*4.2. Degree of Anticoherence and Entanglement*

For the uncoupled modes *au* and *<sup>b</sup>*, mutual coherence function *a*† *ub* is equal zero, and therefore the modes are mutually incoherent. Although the mutual coherence between the modes is equal to zero, it must not be thought that all correlations between the modes are zero. In fact, there are correlations present, but they are reflected by nonzero values of the correlation function *aub*. This happens when *δn* = 0. Note that *aub* = 0 is accompanied by *a*† *ub* = 0. Following Mandel [19], the correlation function *aub* is called the anticoherence function, and to quantify the degree of anticoherence, he introduced the measure of anticoherence

$$
\gamma\_{ub}^{(2)} = \frac{|\langle a\_{\boldsymbol{u}} b \rangle|}{\sqrt{\langle a\_{\boldsymbol{u}}^{\dagger} b^{\dagger} a\_{\boldsymbol{u}} b \rangle}}.\tag{14}
$$

The values of *γ*(2) *ub* lie between 0 and 1. When the modes obey the Gaussian statistics, then [41]

$$
\langle a\_{\mu}^{\dagger}b^{\dagger}a\_{\mu}b\rangle = \langle a\_{\mu}^{\dagger}b^{\dagger}\rangle \langle a\_{\mu}b\rangle + \langle a\_{\mu}^{\dagger}b\rangle \langle b^{\dagger}a\_{\mu}\rangle + \langle a\_{\mu}^{\dagger}a\_{\mu}\rangle \langle b^{\dagger}b\rangle.\tag{15}
$$

Since *a*† *ub* = 0, Equation (14) then gives

$$
\gamma\_{ub}^{(2)} = \frac{\eta\_{ub}}{\sqrt{\eta\_{ub}^2 + 1}} \,\tag{16}
$$

where

$$\eta\_{ub} = \frac{|\langle a\_{\boldsymbol{u}} b \rangle|}{\sqrt{\langle a\_{\boldsymbol{u}}^{\dagger} a\_{\boldsymbol{u}} \rangle \langle b^{\dagger} b \rangle}}. \tag{17}$$

is the normalized anomalous correlation function. Thus, the

The nonvanishing anticoherence corresponds to a situation in which the modes could be entangled. In order to connect anticoherence to entanglement, we consider the Cauchy– Schwarz inequality, which is often used to identify entanglement [3]. The Cauchy–Schwarz inequality for the modes *au* and *b* is verified by reference to the so-called Cauchy–Schwartz parameter *χub* involving the second-order correlation functions

$$\chi\_{ub} = \frac{\mathcal{S}\_u^{(2)} \mathcal{S}\_b^{(2)}}{\left(\mathcal{S}\_{ub}^{(2)}\right)^2} \tag{18}$$

where

$$\mathcal{g}\_{ub}^{(2)} = \frac{\langle a\_{\mu}^{\dagger} b^{\dagger} a\_{\nu} b \rangle}{\langle a\_{\mu}^{\dagger} a\_{\nu} \rangle \langle b^{\dagger} b \rangle} \tag{19}$$

is the normalized second-order cross correlation function, and

$$\mathcal{g}\_{\mu}^{(2)} = \frac{\langle a\_{\mu}^{\dagger 2} a\_{\mu}^{2} \rangle}{\langle a\_{\mu}^{\dagger} a\_{\mu} \rangle^{2}}, \quad \mathcal{g}\_{b}^{(2)} = \frac{\langle b^{\dagger 2} b^{2} \rangle}{\langle b^{\dagger} b \rangle^{2}}. \tag{20}$$

are the normalized intensity autocorrelation functions of the modes *au* and *b*, respectively.

Using the Gaussian-mode decomposition (15), the correlation functions can be readily related to the coherence functions

$$\begin{array}{lcl} \mathcal{g}\_{i}^{(2)} &=& 2 + \eta\_{ii'}^{2} & i = u, b, \\ \mathcal{g}\_{ub'}^{(2)} &=& 1 + \left(\gamma\_{ub}^{(1)}\right)^{2} + \eta\_{ub}^{2}. \end{array} \tag{21}$$

Since in our case, *<sup>η</sup>uu* <sup>=</sup> *<sup>η</sup>bb* <sup>=</sup> *<sup>γ</sup>*(1) *ub* = 0, the Cauchy–Schwarz parameter takes the form

$$\chi\_{ub} = \frac{4}{\left(1 + \eta\_{ub}^2\right)^2}.\tag{22}$$

which can be expressed in terms of the degree of the anticoherence as

$$\chi\_{ub} = 4\left[1 - \left(\gamma\_{ub}^{(2)}\right)^2\right]^2. \tag{23}$$

To examine the occurrence of entanglement, we must check whether the Cauchy– Schwarz inequality (*χub* > 1) is violated. From Equation (23), we see that the condition that the modes are anticoherent, i.e., *γ*(2) *ub* is a necessary but not sufficient condition for entanglement between the modes. In other words, the modes could be anticoherent but not enough to obtain *χub* < 1. It is easily verified that for the Cauchy–Schwarz inequality to be violated, it is necessary that *γ*(2) *ub* > 1/ <sup>√</sup>2. Thus, for two modes to be entangled, they should be anticoherent to a degree about 71%.

Figure 7a shows the Cauchy–Schwarz parameter *χub* as a function of *δn* and *θ*. It is clearly seen that the parameter *χub* is reduced below its maximal value *χub* = 4 when *δn* = 0. The parameter *χub* decreases to a minimum value at *δn* = *n*, but unfortunately the minimal value is not smaller than the threshold for entanglement (*χub* = 1). This indicates that the anticoherence between the modes is not strong enough for the modes *au* and *b* to be entangled.

**Figure 7.** Variation of the Cauchy–Schwarz parameters (**a**) *χub* and (**b**) *χwb* with *δn* and *θ* for *γ* = *κ*, *n* = 3, *nb* = 0.1 and *g* = *κ*.

Although modes *au* and *b* are not entangled, there could be entanglement between modes *aw* and *b*, which are directly coupled to each other. The results for the Cauchy– Schwarz parameter *χwb* are shown in Figure 7b. It is seen that for certain values of *δn* and *θ*, the parameter *χwb* can be reduced below the threshold for entanglement. It was noticed before that in the case when the cavity modes are affected by thermal fields of the same photon numbers (*n*<sup>1</sup> = *n*<sup>2</sup> = *n*), entanglement between cavity mode and the membrane mode is restricted to very small values of *n* < 1/2. The results shown in Figure 7b are in sharp contrast to the case of equal number of thermal excitations, where entanglement is restricted to very small values of *n* and indicate quite clearly that in the case of unequal photon numbers (*n*<sup>1</sup> = *n*2), entanglement between the modes can be observed, even for large values of *n*.

In physical terms, we may attribute the appearance of entanglement between modes *aw* and *b* when *n*<sup>1</sup> = *n*<sup>2</sup> to the fact that a part of the population of the modes, which has a destructive effect on entanglement, is transferred and stored in the decoupled mode *au*.

Before concluding, we note that although we have discussed and graphically illustrated the coherence and anticoherence properties of the modes only for the case of equal damping rates of the modes, *γ* = *κ*, analogous results are obtained in the experimentally realistic case of *γ κ* [27–29].

As an illustration, in Figure 8, we plot *γ*(1) *uw* and *χwb* for *γ* = 0.01*κ*. Comparing the results with those presented in Figures 4b and 7b we saw that *γ*(1) *uw* and *χwb* behave in qualitatively the same manner as for *γ* = *κ*. While the maximal value of the coherence *γ*(1) *uw* between uncoupled modes is reduced for *γ κ* compared with Figure 4b, it is still nonzero over the entire range of *δn* = 0. Similarly, although the parameter *χwb* has risen for *γ κ* compared with Figure 7b, the region near *δn* = *n* still shows reduction of *χwb* below the threshold for entanglement.

**Figure 8.** (**a**) Variation of the degree of coherence *<sup>γ</sup>*(1) *uw* with *δn* and *θ* for *γ* = 0.01*κ*, *n* = 1, *nb* = 0.1 and *g* = 0.19*κ*. (**b**) Variation of the Cauchy-Schwarz parameter *χwb* with *δn* and *θ* for *γ* = 0.01*κ*, *n* = 3, *nb* = 0.1 and *g* = 0.19*κ*.

## **5. Conclusions**

We considered coherence properties between modes of a three-mode optomechanical system composed of two cavity modes simultaneously coupled to a membrane mode. We obtained analytical solutions for the steady-state populations of the modes and correlation functions describing coherence effects between the modes. Working in terms of linear superpositions of the cavity modes, we showed that one of the linear superpositions can be completely decoupled from the remaining modes. In spite of this, we found that the decoupled superposition can be completely coherent with the other superposition modes and can simultaneously exhibit anticoherence with the membrane mode. A detailed analysis showed that these correlation effects can happen only when the cavity modes are affected by the external input modes being in thermal states of unequal average photon numbers. In particular, we found that the coherences have a substantial effect on population distribution between the modes such that the population can be reversibly transferred between the superposition modes. The transfer of the population can lead to lowering of the population of the membrane mode. Therefore, the system can be considered as an alternative way to cool modes to lower temperatures. We also showed that a difference of the average numbers of photons in the thermal fields may affect entanglement between the superposition mode directly coupled to the membrane mode such that it may occur in a less restricted range of the number of thermal photons. In other words, the modes could be entangled, even with large numbers of thermal photons.

**Author Contributions:** Conceptualization, L.S., W.Y. and Z.F.; methodology, L.S. and Z.F.; software, Y.L., C.L. and K.Z.; validation, Y.L., C.L. and K.Z.; formal analysis, L.S., W.Y. and Z.F.; investigation, L.S., Y.L. and C.L.; writing—original draft preparation, Z.F.; writing—review and editing, L.S. and Z.F. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by National Science Foundation (NSF) of China (Grant Nos. 11374050, 11774054, and 12075036).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **Appendix A. Evaluation of the Steady-State Population of the Membrane Mode**

In this Appendix, we provide some details of the derivation of the analytical expression for the steady-state population of the mode *b*. Using Equation (7), we find that a former integration of the equations for *b* and *aw* leads to

$$b(t) = b(0)e^{-\gamma t} + \frac{1}{2}ig e^{-\gamma t} \int\_0^t dt' a\_w^\dagger(t')e^{\gamma t'} + \sqrt{2\gamma}e^{-\gamma t} \int\_0^t dt' b^{in}(t')e^{\gamma t'},\tag{A1}$$

$$a\_{\rm av}(t) = a\_{\rm av}(0)e^{-\rm xt} + \frac{1}{2}ig e^{-\rm xt} \int\_0^t dt' b^\dagger(t')e^{\rm xt'} + \sqrt{2\kappa} \, e^{-\rm xt} \int\_0^t dt' a\_{\rm av}^{\rm in}(t')e^{\rm xt'}.\tag{A2}$$

Substituting the expression for *a*† *<sup>w</sup>* into Equation (A1) and using the double integration rule

$$\int\_0^t dt' A(t') \int\_0^t dt'' B(t'') = \int\_0^{t'} dt' B(t') \int\_{t'}^t dt'' A(t'') \tag{A.3}$$

we find that the expression for *b*(*t*) can be written as

$$b(t) = y(t) + \int\_0^t dt' K(t, t') b(t'),\tag{A4}$$

where *K*(*t*, *t* ) is the kernel of the integral of the form

$$K(t, t') = \frac{g^2}{4(\gamma - \kappa)} \left( e^{-\chi(t - t')} - e^{-\gamma(t - t')} \right),\tag{A5}$$

and the term *y*(*t*) has the form

$$\begin{array}{ll} y(t) &=& b(0)e^{-\gamma t} - \frac{\overline{\chi} a\_{\overline{w}}^{\dagger}(0)}{2(\gamma - \kappa)} \left[ e^{-\kappa t} - e^{-\gamma t} \right] + \sqrt{2\gamma} \int\_{0}^{t} dt' b^{\text{int}}(t')e^{-\gamma(t - t')} \\ &+ \quad i\frac{\sqrt{2\kappa}g}{2(\gamma - \kappa)} \int\_{0}^{t} dt' a\_{\overline{w}}^{\dagger}(t') \left[ e^{-\kappa(t - t')} - e^{-\gamma(t - t')} \right]. \end{array} \tag{A6}$$

It is seen that the kernel *K*(*t*, *t* ) depends only on the time difference *t* − *t* , and may be written in the form

$$K(t, t') = \frac{g^2}{4(\gamma - \kappa)} H(t, t') = \lambda H(t, t'), \tag{A7}$$

where *<sup>λ</sup>* = *<sup>g</sup>*2/4(*<sup>γ</sup>* − *<sup>κ</sup>*). The integral Equation (A4) can be solved using the Laplace transformation. Thus if

$$\begin{aligned} \int\_0^t dt' H(t') e^{-pt'} &= H(p)\_{\prime} \\ \int\_0^t dt' y(t') e^{-pt'} &= y(p)\_{\prime} \\ \int\_0^t dt' b(t') e^{-pt'} &= b(p)\_{\prime} \end{aligned} \tag{A8}$$

we obtain from Equation (A4)

$$b(p) = \frac{y(p)}{1 - \lambda |H(p)|},\tag{A9}$$

where

$$H(p) = \frac{1}{p + \kappa} - \frac{1}{p + \gamma},\tag{A10}$$

and *y*(*p*) is

$$\begin{array}{rcl} y(p) &=& \left\{ b(0) + \sqrt{2\gamma} \mathcal{B}(p) + \frac{\mathcal{Z}}{2(\gamma - \kappa)} \left[ a\_w^\dagger(0) - i\sqrt{2\kappa} A\_w^\dagger(p) \right] \right\} \frac{1}{p + \gamma} \\ &-& \frac{\mathcal{Z}}{2(\gamma - \kappa)} \left[ a\_w^\dagger(0) + i\sqrt{2\kappa} A\_w^\dagger(p) \right] \frac{1}{p + \kappa} \end{array} \tag{A11}$$

with

$$\begin{array}{rcl}B(p) &=& \int\_0^t dt' b^{in}(t') \varepsilon^{-pt'},\\A^\dagger(p) &=& \int\_0^t dt' a\_{\overline{w}}^{in\dagger}(t') e^{-pt'}.\end{array} \tag{A12}$$

Substituting the solution (A10) for *H*(*p*) into Equation (A9), we readily find

$$b(p) = \frac{y(p)(p+\kappa)(p+\gamma)}{(p+\kappa)(p+\gamma)-\lambda}.\tag{A13}$$

Having available the Laplace transform *b*(*p*), we find *b*(*t*) simply by taking the inverse of the Laplace transformation (A13). We then obtain

$$\begin{array}{lcl} b(t) &=& \sum\_{i=1}^{2} (p - p\_i) b(p\_i) e^{\eta t} \\ &=& b(0) \left[ \frac{(\mathbf{x} - \boldsymbol{\gamma})}{\Delta} \sinh\left(\frac{1}{2}\Delta t\right) + \cosh\left(\frac{1}{2}\Delta t\right) \right] e^{-\frac{1}{2}(\mathbf{x} + \boldsymbol{\gamma})t} + a\_w^{\dagger}(0) \frac{i\mathbf{g}}{\Delta} \sinh\left(\frac{1}{2}\Delta t\right) e^{-\frac{1}{2}(\mathbf{x} + \boldsymbol{\gamma})t} \\ &+& \frac{i\boldsymbol{\gamma}\sqrt{\Delta}t}{2\Delta} \left[ A^{\dagger}(p\_1) e^{\frac{1}{2}\Delta t} - A^{\dagger}(p\_2) e^{-\frac{1}{2}\Delta t} \right] e^{-\frac{1}{2}(\mathbf{x} + \boldsymbol{\gamma})t} \\ &+& \frac{\sqrt{2}\gamma}{2\Delta} \left\{ [(\mathbf{x} - \boldsymbol{\gamma}) + \Delta] B(p\_1) e^{\frac{1}{2}\Delta t} - [(\mathbf{x} - \boldsymbol{\gamma}) - \Delta] B(p\_2) e^{-\frac{1}{2}\Delta t} \right\} e^{-\frac{1}{2}(\mathbf{x} + \boldsymbol{\gamma})t},\end{array} \tag{A14}$$

where

$$p\_{1,2} = \frac{1}{2}(\kappa + \gamma) \pm \frac{1}{2}\sqrt{(\kappa - \gamma)^2 + g^2} \tag{A15}$$

are roots of the quadratic equation

$$(p+\kappa)(p+\gamma) - \frac{1}{4}g^2 = 0,\tag{A16}$$

and Δ = -(*κ* − *γ*)<sup>2</sup> + *g*2.

We can use the solution *b*(*t*) to find the population of the mode *b* simply multiplying *b*(*t*) from the left by *b*†(*t*) and then taking the expectation value. We thus find

$$\begin{array}{lcl} \left(b^{\dagger}(t)b(t)\right) &=& \langle b^{\dagger}(0)b(0)\rangle \left[\frac{(\kappa-\gamma)}{\Lambda}\sinh\Big(\frac{1}{2}\Delta t\right) + \cosh\Big(\frac{1}{2}\Delta t\Big)\right]^{2}e^{-(\kappa+\gamma)t} \\ &+& \left[\langle a^{\dagger}\_{w}(0)a\_{w}(0)\rangle+1\right]\frac{\rho^{2}}{\Lambda^{2}}\sinh^{2}\Big(\frac{1}{2}\Delta t\Big)e^{-(\kappa+\gamma)t} \\ &+& \frac{\kappa\overline{\rho}^{2}}{2\Delta t}\left[\langle A(p\_{1})A^{\dagger}(p\_{1})\ranglee^{\Delta t}+\langle A(p\_{2})A^{\dagger}(p\_{2})\ranglee^{-\Delta t} \\ &-& \langle A(p\_{1})A^{\dagger}(p\_{2})\rangle-\langle A(p\_{2})A^{\dagger}(p\_{1})\rangle\big]e^{-(\kappa+\gamma)t} \\ &+& \frac{\gamma}{2\Delta t}\left[\langle(\kappa-\gamma)+\Delta]^{2}(\mathcal{B}^{\dagger}(p\_{1})\mathcal{B}(p\_{1})\ranglee^{\Delta t}+[(\kappa-\gamma)-\Delta]^{2}\langle\mathcal{B}^{\dagger}(p\_{2})B(p\_{2})\ranglee^{-\Delta t} \\ &+& \mathcal{B}^{\dagger}\big(\mathcal{B}^{\dagger}(p\_{1})B(p\_{2})\rangle+\langle\mathcal{B}^{\dagger}(p\_{2})B(p\_{1})\rangle\big)\right]e^{-(\kappa+\gamma)t}, \end{array} \tag{A17}$$

where

$$
\langle A(p\_i)A^\dagger(p\_i)\rangle = \int\_0^t dt' \int\_0^t dt'' \langle a\_w^{\rm in}(t')a\_w^{\rm in\dagger}(t'')\rangle e^{-p\_i(t'+t'')}, \; i=1,2,\tag{A18}
$$

and

$$
\langle B^\dagger(p\_i)B(p\_i)\rangle = \int\_0^t dt' \int\_0^t dt'' \langle b^{\text{in}\dagger}(t')b^{\text{in}}(t'')\rangle e^{-p\_i(t'+t'')}, \; i=1,2. \tag{A19}
$$

Since *bin*†(*<sup>t</sup>* )*bin*(*t* ) = *nbδ*(*t* − *t* ), we get

$$\langle B^\dagger(p\_i)B(p\_i)\rangle = n\_b \int\_0^t dt' e^{-2p\_i t'} = \frac{n\_b}{2p\_i} \left(1 - e^{-2p\_i t}\right), \; i = 1, 2,\tag{A20}$$

and

$$
\langle B^\dagger(p\_1)B(p\_2)\rangle = \langle B^\dagger(p\_2)B(p\_1)\rangle = \frac{n\_b}{p\_1+p\_2} \left[1 - e^{-(p\_1+p\_2)t}\right].\tag{A21}
$$

Similarly, since

$$<\langle a\_{\textit{uv}}^{\dot{\textit{in}}}(t')a\_{\textit{uv}}^{\dot{\textit{in}}\dagger}(t'')\rangle = (n\_{\textit{uv}}+1)\delta(t'-t''),\tag{A.22}$$

where *nw* = (*g*<sup>2</sup> <sup>1</sup>*n*<sup>1</sup> + *<sup>g</sup>*<sup>2</sup> <sup>2</sup>*n*2)/*g*2, we get

$$
\langle A(p\_1) A^\dagger(p\_1) \rangle = (n\_w + 1) \int\_0^t dt' e^{-2p\_1 t'} = \frac{(n\_w + 1)}{(\kappa + \gamma) - \Delta} \Big( e^{(\kappa + \gamma - \Delta)t} - 1 \Big), \tag{A23}
$$

and

$$
\langle A(p\_2)A^\dagger(p\_2)\rangle = \frac{(n\_{\text{w}}+1)}{(\kappa+\gamma)+\Delta} \Big(e^{(\kappa+\gamma+\Delta)t}-1\Big),
$$

$$
\langle A(p\_1)A^\dagger(p\_2)\rangle = \langle A(p\_2)A^\dagger(p\_1)\rangle = \frac{(n\_{\text{w}}+1)}{(\kappa+\gamma)} \Big(e^{(\kappa+\gamma)t}-1\Big). \tag{A24}
$$

Substituting these results for the correlation functions into Equation (A17), and taking the limit of *t* → ∞, we obtain

$$\lim\_{t \to \infty} \langle b^\dagger(t)b(t) \rangle = n\_b + \frac{\kappa (n\_b + 1)g^2}{(\kappa + \gamma)(4\kappa\gamma - g^2)} + \frac{\kappa g^2}{(\kappa + \gamma)(4\kappa\gamma - g^2)} n\_w. \tag{A25}$$

Writing *nw* in terms of *n* = (*n*<sup>1</sup> + *n*2)/2, *δn* = (*n*<sup>1</sup> − *n*2)/2, and tan *θ* = *g*2/*g*1, we obtain the expression for the population of the mode *b* given in Equation (10).

#### **References**

