# General integrable n-level, many-mode Janes-Cummings-Dicke models and classical r-matrices with spectral parameters

## Abstract

Using the technique of classical r-matrices and quantum Lax operators, we construct the most general form of the quantum integrable “n-level, many-mode” spin-boson Jaynes-Cummings-Dicke-type hamiltonians describing an interaction of a molecule of N n-level atoms with many modes of electromagnetic field and containing, in general, additional non-linear interaction terms. We explicitly obtain the corresponding quantum Lax operators and spin-boson analogs of the generalized Gaudin hamiltonians and prove their quantum commutativity. We investigate symmetries of the obtained models that are associated with the geometric symmetries of the classical r-matrices and construct the corresponding algebra of quantum integrals. We consider in detail three classes of non-skew-symmetric classical r-matrices with spectral parameters and explicitly obtain the corresponding quantum Lax operators and Jaynes-Cummings-Dicke-type hamiltonians depending on the considered r-matrix.

- Authors:

- Universita degli Studi di Milano-Bicocca, via Roberto Cozzi, 53, 20125 Milano, Italy and Bogoliubov Institute for Theoretical Physics, Metrologichna st.14-b, 03143 Kiev (Ukraine)

- Publication Date:

- OSTI Identifier:
- 22403115

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 2; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ELECTROMAGNETIC FIELDS; HAMILTONIANS; INTEGRAL CALCULUS; LAX THEOREM; NONLINEAR PROBLEMS; R MATRIX; SYMMETRY

### Citation Formats

```
Skrypnyk, T., E-mail: taras.skrypnyk@unimib.it, E-mail: tskrypnyk@imath.kiev.ua.
```*General integrable n-level, many-mode Janes-Cummings-Dicke models and classical r-matrices with spectral parameters*. United States: N. p., 2015.
Web. doi:10.1063/1.4907980.

```
Skrypnyk, T., E-mail: taras.skrypnyk@unimib.it, E-mail: tskrypnyk@imath.kiev.ua.
```*General integrable n-level, many-mode Janes-Cummings-Dicke models and classical r-matrices with spectral parameters*. United States. doi:10.1063/1.4907980.

```
Skrypnyk, T., E-mail: taras.skrypnyk@unimib.it, E-mail: tskrypnyk@imath.kiev.ua. Sun .
"General integrable n-level, many-mode Janes-Cummings-Dicke models and classical r-matrices with spectral parameters". United States.
doi:10.1063/1.4907980.
```

```
@article{osti_22403115,
```

title = {General integrable n-level, many-mode Janes-Cummings-Dicke models and classical r-matrices with spectral parameters},

author = {Skrypnyk, T., E-mail: taras.skrypnyk@unimib.it, E-mail: tskrypnyk@imath.kiev.ua},

abstractNote = {Using the technique of classical r-matrices and quantum Lax operators, we construct the most general form of the quantum integrable “n-level, many-mode” spin-boson Jaynes-Cummings-Dicke-type hamiltonians describing an interaction of a molecule of N n-level atoms with many modes of electromagnetic field and containing, in general, additional non-linear interaction terms. We explicitly obtain the corresponding quantum Lax operators and spin-boson analogs of the generalized Gaudin hamiltonians and prove their quantum commutativity. We investigate symmetries of the obtained models that are associated with the geometric symmetries of the classical r-matrices and construct the corresponding algebra of quantum integrals. We consider in detail three classes of non-skew-symmetric classical r-matrices with spectral parameters and explicitly obtain the corresponding quantum Lax operators and Jaynes-Cummings-Dicke-type hamiltonians depending on the considered r-matrix.},

doi = {10.1063/1.4907980},

journal = {Journal of Mathematical Physics},

number = 2,

volume = 56,

place = {United States},

year = {Sun Feb 15 00:00:00 EST 2015},

month = {Sun Feb 15 00:00:00 EST 2015}

}