Aephraim M. Steinberg Centre for Q. Info. & Q. Control

Institute for Optical SciencesDept. of Physics, U. of Toronto

Measuring & manipulating quantum states(Fun With Photons and Atoms)

GA Tech, March 2007

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DRAMATIS PERSONÆ

Toronto quantum optics & cold atoms group: Postdocs: Matt Partlow ( ...) Morgan Mitchell ( ICFO)

An-Ning Zhang( IQIS) Marcelo Martinelli ( USP)

Optics: Rob Adamson Kevin Resch(Wien UQIQC)

Lynden(Krister) Shalm Jeff Lundeen (Oxford)

Xingxing Xing

Atoms: Jalani Fox (Imperial) Stefan Myrskog (BECECE)

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: Daniel James, János Bergou, Pete Turner, John Sipe, Paul Brumer, Howard Wiseman, Michael Spanner,...

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

OUTLINE

The grand unified theory of physics talks:“Never underestimate the pleasure people

get from being told something they already know.”

OUTLINE

Something slightly insane, in case you foundthe rest of the talk too boring...

Some things you (may) already know

Some things you probably haven’t seen before...Quantum info, entanglement, nonunitary op’s

Preparing entangled states of N photons Subtleties of measuring multi-photon states Motional-state tomography on trapped atomsDecoherence & progress on echoes

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?

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?

(One partial answer...)

(NB: Product states only require 2n coeff’s, butnon-entangled quantum computation couldequally well be performed with classical waves)

How to build a quantum computerPhotons don't interact

(good for transmission; bad for computation)

Try: atoms, ions, molecules, ...or just be clever with photons!

Nonlinear optics: photon-photon interactionsGenerally exceedingly weak.

Potential solutions:Cavity QEDBetter 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

special |i >

a|0> + b|1> + c|2> a|0> + b|1> – c|2>

Measurement as a tool: KLM...INPUT STATE

ANCILLA TRIGGER (postselection)

OUTPUT STATE

particular |f >

Knill, Laflamme, Milburn Nature 409, 46, (2001); and others since.Experiments by Franson et al., White et al., Zeilinger et al...

MAGIC MIRROR:Acts differently if there are 2 photons or only 1.In other words, can be a “transistor,” or “switch,”or “quantum logic gate”...

Building up entanglement photon by photonby using post-selective nonlinearity

1

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!

States such as |n,0> + |0,n> ("noon" states) have been proposed for high-resolution interferometry – related to "spin-squeezed" states.

M.W. Mitchell et al., Nature 429, 161 (2004)

Trick #1

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

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

"mode-mashing"

Post-selective nonlinearity

Trick #2

Okay, we don't even have single-photon sources*.

But we can produce pairs of photons in down-conversion, andvery weak coherent states from a laser, such that if we detectthree photons, we can be pretty sure we got only one from thelaser and only two from the down-conversion...

SPDC

laser

|0> + |2> + O(2)

|0> + |1> + O(2)

|3> + O(3) + O(2) + terms with <3 photons

* But 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 apolarizer, it's no longer at 45o. If it gets through a partial polarizer, it could be anywhere...

(or nothing)

(or <2 photons)

(or nothing)

Experimental Setup

It works!

Singles:

Coincidences:

Triplecoincidences:

Triples (bgsubtracted):

M.W. Mitchell, J.S. Lundeen, and A.M. Steinberg, Nature 429, 161 (2004)

Complete characterisationwhen you have incomplete information

4b1b

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?

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

€

ρHH ,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:

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)

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

Experimental ResultsNo Distinguishing Info Distinguishing Info

HH + VVMixture of45–45 and –4545

A typical 3-photon density matrix

My favorite cartoon about Alice (almost) and Bob

A better description thandensity matrices?

4b1c

Wigner distributions on the Poincaré sphere ?

Any pure state of a spin-1/2 (or a photon) can be represented as a pointon the surface of the sphere – it is parametrized by a single amplitude anda single relative phase.

(Consider a purely symmetric state: N photons act like a single spin-N/2)

This is the same as the description of a classical spin, or the polarisation(Stokes parameters) of a classical light field.

0

0.1

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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

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

, while for higher spin it shrinks.

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?

-0.2

-0.1

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“Coherent state” = N identically polarized photons

“Spin-squeezed state” trades offuncertainty in H/V projection formore precision in phase angle.

Dowling, Agarwal, & Schleich, PRA 49, 4101 (1993).

Beyond 1 or 2 photons...

-1

-0.8

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15-noon

-0.2

-0.1

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squeezed state

2 photons:4 param’s: Euler angles + squeezing (eccentricity) + orientation

-0.8

-0.6

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0

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0.83-noon

3 photons:6 parameters: Euler angles + squeezing (eccentricity) + orientation + more complicated stuff

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A 1-photon pure state may be represented by a point on the surfaceof the Poincaré sphere, because there are only 2 real parameters.

Ideal N00N state N00N state + background

THEORYMeasured Wigner Function

SqueezingMeasured density matrix

Preliminary Experimental Results

EXPERIMENT

NOTE: To study a broad range of entangled states, a more flexible “mode-masher” is needed – the tradeoff is lower efficiency, which decreases SNR

Quantum CAT scans

2

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

[Myrkog et al., PRA 72, 013615 (05)Kanem et al., J. Opt. B7, S705 (05)]

Complete characterisation ofprocess on arbitrary inputs?

First task: measuring state populations

Time-resolved quantum states

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) = (-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

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.)

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

0

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0 50 100 150 200 250

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

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

(bang!)

Cf. Hannover experiment

Buchkremer, Dumke, Levsen, Birkl, and Ertmer, PRL 85, 3121 (2000).

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.

Echo from compound pulse

Ongoing: More parameters; find best pulse.E.g., combine amplitude & phase mod.

Also: optimize # of pulses.

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?

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?

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.

3D lattice (first data)

Except for one minor disturbing feature:

We see similar plateaux with both 1D and 3D lattices; temporal (laser) noise 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)

Prepare a particle in |i> …try to "measure" some observable A…postselect the particle to be in |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.

ii ffMeasurement of A

Reconciliation: measure A "weakly."Poor resolution, but little disturbance.

…. can be quite odd …

if

iAfA =w

AAV, PRL 60, 1351 ('88)

“Quantum Seeing in the Dark”

3a

Problem:Consider a collection of bombs so sensitive that

a collision with any single particle (photon, electron, etc.)is guarranteed to trigger it.

Suppose that certain of the bombs are defective,but differ in their behaviour in no way other than thatthey will not blow up when triggered.

Is there any way to identify the working bombs (orsome of them) without blowing them up?

" 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)

BS1

BS2

DC

Bomb absent:Only detector C fires

Bomb present:"boom!" 1/2 C 1/4 D 1/4

The bomb must be there... yetmy photon never interacted with it.

BS1-

e-

BS2-

O-

C-D-

I-

BS1+

BS2+

I+

e+

O+

D+C+

W

Outcome Prob

D+ and C- 1/16

D- and C+ 1/16

C+ and C- 9/16

D+ and D- 1/16

Explosion 4/16

Hardy's Paradox(for Elitzur-Vaidman “interaction-free measurements”)

D- –> e+ was in

D+D- –> ?

But … if they wereboth in, they shouldhave annihilated!

D+ –> e- was in

The two-photon switch...OR: Is SPDC really the time-reverse of SHG?

The probability of 2 photons upconverting in a typicalnonlinear crystal is roughly 10 (as is the probabilityof 1 photon spontaneously down-converting).

(And if so, then why doesn't it exist in classical e&m?)

Quantum Interference

(57% visibility)

Suppression/Enhancementof Spontaneous Down-Conversion

H Pol DC

V Pol DC

407 nm Pump

Using a “photon switch” to implement Hardy’s Paradox

Probabilities e- in e- out

e+ in 1

e+ out 0

1 0

0 1

1 1

But what can we say about where the particles were or weren't, once D+ & D– fire?

In fact, this is precisely what Aharonov et al.’s weak measurementformalism predicts for any sufficiently gentle attempt to “observe”these probabilities...

N(I-) N(O)

N(I+) 0 1 1N(O+) 1 1 0

1 0

0.039±0.0230.925±0.024

0.087±0.0210.758±0.0830.721±0.074N(O+)

0.882±0.0150.663±0.0830.243±0.068N(I+)

N(O)N(I-)

Experimental Weak Values (“Probabilities”?)

Ideal Weak Values

Weak Measurements in Hardy’s Paradox

The moral of the story1. Post-selected systems often exhibit surprising behaviour

which can be probed using weak measurements.

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

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