Adiabatic Perturbation Theory Research Articles

Analysis of Timing Jitter for Ultrashort Soliton Communication Systems Using Perturbation Methods

We analyse timing jitter of ultrashort soliton systems taking into account the major higher order effects, namely, intrapulse Raman scattering and third order dispersion and using adiabatic perturbation theory. We obtain an expression for the soliton arrival time variance that depends on the quintic power of propagation distance. This timing jitter could be partly overcome by use of bandwidth-limited amplification as it happens in systems using longer solitons. In this case the quintic dependence with distance is reduced to a linear one.

Analysis of Timing Jitter for Ultrashort Soliton Communication Systems Using Perturbation Methods

Abstract We analyse timing jitter of ultrashort soliton systems taking into account the major higher order effects, namely, intrapulse Raman scattering and third order dispersion and using adiabatic perturbation theory. We obtain an expression for the soliton arrival time variance that depends on the quintic power of propagation distance. This timing jitter could be partly overcome by use of bandwidth-limited amplification as it happens in systems using longer solitons. In this case the quintic dependence with distance is reduced to a linear one.

Quantum lattice fluctuations in a frustrated Heisenberg spin-Peierls chain

As a simple model for spin-Peierls systems we study a frustrated Heisenberg chain coupled to optical phonons. In view of the anorganic spin-Peierls compound CuGeO${}_{3}$ we consider two different mechanisms of spin-phonon coupling. Combining variational concepts in the adiabatic regime and perturbation theory in the antiadiabatic regime we derive effective spin Hamiltonians which cover the dynamical effect of phonons in an approximate way. Ground-state phase diagrams of these models are determined, and the effect of frustration is discussed. Comparing the properties of the ground state and low-lying excitations with exact diagonalization data for the full quantum spin-phonon models, good agreement is found especially in the antiadiabatic regime.

Imbeddings of integral submanifolds and associated adiabatic invariants of slowly perturbed integrable Hamiltonian systems

A new method is developed for characterizing the evolution of invariant tori of slowly varying perturbations of completely integrable (in the sense of Liouville-Arnold [1–3]) Hamiltonian systems on cotangent phase spaces. The method is based on Cartan's theory of integral submanifolds, and it provides an algebro-analytic approach to the investigation of the embedding [4–10] of the invariant tori in phase space that can be used to describe the structure of quasi-periodic solutions on the tori. In addition, it leads to an adiabatic perturbation theory [3,12,13] of the corresponding Lagrangian asymptotic submanifolds via the Poincaré-Cartan approach [4], a new Poincaré-Melnikov type [5,11,14] procedure for determining stability, and fresh insights into the existence problem for adiabatic invariants [2,3] of the Hamiltonian systems under consideration.

Exact high-density limit of correlation potential for two-electron density

Present approximations to the correlation energy, Ec[n], in density functional theory yield poor results for the corresponding correlation potential, vc([n];r)=δEc[n]δ/n(r). Improvements in vc([n];r), are especially needed for high-quality Kohn–Sham calculations. For a two-electron density, the exact form of vc([n];r) in its high-density limit is derived in terms of the density of the system and the first-order wave function from the adiabatic perturbation theory. Our expression leads to a formula for the difference 2Ec[n]−∫vc([n];r)n(r)dr, valid for any two-electron density in the high-density limit, thus generalizes previous results. Numerical results (both exact and approximate) are presented for both Ec[n] and ∫vc([n];r)n(r)dr in this limit for two electrons in a harmonic oscillator external potential (Hooke’s atom).

Quasiparticle dynamics in ballistic weak links under weak voltage bias: an elementary treatment

A simple one-dimensional model for SNS weak links in the ballistic limit is presented. In the presence of a bias voltage, the quasiparticle state at any given instant of time is described as a superposition of that particular set of phase-dependent Andreev bound states that belongs to the specific phase difference present at that instant between the superconducting banks. The treatment—basically a form of adiabatic perturbation theory—has a strong formal similarity to the treatment of the k -space dynamics of an electron in a periodic potential under perturbation by an external electric field, sufficiently strong to cause transitions across the energy gaps between bands (Zener tunneling). It is shown that the quasiparticle wavefunction retains its phase information during analogous transitions between Andreev bands. The experimental observation of Shapiro steps at one-half the canonical voltage follows naturally from the model, along with some of the experimental properties of these steps, especially their much weaker temperature dependence, compared to the canonical steps.

Soliton dynamics in two coupled-ring Josephson junctions

The interaction of nonrelativistic (anti-) fluxons in two coupled-ring Josephson junctions under the action of external currents is considered by using adiabatic perturbation theory. The asymptotic static and dynamic states and conditions for their coexistence are found from qualitative analysis of the system in the phase space. These states correspond to separate stable branches in the current-voltage characteristics, so that hysteresis occurs in a system. Comparison with numerical calculations shows a qualitative agreement with analytical predictions.

Floquet theory for short laser pulses

We develop adiabatic perturbation theory for quantum systems responding to short laser pulses, with or without a frequency chirp. Our approach rests on lifting the time-dependent Schrodinger equation to an extended Hilbert space, then applying standard perturbational techniques to Floquet states in this extended space, and finally projecting back to the physical Hilbert space. The same strategy also allows us to construct superadiabatic bases for monitoring the quantum evolution in the course of a pulse. These bases provide a diagnostic tool for improving the efficiency of pulse-induced population transfer. The formalism is applied to the selective excitation of molecular vibrational states by chirped laser pulses, which exploit either successive single-photon resonances or a multiphoton resonance, and by a STIRAP-like process.

Longitudinal force on a moving potential

We show a formal result of the longitudinal force acting on a moving potential. The potential can be velocity-dependent, which appears in various interesting physical systems, such as electrons in the presence of a magnetic flux-line, or phonons scattering off a moving vortex. By using the phase-shift analysis, we are able to show the equivalence between the adiabatic perturbation theory and the kinetic theory for the longitudinal force in the dilute gas limit.

Importance of Different Multipole Interactions in Fast Reactions at Low Temperature

For fast reactions dominated by long-range multipole interactions, the adiabatic capture model has been shown to work well for the calculation of rate constants. The contributions of different multipole interactions to the rate constants of fast diatom-diatom reactions at low temperatures have been studied by employing the rotational adiabatic second-order perturbation theory (RA2PT). It is found that at ultralow temperatures the weak dipole-induced dipole or quadrupole-quadrupole interactions will probably become significant and even tend to dominate the reaction rate due to the properties of the rotational adiabatic potentials. Calculations for the CN(X) + O-2 reaction and the CH(A) + CO quenching process have been performed to demonstrate this behavior.

Heavy-particle collisions and quantum optics: The parabolic noncrossing model

The problem of deriving analytic formulas for transition probabilities in two-level systems is studied. The two-level systems are described by a pair of first-order differential equations coupled by a time-dependent potential. One such model is given by ${\mathrm{da}}_{m}/dt=\ensuremath{-}i\ensuremath{\beta}{f(t)a}_{n}{e}^{(\ensuremath{-}{1)}^{n}i\ensuremath{\alpha}t}$ $(m,n=1,2$; $m\ensuremath{\ne}/n)$, which describes certain types of ion-atom collisions and some quantum-optics two-level problems. It will be shown that the correct approach in solving the coupled equations is to adopt a Zwaan-Stueckelberg phase-integral analysis of the four-transition-point problem based on the parabolic noncrossing model of Crothers [J. Phys. B 9, 635 (1976)]. Alternatively, one may obtain an approximation by employing adiabatic perturbation theory, but such an approach can at best provide only weak-coupling solutions and can never guarantee unitarity in the probability amplitudes. The advantage of the phase-integral method is that it produces a strong-coupling approximation by embracing the appropriate asymptotic expansions for cylinder functions of large order and argument [D. S. F. Crothers, J. Phys. A 5, 1680 (1972)] and it also ensures analyticity, unitarity, and symmetry.

'Two-stage' perturbation theory for bandwidth-limited amplification of optical solitons near the zero-dispersion point

We show that soliton propagation in transmission lines where bandwidth-limited amplification, nonlinear gain or loss, and a strong third-order dispersion are present can be reasonably described in terms of a ‘‘twostage’’ perturbation theory. In contrast to the adiabatic soliton perturbation theory which is known to fail beyond a critical strength of third-order dispersion, a modulation of the soliton phase caused by third-order dispersion is taken into account in this approach. @S1063-651X~97!07302-9# PACS number~s!: 42.81.Dp Among others, there are two fundamental problems in using solitons in optical fiber communication lines: to reduce the fiber dispersion ~however, keeping it anomalous! ,i n order to allow the weak Kerr nonlinearity to produce a sufficiently narrow soliton, and to compensate the dissipative losses in long-haul systems @1#, simultaneously suppressing the noise. In order to cope with the former problem, a carrier wavelength near the zero-dispersion point ~ZDP! can be used, which in addition leads to the reduction of the soliton peak power. The latter problem can be resolved by means of bandwidth-limited amplification ~BLA!, i.e., the combined action of optical amplifiers and filters. A complicating factor is that third-order dispersion ~TOD! has usually to be taken into account near the ZDP. Although TOD is a Hamiltonian perturbation it gives rise to the emission of radiation where the so-called resonance radiation separates from the soliton and can entail a complete destruction of the soliton @2‐5#. Moreover, TOD leads to the breakup of the two-soliton bound state @2,6‐8#. But it is also known that other perturbations can be exploited to suppress these detrimental effects of TOD. It was shown in @9# that, e.g., the decay of the two-soliton bound state can be avoided if BLA comes into play. Moreover, it has been predicted @8,10# and experimentally verified lately @11# that BLA may absorb the emitted resonance radiation, thus lending the soliton a much better stability. Thus in view of these observations it is necessary to analyze the dynamical properties of solitons in the presence of several qualitatively different perturbations. The effect of TOD on the dynamical behavior of a single soliton has been intensively studied @12,13,6,14,4,5#. The soliton velocity and frequency shifts which appear due to the presence of TOD can correctly be described by transforming

An adiabatic exponential perturbation theory for rotationally inelastic scattering

We develop a perturbation theory to treat rotationally inelastic scattering using an adiabatic basis set. The results for Ar+N2 are twice as accurate as those using a diabatic basis set. The theory can be trivially extended to include closed channels. It can also be simply recast into the exact integration of a set of semiclassical coupled equations. In this mode it agrees to better than 1% with the exact quantal results.

Timing jitter of ultrashort solitons in high-speed communication systems I General formulation and application to dispersion-decreasing fibers

By using adiabatic perturbation theory, we calculate the timing jitter generated by fluctuations in soliton amplitude, frequency, and position that are induced by the amplifiers noise in high-speed soliton communication systems. The analysis is applied to dispersion-tailored fibers which, in contrast with conventional constant-dispersion fibers, allow ultrashort solitons (width <10 ps) to propagate over long distances with minimum production of dispersive waves. We show that a transition from a regime in which amplifier noise-induced frequency fluctuations dominate the timing jitter (Gordon–Haus jitter) to a regime in which amplitude fluctuations dominate the timing jitter occurs when solitons become shorter than 2–7 ps, depending on the total distance of transmission. The latter source of jitter arises because fluctuations in the soliton amplitude are converted to frequency fluctuations by the Raman effect, which in turn are converted to position fluctuations by the group-velocity dispersion. The contribution of third-order dispersion to the timing jitter is evaluated and discussed. We provide an estimate of the distance at which soliton-control elements (such as synchronous modulators) should be inserted to reduce the timing jitter and extend the transmission distance.

Timing jitter of ultrashort solitons in high-speed communication systemsII Control of jitter by periodic optical phase conjugation

We consider the use of periodic optical phase conjugation for reducing the timing jitter in high-speed fiber-optic communication systems employing ultrashort solitons (width &lt;10 ps) in dispersion-decreasing fibers. Using adiabatic perturbation theory, we derive analytically an expression for the trajectory of a periodically conjugated ultrashort soliton in a communication link after including the effect of amplifier noise and use it to derive the timing jitter in the soliton arrival time at the end of a transmission line. The analysis takes into account not only the group-velocity dispersion but also the Raman effect and the third-order dispersion. We show that the timing jitter can be minimized by using an optimized amplifier spacing (∼65–80 km) for a specific value of average fiber dispersion. The use of shorter amplifier spacings increases timing jitter because of third-order dispersion while for larger amplifier spacings the increased jitter originates from the Raman effect. For high-bit rate systems considered in this paper the Gordon–Haus contribution to the timing jitter is negligible. Under optimized conditions, nearly error-free transmission can be realized at a bit rate of ∼100 Gb/s over a distance of 1200 km even in the absence of optical filters. We discuss the role of optical filters for improving system performance and the impact of fiber-dispersion fluctuations on the periodic filtering of ultrashort solitons.

Unification of binary and LCP fission processes

This paper discusses the ambiguities of the parameters of the models used to calculate the yields of binary and light-charged-particle-accompanied (LCP) ternary fission processes. A model based on the adiabatic perturbation theory is set up. It removes these ambiguities and helps to treat the two processes in a unified way.

Timing jitter analysis for optical communication systems using ultrashort solitons and dispersion-decreasing fibers

We use adiabatic perturbation theory to calculate the timing jitter induced by fluctuations in solitons amplitude, frequency and position due to amplifiers noise when ultrashort solitons (∼ 1 ps) are propagated in dispersion-decreasing fibers. The result is applied to a high-speed soliton communication system with amplifiers spacing of 50–100 km. We show that the transition from a regime where frequency fluctuations dominate the timing jitter (Gordon-Haus jitter) to the regime where amplitude fluctuations dominate (Raman-induced jitter) occurs for soliton widths of ∼ 5 ps; the precise value is determined by the total distance of transmission. The contribution of third-order dispersion to the timing jitter is included in our analysis. We also provide an upper estimate of the distance where a soliton-control device (e.g., optical filters, modulators) needs to be inserted to control the timing jitter.

Adiabatic density-functional perturbation theory.

The treatment of adiabatic perturbations within density-functional theory is examined, at arbitrary order of the perturbation expansion. Due to the extremal property of the energy functional, standard variation-perturbation theorems can be used. The different methods (Sternheimer equation, extremal principle, Green's function, and sum over state) for obtaining the perturbation expansion of the wave functions are presented. The invariance of the Hilbert space of occupied wave functions with respect to a unitary transformation leads to the definition of a ''parallel-transport-gauge'' and a ''diagonal-gauge'' perturbation expansion. Then, the general expressions are specialized for the second, third, and fourth derivative of the energy, with an example of application of the method up to third order.

Splitting of high-order spatial solitons under the action of two-photon absorption

We analyze the effect of two-photon absorption on the propagation of high-order spatial optical solitons. The main focus of the paper is the splitting of the soliton bound state which has been recently observed experimentally. Using the results of the adiabatic perturbation theory and the information extracted from the Zakharov-Shabat eigenvalue problem we investigate the evolution of the soliton amplitudes and demonstrate that such a splitting can be understood as resulting from a sequence of the amplitude jumps which finally lead to a structural bifurcation. Different regimes of soliton splitting and decay are analyzed in detail.

To the theory of electron scattering by polyatomic molecules

Within the Born-Oppenheimer adiabatic perturbation theory, an equation was obtained for the intensity of fast electron scattering by polyatomic molecules. All of its parameters are explicitly defined by vibronic interaction in a molecule; due to this, quantum chemical calculations are fully applicable to electron diffraction studies of molecular geometries.