Self-bound liquids in Bose-Fermi mixture in low dimensions

DEBRAJ RAKSHIT Institute of Physics, Polish Academy of Sciences, Warsaw

05-12-2018

QIPA-18, India

In collaboration with

Tomasz Karpiuk (UB, Bialstok)Mirosław Brewczyk (UB, Bialstok)Maciej Lewenstein (ICFO, Barcelona)Mariusz Gajda (IF PAN, Warsaw)

(arXiv:1808:04793)

Droplets

Liquids within van dar Waals paradigm

Droplets

Liquids within van dar Waals paradigm

Low temperature gas mixture in a trap

Droplets

Liquids within van dar Waals paradigm

Self-bound!! Counter-intuitive!!

Liquids beyond van dar Waals paradigm

Most dilute ever!!

Motivations/Overview

•  Droplet formation in 2D Bose-Fermi systemsü  Thermodynamic limitü  Finite systems

•  Low-dimensional dropletsü  Quantum fluctuations are also strongly modified in reduced dimension (distinct scattering properties)ü  Reduced three-body loss …ü  Quadratic scaling of kinetic energy on fermionic density in 2D (a close analogy with Bose-Bose droplets)

•  Story so farü  Bose-Bose droplets (D. Petrov, PRL, 2016)ü  Bose-Fermi droplets in three-dimension (Rakshit et. al., Under review, 2018)

Neutral atoms at low-temperatureü  They can be cooled at extremely low temperature

ü  Dilute atomic gas (n ~ 1012-1015 atoms/cm3)

•  Two-body collisions play important role

•  Three-body collisions are rare

ü  Universality (Average inter-particle spacing )

•  At low-temperature de-Broglie wavelength is much larger in comparison to interaction range

•  Details of potential do not matter for many applications

ü  Statistics

•  Neutral atoms can be composite bosons or composite fermions

dint << λdb;λdb ∝1T

Neutral atoms at low-temperature

a3daa

No bound state

Bound state

a3daa > 0a3d

aa < 0

Feshbach resonance

B-field

V(r)

r

ü  Atom-atom interaction can be manipulated precisely

•  s-wave scattering length

•  For remainder of talk, we assume, only the s-wave scattering length to have a significant contribution at low temperature

Ultradilute liquid droplets

C. R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas, P. Cheiney, and L. Tarruell, Quantum liquid droplets in a mixture of Bose-Einstein condensates, Science 359, 301 (2018)

Bose-Bose Mixture(39k in two hyperfine states)

Single component Bose system

δa = − a12 + a11a22

INTRASPECIES(Effective repulsive interaction)

INTERSPECIES(Effective attractive interaction)

Liquid Bose-Bose dropletsMechanical stability: Two component Bose gas in a box with uniform densities n1 and n2

εBB =12g11n1

2 +12g22n2

2 − g12 n1n2

System becomes unstable if − | g12 |+ g11g22 < 0;δg < 0

∂2ε∂n1

2

⎛

⎝⎜

⎞

⎠⎟∂2ε∂n2

2

⎛

⎝⎜

⎞

⎠⎟−

∂2ε∂n1∂n2

⎛

⎝⎜

⎞

⎠⎟

2

≥ 0

Liquid Bose-Bose droplets

Collapse!!

Mechanical stability:

System would prefer to increase densities of both the species while keeping

εBB =12g11n1

2 +12g22n2

2 − g12 n1n2 = λ+n+2 +λ−n−

2

λ+ = o( g11g22 )> 0λ− = o(δg)< 0

δg = − g12 + g11g22n± = ( g22n1 ∓ g11n2 ) / g11 + g22

Liquid Bose-Bose dropletsQuantum mechanical stabilization of collapsing Bose-Bose mixture:

They balance at certain ‘equilibrium’ atomic densities and leads to formation of liquid droplets.

Highly dilute systems!!

εBBMF ∝n2δg < 0

εLHY ∝n5/2 g11g22 > 0

Liquid Bose-Fermi droplets: 3D caseBose-Fermi mixture:

•  Mean field treatment does not support formation of a stable droplet

•  Mean-field and corrections balance at certain ‘equilibrium’ atomic densities and leads to formation of liquid droplets (arXiv:1801.00346)

•  Higher-order correction in Bose-Fermi interaction plays a crucial role

Highly dilute self-bound systems!!

ε3d =κknF5/3 +

12gBnB

2 + g BF nBnF⎛

⎝⎜

⎞

⎠⎟+ εLHY +δεBF( )

Mean-field Higher-order corrections

aB3nB <<1;aBF

3 nF <<1

How to calculate the equilibrium densities of the droplets?

H = HB +HF +HBF

Bose-Fermi mixture•  We consider two-dimensional Bose-Fermi mixture at zero temperature

•  Fermions do not interact with each other

•  Bosons interact with each other and with fermions

Hamiltonian: H = HB +HF +HBF

Interaction potential

σ

σ 'vBB

HB =!2k2

2mB

b̂k+

k∑ b̂k +

12V

b̂+pb̂p+kvBB (k)p,q,k∑ b̂+qb̂q−k

3D-vs-2D Interaction potential

σ

κ

gσ ,σ ' =2π!2

µσ ,σ '

1ln ε / (a

σ ,σ '

2 κ 2 )( )ε = 4exp(−2γ ) Popov, Theor. Math. Phys. 11, 565

vBB (k) = gBB

vσσ ' (k) = gσσ ' 1+gσσ 'V

m!2k2k≠0

∑⎛

⎝⎜

⎞

⎠⎟

gσσ ' = 2π!2a

σσ '/µ

σσ '

µσ ,σ ' =mσmσ '

(mσ +mσ ' )

3D

3D

HB =!2k2

2mB

b̂k+

k∑ b̂k +

12V

b̂+pb̂p+kvBB (k)p,q,k∑ b̂+qb̂q−k

2D

2D Bose-Fermi mixture: Energy density

Consider leading order contributions [collect all possible quadratic terms] and diagonalize via Bogoliubov transformation:

Energy contribution due to HB

HB =!2k2

2mB

b̂k+

k∑ b̂k +

12V

b̂+pb̂p+kVBB (k)p,q,k∑ b̂+qb̂q−k

εB = gBBnB2 +εLHY

Mean-field LHY correction

εLHY =gBB2 nB

2mB

8π!2ln gBBnB e!2κ 2 /mB

⎛

⎝⎜

⎞

⎠⎟

2D Bose-Fermi mixture: Energy densityEnergy contribution due to Bose-Fermi interaction HBF

Coupling between density fluctuations in the two species

δεBF =4πmBnBgBF

2 kF2

(2π )4!2Ic (ω,α)

kF = 4πnFα = 2ω gBBnB /εF( )ω =mB /mF

In 2D:

εBF = gBFnBnF +δεBF

Energy contribution due to HF εF =12βnF

2 ;β = 2π!2

mF

Kinetic energy

2D Bose-Fermi mixture: Energy densityTotal energy density of Bose-Fermi mixture :

ε =β2nF2 +

gBB2nB2 + gBFnBnF +εLHY +δεBF

Mean-field energy Beyond mean-field higher-order corrections

Diluteness conditions:

aBB2 nB,F <<1

aBF2 nB,F >>1

Weakly repulsive (intraspecies) Weakly attractive (interspecies)

Cut-off dependence

εB = εB (κ ) εBF = εBF (κ )

ε =β2nF2 +

gBB2nB2 +εLHY

⎛

⎝⎜

⎞

⎠⎟+ gBFnBnF +δεBF( )

∂εB (κ )∂κ 2 ≈ 0 Negligible dependence

on the cut-off momentum (can be numerically checked)

Cut-off momentum

∂2εMF∂nB

2

⎛

⎝⎜

⎞

⎠⎟∂2εMF∂nF

2

⎛

⎝⎜

⎞

⎠⎟−

∂2εMF∂nB∂nF

⎛

⎝⎜

⎞

⎠⎟

2

= 0

εMF =β2nF2 +

gBB2nB2 + gBFnBnF

Choose the momentum cut-off from the bare minimum stability condition of the system

2 ln2 4exp(−2γ )aBF2 κc

2

⎛

⎝⎜

⎞

⎠⎟=(1+ω)2

ωln 4exp(−2γ )

aBB2 κc

2

⎛

⎝⎜

⎞

⎠⎟

Solve for κc

Mechanical stabilization of self-bound mixtureVanishing pressure …

Outside of the droplet the pressure must be equal to zero (free-space). The pressure related to surface vanishes in thermodynamic limit.

E(NB,NF,V ) = ε(nB,nF )V

nB,F =NB,F

V

P = dEdV⎛

⎝⎜

⎞

⎠⎟NB ,NF

p(nB,nF ) = nB∂ε∂nB

+ nF∂ε∂nF

−ε(nB,nF ) = 0

Equilibrium density of Stable Bose-Fermi droplets

6Li - 133Cs mixture (weakly interacting dilute gas)

0 2 4 6 8nBaBB[10

-5]

0

0.2

0.4

n Fa BB[10-5 ]

0.5 1 1.5 2 2.5 3 3.5 κ / κc

-8

-4

0

4

ε2D/|ε0|

0 10 20 30nB/nF

-1

ε 2D/| ε 0|

(a)

(b)

2

2

Zero-pressure line

DROPLET

Due to the quadratic form of mean-field energy, equilibrium density ratio can be shown to be approximately

nB / nF ≈ β / gBB

εMF =β2nF2 +

gBB2nB2 + gBFnBnF

aBF / aB =104

Mechanical stability – thermodynamic limit Minimize energy constrained to vanishing pressure

µB∂p∂nF

−µF∂p∂nB

= 0

p(nB,nF ) = nB∂ε∂nB

+ nF∂ε∂nF

−ε(nB,nF ) = 0

µB =∂ε∂nB

;µF =∂ε∂nF

;

Solution gives equilibrium densities

0 2 4 6 8nBaBB[10

-5]

0

0.2

0.4n Fa BB[10-5 ]

0.5 1 1.5 2 2.5 3 3.5 κ / κc

-8

-4

0

4

ε2D/|ε0|

0 10 20 30nB/nF

-1

ε 2D/| ε 0|

(a)

(b)

2

2

Variance with cut-off momentum

0 2 4 6 8nBaBB[10

-5]

0

0.2

0.4

n Fa BB[10-5 ]

0.5 1 1.5 2 2.5 3 3.5 κ / κc

-8

-4

0

4

ε2D/|ε0|

0 10 20 30nB/nF

-1

ε 2D/| ε 0|

(a)

(b)

2

2

Variation of the energy densities as a function of the cut-off momentum around the chosen cut-off momentum

Contribution from Bose-Fermi interactions

Contribution from Bose-Bose interactions

Total energy density

Kinetic energy contribution

6Li - 133Cs mixture ( ) aBF / aB =104

Bose-Fermi mixture: Effects of interaction strengthEquilibrium densities as a function of aBF /aB

0 2 4 6 8 10 12aBF/aBB[104]

8

10

12

n B/nF

0 2 4 6 8aBF/aBB[104]

02468

a B nB[1

0-5]

2

6Li - 133Cs mixture

nB / nF ≈ β / gBB

Nonuniform finite droplets

∂∂tnF = −∇(nF

!vF )

mF∂∂t!vF = −∇

δTδnF

+mF

2!vF2 + gBFnB +

δεBFδnF

⎛

⎝⎜

⎞

⎠⎟

∂T∂nF

= βnF −ξ!2

2mF

∇2 nFnF

Ø  To include surface effects, energy terms related to density gradient has to be added

Ø  Hydrodynamic equation of motion

Fermionic component

Nonuniform finite dropletsInverse-Madelung transformation and Schrodinger-like equation of motion

Nonuniform finite droplets

The case of 133Cs-6Li mixture for aBF /aB = 10^4 and the initial number of bosons (fermions) equal to 1000 (100), 4000 (400), and 10000 (1000)

The density does not change, only volume increases …

Equilibrium densities in thermodynamic limit

NB!1000 NF!100NB!4000 NF!400NB!10000 NF!1000nB!4.8e"5 n F!5.0e"6

0 2000 4000 6000 8000 10000 120000

0.00001

0.00002

0.00003

0.00004

0.00005

0.00006

0.00007

Radius!aB

Density!a B"2

Solution obtained by reworking hydrodynamic equation into a form of Schrodinger-like equation

Summaryq  Low-dimensional Bose-Fermi droplets … (We find that a three-dimensional stable Bose-Fermi mixture in gaseous phase can be liquefied by introducing a transverse confinement)

q  Peculiar scattering processes in low-dimensions significantly modifies the higher-order quantum fluctuation terms, which are crucial ingredients for these droplet formation

q  Quadratic scaling of fermionic kinetic energy makes 2D systems closely analogous to the Bose-Bose droplets

q  In contrary to the 3D case, 2D droplets are formed near vanishing mean-field energy

q  Conditions for droplet formation in 1D were obtained as well (not discussed here; See arXiv:1808.04793)

q  Low-dimensional droplets are most promising from experimental point of view due toreduced three-body losses

Outlook …

q  Across dimensional crossover …

q  Droplets + vapor … q  Droplet collisions …

q  Droplets in strongly interacting regime …

q  A perfect laboratory for studying various processes – polaron physics, stability of astronomical objects …