A. de Paz (PhD), A. Chotia, A. Sharma, B. Laburthe-Tolra, E. - - PowerPoint PPT Presentation

Have left: B. Pasquiou (PhD), G. Bismut (PhD), M. Efremov , Q. Beaufils, J. C. Keller, T. Zanon, R. Barbé, A. Pouderous, R. Chicireanu Collaborators: Anne Crubellier (Laboratoire Aimé Cotton), J. Huckans, M. Gajda

• A. de Paz (PhD), A. Chotia, A. Sharma,
• B. Laburthe-Tolra, E. Maréchal, L. Vernac,
• P. Pedri (Theory),
• O. Gorceix (Group leader)

Dipolar chromium BECs, and magnetism

Small solitons Stable condensate Phonon spectrum Attractive interactions Effect of interactions on condensates Repulsive interactions Implosion of BEC for large atom number Superfluidity Spin dependent interactions Magnetism

Rice… ENS, JILA… Berkeley…

Dipole-dipole interactions

  



2 2 2 3

1 1 3cos ( ) 4

dd J B

V S g R      

Anisotropic Long range

Chromium (S=3): Van-der-Waals plus dipole-dipole interactions

R



B

d  6 

Partially attractive, partially repulsive Interactions couple spin and

• rbital degrees of freedom

Different dipolar systems « Magnetic atom » Molecule with (field induced-) electric dipole moment Rydberg atoms

2

d n ea  d ea 

B

d  

2 2

1 137   

4 8

10 n  

Dipole-dipole interactions

2 2

12

m dd dd VdW

m V a V      

Relative strength of dipole-dipole and Van-der-Waals interactions

Stuttgart: d-wave collapse, PRL 101, 080401 (2008) See also Er PRL, 108, 210401 (2012) See also Dy, PRL, 107, 190401 (2012) … and Dy Fermi sea PRL, 108, 215301 (2012) … and heteronuclear molecules… Anisotropic explosion pattern reveals dipolar coupling. Stuttgart: Tune contact interactions using Feshbach resonances (Nature. 448, 672 (2007))

1

dd

 

BEC collapses

0.16

dd

 

Cr:

R 

1

dd

 

BEC stable despite attractive part of dipole-dipole interactions

Polarized (« scalar ») BEC Hydrodynamics Collective excitations, sound, superfluidity

  



2 2 2 3

1 1 3cos ( ) 4

dd J B

V S g R      

Anisotropic Long-ranged R



Hydrodynamics: non-local mean-field Magnetism: Atoms are magnets Chromium (S=3): involve dipole-dipole interactions Multicomponent (« spinor ») BEC Magnetism Phases, spin textures…

Oven at 1500 °C Zeeman slower Evaporative cooling 100 nK 104 atoms

Vacuum 4 10-11 mbar

MOT 100 µK 106 atoms

52Cr BEC experiment

Small condensates (104 atoms) Oven at 1500 °C Many lasers ! Magnetic field control < 100 µG

1 – Hydrodynamic properties of a weakly dipolar BEC

• Collective excitations
• Bragg spectroscopy

2 – Magnetic properties of a dipolar BEC

• Spinor physics of a Bose gas with free magnetization
• (Quantum) magnetism in opical lattices

Interaction-driven expansion of a BEC

A lie: Imaging BEC after time-of-fligth is a measure of in-situ momentum distribution

Cs BEC with tunable interactions (from Innsbruck))

Self-similar, (interaction-driven) Castin-Dum expansion

• Phys. Rev. Lett. 77, 5315 (1996)

TF radii after expansion related to interactions

Pfau,PRL 95, 150406 (2005)

Modification of BEC expansion due to dipole-dipole interactions

TF profile Eberlein, PRL 92, 250401 (2004) Striction of BEC (non local effect)

3

( ) ( ') ( ') '

dd dd

r V r r n r d r   



(similar results in

• ur group)

Frequency of collective excitations

2 2

. d H dt   

Consider small oscillations, then

2 2 2 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3

3 3 3 H                             

with In the Thomas-Fermi regime, collective excitations frequency independent of number of atoms and interaction strength: Pure geometrical factor (solely depends on trapping frequencies) (Castin-Dum)

Observation of one collective mode

(parametric excitation)

22 20 18 16 14 8 6 4 2 20 15

temps (ms) Rayon (Ox) Rayon (Oy)

Time (ms) (Ox) Radius (µm) (Oy) Radius (µm)

1.2 1.0 0.8 0.6 20 15 10 5

Aspect ratio

Collective excitations of a dipolar BEC

Repeat the experiment for two directions of the magnetic field (differential measurement) Parametric excitations

( ) t ms

PRL 105, 040404 (2010) A small, but qualitative, difference (geometry is not all) Due to the anisotropy of dipole-dipole interactions, the dipolar mean-field depends on the relative orientation of the magnetic field and the axis of the trap

dd

    

Note : dipolar shift very sensitive to trap geometry : a consequence of the anisotropy of dipolar interactions

Bragg spectroscopy

Probe dispersion law Quasi-particles, phonons

• Rev. Mod. Phys. 77, 187 (2005)

( ) E k ck 

c is sound velocity c is also critical velocity Landau criterium for superfluidity

 healing length

( 2 )

k k k c

E E n g   

Bogoliubov spectrum

1 k 

 d  Moving lattice on BEC Lattice beams with an angle. Momentum exchange

2 sin( / 2)

L

k k  

Anisotropic speed of sound

0.15 0.10 0.05 0.00 3000 2000 1000

Frequency difference (Hz) Fraction of excited atoms Width of resonance curve: finite size effects (inhomogeneous broadening) Speed of sound depends on the relative angle between spins and excitation

2

( 2 ( (3cos 1))

k k k c d k

E E n g g      

2 2

4 ( ) (3cos 1) 3

k

d V k    

k

 k B

A 20% effect, much larger than the (~2%) modification of the mean-field due to DDI

Anisotropic speed of sound

An effect of the momentum-sensitivity of DDI

(See also prediction of anisotropic superfluidity of 2D dipolar gases : Phys.

• Rev. Lett. 106, 065301 (2011))

c (mm/s)

Theo Exp Parallel 3.6 3.4 Perpendicular 3 2.8 Good agreement between theory and experiment; Finite size effects at low q

1.2 1.0 0.8 0.6 20 15 10 5

Aspect ratio

Villetaneuse, PRL 105, 040404 (2010)

Stuttgart, PRL 95, 150406 (2005)

Collective excitations Striction Anisotropic speed of sound

0.15 0.10 0.05 0.00 3000 2000 1000

Frequency difference (Hz) Fraction of excited atoms

Bragg spectroscopy Villetaneuse Accepted in PRL (2012)

Conclusions (1) Hydrodynamic properties with weak dipole-dipole interactions

Interesting but weak effects in a scalar Cr BEC (far from Feshbach resonance)

Much more to come with… Cr ? Er ? Dy ? Molecules ? Induced dipoles (Rydberg atoms) ? Examples:

• rotonic excitation spectrum, associated instabilities
• solitons
• New vortex lattice structures
• New quantum phases in optical lattices (supersolidity, checkerboard)
• …

1 – Hydrodynamic properties of a weakly dipolar BEC

• Collective excitations
• Bragg spectroscopy

2 – Magnetic properties of a dipolar BEC

• Spinor physics of a Bose gas with free magnetization
• (Quantum) magnetism in opical lattices

Exchange energy Coherent spin oscillation Chapman, Sengstock…

 

1 0,0 1, 1 1,1 2    

Quantum effects! Klempt Stamper- Kurn Domains, spin textures, spin waves, topological states Stamper-Kurn, Chapman, Sengstock, Shin… Quantum phase transitions Stamper-Kurn, Lett

Introduction to spinor physics

Main ingredients for spinor physics Spin-dependent contact interactions Spin exchange

2 2

4 (a a m          

, 3 1 , 2 3 2 ,        

tot tot S S

m S m S m m

Quadratic Zeeman effect S=1,2,… Main new features with Cr S=3 7 Zeeman states 4 scattering lengths New structures Purely linear Zeeman effect Engineer artificial quadratic effect using tensor light shift Strong spin-dependent contact interactions

And Dipole-dipole interactions

• 1

1

Dipolar interactions introduce magnetization-changing collisions

Dipole-dipole interactions

  



2 2 2 3

1 1 3cos ( ) 4

dd J B

V S g R      

R



• 1

1

• 1

1

• 2
• 3

2 3

without

dd

V

with

dd

V

• 3
• 2
• 1

1 2 3

• 3 -2 -1 0 1 2 3

dd

V  

 

2 dd f B

V g B      

B=0: Rabi In a finite magnetic field: Fermi golden rule (losses) (x1000 compared to alkalis)

Important to control magnetic field Rotate the BEC ? Spontaneous creation of vortices ? Einstein-de-Haas effect

2  

B g m E

B S 

  

   

l S

m m

Angular momentum conservation

Dipolar relaxation and rotation

• 3
• 2
• 1

1 2 3

 

3 , 2 2 , 3 2 1 3 , 3  

Ueda, PRL 96, 080405 (2006) Santos PRL 96, 190404 (2006) Gajda, PRL 99, 130401 (2007)

• B. Sun and L. You, PRL 99, 150402 (2007)

• 3
• 2
• 1

1 2 3

B=1G Particle leaves the trap B=10 mG Energy gain matches band excitation in a lattice B=.1 mG Energy gain equals to chemical potential in BEC

Magnetic field

2 2

2 ) 1 ( ) ( R l l R Veff    

f J B

g B   

 l 2  l

c

R

Interpartice distance Energy

 

1 3,2 2,3 2  3,3

2

( )

in c

R   

B mg l l R

B S C



2

) 1 (   

From the molecular physics point of view: a delocalized probe

PRA 81, 042716 (2010)

2-body physics

1/3 c

R n 

c vdW

R R 

B = 3 G B = .3 mG many-body physics

Distance r (nm)

g’ (r)

2

3 4 5 6

0.1

2 3 4 5 6

1

4 5 6 7 8 9

10

2 3 4 5 6 7 8 9

100

S=3 Spinor physics with free magnetization

Alkalis :

• S=1 and S=2 only
• Constant magnetization

(exchange interactions)  Linear Zeeman effect irrelevant New features with Cr:

• S=3 spinor (7 Zeeman states, four

scattering lengths, a6, a4, a2, a0)

• No hyperfine structure
• Free magnetization

Magnetic field matters !

Technical challenges : Good control of magnetic field needed (down to 100 G) Active feedback with fluxgate sensors Low atom number – 10 000 atoms in 7 Zeeman states

1 Spinor physics of a Bose gas with free magnetization 2 (Quantum) magnetism in opical lattices S=3 Spinor physics with free magnetization

Alkalis :

• S=1 and S=2 only
• Constant magnetization

(exchange interactions)  Linear Zeeman effect irrelevant New features with Cr:

• S=3 spinor (7 Zeeman states, four

scattering lengths, a6, a4, a2, a0)

• No hyperfine structure
• Free magnetization

Magnetic field matters !

Spin temperature equilibriates with mechanical degrees of freedom

Time of flight Temperature ( K) 

Spin Temperature ( K) 

1.4 1.2 1.0 0.8 0.6 0.4 0.2 1.2 1.0 0.8 0.6 0.4 0.2

We measure spin-temperature by fitting the mS population (separated by Stern-Gerlach technique)

At low magnetic field: spin thermally activated

• 1

1

• 2
• 3

2 3

B B

g B k T  

• 3 -2 -1 0 1 2 3

Related to Demagnetization Cooling expts, Pfau, Nature Physics 2, 765 (2006)

1.0 0.8 0.6 0.4 0.2 0.0 1.2 1.0 0.8 0.6 0.4 0.2

• 3.0
• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

1.2 1.0 0.8 0.6 0.4 0.2 0.0

Temperature ( K)  Temperature ( K)  Magnetization Condensate fraction

Spontaneous magnetization due to BEC

BEC only in mS=-3 (lowest energy state) Cloud spontaneously polarizes !

900 B G  

Thermal population in Zeeman excited states

A non-interacting BEC is ferromagnetic New magnetism, differs from solid-state (singlet pairing)

PRL 108, 045307 (2012) T>Tc T<Tc a bi-modal spin distribution

• 3 -2 -1 0 1 2 3
• 3 -2 -1 0 1 2 3

Below a critical magnetic field: the BEC ceases to be ferromagnetic !

Temperature ( K)

 Magnetization

• 3.0
• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0 1.2 0.8 0.4 0.0 1.0 0.8 0.6 0.4 0.2 0.0 0.5 0.4 0.3 0.2 0.1

Temperature ( K)  Condensate fraction

• Magnetization remains small even when the

condensate fraction approaches 1 !! Observation of a depolarized condensate !!

Necessarily an interaction effect B=100 µG B=900 µG

PRL 108, 045307 (2012)

Santos PRL 96, 190404 (2006)

• 2
• 1
• 3
• 3
• 2
• 1

2

1

3



 

2 6 4

2

J B c

n a a g B m    

• 2
• 1

0 1 2 3

• 3

Large magnetic field : ferromagnetic Low magnetic field : polar/cyclic

Ho PRL. 96, 190405 (2006)

• 2
• 3

4

" " 

6

" " 

Cr spinor properties at low field

PRL 106, 255303 (2011)

Density dependent threshold

 

2 6 4

2

J B c

n a a g B m    

BEC Lattice Critical field 0.26 mG 1.25 mG 1/e fitted 0.3 mG 1.45 mG

Load into deep 2D optical lattices to boost density. Field for depolarization depends on density

1.0 0.8 0.6 0.4 0.2 0.0 5 4 3 2 1 Magnetic field (mG)

BEC BEC in lattice

Final m=-3 fraction

Note: Possible new physics in 1D: Polar phase is a singlet-paired phase Shlyapnikov-Tsvelik NJP, 13, 065012 (2011)

On-going discussions with

• M. Brewczyk and M. Gajda

Dynamics analysis

Meanfield picture : Spin(or) precession

Ueda, PRL 96, 080405 (2006)

 

2 1/3 2

( ) 4

dd J B

V r n S g n   



 

Natural timescale for depolarization:

• 3.0
• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0 Magnetization 250 200 150 100 50 Time (ms)

Bulk BEC In 2D lattice

PRL 106, 255303 (2011)

Rapidly lower magnetic field

Open questions about equilibrium state

Santos and Pfau PRL 96, 190404 (2006) Diener and Ho

• PRL. 96, 190405 (2006)

Phases set by contact interactions, magnetization dynamics set by dipole-dipole interactions

• Operate near B=0. Investigate absolute

many-body ground-state

• We do not (cannot ?) reach those new

ground state phases

• Quench should induce vortices…
• Role of thermal excitations ?

!! Depolarized BEC likely in metastable state !!

Demler et al., PRL 97, 180412 (2006)

• 3
• 2
• 1

1 2 3 (a) (b) (c) (d)

Polar Cyclic

 

1 1,0,0,0,0,0,1 2

 

1 1,0,0,0,0,1,0 2

Magnetic field

Spontaneous Magnetization of the cloud at BEC New magnetism New spinor phases at extremely low magnetic fields Interplay between magnetic field, contact interactions and dipolar interactions

• 3
• 2
• 1

1 2 3 (a) (b) (c) (d)

Conclusions (2) Spinor physics with free magnetization

Temperature ( K)  Magnetization

• 3.0
• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0 1.2 0.8 0.4 0.0

Magnetism !

Quasi- Boltzmann distribution Bi-modal spin distribution

Phase diagram adapted from J. Phys. Soc. Jpn, 69, 12, 3864 (2000) See also PRA, 59, 1528 (1999)

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

• 3.0
• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0

Magnetization

T/Tc

A B C

c

B B 

c

B B 

Spinor condensate

Phase diagram

1 Spinor physics of a Bose gas with free magnetization

• Thermodynamics: Spontaneous magnetization of the gas due to

ferromagnetic nature of BEC

• Spontaneous depolarization of the BEC due to spin-dependent interactions

2 Magnetism in 3D optical lattices

• Spin and magnetization dynamics
• Depolarized ground state at low magnetic field

Magnetization changing collisions

1 ( . ) 2 4

i j ij i j i j

n n H J S S



 



1 ( . ) 2

zz z z ij i j i j

H J S S



  1 ( . . ) 2

xy ij i j i j i j

H J S S S S

    

 



Study quantum magnetism with dipolar gases ?

 

3 2 1 2 1 2

) . )( . ( 3 . 4 R u S u S S S g V

R R B J dd

            

           

      

2 2 2 1 1 1 2 1 2 1 2 1

2 . 2 4 3 2 1 S r S r zS S r S r zS S S S S S S

z z z z

Dipole-dipole interactions between real spins Hubard model at half filling, Heisenberg model of magnetism (effective spin model)

  2 1 S

S

Anisotropy Does not rely on Mott physics

Magnetization dynamics resonance for two atoms per site (~15 mG)

Magnetic field (kHz) m=3 fraction 0.8 0.7 0.6 0.5 0.4 46 44 42 40 38 36

• 3
• 2
• 1

1 2 3

Dipolar resonance when released energy matches band excitation Towards coherent excitation of pairs into higher lattice orbitals ? (Rabi oscillations) Mott state locally coupled to excited band

  2 1 S

S

14x103 12 10 8 6 4

200 180 160 140 120 100 80 Magnetic field (kHz) 60 40 A t

• m

n u m b e r

Strong anisotropy of dipolar resonances

Anisotropic lattice sites

2 2 5

3 ( ) 2

r

x iy V Sd r  

See also PRL 106, 015301 (2011)

At resonance May produce vortices in each lattice site (Einstein-de-Haas effect)

• Coll. M. Gajda

(problem of tunneling)

  2 1 S

S

Note: Lineshape of dipolar resonances probes number of atoms per site

3 and more atoms per sites loaded in lattice for faster loading B(kHz) Fraction in m=+3

1.2 1.0 0.8 0.6 0.4 0.2 0.0

44 42 40 38 36 0.6 0.4 0.2 0.0

2 atoms per site 3 atoms per site

Few-body physics ! The 3-atom state which is reached has entangled spin and orbital degrees of freedom Probe of atom squeezing in Mott state

, , 2 3 , 3 , 2 , , 3 , 3 , 3   



spin

• rbit

  2 1 S

S

From now on : stay away from dipolar magnetization dynamics resonances, Spin dynamics at constant magnetization (<15mG) Control the initial state by a tensor light-shift

A s polarized laser Close to a JJ transition (100 mW 427.8 nm) In practice, a  component couples mS states

Magnetic field (kHz)

150 120 90 60 30

• 1

1

• 2
• 3
• 3 -2 -1 0 1 2 3

Energy

 mS

2

Quadratic effect allows state preparation

3 

S

m

Adiabatic state preparation in 3D lattice t

• 2
• 3

 

1 2, 2 3, 1 1, 3 2        

2, 2 6 5 6, 4 4, 4 11 11

S S tot tot

m m S m S m            

 

2 6 4

4

c

B B n a a m    



Initiate spin dynamics by removing quadratic effect

(2 atomes / site)

 

2 6 4

4

c

B B n a a m    



0.9 0.8 0.7 0.6 0.5

0.5 0.4 0.3 0.2 0.1 0.0

• 2.0
• 1.5

temps (ms) magnétization Fraction dans m=-2

On-site spin oscillations

• 3
• 2
• 1

( 250 µs)

vary time

(due to contact

• scillations)

(perdiod  220 µs) Up to now unknown source of damping

Time (ms) populations 0.6 0.5 0.4 0.3 0.2 20 15 10 5

m=-3 m=-2

Long time-scale spin dynamics in lattice

vary time

Sign for intersite dipolar interaction ? (two orders of magnitude slower than on-site dynamics)

 

   



2 1 2 1

2 1 S S S S

• 3.0
• 2.5
• 2.0
• 1.5

15 10 5

Magnetic field (kHz) Magnetization

At extremely low magnetic field (<1.5 mG): Spontaneous demagnetization of atoms in a 3D lattice

 

2 6 4

4

J B c

g B n a a m    

3D lattice Critical field 4kHz Threshold seen 5kHz

6, 6 S m    4, 4 S m   

• 3
• 2

4

" " 

6

" " 

1.2 1.0 0.8 0.6 0.4 0.2 0.0

44 42 40 38 36 0.6 0.4 0.2 0.0

2 atoms per site 3 atoms per site

Away from resonances: spin oscillations at constant magnetization Spin-exchange Dipolar exchange

1.4 1.2 1.0 0.8 0.6 0.4 20 15 10 5

pop(m=-3)/pop(m=-2) time (ms)

Resonant magnetization dynamics Towards Einstein-de-Haas effect Anisotropy Few body vs many-body physics Spontaneous depolarization at low magnetic field Towards low-field phase diagram

• 3.0
• 2.5
• 2.0
• 1.5

15 10 5

Magnetic field (kHz) Magnetization

Conclusions (3) Magnetism in optical lattices

Dipolar BECs: A non-standard superfluid Anisotropic properties Spinor Dipolar BECs: Study magnetism New spinor phases Spins in lattices Study quantum magnetism Spin dynamics

• A. de Paz, A. Chotia, A. Sharma B. Pasquiou, G. Bismut,
• B. Laburthe-Tolra, E. Maréchal, L. Vernac,
• P. Pedri, M. Efremov, O. Gorceix