Transcript Diapositiva 1 - Centro Fermi

Museo Storico della Fisica e Centro Studi e Ricerche
Enrico Fermi
Marco Fattori
Universal Quantum Simulator with cold atoms
in optical lattices
Conferenza e Mostra Centro Fermi 29-30 Novembre 2007, Roma
Universal Quantum Simulator
• Our understanding of phenaomena in nature is based on our ability of
1. conjecturing some Hamiltonian H
2. see if it is able to successfully describe the properties of the system under exam
• Simulating quantum systems on a classical computer can be very hard
• For example, let’s consider N spin ½
....
2N numbers are neededd to address the state of the system
N
.........
a 2N 2N matrix is needed to describe the evolution of the
system
.........
Solving this problem in general can be very hard!!
Universal Quantum Simulator
The rule of simulation that I would like to have is that the number of
computer elements required to simulate a large physical system is
only to be proportional to the space-time volume of the physical
system. I don’t want to have an explosion!!
R. P. Feynman
I would like to have an exact simulation, in other words, that the computer will
do “exactly” the same as nature
R. P. Feynman
Some Hamiltonians used to describe physical systems
•
Hubbard model
for fermions
for bosons
1. simplest model of interacting particles (electrons) in a lattice
2. generalization beyond the band theory description of solids
3. it incorporates the short range part of the Coulomb interaction,
avoiding the high complexity of the long range Coulomb force.
4. Used to describe electronic properties of solids with narrow band. It
predicts metal-Mott insulator transition in metal oxides and for 4He
in porus material.
5. Resolved in 1D, in higher dimensions is not known for most of the
space of parameters, coupling constants, electron concentration,
temperature, etc..
Some Hamiltonians used to describe physical systems
• t-J Model
1. Used to explain how magnetic order in antiferromagnetic
insulator, if doped, gives way to superconductivity
2. Describes the motion of holes in an antiferromagnet
3. Describes High Tc superconductor containing copper oxides
planes
• Anisotropic Heisenberg model (XXZ)
1. It is used to describes quantum magnetism in condensed
matter systems
Ultracold atoms in optical lattices
A periodic potential for an atomic system may be easily obtained from the
interference of two counterpropagating off-resonant laser beams
For sufficiently strong periodic potentials and low temperatures, the
atoms will be confined to the lowest Bloch band and will evolve exactly
according to the Hubbard model
Atoms interacting with a “crystal of light” are a candidate to realize a
UQS to simulate condensed matter systems.
The two Feynman statement are satisfied.
UQS with ultracold atoms in optical lattices
Designing the optical potential
tuning potential strength
time-dependent potentials
tuning lattice spacing
designing complex/disordered structures
UQS with ultracold atoms in optical lattices
Designing the optical potential
tuning potential strength
time-dependent potentials
tuning lattice spacing
designing complex/disordered structures
UQS with ultracold atoms in optical lattices
Designing the optical potential
tuning potential strength
time-dependent potentials
tuning lattice spacing
designing complex/disordered structures
UQS with ultracold atoms in optical lattices
Designing the optical potential
tuning potential strength
time-dependent potentials
tuning lattice spacing
designing complex/disordered structures
UQS with ultracold atoms in optical lattices
Designing the optical potential
tuning potential strength
time-dependent potentials
tuning lattice spacing
designing complex/disordered structures
UQS with ultracold atoms in optical lattices
• Degrees of freedom can be frozen arbitrarily, so physical situations with
reduced dimensionality are accessible
3D
1D
2D
• Boson or fermions can be used
• No crystal vibration, so long simulation times
• Interaction strength can be tuned, tuning the atomic scattering length
(Fano-Feshabch resonances)
K at Florence
41K
(Boson): Giovanni Modugno, et al., Bose-Einstein condensation
of potassium atoms by sympathetic cooling, Science, 294, 1320
(2001)
40K
(Fermion): Giovanni Modugno, et al. Collapse of a Degenerate
Fermi Gas, Science, Vol 297, pp 2240 (2002)
Density distribution
…only recently we have achieved BEC of 39K in the |F=1, mF=1> state
3 s
-200
3.5 s
3.2 s
0
200
-200
0
200
Horizontal position (m)
time
G. Roati, et al. 39K Bose Einstein Condensate with tunable interaction
Phys. Rev. Lett. 99, 010403 (2007)
-200
0
200
Tuning of the interaction in 39K via Fano-Feshbach resonances
• At first glance 39K has unfavorable collisional properties: negative scattering length (aKK=-33 a0)
• Fano-Feshbach resonance
VS’(R)
VS(R)
Ms’ ≠ Ms + B field
Vc
Ec
• The BEC can be stabilized against collapse
•The interaction energy U in the Hubbard model can be tuned at will
Feshbach spectroscopy on 39K
Resonance in the F=1 manifold
K
(mFa, mFb)
Bth
Bexp (G)
(1,1)
25.9
25.85 (10)
C. D’Errico, et al, New Journal of Physics 9 223 (2007)
mF=1
(1,1)
402.4
403.4 (7)
(1,1)
752.4
752.3(1)
(0,0)
58.8
59.3 (6)
(0,0)
65.6
66.0 (9)
(0,0)
471
(0,0)
490
(0,0)
832
(c,c)
33.6
32.6 (1.5)
(c,c)
162.3
162.8 (9)
(c,c)
560.7
562.2 (1.5)
Broad Feshbach resonance D=50 G
Huge degree of tunability
Knowledge of the scattering length vs B for all the three sublevels mF=-1,0,1
Bloch oscillations with a tunable 39K BEC
• We can use a known quantum model to test our ability to control the
scattering length
• Quantum transport in a periodic potential in the presence of an external force
Bloch Oscillations
z
z
pz
After a time T
of Bloch
Oscillations
Interference contrast is destroyed by the interaction
z
pz
Bloch oscillations with tunable 39K BEC
Lattice parameters: l = 1032 nm, slattice= 6, nradial=40 Hz
100 a0 vs 1 a0
100 a0
1 a0
0.4 ms
0.8 ms
1.2 ms
1.6 ms
2 ms
2.4 ms
2.8 ms
900
900
800
800
700
700
600
600
500
T =4 ms
400
300
Atomic density (a.u.)
T =0 ms
Atomic density (a.u.)
0 ms
3.2 ms
3.6 ms
500
400
300
200
200
100
100
-100
-50
0
z axis (px)
50
100
-100
-50
0
z axis (px)
50
100
4 ms
Bloch oscillations with tunable 39K BEC
Peak width vs time
1,0
Peak Width (2hk)
0,8
0,6
0,4
29 ao
13 ao
6 ao
2.5 ao
1.3 ao
0,2
0,0
-20
0
20
40
60
80
100
120
140
160
Time (ms)
Rate of decoherence vs scatteing length
40
D. Witthaut et al., Phys. Rev. E 71, 036625
(2004)
Decoherence rate (
-1
D p/p bragg s )
35
30
25
Good agreement exp vs theory
20
15
M. Fattori et al. submitted to PRL
arXiv:0710.5131v1
10
5
0
0
5
10
15
20
scattering length(a )
25
30
Bloch oscillations with tunable 39K BEC
• Minimum of the decoherence coincide with the zero crossing predicted by our
Feshbach spectroscopy analysis.
• We have an optimum control of the
scattering length
•We are actually performing trapped
atom interferometry with an almost
ideal gas
• We address lattice noise induced
decoherence
• We can simulate electron dynamics in periodic lattice. Note that B. O.
have been nevere observed in a natural crystal because the scattering
time is much shorter than the Bloch period. Only observed in
semiconductor superlattices where tbloch is 600 fs.
Summary
• Why a UQS could be a powerfull device for the study of condensed matter systems
• Atoms in optical lattices are optimum candidate for realizing a UQS
• BEC of 39K, Feshbach spectroscopy and Bloch oscillations with a 39K BEC
Outlook
• Study Bloch oscillations in disordered potentials
• Study the effect of the interaction on Anderson localization of an atomic wave in a
disordered potential
• 3D lattice with a more stable laser
• Single atom addressing
• Production of cold molecules in the ground state to add long range interaction
Interferometric scheme: Bloch Oscillations
___________________________________________________________________
• External force F (in our case gravity)
• The condensate is loaded in an array of potential wells
y
p2
4 2 a
2
i
(
 Vo Cos ( kz)  Fz 
y
t
2m
m
2
)y
• The condensate y is put in a coherent superposition of Wannier Stark fi states
parametrized with the lattice site index.
z
pz
z
y   cifi   i e fi
ii
i
i
After a time T
of Bloch
Oscillations
with interaction
z
pz
 i  const
 (mg
i / 2li /g2)T
)T / h
 (lmg
i 
i  /i h
i
Interference contrast is destroyed by the interaction.