Kerr-like nonlinearities in an optomechanical system with an asymmetric anharmonic mechanical resonator
aa r X i v : . [ qu a n t - ph ] D ec Bogoliubov averaging and effective Hamiltonians for optomechanical systems withasymmetric anharmonic mechanical resonators
A.P. Saiko ∗ and S.A. Markevich Scientific-Practical Material Research Centre, Belarus National Academy of Sciences, 19 P. Brovka str., Minsk 220072 Belarus
R. Fedaruk
Institute of Physics, University of Szczecin, 15 Wielkopolska str., 70-451, Szczecin, Poland (Dated: December 18, 2020)A typical optomechanical system with a mechanical resonator realizing anharmonic oscillationsin linear and cubic potentials is studied. Using the Bogoliubov averaging method in the non-secular perturbation theory, the effective Hamiltonian of the system is constructed. The cross-Kerrinteraction of photons and vibration quanta as well as the Kerr-like mechanical self-interaction arisesin the Hamiltonian. The Kerr and Kerr-like interactions are induced by both the cubic nonlinearityof oscillations of the mechanical resonator and the cavity-resonator interaction linear in mechanicaldisplacements. This approach correctly describes also the hybrid system consisting of a quantum dotand a nanocavity mediated by a mechanical resonator without leading to non-Hermitian terms inthe effective Hamiltonian. The obtained results offer new possibilities for describing optomechanicalsystems with asymmetric mechanical oscillations.
PACS numbers: 42.50.Pq, 85.85.+j, 85.25.Sp, 45.80.+r
I. INTRODUCTION
Optomechanical systems offer the possibility for con-trol of light by mechanical motion and vice versa. Thecoupling between light and mechanical resonator vibra-tions is usually achieved via light pressure. The cou-pling can change the resonant frequency of the mechan-ical resonator and its damping. The latter can be usedfor cooling [1–3] or amplification [4]. The nonlinearity ofthe optomechanical interaction enables the realization ofquantum squeezed states [5]. These states may be cre-ated only in the strong-coupling regime [6], where a cou-pling constant is larger than the decay rates of the cav-ity and the mechanical resonator. Experimental resultsdemonstrate that the single-photon optomechanical cou-pling cannot be operated in this regime [6]. Instead, themulti-photon strong coupling regime is accessible under astrong driving field on the cavity mode and the majorityof the observed physical phenomena can be understoodusing a linear description [6]. In this case, nonclassi-cal states [7, 8], quantum entanglement [9–11], quantumstate transfer [12–14], optomechanically induced trans-parency [15–17], and normal-mode splitting [18, 19] havebeen investigated. The optomechanical interaction is in-trinsically nonlinear [6]. In experiments this nonlinearityso far has played only a role in the classical regime oflarge amplitude light and mechanical oscillations.The cavity-resonator interaction can be enhanced, forexample, via the non-linearity of the Josephson effect[20]. This non-linearity leads to an additional nonlinearinteraction, namely, a cross-Kerr coupling between thecavity and the resonator. The higher-order interactionsin the displacement have been investigated in differentsetups such as the membrane in the middle geometries [21–23]. The cross-Kerr coupling induces a change to therefractive index of the cavity depending on the number ofvibration quanta in the resonator, whereas the radiationpressure coupling gives rise to the Kerr effect dependingon the displacement of the resonator. In nonlinear optics[24], the Kerr effect usually appears in nonlinear disper-sive media due to third-order matter-light interactions. Itwas shown [25] that the Kerr nonlinear Hamiltonian canbe formulated explicitly using the Born-Oppenheimer ap-proximation for a standard Hamiltonian of optomechan-ical system (in the frame rotating with the driving fieldfrequency). The same result was obtained using polaron-like transformation [26, 27].Mechanical resonators of optomechanical systems areusually modeled by harmonic oscillators or rarely by non-linear Duffing oscillators with fourth-order nonlinearity,which generally is very weak. However, nonlinearities ofmechanical resonators can significantly be increased bytechnological improvements of used materials, geometryof the system and additional adaptations [28–30]. Formechanical resonators, the role of asymmetric potentials,which are non-invariant under reversal of displacementsign, has been less studied until now. We mean oscil-lators with potentials including linear and cubic termsin displacement. Optomechanical systems, in which themechanical resonator is modeled by the harmonic oscilla-tor with the Coulomb-interaction-dependent linear forc-ing term, have been considered in [31–33]. In the frame ofthis model in optomechanics, the possibility of precisionmeasuring electrical charge with optomechanically in-duced transparency [31], Coulomb-interaction-dependenteffect of high-order sideband generation [32] as well asforce-induced transparency and conversion between slowand fast lights [33] have been studied. Recently, the elec-tromagnetically induced transparency with a cubic non-linear movable mirror has been considered [34]. Steady-state mechanical squeezing via Duffing and cubic nonlin-earities was analyzed [35]. Probing Duffing and cubic an-harmonicities of quantum oscillators in an optomechani-cal cavity was also studied in [36].In the present paper, we show that anharmonic me-chanical oscillations, linear and cubic in displacements,together with the cavity-resonator interaction, linear indisplacements, directly result in the Kerr and cross-Kerreffects in optomechanical systems. The qualitative andquantitative description of these effects is significant forunderstanding physical behavior of such systems. It isassumed that the frequencies of mechanical oscillationsexceed the parameters, characterizing the interaction be-tween the optical and mechanical subsystems, and thevalue of asymmetric mechanical anharmonicities. Thismakes it possible to replace the original system withsome effective one described by the corresponding effec-tive Hamiltonian. Indeed, we show that using the Bogoli-ubov averaging method in the frame of the non-secularperturbation theory, the original system can be replacedby the effective one described by an approximately diag-onal or diagonal effective Hamiltonian (Section II). Thisapproach is also applied to study the hybrid system con-sisting of a quantum dot and a nanocavity mediated bya mechanical resonator (Section III).
II. MECHANICAL RESONATOR WITH AASYMMETRIC ANHARMONICITY
The optomechanical system under study is expectedto generate the Kerr and cross-Kerr effects in a Fabry-Perot cavity, which has an oscillating mirror in one end.The movable mirror is modeled by a weakly nonlinearquantum mechanical oscillator in a cubic potential. Theoscillations of the mirror are controlled by the cavity field(and the driving laser field) via radiation pressure. Thiscoupling is proportional to the field intensity and approx-imately linear in the displacement.The basic physics of the optomechanical system can becaptured in the following typical Hamiltonian [6]: H = H + V, (1) H = ω c a † a + Ω b † b, V = − ga † a ( b † + b ) , where ω c is the cavity frequency, Ω is the mechanicalfrequency, g is the optomechanical coupling, and a ( b )represents the cavity’s photon (vibration quantum) an-nihilation operator (we take the Planck constant ~ = 1).A photon that has entered the cavity decays at a rate κ either by transmission at one of the mirrors or due to ab-sorptive losses inside the cavity. The decay of mechanical energy is characterized by a damping rate γ . We con-sider oscillations of the mechanical resonator in a poten-tial with weak linear and cubic terms in displacements.It is taken into account by the following Hamiltonian H : V anh = F ( b + b † ) + 16 Λ( b + b † ) , (2)where F = f (1 / m Ω) / , Λ = λ (1 / m Ω) / , m is themass of mechanical resonator, and f and λ are parame-ters describing the values of linear and cubic anharmonic-ities, respectevily. These anharmonicities can be real-ized by specific construction of the mechanical resonator.In the interaction representation, nonlinear members aregiven by e iH t ( V + V anh ) e − iH t = − ga † a ( b † e i Ω t + H.c. )++ 16 Λ( b † b † b † e i t + b † bb † e i Ω t + H.c. )++ F ( b † e i Ω t + H.c. ) ≡ H int ( t ) . (3)Since for real optomechanical systems unequalitiesΩ ≫ g, F, Λ are well fulfilled, we can use the non-secularperturbation theory for averaging fast oscillations e ± i Ω t and e ± i t in the time-dependent interaction Hamilto-nian (3) and to obtain an approximately diagonal or di-agonal time-independent effective Hamiltonian. In thecanonical form it can be realized using the Bogoliubovaveraging method [37–39]. Averaging up to the secondorder in small parameters g/ Ω, F/ Ω, Λ / Ω we obtain H int → H eff int = H effint, + H effint, , where H effint, = < H int ( t ) >,H effint, = i h [ Z t dτ ( H int ( τ ) − < H int ( τ ) > ) , H int ( t )] i . (4)Here the symbol h ... i denotes time averaging over rapidoscillations of the type e ± i Ω t , e ± i t given by h O ( t ) i = Ω2 π R π /Ω0 O ( t ) dt and the upper limit t of the indefiniteintegral indicates the variable on which the result of theintegration depends, and square brackets denote the com-mutation operation.Calculations based on Eq. (4) give H effint, = 0 ,H effint, = g (Λ + 2 F )Ω a † a − / F Ω b † b −− g Ω a † aa † a + 2 g ΛΩ a † ab † b − b † bb † b. (5)In the frame rotating with some frequency ω d relativeto the laboratory frame, the total effective Hamiltonian H eff is H eff = (∆ + g Λ + 2 gF Ω ) a † a ++ (Ω − / F Ω ) b † b −− g Ω a † aa † a + 2 g ΛΩ a † ab † b −− b † bb † b. (6)The frequency ω d can be the optical driving field fre-quency and usually ω c − ω d ≡ ∆ ∼ Ω. The first andsecond terms in Eq. (6) present the free Hamiltoniansof the cavity in the rotating frame and the mechani-cal resonator with renormalized frequencies due to linearand cubic anharmonicities of the oscillations of the me-chanical subsystem. The third term describes the Kerrinteraction of photons in the cavity. The fourth termrepresents the cross-Kerr interaction of photons and vi-bration quanta induced by interference contribution ofboth the cubic nonlinearity of oscillations of the me-chanical resonator and the cavity-resonator interactionlinear in mechanical displacements. The fifth term de-scribes the Kerr-like mechanical self-interaction of theresonator. Hence, the back-action of linear oscillationsof the mirror (the mechanical resonator) in the Fabry-Perot cavity results in the Kerr effect for the cavity field( g / Ω) a † aa † a , with g / Ω the Kerr frequency shift perphoton. At the same time, even weak asymmetric an-harmonicities induce the cross-Kerr effect of the cavityand the resonator(2 g Λ / Ω) a † ab † b , with 2 g Λ / Ω the cross-Kerr frequency shift per photon or per vibration quan-tum. The frequency shifts due to the induced cross-Kerreffect depend on the number of photons in the cavity andof vibration quanta in the resonator.Taking into account driving and dissipation, the quan-tum master equation for the density matrix ρ of our sys-tem is dρdt = − i [ H eff + H d , ρ ] + L c ρ + L m ρ (7)with the driving Hamiltonian H d = iε ( a † e − iω d t − H.c. ).The incoherent coupling of the system with its environ-ment is modeled [40] by the Lindblad dissipators L c ρ = κ aρa † − a † aρ − ρa † a ) ,L m ρ = γ N + 1)(2 bρb † − b † bρ − ρb † b )++ γ N (2 b † ρb − bb † ρ − ρbb † ) , (8) where κ and γ are the decay rates of the cavity and themechanical oscillator, ε = p P κ/ ~ ω c with P being theinput power of the driving field. We assume here a zero-temperature bath for the optical cavity and a mean ther-mal occupation ¯ N = [exp(Ω /k B T ) − − of the mechan-ical resonator bath at the frequency Ω. Using Eqs.(6)-(8), the equations of motion for mean values of dynamicalvariables in the frame rotating with the driving field fre-quency can be written as d h a i dt = − i (∆ + g Λ + 2 gF Ω − g Ω ) h a i − κ h a i ++ i g Ω h a † aa i − i g ΛΩ h ab † b i + ε,d h b † b i dt = − γ ( h b † b i − ¯ N ) , (9)where ∆ = ω c − ω d . In the steady state we have0 = − i (∆ + g Λ + 2 gF Ω − g Ω ) h a i − κ h a i ++ i g Ω h a † aa i − i g ΛΩ h ab † b i + ε, − γ ( h b † b i − ¯ N ) . (10)As an example, consider the semiclassical approximation h a i = α , h a † aa i = | α | α , h ab † b i = α h b † b i . Then it isstraightforward to show that the mean photon number¯ n = | α | satisfies the following nonlinear equation: ( κ (cid:20) ∆ + 2 gF Ω + 2 g ΛΩ ( ¯ N + 12 ) − g Ω (¯ n + 12 ) (cid:21) ) ×× ¯ n = ε . (11)Thus, due to the cross-Kerr interaction,the temperature-dependent occupation ¯ N =[exp(Ω /k B T ) − − of the mechanical resonatorbath appears in Eq. (11) for the averaged photonnumber. As a result, there is the parameter in Eq.(11) with the evident dependence on the thermostattemperature. In special cases, this third order non-linearequation relative to ¯ n can result in a hysteresis behaviorof the mean photon number vs power of the driving field. III. A QUANTUM DOT AND A NANOCAVITYMEDIATED BY A MECHANICAL RESONATOR
A hybrid system [41] composed of a quantized singlemode of a mechanical resonator interacting linearly withboth a single mode nanocavity and a two-level singlequantum dot was studied in [42]. It was assumed thata linear coupling is between the nanocavity and the me-chanical resonator as well as between the single-excitonquantum dot and the mechanical resonator, but thereis no direct interaction between the nanocavity and thequantum dot. In the limit of large cavity-resonator fre-quency detuning and at the emitter-cavity resonance, theeffective Hamiltonian was obtained. Such Hamiltonianis able to retrieve much of the physics of the conven-tional Jaynes-Cummings model. This means that theoriginal model makes it possible to achieve the indirectlight-matter strong coupling through the mediation of vi-bration quanta. The effective Hamiltonian was obtainedin [42] using a formal solution of the Schrodinger equationby means the evolution operator, its expansion in a seriesin the frame of perturbation theory and performing time-ordering integration. The obtained effective Hamiltonianis the Hermitian one only when the cavity and excitonrecombination frequencies are equal to each other [42].Here we show that consistent application of the Bogoli-ubov averaging method allows us to obtain the Hermitianeffective Hamiltonian for this model without restrictionsby the foregoing resonant condition.A Hamiltonian of the hybrid tripartite system com-posed of a mechanical resonator interacting linearly withboth a single mode nanocavity and a two-level singlequantum dot [42] is H = H + H int , (12)where H = ω c a † a + ω a σ + σ − + ω m b † b, (13) H int = g cm ( ab † + a † b ) + g am ( σ − b † + σ + b ) , (14) ω c , ω a and ω m are the cavity, exciton recombination andmechanical frequencies, respectively; a ( a † ) and b ( b † ) arethe annihilation (creation) bosonic operators for photonsand vibration quantum; σ ± , σ z are Pauli operators. Inthe interaction representation the Hamiltonian (14) canbe written as H int → H int ( t ) = e iH t H int e − iH t == g cm ( a † be i ∆ c t + H.c. ) + g am ( σ + be i ∆ a t + H.c. ) , (15)where ∆ c = ω c − ω m and ∆ a = ω a − ω m . We as-sume that ω m /ω c ∼ − , g am /ω a ≈ g cm /ω c > − , | ω a − ω c | < g am , g cm . Then, the terms with e ± i ∆ c t and e ± i ∆ a t in Eq. (15) are fast oscillating and the Bogoliubovaveraging method is applicable to construct the effectiveHamiltonian using Eq. (4). The first order of the non-secular perturbation theory gives zero contribution to theeffective Hamiltonian. In the second order of the theorywe obtain in the laboratory frame H eff = H ′ + H ′ int , (16) H ′ = ( ω c + g cm ∆ c ) a † a + ( ω a + g am ∆ a ) σ + σ − + ( ω m − g cm ∆ c ) b † b, (17) H ′ int = 12 g am g cm ( 1∆ a + 1∆ c )( a † σ − + aσ + ) + g am ∆ a σ z b † b. (18)There is no non-Hermitian contribution in the effectiveHamiltonian obtained by means of the Bogoliubov aver-aging method. In the previous theoretical approach [42],the non-Hermitian term g am g cm ( a † σ − − aσ + ) bb † (1 / ∆ a − / ∆ c ) appears and can be eliminated only at ω a = ω c . Our expressions are also correct at ω a = ω c for | ω a − ω c | < g am , g cm . At the exact resonance ω a = ω c ,our results agree with those obtained in [42]. IV. CONCLUSION
In a typical optomechanical system, mechanical oscil-lations can be realized in asymmetric linear and cubicpotentials created by the pressure of light on the movingmechanical resonator (mirror) of special design. Whenthe frequencies of the mechanical oscillations exceed theparameters, characterizing the interaction between theoptical and mechanical subsystems and their decay rates,as well as the values of linear and cubic anharmonicities,this system can be replaced by an effective one using theBogoliubov averaging method. We have found that in theHamiltonian for the effective system, there is the Kerr in-teraction of photons in the optical cavity as well as thecross-Kerr interaction of photons and vibration quanta,induced by the oscillations in asymmetric anharmonicpotentials of the mechanical resonator. In addition, theKerr-like self-interaction of the mechanical resonator oc-curs. We have shown that the Bogoliubov averagingmethod as a variant of the non-secular perturbation the-ory correctly describes also the hybrid system consistingof a quantum dot and a nanocavity mediated by a me-chanical resonator. In contrast to the previous theory,in our approach the non-Hermitian terms do not appearin the effective Hamiltonian of the hybrid system. Fur-ther studies, both theoretical and experimental, wouldprovide more insight to the behavior of optomechanicalsystems with asymmetric mechanical oscillations. ∗ [email protected][1] F. Marquardt, J.P. Chen, A.A. Clerk et al., Phys. Rev.Lett. , 93902 (2007).[2] A. Schliesser, P. Del’Haye, N. Nooshi et al., Phys. Rev.Lett. , 243905 (2006).[3] J.D. Teufel, J.W. Harlow, C.A. Regal et al., Phys. Rev.Lett. , 197203 (2008).[4] F. Massel, T.T. Heikkil¨a, J.-M. Pirkkalainen et al., Na-ture , 351 (2011). [5] A.A. Clerk, F. Marquardt, and K. Jacobs, New J. Phys. , 95010 (2008).[6] M. Aspelmeyer, T.J. Kippenberg, and F. Marquardt,Rev. Mod. Phys. , 1391 (2014).[7] S. Mancini, V.I. Man’ko, and P. Tombesi, Phys. Rev. A55, 3042 (1997).[8] X.-W. Xu, H. Wang, J. Zhang et al., Phys. Rev. A (2013).[9] M. Paternostro, D. Vitali, S. Gigan et al., Phys. Rev.Lett. , 250401 (2007).[10] D. Vitali, S. Gigan, A. Ferreira et al., Phys. Rev. Lett. , 30405 (2007).[11] L. Tian, Phys. Rev. Lett. , 233602 (2013).[12] L. Tian, Phys. Rev. Lett. , 153604 (2012).[13] Y.-D. Wang, A.A. Clerk, Phys. Rev. Lett. , 153603(2012).[14] J. Bochmann, A. Vainsencher, D.D. Awschalom et al.,Nature Phys , 712 (2013).[15] G.S. Agarwal, S. Huang, Phys. Rev. A (2010).[16] H. Jing, S¸.K. ¨Ozdemir, Z. Geng et al., Scientific reports , 9663 (2015).[17] P.-C. Ma, J.-Q. Zhang, Y. Xiao et al., Phys. Rev. A (2014).[18] J.M. Dobrindt, I. Wilson-Rae, and T.J. Kippenberg,Phys. Rev. Lett. , 263602 (2008).[19] S. Huang, G.S. Agarwal, Phys. Rev. A (2009).[20] T.T. Heikkil¨a, F. Massel, J. Tuorila et al., Phys. Rev.Lett. (2014).[21] M. Bhattacharya, P. Meystre, Phys. Rev. Lett. 99, 73601(2007).[22] A. Xuereb, M. Paternostro, Phys. Rev. A , 653 (2013).[23] J.D. Thompson, B.M. Zwickl, A.M. Jayich et al., Nature , 72 (2008).[24] Y.R. Shen, The principles of nonlinear optics (Wiley-Interscience, New York, Hoboken, 2003). [25] Z.R. Gong, H. Ian, Y.-x. Liu et al.,Phys. Rev. A 80(2009).[26] P. Rabl, Phys. Rev. Lett. , 63601 (2011).[27] A. Nunnenkamp, K. Børkje, and S.M. Girvin, Phys. Rev.Lett. , 63602 (2011).[28] V. Kaajakari, T. Mattila, A. Oja et al., J. Microelec-tromech. Syst. , 715 (2004).[29] P. Huang, J. Zhou, L. Zhang et al., Nature communica-tions , 11517 (2016).[30] K. Jacobs, A.J. Landahl, Phys. Rev. Lett. , 67201(2009).[31] J.-Q. Zhang, Y. Li, M. Feng et al.,Phys. Rev. A (2012).[32] C. Kong, H. Xiong, and Y. Wu, Phys. Rev. A , 50302(2017).[33] Z. Wu, R.-H. Luo, J.-Q. Zhang et al., Phys. Rev. A (2017).[34] S. Huang, H. Hao, and A. Chen, Applied Sciences ,5719 (2020).[35] X.-Y. L¨u, J.-Q. Liao, L. Tian et al., Phys. Rev. A ,2943 (2015).[36] L. Latmiral, F. Armata, M.G. Genoni et al., Phys. Rev.A Asymptotic Meth-ods in the Theory of Nonlinear Oscillations (Gordon andBreach, New York, 1961).[38] A.P. Saiko, S.A. Markevich, and R. Fedaruk, Phys. Rev.A (2016).[39] A.P. Saiko, S.A. Markevich, and R. Fedaruk, Phys. Rev.A (2018).[40] H.J. Carmichael, Statistical Methods in Quantum Optics2: Non-Classical Fields (Springer-Verlag, Berlin, Heidel-berg, 2008).[41] Z.-L. Xiang, S. Ashhab, J.Q. You et al., Rev. Mod. Phys. , 623 (2013).[42] J.E. Ram`ırez-Mun¯oz, J.P. Restrepo Cuartas, and H.Vinck-Posada, Phys. Lett. A382