梁兆新 Can One Hear the Shape of a Drum? 中科院金属所 (IMR, CAS) 2012.08.17 金华.
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Transcript of 梁兆新 Can One Hear the Shape of a Drum? 中科院金属所 (IMR, CAS) 2012.08.17 金华.
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梁兆新
Can One Hear the Shape of a Drum?
中科院金属所 (IMR, CAS)
2012.08.17 金华
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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)
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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?Can one hear the shape of a drum?
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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).
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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?
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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
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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
WGeometry-Dependent !(几何相
关) 电阻
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Hierarchical 3D-2D crossovers
Experimentally, Quasi-2D ultra-cold atomic systems are created using tight traps
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Homogeneous Bose Gas
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3D Bose Gases in an Optical Lattice
PRL84_2551
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Beyond-mean-field Ground State Energy
K. Z. Zhou, Y. Hu, Z. X. Liang, Phys. Rev. A 82, 043609 (2010)
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Hierarchical 3D-2D Dimensional Crossover
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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
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Arising questions?
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2D: Scale Invariance, PR Symmetry, Universal Breathing mode
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2D: Scale Invariance, PR Symmetry, Universal Breathing mode
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2D: Scale Invariance, PR Symmetry, Universal Breathing mode
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However…..
• Previously, a universal breathing frequency is dictated by PR symmetry;
• However, quantum effects can break the PR symmetry and shift the breathing frequency !
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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)
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Broken PR Symmetry in Pure 2D Quantum Gases: Mechanism (I)
Mechanism one: Quantum anomaly arises directly from quantization
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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)
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Visualize Shifted Breathing Frequencies Due To 2D Effects in Scattering
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Shifted Breathing Frequency: Two Regimes
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Revisit Related Theoretical and Experimental Work (I)
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Revisit Related Theoretical and Experimental Work (I)
Our understanding
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Revisit Related Theoretical and Experimental Work (II)
Our understanding
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Experiments in Fermi gas needs more analysis!!!
Revisit Related Theoretical and Experimental Work (III)
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Experimental Considerations: Finite Size Effect
1. Sum rule approach to such frequency shift
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Experimental Considerations: Nonlinear Effect
2. Frequency shift due to effect of nonlinearity
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Experimental Considerations: Vortex
3. The issue of vortex is closely related to operational anisotropy in excitation schemes
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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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