Collapse Revival Research Articles

Quantum Floquet spectra of surface and bulk state of topological insulator

Topological insulator is a new class of material, which consists of insulating bulk and conducting surface state. In this study, we investigated the behaviour of bulk and surface state of topological insulator under a high frequency electromagnetic field, produce a kind of oscillation called as Floquet oscillation. We successfully identified the classical and quantum behavior of Floquet oscillation in the prospective of collapse revival spectra. This work shows the bulk and surface states behaviour of a topological insulator with the help of Floquet and Rabi frequency and describes the dominancy of Floquet oscillation in the low energy physics. Numerical simulation has been applied for verifying the Floquet approximation.

On the collapse revival of entanglement in a two-dimensional noncommutative harmonic oscillator: An algebraic approach

In this paper, we employ an algebraic approach to investigate the dynamics of entanglement in a two-dimensional noncommutative harmonic oscillator. We start by using the so-called the Bopp shift to convert the Hamiltonian describing the system into an equivalent commutative one. This allows us to take advantage of the Schwinger oscillator construction of [Formula: see text] Lie algebra to reveal through the time evolution operator that there is a connection between the noncommutativity and entanglement. The degree of entanglement between the states is evaluated by using the purity function. The remarkable emerging feature is that, as long as the noncommutativity parameter is nonvanishing, the system exhibits the phenomenon of collapse and revival of entanglement.

Signature of structural parameter in Floquet spectra of [formula omitted] lattice model

The impact of structural parameter is described in the Floquet spectra of α−T3 lattice model. There is a comparison between classical and quantum Floquet frequency by using the Jaynes–Cummings model. The collapse–revival behaviour of Floquet oscillations has been studied and found that structural parameter has a major impact on collapse–revival time. The pump–probe experiment has been studied theoretically for the detection of Floquet oscillation. The study of Rabi oscillation has been performed. There is a modification in the Rabi frequency after including counter-rotating term i.e. Bloch–Siegert shift. The Bloch–Siegert shift has been studied in the perspective of structural parameter. The band tuning is described with help of Floquet frequency. To justify analytical expression of Floquet frequency, a numerical simulation has been performed.

Population dynamics of ultra-cold atoms interacting with radiation fields in the presence of inter-atomic collisions

We investigate the dynamics of a system that consists of ultra-cold three-level atoms interacting with radiation fields. We derive the analytical expressions for the population dynamics of the system, particularly, in the presence and absence of nonlinear collisions by considering the rotating wave approximation (RWA). We also reanalyze the dynamics of the system beyond RWA and obtain the state vector of the system to study and compare the time behavior of population inversion. Our results show that the system undergoes two pure quantum phenomena, i.e., the collapse–revival and macroscopic quantum self-trapping due to nonlinear collisional interactions. The occurrence of such phenomena strongly depends on the number of atoms in the system and also the ratio of interaction strengths in the considered system. Finally, we show that the result of the perturbed time evolution operator up to the second-order is in agreement with the numerical solution of the Schrodinger equation. The results presented in the paper may be useful for the design of devices that produce a coherent beam of bosonic atoms known as an atom laser.

Dynamics of an atomic Bose–Einstein condensate interacting with nonlinear quantized field under the influence of Stark effect

In this paper we propose a generalized f-deformed Dicke model for the dynamics of an atomic Bose–Einstein condensate (BEC) in the presence of collisional interactions and Stark effect. The BEC contains N two-level atoms and the nonlinear characteristics of quantized field originates from the f-deformation of the associated raising and lowering operators. After solving the time dependent Schrödinger equation corresponding to the model Hamiltonian, we investigate the dynamics of the system under various physical situations; in particular, two different initial states for the quantized field i.e., coherent and squeezed field. In this regard, at first, we obtain some analytical expressions correspond to the energy transfer and non-classical properties of the constituents of system and then proceed to analyze their dynamics, numerically. Our results show that the dynamical evolution of system can be governed by tuning the parameters describing the considered interactions. In particular, the energy levels can be controlled and shifted by changing the Stark effect. Collapse–revival phenomenon occurs in the patterns of atomic inversion, specially in the case of initial coherent state. However, with initial squeezed field typical revivals accompanied with short time collapses may be observed; moreover, the general behavior of atomic inversion shows irregular oscillatory manner. Short time squeezing takes place and can be exchanged between the field quadratures in the absence or presence of Stark effect. Also, the amount of entanglement depends on the nonlinearities which specify the interactions among the atoms in BEC and photons in the quantized field. Interestingly, the general pattern of Von Neumann entropy as a measure of entanglement is independent of the chosen initial states. However, the amplitude of entropy reduces in the case of squeezed state which means that the system possesses more separable states in this condition. In addition, the time evolution of Bloch sphere confirms irregular behavior of entanglement in the system.

Quantum floquet oscillation in borophane

The hydrogenated borophene is known as borophane, a two-dimensional material which has Dirac characteristics. In this work, it is described how anisotropy of borophane, i.e. wave vector angle, plays a significant role in the Floquet frequency and collapse–revival phenomenon. The various mathematical techniques have been described for the formulation of Floquet frequency. The role of anisotropy is justified by using numerical simulation. The bandgap of borophane can be opened by using Floquet frequency. The Rabi oscillation and Bloch–Siegert shift have also been studied in the perspective of anisotropy.

Probing relaxation dynamics of a few strongly correlated bosons in a 1D triple well optical lattice

The relaxation process of a few strongly interacting bosons in a triple well optical lattice is studied from the first principle using the multiconfigurational time-dependent Hartree method for bosons. We report the contrasting response of the system under two independent quench processes: an interaction quench and a lattice depth quench. We analyze the evolution of the reduced one-body density matrix, two-body density and the Shannon information entropy for a wide range of lattice depth and interaction strength parameters. For the strong interaction quench, we observe a very fast relaxation to the steady state. In contrast, for the lattice depth quench, we observe collapse–revival dynamics in all the key measures. We also provide the best fitting formulas for relaxation and revival time which follow power law decay.

Correlation Dynamics of Dipolar Bosons in 1D Triple Well Optical Lattice

The concept of spontaneous symmetry breaking and off-diagonal long-range order (ODLRO) are associated with Bose–Einstein condensation. However, as in the system of reduced dimension the effect of quantum fluctuation is dominating, the concept of ODLRO becomes more interesting, especially for the long-range interaction. In the present manuscript, we study the correlation dynamics triggered by lattice depth quench in a system of three dipolar bosons in a 1D triple-well optical lattice from the first principle using the multiconfigurational time-dependent Hartree method for bosons (MCTDHB). Our main motivation is to explore how ODLRO develops and decays with time when the system is brought out-of-equilibrium by a sudden change in the lattice depth. We compare results of dipolar bosons with contact interaction. For forward quench ( V f > V i ) , the system exhibits the collapse–revival dynamics in the time evolution of normalized first- and second-order Glauber’s correlation function, time evolution of Shannon information entropy both for the contact as well as for the dipolar interaction which is reminiscent of the one observed in Greiner’s experiment [Nature, 415 (2002)]. We define the collapse and revival time ratio as the figure of merit ( τ ) which can uniquely distinguish the timescale of dynamics for dipolar interaction from that of contact interaction. In the reverse quench process ( V i > V f ) , for dipolar interaction, the dynamics is complex and the system does not exhibit any definite time scale of evolution, whereas the system with contact interaction exhibits collapse–revival dynamics with a definite time-scale. The long-range repulsive tail in the dipolar interaction inhibits the spreading of correlation across the lattice sites.

The amplitude of the cavity pump field and dissipation effects on the entanglement dynamics and statistical properties of an optomechanical system

In this paper we consider a cavity optomechanics consists of a two-level atom that interacts with a single-mode quantized field in the presence of a strong driving laser. By evaluating the effective Hamiltonian of the whole system which contains the field dissipation, we derive the time-dependent state vector of the system corresponding to particular initial conditions for the cavity field and oscillatory mirror. In the continuation, we pay our attention to the dynamical properties of the considered system by choosing different initial conditions for the atom. In particular, the effects of amplitude of the pump field, various initial atomic states and cavity loss rate on the entanglement, atomic population inversion and photon and phonon statistical properties are numerically analysed. According to the obtained results, maximal entanglement for all chosen initial states of the system is visible as time develops. Moreover, the presence of dissipative field leads to decrease the entanglement and finally tends to a fixed positive value, while the Mandel parameter as well as the population inversion reduce and tend to a fixed value in the negative region. Also, the appearance of non-classicality feature such as typical collapse–revival phenomenon in the (negative values of) Mandel parameter is noticeable.

Gaussian, Shannon, and Tsallis entropies of bound magnetopolaron in Gaussian and asymmetric quantum qubit

This paper compared the results of Gaussian, Shannon, and Tsallis entropies in the study of the decoherence of magnetopolaron in nanostructures in the presence of an electric field. The unitary transformation and linear combination method are used to derive the energy spectrum of the magnetopolaron in Gaussian and asymmetric quantum dot. Results suggest that Shannon entropy is suitable for the faster transmission of information about the system. With the increase in the strength of electric field and cyclotron frequency, the collapse revival in entropy is observed. With the asymmetric quantum dot, one realizes that the amount of information indicated by Shannon entropy is efficient than with other entropies. With the enhancement of the electric field strength, the entropies evolve as a wave envelop, and the information is transmitted as a wave bundle.

Application of entropies to the study of the decoherence of magnetopolaron in 0-D nanosystem

Many methods have been experimented to study decoherence in quantum dot (QD). Tsallis, Shannon and Gaussian entropy have been used to study decoherence separately; in this paper, we compared the results of the Gaussian, Shannon, and Tsallis entropies in 0-D nanosystem. The linear combination operator and the unitary transformation was used to derive the magnetopolaron spectrum that strongly interacts with the LO phonons in the presence of an electric field in the pseudoharmonic and delta quantum dot. Numerical results revealed for the quantum pseudo dot that: (i) the amplitude of Gauss entropy is greater than the amplitude of Tsallis entropy which in turn is greater than the amplitude of Shannon entropy. The Tsallis entropy is not more significant in nanosystem compared to Shannon and Gauss entropies, (ii) with an increase of the zero point, the dominance of the Gauss entropy on the Shannon entropy was observed on one hand and the dominance of the Shannon entropy on the Tsallis entropy on the other hand; this suggested that in nanosystem, Gauss entropy is more suitable in the evaluation of the average of information in the system, for the delta quantum dot it was observed that (iii) when the Gauss entropy is considered, a lot of information about the system is missed. The collapse revival phenomenon in Shannon entropy was observed in RbCl and GaAs delta quantum dot with the enhancement of delta parameter; with an increase in this parameter, the system in the case of CsI evolved coherently; with Shannon and Tsallis entropies, information in the system is faster and coherently exchanged; (iv) the Shannon entropy is more significant because its amplitude outweighs the others when the delta dimension length enhances. The Tsallis entropy involves as wave bundle; which oscillate periodically with an increase of the oscillation period when delta dimension length is improved.

Dynamic evolution of double $$\Lambda $$ Λ five-level atom interacting with one-mode electromagnetic cavity field

In this paper, the model describing a double $$\Lambda $$ five-level atom interacting with a single mode electromagnetic cavity field in the (off) non-resonate case is studied. We obtained the constants of motion for the considered model. Also, the state vector of the wave function is given by using the Schrodinger equation when the atom is initially prepared in its excited state. The dynamical evolutions for the collapse revivals, the antibunching of photons and the field squeezing phenomena are investigated when the field is considered in a coherent state. The influence of detuning parameters on these phenomena is investigated. We noticed that the atom–field properties are influenced by changing the detuning parameters. The investigation of these aspects by numerical simulations is carried out using the Quantum Toolbox in Python (QuTip).

Linear entropy and collapse–revival phenomenon for a general formalism N-type four-level atom interacting with a single-mode field

In this paper, the linear entropy and collapse–revival phenomenon through the relation ( $$\langle {\hat{a}}^{+}{\hat{a}} \rangle -{\bar{n}}$$ ) in a system of N-configuration four-level atom interacting with a single-mode field with additional forms of nonlinearities of both the field and the intensity-dependent atom-field coupling functional are investigated. A factorization of the initial density operator is assumed, considering the field to be initially in a squeezed coherent states and the atom initially in its most upper excited state. The dynamical behavior of the linear entropy and the time evolution of ( $$\langle {\hat{a}}^{+} {\hat{a}} \rangle -{\bar{n}}$$ ) are analyzed. In particular, the effects of the mean photon number, detuning, Kerr-like medium and the intensity-dependent coupling functional on the entropy and the evolution of ( $$\langle {\hat{a}}^{+} {\hat{a}} \rangle -{\bar{n}}$$ ) are examined.

Economic Growth and Recovery in the United States: 1919-1941

This paper has two main sections and an appendix. The first provides an overview of what lay behind record productivity growth in the US economy between 1929 and 1941. The second considers the role of rigidities and other negative supply conditions in worsening the downturn and slowing recovery. While I argue consistently that the overarching explanation of the Great Depression will and should continue to emphasize a collapse and slow revival of the growth of aggregate demand, I spend relatively little time on what drives this, since the literature deals extensively with this elsewhere. The paper instead concentrates on the aggregate supply side - both the broad array of positive shocks that I argue propelled potential and, eventually, actual output forward, and the negative conditions which, in interaction with aggregate demand, may have increased the size of the output gap and prolonged its persistence. An appendix offers detail discussion and updated calculations of productivity growth rates for the critical period 1929 through 1941.

Damping, field–field correlation and dipole–dipole interaction effects on the entanglement and atomic inversion dynamics

In this paper we study the interaction between two two-level atoms with a two-mode quantized field in the presence of damping. Dipole–dipole interaction between the two atoms and the correlation between the two modes of field are also taken into account. To solve the model, using appropriate transformations, we reduce the considered model to a well-known Jaynes–Cummings model. After finding the analytical solution for the atom–field system, the effects of damping, field–field correlation and atomic dipole–dipole interaction on the entanglement between atoms and population inversion are investigated, numerically. It is observed that the dynamical behavior of the degree of entanglement for damped systems, in relatively large domains of time, takes a low but constant value adequately far from the beginning of the interaction. In addition, it is found that the value of population inversion after the initial oscillations takes negative values for damped systems and eventually vanishes by increasing time. Also, it is seen that simultaneous presence of both dipole–dipole interaction and field–field correlation provides typical collapse–revival phenomenon in the time-behavior of atomic inversion.

Periodic behavior of collapse–revival phenomenon in the full nonlinear interaction between a three-level atom and a single-mode field

In this paper based on a generalization of the Jaynes–Cummings model we solve the dynamical Hamiltonian describing the interaction between a (Λ or V-type) three-level atom and a single-mode field in the “full nonlinear regime” and then the analytical form of state vector of the system is explicitly obtained. In this manner, we encountered with “intensity-dependent detuning” as well as “intensity-dependent atom–field coupling” in our two models. Via choosing an appropriate deformation function (which imposes nonlinearity to the system) we consider the influence of Kerr-like medium from which the resonance condition for a selected number of quanta is achieved (selective transition is occurred). Furthermore, by these considerations, we may find the optimum values for atom–field coupling constants which provide a regular periodic behavior of probability amplitudes for the two considered atomic systems. Moreover, to show this periodic time behavior, the temporal evolution of the probability of the allowed atomic transitions as well as the Mandel parameter (as a non-classical sign) is depicted for various circumstances. As is observed, complete revivals may appear in some particular situations.

Collapse–revival of squeezing of two atoms in dissipative cavities**Project supported by the Science and Technology Plan of Hunan Province, China (Grant No. 2010FJ3148), the National Natural Science Foundation of China (Grant No. 11374096), and the Doctoral Science Foundation of Hunan Normal University, China.

Based on the time-convolutionless master-equation approach, we investigate the squeezing dynamics of two atoms in dissipative cavities. We find that the atomic squeezing is related to initial atomic states, atom–cavity couplings, non-Markovian effects and resonant frequencies of an atom and its cavity. The results show that a collapse–revival phenomenon will occur in the atomic squeezing and this process is accompanied by the buildup and decay of entanglement between two atoms. Enhancing the atom–cavity coupling can increase the frequency of the collapse–revival of the atomic squeezing. The stronger the non-Markovian effect is, the more obvious the collapse–revival phenomenon is. In particular, if the atom–cavity coupling or the non-Markovian effect is very strong, the atomic squeezing will tend to a stably periodic oscillation in a long time. The oscillatory frequency of the atomic squeezing is dependent on the resonant frequency of the atom and its cavity.

Effect of the time-dependent coupling on a superconducting qubit-field system under decoherence: Entanglement and Wehrl entropy

The dynamics of a superconducting (SC) qubit interacting with a field under decoherence with and without time-dependent coupling effect is analyzed. Quantum features like the collapse–revivals for the dynamics of population inversion, sudden birth and sudden death of entanglement, and statistical properties are investigated under the phase damping effect. Analytic results for certain parametric conditions are obtained. We analyze the influence of decoherence on the negativity and Wehrl entropy for different values of the physical parameters. We also explore an interesting relation between the SC-field entanglement and Wehrl entropy behavior during the time evolution. We show that the amount of SC-field entanglement can be enhanced as the field tends to be more classical. The studied model of SC-field system with the time-dependent coupling has high practical importance due to their experimental accessibility which may open new perspectives in different tasks of quantum formation processing.

Wave packet dynamics in various two-dimensional systems: A unified description

In this article we present an exact and unified description of wave packet dynamics in various 2D systems in the presence of a transverse magnetic field. We consider an initial minimum-uncertainty Gaussian wave packet and find that its long-term dynamics displays the universal phenomena of spontaneous collapse and quantum revival. We estimate the timescales associated with these phenomena based on very general arguments for various materials, whose carrier dynamics is described either by the Schrödinger equation or by the Dirac equation.

Collapse revival behaviour of the entanglement between V-type three-level atoms and two-mode photons in nonlinear Jaynes–Cummings model

In this paper the time evolution of von Neumann entropy, as a measure of entanglement between V-type three-level atoms and the union of a two-mode field, is studied. The atom–field interaction is assumed to occur in a Kerr-type medium with an intensity-dependent coupling. Introducing a Casmir operator whose eigenvalues, N, give total excitations in the system and commutes with the governing Hamiltonian, it is concluded that the latter is block-diagonal with ever growing dimensions. As we shall show, however, each block consists of two 2 × 2 blocks while all the others, (N −1) in number, are 3 × 3. We then proceed to analytically calculate the time-evolution operator which is also block-diagonal, each block with the same properties as that of the Hamiltonian. Our calculations show that, as expected, the atom–field entanglement oscillates which, depending upon the initial state, exhibits the phenomenon of collapse revivals. It is further shown that collapse revivals occur whenever both 2 × 2 blocks are involved in the time evolution of the system. Properties of such behaviour in the entanglement are also discussed in detail.