Excitations%20in%20Bose-Einstein%20condensates

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Excitations in Bose-Einstein condensates
Trento, 2 May 2006
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Excitations in Bose-Einstein condensatesa
long story

• Collective excitations and hydrodynamic
  equations
• Collective vs. single-particle
• Excitations in low dimensions
• Collapse, expansion and nonlinear dynamics
• Solitons

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Excitations in Bose-Einstein condensatesthe
first 2 years _at_

• Response of a condensate to a Bragg pulse
• Evaporation of phonons in a free expansion
• Landau damping of collective excitations

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Excitations in Bose-Einstein condensatesthe
most recent results _at_

• Parametric resonances in optical lattices
• Pattern formation in toroidal condensates
• Stability of solitons in 2D

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Excitations in Bose-Einstein condensatesthe
most recent results _at_

• Parametric resonances in optical lattices
• Pattern formation in toroidal condensates
• Stability of solitons in 2D

Parametric excitation of a Bose-Einstein
condensate in a 1D optical lattice M. Kraemer,
C. Tozzo and F. Dalfovo, Phys. Rev. A 71,
061602(R) (2005) Stability diagram and growth
rate of parametric resonances in Bose-Einstein
condensates in 1D optical lattices C. Tozzo, M.
Kraemer, and F. Dalfovo, Phys. Rev. A 72, 023613
(2005)
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Starting point experiments by Esslinger et al.
T.Stoeferle, et al., PRL 92, 130403 (2004)
M.Koehl et al., JLTP 138, 635 (2005) C.Schori et
al., PRL 93, 240402 (2004).
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Starting point experiments by Esslinger et al.
T.Stoeferle, et al., PRL 92, 130403 (2004)
M.Koehl et al., JLTP 138, 635 (2005) C.Schori et
al., PRL 93, 240402 (2004).
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Gross-Pitaevskii simulations
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Axial width after expansion
Fraction of atoms with q close to resonance
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Simulations vs. experiments
expt
GP
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What kind of resonance?
?q O /2
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?q O /2
Its a parametric resonance
Classical example the vertically driven
pendulum. Stationary solutions  f  0 and f
 180. In the undriven case, these solutions are
always stable and unstable, respectively. But
vertical driving can change stability into
instability and vice versa. The dynamics is
governed by the Mathieu equation
 (t)  c(t)exp(  t), where c(t1/f) c(t).
Floquet exponent. If is is real and
positive, then the oscillator is parametrically
unstable.
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Parametric resonances
Very general phenomenon (classical oscillators,
nonlinear optics, systems governed by a
Non-Linear Schroedinger Equation, Hamiltonian
chaotic systems, etc.)
Previously mentioned in the context of BEC by
Castin and Dum, Kagan and Maksimov, Kevrekidis
et al., Garcia-Ripoll et al., Staliunas et al.,
Salasnich et al., Salmond et al., Haroutyunyan
and Nienhuis, Rapti et al.).
Very recent experiments Parametric Amplification
of Matter Waves in Periodically Translated
Optical Lattices N. Gemelke, E. Sarajlic, Y.
Bidel, S. Hong, and S. Chu Phys. Rev. Lett. 95,
170404 (2005) Parametric Amplification of
Scattered Atom Pairs Gretchen K. Campbell,
Jongchul Mun, Micah Boyd, Erik W. Streed,
Wolfgang Ketterle, and David E. Pritchard Phys.
Rev. Lett. 96, 020406 (2006)
Important remark in order to be parametrically
amplified, the resonant mode must be present at
t0 (seed excitation). The parametric
amplification is sensitive to the initial quantum
and/or thermal fluctuations.
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A deeper theoretical analysis in a simpler case
no axial trap, infinite condensate, Bloch
symmetry
GP equation
with
Ground state fluctuations
j band index k quasimomentum
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Using Bloch theorem
Bogoliubov quasiparticle amplitudes
Bogoliubov equations
with
and
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Bogoliubov spectrum in a stationary lattice
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Dynamics in a periodically modulated lattice
is small.
Assume the order parameter to be still of the
form
where
at time t, is the solution of the stationary GP
equation for s(t)
is small.
and
Linearized GP gives
This term is a source of excitations in the
linear response regime. It is negligible in the
range of O we are interested in.
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Linearized GP gives
Bloch wave expansion
Floquet analysis Look for unstable regions in
the (O,k)-plane. Calculate the growth rate
The lattice modulation enters here (this
equation is the analog of Mathieu equation of
classical oscillators)
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Stability diagram
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Remarks on thermal and quantum seed
In GP simulations the seed is numerical noise or
some extra noise added by hand to simulate the
actual noise.
In the experimental BECs, the seed can be
Excitations due to non-adiabatic loading of BEC
in the lattice Imprinted ad-hoc
excitations Thermal fluctuations Quantum
fluctuations
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Remarks on thermal and quantum seed
GP theory
Excitations due to non-adiabatic loading of BEC
in the lattice yes
Imprinted ad-hoc excitations

yes Thermal fluctuations

no
Quantum fluctuations

no
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Remarks on thermal and quantum seed
Possible approach beyond GP use the full
Bogoliubov expansion with operators, not
c-numbers. Use the Wigner representation of
quantum fields. In this way, the dynamics is
still governed by classical
Bogoliubov-like equations the depletion is
included through a stochastic distribution of the
coefficients cjk. Exact results can be
obtained by averaging over many different
realizations of the condensate in the same
equilibrium conditions. One has
Thermal fluctuations
Quantum fluctuations
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Remarks on thermal and quantum seed
Two limiting cases
Thermal fluctuations. Possible measurement of T,
even when the thermal cloud is not visible
(selective amplification of thermally excited
modes).
Amplification of quantum fluctuations. Analogous
to parametric down-conversion in quantum
optics. Source of entangled counter-propagating
quasiparticles. example Dynamic Casimir
effect the environment in which quasiparticles
live is periodically modulated in time and this
modulation transforms virtual quasiparticles
into real quasiparticles (as photons in
oscillating cavities).
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Excitations in Bose-Einstein condensatesthe
most recent results _at_

• Parametric resonances in optical lattices
• Pattern formation in toroidal condensates
• Stability of solitons in 2D

Detecting phonons and persistent currents in
toroidal Bose-Einstein condensates by means of
pattern formation M. Modugno, C.Tozzo and
F.Dalfovo, to be submitted (today!)
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Bose-Einstein condensates have recently been
obtained with ultracold gases in a ring-shaped
magnetic waveguide (Stamper-Kurn et al.) Other
groups are proposing different techniques to get
toroidal condensates. Main purpose create a
system in which fundamental properties, like
quantized circulation and persistent currents,
matter-wave interference, propagation of sound
waves and solitons in low dimensions, can be
observed in a clean and controllable way. An
important issue concerns also the feasibility of
high-sensitivity rotation sensors.
Our approach Parametric resonances as a tool to
measure the excitation spectrum and
rotations. Advantage of toroidal geometry Clean
response nonlinear mode-mixing suppressed
periodic pattern formation.
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• Procedure
• The condensate is initially prepared in the
  torus.
• The transverse harmonic potential is periodically
  modulated in time.
• Both the trap and the modulation are switched off
  and the condensate expands.

We solve numerically the time dependent GP
equation, using the Wigner representation for
fluctuations at equilibrium at step (i).
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GP simulations (with seed)
no modulation
modulation
in trap
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Pattern visibility (in trap)
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GP simulations (with seed)
no modulation
modulation
in trap
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GP simulations (with seed)
no modulation
modulation
in trap
after expansion
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Mean-field effects in the expansion
without
with
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Sensitive rotation sensor
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• a periodic modulation of the confining
  potential of a toroidal condensate induces a
  spontaneous pattern formation through the
  parametric amplification of counter-rotating
  Bogoliubov excitations.
• This can be viewed as a quantum version of
  Faraday's instability for classical fluids in
  annular resonators.
• The occurrence of this pattern in both density
  and velocity distributions provides a tool for
  measuring fundamental properties of the
  condensate, such as the excitation spectrum, the
  amount of thermal and/or quantum fluctuations and
  the presence of quantized circulation and
  persistent currents.

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Excitations in Bose-Einstein condensatesthe
most recent results _at_

• Parametric resonances in optical lattices
• Pattern formation in toroidal condensates
• Stability of solitons in 2D

Work in progress Shunji Tsuchiya, L.Pitaveskii,
F. Dalfovo, C.Tozzo
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Starting point Motion in a Bose condensate
Axisymmetric solitary waves Jones and Roberts, J.
Phys. A 15, 2599 (1982) Numerical solutions of
GP equation. A continuous family of solitary
waves solutions is obtained. At small velocity
a pair of antiparallel vortices, mutually
propelling in obedience to Kelvins theorem.
At large velocity rarefaction pulse of
increasing size and decreasing amplitude.
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U 0.2
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U 0.5
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U 0.2
U 0.5
( Sound speed 1/v2 )
U 0.7
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In 3D
Crow instability of antiparallel vortex
pairs Berloff and Roberts, J. Phys. A 34, 10057
(2001)
1D soliton in 2D
Instability against transverse modulations
(self-focusing) Kuznetsov and Turitsyn , Zh.
Eksp. Teor. Fiz. 94, 119 (1988)
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Stability or instability of Jones-Roberts soliton
in 2D
Our approach calculate the real and imaginary
(if any) eigenfrequencies of the linearized GP
equation (Bogoliubov spectrum)
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When U approaches the speed of sound
Kadomtsev-Petviashvili equation
Linearized Kadomtsev-Petviashvili equation
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Localized excited states
Work in progress
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Thank you
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(instantaneous) Bogoliubov quasiparticle basis
Multi-mode coupling, induced by s(t)
Coupling parameters
j-jbands, same k Nk constant
j-jbands, opposite k Exponential growth of Nk
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Assumption coupling by pairs.
Two-mode approximation
Replace sum over j with a single j and keep
leading terms (small A)
with
and
growth rate
Resonance condition
Growth rate on resonance
with
Seed