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Explicit form of the evolution operator for the Tavis-Cummings model and quantum diagonalization method (Dynamical Systems and Differential Geometry)

達夫 鈴木

Published Dec 1, 2004• 0 citations
OpenAlex

Abstract

非可換成分を持った行列を $A=A_{n}=S_{+}\otimes a+S_{-}\otimes a^{\dagger}$(1.1)

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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)
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.
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.