Transcript Document

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
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
Density ripples in expanding
low-dimensional condensates
Fluctuations in 1d BEC
Thermal fluctuations
Thermally energy of the superflow velocity
Quantum fluctuations
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
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.
Density ripples in expanding anisotropic 3d condensates
Fluctuations in 1D condensates and density ripples
New generation of low dimensional condensates.
Tight transverse confinement leads to essentially collision-less
expansion.
A pair of 1d condensates on
a microchip.
J. Schmiedmayer et al.
1d tubes created with
optical lattice potentials
I. Bloch et al.
Assuming ballistic expansion we can find direct relation
between density ripples and fluctuations before expansion.
Density ripples: Bogoliubov theory
Expansion during time t
Density
after
expansion
Density
correlations
Density ripples: Bogoliubov theory
Spectrum of density ripples
Density ripples: Bogoliubov theory
Non-monotonic dependence on momentum.
Matter-wave near field diffraction: Talbot effect
Maxima at
Minima at
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
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
Quasicondensates
• One dimensional systems with
• Two dimensional systems below BKT transition
Factorization of higher order correlation functions
One dimensional quasicondensate, Mora and Castin (2003)
Density ripples in 1D for weakly interacting Bose gas
Thermal correlation length
T/m =1, 0.67,
0.3, 0
A single peak in the spectrum after
Different times
of flight. T/m=0.67
Density ripples in expanding cloud:
Time-evolution of g2(x,t)
Sufficient spatial
resolution required to
resolve oscillations in g2
Density ripples in expanding cloud:
Time-evolution of g2(x,t) for hard core bosons
T=0. Expansion times
“Antibunching” at short distances is
rapidly suppressed during expansion
Finite temperature
T/m=1
Density ripples in 2D
Quasicondensates in 2D below BKT transition
For weakly interacting Bose gas
Below Berezinsky-Kosterlitz-Thouless transition at hc=1/4
is a universal dimensionless function
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
Applications of density ripples
Thermometry at low temperatures
T/m =1, 0.67,
0.3, 0
Probe of roton softening
Analysis of non-equilibrium states?
Phase sensitive measurements
of order parameters in
many-body systems
of ultracold atoms
d-wave pairing
Fermionic Hubbard model
Possible phase diagram of the Hubbard model
D.J.Scalapino Phys. Rep. 250:329 (1995)
Non-phase sensitive probes of d-wave pairing:
dispersion of quasiparticles
+
-
Superconducting gap
+
Normal state dispersion of quasiparticles
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
Phase sensitive probe of d-wave pairing in
high Tc superconductors
Superconducting quantum
interference device (SQUID)
F
Van Harlingen, Leggett et al, PRL 71:2134 (93)
From noise correlations to
phase sensitive measurements
in systems of ultra-cold atoms
Quantum noise analysis in
time of flight experiments
Second order coherence
Second order coherence in the insulating state of bosons.
Hanburry-Brown-Twiss experiment
Experiment: Folling et al., Nature 434:481 (2005)
Theory: Altman et al., PRA 70:13603 (2004)
Second order coherence in the insulating state of fermions.
Hanburry-Brown-Twiss experiment
Experiment: Tom et al. Nature 444:733 (2006)
Second order interference from the BCS superfluid
Theory: Altman et al., PRA 70:13603 (2004)
n(k)
n(r’)
kF
k
n(r)
BCS
BEC
n(r, r' )  n(r)  n(r' )
n(r,r) BCS = 0
Momentum correlations in paired fermions
Experiments: Greiner et al., PRL 94:110401 (2005)
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
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
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?
Bragg + Noise
Bragg pulse is applied in the beginning of expansion
p
G
k
G
-k
Assuming mixing between k and p states only
-p
Coincidence count
Common mode propagation after the pulse. We do not need to worry about the
phase accumulated during the expansion.
Many-body BCS state
BCS wavefunction
p
G
k
Strong Bragg pulse: mixing of
many momentum eigenstates
G
-k
-p
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
Noise correlations in the BCS state
Interference between different components of the Cooper pair
Noise correlations in the BCS state
V0t controls Rabi angle b
Compare to
Bragg pulse phases control c’s
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
Summary
Density ripples in expanding low-dimensional condensates
Different times
of flight. T/m=0.67
T/m =1,
0.67, 0.3, 0
Phase sensitive measurements of order parameters
in many body systems of ultra-cold atoms
p
G
k
G
-k
-p
Detection of spin superexchange
interactions and antiferromagnetic
states
Spin noise analysis
Bruun, Andersen, Demler, Sorensen, PRL (2009)
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
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
Two particle interference
Beam splitters perform Rabi rotation
Molecule wavefunction
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