Continuous configuration-interaction for condensates in a ring

A continuous configuration-interaction approach for condensates in a ring is introduced. In its simplest form this approach utilizes for attractive condensates the Gross-Pitaevskii symmetry-broken solution and arrives at a groundstate of correct symmetry. Furthermore, the energy found is lower than the Gross-Pitaevskii one and, with increasing number of particles and/or strength of inter-particle interaction, is even lower than that accessed by tractable diagonalization of the many-body Hamiltonian. Implications also to excited states are briefly discussed. PACS numbers: 03.75.Hh, 05.30.Jp Typeset using REVTEX ∗E-mail: [email protected] 1 The physics of bosons in low dimensions has regained much attention recently following the realization of Bose-Einstein condensates in effectively lower-dimension trap potentials [1] and observation of bright solitons for attractive condensates in a quasi-one-dimensional (1D) trap [2] and optical waveguide [3]. Attractive condensates in 1D do not undergo collapse and have been a subject of long interest, see, e.g., Refs. [4–10]. For a system of two [4] and three [6] attractive bosons in a ring, although it is quite involved, it is possible to obtain expressions for the exact ground-state energy and wavefunction. However, for more than a few bosons and as the inter-particle interaction is increased, it becomes impossible to compute the exact ground-state energy and many-body wavefunction, and thus approximations are a must. The Gross-Pitaevskii (GP) equation has been a very successful approximation for BoseEinstein condensates and can explain many experiments with them, see, e.g., Refs. [11,12] and references therein. For attractive condensates in a ring, due to the continuous axial symmetry, the GP equation has been solved analytically [8]. Here, for weakly-interacting bosons the ground-state solution is symmetry-preserving, i.e., it possesses the rotational symmetry of the full many-body problem. For stronger (attractive) interactions, the GP solution lowest in energy becomes symmetry-broken and the gain in energy in comparison to the symmetry-preserving solution is substantial, see inset of Fig. 1. Because the GP orbital breaks the symmetry of the many-body system, it cannot properly describe the true ground-state. In this Letter, a continuous configuration-interaction (CI) approach for condensates in a ring is introduced. For the attractive condensate, we will start from the GP symmetry-broken solution and obtain a many-body wavefunction of correct symmetry and energy lower than the GP (symmetry-breaking) one. In addition as we explain and demonstrate below, with increasing number of particles and/or inter-particle interaction, our simple approach leads to an energy which is even lower than that accessed by tractable diagonalization of the manybody Hamiltonian. While solving explicitly for the ground state of attractive condensates in a ring, implication of our approach also to excited states and in higher dimensions will be discussed in brief. 2 Our starting point is the dimensionless many-body Hamiltonian describing N interacting bosons in a ring [10]: