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Phonon-phonon interaction and parametric down-conversion generation in multimode optomechanical systems

  • Zhen-Yang Peng 1, 2 ,
  • Ying-Dan Wang , 1, 2
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  • 1CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
  • 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received date: 2022-10-26

  Revised date: 2022-12-10

  Accepted date: 2022-12-12

  Online published: 2023-03-17

Copyright

© 2023 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

We studied the process of polariton conversion in a 3-mode nonlinear optomechanical system. Compared with the standard 2-mode optomechanical system, we find a much larger conversion rate of polariton modes can be achieved under typical dissipation conditions. To obtain a transparent understanding of the relevant physical process, we show that in the large detuned case, the cavity can be eliminated adiabatically, resulting in a parametric down-conversion (PDC) interaction between two phononic polariton modes. By tuning cavity detuning, the nonlinear interaction can be enhanced with the frequency-matching condition. Results from analytical treatment based on the effective PDC model agree with the numerical simulation. Such a system provides potential applications in nonlinear phononics.

Cite this article

Zhen-Yang Peng , Ying-Dan Wang . Phonon-phonon interaction and parametric down-conversion generation in multimode optomechanical systems[J]. Communications in Theoretical Physics, 2023 , 75(3) : 035101 . DOI: 10.1088/1572-9494/acaa97

1. Introduction

Optomechanical systems provide a rich platform to manipulate the quantum states and quantum interactions between mechanical oscillators and photons [1, 2] for quantum state control and quantum information processing [3, 4]. Plenty of works have studied optomechanical systems theoretically and experimentally. For example, mechanical ground state cooling [5-7] and optomechanical amplification [8-10] have been discussed in the linear regime since the nonlinear interaction is usually weak enough to be neglected.
On the other hand, the intrinsic optomechanical nonlinearity could lead to interesting nonlinear phenomena such as cavity density of states splitting [11-13], photon blockade [14, 15], non-classical mechanical states [16], parametric amplifiers [17, 18], nonlinear entanglement [19], etc. These nonlinear effects were proposed in either a strong coupling strength regime [20, 21], or a parametrically driven system [17, 22].
Multimode mechanical oscillators that couple with cavities generate quantum effects and quantum states that transition between mechanical modes [23-26], which extended the controlling and manipulating to mechanics via operating the optical cavity modes. Recently, nonlinear effects between two separate mechanical oscillators have been proposed as a parametric down-converter [27]. However, this mechanical nonlinearity originated from the dynamical Casimir effect, which requires ultrastrong coupling, g, and ultra-high frequencies of mechanical oscillators, with ωmi being the order of cavity eigenfrequency ωc. This will be a challenge in experiments.
As a well-known nonlinear optical model, parametric down-conversion (PDC) generated from nonlinear media has been widely used in quantum optics experiments to prepare single-photon sources and indistinguishable photons and generate entangled photon pairs [28]; the degenerate PDC process has also been used for the second harmonic generation, which has wide applications in quantum measurement [29]. These quantum effects might be realized in mechanical oscillators via optomechanical interactions by controlling optical modes. Experimentally, mechanical oscillators usually have much smaller dissipation rates, it is, therefore worth discussing the similar quantum effects and their applications to mechanical oscillators in quantum technologies.
In this paper, we have proposed a realization for PDC between two independent mechanical oscillators coupled with one cavity. The cavity detuning is modified by the driven laser frequency to obtain the effective phonon-phonon interaction and to resonantly enhance the nonlinear scattering process. We then discuss the advantages of our three-mode optomechanical model in generating PDC, comparing it with the two-mode system discussed in [21]. Finally, an effective conversion rate is discussed under the ‘two-level atom' approximation.
This paper is organized as follows: In section 2, we introduced our theoretical model and the treatment of adiabatic elimination; in section 3, we analyzed the generation and the observation of parametric down-conversion between two mechanical oscillators, then the comparison between our model and two-mode optomechanical system, which would generate down-conversion between mixed photon-phonon polariton modes, is investigated; in section 4, we derived an approximated expression for down-conversion rates. We present our conclusions and discussions in section 5 and some technical details are given in the appendices.

2. Model

We consider an optomechanical system with two membranes inside a Fabry-Perot cavity under a single monochromatic laser drive [30]. Assuming both of the two membranes have high transmission and oscillate at their equilibrium positions with coupling to the intracavity photons [31], the system Hamiltonian is given as (let = 1)
$\begin{eqnarray}\begin{array}{rcl}H & = & {\omega }_{c}{a}^{\dagger }a+\displaystyle \sum _{i=1,2}{\omega }_{{\rm{mi}}}{b}_{i}^{\dagger }{b}_{i}+{g}_{i}{a}^{\dagger }a({b}_{i}+{b}_{i}^{\dagger })\\ & & +{H}_{\mathrm{dri}}+{H}_{\mathrm{diss}},\end{array}\end{eqnarray}$
where ${H}_{\mathrm{dri}}={{\rm{\Omega }}}_{d}a{{\rm{e}}}^{{\rm{i}}{\omega }_{L}t}+{{\rm{\Omega }}}_{d}^{* }{a}^{\dagger }{{\rm{e}}}^{-{\rm{i}}{\omega }_{L}t}$ is the monochromatic driving term, Hdiss is system-bath interactions, a (bi) are the photon (phonon) annihilation operators. The parameter gi is single-photon optomechanical coupling strengths, and α is the classical drive amplitude at the frequency ωL. Transforming into the rotating frame, displacing the cavity field by a = α + d and making the standard linearization, the system Hamiltonian can be divided into a linearized part and a nonlinear interacting term as [11-13, 30]
$\begin{eqnarray}H={H}_{{L}}+{H}_{\mathrm{NL}}+{H}_{\mathrm{diss}},\end{eqnarray}$
$\begin{eqnarray}{H}_{{L}}={\rm{\Delta }}{d}^{\dagger }d+\displaystyle \sum _{i}{\omega }_{{\rm{mi}}}{b}_{i}^{\dagger }{b}_{i}+{G}_{i}(d+{d}^{\dagger })({b}_{i}+{b}_{i}^{\dagger }),\end{eqnarray}$
$\begin{eqnarray}{H}_{\mathrm{NL}}=\displaystyle \sum _{i}{g}_{i}{d}^{\dagger }d({b}_{i}+{b}_{i}^{\dagger }).\end{eqnarray}$
In the above, we have defined the laser detuning Δ = ωcωL + ∣G12∣/ωm1 + ∣G22/ωm2 where the collective optomechanical couplings have been taken into account; Gi = giα is the collective optomechanical coupling strength. Without loss of generality, we take α, gi > 0. For simplicity, the equal coupling case g1 = g2 = g (G1 = G2 = G) is discussed throughout our discussions. Nevertheless, the general case can be generalized similarly.
In the most studied regime of optomechanical systems that α ≫ 1 and gκ, the term HNL can be neglected. Nevertheless, when the nonlinear coupling is sufficiently strong (gκ, as discussed later), the nonlinear term should be taken into consideration. To study these nonlinear effects, our general approach to discuss the nonlinear interaction-induced effects is to treat the nonlinear interaction HNL as perturbations [11, 13, 30, 32]. To keep the linear Hamiltonian in a stable range to avoid optomechanical instability [1], we focus on the red detuning case (Δ > 0) and the large detuning limit Δ ≫ ωmi, G. If the cavity field is close to its ground state at the initial time, we may suggest the cavity emitting into the excited state via the optomechanical couplings will have small probabilities, the cavity mode d can be eliminated adiabatically from the full Hamiltonian [30, 33], and will provide an effective phonon-phonon coupling ${H}^{\mathrm{eff}}={H}_{{L}}^{\mathrm{eff}}+{H}_{\mathrm{NL}}^{\mathrm{eff}}$ with
$\begin{eqnarray}{H}_{{L}}^{\mathrm{eff}}=\displaystyle \sum _{i=1,2}{\omega }_{{\rm{mi}}}{b}_{i}^{\dagger }{b}_{i}-\displaystyle \frac{{G}^{2}}{{\rm{\Delta }}}{\left[({b}_{1}+{b}_{1}^{\dagger })+({b}_{2}+{b}_{2}^{\dagger })\right]}^{2},\end{eqnarray}$
$\begin{eqnarray}{H}_{\mathrm{NL}}^{\mathrm{eff}}=\displaystyle \frac{{{gG}}^{2}}{{{\rm{\Delta }}}^{2}}{\left[({b}_{1}+{b}_{1}^{\dagger })+({b}_{2}+{b}_{2}^{\dagger })\right]}^{3}.\end{eqnarray}$
The adiabatic elimination of photon mode generates an effective phonon-phonon interaction, whereas the original cavity-bath interaction leads to equivalent mechanical-bath interactions. This is treated with effective dissipation rates and effective thermal occupation numbers for mechanical oscillators, as calculated below.
Consider first the cavity and two mechanical oscillators are coupled to independent Markovian baths, respectively. The two mechanics have the thermal occupations ${\bar{n}}_{\mathrm{th},i}={n}_{B}({\omega }_{{\rm{mi}}},{T}_{M})$ in the mechanical-heat bath, where TM is physical temperature and nB(ω, T) is the Bose-Einstein distribution. The cavity-bath interaction Hamiltonian under the rotating-wave approximation takes the form
$\begin{eqnarray}{H}_{\kappa }^{\mathrm{int}}={\rm{i}}\sqrt{\displaystyle \frac{\kappa }{2\pi {\rho }_{c}}}\displaystyle \sum _{j}({f}_{j}^{\dagger }d-{f}_{j}{d}^{\dagger }),\end{eqnarray}$
where fj is the annihilation operator for cavity-bath mode j, κ is the damping rate of photons inside the cavity, and ρc is the density of states of the bath. As the heat bath is Markovian, κ and ρc are both frequency-independent. Under the adiabatic elimination, the modification of mechanical damping rates originates from the cavity-bath interaction. Therefore, the dynamics of two mechanical oscillators can be described via the following master equation,
$\begin{eqnarray}\dot{\rho }=-{\rm{i}}[{H}_{\mathrm{eff}},\rho ]+\displaystyle \sum _{i=1,2}\displaystyle \frac{{\tilde{\gamma }}_{i}}{2}({\bar{n}}_{1}+1)D[{b}_{i}]\rho +\displaystyle \frac{{\tilde{\gamma }}_{i}}{2}{\bar{n}}_{i}D[{b}_{i}^{\dagger }]\rho ,\end{eqnarray}$
with the effective damping of two mechanical oscillators
$\begin{eqnarray}{\tilde{\gamma }}_{i}=\gamma +\kappa \displaystyle \frac{4{G}^{2}{\omega }_{{\rm{mi}}}}{{{\rm{\Delta }}}^{3}},\end{eqnarray}$
and the effective steady-state occupation numbers
$\begin{eqnarray}{\bar{n}}_{i}=\displaystyle \frac{\kappa {\left(1-\tfrac{{\omega }_{{mi}}}{{\rm{\Delta }}}\right)}^{2}\tfrac{{G}^{2}}{{{\rm{\Delta }}}^{2}}+{\bar{n}}_{\mathrm{th},i}\gamma }{{\tilde{\gamma }}_{i}}.\end{eqnarray}$
Here we chose γ1 = γ2 = γ as the original mechanical damping. The condition ωmiκγ is required to ensure the resolved sideband condition.

3. PDC in two effective mechanical oscillators

The effective interaction described by equation (3b) includes various nonlinear interactions. To enhance the PDC between the two phonon modes, a resonant condition is required. Below we will discuss the relevant frequencies matching condition and enhanced parametric conversion process.

3.1. Effective mixed mechanical modes

In order to select the strongest nonlinear scattering process, we first diagonalize the linear Hamiltonian (equation (3a)) to the mixed mechanical base via a Bogoliubov transformation ${\left({b}_{1}\ {b}_{2}\ {b}_{1}^{\dagger }\ {b}_{2}^{\dagger }\right)}^{{\rm{T}}}=V{\left({b}_{-}\ {b}_{+}\ {b}_{-}^{\dagger }\ {b}_{+}^{\dagger }\right)}^{{\rm{T}}}$ [11, 13] (The details are in appendix B):
$\begin{eqnarray}{H}_{{L}}^{\mathrm{eff}}=\displaystyle \sum _{\sigma =\pm }{\omega }_{\sigma }{b}_{\sigma }^{\dagger }{b}_{\sigma },\end{eqnarray}$
where ω± are the eigenfrequencies and we assume ω+ > ω. To enhance the effect of the parametric conversion ${b}_{+}{b}_{-}^{\dagger 2}+h.c.$ , one needs to set the resonant condition ω+ = 2ω, by modifying the detuning to the value ${{\rm{\Delta }}}_{\mathrm{res}}$. In addition, keeping the resonant detuning in the red-detuning regime requires that ωm2 ∼ 2ωm1. Applying the rotating-wave approximation and neglecting the high-frequency terms in equation (3b), the effective nonlinear Hamiltonian takes the form:
$\begin{eqnarray}{H}_{\mathrm{NL}}^{\mathrm{eff}}\simeq {g}_{\mathrm{nl}}({b}_{+}{b}_{-}^{\dagger 2}+{b}_{+}^{\dagger }{b}_{-}^{2}),\end{eqnarray}$
which corresponds to a PDC Hamiltonian; the annihilation of one phonon in the ‘pump' mode b+ generates a pair of phonons in the ‘signal' mode b, with the effective nonlinear interacting strength gnl (The complete expression is equation (B8)).
In figure 1(a) we plot the nonlinear coupling strength gnl and ${{\rm{\Delta }}}_{\mathrm{res}}$ as a function of ωm2. It is noteworthy that in the regime ωm2 ∼ 2ωm1, ${{\rm{\Delta }}}_{\mathrm{res}}$ is indeed much larger than ωmi and G, which justifies the adiabatic eliminating approximation. Moreover, under this condition the mode b+ is almost b2 -like, b is almost b1 -like, as seen from the transformation coefficients plotted in figure 1(b). Whereas in the case that two original mechanics are degenerate (ωm2 = ωm1), there exists a mechanical dark mode decoupled from the optomechanical couplings [34, 35], which removed the effective phonon-phonon couplings. Likewise, this dark mode effect is shown in figure 1(a).
Figure 1. (a) Effective nonlinear coupling strength (cf equation (9)) and the value of detuning ${{\rm{\Delta }}}_{\mathrm{res}}$, as a function of ωm2. (b) Coefficients relating the mixed phonon modes to original mechanical modes, as a function of ωm2 with ${\rm{\Delta }}={{\rm{\Delta }}}_{\mathrm{res}}$. For each plot, ωm1 = 50κ and G = 0.6ωm1.
For further insight, it is useful to derive an approximate Lindblad master equation to describe the dynamics of mixed phonon modes b±. By adiabatically eliminating the cavity mode, the two mechanical modes are coupled with different heat baths [13], with thermal occupation ${\bar{n}}_{i}={n}_{B}({\omega }_{{\rm{mi}}},{T}_{i})(i=1,2)$, where Ti is the effective temperature for the corresponding mechanical bath. By transforming into modes b±, both the damping rates and occupation numbers have two parts, being from the heat baths of modes b1 and b2. The effective damping rates of the mixed modes take the form:
$\begin{eqnarray}{\gamma }_{-}={\tilde{\gamma }}_{1}{\left({V}_{11}+{V}_{13}\right)}^{2}+{\tilde{\gamma }}_{2}{\left({V}_{21}+{V}_{23}\right)}^{2},\end{eqnarray}$
$\begin{eqnarray}{\gamma }_{+}={\tilde{\gamma }}_{1}{\left({V}_{12}+{V}_{14}\right)}^{2}+{\tilde{\gamma }}_{2}{\left({V}_{22}+{V}_{24}\right)}^{2}.\end{eqnarray}$
Here V is the transforming matrix, where the complete form is in appendix B. The corresponding effective occupations are
$\begin{eqnarray}{\bar{n}}_{\pm }^{\mathrm{th}}=\displaystyle \frac{{\bar{n}}_{\pm ,1}^{\mathrm{th}}{\gamma }_{\pm ,1}+{\bar{n}}_{\pm ,2}^{\mathrm{th}}{\gamma }_{\pm ,2}}{{\gamma }_{\pm }},\end{eqnarray}$
where ${\gamma }_{-,1/2}={\tilde{\gamma }}_{1/2}{\left({V}_{1/\mathrm{2,1}}+{V}_{1/\mathrm{2,3}}\right)}^{2}$, ${\gamma }_{+,1/2}={\tilde{\gamma }}_{1/2}{\left({V}_{1/\mathrm{2,2}}+{V}_{1/\mathrm{2,4}}\right)}^{2}$, ${\bar{n}}_{\pm ,1/2}^{\mathrm{th}}={n}_{B}({\omega }_{\pm },{T}_{1/2})$, indicating the corresponding contributions from the two independent heat baths. The effective Lindblad master equation is also obtained as follows:
$\begin{eqnarray}\begin{array}{l}\dot{\rho }=-{\rm{i}}[{H}_{\mathrm{eff}},\rho ]+\displaystyle \sum _{\sigma =\pm }\displaystyle \frac{{\gamma }_{\sigma }}{2}({\bar{n}}_{\sigma }^{\mathrm{th}}+1)D[{b}_{\sigma }]\rho \\ \quad +\displaystyle \frac{{\gamma }_{\sigma }}{2}{\bar{n}}_{\sigma }^{\mathrm{th}}D[{b}_{\sigma }^{\dagger }]\rho ,\end{array}\end{eqnarray}$
where $D[{b}_{\sigma }]\rho =2{b}_{\sigma }\rho {b}_{\sigma }^{\dagger }-{b}_{\sigma }^{\dagger }{b}_{\sigma }\rho -\rho {b}_{\sigma }^{\dagger }{b}_{\sigma }$ is the Lindblad superoperator.
Equation (9) suggests that the mixed phonon modes can be operated as a mechanical parametric down converter. This converter mixes the conversion process between the two real mechanical modes. However, when ωm2 ∼ 2ωm1, the nonlinear dynamics lead to a nearly pure scattering process in which an excitation of b2 converts into two excitations of b1. The numerical simulation for the time evolution of phonon modes is plotted in figure 2 [36]. The contribution of phonon occupations includes two parts, the first is from the conversion process. As figure 2 illustrated, with the single-phonon occupation of b2 and vacuum of b1 initially, the conversion has the maximal contribution when the occupation of the ‘signal' mode is at the peak value. Another part that affects the time evolution of occupations is the heat baths, which will be discussed later.
Figure 2. Time evolution of average occupations of ‘mixed' modes b±and original mechanics b1,2. Figure 2(a) is plotted under the parameter ωm2 = 1.6ωm1 while figure 2(b) described ωm2 = 1.8ωm1. Other parameters are chosen as g = 0.5κ, ωm1 = 50κ, G = 0.6ωm1, γ = 10−5κ and ${\bar{n}}_{\mathrm{th},i}=0$.
The dynamical time evolutions for our model under different parameter conditions are plotted in figure 2. The oscillating amplitudes are reflected in the peak value for the ‘signal' modes and the corresponding positions are influenced by the effective nonlinear coupling strength gnl, the thermal occupations ${\bar{n}}_{\pm }^{\mathrm{th}}(t)={\bar{n}}_{\pm }^{\mathrm{th}}(1-\exp (-{\gamma }_{\pm }t))$ and the effective dissipation rates γ±. An initial state in a superposition of either pump or signal mode will suppress the oscillating amplitudes. As a result, the peak value for the ‘signal' mode in figure 2(b) is larger than that in figure 2(a), because the initial state in b±base is closer to the exact Fock state under b1,2 base, which has been shown in figure 1(b). Meanwhile, the ${{\rm{\Delta }}}_{\mathrm{res}}$ increases monotonically as ωm2 increases. Thus keeping the resonant condition in smaller ωm2 has a larger influence on the adiabatic eliminating approximation and finally depresses the PDC process between the two mixed phononic modes.

3.2. Comparison to the two-mode system

In this section, we compare the conversion efficiency of the PDC process in the 3-mode OM system with the parametric conversion between photonic and phononic polaritons in a 2-mode OM system [21].
To quantify the efficiency of the conversion process, the conversion rate is defined by the rescaled ratio of the population increase of the ‘pump' and ‘signal' mode, due to the conversion Hamiltonian:
$\begin{eqnarray}\eta =\displaystyle \frac{1}{2}\displaystyle \frac{{n}_{\mathrm{conv}}^{-}}{{n}_{\mathrm{conv}}^{+}}=\displaystyle \frac{1}{2}\displaystyle \frac{\langle {n}_{-}(t)\rangle -\langle {n}_{-}(0)\rangle -{\bar{n}}_{-}^{\mathrm{th}}(t)}{\langle {n}_{+}(0)\rangle -\langle {n}_{+}(t)\rangle +{\bar{n}}_{+}^{\mathrm{th}}(t)},\end{eqnarray}$
where ⟨n±(t)⟩ is the total occupation of the polariton modes and it includes the contributions from the initial population and thermalization, besides the converted population:
$\begin{eqnarray}\langle {n}_{\pm }(t)\rangle =\langle {n}_{\pm }(0)\rangle +{\bar{n}}_{\pm }^{\mathrm{th}}(t)\mp {n}_{\mathrm{conv}}^{\pm }.\end{eqnarray}$
Here ${n}_{\mathrm{conv}}^{\pm }$ indicates the part converted from the other polariton mode, ⟨n±(0)⟩ is the initial value for the corresponding mode, ${\bar{n}}_{\pm }^{\mathrm{th}}(t)$ is the thermal contribution. The factor 1/2 in equation (13) is to normalize the conversion rate to be 1 in the ideal case when 1 pump polariton generates 2 signal polaritons.
It has been noted that in the insert of figure 3(b), when the contribution of the thermal bath is dominant, ${n}_{\mathrm{conv}}^{-}$ can be negative, and the occupation of ‘signal' mode transformed via the PDC process is suppressed, which shows the conversion process breaks down.
Figure 3. Conversion rates for two-mode and three-mode optomechanical systems as a function of (a) mechanical dissipating rate γ and (b) single photon optomechanical coupling g. The red lines refer to the two-mode system in [21, 32], while the blue lines indicate the 3-mode system. In the two (three)-mode system, $G={G}_{\mathrm{res}}({\rm{\Delta }}={{\rm{\Delta }}}_{\mathrm{res}})$. We choose ωm2(Δ) = 1.6ωm1(ωM) in three(two)-mode systems, other parameters are chosen the same as in figure 2. Inset: the occupations of b mode in the 3-mode system, the black solid line is ⟨n(t)⟩ − ⟨n(0)⟩, the dotted gray line is the thermal occupation ${\bar{n}}_{\mathrm{th}}^{-}(t)$.
The photon mode in the 2-mode optomechanical system can be considered as either ‘pump' or ‘signal' mode with cavity detuning Δ ∼ 2ωM or Δ ∼ ωM/2 respectively (here ωM is the mechanical frequency) [21, 32]. To be specific, we assume Δ ∼ 2ωM and the other regime can be discussed in the same manner. As shown in figure 3, the conversion efficiency in the 3-mode case is much higher than that of the 2-mode case. This is because the effective dissipation in the 3-mode system is much smaller than that in the 2-mode case. In the typical OM setup, the photon damping is much stronger than the mechanical damping κγ. In the 2-mode case, the effective damping rate of at least one polariton mode is comparable to κ. Therefore, it dissipates into the steady state in a short time without any oscillation unless the nonlinear interaction g is much stronger than κ. Whereas in the 3-mode case, both modes involved in the parametric conversion are phononic polaritons with effective damping on the order of the mechanical damping γ, which can be smaller or comparable to the nonlinear interaction. Hence, the conversion rate for the 3-mode system is significantly larger than the 2-mode case, or in another word, the PDC process in the 3-mode system allows a more realistic value of the single photon optomechanical coupling.

4. Effective conversion rates

So far the down-conversion rates obtained in the previous section are computed numerically. However, when ωm2 → 2ωm1, the effective thermal occupation (equation (11)) grows rapidly, which requires much larger truncated Hilbert spaces for each mixed phonon mode to solve equation (12) numerically.
Fortunately, as shown in figure 1(a), in the regime of ωm2 → 2ωm1, gnl is very small such that the system can hardly be excited into higher states. In that case, if the initial state is ∣1, 0⟩, the nonlinear interaction only involves ∣0, 2⟩, where ∣n+, n⟩ indicates the state with corresponding emitting numbers. Therefore, the system is equivalent to a ‘two-level atom' system, with Hamiltonian
$\begin{eqnarray}{H}_{\mathrm{TLS}}=\displaystyle \frac{1}{2}{\omega }_{A}({\sigma }_{+}^{z}+{\sigma }_{-}^{z})+\sqrt{2}{g}_{\mathrm{nl}}({\sigma }_{+}^{+}{\sigma }_{-}^{-}+{\sigma }_{+}^{-}{\sigma }_{-}^{+}).\end{eqnarray}$
Here ωA = ω+ = 2ω, and ${\sigma }_{\pm }^{z,\pm }$ are Pauli matrices. One can obtain analytical results based on this effective Hamiltonian. The corresponding master equation is
$\begin{eqnarray}\begin{array}{l}\dot{\rho }=-{\rm{i}}[{H}_{\mathrm{TLS}},\rho ]+\displaystyle \sum _{\sigma =\pm }\displaystyle \frac{{\gamma }_{\sigma }}{2}({\bar{n}}_{\sigma }^{\mathrm{th}}+1)D[{\sigma }_{\sigma }^{-}]\rho \\ \quad +\displaystyle \frac{{\gamma }_{\sigma }}{2}{\bar{n}}_{\sigma }^{\mathrm{th}}D[{\sigma }_{\sigma }^{+}]\rho .\end{array}\end{eqnarray}$
If we label the states as ∣1⟩ ≡ ∣0, 0⟩, ∣2⟩ ≡ ∣1, 0⟩, ∣3⟩ ≡ ∣0, 2⟩, then the optical Bloch equations ${\rm{d}}{\rho }_{{ij}}/{\rm{d}}t=\langle i| \dot{\rho }| j\rangle $ can be written as
$\begin{eqnarray}\dot{\vec{{\boldsymbol{v}}}}=-{\boldsymbol{M}}\vec{{\boldsymbol{v}}}+\vec{{\boldsymbol{A}}},\end{eqnarray}$
where $\vec{{\boldsymbol{v}}}={\left({\rho }_{22},{\rho }_{33},{\rho }_{32}-{\rho }_{23}\right)}^{{\rm{T}}}$. The detailed form of M and $\vec{{\boldsymbol{A}}}$ are in appendix C. The above equation has the general solution
$\begin{eqnarray}\vec{{\boldsymbol{v}}}(t)={{\rm{e}}}^{-{\boldsymbol{M}}t}\vec{{\boldsymbol{v}}}(0)+(1-{{\rm{e}}}^{-{\boldsymbol{M}}t}){{\boldsymbol{M}}}^{-1}\vec{{\boldsymbol{A}}}.\end{eqnarray}$
The conversion rate under the approximation is defined as η = ρ33/(1 − ρ22). Similarly, the contribution of thermal fluctuation here has been neglected and the optimal evolution time topt has been chosen. The approximated form of the conversion rate is as follows:
$\begin{eqnarray}\eta =\displaystyle \frac{16{g}_{\mathrm{nl}}^{2}\exp [-({\gamma }_{+}+{\gamma }_{-}){t}_{\mathrm{opt}}/2]}{{{\rm{\Omega }}}^{2}-(16{g}_{\mathrm{nl}}^{2}-{\rm{\Omega }}{\rm{\Gamma }})\exp [-({\gamma }_{+}+{\gamma }_{-}){t}_{\mathrm{opt}}/2]},\end{eqnarray}$
where ${\rm{\Omega }}=\sqrt{32{g}_{\mathrm{nl}}^{2}-{{\rm{\Gamma }}}^{2}}$, Γ = γγ+ and topt = π/Ω. This so-called pseudo-spin approximation provides an alternative description for the dynamics of the degenerate PDC between the two mechanical modes when ωm2 is close to 2ωm1. Under this condition, the effective nonlinear coupling gnl is small enough to restrict the system inside the low-energy subspace. Figure 4 shows that the conversion rate calculated via the pseudo-spin approximation (equation (19)) is close to the numerical results obtained from equation (13) as ωm2/ωm1 approaches 2. This justifies the pseudo-spin approximation near ωm2 ∼ 2ωm1. Out of this regime, gnl is much larger such that the higher energy-level populations can not be neglected. As a result, the conversion rate under pseudo-spin approximation is higher than the more accurate numerical results because the former neglects the leakage due to the nonlinear scattering to all the higher levels.
Figure 4. Conversion rates for the exact numerical results (equation (13)) and pseudo-spin approximation(equation (19)) in different ωm2/ωm1, the parameters are chosen as ωm1 = 50κ, G = 0.6ωm1, g = 0.5κ and γ = 10−5κ. For the different value of ωm2, the cavity detuning is tuned to ${{\rm{\Delta }}}_{\mathrm{res}}$ to keep the resonant conditions.
The success of the PDC process mainly depends on the ratio between the nonlinear interaction strength gnl and the damping rate of γ±. As an example, we only discuss the ratio of gnl/γ below. Under the condition ωm2 ≃ 2ωm1, the approximate result of gnl/γ reads,
$\begin{eqnarray}\displaystyle \frac{{g}_{\mathrm{nl}}}{{\gamma }_{-}}\simeq \displaystyle \frac{3{{gG}}^{2}}{\gamma {{\rm{\Delta }}}^{2}+4{G}^{2}{\omega }_{m1}\kappa /{\rm{\Delta }}}.\end{eqnarray}$
Using the resonant condition Δ ≃ 2G2/(2ωm1ωm2) [30], one can find the optimal value of ωm2 to maximize the ratio of gnl/γ
$\begin{eqnarray}{\omega }_{m2}^{\mathrm{opt}}\simeq 2{\omega }_{m1}-{\left(\displaystyle \frac{4{G}^{4}\gamma }{\kappa {\omega }_{m1}}\right)}^{\tfrac{1}{3}},\end{eqnarray}$
and the maximum gnl/γ under the corresponding value of ${\omega }_{m2}^{\mathrm{opt}}$ takes the form
$\begin{eqnarray}{\left(\displaystyle \frac{{g}_{\mathrm{nl}}}{{\gamma }_{-}}\right)}_{\max }\sim \displaystyle \frac{3g{\left(2{\omega }_{m2}-{\omega }_{m1}\right)}^{2}}{4{G}^{2}\gamma +\tfrac{2\kappa {\omega }_{m1}{\left(2{\omega }_{m2}-{\omega }_{m1}\right)}^{3}}{{G}^{2}}}.\end{eqnarray}$
The ratio of gnl/γ and the optimal value of ωm2 when both the gnl/γ and η at the maximum are shown in figure 5. It should be noted that the ${\omega }_{m2}^{\mathrm{opt}}$ of the maximal gnl/γ and η have a negative correlation with G (equation (21)). This is because the ${\omega }_{m2}^{\mathrm{opt}}$ corresponds to the condition that γG2ωm1κ3 in equation (6), the induced optical damping and the physical mechanical damping are of the same order of magnitude. When ωm2 is approximately equal to 2ωm1, γ is in the dominant part of ${\tilde{\gamma }}_{i}$ (i = 1, 2), γγ while gnl still decreases as ωm2 increases. The ratio gnl/γ drops rapidly, as figure 5(a) suggested. Therefore, as G increases, the ${\omega }_{m2}^{\mathrm{opt}}$ decreases.
Figure 5. (a) The solid line is the exact numerical results, which are calculated from the exact solution equations (B8) and (10b), while the dashed line is the approximated analytical results (equation (20)). (b) The optimal values of ωm2 which for the maximal value of gnl/γ and conversion rate η. The solid line indicates equation (21); the dashed line reflects the value of ωm2 when equation (19) is at the maximum, while the dotted line is the corresponding conversion rate. All the other parameters for both plots are chosen the same as the figure 4.

5. Conclusion

In this paper, we investigate the PDC process generated in a 3-mode optomechanical system, with two mechanical oscillators inside a cavity. The cavity mode is eliminated such that the optomechanical coupling indices an effective phonon-phonon interaction. To resonantly enhance the PDC between two mechanical oscillators, the cavity detuning is tuned to the resonant value to obtain the optimal collective coupling G. The optimal value of ωm2/ωm1 to obtain the optimal PDC process is discussed. In the normal dissipating regime [8], mechanical dissipation is much smaller than cavity damping. Therefore, the PDC process between two mechanical oscillators can be achieved in more feasible parameter regimes, in comparison with the PDC process between photon-phonon polariton modes [21], or photon-photon modes.

Acknowledgments

YDW acknowledges support from the NSFC (Grant No. 12275331), the Penghuanwu Innovative Research Center (Grant No. 12047503) and the Innovation Program for Quantum Science and Technology (Grant No. 2021ZD031602)

Appendix A. Adiabatic elimination and effective Hamiltonian

The effective Hamiltonian is derived by eliminating cavity mode adiabatically. First, consider the Heisenberg equations of motion deriving from equation (2b):
$\begin{eqnarray}\begin{array}{rcl}\dot{d} & = & -{\rm{i}}({\rm{\Delta }}d+G({b}_{1}+{b}_{1}^{\dagger })+G({b}_{2}+{b}_{2}^{\dagger })),\\ {\dot{d}}^{\dagger } & = & {\rm{i}}({\rm{\Delta }}{d}^{\dagger }+G({b}_{1}+{b}_{1}^{\dagger })+G({b}_{2}+{b}_{2}^{\dagger })),\\ {\dot{b}}_{1} & = & -{\rm{i}}({\omega }_{m1}{b}_{1}+G(d+{d}^{\dagger })),\\ {\dot{b}}_{2} & = & -{\rm{i}}({\omega }_{m2}{b}_{2}+G(d+{d}^{\dagger })).\end{array}\end{eqnarray}$
Adiabatic elimination requires $\dot{d}\simeq 0$, $\dot{{d}^{\dagger }}\simeq 0$, which approximated to the lowest-order in g, gives
$\begin{eqnarray}d=-\displaystyle \frac{G}{{\rm{\Delta }}}({b}_{1}+{b}_{1}^{\dagger }+{b}_{2}+{b}_{2}^{\dagger }).\end{eqnarray}$
Substitute back to the Hamiltonian, we then have the effective Hamiltonian as equation (3a), (3b).
The above derivation works well in a closed system. However, if we want to discuss the effective damping rates for mechanical modes by eliminating cavity mode adiabatically, we should approximate $\dot{d}$ to the leading order, which gives
$\begin{eqnarray}\begin{array}{rcl}\dot{d} & = & -\displaystyle \frac{G}{{\rm{\Delta }}}({\dot{b}}_{1}+{\dot{b}}_{1}^{\dagger }+{\dot{b}}_{2}+{\dot{b}}_{2}^{\dagger })+\delta \dot{d}\\ & \backsimeq & {\rm{i}}\displaystyle \frac{G}{{\rm{\Delta }}}\displaystyle \sum _{i}{\omega }_{{\rm{mi}}}({b}_{i}-{b}_{i}^{\dagger })\\ & = & -{\rm{i}}{\rm{\Delta }}d-{\rm{i}}G\displaystyle \sum _{i}({b}_{i}+{b}_{i}^{\dagger }).\end{array}\end{eqnarray}$
Finally, we got the expression of d in leading-order approximation:
$\begin{eqnarray}d=\displaystyle \frac{G}{{\rm{\Delta }}}\displaystyle \sum _{i}({b}_{i}+{b}_{i}^{\dagger })+\displaystyle \frac{G}{{{\rm{\Delta }}}^{2}}\displaystyle \sum _{i}{\omega }_{{mi}}({b}_{i}-{b}_{i}^{\dagger }).\end{eqnarray}$
Now we consider the contribution of cavity-bath to the effective damping rates. Substituting equation (A4) into the cavity-bath interacting Hamiltonian (equation (4)), we can see the influence on each mechanical mode:
$\begin{eqnarray}\begin{array}{rcl}{H}_{\kappa ,i}^{\mathrm{int}} & = & {\rm{i}}\sqrt{\displaystyle \frac{\kappa }{2\pi {\rho }_{c}}}\displaystyle \sum _{i,j}-\displaystyle \frac{G}{{\rm{\Delta }}}\left\{\left(1-\displaystyle \frac{{\omega }_{{mi}}}{{\rm{\Delta }}}\right)({f}_{j}^{\dagger }{b}_{i}^{\dagger }-{f}_{j}{b}_{i})\right.\\ & & \left.+\left(1+\displaystyle \frac{{\omega }_{{mi}}}{{\rm{\Delta }}}\right)({f}_{j}^{\dagger }{b}_{i}-{f}_{j}{b}_{i}^{\dagger })\right\}.\end{array}\end{eqnarray}$
By letting ${A}_{c,i}^{\pm }$ for upward (downward) transition rates in the mechanical oscillator bi, we have
$\begin{eqnarray}{A}_{c,i}^{\pm }=\kappa {\left(1\mp \displaystyle \frac{{\omega }_{{mi}}}{{\rm{\Delta }}}\right)}^{2}\displaystyle \frac{{G}^{2}}{{{\rm{\Delta }}}^{2}}.\end{eqnarray}$
The contribution to effective damping rates of cavity-bath for the corresponding mechanical oscillators is ${A}_{c,i}^{-}-{A}_{c,i}^{+}$. Thus the effective damping rate is given as follows:
$\begin{eqnarray}{\tilde{\gamma }}_{i}=\gamma +({A}_{c,i}^{-}-{A}_{c,i}^{+}),\end{eqnarray}$
with γ indicating the regular mechanical bath. Substituting equation (A6) we finally obtained equation (6).
The average occupation numbers follow the equations:
$\begin{eqnarray}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}\langle {\bar{n}}_{i}\rangle =({\bar{n}}_{i}+1)({A}_{c,i}^{+}+{n}_{\mathrm{th},i}\gamma )-{\bar{n}}_{i}\left[{A}_{c,i}^{-}+({n}_{\mathrm{th},i}+1)\gamma \right],\end{eqnarray}$
and the steady-state occupations (equation (7)) can be obtained by requiring $\langle {\dot{\bar{n}}}_{i}\rangle =0$.

Appendix B. Diagonalization of effective linear Hamiltonian

In order to get equation (8), we first write the Hamiltonian (equation (3a)) into quadrature operators
$\begin{eqnarray}{H}_{{L}}^{\mathrm{eff}}=\displaystyle \frac{1}{2}\displaystyle \sum _{i,j}{x}_{i}{M}_{{ij}}{x}_{j}+\displaystyle \sum _{i}\displaystyle \frac{1}{2}{p}_{i}^{2},\end{eqnarray}$
where ${x}_{i}=({b}_{i}+{b}_{i}^{\dagger })/\sqrt{2{\omega }_{{mi}}}$, ${p}_{i}=-{\rm{i}}({b}_{i}-{b}_{i}^{\dagger })\sqrt{{\omega }_{{mi}}/2}$, and
$\begin{eqnarray}M=\left(\begin{array}{cc}{\omega }_{m1}^{2}-{B}_{11}^{2} & -{B}_{12}^{2}\\ -{B}_{12}^{2} & {\omega }_{m2}^{2}-{B}_{22}^{2}\end{array}\right),\end{eqnarray}$
with ${B}_{{ij}}^{2}=4{G}^{2}\sqrt{{\omega }_{{\rm{mi}}}{\omega }_{{\rm{mj}}}}/{\rm{\Delta }}$.
The eigenfrequencies of the linear Hamiltonian are obtained by diagonalizing M via a unitary transformation U, which has the form
$\begin{eqnarray}\begin{array}{rcl}{\omega }_{\pm }^{2} & = & \displaystyle \frac{1}{2}\{[{\omega }_{m1}^{2}+{\omega }_{m2}^{2}-{B}_{11}^{2}-2{B}_{22}^{2}]\\ & & \pm \sqrt{{\left[({\omega }_{m1}^{2}-{B}_{11}^{2})-({\omega }_{m2}^{2}-{B}_{22}^{2})\right]}^{2}+4{B}_{12}^{4}}\}.\end{array}\end{eqnarray}$
The unitary transformation U has the form:
$\begin{eqnarray}U=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right),\end{eqnarray}$
where $\tan 2\theta =2{B}_{12}^{2}/\left[({\omega }_{m1}^{2}-{B}_{11}^{2})-({\omega }_{m2}^{2}-{B}_{22}^{2})\right]$.
The transformation matrix V has the exact form
$\begin{eqnarray}V=\left(\begin{array}{cc}{V}_{+} & {V}_{-}\\ {V}_{-} & {V}_{+}\end{array}\right),\end{eqnarray}$
where
$\begin{eqnarray}{V}_{\pm }=\left(\begin{array}{cc}{U}_{11}{f}_{\pm }\left(\displaystyle \frac{{\omega }_{m1}}{{\omega }_{-}}\right) & {U}_{12}{f}_{\pm }\left(\displaystyle \frac{{\omega }_{m1}}{{\omega }_{+}}\right)\\ {U}_{21}{f}_{\pm }\left(\displaystyle \frac{{\omega }_{m2}}{{\omega }_{-}}\right) & {U}_{22}{f}_{\pm }\left(\displaystyle \frac{{\omega }_{m2}}{{\omega }_{+}}\right)\end{array}\right),\end{eqnarray}$
where we defined ${f}_{\pm }(x)=(\sqrt{x}\pm \sqrt{1/x})/2$.
From equation (B3) we can get the optimal detuning Δopt, which is tuned to keep the resonant condition ω+ = 2ω, and has the form
$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Delta }}}_{\mathrm{opt}} & = & 32{G}^{2}({\omega }_{m1}+{\omega }_{m2})\{(8{\omega }_{m1}^{2}-25{\omega }_{m1}{\omega }_{m2}+8{\omega }_{m2}^{2})\\ & & +5\sqrt{-{\omega }_{m1}{\omega }_{m2}(16{\omega }_{m1}^{2}-41{\omega }_{m1}{\omega }_{m2}+16{\omega }_{m2}^{2})}\}{}^{-1}.\end{array}\end{eqnarray}$
Finally, the exact form of effective nonlinear coupling strength gnl in equation (9) can be derived as
$\begin{eqnarray}\begin{array}{rcl}{g}_{\mathrm{nl}} & = & \displaystyle \frac{3{{gG}}^{2}}{{{\rm{\Delta }}}^{2}}{\left({V}_{11}+{V}_{13}+{V}_{21}+{V}_{23}\right)}^{2}\\ & & \times ({V}_{12}+{V}_{14}+{V}_{22}+{V}_{24}).\end{array}\end{eqnarray}$

Appendix C. Optical Bloch equations

Based on the effective Hamiltonian (15),
$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{+}^{+} & = & | 2\rangle \langle 1| ,{\sigma }_{-}^{+}=| 3\rangle \langle 1| ,\\ {\sigma }_{+}^{-} & = & | 1\rangle \langle 2| ,{\sigma }_{-}^{-}=| 1\rangle \langle 3| ,\\ {\sigma }_{+}^{z} & = & | 2\rangle \langle 2| -| 1\rangle \langle 1| ,\\ {\sigma }_{-}^{z} & = & | 3\rangle \langle 3| -| 1\rangle \langle 1| .\end{array}\end{eqnarray}$
Then we can derive the equations of motion for each element of the density matrix, noting that ρ11 + ρ22 + ρ33 = 1, we find the main equations are
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}}{\mathrm{dt}}{\rho }_{22} & = & -{\gamma }_{+}(2{\bar{n}}_{+}+1){\rho }_{22}-{\gamma }_{+}{\bar{n}}_{+}{\rho }_{33}\\ & & -{\rm{i}}\sqrt{2}{g}_{\mathrm{nl}}({\rho }_{32}-{\rho }_{23})+{\gamma }_{+}{\bar{n}}_{+},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}}{\mathrm{dt}}{\rho }_{33} & = & -{\gamma }_{-}{\bar{n}}_{-}{\rho }_{22}-{\gamma }_{-}(2{\bar{n}}_{-}+1){\rho }_{33}\\ & & +{\rm{i}}\sqrt{2}{g}_{\mathrm{nl}}({\rho }_{32}-{\rho }_{23})+{\gamma }_{-}{\bar{n}}_{-},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{d}}}{\mathrm{dt}}({\rho }_{32}-{\rho }_{23})=-{\rm{i}}2\sqrt{2}{g}_{\mathrm{nl}}({\rho }_{22}-{\rho }_{33})\\ \quad -\left[\displaystyle \frac{{\gamma }_{+}}{2}({\bar{n}}_{+}+1)+\displaystyle \frac{{\gamma }_{-}}{2}({\bar{n}}_{-}+1)\right]({\rho }_{32}-{\rho }_{23}).\end{array}\end{eqnarray}$
These equations can be written as the matrix form as in equation (17), with $\vec{A}={\left({\gamma }_{+}{\bar{n}}_{+},{\gamma }_{-}{\bar{n}}_{-},0\right)}^{{\rm{T}}}$ and
$\begin{eqnarray}{\boldsymbol{M}}=\left(\begin{array}{ccc}{\gamma }_{+}(2{\bar{n}}_{+}+1) & {\gamma }_{+}{\bar{n}}_{+} & {\rm{i}}\sqrt{2}{g}_{\mathrm{nl}}\\ {\gamma }_{-}{\bar{n}}_{-} & {\gamma }_{-}(2{\bar{n}}_{-}+1) & -{\rm{i}}\sqrt{2}{g}_{\mathrm{nl}}\\ {\rm{i}}2\sqrt{2}{g}_{\mathrm{nl}} & -{\rm{i}}2\sqrt{2}{g}_{\mathrm{nl}} & \displaystyle \frac{{\gamma }_{+}}{2}({\bar{n}}_{+}+1)+\displaystyle \frac{{\gamma }_{-}}{2}({\bar{n}}_{-}+1)\end{array}\right).\end{eqnarray}$
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