1. Nonequilibrium dynamics of ultracold atoms in optical lattices
Nonequilibrium dynamics of ultracold atoms in optical lattices. Lattice modulation experiments and more
David Pekker, Rajdeep Sensarma, Ehud Altman,
Takuya Kitagawa, Susanne Pielawa, Adilet Imambekov,
Mikhail Lukin, Eugene Demler
Harvard University
Collaboration with experimental groups of
I. Bloch, T. Esslinger, J. Schmiedmayer
$$ NSF, AFOSR, MURI, DARPA,

2. Outline Lattice modulation experiments Fermions in optical lattice.
Probe of antiferromagnetic order
Fermions in optical lattice.
Decay of repulsively bound pairs.
Nonequilibrium dynamics of
Hubbard model
Ramsey interferometry and many-body decoherence.
Quantum noise as a probe of dynamics

3. Lattice modulation experiments with fermions in optical lattice.
Probing the Mott state of fermions
Sensarma, Pekker, Lukin, Demler, arXiv:
Related theory work: Kollath et al., PRA 74:416049R (2006)
Huber, Ruegg, arXiv:0808:2350

4. Same microscopic model
Antiferromagnetic and superconducting Tc
of the order of 100 K
Atoms in optical lattice
Antiferromagnetism and
pairing at sub-micro Kelvin
temperatures
Same microscopic model

5. Probing many-body states
Electrons in solids
Fermions in optical lattice
Thermodynamic probes
i.e. specific heat
System size, number of doublons
as a function of entropy, U/t, w0
X-Ray and neutron
scattering
Bragg spectroscopy,
TOF noise correlations
Optical conductivity
STM
Lattice
modulation
experiments
Beyond analogies with condensed matter systems:
Far from equilibrium quantum many-body dynamics

6. Signatures of incompressible Mott state of fermions in optical lattice
Suppression of double occupancies
Jordens et al., Nature 455:204 (2008)
Compressibility measurements
Schneider et al., Science 5:1520 (2008)

7. Lattice modulation experiments
Probing dynamics of the Hubbard model
Modulate lattice potential
Measure number of doubly
occupied sites
Main effect of shaking: modulation of tunneling
Doubly occupied sites created when frequency w matches Hubbard U

8. Lattice modulation experiments
Probing dynamics of the Hubbard model
R. Joerdens et al., Nature 455:204 (2008)

9. Mott state Regime of strong interactions U>>t.
Mott gap for the charge forms at
Antiferromagnetic ordering at
“High” temperature regime
All spin configurations are equally likely.
Can neglect spin dynamics.
“Low” temperature regime
Spins are antiferromagnetically ordered
or have strong correlations

10. Schwinger bosons and Slave Fermions
Constraint :
Singlet Creation
Boson Hopping

11. Schwinger bosons and slave fermions
Fermion hopping
Propagation of holes and doublons is coupled to spin excitations.
Neglect spontaneous doublon production and relaxation.
Doublon production due to lattice modulation perturbation
Second order perturbation theory. Number of doublons

12. “Low” Temperature d h = + Self-consistent Born approximation
Schwinger bosons Bose condensed d h
Propagation of holes and doublons strongly
affected by interaction with spin waves
Assume independent propagation
of hole and doublon (neglect vertex corrections)
Self-consistent Born approximation
Schmitt-Rink et al (1988), Kane et al. (1989) = +
Spectral function for hole or doublon
Sharp coherent part:
dispersion set by J, weight by J/t
Incoherent part:
dispersion

13. Propogation of doublons and holes
Spectral function:
Oscillations reflect shake-off processes of spin waves
Comparison of Born approximation and exact diagonalization: Dagotto et al.
Hopping creates string of altered spins: bound states

14. “Low” Temperature Rate of doublon production Spin wave shake-off peaks
Sharp absorption edge due to coherent quasiparticles
Broad continuum due to incoherent part
Spin wave shake-off peaks

15. “High” Temperature Atomic limit. Neglect spin dynamics.
All spin configurations are equally likely.
Aij (t’) replaced by probability of having a singlet
Assume independent propagation of doublons and holes.
Rate of doublon production
Ad(h) is the spectral function of a single doublon (holon)

16. Propogation of doublons and holes
Hopping creates string of altered spins
Retraceable Path Approximation Brinkmann & Rice, 1970
Consider the paths with no closed loops
Spectral Fn. of single hole
Doublon Production Rate
Experiments

17. Lattice modulation experiments. Sum rule
Ad(h) is the spectral function of a single doublon (holon)
Sum Rule :
Experiments:
Most likely reason for
sum rule violation:
nonlinearity
The total weight does not scale
quadratically with t

18. Fermions in optical lattice. Decay of repulsively bound pairs
Experiment: ETH Zurich, Esslinger et al.,
Theory: Sensarma, Pekker, Altman, Demler

19. Fermions in optical lattice. Decay of repulsively bound pairs
Experiments: T. Esslinger et. al.

20. Relaxation of repulsively bound pairs in the Fermionic Hubbard model
U >> t
For a repulsive bound pair to decay,
energy U needs to be absorbed
by other degrees of freedom in the system
Relaxation timescale is important for quantum
simulations, adiabatic preparation

21. Relaxation of doublon hole pairs in the Mott state
Energy U needs to be
absorbed by
spin excitations
Relaxation requires
creation of ~U2/t2
spin excitations
Energy carried by
spin excitations
~ J =4t2/U
Relaxation rate
Very slow Relaxation

22. Doublon decay in a compressible state
Excess energy U is
converted to kinetic
energy of single atoms
Compressible state: Fermi liquid description U
p-p
p-h
Doublon can decay into a
pair of quasiparticles with
many particle-hole pairs

23. Doublon decay in a compressible state
Perturbation theory to order n=U/t
Decay probability
To calculate the rate: consider processes which maximize the number of particle-hole excitations

24. Doublon decay in a compressible state
Single fermion
hopping
Doublon decay
Doublon-fermion
scattering
Fermion-fermion
scattering due to
projected hopping

25. Fermi’s golden rule G = Neglect fermion-fermion scattering gk1 gk2 2 +
+ other spin combinations
Particle-hole emission
is incoherent: Crossed diagrams unimportant
gk1
gk2
gk = cos kx + cos ky + cos kz

26. Self-consistent diagrammatics
Calculate doublon lifetime from Im S
Neglect fermion-fermion scattering
Emission of particle-hole pairs is incoherent:
Crossed diagrams are not important
Suppressed by vertex functions

27. Self-consistent diagrammatics
Neglect fermion-fermion scattering
Comparison of Fermi’s Golden
rule and self-consistent diagrams
Need to include fermion-fermion scattering

28. Self-consistent diagrammatics
Including fermion-fermion scattering
Treat emission of particle-hole pairs as incoherent
include only non-crossing diagrams
Analyzing particle-hole emission as coherent process requires adding decay amplitudes and then calculating net decay rate. Additional diagrams in self-energy need to be included
No vertex functions to justify
neglecting crossed diagrams

29. Including fermion-fermion scattering
Correcting for missing diagrams
type present
type missing
Assume all amplitudes for particle-hole pair production
are the same. Assume constructive interference
between all decay amplitudes
For a given energy diagrams of a certain order dominate.
Lower order diagrams do not have enough p-h pairs to absorb energy
Higher order diagrams suppressed by additional powers of (t/U)2
For each energy count number of missing crossed diagrams
R[n0(w)] is renormalization
of the number of diagrams

30. Doublon decay in a compressible state
Doublon decay with generation of particle-hole pairs

31. Ramsey interferometry and
many-body decoherence
Quantum noise as a probe of
non-equilibrium dynamics

32. Interference between fluctuating condensates
high T
L [pixels]
0.4
0.2 10 20 30
middle T
low T
high T
BKT x z
Time of flight
low T
2d BKT transition: Hadzibabic et al, Claude et al
1d: Luttinger liquid, Hofferberth et al., 2008

33. Distribution function of interference fringe contrast
Hofferberth, Gritsev, et al., Nature Physics 4:489 (2008)
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
First demonstration of quantum fluctuations for 1d ultracold atoms
First measurements of distribution function of quantum
noise in interacting many-body system

34. Can we use quantum noise as a probe of dynamics?
Example:
Interaction induced collapse of Ramsey fringes
A. Widera, V. Gritsev, et al. PRL 100: (2008),
T. Kitagawa, S. Pielawa, A. Imambekov, J. Schmiedmayer, E. Demler

35. Ramsey interference Atomic clocks and Ramsey interference:1
Ramsey interference is basically Larmor precession for spin ½ particles. If we take a state which is a superposition of spin
up and spin down state, the two states will evolve with different energies. So the spin rotates in the plane with the frequency determined
by the energy difference. Ramsey precession underlies atomic clocks and is done not with one atoms but with many.
Atomic clocks and Ramsey interference:
Working with N atoms improves
the precision by

36. Interaction induced collapse of Ramsey fringes
Two component BEC. Single mode approximation.
Kitagawa, Ueda, PRA 47:5138 (1993)
Ramsey fringe visibility
time
Experiments in 1d tubes:
A. Widera et al. PRL 100: (2008)
What is the role of interaction on Ramsey precession?

37. Spin echo. Time reversal experiments
Single mode analysis
The Hamiltonian can be reversed by changing a12
Predicts perfect spin echo

38. Spin echo. Time reversal experiments
Expts: A. Widera et al., PRL (2008)
Experiments done in array of tubes.
Strong fluctuations in 1d systems.
Single mode approximation does not apply.
Need to analyze the full model
No revival?

39. Interaction induced collapse of Ramsey fringes. Multimode analysis
Low energy effective theory: Luttinger liquid approach
Luttinger model
Changing the sign of the interaction reverses the interaction part
of the Hamiltonian but not the kinetic energy
Time dependent harmonic oscillators
can be analyzed exactly

40. Ramsey interferometry
Interaction induced collapse of Ramsey fringes in one dimensional systems
Only q=0 mode shows complete spin echo
Finite 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
Fundamental limit on
Ramsey interferometry
How to distinquish decoherence due to many-body dynamics?

41. Interaction induced collapse of Ramsey fringes
Single mode analysis
Kitagawa, Ueda, PRA 47:5138 (1993)
Multimode analysis
evolution of spin distribution functions
T. Kitagawa, S. Pielawa, A. Imambekov, J. Schmiedmayer, E. Demler

42. Summary Lattice modulation experiments as a probe of AF order
Fermions in optical
lattice. Decay of repulsively
bound pairs
Ramsey interferometry
in 1d. Luttinger liquid
approach to many-body
decoherence

43. Thanks to
Harvard-MIT