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arXiv:0811.4633 · Physical Review A

Breakdown of the rotating-wave approximation in the description of entanglement of spin-anticorrelated states

Jun Jing, Zhiguo Lü, Z. Ficek

Published Apr 30, 2009• 35 citations
arXivHTMLPDF

Abstract

It is well established that an entanglement encoded in the Bell states of a two-qubit system with correlated spins exhibits completely different evolution properties from that encoded in states with the anticorrelated spins. A complete and abrupt loss of the entanglement, called the entanglement sudden death, can be found to occur for the spin-correlated states, but the entanglement evolves without any discontinuity or decays asymptotically for the spin-anticorrelated states. We consider the evolution of an initial entanglement encoded in the spin-anticorrelated states and demonstrate that the asymptotic behavior predicted before occurs only in the weak-coupling limit or equivalently when the rotating-wave approximation (RWA) is made on the interaction Hamiltonian of the qubits with the field. If we do not restrict ourselves to the RWA, we find that the entanglement undergoes a discontinuity, the sudden-death phenomenon. We illustrate this behavior by employing an efficient scheme for entanglement evolution between two cold-trapped atoms located inside a single-mode cavity. Although only a single excitation is initially present in the system, we find that the two-photon excited state, which plays the key role for the discontinuity in the behavior of the entanglement, gains a population over a short time of the evolution. When the RWA is made on the interaction, the two-photon excited state remains unpopulated for all times and the discontinuity is absent. We attribute this phenomenon to the principle of complementarity between the evolution time and energy, and the presence of the counter-rotating terms in the interaction Hamiltonian.

DOI
10.1103/physreva.79.044305

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Physica Scripta•Sep 1, 2010•13 citations
It is well established that an entanglement encoded in the Bell states of a two-qubit system with correlated spins exhibits completely different evolution properties from that encoded in states with the anticorrelated spins. A complete and abrupt loss of the entanglement, called the entanglement sudden death, can be found to occur for the spin-correlated states, but the entanglement evolves without any discontinuity or decays asymptotically for the spin-anticorrelated states. We consider the evolution of an initial entanglement encoded in the spin-anticorrelated states and demonstrate that the asymptotic behavior predicted before occurs only in the weak-coupling limit or equivalently when the rotating-wave approximation (RWA) is made on the interaction Hamiltonian of the qubits with the field. If we do not restrict ourselves to the RWA, we find that the entanglement undergoes a discontinuity, the sudden-death phenomenon. We illustrate this behavior by employing an efficient scheme for entanglement evolution between two cold-trapped atoms located inside a single-mode cavity. Although only a single excitation is initially present in the system, we find that the two-photon excited state, which plays the key role for the discontinuity in the behavior of the entanglement, gains a population over a short time of the evolution. When the RWA is made on the interaction, the two-photon excited state remains unpopulated for all times and the discontinuity is absent. We attribute this phenomenon to the principle of complementarity between the evolution time and energy, and the presence of the counter-rotating terms in the interaction Hamiltonian.
The entanglement property of two identical atoms, initially entangled in Bell states, coupled to a single-mode cavity is considered. Based on the reduced non-perturbative quantum master equation method, the entanglement evolution of the two atoms with decay is investigated beyond the conventional rotating-wave approximation. We show that the counter-rotating wave terms, usually neglected, have a great influence on the disentanglement behaviour of the system. The phenomena of entanglement sudden death and entanglement sudden birth will occur. In addition, we show that the entanglement can be strengthened by introducing the dipole-dipole interaction of the two atoms.
Entanglement evolution of two independent Jaynes-Cummings atoms without the rotating-wave approximation (RWA) is studied by a numerically exact approach. Previous results based on the RWA are essentially modified in the strong-coupling regime ($g\ensuremath{\geqslant}0.1$), which has been reached in the recent experiments on the flux qubit coupled to the LC\mathit{LC}LC resonator. For the initial Bell state with anticorrelated spins, entanglement sudden death (ESD) is absent in the RWA but does appear in the present numerical calculation without the RWA. Aperiodic entanglement evolution in the strong-coupling regime is observed. The strong atom-cavity coupling facilitates the ESD. The sign of the detuning plays an essential role in the entanglement evolution for strong coupling, which is irrelevant in the RWA. Analytical results based on an unitary transformation are also given, which could not modify the RWA picture essentially. It is suggested that the activation of the photons may be the origin of ESD in this system.
We investigate the time evolution and asymptotic behavior of a system of two two-level atoms (qubits) interacting off-resonance with a single mode radiation field. The two atoms are coupled to each other through dipole-dipole as well as Ising interactions. An exact analytic solution for the system dynamics that spans the entire phase space is provided. We focus on initial states that cause the system to evolve to entanglement sudden death (ESD) between the two atoms. We find that combining the Ising and dipole-dipole interactions is very powerful in controlling the entanglement dynamics and ESD compared with either one of them separately. Their effects on eliminating ESD may add up constructively or destructively depending on the type of Ising interaction (Ferromagnetic or anti-Ferromagnetic), the detuning parameter value, and the initial state of the system. The asymptotic behavior of the ESD is found to depend substantially on the initial state of the system, where ESD can be entirely eliminated by tuning the system parameters except in the case of an initial correlated Bell state. Interestingly, the entanglement, atomic population and quantum correlation between the two atoms and the field synchronize and reach asymptotically quasi-steady dynamic states. Each one of them ends up as a continuous irregular oscillation, where the collapse periods vanish, with a limited amplitude and an approximately constant mean value that depend on the initial state and the system parameters choice. This indicates an asymptotic continuous exchange of energy (and strong quantum correlation) between the atoms and the field takes place, accompanied by diminished ESD for these chosen setups of the system. This system can be realized in spin states of quantum dots or Rydberg atoms in optical cavities, and superconducting or hybrid qubits in linear resonators.
We consider the time evolution of quantum entanglement in the context of interactions between atoms and the electromagnetic field. We explore the influence of interatomic separation and the degree to which this can change the qualitative character of those dynamics, including entanglement generation, protection, and sudden death (SD). We find that the qualitative features of entanglement dynamics can be changed entirely in few-mode models when atomic spacing is varied, allowing for particular choices of configuration that are favorable for maintaining entanglement. We also examine the inaccuracies introduced by the use of common approximations: We characterize unexpected errors that result from using perturbative master equations as well as those that result from using the rotating-wave approximation (RWA). We find that in dissipative systems the errors introduced by these approximations can lead to an incorrect picture of late-time dynamics. Standard perturbative master equations using the RWA are constrained to predict that late-time SD occurs to only some initial states at zero temperature, but this is merely an artifact of those approximation and generally not correct. The same master equations predict that at finite temperature all states are separable asymptotically at late times and must undergo SD. In fact a proper accounting of environmentally-induced corrections to the steady state of the system shows that for low temperatures it is possible to have asymptotic entanglement in some cases. We derive a master equation for two atoms interacting with the free field without using the RWA and solve it to obtain the dynamics, including the effects of distance. From these dynamics we find that, in fact, all initial states of atoms separated by any positive distance undergo SD even at zero temperature, though there are sub-radiant states that can be quite long-lived for closely spaced atoms.
The problem of the creation of entanglement exclusively from the presence of the counter-rotating terms in the interaction Hamiltonian between two atoms and a single-mode cavity field is studied. The time evolution of the concurrence between the atoms is discussed for two different initial separable states and arbitrary strengths of the coupling constants of the atoms to the cavity mode. We show that a transient entanglement can be created between the atoms only if the counter-rotating terms are taken into account in the interaction Hamiltonian. The source of the entanglement is identified, and the entanglement features are explained in terms of the properties of the reduced density operator of the atoms. In particular, we find that the main source of the entanglement is in the two-photon coherence induced by the counter-rotating terms that cause two-photon transitions through virtual states.