Are there two deadweight welfare losses associated with a Pigovian tax?
The Pigovian tax is responsible for neither of the deadweight losses in your diagram.

The Pigovian tax has partially, but not wholly, corrected a deadweight loss that was caused by the negative externality.

There is a deadweight loss associated with Pigovian taxes: that is the administrative cost of collecting the tax. This is not pictured in your diagram.