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

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

Aephraim Steinberg

Dept. of Physics, University of Toronto

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

AcknowledgmentsOUTLINE

Something you already know

Introduction to quantum information with optics

Something you may have known...

but may have forgotten by now

All good talks are alike...

every bad talk is bad in its own way.

Making a strong effective interaction between

two photons

Something you most likely haven\'t heard before

Quantum state and process tomography for q. info.

Something you may not even buy

Weak measurements -- Hardy\'s Paradox et cetera:

"How much can we know about a photon?"

pros & cons of optical schemes...

Quantum Information

What\'s so great about it?

Quantum Information

What\'s so great about it?

Quantum Interference for effective

single-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 l3 regime, but also resonant nonlinearity?

- Cf. talks by Walmsley, Fejer, Gaeta,...

Cavity QED (example of l3 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) ?

The germ of the KLM idea

INPUT STATE

OUTPUT STATE

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

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

TRIGGER (postselection)

ANCILLA

|1>

|1>

In particular: with a similar but somewhat more complicated

setup, one can engineer

a |0> + b |1> + c |2> a |0> + b |1> – c |2> ;

effectively a huge self-phase modulation (p 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

atom 1

w1

w2

atom 2

w2

w1

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

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)?

Appears to violate local energy conservation... but consists of perfectly

reasonable 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.

John Sipe\'s suggestion

Two-photon absorption (by these

single-photon absorbers) is inter-

ferometrically enhanced if the

photons begin distinguishable, but

are indistinguishable to the absorber:

T2 > t > tc

Franson\'s proposal to harness photon-exchange terms investigates the

effect 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-photon

absorption 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) !

Ugly data,but it works.

Resch et al. quant-ph/0306198

Roughly a 4% drop observed in 2-photon transmission when

the photons are delayed relative to one another.

Complicated by other effects due to straightforward frequency

correlations between photons (cf. Wong, Sergienko, Walmsley,...),

as well as correlations between spatial and spectral mode.

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?

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

The probability of 2 photons upconverting in a typical

nonlinear crystal is roughly 10-10 (as is the probability

of 1 photon spontaneously down-converting).

Suppression/Enhancementof Spontaneous Down-Conversion

(57% visibility)

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 switch

Resch et al, Phys. Rev. Lett. 89, 037914 (2002)

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 or

not 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...).

"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...

Quantum State/Process TomographyTwo-photon Process Tomography(Mitchell et al., quant-ph/0305001)

"Black Box" 50/50

Beamsplitter

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

Hong-Ou-Mandel Interference

r

t

+

t

r

How often will both detectors fire together?

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 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.

Comparison to ideal filter

Measured superoperator,

in Bell-state basis:

Superoperator after transformation

to correct polarisation rotations:

A singlet-state filter would have

a single peak, indicating the one

transmitted state.

Dominated by a single peak;

residuals allow us to estimate

degree of decoherence and

other errors.

Tomography in Optical Lattices

Atoms 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…

Quantum state reconstruction

D

x

Wait…

Shift…

Measure ground

state population

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

Oscillations in lattice wells

Ground-state population vs. time bet. translations

Fancy NLO interpretation:

Raman pump-probe study of vibrational states

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

initial state

displaced

delayed & displaced

left in

ground band

tunnels out

during adiabatic

lowering

(escaped during

preparation)

|c0 + i c1 |2

|c0|2

|c0 + c1 |2

|c1|2

Atom superoperators

sitting in lattice, quietly

decohering…

being shaken back and

forth resonantly

Initial Bloch sphere

CURRENT PROJECTS:

On atoms, incorporate "bang-bang" (pulse echo) to

preserve coherence & measure homog. linewidth.

With photons, study "tailored" quantum error

correction (adaptive encodings for collective noise).

Conditional measurements(Aharonov, Albert, and Vaidman)

Measurement

of A

AAV, PRL 60, 1351 (\'88)

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.

Final Pointer Readout

Hint=gApx

System-pointer

coupling

x

x

Well-resolved states

System and pointer become entangled

Decoherence / "collapse"

Large back-action

A (von Neumann) Quantum Measurement of A A Weak Measurement of A

Initial State of Pointer

Final Pointer Readout

Hint=gApx

System-pointer

coupling

x

x

Poor resolution on each shot.

Negligible back-action (system & pointer separable)

Mean pointer shift is given by

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

"Interaction-Free Measurements"

D

C

BS2

BS1

(AKA: The Elitzur-Vaidman bomb experiment)

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

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 that

they will not blow up when triggered.

Is there any way to identify the working bombs (or

some of them) without blowing them up?

Bomb absent:

Only detector C fires

Bomb present:

"boom!" 1/2

C 1/4

D 1/4

Hardy Cartoon

D+

D-

C+

C-

BS2+

BS2-

I+

I-

O-

O+

W

BS1+

BS1-

e-

e+

Hardy’s Paradox

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

D+ e- was in

D- e+ was in

D+D- ?

But …

Det. V (D+)

Det. H (D-)

50-50

BS2

CC

PBS

50-50

BS1

PBS

GaN

Diode Laser

(W)

CC

V

H

DC BS

DC BS

Switch

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

0

1

1

-1

Upcoming experiment: demonstrate that "weak

measurements" (à la Aharonov + Vaidman) will

bear out these predictions.

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!

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