Dissipation-Assisted Matrix Product Factorization

Alejandro D. Somoza, Oliver Marty, James Lim, Susana F. Huelga, and Martin B. Plenio

Phys. Rev. Lett. 123, 100502 – Published 3 September 2019

Abstract

Charge and energy transfer in biological and synthetic organic materials are strongly influenced by the coupling of electronic states to a highly structured dissipative environment. Nonperturbative simulations of these systems require a substantial computational effort, and current methods can only be applied to large systems if environmental structures are severely coarse grained. Time evolution methods based on tensor networks are fundamentally limited by the times that can be reached due to the buildup of entanglement in time, which quickly increases the size of the tensor representation, i.e., the bond dimension. In this Letter, we introduce a dissipation-assisted matrix product factorization (DAMPF) method that combines a tensor network representation of the vibronic state within a pseudomode description of the environment where a continuous bosonic environment is mapped into a few harmonic oscillators under Lindblad damping. This framework is particularly suitable for a tensor network representation, since damping suppresses the entanglement growth among oscillators and significantly reduces the bond dimension required to achieve a desired accuracy. We show that dissipation removes the “time-wall” limitation of existing methods, enabling the long-time simulation of large vibronic systems consisting of 10–50 sites coupled to 100–1000 underdamped modes in total and for a wide range of parameter regimes. For these reasons, we believe that our formalism will facilitate the investigation of spatially extended systems with applications to quantum biology, organic photovoltaics, and quantum thermodynamics.

Received 3 April 2019
Revised 24 June 2019

DOI:https://doi.org/10.1103/PhysRevLett.123.100502

Physics Subject Headings (PhySH)

Dissipative dynamics Open quantum systems & decoherence Quantum effects in biological systems Quantum simulation

Polymers

Non-Markovian processes Tensor network methods

Polymers & Soft MatterQuantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

Alejandro D. Somoza, Oliver Marty, James Lim, Susana F. Huelga, and Martin B. Plenio

Institut für Theoretische Physik and IQST, Albert-Einstein-Allee 11, Universität Ulm, D-89069 Ulm, Germany

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Vol. 123, Iss. 10 — 6 September 2019

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Images

Figure 1
(a) The combined “system + environment” unitary evolution of an open quantum system (left), characterized by a continuous spectral density of the environment $J (ω)$ , is numerically impractical for systems consisting of many sites ( $N ≫ 2$ ). We can bypass this very costly approach, introducing a small set of damped oscillators (pseudomodes) that are coupled to the system (right). The mathematical equivalence between these two models has been recently demonstrated [18]. (b) Thanks to the decorrelating action of the baths ( ${BATH}_{q_{i}}$ ), the growth of correlations among oscillators is limited, enabling the simulation of large systems ( $N \sim 50$ ) while modeling faithfully the action of the environment on the system. A numerical example of the compression error $E (t)$ with $χ = 3$ that corresponds to Fig. 4 is shown.
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Figure 2
(a) Schematic of a vibronic network consisting of $N$ electronically interacting sites where each site is locally coupled to $Q$ oscillators under Lindblad damping. $M = N Q$ denotes the total number of the modes. A bond dimension $χ$ determines how accurately the correlations between oscillators are described using MPOs. (b) Illustration of the MPO representation of the quantum state of a vibronic system. For every operator $| m ⟩ ⟨ n |$ of the electronic subsystem there is a MPO describing the conditional state ${\hat{O}}_{m, n}$ of the nuclear degrees of freedom [Eqs. (4)–(6)]. $N_{b}$ is the number of oscillator levels employed to describe each vibrational mode. (c) The electronic interaction $U_{e} = e^{- i [{\hat{H}}_{e}, \cdot] Δ t}$ creates correlations between oscillators, mediated by the sum of ${\hat{O}}_{k, l}$ and ${\hat{O}}_{k^{'}, l^{'}}$ [see Eq. (8)]. This sum doubles the bond dimension and a compression scheme is required. Crucially, the local environment at each site [ $D_{L}$ , Eq. (2)] suppresses these correlations and helps one to keep the bond dimension under control.
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Figure 3
(a) Model spectral density consisting of broad background centered at around $700 {cm}^{- 1}$ and a narrow Lorentzian peak centered at $1500 {cm}^{- 1}$ . (b) The target BCF computed based on the model spectral density, shown in black, is fitted (in red) by using the effective BCF of 21 damped oscillators (see SM [43]). (c) Absorption line shape of a monomer with $Ω_{1} = 600 nm$ coupled to the model spectral density of (a). The analytical solution is shown in black.
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Figure 4
Time evolution of populations $P_{k}$ of site $k$ in a vibronic chain consisting of (a) $N = 20$ and (b) 50 sites with $Ω_{m} = Ω_{n}$ , $J_{n, n + 1} = 400 {cm}^{- 1}$ , and two oscillators coupled to each site: $ω_{1} = 1500 {cm}^{- 1}$ , $s_{1} = 0.5$ , $γ_{1} = (1 ps)^{- 1}$ , $T_{1} = 300 K$ , $N_{b, 1} = 8$ , and $ω_{2} = 500 {cm}^{- 1}$ , $s_{2} = 0.1$ , $γ_{2} = (50 fs)^{- 1}$ , $T_{2} = 300 K$ , $N_{b, 2} = 4$ . Panels (c) and (d) show the population dynamics of a composite system of $N = 10$ sites with (c) 24 and (d) 48 modes coupled to each site and $Ω_{m} = Ω_{n}$ , $J_{n, n + 1} = 200 {cm}^{- 1}$ . The vibrational frequencies are uniformly distributed between 400 and $1500 {cm}^{- 1}$ , with uniform $s_{q} = 0.05$ and $γ_{q} = (200 fs)^{- 1}$ , except for the $400 {cm}^{- 1}$ mode that is overdamped by $γ^{'} = (50 {fs)}^{- 1}$ with $s^{'} = 0.1$ . All the modes are at room temperature $T_{q} = 300 K$ and simulated with $N_{b} = 5$ . In all the simulations, an electronic excitation is initially localized at site 1. Converged results correspond to solid black curves.
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