Anatoli Polkovnikov Boston University
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Anatoli Polkovnikov Boston UniversityEhud Altman WeizmannVladimir Gritsev HarvardMikhail Lukin HarvardEugene Demler Harvard
Thanks to: J. Schmiedmayer, M. Oberthaler, V. Vuletic, M. Greiner, M. Oshikawa
Strongly Correlated Systems of Cold AtomsDetection of many-body quantum phases
by measuring correlation functions
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Bose-Einstein condensation
Cornell et al., Science 269, 198 (1995)
Ultralow density condensed matter system
Interactions are weak and can be described theoretically from first principles
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New Era in Cold Atoms ResearchFocus on Systems with Strong Interactions
• Atoms in optical lattices
• Feshbach resonances
• Low dimensional systems
• Systems with long range dipolar interactions
• Rotating systems
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Feshbach resonance and fermionic condensates Greiner et al., Nature 426:537 (2003); Ketterle et al., PRL 91:250401 (2003)
Ketterle et al.,Nature 435, 1047-1051 (2005)
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One dimensional systems
Strongly interacting regime can be reached for low densities
One dimensional systems in microtraps.Thywissen et al., Eur. J. Phys. D. (99);Hansel et al., Nature (01);Folman et al., Adv. At. Mol. Opt. Phys. (02)
1D confinement in optical potentialWeiss et al., Science (05);Bloch et al., Esslinger et al.,
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Atoms in optical lattices
Theory: Jaksch et al. PRL (1998)
Experiment: Kasevich et al., Science (2001); Greiner et al., Nature (2001); Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004); and many more …
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Bose Hubbard model
tunneling of atoms between neighboring wells
repulsion of atoms sitting in the same well
U
t
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4
Bose Hubbard model. Mean-field phase diagram
1n
U
02
0
M.P.A. Fisher et al.,PRB40:546 (1989)
MottN=1
N=2
N=3
Superfluid
Superfluid phase
Mott insulator phase
Weak interactions
Strong interactions
Mott
Mott
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Superfluid to insulator transition in an optical lattice
Greiner et al., Nature 415 (2002)See also Ketterle et al. cond-mat/0507288
U
1n
t/U
SuperfluidMott insulator
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New Era in Cold Atoms ResearchFocus on Systems with Strong Interactions
Goals
• Resolve long standing questions in condensed matter physics (e.g. origin of high temperature superconductivity)
• Resolve matter of principle questions (e.g. existence of spin liquids in two and three dimensions)
• Study new phenomena in strongly correlated systems (e.g. coherent far from equilibrium dynamics)
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This talk:
Detection of many-body quantum phases by measuring correlation functions
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Outline
Measuring correlation functions in intereference experiments 1. Interference of independent condensates 2. Interference of interacting 1D systems 3. Full counting statistics of intereference experiments. Connection to quantum impurity problem 4. Interference of 2D systems
Quantum noise interferometry in time of flight experiments
1. Detection of magnetically ordered Mott states in optical lattices 2. Observation of fermion pairing
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Measuring correlation functions in intereference experiments
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Interference of two independent condensates
Andrews et al., Science 275:637 (1997)
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Interference of two independent condensates
1
2
r
r+d
d
r’
Clouds 1 and 2 do not have a well defined phase difference.However each individual measurement shows an interference pattern
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Amplitude of interference fringes, , contains information about phase fluctuations within individual condensates
y
x
Interference of one dimensional condensates
x1
d Experiments: Schmiedmayer et al., Nature Physics 1 (05)
x2
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Interference amplitude and correlations
For identical condensates
Instantaneous correlation function
L
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For impenetrable bosons and
Interference between Luttinger liquidsLuttinger liquid at T=0
K – Luttinger parameter
Luttinger liquid at finite temperature
For non-interacting bosons and
Luttinger parameter K may be extracted from the L or T dependence of
L
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Luttinger parameter K may beextracted from the angular dependence of
Rotated probe beam experiment
For large imaging angle, ,
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Higher moments of interference amplitude
L Higher moments
Changing to periodic boundary conditions (long condensates)
Explicit expressions for are available but cumbersome Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995)
is a quantum operator. The measured value of will fluctuate from shot to shot.Can we predict the distribution function of ?
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Full counting statistics of interference experiments
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Impurity in a Luttinger liquid
Expansion of the partition function in powers of g
Partition function of the impurity contains correlation functions taken at the same point and at different times. Momentsof interference experiments come from correlations functionstaken at the same time but in different points. Lorentz invarianceensures that the two are the same
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Relation between quantum impurity problemand interference of fluctuating condensates
Distribution function of fringe amplitudes
Distribution function can be reconstructed fromusing completeness relations for Bessel functions
Normalized amplitude of interference fringes
Relation to the impurity partition function
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is related to a Schroedinger equation Dorey, Tateo, J.Phys. A. Math. Gen. 32:L419 (1999) Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102:567 (2001)
Spectral determinant
Bethe ansatz solution for a quantum impurity can be obtained from the Bethe ansatz followingZamolodchikov, Phys. Lett. B 253:391 (91); Fendley, et al., J. Stat. Phys. 79:799 (95)Making analytic continuation is possible but cumbersome
Interference amplitude and spectral determinant
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0 1 2 3 4
P
roba
bilit
y P
(x)
x
K=1 K=1.5 K=3 K=5
Evolution of the distribution function
Narrow distributionfor .Distribution widthapproaches
Wide Poissoniandistribution for
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When K>1, is related to Q operators of CFT with c<0. This includes 2D quantum gravity, non-intersecting loop model on 2D lattice, growth of randomfractal stochastic interface, high energy limit of multicolor QCD, …
Yang-Lee singularity
2D quantum gravity,non-intersecting loops on 2D lattice
correspond to vacuum eigenvalues of Q operators of CFT Bazhanov, Lukyanov, Zamolodchikov, Comm. Math. Phys.1996, 1997, 1999
From interference amplitudes to conformal field theories
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Interference of two dimensional condensates
Ly
Lx
Lx
Experiments: Stock et al., cond-mat/0506559
Probe beam parallel to the plane of the condensates
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Interference of two dimensional condensates.Quasi long range order and the KT transition
Ly
LxBelow KT transition
Above KT transition
One can also use rotatedprobe beam experiments to extract from the angulardependence of
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Quantum noise interferometry in time of flight experiments
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Atoms in an optical lattice.Superfluid to Insulator transition
Greiner et al., Nature 415:39 (2002)
U
1n
t/U
SuperfluidMott insulator
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Time of flight experiments
Quantum noise interferometry of atoms in an optical lattice
Second order coherence
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Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment
Theory: Altman et al., PRA 70:13603 (2004)
Experiment: Folling et al., Nature 434:481 (2005)
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Hanburry-Brown-Twiss stellar interferometer
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Second order coherence in the insulating state of bosons
Bosons at quasimomentum expand as plane waves
with wavevectors
First order coherence:
Oscillations in density disappear after summing over
Second order coherence:
Correlation function acquires oscillations at reciprocal lattice vectors
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Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment
Theory: Altman et al., PRA 70:13603 (2004)
Experiment: Folling et al., Nature 434:481 (2005)
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0 200 400 600 800 1000 1200
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Interference of an array of independent condensates
Hadzibabic et al., PRL 93:180403 (2004)
Smooth structure is a result of finite experimental resolution (filtering)
0 200 400 600 800 1000 1200-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
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Applications of quantum noise interferometry
Spin order in Mott states of atomic mixtures
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t
t
Two component Bose mixture in optical latticeExample: . Mandel et al., Nature 425:937 (2003)
Two component Bose Hubbard model
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Two component Bose mixture in optical lattice.Magnetic order in an insulating phase
Insulating phases with N=1 atom per site. Average densities
Easy plane ferromagnet
Easy axis antiferromagnet
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Quantum magnetism of bosons in optical lattices
Duan, Lukin, Demler, PRL (2003)
• Ferromagnetic• Antiferromagnetic
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Probing spin order of bosons
Correlation Function Measurements
Extra Braggpeaks appearin the secondorder correlationfunction in theAF phase
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Applications of quantum noise interferometry
Detection of fermion pairing
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Fermionic atoms in an optical lattice
Kohl et al., PRL 94:80403 (2005)
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Fermions with repulsive interactions
t
U
tPossible d-wave pairing of fermions
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Picture courtesy of UBC Superconductivity group
High temperature superconductors
Superconducting Tc 93 K
Hubbard model – minimal model for cuprate superconductors
P.W. Anderson, cond-mat/0201429
After twenty years of work we still do not understand the fermionic Hubbard model
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Positive U Hubbard model
Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)
Antiferromagnetic insulator
D-wave superconductor
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Second order interference from a BCS superfluid
)'()()',( rrrr nnn
n(r)
n(r’)
n(k)
k
0),( BCSn rr
BCS
BEC
kF
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Momentum correlations in paired fermionsTheory: Altman et al., PRA 70:13603 (2004)Experiment: Greiner et al., PRL 94:110401 (2005)
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Fermion pairing in an optical lattice
Second Order InterferenceIn the TOF images
Normal State
Superfluid State
measures the Cooper pair wavefunction
One can identify unconventional pairing
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Conclusions
We understand well: electron systems in semiconductors and simple metals.Interaction energy is smaller than the kinetic energy. Perturbation theory works
We do not understand: strongly correlated electron systems in novel materials.Interaction energy is comparable or larger than the kinetic energy.Many surprising new phenomena occur, including high temperaturesuperconductivity, magnetism, fractionalization of excitations
Our big goal is to develop a general framework for understanding stronglycorrelated systems. This will be important far beyond AMO and condensed matter
Ultracold atoms have energy scales of 10-6K, compared to 104 K forelectron systems. However, by engineering and studying strongly interactingsystems of cold atoms we should get insights into the mysterious propertiesof novel quantum materials
Developing new detection methods is an important problem in the area of strongly correlated atoms. Interference experiments and analysis of quantumnoise in time of flight experiments are powerful tools for analyzing many-body states