Generalized Ultrastrong Optomechanics

Generalized Ultrastrong Optomechanics

Jie-Qiao Liao jqliao@hunnu.edu.cn Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China    Jin-Feng Huang Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China    Lin Tian School of Natural Sciences, University of California, Merced, California 95343, USA    Le-Man Kuang Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China    Chang-Pu Sun Beijing Computational Science Research Center, Beijing 100193, China Graduate School of Chinese Academy of Engineering Physics, Beijing 100084, China
July 19, 2019
Abstract

We propose a reliable scheme to realize a generalized ultrastrong optomechanical coupling in a two-mode cross-Kerr-type coupled system, where one of the bosonic modes is strongly driven. The effective optomechanical interaction takes the form of a product of the photon number operator of one mode and the quadrature operator of the other mode. The coupling strength and quadrature phase are both tunable via the driving field. The coupling strength can be strongly enhanced to reach the ultrastrong-coupling regime, where the few-photon optomechanical effects such as photon blockade and macroscopically distinct quantum superposition become accessible. The presence of tunable quadrature phase also enables the implementation of geometric quantum operations. Numerical simulations show that this method works well in a wide parameter space. We also present an analysis of the experimental implementation of this scheme.

Introduction.—Light-matter interaction is at the heart of cavity optomechanics Kippenberg2008rev (); Aspelmeyer2012rev (); Aspelmeyer2014 () and is the root of various quantum coherence effects in optomechanical systems. The studies of cavity optomechanics focus primarily on the understanding, manipulation, and exploitation of the optomechanical couplings, and aim to explore both the fundamentals of quantum theory and modern quantum technology. Of particular interest is the study of optomechanics at the few-photon level Nunnenkamp2011 (); Rabl2011 (); Liao2012 (); Liao2013 (); Hong2013 (); Xu2013 (); Marshall2003 (); Liao2016 (). This is because the nonlinear optomechanical interaction is an intrinsic characteristic of optomechanics. Many interesting effects appear in this regime, such as phonon sideband spectrum Nunnenkamp2011 (); Liao2012 (), photon blockade in the cavity driven by a continuous wave Rabl2011 () or a wave packet Liao2013 (), and macroscopic quantum coherence Marshall2003 (); Liao2016 (). However, the few-photon optomechanical effects have not been observed in experiments because the single-photon optomechanical coupling is too weak to be resolved from the environmental noise. How to enhance the optomechanical coupling remains an important challenge in this field. Until now, people proposed several methods to enhance the single-photon optomechanical coupling. These methods include the construction of an array of mechanical resonators Xuereb2012 (), the use of the nonlinearity in Josephson junctions Rimberg2014 (); Heikkila2014 (); Pirkkalainen2015 (), the modulation of the couplings Liao2014 (), and the utilization of quantum squeezing resources Lue2015 (); Li2016 (), and mechanical amplification Lemonde2016 ().

In this Letter, we propose an efficient approach to realize ultrastrong optomechanical coupling in the few-photon regime seeSM (). Here ultrastrong coupling is defined as the strength of the single-photon optomechanical coupling is a considerable fraction of the mechanical frequency Hu2015 (). Our scheme is realized by applying strong driving on one of the two bosonic modes coupled by the cross-Kerr interaction. Note that the cross-Kerr interaction has been widely used in quantum state preparation Paternostro2003 (); Kuang2003 (), quantum information protocols Milburn1989PRL (); Vitali2000 (); Kok2007 (); Nemoto2004 (), quantum nondemolition photon measurement Imoto1985 (); Grangier1998 (), and phonon counting Ding2017 (). In particular, the generalized optomechanical coupling takes the form of the product of the occupation number operator of one mode and the quadrature operator of the other mode. Here, the strength of single-photon optomechanical coupling is enhanced by the driving to reach the single-photon strong coupling regime. Our scheme has the following features. (i) The driving field enhances the optomechanical coupling strength to reach the ultrastrong coupling regime, and the generalized optomechanical coupling can be used to implement geometric quantum operations with proper quadrature angle sequences. (ii) This method works for both steady-state and transient displacements, which correspond to constant and modulated optomechanical coupling cases. (iii) In the displacement representation, the driving detuning plays the role of the effective mechanical frequency, and hence it is possible to choose a high natural frequency of the mechanical mode to suppress its thermal noise.

Model.—We consider two bosonic modes and coupled by a cross-Kerr interaction. One of the modes (for instance mode ) is driven by a monochromatic field with frequency . In a rotating frame with respect to , the Hamiltonian of this system reads ()

(1)

where () and () are the annihilation (creation) operators of the two bosonic modes, with the corresponding resonance frequencies and . The parameter is the detuning of the resonance frequency of mode with respect to the driving frequency , and the parameter is the driving amplitude. The two modes are coupled to each other through a cross-Kerr interaction, with the coupling strength .

To treat the damping and noise in this system, we assume that the two bosonic modes are coupled to two independent Markovian environments, the evolution of the system is hence governed by the quantum master equation

(2)

where is the standard Lindblad superoperator for bosonic-mode damping, () and () are the damping rate and environment thermal excitation occupation of mode (), respectively.

Generalized ultrastrong coupling.—Our motivation in this work is to obtain an ultrastrong optomechanical coupling between the two modes. Under strong driving, the mode is excited with large occupation number, and the operator can be written as a summation of its mean value and a quantum fluctuation , and similarly . Note that the occupation number of mode is independent of the driving on mode because the operator is a conserved quantity. The cross-Kerr interaction then becomes . Here the first term is a frequency shift on mode , the second term is the generalized optomechanical coupling with a coupling strength enhanced by a factor , and the third term is the cross-Kerr interaction between mode and the fluctuation of mode .

To prove the above analysis, we perform the transformation to the quantum master equation (2), where is the mean displacement of mode . By performing this transformation, we obtain the equation of motion of the displacement as . We consider the case where the time scale of system relaxation is much shorter than other time scales. The steady-state displacement reads , which is a tunable complex number by choosing proper and . In the displacement representation, the quantum master equation takes the same form as Eq. (2) under the replacement and , where the transformed Hamiltonian is given by , with the frequency . In this Hamiltonian, the cross-Kerr term is an effective frequency shift for the two modes. When this frequency shift associated with excitations in mode is much smaller than the effective frequency of mode , namely , we can neglect the cross-Kerr interaction term safely. In this case, a generalized ultrastrong optomechanical interaction can be obtained.

In this work, we focus on few-photon optomechanics and hence consider the regime and , then the cross-Kerr interaction term in can be safely discarded and we obtain the generalized optomechanical Hamiltonian

(3)

where is the single-photon optomechanical coupling strength. The Hamiltonian (3) possesses three features: (i) The effective resonance frequency of the mechanical mode is tunable by choosing proper driving frequency . Therefore, we can choose a small such that a near-resonant displacement interaction is obtained and further the displacement effect of single photons is enhanced. (ii) The single-photon optomechanical coupling between the two modes is enhanced by a factor of the displacement amplitude , which is determined by the driving amplitude . Therefore, the coupling strength can be enhanced to be larger than the decay rate of mode and even the resonance frequency when we take , and consequently the system can enter the ultrastrong coupling regime. (iii) The phase angle of the quadrature operator mode can be controlled by choosing proper driving phase in . This feature can be used to implement various geometric quantum operations such as the Kerr interaction and quantum gates.

The only approximation in the above derivation is the omission of the cross-Kerr interaction in the transformed Hamiltonian in the regime of . To evaluate the adequacy of this approximation seeSM (), we conduct numerical simulation of this system with the full Hamiltonian and the approximated Hamiltonian (3). To avoid the crosstalk of the dissipations on the approximation, we consider the closed system case seeSM () by numerically solving the Schrödinger equation. We then calculate the fidelity between the exact state and the approximate state from the simulation of (3). We choose the initial state of the system as . In this case, the fidelity can be obtained for the case of as , with .

In Fig. 1(a), we show the fidelity as a function of the time with and , , and . We see that the fidelity decreases for larger values of . This can be explained from the expression of that the exponential decreasing rate is proportional to in this case. Nevertheless, the fidelity can be very high because of . In Fig. 1(b), we display the fidelity at time (the time for generation of cat state in mode ) as a function of and . Here the fidelity is large in a wide parameter space and it is higher for smaller . For a given value of , is higher for a smaller value of . With the parameters for creating moderate displacement, for example , the fidelity could be larger than . Note that this fidelity is independent of because the term commutates with other terms in the Hamiltonian.

Figure 1: (Color online) (a) The fidelity as a function of the time when and , , and , which correspond to , , and , respectively. (b) The fidelity at time as a function of the parameters and .

Photon blockade—One important application of the optomechanical interaction in the ultrastrong coupling regime is the photon blockade effect Rabl2011 (). The photon blockade effect can be seen from the dressed Kerr nonlinearity in the diagonalized Hamiltonian , with  seeSM (). To observe the photon blockade effect, the magnitude of the self-Kerr nonlinearity should be much larger than the decay rate, namely , such that the anharmonicity in the energy levels can be resolved. In our scheme, the single-photon optomechanical coupling strength is enhanced by the large coherent displacement and a small driving detuning . Here we should point out that the small detuning will not affect the thermal occupation number because is determined by the natural frequency of mode . In Fig. 2(a), we plot the equal-time second-order correlation function as a function of the enhanced factor at various values of . Here operator averages are for the steady state of the system numericalcal (). We can see that the photon blockade effect (corresponding to ) can be observed in the resolved-sideband limit . The decay of mode will harm the photon blockade effect, as shown in the inset, where we display as a function of at , which corresponds to the optimal for photon blockade.

Figure 2: (Color online) (a) The equal-time second-order correlation function of mode as a function of when , , and and under the single-photon resonance driving . Other parameters used in panel (a) are , , and . The inset shows the correlation function at as a function of . (b) The dynamics of the average excitation number in mode when and the enhanced factor takes various values , , and . The inset shows the average excitation number at time as a function of in both the closed- and open-system cases. Other parameters used in panel (b) are and .

Macroscopic mechanical coherence—Another important application of the optomechanical interaction in the ultrastrong coupling regime is the generation of the Schrödinger cat state Marshall2003 () in mode . The dynamical evolution of this system can be used to create the Schrödinger cat states for both mode  Mancini1997 (); Bose1997 (); Bose1999 () and mode  Marshall2003 (). Up to the free evolution , the unitary evolution operator associated with the generalized optomechanical coupling can be written as , where the Kerr parameter is given by and is the conditional displacement operator for mode with the conditional excitation number in mode . At specific times for natural numbers , the two modes decouple and then the dynamics of mode corresponds to a Kerr interaction, which can used to create cat states. The conditional displacement for mode can also be used to create macroscopically distinct superposition. To this end, we consider an initial state of the system . The state at time of the system can be obtained as , where , , and . The maximal displacement is obtained at for natural numbers . When , the coherent state can be approximately distinguished from the vacuum state , and hence macroscopically distinct superposed coherent states in mode can be generated by measuring mode in states . The single-photon displacement of mode can be seen by calculating the average excitation in mode . In Fig. 2(b), we show the dynamics of for several values of in the presence of dissipations. The plots show that a larger maximal accessible displacement can be obtained for a larger , and that the dissipations will decrease the peak value of the displacement. In the inset, we plot the variable , which corresponds to the maximal displacement , as a function of . We see that the maximal displacement could be larger than the zero-point fluctuation of mode (i.e., ). This means that a quantum superposition of macroscopically distinct states in mode can be prepared with this method seeSM ().

Geometrical quantum operations—The generalized nonlinear interaction between the two modes and in Hamiltonian can be used to create a self-Kerr nonlinear interaction of mode via a sequence of operations. Consider the resonant driving case , and the corresponding unitary evolution operator becomes , which takes the form of an evolution operator associated with the conditional quadrature operator . With the above unitary evolution operator, a self-Kerr interaction of mode can be obtained by designing a chain of unitary evolution based on the unconventional geometrical phase effect Zhu2003 () as . The unitary evolution operator represents a pure self-Kerr interaction of mode , and it is different from the transformed Kerr nonlinearity associated with the optomechanical coupling . The pure self-Kerr interaction is independent of the phonon states and hence the two modes are decoupled from each other with no phonon sidebands. However, in the optomechanical interactions, the eigenstates are the number state for mode dressed by the displaced number states for mode  Liao2012 (). Moreover, the phase shift associated with the Kerr interaction is continuously tunable and it can reach which is needed for realization of logic gate for quantum computation. The Kerr interaction in only works at time .

Figure 3: (Color online) The Wigner functions of the generated states at and in the presence of dissipation. The corresponding states in the idea case are the cat state and the kitten state , respectively. Other parameters are given by , , and .

The geometric Kerr interaction can also be used to create the Schrödinger cat and kitten states Birula1968 (); Miranowicz1990 (). For simplicity, we express the unitary evolution operator as , with and . For the initial state , we consider the case of some specific times with two coprime integers and , the state of mode can be expressed as , where and the coefficients are given by . Here we omit the free evolution because this operator corresponds to a whole rotation of the state in phase space. For example, we choose and , the state becomes . When and , we obtain a kitten state with three superposition components as . In Fig. 3 we plot the Wigner function  Barnettbook () of the exact generated states, where is a complex variable, is the density matrix of the state, and is the usual displacement operator for mode . We can observe quantum interference pattern, which is a clear signature of quantum superposition seeSM ().

Discussions—Though we focus on the steady-state displacement in the above discussions, this method works for both the steady-state and the transient displacements and . In the latter case, we can obtain a modulated optomechanical coupling . For example, we can choose a proper driving amplitude such that a sinusoidal enhancement is obtained, where is the modulation frequency. It has been proved that the modulated optomechanical coupling can be used to enhance the photonic nonlinearity and to generate macroscopic superposition states Liao2014 ().

Our scheme can be implemented either by two electromagnetic field modes or by one electromagnetic mode and one mechanical mode coupled by the cross-kerr interaction seeSM (). The requirements on the parameters are: , , and . We can choose proper driving frequency and amplitude such that and . For two cross-Kerr coupled microwave field modes, are of the order of - Hz Aspelmeyer2014 (). For one microwave field and one mechanical mode, the decay rates can be - Hz and - Hz Aspelmeyer2014 (). In these two cases, we choose . Corresponding to , we can choose - Hz and . These parameters are accessible with current experimental technology. Note that the Kerr-type interactions in various quantum optical systems have been evaluated Rebic2009 (); Hu2011 (); Nigg2012 (); Hoi2013 (); Holland2015 (); Bourassa2012 (); Majer2007 (); Thompson2008 (); Sankey2010 (); Karuza2013 (); Gong2009 (); Semiao2005 (); Maurer2004 ().

Conclusions.—We proposed a practical method to realize a generalized ultrastrong optomechanical coupling. This is achieved by driving one of the two bosonic modes coupled through a cross-Kerr interaction. We analyzed the parameter conditions under which this proposal works. We also studied the application of this scheme on the photon blockade effect, the cat state generation, and the implementation of geometric gates. This proposal provide a reliable method for studying few-photon optomechanics or simulating the optomechanical-type interactions between two electromagnetic fields with current experimental techniques.

Acknowledgments.—J.-Q.L. is supported in part by NSFC Grant No. 11774087 and HNNSFC Grant No. 2017JJ1021. J.-F.H. is supported by the NSFC Grant No. 11505055. L.T. is supported by the NSF (USA) under Award No. PHY-1720501. L.-M.K. is supported by the NSFC Grants No. 11375060, No. 11434011, and No. 11775075. C.P.S. is supported by the National Basic Research Program of China Grants No. 2014CB921403 and No. 2016YFA0301201, the NSFC Grants No. 11421063 and No. 11534002, and the NSAF Grant No. U1530401.

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Supplementary materials for “Generalized Ultrastrong Optomechanics”

This document consists of four parts: (I) Analyses of the parameter space of the optomechanical model; (II) Derivation of the approximate Hamiltonian and evaluation of the parameter condition of the approximation; (III) Detailed calculations of the applications of the generalized optomechanical coupling; (IV) Discussions on the experimental implementation.

Appendix A I. Analyses of the parameter space of the optomechanical model

In this section, we present some analyses on the parameter space of a standard cavity optomechanical system driven by a monochromatic field. First of all, we want to point out that the notations in this section are independent of the notations used in the main text and other sections in this supplemental material. This is because the motivation of this section is just to discuss the parameter space of a typical optomechanical system without other additional interaction terms.

For a typical optomechanical model, it is formed by a single-mode cavity field coupled to a single-mode mechanical oscillation via a radiation-pressure interaction (i.e., the optomechanical coupling). In order to manipulate this coupled system, a monochromatic laser field is usually introduced to drive the cavity field. The Hamiltonian of this system reads

(S1)

where () and () are the annihilation (creation) operators of the cavity field and the mechanical mode, respectively, with the corresponding resonance frequencies and . The parameter is the single-photon optomechanical-coupling strength between the cavity field and the mechanical mode. The parameters and are the driven frequency and driving amplitude, respectively. By performing a rotating transformation with respect to , the time factor in Hamiltonian (S1) can be eliminated and then the Hamiltonian becomes

(S2)

where the driving detuning is introduced.

To include the dissipations in this system, we assume that the cavity mode is coupled to a vacuum bath and the mechanical mode is coupled to a heat bath at a finite temperature. In this case, the evolution of the optomechanical system is governed by the quantum master equation

(S3)

where is the standard Lindblad superoperator for bosonic-mode damping. The parameters and are the damping rates of the cavity mode and the mechanical mode, respectively. The parameter is the thermal excitation occupation number of mode ’s heat bath.

In a typical open cavity optomechanical system, the relating parameters can be listed as:

(S4)

Below we will analyze the relationship among these parameters. In this system, the cavity frequency is usually sufficient large such that the thermal excitation occupation number in the cavity’s bath is negligible. From the view point of energy level transition, the cavity driving detuning is a more important parameter to affect the dynamics of the system, and is a tunable parameter via changing the driving frequency . The mechanical frequency is an important parameter in this system because the ratio is the sideband-resolution condition. This condition decides if the phonon sidebands can be resolved from the cavity emission spectrum. The single-photon optomechanical-coupling strength is also a very important parameter. This is because, on one hand, this ratio is used to characterize the single-photon strong-coupling regime in optomechanics. Only when , the optomechanical phenomenon induced by a single photon can be observed in this system. On the other hand, the optomechanical coupling describes a constant force performed on the mechanical resonator and hence this is an unresonant interaction with a detuning . In order to create quantum superposition of macroscopically distinct states, the relation should be satisfied. In few-photon optomechanics, the involved photon number is small and hence the driving magnitude should be much smaller than the cavity-field decay rate, i.e., . As described above, the cavity-field decay rate is an important parameter because this quantity determines the condition for the sideband resolution and the single-photon strong coupling. In most optomechanical systems, the decay rate of the mechanical mode is very small. However, the thermal excitation number in mode is an important parameter. This is because the thermal noise will prevent the observation of quantum effect in this system. As a result, the system should be cooled in advance to approach its ground state for observing quantum effects.

Figure S1: (Color online) A parameter space diagram for cavity optomechanics in the weak-driving case. The parameter space is described by the single-photon optomechanical-coupling strength , the photon decay rate , the resonance frequency of the mechanical resonator.

To analyze the parameter space, below we will focus on these three important parameters , , and . Usually, there are six cases of distribution for the three parameters, as shown in Fig. S1(a). Relating to the single-photon optomechanical-coupling strength , there exist three important parameter regimes: (i) The single-photon strong-coupling condition  Nunnenkamp2011 (); Rabl2011 (), which is also related to the quantum parameter  Ludwig2008 (); Qian2012 (), this condition guarantees that the cavity frequency shift caused by a zero-point fluctuation of the mechanical resonator can be resolved from the cavity spectrum. (ii) The strong dispersive coupling condition  Rabl2011 (), which shows the condition for resolving the photonic Kerr nonlinear energy nonharmonicity from the cavity spectrum when . (iii) The deep-strong coupling condition , this condition depicts if the mechanical displacement forced by a single photon can be distinguished from the mechanical vacuum state. In addition, we should mention the resolved-sideband limit , which is not related to the coupling strength, but it is also very important in nonlinear optomechanics. This limit guarantees that the phonon sidebands can be well resolved in the cavity emission spectrum. Though the former three parameter regimes are very important in nonlinear optomechanics, they are not accessible by current experimental technologies. Based on the above analyses, we can introduce Fig. S1(b) to describe the parameter space of an optomechanical system. The line characterizes the sideband resolution condition. In Fig. S1(b), the line describes the condition for creation of quantum superposition of macroscopically distinct states. The diagonal line confirms that if the system satisfies the single-photon strong-coupling condition. In addition, the curve is determined by the relation . We should emphasize that the parameter condition for evaluation of the photon blockade effect only works in the case of . This is because the phonon sideband states will participate the photon transitions in the optomechanical system. The photon blockade effect is not monotonously stronger for a larger value of . In particular, at the photon sideband resonance for positive integers , there is no photon blockade effect because the second photon transition is resonant Liao2013 (). In typical few-photon optomechanics, there are two important tasks. One is realization of the photon blockade effect, and the other is the generation of mechanical cat states. For observing photon blockade in optomechanics, the two conditions and (also ) should be satisfied. It means that region and part of region are ok for observing photon blockade effect. To create a macroscopic mechanical cat state, the conditions and should be satisfied. Therefore, only the region is ok. Note the the condition guarantees that the maximal displacement has been created before the single photon emits out of the cavity.

Appendix B II. Derivation of the approximate Hamiltonian and evaluation of the parameter condition of the approximation

In this section, we present a detailed derivation of the approximate Hamiltonian . Hereafter, the notations are consistent with those used in the main text. We consider two bosonic modes and coupled via the cross-Kerr interaction. One of the two modes (for instance mode ) is driven by a monochromatic field. The Hamiltonian of the system reads

(S5)

where () and () are the annihilation (creation) operators of the two bosonic modes, with the corresponding resonance frequencies and . The parameter is the coupling strength of the cross-Kerr interaction between the two modes. The mode is driven by a monochromatic field, with and being the driving frequency and amplitude, respectively.

In a rotating frame with respect to , the Hamiltonian becomes

(S6)

where we introduce the driving detuning .

In the presence of dissipations, the evolution of the system is governed by the quantum master equation

(S7)

where () and () are the damping rate and environment thermal excitation occupation number of mode (), respectively. In the strong-driving regime, the excitation number in mode is large and then mode contains a coherent part. This coherent part can be seen by performing the following displacement transformation

(S8)

where is the density matrix of the two-mode system in the displacement representation, is the displacement operator, and is the coherent displacement amplitude, which needs to be determined in the transformed master equation. Under the displacement transformation, the left-hand side of the master equation becomes

(S9)

In terms of the relations

(S10)

the left-hand side of the master equation can be calculated as

(S11)

Using the relations

(S12)

we proceed to derive the transformed master equation as

(S13)

The coherent part can be determined in the displacement representation when the coherent displacement amplitude obeys the equation

(S14)

Then the quantum master equation in the displacement representation becomes

(S15)

where the transformed Hamiltonian in the displacement representation becomes

(S16)

Based on the tasks, we consider two cases of displacement: the steady-state displacement and the transient displacement. In the former case, the steady-state displacement amplitude can be obtained as

(S17)

It can be seen from Eq. (S17) that the coherent displacement amplitude is tunable by choosing proper parameter and . The value of could be very large in the strong-driving case . For the transient-solution case, the optomechanical coupling becomes a time-dependent interaction. In particular, the interaction strength is tailorable because we can obtain a desired by designing a proper driving amplitude .

In this work, we mainly focus on the steady-state displacement case, in which the time scale of the system approaching to its steady state is much shorter than other evolution time scales. In this case, the Hamiltonian becomes

(S18)

where we introduced the normalized frequency , the enhanced coupling strength

(S19)

and the phase angle of the quadrature operator of mode , which is defined by .

The motivation of ultrastrong optomechanics is to study the few-photon physics in optomechanical system, then we focus the few-photon regime, and under the condition

(S20)

with being the largest photon number involved in the system, we can neglect the cross-Kerr interaction term to obtain the approximate Hamiltonian as

(S21)

This approximate Hamiltonian is the main result of this work. Here we can see that the effective frequency of mode is given by , and that the effective coupling strength of the generalized optomechanical coupling is given by . The effective frequency is controllable by tuning the driving frequency , and the generalized coupling strength could be largely enhanced to enter the ultrastrong coupling regime by choosing a proper driving amplitude . The form of the optomechanical coupling is generalized because this coupling takes the form of a product of the occupation number operator of mode and the quadrature operator of mode . The quadrature angle can be tuned by choosing the driving frequency and amplitude .

In the derivation of the approximate Hamiltonian , the only approximation is the omission of the cross-Kerr interaction term in the transformed Hamiltonian . The condition under which the approximation is justified is that the frequency shift of mode induced by the cross-Kerr interaction should be much smaller than its effective frequency in the displacement representation. Blow, we evaluate the reasonability of this approximation by calculating the fidelity between the exact state and the approximate state. To avoid the crosstalk from the dissipations, we first consider the closed-system case, in which the evolutions of the exact state and the approximate state are governed by the exact Hamiltonian and the approximate Hamiltonian , respectively. We assume that the initial state of the system is so that we can calculate the exact state and the approximate state analytically.

Figure S2: (Color online) The fidelity defined by Eq. (S28) as a function of the evolution time when the parameters and take different values: (a) and , (b) and , and (c) and . The initial state of the system is .

Based on the exact Hamiltonian (S16), the exact state of the system at time can be obtained as

(S22)

where the phase and the displacement amplitude are defined by