Transcript Production of the random potential

LENS
European Laboratory for Nonlinear Spectroscopy
Università di Firenze
J. E. Lye,, L. Fallani, M. Modugno, D. Wiersma, C. Fort, M. Inguscio
Bose-Einstein condensates in random potentials
Les Houches, February 2005
Outlook
Why a random potential?
How to produce a random potential
First results from a BEC in a speckle potential
Conclusions
Chiara Fort
Jessica Lye
Leonardo Fallani
Michele Modugno
Massimo Inguscio
Diederik Wiersma
Why random potentials?
Examples of existing systems with random media
Suppression of superfluidity of 4He in porous media with disorder
Anderson Localisation of photons in strongly scattering semiconductor powders
Disruption of electron transport due to defects in a solid – Anderson Localisation?
Bose-Einstein condensates in random potentials …
Long coherence length coupled with a controllable system
Exploring the role of interactions without loss of coherence
Control of dimensionality
Engineering new quantum phases (Bose glass) and Anderson localization
Transport/superfluid properties in the presence of disorder
BEC in microtraps
Fragmentation caused by imperfections of the microchip
Modification of superfluid properties?
Quantum phase transitions
At zero temperature, when quantum fluctuations become important, a BEC in an optical
lattice in the tight-binding regime is well-described by the Bose-Hubbard model:
Bose-Hubbard Hamiltonian
1
H   J  aˆi†aˆ j    i nˆi  U  nˆi (nˆi  1)
2 i
i, j
i
hopping energy
disorder
interaction energy
J
D
U
U
J
D
i
Superfluid/Mott insulator transition
Quantum fluctuations can induce a phase transition from a superfluid phase to a Mott
insulator phase. The transition is induced by a competition between two energy scales:
hopping energy
J
E
<>
interaction energy
U
U
SUPERFLUID PHASE ( J > U)
MOTT INSULATOR PHASE (U > J)
1. Long-range phase coherence
2. High number fluctuations
3. No gap in the excitation spectrum
1.
2.
3.
4.
No phase coherence
Zero number fluctuations
Gap in the excitation spectrum
Vanishing superfluid fraction
Mott insulator / Bose Glass transition
With sufficient disorder, a quantum phase transition to the Bose Glass state occurs:
disorder
>
D
interaction energy
U
hopping energy
>
J
U
E
D
BOSE-GLASS PHASE (BG)
MOTT INSULATOR PHASE (MI)
1.
2.
3.
4.
1.
2.
3.
4.
No phase coherence
Low number fluctuations
No gap in the excitation spectrum
Vanishing superfluid fraction
No phase coherence
Zero number fluctuations
Gap in the excitation spectrum
Vanishing superfluid fraction
Anderson Localisation
1
†
ˆ
ˆ
ˆ
H   J  ai a j    i ni  U  nˆi (nˆi  1)
2 i
i, j
i
Scattering model
Anderson Hopping model
disorder
D
>
hopping energy
J
D. Wiersma et al. Nature 390 671 (1997)
ANDERSON LOCALISATION
1.
2.
3.
4.
Long-range phase coherence
High number fluctuations
No gap in the excitation spectrum
Vanishing superfluid fraction
* Phase coherence is maintained, but
hopping is inhibited by lattice topology
• With sufficient scattering, the light waves
can follow a random light path back to the
source
• The waves can propagate in two opposite
directions along the looped path, each
acquiring the same phase, and interfere
constructively at the source, hence there is
a higher probability of the wave returning
to the source, and a lower probability of
propagating away.
Phase diagram
ANDERSON LOCALISATION
BOSE-GLASS PHASE (BG)
1.
2.
3.
4.
1.
2.
3.
4.
Long-range phase coherence
High number fluctuations
No gap in the excitation spectrum
Vanishing superfluid fraction
No phase coherence
Low number fluctuations
No gap in the excitation spectrum
Vanishing superfluid fraction
(R. Roth and K. Burnett,
PRA 68, 023604 (2003))
SUPERFLUID PHASE
1. Long-range phase coherence
2. High number fluctuations
3. No gap in the excitation spectrum
U/J
MOTT INSULATOR PHASE (MI)
1.
2.
3.
4.
No phase coherence
Zero number fluctuations
Gap in the excitation spectrum
Vanishing superfluid fraction
A possible route to Bose-Glass…
First, to reach a Mott-Insulator phase
with a regular lattice
Second, to add disorder to the lattice
B. Damski et al. PRL 91
080403 (2003)
R. Roth and K. Burnett,
PRA 68, 023604 (2003)
U/J
The amount of disorder necessary to enter the Bose Glass phase is relatively small, being of
the order of the interaction energy U  ER
Or Anderson Localisation…
Reduce interactions through expansion?
in the random potential alone?
E
The random potential
Two possible solutions to add disorder to the system:
Speckle pattern
Bichromatic lattice (pseudorandom)
How we produce a random potential
Production of the random potential
The random potential is produced by shining an off-resonant laser beam onto a diffusive plate
and imaging the resulting speckle pattern on the BEC.
speckle pattern
The BEC is illuminated by the speckle beam
in the same direction as the imaging beam.
With the same imaging setup we can detect
both the BEC and the speckle pattern.
400 mm
BEC
What the random potential looks like
FFT
The speckle pattern is
in good approximation a
random “white” noise.
However, due to the
finite resolution of our
system, the interspeckle
distance starts from 
10 mm.
Vsp  2
 V ( x )  V 
9.6 mm
9.6 mm
2
i
i
N 1
We define the average speckle height VSP as twice
the standard deviation of the potential profile:
A comment
NOTE on length scales:
• With a site separation of 10 mm, the tunnelling time in the tight binding limit is
far greater than the time scale of the experiment, thus by simply increasing the
height of the speckle potential alone we cannot reach the Bose Glass regime.
• If the interactions are sufficiently low this could be a suitable length scale to
see Anderson Localisation?
• This length scale is comparable to that seen in microtrap experiments
First results from a BEC in a speckle potential
Expansion from the speckle potential
We adiabatically ramp the intensity of the speckle pattern on the
trapped BEC, then we suddenly switch off both the magnetic trap
and the speckle field and image the atomic cloud after expansion:
VSP = 30 Hz
VSP = 100 Hz
Releasing the BEC from the weak speckle (VSP < m ~ 1kHz)
potential we observe some irregular stripes in the expanded cloud.
VSP = 200 Hz
Releasing the BEC from the strong speckle (VSP > m potential we
observe the disappearance of the fringes and the appareance of a
broader gaussian unstructured distribution.
VSP = 2000 Hz
speckle intensity
VSP = 10 Hz
Expansion from the speckle potential
In order to check if the observed density distribution was simply caused by heating, we have
checked the adiabaticity of the procedure by applying a reverse ramp on the speckle intensity.
A
B
C
Transport in the speckle potential
Dipole mode
Sudden displacement of the magnetic trap
center along the x direction.
Interference from a finite number of point-like emitters
high contrast
regular spacing
coherent sources
regular spacing
incoherent sources
lower contrast
Expansion of a coherent array of BECs
P. Pedri et al., Phys. Rev. Lett. 87, 220401 (2001)
Detecting a Bose-Glass phase...
Interference of an array of independent BECs
No interference fringes in a randomly
spacedetsample
Z. Hadzibabic
al., PRLeven
93 180403 (2004)
without a phase transition
disorderd spacing
coherent sources
no interference
Interference from randomly spaced
BECs located at different sites
Expansion from the speckle potential
No disorder
speckle intensity
VSP = 0
Moderate disorder (VSP < m):
• long wavelength modulations
• breaking phase uniformity?
• strong damping of the dipole mode
VSP = 200 Hz
Strong disorder (VSP > m):
• broad unstructured density profile because expansion
from randomly spaced array
• classically localized condensates in the speckles sites
S
VSP = 1700 Hz
Dynamical of
instability
of a BEC in a
lattice
Observation
Phase Fluctuations
inmoving
Elongated
BECs
L.
Fallani
et
al.,
Phys.
Rev.
Lett.
93,
140406
(2004)
S. Dettmer et al., Phys. Rev. Lett. 87, 160406 (2001)
Collective excitations in the random potential
After producing the BEC, we adiabatically load the BEC in the disordered potential
Then we excite collective modes in the harmonic + random potential:


1
1
2 2
Vtot  m x x  m2 y 2  z 2  Vopt  x, y 
2
2
Quadrupole mode
5
? x
2
Resonant modulation of the radial trapping
frequency (via the magnetic bias field)
in the case of ordinary fluids:
noninteracting
gas
Dipole
mode
Sudden displacement of the magnetic
strongly
trap
center interacting
along the xgas
direction.
peculiar of superfluid behavior
?x
Collective excitations in the weak speckle potential
We investigate the weak disorder regime, where the speckle field produces a weak
perturbation of the harmonic trapping field and the system is not trapped in individual
speckle wells.
P = 5 mW
VSP = 100 Hz
m
Collective excitations in the weak speckle potential
dipole (0 mW)
quadrupole (0 mW)
dipole (3 mW) – VSP = 60 Hz
quadrupole (2 mW) – VSP = 40 Hz
Frequency shift in the quadrupole mode
We see small frequency shifts to both the blue and the red, depending on the particular
speckle realization, that becomes stronger increasing the speckle power.
Collective excitations in the weak speckle potential
Using the sum-rules approach, and treating the
speckle potential as a small perturbation :
For a non-harmonic potential, shifts in the quadrupole frequency are not necessary
correlated to shifts in the dipole frequency.
This effect could mask any other possible changes in the excitation modes.
Summary
How we produce a random potential
Results from the BEC in a random potential
Stripes in the density profile at moderate disorder, with strong damping
of the dipole mode.
Gaussian distribution at strong disorder, atoms classically localized in
randomly spaced speckle wells.
frequency shift of the quadrupole mode uncorrelated to a frequency shift
in the dipole mode due to anharmonic speckle potential.
Future projects
Study of localization effects:
Combining speckle potential with optical lattice standing wave:
Mott-Insulator Bose Glass
Anderson localization with speckle potential alone, reducing interactions through
expansion
Expansion from the speckle potential
Observation of the Mott insulator phase
(M. Greiner et al., Nature 415, 39 (2002))
The Mott insulator phase has been first
obtained in a BEC trapped in a 3D
optical lattice increasing the lattice
height above a critical value
Interference pattern of an interacting
BEC released from a 3D optical lattice
approaching the quantum transition:
Increasing the lattice height
J decreases
U increases
• Vanishing of 3D interference pattern
U / J increases
loss of long range coherence, phase fluctuations
• Applying a magnetic field gradient, the excitation spectrum was measured and the distinctive
energy gap of the Mott-insulator was seen
Production of the random potential
The random potential is produced by shining an off-resonant laser beam onto a diffusive plate
and imaging the resulting speckle pattern on the BEC.
3 c 2 
V ( x, y ) 
I ( x, y )
3
20 D
Optical dipole potential
stationary in time
randomly varying in space