Download Measuring And Manipulating Coherence In Photonic And Atomic

Measuring & manipulating coherence
in photonic & atomic systems
Aephraim Steinberg
Centre for Quantum Info. & Quantum Control
Institute for Optical Sciences
Department of Physics
University of Toronto
PITP/CQIQC Workshop:
“Decoherence at the Crossroads”
DRAMATIS PERSONAE
Toronto quantum optics & cold atoms group:
Postdocs: Morgan Mitchell ( ICFO)
Matt Partlow
Optics: Rob Adamson
Lynden(Krister) Shalm
Xingxing Xing
An-Ning Zhang
Kevin Resch(Zeilinger 
Masoud Mohseni (Lidar)
Jeff Lundeen (Walmsley)
)
Atoms: Jalani Fox (...Hinds)
Stefan Myrskog (Thywissen)
Ana Jofre(Helmerson) Mirco Siercke
Samansa Maneshi
Chris Ellenor
Rockson Chang
Chao Zhuang
Some helpful theorists:
QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.
Daniel Lidar, János Bergou, Pete Turner, John Sipe, Paul Brumer, Howard Wiseman, Michael Spanner,...
OUTLINE
“Never underestimate the pleasure people get from hearing something they already know”
Some things you may already know
A few words about Quantum Information,
about photons, and about state & process tomography
Some things you probably haven’t heard...
but on which we’d love (more) collaborators!
Two-photon process tomography
How to avoid quantum state & process tomography?
Tomography of trapped atoms, and attempts at control
Complete characterization given incomplete
experimental capabilities
How to draw Wigner functions on the Bloch sphere?
0
Quantum tomography: why?
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
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!)
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>+...c11..1|11..1>;
this is exponentially more information than the 2n coefficients it
would take to describe n independent (e.g., classical) bits.
We need to understand
nature of quantum information itself.
It is also exponentially
sensitive the
to decoherence.
characterize
and compareinformation-quantum states?they
Photons are How
idealtocarriers
of quantum
to most fully
describe theirand
evolution
in a given
can be easilyHow
produced,
manipulated,
detected,
andsystem?
How
to manipulatewith
them?
don't interact
significantly
the environment. They
are already used to transmit quantum-cryptographic
The danger of errors
& decoherence
grows
exponentially
information
through
fibres under
Lake
Geneva,with
andsystem
soonsize.
across the Danube
The only hope
for QI
error correction.
through
the air
upistoquantum
satellites.
We must learn how to measure what the system is doing, and then correct it.
more!)
Unfortunately, they don't interact with(...Another
each othertalk,
veryormuch
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, VV, or any superpositions.
State has four coefficients.
Density matrix has 4x4 = 16 coefficients.
Superoperator has 16x16 = 256 coefficients.
1
Quantum process tomography
on photon pairs
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.
Two-photon Process Tomography
[Mitchell et al., PRL 91, 120402 (2003)]
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
“Measuring” the superoperator
Coincidencences
Output DM
}
}
}
}
16
input
states
Input
HH
HV
etc.
VV
16 analyzer settings
VH
“Measuring” the superoperator
Input
Superoperator
Output DM
HH
HV
VV
VH
etc.
Input
Output
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.
2
Can we avoid doing tomography?
Polynomial Functions of a
Density Matrix
(T. A. Brun, e-print: quant-ph/0401067)
•
•
Often, only want to look at a single figure of merit of a state (i.e. tangle, purity,
etc…)
Would be nice to have a method to measure these properties without needing
to carry out full QST.
• Todd Brun showed that mth degree polynomial functions of a density
matrix fm() can be determined by measuring a single joint observable
involving m identical copies of the state.
Linear Purity of a Quantum State
• For a pure state, P=1
• For a maximally mixed state, P=(1/n)
• Quadratic  2-particle msmt needed
Measuring the purity of a qubit
• Need two identical copies of the state
• Make a joint measurement on the two copies.
• In Bell basis, projection onto the singlet state
P = 1 – 2  –  – 
Singlet-state probability can be
measured by a singlet-state filter (HOM)
HOM as Singlet State Filter
Pure State on either side = 100% visibility
HH H
H HH
H
+
Mixed State = 50% visibility
HV H H
V V
H V
+
HOM Visibility = Purity
Experimentally Measuring the Purity
of a Qubit
•Use Type 1 spontaneous parametric downconversion to prepare two
identical copies of a quantum state
•Vary the purity of the state
•Use a HOM to project onto the singlet
•Compare results to QST
 /2
Single Photon
Detector
Quartz
Slab
Type 1 SPDC
Crystal

Singlet
Filter
 /2

Coincidence
Circuit
Quartz
Slab
Single Photon
Detector
Results For a Pure State
Measuring +45 +45
Prepared the state |+45>
3500
Measured Purity
from Singlet State
Measurement
P=0.92±0.02
Counts per 30 s
3000
2500
2000
1500
1000
500
0
0
50
100
150
200
Delay (um)
Measured Purity
from QST
P=0.99±0.01
250
300
350
Preparing a Mixed State
Can a birefringent delay decohere polarization (when we trace over timing info) ?
[cf. J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, and P. G. Kwiat, Phys. Rev. Lett., 90, 193601 ]
Case 1: Same birefringence in each arm
 /2
H

Visibility = (90±2) %
V
V
 /2
H

100% interference
Case 2: Opposite birefringence in each arm
H and V Completely Decohered Due to Birefringence
1800
1600

H
1400
V
H
 /2

V
25% interference
Counts per 30s
 /2
1200
1000
800
600
Visibility = (21±2) %
400
200
0
0
50
The HOM isn’t actually insensitive to timing information.
100
150
200
250
Delay (um)
300
350
400
450
Not a singlet filter, but an
“Antisymmetry Filter”
• The HOM is not merely a polarisation singlet-state filter
• Problem:
• Used a degree of freedom of the photon as our bath instead of some
external environment
• The HOM is sensitive to all degrees of freedom of the photons
• The HOM acts as an antisymmetry filter on the entire photon state
• Y Kim and W. P. Grice, Phys. Rev. A 68, 062305 (2003)
• S. P. Kulik, M. V. Chekhova, W. P. Grice and Y. Shih, Phys. Rev. A 67,01030(R) (2003)
Preparing a Mixed State
Randomly rotate the half-waveplates to produce |45> and |-45>
|45>
Preliminary results
 /2

No Birefringence, Even Mixture of +45/+45 and +45/-45
3500
3000

|45> or |-45>
Currently setting up LCD waveplates which
will allow us to introduce a random phase shift
between orthogonal polarizations to produce a
variable degree of coherence
Could produce a “better”
maximally mixed state by using
four photons. Similar to Paul
Kwiat’s work on Remote State
Preparation.
Counts per 30 s
 /2
2500
2000
1500
1000
Visibility = (45±2) %
500
0
0
50
100
150
200
250
Delay (um )
 /2
Coincidence
Circuit

 /2

300
350
3
Tomography in optical lattices,
and steps towards control...
Tomography in Optical Lattices
[Myrskog et al., PRA 72, 103615 (’05)
Kanem et al., J. Opt. B 7, 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?
Towards QPT:
Some definitions / remarks
• "Qbit" = two vibrational states of atom in a well of a 1D lattice
• Control parameter = spatial shifts of lattice (coherently couple
states), achieved by phase-shifting optical beams (via AO)
• Initialisation: prepare |0> by letting all higher states escape
• Ensemble: 1D lattice contains 1000 "pancakes", each with
thousands of (essentially) non-interacting atoms.
No coherence between wells; tunneling is a decoherence mech.
• Measurement in logical basis: direct, by preferential tunneling
under gravity
• Measurement of coherence/oscillations: shift and then measure.
• Typical experiment:
• Initialise |0>
• Prepare some other superposition or mixture (use shifts, shakes, and delays)
• Allow atoms to oscillate in well
• Let something happen on its own, or try to do something
• Reconstruct state by probing oscillations (delay + shift +measure)
First task: measuring state
populations
Time-resolved quantum states
Recapturing atoms after setting
final vs midterm, both adjusted to 70 +/- 15
them
intoto 70oscillation...
+/- 15
both adjusted
final vs midterm,
Series1
...or failing to recapture them
final vs midterm, both adjusted to 70 +/- 15
ifboth
you're
too
15
70 +/-impatient
adjusted to
final vs midterm,
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.)
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  (-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.
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
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:
pulsecomparing
echooscillations
indicates
T2 ≈ 1 ms...
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)
decay of coherence introduced by echo pulses
themselves (since they are not perfect p-pulses)
Why does our echo decay?
Finite bath memory time:
So far, our atoms are free to move in the directions transverse to
our lattice. In 1 ms, they move far enough to see the oscillation
frequency change by about 10%... which is about 1 kHz, and hence
enough to dephase them.
Inter-well tunneling should occur on a few-ms timescale... should one think
of this as homogeneous or inhomogeneous? “How conserved” is
quasimomentum?
le shift-back
e
1
Echo from compound pulse
Pulseamplitude
900 us for
after
stateshift-back
preparation,
Echo
a single
vs.
a pulse (shift-back,
shift) at 900 us
and track delay,
oscillations
0.9
single-shift echo
(≈10% of initial oscillations)
0.8
0.7
0.6
double-shift echo
(≈30% of initial oscillations)
0.5
0.4
0.3
0
200
400
600
800
1000 1200 1400 1600
time ( microseconds)
Future: More parameters; find best pulse.
Step 2 (optional): figure out why it works!
Also: optimize # of pulses (given imperfection of each)
What if we try “bang-bang”?
(Repeat pulses before the bath gets amnesia; trade-off since each pulse
is imperfect.)
Some coherence out to > 3 ms now...
How to tell how much of the
coherence is from the initial state?
The superoperator for a second-order echo:
Some future plans...
• Figure out what quantity to optimize!
• Optimize it... (what is the limit on echo amp. from such pulses?)
• Tailor phase & amplitude of successive pulses to cancel
out spurious coherence
• Study optimal number of pulses for given total time.
(Slow gaussian decay down to exponential?)
• Complete setup of 3D lattice. Measure T2 and study
effects of tunneling
• BEC apparatus: reconstruct single-particle wavefunctions
completely by “SPIDER”-like technique?
• Generalize to reconstruct single-particle Wigner functions?
• Watch evolution from pure single-particle functions (BEC)
to mixed single-particle functions due to inter-particle
interactions (free expansion? approach to Mott? etc?)
4a
Measurement as a tool:
Post-selective operations for the construction of
novel (and possibly useful) entangled states...
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)
Postselective nonlinearity
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.
It works!
Singles:
Coincidences:
Triple
coincidences:
Triples (bg
subtracted):
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?
LeftArnold RightDanny
OR –Arnold&Danny ?
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.
Experimental Results (2 photons)
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
More Photons…
If you have a collection of spins, what
are the permutation-blind observables
that describe the system?
They correspond to measurements
of angular momentum operators
J and mj ... for N photons, J runs to N/2
So the total number of operators accessible to measurement is
N /2
Number of ordering - blind ops   2 j  1  N  3N  2N  1 / 6
2
j
Total # of projectors  4 N
Total # of projectors onto symmetric states  N  1
2
Wigner distributions
on the Poincaré sphere
[Following recipe of Dowling, Agarwal, & Schleich, PRA 49, 4101 (1993).]
Some polarisation states of the fully symmetric triphoton
(theory– for the moment), drawn on the J=3/2 Bloch sphere:
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
movie of the
evolution from 3noon state to phasesqueezed, coherent,
and “number”squeezed states...
QuickTime™ and a
YU V420 codec decompressor
are needed to see this pict ure.
3H,0V 2H,1V 1H,2V 0H,1V
a.
a slightly “number”-squeezed state
b. a highly phase-squeezed state
c.
the “3-noon” state
Conclusions Plea For Help
1.
Quantum process tomography can be useful for characterizing and
"correcting" quantum systems (ensemble measurements).
2.
It’s actually quite “expensive” – there is still much to learn about other
approaches, such as “adaptive” tomography, and “direct” measurements of
quantities of interest.
3.
Much work remains to be done to optimize control of systems such as optical
lattices, where a limited range of operations may be feasible, and multiple
sources of decoherence coexist.
4.
Can we do tomography on condensed atoms, e.g., in a lattice? In what
regimes will this help observe interesting (entangling) dynamics?
5.
The full characterisation of systems of several “indistinguishable” photons
offers a number of interesting problems, both for density matrices and for
Wigner distributions.