## Partial destruction of sensitivity in non-ideal interferometric detection of gravitational waves

Optics Express, Vol. 3, Issue 4, pp. 131-140 (1998)

http://dx.doi.org/10.1364/OE.3.000131

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### Abstract

We have studied the interferometric sensitivity for gravitational wave detection explicitly including the photodetector efficiency. We show that the sensitivity is very strongly affected by non-ideal pho-todetector efficiency when we inject a squeezed signal, as compared to the ordinary vacuum case. Quantum limits and resonance are also discussed for short time detections.

© Optical Society of America

## 1. Introduction

5. A. F. Pace, M. J. Collett, and D. F. Walls, Phys. Rev. A **47**, 3173 (1993). [CrossRef] [PubMed]

## 2. Radiation pressure and output signal

**a**and

**b**are the annihilation operators associated to one mode of the internal electromagnetic field, with optical frequency

*ω*

_{0}, and one vibrational mode associated to the motion of the mirror, with mechanical frequency Ω and mass

*m*, respectively. The first term is the free field in the interaction picture, with (Δ =

*ω*

_{0}-

*ω*

_{l}),

*ω*

_{l}being the pump field frequency. The second term corresponds to the oscillation of the movable mirror. The third one describes the adiabatic interaction associated to the radiation pressure, and the fourth one includes the perturbation on the mirror by the gravitational wave which we want to detect. The coupling constants in Eq. (1) have the following values

*γ*

_{a}is the coupling constant between the two fields. Each operator can be decomposed as:

*a*

_{0}and

*b*

_{0}are the steady state solutions,

*δa*=

*δx*+

*iδy*and

*δb*=

*δq*+

*iδp*the time dependent semiclassical fluctuations that depend on

*S*(

*t*), and

*δ*a =

*δ*x +

*iδ*y and

*δ*

**b**=

*δ*

**q**+

*iδ*

**p**the quantum fluctuations.

*S*(

*ω*) and do depend on the pump signal and the mechanical dissipation of the mirrors

## 3. Quantum limits

*ω*

_{g}) in which an external classical force produces a modification on the mirror’s position:

*F*

_{0}=

*m*

*Lh*

_{g}, and the second one is the resonant case (Ω ≃

*ω*

_{g}) in which the same force produces the following perturbation :

*γ*is the damping constant of the oscillator. We write down the sensitivity in the two cases:

*h*

_{g,min}refers to the value of the gravitational amplitude when (Δ

*x*)

_{ext}is of the order of the standard quantum limits given by Eq. (6). These two values will be compared with the values obtained from the signal to noise ratio in the next sections.

## 4. Photocurrent statistics

*t*) = -

*t*), because the gravitational wave pushes the masses of one arm of the interferometer together and the other ones appart [9

9. Kip S Thorne, Rev. Mod. Phys. **52**285 (1980). [CrossRef]

8. P. L. Kelley and W. H. Kleiner, Phys. Rev. **136**, 316 (1964). [CrossRef]

*η*is the efficiency of both photodetectors and Δ

*t*is the detection time. This is the probability of counting

*n*

_{1}and

*n*

_{2}photoelectrons simultaneously in the detectors 1 and 2 respectively.

*n*

_{1}and

*n*

_{2}are the possible values of the random variables

*N*

_{1}and

*N*

_{2}. The photocurrent difference, calculated in terms of these variables, is:

*e*is the charge of the electron. Taking the average of Eq. (18) and using the photocount distribution, we get the photocurrent

*t*is much smaller than the period of the gravitational wave, we find

*S*

_{rp}(

*ω*) depends only on the state of the input field being pumped at the port 2, and in particular for a squeezed state with compression factor

*r*and phase

*ϕ*, is

*η*= 1) we recover the results obtained previously [5

5. A. F. Pace, M. J. Collett, and D. F. Walls, Phys. Rev. A **47**, 3173 (1993). [CrossRef] [PubMed]

## 5. Signal to Noise Ratio and Sensitivity

*ω*= -∞ to +∞, since the width of the integrands is orders of magnitude smaller than the detection band. The signal to noise ratio obtained in terms of power

*is*

**P***h*

_{g}. Because the final expression is a bit cumbersome, we presents only numerical results. The parameters used in the calculation are the usual ones in the smaller working prototypes [5

5. A. F. Pace, M. J. Collett, and D. F. Walls, Phys. Rev. A **47**, 3173 (1993). [CrossRef] [PubMed]

*ω*

_{g}≫ Ω) and resonant cases (

*ω*

_{g}= Ω ) for ideal measurement. In both cases we took no squeezing and the detection time is a tenth of the gravitational wave period. These two curves show that the minimum possible detectable gravitational amplitude is of the order of the corresponding standard quantum limits. However, it is interesting to observe that in the resonant case ( curve 2-b ) the minimum is three orders of magnitude lower than in the non-resonant case.

*η*= 1 and

*η*= 0.9 (

*r*= 4,

*ϕ*= 0 in both cases ). The surfaces with no squezing are not shown because they practically coincide. Obviously the minimum detectable amplitude ( vertical axis ) shows a minimum at the resonance frequency in both cases. Again less efficiency implies a reduction in sensitivity of about two orders of magnitude for a moderate squeezing factor.

## 6. Discussion.

*η*, while the shot noise term is linear. So, if we decrease the efficiency, this has a larger effect on the radiation pressure, thus requiring a higher power to achieve the minimum. On the other hand, the detected signal is affected both by the efficiency of the detector and the input power. However, the increase in input power does not totally compensate for the less efficiency, and as a result, the sensitivity of the detection system is diminished. Obviously this effect is greatly amplified with a high squeezing parameter.

## Acknowledgments.

## References

1. | C. M. Caves, Phys. Rev. Lett. |

2. | W. A. Edelstein, J. Hough, J. R. Pugh, and W. Martin, J. Phys. E |

3. | C. M. Caves, Phys. Rev. D |

4. | R. S. Bondurant and J. H. Shapiro, Phys. Rev. D |

5. | A. F. Pace, M. J. Collett, and D. F. Walls, Phys. Rev. A |

6. | Walls D.F, |

7. | M. J. Collett and C. W. Gardiner, Phys. Rev. A |

8. | P. L. Kelley and W. H. Kleiner, Phys. Rev. |

9. | Kip S Thorne, Rev. Mod. Phys. |

10. | M. Ozawa in |

**OCIS Codes**

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

(270.5570) Quantum optics : Quantum detectors

**ToC Category:**

Focus Issue: Quantum noise reduction in optical systems

**History**

Original Manuscript: June 5, 1998

Published: August 17, 1998

**Citation**

D. Mundarain and Miguel Orszag, "Partial destruction of sensitivity in non-ideal interferometric detection of gravitational waves.," Opt. Express **3**, 131-140 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-4-131

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### References

- C. M. Caves, Phys. Rev. Lett. 45, 75 (1980). [CrossRef]
- W. A. Edelstein, J. Hough, J. R. Pugh and W. Martin, J. Phys. E 11, 710 (1978). [CrossRef]
- C. M. Caves, Phys. Rev. D 23, 1693 (1981). [CrossRef]
- R. S. Bondurant and J. H. Shapiro, Phys. Rev. D 30, 2548 (1984). [CrossRef]
- A. F. Pace, M. J. Collett, D. F. Walls, Phys. Rev. A 47, 3173 (1993). [CrossRef] [PubMed]
- D. F Walls , Quantum Optics (Springer, Berlin, 1994).
- M. J. Collett, C. W. Gardiner, Phys. Rev. A 30, 1386 (1984). [CrossRef]
- P. L. Kelley, W. H. Kleiner, Phys. Rev. 136, 316 (1964). [CrossRef]
- Kip S. Thorne, Rev. Mod. Phys. 52 285 (1980). [CrossRef]
- M. Ozawa in Squeezed and non classical States, edited by P. Tombesi and E. R. Pike (Plenum Press, N. Y., 1989).

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