28) ω m,j , | ω m, − ω m, | . (C7)Finally, we note that for the isolator case we deal with two cavities coupled two mechanical modes, which mathemat-ically is equivalent to set G = G = F = F = 0 in our general model. In this special case, the Hamiltonian (C4)reduces to the Hamiltonian (1) presented in the paper H = − (cid:88) i =1 δ ,i b † i b i + (cid:88) i,j =1 (cid:16) G ij a † i b j + G ∗ ij a i b † j (cid:17) + H off . (C8)with H off = (cid:88) i,j =1 (cid:104) F ij a † i b j e ( − j − iδω m t + H.c. (cid:105) . (C9)1
2. Equations of motion and effective model
The full quantum treatment of the system can be given in terms of the quantum Langevin equations where we addto the Heisenberg equations the quantum noise acting on the mechanical resonators b in ,i with damping rates γ i aswell as the cavities input fluctuations a in ,i with damping rates κ ext ,i . The resulting Langevin equations, including theoff-resonate terms, for the cavity modes and mechanical resonators are˙ a i = − κ i a i − i (cid:88) j =1 G ij b j − i (cid:88) j =1 F ij b j e ( − j − iδω m t + √ κ ext ,i a in ,i , (C10)˙ b j = (cid:16) iδ ,j − γ m,j (cid:17) b j − i (cid:88) i =1 G ∗ ij a i − i (cid:88) i =1 F ∗ ij a i e ( − j iδω m t + √ γ m,j b in ,j , where i = 1 , , j = 1 , O ( n δω m ; δω nm ) with n ≥
2, which yields˙ a i = − κ i a i − i (cid:88) j =1 G ij b j − i (cid:88) j =1 F ij B j + √ κ ext ,i a in ,i , ˙ b = (cid:16) iδ , − γ m, (cid:17) b − i (cid:88) i =1 G ∗ i a i − i (cid:88) i =1 F ∗ i A − i + √ γ m, b in , , ˙ b = (cid:16) iδ , − γ m, (cid:17) b − i (cid:88) i =1 G ∗ i a i − i (cid:88) i =1 F ∗ i A + i + √ γ m, b in , , (C11)˙ A + i = ( iδω m − κ i A + i − i (cid:16) F i b + G i B (cid:17) , ˙ A − i = − ( iδω m + κ i A − i − i (cid:16) F i b + G i B (cid:17) , ˙ B = (cid:16) i [ δω m + δ , ] − γ m, (cid:17) B − i (cid:88) i =1 (cid:16) F ∗ i a i + G ∗ i A + i (cid:17) , ˙ B = − (cid:16) i [ δω m − δ , ] + γ m, (cid:17) B − i (cid:88) i =1 (cid:16) F ∗ i a i + G ∗ i A − i (cid:17) , where i = 1 , ,
3. The auxiliary modes A ± i = a i e ± iδω m t , B = b e iδω m t and B = b e − iδω m t describe the off-resonantcomponents of the equations of motion. Here, we take δω m to be much larger than the relevant system frequencies, i.e. δω m (cid:29) γ m,i , δ ,i , ω , and can thus adiabatically eliminate the auxiliary modes by taking ˙ B j = ˙ A ± i = 0 in Eqs. (C11),which results in the following equations for the auxiliary modes A + i = i (cid:16) F i b + G i B (cid:17) ( iδω m − κ i ) ,A − i = − i (cid:16) F i b + G i B (cid:17) ( iδω m + κ i ) , (C12) B = i (cid:80) i =1 (cid:16) F ∗ i a i + G ∗ i A + i (cid:17)(cid:16) i [ δω m + δ , ] − γ m, (cid:17) ,B = − i (cid:80) i =1 (cid:16) F ∗ i a i + G ∗ i A − i (cid:17)(cid:16) i [ δω m − δ , ] + γ m, (cid:17) , In the limit of δω m → ∞ , the contribution of all auxiliary modes can be totally neglected in the dynamics of thesystem, i.e. { B j , A ± i } →
0. In this case the off-resonant interactions between the mechanical modes and cavities are2negligible and we can safely ignore the time-dependent components of the Hamiltonian (i.e. H off = 0). However, inour system due to finite value of δω m ≈ κ i /
2, we cannot ignore these off-resonant interactions.We can simply further the equations of motion for the main modes by substituting Eqs. (C12) into the equationsof motion for a i and b j in Eqs. (C11) and assuming δω m , κ i (cid:29) (cid:110) | δ ,j | , γ m,j , | G ij | , | F ij | (cid:111) ,˙ a i ≈ − κ i a i − i (cid:88) j =1 G ij b j + √ κ ext ,i a in ,i , ˙ b ≈ (cid:16) iδ − Γ m, (cid:17) b − i (cid:88) i =1 G ∗ i a i + √ γ m, b in , , (C13)˙ b ≈ (cid:16) iδ − Γ m, (cid:17) b − i (cid:88) i =1 G ∗ i a i + √ γ m, b in , , where δ j and Γ m,j are the effective detuning and damping rates of the mechanical modes, respectively, and they aregiven by δ = δ , + δω m (cid:88) i =1 | F i | δω m + κ i ,δ = δ , − δω m (cid:88) i =1 | F i | δω m + κ i , (C14)Γ m, = γ m, + (cid:88) i =1 κ i | F i | δω m + κ i , Γ m, = γ m, + (cid:88) i =1 κ i | F i | δω m + κ i . Note that in the derivation of Eqs. (C13) we assume that the off-resonant interaction does not considerably modify theself-interaction and damping rate of the cavity modes. Inspection of Eqs. (C13) reveals that the off-resonant couplingbetween mechanical modes and cavities shifts the resonance frequency and damps/cools the mechanical modes byintroducing a cross-damping between them. The strength of the frequency shift and the cross-damping is given bythe off-resonant optomechanical coupling parameters F ij , which indicates that the drive tones creates an effectivecoupling between the two mechanical modes. In the weak coupling regime and for very large δω m this cross-couplingis negligible, thus δ j ≈ δ ,j and Γ m,j ≈ γ m,j .We can solve the Eqs. (C13) in the Fourier domain to obtain the microwave cavities’ variables. Eliminating themechanical degrees of freedom from the equations of motion (C13) and writing the remaining equations in the matrixform, we obtain (cid:16) M − iω I (cid:17) a a a = √ κ ext , a in , − iG χ m, ( ω ) √ γ m, b in , − iG χ m, ( ω ) √ γ m, b in , √ κ ext , a in , − iG χ m, ( ω ) √ γ m, b in , − iG χ m, ( ω ) √ γ m, b in , √ κ ext , a in , − iG χ m, ( ω ) √ γ m, b in , − iG χ m, ( ω ) √ γ m, b in , , (C15)where χ − j ( ω ) = Γ m,j / − i ( ω + δ j ) is the mechanical susceptibility for mode j and we introduced the drift matrix M = κ + χ m, ( ω ) | G | + χ m, ( ω ) | G | χ m, ( ω ) G G ∗ + χ m, ( ω ) G G ∗ χ m, ( ω ) G G ∗ + χ m, ( ω ) G G ∗ χ m, ( ω ) G ∗ G + χ m, ( ω ) G ∗ G κ + χ m, ( ω ) | G | + χ m, ( ω ) | G | χ m, ( ω ) G ∗ G + χ m, ( ω ) G ∗ G χ m, ( ω ) G ∗ G + χ m, ( ω ) G ∗ G χ m, ( ω ) G ∗ G + χ m, ( ω ) G ∗ G κ + χ m, ( ω ) | G | + χ m, ( ω ) | G | . By substituting the solutions of Eq. (C15) into the corresponding input-output formula for the cavities variables,i.e. a out ,j = √ κ ext ,j a j − a in ,j , we obtain a out , a out , a out , = T . (cid:16) M − iω I (cid:17) − . √ κ ext , a in , − iG χ m, ( ω ) √ γ m, b in , − iG χ m, ( ω ) √ γ m, b in , √ κ ext , a in , − iG χ m, ( ω ) √ γ m, b in , − iG χ m, ( ω ) √ γ m, b in , √ κ ext , a in, − iG χ m, ( ω ) √ γ m, b in , − iG χ m, ( ω ) √ γ m, b in , − a in , a in , a in , , (C16)where we defined T = Diag (cid:2) √ κ ext , , √ κ ext , , √ κ ext , (cid:3) .3
3. Scattering matrix and nonreciprocity for a two-port device
In this section, we verify the details of our analysis in the isolator section of the main paper and we examine ourmodel to see how the nonreciprocity arises in a two-port electromechanical system. Here, we are only interested inthe response an electromechanical system comprised of two microwave cavities and two mechanical modes. Therefore,by setting G j → δ = − δ = δ in Eq. (C16) and assuming φ = φ, φ = φ = φ = 0, we can find the ratioof backward to forward transmission λ := S ( ω ) S ( ω ) = √ C C Σ m, ( ω ) + √ C C Σ m, ( ω ) e iφ √ C C Σ m, ( ω ) + √ C C Σ m, ( ω ) e − iφ , (C17)as specified in Eq. (2) of the paper. Here, Σ m,j = 1 + 2 i (cid:2) ( − j δ − ω (cid:3) / Γ m,j is the inverse of the mechanical suscepti-bility divided by the effective mechanical linewidth Γ m,j . Examination of Eq. (C17) shows that the nominator anddenominator of this equation are not equal and they possess different relative phase. This asymmetry is the mainsource of the nonreciprocity and appearance of isolation in the system. In particular, at a phase e iφ = − (cid:114) C C C C Σ m, ( ω )Σ m, ( ω ) , (C18)the nominator of the Eq. (C17) will be zero, therefore, backward transmission S is canceled while forward trans-mission S is non-zero. Rewriting Eq. (C18) givestan[ φ ( ω )] = δ (Γ m, + Γ m, ) + ω (Γ m, − Γ m, )Γ m, Γ m, / − δ − ω ) . (C19)By neglecting the contribution of the off-resonant term in the response of the system, i.e. Γ m,j → γ m,j the Eq. (C19)reduces to Eq. (3) of the paper. At the optimum phase (C18) and at cavity resonance, the transmission in the forwarddirection is given by S = − √ η η (cid:2) Σ m, (0)Σ m, (0) (cid:3)(cid:0) √ C C Σ m, (0) + √ C C Σ m, (0) e − iφ (cid:1)(cid:2) C Σ m, (0) + C Σ m, (0) + Σ m, (0)Σ m, (0) (cid:3)(cid:2) C Σ m, (0) + C Σ m, (0) + Σ m, (0)Σ m, (0) (cid:3) . For equal mechanical damping Γ m, = Γ m, = Γ (equivalent to γ m, = γ m, = γ of the main text) and at equalcooperativities for all four optomechanical couplings ( C ij = C ) the above equation reduces to S = −√ η η (cid:104) i δ (1 − iδ/ Γ) C Γ(1 + δ / Γ C ) (cid:105) (C20)as specified in Eq. (4) of the paper. For the particular cooperativity 2 C = 1 + 4 δ / Γ , the power transmission inforward direction is given by | S | = η η (cid:16) − C (cid:17) . (C21)By neglecting the off-resonant interaction all damping rates reduce to Γ m,j ≈ γ m,j which is consistent with our notationin the main text. We also note that the frequency shifts due to off-resonant interaction for the isolator system discussedin the main text are given by ( δ , δ ) / π = ( − , m, , Γ m, ) / π =(190 ,
4. Theoretical model for the circulator
The theoretical model, we presented in Eqs. (C11), or equivalently Eq. (C16), fully describes the nonreciprocalbehavior of the system for the case of the circulator. In order to check this, in Fig. 5 we show both measuredexperimental data and the theoretical prediction. The theoretical model is in excellent agreement with the experimentand can perfectly describe the nonreciprocity of photon transmission for both forward and backward circulation.4 P u m p p h a s e ( d e g ) -180-90090180-180-90090180 Probe detuning (Hz)-700 -350 350 7000 -700 -350 350 7000 -700 -350 350 7000Data Data DataTheory Theory Theory|S | |S | |S | |S | |S | |S | dB P u m p p h a s e ( d e g ) -180-90090180-180-90090180 Theory Theory Theory|S | |S | |S | Data Data Data|S | |S | |S | dB ab FIG. 5.
Full scattering parameters of the circulator. a , Measured power transmission and theoretical model in forwarddirection ( | S | , | S | , and | S | ) as a function of detuning and pump phase. b , Measured power transmission and theoreticalmodel in backward direction ( | S | , | S | , and | S | ) as a function of detuning and pump phase. Appendix D: Added noise
In this section, we discuss the noise properties of the system and present data for the added noise during thefrequency conversion when operated as a circulator.Equation (C16) explains that due to the linear nature of the input-output theorem and in the absence of theinput coherent signal, the output of each cavity is a linear combination of the electromagnetic input noise a in ,i andmechanical noise b in ,j . Therefore, Eq. (C16) can be rewritten in the following general form a out ,i = (cid:88) j =1 S i,j a in , j + (cid:88) j =1 T i,j b in , j , (D1)where S i,j and T i,j are the scattering matrices. Operating under the white noise assumption, the zero-mean quantumfluctuations a in ,i and b in ,j satisfy the correlations (cid:104) O in ,i/j ( t ) O † in ,i/j ( t (cid:48) ) (cid:105) = ( ¯ N i/j + 1) δ ( t − t (cid:48) ), (cid:104) O † in ,i/j ( t ) O in ,i/j ( t (cid:48) ) (cid:105) =¯ N i/j δ ( t − t (cid:48) ), and (cid:104) O in ,i/j ( t ) O in ,i/j ( t (cid:48) ) (cid:105) = 0 where i = 1 , , O = a , and j = 1 , O = b ) and ¯ N i/j =1 (cid:14)(cid:8) exp (cid:2) ¯ hω i / ( k B T i ) (cid:3) − (cid:9) ( ¯ N m,j = 1 (cid:14)(cid:8) exp (cid:2) ¯ hω m,j / ( k B T j ) (cid:3) − (cid:9) ) are the thermal photon (phonon) occupancies of the5 Detuning from cavity(Hz) -700 -350 0
350 700 a dd e d n o i s e p h o t o n s S S S S S -700 -350 0
350 700 -700 -350 0
350 700 S background noise background noise background noise FIG. 6.
Added noise photons of the circulator.
Measured circulator noise properties in forward direction ( | S | , | S | ,and | S | in red) and backward direction ( | S | , | S | , and | S | in blue) as well as the measured background noise as afunction of detuning. cavities (mechanical resonator) for i = 1 , , j = 1 ,
2) at temperature T i . The output of the cavities are then sentthrough a chain of amplifiers. The electromagnetic modes at the output of the amplifiers are given by A out ,i = (cid:0)(cid:112) G i a out ,i + (cid:112) G i − c † amp ,i (cid:1) , (D2)where G i is the effective gain of the amplifier chain at port i and c amp ,i is the added noise operator of the amplifiers.We can now write the expression for the single sided power spectral density as measured by a spectrum analyzer, inthe presence of all relevant noise sources S noise ,i ( ω ) = ¯ hω (cid:90) ∞−∞ dω (cid:48) (cid:104) A † out ,i ( ω (cid:48) ) A out ,i ( ω (cid:48) ) (cid:105) . (D3)Substituting Eqs. (D1) and (D2) into Eq. (D3), assuming G i ≈ G i − G i / where G i is the gain in dB, andusing the white correlation functions for the noise operators, we find S noise ,i ( ω ) = ¯ hω G i / (1 + n amp ,i + n add ,ij ) , (D4)where n amp ,ij is the total noise added by the amplifier chains and n add ,i is the total noise added by the cavities andmechanical resonators associated with the photon conversion from cavity j to cavity i .Measuring the output noise spectrum and having calibrated the gain of the amplifiers at each port ( G , G , G ) =(67 . , , .
5) dB, we can accurately infer the amplifiers added noise quanta at each port ( n amp , , n amp , , n amp , ) =(23 , , ±
2. The only remaining unknown parameter in Eq. (D4) is n add ,ij which can be found by measuring thenoise properties of the three cavities when all six pumps are on and compare them to the case when the pumps areoff. In the Fig. 6 we show the measured added noise photons for all six transmission parameters of the circulator.On resonance where the directionality is maximized we find ( n add , , n add , , n add , ) = (4 , . , .
6) in the forwarddirection and ( n add , , n add , , n add , ) = (4 , , ..