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Explore the fascinating world of quantum mechanics with a focus on manipulating and measuring the quantum state of photons and atoms. Dive into quantum tomography, entangled photon pairs, process tomography, and more. Learn about density matrices, superoperators, and weak measurements in this cutting-edge field of research.
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Manipulating and Measuring the Quantuum State of Photons and Atoms 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 theendof our funding!) Aephraim M. Steinberg Centre for Q. Info. & Q. Control Institute for Optical Sciences Dept. of Physics, U. of Toronto CANADA QUEST 05, Santa Fe
DRAMATIS PERSONAE Toronto quantum optics & cold atoms group: Postdocs: Morgan Mitchell ( Barcelona) Matt Partlow Optics: Jeff Lundeen Kevin Resch(Zeilinger ) Lynden(Krister) Shalm Masoud Mohseni (Lidar) Rob Adamson Atoms: Jalani Fox Stefan Myrskog (Thywissen) Ana Jofre(NIST) Mirco Siercke Samansa Maneshi Chris Ellenor Some theory collaborators: ...Daniel Lidar, Pete Turner, János Bergou, Mark Hillery, Paul Brumer, Howard Wiseman,...
There are many types of measurement! A few to keep in mind: • Projective measurement (or von Neumann); postselection • Quantum state “tomography” (reconstruction of , W, etc) • standard, adaptive, ... • incomplete? • in the presence of inaccessible information • Quantum process “tomography” (CP map from ) • standard, ancilla-assisted, “direct”,... • POVMs • “Direct” measurement of functions of • “Interaction-free” measurements • “Weak measurements” (various senses) • Aharonov/Vaidman application to postselection
OUTLINE Tomography – characterizing quantum states & processes... brief review Entangled photon pairs 2-photon process tomography Direct measurement of purity Generating entanglement by postselection Characterizing states with “inaccessible” info Motional states of atoms in optical lattices Process tomography Pulse echo Inverted states, negative Wigner functions,... Bonus topic if you don’t interrupt me enough: Weak measurements and “paradoxes” (which-path debate; Hardy’s paradox)
0 Quantum tomography: what & why? • Characterize unknown quantum states & processes • Compare experimentally designed states & processes to design goals • Extract quantities such as fidelity / purity / tangle • Have enough information to extract any quantities defined in the future! • or, for instance, show that no Bell-inequality could be violated • Learn about imperfections / errors in order to figure out how to • improve the design to reduce imperfections • optimize quantum-error correction protocols for the system
Density matrices and superoperators Two photons: HH, HV, VH, VV, or any superpositions. State has four coefficients. Density matrix has 4x4 = 16 coefficients. Superoperator has 16x16 = 256 coefficients.
Wigner function of an ion in the excited state Liebfried, Meekhof, King, Monroe, Itano, Wineland, PRL 77, 4281 (96)
Some density matrices... Much work on reconstruction of optical density matrices in the Kwiat group; theory advances due to Hradil & others, James & others, etc...; now a routine tool for characterizing new states, for testing gates or purification protocols, for testing hypothetical Bell Inequalities, etc... Spin state of Cs atoms (F=4), in two bases Polarisation state of 3 photons (GHZ state) Klose, Smith, Jessen, PRL 86 (21) 4721 (01) Resch, Walther, Zeilinger, PRL 94 (7) 070402 (05)
QPT of QFT Weinstein et al., J. Chem. Phys. 121, 6117 (2004) To the trained eye, this is a Fourier transform... From those superoperators, one can extract Kraus operator amplitudes, and their structure helps diagnose the process.
Ancilla-assisted process tomography Proposed in 2000-01 (Leung; D’Ariano & Lo Presti; Dür & Cirac): one member of an maximally-entangled pair could be collapsed to any given state by a measurement on the other; replace multiple state preparations with coincidence measurement. Altepeter et al, PRL 90, 193601 (03) 2-qbit state tomography with the entangled input is equivalent to 1-qbit process tomography using 4 different inputs (and both require 16 measurements). 1-qbit processes represented as deformations of Bloch sphere ...some unphysical results with engineered decoherence.
1 Quantum tomography experiments on photons, and how to avoid them
Two-photon Process Tomography[Mitchell et al., PRL 91, 120402 (2003)] "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.
“Measuring” the superoperatorof a Bell-state filter Coincidencences Output DM Input } HH } } 16 input states } HV etc. VV 16 analyzer settings VH [Mitchell et al., PRL 91, 120402 (2003)]
“Measuring” the superoperator Superoperator Input Output DM HH HV VV VH Output Input etc.
Superoperator provides informationneeded to correct & diagnose operation Measured superoperator, in Bell-state basis: Superoperator after transformation to correct polarisation rotations: Dominated by a single peak; residuals allow us to estimate degree of decoherence and other errors. The ideal filter would have a single peak. Leading Kraus operator allows us to determine unitary error. (Experimental demonstration delayed for technical reasons; now, after improved rebuild of system, first addressing some other questions...)
Some vague thoughts... QPT is incredibly expensive (16n msmts for n qbits) Both density matrices and superoperators we measure typically are very sparse... a lot of time is wasted measuring coherences between populations which are zero. (a) If aiming for constant errors, can save time by making a rough msmt of a given rate first and then deciding how long to acquire data on that point. (b) Could also measure populations first, and then avoid wasting time on coherences which would close to 0. (c) Even if r has only a few significant eigenvalues, is there a way to quickly figure out in which basis to measure? If one wants to know some derived quantity, are there short-cuts? (a) Direct (joint) measurements of polynomial functions (b) Optimize counting procedure based on a given cost function (c) Adaptive search E.g.: suppose you would like to find a DFS within a larger Hilbert space, but need not characterize the rest.
A sample error model:the "Sometimes-Swap" gate Consider an optical system with stray reflections – occasionally a photon-swap occurs accidentally: Two subspaces are decoherence-free: 1D: 3D: Experimental implementation: a slightly misaligned beam-splitter (coupling to transverse modes which act as environment) TQEC goal: let the machine identify an optimal subspace in which to compute, with no prior knowledge of the error model.
# of inputs tested purity of best 2D DFS found standard tomography random tomography adaptive tomography # of input states used Best known 2-D DFS (average purity). averages Some strategies for a DFS search (simulation; experiment underway) Our adaptive algorithm always identifies a DFS after testing 9 input states, while standard tomography routinely requires 16 (complete QPT). Surprise: in the absence of noise, simulations show that essentially any 2 input states suffice to identify the DFS (required max-lik to work). Project to revisit: add noise, do experiment, study scaling,...
Often, only want to look at a single figure of merit of a state (i.e. tangle, purity, etc…) Would be nice to have a method to measure these properties without needing to carry out full QST. Polynomial Functions of a Density Matrix (T. A. Brun, e-print: quant-ph/0401067) • Todd Brun showed that mth degree polynomial functions of a density matrix fm() can be determined by measuring a single joint observable involving m identical copies of the state.
Linear Purity of a Quantum State HOM as Singlet State Filter Pure State on either side = 100% visibility + H H H H Mixed State = 50% visibility H H H V H H H + V V H V HOM Visibility = Purity • For a pure state, P=1 • For a maximally mixed state, P=(1/n) • Quadratic 2-particle msmt needed Measuring the purity of a qubit • Need two identical copies of the state • Make a joint measurement on the two copies. • In Bell basis, projection onto the singlet state P = 1 – 2 – – Singlet-state probability can be measured by a singlet-state filter (HOM)
Experimentally Measuring the Purity of a Qubit • Use Type 1 spontaneous parametric downconversion to prepare two identical copies of a quantum state • Vary the purity of the state • Use a HOM to project onto the singlet • Compare results to QST Single Photon Detector Quartz Slab Type 1 SPDC Crystal Singlet Filter Coincidence Circuit Quartz Slab Single Photon Detector
Results For a Pure State Prepared the state |+45> Measured Purity from Singlet State Measurement P=0.92±0.02 Measured Purity from QST P=0.99±0.01
Preparing a Mixed State Case 1: Same birefringence in each arm Visibility = (90±2) % H V 100% interference Case 2: Opposite birefringence in each arm V H V H H Visibility = (21±2) % V 25% interference Can a birefringent delay decohere polarization (when we trace over timing info) ? [cf. J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, and P. G. Kwiat, Phys. Rev. Lett., 90, 193601 ] The HOM isn’t actually insensitive to timing information.
Not a singlet filter, but an “Antisymmetry Filter” • The HOM is not merely a polarisation singlet-state filter • Problem: • Used a degree of freedom of the photon as our bath instead of some external environment • The HOM is sensitive to all degrees of freedom of the photons • The HOM acts as an antisymmetry filter on the entire photon state • Y Kim and W. P. Grice, Phys. Rev. A68, 062305 (2003) • S. P. Kulik, M. V. Chekhova, W. P. Grice and Y. Shih, Phys. Rev. A 67,01030(R) (2003)
Preparing a Mixed State |45> |45> or |-45> Could produce a “better” maximally mixed state by using four photons. Similar to Paul Kwiat’s work on Remote State Preparation. Coincidence Circuit Randomly rotate the half-waveplates to produce |45> and |-45> Preliminary results Currently setting up LCD waveplates which will allow us to introduce a random phase shift between orthogonal polarizations to produce a variable degree of coherence Visibility = (45±2) %
2 When the distinguishable isn’t…
Highly number-entangled states("low-noon" experiment). M.W. Mitchell et al., Nature 429, 161 (2004); and cf. P. Walther et al., Nature 429, 158 (2004). The single-photon superposition state |1,0> + |0,1>, which may be regarded as an entangled state of two fields, is the workhorse of classical interferometry. The output of a Hong-Ou-Mandel interferometer is |2,0> + |0,2>. States such as |n,0> + |0,n> ("high-noon" states, for n large) have been proposed for high-resolution interferometry – related to "spin-squeezed" states. Multi-photon entangled states are the resource required for KLM-like efficient-linear-optical-quantum-computation schemes. A number of proposals for producing these states have been made, but so far none has been observed for n>2.... until now!
Practical schemes? [See for example H. Lee et al., Phys. Rev. A 65, 030101 (2002); J. Fiurásek, Phys. Rev. A 65, 053818 (2002)] ˘ + = 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! Important factorisation:
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., ...
Trick #1 SPDC laser Okay, we don't even have single-photon sources. But we can produce pairs of photons in down-conversion, and very weak coherent states from a laser, such that if we detect three photons, we can be pretty sure we got only one from the laser and only two from the down-conversion... |0> + e |2> + O(e2) |3> + O(3) + O(2) + terms with <3 photons |0> + |1> + O(2)
Trick #2 "mode-mashing" Yes, it's that easy! If you see three photons out one port, then they all went out that port. How to combine three non-orthogonal photons into one spatial mode?
Trick #3 (or nothing) (or nothing) (or <2 photons) 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 a polarizer, it's no longer at 45o. If it gets through a partial polarizer, it could be anywhere...
It works! Singles: Coincidences: Triple coincidences: Triples (bg subtracted):
Generating / measuring other states With perfect detectors and perfect single-photon sources, such schemes can easily be generalized. With one or the other (and typically some feedback), many states may be synthesized by iteratively adding or subtracting photons, and in some cases implementing appropriate unitaries. Postselection has also been used to generate GHZ, W, and cluster states (to various degrees of fidelity). Photon subtraction can be used to generate non-gaussian states. Postselection is also the heart of KLM and competing schemes, and can be used to implement arbitrary unitaries, and hence to entangle anything. “Continuous” photon subtraction (& counting) can be used, even with inefficient detectors, to reconstruct the entire photon-number distribution.
Fundamentally Indistinguishablevs.Experimentally Indistinguishable LeftArnold RightDanny OR–Arnold&Danny ? But what if when we combine our photons, there is some residual distinguishing information: some (fs) time difference, some small spectral difference, some chirp, ...? This will clearly degrade the state – but how do we characterize this if all we can measure is polarisation?
Quantum State Tomography Distinguishable Photon Hilbert Space Indistinguishable Photon Hilbert Space ? Yu. I. Bogdanov, et al Phys. 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, given that we don’t know how to tell the particles apart ?
The Partial Density Matrix The answer: there are only 10 linearly independent parameters which are invariant under permutations of the particles. One example: Inaccessible information Inaccessible information The sections of the density matrix labelled inaccessible correspond to information about the ordering of photons with respect to inaccessible degrees of freedom.
Experimental Results No Distinguishing Info Distinguishing Info When distinguishing information is introduced the HV-VH component increases without affecting the state in the symmetric space Mixture of 45–45 and –4545 HH + VV
More Photons… If you have a collection of spins, what are the permutation-blind observables that describe the system? They correspond to measurements of angular momentum operators J and mj ... for N photons, J runs to N/2 So the total number of operators accessible to measurement is
3 Tomography in optical lattices, and steps towards control...
Tomography in Optical Lattices [Myrkog et al., quant-ph/0312210 Kanem et al., quant-ph/0506140] Rb atom trapped in one of the quantum levels of a periodic potential formed by standing light field (30GHz detuning, 10s of mK depth) Complete characterisation of process on arbitrary inputs?
Towards QPT:Some definitions / remarks • "Qbit" = two vibrational states of atom in a well of a 1D lattice • Control parameter = spatial shifts of lattice (coherently couple states), achieved by phase-shifting optical beams (via AO) • Initialisation: prepare |0> by letting all higher states escape • Ensemble: 1D lattice contains 1000 "pancakes", each with thousands of (essentially) non-interacting atoms. • No coherence between wells; tunneling is a decoherence mech. • Measurement in logical basis: direct, by preferential tunneling under gravity • Measurement of coherence/oscillations: shift and then measure. • Typical experiment: • Initialise |0> • Prepare some other superposition or mixture (use shifts, shakes, and delays) • Allow atoms to oscillate in well • Let something happen on its own, or try to do something • Reconstruct state by probing oscillations (delay + shift +measure)