Mass-imbalanced mixtures of several ultra-cold fermions in one-dimensional traps

Tomasz Sowiński

Institute of Physics of the Polish Academy of Sciences

Mass-imbalanced mixtures

of several ultra-cold fermions

in one-dimensional traps

FEW-BODY PROBLEMS THEORY GROUP

FEW-BODY PROBLEMS THEORY GROUP

FEW-BODY PROBLEMS THEORY GROUP FEW-BODY PROBLEMS THEORY GROUP FEW-BODY PROBLEMS THEORY GROUP

the group

PostDocs

•

Marcin Płodzień

•

Elnaz Darsheshda

PhD Students

•

Daniel Pęcak

•

Jacek Dobrzyniecki

•

?? (open 2017)

•

?? (open 2018)

Mentors (distinguished professors)

FEW-BODY PROBLEMS THEORY GROUP

FEW-BODY PROBLEMS THEORY GROUP

FEW-BODY PROBLEMS THEORY GROUP FEW-BODY PROBLEMS THEORY GROUP FEW-BODY PROBLEMS THEORY GROUP

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 = 3/2i

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

message from experiments

• two flavors of spinless fermions

• well controlled number of particles

• one-dimensional confinement

• strong interactions between flavors

…what for???

• extreme control on the system

• extreme precision of measurements

• from ’one' to 'many' crossover

FEW-BODY PROBLEMS THEORY GROUP FEW-BODY PROBLEMS THEORY GROUP FEW-BODY PROBLEMS THEORY GROUP

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 = 3/2i

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

message from experiments

• two flavors of spinless fermions

• well controlled number of particles

• one-dimensional confinement

• strong interactions between flavors

…what for???

• extreme control on the system

• extreme precision of measurements

• from ’one' to 'many' crossover

m _" /m _# 6= 1

FEW-BODY PROBLEMS THEORY GROUP

message from experiments

Figure from S. Jochim’s group paper

FEW-BODY PROBLEMS THEORY GROUP

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 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)

 ~

2m

d

dx

+ V (x) ˆ (x)

+ g

Z

dx ˆ

(x) ˆ

(x) ˆ

(x) ˆ

(x)

FEW-BODY PROBLEMS THEORY GROUP

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 = ˆ X Z

dx ˆ

(x)

 ~

2m

d

dx

+ V (x) ˆ (x)

+ g

Z

dx ˆ

(x) ˆ

(x) ˆ

(x) ˆ

(x)

numerical method

FEW-BODY PROBLEMS THEORY GROUP

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

N = ˆ

Z

dx ˆ

(x) ˆ (x)

H = ˆ X Z

dx ˆ

(x)

 ~

2m

d

dx

+ V (x) ˆ (x)

+ g

Z

dx ˆ

(x) ˆ

(x) ˆ

(x) ˆ

(x)

numerical method

FEW-BODY PROBLEMS THEORY GROUP

numerical method

H = ˆ X Z

dx ˆ

(x)

 ~

2m

d

dx

+ V (x) ˆ (x)

+ g

Z

dx ˆ

(x) ˆ

(x) ˆ

(x) ˆ

(x)

the model

FEW-BODY PROBLEMS THEORY GROUP

numerical method

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

• we decompose the field operators in the single-particle basis

(x) = ˆ

X

n=1

ˆ

a

(x)

• we calculate all matrix elements of the Hamiltonian

• we perform an exact diagonalization

H = ˆ X Z

dx ˆ

(x)

 ~

2m

d

dx

+ V (x) ˆ (x)

+ g

Z

dx ˆ

(x) ˆ

(x) ˆ

(x) ˆ

(x)

 ~² 2m

d²

dx² + V (x) _n(x) = E _{i n}(x)

the model

FEW-BODY PROBLEMS THEORY GROUP

N^↑ dim(H) #of non-zero

elements sparsity

1 144 10 512 0.51

2 4 356 1 058 580 5.6•10^-2 3 48 400 21 835 600 9.3•10^-3 4 245 025 159 167 025 2.7•10^-3

N

N

M = 12

numerical complexity

harmonic

confinement

 ~

2m

d

dx

+ m ⌦

2 x

(x) = E

(x)

Natural units:

Energy : ~⌦

Length :

r ~

m ⌦

FEW-BODY PROBLEMS THEORY GROUP

FEW-BODY PROBLEMS THEORY GROUP

equal mass system

T. Sowiński, T. Grass, O. Dutta, M. Lewenstein Phys. Rev. A 88 033607 (2013)

FEW-BODY PROBLEMS THEORY GROUP

equal mass system

0 1

−3 −2 −1 0 1 2 3

g = 0.1

Density

0 1 2

−3 −2 −1 0 1 2 3

Position (osc. u.)

Density

0 1

−3 −2 −1 0 1 2 3

g = 1

0 1 2

−3 −2 −1 0 1 2 3 Position (osc. u.)

0 1

−3 −2 −1 0 1 2 3

g = 10

µ=1

0 1 2

−3 −2 −1 0 1 2 3

Position (osc. u.)

µ=40/6

⇢ (x) = hG| ˆ

(x) ˆ (x) |Gi

E. Lindgren et al.

New J. Phys. 16 063003 (2014)

Single-particle density profile in the ground-state

FEW-BODY PROBLEMS THEORY GROUP

N

=3

N

=1

different mass fermions

D. Pęcak, M. Gajda, T. Sowiński New J. Phys. 18 013030 (2016)

m

/m

= 40/6

m

/m

= 1

FEW-BODY PROBLEMS THEORY GROUP

different mass fermions

Single-particle density profile in the ground-state

⇢ (x) = hG| ˆ

(x) ˆ (x) |Gi

D. Pęcak, M. Gajda, T. Sowiński New J. Phys. 18 013030 (2016)

0 1

−3 −2 −1 0 1 2 3

g = 0.1

Density

0 1 2

−3 −2 −1 0 1 2 3

Position (osc. u.)

Density

0 1

−3 −2 −1 0 1 2 3

g = 1

0 1 2

−3 −2 −1 0 1 2 3 Position (osc. u.)

0 1

−3 −2 −1 0 1 2 3

g = 10

µ=1

0 1 2

−3 −2 −1 0 1 2 3

Position (osc. u.)

µ=40/6

N

=3

N

=1

FEW-BODY PROBLEMS THEORY GROUP

D. Pęcak, M. Gajda, T. Sowiński New J. Phys. 18 013030 (2016)

0 0.5 1 1.5

−3 −2 −1 0 1 2 3

N_↑=1 N↓=2

Density

0 1 2

−3 −2 −1 0 1 2 3

N↑=2 N_↓=1

Density

0 1 2

−3 −2 −1 0 1 2 3

N_↑=3 N↓=1

Density

0 1 2

−5−4−3−2−1 0 1 2 3 4 5

N↑=1 N↓=9

Position (osc. u.)

Density

0 0.5 1 1.5

−3 −2 −1 0 1 2 3

N_↑=1 N↓=3

0 1 2

−3 −2 −1 0 1 2 3

N↑=2 N_↓=2

0 1 2

−3 −2 −1 0 1 2 3

N_↑=3 N↓=2

0 1 2 3

−5−4−3−2−1 0 1 2 3 4 5

N↑=5 N↓=5

Position (osc. u.)

0 0.5 1 1.5

−3 −2 −1 0 1 2 3

N_↑=1 N↓=4

0 1 2

−3 −2 −1 0 1 2 3

N↑=2 N_↓=3

0 1 2

−3 −2 −1 0 1 2 3

N_↑=4 N↓=1

0 1 2 3 4

−3 −2 −1 0 1 2 3

N↑=9 N↓=1

Position (osc. u.)

different mass fermions

Separation in the lighter component

rectangular

box

Natural units:

V (x) =

⇢ 0 if |x| < L

1 if |x| > L,

Energy : ~

⇡

8m L

Length : L

FEW-BODY PROBLEMS THEORY GROUP

FEW-BODY PROBLEMS THEORY GROUP

D. Pęcak, T. Sowiński Phys. Rev. A 94 042118 (2016)

different mass fermions

Separation in the heavier component

Uniform Box

0 0.5 1 1.5

-4 -2 0 2 4

N_↑=1 N_↓=2

Density

0 0.5 1 1.5

-4 -2 0 2 4

N_↑=2 N_↓=1

Density

0 0.5 1 1.5

-4 -2 0 2 4

N_↑=1 N_↓=3

Density

0 0.5 1 1.5

-4 -2 0 2 4

N_↑=1 N_↓=6

Position

Density

0 0.5 1 1.5

-4 -2 0 2 4

N_↑=2 N_↓=2

0 0.5 1 1.5

-4 -2 0 2 4

N_↑=3 N_↓=1

0 0.5 1 1.5

-4 -2 0 2 4

N_↑=1 N_↓=4

0 0.5 1 1.5

-4 -2 0 2 4

N_↑=3 N_↓=4

Position

0 0.5 1 1.5

-4 -2 0 2 4

N_↑=2 N_↓=3

0 0.5 1 1.5

-4 -2 0 2 4

N_↑=3 N_↓=2

0 0.5 1 1.5

-4 -2 0 2 4

N_↑=4 N_↓=1

0 0.5 1 1.5

-4 -2 0 2 4

N_↑=6 N_↓=1

Position

the transition

V (x, ) =

⇢

2

m ⌦

x

if |x| < L

1 if |x| > L,

0 0.2 0.4

-1 -0.5 0 0.5 1 λ = 0

V σ/(m σω2 L2 )

0 0.2 0.4

-1 -0.5 0 0.5 1 λ = 0.5

Position x/L V σ/(m σω2 L2 )

0 0.2 0.4

-1 -0.5 0 0.5 1 λ = 0.25

0 0.2 0.4

-1 -0.5 0 0.5 1 λ = 1

Position x/L

the transition

V (x, ) =

⇢

2

m ⌦

x

if |x| < L

1 if |x| > L,

0 0.2 0.4

-1 -0.5 0 0.5 1 λ = 0

V σ/(m σω2 L2 )

0 0.2 0.4

-1 -0.5 0 0.5 1 λ = 0.5

Position x/L V σ/(m σω2 L2 )

0 0.2 0.4

-1 -0.5 0 0.5 1 λ = 0.25

0 0.2 0.4

-1 -0.5 0 0.5 1 λ = 1

Position x/L 0

0.3 0.6 0.9

-4 -2 0 2 4

Density

µ = 1

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

0 0.3 0.6 0.9

-4 -2 0 2 4

Position

Density

0 0.4 0.8 1.2

-4 -2 0 2 4

λ = 0

µ = 40/6

0 0.3 0.6 0.9

-4 -2 0 2 4

λ = 0.0057

0 0.6 1.2 1.8 2.4

-4 -2 0 2 4

λ = 1

Position

1

0

FEW-BODY PROBLEMS THEORY GROUP

„order” parameter

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

µ = 1

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

0 0.3 0.6 0.9

-4 -2 0 2 4

Position

Density

0 0.4 0.8 1.2

-4 -2 0 2 4

λ = 0

µ = 40/6

0 0.3 0.6 0.9

-4 -2 0 2 4

λ = 0.0057

0 0.6 1.2 1.8 2.4

-4 -2 0 2 4

λ = 1

Position

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

µ = 1

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

0 0.3 0.6 0.9

-4 -2 0 2 4

Position

Density

0 0.4 0.8 1.2

-4 -2 0 2 4

λ = 0

µ = 40/6

0 0.3 0.6 0.9

-4 -2 0 2 4

λ = 0.0057

0 0.6 1.2 1.8 2.4

-4 -2 0 2 4

λ = 1

Position

FEW-BODY PROBLEMS THEORY GROUP

„order” parameter

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

µ = 1

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

0 0.3 0.6 0.9

-4 -2 0 2 4

Position

Density

0 0.4 0.8 1.2

-4 -2 0 2 4

λ = 0

µ = 40/6

0 0.3 0.6 0.9

-4 -2 0 2 4

λ = 0.0057

0 0.6 1.2 1.8 2.4

-4 -2 0 2 4

λ = 1

Position

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

µ = 1

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

0 0.3 0.6 0.9

-4 -2 0 2 4

Position

Density

0 0.4 0.8 1.2

-4 -2 0 2 4

λ = 0

µ = 40/6

0 0.3 0.6 0.9

-4 -2 0 2 4

λ = 0.0057

0 0.6 1.2 1.8 2.4

-4 -2 0 2 4

λ = 1

Position

0 1

FEW-BODY PROBLEMS THEORY GROUP

„order” parameter

M(x) = ⇢

(x) ⇢

(x)

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

µ = 1

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

0 0.3 0.6 0.9

-4 -2 0 2 4

Position

Density

0 0.4 0.8 1.2

-4 -2 0 2 4

λ = 0

µ = 40/6

0 0.3 0.6 0.9

-4 -2 0 2 4

λ = 0.0057

0 0.6 1.2 1.8 2.4

-4 -2 0 2 4

λ = 1

Position

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

µ = 1

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

0 0.3 0.6 0.9

-4 -2 0 2 4

Position

Density

0 0.4 0.8 1.2

-4 -2 0 2 4

λ = 0

µ = 40/6

0 0.3 0.6 0.9

-4 -2 0 2 4

λ = 0.0057

0 0.6 1.2 1.8 2.4

-4 -2 0 2 4

λ = 1

Position

Magnetization-like distribution

0 1

FEW-BODY PROBLEMS THEORY GROUP

„order” parameter

M(x) = ⇢

(x) ⇢

(x)

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

µ = 1

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

0 0.3 0.6 0.9

-4 -2 0 2 4

Position

Density

0 0.4 0.8 1.2

-4 -2 0 2 4

λ = 0

µ = 40/6

0 0.3 0.6 0.9

-4 -2 0 2 4

λ = 0.0057

0 0.6 1.2 1.8 2.4

-4 -2 0 2 4

λ = 1

Position

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

µ = 1

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

0 0.3 0.6 0.9

-4 -2 0 2 4

Position

Density

0 0.4 0.8 1.2

-4 -2 0 2 4

λ = 0

µ = 40/6

0 0.3 0.6 0.9

-4 -2 0 2 4

λ = 0.0057

0 0.6 1.2 1.8 2.4

-4 -2 0 2 4

λ = 1

Position

Magnetization-like distribution

0 1

FEW-BODY PROBLEMS THEORY GROUP

„order” parameter

M(x) = ⇢

(x) ⇢

(x)

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

µ = 1

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

0 0.3 0.6 0.9

-4 -2 0 2 4

Position

Density

0 0.4 0.8 1.2

-4 -2 0 2 4

λ = 0

µ = 40/6

0 0.3 0.6 0.9

-4 -2 0 2 4

λ = 0.0057

0 0.6 1.2 1.8 2.4

-4 -2 0 2 4

λ = 1

Position

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

µ = 1

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

0 0.3 0.6 0.9

-4 -2 0 2 4

Position

Density

0 0.4 0.8 1.2

-4 -2 0 2 4

λ = 0

µ = 40/6

0 0.3 0.6 0.9

-4 -2 0 2 4

λ = 0.0057

0 0.6 1.2 1.8 2.4

-4 -2 0 2 4

λ = 1

Position

Magnetization-like distribution

… and its second-moment

=

Z

L

dx x

M(x)

0 1

FEW-BODY PROBLEMS THEORY GROUP

„order” parameter

M(x) = ⇢

(x) ⇢

(x)

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

µ = 1

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

0 0.3 0.6 0.9

-4 -2 0 2 4

Position

Density

0 0.4 0.8 1.2

-4 -2 0 2 4

λ = 0

µ = 40/6

0 0.3 0.6 0.9

-4 -2 0 2 4

λ = 0.0057

0 0.6 1.2 1.8 2.4

-4 -2 0 2 4

λ = 1

Position

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

µ = 1

0 0.3 0.6 0.9

-4 -2 0 2 4

Density

0 0.3 0.6 0.9

-4 -2 0 2 4

Position

Density

0 0.4 0.8 1.2

-4 -2 0 2 4

λ = 0

µ = 40/6

0 0.3 0.6 0.9

-4 -2 0 2 4

λ = 0.0057

0 0.6 1.2 1.8 2.4

-4 -2 0 2 4

λ = 1

Position

Magnetization-like distribution

… and its second-moment

=

Z

L

dx x

M(x)

g _red > g _blue

0 1

FEW-BODY PROBLEMS THEORY GROUP

-10

-7

-4

-1

0.3 0.5 0.7 0.9 1.1

10 ² λ

σ

N _↑ =1

N _↓ =3

4.0 4.2

4.4 4.6

4.8 5.0

-12

-6

0

6

0.3 0.5 0.7 0.9 1.1

10 ² λ

σ

N _↑ =2

N _↓ =3

4.0 4.2

4.4 4.6

4.8 5.0

-8

-4

0

4

8

12

0.3 0.42 0.54 0.66

10 ² λ

σ

N _↑ =2

N _↓ =2

4.0 4.2

4.4 4.6

4.8 5.0

-6

0

6

12

0.3 0.5 0.7 0.9

10 ² λ

σ

N _↑ =3

N _↓ =2

4.0 4.2

4.4 4.6

4.8 5.0

6

9

12

0.2 0.3 0.4 0.5

10 ² λ

σ

N _↑ =3

N _↓ =1

4.0 4.2

4.4 4.6

4.8 5.0

-15

-12

-9

-6

-3

0.5 0.7 0.9 1.1

10 ² λ

σ

N _↑ =1

N _↓ =4

„order” parameter

FEW-BODY PROBLEMS THEORY GROUP

-10 -7 -4 -1

0.3 0.5 0.7 0.9 1.1 10² λ

σ

N_↑=1 N_↓=3

4.04.2 4.44.6 4.85.0

-12 -6 0 6

0.3 0.5 0.7 0.9 1.1 10² λ

σ

N_↑=2 N_↓=3

4.04.2 4.44.6 4.85.0

-8 -4 0 4 8 12

0.3 0.42 0.54 0.66 10² λ

σ

N_↑=2 N_↓=2

4.04.2 4.44.6 4.85.0

-6 0 6 12

0.3 0.5 0.7 0.9 10² λ

σ

N_↑=3 N_↓=2

4.04.2 4.44.6 4.85.0

6 9 12

0.2 0.3 0.4 0.5 10² λ

σ

N_↑=3 N_↓=1

4.04.2 4.44.6 4.85.0

-15 -12 -9 -6 -3

0.5 0.7 0.9 1.1 10² λ

σ

N_↑=1 N_↓=4

„order” parameter

FEW-BODY PROBLEMS THEORY GROUP

the susceptibility

( ) = d ( )

d

A response of the order

parameter to changes of the

control parameter

FEW-BODY PROBLEMS THEORY GROUP

the susceptibility

0 15 30 45 60 75

0.2 0.5 0.8 1.1

χ

4.04.2 4.44.6 4.85.0

0 100 200 300 400

0.45 0.5 0.55 0.6 0.65

χ

4.04.2 4.44.6 4.85.0

0 30 60 90

0.2 0.3 0.4 0.5 0.6

10² λ

χ

4.04.2 4.44.6 4.85.0

0 2 4 6

-8 -6 -4 -2 0

g-γ/ν χ

λ_c = 0.0097 γ = 0.53 ν = 0.39

4.04.2 4.44.6 4.85.0

0 0.03 0.06 0.09 0.12

-120 -60 0 60 120

g-γ/ν χ

λ_c = 0.0054 γ = 0.23 ν = 0.05

4.04.2 4.44.6 4.85.0

0 1 2 3 4

-20 0 20 40 60 80

g^1/γ τ

g-γ/ν χ

λ_c = 0.0029 γ = 0.33 ν = 0.17

4.04.2 4.44.6 4.85.0

-10 -7 -4 -1

-8 -6 -4 -2 0

σ N ↓=3, N ↑=1

γ = 0.53 λ_c = 0.0097

4.04.2 4.44.6 4.85.0

-8 -4 0 4 8 12

-120 -60 0 60 120

σ N ↓=2, N ↑=2

γ = 0.23 λ_c = 0.0054

4.04.2 4.44.6 4.85.0

6 9 12

-20 0 20 40 60 80

g^1/γ τ

σ N ↓=1, N ↑=3

γ = 0.29 λ_c = 0.0033

4.04.2 4.44.6 4.85.0

N↑=3 N↓=1N↑=2 N↓=2N↑=1 N↓=3

( ) = d ( )

d

A response of the order

parameter to changes of the

control parameter

FEW-BODY PROBLEMS THEORY GROUP

the hypothesis

In the limit of infinitely strong

interactions the system undergoes

a non-analytic transition

between the two orderings.

The transition is driven

by an adiabatic change

of the external confinement.

FEW-BODY PROBLEMS THEORY GROUP

scaling hypothesis

(⌧, g) = g ^/⌫ ˜(g ^1/⌫ ⌧ )

⌧ =

c

c

Critical shape of the trap

Dimensionless distance to the critical point

Scaling ansatz

There exists an appropriate choice of critical

exponents for which all numerical data points

form the unique universal curve

FEW-BODY PROBLEMS THEORY GROUP

finite-size scaling

(⌧, g) = g

˜(g

⌧ )

FEW-BODY PROBLEMS THEORY GROUP

conclusions

The system:

1. of a few fermions of different masses

2. in the limit of infinitely strong repulsions

3. in a one-dimensional confinement

undergoes

a non-analytic transition between the

two orderings of the density profile

D. Pęcak, M. Gajda, T. Sowiński New J. Phys. 18 013030 (2016)

D. Pęcak, T. Sowiński Phys. Rev. A 94 042118 (2016) T. Sowiński, T. Grass, O. Dutta, M. Lewenstein

Phys. Rev. A 88 033607 (2013) D. Pęcak, M. Gajda, T. Sowiński

ArXiv:1703.08116