A set of nearly good real numbers to specify the ground states associated with a Hamiltonian containing non-commutable terms and the effect of the odd-channel of a pair of different bosons emerging in multi-species systems

Abstract

A distinguishing feature of multi-species boson systems is the appearance of the odd channel, in which the spins of two different bosons are coupled to an odd integer. Through exact numerical solutions of the Schrodinger equation for a medium-body cold system containing two kinds of spin-1 atoms, the effect of the odd channel on the ground state (GS) has been studied. It was found that the odd-channel causes two types of fluctuation (a mixing of various components). (i) coherent mixing, where all the components have the same sign. In this way, the probability of an odd-pair emerging in the spin-state would be smaller; thus, this way would be adopted by the GS when the odd channel is repulsive. (ii) cyclic mixing, where half selected components have the + sign while the other half have the - sign. In this way, the probability of an odd-pair is larger; thus, this way would be adopted by the GS when the odd channel is attractive. It was further found that the terms in the Hamiltonian are no longer all commutable. Accordingly, the spin of a single species S_X (X=A, B) is no longer conserved. However, its average \overline{S_X} is well defined. It turns out that S_X and \overline{S_X} vary with the strengths in a similar way. The former jumps step-by-step from a good quantum number (an even integer) to the next good quantum number, the latter jumps also in a step-by-step way, but from a "real number" to another well-separated "real number". Exactly speaking, each of these real numbers is not exactly a number but an interval with a very narrow width at the real axis. Thus, the GS can be specified by these real numbers. It was found that, when the strengths of the two intraspecies interactions are not remarkably different, and/or the particle numbers are larger, the widths of the intervals are narrower, the above picture holds more nicely, and the GS can be well-specified by these real numbers.