Double Electromagnetically Induced Transparency and Narrowing of Probe Absorption in a Ring Cavity with Nanomechanical Mirrors
aa r X i v : . [ qu a n t - ph ] D ec Double Electromagnetically Induced Transparency and Narrowing of ProbeAbsorption in a Ring Cavity with Nanomechanical Mirrors
Sumei Huang , Department of Physics, University of California, Merced, California 95343, USA Department of Electrical and Computer Engineering,National University of Singapore, 4 Engineering Drive 3, Singapore 117583 (Dated: August 22, 2018)We study the effect of a strong coupling field on the absorptive property of a ring cavity withtwo mirrors oscillating at slightly different frequencies to a weak probe field. We observe doubleelectromagnetically induced transparency windows separated by an absorption peak at line centerin the output probe field under the action of a strong coupling field. We find that increasingdriving power can broaden the two transparency windows, which results in narrowing of the centralabsorption peak. At high driving power, the linewidth of the sharp central absorption peak isapproximately equal to the mechanical linewidth. We show the normal mode splitting in both theoutput probe field and the generated Stokes field. We also find that the suppression of the four-wavemixing process can be achieved on resonance.
I. INTRODUCTION
It is well-known that a Λ-type three-level atomicmedium can become transparent to a weak probe field byapplying a strong coupling field, which is the result of thedestructive interference between two different excitationpathways to the upper level. This is the phenomenonof electromagnetically induced transparency (EIT) [1–4]. The EIT has been shown to be important for var-ious applications such as slow light [5], light storage [6],and so on. Besides, the studies of EIT have been ex-tended to multi-level atomic systems. The double EITwindows separated by a narrow absorption peak in theprobe absorption spectrum have been observed in thefour-level atomic systems [7–11]. Recently, the EIT ef-fect has been reported in the macroscopic optomechani-cal systems. The analogy of EIT in optomechanical sys-tems has been shown theoretically [12], and observed ina number of experiments in optical cavities [13–15] andmicrowave cavities [16, 17]. Moreover EIT in optome-chanical systems in the nonlinear regime was analyzed[18–20]. Additionally, the electromagnetically inducedabsorption, the opposite effect to EIT, was discussed ina two-cavity optomechanical system [21]. In addition,it has been proven that a strong dispersive coupling be-tween the optical mode and the mechanical mode whenthe effective optomechanical coupling rate exceeds theoptical and mechanical decay rates leads to normal modesplitting [22–24]. The effective optomechanical couplingrate can be enhanced by increasing the power of the driv-ing laser.In this paper, we investigate the nonlinear response ofa ring cavity with two moving mirrors having two closefrequencies to a weak probe field in the absence and thepresence of a strong coupling field. We find that thereare two transparency windows and an absorption peak inthe transmitted probe field in the presence of the couplingfield. The pump-induced broadening of the two EIT dipsleads to narrowing the central absorption peak. And thenarrow absorption peak associated with double EIT win- dows may have potential application in high-resolutionlaser spectroscopy [7–9]. We also observe the normalmode splitting in the output probe field at high driv-ing power. In addition, we show that the normal modesplitting occurs in the Stokes field generated by means ofthe four-wave mixing process. And the four-wave mixingprocess is completely suppressed on resonance. However,if two movable mirrors in a ring cavity are oscillating atidentical frequencies, the EIT-like dip can be observed inthe output probe field, and the four-wave mixing processis not suppressed on resonance.The paper is organized as follows. In Sec. II, we intro-duce the system, give the time evolutions of the expecta-tion values of the system operators, and solve them. InSec. III, the expressions for the components of the out-put field at the probe frequency and the Stokes frequencyare given. In Sec. IV, we present the numerical resultsfor the output probe field without or with the couplingfield, and compare it with that from a ring cavity withtwo movable mirrors having equal frequencies. In Sec. V,we show the numerical results for the output Stokes fieldfrom a ring cavity with two moving mirrors having dif-ferent or equal frequencies, and compare them. Finallyin Sec. VI, we conclude the paper.
II. MODEL
We consider a ring cavity with round-trip length L formed by three mirrors, as shown in Fig. 1 [25]. Oneof them is not movable and partially transmitting, whilethe other two are allowed to vibrate and assumed to have100% reflectivity. The cavity field at the resonance fre-quency ω is driven by a strong coupling field with am-plitude ε at frequency ω c . Meanwhile a weak probe fieldwith amplitude ε p at frequency ω p is sent into the cavity.The coupling field and the probe field are treated clas-sically here. The mechanical motions of both movablemirrors are coupled to the cavity field through the radia-tion pressure force exerted by the photons in the cavity.The movable mirrors’ dynamics can be approximated asthose of a single harmonic oscillator, with resonance fre-quency ω j , effective mass m j , damping rate γ j ( j = 1 , ω j ( j = 1 ,
2) are muchsmaller than the cavity free spectral range c/L ( c is thespeed of light in vacuum). The adiabatic limit ω j ≪ c/L ( j = 1 ,
2) implies that the mechanical frequencies ω j ( j = 1 ,
2) are very small compared to the cavity reso-nance frequency ω , so the moving mirrors are movingso slow that the retardation effect, the Casimir effect,and the Doppler effect become completely negligible [26–28]. Hence the radiation pressure force does not dependon the velocity of the movable mirrors. Assuming that k is the wave vector of the cavity field with k = ω /c ,the radiation pressure force exerted by the light field onthe movable mirror can be calculated from the momen-tum exchange between the cavity field and the movablemirror, which is F = ~ k cos( θ/ L/c c † c = ~ ω L c † c cos( θ/ θ is the angle between the incident light and thereflected light at the surfaces of the movable mirrors, c † c is the photon number operator of the cavity field, c † and c are the creation and annihilation operators of the cavityfield, obey the standard commutation relation [ c, c † ] = 1.Note that the radiation pressure forces acting on the mov-able mirrors vary linearly with the instantaneous photonnumber in the cavity. Under the action of the radiationpressure force, the movable mirrors make small oscilla-tions. Then the small motion of the mirrors changes thelength of the optical cavity, and alters the intensity of thecavity field, which in turn modifies the radiation pressureforce acting on the mirrors. Therefore the interaction be-tween the cavity field and the mechanical motion in thering cavity is nonlinear. qq x y Coupling field c w Probe field p w Fixed mirror Movable mirror 1 w Movable mirror 2 w FIG. 1: The scheme of the optomechanical system. A strongcoupling field at frequency ω c and a weak probe field at fre-quency ω p are injected into the ring cavity via the fixed mirrorand interact with two movable mirrors whose resonance fre-quencies ( ω , ω ) are a little different. In a frame rotating at the driving frequency ω c , the Hamiltonian of the whole system takes the form H = ~ ( ω − ω c ) c † c + ~ ω Q + P ) + ~ ω Q + P )+ ~ ( g Q − g Q ) c † c cos θ i ~ ε ( c † − c )+ i ~ ( ε p c † e − iδt − ε ∗ p ce iδt ) . (1)Here the first three terms are the free energies of thecavity field and two mechanical oscillators, respectively,( Q j , P j ) denote the dimensionless position and momen-tum quadratures of the two mirrors, Q j = q m j ω j ~ q j , P j = q m j ~ ω j p j ( j = 1 , Q j , P k ] = iδ j,k ( j, k =1 , g j = ω L q ~ m j ω j ( j = 1 ,
2) is the op-tomechanical coupling strength. The last two terms givethe interactions of the cavity field with the coupling fieldand the probe field, respectively, ε is related to the power ℘ of the coupling field by ε = q κ℘ ~ ω c , where κ is the decayrate of the cavity due to the transmission losses throughthe fixed mirror, ε p is related to the power ℘ p of theprobe field by | ε p | = q κ℘ p ~ ω p , δ = ω p − ω c is the detuningof the probe field from the coupling field.Starting from the Heisenberg equations of motion, tak-ing into account the dissipations of the cavity field andthe mechanical oscillators, and neglecting quantum noiseand thermal noise, we obtain the time evolutions of theexpectation values of the system operators h ˙ Q i = ω h P i , h ˙ P i = − ω h Q i − g h c † ih c i cos θ − γ h P i , h ˙ Q i = ω h P i , h ˙ P i = − ω h Q i + g h c † ih c i cos θ − γ h P i , h ˙ c i = − i [ ω − ω c + ( g h Q i − g h Q i ) cos θ h c i + ε + ε p e − iδt − κ h c i , (2)where we have used the mean field assumption h c † c i ≃h c † ih c i and h Q j c i ≃ h Q j ih c i ( j = 1 , | ε p | << ε ),the steady-state solution to Eq. (2) can be approximatedto the first order in the probe field ε p . In the long timelimit, the solution to Eq. (2) can be written as h s i = s + s + ε p e − iδt + s − ε ∗ p e iδt , (3)where s = Q , P , Q , P , or c . The solution containsthree components, which in the original frame oscillateat ω c , ω p , 2 ω c − ω p , respectively. Substituting Eq. (3)into Eq. (2), equating coefficients of e and e ± iδt , we findthe following analytical expressions Q = − G ω | c | , P = 0 ,Q = G ω | c | , P = 0 ,c = εκ + i ∆ ′ ,c + = 1 d ( δ ) (cid:8) [ κ − i (∆ ′ + δ )]( ω − δ − iγ δ ) × ( ω − δ − iγ δ ) + i [ G ω ( ω − δ − iγ δ )+ G ω ( ω − δ − iγ δ )] (cid:9) ,c − = ic | c | d ( δ ) ∗ [ G ω ( ω − δ + iγ δ )+ G ω ( ω − δ + iγ δ )] , (4)where G = g | c | cos θ and G = g | c | cos θ are theeffective optomechanical coupling rates, ∆ ′ = ω − ω c +( g Q − g Q ) cos θ is the effective detuning of the cou-pling field from the cavity resonance frequency, includingthe frequency shift induced by the radiation pressure, and d ( δ ) = [ κ + i (∆ ′ − δ )][ κ − i (∆ ′ + δ )]( ω − δ − iγ δ ) × ( ω − δ − iγ δ ) − ′ [ G ω ( ω − δ − iγ δ )+ G ω ( ω − δ − iγ δ )] . (5) III. THE OUTPUT FIELD
The output field can be obtained by using the input-output relation ε out ( t ) = 2 κ h c i . In analogy with Eq. (3),we expand the output field to the first order in the probefield ε p , ε out ( t ) = ε out + ε out + ε p e − iδt + ε out − ε ∗ p e iδt , (6)where ε out , ε out + , and ε out − are the components of theoutput field oscillating at frequencies ω c , ω p , 2 ω c − ω p .Here ε out − is called a Stokes field, and it is generatedvia the nonlinear four-wave mixing process, in which twophotons at frequency ω c interact with a single photon atfrequency ω p to create a new photon at frequency 2 ω c − ω p . Thus we find that the components of the output fieldat the probe frequency and the Stokes frequency are ε out + = 2 κc + ,ε out − = 2 κc − , (7)respectively. In the absence of the coupling field ( ℘ = 0),the components of the output field at the probe frequencyand the Stokes frequency are given by ε out + = 2 κκ + i (∆ ′ − δ ) ,ε out − = 0 . (8)These are not unexpected results. Let us write the realpart of ε out + as ν p , which exhibits the absorption char-acteristic of the output field at the probe frequency. Itcan be measured by the homodyne technique [29]. IV. NUMERICAL RESULTS OF THE OUTPUTPROBE FIELD
In this section, we numerically evaluate how the cou-pling field modifies the absorption of the ring cavity tothe probe field. Ã H mW L R e H ∆ L (cid:144) Ω m FIG. 2: (Color online) The dependence of the real parts ofthe roots of d ( δ ) in the domain Re( δ ) > ℘ ofthe coupling field. Ã H mW L - I m H ∆ L (cid:144) Ω m FIG. 3: (Color online) The dependence of the imaginary partsof the roots of d ( δ ) on the power ℘ of the coupling field. The values of the parameters chosen are similar tothose in [13]: the wavelength of the coupling field λ = 2 πc/ω c = 775 nm, the coupling constants g =2 π ×
12 GHz/nm × p ~ / ( m ω ), g = 2 π ×
12 GHz/nm × p ~ / ( m ω ), the masses of the movable mirrors m = m = 20 ng, the frequencies of the movable mirrors ω = ω m + 0 . ω m , ω = ω m − . ω m , where ω m =2 π × . κ = 2 π × κ/ω m ≃ . < γ = γ = γ = 2 π × . Q ′ = ω /γ ≃ Q ′ = ω /γ ≃ θ = π/
3. And the coupling field is tuned close tothe red sideband of the cavity resonance ∆ ′ = ω m . Theparameters chosen ensure the system operating in thestable regime.We note that the structure of the quadrature of theoutput probe field ν p is determined by d ( δ ). The roots δ of d ( δ ) are complex values. The real parts Re( δ ) of theroots determine the positions of the normal modes of theoptomechanical system; the imaginary parts Im( δ ) of theroots describe their widths. Figure 2 shows the real partsof the roots of d ( δ ) in the domain Re( δ ) > d ( δ ) are Re( δ ) = 0 . ω m , ω m , . ω m at low drivingpower. At high driving power, one of the real parts is still ω m (dotdashed curve), not changing with increasing thepower of the coupling field, the difference between theother two (dotted curve and dashed curve) increases withincreasing the power of the coupling field, which impliesthe normal mode spitting [22–24] in the output probefield. Figure 3 shows the imaginary parts of the roots of d ( δ ) versus the coupling beam power. We see that theimaginary parts of the roots of d ( δ ) have three differentvalues at low driving power. At high driving power, theimaginary parts of the roots of d ( δ ) have three values, twoof them are identical, the other one is small. Moreover,in Figs. 2 and 3, the dotdashed curves represent thereal part and the imaginary part of one of the roots of d ( δ ), respectively, similarly for dotted curves and dashedcurves. Hence, at high driving power, the central peakin the quadrature ν p is narrow, two side peaks in thequadrature ν p have the same broad linewidths. ∆ (cid:144) Ω m Υ p FIG. 4: (Color online) The quadrature of the output probefield ν p as a function of the normalized probe detuning δ/ω m for ω = 1 . ω m and ω = 0 . ω m . The black solid, blue dot-dashed, and red dotted curves correspond to ℘ = 0, 2 mW,and 15 mW, respectively. In Fig. 4, we plot the quadrature of the output probefield ν p as a function of the normalized probe detuning δ/ω m for three different powers of the coupling field.In the absence of the coupling field, ν p (solid curve)has a standard Lorentzian absorption peak with a fullwidth at half maximum (FWHM) of 2 κ at the line cen-ter δ/ω m = 1. However, in the presence of the couplingfield with power 2 mW, it is seen that the dotdashed curve exhibits two symmetric narrow EIT dips centeredat δ/ω m = 0 . , . δ/ω m = 1. The FWHM of the two EIT dips are about( γ + G κ ) and ( γ + G κ ), respectively. The FWHM of thecentral absorption peak is about ( ω − ω ) − ( γ + G + G κ ).The two transparency dips display that the input probefield could be simultaneously transparent at two sym-metric frequencies, which is the result of the destructiveinterferences between the probe field and the anti-Stokesfields at frequencies ω c +0 . ω m and ω c +1 . ω m generatedby the interactions of the coupling field with the mov-able mirrors. The absorption peak at the line center im-plies that the incident probe field is almost fully absorbedby the optomechanical system. Moreover, increasing thepower ℘ of the coupling field, the two EIT dips becomebroader, which results in narrowing of the central absorp-tion peak. Further, from the dotted curve in Fig. 4, onecan see that increasing the power of the coupling field to15 mW results in a larger splitting between the left andright side peaks with the same linewidths, it also resultsin a narrow central peak. Our calculations show thatin the strong coupling limit 2( G + G ) >> ( κ − γ ) ,the three dressed modes of the system correspondingto the three absorption peaks are δ ≈ ω m − iγ/
2, and δ ≈ ω m ± p G + G ) − i ( κ + γ ). Note that theFWHM for the narrow central peak at δ = ω m is about γ . In addition, the FWHM for the right and left peaksat δ ≈ ω m ± p G + G ) are the same ( κ + γ ), thesplitting between the side peaks is about p G + G ),which is proportional to the power of the coupling field.These results are consistent with those in Figs. 2 and 3. ∆ (cid:144) Ω m Υ p FIG. 5: (Color online) The quadrature of the output probefield ν p as a function of the normalized probe detuning δ/ω m for ω = ω = ω m . The black solid, blue dotdashed, andred dotted curves correspond to ℘ = 0, 2 mW, and 15 mW,respectively. For comparison, we consider the same previous system,but the two mechanical oscillators have the same frequen-cies ω = ω = ω m , thus their effective optomechanicalcoupling rates are equal, we assume G = G = G . Weplot the quadrature of the output probe field ν p as a func-tion of the normalized probe detuning δ/ω m for three dif-ferent powers of the coupling field, as shown in Fig. 5.When the coupling field is not present, the quadrature ν p (solid curve) has a Lorentzian absorption lineshape.However, the presence of the control field with power2 mW leads to a narrow EIT-like dip at the line cen-ter (dotdashed curve). Thus the probe field can almostcompletely propagate through the ring cavity on reso-nance with almost no absorption. The FWHM of theEIT-like dip is about γ + G κ . The EIT-like dip is at-tributed to the destructive interference between the inputweak probe field and the scattering quantum fields at theprobe frequency ω p generated by the interactions of thecoupling field with two mirrors having identical frequen-cies. Moreover, in the strong coupling limit 2 G >> κ ,the normal mode splitting exhibits in the quadrature ν p (dotted curve), the two peaks have the same FWHM of κ + γ , their positions are δ ≈ ω m ± G , the separationbetween them is about 2 G .In order to understand the narrow central peak in Fig.4 and the EIT-like dip in Fig. 5, let us introduce the rela-tive coordinates ( Q a , P a ) and center of mass coordinates( Q s , P s ) of the two movable mirrors as Q a = g Q − g Q p g + g , P a = g P − g P p g + g ,Q s = g Q + g Q p g + g , P s = g P + g P p g + g . (9)With these new coordinates we can write the Hamilto-nian Eq. (1) as H = ~ ( ω − ω c ) c † c + ~ ω Q a + P a ) + ~ ω Q s + P s )+ ~ g + g )( ω g − ω g )( Q a Q s + P a P s )+ ~ q g + g Q a c † c cos θ i ~ ε ( c † − c )+ i ~ ( ε p c † e − iδt − ε ∗ p ce iδt ) , (10)where ω = ( g + g )( ω g + ω g ). The fourth term inEq. (10) describes the interaction between the relativeand center of mass coordinates. The fifth term in Eq.(10) shows that the interaction between the relative co-ordinate and the cavity field. When the two mechanicalfrequencies are equal ω = ω , the fourth term vanishes,the center of mass coordinate is decoupled from the rela-tive coordinate, and it is also decoupled from the cavityfield, so only the relative coordinate is coupled to the cav-ity field via radiation pressure, which generates a trans-parency dip in the output probe field, as shown in Fig.5. When the two mechanical frequencies are not equal ω = ω , it is seen that not only the relative coordinateand the cavity field interact with each other, but alsothe relative and center of mass coordinates interact witheach other, which leads to a narrow central peak in theoutput probe field, as shown in Fig. 4. V. NUMERICAL RESULTS OF THE OUTPUTSTOKES FIELD
In this section, we numerically examine the effect ofthe coupling field on the output Stokes field generatedvia the four-wave mixing process. ∆ (cid:144) Ω m È ¶ ou t - FIG. 6: (Color online) The intensity of the Stokes field | ε out − | as a function of the normalized probe detuning δ/ω m for ω = 1 . ω m and ω = 0 . ω m . The black solid, blue dot-dashed, and red dotted curves correspond to ℘ = 0, 2 mW,and 15 mW, respectively. Figure 6 shows the intensity of the generated Stokesfield | ε out − | as a function of the normalized probe detun-ing δ/ω m for several values of the driving power when thefrequencies of the two mirrors are not equal ( ω = 1 . ω m and ω = 0 . ω m ). It is seen that | ε out − | = 0 (solidcurve) in the absence of the coupling field. However, inthe presence of the coupling field, the | ε out − | (dotdashedcurve and dotted curve) exhibits the normal mode split-ting, the peak separation increases with the power of thecoupling field. Moreover, the maximum value of | ε out − | is increased with the power of the coupling field, the max-imum value of | ε out − | is about 0.19 when ℘ = 15 mW.Note that the intensity of the Stokes field goes to zerowhen the detuning δ/ω m = 1. Hence, the four-wave mix-ing effect is completely suppressed on resonance so thatthere is only the probe field stored inside the optome-chanical system. The suppression of the four-wave mix-ing effect on resonance is due to the destructive interfer-ence between the Stokes fields produced by the couplingfield interacting with the two movable mirrors oscillatingat frequencies ω = 1 . ω m and ω = 0 . ω m .For comparison, we also consider the case of a ring cav-ity in which the frequencies of the two mirrors are equal( ω = ω = ω m ), the intensity | ε out − | of the outputStokes field versus the normalized probe detuning δ/ω m for several values of the driving power is shown in Fig.7. The significant difference between Fig. 7 and Fig. 6is that the intensity of the Stokes field in Fig. 7 is nonzero at δ/ω m = 1 in the presence of the coupling field.Thus the four-wave mixing process is not suppressed at ∆ (cid:144) Ω m È ¶ ou t - FIG. 7: (Color online) The intensity of the Stokes field | ε out − | as a function of the normalized probe detuning δ/ω m for ω = ω = ω m . The black solid, blue dotdashed, andred dotted curves correspond to ℘ = 0, 2 mW, and 15 mW,respectively. δ/ω m = 1, which arises from the constructive interfer-ence between the Stokes fields produced by the couplingfield interacting with the two movable mirrors oscillatingat the same frequencies ω = ω = ω m . So there is noapparent normal mode splitting in this case. VI. CONCLUSIONS
To summarize, we have demonstrated how a strongcoupling field affects the propagation of a weak probe field in a ring cavity with two movable mirrors whose fre-quencies are close to each other. We find double EIT dipsand a central absorption peak in the output probe fieldin the presence of the coupling field. Thus this systemcan become transparent at two different frequencies of aweak probe field. Increasing the pump power gives riseto broadening two EIT dips and narrowing the centralabsorption peak. In addition, we show that the couplingfield induces the normal mode splitting in the outputprobe field and the output Stokes field. Moreover, thefour-wave mixing suppression takes place in this optome-chanical system when the probe detuning is one-half thesum of the two mechanical frequencies. However, whentwo movable mirrors in a ring cavity have identical fre-quencies, the response of the ring cavity to a weak probefield in the presence of a strong coupling field would bedifferent. We observe a narrow transparency dip in theoutput probe field, and no four-wave mixing suppressionwhen the probe detuning is equal to the mechanical fre-quency.The author thanks Prof. G. S. Agarwal for helpfuldiscussions. The author also thanks Prof. Lin Tian andProf. Mankei Tsang for support. [1] Harris S E 1997
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Nature (London) [25] Huang S and Agarwal G S 2009