Physica A 391 (2012) 82–86 Contents lists available at SciVerse ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Reconciling phase diffusion and Hartree–Fock approximation in condensate systems Gian Luca Giorgi a,∗ , Ferdinando de Pasquale b a IFISC (UIB-CSIC), Instituto de Física Interdisciplinar y Sistemas Complejos, UIB Campus, E-07122 Palma de Mallorca, Spain b Dipartimento di Fisica, Università di Roma La Sapienza, Piazzale A. Moro 2, 00185 Roma, Italy article info Article history: Received 15 June 2011 Available online 23 August 2011 Keywords: Bose–Einstein condensation Quantum phase diffusion abstract Despite the weakly interacting regime, the physics of Bose–Einstein condensates is widely affected by particle–particle interactions. They determine quantum phase diffusion, which is known to be the main cause of loss of coherence. Studying a simple model of two interacting Bose systems, we show how to predict the appearance of phase diffusion beyond the Bogoliubov approximation, providing a self-consistent treatment in the framework of a generalized Hartree–Fock–Bogoliubov perturbation theory. © 2011 Elsevier B.V. All rights reserved. 1. Introduction Bose–Einstein condensation (BEC) of weakly interacting gases, since the experimental observations of condensation of trapped alkali atoms [1–3], is used to reproduce the behavior of different condensed-matter models. Recently, the problem of atom interferometry has been widely discussed [4–6]. Even if BEC only appears in the weakly interacting regime, the interaction between particles plays a fundamental role [7]. In particular, number fluctuations determine the instability of the state causing phase diffusion, which limits the observation of coherence much more than other decoherence effects. Phase diffusion (PD), introduced in the seminal work of Lewenstein and You [8], is expected to take place in systems in restricted geometries, where a real symmetry-breaking phase cannot occur. In Ref. [8], the existence of PD has been predicted through the introduction of a gapless approximation, the Bogoliubov or the Hartree–Bogoliubov (HB) perturbation theory. It has been also noted that a complete self-consistent approximation, the Hartree–Fock–Bogoliubov method (HFB), has a gapped spectrum which would prevent PD. This latter approximation is however unsuitable because it does not satisfy the number conservation law. In spite of the weakness of the theoretical background, the existence of PD is widely accepted and used to explain several experimental and theoretical results [9–11]. The Bogoliubov transformation consist of a rotation connecting particle and quasi-particles. Within the HB perturbation theory, the rotation angle becomes infinite, while it is regularized in the HFB approximation [12,13]. The search for self-consistent equations compatible with the continuity equation is generally unsolved and has been representing one of the open problems of the physics of interacting bosons of the last sixty years. A review of these different theoretical approximation methods in the context of Bose gases can be found in Ref. [14]. In this paper, we address this problem introducing the simplest model of physical interest which exhibits condensation and phase diffusion, i.e. two interaction Bose modes in the presence of gauge symmetry-breaking field. Here, we want to show that a self-consistent approximation can be introduced which satisfies both the requirements of finite rotation and the absence of a gap in the excitation spectrum. This latter feature is a direct consequence of the validity of the continuity ∗ Corresponding author. E-mail addresses: gianluca@ifisc.uib-csic.es, gianluca0giorgi@gmail.com (G.L. Giorgi). 0378-4371/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2011.08.031