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arXiv:quant-ph/0501022 · Physical Review A

Stimulated wave of polarization in a one-dimensional Ising chain

Jae‐Seung Lee, Anatoly K. Khitrin

Published Jun 28, 2005• 32 citations
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Abstract

It is demonstrated that in a one-dimensional Ising chain with nearest-neighbor interactions, irradiated by a weak resonant transverse field, a stimulated wave of flipped spins can be triggered by a flip of a single spin. This analytically solvable model illustrates mechanisms of quantum amplification and quantum measurement.
DOI
10.1103/physreva.71.062338

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Stimulated wave of polarization, triggered by a flip of a single spin, presents a simple model of quantum amplification. Previously, it has been found that such a wave can be excited in a one-dimensional Ising chain with nearest-neighbor interactions, irradiated by a weak resonant transverse field. Here we explore models with more realistic Hamiltonians, in particular, with natural dipole-dipole interactions. Results of simulations for one-dimensional spin chains and rings with up to nine spins are presented.
The evolution of entanglement in a one-dimensional Ising chain is numerically studied under various initial conditions. We analyze two problems concerning the dynamics of entanglement: (i) generation of the entanglement from the pseudopure separable state and (ii) transportation of the entanglement from one end of the chain to the other. The model investigated is a one-dimensional Ising spin-1/2 chain with nearest-neighbor interactions placed in an external magnetic field and irradiated by a weak resonant transverse field. The possibility of selective initialization of partially entangled states is considered. It was shown that, in spite of the use of a model with direct interactions between the nearest neighbors, entanglement between remote spins is generated.
Controlled measurement processes with high fidelity and robustness will be of substantial interest to fundamental quantum studies as well as quantum-information processing. This work proposes a simple scheme to realize controlled measurements of the state of a single spin, via controlled quantum signal amplification in a one-dimensional spin-chain model. It is shown that, by adiabatically moving an external field applied to a spin chain that is also irradiated with a transverse driving field, a robust spin amplifier may be realized with little dispersion. Our results may find applications in spin-based quantum-information processing and in controlled growth of ``Schr\"odinger cat'' states.
We study the effect of antiferromagnetic interactions on the single spin-flip Glauber dynamics of two different one-dimensional Ising models with spin $\ifmmode\pm\else\textpm\fi{}1$. The first model is an Ising chain with antiferromagnetic exchange interaction limited to nearest neighbors and subject to an oscillating magnetic field. The system of master equations describing the time evolution of sublattice magnetizations can easily be solved within a linear field approximation and a long time limit. Resonant behavior of the magnetization as a function of temperature (stochastic resonance) is found, at low frequency, only when spins on opposite sublattices are uncompensated owing to different gyromagnetic factors (i.e., in the presence of a ferrimagnetic short range order). The second model is the axial next-nearest-neighbor Ising (ANNNI) chain, where an antiferromagnetic exchange between next-nearest neighbors is assumed to compete with a nearest-neighbor exchange interaction of either sign. The long time response of the model to a weak, oscillating magnetic field is investigated in the framework of a decoupling approximation for three-spin correlation functions, which is required to close the system of master equations. The calculation, within such an approximate theoretical scheme, of the dynamic critical exponent zzz, defined as $1∕\ensuremath{\tau}\ensuremath{\approx}{(1∕\ensuremath{\xi})}^{z}$ (where $\ensuremath{\tau}$ is the longest relaxation time and $\ensuremath{\xi}$ is the correlation length of the chain), suggests that the T=0T=0T=0 single spin-flip Glauber dynamics of the ANNNI chain is in a different universality class than that of the unfrustrated Ising chain.
Most of the existing dynamical studies in one dimension on magnetic insulators have considered the simplest spin models with nearest-neighbor interactions. In real systems, however, it is possible that longer range interactions are not entirely negligible. It is expected that the inclusion of next-nearest-neighbor interactions between spins in one-dimensional spin models will introduce a multitude of new frequencies in addition to the ones already present in the dynamics that arises due to nearest-neighbor interactions. We first present an exact solution for the dynamical xx-spin-pair correlations in an Ising chain with both nearest- and next-nearest-neighbor interactions to confirm our expectation. We next show, via an approximate analytical calculation, that the dynamical zz-spin-pair correlations in the next-nearest-neighbor transverse Ising chain when plotted as a function of time is noticeably different with respect to the exactly solvable nearest-neighbor transverse Ising chain at T\ensuremath{\rightarrow}\ensuremath{\infty} when the next-nearest-neighbor interaction is \ensuremath{\gtrsim}1/2 of the magnitude of the nearest-neighbor interaction. The effects could be fairly subtle in the time domain representation and in the spectral function when these additional interactions are weak (i.e., 1/2 of the nearest-neighbor interaction magnitude). The general conclusions reached in this work are expected to be valid for other simple quantum spin models such as the XY and XXZ models in one dimension. \textcopyright{} 1996 The American Physical Society.