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arXiv:quant-ph/0403008

Explicit Form of Solution of Two Atoms Tavis-Cummings Model

Kazuyuki Fujii, Kyoko Higashida, Ryosuke Kato, Yukako Wada

Published Mar 1, 2004• 11 citations
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Abstract

In this paper we consider the two atoms Tavis--Cummings model and give an explicit form to the solution of this model which will play a central role in quantum computation based on atoms of laser--cooled and trapped linearly in a cavity. We also present a problem of three atoms Tavis--Cummings model which is related to the construction of controlled--controlled NOT operation (gate) in quantum computation.

DOI
10.48550/arxiv.quant-ph/0403008

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In this paper we consider the two atoms Tavis--Cummings model and give an explicit form to the solution of this model which will play a central role in quantum computation based on atoms of laser--cooled and trapped linearly in a cavity. We also present a problem of three atoms Tavis--Cummings model which is related to the construction of controlled--controlled NOT operation (gate) in quantum computation.
In this letter the explicit form of evolution operator of the Tavis–Cummings model with three and four atoms is given. This is an important progress in quantum optics or mathematical physics. The purpose of this paper is to give an explicit form to the evolution operator of Tavis–Cummings model ([1]) with some atoms. This model is a very important one in Quantum Optics and has been studied widely, see [2] as general textbooks in quantum optics. We are studying a quantum computation and therefore want to study the model from this point of view, namely a quantum computation based on atoms of laser–cooled and trapped linearly in a cavity. We must in this model construct the controlled NOT gate or other controlled unitary gates to perform a quantum computation, see [3] as a general introduction to this subject. For that aim we need the explicit form of evolution operator of the models with one, two, three and four atoms (at least). As to the model of one atom or two atoms it is more or less known (see [4]), while as to the case of three and four atoms it has not been given as far as we know. Since we succeeded in finding the explicit form for three and four atoms cases we report it ([5]).
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.
In this paper we consider a model of quantum computation based on n\mathrm{n}n atoms of laser-pooled and trapped linearly in a cavity and realize it as the n\mathrm{n}n atoms Tavis- Cummings Hamiltonian interacting with n\mathrm{n}n external (laser) fields.We solve the Schrodinger equation of the model in the case of n=2\mathrm{n}=2n=2 and construct the controlled NOT gate by making use of a resonance condition and rotating wave approximation associated to it.We also present an idea of the construction of three controlled NOT gates in the case of n=3\mathrm{n}=3n=3 , which gives the controlled-controlled NOT gate.
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.