Uniform process tensor approach for the calculation of multi-time correlation functions of non-Markovian open systems

Abstract

The process tensor framework to open quantum systems provides the most general description of multi-time correlations in non-Markovian quantum dynamics. A compressed representation of a process tensor in terms of matrix product operators (MPO) can be used for numerically exact calculations of multi-time correlation functions in systems strongly coupled to a non-Markovian reservoir. We show here that the numerical scaling for computing multi-dimensional spectra can be significantly improved using a time-translation invariant MPO representation of the process tensor obtained from the uniform time-evolving matrix product operator (uniTEMPO) method. In particular, this approach provides a spectral representation of the non-Markovian dynamics that gives direct access to correlation functions in Fourier-space, avoiding explicit real-time evolution. We calculate linear and 2D electronic spectra for an example system and discuss the performance and numerical scaling of our simulations.

Figures (4)

• Figure 1: Tensor network diagrams. a) Vectorization of a density matrix. The matrix indices are combined to a single superindex. Operators acting on the density matrix are transformed to superoperators. b) A 4-point correlation for a Markovian system. c) Non-Markovian 4-point correlation expressed in an extended state space via uniTEMPO. d) The spectral decomposition of the non-Markovian propagator used in this work.
• Figure 2: Linear and third order response functions for three different sets of parameters, as indicated on top of each column. The upper row shows $\Re{L(\omega)}$ calculated according to Eq. (\ref{['eq:Lin_spek']}). In the four bottom rows 2D-spectra for waiting time $T=0$ are shown. These spectra show $\Re{S_j(\omega_t,0,\omega_\tau)}$, as defined in Eq. (\ref{['eq:S_j(R_j)']}), and are calculated according to Eq. (\ref{['eq:2d-spectrum-T']}). The propagators $\mathsf{Q}$ for each column are computed with a time step $\Delta=0.025 \text{ ps}$, yielding auxiliary space dimensions $\chi =21$, $98$, and $301$, respectively.
• Figure 3: 2D-spectra for different waiting times $T$. The parameters are the same as in the middle column of Fig. \ref{['fig:2d_keeling']}, i.e. $\lambda=0.6\,\mathrm{ps}^{-1}$ and $\Omega=2.0\,\mathrm{ps}^{-1}$.
• Figure 4: Convergence of the uniTEMPO calculation for the linear spectrum shown in Fig. \ref{['fig:2d_keeling']} (intermediate coupling, center panel), for a frequency interval $\omega \in [-15,15]$ with $n_\omega = 5000$ points. The black dashed line indicates $O(\chi^{-(r+1)})$ scaling with $r =3$. Left panel: difference of the linear spectrum with respect to the next-larger bond dimension. The Trotter time step is held fixed at $\Delta = 0.025$. Right panel: difference of the linear spectrum with respect to the next-larger time step at bond dimension $\chi \approx 170$. The black dashed line indicates linear scaling.