For real inverse temperature beta, the canonical partition function is always positive, being a sum of positive terms. There are zeros, however, on the complex beta plane that are called Fisher zeros. In the thermodynamic limit, the Fisher zeros coal
esce into continuous curves. In case there is a phase transition, the zeros tend to pinch the real-beta axis. For an ideal trapped Bose gas in an isotropic three-dimensional harmonic oscillator, this tendency is clearly seen, signalling Bose-Einstein condensation (BEC). The calculation can be formulated exactly in terms of the virial expansion with temperature-dependent virial coefficients. When the second virial coefficient of a strongly interacting attractive unitary gas is included in the calculation, BEC seems to survive, with the condensation temperature shifted to a lower value for the unitary gas. This shift is consistent with a direct calculation of the heat capacity from the canonical partition function of the ideal and the unitary gas.
We develop an approach to solving numerically the time-dependent Schrodinger equation when it includes source terms and time-dependent potentials. The approach is based on the generalized Crank-Nicolson method supplemented with an Euler-MacLaurin exp
ansion for the time-integrated nonhomogeneous term. By comparing the numerical results with exact solutions of analytically solvable models, we find that the method leads to precision comparable to that of the generalized Crank-Nicolson method applied to homogeneous equations. Furthermore, the systematic increase in precision generally permits making estimates of the error.
The equation of state of a system at equilibrium may be derived from the canonical or the grand canonical partition function. The former is a function of temperature T, while the latter also depends on the chemical potential mu for diffusive equilibr
ium. In the literature, often the variables beta=(k_BT)^{-1} and fugacity z=exp(beta mu) are used instead. For real beta and z, the partition functions are always positive, being sums of positive terms. Following Lee, Yang and Fisher, we point out that valuable information about the system may be gleaned by examining the zeros of the grand partition function in the complex z plane (real beta), or of the canonical partition function in the complex beta plane. In case there is a phase transition, these zeros close in on the real axis in the thermodynamic limit. Examples are given from the van der Waal gas, and from the ideal Bose gas, where we show that even for a finite system with a small number of particles, the method is useful.
Phase matching has been studied for the Grover algorithm as a way of enhancing the efficiency of the quantum search. Recently Li and Li found that a particular form of phase matching yields, with a single Grover operation, a success probability great
er than 25/27 for finding the equal-amplitude superposition of marked states when the fraction of the marked states stored in a database state is greater than 1/3. Although this single operation eliminates the oscillations of the success probability that occur with multiple Grover operations, the latter oscillations reappear with multiple iterations of Li and Lis phase matching. In this paper we introduce a multi-phase matching subject to a certain matching rule by which we can obtain a multiple Grover operation that with only a few iterations yields a success probability that is almost constant and unity over a wide range of the fraction of marked items. As an example we show that a multi-phase operation with six iterations yields a success probability between 99.8% and 100% for a fraction of marked states of 1/10 or larger.
