## Physics & Astronomy ETDs

#### Publication Date

2-1-2012

#### Abstract

Quantum state tomography is a fundamental tool in quantum information processing tasks. It allows us to estimate the state of a quantum system by measuring different observables on many identically prepared copies of the system. Usually, one makes projective measurements of an informationally complete set of observables and repeats them enough times so that good estimates of their expectation values are obtained. This is, in general, a very time-consuming task that requires a large number of measurements. There are, however, systems in which the data acquisition can be done more efficiently. In fact, an ensemble of quantum systems can be prepared and manipulated by external fields while being continuously probed collectively, producing enough information to estimate its state. This provides a basis for continuous measurement quantum tomography, and is the main topic of this dissertation. This method, based on weak continuous measurement, has the advantage of being fast, accurate, and almost nonperturbative. In this work, we present a extensive discussion and a generalization of the protocol proposed in [1], which was experimentally achieved in [2] using cold cesium atoms. In this protocol, an ensemble of identically prepared systems is collectively probed and controlled in a time-dependent manner so as to create an informationally complete continuous measurement record. The measurement history is then inverted to determine the state at the initial time. To achieve this, we use two different estimation methods: the widely used maximum likelihood and the novel compressed sensing algorithms. The general formalism is applied to the case of reconstruction of the quantum state encoded in the magnetic sub-levels of a large-spin alkali atom, 133Cs. We extend the applicability of the protocol in [1] to the more ambitious case of reconstruction of states in the full 16-dimensional electronic-ground subspace (F = 3 \u2295 F = 4), controlled by microwaves and radio-frequency magnetic fields. We give detailed derivations of all physical interactions, approximations, numerical methods, and fitting procedures, tailored to the realistic experimental setting. In addition, we numerically study the reconstruction algorithms and determine their applicability and appropriate use. Moreover, in collaboration with the lab of Prof. P. Jessen at the University of Arizona, we present an experimental demonstration of continuous measurement quantum tomography in an ensemble of cold cesium atoms with full control of its 16-dimensional Hilbert space. In this case, we show the exquisite level of control achieved in the lab and the excellent agreement between the theory discussed in this dissertation and the experimental results. This allows us to achieve fidelities > 95% for low complexity quantum states, and > 92% for arbitrary random states, which is a formidable accomplishment for a space of this size. To conclude this work, we study quantum tomography in an abstract system driven by random dynamics and show the conditions for high-fidelity estimation when a single parameter defines the dynamics of the system. This study helps elucidate the reconstruction algorithm and gives rise to interesting questions about the geometry of quantum states.

#### Degree Name

Physics

#### Level of Degree

Doctoral

#### Department Name

Physics & Astronomy

#### First Advisor

Deutsch, Ivan

#### First Committee Member (Chair)

Deutsch, Ivan

#### Second Committee Member

Caves, Carlton

#### Third Committee Member

Jessen, Poul

#### Fourth Committee Member

Jayaweera, Sudharman

#### Language

English

#### Keywords

Quantum computers, Quantum entanglement.

#### Document Type

Dissertation

#### Recommended Citation

Riofrío, Almeida Carlos. "Continuous measurement quantum state tomography of atomic ensembles." (2012). https://digitalrepository.unm.edu/phyc_etds/59