The investigation has been based on Hamiltonian quadratic in Fermi amplitudes or linear in operators representing amplitude and phase of condensate of pair (parallel spins).The latter represents the energy of a \spin" 1=2 L in some eective eld.Simple variational procedure gives for anisotropic ABM and \polar" phases single particle energy with a suitable form of energy gap.Amplitude and phase oscillations are connected with frequencies != 0; 2 .The rst one represents Goldstone{Anderson mode.\Spins" dependence on time ( L(i) (t) = Rij(t) L(j) (0)) is given by rotation matrix Rij(t) used in He 3 {B theory.Here Rij describes rotation of a \spin" around the eective eld.
The investigation has been based on Hamiltonian quadratic in Fermi amplitudes or linear in operators representing amplitude and phase of condensate of pair (parallel spins).The latter represents the energy of a \spin" 1=2 L in some eective eld.Simple variational procedure gives for anisotropic ABM and \polar" phases single particle energy with a suitable form of energy gap.Amplitude and phase oscillations are connected with frequencies != 0; 2 .The rst one represents Goldstone{Anderson mode.\Spins" dependence on time ( L(i) (t) = Rij(t) L(j) (0)) is given by rotation matrix Rij(t) used in He 3 {B theory.Here Rij describes rotation of a \spin" around the eective eld.
The Fermi-liquid approach generalized to superfluid systems is used to derive general formulas for the normal and abnormal distribution functions (DF) for quasiparticles of a superfluid Fermi liquid (SFL) consisting of electrically neutral fermions with spin-triplet pairing (the spin of a pair s=1, the orbital angular momentum l of the pair being an arbitrary odd number) in a static uniform magnetic field, taking into account the Landau spin-exchange Fermi-liquid interaction at temperatures 0<T<Tc (Tc is the temperature of phase transition from the normal to the superfluid state). In deriving the DF, the explicit form of the energy functional (EF) for SFL is not used. A system of coupled equations is obtained for the order parameter, energy of quasiparticles, and effective magnetic field for the SFL under investigation in the case of an EF quadratic in DF. The obtained DF and the system of equations are applicable for describing both unitary and nonunitary phases of superfluid He3 in a magnetic field, e.g., for determining the magnetization and magnetic susceptibility of these phases.
The low-energy modes of a superfluid atomic Fermi gas at zero temperature are investigated. The Bose-Einstein-condensate (BEC) side of the superfluid phase is studied in detail. The atoms are assumed to be in only one internal state, so that for a sufficiently diluted gas the pairing of fermions can be considered effective in the l=1 channel only. In agreement with previous works on p-wave superfluidity in Fermi systems, it is found that the px+ipy phase represents the lowest energy state in both the Bardeen-Cooper-Schrieffer (BCS) and BEC sides. Our calculations show that at low energy three branches of collective modes can emerge, with different species of dispersion relations: a phonon-like mode, a single-particle-like mode and a gapped mode. A comparison with the Bogoliubov excitations of the corresponding spinor Bose condensate is made. They reproduce the dispersion relations of the excitation modes of the p-wave superfluid Fermi gas to a high accuracy.
We present a formalism for the fermionic quasiparticle propagator in a superfluid fermionic system. Starting from a general many-body Hamiltonian confined by the two-body instantaneous interaction, the equation of motion for the fermionic propagator is obtained in the Dyson form. Before making any approximation, the interaction kernel is found to be decomposed into static and dynamical (time-dependent) contributions, where the latter translates to the energy-dependent and the former maps to the energy-independent terms in the energy domain. The three-fermion correlation function, being the heart of the dynamical part of the kernel, is factorized into the two-fermion and one-fermion ones. With the relaxed particle number constraint, the normal propagator is coupled to the anomalous one via both the static and dynamical kernels, which is formalized by introducing the generalized quasiparticle propagator of the Gor'kov type. The dynamical kernel in the factorized form is associated with the quasiparticle-vibration coupling (QVC), with the vibrations unifying both the normal and pairing phonons. The QVC vertices are related to the variations of the Hamiltonian of the Bogoliubov quasiparticles, which can be obtained by the finite amplitude method.
Although our discussion in this section will be largely phenomenological we must anticipate the major result of the microscopic theory which will be developed in Parts II and III; in particular we must make use of the fact that superfluidity in Fermi systems arises from the formation of a special kind of bound state in which electrons (or He quasiparticles) on (or near) the Fermi surface having oppositely directed momenta, + p and – p , form (collectively) bound pairs. For the vast majority of superfluid electron systems (metals), the pairs form in a state with no net orbital angular momentum, an l = 0 or s-wave state which we will refer to as conventional fermion superfluidity; by the Pauli principle the electrons must then be in a singlet spin state (s = 0). In the decade or so following the development of the BCS theory, and especially after the experimental discovery of the superfluid phases of liquid He, theorists explored various types of pairing in systems with a more complicated orbital and spin structure in which the gap can vary, in both magnitude and phase, with position on the Fermi surface. In the case of odd l pairing, the pairs form (again due to the Pauli principle) in a triplet s = 1 rather than a singlet spin state. The energy gap associated with such states may have zeros (i.e., vanish) at points or lines on the Fermi surface, and as a result the number of excitations at low temperatures varies as a power of the temperature, rather than exponentially.
Based on the analysis of the con guration space of an one{particle system, the presence of the quantum interference is shown to result in the eective interaction of an exchange{type.The quenched disorder is represented by analogy with the thermodynamic uctuations for which the halfwidth of scattering of the particle levels plays the role of temperature.With regard to fermions within the framework of Anderson locator approach, the gap in the single excitation spectrum caused by the condensation of pairs of the coupled sites is shown to determine the density of the extended states.Its dependence on the level scattering width and the chemical potential shift from the band center is found.The law of collective mode dispersion is established.For the charged fermions, the collective mode is of an ordinary diusion type, but in the case of the neutral ones (quantum crystal), the zero{sound mode of oscillations of the extended fermion density appears within the long{wave limit.The dependencies of the zero{sound velocity and typical values of its frequency and wave number on the temperature and parameter of the quantum dilatation are examined.The dependencies of the extended state condensate density and eective interaction parameter on the level scattering width W are determined for Bose case.The collective excitations are demonstrated to reduce to the rst sound which is transformed, as the value W decreases, into a pure dissipative mode, and the second sound the velocity of which critically depends on W.
In this work we discuss three problems connected to the damping of oscillations in quantum systems: dynamics and decay in a random matrix model with conserved quantities, energy redistribution and decay of the amplitude mode in a superconductor, and the decay of the gapped mode in a molecular condensate, a bosonic analog to the superconductor. These problems highlight different aspects of damping and energy redistribution in quantum dynamics, while being simple enough that analytic control is possible. The first system captures the idea of a "partially conserved" quantity: in ergodic quantum systems, physical observables have a non-relaxing component if they overlap with a conserved quantity, but how to isolate the non-relaxing component is in general unclear. We compute exact dynamical correlators governed by a Hamiltonian composed of two large interacting random matrices, H=A+B, and we analytically obtain the late-time value of ⟨A(t) A(0)⟩, which quantifies the non-relaxing part of the observable A. We show that the relaxation to this value is governed by a power-law determined by the spectrum of the Hamiltonian H, independent of the observable A, while the long--time value and the amplitude of the oscillations depend on the trace--overlap between the operator and the Hamiltonian. For Gaussian matrices, we further compute out-of-time-ordered-correlators (OTOCs) and find that the existence of a non-relaxing part of A leads to modifications of the late time values and exponents. Our results follow from exact resummation of a diagrammatic expansion and hyperoperator techniques. The above problem deals with energy redistribution in a system with a complex internal structure, but without any spatial dependence nor many--body effects. In the second part of this work we discuss energy relaxation in a system with both: a BCS superconductor. In particular, we study the excitation of the collective Higgs oscillations of the order parameter by incoherent short pulses of light with frequency much larger than the superconducting gap. We find that the excitation amplitude of the Higgs mode is controlled by a single parameter, determined by the total number of quasiparticles excited by the pulse, which we trace back to the universality of the shape of the light-induced quasiparticle cascade at energy below the Debye frequency and above the gap. Our analysis is primarily based on the Keldysh technique for non--equilibrium field theory and the Boltzmann kinetic equation. Finally, we study the damping of the gapped mode in a molecular Bose--Einstein condensate, the Bosonic analogue of a BCS superconductor. This system has the advantage of giving the experimentalist fine control over the interatomic interactions using Feshbach resonances, and is the object of renewed interest as the molecular superfluid phase has only very recently been realized in the lab. We discuss damping in the nontrivial thermodynamic phases: in the molecular superfluid phase the gapped excitation is protected by parity, and is damped only above a threshold momentum --as in the Cherenkov effect--, while in the atomic superfluid phase the gapped mode is damped at all momentum scales. We propose a class of experiments where our results are measurable: transmission (and reflection) of an atom beam through a molecular condensate cloud.