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recently , experimental and theoretical investigations have been carried out to demonstrate the coherent optomechanical coupling between a quantized cavity field and the mechanical motion of a mirror located at one end of an optical cavity ( e.g. , see refs .
meystre , optomechanical-1,optomechanical-2,optomechanical-3 ) .
many efforts have been made to propose new devices and explore the quantum - classical transition ( e.g. , see refs .
@xcite ) based on such optomechanical systems .
it was also discovered that such systems possess nonlinear optical properties .
our study here focuses on another aspect of optomechanical coupling that can realize the kerr nonlinear optical effect in a vacuum , rather than in a conventional dispersive medium  kerr-2,kerr-3 .
the kerr effect usually appears in nonlinear dispersive media  nonlinear , due to the third - order matter - light interaction . instead of resulting from higher - order light - matter interactions , we now realize _ the nonlinear kerr effect _ from the radiation pressure on an oscillating mirror .
this oscillating mirror , located at one end of a fabry - prot ( fp ) cavity , is driven by an input laser field at the other end .
the oscillating mirror is modeled as a quantum mechanical resonator , whose tiny oscillations are controlled by the radiation pressure of the cavity field .
indeed , it has been pointed out  @xcite that the vacuum cavity with a movable mirror might mimic a kerr medium when the cavity field is driven by a coherent light field .    in this article
, we use the born - oppenheimer ( bo ) approximation to derive a kerr - medium - like hamiltonian , which shows the underlying nonlinear mechanism more clearly than other ( equivalent ) approaches .
we shall first point out first that , in the conventional bo approximation for a molecule the ( slower ) nuclear variables are adiabatically separated from the ( faster ) electronic variables .
the stability of the molecular configuration requires the effective potential to have a minimum value .
however , the generalized bo theory  @xcite for spin - orbit systems or cavity qed systems  @xcite does _ not _ have this requirement , and the effective force could be either attractive or repulsive .
as shown in fig .  [ fig : fig1](a ) , we study a fabry - prot ( fp ) cavity with an oscillating mirror at one end , acting as a quantum - mechanical harmonic oscillator . the cavity is driven by a laser field with frequency @xmath0 .
we first study an ideal case , without considering the losses of both the cavity field and the oscillating mirror .
the total hamiltonian @xmath1contains three parts as    @xmath2    the cavity field with frequency @xmath3 is described by bosonic operators @xmath4 and @xmath5 . the symbols @xmath6 , @xmath7 , @xmath8 , and @xmath9 denote , respectively , the mass , frequency , displacement , and momentum of the oscillating mirror ( hereafter , we just call it the mirror " ) .
the coupling strength @xmath10 between the cavity field and the driving laser field is related to the input laser power @xmath11 and the decay rate @xmath12 of the cavity field via the relation @xmath13 .
the interaction constant @xmath14 between the cavity field and the mirror stems from a very small change @xmath8 of the fp cavity length @xmath15 .     around their equilibrium positions.,width=307 ]    usually , the characteristic frequency of the cavity field is about  @xmath16 hz , which is much higher than the nanomechanical resonator frequency  @xmath17 hz achieved by current experiments .
the cavity field would have _ no _ nonlinear effect under the bo approximation when there is no driving field ; nonlinear effects appear when a classical driving field is applied to the cavity .
let us consider the case when the driving field frequency @xmath18 is close to the cavity field frequency @xmath3 .
we also use the rotating frame of referencedefined by a unitary transformation @xmath19 , which is very similar to the nmr experiments used to demonstrate the berry phase  @xcite . in the rotating frame of reference , the effective form @xmath20 of the hamiltonian @xmath21 in eq .
( [ 2 - 1 ] ) reads    @xmath22    with the effective hamiltonians @xmath23and @xmath24here , the detuning @xmath25 is the effective frequency of the cavity field in the new frame .
clearly , @xmath26 can be controlled by tuning the frequency @xmath27 of the driving field .
therefore , the effective frequency @xmath26 of the cavity field can be tuned to be much smaller than that of the mechanical resonator . under such condition ,
the mechanical resonator can be treated as the fast variable and the bo approximation can be employed .
we first study the hamiltonian  ( [ 2 - 5 ] ) of the fast variables @xmath8 and @xmath9 of the mirror ( in the rotating frame ) by taking the slow variables  @xmath4 and @xmath5 of the cavity field as constants ( in the rotating frame ) .
then the hamiltonian  ( [ 2 - 5 ] ) can be rewritten as @xmath28where the creation operator @xmath29 of the cavity field is defined by @xmath30with @xmath31 .
( [ 2 - 6 ] ) shows that the mirror variables are shifted by the amount @xmath32 due to its interaction with the cavity field .
it is clear that the ground state of the effective hamiltonian in eq .
( [ 2 - 6 ] ) can be obtained via the ground state of the hamiltonian @xmath33 in eq .
( [ 2 - 2 - 1 ] ) for a harmonic oscillator with displacement operator @xmath34the eigenvectors and the eigenvalues of the mirror , corresponding to the hamiltonian  ( [ 2 - 6 ] ) , are , respectively , @xmath35and @xmath36eqs .
( [ 2 - 9 ] ) and ( [ 2 - 10 ] ) show that @xmath37 and @xmath38 are functions of the slow variables of the cavity field .
( [ 2 - 9 ] ) also shows that the ground state of the fast variables ( @xmath8 and @xmath9 ) of the mirror is a coherent state @xmath39 , resulting from radiation pressure .    according to the lowest - order generalized bo approximation
, the total eigenfunction @xmath40 of the coupled system of the cavity field and the mirror can be factorized as @xmath41 , where @xmath42 satisfies the schrdinger equation with the effective hamiltonian @xmath43the bo adiabatic separation provides an effective potential @xmath44 for the slow  motion of the cavity field .
this potential contains a typical @xmath45where the parameter @xmath46plays the role of _ the phenomenological third - order susceptibility _ as in usual kerr media .
versus normalized detuning @xmath47 for the value @xmath48 .
analytical results for the full hamiltonian are shown with a red dashed curve , while the bo approximation results are shown with a blue solid curve .
note that squeezing occurs when @xmath49 , and the baseline @xmath50 is shown with a horizontal black dot - dashed line.,width=307 ]    we now verify the validity of the generalized bo approximation applied to the optomechanical system through the squeezing effect , which is induced by the kerr interaction .
this squeezing effect can be demonstrated by the output intensity spectrum @xmath51 . following the definition of the @xmath51 in ref .
@xcite and the linearization technique of the langevin equations governed by the hamiltonian in eq .
( [ 2 - 11 ] ) , we obtain the intensity spectrum under the bo approximation @xmath52where    @xmath53    with renormalized detuning @xmath54 and the steady - state value @xmath55 of the amplitude of the cavity field . here , the cavity loss @xmath12 has been taken into account .
the bo approximation requires that the photons in the cavity can survive a sufficiently long time , which is equivalent to the condition @xmath56    if there is no kerr interaction induced by the radiation pressure ( @xmath57 ) , @xmath58 can be simplified to @xmath59 .
when the kerr interaction is induced ( @xmath60 ) , the intensity spectrum can be less than @xmath61 , which displays a squeezing effect  @xcite of the cavity field .
here , @xmath62 plays an important role in the reduction of the output intensity fluctuation .
hence , the squeezing effect could be observed experimentally by measuring the intensity spectrum of the output laser
.    fig .
[ fig : fig2 ] shows the squeezing of the cavity field in the intensity spectrum with a blue solid curve , where the maximum squeezing occurs in the vicinity of @xmath63 .
we consider that the oscillating mirror has mass @xmath64 ng , frequency @xmath65 , and a normalized damping rate @xmath66 .
[ fig : fig2 ] is plotted for the driving field frequency @xmath67 thz , the optical cavity length @xmath68 m , finesse @xmath69 , and decay rate @xmath70 s@xmath71 ; a driving laser wavelength @xmath72 nm , normalized frequency @xmath73 and power @xmath74 @xmath75 .
the temperature @xmath76 of the cavity field and the mechanical resonator is assumed to be zero , as in ref .
@xcite . then at a particular frequency @xmath77 ( chosen at the position where @xmath58 is minimum ) , the intensity spectrum has the squeezing effect shown in fig .
[ fig : fig2 ] .
now , we also study the intensity spectrum without the bo approximation .
starting from the total hamiltonian in eq .
( [ 2 - 3 ] ) , a full derivation shows that the intensity spectrum has a similar form as in eqs .
( 15 ) except that the renormalized detuning @xmath78 and the kerr interaction strength @xmath79 are replaced , respectively , by @xmath80 and @xmath81 here , @xmath82 is the steady - state value of the oscillation amplitude of the mirror and    @xmath83    is the mechanical susceptibility of the oscillating mirror . the maximum squeezing effect in the intensity spectrum , plotted by the red dashed curve in fig .  [ fig : fig2 ] ,
can also be observed in the vicinity of @xmath84 .
we use the same parameters for calculating the intensity spectrum under the bo approximation .
therefore , the bo approximation is valid when the frequency of the oscillating mirror @xmath85 is much larger than the frequency of the light field @xmath86    the effect of the bo approximation can also be understood via @xmath87 .
if the mirror frequency @xmath7 is much larger than the detuning @xmath88 of the cavity field in the rotating frame of reference , @xmath89 plays an important role in the vicinity of @xmath90 .
when the bo approximation is valid under the condition @xmath91 , and when the macroscopic displacement @xmath92 is extremely small , the mechanical susceptibility is approximately equal to @xmath93@xmath94 which leads the intensity spectrum to have the same form as in eq .
( 3 - 1 ) .
within the bo approximation , the cavity field inside the flexible fabry - prot cavity , driven by an input laser , exhibits a kerr - like nonlinear property .
it is not difficult to generate a two - mode induced kerr interaction , which is useful for quantum nondemolition ( qnd ) measurements  @xcite .    as shown in fig .
[ fig : fig1](b ) , we consider two fp cavities , referred to below as the left and the right cavities with subindexes @xmath15 and @xmath95 , sharing one common oscillating mirror . this mirror is assumed to oscillate with a very small displacement @xmath8 around its equilibrium position .
thus two cavity fields , with frequencies @xmath96 and @xmath97 in the laboratory reference frame , indirectly interact with each other via this oscillating mirror .    in the rotating frame of reference , using the bo approximation discussed above , we can derive the induced effective interaction between the two cavity fields @xmath98where the @xmath99 and @xmath100 with @xmath101 denote the photon number operator of the left and the right cavity fields . to perform a qnd measurement , the self - modulation term @xmath102 of the probe field
can be ignored  @xcite .
therefore , the system hamiltonian for the qnd measurement can be written as @xmath103where @xmath104 and @xmath105 are the detunings of the left and right cavity fields , relevant to the frequency @xmath0 of the driving field .
the kerr interaction in eq .
( [ 4 - 2 ] ) satisfies the qnd measurement conditions  @xcite .
since this two - mode kerr interaction commutes with the free hamiltonians of both cavities , we can nondestructively measure the photon number by observing the other cavity s conjugate observable .
we have studied the kerr nonlinearity in the vacuum induced by the radiation pressure in typical optomechanical systems .
such nonlinear interaction can be explicitly obtained via a bo approximation for an ideal case . through a squeezing effect , which should be experimentally observable via the intensity spectrum
, we verify the validity of the generalized bo approximation by taking into account the dissipation and fluctuation of the cavity field .
furthermore , we propose a two - mode kerr interaction between two cavity fields for quantum nondemolition measurements , which can be realized by using two cavities sharing one oscillating mirror .
it is possible to nondestructively measure a cavity photon number by observing another cavity s conjugate observable .
we are grateful to dr .
lan zhou for helpful discussions .
sun was supported by the nsfc with grant nos.90503003 , nfrpc with grant nos .
2006cb921205 and 2005cb724508 .
fn was supported in part by the nsa , lps , aro , and nsf grant no .
using the born - oppenheimer approximation , we derive an effective hamiltonian for an optomechanical system that leads to a nonlinear kerr effect in the system s vacuum . 
 the oscillating mirror at one edge of the optomechanical system induces a squeezing effect in the intensity spectrum of the cavity field . 
 a near - resonant laser field is applied at the other edge to drive the cavity field , in order to enhance the kerr effect . 
 we also propose a quantum - nondemolition - measurement setup to monitor a system with two cavities separated by a common oscillating mirror , based on our effective hamiltonian approach .