## Optically trapped mirror for reaching the standard quantum limit |

Optics Express, Vol. 22, Issue 11, pp. 12915-12923 (2014)

http://dx.doi.org/10.1364/OE.22.012915

Acrobat PDF (4047 KB)

### 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

## 1. Introduction

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, and Y. Chen, “Quantum back-action in measurements of zero-point mechanical oscillations,” Phys. Rev. A **86**, 033840 (2012). [CrossRef]

*ω*is the angular frequency,

*h̄*the reduced Planck constant,

*m*the mass of the oscillator,

*γ*

_{m}the amplitude mechanical decay rate (i.e., the mechanical quality factor

*Q*

_{m}is given by

*Q*

_{m}=

*ω*

_{m}/2

*γ*

_{m}) and

*χ*

_{m}the mechanical susceptibility. Also, theoretical analysis has proven that there is a connection between reaching the SQL imposed on the free mass (so-called free-mas SQL) and the generation of entanglement states, even between massive mechanical oscillators such as suspended mirrors [5

5. H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, and 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, and Y. Chen, “Probing macroscopic quantum states with a sub-Heisenberg accuracy,” Phys. Rev. A **81**, 012114 (2010). [CrossRef]

*κ*is the total decay rate of the cavity,

*χ*

_{c}the cavity susceptibility, and

*G*

_{opt}the light-enhanced optomechanical coupling constant. Also, we naturally assume that the linewidth of the cavity is sufficiently larger than sum of the sideband frequency and the cavity detuning. The quantum back-action is a measurement-disturbance derived from the Heisenberg uncertainty principle (HUP) [7

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

*k*

_{B}is the Boltzmann constant and

*T*is the temperature. Thus, reaching the SQL needs the condition: where

*n*

_{th}is the phonon occupation number. To reduce the thermal noise, one can freely suspend a massive mirror in order to allow the mirror to be isolated from the environment. The pendulum motion of the suspended mirror is dominantly trapped by the gravitational potential, and thus the dissipation of the pendulum is gravitationally diluted by a factor of

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

*k*

_{grav}and

*k*

_{el}are the gravitational and elastic spring constants of the pendulum,

*r*is the radius of the wire,

*l*is the length of the wire,

*m*is the mass of the mirror,

*Y*is the Young’s module of the wire, and

*g*is the gravitational acceleration. From Eq. (3), any reduction of the dissipation results in a reduction of a thermal fluctuation force, which also drives the mechanical motion similarly to the quantum back-action, by a factor of

*k*

_{grav}/

*k*

_{el}.

*κ*

_{opt}= −

*P*

_{circ}

*L*

_{round}/

*c*, in conventional experiments utilizing a linear optical cavity [9

9. J. A. Sidles and 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, and S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D **81**, 064023 (2010). [CrossRef]

*P*

_{circ}is the intra-cavity power,

*L*

_{round}the round-trip length of the cavity, and c the speed of light. The stable condition concerning both the mechanical restoring force of the wire

*κ*

_{wire}and the optical anti-restoring force

*κ*

_{opt}is given by Thus,

*πGc*/(2

*lL*

_{round}) >

*P*

_{circ}/

*r*

^{4}should be satisfied in the case of using a single wire to suspend, where

*G*is the modulus of rigidity of the wire. This technical limitation becomes a significant issue, because a fundamental compromise between the stability and the sensitivity is generated; sufficient tolerance with firm suspension increases dissipation of the pendulum through the decrease of the gravitational dilution, which results in the increase of the thermal fluctuating force.

*R*

_{s}due to the gravitational dilution is limited by Here,

*Q*

_{en}is the Q enhancement factor, and

*τ*the round-trip time,

*σ*the Poisson’s ratio, and

*P*

_{in}the input laser power. The ratio of

*R*

_{s}in the case of using the linear cavity is plotted as a function both of the input laser power and the Q enhancement factor in Fig. 2(a).

## 2. Optical torsional spring effect in the triangular cavity

*positive*torsional spring effect. Figure 1 enables us to intuitively and visually understand the difference between the linear and triangular optical cavities. The positive torsional spring effect overcomes the trade-off relationship written by Eq. (6), as shown in Fig. 2(b). Although one can calculate the optical positive torsional effect using the result described in Ref. [11

11. F. Kawazoe, R. Schilling, and 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).

## 3. Model of a triangular cavity

_{a}and M

_{c}, and a curved mirror, labeled M

_{b}, as shown in Fig. 3. We decompose the rotations of the two flat mirrors into two basis modes: the common-mode (same the rotation direction, the same amount) and the differential-mode (opposite rotation direction, the same amount). Any misalignment state of the two mirrors can be expressed as a linear combination of these two basis modes. In this picture, the relationship between the misalignment angle, Δ

*α*, of the basis modes and the change in beam position on each of the mirror, Δ

*x*, is given by [12

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

*L*is the distance between the curved mirror and the flat mirror,

*d*is half the distance between two flat mirrors,

*R*is the radius of curvature of the mirror M

_{b}, and

*β*is the incident angle on the flat mirror. The torque,

*N*

_{rad}, on each mirror induced by the radiation pressure is given by with where

*θ*is the incident angle on the curved mirror, and

*β*is the incident angle on the flat mirror.

_{a}is movable and others are fixed. In this case, the equations of motion are given by where,

*I*

_{a}is the moment of inertia about the wire axis of mirror M

_{a},

*κ*

_{opt}is the angular spring constant of mirror M

_{a}induced by the radiation pressure, and

*κ*

_{wire}is the mechanical torsional spring constant of mirror M

_{a}in yaw. Under the self-consistent condition of the cavity, which is given by 0 <

*d*+

*L*<

*R*cos(

*θ*), Eq. (12) is always positive. Thus, this configuration has intrinsic stability in the yaw direction.

## 4. Experiment

*κ*

_{opt}, we expect the resonant frequency to change according to Eq. (13).

13. 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 resonator,” Appl. Phys. B **31**, 97–105 (1983). [CrossRef]

*κ*

_{in}/

*κ*= 0.48, i.e., intra-cavity power gain is 69, where

*κ*

_{in}is the decay rate for the input coupler.) was composed of two flat mirrors and a fixed curved mirror with a radius of curvature of 75 mm. One of the two flat mirrors was a half-inch fused silica mirror suspended by a tungsten wire of 20

*μ*m diameter and 40 mm length. The suspended mirror was attached to an oxigen-free copper cylinder of 3 mm diameter and 3 mm thickness, which was damped by an eddy-current using a doughnut-shaped magnet. Because of its shape, the magnet damps only the pendulum motion without decreasing the mechanical quality factor of the yaw motion. The resonant frequency of the yaw motion was measured to be 369 mHz, by optical shadow sensing. The curved mirror was fixed, and was mounted on a piezoelectric transducer (PZT; NEC Tokin, AE0505D08F), which was used as an actuator to keep the cavity in resonance with the laser. The triangular cavity and photodetectors (HAMAMATSU, G10899-01K, InGaAs photodiode) were placed in a vacuum desiccator (AS ONE, 1-070-01) for acoustic shielding.

## 5. Results

*l*and

*L*. The dashed red curves are the theoretical predictions, obtained from Eqs. (12) and (13) with

*L*= (4.4 ± 0.1) × 10 mm,

*d*= (1.0 ± 0.1) × 10 mm,

*β*= 0.7 ± 0.1 rad, and

*κ*

_{opt}= (3.9 ± 0.2) × 10

^{−10}×

*P*

_{circ}Nm/rad. The theoretically calculated values show good agreement with the experimental results, which suggests that Eq. (12) is suitable for modeling the torsional spring effect caused by the optical restoring force.

## 6. Discussions

*negative*-g condition (i.e., both focal points are inside the cavity; in other words, both mirrors have a concaved structure). However, those induce: (i) a reduction of the quantum back-action; (ii) an increase of the linewidth of the cavity (i.e., reduction of laser frequency noise); however, in practice it is insufficient only by it; (iii) a reduction of the gravitational dilution (i.e., increasing the suspension thermal noise); (iv) introducing an unexpected thermal noise through the unexpected normal mode generated by the complicated suspension system [14

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

*intrinsic*stability in yaw direction. As a result, one can conclude that the triangular cavity overcomes the fundamental compromise.

## 7. Toward reaching the SQL

*μ*m in diameter and 20 mm in length). Here, we suppose the dissipation mechanism is due to internal friction (frequency independent internal friction). In addition, we suppose that the intrinsic quality factor of the wire is 3,800, since the mechanical quality factor of the tungsten wire with 3

*μ*m in diameter was measured to be 3,800 [16]. The quality factor of pendulum can be improved up to 2 × 10

^{7}due to the gravitational dilution. In this case, suspension thermal noise is shown in Fig. 6 as a red line. Mechanical quality factors of the coating and the silica structure of the movable mirror are estimated to be 1 × 10

^{4}and 1 × 10

^{6}, respectively. The mirror thermal noise (sum of the coating and the substrate thermal noise) is shown in Fig. 6 as a green line. The blue line shows the quantum noise (i.e., the quantum back-action and the shot noise). The input laser power is 5 mW, which results in the intra-cavity power of 3.2 W. Although the laser power is about 10,000-times higher than the instability limit for the linear cavity, it is relatively weak. This is because that the frequency, where the SQL equals the sum of the quantum noises (

## 8. Conclusion

## Acknowledgments

## References and links

1. | M. Aspelmeyer, T. J. Kippenberg, and 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. |

3. | K. Somiya, “Detector configuration of KAGRA-the Japanese cryogenic gravitational-wave detector,” Class. Quantum Grav. |

4. | F. Ya. Khalili, H. Miao, A. H. Safavi-Naeini, O. Painter, and Y. Chen, “Quantum back-action in measurements of zero-point mechanical oscillations,” Phys. Rev. A |

5. | H. Müller-Ebhardt, H. Rehbein, R. Schnabel, K. Danzmann, and Y. Chen, “Entanglement of Macroscopic Test Masses and the Standard Quantum Limit in Laser Interferometry,” Phys. Rev. Lett. |

6. | H. Miao, S. Danilishin, H. Müller-Ebhardt, H. Rehbein, K. Somiya, and Y. Chen, “Probing macroscopic quantum states with a sub-Heisenberg accuracy,” Phys. Rev. A |

7. | W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Phys. |

8. | P. R. Saulson, “Thermal noise in mechanical experiments,” Phys. Rev. D |

9. | J. A. Sidles and D. Sigg, “Optical torques in suspended Fabry-perot interferometers,” Phys. Lett. A |

10. | S. Sakata, O. Miyakawa, A. Nishizawa, H. Ishizaki, and S. Kawamura, “Measurement of angular antispring effect in optical cavity by radiation pressure,” Phys. Rev. D |

11. | F. Kawazoe, R. Schilling, and H. Lück, ”Eigenmode changes in a misaligned triangular optical cavity,” J. Opt. |

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, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B |

14. | A. R. Neben, T. P. Bodiya, C. Wipf, E. Oelker, T. Corbitt, and N. Mavalvala, “Structural thermal noise in gram-scale mirror oscillators,” New J. Phys. |

15. | S. Kawamura, (personal communication). |

16. | (to be submitted) |

**OCIS Codes**

(270.5570) Quantum optics : Quantum detectors

(270.5585) Quantum optics : Quantum information and processing

(120.4880) Instrumentation, measurement, and metrology : Optomechanics

**ToC Category:**

Sensors

**History**

Original Manuscript: April 8, 2014

Revised Manuscript: May 9, 2014

Manuscript Accepted: May 12, 2014

Published: May 20, 2014

**Citation**

Nobuyuki Matsumoto, Yuta Michimura, Yoichi Aso, and Kimio Tsubono, "Optically trapped mirror for reaching the standard quantum limit," Opt. Express **22**, 12915-12923 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-12915

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### References

- M. Aspelmeyer, T. J. Kippenberg, F. Marquardt, “Cavity Optomechanics,” arXiv:1303.0733 (2013).
- 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]
- 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. 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]
- 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]
- W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Phys. 43, 172–198 (1927). [CrossRef]
- P. R. Saulson, “Thermal noise in mechanical experiments,” Phys. Rev. D 42, 2437 (1990). [CrossRef]
- J. A. Sidles, D. Sigg, “Optical torques in suspended Fabry-perot interferometers,” Phys. Lett. A 354, 167–172 (2006). [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]
- F. Kawazoe, R. Schilling, H. Lück, ”Eigenmode changes in a misaligned triangular optical cavity,” J. Opt. 13, 055504 (2011). [CrossRef]
- D. Sigg, “Angular stability in a triangular fabry-perot cavity,” LIGO-T030275-00, www.ligo.caltech.edu/docs/T/T030275-00.pdf (2003).
- 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]
- 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]
- S. Kawamura, (personal communication).
- (to be submitted)

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