Diagonal invariance and quasi-trapped states in the micromaser model based on N-atom clusters

The model of a micromaser based on unpolarized N-atom clusters described by the Tavis-Cummings Hamiltonian is considered. The properties of the evolution superoperator Γ(t) that acts on the reduced density matrix of the field are investigated. This superoperator was found to have an integral of motion that leads to a specific type of the matrix element dynamics called diagonal invariance. The same property also holds for the Liouvillian of the interaction between the field and the thermal bath. Within the framework of the considered micromaser model, the anomalous dependence of the average photon number inside the cavity on the quality factor of the cavity and its temperature was discovered. These features of the field evolution were explained by the existence of eigenstates of Γ(t), which are localized in the Fock basis and have eigenvalues close to unity. These states were called the quasi-trapped states.     

Quantum damped oscillator I: Dissipation and resonances

Quantization of a damped harmonic oscillator leads to so called Bateman’s dual system. The corresponding Bateman’s Hamiltonian, being a self-adjoint operator, displays the discrete family of complex eigenvalues. We show that they correspond to the poles of energy eigenvectors and the corresponding resolvent operator when continued to the complex energy plane. Therefore, the corresponding generalized eigenvectors may be interpreted as resonant states which are responsible for the irreversible quantum dynamics of a damped harmonic oscillator.     

Engineering transient spectrum for a two-level system interacting with two-mode intelligent states

The problem of a single two-level atom in its excited state, placed in a bimodal cavity in which an arbitrary form of the intensity-dependent atom–field coupling is present, is solved exactly. Expressions for the energy eigenvalues and eigenvectors are derived, and an exact representation for the time-dependent unitary evolution operator of the system is given, by means of which we analyze the analytic form of the fluorescence spectrum using the transitions among the dressed states of the system. The influence of parameter misfits on the quantum system is discussed. An interesting relation between the fluorescence spectrum and the dynamical evolution is found when the initial field states are prepared in properly intelligent states. By comparison with exact numerical results, we demonstrate that features of the fluorescence spectrum are influenced significantly by the different kinds of the nonlinearity of the intensity-dependent atom–field coupling. We provide numerous examples to illustrate this correspondence, and provide evidence of its usefulness. The present work considerably extends earlier results for two-level systems.     

Fast time-evolution method for dynamical systems

A fast time-evolution method is developed for systems for which the dynamical behavior can be reduced to the eigenvector/eigenvalue problem. The method does not use the eigenvectors/eigenvalues themselves and is based on a polynominal expansion of the formal operator solution in the eigenfrequency domain. It is complementary to the standard time-integration approaches and allows one to calculate or simulate the state of a system at arbitrary times. The time evolution of, e.g., classical harmonic atomic systems and quantum systems described by linear Hamiltonians can be treated by this method.     

Analytical study of the splitting process of a multiply-quantized vortex in a Bose–Einstein condensate and collaboration of the zero and complex modes

We study the dynamics of a trapped Bose–Einstein condensate with a multiply-quantized vortex, and investigate the roles of the fluctuations in the dynamical evolution of the system. Using the perturbation theory of the external potential, and assuming the situation of the small coupling constant of self-interaction, we analytically solve the time-dependent Gross–Pitaevskii equation. We introduce the zero mode and its adjoint mode of the Bogoliubov–de Gennes equations. Those modes are known to be essential for the completeness condition. We confirm how the complex eigenvalues induce the vortex splitting. It is shown that the physical role of the adjoint zero mode is to ensure the conservation of the total condensate number. The contribution of the adjoint mode is exponentially enhanced in synchronism with the exponential growth of the complex mode, and is essential in the vortex splitting.     

Almost-invariant sets and invariant manifolds — Connecting probabilistic and geometric descriptions of coherent structures in flows

We study the transport and mixing properties of flows in a variety of settings, connecting the classical geometrical approach via invariant manifolds with a probabilistic approach via transfer operators. For non-divergent fluid-like flows, we demonstrate that eigenvectors of numerical transfer operators efficiently decompose the domain into invariant regions. For dissipative chaotic flows such a decomposition into invariant regions does not exist; instead, the transfer operator approach detects almost-invariant sets. We demonstrate numerically that the boundaries of these almost-invariant regions are predominantly comprised of segments of co-dimension 1 invariant manifolds. For a mixing periodically driven fluid-like flow we show that while sets bounded by stable and unstable manifolds are almost-invariant, the transfer operator approach can identify almost-invariant sets with smaller mass leakage. Thus the transport mechanism of lobe dynamics need not correspond to minimal transport.The transfer operator approach is purely probabilistic; it directly determines those regions that minimally mix with their surroundings. The almost-invariant regions are identified via eigenvectors of a transfer operator and are ranked by the corresponding eigenvalues in the order of the sets’ invariance or “leakiness”. While we demonstrate that the almost-invariant sets are often bounded by segments of invariant manifolds, without such a ranking it is not at all clear which intersections of invariant manifolds form the major barriers to mixing. Furthermore, in some cases invariant manifolds do not bound sets of minimal leakage.Our transfer operator constructions are very simple and fast to implement; they require a sample of short trajectories, followed by eigenvector calculations of a sparse matrix.     

Completing Bethe's Equations at Roots of Unity

In a previous paper we demonstrated that Bethe's equations are not sufficient to specify the eigenvectors of the XXZ model at roots of unity for states where the Hamiltonian has degenerate eigenvalues. We here find the equations which will complete the specification of the eigenvectors in these degenerate cases and present evidence that the sl ₂ loop algebra symmetry is sufficiently powerful to determine that the highest weight of each irreducible representation is given by Bethe's ansatz.     

Entangled Two Two-Level Atom in the Presence of External Classical Fields

In this contribution, we investigate a TTLAs (two two-level atoms) in interaction with an electromagnetic field in presence of the external classical fields. The general solution of the time evolution operator is obtained and used to derive the density matrix operator. The temporal evolution of the atomic inversion, the degree of entanglement measured by the negativity, as well as the single atom entropy squeezing are discussed. We consider the atomic system at either the upper or Bell states, while the field in the coherent state. It has been shown that the coupling parameter g (the coupling of the external classical fields) gets more effective for the case in which the g is not equal to zero. Also for a strong coupling parameter g the superstructure phenomenon can be reported. The results shown that for increase the value of the classical external fields parameter leads to the entanglement between the atoms and the fields gets stronger. Also it has shown that for specific value of the classical external fields the system never reaches the pure state except during the revival periods.     

A new formalism for the statistical dynamics of vector spin systems

The relaxational dynamics of a classical vector Heisenberg spin system is studied using the Fokker-Planck equation. To calculate the eigenvalues of the Fokker-Planck operator, a new approach is introduced. In this connection, a number space repesentation is introduced, which enables us to visualize the eigenvalue structure of the Fokker-Planck operator. The mean field approximation is derived and a systematic method to improve the mean field approximation is presented.     

Entangling Two Multi-atomic Clusters Through a Cavity Field

Two atomic clusters, which have NA and Ns two-level atoms, respectively, are placed in a cavity but separated spatially. There is no direct interaction between the atoms. All the atoms interact with a single-mode of the cavity field. Quantum entanglement between the two atomic clusters is investigated for various initial states of the two atomic clusters and the field. When the cavity field is initially in a Fock state, we find that the time evolution of entanglement quasi-periodically oscillates regardless of the initial states of atoms. The oscillation period increases as the initial photon number increases. When all the atoms in both of the atomic clusters are initially in the excited state, we show that there is no entanglement between the atomic clusters with NA = NB = 1 regardless the initial state of the cavity field. However, when either NA or NB is larger than one, we find that the entanglement always exists even for a strong thermal field. In cases with different initial states of the atomic clusters, we notice that the entanglement becomes stronger as number of the atoms increases. When all the atoms in both of the clusters in the ground state, we also find that the entanglement can be enhanced even by a thermal field. We also notice that a single qubit can be entangled with multi-atoms which are initially in the ground state by the cavity field initially being in vacuum, thermal, coherent, and squeezed states.     

An operatorially solved model of massless two-dimensional quantum chromodynamics

An operator solution is constructed in (1,1) dimensions to the massless quantum chromodynamics of n fermion quarks and n² ? 1 vector boson gluons with local colour SU(n) symmetry. The interacting quark field is a confined SU(n) Thirring field with zero Abelian coupling. The colour gluons are dependent Lie fields obeying the gluon-free fermionic current identity. Explicit local infinitesimal operator colour transformations (with an arbitrary coordinate-independent Lorentz vector coefficient defining the gauge) are given and the requirement of proper colour covariance linked to the vanishing of the coloured quark source currents and hence to the absence of coloured quark-composite states. The status of Noether's theorem is also clarified.     

High order corrections to the time-independent Born-Oppenheimer approximation II: Diatomic Coulomb systems

We study the bound states of diatomic molecular systems. We prove that if the nuclear masses are proportional to ε^?4 then certain eigenvalues and eigenvectors of the Hamiltonian have asymptotic expansions to arbitrarily high order in powers of ε, as ε→0. The zeroth through fourth order terms in the expansions for the eigenvalues are those of the well-known Born-Oppenheimer approximation. The fifth order term is zero.     

Evolution of collective N atom states in single photon superradiance: Effect of virtual Lamb shift processes

We present analytical solutions for the evolution of collective states of N atoms. On the one hand is a (timed) Dicke state prepared by the conditional absorption of a single photon and exhibiting superradiant decay. This is in strong contrast to the evolution of a symmetric Dicke state which is trapped for large atomic clouds. We show that virtual processes yield only a small effect on the evolution of the rapidly decaying timed Dicke state. However, they change the long time dynamics from exponential decay into a power-law behavior which can be observed experimentally. For trapped states virtual processes are much more important and provide new decay channels resulting in a slow decay of the otherwise trapped state.     

Pure Point Spectrum for Two-Level Systems in a Strong Quasi-Periodic Field

We consider two-level atoms in a strong external quasi-periodic field with Diophantine frequency vector. We show that if the field is an analytic function with zero average, then for a large set of values of its frequency vector, characterized by imposing infinitely many Diophantine conditions, the spectrum of the quasi-energy operator is pure point, as in the case of nonzero average which was already known in literature.     

Anharmonic Parameters of Vibrationally Excited Molecules

An algebraic method of perturbation theory for eigenvalues and eigenvectors is used to examine the intramolecular parameters of excited vibrational states. Based on this method, formulas for higher-order coefficients of the matrix of vibrational modes, changes in the Cartesian coordinates of vibrating atoms, and anharmonic elements of the tensor of inertia moment are derived.     

Multiscale finite element methods for high-contrast problems using local spectral basis functions

In this paper we study multiscale finite element methods (MsFEMs) using spectral multiscale basis functions that are designed for high-contrast problems. Multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem. This local spectral problem strongly depends on the choice of initial partition of unity functions. The resulting space enriches the initial multiscale space using eigenvectors of local spectral problem. The eigenvectors corresponding to small, asymptotically vanishing, eigenvalues detect important features of the solutions that are not captured by initial multiscale basis functions. Multiscale basis functions are constructed such that they span these eigenfunctions that correspond to small, asymptotically vanishing, eigenvalues. We present a convergence study that shows that the convergence rate (in energy norm) is proportional to (H/Λ₁)^1/2, where Λ₁ is proportional to the minimum of the eigenvalues that the corresponding eigenvectors are not included in the coarse space. Thus, we would like to reach to a larger eigenvalue with a smaller coarse space. This is accomplished with a careful choice of initial multiscale basis functions and the setup of the eigenvalue problems. Numerical results are presented to back-up our theoretical results and to show higher accuracy of MsFEMs with spectral multiscale basis functions. We also present a hierarchical construction of the eigenvectors that provides CPU savings.     

Causodynamics of autowave patterns

Perturbative dynamics of spiral and scroll waves involves the "response functions," i.e., critical eigenvectors of the adjoint linearized operator, dual to the Goldstone modes. A well known method of calculating the Goldstone modes is time integration of the linearized equation. We suggest that backward-time integration of the adjoint linearized equation, which we call causodynamics, can be used to calculate the response functions. This new method is more robust and easier to implement than existing methods. We illustrate how it works for propagating and rotating autowaves in reaction-diffusion systems. The method reveals unexpected qualitative difference between similarly looking regimes.     

Quantum Dynamics of Cooled Atoms in the Presence of Bose—Einstein Condensates

Under the Markov approximation,the quantum dynamics of cooled atoms in the presence of Bose-Einstein condensates is studied.A master equation governing the evolution of such a system is derved.Using this master equation,the distribution of the atoms in the excited states at finite temperature and the dynamics of the excited atom at zero temperature are given and discussed.     

Emergence and stability of vortex clusters in Bose-Einstein condensates: A bifurcation approach near the linear limit

We study the existence and stability properties of clusters of alternating charge vortices in repulsive Bose-Einstein condensates. It is illustrated that such states emerge from cascades of symmetry-breaking bifurcations that can be analytically tracked near the linear limit of the system via weakly nonlinear few-mode expansions. We present the resulting states that emerge near the first few eigenvalues of the linear limit, and illustrate how the nature of the bifurcations can be used to understand their stability. Rectilinear, polygonal and diagonal vortex clusters are only some of the obtained states while mixed states, consisting of dark solitons and vortex clusters, are identified as well. We also explore the evolution of unstable states and their transient dynamics exploring configurations of nearby bifurcation branches.