Irreversible loading of optical lattices

1
Irreversible loading of optical lattices
Rotation of cold atoms
Chris Foot
University of Oxford
2
Outline

• Superfluidity tested by the response to
  rotation
• TOP trap ? rotating elliptical potential
• Observation of the scissors mode ? damping
• Nucleation of vortices
• Rf-modified magnetic trapping ? toroidal trap
  (persistent current)
• Rotation of optical lattice ? artificial B-field
• Theory
• Proposed experiment

3
Magnetic coils and vacuum cell
TOP trap Time-orbiting potential
BEC 105 rubidium atoms. Temperature 50
nK Density 1014 cm-3
4
Shape of BEC in a TOP trap
Pancake (oblate) rather than a cigar (prolate) as
in Ioffe traps.
5
Quantised circulation in a quantum fluid
h

• The velocity field ? phase gradient
• Hence velocity field is irrotational
• Circulation around a closed contour is quantised

h
h

• Zero circulation irrotational flow
• Non-zero circulation vortices

6
Types of flow
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Excitation of scissors mode
?
Trap tilted adiabatically to angle ?
Trap suddenly rotated by -2?
2?
Cloud oscillates about new equilibrium position
c.f. torsion pendulum
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Scissors mode results
Described in Pitaevskii Stringari Oxford
University Press 2003
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Scissors mode irrotational flow
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Rotation of the confining magnetic potential
Impart angular momentum using rotating elliptical
potential
BEC
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Nucleation of a single vortex
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Thresholds for vortex nucleation
Eleanor Hodby et al

• Critical frequency Wc 1/?2
• Line II stability boundary for the quadrupole
  II branch.
• Vortices nucleated below Wc.

13
Rotational also introduces centrifugal term
into the Hamiltonian

• Radial harmonic potential

Centrifugal term
Radial trapping decreases as ? ? ??
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Thresholds for vortex nucleation

• Critical frequency Wc 1/?2
• Line II stability boundary for the quadrupole
  II branch.
• Vortices nucleated below Wc.

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Nucleation of a single vortex
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Nucleation of an array of vortices
c.f. other expts at ENS, MIT, JILA
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Observing the Tilting Mode ( side view of the
vortex array )
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Precession of angle of condensate with vortex
lattice
27.75 Hz
lt lz gt 8.4 0.4 h
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Spectrum of BEC modes in modified TOP trap
2v2
down-conversion
v2
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Beliaev damping of the scissors mode
Excite scissors mode
Energy in m2 quadrupole mode
Energy in scissors mode
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Resonance of the Beliaev process
Beliaev damping resonantly enhanced for certain
geometries
G
Scissors mode frequency 2 (quadrupole mode
frequency)
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Two-dimensional trapping of Bose gas

• Weak radial confinement by the magnetic trap
• Squeeze atoms between two sheets of light
• Creates a thin sheet of atoms 2D Bose gas

z
BEC
?z 2 kHz ?? 10 Hz
Physics of 2-D systems, BKT transition
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Combined optical and magnetic trap

• Contours of the magnetic potential

z
Light sheets
x
magnetic potential
MF -1
MF 0
x
MF 1
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Modification of 2-D trap by radio frequency
radiation
dressed-atom potential
magnetic potential
MF -1
rf
rf
MF 0
x
x
rf
rf
MF 1
dressed-atom picture
Proposed by Zobay and Garraway PRL (2001)
rf
rf
rf
rf
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TOP trap
Apply 1.15 MHz field using TOP coils
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Trapping potential Static RF fields
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Ring shaped cloud of atoms (March 2007)

• Will Heathcote, Eileen Nugent, Ben Sheard
• application to persistent currents

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Trap frequencies as function of Bias field
Magnetic trap, no rf
Static rf fields

• March 2007

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Method to excite a persistent current ?
Toroidal potential
Introduce angle dependence in rotating frame
Potential at r const.
Potential at r const.
?
?
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Overview of cold atoms/molecules
Atoms in optical lattices Physics of strongly
correlated systems
Dilute quantum gases
BEC
Fermi gas
Cold molecules
Quantum Information Processing
Condensed Matter Physics
Quantum fluids superfluid helium
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Simulation of Condensed Matter Systems

• Hamiltonian of atoms in optical lattice
  Hamiltonian of CMP system

E.g. Fraction Quantum Hall Effect
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Mathematical equivalence of rotation on cold
atoms and the effect of a magnetic field on
charged particles (electrons)
?

• Coriolis force F 2m v x ?
• Lorentz force F q(E v x B )
• q Beff 2m ?
• For electron, q -e
• Cyclotron frequency, ??c eB 2?rot

m
Analogous transformation of the quantum
mechanical Hamiltonian. Momentum, p ? p e A
where the magnetic vector potential eA - m ?
? r
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Rotational also introduces centrifugal term
into the Hamiltonian

• Radial harmonic potential

Centrifugal term
Radial trapping decreases as ? ? ??
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Energy levels of a 2-D harmonic oscillator
m 1
-1
m 0

• Near degeneracy as ? ? ??
• Interactions mix states single particle states ?
  strongly correlated multi-particle states

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Fractional quantum Hall states
FQHE states predicted in BEC at fast rotation
frequencies Wilkin and Gunn, Ho, Paredes et al.,
Cooper et al,
N Number of atoms
( cf. filling factor, ? )
Nv Number of vortices
Zoo of strongly correlated states
Lindemann criterion suggests that the vortex
lattice melts when
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Atoms in a rotating lattice. Define new parameter
?
Theory R. Palmer D. Jaksch, Phys. Rev. Lett.
96, 180407 (2006)

• Phase shift from hopping around one lattice cell
  is ????

? flux through loop
?
d
?? ? 2m ?d2 eBeff d2 ???
h
h/e
h
h/e flux quantum
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Particles in a lattice have fractal distribution
of eigenenergies
E
Hofstadter Butterfly
Reach much higher fields than in conventional CMP
experiments
Continuum Landau levels
En h eB ( n 1/2)
m
E
B
? ? B
0
1
0.5
B
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High-field FQHE

• The optical lattice setup allows to explore
  parameter regimes which
• are not accessible otherwise ? beyond mimicking
  condensed matter

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Experiment in Oxford
Microscope for quantum matter.
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Two-dimensional rotating optical lattice
Confinement along z by two sheets of laser light
(not shown).
High NA lens
Funded by ESF EuroQUAM programme
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Summary

• Scissors mode and vortices
• Superfluidity?
• Measurements of frequency and damping of modes
• probe BEC
• Combined optical and magnetic trap
• 2-D Bose gas
• Magnetic trap rf toroidal potential for
  dressed-state
• persistent current (superposition of different
  states of rotation)
• Rotating optical lattice gives term in atomic
  Hamiltonian analogous to an applied magnetic
  field of a charged particle (e.g. electron)
• Highly correlated quantum states as in Fractional
  Quantum Hall Effect

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Acknowledgments

• People
• Chris Foot
• Eileen Nugent
• Will Heathcote
• Ben Sheard
• Ben Fletcher
• Andrian Harsono
• Ross Williams
• Amita Deb
• Giuseppe Smirne
• Min Sung Yoon
• Herbert Crepaz

• Funding
• EPSRC
• ESF

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Phase diagram for vortex nucleation
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Rate of Beliaev process proportion to Temperature?
Unpublished data (theory does not predict a
straight line?)
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Scissors mode vortex Superfluid gyroscope
Nilsen, McPeake McCann Queens University,
Belfast
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Hall Effect
Pictures from David Leadley, Warwick.
Integer quantum Hall effect in a GaAs-GaAlAs
heterojunction, at 30mK.
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Hall current measurements
R.N. Palmer and D. Jaksch, Phys. Rev. Lett. 96,
180407 (2006)

• A linear potential V(x,y) -may superimposed on
  a harmonic trap induces a particle velocity
• The Hall current plateaus are affected by the
  trap (?1)

Id dimensionless Hall current Nd particles
per length
The curves show different degrees and types of
disorder
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Explanation of the Quantum Hall Effect
The zeros and plateaux in the two components of
the resistivity tensor are intimately connected
and both can be understood in terms of the Landau
levels (LLs) formed in a magnetic field. In the
absence of magnetic field the density of states
in 2D is constant as a function of energy, but in
field the available states clump into Landau
levels separated by the cyclotron energy, with
regions of energy between the LLs where there are
no allowed states. As the magnetic field is swept
the LLs move relative to the Fermi energy. When
the Fermi energy lies in a gap between LLs
electrons can not move to new states and so there
is no scattering. Thus the transport is
dissipationless and the resistance falls to zero.
The classical Hall resistance was just given by
B/Ne. However, the number of current carrying
states in each LL is eB/h, so when there are i
LLs at energies below the Fermi energy completely
filled with ieB/h electrons, the Hall resistance
is h/ie2. At integer filling factor this is
exactly the same as the classical case. The
difference in the QHE is that the Hall resistance
can not change from the quantised value for the
whole time the Fermi energy is in a gap, i.e
between the fields (a) and (b) in the diagram,
and so a plateau results. Only when case (c) is
reached, with the Fermi energy in the Landau
level, can the Hall voltage change and a finite
value of resistance appear. This picture has
assumed a fixed Fermi energy, i.e fixed carrier
density, and a changing magnetic field. The
QHE can also be observed by fixing the magnetic
field and varying the carrier density, for
instance by sweeping a surface gate.
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Dirt and disorder

• Although it might be thought that a perfect
  crystal would give the strongest effect, the
  QHE actually relies on the presence of dirt in
  the samples. The effect of dirt and disorder can
  best be though of as creating a background
  potential landscape, with hills and valleys, in
  which the electrons move. At low temperature each
  electron trajectory can be drawn as a contour in
  the landscape. Most of these contours encircle
  hills or valleys so do not transfer an electron
  from one side of the sample to another, they are
  localised states. A few states (just one at T0)
  in the middle of each LL will be extented across
  the sample and carry the current. At higher
  temperatures the electrons have more energy so
  more states become delocalised and the width of
  extended states increases.
• The gap in the density of states that gives rise
  to QHE plateaux is the gap between extended
  states. Thus at lower temperatures and in dirtier
  samples the plateaus are wider. In the highest
  mobility semiconductor heterojunctions the
  plateaux are much narrower.

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Energy bands

• Fractal energy bands
• Important for investigating magnetically induced
  effects
• quantum Hall effect
• fractional quantum Hall effect
• Atomic systems allow detailed study of the energy
  bands
• Interaction effects are controllable

? 1/2
? 1/3
The optical lattice setup allows to explore
parameter regimes which are not accessible
otherwise ? beyond mimicking condensed matter
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Calculations of eigenstates/ eigenenergies

• R.N. Palmer and D. Jaksch, Phys. Rev. Lett. 96,
  180407 (2006)
• For a harmonic trap with dimensionless trap
  strength ?md2?/?

solid 99 overlap dotted 99.9
overlap dashed analytical estimate
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upconversion