Eur. Phys. J. D 33, 297–300 (2005) DOI: 10.1140/epjd/e2005-00060-6 T HE EUROPEAN P HYSICAL JOURNAL D Creating the multidimensional entangled coherent states of two cavity modes X.B. Zou a , K. Pahlke, and W. Mathis Electromagnetic Theory Group at THT, Department of Electrical Engineering, University of Hannover, Germany Received 24 July 2004 / Received in final form 25 November 2004 Published online 3 May 2005 – c  EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2005 Abstract. We propose a scheme for creating the multidimensional entangled coherent states of two cavity modes in context of cavity quantum electrodynamics(QED). It is pointed out that under certain condition such superposition states can approximate pair coherent states and pair cat states with a high degree of accuracy. PACS. 42.50.Dv Nonclassical states of the electromagnetic field, including entangled photon states; quan- tum state engineering and measurements The preparation of nonclassical states in well controlled condition is the subject of an intense experimental activ- ity. The manipulation of these states leads to a better un- derstanding of basic quantum phenomena. Recently, there has been increasing interest in the preparation of two- mode nonclassical state in order to test quantum mechan- ics against local hidden theory [1] and implement quantum information processing protocols [2]. Cavity QED, with Rydberg atoms crossing supercon- ducting cavities, offers an almost ideal system for the gen- eration of entangled states and implementation of small scale quantum information processing [3]. In the context of cavity QED, numerous theoretical schemes for gen- erating entangled states of many atoms and nonclassi- cal states of cavity fields have been proposed [4], which led to experimental realization of the Einstein-Podolsky- Rosen (EPR) state [5] of two atoms, Greenberger-Horne- Zeilinger (GHZ) state [6] of three parties (two atoms plus one cavity mode), Schr¨ odinger cat state [7] and Fock state [8] of single-mode cavity field. Most of the schemes are based on the interaction of atoms and single-mode cav- ity field. An experiment has been reported for preparing two modes of a superconducting cavity in a maximally entangled state by using a sequence of interactions of a atom with two cavity modes [9]. This experiment opens up a new possibility for quantum state engineering and quantum information processing using multiple modes in a superconducting cavity. In reference [10], Ikram et al. proposed a scheme for generation of Bell states between two cavity modes. In reference [11], Solano et al. proposed a scheme to generate two-mode entangled coherent state in a cavity. a e-mail: zou@tet.uni-hannover.de In this paper, we propose a scheme to prepare two modes of a superconducting cavity in the multidimen- sional entangled coherent states of the form   Ψ N  = N-1  j=0 C j   αe -ij2π/N  a   αe ij2π/N  b (1) which can be considered as the multi-dimensional gener- alization of entangled coherent state [12]. Recently, it is shown that such superposition states have larger amount of entanglement than entangled coherent states and find applications in quantum information processing [13]. Fur- thermore, we also point out that under certain condition, such superposition states can approximate pair coherent states [14] and superposition of pair coherent states [15] with a high degree of accuracy. Pair coherent states are regarded as an important type of correlated two-mode states, which can exhibit various nonclassical proper- ties [14]. In reference [15], authors showed that pair cat states are characterized by additional nonclassical features beyond those of the pair coherent state. The experimen- tal realization of these nonclassical states is of practical importance. In references [16,17], schemes have been pro- posed for generation of motional pair coherent state and pair cat states in a two-dimensional ion trap. In refer- ence [18], Solano et al. propose a scheme for entangled coherent states in trapped ions. No schemes are proposed for generation of pair coherent states and superposition of pair coherent states in microwave cavity QED. In order to present the principle idea of our scheme to generate entangled states (1), we rewritten equation (1) as follows   Ψ N  = N-1  j=0 C j exp  - ij 2π N ( a † a - b † b )    α  a   α  b . (2)