Transcript Document

Bose-Fermi mixtures in random
optical lattices:
From Fermi glass to
fermionic spin glass
and quantum percolation
Anna Sanpera.
University Hannover
Cozumel 2004
Theoretical Quantum Optics
Cold atoms and cold gases:
• Weakly interacting Bose and Fermi gases (solitons, vortices,
phase fluctuations, atom optics, quantum engineering)
• Dipolar Bose and Fermi gases
• Collective cooling, CW atom laser, quantum master equation
• Strongly correlated systems in AMO physics
Quantum Information:
•Quantification and classification of entanglement
•Quantum cryptography and communications
•Implementations in quantum optics
V. Ahufinger,
B. Damski,
L. Sanchez-Palencia,
A. Kantian,
A. Sanpera
M. Lewenstein
Atomic physics meets condensed
matter physics
or
Atomic physics beats condensed
matter physics ????
Outline
OUTlLINE
1 Bose-Fermi (BF) mixtures in optical lattices
2 Disorder and frustration in BF mixtures
Bose gas in an optical lattice
Idea: D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner and P. Zoller
Superfluid
By courtesy of M. Greiner, I. Bloch, O. Mandel, and T. Hänsch
Mott insulator
Before talking about disorder, let us define order:
an optical lattice with atoms loaded on it.
First band
Tunneling
On site interactions
Bose-Hubbard model
1
1
H  U  n (n  1)  J  b  b  h.c.    ni
2 i i i
2  ij  i j
Some facts about Fermi-Bose Mixtures
Fermi-Bose mixtures in optical lattices:
 Fermions and bosons on equal footing in a lattice:
 Atomic physics “beats” condensed matter physics!!!
 Novel quantum phases and novel kinds of pairing:
Fermion-boson pairing!!!
 Novel possibilites of control of the system
Some of the people working on the subject (theory):
 A. Albus, J. Eisert (Potsdam), F. Illuminati (Salerno), H.P. Büchler
(Innsbruck), G. Blatter (ETH), A.B. Kuklov, B.V. Svistunov
(Amherst/Kurchatov), M.Yu. Kagan, D.V. Efremov,
A.V. Klaptsov (Kapitza), M.-A. Cazalilla (Donostia),
A.F. Ho (Birmingham)
Quantum phases of the Bose-Fermi
Hubbard model
Phase III–
I – Mott(2)+
II–
Fermion-hole
Fermion-2
Fermi
holes
pairing
sea
pairing
Description:
i)
i)
ii)
iii)
Bose-Fermi Hubbard model
Phase I – Mott (n) plus Fermi gas of fermions with NN interactions
Phase II – Interacting composite fermions (fermion + bosonic hole)
Phase III – Interacting composite fermions (fermion + 2 bosonic holes)
1
H   J b  b b  J f  f  f h.c.   U n (n  1)   V n mi   i ni
i j
i j
i i
i
2
i
 ij 
 ij 
i
i
Lewenstein et al. PRL (2003), Ferhman et al. Optics Express (2004)
Lattice gases: Bose-Fermi mixtures
• Low tunneling J<< Ubb,Ubf
• Effective Fermi-Hubbard Hamiltonian



H eff    J eff Ci C j  h.c.  K eff N i N j
ij 
Ubf/Ubb
IIAD
2
1
0
IIAS
IIRF
IRD
IRF
IIRD
IRD
IIRF
-1
IIRF IIAS
.
.
-2 0 IIRD
1
b/Ubb
Composite interactions
Different quantum phases
Attractive: Superfluid
fermionic Domains
Repulsive: Fermi liquid
Density modulations

2
Keff/Jeff
Ubb=1
Jb=Jf=0.02
b=10-7
f=5x10-7

0
1
-2
2
3
4
5
6
7
8
-4
Nf=40
Nb=60
-6
-8
-10
|f1f,0b|
2
-12
|f1f,0b|
2
-14
No composites
 = 0 2No composites
|f0f0b |2
0.8
|f0f1b |2
0.6
0.4
|f1f0b
0.2
2
2
|f1f,0b|
|f1f0b|
1.0
|2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
6
4
i
|f1f1b
6
8
4
|2
2
0
2
4
j
2
0
|f1f,0b|
|f1f0b |2
0.2
8
0
|f0f1b |2
Fermionic domain
Composites
Fermi liquid |f |2
1f,0b
i
6
8
0
2
4
8
6
0
j
2
4
6
i
8
0
2
4
8
6
j
2. DISORDER AND FRUSTRATION
IN ULTRACOLD ATOMIC
GASES
B. Damski, et al. Phys. Rev. Lett. 91, 080403 (2003)
A. Sanpera, et al.. cond-mat/0402375, Phys. Rev. Lett. 93, 040401 (2004)
V. Ahufinger et al. (a review of AMO disordered systems – work in progress)
What are spin glasses?
Spin glasses are disordered systems with competing
ferromagnetic ( )and antiferromagnetic ( )
interactions, which generates FRUSTRATION.
Frustration:
if we only have 2 possible spin orientations and the
interactions are random, no spin configuration can
simultaneously satisfy all couplings.
Ferromagentic (J=1)
Antiferromagnetic (J=-1)
?
J
i, j
i, j
 1
Spin Glasses
70‘s Edwards & Anderson: Essential physics of spin glasses lay not
in the details of their microscopic interaction but rather in the
competetion between quenched ferromagentic and
antiferromagentic interactions. It is enough to study:
H EA   J ij i j  h i
i, j
i
i- site of a d-dimensional lattice
 i  1 Ising classical spins
Jij  0, Jij  0
h
Independent Gaussian random variables with
zero mean and variance 1.
External magnetic Field
Spin glasses= quenched disorder + frustration.
Mean Field Theory: Sherrington-Kirkpatrick model 75
H SK 
J  
1i  j  N
ij
i
j
 h i
i
KT/J
PARAMAG.
FERROMAG
1
SPIN GLASS
0
1
J
Mean Field (infinite range) Sherrington-Kirkpatrick model:
Use replicas:
…….
n-replicas
Solution: Parisi 80‘s: Breaking the replica symmetry.
The spin glass phase is characterized by an infinite
number of pure states organized in an ultrametric
structure, and a phase transition occurring in a magnetic
field.
Order parameter= „overlap between replicas“
REAL SYSTEMS (short range interactions)
1.
How many pure states are in a spin glass at low temperature?
2. Which is the nature and complexity of the glassy phase?
3. Does exist a transition in a non zero magnetic external field?
Despite 30 years of effort on the subject
No consensus has been reached for real systems !
Alternative: Droplet model:
phase glass consist in two pure states related by global inversion of the
spins and no phase transition occurring in a magnetic field.
From Bose-Fermi mixtures
in optical lattices to spin glasses:
From disordered Bose-Fermi Hubbard Hamiltonian:
1
H   J b  b b  J f  f  f h.c.   U n (n  1)   V n mi   i ni
i j
i j
i i
i
2i
i
 ij 
 ij 
i
to spin glass Hamiltonian
H    J i , j i  h  i
j
i
 ij 
• Low tunneling J  Ubf ,Ubb
• Low Temperature
• DISORDER (chemical potential varies site to site)
• Effective Fermi-Hubbard Hamiltonian (second order
perturbation theory)




H eff    J ij Fi  Fj  h.c.  Kij M i M j   ~i M i
ij 
HOPPING of
COMPOSITES
i
INTERACTIONS
between
COMPOSITES
J=0, no tunneling of fermions or bosons
Depending on the disorder 2 types of lattice
sites:
A-sites
B-sites
i 
U
0
V
U
i   0
V
n=1,m=1
n=0,m=1
Disorder: Speckle radiation or supperlattices or…
Tunneling
On site interactions
Damski et al. PRL 2003
How to make a quantum SG with atomic lattice gas?
1. Use spinless fermions or bosons with strong repulsive interactions:




There can be Ni = 1 or = 0 atoms at a site!!
We can define Ising spins si = 2 Mi – 1.
What we need are:
RANDOM NEXT NEIGHBOUR INTERACTIONS,
HQSG = 1/4 Kij sisj + quantum tunneling terms + ...
Composites
Effective n.n. coulings in FB mixture in a random
optical lattice
U

V
Here ij = i - j
SPIN GLASS !
Physics of Fermi-Bose mixtures in random optical lattices
Regime of small disorder (weak randomness of on-site potential)
 With weak repulsive interactions we deal essentially with a
Fermi glass (i.e. an analog of Fermi liquid, but with Anderson
localized quasi-particle states)
 With attractive interactions we deal with the interplay of
superfluidity and disorder
 Both situations might occur simultaneously with
quantum site percolation (some sites might be „blocked“)
Regime of strong disorder
 Using the superlattices method we may make local potential to

fluctuate on n.n. sites strongly, being zero on the mean.
This leads to quantum fermionic spin glass
There is a possibility of novel metallic phases at the interplay
between disorder, hopping and n.n. interactions
SUMMARY OF Bose-Fermi Mixtures
Fermionic spin glasses in optical lattices:
 Spin glasses (SG) are spin systems with random
(disoredered) interactions: equally probable to be ferro- or
antiferromagnetic. The spin behaviour is dominated by
frustration!!! The nature of ordering in SG poses one the
most outstanding open questions of classical (sic!) and
quantum statistical mechanics.
 COLD ATOMIC BOSE-FERMI (BOSE-BOSE) MIXTURES
in optical lattices with disorder can be used to study in
“vivo” the nature of short range spin glasses. (real replicas)
Many novel phases related to composite fermions in
disorder lattices are expected! NEW & RICH PHYSICS
Transition from Fermi liquid to Fermi glass “in vivo”
y
y
nFj
nFj
x
x
Here composite fermions = a fermion + a bosonic hole
Question: Can AMO physics help?
1. Can cold atoms or ions be used to model complex systems?
YES!
• Bose gas in a disordered optical lattice: From Anderson to Bose glass
• Fermi-Bose mixtures in random lattices: From Fermi glass to
fermionic spin glass and quantum percolation
• Trapped ions with engineered interactions: Spin chains with
long range interactions and neural networks
• Atomic lattice gases in non-abelian gauge fields: From Hofstadter
butterfly to Osterloh cheese
2. Can cold atoms and ions be used as quantum simulators of
complex systems?
YES!
3. Can cold atoms and ions be used for quantum information
processing in complex systems?
YES?