Real Time Imaging of Quantum and Thermal Fluctuations (A pinch of quantum mechanics, a drop of probability, ...) D.B. with M. Bauer, and (in part) T. Benoist & A. Tilloy. arXiv:1106.4953, arXiv:1210.0425, arXiv:1303.6658, to appear (hopefully soon...) Kyoto, July 2013 mercredi 31 juillet 2013
Non-demolition measurements and Q-jumps Quantum jumps of light recording the birth and death of a photon in a cavity ´bastien Gleyzes 1 , Stefan Kuhr 1 { , Christine Guerlin 1 , Julien Bernu 1 , Samuel Dele ´glise 1 , Ulrich Busk Hoff 1 , Se Michel Brune 1 , Jean-Michel Raimond 1 & Serge Haroche 1,2 Time (s) 0.90 0.95 1.00 1.05 1.10 1.15 1.20 e Creation-Annihilation of a thermal photon. g a Figure 2 | Birth, life and death of a photon. a , QND detection of a single photon. Red and blue bars show the raw signal, a sequence of atoms detected e State in e or g , respectively (upper trace). The inset zooms into the region where the statistics of the detection events suddenly change, revealing the quantum g jump from | 0 æ to | 1 æ . The photon number inferred by a majority vote over 1 n Courtesy of LKB-ENS. 0 0.0 0.5 1.0 1.5 2.0 2.5 b -- How to measure photons without destroying them ? -- How to record the cavity states ? -- How to observe quantum jumps? Are they detector dependent ? -- What determines the Q-jump dynamics ? The photon system is probed indirectly via another quantum system. mercredi 31 juillet 2013
Cavity QED experiments -- Testing light/photon (the quantum system) with matter (the quantum probes). System (S)= photons in a cavity. Probes (P)= Rydberg atoms (two state systems) Probe measurement Preparation apparatus of the probes Photons in a cavity Courtesy of LKB-ENS. -- Indirect measurements: Direct (Von Neumann) measurements on an auxiliary system (the probes). No direct observation of the cavity (the system). -- Probe like gyroscope (spin half system). U = exp[ i θ σ z N photon ] -- Interaction like rotation of the gyroscope (with an angle depending on the number of photons) mercredi 31 juillet 2013
Progressive field-state collapse and quantum non-demolition photon counting Christine Guerlin 1 , Julien Bernu 1 , Samuel Dele ´glise 1 , Cle ´ment Sayrin 1 , Se ´bastien Gleyzes 1 , Stefan Kuhr 1 { , Michel Brune 1 , Jean-Michel Raimond 1 & Serge Haroche 1,2 Figure 2 | Progressive collapse of field into photon number state. (integral of P N ( n ) normalized to unity). c , Photon number probabilities plotted versus photon and atom numbers n and N . The histograms evolve, as N increases from 0 to 110, from a flat distribution into n 5 5 and n 5 7 peaks. Courtesy of LKB-ENS. Number of indirect measurements P.d.f. of the photon numbers --- Why does the p.d.f. change after each indirect measurement ? --- How does it evolve? why does it become peaked (collapsed) ? --- How does it represent the cavity state ? --- What does continuous-in-time quantum measurement mean ? (Here: a `discrete version’ of time continuous measurement) mercredi 31 juillet 2013
Outline : Classical probability theory with a bit of quantum mechanics, Part I: (or the reverse). -- A view on (quantum) noise. (repeated interactions) -- Repeated (quantum) measurements, Bayes’ law and collapses. (via the martingale convergence theorem). -- Pointer states, exchangeability and de Finetti’s theorem. (objective or subjective probabilities) Part II: Making real «virtual» fluctuations -- Time continuous measurement and Q-jumps. (Bi-stability (multi-stability) and Q-jumps) -- Real time imaging of thermal and quantum fluctuations. (observing quantum fluctuations continuously in time.) mercredi 31 juillet 2013
Preparation Probe measurement of the probes apparatus Quantum non-demolition measurement and random bayesian updating. Photons in a cavity Courtesy of LKB-ENS. -- Repeated cycles of interaction plus probe measurement: Partial gain of information at each iteration, because of system-probe entanglement. -- Probe measurements give values + or - ; ω = (+ , + , − , + , · · · ) = ( ǫ 1 , ǫ 2 , ǫ 2 , · · · ) Recursion for the photon number p.d.f. from data of sequences up-dating of the trial/estimated p.d.f. at each step, using an a priori model for the output conditional probabilities � state ) = P a priori ( probe � N photon ) P trial ( N photon ) state � probe � P estimated ( N photon P normalisation ( probe state ) -- Bayesian approach (encoded into Q-mechanics) : And the updating is random (because of Q-mechanics) -- A two step analysis : -- What happens during one cycle ? -- What happens for an iteration of cycles ? mercredi 31 juillet 2013
Quantum mechanics implies «classical» Bayes’ rules Or what happens during one interaction + measurement cycle? | φ � -- probe: -- Preparation : Q 0 ( α ) = | C ( α ) | 2 -- system: , | ψ � = � α C ( α ) | α � -- Interaction : A delicate point : we suppose that there is a basis of system states *preserved* by the probe-system interaction, i.e.: for U the evolution operator of the probe-system interaction U | α � ⊗ | φ � = | α � ⊗ U α | φ � This will be related to the existence of *pointer states*, to exchangeability,... -- probe + system : -- After interaction: � α C ( α ) | α � ⊗ U α | φ � -- After probe measurement : If output probe measurement is | i � � � � -- probe + system : ∝ α C ( α ) � i | U α | φ � | α � ⊗ | i � -- New state distribution : Q new ( α ) = p ( i | α ) Q 0 ( α ) ... and new system state. with p ( i | α ) = | � i | U α | α � | 2 Z ( i ) i.e. Bayes’ rules mercredi 31 juillet 2013
Evolution of the probability distributions: Random Bayesian up-dating... Pick a basis a of states of the Q-system . � Q 0 ( α ) , Q 0 ( α ) = 1 , Start with a probability distribution (initial system state): α Let i be the output measurements on the probes. � Data (probe-system interaction) are probabilities to measure i p ( i | α ) , p ( i | α ) = 1 conditioned on the Q-system to be in state a . i P : = Probes Out-going probes after interaction with the Q-system Iterations... outputs.... S : = Quantum System Let Q_{n-1}(a) be the probability distribution of the Q-system after (n-1) cycles, π n ( i ) The output of the n-th probe measurement is i_n with probability : Then , Q n ( α ) = 1 � p ( i n | α ) Q n − 1 ( α ) , with Z n = p ( i n | α ) Q n − 1 ( α ) Z n α mercredi 31 juillet 2013
Collapse of the p.d.f. Q n ( α ) (A classical statement...) as n → ∞ Q n ( α ) = p ( i n | α ) Q n − 1 ( α ) � with proba π n := Z n = p ( i n | α ) Q n − 1 ( α ) , Z n α Claim: * Peaked distributions are stable (stability of the pointer states): Q n ( α ) = δ α ; γ are solutions. Then, outputs i_n are i.i.d. with probability: p ( i n | γ ) * Probability distributions converge a.s. (and in L1) towards peaked distributions (collapse of the wave function): n →∞ Q n ( α ) = δ α , γ ω lim with a realisation dependent target γ ω (Von Neumann rules for quantum measurements) P rob[ γ ω = β ] = Q 0 ( β ) * The convergence is exponential : Q n ( α ) ≃ exp[ − n S ( γ ω | α )] ( α � = γ ω ) p ( i | γ ) log p ( i | α ) � with a relative entropy. S ( γ | α ) = − p ( i | γ ) i mercredi 31 juillet 2013
Mesoscopic collapses : Quantum to classical transition. Mesoscopic measurements. "Collapse is nothing more � than the updating of that calculational device on the basis of additional experience". D. Mermin, arXiv:1301.6551 (posterior to these works) ... A Bayesian point of view. ... and some robustness or universality in quantum measurements What about if we don’t record the probe outputs? -- statistical ensemble of peaked distribution = diagonal density matrix Progressive decoherence: � α | ρ n | α � = Q 0 ( α ) = const . � n � α | ρ 0 | β � → 0 � U † � � α | ρ n | β � = α U β � Elements of a proof : -- Q n ( α ) are bounded martingales, i.e. E [ Q n ( α ) |F n − 1 ] = Q n − 1 ( α ) as such they converge a.s. and in L 1 -- Asymptotically, the outputs are i.i.d. with asymptotic frequencies : N n ( i ) ≃ n →∞ n p ( i | γ ω ) -- The limit is independent of the initial trial distribution. (Important for the experiment). mercredi 31 juillet 2013
Quantum noise and repeated Preparation Probe of the measurement quantum interaction. probes apparatus e.g. as in cavity QED experiments.... Photons in Courtesy of LKB-ENS. a cavity S : = Quantum System P : = Probes Out-going probes with after interaction measurements with the Q-system Iterations... or not? -- Hilbert space: H = H s ⊗ H 1 ⊗ · · · ⊗ H n ⊗ · · · -- Algebras of observable: B n := A s ⊗ A 1 ⊗ · · · ⊗ A n ⊗ I -- Filtration: A s ⊂ B n ⊂ B m , for n < m -- Gain of information : by testing output observable on the n-th first probes, but a probabilistic gain because of Q.M. -- Measure some observable on the ( n - th) output probes (not on the Q-system): The quantum filtration is reduced to a classical filtration. (Quantum Trajectories) (classical random process, the events are the out-put measurements) mercredi 31 juillet 2013
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