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arXiv:0904.1059 · Physical Review A

Entanglement production due to quench dynamics of an anisotropic X Y chain in a transverse field

K. Sengupta, Diptiman Sen

Published Sep 2, 2009• 45 citations
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Abstract

We compute concurrence and negativity as measures of two-spin entanglement generated by a power-law quench (characterized by a rate ${\ensuremath{\tau}}^{\ensuremath{-}1}$ and an exponent $\ensuremath{\alpha}$) which takes an anisotropic $XY$ chain in a transverse field through a quantum critical point (QCP). We show that only spins separated by an even number of lattice spacings get entangled in such a process. Moreover, there is a critical rate of quench, ${\ensuremath{\tau}}_{c}^{\ensuremath{-}1}$, above which no two-spin entanglement is generated; the entire entanglement is multipartite. The ratio of the entanglements between consecutive even neighbors can be tuned by changing the quench rate. We also show that for large $\ensuremath{\tau}$, the concurrence (negativity) scales as $\sqrt{\ensuremath{\alpha}/\ensuremath{\tau}}$ $(\ensuremath{\alpha}/\ensuremath{\tau})$, and we relate this scaling behavior to defect production by the quench through a QCP.

DOI
10.1103/physreva.80.032304

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We compute concurrence and negativity as measures of two-spin entanglement generated by a power-law quench (characterized by a rate \ensuremathτ\ensuremath−1{\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremathτ\ensuremath−1 and an exponent \ensuremathα\ensuremath{\alpha}\ensuremathα) which takes an anisotropic XYXYXY chain in a transverse field through a quantum critical point (QCP). We show that only spins separated by an even number of lattice spacings get entangled in such a process. Moreover, there is a critical rate of quench, \ensuremathτc\ensuremath−1{\ensuremath{\tau}}_{c}^{\ensuremath{-}1}\ensuremathτc\ensuremath−1​, above which no two-spin entanglement is generated; the entire entanglement is multipartite. The ratio of the entanglements between consecutive even neighbors can be tuned by changing the quench rate. We also show that for large \ensuremathτ\ensuremath{\tau}\ensuremathτ, the concurrence (negativity) scales as \ensuremathα/\ensuremathτ\sqrt{\ensuremath{\alpha}/\ensuremath{\tau}}\ensuremathα/\ensuremathτ​ (\ensuremathα/\ensuremathτ)(\ensuremath{\alpha}/\ensuremath{\tau})(\ensuremathα/\ensuremathτ), and we relate this scaling behavior to defect production by the quench through a QCP.
We study the impact of noise on the dynamics of entanglement in the transverse-field Ising chain, with the field quenched linearly across one or both of the quantum critical points of the model. Taking concurrence as a measure of entanglement, we find that a quench generates entanglement between nearest- and next-nearest-neighbor spins, with noise reducing the amount of entanglement. Focusing on the next-nearest-neighbor concurrence, known to exhibit Kibble-Zurek scaling with the square root of the quench rate in the noiseless case, we find a different result when noise is present: The concurrence now scales logarithmically with the quench rate, with a noise-dependent amplitude. This is also different from the “anti-Kibble-Zurek” scaling of defect density with quench rate when noise is present, suggesting that noisy entanglement generation is largely independent of the rate of defect formation. Intriguingly, the critical timescale beyond which no entanglement is produced by a noisy quench scales as a power law with the strength of noise, with the same exponent as that which governs the optimal quench time for which defect formation is at a minimum in a standard quantum annealing scheme.
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We study the quantum dynamics of a one-dimensional spin-12\frac{1}{2}21​ anisotropic XYXYXY model in a transverse field when the transverse field or the anisotropic interaction is quenched at a slow but uniform rate. The two quenching schemes are called transverse and anisotropic quenching, respectively. Our emphasis in this paper is on the anisotropic quenching scheme, and we compare the results with those of the other scheme. In the process of anisotropic quenching, the system crosses all the quantum critical lines of the phase diagram where the relaxation time diverges. The evolution is nonadiabatic in the time interval when the parameters are close to their critical values, and is adiabatic otherwise. The density of defects produced due to nonadiabatic transitions is calculated by mapping the many-particle system to an equivalent Landau-Zener problem and is generally found to vary as $1∕\sqrt{\ensuremath{\tau}}$, where $\ensuremath{\tau}$ is the characteristic time scale of quenching, a scenario that supports the Kibble-Zurek mechanism. Interestingly, in the case of anisotropic quenching, there exists an additional nonadiabatic transition, in comparison to the transverse quenching case, with the corresponding probability peaking at an incommensurate value of the wave vector. In the special case in which the system passes through a multicritical point, the defect density is found to vary as $1∕{\ensuremath{\tau}}^{1∕6}$. The von Neumann entropy of the final state is shown to maximize at a quenching rate around which the ordering of the final state changes from antiferromagnetic to ferromagnetic.
We consider a generalized central spin model, consisting of two central qubits and an environmental spin chain (with periodic boundary condition) to which these central qubits are locally and weakly connected either at the same site or at two different sites separated by a distance ddd. Our purpose is to study the subsequent temporal generation of entanglement, quantified by concurrence, when initially the qubits are in an unentangled state. In the equilibrium situation, we show that the concurrence survives for a larger value of ddd when the environmental spin chain is critical. Importantly, a common feature observed both in the equilibrium and the nonequilibrium situations while the latter is created by a sudden but global change of the environmental transverse field is that the two qubits become maximally entangled for the critical quenching. Following a nonequilibrium evolution of the spin chain, our study for $d\ensuremath{\ne}0$ indicates that there exists a threshold time above which concurrence attains a finite value. Additionally, we show that the number of independent decohering channels (DCs) is determined by ddd as well as the local difference of the transverse field of the two underlying Hamiltonians governing the time evolution; the concurrence can be enhanced by a higher number of independent channels. The qualitatively similar behavior displayed by the concurrence for critical and off-critical quenches, as reported here, is characterized by analyzing the nonequilibrium evolution of these channels. The concurrence is maximum when the decoherence factor or the echo associated with the most rapidly DC decays to zero; on the contrary, the condition when the concurrence vanishes is determined nontrivially by the associated decay of one of the intermediate DCs. Analyzing the reduced density of a single qubit, we also explain the observation that the dephasing rate is always slower than the unentanglement rate. We further establish that the maximally and minimally decohering channels show singular behavior which can be explained by invoking upon a quasiparticle picture.
We study the dynamics of a transverse-field XY\mathrm{XY}XY chain driven across quantum critical points by noisy control fields. We characterize the defect density as a function of the quench time and the noise strength and demonstrate that the defect productions for three quench protocols with different scaling exponents exhibit the anti-Kibble-Zurek behavior, whereby slower driving results in more defects. The protocols are quenching through the boundary line between paramagnetic and ferromagnetic phases, quenching across the isolated multicritical point, and quenching along the gapless line, respectively. We also show that the optimal quench time to minimize defects scales as a universal power law of the noise strength in all three cases. Furthermore, by using quantum simulation of the quench dynamics in the spin system with well-designed Landau-Zener crossings in pseudomomentum space, we propose an experimentally feasible scheme to test the predicted anti-Kibble-Zurek behavior of this noisy transverse-field XY\mathrm{XY}XY chain with two-level systems under controllable fluctuations.
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