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 Acknowledgments 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 OUTLINE Something you already know Introduction to quantum information with optics Something you may have known... but may forgotten by now All good

talkshave are alike... Making a every strongbad effective between talk isinteraction bad in its own way. two photons omething you most likely haven't heard befor 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?"

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? 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 takes n |n>|0> to n |n>|f(n)>; it calculates f(n)(n)>; it calculates f(n)n)>; it calculates f(n)>; it calculates f(n)(n)>; it calculates f(n)n) f(n)>; it calculates f(n)or every possible input simultaneously. A clever measurement may determine some global

property of f(n) even though the computer has only run once... A not-clever measurement "collapses" n to some random value, and yields f(that value). The rub: any interaction with the environment leads to "decoherence," which can be thought of as continual unintentional measurement of n. Quantum Computer Scientists What makes a computer quantum? If a quantum "bit" is described by two numbers: |> = 0c|0> + 1c|1>, then n quantum bits are describedn by coef(n)>; it calculates f(n)2f(n)>; it calculates f(n)'s:s:

|> = 00..0 c |00..0>+c00..1|00..1>+...c11..1|11..1>; this is exponentially more inf(n)>; it calculates f(n)ormation than the 2n coef(n)>; it calculates f(n)f(n)>; it calculates f(n)ic would take to describe n independent (n)>; it calculates f(n)e.g., classical) bits.., classical) bits. It is also exponentially sensitive to decoherence. Photons are ideal carriers of(n)>; it calculates f(n) quantum inf(n)>; it calculates f(n)ormation-- they can be easily produced, manipulated, and detected, and don's:t interact sig., classical) bits.nif(n)>; it calculates f(n)icantly with the environment. They are already used to transmit quantum-cryptog., classical) bits.raphic inf(n)>; it calculates f(n)ormation throug., classical) bits.h f(n)>; it calculates f(n)ibres under Lake Geneva, and soon throug., classical) bits.h the air up to satellites. Unf(n)>; it calculates f(n)ortunately, they don's:t interact with each other very m either! How to make a log., classical) bits.ic g., classical) bits.ate? Quantum Interference for effective

single-photonsingle-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 a|0> + b|1> + c|2> ANCILLA |1> OUTPUT STATE a'|0> + b'|1> + c'|2>

TRIGGER (postselection) |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 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)? atom 1 w1 w2 atom 2 w2 w1 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 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)? Two-photon absorption (by these single-photon absorbers) is interferometrically enhanced if the photons begin distinguishable, but are indistinguishable to the absorber: T2 > t > t c 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). Quantum Interference Suppression/Enhancement

of 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/cm 2). The blue pump serves as a catalyst, enhancing the interaction by 10 10. Controlled-phase switch Resch 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 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...). 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 superoperators One photon: H or V. State: two coefficients () ( ) CH CV Density matrix: 2x2=4 coefficients CHH CVH CHV

CVV Measure intensity of horizontal intensity of vertical intensity of 45 o intensity 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. Two-photon Process Tomography

(Mitchell et al., quant-ph/0305001) Two waveplates per photon for state preparation HWP QWP HWP Detector A PBS QWP SPDC source

"Black Box" 50/50 Beamsplitter QWP HWP QWP PBS HWP Detector B Argon Ion Laser Two waveplates per

photon for state analysis Hong-Ou-Mandel Interference r r + t t 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 Time-resolved quantum states Quantum state reconstruction p

p wt x x x Wait Shift p x

Q(0,0) = P g x Measure ground state population W(0,0) = (n)>; it calculates f(n)-1)n Pn (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 QuickTime and a Photo - JPEG decompressor are needed to see this picture. Fancy NLO interpretation: Raman pump-probe study of vibrational states Exp't:"W" or [Pg-Pe](x,p) QuickTime and a Photo - JPEG decompressor are needed to see this picture.

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|2

|c1|2 |c0 + c1 |2 |c0 + i c1 |2 Time-evolution of some states input density matrices output density matrices Atom superoperators sitting in lattice, quietly decohering QuickTime and a

Photo - JPEG decompressor are needed to see this picture. QuickTime and a Photo - JPEG decompressor are needed to see this picture. 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).

QuickTime and a Photo - JPEG decompressor are needed to see this picture. Can we talk about what goes on behind closed doors? Pick a box, any box... A+B+C A +BC What are the odds that the particle was in a given box? Conditional measurements (Aharonov, Albert, and Vaidman)

AAV, PRL 60, 1351 ('88) Prepare a particle in |i> try to "measure" some observable A postselect the particle to be in |f> i i Measurement of A f f Does depend more on i or f, or equally on both? Clever answer: both, as Schrdinger time-reversible. Conventional answer: i, because of collapse.

A (von Neumann) Quantum Measurement of A Initial State of Pointer Final Pointer Readout Hint=gApx x System-pointer coupling x Well-resolved states

System and pointer become entangled Decoherence / "collapse" Large back-action A Weak Measurement of A Initial State of Pointer Final Pointer Readout Hint=gApx x System-pointer coupling x

Poor resolution on each shot. Negligible back-action (system & pointer separable) Mean pointer shift is given by A w = f Ai f i Has many odd properties, as we shall see... "Interaction-Free Measurements" (AKA: The Elitzur-Vaidman bomb experiment) A. C. Elitzur, and L. Vaidman, Found. Phys. 23, 987 (1993) Problem: D

C Consider a collection of bombs so sensitive that a collision with any single particle (photon, electron, etc.) Bomb absent: is guarranteed to trigger it. Only detector C fires BS2that certain of Suppose the bombs are defective, but differ in their behaviour in no way other than that Bomb present: they will not blow up when triggered. "boom!" 1/2 bombs (or Is there any way to identify the working

C up? 1/4 some of them) without blowing them BS1 D 1/4 Hardys Paradox L. Hardy, Phys. Rev. Lett. 68, 2981 (1992) C+ D+ D-

BS2+ COutcome Prob D+ e- C was in D+ and 1/16 D- e+ was in D- and C+ 1/16 BS2I+ I-

O+ W BS1+ e+ OBS1e- C+ and C?- 9/16 D+DD+ and D- 1/16 But Explosion 4/16 Experimental Setup C+

Det. V (D+) Det. H (D-) D+ D- BS2+ BS2I+ I- O+ W BS1+

e+ 50-50 BS2 CC PBS PBS GaN Diode Laser DC BS 50-50 BS1 (W)

CC V H Switch DC BS C- OBS1e- Switch Visibility

400 Vis = 85.7 +/- 0.1 % 350 300 250 200 150 100 50 0 -1 -0.5

0 0.5 1 1.5 Delay (microns) 2 2.5 3 Vertical Polarization Mach-Zehnder 70

Vis = 90 +/- 1% 60 50 40 30 20 10 0 0 1 2 3

4 Delay (microns) 5 6 But what can we say about where the particles were or weren't, once D+ & D fire? Probabilities e- in e+ in 0 e+ out 1

1 e- out 1 1 -1 0 0 Upcoming experiment: demonstrate that "weak measurements" ( 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/0302003) No matter what the Silicon crowd thinks, there's a lot of mileage left in (nonlinear/quantum) optics!