Multiphonon-resonance quantum Rabi model and adiabatic passage in a cavity-optomechanical system

Abstract

In this paper, we propose a scheme to achieve a multiphonon-resonance quantum Rabi model and adiabatic passage in a strong-coupling cavity optomechanical system. In the scheme, when the driving bichromatic laser beam is adjusted to the off-resonant j-order red- and blue-sideband, the interaction between the cavity and mechanical oscillator leads to a j-phonon resonance quantum Rabi model. Moreover, we show that there exists a resonant multi-phonon coupling via intermediate states connected by counter-rotating processes when the frequency of the simulated bosonic mode is near a fraction of the transition frequency of the simulated two-level system. As a typical example, we theoretically analyze the two-phonon resonance quantum Rabi model, and derive an effective Hamiltonian of the six-phonon coupling. Finally, we present a method of six-phonon generation based on adiabatic passage across the resonance. Numerical simulations confirm the validity of the proposed scheme. Theoretically, the proposed scheme can be extended to the realization of 3j-phonon state.

Graphical abstract

Keywords

quantum Rabi model / cavity-optomechanical system / adiabatic passage

1 Introduction

The quantum Rabi model (QRM) [1], which describes the dipolar coupling between a two-level system and a quantized bosonic field, has played an important role in quantum optics and condensed matter systems for the last few decades. The QRM requires to take the counter-rotating term into account in the interaction Hamiltonian that is neglected in the Jaynes–Cummings model (JCM), which is a solvable system where the rotating-wave approximation (RWA) is applied. Under RWA, this system can exhibit various quantum effects, such as quantum Rabi oscillations [1] and anti-Zeno effect [2, 3]. Typically, the RWA is valid when and only when the coupling strength is small enough. With the recent advances in quantum technology, in several systems, the coupling strength has been extended to the ultrastrong coupling (USC) [4, 5] and deep strong coupling (DSC) regimes [6–8], which indicated that the RWA is no longer valid. In recent years, a variety of interesting results have been investigated in the QRM, such as quantum vacuum radiation [9, 10], Quantum phase transition and quench dynamics [11], super-radiance transition [12], collapse and revival dynamics in the deep strong coupling regime [6], and scalable quantum memory [13]. Noted that several proposals and experimental realizations of the QRM in different quantum system have been put forward, such as optical lattices [14], circuit QED [15], as well as trapped ions [16, 17].

In QRM, one interesting feature is that when considering the counter-rotating terms, the number of excitations in the cavity-emitter system is no longer conserved, and the virtual transitions should not be excluded. Thus, the higher order resonant transitions via intermediate states will occur. A recent research shows that a resonant three-photon coupling can be achieved when the frequency of the cavity field is near one-third of the atomic transition frequency [18]. Moreover, it is shown that a system consisting of a single qubit ultrastrongly coupled to a resonator can exhibit anomalous vacuum Rabi oscillations where two or three photons are jointly emitted by the qubit into the resonator and reabsorbed by the qubit in a reversible and coherent process [19].

Cavity-optomechanical system [20], where a cavity field couples to a mechanical oscillator via the radiation-pressure force interaction, is a promising venue to realize quantum entanglement and quantum information processing in the mesoscopic field. Based on this system, schemes for entanglement generation [20–29] and high-precision measurements [30, 31], as well as quantum interfaces [32–34] have been put forward. On the other hand, theoretical studies on cavity-optomechanical system had been focused on the single-photon strong coupling regime, in which the radiation pressure of a single photon can produce observable effects and a strong photon nonlinearity, such as photon blockade effect [35–45]. Although great progress has been made in the theoretical study of the optomechanical system, the realization of QRM of this system remains a challenge [46, 47]. The main obstacle to achieve the QRM is that it is difficult to choice the actors of qubit in cavity-optomechanical system.

In this paper, we show that the multiphonon resonance QRM and adiabatic passage can be realized in a cavity optomechanical system. Our proposal focuses on the single-photon strong-coupling regime. In the scheme, with suitable setting of parameters of the driving laser, the dynamics of the system can be treated as the multiphonon resonance QRM between the simulated two-level system and the simulated bosonic mode. In addition, we also show that this resonance QRM can be exploited to generate multiphonon states through adiabatic passage. As a typical example, we theoretically analyze the two-phonon resonance QRM, and show the generation of six-phonon states through adiabatic passage by using this two-phonon resonance QRM. Numerical simulations show that the proposed scheme is available. In principle, this scheme can be used to prepare 3j-phonon state in cavity optomechanical system.

Our work is organized as follows. In Section 2, we present the theoretical model, and show that the interaction between the cavity and the mechanical oscillator can be treated as a multiphonon resonance QRM by suitably adjusting the detuning of the driving laser. Also, we discuss the nature of energy eigenvectors across the resonance and analyze the position of the resonance from the effective Hamiltonian. In Section 3, we discuss a scheme for six-phonon generation via adiabatic passage and present numerical simulations of the fidelities of the target state. Then, a wigner function of targeted state is numerically simulated and is compared to that of the ideal six-phonon state. A brief discussion of the experimental feasibility and summary appear in Section 4.

2 Theoretical model and numerically simulation

Our proposal is based on a single-mode cavity with a movable perfectly-reflecting microsize mirror (treated as mechanical oscillator). Also, the cavity mode couples to the mechanical oscillators through the radiation-pressure interaction induced by a driving bichromatic laser. It is necessary to point out that the bichromatic laser has been successfully used for different purposes in trapped ion system [48–50]. The total Hamiltonian of this system can be expressed in the form (\hslash =1)

\begin{array}{}\text{(1)}& H=H_0+H_{\text{OM}}+H_{\text{D}}\mathrm{,}\end{array}

where

\begin{array}{}\text{(2)}& H_0={\omega }_c{a}^{†}a+{\omega }_m{b}^{†}b\end{array}

is the free Hamiltonian,

\begin{array}{}\text{(3)}& H_{\text{OM}}=g_0{a}^{†}a(b^{†}+b)\end{array}

is the standard optomechanical Hamiltonian, and

\begin{array}{}\text{(4)}& H_{\text{D}}={\displaystyle \sum _{d=l\mathrm{,}r}\text{Ω}\{a^{†}\exp [-\text{i}({\omega }_{d}t+{\varphi }_d)]+\text{H}\text{.c}\}\mathrm{,}}\end{array}

is the the laser driven term. Here a (a^†) is the annihilation (creation) operator for the cavity, and b (b^†) is the annihilation (creation) operator for the mechanical oscillator. The parameters ω_c and ω_m are the frequencies of the cavity and the mechanical oscillator, while the other parameters Ω, ω_d, and ϕ_d (d = l, r) are the Rabi frequency, frequency and initial phase of the driving laser field, respectively. In addition, g₀ = ω_cx_zpf/L is the single-photon optomechanical coupling strength, where x_zpf =\sqrt{1/(2M{\omega }_m)} is the zero-point fluctuation of the moving mirror with mass M, and L is the length of the cavity.

To solve the Hamiltonian H, we firstly apply a unitary transformation V_1=\text{exp}[(g_0/{\omega }_m)a^{†}a(b^{†}-b)] to transform the Hamiltonian H to H_1=V_{1}H{V}_1^{†}. Thus, we obtain a new Hamiltonian

\begin{array}{}\text{(5)}& \begin{array}{ll}{H}_1=\hfill & ({\omega }_c-{\text{Δ}}_0{a}^{†}a)a^{†}a+{\omega }_m{b}^{†}b\hfill \\ \hfill & +{\displaystyle \sum _{d=l\mathrm{,}r}}\text{Ω}\left[a^{†}{\text{e}}^{-\text{i}({\omega }_{d}t+{\text{Φ}}_d)}{\text{e}}^{\eta (b^{†}-b)}+\text{H}\text{.c}\right]\mathrm{,}\hfill \end{array}\end{array}

where {\text{Δ}}_0=g_0^2/{\omega }_m characterizes the nonlinearity of the cavity field induced by mechanical oscillators, and the parameter η = g₀/ω_m can be treated as the Lamb-Dicke parameter similar to that used in the analysis of trapped ions [16, 17]. In the scheme, we consider the strong optomechanical coupling, which guarantees a big nonlinear parameter ∆₀ so that the photon blockade can occur [43, 51]. In this case, the cavity field is reduced to the two lowest-energy levels, |0⟩ and |1⟩. Then we can rewrite the creation operator of the cavity field as a^† = |1⟩⟨0| = σ₊, and further introduce a new operator σ_z = |1⟩⟨1|−|0⟩⟨0|. Thus, the Hamiltonian H₁ can be rewritten as

\begin{array}{}\text{(6)}& H_2=H_0^1+H_I^{\prime }\mathrm{,}\end{array}

where

\begin{array}{}\text{(7)}& H_0^1=\frac{{\omega }_0}{2}{\sigma }_z+{\omega }_m{b}^{†}b\mathrm{,}\end{array}

and

\begin{array}{}\text{(8)}& \begin{array}{ll}{H}_I^{\text{'}}\hfill & ={\displaystyle \sum _{d=l\mathrm{,}r}}\text{Ω}{\sigma }_{+}{\text{e}}^{-\text{i}({\omega }_{d}t+{\text{Φ}}_d)}\exp [\eta (b^{†}-b)]+\text{H}\mathrm{.}\text{c}\mathrm{.}\hfill \\ \hfill & ={\displaystyle \sum _{d=l\mathrm{,}r}}\text{Ω}{\sigma }_{+}{\text{e}}^{\frac{-{\eta }^2}{2}}{\text{e}}^{-\text{i}({\omega }_{d}t+{\text{Φ}}_d)}{\displaystyle \sum _{m\mathrm{,}n}}\frac{{\eta }^{(m+n)}}{m!n!}{b}^{†m}{b}^n+\text{H}\mathrm{.}\text{c}\mathrm{.}\hfill \end{array}\end{array}

Here, ω₀ = ω_c−∆₀. In the interaction picture, with H_I={\text{e}}^{\text{i}{H}_0^{1}t}{H}_I^{\prime }{\text{e}}^{-\text{i}{H}_0^{1}t}, we have a new Hamiltonian, which reads

\begin{array}{}\text{(9)}& \begin{array}{ll}{H}_I=\hfill & \text{Ω}{\sigma }_{+}{\text{e}}^{\frac{-{\eta }^2}{2}}{\displaystyle \sum _{m\mathrm{,}n}}{\displaystyle \sum _{d=l\mathrm{,}r}}{\text{e}}^{-\text{i}{\text{Φ}}_d}{\text{e}}^{-\text{i}({\omega }_d-{\omega }_0)t}\frac{{\eta }^{(m+n)}}{m!n!}\hfill \\ \hfill & \times b^{†m}{b}^n{\text{e}}^{\text{i}(m-n){\omega }_{m}t}+\text{H}\text{.c}\text{.}\hfill \end{array}\end{array}

We here consider the case where the laser is adjusted to the off-resonant j-order red- and blue-sideband excitations, with corresponding detunings are δ_r and δ_b, respectively, i.e., ω_b = ω₀−jω_m + δ_r, and ω_r = ω₀ + jω_m + δ_b. In this case, the interaction Hamiltonian for such a system is

\begin{array}{}\text{(10)}& H_I^j=\lambda {\sigma }_{+}(b^{†}{}^j{\text{e}}^{-\text{i}{\delta }_{r}t}+b^j{\text{e}}^{-\text{i}{\delta }_{b}t})+\text{H}\text{.c}\mathrm{.,}\end{array}

where \lambda =\text{Ω}{f}_j(\widehat{n}) is the effective coupling strength, which is modulated by the nonlinear parameter η and nonlinear function f_j(\widehat{n}) [52],

\begin{array}{}\text{(11)}& f_j(\widehat{n})={\text{e}}^{-\frac{{\eta }^2}{2}}{\displaystyle \sum _m\frac{{\left(i\eta \right)}^{2m}}{m!(m+j)!}{b}^{†}{}^m{b}^m\mathrm{,}}\end{array}

with n = m + j and b^{†}{}^m{b}^m=\frac{n!}{(n-m)!}. Here, the initial phase Φ_d (d = l, r) are set to zero. The above Hamiltonian (10) corresponds to the interaction picture Hamiltonian of j-phonon quantum Rabi model with respect to the free Hamiltonian H_0=\frac{1}{4}({\delta }_r+{\delta }_b){\sigma }_z+\frac{1}{2j}({\delta }_r-{\delta }_b)b^{†}b. Therefore, after applying an undoing transformation to Hamiltonian (10), we have

\begin{array}{}\text{(12)}& H_R^j={\omega }_0^R\frac{{\sigma }_z}{2}+{\omega }^R{b}^{†}b+\lambda (b^{†}{}^j+b^j)({\sigma }^{+}+{\sigma }^{-})\mathrm{,}\end{array}

where {\omega }_0^R=-\frac{1}{2}({\delta }_r+{\delta }_b) and {\omega }^R=\frac{1}{2j}({\delta }_r-{\delta }_b) represent the qubit transition frequency of the simulated two-level system and the frequency of the simulated bosonic mode, respectively. Next, we focus on the situation where the frequency detuning between the simulated two-level system and the simulated bosonic mode is large. Specifically, if we consider the condition {\omega }_0^R\simeq 3j{\omega }^R, and the corresponding detuning \text{Δ}=\frac{3j-1}{3j}{\omega }_0^R\gg \lambda. Under this parameter region, if the simulated two-level system is prepared in the excited state |e⟩ and the simulated bosonic mode is in the vacuum state |0⟩, i.e, |e, 0⟩. According to the Hamiltonian (12), the term of RWA makes the system evolve to state |g, j⟩, while the counter-rotating terms would enable a virtual transition from |g, j⟩ to |e, 2j⟩, and subsequently the system can evolve to the final state |g, 3j⟩. Since {\omega }_0^R\simeq 3j{\omega }^R, thus |g, j⟩ and |e, 2j⟩ are the intermediate states, while |e, 0⟩ and |g, 3j⟩ can resonantly coupled with each other.

In order to confirm the validity of all our above derivations, we numerically simulate the dynamics governed by the effective model in Eq. (12) and compare it to the dynamics governed by the full Hamiltonian in Eq. (1). We set the parameters as: j = 2, ω_m = 0.1ω_c, Ω = 0.3ω_c and g₀ = 1.6ω_m. In Fig. 1, the red curve describes the time evolution of the population of state |0, 0⟩ (the cavity mode and the mechanical oscillator are all in the vacuum states) governed by full Hamiltonians (1), while the black curve describes time evolution of the population of state |e, 0⟩ (the simulated two-level system is initially in the excited state and the simulated bosonic mode are all in the vacuum states) governed by the effective Hamiltonians (12). It is obvious that the approximations adopted when deriving the effective Hamiltonian (12) are valid, since the two curves described by the full and effective Hamiltonians nearly coincide.

On the other hand, in an isolated system, the radiation pressure force applied to the oscillator is proportional to the photon number n_c and thereby lowers the energy of this photon state by n_c\times g_0^2/{\omega }_m and leads to different resonance conditions for the first and the second laser photons exciting the cavity under the strong single-photon coupling coefficient g₀. Thus, if the driving laser is on resonance with the |0⟩→|1⟩ transition, i.e., {\omega }_c-{\omega }_m+g_0^2/{\omega }_m=0, the transition |1⟩→|2⟩ is detuned by 2\times g_0^2/{\omega }_m and will be suppressed because of under the large detuning condition 2\times g_0^2/{\omega }_m ≫1. Thus, under this condition, the photon number in the cavity reduces to the two lowest-energy levels, |0⟩ and |1⟩. To make our analysis more convincing, we use Hamiltonian (1) to plot the photon number in the cavity, as is shown in Fig. 2. Figure 2 shows that the occupation probability of photon number states |0⟩, |1⟩, |2⟩, |3⟩, |4⟩, |5⟩, |6⟩, |7⟩ in the cavity are 0.581, 0.385, 0.031, 0.002, 0.0003, 0, 0 and 0, respectively. It shows that the probability of state |2⟩ is smaller than that of states |0⟩ and |1⟩, and thus the system can be approximately confined in the subspace {|0⟩, |1⟩}.

To make this physical process more clear, we now consider the condition that j = 2, and the corresponding Hamiltonian is

\begin{array}{}\text{(13)}& \begin{array}{l}{H}_I^2=\text{Ω}[f_2(\widehat{n})b^2{\sigma }^{+}{\text{e}}^{-\text{i}{\delta }_{r}t}+f_2(\widehat{n})b^{†}{}^2{\sigma }^{+}{\text{e}}^{-\text{i}{\delta }_{b}t}]+\text{H}\text{.c,}\hfill \end{array}\end{array}

where f_2(\widehat{n}) is introduced in Eq. (12). The Hamiltonian (13) corresponds to an interaction picture Hamiltonian of the two-phonon quantum Rabi model with respect to the free Hamiltonian \begin{array}{l}{H}_0=\frac{1}{4}({\delta }_r+{\delta }_b){\sigma }_z+\frac{1}{4}({\delta }_r-{\delta }_b)b^{†}b\hfill \end{array}. Therefore, after applying an undoing transformation to Hamiltonian (13), we have

\begin{array}{}\text{(14)}& H_R^2={\omega }_0^R\frac{{\sigma }_z}{2}+{\omega }^R{b}^{†}b+g(b^{†}{}^2+b^2)({\sigma }^{+}+{\sigma }^{-})\mathrm{,}\end{array}

where {\omega }_0^R=-\frac{1}{4}({\delta }_r+{\delta }_b) and {\omega }^R=\frac{1}{4}({\delta }_r-{\delta }_b) represent the qubit transition frequency of the simulated two-level system and the frequency of the simulated bosonic mode, respectively, and g=\text{Ω}{f}_2(\widehat{n}) is the coupling strength between them. Since the frequency {\omega }_0^R is six times of the frequency ω^R, and the corresponding detuning is \text{Δ}=\frac{5}{6}{\omega }_0^R\gg g, thus, the Hamiltonian in Eq. (14) represents a off-resonant two-phonon QRM. Under this parameter region, if the system is initially in the state |e, 0⟩, according to the Hamiltonian (14), the term of RWA makes the system evolve to state |g, 2⟩, while the counter-rotating terms would enable a virtual transition from state |g, 2⟩ to state |e, 4⟩, and subsequently the system can evolve to the final state |g, 6⟩. Because of the large detuning interaction, the states |g, 2⟩ and |e, 4⟩ are the intermediate states, while |e, 0⟩ and |g, 6⟩ can resonantly coupled with each other. The corresponding transitions are depicted in Fig. 3. In the figure, the excitation-number conserving and nonconserving process are represented by the arrowed red-solid and blue-dashed lines, respectively, while the green-dashed line represented the resonant coupling between the states |e, 0⟩ and |g, 6⟩. The parameters \sqrt{2}g, \sqrt{12}g, and \sqrt{30}g are the corresponding transition matrix elements.

In order to further illustrate the above analysis, we find out the energy spectrum as a function of {\omega }^R/{\omega }_0^R with g=0.06{\omega }_0^R under the large detuning condition based on the time-independent Schrodinger equation H_R^2|{\text{Ψ}}_n\rangle = E_n |Ψ_n⟩, with n = 1, 2, 3, 4. For simplicity, the effective coupling coefficient is set as a constant in the numerical simulation. When consider a more realistic situation, the effective coupling coefficient is not a constant, thus the Hamiltonian (13) can be considered as a nonlinear quantum Rabi model. Although the nonlinear term f_2(\widehat{n}) will suppress the collapses and revivals for a large Fock state |n⟩ whenever f_2(\widehat{n})|n\rangle = 0 (i.e., n = 17), it is still can be applied to generate n-phonon state [49]. The result of numerical simulation is shown in Fig. 4. Here, Fig. 4(b) is the enlarged view of the spectrum delimited by a dotted square frame in Fig. 4(a). As seen from Fig. 4(b), at {\omega }^R=0.168{\omega }_0^R, the two energy levels exhibit an avoided crossing, which serves as a signature of a resonance. At the avoided crossing point, the frequency of the simulated bosonic mode is equal to one- sixth of the frequency of the simulated two-level system. When the frequency of simulated bosonic mode across the avoided crossing point, the major component of |Ψ₃⟩ changes from state |e, 0⟩ to state |g, 6⟩. While for state |Ψ₄⟩, the trend of variety is the reversed, which indicates that the components of the two eigenvectors are exchanged across the resonance. Obviously, this avoided crossing corresponds to a six-phonon resonant coupling between the state |e, 0⟩ and |g, 6⟩, while such a resonant coupling does not appear in the traditional Jaynes–Cummings model. In this resonant two-phonon Rabi model, at the avoided crossing point, the states |Ψ₃⟩ and |Ψ₄⟩ can be approximated as the superposition state of |e, 0⟩ and |g, 6⟩, such as \frac{1}{\sqrt{2}}(|e\mathrm{,0}\rangle +|g\mathrm{,6}\rangle ) and \frac{1}{\sqrt{2}}(|e\mathrm{,0}\rangle -|g\mathrm{,6}\rangle ), respectively.

To investigate the effective coupling analytically, we start by reducing the Hamiltonian in the truncated Hilbert space spanned by the bare states |e, 0⟩, |g, 2⟩, |e, 4⟩, |g, 6⟩ and |e, 8⟩. As the parameters we used before, the detuning ∆ is larger than the coupling strength g, thus, the above bare states are nearly decoupled from states outside the truncated Hilbert space. The matrix form of the reduced Hamiltonian is given by

\begin{array}{}\text{(15)}& \left(\begin{array}{ccccc}\frac{{\omega }_0^R}{2}& \sqrt{2}g& 0& 0& 0\\ \sqrt{2}g& 2\frac{{\omega }_0^R}{2}-\frac{{\omega }_0^R}{2}& \sqrt{12}g& 0& 0\\ 0& \sqrt{12}g& 4\frac{{\omega }_0^R}{2}+\frac{{\omega }_0^R}{2}& \sqrt{30}g& 0\\ 0& 0& \sqrt{30}g& 6\frac{{\omega }_0^R}{2}-\frac{{\omega }_0^R}{2}& \sqrt{56}g\\ 0& 0& 0& \sqrt{56}g& 8\frac{{\omega }_0^R}{2}+\frac{{\omega }_0^R}{2}\end{array}\right)\mathrm{.}\end{array}

Here, the order of the column (row) is |e, 0⟩, |g, 2⟩, |e, 4⟩, |g, 6⟩ and |e, 8⟩. We have included state |e, 8⟩ to contributes to the Stark shift in the energy level for |g, 6⟩. In the six-phonon resonance, both |g, 2⟩ and |e, 4⟩ are the intermediate states which can be adiabatically eliminated. Thus, the effective Hamiltonian which describes a six-phonon coupling between the states |e, 0⟩ and |g, 6⟩ is

\begin{array}{}\text{(16)}& \begin{array}{ll}{H}_{\text{eff}}=\hfill & (\frac{{\omega }_0^R}{2}+2\frac{g^2}{\text{Δ}})|e\mathrm{,0}\rangle \langle e\mathrm{,0}|\hfill \\ \hfill & +(6{\omega }^R-\frac{{\omega }_0^R}{2}-36\frac{g^2}{{\omega }_0^R}-56\frac{g^2}{5{\omega }^R})|g\mathrm{,6}\rangle \langle g\mathrm{,6}|\hfill \\ \hfill & +{\text{Ω}}_{\text{eff}}(|e\mathrm{,0}\rangle \langle g\mathrm{,6}|+\text{H}\text{.c}\mathrm{.})\mathrm{.}\hfill \end{array}\end{array}

In the above effective Hamiltonian, the first two terms describe the Stark-shift of the states |e, 0⟩ and |g, 6⟩, respectively, while the last term describes the effective six-phonon coupling between these two states, with the corresponding coupling strength is {\text{Ω}}_{\text{eff}}=72\sqrt{5}{g}^3/5({\omega }_0^R{)}^2.

3 Six-phonon generation based on adiabatic passage across the resonance

In this section, we will show the six-phonon state generation by slowly sweeping the vibration mode frequency across the resonance. Initially, suppose the system is prepared in the state |e, 0⟩, then the system would remain in this state when ω^R is far away from the resonant frequency 6{\omega }_0^R. Subsequently, we adiabatically increase the bosonic mode frequency ω^R(t), after passing through the resonant frequency, a large population in state |g, 6⟩ can be obtained, which means that six-phonon state can be generated.

To verify the validity of the above adiabatic passage process, we perform a numerical simulation of the fidelity of the |Ψ(t)⟩ with respect to the target state and the expectation value of the phonon number of the simulated bosonic mode. Noted that the adiabatic passage, has the advantage that it is insensitive to moderate fluctuations of experimental parameters [53–55]. Thus, it is considered as a powerful tool to manipulate quantum systems which has enjoyed widespread applications in many fields across quantum physics and in nonclassical state generation, i.e., has been used to create photon states [56–58] and kinds of entangled state [53, 54, 59–61]. With the coupling between the vibrational modes and the reservoir being included, the master equation at zero temperature is [9, 62]

\begin{array}{}\text{(17)}& \begin{array}{lll}\frac{\text{d}\rho }{\text{d}t}=\hfill & -\text{i}\left[H_R^2(t)\mathrm{,}\rho \right]+\frac{\text{Γ}}{2}\left(2{\sigma }^{-}\rho {\sigma }^{+}-{\sigma }^{+}{\sigma }^{-}\rho -\rho {\sigma }^{+}{\sigma }^{-}\right)\hfill & \hfill \\ \hfill & +\frac{{\gamma }_m}{2}\left(2b\rho b^{†}-b^{†}b\rho -\rho b^{†}b\right)\mathrm{,}\hfill & \hfill \end{array}\end{array}

where Γ and γ_m are the decaying rates of stimulated two-level system and stimulated bosonic mode, respectively. The Hamiltonian H_R^2(t) described as

\begin{array}{}\text{(18)}& H_R^2(t)={\omega }_0^R\frac{{\sigma }_z}{2}+{\omega }^R(t)b^{†}b+g(b^{†}{}^2+b^2)({\sigma }^{+}-{\sigma }^{-})\mathrm{,}\end{array}

where ω^R(t) is time dependent and in the form

\begin{array}{}\text{(19)}& {\omega }^R(t)={\omega }^R(0)+v_{c}t\mathrm{,}0\le t\le t_f\mathrm{,}\end{array}

here v_c serves as the sweeping speed, and controls the adiabaticity of the process. Thus, in a certain time, ω^R(t) can sweep across the resonance frequency, which indicates that the Hamiltonian in Eq. (18) is the same as that in Eq. (14). We set g=0.06{\omega }_0^R, v_c={10}^{-6}{\omega }_0^{R^2}, {\omega }^R=0.168{\omega }_0^R, and Γ = γ_m = 0.0005g. Figure 5(a) shows the fidelity of the states |ψ_±⟩, defined as F = ⟨ψ_±| ρ |ψ_±⟩, as a function of the evolution time, where ρ is the density operator of the system, |ψ₊⟩ = |e, 0⟩, and |ψ₋⟩ = |g, 6⟩, respectively. The numerical result shows that the fidelity of state |g, 6⟩ approaches F≃0.937, and at the same time the fidelity of state |e, 0⟩ is 0. The deviation of the fidelity of state |g, 6⟩ is understood because the states |e, 4⟩ and |e, 8⟩ have a low but nonzero probability, although it’s very small enough compared with that of state |g, 6⟩. In Fig. 5(b), we display numerical simulation of the expectation value of the phonon number ⟨b^†b⟩, which also indicates the six-phonon generation. Noted that the six-phonon state is known as a nonclassical state and exist quantum interference features in phase space, also, has practical applications in quantum phononics, including the phonon lasers [63–68], phononic quantum networks [69, 70], quantum acoustic devices [71, 72] and ultrasensitive biodetection [73]. Noted that proposed proposal focus on the generation of multiphonon states, offering the possibility for realizing pulsed phonon laser and single phonon source. In contrast, the previous schemes dealt with implementation of continuous output phonon laser [63–68]. Schemes have been proposed to generate multi-phonon state in a micromechanical oscillator [74], in a superconducting quantum interference device [75], in a bulk acoustic-wave resonator [76], in trapped ion [77], and in quantum dot system [78].

To further demonstrate the production of the desired six-phonon state, we now perform the full Wigner tomography at the time \begin{array}{l}gt\simeq 1.1\times {10}^{-3}{\omega }_o^{R}t\hfill \end{array}. The Wigner function, defined as the quasiprobability distribution in phase space

\begin{array}{}\text{(20)}& W(\alpha )=\frac{2}{\pi }\text{Tr}[\rho D(\alpha )(-1{)}^{a^{†}a}D(-\alpha )]\mathrm{,}\end{array}

where D(\alpha )={\text{e}}^{\alpha a^{†}-{\alpha }^{*}a} is the displacement operator for the vibrational mode, with α is the phase-space amplitude. The plane-cut of the Wigner function for the targeted six-phonon state is shown in Fig. 6(a), while that of the ideal six-phonon state is shown in Fig. 6(b). As expected, the Wigner function obtained from prepared state is in well agreement with the result for the ideal six-phonon state.

4 Discussion and conclusions

It is necessary to present a brief discussion of the experimental feasibility of the proposed model. The proposal should work in the single-photon strong-coupling regime, which require the single-photon coupling strength g₀ is much lager than the decay rates of the cavity field γ_c and the mechanical mode γ_m. On the other hand, the two-level approximation of the cavity filed is guaranteed by the photon blockade effect, which requires the nonlinear parameter {\text{Δ}}_0=g_0^2/{\omega }_m should be larger than the decay rate of the cavity field γ_c. According to the current typical experimental parameters, the most promising candidates for realizing our proposal might be the optomechanical crystal devices [79], or the ultracold atoms in optical resonators [80, 81]. However, the parameters used for current experiments still need to be improved for our proposal due to the weak coupling coefficient g₀. Fortunately, a recent report shows that the optomechanical coupling strength can be effectively enhanced to the single-photon strong-coupling regime even when the system is originally in the weak-coupling regime [82]. Another candidate to improve the coupling coefficient is suggested by Pirkkalainen et al. [83], where a quantum two-level system (qubit) is added into the optomechanical system, which can effectively increase the coupling strength and allow for rich physics. Thus, with the rapid development of experimental technology, our proposal can be realized in the near future.

In conclusion, we have proposed a scheme to achieve a resonant multi-phonon QRM and adiabatic passage in a cavity mechanical system. In our scheme, the mechanical oscillator is driven by a bichromatic laser which is adjusted to off resonant j-order red- and blue-sideband. In the single-photon strong-coupling regime, the dynamics of this cavity optomechanical system can be treated as a j-phonon resonance quantum Rabi model. Then, we show that such a resonance appears as an avoided crossing of energy curves at the frequency of the simulated bosonic mode is near a fraction of the transition frequency of the simulated two-level system. As a typical example, we theoretically analyze the two-phonon resonance quantum Rabi model, and derive an effective Hamiltonian of the six-phonon coupling. Also, we present a method of six-phonon generation based on adiabatic passage across the resonance. Numerical simulations confirm the validity of the proposed scheme. Finally, with suitable setting the parameters of the driving laser, this method can be extended to the generation of 3j-phonon state in cavity optomechanical system.

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