Part II. Algebraic tails in three-dimensional quantum plasmas

We review various exact results concerning the presence of algebraic tails in three-dimensional quantum plasmas. First, we present a solvable model of two quantum charges immersed in a classical plasma. The effective potential between the quantum charges is shown to decay as 1/r ⁶ at large distances r. Then, we mention semiclassical expansions of the particle correlations for charged systems with Maxwell-Boltzmann statistics and short-ranged regularization of the Coulomb potential. The quantum corrections to the classical quantities, from orderh ⁴ on, also decay as 1/r ⁶. We also give the result of an analysis of the charge correlation for the one-component plasma in the framework of the usual many-body perturbation theory; some Feynman graphs beyond the random phase approximation display algebraic tails. Finally, we sketch a diagrammatic study of the correlations for the full many-body problem with quantum statistics and pure 1/r interactions. The particle correlations are found to decay as 1/r ⁶, while the charge correlation decays faster, as 1/r ¹⁰. The coefficients of these tails can be exactly computed in the low-density limit. The absence of exponential screening arises from the quantum fluctuations of partially screened dipolar interactions.     

Quasiclassical Statistical Thermodynamics and New Padé Approximations for the Free Charges in Strongly-Coupled Plasma

HNC equations in combination with effective quasi-classical potentials are used to calculate correlation functions and the thermodynamic properties of the free charges in semi-classical non-degenerate quantum plasmas. The interactions of the free particles are taken into account via effective potentials obtained from the Slater sum method. Analytical formulae reproducing the known limits and the HNC-results are constructed. Finally quantum effects are included as corrections by using known analytical results. This method is used to develope new Padé approximations for the subsystem of the free charges in mass-symmetrical as well as for mass-unsymmetrical hydrogen-like plasmas. The most essential result of our investigations is, that in the classical limit the scaling properties correspond to the OCP, e.g. the thermodynamic functions follow for large coupling strength Γ a Berlin-Montroll-Rosenfeld asymptotics via a Γ + b Γ^v + c ln Γ + d. Including quantum effects, the coefficients depend on the temperature, e.g. the slope a(T) increases with decreasing T converging to the classical limit. The new formulae are compared with earlier variants.     

Classical coulomb systems near a plane wall. I

The equilibrium structure of classical Coulomb systems bounded by a plane wall is studied near that wall. Several models are considered: the two-dimensional one-component plasma at a special value of the coupling constant (which makes the model exactly soluble), the two-dimensional and three-dimensional one-component and two-component plasmas in the weak-coupling limit (a Debye-Hückel type of approach is then used). Along a wall, the pair correlation functions decay only as an inverse power of the distancer, namely, asr ^–v for av-dimensional system (v=2,3). The one-body densities are also studied; the first BGY equation is used.     

Distribution functions for a two-dimensional non-interacting quantum electron gas in an external magnetic field

The exact n-body distribution functions are calculated for a two-dimensional, non-interacting quantum electron gas in an external magnetic field for any temperature and density. At low tempertures and filled lowest Landau level (LLL), these functions are identical to the exact distribution functions obtained by Jancovici [1981, Phys. Rev. Lett., 46, 386] for the classical two-dimensional one-component plasma (2DOCP) at the special plasma parameter Γ = 2, thus establishing that the 2DOCP provides an exact classical Boltzmann factor which describes the ideal LLL quantum state associated with the integral quantum Hall effect.     

Universality in some classical Coulomb systems of restricted dimension

Coulomb systems in which the particles interact through thed-dimensional Coulomb potential but are confined in a flat manifold of dimensiond–1 are considered. The actual Coulomb potential acting is defined by particular boundary conditions involving a characteristic macroscopic distanceW in the direction perpendicular to the manifold: either it is periodic of periodW in that direction, or it vanishes on one ideal conductor wall parallel to the manifold at a distanceW from it, or it vanishes on two parallel walls at a distanceW from each other with the manifold equidistant from them. Under the assumptions that classical equilibrium statistical mechanics is applicable and that the system has the macroscopic properties of a conductor, it is shown that the suitably smoothed charge correlation function is universal, and that the free energy and the grand potential have universal dependences onW (universal means independent of the microscopic detail). The casesd=2 are discussed in detail, and the generic results are checked on an exactly solvable model. The cased=3 of a plane parallel to an ideal conductor is also explicitly worked out.Laboratoire associé au Centre National de la Recherche Scientifique-URA D0063.     

Identification of lagrangian coherent structures in the turbulent boundary layer

Using Finite-Time Lyapunov Exponents (FTLE) method, Lagrangian coherent structures (LCSs) in a fully developed flat-plate turbulent boundary layer are successfully identified from a two-dimensional (2D) velocity field obtained by time-resolved 2D PIV measurement. The typical LCSs in the turbulent boundary layer are hairpin-like structures, which are characterized as legs of quasi-streamwise vortices extending deep into the near wall region with an inclination angle θ to the wall, and heads of the transverse vortex tube located in the outer region. Statistical analysis on the characteristic shape of typical LCS reveals that the probability density distribution of θ accords well with t-distribution in the near wall region, but presents a bimodal distribution with two peaks in the outer region, corresponding to the hairpin head and the hairpin neck, respectively. Spatial correlation analysis of FTLE field is implemented to get the ensemble-averaged inclination angle θ _R of typical LCS. θ _R first increases and then decreases along the wall-normal direction, similar to that of the mean value of θ. Moreover, the most probable value of θ saturates at y ⁺=100 with the maximum value of about 24°, suggesting that the most likely position where hairpins transit from the neck to the head is located around y ⁺=100. The ensemble- averaged convection velocity U _c of typical LCS is finally calculated from temporal-spatial correlation analysis of FTLE field. It is found that the wall-normal profile of the convection velocity U _c(y) accords well with the local mean velocity profile U(y) beyond the buffer layer, evidencing that the downstream convection of hairpins determines the transportation properties of the turbulent boundary layer in the log-region and beyond. Supported by the National Natural Science Foundation of China (Grant Nos. 10425207 and 10832001)     

Static structure factor of electrons around a heavy positively charged impurity in a one-component quantum rare plasma

An expression for the static structure factor,g ⁺⁻ (r), of electrons at a distancer from an infinitely heavy positively charged particle in a one component quantum rare plasma has been obtained in linear response theory using an appropriate quantum dielectric function of the rare plasma. The expression is a complicated function of the electron plasma frequency, Debye screening length andr, but reduces to that of classical plasma when quantum corrections are neglected. Forr<r _s (2r _s being the mean distance between two electrons), the temperature dependentg ⁺⁻ (r) has larger values in quantum case in comparison to that in classical situation and keeps increasing with decrease inr, more so at low temperatures when de-Broglie wavelength becomes larger and a considerable fraction ofr _s.     

Resonant amplification of quantum fluctuations in a spinor gas

Bose-Einstein condensates of atoms with non-zero spin are known to constitute an ideal system to investigate fundamental properties of magnetic superfluids. More recently it was realized that they also provide the fascinating opportunity to investigate the macroscopic amplification of quantum and classical fluctuations. This is strikingly manifested in a sample initially prepared in the m _F = 0 state, where spin-changing collisions triggered by quantum fluctuations may lead to the creation of correlated pairs in m _F = ±1. We show that the pair creation efficiency is strongly influenced by the interplay between the external trapping potential and the Zeeman effect. It thus reflects the confinement-induced magnetic field dependence of elementary spin excitations of the condensate. Remarkably, pair production in our experiments is therefore characterized by a multi-resonant dependence on the magnetic field. Pair creation at these resonances acts as strong parametric matter-wave amplifier. Depending on the resonance condition, this amplification can be extremely sensitive or insensitive to the presence of seed atoms. We show that pair creation at a resonance which is insensitive to the presence of seed atoms is triggered purely by quantum fluctuations and thus the system acts as a matter-wave amplifier for the vacuum state.     

Aspects of supersymmetric quantum mechanics

We review the properties of supersymmetric quantum mechanics for a class of models proposed by Witten. Using both Hamiltonian and path integral formulations, we give general conditions for which supersymmetry is broken (unbroken) by quantum fluctuations. The spectrum of states is discussed, and a virial theorem is derived for the energy. We also show that the euclidean path integral for supersymmetric quantum mechanics is equivalent to a classical stochastic process when the supersymmetry is unbroken (E₀ = 0). By solving a Fokker-Planck equation for the classical probability distribution, we find P_c(y) is identical to |Ψ₀(y)|² in the quantum theory.     

Frequency and density dependent radiative recombination processes in III–V semiconductor quantum wells and superlattices

In this paper we review the radiative recombination processes occurring in semiconductor quantum wells and superlattices under different excitation conditions. We consider processes whose radiative efficiency depends on the photogenerated density of elementary excitations and on the frequency of the exciting field, including luminescence induced by multiphoton absorption, exciton and biexciton radiative decay, luminescence arising from inelastic excitonic scattering, and electron-hole plasma recombination.

Semiconductor quantum wells are ideal systems for the investigation of radiative recombination processes at different carrier densities owing to the peculiar wavefunction confinement which enhances the optical non-linearities and the bistable behaviour of the crystal. Radiative recombination processes induced by multi-photon absorption processes can be studied by exciting the crystal in the transparency region under an intense photon flux. The application of this non-linear spectroscopy gives direct access to the excited excitonic states in the quantum wells owing to the symmetry properties and the selection rules for artificially layered semiconductor heterostructures.

Different radiative recombination processes can be selectively tuned at exciting photon energies resonant with real states or in the continuum of the conduction band depending on the actual density of photogenerated carriers. We define three density regimes in which different quasi-particles are responsible for the dominant radiative recombination mechanisms of the crystal: (i) The dilute boson gas regime, in which exciton density is lower than 10¹⁰ cm^-2. Under this condition the decay of free and bound excitons is the main radiative recombination channel in the crystal. (ii) The intermediate density range (n < 10¹¹ cm^-2) at which excitonic molecules (biexcitons) and inelastic excitonic scattering processes contribute with additional decay mechanisms to the characteristic luminescence spectra. (iii) The high density range (n ?10¹² cm^-2) where screening of the Coulomb interaction leads to exciton ionization. The optical transitions hence originate from the radiative decay of free-carriers in a dense electron-hole plasma.

The fundamental theoretical and experimental aspects of the radiative recombination processes are discussed with special attention to the GaAs/Al _x Ga_1-x As and Ga _x In_1-x As/Al _y In_1-y As materials systems. The experimental investigations of these effects are performed in the limit of intense exciting fields by tuning the density of photogenerated quasi-particles and the frequency of the exciting photons. Under these conditions the optical response of the quantum well strongly deviates from the well-known linear excitonic behaviour. The optical properties of the crystal are then no longer controlled by the transverse dielectric constant or by the first-order dielectric susceptibility. They are strongly affected by many-body interactions between the different species of photogenerated quasi-particles, resulting in dramatic changes of the emission properties of the semiconductor.

The systematic investigation of these radiative recombination processes allows us to selectively monitor the many-body induced changes in the linear and non-linear optical transitions involving quantized states of the quantum wells. The importance of these effects, belonging to the physics of highly excited semiconductors, lies in the possibility of achieving population inversion of states associated with different radiative recombination channels and strong optical non-linearities causing laser action and bistable behaviour of two-dimensional heterostructures, respectively.     

Classical Electromagnetic Interaction of a Point Charge and a Magnetic Moment: Considerations Related to the Aharonov–Bohm Phase Shift

A fundamentally new understanding of the classical electromagnetic interaction of a point charge and a magnetic dipole moment through order v ² /c ² is suggested. This relativistic analysis connects together hidden momentum in magnets, Solem's strange polarization of the classical hydrogen atom, and the Aharonov–Bohm phase shift. First we review the predictions following from the traditional particle-on-a-frictionless-rigid-ring model for a magnetic moment. This model, which is not relativistic to order v ² /c ² , does reveal a connection between the electric field of the point charge and hidden momentum in the magnetic moment; however, the electric field back at the point charge due to the Faraday-induced changing magnetic moment is of order 1/c ⁴ and hence is negligible in a 1/c ² analysis. Next we use a relativistic magnetic moment model consisting of many superimposed classical hydrogen atoms (and anti-atoms) interacting through the Darwin Lagrangian with an external charge but not with each other. The analysis of Solem regarding the strange polarization of the classical hydrogen atom is seen to give a fundamentally different mechanism for the electric field of the passing charge to change the magnetic moment. The changing magnetic moment leads to an electric force back at the point charge which (i) is of order 1/c ² , (ii) depends upon the magnetic dipole moment, changing sign with the dipole moment, (iii) is odd in the charge q of the passing charge, and (iv) reverses sign for charges passing on opposite sides of the magnetic moment. Using the insight gained from this relativistic model and the analogy of a point charge outside a conductor, we suggest that a realistic multi-particle magnetic moment involves a changing magnetic moment which keeps the electromagnetic field momentum constant. This means also that the magnetic moment does not allow a significant shift in its internal center of energy. This criterion also implies that the Lorentz forces on the charged particle and on the point charge are equal and opposite and that the center of energy of each moves according to Newton's second law F=Ma where F is exactly the Lorentz force. Finally, we note that the results and suggestion given here are precisely what are needed to explain both the Aharonov–Bohm phase shift and the Aharonov–Casher phase shift as arising from classical electromagnetic forces. Such an explanation reinstates the traditional semiclassical connection between classical and quantum phenomena for magnetic moment systems.     

Quantum and classical correlations for a two-qubit X structure density matrix

We derive explicit expressions for quantum discord and classical correlation for an X structure density matrix. Based on the characteristics of the expressions, the quantum discord and the classical correlation are easily obtained and compared under different initial conditions using a novel analytical method. We explain the relationships among quantum discord, classical correlation, and entanglement, and further find that the quantum discord is not always larger than the entanglement measured by concurrence in a general two-qubit X state. The new method, which is different from previous approaches, has certain guiding significance for analysing quantum discord and classical correlation of a two-qubit X state, such as a mixed state.     

Quantum Moment Hydrodynamics and the Entropy Principle

This paper presents how a non-commutative version of the entropy extremalization principle allows to construct new quantum hydrodynamic models. Our starting point is the moment method, which consists in integrating the quantum Liouville equation with respect to momentum p against a given vector of monomials of p. Like in the classical case, the so-obtained moment system is not closed. Inspired from Levermore's procedure in the classical case,⁽²⁶⁾ we propose to close the moment system by a quantum (Wigner) distribution function which minimizes the entropy subject to the constraint that its moments are given. In contrast to the classical case, the quantum entropy is defined globally (and not locally) as the trace of an operator. Therefore, the relation between the moments and the Lagrange multipliers of the constrained entropy minimization problem becomes nonlocal and the resulting moment system involves nonlocal operators (instead of purely local ones in the classical case). In the present paper, we discuss some practical aspects and consequences of this nonlocal feature.     

Generators of KMS Symmetric Markov Semigroups on {\mathcal{B}({\rm h})} Symmetry and Quantum Detailed Balance

We find the structure of generators of norm-continuous quantum Markov semigroups on B(h){\mathcal{B}({\rm h})} that are symmetric with respect to the scalar product tr (ρ ^1/2 x*ρ ^1/2 y) induced by a faithful normal invariant state ρ and satisfy two quantum generalisations of the classical detailed balance condition related with this non-commutative notion of symmetry: the so-called standard detailed balance condition and the standard detailed balance condition with an antiunitary time reversal.     

Wavevector and frequency dependent dielectric function dynamic structure factor and the instability of plasma waves in two component hot rare quantum and classical plasmas

The full wavevector and frequency dependent complex dielectric function for two component classical and quantum rare hot plasmas have been derived. The real part of dielectric function is obtained in the form of a series. Difference between quantum and classical real and imaginary parts of dielectric function have been brought out by making explicit calculations. The quantum nature of the plasma brings about significant changes in both parts depending upon the magnitude of quantum parameter,R (= 8.93(λ_th)/λ). Expressions for the dynamic structure factors for both two component classical and quantum plasma have been evaluated for different values of the mass of the positive componentm ₊, temperature T₊ and wavevector k. It is found that the plasma exhibits well defined collective modes for certain values of |k| accompanied by varying disorder which depends upon the values of m₊ as well as on |k| and T₊. For the quantum case the collective modes are less well defined as compared to the corresponding classical case, thus proving that quantum nature introduces inherent disorder in the system. But for both the cases, increase in temperature destroys collective modes. Another feature is the appearance of a hump near Ω = 0 which becomes smaller and vanishes as the quantum parameter is decreased. Instability of plasma modes in the presence of constant electric field has also been worked out for the quantum case.     

Crossed Products of C*-Algebras and Spectral Analysis of Quantum Hamiltonians

We study spectral properties of a hamiltonian by analyzing the structure of certain C ^*-algebras to which it is affiliated. The main tool we use for the construction of these algebras is the crossed product of abelian C ^*-algebras (generated by the classical potentials) by actions of groups. We show how to compute the quotient of such a crossed product with respect to the ideal of compact operators and how to use the resulting information in order to get spectral properties of the hamiltonians. This scheme provides a unified approach to the study of hamiltonians of anisotropic and many-body systems (including quantum fields). Received: 5 November 2001 / Accepted: 10 March 2002     

Derivation and classical limit of the mean-field equation for a quantum Coulomb system: Maxwell-Boltzmann statistics

The mean-field density matrix of a changed plasma of quantum particles with Maxwell-Boltzmann statistics in a confining external potential is obtained as a limit of theN-body canonical states for suitably scaled charges. Also, it is shown that the density profile of the quantum mean-field theory converges to the solution of the classical mean-field equation when the Planck's constant tends to zero.