The a.c. and d.c. Josephson effects in a Bose–Einstein condensate

Steinhauer, Jeff; Levy, Shahar; Lahoud, E.; Shomroni, Itay

doi:10.1038/nature06186

Cited by 545 publications

(616 citation statements)

References 30 publications

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“…This Hamiltonian has the form of a classical non-rigid pendulum and is similar to the double-well Bose-Hubbard model that has been used to study Josephson effects in condensates 3,5 . The Hamiltonian can also be written using a phase space of the spin vector S i and nematic (quadrupole) tensor Q ij matrix operators for the spin-1 system:…”

Section: Discussionmentioning

confidence: 99%

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Non-equilibrium dynamics of an unstable quantum pendulum explored in a spin-1 Bose–Einstein condensate

et al. 2012

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A pendulum prepared perfectly inverted and motionless is a prototype of unstable equilibrium and corresponds to an unstable hyperbolic fixed point in the dynamical phase space. Here, we measure the non-equilibrium dynamics of a spin-1 Bose-Einstein condensate initialized as a minimum uncertainty spin-nematic state to a hyperbolic fixed point of the phase space. Quantum fluctuations lead to non-linear spin evolution along a separatrix and non-Gaussian probability distributions that are measured to be in good agreement with exact quantum calculations up to 0.25 s. At longer times, atomic loss due to the finite lifetime of the condensate leads to larger spin oscillation amplitudes, as orbits depart from the separatrix. This demonstrates how decoherence of a many-body system can result in apparent coherent behaviour. This experiment provides new avenues for studying macroscopic spin systems in the quantum limit and for investigations of important topics in non-equilibrium quantum dynamics.

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Section: Discussionmentioning

confidence: 99%

“…Even at zero temperature, unavoidable quantum fluctuations would lead to evolution 1,2 . Although mechanical pendulums operating at the quantum limit are currently unavailable in the lab, it is possible to study quantum many-body systems that have similar dynamical behaviour [3][4][5] .…”

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confidence: 99%

Non-equilibrium dynamics of an unstable quantum pendulum explored in a spin-1 Bose–Einstein condensate

et al. 2012

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“…Nonlinear Josephson physics was first observed in superfluid helium 5 and atomic condensates 6,7 , but it has remained inaccessible in photonic systems because it requires large photon-photon interactions. Here we report on the observation of nonlinear Josephson oscillations of two coupled polariton condensates confined in a photonic molecule formed by two overlapping micropillars etched in a semiconductor microcavity 8 .…”

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confidence: 99%

“…This is the main principle of the bosonic Josephson effect, which manifests in an ensemble of oscillatory regimes. In the absence of interactions, sinusoidal oscillations take place 4,7 with a frequencȳ Josephson physics shows the most spectacular phenomena in the nonlinear regime, when the interaction energy (U |ψ| 2 ) is greater than the coupling J . The transfer of particles from one site to the other gives rise to a dynamical renormalization of the energy in each site, resulting in anharmonic oscillations.…”

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confidence: 99%

Macroscopic quantum self-trapping and Josephson oscillations of exciton polaritons

et al. 2013

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The coupling of two macroscopic quantum states through a tunnel barrier gives rise to Josephson phenomena 1 such as Rabi oscillations 2 , the a.c. and d.c. effects 3 , or macroscopic self-trapping, depending on whether tunnelling or interactions dominate 4 . Nonlinear Josephson physics was first observed in superfluid helium 5 and atomic condensates 6,7 , but it has remained inaccessible in photonic systems because it requires large photon-photon interactions. Here we report on the observation of nonlinear Josephson oscillations of two coupled polariton condensates confined in a photonic molecule formed by two overlapping micropillars etched in a semiconductor microcavity 8 . At low densities we observe coherent oscillations of particles tunnelling between the two sites. At high densities, interactions quench the transfer of particles, inducing the macroscopic self-trapping of polaritons in one of the micropillars 9,10 . The finite lifetime results in a dynamical transition from self-trapping to oscillations with π phase. Our results open the way to the experimental study of highly nonlinear regimes in photonic systems, such as chaos 11-13 or symmetry-breaking bifurcations 14,15 .A bosonic Josephson junction is a device in which two macroscopic ensembles of bosons, each of them occupying a single quantum state, are coupled by a tunnel barrier. The system can be described by the following coupled nonlinear Schrödinger equations 1 :where ψ L,R are the bosonic wavefunctions with particle densities |ψ L,R | 2 localized to the left (L) and to the right (R) of the barrier, E 0 L,R is the single particle energy of the quantum states, U is the particle-particle interaction strength and J is the tunnel coupling constant. In the absence of interactions, equations (1a) and (1b) can be diagonalized in a basis of bonding (). An initial state prepared in a linear combination of these two (for instance, all particles in the left site) will result in density oscillations between the two sites. This is the main principle of the bosonic Josephson effect, which manifests in an ensemble of oscillatory regimes. In the absence of interactions, sinusoidal oscillations take place 4,7 with a frequencȳ Josephson physics shows the most spectacular phenomena in the nonlinear regime, when the interaction energy (U |ψ| 2 ) is greater than the coupling J . The transfer of particles from one site to the other gives rise to a dynamical renormalization of the energy in each site, resulting in anharmonic oscillations. If interactions are strong enough (U |ψ| 2 J ), the self-induced energy renormalization quenches the tunnelling, and most of the particles remain localized in one of the sites. This out of equilibrium metastable regime is called macroscopic quantum self-trapping.A number of bosonic systems have demonstrated Josephson physics. Harmonic oscillations in the linear regime have been observed in superconductor junctions 2 or in nanoscale apertures connecting superfluid helium vessels 5 . Bose-Einstein condensates of ultracold atoms in cou...

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“…Rich physics has been explored with this simple model, including the Josephson effect [1,2] and self-trapping phenomenon for the attractive interaction [3,4]. Furthermore, this model is a prototype of the Bose-Hubbard model in periodic potentials, and can offer clues to phenomenon like superfluid to Mott insulator transition that occurs in the thermodynamic limit [5].…”

Section: Introductionmentioning

confidence: 99%

Extended two-site Bose–Hubbard model with pair tunneling: spontaneous symmetry breaking, effective ground state and fragmentation

Zhu

Zhang

2015

J. Phys. B: At. Mol. Opt. Phys.

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The extended Bose-Hubbard model for a double-well potential with pair tunneling is studied through both exact diagonalization and mean field theory (MFT). When pair tunneling is strong enough, the ground state wavefunction predicted by the MFT is complex and doubly degenerate while the quantum ground state wavefunction is always real and unique. The time reversal symmetry is spontaneously broken when the system transfers from the quantum ground state into one of the mean field ground states upon a small perturbation. As the gap between the lowest two levels decreases exponentially with particle number, the required perturbation inducing the spontaneous symmetry breaking (SSB) is infinitesimal for particle number of typical cold atom systems. The quantum ground state is further analyzed with the Penrose-Onsager criterion, and is found to be a fragmented condensate. The state also develops the pair correlation and has non-vanishing pair order parameter instead of the conventional single particle order parameter. When this model is generalized to optical lattice, a pair superfluid can be generated. The mean field ground state can be regarded as effective ground state in this simple model. The detailed computation for this model enables us to offer an in-depth discussion of the relation between SSB and effective ground state, giving a glimpse on how nonlinearity arises in the SSB of a quantum system.

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The a.c. and d.c. Josephson effects in a Bose–Einstein condensate

Cited by 545 publications

References 30 publications

Non-equilibrium dynamics of an unstable quantum pendulum explored in a spin-1 Bose–Einstein condensate

Non-equilibrium dynamics of an unstable quantum pendulum explored in a spin-1 Bose–Einstein condensate

Macroscopic quantum self-trapping and Josephson oscillations of exciton polaritons

Extended two-site Bose–Hubbard model with pair tunneling: spontaneous symmetry breaking, effective ground state and fragmentation

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