**1. Introduction**

The problem of the creation of coherence and correlations between quantum systems has attracted considerable interest over the years not only because of a basic desire to understand how coherence and correlations could be created but also because of their importance in determination of nonclassical states of quantum systems [1–4]. Various types of correlations can exist between quantum systems, and their importance in understanding properties of quantum systems is often discussed in connection with different phenomena. For example, interference and quantum beats are among the simplest examples of phenomena resulting from the presence of mutual coherence, the so-called first-order correlation between quantum systems. Nonclassical phenomena, such as squeezing and entanglement, result from the presence of a different kind of correlation, often referred to as anomalous correlations [5,6]. The mutual coherence resulting from the first-order correlation is produced by a constant or nearly constant phase difference between quantum systems [3,4,7,8]. There are, however, coherence effects resulting from higher-order correlations, e.g., the intensity correlations, which are possible even when the phase difference between systems is random [9–12].

Anomalous and intensity correlations are the natural products of a range of twophoton processes in which simultaneous or nearly simultaneous pairs of photons are produced [13,14]. Because each photon in the pair has no definite phase, there is no constant phase relation between them. Therefore, photons in the pair behave as mutually incoherent. This property has been observed experimentally in the process of parametric

down conversion where pairs of photons, called the signal and idler photons, are produced [15,16]. Although the signal and idler photons are mutually incoherent, they are found in an entangled state which results from the anomalous correlation between the photons [17,18]. This observation suggests that the first-order correlation, which is responsible for the coherence and the anomalous correlation, are mutually exclusive. Following this observation, Mandel [19] proposed to call quantum systems exhibiting anomalous correlation as anticoherent.

The purpose of the present paper is to explore further possibilities to create coherence and anticoherence in a multipartite system. We consider a tripartite system composed of three coupled bosonic modes and investigate their coherence and anicoherence properties in an example of a three-mode optomechanical system, which consists of two cavity modes simultaneously coupled to the mode of a vibrating membrane. We assume that the cavity modes are affected by external input modes, which are in thermal states of unequal mean photon numbers. The difference in the mean number of photons of the input thermal fields constitutes an important and essential aspect of the work presented here. We will show how the populations of the modes and the correlations between them are sensitive to the population of the external thermal modes. When the external modes are in thermal states of different mean number of photons, we find that the steady-state populations of the modes can be dramatically altered, even to the point of the complete transfer of the population between the modes. Moreover, coherence and anticoherence, which may lead to entanglement between modes, can be established between modes which are completely decoupled from each other. This is certainly a surprising result since one would expect no correlations between decoupled modes affected by external thermal fields.

The paper is organized as follows. In Section 2, we introduce our model and the method of the evaluation of the dynamics of the system´s modes using an optomechanical system as an illustration. In Section 3, we study the properties of the steady-state population distribution between the modes. Section 4 is devoted to studying the correlations between the modes. We finish in Section 5 with the conclusion. In Appendix A, we present, as an illustration, a detailed derivation of the analytical expression for the steady-state population of the membrane mode.

#### **2. Three-Mode System**

The system we study consists of three parts; two modes whose fields are described by annihilation operators *a*<sup>1</sup> and *a*2, coupled to a third mode whose field is described by an annihilation operator *b*. The modes *a*<sup>1</sup> and *a*<sup>2</sup> are coupled to mode *b* through the nonlinear (parametric) squeezing-type interactions. There is no direct coupling between modes *a*<sup>1</sup> and *a*2. The Hamiltonian interaction for the three coupled modes is taken to be

$$H = \hbar g\_1 \left( a\_1^\dagger b^\dagger + a\_1 b \right) + \hbar g\_2 \left( a\_2^\dagger b^\dagger + a\_2 b \right),\tag{1}$$

where *g*<sup>1</sup> and *g*<sup>2</sup> are the coupling constants between modes *a*<sup>1</sup> and *b*, and *a*<sup>2</sup> and *b*, respectively. The nonlinear squeezing-type interactions, as described by the Hamiltonian (1) can be created in a variety of systems. For example, squeezing-type interactions between several modes have been realized in linear optical schemes involving external source of squeezed light and networks of beamsplitters [20]. Another example where this type of interaction can be created is a ring cavity containing an atomic ensemble coupled to counter-propagating modes of the cavity [21,22].

A good example of such a system is an optomechanical system consisting of two single-mode cavities sharing an oscillating mirror [23], or a single-mode optical cavity coupled to two mechanical modes of a vibrating membrane [24–26]. The method of how to achieve the parametric-type interaction between cavity modes and mechanical (mirror or membrane) mode has been discussed in several review papers [27–29]. In what follows, we consider an optomechanical system similar to that considered by Paternostro et al. [23] where entanglement properties between the modes were studied, assuming that only

the mirror mode is affected by external thermal fluctuations, i.e., the cavity modes were assumed to be in the ordinary vacuum states. This a a common practice in the study of the dynamics of optomechanical systems to assume that only the oscillating mirror or membrane is in contact with external modes (reservoir), being in a thermal state [30–35]. The ordinary vacuum states of the cavity modes are achieved by the coupling of the modes to an input (external) zero temperature modes. In practice, external modes are not in the ordinary vacuum but rather in non-zero temperature thermal states. Therefore, in what follows, we explore some correlation properties of a three-mode system, illustrated in Figure 1, assuming that the input modes to each of the cavities are in thermal states of unequal mean numbers of photons. The correlation properties of the modes affected by input thermal fields of unequal number of photons is the key point of the present work.

**Figure 1.** Schematic diagram of the system composed of two single-mode cavities sharing a vibrating membrane. The input fields to the cavities are in thermal states of unequal mean photon numbers.
