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Ultracold Bosonic and FermionicQuantum Gases in Optical LatticesUlrich Schneider, Lucia Hackermüller,Sebastian Will, Thorsten Best, Simon Braun, Philipp Ronzheimer,KC Fong, Tim Rom
Stefan Trotzky, Yuao Chen, Jeff Thompson, Ute Schnorrberger, Simon Fölling, Fabrice Gerbier,Artur Widera
Stefan Kuhr, Jacob Sherson, Marc Cheneau, Christof Weitenberg, Manuel Endres, Rosa Glöckner, Ralf Labouvie
Theory: Belén Paredes, Mariona Moreno
I.B.
Johannes Gutenberg-Universität, MainzMax-Planck-Institut für Quantenoptik, GarchingLudwig-Maximilians Universität, München
funding by€ DFG, European Union,$ AFOSR, DARPA (OLE) www.quantum.physik.uni-mainz.de
Theory collaboration:Achim Rosch, Theo Costi, David Rasch, Rolf Helmes,
T. Kitagawa, E. Demler,
N. Prokof’ev, B. Svistunov, L.Pollet, M Troyer, F. Gerbier
Tuesday, June 30, 2009
Strongly Correlated Atoms in Optical Lattices
• Optical Lattices
Setup, Detection Techniques
• Strong Correlated Bosons
Bosonic Mott Insulators
Second Order Tunneling/Superexchange Interactions
• Strongly Correlated Fermions
Repulsive Interactions - Fermionic Mott
Attractive Interactions
Transport of Interacting Fermi Gases
• Bose-Fermi Mixtures
Multi Orbital Quantum Phase Diffusion
Bose-Fermi Mixtures in Optical Lattices
• Noise Correlations
• OutlookTuesday, June 30, 2009
Introduction
• Controlling Single Quantum Systems
• New challenges ahead: control, engineer and understand complex quantum system quantum computers, quantum simulators, novel (states of) quantum matter, advanced materials, multi-particle entanglement
R. P. Feynman‘s Vision
A Quantum Simulator to study the quantum dynamics
of another system.
Single Atoms and Ions Photons Quantum Dots
R.P. Feyman, Int. J. Theo. Phys. (1982)R.P. Feynman, Found. Phys (1986)
Tuesday, June 30, 2009
Our Starting Point – Ultracold Quantum Gases
Parameters: Densities: 1015 cm-3
Temperatures: Nano KelvinAtom Numbers 106
Bose-Einstein Condensates e.g. 87Rb
Degenerate Fermi Gasese.g. 40K
Ground States at T=0
Tuesday, June 30, 2009
Optical Lattice Potential – Perfect Artificial Crystals
λ/2= 425 nm
Laser Laser
optical standing wave
Periodic intensity pattern creates 1D,2D or 3D light crystals for atoms (Here shown for small polystyrol particles).
Perfect model systems for a fundamental understanding of quantum many body systems
Tuesday, June 30, 2009
1D, 2D & 3D Lattices
2D LatticesArray of one-dimensional quantum systems
3D LatticesArray of quantum dots
Tuesday, June 30, 2009
From Artificial Quantum Matter to Real Materials
e.g. High-Tc Superconductors (YBCO)
•Densities: 1024-1025/cm3
•Temperatures: mK – several hundred K
•Crystal Structures and Material Parameters given by Material(Tuning possible via e.g. external parameters like e.g. pressure, B-fields or via synthesis)
Real MaterialsUltracold Quantum Gases in Optical Lattices
•Densities: 1014/cm3
(100000 times thinner than air)
•Temperatures: few nK(100 millionen times lower than outer space)
•Crystal Structures and Material Parameters canbe changed dynamically and in-situ.
New tunable model systems for many body systems!
Tuesday, June 30, 2009
From Artificial Quantum Matter to Real Materials
e.g. High-Tc Superconductors (YBCO)
•Densities: 1024-1025/cm3
•Temperatures: mK – several hundred K
•Crystal Structures and Material Parameters given by Material(Tuning possible via e.g. external parameters like e.g. pressure, B-fields or via synthesis)
Real MaterialsUltracold Quantum Gases in Optical Lattices
•Densities: 1014/cm3
(100000 times thinner than air)
•Temperatures: few nK(100 millionen times lower than outer space)
•Crystal Structures and Material Parameters canbe changed dynamically and in-situ.
New tunable model systems for many body systems!
Low densities require us to work at even lower
temperaturesbut
we gain the control & manipulations techniques of the atomic physics
toolbox
Tuesday, June 30, 2009
Bose-Hubbard Hamiltonian
Expanding the field operator in the Wannier basis of localized wave functions on each lattice site, yields :
Bose-Hubbard Hamiltonian
Tunnelmatrix element/Hopping element Onsite interaction matrix element
M.P.A. Fisher et al., PRB 40, 546 (1989); D. Jaksch et al., PRL 81, 3108 (1998)
Mott Insulators now at: Mainz, NIST, ETHZ, Texas, Innsbruck, MIT, Munich
Tuesday, June 30, 2009
Time of flight interference pattern
• Interference between all waves coherently
emitted from each lattice site
Tim
e of
flig
ht
Periodicityofthereciprocallattice
20 ms
Wannierenvelope
Grating-likeinterference
Tuesday, June 30, 2009
Momentum Distributions – 1D
Momentum distribution can be obtained by Fourier transformation of the macroscopic wave function.
!(x) =!
i
A(xj) · w(x! xj) · ei!(xj)
Tuesday, June 30, 2009
!!j = (V !"/2) !t
!! = 0 !! = "
Preparing Arbitrary Phase Differences Between Neighbouring Lattice Sites
Phase difference between neighboring lattice sites
(cp. Bloch-Oscillations)
But: dephasing if gradient is left on for long times !
Tuesday, June 30, 2009
Mapping the Population of the Energy Bands onto the Brillouin Zones
Crystal momentum
Free particlemomentum
Population of nth band is mapped onto nth Brillouin zone !
Crystal momentum is conserved while lowering the lattice depth adiabatically !
A. Kastberg et al. PRL 74, 1542 (1995)M. Greiner et al. PRL 87, 160405 (2001)
Tuesday, June 30, 2009
Experimental Results
Piet
Mon
dria
n
Brillouin Zones in 2DMomentum distribution of a dephased condensate after turning off the lattice potential adiabtically
2D
Tuesday, June 30, 2009
Experimental Results
Piet
Mon
dria
n
Brillouin Zones in 2DMomentum distribution of a dephased condensate after turning off the lattice potential adiabtically
2D
Tuesday, June 30, 2009
Experimental Results
Piet
Mon
dria
n
Brillouin Zones in 2DMomentum distribution of a dephased condensate after turning off the lattice potential adiabtically
2D
3D
Tuesday, June 30, 2009
Populating Higher Energy Bands
Stimulated Raman transitions between vibrational levels are used to populate higher energy bands.
Single lattice site Energy bands
Measured Momentum Distribution !
Tuesday, June 30, 2009
From a Conductor to a Band Insulator
Fermi Surfaces become directly visible!
M. Köhl et al. PRL (2005)
Tuesday, June 30, 2009
Entering the Strongly Interacting Regime
Tuesday, June 30, 2009
Entering the Strongly Interacting Regime
Use Feshbach resonances toincrease U
Tuesday, June 30, 2009
Entering the Strongly Interacting Regime
Use Feshbach resonances toincrease U
Increase lattice depth anddecrease J
Tuesday, June 30, 2009
Superfluid to Mott Insulator Transition
H = !J !"i, j#
a†i a j +!
i!ini +
12
U !i
ni(ni !1)
M.P.A. Fisher et al., PRB 40, 546 (1989); D. Jaksch et al., PRL 81, 3108 (1998)Bosonic Mott Insulators now at: Munich,Mainz, NIST, ETHZ, MIT, Innsbruck, Florence, Garching...
Coherence
Number statisticsIn trap density distribution“shell structure”
Particle hole admixture
Criticalmomentum
Tuesday, June 30, 2009
Bose-Hubbard Hamiltonian
Expanding the field operator in the Wannier basis of localized wave functions on each lattice site, yields :
Bose-Hubbard Hamiltonian
Tunnelmatrix element/Hopping element Onsite interaction matrix element
M.P.A. Fisher et al., PRB 40, 546 (1989); D. Jaksch et al., PRL 81, 3108 (1998)Mott Insulators now at: NIST, ETHZ, MIT, Innsbruck, Florence, Garching...
J = !!
d3xw(x! xi)"! !2
2m! + Vlat(x)
#w(x! xj) U =
4!!2a
m
!d3x|w(x)|4
H = !J !"i, j#
a†i a j +!
i!ini +
12
U !i
ni(ni !1)
Tuesday, June 30, 2009
Superfluid Limit
Atoms are delocalized over the entire lattice !Macroscopic wave function describes this state very well.
Poissonian atom number distribution per lattice site
n=1
Atom number distribution after a measurement
!ai"i #= 0|!SF !U=0 =
!M"
i=1
a†i
#N
|0!
Tuesday, June 30, 2009
“Atomic Limit“ of a Mott-Insulator
n=1
Atoms are completely localized to lattice sites !
Fock states with a vanishing atom number fluctuation are formed.
|!Mott!J=0 =M!
i=1
"a†i
#n|0! !ai"i = 0
Tuesday, June 30, 2009
“Atomic Limit“ of a Mott-Insulator
n=1
Atoms are completely localized to lattice sites !
Fock states with a vanishing atom number fluctuation are formed.
Atom number distribution after a measurement
|!Mott!J=0 =M!
i=1
"a†i
#n|0! !ai"i = 0
Tuesday, June 30, 2009
Quantum Phase Transition (QPT) from a Superfluid to a Mott-Insulator
At the critical point gc the system will undergo a phase transition from a superfluid to an insulator !
This phase transition occurs even at T=0 and is driven by quantum fluctuations !
U/J = z 5.8
Critical ratio for:
see Subir Sachdev, Quantum Phase Transitions, Cambridge University Press
Tuesday, June 30, 2009
Quantum Phase Transition (QPT) from a Superfluid to a Mott-Insulator
At the critical point gc the system will undergo a phase transition from a superfluid to an insulator !
This phase transition occurs even at T=0 and is driven by quantum fluctuations !
U/J = z 5.8
Critical ratio for:
see Subir Sachdev, Quantum Phase Transitions, Cambridge University Press
Tuesday, June 30, 2009
Quantum Phase Transition (QPT) from a Superfluid to a Mott-Insulator
At the critical point gc the system will undergo a phase transition from a superfluid to an insulator !
This phase transition occurs even at T=0 and is driven by quantum fluctuations !
Characteristic for a QPT
•Excitation spectrum is dramatically modified at the critical point.
•U/J < gc (Superfluid regime) Excitation spectrum is gapless
•U/J > gc (Mott-Insulator regime) Excitation spectrum is gapped U/J = z 5.8
Critical ratio for:
see Subir Sachdev, Quantum Phase Transitions, Cambridge University Press
Tuesday, June 30, 2009
Superfluid – Mott-Insulator Phase Diagram
Tuesday, June 30, 2009
Superfluid – Mott-Insulator Phase Diagram
For an inhomogeneous system an effective local chemical potential can be introduced
Jaksch et al. PRL 81, 3108 (1998)
Tuesday, June 30, 2009
Ground State of an Inhomogeneous System
From Jaksch et al. PRL 81, 3108 (1998)
From M. Niemeyer and H. Monien (private communication)
Tuesday, June 30, 2009
Momentum Distribution for Different Potential Depths0 Erecoil
22 Erecoil
Tuesday, June 30, 2009
a = ! +!a
a†i a j = !a†
i "!a j"+ !a†i "!a j +!a†
i !a j"= !a†
i "a j + a†i !a j"#!a†
i "!a j"
!ai" =#
ni = !
H = !J !"i, j#
a†i a j +
12
U !i
ni(ni !1)!µ !i
ni
Describing the Phase Transition (1)
Usual Bogoliubov replacement does NOT capture SF-MI transition!(However can describe Quantum Depletion due to interactions)
Self consistent mean field approximation (decoupling approx.)
K. Sheshadri et al., EPL 22, 257 (1993)D. van Oosten, P. van der Straten & H. Stoof, PRA 63, 053601 (2001)
Tuesday, June 30, 2009
H =!zJ! !i
!a†
i + ai
"+ zt!2Ns +
12 ! ni(ni!1)!µ !
ini
Hi =12
U ni(ni!1)! µ ni!!!
a†i + ai
"+!2 U = U/zJ
µ = µ/zJ
H = H(0) +!V
H(0) =12
U n(n!1)! µ n+!2
V =!(a† + a)
Describing the Phase Transition (2)
Is diagonal in site index i, so we can use an effective on-site Hamiltonian
Can diagonalize Hamiltonian in occupation number basis!or use perturbation theory with tunnelling term to find phase diagram analytically....
D. van Oosten, P. van der Straten & H. Stoof, PRA 63, 053601 (2001)
Tuesday, June 30, 2009
E(2)n =
nU(n!1)! µ
+n+1
µ!Un
E(2)n = !2 !
n! "=n
|#n|V |n!$|2
E(0)n %E(0)
n!
Eg(!) = a0 +a2!2 +O(!4)
a2 > 0! ! = 0a2 < 0! ! "= 0
Describing the Phase Transition (3)
D. van Oosten, P. van der Straten & H. Stoof, PRA 63, 053601 (2001)
For our initial state (with fixed particle number), only second order perturbation gives a first correction.
a2 = 0 U/zJ ! n"5.83Phase transition for
Tuesday, June 30, 2009
Phase coherence of a Mott insulator
Does a Mott insulator produce an interference pattern ?
F. Gerbier et al., PRL (2005)
Theory : V. N. Kashurnikov et al., PRA 66, 031601 (2002). R. Roth & K. Burnett, PRA 67, 031602 (2003).
Tuesday, June 30, 2009
Quantitative Analysis of Interference Pattern
Visibilitymeasures coherence
Tuesday, June 30, 2009
Quantitative Analysis of Interference Pattern
Visibilitymeasures coherence
nmax
Tuesday, June 30, 2009
Quantitative Analysis of Interference Pattern
Visibilitymeasures coherence
nmin
Tuesday, June 30, 2009
Quantitative Analysis of Interference Pattern
Visibilitymeasures coherence
nmin
Visibility decaysslowly withincreasing
latticedepth!
Tuesday, June 30, 2009
Excitations in the zero tunneling limit
Perfect Mott insulator ground state
• Low energy excitations :
• Particle/hole pairs couples to the ground state :
Energy E0
Energy E0+U, separated from the ground state by an interaction gap U
n0: filling factorHere n0=1
Tuesday, June 30, 2009
Ground state for J=0 :
``perfect´´ Mott insulator
Ground state for finite J<<U :
treat the hopping term Hhop in 1st order perturbation
=
Coherent admixture of particle/holes at finite J/U
Deviations from the perfect Mott Insulator
JU
Tuesday, June 30, 2009
Predictions for the visibility
Perfect MI
MI with
particle/hole pairs
Perturbation approach predicts a finite visibility, scaling as (U/J)-1
Tuesday, June 30, 2009
Comparison with experiments
Average slope measured to be -0.97(7)
Tuesday, June 30, 2009
Tuesday, June 30, 2009
Dissecting a Mott Insulator
S. Fölling, PRL 97, 060403 (2006)
Tuesday, June 30, 2009
Dissecting a Mott Insulator
High spatial resolution of up to 1 µm can be achieved!
S. Fölling, PRL 97, 060403 (2006)
Tuesday, June 30, 2009
Dissecting a Mott Insulator
High spatial resolution of up to 1 µm can be achieved!
S. Fölling, PRL 97, 060403 (2006)
Tuesday, June 30, 2009
Dissecting a Mott Insulator
High spatial resolution of up to 1 µm can be achieved!
S. Fölling, PRL 97, 060403 (2006)
Tuesday, June 30, 2009
Imaging a 2D Mott Insulator
N. Gemelke, X. Zhang, C.-L. Hung, and C. Chin, arXiv:0904.1532
Creating a single 2D lattice plane with Cs(C. Chin, Univ. of Chicago)
Tuesday, June 30, 2009
Imaging a 2D Mott Insulator
N. Gemelke, X. Zhang, C.-L. Hung, and C. Chin, arXiv:0904.1532
Creating a single 2D lattice plane with Cs(C. Chin, Univ. of Chicago)
Tuesday, June 30, 2009
Imaging a 2D Mott Insulator
N. Gemelke, X. Zhang, C.-L. Hung, and C. Chin, arXiv:0904.1532
Creating a single 2D lattice plane with Cs(C. Chin, Univ. of Chicago)
Tuesday, June 30, 2009