# PT-symmetric sinusoidal optical lattices at the symmetry-breaking threshold

## Abstract

The PT-symmetric potential V{sub 0}[cos(2{pi}x/a)+i{lambda}sin(2{pi}x/a)] has a completely real spectrum for {lambda}{<=}1 and begins to develop complex eigenvalues for {lambda}>1. At the symmetry-breaking threshold {lambda}=1 some of the eigenvectors become degenerate, giving rise to a Jordan-block structure for each degenerate eigenvector. In general this is expected to result in a secular growth in the amplitude of the wave. However, it has been shown in a recent paper by Longhi, by numerical simulation and by the use of perturbation theory, that for a broad initial wave packet this growth is suppressed, and instead a saturation leading to a constant maximum amplitude is observed. We revisit this problem by explicitly constructing the Bloch wave functions and the associated Jordan functions and using the method of stationary states to find the dependence on the longitudinal distance z for a variety of different initial wave packets. This allows us to show in detail how the saturation of the linear growth arises from the close connection between the contributions of the Jordan functions and those of the neighboring Bloch waves.

- Authors:

- Mathematics Department, Imperial College, London SW7 2BZ (United Kingdom)
- Physics Department, Imperial College, London SW7 2BZ (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 22058777

- 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; AMPLITUDES; COMPUTERIZED SIMULATION; EIGENVALUES; EIGENVECTORS; PERTURBATION THEORY; POTENTIALS; SYMMETRY; SYMMETRY BREAKING; WAVE FUNCTIONS; WAVE PACKETS

### Citation Formats

```
Graefe, Eva-Maria, and Jones, H. F..
```*PT-symmetric sinusoidal optical lattices at the symmetry-breaking threshold*. United States: N. p., 2011.
Web. doi:10.1103/PHYSREVA.84.013818.

```
Graefe, Eva-Maria, & Jones, H. F..
```*PT-symmetric sinusoidal optical lattices at the symmetry-breaking threshold*. United States. doi:10.1103/PHYSREVA.84.013818.

```
Graefe, Eva-Maria, and Jones, H. F.. Fri .
"PT-symmetric sinusoidal optical lattices at the symmetry-breaking threshold". United States. doi:10.1103/PHYSREVA.84.013818.
```

```
@article{osti_22058777,
```

title = {PT-symmetric sinusoidal optical lattices at the symmetry-breaking threshold},

author = {Graefe, Eva-Maria and Jones, H. F.},

abstractNote = {The PT-symmetric potential V{sub 0}[cos(2{pi}x/a)+i{lambda}sin(2{pi}x/a)] has a completely real spectrum for {lambda}{<=}1 and begins to develop complex eigenvalues for {lambda}>1. At the symmetry-breaking threshold {lambda}=1 some of the eigenvectors become degenerate, giving rise to a Jordan-block structure for each degenerate eigenvector. In general this is expected to result in a secular growth in the amplitude of the wave. However, it has been shown in a recent paper by Longhi, by numerical simulation and by the use of perturbation theory, that for a broad initial wave packet this growth is suppressed, and instead a saturation leading to a constant maximum amplitude is observed. We revisit this problem by explicitly constructing the Bloch wave functions and the associated Jordan functions and using the method of stationary states to find the dependence on the longitudinal distance z for a variety of different initial wave packets. This allows us to show in detail how the saturation of the linear growth arises from the close connection between the contributions of the Jordan functions and those of the neighboring Bloch waves.},

doi = {10.1103/PHYSREVA.84.013818},

journal = {Physical Review. A},

issn = {1050-2947},

number = 1,

volume = 84,

place = {United States},

year = {2011},

month = {7}

}