Expériences de Tomographie et d'Échos Quantiques ...entre autres Aephraim Steinberg Les 3 Informaticiens Quantiques: ne regardez pas (évitez le "collapse"!) n'écoutez pas (même principe) ne parlez pas.
Download ReportTranscript Expériences de Tomographie et d'Échos Quantiques ...entre autres Aephraim Steinberg Les 3 Informaticiens Quantiques: ne regardez pas (évitez le "collapse"!) n'écoutez pas (même principe) ne parlez pas.
Expériences de Tomographie et d'Échos Quantiques ...
entre autres Aephraim Steinberg Les 3 Informaticiens Quantiques: ne regardez pas (évitez le "collapse"!) n'écoutez pas (même prin cipe) ne parlez pas (si l'on ad met que ce truc ne marchera ja mais, les subventions ris queront de disparaître)
Séminaire à l'IOTA, mai 2004
DRAMATIS PERSONAE Équipe d'optique quantique et d'atomes froids de Toronto: Postdocs
: Morgan Mitchell ( Barcelona) Marcelo Martinelli ( São Paulo);
TBA (contactez-nous!) Optique
: Jeff Lundeen Lynden(Krister) Shalm Rob Adamson Masoud Mohseni ( Lidar ) Reza Mir ( ?)
Atomes
: Karen Saucke ( Munich) QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.
Jalani Fox Stefan Myrskog ( Thywissen) Ana Jofre( NIST) Kevin Resch( Zeilinger) Mirco Siercke Samansa Maneshi Chris Ellenor
Quelques théoriciens amis
: Daniel Lidar, János Bergou, Mark Hillery, John Sipe, Paul Brumer, Howard Wiseman,...
Plan du séminaire
0. Introduction: informatique quantique, tomographies d'état et de processus 1. Tomographie quantique a. Photons intriqués et filtre à état de Bell b. Atomes dans un réseau optique i – Reconstruction de la fonction de Wigner ii – Reconstruction de la matrice de densité iii –Reconstruction du super-opérateur iv –Echos (correcteurs d'erreurs quantiques?) 2. Quelque chose de bien plus farfelu a. Des «mesures faibles»: est-ce que la mécanique quantique nous permet de parler du passé?
b. Paradoxe de Hardy – résolution théorique(?), et vérification expérimentale imminente...
0
Pourquoi faire la tomographie?
Le Grave Problème Pour l'IQ
• Le risque d'erreurs croît de façon exponentielle avec la taille du système quantique.
• Sans des techniques de correction d'erreurs, l'ordinateur quantique serait un rêve irréaliste. (Il l'est peut-être quand même).
• Une tâche primordiale est d'apprendre comment caractériser complètement l'évolution (et la décohérence) de systèmes quantiques physiques afin de concevoir et de perfectionner des systèmes de contrôle d'erreurs.
• Les outils nécessaires sont la «tomographie d'état quantique» et la «tomographie de processus quantique»: caractérisation totale de la matrice de densité ou de la fonction de Wigner, et du «
$
uperopérateur» qui gère leur évolution temporelle.
• Notre projet de recherche en collaboration avec Daniel Lidar: mettre au point un dispositif de correction quantique d'erreurs «adaptive», afin de pouvoir permettre d'amener les erreurs dans de systèmes réels en dessous du «seuil» critique pour le calcul quantique à grande échelle.
Matrices de densité et superopérateurs
Un photon: H or V.
État: 2 coefficients
( )
C V Matrice de densité: 2x2 = 4 coefficients
( )
C HV C VH C VV Mesurer l'intensité de l'horizontal l'intensité du vertical l'intensité du 45 o l'intensité du circulaire Propagateur (superopérateur): 4x4 = 16 coefficients.
Deux photons: HH, HV, VH, HV, ou toute superposition.
L'état dépend de quatre coefficients.
La matrice de densité de 4x4 = 16 coefficients.
Le superopérateur de 16x16 = 256 coefficients.
La tomographie nous permet d'apprendre:
(et avec un peu de chance, comment lui sauver la vie!)
(cartoon from Jonathan Oppenheim's web pg)
1
Tomographie quantique d'états intriqués de 2 photons
Tomographie de processus à 2 photons
[Mitchell et al., PRL 91, 120402 (2003)] Deux retardateurs préparent chaque photon Détecteur A
l
/2
l
/4
l
/4
l
/2 Polariseur Source à SPDC "Boîte noire": Séparatrice 50/50 Laser Argon
l
/2
l
/4
l
/4
l
/2 Polariseur Deux retardateurs servent pour l'analyse de chaque photon Détecteur B
L'Effet Hong-Ou-Mandel
r r + t t Quel est le taux de détections coïncidentes?
r 2 +t 2 = 0; interf. déstructive totale (pour photons indifférenciables).
Si les photons sont dans un état symmétrique , pas de coïncidences.
{Effet d'échange; imaginez le comportement des fermions dans le dispositif analogue!} Le seul état
antisymmétrique
est l'état «singlet» |HV> – |VH>, dans lequel chaque photon est non-polarisé, but les deux sont orthogonaux.
Ce dispositif constitue un «filtre à états-Bell», crucial pour la téléportation quantique et d'autres applications.
Notre but: caractériser ce filtre à l'aide de la tomographie.
La mesure du superopérateur (I)
Coïncidencences Matrice de Entrée densité } } } HH 16 états "in" HV etc.
VV 16 projections d'analyse VH
La mesure du superopérateur (II)
Entrée HH Sortie ( ) Superopérateur HV VV VH Entrée Sortie etc.
Testing the superoperator
LL =
input state Predicted N photons = 297 ± 14
Testing the superoperator
LL =
input state Predicted N photons = 297 ± 14
BBO two-crystal downconversion source.
Argon Ion Laser Detector A HWP QWP HWP QWP PBS "Black Box" 50/50 Beamsplitter QWP HWP QWP HWP PBS Detector B
Observed N photons = 314
Alors, il est comment, notre filtre?
État «singlet» de Bell: = (HV-VH)/√2
1/2 -1/2 -1/2 1/2
État observé , mais un
autre
état maximalement intriqué:
Comparaison au filtre idéal
Superopérateur, dessiné dans la base des états-Bell: Après transformation pour corriger erreur de polarisation: (Un filtre à états singlets n'aurait qu'un seul pic.) Décomposition en opérateurs de Kraus nous permet d'identifier l'erreur dominant.
(Problème lié à l'asymmétrie de lames séparatrices réelles; facile à résoudre en pratique.)
2
Tomographie de l'état motionnel des atomes piégés et tentatives de «corriger» la décohérence (échos)
Tomography in Optical Lattices
85 Rb atom trapped in one of the quantum levels of a periodic potential formed by standing light field ( ≈ 30GHz detuning, ≈ 7
m
K depth) Complete characterisation of process on arbitrary inputs?
Tailoring of error-correction approaches?
Tomography in Optical Lattices
Atoms trapped in standing waves of light are a promising medium for QIP.
(Deutsch/Jessen, Cirac/Zoller, Bloch,...) First task: learn how to measure state populations
Time-resolved quantum states
Quantum state reconstruction
p p
t
x x x Wait… Shift… p
x x Q(0,0) = Pg W(0,0) =
(-1)n Pn Measure ground state population (OR: can now translate in x and p directly...)
Create a coherent state by shifting
Series1 -1 -0.2
corr midterm
A different value of the delay
final vs midterm, both adjusted to 70 +/- 15 final vs midterm, both adjusted to 70 +/- 15
Series1 -1 -0.2
corr midterm
Oscillations in lattice wells
Ground-state population vs. time bet. translations
QuickTime™ and a Photo - JPEG decompressor are needed to see this picture.
Fancy NLO interpretation: Double-Raman (Ramsey) pump-probe study of vibrational states
Q(x,p) for a coherent H.O. state?
QuickTime™ and a Photo - JPEG decompressor are needed to see this picture.
Radial axis: From amplitude of translation Azimuthal axis: From delay (time axis in previous plot)
Exp't:"W" or [P
g
-P
e
](x,p)
QuickTime™ and a Photo - JPEG decompressor are needed to see this picture.
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)
Atomic state measurement (for a 2-state lattice, with c
0
|0> + c
1
|1>)
initial state displaced delayed & displaced left in ground band tunnels out during adiabatic lowering (escaped during preparation) |c 1 | 2 |c 0 | 2 |c 0 + c 1 | 2 |c 0 + i c 1 | 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
m
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
Superoperator for resonant drive
Operation:
x (resonantly couple 0 and 1 by modulating lattice periodically) Measure superoperator to diagnose single-qubit operation (and in future, to correct for errors and decoherence) Observed Bloch sphere Bloch sphere predicted from truncated harmonic-oscillator
plus
decoherence as measured previously.
Upcoming goals: generate tailored pulse sequences to preserve coherence; determine whether decoherence is Markovian; et cetera.
1.5
2
Towards bang-bang error-correction
Different atoms see different well depths.
If we could do and create a "spin echo". Then we will be able to study tunneling and other "true" decoherence.
We can't do
p p
-pulses, we could reverse the phase evolution -pulses, but we can abruptly shift the lattice, coupling |0> and |1> somewhat.
original oscillations 1 0.5
kick after 800 m s kick after 1200 m s
Some echo is visible, and indicates T 2 around 1ms.
Still shorter than expected; bath correlation time limited by transverse motion in 1D lattice?
kick after 1500 m s 0
500
m
s 1000
m
s
t(10us)
1500
m
s 2000
m
s
250
L'écho de Hannover
Buchkremer, Dumke, Levsen, Birkl, and Ertmer, PRL
85
, 3121 (2000).
time
A better "bang" pulse for QEC?
position shift (previous slide) double shift (similar to a momentum shift)
initial state T = 900 m s initial state T = 900 m s A = –60 ° t = 0 A = –60 ° variable hold delay = t
pulse
t = 0 measurement
t
measurement
t In the impulsive regime (and harmonic-oscillator limit and well-localized atoms and 90-degree phase shift), this pulse amounts to a momentum shift.
In a harmonic-oscillator, this would be equivalent to a position shift; in a periodic potential, one might expect it to be slightly better.
In practice: the most important thing is to have a control parameter, and none of these approximations are truly valid, so the pulse does something much more complicated.
Optimising the pulse
amp
echo amplitudes (extracted from a 3-point partial-tomography measurement) 0.14
0.12
0.1
0.08
0.06
20 40 60 80 100
shift-back delay t ( m s)
120 140
Echo from optimized pulse
Single shift-back pulse Pulse 900 us after state preparation, and track oscillations 1 0.9
0.8
single-shift echo
0.7
0.6
0.5
0.4
0.3
0
double-shift echo
200 400 600 800 1000
time ( microseconds )
1200 1400 1600
Project: with newly automated system, vary all pulse parameters (use genetic algorithms?) in order to optimize final coherence.
Step 2 (optional): figure out why it works!
Est-ce que la mécanique quantique nous permet de parler de ce qui se passe derrière les portes?
Predicting the past...
B+C What are the odds that the particle was in a given box (e.g., box B)?
It had to be in B, with 100% certainty.
Consider some redefinitions...
In QM, there's no difference between a box and any other state (e.g., a superposition of boxes).
What if A is really X + Y and C is really X - Y?
A + B = X+B+Y
X Y
B + C = X+B-Y
A redefinition of the redefinition...
So: the very same logic leads us to conclude the particle was definitely in box X.
X + B' = X+B+Y
X Y
X + C' = X+B-Y
What does this mean?
Then we conclude that if you prepare in (X + Y) + B and postselect in (X - Y) + B, you know the particle was in B.
But this is the same as preparing (B + Y) + X and postselecting (B - Y) + X, which means you also know the particle was in X.
If P(B) = 1 and P(X) = 1, where was the particle really?
But back up: is there any physical sense in which this is true?
What if you try to
observe
where the particle is?
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 depend more on i or f, or equally on both?
Clever answer: both, as Schrödinger time-reversible.
Conventional answer: i, because of collapse.
The Rub
A (von Neumann) Quantum Measurement of A
Initial State of Pointer Final Pointer Readout H int =gAp x
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 H int =gAp x
System-pointer coupling
x x
Poor resolution on each shot. Negligible back-action (system & pointer separable)
Mean pointer shift is given by A
w
f A i f i
Has many odd properties, as we shall see...
The 3-box problem: weak msmts
Prepare a particle in a symmetric superposition of three boxes: A+B+C.
Look to find it in this other superposition: A+B-C.
Ask: between preparation and detection, what was the probability that it was in A? B? C?
A
w
f A i f i
P A P B = < |A>
1.
Questions: were these postselected particles really all in A
and
can this negative "weak probability" be observed?
all in B?
[Aharonov & Vaidman, J. Phys. A 24, 2315 ('91)]
A Gedankenexperiment...
e e e e -
l /2 BS1, PBS l /2 BS2, PBS
The implementation – A 3-path interferometer
(Resch et al., Phys Lett A 324, 125('04))
Diode Laser Spatial Filter: 25um PH, a 5cm and a 1” lens GP A GP B MS, f A GP C MS, f C l /2 BS4, 50/50 BS3, 50/50 PD Screen CCD Camera
Data for P
A
, P
B
, and P
C
...
2 1 Rails A and B 0 Rail C -1 -2 -3 STRONG WEAK STRONG -2 -1 0 1 2 Displacement of Individual Rail (Units of RMS Width) 3
"Interaction-Free Measurements"
(AKA: The Elitzur-Vaidman bomb experiment)
A. C. Elitzur, and L. Vaidman, Found. Phys.
23
, 987 (1993) C D
Consider a collection of bombs so sensitive that is guarranteed to trigger it.
but differ in their behaviour in Bomb absent: Only detector C fires
no way
other than that Bomb present: "boom!" C 1/2 1/4 D 1/4
C + D +
Hardy’s Paradox
L. Hardy, Phys. Rev. Lett.
68
, 2981 (1992) D C O + BS2 + I + I BS2 O W BS1 + BS1 -
Outcome Prob
D+ e- was in D + and C and C + 1/16 C + and C 9/16 D + and D 1/16 Explosion 4/16 e + e -
Experimental Setup Det. V (
D+
) Det. H (
D-
) 50-50
BS2
CC GaN PBS Diode Laser PBS DC BS 50-50
BS1
(
W
) V CC Switch H DC BS
But what can we say about where the particles were or weren't, once D+ & D– fire?
Probabilities e- in e+ in
0
e- out 1 1 e+ out 1
1
0 1 0
Upcoming experiment: demonstrate that "weak measurements" (à la Aharonov + Vaidman) will bear out these predictions.
Merci de votre attention!
(dessin de Jon Dowling)