We study quantum phase-slip (QPS) processes in a superconducting ring containing
N Josephson junctions and threaded by an external static magnetic flux ${\ensuremath{\Phi}}_{B}$. In such a system, a QPS consists of a quantum tunneling event connecting two distinct classical states of the phases with different persistent currents [Matveev et al., Phys. Rev. Lett. 89, 096802 (2002)]. When the Josephson coupling energy
EJ of the junctions is larger than the charging energy
EC=e2/2C, where
C is the junction capacitance, the quantum amplitude for the QPS process is exponentially small in the ratio
EJ/EC. At given magnetic flux, each QPS can be described as the tunneling of the phase difference of a single junction of almost $2\ensuremath{\pi}$, accompanied by a small harmonic displacement of the phase difference of the other $N\ensuremath{-}1$ junctions. As a consequence, the total QPS amplitude ${\ensuremath{\nu}}_{\mathrm{ring}}$ is a global property of the ring. Here, we study the dependence of ${\ensuremath{\nu}}_{\mathrm{ring}}$ on the ring size
N, taking into account the effect of a finite capacitance
C0 to ground, which leads to the appearance of low-frequency dispersive modes. Josephson and charging effects compete and lead to a nonmonotonic dependence of the ring's critical current on
N. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$, the system converges either towards a superconducting or an insulating state, depending on the ratio between the charging energy
E0=e2/2C0 and the Josephson coupling energy
EJ.
The persistent current in a lattice model of a one-dimensional interacting electron system is systematically studied using a complex version of the density-matrix renormalization group algorithm and the functional renormalization group method. We mainly focus on the situation where a single impurity is included in the ring penetrated by a magnetic flux. Due to the interplay of the electron-electron interaction and the impurity the persistent current in a system of N lattice sites vanishes faster than
1/N. Only for very large systems and large impurities our results are consistent with the Bosonization prediction obtained for an effective field theory. The results from the density-matrix renormalization group and the functional renormalization group agree well for interactions as large as half the bandwidth, even though as an approximation in the latter method the flow of the two-particle vertex is neglected. This confirms that the functional renormalization group method is a very powerful tool to investigate correlated electron systems. The method will become very useful for the theoretical description of the electronic properties of small conducting ring molecules.
The ground-state persistent current and electron addition spectrum in two-dimensional quantum dot arrays and one-dimensional quantum dot rings, pierced by an external magnetic flux, are investigated using the extended Hubbard model. The collective multidot problem is shown to map exactly into the strong-field noninteracting finite-size Hofstadter butterfly problem at the spin polarization transition. The finite-size Hofstadter problem is discussed, and an analytical solution for limiting values of flux is obtained. In weak fields we predict interesting flux periodic oscillations in the spin component along the quantization axis with a periodicity given by $\ensuremath{\nu}h/e$ $(\ensuremath{\nu}<~1)$. The sensitivity of the calculated persistent current to interaction and disorder is shown to reflect the intricacies of various Mott-Hubbard quantum phase transitions in two-dimensional systems: the persistent current is suppressed in the antiferromagnetic Mott-insulating phase governed by intradot Coulomb interactions; the persistent current is maximized at the spin density wave--charge density wave transition driven by the nearest-neighbor interdot interaction; the Mott-insulating phase persistent current is enhanced by the long-range interdot interactions to its noninteracting value; the strong suppression of the noninteracting current in the presence of random disorder is seen only at large disorder strengths; at half-filling even a relatively weak intradot Coulomb interaction enhances the disordered noninteracting system persistent current; in general, the suppression of the persistent current by disorder is less significant in the presence of the long-range interdot Coulomb interaction.
We study the quantum dynamics of Josephson-junction arrays. We find isolated groups of low-entanglement eigenstates that persist even when the Josephson interaction is strong enough to destroy the overall organization of the spectrum in multiplets, and a perturbative description is no longer possible. These eigenstates lie in the inner part of the spectrum, far from the spectral edge, and provide a weak ergodicity breaking, reminiscent of the quantum scars. Due to the presence of these eigenstates, initializing with a charge-density-wave state, the system does not thermalize and the charge-density-wave order persists for long times. Considering global ergodicity probes, we find that the system tends toward more ergodicity for increasing system size: The parameter range where the bulk of the eigenstates look nonergodic shrinks for increasing system size. We study two geometries, a one-dimensional chain and a two-leg ladder. In the latter case, adding a magnetic flux makes the system more ergodic.
. In circuits of underdamped, small-capacitance Josephson junctions, the charge and phase variables behave quantum mechanically. Superpositions of charge states and flux states have been demonstrated, as well as microwave induced quantum dynamics (see e.g. refs [3, 4]). These e#ects have their origin in the conjugation of the phase and charge variables of Josephson circuits. The qubit design that we present here also contains small-capacitance Josephson junctions. However, even though charge-phase conjugation underlies the quantum behavior of the qubit, it was explicitly designed to be very insensitive to charge e#ects. In practice, random background charges and 1/f charge noise are always present in Josephson circuits, but these charge e#ects have negligible influence on the qubit. The qubit loop contains three Josephson junctions. Two junctions have Josephson coupling E J , the third junction has a coupling of #E J .Weaim
We review experiments on small-capacitance Josephson junctions. When the Josephson junction is fabricated in the configuration of dc superconducting quantum interference device (SQUID), the Josephson coupling can be tuned IN SITU with an external magnetic field. The electrical transport properties of one-dimensional arrays of small-capacitance dc SQUIDs have been investigated in order to study the superconductor-insulator transition. The arrays have also been used to bias a single Josephson junction, and a clear Coulomb blockade of Cooper-pair tunneling has been observed in the single junction.