Petrov (PDF)

Quantum droplets of ultracold bosons

Dmitry Petrov
Laboratoire Physique Théorique et Modèles Statistiques
(Orsay, upstairs)

GAS requires trapping or collapses

E/N

E/N

n

n

?
E/N
n0
n

E / N ∝ g 2 n+ g α +1 nα , α>1

The gas should remain dilute, otherwise short lifetime!
E / N ∝a n+ L3 α−2 n α , α>1

E/N
n0
n

1
1 a α−1
n0 ∼ 3 ( )
L L

Dilute = simultaneously small a and large L
and prefer small α

3-body

(α=2)

Lee-Huang-Yang

(α=3/2)

Beyond-mean-field QUANTUM MECHANISM!
Resonant (Efimov)
3-body force
(Bulgac'02)
lossy :(

Non-resonant
3-body force
(DP'14)
technically
complicated :(

Realized with dipoles and bosonic mixtures
Dy (Stuttgart), Er (Innsbruck), K (Barcelona)
LHY corection depends on dimensionality,
properties of Bogoliubov spectrum
(gapped/nongapped)...
lots of prospects

Quantum stabilization idea
U ( x , y )=−α x 2 +(Ω2 +ϵ x 2 ) y 2

Stable for sufficiently large ϵ/ α
y

x

Classically unstable degree of
freedom stabilized by quantum
fluctuations in another degree of
freedom!

Lee-Huang-Yang term
For spinless BEC:

2
g
n
E
128
2
=
1+
Volume
2
15

(

√

3

na
π +...

)

LHY correction is UNIVERSAL (depends only on the scattering
length) and QUANTUM (zero-point energy of Bogoliubov phonons)!
Observed in ultracold gases where the scattering
length is tunable by using Feshbach resonances
(Innsbruck, MIT, ENS, JILA, Rice)
Navon et al.'11

Unfortunately, the effect is perturbative and the LHY
term is smaller than the mean-field one!

Bose-Bose mixture, mean field
Mean-field energy density:

g 12

g 12 > √ g11 g22

E MF
g11 n12 + g 22 n22 +2 g12 n1 n2
=
Volume
2

phase separation

mean-field stability

g 11>0, g 22 >0, and g212 < g 11 g22

g 12 <−√ g 11 g 22

collapse

39

K: |F=1,mF=0> and |F=1,mF=-1>

a 11

a 22
a
aBohr

a 12
a 12=−√ a11 a22
B [G]

Data from A. Simoni

LHY correction
Bogoliubov theory
2
2
g
n
+
g
n
±
(
g
n
−g
n
)
+
4
g
√
2 2
4
2
11 1
22 2
11 1
22 2
12 n1 n2
E± (k )= √ c± k + k / 4 ; c± =
2
2
2
g
n
+
g
n
E
1
11 1
22 2 +2 g12 n1 n2
=
+ ∑ ∑ [ E± (k )−k 2 /2−c ±2 ]=
Volume
2
2 ± k

g11 n21 + g22 n 22 +2 g12 n1 n 2
8
5
5
=
+
(c
+c
+
−)
2
2
15 π
MF ∝ g n2

LHY ∝( g n)5/ 2 (Larsen'63)

In contrast to one-component case, MF and LHY depend on
(different) combinations of g σ σ ' nσ
...and, thus, can be independently controlled!

LHY correction
2

2

g11 n1 + g22 n2 +2 g12 n1 n2
E
8 m4 5 5
=
+
(c + + c − )+...
2
3
Volume
2
15 π ℏ

g 12

g 12 > √ g11 g22

phase separation

E± (k )
c+ k
c− k
k

δ g=g 12 + √ g11 g 22
g 12 <−√ g 11 g 22

E± (k )
c+ k

collapse

c− k
k