KBlueLeaf/taica-nlp-hw1 · Datasets at Hugging Face

text string
parameter @xmath23 stands for optical relaxation rate , coupling coefficient @xmath24 is proportional to pump power , @xmath25 is the coefficient of feedback
. the right parts of equations represent fluctuational forces acting on corresponding degree of freedom . as of optical one
the force @xmath26 represents input vacuum fluctuations and for mechanical one @xmath27 describes vacuum fluctuations reflected by the cavity and transmitted by feedback .
we neglect all other fluctuational forces including thermal and seismic ones .
depending on device in consideration this first step assumption could be either realistic or not .
some calculations of thermal noise influence is presented in appendix [ app : thermal ] . in particular
they show that in configuration of advanced ligo interferometer the impact of thermal noises is only slightly below the one of quantum fluctuations .
the equations look similar to ones for oscillator coupled to a free mass ( or two coupled oscillators one of which has partial frequency equal to zero ) .
however we would note the difference in signs in front of the coupling terms ( proportional to @xmath24 ) : in the case of conventional coupled oscillators these signs coincide ; in our case the signs differ .
this is the reason of instability which appears in absence of feedback and this is also the reason of possibility for eigen frequencies of such system to coincide ( so called double resonance case @xcite let us remind that in case of two coupled oscillators coincidence of eigen frequencies is impossible ) .
as in the case of conventional coupled oscillators one can treat the system in terms of eigen modes . the evolution of oscillator system could be written as a sum of eigen oscillations : @xmath28 index in the sum runs from 1 to the number of system degrees of freedom ( 2 for set ) .
this equation could be treated as transformation from conventional coordinates @xmath29 to the eigen ones . as in eqn .
one can present evolution of each of eigen modes as combination of oscillations at corresponding eigen frequencies @xmath30 and slow ( compared to these oscillations ) changing of amplitude . in case of free evolution ( zero right parts in eqns . )
quantities @xmath31 i.e. they freely decay with relaxation rates @xmath32 into the equilibrium values .
vectors @xmath33 represent distribution of amplitudes in corresponding eigen modes .
suppose that we have a possibility to modulate the power of pumping laser .
this will result in modulation of coupling coefficient and hence it will cause shifting of eigen frequencies of system .
it is well known @xcite that modulation of eigen frequency of oscillator with frequency twice bigger than its own one results in squeezing of noise in its quadratures . in this paper
we show that modulation of the pumping power performs squeezing of noise in quadratures of eigen modes amplitudes .
consider we harmonically modulate the pumping power at frequency @xmath34 so coupling coefficient in depends on time : @xmath35 .
assuming the quantities @xmath36 to be _ slow _ ( i.e. not to significantly change on times compared to mechanical periods ) one can plug the expression into system and to get rid of rapidly oscillating terms by averaging over a period @xmath37 .
this procedure results in shortened equations for amplitudes @xmath36 . for one with number @xmath38 ( @xmath39 )
the equation takes form : @xmath40 + \|m\| \sum_i g_i^\dagger ( \vec \pi_j \vec w_i^ * ) e^ { i ( \omega_i + \omega_j - 2 p ) t - i \phi } = \\ = ( \vec \pi_j { \underline{\nu}_j } ) e^{i \omega_j t } .
\end{gathered}\ ] ] here we have introduced a set of vectors @xmath41 built to be orthogonal to all @xmath42 except one : @xmath43 , where @xmath44 is a kronecker delta ( in the case of two conventional coupled oscillators vectors @xmath33 are itself orthogonal and system of @xmath45 is unnecessary ) .
vectors @xmath46 are defined as follows : @xmath47 underlining used for @xmath48 is to emphasise that this quantity is obtained in right part of equation by procedure of keeping only _ slow _ ( in comparison to terms oscillating with frequency @xmath49 ) quantities . to make @xmath50 fulfilling this criterion one should keep in @xmath51 only spectral components close to @xmath52 .
from now on we will focus on the amplitude of one of modes taking into account that all the results obtained for this mode are similar for other ones . for clarity let us set
@xmath53 thus considering the first mode .
also we suppose the modulation to happen at frequency twice bigger than the one of first mode : @xmath54 .
first let us consider the simplest case when difference between eigen frequencies @xmath55 is large in respect to decay rates @xmath32 . in this case the exponential multipliers in eqn .
should be considered as fast oscillating ones and the equations for amplitudes @xmath36 decouple . in this case
the equation for @xmath56 takes the following form in spectral domain @xmath57 g_1 ( x ) + \epsilon_{11 } g_1^\dagger ( -x ) = \frac
{ i ( \vec \pi_1 \vec \nu_1 ) e^ { i \omega_1 t } } { 2 \omega_1 } \equiv f_1 ( x).\ ] ] here @xmath58 stands for normalized spectral frequency ( see appendix [ app : notations ] ) , @xmath59 is the decay rate of first mode ; @xmath60 is a quantity proportional to modulation strength , the general expression for it could be easily deduced from eqn .
: @xmath61
we also write the equation for @xmath62 by hermitian conjugation of eqn .
[ eq : g1 ] and replacement @xmath63 .
taking proper combinations of these equations yields the expressions for corresponding quadratures of @xmath56 .
for simplicity let @xmath64 which is achievable by proper choosing @xmath65 , in this case the mentioned combinations reduce to sum or difference : @xmath66 = f_1 + f_1^\dagger ; \\
\label{eq : difq } \big(g_1 - g_1^\dagger \big ) \big [ \gamma_1 - \|\epsilon_{11 } \| - ix \big ] = f_1 - f_1^\dagger ; \end{aligned}\ ] ] one can conclude from these equations that sum quadrature is squeezed due to parametric modulation and the difference quadrature is antisqueezed .
the measure of squeezing is the spectral density which for arbitrary quantity @xmath67 is given by expression @xmath68 where the angle brackets mean averaging .
obvious calculations prove that spectral densities of right parts of both equations and coincide and the equations differ by additional damping @xmath69 ( positive or negative for different quadratures ) introduced by modulation . in absence of modulation @xmath70 and
there is no discrepancy between quadratures .
if modulation is applied then @xmath60 has nonzero value and quadratures spectral densities differ .
one can also estimate the critical level of modulation characterized by value @xmath71 , this is a level when negative damping added to the differential quadrature @xmath72 becomes equal to its own damping : @xmath73 thus making quadrature instable .
given this value of modulation coefficient damping in the sum quadrature is twice bigger than its own damping @xmath59 .
this means that quadrature noise squeezing is limited by factor of two .
in general case of arbitrary eigen frequencies @xmath74 the equations could not be solved in such an obvious way , but in this case one can rewrite these equations to switch from spectral components @xmath75 to the same components but shifted by frequency , namely @xmath76 .
the equations take form @xmath77 + \\ + \sum_i g_i^\dagger ( - x - ( \omega_i - \omega_1 ) ) \epsilon^*_{ji } = \frac { i } { 2 \omega_j } ( \vec \pi_j \vec { \underline{\nu}}_j ( p + x ) ) .\end{gathered}\ ] ] this set represents a system of linear algebraic equations and could be solved by applying kramer s rule . using the solution we obtain the expression for plus or minus quadratures of @xmath38-th mode @xmath78 defined as follows @xmath79 and calculate spectral densities of these quadratures . in coincidence with expectations in absence of parametric modulation both plus and minus quadratures of first mode
have equal spectral densities .
if modulation is applied the plus quadrature appears to be squeezed and the minus quadrature to be antisqueezed .
the bigger is modulation coefficient the more significant the ( anti- ) squeezing effect is .
it could be easily shown that squeezing is also limited by a factor of two in this case .
the important question is whether it is possible to see this squeezing in output light .
the answer to this question is positive .
output fluctuations @xmath16 are defined by input fluctuations @xmath15 and light inside cavity @xmath18 , in particular for phase quadrature of @xmath16 the following expression is valid : @xmath80 plugging the solution obtained in previous section for @xmath81 into the equation we can thus express the output fluctuations through quantities @xmath36 . it is easy to show ( see appendix [ app : readout ] ) that quantity @xmath82 is proportional to the quadrature @xmath83 . in time domain measurement of this quantity is equivalent to multiplication of measured phase quadrature with cosine of frequency @xmath52 with proper phase : @xmath84 setting @xmath85 or @xmath86 we obtain @xmath87 or @xmath88 .
estimations of @xmath89 spectral density demonstrate that it is possible to see squeezing in output light .
the spectral densities of these quantities are plotted in fig .
[ fig : output ] being normalized to the shot noise level .
the limitation of squeezing by factor of two shows up in output light too . in output field
we measure phase quadrature @xmath151 which is proportional to @xmath19 that carries information about squeezing .
further we discuss which combination of spectral components of @xmath19 contains squeezing evidence taking into account that corresponding combination of @xmath151 components represents the same combination with some input fluctuations @xmath152 added . for spectral components of @xmath19 shifted in frequency domain by amount of @xmath52 one can write the following expression using eqn .
: @xmath153 ) + g_i^\dagger ( - x - [ \omega_i + \omega_j ] ) ) .\ ] ] here we use freedom in definition of @xmath33 and let @xmath154 . an equation similar to one above could be written for @xmath155 : @xmath156 ) + g_i ( x - [ \omega_i + \omega_j ] ) \big).\ ] ] we now take sum or difference of these quantities and obtain expression containing sum of quadratures @xmath78 that are of our interest @xmath157 = \\ = \sum_i v_i \big [ g_i^{(\pm ) } ( x - ( \omega_i - \omega_j ) ) + g_i^{(\pm ) } ( x - ( \omega_i + \omega_j ) ) \big ] .
\end{gathered}\ ] ] remind that we assumed @xmath36 to be slow amplitudes which means that spectral components of these quantities are situated close to zero frequency . note that due to resonant multiplyers ( expressions in square brackets ) in eqn .
one can estimate the width of band containing @xmath36 spectra with order of corresponding mode decay rate @xmath32 which is much smaller than difference between eigen frequencies @xmath158 ( for @xmath159 ) .
this consideration is also valid for quadratures @xmath160 hence in definition of all summands differ from zero in different frequency bands ( not overlapping because of narrowness ) , or in other words in each frequency band there is not more than one non - zero summand in this definition . in particular at frequencies close to zero one can use quite exact expression @xmath161 as we actually measure @xmath17 we need to consider a combination of spectral components of this quantity containing @xmath162 .
it is obvious that corresponding combination is given by the following expression @xmath163 which represents a sum of desired quadrature and input fluctuations .
in this paper we have shown the possibility of quadrature noise squeezing of eigen modes amplitudes in an optomechanical system . as a model of latter
we have used a fabry - perot cavity with movable mirror and a detuned pump .
the eigen modes in this system are defined by interaction between optical and mechanical degrees of freedom .
detuned pump transforms the mirror which is initially a free - mass into a harmonic oscillator with spring coefficient provided by means of optical rigidity . in comparison with material rigidity peculiar to usual optomechanical devices such as microtoroids or membranes etc @xcite optical rigidity
is characterized by very low noises level .
the another advantage of using optical springs is the very ability of spring constant manipulation by means of pump power modulation .
comparison of impact exerted by quantum and thermal noises upon quadratures ( see appendix [ app : thermal ] ) reveals that both fluctuations have their affects of same order .
this gives us an ability to talk about quantum noise squeezing of optomechanic quadratures in gravitational wave detectors ( advanced ligo @xcite , einstein telescope @xcite ) and in prototypes ( glasgow university prototype @xcite , aei hannover prototype @xcite and gingin facility @xcite ) despite the fact that we consider a feedback in our model , the feedback itself is unnecessary to achieve the squeezing as it serves only to stabilize the eigen frequencies of the system .
the stabilization could be performed by means of an auxiliary pump with power significantly lower than one in main pump . however
implementation of feedback is more feasible in small - scale experimental setups like gravitational - wave antenna prototypes @xcite . realization of proposed scheme in experiment could demonstrate non - classical behavior of optomechanical mode which could be useful for tests of quantum mechanics applied to macroscopic mechanical objects .
we would like to underline it is not pure optical or pure mechanical , but an _ optomechanical _ degree of freedom that may exhibit quantum behavior .
also a quantum state if created in an eigen mode could be transmitted into another one ( for example , by modulation of pump with difference frequency @xmath90 ) .
this gives us a potential playground for creation and transmission of quantum states between optomechanical modes which could be useful for problems of quantum information .
the authors would thank stefan hild for stimulating discussions .
we also are grateful to yanbei chen , farid khalili and stefan danilishin .
authors are supported by the russian foundation for basic research grant no .
08 - 02 - 00580-a and nsf grant phy-0967049 .
the hamiltonian of considered system ( without feedback ) could be written as follows @xcite : @xmath91 here @xmath92 ( @xmath93 ) is annihilation ( creation ) operator of optical mode , @xmath20 ( @xmath49 ) is position ( momentum ) operator of movable mirror .
also @xmath94 is the frequency of cavity fundamental mode closest to the pump frequency @xmath95 .
the difference between these frequencies is denoted as @xmath13 , please note the sign of this detuning : for the pump tuned onto the right slope of optical resonance curve ( blue detuned pump ) @xmath13 is _
positive_. @xmath96 is the coupling coefficient between light mode and mirror motion .
@xmath97 is the hamiltonian describing interaction of the system with bath and the bath itself .
@xmath98 is the same for pump .
if we then suppose the optical operators to consist of strong classical part @xmath99 ( which is real positive number characterizing mean optical power @xmath14 inside cavity : @xmath100 ) and quantum fluctuations @xmath18 so that @xmath101 and switch to the frame rotating with frequency @xmath95 , then the linearized hamiltonian should take the form @xmath102 one then should write down heisenberg equations for position and momentum of mirror and amplitude ( @xmath19 ) and phase ( @xmath103 ) operators of light defined in following way : @xmath104/ \sqrt 2 ; \qquad b_2 \equiv - i [ b - b^+ ] / \sqrt 2.\ ] ] in consideration above we have not given concrete expressions to the bath and pump hamiltonians so in order to perform inclusion of damping , feedback and optical fluctuational forces in a correct way we use semiclassical approach writing down the equations of motion in spectral form @xcite : @xmath105 +
y \big [ \sqrt { \frac { p \omega_0 } { \hbar l^2 } } \delta \big ] = \\ & = - \sqrt { \frac { \gamma } { \tau } } \left [ a_1^{i } ( \gamma - i \omega ) + a_2^{i } \delta \right ] ; \\ - \omega^2 y - \frac { 2 } { \mu c } & \sqrt { p \hbar \omega_0 } b_1 = i \omega \alpha_\text{fb } ( - a_2^{i } - 2 \sqrt { \gamma \tau } b_1 ) .\end{aligned}\ ] ] here @xmath106 is spectral frequency , @xmath107 is the time it takes light to travel between mirrors ( one way ) , @xmath108 is a coefficient of feedback , i.e. the force fed to the mirror is equal to @xmath109 .
the dimensionless parameters are defined as follows .
@xmath110 switching to dimensionless parameters from dimensional ones using the definitions listed above yields the system @xmath111 , \\ - a b_1 + & i x \alpha b_1 - x^2 z = \nu_2 \equiv \frac { i x \alpha } { 2 \sqrt { \gamma \tau } } a_2^i . \end{aligned}\ ] ] finally by transferring from frequency domain to time one following rule @xmath112 one can write down the system
in order to account thermal fluctuations in the model one should include corresponding fluctuational force in the right part of the equation of motion for mechanical degree of freedom .
this results in formal replacement @xmath114 here @xmath115 is thermal fluctuational force . if we consider coating brownian noise as the source of this force ( in advanced ligo coating fluctuations are dominant thermal noises ) , then its spectral density is given by expression @xcite : @xmath116 , \\
\phi_\bot & = \frac{y}{\sqrt \pi w ( 1-\sigma^2)}\left [ \frac{(1+\sigma_1)(1 - 2\sigma_1)d_1\phi_1}{y_1(1-\sigma_1 ) } + \right .
\nonumber \\ \notag & \qquad + \left .
\frac{(1+\sigma_2)(1 - 2\sigma_2)d_2\phi_2}{y_2(1-\sigma_2)}\right].\end{aligned}\ ] ] here @xmath117 is spectral frequency ( measured in hz ) , @xmath118 is a radius of a beam spot on mirror s surface , @xmath119 are young modulus and poisson ratio of substrate respectively , @xmath120 are the same quantities for alternating layers , @xmath121 are total thicknesses of quater wavelength layers , n is the number of layers pairs , @xmath122 optical wavelength , @xmath123 are refraction indices of layers .
@xmath9 is the temperature of mirrors coating , @xmath124 is bolzmann s constant . the numerical parameters that we use for estimations are listed in table [ param ] .
+ @xmath9 , k & + @xmath122 , m & + @xmath126 & - & @xmath127 & @xmath127 + @xmath128 & 1.45 & 2.035 & 1.45 + @xmath129 , pa & @xmath130 & @xmath131 & @xmath130 + @xmath132 & @xmath133 & @xmath134 & @xmath133 + @xmath65 & @xmath135 & @xmath136 & @xmath137 + as the thermal noise is not correlated to vacuum noises , inclusion of the last term in equation reveals in estimations of spectral densities of quadratures only by additional term in expression for spectral density of @xmath27 : @xmath138 the convenient factor of thermal noise influence on eigen mode amplitude spectral density is the following ratio : @xmath139 here @xmath140 is the spectral density of output field quadrature @xmath141 in case of only thermal noises present and @xmath142 is the spectral density of the same quadrature provided by quantum noises .
both spectral densities are calculated for the case of absent modulation .

parameter @xmath23 stands for optical relaxation rate , coupling coefficient @xmath24 is proportional to pump power , @xmath25 is the coefficient of feedback

. the right parts of equations represent fluctuational forces acting on corresponding degree of freedom . as of optical one

the force @xmath26 represents input vacuum fluctuations and for mechanical one @xmath27 describes vacuum fluctuations reflected by the cavity and transmitted by feedback .

we neglect all other fluctuational forces including thermal and seismic ones .

depending on device in consideration this first step assumption could be either realistic or not .

some calculations of thermal noise influence is presented in appendix [ app : thermal ] . in particular

they show that in configuration of advanced ligo interferometer the impact of thermal noises is only slightly below the one of quantum fluctuations .

the equations look similar to ones for oscillator coupled to a free mass ( or two coupled oscillators one of which has partial frequency equal to zero ) .

however we would note the difference in signs in front of the coupling terms ( proportional to @xmath24 ) : in the case of conventional coupled oscillators these signs coincide ; in our case the signs differ .

this is the reason of instability which appears in absence of feedback and this is also the reason of possibility for eigen frequencies of such system to coincide ( so called double resonance case @xcite let us remind that in case of two coupled oscillators coincidence of eigen frequencies is impossible ) .

as in the case of conventional coupled oscillators one can treat the system in terms of eigen modes . the evolution of oscillator system could be written as a sum of eigen oscillations : @xmath28 index in the sum runs from 1 to the number of system degrees of freedom ( 2 for set ) .

this equation could be treated as transformation from conventional coordinates @xmath29 to the eigen ones . as in eqn .

one can present evolution of each of eigen modes as combination of oscillations at corresponding eigen frequencies @xmath30 and slow ( compared to these oscillations ) changing of amplitude . in case of free evolution ( zero right parts in eqns . )

quantities @xmath31 i.e. they freely decay with relaxation rates @xmath32 into the equilibrium values .

vectors @xmath33 represent distribution of amplitudes in corresponding eigen modes .

suppose that we have a possibility to modulate the power of pumping laser .

this will result in modulation of coupling coefficient and hence it will cause shifting of eigen frequencies of system .

it is well known @xcite that modulation of eigen frequency of oscillator with frequency twice bigger than its own one results in squeezing of noise in its quadratures . in this paper

we show that modulation of the pumping power performs squeezing of noise in quadratures of eigen modes amplitudes .

consider we harmonically modulate the pumping power at frequency @xmath34 so coupling coefficient in depends on time : @xmath35 .

assuming the quantities @xmath36 to be _ slow _ ( i.e. not to significantly change on times compared to mechanical periods ) one can plug the expression into system and to get rid of rapidly oscillating terms by averaging over a period @xmath37 .

this procedure results in shortened equations for amplitudes @xmath36 . for one with number @xmath38 ( @xmath39 )

the equation takes form : @xmath40 + |m| \sum_i g_i^\dagger ( \vec \pi_j \vec w_i^ * ) e^ { i ( \omega_i + \omega_j - 2 p ) t - i \phi } = \\ = ( \vec \pi_j { \underline{\nu}_j } ) e^{i \omega_j t } .

\end{gathered}\ ] ] here we have introduced a set of vectors @xmath41 built to be orthogonal to all @xmath42 except one : @xmath43 , where @xmath44 is a kronecker delta ( in the case of two conventional coupled oscillators vectors @xmath33 are itself orthogonal and system of @xmath45 is unnecessary ) .

vectors @xmath46 are defined as follows : @xmath47 underlining used for @xmath48 is to emphasise that this quantity is obtained in right part of equation by procedure of keeping only _ slow _ ( in comparison to terms oscillating with frequency @xmath49 ) quantities . to make @xmath50 fulfilling this criterion one should keep in @xmath51 only spectral components close to @xmath52 .

from now on we will focus on the amplitude of one of modes taking into account that all the results obtained for this mode are similar for other ones . for clarity let us set

@xmath53 thus considering the first mode .

also we suppose the modulation to happen at frequency twice bigger than the one of first mode : @xmath54 .

first let us consider the simplest case when difference between eigen frequencies @xmath55 is large in respect to decay rates @xmath32 . in this case the exponential multipliers in eqn .

should be considered as fast oscillating ones and the equations for amplitudes @xmath36 decouple . in this case

the equation for @xmath56 takes the following form in spectral domain @xmath57 g_1 ( x ) + \epsilon_{11 } g_1^\dagger ( -x ) = \frac

{ i ( \vec \pi_1 \vec \nu_1 ) e^ { i \omega_1 t } } { 2 \omega_1 } \equiv f_1 ( x).\ ] ] here @xmath58 stands for normalized spectral frequency ( see appendix [ app : notations ] ) , @xmath59 is the decay rate of first mode ; @xmath60 is a quantity proportional to modulation strength , the general expression for it could be easily deduced from eqn .

: @xmath61

we also write the equation for @xmath62 by hermitian conjugation of eqn .

[ eq : g1 ] and replacement @xmath63 .

taking proper combinations of these equations yields the expressions for corresponding quadratures of @xmath56 .

for simplicity let @xmath64 which is achievable by proper choosing @xmath65 , in this case the mentioned combinations reduce to sum or difference : @xmath66 = f_1 + f_1^\dagger ; \\

\label{eq : difq } \big(g_1 - g_1^\dagger \big ) \big [ \gamma_1 - |\epsilon_{11 } | - ix \big ] = f_1 - f_1^\dagger ; \end{aligned}\ ] ] one can conclude from these equations that sum quadrature is squeezed due to parametric modulation and the difference quadrature is antisqueezed .

the measure of squeezing is the spectral density which for arbitrary quantity @xmath67 is given by expression @xmath68 where the angle brackets mean averaging .

obvious calculations prove that spectral densities of right parts of both equations and coincide and the equations differ by additional damping @xmath69 ( positive or negative for different quadratures ) introduced by modulation . in absence of modulation @xmath70 and

there is no discrepancy between quadratures .

if modulation is applied then @xmath60 has nonzero value and quadratures spectral densities differ .

one can also estimate the critical level of modulation characterized by value @xmath71 , this is a level when negative damping added to the differential quadrature @xmath72 becomes equal to its own damping : @xmath73 thus making quadrature instable .

given this value of modulation coefficient damping in the sum quadrature is twice bigger than its own damping @xmath59 .

this means that quadrature noise squeezing is limited by factor of two .

in general case of arbitrary eigen frequencies @xmath74 the equations could not be solved in such an obvious way , but in this case one can rewrite these equations to switch from spectral components @xmath75 to the same components but shifted by frequency , namely @xmath76 .

the equations take form @xmath77 + \\ + \sum_i g_i^\dagger ( - x - ( \omega_i - \omega_1 ) ) \epsilon^*_{ji } = \frac { i } { 2 \omega_j } ( \vec \pi_j \vec { \underline{\nu}}_j ( p + x ) ) .\end{gathered}\ ] ] this set represents a system of linear algebraic equations and could be solved by applying kramer s rule . using the solution we obtain the expression for plus or minus quadratures of @xmath38-th mode @xmath78 defined as follows @xmath79 and calculate spectral densities of these quadratures . in coincidence with expectations in absence of parametric modulation both plus and minus quadratures of first mode

have equal spectral densities .

if modulation is applied the plus quadrature appears to be squeezed and the minus quadrature to be antisqueezed .

the bigger is modulation coefficient the more significant the ( anti- ) squeezing effect is .

it could be easily shown that squeezing is also limited by a factor of two in this case .

the important question is whether it is possible to see this squeezing in output light .

the answer to this question is positive .

output fluctuations @xmath16 are defined by input fluctuations @xmath15 and light inside cavity @xmath18 , in particular for phase quadrature of @xmath16 the following expression is valid : @xmath80 plugging the solution obtained in previous section for @xmath81 into the equation we can thus express the output fluctuations through quantities @xmath36 . it is easy to show ( see appendix [ app : readout ] ) that quantity @xmath82 is proportional to the quadrature @xmath83 . in time domain measurement of this quantity is equivalent to multiplication of measured phase quadrature with cosine of frequency @xmath52 with proper phase : @xmath84 setting @xmath85 or @xmath86 we obtain @xmath87 or @xmath88 .

estimations of @xmath89 spectral density demonstrate that it is possible to see squeezing in output light .

the spectral densities of these quantities are plotted in fig .

[ fig : output ] being normalized to the shot noise level .

the limitation of squeezing by factor of two shows up in output light too . in output field

we measure phase quadrature @xmath151 which is proportional to @xmath19 that carries information about squeezing .

further we discuss which combination of spectral components of @xmath19 contains squeezing evidence taking into account that corresponding combination of @xmath151 components represents the same combination with some input fluctuations @xmath152 added . for spectral components of @xmath19 shifted in frequency domain by amount of @xmath52 one can write the following expression using eqn .

: @xmath153 ) + g_i^\dagger ( - x - [ \omega_i + \omega_j ] ) ) .\ ] ] here we use freedom in definition of @xmath33 and let @xmath154 . an equation similar to one above could be written for @xmath155 : @xmath156 ) + g_i ( x - [ \omega_i + \omega_j ] ) \big).\ ] ] we now take sum or difference of these quantities and obtain expression containing sum of quadratures @xmath78 that are of our interest @xmath157 = \\ = \sum_i v_i \big [ g_i^{(\pm ) } ( x - ( \omega_i - \omega_j ) ) + g_i^{(\pm ) } ( x - ( \omega_i + \omega_j ) ) \big ] .

\end{gathered}\ ] ] remind that we assumed @xmath36 to be slow amplitudes which means that spectral components of these quantities are situated close to zero frequency . note that due to resonant multiplyers ( expressions in square brackets ) in eqn .

one can estimate the width of band containing @xmath36 spectra with order of corresponding mode decay rate @xmath32 which is much smaller than difference between eigen frequencies @xmath158 ( for @xmath159 ) .

this consideration is also valid for quadratures @xmath160 hence in definition of all summands differ from zero in different frequency bands ( not overlapping because of narrowness ) , or in other words in each frequency band there is not more than one non - zero summand in this definition . in particular at frequencies close to zero one can use quite exact expression @xmath161 as we actually measure @xmath17 we need to consider a combination of spectral components of this quantity containing @xmath162 .

it is obvious that corresponding combination is given by the following expression @xmath163 which represents a sum of desired quadrature and input fluctuations .

in this paper we have shown the possibility of quadrature noise squeezing of eigen modes amplitudes in an optomechanical system . as a model of latter

we have used a fabry - perot cavity with movable mirror and a detuned pump .

the eigen modes in this system are defined by interaction between optical and mechanical degrees of freedom .

detuned pump transforms the mirror which is initially a free - mass into a harmonic oscillator with spring coefficient provided by means of optical rigidity . in comparison with material rigidity peculiar to usual optomechanical devices such as microtoroids or membranes etc @xcite optical rigidity

is characterized by very low noises level .

the another advantage of using optical springs is the very ability of spring constant manipulation by means of pump power modulation .

comparison of impact exerted by quantum and thermal noises upon quadratures ( see appendix [ app : thermal ] ) reveals that both fluctuations have their affects of same order .

this gives us an ability to talk about quantum noise squeezing of optomechanic quadratures in gravitational wave detectors ( advanced ligo @xcite , einstein telescope @xcite ) and in prototypes ( glasgow university prototype @xcite , aei hannover prototype @xcite and gingin facility @xcite ) despite the fact that we consider a feedback in our model , the feedback itself is unnecessary to achieve the squeezing as it serves only to stabilize the eigen frequencies of the system .

the stabilization could be performed by means of an auxiliary pump with power significantly lower than one in main pump . however

implementation of feedback is more feasible in small - scale experimental setups like gravitational - wave antenna prototypes @xcite . realization of proposed scheme in experiment could demonstrate non - classical behavior of optomechanical mode which could be useful for tests of quantum mechanics applied to macroscopic mechanical objects .

we would like to underline it is not pure optical or pure mechanical , but an _ optomechanical _ degree of freedom that may exhibit quantum behavior .

also a quantum state if created in an eigen mode could be transmitted into another one ( for example , by modulation of pump with difference frequency @xmath90 ) .

this gives us a potential playground for creation and transmission of quantum states between optomechanical modes which could be useful for problems of quantum information .

the authors would thank stefan hild for stimulating discussions .

we also are grateful to yanbei chen , farid khalili and stefan danilishin .

authors are supported by the russian foundation for basic research grant no .

08 - 02 - 00580-a and nsf grant phy-0967049 .

the hamiltonian of considered system ( without feedback ) could be written as follows @xcite : @xmath91 here @xmath92 ( @xmath93 ) is annihilation ( creation ) operator of optical mode , @xmath20 ( @xmath49 ) is position ( momentum ) operator of movable mirror .

also @xmath94 is the frequency of cavity fundamental mode closest to the pump frequency @xmath95 .

the difference between these frequencies is denoted as @xmath13 , please note the sign of this detuning : for the pump tuned onto the right slope of optical resonance curve ( blue detuned pump ) @xmath13 is _

positive_. @xmath96 is the coupling coefficient between light mode and mirror motion .

@xmath97 is the hamiltonian describing interaction of the system with bath and the bath itself .

@xmath98 is the same for pump .

if we then suppose the optical operators to consist of strong classical part @xmath99 ( which is real positive number characterizing mean optical power @xmath14 inside cavity : @xmath100 ) and quantum fluctuations @xmath18 so that @xmath101 and switch to the frame rotating with frequency @xmath95 , then the linearized hamiltonian should take the form @xmath102 one then should write down heisenberg equations for position and momentum of mirror and amplitude ( @xmath19 ) and phase ( @xmath103 ) operators of light defined in following way : @xmath104/ \sqrt 2 ; \qquad b_2 \equiv - i [ b - b^+ ] / \sqrt 2.\ ] ] in consideration above we have not given concrete expressions to the bath and pump hamiltonians so in order to perform inclusion of damping , feedback and optical fluctuational forces in a correct way we use semiclassical approach writing down the equations of motion in spectral form @xcite : @xmath105 +

y \big [ \sqrt { \frac { p \omega_0 } { \hbar l^2 } } \delta \big ] = \\ & = - \sqrt { \frac { \gamma } { \tau } } \left [ a_1^{i } ( \gamma - i \omega ) + a_2^{i } \delta \right ] ; \\ - \omega^2 y - \frac { 2 } { \mu c } & \sqrt { p \hbar \omega_0 } b_1 = i \omega \alpha_\text{fb } ( - a_2^{i } - 2 \sqrt { \gamma \tau } b_1 ) .\end{aligned}\ ] ] here @xmath106 is spectral frequency , @xmath107 is the time it takes light to travel between mirrors ( one way ) , @xmath108 is a coefficient of feedback , i.e. the force fed to the mirror is equal to @xmath109 .

the dimensionless parameters are defined as follows .

@xmath110 switching to dimensionless parameters from dimensional ones using the definitions listed above yields the system @xmath111 , \\ - a b_1 + & i x \alpha b_1 - x^2 z = \nu_2 \equiv \frac { i x \alpha } { 2 \sqrt { \gamma \tau } } a_2^i . \end{aligned}\ ] ] finally by transferring from frequency domain to time one following rule @xmath112 one can write down the system

in order to account thermal fluctuations in the model one should include corresponding fluctuational force in the right part of the equation of motion for mechanical degree of freedom .

this results in formal replacement @xmath114 here @xmath115 is thermal fluctuational force . if we consider coating brownian noise as the source of this force ( in advanced ligo coating fluctuations are dominant thermal noises ) , then its spectral density is given by expression @xcite : @xmath116 , \\

\phi_\bot & = \frac{y}{\sqrt \pi w ( 1-\sigma^2)}\left [ \frac{(1+\sigma_1)(1 - 2\sigma_1)d_1\phi_1}{y_1(1-\sigma_1 ) } + \right .

\nonumber \\ \notag & \qquad + \left .

\frac{(1+\sigma_2)(1 - 2\sigma_2)d_2\phi_2}{y_2(1-\sigma_2)}\right].\end{aligned}\ ] ] here @xmath117 is spectral frequency ( measured in hz ) , @xmath118 is a radius of a beam spot on mirror s surface , @xmath119 are young modulus and poisson ratio of substrate respectively , @xmath120 are the same quantities for alternating layers , @xmath121 are total thicknesses of quater wavelength layers , n is the number of layers pairs , @xmath122 optical wavelength , @xmath123 are refraction indices of layers .

@xmath9 is the temperature of mirrors coating , @xmath124 is bolzmann s constant . the numerical parameters that we use for estimations are listed in table [ param ] .

+ @xmath9 , k & + @xmath122 , m & + @xmath126 & - & @xmath127 & @xmath127 + @xmath128 & 1.45 & 2.035 & 1.45 + @xmath129 , pa & @xmath130 & @xmath131 & @xmath130 + @xmath132 & @xmath133 & @xmath134 & @xmath133 + @xmath65 & @xmath135 & @xmath136 & @xmath137 + as the thermal noise is not correlated to vacuum noises , inclusion of the last term in equation reveals in estimations of spectral densities of quadratures only by additional term in expression for spectral density of @xmath27 : @xmath138 the convenient factor of thermal noise influence on eigen mode amplitude spectral density is the following ratio : @xmath139 here @xmath140 is the spectral density of output field quadrature @xmath141 in case of only thermal noises present and @xmath142 is the spectral density of the same quadrature provided by quantum noises .

both spectral densities are calculated for the case of absent modulation .