The asymmetric atomoptics deltakicked rotor

1
The asymmetric atom-optics delta-kicked rotor

• Philip Jones
• University College London

Messina, 22 September 2005
2
Overview

• Intro Quantum Chaos
• The Atom-Optics d-kicked Rotor
• The weakly kicked rotor
• Finite temporal width kicks
• Unequal period kicked rotor
• The 2d-kicked rotor
• Conclusion

3
Chaos

• it may happen that small differences in the
  initial conditions produce very great ones in the
  final phenomena. A small error in the former will
  produce an enormous error in the latter.
  Prediction becomes impossible, and we have the
  fortuitous phenomenon.
• Henri Poincaré (1903)

4
Kicked Rotor

• Standard system for investigation of chaotic
  dynamics
• Hamiltonian H r2 Kcos(f)Sd(t n)
• Dynamics described by K, the stochasticity
  parameter
• Diffusive energy growth D K2 / 4

5
The Standard Map

• rn1 rn Ksin(fn)
• fn1 fn rn1
• Energy

6
Phase Space

• Stable trajectories break up as K increases

7
Kicked Rotor

• Correlations between kicks lead to corrections to
  D(K)

8
Cold Atoms

• Compare to Hamiltonian of cold atoms in a pulsed
  optical lattice

9
Dimensionless Units

• r 2p ? 2Tp/Ml
• f 2p ? 2x/l
• f,r i8wrT

K (V0/h) ? tp ? ?
10
Experimental Realisation

• Retro-reflected laser beam
• 1-D lin lin optical lattice
• Sinusoidal potential

11
Experimental Realisation

• Divide kicking beam into two
• Control with separate AOMs
• Lattice moves at lDf

12
Laser Cooling Lab
13
Step 1 Magneto Optical Trap
14
Step 2 Optical Molasses
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Step 3 Kicking
16
Step 4 Imaging
17
Dynamical Localisation

• Quantum interference phenomenon
• Energy diffusion for finite time
• Momentum freezes after break time, t
• Momentum distribution becomes exponential

18
Dynamical Localisation

• Dynamical Localisation shows characteristic
  exponential profile

e.g. Moore et al. Phys. Rev. Lett. 73 2974 (1994)
19
The weakly kicked rotor

• Use regularly spaced kicks, but Kltlt4
• Large islands in phase space
• Use moving lattice to explore stable vs chaotic
  regions

20
Results Mixed Phase Space

• Resolve stable islands and chaotic regions
• Resolution depends on ?

21
Mixed Phase Space
M. Goonasekera et al, in preparation (2005)
22
Finite Temporal Width Kicks

• Finite width causes momentum boundary
• Atom travels l/2 in time tp
• sinc2 envelope to diffusion constant
• rb l ?

23
Results cut-off distribution
24
Results moving lattice

• Large asymmetry develops due to proximity of the
  momentum boundary
• Quantify by first moment of the momentum
  distribution

25
Results moving lattice

• Plot asymmetry as a function of momentum in the
  lattice frame

P. H. Jones et al, Europhys. Lett. 67 (6) 928
(2004)
26
Unequally spaced kicks

• Break time symmetry of kicks
• Use two period cycle
• T(1b) T(1-b)
• Two-kick correlations now give rise to
  momentum-dependent diffusion constant

T. Jonckheere et al. Phys. Rev. Lett. 91 253003
(2004)
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Effect of C2(r)

• ltrgtnp/2b
• Symmetric
• diffusion

28
Effect of C2(r)
ltrgt?np/2b Asymmetric diffusion
29
Results momentum asymmetry
30
Asymmetry Oscillates

• Current oscillates with expected period p / b
• Top panel, b 1/16
• Lower, b 1/8
• K 3.3, ? 1

P. H. Jones et al, arXivphysics/0504096 (2005)
31
Break Spatial Symmetry

• Include a potential gradient of alternating sign
• Adds a phase shift to C2(r)
• Achieve by accelerating the lattice

T. Jonckheere et al. Phys. Rev. Lett. 91 253003
(2004)
32
Accelerating Lattice

• Adds an inertial term to the Hamiltonian (in the
  accelerating frame)
• Change the frequency difference by df in the time
  T
• Scaled gradient, A 2ptpdf
• Can easily control magnitude of A

33
Results asymmetry

• Asymmetry oscillates with period 2p
• K 2.6, ? 1, b 1/16

P. H. Jones et al, arXivphysics/0504096 (2005)
34
Results distribution

• Example of the momentum distribution for A 0,
  p/2, -p/2

35
The 2d-kicked rotor

• Use closely-spaced pairs of kicks
• long period, t, short period e
• Again modifies kick-to-kick correlations

36
2d-kicked rotor results

• Atom travels half a period between kicks of the
  pair
• Effect of kicks cancels out
• e 0.045

37
2d-kicked rotor phase space

• Phase space shows momentum trapping regions
• At r (2m1)(p/e) atoms absorb little energy
• Maxima at r 2m(p/e)

38
2d-kicked rotor results

• Narrow dips appear for slightly larger e
• e 0.16
• Consequence of long-range global correlations

P. H. Jones et al. Phys. Rev. Lett. 93 223002
(2004)
39
2d-KR correlations

• C1 ltK2sin(fm)sin(fm1)gt ? cos(re)
• couples nearest-neighbour kicks, acts over short
  timescales
• CG1 ltK2sin(fm)sin(fj)gt ? - cos(re)
• couples kick j with all preceding kicks,
  accumulates to dominate at long times

M. Stocklin et al, arXivphysics/0408088 (2004)
40
2d-KR correlations

• At intermediate times C1 and CG1 cancel
• Correlation CPn ? cos(nre), n 2,3,4 becomes
  evident
• From correlations between kick j and j n
• Sum to give sharp dips in energy growth
• Poisson sum rule
• S(-1)n cos(nre) S d(re (2m1)p)

M. Stocklin et al, arXivphysics/0408088 (2004)
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Compare theory experiment

• Compare experimental data with theory of Stocklin
  et al
• As e increases observe change from dominant C1 to
  CG1, with narrow dips evidence of CPn

42
Conclusions

• Have investigated variations on simple
  delta-kicked rotor
• Verified effect of finite temporal width kicks
• Demonstrated asymmetric diffusion due to chaotic
  dynamics only (chaotic ratchet)
• Observed new global correlations in 2d-kicked
  rotor (momentum filter)
• Prepare atoms in stable islands of chaotic sea

43
Acknowledgements

• Experiment
• David Meacher (EPO, Munich)
• Malika Goonasekera
• Harry Saunders-Singer (Channel 4)
• Ferruccio Renzoni

• Theory
• Tania Monteiro
• Thibaut Jonckheere (Marseille)
• Charles Creffield
• Nic Hutchings (ABN Amro)
• Matt Isherwood (FRM)
• Gwangok Hur
• Mischa Stocklin

• Funding
• EPSRC (UK)

44
Laser Cooling Group (experiments)
David Meacher
Phil Jones
Harry Singer
Malika Goonasekera
Quantum Chaos Group (theory)
Matt Isherwood
Tania Monteiro
Nic Hutchings
Thibaut Jonckheere
Gwangok Hur
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Further work

• Effects of decoherence destroys dynamical
  localisation
• Momentum filtering

46
Results Time Evolution

• Current grows linearly with time and saturates

47
Results Time Evolution

• Ratchet time, tRat ? 1/Db2

48
Results Pulse Shape
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Results Mixed Phase Space

• Mixed phase space with 2p periodicity
• Start on islands no diffusion