*2.1. Time Evolution of the Modes*

We start by writing a complete set of the quantum Langevin equations for the system which can be easily obtained from the Hamiltonian (1) when taking into account dissipation (damping) of the modes and coupling of the modes to external input modes. In the rotating frame, the equations are of the form

$$\begin{aligned} \dot{b}^{\dagger} &= -\gamma \, b + \frac{1}{2}i \left( \lg\_1 a\_1^{\dagger} + \lg\_2 a\_2^{\dagger} \right) + \sqrt{2\gamma} \, b^{\dot{m}}, \\ \dot{a}\_1 &= -\kappa a\_1 + \frac{1}{2} i \lg\_1 b^{\dagger} + \sqrt{2\kappa} \, a\_1^{\dot{m}}, \\ \dot{a}\_2 &= -\kappa a\_2 + \frac{1}{2} i \lg\_2 b^{\dagger} + \sqrt{2\kappa} \, a\_2^{\dot{m}}, \end{aligned} \tag{2}$$

where *γ* is the decay rate of the membrane mode, and we have assumed the same decay rate *κ* for both cavity modes. Throughout Equation (3), operators *ain* <sup>1</sup> , *<sup>a</sup>in* <sup>2</sup> and *<sup>b</sup>in* are the input noise operators arising from the coupling of the modes to external modes (reservoirs). Here, we assume that the external modes are statistically independent, *δ* correlated, Gaussian, and in thermal states with

$$\begin{array}{rclcrcl}\langle a\_i^{in}(t)a\_i^{in\dagger}(t')\rangle &=& (n\_i+1)\delta(t-t'),\\\langle a\_i^{in\dagger}(t)a\_i^{in\dagger}(t')\rangle &=& n\_i\delta(t-t'),\\\langle b^{in}(t)b^{in\dagger}(t')\rangle &=& (n\_b+1)\delta(t-t'),\\\langle b^{in\dagger}(t)b^{in}(t')\rangle &=& n\_b\delta(t-t'),\end{array} \tag{3}$$

where *ni* = (exp{*h*¯ *<sup>ω</sup>*/*kBTi*} − <sup>1</sup>)−<sup>1</sup> is the average number of photons in the external modes coupled to the *i*-th cavity mode of frequency *ω* and temperature *Ti*, and *nb* = (exp{*h*¯ *<sup>ω</sup>*/*kBTb*} − <sup>1</sup>)−<sup>1</sup> is the average number of photons in the external modes of temperature *Tb* coupled to the membrane mode. Thus, in the absence of coupling to the membrane mode the cavity modes, *a*<sup>1</sup> and *a*<sup>2</sup> are in thermal states with mean numbers of photons *n*<sup>1</sup> and *n*2, respectively, whereas the membrane is in thermal state with mean number of photons *nb*.

#### *2.2. Linear Superpositions of the Modes*

It is seen from Equation (3) that mode *b* interacts simultaneously with both cavity modes. When a mode interacts simultaneously with two other modes, they may act collectively on the given mode. Therefore, it is more convenient to describe the dynamics of the system under consideration in terms of linear superpositions of the cavity modes. Thus, we can transform cavity annihilation operators to linear superpositions *aw* and *au* of the form

$$\begin{aligned} a\_w &= a\_1 \cos \theta + a\_2 \sin \theta, \\ a\_\theta &= a\_1 \sin \theta - a\_2 \cos \theta, \end{aligned} \tag{4}$$

and similarly, for the annihilation operators of the external input fields

$$\begin{cases} a\_{\overline{\omega}}^{in} = a\_1^{in} \cos \theta + a\_2^{in} \sin \theta, \\ a\_{\mu}^{in} = a\_1^{in} \sin \theta - a\_2^{in} \cos \theta, \end{cases} \tag{5}$$

where the mixing angle *θ* is given by tan *θ* = *g*2/*g*1. Hence in terms of the superposition modes, Equation (3) assumes the simplified form

$$\begin{aligned} \dot{b}\_{\text{uv}} &= -\gamma b + \frac{1}{2} i \xi a\_{\text{uv}}^{\dagger} + \sqrt{2\gamma} \, b^{in}, \\ \dot{a}\_{\text{uv}} &= -\kappa a\_{\text{uv}} + \frac{1}{2} i \xi b^{\dagger} + \sqrt{2\kappa} \, a\_{\text{uv}}^{in}, \\ \dot{a}\_{\text{u}} &= -\kappa a\_{\text{u}} + \sqrt{2\kappa} \, a\_{\text{u}}^{in} \end{aligned} \tag{6}$$

where *g* is the effective coupling strength between the modes, *g* = *g*2 <sup>1</sup> + *<sup>g</sup>*<sup>2</sup> 2.

For both analytical and numerical analyses, it is convenient to write the set of differential Equation (7) in a matrix form

$$
\dot{\mathbf{v}} = \mathbf{A}\mathbf{v} + \mathbf{f}\_{in\prime} \tag{7}
$$

where **v***<sup>T</sup>* = [*b*, *a*† *<sup>w</sup>*, *au*],**f***<sup>T</sup> in* = [√2*γbin*, <sup>√</sup>2*κ*(*ain <sup>w</sup>* )†, <sup>√</sup>2*κain <sup>u</sup>* ], and the drift matrix **A** is given by

$$\mathbf{A} = \begin{pmatrix} -\gamma & \frac{1}{2}ig & 0\\ -\frac{1}{2}ig & -\kappa & 0\\ 0 & 0 & -\kappa \end{pmatrix}. \tag{8}$$

From Equation (4) we see that the superpositions of the modes can be controlled through *θ* by changing the relationship between coupling constants *g*<sup>1</sup> and *g*2. However, the most important property seen from Equation (7) is that the superposition mode determined by the annihilation operator *au* is effectively decoupled from modes *aw* and *b*. On the other hand, the mode *aw* is coupled to the membrane mode *b* with the effective coupling constant *g*. Despite the lack of the coupling of the *au* mode to the remaining modes, we will show that the mode *au* can exhibit first-order coherence with the mode *aw* and the so-called anticoherence with the mode *b*. The coupling configurations between different modes is shown in Figure 2.

Although the time-dependent solution of Equation (7) is complicated, see Appendix A, the steady-state solution is simple and easily obtained. Therefore, we will focus on the steady-state populations of the modes and correlations between them. We note that the solutions for the populations and correlation functions can be obtained from Equation (7) without approximations by a direct integration of the equations of motion. In the Appendix A, we present a detailed derivation of the steady-state population of the membrane mode.

**Figure 2.** Coupling configurations between modes of the system. (**a**) Couplings between the mirror mode *b* and the cavity modes *a*<sup>1</sup> and *a*2. (**b**) Couplings between the mode *b* and the superposition modes *aw* and *au*. (**c**) Illustration that the decoupled mode *au* can be coherent with mode *aw* and anticoherent with mode *b*.

#### **3. Populations of the Modes**

Let us first examine how different modes are populated in the presence of thermal fields of different mean photon numbers *ni*. Solving Equation (7) for the steady-state, we find that the populations of the modes are

$$\begin{aligned} \langle b^{\dagger}b \rangle &= \quad n\_{b} + \frac{\kappa(n\_{b}+1)\mathfrak{g}^{2}}{(\kappa+\gamma)(4\kappa\gamma-\mathfrak{g}^{2})} + \frac{\mathfrak{x}\mathfrak{g}^{2}}{(\kappa+\gamma)(4\kappa\gamma-\mathfrak{g}^{2})}(n+\delta n \cos 2\theta), \\ \langle a\_{w}^{\dagger}a\_{w}\rangle &= \quad \frac{\gamma(n\_{b}+1)\mathfrak{g}^{2}}{(\kappa+\gamma)(4\kappa\gamma-\mathfrak{g}^{2})} + \left[1+\frac{\gamma\mathfrak{g}^{2}}{(\kappa+\gamma)(4\kappa\gamma-\mathfrak{g}^{2})}\right](n+\delta n \cos 2\theta), \\ \langle a\_{\mu}^{\dagger}a\_{\mu}\rangle &= \quad n-\delta n \cos 2\theta, \end{aligned} \tag{9}$$

where *n* = (*n*<sup>1</sup> + *n*2)/2 is the average number of photons, and *δn* = (*n*<sup>1</sup> − *n*2)/2 is a difference between the average number of photons in the thermal fields coupled to the cavity modes. Note that *δn* can vary from −*n* to +*n*. The populations depend also on the coupling constant *g*, which cannot be arbitrarily large. The values of *g* are restricted to those at which the steady-state solutions for the populations are stable, i.e., are positive. It is easily seen from Equation (10) that the positivity of the populations requires *<sup>g</sup>* <sup>&</sup>lt; <sup>√</sup>4*κγ*. Alternatively, conditions for the stability of the steady-state solutions (10) can be determined by applying the Routh–Hurwitz criterion [36] to Equation (7), which says that the components of vector **v** decay to stable steady-state values when the determinant of the drift matrix **A** is negative. It is easily verified from Equation (8) that det (**A**) <sup>&</sup>lt; 0 when *<sup>g</sup>* <sup>&</sup>lt; <sup>√</sup>4*κγ*.

The first important fact we can derive from Equation (10) is that in the case of *δn* = 0, the populations depend only on the effective coupling constant *g*. The difference *δn* = 0 induces a variation of the populations with the ratio of the coupling constants *g*<sup>1</sup> and *g*2, determined by the mixing angle *θ*. This means that in the case of *δn* = 0, by changing the ratio *g*2/*g*1, i.e., by varying the mixing angle *θ*, one can change the population of the mode *au* which is decoupled from the remaining modes *aw* and *b*. The transfer rate is proportional to *δn*, the difference of the thermal occupation of the modes *a*<sup>1</sup> and *a*2. Thus, if only one of the cavity modes is subjected to thermal excitation and the other mode is in a vacuum state, then *δn* = ±*n*, indicating that the thermal excitation of the cavity mode can be completely and reversibly transferred from modes *b* and *aw* to mode *au*.

The results of our discussion of variations of the populations with *θ* when the difference *δn* = 0 are illustrated in Figure 3. We present here variations of the populations with *θ* for two different values of the effective coupling constant *g*. As it is seen, for a weak coupling *g κ*, the transfer of the population occurs between modes *aw* and *au* only. The population of the mode *b* remains constant. Note the symmetry of the transfer process about *θ* = *π*/4 corresponding to *g*<sup>1</sup> = *g*2. For a strong coupling *g*, the transfer of the populations between the superposition modes is asymmetric about *θ* = *π*/4 (*g*<sup>1</sup> = *g*2) and is seen to be accompanied by a reduction of the population of mode *b*. In this case, the population is transferred to mode *au* not only from mode *aw*, but also from mode *b*. Lowering the population of the mode *b* implies that the system can be employed to cool the mode to a lower temperature. Thus, when *δn* = 0, it is possible to obtain dramatically reduced populations of the modes. In other words, keeping modes *a*<sup>1</sup> and *a*<sup>2</sup> at levels of different thermal occupations (*n*<sup>1</sup> = *n*2) can work as a mechanism for the cooling of the membrane mode.

**Figure 3.** Populations of the modes plotted as a function of *θ* for *γ* = *κ*, *n* = 1, *δn* = 1, *nb* = 0.1 and two different values of the coupling strength *g*: (**a**) *g* = 0.1*κ*, and (**b**) *g* = 1.5*κ*. Black solid line shows *b*†*b*, dashed red line *a*† *waw*, and dashed-dotted blue line *a*† *uau*.

#### **4. Correlations between the Modes**

We now investigate the coherence and correlation effects between the modes when the modes are influenced by thermal fields. We assume that the thermal fields coupled to the cavity modes are of unequal numbers of thermal photons *n*<sup>1</sup> = *n*2, and the mirror mode is coupled a thermal state with the mean number of phonons *nb*.

Different kinds of correlations can exist between the modes. Since the modes are in Gaussian states, which arises from the fact that the Hamiltonian (1) is quadratic, we consider only correlation functions up to a second order only. The correlation functions are expectation values of any combination of operators of two different modes. It is not difficult to show, using Equation (7), that in the steady state, there are the following non-zero correlation functions

$$\begin{aligned} \langle a\_{\mu}^{\dagger} a\_{\text{w}} \rangle &= \quad \left[ 1 + \frac{\text{g}^{2}}{8\kappa(\kappa + \gamma) - \text{g}^{2}} \right] \delta n \sin 2\theta, \\\langle a\_{\text{w}} b \rangle &= \quad \frac{2i\kappa\gamma\text{g}}{(\kappa + \gamma)(4\kappa\gamma - \text{g}^{2})} (n + n\_{b} + 1 + \delta n \cos 2\theta), \\\langle a\_{\mu} b \rangle &= \quad \frac{4i\kappa g}{8\kappa(\kappa + \gamma) - \text{g}^{2}} \delta n \sin 2\theta. \end{aligned} \tag{10}$$

and *a*† *wb* = *a*† *ub* = *awau* = 0. It is seen that the thermal fields of unequal photon numbers *δn* = 0 induce the first-order coherence between the superposition modes *au* and *aw* determined by the function *a*† *uaw*, and a correlation between between *au* and *b* modes determined by the function *aub*, usually called an anomalous correlation function [5,6], or, after Mandel, called anticoherence [19]. As we already mentioned, the nonvanishing correlation function *a*† *uaw* is the signature of the first-order coherence, which may lead to interference effects between the modes. It is well known that the nonvanishing anticoherence correlation functions *awb* and *aub* may lead to entanglement between the involved modes.

It is interesting that the mode *au* which is decoupled from the other modes can exhibit first-order coherence with the mode *aw* and anticoherence with mode *b*. According to Equation (11), this can happen only when *δn* = 0. To demonstrate this, we examine in detail measures of the degree of coherence and anticoherence.

#### *4.1. Degree of Coherence and Visibility*

We already saw that the cross-correlation or mutual coherence function *a*† *uaw* is different from zero when *δn* = 0. Therefore, the modes can be described as mutually coherent. The degree of coherence of the modes *au* and *aw* is defined by the normalized quantity

$$\gamma\_{\mu\nu}^{(1)} = \frac{|\langle a\_{\mu}^{\dagger} a\_{w} \rangle|}{\sqrt{\langle a\_{\mu}^{\dagger} a\_{\mu} \rangle \langle a\_{w}^{\dagger} a\_{w} \rangle}} \tag{11}$$

whose values lie between 0 and 1.

In Figure 4, we plot the degree of coherence as a function of *δn* and *θ*. Notice that at *δn* = 0, the modes are mutually incoherent, regardless of the value of *θ*. When *δn* = 0, the modes become mutually coherent. It is clearly seen that for a weak coupling between the modes *g*/*κ*  1, illustrated in Figure 4a, the first-order coherence function is symmetric about *θ* = *π*/4, and becomes asymmetric when *g*/*κ* > 1, the case corresponding to a strong coupling between modes, illustrated in Figure 4b. In this case, the degree of coherence is reduced in magnitude as *θ* increases. In the case of a weak coupling, an interesting situation is reached where the coherence attains its maximal value, i.e., the modes become mutually perfectly coherent when *δn* = *n*, i.e., when either *n*<sup>1</sup> or *n*<sup>2</sup> is equal to zero. On the other hand, in the strong coupling regime, the degree of coherence is always less than unity.

**Figure 4.** Variation of the degree of coherence between modes *au* and *aw* with *δn* and *θ* for *γ* = *κ*, *n* = 1, *nb* = 0.1 and two different values of the coupling strength *g*: (**a**) *g* = 0.1*κ*, and (**b**) *g* = 1.5*κ*.

One can notice from Figure 4a that in the limit of *δn* = *n*, the modes are perfectly coherent when *θ* = 0, but are completely incoherent when *θ* = *π*/2. The perfect coherence arises because the definite phase relationship between the modes *a*<sup>1</sup> and *a*<sup>2</sup> through the common coupling to the mode *b*.

Comparing the variation of *γ*(1) *uw* with the variation of the populations of the modes, shown in Figure 3, we see that *γ*(1) *uw* can be of equal unity regardless of the distribution of the population between the modes. This surprising behavior has been noticed before in systems of couple parametric downconverters [15,16,37–40], where interference effects were observed between the signal fields of the two downconverters with the degree of coherence *γ*(1) *ij* = 1.

We saw that the modes can be perfectly mutually coherent regardless of the distribution of the population between them. However, the distribution of the population between the modes has an effect on the visibility of the interference pattern and distinguishability of the modes. The visibility V is determined by the coherence function

$$|\mathcal{V}| = \frac{2|\langle a\_{\boldsymbol{u}}^{\dagger} a\_{\boldsymbol{w}} \rangle|}{\langle a\_{\boldsymbol{u}}^{\dagger} a\_{\boldsymbol{u}} \rangle + \langle a\_{\boldsymbol{w}}^{\dagger} a\_{\boldsymbol{w}} \rangle} \,' \tag{12}$$

whereas distinguishability is determined by the populations of the modes

$$|\mathcal{D}| = \frac{|\langle a\_{\boldsymbol{u}}^{\dagger} a\_{\boldsymbol{u}} \rangle - \langle a\_{\boldsymbol{w}}^{\dagger} a\_{\boldsymbol{w}} \rangle|}{\langle a\_{\boldsymbol{u}}^{\dagger} a\_{\boldsymbol{u}} \rangle + \langle a\_{\boldsymbol{w}}^{\dagger} a\_{\boldsymbol{w}} \rangle},\tag{13}$$

The visibility and distinguishability obey the complementarity relation |V|<sup>2</sup> + |D|<sup>2</sup> ≤ 1, in which the equality holds when the system is described by a pure state. When |D| = 0, the modes are indistinguishable. On the other hand, when |D| = 1, the modes are perfectly distinguished.

The distinguishability |*D*| is plotted in Figure 5 as a function of *δn* and *θ*. For *δn* = 0, the distinguishability |*D*| = 0 for all values of *θ*, indicating that in the case the cavity modes are affected by thermal fields of the same number of photons, and the superposition modes *aw* and *au* are undistinguishable independent of the ratio *g*2/*g*1. For a weak coupling and *δn* = 0, illustrated in Figure 5a, the distinguishability varies between its minimal value |*D*| = 0 at *θ* = *π*/4 to its maximal values at *θ* = 0 and *θ* = *π*/2. In the completely asymmetric case where *δn* = ±*n*, the distinguishability |*D*| = 1. More precisely, the modes can be perfectly distinguishable (|D| = 1) only if *δn* = *n* and either *g*<sup>1</sup> or *g*<sup>2</sup> is equal to zero. Thus, in the case of weak and equal coupling constants, *θ* = *π*/4, the modes are completely non-distinguishable, independent of *δn*. It is easy to understand if we refer to the fact that in the case of *θ* = *π*/4, the superpositions *aw* and *au* are equally weighted, so that one can not predict from which mode a detected photon came from.

**Figure 5.** Dependence of the distinguishability |*D*| on *δn* and *θ* for *γ* = *κ*, *n* = 1, *nb* = 0.1 and two different values of the coupling strength *g*: (**a**) *g* = 0.1*κ*, and (**b**) *g* = 1.5*κ*.

In the case of a strong coupling *g*, illustrated in Figure 5b, the distinguishability is strongly dependent on the relationship between *g*<sup>1</sup> and *g*2. We see that the modes are always at least partly indistinguishable, except for *δn* = *n* and *θ* = 0 at which |*D*| = 1. Moreover, the modes are perfectly indistinguishable at *θ* = *π*/4, i.e., when the modes are coupled to the membrane mode with unequal coupling strengths, *g*<sup>1</sup> = *g*2.

A close-up view of the variation of the distinguishability |*D*| with *θ* at *δn* = *n* is shown in Figure 6. We also plot the visibility |*V*| and the complementarity *S* = |*V*| <sup>2</sup> + |*D*| 2. The visibility vanishes only when *g*<sup>1</sup> = 0 or *g*<sup>2</sup> = 0, i.e., when one of the cavity modes is decoupled from the membrane mode. In the limit of a weak coupling, *g κ*, the visibility and distinguishability are perfectly mutually exclusive, and *S* = 1 for all values of *θ*, indicating that independent of the ratio *g*2/*g*1, the system is in a pure state. On the other hand, in the limit of a strong coupling *g* > *κ*, they are no longer perfectly mutually exclusive, i.e., the visibility is greatest for *g*<sup>1</sup> = *g*<sup>2</sup> and the maximum of the visibility does not correspond to the minimum of the distinguishability. Additionally, in this case, *<sup>V</sup>*<sup>2</sup> + |*D*| <sup>2</sup> < 1, except *θ* = 0 at which the modes are perfectly distinguishable. Thus, except *θ* = 0, the system is in a mixed state. The mixed state results from the fact that in the strong coupling regime, not only the population from mode *aw*, but also a population from the membrane mode *b* is transferred to mode *au*, as it is seen in Figure 3b.

**Figure 6.** Close-up view of the variation of the distinguishability |*D*| (red dashed line) with *θ* at *δn* = *n* shown in Figure 5 together with the visibility |*V*| (blue solid line) and complementarity *S* = |*V*| <sup>2</sup> <sup>+</sup> <sup>|</sup>*D*<sup>|</sup> <sup>2</sup> (black dashed-dotted line) for *γ* = *κ*, *n* = 1, *nb* = 0.1 and two different values of the coupling *g*: (**a**) *g* = 0.1*κ*, and (**b**) *g* = 1.5*κ*.
