[Paper Review] Chain representations of open quantum systems and their numerical simulation with time-adapative density matrix renormalisation group methods

Alex W. Chin, Susana F. Huelga+1 more|arXiv (Cornell University)|Dec 29, 2011

Quantum many-body systems9 references3 citations

TL;DR

This paper introduces an exact chain mapping technique that transforms open quantum systems with structured environments into one-dimensional Hamiltonians with nearest-neighbor interactions, enabling accurate simulation of non-Markovian and non-perturbative dynamics using time-adaptive density matrix renormalization group (t-DMRG) methods. The method reveals that discrete resonances in spectral functions can induce long-lasting quantum coherences, matching experimental observations in photosynthetic complexes such as the FMO complex with coherence lifetimes up to 1.5 ps.

ABSTRACT

This chapter gives a self-contained review of the how standard open quantum system Hamiltonians can be mapped analytically onto representations in which the environments appear as one dimensional harmonic chains with nearest neighbour interactions. This mapping, carried out rigorously using orthogonal polynomial theory, then allows the full evolution of the system and environment to be simulated using time-adaptive density matrix renormalisation group methods. With the combination of these two techniques, numerically-exact results can be obtained for dissipative quantum systems in the presence of arbitrarily complex environmental spectral functions, and the correlations and processes in the environment which drive the effectively irreversible dynamics of the reduced state of the quantum system can be explored in real time. The chain representation also reveals a number of universal features of harmonic environments characterised by a spectral density which are discussed here.

Motivation & Objective

• To develop a systematic, exact mapping of open quantum systems with structured environments into one-dimensional chain Hamiltonians with nearest-neighbor interactions.
• To enable accurate numerical simulation of non-perturbative, non-Markovian dynamics in complex quantum systems such as photosynthetic pigment-protein complexes.
• To investigate the microscopic origin of long-lived electronic and electro-vibronic coherences observed in FMO complexes and related systems.
• To extend t-DMRG methods to handle multi-site systems with independent or correlated baths, including long-range environmental couplings.
• To provide a framework for studying dynamical entanglement and correlation generation between system and environment in realistic open quantum systems.

Proposed method

• The chain mapping is constructed using orthogonal polynomials to transform the system-bath interaction Hamiltonian into a one-dimensional chain with nearest-neighbor couplings.
• The method is exact and preserves the full many-body quantum dynamics, avoiding perturbative or Markovian approximations.
• The resulting 1D chain Hamiltonian is simulated using time-adaptive density matrix renormalization group (t-DMRG) algorithms, which efficiently handle large system sizes and long-time dynamics.
• The approach allows for the inclusion of finite-temperature effects through recent developments in mixed-state t-DMRG.
• For multi-site systems, the chain representation is extended to handle independent baths (Fig. 1.6a) and correlated environmental couplings (Fig. 1.6b), including long-range interactions.
• Numerical validation is performed on model systems with structured spectral functions, including a damped oscillator bath, to demonstrate the emergence of long-lasting coherent oscillations.

Experimental results

Research questions

• RQ1 Can structured, non-Markovian environments give rise to long-lived quantum coherences in open quantum systems?
• RQ2 How do discrete resonances in the spectral function influence the dynamics of excitation energy transfer in photosynthetic complexes?
• RQ3 To what extent can the t-DMRG method accurately simulate non-perturbative, non-Markovian dynamics in open quantum systems with complex bath structures?
• RQ4 What is the role of spatial correlations in environmental fluctuations in sustaining long-lived quantum coherences?
• RQ5 Can the chain mapping technique be generalized to multi-site networks with independent or correlated baths while preserving computational efficiency?

Key findings

• The chain mapping provides an exact transformation of open quantum systems into 1D Hamiltonians with nearest-neighbor interactions, enabling rigorous study of non-Markovian dynamics.
• Numerical simulations with t-DMRG reveal that discrete resonances in the spectral function can induce long-lasting coherent oscillations with lifetimes up to 1.5 ps, matching experimental observations in the FMO complex.
• The method captures the persistence of inter-exciton coherences over a significant fraction of the total transport time (~5 ps), suggesting a quantum origin for efficient energy transfer.
• The presence of a resonant discrete mode in the bath leads to damping of long-lived oscillations, confirming that these features arise from the quantum nature of the system-bath coupling.
• The approach enables the study of dynamical entanglement and correlation generation between system and bath, supporting the investigation of bath reduction and universality in open quantum systems.
• Extensions to finite temperatures and long-range environmental couplings are feasible using recent advances in mixed-state t-DMRG and long-range t-DMRG algorithms.