Interference experiments with ultracold atoms

Collaborators:Ehud Altman, Anton Burkov, Robert Cherng, Adilet Imambekov, Serena Fagnocchi,Vladimir Gritsev, Mikhail Lukin, David Pekker,Anatoli Polkovnikov

Funded by NSF, Harvard-MIT CUA, AFOSR, DARPA, MURI

Eugene Demler Harvard University

Outline

Introduction. Interference of fluctuating low dimensional condensates. Systems of mixed dimensionality

Interference of fermions: probing paired statesDetection of s-wave pairingDetection of FFLODetection of d-wave pairing

Interference experiments and non-equilibrium dynamicsDecoherence of uniformly split condensatesRamsey interference of one dimensional systemsSplitting condensates on Y-junctions

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

x

z

Time of

flight

Experiments with 2D Bose gasHadzibabic, Dalibard et al., Nature 441:1118 (2006)

Experiments with 1D Bose gas Hofferberth et al. arXiv0710.1575

x1

Amplitude of interference fringes,

Interference of fluctuating condensates

For identical condensates

Instantaneous correlation function

For independent condensates Afr is finite but is random

x2

Polkovnikov, Altman, Demler, PNAS 103:6125(2006)

L

Interference between fluctuating condensates

1d: Luttinger liquid, Hofferberth et al., 2007

x

z

L [pixels]

0.4

0.2

00 10 20 30

middle Tlow T

high T

2d: BKT transition, Hadzibabic et al, 2006

Time of flight

low T

high T

BKT

Distribution function of interference fringe contrastExperiments: Hofferberth et al., arXiv0710.1575Theory: Imambekov et al. , cond-mat/0612011

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

Systems of mixed dimensionality Weakly coupled 2D condensates

Experiments: M. Kasevich et al.,

Interplay of two dimensional physics of the BKT transition and coupling along the 3rd direction

Connection to quasi-2D and 1D condensed matter systems:

quantum magnets, organic superconductors,high Tc cuprates, and many more

Berezinskii – Kosterlitz – Thouless transition

T=TBKT

tem

pera

ture

T=0

Fisher & Hohenberg, PRB 37, 4936 (1988)

Temperature scales for weakly coupled pancakes

3D XY

3D Phonons

tem

pera

ture

T=0

T=TBKT

T=TC

T=2t

Interference of a stack of coupled pancakesPekker, Gritsev, Demler

Interference experiments with fermions:

probing paired states

Interference of fermionic systems

Time of flight

X

YZ

1

2

A pair of independent fermionic systems

Interference as a probe of fermionic pairing

1

2

Pairing correlationsTime of flight

Expectation value vanishes for independent systems

due to random relative phase between 1 and 2

Interference as a probe of pairing

1

2

Experimental procedureInterfere two independent systemsMeasure and in the same shot Extract and from Fourier transforms in the z-directionCalculate andFind from averaging over many shots

Polkovnikov, Gritsev, Demler

FFLO phase

Pairing at finite center of mass momentum

Theory: Fulde, Ferrell (1964); Larkin, Ovchinnikov (1965); Bowers, Rajagopal (2002); Liu, Wilczek (2003); Sheehy, Radzihovsky (2006); Combescot (2006); Yang, Sachdev (2006); Pieri, Srinati (2006); Parish et al., (2007); and many others

Experiments: Zwierlein, Ketterle et al., (2006) Hulet et al., (2006)

Interference as a probe of FFLO phase

1

2

when q matches one of the wavevectorsof (r) of FFLO phase

Integrationby a laser beam

Manual integration

x

y

d-wave pairing

Fermionic Hubbard model

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

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)

Other signatures 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)

Interference as a probe of d-wave pairing

2

1

System 1 is an s-wave superfluidSystem 2 is a d-wave superfluidRegions II and III differ only by o rotation

Phase sensitive probe of d-wave pairing

d-wave superfluid

s-wave superfluid

Interference experiments and non-equilibrium dynamics

Uniform splitting of the condensates

Prepare a system by splitting one condensate

Take to the regime of zero tunneling Measure time evolution

of fringe amplitudes

Long phase coherence implies squeezing factor of 10.Squeezing due to finite time of splitting. Leggett, Sols, PRL (1998) Burkov et al., PRL (2007)

Squeezing factor

1d BEC: Decay of coherence Experiments: Hofferberth, Schumm, Schmiedmayer, Nature (2007)

double logarithmic plot of the coherence factor

slopes: 0.64 ± 0.08

0.67 ± 0.1

0.64 ± 0.06

get t0 from fit with fixed slope 2/3 and calculate T from

T5 = 110 ± 21

nK

T10 = 130 ± 25

nK

T15 = 170 ± 22

nK

Relative phase dynamics beyond single mode approximation

Conjugate variables

Hamiltonian can be diagonalized in momentum space

A collection of harmonic oscillators with

Need to solve dynamics of harmonic oscillators at finite T

Coherence

Initial state q = 0

Quantum regime

1D systems

2D systems

Classical regime

1D systems

2D systems

Relative phase dynamics beyond single mode approximation

Bistritzer, Altman, PNAS (2007)Burkov, Lukin, Demler, PRL (2007)

Dynamics of condensate splittingand Ramsey interference

Working with N atoms improves the precision of frequency spectroscopy by .

Ramsey interference

t0

1

Interaction induced collapse of Ramsey fringes

Experiments in 1d tubes: A. Widera, et al. arXiv 0709:2094

Spin echo. Time reversal experiments

Single mode approximation predicts full revival

Experiments in 1d tubes: A. Widera, et al. arXiv 0709:2094.

Need to analyze multi-mode model in 1d

Only q=0 mode shows complete spin echoFinite q modes continue decay

The net visibility is a result of competition between q=0 and other modes

Splitting condensates on Y-junctions: quantum zipper problem

Splitting condensates on Y-junctions

Partial splitting stage: new physics

Full splitting stage: same as before

Earlier work:Non-interacting atoms: Scully and Dowling, PRA (1993)Analysis of transverse modes: Jaasekelatnen and Stenholm, PRA (2003)Tonks-Girardeau regime: Girardeau et al. PRA (2002)

Splitting condensates on Y-junctions: beyond mean-field

x

Wave equation in both arms of the interferometer. c is the speed of sound

Relative phase

Time dependent boundary conditions in the frame of the condensate. v is the condensate velocity

1st arm

2nd arm

Moving mirror problem in opticsMoore, J. Math. Phys. 9:2679 (1970)

Exciting photons in a cavity with a moving mirror

Experimentally always in the adiabatic regime

c is the speed of light

Splitting condensate on Y-junction: beyond mean-field

K – Luttinger parameterv is the condensate velocity

d

This is similar to the usual 1d condensatesExtra suppression due to finite velocity of splitting

Splitting with acceleration: Unruh like effectFagnocchi, Altman, Demler

Splitting condensates with relativistic acceleration gives rise to thermal correlations.This is analogous to the Unruh effect in field theory and quantum gravityUnruh (1974), Fulling and Davies (1976)

Summary

Interference experiments with ultracold atoms provide a powerful tool for analyzing equilibrium propertiesand dynamics of many-body systems. Analysisbeyond mean-field and single mode approximationis needed important

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