Self-consistent field theory of collisions: Orbital equations with asymptotic sources and self-averaged potentials

The self-consistent field theory of collisions is formulated, incorporating the unique dynamics generated by the self-averaged potentials. The bound state Hartree–Fock approach is extended for the first time to scattering states, by properly resolving the principal difficulties of non-integrable continuum orbitals and imposing complex asymptotic conditions. The recently developed asymptotic source theory provides the natural theoretical basis, as the asymptotic conditions are completely transferred to the source terms and the new scattering function is made fullyintegrable. The scattering solutions can then be directly expressed in terms of bound state HF configurations, establishing the relationship between the bound and scattering state solutions. Alternatively, the integrable spin orbitals are generated by constructing the individual orbital equations that contain asymptotic sources and self-averaged potentials. However, the orbital energies are not determined by the equations, and a special channel energy fixing procedure is developed to secure the solutions. It is also shown that the variational construction of the orbital equations has intrinsic ambiguities that are generally associated with the self-consistent approach. On the other hand, when a small subset of open channels is included in the source term, the solutions are only partiallyintegrable, but the individual open channels can then be treated more simply by properly selecting the orbital energies. The configuration mixing and channel coupling are then necessary to complete the solution. The new theory improves the earlier continuum HF model.     

The relativistic bound and scattering states of the Manning-Rosen potential with an improved new approximate scheme to the centrifugal term

The approximately analytical bound and scattering state solutions of the arbitrary l-wave Klein-Gordon equation for the mixed Manning-Rosen potentials are carried out by an improved new approximation to the centrifugal term. The normalized analytical radial wave functions of the l-wave Klein-Gordon equation with the mixed Manning-Rosen potentials are presented and the corresponding energy equations for bound states and phase shifts for scattering states are derived. It is shown that the energy levels of the continuum states, reduce to the bound states of those at the poles of the scattering amplitude. Some useful figures are plotted to show the improved accuracy of our results and the special case for wave is studied briefly.      

Analytic continuation of scattering data to the region of negative energies for systems that have one and two bound states

An exactly solvable potential model is used to study the possibility of deducing information about the features of bound states for the system under consideration (binding energies and asymptotic normalization coefficients) on the basis of data on continuum states. The present analysis is based on an analytic approximation and on the subsequent continuation of a partial-wave scattering function from the region of positive energies to the region of negative energies. Cases where the system has one or two bound states are studied. The α+d and α+¹²C systems are taken as physical examples. In the case of one bound state, the scattering function is a smooth function of energy, and the procedure of its analytic continuation for different polynomial approximations leads to close results, which are nearly coincident with exact values. In the case of two bound states, the scattering function has two poles—one in the region of positive energies and the other in the region of negative energies between the energies corresponding to the two bound states in question. Padéapproximants are used to reproduce these poles. The inclusion of these poles proves to be necessary for correctly describing the properties of the bound states.     

Borromean three-body heteroatomic resonances

We present an approach to analyze recent experimental evidences of Efimov resonant states in mixtures of ultracold gases, by considering two-species three-body atomic systems bound in a Borromean configuration, where all the two-body interactions are unbound. For such Borromean three-body systems, it is shown that a continuum three-body s-wave resonance emerges from an Efimov state as a scattering length or a three-body scale is moved. The energy and width of the resonant state are determined from a scaling function with arguments given by dimension-less energy ratios relating the two-body virtual state subsystem energies with the shallowest three-body bound state. The peculiar behavior of such resonances is that their peaks are expected to move to lower values of the scattering length, with increasing width, as one raises the temperature. For Borromean systems, two resonant peaks are expected in ultralow-temperature regimes, which will disappear at higher energies. It is shown how a Borromean-Efimov excited bound state turns out to a resonant state by tuning the virtual two-body subsystem energies or scattering lengths, with all energies written in units of the next deeper shallowest Efimov state energy. The resonance position and width for the decay into the continuum are obtained as universal scaling functions (limit cycle) of the dimensionless ratios of the two and three-body scales, which are calculated numerically within a zero-range renormalized three-body model.     

Quantum Monte Carlo simulation of exciton–exciton scattering in a GaAs/AlGaAs quantum well

We present a computer simulation of exciton–exciton scattering in a quantum well. Specifically, we use quantum Monte Carlo techniques to study the bound and continuum states of two excitons in a 10 nm wide GaAs/Al_0.3Ga_0.7As quantum well. From these bound and continuum states we extract the momentum-dependent phase shifts for s-wave scattering. A surprising finding of this work is that a commonly studied effective-mass model for excitons in a 10 nm quantum well actually supports two bound biexciton states. The second, weakly bound state may dramatically enhance exciton–exciton interactions. We also fit our results to a hard-disk model and indicate directions for future work.     

Metastable States in Parametrically Excited Multimode Hamiltonian Systems

 Consider a linear autonomous Hamiltonian system with m time periodic bound state solutions. In this paper we study their dynamics under time almost periodic perturbations which are small, localized and Hamiltonian. The analysis proceeds through a reduction of the original infinite dimensional dynamical system to the dynamics of two coupled subsystems: a dominant m-dimensional system of ordinary differential equations (normal form), governing the projections onto the bound states and an infinite dimensional dispersive wave equation. The present work generalizes previous work of the authors, where the case of a single bound state is considered. Here, the interaction picture is considerably more complicated and requires deeper analysis, due to a multiplicity of bound states and the very general nature of the perturbation's time dependence. Parametric forcing induces coupling of bound states to continuum radiation modes, of bound states directly to bound states, as well as coupling among bound states, which is mediated by continuum modes. Our analysis elucidates these interactions and we prove the metastability (long life time) and eventual decay of bound states for a large class of systems. The key hypotheses for the analysis are: appropriate local energy decay estimates for the unperturbed evolution operator, restricted to the continuous spectral part of the Hamiltonian, and a matrix Fermi Golden rule condition, which ensures coupling of bound states to continuum modes. Problems of the type considered arise in many areas of application including ionization physics, quantum molecular theory and the propagation of light in optical fibers in the presence of defects. Received: 13 March 2002 / Accepted: 2 January 2003 Published online: 14 April 2003 Communicated by J.L. Lebowitz     

Allowing for Coulomb effects in the effective range expansion for two coupled channels

A Coulomb-modified matrix of scattering amplitudes (an [(F)\tilde]\tilde F matrix) is considered for the case of two coupled channels of elastic scattering of charged particles with different orbital angular momenta (l ₁ and l ₂ = l ₁ + 2). Matrix elements of the [(F)\tilde]\tilde F matrix are expressed in terms of the matrix elements of a [(K)\tilde] ^{- 1}\tilde K^{ - 1} matrix inverse to a modified reaction K matrix. The elements of the [(K)\tilde] ^{- 1}\tilde K^{ - 1} matrix are written as expansions that are generalizations of single-channel effective range expansion with allowance for the Coulomb interaction. If a system of colliding particles involves a bound state, the analytic continuation of these expansions into the region of negative energies makes it possible to obtain both the position of the pole corresponding to the bound state and the scattering amplitude residues in this pole, in terms of which the corresponding vertex constants and asymptotic normalization coefficients are expressed.     

Study of theS-matrix near bound states in the continuum

The behaviour ofS-matrix for potentials generating bound states in continuum in the neighbourhood of the positive bound state energies is studied. It is shown that unlike the case of usual negative energy bound states, theS-matrix does not have a pole at the positive bound state energy but becomes unity at the energy corresponding to bound states in continuum. Calculations ofS-waveS-matrix for a local potential constructed by Stillinger and Herrick and a separable nonlocal potential constructed by the present authors verify these results. Our results indicate that the bound states embedded in continuum constructedvia the von Neumann and Wigner procedure cannot be interpreted as resonances with zero width.     

On the Distinguishability of Random Quantum States

We develop two analytic lower bounds on the probability of success p of identifying a state picked from a known ensemble of pure states: a bound based on the pairwise inner products of the states, and a bound based on the eigenvalues of their Gram matrix. We use the latter, and results from random matrix theory, to lower bound the asymptotic distinguishability of ensembles of n random quantum states in d dimensions, where n/d approaches a constant. In particular, for almost all ensembles of n states in n dimensions, p > 0.72. An application to distinguishing Boolean functions (the “oracle identification problem”) in quantum computation is given.     

Self-consistent field theory of collisions I. Scattering channels

The conventional Hartree and Hartree-Fock approaches for treating many-electron bound systems have been extended recently to positive energy scattering problems, in which both the bound and continuum orbitals are determined by the requirement of full self-consistency. Serious consequences of such a theory are that the target orbitals become energy dependent and the asymptotic boundary conditions are satisfied only approximately, in lowest order. It is important therefore to test the theory for its convergence under configuration mixing. This self-consistent field (SCF) theory for scattering has been tested here for scattering from hydrogenic target as a model where the target function is determined dynamically. Penetration of the projectile inside the bound target orbital is manifest through the SCF for the bound state. Our results show that the theory converges to the correct amplitudes and to the exact boundary conditions as more configurations are added. The use of the amputated functions and the weak asymptotic condition (WAC) upon which the SCF theory is based, is justified as the WAC converges to the correct limit. It is then applied to the positron-helium and electron-helium scattering systems where the helium function is calculated simultaneously together with the scattering function. The resulting phase shifts and the SCF target functions are compared with those obtained with the pre-determined target functions in the conventional approaches. Received 22 September 1999     

Spectral properties and bound-state scattering for weakly coupled λP(ϕ)2 models

We analyze the particle structure in weakly coupled λP(?)₂ models. Topics include existence of bound states and resonances, asymptotic expansions for bound-state masses, low-energy asymptotic completeness, and perturbation theory for scattering events involving bound states. The analysis is based on analyticity properties and perturbation theory for the Bethe-Salpeter kernel.     

Ladder approximation of the equation of state for degenerate fermi systems

The equation of state is discussed for a system of degenerate fermions. The formation of bound states is allowed for; the latter may exhibit the Bose-Einstein condensation. A complex representation is given which includes bound and scattering two-particle states as well. Different (generalized) representations in terms of scattering quantities are given for the continuum contribution.     

The Relativistic Scattering States of the Hulthén Potential with an Improved New Approximate Scheme to the Centrifugal Term

The approximately analytical scattering state solutions of the l-wave Klein-Gordon equation with the unequal scalar and vector Hulthén potentials are carried out by an improved new approximate scheme to the centrifugal term. The normalized analytical radial wave functions of the l-wave Klein-Gordon equation with the mixed Hulthén potentials are presented and the corresponding calculation formula of phase shifts is derived. It is well shown that the energy levels of the continuum states reduce to those of the bound states at the poles of the scattering amplitude. Some useful figures are plotted to show the improved accuracy of our results and two special cases for s-wave (l=0) and for l=0 and equal scalar and vector Hulthén potentials are also studied briefly.     

Resonance scattering with modified continuum states: (II). Application of the unified model with one particle in the continuum to the 12C + n system

Starting from the conventional shell model, different methods for the treatment of the continuum are given. Using the modified continuum states of part I together with the reaction operator formalism, the present approach unifies the numerical features of the R-matrix theory and the physical picture of the shell model. As it treats the bound states and the scattering states in the same basis we calculated both for the ¹²C + n system as a test case. It is found that the results resemble the experimental data quite well, including for the first time the negative-parity resonances.     

Coherent Population Trapping in a Λ-Configuration Photoionization System in the Presence of an Autoionizing State

We investigate the atomic coherent population trapping in the system of two-mode quantized field interacting with a A-configuration atom with two lower bound states, one autoionizing state and a flat atomic continuum. The states of the field which traps the atom in its two lower bound states are given and the influence of the field induced continuum structure is examined.     

Quasi-elastic scattering of the proton drip line nucleus 17F on 12C at 60 MeV

The quasi-elastic scattering angular distribution of the proton drip line nucleus ¹⁷F on a ¹²C target was measured at 60 MeV. The experimental data have been compared with the theoretical analysis based onto optical model and continuum discretized coupled channels (CDCC). The couplings between breakup and elastic scattering channels, and between inelastic and elastic scattering channels resulted very weak. In order to explore the breakup effects the total reaction cross-section was deduced from the angular distribution of the quasi-elastic scattering data, and then compared with the existing data for the other weakly and tightly bound nuclei on ¹²C target using a universal function. From this comparison, we concluded that the breakup effect is not important for weakly bound projectiles on the light target as obtained also with the CDCC analysis.     

Quantum manifestation of systems on the macro-scale — the concept of transition state and transition amplitude wave

Quantum effects which have usually been associated with micro-scale phenomena can also arise on the macro-scale in situations other than the well-known macro-quantum phenomena of superconductivity and superfluidity. Such situations have been shown here to arise in processes involving inelastic scattering with bound or partially bound systems (not bound in all degrees of freedom), and the macro-quantum behaviour is associated with the state of the total system in transition in the process of scattering. Such a state is designated as a ‘transition-state’. It is pointed out that we have already observed such manifestations for a particular system, the charged particles in a magnetic field where interference effects involving macro-scale matter waves along the magnetic field have been reported [R K Varma et al, Phys. Rev. E65, 026503 (2002)]      

Coupled-channel effective field theory and proton- <Superscript>7</Superscript>Li scattering

We apply the renormalisation group (RG) to analyse scattering by short-range forces in systems with coupled channels. For two S -wave channels, we find three fixed points, corresponding to systems with zero, one or two bound or virtual states at threshold. We use the RG to determine the power countings for the resulting effective field theories. In the case of a single low-energy state, the resulting theory takes the form of an effective-range expansion in the strongly interacting channel. We also extend the analysis to include the effects of the Coulomb interaction between charged particles. The approach is then applied to the coupled p + ^7Li