Simulation and Detection of Relativistic Effects with Ultra-Cold Atoms

1. Simulation and Detection of Relativistic Effects with Ultra-Cold Atoms Shi-Liang Zhu (朱诗亮) slzhu@scnu.edu.cn School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou, China The 3rd International Workshop on Solid-State Quantum Computing & the Hong Kong Forum on Quantum Control 12 - 14 December, 2009

2. Collaborators: Lu-Ming Duan (Michigan Univ.) Z. D. Wang (HKU) Bai-Geng Wang (Nanjing Univ.) Dan-Wei Zhang (South China Normal Univ.) • References: • Delocalization of relativistic Dirac particles in disordered one-dimensional systems and • its implementation with cold atoms. • S.L.Zhu, D.W.Zhang, and Z.D.Wang, Phys.Rev.Lett.102,210403 (2009). • 2) Simulation and Detection of Dirac Fermions with Cold Atoms in an Optical Lattice • S.L.Zhu, B.G.Wang, and L.M.Duan, Phys. Rev. Lett. 98, 260402 (2007)

3. Outline • Introduction: two typical relativistic effects: Klein tunneling and Zitterbewegung • Two approaches to realize Dirac Hamiltonian with tunable parameters Honeycomb lattice and Non- Abelian gauge fields • Observation of relativistic effects with ultra-cold atoms

4. 一、Introduction: quantum Tunneling Rectangular potential barrier V(x) a T Transmission coefficient T a

5. V(x) V E x 0 一、Introduction: Klein Paradox Klein paradox (1929) Dirac eq. in one dimension

6. V(x) V E a 0 Klein tunneling 1 Transmission coefficient Quantum tunneling 0 a Klein tunneling Quantized energies of antiparticle states • Scattering off a square potential barrier V>E • Totally reflection (classical) • Quantum tunneling (non-relativistic QM) • Klein tunneling (relativistic QM) x

7. Challenges in observation of klein tunneling In the past eighty years, Klein tunneling has never been directly observed for elementary particles. E Rest energy Compton length It is not feasible to create such a barrier for free electrons due to the enormous electric fields required. Overcome: Masseless particles or particles with ultra-slow speed

8. Klein paradox in Graphene M.I.Katsnelson et al., Nature Phys.2,620 (2006) A.F.Young and P. Kim, Nature Phys. Phys.(2009) N.Stander et al., PRL102,026807 (2009)

9. Klein tunneling in graphene • Theory: • Experimental evidences: Graphene hetero-junction: Phys. Rev.Lett. 102, 026807 (2009). Nature Phys. 2, 222 (2009) Nature Phys. 2, 620 (2006). • disadvantages: • Disorder, hard to realize full ballistic transport • Massive cases can’t be directly tested • 2D system, hard to distinguish perfect from near-perfect transmission Phys. Rev. B 74, 041403(R) (2006). The transmission probability crucially depends on the incident angle

10. 一、Introduction: Zitterbewegung effect The trajectory of a free particle Newton Particles Non-relativistic quantum particles Zitterbegwegung (trembling motion) Schrodinger (1930) （free electron） The order of the Compton wavelength

11. Dirac-Like Equation with tunable parameters in Cold Atoms Implementation of a Dirac-like equation by using ultra-cold atoms where can be well controllable

12. 三、Realization of Dirac equation with cold atoms • honeycomb lattice • NonAbelian gauge field Interesting results: the parameters in the effective Hamiltonian are tunable masse less and massive Dirac particles

13. Simulation and detection of Relativistic Dirac fermions in an optical honeycomb lattice S. L. Zhu, B. G. Wang, and L. M. Duan, Phys. Rev. Lett.98,260402 (2007)

14. Single-component fermionic atoms in the honeycomb lattice

15. Roughly one atom per unit cite and in the low-energy Massless: Massive: The Dirac Eq.

16. The method of Detection (1) : Density profile Local density approximation The local density profile n(r) is uniquely determined by n(m)

18. The method of Detection (2) : The Bragg spectroscopy Linear quadratic Atomic transition rate ~~~ dynamic structure factor

19. 三、Dirac-like equation with Non-Abelian gauge field x In the k space, G. Juzeliunas et al, PRA (2008); S.L.Zhu,D.W.Zhang and Z.D.Wang,PRL102, 210403 (2009).

20. or If and in one-dimensional case For Rubidium 87 The effective massis Tripod-level configuration of x

21. Tunneling with a Gaussian potential

22. Anderson localization in disordered 1D chains Scaling theory monotonic nonsingular function For non-relativistic particles: All states are localized for arbitrary weak random disorders

23. Two results: (1) a localized state for a massive particle (2) However, for a massless particle for a massless particle, all states are delocalized break down the famous conclusion that the particles are always localized for any weak disorder in 1D disordered systems. S.L.Zhu,D.W.Zhang and Z.D.Wang,PRL102, 210403 (2009).

24. The chiral symmetry The chiral operator The chirality is conserved for a massless particle. Note that

25. must be zero for a massless particle

26. Detection of Anderson Localization Nonrelativistic case: non-interacting Bose–Einstein condensate Billy et al., Nature 453, 891 (2008) BEC of Rubidium 87 Relativistic case: three more laser beams

27. Observation of Zitterbewegung with cold atoms J.Y.Vaishnav and C.W.Clark, PRL100,153002 (2008)

28. Summary (1) Two approaches to realize Dirac Hamiltonian where can be well controllable (2)

29. The end Thank you for your attention