OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 22 — Oct. 25, 2010
  • pp: 23236–23246
« Show journal navigation

Determination of the vacuum optomechanical coupling rate using frequency noise calibration

M. L. Gorodetksy, A. Schliesser, G. Anetsberger, S. Deleglise, and T. J. Kippenberg  »View Author Affiliations


Optics Express, Vol. 18, Issue 22, pp. 23236-23246 (2010)
http://dx.doi.org/10.1364/OE.18.023236


View Full Text Article

Acrobat PDF (2038 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

The strength of optomechanical interactions in a cavity optomechanical system can be quantified by a vacuum coupling rate g0 analogous to cavity quantum electrodynamics. This single figure of merit removes the ambiguity in the frequently quoted coupling parameter defining the frequency shift for a given mechanical displacement, and the effective mass of the mechanical mode. Here we demonstrate and verify a straightforward experimental technique to derive the vacuum optomechanical coupling rate. It only requires applying a known frequency modulation of the employed electromagnetic probe field and knowledge of the mechanical oscillator’s occupation. The method is experimentally verified for a micromechanical mode in a toroidal whispering-gallery-resonator and a nanomechanical oscillator coupled to a toroidal cavity via its near field.

© 2010 Optical Society of America

1. Introduction

Optomechanical systems are being explored for a variety of studies, ranging from exploring quantum limits of displacement measurements [1

1. V. B. Braginsky and Y. I. Vorontsov, “Quantum-mechanical limitations in macroscopic experiments and modern experimental technique,” Sov. Phys. Usp. 17, 644–650 (1975). [CrossRef]

,2

2. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693 (1981). [CrossRef]

], ground-state cooling of mechanical oscillators [3

3. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007). [CrossRef] [PubMed]

,4

4. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007). [CrossRef] [PubMed]

] to low-noise radiation-pressure driven oscillators [5

5. T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95, 033901 (2005). [CrossRef] [PubMed]

8

8. K. J. Vahala, “Back-action limit of linewidth in an optomechanical oscillator,” Phys. Rev. A 78, 023832 (2008). [CrossRef]

]. In this new research field of cavity optomechanics [9

9. T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321, 1172–1176 (2008). [CrossRef] [PubMed]

, 10

10. F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009). [CrossRef]

], rapid progress towards these goals has recently been enabled by the 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 microwave resonator (for an overview of recently studied systems, see e. g. reference [11

11. A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with silica microresonators,” in “Advances in atomic, molecular and optical physics,” vol. 58, E. Arimondo, P. Berman, and C. C. Lin, eds. (ElsevierAcademic Press, 2010), chap. 5, pp. 207–323.

]). Thereby, the resonance frequency ωc of the electromagnetic cavity is changed upon a mechanical displacement x according to ωc(x) = ωc(0) + Gx, with the coupling parameter
Gdωcdx.
(1)

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

12. C. K. Law, “Interaction between a moving mirror and radiation pressure: A Hamiltonian formulation,” Phys. Rev. A 51, 2537–2541 (1995). [CrossRef] [PubMed]

],
H^int=h¯Gx^n^o=h¯Gxzpf(a^m+a^m)a^oa^o,
(2)
where x^=xzpf(a^m+a^m) is the mechanical displacement operator, n^o=a^oa^o the photon occupation of the optical cavity, and âo, a^o, âm 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 mechanical oscillator’s zero-point fluctuations
xzpf=h¯2meffΩm
(3)
is determined by its resonance frequency Ωm and its effective mass meff as discussed in more detail below.

The canonical example of such a system would be a Fabry-Pérot resonator, one mirror of which is mounted as a mass on a spring moving as a whole along the symmetry axis of the cavity. In this case, the coupling parameter is given by G = –ωc/L, where L is the cavity length, and the oscillator’s effective mass simply is the moving mirror’s mass. In most real optomechanical systems, however, internal mechanical modes constitute the mechanical degree of freedom of interest (for simplicity, our discussion will be restricted to a single mechanical mode only). In general, the displacement of such a mode has to be described by a three-dimensional vector field u⃗(r⃗). For every volume element located at r⃗, it yields the displacement u⃗(r⃗) from its equilibrium position. In order to retain the simple one-dimensional description of equation (2), it is then necessary to map the vector field u⃗(r⃗) to a scalar displacement x.

In all cases, the only physically significant relation is the induced resonance frequency shift Δωc upon a displacement u⃗(r⃗), which in a perturbation theory treatment, can be calculated according to [22

22. For open systems, the integration boundaries should be chosen close to the surface of the resonator, to avoid divergence of the integral in the denominator. Alternatively, radiating fields escaping from the volume of interest have to be taken into account by a surface integral.

, 23

23. H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990). [CrossRef] [PubMed]

]
Δωcωc12|E(r)|2(ɛ(r+u(r))ɛ(r))dr3|E(r)|2ɛ(r)dr3
(4)
This expression is uniquely determined by the distribution of the dielectric constant ɛ and the (unperturbed) electric field E⃗ and can, in most cases, be linearized in the small displacements u⃗ and reduced to a surface integral over the relevant resonator boundaries [13

13. A. Gillespie and F. Raab, “Thermally excited vibrations of the mirrors of laser interferometer gravitational wave detectors,” Phys. Rev. D 52, 577–585 (1995). [CrossRef]

, 24

24. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002). [CrossRef]

].

In contrast to the unambiguous relation (4), the equivalent displacement x which is a-priori not a well-defined quantity, may be defined in an arbitrary way as discussed above. An important consequence of this ambiguity in the choice of x is the necessity to introduce an effective mass meff of the mechanical mode [13

13. A. Gillespie and F. Raab, “Thermally excited vibrations of the mirrors of laser interferometer gravitational wave detectors,” Phys. Rev. D 52, 577–585 (1995). [CrossRef]

, 25

25. M. Pinard, Y. Hadjar, and A. Heidmann, “Effective mass in quantum effects of radiation pressure,” Eur. Phys. J. D 7, 107–116 (1999).

] which retains the correct relation for the energy U stored in the mechanical mode given a displacement x,
U=12meffΩm2x2.
(5)
Evidently, rescaling xαx implies GG/α and meffmeff2, so that the frequently quoted figures of merit G and meff are individually only of very limited use. It is possible to eliminate this ambiguity by referring to the vacuum optomechanical coupling rate
g0Gxzpf
(6)
using terminology analogous to cavity quantum electrodynamics [26

26. H. J. Kimble, “Structure and dynamics in cavity quantum electrodynamics,” in “Cavity Quantum Electrodynamics,” P. R. Berman, ed. (Academic Press, 1994), Advances in Atomic, Molecular and Optical Physics.

]. As expected from the Hamiltonian description (2), all optomechanical dynamics can be fully described in terms of this parameter. As a consequence, parameters such as the cooling rate or optical spring effect can be derived upon knowledge of g0. For instance, the cooling rate in the resolved sideband regime is given by Γ=4n^og02/κ, while the power to reach the SQL scales with κ2/g02 with the cavity loss rate κ. Furthermore, the regime where g0n^oκ, Γmm is the mechanical damping rate) denotes the strong optomechanical coupling regime, in which parametric normal mode splitting occurs [3

3. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007). [CrossRef] [PubMed]

, 27

27. J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett. 101, 263602 (2008). [CrossRef] [PubMed]

, 28

28. S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009). [CrossRef] [PubMed]

].

2. Determination of the vacuum optomechanical coupling rate

Our intention here is to detail and verify an experimental technique to determine g0 in a simple and straightforward manner for an arbitrary optomechanical system. The technique we present relies on the fluctuation-dissipation theorem, from which the double-sided (symmetrized) spectral density of displacement fluctuations of a mechanical oscillator in contact with a thermal bath at temperature T can be derived [13

13. A. Gillespie and F. Raab, “Thermally excited vibrations of the mirrors of laser interferometer gravitational wave detectors,” Phys. Rev. D 52, 577–585 (1995). [CrossRef]

, 29

29. H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev. 83, 34–40 (1951). [CrossRef]

, 30

30. P. R. Saulson, “Thermal noise in mechanical experiments,” Phys. Rev. D 42, 2437–2445 (1990). [CrossRef]

],
Sxx(Ω)=Γmh¯Ωmeffcoth(h¯Ω2kBT)(Ω2Ωm2)2+Γm2Ω21meff2ΓmkBT(Ω2Ωm2)2+Γm2Ω2,
(7)
where meff is the effective mass corresponding to the choice of the normalization of x, as derived from Eq. (5), and the approximation is valid for large mechanical occupation numbers 〈nm〉 ≈ kBT/Ωm ≫ 1. The displacement fluctuations are not directly experimentally accessible, but cause fluctuations of the cavity resonance frequency ωc according to
Sωω(Ω)=G2Sxx(Ω)g022Ωmh¯2ΓmkBT(Ω2Ωm2)2+Γm2Ω2.
(8)
If such a spectrum can be measured, it is straightforward to integrate it over Fourier frequencies resulting in the interesting relation
δωc2=+Sωω(Ω)dΩ2π=Sωω(Ωm)Γm2=2nmg02,
(9)
which implies that the knowledge of the mechanical occupation number and the cavity resonance frequency fluctuation spectrum are sufficient to directly derive g0. The former is usually a straightforward task (provided dynamical backaction [31

31. V. B. Braginsky, S. E. Strigin, and V. P. Vyatchanin, “Parametric oscillatory instability in Fabry-Perot interferometer,” Phys. Lett. A 287, 331–338 (2001). [CrossRef]

,32

32. V. B. Braginsky and S. P. Vyatchanin, “Low quantum noise tranquilizer for Fabry-Perot interferometer,” Phys. Lett. A 293, 228–234 (2002). [CrossRef]

] is ruled out, for example, by low probing powers), whereas the latter has to be implemented using a phase-sensitive measurement scheme as shown in figure 1, since the required changes of the optical resonance frequency change the phase of the internal and hence output field.

Fig. 1 Generic scheme for the determination of the vacuum optomechanical coupling rate. The cavity optomechanical system is probed with a monochromatic source of electromagnetic waves which is frequency-modulated by a known amount at a Fourier frequency close to 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 detector measures the phase difference between the arms in an interferometer, one of which accommodates the optomechanical system (shutter S open). If the shutter S is closed, the transmission signal I of an optomechanical system is amplitude-dependent when the driving wave is detuned with respect to the optical resonance.

Various methods for such measurements are available, such as direct detection when the probing laser is detuned, frequency modulation [33

33. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonators,” Appl. Phys. B 31, 97–105 (1983). [CrossRef]

], polarization [34

34. T. W. Hänsch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35, 441–444 (1980). [CrossRef]

], or homodyne spectroscopy [35

35. Y. Hadjar, P. F. Cohadon, C. G. Aminoff, M. Pinard, and A. Heidmann, “High-sensitivity optical measurement of mechanical Brownian motion,” Europhys. Lett. 47, 545–551 (1999). [CrossRef]

]. In general, cavity resonance frequency fluctuations are transduced into the signal in a frequency-dependent manner, such that the spectrum of the measured signal (e.g. a photocurrent) I is given by
SII(Ω)=K(Ω)Sψψ(Ω)=K(Ω)Ω2Sωω(Ω),
(10)
where ψ is the phase of the intracavity field, which gets modulated upon a change of the resonance frequency. Importantly, to exploit the relation (9), it is necessary to convert the spectrum SII(Ω) back into a frequency noise spectrum, so the function K(Ω) = K(Ω, Δ, ηc, κ, P) (where Δ = ωlωc is the detuning from resonance, and ηc = κex/κ is the coupling parameter quantifying the ratio of the cavity loss rate due to input coupling κex compared to the total loss rate κ) has to be calibrated in absolute terms. Experimentally, this can be accomplished by frequency-modulating the source that is used for measuring SII [35

35. Y. Hadjar, P. F. Cohadon, C. G. Aminoff, M. Pinard, and A. Heidmann, “High-sensitivity optical measurement of mechanical Brownian motion,” Europhys. Lett. 47, 545–551 (1999). [CrossRef]

37

37. A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008). [CrossRef]

]. Under correct experimental conditions (cf. appendix) the phase modulation of the input field undergoes the exactly same 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 of monochromatic input field phase modulation (Ele–iϕ0 cos Ωmodt), the equivalent spectral density is given by Sϕϕ=(Ω)=2π12(δ(ΩΩmod)+δ(Ω+Ωmod))ϕ02/2.

A laboratory spectrum analyzer will, of course, not resolve this spectrum but convolve it with 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 given by the squared modulus of the Fourier transform of the time-domain windowing function. We then have
SIImeas(Ω)=2F(Ω)*(K(Ω)(Sψψ(Ω)+Sϕϕ(Ω))),
(12)
where a factor of 2 was introduced to account for the fact that most analyzers conventionally display single-sided spectra (this factor is irrelevant for our final result, however). The filter 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

38. G. Heinzel, A. Rudiger, and R. Schilling, “Spectrum and spectral density estimation by the discrete fourier transform (DFT), including a comprehensive list of window functions and some new at-top windows,” Tech. rep., Max Planck Society (2002). http://pubman.mpdl.mpg.de/pubman/item/escidoc:152164:1.

]. Note, however, that this description is still idealized. In reality, correction factors may apply depending on details of the signal processing (such as averaging of logarithmic noise power readings) in the respective analyzer used, see e.g. [39

39. Agilent Technologies, Agilent Application Note 1303: Spectrum and Signal Analyzer Measurements and Noise .

].

If the effective noise bandwidth is chosen much narrower than all spectral features in Sψψ (Ω), the measured spectrum is approximately
SIImeas(Ω)2K(Ω)Sψψ(Ω)+ϕ022K(Ωmod)(F(ΩΩmod)+F(Ω+Ωmod)).
(14)
Assuming that the signal at Ωmod is dominated by laser modulation, the transduction function K can then be calibrated in absolute terms at this frequency, as in the relation
SIImeas(±Ωmod)ϕ022K(Ωmod)F(0)=ϕ022K(Ωmod)ENBW
(15)
all parameters except Kmod) are known. The spectral shape of K is then needed to derive the spectrum Sψψ of interest. This task can be accomplished either by repeating the measurement for all modulation frequencies of interest, or by theoretical analysis, relating it to the conditions under which the measurements are taken.

We would finally like to emphasize that in many experimental situations, the transduction function K(Ω) is sufficiently constant over the range of frequencies, in which the mechanical response in (9) is significant (several mechanical linewidths around Ωm). In this case, it can be simply approximated to be flat, K(Ω) ≈ Km) ≈ Kmod), and we get from Eqs. (9), (10), (14) and (15)
g0212nmϕ02Ωmod22SIImeas(Ωm)Γm/4SIImeas(Ωmod)ENBW.
(20)
The last fraction in expression (20) can be interpreted as the ratio of the integrals of the displacement thermal noise and the calibration peak as broadened by the filter function. Note also that according to its definition the ENBW is given in direct frequency and not angular frequency as is Γm.

3. Application of the method

As two examples, we have applied the discussed method to two different optomechanical systems shown in Fig. 2. We first consider the case of a doubly-clamped strained silicon nitride string (30 × 0.7 × 0.1 μm3) placed in the near-field of a silica toroidal resonator [41

41. G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009). [CrossRef]

]. This scheme allows sensitive transduction of the nanomechanical motion with an imprecision below the level of the standard quantum limit [42

42. G. Anetsberger, E. Gavartin, O. Arcizet, Q. P. Unterreithmeier, E. M. Weig, M. L. Gorodetsky, J. P. Kotthaus, and T. J. Kippenberg, “Measuring nanomechanical motion with an imprecision far below the standard quantum limit,” arXiv:1003.3752 (2010).

].

Fig. 2 Measured (single-sided) frequency noise spectra Sννmeas (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 laser frequency modulation at frequencies of 8 and 71.38MHz, respectively. Orange lines are Lorentzian fits, the gray area underneath them yield the optomechanical coupling rates multiplied with the occupation number of the oscillator.

Note that the coupling rate in such a configuration depends on the strength of the evanescent field at the location of the nanomechanical oscillator, which increases exponentially if the oscillator is approached to the toroid. The data in Fig. 2 show the single-sided cavity frequency noise spectral density Sννmeas(Ω)SIImeas(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 homodyne detection with Δ = 0 is suffciently flat around the mechanical peak, Γmκ. However, the cavity frequency noise spectrum as measured here contains also contributions from other noise mechanisms present in the silica toroidal oscillator, in particular thermorefractive noise [36

36. M. L. Gorodetsky and I. S. Grudinin, “Fundamental thermal fluctuations in microspheres,” J. Opt. Soc. Am. B 21, 697–705 (2004). [CrossRef]

] known to dominate frequency noise at lower (≲ 10 MHz) Fourier frequencies. The integration of the spectrum is therefore exclusively performed on the Lorentzian corresponding to the mechanical mode (gray area), which can be easily isolated by fitting the noise spectrum with a model accommodating all known noise mechanisms including thermorefractive noise [42

42. G. Anetsberger, E. Gavartin, O. Arcizet, Q. P. Unterreithmeier, E. M. Weig, M. L. Gorodetsky, J. P. Kotthaus, and T. J. Kippenberg, “Measuring nanomechanical motion with an imprecision far below the standard quantum limit,” arXiv:1003.3752 (2010).

, 43

43. A. Schliesser, G. Anetsberger, R. Rivière, O. Arcizet, and T. J. Kippenberg, “High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators,” N. J. Phys. 10, 095015 (2008). [CrossRef]

]. The result of this evaluation is a vacuum coupling rate of g0 = 2π · 520Hz.

Silica WGM microresonators such as spheres or toroids actually support structure-inherent micromechanical modes themselves [43

43. A. Schliesser, G. Anetsberger, R. Rivière, O. Arcizet, and T. J. Kippenberg, “High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators,” N. J. Phys. 10, 095015 (2008). [CrossRef]

, 44

44. R. Ma, A. Schliesser, P. Del’Haye, A. Dabirian, G. Anetsberger, and T. Kippenberg, “Radiation-pressure-driven vibrational modes in ultrahigh-Q silica microspheres,” Opt. Lett. 32, 2200–2202 (2007). [CrossRef] [PubMed]

]. Figure 2 also shows an equivalent cavity frequency noise spectrum measured on such a cavity using direct detection. Again we neglect the frequency-dependence of K due to the very narrow mechanical peak (Γmκ, Ωm). For the evaluation of g0, we remove backgrounds due to thermal noise from other modes, thermorefractive 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 coupling rate of g0 = 2π · 570 Hz is obtained, which is interestingly very similar to the vacuum coupling rate of the nanomechanical string.

4. Conclusion

In summary we have presented a simple method which allows direct measurements of the vacuum coupling rate g0 in optomechanical systems using frequency-modulation. We have applied 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 optomechanical systems, for example, we estimate g0/2π ∼ 10–2 Hz for (present) microwave systems [18

18. C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nat. Phys. 4, 555–560 (2008). [CrossRef]

], g0/2π ∼ 100 Hz in lever-based optical systems [15

15. O. Arcizet, C. Molinelli, P.-F. Briant, T. anf Cohadon, A. Heidmann, J.-M. Mackowksi, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. 10, 125021 (2008). [CrossRef]

,28

28. S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009). [CrossRef] [PubMed]

], and up to g0/2π ∼ 105 Hz in integrated optomechanical crystals [20

20. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009). [CrossRef] [PubMed]

]. Together with the optical and mechanical dissipation rates and resonance frequencies (which do also vary widely), the coupling rate provides a convenient way to assess the potential of optomechanical platforms for the different phenomena and experiments within the setting of cavity optomechanics.

A. Transduction coefficients

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

Assume that the mechanical degree of freedom of the cavity harmonically oscillates so that
x(t)=x0cos(Ωmt).
(22)
These oscillations will produce a periodic change of the optical eigenfrequency and hence the phase of the internal field with ψ0 = x0Gm. If these oscillations are small (|ψ0| ≪ 1), the spectral components of the internal fields in the cavity and output field in the coupler will be:
ax=sinηcκ(0)(1iψ0Ωm(+Ωm)2eiΩmtiψ0Ωm(Ωm)2e+iΩmt),sx,out=sinηcκax,(Ω)=1i(Δ+Ω)+κ/2.
(23)
In the same way if the input field is weakly phase modulated (sine0 cos(Ωmodt)) to obtain a calibration signal:
Sϕ,in=sin(1iϕ02eiΩmodtiϕ02e+iΩmodt),aϕ=sinηcκ((0)iϕ0(+Ωmod)2eiΩmodtiϕ0(+Ωmod)2e+iΩmodt).
(24)
Sϕ,out=Sϕ,inηcκaϕ.
(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 looking at optical power modulation on a detector with appropriate detuning Δ or by measuring phase oscillations using homodyne detection at zero detuning, with the local oscillator signal obtained by splitting the input power. From these relations it is possible to calculate the spectral densities of the power detected directly at the output of the detector, and the homodyne signal, once the spectral densities of mechanical (Sxx(Ω)) or laser phase (Sϕϕ(Ω)) are known.

A.1. Direct detection

Directly calculating the modulation amplitude of the output optical power on a detector Pout(Ω)=h¯ω0|sout|Ω2 (omitting detector efficiency to convert to a photocurrent) we find that
KD(Ω)=SPP(Ω)Pin2Sψψ(Ω)=SPP(Ω)Pin2Sϕϕ(Ω)==4ηc2κ2Δ2Ω2(Ω2+κ2(1ηc)2)((Ω+Δ)2+κ2/4)((ΩΔ)2+κ2/4)(Δ2+κ2/4)2.
(26)

In the resolved sideband regime when Ω>2κ the signal at frequency Ω is maximized when the pump is either tuned on the slope or on one of the mechanical sidebands:
Δ=±122Ω2κ22Ω42κ2Ω2±κ2(1+κ24Ω2),Δ=±122Ω2κ2+2Ω42κ2Ω2±Ω(1+3κ28Ω2),
(27)
producing the same signal
KD(Ω)=4ηc2(1+κ2(1ηc)2)Ω2).
(28)

When Ω<2κ, the output is maximized when
Δ=±1612Ω2+3κ2,KD(Ω)=27Ω2ηc2κ2(Ω2+κ2(1ηc)2)(Ω2+κ2)3.
(29)

A.2. Balanced homodyne detection

In this case, the signal on the detector is proportional to the field amplitudes sLO and sin falling on the output beam splitter,
Hh¯ω0=|i2sout12sLO|2|i2sLO12sout|2=i(sLOsout*sLO*sout)=i(sLOsin*sLO*sin)+iηcκ(asLO*a*sLO)=2|sLO||Sin|sin(ϕLOϕin)+iηcκ(asLO*a*sLO)
(30)

It is important to note here that if the homodyne phase difference ϕLOϕin = 0 then this method 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 phase difference in an experiment is usually locked to have zero d.c. voltage. The local oscillator field is obtained by splitting the input power and is equal to:
sx,LO=sLOsinsx,ineiϕLO=sLOeiϕLO,
(31)
sϕ,LO=sLOsinsϕ,ineiϕLO=sLOeiϕLO(1iϕ02eiΩmodtiϕ02e+iΩmodt),
(32)
where ϕLO is chosen to suppress any d.c. voltage,
ϕLO=arctan(ηcκΔΔ2+(12ηc)κ2/4).
(33)
In this case, again
SHHPinPLOSψψ=SHHPinPLOSϕϕ
(34)
and
KH(Ω)=4κ2ηc2Ω(Ω2(12ηc)2κ2/4+(Δ2(12ηc)κ2/4)2)((ΔΩ)2+κ2/4)((Δ+Ω)2+κ2/4)(Δ2+(12ηc)2κ2/4)(Δ2+κ2/4).
(35)
If Δ = 0 which appears the most favorable case for homodyne detection, then:
KH(Ω)=16ηc2Ω2Ω2+κ2/4.
(36)
Another interesting case is Δ = ±Ωm in case of resolved sideband regime. In this case
KH(±Ωm)4ηc2(1+κ2(16ηc13)16Ωm2).
(37)

Acknowledgments

M. G. acknowledges support from the Dynasty foundation. S. D. is funded by a Marie Curie Individual 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 of Quantum Photonics.

References and links

1.

V. B. Braginsky and Y. I. Vorontsov, “Quantum-mechanical limitations in macroscopic experiments and modern experimental technique,” Sov. Phys. Usp. 17, 644–650 (1975). [CrossRef]

2.

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693 (1981). [CrossRef]

3.

F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99, 093902 (2007). [CrossRef] [PubMed]

4.

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99, 093901 (2007). [CrossRef] [PubMed]

5.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. 95, 033901 (2005). [CrossRef] [PubMed]

6.

H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, “Radiation-pressure-driven micro-mechanical oscillator,” Opt. Express 13, 5293–5301 (2005). [CrossRef] [PubMed]

7.

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94, 223902 (2005). [CrossRef] [PubMed]

8.

K. J. Vahala, “Back-action limit of linewidth in an optomechanical oscillator,” Phys. Rev. A 78, 023832 (2008). [CrossRef]

9.

T. J. Kippenberg and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science 321, 1172–1176 (2008). [CrossRef] [PubMed]

10.

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009). [CrossRef]

11.

A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with silica microresonators,” in “Advances in atomic, molecular and optical physics,” vol. 58, E. Arimondo, P. Berman, and C. C. Lin, eds. (ElsevierAcademic Press, 2010), chap. 5, pp. 207–323.

12.

C. K. Law, “Interaction between a moving mirror and radiation pressure: A Hamiltonian formulation,” Phys. Rev. A 51, 2537–2541 (1995). [CrossRef] [PubMed]

13.

A. Gillespie and F. Raab, “Thermally excited vibrations of the mirrors of laser interferometer gravitational wave detectors,” Phys. Rev. D 52, 577–585 (1995). [CrossRef]

14.

S. Gigan, H. R. Böhm, M. Paternosto, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiation pressure,” Nature 444, 67–70 (2006). [CrossRef] [PubMed]

15.

O. Arcizet, C. Molinelli, P.-F. Briant, T. anf Cohadon, A. Heidmann, J.-M. Mackowksi, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. 10, 125021 (2008). [CrossRef]

16.

J. Hofer, A. Schliesser, and T. J. Kippenberg, “Cavity optomechanics with ultra-high Q crystalline micro-resonators,” Phys. Rev. A 82, 031804 (2010). [CrossRef]

17.

V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, “Coupling and tunability of optical whispering-gallery modes: a basis for coordinate meter,” Opt. Commun. 107, 41–48 (1994). [CrossRef]

18.

C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion with a microwave cavity interferometer,” Nat. Phys. 4, 555–560 (2008). [CrossRef]

19.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. Thomson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008). [CrossRef]

20.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009). [CrossRef] [PubMed]

21.

V. B. Braginsky, M. L. Gorodetsky, V. S. Ilchenko, and S. P. Vyatchanin, “On the ultimate sensitivity in coordinate measurements,” Phys. Lett. A 179, 244–248 (1993). [CrossRef]

22.

For open systems, the integration boundaries should be chosen close to the surface of the resonator, to avoid divergence of the integral in the denominator. Alternatively, radiating fields escaping from the volume of interest have to be taken into account by a surface integral.

23.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990). [CrossRef] [PubMed]

24.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002). [CrossRef]

25.

M. Pinard, Y. Hadjar, and A. Heidmann, “Effective mass in quantum effects of radiation pressure,” Eur. Phys. J. D 7, 107–116 (1999).

26.

H. J. Kimble, “Structure and dynamics in cavity quantum electrodynamics,” in “Cavity Quantum Electrodynamics,” P. R. Berman, ed. (Academic Press, 1994), Advances in Atomic, Molecular and Optical Physics.

27.

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett. 101, 263602 (2008). [CrossRef] [PubMed]

28.

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009). [CrossRef] [PubMed]

29.

H. B. Callen and T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev. 83, 34–40 (1951). [CrossRef]

30.

P. R. Saulson, “Thermal noise in mechanical experiments,” Phys. Rev. D 42, 2437–2445 (1990). [CrossRef]

31.

V. B. Braginsky, S. E. Strigin, and V. P. Vyatchanin, “Parametric oscillatory instability in Fabry-Perot interferometer,” Phys. Lett. A 287, 331–338 (2001). [CrossRef]

32.

V. B. Braginsky and S. P. Vyatchanin, “Low quantum noise tranquilizer for Fabry-Perot interferometer,” Phys. Lett. A 293, 228–234 (2002). [CrossRef]

33.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonators,” Appl. Phys. B 31, 97–105 (1983). [CrossRef]

34.

T. W. Hänsch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35, 441–444 (1980). [CrossRef]

35.

Y. Hadjar, P. F. Cohadon, C. G. Aminoff, M. Pinard, and A. Heidmann, “High-sensitivity optical measurement of mechanical Brownian motion,” Europhys. Lett. 47, 545–551 (1999). [CrossRef]

36.

M. L. Gorodetsky and I. S. Grudinin, “Fundamental thermal fluctuations in microspheres,” J. Opt. Soc. Am. B 21, 697–705 (2004). [CrossRef]

37.

A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008). [CrossRef]

38.

G. Heinzel, A. Rudiger, and R. Schilling, “Spectrum and spectral density estimation by the discrete fourier transform (DFT), including a comprehensive list of window functions and some new at-top windows,” Tech. rep., Max Planck Society (2002). http://pubman.mpdl.mpg.de/pubman/item/escidoc:152164:1.

39.

Agilent Technologies, Agilent Application Note 1303: Spectrum and Signal Analyzer Measurements and Noise .

40.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” arXiv:1007.0565 (2010).

41.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009). [CrossRef]

42.

G. Anetsberger, E. Gavartin, O. Arcizet, Q. P. Unterreithmeier, E. M. Weig, M. L. Gorodetsky, J. P. Kotthaus, and T. J. Kippenberg, “Measuring nanomechanical motion with an imprecision far below the standard quantum limit,” arXiv:1003.3752 (2010).

43.

A. Schliesser, G. Anetsberger, R. Rivière, O. Arcizet, and T. J. Kippenberg, “High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators,” N. J. Phys. 10, 095015 (2008). [CrossRef]

44.

R. Ma, A. Schliesser, P. Del’Haye, A. Dabirian, G. Anetsberger, and T. Kippenberg, “Radiation-pressure-driven vibrational modes in ultrahigh-Q silica microspheres,” Opt. Lett. 32, 2200–2202 (2007). [CrossRef] [PubMed]

OCIS Codes
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(270.2500) Quantum optics : Fluctuations, relaxations, and noise
(230.4685) Optical devices : Optical microelectromechanical devices

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: September 22, 2010
Manuscript Accepted: October 11, 2010
Published: October 19, 2010

Citation
M. L. Gorodetsky, A. Schliesser, G. Anetsberger, S. Deleglise, and T. J. Kippenberg, "Determination of the vacuum optomechanical coupling rate using frequency noise calibration," Opt. Express 18, 23236-23246 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-23236


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. V. B. Braginsky and Y. I. Vorontsov, "Quantum-mechanical limitations in macroscopic experiments and modern experimental technique," Sov. Phys. Usp. 17, 644-650 (1975). [CrossRef]
  2. C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693 (1981). [CrossRef]
  3. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, "Quantum theory of cavity-assisted sideband cooling of mechanical motion," Phys. Rev. Lett. 99, 093902 (2007). [CrossRef] [PubMed]
  4. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, "Theory of ground state cooling of a mechanical oscillator using dynamical back-action," Phys. Rev. Lett. 99, 093901 (2007). [CrossRef] [PubMed]
  5. T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, "Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity," Phys. Rev. Lett. 95, 033901 (2005). [CrossRef] [PubMed]
  6. H. Rokhsari, T. J. Kippenberg, T. Carmon, and K. J. Vahala, "Radiation-pressure-driven micro-mechanical oscillator," Opt. Express 13, 5293-5301 (2005). [CrossRef] [PubMed]
  7. T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, "Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode," Phys. Rev. Lett. 94, 223902 (2005). [CrossRef] [PubMed]
  8. K. J. Vahala, "Back-action limit of linewidth in an optomechanical oscillator," Phys. Rev. A 78, 023832 (2008). [CrossRef]
  9. T. J. Kippenberg and K. J. Vahala, "Cavity Optomechanics: Back-Action at the Mesoscale," Science 321, 1172-1176 (2008). [CrossRef] [PubMed]
  10. F. Marquardt and S. M. Girvin, "Optomechanics," Physics 2, 40 (2009). [CrossRef]
  11. A. Schliesser, and T. J. Kippenberg, "Cavity optomechanics with silica microresonators," in "Advances in atomic, molecular and optical physics," vol. 58, E. Arimondo, P. Berman, and C. C. Lin, eds. (Elsevier Academic Press, 2010), chap. 5, pp. 207-323.
  12. C. K. Law, "Interaction between a moving mirror and radiation pressure: A Hamiltonian formulation," Phys. Rev. A 51, 2537-2541 (1995). [CrossRef] [PubMed]
  13. A. Gillespie and F. Raab, "Thermally excited vibrations of the mirrors of laser interferometer gravitational wave detectors," Phys. Rev. D Part. Fields 52, 577-585 (1995). [CrossRef]
  14. S. Gigan, H. R. Böhm, M. Paternosto, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, "Self-cooling of a micromirror by radiation pressure," Nature 444, 67-70 (2006). [CrossRef] [PubMed]
  15. O. Arcizet, C. Molinelli, P.-F. Briant, T. anf Cohadon, A. Heidmann, J.-M. Mackowksi, C. Michel, L. Pinard, O. Francais, and L. Rousseau, "Experimental optomechanics with silicon micromirrors," N. J. Phys. 10, 125021 (2008). [CrossRef]
  16. J. Hofer, A. Schliesser, and T. J. Kippenberg, "Cavity optomechanics with ultra-high Q crystalline microresonators," Phys. Rev. A 82, 031804 (2010). [CrossRef]
  17. V. S. Ilchenko, M. L. Gorodetsky, and S. P. Vyatchanin, "Coupling and tunability of optical whispering-gallery modes: a basis for coordinate meter," Opt. Commun. 107, 41-48 (1994). [CrossRef]
  18. C. A. Regal, J. D. Teufel, and K. W. Lehnert, "Measuring nanomechanical motion with a microwave cavity interferometer," Nat. Phys. 4, 555-560 (2008). [CrossRef]
  19. A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. Thomson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, "Dispersive optomechanics: a membrane inside a cavity," N. J. Phys. 10, 095008 (2008). [CrossRef]
  20. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, "Optomechanical crystals," Nature 462, 78-82 (2009). [CrossRef] [PubMed]
  21. V. B. Braginsky, M. L. Gorodetsky, V. S. Ilchenko, and S. P. Vyatchanin, "On the ultimate sensitivity in coordinate measurements," Phys. Lett. A 179, 244-248 (1993). [CrossRef]
  22. For open systems, the integration boundaries should be chosen close to the surface of the resonator, to avoid divergence of the integral in the denominator. Alternatively, radiating fields escaping from the volume of interest have to be taken into account by a surface integral.
  23. H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, "Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets," Phys. Rev. A 41, 5187-5198 (1990). [CrossRef] [PubMed]
  24. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, "Perturbation theory for Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066611 (2002). [CrossRef]
  25. M. Pinard, Y. Hadjar, and A. Heidmann, "Effective mass in quantum effects of radiation pressure," Eur. Phys. J. D 7, 107-116 (1999).
  26. H. J. Kimble, "Structure and dynamics in cavity quantum electrodynamics," in "Cavity Quantum Electrodynamics," P. R. Berman, ed. (Academic Press, 1994), Advances in Atomic, Molecular and Optical Physics.
  27. J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, "Parametric normal-mode splitting in cavity optomechanics," Phys. Rev. Lett. 101, 263602 (2008). [CrossRef] [PubMed]
  28. S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, "Observation of strong coupling between a micromechanical resonator and an optical cavity field," Nature 460, 724-727 (2009). [CrossRef] [PubMed]
  29. H. B. Callen and T. A. Welton, "Irreversibility and generalized noise," Phys. Rev. 83, 34-40 (1951). [CrossRef]
  30. P. R. Saulson, "Thermal noise in mechanical experiments," Phys. Rev. D Part. Fields 42, 2437-2445 (1990). [CrossRef]
  31. V. B. Braginsky, S. E. Strigin, and V. P. Vyatchanin, "Parametric oscillatory instability in Fabry-Perot interferometer," Phys. Lett. A 287, 331-338 (2001). [CrossRef]
  32. V. B. Braginsky and S. P. Vyatchanin, "Low quantum noise tranquilizer for Fabry-Perot interferometer," Phys. Lett. A 293, 228-234 (2002). [CrossRef]
  33. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser Phase and Frequency Stabilization Using an Optical Resonators," Appl. Phys. B 31, 97-105 (1983). [CrossRef]
  34. T. W. Hänsch and B. Couillaud, "Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity," Opt. Commun. 35, 441-444 (1980). [CrossRef]
  35. Y. Hadjar, P. F. Cohadon, C. G. Aminoff, M. Pinard, and A. Heidmann, "High-sensitivity optical measurement of mechanical Brownian motion," Europhys. Lett. 47, 545-551 (1999). [CrossRef]
  36. M. L. Gorodetsky, and I. S. Grudinin, "Fundamental thermal fluctuations in microspheres," J. Opt. Soc. Am. B 21, 697-705 (2004). [CrossRef]
  37. A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. Kippenberg, "Resolved-sideband cooling of a micromechanical oscillator," Nat. Phys. 4, 415-419 (2008). [CrossRef]
  38. G. Heinzel, A. Rüdiger, and R. Schilling, "Spectrum and spectral density estimation by the discrete Fourier transform (DFT), including a comprehensive list of window functions and some new at-top windows," Tech. rep., Max Planck Society (2002). http://pubman.mpdl.mpg.de/pubman/item/escidoc:152164:1.
  39. Agilent Technologies, Agilent Application Note 1303: Spectrum and Signal Analyzer Measurements and Noise.
  40. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, "Optomechanically induced transparency," arXiv:1007.0565 (2010).
  41. G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, "Near-field cavity optomechanics with nanomechanical oscillators," Nat. Phys. 5, 909-914 (2009). [CrossRef]
  42. G. Anetsberger, E. Gavartin, O. Arcizet, Q. P. Unterreithmeier, E. M. Weig, M. L. Gorodetsky, J. P. Kotthaus, and T. J. Kippenberg, "Measuring nanomechanical motion with an imprecision far below the standard quantum limit," arXiv:1003.3752 (2010).
  43. A. Schliesser, G. Anetsberger, R. Rivière, O. Arcizet, and T. J. Kippenberg, "High-sensitivity monitoring of micromechanical vibration using optical whispering gallery mode resonators," N. J. Phys. 10, 095015 (2008). [CrossRef]
  44. R. Ma, A. Schliesser, P. Del’Haye, A. Dabirian, G. Anetsberger, and T. Kippenberg, "Radiation-pressure-driven vibrational modes in ultrahigh-Q silica microspheres," Opt. Lett. 32, 2200-2202 (2007). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited