Abstract

Interaction with a thermal environment decoheres the quantum state of a mechanical oscillator. When the interaction is sufficiently strong, such that more than one thermal phonon is introduced within a period of oscillation, quantum coherent oscillations are prevented. This is generally thought to preclude a wide range of quantum protocols. Here we show that the combination of pulsed optomechanics techniques with coherent control can overcome this limitation, allowing ground-state cooling, general linear quantum nondemolition measurements, optomechanical state swaps, and quantum-state preparation and tomography without requiring quantum coherent oscillations. Finally, we show how the protocol can break the usual thermal limit for classical sensing of impulse forces.

© 2017 Optical Society of America

1. INTRODUCTION

Quantum optomechanics uses an optical or microwave field to prepare, control, and characterize the quantum states of a mesoscopic to a macroscopic mechanical oscillator, typically using a cavity to enhance the interaction [16]. Optomechanical systems have been proposed for quantum information applications [79] and tests of foundational physics [10,11] and are currently used for state-of-the-art sensors [1214], with each application requiring different optomechanical regimes. The significant progress in devices and technology of the last decade [2] suggest some of these aspirational targets will be realizable in the near future. With the exception of position nondemolition measurements and their derivatives [15], it is generally considered that operation in the quantum coherent oscillation (QCO) regime—where on average less than one thermal phonon is exchanged per mechanical period and the oscillator remains coherent for at least a single oscillation—is a minimum requirement for such experiments [2,1619]. This notion is reflected in recent theoretical and experimental results [11,2029].

In the high-temperature limit, the QCO regime is achieved when QωM>kBT/ with Q as the quality factor of the mechanical oscillator of frequency ωM at temperature T. This places stringent constraints on both the temperature and mechanical resonance frequency for which optomechanics protocols can be implemented. Highly desirable room temperature operation requires ωMQ4×1013 [18], which has been achieved at megahertz [19,30] frequencies but remains beyond current technology for low-frequency (ωM/2π100  kHz) oscillators [2].

Here we show that combining pulsed optomechanics with quantum coherent control enables ground-state cooling, general linear back-action-evading measurements, state swaps, nonclassical state preparation, and quantum tomography outside the QCO regime. This substantially extends the parameter regime for which quantum protocols are applicable, providing a pathway toward macroscopic tests of quantum mechanics and quantum-enhanced precision force sensing [12] at room temperature, among other applications. Furthermore, when applied in the classical regime, our technique can evade thermal force noise, which limits current state-of-the-art force sensors. Such sensors operate in the non-QCO regime using a “time of flight” method to translate a force signal—coupled directly to momentum—into position, which can be optically read out [13,31]. The necessary time delay introduces thermal force noise, reducing the sensitivity of the measurement. Implementing the measurement over a small fraction of a period reduces the thermal noise, increasing the measurement sensitivity.

Our results extend previous work on speed meters, which have been proposed for gravitational wave detectors [32,33]. Speed meters can achieve quantum nondemolition measurements of the relative momentum of two oscillators. Unlike our approach, they have generally been studied for detecting oscillating forces in a frequency band far from the mechanical resonance [3437] and within the free mass approximation.

2. MODEL

Pulsed optomechanics [15,38] generates optomechanical correlations over a short time scale compared to free mechanical evolution. These short interactions can be used to manipulate the state of a mechanical oscillator, greatly reducing mechanical thermalization for a given protocol. The interaction Hamiltonian for such a system is given by HI=g0aa(b+b), where a (b) is the annihilation operator for the optical (mechanical) mode and g0 is the bare optomechanical interaction rate. We restrict the analysis to the linearized interaction where the optical field is linearized, aα+a, about a time-dependent amplitude α(t)=a(t), where without loss of generality we define α to be real [39]. The pulse envelope α(t) is chosen to be a Gaussian with pulse width τ, which may have negative amplitude. This can be achieved, for instance, through appropriate displacements of the pulses, as discussed later. Expanding to first order in a, the interaction Hamiltonian is given by

HI/=g0α(a+a)(b+b)+g0α2(b+b).
The first term is the linearized optomechanical interaction, and the second term is a coherent momentum displacement. This displacement is deterministic and can be canceled by applying an opposite classical displacement to the oscillator; we therefore neglect it henceforth. Outside the single-photon strong-coupling regime, the second-order term, g0aa(b+b), which has also been neglected, remains negligibly small, even when the envelope α(t)0, as the mechanical dynamics are dominated by Brownian motion [40].

The cavity is modeled as a single-sided cavity with decay rate κ, which is large enough for the optical pulse to adiabatically interact with the cavity τκ1. In this regime, the intracavity field is proportional to the input field α(t)=αin(t)2/κ with the input field is normalized |αin|2dt=N¯, where N¯ is the mean photon number in the pulse. Other pulsed optomechanics protocols [4042] assume the oscillator is frozen during a single-pulsed interaction, so that τ1/ωM with the optomechanical system necessarily operating in the unresolved sideband limit, κωM. However in this work, we require the stricter condition κn¯γ (this condition is only stricter outside the QCO regime) in order to approximate a single-pulsed optomechanical interaction as unitary. The short duration of the pulse means that any addition mechanical modes that couple to the optical field cannot be spectrally resolved. This is an issue faced by many pulsed protocols and can be avoided by using specifically designed mechanical oscillators where only a single mode is optically coupled, such as those used in Ref. [41].

Under these conditions, the unitary describing the total pulsed interaction generated by the Hamiltonian in Eq. (1) is given by U(XMXL)=exp[iλXMXL], where 2XM=b+b and 2XL=a+a are the mechanical position and optical amplitude quadratures, respectively. The dimensionless constant λ=g0α(t)dt=sign(α)4(2π)1/4g0τN¯/κ is the optomechanical interaction strength [15]. Using a pulse shape other than a Gaussian simply changes the proportionality constant between λ and g0τN¯/κ (where τ is now some characteristic time of the pulse shape). In the adiabatic limit, the pulse shape remains constant throughout the protocol [up to order O(κτ)1], and the optimal pulse shape is one that maximizes |α(t)|dt subject to the constraints |α(t)|2dt=1 and τκ1. The unitary correlates the optical phase quadrature (PL) and mechanical position as UPLU=PLλXM at the expense of adding back action to the momentum (PM) of the oscillator UPMU=PMλXL. We will now show that a sequence of such pulsed interactions allows arbitrary mechanical quadrature measurements that are sufficient for quantum-state tomography, squeezed-state preparation, and pulsed optomechanical state swaps [42]. Outside the QCO regime, a single-pulsed measurement scheme cannot achieve these important goals.

In our back-action-evading protocol, a single optical pulse interacts with the mechanical element twice (Fig. 1). This process is represented by an initial optomechanical interaction, free mechanical evolution θ=ωMt, and finally the second optomechanical interaction with the same optical field but different interaction strength. Neglecting decoherence (which will be included in the next section), the protocol is described by the overall unitary as

U=exp[iλ2XMXL]eiθbbexp[iλ1XMXL]=R(θ)χ(λ1λ2)U(XLXMϕ),
where, without loss of generality, we have assumed a negative envelope α(t) for the first interaction and a positive envelope for the second. This unitary, acting from right to left, can be understood as follows. The first term, U(XLXMϕ)=exp(iGXLXMϕ), is the optomechanical interaction between a rotated mechanical quadrature XMϕ=XMcosϕ+PMsinϕ and the amplitude quadrature XL of the light. The (dimensionless) interaction strength is G=(λ12+λ222λ1λ2cosθ)1/2>0, and the rotated quadrature angle is tanϕ=λ2sinθ(λ2cosθλ1)1. We therefore see that by applying two interactions, it is possible to generate a single effective XLXMϕ interaction for an arbitrary mechanical quadrature ϕ. This interaction contains terms of the form ab+ba and will therefore result in entanglement between the optical and mechanical states. For a given ϕ, the natural choice of free parameters (λ1,λ2,θ) are the set that maximizes G. The second unitary in Eq. (2), χ(λ1λ2)=exp[i2λ1λ2XL2sinθ], is an optical Kerr nonlinearity, the pulsed analog of the effects reported in Refs. [43,44]. The final term R(θ)=eiθbb is the mechanical rotation due to the delay between pulses and must be accounted for in the final mechanical state. Different interaction strengths λ1 and λ2, with positive or negative signs, may be chosen by using a displacement pulse to change the mean photon number in between the two interactions.

 

Fig. 1. Schematic of the protocol. Half- and quarter-wave plates (HWP and QWP, respectively), a polarizing beam splitter (PBS), and a switchable beam splitter (SBS) are used to initially direct the interaction pulse along paths 1–4, along which the first optomechanical interaction takes place. When the interaction pulse reaches the top port of the highly reflective beam splitter (RBS), its coherent amplitude is changed by the displacement pulse. Since almost all of the interaction pulse reflects from the RBS, any quantum correlations between it and the oscillator remain, as the pulse interacts with the mechanical oscillator a second time. After the second optomechanical interaction, the SBS switches out the pulse to a homodyne measurement device instead of directing it along path four a second time.

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A displacement pulse is realized by the highly reflective beam splitter (Fig. 1). After the beam splitter, almost all of the light incident on the top port is reflected. Therefore, any correlations between this light and the oscillator remain as the optical pulse follows paths 1–3, interacting with the oscillator a second time. This recycling of the pulse is essential to negate the quantum noise on the oscillator. The coherent amplitude of the displacement pulse is chosen to control α(t) [from Eq. (1)], and since it is predominantly reflected, it contributes negligible quantum noise to the recycled pulse [45]. The phase between the displacement pulse and the interaction pulse may be chosen such that the coherent amplitude of the correlated light is completely negated or reversed sign. The total unitary couples the optical and mechanical states as

XMXMcosθ+PMsinθXLλ1sinθ,
PMPMcosθXMsinθXL[λ2λ1cosθ],
XLXL,
PLPLGXMϕ+XLλ2λ1sinθ.
The cancellation of noise in Eq. (3b) can be intuitively understood by observing that the back-action noise arises from amplitude fluctuations in the light. The coherent amplitude acts as a gain in the linearized Hamiltonian [Eq. (1)], and changing its sign changes the sign of the amplitude fluctuations that kick the oscillator’s momentum. The cancellation is only made possible by the fact that the first pulse is recycled, so that the same amplitude fluctuations kick the resonator with the reversed sign.

For a single optomechanical interaction (λ2=θ=0) the back-action noise (XL) is necessarily imparted onto the momentum of the oscillator. However, if the interaction strengths are chosen such that λ2=λ1cosθ, the back action on the momentum exactly cancels and a homodyne measurement of the optical field can be used to conditionally prepare a momentum squeezed state. This can be done in an arbitrarily short time, θ0, at the expense of decreasing the overall interaction strength G. It can therefore introduce arbitrarily low levels of the thermal noise that precludes momentum squeezing with other protocols outside the QCO regime. By varying the interaction strengths for the two pulses (λ1,λ2), one may deterministically choose which mechanical quadrature the back action is added to and which quadrature is back-action free.

For the unitary case, or indeed, in the QCO regime where a negligible amount of thermal noise is introduced after a single oscillation, there is no need to implement the two-pulse protocol. A momentum measurement can be achieved with θ=π/2 and λ1=0 (wait then measure), correlating the phase quadrature with the initial momentum of the oscillator. Alternatively, momentum state preparation can be achieved with θ=π/2 and λ2=0 (measurement then wait), preparing a conditional position squeezed state that rotates into momentum a quarter cycle later. However, outside the QCO regime phonon exchange during the necessary θ=π/2 delay thermalizes the mechanical state, resulting in additional measurement noise and degrading conditional state preparation, even in the limit of arbitrarily large optomechanical interaction strengths. Equations (3a)–(3d) illustrate how, with our protocol, one can reduce the duration of the protocol θ—and thus the thermalization—while still generating the optomechanical correlations necessary to prepare and measure an arbitrary mechanical quadrature. Note that the unitary in Eq. (2) can be used to extend the applicability of the state swap protocol in Ref. [42] and cooling protocol of Ref. [15], which are currently limited to the QCO regime. Both of these protocols require a pulsed momentum measurement, which the authors propose to have achieved via a quarter-period rotation of the mechanical state, followed by a position measurement. However, by replacing these rotations and position interactions with the direct momentum interaction in Eq. (2) (with ϕ=π/2), one can implement the state swap and cooling protocols within a small fraction of one mechanical period, thereby avoiding the requirement for coherent quantum evolution of the quarter-period time scale.

3. NONUNITARY EVOLUTION

In this section, the description of the protocol is extended to include mechanical dissipation and optical losses [46,47]. During each pulsed interaction, it is still assumed that the mechanical oscillator remains effectively frozen only being free to evolve during the wait time between pulses. The cavity is treated as highly overcoupled such that κκexκ0, where κex and κ0 are the extrinsic and intrinsic cavity decay rates, respectively. In this case, the only significant source of optical loss occurs during the storage time between the two interactions and each optomechanical interaction is well described by the unitary operator in Eq. (2). Thermalization is included during the free evolution of the oscillator by the Langevin equation of motion [48]

X˙M=ωMPM,
P˙M=ωMXMγPM+2γξ,
where γ is the oscillator decay rate and ξ(t) is a zero-mean white noise operator with correlation function ξ(t)ξ(t)=(n¯+12)δ(tt), where n¯kBT/ωM in the high-temperature limit. The thermal force satisfies [XM(t),ξ(t)]=γ/2i to preserve the commutation relations. Optical losses are modeled as a single beam splitter unitary Bη with intensity loss η after each optomechanical interaction. The protocol is then given by BηU2BηMθU1, where Ui is the ith optomechanical unitary, and Mθ is a nonunitary map that describes the dissipative mechanical evolution given by Eqs. (4a) and (4b) over the rotation angle θ. After the full protocol, the optical and mechanical quadratures are given by
XM,out=XMcosθ+PMsinθ+ξXλ1XLsinθ,
PM,out=PMcosθXMsinθ+ξPXL(ηλ2λ1cosθ)λ21ηδX1,
XL,out=ηXL+ηη2δX1+1ηδX2,
PL,out=ηPL+ηη2δP1+1ηδP2+ηλ1λ2sinθXLGXMϕηλ2ξX,
where ξX(P) is the thermal noise added to the mechanical position (momentum) during the nonunitary evolution and δXi (δPi) is the amplitude (phase) quadrature vacuum noise entering from the ith optical loss channel. With decoherence included, the new measured quadrature and measurement strength are given by ϕ=arctan[λ2sinθ(λ2cosθηλ1)1] and G=(η2λ12+ηλ222λ1λ2η3/2cosθ)1/2, respectively. Equation (5d) highlights how the scheme works in the presence of mechanical thermalization. A measurement of PL,out provides information about GXMϕ with all other terms contributing zero-mean Gaussian noise. For small θ, the thermal noise in the position increases as θ3 scaling (see Ref. [42] for details). Consequently, reducing the duration of the protocol cubicly reduces this noise term, which is generally the dominant noise source outside the QCO regime where n¯/Q1. Due to correlations between the phase and amplitude quadratures, i.e., the Kerr term η2λ1λ2sinθXL in Eq. (5d), extra noise is added to the phase quadrature. Since this noise is correlated to the optical amplitude quadrature, XL, it can be reduced by measuring a rotated optical quadrature XLφ=XLcosφ+PLsinφ, where the optimal angle φ is determined numerically. The effect of optical loss is detrimental in two ways. First, it reduces the effective interaction strength G, degrading the signal of XMϕ in the phase quadrature. Second, it adds an extra irreversible noise term (δX1) to PM,out. In the following section, we show how the correlations in Eqs. (5a)–(5d) can be used for state preparation, measurement, and force sensing.

4. MEASUREMENT AND TOMOGRAPHY

For quantum measurement and tomography, the aim is to measure the statistics of the a priori mechanical state, XMϕ, for each ϕ[0,π), without making any assumptions on the statistics of XMϕ. From Eq. (5d), a measurement of PL,out (or indeed any optical quadrature XLφXL) is a measurement of XMϕ with all other terms contributing zero-mean Gaussian noise. The uncertainty in a Gaussian variable A given a measurement of B is given in general by the conditional variance, V(A|B)=V(A)C(A,B)2/V(B), where C(A,B)12AB+BAAB is the correlation between A and B. When the conditional variance of the measured quadrature V(XMϕ|XLφ) is below the ground-state variance (of 1/2), full-quantum-state tomography can be performed efficiently and Wigner negativity can be directly observed [49,50]. Outside the QCO regime, direct observation of Wigner negativity is not possible, using either single-pulse or stroboscopic measurement, while state tomography becomes increasingly challenging due to the convolution of the thermal noise in the measurement results. Both can be achieved, in principle, to an arbitrary degree of accuracy here. To the best of our knowledge, this is the first protocol capable of efficient state tomography for oscillators outside the QCO regime.

Figure 2(a) compares the two approaches for realistic experimental parameters. The parameters are chosen as ωM/2π=100  kHz and γ/2π=1  Hz (Q=105), similar to the silicon carbide resonators in Ref. [51]. At a temperature of 1 K, 2πn¯/Q13 phonons enter per oscillation, residing well outside of the QCO regime. The bare optomechanical coupling rate is conservatively set at g0/2π=1  Hz with a cavity decay rate of κ/2π=1  GHz (optical Q2.5×105 at 1064 nm). To ensure a fair comparison, we choose to relate the single-pulse interaction strength λ to the two-pulse interaction strengths via λ=λ12+λ22 such that the mean photon number in the single pulse is equal to the sum of mean photon numbers in the two-pulse scheme. As shown in Fig. 2(a), there is lower bound for the conditional variance using a single-pulsed interaction, independent of the interaction strength. This lower bound is removed using our protocol, allowing sub-ground-state resolution outside the QCO regime. Figures 2(b)2(d) show that sub-ground-state resolution is possible for all quadrature angles, allowing for full-quantum-state tomography for λ10. In order to achieve this sub-ground-state resolution, the two pulses must interact with the mechanical oscillator within the thermal decoherence time.

 

Fig. 2. (a) Conditional variance for the measurement of different quadratures (ϕ=π/2,π/4, and π/8), using the double (solid lines) or single (dashed lines) interaction protocols. (b)–(d) Conditional variance as a function of mechanical quadrature angle, ϕ, for the double (solid lines) and single (dashed lines) interaction protocols for λ=10, λ=1, and λ=0.1 respectively. (a)–(d) Assume a bath temperature of 1 K, ωM/2π=100  kHz, and γ/2π=1  Hz. Gray dashed lines indicate the ground-state variance.

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An interaction strength of λ=10 requires a high pulse energy, E=ωκλ2/(g02τ162π)14  mJ in a 5 ns pulse (assuming a temperature of 100 K with τn¯γ). However this is a consequence of our conservative choice of coupling strength g0/2π=1  Hz. Coupling rates of kilohertz [52] and even up to 20 kHz [53] have been observed, and taking g0/2π=2  kHz significantly reduces the pulse energy to 3.7 nJ for λ=10 or 370 nJ for λ=100. Combining this higher coupling rate with cryogenic (1 K) operation will enable longer optical pulses (increasing τ to 500 ns), further reducing the pulse energy 37 fJ for λ=10, as is required for ground-state resolution at 1 K (Fig. 2). We note that a 5 ns, 3.7 nJ pulse corresponds to 0.74 W of peak power, or 3.7 μW average power at a 1 kHz repetition rate, which can be routinely achieved in optical microcavities [54,55].

5. SQUEEZED-STATE PREPARATION

We now turn to how the protocol can be used to prepare a squeezed mechanical state in the presence of thermalization. Mechanical squeezing, such as ponderomotive squeezing [56] or reservoir engineering [57], requires many oscillations of the mechanical oscillator and therefore cannot be implemented within the QCO regime. A single-pulsed interaction can be used to generate squeezing outside the QCO regime but is best suited to position squeezing [15]. Our scheme allows squeezing of an arbitrary mechanical quadrature outside the QCO regime. Due to the finite rotation during the protocol, there is a subtle distinction between squeezed-state preparation and quantum measurement. For state preparation, the aim is to condition the variance of the a posteriori quantum state, V(XM,outϕ|PL), instead of the a priori state, V(XMϕ|PL). Figure 3(a) shows the a posteriori conditional variance for a momentum measurement at different temperatures, comparing the double interaction protocol introduced here with a single-pulsed interaction. It shows arbitrary quadrature squeezing is, in principle, possible if the interaction strength is high enough and is necessarily achievable within a fraction of an oscillation. Our scheme can achieve significantly lower variance at higher temperatures, even for small–moderate interaction strengths. As with quantum tomography, our scheme is able to prepare states with variance below the thermalization line that bounds the single-pulsed scheme, meaning the conditional variance can be arbitrarily reduced by increasing the interaction strength. This is due to the finite time taken for the position squeezed state to rotate to a momentum squeezed state. From Eqs. (5a)–(5d), it can be seen that the optimal measured quadrature is not orthogonal to the back-action quadrature. As a result, any squeezing will be accompanied by antisqueezing larger than the lower bound set by the Heisenberg limit. Combined with optical losses and the mechanical thermalization, this effect reduces the purity of the final mechanical state.

 

Fig. 3. Conditional variance of the final oscillator momentum using the double (solid lines) and single (dashed lines) interaction protocols at different temperatures. A temperature of 100 K corresponds to 1.3×103 phonons exchanged per cycle for ωM/2π=100  kHz and γ/2π=1  Hz.

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Note that direct measurement and manipulation of the momentum of an oscillator over a short time scale (θ/ωM) may be used to directly measure impulse forces—which couple directly to momentum through dp=Fdt—at the thermal limit. Using our pulsed protocol, the momentum state preparation (measurement) can be made immediately before (after) the impulse force. Therefore, the only thermal force in the signal is that which enters during the duration of the impulse. In contrast, a position measurement must necessarily wait an additional time period while the force evolves into a displacement, during which, extra thermal noise is added to the signal. For example, taking the detectable momentum change to be of the order of the conditional momentum standard deviation of 200mω/2 at 100 K (from Fig. 3 with λ=1) corresponds to a thermally limited impulsed force sensitivity of 8 pN over 150th of an oscillation period (200 ns), compared with 30 pN over 1/4th of an oscillation period (2.5 μs) for the single-pulsed protocol.

6. SUMMARY

We have introduced a protocol to realize quantum optomechanics beyond the QCO regime, where quantum-state preparation and direct tomography are possible within a fraction of a mechanical period. While it may seem that a fraction of a mechanical period is an extremely short time scale, the low-frequency oscillations mean that in absolute terms, it is comparable to the state lifetime of some of the best cryogenic quantum optomechanics experiments. For example, the resonator in the mechanical Fock state preparation and measurement experiment of Ref. [29] has a decoherence rate of 14 kHz, which is only two orders of magnitude smaller than the single-phonon thermalization time n¯γ=1.3  MHz (at 1 K) for the oscillators considered here. By using a high-Q low-frequency oscillator, the lifetime of a quantum state 1/n¯γ=Q/kBT can be made comparable to the high-frequency oscillators operating at cryogenic temperatures, while still remaining outside the QCO regime. This allows for the possible observation of quantum effects in a new realm of mechanical resonators, such as low-frequency, high-temperature, or high-mass systems by removing the requirement of coherent oscillations. Furthermore, even in the classical regime, the reduction of thermal noise in our measurement protocol results in significantly less measurement uncertainty in pulsed measurements of arbitrary mechanical quadratures.

Funding

Australian Research Council (ARC) (CE110001013); Discovery Project (DP140100734); Future Fellowship (FT140100650); Engineering and Physical Sciences Research Council (EPSRC) (EP/N014995/1); Air Force Office of Scientific Research (AFOSR) (FA2386-14-1-4046).

Acknowledgment

The authors would like to thank Gerard J. Milburn for helpful discussions throughout this project.

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27. R. Riviere, S. Deleglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83, 063835 (2011). [CrossRef]  

28. G. Anetsberger, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Cavity optomechanics and cooling nanomechanical oscillators using microresonator enhanced evanescent near-field coupling,” C. R. Phys. 12, 800–816 (2011). [CrossRef]  

29. S. Hong, R. Riedinger, I. Marinkovic, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” arXiv:1706.03777 (2017).

30. A. H. Ghadimi, D. J. Wilson, and T. J. Kippenberg, “Radiation and internal loss engineering of high-stress silicon nitride nanobeams,” Nano Lett. 17, 3501–3505 (2017). [CrossRef]  

31. C. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45, 75–79 (1980). [CrossRef]  

32. V. B. Braginsky and F. J. Khalili, “Gravitational wave antenna with QND speed meter,” Phys. Lett. A 147, 251–256 (1990). [CrossRef]  

33. C. Graf, B. W. Barr, A. S. Bell, F. Campbell, A. V. Cumming, S. L. Danilishin, N. A. Gordon, G. D. Hammond, J. Hennig, E. A. Houston, S. H. Huttner, R. A. Jones, S. S. Leavey, H. Luck, J. Macarthur, M. Marwick, S. Rigby, R. Schilling, B. Sorazu, A. Spencer, S. Steinlechner, K. A. Strain, and S. Hild, “Design of a speed meter interferometer proof-of-principle experiment,” Class. Quantum Grav. 31, 215009 (2014). [CrossRef]  

34. S. L. Danilishin, “Sensitivity limitations in optical speed meter topology of gravitational-wave antennas,” Phys. Rev. D 69, 102003 (2004). [CrossRef]  

35. P. Purdue and Y. Chen, “Practical speed meter designs for quantum nondemolition gravitational-wave interferometers,” Phys. Rev. D 66, 122004 (2002). [CrossRef]  

36. P. Purdue, “Analysis of a quantum nondemolition speed-meter interferometer,” Phys. Rev. D 66, 022001 (2002). [CrossRef]  

37. Y. Chen, “Sagnac interferometer as a speed-meter-type, quantum-nondemolition gravitational-wave detector,” Phys. Rev. D 67, 122004 (2003). [CrossRef]  

38. V. Braginskii, Y. I. Vorontsov, and F. Y. Khalili, “Optimal quantum measurements in detectors of gravitation radiation,” JETP Lett. 27, 296 (1978).

39. W. P. Bowen and G. J. Milburn, Quantum Optomechanics, 1st ed. (CRC Press, Taylor & Francis Group, 2016).

40. K. E. Khosla, M. R. Vanner, W. P. Bowen, and G. J. Milburn, “Quantum state preparation of a mechanical resonator using an optomechanical geometric phase,” New J. Phys. 15, 043025 (2013). [CrossRef]  

41. M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, “Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,” Nat. Commun. 4, 3295 (2013). [CrossRef]  

42. J. S. Bennett, K. Khosla, L. S. Madsen, M. R. Vanner, H. Rubinsztein-Dunlop, and W. P. Bowen, “A quantum optomechanical interface beyond the resolved sideband limit,” New J. Phys. 18, 053030 (2016). [CrossRef]  

43. T. P. Purdy, P.-L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X 3, 031012 (2013).

44. A. H. Safavi-Naeini, S. Groblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2013). [CrossRef]  

45. The optical annihilation operator after the beam splitter is af=0.99ai+0.01(di+β)ai+0.1β, where ai is the annihilation operator of the light in the top port, and di is the annihilation operator of the displacement point with coherent amplitude β. We have neglected the input vacuum noise (di) since it contributes only 0.01 vacuum noise to the final operator af.

46. A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587–616 (1983). [CrossRef]  

47. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd ed. (Springer, 1983).

48. R. Benguria and M. Kac, “Quantum Langevin equation,” Phys. Rev. Lett. 46, 1–4 (1981). [CrossRef]  

49. A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402 (2001). [CrossRef]  

50. M. R. Vanner, I. Pikovski, and M. S. Kim, “Towards optomechanical quantum state reconstruction of mechanical motion,” Ann. Phys. 527, 15–26 (2015). [CrossRef]  

51. A. R. Kermany, J. S. Bennett, G. A. Brawley, W. P. Bowen, and F. Iacopi, “Factors affecting the f x Q product of 3c-SiC microstrings: what is the upper limit for sensitivity?” J. Appl. Phys. 119, 055304 (2016). [CrossRef]  

52. E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482, 63–67 (2012). [CrossRef]  

53. D. J. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. J. Kippenberg, “Measurement-based control of a mechanical oscillator at its thermal decoherence rate,” Nature 524, 325–329 (2015). [CrossRef]  

54. P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011). [CrossRef]  

55. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011). [CrossRef]  

56. C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994). [CrossRef]  

57. K. Jahne, C. Genes, K. Hammerer, M. Wallquist, E. S. Polzik, and P. Zoller, “Cavity-assisted squeezing of a mechanical oscillator,” Phys. Rev. A 79, 063819 (2009). [CrossRef]  

References

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  8. S. A. McGee, D. Meiser, C. A. Regal, K. W. Lehnert, and M. J. Holland, “Mechanical resonators for storage and transfer of electrical and optical quantum states,” Phys. Rev. A 87, 053818 (2013).
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  19. Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nano 12, 776–783 (2017).
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  20. A. Naik, O. Buu, M. D. LaHaye, A. D. Armour, A. A. Clerk, M. P. Blencowe, and K. C. Schwab, “Cooling a nanomechanical resonator with quantum back-action,” Nature 443, 193–196 (2006).
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  21. T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
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  23. A. Kronwald, F. Marquardt, and A. A. Clerk, “Arbitrarily large steady-state bosonic squeezing via dissipation,” Phys. Rev. A 88, 063833 (2013).
  24. M. J. Woolley and A. A. Clerk, “Two-mode back-action-evading measurements in cavity optomechanics,” Phys. Rev. A 87, 063846 (2013).
  25. J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
    [Crossref]
  26. J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
    [Crossref]
  27. R. Riviere, S. Deleglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83, 063835 (2011).
    [Crossref]
  28. G. Anetsberger, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Cavity optomechanics and cooling nanomechanical oscillators using microresonator enhanced evanescent near-field coupling,” C. R. Phys. 12, 800–816 (2011).
    [Crossref]
  29. S. Hong, R. Riedinger, I. Marinkovic, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” arXiv:1706.03777 (2017).
  30. A. H. Ghadimi, D. J. Wilson, and T. J. Kippenberg, “Radiation and internal loss engineering of high-stress silicon nitride nanobeams,” Nano Lett. 17, 3501–3505 (2017).
    [Crossref]
  31. C. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45, 75–79 (1980).
    [Crossref]
  32. V. B. Braginsky and F. J. Khalili, “Gravitational wave antenna with QND speed meter,” Phys. Lett. A 147, 251–256 (1990).
    [Crossref]
  33. C. Graf, B. W. Barr, A. S. Bell, F. Campbell, A. V. Cumming, S. L. Danilishin, N. A. Gordon, G. D. Hammond, J. Hennig, E. A. Houston, S. H. Huttner, R. A. Jones, S. S. Leavey, H. Luck, J. Macarthur, M. Marwick, S. Rigby, R. Schilling, B. Sorazu, A. Spencer, S. Steinlechner, K. A. Strain, and S. Hild, “Design of a speed meter interferometer proof-of-principle experiment,” Class. Quantum Grav. 31, 215009 (2014).
    [Crossref]
  34. S. L. Danilishin, “Sensitivity limitations in optical speed meter topology of gravitational-wave antennas,” Phys. Rev. D 69, 102003 (2004).
    [Crossref]
  35. P. Purdue and Y. Chen, “Practical speed meter designs for quantum nondemolition gravitational-wave interferometers,” Phys. Rev. D 66, 122004 (2002).
    [Crossref]
  36. P. Purdue, “Analysis of a quantum nondemolition speed-meter interferometer,” Phys. Rev. D 66, 022001 (2002).
    [Crossref]
  37. Y. Chen, “Sagnac interferometer as a speed-meter-type, quantum-nondemolition gravitational-wave detector,” Phys. Rev. D 67, 122004 (2003).
    [Crossref]
  38. V. Braginskii, Y. I. Vorontsov, and F. Y. Khalili, “Optimal quantum measurements in detectors of gravitation radiation,” JETP Lett. 27, 296 (1978).
  39. W. P. Bowen and G. J. Milburn, Quantum Optomechanics, 1st ed. (CRC Press, Taylor & Francis Group, 2016).
  40. K. E. Khosla, M. R. Vanner, W. P. Bowen, and G. J. Milburn, “Quantum state preparation of a mechanical resonator using an optomechanical geometric phase,” New J. Phys. 15, 043025 (2013).
    [Crossref]
  41. M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, “Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,” Nat. Commun. 4, 3295 (2013).
    [Crossref]
  42. J. S. Bennett, K. Khosla, L. S. Madsen, M. R. Vanner, H. Rubinsztein-Dunlop, and W. P. Bowen, “A quantum optomechanical interface beyond the resolved sideband limit,” New J. Phys. 18, 053030 (2016).
    [Crossref]
  43. T. P. Purdy, P.-L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X 3, 031012 (2013).
  44. A. H. Safavi-Naeini, S. Groblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2013).
    [Crossref]
  45. The optical annihilation operator after the beam splitter is af=0.99ai+0.01(di+β)≈ai+0.1β, where ai is the annihilation operator of the light in the top port, and di is the annihilation operator of the displacement point with coherent amplitude β. We have neglected the input vacuum noise (di) since it contributes only 0.01 vacuum noise to the final operator af.
  46. A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587–616 (1983).
    [Crossref]
  47. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd ed. (Springer, 1983).
  48. R. Benguria and M. Kac, “Quantum Langevin equation,” Phys. Rev. Lett. 46, 1–4 (1981).
    [Crossref]
  49. A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402 (2001).
    [Crossref]
  50. M. R. Vanner, I. Pikovski, and M. S. Kim, “Towards optomechanical quantum state reconstruction of mechanical motion,” Ann. Phys. 527, 15–26 (2015).
    [Crossref]
  51. A. R. Kermany, J. S. Bennett, G. A. Brawley, W. P. Bowen, and F. Iacopi, “Factors affecting the f x Q product of 3c-SiC microstrings: what is the upper limit for sensitivity?” J. Appl. Phys. 119, 055304 (2016).
    [Crossref]
  52. E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482, 63–67 (2012).
    [Crossref]
  53. D. J. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. J. Kippenberg, “Measurement-based control of a mechanical oscillator at its thermal decoherence rate,” Nature 524, 325–329 (2015).
    [Crossref]
  54. P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
    [Crossref]
  55. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
    [Crossref]
  56. C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
    [Crossref]
  57. K. Jahne, C. Genes, K. Hammerer, M. Wallquist, E. S. Polzik, and P. Zoller, “Cavity-assisted squeezing of a mechanical oscillator,” Phys. Rev. A 79, 063819 (2009).
    [Crossref]

2017 (2)

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nano 12, 776–783 (2017).
[Crossref]

A. H. Ghadimi, D. J. Wilson, and T. J. Kippenberg, “Radiation and internal loss engineering of high-stress silicon nitride nanobeams,” Nano Lett. 17, 3501–3505 (2017).
[Crossref]

2016 (2)

J. S. Bennett, K. Khosla, L. S. Madsen, M. R. Vanner, H. Rubinsztein-Dunlop, and W. P. Bowen, “A quantum optomechanical interface beyond the resolved sideband limit,” New J. Phys. 18, 053030 (2016).
[Crossref]

A. R. Kermany, J. S. Bennett, G. A. Brawley, W. P. Bowen, and F. Iacopi, “Factors affecting the f x Q product of 3c-SiC microstrings: what is the upper limit for sensitivity?” J. Appl. Phys. 119, 055304 (2016).
[Crossref]

2015 (2)

M. R. Vanner, I. Pikovski, and M. S. Kim, “Towards optomechanical quantum state reconstruction of mechanical motion,” Ann. Phys. 527, 15–26 (2015).
[Crossref]

D. J. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. J. Kippenberg, “Measurement-based control of a mechanical oscillator at its thermal decoherence rate,” Nature 524, 325–329 (2015).
[Crossref]

2014 (3)

C. Graf, B. W. Barr, A. S. Bell, F. Campbell, A. V. Cumming, S. L. Danilishin, N. A. Gordon, G. D. Hammond, J. Hennig, E. A. Houston, S. H. Huttner, R. A. Jones, S. S. Leavey, H. Luck, J. Macarthur, M. Marwick, S. Rigby, R. Schilling, B. Sorazu, A. Spencer, S. Steinlechner, K. A. Strain, and S. Hild, “Design of a speed meter interferometer proof-of-principle experiment,” Class. Quantum Grav. 31, 215009 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Phys. Rev. A 90, 043848 (2014).
[Crossref]

2013 (8)

S. A. McGee, D. Meiser, C. A. Regal, K. W. Lehnert, and M. J. Holland, “Mechanical resonators for storage and transfer of electrical and optical quantum states,” Phys. Rev. A 87, 053818 (2013).
[Crossref]

T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
[Crossref]

A. Kronwald, F. Marquardt, and A. A. Clerk, “Arbitrarily large steady-state bosonic squeezing via dissipation,” Phys. Rev. A 88, 063833 (2013).

M. J. Woolley and A. A. Clerk, “Two-mode back-action-evading measurements in cavity optomechanics,” Phys. Rev. A 87, 063846 (2013).

T. P. Purdy, P.-L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X 3, 031012 (2013).

A. H. Safavi-Naeini, S. Groblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2013).
[Crossref]

K. E. Khosla, M. R. Vanner, W. P. Bowen, and G. J. Milburn, “Quantum state preparation of a mechanical resonator using an optomechanical geometric phase,” New J. Phys. 15, 043025 (2013).
[Crossref]

M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, “Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,” Nat. Commun. 4, 3295 (2013).
[Crossref]

2012 (5)

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482, 63–67 (2012).
[Crossref]

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref]

I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and C. Brukner, “Probing Planck-scale physics with quantum optics,” Nat. Phys. 8, 393–397 (2012).
[Crossref]

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108, 120801 (2012).
[Crossref]

E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nat. Nano 7, 509–514 (2012).
[Crossref]

2011 (9)

K. Stannigel, P. Rabl, A. S. Sorensen, M. D. Lukin, and P. Zoller, “Optomechanical transducers for quantum-information processing,” Phys. Rev. A 84, 042341 (2011).
[Crossref]

G. Milburn and M. Woolley, “An introduction to quantum optomechanics,” Acta Phys. Slovaca 61, 483–601 (2011).
[Crossref]

M. R. Vanner, I. Pikovski, G. D. Cole, M. S. Kim, C. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, “Pulsed quantum optomechanics,” Proc. Natl. Acad. Sci. USA 108, 16182–16187 (2011).
[Crossref]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref]

R. Riviere, S. Deleglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83, 063835 (2011).
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G. Anetsberger, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Cavity optomechanics and cooling nanomechanical oscillators using microresonator enhanced evanescent near-field coupling,” C. R. Phys. 12, 800–816 (2011).
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P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[Crossref]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

2010 (2)

J. B. Hertzberg, T. Rocheleau, T. Ndukum, M. Savva, A. A. Clerk, and K. C. Schwab, “Back-action-evading measurements of nanomechanical motion,” Nat. Phys. 6, 213–217 (2010).
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J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
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2009 (3)

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
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M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
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K. Jahne, C. Genes, K. Hammerer, M. Wallquist, E. S. Polzik, and P. Zoller, “Cavity-assisted squeezing of a mechanical oscillator,” Phys. Rev. A 79, 063819 (2009).
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2008 (2)

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref]

A. A. Clerk, F. Marquardt, and K. Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008).
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2006 (1)

A. Naik, O. Buu, M. D. LaHaye, A. D. Armour, A. A. Clerk, M. P. Blencowe, and K. C. Schwab, “Cooling a nanomechanical resonator with quantum back-action,” Nature 443, 193–196 (2006).
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2004 (1)

S. L. Danilishin, “Sensitivity limitations in optical speed meter topology of gravitational-wave antennas,” Phys. Rev. D 69, 102003 (2004).
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2003 (2)

Y. Chen, “Sagnac interferometer as a speed-meter-type, quantum-nondemolition gravitational-wave detector,” Phys. Rev. D 67, 122004 (2003).
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W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
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2002 (2)

P. Purdue and Y. Chen, “Practical speed meter designs for quantum nondemolition gravitational-wave interferometers,” Phys. Rev. D 66, 122004 (2002).
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P. Purdue, “Analysis of a quantum nondemolition speed-meter interferometer,” Phys. Rev. D 66, 022001 (2002).
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2001 (1)

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402 (2001).
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C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
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1983 (1)

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1981 (1)

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1980 (1)

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A. Naik, O. Buu, M. D. LaHaye, A. D. Armour, A. A. Clerk, M. P. Blencowe, and K. C. Schwab, “Cooling a nanomechanical resonator with quantum back-action,” Nature 443, 193–196 (2006).
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C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
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W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
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A. R. Kermany, J. S. Bennett, G. A. Brawley, W. P. Bowen, and F. Iacopi, “Factors affecting the f x Q product of 3c-SiC microstrings: what is the upper limit for sensitivity?” J. Appl. Phys. 119, 055304 (2016).
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V. B. Braginsky and F. J. Khalili, “Gravitational wave antenna with QND speed meter,” Phys. Lett. A 147, 251–256 (1990).
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A. R. Kermany, J. S. Bennett, G. A. Brawley, W. P. Bowen, and F. Iacopi, “Factors affecting the f x Q product of 3c-SiC microstrings: what is the upper limit for sensitivity?” J. Appl. Phys. 119, 055304 (2016).
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I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and C. Brukner, “Probing Planck-scale physics with quantum optics,” Nat. Phys. 8, 393–397 (2012).
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M. R. Vanner, I. Pikovski, G. D. Cole, M. S. Kim, C. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, “Pulsed quantum optomechanics,” Proc. Natl. Acad. Sci. USA 108, 16182–16187 (2011).
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A. Naik, O. Buu, M. D. LaHaye, A. D. Armour, A. A. Clerk, M. P. Blencowe, and K. C. Schwab, “Cooling a nanomechanical resonator with quantum back-action,” Nature 443, 193–196 (2006).
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A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587–616 (1983).
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M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
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C. Graf, B. W. Barr, A. S. Bell, F. Campbell, A. V. Cumming, S. L. Danilishin, N. A. Gordon, G. D. Hammond, J. Hennig, E. A. Houston, S. H. Huttner, R. A. Jones, S. S. Leavey, H. Luck, J. Macarthur, M. Marwick, S. Rigby, R. Schilling, B. Sorazu, A. Spencer, S. Steinlechner, K. A. Strain, and S. Hild, “Design of a speed meter interferometer proof-of-principle experiment,” Class. Quantum Grav. 31, 215009 (2014).
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C. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45, 75–79 (1980).
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A. H. Safavi-Naeini, S. Groblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2013).
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J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
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Y. Chen, “Sagnac interferometer as a speed-meter-type, quantum-nondemolition gravitational-wave detector,” Phys. Rev. D 67, 122004 (2003).
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P. Purdue and Y. Chen, “Practical speed meter designs for quantum nondemolition gravitational-wave interferometers,” Phys. Rev. D 66, 122004 (2002).
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M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, “Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,” Nat. Commun. 4, 3295 (2013).
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The optical annihilation operator after the beam splitter is af=0.99ai+0.01(di+β)≈ai+0.1β, where ai is the annihilation operator of the light in the top port, and di is the annihilation operator of the displacement point with coherent amplitude β. We have neglected the input vacuum noise (di) since it contributes only 0.01 vacuum noise to the final operator af.

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Figures (3)

Fig. 1.
Fig. 1. Schematic of the protocol. Half- and quarter-wave plates (HWP and QWP, respectively), a polarizing beam splitter (PBS), and a switchable beam splitter (SBS) are used to initially direct the interaction pulse along paths 1–4, along which the first optomechanical interaction takes place. When the interaction pulse reaches the top port of the highly reflective beam splitter (RBS), its coherent amplitude is changed by the displacement pulse. Since almost all of the interaction pulse reflects from the RBS, any quantum correlations between it and the oscillator remain, as the pulse interacts with the mechanical oscillator a second time. After the second optomechanical interaction, the SBS switches out the pulse to a homodyne measurement device instead of directing it along path four a second time.
Fig. 2.
Fig. 2. (a) Conditional variance for the measurement of different quadratures (ϕ=π/2,π/4, and π/8), using the double (solid lines) or single (dashed lines) interaction protocols. (b)–(d) Conditional variance as a function of mechanical quadrature angle, ϕ, for the double (solid lines) and single (dashed lines) interaction protocols for λ=10, λ=1, and λ=0.1 respectively. (a)–(d) Assume a bath temperature of 1 K, ωM/2π=100  kHz, and γ/2π=1  Hz. Gray dashed lines indicate the ground-state variance.
Fig. 3.
Fig. 3. Conditional variance of the final oscillator momentum using the double (solid lines) and single (dashed lines) interaction protocols at different temperatures. A temperature of 100 K corresponds to 1.3×103 phonons exchanged per cycle for ωM/2π=100  kHz and γ/2π=1  Hz.

Equations (12)

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HI/=g0α(a+a)(b+b)+g0α2(b+b).
U=exp[iλ2XMXL]eiθbbexp[iλ1XMXL]=R(θ)χ(λ1λ2)U(XLXMϕ),
XMXMcosθ+PMsinθXLλ1sinθ,
PMPMcosθXMsinθXL[λ2λ1cosθ],
XLXL,
PLPLGXMϕ+XLλ2λ1sinθ.
X˙M=ωMPM,
P˙M=ωMXMγPM+2γξ,
XM,out=XMcosθ+PMsinθ+ξXλ1XLsinθ,
PM,out=PMcosθXMsinθ+ξPXL(ηλ2λ1cosθ)λ21ηδX1,
XL,out=ηXL+ηη2δX1+1ηδX2,
PL,out=ηPL+ηη2δP1+1ηδP2+ηλ1λ2sinθXLGXMϕηλ2ξX,

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