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 MitchellMarcelo Martinelli Optics: Kevin Resch(  Zeilinger)Jeff Lundeen Chris Ellenor Masoud Mohseni (  Lidar) Reza MirRob Adamson Karen Saucke (visiting from Munich) Atom Traps: Stefan MyrskogJalani Fox Ana JofreMirco Siercke Samansa ManeshiSalvatore 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 between two 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 Information What's so great about it?

Quantum Information What's so great about it?

Quantum Computer Scientists

What makes a quantum computer?

What makes a computer quantum?

Conventional Answers 1.Computers are made from Silicon, not photons. 2.Maybe trapped atoms/ions have some of the advantages of photons without the disadvantages. 3.Maybe SQUIDs or quantum dots or something else will prove the right technology instead. 4.Maybe using quantum measurement and postselection as an "effective interaction" will save the day for optics. 5.Maybe photons can be made to interact better after all…

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 (10 10 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 (K nill L aflamme M ilburn ) - 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 idea INPUT STATE ANCILLA TRIGGER (postselection) OUTPUT STATE |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 (  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 N 2 enhancement (where N might be )? atom 1 atom 2 11 11 22 22 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. J.D. Franson, Phys. Rev. Lett 78, 3852 (1997)

John Sipe's suggestion 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 N 2,... but N is the photon number (2) and not the atom number (10 13 ) ! Two-photon absorption (by these single-photon absorbers) is inter- ferometrically enhanced if the photons begin distinguishable, but are indistinguishable to the absorber: T 2 >  >  c

Ugly data,but it works. 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. Resch et al. quant-ph/

What was the setup? Type-II SPDC + birefringent delay + 45 o 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.

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.

Another approach to 2-photon interactions... Ask: Is SPDC really the time-reverse of SHG? The probability of 2 photons upconverting in a typical nonlinear crystal is roughly 10  (as is the probability of 1 photon spontaneously down-converting). (And if so, then why doesn't it exist in classical e&m?)

Quantum Interference

Type-II down-conversion

2-photon "Switch": experiment

(57% visibility) Suppression/Enhancement of Spontaneous Down-Conversion

Switchiness ("Nonlinearity")

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/cm 2 ). The blue pump serves as a catalyst, enhancing the interaction by

Controlled-phase switch Resch et al, Phys. Rev. Lett. 89, (2002)

Fringe data with and w/o postsel.

...but it actually is true

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

Characterisation of quantum processes in QI systems

The Serious Problem For QI The danger of errors grows exponentially with the size of the quantum system. Without error-correction techniques, quantum computation would be a pipe dream. A major goal is to learn to completely characterize the evolution (and decoherence) of physical quantum systems in order to design and adapt error-control systems. $The tools are "quantum state tomography" and "quantum process tomography": full characterisation of the density matrix or Wigner function, and of the " $ uperoperator" which describes its time-evolution.

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 superoperators

HWP QWP PBS Argon Ion Laser Beamsplitter "Black Box" 50/50 Detector B Detector A Two waveplates per photon for state preparation Two waveplates per photon for state analysis SPDC source Two-photon Process Tomography (Mitchell et al., quant-ph/ )

Hong-Ou-Mandel Interference How often will both detectors fire together? r r t t + r 2 +t 2 = 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.

“Measuring” the superoperator } Output DM Input HH HV VV VH } } } etc. 16 analyzer settings 16 input states Coincidencences

“Measuring” the superoperator Input Output DM HH HV VV VH etc. Superoperator Input Output

“Measuring” the superoperator Input Output DM HH HV VV VH etc. Input Output Superoperator

Testing the superoperator LL = input state Predicted N photons = 297 ± 14

Testing the superoperator LL = input state Predicted Observed N photons = 297 ± 14 N photons = 314

So, How's Our Singlet State Filter? Observed   , but a different maximally entangled state: Bell singlet state:   = (HV-VH)/√2 1/2 -1/2

Model of real-world beamsplitter 45° “unpolarized” 50/50 dielectric beamsplitter at 702 nm (CVI Laser) birefringent element + singlet-state filter + birefringent element Singlet filter AR coating multi-layer dielectric

Comparison to ideal filter Measured superoperator, in Bell-state basis: A singlet-state filter would have a single peak, indicating the one transmitted state. Superoperator after transformation to correct polarisation rotations: 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…

Time-resolved quantum states

Lattice experimental setup Setup for lattice with adjustable position & velocity

Wait… Quantum state reconstruction Measure ground state population Shift…  x (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 Ground-state population vs. time bet. translations Fancy NLO interpretation: Raman pump-probe study of vibrational states

Q(x,p) for a coherent H.O. state?

Quasi-Q for a mostly-excited state in a 2-state lattice

Theory for 80/20 mix of e and g

Exp't:"W" or [P g -P e ](x,p)

W(x,p) for 80% excitation

Atomic state measurement (for a 2-state lattice, with c 0 |0> + c 1 |1>) left in ground band tunnels out during adiabatic lowering (escaped during preparation) initial statedisplaceddelayed & displaced |c 0 | 2 |c 0 + c 1 | 2 |c 0 + i c 1 | 2 |c 1 | 2

input density matrices output density matrices Time-evolution of some states

Atom superoperators Initial Bloch sphere sitting in lattice, quietly decohering… being shaken back and forth resonantly 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).

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 particle was 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 depend more on i or f, or equally on both? Clever answer: both, as Schrödinger time-reversible. Conventional answer: i, because of collapse. Measurement of A AAV, PRL 60, 1351 ('88)

The Rub

A (von Neumann) Quantum Measurement of A Well-resolved states System and pointer become entangled Decoherence / "collapse" Large back-action Initial State of Pointer x x H int =gAp x System-pointer coupling Final Pointer Readout

A Weak Measurement of A H int =gAp x System-pointer coupling 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 by 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 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? "Interaction-Free Measurements" (AKA: The Elitzur-Vaidman bomb experiment) A. C. Elitzur, and L. Vaidman, Found. Phys. 23, 987 (1993) BS1 BS2 D C Bomb absent: Only detector C fires Bomb present: "boom!"1/2 C1/4 D1/4

H Pol DC V Pol DC 407 nm Pump

BS1 - e-e- BS2 - O-O- C-C- D-D- I-I- BS1 + BS2 + I+I+ e+e+ O+O+ D+D+ C+C+ W OutcomeProb D + and C - 1/16 D - and C + 1/16 C + and C - 9/16 D + and D - 1/16 Explosion4/16 Hardy’s Paradox L. Hardy, Phys. Rev. Lett. 68, 2981 (1992) Hardy Cartoon D- e+ was in D+D- ? But … D+ e- was in

GaN Diode Laser PBS Det. A Det. B Cf. Torgerson et al., Phys. Lett. A. 204, 323 (1995) DC BS BS BS2 Switch (W) H V CC Hardy's Paradox: Setup

GaN Diode Laser PBS Det. H (D-)Det. V (D+) DC BS BS BS2 Switch H V CC PBS Experimental Setup (W)(W)

Probabilitiese- ine- out e+ in1 e+ out 11 But what can we say about where the particles were or weren't, once D+ & D– fire? Upcoming experiment: demonstrate that "weak measurements" (à la Aharonov + Vaidman) will bear 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/ ) No matter what the Silicon crowd thinks, there's a lot of mileage left in (nonlinear/quantum) optics!