1. CLARENDON LABORATORY PHYSICS DEPARTMENT UNIVERSITY OF OXFORD and CENTRE FOR QUANTUM TECHNOLOGIES NATIONAL UNIVERSITY OF SINGAPORE Quantum Simulation Dieter Jaksch

2. Outline  Lecture 1: Introduction  What defines a quantum simulator? Quantum simulator criteria. Strongly correlated quantum systems.  Lecture 2: Optical lattices  Bose-Einstein condensation, adiabatic loading of an optical lattice. Hamiltonian  Lecture 3: Quantum simulation with ultracold atoms  Analogue simulation: Bose-Hubbard model and artificial gauge fields. Digital simulation: using cold collisions or Rydberg atoms.  Lecture 4: Tensor Network Theory (TNT)  Tensors and contractions, matrix product states, entanglement properties  Lecture 5: TNT applications  TNT algorithms, variational optimization and time evolution

3. Remarks  Lattice systems/crystals: strong correlations can be achieved at any density by quenching the kinetic energy  Continuum systems/gases: for finite range interactions the system will become weakly interacting if mean separation between particles much larger than range of the interaction  After one more discussion with Prof J. Walraven  The continuum argument only holds in 3D (three spatial dimensions)  1D Tonks Girardeau gas is an example of a system with increasingly strong correlations as the density is reduced  In 2D the situation depends on the details of the interaction

4. THE BOSE HUBBARD MODEL Analogue quantum simulation

5. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. The Bose-Hubbard Hamiltonian Occupy lowest band only Substitute into Hamiltonian With parameters

6. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Dominant contributions

7. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Hopping and interaction terms g g

8. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Tight binding Hubbard model

9. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Changing the lattice potential U 4J U D.J., C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998). M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

10. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford.

13. Gutzwiller ansatz

14. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. 0 1 2 3 n

15. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. 0 1 2 3 n > 0 Mott phase = 0 critical point minimum

16. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. The Mott insulator– loading from a BEC Theory: D.J, et al. 1998, Experiment: M. Greiner, et al., 2002 Mott n=1 n=2 n=3 superfluid  /U quantum freezing super fluid Mott melting

17. ARTIFICIAL GAUGE FIELDS Analogue quantum simulation

18. Introduction: Huge magnetic fields  Effect of a magnetic field  The wave function accumulates a phase characterized by  when hopping around a plaquette.  Phase proportional to enclosed magnetic flux  Resulting energy spectrum  c =1/2  c =1/3  JJ

19. Ultracold atoms in rotating lattices  Effective magnetic field via rotation  N.K. Wilkin et al. PRL 1998  B. Paredes et al. PRL 2001  Experiment: J. Dalibard, ENS  Experiment: C. Foot, Oxford  Alternative ways for realizing artificial magnetic fields, e.g.  A.S. Sorensen et al. PRL 2005  G. Juzeliunas et al. PRL 2004  E.J. Mueller, PRA 2004  D.J et al. New J. Phys. 2003

20. Magnetic field vs rotating system  Hamiltonian of the form with vector potential for a magnetic field along the z-direction leading to terms plus potential terms.  Compare with system of neutral particles rotating around z-axis with angular momentum operator  In both cases a force orthogonal to the direction of motion acts on the particle.  Quantum mechanically this leads to an energy and thus phase difference when one and the same path is travelled in two different directions  broken symmetry.

21. Artificial magnetic field on a lattice  For a lattice geometry rotation or a magnetic field leads to the following properties  When hopping from one lattice site to the next a phase is acquired.  When a closed path is travelled the wave function should get a phase proportional to the surrounded area (i.e. the enclosed flux).  When discretizing the Hamiltonian a Peierl’s transformation can be used to bring the Hamiltonian into a form which obviously fulfils these properties (in Landau gauge)

22. Energy bands  Fractal energy bands Hofstaedter butterfly  Investigate magnetically induced effects  quantum Hall effect  fractional quantum Hall effect  Atomic systems allow detailed study of the energy bands  Interaction effects are controllable The optical lattice setup allows to explore exotic parameter regimes  novel effects?  c =1/2  c =1/3  JJ

23. Alternative methods?  Rotating the lattice creates centrifugal terms in the potential part of the Hamiltonian  These need to be precisely balanced by a trapping potential which is experimentally difficult  Use alternative methods to create an artificial magnetic field  Laser induced hopping along the x direction DJ et al., New J. Phys. 5, 56 (2003).  By immersing the lattice into a rotating BEC A. Klein and DJ, EuroPhys. Lett. 85, 13001 (2009). r

24. Laser induced magnetic field  Two component optical lattice  trapping two internal states in different columns  The polarization of the lasers determines the position of the lattice sites U … onsite interactions J … hopping rate  … trap potential  eg … atomic energy difference

25. Acceleration, laser induced hopping  Acceleration or inhomogeneous electric field yielding offset   Apply two Raman lasers with detunings  and Rabi frequency   which induces hopping along x direction  The phases  1,2 =  e iqy of the lasers determine the phase imprinted on the atoms Raman lasers

26. Resulting setup

27. Laser imprinted hopping phases Proposal: DJ et al., New J. Phys. 2003 M Aidelsburger et al., PRL. 2011 x y z K. Jimenez-Garca et al., PRL. 2012

28. M. Aidelsburger et al., arXiv:1407.4205 M. Aidelsburger et al., PRL 111, 185301 (2013) H. Miyake et al., PRL 111, 185302 (2013)

29. GATE OPERATIONS Digital quantum simulation

30. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Two-level atoms and their manipulation Single atom as a two level system Use hyperfine states e.g. 87 Rb Single qubit manipulation Focussing the laser to a single atom position is challenging qubit in long-lived internal states laser addressing qubits laser F=1 F=2

31. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Cold controlled collisionsRydberg atoms atoms V(R) DJ et al. PRL 82, 1975 (1999); DJ et al. PRL 85, 2208 (2000) Controlled interactions Exp: Grangier, Saffman, 2008/09Exp: Bloch, Greiner, 2002/03

32. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Engineering a Cluster-state ii+1

33. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Engineering a Cluster-state ii+1

34. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Engineering a Cluster-state ii+1

35. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Engineering a Cluster-state ii+1

36. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Engineering a Cluster-state ii+1

37. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Engineering a Cluster-state ii+1

38. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Dipole-dipole interactions Electric Field

39. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Rydberg atoms: Internal states 11 Atom 1 22 Atom 2

40. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. 11 11 Fast phase gate - excitation U  22  Laser pulse: 11 22 11

41. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Fast phase gate - blockade   Laser pulse: U 11 22 11 22 22 No excitation! „-“ 22

42. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Fast phase gate – de-excitation  22 Laserpuls: U 11 22 11 11 11 „-“ 

43. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Adiabatic gate – no addressing U detuned by large interaction

44. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Dressed states picture

45. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Optical addressing High resolution optical imaging systems Strathclyde: S Kuhr et al. Oxford: C Foot et al. Bonn: D Meschede et al. Harvard: Greiner et al. Munich: Bloch et al. Greiner Lab, Science 2010 Bloch Lab, Science 2011

46. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Single site addressing Scanning electron microscopy Mainz: H Ott et al.

47. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Quantum simulator criteria Quantum system Large number of degrees of freedom, lattice system or confined in space Initialization Prepare a known quantum state, pure or mixed, e.g. thermal Hamiltonian engineering Set of interactions with external fields or between different particles Interactions either local or of longer range Detection Perform measurement on the system, particles individually or collectively. Single shot which can be repeated several times. Verification Increase confidence about result, benchmark by running known limiting cases, run backward and forward, adjust time in adiabatic simulations. J. Ignacio Cirac and Peter Zoller, Nature Physics 8, 264 (2010)

48. DETAILS (IF TIME PERMITS ONLY) Quantum simulation

49. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. State selective potential Lin angle Lin laser configuration Electrical field

50. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Atomic Level Structure Alkali atoms Qubit states

51. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Stark shift Fine structure shift Hyperfine structure (Clebsch Gordon)

52. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Moving harmonic potentials Retain motional ground state

53. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Accumulated phases

54. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Evolution truth table

55. Centre for Quantum Physics & Technology, Clarendon Laboratory, University of Oxford. Gate fidelity Consider entanglement between motional and internal degrees of freedom as source of infidelity