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We derive a set of hierarchical equations for qubits interacting with a Lorentz-broadened cavity mode at zero temperature, without using the rotating-wave, Born, and Markovian approximations. We use this exact method to reexamine the entanglement dynamics of two qubits interacting with a common bath, which was previously solved only under the rotating-wave and single-excitation approximations. With the exact hierarchy equation method used here, we observe significant differences in the resulting physics, compared to the previous results with various approximations. Double excitations due to counter-rotating-wave terms are also found to have remarkable effects on the dynamics of entanglement.
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We study the non-equilibrium dynamics of a pair of qubits made of two-level atoms separated in space with distance $r$ and interacting with one common electromagnetic field but not directly with each other. Our calculation makes a weak coupling assumption but no Born or Markov approximation. We write the evolution equations of the reduced density matrix of the two-qubit system after integrating out the electromagnetic field modes. We study two classes of states in detail: Class A is a one parameter family of states which are the superposition of the highest energy and lowest energy states, and Class B states which are the linear combinations of the symmetric and the antisymmetric Bell states. Our results for an initial Bell state are similar to those obtained before for the same model derived under the Born-Markov approximation. However, in the Class A states the behavior is qualitatively different: under the non-Markovian evolution we do not see sudden death of quantum entanglement and subsequent revivals, except when the qubits are sufficiently far apart. We provide explanations for such differences of behavior both between these two classes of states and between the predictions from the Markov and non-Markovian dynamics. We also study the decoherence of this two-qubit system.
We derive the stochastic equations and consider the non-Markovian dynamics of a system of multiple two-level atoms in a common quantum field. We make only the dipole approximation for the atoms and assume weak atom-field interactions. From these assumptions we use a combination of non-secular open- and closed-system perturbation theory, and we abstain from any additional approximation schemes. These more accurate solutions are necessary to explore several regimes: in particular, near-resonance dynamics and low-temperature behavior. In detuned atomic systems, small variations in the system energy levels engender timescales which, in general, cannot be safely ignored, as would be the case in the rotating-wave approximation (RWA). More problematic are the second-order solutions, which, as has been recently pointed out, cannot be accurately calculated using any second-order perturbative master equation, whether RWA, Born-Markov, Redfield, etc.. This latter problem, which applies to all perturbative open-system master equations, has a profound effect upon calculation of entanglement at low temperatures. We find that even at zero temperature all initial states will undergo finite-time disentanglement (sometimes termed sudden death), in contrast to previous work. We also use our solution, without invoking RWA, to characterize the necessary conditions for Dickie subradiance at finite temperature. We find that the subradiant states fall into two categories at finite temperature: one that is temperature independent and one that acquires temperature dependence. With the RWA there is no temperature dependence in any case.
We study the relation between the sudden death and revival of the entanglement of two qubits in a common squeezed reservoir, and the normal decoherence, by getting closer to the Decoherence Free Subspace and calculating the effect on the death and revival times.
A procedure to obtain the dynamics of $N$ independent qudits ($d$-level systems) each interacting with its own reservoir, for any arbitrary initial state, is presented. This is then applied to study the dynamics of the entanglement of two qubits, initially in an extended Werner-like mixed state with each of them in a zero temperature non-Markovian environment. The dependence of the entanglement dynamics on the purity and degree of entanglement of the initial states and on the amount of non-Markovianity is also given. This extends the previous work about non-Markovian effects on the two-qubit entanglement dynamics for initial Bell-like states [B. Bellomo textit{et al.}, Phys. Rev. Lett. textbf{99}, 160502 (2007)]. The effect of temperature on the two-qubit entanglement dynamics in a Markovian environment is finally obtained.
We study the non-Markovian entanglement dynamics of two qubits in a common squeezed bath. We see remarkable difference between the non-Markovian entanglement dynamics with its Markovian counterpart. We show that a non-Markovian decoherence free state is also decoherence free in the Markovian regime, but all the Markovian decoherence free states are not necessarily decoherence free in the non-Markovian domain. We extend our calculation from squeezed vacuum bath to squeezed thermal bath, where we see the effect of finite bath temperatures on the entanglement dynamics.
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