1. 梁兆新 Can One Hear the Shape of a Drum? 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……. 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? Can one hear the shape of a drum?

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

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

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. Our understanding

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

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

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 1.Crossover in kinematics (Qausi-2D regime) 2.Crossover in two-body scattering: 3D scattering to 2D scattering  Visualize such dimensional effects in collective breathing frequencies

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