23 April 2001 Physics Letters A 282 (2001) 235–244 www.elsevier.nl/locate/pla Production of two-Fock states superpositions from even circular states and their decoherence A.L. Souza Silva a,1 , W. Duarte José b , V.V. Dodonov a,∗ , S.S. Mizrahi a a Departamento de Física, CCET, Universidade Federal de São Carlos, Rod. Washington Luiz km 235, São Carlos, 13565-905, SP, Brazil b Departamento de Ciências Exatas e Tecnologicas, Universidade Estadual de Santa Cruz, Rod. Ilhéus-Itabuna km 16, Ilhéus, BA, Brazil Received 23 February 2001; accepted 9 March 2001 Communicated by V.M. Agranovich Abstract We analyze the even circular state (ECS — a superposition of N coherent states |α〉 uniformly distributed with equal amplitudes on a circle) that can be engineered for the center of mass vibrational motion of a trapped ion. As a result of interference effects in phase space this state shows interesting features: depending on the value of |α| it may show up as the vacuum state |0〉, the number state |N 〉 or as two-state superposition, |0〉+|N 〉, |N 〉+|2N 〉. Here we discuss the statistics of the ECS, the probability for its production, its evolution under the effects of the environment and decoherence.  2001 Published by Elsevier Science B.V. PACS: 42.50.Vk; 42.50.Ct; 32.90.+a; 03.65.Bz Keywords: Even circular states; Trapped ion; Decoherence; Relaxation 1. Introduction Engineering of quantum states [1] that do not exist in the natural world is an exciting problem of nowa- days quantum mechanics and recent advances in quan- tum optics have allowed the realization of this chal- lenge with much success. A variety of states have been produced already, both for the modes of electromag- netic field in QED superconducting cavities [2] and for the vibrational modes of trapped ions [3]. They include, in particular, squeezed states and different * Corresponding author. E-mail addresses: palus@df.ufscar.br (A.L. Souza Silva), wjose@jacaranda.uescba.com.br (W. Duarte José), vdodonov@df.ufscar.br (V.V. Dodonov), salomon@df.ufscar.br (S.S. Mizrahi). 1 On leave from Universidade Federal de Rondônia, Brazil. kinds of the “Schrödinger cat” states. An interesting problem is the generation of the “most nonclassical” Fock states |n〉 and their finite superposition, such as |n 1 〉+|n 2 〉. Many different schemes have been pro- posed for this purpose [4]. In this Letter, we consider the possibilities of constructing the Fock states via the so-called circular states [5], (1)   Ψ N (α 0 )  = N  k=1 C k |α k 〉,α k = α 0 e i 2πk/N , where the coherent-states components are given by |α k 〉= e -|α k | 2 /2 ∞  n=0 α n k √ n! |n〉. For the first time the superposition of N harmonic oscillator coherent states |α k 〉 (having same modulus |α k |≡ √ a and uniformly distributed along a circle of 0375-9601/01/$ – see front matter  2001 Published by Elsevier Science B.V. PII:S0375-9601(01)00195-5