Measuring & manipulating quantum states
(Fun With Photons and Atoms)
Aephraim M. Steinberg
Centre for Q. Info. & Q. Control
Institute for Optical Sciences
Dept. of Physics, U. of Toronto
GA Tech, March 2007
DRAMATIS PERSONÆ
Toronto quantum optics & cold atoms group:
Postdocs: Matt Partlow ( ...)
Morgan Mitchell ( ICFO)
An-Ning Zhang( IQIS) Marcelo Martinelli ( USP)
Optics: Rob Adamson
Lynden(Krister) Shalm
Xingxing Xing
Kevin Resch(Wien UQIQC)
Jeff Lundeen (Oxford)
Stefan Myrskog (BEC ECE)
Ana Jofre(NIST)
Mirco Siercke ( ...)
Samansa Maneshi
Chris Ellenor
Rockson Chang
Chao Zhuang
Xiaoxian Liu
Recent ug’s: Shannon Wang, Ray Gao, Sabrina Liao, Max Touzel, Ardavan Darabi
Some helpful theorists:
Atoms: Jalani Fox (Imperial)
QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.
Daniel James, János Bergou, Pete Turner, John Sipe, Paul Brumer, Howard Wiseman, Michael Spanner,...
Quantum Computer Scientists
The 3 quantum computer scientists:
see nothing (must avoid "collapse"!)
hear nothing (same story)
say nothing (if any one admits this thing
is never going to work,
that's the end of our
funding!)
OUTLINE
The grand unified theory of physics talks:
“Never underestimate the pleasure people
get from being told something they already know.”
OUTLINE
Some things you (may) already know
Quantum info, entanglement, nonunitary op’s
Some things you probably haven’t seen before...
Preparing entangled states of N photons
Subtleties of measuring multi-photon states
Motional-state tomography on trapped atoms
Decoherence & progress on echoes
Something slightly insane, in case you found
the rest of the talk too boring...
Ask me after the talk if you want to know about:
Dr. Aharonlove OR
“How I learned to stop worrying and love
negative (& complex) probabilities...”
What makes a computer quantum?
(One partial answer...)
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
(NB: Product states only require 2n coeff’s, but
non-entangled
quantum computation could
exponentially sensitive
to
decoherence.
equally well be performed with classical waves)
We need to understand the nature of quantum
Photons are ideal carriers of
quantumitself.
information-- they
information
can be easily produced, manipulated, and detected, and
characterize
and compare
don't interact significantlyHow
withtothe
environment.
They quantum states?
are already used to transmit quantum-cryptographic
to most
fully
describeand
theirsoon
evolution in a
information through fibresHow
under
Lake
Geneva,
given system?
through the air up to satellites.
across thethem?
Danube
How to manipulate
Unfortunately, they don't interact with each other very much
either! How to make a logic gate?
(...Another talk, or more!)
How to build a quantum computer
Photons don't interact
(good for transmission; bad for computation)
Try: atoms, ions, molecules, ...
or just be clever with photons!
Nonlinear optics: photon-photon interactions
Generally exceedingly weak.
Potential solutions:
Cavity QED
Better materials (1010 times better?!)
Measurement as nonlinearity (KLM)
Novel effects (slow light, EIT, etc)
Photon-exchange effects (à la Franson)
Interferometrically-enhanced nonlinearity
“Moderate” nonlinearity + homodyne measurement
Measurement as a tool: KLM...
INPUT STATE
a|0> + b|1> + c|2>
OUTPUT STATE
a|0> + b|1> – c|2>
MAGIC MIRROR:
Acts differently if there are 2 photons or only 1.
TRIGGER (postselection)
ANCILLA
In other words, can be a “transistor,” or “switch,”
or “quantum
special
|i > logic gate”...
particular | >
f
Knill, Laflamme, Milburn Nature 409, 46, (2001); and others since.
Experiments by Franson et al., White et al., Zeilinger et al...
1
Building up entanglement photon by photon
by using post-selective nonlinearity
Highly number-entangled states
("low-noon" experiment).
M.W. Mitchell et al., Nature 429, 161 (2004)
States such as |n,0> + |0,n> ("noon" states) have been proposed for
high-resolution interferometry – related to "spin-squeezed" states.
Important factorisation:
+
=
A "noon" state
A really odd beast: one 0o photon,
one 120o photon, and one 240o photon...
but of course, you can't tell them apart,
let alone combine them into one mode!
Theory: H. Lee et al., Phys. Rev. A 65, 030101 (2002); J. Fiurásek, Phys. Rev. A 65, 053818 (2002)
Trick #1
How to combine three non-orthogonal photons into one spatial mode?
"mode-mashing"
Yes, it's that easy! If you see three photons
out one port, then they all went out that port.
Post-selective nonlinearity
Trick #2
Okay, we don't even have single-photon sources*.
But we can produce pairs of photons in down-conversion, and
very weak coherent states from a laser, such that if we detect
three photons, we can be pretty sure we got only one from the
laser and only two from the down-conversion...
SPDC
|0> + e |2> + O(e2)
laser
* But
|0> +  |1> + O(2)
e |3> + O(3) + O(e2)
+ terms with <3 photons
we’re working on it (collab. with Rich Mirin’s quantum-dot group at NIST)
Trick #3
But how do you get the two down-converted photons to be at 120o to each other?
More post-selected (non-unitary) operations: if a 45o photon gets through a
polarizer, it's no longer at 45o. If it gets through a partial polarizer, it could be
anywhere...
(or nothing)
(or nothing)
(or <2 photons)
Experimental Setup
It works!
Singles:
Coincidences:
Triple
coincidences:
Triples (bg
subtracted):
M.W. Mitchell, J.S. Lundeen, and A.M. Steinberg, Nature 429, 161 (2004)
1b
4b
Complete characterisation
when you have incomplete information
Fundamentally Indistinguishable
vs.
Experimentally Indistinguishable
But what if when we combine our photons,
there is some residual distinguishing information:
some (fs) time difference, some small spectral
difference, some chirp, ...?
This will clearly degrade the state – but how do
we characterize this if all we can measure is
polarisation?
Quantum State Tomography
Indistinguishable
Photon Hilbert Space
2
H
,0V , 1H ,1V , 0 H , 2V
 HH
, HV  VH , VV


?
Distinguishable Photon
Hilbert Space
 H1H 2 , V1H 2 , H1V2 , V1V2

Yu. I. Bogdanov, et al
Phys. Rev. Lett. 93, 230503 (2004)
If we’re not sure whether or not the particles are distinguishable,
do we work in 3-dimensional or 4-dimensional Hilbert space?
If the latter, can we make all the necessary measurements, given
that we don’t know how to tell the particles apart ?
The Partial Density Matrix
The answer: there are only 10 linearly independent parameters which
are invariant under permutations of the particles. One example:
  HH, HH
 HV VH, HH

 HH, HV VH  HV VH, HV VH

 HV VH,VV
  HH,VV

Inaccessible

VV , HH 

VV , HV VH 

VV ,VV 


information



 HV VH, HV VH 
Inaccessible
information
The sections of the density matrix labelled “inaccessible” correspond to
information about the ordering of photons with respect to inaccessible
degrees of freedom.
For n photons, the # of parameters scales as n3, rather than 4n
Note: for 3 photons, there are 4 extra parameters – one more
than just the 3 pairwise HOM visibilities.
R.B.A. Adamson, L.K. Shalm, M.W. Mitchell, and A.M. Steinberg, PRL 98, 043601 (07)
Experimental Results
No Distinguishing Info
Distinguishing Info
When distinguishing
information is introduced the
HV-VH component increases
without affecting the state in
the symmetric space
HH + VV
Mixture of
45–45 and –4545
A typical 3-photon density matrix
My favorite cartoon about
Alice (almost) and Bob
1c
4b
A better description than
density matrices?
Wigner distributions
on the Poincaré sphere ?
(Consider a purely symmetric state: N photons act like a single spin-N/2)
Any pure state of a spin-1/2 (or a photon) can be represented as a point
on the surface of the sphere – it is parametrized by a single amplitude and
a single relative phase.
This is the same as the description of a classical spin, or the polarisation
(Stokes parameters) of a classical light field.
Of course, only one basis yields a definite result, so a better description
would be some “uncertainty blob” about that classical point... for spin-1/2,
this uncertainty covers a hemisphere , while for higher spin it shrinks.
0.9
2
0.8
1.8
0.7
0.6
1.6
1.4
1.2
0.5
1
0.4
0.8
0.3
0.6
0.2
0.1
0.4
0.2
Wigner distributions
on the Poincaré sphere
Can such quasi-probability distributions over the “classical”
polarisation states provide more helpful descriptions of the
“state of the triphoton” than density matrices?
“Coherent state” = N identically polarized photons
0.8
0.9
0.7
0.8
0.6
0.7
0.5
0.6
0.4
0.5
0.3
0.4
0.2
0.3
0.1
0.2
0
0.1
0
-0.1
-0.2
“Spin-squeezed state” trades off
uncertainty in H/V projection for
more precision in phase angle.
Dowling, Agarwal, & Schleich, PRA 49, 4101 (1993).
Beyond 1 or 2 photons...
A 1-photon pure state may be represented by a point on the surface
of the Poincaré sphere, because there are only 2 real parameters.
squeezed state
3-noon
0.8
15-noon
0.6
0.8
0.7
0.4
0.6
0.5
0.2
0.4
0.3
0
0.2
0.1
-0.2
0
-0.4
-0.1
-0.2
3 photons:
2 photons:
6 parameters:
4 param’s:
Euler angles
Euler angles
+ squeezing (eccentricity) + squeezing (eccentricity)
+ orientation
+ orientation
+ more complicated stuff
-0.6
-0.8
QuickTime™ and a
YUV420 codec decompressor
are needed to see t his picture.
Preliminary Experimental Results
NOTE: To study a broad range of entangled states, a more flexible “mode-masher”
is needed – the tradeoff is lower efficiency, which decreases SNR
THEORY
EXPERIMENT
Measured Wigner Function
Ideal N00N state
N00N state + background
Measured density matrix
Squeezing
2
Quantum CAT scans
Tomography in Optical Lattices
[Myrkog et al., PRA 72, 013615 (05)
Kanem et al., J. Opt. B7, S705 (05)]
Rb atom trapped in one of the quantum levels
of a periodic potential formed by standing
light field (30GHz detuning, 10s of mK depth)
Complete characterisation of
process on arbitrary inputs?
First task: measuring state
populations
Time-resolved quantum states
Quantum state reconstruction
p
p
t
Dx
x
Wait…
x
Shift…
p
Dx
x
Measure ground
state population
Q(0,0) = 1p Pg
W(0,0) = 1p S (-1)n Pn
(former for HO only; latter requires only symmetry)
Cf. Poyatos,Walser,Cirac,Zoller,Blatt, PRA 53, 1966 ('96)
& Liebfried,Meekhof,King,Monroe,Itano,Wineland, PRL77, 4281 ('96)
Husimi distribution of coherent state
Data:"W-like" [Pg-Pe](x,p) for
a mostly-excited incoherent mixture
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
Recapturing atoms after setting
them
into
final
vs midterm,
bothoscillation...
adjusted to 70 +/- 15
final vs midterm, both adjusted to 70 +/- 15
Series1
...or failing to recapture them
if you're too impatient
final vs midterm, both adjusted to 70 +/- 15
final vs midterm, both adjusted to 70 +/- 15
Series1
Oscillations in lattice wells
(Direct probe of centre-of-mass oscillations in 1mm wells;
can be thought of as Ramsey fringes or Raman pump-probe exp’t.)
Extracting a superoperator:
prepare a complete set of input states and measure each output
Likely sources of decoherence/dephasing:
Real photon scattering (100 ms; shouldn't be relevant in 150 ms period)
Inter-well tunneling (10s of ms; would love to see it)
Beam inhomogeneities (expected several ms, but are probably wrong)
Parametric heating (unlikely; no change in diagonals)
Other
Towards bang-bang error-correction:
pulse echo indicates T2 ≈ 1 ms...
comparing oscillations for shift-backs
applied after time t
2
Free-induction-decay signal for comparison
1.5
1/(1+2)
echo after “bang” at 800 ms
1
echo after “bang” at 1200 ms
0.5
echo after “bang” at 1600 ms
0
00
(bang!)
50
500
ms
100
1000
ms
150
1500
ms
200
2000
ms
250
t(10us)
coherence introduced by echo pulses themselves
(since they are not perfect p-pulses)
Cf. Hannover experiment
Far smaller echo, but far better signal-to-noise ("classical" measurement of <X>)
Much shorter coherence time, but roughly same number of periods
– dominated by anharmonicity, irrelevant in our case.
Buchkremer, Dumke, Levsen, Birkl, and Ertmer, PRL 85, 3121 (2000).
Echo from compound pulse
Instead of a single abrupt phase shift, try different shapes.
Perhaps a square (two shifts) can provide destructive interference into lossy states?
Perhaps a “soft” (gaussian) pulse has fewer dangerous frequency components?
Ongoing: More parameters;
find best pulse.
E.g., combine amplitude &
phase mod.
Also: optimize # of pulses.
Optimizing inter-band couplings
Optimized 1 -> 2
coupling versus
size of shift, for
three pulse
shapes:
Even with relatively simple pulse
shapes, could we get 65%
excitation by choosing the
optimal lattice depth? (Not in a
vertical geometry, as the states
become unbound.)
Why does our echo decay?
3D lattice
(first data)
Finite bath memory time:
Except
foratoms
one minor
feature:
So far, our
are freedisturbing
to move in the
directions transverse to
our lattice.
In 1 similar
ms, they plateaux
move far enough
to see 1D
the and
oscillation
We see
with both
3D lattices;
frequency change by about 10%... which is about 1 kHz, and hence
temporal
enough to (laser)
dephase noise
them. in addition to spatial fluctuations?
Ongoing work to eliminate what noise we can & understand
the rest!
Can we talk about what goes on behind closed doors?
(“Postselection” is the big new buzzword in QIP...
but how should one describe post-selected states?)
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.
Reconciliation: measure A "weakly."
Poor resolution, but little disturbance.
Aw 
f Ai
f i
…. can be quite odd …
3a
“Quantum Seeing in the Dark”
" Quantum seeing in the dark "
(AKA: “Interaction-free” measurement)
A. Elitzur, and L. Vaidman, Found. Phys. 23, 987 (1993)
P.G. Kwiat, H. Weinfurter, and A. Zeilinger, Sci. Am. (Nov., 1996)
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
The bomb must be there... yet
my photon never interacted with it.
Hardy's Paradox
(for Elitzur-Vaidman “interaction-free measurements”)
C+
D+
D-
CD+
Outcome
–> e- was
Prob
in
Din
D+–>
ande+
C-was
1/16
BS2+
BS2I+
I-
O-
O+
W
BS1+
e+
BS1e-
D- and C+ 1/16
D+D- –> ?
C+ and C- 9/16
But
D+ …
and
if they
D- 1/16
were
both
in, they 4/16
Explosion
should
have annihilated!
The two-photon switch...
OR: 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)
Using a “photon switch” to
implement Hardy’s Paradox
H Pol DC
V Pol DC
407 nm Pump
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
In fact, this is precisely what Aharonov et al.’s weak measurement
formalism predicts for any sufficiently gentle attempt to “observe”
these probabilities...
Weak Measurements in Hardy’s Paradox
Ideal Weak Values
N(I-)
N(O)
N(I+)
N(O+)
0
1
1
1
1
0
1
0
Experimental Weak Values (“Probabilities”?)
N(I-)
N(O)
N(I+)
0.243±0.068
0.663±0.083
0.882±0.015
N(O+)
0.721±0.074
0.758±0.083
0.087±0.021
0.925±0.024
0.039±0.023
The moral of the story
1. Post-selected systems often exhibit surprising behaviour
which can be probed using weak measurements.
QuickTime™ and a
YUV420 codec decompressor
are needed to see t his picture.
2. Post-selection can also enable us to generate novel “interactions”
(KLM proposal for quantum computing), and for instance to
produce useful entangled states.
• Multi-photon entangled states may be built “from the
ground up” – no need for high-frequency parent photons
3. Multi-photon entangled states may be built “from the ground
up” – no need for high-frequency parent photons
4. A modified sort of tomography is possible on “practically
indistinguishable” particles; there remain interesting questions
about the characterisation of the distinguishability of >2 particles.
5. There are many exciting open issues in learning how to optimize
control and error correction of quantum systems