Specifying the Unitary Evolution of a Qudit for a General Nonstationary Hamiltonian via the Generalized Gell-Mann Representation

• Main Authors: Elena R. Loubenets, Christian Käding
• Format: Article
• Language: English
• Published: MDPI AG 2020-05-01
• Series: Entropy
• Subjects: unitary evolution of a qudit; nonstationary Hamiltonian; exponential representation; Bloch-like vector space; analytical solutions
• Online Access: https://www.mdpi.com/1099-4300/22/5/521

Optimal realizations of quantum technology tasks lead to the necessity of a detailed analytical study of the behavior of a <i>d</i>-level quantum system (qudit) under a time-dependent Hamiltonian. In the present article, we introduce a new general formalism describing the unitary evolution of a qudit <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>d</mi> <mo>≥</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> in terms of the Bloch-like vector space and specify how, in a general case, this formalism is related to finding time-dependent parameters in the exponential representation of the evolution operator under an arbitrary time-dependent Hamiltonian. Applying this new general formalism to a qubit case <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, we specify the unitary evolution of a qubit via the evolution of a unit vector in <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="double-struck">R</mi> <mn>4</mn> </msup> </semantics> </math> </inline-formula>, and this allows us to derive the precise analytical expression of the qubit unitary evolution operator for a wide class of nonstationary Hamiltonians. This new analytical expression includes the qubit solutions known in the literature only as particular cases.