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 OpticsTheoretical Quantum Opticsat the University of

Hannover

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 latticeIdea: D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner and P.

Zoller

By courtesy of M. Greiner, I. Bloch, O. Mandel, and T. Hänsch

Superfluid 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

inchij

jbibJ

in

in

iUH ..

2

1)1(

2

1

Some facts about Fermi-Bose Mixtures

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

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)

Fermi-Bose mixtures in optical lattices:

Some of the people working on the subject (theory):

Quantum phases of the Bose-Fermi Hubbard model

Description: i) Bose-Fermi Hubbard modeli) Phase I – Mott (n) plus Fermi gas of fermions with NN interactionsii) Phase II – Interacting composite fermions (fermion + bosonic hole)iii) Phase III – Interacting composite fermions (fermion + 2 bosonic holes)

Phase I – Mott(2)+ Fermi seaPhase II– Fermion-hole pairingPhase III– Fermion-2 holes pairing

Lewenstein et al. PRL (2003), Ferhman et al. Optics Express (2004)

ii

iifb nm

in

iV

inin

iUch

ijjf

ifJ

ijjbibJH

)1(2

1..

• Low tunneling J<< Ubb,Ubf

• Effective Fermi-Hubbard Hamiltonian

Lattice gases: Bose-Fermi mixtures

Ubf/Ubb

0

1

2

-1

-2

IIAD

IIAS

IIRF

IRF

IRD

IRDIIRD

IIAS

IIRF

IIRD

IIRF

0 1bUbb

.

.

Composite interactions Different quantum phases

Attractive: Superfluid fermionic Domains

Repulsive: Fermi liquid Density modulations

ij

jieffjieffeff NNKchCCJH ..

|f0f0b |2

|f0f1b |2

|f1f0b |2

|f1f1b |2

= 0 No composites

Ubb=1Jb=Jf=0.02b=10-7

f=5x10-7

Nf=40Nb=60

No composites

|f0f1b |2

|f1f0b |2

CompositesFermi liquid Fermionic domain

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)

?

1,

, ji

jiJ

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:

i

ijiji

ijEA hJH ,

i- site of a d-dimensional lattice

1i Ising classical spins

Independent Gaussian random variables with zero mean and variance 1.

0,0 ijij JJ

h External magnetic Field

Spin Glasses

Spin glasses= quenched disorder + frustration.

Mean Field Theory: Sherrington-Kirkpatrick model 75

PARAMAG.FERROMAG

SPIN GLASS

KT/J

1

10J

i

ijNji

iijSK hJH 1

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“

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 !

REAL SYSTEMS (short range interactions)

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:

ii

iifb nm

in

iV

inin

iUch

ijjf

ifJ

ijjbibJH

)1(2

1..

From disordered Bose-Fermi Hubbard Hamiltonian:

to spin glass Hamiltonian

i

iiji hij

jJH ,

• Low tunneling

• Low Temperature

• DISORDER (chemical potential varies site to site)

• Effective Fermi-Hubbard Hamiltonian (second order perturbation theory)

bbbf UUJ ,

HOPPING ofCOMPOSITES

INTERACTIONSbetweenCOMPOSITES

iii

ijjiijjiijeff MMMKchFFJH ~..

J=0, no tunneling of fermions or bosonsDepending on the disorder 2 types of latticesites:

0V

UiA-sites n=1,m=1

B-sites 0V

Ui n=0,m=1

Speckle radiation or supperlattices or…Disorder:

Tunneling

On site interactions

Damski et al. PRL 2003

How to make a quantum SG with atomic lattice gas?

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 + ...

1. Use spinless fermions or bosons with strong repulsive interactions:

Composites

Effective n.n. coulings in FB mixture in a random optical lattice

Here ij = i - j

V

U

SPIN GLASS !

Physics of Fermi-Bose mixtures in random optical lattices

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“)

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

Regime of small disorder (weak randomness of on-site potential)

Regime of strong disorder

SUMMARY OF Bose-Fermi Mixtures

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

Fermionic spin glasses in optical lattices:

nFj

y

x

nFj

y

x

Transition from Fermi liquid to Fermi glass “in vivo”

Here composite fermions = a fermion + a bosonic hole

Question: Can AMO physics help?

2. Can cold atoms and ions be used as quantum simulators of complex systems?

3. Can cold atoms and ions be used for quantum information processing in complex systems?

1. Can cold atoms or ions be used to model complex systems?

• 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

YES!

YES!

YES?