Abstract

The preparation of a mechanical oscillator driven by quantum back-action is a fundamental requirement to reach the standard quantum limit (SQL) for force measurement, in optomechanical systems. However, thermal fluctuating force generally dominates a disturbance on the oscillator. In the macroscopic scale, an optical linear cavity including a suspended mirror has been used for the weak force measurement, such as gravitational-wave detectors. This configuration has the advantages of reducing the dissipation of the pendulum (i.e., suspension thermal noise) due to a gravitational dilution by using a thin wire, and of increasing the circulating laser power. However, the use of the thin wire is weak for an optical torsional anti-spring effect in the cavity, due to the low mechanical restoring force of the wire. Thus, there is the trade-off between the stability of the system and the sensitivity. Here, we describe using a triangular optical cavity to overcome this limitation for reaching the SQL. The triangular cavity can provide a sensitive and stable system, because it can optically trap the mirror’s motion of the yaw, through an optical positive torsional spring effect. To show this, we demonstrate a measurement of the torsional spring effect caused by radiation pressure forces.

© 2014 Optical Society of America

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References

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  1. M. Aspelmeyer, T. J. Kippenberg, F. Marquardt, “Cavity Optomechanics,” arXiv:1303.0733 (2013).
  2. G. M. Harry, (for the LIGO Scientific Collaboration). “Advanced LIGO: the next generation of gravitational wave detectors,” Class. Quantum Grav. 27, 084006 (2010).
    [CrossRef]
  3. K. Somiya, “Detector configuration of KAGRA-the Japanese cryogenic gravitational-wave detector,” Class. Quantum Grav. 29, 124007 (2012).
    [CrossRef]
  4. F. Ya. Khalili, H. Miao, A. H. Safavi-Naeini, O. Painter, Y. Chen, “Quantum back-action in measurements of zero-point mechanical oscillations,” Phys. Rev. A 86, 033840 (2012).
    [CrossRef]
  5. H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, Y. Chen, “Entanglement of Macroscopic Test Masses and the Standard Quantum Limit in Laser Interferometry,” Phys. Rev. Lett. 100, 013601 (2008).
    [CrossRef] [PubMed]
  6. H. Miao, S. Danilishin, H. Müller-Ebhardt, H. Rehbein, K. Somiya, Y. Chen, “Probing macroscopic quantum states with a sub-Heisenberg accuracy,” Phys. Rev. A 81, 012114 (2010).
    [CrossRef]
  7. W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Phys. 43, 172–198 (1927).
    [CrossRef]
  8. P. R. Saulson, “Thermal noise in mechanical experiments,” Phys. Rev. D 42, 2437 (1990).
    [CrossRef]
  9. J. A. Sidles, D. Sigg, “Optical torques in suspended Fabry-perot interferometers,” Phys. Lett. A 354, 167–172 (2006).
    [CrossRef]
  10. S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D 81, 064023 (2010).
    [CrossRef]
  11. F. Kawazoe, R. Schilling, H. Lück, ”Eigenmode changes in a misaligned triangular optical cavity,” J. Opt. 13, 055504 (2011).
    [CrossRef]
  12. D. Sigg, “Angular stability in a triangular fabry-perot cavity,” LIGO-T030275-00, www.ligo.caltech.edu/docs/T/T030275-00.pdf (2003).
  13. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [CrossRef]
  14. A. R. Neben, T. P. Bodiya, C. Wipf, E. Oelker, T. Corbitt, N. Mavalvala, “Structural thermal noise in gram-scale mirror oscillators,” New J. Phys. 14, 115008 (2012).
    [CrossRef]
  15. S. Kawamura, (personal communication).
  16. (to be submitted)

2012 (3)

K. Somiya, “Detector configuration of KAGRA-the Japanese cryogenic gravitational-wave detector,” Class. Quantum Grav. 29, 124007 (2012).
[CrossRef]

F. Ya. Khalili, H. Miao, A. H. Safavi-Naeini, O. Painter, Y. Chen, “Quantum back-action in measurements of zero-point mechanical oscillations,” Phys. Rev. A 86, 033840 (2012).
[CrossRef]

A. R. Neben, T. P. Bodiya, C. Wipf, E. Oelker, T. Corbitt, N. Mavalvala, “Structural thermal noise in gram-scale mirror oscillators,” New J. Phys. 14, 115008 (2012).
[CrossRef]

2011 (1)

F. Kawazoe, R. Schilling, H. Lück, ”Eigenmode changes in a misaligned triangular optical cavity,” J. Opt. 13, 055504 (2011).
[CrossRef]

2010 (3)

H. Miao, S. Danilishin, H. Müller-Ebhardt, H. Rehbein, K. Somiya, Y. Chen, “Probing macroscopic quantum states with a sub-Heisenberg accuracy,” Phys. Rev. A 81, 012114 (2010).
[CrossRef]

G. M. Harry, (for the LIGO Scientific Collaboration). “Advanced LIGO: the next generation of gravitational wave detectors,” Class. Quantum Grav. 27, 084006 (2010).
[CrossRef]

S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D 81, 064023 (2010).
[CrossRef]

2008 (1)

H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, Y. Chen, “Entanglement of Macroscopic Test Masses and the Standard Quantum Limit in Laser Interferometry,” Phys. Rev. Lett. 100, 013601 (2008).
[CrossRef] [PubMed]

2006 (1)

J. A. Sidles, D. Sigg, “Optical torques in suspended Fabry-perot interferometers,” Phys. Lett. A 354, 167–172 (2006).
[CrossRef]

1990 (1)

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

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

1927 (1)

W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Phys. 43, 172–198 (1927).
[CrossRef]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, F. Marquardt, “Cavity Optomechanics,” arXiv:1303.0733 (2013).

Bodiya, T. P.

A. R. Neben, T. P. Bodiya, C. Wipf, E. Oelker, T. Corbitt, N. Mavalvala, “Structural thermal noise in gram-scale mirror oscillators,” New J. Phys. 14, 115008 (2012).
[CrossRef]

Chen, Y.

F. Ya. Khalili, H. Miao, A. H. Safavi-Naeini, O. Painter, Y. Chen, “Quantum back-action in measurements of zero-point mechanical oscillations,” Phys. Rev. A 86, 033840 (2012).
[CrossRef]

H. Miao, S. Danilishin, H. Müller-Ebhardt, H. Rehbein, K. Somiya, Y. Chen, “Probing macroscopic quantum states with a sub-Heisenberg accuracy,” Phys. Rev. A 81, 012114 (2010).
[CrossRef]

H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, Y. Chen, “Entanglement of Macroscopic Test Masses and the Standard Quantum Limit in Laser Interferometry,” Phys. Rev. Lett. 100, 013601 (2008).
[CrossRef] [PubMed]

Corbitt, T.

A. R. Neben, T. P. Bodiya, C. Wipf, E. Oelker, T. Corbitt, N. Mavalvala, “Structural thermal noise in gram-scale mirror oscillators,” New J. Phys. 14, 115008 (2012).
[CrossRef]

Danilishin, S.

H. Miao, S. Danilishin, H. Müller-Ebhardt, H. Rehbein, K. Somiya, Y. Chen, “Probing macroscopic quantum states with a sub-Heisenberg accuracy,” Phys. Rev. A 81, 012114 (2010).
[CrossRef]

Danzmann, K.

H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, Y. Chen, “Entanglement of Macroscopic Test Masses and the Standard Quantum Limit in Laser Interferometry,” Phys. Rev. Lett. 100, 013601 (2008).
[CrossRef] [PubMed]

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Harry, G. M.

G. M. Harry, (for the LIGO Scientific Collaboration). “Advanced LIGO: the next generation of gravitational wave detectors,” Class. Quantum Grav. 27, 084006 (2010).
[CrossRef]

Heisenberg, W.

W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Phys. 43, 172–198 (1927).
[CrossRef]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Ishizaki, H.

S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D 81, 064023 (2010).
[CrossRef]

Kawamura, S.

S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D 81, 064023 (2010).
[CrossRef]

S. Kawamura, (personal communication).

Kawazoe, F.

F. Kawazoe, R. Schilling, H. Lück, ”Eigenmode changes in a misaligned triangular optical cavity,” J. Opt. 13, 055504 (2011).
[CrossRef]

Kippenberg, T. J.

M. Aspelmeyer, T. J. Kippenberg, F. Marquardt, “Cavity Optomechanics,” arXiv:1303.0733 (2013).

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Lück, H.

F. Kawazoe, R. Schilling, H. Lück, ”Eigenmode changes in a misaligned triangular optical cavity,” J. Opt. 13, 055504 (2011).
[CrossRef]

Marquardt, F.

M. Aspelmeyer, T. J. Kippenberg, F. Marquardt, “Cavity Optomechanics,” arXiv:1303.0733 (2013).

Mavalvala, N.

A. R. Neben, T. P. Bodiya, C. Wipf, E. Oelker, T. Corbitt, N. Mavalvala, “Structural thermal noise in gram-scale mirror oscillators,” New J. Phys. 14, 115008 (2012).
[CrossRef]

Miao, H.

F. Ya. Khalili, H. Miao, A. H. Safavi-Naeini, O. Painter, Y. Chen, “Quantum back-action in measurements of zero-point mechanical oscillations,” Phys. Rev. A 86, 033840 (2012).
[CrossRef]

H. Miao, S. Danilishin, H. Müller-Ebhardt, H. Rehbein, K. Somiya, Y. Chen, “Probing macroscopic quantum states with a sub-Heisenberg accuracy,” Phys. Rev. A 81, 012114 (2010).
[CrossRef]

Miyakawa, O.

S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D 81, 064023 (2010).
[CrossRef]

Müller-Ebhardt, H.

H. Miao, S. Danilishin, H. Müller-Ebhardt, H. Rehbein, K. Somiya, Y. Chen, “Probing macroscopic quantum states with a sub-Heisenberg accuracy,” Phys. Rev. A 81, 012114 (2010).
[CrossRef]

H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, Y. Chen, “Entanglement of Macroscopic Test Masses and the Standard Quantum Limit in Laser Interferometry,” Phys. Rev. Lett. 100, 013601 (2008).
[CrossRef] [PubMed]

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Neben, A. R.

A. R. Neben, T. P. Bodiya, C. Wipf, E. Oelker, T. Corbitt, N. Mavalvala, “Structural thermal noise in gram-scale mirror oscillators,” New J. Phys. 14, 115008 (2012).
[CrossRef]

Nishizawa, A.

S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D 81, 064023 (2010).
[CrossRef]

Oelker, E.

A. R. Neben, T. P. Bodiya, C. Wipf, E. Oelker, T. Corbitt, N. Mavalvala, “Structural thermal noise in gram-scale mirror oscillators,” New J. Phys. 14, 115008 (2012).
[CrossRef]

Painter, O.

F. Ya. Khalili, H. Miao, A. H. Safavi-Naeini, O. Painter, Y. Chen, “Quantum back-action in measurements of zero-point mechanical oscillations,” Phys. Rev. A 86, 033840 (2012).
[CrossRef]

Rehbein, H.

H. Miao, S. Danilishin, H. Müller-Ebhardt, H. Rehbein, K. Somiya, Y. Chen, “Probing macroscopic quantum states with a sub-Heisenberg accuracy,” Phys. Rev. A 81, 012114 (2010).
[CrossRef]

H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, Y. Chen, “Entanglement of Macroscopic Test Masses and the Standard Quantum Limit in Laser Interferometry,” Phys. Rev. Lett. 100, 013601 (2008).
[CrossRef] [PubMed]

Safavi-Naeini, A. H.

F. Ya. Khalili, H. Miao, A. H. Safavi-Naeini, O. Painter, Y. Chen, “Quantum back-action in measurements of zero-point mechanical oscillations,” Phys. Rev. A 86, 033840 (2012).
[CrossRef]

Sakata, S.

S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D 81, 064023 (2010).
[CrossRef]

Saulson, P. R.

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

Schilling, R.

F. Kawazoe, R. Schilling, H. Lück, ”Eigenmode changes in a misaligned triangular optical cavity,” J. Opt. 13, 055504 (2011).
[CrossRef]

Schnabel, R.

H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, Y. Chen, “Entanglement of Macroscopic Test Masses and the Standard Quantum Limit in Laser Interferometry,” Phys. Rev. Lett. 100, 013601 (2008).
[CrossRef] [PubMed]

Sidles, J. A.

J. A. Sidles, D. Sigg, “Optical torques in suspended Fabry-perot interferometers,” Phys. Lett. A 354, 167–172 (2006).
[CrossRef]

Sigg, D.

J. A. Sidles, D. Sigg, “Optical torques in suspended Fabry-perot interferometers,” Phys. Lett. A 354, 167–172 (2006).
[CrossRef]

D. Sigg, “Angular stability in a triangular fabry-perot cavity,” LIGO-T030275-00, www.ligo.caltech.edu/docs/T/T030275-00.pdf (2003).

Somiya, K.

K. Somiya, “Detector configuration of KAGRA-the Japanese cryogenic gravitational-wave detector,” Class. Quantum Grav. 29, 124007 (2012).
[CrossRef]

H. Miao, S. Danilishin, H. Müller-Ebhardt, H. Rehbein, K. Somiya, Y. Chen, “Probing macroscopic quantum states with a sub-Heisenberg accuracy,” Phys. Rev. A 81, 012114 (2010).
[CrossRef]

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Wipf, C.

A. R. Neben, T. P. Bodiya, C. Wipf, E. Oelker, T. Corbitt, N. Mavalvala, “Structural thermal noise in gram-scale mirror oscillators,” New J. Phys. 14, 115008 (2012).
[CrossRef]

Ya. Khalili, F.

F. Ya. Khalili, H. Miao, A. H. Safavi-Naeini, O. Painter, Y. Chen, “Quantum back-action in measurements of zero-point mechanical oscillations,” Phys. Rev. A 86, 033840 (2012).
[CrossRef]

Appl. Phys. B (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Class. Quantum Grav. (2)

G. M. Harry, (for the LIGO Scientific Collaboration). “Advanced LIGO: the next generation of gravitational wave detectors,” Class. Quantum Grav. 27, 084006 (2010).
[CrossRef]

K. Somiya, “Detector configuration of KAGRA-the Japanese cryogenic gravitational-wave detector,” Class. Quantum Grav. 29, 124007 (2012).
[CrossRef]

J. Opt. (1)

F. Kawazoe, R. Schilling, H. Lück, ”Eigenmode changes in a misaligned triangular optical cavity,” J. Opt. 13, 055504 (2011).
[CrossRef]

New J. Phys. (1)

A. R. Neben, T. P. Bodiya, C. Wipf, E. Oelker, T. Corbitt, N. Mavalvala, “Structural thermal noise in gram-scale mirror oscillators,” New J. Phys. 14, 115008 (2012).
[CrossRef]

Phys. Lett. A (1)

J. A. Sidles, D. Sigg, “Optical torques in suspended Fabry-perot interferometers,” Phys. Lett. A 354, 167–172 (2006).
[CrossRef]

Phys. Rev. A (2)

F. Ya. Khalili, H. Miao, A. H. Safavi-Naeini, O. Painter, Y. Chen, “Quantum back-action in measurements of zero-point mechanical oscillations,” Phys. Rev. A 86, 033840 (2012).
[CrossRef]

H. Miao, S. Danilishin, H. Müller-Ebhardt, H. Rehbein, K. Somiya, Y. Chen, “Probing macroscopic quantum states with a sub-Heisenberg accuracy,” Phys. Rev. A 81, 012114 (2010).
[CrossRef]

Phys. Rev. D (2)

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

S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D 81, 064023 (2010).
[CrossRef]

Phys. Rev. Lett. (1)

H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, Y. Chen, “Entanglement of Macroscopic Test Masses and the Standard Quantum Limit in Laser Interferometry,” Phys. Rev. Lett. 100, 013601 (2008).
[CrossRef] [PubMed]

Z. Phys. (1)

W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Phys. 43, 172–198 (1927).
[CrossRef]

Other (4)

M. Aspelmeyer, T. J. Kippenberg, F. Marquardt, “Cavity Optomechanics,” arXiv:1303.0733 (2013).

D. Sigg, “Angular stability in a triangular fabry-perot cavity,” LIGO-T030275-00, www.ligo.caltech.edu/docs/T/T030275-00.pdf (2003).

S. Kawamura, (personal communication).

(to be submitted)

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Figures (6)

Fig. 1
Fig. 1

Schematic of optical torsional effects. The schematic responses of the optical axes to the angular motion of the movable mirror are shown. The detailed response for the triangular cavity is given in Ref. [11]. An optical torque occurs through the stationary radiation pressure. a: In the case of the linear optical cavity, the optical torque occurs in the same direction as the angular motion. This results in an anti-restoring force. b: In the case of the triangular optical cavity, the optical torque occurs in the opposite direction as the angular motion. This results in a restoring force.

Fig. 2
Fig. 2

The square root of the ratio of the quantum back-action to the thermal fluctuating force. a: Trade-off relationship between the sensitivity and the stability is generated, in the case of using a linear cavity. A white domain represents the area, where κwire + κopt < 0. Parameters: l = 0.02 m, m = 14.7 mg, σ = 0.28, T = 300 K, and κ/(2π) = 1.39 MHz. Here, we suppose that each dissipation mechanisms are due to viscous friction (frequency dependent friction), for simplicity. In addition, we suppose that the intrinsic quality factor of the wire of Qin is 3,800 (i.e., Qm = Qen × Qin), and it has no dependence of the radius of the wire. b: As for a triangular cavity, the square root of the ratio is plotted. The ratio of Rs becomes one at the boundary of blue and red domains. In the blue domain, there is a possibility to reach the SQL. The same parameters as in Fig. 2(a) are used, but there is no unstable domain. In addition, the incident angle of β is supposed to be 0.64 rad.

Fig. 3
Fig. 3

Schematic of the triangular cavity. This figure represents the layout of the triangular cavity. The triangular cavity formed by two flat mirrors, labeled Ma (the movable mirror) and Mc, and a curved mirror, labeled Mb. L, represents the distance between the curved mirror and the flat mirror; d is half the distance between the two flat mirrors, R is the radius of curvature of mirror Mb, θ is the incident angle on the curved mirror, and β is the incident angle on the flat mirror.

Fig. 4
Fig. 4

The detailed experimental setup for observing optical torsional spring effect. The laser beam (red line) was fed into the triangular cavity. An electro-optical modulator (EOM) was used to apply frequency sidebands for a Pound-Drever-Hall (PDH) method [13]. Light was detected at various points using photodetectors (PD). HWP, Half-Wave Plate; QWP, Quarter-Wave Plate; FI, Faraday Isolator.

Fig. 5
Fig. 5

Measurement of the optical torsional spring in the triangular cavity. a: The observed power spectral density (PSD) of the feedback signal. A peak represents the yaw motions of the suspended mirror at the intra-cavity powers [4 W (blue), 32 W (red), 46 W (green), and 69 W (cyan)]. The yaw resonant frequency is shifted to the higher one with the increased power. b: Angular resonant frequency of the mirror suspension against the intra-cavity power. The blue circles are the measurement data and the blue horizontal lines are statistical errors. The red curve is the theoretical one from Eq. (13), and the dashed red curve shows systematic error.

Fig. 6
Fig. 6

Design sensitivity for reaching the SQL. The quantum noise (blue), the suspension thermal noise (red), the mirror thermal noise (green), and the SQL (black) are shown. The same parameters as in Fig. 2 are used. The peak at around 10 Hz is the rocking mode and at around 2 kHz is the 1-st violin mode of the wire, respectively.

Equations (13)

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S FF , SQL ( 2 ) ( ω ) = h ¯ | χ m ( ω ) | 1 + 2 h ¯ ω m γ m m .
S FF , q ( 2 ) = 2 h ¯ 2 κ | G opt | 2 | χ c ( ω ) | 2 .
S FF , th ( 2 ) = 4 k B T γ m m .
R s = S FF , q ( 2 ) S FF , th ( 2 ) = | G opt | 2 / ( n th κ γ m ) > 1 ,
κ wire + κ opt > 0 .
Q en < G Y 8 lmg 1 τ P circ = 2 lmg ( 1 + σ ) κ P in .
Δ x = L K h Δ α
K h = 1 L ( d + L R ) ( 2 d ( L R ) cos β 0 d R cos β 0 2 L ( d + L R ) cos β 0 d R 0 ( d + L ) R )
N rad = 2 P circ c L T K h
T = ( cos β 0 0 0 cos β 0 0 0 cos θ )
I a α ¨ a = ( κ opt + κ wire ) α a ,
κ opt = 2 P circ L c ( d + L ) [ R ( L + d L / ( d + L ) ) ] L ( d + L R ) ,
f a = 1 2 π κ opt + κ wire I a

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