# Pythagorean coupling: Complete population transfer in a four-state system

## Abstract

Complete population transfer in a four-coupled-modes system is analyzed from a geometrical point of view. An analytical solution of the dynamics is written by the use of two distinct frequencies, the generalization of the single Rabi frequency of the two-state dynamics. We also present its visualization on two separate Bloch spheres with two independent torque equations. With this scheme we analytically derive the requirements for complete population transfer in a four-state quantum system. Interestingly, the solutions are found to be linked to fundamental number theory, whereas complete population transfer occurs only if the ratios between coupling coefficients exactly match a set of Pythagorean triples.

- Authors:

- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot IL-76100 (Israel)
- (United States)
- Department of Physics and Engineering Physics, Tulane University, New Orleans, Louisiana 70118 (United States) and Brescia University, Division of Mathematics and Natural Sciences, Owensboro, Kentucky 42301 (United States)

- Publication Date:

- OSTI Identifier:
- 22051325

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review. A

- Additional Journal Information:
- Journal Volume: 84; Journal Issue: 1; Other Information: (c) 2011 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1050-2947

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 74 ATOMIC AND MOLECULAR PHYSICS; ANALYTICAL SOLUTION; BLOCH THEORY; COUPLING; EQUATIONS; QUANTUM STATES

### Citation Formats

```
Suchowski, Haim, Kavli Institute for Theoretical Physics, Santa Barbara, California 93106, Silberberg, Yaron, Uskov, Dmitry B., and Kavli Institute for Theoretical Physics, Santa Barbara, California 93106.
```*Pythagorean coupling: Complete population transfer in a four-state system*. United States: N. p., 2011.
Web. doi:10.1103/PHYSREVA.84.013414.

```
Suchowski, Haim, Kavli Institute for Theoretical Physics, Santa Barbara, California 93106, Silberberg, Yaron, Uskov, Dmitry B., & Kavli Institute for Theoretical Physics, Santa Barbara, California 93106.
```*Pythagorean coupling: Complete population transfer in a four-state system*. United States. doi:10.1103/PHYSREVA.84.013414.

```
Suchowski, Haim, Kavli Institute for Theoretical Physics, Santa Barbara, California 93106, Silberberg, Yaron, Uskov, Dmitry B., and Kavli Institute for Theoretical Physics, Santa Barbara, California 93106. Fri .
"Pythagorean coupling: Complete population transfer in a four-state system". United States. doi:10.1103/PHYSREVA.84.013414.
```

```
@article{osti_22051325,
```

title = {Pythagorean coupling: Complete population transfer in a four-state system},

author = {Suchowski, Haim and Kavli Institute for Theoretical Physics, Santa Barbara, California 93106 and Silberberg, Yaron and Uskov, Dmitry B. and Kavli Institute for Theoretical Physics, Santa Barbara, California 93106},

abstractNote = {Complete population transfer in a four-coupled-modes system is analyzed from a geometrical point of view. An analytical solution of the dynamics is written by the use of two distinct frequencies, the generalization of the single Rabi frequency of the two-state dynamics. We also present its visualization on two separate Bloch spheres with two independent torque equations. With this scheme we analytically derive the requirements for complete population transfer in a four-state quantum system. Interestingly, the solutions are found to be linked to fundamental number theory, whereas complete population transfer occurs only if the ratios between coupling coefficients exactly match a set of Pythagorean triples.},

doi = {10.1103/PHYSREVA.84.013414},

journal = {Physical Review. A},

issn = {1050-2947},

number = 1,

volume = 84,

place = {United States},

year = {2011},

month = {7}

}

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