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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 ([email protected])
Jussi Kajalanow at