Cryptographic constructions based on hard lattice problems have emerged as a front runner for the standardization of post quantum public key cryptography. As the standardization process takes place, optimizing specific parts of proposed schemes becomes a worthwhile endeavor. Gaussian sampling over the integers is one of the fundamental building blocks of latticed-based cryptography. In this work, we propose a new integer Gaussian sampler based on polar codes, dubbed “polar sampler". The polar sampler is asymptotically information theoretically optimum in the sense that the number of uniformly random bits it uses approaches the entropy bound. It also features quasi-linear complexity and constant-time implementation. Our algorithm becomes effective when sufficiently many samples are required at each query to the sampler. Security analysis is given based on the statistical distance, Kullback-Leibler divergence and R\'enyi divergence. A comparison between the polar sampler and the Knuth-Yao sampler verifies its time efficiency and the memory cost can be further optimized if space-efficient successive-cancellation decoding is adopted.