* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download First lecture, 7.10.03
Atomic theory wikipedia , lookup
Density matrix wikipedia , lookup
Quantum fiction wikipedia , lookup
Quantum field theory wikipedia , lookup
Spin (physics) wikipedia , lookup
Quantum computing wikipedia , lookup
Renormalization group wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Hydrogen atom wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Coherent states wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Ensemble interpretation wikipedia , lookup
Elementary particle wikipedia , lookup
Quantum machine learning wikipedia , lookup
Renormalization wikipedia , lookup
Quantum group wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Path integral formulation wikipedia , lookup
Identical particles wikipedia , lookup
History of quantum field theory wikipedia , lookup
Wave function wikipedia , lookup
Bell test experiments wikipedia , lookup
Probability amplitude wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Quantum key distribution wikipedia , lookup
Delayed choice quantum eraser wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Wheeler's delayed choice experiment wikipedia , lookup
Canonical quantization wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Quantum entanglement wikipedia , lookup
Quantum teleportation wikipedia , lookup
Bell's theorem wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Particle in a box wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Quantum state wikipedia , lookup
Wave–particle duality wikipedia , lookup
Double-slit experiment wikipedia , lookup
Matter wave wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Hidden variable theory wikipedia , lookup
Experimental Issues in Quantum Measurement • Today, 7.10.03: OVERVIEW – a survey of some important situations in q. msmt. theory (“why bother coming to these lectures?”) Being a quantum physicist is like being an alcoholic. • 13.10.03: SOME TECHNICAL BACKGROUND – introduction to “standard” quantum measurement theory (measurement postulate, collapse, von Neumann msmts, density matrices and is entanglement,...) ...the first step to admit you have a problem. • 20.10.03: THE QUANTUM ERASER – Bohr-Einstein debates – Scully, Englert, Walther: complementarity vs uncertainty – Two-photon experiments – Alternate pictures (collapse vs correlations) • 27.10.03–: OTHER MODERN EXPERIMENTS... FIRST TOPIC: Interruptions MAKE THEM! VALID REASONS TO INTERRUPT ME: I’m going too fast. I’m going too slow. You want to correct my grammar. You disagree with something I said. I seem to disagree with something I’ve said. You have a question about something I’ve said. You have a question about something completely unrelated. Some references I will not be following any particular textbook, but for obvious reasons, will draw disproportionately from experiments I myself have worked on... Appropriate references will be provide as the lectures progress. General references on quantum mechanics: Your favorite QM text + Shankar’s Principles of QM Background on the quantum measurement problem: Wheeler & Zurek’s Quantum Theory and Measurement Bell’s Speakable and Unspeakable in Quantum Mechanics My general perspective on these issues: References on my web page, http://www.physics.utoronto.ca/~aephraim/aephraim.html {where slides from these lectures will be too, eventually} “Speakable and Unspeakable, Past and Future” http://lanl.arxiv.org/abs/quant-ph/0302003 The Copenhagen Viewpoint (Toronto description of) Bohr, Heisenberg: We must only discuss the outcomes of measurements. An experiment described to measure wave properties will measure wave properties. An experiment described to measure particle properties will measure particle properties. In an experiment which measures wave properties, a question about particle properties is not a question about the outcome of real measurements – it is “not a proper question.” Wave and particle descriptions are “complementary” – they can never both be observed in a single experiment. The Bohr-Einstein debates Inserting Weg” detectors Two-slit“Welcher interference: destroys our ignorance the prototypical wave – and thus the interference. phenomenon. Each particle seems to “go through both slits”; we can’t ask which one it came from. “Heisenberg microscope”: photons which allow you to look at the particle bounce off it, disturbing its momentum. Feynman’s Rules for interference If two or more indistinguishable processes can lead to the same final event (particle could go through either slit and still get to the same spot on the screen), then add their complex amplitudes and square, to find the probability: P = |A1+A2 |2 ≈ |eikL1 + eikL2 |2 ≈ 1 + cos k(L1-L2) If multiple distinguishable processes occur, find the real probability of each, and then add: P = |A1|2 + |A2|2 ≈ |eikL1 |2 + |eikL2 |2 ≈ 1 If there is any way – even in principle – to tell which process occurred, then there can be no interference (if you knew which slit the particle came from, you’d see a 1-slit pattern) ! The quantum eraser spin-up () particles Waveplate: flips the spin of particles passing slit 2, without affecting linear momentum. Still no interference – because we could check the spin of the particle, and discover which slit it had traversed. Must there be a disturbance? Bohr: Measurement of X disturbs P; et cetera Measurement means amplification of a quantum phenomenon by interaction with some “large” (classical) device Msmt involves some uncontrollable, irreversible disturbance We must treat the measuring device classically. Wigner: Why must we? What will happen to us if we don’t? Scully, Englert, Walther: Complementarity is more fundamental than uncertainty. We can use information to destroy interference, without disturbing the momentum. Storey, Tan, Collett, Walls: No. Any such measurement always disturbs the momentum. Wiseman (+ Toronto experiment): They’re both right. And we can measure how much the momentum is disturbed. The EPR “Paradox” –– and superluminal signalling? Particle with 0 angular RECALL: momentum decays... Spin-projections along different axes are “incompatible” (can’t be measured simultaneously -- like X & P) If you find Sz = +1 (spin ), and then measure Sx, here implies... ... here... Sx = +1 (spin ) and Sx= –1 (spin ) are equally likely. but here implies... here. Then...if you find one of those, and become equally likely. EPR: we could measure Sx on particle 1, but simultaneously know what we would have Bohr & Heisenberg tell us weundoubtedly must choose:gotten if wewe hadcan measured Sz;but aren’t real? of Sx... know Sz, givethese up allboth knowledge Copenhagen:or noknow waveSx function hasup both properties defined – and give all those knowledge of Sz... and the wave function is all you can possibly know. EPR are cheating, discussing measurements they didn’t do. Some important lessons The first one (only 30 years... or maybe 50, or 70+): QUANTUM MECHANICS IS NOT LOCAL (i.e.: it is not always possible to describe what happens in Vienna without simultaneously taking into account what is going on in Toronto– even for times so short that even John Bell at the speed of light, no signal could have connected the two.) Q ui ck Ti m e ™ an d a T I FF ( U nc om p r es se d) de co m pr e ss or ar e n ee de d t o se e t hi s p i ct ur e . Q ui ck Ti m e ™ an d a T I FF ( U nc om p r es se d) de co m pr e ss or ar e n ee de d t o se e t hi s p i ct ur e . Q ui ck Ti m e ™ an d a T I FF ( U nc om p r es se d) de co m pr e ss or ar e n ee de d t o se e t hi s p i ct ur e . Q ui ck Ti m e ™ an d a T I FF ( U nc om p r es se d) de co m pr e ss or ar e n ee de d t o se e t hi s p i ct ur e . One of the more subtle ones: You can extract very limited information from a single particle. In fact, to duplicate the particle, you must destroy it – information in QM is never gained or lost. NO CLONING! (...and yet, recent “quantum cloning” experiments...) “Distinguishing the indistinguishable” • Non-orthogonal quantum states cannot be distinguished with certainty. • This is one of the central features of quantum information which leads to secure (eavesdrop-proof) communications. • Crucial element: we must learn how to distinguish quantum states as well as possible -- and we must know how well a potential eavesdropper could do. If it gets through an H polarizer, ...it could still have been 45, and it’s too late to tell. If it gets through a 45 polarizer, same story. H-polarized photon 45o-polarized photon BUT: a clever measurement can tell with certainty, 25% of the time. BUT BUT: a non-standard quantum measurement can do better! A 14-path interferometer for arbitrary 2-qubit unitaries... Success! "Definitely 3" "Definitely 2" "Definitely 1" "I don't know" The correct state was identified 55% of the time-Much better than the 33% maximum for “standard measurements” ( = everything in your textbook). " Quantum seeing in the dark " (AKA: The Elitzur-Vaidman bomb experiment) A. Elitzur, and L. Vaidman, Found. Phys. 23, 987 (1993) P.G. Kwiat, H. Weinfurter, and A. Zeilinger, Sci. Am. (Nov., 1996) 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 BS2 that 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 The bomb must be there... yet my photon never interacted with it. Quantum CAT scans If you measure momentum P... you don’t know anything about X. If you measure position X... you don’t know anything about P. But in real life, don’t I know something about each? Don’t I also know that if a car left this morning and is already in Budapest, it’s going faster than if it’s still on Währingerstr.? Wigner function: W(x,p) is like the probability for a particle to be at x and have momentum p. Its integrals correctly predict P(x), P(p), and everything else you want. Of course, you must study a large ensemble of particles to get so much information: “quantum state tomography” The Wigner quasiprobability function for an atom trapped in a light wave QuickTime™ and a Photo - JPEG decompressor are needed to see this picture. P(0,0) < 0 ?! Predicting the past ? Standard recipe of quantum mechanics: 1. Prepare a state |i> (by measuring a particle to be in that state; see 4) 2. Let Schrödinger do his magic: |i> |f>=U(t) |i>, deterministically 3. Upon a measurement, |f> some result |n> , randomly 4. Forget |i>, and return to step 2, starting with |n> as new state. Aharonov’s objection (as I read it): No one has ever seen any evidence for step 3 as a real process; we don’t even know how to define a measurement. Step 2 is time-reversible, like classical mechanics. Why must I describe the particle, between two measurements (1 & 4) based on the result of the first, propagated forward, rather than on that of the latter, propagated backward? Predicting the past... A+B B+C What are the odds that the particle was in a given box (e.g., box B)? Pick a box, any box... A+B+C A+B–C We’ll see that applying similar logic here lets us conclude: P(A) = 100% P(B) = 100% and then, necessarily: P(C) = –100% (?!) ....and that real measurements agree (somehow!) Measurement as a tool: KLM... INPUT STATE a|0> + b|1> + c|2> OUTPUT STATE a|0> + b|1> – c|2> MAGIC MIRROR: Acts differently if there are 2 photons or only 1. TRIGGER (postselection) ANCILLA In other words, can be a “transistor,” or “switch,” or “quantum special |i > logic gate”... particular | > f Knill, Laflamme, Milburn Nature 409, 46, (2001); and others since. Experiments by Franson et al., White et al., Zeilinger et al... Summary: the kinds of things we’ll cover... • Why does one thing happen and not another? • When is a quantum measurement? • Does a measurement necessarily disturb the system, and how? • What can we say about an observable before we measure it? • Does a wave function describe a single particle, or only an ensemble? • Is a wave function a complete description of a single particle? • Can we predict the past?