## 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 (x10^{8})

Ultracold Fermi gases

40K (x10^{8})

**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**

**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**

**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**

**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

^{B}

### 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*

_{π}

*t*

Time
### π

}

### δ

**b** **a**

### 1

### 0 0.5

### 0 10 20

### 10 6 Γ( δ)

### ħδ/E _{F}

_{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}

_{F}

### x10

### π ^{/2}

**|+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 6 Γ( δ)

### ħδ/E _{F}

_{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}

_{F}

### x10

### π ^{/2}

**|+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 6 Γ (δ )

### ħδ/E ^{F}

^{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}

^{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}

_{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

_{M}

### t)

^{3}

### /12]

### -Observe correct B’ scaling: ^{B}

^{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

### 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

^{/2}

**|+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

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}

_{F}

x10

### π

^{/2}

**|+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

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}

_{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)

_{M}

*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

*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*

_{⊥}

*|* 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*

_{π}

*t*

Time
### π

}

### δ

**b** **a**

### 1

### 0 0.5

### 0 10 20

### 10 6 Γ( δ)

### ħδ/E _{F}

_{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}

_{F}

### x10

### |+zi

### | zi

### |pi

### Initiate dynamics

### Time Image

### Jump B to 209 G, open trap

### Set B, B’

### ramp to V

0### 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!**

### "

_{B}

### = ~

^{2}

### ma

^{2}

_{2D}

### 3D: f (k) = 1

### 1/a

_{3D}

### ik f (k) = 4⇡

### ln(1/k

^{2}

### a

^{2}

_{2D}

### ) + i⇡

### Adhikari 1986

### • typical scale k=k

F### : expansion parameter g=-1/ln(k

F### a

2D### )

### • 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*

_{F}*a*

_{2 D}### )

### E / *E*

FG **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}

_{F}

[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