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, Volume 7 (3) – Jul 1, 2017

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- The American Physical Society
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- Copyright © Published by the American Physical Society
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- 2160-3308
- D.O.I.
- 10.1103/PhysRevX.7.031001
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PHYSICAL REVIEW X 7, 031001 (2017) G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA (Received 22 March 2017; revised manuscript received 2 May 2017; published 6 July 2017) The ability to engineer nonreciprocal interactions is an essential tool in modern communication technology as well as a powerful resource for building quantum networks. Aside from large reverse isolation, a nonreciprocal device suitable for applications must also have high efficiency (low insertion loss) and low output noise. Recent theoretical and experimental studies have shown that nonreciprocal behavior can be achieved in optomechanical systems, but performance in these last two attributes has been limited. Here, we demonstrate an efficient, frequency-converting microwave isolator based on the optomechanical interactions between electromagnetic fields and a mechanically compliant vacuum-gap capacitor. We achieve simultaneous reverse isolation of more than 20 dB and insertion loss less than 1.5 dB. We characterize the nonreciprocal noise performance of the device, observing that the residual thermal noise from the mechanical environments is routed solely to the input of the isolator. Our measurements show quantitative agreement with a general coupled-mode theory. Unlike conventional isolators and circulators, these compact nonreciprocal devices do not require a static magnetic field, and they allow for dynamic control of the direction of isolation. With these advantages, similar devices could enable programmable, high-efficiency connections between disparate nodes of quantum networks, even efficiently bridging the microwave and optical domains. DOI: 10.1103/PhysRevX.7.031001 Subject Areas: Acoustics, Condensed Matter Physics, Quantum Physics Many branches of physics and engineering employ enabled much of the progress in classical and quantum nonreciprocal devices to route signals along desired paths signal processing, overcoming their limitations could of measurement networks. Conceptually, the simplest lead to exciting new developments in both areas. For nonreciprocal element is the isolator, a two-port device example, these components are typically bulky, not chip that transmits signals from the first to the second port but compatible, and incompatible with superconducting tech- strongly attenuates in the reverse direction [1]. Placing an nology because they require strong magnetic fields. Signal ideal isolator (or its close relative, the circulator) between losses due to these conventional nonreciprocal devices have now become the bottleneck for the overall efficiency of, for two systems allows the first system to influence the second example, state-of-the-art microwave measurements [6–8]. but not vice versa. This nonreciprocal functionality enables, In recent years, there has been interest in developing for example, telecommunication antennas to transmit and receive signals at the same time. Another example relevant nonmagnetic nonreciprocal devices to replace conventional for future applications is quantum signal processing, where isolators and overcome the limitations discussed above for the strict demands of quantum measurement require iso- superconducting microwave applications [9–15] as well as lators with high performance in several metrics, including limitations that arise in optical and room temperature not only large isolation, but also high efficiency and low isolation [16]. Schemes based on coupled-mode physics noise [2]. can break reciprocity without a static magnetic field Well-established technology uses magnetic materials to if the coupling is parametrically modulated in time [1]. achieve nonreciprocity for both microwave and optical Producing isolation further requires the coherent interfer- frequencies [3–5]. While these conventional devices have ence of two paths from one port to another, as well as a reservoir to absorb the backward-propagating power * [15,17]. These schemes are particularly promising because This article is a contribution of the U.S. Government, not they can naturally integrate with existing chip-based super- subject to U.S. copyright. conducting technology [10,11,17–20]. Corresponding author. One route for efficient parametric nonreciprocity in the john.teufel@nist.gov microwave domain is to use Josephson junctions to couple Published by the American Physical Society under the terms of superconducting circuits [13,20]. More recently, theoretical the Creative Commons Attribution 4.0 International license. proposals [15,17,21,22] and experiments [23–25] have Further distribution of this work must maintain attribution to begun exploring the parametric coupling between an the author(s) and the published article’s title, journal citation, and DOI. electromagnetic cavity and a mechanical oscillator as an 2160-3308=17=7(3)=031001(10) 031001-1 Published by the American Physical Society G. A. PETERSON et al. PHYS. REV. X 7, 031001 (2017) alternative mode-coupling mechanism for nonreciprocity. nonreciprocal frequency conversion between the two These optomechanical systems are attractive because of cavities in Fig. 1(a). their wide applicability beyond microwave frequencies and To understand the optomechanical isolator, we begin cryogenic environments. For example, efficient, reciprocal with the fundamental parametric interaction between an frequency conversion using optomechanics has already electromagnetic cavity and a mechanical oscillator [30].A been demonstrated in both the microwave [26] and optical general multimode cavity-optomechanical system consists 27,28]] frequency bands, as well as in conversion between of a set of cavity resonances and mechanical modes. Consider a given cavity mode j with resonant frequency the two [29]. Nonreciprocal optomechanical devices, how- ω and linewidth κ and a given mechanical mode k with ever, have yet to show the efficiencies and noise properties j j needed for most applications. frequency Ω and intrinsic linewidth Γ , obeying k k Combining two independent optomechanical frequency Γ ≪ κ < Ω ≪ ω . The position of the mechanical oscil- k j k j converters gives a natural way to achieve the interference lator tunes the cavity frequency, providing the mechanism needed for nonreciprocity. Here, we realize this interference of coupling. Analysis of the equations of motion for the by simultaneously coupling two electromagnetic cavity cavity and mechanical mode annihilation operators, a ˆ and modes to two distinct vibrational modes of a mechanical b , shows that a strong electromagnetic field (the drive) membrane. We illustrate this concept for achieving applied at a frequency near the red sideband (defined by ω ¼ ω − Ω ) induces an effective beam splitter jk j k interaction. The interaction Hamiltonian is ℏðg a ˆ b þ jk j g a ˆ b Þ, where ℏ is the reduced Planck constant, and jk j the coupling rate g is a complex number with phase and jk amplitude set by the drive. We parametrize the coupling strength in terms of the cooperativity C ¼ 4jg j =ðκ Γ Þ. jk jk j k Our optomechanical isolator is fully described by the general theory of linear coupled-mode systems [17,20,31]. In the quantum input-output formalism [32], each mode a ˆ couples to its environmental input and output operators a ˆ and a ˆ through the standard input-output boundary j;in j;out conditions. The scattering matrix elements are defined as the ratios of output to input field amplitudes, S ¼ jk ha ˆ i=ha ˆ i, where h·i indicates expectation value. j;out k;in Demonstrating an efficient isolator requires maximizing the forward transmission jS j while minimizing the jk reverse transmission jS j . kj We experimentally create a system consisting of two cavity modes and two mechanical modes by designing and fabricating a superconducting circuit of aluminum on a sapphire substrate [33–35], as shown and characterized in Figs. 1(b)–1(d). A vacuum-gap capacitor combined with an FIG. 1. Concept and experimental realization. (a) Mode-cou- inductive network defines two microwave cavities with pling diagrams for the optomechanical isolator. Optomechanical resonant frequencies ω =2π ¼ 6.528 GHz and ω =2π ¼ interactions (double-sided arrows) between two cavity modes (a ˆ 1 2 ˆ ˆ 6.733 GHz and linewidths κ =2π ¼ 1.3 MHz and and a ˆ ) and two mechanical modes (b and b ) induce directional 1 2 1 2 κ =2π ¼ 2.0 MHz. We design the cavities to be highly scattering between the two cavities when the parametric loop phase is equal to its optimal values ϕ . (b) Microscope images opt overcoupled so that the intentional inductive coupling rate of the device. A microfabricated vacuum-gap capacitor (inset) to the measurement line κ dominates the total dissipation ext resonates with spiral inductors to produce two electromagnetic rate of each cavity κ . The coupling efficiencies for tot cavities. (c) Schematic of the optomechanical circuit. Input each cavity, defined as η ≡ κ =κ , are measured to j j;ext j;tot signals from microwave generators couple inductively to the be η ≃ 0.99 and η ≃ 0.98. The vacuum-gap capacitor has 1 2 device and reflect back through the amplification chain to be a mechanically compliant top plate that vibrates with measured by a network or spectrum analyzer. (d) Frequency several spectrally distinct mode frequencies. In this experi- space diagram. Mode susceptibilities are plotted versus fre- ment, we use the two lowest-frequency vibrational modes quency. Two mechanical modes and two cavity modes are at Ω =2π ¼ 6.7 MHz and Ω =2π ¼ 9.4 MHz with intrin- characterized by their resonant frequencies (Ω and ω ) and 1 2 k j sic linewidths Γ =2π ¼ 15 Hz and Γ =2π ¼ 19 Hz, as their linewidths (Γ and κ ), and the cavities are further charac- k j 1 2 terized by their coupling efficiencies η . determined by independent measurements of the energy 031001-2 DEMONSTRATION OF EFFICIENT NONRECIPROCITY IN … PHYS. REV. X 7, 031001 (2017) dissipation rate. We place the device in a dilution cryostat In Fig. 2(c), we show the reciprocal transmission from with a base temperature of 19 mK and interrogate the one cavity to the other as a function of detuning from the circuit with signals routed from microwave generators and cavity center frequencies. We calibrate the scattering a vector network analyzer. From room temperature com- parameters using methods described previously [26,29]. ponents, input signals pass through attenuators, reflect off A drive power of approximately 1 nW damps mechanical the device at a circulator, and pass through a cryogenic mode 1 (left) to about 70 kHz and mode 2 (right) to 7 kHz. high-electron-mobility transistor amplifier, with more These damping rates are comparable to those we use later in amplification at room temperature. We operate the device the nonreciprocal scheme. We achieve transmission above as a single physical port measured in reflection; ports 1 and −0.6 dB through each mode, limited by cavity loss and 2 used hereafter refer to input or output signals near the drive strength imbalance. At our highest drive powers, the resonant frequencies of cavities 1 and 2. bandwidths of frequency conversion through the mechani- As reciprocal frequency conversion forms the basis for the cal modes reach 150 and 35 kHz. Our frequency converter optomechanical isolator, we first demonstrate this process operates in the high cooperativity limit, as evidenced by the through each mechanical mode (Fig. 2). In this scheme, one large ratios of damped mechanical linewidths to intrinsic microwave drive is applied at each cavity’s red sideband linewidths and the plateau in peak transmission versus with respect to a single mechanical mode; a signal entering input power, shown in Fig. 2(c). one cavity down-converts to the mechanical mode and then Now, to realize the optomechanical isolator, we up-converts to the other cavity [Figs. 2(a) and 2(b)]. drive two branches of mechanically mediated frequency conversion simultaneously. Figure 3(a) shows the FIG. 2. Reciprocal mechanically mediated frequency conver- sion. (a) Mode-connection diagrams. Double-sided arrows in- dicate driven optomechanical interactions. (b) Frequency-space diagrams. A red-detuned drive applied at each cavity induces frequency conversion through one mechanical mode. Dashed FIG. 3. Optomechanical isolation. (a) Frequency space dia- lines indicate frequencies of microwave drives. (c) Measured gram. Four drives (dashed lines) induce frequency conversion magnitude of reciprocal transmission from cavity 2 to cavity 1 as between the two cavities through both mechanical modes simul- a function of the probe detuning from cavity center for a taneously. (b) Measured magnitude of transmitted signal received particular drive power. Frequency conversion through the first at cavity 1 (left) and cavity 2 (right) for two choices of loop phase. mechanical mode is shown in red on the left and through the At ϕ ¼þ0.21π (solid blue) signals are transmitted from cavity 2 second mechanical mode in orange on the right. Solid lines are to cavity 1 and attenuated in the reverse direction. The behavior fits to Lorentzian line shapes. (d) Maximum transmission as a reverses at ϕ ¼ −0.21π (dashed green line). Solid lines are fits to function of total input drive power for the first (red) and second the expanded coupled-mode theory model described in the text. (orange) mechanical modes. Solid lines are fits to a model (c) Transmission (color scale) as a function of detuning and loop described in Ref. [26]. The arrow indicates the drive power phase. Lines show the source of data shown in (b). (d) Result of used in (c). the least-squares fit of the two-dimensional data in (c). 031001-3 G. A. PETERSON et al. PHYS. REV. X 7, 031001 (2017) frequency-space diagram of the experiment, with dashed In contrast to reciprocal frequency conversion, the lines indicating the frequencies of the four drives. Ideal mechanical dissipation plays a key role in the nonreciprocal isolation maximizes the magnitude of the transmission behavior of the device. This is a consequence of power 2 2 conservation; isolation can occur only if power entering a difference, defined as ΔT ¼jS j − jS j . The transmis- 12 21 cavity mode can be completely routed into the mechanical sion difference lies between −1 and 1, making it a useful environments. The mechanical modes are coupled to their metric because it simultaneously favors high reverse iso- environment with fixed rates Γ . So, while the bandwidth lation and low insertion loss, both important for quantum signals applications. Γ of reciprocal mechanically mediated frequency con- To achieve ideal isolation at the cavity resonances, the version increases with cooperativity as Γ ¼ Γ ð1 þ 2CÞ R j powers, frequencies, and relative phases of the four drives [26], the nonreciprocal bandwidth Γ for the isolating NR must be tuned to optimal values. Assuming the cavity system in the high-cooperativity limit is Γ ¼ 4Γ Γ = NR 1 2 linewidths are much larger than the mechanical mode ðΓ þ Γ Þ, involving only the intrinsic mechanical line- 1 2 linewidths and the optomechanical cooperativities are widths, independent of cooperativity. As we explore below, large, we can derive simple closed-form solutions for damping processes that occur outside the nonreciprocal the optimal drive parameters and the scattering matrix loop produce effective mechanical linewidths and, there- by analytically maximizing the function ΔT (see fore, allow the nonreciprocal bandwidth to increase. Appendix). First, the drive powers should be such that Before describing the data, it is necessary to include an the cooperativities for all four optomechanical couplings important deviation of our device from the simple system of are equal (let their shared value be C). The isolation four modes described thus far. Ideally, a given parametric performance increases with this cooperativity as ΔT ¼ drive couples a single mechanical mode to a single cavity −1 η η ½1 − ð2CÞ . The second condition sets the drive mode. In practice, however, this drive also couples the other 1 2 frequencies. One might expect that tuning the four drives mechanical mode to the cavity off-resonantly. This residual to the exact red sideband frequencies would be ideal. coupling damps and cools the mechanical modes. These In fact, this configuration leads to reciprocal behavior effects can be rigorously accounted for in the coupled precisely at the cavity center frequencies. Permitting equations of motion by expanding the mode basis to detuning of the drive pairs from the red sidebands include all interactions (see Appendix). Modeling these allows nonreciprocal transmission to occur on resonance processes as additional modes allows us to accurately map with the cavities. The optimal drive detunings are the experimental system to the simpler system of four pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ δ ¼ð−1Þ Γ 2C − 1=2, where δ is the detuning from modes with effective mechanical linewidths and effective j j j cooperativities. By damping the mechanical modes to the red sideband of the drives accessing the jth mechanical widths much larger than the intrinsic mechanical line- mode. The third important condition relates to the optimal widths, these off-resonant terms greatly enhance the band- relative drive phases. A signal traversing the loop in mode width and noise performance of the isolator, but they also space acquires a phase called the loop phase ϕ. Because the reduce the effective cooperativities attainable. Modeling the frequency conversion processes are parametric, this phase extra damping terms gives us a predictive theory with is related to the sum of the relative phases of the four drives, which to tune the device and arrive at ideal performance making it a dynamically tunable parameter. Under the parameters. assumptions mentioned above, the optimal value of the Figure 3(b) shows the measured transmission from cavity loop phase is ϕ ¼ arccosð1 − 1=CÞ. opt 2 to cavity 1 (left) and from 1 to 2 (right) at two loop phases After substituting these optimized drive parameters, and for a particular drive configuration found from the tuning further letting η ¼ η ¼ 1 and taking the large C limit, the 1 2 process. On cavity resonance at ϕ ¼þ0.21π (solid blue full scattering matrix becomes line), we see high transmission (insertion loss of 1.5 dB) 0 1 from cavity 2 to cavity 1 but low transmission (isolation 01 0 0 above 20 dB) from cavity 1 to 2 with a 3-dB bandwidth of B C 00 1=21=2 B C jSj ¼ ; ð1Þ 5kHz.At ϕ ¼ −0.21π (dashed green line), the behavior B C @ A 1=20 1=41=4 reverses. We collect data at many loop phases, shown in Fig. 3(c) with horizontal lines indicating the cuts shown in 1=20 1=41=4 Fig. 3(b). We fit the data to the expanded coupled-mode ˆ ˆ model using a two-dimensional nonlinear least-squares fit, where the mode basis is ordered ða ˆ ; a ˆ ; b ; b Þ. We see 1 2 1 2 the result of which is shown in Fig. 3(d), demonstrating that the upper left-hand corner defines the ideal 2 × 2 isolator, perfectly isolating cavity 2 from cavity 1. The excellent agreement with the data. Mapping our expanded other matrix elements describe scattering of signals input to model onto the four-mode system gives the effective the mechanical modes. At the opposite loop phase, the system parameters. The effective mechanical linewidths scattering matrix becomes the transpose of that shown are Γ =2π ¼ 1.6 kHz and Γ =2π ¼ 7.5 kHz, in agree- 1;eff 2;eff above, isolating cavity 1 from cavity 2. ment with the nonreciprocal bandwidth of 5 kHz. The 031001-4 DEMONSTRATION OF EFFICIENT NONRECIPROCITY IN … PHYS. REV. X 7, 031001 (2017) four effective cooperativities are ðC ;C ;C ;C Þ¼ 11 12 21 22 ð5.4; 5.7; 2.9; 2.0Þ, where the notation C indicates the jk cooperativity coupling cavity j to effective mechanical mode k. While the loop phases of ϕ ¼0.21π give good balance between the goals of high reverse isolation and low insertion loss, other loop phases can maximize these metrics individually. For the drive configuration shown here, the insertion loss can be as low as 1.16 dB (≈77% efficiency) at ϕ ¼0.35π at the expense of reducing reverse isolation to 9.2 dB. Alternatively, the reverse isolation can be tuned arbitrarily high near ϕ ¼0.11π at the expense of slightly increasing the insertion loss. In our system, we observe isolation at a single frequency as high 49 dB with corresponding insertion loss of 1.9 dB. An ideal isolator for applications to signal processing and quantum information would be both efficient and noiseless. To characterize the noise properties of the device while the four drives are on, we measure the noise spectrum at the cavity outputs. In Fig. 4(a), we show the signal flow diagrams corresponding to the ideal scattering matrix [Eq. (1)] at the two optimal loop phases. Importantly, the power input to the mechanical modes (namely, thermal noise) should appear at the isolated cavity but not the other cavity. The measured noise spectra shown in Fig. 4(b) demonstrate this behavior. At the loop phase that isolates cavity 1 from cavity 2 (near −0.21π in green), a noise peak of about 7 photons appears at cavity 1. The behavior reverses at the opposite loop phase. Data as a function of frequency and loop phase are shown in Fig. 4(c), with horizontal lines indicating the cuts used in Fig. 4(b). We fit the noise spectra to our expanded model using the FIG. 4. Noise performance of the optomechanical isolator. parameters determined from the driven response fit as fixed (a) Graphical representation of signal flow. The mode-connection inputs [Fig. 4(d)]. The only remaining free parameters are diagram (left) induces signal flow diagrams (right) at the optimal loop phases ϕ . Arrow widths are proportional to their the thermal occupation numbers of the two mechanical opt corresponding scattering matrix element [Eq. (1)]. (b) Measured environments, n and n . Equation (1) predicts the output 1 2 output noise at cavities 1 (left) and 2 (right) near loop phases noise of the isolated port to be the average of these two þ0.21π (solid blue line) and −0.21π (dashed green line). We occupation numbers. In our system, off-resonant inter- subtract constant noise offsets of 31.5 and 22.8 photons due to the actions naturally damp and cool the mechanical modes, measurement chain at the two cavity frequencies. (c) Output noise yielding lower effective occupation numbers of the envi- data (color scale) as a function of detuning from the cavity ronment n ¼ Γ n =Γ , measured to be n ¼ j;eff j j j;eff 1;eff frequencies and loop phase. Lines indicate the cuts shown in (a). 0.89 0.09 and n ¼ 12 1. The occupancies of the (d) Fit of the data in (c) to a coupled-mode theory with the 2;eff mechanical modes themselves depend on the loop phase, mechanical environment occupation numbers as free parameters. with their maxima and minima occurring at ϕ ¼ 0 and ϕ ¼ π, respectively. From the fit to the data in Fig. 4,we optomechanical resources. We derive closed-form expres- infer that these mechanical occupancies range from 0.13 to sions for the optimal drive conditions required for ideal 0.60 phonons in mode 1 and from 1.5 to 3.7 phonons in isolation and experimentally implement them in a micro- mode 2. Future implementations of the optomechanical wave optomechanical circuit. We fully characterize the isolator could reduce the output noise by starting with nonreciprocal performance of the device, both in the lower effective mechanical environment occupation num- scattering parameters and the output noise. Although recent bers, for example, by introducing additional beam splitter optomechanical experiments have demonstrated large rel- interactions to further damp and cool the mechanical modes ative contrast between forward and reverse transmission outside the nonreciprocal loop. [23–25], applications in signal processing and quantum The device we report here represents a significant information will also require simultaneously high effi- advancement of nonreciprocal technology using ciency. Our ability to reach high cooperativity combined 031001-5 G. A. PETERSON et al. PHYS. REV. X 7, 031001 (2017) with the use of an expanded coupled-mode model to fit the where Δ ¼ðω − ω Þ=γ þ i=2 is the normalized complex j j j j pﬃﬃﬃﬃﬃﬃﬃﬃ data and tune parameters has allowed us to greatly improve detuning of mode j, and β ¼ g = γ γ is the normalized jk jk j k the efficiency, isolation, and noise performance of an complex coupling strength between modes j and k. (Note optomechanical isolator, approaching the stringent require- that the definition of β differs from that in Ref. [20] by a ments of quantum information processing. In addition, the factor of 2 to coincide with the conventional definition of g quantitative agreement between data and theory we show in the optomechanics literature.) In our system, modes 1 here will be crucial for further optimizing performance and 2 are microwave cavities and modes 3 and 4 are within experimental constraints as well as developing more mechanical. The normalized magnitude of susceptibility for complex multimode systems. While we have pursued ideal mode j plotted in Fig. 1 is 1=jΔ j . To clarify the analytic isolation, which preserves quantum signals, the parameters results, we assume jβ j¼jβ j≡β and jβ j¼jβ j≡β ; 13 23 3 14 24 4 we demonstrate here are also well suited for implementing that is, each mechanical mode is equally coupled to both nonreciprocal amplification schemes [11,15,18,20,25,36]. iϕ cavity modes. We also put an explicit e on β for the loop Looking forward, the scheme we employ can be phase, so that the mode-coupling matrix becomes straightforwardly applied to other optomechanical sys- 0 1 iϕ tems, including those at optical frequencies. The addition Δ 0 β β e 1 3 4 of optomechanical systems to the nonreciprocal parametric B C 0 Δ β β B 2 3 4 C toolbox offers the new possibility to directionally route M ¼ B C: ðA2Þ @ A β β Δ 0 acoustic signals and could enable nonreciprocal micro- 3 3 3 −iϕ wave-to-optical transduction. Because the theory of the β e β 0 Δ 4 4 4 device applies generally beyond optomechanical systems, −1 The scattering matrix is found from S ¼ iHM H − 1, the nonreciprocal behavior we describe here could pﬃﬃﬃﬃ where H ¼ δ η . We require nonreciprocity to occur at jk jk j also be explored in other parametric systems including the cavity resonance frequencies. This demand lets us set microwave resonators coupled through Josephson junc- Δ ¼ Δ ¼ i=2. On resonance, the real parts of the tions. Parametric nonreciprocity is a promising and 1 2 mechanical detunings are equal to the detunings of the quickly developing field, which may soon enable previ- drives from the red sidebands: Δ ¼ δ þ i=2, where δ ously unattainable efficiencies for both measurement and 3;4 3;4 j is a normalized detuning such that the drive frequency is control of classical and quantum systems. ω ¼ ω − ω þ γ δ , for j ∈ f1; 2g and k ∈ f3; 4g. jk j k k k We first require impedance matching (S ¼ S ¼ 0)on 11 22 resonance. In the η ¼ 1 limit, impedance matching results ACKNOWLEDGMENTS j in the condition δ ¼ −δ and gives the optimal detuning as 3 4 Official contribution of the National Institute of Standards and Technology. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ δ ¼ −δ ¼ 2C C ð1 − cos ϕÞ − 1; ðA3Þ 3;opt 4;opt 3 4 Note added.—Recently, we became aware of another work where C ¼ 4β is the cooperativity associated with the j j using a similar method to demonstrate optomechanical optomechanical interaction involving mode j ∈ f3; 4g. nonreciprocity [37]. We parametrize isolation in the system by the trans- 2 2 mission difference ΔT ¼jS j − jS j . At the optimal 12 21 APPENDIX drive detuning, 1. General theory of a four-mode isolator pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4η η sinϕ 2C C ð1−cosϕÞ−1 1 2 3 4 We use the framework established in Ref. [17] with the ΔT ¼ : 2 2 2þð1−cosϕÞðC þC þ2C þ2C −2C C cosϕÞ notational conventions used in Ref. [20] to analyze a four- 3 4 3 4 3 4 mode isolator. We characterize each mode, regardless of its ðA4Þ physical manifestation, by a natural frequency ω , a (full width at half maximum) linewidth γ , and an input signal j Maximizing transmission difference over phase, we find pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ frequency ω . Modes j and k can be coupled with a the optimal loop phase ϕ ¼ arccosð1 − 1= C C Þ, with opt 3 4 complex coupling rate g . We describe the four-mode jk which the transmission difference becomes system by a mode-coupling matrix, pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 C C − 4 3 4 0 1 ΔT ¼ η η pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ : ðA5Þ 1 2 Δ 0 β β 2 2 1 13 14 ðC − C Þ þ 2ð C þ C Þ 3 4 3 4 B C 0 Δ β β B 2 23 24 C M ¼ B C; ðA1Þ At high cooperativity, maximizing this function yields @ A β β Δ 0 13 23 3 C ¼ C ≡ C with corrections at order 1=C, simplifying 3 4 −1 β β 0 Δ 14 24 the transmission difference to ΔT ¼ η η ½1 − ð2CÞ . 1 2 031001-6 DEMONSTRATION OF EFFICIENT NONRECIPROCITY IN … PHYS. REV. X 7, 031001 (2017) With these conditions applied, the scattering matrix at high diagram, like-colored arrows indicate interactions driven by the same microwave drive. Modes 1–4 are the four modes cooperativity and η ¼ 1 becomes appearing in the simplified four-mode model discussed 0 1 01 0 0 above. Modes 5–10 are auxiliary modes evaluated at the B C relevant off-resonant frequencies determined by the drives. 00 1=21=2 B C s s jSj ¼ : ðA6Þ B C For example, the signal frequency of mode 7 is ω ¼ ω − 7 1 @ A 1=20 1=41=4 s s ω þ ω , while that of mode 8 is ω ¼ ω − ω þ ω . 13 14 8 2 23 24 1=20 1=41=4 As our analysis takes place in the Fourier domain, each of these distinct coupled frequencies acts as another mode, Choosing ϕ ¼ −ϕ transposes the above matrix. opt even if it resides in the same physical oscillator as another To find the bandwidth of nonreciprocity, we calculate the mode. For this reason, modes 1, 7, and 9 share the transmission difference as a function of the detuning δω resonance frequency and linewidth of cavity 1, and likewise from the cavity centers with the approximation that the for modes 2, 8, and 10 in cavity 2 and for the mechanical cavity widths are much larger than the mechanical widths. mode pairs f3; 5g and f4; 6g. With the above optimizations for drive detunings, loop A note is needed to justify the presence of the off- phase, and cooperativities, the result is resonant mechanical modes 5 and 6. In general, these extra modes are needed to maintain common linewidths and −1=2 ΔTðωÞ¼ η η þ OðC Þ; ðA7Þ frequencies of all the auxiliary cavity modes. This effect is 1 2 2 2 γ þ 4ðδωÞ typically negligible in optomechanics because the cavity linewidths are so much larger than the mechanical line- where γ ¼ 4γ γ =ðγ þ γ Þ. The above shows that in the 3 4 3 4 widths. Another reason for including modes 5 and 6, high cooperativity limit, the bandwidth of nonreciprocity is however, is to be able to model the scattering parameters independent of cooperativity and equal to over wide spans that include both resonant and off-resonant γ γ 3 4 structure. We therefore include the off-resonant mechanical Γ ¼ 4 : ðA8Þ NR γ þ γ 3 4 terms to fit wide scans of scattering parameters. In total, these considerations lead to our system of ten 2. Off-resonant damping and expanded modes that quantitatively accounts for the off-resonant coupled-mode theory damping. Notably, we ignore all amplification processes occurring at the blue sidebands. This is a reasonable Because of the off-resonant coupling terms we discuss in approximation because the damping effects from these the main text, each mechanical mode can respond to all four 2 2 terms are smaller by a factor of κ =ð16Ω Þ < 1%. drives. To predict the effect of changing the drive powers We justify above the need for an expanded model and and frequencies, these extra interactions must be included show how to find the signal frequencies of the modes. The in the model. Expanding our mode basis allows us to fit the last part needed before calculating the scattering matrix is experimental data using the intrinsic mechanical properties the couplings involving the auxiliary modes. These are and also predict the needed drive parameters to obtain found by relating all 16 couplings to the four original optimal performance. couplings by multiplying by ratios of vacuum optome- The expanded mode basis needed, diagrammed in Fig. 5, chanical coupling rates and intrinsic mechanical linewidths. comes directly from the coupled equations of motion. In the With the mode-coupling matrix fully determined, we proceed to calculate the scattering matrix as above. We use Cavity 1: 7 1 9 this expanded model for the scattering parameters to fit the data shown in the main text and to predict the needed drive parameters to maximize the transmission difference func- tion. Figure 6 shows the data and theory fit for the full scattering matrix including the reflection coefficients. Mechanical modes: 6 3 4 5 The ten-mode graph can be reduced to obtain an effective four-mode graph. By allowing the inputs for the auxiliary modes to be exactly zero, one can derive the effective mode-coupling matrix describing the reduced system. This Cavity 2: 8 2 10 reduction procedure is a classical approximation, so care must taken in its application to quantum noise calculations. In general, to reduce mode k from the matrix M,we FIG. 5. Ten-mode graph diagram. Like-colored double-sided perform the transformation arrows indicate optomechanical coupling driven by the same microwave drive. Modes 1–4 are the four modes in the simplified M M ik kj four-mode model. Modes 5–10 are duplicates evaluated off M ¼ M − ; ðA9Þ ij ij kk resonance. 031001-7 G. A. PETERSON et al. PHYS. REV. X 7, 031001 (2017) FIG. 6. Full scattering matrix including transmission and reflection. Measured scattering parameters are shown on the top row and the results of a least-squares fit to our expanded coupled-mode theory are shown on the bottom row. which results in a new matrix M with one less dimension. X N ½ω¼ ℏωG 1 þ n þ jS j n ; ðA13Þ Reducing each auxiliary mode in turn results in the j j j;amp jk k;th k¼1 effective four-mode model. Incidentally, the mode reduc- tion formula encodes the meaning of the rotating wave where n ≥ 0 is the noise from the amplifier and j;amp approximation in Fourier space; if the correction to element n ≥ 0 is the thermal occupation number for the input k;th M is negligible for all signal frequencies of interest, the ij −1 field at port k. We measure N ½ω in units of W Hz . dynamics of mode k can be safely ignored. Knowing the system gain and added noise allows us to convert the spectrum to units of output photons from the 3. Calculation and calibration of output noise device. When the set of n (and possibly the n ) are k;th j;amp Here, we calculate a model for the output noise given the the only fit parameters, the model is linear and can therefore (10 × 10) scatting matrix calculated above. We start with be fit to the data using linear least-squares fitting methods. the system output amplitude, then model the amplifier The third term in the above equation is what we refer to as chain and the spectrum analyzer. the output noise of the device. We measure the amplifica- The output amplitude for mode j in terms of the N input tion noise at the two cavity frequencies to be n ¼ 1;amp amplitudes is 30 3 and n ¼ 22 2. 2;amp We calibrate the output noise by heating the cryostat to a ˆ ¼ S a ˆ : ðA10Þ j;out jk k;in 100 mK and measuring single-drive optomechanical spec- k¼1 tra [39]. 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Physical Review X – American Physical Society (APS)

**Published: ** Jul 1, 2017

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