Temporal and diffraction effects in entanglement creation in an optical cavity

A practical scheme for entanglement creation between distant atoms located inside a single-mode optical cavity is discussed. We show that the degree of entanglement and the time it takes for the entanglement to reach its optimum value is a sensitive function the initial conditions and the position of the atoms inside the cavity mode. It is found that the entangled properties of the two atoms can readily be extracted from dynamics of a simple two-level system. Effectively, we engineer two coupled qubits whose the dynamics are analogous to that of a driven single two-level system. It is found that spatial variations of the coupling constants actually help to create transient entanglement which may appear on the time scale much longer than that predicted for the case of equal coupling constants. When the atoms are initially prepared in an entangled state, they may remain entangled for all times. We also find that the entanglement exhibits an interesting phenomenon of diffraction when the the atoms are located between the nodes and antinodes of the cavity mode. The diffraction pattern of the entanglement varies with time and we explain this effect in terms of the quantum property of complementarity, which is manifested as a tradeoff between the knowledge of energy of the exchanged photon versus the evolution time of the system.Comment: Phys. Rev. A75, 042307 (2007

Dark periods and revivals of entanglement in a two qubit system

In a recent paper Yu and Eberly [Phys. Rev. Lett. {\bf 93}, 140404 (2004)] have shown that two initially entangled and afterwards not interacting qubits can become completely disentangled in a finite time. We study transient entanglement between two qubits coupled collectively to a multimode vacuum field and find an unusual feature that the irreversible spontaneous decay can lead to a revival of the entanglement that has already been destroyed. The results show that this feature is independent of the coherent dipole-dipole interaction between the atoms but it depends critically on whether or not the collective damping is present. We show that the ability of the system to revival entanglement via spontaneous emission relies on the presence of very different timescales for the evolution of the populations of the collective states and coherence between them.Comment: 4 pages, 3 figure

Delayed (sudden) birth of entanglement

The concept of time delayed creation of entanglement by the dissipative process of spontaneous emission is investigated. A threshold effect for the creation of entanglement is found that the initially unentangled qubits can be entangled after a finite time despite the fact that the coherence between the qubits exists for all times. This delayed creation of entanglement, that we call sudden birth of entanglement, is opposite to the currently extensively discussed sudden death of entanglement and is characteristic for transient dynamics of one-photon entangled states of the system. We determine the threshold time for the creation of entanglement and find that it is related to time at which the antisymmetric state remains the only excited state being populated. It is shown that the threshold time can be controlled by the distance between the qubits and the direction of initial excitation relative to the interatomic axis. This effect suggests a new alternative for the study of entanglement and provides an interesting resource for creation on demand of entanglement between two qubits.Comment: References added, version accepted for publication in PR

Entanglement induced by spontaneous emission in spatially extended two-atom systems

We investigate the role of the collective antisymmetric state in entanglement creation by spontaneous emission in a system of two non-overlapping two-level atoms. We calculate and illustrate graphically populations of the collective atomic states and the Wootters entanglement measure (concurrence) for two sets of initial atomic conditions. Our calculations include the dipole-dipole interaction and a spatial separation between the atoms that the antisymmetric state of the system is included throughout even for small interatomic separations. It is shown that spontaneous emission can lead to a transient entanglement between the atoms even if the atoms were prepared initially in an unentangled state. We find that the ability of spontaneous emission to create the transient entanglement relies on the absence of population in the collective symmetric state of the system. For the initial state of only one atom excited, the entanglement builds up rapidly in time and reaches a maximum for the parameter values corresponding roughly to zero population in the symmetric state. On the other hand, for the initial condition of both atoms excited, the atoms remain unentangled until the symmetric state is depopulated. A simple physical interpretation of these results is given in terms of the diagonal states of the density matrix of the system. We also study entanglement creation in a system of two non-identical atoms of different transition frequencies. It is found that the entanglement between the atoms can be enhanced compared to that for identical atoms, and can decay with two different time scales resulting from the coherent transfer of the population from the symmetric to the antisymmetric state. In addition, we find that a decaying initial entanglement between the atoms can display a revival behaviour.Comment: 14 pages, 6 figure

Entangling two atoms via spontaneous emission

We discuss the creation of entanglement between two two-level atoms in the dissipative process of spontaneous emission. It is shown that spontaneous emission can lead to a transient entanglement between the atoms even if the atoms were prepared initially in an unentangled state. The amount of entanglement created in the system is quantified by using two different measures: concurrence and negativity. We find analytical formulas for the evolution of concurrence and negativity in the system. We also find the analytical relation between the two measures of entanglement. The system consists of two two-level atoms which are separated by an arbitrary distance $r_{12}$ and interact with each other via the dipole-dipole interaction, and the antisymmetric state of the system is included throughout, even for small inter-atomic separations, in contrast to the small sample model. It is shown that for sufficiently large values of the dipole-dipole interaction initially the entanglement exhibits oscillatory behaviour with considerable entanglement in the peaks. For longer times the amount of entanglement is directly related to the population of the slowly decaying antisymmetric state.Comment: 13 pages, 5 figure

Comment on "Discrepancies in the resonance-fluorescence spectrum calculated with two methods"

There are two alternative methods used in literature to calculate the incoherent part of the spectrum of light scattered by an atomic system. In the first, one calculates the spectrum of the total light scattered by the system and obtains the incoherent part by subtracting the coherent part. In the second method, one introduces the fluctuation operators and obtains the incoherent part of the spectrum by taking the Fourier transform of the two time correlation function of the fluctuation operators. These two methods have been recognized for years as two completely equivalent for evaluating the incoherent part of the spectrum. In a recent paper Qing Xu et al. [Phys. Rev. A 78, 013407 (2008)] have shown that there are discrepancies between the incoherent parts of the stationary spectrum of a three-level Lambda-type system calculated with these two methods. The predicted discrepancies can be severe that over a wide rage of the Rabi frequencies and atomic decay rates, the spectrum calculated with the variance method can have negative values. This is obviously unphysical result since the fluorescence spectrum is a positively defined quantity. It represents the frequency distribution of light incoherently scattered by the atomic system. Therefore, the calculated spectrum should be positive for all frequencies independent of values of the Rabi frequencies and the damping rates. In this comment, we show that there are no discrepancies between these two methods. The equivalence of these two alternative methods leads to the same incoherent spectra that are positive for all frequencies independent of values of the parameters involved. The analytical analysis is supported by simple numerical calculations.Comment: Comments on the comment are welcom

Radiative properties of a linear chain of coupled qubits

We calculate the radiative properties for a linear dipole-coupled chain of qubits. Using the explicit energy eigenstates of the system, we find the radiation patterns for spontaneous transitions from the one-photon eigenstates to the ground state of the system. We show that depending on the excitation of a specific atom, the radiation tends to be focused either along or perpendicular to the chain. We conclude with a derivation of the total decay rate of the one-photon eigenstates, and find the interesting result that for systems where the photon wavenumber is not much larger than the interatomic spacing, up to 94% of the eigenstates are subradiant, that is, they decay significantly slower than a single atom in isolation.Comment: 20 pages, 11 figure

Spontaneous emission from two atoms interacting with a broadband squeezed vacuum

Spontaneous emission from two-level atoms interacting with a squeezed vacuum field is examined, taking account of possible different interatomic separations. It is shown that the steady-state atomic population depends on whether the interatomic separations are comparable to or much smaller than the resonant wavelength. For the extended system the squeezed vacuum introduces a dependence of the atomic population on the interatomic separations. For large interatomic separations the atomic population is identical to that for the thermal field, which is the same for two independent atoms. For small interatomic separations the atomic population is higher than that for the thermal field. This is in contrast to the small-sample model in which the interatomic separation is ignored. In this case the final atomic population in the squeezed vacuum is the same as that for the independent atoms in the squeezed vacuum. Moreover, this population differs from that for the thermal field for which the atomic population is lower than that for the independent atoms. This difference is due to the interatomic correlations whose presence depends on whether the interatomic separation is or is not included. For the extended system the interatomic correlations are induced by the squeezed field and vanish for the thermal field. For the small-sample model, however, the interatomic correlations are induced by the thermal field and the squeezed vacuum changes these correlations in such a way that for the minimum-uncertainty squeezed states these correlations vanish. We also discuss the effect of the interatomic separation on the two-photon transitions and the normalized intensity correlation function in the two-atom system interacting with the squeezed vacuum field

Effect of retardation on the dynamics of entanglement between atoms

The role of retardation in the entanglement dynamics of two distant atoms interacting with a multi-mode field of a ring cavity is discussed. The retardation is associated with a finite time required for light to travel between the atoms located at a finite distance and between the atoms and the cavity boundaries. We explore features in the concurrence indicative of retardation and show how these features evolve depending on the initial state of the system, distance between the atoms and the number of modes to which the atoms are coupled. In particular, we consider the short-time and the long time dynamics for both the multi- and sub-wavelength distances between the atoms. It is found that the retardation effects can qualitatively modify the entanglement dynamics of the atoms not only at multi- but also at sub-wavelength distances. We follow the temporal evolution of the concurrence and find that at short times of the evolution the retardation induces periodic sudden changes of entanglement. To analyze where the entanglement lies in the space spanned by the state vectors of the system, we introduce the collective Dicke states of the atomic system that explicitly account for the sudden changes as a periodic excitation of the atomic system to the maximally entangled symmetric state. At long times, the retardation gives rise to periodic beats in the concurrence that resemble the phenomenon of collapses and revivals in the Jaynes-Cummings model. In addition, we identify parameter values and initial conditions at which the atoms remain separable or are entangled without retardation during the entire evolution time, but exhibit the phenomena of sudden birth and sudden death of entanglement when the retardation is included.Comment: 16 pages, 14 figure

Spin squeezing as a measure of entanglement in a two qubit system

We show that two definitions of spin squeezing extensively used in the literature [M. Kitagawa and M. Ueda, Phys. Rev. A {\bf 47}, 5138 (1993) and D.J. Wineland {\it et al.}, Phys. Rev. A {\bf 50}, 67 (1994)] give different predictions of entanglement in the two-atom Dicke system. We analyze differences between the definitions and show that the Kitagawa and Ueda's spin squeezing parameter is a better measure of entanglement than the commonly used spectroscopic spin squeezing parameter. We illustrate this relation by examining different examples of a driven two-atom Dicke system in which spin squeezing and entanglement arise dynamically. We give an explanation of the source of the difference in the prediction of entanglement using the negativity criterion for entanglement. For the examples discussed, we find that the Kitagawa and Ueda's spin squeezing parameter is the sufficient and necessary condition for entanglement.Comment: 5 pages, 4 figure