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Spin Diffusion and Dynamical Defects in a Strongly interacting Fermi gas
ECT Workshop, May 13th, 2014
Tarik Yefsah Massachusetts Institute of Technology
Length scales
10-15 m
White dwarf
Superfluid Helium
High-Tc Superconductors
Nuclei
1 m 107 m
Small Extremely difficult Far
Many-body physics
Can we provide a more accessible playground to benchmark theories?
Can we realize clean tunable experimental systems that can directly be compared to N-Body theories?
Ultracold gases as practical model systems
Feynman’s “quantum simulator”
Potential
Interactions
…and more
Composition
The diluteness of the gas is a key!
𝑛~1014 atoms/cm3 or 100 atoms/𝜇m3
→ a million times thinner than air
When does wave mechanics matter?
Interparticle distance
1/3d n
When does wave mechanics matter?
x
1/3d n
When does wave mechanics matter?
x
1/3d n
When does wave mechanics matter?
x
1/3d n
How cold is ultracold?
Your
living
room
Supernova
explosion
Center
of the
sun
Outer
space
109 10-3 106 103 10-6 10-9 1 T(K)
~ 100 m/s ~ 106 m/s
Paris in 10
seconds
~ 1 mm/s
Atoms move at:
Ultracold atom
experiments
From BEC to BCS
Weakly Interacting Bosons
Strongly Interacting Bosons
Strongly Interacting Fermions
Weakly Interacting Fermions
Interparticle Distance
Scattering Length =
What can experiments teach us on strongly interacting Fermi gases?
Superfluidity And
Coherence
MIT (2005)
BEC-BCS Crossover Leggett (1980) Nozières & Schmitt-Rink (1985)
Boulder, ENS, Innsbruck, MIT (2003)
Experiments
1/𝑘𝐹𝑎 −∞ +∞
What can experiments teach us on strongly interacting Fermi gases?
e.g.: ground-state energy:
(at unitarity)
N. Navon, S. Nascimbène, F. Chevy, C. Salomon Science 328, 729-732 (2010)
M. Ku, A. Sommer, L. Cheuk, M. Zwierlein, Science 335, 563-567 (2012)
𝜉Mean-Field=0.59
𝜉Experiment=0.37(1)
MIT (2012)
𝜉advanced-theories= 0.36-0.46
Leggett Ansatz T-Matrix (Zwerger et al.) Epsilon expansion (Arnold, Drut & Son) Fixed-node Monte-Carlo (Giorgini et al.)
M. Ku, A. Sommer, L. Cheuk, M. Zwierlein, Science 335, 563-567 (2012)
K. Van Houcke, F. Werner, E. Kozik, N. Prokofev, B. Svistunov,
M. Ku, A. Sommer, L. Cheuk, A. Schirotzek, M. Zwierlein, Nature Physics 8, 366 (2012)
Experiments as a Quantum Simulator
Density equation of state
Meanfield
Non-
Interacting
Gas
Unitary
Gas
(Expt.)
3
B
n fk T
We know the equilibrium properties
But: How about dynamics?
Spin Diffusion in a Strongly interacting Fermi gas
Large Hadron Collider (LHC)
4 ft
A Fermi gas collides with a cloud with resonant interactions
Little Fermi Collider (LFC)
Harmonic
Trap
Collision at 100 peV
23 orders of magnitude smaller than LHC
A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011)
Little Fermi Collider (LFC)
Without Interactions
A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011)
Little Fermi Collider (LFC)
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Dis
tan
ce
[m
m]
6050403020100
Time [ms]
Without Interactions
A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011)
Little Fermi Collider (LFC)
With resonant interactions
A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011)
1.3
mm
Time (1ms per frame)
The bouncing gas First collision
1.3
mm
1.3
mm
Difference
density
Total
density
-1.0
-0.5
0.0
0.5
1.0
Dis
tan
ce
[m
m]
200150100500
Time [ms]
Later times
A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011)
-1.0
-0.5
0.0
0.5
1.0
Dis
tance [m
m]
10008006004002000
Time [ms]
Much later times
A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011)
95 ms
-1.0
-0.5
0.0
0.5
1.0
Dis
tance [m
m]
10008006004002000
Time [ms]
450 ms
Much later times
A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011)
Diffusion constant: ~ mean free path average velocity D
Mean free path ~ Interparticle spacing
Quantum Limit of Diffusion
~Dm
Planck’s constant
Particle mass =
(0.1 mm)2
1s =
Quantum limit of spin diffusion
2
~c
DT
In a hot relativistic fluid (e.g. Quark-Gluon Plasma): 2mc T
d
d md
Quantum Limit of Spin Diffusion
Spin Diffusion vs Temperature
Spin current = -D Spin density gradient
Universal high-T behavior:
A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 201 (2011)
6.3(3)ℏ
𝑚
5.8(2)ℏ
𝑚
𝑇
𝑇𝐹
3/2
s tot tot sd s
dn dnj n d n d D
dx dx
Dynamical defects in a Fermionic Superfluid
What kind of “strong” excitation ?
Dark Soliton Vortex
• “Far” from the ground state non-linear excitation • Long-lived state stationary or topological excitation • “One-defect” state Particle-like (energy E and effective mass M* )
J. Dalibard group (2000)
2𝜋
2𝜋 phase widing W.D Phillips’ group (2000)
0 − 𝜋 phase jump
0
𝜋
Experiments with Fermionic Superfluids
Phase imprinting
superfluid
Pulse
off-resonant light
Solitons in BECs by phase imprinting: Hamburg, NIST,...
After the pulse: ∆𝜑 =2𝑈𝑡
ℏ
Needs to be fast enough:
𝑡 <ℏ
𝜇~100𝜇s
Solitons in BECs by phase imprinting: Hamburg, NIST,... superfluid
Pulse
off-resonant light After time-of-flight
+ ramp to the BEC side of the Feshbach resonance
300𝜇
m
20𝜇
m
Phase imprinting
Phase imprinting
Solitons in BECs by phase imprinting: Hamburg, NIST,... superfluid
Pulse
off-resonant light
piège harmonique 𝑇𝑧 = 0.1𝑠
After time-of-flight + ramp to the BEC side of the Feshbach resonance
300𝜇
m
20𝜇
m
1s 2s 3s 4s
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5
Time (in s)
Long-lived solitary wave oscillate in a superfluid
Str
ong
er
Inte
ractio
ns
700 G
760 G
815 G
832 G
Regime of strongly interacting
Molecular BEC
4.4 2s z zT T T
BEC BCS
Heavy Defects
BdG
theory for
planar
soliton
• Long-lived solitary waves in a fermionic superfluid • Inconsistent with current theories for planar solitons
An experimental riddle
… what did we observe ?
In the following months …
Also supporting the vortex ring scenario • W. Wen, C. Zhao, X. Ma, arXiv:1309.7408 • M. D. Reichl, E. J. Mueller arXiv:1309.7012 • A. Bulgac, M. McNeil Forbes arXiv:1312.2563 Against the vortex ring scenario • L. Pitaevskii arXiv:1311.4693
From Eric Mueller’s webpage
Bulgac et al. arXiv:1306.4266
slice from numerical simulation
Measured integrated density profile
Probe Z axis
Z axis
Probe
Origin of the debate : Integrated profile vs slice
Slicing our Cloud
Z axis
Blaster
What do we see from top ?
Probe
After slicing…
It is not a vortex ring
Mask
~Central slice
Top Slice
Bottom Slice
Tomography
This is not a planar but a linear defect
Can we infer the gyroscopic nature of the defect? • The trap exerts an expelling force −𝛻𝑉𝑡𝑟𝑎𝑝
• As a gyroscope the vortex experiences a Magnus force
follows a trajectory along 𝑥 × −𝛻𝑉𝑡𝑟𝑎𝑝
superfluid
Axe z
M. Ku, W. Ji, B. Mukherjee, E. Guardado-Sanchez, L.W. Cheuk, T. Yefsah, M.W. Zwierlein, arXiv:1402.7052 (2014)
Can we infer the gyroscopic nature of the defect? • The trap exerts an expelling force −𝛻𝑉𝑡𝑟𝑎𝑝
• As a gyroscope the vortex experiences a Magnus force
follows a trajectory along 𝑥 × −𝛻𝑉𝑡𝑟𝑎𝑝
superfluid
Axe z
Precession as the signature of a vortex
M. Ku, W. Ji, B. Mukherjee, E. Guardado-Sanchez, L.W. Cheuk, T. Yefsah, M.W. Zwierlein, arXiv:1402.7052 (2014)
Theoretical model for the vortex motion Hydrodynamic picture : 𝑣 = ℏ𝛻𝜙/2𝑚 ( 𝜙 the phase of the order parameter)
Effective potential acting on the vortex
𝐸𝑣 =𝜋ℏ2
4𝑚𝑛 𝑠 𝑟𝑣 ln(
𝑅⊥
𝜉)
(valid in the limit ln(𝑅⊥
𝜉) ≫ 1 )
For BEC
Lundh, Ao, PRA 2000
Fetter, Kim, JLTP 2001
(𝜉 vortex core size)
Ω
𝜔𝑎𝑥=
2𝛾+1
8 ℏ𝜔⊥
𝜇ln(
𝑅⊥
𝜉) We find
At unitarity 𝛾 =3
2 and 𝜉~
1
𝑘𝐹
BEC BCS
Heavy Defects
BEC BCS
Heavy Defects
Hydrodynamic model
Decay cascade of non linear excitations
Right after imprint:
1. we observe the propagation of sound waves … 2. ..then emerges a dark soliton that breaks into 3. .. a Vortex rings 4. ..and then the vortex ring breaks into vortices 5. Eventually, one vortex remains
150𝜇m
Soliton ?
Vortex ring ?
Conclusion
• T. Yefsah, A. Sommer, M.Ku, L. Cheuk, W. Ji, W. Bakr, M. Zwierlein,
Nature 499, 426–430 (2013)
• M. Ku, W. Ji, B. Mukherjee, E. Guardado-Sanchez, L. Cheuk, T. Yefsah, M. Zwierlein
arXiv:1402.7052
• We observed long-lived “heavy defects”
• Excluded the vortex ring scenario
• It is a vortex which results from a decay cascade
• Benchmark for out-of-equilibrium dynamics
• Controlled creation of Topological excitations
• Dynamical instabilities and turbulence
The MIT Team
Martin Zwierlein Waseem Bakr
Ariel Sommer
(PhD 2013)
Mark Ku
Lawrence Cheuk Wenjie Ji
T. Yefsah, A. Sommer, M. J.-H. Ku, L. Cheuk, W. Ji, W. Bakr, M. Zwierlein,
Nature 499, 426–430 (2013)
Biswaroop
Mukherjee
Type equation here.
A new kind of excitation
Komineas & Papanicolaou PRA 68, 043617 (2003)
ℏ𝜔⊥ 𝑟
𝑉(𝑟)
𝜔⊥
Ener
gy o
f th
e d
efec
t
more 3D
𝜇/ℏ𝜔⊥
𝝁
Type equation here.
A new kind of excitation
Komineas & Papanicolaou PRA 68, 043617 (2003)
ℏ𝜔⊥ 𝑟
𝑉(𝑟)
𝜔⊥
Ener
gy o
f th
e d
efec
t
more 3D
𝜇/ℏ𝜔⊥
𝝁
Type equation here.
A new kind of excitation
Komineas & Papanicolaou PRA 68, 043617 (2003)
ℏ𝜔⊥ 𝑟
𝑉(𝑟)
𝜔⊥
Ener
gy o
f th
e d
efec
t
more 3D
𝜇/ℏ𝜔⊥
𝝁
Soliton
Solitonic
Vortex
Ground state of 2D BEC via
GP equation after “imprint”:
n mod 2
0
0
0
0 0
1.17
2.89
6.67
• The S-vortex obeys a current-phase relation just like the soliton • At rest the S-vortex has the far-field phase profile of a dark soliton
A new kind of excitation
More slides on Fermionic Superfluid
Speed of Sound
mm8.8
s 3s Fv v
B = 815G
Making Solitons by phase imprinting
What is a soliton ?
Dispersion
(broadening)
Non-linearity
(localization)
Localized wave-packet
Maintains its shape during propagation
Solitons in a Fermionic Superfluid
What kind of “strong” excitation ?
Dark Soliton Vortex
W.D Phillips’ group (2000)
J. Dalibard group (2000)
Vortex ring
A vortex ring maker
Solitonic vortex
Brand & Reinhardt (2001)
• “Far” from the ground state non-linear excitation • Long-lived state stationary or topological excitation • “One-defect” state Particle-like (energy E and effective mass M* )
BEC BCS
(𝑘𝐹𝑎)−1
What is the mass of a soliton in a Fermionic Superfluid ?
We do not know the current-phase relation
GP equation still valid
Bogoliubov-de Gennes
equations
Wave function = pairing gap Δ(z)
Soliton mass in the BCS limit
Soliton almost completely filled with “normal” fluid
1.0
0.5
0.0
n(x
)
-2 0 2
x/
No Cooper pairing inside soliton
But fraction of Cooper pairs is very small
n(z
)
z/ξ
Bare mass (= mass of missing atoms)
𝑧/𝜉
Dark Soliton in a Fermionic Superfluid ∆(𝑧
)/∆
0
Andreev bound state
fill the soliton
Dark Soliton in a Fermionic Superfluid
𝑧/𝜉
∆(𝑧
)/∆
0
Effective Mass vs Interaction Strength Solitons are filled in
Effective mass M* >> M bare mass
Period of oscillations should get much longer
Scott, Dalfovo, Pitaevskii, Stringari, Phys. Rev. Lett. 106, 185301 (2011)
BCS 1 0.5 0 -0.5
BEC
On Resonance:
Brand, Liao, PRA 83, 041604 (2011)
Imaging Solitons
Ramp to
Final
Field
The cooling methods
• Laser Cooling ~ 1 mK
• Evaporative Cooling ~ 10 nK
Laser Cooling
Hot atomic beam from oven, 400 °C Laser cooled cloud
Less than 1 mK
Laser beams
“Zeeman Slower”
Chu, Cohen-Tannoudji, Phillips, Pritchard, Ashkin, Lethokov, Hänsch, Schawlow, Wineland …
Evaporative Cooling
Invented for spin-polarized H: Hess, Kleppner, Greytak, Silvera, Walraven
http://www.colorado.edu/physics/2000/applets/bec.html
Atom cloud Lens
CCD
Camera Laser beam
Shadow image
of the cloud
Trapped Expanded
1 mm
Observation of the atom cloud
Expanded
Atom cloud Lens
CCD
Camera Laser beam
Shadow image
of the cloud
1 mm
Observation of the atom cloud
Expanded
Atom cloud Lens
CCD
Camera Laser beam
Shadow image
of the cloud
1 mm
Observation of the atom cloud
BEC @ JILA, Juni ‘95 (Rubidium)
BEC @ MIT, Sept. ‘95 (Sodium)
1 K 100 K 104 K 106 K 1 mK 1 K T
Room temperature
Sun (center)
Laser cooling
Nobel Prize 1997
S. Chu, C. Cohen-Tannoudji, W. Phillips
Nobel Prize 2001
E. Cornell, W. Ketterle, C. Wieman
Evaporative Cooling to BEC
The cooling methods on the temperature scale
Solitons as “strong” excitation
Y(x)
x
Order Parameter
A localized, long-lived, highly non-linear excitation
An excellent probe for the medium in which it propagates
Aspect ratio = 6
Aspect ratio = 15
BEC BCS
Heavy Solitons
BdG theory for
planar soliton
Aspect ratio = 6
Aspect ratio = 15
Aspect ratio = 3
BEC BCS
Heavy Solitons
BdG theory for
planar soliton
Werner
Heisenberg xp mT
Temperature = Uncertainty of velocity2
Size of a wave packet = Uncertainty of position
Louis
de Broglie Velocity Mass
Planck’s constant h
mv
109 10-3 106 103 10-6 10-9 1 T(K)
When does wave mechanics matter?
Werner
Heisenberg xp mT
Temperature = Uncertainty of velocity2
Size of a wave packet = Uncertainty of position
109 10-3 106 103 10-6 10-9 1 T(K)
How much do we need to cool?
Anti-Damping of Soliton Oscillations
Anti-Damping
M*<0 T
Phase fluctuations
Solitons at finite temperature
Period
Anti-Damping
Time
Lifetime
Long period and large M*/M is a Quantum Effect
Bosons versus Fermions Fermions (unsociable): • Half-Integer Spin • Pauli blocking Fermi sea • No phase transition at low Temperature
Bosons (sociable): • Integer Spin • Can share quantum states • At low temperatures: Bose-Einstein condensation
Bosons
N bosons sharing one and the same macroscopic matter wave
(Artist’s conception)
Fermions
N fermions avoiding each other
(Artist’s conception)
Soliton effective mass phase change across soliton
implies superfluid backflow Scott, Dalfovo, Pitaevskii, Stringari, PRL 106, 185301 (2011)
1D 1D 1D
1 d d
2 2SP mn v x mn x n
m
Effective Mass:
*
1D
1
2
S
s s
PM M n
u u
cos2
su
c
Requires Josephson Current-Phase Relation
* 2M MFor BEC:
Local density approximation (LDA) How to get the thermodynamics of the homogeneous gas from trapped samples?
→ A single profile contains the full EoS from 𝜇
𝑘𝐵𝑇= −∞ to
𝜇0
𝑘𝐵𝑇
Trapped gas Homogeneous equivalent
The trapping potential tunes for us the chemical potential:
Comparison to the data
𝜇/ℏ𝜔⊥
T v/T ax
ial
Comparison to the data
𝜇/ℏ𝜔⊥
T v/T ax
ial
For BEC
Lundh, Ao, PRA 2000
Fetter, Kim, JLTP 2001
𝑇𝑣𝑇𝑎𝑥
=8
3
𝜇
ℏ𝜔⊥
1
ln𝑅⊥𝜉
+34
𝑅⊥
𝜉= 4
𝜇
ℏ𝜔⊥
Comparison to the data
𝜇/ℏ𝜔⊥
T v/T ax
ial
For BEC
Lundh, Ao, PRA 2000
Fetter, Kim, JLTP 2001
𝑇𝑣𝑇𝑎𝑥
=8
3
𝜇
ℏ𝜔⊥
1
ln𝑅⊥𝜉
+34
𝑅⊥
𝜉= 4
𝜇
ℏ𝜔⊥
𝑇𝑣𝑇𝑎𝑥
= 2 𝜇
ℏ𝜔⊥
1
ln𝑅⊥𝜉
𝑅⊥
𝜉= 1 ×
1
0.37
𝜇
ℏ𝜔⊥
Logarithmic corrections unknown
Comparison to the data
𝜇/ℏ𝜔⊥
T v/T ax
ial
For BEC
Lundh, Ao, PRA 2000
Fetter, Kim, JLTP 2001
𝑇𝑣𝑇𝑎𝑥
=8
3
𝜇
ℏ𝜔⊥
1
ln𝑅⊥𝜉
+34
𝑅⊥
𝜉= 4
𝜇
ℏ𝜔⊥
𝑇𝑣𝑇𝑎𝑥
= 2 𝜇
ℏ𝜔⊥
1
ln𝑅⊥𝜉
𝑅⊥
𝜉= 0.5 ⋯2 ×
1
0.37
𝜇
ℏ𝜔⊥
Logarithmic corrections unknown
The vortex ring scenario
T/Tz~1.9
Snake instability more effective than observed
A. Bulgac, M. McNeil Forbes, M. M. Kelley, K. J. Roche, G. Wlazłowski Phys. Rev. Lett (2014)
Origin of the debate : Integrated profile vs slice
Bulgac et al. arXiv:1306.4266
slice from numerical simulation
Measured integrated density profile
Probe Z axis
Z axis
Why the data does not speak for vortex rings
815 G
760 G
1. The soliton to vortex ring decay is a stochastic mechanism
2. Bulgac et al. : Ts/Tz ~10 requires small vortex rings (R ~ 0.3 Rx)
But the trajectory we measure is deterministic
But the depletion we observe goes through the whole transverse size
Spin relaxation gives spin drag coefficient:
d(t) ~ e-gt
Spin relaxation rate 2 0sdd d d Magnetization d decays
due to spin drag
214.8 / sd
sg
2 3 1/ 10 s ~ /sd FE g
Spin drag vs Temperature
Fsd
F
E Tf
T
On resonance, spin drag must be universal:
Control temperature by cooling after collision For a 50/50 mixture:
22.8 Hz
11.2 Hz
37.5 Hz