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Analyses of the transmission of the disorder from a disturbed environment to a spin chain

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Abstract

We study spin chains submitted to disturbed kick trains described by classical dynamical processes. The spin chains are described by Heisenberg and Ising models. We consider decoherence, entanglement and relaxation processes induced by the kick irregularity in the multipartite system (the spin chain). We show that the different couplings transmit the disorder along the chain differently and also to each spin density matrix with different efficiencies. In order to analyze and to interpret the observed effects, we use a semi-classical analysis across the Husimi distribution. It consists to consider the classical spin orientation movements. A possibility of conserving the order into the spin chain is finally analyzed.

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Notes

  1. In this analysis, a spin is viewed as a classical magnetic moment vector, inducing a local magnetic field \(\mathbf {B}_{loc} \propto \langle \mathbf {S} \rangle = (\rho \mathbf {S})\) (where \(\mathbf {S}\) are the spin operators and \(\rho \) is spin density matrix) which is felt by their neighbors. In this framework, we talk about the (classical) spin orientation in place of the (quantum) spin state (a quantum spin state \(\alpha |\uparrow \rangle + \beta |\downarrow \rangle \) being equivalent to the classical spin orientation \(\theta = 2 \arctan \left| \frac{\beta }{\alpha } \right| \) and \(\varphi = \arg \beta - \arg \alpha \), or in other words, we identify the Bloch sphere (the space of the spin normalized states without global phase) characterized by \(S^3 = \{ z \in \mathbb {C}^2, ||z|| = 1\}\), with a sphere of classical vector directions).

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Authors and Affiliations

  1. CNRS UMR 6213, Institut UTINAM, Université de Bourgogne-Franche-Comté, Observatoire de Besançon, 41bis Avenue de l’Observatoire, BP 1615, 25010, Besançon Cedex, France

    Lucile Aubourg & David Viennot

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  1. Lucile Aubourg
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  2. David Viennot
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Correspondence to Lucile Aubourg.

Appendix: Discussion about the size of the chain

Appendix: Discussion about the size of the chain

We have seen the evolution of a ten spin chain coupled by the Heisenberg, Ising-Z or Ising-X interaction and kicked with different initial dispersions. But now, an useful question is to know the influence of the spin number into the chain. To analyze this, 6, 8, 10 and 12 spins have been taken with a medium interaction and with a medium initial dispersion parameters. The coherence is represented for these conditions and for the different couplings Figs. 26, 27 and 28. We see on these figures that there is nearly no variation with the spin number.

Fig. 26
figure 26

Evolution of the coherence with various spin numbers into the chain. The spins are coupled to their neighbors by an Heisenberg interaction. Each spin is submitted to a drift classical kick bath. The graphics associated with \(\rho _\mathrm{tot}\) are for the average chain and ones with \(\rho _4\) are about the fourth spin of the chain

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Fig. 27
figure 27

Evolution of the coherence with various spin numbers into the chain. The spins are coupled to their neighbors by an Ising-Z interaction. Each spin is submitted to a drift classical kick bath. The graphics associated with \(\rho _\mathrm{tot}\) are for the average chain and ones with \(\rho _4\) are about the fourth spin of the chain

Full size image
Fig. 28
figure 28

Evolution of the coherence with various spin numbers into the chain. The spins are coupled to their neighbors by an Ising-X interaction. Each spin is submitted to a drift classical kick bath. The graphics associated with \(\rho _\mathrm{tot}\) are for the average chain and ones with \(\rho _4\) are about the fourth spin of the chain

Full size image

We can suppose that there is no modification of the spin behavior with the number of spins. So, the analyses will be the same with a large number of spins in the chain.

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Aubourg, L., Viennot, D. Analyses of the transmission of the disorder from a disturbed environment to a spin chain. Quantum Inf Process 14, 1117–1150 (2015). https://doi.org/10.1007/s11128-014-0885-9

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