Measuring

1
Measuring manipulating coherencein 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
2
DRAMATIS PERSONAE Toronto quantum optics cold
atoms group Postdocs Morgan Mitchell (?
ICFO) Matt Partlow An-Ning Zhang
Optics Rob Adamson Kevin Resch(?Zeilinger
??????) Lynden(Krister) Shalm Masoud Mohseni
(?Lidar) Xingxing Xing 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 Daniel Lidar,
János Bergou, Pete Turner, John Sipe, Paul
Brumer, Howard Wiseman, Michael Spanner,...
3
OUTLINE
Never underestimate the pleasure people get from
hearing something they already know
4
Quantum tomography why?
5
Quantum Information
What's so great about it?
6
Quantum Information
What's so great about it?
7
Quantum Computer Scientists
8
What makes a computer quantum?
We need to understand the nature of quantum
information itself. How to characterize and
compare quantum states? How to most fully
describe their evolution in a given system? How
to manipulate them?
The danger of errors decoherence grows
exponentially with system size. The only hope for
QI is quantum error correction. We must learn how
to measure what the system is doing, and then
correct it.
across the Danube
(...Another talk, or more!)
9
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.

10
Density matrices and superoperators
11
Quantum process tomography on photon pairs
12
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.
13
Two-photon Process TomographyMitchell et al.,
PRL 91, 120402 (2003)
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
14
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.
15
Measuring the superoperator
Coincidencences
Output DM Input

HH



16 input states
HV
etc.
VV
16 analyzer settings
VH
16
Measuring the superoperator
Superoperator
Input Output DM
HH
HV
VV
VH
Output
Input
etc.
17
(No Transcript)
18
Comparison to ideal filter
19
Can we avoid doing tomography?
20
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.

21
Linear Purity of a Quantum State

• For a pure state, P1
• 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)
22
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

Single Photon Detector
Quartz Slab
Type 1 SPDC Crystal
Singlet Filter
Coincidence Circuit
Quartz Slab
Single Photon Detector
23
Results For a Pure State
Prepared the state 45gt
Measured Purity from Singlet State
Measurement P0.920.02
Measured Purity from QST P0.990.01
24
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
The HOM isnt actually insensitive to timing
information.
25
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)
26
Preparing a Mixed State
Randomly rotate the half-waveplates to produce
45gt and -45gt
Preliminary results
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
Visibility (452)
27
Tomography in optical lattices, and steps towards
control...
28
Tomography in Optical Lattices
Myrskog et al., PRA 72, 103615
(05)Kanem et al., J. Opt. B 7, S705 (05)
Complete characterisation of process on arbitrary
inputs?
29
Towards QPTSome 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 0gt 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 0gt
• 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)

30
First task measuring state populations
31
Time-resolved quantum states
32
Recapturing atoms after setting them into
oscillation...
33
...or failing to recapture themif you're too
impatient
34
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 expt.)
35
Quantum state reconstruction
Cf. Poyatos,Walser,Cirac,Zoller,Blatt, PRA 53,
1966 ('96) Liebfried,Meekhof,King,Monroe,Itano,W
ineland, PRL77, 4281 ('96)
36
Husimi distribution of coherent state
37
Data"W-like" Pg-Pe(x,p) for a mostly-excited
incoherent mixture
38
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
39
Extracting a superoperatorprepare 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 ?s 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
40
Towards bang-bang error-correctionpulse echo
indicates T2 1 ms...
Free-induction-decay signal for comparison
echo after bang at 800 ms
echo after bang at 1200 ms
echo after bang at 1600 ms
(bang!)
41
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?
42
Cf. Hannover experiment
Far smaller echo, but far better signal-to-noise
("classical" measurement of ltXgt) 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).
43
A better "bang" pulse for QEC?
Under several (not quite valid) approximations,
the double-shift is a momentum displacement. We
expected a momentum shift to be at least as good
as a position shift. In practice we want to
test the idea of letting learning
algorithms search for the best pulse shape on
their own, and this is a first step.
44
Echo from compound pulse
Pulse 900 us after state preparation, and track
oscillations
single-shift echo (10 of initial oscillations)
double-shift echo (30 of initial oscillations)
Future More parameters find best pulse. Step 2
(optional) figure out why it works! Also
optimize of pulses (given imper- fection of
each)
time ( microseconds)
45
A pleasant surprise from tomography
To characterize processes such as our echo
pulses, we extract the completely positive map or
superoperator, shown here in the Choi-matrix
representation
Ironic fact when performing tomography, none of
our inputs was a very pure ground state, so in
this extraction, we never saw Pe gt 55 or so,
though this predicts 70 upon observing this
superoperator, we went back and confirmed that
our echo can create 70 inversion!
46
What if we try bang-bang?
(Repeat pulses before the bath gets amnesia
trade-off since each pulse is imperfect.)
47
Some coherence out to gt 3 ms now...
48
How to tell how much of the coherence is from the
initial state?
The superoperator for a second-order echo
49
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?)
50
Measurement as a tool Post-selective operations
for the construction of novel (and possibly
useful) entangled states...
51
Highly number-entangled states("low-noon"
experiment).
M.W. Mitchell et al., Nature 429, 161 (2004)
States such as n,0gt 0,ngt ("noon" states) have
been proposed for high-resolution interferometry
related to "spin-squeezed" states.
Important factorisation
52
Trick 1
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...
0gt e 2gt O(e2)
?? 3gt O(?3) O(?2) terms with lt3 photons
0gt ? 1gt O(?2)
But were working on it (collab. with Rich
Mirins quantum-dot group at NIST)
53
Postselective nonlinearity
How to combine three non-orthogonal photons into
one spatial mode?
54
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...
55
The basic optical scheme
56
It works!
Singles
Coincidences
Triple coincidences
Triples (bg subtracted)
57
Complete characterisation when you have
incomplete information
58
Fundamentally Indistinguishablevs.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?
59
Quantum State Tomography
If were 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 dont know how to tell the
particles apart ?
60
The Partial Density Matrix
The sections of the density matrix labelled
inaccessible correspond to information about
the ordering of photons with respect to
inaccessible degrees of freedom.
61
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

Mixture of ?45??45? and ?45??45?
?H??H? ?V??V?
62
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
63
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
J3/2 Bloch sphere
movie of the evolution from 3-noon state to
phase-squeezed, coherent, and number-squeezed
states...

• a slightly number-squeezed state
• a highly phase-squeezed state
• the 3-noon state

64
Conclusions Plea For Help

• Quantum process tomography can be useful for
  characterizing and "correcting" quantum systems
  (ensemble measurements).
• Its actually quite expensive there is still
  much to learn about other approaches, such as
  adaptive tomography, and direct measurements
  of quantities of interest.
• 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.
• 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.