Diapositive 1

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Entanglement of Microcavity-Polaritons
Fabrice P. LAUSSY University of Sheffield, LDSD
group. F.P.Laussy_at_sheffield.ac.uk

• Outline
• I Quantum Information
• The qubit
• EPR pairs
• Bells inequalities
• Applications (cryptography computation)
• II Microcavities
• The basics polaritons
• III Quantum Information with cQED
• Work in progress and state of the art
• Our motivations will serve as our
  conclusions

Applied Mathematics seminar of Southampton
University. 25 October 2005
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The Qubit (Quantum Bit)
or how to encode information with a quantum
system...

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The Postulates of Quantum Mechanics illustrated
on a qubit
I The qubit is described by a two-dimensional
vector in a complex Hilbert space.
We note the canonical basis, in Dirac notation,
as
The most general qubit is a superposition
with
and to later link with the statistical
interpretation
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Physical realization of a qubit with spin ½
particle
II An observable on a qubit is a 2x2 hermitean
matrix. The result of the associated observation
is an eigeinvalue of this matrix with probability
the square modulus of the state projection on the
associated eigenvector, on which the system
collapses upon observation.
The spin
where
measured on axis yields observable
For all
there are such that
with probability
collapses onto
with probability
If measured on x axis
collapses on
or
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Physical realization of a qubit with polarization
Qubit 0 is now associated to horizontal
polarization
and qubit 1 with the orthogonal state, vertical
polarization
The linear superposition of which yields other
polarizations which form other convenient bases,
e.g., is the canonical basis,
and
or the circularly polarized states
the orthogonal basis of diagonal polarizations
(which correspond to spin-up and spin-down states
of the photon)
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An application already, in Quantum Cryptography
BB84 protocol
C. H. Bennett and G. Brassard. Quantum
cryptography Public key distribution and coin
tossing. Proc. IEEE, page 175, 1984.
An unbreakable way to encrypt data is to use a
single-time cypher pad (a key) which has the
length of the message. The practical problem is
the distribution of this key.
It can be done if you can exchange single photons
with the recipient.
Secret key
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Einstein-Podolsky-Rosen
This wavefunction does not tell it all!

• Reality values of observables exist or are
  defined before they are measured
• Completeness these values have variable (even
  if unknown or hidden) to describe them
• Locality There is no action at distance

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Hidden variables in a generalized EPRB
We call
the product of polarization measurements
for QM
which knew beforehand the
Let us assume there are hidden variables
outcome of the measurements at each branch, in
addition to the information contained in the
usual wavefunction.
If these variables are local
and
In case of full correlations
Whatever theory models
with a distribution
A very deep result lie behind all this
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By substitution,
Introducing a third angle
as
and also
By triangular inequalities
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Bells Inequalities
A complete, local theory is not consistent with
Quantum Mechanics
gt
0.7
0.3
A very deep result indeed
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CHSH inequalities
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The single-channel version of the experiment CH
inequalities, Phys. Rev. D, 10, 526 (1974)
The inequality reads
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Better Cryptography
There is perfect anticorrelations for same
angles which defines the key, while checking CHSH
ensures there was no eavesdropping.
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Quantum Computation Dense Coding
Entanglement allows to send 2 classical bits x
and y through a single qubit.
EPR Source

• To send
• Bits 00, do nothing
• Bits 01, apply X
• Bits 11, apply Y
• Bits 10, apply Z

H
With knowledge of first bit
With knowledge of second bit
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Microcavities
Schematic view (Khitrova et al.)
Scanning Electron Microscopy of a planar
microcavity (Yamamoto et al.)
Pillar cavities with Quantum Dots (Sheffield)
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Exciton dispersion
Photon dispersion
Dispersion of polaritons in a typical pump/probe
experiment (Savvidis et al.)
Light-Matter coupling in a semiconductor
At low densities of excitons,
where
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Quantum Information with MCs Pioneering work by
Ciuti (ENS Paris)
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Ciuti's proposal
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Signal is not entangled with idler(s)!
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Entanglement of polaritons

• The field is emerging and is a very specific
  system, without analogies
• In conventional systems, e.g., nonlinear
  crystals, EPR pairs are generated directly their
  coupling to the environment is a side effect
  (although very important and with fundamental
  physics).
• But with microcavities, polaritons generate the
  entanglement, and it is manifested through
  photons outside the cavity. One needs to study
  the transfert of entanglement between intra and
  external fields.
• In atomic cavities, the cavity is a medium of
  interaction and factors out of the system.
• But with microcavities, because of strong
  coupling, the cavity mode is part of the
  entanglement and plays a key role in the
  properties of the system. One has to design
  working geometries and configurations, and take
  into account the dynamics, which is a key
  ingredient of the polariton physics.