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Squeezing
eigenmodes
Entanglement-enhanced
in
parametric downcommunication
over
conversion
a correlated-noise channel
Konrad Banaszek
Nicolaus Copernicus University
Toruń, Poland
Andrzej Dragan
Wojciech Wasilewski
Czesław Radzewicz
Warsaw University
Jonathan
Ball
Alex Lvovsky
University
Oxford
University
ofofCalgary
National Laboratory for Atomic, Molecular, and Optical Physics, Toruń, Poland
Mutual information:
Channel capacity:
Receiver
Sender
All that jazz
Depolarization in an optical fibre
Photon in a
polarization
state
Independently of
the averaged
output state has the form:
Random polarization
transformation
V
H
1/2
1
1/2
H
1/2
2
1/2
V
Capacity of coding in
the polarization state of
a single photon:
Information coding
Sender:
V
H
V
Probabilities of
measurement outcomes:
2/3
1/3
H
H&H, V&V
1/3
2/3
H&V, V&H
Capacity per
photon pair:
Collective detection
Probabilities of
measurement outcomes:
1
2&0, 0&2
1/2
1/2
1&1
Capacity:
Entangled states are useful!
Probabilities of
measurement outcomes:
1
1
2&0, 0&2
1&1
Capacity:
Proof-of-principle experiment
Separable ensemble:
1
1/2
1/2
2&0, 0&2
1&1
Entangled ensemble:
1
1
2&0, 0&2
1&1
These are optimal ensembles for
separable and entangled inputs
(assuming collective detection),
which follows from optimizing
the Holevo bound.
J. Ball, A. Dragan, and K.Banaszek,
Phys. Rev. A 69, 042324 (2004)
Source of polarization-entangled pairs
P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum,
and P. H. Eberhard, Phys. Rev. A 60, R773 (1999)
For a suitable polarization of the pump
pulses, the generated two-photon state has
the form:
With a half-wave plate in one arm it can be transformed into:
or
Experimental setup
K. Banaszek, A. Dragan, W. Wasilewski, and C. Radzewicz, Phys. Rev. Lett. 92, 257901 (2004)
Triplet events:
D1 & D2
D3 & D4
Singlet events:
D1 & D3 D2 & D3
D2 & D3 D2 & D4
Experimental results
Dealing with collective depolarization
1) Phase encoding in time bins:
fixed input polarization,
polarization-independent
receiver.
J. Brendel, N. Gisin, W. Tittel, and H. Zbinden,
Phys. Rev. Lett. 82, 2594 (1999).
2) Decoherence-free subspaces
for a train of single photons.
J.-C. Boileau, D. Gottesman, R. Laamme,
D. Poulin, and R. W. Spekkens, Phys. Rev.
Lett. 92, 017901 (2004).
General scenario
Physical system:
• arbitrarily many photons
• N time bins that encompass
two orthogonal polarizations
• How many distinguishable states can we send via the channel?
• What is the biggest decoherence-free subspace?
Mathematical model
General transformation:
where:
– the entire quantum state of light across N time bins
– element of U(2) describing the transformation
of the polarization modes in a single time bin.
– unitary representation of W in a single time bin
We will decompose
and
with
Schwinger representation
Ordering Fock states in a single time bin according to the combined number
of photons l:
...
Representation of W:
...
Here
Consequently
is (2j+1)-dimensional representation of SU(2).
has the explicit decomposition in the form:
Decomposition
Decomposition into irreducible representations:
Integration over a removes coherence between subspaces with different
total photon number L. Also, no coherence is left between subspaces with
different j.
tells us:
• how many orthogonal states can be sent in the subspace j
• dimensionality of the decoherence-free subsystem
Recursion formula for
:
J. L. Ball and K. Banaszek,
quant-ph/0410077;
Open Syst. Inf. Dyn. 12, 121 (2005)
Biggest decoherence-free subsystems have usually hybrid character!
Questions
• Relationship to quantum reference frames for spin systems
[S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Phys. Rev. Lett. 91, 027901 (2003)]
• Partial correlations?
• Linear optical implementations?
• How much entanglement is needed for implementing decoherence-free
subsystems?
• Shared phase reference?
• Self-referencing schemes?
[Z. D. Walton, A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich,
Phys. Rev. Lett. 91, 087901 (2003)]
• Other decoherence mechanisms, e.g. polarization mode dispersion?
Multimode squeezing
SHG
Single-mode model:
PDC
Brutal reality (still simplified):
[See for example: M. Matuszewski, W. Wasilewski, M. Trippenbach, and Y. B. Band,
Opt. Comm. 221, 337 (2003)]
–
Perturbative regime
The wave function up to the two-photon term:
W. P. Grice and I. A. Walmsley, Phys. Rev. A 56, 1627 (1997);
T. E. Keller and M. H. Rubin, Phys. Rev A 56, 1534 (1997)
Schmidt decomposition for a symmetric
two-photon wave function:
C. K. Law, I. A. Walmsley, and J. H. Eberly,
Phys. Rev. Lett. 84, 5304 (2000)
We can now define eigenmodes
The spectral amplitudes
which yields:
characterize pure squeezing modes
Decomposition for arbitrary pump
As the commutation relations for the output field operators must be preserved,
the two integral kernels can be decomposed using the Bloch-Messiah theorem:
S. L. Braunstein, quant-ph/9904002;
see also R. S. Bennink and R. W. Boyd,
Phys. Rev. A 66, 053815 (2002)

Squeezing modes
The Bloch-Messiah theorem allows us to introduce eigenmodes for input and
output fields:
which evolve according to
•
•
describe modes that are described by pure squeezed states
tell us what modes need to be seeded to retain purity
Some properties:
• For low pump powers, usually a large number of modes becomes squeezed
with similar squeezing parameters
• Any superposition of these modes (with right phases!) will exhibit squeezing
• The shape of the modes changes with the increasing pump intensity!
This and much more in a poster by Wojtek Wasilewski
The End
Theory
Everything that emerges are Werner states
One-dimensional optimization problem for
the Holevo bound
What about phase encoding?
Recursion formula
Decompostion of
the corresponding
su(2) algebra:
If we subtract
one time bin:
Direct sum
The product of two angular
momentum algebras has the
standard decomposition as:
Therefore the algebra for L
photons in N time bins can
be written as a triple direct
sum:
Decoherence-free subsystems
Rearranging the summation
order finally yields:
Underlined entries with
correspond to
pure phase encoding (with
all the input pulses having
identical polarizations)
– in most cases we can do
better than that!