Transcript B c

Spontaneous demagnetization of ultra cold chromium atoms
at low magnetic fields
B. Pasquiou (PhD), G. Bismut (PhD)
B. Laburthe, E. Maréchal, L. Vernac,
P. Pedri, O. Gorceix (Group leader)
We study the effects of Dipole-Dipole Interactions (DDIs) in a 52 Cr BEC
Spontaneous demagnetization of ultra cold chromium atoms
at low magnetic fields
I- Dipole – dipole interactions
how have they been evidenced in a polarized BEC
what we are seeing new when going to low B fields
II- Existence of a critical magnetic field
below Bc the BEC is not ferromagnetic anymore
a quantum phase transition due to contact interactions
III- Thermodynamics of a spin 3 gas
how thermodynamics is modified when the spin degree of freedom is released
Different interactions in a BEC

GPE :
Van-der-Waals interactions
g

2
2

  Vext  g   dd    
2m
4  2
m
polarized BEC
m1
Dipole-dipole interactions (DDIs)

 




2 1  3 cos 
Vdd (r )  0  m
4
r3
2
Isotropic
Short Range
m2

dd (r )   Vdd (r  r ' ) n(r ' )d 3r '
as


r
R
m  J g J B
Anisotropic
Long Range
Relative strength of dipole-dipole and Van-der-Waals interactions
0 m2 m Vdd
 dd 

2
12 a VVdW
alkaline
 dd  1
 dd  0.01 for 87Rb
for
 dd  1
chromium  dd  0.16
 m  6 B
the BEC is unstable
dysprosium
 dd  1
 m  10 B
polar   1
dd
molecules
Some effects of DDIs on BECs for dd < 1 (Cr)
TF profile
 dd  0.16
dd adds a non local
B
Striction of the BEC
(non local effect)
anisotropic mean-field
Eberlein, PRL 92, 250401 (2004)
DDIs change in
the few % range
the physics of
polarized BEC
Modification of the
BEC expansion
Pfau,PRL 95, 150406 (2005)
all the atoms are in the
same Zeeman state
Collective excitations
of a dipolar BEC
The effects of DDIs
are experimentally
evidenced by
differential
measurements,
for two orthogonal
orientations of the B field
Aspect ratio
Bismut et al., PRL 105, 040404 (2010)
1.2
1.0
0.8
0.6
5
10
15
20
t (ms)
Other effect of DDIs: they can change magnetization of the atomic sample
Dipole-dipole interaction potential with spin operators:
Induces several types of collision:
Elastic collision
S1z 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 
r/   x  iy
change in magnetization:
Spin exchange
mS  mS1  mS 2  f  mS1  mS 2 i
mS  0
Inelastic collisions
mS  1,2
+3
+2
rotation induced
mS  ml  0
=> Einstein-de-Haas effect
1
0
-1
+1
Cr BEC in -3

Optical trap
-1
Cr
-2
-3
magnetization
becomes free
S=3 Spinor physics with free magnetization
- Up to now, spinor physics with S=1 and S=2 only
- Up to now, all spinor physics at constant magnetization
exchange interactions (VdW), no DDIs
- The ground state for a given magnetization was investigated
-> Linear Zeeman effect irrelevant
New features with Cr
- First S=3 spinor (7 Zeeman states, four scattering
lengths, a6, a4, a2, a0)
- Dipole-dipole interactions free total magnetization
- We can investigate the true ground state of the
system (need very small magnetic fields)
Two different B regims:
B > Bc : ferromagnetic BEC and B < Bc non-ferromagnetic BEC
1
0
-1
3
2
1
0
-1
-2
-3
Spontaneous demagnetization of ultra cold chromium atoms
at low magnetic fields
I- Dipole – dipole interactions
how have they been evidenced in a polarized BEC
what we are seeing new when going to low B fields
II- Existence of a critical magnetic field
below Bc the BEC is not ferromagnetic anymore
a quantum phase transition due to contact interactions
III- Thermodynamics of a spin 3 gas
how thermodynamics is modified when the spin degree of freedom is released
S=3 Spinor physics below Bc: emergence of new quantum phases
S=3: four scattering lengths, a6, a4, a2, a0
3

-2
(1,0,0,0,0,0,0)
Magnetic field
0
1
-1
-3
Critical magnetic field Bc
2  (a6  a4 )
m
Bc
n0
at Bc, it costs no energy to go from mS=-3
to mS=-2 : stabilization in interaction
energy compensates for the Zeeman
energy excitation
below Bc a non-feromagnetic phase is favored
(a ,0,0,0,0 ,b,0 )
(a,0,0,0,0,0,b )
polar
phase
All
populated
(a ,0,b,0,g,0,d )
0
2
g J  B Bc  0.7
ferromagnetic
i.e. polarized in
lowest energy single
particle state
2
All populated
-10
0
Santos et Pfau PRL 96,
190404 (2006)
Diener et Ho PRL 96,
190405 (2006)
-10
a0 (Bohr radius)
Quantum phases are set by contact
interactions
(a6, a4, a2, a0)
and differ by total magnetization
DDIs ensure the coupling between states with
different magnetization
S=3 Spinor physics below Bc: spontaneous demagnetization of 3D and 1D quantum gases
Experimental procedure:
Rapidly lower magnetic field below Bc
measure spin populations with Stern Gerlach experiment
Bi>>Bc
1 mG
BEC in m=-3
(a)
0.5 mG (b)
0.25 mG (c)
« 0 mG » (d)
B=Bc
-3
Bf < Bc
Magnetic field control below .5 mG
dynamic lock, fluxgate sensors
reduction of 50 Hz noise fluctuations, earth
Magnetic field, "elevators"
Performances: 0.1 mG stability
without magnetic shield,
up to 1Hour stability
BEC in
all Zeeman
components !
+ Nthermal << Ntot
Pasquiou et al., PRL 106, 255303 (2011)
-2
-1
0
1
2
3
S=3 Spinor physics below Bc: local density effect
Final m=-3 fraction
1.0
g J  B Bc 
0.8
0.6
2
2
n0  a6  a4 
m
3D BEC
1D
Quantum gas
0.4
BEC
BEC in lattice
0.2
Critical field
0.26 mG
1.25 mG
1/e fitted
0.3 mG
1.45 mG
0.0
0
1
2
3
4
Magnetic field (mG)
Bc depends on density
2D Optical lattices increase the peak
density by about 5
5
Note
Spinor Physics in 1 D can
be qualitatively different
see Shlyapnikov and Tsvelik
New Journal of Physics 13 065012 (2011)
Pasquiou et al., PRL 106, 255303 (2011)
S=3 Spinor physics below Bc: dynamic analysis
Simple model
Bulk BEC
At short times, transfert
between mS = -3 and mS = -2
~ a two level system coupled by Vdd
Corresponding timescale for
demagnetization:
2D optical lattices
good agreement with experiment
both for bulk BEC (t =3 ms)
and 1 D quantum gases (t = 10 ms)
But dynamics still unaccounted for :
In lattices (in our experimental configuration), the
volume of the cloud is multiplied by 3
Mean field due to dipole-dipole
interaction is reduced
Slower dynamics,
even with higher
peak densities
Bc  B  Bdd  Vdd / g J  B B
Non local character
of DDIs
Pasquiou et al., PRL 106, 255303 (2011)
Spontaneous demagnetization of ultra cold chromium atoms
at low magnetic fields
I- Dipole – dipole interactions
how have they been evidenced in a polarized BEC
what are we seeing new when going to low B fields
II- Existence of a critical magnetic field
below Bc the BEC is not ferromagnetic anymore
a quantum phase transition due to contact interactions
III- Thermodynamics of a spin 3 gas
how thermodynamics is modified when the spin degree of freedom is released
Summarize of our experimental situation
g J  B B  k BT
Ultra cold gas of spin 3 52Cr atoms at low magnetic fields
spin degree of freedom unfrozen
B3
2

  Vext  VVdW  dd  VZeeman     
2m
mG at 400 nK
  7 components spinor
Optical trap, (almost) same trapping potential for the 7 Zeeman states
Single component Bose thermodynamics
Multi-component Bose thermodynamics
g J  B B  k BT
g J  B B  k BT
N th  N tot  N c 
N th b i  
 exp b n
x

x   y n y   z n z   1
nx ,n y ,nz
b  1/ kBT
k BTc 0  0.94  N at
1
 exp b n
x
nx ,n y ,nz
i    g J  B mSi B
x

  y n y   z nz   bi  1
1
Simkin and Cohen, PRA, 59, 1528 (1999)
Isoshima et al., J. Phys. Soc. Jpn, 69, 12, 3864 (2000)
1
Tc 0
B 0 ( 2 S  1)1 / 3
Tc 
1/ 3
average trap frequency
3
2
1
0
-1
-2 
-3
3
2
1
0
-1
-2
-3
Similar to
M. Fattori et al., Nature Phys. 2, 765 (2006)
at large B fields and in the thermal regime
S=3 Spinor physics above Bc: magnetization versus T
B = 0.9 mG > Bc
Above Bc, the BEC is ferromagnetic:
only atoms in mS=-3 condense
(i.e. in the absolute ground state of the system)
BEC in
m=-3
thermal gas
the kink in magnetization
reveals BEC
Tc1
Solid line: results of theory
without interactions and
free magnetization
Tc1 is the critical temperature
for BEC of the spinor gas
(in the mS=-3 component)
B 0
1
Tc 0  Tc1  Tc 0
(2S  1)1/ 3
B 
The good agreement shows that
the system behaves as if there
were no interactions
(expected for S=1)
S=3 Spinor physics above Bc: spin populations and thermometry
A new thermometry
BEC in mS= -3
8
6
4
Boltzmanian fit
2
6000
1000
8
6
4
4000
-3
-2
-1
0
1
2
3
Spin Temperature (K)
population
8000
1.5
1.0
0.5
2000
Tspin more
accurate at low T
-3
-2
-1
0
mS
1
2
3
depolarized thermal gas
« bi-modal » spin distribution
0.0
0.0
0.4
0.8
1.2
Time of flight Temperature (K)
bimodal distribution
Only thermal gas depolarizes
Cooling scheme if selective
losses for mS > -3
e.g. field gradient
S=3 Spinor physics below Bc: thermodynamics change
B < Bc
B > Bc
B < Bc
B >> Bc
g J  B B  k BT
T 

 1  
 Tc 0 
for T < Tc2
BEC in all mS !
for B < Bc, magnetization remains constant
after the demagnetization process
independant of T
This reveals the non-ferromagnetic
nature of the BEC below Bc
Tc2
B=Bc(Tc2)
3
Tc1
for Tc2 < T < Tc1
BEC only in mS = -3
Pasquiou et al., ArXiv:1110.0786 (2011)
Thermodynamics of a spinor 3 gas:
Results of theory with fixed magnetization and no interactions
T
Tc 0
A phase
1.0
A double
phase transition
0.8
Evolution for a free
magnetization
(normal)
Tc1(M)
Evolution at fixed
magnetization
0.6
1
(2 S  1)1/ 3
B phase
BEC in mS=-3
0.4
C phase
0.2
0
BEC in each component
-1
Tc2(M)
-2
-3
Magnetization
Thermodynamics of a spinor 3 gas: outline of our results
evolution for M fixed
(exp: for B < Bc)
T
Tc 0
evolution for free M
(exp: for B > Bc)
In green:
Results of a theory with
no interactions and
constant magnetization
A phase: normal (thermal)
B phase: BEC in one component
C phase: multicomponent BEC
In purple: our data
measurement of Tc1(M),
by varying B
histograms: spin populations
Pasquiou et al., ArXiv:1110.0786 (2011)
Conclusion: what does free magnetization bring ?
Above Bc
- Spinor thermodynamics with free magnetization of a ferromagnetic gas
- Application to thermometry / cooling
Below Bc
A quench through a (zero temperature
quantum) phase transition
first steps towards spinor ground state
The non ferromagnetic phase is set
by contact interactions,
but magnetization dynamics is set by
dipole-dipole interactions
-
We do not (cannot ?) reach the new ground state
phase
- Thermal excitations probably dominate but…
- … effects of DDIs on the quantum phases have to be
evaluated
Thank you for your attention
… PhD student welcome in our group…
When dipolar mean field beats local contact meanfield
 dd  1
Contact interactions can be tuned using Feshbach resonances: dipolar interactions
then can get larger than contact interactions
T.Lahaye et al, Nature. 448, 672 (2007)
Anisotropic d-wave collapse of (spherical) BEC when scattering length is
reduced (Feshbach resonance) => reveals dipolar coupling Pfau, PRL 101, 080401 (2008)
And: roton, vortices, Mott physics, 1D or 2D physics, breakdown of
integrability in 1D…
With Cr, Dy, Dipolar molecules ?