Two-level Case Research Articles

Multilevel Item Response Models: An Approach to Errors in Variables Regression

In this article we show how certain analytic problems that arise when one attempts to use latent variables as outcomes in regression analyses can be addressed by taking a multilevel perspective on item response modeling. Under a multilevel, or hierarchical, perspective we cast the item response model as a within-student model and the student population distribution as a between-student model. Taking this perspective leads naturally to an extension of the student population model to include a range of student-level variables, and it invites the possibility of further extending the models to additional levels so that multilevel models can be applied with latent outcome variables. In the two-level case, the model that we employ is formally equivalent to the plausible value procedures that are used as part of the National Assessment of Educational Progress (NAEP), but we present the method for a different class of measurement models, and we use a simultaneous estimation method rather than two-step estimation. In our application of the models to the appropriate treatment of measurement error in the dependent variable of a between-student regression, we also illustrate the adequacy of some approximate procedures that are used in NAEP.

Mechanical potentials due to Raman transitions.

We investigate the mechanical potential of an atom due to a Raman transition. This is a generalization of the usual two-level dipole potential. With sufficiently large one- and two-photon detunings the forms of the potential and excitation probability are the same as for the two-level case, but with a scaled detuning. This allows greater effective blue detunings to be accessible with the Raman scheme. This means that with the same available laser power, atoms can move coherently in the light field for many orders of magnitude longer without spontaneously emitting. Furthermore the sign of the potential may be controlled independently of the one-photon detuning. \textcopyright{} 1996 The American Physical Society.

Validatable nonrobust delay-fault testable circuits via logic synthesis

The authors advocate a synthesis approach to delay-fault testing, wherein completely path-delay-fault testable circuits are automatically synthesized, meeting area and performance requirements. They give necessary and sufficient conditions for validatable nonrobust fault testability of paths in arbitrary multilevel networks. Validatable nonrobust testing as opposed to robust testing offers degrees of freedom that enable the development of efficient synthesis procedures that target delay-fault testability, and also provides a means of producing compact test vector sets. The authors then focus on the development of synthesis procedures that produce networks that are fully testable under the nonrobust fault model. They show that primality and irredundancy are both a necessary and sufficient condition for complete validatable nonrobust testability in the two-level case. They prove that synthesizing a multilevel network using algebraic factorization retains complete validatable nonrobust testability. Preliminary results that verify the procedures are reported. >

Deflection of resonant multilevel particles in a standing wave light field

We study the deflection of an atomic system with a resonant multilevel energy structure in classical and quantized wave light fields. We obtain that the shape of the momentum distribution is very sensitive to the energy level structure, whereas its moments are not. Comparison with a two-level case is made.

Intramanifold level mixing by time-dependent electric fields: Multilevel Landau-Zener effect.

The mixing of substates of an isolated Rydberg manifold (constant n,m) is studied using time-evolution matrices U(t,t0). Coherent mixing is effected by a linearly ramped electric field, F(t)=F\ifmmode \dot{}\else \.{}\fi{}t. The spherical l basis, diabatic (hydrogenic) p basis, and adiabatic (nonhydrogenic) q basis are considered. In the linear Stark regime, the diabatic levels are coupled by F(t) through the atomic core, parametrized by quantum defects. The resulting adiabatic levels EqF in this model group into a submanifold degenerate at F=0 plus shifted levels split off from the adiabatic manifold. Analytical expressions for the q levels Eq\ensuremath{\tau} and their coupling \ensuremath{\Gamma}qq\ensuremath{'}\ensuremath{\tau} are derived as a function of rescaled field or time \ensuremath{\tau}. The time evolution of the levels' populations is studied first for the two-level case---the usual Landau-Zener effect (LZE)---which is generalized to a multilevel Landau-Zener effect (MLZE) for any n manifold. The matrices U(t,t0) are constructed in the Riemann product representation, including Magnus corrections up to fourth order. The probability Pq\ensuremath{\rightarrow}q\ensuremath{'}(Vl) of making a transition from state q to q\ensuremath{'} upon a one-way pass across the manifold depends on a single parameter Vl=[2\ensuremath{\mu}l2/(3F\ifmmode \dot{}\else \.{}\fi{}n9)]1/2 (when one \ensuremath{\mu}l\ensuremath{\ne}0). Numerical results for l=m=0 indicate that either diabatic (p\ensuremath{\rightarrow}p\ensuremath{'}=p) or adiabatic (q\ensuremath{\rightarrow}q\ensuremath{'}=q) transitions always predominate, no matter what the ramp rate F\ifmmode \dot{}\else \.{}\fi{}. Strong evidence supports the conjecture that outer diabatic transitions between edge states of a manifold obey Pndiab(V)=exp(-\ensuremath{\beta}n\ensuremath{\pi}V2) for l=m=0, with \ensuremath{\beta}n=1 for n=2 (the LZE) and \ensuremath{\beta}n\ensuremath{\approx}lnn for n\ensuremath{\gg}2 (the MLZE), and that a similar analytical result holds for arbitrary {\ensuremath{\mu}l} and m.

Generalized Bloch equations for time-dependent N-level systems

The maximum entropy principle formalism is applied in order to solve time-dependent N-level systems. The formalism, related to a quasi-Lie algebra, allows us to find not only the dynamical equations of the relevant operators for any temporal dependence of the interaction between levels, but also to diagonalize the associated off-equilibrium (temperature dependent) density matrix. A set of generalized Bloch equations, in terms of the relevant operators, is given. The time-dependent two-level case is analyzed as a particular example.

Spontaneous emission and absorption properties of a driven three-level system.

We investigate the steady-state spontaneous emission spectrum of a three-level atom driven by two coherent fields and the absorption spectrum of a weak probe passing through a collection of such driven atoms. We find that the fluorescence spectrum is strongly affected by the decay rates of all the levels involved in the atomic evolution and not just by the decay parameters of the specific transition whose emission spectrum is being monitored. In particular, the spectral components can acquire very different widths and peak heights relative to the case of the standard resonance fluorescence in which a two-level system is driven by a single near-resonant field. An external probe signal passing through the gas of three-level atoms may be absorbed or amplified, as in the standard two-level case, but under specific operating conditions, amplification (or absorption) occurs over the entire range where the atomic response is appreciable. In this case the existence of amplification or absorption is controlled solely by the population difference between two dressed states of the system. We provide numerical results in support of our arguments for arbitrary values of the atomic and field parameters and also develop an analytic description in the limit of strong driving fields that leads to explicit line-shape and linewidth formulas.

A thermodynamic cycle for photovoltaic energy conversion

The possibility of creating a Carnot cycle for photovoltaic energy conversion is explored. The thermodynamic functions for an electron-hole gas are derived and they are used to show that a conventional Carnot cycle involving two heat reservoirs can be constructed. The additional problems which arise from radiative energy transfers are show to lead to the result that only if the electron-hole plasma is simplified to the two-level case can a reversible cycle occur. Since this cycle requires an infinity of reservoirs it is not a Carnot cycle in the sense of classical thermodynamics.

Theory of phase factors for a one-body potential of arbitrary symmetry

A general discussion of the single-particle density matrix in terms of the square root of the electron density ϱ( r) and phase factor for a given one-body potential V( r) of arbitrary symmetry for n occupied levels is given. The defining equations for the phase factor were obtained explicitly for the two-level case (e.g. LiH). By eliminating the one-body potential, a partial differential equation for the phase was obtained in terms of ϱ( r). Using the solution of this equation, the one-body potential can be expressed solely in terms of the ground-state density, to within an additive constant. The interest of these results with regard to density functional theory is emphasized.

Decomposition and factorization of sequential finite state machines

Algorithms are proposed for decomposing a finite-state machine into smaller interacting machines so as to optimize area and performance of the eventual logic implementation. Cascade decomposition algorithms, which decompose a given machine into independent and dependent components, have been proposed in the past. The authors propose a more powerful form of decomposition where both components of the decomposed machine interact with each other. Experimental results indicate that this decomposition technique for state machine decomposition is superior to cascade decomposition techniques. It is the premise of this study that optimal state assignment corresponds to finding an optimal multiple general decomposition of a finite-state machine. State assignment techniques that target two-level and multilevel implementations based on state machine factorization algorithms followed by state assignment algorithms are presented. It is rigorously proved that one-hot encoding a nontrivially factored machine is guaranteed to produce a better result than one-hot encoding the original machine for the two-level case.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Theory of resonance fluorescence from vibronic systems: stationary spectrum

The stationary second-order spectrum of the resonance-fluorescence light from a vibronic system is studied in a nonperturbative manner. In the model used the potential energy surfaces for the vibrational motion in the electronic ground state and in the excited state are assumed to be displaced and distorted relative to each other. The calculations are performed for the case when the frequency of the driving laser field is tuned to the vibrationless electronic transition. In particular, in the high-driving-field limit the vibrationless transition exhibits a triplet structure, the sidebands being narrower and higher in comparison with the well-known two-level case. The Raman lines are shifted and split into doublets, the linewidths being determined by the electronic as well as the vibrational relaxation rates.

Effects of separated absorption and emission spectra on degenerate four-wave mixing in organic dyes

Degenerate four-wave mixing in a separated absorption-gain system, such as that of an organic dye, is studied by using a four-level model. The effects of the nonlinearity arising from the displaced emission spectrum are found and compared with the results of the two-level model of Abrams and Lind [ Opt. Lett.2, 94 ( 1978); Opt. Lett.3, 205 ( 1978)]. A nonsymmetric saturation behavior over the width of the absorption line is shown to exist that can be detrimental to the phase-conjugate reflectivity. Otherwise, minor differences are seen from the two-level case.

Theory of Adiabatic Rapid Passage for Three Equally Spaced Levels

In experiments reported elsewhere, adiabatic rapid passage (ARP) has been performed on multilevel electron-paramagnetic crystals having three equally spaced energy levels separated by $\ensuremath{\hbar}{\ensuremath{\omega}}_{0}$. All transitions among the three levels are allowed due to crystalline electric-field mixing of the pure spin states. Simultaneous ARP inversion of the two coincident transitions at ${\ensuremath{\omega}}_{0}$ produces a negative temperature simultaneously on all three transitions, at ${\ensuremath{\omega}}_{0}$ and $2{\ensuremath{\omega}}_{0}$. The present paper explains these results through an analysis of adiabatic rapid passage in a spin system having an arbitrary spin-Hamiltonian with three equally spaced energy levels. Previous treatments of ARP have been limited either to the multilevel case with pure spin states or to the two-level case with mixed states. In the present analysis, the 3\ifmmode\times\else\texttimes\fi{}3 density matrix is transformed to an equivalent matrix whose equation of motion contains a Hamiltonian which is real and constant, even with the rf perturbation applied. The transformed matrix is expanded in terms of nine orthonormal Hermitian basis matrices. Three of the expansion coefficients are directly related to state populations. The time variation of the expansion coefficients is calculated for ARP conditions. The transition probabilities of the two ${\ensuremath{\omega}}_{0}$ transitions are not required to be equal. The final values of the three relevant expansion coefficients indicate negative temperatures on all three transitions.