Nonlinear optics at the quantum level and quantum information in optical systems

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Nonlinear optics at the quantum level and
quantum information in optical systems
Aephraim Steinberg Dept. of Physics, University
of Toronto
2003 GRC on Nonlinear Optics Lasers
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Acknowledgments
U of T quantum optics laser cooling
group PDFs Morgan Mitchell Marcelo
Martinelli Optics Kevin
Resch(?Zeilinger) Jeff Lundeen Chris Ellenor
Masoud Mohseni (?Lidar) Reza Mir Rob
Adamson Karen Saucke (visiting from Munich)
Atom Traps Stefan Myrskog Jalani Fox Ana
Jofre Mirco Siercke Samansa Maneshi Salvatore
Maone (? real world) Some of our theory
friends Daniel Lidar, Janos Bergou, Mark
Hillery, John Sipe, Paul Brumer, Howard Wiseman
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OUTLINE
Something you already know
Introduction to quantum information with optics
Something you may have known... but may have
forgotten by now
All good talks are alike... every bad talk is
bad in its own way.
Making a strong effective interaction
between two photons
Something you most likely haven't heard before
Quantum state and process tomography for q. info.
Something you may not even buy
Weak measurements -- Hardy's Paradox et
cetera "How much can we know about a photon?"
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Intro to Quantum Info -- pros cons of optical
schemes...
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Quantum Information
What's so great about it?
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Quantum Information
What's so great about it?
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Quantum Computer Scientists
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What makes a quantum computer?
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What makes a computer quantum?
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Conventional Answers

1. Computers are made from Silicon, not photons.
2. Maybe trapped atoms/ions have some of the
   advantages of photons without the disadvantages.
3. Maybe SQUIDs or quantum dots or something else
   will prove the right technology instead.
4. Maybe using quantum measurement and postselection
   as an "effective interaction" will save the day
   for optics.
5. Maybe photons can be made to interact better
   after all

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Quantum Interference for effective single-photons
ingle-photon interactions...?
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Can we build a two-photon switch?
Photons don't interact(good for transmission bad
for computation) Nonlinear optics photon-photon
interactions generally exceedingly
weak. Potential solutions Better materials
(1010 times better?!) - Want l3 regime, but
also resonant nonlinearity? - Cf. talks by
Walmsley, Fejer, Gaeta,... Cavity QED (example
of l3 regime plus resonance) - Kimble, Haroche,
Walther, Rempe,... EIT, slow light, etc... -
Lukin, Fleischhauer, Harris, Scully,
Hau,... Measurement as nonlinearity
(KnillLaflammeMilburn) - KLM Franson,
White,... Other quantum interference
effects? - Exchange effects in quantum NLO
(Franson) ? - Interferometrically-enhanced SHG,
etc (us) ?
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The germ of the KLM idea
INPUT STATE
OUTPUT STATE
a0gt b1gt c2gt
a'0gt b'1gt c'2gt
TRIGGER (postselection)
ANCILLA
1gt
1gt
In particular with a similar but somewhat more
complicated setup, one can engineer a 0gt b
1gt c 2gt ??a 0gt b 1gt c 2gt
effectively a huge self-phase modulation (p per
photon). More surprisingly, one can efficiently
use this for scalable QC.
KLM Nature 409, 46, (2001) Cf. experiments by
Franson et al., White et al., ...
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The mad, mad idea of Jim Franson
J.D. Franson, Phys. Rev. Lett 78, 3852 (1997)
Nonlinear coefficients scale linearly with the
number of atoms. Could the different atoms'
effects be made to add coherently, providing an
N2 enhancement (where N might be 1013)?
Appears to violate local energy conservation...
but consists of perfectly reasonable Feynman
diagrams, with energy conserved in final
state. Controversy regarding some magic
cancellations.... Each of N(N-1)/2 pairs of
atoms should contribute. Franson proposes that
this can lead to immense nonlinearities. No
conclusive data.
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John Sipe's suggestion
Franson's proposal to harness photon-exchange
terms investigates the effect on the real index
of refraction (virtual intermediate state). Why
not first search for such effects on real
intermediate states (absorption)?
Conclusion exchange effects do matter
Probability of two-photon absorption may be
larger than product of single-photon abs.
prob's. Caveat the effect indeed goes as N2,
... but N is the photon number (2)
and not the atom number (1013) !
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Ugly data,but it works.
Resch et al. quant-ph/0306198
Roughly a 4 drop observed in 2-photon
transmission when the photons are delayed
relative to one another.
Complicated by other effects due to
straightforward frequency correlations between
photons (cf. Wong, Sergienko, Walmsley,...), as
well as correlations between spatial and spectral
mode.
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What was the setup?
Type-II SPDC birefringent delay 45o polarizer
produces delayed pairs. Use a reflective notch
filter as absorbing medium, and detect remaining
pairs.

• This is just a Hong-Ou-Mandel interferometer,
  with detection in a complementary mode.
• Although the filter is placed after the output,
  this is irrelevant for a linear system.
• Interpretations
• Our "suppressed" two-photon reflection is
  merely the ratio of two different interference
  patterns the modified spectrum broadens the
  pattern.
• Yet photons which reach the filter in pairs
  really do not behave independently. The
  HOM interference pattern is itself a
  manifestation of photon exchange effects.

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Entangled photon pairs(spontaneous parametric
down-conversion)
The time-reverse of second-harmonic
generation. A purely quantum process (cf.
parametric amplification) Each energy is
uncertain, yet their sum is precisely
defined. Each emission time is uncertain, yet
they are simultaneous.
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Another approach to 2-photon interactions...Ask
Is SPDC really the time-reverse of SHG?
(And if so, then why doesn't it exist in
classical em?)
The probability of 2 photons upconverting in a
typical nonlinear crystal is roughly 10-10 (as
is the probability of 1 photon spontaneously
down-converting).
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Quantum Interference
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Type-II down-conversion
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2-photon "Switch" experiment
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Suppression/Enhancementof Spontaneous
Down-Conversion
(57 visibility)
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Switchiness ("Nonlinearity")
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Photon-photon transmission switch
On average, less than one photon per pulse. One
photon present in a given pulse is sufficient to
switch off transmission. The photons upconvert
with near-unit eff. (Peak power approx.
mW/cm2). The blue pump serves as a catalyst,
enhancing the interaction by 1010.
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Controlled-phase switch
Resch et al, Phys. Rev. Lett. 89, 037914 (2002)
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Fringe data with and w/o postsel.
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...but it actually is true
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So why don't we "rule the world"?
N.B. This switch relies on interference. Input
state must have specific phase. Single photons
don't have well-defined phase. The switch does
not work on Fock states. The phase shifts if and
only if a control photon is present-- so long as
we make sure not to know in advance whether
or not it is present. Another example
of postselected logic. Nonetheless Have shown
theoretically that a polarisation version could
be used for Bell-state determination (and, e.g.,
dense coding) a task known to be impossible
with LO. Resch et al., quant-ph/0204034 Presen
t "application," however, is to a novel test of
QM (later in this talk, with any luck...).
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Characterisation of quantum processes in QI
systems
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The Serious Problem For QI

• The danger of errors grows exponentially with the
  size of the quantum system.
• Without error-correction techniques, quantum
  computation would be a pipe dream.
• A major goal is to learn to completely
  characterize the evolution (and decoherence) of
  physical quantum systems in order to design and
  adapt error-control systems.
• The tools are "quantum state tomography" and
  "quantum process tomography" full
  characterisation of the density matrix or Wigner
  function, and of the "uperoperator" which
  describes its time-evolution.

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Quantum State/Process Tomography

• "Pre"-QI Wigner function for nonclassical light
  (Raymer et al), molecules (Walmsley et al), et
  cetera
• Kwiat/White et al. tomography of entangled
  photons entanglement-assisted tomography
• Jessen et al. density matrix reconstruction for
  high-spin state (9x9 density matrix in F4 Cs)
• Cory et al. use of superoperator to design QEC
  pulse sequences for NMR (QFT etc)
• Many, many people I've omitted...

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Density matrices and superoperators
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Two-photon Process Tomography(Mitchell et al.,
quant-ph/0305001)
Two waveplates per photon for state preparation
Detector A
HWP
HWP
PBS
QWP
QWP
SPDC source
QWP
QWP
PBS
HWP
HWP
Detector B
Argon Ion Laser
Two waveplates per photon for state analysis
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Hong-Ou-Mandel Interference
How often will both detectors fire together?
r2t2 0 total destructive interf. (if photons
indistinguishable). If the photons begin in a
symmetric state, no coincidences. Exchange
effect cf. behaviour of fermions in analogous
setup! The only antisymmetric state is the
singlet state HVgt VHgt, in which each photon
is unpolarized but the two are orthogonal. This
interferometer is a "Bell-state filter,"
needed for quantum teleportation and other
applications.
Our Goal use process tomography to test this
filter.
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Measuring the superoperator
Coincidencences
Output DM Input

HH



16 input states
HV
etc.
VV
16 analyzer settings
VH
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Measuring the superoperator
Superoperator
Input Output DM
HH
HV
VV
VH
Output
Input
etc.
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Measuring the superoperator
Superoperator
Input Output DM
HH
HV
VV
VH
Output
Input
etc.
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Testing the superoperator
LL input state
Predicted
Nphotons 297 14
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Testing the superoperator
LL input state
Predicted
Nphotons 297 14
Observed
Nphotons 314
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So, How's Our Singlet State Filter?
Bell singlet state ?? (HV-VH)/v2
Observed ? ??, but a different maximally
entangled state
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Model of real-world beamsplitter
Singlet filter
multi-layer dielectric
AR coating
45 unpolarized 50/50 dielectric beamsplitter
at 702 nm (CVI Laser)
birefringent element singlet-state
filter birefringent element
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Comparison to ideal filter
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Tomography in Optical Lattices
Atoms trapped in standing waves of light are a
promising medium for QIP. (Deutsch/Jessen,
Cirac/Zoller, Bloch,...) We would like to
characterize their time-evolution
decoherence. First must learn how to measure
state populations in a lattice
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Time-resolved quantum states
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Lattice experimental setup
Setup for lattice with adjustable position
velocity
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Quantum state reconstruction
(OR can now translate in x and p directly...)
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Create a coherent state by shifting lattice
delay and shift to measure W.
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A different value of the delay
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Oscillations in lattice wells
Ground-state population vs. time bet. translations
Fancy NLO interpretation Raman pump-probe study
of vibrational states
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Q(x,p) for a coherent H.O. state?
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Quasi-Q for a mostly-excited statein a 2-state
lattice
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Theory for 80/20 mix of e and g
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Exp't"W" or Pg-Pe(x,p)
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W(x,p) for 80 excitation
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Atomic state measurement(for a 2-state lattice,
with c00gt c11gt)
initial state
displaced
delayed displaced
left in ground band

tunnels out during adiabatic lowering
(escaped during preparation)
c0 i c1 2
c02
c0 c1 2
c12
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Time-evolution of some states
input density matrices
output density matrices
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Atom superoperators
sitting in lattice, quietly decohering
being shaken back and forth resonantly
Initial Bloch sphere
CURRENT PROJECTS On atoms, incorporate
"bang-bang" (pulse echo) to preserve coherence
measure homog. linewidth. With photons, study
"tailored" quantum error correction (adaptive
encodings for collective noise).
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Can we talk about what goes on behind closed
doors?
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Pick a box, any box...
A
BC
What are the odds that the particle was in a
given box?
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Conditional measurements(Aharonov, Albert, and
Vaidman)
AAV, PRL 60, 1351 ('88)
Prepare a particle in igt try to "measure" some
observable A postselect the particle to be in fgt
Does ltAgt depend more on i or f, or equally on
both? Clever answer both, as Schrödinger
time-reversible. Conventional answer i, because
of collapse.
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The Rub
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A (von Neumann) Quantum Measurement of A
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A Weak Measurement of A
Has many odd properties, as we shall see...
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"Interaction-Free Measurements"
(AKA The Elitzur-Vaidman bomb experiment) A. C.
Elitzur, and L. Vaidman, Found. Phys. 23, 987
(1993)
Problem Consider a collection of bombs so
sensitive that a collision with any single
particle (photon, electron, etc.) is guarranteed
to trigger it. Suppose that certain of the bombs
are defective, but differ in their behaviour in
no way other than that they will not blow up when
triggered. Is there any way to identify the
working bombs (or some of them) without blowing
them up?
Bomb absent Only detector C fires
Bomb present "boom!" 1/2 C 1/4
D 1/4
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Hardy Cartoon
Hardys Paradox L. Hardy,
Phys. Rev. Lett. 68, 2981 (1992)
Outcome Prob
D and C- 1/16
D- and C 1/16
C and C- 9/16
D and D- 1/16
Explosion 4/16
D- e was in DD- ? But
D e- was in
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Hardy's Paradox Setup
Det. A
Det. B
CC
50-50 BS2
PBS
50-50 BS1
GaN Diode Laser
CC
V
H
DC BS
DC BS

Switch (W)
Cf. Torgerson et al., Phys. Lett. A. 204, 323
(1995)
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But what can we say about where the particles
were or weren't, once D D fire?
Probabilities e- in e- out
e in 1
e out 0
1 0
0
1
1
-1
Upcoming experiment demonstrate that
"weak measurements" (à la Aharonov Vaidman)
will bear out these predictions.
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PROBLEM SOLVED!(?)
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SUMMARY

• Quantum interference allows huge enhancements
  of effective optical nonlinearities. How
  do they relate to"real" nonlinearities? What are
  or aren't they good for?
• Two-photon switch useful for studies of quantum
  weirdness
• (Hardy's paradox, weak measurement), and
  Bell-state detection.
• Two-photon process tomography useful for
  characterizing
• various candidate QI systems.
• Next round of experiments on tailored quantum
  error correction
• (w/ D. Lidar et al.).
• As we learn to control individual quantum
  systems, more and more applications of
  postselection appear need to learn how to think
  about postselected subensembles (weak
  measurement, conditional logic, et
  cetera). (see Steinberg, quant-ph/0302003)
• No matter what the Silicon crowd thinks,
  there's a lot of mileage left in
  (nonlinear/quantum) optics!