Transcript Superfluid BEC dynamics in peridioc potentials
Superfluid dynamics of BEC in a periodic potential Augusto Smerzi
INFM-BEC & Department of Physics, Trento LANL, Theoretical Division, Los Alamos
Collaboration with:
Chiara Menotti INFM-BEC & Department of Physics, Universita` di Trento Andrea Trombettoni INFM & Department of physics, Universita` di Parma
BEC trapped in a periodic potential
i
(
r
,
t
)
t
2 2
m
2
V ext
(
r
)
g
0 2 The trapping
Lattice field Driving field
interatomi c interactio n
V L
(s (
r
)
V D g
V
0 sin ( 0
r
) 4
m
2
m r
2
a
2 (
k x
2
x
)
m
2 2
a r
(
y
2 0
z
) ion) repulsive
a
0 attractive
BEC expanding in a 1D optical lattice No interaction --> a = 0
Density profile Trapping potential Momentum distribution
d
q B intersite Bragg spacing momentum
BEC expanding in a 1D optical lattice A. Trombettoni and A. Smerzi, PRL 86, 2353 (2000)
a N
0
BEC expanding in a 1D optical lattice Interacting atoms --> a > 0
Density profile Trapping potential Momentum distribution
Preliminary experimental evidences of self-trapping B. Eiermann, M. Albiez, M. Taglieber, M. Oberthaler University of Konstanz
BEC expanding in a 1D optical lattice Interacting atoms --> a > 0 Array of
weakly coupled
BEC Dynamical
n
(
t
)
N n
variables (
t
)
e i
n
(
t
) Tunneling
k
n
(
x
) rate
T
V
n
1 (
x
)
Josephson oscillations
1. Atoms are condensed in the optical and magnetic fields. 2. The harmonic confinement is instantaneously shifted along the x direction.
N
10 5
Rd atoms
x
r
2 2
V
0
d
5 0 .
5
E R
10 100
m Hz Hz
Array of Josephson junctions driven by a harmonic external field
Josephson oscillations
center relative of mass phases position
:
p
:
n
1
n
n n N n
d dt
(
t
)
d dt
2
K
sin
p
(
t
)
p
(
t
)
m
x
2 ( 2 ) 2 (
t
) Oscillations of the three peaks of the interferogram. Blue circles: no periodic potential The array is governed by a pendulum equation F.S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, M. Inguscio, Science 293, 843 (2001)
Small amplitude pendulum oscillations
Jos
x d
2
m K
Triangles: GPE; stars: variational calculation of K Circles: experimental results Relation between the oscillation frequency and the tunneling rate
Breakdown of Josephson oscillations
The interwell phase coherence breaks down for a large initial displacement of the BEC center of mass
Questions:
1) Why the interaction can break the inter-well phase coherence of a condensate at rest confined in a periodic potential ?
2) Why a large velocity of the BEC center of mass can break the inter-well phase coherence of a condensate confined in a periodic potential and driven by a harmonic field ?
Which are the transport properties of BEC in periodic potentials ?
The discrete nonlinear equation (DNL)
i
k
t
n
(
n
V D n
2 (
n
1 2
n
) *
n
1
n
1
n
k
*
n
1
c
.
c
.) ( 2
n
n
U n
1 2 )
n
n
1
n
Tunneling Dynamical
and
on the depends variables interactio on n the
k n
(
t
)
n
(
x
)
T n
(
t V
)
e i
n n
1 ( (
t x
) ) height
g
0 3
n
(
x
)
n
1 (
x
)
a
of the interwell barriers
Newtonian Dynamics of a wave-packet
n
(
t
)
C f
n
(
t
) (
t
) exp
ip
[
n
(
t
)]
i
(
t
) [
n
2 (
t
)] 2 The collective satisfy coordinate s
ξ(t), p(t),σ(t), δ(t)
variationa l Euler Lagrange equations
v g
m E
1
V D
( ) sin
p m E
1
n
1
m E
(
n N
n n
) 2
n
2
Bloch energies & effective masses
n
(
t
)
N
0
e i
(
p n
t
) are eigenstate s of the DNL
E
E loc
loc
1
m
N
0 cos
m E
1 cos
p p
Effective masses depend on the height of the inter-well barriers and on the density
m
1 (
N
0 ,
V
0 ) 2 2
p
2
p
0
m E
1 (
N
0 ,
V
0 ) 1
N
0 2
E
2
p
2
p
0 In the limit
V
0 0 ,
m
m E
m
See also M. Kramer, C. Menotti, L. Pitaevskii and S. Stringari, unpublished
Bogoliubov spectrum
n
(
t
)
N
0
e i
(
p n
t
)
n
(
t
)
n
(
t
)
q u q e i
q t e i
[(
p
q
)
n
t
] ,
N
0 1 p quasi momentum q quasi momentum of the large amplitude traveling of the perturbati on mode wave 1. Replace in the DNL 2. After linearization, retrieve the dispersion relation (
p
,
q
,
N
0 )
Bogoliubov spectrum
B
sin
m
p
sin
q
2 cos
m
2 2
p
sin 4
q
2 cos
p m E
N
0
N
0 sin 2
q
2
O
m
1
m
1
m E
1 2 Free limit
B
(periodic potential
OFF
)
p m q
1
m
2
q
4 4 1
m
N
0
N
0
q
2
Sound-wave & energetic instability
sound velocity v s
,
q B
(
q
0 )
m E m
v g
c c
1
m E
N
0
N
0 cos
p v g
sin
m E p Energetic instabilit y
B
0
c
2
m E
2
m
2
v g
2 Landau criteria in free space : for breakdown of superfluid
c
2
v g
2 ity Cfr. with B. Wu and Q. Niu, PRA64, 061603R (2001)
Dynamical instability
The system becomes modulation ally unstable when
ω B becomes complex c
2 1
m E
N
0
N
0 cos
p
0 In ω B the v free g q (V 1 m 2 0) q limit 4 1 the m Bogoliubov μ N 0 N 0 q 2 spectrum is always real (when a No dynamical instabilities ( 1
q
,
p
) 2 sin
q m
1 2 cos 2
p
sin 2
q
2
m
1
E
N
0
N
0 cos
p
0) in a periodic potential
Comparison between analytical and numerical dispersion relation
p
1 4 Full line: analytical (DNL) Dots: numerical (GPE)
p
3 4 Dashed line :
m
m E
Breakdown of superfluidity for a BEC driven by a harmonic field Density at t=0,20,40 ms as a function of the Quasi-momentum vs. time for three different initial displacements: 40, 80, 90 sites Position. Initial displacements: 50, 120 sites
v g
m E
1
V D
( ) sin
p The system critical becomes group dynamicall y velocity v g
A. Smerzi, A. Trombettoni, P.G. Kevrekidis, A.R. Bishop, PRL 89, 170402 (2002)
m
1
unstable at the
The Frontier
• • • • •
Technological applications
Interferometry at the Heisenberg limit Quantum information
Foundational problems
Quantum – classical correspondence principle Schroedinger cats, entanglement
Tools
Quantum many-body dynamical theory