Probing many-body systems of ultracold atoms

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)

Eugene Demler Harvard University

Collaboration with experimental group of I. Bloch

Outline

Density ripples in expanding low-dimensional condensates

Review of earlier workAnalysis of density ripples spectrum1d systems2d systems

Phase sensitive measurements of order parametersin many body system of ultra-cold atoms

Phase sensitive experiments in unconventional superconductorsNoise correlations in TOF experimentsFrom 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 anddensity 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-lessexpansion.

Assuming ballistic expansion we can find direct relationbetween density ripples and fluctuations before expansion.

A pair of 1d condensates on a microchip.J. Schmiedmayer et al.

1d tubes created withoptical lattice potentialsI. Bloch et al.

Density ripples: Bogoliubov theory

Expansion during time t

Density after expansion

Densitycorrelations

Density ripples: Bogoliubov theory

Spectrum of density ripples

Density ripples: Bogoliubov theory

Maxima at

Minima at

The amplitude of the spectrum is dependent on temperatureand interactions

Non-monotonic dependence on momentum.Matter-wave near field diffraction: Talbot effect

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/ =1, 0.67, 0.3, 0

Different timesof flight. T/=0.67

A single peak in the spectrum after

Density ripples in expanding cloud:Time-evolution of g2(x,t)

Sufficient spatial resolution required toresolve oscillations in g2

Density ripples in expanding cloud:Time-evolution of g2(x,t) for hard core bosons

T=0. Expansion times

T/

“Antibunching” at short distances is rapidly suppressed during expansion

Finite temperature

Density ripples in 2D

Quasicondensates in 2D below BKT transition

For weakly interacting Bose gas

is a universal dimensionless function

Below Berezinsky-Kosterlitz-Thouless transition at c=1/4

Density ripples in 2D

Expansion times

Fixed time of flight.Different temperatures

87Rb

t = 4, 8, 12 ms

= 0.1, 0.15, 0.25

Applications of density ripples Thermometry at low temperatures

T/ =1, 0.67, 0.3, 0

Probe of roton softening

Analysis of non-equilibrium states?

Phase sensitive measurements of order parameters inmany-body systems of ultracold atoms

d-wave pairing

Fermionic Hubbard model

Possible phase diagram of the Hubbard modelD.J.Scalapino Phys. Rep. 250:329 (1995)

Non-phase sensitive probes of d-wave pairing:

dispersion of quasiparticles

++-

-Quasiparticle energies

Superconducting gap

Normal state dispersion of quasiparticles

Low energy quasiparticles correspond to four Dirac nodes

Observed in:

• Photoemission• Raman spectroscopy• T-dependence of thermodynamic

and transport properties, cV, , L

• STM• and many other probes

Phase sensitive probe of d-wave pairing in high Tc superconductors

Superconducting quantum interference device (SQUID)

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

)'()()',( rrrr nnn

n(r)

n(r’)

n(k)

k

0),( BCSn rr

BCS

BEC

kF

Theory: Altman et al., PRA 70:13603 (2004)

Momentum correlations in paired fermionsExperiments: Greiner et al., PRL 94:110401 (2005)

Fermion pairing in an optical lattice

Second Order InterferenceIn the TOF images

Normal State

Superfluid State

Measures the absolute value of theCooper pair wavefunction.Not a phase sensitive probe

P-wave molecules

How to measure the non-trivial symmetry of (p)?

We want to measure the relative phase between components of the molecule at different wavevectors

Two particle interferenceBeam splitters perform Rabi rotation

Coincidence count

Coincidence count is sensitive to the relative phase between different components of themolecule wavefunction

Questions: How to make atomic beam splitters and mirrors? Phase difference includes phase accumulated during free expansion. How to control it?

Bragg + Noise

Gk

-k

p

-p

G

Bragg pulse is applied in the beginning of expansion

Coincidence count

Assuming mixing between k and p states only

Common mode propagation after the pulse. We do not need to worry about thephase accumulated during the expansion.

Gk

-k

p

-p

G

Many-body BCS stateBCS 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

Noise correlations in the BCS state

Interference between different components of the Cooper pair

Noise correlations in the BCS state

Compare to

V0 controls Rabi angle Bragg pulse phases control ’s

Systems with particle-hole correlations

D-density wave state

Suggested as a competing order in high Tc cuprates

Phase sensitive probe of DDWorder parameter

G k

-k

p

-p

G

SummaryDensity ripples in expanding low-dimensional condensates

Phase sensitive measurements of order parametersin many body systems of ultra-cold atoms

T/ =1, 0.67, 0.3, 0

Different timesof flight. T/=0.67

Detection of spin superexchange interactions and antiferromagnetic

statesSpin noise analysis

Bruun, Andersen, Demler, Sorensen, PRL (2009)

Spin shot noise as a probe of AF orderMeasure net spin in a part of the system.Laser beam passes through the sample. Photons experience phaseshift 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 dueto spin superexchange interactions can beobserved at temperatures well above theNeel ordering transition

Two particle interferenceBeam splitters perform Rabi rotation

Molecule wavefunction

Two particle interferenceCoincidence count

Coincidence count is sensitiveto the relative phase betweendifferent components of themolecule wavefunction

Phase difference includes phase accumulatedduring free expansion