Quantum carpets: efficiently probing fractional revivals in position-dependent mass systems

Position-dependent-mass systems are of great importance in many physical situations,such as the transport of charge carriers in semiconductors with non-uniform composition and in the theory of many-body interactions in condensed matter.Here we investigate,numerically and analytically,the phenomenon of fractional revivals in such systems,which is a generic characteristic manifested by the wave-packet evolution in bounded Hamiltonian systems.Identifying the fractional revivals using specific probe...     

Three-dimensional molecular wave packets: Calculation of revival times from periodic orbits

We present a theoretical analysis of revivals and fractional revivals of three-dimensional wave packets, which describe the coupled vibrational motion of phosphaethyne (HCP) in its ground electronic state. The wave packets studied are chosen to evolve along the periodic orbits, which quantize the states in the three fundamental progressions. The revival times T_rev are found to depend strongly on the particular mode excited and on the mean excitation energy. Based on a semiclassical analysis, T_rev is shown to be determined by the dependence of the period of the orbits on the classical action along them.     

A method of reduction of Einstein's equations of evolution and a natural symplectic structure on the space of gravitational degrees of freedom

A program is outlined which addresses the problem of thereduction of Einstein's equations, namely, that of writing Einstein's vacuum equations in (3+1)-dimensions as anunconstrained dynamical system where the variables are thetrue degrees of freedom of the gravitational field. Our analysis is applicable for globally hyperbolic Ricci-flat spacetimes that admit constant mean curvature compact orientable spacelike Cauchy hypersurfaces M with degM=0 andM not diffeomorphic toF ₆, the underlying manifold of a certain compact orientable flat affine 3-manifold. We find that for these spacetimes, modulo the extended Poincaré conjecture and the use of local cross-sections rather than a global cross-section, (3+1)-reduction can be completed much as in the (2+1)-dimensional case. In both cases, one gets as the reduced phase space the cotangent bundleT ^* T _M of theTeichmüller space T _M of conformal structures onM, whereM is a given initial constant mean curvature compact orientable spacelike Cauchy hypersurface in a spacetime (V, g _V ), and one gets reduction of the full classical non-reduced Hamiltonian system with constraints to a reduced Hamiltonian system without constraints onT ^* T _M . For these reduced systems, the time parameter is the parameter of a family of monotonically increasing constant mean curvature compact orientable spacelike Cauchy hypersurfaces in a neighborhood of a given initial one. In the (2+1)-dimensional case, the Hamiltonian is the area functional of these hypersurfaces, and in the (3+1)-dimensional case, the Hamiltonian is the volume functional of these hypersurfaces.     

Extending $\mathcal{P}\mathcal{T}$ Symmetry from Heisenberg Algebra to E2 Algebra

The E2 algebra has three elements, J, u, and v, which satisfy the commutation relations [u,J]=iv, [v,J]=−iu, [u,v]=0. We can construct the Hamiltonian H=J ²+gu, where g is a real parameter, from these elements. This Hamiltonian is Hermitian and consequently it has real eigenvalues. However, we can also construct the PT\mathcal{P}\mathcal{T}-symmetric and non-Hermitian Hamiltonian H=J ²+igu, where again g is real. As in the case of PT\mathcal{P}\mathcal{T}-symmetric Hamiltonians constructed from the elements x and p of the Heisenberg algebra, there are two regions in parameter space for this PT\mathcal{P}\mathcal{T}-symmetric Hamiltonian, a region of unbroken PT\mathcal{P}\mathcal{T} symmetry in which all the eigenvalues are real and a region of broken PT\mathcal{P}\mathcal{T} symmetry in which some of the eigenvalues are complex. The two regions are separated by a critical value of g.     

Revisiting non-degenerate parametric down-conversion

The quantum dynamics of a two-mode non-resonant parametric down-conversion process is studied by recasting the time evolution equations for the basic operators in an equivalent spin equation form with simpler exact solutions for a pump field with harmonic time dependence. Expectation values of suitable operators for studying important features such as squeezing and quantum revivals are presented in simple forms.      

Ab initio ro-vibrational Hamiltonian in irreducible tensor formalism: a method for computing energy levels from potential energy surfaces for symmetric-top molecules

A theoretical approach to study ro-vibrational molecular states from a full nuclear Hamiltonian expressed in terms of normal-mode irreducible tensor operators is presented for the first time. Each term of the Hamiltonian expansion can thus be cast in the tensor form in a systematic way using the formalism of ladder operators. Pyramidal XY₃ molecules appear to be good candidates to validate this approach which allows taking advantage of the symmetry properties when doubly degenerate vibrational modes are considered. Examples of applications will be given for PH₃ where variational calculations have been carried out from our recent potential energy surface [Nikitin et al., J. Chem. Phys. 130, 244312 (2009)].     

The goodness of ergodic adiabatic invariants

For a slowly time-dependent Hamiltonian system exhibiting chaotic motion that ergodically covers the energy surface, the phase space volume enclosed inside this surface is an adiabatic invariant. In this paper we examine, both numerically and theoretically, how the error in this ergodic adiabatic invariant scales with the slowness of the time variation of the Hamiltonian. It is found that under certain circumstances, the error is diffusive and scales likeT ^–1/2, whereT is the characteristic time over which the Hamiltonian changes. On the other hand, for other cases (where motion in the Hamiltonian has a long-time 1/t tail in a certain correlation function), the error scales like [T ^–1 ln(T)]^1/2. Both of these scalings are verified by numerical experiments. In the situation where invariant tori exist amid chaos, the motion may not be fully ergodic on the entire energy surface. The ergodic adiabatic invariant may still be useful in this case and the circumstances under which this is so are investigated numerically (in particular, the islands have to be small enough).     

Inelastic scattering of phonons by quadrupole defects with internal degrees of freedom

We solve the quantum mechanical problem of the inelastic scattering of phonons by a quadrupole defect in a crystal lattice for the case of solid parahydrogen whose matrix contains pair complexes of H₂ orthomolecules. By employing the pseudospin approximation for the operator of the energy of quadrupole-quadrupole interaction of the molecules in an orthopair we derive an effective Hamiltonian that describes the interaction of phonons with a pair quadrupole orthodefect in the lattice. We set up the scattering matrix and calculate the effective phonon relaxation time τ(ω, T) as a function of the frequency ω and the crystal temperature T. We also find that a pair quadrupole defect, which has a complicated system of levels, can be replaced by an effective two-level system with temperature-dependent parameters. The fact that a pair quadrupole orthocluster has internal degrees of freedom results in a resonant scattering peak near a certain critical temperature T ₀. Our estimates for H₂ yield T ₀≃ 6–7 K. Finally, we discuss the contribution of this mechanism to the low-temperature thermal conductivity of solid hydrogen. Zh. éksp. Teor. Fiz. 114, 555–569 (August 1998)     

A modified fractional model to describe the viscoelastic behavior of solid amorphous polymers: The effect of physical aging

Amorphous polymers are viscoelastic materials. When they are subjected to dynamical loads, their behavior can be modeled by transform functions of stress and strain in the complex plane. In our work, a model based on the fractional calculus concept is proposed in order to predict the viscoelastic behavior of polymethylmethylacrylate (PMMA) over a wide temperature range between (T_g -190°C) and (T_g +15°C). The extended fractional solid model is shown to be capable of describing experimentally observed dynamic viscoelastic behavior over a wide temperature range, including multiple relaxations, using a limited number of free parameters. Structural recovery of PMMA was studied by dynamic mechanical spectrometry, and its effect on the different parameters is also discussed. Furthermore, from the fractional differential and fractional integral formulations. most of the relevant viscoelastic functions that quantify the degree of molecular mobility of amorphous polymers, like E(t), E?(τ). and H(t), can be derived analytically.     

Quantum revivals,geometric phases and circle map recurrences

Revivals of the coherent states of a deformed, adiabatically and cyclically varying oscillator Hamiltonian are examined. The revival time distribution is exactly that of Poincaré recurrences for a rotation map: only three distinct revival times can occur, with specified weights. A link is thus established between quantum revivals and recurrences in a coarse-grained discrete-time dynamical system.     

Diagrammatic structure of the general coupled cluster equations

The general formula for the number of diagrammatic terms occurring in the T_n equation within a particular coupled cluster model is derived. Both the antisymmetrized and Goldstone diagrams are considered. In addition to the full coupled cluster equation approximate approaches are discussed, and for each the general formula for the number of terms is given. Analogous expressions are presented for the number of diagrammatic terms contributing to the elements of the transformed Hamiltonian [Hbar] = e^?T He^T .     

Perturbation theory with higher derivative couplings

The formalism of partial differential equations with respect to coupling constants is used to develop a covariant perturbation theory for the interpolating fields and theS matrix when the coupling terms in the Larangian density involve arbitrary (first and higher) derivatives. Through the notion of pure noncovariant contractions, the free-fieldT and the (covariant)T ^* products can be related to each other, allowing us to avoid the Hamiltonian density altogether when dealing with theS matrix. The important ingredients in our approach are (1) the adiabatic switching on and off of the interactions in the infinite past and future, respectively, and (2) the vanishing of four-dimensional delta functions and their derivatives at zero space-time points. The latter ingredient is a prerequisite that our formalism and the canonical formalism be consistent with each other, and on the other hand, it is supported by the dimensional regularization. Corresponding to any Lagrangian, the generalized interaction Hamiltonian density is defined from the covariantS matrix with the help of the pure noncovariant contractions. This interaction Hamiltonian density reduces to the usual one when the Lagrangian density depends on just first derivatives and when the usual canonical formalism can be applied.     

Fractional Heisenberg equation

Fractional derivative can be defined as a fractional power of derivative. The commutator (i/?)[H,⋅], which is used in the Heisenberg equation, is a derivation on a set of observables. A derivation is a map that satisfies the Leibnitz rule. In this Letter, we consider a fractional derivative on a set of quantum observables as a fractional power of the commutator (i/?)[H,⋅]. As a result, we obtain a fractional generalization of the Heisenberg equation. The fractional Heisenberg equation is exactly solved for the Hamiltonians of free particle and harmonic oscillator. The suggested Heisenberg equation generalize a notion of quantum Hamiltonian systems to describe quantum dissipative processes.     

On the relationship betweenT 1 andT 2 for stochastic relaxation models

We consider the relaxation dynamics of two quantum levels coupled to a stochastic bath. We emphasize that even if the matrix elements of the fluctuating Hamiltonian are Gaussian, a second-order cumulant truncation is not exact. For various stochastic models, including the case of a spin-1/2 particle in a fluctuating magnetic field, we calculate 1/T ₁, the population relaxation rate, and 1/T ₂, the phase relaxation rate, up to fourth order in perturbation theory. We show that unlike the commonly accepted second-order result that 1/T ₂1/2T ₁, when fourth-order terms are included, in some instances 1/T ₂<1/2T ₁.     

Reduction of constrained mechanical systems and stability of relative equilibria

A mechanical system with perfect constraints can be described, under some mild assumptions, as a constrained Hamiltonian system(M, , H, D, W): (M, ) (thephase space) is a symplectic manifold,H (theHamiltonian) a smooth function onM, D (theconstraint submanifold) a submanifold ofM, andW (theprojection bundle) a vector sub-bundle ofT _D M, the reduced tangent bundle alongD. We prove that when these data satisfy some suitable conditions, the time evolution of the system is governed by a well defined differential equation onD. We define constrained Hamiltonian systems with symmetry, and prove a reduction theorem. Application of that theorem is illustrated on the example of a convex heavy body rolling without slipping on a horizontal plane. Two other simple examples show that constrained mechanical systems with symmetry may have an attractive (or repulsive) set of relative equilibria.     

Thermal State for the Capacitance Coupled Mesoscopic Circuit with a Power Source

The Schrödinger equation of the mesoscopic capacitance coupled circuit with an arbitrary power source is solved by means of two step unitary transformation. The original Hamiltonian transformed to a very simple form by unitary operators so that it can be easily treated. We derived the exact full wave functions in Fock state. By making use of these wave functions and introducing the Lewis--Riesenfeld invariant operator, the thermal state have been constructed. The fluctuations of charges and currents are evaluated in thermal state. For T→ 0, the uncertainty products between charges and currents in thermal state recovers exactly to that of Fock state with n, m=0.     

Application of the contact transformation method to torsional problems in methyl silane

The effect of the torsional degree of freedom on redundancies in the Hamiltonian and on the dipole operator has been investigated for methyl silane-like molecules. By applying a rotational contact transformation to the torsion-rotation Hamiltonian H_TR for the ground vibrational state, a systematic method is demonstrated for treating the redundancies that relate different terms in H_TR. In general, with this method, the experimentally accessible molecular parameters in the reduced Hamiltonian can be related to the physically significant molecular parameters in the untransformed Hamiltonian. It is shown that H_TR contains a new term which has matrix elements with selection rules (ΔK = ±3), (Δσ = 0), and Δv_T arbitrary, where v_T and σ label the torsional levels and sublevels, respectively. As a result of this term, the distortion dipole constant μ_D which characterizes (ΔK = ±3) matrix elements in C_3v molecules cannot, in systems like CH₃SiH₃, be ascribed entirely to centrifugal distortion but can contain a significant contribution from torsional effects. Furthermore, new transitions can appear in the pure torsional bands which may be strong enough to observe in low barrier molecules. By applying a vibrational contact transformation, the form is derived of the leading torsional terms in the dipole moment expansion. The four dipole distortion constants μ₀^T, μ₂^T, μ_|;^T, and μ_Λ^T which characterize these terms are related to the molecular parameters that enter the Coriolis, centrifugal distortion, and anharmonicity contributions to the vibration-torsion-rotation Hamiltonian.     

Oscillations of charged particles in an external magnetic field about steady motion

We develop a Hamiltonian formalism that can be used to study the particle dynamics near stable equilibria. The construction of an original canonical transformation allowed us to prove the conservation of the linear momentum P₃, which permitted the expansion of the Hamiltonian about a fixed point. The definition of the rotational variable h whose Poisson algebra properties played the essential role in the diagonalization of the quadratic Hamiltonian yielding two uncoupled oscillators with definite frequencies and amplitudes. It is through applying this variable near a fixed point that come to light Heisenberg's and Harmonic Oscillator equations of motion of the particles, leading thus the association of the fixed point trajectories with arbitrary trajectories in its immediate neighborhood. The present formalism succeeded to treat the problem of free-electron laser dynamics and may be applied to similar cases. Received 20 October 2001     

Microscopic approach to the statics and dynamics of solid electrolytes near the phase transition

The statics and local dynamics of solid electrolytes near the transition temperature is investigated within mean field approach and Mori theory, starting from an Hamiltonian which takes into account the mutual interactions of the mobile ions and their coupling to the phonons of the cage. The Hamiltonian is written as function of the local density of the mobile ions on their available sites and as function of the displacements of the cage ions. The correlation functions of these variables are calculated within MFA. Those correlations which are connected with the order parameter diverge at the stability limits but they finite atT _c . As a consequence, the sound velocities of those acoustic modes which are coupled to the mobile ions show a softening atT _c only when the coupling strength is sufficient large. It is shown that this softening can be observed by ultrasonic methods, but not by neutron scattering, in agreement with experiments. The experimental relevance of aZ-branch, which may appear in the dispersion of the relaxational peak nearT _c , is discussed. The coupling between optical modes and mobile ions is also considered. It is shown that a softening of optical modes nearT _c is not expected in solid electrolytes, in agreement with the experimental situation.