[Paper Review] Mirror-field-atom interaction: Hamiltonian diagonalization

D. Rodríguez-Méndez, O. Aguilar–Loreto+1 more|arXiv (Cornell University)|Mar 7, 2013

Catalysis and Oxidation Reactions1 citations

TL;DR

This paper presents an exact transformation using Susskind-Glogower operators to simplify the Hamiltonian of a cavity field, a movable mirror, and a two-level atom. By eliminating the field degrees of freedom exactly, the system reduces to an effective atom-mirror interaction that is diagonalized via small rotations, enabling exact solution of the dynamics and paving the way for studying entanglement and quantum state reconstruction in cavity QED systems.

ABSTRACT

We show that the interaction between a movable mirror with a quantized field that interacts with a two-level atom may be simplified via a transformation that involves Susskind-Glogower operators (SGO). By using this transformation it is easy to show that we can cast the Hamiltonian, after a series of small rotations, into an effective Hamiltonian that may be solved. We would like to stress that the transformation in terms of SGO already simplifies enough the Hamiltonian in the sense that, in an exact way, it "eliminates" one of the three-subsystems, namely the quantized field.

Motivation & Objective

• To simplify the Hamiltonian of a quantized field, movable mirror, and two-level atom system.
• To eliminate the field degrees of freedom exactly using Susskind-Glogower operators.
• To derive an effective, diagonal Hamiltonian for the atom-mirror subsystem via small unitary rotations.
• To enable exact solution of the time evolution and facilitate future study of entanglement and state reconstruction.
• To provide a framework for reconstructing mirror-field interactions through atomic measurements.

Proposed method

• Utilizes Susskind-Glogower operators V and V† to transform the Hamiltonian, effectively decoupling the field from the atom-mirror subsystem.
• Applies a unitary transformation M and M† to rewrite the Hamiltonian in a form where field operators commute with other subsystems.
• Introduces an exact rotation using a matrix R to simplify the initial Hamiltonian structure.
• Applies small unitary rotations U1 and U2 with small parameters ξ1 and ξ2 to approximate the system into a diagonal form.
• Chooses ξ1 and ξ2 to cancel non-diagonal terms in the effective Hamiltonian, achieving a dispersive-like form.
• Neglects higher-order terms in ξ1 and ξ2 due to their smallness, resulting in a solvable effective Hamiltonian.

Experimental results

Research questions

• RQ1 Can the full mirror-field-atom Hamiltonian be simplified without approximations?
• RQ2 Can the field degrees of freedom be exactly eliminated from the dynamics using non-perturbative transformations?
• RQ3 How can the atom-mirror interaction be effectively diagonalized despite the presence of non-linear couplings?
• RQ4 What is the role of Susskind-Glogower operators in decoupling the field from the rest of the system?
• RQ5 Can small rotations lead to a solvable effective Hamiltonian for the atom-mirror subsystem?

Key findings

• The Susskind-Glogower transformation exactly eliminates the field operators from the effective dynamics, reducing the system to an atom-mirror interaction.
• The transformed Hamiltonian allows the field operators to be treated as classical numbers due to their commutativity with other subsystems.
• After small rotations, the effective Hamiltonian becomes diagonal and analytically solvable.
• The final effective Hamiltonian is approximated as H2 ≈ νN̂ + χ(b̂n + 1/2)(b̂ + b̂†) + λ√(b̂n + 1)σz + (χ²/λ√(b̂n + 1))(b̂ + b̂†)².
• The non-diagonal terms are canceled by choosing ξ1 and ξ2 appropriately, and higher-order corrections are negligible due to smallness of ξ1, ξ2.
• The evolution operator is expressed as U(t) = M† exp(−iĤeff t) M exp(−iρ̂₀²² t), enabling exact time evolution calculation.