Interplay of different environments in open quantum systems: Breakdown of the additive approximation

We analyze an open quantum system under the influence of more than one environment: a dephasing bath and a probability-absorbing bath that represents a decay channel, as encountered in many models of quantum networks. In our case, dephasing is modeled by random fluctuations of the site energies, while the absorbing bath is modeled with an external lead attached to the system. We analyze under which conditions the effects of the two baths can enter additively the quantum master equation. When such additivity is legitimate, the reduced master equation corresponds to the evolution generated by an effective non-Hermitian Hamiltonian and a Haken-Strobl dephasing super-operator. We find that the additive decomposition is a good approximation when the strength of dephasing is small compared to the bandwidth of the probability-absorbing bath.

Figure 1

A ringlike network interacts with a dephasing bath (light red) and a probability-absorbing bath (light blue). The on-site energies of each of the $N_R$ sites of the ring, connected with nearest-neighbor coupling ${\mathrm{\Omega }}_R$, undergo white-noise fluctuations with dephasing strength ${\sigma }_R^2$. The probability-absorbing bath is modeled by a lead of $N_L$ sites, connected with nearest-neighbor coupling ${\mathrm{\Omega }}_L$. The ring sites are equally coupled to the first lead site with tunneling amplitude ${\mathrm{\Omega }}_{RL}$ (dashed lines). In general, white-noise fluctuations of intensity ${\sigma }_L^2$ could be present also on the lead (see Appendix pp1), but only the case ${\sigma }_L^2=0$ is considered in the main text.

Figure 2

The presence of dephasing noise on the ring first destroys superradiance and then leads to the breakdown of the additive approximation. By considering the evolution of the ring population $P\left(t^{*}\right)$, in terms of the rescaled time $t^{*}=\gamma t/\hslash$, we observe that, for dephasing strengths ${\sigma }_R^2$ smaller than the coupling ${\mathrm{\Omega }}_L$, the evolution of the reduced model (curves) agrees with that of the extended model (symbols) up to the insurgence of numerical finite-size effects (vertical dotted line), see also the discussion in Ref. [24]. For dephasing strengths larger than ${\mathrm{\Omega }}_L$, there is no agreement between reduced and extended model (the decay of solid lines is markedly faster than that of symbols). Employed values: ${\mathrm{\Omega }}_R=0$, $N_R=10$, $\gamma =2{\mathrm{\Omega }}_{RL}^2/{\mathrm{\Omega }}_L=1$, ${\mathrm{\Omega }}_L=100$, $N_L=40$.

Figure 3

When the dephasing strength ${\sigma }_R^2$ equals the lead coupling ${\mathrm{\Omega }}_L$, the agreement between reduced and extended model is completely lost. The decay width $\mathrm{\Lambda }$ of an excitation initialized on the superradiant state, normalized by the single-site decay width $\gamma$, is plotted as a function of the dephasing rate ${\sigma }_R^2$ affecting the ring for different values of the coupling ${\mathrm{\Omega }}_L$ (see legend). Symbols correspond to the extended model, while solid curves are obtained by the reduced model. The agreement is lost when ${\sigma }_R^2\approx {\mathrm{\Omega }}_L$ (vertical dashed lines), after which point the decay rates of the extended model vanish, while those of the reduced model converge to the single-site decay (horizontal dotted line). Employed values: ${\mathrm{\Omega }}_R=0$, $N_R=10$, $\gamma =1$, $N_L=40$. In addition, data obtained by setting ${\mathrm{\Omega }}_R=0.1{\mathrm{\Omega }}_L$ (black crosses) show that a coupling among ring sites small compared to ${\mathrm{\Omega }}_L$ has a negligible effect on the behavior of the system.