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arXiv:quant-ph/0410003 · International Journal of Geometric Methods in Modern Physics

QUANTUM DIAGONALIZATION METHOD IN THE TAVIS–CUMMINGS MODEL

KAZUYUKI FUJII, KYOKO HIGASHIDA, RYOSUKE KATO, TATSUO SUZUKI, YUKAKO WADA

Published Jun 1, 2005• 3 citations
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Abstract

To obtain the explicit form of evolution operator in the Tavis–Cummings model we must calculate the term e -itg(S+⊗a+S-⊗ a† explicitly which is very hard. In this paper we try to make the quantum matrix A ≡ S+ ⊗ a +S- ⊗ a† diagonal to calculate e -itgA and, moreover, to know a deep structure of the model. For the case of one, two and three atoms we give such a diagonalization which is the first nontrivial examples as far as we know, and reproduce the calculations of e -itgA given in quant-ph/0404034. We also give a hint to an application to non-commutative differential geometry. However, a quantum diagonalization is not unique and is affected by some ambiguity arising from the non-commutativity of operators in quantum physics. Our method may open a new point of view in mathematical or quantum physics.

DOI
10.1142/s021988780500065x

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To obtain the explicit form of evolution operator in the Tavis–Cummings model we must calculate the term e -itg(S+⊗a+S-⊗ a† explicitly which is very hard. In this paper we try to make the quantum matrix A ≡ S+ ⊗ a +S- ⊗ a† diagonal to calculate e -itgA and, moreover, to know a deep structure of the model. For the case of one, two and three atoms we give such a diagonalization which is the first nontrivial examples as far as we know, and reproduce the calculations of e -itgA given in quant-ph/0404034. We also give a hint to an application to non-commutative differential geometry. However, a quantum diagonalization is not unique and is affected by some ambiguity arising from the non-commutativity of operators in quantum physics. Our method may open a new point of view in mathematical or quantum physics.
To obtain the explicit form of evolution operator in the Tavis–Cummings model we must calculate the term e−itg(S+⊗a+S−⊗a †) explicitly which is very hard. In this paper we try to make the quantum matrix A ≡ S+ ⊗ a + S − ⊗ a † diagonal to calculate e −itgA and, moreover, to know a deep structure of the model. For the case of one, two and three atoms we give such a diagonalization which is first nontrivial examples as far as we know, and reproduce the calculations of e−itgA given in quant-ph/0404034. We also give a hint to an application to a noncommutative differential geometry. However, a quantum diagonalization is not unique and is affected by some ambiguity arising from the noncommutativity of operators in quantum physics.
非可換成分を持った行列を A=An=S+⊗a+S−⊗a†A=A_{n}=S_{+}\otimes a+S_{-}\otimes a^{\dagger}A=An​=S+​⊗a+S−​⊗a†(1.1)
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]).