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arXiv:1008.3057 · Physical Review Letters

Classical Bifurcation at the Transition from Rabi to Josephson Dynamics

Tilman Zibold, E. Nicklas, Christian Groß, Markus K. Oberthaler

Published Nov 12, 2010• 406 citations
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Abstract

We report on the experimental demonstration of the internal bosonic Josephson effect in a rubidium spinor Bose-Einstein condensate. The measurement of the full time dynamics in phase space allows the characterization of the theoretically predicted π-phase modes and quantitatively confirms analytical predictions, revealing a classical bifurcation. Our results suggest that this system is a model system which can be tuned from classical to the quantum regime and thus is an important step towards the experimental investigation of entanglement generation close to critical points.

DOI
10.1103/physrevlett.105.204101

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The Josephson junction model is applied to the experimental implementation of classical bifurcation in a quadrupolar nuclear magnetic resonance system. There are two regimes, one linear and one nonlinear, which are implemented by the radio-frequency and the quadrupolar terms of the Hamiltonian of a spin system, respectively. These terms provide an explanation of the symmetry breaking due to bifurcation. Bifurcation depends on the coexistence of both regimes at the same time in different proportions. The experiment is performed on a lyotropic liquid crystal sample of an ordered ensemble of 133{}^{133}133Cs nuclei with spin I=7/2I=7/2I=7/2 at room temperature. Our experimental results confirm that bifurcation happens independently of the spin value and of the physical system. With this experimental spin scenario, we confirm that a quadrupolar nuclei system could be described analogously to a symmetric two-mode Bose-Einstein condensate.
We theoretically study the internal Josephson effect, which is driven by spin-exchange interactions and magnetic dipole-dipole interactions, in a three-level system for spin-1 Bose--Einstein condensates, obtaining novel spin dynamics. We introduce single spatial mode approximations into the Gross--Pitaevskii equations and derive the Josephson-type equations, which are analogous to tunneling currents through three junctions between three superconductors. From an analogy with two interacting nonrigid pendulums, we identify unique varied oscillational modes, called the $0--\ensuremath{\pi}$, 0−−running0--\mathit{running}0−−running, running−−running\mathit{running}--\mathit{running}running−−running, $2n\ensuremath{\pi} & \mathit{running}--2\ensuremath{\pi}$, singlenonrigidpendulum\mathit{single nonrigid pendulum}singlenonrigidpendulum, and tworigidpendulums\mathit{two rigid pendulums}tworigidpendulums phase modes. These Josephson modes in the three states are expected to be found in real atomic Bose gas systems.
A dynamical quantum phase transition can be characterized by a nonanalytic change in the quench dynamics when a parameter in the governing Hamiltonian is varied. Such a transition typically only shows up in long-time dynamics for extensive systems with short-range couplings. We analyze a model Hamiltonian of spin-1 particles with effectively infinite-range couplings and demonstrate that for this system the nonanalytic transition occurs for local observables in short-time durations even when the system is of a large size. We experimentally realize this model Hamiltonian and observe the dynamical quantum phase transition in an antiferromagnetic spinor Bose-Einstein condensate of around 105{10}^{5}105 sodium atoms. Our observations agree well with the theoretical prediction. We also analyze the scaling exponent near the dynamical phase transition and discuss its relation with the excited-state spectrum of the system.