Determination of the vacuumoptomechanical coupling rate using

frequency noise calibration

M. L. Gorodetksy,1,2 A. Schliesser,1,3 G. Anetsberger,3 S. Deleglise,1

and T. J. Kippenberg1,3,∗1Ecole Polytechnique Federale de Lausanne, EPFL, 1015 Lausanne, Switzerland

2Faculty of Physics, Moscow State University, Moscow, 119991, Russia3Max-Planck-Institut fur Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany

*tobias.kippenberg@epfl.ch

Abstract: The strength of optomechanical interactions in a cavity optome-chanical system can be quantified by a vacuum coupling rate g0 analogousto cavity quantum electrodynamics. This single figure of merit removesthe ambiguity in the frequently quoted coupling parameter defining thefrequency shift for a given mechanical displacement, and the effective massof the mechanical mode. Here we demonstrate and verify a straightforwardexperimental technique to derive the vacuum optomechanical coupling rate.It only requires applying a known frequency modulation of the employedelectromagnetic probe field and knowledge of the mechanical oscillator’soccupation. The method is experimentally verified for a micromechanicalmode in a toroidal whispering-gallery-resonator and a nanomechanicaloscillator coupled to a toroidal cavity via its near field.

© 2010 Optical Society of America

OCIS codes: (120.5050) Phase measurement; (230.4685) Optical microelectromechanical de-vices; (270.2500) Fluctuations, relaxations, and noise.

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1. Introduction

Optomechanical systems are being explored for a variety of studies, ranging from exploringquantum limits of displacement measurements [1,2], ground-state cooling of mechanical oscil-lators [3,4] to low-noise radiation-pressure driven oscillators [5–8]. In this new research field ofcavity optomechanics [9, 10], rapid progress towards these goals has recently been enabled bythe conception and realization of a large variety of nano- and micro-optomechanical systems,which implement parametric coupling of a mechanical oscillator to an optical or microwaveresonator (for an overview of recently studied systems, see e. g. reference [11]). Thereby, theresonance frequency ωc of the electromagnetic cavity is changed upon a mechanical displace-ment x according to ωc(x) = ωc(0)+Gx, with the coupling parameter

G ≡ dωc

dx. (1)

The resulting optomechanical interaction can be described by the interaction Hamiltonian[12],

Hint = hG x no = hGxzpf(a†m + am) a†

oao, (2)

where x = xzpf(a†m + am) is the mechanical displacement operator, no = a†

oao the photon occu-pation of the optical cavity, and ao, a†

o, am and a†m are the usual ladder operators of the optical

(o) and mechanical (m) degrees of freedom. The root-mean-square amplitude of the mechanicaloscillator’s zero-point fluctuations

xzpf =

√h

2meffΩm(3)

is determined by its resonance frequency Ωm and its effective mass meff as discussed in moredetail below.

The canonical example of such a system would be a Fabry-Perot resonator, one mirror ofwhich is mounted as a mass on a spring moving as a whole along the symmetry axis of the cav-ity. In this case, the coupling parameter is given by G=−ωc/L, where L is the cavity length, andthe oscillator’s effective mass simply is the moving mirror’s mass. In most real optomechanicalsystems, however, internal mechanical modes constitute the mechanical degree of freedom ofinterest (for simplicity, our discussion will be restricted to a single mechanical mode only). Ingeneral, the displacement of such a mode has to be described by a three-dimensional vectorfield �u(�r). For every volume element located at�r, it yields the displacement �u(�r) from its equi-librium position. In order to retain the simple one-dimensional description of equation (2), it isthen necessary to map the vector field�u(�r) to a scalar displacement x.

A frequently adapted procedure to address this mapping is to deliberately choose a couplingparameter G adequate to the overall geometry of the system, such as G = −ωc/L for a Fabry-Perot-type cavity of length L [13–15], or G = −ωc/R for a whispering-gallery-mode (WGM)resonator of major radius R [11,16]. The equivalent displacement x is then defined by the meas-ured resonance frequency shift Δωc induced by the displacement field �u(�r), i. e. x ≡ Δωc/G.Following a different approach, the equivalent displacement x is defined, for example, as the

#135382 - $15.00 USD Received 22 Sep 2010; accepted 11 Oct 2010; published 19 Oct 2010(C) 2010 OSA 25 October 2010 / Vol. 18, No. 22 / OPTICS EXPRESS 23238

displacement of the structure at a given position, x ≡ |�u(�r0)| or the maximum displacement ofa mode [17–20]. Then, the coupling parameter G is calculated via the relation x = Δωc/G. Insome cases [17,20,21], the coupling parameter is expressed as the effective length L−1

eff =1

ωc

dωcdx

of a resonator for which |G|= ωc/Leff. In this approach, the coupling parameter may be definedindividually for each mechanical mode.

In all cases, the only physically significant relation is the induced resonance frequency shiftΔωc upon a displacement �u(�r), which in a perturbation theory treatment, can be calculatedaccording to [22, 23]

Δωc

ωc≈ 1

2

∫ |�E(�r)|2 · (ε (�r+�u(�r))− ε (�r)) dr3∫ |�E(�r)|2 · ε(�r)dr3. (4)

This expression is uniquely determined by the distribution of the dielectric constant ε and the(unperturbed) electric field �E and can, in most cases, be linearized in the small displacements�u and reduced to a surface integral over the relevant resonator boundaries [13, 24].

In contrast to the unambiguous relation (4), the equivalent displacement x which is a-priorinot a well-defined quantity, may be defined in an arbitrary way as discussed above. An impor-tant consequence of this ambiguity in the choice of x is the necessity to introduce an effectivemass meff of the mechanical mode [13, 25] which retains the correct relation for the energy Ustored in the mechanical mode given a displacement x,

U =12

meffΩ2mx2. (5)

Evidently, rescaling x → αx implies G → G/α and meff → meff/α2, so that the frequentlyquoted figures of merit G and meff are individually only of very limited use. It is possible toeliminate this ambiguity by referring to the vacuum optomechanical coupling rate

g0 ≡ G · xzpf (6)

using terminology analogous to cavity quantum electrodynamics [26]. As expected from theHamiltonian description (2), all optomechanical dynamics can be fully described in terms ofthis parameter. As a consequence, parameters such as the cooling rate or optical spring effectcan be derived upon knowledge of g0. For instance, the cooling rate in the resolved sidebandregime is given by Γ = 4〈no〉g2

0/κ , while the power to reach the SQL scales with κ2/g20 with

the cavity loss rate κ . Furthermore, the regime where g0√〈no〉 � κ ,Γm (Γm is the mechanical

damping rate) denotes the strong optomechanical coupling regime, in which parametric normalmode splitting occurs [3, 27, 28].

2. Determination of the vacuum optomechanical coupling rate

Our intention here is to detail and verify an experimental technique to determine g0 in a simpleand straightforward manner for an arbitrary optomechanical system. The technique we presentrelies on the fluctuation-dissipation theorem, from which the double-sided (symmetrized) spec-tral density of displacement fluctuations of a mechanical oscillator in contact with a thermalbath at temperature T can be derived [13, 29, 30],

Sxx(Ω) =ΓmhΩmeff

coth(

hΩ2kBT

)(Ω2 −Ω2

m)2 +Γ2

mΩ2 ≈ 1meff

2ΓmkBT(Ω2 −Ω2

m)2 +Γ2

mΩ2 , (7)

where meff is the effective mass corresponding to the choice of the normalization of x, as derivedfrom Eq. (5), and the approximation is valid for large mechanical occupation numbers 〈nm〉 ≈

#135382 - $15.00 USD Received 22 Sep 2010; accepted 11 Oct 2010; published 19 Oct 2010(C) 2010 OSA 25 October 2010 / Vol. 18, No. 22 / OPTICS EXPRESS 23239

kBT/hΩm � 1. The displacement fluctuations are not directly experimentally accessible, butcause fluctuations of the cavity resonance frequency ωc according to

Sωω(Ω) = G2Sxx(Ω)≈ g20 ·

2Ωm

h2ΓmkBT

(Ω2 −Ω2m)

2 +Γ2mΩ2 . (8)

If such a spectrum can be measured, it is straightforward to integrate it over Fourier frequenciesresulting in the interesting relation

〈δω2c 〉=

∫ +∞

−∞Sωω(Ω)

dΩ2π

= Sωω(Ωm)Γm

2= 2〈nm〉g2

0, (9)

which implies that the knowledge of the mechanical occupation number and the cavity reso-nance frequency fluctuation spectrum are sufficient to directly derive g0. The former is usuallya straightforward task (provided dynamical backaction [31,32] is ruled out, for example, by lowprobing powers), whereas the latter has to be implemented using a phase-sensitive measurementscheme as shown in figure 1, since the required changes of the optical resonance frequencychange the phase of the internal and hence output field.

+

- in

+

- in

Monochromaticsource

Frequencymodulation

Optomechanicalsystem

Phase-sensitivedetection

Spectralanalysis

S I

Fig. 1. Generic scheme for the determination of the vacuum optomechanical coupling rate.The cavity optomechanical system is probed with a monochromatic source of electromag-netic waves which is frequency-modulated by a known amount at a Fourier frequency closeto the mechanical resonance frequency. A phase-sensitive detector generates a signal I,which is analyzed with a spectrum analyzer. As two possible examples, a homodyne de-tector measures the phase difference between the arms in an interferometer, one of whichaccommodates the optomechanical system (shutter S open). If the shutter S is closed, thetransmission signal I of an optomechanical system is amplitude-dependent when the driv-ing wave is detuned with respect to the optical resonance.

Various methods for such measurements are available, such as direct detection when theprobing laser is detuned, frequency modulation [33], polarization [34], or homodyne spec-troscopy [35]. In general, cavity resonance frequency fluctuations are transduced into the signalin a frequency-dependent manner, such that the spectrum of the measured signal (e.g. a pho-tocurrent) I is given by

SII(Ω) = K(Ω)Sψψ(Ω) =K(Ω)

Ω2 Sωω(Ω), (10)

where ψ is the phase of the intracavity field, which gets modulated upon a change of the reso-nance frequency. Importantly, to exploit the relation (9), it is necessary to convert the spectrumSII(Ω) back into a frequency noise spectrum, so the function K(Ω)≡ K(Ω,Δ,ηc,κ ,P) (where

#135382 - $15.00 USD Received 22 Sep 2010; accepted 11 Oct 2010; published 19 Oct 2010(C) 2010 OSA 25 October 2010 / Vol. 18, No. 22 / OPTICS EXPRESS 23240

Δ = ωl −ωc is the detuning from resonance, and ηc = κex/κ is the coupling parameter quan-tifying the ratio of the cavity loss rate due to input coupling κex compared to the total lossrate κ) has to be calibrated in absolute terms. Experimentally, this can be accomplished byfrequency-modulating the source that is used for measuring SII [35–37]. Under correct experi-mental conditions (cf. appendix) the phase modulation of the input field undergoes the exactlysame transduction, so that

SII(Ω) = K(Ω)Sφφ (Ω), (11)

where Sφφ (Ω) is the spectrum of input field phase modulation. For the typical case ofmonochromatic input field phase modulation (El ∝ e−iφ0 cosΩmodt), the equivalent spectral den-sity is given by Sφφ (Ω) = 2π 1

2 (δ (Ω−Ωmod)+δ (Ω+Ωmod))φ 20 /2.

A laboratory spectrum analyzer will, of course, not resolve this spectrum but convolve itwith its bandwidth filter lineshape F(Ω), for example a Gaussian centered at Ω = 0. If a time-domain trace is analyzed using a discrete Fourier transform, the filter function would be givenby the squared modulus of the Fourier transform of the time-domain windowing function. Wethen have

SmeasII (Ω) = 2 ·F(Ω)∗ (K(Ω)

(Sψψ(Ω)+Sφφ (Ω)

)), (12)

where a factor of 2 was introduced to account for the fact that most analyzers convention-ally display single-sided spectra (this factor is irrelevant for our final result, however). Thefilter function is usually normalized and can be characterized by the effective noise bandwidth(ENBW), for which

F(0) ·ENBW =

∫ +∞

−∞F(Ω)

dΩ2π

= 1, (13)

essentially absorbing differences in hardware or software implementations of spectral analysis[38]. Note, however, that this description is still idealized. In reality, correction factors mayapply depending on details of the signal processing (such as averaging of logarithmic noisepower readings) in the respective analyzer used, see e.g. [39].

If the effective noise bandwidth is chosen much narrower than all spectral features inSψψ(Ω), the measured spectrum is approximately

SmeasII (Ω)≈ 2 ·K(Ω)Sψψ(Ω)+

φ 20

2K(Ωmod)(F(Ω−Ωmod)+F(Ω+Ωmod)) . (14)

Assuming that the signal at Ωmod is dominated by laser modulation, the transduction functionK can then be calibrated in absolute terms at this frequency, as in the relation

SmeasII (±Ωmod)≈ φ 2

0

2K(Ωmod)F(0) =

φ 20

2K(Ωmod)

ENBW(15)

all parameters except K(Ωmod) are known. The spectral shape of K is then needed to derive thespectrum Sψψ of interest. This task can be accomplished either by repeating the measurementfor all modulation frequencies of interest, or by theoretical analysis, relating it to the conditionsunder which the measurements are taken.

As the simplest example, we may consider the case of direct detection, where the signal isthe power P “transmitted” through the optomechanical system. For a given input power Pin, wefind (see appendix) the transduction function:

KD(Ω) =SPP

P2inSψψ

=SPP

P2inSφφ

=4η2

c κ2Δ2Ω2(Ω2 +κ2(1−ηc)2)

((Ω+Δ)2 +κ2/4)((Ω−Δ)2 +κ2/4)(Δ2 +κ2/4)2 . (16)

#135382 - $15.00 USD Received 22 Sep 2010; accepted 11 Oct 2010; published 19 Oct 2010(C) 2010 OSA 25 October 2010 / Vol. 18, No. 22 / OPTICS EXPRESS 23241

We assumed in this expression sufficiently low pump powers such that effects related to optome-chanically induced tranparency (OMIT) [40] are negligible. For the more advanced homodynespectroscopy we calculate (see appendix),

KH(Ω) =SHH

PinPLOSψψ=

SHH

PinPLOSφφ=

=4κ2η2

c Ω2(Ω2(1−2ηc)2κ2/4+(Δ2 − (1−2ηc)κ2/4)2)

((Δ−Ω)2 +κ2/4)((Δ+Ω)2 +κ2/4)(Δ2 +(1−2ηc)2κ2/4)(Δ2 +κ2/4). (17)

Advantageously, this method can also be applied with the laser tuned to resonance (Δ = 0),ruling out effects of dynamical backaction [31, 32] even for larger laser powers. In this specialcase

KH(Ω) =16η2

c Ω2

Ω2 +κ2/4. (18)

Another interesting case is Δ=±Ωm in the resolved sideband regime [37] with Ωm � κ , where

KH(Ωm) 4η2c

(1+

κ2(16ηc −13)16Ω2

m

). (19)

We would finally like to emphasize that in many experimental situations, the transductionfunction K(Ω) is sufficiently constant over the range of frequencies, in which the mechanicalresponse in (9) is significant (several mechanical linewidths around Ωm). In this case, it can besimply approximated to be flat, K(Ω) ≈ K(Ωm) ≈ K(Ωmod), and we get from Eqs. (9), (10),(14) and (15)

g20 ≈

12〈nm〉

φ 20 Ω2

mod

2Smeas

II (Ωm) ·Γm/4Smeas

II (Ωmod) ·ENBW. (20)

The last fraction in expression (20) can be interpreted as the ratio of the integrals of the dis-placement thermal noise and the calibration peak as broadened by the filter function. Note alsothat according to its definition the ENBW is given in direct frequency and not angular frequencyas is Γm.

3. Application of the method

As two examples, we have applied the discussed method to two different optomechanical sys-tems shown in Fig. 2. We first consider the case of a doubly-clamped strained silicon nitridestring (30 × 0.7 × 0.1 μm3) placed in the near-field of a silica toroidal resonator [41]. Thisscheme allows sensitive transduction of the nanomechanical motion with an imprecision belowthe level of the standard quantum limit [42].

Note that the coupling rate in such a configuration depends on the strength of the evanescentfield at the location of the nanomechanical oscillator, which increases exponentially if the oscil-lator is approached to the toroid. The data in Fig. 2 show the single-sided cavity frequency noisespectral density Smeas

νν (Ω)≡ SmeasII (Ω) ·(Ω/2π)2/K(Ω) recorded using the homodyne technique.

The spectrum was calibrated by laser frequency modulation at the Fourier frequency of 8MHz.The spectrum needs not be processed further, since the transduction function for homodynedetection with Δ = 0 is suffciently flat around the mechanical peak, Γm κ . However, thecavity frequency noise spectrum as measured here contains also contributions from other noisemechanisms present in the silica toroidal oscillator, in particular thermorefractive noise [36]known to dominate frequency noise at lower (� 10MHz) Fourier frequencies. The integration

#135382 - $15.00 USD Received 22 Sep 2010; accepted 11 Oct 2010; published 19 Oct 2010(C) 2010 OSA 25 October 2010 / Vol. 18, No. 22 / OPTICS EXPRESS 23242

of the spectrum is therefore exclusively performed on the Lorentzian corresponding to the me-chanical mode (gray area), which can be easily isolated by fitting the noise spectrum with amodel accommodating all known noise mechanisms including thermorefractive noise [42, 43].The result of this evaluation is a vacuum coupling rate of g0 = 2π ·520Hz.

Frequency (MHz)

105

107

106

104

108

103

Freq

uenc

y no

ise

spec

tral d

ensi

ty (H

z2 /Hz)

71.25 71.30 71.35

105

107

106

104

8.0 8.2 8.4 8.6

Freq

uenc

y no

ise

spec

tral d

ensi

ty (H

z2 /Hz)

Frequency (MHz)

Fig. 2. Measured (single-sided) frequency noise spectra Smeasνν (Ω) (blue lines, ν ≡ ωc/2π)

induced by a nanomechanical string oscillator (8.3 MHz) coupled to a silica microtoroid[42] (left) and a radial breathing mode of a silica microtoroid (right), calibrated using laserfrequency modulation at frequencies of 8 and 71.38MHz, respectively. Orange lines areLorentzian fits, the gray area underneath them yield the optomechanical coupling ratesmultiplied with the occupation number of the oscillator.

Silica WGM microresonators such as spheres or toroids actually support structure-inherentmicromechanical modes themselves [43, 44]. Figure 2 also shows an equivalent cavity fre-quency noise spectrum measured on such a cavity using direct detection. Again we neglect thefrequency-dependence of K due to the very narrow mechanical peak (Γm κ ,Ωm). For theevaluation of g0, we remove backgrounds due to thermal noise from other modes, thermore-fractive fluctuations or shot noise, by fitting a Lorentzian model with a flat background term.Only the peak amplitude of the Lorentzian is then used with the relation (20). A vacuum cou-pling rate of g0 = 2π · 570Hz is obtained, which is interestingly very similar to the vacuumcoupling rate of the nanomechanical string.

4. Conclusion

In summary we have presented a simple method which allows direct measurements of the vac-uum coupling rate g0 in optomechanical systems using frequency-modulation. We have ap-plied the method to two different devices, namely doubly-clamped strained silicon nitride string(g0/2π ≈ 103 Hz) and the radial breathing mode in a silica toroidal resonator (g0/2π ≈ 103 Hz).It is interesting to note how widely this coupling rate can vary in different optomechanicalsystems, for example, we estimate g0/2π ∼ 10−2 Hz for (present) microwave systems [18],g0/2π ∼ 100 Hz in lever-based optical systems [15,28], and up to g0/2π ∼ 105 Hz in integratedoptomechanical crystals [20]. Together with the optical and mechanical dissipation rates andresonance frequencies (which do also vary widely), the coupling rate provides a convenientway to assess the potential of optomechanical platforms for the different phenomena and ex-periments within the setting of cavity optomechanics.

#135382 - $15.00 USD Received 22 Sep 2010; accepted 11 Oct 2010; published 19 Oct 2010(C) 2010 OSA 25 October 2010 / Vol. 18, No. 22 / OPTICS EXPRESS 23243

A. Transduction coefficients

In this Appendix we show by direct calculations that the amplitude transduction coefficients ofthe input phase modulation and of the mechanical motion are the same both in the case of eitherphase homodyne or direct amplitude detection.

The canonical equations describing dynamical behavior of the optomechanical system arethe following:

a = (iΔ− iGx(t)−κ/2)a(t)+√

ηcκsin(t), (21)

x(t)+Γmx(t)+Ω2mx(t) =−hG|a|2,

where a is the internal field in the cavity normalized so that |a|2 is the number of photons inthe mode, sin is the input field related to the input power as Pin = hωo|sin|2, Δ is the detuningof the pump from optical resonance, κ and Γm are total optical and mechanical damping rates,ηc is the coupling strength equal to the ratio of coupling and total optical losses and Ωm ismechanical resonance frequency. We neglect in the calculations below the effects of dynami-cal and quantum backaction of the optical probe field on the mechanical subsystem, implyingsufficiently low input power.

Assume that the mechanical degree of freedom of the cavity harmonically oscillates so that

x(t) = x0 cos(Ωmt) . (22)

These oscillations will produce a periodic change of the optical eigenfrequency and hence thephase of the internal field with ψ0 = x0G/Ωm. If these oscillations are small (|ψ0| 1), thespectral components of the internal fields in the cavity and output field in the coupler will be:

ax = sin√

ηcκL (0)

(1− iψ0ΩmL (+Ωm)

2e−iΩmt − iψ0ΩmL (−Ωm)

2e+iΩmt

), (23)

sx,out = sin −√ηcκax,

L (Ω) = 1−i(Δ+Ω)+κ/2 .

In the same way if the input field is weakly phase modulated (sin ∝ e−iφ0 cos(Ωmodt)) to obtain acalibration signal:

sφ ,in = sin

(1− iφ0

2e−iΩmodt − iφ0

2e+iΩmodt

), (24)

aφ = sin√

ηcκ(

L (0)− iφ0L (+Ωmod)

2e−iΩmodt − iφ0L (−Ωmod)

2e+iΩmodt

).

sφ ,out = sφ ,in −√ηcκaφ . (25)

Dynamically these two modulations are not equivalent, but surprisingly produce the same trans-duction coefficient in terms of spectral densities.

We discuss here two different methods to probe the output field, either by directly lookingat optical power modulation on a detector with appropriate detuning Δ or by measuring phaseoscillations using homodyne detection at zero detuning, with the local oscillator signal obtainedby splitting the input power. From these relations it is possible to calculate the spectral densitiesof the power detected directly at the output of the detector, and the homodyne signal, once thespectral densities of mechanical (Sxx(Ω)) or laser phase (Sφφ (Ω)) are known.

#135382 - $15.00 USD Received 22 Sep 2010; accepted 11 Oct 2010; published 19 Oct 2010(C) 2010 OSA 25 October 2010 / Vol. 18, No. 22 / OPTICS EXPRESS 23244

A.1. Direct detection

Directly calculating the modulation amplitude of the output optical power on a detectorPout(Ω) = hω0|sout|2Ω (omitting detector efficiency to convert to a photocurrent) we find that

KD(Ω) =SPP(Ω)

P2inSψψ(Ω)

=SPP(Ω)

P2inSφφ (Ω)

=

=4η2

c κ2Δ2Ω2(Ω2 +κ2(1−ηc)2)

((Ω+Δ)2 +κ2/4)((Ω−Δ)2 +κ2/4)(Δ2 +κ2/4)2 . (26)

In the resolved sideband regime when Ω >√

2κ the signal at frequency Ω is maximizedwhen the pump is either tuned on the slope or on one of the mechanical sidebands:

Δ =±12

√2Ω2 −κ2 −2

√Ω4 −2κ2Ω2 ±κ

2

(1+

κ2

4Ω2

), (27)

Δ =±12

√2Ω2 −κ2 +2

√Ω4 −2κ2Ω2 ±Ω

(1− 3κ2

8Ω2

),

producing the same signal

KD(Ω) = 4η2c

(1+

κ2(1−ηc)2)

Ω2

). (28)

When Ω <√

2κ , the output is maximized when

Δ =±16

√12Ω2 +3κ2, (29)

KD(Ω) =27Ω2η2

c κ2(Ω2 +κ2(1−ηc)2)

(Ω2 +κ2)3 .

A.2. Balanced homodyne detection

In this case, the signal on the detector is proportional to the field amplitudes sLO and sin fallingon the output beam splitter,

Hhω0

= | i√2

sout − 1√2

sLO|2 −| i√2

sLO − 1√2

sout|2

= i(sLOs∗out − s∗LOsout) = i(sLOs∗in − s∗LOsin)+ i√

ηcκ(as∗LO −a∗sLO)

= 2|sLO||sin|sin(φLO −φin)+ i√

ηcκ(as∗LO −a∗sLO) (30)

It is important to note here that if the homodyne phase difference φLO − φin = 0 then thismethod directly probes the phase of the internal field and hence the mechanical oscillations.This regime can be provided in case of Δ = 0. If the detuning is not equal to zero, the phasedifference in an experiment is usually locked to have zero d.c. voltage. The local oscillator fieldis obtained by splitting the input power and is equal to:

sx,LO =sLO

sinsx,ineiφLO = sLOeiφLO , (31)

sφ ,LO =sLO

sinsφ ,ineiφLO = sLOeiφLO

(1− iφ0

2e−iΩmodt − iφ0

2e+iΩmodt

),

(32)

#135382 - $15.00 USD Received 22 Sep 2010; accepted 11 Oct 2010; published 19 Oct 2010(C) 2010 OSA 25 October 2010 / Vol. 18, No. 22 / OPTICS EXPRESS 23245

where φLO is chosen to suppress any d.c. voltage,

φLO =−arctan

(ηcκΔ

Δ2 +(1−2ηc)κ2/4

). (33)

In this case, againSHH

PinPLOSψψ=

SHH

PinPLOSφφ(34)

and

KH(Ω) =4κ2η2

c Ω2(Ω2(1−2ηc)2κ2/4+(Δ2 − (1−2ηc)κ2/4)2)

((Δ−Ω)2 +κ2/4)((Δ+Ω)2 +κ2/4)(Δ2 +(1−2ηc)2κ2/4)(Δ2 +κ2/4). (35)

If Δ = 0 which appears the most favorable case for homodyne detection, then:

KH(Ω) =16η2

c Ω2

Ω2 +κ2/4. (36)

Another interesting case is Δ =±Ωm in case of resolved sideband regime. In this case

KH(±Ωm) 4η2c

(1+

κ2(16ηc −13)16Ω2

m

). (37)

Acknowledgements

M. G. acknowledges support from the Dynasty foundation. S. D. is funded by a Marie CurieIndividual Fellowship. This work was funded by an ERC Starting Grant SiMP, the EU (Minos)and by the DARPA/MTO ORCHID program through a grant from AFOSR and the NCCR ofQuantum Photonics.

#135382 - $15.00 USD Received 22 Sep 2010; accepted 11 Oct 2010; published 19 Oct 2010(C) 2010 OSA 25 October 2010 / Vol. 18, No. 22 / OPTICS EXPRESS 23246