Dynamics of the inhomogeneous Dicke model for a single-boson mode coupled to a bath of nonidentical spin-1/2 systems

We study the time dynamics of a single boson coupled to a bath of two-level systems (spins 1/2) with different excitation energies, described by an inhomogeneous Dicke model. Analyzing the time-dependent Schrödinger equation exactly, we find that at resonance the boson decays in time to an oscillatory state with a finite amplitude characterized by a single Rabi frequency if the inhomogeneity is below a certain threshold. In the limit of small inhomogeneity, the decay is suppressed and exhibits a complex (mainly Gaussian-like) behavior, whereas the decay is complete and of exponential form in the opposite limit. For intermediate inhomogeneity, the boson decay is partial and governed by a combination of exponential and power laws.

Figure 1

Time evolution of the boson $⟨n\left(t\right)⟩={\left|\alpha \left(t\right)\right|}^2$ obtained from numerical evaluation of Eqs. (7, 8) (full lines). Period of oscillation is $T=2\pi /s_0$ and gray bars are ${\left|{\alpha }_p\left(0\right)\right|}^2$. Main plot illustrates the intermediate regime with a partial decay and a combination of exponential and power decay laws: $\Delta /\Omega =2.2$; dashed line is the asymptote, ${\left|{\alpha }_p\left(t\right)+{\alpha }_c\left(t\right)\right|}^2$, from Eqs. (7, 9), $s_0=0.57\Delta$, and ${\left|{\alpha }_p\left(0\right)\right|}^2=0.26$. Inset illustrates the small inhomogeneity regime with a parametrically small decay and a Gaussian-like decay law: $\Delta /\Omega =0.2$; dashed line is the asymptote, ${\left|{\alpha }_p\left(0\right)\pm {\alpha }_c\left(t\right)\right|}^2$, from Eqs. (7, 10). The decay of ${\left|\alpha \left(t\right)\right|}^2$ is small, $\left({\left|{\alpha }_p\left(0\right)\right|}^2=1-{\Delta }^2/6{\Omega }^2\right)$, and the main contribution comes from ${\alpha }_p$.