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