Superfluidity Of Bose-Einstein Condensates In Ring Traps

2018/08/21 Seminar at Kochi University of TechnologySuperfluidity of Bose-Einsteincondensates in ring trapsYukawa Institute for Theoretical Physics,Kyoto UniversityMasaya Kunimi (國見 昌哉)References:MK and Y. Kato, Phys. Rev. A 91, 053608 (2015).MK and I. Danshita, arXiv:1712.09403.Collaborator：Yusuke Kato (University of Tokyo)Ippei Danshita (Kindai University)1

Contents Introduction to cold atomic gases and superfluidity Part 1. Multiple-swallowtail structures in 2D Bose-Einsteincondensate : MK and Y. Kato, Phys. Rev. A 91, 053608 (2015). Part 2. Superflow decay of Bose-Einstein condensate in the ringtraps : MK and I. Danshita, arXiv:1712.09403.2

Introduction : Cold Atomic GasesCold atomic gasesAtoms (typically alkali atoms) are trapped in a vacuum byusing a magnetic field or a laser beam.Bose-Einstein condensation (BEC, 1995) andFermi superfluid (2004) were experimentally realized.Important featuresT Tc Highly controllability Very clean systems.T TcT Tc Ideal quantum many-body systems!3W. Ketterle, Rev. Mod. Phys. 74, 1131 (2002).

Introduction : Cold Atomic GasesTypical scales of cold atomic gasesTemperature：10nK〜500nK TcBEC O(100nK)Density : 10-5 times the density of the airLength : 0.1µm〜100µmI. Bloch, Rev. Mod. Phys. 80, 885 (2008).Particle number：103〜107What we can do in cold atomic gasesTune of the inter-particle interaction BCS-BEC crossover, supersolidControl of spatial dimension and shape of the trap potential Ring trapControl of disorder Anderson localization, tuning the defect strengthSpin-orbit coupling Topological physicsQuantum gas microscope single site imaging, measurements of entanglement4

Methods : Hamiltonian of dilute Bose gasesHamiltonian for dilute Bose gasesĤ Z 2g††2ˆ2†ˆˆˆˆˆdrr (r) · r (r) U (r) (r) (r) (r) (r)2m2ˆ(r) ：Field operator of bosons U (r) ：external potentialV (r0r ) g (r4 2 ag m0r )：Contact interactiona ：s-wave scattering lengtha 0 : repulsive interactiona 0 : attractive interaction latexit sha1 base64 "0Pq82NZEATMCdfX9tFjb7Eq45AU " qKh3mJkzZ u6riCkM3RdHTPUPs2o5QaqohdtTqZu9 qxe FGarvFb4VA5Yk8 9mq6v1vgVg6fDyjiS9ES31KVHuqNnev hedvdk 03Op9es3Ttrdwlo UuYDyKuQWg /latexit Heisenberg equation latexit sha1 base64 "q6BuQaL3krYeMjAC2iwq1kplclU " qKh3mJkzZ u6riCkM3RdHTPUPs2o5QaqohdtTqZu9 qxe FGarvFb4VA5Yk8 9mq6v1vgVg6fDyjiS9ES31KVHuqNnev hedvdk 03Op9es3Ttrdwlo UuYDxGmQWA /latexit @ ˆi (r, t) [ ˆ(r, t), Ĥ]@t5See, for example, L. Pitaevskii and S. Stringari,Bose-Einstein Condensation(Oxford University Press, Oxford, UK, 2003)

Introduction : Gross-Pitaevskii Equationh ˆ(r, t)i (r, t) Neglecting the fluctuations ˆ(r, t) !(r, t)Gross-Pitaevskii (GP) equation (Mean-field theory)@i (r, t) @t 2 22r (r, t) U (r, t) (r, t) g (r, t) (r, t)2mThis approximation is justified under the low temperature and the diluteconditions.n3dB3na 11 Thermal de Brogile wave length mean particle distances-wave scattering length mean particle distanceThe GP equation can correctly describe the static and dynamicalproperties of the BEC.6

Introduction : Mean-field ApproximationPhysical quantities can be obtained by the order parameter.(r, t) pn(r, t)ei'(r,t)：Order parameter2：local particle densityn(r, t) (r, t) ：local velocity fieldv(r, t) r'(r, t)mFrom the single-valuedness of the order parameter,the circulation is quantized :Quantization of the circulation latexit sha1 base64 "yR9gXPkdUgV jzZAYhsXLxzbcWk " biJ4 4GiwXib6jp17j236nC92NeJZexgxjlx8tTpM7NnG fOX7g417x0eTOJJkaqroz8yGx5IlG DlXXauurrdgoEXi 6nm7K0W uk4v8lj/Kun1m23WYWW0jgK3Bm3UsRo134NjgAgSEwRQCGEJ 9k1It6RafliFlC/PsM3vLpuwje8e sMN/9srKHsVb9mj3Kq2K 2Udd22pvXy/HtUsruE6Fmget7GMh1hFl 79hCkO8cMJnWfOvvOiKnVmas0V/BHOm582mLxt /latexit I dr · v(r, t) mCI2 d' WmcW 2 Z : Winding number latexit sha1 base64 "lMLOwCiZXqUtNd/hO4ZvisTm2cE " miK8fOOp6Du7PL7twRs9wfCPYWqSJYhHQ2FtrZ 4pANsTe dIz W87lXM83VPr8S8ExYaWOKrugr3dAlfaPv9Ou/tbK8Ru8v 7yLvlbGzdFP47WfD6pC3g12/6juUQjOvt RS6jzuwc4wRnOrVmrZm1Ybj/VGig0Y/grLPkbCsqfzA /latexit 7

Introduction : SuperfluidityThe winding number is a topological number. The topological number can not be changed by a perturbation. The finite winding number states have long lifetime. Persistent current ! TLifetimeTexperimentSimply connected systems(Typical setup of cold atoms)Wallcore ofquantum vortex 0Multiply connected systems(Ring-shaped trap)no vorticesbut flow 0yxflow8

Introduction : BEC in a ring trapRing traps are ideal systems for studying superfluidity.Bose-Einstein condensates confined in a ring traphave been investigated by several experimental groups.Experimental studies for the ring trapRyu et al., PRL 99, 260401 (2007) : Persistent currentRamanathan et al., PRL 106, 130401 (2011) : Persistent currentMoulder et al., PRA 86, 013629 (2012) : Persistent currentBeattie et al., PRL 110, 025301 (2013) : Two-component BECWright et al., PRL 111, 025302 (2013) : RotationNeely et al., PRL 111, 235301 (2013) : TurbulenceRyu et al., New J. Phys. 16, 013046 (2014) : RotationEckel et al., Nature 506, 200 (2014) : HysteresisWright et al., PRL 111, 025302 (2013).Eckel et al., PRX 4, 031052 (2014) : Current-phase relationsCorman et al., PRL 113, 135302 (2014) : Kibble-Zurek mechanismKumar et al., PRA 95, 021602(R) (2017) : Temperature dependent Decay of persistent currentRed : Persistent current (no stirring)Blue : stirring the BEC in the ring trap9

Contents Introduction to cold atomic gases and superfluidity Part 1. Multiple-swallowtail structures in 2D Bose-Einsteincondensate : MK and Y. Kato, Phys. Rev. A 91, 053608 (2015). Part 2. Superflow decay of Bose-Einstein condensate in the ringtraps : MK and I. Danshita, arXiv:1712.09403.10

Part 1 : NIST experimentThe NIST group observed hysteresis in stirred BEC.Laser beam(repulsive defect)Density profiles of BECWinding numberBECHow do we understand thisphenomenon?010.5Angular velocity of rotation [Hz] The swallowtail structure isa key concept tounderstand superfluidity.Eckel et al., Nature 506, 200 (2014)11

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Swallowtail structure in 1D ringEnergy vs rotation velocity (non-interacting case)g 0 and U 6 0g 0 and U 0 latexit sha1 base64 "lLJKWLnneWE6bpjxGVEAS8y3ylU " UkcazBNAnJtFiLP sRWWlJW Q9X glwxY542CpcZ rCD cz8gK7wWOEWNtmbMEZ YY3GX6q8G02t8AoDJyPrb/ qSrxL7H T3SHTt0Kq/m1ZrInCHGDdK t Mn2JxOapTU1mYSqcWoVV0YwzimuB zSGEFaWT53SrOcY26MqqklGVltZmqtESaYXwJJfMBfSCWng /latexit W 1EE latexit sha1 base64 "ifz89V hqdFT6OPT89bvKz/zGyY " Rb3DzJw5c8 IACl0cRMgg2cyU0mHMY6cG9wAmirK1yluAMhdkjXg/4VAxZk89 Dp Pb7v6oK7x4Ov1R/KFTO/tuThzJSgRedvdkB47vUWvVrx lNML8dX1sFURTGAKM9yPJaxiE1vI8ruHOMM5LqQ aU5alFKtVKkj1IzhW0hrnzlKkIc /latexit latexit sha1 base64 "AfXEGAT3 nIakiTUjm

2 Contents Introduction to cold atomic gases and superfluidity Part 1. Multiple-swallowtail structures in 2D Bose-Einstein condensate : MK and Y. Kato, Phys. Rev. A 91, 053608 (2015). Part 2. Superflow decay of Bose-Einstein condensate in the ring