Transcript B c

Excitation of a dipolar BEC and Quantum Magnetism
E. Maréchal, O. Gorceix , P. Pedri, Q. Beaufils,
B. Laburthe, L. Vernac,
B. Pasquiou (PhD), G. Bismut (PhD)
M. Efremov
IFRAF post doc
A. de Paz
1st year PhD student
Radu Chicireanu
former PhD student
J.C. Keller
R. Barbé
former members
A. Crubelier
collaboration (theory)
We study the effects of Dipole-Dipole Interactions (DDIs) in a 52 Cr BEC
Specificities of chromium
6 electrons in the outer shells
permanent magnetic moment 6 µB
S=3 in the ground state
paramagnetic gas with rather
strong dipole-dipole interactions
(DDIs)
DDIs change the physics of a polarized BEC
(all atoms in the same Zeeman substate)
DDIs allow the cold gas magnetization to change
Obtaining a chromium BEC
Preliminary works of J. Mc Clelland (NIST) - Tilman Pfau, J. Mlynek
High temperature oven (T=1350°C in our case)
choice of the materials
Trapping Transition at 425 nm
needs to double a Ti:Sa laser
High inelastic loss rates due to light assisted collisions
low atom number MOT
Experiment at Stuttgart (Tilman Pfau)
High dipolar relaxation rate
BEC only possible in an optical trap
Accumulation in metastable states is efficient
Red light repumpers required
Our way to BEC: direct loading of an optical trap in metastable states
optimization of the loading with depumping to a new metastable state + use of RF sweep
Physics of a dipolar BEC at Villetaneuse
BEC excitations (phonons, free particle,…)
Bragg
spectroscopy
Spin Flip
polarized BEC
in the excited state
mS=+3
Study of dipolar relaxation
in 3D, 2D, 1D, and 0D
polarized BEC
in the ground state
mS=-3
trap
modulation
BEC excitations
("quadrupole" mode)
D wave Fescbach resonance
lower
B field
unpolarized ultra cold gas
RF association of molecules
S=3 spinor gas with
free magnetization
Ground State RF induced degeneracy
multi-component BEC
Summary of the talk
I- Dipole – dipole interactions in a polarized chromium BEC
how DDIs have been evidenced in ground state BECs
why larger effects are expected with the excitations (phonons, free particles, …)
how do we observe them
II- Demagnetization of ultracold chromium gases at "ultra" low magnetic field
study of S=3 spinor gas with free magnetization
how thermodynamics is modified when the spin degree of freedom is released
observation of a phase transition due to contact interactions:
below a critical B field we observe a multi-component BEC
Different interactions in a polarized Cr BEC

GPE :
Van-der-Waals interactions
gc 

2
2

  Vext  g c   dd    
2m
4  2
m
B
m1
 


dd (r )   Vdd (r  r ' ) n(r ' )d 3r '
as


Rr


2 1  3 cos 
Vdd (r )  0  m
4
r3
2
Isotropic
Short Range
m2

Dipole-dipole interactions (DDIs)


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 Cr BECs
Striction of the BEC
(non local effect)
 dd  0.16
B
DDIs
Eberlein, PRL 92, 250401 (2004)
B
dd adds a non local
anisotropic mean-field
TF profile
Modification of the
BEC expansion
J. Stuhler, PRL 95, 150406 (2005)
The effects of DDIs
are experimentally
evidenced by
differential measurements,
for two orthogonal
orientations of the B field
Some effects of DDIs on Cr BECs
Shift of the
quadrupole
mode frequency
(%)
Collective excitations
of a dipolar BEC
Shift of the
aspect ratio
(%)
Rx (t )  Rx 0  a cos(t )
R y (t )  R y 0  b cos(t )
Rz (t )  Rz 0  c cos(t )
Aspect ratio
1.2
1.0
0.8
0.6
5
10
Bismut et al., PRL 105, 040404 (2010)
DDIs change in the few % range the
ground state physics of a polarized BEC
15
t (ms)
20
Trap anisotropy
DDIs induce changes smaller than dd !
A new and larger effect of DDIs: modification of the
excitation spectrum of a Cr BEC
 dd  0.16
In the BEC ground state the effects of DDIs are averaged due to their anisotropic nature
the dipolar mean field depends on trap geometry
attractive and repulsive contributions of DDIs almost compensate
New idea: probe the effects of DDIs on other kind of excitations of the BEC
the excitation spectrum is given by the Fourier Transform of the interactions
~  0 2
Vdd (k ) 
 m 3 cos 2  k  1
3

B
k

k
all dipoles contribute
in the same way

Experiment: probe dispersion law
Quasi-particles, phonons
k  c  k
c is the sound velocity
c depends on
measure the modification of c due to DDIs: 15% ?
 k
Excitation spectrum of a BEC with pure contact interactions
Bogoliubov spectrum:
 k  Ek ( Ek  2n0 gc )
Rev. Mod. Phys. 77, 187 (2005)
 2k 2
Ek 
2 m
Ek  2n0 gc  k  1 / 

k  1
k  1
Quasi-particles, phonons
Free particles
k  c  k
 k  Ek
is the healing length
c is the sound velocity
c is also the critical velocity
Excitation spectrum of a BEC in presence of DDIs
An effect of the momentum-sensitivity of DDIs:
~  0 2
Vdd (k ) 
 m 3 cos 2  k  1
3

 k  Ek ( Ek  2n0 gc )

B
k
becomes:
 k  Ek Ek  2n0 gc 1   dd 3 cos 2 k 1
if
 k  0 , c  c//
c// / c 

k
 dd  0.16
and if  k   / 2 , c  c
1  2 dd
 1.2
1   dd
A 20% shift due to DDIs expected on the speed of sound !
much larger than the (~3%) effects measured on the ground state and the "quadrupole" mode
Excitation spectrum of a BEC: the local density approximation (LDA)
= the theory giving predictions that you can compare with
* the BEC is trapped, the density is not uniform

n(r )  n0 1  x 2 / RTFx  y 2 / RTFy  z 2 / RTFz
2
2
LDA = consider the gas locally uniform
validity of LDA:
  2 / k  RTFz
with
2


k
 
k // u z
LDA not valid at small k
* the BEC has a non zero width momentum distribution
k z  1 / RTFz
two sources of broadening: the excitation spectrum of the BEC has a non zero width
and the effect of DDIs is going to be less than naively expected…
Excitation of a BEC: principle of Bragg Spectroscopy
Two laser beams detuned:
Momentum and energy transfer

k1


E
Bragg beams very far detuned
from atomic resonances

k2
d
  
k  k1  k 2
k  2 k L sin(  / 2)
k1  k2  kL
L  532 nm
d = 100 Hz to 100 kHz
d
2 k L sin(  / 2)
k
For a given , tune d to find a good excitation,
and register the excitation spectrum
Bragg Spectroscopy: experimental realization
Two lasers "in phase" are required
1 (t )   2 (t )  d t
We use two AOMs driven by a digital double RF
source providing two RF signals in phase

k1



k2
d
For given (accessible) values of , we register excitation spectra
optical access
6° to 14°, 28°, 83°
we measure the excited fraction
for a given d
excited and non-excited parts
spatially separated by momentum transfer
Bragg Spectroscopy: experimental difficulties
* poor spatial separation of the excited fraction at low k
kz  1/ RTFz  k
k
non
excited
to have a good spatial separation after expansion
excited
k  1 /  becomes hard to reach in our case
(we don't work with an elongated BEC)
k z
* choice of t = the excitation duration (of the Bragg pulse)
if t is too small, we add a Fourier broadening
t >> 1 / f
if t is too large, the mechanical effect of the trap comes into play
t << Ttrap / 4
excited
fraction
not quite possible at low k…
non excited
fraction
Bragg Spectroscopy: experimental difficulties
a)
no excitation
b)
 = 6°
poor separation of
the excited fraction
at low k !
c)
 = 14°
d)
 = 83°
data analysis
complicated,
noisy data
Bragg Spectroscopy of a dipolar BEC: experimental results
Excitation spectra at  =14°
Fraction of excited atoms
  
k  B,  i
0.15
  
k // B, i
From the different spectra,
registered for a given ,
we deduce the value of:
f //  f 
 f //  f   / 2
0.10
= shift of the excitation
spectrum due to DDIs
0.05
0.00
0
1000
f  f //
2000
Frequency difference (Hz)
3000
d
Width of resonance curve: finite size effects (inhomogeneous broadening)
The excitation spectra depends on the relative angle between spins and excitation
Bragg Spectroscopy of a dipolar BEC: experimental results
f //  f 
 f //  f   / 2
0.20.20
x10
-3
14
0.15
8
2
4.2
0.10.10
1
0.05
00.00
k
-0.05
0
1
1
2
21
q
3
3
4
4
Conclusion: a 52Cr BEC is a "non-standard superfluid"
 dd  0.16
Expansion
Striction
Stuttgart
Aspect ratio
1.2
Collective excitations
1.0
0.8
0.6
Anisotropic
speed of sound
Fraction of excited atoms
5
10
15
0.15
Villetaneuse
0.10
0.05
0.00
0
1000
2000
Frequency difference (Hz)
3000
20
Summary of the talk
I- Dipole – dipole interactions in a polarized chromium BEC
how DDIs have been evidenced in ground state BECs
why larger effects are expected with the excitations
how do we observe them
II- Demagnetization of ultracold chromium gases at "ultra" low magnetic field
Study of S=3 spinor gas with free magnetization
how thermodynamics is modified when the spin degree of freedom is released
observation of a quantum phase transition due to contact interactions:
below a critical B field we observe a multi-component BEC
DDIs can change the magnetization of the atomic sample
Dipole-dipole interaction potential with spin operators:
1
0
-1
Induces several types of collision:
Elastic collision
S1z S 2 z 

Spin exchange
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
mS tot  0
Inelastic collisions
mS tot  1,2
+3
+2
-1
Cr
-2
change in magnetization:
mS tot  mS1  mS 2  f  mS1  mS 2 i
+1
Cr BEC in -3

Optical trap
-3
magnetization
becomes free
DDIs can change the magnetization of the atomic sample
Dipole-dipole interaction potential with spin operators:
Induces several types of 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
rotation induced
mStot  ml  0
=> Einstein-de-Haas effect
Inelastic collisions
mStot  1,2
+3
+2
+1
Cr BEC in -3
-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)
1
0
-1
3
2
1
0
-1
-2
-3
Ultra cold gas of spin 3 52Cr atoms at "ultra" low magnetic fields
-1
The spin degree of freedom is unfrozen when:
g J  B B  k BT
B  3 mG at 400 nK
2

  Vext  VVdW  dd  VZeeman     
2m
Optical trap, (almost) same
trapping potential for the 7
Zeeman states
-2
g J B B
-3
  7 components spinor
allows the magnetization
to change
Two different B regims for the ground state are predicted:
when B > Bc , the Zeeman interactions dominate: one component (ferromagnetic) BEC
when B < Bc , the contact interactions dominate: multi-component (non-ferromagnetic) BEC
Above Bc: is 52Cr close to a non-interacting S=3 gas with free magnetization ?
Why Bc ? What do we observe below Bc ?
S=3 spinor gas: the non interacting picture (I)
Tc is lowered
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 
 exp  n
x

x   y n y   z n z   1
nx ,n y ,nz
  1/ kBT
k BTc 0  0.94  N at
1/ 3
1
N th  i  
 exp  n
x
x

  y n y   z nz   i  1
1
nx ,n y ,nz
i    g J  B mSi B
1
Tc 0
B 0 ( 2 S  1)1/ 3
Tc 
Simkin and Cohen, PRA, 59, 1528 (1999)
Isoshima et al., J. Phys. Soc. Jpn, 69, 12, 3864 (2000)
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 gas: the non interacting picture (II)
Evolution for a free
magnetization
T
Tc 0
1.0
Tc1(M)
Evolution at fixed
magnetization
A phase
(normal)
0.8
0.6
1
(2 S  1)1/ 3
For Cr:
One BEC component,
in mS= -3
the absolute ground
state of the system
For Na:
a double phase
transition expected
B phase
BEC in mS=-3
0.4
C phase
0.2
0
BEC in each
component
-1
Tc2(M)
-2
Magnetization
Isoshima et al., J. Phys. Soc. Jpn, 69, 12, 3864 (2000)
-3
Our results: magnetization versus T
B > Bc
B = 0.9 mG > 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
Tc1
the kink in magnetization
reveals BEC
Pasquiou et al., ArXiv:1110.0786 (2011)
Solid line: results of theory
without interactions and
free magnetization
Tc1 is the critical temperature
for condensation of the spinor gas
(in the mS=-3 component)
B 0
1
T  Tc1  Tc 0
1/ 3 c 0
(2S  1)
The good agreement shows that
the system behaves as if
there were no interactions
(expected for S=1)
B 
Isoshima et al., J. Phys. Soc. Jpn, 69, 12, 3864 (2000)
Our results (II): measurements of Tc1
B > Bc
T
Tc 0
1.4
A
1.2
Tc1(M)
measurement of Tc1(M),
by varying B
1.0
histograms: spin populations
0.8
0.6
B
0.4
0.2
The good agreement shows that
the system behaves as if there
were no interactions
C
Tc2(M)
0.0
0.0
-0.5
-1.0
-1.5
-2.0
Magnetization
Pasquiou et al., ArXiv:1110.0786 (2011)
-2.5
-3.0
(expected for S=1)
Isoshima et al., J. Phys. Soc. Jpn, 69, 12, 3864 (2000)
Our results (III): spin populations and thermometry
BEC in mS= -3
1.5
Boltzmanian fit
2
6000
1000
8
6
4
4000
-3
-2
-1
0
1
2
3
Spin Temperature (K)
population
A new thermometry
8
6
4
8000
B > Bc
1.0
0.5
2000
Tspin more
-3
-2
-1
0
mS
1
2
3
accurate at
low T !
depolarized thermal gas
« bi-modal » spin distribution
Pasquiou et al., ArXiv:1110.0786 (2011)
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: emergence of new quantum phases
Below Bc
Above Bc
3
the BEC is
ferromagnetic
i.e. polarized in lowest
energy single particle
state
0
-2
1
the BEC is non
ferromagnetic
2
3
i.e. it is a
multicomponent BEC
-1
0
-2
-3
1
2
-1
-3
All the atoms in mS= -3
interactions only in the
molecular potential Stot= 6
because ms tot = -6
If atoms are transferred in mS= -2
then they can interact in the molecular
potential Stot= 4 because ms tot = -4
The repulsive contact interactions are
set by a6 and a4
The repulsive contact
interactions set by a6
As a6 > a4 , it costs no energy at Bc to go from mS=-3 to mS=-2 : the stabilization
in interaction energy compensates for the Zeeman energy excitation
g J  B Bc  0.7
2  2 (a6  a4 )
m
n0
S=3 Spinor physics below Bc: new quantum phases
For an S=3 BEC, contact interactions are set by four scattering lengths, a6 , a4 , a2 , a0
Quantum phases are results of an interplay between Zeeman and contact interactions
unknown
ferromagnetic
i.e. polarized in
lowest energy single
particle state
Critical magnetic field Bc
g J  B Bc  0.7
2  2 (a6  a4 )
m
n0
Magnetic field
(1,0,0,0,0,0,0)
Bc
(a ,0,0,0,0 ,,0 )
(a,0,0,0,0,0, )
polar
phase
Santos et Pfau PRL 96,
190404 (2006)
Diener et Ho PRL 96,
190405 (2006)
All
populated
(a ,0,,0,g,0,d )
0
nematic
phase
All populated
-10
0
-10
a0 (Bohr radius)
Quantum phases are set by contact
interactions and differ by total magnetization
DDIs ensure the coupling between states with
different magnetization
S=3 Spinor physics below Bc: spontaneous demagnetization of the BEC
Experimental procedure:
Rapidly lower magnetic field below Bc
measure spin populations with Stern Gerlach experiment
Bi>>Bc
1 mG
BEC in mS=-3
(a)
0.5 mG (b)
0.25 mG (c)
« 0 mG » (d)
B=Bc
-3
-2
-1
Pasquiou et al., PRL 106, 255303 (2011)
Bf < Bc
Magnetic field control below .5 mG (!!)
dynamic lock, fluxgate sensors
reduction of 50 Hz noise fluctuations
feedback on earth magnetic field, "elevators"
Performances: 0.1 mG stability
without magnetic shield,
up to 1 Hour stability
BEC in
all Zeeman
components !
+ Nthermal << Ntot
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
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
Bc expected
0.26 mG
1.25 mG
1/e fitted
0.3 mG
1.45 mG
Pasquiou et al., PRL 106, 255303 (2011)
Note
Spinor Physics in 1 D can
be qualitatively different
see Shlyapnikov and Tsvelik
New Journal of Physics 13 065012 (2011)
S=3 Spinor physics below Bc: dynamic of the demagnetization
Bc
Simple model
Bulk BEC
At short times, transfer
between mS = -3 and mS = -2
~ a two level system coupled by Vdd
Corresponding timescale for
demagnetization: t   / V
dd
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)
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
independent of T
This reveals the non-ferromagnetic
nature of the BEC below Bc
hint for double
phase transition
Tc2
B=Bc(Tc2)
Tc1
for Tc2 < T < Tc1
BEC only in mS = -3
Pasquiou et al., ArXiv:1110.0786 (2011)
3
Thermodynamics of a spinor 3 gas: outline of our results
evolution
for B < Bc
T
Tc 0
evolution
for B > Bc
1.4
A phase: normal (thermal)
B phase: BEC in one component
1.2
Tc1(M)
A
C phase: multi-component BEC
1.0
0.8
Bc
reached
0.6
In purple: our data
B
measurement of Tc1(M),
by varying B
0.4
0.2
C
histograms: spin populations
Tc2(M)
0.0
0.0
-0.5
-1.0
-1.5
-2.0
Magnetization
-2.5
-3.0
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 exotic 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…