Temporal Dynamics and Nonclassical Photon Statistics of Quadratically Coupled Optomechanical Systems
NNoname manuscript No. (will be inserted by the editor)
Temporal dynamics and nonclassical photon statistics ofquadratically coupled optomechanical systems
Shailendra Kumar Singh · S. V. Muniandy the date of receipt and acceptance should be inserted later
Abstract
Quantum Optomechanical system serves as an interface for coupling be-tween photons and phonons due to mechanical oscillations. We use the Heisenberg-Langevin approach under Markovian white noise approximation to study a quadrat-ically coupled optomechanical system which contains a thin dielectric membranequadratically coupled to the cavity field. A decorrelation method is employed tosolve for a larger number of coupled equations. Transient mean number of cavityphotons and phonons that provide dynamical behaviour are computed under differ-ent coupling regime. We have also obtained the two-boson second-order correlationfunctions for the cavity field, membrane oscillator and their cross correlations thatprovide nonclassical properties governed by quadratic optomechanical system.
Recent years have seen significant experimental progress in realizing deterministicinteractions between single photons, which has profound importance for future op-tical technologies. Most novel experiments have been explored in cavity quantumelectrodynamics (cQED) [1,2], where photons inherent the saturation of a singletwo-level atom due to strong interactions between the atom and the cavity field.A single atom-cavity system described by well-known Jaynes-Cummings model [3]has been used as an important test bed for implementation of quantum informa-tion algorithms as well as construction of a quantum network with the aim forquantum computation. The strong-coupling regime of the cQED is reached whenthe coupling strength of the atom with the cavity mode dominates over the de-coherence processes, due to spontaneous emission from the excited atomic level
Corresponding Author: Shailendra Kumar SinghInstitute of Nuclear Sciences, Hacettepe University, 06800, Ankara, TurkeyE-mail: [email protected]. MuniandyCenter of Theoretical and Computational Physics, Department of Physics, University ofMalaya, 50603 KualaLumpur, MalaysiaE-mail: msithi.um.edu.my a r X i v : . [ qu a n t - ph ] J un Shailendra Kumar Singh, S. V. Muniandy to ground state and the leakage of photons in the cavity mode. Based on Fabry-Perot interferometry [4], semiconductor microcavities have been developed whereexcitons act as quantum systems [5]. Alternative approaches have been exploredbased on slow-light enhanced Kerr nonlinearites [6], single dye-molecules [7], strongphoton interactions mediated by Rydberg atoms [8], atoms coupled to hollow-corefibers [9] and nanofiber traps [10].Optomechanical system provides a more promising approach towards realiz-ing strong photon interactions [11]. In comparison to atomic systems, for examplemacroscopic mechanical resonators are relatively easy to control and to be inte-grated with other systems. In these systems, an optical cavity, with a movableend mirror [12,13] or with a micromechanical membrane is subjected to mechan-ical effect caused by light through radiation pressure [14]. So, Cavity QuantumOptomechanics has emerged as an interesting area for studying quantum featuresat the mesoscale, where it is possible to control the quantum state of mechani-cal oscillators by their coupling to light field [15]. Recent advances in this areaincludes the realization of quantum-coherent coupling of a mechanical oscillatorwith an optical cavity [16], where the coupling rate exceeds both the cavity andmechanical motion decoherence rate, laser cooling of a nanomechanical oscillatorto its ground state [17].Experimental and theoretical proposals aiming to study quantum aspects ofinteractions between the optical cavities and mechanical objects have focused oncavities in which one of the cavity’s mirror is free to move for example, in responseto radiation pressure exerted by light in the cavity. This nearly includes all op-tomechanical systems described in the literature, including cavities with ’folded’geometries, cavities in which multiple mirrors are free to move [18], and whisper-ing gallery mode resonators [19] in which light is confined to a waveguide. Allthese methods have two important considerations. First, cavity’s detuning is pro-portional to the displacement of a mechanical degree of freedom, which is mirrordisplacement or waveguide elongation. Second, a single device must provide bothoptical confinement and mechanical effects. So, achieving good mechanical andoptical properties both simultaneously has been the main aim in quantum op-tomechanical systems because high-finesse mirrors are not easily integrated intomicromachined devices. Experimental work done so far have achieved sufficient op-tomechanical coupling to laser-cooled mechanical devices [20,21] but have not yetreached to quantum regime due to technical problems mentioned above. A morefundamental challenge is to read out the mechanical element’s energy eigenstate.Displacement measurements cannot determine an oscillator’s energy eigenstateand measurements coupling to quantities other than displacement have been dif-ficult to realize in practice [22,23].In this work, we theoretically study an optomechanical system realized in sem-inal experimental works [24,25] and has potential to resolve both of these above- mentioned challenges. We study a system which contains a thin dielectric mem-brane with quadratic response to the cavity fields. Coupled Heisenberg-Langevinequations are obtained under Markovian white noise approximation, and solutionsup to second-order correlation operators have been developed in Section II. In thesame section, we also discuss the transient dynamics of the system Hamiltonian itle Suppressed Due to Excessive Length 3 in various scenario. Section III discusses the results with the nonclassical photonstatistics of the cavity mode as well as mechanical oscillations mode includingcross-correlation between them. We conclude our results in Section IV.
We consider a quadratically coupled optomechanical resonator formed by a mi-cropillar with moveable Bragg reflectors and a thin dielectric membrane at thenode (or antinode) of the resonator. The mechanical displacement of the mem-brane quadratically couples to the cavity photon number. In addition, we havea monochromatic laser field with frequency ω L applied to drive the cavity. Thecorresponding Hamiltonian for the scheme is given by,ˆ H = ˆ H + ˆ H R , (1)where ˆ H and ˆ H R are given by Eqs. (2) and (3). In our model, ˆ H R is the partwhere all the two subsystem cavity photons, phonons (due to mechanical motionof the membrane) respectively interact with their corresponding reservoirs whichare collection of harmonic oscillators. We write ˆ H = ω c ˆ a † ˆ a + ω M ˆ b † ˆ b + g opt (cid:20) ˆ a † ˆ a (cid:16) ˆ b † + ˆ b (cid:17) (cid:21) + Ω (cid:104) e − iω L t ˆ a † + e iω L t ˆ a (cid:105) (2)where ˆ a and ˆ b correspond to field operators for cavity photons and phonons (dueto mechanical motion of the membrane) with frequencies ω c and ω M , respectively.The third term in R.H.S of Eq. (2) describes the quadratic optomechanical couplingwith strength g opt between the cavity field and the mechanical motion of themembrane and the fourth term describes driving process of the cavity field, with Ω is the Rabi frequency of the driving field. Meanwhile, ˆ H R takes the followingform:ˆ H R = (cid:122) (cid:125)(cid:124) (cid:123) for cavity mode (cid:88) k ω k ˆ m † k ˆ m k + (cid:88) k g k (cid:16) ˆ m † k ˆ a + ˆ a † ˆ m k (cid:17) + (cid:122) (cid:125)(cid:124) (cid:123) for moving membrane (cid:88) k (cid:48) ω k (cid:48) ˆ n † k (cid:48) ˆ n k (cid:48) + (cid:88) k (cid:48) g k (cid:48) (cid:16) ˆ n † k (cid:48) ˆ b + ˆ b † ˆ n k (cid:48) (cid:17) , (3)where the first two terms on the R.H.S of Eq.(3) represent the damping of cavitymode through reservoir of harmonic oscillator operators ˆ m k and ˆ m † k ; while the thirdand fourth terms are responsible for damping of membrane through the harmonicoscillator operators ˆ n k (cid:48) and (cid:16) ˆ n † k (cid:48) (cid:17) . In the rotating frame with the driving frequency ω L , the Hamiltonian ˆ H reduces toˆ H = ∆ c ˆ a † ˆ a + ω M ˆ b † ˆ b + g opt (cid:20) ˆ a † ˆ a (cid:16) ˆ b † + ˆ b (cid:17) (cid:21) + Ω (cid:16) ˆ a † + ˆ a (cid:17) , (4) where ∆ c = ( ω c − ω L ) is the corresponding detuning from the driving laser fre-quency ω L . Expanding the quadratic term g opt (cid:20) ˆ a † ˆ a (cid:16) ˆ b † + ˆ b (cid:17) (cid:21) , we have theHamiltonian ˆ H in the following form:ˆ H = ∆ c ˆ a † ˆ a + ω M ˆ b † ˆ b + g opt (cid:104) ˆ a † ˆ a (cid:16) ˆ b † + ˆ b + 2ˆ b † ˆ b + 1 (cid:17)(cid:105) + Ω (cid:16) ˆ a † + ˆ a (cid:17) (5) Shailendra Kumar Singh, S. V. Muniandy
Coupling strength g opt should satisfy the condition ( ω M + 4 sg opt ) > s is the number of photons inside the cavity.The corresponding Heisenberg equations of motion for the field operators of dif-ferent subsystem (cavity, membrane) are given by Eqs. (6) and (8), whereas thecorresponding equations of motion for reservoir operators are given by Eqs. (7)and (9): ddt ˆ a = − i∆ c ˆ a − ig opt (cid:104) ˆ a (cid:16) ˆ b † + ˆ b + 2ˆ b † ˆ b + 1 (cid:17)(cid:105) − iΩ − i (cid:88) k g k ˆ m k . (6) ddt ˆ m k = − iω k ˆ m k − ig k ˆ a. (7) ddt ˆ b = − iω M ˆ b − ig opt (cid:104) ˆ a † ˆ a (cid:16) ˆ b + ˆ b † (cid:17)(cid:105) − i (cid:88) k (cid:48) g k (cid:48) ˆ n k (cid:48) . (8) ddt ˆ n k (cid:48) = − iω k (cid:48) ˆ n k (cid:48) − ig k (cid:48) ˆ b. (9)Now the corresponding operator equations for different reservoir need to be ex-plicitly solved further to remove the terms of reservoir operators in Eqs (6) and(8). Integrating Eq. (7), one getsˆ m k ( t ) = ˆ m k (0) e − iω k t − ig k (cid:90) t dt (cid:48) ˆ a ( t (cid:48) ) e − iω k ( t − t (cid:48) ) . (10)In Eq. (10), the first term represents the free evolution of the reservoir modes,whereas the second term arises from their interaction with the cavity. The reservoirmodes ˆ m k ( t ) can be eliminated from the Eq. (6) by substituting the value of ˆ m k ( t )from Eq. (10) in Eq. (6). So, we have ddt ˆ a = − i∆ c ˆ a − ig opt (cid:104) ˆ a (cid:16) ˆ b † + ˆ b + 2ˆ b † ˆ b + 1 (cid:17)(cid:105) − iΩ − (cid:88) k g k (cid:90) t dt (cid:48) ˆ a ( t (cid:48) ) e − iω k ( t − t (cid:48) ) + F a ( t ) , (11)where F a ( t ) = − i (cid:80) k g k ˆ m k (0) e − iω k t is the noise operator for the cavity whichdepends on the reservoir variables. The term containing the integral is expressedas (cid:88) k g k (cid:90) t dt (cid:48) ˆ a ( t (cid:48) ) e − iω k ( t − t (cid:48) ) (cid:39) Γ a ˆ a ( t ) . (12)Thus, from Eqs. (11) and (12), we obtain ddt ˆ a = − i∆ c ˆ a − ig opt (cid:104) ˆ a (cid:16) ˆ b † + ˆ b + 2ˆ b † ˆ b + 1 (cid:17)(cid:105) − iΩ − Γ a ˆ a ( t ) + F a ( t ) , (13)where Γ a is the damping constant for cavity mode. Similarly, we have equation formechanical motion of the membrane as follows: ddt ˆ b = − iω M ˆ b − ig opt (cid:104) ˆ a † ˆ a (cid:16) ˆ b + ˆ b † (cid:17)(cid:105) − Γ b ˆ b + F b ( t ); (14)where F b ( t ) and Γ b are noise operator and damping constant for moving membranerespectively, given by F b ( t ) = − i (cid:88) k (cid:48) g k (cid:48) ˆ n k (cid:48) (0) e − iω k (cid:48) t , (15) itle Suppressed Due to Excessive Length 5 and (cid:88) k (cid:48) g k (cid:48) (cid:90) t dt (cid:48) ˆ b ( t (cid:48) ) e − iω k (cid:48) ( t − t (cid:48) ) (cid:39) Γ b ˆ b ( t ) . (16)2.1 Quantum CorrelationsWe have obtained the coupled equations involving the mean of the operatorsˆ a and ˆ b , their corresponding adjoints and their odd and even products. Sinceboth the reservoirs are big as well as always in thermal equilibrium, we have (cid:104) F a ( t ) (cid:105) = (cid:104) F b ( t ) (cid:105) = 0. We also have, (cid:68) F ˆ a † ( t )ˆ b ( t ) (cid:69) = (cid:10) F ˆ b ( t ) ˆ a ( t ) (cid:11) = (cid:68) F a ( t )ˆ b † ( t ) (cid:69) = (cid:10) F ˆ b † ( t ) ˆ a ( t ) (cid:11) = 0 . Similarly, quantum correlations between any subsystem (cavity,moving membrane) with different noise operator other than its own coupled reser-voir’s noise operator vanish altogether. The following set of equations has beenobtained by using the Heisenberg-Langevin approach given in [27]. ddt (cid:104) ˆ a (cid:105) = − i∆ c (cid:104) ˆ a (cid:105) − ig opt (cid:104)(cid:68) ˆ a ˆ b † (cid:69) + (cid:68) ˆ a ˆ b (cid:69) + 2 (cid:68) ˆ a ˆ b † ˆ b (cid:69) + (cid:104) ˆ a (cid:105) (cid:105) − iΩ − Γ a (cid:104) ˆ a (cid:105) . (17) ddt (cid:68) ˆ a † (cid:69) = i∆ c (cid:68) ˆ a † (cid:69) + ig opt (cid:104)(cid:68) ˆ a † ˆ b † (cid:69) + (cid:68) ˆ a † ˆ b (cid:69) + 2 (cid:68) ˆ a † ˆ b † ˆ b (cid:69) + (cid:68) ˆ a † (cid:69)(cid:105) + iΩ − Γ a (cid:68) ˆ a † (cid:69) . (18) ddt (cid:68) ˆ b (cid:69) = − iω M (cid:68) ˆ b (cid:69) − ig opt (cid:104)(cid:68) ˆ a † ˆ a ˆ b (cid:69) + (cid:68) ˆ a † ˆ a ˆ b † (cid:69)(cid:105) − Γ b (cid:68) ˆ b (cid:69) . (19) ddt (cid:68) ˆ b † (cid:69) = iω M (cid:68) ˆ b † (cid:69) + 2 ig opt (cid:104)(cid:68) ˆ a † ˆ a ˆ b † (cid:69) + (cid:68) ˆ a † ˆ a ˆ b (cid:69)(cid:105) − Γ b (cid:68) ˆ b † (cid:69) . (20) ddt (cid:68) ˆ a † ˆ a (cid:69) = − iΩ (cid:16)(cid:68) ˆ a † (cid:69) − (cid:104) ˆ a (cid:105) (cid:17) − Γ a (cid:68) ˆ a † ˆ a (cid:69) + Γ a ¯ n ath . (21) ddt (cid:68) ˆ b † ˆ b (cid:69) = − ig opt (cid:104)(cid:68) ˆ a † ˆ a ˆ b † (cid:69) − (cid:68) ˆ a † ˆ a ˆ b (cid:69)(cid:105) − Γ b (cid:68) ˆ b † ˆ b (cid:69) + Γ b ¯ n bth . (22) ddt (cid:68) ˆ a ˆ b † (cid:69) = i ( ω M − ∆ c ) (cid:68) ˆ a ˆ b † (cid:69) − iΩ (cid:68) ˆ b † (cid:69) −
12 ( Γ a + Γ b ) (cid:68) ˆ a ˆ b † (cid:69) (23)+ ig opt (cid:104) (cid:68) ˆ a † ˆ a ˆ b (cid:69) − (cid:68) ˆ a ˆ b † ˆ b (cid:69) − (cid:68) ˆ a ˆ b † (cid:69) + 2 (cid:16)(cid:68) ˆ a † ˆ a ˆ b † (cid:69) − (cid:68) ˆ a ˆ b † ˆ b (cid:69)(cid:17) − (cid:68) ˆ a ˆ b † (cid:69)(cid:105) .ddt (cid:68) ˆ a † ˆ b (cid:69) = i ( ∆ c − ω M ) (cid:68) ˆ a † ˆ b (cid:69) + iΩ (cid:68) ˆ b (cid:69) −
12 ( Γ a + Γ b ) (cid:68) ˆ a † ˆ b (cid:69) (24)+ ig opt (cid:104)(cid:68) ˆ a † ˆ b † ˆ b (cid:69) + (cid:68) ˆ a † ˆ b (cid:69) − (cid:68) ˆ a † ˆ a ˆ b † (cid:69) + 2 (cid:16)(cid:68) ˆ a † ˆ b † ˆ b (cid:69) − (cid:68) ˆ a † ˆ a ˆ b (cid:69)(cid:17) + (cid:68) ˆ a † ˆ b (cid:69)(cid:105) .ddt (cid:68) ˆ a ˆ b (cid:69) = − i ( ∆ c + ω M ) (cid:68) ˆ a ˆ b (cid:69) − iΩ (cid:68) ˆ b (cid:69) −
12 ( Γ a + Γ b ) (cid:68) ˆ a ˆ b (cid:69) − ig opt (cid:104) (cid:68) ˆ a ˆ b † (cid:69) + (cid:68) ˆ a ˆ b † ˆ b (cid:69) +2 (cid:68) ˆ a † ˆ a ˆ b † (cid:69) + (cid:68) ˆ a ˆ b (cid:69) + 2 (cid:16)(cid:68) ˆ a ˆ b (cid:69) + (cid:68) ˆ a ˆ b † ˆ b (cid:69) + (cid:68) ˆ a † ˆ a ˆ b (cid:69)(cid:17) + (cid:68) ˆ a ˆ b (cid:69)(cid:105) . (25) Shailendra Kumar Singh, S. V. Muniandy ddt (cid:68) ˆ a † ˆ b † (cid:69) = i ( ∆ c + ω M ) (cid:68) ˆ a † ˆ b † (cid:69) + iΩ (cid:68) ˆ b † (cid:69) −
12 ( Γ a + Γ b ) (cid:68) ˆ a † ˆ b † (cid:69) + ig opt (cid:104) (cid:68) ˆ a † ˆ b (cid:69) + (cid:68) ˆ a † ˆ b † ˆ b (cid:69) +2 (cid:68) ˆ a † ˆ a ˆ b (cid:69) + (cid:68) ˆ a † ˆ b † (cid:69) + 2 (cid:16)(cid:68) ˆ a † ˆ b † (cid:69) + (cid:68) ˆ a † ˆ b † ˆ b (cid:69) + (cid:68) ˆ a † ˆ a ˆ b † (cid:69)(cid:17) + (cid:68) ˆ a † ˆ b † (cid:69)(cid:105) . (26) ddt (cid:68) ˆ a (cid:69) = − i∆ c (cid:68) ˆ a (cid:69) − iΩ (cid:104) ˆ a (cid:105)− Γ a (cid:68) ˆ a (cid:69) − ig opt (cid:104)(cid:68) ˆ a ˆ b † (cid:69) + (cid:68) ˆ a ˆ b (cid:69) + 2 (cid:68) ˆ a ˆ b † ˆ b (cid:69) + (cid:68) ˆ a (cid:69)(cid:105) . (27) ddt (cid:68) ˆ a † (cid:69) = 2 i∆ c (cid:68) ˆ a † (cid:69) +2 iΩ (cid:68) ˆ a † (cid:69) − Γ a (cid:68) ˆ a † (cid:69) +2 ig opt (cid:104)(cid:68) ˆ a † ˆ b † (cid:69) + (cid:68) ˆ a † ˆ b (cid:69) + 2 (cid:68) ˆ a † ˆ b † ˆ b (cid:69) + (cid:68) ˆ a † (cid:69)(cid:105) . (28) ddt (cid:68) ˆ b (cid:69) = − iω M (cid:68) ˆ b (cid:69) − Γ b (cid:68) ˆ b (cid:69) − ig opt (cid:104)(cid:68) ˆ a † ˆ a (cid:69) + 2 (cid:68) ˆ a † ˆ a ˆ b † ˆ b (cid:69) + 2 (cid:68) ˆ a † ˆ a ˆ b (cid:69)(cid:105) . (29) ddt (cid:68) ˆ b † (cid:69) = 2 iω M (cid:68) ˆ b † (cid:69) − Γ b (cid:68) ˆ b † (cid:69) +2 ig opt (cid:104)(cid:68) ˆ a † ˆ a (cid:69) + 2 (cid:68) ˆ a † ˆ a ˆ b † ˆ b (cid:69) + 2 (cid:68) ˆ a † ˆ a ˆ b † (cid:69)(cid:105) . (30)where n ath , n bth are the thermal mean number of cavity photons and membraneoscillations respectively.2.2 Coupled Equations and Decorrelation of Higher-Order OperatorsThe above set of equations are not closed, constitute higher order operator prod-ucts, and need to be solved numerically with approximations. We proceed to decor-relate all the higher-order correlations in the above equations. The above set ofequations will then be closed up to second order when we apply the decorrelationmethod. We proceed to decorrelate the higher-(third-and fourth-) order quantumcorrelations present in the above equations like [28], which studied the correlationof photon pairs from a double Raman Amplifier; a hybrid quantum optomechan-ical system containing a single semiconductor quantum well [29]. This approachcorresponds to truncation of higher-order operator products in order to solve forall the second-order correlation functions. A similar kind of approximation has alsobeen used in Ref. [30] to study the dynamics of a two-mode BEC beyond meanfield approximation: (cid:104) ABC (cid:105) ≈ [ (cid:104) A (cid:105) (cid:104) BC (cid:105) + (cid:104) AB (cid:105) (cid:104) C (cid:105) + (cid:104) AC (cid:105) (cid:104) B (cid:105) ] (31)and (cid:104) ABCD (cid:105) ≈ [ (cid:104) AB (cid:105) (cid:104) CD (cid:105) + (cid:104) AC (cid:105) (cid:104) BD (cid:105) + (cid:104) AD (cid:105) (cid:104) BC (cid:105) ] . (32) itle Suppressed Due to Excessive Length 7 (cid:68) ˆ a † ˆ a (cid:69) and phonons due to mechanical motion of moving membrane (cid:68) ˆ b † ˆ b (cid:69) for a set of different detuning parameters. For a detuning of the order ofmechanical frequency ω M , cavity is driven around red sideband associated withthe membrane oscillator and we have photonic oscillations (cid:68) ˆ a † ˆ a (cid:69) with phononsidebands (cid:68) ˆ b † ˆ b (cid:69) . Both mean photonic and phonon excitations are asymmetricin shape also due to strong quadratic optomechanical coupling. As the detuningincreases, cavity is driven far from red sideband regime due to which amplitudeof the oscillations for (cid:68) ˆ a † ˆ a (cid:69) are reduced as shown in Figure 1(c). For a very largedetuned cavity, oscillations for (cid:68) ˆ a † ˆ a (cid:69) is completely reduced to its initial value,whereas for (cid:68) ˆ b † ˆ b (cid:69) we have a regular oscillations over the whole range of time scale.So, for a quadratic coupled optomechanical system cavity should be driven in red(blue) sideband regime of membrane oscillator. This is also required for generationof sub-Poissonian light from the driven cavity in our system Hamiltonian.In Figure 2, we observe the effect of cavity decay on the temporal dynamicsof the system. In the weak cavity decay regime, we have undamped oscillationsfor photonic as well as phonon oscillations as shown in Figure 2(a). However, inthe strong cavity decay regime, amplitude of oscillations for both (cid:68) ˆ a † ˆ a (cid:69) as wellas (cid:68) ˆ b † ˆ b (cid:69) show damping with time as shown in the Figure 2(b). Although, in boththe cases decay rate for phonons Γ b remains the same. For strong cavity decay,photons leak out of the cavity very rapidly and this leads to damped oscillationsfor mean phonon number (cid:68) ˆ b † ˆ b (cid:69) also with time.We also study the temporal dynamics for finite thermal phonon numbers inFigure 3. We can see that the mean phonon number (cid:68) ˆ b † ˆ b (cid:69) due to mechanicalmotion of membrane increases and saturates with time due to thermal noise (forfinite n bth ) even for the strong decay regime of cavity as well as mechanical modeas shown in Figure 3(b). So, a finite thermal noise due to phonons is found toincrease the mean number of excitations for the dielectric membrane. This is dueto dependence of mean excitations number of phonons (cid:16)(cid:68) ˆ b † ˆ b (cid:69)(cid:17) on thermal noiseand can be also seen in Eq.A6. In this case mean excitations number for photonicand phonons also oscillate in opposite phase upto certain range of time. Shailendra Kumar Singh, S. V. Muniandy(a) (b)(c) (d)
Fig. 1: (Color online) Mean number of cavity photons (cid:68) ˆ a † ˆ a (cid:69) (blue line) andphonons of moving membrane (cid:68) ˆ b † ˆ b (cid:69) (green line) for chosen set of parameters g opt /ω M = 1 . Γ a /ω M = 0 . Γ b /ω M = 0 . Ω/ω M = 0 . n ath = n bth = 0;where ω M is frequency of mechanical motion of the membrane chosen to 1 in ournumerical simulations. a) ∆ c /ω M = 0 .
5; b) ∆ c /ω M = 1 .
0; c) ∆ c /ω M = 2 .
0; d) ∆ c /ω M = 5 . itle Suppressed Due to Excessive Length 9(a) (b) Fig. 2: (Color online) Mean number of cavity photons (cid:68) ˆ a † ˆ a (cid:69) (blue line) andphonons of moving membrane (cid:68) ˆ b † ˆ b (cid:69) (green line) for chosen set of parameters ∆ c /ω M = 1 . g opt /ω M = 1 . Γ b /ω M = 0 . Ω/ω M = 0 . n ath = n bth = 0; a)Forweak cavity decay Γ a /ω M = 0 .
01; b) For strong cavity decay Γ a /ω M = 0 . (a) (b) Fig. 3: (Color online) Effects of thermal phonons on the mean number of cavityphotons (cid:68) ˆ a † ˆ a (cid:69) (blue line) and phonons of moving membrane (cid:68) ˆ b † ˆ b (cid:69) (green line) forchosen set of parameters ∆ c /ω M = 1 . g opt /ω M = 1 . Ω/ω M = 0 . Γ a /ω M = Γ b /ω M = 0 .
1; a) For n bth = 0 .
0; b) n bth = 2 . In addition to the mean excitation numbers for cavity (cid:16)(cid:68) ˆ a † ˆ a (cid:69)(cid:17) as well as mov-ing membrane (cid:16)(cid:68) ˆ b † ˆ b (cid:69)(cid:17) , we have computed the normalized two-boson correlationfunctions for the cavity field g (2) a (0), quadratically coupled membrane g (2) b (0) , andthe cross-correlation between them g (2) ab (0) using [27,28,29]. g (2) a (0) = (cid:68) ˆ a † ˆ a (cid:69)(cid:10) ˆ a † ˆ a (cid:11) ≈ (cid:68) ˆ a † ˆ a (cid:69) (cid:68) ˆ a † ˆ a (cid:69) + (cid:68) ˆ a † (cid:69) (cid:10) ˆ a (cid:11)(cid:10) ˆ a † ˆ a (cid:11) . (33) g (2) b (0) = (cid:68) ˆ b † ˆ b (cid:69)(cid:68) ˆ b † ˆ b (cid:69) ≈ (cid:68) ˆ b † ˆ b (cid:69) (cid:68) ˆ b † ˆ b (cid:69) + (cid:68) ˆ b † (cid:69) (cid:68) ˆ b (cid:69)(cid:68) ˆ b † ˆ b (cid:69) . (34) g (2) ab (0) = (cid:68) ˆ a † ˆ b † ˆ b ˆ a (cid:69)(cid:68) ˆ b † ˆ b (cid:69) (cid:10) ˆ a † ˆ a (cid:11) ≈ (cid:68) ˆ a † ˆ b (cid:69) (cid:68) ˆ b † ˆ a (cid:69) + (cid:68) ˆ b † ˆ b (cid:69) (cid:68) ˆ a † ˆ a (cid:69) + (cid:68) ˆ a † ˆ b † (cid:69) (cid:68) ˆ b ˆ a (cid:69)(cid:68) ˆ b † ˆ b (cid:69) (cid:10) ˆ a † ˆ a (cid:11) . (35) If g (2) a (0), g (2)) b (0) and g (2) ab (0) satisfy the inequality g (2) X (0) < X = a, b, ab ,then the statistics of the bosonic systems are referred to as sub-Poissonian . Statis-tics with g (2) X (0) = 1 and g (2) X (0) > Poissonian and super-Poissonian, respectively. Here, we have studied the temporal dynamics ofsecond-order correlation functions, namely the self-correlation and cross correla-tion for cavity and oscillating membrane modes.In photon blockade effect, where coupling of a single photon to the systemhinders the coupling of the subsequent photons, we have g (2) a (0) < g (2) a (0) > g (2) a (0), g (2) b (0) and g (2) ab (0) as well as under different decay regimes for cavity photons aswell as phonons due to micromechanical motion.The second order correlation functions, namely the self- and cross-correlationsfor photonic and membrane oscillations are shown in Figure 4. We study the tem-poral dynamics of g (2) a (0), g (2)) b (0) and g (2) ab (0) for different cavity detunings ∆ c .It can be seen that for a resonant cavity i.e. ∆ c = 0 and for a shorter period oftime ω M t ∼
2, all the three correlations follow sub-Poissonian photon statistics asshown in Figure 4 (a). As ω M t increases, g (2) a (0) tends to become super-Poissonian,whereas g (2) b (0) oscillates from sub-Poissonian to super-Poissonian. For a finite de- tuning ∆ c ∼ ω M , g (2) a (0) as well as g (2)) b (0) both oscillate periodically from sub-Poissonian to super-Poissonian whereas g (2) ab (0) mostly remains sub-Poissonian asshown in Figure 4(b). For a very far off detuned cavity, g (2) a (0) oscillates from Pois-sonian to super-Poissonian and it never becomes sub-Poissonian, whereas g (2)) b (0)behaves almost the same and g (2)) ab (0) oscillates from sub-Poissonian to Poissonian itle Suppressed Due to Excessive Length 11 as shown in Figure 4(c) and Figure 4(d). For a very large cavity detuning, a sin-gle photon can not be resonantly excited into the cavity, while the probability offinding two or more photons resonantly enhanced [26]. So, we do not have photonblockade effect for larger cavity detuning at any period of time ω M t .Similarly, for a finite detuning ∆ c , we study the effect of optomechanical cou-pling strengths g opt on various correlations in Figure 5. For the case g opt compa-rable to driving field Ω , g (2)) a (0) and g (2)) b (0) oscillate from sub-Poissonian tosuper-Poissonian, whereas g (2)) ab (0) remains mostly sub-Poissonian except at somepoints as shown in Figure 5(a) and Figure 5(b). For a very strong optomechanicalcoupling g opt , g (2)) a (0) oscillates from Poissonian to super-Poissonian, whereas g (2)) b (0) remains same and g (2)) ab (0) varies to Poissonian limit as shown in Figure5(d).Furthermore, we examine the effects of photonic and phonon decay rates onvarious two-boson correlations in Figure 6. For very weak decay rates of cavityand membrane oscillation, g (2)) a (0) varies from Poissonian to super-Poissonian andnever becomes sub-Poissonian. Although g (2)) b (0) oscillates from sub-Poissonian tosuper-Poissonian and g (2)) ab (0) remains mostly sub-Poissonian as shown in Figure6(a). For a strong cavity decay rate Γ a , g (2)) a (0) becomes sub-Poissonian over alarge scale of time and even becomes smaller than 0 . Γ a /ω M < Γ a /ω M << Fig. 4: Second-order autocorrelations g (2) a (0) (black solid line) for cavity mode, g (2) b (0) for moving membrane (red dashed line) and g (2) ab (0) (blue dotted line)corresponding cross-correlation between photons and phonons for chosen set ofparameters g opt /ω M = 1 . Γ a /ω M = 0 . Γ b /ω M = 0 . Ω/ω M = 0 . n ath = n bth = 0; a) ∆ c /ω M = 0 .
0; b) ∆ c /ω M = 1 .
3; c) ∆ c /ω M = 2 .
5; d) ∆ c /ω M = 4 . itle Suppressed Due to Excessive Length 13(a) (b)(c) (d) Fig. 5: Second-order autocorrelations g (2) a (0) (black solid line) for cavity mode, g (2) b (0) for moving membrane (red dashed line) and g (2) ab (0) (blue dotted line)corresponding cross-correlation between photons and phonons for chosen set ofparameters ∆ c /ω M = 0 . Γ a /ω M = 0 . Γ b /ω M = 0 . Ω/ω M = 0 . n ath = n bth = 0; a) g opt /ω M = 0 .
8; b) g opt /ω M = 1 .
7; c) g opt /ω M = 3 .
0; d) g opt /ω M = 5 . Fig. 6: Second-order autocorrelations g (2) a (0) (black solid line) for cavity mode, g (2) b (0) for moving membrane (red dashed line) and g (2) ab (0) (blue dotted line)corresponding cross-correlation between photons and phonons for chosen set ofparameters ∆ c /ω M = 1 . g opt /ω M = 1 . Ω/ω M = 0 . n ath = n bth = 0 underdifferent decay regimes; a) Γ a /ω M = 0 . Γ b /ω M = 0 . Γ a /ω M = 0 . Γ b /ω M = 0 . Γ a /ω M = Γ b /ω M = 0 . itle Suppressed Due to Excessive Length 15 We have studied a qudratically coupled optomechanical system formed by a mi-cropillar with moveable Bragg reflectors and a thin dielectric membrane at the node(or antinode) of the resonator through the Heisenberg- Langevin approach. Wehave obtained coupled equations up to the second order without any approxima-tion. We decorrelate the higher-order correlation functions in the coupled equationsto obtain a closed set of equations. This enables the study of temporal dynamicsof the optomechanical system as well as two-boson correlation functions under dif-ferent scenarios. In the resolved sideband regime, mean number of cavity as well asmechanical excitation show oscillations. Furthermore, amplitude of these oscilla-tions changes as we change the cavity detuning. We have also studied the temporaldynamics under the different decay regimes of cavity and membrane oscillations.We have discussed the effect of thermal noise due to phonons on the dynamicsof the optomechanical system. Furthermore, we analyzed the temporal dynam-ics of the second-order correlation functions, namely the self and cross-correlationof the cavity and mechanical modes under different cavity detunings as well asoptomechanical coupling strengths. We obtained the regimes where all the threecorrelations display strong sub-Poissonian photon statistics particularly at verysmall cavity detuning as well as strong cavity decay rate. Our study is useful forcoherent control of photon statistics as well as photon and phonon correlations inquadratically coupled optomechanical systems.
Acknowledgments
The authors thank University of Malaya for financial support under the Universityof Malaya Research Grant (RG231-12AFR and RP006A-13AFR). This work alsoget supported through the postdoctoral fellowship position held by ShailendraKumar Singh in T ¨UB˙ITAK-1001 Grant No. 114F170. SKS would like to gratefullyacknowledge Dr. Mehmet Emre Tasgin for his kind help to introduce the subjectof cavity optomechanics in details. SKS also kindly acknowledge to Hacettepeuniversity colleague Mr. Saidul Alom Mozumdar for his help to improve manuscriptgrammatically.
References
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A Decorrelated and Closed set of Coupled Equations ddt (cid:104) ˆ a (cid:105) = − i∆ c (cid:104) ˆ a (cid:105) − iΩ − Γ a (cid:104) ˆ a (cid:105) − ig opt (cid:104)(cid:110) (cid:104) ˆ a (cid:105) (cid:68) ˆ b † (cid:69) + 2 (cid:68) ˆ a ˆ b † (cid:69) (cid:68) ˆ b † (cid:69)(cid:111) (36)+ (cid:110) (cid:104) ˆ a (cid:105) (cid:68) ˆ b (cid:69) + 2 (cid:68) ˆ a ˆ b (cid:69) (cid:68) ˆ b (cid:69)(cid:111) + 2 (cid:110) (cid:104) ˆ a (cid:105) (cid:68) ˆ b † ˆ b (cid:69) + (cid:68) ˆ a ˆ b † (cid:69) (cid:68) ˆ b (cid:69) + (cid:68) ˆ a ˆ b (cid:69) (cid:68) ˆ b † (cid:69)(cid:111) + (cid:104) ˆ a (cid:105) (cid:105) . itle Suppressed Due to Excessive Length 17 ddt (cid:68) ˆ a † (cid:69) = i∆ c (cid:68) ˆ a † (cid:69) + iΩ − Γ a (cid:68) ˆ a † (cid:69) + ig opt (cid:104)(cid:110)(cid:68) ˆ a † (cid:69) (cid:68) ˆ b † (cid:69) + 2 (cid:68) ˆ a † ˆ b † (cid:69) (cid:68) ˆ b † (cid:69)(cid:111) (37)+ (cid:110)(cid:68) ˆ a † (cid:69) (cid:68) ˆ b (cid:69) + 2 (cid:68) ˆ a † ˆ b (cid:69) (cid:68) ˆ b (cid:69)(cid:111) + 2 (cid:110)(cid:68) ˆ a † (cid:69) (cid:68) ˆ b † ˆ b (cid:69) + (cid:68) ˆ a † ˆ b † (cid:69) (cid:68) ˆ b (cid:69) + (cid:68) ˆ a † ˆ b (cid:69) (cid:68) ˆ b † (cid:69)(cid:111) + (cid:68) ˆ a † (cid:69)(cid:105) .ddt (cid:68) ˆ b (cid:69) = − iω M (cid:68) ˆ b (cid:69) − Γ b (cid:68) ˆ b (cid:69) − ig opt (cid:104)(cid:68) ˆ a † (cid:69) (cid:110)(cid:68) ˆ a ˆ b (cid:69) + (cid:68) ˆ a ˆ b † (cid:69)(cid:111) + (cid:68) ˆ a † ˆ a (cid:69) (cid:110)(cid:68) ˆ b (cid:69) + (cid:68) ˆ b † (cid:69)(cid:111) + (cid:104) ˆ a (cid:105) (cid:110)(cid:68) ˆ a † ˆ b (cid:69) + (cid:68) ˆ a † ˆ b † (cid:69)(cid:111)(cid:105) . (38) ddt (cid:68) ˆ b † (cid:69) = iω M (cid:68) ˆ b † (cid:69) − Γ b (cid:68) ˆ b † (cid:69) + 2 ig opt (cid:104)(cid:68) ˆ a † (cid:69) (cid:110)(cid:68) ˆ a ˆ b (cid:69) + (cid:68) ˆ a ˆ b † (cid:69)(cid:111) + (cid:68) ˆ a † ˆ a (cid:69) (cid:110)(cid:68) ˆ b (cid:69) + (cid:68) ˆ b † (cid:69)(cid:111) + (cid:104) ˆ a (cid:105) (cid:110)(cid:68) ˆ a † ˆ b (cid:69) + (cid:68) ˆ a † ˆ b † (cid:69)(cid:111)(cid:105) . (39) ddt (cid:68) ˆ a † ˆ a (cid:69) = − iΩ (cid:16)(cid:68) ˆ a † (cid:69) − (cid:104) ˆ a (cid:105) (cid:17) − Γ a (cid:68) ˆ a † ˆ a (cid:69) + Γ a ¯ n ath . (40) ddt (cid:68) ˆ b † ˆ b (cid:69) = − ig opt (cid:104)(cid:110)(cid:68) ˆ a † ˆ a (cid:69) (cid:68) ˆ b † (cid:69) + 2 (cid:68) ˆ a † ˆ b † (cid:69) (cid:68) ˆ a ˆ b † (cid:69)(cid:111) − (cid:110)(cid:68) ˆ a † ˆ a (cid:69) (cid:68) ˆ b (cid:69) + 2 (cid:68) ˆ a † ˆ b (cid:69) (cid:68) ˆ a ˆ b (cid:69)(cid:111)(cid:105) − Γ b (cid:68) ˆ b † ˆ b (cid:69) + Γ b ¯ n bth . (41) ddt (cid:68) ˆ a ˆ b † (cid:69) = i ( ω M − ∆ c ) (cid:68) ˆ a ˆ b † (cid:69) − iΩ (cid:68) ˆ b † (cid:69) −
12 ( Γ a + Γ b ) (cid:68) ˆ a ˆ b † (cid:69) + ig opt (cid:104) (cid:110) (cid:68) ˆ a † ˆ a (cid:69) (cid:68) ˆ a ˆ b (cid:69) + (cid:68) ˆ a † ˆ b (cid:69) (cid:10) ˆ a (cid:11)(cid:111) − (cid:110)(cid:68) ˆ a ˆ b † (cid:69) (cid:68) ˆ b (cid:69) + 2 (cid:68) ˆ a ˆ b (cid:69) (cid:68) ˆ b † ˆ b (cid:69)(cid:111) +2 (cid:110)(cid:16) (cid:68) ˆ a † ˆ a (cid:69) (cid:68) ˆ a ˆ b † (cid:69) + (cid:68) ˆ a † ˆ b † (cid:69) (cid:10) ˆ a (cid:11)(cid:17) − (cid:16) (cid:68) ˆ b † ˆ b (cid:69) (cid:68) ˆ a ˆ b † (cid:69) + (cid:68) ˆ a ˆ b (cid:69) (cid:68) ˆ b † (cid:69)(cid:17)(cid:111) − (cid:68) ˆ a ˆ b † (cid:69) (cid:16) (cid:68) ˆ b † (cid:69) + 1 (cid:17)(cid:105) . (42) ddt (cid:68) ˆ a † ˆ b (cid:69) = i ( ∆ c − ω M ) (cid:68) ˆ a † ˆ b (cid:69) + iΩ (cid:68) ˆ b (cid:69) −
12 ( Γ a + Γ b ) (cid:68) ˆ a † ˆ b (cid:69) + ig opt (cid:104)(cid:110)(cid:68) ˆ a † ˆ b (cid:69) (cid:68) ˆ b † (cid:69) + 2 (cid:68) ˆ a † ˆ b † (cid:69) (cid:68) ˆ b † ˆ b (cid:69)(cid:111) − (cid:110)(cid:68) ˆ a † (cid:69) (cid:68) ˆ a ˆ b † (cid:69) + 2 (cid:68) ˆ a † ˆ a (cid:69) (cid:68) ˆ a † ˆ b † (cid:69)(cid:111) +2 (cid:110)(cid:16)(cid:68) ˆ a † ˆ b † (cid:69) (cid:68) ˆ b (cid:69) + 2 (cid:68) ˆ a † ˆ b (cid:69) (cid:68) ˆ b † ˆ b (cid:69)(cid:17) − (cid:16)(cid:68) ˆ a † (cid:69) (cid:68) ˆ a ˆ b (cid:69) + 2 (cid:68) ˆ a † ˆ b (cid:69) (cid:68) ˆ a † ˆ a (cid:69)(cid:17)(cid:111) + (cid:68) ˆ a † ˆ b (cid:69) (cid:16) (cid:68) ˆ b (cid:69) + 1 (cid:17)(cid:105) . (43) ddt (cid:68) ˆ a ˆ b (cid:69) = − i ( ∆ c + ω M ) (cid:68) ˆ a ˆ b (cid:69) − iΩ (cid:68) ˆ b (cid:69) −
12 ( Γ a + Γ b ) (cid:68) ˆ a ˆ b (cid:69) − ig opt (cid:104) (cid:68) ˆ a ˆ b † (cid:69) + 3 (cid:68) ˆ a ˆ b (cid:69) (cid:16)(cid:68) ˆ b (cid:69) + 1 (cid:17)(cid:105) − ig opt (cid:104)(cid:68) ˆ a ˆ b (cid:69) (cid:68) ˆ b † (cid:69) + 2 (cid:68) ˆ a ˆ b † (cid:69) (cid:68) ˆ b † ˆ b (cid:69)(cid:105) − ig opt (cid:104)(cid:16) (cid:68) ˆ a † ˆ a (cid:69) (cid:68) ˆ a ˆ b † (cid:69) + (cid:68) ˆ a † ˆ b † (cid:69) (cid:10) ˆ a (cid:11)(cid:17) + (cid:16)(cid:68) ˆ a ˆ b † (cid:69) (cid:68) ˆ b (cid:69) + 2 (cid:68) ˆ a ˆ b (cid:69) (cid:68) ˆ b † ˆ b (cid:69)(cid:17) + (cid:16) (cid:68) ˆ a † ˆ a (cid:69) (cid:68) ˆ a ˆ b (cid:69) + (cid:68) ˆ a † ˆ b (cid:69) (cid:10) ˆ a (cid:11)(cid:17)(cid:105) . (44) ddt (cid:68) ˆ a † ˆ b † (cid:69) = i ( ∆ c + ω M ) (cid:68) ˆ a † ˆ b † (cid:69) + iΩ (cid:68) ˆ b † (cid:69) −
12 ( Γ a + Γ b ) (cid:68) ˆ a † ˆ b † (cid:69) + ig opt (cid:104) (cid:68) ˆ a † ˆ b (cid:69) + 3 (cid:68) ˆ a † ˆ b † (cid:69) (cid:16)(cid:68) ˆ b † (cid:69) + 1 (cid:17)(cid:105) + ig opt (cid:104)(cid:68) ˆ a † ˆ b † (cid:69) (cid:68) ˆ b (cid:69) + 2 (cid:68) ˆ a † ˆ b (cid:69) (cid:68) ˆ b † ˆ b (cid:69)(cid:105) +2 ig opt (cid:104)(cid:16)(cid:68) ˆ a † (cid:69) (cid:68) ˆ a ˆ b (cid:69) + 2 (cid:68) ˆ a † ˆ a (cid:69) (cid:68) ˆ a † ˆ b (cid:69)(cid:17) + (cid:16)(cid:68) ˆ b † (cid:69) (cid:68) ˆ a † ˆ b (cid:69) + 2 (cid:68) ˆ a † ˆ b † (cid:69) (cid:68) ˆ b † ˆ b (cid:69)(cid:17) + (cid:16) (cid:68) ˆ a † ˆ a (cid:69) (cid:68) ˆ a † ˆ b † (cid:69) + (cid:68) ˆ a ˆ b † (cid:69) (cid:68) ˆ a † (cid:69)(cid:17)(cid:105) . (45)8 Shailendra Kumar Singh, S. V. Muniandy ddt (cid:10) ˆ a (cid:11) = − i∆ c (cid:10) ˆ a (cid:11) − iΩ (cid:104) ˆ a (cid:105) − Γ a (cid:10) ˆ a (cid:11) − ig opt (cid:10) ˆ a (cid:11) (46) − ig opt (cid:20)(cid:10) ˆ a (cid:11) (cid:16)(cid:68) ˆ b † (cid:69) + (cid:68) ˆ b (cid:69) + 2 (cid:68) ˆ b † ˆ b (cid:69)(cid:17) + 2 (cid:18)(cid:68) ˆ a ˆ b † (cid:69) + (cid:68) ˆ a ˆ b (cid:69) + 2 (cid:68) ˆ a ˆ b (cid:69) (cid:68) ˆ a ˆ b † (cid:69)(cid:19)(cid:21) .ddt (cid:68) ˆ a † (cid:69) = 2 i∆ c (cid:68) ˆ a † (cid:69) + 2 iΩ (cid:68) ˆ a † (cid:69) − Γ a (cid:68) ˆ a † (cid:69) + 2 ig opt (cid:68) ˆ a † (cid:69) (47)+2 ig opt (cid:20)(cid:68) ˆ a † (cid:69) (cid:16)(cid:68) ˆ b † (cid:69) + (cid:68) ˆ b (cid:69) + 2 (cid:68) ˆ b † ˆ b (cid:69)(cid:17) + 2 (cid:18)(cid:68) ˆ a † ˆ b † (cid:69) + (cid:68) ˆ a † ˆ b (cid:69) + 2 (cid:68) ˆ a † ˆ b † (cid:69) (cid:68) ˆ a † ˆ b (cid:69)(cid:19)(cid:21) .ddt (cid:68) ˆ b (cid:69) = − iω M (cid:68) ˆ b (cid:69) − Γ b (cid:68) ˆ b (cid:69) − ig opt (cid:68) ˆ a † ˆ a (cid:69) (48) − ig opt (cid:104)(cid:68) ˆ a † ˆ a (cid:69) (cid:16)(cid:68) ˆ b (cid:69) + (cid:68) ˆ b † ˆ b (cid:69)(cid:17) + (cid:68) ˆ a ˆ b (cid:69) (cid:16)(cid:68) ˆ a † ˆ b † (cid:69) + 2 (cid:68) ˆ a † ˆ b (cid:69)(cid:17) + (cid:68) ˆ a † ˆ b (cid:69) (cid:68) ˆ a ˆ b † (cid:69)(cid:105) .ddt (cid:68) ˆ b † (cid:69) = 2 iω M (cid:68) ˆ b † (cid:69) − Γ b (cid:68) ˆ b † (cid:69) + 2 ig opt (cid:68) ˆ a † ˆ a (cid:69) (49)+4 ig opt (cid:104)(cid:68) ˆ a † ˆ a (cid:69) (cid:16)(cid:68) ˆ b † (cid:69) + (cid:68) ˆ b † ˆ b (cid:69)(cid:17) + (cid:68) ˆ a † ˆ b † (cid:69) (cid:16)(cid:68) ˆ a ˆ b (cid:69) + 2 (cid:68) ˆ a ˆ b † (cid:69)(cid:17) + (cid:68) ˆ a † ˆ b (cid:69) (cid:68) ˆ a ˆ b † (cid:69)(cid:105)(cid:69)(cid:105)