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A four-level “Ξ-system”: Exact solvability and effective evolution operators via multiple scales
Abstract
Properties of a four-level atomic system interacting with one and two modes of the electromagneticfield in a “Ξ”-configuration are investigated. By linearization of the Hamiltonians we show that the corresponding mathematical models are exactly solvable. To obtain simpler effective Hamiltonians the perturbative method of multiple scales is applied. The lowest-order corrections to the resulting effective evolution operators are also calculated.
We investigate the atom−field system in the framework of harmonic oscillators based on deformed Heisenberg algebras. We explore the dynamic characteristics of the considered system under the effect of a nonlinear field. In particular, we consider the atomic population, atomic coherence, and atom−field entanglement for a system that comprises a single five−level atom interacting with a single−mode nonlinear field when the deformation effect is taken into account. We examine the time evolution of the quantum quantifiers in the presence of deformation when the initial state of the quantized field is defined to be a nonlinear coherent state (CS) or a superposition state
Recently, Kumar Gerry et al. [Phys. Rev. A 90, 033427 (2014)] studied the coherence control in a six-level atom through solving the Schrödinger equation in the field-interaction representation. In this manuscript, we investigate the interaction between a six-level atomic system (SLAS) and a single-mode field initially prepared in a squeezed coherent state. We extend the Jeans-Cummings model to describe the interaction between the atom and the squeezed field (SF) and the system dynamics. We examine the time evolution of the atomic coherence, non-local correlation, statistical properties within the bipartite system in the presence and absence intensity-dependent coupling (I-DC) for different squeezing regimes of the field.
Entanglement and atomic coherence are investigated for a system that consists of a four-level atom (FLA) coupled with a nonlinear field. We explore the influence of field deformation and photon transition on the dynamical behavior of the quantum measures when the quantized field is initially in a coherent state or in a Schrödinger cat state with and without energy dissipation. The results indicate that the entanglement and coherence in the FLA–field system can be controlled and manipulated through the nature and initial state of the quantized field.
We study the dynamics of the von Neumann entropy, Wehrl entropy, and Wehrl phase distribution for a single four-level ladder-type atom interacting with a one-mode cavity field taking into account the atomic motion. We obtain the exact solution of the model using the Schr¨odinger equation under specific initial conditions. Also we investigate the quantum and classical quantifiers of this system in the nonresonant case. We examine the effects of detuning and the atomic motion parameter on the entropies and their density operators. We observe an interesting monotonic relation between the different physical quantities in the case of nonmoving and moving atoms during the time evolution. We show that both the detuning and the atomic motion play important roles in the evolution of the Wehrl entropy, its marginal distributions, entanglement, and atomic populations.
In this paper, we investigate the collapse-revival phenomenon and the Poissonian statistics of a four-level N-configuration atom interacting with two-mode cavity fields. We assume that the transition between the upper level and the third level of an atom is coupled by the frequency of mode one, while the other transitions are coupled by the frequencies of the two modes or mode two. The interaction is a multi-photon process and a Kerr medium is taken into account in the non-resonant case. We solve the models when the atom is initially prepared in a coherent superposition of the upper and ground states and the field is considered in a coherent state. The effects of both the detuning and the Kerr medium on the temporal behaviour of some statistical aspects are analysed. The conclusions are reached and some features are given.
Decoherence of the Schrödinger-cat states in two-mode cavities are discussed. The cat states are assumed to be produced by the measurement of the energy of a three-level atom which have passed through the cavity. The evolution of the field density matrix is obtained from the Lindblad form of the dissipater at zero temperature. It is shown that the decoherence time of the two-mode cat states critically depends on the degree of entanglement. Implications for decoherence of entanglement in the macroscopic scales are discussed.
In this paper we are interested in studying the entanglement between a single four-level ladder-type atom interacting with
one-mode cavity field when the atomic motion is taken into account. The exact solution of the model is obtained by using Schrodinger
equation for a specific initial conditions. The field entropy of this system is investigated in the non-resonant case. The
effects of the detuning parameter and the atomic motion on the entanglement degree are examined. These investigations show
that both of the detuning and the atomic motion play important roles in the evolution of the von Neumann entropy and atomic
populations. Finally, conclusions and some features are given.
KeywordsFour-level atom–von Neumann entropy–Entanglement
We study the entanglement between a single five-level atom interacting with a one-mode cavity field in the case where the
atomic motion is taken into account. The field entropy of this system is investigated under the nonresonant conditions. The
effects of the detuning parameter and the atomic motion on the degree of entanglement are investigated. We show that both
the detuning and atomic motion play important roles in the evolution of the von Neumann entropy and atomic populations.
Keywordsfive-level atom–von Neumann entropy–atomic motion
WKB theory is a powerful tool for obtaining a global approximation to the solution of a linear differential equation whose highest derivative is multiplied by a small parameter ε; it contains boundary-layer theory as a special case.
We use the multiple-time-scale perturbation technique to treat the long interaction time limit of atomic scattering from a standing light wave, i.e., the atomic Kapitza—Dirac effect. The use of the method is restricted to an exactly resonant interaction and an oblique angle of incidence. We develop the theory in some detail in view of its general applicability to a large class of physical phenomena. The accuracy of the ensuing analytical expressions is tested against exact numerical computations of the effect, and the allowed ranges of the parameters are given with reference to the actual atomic experiments. For moderately large values of the parameters that characterize the field strength and the photon recoil, the method is found to give excellent approximations for the long-time behavior, whereas straightforward perturbation expansions diverge because of unbounded secular terms.
The multiple-linear-time-scale method is used to construct a perturbation theory for N-level quantum systems subjected to time-dependent perturbations. The secular and small-denominator terms which plague conventional time-dependent perturbation theory are avoided; consequently, the theory is useful for treating long-term behavior and resonant interactions. The introduction of a self-energy operator allows the shifts in energy levels to be displayed explicitly. When the perturbation is the dipole interaction with an electromagnetic field, the successive approximations yield the well-known rotating-wave approximation, corrections due to counter rotating terms, and the Bloch-Siegert frequency shift. The formalism is applicable to arbitrary pulse shapes, provided that ∥H1∥≪ℏω̅ , where ∥H1∥ represents the inter-action strength and ω̅ is a characteristic frequency of the field. Rabi oscillations induced by multiphoton resonances are automatically included, and this effect is demonstrated in the case of a square pulse with frequency approximately one half the resonance frequency for a transition between two vibrational levels of a molecule. These calculations are compared to an exact numerical solution in order to find the limits of validity of the approximation. Even at high intensities (I∼1014 W/cm2) the shifted resonance frequency is quite accurate; however, the approximate and exact solutions are slightly out of phase, so that the approximate solution can only be trusted for a few Rabi cycles. For much lower intensities (I∼1010 W/cm2), the approximate solution is valid for many thousands of Rabi cycles.
We develop a systematic method of operator perturbation theory in the Heisenberg picture, which is the formal analog of time-independent stationary-state perturbation theory in the Schrödinger picture. However, we use the eigenoperators and operator eigenfrequencies to calculate the time evolution of the dynamical variables of interest. The spectrum of the unperturbed Hamiltonian is assumed to be discrete. The method is especially well-suited for treating systems where the unperturbed Hamiltonian represents noninteracting fermions and/or bosons, and therefore, we illustrate the method using the exactly solvable model of Jaynes and Cummings: a single two-level atom interacting with a single quantized field mode. The time evolution of these operator solutions is shown to be unitary order by order in powers of the interaction strength. We find that the time evolution of exact operator solutions is well approximated for significantly longer times by the time evolution of this method's operator solutions than by the time evolution of the same operator solutions calculated using the Dyson expansion. We also demonstrate that time-dependent operator solutions are very convenient for computing quantities such as correlation functions.
In this first article of work on radiation processes, a multiple-time-scale expansion method specifically for these processes is developed, based on the two distinct time scales corresponding to the natural frequency and the linewidth of the radiation. The method is then demonstrated by treating systems of one and two two-level atoms. It is seen that the exponential-decay behavior of the excited-state amplitude, assumed a priori by Weisskopf and Wigner, follows naturally from the mathematical procedure of eliminating the secularity in each order of the expansion. The results of two-atom systems also agree with those of previous workers.
We investigate the connection between a multiple-scale approach recently
applied to anharmonic oscillators to avoid secular terms in the
time-dependent perturbation series, and a frequency-operator method
developed earlier for the same purpose.
In this paper we investigate the properties of a four-level atomic system interacting with two modes of the electromagnetic field in a cavity. By linearization of the Hamiltonian we show that the corresponding mathematical model is exactly solvable. To obtain simpler effective Hamiltonians the method of multiple scales is applied. It is shown that under some special values of the detunings and coupling constants, exactly predictable collapses and revivals in the atomic populations appear. Also, due to the specific interactions of the atom with the cavity modes and the ensuing measurement of energy or angular momentum and energy, two-mode Schrödinger-cat states arise in the cavity. Finally, the stabilization and trapping properties of the system are demonstrated in a lossy cavity within the Wigner-Weisskopf approximation. The existence of stationary states is shown and it is also shown that interference effects can cause cancellation of one line in both transmitted light and spontaneous emission spectra.
We review the dynamical interaction of an atom with radiation in the framework of models of the Jaynes-Cummings type. The atom is restricted to a few energy levels and the radiation consists of quantized near-resonant loss-free modes. A general formalism is presented that allows a compact discussion of atom and field dynamics without approximations. Quantum wave packet collapse and revival phenomena, coherent trapping states, and some experimental prospects for two-level and three-level atoms are treated in detail, and some related effects are discussed more briefly.
Applications of the method of multiple scales to the quantum-optical problems are reviewed. After a preliminary study of applications to the classical and quantum anharmonic oscillator, several examples in the spontaneous emission, resonance fluorescence, and cavity quantum electrodynamics are analyzed. A preliminary account of application of the method to a model of Bose–Einstein condensates is given. A common thread throughout the main body of the review is the stabilization of the wave function and population of a level due to the phase modulation by an external agent.
We describe the dynamics of an exactly solvable Raman coupled model interacting with two modes of a quantized cavity electromagnetic field. The model consists of a three-level atom in the Λ configuration whose excited state is considered to be far off resonance and is adiabatically eliminated. The effective Hamiltonian for this system contains products of the transition operators between the two ground-state levels and a field annihilation operator from one mode and a creation operator from the other. This Hamiltonian provides an example of a zero-photon process. We study the atomic inversion and investigate the production of nonclassical effects such as antibunched light, violations of the Cauchy-Schwartz inequality, and squeezing.