Few-fermion thermometry

Tomasz Sowiński

Institute of Physics of the Polish Academy of Sciences

Few-fermion

thermometry

FEW-BODY PROBLEMS

THEORY GROUP

FEW-BODY PROBLEMS

THEORY GROUP

FewBody.ifpan.edu.pl

Marcin Płodzień

Rafał Demkowicz-Dobrzański

Co-authors:

Phys. Rev. A 97, 063619 (2018)

motivation

6 Li

Atomic number (Z) 3

Nucleons (Z+N) 6

Total electronic spin (S) 1/2

Total nuclear spin (I) 1

Hyperfine states (F=S+I) 1/2 or 3/2

| "i = |F = 3/2, m ^F = 3/2 i

| #i = |F = 1/2, m ^F = 1/2 i

message from experiments

• two flavours of spinless fermions

• well controlled number of particles

• one-dimensional confinement

• strong interactions between flavours

…what for???

• extreme control on the system

• extreme precision of measurements

• from ’one' to 'many' crossover

Selim Jochim’s experimental group

in Heidelberg

message from experiments

F. Serwane et al., Science 332, 6027 (2011)

message from experiments

F. Serwane et al., Science 332, 6027 (2011)

the model

(anti-)commutation relations

n (x), ˆ ˆ ^† (x ⁰ ) o

= (x x ⁰ )

n (x), ˆ ˆ (x ⁰ ) o

= 0

• two distinguishable flavours of fermions (↑ and ↓)

• both flavours may have different masses

• both flavours confined in one-dimensional trap

• opposite spins do interact via sort range δ-like potential

H = ˆ X Z

dx ˆ ^† (x)

 ~ ²

2m

d ²

dx ² + V (x) ˆ (x)

+ g

Z

dx ˆ ^† _# (x) ˆ ^† _" (x) ˆ _" (x) ˆ _# (x)

h N ˆ " , ˆ H i

= h

N ˆ # , ˆ H i

= 0

N = ˆ

Z

dx ˆ ^† (x) ˆ (x)

many-body spectrum

Utilise it

for precise thermometry!

Notice the quasi-degeneracy

of the spectrum for strong repulsions!

thermal sensitivity

Let us assume that the system is in thermal equilibrium with some other

system of unknown temperature T

Question: how precisely one can determine

temperature T by performing

measurements on the system?

Answer: it depends on measurements

one has in the arsenal!

ˆ

⇢ _T = 1

Z e ^{H/k ˆ B T} = X

i

P _i |E ⁱ ihE ⁱ | P _i = e ^{E i /k B T}

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P _i = 1

Z e ^{E i /k B T}

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thermal sensitivity

Measurements: Some set of probabilities as functions of T.

The limit is determined by the Fisher information.

{p ⁱ (T ) } ! F({p ⁱ }) = X

i

1

p _i

✓ dp _i

dT

◆ 2

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T ² 1

n F({p ⁱ })

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ˆ

⇢ _T = 1

Z e ^{H/k ˆ B T} = X

i

P _i |E ⁱ ihE ⁱ | P _i = e ^{E i /k B T}

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IF: One can measure Gibbs probabilities

then the Fisher information is the largest possible!

{p ⁱ (T ) } = {P ⁱ (T ) }

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F ^Q (ˆ ⇢ _T ) = H ˆ ²

T ⁴ = Tr[ˆ ⇢ _T H ˆ ² ] Tr[ˆ ⇢ _T H] ˆ ²

T ⁴

F ^Q (ˆ ⇢ _T ) = H ˆ ²

T ⁴ = Tr[ˆ ⇢ _T H ˆ ² ] Tr[ˆ ⇢ _T H] ˆ ²

T ⁴

quantum Fisher information

F _Q

(ˆ ⇢ _T

) = H ˆ ²

T 4 = Tr

[ˆ⇢ _T H ˆ ² ]

Tr [ˆ⇢ _T H] ˆ ²

T 4

10 ¹

10 ²

10 ³

0 0.02 0.04 0.06 0.08

N _↓ = N _↑ = 2

g=14

Temperature T

Fisher Information

the simplest scenario

The probability of finding a particle just above the Fermi level

10 ¹

10 ²

10 ³

0 0.02 0.04 0.06 0.08

N _↓ = N _↑ = 2

g=14

Temperature T

Fisher Information

F _Q

(ˆ ⇢ _T

) = H ˆ ²

T 4 = Tr

[ˆ⇢ _T H ˆ ² ]

Tr [ˆ⇢ _T H] ˆ ²

T 4

Occupations of the lowest orbitals

Probabilities p 0 , p 1 , p 2 , p 3 , p 4 that in two the lowest orbitals one can find

0, 1, 2, 3, or 4 particles.

10 ¹

10 ²

10 ³

0 0.02 0.04 0.06 0.08

N _↓ = N _↑ = 2

g=14

Temperature T

Fisher Information

F _Q

(ˆ ⇢ _T

) = H ˆ ²

T 4 = Tr

[ˆ⇢ _T H ˆ ² ]

Tr [ˆ⇢ _T H] ˆ ²

T 4

{

10 ¹

10 ²

10 ³

0 0.02 0.04 0.06 0.08

N _↓ = N _↑ = 2

g=14

Temperature T

Fisher Information

All occupations are known

F _Q

(ˆ ⇢ _T

) = H ˆ ²

T 4 = Tr

[ˆ⇢ _T H ˆ ² ]

Tr [ˆ⇢ _T H] ˆ ²

T 4

When we know all probabilities

of finding four particles in different configurations

10 ^-1

1

10 ¹

10 ²

10 ³

10 ⁴

10 ^-3 10 ^-2 10 ^-1 1

N _↓ = N _↑ = 2

g = 14

g = 10

g = 5

g = 0

Temperature T

Quantum Fisher Information

10 ^-1

1

10 ¹

10 ²

10 ³

10 ⁴

10 ^-3 10 ^-2 10 ^-1 1

Fundamental bound

the fundamental bound

Fundamental bound when level spacings

can be engineered

L. A. Correa et al.


Phys. Rev. Lett. 114, 220405 (2015).

Tomasz Sowiński

Institute of Physics of the Polish Academy of Sciences

Few-fermion

thermometry

FEW-BODY PROBLEMS

THEORY GROUP

FEW-BODY PROBLEMS

THEORY GROUP

FewBody.ifpan.edu.pl

Marcin Płodzień

Rafał Demkowicz-Dobrzański

Co-authors:

Phys. Rev. A 97, 063619 (2018)