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arXiv:quant-ph/0312060 · The Yokohama mathematical journal = 横濱市立大學紀要. D部門, 数学

A Rabi Oscillation in Four and Five Level Systems

Kazuyuki Fujii, Kyoko Higashida, Ryosuke Kato, Andyukako Wada

Published Dec 6, 2003• 9 citations
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Abstract

In this paper we consider a general model of an atom with n energy levels interacting with n-1 external (laser) fields which is a natural extension in the two and three level systems. We exactly solve the Schr{\" o}dinger equation to obtain a Rabi oscillation when n = 4 and 5, which will constitute a quantum logic gate of Quantum Computation based on four and five level systems.

DOI
10.48550/arxiv.quant-ph/0312060

Related papers

A Generalization of the Preceding Paper ``A Rabi Oscillation in Four and Five Level Systems"

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In the preceding paper quant-ph/0312060 we considered a general model of an atom with n energy levels interacting with n-1 external laser fields and constructed a Rabi oscillation in the case of n =3, 4 and 5. In the paper we present a systematic method getting along with computer to construct a Rabi oscillation in the general case.

arXiv:quant-ph/0605132•May 16, 2006•1 citation

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In this paper we propose a Hamiltonian generalizing the interaction of the two--level atom and both the single radiation mode and external field $...$ a kind of cavity QED. We solve the Schrodinger equation in the strong coupling regime by making use of rotating wave approximation under new resonance conditions containing the Bessel functions and etc, and obtain unitary transformations of four types corresponding to Rabi oscillations which perform quantum logic gates in Quantum Computation.

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This paper is an extension of Fujii et al (quant--ph/0307066) and in this one we again treat a model of atom with n energy levels interacting with n(n-1)/2 external laser fields, which is a natural extension of usual two level system. Then the rotating wave approximation (RWA) is assumed from the beginning. To solve the Schr{\" o}dinger equation we set the {\bf consistency condition} in our terminology and reduce it to a matrix equation with symmetric matrix $Q$ consisting of coupling constants. However, to calculate $\exp(-itQ)$ {\bfexplicitly} is not easy. In the case of three and four level systems we determine it in a complete manner, so our model in these levels becomes realistic. In last, we make a comment on Cavity QED quantum computation based on three energy levels of atoms as a forthcoming target.

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N Level System with RWA and Analytical Solutions Revisited

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In this paper we consider a model of an atom with n energy levels interacting with n(n-1)/2 external (laser) fields which is a natural extension of two level system, and assume the rotating wave approximation (RWA) from the beginning. We revisit some construction of analytical solutions (which correspond to Rabi oscillations) of the model in the general case and examine it in detail in the case of three level system.

arXiv:quant-ph/0307066•Jul 9, 2003•5 citations

Cavity QED and Quantum Computation in the Weak Coupling Regime II : Complete Construction of the Controlled-Controlled NOT Gate

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In this paper we treat a cavity QED quantum computation. Namely, we consider a model of quantum computation based on n atoms of laser-cooled and trapped linearly in a cavity and realize it as the n atoms Tavis-Cummings Hamiltonian interacting with n external (laser) fields. We solve the Schr{\" o}dinger equation of the model in the weak coupling regime to construct the controlled NOT gate in the case of n=2, and to construct the controlled-controlled NOT gate in the case of n=3 by making use of several resonance conditions and rotating wave approximation associated to them. We also present an idea to construct general quantum circuits. The approach is more sophisticated than that of the paper [K. Fujii, Higashida, Kato and Wada, Cavity QED and Quantum Computation in the Weak Coupling Regime, J. Opt. B : Quantum Semiclass. Opt. {\bf 6} (2004), 502]. Our method is not heuristic but completely mathematical, and the significant feature is based on a consistent use of Rabi oscillations.

arXiv:quant-ph/0501046•Jan 11, 2005•15 citations

Cavity QED and quantum computation in the weak coupling regime

Kazuyuki Fujii, Kyoko Higashida, Ryosuke Kato +1 more

In this paper we consider a model of quantum computation based on n atoms, laser cooled and trapped linearly in a cavity, and realize it as the n -atom Tavis–Cummings Hamiltonian interacting with n external (laser) fields. We solve the Schrödinger equation of the model in the case of n = 2 and construct the controlled NOT gate by making use of a resonance condition and the rotating wave approximation associated with it. Our method is not heuristic but completely mathematical, and the significant feature is the consistent use of Rabi oscillations. We also present an idea for the construction of three controlled NOT gates in the case of n = 3 which gives a controlled–controlled NOT gate.

arXiv:quant-ph/0407014•Journal of Optics B Quantum and Semiclassical Optics•Oct 16, 2004•25 citations
In this paper we consider a general model of an atom with n energy levels interacting with n-1 external (laser) fields which is a natural extension in the two and three level systems. We exactly solve the Schr{" o}dinger equation to obtain a Rabi oscillation when n = 4 and 5, which will constitute a quantum logic gate of Quantum Computation based on four and five level systems.
In this paper we propose a Hamiltonian generalizing the interaction of the two--level atom and both the single radiation mode and external field ......... a kind of cavity QED. We solve the Schrodinger equation in the strong coupling regime by making use of rotating wave approximation under new resonance conditions containing the Bessel functions and etc, and obtain unitary transformations of four types corresponding to Rabi oscillations which perform quantum logic gates in Quantum Computation.
This paper is an extension of Fujii et al (quant--ph/0307066) and in this one we again treat a model of atom with n energy levels interacting with n(n-1)/2 external laser fields, which is a natural extension of usual two level system. Then the rotating wave approximation (RWA) is assumed from the beginning. To solve the Schr{\" o}dinger equation we set the {\bf consistency condition} in our terminology and reduce it to a matrix equation with symmetric matrix QQQ consisting of coupling constants. However, to calculate exp⁡(−itQ)\exp(-itQ)exp(−itQ) {\bfexplicitly} is not easy. In the case of three and four level systems we determine it in a complete manner, so our model in these levels becomes realistic. In last, we make a comment on Cavity QED quantum computation based on three energy levels of atoms as a forthcoming target.
In this paper we treat a cavity QED quantum computation. Namely, we consider a model of quantum computation based on n atoms of laser-cooled and trapped linearly in a cavity and realize it as the n atoms Tavis-Cummings Hamiltonian interacting with n external (laser) fields. We solve the Schr{\" o}dinger equation of the model in the weak coupling regime to construct the controlled NOT gate in the case of n=2, and to construct the controlled-controlled NOT gate in the case of n=3 by making use of several resonance conditions and rotating wave approximation associated to them. We also present an idea to construct general quantum circuits. The approach is more sophisticated than that of the paper [K. Fujii, Higashida, Kato and Wada, Cavity QED and Quantum Computation in the Weak Coupling Regime, J. Opt. B : Quantum Semiclass. Opt. {\bf 6} (2004), 502]. Our method is not heuristic but completely mathematical, and the significant feature is based on a consistent use of Rabi oscillations.
In this paper we consider a model of quantum computation based on n atoms, laser cooled and trapped linearly in a cavity, and realize it as the n -atom Tavis–Cummings Hamiltonian interacting with n external (laser) fields. We solve the Schrödinger equation of the model in the case of n = 2 and construct the controlled NOT gate by making use of a resonance condition and the rotating wave approximation associated with it. Our method is not heuristic but completely mathematical, and the significant feature is the consistent use of Rabi oscillations. We also present an idea for the construction of three controlled NOT gates in the case of n = 3 which gives a controlled–controlled NOT gate.