Local quantum coherence and superfluidity

We consider a model of bosons on a regular lattice with a kinetic energy due to hopping among sites and a potential energy due to strong on site interaction. A superfluid phase is expected when the ground state of the local energy is doubly degenerate. We consider a new scheme of simmetry breaking associated to the superfluid phase in which the order parameter is the statistical average of the quantum coherence operator associated to the superposition of the degenerate local ground states. In the strong coupling limit a systematic expansion of the free energy can be performed in terms of the hopping amplitude at constant order parameter. Within such an expansion we obtain a self-consistent equation for the order parameter. The first order approximation gives, in the case of degeneracy between single occupied and empty state, the same result of the standard mean field approximation for the “hard core bosons”. This new approach to the superfluid phase is shown to have a natural application to the implementation of quantum computation on a superfluid. Typeset using REVTEX 1 The achievement of Bose Einstein Condensation experiments [1–3], and the interest to the quantum computing lead naturally to the proposal of using superfluid macroscopic states as qubits [4]. On the other hand trapping of bosons in optical lattices, [5] opens the possibility of new gating procedures. It is therefore of interest to improve the theory of bosonic particles with strong repulsive interaction and hopping among sites of a regular lattice and to investigate local excitations as possible qubits. It is well known that at zero temperature the transition from a superfluid to a Mott insulator phase is expected at a critical value of the ratio between the interaction strength and hopping amplitude except for integer values of the ratio between the chemical potential and the interaction strength μ = Un [6,7]. These values correspond in the atomic limit, i.e. for vanishing hopping amplitude, to a degeneracy of the ground state which corresponds to the same statistical weight of states involving n and n+1 particles. The associated energy can be assumed to vanish for a suitable choice of the zero energy level. A strong coupling expansion around the atomic limit has been proposed in [7]. This approach depends however on a “factorization” assumption. We show that an alternative approach can be introduced which avoids any factorization, and gives, to the first order in the hopping amplitude, the classical mean field result. The key point is a new definition of the superfluid phase in which the order parameter is associated to the statistical average of suitably defined local quantum coherence operator. We define the coherence of a system with respect to a particular state as the probability of finding the system, or part of it, in that state in the ensemble of all possible states. If the system is defined on a lattice, as in our case, and the ensemble is that of the statistical equilibrium, it is possible to define a local quantum coherence probability ( LQCP ) as the average of the projection operator πi = |Ai >< Ai| associated to the local state. ρ(πi) = tr (