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Flow equations for light field interacting with two-level atoms

Zhiming Bai

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

We apply Wegner's flow equation method to study the eigenvalue of two-level atom system acted by light field.Interactions not contained in the original Hamiltonian are generated by this unitary transformation.With an average field approximation,we can close the flow equations.In the flow parameter l→∞ limit with Hamiltonian H(l),the interaction terms vanish and noninteraction terms approach fixed values.In the end,we obtain an explicit expression of the spin atom autocorrelation function.

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We apply Wegner's flow equation method to study the eigenvalue of two-level atom system acted by light field.Interactions not contained in the original Hamiltonian are generated by this unitary transformation.With an average field approximation,we can close the flow equations.In the flow parameter l→∞ limit with Hamiltonian H(l),the interaction terms vanish and noninteraction terms approach fixed values.In the end,we obtain an explicit expression of the spin atom autocorrelation function.
Light is both radiated and absorbed by atoms, and the interaction between the quantized electromagnetic field and an atom represents one of the most fundamental problems in quantum optics. However, real atoms are complicated systems, and even the simplest real atom, the hydrogen atom, has a non-trivial energy level structure. It is therefore often necessary or desirable to approximate the behavior of a real atom by that of a much simpler quantum system. For many purposes only two atomic energy levels play a significant role in the interaction with the electromagnetic field, so that it has become customary in many theoretical treatments to represent the atom by a quantum system with only two energy eigenstates. This is the most basic of all quantum systems, and it generally simplifies the treatment substantially. In a real atom the selection rules limit the allowed transitions between states, so that, in some cases, a certain state may couple to only one other. Moreover, optical pumping techniques have been developed that allow such preferred states to be prepared in the laboratory, and they have been successfully used in experiments (Abate, 1974). The two-level atom approximation is therefore close to the truth and not merely a mathematical convenience in some experimental situations. In the following we begin by developing the algebra for such a two-level atom. Dynamical variables for a two-level atom We consider an atomic quantum system with the two energy levels shown in Fig. 15.1.
We derive and solve the Hamiltonian flow equations for a Dirac particle in an external static potential. The method shows a general procedure for the set up of continuous unitary transformations to reduce the Hamiltonian to a quasidiagonal form. Supported by DFG Grant 436 RUS 113/205/1 on Russian German CooperationThe well known Foldy-Wouthuysen (FW) transformation [1] is a unitary transformation that separates big and small components in the Dirac equation for a particle interacting with an external electromagnetic field. If the external field is stationary then the FW transformation reduces to a unitary transformation of the Hamiltonian H of the Dirac particle H ′ = e iS (⃗α(⃗p − e ⃗ A(⃗x)) + βm + eA0(⃗x))e −iS (1) where S can be expanded in a sequence of hermitean operators in powers of 1/m. Recently a novel method to diagonalize Hamiltonians via continuous unitary transformations has been proposed [2], where the Hamiltonian H and the S are considered to be functions of a continuous variable, the flow parameter l. This new method has been applied to several topics like the n − orbital model [2], superconductivity and the electron phonon interaction [3, 4]. We want to apply it here to the Dirac particle in an external field in order to explore its possibilities with a well known problem. Thereby we can gain experience in solving problems like the elimination of negative energy states in near light cone QCD [5], where the solution has yet to be discovered. For infinitesimal steps exp(η(l)dl) a continuous unitary transformation of the Hamiltonian H leads to the so-called Hamiltonian flow equation dH(l) dl = [η(l), H(l)] (2) with initial condition H(0) = H. The transformation matrix η(l) is antihermitian and related to S as follows η(l) = ( d dl eiS(l))e −iS(l). The parameter l has dimension of energy −2 and characterizes the amount of diagonalization which has been achieved. In the energy representation the matrix elements of the Hamiltonian H will be grouped in a band of width
Abstract The Wegner flow equation is a differential equation for a family of Hamiltonians and can be considered as a continuous unitary transformation. In essence, the transformation continuously decouples degrees of freedom and gradually converts the transformed Hamiltonian into the near-diagonal form. The Wegner flow equation has so far been applied to many areas of physics, but mainly in perturbative calculations. However, the exact solutions are known only for the simplest of two-dimensional case and this certainly limits the practical application of the method. On the other hand, the knowledge of exact solutions is essential for many physical problems as well as it can help in constructing improved approximated approaches. In this paper we present the exact solution of the Wegner flow equation with the Mielke generator for 3 × 3 Hermitian matrices, the general solutions for N × N tridiagonal Hermitian matrices and partially for 4 × 4 real symmetric matrices.
A quantum optics model of the asymmetric case of the interaction between two two-level atoms and quantum field, which obeys SU(1,1) Lie group, is proposed. The atom-atom interaction and the rotating wave approximation are suggested in the Hamiltonian operator. Our aim is to obtain the time-dependent wave function of asymmetric case analytically. The analytical method is based on the eigenvalues and corresponding eigenvectors of the coefficient matrix of the interaction Hamiltonian operator. The SU(1,1) quantum system is initially in the Perelomov coherent state. Therefore, the atomic inversion is obtained and discussed for different values of model parameters such as initial atomic angles, Perelomov coherent, the Bargmann index and the detuning parameters. Observe that the quantum optics model is sensitive to the variation in both the Perelomov coherent parameter and the Bargmann index. In addition, there are nonclassical properties of the proposed quantum model in the presence the detuning parameter changes.
We investigate the non-Hermitian Hamiltonian which governs the system including two-level atom and electromagnetical field with a circular polarization vector. We find the Hamiltonian by using dipole approximation and rotating wave approximation (RWA), which lead to a non-Hermitian Hamiltonian. We solve the time-independent non-Hermitian Hamiltonian and obtain real eigenvalues of energy for this system. Finally, by solving the Schrodinger equation in the pseudo interaction picture, we show that our results and the results of Jaynes‐Cummings (JC) Hamiltonian are in excellent agreement.