  The Nambu Jona-Lasinio model with a Polyakov loop is extended to finite
isospin chemical potential case, which is characterized by simultaneous
coupling of pion condensate, chiral condensate and Polyakov loop. The pion
condensate, chiral condensate and the Polyakov loop as functions of temperature
and isospin chemical potential are investigated by minimizing the thermodynamic
potential of the system. The resulting $(T,\mu_I)$ phase diagram is studied
with emphasis on the critical point and Polyakov loop dynamics. The tricritical
point for the pion superfluidity phase transition is confirmed and the phase
transition for isospin symmetry restoration in high isospin chemical potential
region perfectly coincides with the crossover phase transition for Polyakov
loop. These results are in agreement with the Lattice QCD data.
