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Basing our model and method on the microscopic theory we formulate a quantum-mechanical description for the relevant variable in a superconducting tunnel junction, i.e. , the phase difference across the junction. The quasiparticle degrees of freedom are responsible for dissipation and noise in the system. Because of the discreteness of the charge-transfer process, the noise is shot noise. The energy gaps in the superconductors lead to further interesting features. We discuss the consequences of these physical effects on macroscopic quantum phenomena.

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Using the density matrix renormalization group algorithm, we investigate the lattice model for spinless fermions in one dimension in the presence of a strong interaction and disorder. The phase sensitivity of the ground state energy is determined with high accuracy for systems up to a size of 60 lattice constants. This quantity is found to be log-normally distributed. The fluctuations grow algebraically with system size with a universal exponent of ≈ 2/3 in the localized region of the phase diagram. Surprisingly, we find, for an attractive interaction, a delocalized phase of finite extension. The boundary of this delocalized phase is determined.PACS numbers: 71.30.+h, 72.15.RnThe influence of electron-electron interaction on Anderson localization has attracted a lot of interest for several years. Many recent studies were motivated by the experimental observation of persistent currents in mesoscopic rings [1]. Motivated by an early suggestion [2] that the interaction between the electrons may give a significant contribution to the average persistent current, this phenomenon in the presence of both interaction and disorder has been investigated by various methods [3][4][5][6][7][8]. Nevertheless, the magnitude of the effect is still not well understood.In one dimension, interacting systems in the absence of disorder [9][10][11], as well as for disordered systems in the absence of interactions [12] are well studied. However, a clear understanding of the interplay between interaction and disorder has not yet been obtained. In this Letter, we present novel results of a detailed study of a simple interacting-fermion model with disorder. We determine the ground state phase sensitivity with high accuracy for a wide range of parameters and system sizes up to 60 lattice constants. Our main results are (i) a universal behavior of the rms-value of the logarithmic phase sensitivity, which grows with system size, M , proportional to M 2/3 in the localized region, and (ii) the zero-temperature phase diagram, which shows, for an attractive interaction a delocalized phase of finite extension.The numerical results are obtained with the density matrix renormalization group algorithm (DMRG) [13], which allows calculation of ground state properties of disordered, interacting fermion systems with an accuracy which is comparable to exact diagonalization, but for much larger systems [14,15]. In our implementation of the DMRG we perform 5 finite lattice sweeps keeping up to 750 states per block.We consider a chain of spinless fermions with nearestneighbor interaction and disorder,and twisted boundary conditions, c 0 = e iφ c M . The length of the chain is denoted by M , and the particle number is N . For simplicity, we will set t = 1 in some of the formulas below.The ground state energy E(φ) depends on the phase φ. The energy difference between periodic and anti-periodic boundary conditions, ∆E = (−), and the charge stiffness, D ∼ E ′′ (φ = 0), are a measure of the phase sensitivity of the system. In the clean limit, i.e. ǫ n = 0 for al...

Motivated by the experiment of two-component Bose-Einstein condensates produced in magnetically trapped 87 Rb, we study one dimensional Boson systems with repulsive δ-function interaction in the presence of SU (2)