## 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 [1–6]. Optomechanical systems have been proposed for quantum information applications [7–9] and tests of foundational physics [10,11] and are currently used for state-of-the-art sensors [12–14], 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 [2,16–19]. This notion is reflected in recent theoretical and experimental results [11,20–29].

In the high-temperature limit, the QCO regime is achieved when $Q{\omega}_{\mathrm{M}}>{k}_{\mathrm{B}}T/\hslash $ with $Q$ as the quality factor of the mechanical oscillator of frequency ${\omega}_{\mathrm{M}}$ 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 ${\omega}_{\mathrm{M}}Q\gtrsim 4\times {10}^{13}$ [18], which has been achieved at megahertz [19,30] frequencies but remains beyond current technology for low-frequency (${\omega}_{\mathrm{M}}/2\pi \lesssim 100\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$) 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 [34–37] 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 ${H}_{\mathrm{I}}=\hslash {g}_{0}{a}^{\u2020}a(b+{b}^{\u2020})$, where $a$ ($b$) is the annihilation operator for the optical (mechanical) mode and ${g}_{0}$ is the bare optomechanical interaction rate. We restrict the analysis to the linearized interaction where the optical field is linearized, $a\to \alpha +a$, about a time-dependent amplitude $\alpha (t)=\u27e8a(t)\u27e9$, where without loss of generality we define $\alpha $ to be real [39]. The pulse envelope $\alpha (t)$ is chosen to be a Gaussian with pulse width $\tau $, 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

The cavity is modeled as a single-sided cavity with decay rate $\kappa $, which is large enough for the optical pulse to adiabatically interact with the cavity $\tau \gg {\kappa}^{-1}$. In this regime, the intracavity field is proportional to the input field $\alpha (t)={\alpha}_{\mathrm{in}}(t)\sqrt{2/\kappa}$ with the input field is normalized $\int {|{\alpha}_{\mathrm{in}}|}^{2}\mathrm{d}t=\overline{N}$, where $\overline{N}$ is the mean photon number in the pulse. Other pulsed optomechanics protocols [40–42] assume the oscillator is frozen during a single-pulsed interaction, so that $\tau \ll 1/{\omega}_{\mathrm{M}}$ with the optomechanical system necessarily operating in the unresolved sideband limit, $\kappa \gg {\omega}_{\mathrm{M}}$. However in this work, we require the stricter condition $\kappa \gg \overline{n}\gamma $ (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 $U({X}_{\mathrm{M}}{X}_{\mathrm{L}})=\mathrm{exp}[-i\lambda {X}_{\mathrm{M}}{X}_{\mathrm{L}}]$, where $\sqrt{2}{X}_{\mathrm{M}}=b+{b}^{\u2020}$ and $\sqrt{2}{X}_{\mathrm{L}}=a+{a}^{\u2020}$ are the mechanical position and optical amplitude quadratures, respectively. The dimensionless constant $\lambda ={g}_{0}\int \alpha (t)\mathrm{d}t=\mathrm{sign}(\alpha )4{(2\pi )}^{1/4}{g}_{0}\sqrt{\tau \overline{N}/\kappa}$ is the optomechanical interaction strength [15]. Using a pulse shape other than a Gaussian simply changes the proportionality constant between $\lambda $ and ${g}_{0}\sqrt{\tau \overline{N}/\kappa}$ (where $\tau $ is now some characteristic time of the pulse shape). In the adiabatic limit, the pulse shape remains constant throughout the protocol [up to order $\mathcal{O}{(\kappa \tau )}^{-1}$], and the optimal pulse shape is one that maximizes $\int |\alpha (t)|\mathrm{d}t$ subject to the constraints $\int {|\alpha (t)|}^{2}\mathrm{d}t=1$ and $\tau \gg {\kappa}^{-1}$. The unitary correlates the optical phase quadrature (${P}_{\mathrm{L}}$) and mechanical position as ${U}^{\u2020}{P}_{\mathrm{L}}U={P}_{\mathrm{L}}-\lambda {X}_{\mathrm{M}}$ at the expense of adding back action to the momentum (${P}_{\mathrm{M}}$) of the oscillator ${U}^{\u2020}{P}_{\mathrm{M}}U={P}_{\mathrm{M}}-\lambda {X}_{\mathrm{L}}$. 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 $\theta ={\omega}_{\mathrm{M}}t$, 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

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 $\alpha (t)$ [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

For a single optomechanical interaction (${\lambda}_{2}=\theta =0$) the back-action noise (${X}_{\mathrm{L}}$) is necessarily imparted onto the momentum of the oscillator. However, if the interaction strengths are chosen such that ${\lambda}_{2}={\lambda}_{1}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta $, 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, $\theta \to 0$, at the expense of decreasing the overall interaction strength $\mathcal{G}$. 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 $({\lambda}_{1},{\lambda}_{2})$, 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 $\theta =\pi /2$ and ${\lambda}_{1}=0$ (wait then measure), correlating the phase quadrature with the initial momentum of the oscillator. Alternatively, momentum state preparation can be achieved with $\theta =\pi /2$ and ${\lambda}_{2}=0$ (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 $\theta =\pi /2$ 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 $\theta $—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 $\varphi =\pi /2$), 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 $\kappa \approx {\kappa}_{\mathrm{ex}}\gg {\kappa}_{0}$, where ${\kappa}_{\mathrm{ex}}$ and ${\kappa}_{0}$ 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]

where $\gamma $ is the oscillator decay rate and $\xi (t)$ is a zero-mean white noise operator with correlation function $\u27e8\xi (t)\xi ({t}^{\prime})\u27e9=(\overline{n}+\frac{1}{2})\delta (t-{t}^{\prime})$, where $\overline{n}\approx {k}_{\mathrm{B}}T/\hslash {\omega}_{\mathrm{M}}$ in the high-temperature limit. The thermal force satisfies $[{X}_{\mathrm{M}}(t),\xi (t)]=\sqrt{\gamma /2}i$ to preserve the commutation relations. Optical losses are modeled as a single beam splitter unitary ${\mathcal{B}}_{\eta}$ with intensity loss $\eta $ after each optomechanical interaction. The protocol is then given by ${\mathcal{B}}_{\eta}{U}_{2}{\mathcal{B}}_{\eta}{\mathcal{M}}_{\theta}{U}_{1}$, where ${U}_{i}$ is the $i$th optomechanical unitary, and ${\mathcal{M}}_{\theta}$ is a nonunitary map that describes the dissipative mechanical evolution given by Eqs. (4a) and (4b) over the rotation angle $\theta $. After the full protocol, the optical and mechanical quadratures are given by## 4. MEASUREMENT AND TOMOGRAPHY

For quantum measurement and tomography, the aim is to measure the statistics of the *a priori* mechanical state, ${X}_{\mathrm{M}}^{\varphi}$, for each $\varphi \in [0,\pi )$, without making any assumptions on the statistics of ${X}_{\mathrm{M}}^{\varphi}$. From Eq. (5d), a measurement of ${P}_{\mathrm{L},\mathrm{out}}$ (or indeed any optical quadrature ${X}_{\mathrm{L}}^{\phi}\ne {X}_{\mathrm{L}}$) is a measurement of ${X}_{\mathrm{M}}^{\varphi}$ 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 $C(A,B)\equiv \frac{1}{2}\u27e8AB+BA\u27e9-\u27e8A\u27e9\u27e8B\u27e9$ is the correlation between $A$ and $B$. When the conditional variance of the measured quadrature $V({X}_{\mathrm{M}}^{\varphi}|{X}_{\mathrm{L}}^{\phi})$ 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 ${\omega}_{\mathrm{M}}/2\pi =100\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$ and $\gamma /2\pi =1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{Hz}$ ($Q={10}^{5}$), similar to the silicon carbide resonators in Ref. [51]. At a temperature of 1 K, $2\pi \overline{n}/Q\approx 13$ phonons enter per oscillation, residing well outside of the QCO regime. The bare optomechanical coupling rate is conservatively set at ${g}_{0}/2\pi =1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{Hz}$ with a cavity decay rate of $\kappa /2\pi =1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{GHz}$ (optical $Q\approx 2.5\times {10}^{5}$ at 1064 nm). To ensure a fair comparison, we choose to relate the single-pulse interaction strength $\lambda $ to the two-pulse interaction strengths via $\lambda =\sqrt{{\lambda}_{1}^{2}+{\lambda}_{2}^{2}}$ 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 $\lambda \gtrsim 10$. In order to achieve this sub-ground-state resolution, the two pulses must interact with the mechanical oscillator within the thermal decoherence time.

An interaction strength of $\lambda =10$ requires a high pulse energy, $E=\hslash \omega \kappa {\lambda}^{2}/({g}_{0}^{2}\tau 16\sqrt{2\pi})\approx 14\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mJ}$ in a 5 ns pulse (assuming a temperature of 100 K with $\tau \ll \overline{n}\gamma $). However this is a consequence of our conservative choice of coupling strength ${g}_{0}/2\pi =1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{Hz}$. Coupling rates of kilohertz [52] and even up to 20 kHz [53] have been observed, and taking ${g}_{0}/2\pi =2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$ significantly reduces the pulse energy to 3.7 nJ for $\lambda =10$ or 370 nJ for $\lambda =100$. Combining this higher coupling rate with cryogenic (1 K) operation will enable longer optical pulses (increasing $\tau $ to 500 ns), further reducing the pulse energy 37 fJ for $\lambda =10$, 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, $\mathrm{V}({X}_{\mathrm{M},\mathrm{out}}^{\varphi}|{P}_{\mathrm{L}})$, instead of the *a priori* state, $\mathrm{V}({X}_{\mathrm{M}}^{\varphi}|{P}_{\mathrm{L}})$. 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.

Note that direct measurement and manipulation of the momentum of an oscillator over a short time scale ($\theta /{\omega}_{\mathrm{M}}$) 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 $\sqrt{200}\sqrt{\hslash m\omega /2}$ at 100 K (from Fig. 3 with $\lambda =1$) corresponds to a thermally limited impulsed force sensitivity of 8 pN over $\frac{1}{50}$th 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 $\overline{n}\gamma =1.3\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{MHz}$ (at 1 K) for the oscillators considered here. By using a high-$Q$ low-frequency oscillator, the lifetime of a quantum state $1/\overline{n}\gamma =\hslash Q/{k}_{\mathrm{B}}T$ 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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