Spontaneously induced atom–radiation entanglement in an ensemble of trapped two-level atoms

Analysis of the effects of the spontaneously induced correlation on atom–radiation entanglement in an ensemble of two-level atoms initially prepared in the upper energy level and then trapped in a cavity containing a source of a squeezed radiation employing the method of evaluating the coherent-state propagator is presented. It is found that the cavity radiation exhibits squeezing which is directly attributed to the squeezed radiation initially present in the cavity. The intensity of the cavity radiation increases with the squeeze parameter and interaction time. It is also shown that a substantial degree of entanglement between the atomic state and radiation mode exits at a particular time which depends on the coupling constant and squeeze parameter. It is understood that although the squeezed radiation is directly accountable for the cavity squeezing, it significantly destroys the atom–radiation entanglement induced by the correlation between spontaneously emitted radiation and the atoms.     

Effect of the Stark shift on entanglement in a double two-photon JC model

Considering a double two-photon JC model, we investigate the entanglement between the two two-level atoms and that between the two cavity fields, and study the effect of the Stark shift on entanglement. The results show that, on the one hand the atom–atom and cavity–cavity concurrences evolve periodically with time and the periods are affected by the Stark shift; on the other hand, the two atoms are not disentangled at any time when the Stark shift is considered, and for large values of the Stark shift parameter, the two atoms can remain in a stationary entangled state. In addition, we find that the so-called entanglement sudden death can occur under appropriate conditions on the dynamic Stark shift for a certain initial state of the system.     

Quantum state transfer between atoms located in coupled optical cavities

We investigated the interaction between two coupled cavities, each one of them interacting with a two-level atom in its interior. We observed that if one of the atoms is in a superposition state and the other parts of the system are in their fundamental states, it is possible to transfer this state to the atom in the other cavity through the temporal evolution of the system. The time-evolution behaviour of the system during this transfer was studied and we observed its dependence with the frequency of the atom and the coupling constant between the atom and its respective cavity.     

The standard model in cavity quantum electrodynamics. II. Coupling constants and atom field interaction

Abstract

Specific forms of the travelling and trapped vector mode functions for a three-dimensional Fabry-Perot cavity are developed from the general results of the preceding paper, with parameters describing the output cavity mirror chosen for a typical high Q cavity case. Cavity and external quasi-mode functions associated with the quasi-mode theory of macroscopic canonical quantization are then obtained via an idealized choice of output mirror parameters. The coupling constants describing photon exchange processes between the single cavity quasi-mode associated with each Fabry-Perot resonance and various external quasi-modes are calculated, and their slow dependence on the external quasi-mode frequency shows that the conditions for irreversible Markovian damping of the cavity quasi-mode are satisfied. For radiative atoms placed in the cavity the coupling constants for energy exchange processes with sideways travelling external quasi-modes also vary slowely, so that Markovian spontaneous emission damping occurs for the radiative atoms. However, their coupling with the isolated cavity quasi-modes is associated with reversible photon exchanges as represented via one photon Rabi frequencies. The standard model in cavity quantum electrodynamics, in which the basic processes are described by a cavity damping rate, a radiative atom spontaneous decay rate and an atom-cavity mode coupling constant has now been justified in terms of the quasi-mode theory of macroscopic canonical quantization.     

Entanglement-induced population trapping by Schrödinger's cat

Abstract

We propose an experiment that is a variation of the Schrödinger's cat ′paradox' wherein the entanglement between a microscopic system and a macroscopic system is of primary interest. The experiment involves tunable entanglement and serves as a model for controllable decoherence in the context of cavity quantum electrodynamics where atoms interact dispersively with a cavity field initially in a coherent state. The interaction produces an entanglement between the atom and the field, and the degree of entanglement can be probed by subjecting the atom to resonant classical radiation after it leaves the cavity. The amplitude of the resulting Rabi oscillations reflects the degree of the entanglement, there being no Rabi oscillations when the entanglement is maximum. We show that the cavity damping does not affect the experiment.     

Violation of Mermin–Ardehali–Belinksii–Klyshko inequality in the three-Jaynes–Cummings model

In this paper, we consider the situation that three identical two-level atoms are separately trapped in the three single-mode cavities. Each atom resonantly interacts with cavity via a one-photon hopping. The dynamics of nonlocality in the system is investigated via Mermin–Ardehali–Belinksii–Klyshko inequality. The results show that when three atoms are initially in W state and three-cavity fields are in vacuum states both the quantum state of three atoms and that of three cavities all display nonlocality On the other hand, when three atoms are initially in Greenberger–Horne–Zeilinger state and three-cavity fields are in vacuum states, the quantum state of three atoms and that of three cavities all do not display nonlocality.     

Meeting Report

Abstract

The cooperative dynamics of a microlaser consisting of two threelevel atoms interacting with a pump field and two quantized cavity modes forming a radiative cascade are studied. Adiabatic elimination of one mode leads to a strong dynamical entanglement between the internal states of the atoms which allows us to study the effects of a cavity-mediated dipole-dipole interaction. We show that the coherent dynamics of the two-atom system will preferentially couple symmetrical linear combinations of internal states. If this coupling dominates the dynamics, the two-atom system will behave like a single atom with correspondingly larger dipole moment, that is a superradiant two-atom system. Even very small spontaneous decay causes transitions from symmetrical to antisymmetrical states and conversely. The hopping between two subsets of the state space can give rise to intriguing phenomena such as bistability of the laser mode intensity. By a randomization of the two coupling phases we recover the standard independent-atom laser theory.     

Quasi mode theory of macroscopic canonical quantization in quantum optics and cavity quantum electrodynamics

Abstract

A macroscopic, canonical quantization of the EM field and radiating atom system in quantum optics and cavity QED involving classical, linear optical devices, based on expanding the vector potential in terms of quasi mode functions is presented. The quasi mode functions approximate the true mode functions for the device, and are obtained by solving the Helmholtz equation for an idealized spatially dependent electric permittivity function describing the device. The Hamiltonian for the EM field and radiating atom system is obtained in multipolar form and the quantum EM field is found to be equivalent to a set of quantum harmonic oscillators, one oscillator per quasi mode. However, unlike true mode theory where the quantum harmonic oscillators are uncoupled, in the quasi mode theory they are coupled and photon exchange processes can occur. Explicit expressions for the coupling constants are obtained. The interaction energy between the radiative atoms and the quantum EM field depends on the amplitudes of the quasi mode functions at the positions of the radiating atoms, similar to that for the true mode approach. The simpler forms for the quasi mode functions enable the atom-field interaction energy to be written in a form in which the atoms are only coupled to certain types of modes—for example cavity quasi modes, which are large inside the optical cavity. In such cases the escape of energy from excited atoms in the cavity can be pictured in quasi mode theory as a two step process—the atom de-excites and creates a photon in a cavity quasi mode, the photon in the cavity quasi mode is then lost and appears as a photon in an external quasi mode. In this process the first step occurs via the atom-cavity quasi mode interaction, the second through coupling between cavity and external quasi modes. This may be contrasted with the true mode approach, where the excited atom loses its energy and the photon is created in one of the true modes. As all true modes have non-zero amplitudes outside as well as inside the cavity, the escape of energy from excited atoms in the cavity is seen as a one step process. An application of the quasi mode theory to the quantum theory of the beam splitter is outlined. The unitary operator used to describe this device is a scattering operator, relating initial and long time values of annihilation, creation operators for pairs of incident and reflected modes, interpreted here as quasi modes.     

Entanglement sudden birth and sudden death in a system of two distant atoms coupled via an optical element

An investigation is reported of the collective effects and the dynamics of atom–atom entanglement in a system of two distant two-level atoms which are coupled via an optical element. In the system under consideration, the two atoms, which are trapped in the foci of a lens, are coupled to a common environment being in the vacuum state and they emit photons spontaneously. A fraction of the emitted photons from each atom is thus focused on the position of the other atom. The presence of optical element between two distant atoms leads to the occurrence of delayed collective effects, such as delayed dipole–dipole interaction and delayed collective spontaneous emission, which play the crucial role in the dynamical behaviour of the entanglement. We discuss the phenomena of entanglement sudden birth, entanglement sudden death, and revival of entanglement for both cases of initial one-photon and initial two-photon unentangled atomic states. We show that the evolution of the entanglement is sensitive not only to the interatomic distance but also to the initial state of the system as well as to the properties of the optical element.     

The dynamics of entanglement and quantum discord of two atoms in coupled cavities

We investigate the dynamics of quantum correlations such as entanglement and quantum discord between two noninteracting atoms, each of which is trapped inside one of two coupled cavities. We find that the cavity decay can induce both entanglement and quantum discord between the two atoms when they are initially prepared in doubly excited state. The result shows the sudden death and sudden birth of entanglement and robustness of the quantum discord to sudden death. It is also found that the doubly excited state is responsible for the sudden death of entanglement. Moreover, the sudden death of entanglement can be controlled by the intercavity hopping rate.     

Tripartite entanglement generation using cavity-QED and its dynamics in dissipative environments

We investigate the generation of tripartite field states inside the high-Q cavities using the cavity QED. The main goal is to successfully generate the entanglement in tripartite systems by passing two-level atoms through three identical high-Q cavities. Our scheme gives the successful generation of entangled tripartite W and GHZ states for pre-determined interaction times of atoms with the cavity fields. The dynamics of initial entangled states is studied as the system evolves in the dissipative environments.     

Implementation of three-qubit Toffoli gates via the Heisenberg XY model in coupled cavities

We propose a distributed quantum architecture (three atoms trapped in three coupled cavities) to construct an asymmetric spin 1/2 Heisenberg XY model. The effective atom–atom coupling is obtained with the cavity modes and the atomic excited states being virtually populated. We then demonstrate that the model can be used for implementing a three-qubit Toffoli gate with high fidelity. The feasibility and effectiveness of our scheme as well as the influences of decoherence effects (the atomic spontaneous emission and the cavity decay) on the gate fidelity are finally discussed. The scheme can be generalized for distributed quantum computation with the Heisenberg XY model.     

Dynamics of quantum steerability for two atoms in coupled cavities

We study the dynamics of quantum steerability between two non-interacting atoms, each of which is trapped inside one of two coupled cavities. Compared with entanglement, quantum steerability manifests sudden birth and sudden death phenomenon during the time evolution. We find that the cavity decay plays a destruction role for both steerability and entanglement. It is also shown that the survival time as well as the maximal value of steerability are sensitive to the asymmetry of the cavities. Moreover, it is found the sudden death of steerability can be controlled by the hopping rate of the coupled cavities.     

Local entanglement in a bimodal high- Q cavity: Production and utilization

A new conditional scheme for entangling the two quantized modes of a bimodal high-Q cavity field is presented. We show that, injecting one at time k atoms inside the cavity, it is possible to guide the field toward k-dependent linear combinations of k + 1 bimodal Fock states, each one possessing the same total number of photons. The two simple cases corresponding to the passage of one or two atoms only through the resonator are considered. Their practical feasibility against cavity losses, spontaneous emission and other sources of imprecision of the experimental set-up is discussed. Two examples illustrating the usefulness of and the interest toward the creation of such a local entanglement, are reported. The first one relates the specific features of these bimodal entangled states to the occurrence of a non-classical correlation effect in the dynamics of a two-level atom interacting with the entangled cavity. The second one demonstrates that the peculiar entanglement initially stored in the cavity in accordance with our method, provides an effectively exploitable resource to entangle two spatially separated cavities.     

Squeezing of light field in a dissipative Jaynes–Cummings model

Based on the time-convolutionless master-equation approach, we investigate squeezing of light field in a dissipative Jaynes–Cummings model. The results show that squeezing light can be generated when the atom transits to a ground state from an excited state, and then a collapse-revival phenomenon will occur in the squeezing of light field due to atom-cavity coupling. Enhancing the atom-cavity coupling can increase the frequency of the collapse-revival of squeezing. The stronger the non-Markovian effect is, the more obvious the collapse-revival phenomenon is. The oscillatory frequency of the squeezing is dependent on the resonant frequency of the atom-cavity.     

A simple model for a minimal environment: the two-atom Tavis–Cummings model revisited

Individual quantum systems may be interacting with surrounding environments having a small number of degrees of freedom. Here we discuss a simple toy model: a system constituted by a two-level atom (atom 1) interacting with a single mode cavity field which is (weakly) coupled to a small environment (atom 2). We investigate the influence of the minimal environment on the dynamics of the linear entropy and the atomic dipole squeezing of atom 1, as well as the entanglement between atom 1 and the field. We also obtain the full analytical solution of the two-atom Tavis–Cummings model for both arbitrary coupling strengths and frequency detunings, necessary to analyse the influence of the field-environment detuning on the evolution of the system’s quantum properties. For complementarity, we discuss the role of the degree of mixedness of the environment by analysing the time-averaged linear entropy of atom 1.     

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

Abstract

The role of the collective antisymmetric state in entanglement creation by spontaneous emission in a system of two non-overlapping two-level atoms has been investigated. Populations of the collective atomic states and the Wootters entanglement measure (concurrence) for two sets of initial atomic conditions are calculated and illustrated graphically. 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. It is found that the ability of spontaneous emission to create 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, entanglement builds up rapidly in time and reaches a maximum for 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, it was found that a decaying initial entanglement between the atoms can display a revival behaviour.     

The standard model in cavity quantum electrodynamics. I. General features of mode functions for a Fabry-Perot cavity

Abstract

In the present and the accompanying paper a justification of the standard model of cavity quantum electrodynamics is given in terms of a quasi-mode theory of macroscopic canonical quantization. The coupling of the cavity quasi-mode to external quasi-modes is treated for the representative case of the three-dimensional Fabry-Perot cavity. The general form of the travelling and trapped mode functions for this cavity are derived in this paper and the mode-mode coupling constants are calculated in the accompanying paper. The slow dependence of the coupling constants with the mode frequency difference demonstrates that the conditions for Markovian damping of the cavity quasimode are satisfied. As also discussed in the accompanying paper, the interaction of radiative atoms with cavity quasi-modes is associated with reversible energy exchanges between atom and cavity and represented by Rabi coupling constants. The interaction of radiative atoms located within the cavity with sideways travelling external quasi-modes involves slowly varying coupling constants and is associated with irreversible spontaneous emission dampling. The basic processes represented in the standard cavity quantum electrodynamics model and the associated coupling constant and decay rates thereby follow from the quasi-mode theory.     

Controlling tripartite entanglement among optical cavities by reservoir engineering

We study how to control the dynamics of tripartite entanglement among optical cavities using non-Markovian baths. In particular, we demonstrate how the reservoir engineering through the utilization of non-Markovian baths with different types of Lorentzian and ohmic spectral densities can lead to an entanglement survival for longer times and in some cases considerable regain of seemingly lost entanglement. Both of these behaviours indicate a better sustainability of entanglement (in time) compared to the usual Markovian bath situations which assumes a flat spectrum of the bath around the system resonant frequency. Our scheme shows these effects in the context of optical cavities starting off in a maximally entangled W and Greenberger–Horne–Zeilinger tripartite states. In Lorentzian cases, we find that the far detuned double Lorentzian baths with small coupling strengths and for ohmic-type baths super-ohmic environments with smaller cutoff frequencies are the best candidates for preserving entanglement among cavities for significant amount of time. A non-Markovian quantum jump approach is employed to understand the entanglement dynamics in these cases, especially to recognize the collapse and revival of the entanglement in both W and GHZ states.     

Nonreciprocal behaviour between photon and phonon in coupled optomechanical systems

It is shown that the asymmetry coupling between two coupled optomechanical cavities leads to special class of PT-symmetric model for optomechanical structure. Under these conditions, Hamiltonian is considered in blue and red sideband regime. In these cases, the asymmetric coupling between two cavities has been transferred such that the asymmetric beam-splitter or squeezing interaction is generated between optical and mechanical modes. Then, the amount of entanglement between the different optical and mechanical modes is calculated. The results define that PT-symmetry can improve the entanglement in special conditions. The proposed system provides good condition to investigate the nonreciprocal interaction between photon and phonon.