Nonlinear optomechanical measurement of mechanical motion

Precision measurement of nonlinear observables is an important goal in all facets of quantum optics. This allows measurement-based non-classical state preparation, which has been applied to great success in various physical systems, and provides a route for quantum information processing with otherwise linear interactions. In cavity optomechanics much progress has been made using linear interactions and measurement, but observation of nonlinear mechanical degrees-of-freedom remains outstanding. Here we report the observation of displacement-squared thermal motion of a micro-mechanical resonator by exploiting the intrinsic nonlinearity of the radiation-pressure interaction. Using this measurement we generate bimodal mechanical states of motion with separations and feature sizes well below 100 pm. Future improvements to this approach will allow the preparation of quantum superposition states, which can be used to experimentally explore collapse models of the wavefunction and the potential for mechanical-resonator-based quantum information and metrology applications.

Precise measurement of non-linear observables is an important goal in all facets of quantum optics.This allows measurement-based nonclassical state preparation, which has been applied to great success in various physical systems [1][2][3][4], and provides a route for quantum information processing with otherwise linear interactions [5].In cavity optomechanics much progress has been made using linear interactions and measurement [6][7][8], but observation of nonlinear degrees-of-freedom, such as phonon number [9,10], remains outstanding.Here we report the observation of displacement-squared thermal motion of a micro-mechanical resonator by exploiting the optical non-linearity of the radiation pressure interaction [11].Using this measurement, we conditionally prepare bimodal mechanical states of motion with separations and feature sizes well below 100 pm.Future improvements to this approach will allow the preparation of quantum superposition states [12], which can be used to experimentally explore collapse models of the wavefunction [11,[13][14][15][16][17] and the potential for mechanical-resonator based quantummetrology applications.
In cavity optomechanics light circulating inside an optical resonator is used to manipulate and measure the motion of a mechanical element via the radiationpressure interaction.The Hamiltonian for such a cavityoptomechanical system is which includes the energy of the mechanical harmonic oscillator, the radiation-pressure interaction, and a continuous resonant optical drive in a frame rotating at the cavity resonance frequency.Here, a (b) is the optical (mechanical) annihilation operator, ω M is the mechanical angular frequency, κ is the optical amplitude decay rate of the cavity, N is the equalibrium mean photon number and g 0 is the zero-point optomechanical coupling rate.Quite generally, optomechanics experiments thus far [6][7][8] can be described with a linearised model of the interaction, where mechanical displacements give rise to displacements of the optical phase quadrature and the optical amplitude quadrature is unchanged.Fundamentally however, the radiation-pressure interaction is non-linear, and generates mechanical position dependent rotations of the intracavity optical quadratures, as is illustrated in Fig. 1(a).For small changes in the mechanical position, the optical phase quadrature changes linearly in proportion to the mechanical displacement, and the optical amplitude quadrature reduces in proportion to the mechanical displacement squared.By choosing which optical quadrature to observe with homodyne detection, one may selectively measure the mechanical position or displacement-squared [11].When performing an optical amplitude quadrature measurement, linear information about the mechanical position is suppressed, and provided sufficient displacement-squared measurement sensitivity, superposition states can be generated by the nonlinear measurement [12].
In a regime where κ ω M the intracavity field can be approximated as a √ N /(1 − iλx M ) + ξ, where λ = √ 2g 0 /κ quantifies the optomechanical interaction strength, ξ is the intracavity noise term and x M is the mechanical displacement in units of quantum noise.Taylor expanding this expression, the time-dependent optical output quadratures are then where X L = (a + a † )/ √ 2, and Conventionally, experimental optomechanics has focused on the leading, linear term in the expansion of the phase quadrature.However the higher-order terms in mechanical displacement can be associated with spectral peaks at the respective multiples of the mechanical resonance frequency.Precision measurement of these higher-order terms can enable conditional non-Gaussian state preparation.Here we use the quadratic term to conditionally prepare classical bimodal states from an initial roomtemperature thermal state of mechanical motion.
A schematic of our non-linear optomechanics experiment is shown in Fig. 1(b).We use a near-field cavity optomechanical setup [18], where a mechanical SiN nanostring oscillator [19] is placed in close proximity to a 60 µm diameter optical microsphere resonator and interacts with the optical cavity field via the optical evanescent field (Fig. 1(c)).The nanostring has dimen- The intrinsic optical non-linearity of an optomechanical interaction gives rise to rotations of the optical field in phasespace that can be observed in both the phase (PL) and amplitude (XL) quadratures.Conventionally this interaction is linearised in the weak coupling regime, leading to optical phase quadrature displacements only.Our optical setup (b) can measure an arbitrary optical quadrature of light from the optomechanical system using homodyne interferometry and is capable of observing the higher-order terms in displacement, described by Eq. ( 2).The optomechanical system consists of a nanostring mechanical resonator evanescently coupled to an optical microsphere resonator (c) (not shown to scale), which is mounted in a high vacuum chamber (< 10 −6 mbar).The drive laser is stabilised to the cavity resonance using the Pound Drever Hall technique.Polarization control is not shown for clarity.
sions 1000 × 10 × 0.054 µm (length×width×thickness) and a fundamental mechanical resonance frequency of ω M /2π = 100.2kHz.From the known dimensions and density we estimate an effective mass of m = 0.86 ng.A continuous-wave fibre laser, operating at 1,559 nm, is locked on resonance with a whispering-gallery mode of the microsphere.We measure an optical amplitude decay rate of κ/2π = 25.6 MHz and mechanical linewidth of γ/2π = 0.7 Hz.An evanescent optomechanical coupling of g = 7.6 MHz/nm was determined (see methods), corresponding to a coupling rate of g 0 /2π = 75 Hz.We use approximately 2 µW of optical drive power resulting in an intracavity photon number N = 2.4 × 10 4 .A fibrebased Mach-Zehnder interferometer is used to perform homodyne detection and thereby selectively measure a quadrature of the optical output field.
Fig. 2(a) and (b) show the observed homodyne noise power spectra for both optical phase and amplitude quadratures at the mechanical frequency and the second harmonic, respectively.At ω M (Fig. 2(a)) we observe a Lorentzian peak on the phase quadrature from the thermal motion of the oscillator, which corresponds to an RMS displacement of 124 pm.The thermal noise is resolved with 85 dB of signal relative to the homodyne noise power when the signal is blocked, which corresponds to an ideal displacement sensitivity of 1.3 × 10 −15 m/ √ Hz.In practice, the signal beam is not shot noise limited due to cavity, acoustic, and laser noise which raise the measurement imprecision by roughly an order of magnitude.By setting the interferometer phase to measure the optical amplitude quadrature, the linear measurement of mechanical motion is suppressed by approximately 45 dB.At this quadrature, information about the displacement squared mechanical motion is found in a frequency band centred at 2ω M (Fig. 2(b)).We observe a Lorentzian peak with a linewidth of 1.5 Hz, which to within the measurement uncertainty, is equal to twice the linewidth at ω M .The signal-to-noise at this frequency is 65 dB relative to the homodyne noise, which corresponds to a calibrated ideal displacement squared sensitivity of 3.3 × 10 −24 m 2 / √ Hz.Fig. 2(c) shows the band power in the first and second harmonics as a function of the interferometer phase.The powers in each band are expected to follow sine and cosine squared functions (fitted).The observed suppression of the linear measurement allows a lower bound to be placed on the phase stability of the cavity and interferometer locks of at least 5 × 10 −3 rad.Fig. 2(d) shows the observed relative noise powers up to the 4 th harmonic of the mechanical frequency.The expected noise powers can be computed with the Isserlis-Wick theorem, showing excellent agreement with experiment.
Of primary interest in this work is the lowest order nonlinear measurement term in the optical amplitude quadrature, proportional to x 2 M .To describe this quantitatively we introduce the slowly varying quadratures of motion, X and Y , defined via x M (t) = X(t) cos ω M t + Y (t) sin ω M t.The mechanical displacement-squared signal can then be written as where the displacement-squared quadratures of motion . By inspection, it can be seen that the quadratic measurement has spectral components both at DC and 2ω M .Higher order terms in the expansion Eq.( 2) can in principle contribute power at 2ω M , however due to small displacements and weak coupling, the quadratic term is the only term to contribute substantial power at 2ω M .Consequently, linear and quadratic information can be spectrally separated and therefore at an appropriate homodyne angle measured simultaneously.
We employ this measurement protocol experimentally by choosing a homodyne angle of π/4, allowing simultaneous high fidelity linear and quadratic measurements of the mechanical motion.The photocurrent generated at the homodyne output is then digitized into 4 second blocks at 5 MS/sec, which are filtered in postprocessing at ω M and 2ω M and decomposed into sine and cosine components to obtain the respective quadratures of motion in each frequency band.In the limit of large displacement, the squares of each quadrature of motion can be estimated from the measured displacementsquared quadratures as X 2 X 2 2ω = P 2 + Q 2 + P and Y 2 Y 2 2ω = P 2 + Q 2 − P , where the tildes indicate noise-inclusive measurement outcomes of the respective quantity.This transformation allows recovery of a classical estimate of x 2 M without knowledge of the signal at DC. Fig. 3(a) plots the X 2 2ω estimates thus obtained from the 2ω M signal against the cosine mechanical position quadrature, X, obtained from the measurement at ω M .A clear quadratic relationship between the two measurements is observed, validating the displacementsquared nature of the 2ω M peak.
Conditioning based upon the outcome of the quadratic measurement can be used to prepare non-Gaussian states.In the most basic approach, conditioning Q = 0 and P = C, for some constant C, will produce a bimodal state with a separation of 2 √ 2C.However, to make more efficient use of the available data, we additionally perform a phase rotation for each sample in the measurement record.Firstly, at each discrete sample we find a rotation by a phase angle 2φ such that the new rotated variable Q 2φ = Q cos(2φ) − P sin(2φ) is equal to zero.In this rotated frame, conditional correlation between the two mechanical quadratures X and Y is eliminated.Secondly, we condition on a particular magnitude of P 2φ = P cos(2φ) + Q sin(2φ) (see methods).This operation localizes the phase space distribution of the reconstructed state to two small regions as shown in Fig. 3(b-d), with a separation dependent upon the conditioning value.These states, although classical, are evidently bimodal and non-Gaussian.Extending this protocol to a regime where the quadratic measurement rate dominates all decoherence processes, a macroscopic quantum superposition state could be generated [11].
The primary route towards measurement of nonlinear observables of mechanical motion to date has been the 'membrane-in-the-middle' (MiM) approach [9].In this approach, a mechanically vibrating element is appropriately placed within an optical standing wave in a cavity to give a displacement-squared dispersive coupling of the form , where µ 0 is the zero-point quadratic coupling rate.When operating in the resolved sideband regime, this interaction allows for the observation of quantum jumps in the mechanical phonon number.However, due to the fundamentally linear origin of the quadratic interaction with MiM systems, strong single photon coupling is also required [10].Experiments exhibiting this type of coupling include dielectric membrane systems [9,20,22], trapped cold atoms [21], trapped microspheres [23], or double-disk structures [24].However it should be noted that in these systems, quadratic coupling rates µ 0 are typically orders of magnitude smaller than attainable linear coupling rates g 0 .
contrast to a displacement-squared Hamiltonian coupling, the scheme employed here gives an effective quadratic coupling rate of g 2 0 /κ [11], which should be compared to µ 0 defined above.For the modest linear coupling achieved in the present work, we have a g 2 0 /2πκ = 2.2 × 10 −4 Hz.Crucially, since the coupling rate in our scheme scales as g 2 0 , substantial gains are possible by improving g 0 .For instance, the coupling rate for a state-ofthe-art evanescently coupled nanostring-microcavity system as described in [25] is g 2 0 /2πκ 5 × 10 −2 Hz.Furthermore, other systems with exceptional linear coupling rates [26], should have quadratic coupling rates using our scheme as high as 160 Hz.In comparison, a state-of-theart MiM system as described in Ref. [22] has a quadratic coupling rate of µ 0 /2π = 6.0 × 10 −3 Hz.Thus a substantial gain in in effective quadratic coupling is possible using our protocol.
It should be noted that while measurement of the optical amplitude quadrature erases linear information about the mechanical position, the radiation-pressure back-action remains linear.For a continuous interaction, there is then a standard quantum limit in displacementsquared measurement via heating of the mechanical oscillator due to the optical intensity noise.However, in- FIG. 3: Bimodal state preparation via non-linear measurement.Using a homodyne angle of π/4, a highfidelity measurement of both linear and quadratic motion of the mechanical oscillator at frequencies of ωM and 2ωM, respectively, is obtained.In (a) the quadratic measurement outcomes ( X 2 2ω ) obtained from the 2ωM signal are plotted against the linear outcomes ( X) obtained from the signal at ωM.The histogram of X measurements (above) is well described by a Gaussian thermal distribution with standard deviation σ th = 124 pm, while the histogram of the X 2 2ω measurements (right) forms a chi-squared distribution.Fig- ures (b-d) show the phase-space distributions (and associated histograms) of states conditionally prepared using data at 2ωM and obtained using a read-out at ωM.The conditionally rotated read-out data decomposed into conjugate quadratures, labelled X φ and Y φ .The chosen quadratic conditioning values are (b) (2 P 2φ 0.2), (c) (2 P 2φ = 1.0), and (d) (2 P 2φ = 2.0).The same quadratic conditioning values are indicated as overlay histograms in (a).The histograms in (bd) are normalised to their peak value to more easily allow the width of the features to be compared to the initial thermal state (orange dash-dot curve).All the data presented here have been generated from three 4 second blocks of sampled homodyne output.
formation about the back-action noise to the mechanical oscillator is available in the optical amplitude quadrature in the band centred around ω M .In the limit of a perfect amplitude quadrature measurement, the linear back-action noise can then be eliminated by applying appropriate feedback to the mechanical motion.In this case, the mechanical dynamics reduce to the same form as with the displacement-squared dispersive Hamiltonian coupling [27].
To reach a quantum regime of state preparation via measurement, decoherence due to interaction with the environment will have to be overcome.A minimum requirement in our case is that the coherent field enhanced effective quadratic measurement rate is greater than rethermalization i.e.
√ N g 2 0 /κ > γ n.For our parameters at present, this is satisfied at a bath temperature lower than 2.5 µK.However, for a state of the art system [26] this requirement would be satisfied below 85 mK, well above standard dilution refrigerator base temperatures.A further requirement is that linear information lost to the environment or obtained by measurement does not destroy the quantum state produced.For a MiM optomechanical system this problem can arise, for instance, due to inaccurate positioning of the membrane within the standing wave.Moreover, as is discussed in Ref. [10], even with perfect positioning, linear position information is attained by an optical probe field due to the mechanical position dependent coupling between the optical normal modes and the finite cavity decay.With the approach employed here, fluctuations or offsets in the interferometer phase or cavity lock will result in leakage of linear information.Optical loss prior to measurement will additionally allow leakage of linear information to the environment.Linear information about the mechanical mode of interest can also undesirably enter the measurement due to the presence of other mechanical modes in the system, which mix via the optical nonlinearity with the mode of interest.This experimentally relevant effect can be seen in the higher frequency peak in the optical amplitude measurement in Fig. 2(b), which is the sum beat between the two mechanical vibrational modes seen in the phase measurement in Fig. 2(a).Unwanted modemixing of this type will generally be present in schemes to non-linearly measure mechanical motion.To minimise mechanical mode mixing one could tailor the optical intensity profile to match a desired vibrational mode [28] or engineer a mechanical structure with a single dominant vibrational mode [29].Finally, we recognise that the currently un-utilised DC component in our experiment is an additional decoherence channel.
To summarize, by exploiting the optical non-linearity of the canonical optomechanical interaction, which couples photon number to mechanical displacement, we have performed a measurement of the displacement-squared thermal motion of a nanostring mechanical oscillator at room temperature.Furthermore, by conditioning on the displacement-squared outcomes we prepare bimodal states of mechanical motion.Favourable scaling of the coupling rate in our approach makes realistic the possibility of observing the displacement squared fluctuations of the mechanical ground state in the near future.This would allow for mechanical quantum superposition state preparation without the need for strong single-photon optomechanical coupling.As a result, this experiment paves the way for quantum non-Gaussian state preparation of mechanical motion via measurement with applicability to a number of other physical systems, such as atomic spin ensembles [30] and superconducting microwave circuits [31,32].

Methods Summary
Linear Calibration Procedure.We determine the evanescent optomechanical coupling by displacing the nanostring by a known distance using a piezoelectric element and measuring the resulting frequency shift on the optical resonance.The frequency shift is calibrated via modulation of known frequency applied to the laser.We then establish the response of the homodyne by sweeping the laser detuning over the optical resonance and measuring the slope of the phase response.This parameter combined with the previously determined optomechanical coupling rate gives the total response of the combined cavity interferometer system in [V/nm], allowing direct calibration of the time domain data in [nm].We calibrate the response of our spectrum analyser by applying a test tone of known amplitude, which using time domain calibration gives a spectral peak of known displacement spectral density.Quadratic Calibration Procedure.Frequency calibration of the quadratic measurement is performed by ensuring the calibrated RMS displacement, obtained from the linear measurement, is consistent with the noise power of the 2ω M peak, in accordance with the Isserlis-Wick theorem.In the time domain, a simple regression is used between the square of the linear measurement ( X) and the quadratic measurement ( X 2 2ω ).We verify that these procedures are consistent, to within known uncertainties, with one and with the value of λ 2 n computed from the independently measured system parameters.
State Conditioning.At each discrete time step, we rotate the vector { P , Q} by an angle 2φ, such that a new vector , 0} is obtained.The simultaneously acquired linear data { X, Y } is then rotated through the half angle, φ, to obtain { X φ , Y φ } = { X cos(φ)+ Y sin(φ), Y cos(φ)− X sin(φ)}.For state preparation, the rotated linear data is conditioned upon the value of P 2φ , which is proportional to 1  2 ( X φ ) 2 .We choose a conditioning window 4 times smaller than the quadratic measurement uncertainty.When the conditioning criterion is satisfied, the state is read-out using the rotated linear data { X φ , Y φ }.

FIG. 1 :
FIG. 1: Concept and experimental apparatus.(a)The intrinsic optical non-linearity of an optomechanical interaction gives rise to rotations of the optical field in phasespace that can be observed in both the phase (PL) and amplitude (XL) quadratures.Conventionally this interaction is linearised in the weak coupling regime, leading to optical phase quadrature displacements only.Our optical setup (b) can measure an arbitrary optical quadrature of light from the optomechanical system using homodyne interferometry and is capable of observing the higher-order terms in displacement, described by Eq. (2).The optomechanical system consists of a nanostring mechanical resonator evanescently coupled to an optical microsphere resonator (c) (not shown to scale), which is mounted in a high vacuum chamber (< 10 −6 mbar).The drive laser is stabilised to the cavity resonance using the Pound Drever Hall technique.Polarization control is not shown for clarity.

FIG. 2 :
FIG. 2:Observation of linear and quadratic motion of a mechanical oscillator.(a) Measurement of the optical phase quadrature (red trace) at the mechanical frequency, ωM, shows a Lorentzian mechanical displacement spectrum, which is strongly suppressed when measuring the optical amplitude quadrature (blue trace).Sidebands appear in the amplitude quadrature measurement due to mix-up of low frequency noise.(b) Measurement of the optical amplitude quadrature at 2ωM (blue trace), shows the Lorentzian mechanical displacement squared noise, which is suppressed when measuring the optical phase quadrature (red trace).In (a-b) the homodyne noise floor is shown as a grey dashed line, and a measurement bandwidth of 20 Hz was used.Plot (c) shows the optical noise power measured over a 51 Hz bandwidth at ωM (purple) and 2ωM (yellow) as a function of the optical homodyne angle.(d) The relative observed powers in each of the mechanical harmonics when measuring the optical phase (red) and amplitude quadratures (blue); bars -theory; dots -experimental data.