Localising Relative Parameters QCMC 04 Hugo Cable, Terry Rudolph, Peter L Knight Department of Quant

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Localising Relative ParametersQCMC 04Hugo
Cable, Terry Rudolph, Peter L KnightDepartment
of Quantum Optics and Laser Science
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INTRODUCTORY REMARKS

• MEASUREMENT INDUCED LOCALISATION OF RELATIVE
  PARAMETERS
• e.g. Optical fields (relative optical phase),
  BECs (relative order parameter phase), massive
  particles scattering light (relative position).
• Applications
• - relational decoherence natural pointer
  states simple interaction with the environment,
• - symmetry breaking interference without
  assuming prior phase.
• What is new
• - analogous calculations of relative
  localisation for different systems,
• - mixed states,
• - simpler cases treated analytically.

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Relative Phase Optical Fock States

• Basis of Glauber coherent states
• minimum uncertainty, robust,
• maximally classical

• Interference of photon number states
• Initial state NiNi,
• Cavities leak ?,
• 5050 beam splitter,
• After first click

Optical coherence A convenient fiction, Klaus
Mølmer, PRA 55, 3195 (1997).
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Relative Phase Optical Fock States

• Entangled coherent state representation,

after l clicks at left detector and r at right
detector
where,
??-?
peaked at
Photon-number superselection and the entangled
coherent-state representation, Sanders,
Bartlett, Rudolph, Knight PRA 68, 042329 (2003).
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Relative Phase Optical Fock States
for rl0,1,2,5,15 counts
Rapid localisation of relative phase BUT fragile
relational Schrödinger Cat state!
A small asymmetry e.g. small frequency difference
between the two modes can prevent multiple peaks.
??-?
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Quantifying Localisation of Relative Optical Phase

• VISIBILITY
• Mode 2 undergoes variable phase
• shift of ?.
• Modes mixed at 5050 beamsplitter.
• I(?) denotes intensity at one port.

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Poissonian and Thermal Optical Fields

• Poissonian States e.g. laser cavity

Initial states of both modes Poissonian with same
mean photon number . Localisation of relative
phase as for initial Fock states NiNi when N
large.
However state remains separable at every stage in
process
where
is a localising function.
Analytic expression for the visibility

• Thermal states

Analytic expression for the visibility
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Expected Visibilities for Poissonian and Thermal
Optical Fields
initial states Poissonian o initial states
thermal
Expected number of detected photons

• Localisation of relative phase faster for two
  Poissonian states than
• two thermal states.

• V!0.5 artefact of multiple peaks.

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Interference of Two Independent Bose-Einstein
Condensates

• Two independent BECs made to overlap express
  interference when some or all atoms are detected.
• Elementary approach assigns a macroscopic
  wavefunction (order parameter) to a BEC.
  Corresponds to a Glauber coherent state, a
  superposition of different atom numbers.
• Interference can also be predicted without
  assuming prior symmetry breaking, assuming
  instead initial atom number states e.g.
  Javanainen and Yoo PRL 76, 161 (1996) Castin and
  Dalibard PRA 55, 4330 (1997)
• Interference also predicted if a condensate, in
  equilibrium with uncondensed atoms, described by
  a Poissonian state.
• Two thermal clouds would also acquire relative
  phase though interference pattern would have less
  contrast!

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Localisation of Relative Distance
Extracts from Measurement-Induced
Relative-Position Localization Through
Entanglement A. V. Rau, J. A. Dunningham, K.
Burnett (SCIENCE VOL 301 22 AUGUST 2003)
(i) Rubber Cavity
(ii) Two free particles
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Spatial Relative Localisation in Analogy
Gaussian states k,a,di
Thermal particles with (classical) Maxwell
distribution of momentum
Two length scales determined by thermal
distribution and wavelength of incident light
Scattering operators lead to flat superpositions
over a relative momentum variable and therefore
well defined relative position
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Localisation of Relative Position of Two Mirrors
in a Rubber Cavity

• Scattering Kraus operators
• cmp optical operators

Initially uniform probability density for
relative displacements between 20? and
40?. d0.5?, 2?
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Localisation of Relative Position of Two Free
Particles

• Two particles initially delocalised in
  one-dimensional region.
• Incident photons either scattered or continue in
  the forward direction. Non-scattering events
  yield information about particles and therefore
  contribute to localisation process.
• In the case of scattering all angles are mixed
  over.
• Contrary to RDB see interference patterns rather
  than sharp localisation!

Initially uniform probability density for
relative displacements between -10? and
10?. d0.2?, 0.2?
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Conclusions

• Investigating in depth the localisation of
  relative parameters for a
• variety of physical systems and for realistic
  initial quantum states (both
• pure and mixed).
• Examining possibility that relational
  decoherence may overcome
• the pointer basis issues of the traditional
  models.