1. Efimovian Expansion in Scale Invariant Quantum Gases
Speaker: Ran Qi （齐燃）
Department of Physics, Renmin University of China
Collaborators: Zhe-Yu Shi, Hui Zhai, Haibin Wu et. al.
Ref number, arXiv: , accepted by Science
Venue: Zhejiang University
Date: June 29, 2016
2016 Hangzhou Symposium on Degenerate Fermi Gases

2. Outline What problem are we looking at?
Brief review of Efimov effect in few-body physics
Dynamic Efimovian expansion in isotropic and anisotropic harmonic trap
Experimental observations

3. Gas hold in a trap

4. Turn off the trap

5. Expansion

6. Turn-off the trap:
decreasing the trap frequency
Box: a harmonic trap
Scale Invariant Quantum Gas

7. Turn-off the trap:
decreasing the trap frequency
Box: a harmonic trap
Scale Invariant Quantum Gas
Harmonic length:
Naive expectation
By dimension analysis:
Is this true?

8. Turn-off the trap:
decreasing the trap frequency
Box: a harmonic trap
Scale Invariant Quantum Gas
Harmonic length:
Naive expectation
By dimension analysis:
Is this true? No!

9. Turn-off the trap:
decreasing the trap frequency
Box: a harmonic trap
Scale Invariant Quantum Gas

10. decreasing the trap frequency Box: a harmonic trap
Turn-off the trap:
decreasing the trap frequency
Box: a harmonic trap
Scale Invariant Quantum Gas
A natural guess: should has something to do with Efimov effect??
Universal Discrete Scaling Symmetry

11. A short review of Efimov effect in few-body physics

12. Continuous Scaling Symmetry
Unitary Quantum Gases
Finite
no scaling symmetry
continuous scaling symmetry

13. Discrete Scaling Symmetry
The Efimov Problem
Three bosons
1970
An extra length scale for the three-body short-range cut-off
Discrete Scaling Symmetry

14. The Efimov Problem
Infinite many eigen-states with geometric scaling symmetry
Many experiments have confirmed the Efimov effect

15. Interacting System Unitary Quantum Gases Finite no scaling symmetry
continuous scaling symmetry
Exist but trivial!

16. Dynamic Efimovian expansion

17. Continuous Scaling Symmetry
This scaling symmetry exists only if
Simplest non-trivial form!

18. Equation of motion (isotropic case)
Define: and we have:
is the generator of a spatial scaling transformation

19. Equation of motion (isotropic case)
Define: and we have:
is the generator of a spacial scaling transformation
Here we have used the fact: , which comes from scale invariance of

20. Equation of motion (isotropic case)
Define: and we have:
is the generator of a spacial scaling transformation
Here we have used the fact: , which comes from scale invariance of
At last, from Hellmann-Feynman theorem:
We finally get the equation of motion for :
Initial conditions:

21. Two different kinds of solutions
at t>>t0,
No discrete scaling symmetry!

22. Two different kinds of solutions
at t>>t0,
No discrete scaling symmetry!
, discrete scaling symmetry!

23. Two different kinds of solutions
at t>>t0,
No discrete scaling symmetry!
, discrete scaling symmetry!

24. Two different kinds of solutions
at t>>t0,
No discrete scaling symmetry!
, discrete scaling symmetry!

25. Two different kinds of solutions
at t>>t0,
No discrete scaling symmetry!
, discrete scaling symmetry!

26. Dynamic V.S. Real Space Efimov effect

27. Unitary Fermi gas in anisotropic expansion
Inspired from the isotropic solution, we perform a scaling transformation:

28. Unitary Fermi gas in anisotropic expansion
Inspired from the isotropic solution, we perform a scaling transformation:
The new wave function, after transformation satisfies a static Hamiltonian

29. Unitary Fermi gas in anisotropic expansion
Inspired from the isotropic solution, we perform a scaling transformation:
The new wave function, after transformation satisfies a static Hamiltonian
The initial state, however, is equilibrium state of another Hamiltonian:

30. Longitudinal expansion
We consider the experimental relevant case: where the trap has a tube shape. The problem is now converted to a breathing mode problem (arXiv: ).
We find that the expansion in the longitudinal direction is described by the same analytic formula as given in the isotropic case:
But now the scaling factor is given as for unitary Fermi gas.

31. Experimental observations

32. Experimental Observation
by Haibin Wu in East China Normal University
Curves corresponds to our analytic formula:
Expansion nearly stops at the plaque!

33. Experimental Observation
by Haibin Wu in East China Normal University

34. Experimental Observation
by Haibin Wu in East China Normal University
Unitary
Non-interacting
Universal Dynamics for ALL Scale Invariant Quantum Gases

35. Conclusion
We proposal an interesting setting and we predict its quantum dynamics exhibits discrete scaling symmetry, which has been observed experimentally
❒ Generalization of the Efimov effect to time domain
❒ Opens up new possibility to simulate real space Shcrodinger Eq in the time domain (more results will be given in Wuhan!)
❒ Dynamic detection of scaling symmetry
Theory: Zheyu Shi, Hui Zhai, and RQ
Experiment: Haibin Wu in East China Normal University

36. Thank you very much for your attention !