Transcript 2 - Lpl

Magnetism with a dipolar condensate: spin dynamics and thermodynamics
B. Naylor (PhD), A. de Paz (PhD), A. Chotia, A. Sharma,
O. Gorceix, B. Laburthe-Tolra, E. Maréchal, L. Vernac,
P. Pedri (Theory), L. Santos (Theory, Hannover)
Have left: A. Chotia, A. Sharma, B. Pasquiou , G. Bismut, M. Efremov, Q. Beaufils, J. C.
Keller, T. Zanon, R. Barbé, A. Pouderous, R. Chicireanu
Collaborators: Anne Crubellier, Mariusz Gajda, Johnny Huckans,
Perola Milman, Rejish Nath
This talk
1 Thermodynamics of a Bose gas with free magnetization
Purification of a BEC by spin-filtering
2 Exotic quantum magnetism in optical lattices
Intersite spin-spin many-body interactions
from Mott to superfluid
Experimental system
Nov 2007 : Chromium BEC
S=3
104 atoms
April 2014 : Chromium Fermi sea
103
atoms
(from only 3.104 atoms in dipole trap !)
Phys. Rev. A 91, 011603(R) (2015)
F=9/2
A “hot” topic : Cold atoms revisit (quantum) magnetism
Interacting spin-less bosons
(effective spin encoded in orbital degrees of freedom)
Greiner: Anti-ferromagnetic (pseudo-)spin chains
I. Bloch,…
Non-interacting spin-less
bosons
Sengstock: classical frustration
Spinor gases:
Large spin bosons (or fermions)
Stamper-Kurn, Lett, Klempt, Chapman,
Sengstock, Shin, Gerbier, ……
Spin ½ interacting Fermions or Bosons
Super-exchange interaction
Esslinger: short range anti-correlations
I. Bloch, T. Porto, W. Ketterle, R. Hulet…
Ion traps: spin lattice models with effective
long-range interactions
C. Monroe
Dipolar gases: long range spin-spin
interactions
J. Ye, this work…
This seminar: magnetism with large spin cold atoms
Optical dipole traps equally trap all Zeeman state of a same atom
Linear (+ Quadratic)
Zeeman effect
E (mS )  mS g  B B   B 2 
Stern-Gerlach separation:
(magnetic field gradient)
3
2
1
0
mS  1
mS  0
mS  1
-1
-2
-3
Spinor physics due to contact interactions:
scattering length depends on molecular channel
Van-der-Waals (contact) interactions
C6
R6
V ( R) 
4
aS  ( R)
m
mS  2, mS  2 
6
5
S  6, mtot  4 
S  4, mtot  4
11
11
0.4
Fractionnal population
V ( R)  
Spin oscillations (exchange)
2
-2
-3
0.3
-1
0.2
0.1
-1
-2
-3
4 2
B  BG
n  a6  a4 
c 
m
( 250 µs)
Magnetism… at constant magnetization
linear Zeeman effect does not matter
0
0.0
0.2
0.4
0.6
time (ms)
0.8
(period  220 µs)
Spin-changing collisions have
no analog in spin ½ systems
Spinor physics driven by interplay between
spin-dependent contact interactions and quadratic Zeeman effect
1
0
-1
-1
 4
G

0
2
Chapman,
Sengstock,
Bloch,
Lett,
Klempt…
(a2  a0 

m

1
Stamper-Kurn,
Lett,
Gerbier
Quantum phase transitions
Domains, spin textures, spin waves, topological states
NB: high spin fermions coming up!
SU(N) physics, spinor physics
Stamper-Kurn, Chapman,
Sengstock, Shin…
third energy scale set by Fermi energy
(Sengstock, Fallani, Bloch, Ye…)
Chromium:
Two
unusually
types oflarge
interactions
dipolar interactions
(large electronic spin)
Dipole-dipole interactions

1
2
Vdd  0 S 2  g J  B  1  3cos 2 ( )  3
4
R

R
Long range
Anisotropic
Van-der-Waals (contact) interactions
C
V ( R)   66
R
4 2
V ( R) 
aS  ( R)
m
Short range
Isotropic
(only few experiments worldwide with non-negligible dipolar interactions
- Stuttgart, Innsbruck, Stanford, Boulder)
Two new features introduced by dipolar interactions:
Free Magnetization
-3 -2 -1 0
1
2
G  Vdd
Non-local coupling between spins
Vdd 
1
R3
3
1st main feature : Spinor physics with free magnetization
Without
dipolar
interactions
1
0
-1
With
anisotropic
Vdd
1
0
-1
Example: spontaneous demagnetization of a dipolar BEC
(a)
Occurs when the change in magnetic
field energy is smaller than the spindependent contact interaction
(b)
(c)
(d)
-3
-2
-1
0
1
2
3
g J  B Bc 
PRL 106, 255303 (2011)
E  mS g B B
Need a very good
control of B
(100 µG)
2
2
n0  a6  a4 
m
Fluxgate
sensors
1st main feature : Spinor physics with free magnetization
Spin-orbit coupling
(conservation of total angular
momentum)
Rotate BEC ?
Vortex ?
Einstein-de-Haas effect
Quantum Hall regime with fermions?
E  mS g B B
Ueda, PRL 96, 080405 (2006)
Santos PRL 96, 190404 (2006)
Gajda, PRL 99, 130401 (2007)
B. Sun and L. You, PRL 99, 150402 (2007)
Buchler, PRL 110, 145303 (2013)
Magnetization changing
processes write an x+iy
intersite phase
Flat bands, topological insulators
XYZ magnetism
Frustration
Carr, New J. Phys. 17 025001 (2015)
Peter Zoller arXiv:1410.3388 (2014)
H.P. Buchler, arXiv:1410.5667 (2014)
engineer
E  0
2nd main feature of dipolar interactions:
Long range-coupling between atoms
Implications for lattice magnetism, spin domains…
0 Introduction to spinor physics
1 Thermodynamics and cooling of a Bose gas with free
magnetization
2 Exotic quantum magnetism in optical lattices
Spin temperature equilibriates with mechanical degrees of freedom
At low magnetic field: spin thermally activated
1.4
3
2
1
0
-1
-2
-3
-3 -2 -1 0 1 2 3
We measure spin-temperature
by fitting the mS population
(separated by Stern-Gerlach
technique)
Spin Temperature ( K)
g  B B  k BT
1.2
1.0
0.8
0.6
0.4
0.2
0.2
Related to Demagnetization Cooling expts,
T. Pfau, Nature Physics 2, 765 (2006)
0.4
0.6
0.8
1.0
Time of flight Temperature ( K)
1.2
T<Tc
-3 -2 -1 0 1 2 3
Thermal
population in
Zeeman excited
states
0.8
0.6
0.4
0.2
0.0
0.2
0.4
Cloud spontaneously
polarizes !
0.6
0.8
1.0
1.2
1.0
1.2
Temperature (K)
-0.5
a bi-modal spin
distribution
BEC only in mS=-3
(lowest energy state)
B  900G
1.0
-3 -2 -1 0 1 2 3
Magnetization
T>Tc
Condensate fraction
Spontaneous magnetization due to BEC
-1.0
-1.5
-2.0
-2.5
-3.0
A non-interacting BEC is ferromagnetic
New magnetism, differs from solid-state
0.0
0.2
0.4
0.6
0.8
Temperature (K)
PRL 108, 045307 (2012)
A new cooling method using the spin degrees of freedom?
60
80
100
120
140
160
180
-1.2
-0.8
Optical depth
3
2
1
0
-1
-2
-3
-1.0
A BEC component only
in the ms=-3 state
-0.6
-0.4
-0.2
ms=-2
Only thermal gas
depolarizes…
BEC
get rid of it ?
(Bragg excitations
or field gradient)
Thermal
Purify the BEC
ms=-1
ms=-3
0.0
ms=-3
ms=-3,
-2,
-1, …
(i) Thermal cloud
depolarizes
(ii) Kill spin-excited
states
A competition between two mechanisms
BEC
Thermal
ms=-3,
ms=-3
-2,
-1, …
(i) Thermal cloud
depolarizes
(ii) BEC melts to resaturate ms=-3 thermal gas
(and cools it)
(iii) Kill spin-excited
states
3
12x10
BEC melts
(a little)
BEC atom number
Thermal atom number
8
6
?
Who
Wins
?
BEC fraction
Atom Number
10
0.8
0.6
final condensate fraction
0.4
4
Losses in
thermal cloud
due to
depolarization
2
0
0.2
0.4
0.6
0.8
1.0
0.2
0.0
0.2
0.4
0.6
0.8
B-field
1.0
1.2
A competition between two mechanisms
T 
 
 Tc  f
1.2
At high T/Tc, BEC
melts
(too few atoms in the
BEC to cool the
thermal gas back to
saturation)
1.0
0.8
0.6
B=1,5mG
0.4
0.3
At low T/Tc, spin
filtering of excited
thermal atoms
efficiently cools the gas
0.4
0.5
0.6
0.7
0.8
0.9
1.0
T 
 
 Tc i
Theoretical model: rate equation based on the
thermodynamics of Bosons with free magentization.
Interactions are included within Bogoliubov
approximation
Summary of the experimental results as a function of B
1.0
Final condensat fraction
0.8
0.6
(large field, no effect)
0.4
0.2
0.0
0
0.0
2,5
0.4
5
0.8
7,5
1.2
Magnetic field (mG)
Theoretical limits for cooling
As T→0, less and less atoms
are in the thermal cloud,
therefore less and less spilling
0.8
T 
 
 Tc  f
0.6
However, all the entropy lies
in the thermal cloud
0.4
Therefore, the gain in entropy is
high at each spilling
provided T~B
0.2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
T 
 
 Tc i
There does not seem to be any limit other
than practical
In principle, cooling is efficient as long as
depolarization is efficient
Entropy compression
Process can be repeated
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.5
1.0
1.5
2.0
Initial entropy per atom
2.5
3.0
Proposal: Extension to ultra-low temperatures for non-dipolar gases
In our scheme, limitation around 25 nK, limited by
(difficult to control below 100 µG)
k BT  g J  B B
g J B B 
2.8kHz / G
Proposal: use Na or Rb at zero magnetization.
Spin dynamics occurs at constant magnetization
1
0
-1
q  70 Hz / G 2
F=1, mF=-1,
0,
1
Related to collision-assisted Zeeman cooling,
J. Roberts, EPJD, 68, 1, 14 (2014)
We estimate that temperatures in the pK regime may be reached
Nota: the spin degrees of freedom may also be used to measure temperature then
0 Introduction to spinor physics
1 Thermodynamics and cooling of a Bose gas with free
magnetization
2 Exotic quantum magnetism in optical lattices
A 52Cr BEC in a 3D optical lattice
Optical lattice: Perdiodic potential made by a standing wave
Our lattice architecture:
(Horizontal 3-beam lattice) x (Vertical retro-reflected lattice)
Rectangular lattice of anisotropic sites
3D lattice  Strong correlations, Mott transition…
Study quantum magnetism with dipolar gases ?
Condensed-matter: effective spin-spin interactions arise due to exchange interactions
t
t2
G
U
1
S1z S2 z   S1 S2  S1 S2 
2
Dipole-dipole interactions
between real spins
S1 z S 2 z 
1
S1 S 2  S1 S 2 
2
3
2 zS1z  r S1  r S1 .
4
2 zS 2 z  r S 2  r S 2 

Heisenberg model of magnetism (effective spin model)
Tentative model for strongly correlated materials, and
emergent phenomena such as high-Tc superconductivity


0
2 S1 .S 2  3( S1 .u R )( S 2 .u R )
g J  B 
Vdd 
4
R3
Magnetization
changing collisions
S1 S 2
Control of magnetization-changing collisions:
Magnetization dynamics resonance for a Mott state with two atoms per site (~15 mG)
3
Magnetization
changing collisions
2
0
S1 S 2
1
-1
-2
-3
0.8
Dipolar resonance when released
energy matches band excitation
m=3 fraction
0.7
0.6
Mott state locally coupled to excited band
Non-linear spin-orbit coupling
0.5
0.4
Phys. Rev. A 87, 051609 (2013)
36
38
40
42
Magnetic field (kHz)
44
46
See also Gajda: Phys. Rev. A 88, 013608 (2013)
From now on : stay away from dipolar magnetization dynamics resonances,
Spin dynamics at constant magnetization (<15mG)
Magnetization
changing collisions
Can be suppressed in
optical lattices
1


2
 S1z .S 2 z  ( S1 S 2  S1 S 2 ) 1  3z 
4


S1 S 2
Ressembles but differs from
Heisenberg magnetism:
S1z S2 z 
1
 S1 S2  S1 S2 
2
Related research with polar molecules:
A. Micheli et al., Nature Phys. 2, 341 (2006).
A.V. Gorshkov et al., PRL, 107, 115301 (2011),
See also D. Peter et al., PRL. 109, 025303 (2012)
See Jin/Ye group Nature (2013)
1
 S1z S2 z    S1 S2  S1 S2 
2
Adiabatic state preparation in 3D lattice
3

0
-2
-3
-3
-2
Initiate spin dynamics by removing quadratic effect
vary time
-1
1
2
Explore spin dynamics in two configurations
(i) Mott state with a core of two atomes per site
(ii) Empty doublons: singly occupied sites, unit filling
Pop (ms =-3)/Pop (ms =-2)
Spin dynamics after emptying doubly-occupied sites:
A proof of inter-site dipole-dipole interaction
1.0
0.8
0.6
0.4
-2
1.0
0.5
0.2
-2
0.0
0.0
-0.5
-0.5
-1.0
0
0
5
10
15
20
Time (ms)
Experiment: spin dynamics after the atoms are
promoted to ms=-2
Theory: exact diagonalization of the t-J model
on a 3*3 plaquette (P. Pedri, L. Santos)
-1
0.5
-1.0
0.0
-3
1.0
200
400
600
800
1000
0
200
400
600
800
Magnetization is constant
Timescale for spin dynamics = 20 ms
Tunneling time = 100 ms
Super-exchange > 10s
!! Many-body dynamics !!
(each atom coupled to many neighbours)
Mean-field theories fail
Phys. Rev. Lett., 111, 185305 (2013)
1000
Spin dynamics in doubly-occupied sites:
Faster dynamics due to larger effective dipole (3+3=6 ?)
Magnetization is constant
Phys. Rev. Lett., 111, 185305 (2013)
A toy many-body model for the dynamics at large lattice depth
Exact diagonalization is excluded with two atoms per site
(too many configurations for even a few sites)
Toy models for singlons
1


2
 S1z .S 2 z  ( S1 S 2  S1 S 2 ) 1  3z 
4


2,2,....,22   Vi , j  2,2,...2,1,2,...2,3,2,2
i , j 
(i)
(j)
G  2
Toy models for doublons: replace S=3 by S=4 or S=6
Measured frequency: 300 Hz
Calculated frequency:
S=4: 220 Hz
S=6: 320 Hz
Toy models seems to qualitatively reproduce oscillation;
see related analysis in Porto, arXiv:1411.7036 (2015)
 V  
2
i , j 
i, j
Observed spin dynamics, from superfluid to Mott
-2
-3
-2
Vdd
J
-1
-2
-2
-1
An exotic magnetism driven by the
competition between three types of exchange
J
-3
Vc
Lower lattice depth: super-exchange may
occur and compete
Dipolar
Spin-dependent contact interactions
Super-exchange
p3 /p2
2.5
Superfluid
3 ER
2.0
1.5
1.0
Large lattice depth: dynamics dominated
by dipolar interactions
Mott
25 ER
1.4
1.2
0.8
1.0
0.4
0.0
0.6
0.0
0
2
4
6
8
10
Time (ms)
0.2
12
0.4
0.6
14
Empirical description, from superfluid to Mott
Spin dynamics mostly exponential at low
lattice depth
Dynamics shows oscillation
at larger lattice depth
Spin dynamics amplitude
1.2
1.0
Amplitude of
exponential behaviour
0.8
0.6
Slow cross-over between
two regimes?
0.4
0.2
Amplitude of
oscillatory behaviour
0.0
0
5
10
15
20
Lattice depth (Er)
25
Observed, and calculated frequencies
-2
1.0
4
0.5
0.5
0.0
0.0
-2
-0.5
-0.5
-1.0
-1.0
0
Frequency (Hz)
200
400
600
800
0 1000
200
400
-1
600
Two-body spin dynamics in
isolated lattice sites
2
1000
GP- mean-field
simulation
-3
1.0
4
8
2
m
a
6
 a4 n
6
4
Many-body spin dynamics
due to intersite couplings
2
2
100
 Vi ,2j
(i , j )
8
6
superexchange
4
0
5
10
15
20
25
Lattice depth (Er)
30
Summary: a slow cross-over between two behaviors
At low lattice depth:
In the Mott regime:
- GP-simulations predict long-lived
oscillations (not seen on the experiment)
Temperature effect ?
Two well separated oscillating
frequencies corresponding to:
-Drift in spin dynamics qualitatively
reproduced by simulation
-On site contact-driven spin-exchange
interactions
-Many-body intersite dipole-dipole
interactions
-The dynamics depends on an
interplay between contact and dipolar
interactions
In the intermediate regime:
p3/p2
-oscillations survive.
- Two frequencies get closer
3.0
7 Er
2.5
2.0
1.5
-2
0
5
10
Time (ms)
15
-1
-2
-2
0.5
Vdd
J
No theoretical model yet
1.0
-3
-2
-3
Vc
-1
J
More probes to caracterize both regimes (1)
Hamiltonian should create entanglement (collaboration Perola Milman; Paris 7 University)
1


2


S
.
S

(
S
S

S
S
)
1

3
z
 1z 2 z

1 2 
1 2 
4


2,2,....,22   Vi , j  2,2,...2,1,2,...2,3,2,2
i , j 
(i)
(j)
We are looking for an entanglement witness based on measurements of global spin variables.
(e.g.
S x  S y  S z  S
for any mixture of separable states)
Idea: adapt this criterion to our observations, measure spin fluctuations
Entanglement may only arrise at high lattice depth
(otherwise BEC-like state)
When does entanglement appear/disappear?
More probes to caracterize both regimes (2)
Mean-field vs « many-body » dynamics
0 2
1
2
2
Vdd 
S  g J  B  1  3cos ( )  3
4
R
At the mean-field level, dipolar interactions cancel out for an homogeneous system
 d V
dd
0
Spin dynamics is a border effect
(low lattice depth)
True many-body Hamiltonian predicts non-vanishing spin dynamics
1


2
S
.
S

(
S
S

S
S
)
 1z 2 z
1 2 
1 2  1  3 z 
4


2,2,....,22   Vi , j  2,2,...2,1,2,...2,3,2,2
i , j 
(i)
(j)
G  2
V  

 
2
i, j
i, j
Deep lattices: spin dynamics occurs in the core
Measure locally could differentiate between regimes
What have we learned (1)?
Bulk Magnetism:
spinor physics with free magnetization
(a)
(b)
(c)
New spinor phases at extremely low magnetic fields
(d)
-3
T 
 
 Tc  f
-2
-1
0
1
2
3
1.2
1.0
New cooling mechanism
to reach very low entropies (in bulk):
Use spin to store and remove entropy
0.8
0.6
Should be applicable to non-dipolar species
pK regime possible
0.4
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
T 
 
 Tc i
What have we learned (2)?
Lattice Magnetism:
1.2
0.4
1.0
0.6
0.4
Populations
0.8
0.6
2 atoms per site
0.4
3 atoms per site
0.2
0.2
0.0
0.3
0.2
0.0
36
38
40
42
mS=-3
mS=-2
mS=-1
mS=0
0.1
44
Magnetization dynamics is resonant
0.0
0
4
6
8
10
12
14
Time (ms)
4
Intersite dipolar spin-exchange
2
Frequency (Hz)
2
1000
8
6
Exotic quantum magnetism, from
Mott to superfluid
4
2
Different types of exchange contribute
100
8
6
Consequences for magnetic ordering ?
4
0
5
10
15
20
25
Lattice depth (Er)
30
What have we learned ? (3)
Truly new phenomena arrise due to dipolar interactions when the spin degrees of
freedom are released.
- Free magnetization. Spin orbit coupling. Also an interesting challenge from
the theoretical point of view.
Carr, New J. Phys. 17 025001 (2015)
Peter Zoller arXiv:1410.3388 (2014)
H.P. Buchler, arXiv:1410.5667 (2014)
- Effective Hamiltonians relevant for quantum magnetism. Some of the
physics is specific to high spin atoms (no analog with electrons or with
heteronuclear molecules)
See M. Wall et al., arXiv 1305.1236
- Large spin atoms in optical lattices: a yet almost unexplored playground for
many-body physics (even without dipolar interactions)
Thank you
A. de Paz (PhD), A. Sharma, A. Chotia, B. Naylor (PhD)
E. Maréchal, L. Vernac,O. Gorceix, B. Laburthe
P. Pedri (Theory), L. Santos (Theory, Hannover)
Bruno
Naylor
Post-doctoral position available
Arijit
Sharma
Aurélie
De Paz
Amodsen
Chotia
This is not the whole picture:
the (small but interesting) effect of demagnetization cooling
Our theoretical results predict that depolarization may induce an increase
of the BEC atom number!!
How is it compatible with the fact that the entropy must increase?
Entropy of a saturated cloud:
Entropy of a non-saturated cloud: is… larger
3
T 
S / N  3.6k B    3.6k B f th
 Tc 
For a fully saturated gas, the entropy is given
by the condensate fraction: you cannot increase
the condensate fraction and entropy at the same
time !
NB: this effect is associated to
demagnetization cooling
T. Pfau, Nature Physics 2, 765 (2006)
The entropy can therefore be stored in the
non-saturated gas, and the BEC atom
number increase, without filtering!
 g J B B 

3k B T   g J  B B  exp  
k BT 

How to characterize the cooling efficiency:
use entropy per particle
As T→0, less and less atoms are in the thermal cloud
3
T 
S / N  3.6k B    3.6k B f th
 Tc 
Therefore less and less spilling
However, all the entropy lies in the thermal cloud
Entropy compression
Therefore, the gain in entropy is high at each spilling
provided T~B
1.2
1.0
0.8
0.6
0.4
0.2
Entropy (given by BEC fraction)
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.0
Initial entropy per atom
2.5
3.0
0.0
0
0.22,5 0.4
0.6
5
0.8
7,5 1.0
Magnetic field (mG)
1.2
Simple two-body Hamiltonian
1
S1z S2 z   S1 S2  S1 S2 
2
Our approach :
Vdd 
0
g J  B 2
4
Complex Many-body physics
Many open questions…
Study magnetism with strongly
magnetic atoms : dipole-dipole
interactions between real spins


S1.S 2  3( S1.u R )( S 2 .u R )
R3

R
1
S1z S 2 z  S1 S 2  S1 S 2 
2
3
 2 zS1z  r S1  r S1 .
4
2 zS 2 z  r S 2  r S 2 
Possibilities for quantum simulation – possibilities for exotic quantum magnetism
Quantum magnetism, some paradigms, from solid-state physics
High-Tc superconductivity
Antiferromagnetism Hubbard model
Spin liquids
Frustrated magnetism
??
Condensed-matter: effective spin-spin interactions arise due to exchange interactions
t
Heisenberg model of magnetism
(real spins, effective spin-spin interaction)
t2
G
U
1
S1z S2 z   S1 S2  S1 S2 
2
Ising
Exchange
Our experiment:
real spin-spin interactions due to dipole-dipole interactions
This is not the whole picture:
the (small but interesting) effect of demagnetization cooling
At finite magnetic field,
depolarization implies a
conversion of kinetic energy in
magnetic energy
g J B B
ms=-3
ms=-3,
-2,
-1, …
T. Pfau, Nature Physics 2, 765 (2006)
 g  B
3k B T   g J  B B  exp   J B 
k BT 

Entropy of a saturated cloud:
At finite magnetic
field, depolarization
may induce an
increase of the BEC
atom number!!
How is it
compatible with the
fact that the
entropy must
increase?
3
T 
S / N  3.6k B    3.6k B f th
 Tc 
For a fully saturated gas, the entropy is given by
the condensate fraction: you cannot increase the
condensate fraction and entropy at the same time !
Entropy of a non-saturated cloud: is… larger
The entropy can therefore be stored in the nonsaturated gas, and the BEC atom number
increase, without filtering!