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arXiv:quant-ph/0208097

Quantum bits and superposition displaced Fock states of the cavity field

L. Arevalo Aguilar, H. M. Moya-Cessa

Published Aug 14, 2002• 3 citations
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Abstract

We study the effects of counter rotating terms in the interaction of quantized light with a two-level atom, by using the method of small rotations. We give an expression for the wave function of the composed system atom plus field and point out one initial wave function that generates a quantum bit of the electromagnetic field with arbitrary amplitudes.

DOI
10.48550/arxiv.quant-ph/0208097

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Fock states and coherent states have been widely applied in analyses of quantum-optics experiments. In this paper, the application of a set of basis functions that consist of a product of displaced harmonic-oscillator states for the electromagnetic field and atomic states that are eigenfunctions of the momentum operator will be explored. For the case of a single mode, these states are the exact eigenfunctions in the limiting case where the electromagnetic-field mode frequency is much larger than the atomic transition frequency. This choice of basis functions will be used to analyze the interaction of a nonrelativistic, ``two-level'' atom with a single-frequency, quantized electromagnetic-field mode. Since in this basis the sign of the displacement of the electromagnetic-field state depends on the state of the atom, these new basis wave functions exhibit explicitly some aspects of atom-field correlations that are intrinsically quantum mechanical in nature. The analysis will be performed in the electric-dipole approximation for an atomic system that obeys the \ensuremath{\Delta}m=0 selection rule. The rotating-wave-approximation (RWA) terms cannot be made negligible through judicious choice of the field polarization for this system. The diamagnetic term, ${\mathit{e}}^{2}$${\mathbf{A}}^{2}$/2${\mathit{mc}}^{2}$, which is of the same order of magnitude as the RWA terms, is taken into account exactly by slightly modifying the traditional approach to quantizing the electromagnetic field. In this way the physical effects of the diamagnetic term are absorbed into a new electromagnetic cavity mode frequency.An approximate expression for the combined system's energy eigenvalues is found up to third order in the ratio of one-half the atomic transition frequency to the mode frequency. This expression indicates that inclusion of the RWA terms adds a small ``ripple'' on top of the smoothly varying energy eigenvalues when they are plotted as a function of the atom-field coupling constant. The ripple is interpreted as ``interference'' between two displaced oscillator-field states that represent the unperturbed state. When the Fock states are used as the basis for expanding the wave function of this system, a formidable five-term recurrence relation results. However, expanding the wave function in terms of the displaced harmonic-oscillator basis results in a less formidable three-term recurrence relation for the quantum coefficients.

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One of the simplest models involving the atom-field interaction is the coupling of a single two-level atom with single-mode optical field. Under the rotating wave approximation, this problem is reduced to a form that can be solved exactly. But the approximation is only valid when the two levels are resonant or nearly resonant with the applied electromagnetic radiation. Here we present an analytical solution without the rotating wave approximation and applicable to general atom-field interaction far away from the resonance. We find that there exists remarkable influence of the initial phase of optical field on the Rabi oscillations and Rabi splitting, and this issue cannot be explored in the context of the rotating wave approximation. Due to the retention of the counter-rotating terms, higher-order harmonic appears during the Rabi splitting. The analytical solution suggests a way to regulate and control the quantum dynamics of a two-level atom and allows for exploring more essential features of the atom-field interaction.

Optics Express•Nov 10, 2014•12 citations
We study the effects of counter rotating terms in the interaction of quantized light with a two-level atom, by using the method of small rotations. We give an expression for the wave function of the composed system atom plus field and point out one initial wave function that generates a quantum bit of the electromagnetic field with arbitrary amplitudes.
ESTUDIAMOS LOS EFECTOS DE LOS TERMINOS CONTRA-ROTANTES EN LA INTERACCION ENTRE UN CAMPO ELECTROMAGNETICO CUANTIZADO Y UN ATOMO DE DOS NIVELES, USANDO EL METODO DE PEQUENAS ROTACIONES. OBTENEMOS UNA EXPRESION PARA LA FUNCION DE ONDA DEL SISTEMA COMPUESTO ATOMO-CAMPO Y ELEGIMOS UN ESTADO INICIAL DE LA FUNCION DE ONDA QUE GENERA UN BIT CUANTICO DEL CAMPO ELECTROMAGNETICO. Abstract WE STUDY THE EFFECTS OF COUNTER ROTATING TERMS IN THE INTERACTION OF QUANTIZED LIGHT WITH A TWO-LEVEL ATOM, BY USING THE METHOD OF SMALL ROTATIONS. WE GIVE AN EXPRESSION FOR THE WAVE FUNCTION OF THE COMPOSED SYSTEM ATOM PLUS FIELD AND POINT OUT ONE INITIAL WAVE FUNCTION THAT GENERATES A QUANTUM BIT OF THE ELECTROMAGNETIC FIELD WITH ARBITRARY AMPLITUDES.
We investigate (non-relativistic) atomic systems interacting with quantum electromagnetic field (QEF). The resulting model describes spontaneous emission of light from a two-level atom surrounded by various initial states of the QEF. We assume that the quantum field interacts with the atom via the standard, minimal-coupling Hamiltonian, with the A2 term neglected. We also assume that there will appear at most single excitations (photons). By conducting the analysis on a general level we allow for an arbitrary initial state of the QEF (which can be for instance: the vacuum, the ground state in a cavity, or the squeezed state). We derive a Volterra-type equation which governs the time evolution of the amplitude of the excited state. The two-point function of the initial state of the QEF, integrated with a combination of atomic wavefunctions, forms the kernel of this equation.
Fock states and coherent states have been widely applied in analyses of quantum-optics experiments. In this paper, the application of a set of basis functions that consist of a product of displaced harmonic-oscillator states for the electromagnetic field and atomic states that are eigenfunctions of the momentum operator will be explored. For the case of a single mode, these states are the exact eigenfunctions in the limiting case where the electromagnetic-field mode frequency is much larger than the atomic transition frequency. This choice of basis functions will be used to analyze the interaction of a nonrelativistic, ``two-level'' atom with a single-frequency, quantized electromagnetic-field mode. Since in this basis the sign of the displacement of the electromagnetic-field state depends on the state of the atom, these new basis wave functions exhibit explicitly some aspects of atom-field correlations that are intrinsically quantum mechanical in nature. The analysis will be performed in the electric-dipole approximation for an atomic system that obeys the \ensuremath{\Delta}m=0 selection rule. The rotating-wave-approximation (RWA) terms cannot be made negligible through judicious choice of the field polarization for this system. The diamagnetic term, e2{\mathit{e}}^{2}e2A2{\mathbf{A}}^{2}A2/2mc2{\mathit{mc}}^{2}mc2, which is of the same order of magnitude as the RWA terms, is taken into account exactly by slightly modifying the traditional approach to quantizing the electromagnetic field. In this way the physical effects of the diamagnetic term are absorbed into a new electromagnetic cavity mode frequency.An approximate expression for the combined system's energy eigenvalues is found up to third order in the ratio of one-half the atomic transition frequency to the mode frequency. This expression indicates that inclusion of the RWA terms adds a small ``ripple'' on top of the smoothly varying energy eigenvalues when they are plotted as a function of the atom-field coupling constant. The ripple is interpreted as ``interference'' between two displaced oscillator-field states that represent the unperturbed state. When the Fock states are used as the basis for expanding the wave function of this system, a formidable five-term recurrence relation results. However, expanding the wave function in terms of the displaced harmonic-oscillator basis results in a less formidable three-term recurrence relation for the quantum coefficients.
Abstract The dynamics of an atom of three levels in Λ configuration interacting with a quantized field of two modes in a cavity is studied within the rotating wave approximation by considering experimental values of accessible hyperfine levels of alkali atoms. The Hamiltonian model is constructed on the basis of the direct product of the atomic states with Fock states of light. This simplified basis allows us to generalize the state of light to any quantum field. The interaction term of the Hamiltonian restricts the system to a 3D Hilbert space. The system’s time evolution results show some similarities with the effective two-level system, such as the presence of collapses and revivals and the atom-field entanglement. By an analysis of the statistics of the photons, we show that when the three-level atom is interacting with a coherent field of two modes, it is possible to control experimentally the statistics of the field to be super-Poissonian or sub-Poissonian by the choice of the initial atomic state.
We investigate the dynamics of an arbitrary atomic system (n-level atom or many n-level atoms) interacting with a resonant quantized mode of an electromagnetic field in a lossless cavity. Initially, the field is taken in a coherent state with a large average photon number. If the initial atomic state is chosen as an eigenstate of the atomic Hamiltonian in the classical field limit (semiclassical eigenstates), then the evolution of the system wave function is approximately factorized into the field and atomic parts. Explicit analytical expressions are given for the factorized wave functions. The time evolution of an arbitrary initial atomic state can be fully described in terms of factorized states. As examples, we discuss trapping states, and collapses and revivals of the atomic energy.
One of the simplest models involving the atom-field interaction is the coupling of a single two-level atom with single-mode optical field. Under the rotating wave approximation, this problem is reduced to a form that can be solved exactly. But the approximation is only valid when the two levels are resonant or nearly resonant with the applied electromagnetic radiation. Here we present an analytical solution without the rotating wave approximation and applicable to general atom-field interaction far away from the resonance. We find that there exists remarkable influence of the initial phase of optical field on the Rabi oscillations and Rabi splitting, and this issue cannot be explored in the context of the rotating wave approximation. Due to the retention of the counter-rotating terms, higher-order harmonic appears during the Rabi splitting. The analytical solution suggests a way to regulate and control the quantum dynamics of a two-level atom and allows for exploring more essential features of the atom-field interaction.