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

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)

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???

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

0 2 4 6 8 10

H ˆ

= ~

2m

d

dx

+ m⌦

2 x

+ exp

✓ m⌦

2 ~ x

◆

shape of a potential spectrum

Position (osc. units)

2

0 x₀/2

-2x₀ -x₀ 0 +x₀ +2x₀

Energy (osc. units)

Parameter x

(osc. units)

0 1 2 3 4 5

0 1 2 3 4

Double-well models

Position (osc. units)

2

0 x₀/2

-2x₀ -x₀ 0 +x₀ +2x₀

Energy (osc. units)

Parameter x

(osc. units)

0 1 2 3 4 5

0 1 2 3 4

H ˆ

= ~

2m

d

dx

+ m⌦

2 ( |x| x

)

shape of a potential spectrum

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???

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???

MANY-BODY PROBLEM

standard commutation relations

BOSONS

MANY-BODY PROBLEM

FERMIONS

anticommutation relations

H = ˆ X Z

dx ˆ

(x) H

(x) + g ˆ

Z

dx ˆ

(x) ˆ

(x) ˆ

(x) ˆ

(x)

MANY-BODY PROBLEM

standard commutation relations

BOSONS

MANY-BODY PROBLEM

standard commutation relations

BOSONS

MANY-BODY PROBLEM

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

0 2 4 6 8 10

NEGLEC

TED

Two-mode 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

0 2 4 6 8 10

NEGLEC

TED

Two-mode 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

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

Two-mode models

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

Then we obtain two-site version

of the standard Bose-Hubbard model

Two-mode models

• Exact model

• Two-mode approximation

• Hubbard-like description

Very high barrier

exact model

complete two-mode model

Simple

Bose-Hubbard model

Shallow barrier

Shallow barrier

exact model

complete two-mode model

Simple

Bose-Hubbard model

Very high barrier

Inter-particle correlations

probability that bosons occupy different wells

Two-mode description is insufficient

to describe inter-particle correlations correctly!

FERMIONS

anticommutation relations

H = ˆ X Z

dx ˆ

(x) H

(x) + g ˆ

Z

dx ˆ

(x) ˆ

(x) ˆ

(x) ˆ

(x)

MANY-BODY PROBLEM

two distinguishable particles

Evolution of the densities

0 0.5 1

0 20 40 60 80 100

g = +0.0010

0 0.5 1

0 200 400 600 800 1000

g = +0.0010

0 0.5 1

0 20 40 60 80 100

g = +1.0000

0 0.5 1

0 200 400 600 800 1000

g = +1.0000

0 0.5 1

0 20 40 60 80 100

g = +2.0000

0 0.5 1

0 200 400 600 800 1000

g = +2.0000

Evolution of the densities

0 0.5 1

0 20 40 60 80 100

g = +0.0010

0 0.5 1

0 200 400 600 800 1000

g = +0.0010

0 0.5 1

0 20 40 60 80 100

g = +1.0000

0 0.5 1

0 200 400 600 800 1000

g = +1.0000

0 0.5 1

0 20 40 60 80 100

g = +2.0000

0 0.5 1

0 200 400 600 800 1000

g = +2.0000

0 0.5 1

0 20 40 60 80 100

g = -2.5000

0 0.5 1

0 200 400 600 800 1000

g = -2.5000

0 0.5 1

0 20 40 60 80 100

g = -1.0000

0 0.5 1

0 200 400 600 800 1000

g = -1.0000

Evolution of the densities

0 0.5 1

0 20 40 60 80 100

g = -2.5000

0 0.5 1

0 200 400 600 800 1000

g = -2.5000

0 0.5 1

0 20 40 60 80 100

g = -1.8000

0 0.5 1

0 200 400 600 800 1000

g = -1.8000

0 0.5 1

0 20 40 60 80 100

g = -1.0000

0 0.5 1

0 200 400 600 800 1000

g = -1.0000 0

0.5 1

0 20 40 60 80 100

g = +0.0010

0 0.5 1

0 200 400 600 800 1000

g = +0.0010

0 0.5 1

0 20 40 60 80 100

g = +1.0000

0 0.5 1

0 200 400 600 800 1000

g = +1.0000

0 0.5 1

0 20 40 60 80 100

g = +2.0000

0 0.5 1

0 200 400 600 800 1000

g = +2.0000

Many-body spectrum

-2 -1 0 1 2 3

-4 -3 -2 -1 0 1 2 3 4

Energy (osc. units)

Interaction

Many-body spectrum (N_↑=N_↓=1)

Many-body spectrum

-2 -1 0 1 2 3

-4 -3 -2 -1 0 1 2 3 4

Energy (osc. units)

Interaction

Many-body spectrum (N_↑=N_↓=1)

0 0.5 1

0 20 40 60 80 100

g = -2.9000

0 0.5 1

0 200 400 600 800 1000

g = -2.9000

0 0.5 1

0 20 40 60 80 100

g = -1.8000

0 0.5 1

0 200 400 600 800 1000

g = -1.8000

0 0.5 1

0 20 40 60 80 100

g = -2.5000

0 0.5 1

0 200 400 600 800 1000

g = -2.5000

two fermionic cloudlets

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

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)