Tomography of a Heralded N00N State with Losses

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Tomography of a Heralded N00N State with Losses

• Brian J. Smith1,2, N. Thomas-Peter2, and I. A.
  Walmsley1
• 1Clarendon Laboratory, University of Oxford,
  Parks Road, Oxford OX1 3PU, UK
• 2Centre for Quantum Technologies, National
  University of Singapore, 117543 Singapore

IQEC IWF2 Wednesday, 3 June 2009
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Why N00N? Precision measurements

• Fundamental interest Measurements are how we
  gain knowledge about the world
• Better precision can re-enforce or reject a
  scientific theory
• New measurement techniques often lead to new and
  unexpected discoveries
• Practical interest
• Precise measurements are conjugate to precision
  control of systems Think electron microscope,
  femto-spectroscopy, etc.
• High precision allows for better engineered
  machinery and equipment, which often perform
  better than their predecessors

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Why N00N? Precision measurements

• Fundamental interest Measurements are how we
  gain knowledge about the world
• Better precision can re-enforce or reject a
  scientific theory
• New measurement techniques often lead to new and
  unexpected discoveries
• Practical interest
• Precise measurements are conjugate to precision
  control of systems Think electron microscope,
  femto-spectroscopy, etc.
• High precision allows for better engineered
  machinery and equipment, which often perform
  better than their predecessors

It is well know that quantum states can increase
measurement precision (N00N-states for example).
V. Giovannetti, S. Lloyd, and L. Maccone, Science
306, 1330-1336 (2004).
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A simple example phase measurement
Quantum optical interferometry
estimated phase
phase
Input light
Detection
uncertainty
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A simple example phase measurement
Quantum optical interferometry
estimated phase
phase
Input light
Detection
uncertainty
Quantum mechanics allows for a better phase
estimate than classical light.
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A simple example phase measurement
Quantum optical interferometry
estimated phase
phase
Input light
Detection
uncertainty
Quantum mechanics allows for a better phase
estimate than classical light.
For a N00N-state input
We get Heisenberg limited phase uncertainty
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A simple example phase measurement
Quantum optical interferometry
estimated phase
phase
Input light
Detection
uncertainty
Quantum mechanics allows for a better phase
estimate than classical light.
For a N00N-state input
We get Heisenberg limited phase uncertainty
This only works in theory - when there are losses
or inefficiencies present (i.e. in a real
experiment), there is a crucial balance.
N00N states are no longer optimal with loss or
non-unit preparation efficiency
U. Dorner, et. al. Phys. Rev. Lett. 102, 040403
(2009).
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Heralding and input state tomography

• Until recently, post-selection on getting the
  N00N state through the interferometer has been
  used.
• Does not properly count all resources used
• Heralding of desired input state is thus
  necessary
• State tomography of the heralded state is
  necessary to assess its utility in precision
  measurements
• For a N00N state this implies all photon numbers
  less than and equal to N.

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Heralded two-photon N00N state
Herald two photons from two individual SPDC
sources
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Heralded two-photon N00N state
Herald two photons from two individual SPDC
sources
Click!
Click!
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Heralded two-photon N00N state
Herald two photons from two individual SPDC
sources
Interfere heralded photons on a beam splitter,
utilizing the HOM effect to produce a two-photon
N00N state
Click!
a
b
Click!
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Heralded two-photon N00N state
Herald two photons from two individual SPDC
sources
Interfere heralded photons on a beam splitter,
utilizing the HOM effect to produce a two-photon
N00N state
Click!
a
b
Click!
Fiber coupling, and other losses will inhibit
heralding efficiency
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Heralded two-photon N00N state
Click!
Click!
Input state
Polarization interferometer
The photons have to be in pure states, i.e.
single mode wave packets, for high visibility
interference.
Two-photon polarization NOON state
2
2
1
1
Note No entanglement to start
Mosley et al, PRL 100, 133601 (2008).
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Heralded two-photon N00N state
Click!
Click!
Input state
Polarization interferometer
The photons have to be in pure states, i.e.
single mode wave packets, for high visibility
interference.
Two-photon polarization NOON state
2
2
1
1
Note No entanglement to start
Mosley et al, PRL 100, 133601 (2008).
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Heralded t00t state

• Real systems contain loss so the state must be
  represented by a density matrix (not a pure
  state).
• Loss is an incoherent process, therefore the
  density matrix can be written in block diagonal
  form (no coherences between different
  photon-number manifolds)

Describes the state of 0, 1, and 2 excitations
(photons) in two modes.
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State Tomography
E

• Replace polarization interferometer with
  polarization tomography detection
• Measure click patterns at 8 wave-plate settings,
  each setting has 5 POVM elements corresponding to
  0, 1 and 2 click events.
• This generalizes the work of Adamson et al

R. B. A. Adamson et al. Phys. Rev. Lett. 98,
043601 (2007)
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Reconstructed state

• State is reconstructed using Maximum Likelihood
  technique

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Reconstructed state

• State is reconstructed using Maximum Likelihood
  technique

• Clearly shows largest contribution is vacuum
  component with 76, only 1.8 comes from two
  photons.
• Post selection on the two photon subspace gives
  74 fidelity with the N00N state (nearly the same
  as four-fould fringe visibility).
• Fidelity of whole state with the ideal state is
  9.9.

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How well can this state perform?

• Calculate the Cramer-Rao bound to give the
  ultimate precision achievable with this state
• In order to compare with a coherent beam, we must
  assume indistinguishability to approximate a two
  mode interferometer, which introduces 10
  error.
• For same average photon number (
  ) the standard interferometric limit (for a
  coherent state input) gives

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Conclusion and Outlook

• Demonstrated full characterization of a heralded
  two-photon polarization N00N state taking into
  account the effect of losses.
• Results highlight the need to completely
  characterize the input state in order to assess
  precision improvements.
• Calculated the Cramer-Rao bound of the heralded
  state and compared to classical scenario.
• Future directions
• Examine schemes to increase heralding efficiency.
• Develop methods to prepare non-N00N states that
  are optimal in the presence of losses.

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Thanks!
Nick Thomas-Peter
Uwe Dorner
Ian Walmsley
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References

• V. Giovannetti, S. Lloyd, and L. Maccone,
  Quantum-enhanced measurements beating the
  standard quantum limit, Science 306, 1330-1336
  (2004).
• U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith,
  J. S. Lundeen, W. Wasilewski, K. Banaszek, and I.
  A. Walmsley, Optimal Quantum Phase Estimation,
  Phys. Rev. Lett. 102, 040403 (2009).
• R. B. A. Adamson, L. K. Shalm, M. W. Mitchell,
  and A. M. Steinberg, Multiparticle State
  Tomography Hidden Differences, Phys. Rev. Lett.
  98, 043601 (2007).

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Cramer-Rao Bound

• Defined in terms of the Fisher information as
• Maximize the Fisher information over all POVM
  sets.