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Measuring correlation functions in interacting systems of cold atoms
Ehud AltmanRyan BarnettMikhail LukinDmitri PetrovAnatoli PolkovnikovEugene Demler
Physics Department,Harvard University
Thanks to: J. Schmiedmayer, M. Oberthaler, V. Vuletic, M. Greiner
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Correlation functions in condensed matter physicsMost experiments in condensed matter physics measure correlation functions
Example: neutron scattering measures spin and density correlation functions
Neutron diffraction patterns for MnO
Shull et al., Phys. Rev. 83:333 (1951)
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Outline1. Measuring correlation functions in intereference experiments
Introduction: interference of independent condensates1D systems: Luttinger liquid behavior2D systems: quasi long range order and the KT transition
2. Quantum noise interferometry of atoms in an optical lattice
3. Applications of quantum noise interferometrySpin order in Mott states of atomic mixtures Polar molecules in optical lattices. Charge and spin order
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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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Interference of one dimensional condensates
x1
Amplitude of the interference fringes, , contains information about phase fluctuations within individual condensates
d
y
x
Similar experimental setup: Schmiedmayer et al.
x2
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Interference of one dimensional condensates
For identical condensates
Instantaneous correlation function
L
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For impenetrable bosons and
Interference of one dimensional condensatesLuttinger liquid at T=0
K – Luttinger parameter
Luttinger liquid at finite temperature
For non-interacting bosons and
L
Luttinger parameter K may be extracted from the L or T dependence of
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Luttinger parameter K may beextracted from the angular dependence of
Interference of one dimensional condensates
θ
Luttinger liquid at T=0. Rotated probe beam experiment
For large imaging angle, ,
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Interference of two dimensional condensates
Ly
LxLx
Similar experimental setup: 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 α from the angulardependence of
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Rapidly rotating two dimensional condensates
Time of flight experiments with rotating condensates correspond to density measurements
Interference experiments measure singleparticle correlation functions in the rotatingframe
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Quantum noise interferometryof atoms in an optical lattice
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Atoms in an optical lattice.Superfluid to Insulator transition
Greiner et al., Nature 415:39 (2002)
Uμ
1−n
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 et al., PRL (2003)
• Ferromagnetic• Antiferromagnetic
Kuklov and Svistunov, PRL (2003)
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Two component Bose mixture in optical lattice.Mean field theory + Quantum fluctuations
2 ndorder line
Hysteresis
1st order
Altman et al., NJP 5:113 (2003)
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Coherent spin dynamics in optical latticesWidera et al., cond-mat/0505492
atoms in the F=2 state
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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
Polar molecules in optical lattices. Charge and spin order
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Extended Hubbard model
- on site repulsion - nearest neighbor repulsion
Checkerboard phase:
Crystal phase of bosons.Breaks translational symmetry
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Extended Hubbard model. Mean field phase diagramvan Otterlo et al., PRB 52:16176 (1995)
Hard core bosons.
Supersolid – superfluid phase with broken translational symmetry
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Dipolar bosons in optical lattices
Goral et al., PRL88:170406 (2002)
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Probing a checkerboard phase
Correlation Function Measurements
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Multicomponent polar molecules in an optical lattice.Long range interactions and quantum magnetism
x z
Electric dipolar interactions. Heteronuclear molecules. Mixture of l=0 and l=0, lz=+1 states.
Barnett, Petrov, Lukin, Demler
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Conclusions
Interference of extended condensates can be usedto probe correlation functions in one and two dimensional systems
Noise interferometry is a powerful tool for analyzingquantum many-body states in optical lattices