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arXiv:quant-ph/0203135 · Journal of Physics A Mathematical and General

Mathematical structure of Rabi oscillations in the strong coupling regime

Kazuyuki Fujii

Published Feb 12, 2003• 6 citations
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Abstract

In this paper, we generalize the Jaynes–Cummings Hamiltonian by making use of some operators based on Lie algebras su (1, 1) and su (2), and study a mathematical structure of Rabi floppings of these models in the strong coupling regime. We show that Rabi frequencies are given by matrix elements of generalized coherent operators (Fujii K 2002 Preprint quant-ph/0202081) under the rotating-wave approximation. In the first half, we make a general review of coherent operators and generalized coherent ones based on Lie algebras su (1, 1) and su (2). In the latter half, we carry out a detailed examination of Frasca (Frasca M 2001 Preprint quant-ph/0111134) and generalize his method, and moreover present some related problems. We also apply our results to the construction of controlled unitary gates in quantum computation. Lastly, we make a brief comment on application to holonomic quantum computation.

DOI
10.1088/0305-4470/36/8/309

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In this paper, we generalize the Jaynes–Cummings Hamiltonian by making use of some operators based on Lie algebras su (1, 1) and su (2), and study a mathematical structure of Rabi floppings of these models in the strong coupling regime. We show that Rabi frequencies are given by matrix elements of generalized coherent operators (Fujii K 2002 Preprint quant-ph/0202081) under the rotating-wave approximation. In the first half, we make a general review of coherent operators and generalized coherent ones based on Lie algebras su (1, 1) and su (2). In the latter half, we carry out a detailed examination of Frasca (Frasca M 2001 Preprint quant-ph/0111134) and generalize his method, and moreover present some related problems. We also apply our results to the construction of controlled unitary gates in quantum computation. Lastly, we make a brief comment on application to holonomic quantum computation.
In this paper we generalize so--called Jaynes--Cummings Hamiltonian by making use of some operators based on Lie algebras su(1,1) and su(2), and study a mathematical structure of Rabi floppings of these models in the strong coupling regime. We show that Rabi frequencies are given by matrix elements of generalized coherent operators under the rotating--wave approximation. In the first half we make a general review of coherent operators and generalized coherent ones based on Lie algebras su(1,1) and su(2). In the latter half we carry out a detailed examination of Frasca (quant--ph/0111134) and generalize his method, and moreover present some related problems. Lastly we make a brief comment on application to Holonomic Quantum Computation.
In this paper we consider some (generalized) models which deal with n--level atom interacting with a single radiation mode and study a general structure of Rabi floppings of these generalized ones in the strong coupling regime. To solve these models we introduce multi cat states of Schr{\" o}dinger in our terminology and extend the result in (quant--ph/0203135) which Rabi frequencies are given by matrix elements of generalized coherent operators (quant--ph/0202081) under the rotating--wave approximation. Our work is a full generalization of Frasca (quant--ph/0111134) and Fujii (quant--ph/0203135). In last we make a brief comment on an application to Quantum Computation on the space of qudits.
The effects of counter-rotating terms (CRTs) on Rabi splitting and the dynamic evolution of atomic population in the Jaynes—Cummings model are studied with a coherent-state approach. When the coupling strength increases, the Rabi splitting becomes of multi-Rabi frequencies for the initial state of an excited atom in a vacuum field, and the collapses and revivals gradually disappear, and then reappear with quite good periodicity. Without the rotating-wave approximation (RWA), the initial excited state contains many eigenstates rather than two eigenstates under the RWA, which results in the multi-peak emission spectrum. An analytical approximate solution for the strong coupling regime is obtained, which gives a new oscillation frequency and explains the recovery of collapses and revivals due to the equal energy spacing.
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.
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.
The Q function in the undamped Jaynes-Cummings model for a coherent initial state in the rotating-wave approximation is investigated. By applying the method of steepest decent and saddle-point integration in the limit of high photon numbers, closed analytic expressions for the Q function at different times are given, from which analytic expressions for the Rabi frequency, revival times, envelope shapes of the inversion, beat frequencies, etc. are derived. As Eiselt and one of us have shown [J. Eiselt and H. Risken, Phys. Rev. A 43, 346 (1991)], the Q function first splits into two peaks which counterrotate in the complex plane and then collide on the opposite side of the plane. This behavior of the Q function leads to an interpretation of the collapses and revivals of the Rabi oscillations and a better understanding of expressions for the Rabi frequency, revival times, envelope shapes of the inversion, beat frequencies, etc. The results are generalized to N-photon processes; in particular, the cases N=2 and 4 are treated in some detail.