Fingerprint and Universal Markovian Closure of Structured Bosonic Environments

We exploit the properties of chain mapping transformations of bosonic environments to identify a finite collection of modes able to capture the characteristic features, or fingerprint, of the environment. Moreover we show that the countable infinity of residual bath modes can be replaced by a universal Markovian closure, namely, a small collection of damped modes undergoing a Lindblad-type dynamics whose parametrization is independent of the spectral density under consideration. We show that the Markovian closure provides a quadratic speedup with respect to standard chain mapping techniques and makes the memory requirement independent of the simulation time, while preserving all the information on the fingerprint modes. We illustrate the application of the Markovian closure to the computation of linear spectra but also to nonlinear spectral response, a relevant experimentally accessible many body coherence witness for which efficient numerically exact calculations in realistic environments are currently lacking.

Figure 1

Schematics of our procedure: (a) The system ($S$) interacting with a bosonic environment ($E$), as described by Eqs. (1)–(4). (b) After the chain-mapping, the system interacts with the primary environment, which interacts in turn with the residual environment. (c) The residual environment is replaced by a finite set of interacting damped modes, i.e., the Markovian closure.

Figure 2

Coefficients, fingerprint, and closure: (a) The TEDOPA chain oscillator frequencies ${\omega }_n$ (blue triangles) and couplings ${\kappa }_n$ (green circles) obtained for the WSCP spectral density $J(\omega )$ described in the main text. In the inset the relative errors between the exact and asymptotic coefficients. (b) The effective spectral density $J^{\prime }(\omega )$ of the TEDOPA chain truncated after 400 sites (red line) and the effective spectral density $J_M^{\prime }(\omega )$ with the asymptotic values $\mathrm{\Omega }/K$ replacing the exact coefficients ${\omega }_n/{\kappa }_n$ for $n>M=80$ (dashed green line). In the inset, the effective spectral densities $J^{\prime }(\omega )$, $J_{80}^{\prime }(\omega )$, $J_{40}^{\prime }(\omega )$ (vertically shifted for improved visibility). (c) The residual spectral density (10) for $[{\omega }_{\min },{\omega }_{\max }]=[-1,1]$ and the approximation provided by the TSO with $N=6$, 8, 10 auxiliary modes; inset: the relative error $|J_{\infty }(\omega )-J_N^{\mathrm{TSO}}(\omega )|/J_{\infty }(\omega )$ for $\omega \in [-1,1]$.

Figure 3

Linear and 2DES optical spectra of WSCP, computational cost: (a) Absorption spectra computed by TEDOPA with Markovian closure (MC), cumulant expansion (CE), reduced vibronic models with Markovian and non-Markovian effects of a broad environmental spectrum $J_{\mathrm{AR}}(\omega )$. 2D electronic spectra at $t_2=500\mathrm{fs}$ computed by (b) MC, and reduced vibronic models with (c) Markovian and (d) non-Markovian effects of the broad noise spectrum. The time (e) and memory scaling (f) as functions of the simulation time $t_{\max }$.