Engineering Superconducting Phase Qubits

The superconducting phase qubit combines Josephson junctions into superconducting loops and defines one of the promising solid state device implementations for quantum computing. While conventional designs are based on magnetically frustrated superconducting loops, here we discuss the advantages offered by $\pi$-junctions in obtaining naturally degenerate two-level systems. Starting from a basic five-junction loop, we show how to construct degenerate two-level junctions and superconducting phase switches. These elements are then effectively engineered into a superconducting phase qubit which operates exclusively with switches, thus avoiding permanent contact with the environment through external biasing. The resulting superconducting phase qubits can be understood as the macroscopic analogue of the `quiet' s-wave-d-wave-s-wave Josephson junction qubits introduced by Ioffe {\it et al.} [Nature {\bf 398}, 679 (1999)].Comment: 8 pages, RevTeX, seven postscript figures incorporated using psfi

Suppression of Geometric Barrier in Type II Superconducting Strips

We study the magnetic response of a superconducting double strip, i.e., two parallel coplanar thin strips of width $2w$, thickness $d \ll w$ and of infinite length, separated by a gap of width $2s$ and subject to a perpendicular magnetic field $H$. The magnetic properties of this system are governed by the presence of a geometric energy barrier for vortex penetration which we investigate as a function of applied field $H$ and gap parameter $s$. The new results deal with the case of a narrow gap $s \ll w$, where the field penetration from the inner edges is facilitated by large flux focusing. Upon reducing the gap width $2s$, we observe a considerable rearrangement of the screening currents, leading to a strong reduction of the penetration field and the overall magnetization loop, with a suppression factor reaching $\sim (d/w)^{1/2}$ as the gap drops below the sample thickness, $2s < d$. We compare our results with similar systems of different shapes (elliptic, rectangular platelet) and include effects of surface barriers as well. Furthermore, we verify that corrections arising from the magnetic response of the Shubnikov phase in the penetrated state are small and can be omitted. Extending the analysis to multiple strips, we determine the specific sequence of flux penetrations into the different strips. Our studies are relevant for the understanding of platelet shaped samples with cracks or the penetration into layered superconductors at oblique magnetic fields.Comment: 26 pages, 19 figure

Probing the pinning landscape in type-II superconductors via Campbell penetration depth

Type-II superconductors owe their magnetic and transport properties to vortex pinning, the immobilization of flux quanta through material inhomogeneities or defects. Characterizing the potential energy landscape for vortices, the pinning landscape (or short, pinscape), is of great technological importance. Besides measurement of the critical current density $j_c$ and of creep rates $S$, the $ac$ magnetic response provides valuable information on the pinscape which is different from that obtained through $j_c$ or $S$, with the Campbell penetration depth $\lambda_{\rm \scriptscriptstyle C}$ defining a characteristic quantity well accessible in an experiment. Here, we derive a microscopic expression for the Campbell penetration depth $\lambda_{\rm \scriptscriptstyle C}$ using strong pinning theory. Our results explain the dependence of $\lambda_{\rm \scriptscriptstyle C}$ on the state preparation of the vortex system and the appearance of hysteretic response. Analyzing different pinning models, metallic or insulating inclusions as well as $\delta T_c$- and $\delta \ell$-pinning, we discuss the behavior of the Campbell length for different vortex state preparations within the phenomenological $H$-$T$ phase diagram and compare our results with recent experiments.Comment: 16 pages, 11 figure

Vortex dynamics in type II superconductors under strong pinning conditions

We study effects of pinning on the dynamics of a vortex lattice in a type II superconductor in the strong-pinning situation and determine the force--velocity (or current--voltage) characteristic combining analytical and numerical methods. Our analysis deals with a small density $n_p$ of defects that act with a large force $f_p$ on the vortices, thereby inducing bistable configurations that are a characteristic feature of strong pinning theory. We determine the velocity-dependent average pinning-force density $\langle F_p(v)\rangle$ and find that it changes on the velocity scale $v_p \sim f_p/\eta a_0^3$, where $\eta$ is the viscosity of vortex motion and $a_0$ the distance between vortices. In the small pin-density limit, this velocity is much larger than the typical flow velocity $v_c \sim F_c/\eta$ of the free vortex system at drives near the critical force-density $F_c = \langle F_p(v=0)\rangle \propto n_p f_p$. As a result, we find a generic excess-force characteristic, a nearly linear force--velocity characteristic shifted by the critical force-density $F_c$; the linear flux-flow regime is approached only at large drives. Our analysis provides a derivation of Coulomb's law of dry friction for the case of strong vortex pinning.Comment: 24 pages, 12 figure

Transport in a One-Dimensional Superfluid: Quantum Nucleation of Phase Slips

We present an analytical derivation for the quantum decay rate of the superflow through a weak link in a one-dimensional Bose-Einstein-condensate. The effective action for the phase difference across the link reduces to that of a massive particle with damping subject to a periodic potential. We find an algebraic flow-pressure relation, characteristic for quantum nucleation of phase slips in the link and show how short-wave length fluctuations renormalizing the interaction between the Bosons remove the quantum phase transition expected in this class of systems.Comment: 4pages, RevTex, 2 Postscript figure

Appearance of Schrodinger Cat States in the Measurement Process

Although quantum mechanics is a mature theory, fundamental problems discussed during its time of foundation have remained with us to this day. These problems are centered on the problematic relation between the quantum and classical worlds. The most famous element is the measurement problem, i.e., the measurement of a quantum system by a classical apparatus, and the concomitant phenomena of wave packet reduction, the appearance of probability, and the problems related to Schr\"odinger cat states. A fundamental question in this context is whether quantum mechanics can bootstrap itself to the classical world: is quantum mechanics self-consistent, such that the measurement process can be understood within quantum mechanics itself, or does this process require additional elements from the realm outside of traditional quantum mechanics? Here, we point to a problematic aspect in the traditional Schr\"odinger cat argument which can be overcome through its extension with a proper macroscopic preparation device; the deliberate creation of a cat state and its identification then turns into a non-trivial problem requiring the determination of the evolution of a quantum system entangled with a macroscopic reservoir. We describe a new type of wave-function correlator testing for the appearance of Schr\"odinger cat states and discuss its implications for theories deriving the wave function collapse from a unitary evolution

Optimal non-invasive measurement of Full Counting Statistics by a single qubit

The complete characterisation of the charge transport in a mesoscopic device is provided by the Full Counting Statistics (FCS) $P_t(m)$, describing the amount of charge $Q = me$ transmitted during the time $t$. Although numerous systems have been theoretically characterized by their FCS, the experimental measurement of the distribution function $P_t(m)$ or its moments $\langle Q^n \rangle$ are rare and often plagued by strong back-action. Here, we present a strategy for the measurement of the FCS, more specifically its characteristic function $\chi(\lambda)$ and moments $\langle Q^n \rangle$, by a qubit with a set of different couplings $\lambda_j$, $j = 1,\dots,k,\dots k+p$, $k = \lceil n/2 \rceil$, $p \geq 0$, to the mesoscopic conductor. The scheme involves multiple readings of Ramsey sequences at the different coupling strengths $\lambda_j$ and we find the optimal distribution for these couplings $\lambda_j$ as well as the optimal distribution $N_j$ of $N = \sum N_j$ measurements among the different couplings $\lambda_j$. We determine the precision scaling for the moments $\langle Q^n \rangle$ with the number $N$ of invested resources and show that the standard quantum limit can be approached when many additional couplings $p\gg 1$ are included in the measurement scheme.Comment: 8 pages, 1 figure, Accepted for publication in PR

Supercurrent Quantization in Narrow Channel SNS Junctions

We determine the quasi-particle excitation spectrum in the normal region of a narrow ballistic superconductor--normal-metal--superconductor (SNS) Josephson contact. Increasing the effective chemical potential in the contact converts the electronic levels into Andreev-levels carrying supercurrent. The opening of these superchannels leads to a supercurrent quantization which exhibits a non-universal behavior in general and we discuss its dependence on the junction parameters.Comment: 4 pages, RevTeX, three postscript figure

Campbell response in type II superconductors under strong pinning conditions

Measuring the $ac$ magnetic response of a type II superconductor provides valuable information on the pinning landscape (pinscape) of the material. We use strong pinning theory to derive a microscopic expression for the Campbell length $\lambda_{\rm \scriptscriptstyle C}$, the penetration depth of the $ac$ signal. We show that $\lambda_{\rm \scriptscriptstyle C}$ is determined by the jump in the pinning force, in contrast to the critical current $j_c$ which involves the jump in pinning energy. We demonstrate that the Campbell lengths generically differ for zero-field-cooled and field-cooled samples and predict that hysteretic behavior can appear in the latter situation. We compare our findings with new experimental data and show the potential of this technique in providing information on the material's pinscape.Comment: 5 pages, 3 figure

Andreev Spectroscopy for Superconducting Phase Qubits

We propose a new method to measure the coherence time of superconducting phase qubits based on the analysis of the magnetic-field dependent dc nonlinear Andreev current across a high-resistance tunnel contact between the qubit and a dirty metal wire and derive a quantitative relation between the subgap I-V characteristic and the internal correlation function of the qubit.Comment: LaTeX 2.09, 11 pages, 2 eps-figures; special LaTeX style file included. Contribution to the Proceedings of "Electron Transport In Mesoscopic Systems", LT22 Satellite Conference in Goteborg, Sweden, 12-15 August 1999. To be published in a special issue of J. Low Temp. Phys. The corresponding Extended Abstract (2 pages) was posted previously at cond-mat/990731