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arXiv:1306.1610 · The Journal of Chemical Physics

Entanglement dynamics of two qubits coupled individually to Ohmic baths

Liwei Duan, Hui Wang, Qing-Hu Chen, Yang Zhao

Published Jul 26, 2013• 25 citations
arXivHTMLPDF

Abstract

Developed originally for the Holstein polaron, the Davydov D1 ansatz is an efficient, yet extremely accurate trial state for time-dependent variation of the spin-boson model [N. Wu, L. Duan, X. Li, and Y. Zhao, J. Chem. Phys. 138, 084111 (2013)]. In this work, the Dirac-Frenkel time-dependent variational procedure utilizing the Davydov D1 ansatz is implemented to study entanglement dynamics of two qubits under the influence of two independent baths. The Ohmic spectral density is used without the Born-Markov approximation or the rotating-wave approximation. In the strong coupling regime finite-time disentanglement is always found to exist, while at the intermediate coupling regime, the entanglement dynamics calculated by Davydov D1 ansatz displays oscillatory behavior in addition to entanglement disappearance and revival.

DOI
10.1063/1.4816122

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Developed originally for the Holstein polaron, the Davydov D1 ansatz is an efficient, yet extremely accurate trial state for time-dependent variation of the spin-boson model [N. Wu, L. Duan, X. Li, and Y. Zhao, J. Chem. Phys. 138, 084111 (2013)]. In this work, the Dirac-Frenkel time-dependent variational procedure utilizing the Davydov D1 ansatz is implemented to study entanglement dynamics of two qubits under the influence of two independent baths. The Ohmic spectral density is used without the Born-Markov approximation or the rotating-wave approximation. In the strong coupling regime finite-time disentanglement is always found to exist, while at the intermediate coupling regime, the entanglement dynamics calculated by Davydov D1 ansatz displays oscillatory behavior in addition to entanglement disappearance and revival.
Dynamics of the sub-Ohmic spin-boson model is examined using three numerical approaches, namely the Dirac-Frenkel time-dependent variation with the Davydov D(1) ansatz, the adaptive time-dependent density matrix renormalization group method within the representation of orthogonal polynomials, and a perturbative approach based on a unitary transformation. In order to probe the validity regimes of the three approaches, we study the dynamics of a qubit coupled to a bosonic bath with and without a local field. Comparison of the up-state population evolution shows that the three approaches are in agreement in the weak-coupling regime but exhibit marked differences when the coupling strength is large. The Davydov D(1) ansatz and the time-dependent density matrix renormalization group can both be reliably employed in the weak-coupling regime, while the former is also valid in the strong-coupling regime as judged by how faithfully the trial state follows the Schrödinger equation. We further explore the bipartite entanglement dynamics between two qubits coupled with individual bosonic baths which reveals entanglement sudden death and revival.
Dynamics of the two-spin spin-boson model in the presence of Ohmic and sub-Ohmic baths is investigated by employing a multitude of the Davydov D1 trial states, also known as the multi-D1 Ansatz. Its accuracy in dynamics simulations of the two-spin SBM is improved significantly over the single D1 Ansatz, especially in the weak to moderately strong coupling regime. Validity of the multi-D1 Ansatz for various coupling strengths is also systematically examined by making use of the deviation vector which quantifies how faithfully the trial state obeys the Schrödinger equation. The time evolution of population difference and entanglement has been studied for various initial conditions and coupling strengths. Careful comparisons are carried out between our approach and three other methods, i.e., the time-dependent numerical renormalization group (TD-NRG) approach, the Bloch-Redfield theory, and a method based on a variational master equation. For strong coupling, the multi-D1 trial state yields consistent results as the TD-NRG approach in the Ohmic regime while the two disagree in the sub-Ohmic regime, where the multi-D1 trial state is shown to be more accurate. For weak coupling, the multi-D1 trial state agrees with the two master-equation methods in the presence of both Ohmic and sub-Ohmic baths, but shows considerable differences with the TD-NRG approach in the presence of a sub-Ohmic bath, calling into question the validity of the TD-NRG results at long times in the literature.
The Dirac-Frenkel time-dependent variation is employed to probe the dynamics of the zero temperature sub-Ohmic spin-boson model with strong friction utilizing the Davydov D1 ansatz. It is shown that initial conditions of the phonon bath have considerable influence on the dynamics. Counterintuitively, even in the very strong coupling regime, quantum coherence features still manage to survive under the polarized bath initial condition, while such features are absent under the factorized bath initial condition. In addition, a coherent-incoherent transition is found at a critical coupling strength α ≈ 0.1 for s = 0.25 under the factorized bath initial condition. We quantify how faithfully our ansatz follows the Schrödinger equation, finding that the time-dependent variational approach is robust for strong dissipation and deep sub-Ohmic baths (s ≪ 1).
The Davydov D1 ansatz, which assigns individual bosonic trajectories to each spin state, is an efficient, yet extremely accurate trial state for time-dependent variation of the sub-Ohmic spin-boson model [N. Wu, L. Duan, X. Li, and Y. Zhao, J. Chem. Phys. 138, 084111 (2013)]. A surface hopping algorithm is developed employing the Davydov D1 ansatz to study the spin dynamics with a sub-Ohmic bosonic bath. The algorithm takes into account both coherent and incoherent dynamics of the population evolution in a unified manner, and compared with semiclassical surface hopping algorithms, hopping rates calculated in this work follow more closely the Marcus formula.
Dynamics of the sub-Ohmic spin-boson model is investigated by employing a multitude of the Davydov D1 trial states, also known as the multi-D1 Ansatz. Accuracy in dynamics simulations is improved significantly over the single D1 Ansatz, especially in the weak system-bath coupling regime. The reliability of the multi-D1 Ansatz for various coupling strengths and initial conditions is also systematically examined, with results compared closely with those of the hierarchy equations of motion and the path integral Monte Carlo approaches. In addition, a coherent-incoherent phase crossover in the nonequilibrium dynamics is studied through the multi-D1 Ansatz. The phase diagram is obtained with a critical point sc = 0.4. For sc < s < 1, the coherent-to-incoherent crossover occurs at a certain coupling strength, while the coherent state recurs at a much larger coupling strength. For s < sc, only the coherent phase exists.
The dynamics of the (sub-)Ohmic spin-boson model under various bath initial conditions is investigated by employing the Dirac–Frenkel time-dependent variational approach with the multiple Davydov D1 Ansatz in the interaction picture. The validity of our approach is carefully checked by comparing the results with those of the hierarchy equations of motion method. By analyzing the features of nonequilibrium dynamics, we identify the phase diagrams for different bath initial conditions. We find that for the spectral exponent s < sc, there exists a transition from coherent to quasicoherent dynamics with increasing coupling strengths. For sc < s ≤ 1, the coherent to incoherent crossover occurs at a certain coupling strength and the quasicoherent dynamics emerges at much larger couplings. The initial preparation of the bath has a considerable influence on the dynamics.