Modulation spectroscopyModulation spectroscopyBEC polaronsQuasicrystals

Everything is work in progress

$$ NSF, MURI ATOMTRONICS, MURI QuISM, DARPA OLE

Harvard-MIT

Lattice modulation experiments as aLattice modulation experiments as a probe of collective mode dispersion

Based on recent discussion with Immanuel BlochRecent results by Michael Knap (Harvard)Earlier results by David Pekker (Harvard,Caltech)

Modulation experiments in optical latticeObservation of the Anderson Higgs modeObservation of the Anderson-Higgs mode

M. Endres et al., Nature 487:454 (2012)

Lattice modulation probe of collective mode dispersion

Modulation performed at zero wavevector.How can this probe finite momentum modes?How can this probe finite momentum modes?

Faraday waves M F d Phil hi l T ti f th R l S i tM. Faraday, Philosophical Transactions of the Royal Society (London), vol. 121, pages 299–318 (1831)

Faraday waves in BEC B li b thBogoliubov theory

Time evolution can be solved exactly

Resonance

Faraday waves in BEC

From wavefunction

we expect the appearance of and

Density acquires ripples

We expect ripples peaked at the resonance wavevector

Faraday waves in BEC. One dimensional caseOne dimensional case

U=0.11n=1

Faraday waves in BEC. Two dimensional caseTwo dimensional case Different driving geometries

Faraday waves in BEC. Two dimensional caseTwo dimensional case

Different driving amplitudes

Faraday waves in BEC. Two dimensional caseTwo dimensional case

Different pulse durations

Modulation spectroscopy

PRA (2006)

Modulation spectroscopyModulation of interaction strength

Modulation spectroscopyInteresting applications

Particle-hole excitations in the Mott state, excitations in the critical region

Probe of spin waves in a ferromagnetic state

Quantum simulations with ultracold atoms

Atoms in optical lattice

Antiferromagnetic and superconducting Tc

Atoms in optical lattice

Antiferromagnetism and pairing at sub-micro Kelvin p g

of the order of 100 K temperatures

Same microscopic model

Positive U Hubbard modelPositive U Hubbard modelPossible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)

Antiferromagnetic insulator

D-wave superconductorp

--

Signature of d-wave pairing of fermions: dispersion of quasiparticlesdispersion of quasiparticles

- Superconducting gap

++-

Normal state dispersion of quasiparticles

Quasiparticle energies

Low energy quasiparticles correspond to four Dirac nodesto four Dirac nodes

Observed in:

• Photoemission• Raman spectroscopy• T-dependence of thermodynamic and transport properties c and transport properties, cV, , L

• STM• and many other probes

Modulation spectroscopyProbe of quasiparticles in d-wave paired state

D. Pekker et al., 

Probing polaronsin bosonic systemsin bosonic systems

Dima Abanin (Harvard Perimeter),Adiya Shashi (Rice/Harvard)Eugene DemlerEugene Demler+ recent discussion with Immanuel Bloch

Polarons in ultracold atoms: fermionic environment

2D Kohl et al., Nature (2012)3D Zwierlein et al., PRL (2009)3D Zwierlein et al., PRL (2009)

Polarons in BEC

BEC regimeg

Polarons in BEC

Dynamics of BEC polarons. Spectral functionRF t f i t t

Interacting

RF spectroscopy of spin states

g

NoninteractingNoninteracting

Transitions of localized impurity atoms

Change of the impuritywavefunction excitesBogoliubov phononsg pin BEC

Dynamics of BEC polarons. Lang-Firsov transformation approachg pp

Lang-Firsov transformation

Dynamics of BEC polarons. Langs-Firsov transformation approachg pp

Conserved total momentum of the system p is now explicitly separated

Interaction between impurity and phonons was traded forinteractions between phonons

Dynamics of BEC polarons. Spectral functionp

Dynamics of BEC polarons. RF absorption for infinite impurity massp p y

d=3

Dynamics of BEC polarons. RF absorption of localized impurityRF absorption of localized impurity

1

Dynamics of BEC polarons. RF b ti f l li d i itRF absorption of localized impurity

Dynamics of BEC polarons. RF absorption for finite impurity massp p y

Signatures of self-trapping transition?

Quantum quasicrystalsof spin orbit coupled bosons

Sarang Gopalakrishnan (Harvard)

I M ti (L Al /A )Ivar Martin (Los Alamos/Argonne)

Eugene Demler (Harvard)

What are quasicrystals

Associated with fivefold (or sevenfold) symmetry: pentagons

Translational order without periodicity( iti f til d t i th ti tili b t th tili i i di )

in 2D, dodecahahedra/icosahedra in 3D

(position of tile determines the entire tiling, but the tiling is aperiodic)

Sharply defined Bragg peaks that are inconsistent with space group(spatial periodicity)(spatial periodicity)

Conjectured in models with multiple incommensurate length scales(e g Mermin Troian PRL 1985)(e.g. Mermin, Troian, PRL 1985)

Origin is still a subject of debate (equilibrium or metastable states)

Phase diagram of Bose-Fermi mixturewith spin orbit coupling for bosons

2kF

p p g

k0q0

One species of fermions Two species of fermions

Spin-orbit coupled BECs

Mean-field phase diagram

Idea: Bose-Fermi mixtures

Use Fermi momentum to engineerUse Fermi momentum to engineermomentum dependent interactions

Make 2kf to provide 5-foldcovering of the spin orbit circlecovering of the spin-orbit circle

Quasicrystals in systems with competing lengthscales Mermin, Troian, PRL 1985

Order parameter y has preferred wave vector q

Order parameters fi have preferred wave vectors kiMany possible phases depending on q, ki

Competition of Bravais lattices and quasicrystals

k k

Competition of Bravais lattices and quasicrystals

k1

q

k k

q q

k1

k2q q

k

q2

Fermion mediated interaction (RKKY)

k pk

k’

p

p’

We need to include renormalization of V(k-k’)

Renormalization of RKKY interaction

k p

e o a a o o e ac o

k

k’

p

p’

k

k’p’

p

T-matrix renormalization of Bose-Fermi interaction

k p

k’ p’

“Stuff” is angle dependent

T-matrix renormalization of Bose-Fermi interaction

Renormalized Bose-Fermi interaction

Renormalized interaction is peaked at 2kf

Mean-field theory with renormalized interactionsy

Example m=5

Mean-field theory with renormalized interactionsyOne species of fermions

2kF

k0q0

Several competing Bravais latticesNo quasicrystals

Mean-field theory with renormalized interactionsyTwo species of fermions

Quasicrystal is the only configuration that can benefit from both

Toward 3d quasicrystals

Interaction renormalization in 3d

I i tt ti VIncreasing attractive V

Renormalized RKKY interaction in in 3dRenormalized RKKY interaction in in 3d

Quantum quasicrystals beyond mean-fieldNew aspects of crystal order intertwined with superfluidity.Example FFLO

Significance of wavefunction signHybrid defects, e.g. half-vortex half-dislocation orhalf vortex half disclinationhalf-vortex half-disclination

Even richer phenomena for quantum quasicrystals

SSummary

Modulation spectroscopy

BEC polaronsBEC polarons

Quasicrystals