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Can One Hear the Shape of a Drum?

Can One Hear the Shape of a Drum?. 梁兆新. 中科院金属所 (IMR, CAS). 2012.08.17 金华. Acknowledgement. Collaborator Dr. Ying Hu International Center for Quantum Materials, Peking University Great Thanks Prof. Biao Wu

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Can One Hear the Shape of a Drum?

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  1. Can One Hear the Shape of a Drum? 梁兆新 中科院金属所 (IMR, CAS) 2012.08.17 金华

  2. Acknowledgement • Collaborator • Dr. Ying Hu • International Center for Quantum Materials, Peking University • Great Thanks • Prof. Biao Wu • International Center for Quantum Materials, Peking University • References: • Y. Hu and Z. X. Liang, Phys. Rev. Lett. 107, 110401 (2011) • Y. Hu and Z. X. Liang, Mod. Phys. Lett. B (Invited review paper)

  3. Let’s Start with an Opening Question……. Can one hear the shape of a drum? • The frequencies at which a drumhead can vibrate depend on its shape. • If we know the shape, the  Helmholtz equation tells us the frequencies, which are the  eigen-values  of the Laplacian in the region. Question: can they tell us the shape of the drum if we know the frequencies?

  4. Hearing the Shape of a Drum M. Kac, Am. Math. Monthly 73, 1 (1966); O. Giraud and K. Thas, Rev. Mod. Phys. 82, 2213 (2010).

  5. Analogous Questions in Ultra-cold Gases • Important effects of dimensionality; • Experimentally, dimension is control using traps; • - Quasi-2D is realized; • - Yet, quasi-2D is different with pure 2D; • How to visualize dimensional crossovers in collective frequencies?

  6. Outline • Important effects of dimensionality • Dimension control using traps: Hierarchical 3D-2D crossovers • - Crossover in kinematics (Qausi-2D regime) • - Crossover in two-body scattering: 3D scattering to 2D scattering • How to visualize dimensional crossovers in collective excitations? • - Scale invariance, PR symmetry and universal breathing mode; • - Quantum anomaly and shifted breathing frequency; • - Manifestation of dimensional crossovers in frequency shift; • - Experimental realization. • Conclusions

  7. Importance of Dimensionality L 电阻 是电阻率 L L Scale-Independent！（尺度不变） L • 电阻 Geometry-Dependent！（几何相关） L • 霍尔电阻 W Scale AND Geometry Independent！（尺度和几何无关） L • 霍尔电导 Topological-Dependent（拓扑相关）

  8. Hierarchical 3D-2D crossovers Experimentally, Quasi-2D ultra-cold atomic systems are created using tight traps

  9. Homogeneous Bose Gas

  10. 3D Bose Gases in an Optical Lattice PRL84_2551

  11. Beyond-mean-field Ground State Energy K. Z. Zhou, Y. Hu, Z. X. Liang, Phys. Rev. A 82, 043609 (2010)

  12. Hierarchical 3D-2D Dimensional Crossover

  13. Trapped Quantum Gases:3D-2D Hierarchy (Summary) Bose gas in a 1D trap Crossover in kinematics: quasi-2D regime Crossover in two-body scattering: 3D scattering to 2D scattering

  14. Arising questions?

  15. 2D: Scale Invariance, PR Symmetry, Universal Breathing mode

  16. 2D: Scale Invariance, PR Symmetry, Universal Breathing mode

  17. 2D: Scale Invariance, PR Symmetry, Universal Breathing mode

  18. However….. • Previously, a universal breathing frequency is dictated • by PR symmetry; • However, quantum effects can break the PR symmetry • and shift the breathing frequency !

  19. Broken PR Symmetry in Pure 2D Quantum Gases: Mechanism (I) The scale invariance is broken and quantum anomaly arises for a pure 2D Bose gas even in the case of constant interaction coupling !!!! L. P. Pitaevskii and A. Rosch, Phys. Rev. A 55, R853 (1997)

  20. Broken PR Symmetry in Pure 2D Quantum Gases: Mechanism (I) Mechanism one: Quantum anomaly arises directly from quantization

  21. Broken PR Symmetry in Quasi-2D Quantum Gases: Mechanism (II) Mechanism two: Scale invariance is violated by quantum fluctuations (Y. Hu and Z. Liang PRL107_110401)

  22. Visualize Shifted Breathing Frequencies Due To 2D Effects in Scattering

  23. Shifted Breathing Frequency: Two Regimes

  24. Revisit Related Theoretical and Experimental Work (I)

  25. Revisit Related Theoretical and Experimental Work (I) Our understanding

  26. Revisit Related Theoretical and Experimental Work (II) Our understanding

  27. Revisit Related Theoretical and Experimental Work (III) Experiments in Fermi gas needs more analysis!!!

  28. Experimental Considerations: Finite Size Effect 1. Sum rule approach to such frequency shift

  29. Experimental Considerations: Nonlinear Effect 2. Frequency shift due to effect of nonlinearity

  30. Experimental Considerations: Vortex 3. The issue of vortex is closely related to operational anisotropy in excitation schemes

  31. Summary: One Can Hear the Shape of a Drum • Experimentally, hierarchical 3D-2D dimensional crossover • Crossover in kinematics (Qausi-2D regime) • Crossover in two-body scattering: 3D scattering to 2D scattering • Visualize such dimensional effects in collective breathing frequencies

  32. www.imr.cas.cn 谢谢!

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