Non-Markovian dynamics of mixed fermionic-bosonic systems: Rotating-wave-approximation coupling

Employing quadratic fermionic and bosonic Hamiltonians for collective and internal subsystems with a linear rotating-wave-approximation coupling, we studied the role of heat-bath statistics on the dynamics of the collective motion. The master equations for the collective occupation number derived directly from the quadratic Hamiltonians and within the Non-Markovian Langevin approach are discussed and their solutions are obtained. Because of the different nature of the heat-bath statistics, the path to equilibrium or the relaxation time is affected as shown in the numerical calculations.

Figure 1

The calculated dependencies of the friction coefficients on time $t$ for the fermionic-fermionic (f-f) and bosonic-bosonic (b-b) and the mixed fermionic-bosonic (f-b) and bosonic-fermionic (b-f) systems at different coupling constants $g_0$ and temperatures $T$ which are indicated in the plots. In the plots (a) and (c), the results for different systems are almost identical.

Figure 2

The same as in Fig. 1, but for the diffusion coefficients. For the better discrimination, the results presented in (a), (c), and (d) are multiplied by the factors ${10}^2,{10}^4$, and ${10}^2$, respectively.

Figure 3

The calculated dependencies of the asymptotic diffusion (a), (b) and friction (c), (d) coefficients on the coupling constant $g_0$. The results for the fermionic-fermionic (f-f) and bosonic-bosonic (b-b) and the mixed fermionic-bosonic (f-b) and bosonic-fermionic (b-f) systems at different temperatures $T$ are indicated. For the better discrimination, the results presented in (a) are multiplied by the factor ${10}^3$. In (c) the results for different systems almost coincide.

Figure 4

The calculated dependencies of the asymptotic diffusion (a), (b) and friction (c), (d) coefficients on the temperature $T$. The results for the fermionic-fermionic (f-f) and bosonic-bosonic (b-b) and the mixed fermionic-bosonic (f-b) and bosonic-fermionic (b-f) systems at different coupling constant $g_0$ are indicated. For the better discrimination, the results presented in (a) and (c) are multiplied by the factor ${10}^2$.

Figure 5

For fermionic-fermionic (solid lines), fermionic-bosonic (dotted lines), bosonic-bosonic (dashed lines), and bosonic-fermionic (dash-dotted lines) systems, the calculated dependencies of the average occupation numbers on time $t$. The results for different coupling constants $g_0$ and temperatures $T$ are indicated in the plots. The plots (a) and (c) [(b) and (d)] correspond to an initially unoccupied, i.e., $n_a(t=0)=0$ [occupied, i.e., $n_a(t=0)=1$] system state. In plots (c) and (d), the results for the fermionic-bosonic (bosonic-fermionic) and bosonic-bosonic (fermionic-fermionic) systems are almost identical.

Figure 6

For the fermionic-fermionic (f-f), fermionic-bosonic (f-b), bosonic-bosonic (b-b), and bosonic-fermionic (b-f) systems, the calculated dependencies of the asymptotic collective occupation numbers on coupling constant $g_0$ at $kT/\left(\hslash \mathrm{\Omega }\right)=0.1$ (a), (b) and $kT/\left(\hslash \mathrm{\Omega }\right)=1$ (c), (d). The Fermi-Dirac and Bose-Einstein occupation numbers are presented by dash-dot-dotted and short dash-dotted lines, respectively. For the better discrimination, the results presented in (c) are multiplied by the factor 2.

Figure 7

For the fermionic-fermionic (f-f), fermionic-bosonic (f-b), bosonic-bosonic (b-b), and bosonic-fermionic (b-f) systems, the calculated dependencies of the asymptotic collective occupation numbers on temperature $T$ at $g_0=0.001$ (c), (d) and $g_0=0.1$ (a), (b). The Fermi-Dirac and Bose-Einstein occupation numbers are presented by dash-dot-dotted and short dash-dotted lines, respectively.