## Physics & Astronomy ETDs

#### Publication Date

8-27-2009

#### Abstract

In this dissertation I analyze Hamiltonian control of d-dimensional quantum systems as realized in alkali atomic spins. Alkali atoms provide an ideal platform for studies of quantum control due to the extreme precision with which the control fields are characterized as well as their isolation from their environment. In many cases, studies into the control of atomic spins restrict attention to a 2-dimesional subspace in order to consider qubit control. The geometry of quantum 2-level systems is much simpler than for any larger dimensional Hilbert space, and so control techniques for qubits often are not applicable to larger systems. In reality, atoms have many internal levels. It seems a shame to throw away most of our Hilbert space when it could in principle be used for encoding information and performing error correction. This work develops some of the tools necessary to control these large atomic spins. Quantum control theory has some very generic properties that have previously been explored in the literature, notably in the work from the Rabitz group. I provide a review of this literature, showing that while the landscape topology of quantum control problems is relatively independent of physical platform, different optimization techniques are required to find optimal controls depending on the particular control task. To this end I have developed two optimal control algorithms for finding unitary maps for the problems of: "state preparation" where we require only that a single fiducial state us taken to a particular target state and "unitary construction" where the entire map is specified. State mapping turns out to be a simple problem to solve and is amenable to a gradient search method. This protocol is not feasible for the task of finding full unitary maps, but I show how we can weave state mappings together to form full unitary maps. This construction of unitary maps is efficient in the dimension of the Hilbert space. The particular system I have used for demonstrating these control techniques is that of alkali atoms, specifically {133}Cs. The state preparation algorithm was used to create a broad range of target states in the 7-dimension F=3 hyperfine manifold in an experiment using a combination of time-dependent magnetic fields and a static tensor light shift. The yields from this experiment were in the range of 0.8-0.9. I have developed another control system for the full hyperfine manifold in the ground-electronic state of {133}Cs, a 16-dimensional Hilbert space, based on applied radio frequency and microwave fields. Numerical studies of the state preparation algorithm find good operating points commensurate with modest laboratory requirements. This system of microwave and rf control also admits a Hamiltonian structure than can be used by my protocol for unitary construction. I demonstrate the performance of this algorithm by creating a standard set of qudit gates using physically realistic control fields, as well as by implementing a simple form of error correction.

#### Degree Name

Physics

#### Level of Degree

Doctoral

#### Department Name

Physics & Astronomy

#### First Advisor

Deutsch, Ivan

#### First Committee Member (Chair)

Caves, Carlton

#### Second Committee Member

Jessen, Poul

#### Third Committee Member

Landahl, Andrew

#### Language

English

#### Keywords

Quantum theory, Control theory, Hamiltonian systems, Quantum computers, Feedback control systems--Dynamics, Nuclear spin, Alkali metals.

#### Document Type

Dissertation

#### Recommended Citation

Merkel, Seth. "Quantum control of d-dimensional quantum systems with application to alkali atomic spins." (2009). https://digitalrepository.unm.edu/phyc_etds/44