Download Superfluid BEC dynamics in peridioc potentials

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts
no text concepts found
Transcript
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
Technological applications
• Interferometry at the Heisenberg limit
• Quantum information
Foundational problems
• Quantum – classical correspondence principle
• Schroedinger cats, entanglement
Tools
• Quantum many-body dynamical theory