1. JQI Post-Docs: Deadline Dec 2007
Pairing and Structure in Trapped Atomic Systems Carl J. Williams Chief, Atomic Physics Division Co-Director, Joint Quantum Institute National Institute of Standards & Technology
20078 NIST NRC Post-Docs
see:
2008 Salary: $60,000 Deadline: Feb 1, 2008
JQI Post-Docs: Deadline Dec 2007
See:

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. 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
States
Degeneracy
Energy/E 1 2
N/2 N

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. Mott-Insulator Experiment
M. Greiner, Mandel, Esslinger, Hänsch, Bloch, Nature 415, 39 (2002)
NIST 2D Mott Transition: I. Spielman, Phillips, Porto, PRL 98, (2007)

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

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

9. ground vibrational level
Results
Clear swap oscillations!
mf = -1
mf = 0
1st vibrational level
ground vibrational level

10. Exchange Oscillations
1.0
0.8
0.6
0.4
0.2
0.0
Population in "1X"
1.5
0.5
MF=0
MF=-1
1.5
1.0
0.5
0.0
150
100 50
Position (pxl)
Time (ms)
1.0
0.8
0.6
0.4
0.2
0.0
Population in "0"
1.5
0.5
Time (ms)
MF=0
MF=-1
Measured Ueg: 3.0 kHz (agrees with other measurements/calculations)
280ms swap time,
700ms total manipulation time
Coherence preserved for >4 ms –dominated by inhomogeneity
I think the main limitation is the initial state. (ie. Holes) We now believe that this is dominated by a slight imbalance in the loading procedure, with high temperature also adding to the holes. The loading seems very, very sensitive to the imbalance, and by improving it we certainly got better performance on the “2U” data. We have not gone back to the swap gate, though.
NIST Exchange Oscillations: P. Lee et al, Nature (2007)

11. 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?

12. 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, (2002)
Mott Insulator
Superfluid
Commensurate
filling of sites
Off diagonal
long range order
Laser Parameters
See: G. Pupillo, Tiesinga, Williams, Phys. Rev. A 68, (2003) – Inhomogeneity
A.M. Rey, Pupillo, Clark, Williams, Phys. Rev. A 72, (2005) – Closed Form
G. Pupillo, Williams, Prokof'ev, Phys. Rev. A 73, (2006) – Finite T
G. Pupillo, Rey, Williams, Clark, New J. Phys. 8, 161 (2006) – Extended Fermionization

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

14. 1D Example of Trap and Lattice

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

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

17. The Phase Diagram
This is the celebrated phase structure of this system. This is at T=0 for 2d optical lattice. We can get a similar phase diagram in 1 and 3D. However this is the example we will consider for rest of the talk unless otherwise stated. The x axis is tunneling t/U and the y axis the chemical potential mu/U. As you can see it has this lobe structure, Inside these lobes we have mott insulator with n atom per site, n being 1, 2, 3 and so on. Outside we have the superfluid region which is typically for large t. 17 17

18. Bose Hubbard Model Set . Boundaries given by:
One feature of this phase diagram which is very easy to see and also very important is the case when t=0. Then the Hamiltonian is exactly diagonal in number basis. The system is in the Mott phase where the onsite energy is given by this where n is the number of atoms per site. n is determined by the chemcal potentlal. The boundary between two mott phases is given by mu=n.

19. Phase Diagram for t=0
To see that let us go back to the phase diagram and look at the case where t=0. Then the local chemical potential varies all the way from mu_r = mu at the center to mu_r = 0 at the edge. This is shown by this red line here. Thus we see that the system should exhibit a mott inulating shell strucuture where 19 19

20. Physics of Mott Shells and Rings
Boundaries at:
n=1

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

22. 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
n=3
n=2
n=1

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

24. Phase diagram for t≠0
So what happens if we slowly increase t.. As you can see in this diagram, the red line shifts towards right and a superfluid region emerges in between the mott insulating region. 24 24

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

26. What happens as we increase t?
This is shown in this slide. As you can see superfluid regions emerge in between the Mott regions at the boundary mu_r = nU. Here the number density of atomos drops from n to n-1. So this is the icing on the cake I was talking about. 26 26

27. 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
Now, examine the Mott region with integer Boson filling n and 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)

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

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

30. Properties of Superfluid Shells/Rings

31. 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.
The superfluidity is different from the usual superfluid described by a weakly interacting bose gas. this is due to the fact that …………………..for eg. this case is different from the superfluid we will get in a ring or shell with weakly interacting bose gas and hard boundary. The boundary conditions are active. 31 31

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

33. Mott Excitation Spectra (t=0)
Ea/U
quasi-particle
quasi-hole

34. Sound velocity in superfluid region
Excitations (t≠0)
c(r)/a
Sound velocity in superfluid region
Like a medium of continuous refractive index
n=3
n=2
n=1
Ea/U
Here I show the excitation in the Mott insulating region. The dashed line corresponds to energy requires to remove a particle or create a hole where as the solid curve shows the energy shows the energy required to add a particle. The red curve is a number conserving exciation of creating an electron hole pair which is the Mott gap. As one can see the Mott gap depends on the mott insulating shell we are in and is lowest for n=3, then n=2 and then n=1. This can also be seen from the phase diagram where the mott gap is given by these red lines for a given value of t. It is smallest for n=3 and largest for n=1. In the superfluid regon we can have sound modes. Here I show the sound velocity for the outermost shell which depends on the order parameter and is also spatially varying. This can be thought of sound propagating in a wave guide with a R.I
Quasiparticle-quasihole excitations in Mott regions
quasi-particle
quasi-hole 34 34

35. 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

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

37. Quantum Statistics: Bosons vs. Fermions

38. Pairing Hamiltonian for Fermions

39. 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

40. Non-interacting Trap Mixtures

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

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

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

44. Interacting Trap Mixtures: w=2w

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

46. Evolution of Structure

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

48. Bosons and Fermions in Traps have Rich Structure
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

49. Contributors 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