A general theory is presented to describe optomechanical interactions of acoustic phonons, having extremely long lifetimes in superfluid He-4, with optical photons in the medium placed in a suitable electromagnetic cavity. The acoustic nonlinearity in the fluid motion is included to consider processes beyond the usual linear process involving the absorption or emission of one phonon at a time. We first apply our formulation to the simplest one-phonon process involving the usual resonant anti-Stokes upconversion of an incident optical mode. However, when the allowed optical cavity modes are such that there is no single-phonon mode in the superfluid, which can give rise to a resonant allowed anti-Stokes mode, we must consider the possibility of two-phonon upconversion. For such a case, we show that the two-step two-phonon process could be dominant. We present arguments for a large two-step process and negligible single-step two-phonon contribution. The two-step process also shows interesting quantum interference among different transition pathways.

We discuss the possible cooling of different phonon modes via three-wave mixing interactions of vibrational and optical modes. Since phonon modes exhibit a variety of dispersion relations or frequency spectra with diverse spatial structures, depending on the shape and size of the sample, we formulate our theory in terms of relevant spatial mode functions for the interacting fields in any given geometry. Our general formulation is applicable to cooling of any low-frequency excitation mode in matter which can give appreciable inelastic light scattering via the modulation of the optical dielectric function. We discuss the possibility of Dicke-like collective effects in phonon cooling and present explicit results for simultaneous cooling of two-phonon modes via the anti-Stokes up-conversions. We show that the bimodal cooling should be observable experimentally.

We show that the ground state of a system of magnetic dipoles, with no electric charge, is a ferromagnetic quantum Fermi liquid at high densities, driven by the dipolar exchange energy. As in the system of classical point dipoles, the direct dipole energy is zero in this case. With decreasing density, there is a transition to an antiferromagnetic lattice state. An addition of short range hard core repulsive potential will arrest the infinite density collapse of the ferromagnetic state, and possible melting of the low density antiferromagnetic lattice state.

We obtain the best upper bound for the ground-state energy of a system of chargeless fermions of mass m, spin s=1/2 , and magnetic moment mus[over ] as a function of its density in the fully spin-polarized Hartree-Fock determinantal state, specified by a prolate spheroidal plane-wave single-particle occupation function n_(k[over ]) , by minimizing the total energy E at each density with respect to the variational spheroidal deformation parameter beta(2),0< or =beta(2)< or =1 . We find that at high densities, this spheroidal ferromagnetic state is the most likely ground state of the system, but it is still unstable towards the infinite-density collapse. This optimized ferromagnetic state is shown to be a stable ground state of the dipolar system at high densities, if one has an additional repulsive short-range hardcore interaction of sufficient strength and nonvanishing range.

We use different determinantal Hartree-Fock (HF) wave functions to calculate true variational upper bounds for the ground state energy of N spin-half fermions in volume V 0, with mass m, electric charge zero, and magnetic moment µ, interacting through magnetic dipole-dipole interaction. We find that at high densities when the average interparticle distance r 0 becomes small compared to the magnetic length r m ≡ 2mµ2/ħ2, a ferromagnetic state with spheroidal occupation function n ↑(\(\vec k\)), involving quadrupolar deformation, gives a lower upper bound compared to the variational energy for the uniform paramagnetic state or for the state with dipolar deformation. This system is unstable towards infinite density collapse, but we show explicitly that a suitable short-range repulsive (hard core) interaction of strength U 0 and range a can stop this collapse. The existence of a stable equilibrium high density ferromagnetic state with spheroidal occupation function is possible as long as the ratio of coupling constants Γcm ≡ (U 0 a 3/µ2) is not very small compared to 1.

The decoherence and fidelity of spin states in a localized single-electron quantum dot in the presence of a dc magnetic field, arising either from the nuclear hyperfine interaction within the dot or due to its coupling with another localized quantum dot, are examined in detail. A general framework for determining the time evolution of the reduced density matrix. for a single dot is presented, which is exact up to the second order in interaction with any reservoir. In particular, it is applied to the problem of nuclear hyperfine coupling, and approximate estimates of coherence decay time are made when the nuclear spins are either polarized or unpolarized and the internal dynamics of nuclear spins is determined mainly by the nuclear magnetic dipole-dipole interaction. The fidelity of a pure unperturbed electronic one-qubit spin state is obtained as a function of time, which is exact even on a very short timescale of logic gate operations. The time variation of the fidelity of the same one-qubit state on the localized dot as a part of the direct product with another one-qubit state on another localized dot arising because of coupling between these quantum dots is also calculated in this paper. In this case, we include both the single-particle tunnelling between the dots as well as the direct and exchange Coulomb interactions, including on-site Coulomb repulsion. This allows for the double occupation of a single dot. It is found that the loss of fidelity of such two-qubit states due to double occupancy and additional phase errors in the presence of appreciable dot-dot coupling can become a more severe limiting factor than that due to the hyperfine interaction in individual dots.

We consider the ground state of a system of chargeless fermions, such as neutrinos, of mass m and magnetic moment p interacting through long-range magnetic dipole interaction, within the framework of a Hartree-Fock variational approach. At high densities the uniform paramagnetic state becomes unstable towards a ferromagnetic state with quadrupolar deformation of the Fermi surface. The exchange energy which is attractive dominates the repulsive kinetic energy. If we let the density be a variable, then above a certain density the system will collapse to an infinite density state unless another short-range interaction stops the collapse. In the case of large deformations, the possibility of a purely dipolar deformation exists.

Using the general formulation for obtaining chemical potentialμ of an ideal Fermi gas of particles at temperature T, with particle rest mass m0 and average density 〈N〉/V, the dependence of the mean square number fluctuation 〈ΔN 2〉/V on the particle mass m0 has been calculated explicitly. The numerical calculations are exact in all cases whether rest mass energym 0c2 is very large (non-relativistic case), very small (ultra-relativistic case) or of the same order as the thermal energy kBT. Application of our results to the detection of the universal very low energy cosmic neutrino background (CNB), from any of the three species of neutrinos, shows that it is possible to estimate the neutrino mass of these species if from approximate experimental measurements of their momentum distribution one can extract, someday, not only the density 〈N v〉/V but also the mean square fluctuation 〈Δ v 2 〉/V. If at the present epoch, the universe is expanding much faster than thermalization rate for CNB, it is shown that our analysis leads to a scaled neutrino massm v instead of the actual massm 0v .

Background thermal noise often becomes the limiting factor while detecting low signals of photon flux from a distant thermal source. This cannot be eliminated by simply cooling the photodetector. A general method is developed here to calculate the background thermal noise and the noise equivalent power (NEP) for background source temperature T, with any given frequency-dependent quantum efficiency function eta(v) for the photodetector. Applications of our analysis to some specific model forms of eta(v), with finite bandwidths, show that earlier calculations are highly inadequate for peak efficiency frequencies v(o) < k(B)T/h, where k(B) is the Boltzmann constant and h is the Planck constant. The NEP, which determines the limit on the lowest signal power that can be detected by any given photodetector, is much higher in these frequency regions compared to earlier approximate estimates.

Anisotropy and the wave-vector dependence of the energy gap function determine many important properties of a superconductor. Stal ting from fist principles, we present here a complete analysis of possible symmetries of the superconducting gap function E-g(k) at the Fermi surface in high-T-c layered superconductors with either a simple orthorhombic or a tetragonal unit cell. This is done within the framework of Gorkov's mean-held theory of superconductivity in the so-called ''layer representation'' introduced by us earlier. For N conducting cuprate layers. J = 1,2,...,N, in each unit cell, the spin-singlet order parameters Delta(JJ')(k) can be expanded in terms of possible basis functions of all the irreducible representations relevant to layered crystals, which are obtained here. In layered materials, the symmetry is restricted to the translational lattice periodicity in the direction perpendicular to the layers and the residual point group and translational symmetries for the two-dimensional unit cell in each layer of the three-dimensional unit cell. We derive an exact general relation to determine different branches of the energy gap function E-g(k) at the Fermi surface in terms of Delta(JJ')(k), which include both intralayer and interlayer order parameters. For N = 2, we also obtain an exact expression for quasiparticle energies E-p(k), p = 1,2. in the superconducting state in the presence of intralayer and complex interlayer order parameters as well as complex tunneling matrix elements between the two layers in the unit cell, which need not be equivalent. The form of the possible basis functions are also listed in terms of cylindrical coordinates k(t),phi,k(z) to take advantage of the orthogonality of functions with respect to phi integrations. In layered materials, with open Fermi surfaces in the k(z) direction, there is orthogonality of basis functions with respect to k(z) also (-pi less than or equal to k(z)d less than or equal to pi). Our results show that in orthorhombic systems, planar d(kx2-ky2)-like (B-1g) and d(kxky)-like (B-2g) symmetries are always mixed, respectively, with the planar s-wave-like (A(1g)) and A(2g)-like symmetries of the corresponding tetragonal system. There is also the possibility of a weak modulation of E-g(k) as a function of k(z)(similar to cos k(z)d). In addition, in the presence of interlayer pairings which may or may not have the same symmetry as the intralayer order parameters, even in tetragonal systems the nodes of the d(kz2-ky2)-like intralayer gap function will be shifted. In view of this, some suggestions for analyzing experimental data are also presented.

The wave-vector dependence of superconducting order parameters and the corresponding transition temperature in three-dimensional layered metals are investigated. It includes planar anisotropic s-wave-like as well as d-wave-like pairing contributions, but for simplicity only intralayer pairing of electrons has been considered for tetragonal and orthorhombic crystal structures, relevant to high-T-c materials. It is shown that for the orthorhombic case order parameters are likely to have mixed symmetry, whereas in tetragonal structures it is possible to have only planar d-wave-like symmetry with a possible correction term with an additional wavevector dependence in the direction perpendicular to the reciprocal layer plane.

At very low temperatures, low-frequency electronic Raman scattering in a superconductor arises from pair-breaking excitations across the energy gap. In principle, this allows the possible determination of the nature of the energy gap, including its symmetry. To examine this carefully, a simplified general expression was derived for the electronic Raman scattering intensity, suitable for high-T-c layered superconductors with widely open Fermi surfaces in the k(z) direction, perpendicular to the k(x), k(y) reciprocal layer plane. The unscreened part of the single-particle scattering from both charge-density fluctuations (CDF) and spin-density fluctuations (SDF) was taken into account. The shape of the resulting electronic Raman spectrum depends crucially on the directions of the incident and scattered light polarizations, e(i) and e(s), the wavevector transfer q = q(l) - q(s), the nature of constant normal-state single-particle energy surfaces near the Fermi energy and the form of the energy gap function Delta(k). A careful analysis of the results for different symmetries of the gap function Delta and interaction vertex functions \gamma(CDF)\ and \gamma(SDF)\, averaged over k(z), for the tetragonal D-4h point group, showed that currently available experimental data for different high-T-c cuprates do not allow one to determine the symmetry of the gap function uniquely. Although d(x2-y2) symmetry for the gap function in these materials may be a strong possibility, a highly anisotropic s-wave type gap function cannot be ruled out completely.

The problem of superconductivity in metal at low temperatures has been of great theoretical interest for a very long time. The first real microscopic understanding of this impotant process became possible only in 1957, with the development of the well-known BCS pairing theory involving a net dynamical attractive interaction between electrons near the metallic Fermi surface, induced by the phonon-exchange mechanism. It is known for sometime now that the net attractive interaction can also arise from the exchange of suitable electronic excitations in the system. However, with the recent discovery of high-temperature superconductivity in layered cuprates and other oxides, in which the electronic motion in their normal metallic state is highly correlated, many questions have been raised regarding the validity of the generalized pairing theory of superconductivity in such materials. Alternative possibilities for superconductivity in highly correlated cuprates based mostly on the two-dimensional one-band repulsive Hubbard model have been investigated extensively in recent years. In this presentation, it will be argued that superconductivity in highly correlated layered metals can still be understood within a suitably generalized framework of the pairing theory developed by us recently. Explicit results for the transition temperature Tc and the anisotropy of the superconductivity order parameter JJ1 will be presented. These are calculated for a general phenomenological model in which both intra-layer and inter-layer couplings of electrons at various conducting layers in each unit cell are included, without specifying the exact exchange mechanism. The effect of an external magnetic field on Tc and the calculation of the temperature-dependence of the uppe critical field Hc2 will also be discussed briefly.

Frontiers in Solid State SciencesSelected Topics in Superconductivity, pp. 165-178 (1993) No AccessSUPERCONDUCTING PAIRING IN LAYERED SUPERCONDUCTORSSUDHANSHU S. JHASUDHANSHU S. JHATata Institute of Fundamental Research, Bombay 400005, Indiahttps://doi.org/10.1142/9789814354660_0008Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: The following sections are included: Introduction Mathematical Outline of the Pairing Theory in Layered Crystals Superconductivity with Phenomenological Intralayer and Interlayer Couplings Concluding Remarks Acknowledgement References FiguresReferencesRelatedDetails Selected Topics in SuperconductivityMetrics History PDF download

The structure of the general effective pairing interaction V(r1,r2;omega) for spin-singlet superconductivity in layered systems is examined in detail in a layer representation for the single-particle states. This interaction, averaged over the layer-plane momentum transfers near the Fermi surface [i.e., (pp)VBAR(q,q',omega)], is expressed in terms of intracell-intralayer, intracell-interlayer, intercell-intralayer, and intercell-interlayer couplings, where q,q' are wave vectors in the direction of the normal to the layers. In our general picture, because of the interaction process, a localized particle can scatter across from one layer to another in the same unit cell, as well as from one vertical unit cell to another cell. The dominant coupling terms are due to the interaction of two particles localized on layers within the same vertical unit cell, both before and after the scattering. The intercell coupling terms correspond to scattering process in which at least one of the particles is localized in a different vertical unit cell either before or after the scattering. Whereas the ratio delta(c) between the intracell and nearest-neighbor intercell couplings is assumed to be small compared with unity in our analysis, no perturbative approach is used to distinguish the magnitudes of interlayer couplings from the corresponding intralayer couplings. General results for T(c) and the superconducting-order-parameter matrix DELTA, are discussed for the case of N layers per unit cell, both in the limit in which the off-diagonal tunneling terms, t(q), appearing in the inverse of the normal-state single-particle N X N Green-function matrix, g-1, in the layer representation, are either neglected or included to the lowest order in \t(q)\/mu. Here, mu is the chemical potential. We find that the q,q' dependence of (pp)V arises only from intercell couplings which in turn implies that the q dependence and anisotropy in DELTA are entirely due to intercell interactions. Explicit results are given for T(c) as well as the order parameters for the case of N = 1 and 2, to leading orders in delta(c) and \t(q)\/mu. We find that anisotropy in the gap parameter in the layer representation always arises due to the intercell interactions, and is present even in the N = 1 case. It is also found that the presence of interlayer couplings tends to enhance T(c) in general. In a representation in which g-1 is diagonal, called the alpha-band representation here, we reformulate the problem in terms of new order parameters, DELTA(alpha) and new effective interaction, (pp)V(alpha)(qq';omega). Although, in this representation there appear additional q,q' dependences in (pp)V(alpha), these do not imply extra anisotropy in the corresponding gap parameter DELTA in the layer representation. This is only an artifact of the use of the alpha representation as revealed by the relation between DELTA(alpha) and DELTA. For interband pairing in the alpha representation, we also present a complete calculation for the resulting order parameter.

A theoretical framework for treating the effects of magnetic fieldH on the pairing theory of superconductivity is considered, where the field is taken in an arbitrary direction with respect to crystal axes. This is applicable to closed, as well as open normal state Fermi surface (FS), including simple layered metals. The orbital effects of the magnetic field are treated semiclassically while retaining the full anisotropic paramagnetic contribution. Explicit calculations are presented in the limits |H| → |H c2(T)|,T ∼ 0 andT →T c(|H|), |H| ∼ 0. Effects of weak nonmagnetic impurity scattering, without vertex corrections, have also been taken into account in a phenomenological way. The final results for the case of open FS and layered materials are found to differ considerably from those of the closed FS. For example, an important parameter,h(T=0)=|Hc2(0)|/[-Tδ|H c2 T|δT]T{s0} for the case of a FS open ink z-direction with thek z-bandwidth, 4t 3, very small compared to the Fermi energy,E F, is close to 0.5906, compared to 0.7273 for the closed FS, in the clean limit. Analytical results are given for the magnetic field dependence ofT c and the temperature dependence of H c2 for a model of layered superconductors with widely open FS. For a set of band structure parameters for YBa2Cu3O7 used elsewhere, we find reasonable values for the upper critical fieldH c2(0), the slope (dH c2/dT)T c0, anisotropic coherence lengths ζi(T=0),i=x, y, z, and (dT c/d|H|)|H| → 0.

Within the framework of a generalized spin-singlet pairing theory of superconductivity in layered metals, the role of intralayer anti-ferromagnetic spin-fluctuations in cuprates has been examined. For this purpose, a two-dimensional one-band Hubbard model is used to obtain the intralayer effective interaction due to spin-fluctuations. It is already known that a simple perturbative calculation involving the exchange of one spin-fluctuation excitation between electrons of opposite spins leads to a repulsive interaction for small energy transfers (omega-->0). This prevents the possibility of nodeless s-wave type of gap function observed experimentally in cuprates. However, in a more exact non-perturbative analysis based on very general considerations, we show that in highly correlated metals like cuprates, the static part of the effective interactioin can become attractive in the presence of strong residual spin-fluctuations close to the magnetic ordering. Thus spin-fluctuations may be playing a leading role in suppressing the long range Coulomb repulsion. In addition. the intralayer interaction is made more attractive because of interlayer interaction between the conducting CuO2 layers and nearby nonconducting layers. Due to significant interlayer interactions among the conducting layers, T(c) is enhanced further if there is more than one such CuO2 layer per unit cell.