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Electronic journal of theoretical physics

Derivation of the Rabi equation by means of the Pauli matrices

Maria Cristina De Sanctis, C. J. Quimbay

Published Jan 1, 2009• 1 citation
OpenAlex

Abstract

Abstract: The general quantum-mechanical properties of two-level systems are very relevant for the study of different physical subjects of great interest, as, for instance, magnetic resonance, nuclear physics, molecular dynamics, masers, neutrino oscillations, quantum computation, etc. In this work we present a nonperturbative approach to the study of the oscillations among two quantum states when an interaction term is considered. First, we show that using a simple parametrization for the superposition of the two states, it is possible to derive easily the Rabi equation. Next, we show that the Pauli matrices can be conveniently used to write the Hamiltonian operator and to derive the Rabi equation, directly applying the time evolution operator to the initial state of the system.

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Quantum dynamics using the time-dependent Schrödinger equation

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Abstract This chapter the reviews essential concepts and methods of time-dependent quantum mechanics. (1) Formal solutions of the time-dependent Schrödinger equation. (2) The two-level system. (3) Interacting systems: The time-dependent self-consistent field approximation and the Ehrenfest approximation. (4) The Born–Oppenheimer approximation and nuclear potential surfaces. (5) Time-dependent Hamiltonians: A two-level system in a time-dependent field. (6) Non-adiabatic dynamics: The surface-hopping method. (7) Expressing the time evolution in terms of the Green operator. (8) Representations (the Schrödinger and Heisenberg representations, the interaction representation). (9) Time-dependent perturbation theory. (10) Quantum dynamics of the free particles. (11) Quantum dynamics of the harmonic oscillator. (12) The raising/lowering operators formalism. (13) Tunneling.

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نموذج نظري للجمل الكمومية بسويتي طاقة

محي الدين نظام, محمد موسى

قدمنا في هذا العمل نموذجاً نظرياً لدراسة اهتزازات جملة كمومية بسويتي طاقة مفترضين وجود تفاعل متبادل بينهما من خلال: أولاً: تمثيل الجملة الكمومية ذات سويتي الطاقة بحفرتي كمون كموميتين لانهائيتين متآثرتين على بعد واحد x ومتناظرتين بالنسبة لمبدأ الإحداثيات o، ثمّ إيجاد المتجهات الخاصة والقيم الخاصة لهما باستخدام معادلة شرودنغر المستقرّة Stationary Schrödinger’s equation. ثانياً: إثبات أنّ التقريب غير الاضطرابي كاف لكتابة مؤثر الهاملتوني لهذه الجملة، ولاشتقاق معادلة رابي للجملة Rabi’s equation، وتطبيقها مباشرة على مؤثر التطور للحالة الابتدائية للجملة. We present a theoretical model for studying the oscillations among two-level quantum system when an interaction term is considered: First, we present the two-level quantum system with a symmetric double square potential well in interaction, in one dimension x, then we find the eigenvectors and the eigenvalues for this system by solving the stationary Schrödinger’s equation. Second, we proved that a nonperturbative approach can be conveniently used to write the Hamiltonian operator and to derive the Rabi’s equation for this system, directly applying the time evolution operator to the initial state of the system.

•Feb 28, 2013•0 citations

On the solvability of p states quantum Rabi Model with Zp -graded parity

Won Sang Chung, Jae Yoon Kim

In this paper the p-level Rabi model with Z p -graded symmetry is discussed. The p-level Rabi Hamiltonian is constructed by introducing the generalized Pauli matrices. The energy and wave function for the p-level Rabi equation are obtained by using the standard perturbation method.

Journal of Physics Conference Series•Oct 1, 2016•0 citations
Abstract: The general quantum-mechanical properties of two-level systems are very relevant for the study of different physical subjects of great interest, as, for instance, magnetic resonance, nuclear physics, molecular dynamics, masers, neutrino oscillations, quantum computation, etc. In this work we present a nonperturbative approach to the study of the oscillations among two quantum states when an interaction term is considered. First, we show that using a simple parametrization for the superposition of the two states, it is possible to derive easily the Rabi equation. Next, we show that the Pauli matrices can be conveniently used to write the Hamiltonian operator and to derive the Rabi equation, directly applying the time evolution operator to the initial state of the system.
The scientific interest in the analytical solution of the quantum Rabi model is due to the widespread use of this simple model in quantum optics, quantum computing, cavity QED, and nanoelectromechanical systems. This interest is related to the need for the theoretical description of the interaction of a two-level system with a quantum oscillator in the case when the rotating wave approximation fails. In this Letter, we present an approach to the exact diagonalization of the quantum Rabi Hamiltonian. This approach is based on the properties of the Pauli operators and allows us to readily solve the stationary Schrodinger equation for a two-level system. First, we demonstrate the applicability of the approach to the Jaynes-Cummings Hamiltonian to get the well-known solution. Then, we obtain the eigenvalues and eigenstates for the quantum Rabi Hamiltonian using the proposed approach. It is shown that the obtained eigenstates can be represented in the basis of the eigenstates of the Jaynes-Cummings Hamiltonian.
Evolution of quantum fidelity for two-level systems is studied in the context of periodic echo. From a general treatment for time independent case, we obtain a simple condition on the governing Hamiltonians under which the systems display periodic quantum echo. For a specific time dependent problem the quantum fidelity is shown to exhibit Rabi oscillation. This may be considered as a simple mechanism to generate periodic echo, except for a specific initial superpositional state in which case the fidelity remains invariant.
We outline a method based on successive canonical transformations which yields a product expansion for the evolution operator of a general (possibly non-Hermitian) Hamiltonian. For a class of such Hamiltonians this expansion involves a finite number of terms, and our method gives the exact solution of the corresponding time-dependent Schrödinger equation. We apply this method to study the dynamics of a general nondegenerate two-level quantum system, a time-dependent classical harmonic oscillator, and a degenerate system consisting of a spin 1 particle interacting with a time-dependent electric field E⃗(t) through the Stark Hamiltonian H=λ(J⃗⋅E⃗)2.
Abstract This chapter the reviews essential concepts and methods of time-dependent quantum mechanics. (1) Formal solutions of the time-dependent Schrödinger equation. (2) The two-level system. (3) Interacting systems: The time-dependent self-consistent field approximation and the Ehrenfest approximation. (4) The Born–Oppenheimer approximation and nuclear potential surfaces. (5) Time-dependent Hamiltonians: A two-level system in a time-dependent field. (6) Non-adiabatic dynamics: The surface-hopping method. (7) Expressing the time evolution in terms of the Green operator. (8) Representations (the Schrödinger and Heisenberg representations, the interaction representation). (9) Time-dependent perturbation theory. (10) Quantum dynamics of the free particles. (11) Quantum dynamics of the harmonic oscillator. (12) The raising/lowering operators formalism. (13) Tunneling.
قدمنا في هذا العمل نموذجاً نظرياً لدراسة اهتزازات جملة كمومية بسويتي طاقة مفترضين وجود تفاعل متبادل بينهما من خلال: أولاً: تمثيل الجملة الكمومية ذات سويتي الطاقة بحفرتي كمون كموميتين لانهائيتين متآثرتين على بعد واحد x ومتناظرتين بالنسبة لمبدأ الإحداثيات o، ثمّ إيجاد المتجهات الخاصة والقيم الخاصة لهما باستخدام معادلة شرودنغر المستقرّة Stationary Schrödinger’s equation. ثانياً: إثبات أنّ التقريب غير الاضطرابي كاف لكتابة مؤثر الهاملتوني لهذه الجملة، ولاشتقاق معادلة رابي للجملة Rabi’s equation، وتطبيقها مباشرة على مؤثر التطور للحالة الابتدائية للجملة. We present a theoretical model for studying the oscillations among two-level quantum system when an interaction term is considered: First, we present the two-level quantum system with a symmetric double square potential well in interaction, in one dimension x, then we find the eigenvectors and the eigenvalues for this system by solving the stationary Schrödinger’s equation. Second, we proved that a nonperturbative approach can be conveniently used to write the Hamiltonian operator and to derive the Rabi’s equation for this system, directly applying the time evolution operator to the initial state of the system.