Des mesures quantiques non-destructives et des bruits quantiques.

Des mesures quantiques non-destructives et des bruits quantiques. (Un peu de mécanique quantique, un soupçon de probabilités,...) with M. Bauer Jan. 2013

intensity, that is, the creation of a thermal photon, which disappears A microscopic quantum system under continuous observation microwave photons. In our experiment, the measurement of the exhibits g at random times sudden jumps between its states. The light shift induced by the field on Rydberg atoms is repeated detection 1 of this quantum feature requires a quantum nondemolition (QND) measurement 1 3 repeated many Time times (s) during photons. more than 100 times within the average decay time of individual the system s evolution. 0.90 Whereas 0.95quantum1.00 jumps of1.05 trapped massive particles e (electrons, ions or molecules 4 8 ) have been observed, is an open cavity C made up of two superconducting niobium 1.10 The 1.15 core of 1.20 the experiment is a photon box (see Fig. 1), which this has proved more challenging for light quanta. Standard mirrors facing each other (the Fabry Perot configuration) 13. The photodetectors 0 absorb light and are thus unable to detect the same cavity is resonant at 51.1 GHz and cooled to 0.8 K. Its damping photon twice. It is therefore necessary to use a transparent counter time, as measured by the ring-down of a classical injected microwave field, is T2.0 c 5 0.129 6 0.003 s, 2.5 corresponding to a light travel that0.0 can see g photons without 0.5 destroying 1.0 them 3. Moreover, the 1.5 blight needs to be stored for durations much longer than the QND distance of 39,000 km, folded in the 2.7 cm space between the detection a time. Here we report an experiment in which we fulfil mirrors. The QND probes are rubidium atoms, prepared in circular these challenging conditions and observe quantum jumps in the Rydberg states 10, travelling along the z direction average. transverse to the photon e enumber. Microwave photons are stored in a superconducting cavity for times up to half a second, and are repeatedly probed velocity v 5 250 m s 21 (see Methods). The cavity C is nearly res- cavity axis. They cross C one at a time, at a rate of 900 s 21 with a by a stream of non-absorbing atoms. An atom interferometer onant with the transition between the two circular states e and g measures g g the atomic dipole phase shift induced by the nonresonant cavity field, so that the final atom state reveals directly (principal quantum numbers 51 and 50, respectively). The position- (z-)dependent atom field coupling V(z) 5 V 0 exp(2z 2 /w 2 ) 1 the1presence of a single photon in the cavity. Sequences of hundreds of atoms, highly correlated in the same state, are interrupted The maximum coupling, V 0 /2p 5 51 khz, is the rate at which the follows the gaussian profile of the cavity mode (waist w 5 6 mm). by sudden state switchings. These telegraphic signals record the field and the atom located at the cavity centre (z 5 0) exchange a birth, life and death of individual photons. Applying a similar quantum of energy, when the initially empty cavity is set at resonance with the e g transition 10. QND procedure 0 to mesoscopic fields with tens of photons should open 0 new 0.0 perspectives 0.5 for the exploration 1.0 of the quantum-toclassical b boundary 9,10. 1.5 2.0 2.5 A0.0 QND detection 1 3 realizes 0.5 an ideal projective 1.0measurement that 1.5 2.0 2.5 leaves the e system in an eigenstate of the measured observable. Time (s) It can therefore be repeated many times, leading to the same result until the S Figure system jumps 2 Birth, into another lifeeigenstate and death under the ofeffect a photon. of an external a, QND detection of a single perturbation. g For a single trapped ion, laser-induced fluorescence photon. provides 1 an Red efficient andmeasurement blue barsofshow the ion sthe internal rawstate signal, 5 7. Thea sequence of atoms detected inion e or scatters g, respectively many photons while (upper evolvingtrace). on a transition Thebetween insetazooms into the region where D ground sublevel and an excited one. This fluorescence stops and the reappears statistics abruptly ofwhen the the detection jumps in events and outsuddenly of a third, metastable level, decoupled from the illumination laser. Quantum jumps change, Brevealing the quantum R jump from have also 0 0æ to 1æ. The photon number inferred by a majority 1 vote over been observed between states of individual molecules 8 R and 2 C eight between consecutive 0.0the cyclotron atoms 0.5 motional is states shown of 1.0a single in the electron lower 1.5in atrace, revealing the birth, life 2.0 2.5 and Penning death trap 4 of. Asana common exceptionally feature, these long experiments Time lived (s) use photon. fields b, Similar signals showing to probe quantum jumps matter. Our experiment realizes for the two first Figure successive time the 2 opposite Birth, single life situation, and photons, in death whichof the separated ajumps photon. of a field a, byqnd oscillator are revealed via QND measurements performed with matter prepared in the circular state g in box B, out of a thermal beam of rubidium a long detection Figure time 1 Experimental interval of a single set-up. withsamples cavity of circular Rydberg atoms are State n n State photon. Red and blue bars show the raw signal, a sequence of atoms detected LETTERS Non-demolition measurements Time (s) a State n n State e 0.90 0.95 1.00 1.05 1.10 1.15 1.20 Quantum jumps of light recording the birth and death of ag photon in a cavity Sébastien Gleyzes 1, Stefan Kuhr 1 {, Christine Guerlin 1, Julien Bernu 1, Samuel Deléglise 1, Ulrich Busk Hoff 1, Michel Brune 1, Jean-Michel Raimond 1 & Serge Haroche 1,2 e in vacuum. atoms, velocity-selected by laser optical pumping. The atoms cross the cavity repetitions of the QND measurement. Methods phase shifts have much larger emission rates, in atom 3. Non-resonant methods in which the dete coupled to the cavity 12 have error rates of the or would require much larger d/v 0 ratios to be co observation of field quantum jumps. In a second experiment, we monitor the decay Fock state prepared at the beginning of each sequ the field in j0æ by first absorbing thermal photon prepared in g and tuned to resonance with the cav photon number,0.003 6 0.003). We then sen single atom in e, also resonant with C. Its interacti so that it undergoes half a Rabi oscillation, exits i j1æ. The QND probe atoms are then sent across C typical single photon trajectory (signal inferred by and Fig. 3b d presents the averages of 5, 15 and 90 The staircase-like feature of single events is prog out into an exponential decay, typical of the evolu -- How to measure photons without destroying them? -- How to record the cavity states? -- How does this fit within standard quantum mechanics textbooks? We have neglected so far the probability of find C. This is justified, to a good approximation, by precise statistical analysis reveals, however, the s two-photon events, which vanishes only at 0 K. W decays towards j0æ with the rate (1 1 n 0 )/T c. T spontaneous (1/T c ) and thermally stimulated annihilation. Thermal fluctuations can also driv photon state j2æ at the rate 2n 0 /T c (the factor o the photon creation operator matrix element be The total escape rate from j1æ is thus (1 1 3n 0 )/ (1 1 3n 0 ) < 0.10 of the quantum jumps out of jumps towards j2æ. In this experiment, the detection does not disti and j0æ. The incremental phase shift W(2,d) 2 W d/2p 5 67 khz. The probability of detecting an is in j2æ is ideally [1 2 cos(0.88p)]/2 5 0.96, indist within the experimental errors. Since the prob

Cavity QED experiments: Probe measurement apparatus Preparation of the probes Courtesy of LKB-ENS. Photons in a cavity Indirect measurements: Direct (Von Neumann) measurements on an auxiliary system.

Progressive field-state collapse and quantum non-demolition photon counting Christine Guerlin 1, Julien Bernu 1, Samuel Deléglise 1, Clément Sayrin 1, Sébastien 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. N 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. P.d.f. of the photon numbers Number of indirect measurements --- why does the pdf change after each indirect measurement? --- and what about the quantum Zeno effect? --- and what does continuous-in-time quantum measurements mean?

Cavity QED experiments: Courtesy of LKB-ENS. Probe measurement apparatus Preparation of the probes Photons in a cavity Indirect measurements: Direct (Von Neumann) measurements on an auxiliary system.

Motivations: Indirect evaluations of probability distribution functions (p.d.f.) of the photon number in a cavity. But how?... Courtesy of LKB-ENS. P.d.f. of the photon numbers Number of indirect measurements The estimated photon number p.d.f. is recalculated after each indirect measurements via Bayesian rules arising from Q.M. : Collapse of the p.d.f.!!?? (with a realization dependent target) «An exercise in probability theory» and in quantum mechanics.

Plan: Classical probability theory with a bit of quantum mechanics. - A view on (quantum) noises. - Repeated (quantum) measurements and Bayes law. (hidden random walks). - Repeated (quantum) measurements and collapses. (via the martingale convergence theorem). - Pointer states (and De Finetti s theorem).

Random walks... and repeated interactions. random walks via iterated kicks and measure or Events: (...) or ω = (+, +,, +, )=(ɛ 1,ɛ 2,ɛ 2, ) Filtration: B ɛ1,ɛ 2,,ɛ n := {ω =(ɛ 1,ɛ 2,,ɛ n, any thing else)} F n := σ algebra generated by all B ɛ1,ɛ 2,,ɛ n Gain of information as n increases. Or... we give ourselves the algebras B_n of F_n measurable functions Quantization: Everything becomes non-commutative...

Quantum repeated interactions:... QUANTUM NOISES... P := Probes S := Quantum System Iterations... Measurements on the probe after interaction with the Q-system -- Hilbert space: -- Algebras of observables: -- Filtration: H = H s H 1 H n B n := A s A 1 A n I Only test the first n probes. A s B n B m, for n<m Gain of informations: Test output probe observables on the n-th first probes, but probabilistic gain because of Q.M. Comparison with classical random walks (or stochastic processes.)

Today s Aim: Pick a basis a of states of the Q-system. Reconstruct the probability distribution for the system to be in state a Q 0 (α), Q 0 (α) = 1, α P := Probes S := Quantum System Measurements!!! ouputs: i in I Measure some observables on the (n-th) output probes: (not on the Q-system) The quantum filtration is reduced to a classical filtration. Possible outputs of the measurements: i in some set I. After n measurements: gain in information (i 1,i 2,,i n ) Events are the infinite sequences of measurement outputs. F n := σ algebra generated by all B i1,i 2,,i n {ω =(i 1,i 2,,i n, any thing else)}

Cavity QED experiments: Repeated non-demolition measurements. Preparation of the probes Probe measurement apparatus Courtesy of LKB-ENS. Photons in a cavity System (S)= photons in a cavity. Probes (P)= Rydberg atoms (two state systems : + or -) Probe measurements give values + or - ; Recursion for the photon number p.d.f. from data of sequences ω = (+, +,, +, )=(ɛ 1,ɛ 2,ɛ 2, ) Back to classical probability, to classical stochastic processes. Gain of information because probe-system entanglement.

Quantum mechanics implies «classical» Bayes rules Or what happens during one interaction + measurement cycle? * Preparation: -- probe: -- system: φ ψ = α C(α) α, Q 0 (α) = C(α) 2 * The only delicate point: we suppose that there is a basis of state a in H_s preserved by the probe-system interaction, i.e.: U α φ = α U α φ for the U the evolution operator during the probe-system interaction * After interaction: -- probe + system : 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 states distribution :... and new system state. Q new (α) = p(i α) Q 0(α) Z(i) with p(i α) = i U α α 2 Gain of information because probe-system entanglement.

Evolution of the probability distributions: Random Bayesian updatings... Pick a basis a of state of the Q-system. Start with a probability distribution (initial system state): Q 0 (α), Q 0 (α) = 1, α Let i be the output measurements on the probe. Data (probe-system interaction) are probabilities to measure i conditioned on the Q-system to in state a. p(i α), p(i α) =1 i Let Q_{n-1}(a) be the probability distribution of the Q-system after (n-1) cycles, The output of the n-th probe measurement is i_n with proba: π n (i) Then, Q n (α) = 1 Z n p(i n α) Q n 1 (α), with Z n = α p(i n α) Q n 1 (α)

Collapse of the p.d.f. Q n (α) as n Q n (α) = 1 Z n p(i n α) Q n 1 (α), with Z n = α p(i n α) Q n 1 (α) * Peaked distributions are stable (stability of the pointer states): Q n (α) =δ α;γ are solutions. Then, output i_n with probability: p(i n γ) * Probability distributions converge a.s. towards peaked distributions (collapse of the wave function): lim n Q n(α) =δ α,γω Prob[γ ω = β] =Q 0 (β) with a realization dependent target γ ω (Von Neumann rules for quantum measurements) * The convergence is exponential : Q n (α) exp[ ns(γ ω α)] (α γ ω ) with S(γ α) = i p(i γ) log p(i α) p(i γ) a relative entropy.

Collapse of the pdf (hence wave function): Courtesy of LKB-ENS. P.d.f. of the photon numbers Number of indirect measurements The estimated photon number p.d.f. is recalculated after each indirect measurements via Bayesian rules arising from Q.M. : Collapse of the p.d.f.!!?? (with a realization dependent target)

Proof: Q_n are bounded martingales. * Q n (α) = 1 Z n p(i n α) Q n 1 (α), with Z n = α p(i n α) Q n 1 (α) with probability π n (i n ) := α p(i n α) Q n 1 (α), thus : E[Q n (α) F n 1 ]= i π n (i) p(i α)q n 1(α) Z n = Q n 1 (α) * By the martingale convergence th., Q_n converges a.s. and in L1: * If all distributions p(i a) are different (for a, a diff.), (by the stationary point of the recursion relation) Q (α) = lim n Q n(α) Q (α) =δ α,γω exists By the martingale property, E[Q (β)] = Q 0 (β) Prob[γ ω = β] =Q 0 (β) * At large n, each i occurs with probability p(i γ ω) frequences: N n (i) n np(i γ ω ) Q n (α) =Q 0 (α) i p(i α)n n(i α) only one term is dominant in the sum Z_n. β Q 0(β) i p(i β)n n(i β) Q 0(α) Q 0 (γ ω ) exp [ ns(γ ω α)] * Independence w.r.t to the initial trial distribution (not Q_0, because the latter is a priori unknown).

Macroscopic measurement apparatus: Measure whether the system is in state a, i.e. measure observable with eigenstates a. Data of the apparatus: the p.d.f. s p(i α) on I, for all α. S := Quantum System For each infinite cycle, the apparatus provide the infinite sequence, (i 1,,i n, ) Reader: Compare the empirical histogram of the output measurements, with the given distributions p(i a) Target state = result of the measure Partial collapse for mesoscopic measurements. But also «classical Bayesian measurement apparatus». Generalizations : with different probes, probe measurements, etc.. randomly chosen Control and state manipulations.

Understanding the collapses of probability distribution functions: An interesting exercise in probability theory, with some quantum applications... Thank you.