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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
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