Bose-Einstein condensates in random potentials

Les Houches, February 2005

LENSEuropean Laboratory for Nonlinear Spectroscopy

Università di Firenze

J. E. Lye,, L. Fallani, M. Modugno, D. Wiersma, C. Fort, M. Inguscio

Outlook

Why a random potential?

How to produce a random potential

First results from a BEC in a speckle potential

Conclusions

Chiara FortJessica LyeLeonardo Fallani Michele ModugnoMassimo Inguscio

Diederik Wiersma

Why random potentials?

Examples of existing systems with random media

Suppression of superfluidity of 4He in porous media with disorderAnderson 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 systemExploring the role of interactions without loss of coherence Control of dimensionalityEngineering new quantum phases (Bose glass) and Anderson localizationTransport/superfluid properties in the presence of disorder

BEC in microtraps

Fragmentation caused by imperfections of the microchipModification of superfluid properties?

Quantum phase transitions

†

,

1ˆ ˆ ˆ ˆ ˆ( 1)

2i j i i i ii j i i

H J a a n U n n Bose-Hubbard Hamiltonian

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:

hopping energy

J

interaction energy

U

disorder

i

U

J

Superfluid/Mott insulator transition

SUPERFLUID PHASE ( J > U)

1. Long-range phase coherence2. High number fluctuations3. No gap in the excitation spectrum

MOTT INSULATOR PHASE (U > J)

1. No phase coherence2. Zero number fluctuations3. Gap in the excitation spectrum4. Vanishing superfluid fraction

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 interaction energy

J U<

UE

Mott insulator / Bose Glass transition

With sufficient disorder, a quantum phase transition to the Bose Glass state occurs:

hopping energyinteraction energy

JU

disorder

> >

BOSE-GLASS PHASE (BG)

1. No phase coherence2. Low number fluctuations3. No gap in the excitation spectrum4. Vanishing superfluid fraction

MOTT INSULATOR PHASE (MI)

1. No phase coherence2. Zero number fluctuations3. Gap in the excitation spectrum4. Vanishing superfluid fraction

UE

Anderson Localisation

hopping energy

J

disorder

>

ANDERSON LOCALISATION

1. Long-range phase coherence2. High number fluctuations3. No gap in the excitation spectrum4. Vanishing superfluid fraction

†

,

1ˆ ˆ ˆ ˆ ˆ( 1)

2i j i i i ii j i i

H J a a n U n n

Anderson Hopping model

* Phase coherence is maintained, but hopping is inhibited by lattice topology

Scattering model

• 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.

D. Wiersma et al. Nature 390 671 (1997)

Phase diagram

BOSE-GLASS PHASE (BG)

1. No phase coherence2. Low number fluctuations3. No gap in the excitation spectrum4. Vanishing superfluid fraction

MOTT INSULATOR PHASE (MI)

1. No phase coherence2. Zero number fluctuations3. Gap in the excitation spectrum4. Vanishing superfluid fraction

/U J

(R. Roth and K. Burnett, PRA 68, 023604 (2003))

SUPERFLUID PHASE

1. Long-range phase coherence2. High number fluctuations3. No gap in the excitation spectrum

ANDERSON LOCALISATION

1. Long-range phase coherence2. High number fluctuations3. No gap in the excitation spectrum4. Vanishing superfluid fraction

A possible route to Bose-Glass…

First, to reach a Mott-Insulator phasewith a regular lattice

Second, to add disorder to the lattice

/U J

The amount of disorder necessary to enter the Bose Glass phase is relatively small, being of the order of the interaction energy RU E

Or Anderson Localisation…

Reduce interactions through expansion? in the random potential alone?

B. Damski et al. PRL 91 080403 (2003)

R. Roth and K. Burnett, PRA 68, 023604 (2003)

E

The random potential

Two possible solutions to add disorder to the system:

Bichromatic lattice (pseudorandom)Speckle pattern

How we produce a random potential

Production of the random potential

With the same imaging setup we can detect both the BEC and the speckle pattern.

The BEC is illuminated by the speckle beam in the same direction as the imaging beam.

speckle pattern BEC

400 m

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.

What the random potential looks like

FFTThe 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 m.

9.6 m9.6 m

2( )2

1

ii

sp

V x VV

N

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 m, 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:

spec

kle

inte

nsity

Releasing the BEC from the weak speckle (VSP < 1kHz) potential we observe some irregular stripes in the expanded cloud.

Releasing the BEC from the strong speckle (VSP > potential we observe the disappearance of the fringes and the appareance of a broader gaussian unstructured distribution.

VSP = 10 Hz

VSP = 30 Hz

VSP = 100 Hz

VSP = 200 Hz

VSP = 2000 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

Sudden displacement of the magnetic trap center along the x direction.

Dipole mode

Interference from a finite number of point-like emitters

regular spacingcoherent sources

regular spacingincoherent sources

disorderd spacingcoherent sources

Expansion of a coherent array of BECsP. Pedri et al., Phys. Rev. Lett. 87, 220401 (2001)

high contrast

lower contrast

no interference

Interference of an array of independent BECsZ. Hadzibabic et al., PRL 93 180403 (2004)

Interference from randomly spacedBECs located at different sites

Detecting a Bose-Glass phase...

No interference fringes in a randomly spaced sample even

without a phase transition

Expansion from the speckle potential

Moderate disorder (VSP < ):

• long wavelength modulations• breaking phase uniformity?• strong damping of the dipole mode

Strong disorder (VSP > ):

• broad unstructured density profile because expansion from randomly spaced array• classically localized condensates in the speckles sites

No disorder

Dynamical instability of a BEC in a moving latticeL. Fallani et al., Phys. Rev. Lett. 93, 140406 (2004)

spec

kle

inte

nsity

VSP = 0

VSP = 1700 Hz

VSP = 200 Hz

S Observation of Phase Fluctuations in Elongated BECsS. 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:

2 2 2 2 21 1,

2 2tot x optV m x m y z V x y

Sudden displacement of the magnetic trap center along the x direction.

Dipole mode

x

Resonant modulation of the radial trapping frequency (via the magnetic bias field)

Quadrupole mode 5

2 x

noninteracting gas

strongly interacting gas

in the case of ordinary fluids:

peculiar of superfluid behavior

?

?

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

Collective excitations in the weak speckle potential

dipole (0 mW)

dipole (3 mW) – VSP = 60 Hz

quadrupole (0 mW)

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

Increasing the lattice height

Interference pattern of an interacting BEC released from a 3D optical lattice approaching the quantum transition:

J decreasesU increases

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

/U J increases

• Vanishing of 3D interference pattern loss of long range coherence, phase fluctuations

(M. Greiner et al., Nature 415, 39 (2002))

• 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.

2

30

3( , ) ( , )

2

cV x y I x y

Optical dipole potential

stationary in timerandomly varying in space