#### Title

Casus Irreducibilis in Atomic States

#### Standing

Undergraduate

#### Type of Proposal

Poster Presentation

#### Challenges Theme

Open Challenge

#### Faculty

Faculty of Science

#### Faculty Sponsor

Dr. Chitra Rangan

#### Abstract/Description of Original Work

In Quantum Computing, one must switch between two quantum states called a 'qubit'. A well-known method for switching between quantum states is known as Stimulated Raman Adiabatic Passage (STIRAP). This method utilizes a specific atomic structure known as a 3-Level Lambda System (3LLS). A numerical investigation of these atomic structures led to the apparent result that the Hamiltonian describing the structure has eigenvalues that are complex-valued. This is problematic because the Hamiltonian that describes this atomic structure is self-adjoint and its eigenvalues can only be real-valued. In this project, these numerically-found, complex eigenvalues were found to be the result of the 'Casus irreducibilis'; a special case when trying to solve polynomials of degree 3 or higher, in which real solutions must be represented as complex numbers, proven by Pierre Wantzel in 1843. The impact of this discovery is that special care must be taken when theoretically & numerically modeling the dynamics of quantum systems.

Casus Irreducibilis in Atomic States

In Quantum Computing, one must switch between two quantum states called a 'qubit'. A well-known method for switching between quantum states is known as Stimulated Raman Adiabatic Passage (STIRAP). This method utilizes a specific atomic structure known as a 3-Level Lambda System (3LLS). A numerical investigation of these atomic structures led to the apparent result that the Hamiltonian describing the structure has eigenvalues that are complex-valued. This is problematic because the Hamiltonian that describes this atomic structure is self-adjoint and its eigenvalues can only be real-valued. In this project, these numerically-found, complex eigenvalues were found to be the result of the 'Casus irreducibilis'; a special case when trying to solve polynomials of degree 3 or higher, in which real solutions must be represented as complex numbers, proven by Pierre Wantzel in 1843. The impact of this discovery is that special care must be taken when theoretically & numerically modeling the dynamics of quantum systems.