pairing in a system of a few ultra-cold attractive fermions in a one-dimensional harmonic trap

pairing in a system of a few

ultra-cold attractive fermions

in a one-dimensional harmonic trap

tomasz sowiński

Institute of Physics of the Polish Academy of Sciences

T. Sowiński, M. Gajda, K. Rzążewski Europhys. Lett. 109, 26005 (2015)

general motivation

• theoretical point of view

• FEW-BODY PROBLEMS

forms poorly explored bridge between two-body and many-body physics

• ONE-DIMENSIONAL PHYSICS

is completely different than physics in higher dimensions

• experimental point of view

• IT IS POSSIBLE

to prepare experiments with a few ultra-cold atoms effectively confined in a one-dimensional harmonic trap

general motivation

• theoretical point of view

• FEW-BODY PROBLEMS

forms poorly explored bridge between two-body and many-body physics

• ONE-DIMENSIONAL PHYSICS

is completely different than physics in higher dimensions

• experimental point of view

• IT IS POSSIBLE

to prepare experiments with

a few ultra-cold atoms confined effectively in a one-dimensional harmonic trap

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 = 3/2i

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

the question

Is it possible to find

any tracks

of Cooper-like pairing

in the one-dimensional system

of a few fermions?

the model

(anti-)commutation relations

n (x), ˆˆ ^† (x⁰)o

= (x x⁰) n (x), ˆˆ (x⁰)o

= 0

h ˆ_"(x), ˆ ^†_#(x⁰)i

= 0 h ˆ

"(x), ˆ _#(x⁰)i

= 0

• the same spins • opposite spins

• two distinguishable flavors of fermions (↑ and ↓)

• both flavors have equal masses

• both in the same one-dimensional harmonic confinement

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

H = ˆ X Z

dx ˆ

(x)

 1

2

d

dx

+ 1

2 x

(x) ˆ

+ g

Z

dx ˆ

(x) ˆ

(x) ˆ

(x) ˆ

(x)

g < 0

the model

(anti-)commutation relations

n (x), ˆˆ ^† (x⁰)o

= (x x⁰) n (x), ˆˆ (x⁰)o

= 0

h ˆ_"(x), ˆ ^†_#(x⁰)i

= 0 h ˆ

"(x), ˆ _#(x⁰)i

= 0

• the same spins • opposite spins

h N ˆ

, ˆ H i

= h

N ˆ

, ˆ H i

= 0

conservation of the number of fermions

H = ˆ X Z

dx ˆ

(x)

 1

2

d

dx

+ 1

2 x

(x) ˆ

+ g

Z

dx ˆ

(x) ˆ

(x) ˆ

(x) ˆ

(x)

N = ˆ

Z

dx ˆ

(x) ˆ (x)

the method

g < 0

the method

the method

• we fix the number of fermions N^↑ and N^↓

• we decompose the field operator in the single-particle basis
 of the harmonic oscillator eigenfunctions

H = ˆ X Z

dx ˆ

(x)

 1

2

d

dx

+ 1

2 x

(x) ˆ

+ g

Z

dx ˆ

(x) ˆ

(x) ˆ

(x) ˆ

(x)

(x) = ˆ

X

n=1

ˆ

a

(x)

2 d²

dx² + 1

2x² _n(x) =

✓

n + 1 2

◆

n(x)

• we calculate all matrix elements of the Hamiltonian

• we perform an exact diagonalization

M ⇠ 12

the model

on the track of pairs…

two-fermion reduced density matrix

⇢

(x

, x

; x

, x

) = X

i

i i

(x

, x

)

(x

, x

)

spectral decomposition of ⇢

⇢

(x

, x

; x

, x

) = hG| ˆ

(x

) ˆ

(x

) ˆ

(x

) ˆ

(x

) |Gi

T. Sowiński, M. Gajda, K. Rzążęwski Europhys. Lett. 109, 26005 (2015)

the dominant orbital…

pairs but not molecules!

one-f er mion reduced density matrix

structure of the orbitals

T. Sowiński, M. Gajda, K. Rzążęwski Europhys. Lett. 109, 26005 (2015)

structure of the orbitals

0

(x

, x

) ⇠ X

j

↵

(j)'

(x

)'

(x

)

k

(x

, x

) ⇠ '

(x

)'

(x

) ± '

(x

)'

(x

)

⇢

(x

; x

) = hG| ˆ

(x

) ˆ (x

) |Gi = X

i

⌘

'

(x

)'

(x

)

one-fermion reduced density matrix

decomposition of the two-body orbitals

in the basis spanned by one-body orbitals

single-fermion orbitals

dominant orbital (strongly correlated)

higher orbitals (trivial correlations)

pairs but not molecules!

take-home message

open access

T. Sowiński, M. Gajda, K. Rzążewski Europhys. Lett. 109, 26005 (2015)

in the system of a few fermions

confined in a one-dimensional

trap Cooper-like pairing

can be observed!