Trapped cold ions and neutral atoms serve as essential experimental platforms for investigating and showcasing fundamental tasks in quantum information processing. Among these, trapped Rydberg ions stand out as highly promising candidates for realizing quantum computation [1]. In trapped ion quantum computation, a pivotal focus lies in achieving manageable interactions among qubits that are spatially distant. This capability is essential for creating qubit entanglement, a cornerstone for quantum information processing, and for facilitating the transfer of quantum information between distant qubits. In this framework, an effective interaction between qubits is often engineered through state-dependent laser coupling, which connects ions to the quantum harmonic oscillations (phonons) of the ion crystal.

The main responsibilities of this post will be to carry out independent research on quantum control of many Rydberg ions and to apply and/or develop new theoretical and numerical methods to effectively steer a many body physical system to a predefined state [2-4].

The research objectives for this Ph.D. proposal encompass the following:

1. Investigate and enhance methods for controlling interactions among many Rydberg ions.
2. Achieve precise quantum control over a many-body system to reach specific predefined quantum properties.
3. Advance Theoretical and Numerical Methods: Apply and innovate theoretical and numerical approaches to steer complex many-body quantum systems.
4. Develop new models and algorithms that can efficiently manage and predict the behaviour of these systems.
5. Facilitate effective transfer of quantum information between distant qubits.

To achieve the research objectives, the following methodology will be employed:

1. Formulate new theoretical frameworks to describe and predict the behavior of many-body Rydberg ion systems.
2. Incorporate these frameworks into computational models to simulate complex interactions.
3. Develop and utilize advanced numerical methods to simulate quantum control processes.
4. Perform extensive simulations to explore different control strategies and optimize them for practical implementation.

The proposed research aims to significantly advance the field of quantum optimal control of Rydberg ions. The expected outcomes of this research include:

1. Development of refined techniques for precise control of many Rydberg ions.
2. Achieving targeted quantum states with high accuracy in many-body systems.
3. Innovative Theoretical Models: Creation of advanced theoretical models that better explain the dynamics of Rydberg ions.
4. Contribution to the theoretical understanding of quantum control in many-body systems.

[1] D. Leibfried et. all, Rev. Mod. Phys, 75, 281 (2003).

[2] W. Zhu, J. Botina and H. Rabitz, J. Chem. Phys. 108, 1953 (1998).

[3] W. Zhu and H. Rabitz, J. Chem. Phys. 109, 385 (1998).

[4] R. Srinivas et. all., Phys. Rev. Lett, 131, 020601 (2023).