Making a spin difference with ultracold gases

Päivi Törmä

Aalto University

Colloquium, Research Training Group 1729HannoverOctober 24th, 2013

ContentsBackground: Fermi condensates

Spin-density imbalance and superfluidity

Is the coexistence of magnetization and superfluiditypossible? Is the exotic FFLO (Fulde-Ferrel-Larkin-Ovchinnikov) state stable?

Pairing in mixed (spin-dependent) geometriesHow is superfluidity affected by having differentconfining geometries for the spins?

Expansion dynamics in one dimension

Does expansion in 1D tell us as much as in higherdimensions – or more?

Background: Fermi condensatesSpin-density imbalance and superfluidity

Is the coexistence of magnetization and superfluiditypossible? Is the exotic FFLO (Fulde-Ferrel-Larkin-Ovchinnikov) state stable?

Pairing in mixed (spin-dependent) geometriesHow is superfluidity affected by having differentconfining geometries for the spins?

Expansion dynamics in one dimension

Does expansion in 1D tell us as much as in higherdimensions – or more?

Bose-Einstein condensates are made of bosonic atoms. Other bosons: e.g. photonsFermions are constituents of matter: electrons, protons, neutrons, some atoms…

Fermi condensates?

r

E

00 r

E

00

01

0

1

Fermions have to make pairs (which are bosons)to condense to the lowest state

BEC-BCS crossoverRelated to, e.g., high temperature superconductivity

Fermi condensates 2004-2005

BEC of molecules (dimers of twoFermions) 2003-2004Grimm, Jin, Ketterle, Salomon

Fermion pairs near the FeshbachResonance 2004Jin, Ketterle

Density profile throughout thecrossoverGrimm 2004

Collective modes Thomas 2004,Grimm 2004

Pairing gap Grimm 2004

0.0

0.4

(d)

(c)

(a)

0.0

0.4

-20 0 20 400.0

0.4

fract

iona

l los

s in

|2>

RF frequency offset (kHz)

0.0

0.4(b)

Heat capasity Thomas 2005

Science and Physics World ranked the observation of Fermi condenScience and Physics World ranked the observation of Fermi condensates sates among the top ten scientific breakthroughs of the year 2004among the top ten scientific breakthroughs of the year 2004

Vortices Ketterle 2005

P. Törmä and P. Zoller, Phys. Rev. Lett. 85, 487 (2000)

C. Chin, M. Bartenstein, A. Altmayer,S. Riedl, S. Jochim, J.H. Denschlag,and R. Grimm, Science 305, 1128, 2004

J. Kinnunen, M. Rodriguez, and P. Törmä, Science 305, 1131, 2004

Background: Fermi condensates

Spin-density imbalance and superfluidityIs the coexistence of magnetization and superfluiditypossible? Is the exotic FFLO (Fulde-Ferrel-Larkin-Ovchinnikov) state stable?

Pairing in mixed (spin-dependent) geometriesHow is superfluidity affected by having differentconfining geometries for the spins?

Expansion dynamics in one dimension

Does expansion in 1D tell us as much as in higherdimensions – or more?

JR Maglev MLX01 – 581 km/h (Japan, 2003)   fMRI: brain imaging

LHC: Go Higgs! Future? Superconducting power grid

Long Island, NY 

SUPERCONDUCTIVITY IN OUR LIVES

What is superconductivity?

Not really: More like:

Cooper pairs of spin up and spin down electrons

BEC BCS

− 1k F a

0

The Fermi surface

EF

(kF)

ky

kx

2yxF k+kE 2∝

YBCO

LaOFeAs

Cuprates

Fe‐based superconductorsFermi surfaces (BaKFeAs)

EPL 83, 47001 (2008).Spin fluctuations

FeSc

Science 328, 441 (2010) 

High-temperature superconductors:complicated structure, spin plays a role (?)

Polarized Fermi gases

Pairing between particles with unequal mass or unequal total number

Related to, e.g., high energy physics (colour superconductivityof quarks)

↓↑

↓↑

+−

=NNNN

P

Polarization

1. Magnetism versus Superconductivity

Chandrasekhar-Clogston limit

critical magnetic field to break superconductivity

Chandrasekhar, APL 1962Clogston, PRL 1962

2. Exotic superconducting phase?

Fulde-Ferrell-Larkin-Ovchinnikov StatesOscillating order parameter (FF)

(LO)

FF, PR 1964LO, JETP 1965

Spin-Population Imbalanced Fermi Gases

Polarized Superfluid StatesSarma, J Phys Chem Solids 1963Liu & Wilczek, PRL 2003; Sheehy & Radzihovsky, PRL 2006,Pao, Wu, Yip, PRB 2006; Parish et al., PRL 2007, Pilati & Giorgini PRL 2008, etc.

fully paired+ excess unpaired

↓↑ ≠ μμ OR

Ketterle 2005 P=0P=1

r

E

Harmonic trap in 3D:The gas is a ball with highestdensity in the middle

00

r0

density nV(r) ‏

The effect of the (harmonic) trappingpotential

)(xVtrapeff −= μμ

Spin-imbalanced fermionsin 3D elongated traps

D.-H. Kim, J. J. Kinnunen, J.-P. Martikainen, PT, PRL 106, 095301 (2011)

Partridge et. al., Science 311, 5760 (2006)

Partridge et. al., PRL 97, 190407 (2006)Shin et. al., PRL 97, 030401 (2006)

Nascimbene et. al., PRL 103, 170402 (2009)

C. Lobo, A. Recati, S. Giorgini, S. Stringari, PRL 97, 200403 (2006)

Experiments:

QMC:

Zwierlein et. al., Science 311, 492 (2006)

DMFT:

Phase separation; no FFLO

Prediction: FFLO is stabilized in lattices! (mean-field calculation)

T.K. Koponen, T. Paananen, J.-P. Martikainen, and P.T., PRL 99, 120403 (2007)

19

Fermi surfaces

Freespace

Lattice

↓↑ −= FF kkqT.K. Koponen, T. Paananen, J.-P. Martikainen, and P.T., PRL 99, 120403 (2007)

FFLO in quasi-1D to 3D lattices

D. H. Kim, PT, PRB 85, 180508(R) (2012)M.O.J. Heikkinen, D.H. Kim, PT, PRB 87, 224513 (2013)

Stronger FFLO signature in (continuum) quasi-1D?

?

In 1D, an exact solution gives FFLO, but no long-range order is possible.

In 3D, the FFLO area may be very narrow.

1D vs. 3D

Somewhere between 3D and 1D: an ideal place for FFLO?

Parish et al., PRL 99, 250403 (2007)

Liao et al., Nature 467, 567 (2010)

Mean-field theory (T=0)

Experiment in 1D (density profiles)

The long-range order may stabilize FFLO.

Finite temperature phase diagram

Confirms the mean-field predictions about stabilityof FFLO in lattices (due to nesting)

In contrast, dimensionality does not play a large role,unlike suggested by mean-field: the FFLO is prominentthroughout the crossover

T.K. Koponen, T. Paananen, J.-P. Martikainen, and PT, PRL 99, 120403 (2007)Y.L. Loh and N. Trivedi, PRL 104, 165302 (2010)

Background: Fermi condensates

Spin-density imbalance and superfluidity

Is the coexistence of magnetization and superfluiditypossible? Is the exotic FFLO (Fulde-Ferrel-Larkin-Ovchinnikov) state stable?

Pairing in mixed (spin-dependent) geometriesHow is superfluidity affected by having differentconfining geometries for the spins?

Expansion dynamics in one dimension

Does expansion in 1D tell us as much as in higherdimensions – or more?

New in ultracold gas experiments: spin-dependent trapping potentials

Different electronic (e.g. hyperfine) states (Bloch, Sengstock,Porto, etc.)

ORDifferent fermionic atoms (e.g. 6Li, 40K) (Grimm, Schreck, Hänsch,Dieckmann, Salomon, Walraven, Zwierlein, Inguscio, etc.)

Theory work with spin-dependence/mass-imbalance and fermions e.g. from the groups:Giamarchi, Liu, Wilczek, Zoller, Hofstetter, Demler, Lukin, Stringari, Rozenberg, Dalmonte, Das Sarma, Stoof, Sa de Melo, Batrouni, Scalettar

Pairing in mixed geometries

• Pairing with spin-imbalance (mass-imbalance)• Chandrasekhar-Clogston limit, FFLO, polarized superfluids, breach pair

superfluids, etc.• Pairing in mixed dimensions

• A couple of theory papers (Tan, Iskin)• Our question: pairing in mixed geometries?

• D.-H. Kim, J.S.J. Lehikoinen, PT, PRL 110, 055301 (2013)• Motivation

• Spin-dependent confinement• Novel superfluids? High Tc? Polarized superfluid?

Our choice of geometry: honeycomb lattice for the up-component, triangular for the down-component

Hubbard-type model, mean-field theory

The non-interacting system

Pairing ansatz at site A

Bogoliubov transform; then minimize energy to find phases

The phase diagram

A new stable polarized superfluid phase: incomplete breach pair (iBP) state

Quantum phase transitions (Lifshitz transitions)between states with different Fermi surfacetopology

Mixed geometry: multiband pairing

E=0

Background: Fermi condensates

Spin-density imbalance and superfluidity

Is the coexistence of magnetization and superfluiditypossible? Is the exotic FFLO (Fulde-Ferrel-Larkin-Ovchinnikov) state stable?

Pairing in mixed (spin-dependent) geometriesHow is superfluidity affected by having differentconfining geometries for the spins?

Expansion dynamics in one dimensionDoes expansion in 1D tell us as much as in higherdimensions – or more?

Expansion of a band insulator stateDynamics of a many-body Fermion systemU. Schneider, L. Hackermuller, J.P. Ronzheimer, S. Will, S. Braun, T. Best, I.

Bloch, E. Demler, S. Mandt, D. Rasch, A. Rosch, Breakdown of diffusion: From collisional hydrodynamics to a continuous quantum walk in ahomogeneous Hubbard model, Nat. Phys. 2012

Core expansion speedas a function of interaction

The system

J. Kajala, F. Massel, PT, PRL 106, 206401 (2011)

Earlier 1D dynamics: Kollath, Schollwöck, Zwerger,Heidrich-Meisner, Rigol, Cirac, Zoller, Shlyapnikov,Daley, Santos, Carr, Pupillo, Tezuka, Ueda, Diehl, our group, etc.

Results (TEBD (~ t-DMRG))

• With TEBD one can also calculate the doublon density

• Here we call doublons excitations of the form

Inspired by two-fluid models in the imbalanced 1D ultracold gascontext: E. Zhao, W. Liu, G. Orso, H. Hu, X. Liu, P. Drummond, etc.

Experiment by Schneider et al.,2D

Our simulation, 1D

• Our explanation for the numerical findings: describe the two last sites by the Hubbard dimer

• We are interested in the paired state <-> singlet time evolution.

• The singlet state•

• Compare to numerics

• In the light of the Dimer analysis, let us take another look at the TEBD results.

Experiment by Schneider et al.,2D

Our simulation, 1D

Expansion of an FFLO state in a 1D lattice

J. Kajala, F. Massel, PT, PRA 106, 206401(R) (2011)

Inspired by the (continuum) 1D experiment: Liao et al., Nature 467, 567 (2010)

FFLO in 1D lattice

Feiguing and Heidrich-Meisner 2007, Batrouni et al. 2008, Rizzi et al. 2008

Consistent with Betheansatz in the large U limit

unpaired

unpaired

q

c.f. C.J. Bolech, F. Heidrich-Meisner, S. Langer, I.P. McCulloch,G. Orso, M. Rigol, PRL 109, 110602 (2012): FFLO correlations lostduring the expansion; however, as we point out the initial FFLO q is imprintedto the fastest majority particles that travel at the edge of the cloud.

Summary: measuring the expansion velocity of the edge(majority particles) gives the FFLO q-vector!

J. Kajala, F. Massel, PT, PRA 106, 206401(R) (2011)

Dynamics of an impurity in a one-dimensional lattice

F. Massel, A. Kantian, A.D. Daley, T. Giamarchi, PT, submitted to New J. Phys. 15, 045018 (2013)

c.f. 1D impurity dynamics experiments for bosons:Inguscio group, Bloch group

Polaronsin 2D/3DGrimm, Zwierlein,Köhl, Salomon,Zwerger,Chevy,Lobo,Bruunetc.

ConclusionsSpin-density imbalance and superfluidity

The FFLO state stable in lattices at finite T (DMFT)

Pairing in mixed (spin-dependent) geometriesNew iBP polarized superfluid state at T=0, Lifshitztransitions, multiband character essential

Expansion dynamics in 1D

Band insulator state expansion explained by a simpletwo-site model

FFLO state directly reflected in the expansion velocityof the majority atoms

Dong-Hee Kim (now assistant professor at GIST, South Korea)

Joel Lehikoinen(now at his own start-up)

Thanks: Academy of Finland

Francesco Massel(now at University

of Helsinki)

Miikka Heikkinen

Ph.D. and postdoc positions, ERC AdG projectstarting late 2013 (paivi.torma@aalto.fi)

Jussi Kajalanow at