Observation of quantum bounds in spin diffusivity

Observation of quantum bounds in spin diffusivity

Joseph H Thywissen 
 University of Toronto

based on:

A. B. Bardon et al., Transverse Demagnetization Dynamics of a Unitary Fermi Gas Science 344, 6185 (2014) S. Trotzky et al., Observation of the Leggett-Rice effect in a unitary Fermi gas PRL 114, 015301 (2015)


C. Luciuk et al., Observation of quantum-limited spin transport in 2D Fermi gases, PRL 118, 130405 (2017)
 T. Enss & JT, Universal Spin Transport and Quantum Bounds for Unitary Fermions, arXiv:1805.05354

Nordita program on 


Bounding Transport and Chaos


29 August 2018

Strongly coupled matter

[adapted from Y. Cao… P. Jarillo-Herrero, Nature 556, 43 (2018)]

critical temperatur e T c (K)

Fermi temperature T F (K)

6Li (x10⁸)

Ultracold Fermi gases

40K (x10⁸)

1. Separation of length scales 2. Control of interactions

3. Optical imaging of individual atoms

4. Slow time scales: dynamics easily accessible 5. Ab initio theory

OPPORTUNITIES AT LOW DENSITY

1. Separation of length scales 2. Control of interactions

✦ Round-trip phase of collision depends on E-E c

✦ Relative energy is tuned using a magnetic field

R

“Feshbach 
 Resonance”

OPPORTUNITIES AT LOW DENSITY

1. Separation of length scales

change magnetic field

[Following work by: Verhaar, Stwalley, Ketterle, Jin, Wieman, …]

1. Separation of length scales

OPPORTUNITIES AT LOW DENSITY

2. Control of interactions

2. Control of interactions

OPPORTUNITIES AT LOW DENSITY 1. Separation of length scales

change magnetic field

[Following work by: Verhaar, Stwalley, Ketterle, Jin, Wieman, …]

50 µm =

0.5 pm =

1/k _F

a (i n Bohr) ⇥ =4 a 2 !

Extreme tuneability of interactions

or

1. Separation of length scales 2. Control of interactions

3. Optical imaging of individual atoms

4. Slow time scales: dynamics easily accessible 5. Ab initio theory

OPPORTUNITIES AT LOW DENSITY

A wonderful playground for many-body physics,


and especially for quantum dynamics

[Reviews: Schäfer 2009; Enss,Haussman, Zwerger 2011; More recent work: Elliot et al PRL (2014); arXiv:1410.4835]

Universality in transport of unitary gases ^{(10 -6 K to 10 14 K)}

⌘/s 0.5 ~/k

Ultracold atoms:

[J. E. Thomas group]

Relativistic ions:

[STAR collaboration]

⌘ s

1 4⇡

~ k _B

KSS conjecture:

(2005)

Longitudinal spin diffusivity

MIT (2011):

3D longitudinal spin diffusion

D ^|| & 6 ~ m

A. Sommer…M. Zwierlein, Nature 472, 201 (2011)


Review: T. Enss & JT, arXiv:1805.05354, to appear in Ann Rev CMP.

Conjectured bounds on transport coefficients

for systems with intrinsic limits due to scattering.

D & ~/m

Simple argument #1, for Fermi gas

` & n ^1/3 ⇠ 1/k ^F

giving bound to O(m/m*)

idea: mfp is at least the inter-particle spacing

[see work by Bruun, Pethick, Enss, Huse, Roche, Heiselberg, Duine, Zaanen, Kovtun, Sachdev, Hartnoll, Maldecena …]

D = v _F ` & ~ k _F m ^⇤

1

k _F = ~

m ^⇤

for systems with intrinsic limits due to scattering.

[see work by Bruun, Pethick, Enss, Huse, Roche, Heiselberg, Duine, Zaanen, Kovtun, Sachdev, Hartnoll, Maldecena …]

D & ~/m

Simple argument #2

“Planckian” conjecture

Idea: transport time reveals

(bounded) local relaxation time

⌧ _r & ~/k ^B T

D ⇠ ⌧ ^r hv ² i & ~ k _B T

k _B T

m ⇠ ~ m

Conjectured bounds on transport coefficients

for systems with intrinsic limits due to scattering.

[see work by Bruun, Pethick, Enss, Huse, Roche, Heiselberg, Duine, Zaanen, Kovtun, Sachdev, Hartnoll, Maldecena …]

D & ~/m

Simple argument #2, for Fermi liquid

“Planckian” conjecture

Idea: transport time reveals

(bounded) local relaxation time

Conjectured bounds on transport coefficients

⌧ _r & ~/E ^F

D ⇠ ⌧ ^r v _F ² & ~

m ^⇤ v _F ² v _F ² = ~

m ^⇤

Transverse spin diffusion

Scattering no longer restricted to Fermi surfaces

• An exception to usual FL 1/T ² behaviour

• Finite damping coefficient at zero temperature!

x

Particles can move without scattering E>>E F +k B T

• Low T: reduced scattering, so longer mfp: D larger

• In 3D, gives the typical 1/T ² Fermi Liquid signature

x

Longitudinal diffusion (no spin coherence)

Transverse diffusion (w/ spin coherence)

“anisotropy
 temperature”

D ^||

D ₀ ^?

Result: “Anisotropy”

at low temperature

due to Pauli blocking


[Mullin,1992]

Spin-echo measurement of magnetisation

irreversible loss of
 magnetization due to spin diffusion spiral in M due to

external B-field gradient

π ^/2

|+z〉

|-z〉 |+y〉 |+z〉

|-z〉

|p〉

RF amplitude

0 t

t

π

}

δ

b a

1

0 0.5

0 10 20

10 6 Γ( δ)

ħδ/E _F

s δ 3/2 Γ( δ)

2 4 6 8

0

1

1.5 2

0 0.5

2.5

Holdtime (ms)

20 10

0

ħδ/E _F

x10

π ^/2

|+z〉

|-z〉 |+y〉 |+z〉

|-z〉

|p〉

RF amplitude

0 t

t

π

}

δ

b a

1

0 0.5

0 10 20

10 6 Γ( δ)

ħδ/E _F

s δ 3/2 Γ( δ)

2 4 6 8

0

1

1.5 2

0 0.5

2.5

Holdtime (ms)

20 10

0

ħδ/E _F

x10

π ^/2

|+z〉

|-z〉 |+y〉

|+z〉

|-z〉

|p〉

RF amplitude

0 t

t

π

}

δ

b

a

1 0 0.5

0 10 20

10 6 Γ (δ )

ħδ/E ^F

s δ 3/2 Γ (δ )

2 4 6 8 0

1

1.5 2 0 0.5

2.5

Holdtime (ms)

20 10

0

ħδ/E ^F

x10

with phase lag:


 initial state is full transverse

polarization

At ,

“...photograph indicates

approximately an exp(-kt ³ /3) decay law for the primary

echo envelope in H 2 0.”

[Purcell, Hahn, Torrey, Slichter, Abragam…]

spin
 spiral

cubic exponential
 decay

R _M ⌘ (D 0 ^? ↵ ² ) ^1/3 M _x + iM _y = M ₀ e ^i↵zt exp [ 1

3 D ₀ ^? ↵ ² t ³ ]

define demagnetization rate

fit to

Demagnetisation at unitarity

B A

τ M (ms)

0 2 4

1 3

10 15 20 25 30

5 0

B-field gradient B’ (G/cm)

0 1 2 3 4

0 0.2 0.4 0.6 0.8 1

|M ⊥ |

Hold time t (ms)

5

Initial temperature (T/T _F ) _i

10

0 2 4 6 8

0 0.2 0.4 0.6 0.8 1

D s ⊥ (ħ /m )

t (ms)

0 1 2 3

|M

|

1 0.5 0

t (ms)

0 1 2

|M

|

1 0.5 0

Ramsey fringe visibility vs time 


(each point is ~20 phases)

exp [ (R

t)

/12]

-Observe correct B’ scaling: ^B

A

τ

(ms)

0 2 4

1 3

10 15 20 25 30

5 0

B-field gradient B’ (G/cm)

0 1 2 3 4

0 0.2 0.4 0.6 0.8 1

|M ⊥|

Hold time t (ms)

5

Initial temperature (T/T

)

10

0 2 4 6 8

0 0.2 0.4 0.6 0.8 1

D s

(ħ /m )

t (ms)

0 1 2 3

|M⊥|

1 0.5 0

t (ms)

0 1 2

|M⊥|

1 0.5 0

1 /R M (m s)

1/R _M ⇠ (B ⁰ ) ^2/3

-Vary gradient:

Time scale is a measure of diffusivity!

D = R ³ _M /↵ ²

-Single-parameter fit to find diffusivity:

D = 1.1(2) ~

m

“Birth of a strongly correlated system”

| zi

|+yi

π/2 pulse

initial state final state

ideal Fermi gas unitary Fermi gas

Bloch sphere

full polarized mixture

growth of correlations

Correlations/interactions: “Contact” dynamics

π

|+z〉

|-z〉|+y〉 |+z〉

|-z〉

|p〉

RF amplitude

0 t_π t Time

π

}δ

b a

1

0 0.5

0 10 20

10

Γ( δ)

ħδ/E

s δ 3/2 Γ( δ)

2 4 6 8

0

1

1.5 2 0 0.5

2.5

Holdtime (ms)

20 10

0

ħδ/E _F

x10

π

|+z〉

|-z〉|+y〉 |+z〉

|-z〉

|p〉

RF amplitude

0 t_π t Time

π

}δ

b a

1

0 0.5

0 10 20

10

Γ( δ)

ħδ/E

s δ 3/2 Γ( δ)

2 4 6 8

0

1

1.5 2 0 0.5

2.5

Holdtime (ms)

20 10

0

ħδ/E _F

x10

t/2

1

2

D C A

0 B

0.5 1 1.5

0 1 2 3 4 5 10 15 20 0 1 2 3 4 5

π-pulse echo time

2.5

0.5 1 1.5 2 3.5 3

2.5 4

1.5 3.5

1 2 3

τ C (ms)

τ _M (ms)

Hold time t (ms) Hold time t (ms)

B’ (G/cm)

0 10 20 30

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

C/C max C /Nk F

|M _⊥ |

2 1

[contact theory: Tan, Braaten & Platter, S Zhang & Leggett, Werner, Tarruell, & Castin, Barth, Zwerger, Combescot, Yu, Bruun, Baym, Drummond, Randeria, E Taylor, Son, H Hu, Romera-Rochin, Mølmer, Q Zhou, …]

[contact reviews by Braaten (2012); by Werner & Castin (2012)]

D C A

0

B

0.5 1 1.5

0 1 2 3 4 5 10 15 20 0 1 2 3 4 5

π-pulse echo time

2.5

0.5 1 1.5 2 3.5 3

2.5 4

1.5 3.5

1 2 3

τ

(ms)

τ

(ms)

Hold time t (ms) Hold time t (ms)

B’ (G/cm)

0 10 20 30

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

C/C

C /Nk

|M

| with echo

no echo

1/R _M (ms)

1 /R C (m s)

Compare rate of contact

growth to rate of M decay:

2D Spin transport

π ^/2

|+z〉

|-z〉 |+y〉 |+z〉

|-z〉

|p〉

RF amplitude

0 t

t

π

}

δ

b a

1

0 0.5

0 10 20

10 6 Γ( δ)

ħδ/E _F

s δ 3/2 Γ( δ)

2 4 6 8

0

1

1.5 2

0 0.5

2.5

Holdtime (ms)

20 10

0

ħδ/E _F

x10

|+zi

| zi

|pi

Initiate dynamics

Time Image

Jump B to 209 G,
 open trap

Set B, B’


ramp to V

Probe

10 ms SG TOF

2 3 1

B B’


Laser

quasi-2D

“crêpes”

otherwise the same procedure:

2D is different

scattering amplitude:

2D:

always bound state!

"

= ~

ma

3D: f (k) = 1

1/a

ik f (k) = 4⇡

ln(1/k

a

) + i⇡

Adhikari 1986

• typical scale k=k

: expansion parameter g=-1/ln(k

a

)

• coupling always energy-dependent

• never scale invariant (quantum anomaly breaks classical scale invariance

Holstein 1993; Pitaevskii & Rosch 1997

thermodynamics:

-2 0 2 4 6

-3 -2 -1 0 1

ln(k

a

)

E / E

E

P

contact quantifies breaking of scale invariance

E = P + C ^2D 4⇡m

[review of 2D by Levinsen & Parish (2015)]

[contact in 2D: Werner, Castin, Combescot, Leyronas, Mølmer, Hofmann, Kohl, Giamarchi, Enss, Gazerlis, …]

2D Demagnetization dynamics

Quantum-limited diffusion in 2D

Contact dynamics in 2D

c

b

D ^? ₀ < ~/m

C. Luciuk et al., PRL (2017)

0.2 0.4 0.6 0.8 1 0

2 4 6 8 10

d=3

d=2 D ₀ ^? & 2 ~

m

D ^|| & 6 ~

m _T/T

[Trotzky et al. PRL 2015]

D ₀ ^? & ~/m

[Luciuk et al. PRL 2017]

[Sommers et al. Nature 2011]

Review: T. Enss & JT, arXiv:1805.05354

Theory Collaborations

Frédéric Chevy (LKB/ENS) Tilman Enss (Heidelberg) Ana Maria Rey (JILA)

Edward Taylor (Toronto)

Zhenhua Yu 俞振华 (中⼭山⼤大学）

Shizhong Zhang 张世忠 (HKU) Rhys Anderson


Kenneth Jackson Scott Smale

Matthew Taylor Vijin Venu

Peihang Xu 许培航

Ben Olsen (-»Yale/NUS)


Fudong Wang 汪福东 (-»南⽅方科技⼤大学) Stefan Trotzky (-»Metamaterials Halifax) Dave McKay (-»IBM Watson)


Alma Bardon (-»Morgan Solar) Scott Beattie (-»NRC Ottawa)

Fabian Böttcher (visit from Stuttgart)

Research team

Department of Physics
 University of Toronto

Canadian Institute


for Advanced Research

NSERC AFOSR ARO

Ph.D. and Postdoctoral position available!