Dynamics of several ultra-cold particles in a double-well potential

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Dynamics of several

ultra-cold particles

in a double-well potential

Tomasz Sowinski

Institute of Physics of the Polish Academy of Sciences

References:

[1] T. Sowiński, M. Gajda, K. Rzążewski: EPL 113, 56003 (2016) [2] J. Dobrzyniecki, T. Sowiński: EPJ D 70, 83 (2016)

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Institute of Physics

Polish Academy of Sciences

Warsaw, Poland

9650 km

Nicolaus Copernicus

was born in Torun (Poland) 9650 km North from Cape Town

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The question

0 a²/2

-2a -a 0 +a +2a

Position (osc. units)

N _" N _#

THE INITIAL STATE

What one can say about the evolution governed by the many-body Hamiltonian

without any approximations???

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Double-well models

λ=0 λ=2 λ=4 λ=6

Position (osc. units)

Potential (osc. units)

0 2 4 6

-4 -2 0 2 4

Energy (osc. units)

Parameter λ

0 2 4 6 8 10

H ˆ

⁰

= ~

²

2m

d

²

dx

²

+ m⌦

²

2 x

²

+ exp

✓ m⌦

2 ~ x

²

◆

shape of a potential spectrum

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Double-well models

Position (osc. units)

2

0 x₀/2

-2x₀ -x₀ 0 +x₀ +2x₀

Energy (osc. units)

Parameter x

₀

(osc. units)

0 1 2 3 4 5

0 1 2 3 4

H ˆ

⁰

= ~

²

2m

d

²

dx

²

+ m⌦

²

2 ( |x| x

₀

)

²

shape of a potential spectrum

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MANY-BODY PROBLEM

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standard commutation relations

BOSONS

MANY-BODY PROBLEM

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FERMIONS

anticommutation relations

H = ˆ X Z

dx ˆ

^†

(x) H

⁰

(x) + g ˆ

Z

dx ˆ

^†_#

(x) ˆ

^†_"

(x) ˆ

_"

(x) ˆ

_#

(x)

MANY-BODY PROBLEM

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BOSONS

MANY-BODY PROBLEM

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BOSONS

MANY-BODY PROBLEM

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λ=0 λ=2 λ=4 λ=6

Potential (osc. units)

0 2 4 6

-4 -2 0 2 4

Energy (osc. units)

Parameter λ

0 2 4 6 8 10

NEGLEC

TED

Two-mode models

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λ=0 λ=2 λ=4 λ=6

0 2 4 6

-4 -2 0 2 4

Energy (osc. units)

Parameter λ

0 2 4 6 8 10

NEGLEC

TED

Two-mode models

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λ=0 λ=2 λ=4 λ=6

0 2 4 6

-4 -2 0 2 4

Energy (osc. units)

Parameter λ

0 2 4 6 8 10

• Two-mode model appears in consequence of neglecting higher single-particle levels

NEGLEC

TED

• Then for TWO bosons only three states are relevant:

Two-mode models

• The Hamiltonian

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λ=0 λ=2 λ=4 λ=6

0 2 4 6

-4 -2 0 2 4

Energy (osc. units)

Parameter λ

0 2 4 6 8 10

NEGLEC

TED

interaction energy when both bosons occupy given site

Two-mode models

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λ=0 λ=2 λ=4 λ=6

0 2 4 6

-4 -2 0 2 4

Energy (osc. units)

Parameter λ

0 2 4 6 8 10

NEGLEC

TED

interaction energy when bosons occupy opposite sites

Two-mode models

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λ=0 λ=2 λ=4 λ=6

0 2 4 6

-4 -2 0 2 4

Energy (osc. units)

Parameter λ

0 2 4 6 8 10

NEGLEC

TED

tunneling of pairs induced by interactions

Two-mode models

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λ=0 λ=2 λ=4 λ=6

0 2 4 6

-4 -2 0 2 4

Energy (osc. units)

Parameter λ

0 2 4 6 8 10

NEGLEC

TED

single boson tunneling INDUCED by interactions

Two-mode models

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Two-mode models

For short-range interactions one can anticipate that some terms can be neglected

Then we obtain two-mode version

of the standard Bose-Hubbard model

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Very high barrier

exact model

complete two-mode model

Simple

Bose-Hubbard model

Shallow barrier

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Shallow barrier

exact model

complete two-mode model

Simple

Bose-Hubbard model

Very high barrier

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Inter-particle correlations

probability that bosons occupy different wells

Two-mode description is insufficient

to describe inter-particle correlations correctly!

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FERMIONS

anticommutation relations

H = ˆ X Z

dx ˆ

^†

(x) H

⁰

(x) + g ˆ

Z

dx ˆ

^†_#

(x) ˆ

^†_"

(x) ˆ

_"

(x) ˆ

_#

(x)

MANY-BODY PROBLEM

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In the case of

FERMIONS

the story is even

more complicated

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Spectrum of the Hamiltonian

3 3.5 4 4.5 5

-3 -2 -1 0 1 2 3

Many-body spectrum (N_↑=N_↓=2)

Energy (osc. units)

Interaction

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Institute of Physics

Polish Academy of Sciences

Warsaw, Poland

Thank you!

References:

[1] T. Sowiński, M. Gajda, K. Rzążewski: EPL 113, 56003 (2016) [2] J. Dobrzyniecki, T. Sowiński: EPJ D 70, 83 (2016)

Dynamics of several ultra-cold particles in a double-well potential