Split-sideband spectroscopy in slowly modulated optomechanics

Optomechanical coupling between the motion of a mechanical oscillator and a cavity represents a new arena for experimental investigation of quantum effects on the mesoscopic and macroscopic scale.The motional sidebands of the output of a cavity offer ultra-sensitive probes of the dynamics. We introduce a scheme whereby these sidebands split asymmetrically and show how they may be used as experimental diagnostics and signatures of quantum noise limited dynamics. We show split-sidebands with controllable asymmetry occur by simultaneously modulating the light-mechanical coupling $g$ and $\omega_M$ - slowly and out of-phase. Such modulations are generic but already occur in optically trapped set-ups where the equilibrium point of the oscillator is varied cyclically. We analyse recently observed, but overlooked, experimental split-sideband asymmetries; although not yet in the quantum regime, the data suggests that split sideband structures are easily accessible to future experiments.


Introduction
Cavity optomechanics offers rich possibilities for experimental investigation of the theory of quantum measurement and the role of quantum noise [1,2]. Several groups have successfully cooled a mechanical oscillator via its coupling to a mode of an electromagnetic cavity [3][4][5] down to its quantum ground state (or very close to it) i.e. mean phonon occupancy  n 1 ph . Read-out of the temperature was achieved by detection of motional sidebands in the cavity output; the theory for quantum sidebands was elucidated in [6,7] . The cavity fields serve a dual purpose: they provide not only the laser cooling but also an ultrasensitive means for detection of displacements on the scale of quantum zero-point fluctuations; this has motivated considerable interest in quantum-limited measurements in this context, following the early pioneering work by Braginsky and Khalili [8].
An important development was the detection of an asymmetry [9][10][11] in the two frequency peaks (sidebands) of the output power of a probe beam detuned to the positive and negative side of the cavity resonance. Albeit indirectly [10,11], the observations mirror an underlying asymmetry in the motional spectrum: an oscillator in its ground state = n 0 ph , can absorb a phonon and down-convert the photon frequency (Stokes process); but it can no longer emit any energy and up-convert a photon (anti-Stokes process).
Sideband asymmetry has become an important tool in optomechanics and has now been used to establish cooling limited by only quantum backaction [12]. Ponderomotive squeezing, whereby narrowband cavity output falls below the technical imprecision noise floor is also of much current interest [13][14][15] though is also observed in oscillators in a high thermal state. A standard optomechanical system comprises a mechanical oscillator interacting with a laser-driven cavity. In the frame rotating with the driving laser, typical experimental regimes using an extraordinarily broad range of physical platforms (cantilevers, microtoroids, membranes, photonic crystals) are well described by the twocoupled oscillator Hamiltonian:ˆˆ(ˆˆ) a ,â are creation and annihilation operators for small fluctuations cavity mode about its equilibrium value α whileˆˆˆ † º + b b x (in appropriately scaled units) represents a small displacement of the mechanical oscillator about its equilibrium position x 0 . Dissipative processes are treated by standard input-output theory, including input noises incident in the optical cavityˆˆ † k k a a , and α) and hence are far off-resonant. In addition, the w g, M modulations are out of phase, in the sense that when the mechanical frequency is a maximum, the magnitude of coupling strength between motion and cavity field is a minimum; and vice versa.
Added impetus to our study is provided by its relevance to optically trapped systems including levitated nanoparticles [22][23][24][25][26][27][28]. We show here that the anti-phase w g, M modulation arises automatically if the mean position of the oscillator varies harmonically ( ) w = x t X t sin 0 d d as occurs for nanoparticles in hybrid electricaloptical traps [16,29]. Such traps to date provide the only experimental realisation of stable trapping and cooling of a nanoparticle at high vacuum, in a cavity. This is illustrated in figure 1(b) where we also show that for these systems, the cavity output (lower panels, detected by a heterodyne method as in [16]) no longer has similar frequency spectrum to the displacement spectrum ( ) |ˆ( )| w w º á ñ S X xx 2 which is peaked at w w =  M ). In contrast, the cavity output modulations are peaked at frequencies w w w =  M d . The theory developed here accounts for this behaviour as well as unexplained asymmetries in the sidebands of levitated oscillators seen previously [29]. x 0 0 (the potential minimum of an optical well of width l 2), since there is no light-matter coupling. Hence, such set-ups [26,29] require an auxiliary field to pull the centre of the mechanical oscillation away from = x 0 0 . In electro-optical traps [29], a slow oscillation is induced such that ( ) w = x t X t sin 0 d d , enabling cooling. For small oscillations, here we find this corresponds to an effective modulation of the optomechanical coupling ( ) w = g t g t 2 sin d and a simultaneous, out-of-phase, modulation of the mechanical frequency and only g is appreciably modulated. In that case, while the power spectral density (PSD) of the displacement spectrum, The measured cavity output spectrum (right panel) detected by a heterodyne method, also exhibits this behaviour.
However, the key objective of the study is to investigate the dynamics, including the quantum spectra, of any set-up that can achieve these anti-phase w g, M modulations. Thus central parts of the study are generic to any system described by the optomechanics Hamiltonian equation (1), but with the additional w g, M modulations. In section 2 we present the theory. The results of section 2, in the appendix and in figures 2 and 3 are generic and make no reference to a specific set-up.
The analytical solutions are more challenging than for the standard case: unlike the former , where closedform solutions are the norm, in the present case only approximate expressions are possible. Here we obtain solutions via an iterative method. Then in section 3, the derived expressions are tested and validated against numerical solutions, of the nonlinear stochastic equations of motion, developed previously to simulate data in [16,29]. In the numerics, the values g and w M , including their modulation properties, are not even specified: they are simply emergent properties of the combined trapping fields. Here we exploit the fact that, when the particle is cold, the nonlinear Langevin numerics agree with linearised expressions, in the classical regime. The validated expressions can then be used to investigate quantum regimes by taking the G  0 M limit. A comparison between the analytical expressions and the numerics is presented in figure 4.

Slowly modulated optomechanics
The equations of motion for the standard set-up equation (1) can be solved [1] in frequency space to obtain quantum spectra of experimental or theoretical interest . In terms of quadrature operatorsˆ( ) w = y 1 2 (ˆ( )ˆ( )) † w w + a a one obtains:ˆ( represents the optical susceptibility. In this well-known form, the first term represents the back-action of the mechanical motion on the cavity field, the second the cavity-filtered incoming quantum noise. The measurable, cavity output spectrum is then obtained from input-output theory out in by considering Instead of the 'twin peak' structure seen in figure 1 for w = 0 2 , the resultant cavity output sidebands display a pair of peaks of asymmetric heights (lower panel). For small g , the ratio between peaks . This asymmetry is distinct from the usual Stokes/antiStokes sideband asymmetry at  w w  M , which is still present. Lower panel shows the Stokes peaks (red). (b) In thermal regimes, the ratio r is insensitive to G ; M however, as G  0 M and the backaction limit is attained, correlations between back-action and incoming noise alters the relative heights of the peaks, mainly since ponderomotive squeezing lowers the height of the w w + M d peak relative to the imprecision floor. For incoming quantum shot noise, significant changes in r arise only if the oscillator is near the ground state. Inset reproduces part of figure 5 of [29] illustrating previously overlooked asymmetries in experimental sidebands which supports our modulatedoptomechanics model.
The same solutions in (b) are now added to incoming imprecision noise to obtain output spectra ( ) w S y y out out . At high phonon occupancies, the shape is unchanged. As  n n ph BA , the ratio above the quantum imprecision floor alters significantly. (c) Shows individual contributions to the PSD; the pure backaction term has the same shape as the thermal split sidebands; its interference with incoming imprecision noise lowers the height of the w w +  the interference with the incoming (imprecision noise, typically shot noise from the laser), sô . However, if we solve the equations of motion including the modulation of g(t) we obtain instead: We elucidate details in the appendix, but the notable difference between the standard case and the modulated optomechanics considered here is that in equation (3) the optical field does not probe the displacement spectrumˆ( ) w x but is, instead, sensitive to the interference between the shifted displacement spectra , the minus sign in equation (3) is not significant: the shifted spectra do not interfere appreciably. The result is a cavity field fluctuation spectrum characterised by a 'twin peaks' structure. However, the effect of the additional frequency modulation, acquires corresponding sidebands which, as illustrated in figure 2(a), cause the two main peaks of the shifted spectraˆ( )ˆ( interfere with each other's sidebands. In this case, the minus sign in equation (3) (and the out-of-phase nature of the modulations) implies that one peak grows by constructive interference, while the other one diminishes.
Further details are in the appendix, but this can be understood from a simple argument. For modest backaction (ḡ small), we can writeˆ( ) w  X in the form: where theX th terms represent incoming thermal noises,Ŷ BA represents the back-action terms driven by imprecision noise. The last two terms are corrections to account for the modulation of w M ; the first comprises thermal effects, the second the corresponding back-action effects. For w = 0 2 and neglecting backaction, the shifted spectra arise mainly from incoming thermal noiseŝ The anti-Stokes sideband for example, is primarily due to the weighted thermal , yielding the 'twin peaks' structure since the ratio of the twin peak weights The main effect of w 2 is to introduce the extra correction from theˆw X 2 term which means replacing the thermal weights: Evaluating the corrections (the terms in square brackets) near the frequency peaks of the noise, we find they are d so the ratio of peaks in the PSD would be: predicting a full cancellation for w w 2 d 2 . For the standard optomechanical equations, equation (2) and itsˆ( ) w X equivalent are solved to obtain ( ) w S xx and ( ) w S yy or ( ) w S y y out out in closed form. However, for the modulated spectra this is not possible:ˆ( ) w y depends on shiftedˆ( ) w X spectra; and the w 2 modulation couplesˆ( ) w x spectra to the displacement spectrâ (3) and (4) are instead solved iteratively, assuming¯ w k w g , , 2 M and retaining terms up to quadratic order in¯w g , 2 (see appendix). In the thermal regime, the resulting equations can be shown to be accurate by testing against numerical stochastic equations, and another 'fast cavity' model, used to simulate optically trapped particles [16,29], as seen in figure 4.
We then take G  0 M which for cooling parameters (red-detuned light) takes the system down to the quantum backaction limit, where the heating is limited by quantum shot noise, When we calculate   S X X (the PSD forˆ( ) w  X ), we find that it differs very little from the thermal spectrum. This indicates that even for G = 0 M , (where the oscillator motion is completely driven by the incoming optical imprecision noise, quantum or classical), the shape, and r for   S X X unchanged as shown in figure 3(a). However, for ( ) w S y y out out , this is not the case: when the same solution used for figure 3(a) is added and interfered with the incoming (imprecision) optical shot noise, the sideband shape is unchanged for the thermal regime but changes significantly in the quantum back-action limit.
The underlying reason for this change can be understood as follows: the total back-action in equation (4),ˆ( , which by itself still yields a ratio of r, develops correlations with the incoming imprecision termsˆ( )ˆˆ The key difference seen between   S X X and ( ) w S y y out out in the quantum limit, arise because: The above two terms are contrasted in figure 3(c). Ponderomotive squeezing originates from such correlations [13][14][15] between backaction and incoming noise and, in the standard optomechanical case, it leads to a Fanolike line experimental profile [13][14][15] and (an often small) dip where the output light spectrum lies below the imprecision floor. However, in the present case, the height of the w w + M d peak is lowered as it overlaps with a ponderomotive squeezing 'dip' of the stronger peak as seen in figure 3(b), leading to a change in r: the sideband structure is more strikingly reshaped and the G M invariance of r is lost. Although ponderomotive squeezing does not require a ground state oscillator, for quantum shot-noise limited spectra, a change in r only becomes appreciable if  n n ph BA , leading to a noticeable decrease in height of the w w + M d peak above the imprecision noise level.

Comparisons with stochastic numerical model
One may test the validity and accuracy of the generic expressions derived above by comparing the calculated spectra with numerical solutions of equations of motion for the specific example of a levitated nanoparticle in a hybrid trap [16,29] since, when linearised, they reduce to the modulated optomechanics case. Provided the particle motion x(t) corresponds to small oscillations about an equilibrium value x 0 (which may be slowly modulated) and the cavity field dynamics a(t) correspond to small fluctuations about a mean field α, comparisons are possible.
As outlined in [16,29], a nanoparticle in a hybrid electrical-optical trap experiences a dipole force potential Aa t k x cos 2 2 from the optical standing wave of a cavity (with axis along x). In [16], the depth of the potential p =Á 26 2 kHz. In the equations of motion, the particle's motion causes an effective change in the length of the cavity by modifying the detuning between the driving laser and the cavity resonance: Neglecting noise, the intracavity field a(t) then is: T where  T is the laser drive amplitude. The cavity photon number | ( )| a t 2 typically fluctuates about a mean value of | | -a » 10 10 2 9 1 0 photons; p l = k 2 with l = 1064 nm. The particle becomes trapped in a given optical well N, with anti-node (potential It experiences also an additional oscillating harmonic potential x c o s . We test our model by comparing with solutions of the equations of motion in these combined potentials, including also damping for the cavity (κ) and for mechanical degrees of freedom (G M ) as well as stochastic Gaussian noise to allow for gas collisions and shot noise. This represents a stringent test of our analytical noise model since, in the numerics,¯w g , M and w 2 are not even input parameters: they are themselves emergent properties of the numerical simulations. In the linearised regime we obtain (taking α to be real): . For a fast cavity, we can assume the fluctuations of the cavity field follow ( ) kx t cos 2 with no delay. In turn, x(t) combines the slow ( ) x t 0 motion with the fast mechanical motion where X M is the variance of the thermal motion, the accumulated phase being ( ) ( ) . We find the Fourier transform of ( ) kx t cos 2 using this ansatz gives a reasonable approximation of the split-sideband spectrum for a fast cavity. Figure 4, corresponding to  k w M , shows that it yields reasonable agreement with numerics and analysis.
More importantly, it describes also scattering of light out of the cavity (illustrated in inset of figure 2(b)) which illustrated suppression of the w w + M d sideband. While not a full demonstration, this classical-regime data does demonstrate the coherent relative phase accumulation and interplay between the slow and fast motions; it indicates that in combination with homodyne or heterodyne detection, split sideband asymmetries may be investigated experimentally once quantum-limited regimes are attained.

Conclusions
In the present paper we have obtained approximate analytical expressions for an optomechanical system subjected to slow, out-of-phase modulation of the optomechanical coupling and spring constant. We show that a generic feature that emerges are split-sidebands in the cavity output spectra: these sidebands are characterised by a dominant pair of peaks of unequal heights. The ratio of the peaks depends on the modulation amplitude. The shape of these structures remains invariant in the thermal regime, as G M is reduced by many orders of magnitude, but changes shape significantly as the quantum-backation limited regime is attained.
Hence split sideband spectroscopy offers potential new experimental signatures of the underlying dynamics and also of the quantum regime; measurement of the ratio r complements Stokes/antiStokes asymmetry between the w M sidebands; it may be combined with that measurement to provide a more reliable quantum signature, since r is well defined and controllable and there is a sharp change in the quantum back-action limit; the double-sidebands may offer an additional diagnostic of Stokes/antiStokes asymmetry as there are two pairs of peaks to compare. It also offers a distinctive alternative experimental probe of ponderomotive squeezing; future studies will investigate whether these may be enhanced with modified techniques of detection (homodyne or general-dyne).
The expressions were tested and validated against stochastic numerics used to simulate current experiments on levitated nanoparticles in hybrid cavity-Paul trap setups. The optomechanical model is valid in both classical and quantum regimes and is shown here to accounts for previously unexplained split-sideband structures already seen in existing experiments. leads to the following equations of motion, [1], describing the motion of two coupled harmonic oscillators:ˆ(ˆˆ)(ˆˆ)ˆ(

Appendix. Calculation of noise spectra
describe small displacements about an equilibrium cavity field α and the equilibrium position of the oscillator. The oscillators are characterised by a mechanical frequency w M and a detuning Δ from the cavity resonance frequency (since the Hamiltonian is given in a frame rotating at the laser frequency). In general, Δ includes a correction term dependent on x 0 (see below).
Whileˆ † a andâ are creation and annihilation operators for small fluctuations cavity mode, in the equations of motion we givex in operator form, thusˆˆˆ † = + X b b (in the usual scalingˆ(ˆˆ) † = + X X b b zpf but here, for convenience, the appropriate length scaling M is absorbed into g). The equations of motion include dissipative terms given by standard input-output theory. k G , M are the cavity and mechanical damping rates. For the optical modes, this is laser noise incident in the optical cavitŷˆ † k k a a , in in . We consider a single-sided cavity here; our results can be adapted to other configurations but are not modified qualitatively. The mechanical modes are subject to thermal noises which input incoming phonons. For the specific case of an optically trapped system, this is Brownian noise [16,25,26] from collisions with roomtemperature gas molecules in the cavity, which in that case, depend on gas pressure P.
For the photon noise bath , we take the correlatorŝ a t a t n t t a t a t n t t where for photon shot noise,¯= n 0 a . However we can also include a finite photon temperature to test for the effects of laser noise. For the thermal noises we take the batĥ respectively. In order to calculate power spectral densities (PSDs) of either the displacement or the cavity fields, one reexpresses the operators equation (15) in terms of noise vectors, thus for example: , , in in in in is a vector comprising the noise terms and where: represents the respective coefficients at a given frequency ω.
The corresponding PSD relevant to heterodyne linear detection of the cavity field is hence: yy is then directly obtained from the noises using the correlators in equations (13) and (14). We note that for experimental analysis, a symmetrised version of equation (19) is typically evaluated [11]. As this does not affect our conclusions, we consider simply the unsymmetrised two-sided ( ) w S yy .

Slowly modulated optomechanics
We now generalise to the case where g, w M are now modulated, but sufficiently slowly  w w d M so that linearisation about equilibrium is still possible; we consider a harmonically modulated optomechanical coupling ( )¯w = g t g t 2 sin d as well as a modulated mechanical frequency ( ) w w w w = + t t 2 cos2 M M 2 d . Introducing the g and w M modulations in equation (12), the corresponding Fourier-transformed spectra become:ˆ( In other words, the optical field does not directly probe the mechanical motion but rather is sensitive to the difference between the displacement spectra at w w  with a similar dependence on shifted spectra. However, in this case there is also a correction due to the frequency modulation: Thermal noise coefficients of ( ) w  X . For the thermal components in equation (28), we can write