Condensed Matter models for many-body systems of ultracold atoms Eugene Demler Harvard University

Collaborators:Ehud Altman, Robert Cherng, Adilet Imambekov, Vladimir Gritsev, Takuya Kitagawa, Susanne Pielawa,David Pekker, Rajdeep Sensarma

Experiments: Bloch et al., Esslinger et al., Schmiedmayer et al., Stamper-Kurn et al.

Harvard-MIT

New Tricks for Old Dogs,Old Tricks for New Dogs

Condensed Matter models for many-body systems of ultracold atoms

Dipolar interactions. Magnetoroton softening, spin textures, supersolid. New issues: averaging over Larmor precession, coupling of spin textures and vorticesR. Cherng, V. Gritsev. In collaboration with D. Stamper-Kurn

Luttinger liquid. Ramsey interferometry and many-body decoherence in 1d. New issues: nonequilibrium dynamics, analysis of quantum noise.V. Gritsev, T. Kitagawa, S. Pielawa. In collaboration with expt. groups of I. Bloch and J. Schmiedmayer

Hubbard model. Fermions in optical lattice.

Decay of repulsively bound pairs. New issues: nonequilibrium dynamics in strongly interacting regime.D. Pekker, R. Sensarma, E. Altman. In collaboration with expt. group of T. Esslinger

Summary

Dipolar interactions in spinor condensates.Magnetoroton softening and spin texturesR. Cherng, V. Gritsev. In collaboration with D. Stamper-Kurn

Roton minimum in 4He

Glyde, J. Low. Temp. Phys. 93 861

Phase diagram of 4He

Possible supersolid phase in 4He

A.F. Andreev and I.M. Lifshits (1969):Melting of vacancies in a crystal due to strong quantum fluctuations.

Also G. Chester (1970); A.J. Leggett (1970)

D. Kirzhnits, Y. Nepomnyashchii (1970),T. Schneider and C.P. Enz (1971).Formation of the supersolid phase due tosoftening of roton excitations

Roton spectrum in pancake polar condensates

Santos, Shlyapnikov, Lewenstein (2000)Fischer (2006)

Origin of roton softening

Repulsion at long distances Attraction at short distances

Stability of the supersolid phase is a subject of debate

Cold atoms: magnetic dipolar interactions

x y

x

y

y

)(4

)(2

rm

arU F

contact

)()(

ˆˆ

4)(

3ySxS

r

rrCrU ji

iiijdd

dipolar

For 87Rb =B and =0.007For 52Cr =6B and =0.16

Menotti et al., arxiv:0711.3422

rr

r

Short-rangedcontact interactions

Long-ranged, anisotropicdipolar interactions

Contact vs. dipolar interactions

Magnetic dipolar interactions in spinor condensates

Interaction of F=1 atoms

Ferromagnetic Interactions for 87Rb

Spin-depenent part of the interaction is small. Dipolar interaction may be important (D. Stamper-Kurn)

a2-a0= -1.07 aB

A. Widera, I. Bloch et al., New J. Phys. 8:152 (2006)

Spinor condensates at Berkeley

M. Vengalattore et al., arXiv:0901.3800

Spinor condensates at Berkeley

Hz5.1,ˆ//ˆ qxB

kIm

Competing energy scales

Quadratic Zeeman (0-20 Hz)

Spin dependentS-wave scattering (gsn=8 Hz)

Dipolar interaction(gdn=0.8 Hz)

Quasi-2D geometry

2

~

FBE

3~ r

spind BF

Precession (115 kHz)

Spin independentS-wave scattering (gsn=215 Hz)

High energy scales

Low energy scales

Dipolar interactions after averaging over Larmor precession

Dipolar interactions

parallel to is preferred

“Head to tail” component dominates

Static interaction

Averaging over Larmor precession

z

perpendicular to is preferred. “Head to tail” component is averaged with the “side by side”

Instabilities: qualitative picture

Stability of systems with static dipolar interactions

Ferromagnetic configuration is robust against small perturbations. Any rotation of the spins conflicts with the “head to tail” arrangement

Large fluctuation required to reach a lower energy configuration

XY components of the spins can lower the energy using modulation along z.

Z components of thespins can lower the energyusing modulation along x

X

Dipolar interaction averaged after precession

“Head to tail” order of the transverse spin components is violated by precession. Only need to check whether spins are parallel

Strong instabilities of systems with dipolar interactions after averaging over precession

X

Instabilities: technical details

Quad ZeemanPrecession

Spin dep.

Dipolar

Spin indep.

Hamiltonian

Effective dipolar interaction:Spatial and time averaging

Larmor precession comoving frame

Gaussian profile

Field Ansatz

Time-averaged Quasi-2D Effective dipolar interaction

Effective dipolar interactionTime-averaged Quasi-2D Effective dipolar interaction

BF ˆ

BF ˆ//

B̂ B̂

B̂ B̂

Bk

BF

ˆ//

ˆ

Bk

BF

ˆ

ˆ//

F F

Collective Modes

Spin ModeδfB – longitudinal magnetizationδφ – transverse orientation

Charge Modeδn – 2D density

δη – global phase

ikxitx k exp~,

Mean Field

Collective Fluctuations(Spin, Charge)

δφ

δfB

δη

δn

Ψ0

Equations of Motion

Instabilities of collective modes

Q measures the strengthof quadratic Zeeman effect

Collective mode phase diagram

Zeeman

quad

ˆ)sin(

ˆ)cos(ˆ

q

x

nB

R

R0

C┴ DBC┴ CB D┴CB

D┴

Berkeley Experiments: checkerboard phase

M. Vengalattore, et. al, PRL 100:170403 (2008)

Spin texture length scales

M. Vengalattore et al., arXiv:0901.3800

Spin axis modulation~30 μm

Spin modulation~10 μm

Most unstable mode• |k|2 cost in kinetic energy• |k| gain in dipolar energy• l ~ 30 μm

kIm

Hz5.1,ˆ//ˆ qxB

Finding a stable ground state

Non-linear sigma model:Spin textures cause phase twists

Spinor “vector potential”Energetic Constraints

Equations of motion for η

Effective kinetic energy

Non-linear sigma model

Topological charge(net vorticity)

Spin gradient

Vortex interaction

Dipolar interaction

Spin Textures

Unit Cell

Top. Charge Q Kinetic EnergyQ<0

Q>0

Min KE

Max KE

Spin Textures: Skyrmion Stripes

Unit Cell

Top. Charge Q Kinetic Energy

Unit Cell

Top. Charge Q Kinetic Energy

Spin Textures: Skyrmion Lattice

Unit Cell

Top. Charge Q Kinetic Energy

Unit Cell

Top. Charge Q Kinetic Energy

Quantum noise as a probe of non-equilibrium dynamics

Ramsey interferometry and many-body decoherence

T. Kitagawa, A. Imambekov, S. Pielawa, J. Schmeidmayer’s group. Continues earlier work with V. Gritsev, M. Lukin, I. Bloch’s group.Phys. Rev. Lett. 100:140401 (2008)

Working with N atoms improves the precision by .

Ramsey interference

t0

1

Atomic clocks and Ramsey interference:

Two component BEC. Single mode approximation

Interaction induced collapse of Ramsey fringes

time

Ramsey fringe visibility

Experiments in 1d tubes: A. Widera et al. PRL 100:140401 (2008)

Spin echo. Time reversal experiments

Single mode approximation

Predicts perfect spin echo

The Hamiltonian can be reversed by changing a12

Spin echo. Time reversal experiments

No revival?

Expts: A. Widera et al., Phys. Rev. Lett. (2008)

Experiments done in array of tubes. Strong fluctuations in 1d systems.Single mode approximation does not apply.Need to analyze the full model

Interaction induced collapse of Ramsey fringes.Multimode analysis

Luttinger model

Changing the sign of the interaction reverses the interaction part of the Hamiltonian but not the kinetic energy

Time dependent harmonic oscillatorscan be analyzed exactly

Low energy effective theory: Luttinger liquid approach

Time-dependent harmonic oscillator

Explicit quantum mechanical wavefunction can be found

From the solution of classical problem

We solve this problem for each momentum component

See e.g. Lewis, Riesengeld (1969) Malkin, Man’ko (1970)

Interaction induced collapse of Ramsey fringesin one dimensional systems

Fundamental limit on Ramsey interferometry

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

Decoherence due to many-body dynamics of low dimensional systems

How to distinquish decoherence due to many-body dynamics?

Single mode analysisKitagawa, Ueda, PRA 47:5138 (1993)

Multimode analysisevolution of spin distribution functions

T. Kitagawa, S. Pielawa, A. Imambekov, et al.

Interaction induced collapse of Ramsey fringes

Noise measurements using BEC on a chipIntereference of independent condensates

Hofferberth et al., Nature Physics 2008

Average contrast

Distributionfunction offringe contrast

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

Fermions in optical lattice.Decay of repulsively bound pairs

Experiment: ETH Zurich, Esslinger et al.,Theory: Sensarma, Pekker, Altman, Demler

Fermions in optical lattice.Decay of repulsively bound pairs

Experiments: T. Esslinger et. al.

Relaxation of repulsively bound pairs in the Fermionic Hubbard model

U >> t

For a repulsive bound pair to decay, energy U needs to be absorbedby other degrees of freedom in the system

Relaxation timescale is important for quantum simulations, adiabatic preparation

Doublon decay in a compressible state

To calculate the rate: consider processes which maximize the number of particle-hole excitations

Perturbation theory to order n=U/tDecay probability

Doublon decay in a compressible state

Doublon decay with generation of particle-hole pairs

Dipolar interactions. Magnetoroton softening and spin textures in spinor condensates.

Luttinger liquid. Ramsey interferometry and many-body decoherence in 1d.

Hubbard model. Fermions in optical lattice. Decay of repulsively bound pairs.

Outline

Thanks to

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