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In this paper we study a quench protocol on thermofield double states in thepresence of time-reversal symmetry that is inspired by the work of Gao,Jafferis and Wall.  The deformation is a product of hermitian operators on the left and rightsystems that are identical to each other and that lasts for a small amount oftime.  We study the linear dependence on the quench to the properties of thedeformation under time reversal. If the quench is time symmetric, then thelinear response after the quench of all T-even operators vanishes. Thisincludes the response of the energy on the left system and all thethermodynamic expectation values (the time averaged expectation values of theoperators). Also, we show under an assumption of non-degeneracy of theHamiltonian that the entanglement entropy between left and right is notaffected to this order. We also study a variation of the quench where aninstantaneous deformation is given by an operator of fixed T-parity and it'stime derivative. It is shown that the sign of the response of the Hamiltonianis correlated with the T-parity of the operator. We can then choose the sign ofthe amplitude of the quench to result in a reduction in the energy. Thisimplies a reduction of the entanglement entropy between both sides.

Quenches on thermofield double states and time reversal symmetry