Probing manybody systems of ultracold atoms

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Probing many-body systems of ultracold atoms
Eugene Demler Harvard University
E. Altman (Weizmann), A. Aspect (CNRS, Paris),
M. Greiner (Harvard), V. Gritsev (Freiburg), S.
Hofferberth (Harvard), A. Imambekov (Yale), T.
Kitagawa (Harvard), M. Lukin (Harvard), S. Manz
(Vienna), I. Mazets (Vienna), D. Petrov (CNRS,
Paris), T. Schumm (Vienna), J. Schmiedmayer
(Vienna)
Collaboration with experimental group of I. Bloch
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Outline
Density ripples in expanding low-dimensional
condensates Review of earlier work Analysis of
density ripples spectrum 1d systems 2d systems
Phase sensitive measurements of order
parameters in many body system of ultra-cold
atoms Phase sensitive experiments in
unconventional superconductors Noise correlations
in TOF experiments From noise correlations to
phase sensitive measurements
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Density ripples in expanding low-dimensional
condensates
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Fluctuations in 1d BEC
Thermal fluctuations
Thermally energy of the superflow velocity
Quantum fluctuations
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Density fluctuations in 1D condensates
In-situ observation of density fluctuations is
difficult. Density fluctuations in confined
clouds are suppressed by interactions. Spatial
resolution is also a problem. When a cloud
expands, interactions are suppressed and density
fluctuations get amplified. Phase fluctuations
are converted into density ripples
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Density ripples in expanding anisotropic 3d
condensates
Dettmer et al. PRL 2001
Hydrodynamics expansion is dominated by
collisions Complicated relation between
original fluctuations and final density
ripples.
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Density ripples in expanding anisotropic 3d
condensates

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Fluctuations in 1D condensates and density ripples
New generation of low dimensional condensates.
Tight transverse confinement leads to
essentially collision-less expansion.
1d tubes created with optical lattice
potentials I. Bloch et al.
A pair of 1d condensates on a microchip. J.
Schmiedmayer et al.
Assuming ballistic expansion we can find direct
relation between density ripples and fluctuations
before expansion.
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Density ripples Bogoliubov theory

Expansion during time t
Density after expansion
Density correlations
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Density ripples Bogoliubov theory

Spectrum of density ripples
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Density ripples Bogoliubov theory

Non-monotonic dependence on momentum. Matter-wave
near field diffraction Talbot effect
The amplitude of the spectrum is dependent on
temperature and interactions
Concern Bogoliubov theory is not applicable to
low dimensional condensates. Need extensions
beyond mean-field theory
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Density ripples general formalism

Free expansion of atoms. Expansion in different
directions factorize
We are interested in the motion along the
original trap. For 1d systems
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Quasicondensates

Factorization of higher order correlation
functions
One dimensional quasicondensate, Mora and Castin
(2003)
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Density ripples in 1D for weakly interacting Bose
gas

Thermal correlation length
A single peak in the spectrum after
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Density ripples in expanding cloud Time-evolution
of g2(x,t)
Sufficient spatial resolution required
to resolve oscillations in g2
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Density ripples in expanding cloud Time-evolution
of g2(x,t) for hard core bosons
Antibunching at short distances is rapidly
suppressed during expansion
Finite temperature
T/m1
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Density ripples in 2D

Quasicondensates in 2D below BKT transition
For weakly interacting Bose gas
Below Berezinsky-Kosterlitz-Thouless transition
at hc1/4
is a universal dimensionless function
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Density ripples in 2D

87Rb
Expansion times
t 4, 8, 12 ms
Fixed time of flight. Different temperatures
h 0.1, 0.15, 0.25
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Applications of density ripples Thermometry at
low temperatures
Probe of roton softening Analysis of
non-equilibrium states?

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Phase sensitive measurements of order parameters
in many-body systems of ultracold atoms
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d-wave pairing
Fermionic Hubbard model
Possible phase diagram of the Hubbard
model D.J.Scalapino Phys. Rep. 250329 (1995)
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Non-phase sensitive probes of d-wave pairing
dispersion of quasiparticles
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Superconducting gap


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Quasiparticle energies
Low energy quasiparticles correspond to four
Dirac nodes

• Observed in
• Photoemission
• Raman spectroscopy
• T-dependence of thermodynamic
• and transport properties, cV, k, lL
• STM
• and many other probes

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Phase sensitive probe of d-wave pairing in high
Tc superconductors
Superconducting quantum interference device
(SQUID)
Van Harlingen, Leggett et al, PRL 712134 (93)
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From noise correlations to phase sensitive
measurements in systems of ultra-cold atoms
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Quantum noise analysis in time of flight
experiments
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 434481 (2005)
Theory Altman et al., PRA 7013603 (2004)
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Second order coherence in the insulating state of
fermions.Hanburry-Brown-Twiss experiment
Experiment Tom et al. Nature 444733 (2006)
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Second order interference from the BCS superfluid
Theory Altman et al., PRA 7013603 (2004)
n(k)
k
BCS
BEC
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Momentum correlations in paired fermions
Experiments Greiner et al., PRL 94110401 (2005)
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Fermion pairing in an optical lattice
Second Order Interference In the TOF images
Normal State
Superfluid State
Measures the absolute value of the Cooper pair
wavefunction. Not a phase sensitive probe
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P-wave molecules
How to measure the non-trivial symmetry of y(p)?
We want to measure the relative phase between
components of the molecule at different
wavevectors
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Two particle interference
Beam splitters perform Rabi rotation
Coincidence count
Coincidence count is sensitive to the relative
phase between different components of
the molecule wavefunction
Questions How to make atomic beam splitters and
mirrors? Phase difference includes phase
accumulated during free expansion. How to control
it?
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Bragg Noise
Bragg pulse is applied in the beginning of
expansion
Assuming mixing between k and p states only
Coincidence count
Common mode propagation after the pulse. We do
not need to worry about the phase accumulated
during the expansion.
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Many-body BCS state
BCS wavefunction
Strong Bragg pulse mixing of many momentum
eigenstates
Noise correlations
Interference term is sensitive to the phase
difference between k and p parts of the Cooper
pair wavefunction and to the phases of Bragg
pulses
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Noise correlations in the BCS state
Interference between different components of the
Cooper pair
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Noise correlations in the BCS state
V0t controls Rabi angle b
Bragg pulse phases control cs
Compare to
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Systems with particle-hole correlations
D-density wave state
Suggested as a competing order in high Tc
cuprates
Phase sensitive probe of DDW order parameter
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Summary
Density ripples in expanding low-dimensional
condensates
Phase sensitive measurements of order
parameters in many body systems of ultra-cold
atoms
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Detection of spin superexchange interactions and
antiferromagnetic statesSpin noise analysis
Bruun, Andersen, Demler, Sorensen, PRL (2009)
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Spin shot noise as a probe of AF order
Measure net spin in a part of the system. Laser
beam passes through the sample. Photons
experience phase shift determined by the net
spin. Use homodyne to measure phase shift
Average magnetization zero
Shot to shot magnetization fluctuations reflect
spin correlations
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Spin shot noise as a probe of AF order
High temperatures. Every spin fluctuates
independently
Low temperatures. Formation of antiferromagnetic
correlations
Suppression of spin fluctuations due to spin
superexchange interactions can be observed at
temperatures well above the Neel ordering
transition
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Two particle interference
Beam splitters perform Rabi rotation
Molecule wavefunction
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Two particle interference
Coincidence count
Coincidence count is sensitive to the relative
phase between different components of
the molecule wavefunction
Phase difference includes phase
accumulated during free expansion