Multimode optomechanical dynamics in a cavity with avoided crossings

Cavity optomechanics offers powerful methods for controlling optical fields and mechanical motion. A number of proposals have predicted that this control can be extended considerably in devices where multiple cavity modes couple to each other via the motion of a single mechanical oscillator. Here we study the dynamical properties of such a multimode optomechanical device, in which the coupling between cavity modes results from mechanically-induced avoided crossings in the cavity's spectrum. Near the avoided crossings we find that the optical spring shows distinct features that arise from the interaction between cavity modes. Precisely at an avoided crossing, we show that the particular form of the optical spring provides a classical analog of a quantum-nondemolition measurement of the intracavity photon number. The mechanical oscillator's Brownian motion, an important source of noise in these measurements, is minimized by operating the device at cryogenic temperature (500 mK).

Optomechanical systems are typically modelled as a single cavity mode whose eigenfrequency is proportional to the displacement of a mechanical oscillator. 1 This "singlemode" model of optomechanics gives an accurate description of devices in which there is a clear separation of frequencies (e.g., between the mechanical frequency and the cavity mode spacings), and when only a single cavity mode is strongly driven. 2 Single-mode optomechanical devices have been used to realize a number of goals in recent years, including demonstrations of quantum effects associated with Gaussian states of the cavity field and/or the mechanical oscillator. 3,4,5,6,7,8,9,10,11,12,13,14 For some optomechanical devices the single-mode description breaks down and more complex behavior can occur. In particular, devices in which multiple cavity modes couple to each other via the oscillator's motion are predicted to offer novel means for controlling and measuring both mechanical motion and electromagnetic fields. 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 Such a mechanical coupling between cavity modes can be produced by applying strong coherent drives to the modes (in which case adiabatic elimination of the mechanical degree of freedom results in an effective coupling between the drives' sidebands). 22,23 This approach can be realized in a very wide range of optomechanical systems, since most cavities possess several modes that can be driven strongly, and whose eigenfrequencies depend upon the oscillator's displacement. Recent experiments have used this approach 24,25 (or a related approach that combines strong drives with a piezoelectric material 26 ) to transfer modulation sidebands between different wavelengths, including from microwave to near-infrared.
A different method for mechanically coupling cavity modes (and one which does not require multiple strong drives) is to employ devices in which the cavity's eigenmodes (rather than eigenfrequencies) depend strongly upon the oscillator's displacement. This situation occurs when the oscillator's displacement causes crossings in the cavity's spectrum: these crossings are typically avoided (owing to broken symmetries within the device), 27,28,29 and in the vicinity of each avoided crossing the cavity's eigenbasis depends strongly upon the oscillator's displacement. 30,29 Theoretical studies of the resulting coupling show that it can offer improved performance over single-mode devices, e.g. in producing squeezed states of the mechanical oscillator and optical field. 19,20,21 Perhaps more importantly, the multimode coupling associated with avoided crossings offers capabilities that are fundamentally distinct from those of single mode devices, with applications in macroscopic matter-wave interferometry 18 and measuring the phonon statistics of a driven mechanical oscillator. 15,16 Avoided crossings are not a generic feature in optomechanical systems, but have been demonstrated in devices based on the membrane-in-the-middle design, 27,29,31,32 ultracold atoms, 33 and whispering gallery mode resonators. 34 , 35 To date, measurements of these systems have mostly focused on static spectroscopy of the cavity modes (i.e., to determine the parameters of the avoided crossings). 27,28,29,31,32,33,34,35 However the utility of the avoided crossings arises from their dynamical effects, which have received much more limited experimental study. 33,34,35 Here we address three outstanding issues related to multimode optomechanical devices based on cavities with avoided crossings. First, we describe thorough measurements of the optomechanical dynamics in the vicinity of avoided crossings. Far from the crossings, we find behavior that is dominated by the conventional dynamical back action 1 of the laser driving the cavity; in contrast, near the crossings the behavior is dominated by the elastic energy stored by the intracavity light. Second, we exploit the elasticity of the intracavity light at the crossings to demonstrate a classical analog of a quantum non-demolition (QND) measurement of the cavity's photon number. Third, the device is operated at temperature T = 500 mK, which minimizes the impact of thermomechanical noise, and should aid in future work directed at observing quantum effects in multimode optomechanical systems.
These results complement earlier studies of classical multimode dynamics in different systems, for example in purely mechanical devices, 36,37 purely electromagnetic devices, 38 and devices in which multiple mechanical modes couple via a single electromagnetic mode. 39,40,41

Experimental setup
The experimental setup is shown in Fig 1a. It consists of a Si 3 N 4 membrane (1 mm × 1mm × 50 nm) placed inside a Fabry-Perot optical cavity and cooled by a 3 He cryostat to T = 500 mK.
The cavity finesse F = 4,000 (linewidth κ/2π = 1 MHz), and the membrane's fundamental mode resonates at ω m /2π = 354.6 kHz with quality factor Q = 100,000. Laser light with wavelength λ = 1064 nm enters the cryostat via an optical fiber. This light is coupled from the fiber to the cavity via cryogenic free-space optics which are aligned in situ using piezoelectric motors.
Similar motors are used to adjust the membrane's position, tip, and tilt within the cavity. An additional piezoelectric element allows for fine displacement of the membrane along the cavity axis, and for excitation of the membrane's vibrational modes.
Two lasers are used to address two cavity modes that are separated by 8.13 GHz (roughly twice the free spectral range). The first laser is the "probe" beam; it is locked to the cavity and detects the membrane's motion via a heterodyne scheme. The second laser is the "control" beam, and is locked to the probe beam with a controllable frequency offset. This control beam produces the multimode optomechanical interactions that are the main focus of this paper. Additional information about the setup is provided in the Supplemental Information.

Static spectroscopy
Figures 1b and c show cavity reflection spectra measured separately by the probe beam (upper plots) and the control beam (lower plots). In each case the reflection was recorded as a function of laser detuning and the membrane's static displacement z dis . The brightest curve corresponds to the TEM 00 mode ('singlet'), while the slightly dimmer curves correspond to the TEM {20,11,02} ('triplet') modes. The triplet modes are nearly degenerate, but can be resolved in the closer view shown in Fig. 1c.
The longitudinal order of the singlet mode differs by one from that of the triplet modes; as a result their resonance frequencies ω cav undergo roughly opposite detuning as a function of z dis , 28 and so appear to cross each other near z dis = 0 nm and z dis = -160 nm. A closer view of the apparent crossing near z dis = 0 nm shows that two of the triplet modes avoid the singlet mode ( Fig. 1d). 29 The optomechanical dynamics that occur near these avoided crossings is the main focus of this paper.
Because the probe and control beams address modes with slightly different wavelength, the avoided crossings for the two beams occur at different values of z dis . This makes it possible to position the membrane so that the probe beam addresses a mode that is not part of an avoided crossing (and so simply provides an efficient readout of the membrane's oscillatory motion z osc (t)) while the control beam addresses modes that undergo an avoided crossing (thereby producing multimode optomechanical coupling). Such a position is indicated in Fig 1c as a dashed white line, which we define as z dis = 0 nm.

Optomechanical dynamics near avoided crossings
To demonstrate the impact of the avoided crossings on the membrane's motion, we first position the membrane at z dis = 0 nm where the detuning of the modes addressed by the control beam is quadratic to lowest order, i.e. cav ∝ osc 2 . In this case, each intracavity photon is predicted 27,42 to produce an optical spring that shifts ω m by an amount 2 = cav ′′ ZP 2 (the primes indicate differentiation with respect to z osc , and z ZP is the amplitude of the membrane's zero-point motion). Fig. 1e plots the power spectral density of the membrane's Brownian motion (recorded by the probe beam) as the control beam's detuning Δ is varied.
This data shows the two qualitative features of quadratic coupling. First, the change in the membrane's resonance frequency δω m is proportional to the number of intracavity photons (i.e., δω m (Δ) has even symmetry about each cavity resonance with an approximately Lorentzian shape). Second, the sign of δω m is set by the sign of cav ′′ (i.e., positive when the laser is tuned to the higher-frequency cavity mode, and negative when the laser is tuned to the lower-frequency mode). In contrast, for conventional single-mode optomechanics (in which the detuning is linear: cav ∝ osc ) δω m (Δ) has odd symmetry about a cavity resonance, and its sign is the same regardless of which cavity mode is excited by the laser. 1 To make a more quantitative comparison with theory, we use multimode optomechanics theory 42 to calculate the cavity reflection, optical spring, and optical damping in the presence of avoided crossings (see Methods and Supplemental Information for more details). The majority of the parameters in this theory are determined by fitting the cavity's static spectrum to expressions that include three cavity modes ( fig. 1d shows a comparison of the measured (left) and fitted (right) reflection). To determine the remaining parameters, and to test the predictions of this model with respect to dynamical behavior, we measured the membrane's Brownian motion at several values of z dis between -1 nm and +1.25 nm. At each value of z dis , the control beam detuning Δ was varied over a range that included both of the cavity modes participating in the avoided crossing. For each value of Δ, the membrane's resonance frequency ω m and mechanical damping rate γ m were determined by fitting the Brownian motion spectrum. Figure 2 shows the changes in these quantities (i.e., the optical spring δω m and the optical damping δγ m ) as a function of Δ for each value of z dis .
When the membrane is furthest from the avoided crossing (i.e., for the uppermost and lowermost curves in Fig. 2), the features in δω m and δγ m show odd symmetry about the cavity resonances (which are indicated by dashed lines), consistent with conventional single-mode optomechanics and linear coupling. As z dis approaches 0 nm, the features in δω m and δγ m decrease in size, consistent with the decreasing slope of the cavity detuning near the avoided crossing. Precisely at the avoided crossing (olive-colored data in Fig. 2) the odd-symmetry feature in δω m is completely absent, and is replaced by an even-symmetry feature (as discussed above in the context of Fig. 1e).
The solid lines in Fig. 2 are calculated from the model described in Methods. These calculations use the parameters determined from the cavity's static spectrum (Fig. 1d), as well as three additional fit parameters. A complete description of the fitting process is given in the Supplementary Information. The agreement between the data and the fits in Fig. 2 indicates that multimode optomechanics theory provides an accurate description of this system, particularly in the vicinity of multiple avoided crossings between cavity modes. Figure 3 shows similar measurements, but carried out at fixed z dis ≈ 0 nm as a function of the control beam power P in . The data are plotted along with the predictions of the model. These predictions use the parameter values taken from the fits in Fig. 2, except for z dis and P in which are used as fit parameters (the fit values of z dis and P in agree well with independent measurements, as described in the Supplementary Information). Figure 3 shows clearly that when z dis ≈ 0 nm, the feature in δω m has even symmetry at each cavity resonance while the feature in δγ m has odd symmetry, in agreement with theory.
Previous measurements of static reflection spectra at room temperature showed that it is possible to tune the avoided crossings by adjusting the membrane's tilt relative to the cavity axis, and its position along the cavity axis 29,32 (see also Ref.  For the uppermost trace, the gap at the avoided crossing is no longer substantially larger than κ, and the two peaks begin to merge. See Table S2 in the Supplementary Information for a full description of the fit results.

Classical analog of a photon QND measurement
Proposals for realizing a QND measurement of the membrane's phonon number or the cavity's photon number make use of the fact that at an avoided crossing, a change in the number of quanta in one oscillator (optical or mechanical) shifts the frequency of the other oscillator by g 2 . Fully realizing these proposals and using them to detect individual quantum jumps requires single-quantum strong coupling, 30 which has not been achieved for optomechanical devices to date. Instead, we demonstrate a classical analog of such a measurement by using the membrane's resonance frequency ω m to monitor classical fluctuations of the intracavity laser power.

Methods
Following the description in Ref.
[42], we represent the cavity field as a superposition of basis modes, which we take to be the cavity's eigenmodes when the membrane is far from the avoided crossings. The amplitudes of these modes, a n , are the cavity's degrees of freedom. The membrane couples these modes and detunes them by an amount that depends upon z dis and z osc (here z dis is the uniform translation of the membrane chip, and z osc is the instantaneous displacement associated with the membrane's oscillatory motion). For the small range of motion considered here, we assume this detuning is linear in both z dis and z osc . These effects can be incorporated into the usual optomechanical equation of motion via the Hamiltonian 1 = where the components of the vector ⃗ are the mode amplitudes a n , b is the amplitude of the mechanical oscillation, and M is a matrix whose diagonal elements represent the detuning of the cavity modes, and whose off-diagonal terms represent the coupling between modes. 42 The optomechanical effects associated with avoided crossings emerge from this model even in the simple case of just two optical modes (n = 1,2); in this case (1) This model allows the detuning associated with z dis to have different coefficients The Supplementary Information provides a more detailed description of this model, and describes how it is used to calculate the optical spring, optical damping, and cavity reflection spectrum. We note that although the restriction to two optical modes (equation (1)     For large negative values of z dis , the lower-frequency cavity mode produces larger optomechanical effects than the higher-frequency cavity due to the fact that it corresponds to the TEM 00 mode, which is more strongly coupled to the driving laser (as can be seen in Fig. 1d). For large positive values of z dis , the situation is reversed.

Laser setup
As shown in Fig. 1a, we used two Nd-YAG 1064 nm lasers (Innolight Prometheus) in this experiment. The first laser, which we call the probe laser, is used for cavity locking and for measurement of the membrane's Brownian motion. To make this possible, a portion of the probe beam is sent through an electro-optic modulator (EOM) to apply 15 MHz phase modulation sidebands for the Pound-Drever-Hall (PDH) locking technique. This portion of the beam, (the "PDH beam") also passes through an acousto-optic modulator (AOM) which shifts it by 80 MHz. The frequency-shifted PDH beam is then combined with the unshifted beam which serves as a local oscillator (LO). Both beams are sent into the cryostat to the experimental cavity. Only the PDH beam has the necessary phase modulation sidebands to lock to the cavity, so when the probe laser is "locked", light from the relatively weak PDH beam enters the cavity and interacts with the membrane. The LO beam, which is detuned from the cavity by 80 MHz, promptly reflects off the input mirror of the cavity. When the reflected PDH and LO beams recombine on the signal photodiode (SPD), they produce a beat note at 80 MHz. The membrane's mechanical Brownian motion appears as a phase modulation of this beat note. To observe the beat note, we use a lock-in amplifier to demodulate the signal from the SPD. Typically, the probe beam has about 20 µW PDH power and several hundred µW LO power.
The control laser is nominally identical to the probe laser, except it is detuned in frequency from the probe laser by two cavity free spectral ranges. This frequency offset is produced by mixing a small amount of light from both lasers on the fast photodiode (FPD) shown in Fig. 1a, and comparing the beat note with a reference tone produced by a signal generator. When both lasers are locked to the TEM 00 mode, they are locked to different longitudinal modes of the cavity, and therefore at a given membrane position, the two lasers may have different couplings to the membrane's motion. This allows us to lock the probe laser to the cavity at a linear point, useful for measurement of the membrane's Brownian motion, and the control laser to the cavity at a quadratic point, useful for producing the effects that we want to study. Figure S1: Schematics of experimental cavity setup inside the 3 He refrigerator.

Cryostat setup
Light from the two lasers is coupled into the cryostat (Janis Research) through a single-mode optical fiber. Light from the fiber passes through a collimator and then continues in free space, hitting two 45° angled mirrors before reaching the input mirror of the cavity. The fiber collimator and one of the angled mirrors are mounted on custom 1" mirror mounts that can be adjusted in situ using commercial piezoelectric actuators (Janssen Precision Engineering, PiezoKnob).
The fiber collimator, mirrors, and cavity are all attached to a titanium stage. The stage is designed to be vibrationally isolated from the outside environment. This is done by suspending the stage on springs inside the cryostat. To reduce oscillatory motion of the stage on the springs, copper eddy current damping fins are attached to the bottom of the stage. Between the fins are strong rare earth magnets. Motion of the stage induces eddy currents in the copper fins, which dissipate the energy as heat. The spring/stage system has a resonance frequency around 2 Hz, and is approximately critically damped by the eddy current dampers. Several hundred flexible gold-coated copper wires (wire diameter of 76 µm) are used for a thermal link between the 3 He pot (T ≈ 300 mK) and the stage and membrane. A schematic of the cold experimental cavity setup is shown in Fig. S1.
To provide further vibration isolation, the cryostat itself is attached to a massive aluminum plate, which is mounted on pneumatic air legs. The air legs sit on additional square aluminum plates, which are each supported by four passive vibration reducing feet. The entire system can be enclosed within an acoustic noise reducing "room", consisting of plastic panels coated with sound absorbing foam, to achieve 13 dB of acoustic noise reduction. However, we determined that this level of acoustic isolation was not necessary for the quadratic optomechanics measurement described in this paper, so the acoustic shield was not used in this measurement.

Phase-locked loop (PLL) measurement
To detect classical laser modulation by way of the optical spring effect, we injected 75 Hz amplitude noise onto the probe laser. This was accomplished by modulating the drive tone of the control beam AOM at 75 Hz with a modulation depth of 0.77.
We then used a piezoelectric element mounted directly beneath the membrane to drive the membrane to an amplitude of 2 nm at its fundamental resonant frequency (~ 354.6 kHz). The 75 Hz amplitude modulation of the control beam causes a 75 Hz modulation of magnitude of the optical spring effect, and therefore modulates the membrane's fundamental frequency at 75 Hz. We used a phase-locked loop (PLL) from a Zurich Instruments HF2LI lock-in amplifier to track the membrane's resonant frequency and detect this 75 Hz modulation, adjusting the frequency of the piezo drive in real time to stay on resonance with the membrane. The output signal of the PLL then contains information about the laser modulation.

Drift subtraction
The membrane's resonant frequency was observed to drift on the order of Hz on a timescale of hours. The amount of drift was sometimes larger than the size of the optical spring shift, which complicated the characterization of the quadratic optomechanical effects. In order to compensate for this drift in our analysis, we always made sure to remeasure the membrane's Brownian motion at selected laser detunings after completing a data run with a given set of parameters. This provided a measurement of the Brownian motion under nominally identical conditions, but at different points in time allowing us to determine the amount by which the membrane's resonant frequency had drifted.
As an example of this process, the membrane's resonant frequency is plotted as a function of laser detuning for 60 µW laser power at z dis = 0 nm (Fig. S2a). This data run took 1 hour and 46 minutes to complete and consisted of a high resolution laser detuning sweep across the avoided crossing (starting at negative detuning), followed by a retaking of selected points in the opposite direction. As can be seen in Fig. S2a, the membrane's mechanical frequency drifts by just under 3 Hz during this time. For laser detunings that were measured in both the forward and backward directions, we plotted the difference in the membrane's mechanical frequency as a function of the time passed between the first and second data point at each detuning (Fig. S2b). From the slope of this plot, we determined the rate of membrane resonant frequency drift, and subtracted this drift from the original spring shift data. The corrected data is shown in Fig. S2c. After correction, the data shows a reasonable amount of repeatability despite the time that passed between the forward and backward runs. For actual fitting and data analysis, we discarded the backward run from the post-drift subtraction data.

System parameters
Our model for predicting optomechanical effects near an avoided crossing depends on a large number of system parameters, including cavity properties, membrane properties, and interaction strengths. When fitting the actual optomechanics data, we would like to minimize the number of free parameters by using independent measurements whenever possible. Our cavity spectrum (as in Fig. S3a) provide an excellent resource for characterizing both the optical properties and some of the interaction strengths in our system.
To completely model the anti-crossing of two optical modes, we need to know the total decay rate of each mode ( L , R ), the decay rate of each mode due to its input mirror ( in,L , in,R ), the linear coupling between each mode and the membrane's displacement ( dis,L ′ , dis,R ′ ), and the membranemediated coupling rate between the two modes, which we describe as , with t and φ real. All of these parameters can be measured from cavity spectroscopy data such as in Fig. S3a. Each vertical slice of the spectrum (e.g. dashed line in Fig. S3a) shows the reflected light intensity measured as the laser driving the cavity is swept over a certain frequency range. Cavity mode resonances appear as Lorentzian peaks whose full width at half maximum (FWHM) is equal to . The 'depth' of the dip provides a measure of in . If we choose a membrane position far from the avoided crossing, then the interaction of the two optical modes can be neglected, and we can make independent measurements of and in for both modes. For the two-mode crossing in Fig. S3, we find L 2 ⁄ = 1.0 MHz, L,in 2 ⁄ = 47 kHz, R 2 ⁄ = 1.3 MHz, and R,in 2 ⁄ = 5 kHz.
While the triplet modes are clearly visible in the color maps of cavity spectrum, the lasers are only weakly coupled to them (by design), and our ability to accurately determine the resonance reflection dip and linewidth is limited. However, by averaging data from different membrane positions, we are able to produce values with sufficient accuracy for use in the theoretical model.
The linear couplings ( dis,L ′ , dis,R ′ ) and tunneling rate (t) determine the exact shape of the anticrossings in the cavity spectra. To measure them, we again fit the Lorentzian peaks at each membrane position and record the center frequencies of each mode (see Fig. S4). The functional dependence of cavity resonant frequency on membrane position near the crossing is given by the eigenvalues of the M matrix in equation (1) Figure S4: Plot of upper and lower mode resonance frequencies near the avoided crossing from Fig. S3, as found from Lorentzian fits. The solid lines are theory fits whose parameters are given in the text.
The final system parameter is the phase factor, . It is perhaps most instructive to think of φ as the phase acquired by a photon as it tunnels from one mode to the other. An alternate interpretation can be seen by removing this complex phase from the tunneling amplitude and instead having each mode couple to the laser drive with a different phase shift. It is physically correct to include both of these phases, but their effects on the model are equivalent, so we group them together as the complex phase of t. This phase shift affects the avoided crossing in measurable ways. The plots in Fig. S5 show the calculated effect of φ on the cavity spectrum near the crossing. We see clearly that when the optical modes hybridize, φ modifies the interference of the two modes and results in different relative coupling strengths. We determined φ for our system by measuring the relative coupling (comparing resonant reflection dips) at z dis = 0 nm. We found φ = 1.6 (approximately π/2, corresponding to equal dips at the quadratic point).
The case in which there are two avoided crossings between nearly-degenerate triplet modes and the singlet can be handled in almost exactly the same way as described above to measure dis,L ′ , dis,R ′ , φ, and t for each of the three modes. However, since the quadratic curvature is poorly resolved for the smallest crossing, we find t for this crossing directly by measuring the gap between the two modes (instead of fitting the quadratic curvature). The result is 2 2 ⁄ = 0.76 MHz and the other results are listed in the Table S1. Note that the larger gap is denoted as the crossing t 1 between modes L and R 1 and the smaller gap as the crossing t 2 between modes L and R 2 . Figure S5: Cavity spectrum (calculated from theory) for three different values of the tunneling phase, φ.
Equally-coupled modes were used here to make the effect more visible.

Fit results
We obtained most of the system parameters from the cavity reflection spectrum. The effective linear coupling, osc ′ , however, is not directly obtained from the spectrum. We include it as a fit parameter when fitting data measured with different membrane displacements and use the average value as a fixed system parameter for the final fit analysis. The average values of osc ′ are listed in Table S1.      Control laser power P in is measured with a power meter at the entrance of the fiber prior to entering the cryostat. We consider ~ 40% power loss through the fiber. Mechanical quality factor Q ≈ 100,000 is obtained from membrane ringdown time ≈ 0.1 s by measuring the decay of the membrane's vibration at 354.6 kHz after the application of a strong piezo drive. The effective mass of the membrane is calculated to be 43 ng based on its size and material properties (i.e. Si 3 N 4 membrane of 1 × mm × 1mm × 50 nm). The system parameters and their values used for Fig. 2-4 in the paper are listed in Table S1 while Table S2 shows the fit results. Some of the results i.e. z dis and P in are compared with control values (Fig. S6a-c). Note that for the data analysis of 'I' in Fig.   4, two optical modes are considered: the singlet and one of the triplet modes. For the rest of the data, however, an additional triplet mode is included. This additional mode forms an avoided crossing nearby with the singlet mode (Fig. S7).

Theory
Here, we outline our model for the optomechanical interactions arising from two coupled optical modes. We begin with a derivation of single (optical) mode optomechanics, then generalize this to two or more coupled optical modes.

Optomechanics of a single optical mode
First, we review the derivation of optomechanics for a system with a single optical mode, in which the Hamiltonian is: The first term describes the optical cavity, while the second accounts for the mechanical motion. In this expression � and ̂ are annihilation operators for the optical and mechanical modes, respectively, is bare cavity resonant frequency, is the linear optomechanical coupling for one phonon ( where = �ℏ 2 ⁄ ) and is the mechanical mode frequency. Mechanical displacement is expressed as ̂= ̂+̂ †. Finally, ℋ � accounts for all coupling to the environment (decays and drives).
This Hamiltonian leads to the following equations of motion: Decay rates of the optical and mechanical modes are denoted as and , respectively. describes the coupling through the input port, which we use to drive the mode, while = − describes coupling to other dissipation channels. � and � are drives through these two channels ( � is just vacuum noise, while � includes any external drives). Finally, ̂ is the thermal drive for the mechanical mode.
For simplicity, we consider the (experimentally relevant) classical case, for which the equations of motion become Next, we introduce an external coherent optical drive detuned by ∆ from the cavity resonance: ( ) = − ( +Δ) , which (if we disregard mechanical motion and the negligible static displacement due to radiation pressure) creates a steady cavity optical field ( ) = 0 − ( +Δ) . The field's amplitude can be expressed as where [ ] is the cavity susceptibility [ ] = ( 2 ⁄ − (Δ + ω)) −1 . We can now linearize our equations of motion around this coherent drive by writing ( ) = ( 0 + ( )) − ( +Δ) where ( ) ≪ 0 : Here, = 0 is the total optomechanical coupling. Taking the Fourier transform of these equations, we find

Optomechanics of coupled optical modes
Consider the case of two crossing optical modes, which we'll call left (L) and right (R). We will disregard mechanical motion for now, but still consider constant membrane displacement (as it provides a way to tune the resonant frequencies of the two optical modes). The Hamiltonian for this system is ℋ � 0 = ℏ� 0 + 0, 0 � � † � + ℏ� 0 + 0, 0 � � † � + ℏ� � † � + − � † � � + ℋ � The first two terms describe the behavior of the left and the right cavity modes. The optomechanical coupling rate of each mode to the membrane displacement is denoted as 0, and 0, (in the notation of the main text, these are equal to , ′ and , ′ multiplied by ). The membrane displacement, 0 , which is a unitless (normalized to ) parameter here, is chosen such that for 0 = 0, the frequencies of both modes are equal to 0 . The third term describes tunneling between the two modes with rate t. Note that we have chosen to use a real coupling term t and explicitly include a complex phase factor . This can be thought of as the phase acquired by a photon tunneling from one mode to another. In addition to this phase factor, we could have chosen to have each mode couple to the input drive with a different phase shift. These two effects, while both physical, have identical effects on the model, so here we choose to only include a tunneling phase.
It is natural now to introduce vector notation for these modes, denoting vectors with a single bar and matrices with a double bar. For later notational convenience, we will also move to a frame rotating at 0 , so that our mode crossing effectively occurs at 0 = 0. Using the definitions the Hamiltonian simplifies to (DC optomechanical coupling is absorbed into � ( 0 ) = � + ̿ 0 0 ).
We now switch to the classical description and express the equations of motion using the vector notation: Here we account for the fact that the bare linewidths ( and ) and input coupling rates ( , and , ) can be different for the two modes. Since the same incident beam couples to both modes, is just a scalar, and the modes only differ in their coupling rates (as noted before, the phases of input coupling coefficients have been absorbed into our definitions of and ). Now we turn on an external drive detuned from the crossing point by ∆, written (in the rotating frame) as ( ) = − Δ . This provides us with a steady state solution