qubit.nist jqi.umd

1. 20078 NIST NRC Post-Docs • see: http://physics.nist.gov/ResOpp/index.html • 2008 Salary: $60,000 Deadline: Feb 1, 2008 JQI Post-Docs: Deadline Dec 2007 See: www.jqi.umd.edu Pairing and Structure inTrapped Atomic SystemsCarl J. Williams Chief, Atomic Physics DivisionCo-Director, Joint Quantum InstituteNational Institute of Standards & Technology http://qubit.nist.gov http://www.jqi.umd.edu

2. Outline • Perspective on QI, QPT, Many-Body problem • Neutral Atoms and the Bose Hubbard Model • NIST approach to Neutral Atom Quantum Computing • What’s been done and what problems remain • Vortices, Antivortices, and Superfluid Shells Separating Mott-Insulating Regions • Two Component Fermi Gases in Imbalanced Traps • Pattern Formation in Light-Heavy Fermion Mixtures • Conclusions

3. N Energy/E N/2 2 1 0 States Degeneracy QI, QPT, and the Many-Body Problem Do QPT’s place a limit on a general purpose QC? • Assume N qubits in M wells where N=M • Let Number States w

4. QI, QPT, and the Many-Body Problem • Does the density of states cause a problem for my QC? • Why not? • What are the consequences? • Finally, what are the signals of exotic many-body states in a quantum simulation? • How do I know they are unique or robust? • Does temperature play a role again?

5. NIST Neutral Atom QC Scheme • Laser cool and capture atoms in a Magneto-Optical Trap (MOT) • Evaporatively cool atoms to form a Bose-Einstein Condensate • Coherently load a Bose-Einstein condensate into a 3D-optical lattice - one atom per lattice site(Jaksch, PRL 81, 3108 (1998); Greiner et al., Nature 415, 39 (2002)) • Entangle adjacent atoms via controlled “coherent” collisions • Demonstrate 1- and 2-qubit operations on demand • Demonstrate quantum error correction B. Brown, P. Lee, N. Lundblad, J. Obrecht, T. Porto, I. Spielman, W. Phillips

6. M. Greiner, Mandel, Esslinger, Hänsch, Bloch,Nature415, 39 (2002) NIST 2D Mott Transition: I. Spielman, Phillips, Porto, PRL 98, 080404 (2007) Mott-Insulator Experiment

7. f ‘l’ ‘l/2’ + = Double Well Lattices • Basic idea: • Combine two different period lattices with adjustable • Intensities • positions • Mott insulator  single atom/site

8. Exchange Gate: projection B A triplet projection exchange split by energy U singlet

9. Summary of Neutral Atom QC • Massive Register Initialization has been shown • Yes, but how good? What are the effects of finite T? • Are Fermions better than Bosons? • Double well lattice allows control and manipulation of atoms in every other site • One- and two-qubit gates have been shown So what is the next logical extension? Sufficient control now exists to use the lattice parallelism to simulate iconic condensed matter Hamiltonians  “a quantum analog simulation” But how do you read out the answer?

10. Mott Insulator Superfluid Commensurate filling of sites Off diagonal long range order Laser Parameters Mott Insulator Transition D. Jaksch, Bruder, Cirac, Gardiner, Zoller, Phys. Rev. Lett. 81, 3108(1998) D. Jaksch, Venturi, Cirac, Williams, Zoller, Phys. Rev. Lett. 89, 040402(2002) See: G. Pupillo, Tiesinga, Williams, Phys. Rev. A68, 063604 (2003) – Inhomogeneity A.M. Rey, Pupillo, Clark, Williams, Phys. Rev. A72, 033616 (2005)– Closed Form G. Pupillo, Williams, Prokof'ev, Phys. Rev. A73, 013408 (2006) – Finite T G. Pupillo, Rey, Williams, Clark, New J. Phys.8, 161 (2006) – Extended Fermionization

11. t U Bose Hubbard Hamiltonian in a Trap harmonic trap hopping interaction

12. 1D Example of Trap and Lattice

13. Occupancy in the Mott Region A.M. Rey, G. Pupillo, C.W. Clark, and CJW, Phys. Rev. A 72, 033616 (2005) G. Pupillo, A.M. Rey, CJW, and C.W. Clark, New J. Phys. 8, 161 (2006)

14. Vortices, Antivortices, and Superfluid Shells Separating Mott-Insulating Regions K. Mitra, C.J. Williams, and C.A.R. Sá de Melo, cond-mat0702156

15. The Phase Diagram 17

16. Set . Boundaries given by: Bose Hubbard Model

17. Phase Diagram for t=0 19

18. n=3 n=3 n=2 n=2 n=1 n=1 n=2 Boundaries at: R n=1 n=1 n=2 Physics of Mott Shells and Rings

19. For t=0, the Mott shell structure is revealed by fixing The local energy is then: Since: Energetics and Effective Hamiltonian Introduce local superfluid order parameter:

20. n=1 n=2 n=3 From Shell Structure to Superfluid (t≠0) For t=0, one can see from that the Mott shell filling fraction changes from n to n+1 at: occurring at

21. Target state but But the Mott State is not Perfect

22. Phase diagram for t≠0 24

23. f n y x 10 10 10 10 0 0 0 0 x/a x/a y/a y/a -10 -10 -10 -10 Superfluid Rings Gutzwiler Ansatz showing superfluid rings in a 2D-Mott Fig. 2: D. Jaksch, Bruder, Cirac, Gardiner, Zoller, Phys. Rev. Lett. 81, 3108(1998)

24. n=3 n=1 n=1 n=2 n=2 What happens as we increase t? n=3 n=2 n=2 n=1 n=1 26

25. Now, examine the Mott region with integer Boson filling nand n+1 and the superfluid shell that emerges between them. When U >> t only the states and contribute (others are smaller by at least the order ) For insight, make a continuum approximation of to second order in a(this assumes r >> a) From Shell Structure to Superfluid (t≠0) For t=0, one can see from that the Mott shell filling fraction changes from n to n+1 at: occurring at

26. Diagonalize: Hamiltonian in Continuum Limit z – coordination number (depends on dimn)

27. Zeroth Order Solution: Requiring this to be non-negative yields: Local Superfluid Order Parameter Eq. Minimize ground state with respect to

28. Properties of Superfluid Shells/Rings

29. A Different Kind of Superfluid • The interlayer shells (rings) emerge as a result of fluctuations due to finite hopping in a Mott insulator and describe superfluid regions amidst insulating Mott states. • In general the order parameter is different from (less than) the number density. The resulting ‘quantum depletion’ is due to atoms being in the Mott state. • The superfluid state is not described in general by the GP equation. 31

30. _ + superfluid Mott core Mott-Insulator Phase • Mott phase: t<<U commensurate filling N=M excitations: gapped ~U U robust! incommensurate filling N / M excitations: ~t Complements P. Zoller

31. n=2 n=3 n=1 Ea/U Mott Excitation Spectra (t=0) quasi-particle quasi-hole

32. n=1 n=2 c(r)/a n=3 Ea/U Sound velocity in superfluid region Like a medium of continuous refractive index Excitations (t≠0) Quasiparticle-quasihole excitations in Mott regions quasi-particle quasi-hole 34

33. Superfluid Excitations: KT Physics Spontaneous formation of vortex-antivortex pairs form indicating a Kosterlitz-Thouless (or BKT) transition In 3D: this creates a 2D superfluid shell (shown) In 2D: this creates a 2D superfluid ring

34. BCS TO BEC SUPERFLUIDITYIN TRAP-IMBALANCED MIXTURES M. Iskin and C.J. Williams, cond-mat07xxxx, PRA (accepted)

35. Quantum Statistics: Bosons vs. Fermions

36. Pairing Hamiltonian for Fermions

37. Evolution from BCS to BEC Superfluidity Goal: Examine evolution from BCS to BEC in imbalanced traps Conclusion: BCS to BEC in imbalance Fermi gases is not a crossover but a Quantum Phase Transition

38. Non-interacting Trap Mixtures

39. Non-interacting Trap Mixtures: w=2w

40. Non-interacting Trap Mixtures: w=2w

41. Weakly Interacting Trap Mixtures: w=2w

42. Interacting Trap Mixtures: w=2w

43. Strongly Interacting Trap Mixtures: w=2w

44. Evolution of Structure

45. Pattern Formation in Light-Heavy Fermion Mixtures K. Mitra, C.J. Williams, and C.A.R. Sá de Melo, in progress

46. Conclusions • Bosons and Fermions in Traps have Rich Structure • Manipulations of 2-componant systems, whether qubits or 2-species, remains interesting • Quantum Phase Transitions Occur • But how robust are they • What are the unique signatures • What are the low level excitations of the system • Do those signatures say anything about T • In a simulation can QPT’s be controlled • How do I know how robust a simulation

47. Students: Kaushik Mitra *** Anzi Hu Former Students: Guido Pupillo Ana-Marie Reyes Post-Docs: Menders Iskin *** Fred Strauch Collaborators: Carlos Sa de Melo James Freericks R. Lemanski M. Maska Permanent Staff - Expt: James (Trey) Porto Ian Spielman Bill Phillips Steve Rolston (Collaborator) Post-Docs - Expt: Marco Anderlini Ben Brown Patty Lee Nathan Lundblad John Obrecht Jennifer Strabley Contributors