Optical Implementations of QIP
Kevin Resch
IQC, Department of Physics
University of Waterloo
Quantum optics and Quantum Info. group
Entanglement
sources
Tomography
Quantum
computing
Optical Imaging
Tests of nonlocality
Interferometry
Goal of the talk
• To understand from basic principles how a
quantum information protocol works in
theory and in practice using optics
• I chose quantum teleportation where we
can understand the discrete (polarization)
and continuous variable versions of this
protocol
Quantum teleportation
• A process for transmitting quantum
information using a classical channel and
shared entanglement
Figure credit: Bouwmeester et al. Nature 390, 575 (1997).
Outline
• Introduction to quantum optics
– photons, encodings, entanglement
• QIP with polarization
– waveplates, CNOT, teleportation
• QIP with continuous variables
– Wigner function, measurements, teleportation
Photons
Field quantization
• A procedure for finding a quantum
description of light
• Starting point is Maxwell’s Equations in
vacuum
• Introducing the potentials and gauge
Field quantization
• From these derive wave equation for the
vector potential
• Spatial mode expansion (exact form
depends on boundary conditions)
Plane wave solutions
Periodic BC, cubic volume
Field quantization
• Also from classical physics, the energy
stored in an EM field
• Energy for a single mode
• Rewriting complex A in terms of real quant
• Gets us onto familiar territory
Classical
Harmonic
Oscillator
(mass = 1)
Field quantization
Field quantization
• Promote the classical parameters to
operators
• Which defines field operators
Field quantization
• And find the energy for each mode
• Which simplifies to
Field quantization
• Harmonic oscillator
“number” operator
q
The excitations of the EM modes are
“photons” – particles of light
Experimental evidence for photons
• Particles can only be detected in one
place
Ca Atomic Cascade
Grangier, Roger, Aspect Europhysics Lett 1, 173 (1986)
Properties of photons
• A single photon has just three properties:
– Colour/energy,
– Polarization,
– Direction/momentum,
• Its quantum state can be described as a
superposition of these properties
Single photon QI encodings
• Spatial modes
• Polarization
+i
|H>
• Time-bin
=
|V>
• Freq. encoding
QIP with optics
• Pros:
– Low decoherence*
– High speed
– Flexible encodings
Ideal for quantum
communication
• Cons:
– Negligible photon-photon interactions
– Loss
– Hard to keep in one place
– Some encodings unsuitable for some
situations, ex., polarization/modes in fibre
*can be susceptible to coupling internal DOF
But an optical mode is more complicated…
• Photons are bosons, so we can have
many per mode
• Important multi-photon states of a single
mode:
– Fock or number state
– Coherent state
– Squeezed state
– Thermal state
• (Things can get very complicated with a
large number of modes and all the DOF)
“Mode” observables: Quadratures
• We can write quadrature operators
analogous to x and p (but do not
correspond to pos/mom of the photon!)
• Since
, there must be an
uncertainty relation
Useful operator identities
• Baker-Campbell-Hausdorff lemma
• Glauber’s identity
valid when [A,[A,B]]= [A,[A,B]]=0.
Phase shift operators
• Phase shift operator (exp free-field)
• Using BCH
• Or
• Free-field evolution converts one
quadrature into the other in the form of a
rotation
Quadratures
• These observables correspond to
components of the electric field
• There is an uncertainty relation between
the E field ‘now’ and the E field a quarter
cycle ‘later’
Coherent states
• Defined as eigenstates of lowering
operator
a is not Hermitian so α can be complex
• Uncertainties in mode variables:
• Min uncertainty, equal between q and p
Displacement operator
• Coherent states can be generated using
the displacement operator:
• This can be seen by rewriting the operator
using Glauber’s identity and comparing
Displacement operator
• Useful identities and properties:
Coherent states in quantum optics
• Coherent states play an important role as
a basis in quantum optics
• But coherent states with different
amplitudes are orthogonal
• And the basis is “overcomplete”
(projectors do not sum to identity)
Entanglement
The characteristic trait of QM
E. Schrödinger
Math. Proc. Camb. Philos. Soc. 31, 555 (1935).
Definition of entanglement
• Any state that can be written,
is said to be separable, otherwise it is
entangled
ÃA B
• Pure states:
Ã(x 1 ; x 2 )
=
=
ÃA ÃB
Ã(x 1 )Ã(x 2 )
are separable, otherwise entangled
Superposition and entanglement
Superposition
Entanglement
The characteristic trait of QM
Quantum Computing
Enhanced Sensors
Phase transitions
http://www.ligo.caltech.edu
http://www.eng.yale.edu/rslab/
Entanglement
Quantum Communication
http://www.quantum-munich.de
Quantum relativistic
effects
Foundations of QM
Figure credit: Rupert Ursin
Jennewein et al. PRL 84, 4729 (2000)
S. Hawking Illustrated Brief History of Time
Nonlinear optics
• Direct photon-photon interactions too
weak
• Instead atoms can mediate interactions
between photons – Nonlinear Optics
• Ex. Second-order nonlinearity
Nonlinear coefficient
Creates pairs of
photons
Destroys pairs of
photons
Nonlinear optics
• Instead of oscillating only at the frequency
of the driving field, the charge can oscillate
at new frequencies
Example: Second
Harmonic Generation
ω
2ω
Χ(2) material
(such as BBO or KTP)
Second-harmonic generation
Entangled photons
• Reverse of SHG
Parametric Down-conversion
“blue” photon
two “red”
photons
c(2)
wpump = ws + wi
Phase matching:
kpump = ks + ki
‘Conservation laws’ constrain the pair without
constraining the individual  entanglement
Also: QD, at. casc
Down-conversion movie
KTP – nonlinear crystal
www.quantum.at
PPKTP source
PPKTP source
Multiphoton sources: Pulsed SPDC
• Down-conversion can sometimes emit two pairs.
• If a short pulse is used for an entangled photon source,
the pair are properly described by a 4-photon state
H = gay ay + gay ay
H1 V2
H 2 j0i
V1 H2
+ h:c:
!
(ay ay + ay ay ) 2 j0i
=
j2H 1 ; 2V2 i + jH 1 ; V1 ; H 2 ; V2 i + j2V1 ; 2H 2 i
H1 V2
V1 H2
GHZ Correlations
• Measured 4-photon coincidences to post-select GHZ
state
j2H 1 ; 2V2 i + jH 1 ; V1 ; H 2 ; V2 i + j2V1 ; 2H 2 i
V
H
• Needs at least 1H and 1V
in mode 1
H
V
Bouwmeester PRL 82, 1345 (1999)
Lavoie NJP 11, 073501 (2009)
H
H
V
V
H a H bVc
H
V
V
Va VbH c
Three-photon GHZ states
• ~4 four-fold coincidence counts per minute (3fold coincidence + trigger)
• Fidelity with target GHZ 84% from tomography
2nd method: Cascaded down-conversion
e
10-11
~1 in a billion years
~1 per day
10-9
~1 in a hundred
thousand years
~3 per hour
10-6
~2 per month
~1 per second
Bulk crystal (BBO)
PPKTP
Waveguide
PPLN
*assuming 106 s-1 primary photons,
no loss, perfect detectors
Experimental cascaded down-conversion
See also Shalm Nature Physics 9, 19 (2013);
Hamel arxiv: 1404.7131
4.7 ± 0.6 counts/hr
Two-mode squeezed vacuum
• Two mode squeezing operator
• Creates or destroys photons in pairs
• Properties
• Warning: can’t use Glauber’s theorem
Two-mode squeezed vacuum
• The interesting properties show up in the
correlations between quadrature obs.
Two-mode squeezed vacuum
• The commutator,
• And so we have the same uncertainty
relation between these joint observables
as the quadratures themselves:
Two-mode squeezed vacuum
• We can calculate the uncertainty in these
observables for the TMSV
• Recall
• To calculate this requires several
applications of the squeeze operator
identities, ex.,
Two-mode squeezed vacuum
• After some algebra
• Choosing
• We can “squeeze” the uncertainty in one
observable at the expense of the other
Einstein Podolsky Rosen correlations
• If we consider a different pair of joint
quadrature observables, ex.
• These operators commute (thus the
uncertainty relation is trivial) and for the
TMSV
Einstein Podolsky Rosen Correlations
• For infinite squeezing, the state is an
eigenstate of both
• Highly entangled state central to:
Two-mode squeezed vacuum
• This state is the most entangled state for a
given amount of energy (its subsystems
are thermal states, which have the highest
entropy for a fixed energy)
• As such it plays the role of the Bell states
in CV protocols