^{1,a)}, Clara D. Christ

^{2,b)}and Irene Burghardt

^{3,c)}

### Abstract

An approach to non-Markovian system-environment dynamics is described which is based on the construction of a hierarchy of coupled effective environmental modes that is terminated by coupling the final member of the hierarchy to a Markovian bath. For an arbitrary environment, which is linearly coupled to the subsystem, the discretized spectral density is replaced by a series of approximate spectral densities involving an increasing number of effective modes. This series of approximants, which are constructed analytically in this paper, guarantees the accurate representation of the overall system-plus-bath dynamics up to increasing times. The hierarchical structure is manifested in the approximate spectral densities in the form of the imaginary part of a continued fraction similar to Mori theory. The results are described for cases where the hierarchy is truncated at the first-, second-, and third-order level. It is demonstrated that the results generated from a reduced density matrix equation of motion and large dimensional system-plus-bath wavepacket calculations are in excellent agreement. For the reduced density matrix calculations, the system and hierarchy of effective modes are treated explicitly and the effects of the bath on the final member of the hierarchy are described by the Caldeira–Leggett equation and its generalization to zero temperature.

We would like to thank Hiroyuki Tamura for useful discussions.

I. INTRODUCTION

II. THEORY

A. System-bath Hamiltonian

B. Effective-mode representation of the bath

C. Transformations of the residual bath

D. Effective-mode chain including Markovian closure

E. Higher-order spectral densities

III. APPLICATIONS AND COMPUTATIONAL DETAILS

IV. RESULTS AND DISCUSSION

A. First-order hierarchy truncation

1. Harmonic oscillator

2. Double-well system

B. Second-order hierarchy truncation

C. Third-order hierarchy truncation

V. CONCLUSION

### Key Topics

- Energy transfer
- 19.0
- Oscillators
- 12.0
- Friction
- 7.0
- Markov processes
- 7.0
- Surface dynamics
- 7.0

## Figures

(a) Energy relaxation for the harmonic oscillator example of Sec. ???. The black curve depicts the energy relaxation computed from the Caldeira–Leggett equation, as described in Eq. (45); the red curve depicts the energy relaxation computed from Eq. (1) by a 31 mode wavepacket calculation, where are depicted in part (b) below. The green curve depicts the energy relaxation computed from Eq. (17) by wavepacket calculations with Ohmic effective-mode coupling to the 30 mode bath. (b) Coupling coefficient defined in Eq. (4) that corresponds to the spectral density of Eq. (19).

(a) Energy relaxation for the harmonic oscillator example of Sec. ???. The black curve depicts the energy relaxation computed from the Caldeira–Leggett equation, as described in Eq. (45); the red curve depicts the energy relaxation computed from Eq. (1) by a 31 mode wavepacket calculation, where are depicted in part (b) below. The green curve depicts the energy relaxation computed from Eq. (17) by wavepacket calculations with Ohmic effective-mode coupling to the 30 mode bath. (b) Coupling coefficient defined in Eq. (4) that corresponds to the spectral density of Eq. (19).

Expectation values associated with the harmonic oscillator example of Sec. ???.

Expectation values associated with the harmonic oscillator example of Sec. ???.

Energy relaxation dynamics , evaluated from the reduced density matrix calculation of Eq. (45), for the harmonic oscillator example of Sec. ???. In one example (black curve) the system harmonic frequency is off resonant with the effective-mode frequency , in the other example (red curve), is resonant with .

Energy relaxation dynamics , evaluated from the reduced density matrix calculation of Eq. (45), for the harmonic oscillator example of Sec. ???. In one example (black curve) the system harmonic frequency is off resonant with the effective-mode frequency , in the other example (red curve), is resonant with .

Energy relaxation for the double-well example of Sec. ???. The black curve depicts the energy relaxation computed from the Caldeira–Leggett equation, as described in Eq. (45); the red curve depicts the energy relaxation computed from Eq. (1) by a 31 mode wavepacket calculation, where are related to the spectral density of Eq. (19). The green curve depicts the energy relaxation computed from Eq. (17) by wavepacket calculations with Ohmic effective-mode coupling to the 30 mode bath. The 2D system and effective-mode calculation illustrated by the blue curve shows that the short-time dynamics are accurately reproduced.

Energy relaxation for the double-well example of Sec. ???. The black curve depicts the energy relaxation computed from the Caldeira–Leggett equation, as described in Eq. (45); the red curve depicts the energy relaxation computed from Eq. (1) by a 31 mode wavepacket calculation, where are related to the spectral density of Eq. (19). The green curve depicts the energy relaxation computed from Eq. (17) by wavepacket calculations with Ohmic effective-mode coupling to the 30 mode bath. The 2D system and effective-mode calculation illustrated by the blue curve shows that the short-time dynamics are accurately reproduced.

(a) Energy relaxation for the double-well example of Sec. ??? using different values of . (b) The corresponding effective mode position expectation values.

(a) Energy relaxation for the double-well example of Sec. ??? using different values of . (b) The corresponding effective mode position expectation values.

(a) Energy relaxation for the harmonic oscillator example of Sec. IV B where the hierarchy is truncated at second order. The black curve depicts the energy relaxation computed from the Caldeira–Leggett equation, as described in Eq. (45); the red curve depicts the energy relaxation computed from Eq. (1) by a 31 mode wavepacket calculation, where are depicted in part (b) below. The green curve depicts the energy relaxation computed from Eq. (17) by wavepacket calculations with Ohmic effective-mode coupling to the 30 mode bath. (b) Coupling coefficient defined in Eq. (4) that corresponds to the spectral density of Eq. (44).

(a) Energy relaxation for the harmonic oscillator example of Sec. IV B where the hierarchy is truncated at second order. The black curve depicts the energy relaxation computed from the Caldeira–Leggett equation, as described in Eq. (45); the red curve depicts the energy relaxation computed from Eq. (1) by a 31 mode wavepacket calculation, where are depicted in part (b) below. The green curve depicts the energy relaxation computed from Eq. (17) by wavepacket calculations with Ohmic effective-mode coupling to the 30 mode bath. (b) Coupling coefficient defined in Eq. (4) that corresponds to the spectral density of Eq. (44).

(a) Energy relaxation computed from the Caldeira–Leggett equation as described in Eq. (45) using for the harmonic oscillator example of Sec. IV B where the hierarchy is truncated at second order. (b) Coupling coefficient defined in Eq. (4) that corresponds to the spectral density of Eq. (44).

(a) Energy relaxation computed from the Caldeira–Leggett equation as described in Eq. (45) using for the harmonic oscillator example of Sec. IV B where the hierarchy is truncated at second order. (b) Coupling coefficient defined in Eq. (4) that corresponds to the spectral density of Eq. (44).

Energy relaxation at different temperatures for the harmonic oscillator example of Sec. IV B computed using a fixed value of for Caldeira–Leggett equation as described in Eq. (45).

Energy relaxation at different temperatures for the harmonic oscillator example of Sec. IV B computed using a fixed value of for Caldeira–Leggett equation as described in Eq. (45).

(a) Energy relaxation dynamics for the harmonic oscillator example of Sec. IV C using spectral densities approximated at the first-, second-, and third-order level of the effective-mode hierarchy. For the bath, a fixed value of was used. Part (b) depicts the couplings that are related to through Eq. (46).

(a) Energy relaxation dynamics for the harmonic oscillator example of Sec. IV C using spectral densities approximated at the first-, second-, and third-order level of the effective-mode hierarchy. For the bath, a fixed value of was used. Part (b) depicts the couplings that are related to through Eq. (46).

## Tables

Parameters, quoted in a.u., used for the examples described in Sec. IV.

Parameters, quoted in a.u., used for the examples described in Sec. IV.

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