Frequency-dependent rigidity in large-scale interferometric gravitational-wave detectors
F.Ya.Khalili
{Moscow 119899, Russia}
Electromagnetic rigidity which exists in large-scale optical resonators if pumping frequency is detuned from the eigenfrequency of resonator have sophisticated spectral dependence which allows to obtain sensitivity better than the Standard Quantum Limits both for the free test mass and the harmonic oscillator.
gr-qc0107084.tex
\documentclass[12pt]{article}
\usepackage{amssymb,amsmath,eucal}
\usepackage{epic}
\usepackage[dvips]{graphicx}
\usepackage{a4wide}

%\input{bigpage}

\title{Frequency-dependent rigidity in large-scale interferometric
gravitational-wave detectors}

\author{
  F.Ya.Khalili \\
  {\it Dept.\,of Physics, Moscow State University,} \\
  {\it Moscow 119899, Russia} \\
  {\it e-mail: farid@mol.phys.msu.su}
}

%\date{Draft of \today}
\date{}

\begin{document}

\maketitle

\begin{abstract}

Electromagnetic rigidity which exists in large-scale optical resonators if
pumping frequency is detuned from the eigenfrequency of resonator have
sophisticated spectral dependence which allows to obtain sensitivity better
than the Standard Quantum Limits both for the free test mass and the harmonic
oscillator.

\end{abstract}

\section{Introduction}

The standard quantum limit (SQL) \cite{Braginsky1967} is one of the most
fundamental factors which prevent the gravitational-wave antennae
\cite{300years} sensitivity from increasing. The basis for this limit is the
uncertainty relation for two kind of noises inherent in position meters: the
measurement noise and the back action noise. In the interferometric position
meters, these noises are proportional to phase fluctuations of the pumping
beam and radiation pressure noise, correspondingly.

Several methods to overcome the SQL has been proposed (see, for example,
articles \cite{OptBar1997, Symphot1998, Vyatchanin1993, Vyatchanin1998,
KLMTV2000}) but most of them encounters serious technological limitation
and/or has some other disadvantages which do not permit implementation of
these methods in the near future. On the other hand, the SQL itself is not an
absolute limit but, in particular, it depends on the dynamic properties of
the test object which is used in the experiment. The well known example is
the harmonic oscillator. Its response to a resonant force is relatively
strong and it allows to use less sensitive meter (with larger measurement
noise and therefore with smaller back action noise). Due to this property the
harmonic oscillator allows to obtain sensitivity better than the SQL for the
free mass when the signal frequency is close to the eigen frequency of the
oscillator $\Omega_m$ \cite{Rigidity1999, Minsk2000e, Rigidity2001,
Buonanno2001}.

In the articles \cite{Rigidity1999, Rigidity2001} it was shown that it is
possible to create very low noise mechanical rigidity using Fabry-Perot
resonators with detuned pumping.  In the article \cite{Buonanno2001} it was
also shown that such a rigidity exists in the signal-recycled topology of the
gravitational-wave antennae and it permits to overcome the SQL for a free
mass in narrow band.

One can imagine the following frequency dependent rigidity:

\begin{equation}
  K(\Omega) = m\Omega^2 \label{K_ideal}
\end{equation}
where $m$ is the mass it is attached to, and $\Omega$ is an {\em arbitrary}
observation frequency. In principle, such a rigidity would allow to obtain
arbitrarily high sensitivity throughout the spectral range where the formula
(\ref{K_ideal}) is valid.  On the other hand, it is well known that the
mechanical rigidity created by parametrical opto-mechanical systems can be
frequency dependent. In this article we show that the mechanical rigidity
frequency dependence in large-scale interferometric meters with bandwidth
comparable to or smaller than the signal frequency can be close to the
formula (\ref{K_ideal}) in some spectral band.

\section{The simple example: Second-order-pole regime}

Consider the simplified interferometric detector scheme presented in
Fig.\,\ref{Fig:scheme}. Here the signal force with the amplitude
$F_\mathrm{signal}$ which had to be detected, acts on the test mass $m$. This
mass serves also as mirror {\sf M1} and together with second mirror {\sf M2}
forms Fabry-Perot resonator.  In this article we suppose that refraction of
the mirror {\sf M1} is equal to unity and there is no absorption in the
mirror {\sf M2}.

The resonator is pumped at the frequency $\omega_\mathrm{pump}$
which is detuned far from its eigen frequency $\omega_o$:

\begin{equation}
  \delta = \omega_\mathrm{pump} - \omega_o\gg\gamma\,,
\end{equation}
where $\gamma$ is the half-bandwidth of the resonator. This detuned pumping
creates a ponderomotive rigidity. One of the reflected beam quadrature
amplitudes is measured, giving information about the mirrors relative
position.

\begin{figure}

\unitlength=1mm

\begin{center}

\begin{picture}(120,30)

\thicklines

\put(40,5){\framebox(5,20)}\put(42.5,26){\makebox(0,0)[cb]{{\sf M2}}}

\put(95,5){\framebox(20,20)}\put(105,26){\makebox(0,0)[cb]{{\sf M1}}}
\put(105,15){\vector(1,0){20}}
\put(117,17){\makebox(0,0)[lb]{$F_\mathrm{signal}$}}

\thinlines

\put(70,7){\vector(1,0){24}}\put(70,7){\vector(-1,0){24}}
\put(70,6){\makebox(0,0)[ct]{$L$}}

\put(70,15){\vector(1,0){15}}\put(85,15){\line(1,0){10}}
\put(70,15){\vector(-1,0){15}}\put(55,15){\line(-1,0){10}}

\put(20,15){\vector(1,0){10}}\put(30,15){\line(1,0){10}}
\put(20,15){\vector(-1,0){10}}\put(10,15){\line(-1,0){10}}

\put(20,10){\makebox(0,0)[cc]{$\omega_{pump}=\omega_o+\delta$}}

\end{picture}

\caption{Simplified scheme of the interferometric detector}\label{Fig:scheme}

\end{center}

\end{figure}

We will refer to this simple scheme in this article but it can be shown that
all results obtained here are valid for the signal-recycled topology
\cite{Meers1988} planned for the second stage of the LIGO program.

By solving this system equations of motion it is easy to show that if
$\gamma\to 0$ then mechanical rigidity created by the optical pumping will be
equal to

\begin{equation}
  K(\Omega) \approx
  \frac{2\omega_o\mathcal{E}\delta}{L^2(\delta^2-\Omega^2)}\,
\end{equation}
where $\Omega$ is the observation frequency, $\mathcal{E}$ is the optical
energy stored in the resonator, and $L$ is the resonator length.

\begin{figure}

\begin{center}

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\caption{Second-order pole ($\gamma\to 0$)}
\label{Fig:plot1}

\end{center}

\end{figure}

Our goal is to set $K(\Omega)$ as close to the ideal frequency
dependence (\ref{K_ideal}) as possible nearby some given value of
$\Omega$, so we require that

\begin{equation}
  K(\Omega) = m\Omega^2
\end{equation}
and

\begin{equation}
  \frac{dK(\Omega)}{d\Omega} = \frac{d(m\Omega^2)}{d\Omega} \,.
\end{equation}
It is easy to show that these conditions can be fulfilled if (and only if)

\begin{equation}
  \Omega = \Omega_2 \equiv \frac{\delta}{\sqrt 2}
\end{equation}
and

\begin{equation}
  \mathcal{E} = \frac{mL^2\delta^3}{8\omega_o}
  = \frac{mL^2\Omega_2^3}{2\sqrt 2\omega_o} \,.
  \label{Energy}
\end{equation}
In this case susceptibility of test object which consists of mass $m$ and
such a rigidity

\begin{equation}
  \chi(\Omega) = \frac{1}{-m\Omega^2+K(\Omega)}
\end{equation}
will have a second-order pole at the frequency $\Omega_2$, {\it i.e} if
$|\Omega-\Omega_2|\ll \Omega$ then we will obtain

\begin{equation}
  -m\Omega^2 + K(\Omega) \approx 4m(\Omega-\Omega_2)^2
\end{equation}
where subscript ``2'' means ``second-order pole'' (see Fig.\,\ref{Fig:plot1},
where second-order-pole point marked by ``$\bullet$''). It means if the
signal force has the form of a sinusoidal train with duration $\tau_F$ and
the mean frequency $\Omega_F\simeq\Omega_2$ then the amplitude of the mass
$m$ oscillations caused by this force will be proportional to\footnote{We
want to remind that the free mass has second-order pole at zero frequency,
and the harmonic oscillator has first-order pole at resonance frequency.}

\begin{equation}
  x_\mathrm{signal} \sim \frac{F_\mathrm{signal}\tau_F^2}{m}
\end{equation}
(in this section we omit all numerical factors of the order of unity). On the
other hand, it can be shown that the Standard Quantum Limit for such a
second-order-pole system is equal to

\begin{equation}
  x_\mathrm{SQL}^{(2)} \sim \sqrt{\frac{\hbar\tau_F}{m}} \,.
\end{equation}
Hence using this system and ordinary position meter, it is possible to detect
the force

\begin{equation}
  F_\mathrm{signal} \sim \frac{m x_\mathrm{SQL}^{(2)}}{\tau_F^2}
  = F_\mathrm{SQL}^{(2)}
  = \sqrt{\frac{\hbar m}{\tau_F^3}} \,.
  \label{F_SQL}
\end{equation}
This value is $\Omega_F\tau_F$ times smaller than the SQL value corresponding
to the free test mass

\begin{equation}
  F_\mathrm{SQL}^\mathrm{free\ mass}
  = \sqrt{\frac{\hbar m\Omega_F^2}{\tau_F}}
\end{equation}
and $\sqrt{\Omega_F\tau_F}$ times smaller than the SQL for the harmonic
oscillator with ordinary frequency-independent rigidity,

\begin{equation}
  F_\mathrm{SQL}^\mathrm{oscillator}
  = \frac{\sqrt{\hbar m\Omega_F}}{\tau_F} \,.
\end{equation}

It had to be noted that the energy (\ref{Energy}) is close to the energy

\begin{equation}
  \mathcal{E} = \frac{mL^2\Omega^3}{2\omega_o} \,.
  \label{E_SQL}
\end{equation}
which is necessary to achieve the SQL using traditional scheme of the
interferometric position meter.

\section{Sensitivity for different regimes of the frequency-dependent rigidity}

In this section we will use spectral approach based on the total net noise of
the meter (see article \cite{Speedmeter2000}). This noise is normalized in
such a way that the signal-to-noise ratio is equal to

\begin{equation}
  \frac{s}{n} = \int_{-\infty}^\infty
     \frac{|F_\mathrm{signal}(\Omega)|^2}{S_\mathrm{total}(\Omega)}\,
     \frac{d\Omega}{2\pi} \,,
\end{equation}
where $F_\mathrm{signal}(\Omega)$ is the spectrum of the signal force, and
$S_\mathrm{total}(\Omega)$ is the spectral density of this noise.

In the case of the interferometric detector (see Fig.\,\ref{Fig:scheme})
spectral density of the net noise is equal to

\begin{equation}
  S_\mathrm{total}(\Omega) = S_F^\mathrm{eff}(\Omega)
    + \chi_\mathrm{eff}^{-2}(\Omega)S_x(\Omega) \,,  \label{S_total}
\end{equation}
where

\begin{equation}
  S_F^\mathrm{eff}(\Omega) = \frac{\hbar^2}{4S_x(\Omega)}
\end{equation}
is the residual back-action noise of the meter ({\it i.e.} part of the
back-action noise $S_F$ which does not correlate with the measurement noise),

\begin{equation} S_x(\Omega) = \frac{\hbar
  L^2}{8\omega_o\mathcal{E}\gamma} \times
    \frac{\Omega^4+2\Omega^2(\gamma^2-\delta^2)+(\gamma^2+\delta^2)^2}
      {\Omega^2+\gamma^2}
  \label{S_x}
\end{equation}
is the measurement noise,

\begin{equation}
  \chi_\mathrm{eff}(\Omega) = \frac{1}{-m\Omega^2+K_\mathrm{eff}(\Omega)}
  \label{chi_eff}
\end{equation}
is the effective susceptibility of the system, and

\begin{equation}
  K_\mathrm{eff} = \frac{2\omega_o\mathcal{E}\delta}{L^2}\times
    \frac{3\gamma^2+\delta^2-\Omega^2}
      {\Omega^4+2\Omega^2(\gamma^2-\delta^2)+(\gamma^2+\delta^2)^2}
  \label{K_eff}
\end{equation}
is the effective rigidity which is the sum of two terms: (i) the real
physical rigidity which exists in the system due to the dependence of the
optical energy in the resonator on the mirrors position, and (ii) the
``virtual'' rigidity introduced by the cross-correlation of the measurement
noise and back-action noise (see article \cite{Syrtsev1994}). It should be
noted that in our case the real rigidity is much larger than the ``virtual''
one.  However, it is the ``virtual'' rigidity that compensates the imaginary
part of the real physical rigidity which describes dynamical instability of
the system. In other words, the instability does exist and must be
compensated by some feed-back scheme but the meter does not ``see'' it.

Expressions (\ref{S_x},\ref{K_eff}) are obtained for the case where the phase
quadrature amplitude of the output optical wave is measured.  Our results
does not depend essentially on which quadrature amplitude is measured but
choosing of the phase quadrature amplitude provides slightly better results
and also allows to simplify the formulae.

The behavior of the $K_\mathrm{eff}(\Omega)$ is rather sophisticated and
allows a several different regimes, depending on the pumping energy, resonator
bandwidth and  detuning: with three first-order poles
[Fig.\,\ref{Fig:poles}(a)]; with one second-order and one first-order poles
[Fig.\,\ref{Fig:poles}(b,c)]; and with one third-order pole
[Fig.\,\ref{Fig:poles}(d)]. Which one should be chosen depends on the signal
form.  Detailed analysis of all of them exceeds the frames of this short
article. One of these regimes was considered in details in the article
\cite{Buonanno2001}. Here we consider two other regimes which for our opinion
are the most interesting ones from both theoretical and ``consumer'' points
of view.

\begin{figure}

\begin{center}
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}}%
\put(2947,541){\makebox(0,0)[l]{$(\Omega/\delta)^2$}}%
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\end{picture}%
\endgroup
%\endinput
\caption{
  Different regimes of the frequency-dependent rigidity:  (a) --- three
  first-order poles; (b,c) --- one second-order and one first-order poles;
  (d) --- one third-order pole}
\end{center}

\label{Fig:poles}

\end{figure}

\paragraph{The second-order-pole regime}

If the bandwidth of the Fabry-Perot resonator is small, $\gamma\ll\delta$,
and the pumping energy is equal to\footnote{The exact expressions are too
cumbersome so we present here only second-order Taylor expansions with
respect to small parameter $\gamma/\delta$}

\begin{equation}
  \mathcal{E} \approx \frac{mL^2\delta^3}{8\omega_o}
    \left(1+\frac{6\gamma^2}{\delta^2}\right) \,,
  \label{E_2}
\end{equation}
then in the narrow vicinity of the frequency

\begin{equation}
  \Omega_2 \approx \sqrt{\frac{\delta^2+11\gamma^2}{2}}
\end{equation}
the formula (\ref{S_total}) can be presented as

\begin{equation}
  S_\mathrm{total}(\Omega) = \hbar m\delta^2\left[\frac{\gamma}{2\delta}
    + \frac{2\delta}{\gamma}\,\frac{(\Omega-\Omega_2)^4}{\Omega_2^4}
  \right] \,.
\end{equation}
The value of $\gamma/\delta$ can be adjusted to provide minimum of this
spectral density at the edges of some given spectral band
$\Omega_2\pm\Delta\Omega/2$:

\begin{equation}
  \frac{\gamma}{\delta} = \frac{1}{2}
    \left(\frac{\Delta\Omega}{\Omega_2}\right)^2 \,.
\end{equation}
In this case there will be

\begin{equation}
  S_\mathrm{total}(\Omega_2\pm\Delta\Omega/2) = \hbar m(\Delta\Omega)^2 \,.
  \label{S_total_2}
\end{equation}
This is the spectral equivalent of the formula (\ref{F_SQL}).

In Fig.\,\ref{Fig:noise2} spectral density of the total noise (\ref{S_total})
is presented for several values of $\gamma$ and for the pumping energy
(\ref{E_2}) corresponding to the second-order pole (dashed line is the SQL
level).

\begin{figure}

\begin{center}

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1297 2492 M
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}}%
\put(800,565){\makebox(0,0)[l]{$\sqrt{\dfrac{\hbar m\Omega^2}{\hbar m\delta^2}}$}}%
\put(800,958){\makebox(0,0)[l]{$\sqrt{\dfrac{S_\mathrm{total}}{\hbar m\delta^2}}$}}%
\put(2910,1286){\makebox(0,0)[l]{$\gamma/\delta=0.1$}}%
\put(2550,738){\makebox(0,0)[l]{$\gamma/\delta=0.01$}}%
\put(2280,410){\makebox(0,0)[l]{$\gamma/\delta=0.001$}}%
\put(1875,50){\makebox(0,0){$\Omega/\delta$}}%
\put(3450,200){\makebox(0,0){1}}%
\put(3000,200){\makebox(0,0){0.9}}%
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%\endinput
\caption{Sensitivity for the second-order-pole regime}
\label{Fig:noise2}

\end{center}

\end{figure}

It is useful to compare sensitivity of this regime with the sensitivity
provided by the usual probe oscillator with frequency independent
rigidity and with eigen frequency $\Omega_m$. In the latter case (see
article \cite{Rigidity2001})

\begin{equation}
  \xi^2 \equiv \frac{S_\mathrm{total}(\Omega_m\pm\Delta\Omega/2)}
    {S_\mathrm{SQL}(\Omega_m)}
  = \frac{\Delta\Omega}{\Omega_m}\,,
  \label{xi_1}
\end{equation}
where $S_\mathrm{SQL}(\Omega) = \hbar m\Omega^2$ is the spectral density
corresponding to the SQL for the free test mass.  In the case of second-order
pole regime will be

\begin{equation}
  \xi^2 \equiv \frac{S_\mathrm{total}(\Omega_2\pm\Delta\Omega/2)}
    {S_\mathrm{SQL}(\Omega_2)}
  = \left(\frac{\Delta\Omega}{\Omega_2}\right)^2
  = \frac{2\gamma}{\delta}  \,.
  \label{xi_2}
\end{equation}

\paragraph{The third-order-pole regime}

Parameters of the scheme can be tuned also to create the third order pole of
the effective susceptibility (\ref{chi_eff}) by setting

\begin{equation}
  \mathcal{E} = \frac{9\sqrt{177}-113}{49}\times\frac{mL^2\delta^3}{\omega_o}
  \approx \frac{0.14mL^2\delta^3}{\omega_o}
  \label{E_3}
\end{equation}
and

\begin{equation}
  \frac{\gamma}{\delta}
  = \frac{\sqrt{280-21\sqrt{177}}}{7}\approx 0.11 \,.
  \label{gamma_3}
\end{equation}
The pole frequency is equal to

\begin{equation}
  \Omega_3 = \sqrt{\frac{22-2\sqrt{177}}{7}}\,\delta \approx 0.81\delta \,.
\end{equation}
In Fig.\,\ref{Fig:noise3} spectral density of the total noise (\ref{S_total})
is presented for the case of the third-order-pole regime.

\begin{figure}

\begin{center}

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/hpt2 hpt 2 mul def
/Lshow { currentpoint stroke M
  0 vshift R show } def
/Rshow { currentpoint stroke M
  dup stringwidth pop neg vshift R show } def
/Cshow { currentpoint stroke M
  dup stringwidth pop -2 div vshift R show } def
/UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def
  /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def
/DL { Color {setrgbcolor Solid {pop []} if 0 setdash }
 {pop pop pop Solid {pop []} if 0 setdash} ifelse } def
/BL { stroke userlinewidth 2 mul setlinewidth } def
/AL { stroke userlinewidth 2 div setlinewidth } def
/UL { dup gnulinewidth mul /userlinewidth exch def
      10 mul /udl exch def } def
/PL { stroke userlinewidth setlinewidth } def
/LTb { BL [] 0 0 0 DL } def
/LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def
/LT0 { PL [] 1 0 0 DL } def
/LT1 { PL [4 dl 2 dl] 0 1 0 DL } def
/LT2 { PL [2 dl 3 dl] 0 0 1 DL } def
/LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def
/LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def
/LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def
/LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def
/LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def
/LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def
/Pnt { stroke [] 0 setdash
   gsave 1 setlinecap M 0 0 V stroke grestore } def
/Dia { stroke [] 0 setdash 2 copy vpt add M
  hpt neg vpt neg V hpt vpt neg V
  hpt vpt V hpt neg vpt V closepath stroke
  Pnt } def
/Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V
  currentpoint stroke M
  hpt neg vpt neg R hpt2 0 V stroke
  } def
/Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M
  0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V closepath stroke
  Pnt } def
/Crs { stroke [] 0 setdash exch hpt sub exch vpt add M
  hpt2 vpt2 neg V currentpoint stroke M
  hpt2 neg 0 R hpt2 vpt2 V stroke } def
/TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M
  hpt neg vpt -1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath stroke
  Pnt  } def
/Star { 2 copy Pls Crs } def
/BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M
  0 vpt2 neg V  hpt2 0 V  0 vpt2 V
  hpt2 neg 0 V  closepath fill } def
/TriUF { stroke [] 0 setdash vpt 1.12 mul add M
  hpt neg vpt -1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath fill } def
/TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V closepath stroke
  Pnt  } def
/TriDF { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V closepath fill} def
/DiaF { stroke [] 0 setdash vpt add M
  hpt neg vpt neg V hpt vpt neg V
  hpt vpt V hpt neg vpt V closepath fill } def
/Pent { stroke [] 0 setdash 2 copy gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath stroke grestore Pnt } def
/PentF { stroke [] 0 setdash gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath fill grestore } def
/Circle { stroke [] 0 setdash 2 copy
  hpt 0 360 arc stroke Pnt } def
/CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def
/C0 { BL [] 0 setdash 2 copy moveto vpt 90 450  arc } bind def
/C1 { BL [] 0 setdash 2 copy        moveto
       2 copy  vpt 0 90 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C2 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 90 180 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C3 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 0 180 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C4 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 180 270 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C5 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 0 90 arc
       2 copy moveto
       2 copy  vpt 180 270 arc closepath fill
               vpt 0 360 arc } bind def
/C6 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 90 270 arc closepath fill
              vpt 0 360 arc closepath } bind def
/C7 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 0 270 arc closepath fill
              vpt 0 360 arc closepath } bind def
/C8 { BL [] 0 setdash 2 copy moveto
      2 copy vpt 270 360 arc closepath fill
              vpt 0 360 arc closepath } bind def
/C9 { BL [] 0 setdash 2 copy moveto
      2 copy  vpt 270 450 arc closepath fill
              vpt 0 360 arc closepath } bind def
/C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill
       2 copy moveto
       2 copy vpt 90 180 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C11 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 0 180 arc closepath fill
       2 copy moveto
       2 copy  vpt 270 360 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C12 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 180 360 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C13 { BL [] 0 setdash  2 copy moveto
       2 copy  vpt 0 90 arc closepath fill
       2 copy moveto
       2 copy  vpt 180 360 arc closepath fill
               vpt 0 360 arc closepath } bind def
/C14 { BL [] 0 setdash 2 copy moveto
       2 copy  vpt 90 360 arc closepath fill
               vpt 0 360 arc } bind def
/C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill
               vpt 0 360 arc closepath } bind def
/Rec   { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto
       neg 0 rlineto closepath } bind def
/Square { dup Rec } bind def
/Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def
/S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def
/S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def
/S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def
/S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def
/S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill
       exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def
/S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def
/S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill
       2 copy vpt Square fill
       Bsquare } bind def
/S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def
/S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def
/S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill
       Bsquare } bind def
/S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill
       Bsquare } bind def
/S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def
/S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
       2 copy vpt Square fill Bsquare } bind def
/S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill
       2 copy exch vpt sub exch vpt Square fill Bsquare } bind def
/S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def
/D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def
/D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def
/D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def
/D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def
/D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def
/D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def
/D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def
/D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def
/D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def
/D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def
/D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def
/D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def
/D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def
/D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def
/D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def
/D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def
/DiaE { stroke [] 0 setdash vpt add M
  hpt neg vpt neg V hpt vpt neg V
  hpt vpt V hpt neg vpt V closepath stroke } def
/BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M
  0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V closepath stroke } def
/TriUE { stroke [] 0 setdash vpt 1.12 mul add M
  hpt neg vpt -1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V closepath stroke } def
/TriDE { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V closepath stroke } def
/PentE { stroke [] 0 setdash gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  closepath stroke grestore } def
/CircE { stroke [] 0 setdash
  hpt 0 360 arc stroke } def
/Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def
/DiaW { stroke [] 0 setdash vpt add M
  hpt neg vpt neg V hpt vpt neg V
  hpt vpt V hpt neg vpt V Opaque stroke } def
/BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M
  0 vpt2 neg V hpt2 0 V 0 vpt2 V
  hpt2 neg 0 V Opaque stroke } def
/TriUW { stroke [] 0 setdash vpt 1.12 mul add M
  hpt neg vpt -1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt 1.62 mul V Opaque stroke } def
/TriDW { stroke [] 0 setdash vpt 1.12 mul sub M
  hpt neg vpt 1.62 mul V
  hpt 2 mul 0 V
  hpt neg vpt -1.62 mul V Opaque stroke } def
/PentW { stroke [] 0 setdash gsave
  translate 0 hpt M 4 {72 rotate 0 hpt L} repeat
  Opaque stroke grestore } def
/CircW { stroke [] 0 setdash
  hpt 0 360 arc Opaque stroke } def
/BoxFill { gsave Rec 1 setgray fill grestore } def
end
%%EndProlog
}}%
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6 13 V
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6 15 V
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6 15 V
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6 4 V
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6 4 V
6 3 V
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6 3 V
6 4 V
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7 3 V
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6 3 V
7 3 V
6 3 V
6 3 V
7 3 V
6 3 V
6 4 V
7 3 V
6 3 V
6 3 V
6 3 V
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\put(800,565){\makebox(0,0)[l]{$\sqrt{\dfrac{\hbar m\Omega^2}{\hbar m\delta^2}}$}}%
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\caption{Sensitivity for the third-order-pole regime}
\label{Fig:noise3}

\end{center}

\end{figure}

It is necessary to note that this third-order-pole regime is ``overpumped'':
the second term in the formula (\ref{S_total}), that is proportional to the
measurement noise, is several orders of magnitude smaller than the first one
(back-action noise) in the frequency area of interest. It is evident from
Fig.\,\ref{Fig:noise3s}, where these two terms are plotted separately.
Usually, in such a situation the reduction of the total noise is possible by
increasing the measurement noise and proportional decreasing the back-action
noise due to using, for example, smaller value of the pumping energy.
Unfortunately, within the framework of our simple scheme it is impossible
because the same optical pumping is used both for measurement and for
creating the rigidity. There is no additional ``degree of freedom'' here:
values of  all parameters are fixed by the equations
(\ref{E_3},\ref{gamma_3}). It is probable, however, that more sophisticated
topologies based on use of separate optical modes for measurement and for
creating rigidity, or/and which eliminate back-action noise by using
variational measurement \cite{Vyatchanin1993,Vyatchanin1998,Danilishin2000},
will allow to create ``well-balanced'' third-order-pole regime with very low
total noise.

\begin{figure}

\begin{center}

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6 23 V
7 23 V
6 24 V
6 23 V
7 24 V
6 24 V
6 25 V
7 24 V
6 25 V
1.000 UL
LT1
397 740 M
263 0 V
300 433 M
6 1 V
7 1 V
6 0 V
6 1 V
7 1 V
6 0 V
6 1 V
7 1 V
6 0 V
6 1 V
6 1 V
7 0 V
6 1 V
6 1 V
7 0 V
6 1 V
6 1 V
7 0 V
6 1 V
6 1 V
7 1 V
6 0 V
6 1 V
7 1 V
6 0 V
6 1 V
6 1 V
7 1 V
6 0 V
6 1 V
7 1 V
6 1 V
6 0 V
7 1 V
6 1 V
6 0 V
7 1 V
6 1 V
6 1 V
7 0 V
6 1 V
6 1 V
6 1 V
7 1 V
6 0 V
6 1 V
7 1 V
6 1 V
6 0 V
7 1 V
6 1 V
6 1 V
7 1 V
6 0 V
6 1 V
7 1 V
6 1 V
6 0 V
6 1 V
7 1 V
6 1 V
6 1 V
7 1 V
6 0 V
6 1 V
7 1 V
6 1 V
6 1 V
7 0 V
6 1 V
6 1 V
7 1 V
6 1 V
6 1 V
6 1 V
7 0 V
6 1 V
6 1 V
7 1 V
6 1 V
6 1 V
7 1 V
6 0 V
6 1 V
7 1 V
6 1 V
6 1 V
7 1 V
6 1 V
6 1 V
6 1 V
7 1 V
6 0 V
6 1 V
7 1 V
6 1 V
6 1 V
7 1 V
6 1 V
6 1 V
7 1 V
6 1 V
6 1 V
7 1 V
6 1 V
6 1 V
6 1 V
7 1 V
6 1 V
6 1 V
7 1 V
6 0 V
6 1 V
7 1 V
6 1 V
6 1 V
7 1 V
6 1 V
6 2 V
7 1 V
6 1 V
6 1 V
6 1 V
7 1 V
6 1 V
6 1 V
7 1 V
6 1 V
6 1 V
7 1 V
6 1 V
6 1 V
7 1 V
6 1 V
6 1 V
7 1 V
6 2 V
6 1 V
6 1 V
7 1 V
6 1 V
6 1 V
7 1 V
6 1 V
6 2 V
7 1 V
6 1 V
6 1 V
7 1 V
6 1 V
6 1 V
7 2 V
6 1 V
6 1 V
6 1 V
7 1 V
6 2 V
6 1 V
7 1 V
6 1 V
6 1 V
7 2 V
6 1 V
6 1 V
7 1 V
6 2 V
6 1 V
7 1 V
6 2 V
6 1 V
6 1 V
7 1 V
6 2 V
6 1 V
7 1 V
6 2 V
6 1 V
7 1 V
6 2 V
6 1 V
7 1 V
6 2 V
6 1 V
7 2 V
6 1 V
6 1 V
6 2 V
7 1 V
6 2 V
6 1 V
7 1 V
6 2 V
6 1 V
7 2 V
6 1 V
6 2 V
7 1 V
6 2 V
6 1 V
7 2 V
6 1 V
6 2 V
6 1 V
7 2 V
6 1 V
6 2 V
7 1 V
6 2 V
6 2 V
7 1 V
6 2 V
6 1 V
7 2 V
6 2 V
6 1 V
7 2 V
6 2 V
6 1 V
6 2 V
7 2 V
6 1 V
6 2 V
7 2 V
6 2 V
6 1 V
7 2 V
6 2 V
6 2 V
7 1 V
6 2 V
6 2 V
7 2 V
6 2 V
6 1 V
6 2 V
7 2 V
6 2 V
6 2 V
7 2 V
6 2 V
6 2 V
7 2 V
6 2 V
6 1 V
7 2 V
6 2 V
6 2 V
7 2 V
6 2 V
6 3 V
6 2 V
7 2 V
6 2 V
6 2 V
7 2 V
6 2 V
6 2 V
7 2 V
6 3 V
6 2 V
7 2 V
6 2 V
6 2 V
7 3 V
6 2 V
6 2 V
6 2 V
7 3 V
6 2 V
6 2 V
7 3 V
6 2 V
6 2 V
7 3 V
6 2 V
6 3 V
7 2 V
6 3 V
6 2 V
7 3 V
6 2 V
6 3 V
6 2 V
7 3 V
6 2 V
6 3 V
7 3 V
6 2 V
6 3 V
7 3 V
6 3 V
6 2 V
7 3 V
6 3 V
6 3 V
7 2 V
6 3 V
6 3 V
6 3 V
7 3 V
6 3 V
6 3 V
7 3 V
6 3 V
6 3 V
7 3 V
6 3 V
6 3 V
7 3 V
6 4 V
6 3 V
7 3 V
6 3 V
6 3 V
6 4 V
7 3 V
6 3 V
6 4 V
7 3 V
6 4 V
6 3 V
7 3 V
6 4 V
6 4 V
7 3 V
6 4 V
6 3 V
7 4 V
6 4 V
6 3 V
6 4 V
7 4 V
6 4 V
6 4 V
7 4 V
6 4 V
6 3 V
7 4 V
6 4 V
6 5 V
7 4 V
6 4 V
6 4 V
7 4 V
6 4 V
6 5 V
6 4 V
7 4 V
6 5 V
6 4 V
7 5 V
6 4 V
6 5 V
7 4 V
6 5 V
6 5 V
7 4 V
6 5 V
6 5 V
7 5 V
6 5 V
6 5 V
6 5 V
7 5 V
6 5 V
6 5 V
7 5 V
6 5 V
6 6 V
7 5 V
6 5 V
6 6 V
7 5 V
6 6 V
6 5 V
7 6 V
6 6 V
6 5 V
6 6 V
7 6 V
6 6 V
6 6 V
7 6 V
6 6 V
6 6 V
7 6 V
6 6 V
6 6 V
7 7 V
6 6 V
6 7 V
7 6 V
6 7 V
6 6 V
6 7 V
7 7 V
6 7 V
currentpoint stroke M
6 6 V
7 7 V
6 7 V
6 7 V
7 7 V
6 8 V
6 7 V
7 7 V
6 7 V
6 8 V
7 7 V
6 8 V
6 7 V
6 8 V
7 8 V
6 7 V
6 8 V
7 8 V
6 8 V
6 8 V
7 8 V
6 8 V
6 8 V
7 8 V
6 9 V
6 8 V
7 8 V
6 9 V
6 8 V
6 9 V
7 8 V
6 9 V
6 9 V
7 8 V
6 9 V
6 9 V
7 9 V
6 8 V
6 9 V
7 9 V
6 9 V
6 9 V
7 9 V
6 9 V
6 9 V
6 9 V
7 9 V
6 9 V
6 9 V
7 9 V
6 9 V
6 9 V
7 9 V
6 9 V
6 9 V
7 9 V
6 9 V
6 9 V
7 9 V
6 9 V
6 8 V
6 9 V
7 9 V
6 8 V
6 9 V
7 8 V
6 8 V
6 9 V
7 8 V
6 8 V
6 7 V
7 8 V
6 8 V
6 7 V
7 8 V
6 7 V
6 7 V
6 7 V
7 6 V
6 7 V
6 6 V
7 6 V
6 6 V
6 6 V
7 6 V
6 5 V
6 5 V
7 5 V
6 5 V
6 4 V
7 4 V
6 4 V
6 4 V
6 3 V
7 4 V
6 2 V
6 3 V
7 3 V
6 2 V
6 2 V
7 1 V
6 1 V
stroke
grestore
end
showpage
}}%
\put(710,740){\makebox(0,0)[l]{$\sqrt{\dfrac{S_F^\mathrm{eff}}{\hbar m\delta^2}}$}}%
\put(710,1133){\makebox(0,0)[l]{$\sqrt{\dfrac{\chi_\mathrm{eff}^{-2}S_x}{\hbar m\delta^2}}$}}%
\put(1875,50){\makebox(0,0){$\Omega/\delta$}}%
\put(3450,200){\makebox(0,0){1}}%
\put(3000,200){\makebox(0,0){0.9}}%
\put(2550,200){\makebox(0,0){0.8}}%
\put(2100,200){\makebox(0,0){0.7}}%
\put(1650,200){\makebox(0,0){0.6}}%
\put(1200,200){\makebox(0,0){0.5}}%
\put(750,200){\makebox(0,0){0.4}}%
\put(300,200){\makebox(0,0){0.3}}%
\put(250,2492){\makebox(0,0)[r]{1}}%
\put(250,2054){\makebox(0,0)[r]{0.8}}%
\put(250,1615){\makebox(0,0)[r]{0.6}}%
\put(250,1177){\makebox(0,0)[r]{0.4}}%
\put(250,738){\makebox(0,0)[r]{0.2}}%
\put(250,300){\makebox(0,0)[r]{0}}%
\end{picture}%
\endgroup
%\endinput
\caption{Measurement noise and back-action noise for the
third-order-pole regime} \label{Fig:noise3s}

\end{center}

\end{figure}

\section{Conclusion}

It is evidently impossible to consider thoroughly in one short article all
the possible variants of the use of the frequency-dependent rigidity which
exists in large-scale optical resonators. Such a consideration has to be
based on \textit{a priori} information about the signal spectrum provided,
for example, by astrophysical predictions in the similar as it had been done
in the article \cite{Buonanno2001}. It is evident, however, that:

\begin{itemize}

  \item The second-order-pole regime allows to ``dive'' deep below the SQL in
  the narrow spectral band $\Delta\Omega$ which is, however, much wider than
  if the usual frequency independent rigidity is used [compare formulae
  (\ref{xi_1}) and (\ref{xi_2})]. The recent achievements in fabrication of
  high-reflectivity mirrors \cite{Kimble1992} allows to expect that it will
  be possible to obtain relaxation time of the large-scale interferometers
  $\gamma^{-1}\gtrsim 1\,\mathrm{s}$ and thus to reach the sensitivity at the
  level of $\xi^2\lesssim 10^{-3}$, if $\Omega\sim\delta\sim
  10^3\,\mathrm{s}^{-1}$. It is important that the pumping energy in this
  regime does not depend on the sensitivity and remains approximately equal
  to the energy (\ref{E_SQL}) which is necessary to achieve the SQL in the
  traditional scheme of the interferometric position meter.

  \item The third-order-pole regime provides sensitivity a few times
  better than the Standard Quantum Limit in relatively wide spectral band and
  at extremely low level of the measurement noise in this band. This regime
  looks as a good candidate for use in advanced topologies of the
  gravitational-wave antennae.

\end{itemize}

\section*{Acknowledgments}

Author thanks V.B.Braginsky and S.P.Vyatchanin for very useful discussions.
This paper was supported in part by the California Institute of Technology,
US National Science Foundation, by the Russian Foundation for Basic Research,
and by the Russian Ministry of Industry and Science.

%\bibliographystyle{phaip}
%\bibliography{sql,khalili}
%\input{rigid3e.bbl}
\begin{thebibliography}{10}

\bibitem{Braginsky1967}
{V.B.Braginsky},
\newblock Sov.\,Phys.\,JETP {\bf 26}, 831 (1968).

\bibitem{300years}
{K.S.Thorne},
\newblock {\em Three Hundred Years of Gravitation},
\newblock Cambridge University Press, 1987.

\bibitem{OptBar1997}
{V.B.Braginsky, M.L.Gorodetsky, F.Ya.Khalili},
\newblock Physics Letters A {\bf 232}, 340 (1997).

\bibitem{Symphot1998}
{V.B.Braginsky, M.L.Gorodetsky, F.Ya.Khalili},
\newblock Physics Letters A {\bf 246}, 485 (1998).

\bibitem{Vyatchanin1993}
{A.B.Matsko, S.P.Vyatchanin},
\newblock JETP {\bf 77}, 218 (1993).

\bibitem{Vyatchanin1998}
{S.P.Vyatchanin},
\newblock Physics Letters A {\bf 239}, 201 (1998).

\bibitem{KLMTV2000}
{H.J.Kimble, Yu.Levin, A.B.Matsko, K.S.Thorne and S.P.Vyatchanin},
\newblock Physical Review D , in press (2001).

\bibitem{Rigidity1999}
{V.B.Braginsky, F.Ya.Khalili},
\newblock Physics Letters A {\bf 257}, 241 (1999).

\bibitem{Minsk2000e}
{F.Ya.Khalili},
\newblock Optics and Spectroscopy {\bf 91}, 550 (2001).

\bibitem{Rigidity2001}
{V.B.Braginsky, F.Ya.Khalili},
\newblock Physics Letters A , in press (2001).

\bibitem{Buonanno2001}
{A.Buonanno, Y.Chen},
\newblock Physical Review D , in press (2001).

\bibitem{Meers1988}
{B.Meers},
\newblock Physical Review D {\bf 38}, 2217 (1988).

\bibitem{Speedmeter2000}
{V.B.Braginsky, M.L.Gorodetsky, F.Ya.Khalili and K.S.Thorne},
\newblock Physical Review D {\bf 61}, 044002 (2000).

\bibitem{Syrtsev1994}
{A.V.Syrtsev, F.Ya.Khalili},
\newblock JETP {\bf 79}, 409 (1994).

\bibitem{Danilishin2000}
{S.L.Danilishin, F.Ya.Khalili, S.P.Vyatchanin},
\newblock Physics Letters A {\bf 278}, 123 (2000).

\bibitem{Kimble1992}
{G.Rempe, R.Tompson, H.J.Kimble},
\newblock Optics Letters {\bf 17}, 363 (1992).

\end{thebibliography}

\end{document}

gr-qc0107084.extracted_bib
\begin{thebibliography}{10}

\bibitem{Braginsky1967}
{V.B.Braginsky},
\newblock Sov.\,Phys.\,JETP {\bf 26}, 831 (1968).

\bibitem{300years}
{K.S.Thorne},
\newblock {\em Three Hundred Years of Gravitation},
\newblock Cambridge University Press, 1987.

\bibitem{OptBar1997}
{V.B.Braginsky, M.L.Gorodetsky, F.Ya.Khalili},
\newblock Physics Letters A {\bf 232}, 340 (1997).

\bibitem{Symphot1998}
{V.B.Braginsky, M.L.Gorodetsky, F.Ya.Khalili},
\newblock Physics Letters A {\bf 246}, 485 (1998).

\bibitem{Vyatchanin1993}
{A.B.Matsko, S.P.Vyatchanin},
\newblock JETP {\bf 77}, 218 (1993).

\bibitem{Vyatchanin1998}
{S.P.Vyatchanin},
\newblock Physics Letters A {\bf 239}, 201 (1998).

\bibitem{KLMTV2000}
{H.J.Kimble, Yu.Levin, A.B.Matsko, K.S.Thorne and S.P.Vyatchanin},
\newblock Physical Review D , in press (2001).

\bibitem{Rigidity1999}
{V.B.Braginsky, F.Ya.Khalili},
\newblock Physics Letters A {\bf 257}, 241 (1999).

\bibitem{Minsk2000e}
{F.Ya.Khalili},
\newblock Optics and Spectroscopy {\bf 91}, 550 (2001).

\bibitem{Rigidity2001}
{V.B.Braginsky, F.Ya.Khalili},
\newblock Physics Letters A , in press (2001).

\bibitem{Buonanno2001}
{A.Buonanno, Y.Chen},
\newblock Physical Review D , in press (2001).

\bibitem{Meers1988}
{B.Meers},
\newblock Physical Review D {\bf 38}, 2217 (1988).

\bibitem{Speedmeter2000}
{V.B.Braginsky, M.L.Gorodetsky, F.Ya.Khalili and K.S.Thorne},
\newblock Physical Review D {\bf 61}, 044002 (2000).

\bibitem{Syrtsev1994}
{A.V.Syrtsev, F.Ya.Khalili},
\newblock JETP {\bf 79}, 409 (1994).

\bibitem{Danilishin2000}
{S.L.Danilishin, F.Ya.Khalili, S.P.Vyatchanin},
\newblock Physics Letters A {\bf 278}, 123 (2000).

\bibitem{Kimble1992}
{G.Rempe, R.Tompson, H.J.Kimble},
\newblock Optics Letters {\bf 17}, 363 (1992).

\end{thebibliography}