QUANTUM-RESONANCE RATCHETS: THEORY AND EXPERIMENT

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QUANTUM-RESONANCE RATCHETSTHEORY AND EXPERIMENT
BH

• I. Dana (Bar-Ilan University)
• Theory ID and V. Roitberg, PRE 76,
  015201(R) (2007)
• Experiment ID, V. Ramareddy, I. Talukdar, and
  G.S. Summy,
• PRL, in press
  (http//arxiv.org/abs/0706.0871 )

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• Classical Hamiltonian Ratchets
• General concept of Ratchet
• Translationally-invariant system in which a
  directed transport can be
• established without a biased force (average force
  0). Usually, the
• directed transport is due to the breaking of a
  spatial/temporal symmetry.
• E.g., molecular motors in biological systems
  with dissipation and
• external noise see, e.g., R.D.
  Astumian and P. Hänggi (2002).
• Classical Hamiltonian Ratchet
• S. Flach et al. (2000), T. Dittrich et al.
  (2000, 2001)
• No dissipation and external noise is replaced by
  deterministic chaos.

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Hamiltonian-Ratchet Maps
pt1 pt f(xt ),
xt1
xt pt1 mod(2p), f(x 2p, t) f(x),
?f(x)? 0. Momentum Current
(acceleration) of phase-space region A

IA A limt?8 ?pt?A /t ?f(x)?
0 ICHAOS IISLANDS 0
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ICHAOS 0
for fully chaotic system.
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Generalized Quantum Kicked Rotor
Quantum Resonances (QRs) Rational t/(2p)
l/q, with a band quasienergy spectrum. Purely
quantum ballistic motion


QR Ratchets For asymmetric , e.g.,
a ratchet
acceleration may arise at QR, even for fully
chaotic classical system (Iclassical 0).
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Quantum Kicked Particle
Translational invariance implies conservation of
quasimomentum ?, 0 ? lt 1, in time-evolution
of Bloch wave
with 2p-periodic At fixed ?,
and x ? ?
?-rotor. General QR Conditions for integers
l, q, r, g, with coprime (l, q)
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Exactly solvable case of main QRs t
2pl. Resonant quasimomenta ? ?r,g r/(lg)
1/2 mod(1). For general potential
and initial wave packet
one finds, for arbitrary ? and defining t?
pl(1 2?),
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• For resonant ? ?r,g , a QR-ratchet
  acceleration is obtained
• with ratchet coefficient

R ? 0 for generic potentials and wave packets.
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• Atom-Optics Experimental Realization of
  QR-Ratchets
• Potential ,
  with symmetry center at .
• Initial wave packet
  , symmetric under
• time reversal and inversion around symmetry
  center at .

Ratchet acceleration for resonant ? with
coefficient
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For a BEC with quasimomenta (initial momenta)
Gaussian distributed with small width ?? ltlt 1
around some given ? , the average QR-ratchet
behavior for arbitrary ? is exactly given by
For resonant ?
Experimental values l 1, only resonant ?
0.5, ?? ? 0.1, ?0 0 .
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• CONCLUSIONS
• QR-Ratchet Purely quantum momentum current
  (acceleration) for resonant quasimomenta ? .
• QR-ratchet effects can emerge also for symmetric
  potentials and initial wave packets if, e.g.,
  their symmetry centers do not coincide. Results
  are totally unaffected by potential high
  harmonics for simple initial wave packets.
• Consideration of arbitrary ? Indispensable for
  taking into account the small but finite
  quasimomentum width of the BEC, leading to a
  suppression of the QR-ratchet acceleration.
  Pronounced ratchet effect near resonant ? .
• Work in progress Experimental realization of
  QR-ratchets in the free-falling frame of quantum
  kicked particle under gravity (linear
  potential).