Measuring mechanical motion with a single spin

We study theoretically the measurement of a mechanical oscillator using a single two level system as a detector. In a recent experiment, we used a single electronic spin associated with a nitrogen vacancy center in diamond to probe the thermal motion of a magnetized cantilever at room temperature {Kolkowitz et al., Science 335, 1603 (2012)}. Here, we present a detailed analysis of the sensitivity limits of this technique, as well as the possibility to measure the zero point motion of the oscillator. Further, we discuss the issue of measurement backaction in sequential measurements and find that although backaction heating can occur, it does not prohibit the detection of zero point motion. Throughout the paper we focus on the experimental implementation of a nitrogen vacancy center coupled to a magnetic cantilever; however, our results are applicable to a wide class of spin-oscillator systems. Implications for preparation of nonclassical states of a mechanical oscillator are also discussed.


Introduction
Recent interest in mechanical oscillators coupled to quantum systems is motivated by quantum device applications and by the goal of observing quantum behavior of macroscopic mechanical objects. The past decade has seen rapid progress studying mechanical oscillators coupled to quantum two-level systems such as superconducting qubits [1,2,3], and single electronic spins [4], and theoretical work has explored strong mechanical coupling to collective atomic spins [5,6]. Recently, it was proposed that a mechanical oscillator could be strongly coupled to an individual spin qubit [7,8]. Experiments based on single spins coupled to mechanical systems have demonstrated scanning magnetometry [9], mechanical spin control [10], and detection of mechanical motion [11,12]. In parallel, pulsed spin control techniques have attracted renewed interest for decoupling a spin from low frequency noise in its environment, extending its coherence [13], while also enhancing the sensitivity of the spin for magnetometry [14,15,16,17].
In this paper we consider pulsed single spin measurements applied to the detection of mechanical motion at the single phonon level. We extend the analysis presented in our recent work [12], providing a detailed theoretical framework and a discussion of measurement backaction. The central concept of our measurement approach is to apply a sequence of control pulses to the spin, synchronizing its dynamics with the period of a magnetized cantilever, thereby enhancing its sensitivity to the motion. By measuring the variance of the accumulated phase imprinted on the spin by the oscillator during a measurement, we directly probe the average phonon number, despite the fact that the oscillator position is linearly coupled to the transition frequency of the spin. We derive the conditions for observing a single phonon using the spin as a detector, and find that these conditions coincide with that of large effective cooperativity, sufficient to perform a two-spin gate mediated by mechanical motion [18]. Further, we consider the backaction arising from sequential measurements and show that this does not prohibit single phonon resolution. Throughout the paper, we focus on the specific spin-oscillator system of a magnetized cantilever coupled to the electronic spin associated with a nitrogen-vacancy (NV) center in diamond. For realistic experimental parameters we find that this system can reach the regime of large cooperative spin-phonon coupling, and the spin may be used to measure and manipulate mechanical motion at the quantum level.
We begin in Sec. 2 by introducing the coupled system and spin control sequences, and calculate the signal due to thermal and driven motion of the oscillator. Then in Sec. 3 we derive the optimal phonon number sensitivity, and show the relation between strong cooperativity and single phonon resolution. Finally, in Sec. 4 we consider the limit of zero temperature and calculate the signal due to zero point motion, including a discussion of backaction heating for sequential measurements.

Model
We consider the setup shown schematically in Fig. 1, in which a magnetized cantilever is coupled to the electronic spin of a single NV center. The magnetic tip generates a field gradient at the location of the NV, and as a result its motion modulates the magnetic field seen by the spin causing Zeeman shifts of its precession frequency. To lowest order in small cantilever motion, the precession frequency depends linearly on the position of the tip and is described by the Hamiltonian (h = 1) whereσ z is the Pauli operator of the spin andâ is the annihilation operator of the oscillator. For a spin associated with an NV center in diamond, we take |↑ = |m s = 1 and |↓ = |m s = 0 in the spin-1 ground state of the NV center, and safely ignore the |m s = −1 state assuming it is far detuned by an applied dc magnetic field. ∆ is the detuning of the microwave pulses used for spin manipulation, which plays no role in what follows and we take ∆ = 0 throughout the paper. The spin-oscillator coupling strength is λ = g e µ B G m x 0 /h, where g e ≈ 2 is the Landé g-factor, µ B is the Bohr magneton, G m is the magnetic field gradient along the NV axis, and x 0 = h/2mω 0 is the zero point motion of the cantilever mode of mass m and frequency ω 0 (we includedh in the definitions of λ and x 0 for clarity). The damped, driven oscillator is described bŷ describes dissipative coupling to a bath of oscillatorsb k , characterized by damping rate γ and temperature T . Finally,Ĥ dr describes a coherent oscillator drive which we consider briefly in Sec. 2.4. Note that in Eq. (1) we have temporarily omitted intrinsic spin decoherence due to the environment; we will include this explicitly in Sec. 3.

Spin echo and multipulse sequences
The motion of the oscillator imprints a phase on the spin as it evolves under Eq. (1), which can be detected using spin echo [3,19], or more generally a multiple pulse measurement. Throughout the paper we focus on Carr-Purcell-Meiboom-Gill (CPMG) type pulse sequences, consisting of equally spaced π pulses at intervals of time τ , as depicted in Fig. 1. After initialization in |↑ , a π/2 pulse prepares the spin in an eigenstate ofσ x , |ψ 0 = 1 2 (|↑ + |↓ ) with ψ 0 |σ x |ψ 0 = 1. The spin is then allowed to interact with the oscillator for time t, accumulating a phase, and during which time we apply a sequence of π pulses which effectively reverse the direction of spin precession. At the end of the sequence, a final π/2 pulse converts the accumulated phase into  Figure 1. (a) Schematic of the setup. A single spin can be used to measure mechanical motion via magnetic coupling. (b) Toggling sign of the interaction describing π pulses flipping the spin. Each sequence begins and ends with π/2 pulses, and π pulses flip the sign of the interaction at regular intervals of time τ . Thin dashed line shows oscillator position, which is synchronized with pulse sequence for τ = π/ω 0 as shown. The total sequence time is t = 2τ for spin echo and t = N τ for CPMG.
a population in |↑ which is then read out. By applying both initial and final π/2 rotations about the same axis, we measure the probability to find the spin in its initial state |ψ 0 at the end of the sequence, given by where angle brackets denote the average over spin and oscillator degrees of freedom. Our choice to measureσ x probes the accumulated phase variance; this is crucial for our purpose because the average phase imprinted by an undriven fluctuating oscillator is zero. In contrast, by applying the first and final π/2 pulses about orthogonal axes one would instead measureσ y , which probes the average accumulated phase. The sensitivity of the spin to mechanical motion is determined by the impact of the oscillator on the spin coherence σ x (t) . The key to maximizing this impact is to synchronize the spin evolution with the mechanical period using a CPMG sequence of π pulses, increasing the accumulated phase variance and improving the sensitivity as discussed in the context of ac magnetometry [20]. Choosing τ = π/ω 0 between the π pulses, we flip the spin every half-period of the oscillator and maximize the accumulated phase variance. At the same time, these pulse sequences decouple the spin from lowfrequency magnetic noise of the environment, extending the spin coherence time T 2 [15,16]. We describe the effects of the applied π pulses using a function f (t, τ ), which flips the sign of the spin-oscillator interaction at regular intervals of time τ as illustrated in Fig. 1. In this toggling frame, the interaction Hamiltonian iŝ whereX =â +â † andX(t) = e iĤosctX e −iĤosct . We calculate the spin coherence, and T denotes time ordering. Since the interaction is proportional toσ z , it leads to pure dephasing and we obtain [21] σ where we used σ x (0) = 1, the average · osc is over oscillator degrees of freedom,T denotes anti-time ordering, and the accumulated phase operator iŝ The spin coherence in Eq. (6) can be calculated using a cumulant expansion, which is vastly simplified by noting that the full Hamiltonian in Eq. (1), including the oscillator drive and ohmic dissipation, is quadratic inX. As a result, the second cumulant-which in general corresponds to a Gaussian approximation-in the present case constitutes the exact result. We use this below to calculate the coherence for both thermal and driven motion.
Another consequence of the fact thatĤ is quadratic inX is that the effect of the pulse sequence is completely characterized by its associated filter function [20,22], The filter function describes how twotime position correlations X (t)X(t ′ ) of the oscillator affect the spin coherence in the second cumulant in the expansion of Eq. (6). For the pulse sequences illustrated in Fig. 1, the corresponding filter functions are Note that phase-alternated versions of CPMG, such as XY4, which vary the axis of π pulse rotation in order to mitigate pulse errors, are also described by the above model in the limit of ideal pulses.

Thermal motion
As discussed above, the spin coherence in Eq. (6) is given exactly by its second order cumulant expansion. Since the total sequence time is t = Nτ , the coherence depends only on the time τ between π pulses, where andS X (ω) = dte iωt 1 2 {X(t),X(0)} is the symmetrized noise spectrum ofX. For the damped thermal oscillator described byĤ osc in the abscence of a drive, the symmetrized spectrum is (k B = 1) where γ = ω 0 /Q is the mechanical damping rate due to coupling to the ohmic environment at temperature T . We plot the spin coherence due to thermal motion in the classical limit T ≫ ω 0 in Fig. 2. The impact of the oscillator is greatest when the pulse sequence is synchronized with the cantilever frequency, τ = (2k + 1)π/ω 0 with k an integer. At times τ = 2kπ/ω 0 , the accumulated phase due the oscillator cancels within each free precession time, so that the accumulated phase variance averages nearly to zero and the coherence revives. We stress that this structure of collapse and revival can arise from purely classical motion; it is simply a consequence of averaging the phase variance accumulated by the spin over Gaussian distributed magnetic field fluctuations with a characteristic frequency. In addition to collapses and revivals, the finite Q of the cantilever also causes dephasing of the spin which leads to an exponential decay factor of the envelope as e −Γ φ τ . In the limit Q ≫ 1 and T > ω 0 , the dephasing rate is given by where η = λ/ω 0 is the dimensionless coupling strength andn th = (e ω 0 /T − 1) −1 is the thermal occupation number of the oscillator. We provide a derivation of Eq. (12) in Appendix A. Increasing Q not only increases the depth of the collapses in spin coherence due to the oscillator, but also decreases the overall spin dephasing resulting in more complete revivals, as shown in Fig. 2. We also see that increasing the temperature increases both the depth of collapse and the dephasing. Below in Sec. 3 we use these results to calculate the lowest temperature motion that can be detected, characterized by the phonon number sensitivity at the optimal pulse timing τ = π/ω 0 .

Driven motion
It is straightforward to include the effects of a classical drive through H dr in Eq. (1). This simply adds a classical deterministic contribution toX(t), and we can decompose the accumulated phase in Eq. (7) it asφ = φ dr +φ th where is the classical accumulated phase due to the drive. Here, A is the dimensionless amplitude of driven motion and θ 0 is its phase at the start of a particular measurement. We assume that the cantilever drive is not phase-locked to the pulse sequence, so θ 0 is random and uniformly distributed between 0 and 2π. Using Eq. (6) and averaging over θ 0 we obtain where J 0 is the zeroth order Bessel function [15], a(τ ) = ηA 2F (ω 0 τ ), and χ N (τ ) is the thermal contribution given by Eq. (10). For a strong drive, thermal fluctuations are unimportant and the signal is given by the Bessel function. For a weak drive, comparable to thermal motion with |A| 2 ∼n th , both thermal and driven contributions may be important as illustrated in Fig. 3 and observed in experiment [12]. In Fig. 3 we see that, unlike thermal motion (see Fig. 2), driven motion can lead to dips in the spin coherence below zero. In the remainder of the paper we focus on detecting undriven thermal or quantum motion with the drive switched off.

Phonon number sensitivity
In this section we discuss the sensitivity limits of the spin used as a detector of undriven mechanical motion. By comparing the signal from thermal motion to the relevant noise sources, we obtain the phonon number sensitivity. We then discuss the sensitivity in several limits relevant to experiments.

Signal
The impact of an undriven thermal oscillator on the spin coherence in a spin echo or CPMG measurement sequence is described by Eqs. (4) and (9). In addition to its coupling to the oscillator, the spin is also coupled to an environment which leads to intrinsic decoherence and degrades the signal. For an NV center, decoherence or T 2 processes are caused by a 1% natural abundance of 13 C nuclear spins in the otherwise 12 C lattice. Flip-flop processes between pairs of these nuclear spins produce low frequency magnetic noise which leads to decoherence of the form e −N (τ /T 2 ) 3 for a CPMG sequence with N pulses [20,22]. Note that T 2 here refers to the decoherence time in a spin echo sequence (i.e. N=1), typically ∼ 100 µs in natural diamond and up to ∼ 2 ms in isotopically pure diamond [23]. An added benefit of multipulse sequences is the enhanced spin coherence time,T 2 = N 2/3 T 2 , due to dynamical decoupling [13]. Finally, spin-lattice relaxation due to phonon processes leads to exponential decay on a timescale T 1 , typically ∼ 1 ms at room temperature and up to ∼ 200 s at 10 K [24]. Including these intrinsic sources of spin decoherence, as well as the oscillator-induced decoherence Γ φ given in Eq. (12), the probability to find the spin in its initial state given in Eq. (4) is modified as where we have isolated the coherent signal due to the oscillator, Note that we have accounted for the oscillator-induced decoherence Γ φ τ which diminishes the coherent signal we are interested in.
We can obtain a simple analytic expression for the signal in the limit Q ≫ 1. In this limit the oscillator spectrum is well-approximated by Lorentzians at ω = ±ω 0 , Using Eq. (17) with Eq. (10) we obtain a compact analytic expression for χ with no further approximation, which we provide in Appendix A. We choose the pulse timing τ to maximize the impact of the oscillator motion on the spin coherence, providing optimal sensitivity. This is achieved by setting τ = π/ω 0 , flipping the spin every half period of the oscillator and resulting in the maximum accumulated phase variance. For N ≫ 1, the filter function with τ = π/ω 0 is well-approximated by a Lorentzian centered at ω 0 of bandwidth bω 0 /N, where b ≃ 1.27. Together with Eq. (17) this yields and substituting this into Eq. (16) we obtain the signal.

Sensitivity
To find the sensitivity we must account for noise. We combine spin projection and photon shot noise into a single parameter K so that the noise averaged over M measurements is σ = 1/K √ M , where M = t tot /Nτ is the number of measurements of duration Nτ that can be performed in a total time t tot . It follows that the minimum number of phonons that we can resolve in a given time t tot is and the corresponding phonon number sensitivity is ξ =n min √ t tot . Using Eqs. (16) and (18) with τ = π/ω 0 we obtain where we have expressed the total spin dephasing in terms of a single pulse number, which combines both intrinsic and oscillator-induced decoherence. Eqs. (20) and (21) reflect the competition between the oscillator damping rate γ = ω 0 /Q, the intrinsic decoherence times T 1 and T 2 of the spin, and the measurement bandwidth bω 0 /N. It is clear from Eq. (18) that increasing the number of pulses increases the coherent signal due to the oscillator; however, this also leads to increased spin decoherence. As a result, the resolvable phonon number is minimized at an optimal number of pulses, Note that the optimal pulse number is always set by the spin decoherence, N opt ∼ N φ , with only a prefactor of order one depending on Q. Neglecting pulse imperfections, the optimized sensitivity is determined by an interplay of Q, T 1 and T 2 in Eq. (20). In practice, the optimal pulse number may be very large due to long spin coherence times, and pulse errors may play a role as discussed further below. In Fig. 4 we plot the sensitivity as a function of pulse number N, and the optimized sensitivity as a function of coupling strength λ. To check the validity of the above approximations it is straightforward to calculate the phonon number sensitivity directly from Eqs. (10) and (11). The numerically exact sensitivity is shown in Fig. 4 in agreement with our analytic results. In the remainder of this section we discuss the sensitivity in several experimentally relevant limits.

Optimal sensitivity and cooperativity
An important limit for current experiments is one where the spin coherence is much longer than the oscillator coherence during the measurement, corresponding to N φ > Q. We assume that the spin coherence is dominated by intrinsic sources described by T 1 and T 2 , and that the oscillator-induced spin decoherence Γ φ can be neglected, well-justified in the limit of weak coupling. Within these limits, the optimal number of pulses is N opt ∼ N φ and the optimized sensitivity is where the cooperativity is opt T 2 is the enhanced spin coherence time due to decoupling. For a large number of pulses, the enhanced spin coherence N 2/3 T 2 may be very long, and ultimately the spin coherence may be limited by T 1 which is not suppressed by decoupling. In this case Eqs. (23) and (24) are simply modified byT 2 → T 1 . The cooperativity parameter C is ubiquitous in quantum optics, and marks the onset of Purcell enhancement in cavity quantum electrodynamics. In the present case, C > 1 is the requirement for a single phonon to strongly influence the spin coherence, leading to a measurable signal despite the relatively short coherence time of the oscillator. The condition C > 1 to resolve a single phonon can be simply understood: if the spin coherence is much longer than the oscillator coherence, i.e. Q ≪ N φ , the accumulated phase variance increases at a rate ∼ λ 2 /γ (see Eq. (18) with sequence time Nτ ∼ N/ω 0 ) and the maximum interrogation time (assuming that oscillator-induced decoherence is negligible) isT 2 .
With feasible experimental parameters,T 2 ∼ T 1 ∼ 10 ms, λ/2π ∼ 150 Hz, ω 0 /2π ∼ 1 MHz and Q ∼ 1000, a cooperativity of C ∼ 1 can be reached. In current experiments, NV centers exhibit a 30% contrast in spin-dependent fluorescence, and collection efficiencies of 5% are realistic [20,25]. These parameters yield K ∼ 0.3 and an optimal phonon number sensitivity of ξ opt ∼ 1/ √ Hz with N ∼ N φ ∼ 15000 pulses. Due to long spin coherence times T 1 and T 2 , the optimal pulse number N φ may be very large, and in practice finite pulse errors may play an important role in limiting the spin coherence. For example if the number of pulses is limited to N ∼ 1000, a sensitivity of ξ ∼ 3/ √ Hz can be reached. We discuss this further below when we calculate the signal due to zero point motion.

Ideal oscillators and ideal spin qubits
While the cooperativity regime describes an important part of parameter space, it is useful to briefly consider two more simple limits that describe features in Fig. 4. First, we consider a harmonic oscillator that remains coherent for a much longer time than the entire pulse sequence, satisfying Q ≫ N. In this limit, the long oscillator coherence time plays plays no role and the optimal sensitivity is limited only by the spin coherence, ξ opt ∼ 1/(Kλ 2T 2 2 ω 0 /N). This limit can be seen on the left side of Fig. 4a, where the sensitivities for different values of Q fall on the same curve at low pulse numbers N.
Finally, we consider the limit of very strong but incoherent coupling where the spin decoherence is dominated by the oscillator, i.e. Γ φ becomes larger than 1/T 1 and 1/T 2 . This limit is reached when either the intrinsic spin decoherence is negligible or for very strong coupling, η 2n th ≫ Q/(ω 0 T 2 ) 3 , Q/ω 0 T 1 . In this limit, the coherent signal is large due to strong coupling, but saturates at a low number of pulses; further increasing the coupling strength only increases the oscillator-induced decoherence, reducing the signal. This is reflected in Fig. 4b, where we see that increasing the coupling strength larger than η 2 > 1/γn th T 1 no longer improves the optimized sensitivity but instead degrades it.

Detecting quantum motion
Above we found that for realistic experimental parameters, a single phonon can be resolved in one second of averaging time. This raises the intriguing question of whether a single spin can be used to sense the quantum zero point motion of an oscillator in its ground state. It also implies that we must consider the effect of measurement backaction, which we have so far ignored in our discussion. To address these questions we analyze the experimentally relevant scenario where the spin is used to detect the motion of a mechanical resonator which is externally cooled close to its ground state.

Measuring a cooled oscillator
Even at cryogenic temperatures, a mechanical oscillator of frequency ω 0 /2π ∼ MHz has an equilibrium occupation numbern th much larger than one. For this reason we assume that the mechanical oscillator is cooled from its equilibrium occupationn th to a much lower valuen 0 ∼ 1 using either optical cooling techniques [26] or the driven spin itself [7,27]. An important consequence of cooling below the environmental temperature is the effective reduction in Q of the oscillator. For an oscillator coupled to both a thermal environment and an external, effective zero temperature source for cooling, the mean phonon number satisfies where γ cool is the cooling rate. The steady state occupation number is and in order to maintainn 0 < 1 we require γ cool > γn th . As a result, the relevant decoherence rate of the oscillator is the rethermalization rate γn th . For this reason, to calculate the signal from a cooled oscillator we replace the equilibrium thermal occupation numbern th by the effective occupationn 0 → 0 in all expressions, while at the same time replacing the intrinsic Q by the reduced, effective quality factor Q eff = ω 0 /γ cool ≈ Q/n th .

Single shot readout
In Sec. 3 we calculated the sensitivity ξ, which reflects the minimum detectable phonon number n min that can be resolved in one second of averaging time. For the following discussion it is useful to convert the sensitivity to a minimum detectable phonon number per single measurement shot,n min,1 = ξ/ (Nπ/ω 0 ), where we have taken the total measurement time to be t tot = Nτ and τ = π/ω 0 . Assuming single shot spin readout (K → 1), which has been demonstrated at low temperature [25], and using Eq. (20) we obtainn where C eff = λ 2T 2 /γn th is the reduced, effective cooperativity. We see that under the assumption N ∼ N opt ≫ Q eff , the ability to resolve ground state fluctuations of a cooled oscillator within a few spin measurements requires C eff > 1, which is the same strong cooperativity condition required to perform a quantum gate between two spins mediated by a mechanical oscillator [18]. Alternatively,n min,1 corresponds to the occupation number required to produce a signal S of order one in Eq. (16). It provides a convenient way to directly compare the sensitivity with the backaction due to sequential measurements, as discussed below.
In Fig. 5 we plot the calculated signal due to zero point motion, assuming that the mechanical oscillator is cooled near its ground staten 0 = 0 and using the reduced quality factor Q eff . These plots show that the intrinsic coherence times typical for NV centers are more than sufficient to resolve single phonons provided enough pulses can be applied to exploit the full spin coherence. In practice, the limiting factor is likely to be finite pulse errors, which limit the absolute number of pulses that can be applied before losing the spin coherence. To estimate the effect of finite pulse errors, we include the calculated signal assuming additional spin decoherence of the form e −N/Nc with a cutoff pulse number N c . Pulse numbers of N ∼ 160 have been demonstrated in experiment [13], and with further improvements this can be increased to more than N ∼ 1000. Based on this we plot the modified signal using N c ∼ 1000 and find that even with a limited number of pulses, zero point motion results in a significant signal for realistic coupling strengths.

Backaction
The result that a single spin magnetometer can resolve the quantum zero point motion of a mechanical oscillator calls for a discussion of measurement backaction. We begin by noting that, despite the linear coupling of the spin to the oscillator position in Eq. (1), the described measurement protocol is sensitive to the variance of the accumulated phase ∼ X 2 , which we obtain by averaging independent spin measurements. As a result, our approach does not correspond to standard continuous position measurement [28], nor does it implement a quantum nondemolition measurement of the phonon number, since the interaction in Eq. (1) does not commute withn. In principle, by cooling between measurements our approach may be used to measure the phonon number with arbitrary precision. Nonetheless, the effect of the spin's backaction on the oscillator is both a practical issue and interesting in itself, and could be used to prepare nonclassical mechanical states. We describe two possible approaches to observe the influence of measurement backaction on the oscillator. First, we consider directly probing the projective nature of the measurement. For simplicity we assume that the oscillator is initially in its ground state and decoupled from the environment, and assume single shot spin readout. In a single measurement sequence, the oscillator experiences a spin-dependent force according to Eq. (5). Measuring σ x = ±1 at the end of the sequence projects the oscillator onto a superposition of coherent states [6,29], where α = Nλ/2ω 0 is the total displacement for a sequence of N ≫ 1 pulses and τ = π/ω 0 . The probabilities to measure |± are given by which shows, consistent with the discussion above, that for a measurement strength α > 1 the oscillator in its ground state can significantly affect the spin dynamics.
To observe the backaction of this measurement on the oscillator we can perform a second spin measurement, which is sensitive to the state of the oscillator conditioned on the first measurement. In principle, by using techniques developed in cavity quantum electrodynamics, this procedure can be used to fully reconstruct the conditionally prepared oscillator state [30].
Let us now consider an alternative, indirect way to observe backaction by performing many successive measurements. Again beginning with the oscillator near its ground state, the first measurement projects the oscillator into one of the states |ψ ± . By averaging over the two possible spin measurement outcomes, the resulting mixed oscillator state is and we see that on average the oscillator energy has increased by |α| 2 . Repeating this measurement many times, without cooling between measurements, the oscillator amplitude undergoes a random walk of stepsize ±α, and on average the phonon number increases approximately linearly in time. This corresponds to backaction heating described by an effective diffusion rate, Combining the measurement backaction with intrinsic mechanical dissipation and external cooling, the average occupation number satisfies and for γ cool ≫ γ the steady state phonon number added due to backaction is We see that increasing the coupling strength not only improves the single shot resolution n min,1 , but also leads to backaction heating of the oscillator. For sufficiently strong coupling, the steady state backaction phonon numbern ba exceeds the phonon number resolution, and the inferred phonon number is determined by backaction. We thus take the sumn meas =n min,1 +n ba as a measure of the minimum inferred phonon number. Note that for simplicity in this discussion we have assumed the limit N ≪ Q eff , in which the oscillator is coherent within each measurement sequence. Within this limit we find The total inferred phonon numbern meas is shown in Fig. 6 as a function of the coupling parameter η and a fixed number of pulses N = Q eff /5. In this casen meas is minimized for η ∼ 1/ √ Q eff , where it reaches a value ofn meas ∼ O(1). Observing this minimum in the phonon number resolution as a function of coupling strength would provide an indirect signature of measurement backaction. This observation may be more feasible in near term experiments than directly observing projective backaction as discussed above.

Summary and conclusions
We have presented the sensitivity limits of a novel position sensor consisting of a single spin. For realistic experimental parameters, we predict that a single NV center in diamond can be used to resolve single phonons in a cooled, magnetized mechanical cantilever. The condition to resolve single phonons is that of strong effective cooperativity, the same condition needed to perform a quantum gate between two spins mediated by a mechanical oscillator. For even stronger coupling, the backaction of the spin on the oscillator can be probed directly or indirectly, and used to prepare nonclassical mechanical states. Appendix A. Analytic signal for thermal motion in high Q limit Here we sketch the derivation of Eqs. (12) and (18). The impact of the oscillator on the spin coherence is given by Eqs. (10) and (11), (A.1) To perform this integral it is useful to decompose the filter function as where s j = (j + 1/2)τ and t j = (N − j)τ . We first consider the high temperature limit, T ≫ ω 0 , in which we can approximate coth(ω/2T ) ≈ 2T /ω. The result is a sum of integrals of the form which can be done exactly. In the limit Q ≫ 1 we obtain q(t) = γt + 1 − e −γt/2 cos(ω 0 t) − 4γ 3ω 0 e −γt/2 sin(ω 0 t).