learning about order from noise - harvard university
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Learning about order from noiseQuantum noise studies of ultracold atoms
Eugene Demler Harvard University
Robert Cherng, Adilet Imambekov, Ehud Altman, Vladimir Gritsev, Anatoli Polkovnikov,
Ana Maria Rey, Mikhail Lukin
Experiments:Bloch et al., Dalibard et al., Greiner et al., Schmiedmayer et al.
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Quantum noiseClassical measurement:
collapse of the wavefunction into eigenstates of x
Histogram of measurements of x
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Probabilistic nature of quantum mechanics
“Spooky action at a distance”
Bohr-Einstein debate:EPR thought experiment
(1935)
Aspect’s experiments with correlated photon pairs:
tests of Bell’s inequalities (1982)
Analysis of correlation functions can be used
to rule out hidden variables theories
+
-
+
-1 2S
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Second order coherence: HBT experimentsClassical theory
Hanburry Brown and Twiss (1954)
Used to measure the angular diameter of Sirius
Quantum theory
Glauber (1963)
For bosons
For fermions
HBT experiments with matter
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Shot noise in electron transportShot noise Schottky (1918)
Variance of transmitted charge
e- e-
Measurements of fractional charge
Current noise for tunneling across a Hall bar on the 1/3
plateau of FQE
Etien et al. PRL 79:2526 (1997)see also Heiblum et al. Nature (1997)
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Analysis of quantum noise:
powerful experimental tool
Can we use it for cold atoms?
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Quantum noise in interference experimentswith independent condensates
Outline
Goal: new methods of detection of quantummany-body phases of ultracold atoms
Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices:HBT experiments and beyond
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Interference experiments
with cold atomsAnalysis of thermal and quantum noise
in low dimensional systems
Theory: For review seeImambekov et al., Varenna lecture notes, c-m/0612011
Experiment2D: Hadzibabic, Kruger, Dalibard, Nature 441:1118 (2006)
1D: Hofferberth et al., Nature Physics 4:489 (2008)
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Interference of independent condensates
Experiments: Andrews et al., Science 275:637 (1997)
Theory: Javanainen, Yoo, PRL 76:161 (1996)
Cirac, Zoller, et al. PRA 54:R3714 (1996)
Castin, Dalibard, PRA 55:4330 (1997)
and many more
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x
z
Time of
flight
Experiments with 2D Bose gasHadzibabic, Kruger, Dalibard et al., Nature 441:1118 (2006)
Experiments with 1D Bose gas Hofferberth et al., Nature Physics 4:489 (2008)
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Interference of two independent condensates
1
2
r
r+d
d
r’
Phase difference between clouds 1 and 2is not well defined
Assuming ballistic expansion
Individual measurements show interference patternsThey disappear after averaging over many shots
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x1
d
Amplitude of interference fringes,
Interference of fluctuating condensates
For identical condensates
Instantaneous correlation function
For independent condensates Afr is finite but ∆φ is random
x2
Polkovnikov, Altman, Demler, PNAS 103:6125(2006)
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Fluctuations in 1d BEC
Thermal fluctuations
Thermally energy of the superflow velocity
Quantum fluctuations
Weakly interactingatoms
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For impenetrable bosons and
Interference between Luttinger liquids
Luttinger liquid at T=0
K – Luttinger parameter
Finite temperature
Experiments: Hofferberth,Schumm, Schmiedmayer
For non-interacting bosons and
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Distribution function of fringe amplitudes
for interference of fluctuating condensates
L
is a quantum operator. The measured value of will fluctuate from shot to shot.
Higher moments reflect higher order correlation functions
Gritsev, Altman, Demler, Polkovnikov, Nature Physics 2006
Imambekov, Gritsev, Demler, Varenna lecture notes, c-m/0703766
We need the full distribution function of
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Distribution function of interference fringe contrastHofferberth et al., Nature Physics 4:489 (2008)
Comparison of theory and experiments: no free parametersHigher order correlation functions can be obtained
Quantum fluctuations dominate:
asymetric Gumbel distribution
(low temp. T or short length L)
Thermal fluctuations dominate:
broad Poissonian distribution
(high temp. T or long length L)
Intermediate regime:
double peak structure
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Interference of two dimensional condensates
Ly
Lx
Lx
Experiments: Hadzibabic et al. Nature (2006)
Probe beam parallel to the plane of the condensates
Gati et al., PRL (2006)
Observation of BTK transition: see talk by Peter Kruger
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Time-of-flight experiments
with atoms in optical lattices
Theory: Altman, Demler, Lukin, PRA 70:13603 (2004)
Experiment: Folling et al., Nature 434:481 (2005); Spielman et al., PRL 98:80404 (2007);
Tom et al. Nature 444:733 (2006);
Guarrera et al., PRL 100:250403 (2008)
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Atoms in optical lattices
Theory: Jaksch et al. PRL (1998)
Experiment: Greiner et al., Nature (2001) and many more
Motivation: quantum simulations of strongly correlated electron systems including quantum magnets andunconventional superconductors. Hofstetter et al. PRL (2002)
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Superfluid to insulator transition in an optical lattice
M. Greiner et al., Nature 415 (2002)
U
µ
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
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
Experiment: Folling et al., Nature 434:481 (2005)
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Second order coherence in the insulating state of fermions.
Hanburry-Brown-Twiss experiment
Experiment: Tom et al. Nature 444:733 (2006)
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Probing spin order in optical lattices
Correlation Function Measurements
Extra Braggpeaks appearin the secondorder correlationfunction in theAF phase
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Detection of fermion pairing
Quantum noise analysis of TOF images
is more than HBT interference
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Second order interference from the BCS superfluid
)'()()',( rrrr nnn −≡∆
n(r)
n(r’)
n(k)
k
0),( =Ψ−∆BCS
n rr
BCS
BEC
kF
Theory: Altman et al., PRA 70:13603 (2004)
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Momentum correlations in paired fermions
Experiments: Greiner et al., PRL 94:110401 (2005)
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Summary
Experiments with ultracold atoms provide a new
perspective on the physics of strongly correlated
many-body systems. Quantum noise is a powerful
tool for analyzing many body states of ultracold atoms
Thanks to:
Harvard-MIT
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Preparation and detection of Mott statesof atoms in a double well potential
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Second order coherenceClassical theoryHanburry-Brown-Twiss
Measurements of the angular diameter of Sirius
Proc. Roy. Soc. (19XX)
Quantum theory
Glauber
For bosons
For fermions
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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