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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 3, Iss. 4 — Aug. 17, 1998
  • pp: 131–140
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Partial destruction of sensitivity in non-ideal interferometric detection of gravitational waves

D. Mundarain and M. Orszag  »View Author Affiliations


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

The study of the quantum noise reduction in optical systems, especially in interfero-metric systems, has been a topic of great interest in the last decades, particularly in connection with the detection of gravitational waves. In this case, the signal is so small that it imposes severe restrictions on the maximum noise allowed in the detection system. Some authors [1

1. C. M. Caves, Phys. Rev. Lett. 45, 75 (1980). [CrossRef]

,2

2. W. A. Edelstein, J. Hough, J. R. Pugh, and W. Martin, J. Phys. E 11, 710 (1978). [CrossRef]

] showed , in the decade of the 80’s, that interferometric devices might possibly achieve the limit imposed by Quantum Mechanics. Caves [3

3. C. M. Caves, Phys. Rev. D 23, 1693 (1981). [CrossRef]

] found that by injecting a squeezed signal in the unused input port of the interferometer the sensitivity could not be improved but a lower power was required to achieve the standard quantum limit. In more recent work [4

4. R. S. Bondurant and J. H. Shapiro, Phys. Rev. D 30, 2548 (1984). [CrossRef]

,5

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

] it has been shown that actually the standard quantum limit can be beaten if one chooses an appropiate phase for the squeezed signal.

In this letter, we study the effects of non-ideal photodetector efficiency on the sensitivity of the gravitational wave detector for a short time measurement. We show that the degrading effect of non-ideal photodetectors increases dramatically when we inject a vacuum squeezed signal.

This letter is organized as follow: in section 2 we discuss the model and fluctuations, using the input-output formalism [5

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

,6

6. Walls D.F, Quantum Optics (Springer, Berlin, 1994).

]. Sections 3 and 4 are devoted to the quantum limits and photodetections respectively. In section 5 we present the main results of this paper and the last section is a brief discussion.

Figure 1. Schematic representation of the optical detector for gravitational radiation.

2. Radiation pressure and output signal

The Hamiltonian of a cavity with one movable mirror has the following structure

H=ħΔaa+ħΩbb+ħκaa(b+b)
+ħkS(t)(b+b),
(1)

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 - ωl ), ωl 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

κ=ω0Lħ2mΩk=Lωgm2ħΩhg,

Any cavity is an open system that can be treated by the standard methods that deal with dissipation in quantum mechanics. In this context, the input-output formulation [6

6. Walls D.F, Quantum Optics (Springer, Berlin, 1994).

,7

7. M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984). [CrossRef]

] allows us to distinguish between the measurable external field, which we denote by a out, and the internal field a, that is not directly accessible to detection. These two fields are correlated due to the boundary condition over the semi-transparent mirrors of the cavities (Fig. 1):

aout+ain=γaa,
(2)

where γa is the coupling constant between the two fields. Each operator can be decomposed as:

a=x+iy=a0+δa+δa,
b=q+ip=b0+δb+δb,

where 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 δa = δx + y and δ b = δ q + p the quantum fluctuations.

If we use the boundary condition (2) and find the internal solutions ( solving the Langevin equations asociated to the Hamiltonian plus baths) we obtain the following semiclasical solutions for the external fields.

δxout(ω)=0
δyout(ω)=γakμΩS(ω)ʌa(ʌb2+Ω2).
(3)

where

ʌa,b=γa,b2+
(4)

Finally we write down the quantum fluctuation of the output signal , i.e. the external field at the time of measurement in terms of the input signal,

δXout(ω)=(γaʌa+1)δXin(ω)
δYout=γaμ2Ωʌa2(ʌb2+Ω2)δXin(ω)(γaʌa+1)δYin(ω)
ʌaʌbμγaγbʌa2(ʌb2+Ω2)δqin(ω)ʌaΩμγaγbʌa2(ʌb2+Ω2)δPin(ω).
(5)

We notice that the above fluctuations do not depend on the gravitational signal S(ω) and do depend on the pump signal and the mechanical dissipation of the mirrors

3. Quantum limits

In the interferometric detection, we infer the perturbation that the gravitational wave produces on the position of the mirrors, by measuring the phase changes of external field operators. In a short time measurement, the precision of this free particle’s position is limited by the standard quantum limit:

ΔxħΔtm.
(6)

This limit will be restricted to measurements in which the state of the system is completely unknown. This type of detection includes all the measurements in which a macroscopic object interacts directly with a quantum system [10

10. M. Ozawa in Squeezed and non classical States, edited by P. Tombesi and E. R. Pike (Plenum Press, N. Y., 1989)

]. In order to do a succesful detection, the change in the position of the mirrors of the interferometer must be greater than the standard quantum limit.

The influence of the gravitational wave can be considered in two important cases: the first one is the non resonant short time detection (Ω ≪ ωg ) in which an external classical force produces a modification on the mirror’s position:

(Δx)ext=F02mΔt2,
(7)

with F 0 = m ωg2 Lhg , and the second one is the resonant case (Ω ≃ ωg ) in which the same force produces the following perturbation :

(Δx)ext=F0mγΔt,
(8)

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

hg,minNR=2ωg2Δt21LħΔtm,
(9)

and

hg,minR=γωg2Δt1LħΔtm,
(10)

4. Photocurrent statistics

In section 2 we showed the semiclassical and quantum fluctuations of the fields just when they leave the cavities; the presence of the beam splitter (B-S) mixes the signals read by the two photodetectors. The relationship between the output fields and the fields measured in the photodetectors (Fig. 1) is

ain1=12(c1+ic2),ain2=12(c1ic2),
d1=12(aout2iaout1),d2=12(aout2+iaout1).
(11)

In getting the Eq. (11), we used a non-symmetric Beam-Splitter. Also for simplicity, we chose:

<c1>=2α0and<c2>=0.
(12)

The two last operators in Eq. (11) represent the light measured directly by the two photodetectors.

One can calculate (and measure) the simultaneous difference and sum of the number of photons from the two output ports of the interferometer:

D(t)=d1(t)d1(t)d2(t)d2(t)
S(t)=d1(t)d1(t)d2(t)d2(t).
(13)

The lowest order non-zero term of the difference is proportional to the perturbation generated by the gravitational wave,

δD(t)=2α0(δyout1(t)δyout2(t))=4α0δyout1(t),
(14)

(yout2(t) = -yout1(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. 52285 (1980). [CrossRef]

]) . The final value of this quantity is calculated from the Fourier transform of Eq. (3). The main contribution for the sum is a DC term:

S(t)=S0=2(α0)2.
(15)

In order to account for the quantum nature of the measurement we need to consider the Kelley-Kleiner formulation of the photodetection [8

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

], which yields the statistics of the photocurrent difference in a simultaneous measurement in both detectors. The photocount distribution for the two detectors is:

P(n1,n2,Δt)=<:(Ω1)n1n1!eΩ1(Ω2)n2n2!eΩ2:>
(16)

with:

Ω1,2=ηtΔttd1,2(t′)d1,2(t′)dt,
(17)

where η is the efficiency of both photodetectors and Δt 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:

I(t)=GeΔt(N1N2),
(18)

where G is the photomultiplier gain factor and e is the charge of the electron. Taking the average of Eq. (18) and using the photocount distribution, we get the photocurrent

i(t)=GeΔt:Ω1(t)Ω2(t):.
(19)

If Δt is much smaller than the period of the gravitational wave, we find

i(t)=ηGe:D(t):=ηGeδD(t).
(20)

If the detection time is much bigger than this period , the detected signal will be zero. In order to get a non-vanishing photocurrent difference, the detection time cannot be larger than half of the gravitational period.

The noise (i.e. the variance of the photocurrent difference Eq. (18)), depending on the quantum fluctuations (Eq. 5), is

Δ2i(t)=(GeΔt)2{η2<:(δD(t))2:>Δt2+η<:S(t):>Δt}
(21)

It is important to notice the origin of the two terms in the above formula. The first one is a purely radiation pressure contribution while the second term is the shot-noise generated by the power injected into the cavities, which is proportional to sum of the detected photons at both sides of the interferometer. Then, the photocurrent spectrum can be separated in two parts

Spctot(ω)=(Ge)2πα02(η2Srp(ω)+η),
(22)

The radiation pressure spectrum Srp (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

Srp(ω)=(16κ2α02Ω)2cosh(2r)+cos(ϕ)sinh(2r)(γa4+ω2)2((γb24+Ω2ω2)2+γb2ω2)
+32κ2α02Ω(γb4+Ω2ω2)sin(ϕ)sinh(2r)(γa4+ω2)2((γb24+Ω2ω2)2+γb2ω2)
+16κ2α02γb(γb4+Ω2ω2)(γa4+ω2)2((γb24+Ω2ω2)2+γb2ω2)
+cosh(2r)cos(ϕ)sinh(2r)1
(23)

In the ideal detector case (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

The signal obtained is the difference in photocurrent delivered by the two detectors in Fig. 1 which is proportional to the gravitational perturbation, and its magnitude is (see Eq. 14 and Eq. 20)

i(t)=ηGeα0216κkΩ(γa24+ωg2)1/2((γb24+Ω2ωg2)2+γb2ωg2)1/2.
(24)

in fact this quantity is modulated by a sinusoidal factor but we make a stroboscopic measurement at the maximum value of the intensity.

Since we are dealing with short detection times, on the frequency domain this is a broadband detection, so we integrate the photocurrent noise spectrum over a finite bandwidth:

Δ2i(t)=ΔωdωSpctot(ω).

For the particular case of a squeezed vacuum input state, the total noise is found to be:

Δ2i(t)=(Ge)2πα02(η2(α04(cosh(2r)+cos(θ)sinh(2r))I0
+α02(I1+sin(θ)sinh(2r)I2)+(cosh(2r)cos(θ)sinh(2r))2πΔt)
+(ηη2)2πΔt)
(25)

where,

I0=(16κ2Ω)21(γa24+ω2)2((γb24+Ω2ω2)2+γb2ω2)
I1=16κ2γbγb24+Ω2+ω2(γa24+ω2)2((γb24+Ω2ω2)2+γb2ω2)
I2=32κ2Ωγb24+Ω2+ω2(γa24+ω2)2((γb24+Ω2ω2)2+γb2ω2).
(26)

The above integrals have been calculated from 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 P is

SNR2=i2(t)Δ2i(t)=APBP2+CP+D’
(27)

with

A=(16ω0ωg2hg)2Δħω0(γa24+ωg2)((γb24+Ω2ωg2)2+γb2ωg2)
B=I02π(ħω0)2(cosh(2r)+cos(θ)sinh(2r))ηΔt
C=12πħω0(I1+I2sin(θ)sinh(2r))ηΔt
D=(cosh(2r)cos(θ)sinh(2r)1)η+1
(28)

We put SNR ≃ 1 and from this we obtain the value for the minimum detectable hg . 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]

].

The figure 2 corresponds to the minimum possible detectable gravitational amplitude in the non-resonant (ωg ≫ Ω) and resonant cases (ωg = 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.

The most interesting effect can be observed in the figure 3. This figure shows the sensitivity in four cases. In the first two, we see the influence of the efficiency when there is no squeezing. For this case, if we have good efficiencies the difference between a non-ideal ( but good ) and an ideal detections is very small. On the other hand, when we inject a squeezed signal in the unused port ( port 2 ), the sensitivity decreases dramatically, even for good ( but not perfect ) detectors. Also the power to achieve the minimum becomes higher.

Figure 2. Minimum detectable gravitational amplitude versus input power for the non-resonant case (curve a), and resonant case (curve b). Notice the comparison with the corresponding standard quantum limits (dashed line) r = 0, Ω = 20πseg-1 (non resonant case) ωg = 2000πseg-1.
Figure 3. Minimum detectable gravitational amplitude versus input power for r = 0, η = 1 (curve a), r = 0 , η = 0.9 (curve b) r = 4 , η = 1 (curve c) and r = 4,η = 0.9 (curve d). In all cases we took ϕ = 0.
Figure 4. Minimum detectable amplitude versus input power and mechanical frequency, near resonance. The upper surface corresponds to η = 0.9 and the lower one to η = 1. In both cases: r = 4, ϕ = 0.

The decrease in sensitivity for non-ideal detection with squeezing can be seen at the resonance as well. In the figure 4 we show two surfaces corresponding to minimimum detectable amplitude of the gravitational wave for 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.

In general, the minimum detectable gravitational amplitude is affected by the efficiency of the detector, the input power, the squeezing and the frequency. An optimum power appears in the sensitivity because of a delicate balance between two effects: the radiation pressure and the shot noise. The noise contribution due to radiation pressure is quadratic in 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.

So, as a final and main conclusion of this work, we see that in an non-ideal photodetection, the advantage of using squeezed signal is strongly limited in beating the standard quantum limit and lowering the optimal power.

Acknowledgments.

This work has been partially supported by the Consejo Nacional de Ciencia y Tecnologìa (Conicyt)(Proyecto FONDECYT 2960009) and the Departamento de Inves-tigacióon de la Pontificia Universidad Catóolica (DIPUC).

References

1.

C. M. Caves, Phys. Rev. Lett. 45, 75 (1980). [CrossRef]

2.

W. A. Edelstein, J. Hough, J. R. Pugh, and W. Martin, J. Phys. E 11, 710 (1978). [CrossRef]

3.

C. M. Caves, Phys. Rev. D 23, 1693 (1981). [CrossRef]

4.

R. S. Bondurant and J. H. Shapiro, Phys. Rev. D 30, 2548 (1984). [CrossRef]

5.

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

6.

Walls D.F, Quantum Optics (Springer, Berlin, 1994).

7.

M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984). [CrossRef]

8.

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

9.

Kip S Thorne, Rev. Mod. Phys. 52285 (1980). [CrossRef]

10.

M. Ozawa in Squeezed and non classical States, edited by P. Tombesi and E. R. Pike (Plenum Press, N. Y., 1989)

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

  1. C. M. Caves, Phys. Rev. Lett. 45, 75 (1980). [CrossRef]
  2. W. A. Edelstein, J. Hough, J. R. Pugh and W. Martin, J. Phys. E 11, 710 (1978). [CrossRef]
  3. C. M. Caves, Phys. Rev. D 23, 1693 (1981). [CrossRef]
  4. R. S. Bondurant and J. H. Shapiro, Phys. Rev. D 30, 2548 (1984). [CrossRef]
  5. A. F. Pace, M. J. Collett, D. F. Walls, Phys. Rev. A 47, 3173 (1993). [CrossRef] [PubMed]
  6. D. F Walls , Quantum Optics (Springer, Berlin, 1994).
  7. M. J. Collett, C. W. Gardiner, Phys. Rev. A 30, 1386 (1984). [CrossRef]
  8. P. L. Kelley, W. H. Kleiner, Phys. Rev. 136, 316 (1964). [CrossRef]
  9. Kip S. Thorne, Rev. Mod. Phys. 52 285 (1980). [CrossRef]
  10. 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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