Revival Phenomenon Research Articles

Two mode theory of Bose–Einstein condensates: interferometry and the Josephson model

This topical review provides an overview of the key theoretical features of Bose–Einstein condensates (BECs) in cold atomic gases at near zero temperature in the situation where all the bosons occupy at most two single particle states or modes. This situation applies to single-component BECs in double well trap potentials and to two-component BEC in single well trap potentials, such as occur when BEC are used in interferometry experiments. The Hamiltonian is introduced in terms of field operators and mode expansions are restricted to a total of two modes. Spin operators and their eigenstates are introduced as the fundamental basis states for describing the two-mode N boson quantum system. The spin states have a macroscopic angular momentum quantum number of N/2 and the magnetic quantum number k specifies the relative number of bosons in the two modes. The treatment presented involves an extensive use of angular momentum theory, including unitary rotation operators. Important states of the two-mode system such as binomial or coherent states, relative phase eigenstates are discussed. Boson position measurements are specified via quantum correlation functions, and the use of these functions in describing coherence properties, interference patterns and fragmentation effects in BECs is presented. The Bloch vector is defined and related to the quantum correlation functions, with quantum fluctuations of the Bloch vector being treated in terms of the covariance matrix. Applications to important two-mode states are made. Spin squeezing is discussed. Based on applying variational principles, the general dynamical behaviour of the two-mode BEC is determined via generalised Gross–Pitaevskii equations for the modes and matrix mechanics equations for the probability amplitudes of the relative number basis states, the mode and amplitude equations being coupled and self-consistent. The single mode equations are also presented. The Hamiltonian is written in terms of the spin operators and the Josephson Hamiltonian obtained as a simplification in which the dynamical behaviour of the mode functions is ignored – for the one-component case the mode functions are also required to be localised and separate. Coefficients in the Josephson Hamiltonian describe tunneling/intercomponent coupling, asymmetry and collisions and these are defined via integrals involving the mode functions. The Josephson model involves using the Josephson Hamiltonian to give simple predictions of the energy states and dynamical behaviour of the two-mode system, dynamical effects on the mode functions being ignored. The three regimes – Rabi, Josephson and Fock are described, and the energy states obtained for the Fock and Rabi regimes. Dynamical behaviour treatments based on the Josephson model are outlined. In the situation where all bosons are in the same single particle state, semi-classical Bloch equations are derived and their solutions given in terms of elliptic functions. The quantum regime is treated using matrix mechanics equations for the probability amplitudes. Two representative applications of the Josephson model dynamics are treated, with graphs showing the results of numerical work being displayed. The first is in describing Heisenberg limited BEC interferometry for a single-component BEC in a double well, the treatment showing collapses and revivals in the probability distribution for the relative phase. The second treats Ramsey interferometry for a two-component BEC in a single well, the study revealing that oscillations of the Bloch vector collapse and revive, with the Bloch vector's departure from the Bloch sphere during the collapse period revealing that the BEC has fragmented. In both cases collisions cause the dephasing effects that result in the collapse, revival phenomena. The review ends with a brief outline of phase space and other approaches that extend the treatment beyond the two-mode theory, enabling decoherence effects associated with bosons in non-condensate modes to be studied. A summary of the review contents is included. Detailed mathematical derivations are included in several appendices, available as online supplementary material.

Fractional Revival of Rydberg Wave Packets in Twice-Kicked One-Dimensional Atoms

We study revival and fractional revival phenomena of wave packets in a one-dimensional Rydberg atom irradiated by two time-delayed half-cycle pulses using an autocorrelation function characterized by electronic transition probabilities as weighting factors rather than modeling them by a Gaussian or Lorentzian distribution. If the momentum (q2) delivered to the atom by the second kick is much smaller than that (q1) imparted by the first one, the times of revival and fractional revival coincide with those of the single kicked atom. For q2 ≥ q1 4 appearance of revival and fractional revival depends on both the values of q2 and time delay t1 between the pulses but more sensitively on t1. The number of fractional revivals tends to become numerous as the value of t1 increases.

CONTROL OF COLLAPSE-REVIVAL PHENOMENON USING A TIME VARYING SQUEEZED FIELD

We have investigated the interaction of two level atom with time varying quadrature squeezed light field. Jaynes-Cummings model is used for solving the atom radiation interaction. Time evolution of the system for different squeezing parameter and phase have been studied. There are no well-defined revivals in population inversion when the squeezed phase is π and the squeezing parameter is greater than 0.5. Using a time varying frequency for the light field, it is found that the randomness of the population inversion and the collapse revival phenomena can be controlled. Frequency modulation of the field can thus be used as a tool for manipulating the squeezed light atom interaction.

Entanglement Dynamics of Three Qubits in the Non-Markovian Environments

We consider the entanglement dynamics of three two-level atoms interacting with independent structured non-Markovian reservoirs. The atoms are initially prepared in a mixed multipartite Greenberger—Horne—Zeilinger (GHZ)-type state. Based on the purity and non-Markovian reservoirs, we show that the reservoirs' entanglement can exhibit not only multipartite entanglement sudden birth but also entanglement sudden death, and the revival of atom entanglement is not always accompanied by the disentanglement of reservoirs. Meanwhile, we derive the quantitative criteria for multipartite revival and death phenomena and find multipartite entanglement sudden death and entanglement sudden birth simultaneous apparition time is independent of the initial three-qubit state.

The solutions of the generalized classical and quantum harmonic oscillators with time dependent mass, frequency, two-photon parameter and external force: The squeezing effects

The exact analytical solutions of a generalized classical harmonic oscillator with time dependent mass, frequency, two-photon parameter and external forces are obtained. By using the invariance property of the scaled Wronskian, these solutions are used to obtain the solutions of the quantum mechanical counterpart of the oscillator under Heisenberg picture. In order to discuss the applications of these solutions of the quantum mechanical oscillator, we calculate the exact analytical expressions for the second order variances of both the canonically conjugate quadratures in terms of the time dependent mass, frequency and two-photon parameter. However, these variances calculated in terms of the initial coherent state do not depend on the time dependent driven terms. We argued that the time dependent frequency is on the way of the exact analytical solutions and hence it is kept constant throughout the investigation. We, however, discuss few situations of physical interest where the mass is varying in time. The special circumstance where all the parameters are time independent is used to discuss the squeezing effect in both the quadratures. It is seen that the parameter g involving the two-photon interaction term produces squeezing effects. With the increase of interaction time, the squeezing in both the quadrature components exhibit collapse and revival phenomena for g < ω (frequency of the oscillator). The squeezing of X-quadrature is completely ruled out for g > ω. However, the squeezing for P-quadrature is possible for small interaction time. The squeezing patterns of the X and P-quadratures are also discussed for pulsating mass, and for mass increasing with time. The squeezing is also discussed when the mass is increasing exponentially in time. It is envisage that the solutions could be used in the investigation of quantum statistical properties of the radiation field coupled to the said oscillator.

Interaction of a two-level atom and a field with a time-varying frequency in a Kerr-like medium

The interactions between a two-level atom and a coherent field with a time-varying frequency in a Kerr-like medium have been investigated. It is found that the average of the atomic population inversion is deformed as the frequency changing with time in sine form. It is interesting that when the amplitude of field modulation and third-order nonlinear susceptibility are adjusted to certain values, the collapse–revival phenomena of population inversion in the standard Jaynes–Cumming model disappear and are replaced by a nearly Rabi-oscillation-like evolution. In other words, population inversion in the initial coherent field behaves like that in the initial Fock state. This provides a new way to manipulate the interaction between the atom and the field, and has potential applications in quantum information and quantum communication.

Experimental Demonstration of Photonic Entanglement Collapse and Revival

We demonstrate the collapse and revival features of the entanglement dynamics of different polarization-entangled photon states in a non-Markovian environment. Using an all-optical experimental setup, we show that entanglement can be revived even after it suffers from sudden death. A maximally revived state is shown to violate a Bell's inequality with 4.1 standard deviations which verifies its quantum nature. The revival phenomenon observed in this experiment provides an intriguing perspective on entanglement dynamics.

Sub-Poissonian photon statistics of light in interaction of two-level atoms in superposed states with a single mode superposed coherent radiation

We study sub-Poissonian photon statistics of light in interaction of a single mode radiation, initially either in a coherent state or in a superposed coherent state with an assembly of two-level atoms using the Hamiltonian, H = ω(a+a + Sz) + g(aS+ + a+ S–) in natural units, where a+ and a are creation and annihilation operators, Sz, S ± are the collective Dicke operators, g is the coupling constant, and ω is the energy of the photons and also the separation between the two atomic levels. We study the cases of (i) a single two-level and (ii) of two two-level atoms interacting with a single mode coherent or superposed coherent radiation. We find that for large coupling time gt, Fano factor shows collapses and revival phenomena, and that the variation is large for small mean number of photons. We also find that in the case of two two-level atoms, photon statistics shows larger sub-Poissonian than the case of a single two-level atom, and that there is no definite relationship between squeezing and sub-Poissonian photon statistics of light.

The nonequilibrium phase transition and the symmetry revival induced by time delay in an asymmetric bistable system with correlated noises

The nonequilibrium phase transition and the symmetry revival induced by time delay in a bistable system are investigated. The stationary probability distribution function (SPDF) of the bistable system with time delay and correlated noises are calculated by an analytical method and stochastic simulation respectively. The analytical and simulative results indicate that: (1) There is a certain value of λ(λ denotes the strength of correlations between the multiplicative and additive noises) to make the SPDF symmetric under some time delay; however, above or below the given value, the symmetry will be broken; (2) With the monotonic change of λ, the unimodal peak structure of SPDF becomes bimodal at the beginning, then it becomes unimodal again; this means that there is a reentrance phenomenon in the process; (3) There is a critical value of delay time, which makes the lower peak of SPDF equal to the higher one under the critical condition. This means that the symmetry revival phenomenon emerges.

Non-Markovian dynamics of entanglement for multipartite systems

Entanglement dynamics for a couple of two-level atoms interacting with independent structured reservoirs is studied using a non-perturbative approach. It is shown that the revival of atom entanglement is not necessarily accompanied by the sudden death of reservoir entanglement, and vice versa. In fact, atom entanglement can revive before, simultaneously or even after the disentanglement of reservoirs. Using a novel method based on the population analysis for the excited atomic state, we present the quantitative criteria for the revival and death phenomena. To give a more physically intuitive insight, the quasimode Hamiltonian method is applied. Our quantitative analysis is helpful for the practical engineering of entanglement.

Quantum rings with time-dependent spin-orbit coupling: Spintronic Rabi oscillations and conductance properties

The strength of the (Rashba-type) spin-orbit coupling in mesoscopic semiconductor rings can be tuned with external gate voltages. Here we consider the case of a periodically changing spin-orbit interaction strength in time as induced by sinusoidal voltages. In a closed one dimensional quantum ring with weak spin-orbit coupling, Rabi oscillations are shown to appear. We find that the time evolution of initially localized wave packets exhibits a series of collapse and revival phenomena. Partial revivals---that are typical in nonlinear systems---are shown to correspond to superpositions of states localized at different spatial positions along the ring. These ``spintronic Schr\"odinger-cat states'' appear periodically, and similarly to their counterparts in other physical systems, they are found to be sensitive to disturbances caused by the environment. The time-dependent spin transport problem, when leads are attached to the ring, is also solved. We show that the ``sideband currents'' induced by the oscillating spin-orbit interaction strength can become the dominant output channel, even in the presence of moderate thermal fluctuations and random scattering events.

Semiclassical investigation of the revival phenomena in a one-dimensional system

In a quantum revival, a localized wave packet re-forms or ‘revives’ into a compact reincarnation of itself long after it has spread in an unruly fashion over a region restricted only by the potential energy. This is a purely quantum phenomenon, which has no classical analog. Quantum revival and Anderson localization are members of a small class of subtle interference effects resulting in a quantum distribution radically different from the classical after long time evolution under classically nonlinear evolution. However, it is not clear that semiclassical methods, which start with the classical density and add interference effects, are in fact capable of capturing the revival phenomenon. Here we investigate two different one-dimensional systems, the infinite square well and Morse potential. In both the cases, after a long time the underlying classical manifolds are spread rather uniformly over phase space and are correspondingly spread in coordinate space, yet the semiclassical amplitudes are able to destructively interfere over most of coordinate space and constructively interfere in a small region, correctly reproducing a quantum revival. Further implications of this ability are discussed.

Atom optics quantum pendulum

We explain the dynamics of cold atoms, initially trapped and cooled in a magneto-optic trap, in a monochromatic stationary standing electromagnetic wave field. In the large detuning limit the system is modeled as a nonlinear quantum pendulum. We show that wave packet evolution of the quantum particle probes parametric regimes in the quantum pendulum which support classical period, quantum mechanical revival and super revival phenomena. Interestingly, complete reconstruction in particular parametric regime at quantum revival times is independent of potential height.

Quantum fluctuations in the mazer

Quantum fluctuations in the mazer are considered, arising either from the atomic motion or from the quantized intracavity field. Analytical results, for both the meza and the hyperbolic secant mode profile, predict for example an attenuation of tunnelling resonances due to such fluctuations. The case of a Gaussian mode profile is studied numerically using a wave packet propagation approach. The method automatically takes into account fluctuations in the atomic motion and the dynamics is especially considered at or adjacent to a tunnel resonance. We find that the system evolution is greatly sensitive to the atom–field detuning, bringing about a discussion about the concept of adiabaticity in this model. Further, a novel collapse–revival phenomenon is demonstrated, originating from the quantum fluctuations in the atomic motion rather from field fluctuations, as is normally the case.

Analysis of Revival Phenomenon for Strong-Field Excitation of Rydberg Atom

Analysis of Revival Phenomenon for Strong-Field Excitation of Rydberg AtomThe autocorrelation function for a one-dimensional Rydberg atom irradiated by half-cycle pulse (HCP) is investigated. At weak field only the full revival is observed, at a medium field strength - a fractional revival, whereas at a strong field (when the continuous energy states are also populated) the revival phenomenon disappears. The contribution of the continuous energy states in the autocorrelation function decreases dramatically with time after the wave packet formation.

Collapse and revival of the Doppler-Rabi oscillations of a moving atom in a cavity

Collapse and revival of the Doppler-Rabi oscillations of a two-level atom moving in a cavity electromagnetic field are analyzed. The coupled atom-field dynamics are predicted accurately by numerical calculation and approximately by using the stationary phase approximation combined with the Poisson summation formula. The collapse and revival patterns are shown to be qualitatively different in the cases of moving atom and atom at rest. In particular, quantum revivals of Doppler-Rabi oscillations occur with a period determined by the Doppler shift of the atomic transition frequency. This regime of Doppler-Rabi oscillations requires that the Rabi frequency and the Doppler shift satisfy the condition ΩR ≪ ΩD. Under the inverse relation, the collapse- revival phenomenon generally does not occur. It is shown that even a small amount of atom-cavity detuning eliminates collapse-revival behavior. The analysis is performed for both coherent and thermal cavity fields.

Entanglement in the anisotropic Heisenberg XYZ model with different Dzyaloshinskii-Moriya interaction and inhomogeneous magnetic field

We investigate the entanglement in a two-qubit Heisenberg XYZ system with different Dzyaloshinskii-Moriya (DM) interaction and inhomogeneous magnetic field. It is found that the control parameters (Dx, Bx and bx) are remarkably different with the common control parameters (Dz, Bz and bz) in the entanglement and the critical temperature, and these x-component parameters can increase the entanglement and the critical temperature more efficiently. Furthermore, we show the properties of these x-component parameters for the control of entanglement. In the ground state, increasing Dx (spin-orbit coupling parameter) can decrease the critical value bxc and increase the entanglement in the revival region, and adjusting some parameters (increasing bx and J, decreasing Bx and Δ) can decrease the critical value Dxc to enlarge the revival region. In the thermal state, increasing Dx can increase the revival region and the entanglement in the revival region (for T or bx), and enhance the critical value Bxc to make the region of high entanglement larger. Also, the entanglement and the revival region will increase with the decrease of Bx (uniform magnetic field). In addition, small bx (nonuniform magnetic field) has some similar properties to Dx, and with the increase of bx the entanglement also has a revival phenomenon, so that the entanglement can exist at higher temperature for larger bx.

Coherent states and quantum oscillations of BEC-BCS systems

We show by using the mean-field approximation that the states of composite Fermi–Bose superfluids created in cold-atom traps via a Feshbach resonance at zero temperature are generalized SU ( 2 ) ⊗ SU ( 1 , 1 ) coherent states. In response to a sudden change of the interaction between fermionic atoms and bosonic molecules, a Cooper pair can exhibit collapse and revival quantum behaviors for an initial generalized coherent state of molecules, and Rabi oscillation for a vacuum molecular state. Occurrence of the collapse and revival phenomenon is thus the manifestation of the formation of the Bose–Einstein condensate.

A moving four-level N-type atom interacting with cavity fields

We study the non-resonant interaction of a moving four-level N-type atom with cavity fields in the presence of a nonlinear Kerr-like medium. The exact solutions of the considered models are obtained when the atom is initially in an upper state, while the field is in a coherent state. We derive an expression for the occupation numbers of the photon state. Also, the momentum increment of the atomic system is obtained. The influence of the detuning parameters or the Kerr medium on the evolution time of the momentum increment is examined numerically. The existence of the detuning parameters leads to the retraction of the revivals for the three atomic systems. On the other hand, the presence of the Kerr medium is conducive to increasing the regularity and periodicity of the collapse–revival phenomenon for the one- and two-mode systems, while chaotic behaviour occurs for the three-mode system.

Charismatic/Pentecostal Christians, Spirituality, and Treatment: The Revival Phenomenon

ABSTRACT This article examines the practice of revival as a means of “treating” people with addictions. Revival is practiced widely in Charismatic/Pentecostal circles and many people believe that that their recovery from addictions would not have been possible without it. The practice is reviewed and suggestions are made as to how to provide evidence-based practice while recognizing the contributions of revival practice.