  Bose-Einstein condensation happens as a gas of bosons is cooled below its
transition temperature, and the ground state becomes macroscopically occupied.
The phase transition occurs in the thermodynamic limit of many particles.
However, recent experimental progress has made it possible to assemble quantum
many-body systems from the bottom up, for example, by adding single atoms to an
optical lattice one at a time. Here, we show how one can predict the
condensation temperature of a Bose gas from the energy fluctuations of a small
number of bosons. To this end, we make use of recent advances in Lee-Yang
theories of phase transitions, which allow us to determine the zeros and the
poles of the partition function in the complex plane of the inverse temperature
from the high cumulants of the energy fluctuations. By increasing the number of
bosons in the trapping potential, we can predict the convergence point of the
partition function zeros in the thermodynamic limit, where they reach the
inverse critical temperature on the real axis. Using less than 100 bosons, we
can estimate the condensation temperature for a Bose gas in a harmonic
potential in two and three dimensions, and we also find that there is no phase
transition in one dimension as one would expect.
