Three-particle Collisions Research Articles

Generation of various classes of entangled states in a two-mode Bose–Einstein condensate under the influence of interatom collisions

In this paper, we generate some new classes of entangled states of a bimodal Bose–Einstein condensate (BEC), a pair of tunnel-coupled BEC, in the presence of two- and three-body elastic as well as mode-exchange collisions. The Hamiltonian of the considered system is very complicated, moreover, it can be fortunately transformed into a simple form using a two-mode displacement operator. After introducing the general form of the time evolved state, various classes of entangled states are generated. Indeed, the influence of different orders of tunneling strengths on the generated entangled states has been studied. Depending on the tunneling strength constants, two-, three- and four-partite entangled states are generated, all of which are superposition states of macroscopic number of BEC atoms. Considering three-particle collision dramatically changes the generated entangled states. Moreover, in particular cases, the resulted states are non-entangled. Also, we show that tunneling and collisional interactions can be manipulated to generate a pair of atomic entangled coherent states (quasi-Bell states). In addition, it is observed that the degree of entanglement for two-partite entangled states can be tuned via the number of BEC atoms, i.e. the corresponding concurrences tend to their maximum value by increasing the atoms in both modes of system.

Entropy production in one-dimensional quantum fluids

We study nonequilibrium thermodynamic properties of a driven one-dimensional quantum fluid by combining nonlinear Luttinger liquid theory with the quantum kinetic equation. In particular, we derive an entropy production consistent with the laws of thermodynamics for a system subject to an arbitrary perturbation varying slowly in space and time. Working in a basis of weakly interacting fermionic quasiparticles, we show that the leading contribution to the entropy production results from three-particle collisions, and we derive its scaling law at low temperatures.

Operating stability analysis of continuous gas–liquid–solid FBR under super-condensed mode

In the gas–liquid–solid fluidized bed reactor (FBR) under super-condensed mode, the injection of liquid causes the particle agglomeration that leads to a complicated and unstable fluidization system. In this work, a probabilistic stability model is developed on the basis of a three-particle collision theory to describe such mode. The criterion of the model is achieved by analyzing the particle interaction, relative velocity distribution, and energy transfer in particle collision. The model error mainly originates from by the simplification of the actual system. The developed model is applicable to the case that a droplet completely spreads on the particle surface, that is, the solid–liquid contact angle is less than 40°. We then systematically verify the current model by comparison with the experimental data from the literature. The developed model provides an application prospect in reactor design and scale-up.

Statistical analysis of dimer formation in supersaturated metal vapor based on molecular dynamics simulation

We discuss the problem of dimer formation during the homogeneous nucleation of atomic metal vapor in an inert gas environment. We simulated nucleation with molecular dynamics and carried out the statistical analysis of double- and triple-atomic collisions as the two ways of long-lived diatomic complex formation. Close pair of atoms with lifetime greater than the mean time interval between atom–atom collisions is called a long-lived diatomic complex. We found that double- and triple-atomic collisions gave approximately the same probabilities of long-lived diatomic complex formation, but internal energy of the resulted state was essentially lower in the second case. Some diatomic complexes formed in three-particle collisions are stable enough to be a critical nucleus.

Interaction effects on thermal transport in quantum wires

We develop a theory of thermal transport of weakly interacting electrons in quantum wires. Unlike higher-dimensional systems, a one-dimensional electron gas requires three-particle collisions for energy relaxation. The fastest relaxation is provided by the intrabranch scattering of comoving electrons which establishes a partially equilibrated form of the distribution function. The thermal conductance is governed by the slower interbranch processes which enable energy exchange between counterpropagating particles. We derive an analytic expression for the thermal conductance of interacting electrons valid for arbitrary relation between the wire length and electron thermalization length. We find that in sufficiently long wires the interaction-induced correction to the thermal conductance saturates to an interaction-independent value.

Thermalization of Nonequilibrium Electrons in Quantum Wires

We study the problem of energy relaxation in a one-dimensional electron system. The leading thermalization mechanism is due to three-particle collisions. We show that for the case of spinless electrons in a single channel quantum wire the corresponding collision integral can be transformed into an exactly solvable problem. The latter is known as the Schrödinger equation for a quantum particle moving in a Pöschl-Teller potential. The spectrum for the resulting eigenvalue problem allows for bound-state solutions, which can be identified with the zero modes of the collision integral, and a continuum of propagating modes, which are separated by a gap from the bound states. The inverse gap gives the time scale at which counterpropagating electrons thermalize.

Interaction-induced corrections to conductance and thermopower in quantum wires

We study transport properties of weakly interacting spinless electrons in one-dimensional single-channel quantum wires. The effects of interaction manifest as three-particle collisions due to the severe constraints imposed by the conservation laws on the two-body processes. We focus on short wires where the effects of equilibration on the distribution function can be neglected and the collision integral can be treated in perturbation theory. We find that interaction-induced corrections to conductance and thermopower rely on the scattering processes that change the number of right- and left-moving electrons. The latter requires transition at the bottom of the band which is exponentially suppressed at low temperatures. Our theory is based on the scattering approach that is beyond the Luttinger-liquid limit. We emphasize the crucial role of the exchange terms in the three-particle scattering amplitude that was not discussed in previous studies.

Transport in partially equilibrated inhomogeneous quantum wires

We study transport properties of weakly interacting one-dimensional electron systems including on an equal footing thermal equilibration due to three-particle collisions and the effects of large-scale inhomogeneities. We show that equilibration in an inhomogeneous quantum wire is characterized by the competition of interaction processes which reduce the electrons total momentum and such which change the number of right- and left-moving electrons. We find that the combined effect of interactions and inhomogeneities can dramatically increase the resistance of the wire. In addition, we find that the interactions strongly affect the thermoelectric properties of inhomogeneous wires and calculate their thermal conductance, thermopower, and Peltier coefficient.

Relaxation dynamics of stochastic long-range interacting systems

Long-range interacting systems, while relaxing towards equilibrium, may get trapped innonequilibrium quasistationary states (QSS) for a time which diverges algebraically withthe system size. These intriguing non-Boltzmann states have been observed underdeterministic Hamiltonian evolution of a paradigmatic system, the Hamiltonian mean-field(HMF) model. We study here the robustness of QSS with respect to stochastic processesbeyond deterministic dynamics within a microcanonical ensemble. To this end, wegeneralize the HMF model by allowing for stochastic three-particle collision dynamics inaddition to the deterministic ones. By analyzing the resulting Boltzmann equation forthe phase space density, we demonstrate that, in the presence of stochasticity,QSS occur only as a crossover phenomenon over a finite time determined by thestrength of the stochastic process. In particular, we argue that the relaxation timeto equilibrium does not scale algebraically with the system size. We propose ascaling form for the relaxation time which is in very good agreement with results ofextensive numerical simulations. The broader validity of these results is tested on adifferent stochastic HMF model involving microcanonical Monte Carlo dynamicalmoves.

Newton’s cradle undone: Experiments and collision models for the normal collision of three solid spheres

Using an apparatus inspired by Newton’s cradle, the simultaneous, normal collision between three solid spheres is examined. Namely, an initially touching, motionless pair of “target” particles (doublet) is impacted on one end by a third “striker” particle. Measurements of postcollisional velocities and collision durations are obtained via high-speed photography and an electrical circuit, respectively. Contrary to intuition, the expected Newton's cradle outcome of a motionless, touching particle pair at the bottom of the pendulum arc is not observed in either case. Instead, the striker particle reverses its direction and separates from the middle particle after collision. This reversal is not observed, however, if the target particles are separated by a small distance (not in contact) initially, although a separation still occurs between the striker and middle particle after the collision, with both particles traveling in the same direction. For the case of initially touching target particles, contact duration measurements indicate that the striker separates from the three particles before the two target particles separate. However, when the targets are slightly separated, a three-particle collision is never observed, and the collision is, in fact, a series of two-body collisions. A subsequent implementation of a variety of hard-sphere and soft-sphere collision models indicates that a three-body (soft-sphere) treatment is essential for predicting the velocity reversal, consistent with the experimental findings. Finally, a direct comparison between model predictions and measurements of postcollisional velocities and contact durations provides a gauge of the relative merits of existing collision models for three-body interactions.

Solving the electron-nuclear Schrödinger equation of helium atom and its isoelectronic ions with the free iterative-complement-interaction method

Our previous paper [J. Chem. Phys. 127, 224104 (2007)] revealed that the Schrodinger equation in the fixed-nucleus approximation could be very accurately solved for helium atom and its isoelectronic ions (Z=1-10) with the free iterative-complement-interaction (ICI) method combined with the variation principle. In this report, the quantum effect of nuclear motion has further been variationally considered by the free ICI formalism for the Hamiltonian including mass-polarization operator. We obtained -2.903 304 557 729 580 294 733 816 943 892 697 752 659 273 965 a.u. for helium atom, which is over 40 digits in accuracy, similarly to the previous result for the fixed-nucleus level. Similar accuracy was also obtained for the helium isoelectronic ions. The present results may be regarded to be the nonrelativistic limits. We have further analyzed the physics of the free ICI wave function by applying it to an imaginary atom called "eneon," [e(-)e(10+)e(-)](8+), in which both of the quantum effect of nuclear motion and the three-particle collisions are differently important from the helium and its isoelectronic ions. This revealed the accurate physics automatically generated by the free ICI formalism.

Solving the Schrödinger equation for helium atom and its isoelectronic ions with the free iterative complement interaction (ICI) method

The Schrodinger equation was solved very accurately for helium atom and its isoelectronic ions (Z=1-10) with the free iterative complement interaction (ICI) method followed by the variational principle. We obtained highly accurate wave functions and energies of helium atom and its isoelectronic ions. For helium, the calculated energy was -2.903,724,377,034,119,598,311,159,245,194,404,446,696,905,37 a.u., correct over 40 digit accuracy, and for H(-), it was -0.527,751,016,544,377,196,590,814,566,747,511,383,045,02 a.u. These results prove numerically that with the free ICI method, we can calculate the solutions of the Schrodinger equation as accurately as one desires. We examined several types of scaling function g and initial function psi(0) of the free ICI method. The performance was good when logarithm functions were used in the initial function because the logarithm function is physically essential for three-particle collision area. The best performance was obtained when we introduce a new logarithm function containing not only r(1) and r(2) but also r(12) in the same logarithm function.

Three-particle collisions in quantum wires: Corrections to thermopower and conductance

We consider the effect of electron-electron interaction on the electron transport through a finite length single-mode quantum wire with reflectionless contacts. The two-particle scattering events cannot alter the electric current and therefore we study the effect of three-particle collisions. Within the Boltzmann equation framework, we calculate corrections to the thermopower and conductance to the leading order in the interaction and in the length of wire $L$. We check explicitly that the three-particle collision rate is identically zero in the case of several integrable interaction potentials. In the general (non-integrable) case, we find a positive contribution to the thermopower to leading order in $L$. The processes giving rise to the correction involve electron states deep in the Fermi sea. Therefore the correction follows an activation law with the characteristic energy of the order of the Fermi energy for the electrons in the wire.

On the soliton states of an atomic Bose-Einstein condensate

Numerical analysis of the one-, two-, and three-dimensional soliton solutions for the modified Gross-Pitaevski equation with regard to the nonlocality of interaction between atoms of a condensate is performed. In calculations of the parameter of interaction nonlocality, the Lennard-Jones potential for collisions between Li atoms in the triplet state was used. It is shown that the nonlocality stabilizes the three-dimensional solitons with sizes in a certain range. The lifetimes of solitons, which are determined by two-and three-particle collisions between atoms, are estimated.

Atomic “needles” and “bullets” in Bose-Einstein condensate and the formation of nanosized structures

The Gross-Pitaevski equation modified through the inclusion of a term accounting for the nonlocality of interatomic interaction was used to demonstrate the occurrence of extremely narrow two-and three-dimensional solitonic states in atomic Bose-Einstein condensates. The estimates of the sizes of these states gave a value of ∼ 20–60 nm (atomic “needles” and “bullets”) for lithium atoms. The soliton lifetimes caused by two-and three-particle collisions were estimated. The limiting possibilities of the formation of nanostructures using needles and bullets were compared with the possibilities of other nanolithographic methods.

Stopping power of nonideal, partially ionized plasmas.

The stopping power of strongly coupled, partially ionized plasmas is investigated for charged beam particles with arbitrary velocities. Our approach is based on kinetic equations of the Boltzmann type that are suitably generalized to describe three-particle collisions. In this way, we consider elastic collisions between the beam and free plasma particles as well as the ionization and excitation of composite plasma particles by beam particle impact. Explicit expressions for both contributions are given in terms of the momentum transfer cross section that has been generalized for three-particle collisions. For fast beam particles, we obtain a generalized Bethe formula that includes correction terms due to the nonideality of the target plasma. Results are shown for hydrogen, carbon, and argon plasmas. Considerable modifications compared to the ideal behavior arise for strongly coupled plasmas. In particular, we are able to describe the Mott transition in the stopping power of dense, partially ionized plasmas.

Enhancing electroluminescence from metal-oxide–silicon tunneling diodes by nano-structures of oxide grown by liquid-phase method

Significant enhancement of electroluminescence (EL) at Si bandgap energy is discovered from metal-oxide–semiconductor tunneling diode on p-type Si with oxide grown by liquid-phase deposition (LPD). The LPD grown oxide has nano-structures with the grain size of 10–20 nm. The nano-structure of oxide causes the simultaneous localization of electrons and holes at the Si/SiO 2 interface, similar to the formation of excitons. This makes the process of the phonon-assisted radiative recombination of electron–hole pair more like two-particle collision than three-particle collision, so increasing the probability of radiative recombination. The measured EL efficiency could be more than 1×10 −6.

Pion propagation at finite temperature.

We calculate the pole shift of the pion up to second order in the density, considering the low-temperature expansion of the axial two-point function. To this order both absorptive and dispersive properties of pions are completely determined by two- and three-particle [ital S]-matrix elements. In particular, we show how the pole shift can be expressed in terms of these scattering amplitudes and discuss the size of corrections from three-particle collisions.

Simulation of the breakup channel in three-particle collisions with pseudoreaction channels.

A pseudochannel extension of the coupled-reaction-channel (CRC) method is investigated formally and numerically as a means of simulating the effect of the breakup channel on the rearrangement amplitudes. The equations of the pseudostate-augmented CRC method are derived within a two-Hilbert-space approach. The connection of the effective CRC transition operators with the postform transition operators of the formal scattering theory is established. Alternative formulations of the method corresponding to different ways of treating the nonorthogonality interaction are given. Convergence with respect to the number and nature of the pseudostates are investigated numerically on a solvable three-particle model. It is observed that the results obtained with six distinct sets of pseudochannels that have considerably different threshold energies converge toward the exact ones.

Cluster expansion for transport coefficients for dense simple fluids

Cluster expansions are presented for the collision bracket integrals for transport cofficients of a dense simple fluid which appear in the kinetic theory of dense fluid reported previously. The density series calculated with the cluster expansions are given in exponential forms and their leading terms consist of a connected three-particle collision operator. The three-particle contributions are discussed in terms of mass-normalized coordinates which make it possible to put them in forms structurally rather similar to the two-particle collision bracket integrals in the Chapman-Enskog theory. The three particle contributions are thus cast into forms suitable for numerical computation.