Quantum Simulation of Tunable and Ultrastrong Mixed-Optomechanics

We propose a reliable scheme to simulate tunable and ultrastrong mixed (first-order and quadratic optomechanical couplings coexisting) optomechanical interactions in a coupled two-mode bosonic system, in which the two modes are coupled by a cross-Kerr interaction and one of the two modes is driven through both the single- and two-excitation processes. We show that the mixed-optomechanical interactions can enter the single-photon strong-coupling and even ultrastrong-coupling regimes. The strengths of both the first-order and quadratic optomechanical couplings can be controlled on demand, and hence first-order, quadratic, and mixed optomechanical models can be realized. In particular, the thermal noise of the driven mode can be suppressed totally by introducing a proper squeezed vacuum bath. We also study how to generate the superposition of coherent squeezed state and vacuum state based on the simulated interactions. The quantum coherence effect in the generated states is characterized by calculating the Wigner function in both the closed- and open-system cases. This work will pave the way to the observation and application of ultrastrong optomechanical effects in quantum simulators.

In this work, we propose a scheme to realize this task by simulating a mixed optomechanical coupling in a coupled two-mode system, in which the two modes are coupled with each other via the cross-Kerr interaction. By introducing both single-and two-excitation drivings to one of the two modes, we can obtain the tunable and ultrastrong mixed-optomechanical interactions. In particular, the first-order (quadratic) optomechanical coupling strength could be greater than the effective frequency of the mechanical-like mode under proper conditions. Therefore, this effective mixed optomechanical model can enter the single-photon strong-coupling even ultrastrong-coupling regimes. As an application of the ultrastrong mixed-optomechanical couplings, we study how to generate the Schrödinger cat states in the mechanical-like mode. We also investigate the quantum coherence effect in the generated states by calculating their Wigner functions.
The remaining part of this paper is organized as follows. In Sec. 2, we show the physical model and the Hamiltonian. In Sec. 3, we derive an approximate Hamiltonian for the mixed optomechanical interactions and evaluate its validity. In Sec. 4, we study the preparation of the Schrödinger cat states in mode and calculate their Wigner functions. In Sec. 5, we present some discussions concerning the experimental implementation. Finally, we present a brief conclusion in Sec. 6.

Model and Hamiltonian
We consider a coupled two-bosonic-mode ( and ) system (see Fig. 1), where the two modes are coupled with each other via a cross-Kerr interaction. In this system, one (e.g., mode ) of the two modes is subjected to both single-and two-excitation drivings. The Hamiltonian of the system readŝ whereˆ (ˆ ) is the annihilation operator of the bosonic mode ( ) with the resonance frequency ( ). The third term in Eq. (1) describes the cross-Kerr interaction with the coupling strength , and amplitude ) and two-excitation driving (with phase , frequency , and amplitude Ω ). A squeezed vacuum bath with the squeezing parameter and reference phase is introduced to mode for suppressing the excitations caused by the two-excitation driving.
. The parameters − and (Ω − and ) are, respectively, the driving amplitude and frequency of the single (two)-excitation driving, with ( ) being the driving phase. In our model, the cross-Kerr interaction describes a general coupling between two bosonic modes [84]. In principle, the involving two modes could be implemented by either two electromagnetic fields or two mechanical modes, and even one electromagnetic field and one mechanical mode. Note that some proposals have been proposed to implement the optomechanical couplings based on two microwave fields [48,77,78]. In this case, though the mechanical-like mode cannot be used to transduce other physical signals for sensing, the generated optomechanical interactions can be used to demonstrate the optomechanical physical effects. The two-excitation driving process can be realized with the degenerate parametric-down-conversion mechanism. In optical system, this process can be implemented by introducing the optical parametric amplifier [85]. In circuit-QED systems, the two-excitation driving can be induced by a cycle three-level system [86], and this interaction has been widely used in various schemes [27,79,80,82,[87][88][89].
In a rotating frame defined by exp(− ˆ 0 /ℏ) withˆ 0 = ℏ ˆ †ˆ + ℏ ˆ †ˆ , the Hamiltonian takes the form aŝ where Δ = − is the single-excitation driving detuning. Hereafter, we consider the case of = 2 . To include the dissipations in this system, we assume that mode is connected with a heat bath and mode is in contact with a squeezed vacuum reservoir [90] with the central frequency . In this case, the evolution of the system in the rotating frame is governed by the quantum master equation Here, are superoperators acting on the density matrixˆ of the system, and are the decay rates of modes and , It can be seen from Eq. (8) that the equations of and are independent of the decay rates of the system. Below we analyze Eqs. (8a) and (8b) firstly. For a chosen two-excitation driving, the parameters Ω and are given. Then the value of and can be determined by Eqs. (8a) and (8b). To obtain a stationary quadratic optomechanical coupling, we consider a case of = and = , which leads to In addition, we only study the steady-state displacement case where the time scale of system relaxation is much shorter than other time scales. Then the steady-state displacement amplitude is obtained as Hereafter, we consider the case of tan = /[2(Δ − 2Ω )] so that ss is a real number for simplicity.
In Fig. 2(a), we show the squeezing amplitude defined in Eq. (9) as a function of Ω /Δ . Here, we can find that increases with the increase of the ratio Ω /Δ . In particular, the value of could be increased by choosing Δ − 2Ω ≃ 0 such that both the first-order and quadratic optomechanical interactions can be enhanced largely. This point is very important for the realization of single-photon strong-coupling regime for the optomechanical interactions. In Fig. 2(b), we show the scaled displacement amplitude ss /( /Δ ) as a function of /Δ and Ω /Δ based on Eq. (10). Here, we can see that the displacement amplitude ss is proportional to the single-excitation driving amplitude . For a given , we can choose a small and a proper value of Ω /Δ (approaching 1/2) to get a large ss . In addition, the point Δ = 2Ω is not a singularity in the presence of dissipation .
Based on the above analyses, we know that the system, under the steady-state displacement ss , is governed by the following quantum master equation, where the transformed Hamiltonianˆ ′ becomeŝ and the effective frequencies of modes and are reduced to˜ = − /2 + 2 ss 2 and˜ = (Δ − 2Ω ) 2 = Δ − 2Ω sinh /cosh . Those effective coupling strengthes in Hamiltonian ′ are given by We can see from Eqs. (9) and (10) that the coupling strengths of both the first-order and quadratic optomechanical interactions can be tuned by controlling the single-and two-excitation driving parameters. Meanwhile, the value of ss could be either larger or smaller than 1/4 which means that the first-order optomechanical coupling could be either stronger or weaker than the quadratic optomechanical coupling. In Eq. (11), the parameters ss and ss are, respectively, the effective thermal occupation number and two-photon-correlation strength in the transformed representation, which take the form It can be seen from Eq. (14) that the parameters ss and ss depend on the three parameters , , and . In principle, when we take a given , then we can plot the parameters ss and ss as functions of and , as shown in Figs. 2(c) and 2(d). Here, we can see that ss and ss become very large in a large parameter range. In the vicinity of = 0 and = , however, the values of ss and ss are very small. In order to understand the nature of ss and ss more clearly, we consider two interesting cases. (i) When = 0, we plot ss and ss as a function of Δ = − in Fig. 2(e). We have ss = 0 and ss = 0 at the point = . (ii) When = , we plot ss and ss as a function of in Fig. 2(f). We also see ss = 0 and ss = 0 at the point = 0. It should be emphasized that ss = 0 and ss = 0 imply that both the thermal noise and the squeezing vacuum noise have been suppressed completely [79]. This working point is very important to observe the optomechanical effects at the single-photon level. This feature is also the motivation for introducing the squeezing bath, i.e., using a well-designed squeezing vacuum bath to suppress the thermal noise and the excitations caused by the two-excitation driving.
Meanwhile, we can safely discard the cross-Kerr term 2ℏ ′ 2ˆ †ˆ ˆ †ˆ because the frequency shift caused by this term for mode is much smaller than the effective frequency of mode . Then Eq. (12) is reduced tô where˜ ′ =˜ + ′ 2 denotes the renormalized frequency of mode . The approximate Hamilto-nianˆ app describes the standard mixed optomechanical model consisting of both the first-order and quadratic optomechanical interactions, where modes and play the role of optical mode and mechanical mode, with the effective resonance frequencies˜ ′ and˜ , respectively. The parameters 1 and 2 are, respectively, the single-photon first-order and quadratic optomechanical coupling strengths.
To characterize these parameters inˆ app , we investigate the magnitude of these related parameters as functions of the detuning Δ of mode . Concretely, Fig. 3 shows the parameter /Δ as a function of Ω /Δ (dotted blue line), we find that the effective frequency˜ of mode decreases with the increase of driving strength Ω . We also plot the ratios 1 /Δ , 2 /Δ , and ′ 2 /˜ as functions of Ω /Δ . Here, we can see that the coupling strengths 1 and 2 increase with the increase of the driving amplitude Ω . In addition, we can see that the coupling strengths 1 and 2 could be a considerable fraction of (and smaller than)˜ (see the inset), which means that both the first-order and quadratic optomechanical couplings could enter the ultrastrong-coupling regime [92]. In particular, when the coupling strengths 1 and 2 are larger than the resonance frequency˜ , the system enters the so-called deep-strong coupling regime. As show in Fig. 3, the ratio ′ 2 /˜ decreases with the increase of Ω /Δ , and its value is much smaller than 1, which means that the two terms 2ℏ ′ 2ˆ †ˆ ˆ †ˆ and −ℏ ′ 2ˆ †ˆ (ˆ †2 +ˆ 2 ) can be safely discarded.
Interestingly, when = 0, 4 ≫ 1, and˜ ≫ 2 max ′ 2 , we can obtain an approximate quadratic optomechanical Hamiltonian Here, 2 could reach a considerable fraction of˜ under proper parameters, which means that the physical system corresponding to Eq. (16) can work in the single-photon strong-coupling even ultrastrong-coupling regimes of the quadratic optomechanical coupling.

Evaluation of the validity of the approximate Hamiltonianˆ app
To study the validity of the approximate Hamiltonianˆ app given in Eq. (15), we adopt the fidelity between the approximate state | app ( ) and exact state | ext ( ) , which are obtained with the approximate Hamiltonian (15) and the exact Hamiltonian (12), respectively. For avoiding the crosstalk from the system dissipations, we first study the closed-system case. In this case, the fidelity can be calculated by ( ) = | ext ( )| app ( ) |. In order to know the validity of the approximate Hamiltonians in the low-excitation regime, we choose the initial state as |1 | with coherent state | . In Fig. 4(a), we plot the fidelity ( ) as a function of the time Δ .
Here, we can see that the fidelity ( ) is high (> 0.96) under the used parameters. Considering that the system can produce the cat states at time = / (˜ + 4 2 )˜ , we show the fidelity ( ) as a function of the parameters /Δ and /Δ in Fig. 4(b), and we also show the fidelity ( ) as a function of the parameters /Δ and Ω /Δ in Fig. 4(c). We can see that the fidelity is high in a wide parameter space.
We also study the fidelity in the realistic case by including the dissipations. In the opensystem case, the state of the system is described by the density matrix. The fidelity between the exact density matrixˆ ext and the approximate density matrixˆ app can be calculated by ( ) =Tr[( ˆ extˆ app ˆ ext ) 1/2 ], where the exact density matrix evolves under Eq. (11) with the exact Hamiltonian, while the approximate density matrix is governed by Eq. (11) under the replacementˆ ′ →ˆ app . In Fig. 4(d), we show the dynamical evolution of the fidelity corresponding to Eq. (15). Here, we can see that, due to the dissipation, the fidelity of the open system approaches gradually to a stationary value in the long-time limit. Therefore, the approximate Hamiltoniansˆ app given by (15) is valid in both the closed-and open-system cases.

Generation of macroscopic quantum superposed states
One of the important applications of the simulated optomechanical interactions is the generation of macroscopic quantum superposed states. Now, we study the generation of a superposition of coherent squeezed state and vacuum state based on the mixed optomechanical interaction. To this end, we consider the initial state as |Ψ(0) = [(|0 + |1 )|0 ]/ √ 2, where |0 and |1 are, respectively, the vacuum state and single-photon state of mode , while |0 is the vacuum state of mode . To see the analytical expression of the generated states, in this section we analytically derive the state evolution in the closed-system case. We also calculate the Wigner function of the generated states in the presence of dissipations by numerically solving the quantum master equations. Based on Eq. (15), the evolution of the system can be calculated, and the state of the system at time becomes where we introduce the squeezing parameter 1 = ln[(˜ + 4 2 )/˜ ]/4 and the displacement According to Eq. (17), we know that the coherent-state component for mode can be generated at time = (2 + 1) / 1 with natural numbers . By expressing mode with the basis states | ± = (|0 ± |1 ) / √ 2, then the state of the system becomes where we introduced the cat states of mode as with the coherent state amplitude 1 ( ) = −2 1 /(4 2 +˜ ) and the normalization constants ± = {2 ± 2 exp[− 2 1 ( )/2] cos( 1,0 )} −1/2 . Mode will collapse into the cat states | ± ( ) when the states | ± are measured with the measuring probabilities ± ( ) = 1/(4 2 + ), respectively. For a relatively large 1 ( ), we have ± ( ) ≈ 1/2.
The quantum coherence and interference in the generated states can be investigated by calculating the Wigner function [93], which is defined by for the density matrixˆ , where is a complex variable. that the coherence effect in the cat states becomes weaker for larger decay rates, which means that the decay of the system will attenuate quantum coherence in the generated cat states.
It can be seen from Eq. (17) that, at time = (2 + 1) /2 1 for natural numbers , a superposition of the coherent squeezed state and vacuum state can be created with the maximum squeezing strength. If we re-express the state of mode with the basis states | ± at time , then the state of the system becomes where we introduce the superposed states of the coherent squeezed state and vacuum state with N ± = {2 ± 2 − | 1 ( ) | 2 /2 Re[ − ( )− * 2 1 ( ) tanh(2 1 )/2 ]/ cosh(2 1 )} −1/2 are the normalization constants. Mode will collapse into the states | ± ( ) when the states | ± are measured with measuring probabilities P ± ( ) = 1/(4N 2 ± ), respectively. For the generated states | ± ( ) , we show the Wigner functions ± ( ) in Figs. 6(a) and 6(e). Here, we also see the clear quantum interference pattern and evidence of macroscopically distinct superposition components. Similar to the state generated at time , here we also study the influence of the system dissipations on the state generation. In Figs. 6(b)-6(d) and 6(f)-6(h), we show the Wigner functions ± ( ) for the generated states (at times ) with various values of the decay rates. The density matrices of the generated states are also obtained by using the same procedure as the state generation case corresponding to the measurement . However, for the present case, the measurement time is . Figures 6(d) and 6(h) indicate that the same conclusion as the previous case. Here, the dissipations will also attenuate the quantum coherence in the generated states.

Discussions on the experimental implementation
In this section, we present some discussions about the experimental implementation of this quantum simulation scheme. The physical model proposed in this paper is general and it consists of four main physical elements: (i) the cross-Kerr interaction between the two modes; (ii) the single-excitation driving on mode ; (iii) the two-excitation driving on mode ; (iv) the squeezed vacuum reservoir of mode . In principle, this scheme can be implemented with physical setups in which the above four physical processes can be realized. Below, we focus our discussions on the superconducting quantum circuits because the above four elements can be well implemented with this setup. In circuit-QED systems, the cross-Kerr interaction has been realized between two superconducting cavities [94][95][96][97]. In particular, the single-photon cross-Kerr interaction can be resolved from the quantum noise. The single-photon driving process can be realized by driving the cavity field by microwave signal, and the two-photon driving process can be obtained by introducing a two-photon parametric process [86,89]. In addition, the squeezed vacuum reservoir of the cavity mode can be implemented by injecting a squeezing vacuum field to the cavity.
Below, we present some analyses on the possible experimental parameters. In this paper, we adopt the detuning Δ as the frequency scale for convenience in the discussions on the resolvedsideband condition. The value of Δ is a controllable value by tuning the driving frequency . For comparison with typical optomechanical systems, we choose Δ ∼ 2 × 1 − 10 MHz. In our scheme, the cross-Kerr interaction magnitude could be smaller or larger than the decay rate of the cavity mode . The two-excitation driving Ω is smaller than Δ /2, and the phase = . The single-excitation driving is a free choosing parameter because the displacement amplitude ss is proportional to . The phase angle is determined by tan = /[2(Δ − 2Ω )]. The parameters = and = 0 of the squeezed vacuum reservoir are chosen for suppressing the excitations caused by the two-photon driving, where is determined by the ratio Ω /Δ . The decay rates of the two modes are taken as /Δ = /Δ ≈ 0.01 − 0.1, which corresponds to , ∼ 2 × 10 kHz − 1 MHz. Based on the above analyses, we know that the present scheme should be within the reach of current or near-future experimental conditions.

Conclusion
In conclusion, we have proposed a reliable scheme to implement a quantum simulation for the tunable and ultrastrong mixed-optomechanical interactions. This was realized by introducing both the single-and two-excitation drivings to one of the two bosonic modes coupled via the cross-Kerr interaction. The validity of the approximate Hamiltonians was evaluated by checking the fidelity between the approximate and exact states. The results indicate that the approximate Hamiltonians work well in a wide parameter space. We have also studied the generation of the Schrödinger cat states based on the simulated mixed optomechanical interactions. The quantum coherence effect in the generated states has been investigated by calculating the Wigner functions in both the closed-and open-system cases. This work will open a route to the observation of ultrastrong optomechanical effects based on quantum simulators and to the applications of optomechanical technologies in modern quantum sciences.

Disclosures. The authors declare no conflicts of interest.
Data availability. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.