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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