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arXiv:1005.1823

Perturbation theory based on Darboux transformation on one-dimensional Dirac equation in the case of cavity quantum electrodynamics and quantum transistor

Agung Trisetyarso

Published May 11, 2010• 0 citations
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Abstract

We present the recent works \cite{trisetyarso2011} on the application of Darboux transformation on one-dimensional Dirac equation related to the field of Quantum Information and Computation (QIC). The representation of physical system in one-dimensional equation and its transformation due to the Bagrov, Baldiotti, Gitman, and Shamshutdinova (BBGS)-Darboux transformation showing the possibility admitting the concept of relativity and the trade-off of concurrent condition of quantum and classical physics play into the area of QIC. The applications in cavity quantum electrodynamics and on the proposal of quantum transistor are presented.

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We present the recent works \cite{trisetyarso2011} on the application of Darboux transformation on one-dimensional Dirac equation related to the field of Quantum Information and Computation (QIC). The representation of physical system in one-dimensional equation and its transformation due to the Bagrov, Baldiotti, Gitman, and Shamshutdinova (BBGS)-Darboux transformation showing the possibility admitting the concept of relativity and the trade-off of concurrent condition of quantum and classical physics play into the area of QIC. The applications in cavity quantum electrodynamics and on the proposal of quantum transistor are presented.
Controlling the time evolution of the population of two states in cavity quantum electrodynamics is necessary by tuning the modified Rabi frequency in which the extra classical effect of electromagnetic field is taken into account. The theoretical explanation underlying the perturbation of potential on spatial regime of bloch sphere is by the use of Bagrov–Baldiotti–Gitman–Shamshutdinova–Darboux transformations [Bagrov et al., “Darboux transformation for two-level system,” Ann. Phys. 14, 390 (2005)] on the electromagnetic field potential in one-dimensional stationary Dirac model in which the Pauli matrices are the central parameters for controlling the collapse and revival of the Rabi oscillations. It is shown that by choosing σ1 in the transformation generates the parabolic potential causing the total collapse of oscillations, while {σ2,σ3} yield the harmonic oscillator potentials ensuring the coherence of qubits.
In this article the Dirac equation is used as a guideline to see the historical emergence of the concept of quanta, associated with the quantum field. In P. Jordan’s approach, electrons as quanta result from the quantization of a classical field described by the Dirac equation. With this quantization procedure – also used for the electromagnetic field – the concept of quanta becomes a central piece in the applications of quantum electrodynamics. This does not seem to avoid the apparent impossibility of using the concept of quanta – and with it the common interpretation of quantum fields – when interacting fields are considered together as one complete system. In this article it is defended that the type of analysis that leads to so drastic conclusions is avoidable if a clear distinction is made between the mathematical framework of the theory and the particular physical models (used in the empirical corroboration of the theory) that are constructed using the theory. When dealing with models there really is no system of complete interacting fields, and what we have is a description of the interactions between distinct fields. In this situation the concept of quanta is central in the description of interacting fields, the Fock space being the natural mathematical structure that permits maintaining the quanta concept when considering the interaction between fields.
We propose a mathematical model of \textit{quantum} transistor in which bandgap engineering corresponds to the tuning of Dirac potential in the complex four-vector form. The transistor consists of nnn-relativistic spin qubits moving in \textit{classical} external electromagnetic fields. It is shown that the tuning of the direction of the external electromagnetic fields generates perturbation on the potential temporally and spatially, determining the type of quantum logic gates. The theory underlying of this scheme is on the proposal of the intertwining operator for Darboux transfomations on one-dimensional Dirac equation amalgamating the \textit{vector-quantum gates duality} of Pauli matrices. Simultaneous transformation of qubit and energy can be accomplished by setting the {control, cyclic}\{\textit{control, cyclic}\}{control, cyclic}-operators attached on the coupling between one-qubit quantum gate: the chose of \textit{cyclic}-operator swaps the qubit and energy simultaneously, while \textit{control}-operator ensures the energy conservation.
In this article the Dirac equation is used as a guideline to the historical emergence of the concept of quanta, associated with the quantum field. In Pascual Jordan’s approach, electrons as quanta result from the quantization of a classical field described by the Dirac equation. With this quantization procedure – also used for the electromagnetic field – the concept of quanta becomes a central piece in quantum electrodynamics. This does not seem to avoid the apparent impossibility of using the concept of quanta when interacting fields are considered together as a closed system. In this article it is defended that the type of analysis that leads to so drastic conclusions is avoidable if we look beyond the mathematical structure of the theory and take into account the physical ideas inscribed in this mathematical structure. In this case we see that in quantum electrodynamics we are not considering a closed system of interacting fields, what we have is a description of the interactions between distinct fields. In this situation the concept of quanta is central, the Fock space being the natural mathematical structure that permits maintaining the concept of quanta when considering the interaction between the fields.
In Session 8, we demonstrated that circuit theory can be applied to the Schrödinger equation, a basic equation in quantum mechanics. In this session, we attempt to apply circuit theory to the Dirac equation using the Riccati differential equation and demonstrate that physical phenomena such as spin 1/2 can be explained on the basis of circuit theory. To this end, we obtain the one-dimensional Dirac equation for the complex equivalent circuit for a physical phenomenon. We show that the Dirac equation can be regarded as a type of low-energy effective field theory in quantum theory and that circuit theory can be applied to the Dirac equation.
This paper examines the Foldy-Wouthuysen and Feynman-Gell-Mann representations of the Dirac equation. The analysis is conducted for electrons and positrons interacting with electromagnetic fields. Versions of quantum electrodynamics are considered both within the scope of perturbation theory and in the nonperturbative case with strong electromagnetic fields. Mathematical artifacts that contradicting the physical premises of the theory are identified in the studied representations of the Dirac equation. These mathematical paradoxes are resolved if the theory only employs amplitude states (real and virtual) with positive energies.