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arXiv:1604.05146 · Physical Review A

Generation of concurrence between two qubits locally coupled to a one-dimensional spin chain

Tanay Nag, Amit Dutta

Published Aug 15, 2016• 7 citations
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Abstract

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 $d$. 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 $d$ 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 $d$ 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.

DOI
10.1103/physreva.94.022316

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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≠0d\ensuremath{\ne}0d\ensuremath=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 consider the three-spin Ising model to study the effect of three-spin interacting term on bi-partitie entanglement between adjacent spins. The three-dominated disordered region has tri-partite entanglement causing a vanishingly small concurrence, while it acquires maximum value around the critical points. Considering the above model as an environment, we construct a generalized central spin model where two central spins, initially in an unentangled pure state, are coupled locally to two distinct sites of the environmental spin chain. We study the generation of mixed state entanglement between the central spins when the transverse field of the environment is kept fixed, and suddenly quenched, referring to equilibrium and non-equilibrium dynamics of the central spins, respectively. For the critical environment in the equilibrium, the concurrence shows a dip-revival structure governed by quasi-particle movement. In the non-equilibrium study, we find an initial growth of concurrence followed by a two-stage fall for the inter-phase quench which is governed by dynamic decoherence channels. The central spins are maximally entangled for a quench in the vicinity of a multicritical point, which arises due to three-spin interaction only. The concurrence becomes long-lived for an intra-phase quench, and this sustainability depends on the strength of the three-spin interaction. Therefore, the three-spin interaction indeed helps in generating bi-partite entanglement in the central spins.
Abstract We study the non-equilibrium evolution of concurrence of a Bell pair constituted of two qubits, through the measurement of Loschmidt echo (LE) under the scope of generalized central spin model. The qubits are locally coupled to a one dimensional disordered Ising spin chain. We first show that in equilibrium situation the derivative of LE is able to detect the extent of Griffiths phase that appeared in presence of disordered transverse field only. While in the non-equilibrium situation, the spin chain requires a temporal window to realize the effect of disorder. We show that within this timescale, LE falls off exponentially and this decay is maximally controlled by the initial states and coupling strength. Our detail investigation suggests that there actually exist three types of exponential decay, a Gaussian decay in ultra short time scale followed by two exponential decay in the intermediate time with two different decay exponents. The effect of the disorder starts appearing in the late time power law fall of LE where the power law exponent is strongly dependent on disorder strength and the final state but almost independent of initial states and coupling strength. This feature allows us to indicate the presence of Griffiths phase. To be precise, continuously varying critical exponent and wide distribution of relaxation time imprint their effect in LE in the late time limit where the power law fall is absent for quenching to a Griffiths phase. Here, LE vanishes following the fast exponential fall. Interestingly, for off-critical quenching LE attains a higher saturation value for increasing disorder strength, otherwise vanishes for a clean spin chain, referring to the fact that disorder prohibits the rapid decay of entanglement in long time limit. Moreover, we show that disorder is also able to destroy the light cone like nature of post quench quasi-particles as LE does not sense the singular time scales appearing for clean spin chain with qubits coupled at symmetric positions.
We study the dynamics of disentanglement of two qubits initially prepared in a Bell state and coupled at different sites to an Ising spin chain in a transverse field (ITF) playing the role of a dynamic spin environment. The initial state of the whole system is prepared in a tensor product state ${\ensuremath{\rho}}_{\mathrm{Bell}}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{\mathrm{chain}}$, where the state of the chain is taken to be given by the ground state $|G({\ensuremath{\lambda}}_{i})\ensuremath{\rangle}$ of the ITF Hamiltonian ${H}_{E}({\ensuremath{\lambda}}_{i})$ with an initial field ${\ensuremath{\lambda}}_{i}$. At time t=0+t={0}^{+}t=0+, the strength of the transverse field is suddenly quenched to a new value, ${\ensuremath{\lambda}}_{f}$, and the whole system (chain +++ qubits) undergoes a unitary dynamics generated by the total Hamiltonian ${H}_{\mathrm{Tot}}={H}_{E}({\ensuremath{\lambda}}_{f})+{H}_{I}$, where HI{H}_{I}HI​ describes a local interaction between the qubits and the spin chain. The resulting dynamics leads to a disentanglement of the qubits, which is described through Wootters' concurrence, due to their interaction with the nonequilibrium environment. The concurrence is related to the Loschmidt echo, which in turn is expressed in terms of the time-dependent covariance matrix associated with the ITF. This permits a precise numerical and analytical analysis of the disentanglement dynamics of the qubits as a function of their distance, bath properties, and quench amplitude. In particular, we emphasize the special role played by a critical initial environment.
We study the dynamical process of disentanglement of two qubits and two qutrits coupled to an Ising spin chain in a transverse field, which exhibits a quantum phase transition. We use the concurrence and negativity to quantify entanglement of two qubits and two qutrits, respectively. Explicit connections between the concurrence (negativity) and the decoherence factors are given for two initial states, the pure maximally entangled state and the mixed Werner state. We find that the concurrence and negativity decay exponentially with fourth power of time in the vicinity of the critical point of the environmental system.
We study the Loschmidt echo (LE) in a central spin model in which a central spin is globally coupled to an environment (EEE) which is subjected to a small and sudden quench at t=0t=0t=0, so that its state at t=0+t={0}^{+}t=0+ remains the same as the ground state of the initial environmental Hamiltonian before the quench; this leads to a nonequilibrium situation. This state now evolves with two Hamiltonians, the final Hamiltonian following the quench and its modified version which incorporates an additional term arising due to the coupling of the central spin to the environment. Using a generic short-time scaling of the decay rate, we establish that in the early-time limit, the rate of decay of the LE close to the quantum critical point (QCP) of EEE is independent of the quenching. We also study the temporal evolution of the LE and establish the presence of a crossover to a situation where the quenching becomes irrelevant. In the limit of large quench amplitude the nonequilibrium initial condition is found to result in a drastic increase in decoherence at large times, even far away from a QCP. These generic results are verified analytically as well as numerically, choosing EEE to be a transverse Ising chain where the transverse field is suddenly quenched.
We study the evolution of a system of two qubits, each of which interacts locally with a spin chain with nontrivial internal Hamiltonian. We present an exact solution to this problem and analyze the dependence of decoherence on the distance between the interaction sites. In the strong coupling regime we find that decoherence increases with increasing distance. In the weak coupling regime the dependence of decoherence with distance is not generic (i.e., it varies according to the initial state). Decoherence becomes independent of distance when the latter is over a saturation length lll. Numerical results for the Ising chain suggest that the saturation scale is related to the correlation length $\ensuremath{\xi}$. For strong coupling we display evidence of the existence of non-Markovian effects (such as environment-induced interactions between the qubits). As a consequence the system can undergo a quasiperiodic sequence of ``sudden deaths and revivals'' of entanglement, with a time scale related to the distance between qubits.