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arXiv:1609.05668 · Journal of Physics Conference Series

Energy spectra of a spin-½ XY spin molecule interacting with a single mode field cavity: Numerical study

H Tonchev, Alexander A. Donkov, Hassan Chamati

Published Oct 1, 2016• 1 citation
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Abstract

In a previous paper [J.Phys.: Conf. Ser. 682 (2016) 012032] we studied analytically the energy spectra of a finite-size spin ½ XY chain (molecule) coupled at an arbitrary spin site to a single mode of an electromagnetic field via the Jaynes-Cummings model. We considered spin rings and open spin molecules with up to 4 spins and an interaction restricted to nearest-neighbours. Here we extend our investigation, addressing numerically the energy spectra of molecules of up to 10 spins with nearest-neighbour or long- range interaction. Furthermore we analyze the behaviour of an invariant operator, constructed by combining the magnetization of the spin-chain and the total number of photons in the system. We found a strong dependence on the number (even or odd) of sites in the molecules. This study is aimed at finding the appropriate combination of the physical parameters that could make the system suitable for use in quantum computations.

DOI
10.1088/1742-6596/764/1/012017

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Energy spectra of a spin-12\frac{1}{2}21​ XY spin molecule interacting with a single mode field cavity
We report results from our continued investigations on a finite-size chain (molecule) of L spin- spins with XY spin exchange interaction coupled at an arbitrary spin site to a single mode of an electromagnetic field via the Jaynes-Cummings model. Here we study the energy spectra in the case of an arbitrary photon fill number n , as compared to the n = 1 limit in our previous work [J. Phys.: Conf. Ser. 682 (2016) 012032] and [J. Phys.: Conf. Ser. 764 (2016) 012017]. By building the appropriate combinations of basis product states involving the photon as well as the up down spins it is possible to simplify the systems that are necessary to diagonalize from n × 2 L to at most 2 L . This results are in tune with those reported in the recent paper by N. Wu [Phys. Rev. B 97 (2018) 014301], and may prove to be useful for the quantum computing aficionados.
In this work we use the fundamental in quantum optics Jaynes-Cummings model to study the response of spin 1/2chain to a single mode of a laser light falling on one of the spins, a focused interaction model between the light and the spin chain. For the spin-spin interaction along the chain we use the XY model. We report here the exact analytical results, obtained with the help of a computer algebra system, for the energy spectrum in this model for chains of up to 4 spins with nearest neighbors interactions, either for open or cyclic chain configurations. Varying the sign and magnitude of the spin exchange coupling relative to the light-spin interaction we have investigated both cases of ferromagnetic or antiferromagnetic spin chains.
This paper describes two cases of interaction between a quantized electromagnetic field and two different XY spin molecules; one with spins ½, and the other with spins 1. Both interact with a quantized electromagnetic field, with one of the spins in the chain interacting with the electromagnetic field. The interaction between the field mode and the spin chain with spins 1 is described by the one- and two-photon Jaynes-Cummings model (JC model). On the other hand, the interaction between the spins ½ and the electromagnetic field is described only by the one-photon Jaynes-Cummings model. Analytical and numerical calculations were made for the case of a different number of photons in the field mode, a different number of spins, and a different position of spin, interacting with the electromagnetic field. The invariant and block structures of such a chain are shown with a comparison made between the evolution of the magnetic moment and the number of photons in both cases.
The new class of exactly solvable models of one-dimensional spin chains is found. Interactions of spins with crystalline electric fields (CEF) and spin-spin coupling between neighboring sites are taken into account for those models. Spin-spin coupling includes higher-order spin operators (the Stevens operators) and describes the interaction between z th components of the spins. We show that essential parts of the Hamiltonians of these models can be written in terms of exactly solvable transverse field Ising model (TFIM), in which certain combinations of the parameters of spin-spin interactions play the role of the TFIM exchange constant, and components of the CEFs play the role of external transverse magnetic field of the TFIM. The obtained results are valid for any values of spin (or total moments for rare-earth ions) and for the wide class of CEFs. The requirement for the exact solvability is that the space of single spin states should be split by the CEF into the subspaces with dimensions ≤ 2 (i.e., the only singlets and doublets are allowed). Cases of integer and half-integer spins are considered. For finite-size chains with integer spins and open ends we show numerically that energy spectra always contain the states whose energies do not depend on the CEF (the magnetic field for the TFIM). The energies (per spin) of these states decay as L − 1 , where L is the chain length. This behavior qualitatively distinguishes these states from known Majorana edge states and strong zero modes, where the energy decreases exponentially in L .
The authors have investigated the spectrum of the emitted light by a single atom interacting with a single mode of the radiation in an ideal cavity via the intensity-dependent coupling (the Jaynes-Cummings model with intensity-dependent coupling see B. Buck, Phys. Lett., vol.A81, p.132 (1981)). From the authors' results it follows that in the resonant case the two-peaked structure of the spectrum associated with the vacuum-field Rabi splitting is preserved for any initial state of the radiation field (e.g. number state, thermal state, coherent state and the squeezed vacuum) with an arbitrary average number of photons.
In this article, the interaction of an arbitrary number of quantum dots, behaving as artificial molecules, with different energy levels and multi-mode electromagnetic field is studied. We make the assumption that each quantum dot can be represented as an atom with zero kinetic energy, and that all excitonic effects except dipole-dipole interactions may be disregarded. We use Jaynes-Cummings-Paul model with applications to quantum systems based on a time-dependent Hamiltonian and entangled states. We obtain a system of equations describing the interaction and present a method to solve the equations analytically for a single mode field within the Rotating-Wave Approximation. As an example of the applicability of this approach, we solve the system of two two-level quantum dots in a lossless cavity with two modes of electromagnetic field. We furthermore study the evolution of entanglement by defining and computing the concurrency.