# Hierarchy of equations of multiple-time correlation functions

## Abstract

In this paper we derive the evolution equations for non-Markovian multiple-time correlation functions of an open quantum system without using any approximation. We find that these equations conform an open hierarchy in which N-time correlation functions are dependent on (N+1)-time correlations. This hierarchy of equations is consistently obtained with two different methods: A first one based on Heisenberg equations of system operators, and a second one based on system propagators. The dependency on higher order correlations, and therefore the open hierarchy structure, only disappears in certain particular cases and when some hypothesis or approximations are considered in the equations. In this paper we consider a perturbative approximation and derive the general evolution equation for N-time correlations. This equation turns to depend only on N-time and lower order correlation functions, conforming a closed hierarchy structure that is useful for computational purposes.

- Authors:

- Departamento de Fisica Fundamental y Experimental, Electronica y Sistemas, Facultad de Fisica, Universidad de La Laguna, La Laguna 38203, Tenerife (Spain)
- Departamento de Fisica Fundamental II, Facultad de Fisica, Universidad de La Laguna, La Laguna 38203, Tenerife (Spain)

- Publication Date:

- OSTI Identifier:
- 20982460

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 75; Journal Issue: 5; Other Information: DOI: 10.1103/PhysRevA.75.052108; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; COMPUTER CALCULATIONS; CORRELATION FUNCTIONS; CORRELATIONS; EQUATIONS; HYPOTHESIS; MARKOV PROCESS; PROPAGATOR

### Citation Formats

```
Alonso, Daniel, and Vega, Ines de.
```*Hierarchy of equations of multiple-time correlation functions*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVA.75.052108.

```
Alonso, Daniel, & Vega, Ines de.
```*Hierarchy of equations of multiple-time correlation functions*. United States. doi:10.1103/PHYSREVA.75.052108.

```
Alonso, Daniel, and Vega, Ines de. Tue .
"Hierarchy of equations of multiple-time correlation functions". United States.
doi:10.1103/PHYSREVA.75.052108.
```

```
@article{osti_20982460,
```

title = {Hierarchy of equations of multiple-time correlation functions},

author = {Alonso, Daniel and Vega, Ines de},

abstractNote = {In this paper we derive the evolution equations for non-Markovian multiple-time correlation functions of an open quantum system without using any approximation. We find that these equations conform an open hierarchy in which N-time correlation functions are dependent on (N+1)-time correlations. This hierarchy of equations is consistently obtained with two different methods: A first one based on Heisenberg equations of system operators, and a second one based on system propagators. The dependency on higher order correlations, and therefore the open hierarchy structure, only disappears in certain particular cases and when some hypothesis or approximations are considered in the equations. In this paper we consider a perturbative approximation and derive the general evolution equation for N-time correlations. This equation turns to depend only on N-time and lower order correlation functions, conforming a closed hierarchy structure that is useful for computational purposes.},

doi = {10.1103/PHYSREVA.75.052108},

journal = {Physical Review. A},

number = 5,

volume = 75,

place = {United States},

year = {Tue May 15 00:00:00 EDT 2007},

month = {Tue May 15 00:00:00 EDT 2007}

}