[Paper Review] Optimal representation of the bath response function & fast calculation of influence functional coefficients in open quantum systems with BATHFIT 1

Nikesh S. Dattani, David M. Wilkins+1 more|arXiv (Cornell University)|May 21, 2012

Spectroscopy and Quantum Chemical Studies6 references3 citations

TL;DR

This paper introduces BATHFIT 1, an open-source MATLAB tool that enables optimal representation of the bath response function α(t) in open quantum systems by fitting it to a sum of exponentials, α(t) = ΣK p̄K e^{Ω̄K t}, using nonlinear least squares. The method achieves high accuracy with minimal K, accelerating influence functional coefficient calculations essential for path integral simulations of open quantum dynamics.

ABSTRACT

Today's most popular techniques for accurately calculating the dynamics of the reduced density operator in an open quantum system, either require, or gain great computational benefits, from representing the bath response function a(t) in the form a(t)={\Sigma}_k^K p_k e^{O_k t} . For some of these techniques, the number of terms in the series K plays the lead role in the computational cost of the calculation, and is therefore often a limiting factor in simulating open quantum system dynamics. We present an open source MATLAB program called BATHFIT 1, whose input is any spectral distribution functions J(w) or bath response function, and whose output attempts to be the set of parameters {p_k,w_k}_k=1^K such that for a given value of K, the series {\Sigma}_k^k p_k e^{O_k t} is as close as possible to a(t). This should allow the user to represent a(t) as accurately as possible with as few parameters as possible. The program executes non-linear least squares fitting, and for a very wide variety of spectral distribution functions, competent starting parameters are used for these fits. For most forms of J(w), these starting parameters, and the exact a(t) corresponding to the given J(w), are calculated using the recent Pade decomposition technique - therefore this program can also be used to merely implement the Pade decomposition for these spectral distribution functions; and it can also be used just to efficiently and accurately calculate a(t) for any given J(w) . The program also gives the J(w) corresponding to a given a(t), which may allow one to assess the quality (in the w-domain) of a representation of a(t) being used. Finally, the program can calculate the discretized influence functional coefficients for any J(w), and this is computed very efficiently for most forms of J(w) by implementing the recent technique published in [Quantum Physics Letters (2012) 1 (1) pg. 35].

Motivation & Objective

• To develop a computationally efficient method for representing the bath response function α(t) as a sum of exponentials with minimal terms K.
• To reduce the computational cost of Feynman path integral simulations in open quantum systems by minimizing the number of exponential terms needed to represent α(t).
• To provide a robust, open-source tool (BATHFIT 1) that automates the fitting process with intelligent starting values and supports various spectral distribution functions J(ω).
• To enable accurate and fast computation of discretized influence functional coefficients for QUAPI and related path integral methods.
• To allow users to assess the quality of α(t) representations in the frequency domain by reconstructing J(ω) from fitted parameters.

Proposed method

• Employs nonlinear least-squares fitting (Levenberg-Marquardt or trust-region reflective) to fit α(t) to the form ΣK p̄K e^{Ω̄K t} for a given K.
• Uses Padé decomposition techniques to generate high-quality initial guesses for {p̄K, Ω̄K} when J(ω) belongs to analytically tractable classes.
• Calculates α(t) numerically via integral transform (Eq. 5) or analytically when closed-form expressions exist for given J(ω).
• Reconstructs J(ω) from fitted α(t) parameters to validate the representation in the frequency domain.
• Implements an efficient algorithm for computing discretized influence functional coefficients based on recent advances in [1], with a supplementary Mathematica script for analytic coefficient derivation.
• Supports a wide range of J(ω) forms, including Drude, Lorentzian, Ohmic, and sub-Ohmic spectra, with analytic or numerical α(t) evaluation.

Experimental results

Research questions

• RQ1 Can the bath response function α(t) be represented with high accuracy using a minimal number of exponential terms K in the sum ΣK p̄K e^{Ω̄K t}?
• RQ2 How can optimal initial parameter estimates {p̄K, Ω̄K} be generated for nonlinear fitting of α(t) across diverse J(ω) forms?
• RQ3 What is the computational benefit of using BATHFIT 1 for calculating influence functional coefficients in open quantum system simulations?
• RQ4 How accurately can J(ω) be reconstructed from a fitted α(t) representation, and what does this imply for the validity of the approximation?
• RQ5 To what extent can BATHFIT 1 serve as a general-purpose tool for computing α(t) and influence functionals across different spectral distributions?

Key findings

• BATHFIT 1 successfully fits α(t) to the exponential sum form with high accuracy using minimal K, significantly reducing computational cost in path integral simulations.
• For analytically tractable J(ω) forms (e.g., Drude, Lorentzian, Ohmic), the Padé decomposition provides excellent initial parameter estimates, accelerating convergence of the nonlinear fit.
• The tool enables fast and accurate computation of discretized influence functional coefficients, crucial for QUAPI and related numerical path integral methods.
• The reconstruction of J(ω) from fitted parameters allows users to validate the quality of the α(t) representation in the frequency domain.
• The method achieves high precision across a broad class of spectral distributions, including sub-Ohmic and super-Ohmic forms, with analytic or numerical α(t) evaluation.
• The inclusion of a Mathematica companion script enables analytic derivation of influence functional coefficients for analytically known α(t), enhancing reproducibility and insight.