doi:10.1038/nature10261 NOISE SPECTRUM OF AN OPTOMECHANICAL SYSTEM â in â out A mechanical degree of freedom that parametrically couples to the resonance frequency of the cavity modifies the power emerging from the cavity by scattering photons to the upper and lower mechanical sidebands [1]. We follow the general method of input-output theory to calculate the noise spectrum of the system [2, 3]. We start by considering the Hamiltonian of the optomechanical system [4]. Ĥ = ω c â â + Ω mˆb ˆb + g0 â âˆb + ˆb) S1) Here, ω c and Ω m are the resonance frequencies of the cavity and the mechanical modes, respectively. â â) and ˆb ˆb) are the creation annihilation) operators for cavity and the mechanical modes, respectively. The coupling between the modes is quantified by the single-photon coupling rate g 0 = Gx zp, where G = dω c /dx, x zp = /2mΩ m and m is the mass. For all the work discussed in this paper, the cavity mode is excited with a strong drive at frequency ω d, which is detuned below the cavity resonance by a frequency =ω d ω c. The strength of the drive can be parameterized in terms of the resulting steady-state number of photons in the cavity due to the drive tone, n d. After shifting all normal coordinates to their steady state values, the Hamiltonian may now be linearized and rewritten in a reference frame co-rotating with the drive [5 7]. This approximation assumes that the drive is much larger than the vacuum fluctuations of the mode such that α = n d 1. Ĥ = â â + Ω mˆb ˆb + gâ +â)ˆb + ˆb) S2) Now the coupling term is symmetric with respect to the two modes and the coupling rate has been parametrically enhanced such that g = g 0 α. We may now write the linearized Heisenberg-Langevin equations of motion for the two modes [5 7]. â = j â κ 2 â jgˆb + ˆb) i κi i ˆb = jω mˆb Γ m 2 ˆb jgâ +â) Γ m m S3) S4) The energy decay rate of the mechanical mode is given by Γ m. The noise m associated with this intrinsic mechanical dissipation satisfies the relations: mt) m 0) = n T mδt) κ ex ω c cavity mode κ o g Ω m mechanical mode Γ m dissipation to environment FIG. S1. Cavity coupling block diagram. and m t) m0) =n T m + 1)δt). Here, n T m is the equilibrium Bose-Einstein occupancy for the mechanical mode given by n T m = [exp Ω m /k B T ) 1] 1. Likewise, the cavity mode is also subject to noise from each of the baths to which it is coupled. The total energy decay rate κ = i κ i is the sum of coupling rates to each port as shown in Fig. S1). Each port is characterized by both a noise operator i and a coupling rate κ i to that dissipative bath. Again, the noise operators obey the commutation relations: i t) i 0) = n i δt) and i t) i 0) =n i + 1)δt). The final cavity occupancy n c is the weighted sum of the occupancy of each bath to which the cavity is coupled n c = i n iκ i /κ. Equations S3 and S4 and their Hermitian conjugates constitute a system of four first-order coupled operator equations, which may be solved by method of Fourier transform. χ 1 c â[ω] =jgˆb [ ω]+ˆb[ω]) κi i [ω] S5) c â [ ω] = jgˆb [ ω]+ˆb[ω]) κi i [ ω] S6) χ 1 m ˆb[ω] =jgâ [ ω]+â[ω]) Γ m m[ω] S7) m ˆb [ ω] = jgâ [ ω]+â[ω]) Γ m m [ ω] S8) In these expressions, we identify the bare response functions of the cavity and the mechanical modes, which are defined as χ 1 c = κ/2+jω + ) S9) c = κ/2 j ω + ) S10) χ 1 m =Γ m /2+jω Ω m ) S11) m =Γ m /2 j ω Ω m ). S12) WWW.NATURE.COM/NATURE 1
RESEARCH SUPPLEMENTARY INFORMATION We may further simplify these expressions with a rotating wave approximation with respect to both the mechanical mode and the cavity mode, by assuming that χ m 2 χ m 2 and χ c 2 χ c 2. As the mechanical oscillator only responds appreciably to forces near Ω m, this first assumption is valid when the loaded quality factor of the mechanical oscillator is much bigger than 1. For the cavity, this assumption is valid because we are considering a drive that is detuned by = Ω m and are in the good cavity limit in which Ω m κ, g. Thus, χ c 2 / χ c 2 κ/4ω m ) 2 1. The solutions to Eqs. S5, S6, S7 and S8 are [8] â[ω] = jgχ mχ c Γm m χ c κi i 1+g 2 χ m χ c S13) ˆb[ω] = χ m Γm m jgχ m χ c κi i 1+g 2 χ m χ c. S14) Having solved for the operators â and ˆb, we may now calculate the output of the cavity which couples to the port that we amplify and measure in the experiment. For our circuit, there are effectively three ways in which energy can enter or leave the cavity as shown in Fig. S1). These are denoted by the subscripts i={l,r,o} representing noise entering from the left, from the right, or from the environment, respectively. Thus, κ o is the intrinsic cavity decay rate, it would have in the absence of coupling to the transmission line, κ ex. The cavity mode can couple to waves propagating either left or right in the transmission line. We quantify the fraction of the mode amplitude that couples to the right with the dimensionless parameter β, such that κ r = βκ ex and κ l = 1 β)κ ex. For our circuit β =1/2, which implies the coupling is symmetric κ r = κ l = κ ex /2). As other circuits could be designed in future experiments to advantageously break this symmetry, we choose to leave the expressions in their general form in terms of β. â out = l + κ r â = 1 χ ) c βκex 1 β)κex 1+g 2 l χ m χ c ) χc βκex βκex 1+g 2 r χ m χ c ) χc βκex κo 1+g 2 o χ m χ c ) jgχm χ c βκex Γm 1+g 2 m χ m χ c S15) S16) S17) S18) S19) Notice that in the absence of any coupling to the cavity κ ex =0), â out = l as we would expect for a simple transmission line. We can now calculate the symmetric power spectral density of the output mode S[ω] in units of W/Hz. S[ω] ω = 1 2 dte jωt â out t)â out0) + â out t)â out 0) S20) = 1 2 â out [ ω]â out[ω]+â out [ω]â out [ ω] S21) = 1 2 + βκexg2 χ m 2 χ c 2 Γ mn T m + βκ ex χ c 2 κn c 1+g 2 χ cχ m 2 S22) In the final expression, we have assumed that the excess cavity occupation is not from the left port of the transmission line so that n l = 0 which implies that κn c = κ r n r + κ o n o. This assumption is supported by the experimental measurements in which we measure the excess cavity noise as a peak not a dip) in the output noise spectrum. In our experiment, we amplify the noise at the cavity frequency and use a microwave mixer to down-convert the noise spectrum to a RF frequency where the power spectral density may be conveniently measured. We choose to plot the noise spectra such that the noise from the mechanical oscillator appears at Ω m. In this frame it is convenient to define the variables, =ω d ω c +Ω m and δ = ω Ω m. In terms of these variables, the power spectral density is as given in Eq. 1 in the main text, S[ω] ω = 1 2 + n add + 4βκ ex Γ 2 m +4δ 2 )κn c 4g 2 + κ +2jδ + ) ) Γ m +2jδ) 2 16βκ ex g 2 Γ m n T m + 4g 2 + κ +2jδ + ) ). Γ m +2jδ) 2 The first term simply represents the vacuum noise of the photon field that sets the fundamental limit on the imprecision and back-action of the measurement. The second term n add is the total added noise of the measurement in units of equivalent number of photons. For an ideal measurement i.e. for a quantum-limited measurement of both quadratures of the light field), n add =1/2. The third term is due to the thermal noise of a cavity with occupancy n c whose spectral weight is distributed over the dressed cavity mode. The dressed cavity mode includes the effect of optomechanically induced transparency [9 11] and reduces to a single Lorentzian lineshape in the limit of weak coupling g κγ m ). The last term is the thermal noise of the mechanical mode with its modified mechanical susceptibility. Unlike previous derivations [12], we have not assumed the weak-coupling regime. Thus, as this equation is valid in both the weak- and strong-coupling regimes, it gives a unified description of the thermal noise spectrum even in the presence of normal-mode splitting. Before the onset of normal-mode splitting, one can directly relate the measured microwave power spectral density S to the displacement spectral density S x. Assuming 2 WWW.NATURE.COM/NATURE
RESEARCH Spectrum analyzer 300 K Filter cavity Variable attenutor & phase shifter 4 K Cryogenic HEMT amplifier 15 mk 6 db 6 db 10 db Superconducting coaxial cable Ω Termination Electromechanical circuit Directional coupler Circulator Josephson parametric amplifier FIG. S2. Detailed schematic diagram. A microwave generator creates a tone at the drive frequency. This signal is filtered with a resonant cavity at room temperature and split into two arms. The first arm excites the cavity through approximately 53 db of cryogenic attenuation. In order to avoid saturating the low-noise amplifier with the microwave drive tone, the second arm is used to cancel the drive before amplification. A computer-controlled variable attenuator and phase shifter are run in a feedback loop to maintain cancellation at the part per million level. A second microwave generator is used to provide the pump tone for the Josephson parametric amplifier JPA) as well as the reference oscillator for the mixer. This pump tone is 1.3 MHz above ω c so that the JPA is operated as a non-degenerate parametric amplifier, which measures both quadratures of the electromagnetic filed at the upper sideband frequency. The last stage of attenuation on all lines occurs inside a directional coupler, which allows us to minimize the microwave power dissipated on the cold stage of the cryostat. The JPA is a reflection amplifier; a signal incident on the strongly coupled port of the JPA is reflected and amplified. A cryogenic circulator is used to separate the incident and reflected waves, defining the input and output ports of the JPA. The other circulators are used to isolate the cavity from the noise emitted from the amplifier s input. =0, n c n m, and κ g, δ, MICROWAVE MEASUREMENT AND CALIBRATION S ω = 1 2 + n add +4β κ ex κ Γ Γ m n T m Γ m + Γ) 2 S23) +4δ 2 = 1 2 + n add + 2βG2 n d κ κ ex κ S x, where Γ=4g 2 /κ is the optomechanical damping rate. S24) The detailed circuit diagram for our measurements is shown in Fig. S2. In order to calibrate the value of g 0 for this device, we applied a microwave drive optimally reddetuned =0) and measured the thermal noise spectrum of the mechanical oscillator as a function of cryostat temperature. Here we restricted n d 3 in order to ensure that radiation pressure effects are negligible. With the value of g 0 now determined, we increase the drive ampli- WWW.NATURE.COM/NATURE 3
RESEARCH SUPPLEMENTARY INFORMATION tude and measure the thermal noise spectrum at each drive power. The noise spectra are recorded and averaged with commercial FFT spectrum analyser. Each spectrum is typically an average of 0 traces with a measurement time of 0.5 s per trace. The cavity response is then measured with a weak probe tone with a vector network analyser to determine precise cavity parameters at each microwave drive power, including the precise detuning and κ. For larger microwave drive powers where the cavity spectrum exhibits optomechanically induced transparency effects [9 11], this spectrum also serves as a direct measure of g. Finally, using additional calibration tones, each noise spectrum is calibrated in units of absolute microwave noise quanta and fit with Eq. 1 from the main text to determine the occupancy of both the cavity and mechanical modes. In order to ensure that the amplitude or phase noise of the signal generator was not responsible for the finite occupancy of the cavity at high drive power, we designed and built a custom filter cavity [12]. As shown in Fig. S3, when the filter cavity is tuned to precisely the frequencies of our circuit, it provides an additional 40 db of noise suppression at the cavity resonance frequency. The phase and amplitude noise of our signal generator alone are specified by the manufacturer to be less than -1 dbc at Fourier frequencies 10 MHz away from the drive. With the addition of filter cavity, we lower this noise to well below the shot-noise level of our microwave drive. Furthermore, even without the filter cavity, we could not resolve an appreciable difference in the cavity occupation. Thus, while we do not know the precise mechanism for this occupancy, we conclude the generator noise is not the cause. Filter Transm ission db) 0-10 -20-30 -40-7.52 ω d /2π ω c /2π 7.53 7.54 Frequency G H z) 7.55 FIG. S3. Measured transmission of filter cavity. A tunable resonant cavity was implemented at room temperature in order to suppress noise 10 MHz above the drive frequency. As shown here, this cavity reduces the noise at the cavity frequency by more than 40 db, ensuring that the phase or amplitude noise of the generator is not responsible for the finite occupancy of the cavity at large drive power. INFERRING CAVITY PARAMETER AND NUMBER OF DRIVE PHOTONS The power at the output of the cavity P o is related to the input power P i by [11] ) κ 2 P o = P o + 4 2 i, S25) κ 2 + 4 2 where = ω d ω c is the difference between the frequency of the drive ω d and the cavity resonance frequency ω c. κ is the total intensity decay rate of the cavity full width at half maximum) with κ = κ o + κ ex. The number of photons in the cavity due to a coherent input drive at detuning may be calculated from the stored energy in the cavity. n d = α 2 = 2P i κ ex ω d κ 2 + 4 2 S26) For our circuit, κ ex =2π 133 khz. Thus, when the drive is optimally detuned such that = Ω m, the input power required to excite the cavity with one photon is P i 2 ω d Ω 2 m/κ ex fw. FUNDAMENTAL LIMITS OF SIDEBAND COOLING Equation 2 in the main text gives an expression for the final occupancy of a mechanical mode, assuming that the microwave drive is optimally detuned = Ω m ). This expression is only the lowest order approximation in the small quantities g/ω m and κ/ω m. Up to second order, the final occupancy is [6] ] n m = n T Γm 4g 2 + κ 2 m )[1+ g2 4g 2 + κγ m κ 4g 2 + κγ m Ω 2 m 4g 2 + κ 2 ] 4g 2 + n c )[1+ 8g2 + κ 2 4g 2 + κγ m 4g 2 + κγ m 8Ω 2 m 4g 2 + 8g2 + κ 2. 16Ω 2 m The last term represents the fundamental limit for sideband cooling and demonstrates the importance of the resolvedsideband regime. For our system, Ω m κ, g; and hence this last term only contributes negligibly to the final occupancy of the mechanical mode < 10 4 quanta). MEASUREMENT IMPRECISION AND BACKACTION In the main text and this supplementary information, we use the single-sided convention for the displacement and force spectral densities. Thus, for example, the meansquare fluctuations of x are x 2 = S 0 x ω) dω. This 2π also yields the familiar classical result that an oscillator coupled to a thermal bath of temperature T will experience 4 WWW.NATURE.COM/NATURE
RESEARCH a random force characterized by the force spectral density S F =4k B TmΓ m. More generally, S F =4 Ω m n T m + 1 ) mγ m, S27) 2 where n T m is the Bose-Einstein occupancy factor given by n T m = [exp Ω m /k B T ) 1] 1. Independent of any convention for defining the spectral density, the visibility of a thermal mechanical peak of given mechanical occupancy above the noise floor of the measurement represents a direct measure of the overall efficiency of the detection. As shown in Fig. S4, ratio of the peak height to the white-noise background allows us to quantify the imprecision of the measurement in units of mechanical quanta [14], n imp Sx imp mω m Γ m/4 ). Inspection of Eq. S23 implies n imp = 1 ) κ 4g 2 + κγ m 1 4β κ ex 4g 2 2 + n add S28) S x arb.) FIG. S4. quanta. 3 2 1 0 n m n imp 10.556 10.557 10.558 Frequency M H z) 10.559 Measurement imprecision in units of mechanical Once the drive is strong enough that g κγ m ), n imp no longer decreases with increasing drive. It is precisely because we are measuring with a detuned drive that also damps the mechanical motion, that n imp asymptotically approaches a constant value [7, 16]. For an ideal measurement β =1,κ = κ ex, and n add =1/2), n imp 1/4. Implicit in obtaining this optimal value for n add and hence n imp is that all the photons exiting the cavity are measured. Any losses between the cavity and the detector can be modeled as a beam-splitter that only transmits a fraction η of the photons to the detector and adds a fraction 1 η) of vacuum noise. So the effective added noise n add accounting for these losses becomes n add = n add η 1 η + η ) 1 2, S29) Thus, shot-noise limited detection of the photons n add = 1/2) is a necessary, but not sufficient, condition for reaching the best possible level of precision. For our measurements, we infer that our entire measurement chain has an effective added noise of n add =2.1. This value is consistent with the independently measured value for the added noise of the JPA n add =0.8) and the 2.5 db of loss between the output of the cavity and the JPA [13, 14]. Quantum mechanics also requires that a continuous displacement measurement must necessarily impart a force back on the measured object. For an optimally detuned drive =0) in the resolved-sideband regime, this backaction force spectral density SF ba approaches a constant value as a function of increasing drive strength and asymptotically approaches SF ba =2 Ω m mγ m. Again, expressing the spectral density in units of mechanical quanta gives n ba SF ba /4 Ω m mγ m) 1/2. Fundamentally, the Heisenberg limit does not restrict the imprecision Sx imp or the backaction SF ba alone, but rather it requires their product has a minimum value [1, 7, 15] Sx imp SF ba =4 n imp n ba. S30) An ideal cavity optomechanical system can achieve this lower limit for a continuous measurement with a drive applied at the cavity resonance frequency. When considering the case where the drive is instead applied detuned below the cavity resonance =0), this product never reaches this lower limit [7, 16] and is at minimum Sx imp SF ba = 2. To estimate these quantities for our measurements, we can infer the total force spectral density experienced by our oscillator as SF tot =4 Ω m mγ mn m +1/2). As this total necessarily includes the backaction, we may make the most conservative assumption that it was solely due to backaction that our oscillator remained at finite occupancy. Hence, n ba n m +1/2. The low thermal occupancies attained in this work allow us to place an upper bound on how large the backaction could possibly be, and hence quantify our measurement in terms of approach to the Heisenberg limit. Thus, Sx imp SF ba =4 n imp n ba 4 n imp n m +1/2). At n d = 3 10 4, we simultaneously achieve n m = 0.36 and n imp = 1.9, corresponding to SF total = 1.6 10 34 N 2 /Hz and Sx imp = 1.7 10 33 m 2 /Hz, respectively. This gives an upper limit on the measured product of backaction and imprecision of 5.1. As stated above, the best possible backactionimprecision product is 2 when using red-detuned excitation; thus our measurement is only a factor of 3.6 above this limit. It may also be noted that this factor would have been 2.6 except that our chosen circuit geometry losses half of the signal back to the input β = 1/2). In future experiments, using a single-port geometry β = 1) WWW.NATURE.COM/NATURE 5
RESEARCH SUPPLEMENTARY INFORMATION will improve this inefficiency. This measured imprecisionbackaction product within a factor of five of the Heisenberg limit is the closest experimental approach of any displacement detector to date [7], surpassing single-electron transistors [17], atomic point contacts [18], SQUIDs [19] and cavity optomechanical systems [16, 20, 21]. [1] Braginsky, V. B. & Khalili, F. Y. Quantum Measurement Cambridge University Press, 1992). [2] Gardiner, C. W. & Collett, M. J. Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation. Phys. Rev. A 31, 3761 3774 1985). [3] Walls, D. F. & Milburn, G. J. Quantum Optics Springer, Berlin, 1994). [4] Law, C. K. Interaction between a moving mirror and radiation pressure: A Hamiltonian formulation. Phys. Rev. A 51, 2537 2541 1995). [5] Marquardt, F., Chen, J. P., Clerk, A. A. & Girvin, S. M. Quantum theory of cavity-assisted sideband cooling of mechanical motion. Phys. Rev. Lett. 99, 093902 2007). [6] Dobrindt, J. M., Wilson-Rae, I. & Kippenberg, T. J. Parametric normal-mode splitting in cavity optomechanics. Phys. Rev. Lett. 101, 263602 2008). [7] Clerk, A. A., Devoret, M. H., Girvin, S. M., Marquardt, F. & Schoelkopf, R. J. Introduction to quantum noise, measurement, and amplification. Rev. Mod. Phys. 82, 1155 1208 2010). [8] Throughout this work, we have used the definition of the Fourier transform in which f[ω] = e jωt ft)dt. [9] Agarwal, G. A. & Huang, S. Electromagnetically induced transparency in mechanical effects of light. Phys. Rev. A 81, 041803 2010). [10] Weis, S. et al. Optomechanically Induced Transparency. Science 330, 1520 1523 2010). [11] Teufel, J. D. et al. Circuit cavity electromechanics in the strong-coupling regime. Nature 471, 204 208 2011). [12] Rocheleau, T. et al. Preparation and detection of a mechanical resonator near the ground state of motion. Nature 463, 72 75 2010). [13] Castellanos-Beltran, M. A., Irwin, K. D., Hilton, G. C., Vale, L. R. & Lehnert, K. W. Amplification and squeezing of quantum noise with a tunable Josephson metamaterial. Nature Physics 4, 929 931 2008). [14] Teufel, J. D., Donner, T., Castellanos-Beltran, M. A., Harlow, J. W. & Lehnert, K. W. Nanomechanical motion measured with an imprecision below that at the standard quantum limit. Nature Nanotechnology 4, 820 823 2009). [15] In many references, the Heisenberg limit for a continuous position detection is stated as S xxs FF /2. This is an equivalent statement to Eq. S20 because S xx and S FF are the double-sided spectral densities, each of which differ from our single-sided convention by a factor of 2. [16] Schliesser, A. et al. Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit. Nature Physics 5, 9 514 2009). [17] Naik, S. et al. Cooling a nanomechanical resonator with quantum back-action. Nature 443, 193 196 2006). [18] Flowers-Jacobs, N. E.Schmidt, D. R. & Lehnert, K. W. Intrinsic Noise Properties of Atomic Point Contact Displacement Detectors. Phys. Rev. Lett. 98, 096804 2007). [19] Etaki, S et al. Motion detection of a micromechanical resonator embedded in a d.c. SQUID. Nature Physics 4, 785 788 2008). [20] Gröblacher, S. et al. Demonstration of an ultracold microoptomechanical oscillator in a cryogenic cavity. Nature Physics 5, 485 488 2009). [21] Rivière, R. et al. Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state. arxiv:1011.0290 2010). 6 WWW.NATURE.COM/NATURE
RESEARCH Symbol Definition Ω m Γ m Γ TABLE S1. Summary of symbols and their definitions. mechanical resonance frequency intrinsic mechanical damping rate additional mechanical damping rate due to radiation pressure effects Γ m total mechanical damping rate, Γ m =Γ m +Γ n T m n m equilibrium occupancy of mechanical mode at cryostat temperature T final occupancy of mechanical mode with radiation pressure cooling x zp zero-point amplitude of the mechanical mode, x zp = /2mΩ m) m ω c κ o κ ex κ β n c ω d P i P o n d G g o g S n add n add S x Sx imp n imp S F SF tot SF ba n ba mass of the mechanical oscillator cavity resonance frequency intrinsic cavity damping rate cavity damping rate due to intentional coupling to the transmission line total cavity damping rate, κ = κ o + κ ex fraction of the noise power the exiting κ ex which couples to the output β=1/2 in this work) noise occupancy of the cavity mode frequency at which the cavity is driven power of the drive incident on the cavity power of the drive at the output of the cavity number of cavity photons due to the coherent drive detuning between the drive and the cavity, =ω d ω c detuning between the upper sideband of the drive and the cavity, =ωd +Ω m ω c cavity frequency shift per unit motion, G = dω c/dx vacuum optomechanical coupling rate, g o = Gx zp linearized optomechanical coupling rate, g = g 0 nd power spectral density noise added by the microwave amplifier in units of microwave quanta effective noise added by the amplifier in units of microwave quanta including loss displacement spectral density imprecision displacement spectral density imprecision in units of mechanical quanta, n imp Sx imp mω mγ m/4 ) force displacement spectral density total force displacement spectral density force displacement spectral density due to measurement backaction backaction in units of mechanical quanta, n ba SF ba /4 ΩmmΓ m) WWW.NATURE.COM/NATURE 7