Aephraim Steinberg Dept. of Physics, University of Toronto

Nonlinear optics at the quantum level and quantum information in optical systems

2003 GRC on Nonlinear Optics & Lasers

U of T quantum optics & laser cooling group:

PDFs: Morgan Mitchell Marcelo Martinelli

Optics: Kevin Resch(Zeilinger) Jeff Lundeen

Chris Ellenor Masoud Mohseni (Lidar)

Reza Mir Rob Adamson

Karen Saucke (visiting from Munich)

Atom Traps: Stefan Myrskog Jalani Fox

Ana Jofre Mirco Siercke

Samansa Maneshi Salvatore Maone ( real world)

Some of our theory friends: Daniel Lidar, Janos Bergou, Mark Hillery, John Sipe, Paul Brumer, Howard Wiseman

Acknowledgments

OUTLINE

Something you already know

Something you may have known...but may have forgotten by now

Something you most likely haven't heard before

Something you may not even buy

Introduction to quantum information with optics

Making a strong effective interaction betweentwo photons

Quantum state and process tomography for q. info.

Weak measurements -- Hardy's Paradox et cetera:"How much can we know about a photon?"

All good talks are alike... every bad talk is bad in its own way.

Intro to Quantum Info -- pros & cons of optical schemes...

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).

Quantum Computer Scientists

What makes a computer quantum?

Quantum Interference for effectivesingle-photon–single-photon interactions...?

Can we build a two-photon switch?Photons don't interact(good for transmission; bad for computation)

Nonlinear optics: photon-photon interactions generally exceedingly weak.

Potential solutions:Better materials (1010 times better?!)

- Want 3 regime, but also resonant nonlinearity?- Cf. talks by Walmsley, Fejer, Gaeta,...

Cavity QED (example of 3 regime plus resonance)- Kimble, Haroche, Walther, Rempe,...

EIT, slow light, etc...- Lukin, Fleischhauer, Harris, Scully, Hau,...

Measurement as nonlinearity (KnillLaflammeMilburn)- KLM; Franson, White,...

Other quantum interference effects?- Exchange effects in quantum NLO (Franson) ?- Interferometrically-enhanced SHG, etc (us) ?

|1>

a|0> + b|1> + c|2> a'|0> + b'|1> + c'|2>

The germ of the KLM ideaINPUT STATE

ANCILLA TRIGGER (postselection)

OUTPUT STATE

|1>

In particular: with a similar but somewhat more complicatedsetup, one can engineer

a |0> + b |1> + c |2> a |0> + b |1> – c |2> ;effectively a huge self-phase modulation ( per photon).More surprisingly, one can efficiently use this for scalable QC.

KLM Nature 409, 46, (2001); Cf. experiments by Franson et al., White et al., ...

The mad, mad idea of Jim Franson

Nonlinear coefficients scale linearly with the number of atoms.Could the different atoms' effects be made to add coherently, providing an N2 enhancement (where N might be 1013)?

atom 1

atom 2

1

1

2

2

Appears to violate local energy conservation... but consists of perfectlyreasonable Feynman diagrams, with energy conserved in final state.

{Controversy regarding some magic cancellations....}

Each of N(N-1)/2 pairs of atoms should contribute. Franson proposes that this can lead to immense nonlinearities. No conclusive data.

J.D. Franson, Phys. Rev. Lett 78, 3852 (1997)

John Sipe's suggestionFranson's proposal to harness photon-exchange terms investigates theeffect on the real index of refraction (virtual intermediate state).

Why not first search for such effects on real intermediate states (absorption)?

Conclusion: exchange effects do matter: Probability of two-photonabsorption may be larger than product of single-photon abs. prob's.

Caveat: the effect indeed goes as N2, ... but N is the photon number (2) and not the atom number (1013) !

Two-photon absorption (by thesesingle-photon absorbers) is inter-ferometrically enhanced if thephotons begin distinguishable, butare indistinguishable to the absorber:

T2 > > c

Ugly data,but it works.

Roughly a 4% drop observed in 2-photon transmission whenthe photons are delayed relative to one another.

Complicated by other effects due to straightforward frequencycorrelations between photons (cf. Wong, Sergienko, Walmsley,...),as well as correlations between spatial and spectral mode.

Resch et al. quant-ph/0306198

What was the setup?Type-II SPDC + birefringent delay + 45o polarizer produces delayed pairs.Use a reflective notch filter as absorbing medium, and detect remaining pairs.

This is just a Hong-Ou-Mandel interferometer, with detection in a complementary mode.Although the filter is placed after the output, this is irrelevant for a linear system.

Interpretations: • Our "suppressed" two-photon reflection is merely the ratio of two different interference patterns; the modified spectrum broadens the pattern.• Yet photons which reach the filter in pairs really do not behave independently. The HOM interference pattern is itself a manifestation of photon exchange effects.

Another approach to 2-photon interactions...Ask: Is SPDC really the time-reverse of SHG?

The probability of 2 photons upconverting in a typicalnonlinear crystal is roughly 100 (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

Photon-photon transmission switch

On average, less than one photon per pulse.One photon present in a given pulse is sufficient to switch off transmission.

The photons upconvert with near-unit eff. (Peak power approx. mW/cm2).The blue pump serves as a catalyst, enhancing the interaction by 1010.

Controlled-phase switchResch et al, Phys. Rev. Lett. 89, 037914 (2002)

Fringe data with and w/o postsel.

So why don't we "rule the world"?

N.B.:This switch relies on interference.Input state must have specific phase.Single photons don't have well-defined phase.

The switch does not work on Fock states.

The phase shifts if and only if a control photon is present--so long as we make sure not to know in advance whether ornot it is present. Another example of postselected logic.

Nonetheless:Have shown theoretically that a polarisation version could be used for Bell-state determination (and, e.g., dense coding)… a task known to be impossible with LO.[Resch et al., quant-ph/0204034]

Present "application," however, is to a novel test of QM(later in this talk, with any luck...).

Characterisation of quantum processes in QI systems

Quantum State/Process Tomography

• "Pre"-QI: Wigner function for nonclassical light (Raymer et al), molecules (Walmsley et al), et cetera

• Kwiat/White et al.: tomography of entangled photons; entanglement-assisted tomography

• Jessen et al.: density matrix reconstruction for high-spin state (9x9 density matrix in F=4 Cs)

• Cory et al.: use of superoperator to design QEC pulse sequences for NMR (QFT etc)

• Many, many people I've omitted...

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, HV, or any superpositions.State has four coefficients.Density matrix has 4x4 = 16 coefficients.Superoperator has 16x16 = 256 coefficients.

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., quant-ph/0305001)

Hong-Ou-Mandel Interference

How often will both detectors fire together?

r

r

tt

+

r2+t2 = 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

|HV> – |VH>, in which each photon isunpolarized but the two are orthogonal.

This interferometer is a "Bell-state filter," neededfor quantum teleportation and other applications.

Our Goal: use process tomography to test this filter.

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.

Tomography in Optical LatticesAtoms trapped in standing waves of light are a promising medium for QIP.

(Deutsch/Jessen, Cirac/Zoller, Bloch,...)We would like to characterize their time-evolution & decoherence.

First: must learn how to measure state populations in a lattice…

Time-resolved quantum states

x

p

t

Wait…

Quantum state reconstruction

Q(0,0) = Pg

W(0,0) = Σ (-)n PnMeasure groundstate population

x

p

Δx

x

p

Shift…

Δx

(OR: can now translate in x and p directly...)

Create a coherent state by shifting lattice; delay and shift to measure W.

A different value of the delay

Oscillations in lattice wells

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Ground-state population vs. time bet. translations

Fancy NLO interpretation: Raman pump-probe study of vibrational states

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Exp't:"W" or [Pg-Pe](x,p)

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

input density matrices output density matrices

Time-evolution of some states

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Atom superoperators

Initial Bloch sphere

sitting in lattice, quietlydecohering…

being shaken back and forth resonantly

CURRENT PROJECTS:

On atoms, incorporate "bang-bang" (pulse echo) topreserve coherence & measure homog. linewidth.

With photons, study "tailored" quantum errorcorrection (adaptive encodings for collective noise).

Can we talk about what goes on behind closed doors?

Pick a box, any box...

A+B+C

A

What are the odds that the particlewas in a given box?

+B–C

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

AAV, PRL 60, 1351 ('88)

A (von Neumann) Quantum Measurement of A

Well-resolved statesSystem and pointer become entangled

Decoherence / "collapse"Large back-action

Initial State of Pointer

x x

Hint=gApx

System-pointercoupling

Final Pointer Readout

A Weak Measurement of A

Hint=gApx

System-pointercoupling

x

Initial State of Pointer

x

Final Pointer Readout

Poor resolution on each shot. Negligible back-action (system & pointer separable)

Mean pointer shift is given byif

iAfA =w

Has many odd properties, as we shall see...

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?

"Interaction-Free Measurements"(AKA: The Elitzur-Vaidman bomb experiment)

A. C. Elitzur, and L. Vaidman, Found. Phys. 23, 987 (1993)

BS1

BS2

DC

Bomb absent:Only detector C fires

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

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 L. Hardy, Phys. Rev. Lett. 68, 2981 (1992)

D- e+ was in

D+D- ?

But …

D+ e- was in

GaNDiode Laser

PBS

Det. H (D-)Det. V (D+)

DC BS DC BS

50-50BS1

50-50BS2

Switch H

V

CC

CC

PBS

BS1-

BS2-

O-

C-

BS1+

BS2+

I+

e+ e-

I-O+

D+C+ D-

WBS1-

BS2-

O-

C-

BS1+

BS2+

I+

e+ e-

I-O+

D+C+ D-

W

Experimental Setup

(W)

0

50

100

150

200

250

300

350

400

-1 -0.5 0 0.5 1 1.5 2 2.5 3

Switch Visibility

Delay (microns)

Vis = 85.7 +/- 0.1 %

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6

Vertical Polarization Mach-Zehnder

Delay (microns)

Vis = 90 +/- 1%

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?

Upcoming experiment: demonstrate that "weakmeasurements" (à la Aharonov + Vaidman) willbear out these predictions.

PROBLEM SOLVED!(?)

SUMMARY• Quantum interference allows huge enhancements of effective optical nonlinearities. How do they relate to"real" nonlinearities? What are or aren't they good for?

• Two-photon switch useful for studies of quantum weirdness(Hardy's paradox, weak measurement), and Bell-state detection.

• Two-photon process tomography useful for characterizing various candidate QI systems.Next round of experiments on tailored quantum error correction (w/ D. Lidar et al.).

• As we learn to control individual quantum systems, more and more applications of postselection appear; need to learn how to think about postselected subensembles (weak measurement, conditional logic, et cetera). (see Steinberg, quant-ph/0302003)

• No matter what the Silicon crowd thinks, there's a lot of mileage left in (nonlinear/quantum) optics!