Parapositronium atom in the field of annihilation and optical photons as a nonlinear quantum system

A parapositronium atom in an optical laser field is described beyond the perturbation theory framework by a closed system of Heisenberg equations on operators of atoms and photons. Wwe consider the annihilation of the parapositronium atom, which starts from one or two quantum states; optical quantum transitions between these states are caused by one or two optical photons. Mean occupation numbers of these states are governed by a system of two nonlinear equations. We investigated particular stationary and nonstationary solutions of this system and find that annihilation photons substantially affect the annihilation process. We show that definite optical laser radiation may stabilize the parapositronium atom and make its lifetime hundreds of times longer than the lifetime of the free parapositronium atom in the 1s state. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 1, pp. 148–168, July, 2000.     

Entropy and entanglement of an effective two-level atom interacting with two quantized field modes in squeezed displaced fock states

We have studied the influences of ac-Stark shifts on the field quantum entropy, with “squeezed displaced Fock states” (SDFSs) basis. By a unitary transformation we derive a Raman-coupled Hamiltonian perturbatively in coupling constants. The exact results are employed to perform a careful investigation of the temporal evolution of entropy. A factorization of the initial density operator is assumed, with the privileged field mode being in the SDFS. We invoke the mathematical notion of maximum variation of a function to construct a measure for entropy fluctuations. The results show that the effect of the SDFS changes the quasiperiod of the field entropy evolution and entanglement between the atom and the field. The Rabi oscillation frequency, the collapse and revival times of the atomic coherence are found to have strikingly different photon-intensity dependent than those found previously. The general conclusions reached are illustrated by numerical results.     

Squeezed states and their applications to quantum evolution

In this paper, we consider quantum multidimensional problems solvable by using the second quantization method. A multidimensional generalization of the Bogolyubov factorization formula, which is an important particular case of the Campbell-Baker-Hausdorff formula, is established. The inner product of multidimensional squeezed states is calculated explicitly; this relationship justifies a general construction of orthonormal systems generated by linear combinations of squeezed states. A correctly defined path integral representation is derived for solutions of the Cauchy problem for the Schrödinger equation describing the dynamics of a charged particle in the superposition of orthogonal constant (E,H)-fields and a periodic electric field. We show that the evolution of squeezed states runs over compact one-dimensional matrix-valued orbits of squeezed components of the solution, and the evolution of coherent shifts is a random Markov jump process which depends on the periodic component of the potential.     

Entropy squeezing of a driven two-level atom in a cavity with injected squeezed vacuum

We derive, the time-dependent solution of the effective master equation for the reduced density matrix operator of a strongly driven atom coupled to a frequency-tunable cavity and damped by a squeezed vacuum. The effects of different parameters on the entropy squeezing factors, the variance squeezing and population inversion of such an atom emitted from the cavity, are discussed.     

Entanglement,fidelity and quantum chaos in cavity QED

Nonlinear dynamics in the fundamental interaction between a two-level atom with recoil and a quantized radiation field in a high-quality microcavity is studied. We consider the strongly coupled atom–field system as a quantum–classical hybrid with dynamically coupled quantum and classical degrees of freedom. We show that, even in the absence of any other interaction with environment, the coupling of quantum and classical degrees of freedom provides the emergence of classical dynamical chaos from quantum electrodynamics. Chaos manifests itself in the atomic external degree of freedom as a random walking of an atom inside a cavity with prominent fractal-like behavior and in the quantum atom–field degrees of freedom as a sensitive dependence of atomic inversion on small variations in initial conditions. It is shown that dependences of variance of quantum entanglement and of the maximum Lyapunov exponent on the detuning of the atom–field resonance correlate strongly. It is shown that the Jaynes–Cummings dynamics can be unstable in the regime of chaotic walking of an atom in the quantized field of a standing wave in the absence of any other interaction with environment. Quantum instability manifests itself in strong variations of quantum purity and entropy and in exponential sensitivity of fidelity of quantum states to small variations in the atom–field detuning. It is quantified in terms of the respective classical maximal Lyapunov exponent that can be estimated in appropriate in–out experiments. This result provides a quantum–classical correspondence in a closed physical system.     

Noise of quantum solitons and their quasi-coherent states

Quantum noise of optical solitons is analysed based on the exact solutions of the quantum nonlinear Schrödinger equation (QNSE) and the construction of the quantum soliton states. The noise limits are obtained for the local photon number and for the local quadrature phase amplitude. They are larger than the vacuum fluctuation. So in the fundamental soliton states the variance of the local photon number and the local quadrature phase amplitude cannot be squeezed. The soliton states with the minimum noise are quasi-coherent states, in which the quantum dispersion effects are negligible.     

The improved quantum switching mechanism based on contention

Quantum switching architecture is one of the promising schemes for transmitting quantum data to its destination. In this paper, we propose an improved quantum switching mechanism for solving the output contention problem. Firstly we analyze the impropriety of Tsai and Kuo’s quantum switching scheme for multicast service. Secondly an improved quantum switching scheme is presented for completing the switching task with output contention. Lastly an illustrative example is given to evaluate our method and the result shows that the present improved mechanism can ensure the data reliable in transmitting and is scalable for quantum switching with output contention.     

Squeezed quantum states on an interval and uncertainty relations for nanoscale systems

We construct families of squeezed quantum states on an interval and analyze their asymptotic behavior. We study the localization properties of a kind of such states constructed on the basis of the theta function. For the coordinate and momentum dispersions of a quantum particle on an interval, we obtain estimates that apply, in particular, to nanoscale systems. Original Russian Text ? I.V. Volovich, A.S. Trushechkin, 2009, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 265, pp. 288–319.     

混合态的不确定关系与压缩效应

在关于混合态的海森堡不确定关系的基础上，研究了纯态和混合态的最小不确定性和压缩效应.虽然最小不确定态必定是纯态，但在某些并非最小不确定态的纯态或混合态中，依然可 以实现力学量不确定度的压缩.还给出了普通统计学的不确定关系，它们不涉及量子相干性却与量子力学的海森堡不确定关系具有相似的数学结构.     

Atom-photon interactions with respect to quantum computation: A three-level atom in a two-mode field

In this paper, analysis of a quantum optical system—three-level atom in a quantum electromagnetic field is given. Evolution operators are constructed in closed form. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 44, Quantum Computing, 2007.     

Adaptive grid refinement for a model of two confined and interacting atoms

We have applied adaptive grid refinement to solve a two-dimensional Schrödinger equation in order to study the feasibility of a quantum computer based on extremely-cold neutral alkali-metal atoms. Qubits are implemented as motional states of an atom trapped in a single well of an optical lattice of counter-propagating laser beams. Quantum gates are constructed by bringing two atoms together in a single well leaving the interaction between the atoms to cause entanglement. For special geometries of the optical lattices and thus shape of the wells, quantifying the entanglement reduces to solving for selected eigenfunctions of a Schrödinger equation that contains a two-dimensional Laplacian, a trapping potential that describes the optical well, and a short-ranged interaction potential. The desired eigenfunctions correspond to eigenvalues that are deep in the interior of the spectrum where the trapping potential becomes significant. The spatial range of the interaction potential is three orders of magnitude smaller than the spatial range of the trapping potential, necessitating the use of adaptive grid refinement.     

Dynamics of an n level system of atoms interacting with laser fields

This paper is an extension of Fujii et al. (quant—ph/0307066), and in this one we again treat a model of an atom with n energy levels interacting with n(n ? 1)/2 external laser fields, which is a natural extension of the usual two-level system. Then the rotating wave approximation (RWA) is assumed from the beginning. To solve the Schrödinger equation we set the consistency condition in our terminology and reduce it to a matrix equation with symmetric matrix Q consisting of coupling constants. However, to calculate exp(?itQ) explicitly is not easy. In the case of three-and four-level systems we determine it in a complete manner, so our model in these levels becomes realistic. Lastly, we make a comment on cavity QED quantum computation based on three energy levels of atoms as a forthcoming target.     

Generic initial ideals and squeezed spheres

In 1988 Kalai constructed a large class of simplicial spheres, called squeezed spheres, and in 1991 presented a conjecture about generic initial ideals of Stanley-Reisner ideals of squeezed spheres. In the present paper this conjecture will be proved. In order to prove Kalai's conjecture, based on the fact that every squeezed (d−1)-sphere is the boundary of a certain d-ball, called a squeezed d-ball, generic initial ideals of Stanley-Reisner ideals of squeezed balls will be determined. In addition, generic initial ideals of exterior face ideals of squeezed balls are determined. On the other hand, we study the squeezing operation, which assigns to each Gorenstein* complex Γ having the weak Lefschetz property a squeezed sphere Sq(Γ), and show that this operation increases graded Betti numbers.     

Nonadiabatic quantum chaos in atom optics

Coherent dynamics of atomic matter waves in a standing-wave laser field is studied. In the dressed-state picture, wave packets of ballistic two-level atoms propagate simultaneously in two optical potentials. The probability to make a transition from one potential to another one is maximal when centroids of wave packets cross the field nodes and is given by a simple formula with the single exponent, the Landau-Zener parameter κ. If κ ? 1, the motion is essentially adiabatic. If κ ? 1, it is (almost) resonant and periodic. If κ ? 1, atom makes nonadiabatic transitions with a splitting of its wave packet at each node and strong complexification of the wave function as compared to the two other cases. This effect is referred as nonadiabatic quantum chaos. Proliferation of wave packets at κ ? 1 is shown to be connected closely with chaotic center-of-mass motion in the semiclassical theory of point-like atoms with positive values of the maximal Lyapunov exponent. The quantum-classical correspondence established is justified by the fact that the Landau-Zener parameter κ specifies the regime of the semiclassical dynamical chaos in the map simulating chaotic center-of-mass motion. Manifestations of nonadiabatic quantum chaos are found in the behavior of the momentum and position probabilities.     

Entropy evolution of the bimodal field interacting with an effective two-level atom via the Raman transition in Kerr medium

For a general class of two-mode, simple analytic expressions are derived for the evolution of the field quantum entropy in the bimodal field interacting with an effective two-level atom via the Raman transition, with an additional Kerr-like medium. The effect of a Kerr-like medium on the entropy is analyzed. It is shown that the addition of the Kerr medium has an important effect on the properties of the entropy and the entanglement. The results show that the effect of the Kerr medium changes the quasi-period of the field entropy evolution and entanglement between the atom and the field. The general conclusions reached are illustrated by numerical results.     

Parton version of the quantum model with memory and the Zeno effect

A parton version of the model with memory is proposed to describe quantum objects. These objects are assumed to consist of valence subpartons and “sea subpartons,” which are carriers of the particle and wave properties, respectively. Subpartons are similar, but not identical, to quantum chromodynamic partons. An interpretation of the quantum Zeno effect is given in the framework of the model. This effect is shown to be caused by a specific interaction of the quantum atomic system with classical fields. This interaction breaks down the coherence of “sea subpartons” localized at different energy levels of the atom, which can be considered as the physical reason behind the wave-function collapse. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 298–307, February, 1997.     

On an approach to the study of the Jaynes–Cummings sum in quantum optics

A new approach to the study of the Jaynes–Cummings sum, which determines the atomic inversion in quantum model of a single two-level atom interacting with a single mode of the quantized radiation field, based on the number theory theorems on approximation of trigonometric sums is presented.      

Normal forms,inner products,and Maslov indices of general multimode squeezings

In this paper, we present a purely algebraic construction of the normal factorization of multimode squeezed states and calculate their inner products. This procedure allows one to orthonormalize bases generated by squeezed states. We calculate several correct representations of the normalizing constant for the normal factorization, discuss an analog of the Maslov index for squeezed states, and show that the Jordan decomposition is a useful mathematical tool for problems with degenerate Hamiltonians. As an application of this theory, we consider a nontrivial class of squeezing problems which are solvable in any dimension.