the interaction.In other words, the correlations lower the ground-state energy relatively far more than Cooper-pair condensation for g ≪ 1.This fact implies that, formally speaking, Cooper-pair condensation should be studied only after the correlations effects have been incorporated.We incorporate the correlation effects to show explicitly that the correlations produce finite non-condensed particles in the ground-state, which work as a particle reservoir for the condensate to naturally yield the superposition, in exactly the same way as in the case of interacting Bose-Einstein condensates.[10] Thus, the superposition is a real physical entity that exists in any isolated superconductor or superfluid.Note in this context that the superposition and coherence have so far been discussed mostly in terms of condensed particles alone.[5,7,11,12] This thesis is organized as follows; We will discuss the improved ground-wave function with many-body correlations.Section 2 presents formulation.Section 3 gives numerical results.Section 4 presents concluding remarks.We also present details that deriving the formalism from Appendix.
the interaction.In other words, the correlations lower the ground-state energy relatively far more than Cooper-pair condensation for g ≪ 1.This fact implies that, formally speaking, Cooper-pair condensation should be studied only after the correlations effects have been incorporated.We incorporate the correlation effects to show explicitly that the correlations produce finite non-condensed particles in the ground-state, which work as a particle reservoir for the condensate to naturally yield the superposition, in exactly the same way as in the case of interacting Bose-Einstein condensates.[10] Thus, the superposition is a real physical entity that exists in any isolated superconductor or superfluid.Note in this context that the superposition and coherence have so far been discussed mostly in terms of condensed particles alone.[5,7,11,12] This thesis is organized as follows; We will discuss the improved ground-wave function with many-body correlations.Section 2 presents formulation.Section 3 gives numerical results.Section 4 presents concluding remarks.We also present details that deriving the formalism from Appendix.
We improve the Bardeen-Cooper-Schrieffer wave function with a fixed particle number so as to incorporate many-body correlations beyond the mean-field treatment. It is shown that the correlations lower the ground-state energy far more than Cooper-pair condensation in the weak-coupling region. Moreover, they naturally bring a superposition over the number of condensed particles. Thus, Cooper-pair condensation is special among the various bound-state formations of quantum mechanics in that number fluctuations are necessarily present in the condensate through the dynamical exchange of particles with the non-condensate reservoir. On the basis of this result, we propose ΔNcon⋅Δϕ≳1 as the uncertainty relation relevant to the number-phase fluctuations in superconductors and superfluids, where the number of condensed particles Ncon is used instead of the total particle number N. The formula implies that a macroscopic phase ϕ can be established even in number-fixed superconductors and superfluids since ΔNcon≫1.
Abstract Chapter 13 addresses Bose condensation in superfluids (and superconductors), which involves the field operator ψ having a c-number component (<ψ(x,t)>≠0), challenging number conservation. The nonlinear Gross-Pitaevskii equation is derived for this condensate wave function <ψ>=ψ−ψ˜, facilitating identification of the coherence length and the core region of vortex motion. The noncondensate Green’s function G˜1(1,1′)=−i<(ψ˜(1)ψ˜+(1′))+> and the nonvanishing anomalous correlation function F˜∗(2,1′)=−i<(ψ˜+(2)ψ˜+(1′))+> describe the dynamics and elementary excitations of the non-condensate states and are discussed in conjunction with Landau’s criterion for viscosity. Associated concepts of off-diagonal long-range order and the interpretation of <ψ> as a superfluid order parameter are also introduced. Anderson’s Bose-condensed state, as a phase-coherent wave packet superposition of number states, resolves issues of number conservation. Superconductivity involves bound Cooper pairs of electrons capable of Bose condensation and superfluid behavior. Correspondingly, the two-particle Green’s function has a term involving a product of anomalous bound-Cooper-pair condensate wave functions of the type F(1,2)=−i<(ψ(1)ψ(2))+>≠0, such that G2(1,2;1′,2′)=F(1,2)F+(1′,2′)+G˜2(1,2;1′,2′). Here, G˜2 describes the dynamics/excitations of the non-superfluid-condensate states, while nonvanishing F,F+ represent a phase-coherent wave packet superposition of Cooper-pair number states and off-diagonal long range order. Employing this form of G2 in the G1-equation couples the condensed state with the non-condensate excitations. Taken jointly with the dynamical equation for F(1,2), this leads to the Gorkov equations, encompassing the Bardeen–Cooper–Schrieffer (BCS) energy gap, critical temperature, and Bogoliubov-de Gennes eigenfunction Bogoliubons. Superconductor thermodynamics and critical magnetic field are discussed. For a weak magnetic field, the Gorkov-equations lead to Ginzburg–Landau theory and a nonlinear Schrödinger-like equation for the pair wave function and the associated supercurrent, along with identification of the Cooper pair density. Furthermore, Chapter 13 addresses the apparent lack of gauge invariance of London theory with an elegant variational analysis involving re-gauging the potentials, yielding a manifestly gauge invariant generalization of the London equation. Consistency with the equation of continuity implies the existence of Anderson’s acoustic normal mode, which is supplanted by the plasmon for Coulomb interaction. Type II superconductors and the penetration (and interaction) of quantized magnetic flux lines are also discussed. Finally, Chapter 13 addresses Josephson tunneling between superconductors.
1. Define Bose-Einstein condensation in an interacting boson system in term of macroscopic occupation. 2. Introduce the Gross-Pitaevskii equation and its applicable conditions. 3. Derive the hydrodynamic equations and various physical consequences of the hydrodynamics with or without harmonic trap. 4. Introduce sound wave and emphasize the difference between the wave behavior and the diffusive behavior. 5. Introduce the criterion for superfluidity and determine the superfluid critical velocity. 6. Introduce two-fluid hydrodynamics. 7. Introduce the Bogoliubov theory. 8. Introduce various concepts related to quasi-particle, including ground state as vacuum of quasi-particles, the zero-point energy of quasi-particle, quasi-particle lifetime and vacuum fluctuation. 9. Introduce healing length as the characteristic length scale for low-energy physics of an interacting Bose condensate. 10. Summarize the difference between an interacting and non-interacting Bose condensate. 11. Introduce the Bragg spectroscopy. 12. Show that the Bogoliubov theory fails in one-dimension due to the infrared divergence. 13. Discuss the basic idea of the Bethe-ansatz solution and the Tonks-Girardeau limit. 14. Introduce scale invariant quantum gases. 15. Introduce the Josephson effect and the self-trapping as two different transport phenomena driven by phase coherence. 16. Discuss the conjugate relation between the relative phase fluctuation and the relative particle number fluctuation, and show strong enough phase fluctuation renders the system into a Fock state. 17. Discuss an example of quantum measurement that can project two independent condensates to a coherent superposition state. 18. Discuss interference effects in density and density-density correlations. 19. Discuss the relation between the instability of the Sch\"odinger Cat state and the stability of symmetry breaking.
In no wave function of the type (1) [product of single determinants for each spin] is there a statistical correlation between the positions of the electrons with antiparallel spin. The purpose of the aforementioned generalization of (1) is to allow for such correlations. This will lead to an improvement of the wave function and, therefore, to a lowering of the energy value. E. Wigner, Phys. Rev. 46, 1002 (1934) Summary Although the exact many-body wavefunction cannot be written down in general, we do know of some of its properties. For example, there are differences between the wavefunctions of insulators and metals and the cusp condition gives the behavior as any two charges approach each other. In this chapter we also discuss approximate wavefunctions, ways to judge their accuracy and how to include electronic correlation. Examples of many-body wavefunctions are the Slater–Jastrow (pair product) wavefunction and its generalization to pairing and backflow wavefunctions. In other places in this book, we argue that it is not necessary to look at the many-body wavefunctions explicitly because they are unwieldy; the one- and two-body correlation functions discussed in Ch. 5 are sufficient to determine the energy and give information on the excitation spectra. However, these correlation functions do not always contain all information of interest. In principle, the ground-state energy of a many-electron system is a functional of the density, but the very derivation of the theorem invokes the many-body wavefunction, as expressed explicitly in Eq. (4.16). The effects of antisymmetry are manifest in the correlation functions, but antisymmetry is most simply viewed as a property of the wavefunction; electronic correlation is fundamentally a result of properties of the many-body wavefunction. Studying many-body wavefunctions provides a very useful, different point of view of many-body physics from the approaches based on correlation functions. Many of the most important discoveries in physics have come about by understanding the nature of wavefunctions, such as the Laughlin wavefunction for the fractional quantum Hall effect, the BCS wavefunction for superconductors, p-wave pairing in superfluid 3 He, and the Heitler– London approach for molecular binding. The role of Berry's phases has brought out the importance of the phase of the wavefunction in determining properties of quantum systems. This has led to new understanding of the classification of insulators, metals, superconductors, vortices, and other states of condensed matter.
The impact of interparticle correlations on the behavior of Bose-Einstein Condensates (BECs) is discussed using two approaches. In the first approach, the wavefunction of a BEC is encoded in the N-particle sector of an extended "catalytic state". Going to a time-dependent interaction picture, we can organize the effective Hamiltonian by powers of N−1/2. Requiring the terms of order N1/2 to vanish, we get the Gross-Pitaevskii Equation. Going to the next order, N0, we obtain the number-conserving Bogoliubov approximation. Our approach allows one to stay in the Schrödinger picture and to apply many techniques from quantum optics. Moreover, it is easier to track different orders in the Hamiltonian and to generalize to the multi-component case. In the second approach, I consider a state of N=l×n bosons that is derived by symmetrizing the n-fold tensor product of an arbitrary l-boson state. Particularly, we are interested in the pure state case for l=2, which we call the Pair-Correlated State (PCS). I show that PCS reproduces the number-conserving Bogoliubov approximation; moreover, it also works in the strong interaction regime where the Bogoliubov approximation fails. For the two-site Bose-Hubbard model, I find numerically that the error (measured by the trace distance of the two-particle reduced density matrices) of PCS is less than two percent over the entire parameter space, thus making PCS a bridge between the superfluid and Mott insulating phases. Amazingly, the error of PCS does not increase, in the time-dependent case, as the system evolves for longer times. I derive both time-dependent and -independent equations for the ground state and the time evolution of the PCS ansatz. The time complexity of simulating PCS does not depend on N and is linear in the number of orbitals in use.
The condensation of Cooper pairs, originating from the Fermi-surface instability due to a weakly attractive interaction between two fermions, opened a new frontier for exploring many-body physics in interdisciplinary contexts. In this work, we discuss the possible condensation of Cooper triples, which are three-body counterparts of Cooper pairs for three-component fermions with a three-body attraction. Although each composite trimer-like state obeys the Fermi-Dirac statistics, its aggregate can form a condensate at zero center-of-mass momentum in the presence of the internal degrees of freedom associated with the relative momenta of constituent particles of momenta close to the Fermi surface. Such condensation can be regarded as bosonization in infinite-component fermions. We propose a variational wave function for the condensate of Cooper triples in analogy with the Bardeen-Cooper-Schrieffer ground state, and we obtain the ground-state energy.