# Exact solutions of the coherent time evolution on a grid of Landau-Zener anticrossings

## Abstract

In a model of interacting Rydberg manifolds, two sets of parallel energy levels, E{sub m}(t) = pt - m{epsilon} and E{sub m{prime}}(t)=p{prime}t + m{prime}{epsilon}{prime} (m,m{prime} = 0, {plus_minus}1,...;p{prime} {ne} p), are linearly shifted by an electric field F(t) = {dot F}t. Atomic core coupling V occurs only at the point of each pair of levels intersection, with a Landau-Zener diabatic transition probability D = d{sup 2} = e{sup {minus}2}{pi}V{sup 2}/{vert_bar}p{minus}p{prime}{vert_bar}. Adiabatic phases depend on the unit of action, {var_phi} = {contour_integral} dt E(t) = {epsilon}{epsilon}{prime}/{vert_bar}p{minus}p{prime}{vert_bar}. The amplitudes for approaching the intersection of levels m and m{prime} from an initial m = 0 at m{prime} = 0 are found from recursion relations, similar to an earlier model but including paths` interference via operators {xi} and {eta}, where {eta}{xi} = {xi}{eta}e{sup i{var_phi}}. Coefficients of ordered power {xi}{sup m}{eta}{sup m {prime}} in the generating functions 1 + {eta}[1 + d({eta}{minus}{xi}) {minus} {xi}{eta}]{sup {minus}1}({xi} {minus} d) and {alpha}{xi}[1 + d({eta} {minus} {xi}) {minus} {eta}{xi}]{sup {minus}1} yield the amplitudes for m- and m{prime}-levels. Another exact form for the amplitudes is obtained from integral representations. In the adiabatic limit (d {r_arrow} 0) the one for m-levels reduces to J{sub m{minus}m{prime}} (2d sin [1/2m{prime}{var_phi}]/sin[1/2{var_phi}]); in the diabaticmore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 394156

- Report Number(s):
- CONF-9605105-

Journal ID: BAPSA6; ISSN 0003-0503; TRN: 96:028821

- Resource Type:
- Journal Article

- Journal Name:
- Bulletin of the American Physical Society

- Additional Journal Information:
- Journal Volume: 41; Journal Issue: 3; Conference: 27. annual meeting of the Division of Atomic, Molecular and Optical Physics (DAMOP) of the American Physical Society (APS), Ann Arbor, MI (United States), 15-18 May 1996; Other Information: PBD: May 1996

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 66 PHYSICS; RYDBERG STATES; CONFIGURATION MIXING; COLLISIONS; ENERGY LEVELS; ELECTRIC FIELDS

### Citation Formats

```
Harmin, D A.
```*Exact solutions of the coherent time evolution on a grid of Landau-Zener anticrossings*. United States: N. p., 1996.
Web.

```
Harmin, D A.
```*Exact solutions of the coherent time evolution on a grid of Landau-Zener anticrossings*. United States.

```
Harmin, D A. Wed .
"Exact solutions of the coherent time evolution on a grid of Landau-Zener anticrossings". United States.
```

```
@article{osti_394156,
```

title = {Exact solutions of the coherent time evolution on a grid of Landau-Zener anticrossings},

author = {Harmin, D A},

abstractNote = {In a model of interacting Rydberg manifolds, two sets of parallel energy levels, E{sub m}(t) = pt - m{epsilon} and E{sub m{prime}}(t)=p{prime}t + m{prime}{epsilon}{prime} (m,m{prime} = 0, {plus_minus}1,...;p{prime} {ne} p), are linearly shifted by an electric field F(t) = {dot F}t. Atomic core coupling V occurs only at the point of each pair of levels intersection, with a Landau-Zener diabatic transition probability D = d{sup 2} = e{sup {minus}2}{pi}V{sup 2}/{vert_bar}p{minus}p{prime}{vert_bar}. Adiabatic phases depend on the unit of action, {var_phi} = {contour_integral} dt E(t) = {epsilon}{epsilon}{prime}/{vert_bar}p{minus}p{prime}{vert_bar}. The amplitudes for approaching the intersection of levels m and m{prime} from an initial m = 0 at m{prime} = 0 are found from recursion relations, similar to an earlier model but including paths` interference via operators {xi} and {eta}, where {eta}{xi} = {xi}{eta}e{sup i{var_phi}}. Coefficients of ordered power {xi}{sup m}{eta}{sup m {prime}} in the generating functions 1 + {eta}[1 + d({eta}{minus}{xi}) {minus} {xi}{eta}]{sup {minus}1}({xi} {minus} d) and {alpha}{xi}[1 + d({eta} {minus} {xi}) {minus} {eta}{xi}]{sup {minus}1} yield the amplitudes for m- and m{prime}-levels. Another exact form for the amplitudes is obtained from integral representations. In the adiabatic limit (d {r_arrow} 0) the one for m-levels reduces to J{sub m{minus}m{prime}} (2d sin [1/2m{prime}{var_phi}]/sin[1/2{var_phi}]); in the diabatic limit (d {r_arrow} 1) it reduces to a Whittaker function. These results help explain overall patterns in the population distributions during coherent evolution under field ramping.},

doi = {},

url = {https://www.osti.gov/biblio/394156},
journal = {Bulletin of the American Physical Society},

number = 3,

volume = 41,

place = {United States},

year = {1996},

month = {5}

}