# Tomographic reconstruction of quantum states in N spatial dimensions

## Abstract

Most quantum tomographic methods for massive particles using destructive measurement can only be used for motion in one dimension. We show how to infer the quantum state of a nonrelativistic, N-dimensional harmonic oscillator system by simple inverse Radon transforms. We assume a time-independent Hamiltonian for which the dynamics along each coordinate always occurs simultaneously with the others and measurements on all coordinates are performed simultaneously. This is unlike light systems where different modes can be delayed and phase shifted with respect to each other. A requirement of the procedure is that the angular frequencies of the N harmonic potentials be incommensurable. We discuss the information available if the requirement of incommensurability is not fulfilled and also under what conditions the state can be reconstructed from finite-time measurements. As further examples, we consider free particles in N dimensions with periodic boundary conditions and particles in an N-dimensional box, where we find a similar condition of incommensurability and finite recurrence time.

- Authors:

- QUANTOP, Danish National Research Foundation Center for Quantum Optics, Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C (Denmark)

- Publication Date:

- OSTI Identifier:
- 20787065

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 73; Journal Issue: 4; Other Information: DOI: 10.1103/PhysRevA.73.042105; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; BOUNDARY CONDITIONS; COORDINATES; ENERGY LEVELS; HAMILTONIANS; HARMONIC OSCILLATORS; HARMONIC POTENTIAL; PARTICLES; PERIODICITY; PHASE SHIFT; RADON; TIME MEASUREMENT; TOMOGRAPHY

### Citation Formats

```
Mouritzen, Anders S., and Moelmer, Klaus.
```*Tomographic reconstruction of quantum states in N spatial dimensions*. United States: N. p., 2006.
Web. doi:10.1103/PHYSREVA.73.0.

```
Mouritzen, Anders S., & Moelmer, Klaus.
```*Tomographic reconstruction of quantum states in N spatial dimensions*. United States. doi:10.1103/PHYSREVA.73.0.

```
Mouritzen, Anders S., and Moelmer, Klaus. Sat .
"Tomographic reconstruction of quantum states in N spatial dimensions". United States.
doi:10.1103/PHYSREVA.73.0.
```

```
@article{osti_20787065,
```

title = {Tomographic reconstruction of quantum states in N spatial dimensions},

author = {Mouritzen, Anders S. and Moelmer, Klaus},

abstractNote = {Most quantum tomographic methods for massive particles using destructive measurement can only be used for motion in one dimension. We show how to infer the quantum state of a nonrelativistic, N-dimensional harmonic oscillator system by simple inverse Radon transforms. We assume a time-independent Hamiltonian for which the dynamics along each coordinate always occurs simultaneously with the others and measurements on all coordinates are performed simultaneously. This is unlike light systems where different modes can be delayed and phase shifted with respect to each other. A requirement of the procedure is that the angular frequencies of the N harmonic potentials be incommensurable. We discuss the information available if the requirement of incommensurability is not fulfilled and also under what conditions the state can be reconstructed from finite-time measurements. As further examples, we consider free particles in N dimensions with periodic boundary conditions and particles in an N-dimensional box, where we find a similar condition of incommensurability and finite recurrence time.},

doi = {10.1103/PHYSREVA.73.0},

journal = {Physical Review. A},

number = 4,

volume = 73,

place = {United States},

year = {Sat Apr 15 00:00:00 EDT 2006},

month = {Sat Apr 15 00:00:00 EDT 2006}

}