document: in a recent paper li _ et al . _ 
   @xcite presented a new design for an optomechanical system that consists of a microdisk resonator coupled to a waveguide . 
   this design has several attractive features . 
   besides its universality , it enables one to study the reactive effects @xcite in optomechanical coupling . 
   the origin of the reactive coupling is well explained in ref . 
   its origin lies in the mechanical motion dependence of the extrinsic losses of the disk resonator . 
   further phase - dependent gradient forces lead to reactive coupling . 
   have also argued that this design is more effective in achieving cooling of the system to its ground state . 
   while cooling is desirable for studying quantum effects at the macroscopic scale @xcite , we examine other possibilities , which do not depend on the cooling of the system , to investigate the effects arising from strong reactive coupling . 
   since optomechanical coupling effects are intrinsically nonlinear , we examine the nonlinear response of the microdisk resonator to pump probe fields . 
   we report reactive - coupling - induced normal mode splitting . 
   note that in previous works @xcite on normal mode splitting in optomechanical devices , only dispersive coupling was used . in this paper 
   , we report on normal mode splitting due to reactive effects . 
   the paper is organized as follows . in sec . 
   ii , the physical system is introduced and the time evolutions of the expectation values of the system operators are given and solved . in sec . 
   iii , the expectation value of the output fields is calculated , and the nonlinear susceptibilities for stokes and anti - stokes processes are obtained . in sec . 
   iv , we discuss normal mode splitting in output fields with or without reactive coupling . 
   we find that there is no normal mode splitting in output fields in the absence of reactive coupling . 
   however , normal mode splitting occurs in output fields in the presence of reactive coupling . 
   we consider the system shown in fig . 
   [ fig1 ] , in which a microdisk cavity is coupled to a freestanding waveguide . a strong pump field with frequency @xmath0 and a weak stokes field with frequency @xmath1 enter the system through the waveguide . 
   the waveguide will move along the @xmath2 direction under the action of the optical force exerted by the photons from the cavity . 
   further , considering the dispersive coupling and reactive coupling between the waveguide and the cavity , displacement @xmath3 of the waveguide from its equilibrium position will change the resonant frequency of the cavity field and the cavity decay rate , represented by @xmath4 and @xmath5 , respectively .    in a rotating frame at pump frequency @xmath0 , 
   the hamiltonian of the system is given by @xcite @xmath6c^{\dag}c+\frac{p^2}{2m}+\frac{1}{2}m\omega_{m}^2q^2\vspace{0.1in}\\\hspace{0.3in}+\hbar\frac { l}{c}\tilde{n}_{g}(\omega_{l}\varepsilon_{l}^2+\omega_{s}|\varepsilon_{s}|^2)+i\hbar\sqrt{2\kappa_{e}(q)}\varepsilon_{l}(c^{\dag}-c)\vspace{0.1in}\\\hspace{0.3 in } + i\hbar\sqrt{2\kappa_{e}(q)}(\varepsilon_{s}e^{-i\delta t}c^{\dag}-\varepsilon^{*}_{s}e^{i\delta t}c ) . 
   \end{array}\ ] ] the first term is the energy of the cavity field , whose annihilation ( creation ) operators are denoted @xmath7 . 
   the second and third terms are the energy of the waveguide with mass @xmath8 , frequency @xmath9 , and momentum operator @xmath10 . 
   the fourth term gives the interactions between the waveguide and the incident fields ( the pump field and the stokes field ) , @xmath11 is the length of the waveguide , @xmath12 is the speed of light in vacuum , @xmath13 is the group index of the waveguide optical mode @xcite , @xmath14 and @xmath15 are the amplitudes of the pump field and the stokes field , respectively , and they are related to their corresponding power @xmath16 and @xmath17 by @xmath18 and @xmath19 . 
   the latter two terms describe the coupling of the cavity field to the pump field and the stokes field , respectively . 
   and @xmath20 is the detuning between the stokes field and the pump field . 
   we would study the physical effects by scanning the stokes laser .    for a small displacement @xmath3 , @xmath4 and @xmath5 
   can be expanded to the first order of @xmath3 , @xmath21 thus the quantities @xmath22 and @xmath23 describe the cavity - waveguide dispersive and reactive coupling strength , respectively . 
   further , note that the photons in the cavity can leak out of the cavity by an intrinsic damping rate @xmath24 of the cavity and by a rate of @xmath5 due to the reactive coupling between the waveguide and the cavity . 
   in addition , the velocity of the waveguide is damped at a rate of @xmath25 . applying the heisenberg equation of motion and adding the damping terms , the time evolutions of the expectation values ( @xmath26 , and @xmath27 ) for the system 
   can be expressed as    @xmath28-\gamma_{m}\langle p\rangle,\vspace{0.2in}\\ 
   \displaystyle\langle\dot{c}\rangle=-[\kappa+\langle q\rangle\kappa_{om}+i(\omega_{c}-\omega_{l}+\langle q\rangle\chi)]\langle c\rangle\vspace{0.2in}\\\hspace{0.4in}\displaystyle+\sqrt{\kappa}[1+\langle q\rangle\frac{\kappa_{om}}{\kappa}](\varepsilon_{l}+\varepsilon_{s}e^{-i\delta t } ) , \end{array}\ ] ]    where we have used the mean field assumption @xmath29 , expanded @xmath5 to the first order of @xmath3 , and assumed @xmath30 , where @xmath31 is the half - linewidth of the cavity field . 
   it should be noted that the steady - state solution of eq . 
   ( [ 3 ] ) contains an infinite number of frequencies . since the stokes field @xmath32 is much weaker than the pump field @xmath14 , the steady - state solution of eq . 
   ( [ 3 ] ) can be simplified to first order in @xmath32 only . 
   we find that in the limit @xmath33 , each @xmath34,@xmath35 , and @xmath27 has the form @xmath36 where @xmath37 stands for any of the three quantities @xmath3 , @xmath10 , and @xmath12 . 
   thus the expectation values @xmath38 , and @xmath27 ) oscillate at three frequencies ( @xmath0 , @xmath1 , and 2@xmath39 ) . substituting eq . 
   ( [ 4 ] ) into eq . ( [ 3 ] ) , ignoring those terms containing the small quantities @xmath40 , @xmath41 , @xmath42 , and equating coefficients of terms with the same frequency , respectively , we obtain the following results @xmath43,\vspace{.1in}\\ c_{+}=\displaystyle \frac{1}{d(\delta)}[a(be+fj)-i\hbar\frac{\kappa_{om}}{\sqrt{\kappa}}c_{0}^{*}bf^{*}],\vspace{.1in}\\ c_{-}=\displaystyle \frac{f^{*}}{d^{*}(\delta)}(-aj+i\hbar\frac{\kappa_{om}}{\sqrt{\kappa}}c_{0}v),\vspace{0.1in}\\ q_{+}=\displaystyle \frac{b}{d(\delta)}(-aj^{*}-i\hbar\frac{\kappa_{om}}{\sqrt{\kappa}}c_{0}^{*}v^{*}),\vspace{0.1in}\\ q_{-}=(q_{+})^ { * } , \end{array}\ ] ] where @xmath44 @xmath45 and @xmath46 , @xmath47 , @xmath48 , @xmath49 , @xmath50 , @xmath51 . 
   the approach used in this paper is similar to our earlier work @xcite which dealt with optomechanical systems with dispersive coupling only . 
   to investigate the normal mode splitting of the output fields , we need to calculate their expectation value . 
   it can be obtained by using the input - output relation @xcite @xmath52 . if we write @xmath53 as @xmath54 where @xmath55 is the response at the pump frequency @xmath0 
   , @xmath56 is the response at the stokes frequency @xmath1 , and @xmath57 is the field generated at the new anti - stokes frequency @xmath58 . 
   then we have @xmath59 furthermore , whether there is normal mode splitting in the output fields is determined by the roots of the denominator @xmath60 of @xmath56 . here 
   we examine the roots of @xmath60 given by eq . 
   ( [ 7 ] ) numerically . 
   the response of the system is expected to be especially significant if we choose @xmath1 corresponding to a sideband @xmath61 or @xmath62 , so we consider the case @xmath63 . 
   the other parameters are chosen from a recent experiment focusing on the effect of the reactive force on the waveguide @xcite : the wavelength of the laser @xmath64 nm , @xmath65 mhz / nm , @xmath66 pg ( density of the silicon waveguide , 2.33 g/@xmath67 ; length , 10 @xmath68 m ; width , 300 nm ; height , 300 nm ) , @xmath69 , @xmath70 mhz , and the mechanical quality factor @xmath71 . in the following , we work in the stable regime of the system . 
   figure  [ fig2 ] shows the variation of the real parts of the roots of @xmath60 in the domain re@xmath72 with increasing pump power for no reactive coupling , @xmath73 , and for @xmath74 mhz / nm . for @xmath73 , 
   the interaction of the waveguide with the cavity is purely dispersive ; the cavity decay rate does not depend on the displacement of the waveguide . in this case 
   , the real parts of the roots of @xmath60 always have two equal values with increasing pump power . 
   thus there is no splitting because the dispersive coupling is not strong enough . 
   however , for @xmath74 mhz / nm , the system has both dispersive and reactive couplings , the cavity decay rate depends on the displacement of the waveguide , and the real parts of the roots of @xmath60 will change from two equal values to two different values with increasing pump power . and the difference between two real parts of the roots of @xmath60 in the domain re@xmath72 is increased with increasing pump power . 
   therefore , the reactive coupling between the waveguide and the cavity can result in normal mode splitting of the output fields , and the peak separation becomes larger with increasing pump power . 
   figure  [ fig3 ] shows the variation of the imaginary parts of the roots of @xmath60 with increasing pump power for zero reactive coupling @xmath73 and nonzero reactive coupling @xmath74 mhz / nm . for @xmath73 , 
   the imaginary parts of the roots of @xmath60 do not change with increasing pump power . 
   however , for @xmath74 mhz , the imaginary parts of the roots of @xmath60 change with increasing pump power . we thus conclude that for the present microdisk resonator coupled to a waveguide the normal mode splitting is solely due to the reactive coupling . 
   we now discuss how the output fields depend on the behavior of the roots of @xmath60 . for convenience , 
   we normalize all quantities to the input stokes power @xmath17 . assuming that @xmath32 is real , we express the output power at the stokes frequency @xmath1 in terms of the input stokes power @xmath75 further 
   , we introduce the two quadratures of the stokes component of the output fields by @xmath76 and @xmath77 . 
   one can measure either the quadratures of the output by homodyne techniques or the intensity of the output . for brevity , 
   we only show @xmath78 and @xmath79 as a function of the normalized detuning between the stokes field and the pump field @xmath80 for this model , without reactive coupling ( @xmath23=0 ) and with it ( @xmath74 mhz / nm ) , for different pump powers in figs . 
   [ fig4][fig5 ] . for @xmath23=0 
   , it is found that @xmath78 has a lorentzian lineshape corresponding to the absorptive behavior . 
   note that @xmath78 and @xmath79 exhibit no splitting when @xmath23=0 . 
   however , for @xmath74 mhz / nm , it is clearly seen that normal mode splitting appears in @xmath78 and @xmath79 . 
   therefore reactive coupling can lead to the appearance of normal mode splitting in the output stokes field . 
   and the peak separation increases with increasing pump power @xcite . 
   the dip at the line center exhibits power broadening . 
   we also find that the stokes field can be amplified by the stimulated process . 
   obviously the maximum gain @xmath79 for the stokes field depends on the system parameters . for a pump power @xmath81 @xmath68w , 
   the maximum gain for the stokes field is about 1.3 . 
   note that the nonlinear nature of the reactive coupling generates anti - stokes radiation . in a similar way 
   , we define a normalized output power at the anti - stokes frequency @xmath58 as @xmath82 the plots of @xmath83 versus the normalized detuning between the stokes field and the pump field @xmath80 for this model , without reactive coupling ( @xmath23=0 ) and with it ( @xmath74 mhz / nm ) , for different pump powers are presented in fig . 
   we can see that @xmath84 for @xmath23=0 . 
   the reason is that the dispersive coupling constant @xmath22 is too small . 
   however , for @xmath74 mhz / nm , @xmath83 is not equal to zero . 
   this shows that the optomechanical system can generate an anti - stokes field with frequency @xmath85 due to the reactive coupling . for pump power @xmath81 @xmath68w , 
   the maximum gain defined with reference to the input stokes power for the anti - stokes field is about 0.1 . 
   in conclusion , we have observed normal mode splitting of output fields due to reactive coupling between the waveguide and the cavity . 
   meanwhile , the separation of the peaks increases for larger pump powers . 
   further , the reactive coupling can also cause four - wave mixing , which creates an anti - stokes component generated by the optomechanical system . 
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summary: we study the optomechanical design introduced by m. li _ 
   et al . _ 
   [ phys . 
   rev . lett . * 103 * , 223901 ( 2009 ) ] , which is very effective for investigations of the effects of reactive coupling . 
   we show the normal mode splitting that is due solely to reactive coupling rather than due to dispersive coupling . 
   we suggest feeding the waveguide with a pump field along with a probe field and scanning the output probe for evidence of reactive - coupling - induced normal mode splitting .