Pseudomodes and the corresponding transformation of the temperature-dependent bath correlation function

In open-system approaches with non-Markovian environments, the process of inserting an individual mode (denoted as “pseudomode”) into the bath or extracting it from the bath is widely employed. This procedure, however, is typically performed on basis of the spectral density and does not incorporate temperature. Here, we show how the (temperature-dependent) bath correlation function (BCF) transforms in such a process. We present analytic formulas for the transformed BCF and numerically study the differences between factorizing initial state and global thermal (correlated) initial state of mode and bath, respectively. We find that in the regime of strong coupling of the mode to both system and bath, the differences in the BCFs give rise to pronounced differences in the dynamics of the system.

Figure 1

Illustration of two different ways of performing a system-environment partitioning in the presence of a PM linearly coupled to both system and bath. In (I), the system part SI consists of the relevant system degrees of freedom linearly coupled to the PM, which in turn is coupled to an (unstructured) environment EI whereas in (II) the system SII directly couples to a (structured) environment EII including the PM.

Figure 2

Bath correlation functions $\alpha (t,t^{\prime })/{\left|g\right|}^2$ for different coupling strengths $\eta$ and different reference times $t^{\prime }$. The left column [panels (a) and (c)] shows the BCF for weak PM-bath coupling $\eta =0.25$, whereas the right column [panels (b) and (d)] shows a strong-coupling regime $\eta =1.0$. In the first row [panels (a) and (b)] the reference time of the BCF is set to $t^{\prime }=0$, in the second row [panels (c) and (d)] $t^{\prime }=32.5{\Lambda }^{-1}$. Solid blue lines indicate the real part of the BCF with factorizing initial state [Eq. (23)], dashed green lines the real part of the BCF with diagonal initial state [Eq. (26)]. Dotted lines show the corresponding imaginary parts. The insets in panels (c) and (d) provide a detail of the short-time dynamics, while the inset in (a) shows the SD $J_{\mathrm{EI}}\left(\omega \right)$ (dashed red line) with the position of the PM indicated by a solid vertical line. The BCFs were calculated using $T=46\Lambda$ and $\Omega =1.5\Lambda$.

Figure 3

Spectral density $J_{\mathrm{EII}}\left(\omega \right)$ (black, solid line) for (a) $\eta =0.25$ and (b) $\eta =1.0$ obtained from Fourier transformation of the BCF (for details, see text). As the BCFs for diagonal and factorizing initial state (evaluated at $t_{\mathrm{c}}.\mathrm{m}.=130{\Lambda }^{-1}$) coincide, they yield identical SDs. In both panels, the red, dashed line indicates the original SD of the Ohmic bath, which has been scaled by the factor denoted in red. All other parameters are as in Fig. 2.

Figure 4

Bath correlation function ${\alpha }_1(t,0)/{\left|g\right|}^2$ (solid lines) and system mean occupation number $n_{\mathrm{sys}}\left(t\right)\equiv {\langle d^{†}\left(t\right)d\left(t\right)\rangle }_{\mathrm{EII}}$ (dashed lines) for different initial conditions and different coupling strengths $g$. In the upper panel (a), $g=0.3\Lambda$, whereas the lower panel (b), $g=0.08\Lambda$. Thin blue lines indicate the BCF with factorizing initial state [Eq. (23)], thick green lines indicate the BCF with diagonal initial state [Eq. (26)]. Only the real parts of the BCFs are shown. The inset shows the PM mean occupation number $n_{\mathrm{PM}}\left(t\right)\equiv {\langle b^{†}\left(t\right)b\left(t\right)\rangle }_{\mathrm{EII}}$ for $g=0.3\Lambda$ (dashed line) and $g=0.08\Lambda$ (solid line) for both initial states. Except for $\eta =1.0$ and ${\Omega }_{\mathrm{sys}}=0.46\Lambda$, the parameters of Fig. 2 have been used. Note that ${\alpha }_1(t,0)$ is approximately $\alpha (t,0)/2$ in Fig. 2.