1 Simulation and Detection of Relativistic Effects with Ultra-Cold Atoms Shi-Liang Zhu ( 朱诗亮 ) 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 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: 1)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, (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, (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 V(x) a T a Rectangular potential barrier Transmission coefficient T

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

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

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

8 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, (2009) Klein paradox in Graphene

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

10 一、 Introduction: Zitterbewegung effect （ free electron ） Newton Particles Non-relativistic quantum particles The trajectory of a free particle Zitterbegwegung (trembling motion) Schrodinger (1930) 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, (2007)

14 Single-component fermionic atoms in the honeycomb lattice

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

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

17

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

19 三、 Dirac-like equation with Non-Abelian gauge field In the k space, x G. Juzeliunas et al, PRA (2008); S.L.Zhu,D.W.Zhang and Z.D.Wang, PRL 102, (2009).

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

21 Tunneling with a Gaussian potential

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

23 Two results: (1) a localized state for a massive particle (2) However, for a massless particle 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, PRL 102, (2009). for a massless particle, all states are delocalized

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, (2008)

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

29 The end