Non-Markovian dynamics of fully coupled fermionic and bosonic oscillators

The non-Markovian Langevin approach is applied to study the dynamics of fermionic (bosonic) oscillator linearly coupled to a fermionic (bosonic) environment. The analytical expressions for occupation numbers in two different types of couplings (rotating-wave approximation and fully coupled) are compared and discussed. The weak-coupling and high- and low-temperature limits are considered as well. The conditions under which the environment imposes its thermal equilibrium on the collective subsystem are discussed. The sameness of the results, obtained with both the Langevin approach and the discretized environment method are shown. Short- and long-time nonequilibrium dynamics of fermionic and bosonic open quantum systems are analyzed both analytically and numerically.

Figure 1

The dependence of $s_2+s_2^{*}$ and $z_2+z_2^{*}$ on the parameter $\gamma$ for the indicated coupling constant $g_0$. The results of calculations for FC and RWA couplings are presented by solid and dashed lines, respectively.

Figure 2

The time dependence of indicated functions at $g_0=0.1, \gamma /\mathrm{\Omega }=12$.

Figure 3

The time dependencies of integral terms in $n_{\mathrm{f},\mathrm{b}}^{\mathrm{FC}}\left(t\right)$ (7) and $n_{\mathrm{f},\mathrm{b}}^{\mathrm{RWA}}\left(t\right)$ (13). The calculations are performed at $g_0=0.1, \gamma /\mathrm{\Omega }=12$, and $kT/\left(\hslash \mathrm{\Omega }\right)=0.1$.

Figure 4

The time dependencies of (a) $n_{\mathrm{f},\mathrm{b}}^{\mathrm{FC},\mathrm{RWA}}\left(t\right)$ and (b) the value of $-(d{n}_{\mathrm{f},\mathrm{b}}^{\mathrm{FC}}/dt)/n_{\mathrm{f},\mathrm{b}}^{\mathrm{FC}}$ of the initially occupied state with $n_{\mathrm{f},\mathrm{b}}^{\mathrm{FC},\mathrm{RWA}}\left(0\right)=1$. The results of calculations for different couplings (FC, RWA) and subsystems (bosonic, fermionic) are indicated. The calculations are performed at $g_0=0.1, \gamma /\mathrm{\Omega }=12$, and $kT/\left(\hslash \mathrm{\Omega }\right)=1$.

Figure 5

Time evolution of (a) $n_{\mathrm{f}}^{\mathrm{FC}}\left(t\right)$ and (b) $n_{\mathrm{b}}^{\mathrm{FC}}\left(t\right)$ obtained with the discretized environment method applied to the FC case (star symbols) and compared with the results of the quantum Langevin approach (solid lines) for $n_{\mathrm{f},\mathrm{b}}^{\mathrm{FC}}\left(0\right)=0, g_0=0.1, \gamma /\mathrm{\Omega }=12$, and $kT/\left(\hslash \mathrm{\Omega }\right)=1$. The results of the two approaches are almost identical.

Figure 6

The dependencies of asymptotic occupation numbers $n_{\mathrm{f}}(t\to \infty )$ on the coupling strength $g_0$ for the FC (solid lines) and RWA (dashed lines) fermionic oscillators. (a) The separate contributions of ${\mathrm{I}}_{\mathrm{f}}^{\mathrm{FC}}(t\to \infty )$ and ${\mathrm{J}}_{\mathrm{f}}^{\mathrm{FC}}(t\to \infty )$ in Eq. (28) are presented by dotted and dash-dotted lines, respectively. (b) The separate contributions of ${\mathrm{I}}_{\mathrm{f}}^T$ and ${\mathrm{I}}^C$ in Eq. (31) are presented by dotted and dash-dotted lines, respectively. The calculations are performed at $\gamma /\mathrm{\Omega }=12$, and $kT/\left(\hslash \mathrm{\Omega }\right)=1$.

Figure 7

The same as in Fig. 6, but for the FC and RWA bosonic oscillators.

Figure 8

The calculated dependencies of temperature parts ${\mathrm{I}}_{\mathrm{f},\mathrm{b}}^T$ of the asymptotic occupation numbers for (a) a FC fermionic oscillator and (b) for a FC bosonic oscillator at temperature $T$ (solid lines). The occupation numbers for the Fermi–Dirac and Bose–Einstein distributions are presented by dashed lines. The leading-order expressions obtained for fermions and bosons in power of $\left(kT\right)/\left(\hslash \omega \right)$, Eqs. (45) and (46), are presented by dotted lines. The calculated dependencies of asymptotic ${\mathrm{I}}_{\mathrm{f},\mathrm{b}}^{\mathrm{RWA}}$ in the case of a RWA oscillator are presented by dash-dotted lines. The value of ${\mathrm{I}}^C=5.7\times {10}^{-4}$ is marked by the horizontal line. The calculations are performed at $g_0=0.001$ and $\gamma /\mathrm{\Omega }=12$.