We study Magnus expansion (ME) approximation scheme for the interaction between an atom and a single quantized cavity mode (Jaynes-Cumming model) in a closed quantum system in resonance or near resonance for a time-dependent coupling coefficient g(t) in both the interaction and rotating picture by implementing a novel numerical method called MG4 and compare out results to the Runge-Kutta 4th (RK4) order solution to demonstrate the conservation of unitary evolution of the ME. A cursory study of open quantum system is given to encourage the study of ME for dissipative systems. Furthermore, we assume that our time-dependent coupling coefficient g(t) can take on two forms, Gaussian and sinusoidal, which are introduced as pulses to study the behavior and response of the cavity. Our results show that ME is a sufficient approximation scheme in our study of closed quantum systems which may have applications in quantum control.
Exact 1/4 mean-rung population and 2 continuum revival-time ratio in the anti-Jaynes--Cummings vertex at f⋆=nˉ+1
At atomic resonance the Jaynes--Cummings (JC) vertex of a single bosonic mode coupled to a two-level emitter is blind to f=ω/λ at zero detuning and Kerr; the anti-Jaynes--Cummings (AJC) vertex is not. We show that the mean-rung excited population of the AJC block equals exactly 1/4 at f⋆=nˉ+1, while the JC population at the same point is 1/2. The Poisson-averaged packet value at nˉ=16 is 0.24650, within 3.5×10−3 of the exact value, with the deviation reproduced by −nˉ/[16(nˉ+1)2]. At f⋆ the modulation depth of the AJC inversion is 1/2 and the relative sensitivity of the continuum revival-time ratio r=tRAJC/tRJC is maximal with value 1/(2nˉ+1). Full diagonalization on 2(nmax+1) states confirms the analytic values. The 1/4 value can be generated by Floquet-engineered counter-rotating coupling provided that the static transverse rate satisfies g0≪g1. Parameter translations for an ideal trapped-ion blue sideband and a flux-tunable circuit-QED coupler are reported.