Part l, Exact solution to the quantum Rabi model (Qrm) D In RWA, the n-th eigenstate isp -/a,In> ,n+1> En=O(n+ Rn at resonance, 0=0 Ern=a(n+o+=R E,n=O(m+)-g√hn+1 6=△-O,Rn=√62+482(n+1) E2n=0(n+3)+gVn+1 -sin e, n) 1(-|n) cos n+1 n+1 cose,n) In 2. sin n+ 1) 2n√2(n+ gvn+ △ cOS (R-6)2+4(m+1) The ground-state 0.个| | 0,1,2. | 1 n n n a n n b n     = =     +  □ In RWA, the n-th eigenstate is The ground-state 0 0 2 | | 0, E  = −  =  ( ) 1, 2 , 2 2 1, 2 , 2 2 1 1 ( ) 2 2 1 1 ( ) 2 2 , 4 ( 1) sin | | cos | 1 cos | | sin | 1 2 1 cos 4 ( 1) n n n n n n n n n n n n n E n R E n R R g n n n n n g n R g n            = + − = + + =  − = + +   −   =   +        =   +    + = − + + at resonance, δ=0 1, 2, 1, 2, 1 ( ) 1 2 1 ( ) 1 2 1 | | 2 | 1 1 | | 2 | 1 n n n n E n g n E n g n n n n n   = + − + = + + +   −   =   +        =   +    Part I, Exact solution to the Quantum Rabi model (QRM)