Real Time Imaging of Quantum and Thermal Fluctuations

Transcription

1 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: , arxiv: , arxiv: , to appear (hopefully soon...) Kyoto, July 2013

2 g Time (s) 1 A microscopic quantum 0.90 system 0.95 under 1.00 continuous 1.05observation 1.10 microwave photons. In our experiment, the measurement of the exhibits at erandom times sudden jumps between its states. The light shift induced by the field on Rydberg atoms is repeated detection of this quantum feature requires a quantum nondemolition (QND) measurement 1 3 repeated many times during photons. more than 100 times within the average decay time of individual 0 the system s evolution. Whereas quantum jumps of trapped massive particles g (electrons, ions or molecules 4 8 ) have been observed, is an open cavity C made up of two 0.0 superconducting 0.5 niobium 1.0of a thermal The core of the experiment is a photon box (see Fig. 1), which Creation-Annihilation 1.5 photon this has proved more challenging for light quanta. Standard mirrors facing each other (the Fabry Perot configuration) a 13. The Time (s) photodetectors 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 Figure of 2 a classical Birth, life injected and death microwave field, is T c photon. s, corresponding Red and blue to a light barstravel show the raw signal, a sequence of atoms detected of a photon. a, QND detection of a single that can e see photons without destroying them 3. Moreover, the light needs to be stored for durations much longer than the QND distance of 39,000 km, foldedinine the or g, 2.7respectively cm space between (upperthe trace). The inset zooms into the region where detection g time. Here we report an experiment in which we fulfil mirrors. The QND probes are rubidium the statistics atoms, ofprepared the detection in circular events suddenly change, revealing the quantum these challenging conditions and observe quantum jumps in the Rydberg states 1 10, travelling along jump the from z direction 0æ to transverse 1æ. Thetophoton the number inferred by a majority vote over photon number. Microwave photons are stored in a superconducting cavity for times up to half a second, and are repeatedly probed velocity v 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 eight consecutive atoms is shown in the lower trace, revealing the birth, life by a stream of non-absorbing atoms. An atom interferometer onant with the transition between and the death twoof circular an exceptionally states e and glong lived photon. b, Similar signals showing measures 0 the atomic dipole phase shift induced by the nonresonant cavity field, so that the final atom state reveals directly tion- (z-)dependent atom field in coupling vacuum. V(z) 5 V 0 exp(2z 2 /w 2 ) (principal quantum numbers 51 two Courtesy and successive 50, respectively). of single LKB-ENS. photons, The posi-separated by a long time interval with cavity the presence 0.0 of a single0.5 photon in the1.0 cavity. Sequences 1.5of hundreds b of atoms, highly correlated in the same state, are interrupted The maximum coupling, V 0 /2p khz, is the rate at which the follows 2.0 the gaussian 2.5 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 e and death of individual photons. Applying a similar quantum of energy, when the initially empty cavity is set at resonance with the e gthem transition QND procedure mesoscopic fields with tens of photons should open -- How new perspectives to measure for the exploration photons of the quantum-toclassical boundary 9,10. without destroying? 10. g -- 1 AHow QND detection to record 1 3 realizes an ideal the projective cavity measurement states that? leaves the system in an eigenstate of the measured observable. It can therefore -- How be repeated to observe many times, leading quantum to the same result jumps? until the Are they detector dependent S? system jumps into another eigenstate under the effect of an external 0 perturbation. -- What For determines a single trapped ion, the laser-induced Q-jump fluorescence dynamics? provides0.0 an efficient measurement 0.5 of the1.0 ion s internal state The ion scatters many photons while evolving on atime transition (s) between a D ground sublevel and an excited one. This fluorescence stops and reappears Figureabruptly 2 Birth, when lifethe and iondeath jumps of in and a photon. out of a third, a, QND metastable The photon. level, photon decoupled Red and from blue system the bars illumination showis theprobed raw laser. signal, Quantum aindirectly sequence jumps of atoms via detected another R 1 quantum system. detectionb of a single have inalso e orbeen g, respectively observed between (upper states trace). of individual The inset molecules zooms 8 R andinto the region where 2 C between the statistics the cyclotron of themotional detectionstates events of suddenly a single electron change, in revealing a the quantum Penning trap jump from 4. As a common feature, all these experiments use fields 0æ to 1æ. The photon number inferred by a majority vote over to probe quantum jumps in matter. Our experiment realizes for the first eight timeconsecutive the opposite situation, atoms isinshown which the in jumps the lower of a field trace, oscil- revealing Figurethe 1 Experimental birth, life set-up. Samples of circular Rydberg atoms are prepared in the circular state g in box B, out of a thermal beam of rubidium State n n State LETTERS e Non-demolition measurements and Q-jumps Quantum jumps of light recording the birth and death 0 of a photon in a cavity b 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 a State n n State g 1 g Fock the f prep phot singl so th j1æ. T typic and The out i avera W C. T prec twodeca spon anni phot the p The (1 1 jump In and d/2p is in with 2007 Nature Publishin

3 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) Preparation of the probes Probe measurement apparatus 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). -- Interaction like rotation of the gyroscope (with an angle depending on the number of photons) U = exp[iθσ z N photon ]

4 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 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)

5 Outline: Part I: Classical probability theory with a bit of quantum mechanics, (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.)

6 Quantum non-demolition measurement and random bayesian updating. Preparation of the probes Probe measurement apparatus -- Repeated cycles of interaction plus probe measurement: Partial gain of information at each iteration, because of system-probe entanglement. up-dating of the trial/estimated p.d.f. at each step, using an a priori model for the output conditional probabilities state Photons in a cavity -- Probe measurements give values + or - ; Recursion for the photon number p.d.f. from data of sequences P estimated (N photon probe state) = P a priori( probe Nphoton ) P trial (N photon ) P normalisation ( probe state) Courtesy of LKB-ENS. ω = (+, +,, +, )=(ɛ 1, ɛ 2, ɛ 2, ) -- 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?

7 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 -- Interaction: A delicate point : we suppose that there is a basis of system states *preserved* by the probe-system interaction, i.e.: U α φ = α U α φ for U the evolution operator of the probe-system interaction This will be related to the existence of *pointer states*, to exchangeability, 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 state distribution :... and new system state. Q with p(i α) = i U α α 2 new (α) = p(i α) Q 0(α) Z(i) i.e. Bayes rules

8 Evolution of the probability distributions: Random Bayesian up-dating... Pick a basis a of states of the Q-system. Start with a probability distribution (initial system state): Let i be the output measurements on the probes. Data (probe-system interaction) are probabilities to measure i conditioned on the Q-system to be in state a. Q 0 (α), p(i α), Q 0 (α) = 1, α p(i α) =1 i P := Probes Iterations... S := Quantum System Out-going probes after interaction with the Q-system outputs... 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 probability : Then, Q n (α) = 1 Z n p(i n α) Q n 1 (α), with Z n = α π n (i) p(i n α) Q n 1 (α)

9 Collapse of the p.d.f. Q n (α) as n (A classical statement...) Q n (α) = p(i n α) Q n 1 (α) Z n, with proba π n := Z n = α p(i n α) Q n 1 (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): lim n Q n(α) =δ α,γω Prob[γ ω = β] =Q 0 (β) with a realisation 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.

10 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: (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. α ρ n β = ( U αu β ) n α ρ0 β 0 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 np(i γ ω ) -- The limit is independent of the initial trial distribution. (Important for the experiment).

11 Quantum noise and repeated quantum interaction. e.g. as in cavity QED experiments... Preparation of the probes Probe measurement apparatus S := Quantum System Photons in a cavity Courtesy of LKB-ENS. P := Probes Iterations... Out-going probes after interaction with the Q-system with measurements or not? -- Hilbert space: -- Algebras of observable: H = H s H 1 H n 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)

12 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. 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, ) i.e. the frequencies N(i α) 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, randomly chosen, etc.. -- continuous in time description, continuous measurements, etc... Applications: e.g. control and state manipulations...

13 Time continuous measurement and Q-jumps: Making real Bohr s «virtual» quantum jumps -- As a model for discrete repeated measurement but with short time interval -- hamiltonian evolution (T=0): ρ U hamilton ρ U hamilton -- measurement (POVM) : ρ (F i ρ F i )/π i, with F i := i U meas. ψ -- If time duration of «probe+system interaction cycles» is small: dρ = i[h, ρ] dt +(dρ) meas Random time continuous measurements, (randomness due to (random) output probe measurements) -- If i ψ 0 (a condition on probe data), these are diffusive like equations (Belavkin s eqs.) For spin 1/2 probes : discrete (+, +,, +,, ) B t := brownian motion. (dρ) meas = L meas (ρ) dt + D meas (ρ) db t (in law) For a Q-bit system : 0, 1 τ collapse γ 2 dq t = γ Q t (1 Q t ) db t, for Q t = 0 ρ t Non-demolition measurement : H and the measured observable commute.

14 Time continuous measurement and Q-jumps (II): -- What happens if the H and the measured observable do not commute? A system (spin half) under continuous measurement. Take H = ω 0 σ 2 and measure S ( z = With: ρ = cos θσ 3 + sin θσ 1) σ 3 dθ t = (ω 0 + γ 2 sin 2θ t )dt 2γ sin θ t db t γ 2 ω 0 Slightly deformed Rabi oscillations (measurement does take place) γ 2 ω 0, τ collpase τ evolution Q-jumps between the two eigen-states τ flip τevolution/τ 2 collapse -- Technically: Kramers like transition for a two well potential random process. Conclusion -- Measuring an observable commuting with H : (progressive) collapse. -- Measuring an observable not commuting with H : Q-jumps.

15 Real Time Imaging of Quantum and Thermal Fluctuations. -- For system in contact with a thermal reservoir and under continuous measurements. Reservoir -- Quantum/Thermal fluctuations and Q-jumps: Quantum System Outputs What are the quantum trajectories? -- Evolution under thermal contact: ρ n ρ n := M therm [ρ n ] := k B k ρ n B k -- Recursive indirect measurements: ρ n ρ n+1 := M i n meas [ ρ n ] := F in ρ n F i n /π in Energy cascade and Q-jumps with probability π in := Tr(F in ρ n F i n )

16 Real Time Imaging of Quantum and Thermal Fluctuations (II). -- Time continuous formulation: dρ =(dρ) therm. +(dρ) mes. Two time scales : τ collapse τ therm. Deterministic thermal evolution (Lindblad) Random time continuous measurements -- For two states systems (with spin half probes): ρ = Q (1 Q) 1 1 (γ 2 λ) dq t = λ [p Q t ] dt + γ Q t (1 Q t ) db t. * Waiting times : T 1 τ therm /p, T 0 τ therm /(1 p), T 0 /T 1 = e β, and exponentially distributed. * Jump times : τ jump τ collapse log(τ therm /τ collapse ) controlled by the measurement process, * Stationary measure: Close to Gibbs but not quite. Q-trajectories and Q-jumps Generalisations with many states and arbitrary probes (OK), and Applications... with more thermal bath (off-equilibrium).

17 Understanding repeated QND measurements and mesoscopic measurements : An interesting exercise in probability/q.m. theory, with some (probable) quantum applications... Thank you.