Reduced quantum dynamics with arbitrary bath spectral densities: Hierarchical equations of motion based on several different bath decomposition schemes

134

237

We consider a quantum mechanical system represented in phase space (referred to hereafter as "Wigner space"), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possesses the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility, can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first-and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and systembath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy. The FORTRAN source codes that we developed, which allow for the treatment of Wigner space dynamics with any potential form (TanimuranFP15 and ImTanimuranFP15), are provided as the supplementary material. C 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Real-time and imaginary-time quantum hierarchal Fokker-Planck equations

Tanimura

2015

134

237

“…25,37,39,40 A multitude of powerful computational methods have been developed to deal with the difficulties faced in modelling strongly dissipative quantum systems. Examples include the hierarchical equations of motion (HEOM), [41][42][43][44][45][46] density matrix renormalisation group (and related) techniques, 25,36,47,48 and those based on the path integral formalism. [49][50][51][52] All can converge to numerically exact results in specific circumstances.…”

Section: Introductionmentioning

confidence: 99%

Energy transfer in structured and unstructured environments: Master equations beyond the Born-Markov approximations

Iles-Smith

Dijkstra

Lambert

et al. 2016

128

164

Criteria for the accuracy of small polaron quantum master equation in simulating excitation energy transfer dynamics J. Chem. Phys. 139, 224112 (2013) (2014)], we go beyond the commonly used Born-Markov approximations to incorporate system-environment correlations and the resultant non-Markovian dynamical effects. We obtain energy transfer dynamics for both underdamped and overdamped oscillator environments that are in perfect agreement with the numerical hierarchical equations of motion over a wide range of parameters. Furthermore, we show that the Zusman equations, which may be obtained in a semiclassical limit of the reaction coordinate model, are often incapable of describing the correct dynamical behaviour. This demonstrates the necessity of properly accounting for quantum correlations generated between the system and its environment when the Born-Markov approximations no longer hold. Finally, we apply the reaction coordinate formalism to the case of a structured environment comprising of both underdamped (i.e., sharply peaked) and overdamped (broad) components simultaneously. We find that though an enhancement of the dimer energy transfer rate can be obtained when compared to an unstructured environment, its magnitude is rather sensitive to both the dimer-peak resonance conditions and the relative strengths of the underdamped and overdamped contributions. C 2016 AIP Publishing LLC. [http://dx

“…[37][38][39][40] In addition, with the extension of the dimension of the hierarchy, this approach is capable of treating a great variety of spectral distribution functions. [41][42][43][44][45][46][47][48] This formalism is valuable because it can treat not only strong system-bath coupling but also quantum coherence between the system and bath, which is essential to calculate nonlinear response functions. The system-bath coherence becomes particularly important if the bath interaction is regarded as non-Markovian, as was found from nonlinear optical measurements in the late 1980s, when laser technology reached the femto-second time scale, which is much shorter than the noise correlation time of environmental molecules.…”

Section: Introductionmentioning

confidence: 99%

Reduced hierarchical equations of motion in real and imaginary time: Correlated initial states and thermodynamic quantities

Tanimura¹

2014

194

244

For a system strongly coupled to a heat bath, the quantum coherence of the system and the heat bath plays an important role in the system dynamics. This is particularly true in the case of nonMarkovian noise. We rigorously investigate the influence of system-bath coherence by deriving the reduced hierarchal equations of motion (HEOM), not only in real time, but also in imaginary time, which represents an inverse temperature. It is shown that the HEOM in real time obtained when we include the system-bath coherence of the initial thermal equilibrium state possess the same form as those obtained from a factorized initial state. We find that the difference in behavior of systems treated in these two manners results from the difference in initial conditions of the HEOM elements, which are defined in path integral form. We also derive HEOM along the imaginary time path to obtain the thermal equilibrium state of a system strongly coupled to a non-Markovian bath. Then, we show that the steady state hierarchy elements calculated from the real-time HEOM can be expressed in terms of the hierarchy elements calculated from the imaginary-time HEOM. Moreover, we find that the imaginary-time HEOM allow us to evaluate a number of thermodynamic variables, including the free energy, entropy, internal energy, heat capacity, and susceptibility. The expectation values of the system energy and system-bath interaction energy in the thermal equilibrium state are also evaluated.