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arXiv:0801.1567 · New Journal of Physics

Semifluxons in superconductivity and cold atomic gases

R Walser, E Goldobin, O Crasser, D Koelle, R Kleiner, W P Schleich

Published Apr 30, 2008• 8 citations
arXivHTMLPDF

Abstract

Josephson junctions and junction arrays are well studied devices in superconductivity. With external magnetic fields one can modulate the phase in a long junction and create traveling, solitonic waves of magnetic flux, called fluxons. Today, it is also possible to device two different types of junctions: depending on the sign of the critical current density, they are called 0- or pi-junction. In turn, a 0-pi junction is formed by joining two of such junctions. As a result, one obtains a pinned Josephson vortex of fractional magnetic flux, at the 0-pi boundary. Here, we analyze this arrangement of superconducting junctions in the context of an atomic bosonic quantum gas, where two-state atoms in a double well trap are coupled in an analogous fashion. There, an all-optical 0-pi Josephson junction is created by the phase of a complex valued Rabi-frequency and we a derive a discrete four-mode model for this situation, which qualitatively resembles a semifluxon.

DOI
10.1088/1367-2630/10/4/045020

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Josephson junctions and junction arrays are well studied devices in superconductivity. With external magnetic fields one can modulate the phase in a long junction and create traveling, solitonic waves of magnetic flux, called fluxons. Today, it is also possible to device two different types of junctions: depending on the sign of the critical current density, they are called 0- or pi-junction. In turn, a 0-pi junction is formed by joining two of such junctions. As a result, one obtains a pinned Josephson vortex of fractional magnetic flux, at the 0-pi boundary. Here, we analyze this arrangement of superconducting junctions in the context of an atomic bosonic quantum gas, where two-state atoms in a double well trap are coupled in an analogous fashion. There, an all-optical 0-pi Josephson junction is created by the phase of a complex valued Rabi-frequency and we a derive a discrete four-mode model for this situation, which qualitatively resembles a semifluxon.
Josephson junctions (JJs) and junction arrays are well-studied devices in superconductivity. With external magnetic fields one can modulate the phase in a long junction and create traveling, solitonic waves of magnetic flux, called fluxons. Today, it is also possible to devise two different types of junctions: depending on the sign of the critical current density , they are called 0- or π-junctions. In turn, a 0–π junction is formed by joining two of these junctions. As a result, one obtains a pinned Josephson vortex of fractional magnetic flux, at the 0–π boundary. Here, we analyze this arrangement of superconducting junctions in the context of an atomic bosonic quantum gas, where two-state atoms in a double well trap are coupled in an analogous fashion. There, an all-optical 0–π JJ is created by the phase of a complex valued Rabi frequency and we derive a discrete four-mode model for this situation, which qualitatively resembles a semifluxon.
We propose to create pairs of semifluxons starting from a flat-phase state in long, optical 0-$\ensuremath{\pi}$-0 Josephson junctions formed with internal electronic states of atomic Bose-Einstein condensates. In this optical system, we can dynamically tune the length aaa of the $\ensuremath{\pi}$ junction, the detuning $\ensuremath{\delta}$ of the optical transition, or the strength ${\ensuremath{\Omega}}_{0}$ of the laser coupling to induce transitions from the flat-phase state to such a semifluxon-pair state. Similarly as in superconducting 0-$\ensuremath{\pi}$-0 junctions, there are two, energetically degenerate semifluxon-pair states. A linear mean-field model with two internal electronic states explains this degeneracy and shows the distinct static field configuration in a phase diagram of the junction parameters. This optical system offers the possibility to dynamically create a coherent superposition of the distinct semifluxon-pair states and observe macroscopic quantum oscillation.
Josephson junctions with a phase drop of $\ensuremath{\pi}$ in the ground state allow us to create vortices of supercurrent carrying only half of the magnetic flux quantum ${\ensuremath{\Phi}}_{0}\ensuremath{\approx}2.07\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}15}\text{ }\text{ }\mathrm{Wb}$. Such semifluxons have twofold degenerate ground states denoted $\ensuremath{\uparrow}$ (with flux $+{\ensuremath{\Phi}}_{0}/2$ and supercurrent circulating clockwise) and $\ensuremath{\downarrow}$ (with flux $\ensuremath{-}{\ensuremath{\Phi}}_{0}/2$ and supercurrent circulating counterclockwise). We investigate a molecule consisting of two coupled semifluxons in a 0-$\ensuremath{\pi}$-0 long Josephson junction. The fluxes (polarities) of semifluxons are measured by two on-chip SQUIDs. By varying the dc bias current applied to the 0-$\ensuremath{\pi}$-0 junction, we demonstrate controllable manipulation and switching between two states, $\ensuremath{\uparrow}\ensuremath{\downarrow}$ and $\ensuremath{\downarrow}\ensuremath{\uparrow}$, of a semifluxon molecule. These results provide a major step towards employing semifluxons as bits or qubits for classical and quantum digital electronics.
Long Josephson tunnel junctions are non-linear transmission lines that allow propagation of current vortices (fluxons) and electromagnetic waves and are used in various applications within superconductive electronics. Recently, the Josephson vortex has been proposed as a new superconducting qubit. We describe a simple method to create a double-well potential for an individual fluxon trapped in a long elliptic annular Josephson tunnel junction characterized by an intrinsic non-uniform width. The distance between the potential wells and the height of the inter-well potential barrier are controlled by the strength of an in-plane magnetic field. The manipulation of the vortex states can be achieved by applying a proper current ramp across the junction. The read-out of the state is accomplished by measuring the vortex depinning current in a small magnetic field. An accurate one-dimensional sine-Gordon model for this strongly non-linear system is presented, from which we calculate the position-dependent fluxon rest-mass, its Hamiltonian density and the corresponding trajectories in the phase space. We examine the dependence of the potential properties on the annulus eccentricity and its electrical parameters and address the requirements for observing quantum-mechanical effects, as discrete energy levels and tunneling, in this two-state system.
Quantum Gases In superconductors, electrons form a macroscopic wave function that has a definite phase. If two superconductors with different wave function phases are placed in contact with each other through an insulating link, a current will flow through this so-called Josephson's junction without any external voltage. Luick et al. and Kwon et al. observed an analogous phenomenon in a setup that involved two reservoirs of superfluid Fermi gases. Both groups measured the so-called current-phase relation: the dependence of the magnitude of the current on the relative phase. By tuning an external magnetic field, they were able to study how the interactions between fermions affected the nature of the superfluid state. Science , this issue p. [89][1], p. [84][2] [1]: /lookup/doi/10.1126/science.aaz2342 [2]: /lookup/doi/10.1126/science.aaz2463
Motivated by possible applications to cold atomic gases in a rotating trap, we give an overview of some important concepts in two-dimensional vortex physics. In the context of Josephson junction arrays, we first discuss the crucial role payed by vortices in both the thermal superconducting to normal metal transition, and the zero temperature superconducting to Mott insulator transition. Important notions on vortex dynamics in these arrays are then presented, with emphasis on experimental results indicating the quantum nature of vortex motion at low temperature. We move then to the Fractional Quantum Hall effect, as observed in electronic systems, and introduce the notions of fractional excitations, generalized statistics, and composite Fermions, which have played a key role in the theoretical description of this remarkable phenomenon. Very likely, cold atomic gases in a rotating trap will soon provide a way to explore the bosonic version of the Fractional Quantum Hall effect, and thus shedding new light on these subtle strongly correlated quantum fluids.