Dynamics of two interacting dipolar two-level systems in a multi-mode electromagnetic cavity: sudden death and revival of the entanglement within the Born-Markov approximation
Interacting dipolar two-level systems form a special class of qubits that interact with a cavity in a particular way. We first prove that the Markovian dynamics of one 1/2-spin in interaction with a quantised magnetic field from a multi-mode cavity at thermal equilibrium is equivalent to a two-level atom interacting in the dipole approximation with the electric field of the cavity. We then use the Born-Markov approximation to study the dynamics of two spins interacting through the antiferromagnetic Heisenberg coupling in the same environment. By solving the GKSL equation, we find the exact expression of the density matrix of the system, with the off-diagonal coherence decay time and spin relaxation time. The concurrence for the stationary state is explicitly derived for any kind of initial state and the role of the singlet state is brought to light. The temporal evolution of the concurrence is numerically computed for different initial states, the phenomenon of sudden death and revival of the entanglement is observed for this dynamics. A detailed analysis of the sudden death and revival of the concurrence is conducted for Werner states, with new analytical results obtained thanks to the solution of the GKSL equation. We finally derive the equations and the stationary concurrence for the XXZ coupling.
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14 pages, 12 figures, reference added, typos corrected, title changed
Interacting dipolar two-level systems form a special class of qubits that interact with a cavity in a particular way. We first prove that the Markovian dynamics of one 1/2-spin in interaction with a quantised magnetic field from a multi-mode cavity at thermal equilibrium is equivalent to a two-level atom interacting in the dipole approximation with the electric field of the cavity. We then use the Born-Markov approximation to study the dynamics of two spins interacting through the antiferromagnetic Heisenberg coupling in the same environment. By solving the GKSL equation, we find the exact expression of the density matrix of the system, with the off-diagonal coherence decay time and spin relaxation time. The concurrence for the stationary state is explicitly derived for any kind of initial state and the role of the singlet state is brought to light. The temporal evolution of the concurrence is numerically computed for different initial states, the phenomenon of sudden death and revival of the entanglement is observed for this dynamics. A detailed analysis of the sudden death and revival of the concurrence is conducted for Werner states, with new analytical results obtained thanks to the solution of the GKSL equation. We finally derive the equations and the stationary concurrence for the XXZ coupling.
We demonstrate an exact mapping of a class of models of two interacting qubits in thermal reservoirs to two separate problems of spin–boson-type systems. Based on this mapping, exact numerical simulations of the qubits dynamics can be performed, beyond the weak system–bath coupling limit and the Markovian approximation. Given the time evolution of the system population and coherences, we study as an application the dynamics of entanglement between the pair of qubits immersed in boson thermal baths, showing a rich phenomenology, including an intermediate oscillatory behavior, the entanglement sudden birth, sudden death and revival. We find that the occurrence of entanglement sudden death in this model depends on the portion of the zero and double excitation states in the subsystem initial state. In the long-time limit, analytic expressions are presented at weak system–bath coupling, for a range of relevant qubit parameters.
We study a model of two coupled two-level atoms (qubits) interacting off-resonance (at non-zero detuning) with a single mode radiation field. This system is of special interest in the field of quantum information processing (QIP) and can be realized in electron spin states in quantum dots or Rydberg atoms in optical cavities and superconducting qubits in linear resonators. We present an exact analytical solution for the time evolution of the system starting from any initial state. Utilizing this solution, we show how the entanglement sudden death (ESD), which represents a major threat to QIP, can be efficiently controlled by tuning atom-atom coupling and non-zero detuning. We demonstrate that while one of these two system parameters may not separately affect the ESD, combining the two can be very effective, as in the case of an initial correlated Bell state. However in other cases, such as a W-like initial state, they may have a competing impacts on ESD. Moreover, their combined effect can be used to create ESD in the system, as in the case of an anti-correlated initial Bell state. A clear synchronization between the population inversion collapse-revival pattern and the entanglement dynamics is observed at all system parameter combinations. Nevertheless, only for initial states that may evolve to ESD, the population inversion revival oscillations, where exchange of energy between the atoms and the field takes place, temporally coincide with the entanglement revival peaks, whereas the population collapse periods match the ESD intervals. The variation of the radiation field intensity has a clear impact on the duration of the ESD at any combination of the other system parameters.
The aim of this topical review is a systematic and concise presentation of the results of a series of theoretical works on the quantum dynamics of two-spin-qubit systems towards the elaboration of a physical mechanism of the quantum information transfer between two spin-qubits. For this purpose the main attention is paid to exactly solvable models of two-spin-qubit systems, since the analytical expressions of the elements of their reduced density matrices explicitly exhibit the mutual dependence of the quantum information encoded into the spin-qubits. The treatment of their decoherence due to the interaction with the environment is performed in the Markovian approximation. Rate equations for axially symmetric systems of two coupled spin-qubits non-interacting, as well as interacting, with the environment are exactly solved. It is shown how the solutions of rate equations demonstrate the physical mechanism of the quantum information exchange between the spin-qubits. This mechanism holds also in all two-spin-qubit systems whose rate equations can be solved only by means of numerical calculations. Exact solutions of rate equations for two uncoupled spin-qubits interacting with two separate environments reveal an interesting physical phenomenon in the time evolution of the qubit-qubit entanglement generated by their interaction with the environments: the entanglement sudden death and revival. A two-spin-qubit system with an asymptotically decoherence free subspace was also explicitly constructed. The presented calculations and reasonings can be extended for application to the study of spin-qubit chains or networks.
We consider the entanglement evolution of two qubits embedded into disordered multiconnected environment. We model the environment and its interaction with qubits by large random matrices allowing for a possibility to describe environments of meso- and even nanosize. We obtain general formulas for the time dependent reduced density matrix of the qubits corresponding to several cases of the qubit-environment interaction and initial condition. We then work out an analog of the Born–Markov approximation to find the evolution of the widely used entanglement quantifiers: the concurrence, the negativity and the quantum discord. We show that even in this approximation the time evolution of the reduced density matrix can be non-Markovian, thereby describing certain memory effects due to the backaction of the environment on qubits. In particular, we find the vanishing of the entanglement (Entanglement Sudden Death) at finite moments and its revivals (Entanglement Sudden Birth). Our results, partly known and partly new, can be viewed as a manifestation of the universality of certain properties of decoherent qubit evolution which have been found previously in various versions of bosonic macroscopic environment.
We examine the dynamics of bipartite quantum entanglement between a two-level atom and the electromagnetic field. We begin with the Jaynes-Cummings model with a single field mode and examine in detail the exact time evolution of entanglement, including cases where the atomic state is initially mixed and the atomic transition is detuned from resonance. We then explore the effects of other nearby modes by calculating the exact time evolution of entanglement in more complex systems with two, three, and five field modes. For these cases we can obtain exact solutions which include the strong-coupling regime. Finally, we consider the entanglement of a two-level atom with the infinite collection of modes present in the intracavity field of a Fabre-Perot cavity. In contrast to the usual treatment of atom-field interactions with a continuum of modes using the Born-Markov approximation, our treatment in all cases describes the full non-Markovian dynamics of the atomic subsystem under general couplings. Only when an analytic expression for the infinite mode case is desired do we need to make a weak coupling assumption which at late times approximates Markovian dynamics.
Abstract Bipartite entanglement is a necessary resource in most processes in quantum information science. Decoherence resulting from the interaction of the bipartite system with environment not only degrades the entanglement, but can result in abrupt disentanglement, known as entanglement sudden death (ESD). In some cases, a subsequent revival of entanglement is also possible. ESD is an undesirable feature for the state to be used as a resource in applications. In order to delay or avoid ESD, it is necessary to understand its origin. In this work we investigate the role of initial coherence on entanglement dynamics of a spatially separated two qubit system in a common vacuum reservoir with dipolar interaction. We construct two classes of X states, namely, states with one photon coherence ( X 1 ) and states with two photon coherence ( X 2 ). Considering them as initial states, we study entanglement dynamics under Markov approximation. We find for states in X 1 , ESD time, revival time and time over which the state remains disentangled increase with increase in coherence. On the other hand for states in X 2 , with increase in coherence ESD time increases, revival time remains same and time of disentanglement decreases. Thus, states with two photon coherence are better resources for applications since their entanglement is robust against decoherence compared to states with one photon coherence.