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
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
Ana Jofre
Samansa Maneshi
Jalani Fox
Mirco Siercke
Salvatore Maone ( real world)
Some of our theory friends:
Daniel Lidar, Janos Bergou, Mark Hillery, John Sipe, Paul Brumer, Howard Wiseman
OUTLINE
Something you already know
Introduction to quantum information with optics
Something you may have known...
but talks
may are
have
forgotten by now
All good
alike...
Making a every
strongbad
effective
between
talk isinteraction
bad in its own
way.
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?"
Intro to Quantum Info -pros & cons of optical schemes...
Quantum Information
What's so great about it?
Quantum Information
What's so great about it?
If a classical computer takes input |n> to output |f(n)>,
an analogous quantum computer takes a state
|n>|0> and maps it to |n>|f(n)> (unitary, reversible).
By superposition, such a computer takes
n |n>|0> to n |n>|f(n)>; it calculates f(n)
for every possible input simultaneously.
A clever measurement may determine some global
property of f(n) even though the computer has
only run once...
A not-clever measurement "collapses" n to some
random value, and yields f(that value).
The rub: any interaction with the environment
leads to "decoherence," which can be thought
of as continual unintentional measurement of n.
Quantum Computer Scientists
What makes a computer quantum?
If a quantum "bit" is described by two numbers:
|> = c0|0> + c 1|1>,
then n quantum bits are described by 2n coeff's:
|> = c00..0|00..0>+c 00..1|00..1>+...c 11..1|11..1>;
this is exponentially more information than the 2n coefficients it
would take to describe n independent (e.g., classical) bits.
It is also exponentially sensitive to decoherence.
Photons are ideal carriers of quantum information-- they
can be easily produced, manipulated, and detected, and
don't interact significantly with the environment. They
are already used to transmit quantum-cryptographic
information through fibres under Lake Geneva, and soon
through the air up to satellites.
Unfortunately, they don't interact with each other very much
either! How to make a logic gate?
Quantum Interference for effective
single-photon–single-photon interactions...?
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) ?
The germ of the KLM idea
INPUT STATE
a|0> + b|1> + c|2>
ANCILLA
|1>
OUTPUT STATE
a'|0> + b'|1> + c'|2>
TRIGGER (postselection)
|1>
In particular: with a similar but somewhat more complicated
setup, one can engineer
a |0> + b |1> + c |2> a |0> + b |1> – c |2> ;
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., ...
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)?
atom 1
w1
w2
atom 2
w2
w1
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.
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)?
Two-photon absorption (by these
single-photon absorbers) is interferometrically enhanced if the
photons begin distinguishable, but
are indistinguishable to the absorber:
T2 > t > t c
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) !
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.
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.
HOM interference pattern is itself a manifestation of photon exchange effects.
The
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 e&m?)
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).
Quantum Interference
Suppression/Enhancement
of Spontaneous Down-Conversion
(57% visibility)
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.
Controlled-phase switch
Resch et al, Phys. Rev. Lett. 89, 037914 (2002)
Fringe data with and w/o postsel.
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]
Present "application," however, is to a novel test of QM
(later in this talk, with any luck...).
Characterisation of quantum processes in QI systems
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 F=4 Cs)
• Cory et al.: use of superoperator to design
QEC pulse sequences for NMR (QFT etc)
• Many, many people I've omitted...
Density matrices and superoperators
()
( )
One photon: H or V.
State: two coefficients
CH
CV
Density matrix: 2x2=4 coefficients
CHH CVH
CHV
CVV
Measure
intensity of horizontal
intensity of vertical
intensity of 45o
intensity of RH circular.
Propagator (superoperator): 4x4 = 16 coefficients.
Two photons: HH, HV, VH, HV, or any superpositions.
State has four coefficients.
Density matrix has 4x4 = 16 coefficients.
Superoperator has 16x16 = 256 coefficients.
Two-photon Process Tomography
(Mitchell et al., quant-ph/0305001)
Two waveplates per photon
for state preparation
HWP
QWP
HWP
Detector A
PBS
QWP
SPDC source
"Black Box" 50/50
Beamsplitter
QWP
HWP
QWP
PBS
HWP
Detector B
Argon Ion Laser
Two waveplates per
photon for state analysis
Hong-Ou-Mandel Interference
r
r
+
t
t
How often will both detectors fire together?
r2+t2 = 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
|HV> – |VH>, 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.
Comparison to ideal filter
Measured superoperator,
in Bell-state basis:
Superoperator after transformation
to correct polarisation rotations:
A singlet-state filter would have
a single peak, indicating the one
transmitted state.
Dominated by a single peak;
residuals allow us to estimate
degree of decoherence and
other errors.
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…
Time-resolved quantum states
Quantum state reconstruction
p
p
wt
x
x
x
Wait…
Shift…
p
x
Q(0,0) = Pg
x
Measure ground
state population
W(0,0) =  (-1)n Pn
(OR: can now translate in x and p directly...)
Create a coherent state by shifting
final vs midterm, both adjusted to 70 +/- 15
lattice;
delay
andtoshift
70 +/- 15 to measure W.
both adjusted
vs midterm,
final
Series1
A different
value
of
the
delay
final vs midterm, both adjusted to 70 +/- 15
final vs midterm, both adjusted to 70 +/- 15
Series1
Oscillations in lattice wells
Ground-state population vs. time bet. translations
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Fancy NLO interpretation:
Raman pump-probe study of vibrational states
Exp't:"W" or [Pg-Pe](x,p)
QuickTime™ and a
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Atomic state measurement
(for a 2-state lattice, with c0|0> + c1|1>)
initial state
displaced
delayed & displaced
left in
ground band
tunnels out
during adiabatic
lowering
(escaped during
preparation)
|c0|2
|c1|2
|c0 + c1 |2
|c0 + i c1 |2
Time-evolution of some states
input density matrices
output density matrices
Atom superoperators
sitting in lattice, quietly
decohering…
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are needed to see this picture.
QuickTime™ and a
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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).
QuickTime™ and a
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are needed to see this picture.
Can we talk about what goes on behind closed doors?
Pick a box, any box...
A+B+C
A +B–C
What are the odds that the particle
was in a given box?
Conditional measurements
(Aharonov, Albert, and Vaidman)
AAV, PRL 60, 1351 ('88)
Prepare a particle in |i> …try to "measure" some observable A…
postselect the particle to be in |f>
i i
Measurement
of A
f f
Does <A> depend more on i or f, or equally on both?
Clever answer: both, as Schrödinger time-reversible.
Conventional answer: i, because of collapse.
A (von Neumann) Quantum
Measurement of A
Initial State of Pointer
Final Pointer Readout
Hint=gApx
System-pointer
coupling
x
x
Well-resolved states
System and pointer become entangled
Decoherence / "collapse"
Large back-action
A Weak Measurement of A
Initial State of Pointer
Final Pointer Readout
Hint=gApx
x
System-pointer
coupling
x
Poor resolution on each shot.
Negligible back-action (system & pointer separable)
Mean pointer shift is given by A w 
f Ai
f i
Has many odd properties, as we shall see...
"Interaction-Free Measurements"
(AKA: The Elitzur-Vaidman bomb experiment)
A. C. Elitzur, and L. Vaidman, Found. Phys. 23, 987 (1993)
Problem:
D
C
Consider a collection of bombs so sensitive that
a collision with any single particle (photon, electron, etc.)
Bomb absent:
is guarranteed to trigger it.
Only detector C fires
BS2 that certain of
Suppose
the bombs are defective,
but differ in their behaviour in no way other than that
Bomb present:
they will not blow up when triggered.
"boom!"
1/2 bombs (or
Is there any way to identify
the working
C up? 1/4
some of them)
without blowing them
BS1
D
1/4
Hardy’s Paradox
L. Hardy, Phys. Rev. Lett. 68, 2981 (1992)
C+
D+
D-
BS2+
C-
BS2I+
I-
O-
O+
W
BS1+
e+
BS1e-
Outcome Prob
D+
e- was
D+ and
C- in
1/16
D- e+ was in
D- and C+ 1/16
C+ and ?C- 9/16
D+DD+ and D- 1/16
But
…
Explosion
4/16
Experimental Setup
Det. V (D+) Det. H (D-)
50-50
BS2
CC
PBS
PBS
GaN
Diode Laser
DC BS
50-50
BS1
(W)
CC
V
H
Switch
DC BS
But what can we say about where the particles
were or weren't, once D+ & D– fire?
Probabilities e- in
e- out
e+ in
0
1
1
e+ out
1
-1
0
1
0
Upcoming experiment: demonstrate that "weak
measurements" (à la Aharonov + Vaidman) will
bear out these predictions.
PROBLEM SOLVED!(?)
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!