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数理解析研究所講究録

Cavity QED and Quantum Computation in the Weak Coupling Regime (Dynamical Systems and Differential Geometry)

Kazuyuki Fujii, Kyoko Higashida, Ryosuke Kato, Yukako Wada

Published Dec 1, 2004• 0 citations
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Abstract

In this paper we consider a model of quantum computation based on $\mathrm{n}$ atoms of laser-pooled and trapped linearly in a cavity and realize it as the $\mathrm{n}$ atoms Tavis- Cummings Hamiltonian interacting with $\mathrm{n}$ external (laser) fields.We solve the Schrodinger equation of the model in the case of $\mathrm{n}=2$ 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 $\mathrm{n}=3$ , which gives the controlled-controlled NOT gate.

Related papers

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

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

Kazuyuki Fujii, Kyoko Higashida, Ryosuke Kato +1 more

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 Strong Coupling Regime II : Two Atoms Case

Kazuyuki Fujii

In this paper we study the behavior of (laser--cooled) m--atoms trapped in a cavity interacting with a photon $...$ Cavity QED $...$ and attempt to solve the Schr{ö}dinger equation of this model in the strong coupling regime. In the case of m = 2 we construct Bell--Schr{\" o}dinger cat states (in our terminology) and obtain with such bases some unitary transformations by making use of the rotating wave approximation under new resonance conditionscontaining the Bessel functions, which will be applied to construct important quantum logic gates in Quantum Computation. Moreover we propose in the case of m = 3 a crucial problem to solve on Quantum Computation.

arXiv:quant-ph/0307168•Jul 24, 2003•0 citations

Cavity QED and Quantum Computation in the Strong Coupling Regime

Kazuyuki Fujii

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.

arXiv:quant-ph/0305155•May 26, 2003•0 citations

Explicit Form of Solution of Two Atoms Tavis-Cummings Model

Kazuyuki Fujii, Kyoko Higashida, Ryosuke Kato +1 more

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.

arXiv:quant-ph/0403008•Mar 1, 2004•11 citations

Study on Dynamics of N Level System of Atom by Laser Fields

Kazuyuki Fujii

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.

arXiv:quant-ph/0512126•Dec 16, 2005•3 citations
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 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 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 study the behavior of (laser--cooled) m--atoms trapped in a cavity interacting with a photon ......... Cavity QED ......... and attempt to solve the Schr{ö}dinger equation of this model in the strong coupling regime. In the case of m = 2 we construct Bell--Schr{\" o}dinger cat states (in our terminology) and obtain with such bases some unitary transformations by making use of the rotating wave approximation under new resonance conditionscontaining the Bessel functions, which will be applied to construct important quantum logic gates in Quantum Computation. Moreover we propose in the case of m = 3 a crucial problem to solve on Quantum Computation.
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.