Abstract
Considering two non-interacting qubits in the context of open quantum systems, it is well known that their common environment may act as an entangling agent. In a perturbative regime the influence of the environment on the system dynamics can effectively be described by a unitary and a dissipative contribution. For the two-spin Boson model with (sub-) Ohmic spectral density considered here, the particular unitary contribution (Lamb shift) easily explains the buildup of entanglement between the two qubits. Furthermore it has been argued that in the adiabatic limit, adding the so-called counterterm to the microscopic model compensates the unitary influence of the environment and, thus, inhibits the generation of entanglement. Investigating this assertion is one of the main objectives of the work presented here. Using the hierarchy of pure states (HOPS) method to numerically calculate the exact reduced dynamics, we find and explain that the degree of inhibition crucially depends on the parameter s determining the low frequency power law behavior of the spectral density J(ω)∼ωse−ω/ωc. Remarkably, we find that for resonant qubits, even in the adiabatic regime (arbitrarily large ωc), the entanglement dynamics is still influenced by an environmentally induced Hamiltonian interaction. Further, we study the model in detail and present the exact entanglement dynamics for a wide range of coupling strengths, distinguish between resonant and detuned qubits, as well as Ohmic and deep sub-Ohmic environments. Notably, we find that in all cases the asymptotic entanglement does not vanish and conjecture a linear relation between the coupling strength and the asymptotic entanglement measured by means of concurrence. Further we discuss the suitability of various perturbative master equations for obtaining approximate entanglement dynamics.
Highlights
Entanglement – this very peculiar kind of correlation, not occurring in the classical world, is known to be a fragile property with respect to environmental influences [1,2,3,4]
With the help of the exact dynamics obtained by the hierarchy of pure states (HOPS) method we examine the applicability of the quantum-optical master equation (QOME) (Born-Markov approximation and RWA) [33, 35], its variation with only a partial rotating wave approximation (PRWA) [48,49,50], the Redfield equation (RFE) [51], the very recent geometric-arithmetic master equation (GAME) (GKSL kind equation based on the RFE) [52] and the coarse-graining approach [39, 40, 53], in the context of two independent qubits coupled to a common environment with sub-Ohmic spectral density (SD)
In the context of an open two-qubit system we address the fundamental question: To what extent does the incorporation of the counterterm into the microscopic model counterbalance the effect of the Lamb shift Hamiltonian and, inhibit the environmentally induced generation of entanglement? The answer requires some prerequisites which we provided and discussed as well
Summary
Entanglement – this very peculiar kind of correlation, not occurring in the classical world, is known to be a fragile property with respect to environmental influences [1,2,3,4]. In the case of two resonant qubits (ωA = ωB), the widely used quantum-optical master equation (QOME) [32, 33] of Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) form [34] reveals that the relaxation dynamics is accompanied by an environmentally induced Hermitian coupling between the two parties [28, 33, 35,36,37]. This effective interaction explains the entanglement generation, even in the absence of a direct coupling [17, 37, 38]. The time-evolving matrix product operator (TEMPO) algorithm [42], an advancement of QUAPI, gives results consistent with ours
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