The Open Quantum Brownian Motion: a Random Walk on Quantum Noise. D.B., Michel Bauer and Antoine Tilloy IAS-Princeton, March 2014 Two facets of Quantum Noise: — Observing, manipulating, controlling quantum systems. These are probabilistic by nature, and if time evolution is taken care of these are stochastic processes. — Extending applications of probability theory by incorporating quantum effects, e.g. quantum stochastic processes (as open quantum systems). Aim: Use the Open Quantum Brownian Motion (the definition will come) to illustrate a few notions/properties of Quantum Noise. — Progressive collapse in non-demolition quantum measurements (as an illustration of the first point above) — Transition in quantum random walks from diffusive to ballistic regimes. (as an illustration of the second point above) NATURE | Vol 448 | 23 August 2007 ARTICLES NATURE | Vol 448 | 23 August 2007 ARTICLES Observing the field-state collapse (Methods). The evolutions of PN(n), displayed as functions of n We have applied this procedure to a coherent microwave field at treated as a continuous variable, are shown in Fig. 2b for N increasing 51.1 GHz stored in an ultrahigh-Q Fabry–Pe´rot cavity made of from 1 to 50. The PN(n) functions converge into narrow distribu- 28 tions whose widths decrease as more information is acquired. These Observingniobium-coated the field-state superconducting collapse mirrors . Our set-up is described(Methods). The evolutions of PN(n), displayed as functions of n functions are determined uniquely by the experimental data. Their We have appliedin ref. 29. this The procedure cavity has a to very a long coherent damping microwave time Tc 5 field0.130 at s. It istreated as a continuous variable, are shown in Fig. 2b for N increasing cooled to 0.8 K (average thermal photon number nt 5 0.05). The field evolution is independent of any a priori knowledge of the initial ´ from 1 to 50. The PN(n) functions converge into narrow distribu- 51.1 GHzis stored prepared in by an coupling ultrahigh- a shortQ Fabry–Pe microwaverot pulse cavity into made C (by way of of photon distribution. The data sequence itself, however, depends of 28 tions whose widths decrease as more information is acquired. These niobium-coateddiffraction superconducting on the mirrors’ mirrors edges28). Its. Our photon set-up number is described distribution the unknown state of the field, which the measurement reveals. functions are determined uniquely by the experimental data. Their in ref. 29. Theand average cavity has photon a very number, long dampingn0 5 3.82 6 time0.04,Tc are5 inferred0.130 s. from It is the Inserting PN(n) into equation (2) and extending the procedure to cooled to 0.8experimental K (average data thermal (see photon below). number Our single-photon-sensitivent 5 0.05). The field spin-evolutionN 5 110, is we independent obtain the evolution of any of a thepriori photon knowledge number histograms of the initial is preparedclocks by coupling are circular a short Rydberg microwave atoms of rubidium. pulse into They C (by cross way C succes- of photonfor these distribution. two realizations The data (Fig. sequence 2c). These itself, histograms however, show depends how our of 24 diffractionsively, on the separated mirrors’ on edges average28). by Its 2.33 photon3 10 numbers. Parameters distribution are adjustedtheknowledge unknown of state the field of the state field, evolves which in a the single measurement measuring sequence, reveals. as to realize a ,p/4 clock shift per photon (Methods), corresponding to inferred from baysian logic. The initial distributions (P0(n) 5 1/8) and average photon number, n 5 3.82 6 0.04, are inferred from the Inserting PN(n) into equation (2) and extending the procedure to 0 are flat because the only knowledge assumed at the beginning of each experimentaleight data positions (see ofbelow). the spin Our on the single-photon-sensitive Bloch sphere (Fig. 1b). spin-This con-N 5 110, we obtain the evolution of the photon number histograms figuration is adapted to count photon numbers between 0 and 7. For sequence is the maximum photon number nmax. Data are analysed clocks are circular Rydberg atoms of rubidium. They cross C succes- forafter these the two experiment, realizations but (Fig.P (n 2c).) could These also histograms be obtained showin real how time. our n0 5 3.82, the probability for n $284 is 3.5%. N sively, separatedFour on phases averagew (i by5 a 2.33, b, c3, d),10 correspondings. Parameters to directions are adjusted pointingknowledgeThe progressive of the field collapse state of evolves the field in into a single a Fock measuring state (here sequence,n 5 5æ or as i j to realize aapproximately,p/4 clock shift along per the photon spin states (Methods), associated corresponding with n 5 6, 7, 0, to 1, areinferredn 5 7 fromæ) is clearly baysian visible. logic. Information The initial extracted distributions from the first (P0( 20n) to5 301/8) j eight positionsused, ofinVol random the 448 spin| 23 August order, on 2007 the for| doi:10.1038/nature06057 Bloch successive sphere atoms (Fig. (Methods). 1b). This A con- sequenceareatoms flat because leaves an the ambiguity only knowledge between assumedtwo competing at the Fock beginning states. After of each figurationof is adaptedj values can to be count decoded photon only numberswhen combined between with 0 the and correspond- 7. For sequence,50 atoms is the (detected maximum within photon,0.012 number s), each distributionnmax. Data has are turned analysed ing phase choices, in analogy with the detection basis reconciliation into a main peak with a small satellite, which becomes totally neg- n0 5 3.82, theI) probability Progressive for n $ 8 is 3.5%. collapse in non-demolitionafter the experiment, but PN(n) couldmeasurements also be obtained in real time. 36 ligible at the end of the two sequences. Four phasesof quantumw (i 5 a key, b, distributionc, d), corresponding protocols to. Figure directions 2a shows pointing the dataThe progressiveARTICLES collapse of the field into a Fock state (here n 5 5æ or from thei first 50 detected atoms, presented as (j, i) doublets, for two j approximately along the spin states associated with n 5 6, 7, 0, 1, are n 5 7æ) is clearly visible. Information extracted from the first 20 to 30 independent detection sequences performed on the same initial field.j Reconstructing photon number statistics used, in randomFrom order, these for real successive data, we computeatoms (Methods). the products A sequence of functionsatomsRepeatedly leaves an preparing ambiguity the between field in the two same competing coherent Fock state, states. we have After of j values can be decoded only when combined with the correspond- ,50 atoms (detected within ,0.012 s), each distribution has turned PN(n) 5PProgressive(k 5 1…N) [A 1 Bcos(nW 2 field-statewi(k) 1 j(k)p)]. The A, B, collapseW analysed 2,000 and independent sequences, each made of 110 (j, i) doublets ing phase choices,and wi values in analogy are given with the by detection Ramsey interferometer basis reconciliation calibrationintorecorded a main within peak withTm < a0.026 small s. This satellite, measuring which time becomes is a compromise. totally neg- of quantum key distribution protocols36. Figure 2a shows the data ligible at the end of the two sequences. Figure 2 | Progressive collapse of from the first 50 detectedquantum atoms, presented non-demolition as (j, i) doublets, for two photon counting a field into photon number state. 1 1 1 Reconstructing1 photon1 number1 statistics independent detectionjChristine 1101111111110011101101111 sequences Guerlin , performed Julien Bernu , on Samuel the same Dele´glise initial,j Cle 0001000110110000001010110´ field.ment Sayrin , Se´bastien Gleyzes , Stefan Kuhr {, a, Sequences of (j, i) data (first 50 1 1 1,2 atoms) produced by two From these realiMichel ddcbccabcdaadaabadddbadbc data, Brune we, Jean-Michelcompute the Raimond products& Serge of Haroche functionsi ddcaddabbccdccbcdaabbccabRepeatedly preparing the field in the same coherent state, we have independent measurements. PN(n) 5P(k 5 1…NThe) [ irreversibleA 1 Bcos( evolutionnW 2 ofw ai(k microscopic) 1 j(k)p)]. system The underA, measurementB, W analysed is a central 2,000 feature independent of quantum theory. sequences, From an each made of 110 (j, i) doublets j 0101001101010101101011111 j 0001010100000100011101101 b, Evolution of PN(n) for the two initial state generally exhibiting quantum uncertainty in the measured observable, the system is projected into a state in and wi values arei dababbaacbccdadccdcbaaacc given by Ramsey interferometer calibrationi bcdaddaabbbbdbdcdccadaadarecorded within Tm < 0.026 s. This measuringsequences displayed time is in aa compromise., when N which this observable becomes precisely known. Its value is random, with a probability determined by the initial system’sincreases from 1 to 50, n being state. The evolution induced by measurement (known as ‘state collapse’) can be progressive, accumulating the effects of b treatedFigure as 2 a| Progressive continuous variable collapse of elementary state changes. Here we report the observation of such a step-by-step collapse by non-destructively measuringfield into photon number state. a the photon number of a field stored in a cavity. Atoms behaving as microscopic clocks cross the cavity successively. By(integral of PN(n) normalized to 1.0 1.0 c j 1101111111110011101101111measuring the light-induced alterations of the clockj rate,0001000110110000001010110 information is progressively extracted, until the initially uncertainunity).a, Sequences, Photon of number(j, i) data (first 50 photon number converges to an integer. The suppression of the photon number spread is demonstrated by correlationsprobabilitiesatoms) produced plotted by versus two photon i ddcbccabcdaadaabadddbadbc i ddcaddabbccdccbcdaabbccab between repeated measurements. The procedure illustrates all the postulates of quantum measurement (state collapse,andindependent atom numbers measurements.n and N.