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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