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: 1512.02044, accepted by Science Venue: Zhejiang University Date: June 29, 2016 2016 Hangzhou Symposium on Degenerate Fermi Gases

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

Gas hold in a trap

Turn off the trap

Expansion

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

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?

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!

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

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

A short review of Efimov effect in few-body physics

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

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

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

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

Dynamic Efimovian expansion

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

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

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

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:

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

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

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

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

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

Dynamic V.S. Real Space Efimov effect

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

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

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:

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: 1302.2258). 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.

Experimental observations

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

Experimental Observation by Haibin Wu in East China Normal University

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

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

Thank you very much for your attention !