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arXiv:1906.05971 · The Journal of Chemical Physics

Open quantum dynamics of a three-dimensional rotor calculated using a rotationally invariant system-bath Hamiltonian: Linear and two-dimensional rotational spectra

Yuki Iwamoto, Yoshitaka Tanimura

Published Jul 26, 2019• 15 citations
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Abstract

We consider a rotationally invariant system-bath (RISB) model in three-dimensional space that is described by a linear rigid rotor independently coupled to three harmonic-oscillator baths through functions of the rotor's Euler angles. While this model has been developed to study the dielectric relaxation of a dipolar molecule in solvation as a problem of classical Debye relaxation, here we investigate it as a problem of open quantum dynamics. Specifically, the treatment presented here is carried out as an extension of a previous work [Y. Iwamoto and Y. Tanimura, J. Chem. Phys 149, 084110 (2018)], in which we studied a two-dimensional (2D) RISB model, to a three-dimensional (3D) RISB model. As in the 2D case, due to a difference in the energy discretization of the total Hamiltonian, the dynamics described by the 3D RISB model differ significantly from those described by the rotational Caldeira-Leggett model. To illustrate the characteristic features of the quantum 3D rotor system described by angular momentum and magnetic quantum numbers, we derive a quantum master equation (QME) and hierarchical equations of motion for the 3D RISB model in the high-temperature case. Using the QME, we compute linear and 2D rotational spectra defined by the linear and nonlinear response functions of the rotor dipole, respectively. The quantum transitions between the angular momentum states and magnetic states arising from polarized Stark fields as well as the system-bath interactions can be clearly observed in 2D rotational spectroscopy.

DOI
10.1063/1.5108609

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We consider a rotationally invariant system-bath (RISB) model in three-dimensional space that is described by a linear rigid rotor independently coupled to three harmonic-oscillator baths through functions of the rotor's Euler angles. While this model has been developed to study the dielectric relaxation of a dipolar molecule in solvation as a problem of classical Debye relaxation, here we investigate it as a problem of open quantum dynamics. Specifically, the treatment presented here is carried out as an extension of a previous work [Y. Iwamoto and Y. Tanimura, J. Chem. Phys 149, 084110 (2018)], in which we studied a two-dimensional (2D) RISB model, to a three-dimensional (3D) RISB model. As in the 2D case, due to a difference in the energy discretization of the total Hamiltonian, the dynamics described by the 3D RISB model differ significantly from those described by the rotational Caldeira-Leggett model. To illustrate the characteristic features of the quantum 3D rotor system described by angular momentum and magnetic quantum numbers, we derive a quantum master equation (QME) and hierarchical equations of motion for the 3D RISB model in the high-temperature case. Using the QME, we compute linear and 2D rotational spectra defined by the linear and nonlinear response functions of the rotor dipole, respectively. The quantum transitions between the angular momentum states and magnetic states arising from polarized Stark fields as well as the system-bath interactions can be clearly observed in 2D rotational spectroscopy.
We consider a two-dimensional rigid rotator system coupled to a two-dimensional heat bath. The Caldeira-Leggett (Brownian) model for the rotator and the spin-Boson model have been used to describe such systems, but they do not possess rotational symmetry, and they cannot describe the discretized rotational bands in absorption and emission spectra that have been found experimentally. Here, to address this problem, we introduce a rotationally invariant system-bath (RISB) model that is described by two sets of harmonic-oscillator baths independently coupled to the rigid rotator as sine and cosine functions of the rotator angle. Due to a difference in the energy discretization of the total Hamiltonian, the dynamics described by the RISB model differ significantly from those described by the rotational Caldeira-Legget model, while both models reduce to the Langevin equation for a rotator in the classical limit. To demonstrate this point, we compute the rotational absorption spectrum defined by the linear response function of a rotator dipole. For this purpose, we derive a quantum master equation for the RISB model in the high-temperature Markovian case. We find that the spectral profiles of the calculated signals exhibit a transition from quantized rotational bands to a single peak after spectrum collapse. This is a significant finding because previous approaches cannot describe such phenomena in a unified manner.
We consider a two-dimensional rigid rotator system coupled to a two-dimensional heat bath. The Caldeira-Leggett (Brownian) model for the rotator and the spin-Boson model have been used to describe such systems, but they do not possess rotational symmetry, and they cannot describe the discretized rotational bands in absorption and emission spectra that have been found experimentally. Here, to address this problem, we introduce a rotationally invariant system-bath (RISB) model that is described by two sets of harmonic-oscillator baths independently coupled to the rigid rotator as sine and cosine functions of the rotator angle. Due to a difference in the energy discretization of the total Hamiltonian, the dynamics described by the RISB model differ significantly from those described by the rotational Caldeira-Legget model, while both models reduce to the Langevin equation for a rotator in the classical limit. To demonstrate this point, we compute the rotational absorption spectrum defined by the linear response function of a rotator dipole. For this purpose, we derive a quantum master equation for the RISB model in the high-temperature Markovian case. We find that the spectral profiles of the calculated signals exhibit a transition from quantized rotational bands to a single peak after spectrum collapse. This is a significant finding because previous approaches cannot describe such phenomena in a unified manner.
We investigate the quantum dynamics of Coulomb potential systems in thermal baths. We study these systems within the framework of open quantum dynamics theory, focusing on preserving the rotational symmetry of the entire system, including the baths. Thus, we employ a three-dimensional rotationally invariant system-bath model to derive numerically "exact" hierarchical equations of motion for atomic orbitals that enable a non-perturbative and non-Markovian treatment of system-bath interactions at finite temperatures. To assess the formalism, we calculated the linear absorption spectrum of an atomic system under an isotropic thermal environment, with systematic variation of system-bath coupling strength and temperature.
An open quantum system refers to a system that is further coupled to a bath system consisting of surrounding radiation fields, atoms, molecules, or proteins. The bath system is typically modeled by an infinite number of harmonic oscillators. This system-bath model can describe the time-irreversible dynamics through which the system evolves toward a thermal equilibrium state at finite temperature. In nuclear magnetic resonance and atomic spectroscopy, dynamics can be studied easily by using simple quantum master equations under the assumption that the system-bath interaction is weak (perturbative approximation) and the bath fluctuations are very fast (Markovian approximation). However, such approximations cannot be applied in chemical physics and biochemical physics problems, where environmental materials are complex and strongly coupled with environments. The hierarchical equations of motion (HEOM) can describe the numerically "exact" dynamics of a reduced system under nonperturbative and non-Markovian system-bath interactions, which has been verified on the basis of exact analytical solutions (non-Markovian tests) with any desired numerical accuracy. The HEOM theory has been used to treat systems of practical interest, in particular, to account for various linear and nonlinear spectra in molecular and solid state materials, to evaluate charge and exciton transfer rates in biological systems, to simulate resonant tunneling and quantum ratchet processes in nanodevices, and to explore quantum entanglement states in quantum information theories. This article presents an overview of the HEOM theory, focusing on its theoretical background and applications, to help further the development of the study of open quantum dynamics.
An open quantum system is a system coupled to an environment that can describe time-irreversible dynamics through which the system evolves toward the thermal equilibrium state.We present a quantum mechanically rigorous theory in order to help the analysis of spectra obtained from the advanced nuclear magnetic resonance (NMR) and muon spin rotation, relaxation or resonance (μSR) techniques.Our approach is based on the numerically "Exact" hierarchical equations of motion (HEOM) approach, which allows us to study the reduced system dynamics for non-perturbative and non-Markovian system-bath interactions at finite temperature even under strong time-dependent perturbations.We demonstrate the present theory to analyze μSR and low-field NMR spectra, as an extension of the Kubo-Toyabe theory focusing on the effects of temperature and anisotropy of a local magnetic field on spectra, to help further the development of these experimental means.
Abstract Discretizing distribution function in a phase space for an efficient quantum dynamics simulation is non-trivial challenge, in particular for a case that a system is further coupled to environmental degrees of freedom. Such open quantum dynamics is described by a reduced equation of motion (REOM) most notably by a quantum Fokker-Planck equation (QFPE) for a Wigner distribution function (WDF). To develop a discretization scheme that is stable for numerical simulations from the REOM approach, we find that a two-dimensional (2D) periodically invariant system-bath (PISB) model with two heat baths is an ideal platform not only for a periodical system but also for a system confined by a potential. We then derive the numerically ''exact'' hierarchical equations of motion (HEOM) for a discrete WDF in terms of periodically invariant operators in both coordinate and momentum spaces. The obtained equations can treat non-Markovian heat-bath in a non-perturbative manner at finite temperatures regardless of the mesh size. The stability of the present scheme is demonstrated in a high-temperature Markovian case by numerically integrating the discrete QFPE with by a coarse mesh for a 2D free rotor and harmonic potential systems for an initial condition that involves singularity.