Inter-Component Correlations in One-Dimensional Mass-Imbalanced Ultra-Cold Few-Fermion Mixtures

(1)

Tomasz Sowiński

Institute of Physics of the Polish Academy of Sciences

in One-Dimensional

Mass-Imbalanced Ultra-Cold

Few-Fermion Mixtures

FEW-BODY PROBLEMS

THEORY GROUP

FEW-BODY PROBLEMS

THEORY GROUP

FewBody.ifpan.edu.pl

D. Pęcak, T. Sowiński

Phys. Rev. A 99, 043612 (2019)

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(3)

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

Figure from:

Strong correlations in few-fermion systems

Andrea Bergschneider, PhD thesis (July 2017)

Ruperto-Carola University of Heidelberg, Germany

T = 250 nK

N ≈ 6•10 ⁴

T/T F ≈ 0.5

T = 250 nK

T F = 7 μK

N ≈ 2 ... 20

T/T F ≈ 0.05

! _r ⇡ 2⇡ ⇥ 370 Hz

! _l ⇡ 2⇡ ⇥ 34 Hz

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! _r ⇡ 2⇡ ⇥ 14 000 Hz

! _l ⇡ 2⇡ ⇥ 1 487 Hz

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motivation

(4)

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

Figure from:

Strong correlations in few-fermion systems

Andrea Bergschneider, PhD thesis (July 2017)

Ruperto-Carola University of Heidelberg, Germany

T = 250 nK

N ≈ 6•10 ⁴

T/T F ≈ 0.5

T = 250 nK

T F = 7 μK

N ≈ 2 ... 20

T/T F ≈ 0.05

! _r ⇡ 2⇡ ⇥ 370 Hz

! _l ⇡ 2⇡ ⇥ 34 Hz

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! _r ⇡ 2⇡ ⇥ 14 000 Hz

! _l ⇡ 2⇡ ⇥ 1 487 Hz

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motivation

(5)

message from the experiment

Figure from:

Strong correlations in few-fermion systems

Andrea Bergschneider, PhD thesis (July 2017)

Ruperto-Carola University of Heidelberg, Germany

(6)

message from the experiment

Figure from:

Strong correlations in few-fermion systems

Andrea Bergschneider, PhD thesis (July 2017)

Ruperto-Carola University of Heidelberg, Germany

m _" /m _# 6= 1

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the model

(anti-)commutation relations

n (x), ˆ ˆ ^† (x ⁰ ) o

= (x x ⁰ )

n (x), ˆ ˆ (x ⁰ ) o

= 0

h ˆ

" (x), ˆ ^† _# (x ⁰ ) i

= 0

h ˆ

" (x), ˆ _# (x ⁰ ) i

= 0

• the same spins • opposite spins

• two distinguishable flavors of fermions (↑ and ↓)

• both flavors may have different masses

• both flavors 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)

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the model

(anti-)commutation relations

n (x), ˆ ˆ ^† (x ⁰ ) o

= (x x ⁰ )

n (x), ˆ ˆ (x ⁰ ) o

= 0

h ˆ

" (x), ˆ ^† _# (x ⁰ ) i

= 0

h ˆ

" (x), ˆ _# (x ⁰ ) i

= 0

• the same spins • opposite spins

• two distinguishable flavors of fermions (↑ and ↓)

• both flavors may have different masses

• both flavors 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

number of fermions is conserved

N = ˆ

Z

dx ˆ ^† (x) ˆ (x)

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in this talk...

Harmonic confinement:

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)

Attractive interactions:

g < 0

V (x) = m ⌦ ²

2 x ²

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Many-body ground-state properties: |Gi

(10)

Equal mass fermions

(11)

inter-particle correlations

two-fermion reduced density matrix

⇢ ⁽²⁾ (x ⁰ ₁ , x ⁰ ₂ ; x ₁ , x ₂ ) = X

i

i i (x ₁ , x ₂ ) ^⇤ _i (x ⁰ ₁ , x ⁰ ₂ )

spectral decomposition of ⇢ ⁽²⁾

T. Sowiński, M. Gajda, K. Rzążęwski

Europhys. Lett. 109, 26005 (2015)

⇢ ⁽²⁾ (x ⁰ ₁ , x ⁰ ₂ ; x ₁ , x ₂ ) = hG| ˆ ^† _" (x ⁰ ₁ ) ˆ ^† _# (x ⁰ ₂ ) ˆ _# (x ₂ ) ˆ _" (x ₁ ) |Gi

(12)

inter-particle correlations

two-fermion reduced density matrix

⇢ ⁽²⁾ (x ⁰ ₁ , x ⁰ ₂ ; x ₁ , x ₂ ) = X

i

i i (x ₁ , x ₂ ) ^⇤ _i (x ⁰ ₁ , x ⁰ ₂ )

spectral decomposition of ⇢ ⁽²⁾

⇢ ⁽²⁾ (x ⁰ ₁ , x ⁰ ₂ ; x ₁ , x ₂ ) = hG| ˆ ^† _" (x ⁰ ₁ ) ˆ ^† _# (x ⁰ ₂ ) ˆ _# (x ₂ ) ˆ _" (x ₁ ) |Gi

T. Sowiński, M. Gajda, K. Rzążęwski

Europhys. Lett. 109, 26005 (2015)

(13)

the dominant orbital…

T. Sowiński, M. Gajda, K. Rzążęwski

Europhys. Lett. 109, 26005 (2015)

(14)

Density distributions

n ⁽²⁾ (x ₁ , x ₂ ) = ⇢ ⁽²⁾ (x ₁ , x ₂ ; x ₁ , x ₂ )

= hG| ˆ ^† _" (x ₁ ) ˆ ^† _# (x ₂ ) ˆ _# (x ₂ ) ˆ _" (x ₁ ) |Gi

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n ⁽²⁾ (p ₁ , p ₂ ) = ⇢ ⁽²⁾ (p ₁ , p ₂ ; p ₁ , p ₂ )

= hG| ˆ ^† _" (p ₁ ) ˆ ^† _# (p ₂ ) ˆ _# (p ₂ ) ˆ _" (p ₁ ) |Gi

<latexit sha1_base64="ynL0f8hadZeF5NfLZxPIVbgDeRE=">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</latexit>

(15)

Density distributions

n ⁽²⁾ (x ₁ , x ₂ ) = ⇢ ⁽²⁾ (x ₁ , x ₂ ; x ₁ , x ₂ )

= hG| ˆ ^† _" (x ₁ ) ˆ ^† _# (x ₂ ) ˆ _# (x ₂ ) ˆ _" (x ₁ ) |Gi

n ⁽²⁾ (p ₁ , p ₂ ) = ⇢ ⁽²⁾ (p ₁ , p ₂ ; p ₁ , p ₂ )

= hG| ˆ ^† _" (p ₁ ) ˆ ^† _# (p ₂ ) ˆ _# (p ₂ ) ˆ _" (p ₁ ) |Gi

(16)

Density distributions

n ⁽²⁾ (x ₁ , x ₂ ) = ⇢ ⁽²⁾ (x ₁ , x ₂ ; x ₁ , x ₂ )

= hG| ˆ ^† _" (x ₁ ) ˆ ^† _# (x ₂ ) ˆ _# (x ₂ ) ˆ _" (x ₁ ) |Gi

n ⁽²⁾ (p ₁ , p ₂ ) = ⇢ ⁽²⁾ (p ₁ , p ₂ ; p ₁ , p ₂ )

= hG| ˆ ^† _" (p ₁ ) ˆ ^† _# (p ₂ ) ˆ _# (p ₂ ) ˆ _" (p ₁ ) |Gi

n ⁽¹⁾ (x) = ⇢ ⁽¹⁾ (x; x)

= hG| ˆ ^† (x) ˆ (x) |Gi

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n ⁽¹⁾ (p) = ⇢ ⁽¹⁾ (p; p)

= hG| ˆ ^† (p) ˆ (p) |Gi

<latexit sha1_base64="uP7g7oFg4S8VkL/V1dYSJd8DE54=">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</latexit>

(17)

Density distributions

n ⁽²⁾ (x ₁ , x ₂ ) = ⇢ ⁽²⁾ (x ₁ , x ₂ ; x ₁ , x ₂ )

= hG| ˆ ^† _" (x ₁ ) ˆ ^† _# (x ₂ ) ˆ _# (x ₂ ) ˆ _" (x ₁ ) |Gi

n ⁽²⁾ (p ₁ , p ₂ ) = ⇢ ⁽²⁾ (p ₁ , p ₂ ; p ₁ , p ₂ )

= hG| ˆ ^† _" (p ₁ ) ˆ ^† _# (p ₂ ) ˆ _# (p ₂ ) ˆ _" (p ₁ ) |Gi

n ⁽¹⁾ (x) = ⇢ ⁽¹⁾ (x; x)

= hG| ˆ ^† (x) ˆ (x) |Gi

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n ⁽¹⁾ (p) = ⇢ ⁽¹⁾ (p; p)

= hG| ˆ ^† (p) ˆ (p) |Gi

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Correlations induced by interactions

G ^⇡ (p ₁ , p ₂ ) = n ⁽²⁾ (p ₁ , p ₂ ) n ⁽¹⁾ _" (p ₁ )n ⁽¹⁾ _# (p ₂ )

G ^⇢ (x ₁ , x ₂ ) = n ⁽²⁾ (x ₁ , x ₂ ) n ⁽¹⁾ _" (x ₁ )n ⁽¹⁾ _# (x ₂ )

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(18)

Several equal mass particles

N _" = N _# = 4

G ^⇡ (p ₁ , p ₂ ) = n ⁽²⁾ (p ₁ , p ₂ ) n ⁽¹⁾ _" (p ₁ )n ⁽¹⁾ _# (p ₂ )

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G ^⇢ (x ₁ , x ₂ ) = n ⁽²⁾ (x ₁ , x ₂ ) n ⁽¹⁾ _" (x ₁ )n ⁽¹⁾ _# (x ₂ )

n ⁽²⁾ (p ₁ , p ₂ )

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(19)

Several equal mass particles

N _" = N _# = 4

G ^⇡ (p ₁ , p ₂ ) = n ⁽²⁾ (p ₁ , p ₂ ) n ⁽¹⁾ _" (p ₁ )n ⁽¹⁾ _# (p ₂ )

G ^⇢ (x ₁ , x ₂ ) = n ⁽²⁾ (x ₁ , x ₂ ) n ⁽¹⁾ _" (x ₁ )n ⁽¹⁾ _# (x ₂ )

n ⁽¹⁾ _" (p ₁ )n ⁽¹⁾ _# (p ₂ )

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n ⁽²⁾ (p ₁ , p ₂ )

(20)

Several equal mass particles

N _" = N _# = 4

G ^⇡ (p ₁ , p ₂ ) = n ⁽²⁾ (p ₁ , p ₂ ) n ⁽¹⁾ _" (p ₁ )n ⁽¹⁾ _# (p ₂ )

G ^⇢ (x ₁ , x ₂ ) = n ⁽²⁾ (x ₁ , x ₂ ) n ⁽¹⁾ _" (x ₁ )n ⁽¹⁾ _# (x ₂ )

G ^⇡ (p ₁ , p ₂ )

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n ⁽¹⁾ _" (p ₁ )n ⁽¹⁾ _# (p ₂ )

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n ⁽²⁾ (p ₁ , p ₂ )

(21)

Inter-particle correlations

• Along with increasing mass ratio 

inter-particle correlations decrease

• Can we describe this behavior qualitatively?

N _" = N _# = 4

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g = 5

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µ = m _" /m _#

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D. Pęcak, T. Sowiński: Phys. Rev. A 99, 043612 (2019)

µ = 2

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µ = 1

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µ = 3

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µ = 4

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G ^⇡ (p ₁ , p ₂ )

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(22)

The Frobenius distance between densities

encodes the total amount of correlations

induced by mutual interactions which can be

extracted from the two-particle densities.

NOTE!

This quantity does not count all the possible correlations

which can be detected in two-particle measurements.

||G ^⇡ || =

✓Z

dp ₁ dp ₂ |G ^⇡ (p ₁ , p ₂ ) | ²

◆ 1/2

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G ^⇡ (p ₁ , p ₂ ) = n ⁽²⁾ (p ₁ , p ₂ ) n ⁽¹⁾ _" (p ₁ )n ⁽¹⁾ _# (p ₂ )

(23)

µ = m _" /m _# D. Pęcak, T. Sowiński: Phys. Rev. A 99, 043612 (2019)

(24)

D. Pęcak, T. Sowiński: Phys. Rev. A 99, 043612 (2019)

Inter-Component Correlations in One-Dimensional Mass-Imbalanced Ultra-Cold Few-Fermion Mixtures

Tomasz Sowiński

Institute of Physics of the Polish Academy of Sciences

in One-Dimensional

Mass-Imbalanced Ultra-Cold

Few-Fermion Mixtures

FEW-BODY PROBLEMS

THEORY GROUP

FEW-BODY PROBLEMS

THEORY GROUP

FewBody.ifpan.edu.pl

D. Pęcak, T. Sowiński

Phys. Rev. A 99, 043612 (2019)

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

Figure from:

Strong correlations in few-fermion systems

Andrea Bergschneider, PhD thesis (July 2017)

Ruperto-Carola University of Heidelberg, Germany

T = 250 nK

N ≈ 6•10 4

T/T F ≈ 0.5

T = 250 nK

T F = 7 μK

N ≈ 2 ... 20

T/T F ≈ 0.05

! r ⇡ 2⇡ ⇥ 370 Hz

! l ⇡ 2⇡ ⇥ 34 Hz

! r ⇡ 2⇡ ⇥ 14 000 Hz

! l ⇡ 2⇡ ⇥ 1 487 Hz

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

Figure from:

Strong correlations in few-fermion systems

Andrea Bergschneider, PhD thesis (July 2017)

Ruperto-Carola University of Heidelberg, Germany

T = 250 nK

N ≈ 6•10 4

T/T F ≈ 0.5

T = 250 nK

T F = 7 μK

N ≈ 2 ... 20

T/T F ≈ 0.05

! r ⇡ 2⇡ ⇥ 370 Hz

! l ⇡ 2⇡ ⇥ 34 Hz

! r ⇡ 2⇡ ⇥ 14 000 Hz

! l ⇡ 2⇡ ⇥ 1 487 Hz

motivation

message from the experiment

Figure from:

Strong correlations in few-fermion systems

Andrea Bergschneider, PhD thesis (July 2017)

Ruperto-Carola University of Heidelberg, Germany

message from the experiment

Figure from:

Strong correlations in few-fermion systems

Andrea Bergschneider, PhD thesis (July 2017)

Ruperto-Carola University of Heidelberg, Germany

m " /m # 6= 1

the model

(anti-)commutation relations

n (x), ˆ ˆ † (x 0 ) o

= (x x 0 )

n (x), ˆ ˆ (x 0 ) o

= 0

h ˆ

" (x), ˆ † # (x 0 ) i

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

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

N ≈ 6•10 ⁴

! _r ⇡ 2⇡ ⇥ 370 Hz

! _l ⇡ 2⇡ ⇥ 34 Hz

! _r ⇡ 2⇡ ⇥ 14 000 Hz

! _l ⇡ 2⇡ ⇥ 1 487 Hz

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

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

N ≈ 6•10 ⁴

! _r ⇡ 2⇡ ⇥ 370 Hz

! _l ⇡ 2⇡ ⇥ 34 Hz

! _r ⇡ 2⇡ ⇥ 14 000 Hz

! _l ⇡ 2⇡ ⇥ 1 487 Hz

m _" /m _# 6= 1

n (x), ˆ ˆ ^† (x ⁰ ) o

= (x x ⁰ )

n (x), ˆ ˆ (x ⁰ ) o

" (x), ˆ ^† _# (x ⁰ ) i

" (x), ˆ _# (x ⁰ ) i

dx ˆ ^† (x)

 ~ ²

d ²

dx ² + V (x) ˆ (x)

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

n (x), ˆ ˆ ^† (x ⁰ ) o

= (x x ⁰ )

n (x), ˆ ˆ (x ⁰ ) o

" (x), ˆ ^† _# (x ⁰ ) i

" (x), ˆ _# (x ⁰ ) i

dx ˆ ^† (x)

 ~ ²

d ²

dx ² + V (x) ˆ (x)

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

h N ˆ _" , ˆ H i

N ˆ _# , ˆ H i

dx ˆ ^† (x) ˆ (x)

dx ˆ ^† (x)

 ~ ²

d ²

dx ² + V (x) ˆ (x)

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

V (x) = m ⌦ ²

2 x ²