We propose a new procedure, for Bayesian experimental design, that performs sequential design optimization while simultaneously providing accurate estimates of successive posterior distributions for parameter inference. The sequential design process is carried out via a contrastive estimation principle, using stochastic optimization and tempered Sequential Monte Carlo (SMC) samplers to maximise the Expected Information Gain (EIG). As larger information gains are obtained for larger distances between successive posterior distributions, this EIG objective worsens classical SMC performance. To handle this issue, tempering is proposed to have both a large information gain and an accurate SMC sampling. %tempering is proposed to improve SMC sampling accuracy. This novel combination of stochastic optimization and tempered SMC allows to jointly handle design optimization and parameter posterior inference. We provide a proof that the obtained optimal design estimators benefit from some consistency property. Numerical experiments confirm the approach potential on various benchmarks where our procedure outperforms other recent existing approaches.