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

D. Mundarain; M. Orszag

doi:10.1364/OE.3.000131

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 [1,2] showed , in the decade of the 80’s, that interferometric devices might possibly achieve the limit imposed by Quantum Mechanics. Caves [3] 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 [4,5] 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 [5,6]. 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.

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2. Radiation pressure and output signal

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

H = ħ Δ a^{†} a + ħ Ω b^{†} b + ħ κ a^{†} a (b + b^{†})

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

κ = - \frac{ω_{0}}{L} \sqrt{\frac{ħ}{2 m Ω}} k = - L ω_{g} \sqrt{\frac{m}{2 ħ Ω}} h_{g},

S(t) is a dimensionless form factor related to the gravitational wave spectrum,which is supposed monomodal, and h_g is its amplitude.

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 [6,7] 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):

a_{out} + a_{in} = \sqrt{γ_{a}} a,

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

a = x + iy = a_{0} + δ_{a} + δ_{a},

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 + iδy and δ b = δ q + iδ 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.

δ x_{out} (ω) = 0

where

ʌ_{a, b} = - \frac{γ_{a, b}}{2} + iω

γ_b is a coefficient of mechanical loss in the movable mirrors. We can see from Eq. (3), that if the external phase quadrature is measured, we can obtain an estimation of the perturbation of the position of the mirrors caused by the gravitational signal. This quantity will be related to the measured photocurrent signal which then depend only on the gravitational perturbation.

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,

δ X_{out} (ω) = - (\frac{γ_{a}}{ʌ_{a}} + 1) δ X_{in} (ω)

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:

Δ x \geq \sqrt{\frac{ħ Δ t}{m}} .

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]. 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 ≪ ω_g ) in which an external classical force produces a modification on the mirror’s position:

{(Δ x)}_{ext} = \frac{F_{0}}{2 m} Δ t^{2},

with F ₀ = m $ω_{g}^{2}$ Lh_g , and the second one is the resonant case (Ω ≃ ω_g ) in which the same force produces the following perturbation :

{(Δ x)}_{ext} = \frac{F_{0}}{m γ} Δ t,

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

h_{g, min}^{N R} = \frac{2}{ω_{g}^{2} Δ t^{2}} \frac{1}{L} \sqrt{\frac{ħ Δ t}{m}},

and

h_{g, min}^{R} = \frac{γ}{ω_{g}^{2} Δ t} \frac{1}{L} \sqrt{\frac{ħ Δ t}{m}},

where h_g,min refers to the value of the gravitational amplitude when (Δx) _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

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

a_{in}^{1} = \frac{1}{\sqrt{2}} (c_{1} + i c_{2}), a_{in}^{2} = \frac{1}{\sqrt{2}} (c_{1} - i c_{2}),

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

< c_{1} > = \sqrt{2} α_{0} ∊ ℜ and < c_{2} > = 0 .

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) = d_{1}^{†} (t) d_{1} (t) - d_{2}^{†} (t) d_{2} (t)

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

δD (t) = 2 α_{0} (δ y_{out}^{1} (t) - δ y_{out}^{2} (t)) = 4 α_{0} δ y_{out}^{1} (t),

( $y_{out}^{2}$ (t) = - $y_{out}^{1}$ (t), because the gravitational wave pushes the masses of one arm of the interferometer together and the other ones appart [9]) . 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) = S_{0} = 2 {(α_{0})}^{2} .

In order to account for the quantum nature of the measurement we need to consider the Kelley-Kleiner formulation of the photodetection [8], which yields the statistics of the photocurrent difference in a simultaneous measurement in both detectors. The photocount distribution for the two detectors is:

P (n_{1}, n_{2}, Δ t) = < : \frac{{(Ω_{1})}^{n_{1}}}{n_{1}!} e^{- Ω_{1}} \frac{{(Ω_{2})}^{n_{2}}}{n_{2}!} e^{- Ω_{2}} : >

with:

Ω_{1,2} = η \int_{t - Δ t}^{t} d_{1,2}^{†} (t′) d_{1,2} (t′) dt',

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) = \frac{G e}{Δ t} (N_{1} - N_{2}),

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) = \frac{G e}{Δ t} 〈 : Ω_{1} (t) - Ω_{2} (t) : 〉 .

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

i_{-} (t) = η G e 〈 : D (t) : 〉 = η G e δ D (t) .

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

Δ^{2} i_{-} (t) = {(\frac{G e}{Δ t})}^{2} {η^{2} < : {(δ D (t))}^{2} : > Δ t^{2} + η < : S (t) : > Δ t}

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

S_{pc}^{tot} (ω) = \frac{{(G e)}^{2}}{π} α_{0}^{2} (η^{2} S_{rp} (ω) + η),

The radiation pressure spectrum 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

S_{rp} (ω) = {(16 κ^{2} α_{0}^{2} Ω)}^{2} \frac{\cosh (2 r) + \cos (ϕ) \sinh (2 r)}{{(\frac{γ_{a}}{4} + ω^{2})}^{2} ({(\frac{γ_{b}^{2}}{4} + Ω^{2} - ω^{2})}^{2} + γ_{b}^{2} ω^{2})}

In the ideal detector case (η = 1) we recover the results obtained previously [5].

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) = \frac{η G e α_{0}^{2} 16 κ k Ω}{{(\frac{γ_{a}^{2}}{4} + ω_{g}^{2})}^{1 / 2} {({(\frac{γ_{b}^{2}}{4} + Ω^{2} - ω_{g}^{2})}^{2} + γ_{b}^{2} ω_{g}^{2})}^{1 / 2}} .

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:

Δ^{2} i_{-} (t) = \int_{Δ ω} d ω S_{pc}^{tot} (ω) .

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

Δ^{2} i_{-} (t) = \frac{{(Ge)}^{2}}{π} α_{0}^{2} (η^{2} (α_{0}^{4} (\cosh (2 r) + \cos (θ) \sinh (2 r)) I_{0}

where,

I_{0} = {(16 κ^{2} Ω)}^{2} \int dω \frac{1}{{(\frac{γ_{a}^{2}}{4} + ω^{2})}^{2} ({(\frac{γ_{b}^{2}}{4} + Ω^{2} - ω^{2})}^{2} + γ_{b}^{2} ω^{2})}

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

{SNR}^{2} = \frac{i_{-}^{2} (t)}{Δ^{2} i_{-} (t)} = \frac{AP}{{BP}^{2} + CP + D’}

with

A = \frac{{(16 ω_{0} ω_{g}^{2} h_{g})}^{2} Δ tη}{ħ ω_{0} (\frac{γ_{a}^{2}}{4} + ω_{g}^{2}) ({(\frac{γ_{b}^{2}}{4} + Ω^{2} - ω_{g}^{2})}^{2} + γ_{b}^{2} ω_{g}^{2})}

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

The figure 2 corresponds to the minimum possible detectable gravitational amplitude in the non-resonant (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.

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) ω_g = 2000πseg^-1.

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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.

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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.

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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.

A first point is related to the capability of the present system to achieve the range of sensitivity required by the limits imposed by the quantum mechanical measurement of an observable. In particular, the quantum standard limits for the non-resonant and resonant cases are different, and in both cases, one can approach the limit. We must emphasize that for non-resonant case and for short time measurement, this limit corresponds to that of a free particle. We should also point out, that in the resonant case, the gain in signal is larger than the amplified noise produced by the resonance, thus a much lower minimum detectable amplitude is obtained. But, the price one pays, is an increase, of several orders of magnitude, in the optimum input power.

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).

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

Abstract

1. Introduction

2. Radiation pressure and output signal

3. Quantum limits

4. Photocurrent statistics

5. Signal to Noise Ratio and Sensitivity

6. Discussion.

Acknowledgments.

References

Cited By

Figures (4)

Equations (48)

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