The standard theoretical basis for understanding superfluidity in Bose systems was formulated by Beliaev in 1957, based on splitting the quantum field operator into a macroscopically occupied condensate component and a non-condensate component. This leads to a description of the condensate in terms of a 'single-particle state', the so-called macroscopic wavefunction. Since the discovery of Bose-condensed gases, an alternative theoretical picture has been developed which is based on a 'coherent band' of classically occupied states. This is often called the classical or c-field approach. The goal of this chapter is to review the differences between the Beliaev broken symmetry and c-field approach, and to argue that the c-field concept of a coherent condensate band of states has problems as a description of Bose superfluidity. However, the c-field idea of treating the lowest energy excitations classically can be used to advantage to simplify calculations within the Beliaev broken-symmetry formalism.
We discover transitions characterized by distinguishability of bosons in nonunitary dynamics with parity-time
(PT) symmetry. We show that
PT-symmetry breaking, a unique transition in non-Hermitian open systems, enhances regions in which bosons can be regarded as distinguishable. This means that classical computers can sample the boson distributions efficiently in these regions by sampling the distribution of distinguishable particles. In a
PT-symmetric phase, we find one dynamical transition upon which the distribution of bosons deviates from that of distinguishable particles, when bosons are initially put at distant sites. If the system enters a
PT-broken phase, the threshold time for the transition is suddenly prolonged, since dynamics of each boson is diffusive (ballistic) in the
PT-broken (
PT-symmetric) phase. Furthermore, the
PT-broken phase also exhibits a notable dynamical transition on a longer time scale, at which the bosons again become distinguishable. This transition, and hence the classical easiness of sampling bosons in long times, is true for generic postselected nonunitary quantum dynamics, while it is absent in unitary dynamics of isolated quantum systems.
PT-symmetry breaking can also be characterized by the efficiency of a classical algorithm based on the rank of matrices, which can (cannot) efficiently compute the photon distribution in the long-time regime of the
PT-broken (
PT-symmetric) phase.