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arXiv:1211.0429 · Physical Review A

Emergence of atomic semifluxons in optical Josephson junctions

M. Grupp, W. P. Schleich, E. Goldobin, D. Koelle, R. Kleiner, R. Walser

Published Feb 13, 2013• 4 citations
arXivHTMLPDF

Abstract

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 $a$ 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.

DOI
10.1103/physreva.87.021602

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We propose to create pairs of semifluxons starting from a flat-phase state in long, optical 0-\ensuremathπ\ensuremath{\pi}\ensuremathπ-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 \ensuremathπ\ensuremath{\pi}\ensuremathπ junction, the detuning \ensuremathδ\ensuremath{\delta}\ensuremathδ of the optical transition, or the strength \ensuremathΩ0{\ensuremath{\Omega}}_{0}\ensuremathΩ0​ of the laser coupling to induce transitions from the flat-phase state to such a semifluxon-pair state. Similarly as in superconducting 0-\ensuremathπ\ensuremath{\pi}\ensuremathπ-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 (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.
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.
Coherent oscillations between two weakly coupled Bose-Einstein Condensates: Josephson effects, πππ-oscillations, and macroscopic quantum self trapping
We discuss the coherent atomic oscillations between two weakly coupled Bose-Einstein condensates. The weak link is provided by a laser barrier in a (possibly asymmetric) double-well trap or by Raman coupling between two condensates in different hyperfine levels. The Boson Josephson Junction (BJJ) dynamics is described by the two-mode non-linear Gross-Pitaevskii equation, that is solved analytically in terms of elliptic functions. The BJJ, being a neutralneutralneutral, isolated system, allows the investigations of new dynamical regimes for the phase difference across the junction and for the population imbalance, that are not accessible with Superconductor Josephson Junctions (SJJ). These include oscillations with either, or both of the following properties: 1) the time-averaged value of the phase is equal to πππ (π−phaseπ-phaseπ−phase oscillations); 2) the average population imbalance is nonzero, in states with ``macroscopic quantum self-trapping'' (MQST). The (non-sinusoidal) generalization of the SJJ `ac' and `plasma' oscillations and the Shapiro resonance can also be observed. We predict the collapse of experimental data (corresponding to different trap geometries and total number of condensate atoms) onto a single universal curve, for the inverse period of oscillations. Analogies with Josephson oscillations between two weakly coupled reservoirs of 3^33He-B and the internal Josephson effect in 3^33He-A are also discussed.
We discuss the coherent atomic oscillations between two weakly coupled Bose-Einstein condensates. The weak link is provided by a laser barrier in a (possibly asymmetric) double-well trap or by Raman coupling between two condensates in different hyperfine levels. The boson Josephson junction (BJJ) dynamics is described by the two-mode nonlinear Gross-Pitaevskii equation that is solved analytically in terms of elliptic functions. The BJJ, being a neutral, isolated system, allows the investigations of dynamical regimes for the phase difference across the junction and for the population imbalance that are not accessible with superconductor Josephson junctions (SJJ's). These include oscillations with either or both of the following properties: (i) the time-averaged value of the phase is equal to $\ensuremath{\pi} (\ensuremath{\pi}$-phase oscillations); (ii) the average population imbalance is nonzero, in states with macroscopic quantum self-trapping. The (nonsinusoidal) generalization of the SJJ ac and plasma oscillations and the Shapiro resonance can also be observed. We predict the collapse of experimental data (corresponding to different trap geometries and the total number of condensate atoms) onto a single universal curve for the inverse period of oscillations. Analogies with Josephson oscillations between two weakly coupled reservoirs of ${}^{3}\mathrm{He}\ensuremath{-}B$ and the internal Josephson effect in ${}^{3}\mathrm{He}\ensuremath{-}A$ are also discussed.
In this thesis, I present the first experimental implementation of a Josephson junction for Bose-Einstein condensates. The weak link between two BECs constitutes the nonlinear generalization of the well known Josephson junction of weakly-coupled superconductors that are separated by a thin insulating barrier. In our experiment, the required overlap of two macroscopic wavefunctions is provided by loading a BEC into an optical double well potential. It is realized by a superposition of a one-dimensional optical lattice with a focused laser beam optical dipole trap. The tunneling dynamics between the two potential wells exhibits two distinct dynamical regimes. For small initial population imbalances of the two wells we observe nearly sinusoidal Josephson tunneling oscillations, which are characterized by an oscillating population and relative phase. The situation changes drastically, if the initial population imbalance is chosen above a critical value. In this case, resonant tunneling between the two wells is prohibited because the difference between the on-site particle interaction energies in the two wells exceeds the tunneling energy splitting. As a consequence, the atomic distribution becomes self-locked and the relative phase evolves unbound in time. This regime of prohibited tunneling, which has no analogon in superconducting Josephson junctions, is called “macroscopic quantum self-trapping”.
We investigate a two-component Bose-Einstein condensate (BEC) strongly coupled to a single optical cavity, effectively described by a mean-field Dicke model supplemented with interatomic nonlinearities. Here, we propose a unified theoretical framework demonstrating that macroscopic quantum self-trapping (MQST) natively emerges between two internal atomic energy levels within a single cavity. By deriving the dimensionless semiclassical Josephson equations (SJE) governing this purely internal-state architecture, we analytically determine the critical nonlinear threshold and intrinsic phase shift mechanism for the phase transition. Based on this framework, we present two approaches for manipulating quantum phase transitions: dynamic in-situ tuning via photon pumping and inducing non-equilibrium dynamical phase transitions (DPT) via real-time parameter quenches. Furthermore, we rigorously prove that the effective charging energy driving this system scales exactly as one-quarter of the effective spin-dependent interaction energy -- the precise parameter governing recent spin-orbit coupled (SOC) BEC experiments. Incorporating realistic 87^{87}87Rb atomic parameters, we substantiate that these single-cavity MQST and transition dynamics are highly feasible for observation under current state-of-the-art cold-atom technologies.