Aephraim Steinberg

Centre for Quantum Info. & Quantum ControlInstitute for Optical Sciences

Department of Physics University of Toronto

Measuring & manipulating coherencein photonic & atomic systems

PITP/CQIQC Workshop: “Decoherence at the Crossroads”

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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,...

OUTLINE

Some things you may already know

Some things you probably haven’t heard...but on which we’d love (more) collaborators!

A few words about Quantum Information,about photons, and about state & process tomography

Two-photon process tomographyHow to avoid quantum state & process tomography?Tomography of trapped atoms, and attempts at controlComplete characterization given incomplete

experimental capabilitiesHow to draw Wigner functions on the Bloch sphere?

“Never underestimate the pleasure people get from hearing something they already know”

Quantum tomography: why?

0

Quantum InformationWhat's so great about it?

Quantum InformationWhat'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 takesΣn | >|0> n toΣn | >| ( )>; ( )n f n it calculates f n

.for every possible input simultaneouslyA clever measurement may determine some globalproperty of f(n) even though the computer hasonly run once...

The rub: any interaction with the environmentleads to "decoherence," which can be thoughtof as continual unintentional measurement of n.

A not-clever measurement "collapses" n to somerandom value, and yields f(that value).

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!)

Quantum Computer Scientists

What makes a computer quantum?

across the Danube

(...Another talk, or more!)

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.

Density matrices and superoperatorsOne photon: H or V.State: two coefficients ()CHCV

( )CHH

CHV

CVH

CVV

Density matrix: 2x2=4 coefficientsMeasure

intensity of horizontalintensity of verticalintensity of 45ointensity 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.

Quantum process tomographyon photon pairs

1

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.

HWP

HWP

HWP

HWP

QWP

QWPQWP

QWPPBS

PBS

Argon Ion Laser

Beamsplitter"Black Box" 50/50

Detector B

Detector ATwo waveplates per photonfor state preparation

Two waveplates per photon for state analysis

SPDC source

Two-photon Process Tomography[Mitchell et al., PRL 91, 120402 (2003)]

“Measuring” the superoperator

}

Output DM Input

HH

HV

VV

VH

}

}}

etc.

16 analyzer settings

16 input states

Coincidencences

“Measuring” the superoperatorInput Output DM

HH

HV

VV

VH

etc.

Superoperator

Input Output

Comparison to ideal filterMeasured superoperator,in Bell-state basis:

A singlet-state filter would havea single peak, indicating the onetransmitted state.

Superoperator after transformationto correct polarisation rotations:

Dominated by a single peak;residuals allow us to estimatedegree of decoherence andother errors.

Can we avoid doing tomography?

2

• 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.

(T. A. Brun, e-print: quant-ph/0401067)

Polynomial Functions of a Polynomial Functions of a Density MatrixDensity Matrix

• 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

HOM as Singlet State Filter

+

Pure State on either side = 100% visibility

HH

HH

HH

H

HOM Visibility = Purity

Mixed State = 50% visibility

+V

HHH

HV V

V

Linear Purity of a Quantum State

P = 1 – 2 – –

Singlet-state probability can be measured by a singlet-state filter (HOM)

•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

Coincidence Circuit

Single PhotonDetector

Single PhotonDetector

€

λ /2

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λ /2

Type 1 SPDC Crystal

Singlet Filter

QuartzSlab

QuartzSlab

Experimentally Measuring the Purity of a Qubit

Prepared the state |+45>

Measured Purity from Singlet State MeasurementP=0.92±0.02

Measured Purity from QSTP=0.99±0.01

Measuring +45 +45

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3500

0 50 100 150 200 250 300 350

Delay (um)

Counts per 30 s

Results For a Pure 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 ]

€

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λ /2

Case 1: Same birefringence in each arm

V

V

H

H

100% interference

Visibility = (90±2) %

€

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Case 2: Opposite birefringence in each arm

V

V

H

H

25% interference

H and V Completely Decohered Due to Birefringence

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Delay (um)

Counts per 30s

Visibility = (21±2) %

Preparing a Mixed State

The HOM isn’t actually insensitive to timing information.

• 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)

Not a singlet filter, but an “Antisymmetry Filter”

Randomly rotate the half-waveplates to produce |45> and |-45>

No Birefringence, Even Mixture of +45/+45 and +45/-45

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Delay (um)

Counts per 30 s

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|45>

|45> or |-45>

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

Could produce a “better” maximally mixed state by using four photons. Similar to Paul Kwiat’s work on Remote State Preparation.

Coincidence Circuit

€

λ /2

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λ /2

Visibility = (45±2) %

Preparing a Mixed State

Tomography in optical lattices, and steps towards control...

3

Rb atom trapped in one of the quantum levelsof a periodic potential formed by standing

light field (30GHz detuning, 10s of K depth)

Tomography in Optical Lattices

[Myrskog et al., PRA 72, 103615 (’05)Kanem et al., J. Opt. B 7, S705 (’05)]

Complete characterisation ofprocess 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 them into oscillation...

...or failing to recapture themif you're too impatient

Oscillations in lattice wells(Direct probe of centre-of-mass oscillations in 1m wells;can be thought of as Ramsey fringes or Raman pump-probe exp’t.)

x

p

ωt

Wait…

Quantum state reconstruction

x

p

Shift…

x

Cf. Poyatos,Walser,Cirac,Zoller,Blatt, PRA 53, 1966 ('96)& Liebfried,Meekhof,King,Monroe,Itano,Wineland, PRL77, 4281 ('96)

Measure groundstate population

x

p

x

(former for HO only; latter requires only symmetry)

Q(0,0) = Pg1

W(0,0) = Σ (-1)n Pn1

Husimi distribution of coherent state

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Data:"W-like" [Pg-Pe](x,p) for a mostly-excited incoherent mixture

Atomic state measurement(for a 2-state lattice, with c0|0> + c1|1>)

left inground band

tunnels outduring adiabaticlowering

(escaped duringpreparation)

initial state displaced delayed & displaced

|c0|2 |c0 + c1 |2 |c0 + i c1 |2

|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 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

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2

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comparing oscillations for shift-backs applied after time t

1/(1+2)

t(10us)

0 500 s 1000 s 1500 s 2000 s

Towards bang-bang error-correction:pulse 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

decay of coherence introduced by echo pulses themselves (since they are not perfect -pulses)

(bang!)

Why does our echo decay?

Finite bath memory time:

So far, our atoms are free to move in the directions transverse toour lattice. In 1 ms, they move far enough to see the oscillationfrequency change by about 10%... which is about 1 kHz, and henceenough to dephase them.

Inter-well tunneling should occur on a few-ms timescale... should one thinkof this as homogeneous or inhomogeneous? “How conserved” is quasimomentum?

time ( microseconds)

single-shift echo(≈10% of initial oscillations)

double-shift echo (≈30% of initial oscillations)

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1

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Echo amplitude for a single shift-back vs. a pulse (shift-back, delay, shift) at 900 us

Single shift-backpulse

ground state ratio

Echo from compound pulse

Future: More parameters; find best pulse.Step 2 (optional): figure out why it works!Also: optimize # of pulses (given imper-fection of each)

Pulse 900 us after state preparation,and track oscillations

What if we try “bang-bang”?(Repeat pulses before the bath gets amnesia; trade-off since each pulseis 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 studyeffects of tunneling

• BEC apparatus: reconstruct single-particle wavefunctionscompletely 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-particleinteractions (free expansion? approach to Mott? etc?)

Measurement as a tool:Post-selective operations for the construction of

novel (and possibly useful) entangled states...

4a

Theory: H. Lee et al., Phys. Rev. A 65, 030101 (2002); J. Fiurásek, Phys. Rev. A 65, 053818 (2002)

Highly number-entangled states("low-noon" experiment).

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!

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.

How to combine three non-orthogonal photons into one spatial mode?

Postselective nonlinearityPostselective nonlinearity

Yes, it's that easy! If you see three photonsout one port, then they all went out that port.

"mode-mashing"

It works!

Singles:

Coincidences:

Triplecoincidences:

Triples (bgsubtracted):

Complete characterisationwhen you have incomplete information

4b

Fundamentally Indistinguishablevs.

Experimentally Indistinguishable

But what if when we combine our photons,there is some residual distinguishing information:some (fs) time difference, some small spectraldifference, some chirp, ...?

This will clearly degrade the state – but how dowe characterize this if all we can measure ispolarisation?

LeftArnold RightDanny

OR –Arnold&Danny ?

Quantum State Tomography

{ }21212121 ,,, VVVHHVHH

Distinguishable Photon Hilbert Space

{ }, ,HH HV VH VV+

{ }2 ,0 , 1 ,1 , 0 ,2H V H V H V

Indistinguishable Photon Hilbert Space

??Yu. I. Bogdanov, et alPhys. 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, giventhat we don’t know how to tell the particles apart ?

The sections of the density matrix labelled “inaccessible” correspond to information about the

ordering of photons with respect to inaccessible degrees of freedom.

The Partial Density Matrix

Inaccessible

information

Inaccessible

information

€

HH ,HH ρ HV +VH ,HH ρVV ,HH

ρ HH ,HV +VH ρ HV +VH ,HV +VH ρVV ,HV +VH

ρ HH ,VV ρ HV +VH ,VV ρVV ,VV

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

ρ HV −VH ,HV −VH( )

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

The answer: there are only 10 linearly independent parameters which are invariant under permutations of the particles. One example:

When distinguishing information is introduced the HV-VH component increases without affecting the state in the symmetric space

Experimental Results (2 photons)No Distinguishing Info Distinguishing Info

HH + VVMixture of45–45 and –4545

More Photons…

So the total number of operators accessible to measurement is

( ) ( )( )( )

( )2

2/2

1states symmetric onto projectors of # Total

4projectors of # Total

6/12312ops blind-ordering ofNumber

+=

=

+++=+= ∑

N

NNNj

N

N

j

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

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Wigner distributions on the Poincaré sphere

a. a slightly “number”-squeezed state

b. a highly phase-squeezed state

c. the “3-noon” state

movie of the evolution from 3-noon state to phase-squeezed, coherent, and “number”-squeezed states...

Some polarisation states of the fully symmetric triphoton (theory– for the moment), drawn on the J=3/2 Bloch sphere:

[Following recipe of Dowling, Agarwal, & Schleich, PRA 49, 4101 (1993).]

3H,0V 2H,1V 1H,2V 0H,1V

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.