Conserved excitation number and U(1)-symmetry operator for the anti-rotating (anti-Jaynes-Cummings) term of the Rabi Hamiltonian

In this article we apply algebraic operator ordering property of the basic atom and field operators to construct and prove conservation of the excitation number operator of the anti-rotating (anti-Jaynes-Cummings) component of the quantum Rabi model. The corresponding U(1) and parity symmetry operators have also been derived in explicit form.