Light interference and localization of strongly driven multiparticle systems

1
Light interference and localization of strongly
driven multi-particle systems

• Mihai Macovei, JÖrg Evers and Christoph H. Keitel
• Max-Planck Institute for Nuclear Physics,
  Heidelberg, Germany

CEWQO 2007, Palermo, Italy
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Outline

• Motivation
• Spatial distribution of spontaneous emission
• Weak and strong-field spatial interference
• Two photon spatial coherences in strongly
  driven regular multi-particle structures.
• Cauchy-Schwarz inequalities
• Localization of atomic ensembles via
  super-fluorescence
• Conclusions

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Motivation

• Fundamental questions

• recovering the first-order spatial
  interference for stronger laser fields,
• improving the spatial light-resolution,
• investigating the spatial behaviors of quantum
  properties of light
• to localize atomic ensembles with a
  sub-wavelength accuracy

• multi-particle systems,
• single- and two-photon interference,
• sub-wavelength localization.

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Spatial distribution of spontaneous emission

• The spontaneous emission of non-interacting
  excited particles in vacuum is isotropic.
• The collective spontaneous emission of an
  excited pencil-shaped two-state system is
  distributed along the samples axis.

M. Gross and S. Haroche, Phys. Rep. 93(5), 301
(1982).
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Spatial quantum interference
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Spatial interference
Double slit with two trapped ions
applications e.g. in lithography?
U. Eichmann et al., Phys. Rev. Lett. 70, 2359
(1993)
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Two-atom collective states
Introduce new state basis
dark-center interference
bright-center interference
Two transition amplitudes add with different
phases
-

bright center
dark center
via symmetric state
via anti-symmetric state
C. Skornia et al., Phys. Rev. A 64, 063801 (2001)
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Weak- field spatial interference. The model
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Analytical methods. Approximations

• Master equation approach
• Dipole and rotating - wave approximations
• Mean-field and Born - Markov approximations
• Dressed - states formalism
• Secular approximation.

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Weak- field spatial interference. Visibility
For weak fields,
the visibility of the interference pattern
is
if
Note that V?0 when ? gtgt ?.
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The strong field case
Spectral decomposition
under strong driving splitting in different
spectral bands
Mollow spectrum
define observables for each spectral band
separately
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Recovering strong-field interference
Ansatz

• modify mode density at dressed-state transition
  frequencies e.g. via cavity
• this changes spontaneous emission and
  redistributes populations
• look at single spectral band

change mode density via cavity
right sideband suppressed
left sideband suppressed
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Strong - field spatial interference. Analytical
treatment
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Strong - field spatial interference. Spectral
-line intensities
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Strong-field spatial interference in a tailored
electromagnetic bath
One can define separately the visibilities V
(Imax-Imin) /(Imax Imin) for each of the
central, left and right spectral lines,
respectively.
Central band visibility VCB as a function of ?
?(?)/?(?-).
no interference in plain vacuum, ? 1
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Strong-field spatial interference. Regular atomic
structures
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The strong-field spatial interference pattern
...
...
Central-band intensity ICB/N2 as function of a1.
Here kLrab20p and VCB 0.9. Blue line N8, red
curve N2.
M. Macovei, J. Evers, G.-x. Li, C. H. Keitel,
Phys. Rev. Lett. 98, 043602 (2007).
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Two-photon spatial coherences in strongly driven
multi-particle structures
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Two-photon spatial coherences
The coherence properties of an electromagnetic
field, at space-point R, can be evaluated with
the help of the second-order coherence functions
where
are the creation (annihilation) operator for
modes i, j .
can be interpreted as a measure for the
probability for detecting one photon emitted in
mode i and another photon emitted in mode j with
delay t.
The quantity
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Second-order correlation functions. Properties
Equal times

• g2( t 0 ) 1 poissonian photon statistics
  (e.g. coherent state)
• g2( t 0 ) lt 1 sub-poissonian photon
  statistics (e.g. Fock state)
• g2( t 0 ) gt 1 super-poissonian photon
  statistics (e.g. thermal light)

Different times

• g2( t gt 0 ) g2( t 0 ) bunching (e.g.
  thermal light)
• g2( t gt 0 ) gt g2( t 0 ) anti-bunching (e.g.
  single atom fluorescence)

Non-classical

• g2( t gt 0 ) gt g2( t 0 )
• g2( t 0 ) lt 1

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Cauchy-Schwarz inequalities
The Cauchy-Schwarz inequalities are violated if
i.e., if the cross-correlation between photons
emitted into two different modes are larger than
the correlation between photons emitted into
individual modes.
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The Cauchy-Schwarz parameters
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Two-photon spatial coherences
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Second-order spatial interference resolution
The central-band second-order correlation
function. The red line depicts the strong-field
limit (Vc0) while the blue curve describes the
weak-field case with N2,
and
M. Macovei, J. Evers, G.-x. Li, C. H. Keitel,
Phys. Rev. Lett. 98, 043602 (2007).
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Summary

• The strong-field first order interference pattern
  can be recovered by tailoring the electromagnetic
  reservoir surrounding the atomic structure
• Second order correlation functions do exhibit
  interference effects even in the standard vacuum.
• The cross-correlations between photons emitted
  in the spectral sidebands violate Cauchy-Schwartz
  inequalities and their emission ordering cannot
  be predicted.

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Localization of atomic ensembles via
super-fluorescence
M. Macovei, J. Evers, C. H. Keitel, M. S.
Zubairy, Phys. Rev. A 75, 033801 (2007)
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Multi-particle sub-wavelength localisation.
Approach
sample of atoms
Localization of ensembles

• atoms are positioned in a standing wave laser
  field
• sample has linear dimensions smaller than
  wavelength
• atoms dipole-dipole interact

Dicke model

• collective operators, e.g.

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The Hamiltonian
Free energies
Atom-laser coupling
Atom-vacuum coupling
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Master equation
-
Coherent part

Spontaneous emission Dipole-dipole coupling
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Steady-state solution
Ansatz for steady state
Bosonic ladder operators
Inserting in Master equation yields analytic
solution
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Steady-state solution
Ansatz for steady state
Bosonic ladder operators
Inserting in Master equation yields analytic
solution
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Fluorescence intensity profile
O / (N?) 100 O / (N?) 50 O / (N?) 25
Od -10 ? ?/(N?) 0.5
width 0.01 ?
node
Intensity dip width can becontrolled
Intensity
Position in standing wave
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Fluorescence intensity profile
2 atoms 4 atoms 8 atoms
O 100 ? Od -5 ? ? 10 ?
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Interpretation
Two atom case

• For small distances, anti-symmetricstate
  decouples
• Minimum dip width aroundposition of symmetric
  state

detuning
position
Many atom case

• Minimum width around positionwhere symmetric
  states are
• No structure at positions ofantisymmetric states
• But positions not exactlydefined because of
  many(anti-) symmetric states

detuning
position
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Scanning-dip spectroscopy

• Change phase of standingwave field to move nodes
• Constantly monitorscattered light intensity
• Identify position of atomsvia the absence of
  lightat the nodes? localized particles
  unperturbed
• accuracy determined bydip width

Intensity dip should be as narrow as possible
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Single-pass localization
Collection flying through cavity

• Only short interaction
• Try to obtain as much information as possible

Measurement scheme

• The total amount of scatteredlight provides
  localization information
• Measured intensity fixeshorizontal cut and
  thusposition range

Large slope of intensity profile desirable over
whole wavelength
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Summary
Localization of atomic ensembles

• ensemble confined to region smaller than
  wavelength
• evaluate fluorescence in standing wave field?
  collective effects

Scanning-dip localization

• relies on the absence of fluorescence? localized
  particle essentially unperturbed
• exploits collectivity and strongfields to tailor
  a narrow dipin fluorescence profile

Single-pass localization

• gain as much position informationas possible in
  single pass
• requires a wide dip in fluorescence profile in
  order to work at all positions