# Reduced dynamics of coupled harmonic and anharmonic oscillators using higher-order perturbation theory

## Abstract

Several techniques to solve a hierarchical set of equations of motion for propagating a reduced density matrix coupled to a thermal bath have been developed in recent years. This is either done using the path integral technique as in the original proposal by Tanimura and Kubo [J. Phys. Soc. Jpn. 58, 101 (1998)] or by the use of stochastic fields as done by Yan et al. [Chem. Phys. Lett. 395, 216 (2004)]. Based on the latter ansatz a compact derivation of the hierarchy using a decomposition of the spectral density function is given in the present contribution. The method is applied to calculate the time evolution of the reduced density matrix describing the motion in a harmonic, an anharmonic, and two coupled oscillators where each system is coupled to a thermal bath. Calculations to several orders in the system-bath coupling with two different truncations of the hierarchy are performed. The respective density matrices are used to calculate the time evolution of various system properties and the results are compared and discussed with a special focus on the convergence with respect to the truncation scheme applied.

- Authors:

- Institut fuer Physik, Technische Universitaet Chemnitz, 09107 Chemnitz (Germany)
- (Germany)

- Publication Date:

- OSTI Identifier:
- 20991224

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Chemical Physics; Journal Volume: 126; Journal Issue: 11; Other Information: DOI: 10.1063/1.2538754; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANHARMONIC OSCILLATORS; COUPLING; DECOMPOSITION; DENSITY MATRIX; EQUATIONS OF MOTION; HARMONIC OSCILLATORS; OSCILLATORS; PATH INTEGRALS; PERTURBATION THEORY; SPECTRAL DENSITY

### Citation Formats

```
Schroeder, Markus, Schreiber, Michael, Kleinekathoefer, Ulrich, Institut fuer Physik, Technische Universitaet Chemnitz, 09107 Chemnitz, and School of Engineering and Science, Jacobs University Bremen, Campus Ring 1, 28759 Bremen.
```*Reduced dynamics of coupled harmonic and anharmonic oscillators using higher-order perturbation theory*. United States: N. p., 2007.
Web. doi:10.1063/1.2538754.

```
Schroeder, Markus, Schreiber, Michael, Kleinekathoefer, Ulrich, Institut fuer Physik, Technische Universitaet Chemnitz, 09107 Chemnitz, & School of Engineering and Science, Jacobs University Bremen, Campus Ring 1, 28759 Bremen.
```*Reduced dynamics of coupled harmonic and anharmonic oscillators using higher-order perturbation theory*. United States. doi:10.1063/1.2538754.

```
Schroeder, Markus, Schreiber, Michael, Kleinekathoefer, Ulrich, Institut fuer Physik, Technische Universitaet Chemnitz, 09107 Chemnitz, and School of Engineering and Science, Jacobs University Bremen, Campus Ring 1, 28759 Bremen. Wed .
"Reduced dynamics of coupled harmonic and anharmonic oscillators using higher-order perturbation theory". United States.
doi:10.1063/1.2538754.
```

```
@article{osti_20991224,
```

title = {Reduced dynamics of coupled harmonic and anharmonic oscillators using higher-order perturbation theory},

author = {Schroeder, Markus and Schreiber, Michael and Kleinekathoefer, Ulrich and Institut fuer Physik, Technische Universitaet Chemnitz, 09107 Chemnitz and School of Engineering and Science, Jacobs University Bremen, Campus Ring 1, 28759 Bremen},

abstractNote = {Several techniques to solve a hierarchical set of equations of motion for propagating a reduced density matrix coupled to a thermal bath have been developed in recent years. This is either done using the path integral technique as in the original proposal by Tanimura and Kubo [J. Phys. Soc. Jpn. 58, 101 (1998)] or by the use of stochastic fields as done by Yan et al. [Chem. Phys. Lett. 395, 216 (2004)]. Based on the latter ansatz a compact derivation of the hierarchy using a decomposition of the spectral density function is given in the present contribution. The method is applied to calculate the time evolution of the reduced density matrix describing the motion in a harmonic, an anharmonic, and two coupled oscillators where each system is coupled to a thermal bath. Calculations to several orders in the system-bath coupling with two different truncations of the hierarchy are performed. The respective density matrices are used to calculate the time evolution of various system properties and the results are compared and discussed with a special focus on the convergence with respect to the truncation scheme applied.},

doi = {10.1063/1.2538754},

journal = {Journal of Chemical Physics},

number = 11,

volume = 126,

place = {United States},

year = {Wed Mar 21 00:00:00 EDT 2007},

month = {Wed Mar 21 00:00:00 EDT 2007}

}