Download 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



(r , t )   2 2
2
i
 
  Vext (r )  g 0     
t
 2m

The interatomi c interactio n
The trapping potential
(s wave scattering length
approximat ion)
m
Lattice field VL (r )  V0 sin 2 (k x)   r2 ( y 2  z 2 )
2 a  0 repulsive
4  2 a
g 0 m 2 2
Driving field VD (r )  m
r x
a  0 attractive
2
BEC expanding in a 1D optical lattice
No interaction --> a = 0
Density
profile
Trapping
potential
Momentum
distribution
d  intersite spacing
q B  Bragg momentum
BEC expanding in a 1D optical lattice
  a N0
A. Trombettoni and A. Smerzi, PRL 86, 2353 (2000)
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
Dynamical variables
 n (t )  N n (t ) ei
n (t )
Tunneling rate
Array of weakly coupled BEC
k   n ( x) 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  105 Rd atoms
 x  2  10 Hz
 r  2  100 Hz
V0  5 E R
d  0.5  m
Array of Josephson junctions driven by a harmonic external field
Josephson oscillations
center of mass position :    n N n
n
relative phases : p   n 1   n
d
  (t )  2 K sin p(t )
dt
d
2  2
 p(t )  m x ( )  (t )
dt
2
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 
 xd

2m
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)
 n

n
*
*
i
 VD   ( n n 1   n n 1  c.c.)  n  U  n  n 
t






 k   ( n   n 1 )  n 1  k   ( n   n 1 )  n 1
2
2
2
2
Tunneling depends on the height of the interwell barriers
Dynamical
variables
and on the interactio n
kn (t )n ( x)NTn (tV)  e n1n( x)
i ( t )
   g 0  3n ( x) n 1 ( x)  a
Newtonian Dynamics of a wave-packet
 n   (t ) 
 (t )

2
 exp ip[n   (t )]  i
 n (t )  C f 
[n   (t )] 
2


  (t ) 
The collective coordinate s ξ(t), p(t),σ(t), δ(t)
satisfy variationa l Euler  Lagrange equations
1

v g    mE sin p
VD ( )
p  

mE1 
1
2
n m ( N )  n
E
n

n
2
n
Bloch energies & effective masses
 n (t )  N 0 e i ( p n t ) are eigenstate s of the DNL
1
EE 
N 0 cos p
mE
1
   loc 
cos p
m
Effective masses depend on the height of
loc
the inter-well barriers and on the density
2
m ( N 0 ,V0 )  2 2
 p
1
1 2E
 m ( N 0 ,V0 ) 
N0  2 p 2
1
E
p 0
In the limit
p 0
V0  0, m  mE  m
See also M. Kramer, C. Menotti, L. Pitaevskii and S. Stringari, unpublished
Bogoliubov spectrum
 n (t )  N 0 ei ( p n  t )   n (t )
 n (t )   uq e
i q t
ei[( p  q ) n  t ] ,

N0  1
q
p  quasi  momentum of the large amplitude traveling wave
q  quasi  momentum of the perturbati on mode
1. Replace in the DNL
2. After linearization, retrieve the dispersion relation
   ( p, q, N 0 )
Bogoliubov spectrum
sin p
cos 2 p
cos p 
4 q
2 q
B 
sin q  2
sin

N 0 sin
2
m
m
2
mE N 0
2
 m 1  m 1  2 

E
 
 O 

1
 
 m
 

Free limit (periodic potential OFF )
p
1 q 4 1 
2
B  q 

N
q
0
2
m
m 4 m N 0
Sound-wave & energetic instability
Energetic instabilit y   B  0
sound velocity
vs , 
 B

q
( q 0 )
mE

vg  c
m
1 
c
N 0 cos p
mE N 0
sin p
vg 
mE
2
m
c 2  E2 vg2
m
Landau criteria for breakdown of superfluid ity
in free space : c 2  vg2
Cfr. with B. Wu and Q. Niu,
PRA64, 061603R (2001)
Dynamical instability
The system becomes modulation ally unstable when ωB becomes complex
c2 
1 
N 0 cos p  0
mE N 0
In the free (V  0) limit the Bogoliubov spectrum
The amplitude of the perturbation modes grows exponentially fast,
dissipating the energy
1 q 4 of 1thelarge
μ amplitude wave-packet
ωB  vg q 
m
2
4

m N 0
N0 q 2
is always real (when a  0)
No dynamical instabilities
1
1 
2
2 q

 2 sin q
cos p sin 
N 0 cos p
2
m breakdown2 ofmsuperfluidity
New mechanism for the
of a BEC
E N 0
1
(q, p)
in a periodic potential
Comparison between analytical and
numerical dispersion relation
1
p 
4
Full line: analytical (DNL)
Dots: numerical (GPE)
3
p 
4
Dashed line : m  mE
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    mE1 sin p
p  
VD ( )

The system becomes dynamically unstable at the
critical group velocity vg  m1
A. Smerzi, A. Trombettoni, P.G. Kevrekidis, A.R. Bishop, PRL 89, 170402 (2002)
The Frontier
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• Interferometry at the Heisenberg limit
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Tools
• Quantum many-body dynamical theory