The Open Quantum Brownian Motion: a Random Walk on Quantum Noise

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. The statisticalP.d.f. of resultsthe and repeatability) and should facilitate studies of non-classical fields trapped in cavities. histograms evolve, as N increases j 0101001101010101101011111 j 0001010100000100011101101 b, Evolution of PN(n) for the two

) ) from 0 to 110, from a flat

0.5 n 0.5 n i dababbaacbccdadccdcbaaaccphoton numbers ( i bcdaddaabbbbdbdcdccadaada ( sequences displayed in a, when N The projection of a microscopic system into an eigenstateN of the during a sequence of repeated measurements.N We have realizeddistribution a into n 5 5 and n 5 7 measured observable reflects the change of knowledgeΠ produced by superconducting cavity with a very long fieldΠ damping time28,peaks.increases and from 1 to 50, n being b the measurement. Information is either acquired in a single step, as in used it to detect repeatedly a single photon29. Here, we demonstratetreated as a continuous variable 1 Courtesy of a Stern–Gerlach spin-component measurement , or in an incremen- with this cavity a general QND photon counting method applied(integral to a of P (n) normalized to tal way, as in spin-squeezing experiments2,3. A projective measure- microwave field containing several photons. It implements a variant LKB-ENS.N 1.0 4–80.0 1.00.0 7 ment is called ‘quantum non-demolition’50 (QND) when7 the of a procedure proposed in refs 30 and50 31, and illustrates allunity). the c, Photon number 6 6 1 collapsed5 state is invariant under the system’s40 free unitary evolution.5 postulates of a projective measurement40 . probabilities plotted versus photon PhotonSequences numb4 of repeated measurements30 then yieldN identical resultsPhotTime and 4 evolutionA stream of atoms crosses the30 cavityN and performs a step-by-step 3 on numb3 jumps between2 different outcomes20 reveal an external perturbation7,8. measurement2 of the photon20 number-dependent alteration ofand the atom numbers n and N. The 10 10 Various QND1 measurements have been realized on massive part- atomic1 transition frequency known as the ‘light shift’. We follow er, 0 Atom number, er, 0 Atom number, histograms evolve, as N increases icles. The motionaln state of a trapped electron has been measured the measurement-inducedn evolution from a coherent state of light through the current induced in the trapping electrodes) 9. The internal into a Fock state of well-defined energy, containing) up to 7 photons.from 0 to 110, from a flat 0.5 n 0.5 n — How to record the( cavity states? Is it faithful representation?(

state of trapped ions has been read out, directlyN by way of laser- Repeating the measurement on the collapsedN state yields the samedistribution into n 5 5 and n 5 7 10 c induced fluorescence , or indirectly throughΠ quantum gate opera- result, until cavity damping makes the photonΠ number decrease. Thepeaks. tions— entangling Why them does to an ancillary the ion p.d.f.11. Collective changes spin states of anwithmeasured time field? energy then decays by quantum jumps along a stair- atomic ensemble have been QND-detected through1.0 its dispersive case-like cascade, ending in vacuum. 1.0 interaction— How with light 12does. it evolve? What’s the dynamics?In this experiment, W lighthy is andoes object ofit investigationbecome repeatedly peaked (collapsed)? QND light measurements are especially0.0 challenging, as photons interrogated by atoms. Its evolution0.0 under continuous non- 7 are detected with photosensitive materials50 that usually7 absorb them. destructive monitoring is directly accessible50 to measurement, making 6 Photon demolition is however40 avoidable13. In non-resonant6 pro- real the stochastic trajectories of40 quantum field Monte Carlo simula- 5 y 5 Photon numb4 cesses, light induces nonlinear30 dispersiveN effects14 in aPhot medium,4 with- tions20,32. Repeatedly counting30 photonsN in a cavity as marbles in a 3 Use (quantum) random walks to illustrate the universality of this phenomenon. bilit on numb3 2 out real transitions.20 Photons can then be detected0.5 without loss. 2 box opens novel perspectives20 for studying0.5 non-classical states of Dispersive1 schemes10 have been applied to detect the fluctuations of radiation.1 10 er, a signal0 light beamAtom by the number, phase shifts it induces on a probe beamer, 0 Atom number, n Proba n Probability interacting with the same medium15,16. Neither these methods, nor An atomic clock to count photons alternative ones based on the noiseless duplication of light by optical To explain our QND method, consider the thought experiment parametric amplifiers17,18, have been able, so far, to pin down photon sketched in Fig. 1a. A photon box, similar to the contraption ima- 0.0 0.0 1 c 7 numbers. 100 7 gined in another context by Einstein100 and Bohr , contains a few 6 Single-photon resolution requires an extremely strong6 light– Ph 5 80 Photon5 numbephotons together with a clock whose80 rate is affected by the light. 4 60 N 4 60 N otonmatter3 coupling, optimally achieved by1.0 confining radiation inside a 3Depending upon the photon numberr, n1.0, the hand of the clock points cavity. numbe This2 is the domain of cavity40 quantum electrodynamics19–21, in in2 different directions after40 a given interaction time with the field. 20 20 which experiments1 attaining single-quantum resolution have been Thisr, time1 is set so that a photon causes a p/q angular shift of the hand r, n 0 n 0 0 performed with optical0 22,23 orAtom microwave number, photons, the latter being (here q is an integer).Atom There numbe are 2q values (0, 1, …2q21) of the coupled either to Rydberg atoms24–26 or to superconducting junc- photon number corresponding to regularly spaced directions of the tions27. In a QND experiment, cavity losses should be negligible hand, spanning 360u (Fig. 1a shows the hand’s positions for q 5 4 and y 1Laboratoire Kastler Brossel, Ecole Normale Supe´rieure, CNRS, Universite´ Pierre et Marie Curie, 24 rue Lhomond, 75231 Paris Cedex 05, France. 2Colle`ge de France, 11 place Marcelin 891 Berthelot, 75231 Paris Cedex 05, France. {Present0.5 address:bilit Johannes Gutenberg© 2 Universita007 N¨t,a Institutture P fu¨ru Physik,blish Staudingerweging Group 7, 55128 Mainz,0.5 Germany. 889 © 2007 Nature Publishing Group

Proba

Probability

0.0 0.0 100 100 7 6 7 6 Ph 5 80 Photon5 numbe 80 4 60 N 4 60 N oton 3 3 r, numbe2 40 2 40 1 20 1 20 r, r, n 0 0 Atom number, n 0 0 Atom numbe

891 © 2007 Nature Publishing Group II) Quantum trajectories and a diffusive to ballistic transition in open quantum Brownian motion. — Open Quantum Brownian Motion: A quantum random motions on the line for a « quantum walker » with internal and orbital degrees of freedom.

A Brownian like regime. A ballistic like regime.

Position

Internal gyroscope

Transition in the behavior of the internal « gyroscope »: from oscillation to bi-stability with jumps (analogous to Bohr quantum jumps)

— The ballistic flips are not due to collision but to the flips of the internal gyroscope. Outline:

— Quantum parallelism on (classical) random walks and measurements. (progressive collapse or `quantum computation’ with random walks) — A glance at Quantum Noise and quantum trajectories. (how analogue to classical processes is it?) — Open quantum random walk trajectories. (a random walk but with an internal quantum gyroscope) — Two regimes and bi-stability (diffusion or ballistics). (gyroscope quantum jumps and flips a la Kramer’s) — How to make OQBM conformally invariant? (if time permits….) Quantum Parallelism on Random Walks.

— Consider a random walk on the line:

Events: ! =(+, +, , +, )=(✏1,✏2,✏2, ) ··· ··· Position: Xn(!)=X0 + ✏1 + + ✏n ··· — We can use Quantum Mechanics to describe it, keeping all possible walks at once:

Let C2 be an auxiliary “coin” Hilbert space, with basis . |±i

1 n := n/2 Xn(!]) [!n] | i 2 [!n] | i⌦| i

P 2 2 with a sum over all walks of n steps. [!n] = ✏1 ✏n C C | i | i⌦···⌦ | i2 ⌦ ···⌦ — There is no more information than in the usual description, all walks appear in the sum with equal weight: simple « book keeping » or « quantum parallelism », keeping track of the position « X », of the history « w ». Quantum Parallelism as dynamical process.

— We view the auxiliary Hilbert space of an auxiliary coin/probe. We consider a collection of probes. Assume that each probes are all initially prepared in a state C2 | i2 — Consider the elementary (unitary) evolution (walker+one probe):

x 1 x +1 + + 1 x 1 | i⌦| i! p2 | i⌦| i p2 | i⌦|i

After e.g. there iterations:

— The occurrence probability of a given walk is the modulus square of the amplitude (« quantum measurement »). A given output measurement projects on a given realization of the random walk. Quantum noise and repeated quantum interaction.

S := Quantum System

P := Probes Out-going probes after interaction with with the Q-system measurements Iterations... or not?

— Hilbert space: = H Hs ⌦H1 ⌦···⌦Hn ⌦··· — Algebras of observable: n := s 1 n I B A ⌦A ⌦···⌦A ⌦ — (Quantum) Filtration: , for n

— Once the auxiliary space is viewed as that of probes we may imagine measuring the « spin » probes in another direction, say at an angle « theta ». u = cos #/2 sin #/2 |± i |±i ± |⌥i Measuring the probes (successively) given a random output +/-, (with probability given by the (square of the) projection of the state on the above vectors). Hence the state of the walker after the probe measurement is random.

— What is this random state evolution? = x , with 2 =1 At time t: | ti x x,t | i x | x,t| 1 Elementary evolution: t P x,t x +1P + + x 1 . | i! p2 x | i⌦| i | i⌦|i Measuring the spin in the « u » direction, we haveP to project the probe on the above states, and the probability of +/- is the modulus square of the amplitude:

1 u u t+1 = x 1,t + + x+1,t x Probe measurement: p2p (t) x | i ± h± | i h± |i | i 2 1 P u u with probability: p (t):= x 1,t + + x+1,t . ± 2 x h± | i h± |i P — The probability distribution of the walker position is then always delocalized. « But » this distribution in momentum space collapses to a peaked measure !! i.e. the Fourier component of the wave function collapse almost surely. 1 2i⇡kx/N 2 — In momentum space x,t = k,t e , with k,t =1 pN k k | | for a walk on ZN P P

— The momentum distribution 2 are random but: | k,t|

Claim: 2 limt k,t = k;k , with k random !1 | | 1 1 2 P[k = p]= p,t=0 1 | |

— Open Quantum Random Walks but wit tilted probe measurements define « progressive » quantum measurements of the momentum. Progressive: the collapse/measurement takes some time, but the collapse is exponential in time. Proof: — The random state evolution in momentum space 2 1 2 k,t+1 = 2p (t) 1 sin # cos 4⇡k/N k,t | | ± ± 1 ⇥ ⇤ 2 p (t)= 1 sin # cos(4⇡l/N) l,t . ± 2 l ± | | ⇣ ⌘ preserving the normalisationP of the momentum⇥ distribution. ⇤ — Why does the momentum distribution converges and collapse?

t 2 are bounded martingales. ! | k,t| as such they converge almost surely and in L1

A martingale is a random process which is conserved 2 2 in mean when conditioned on the past: E[ p,t+1 t]= p,t And here it is tautologically true. | | |F | |

— Bayesian interpretation. The above formula is actually the formula for conditional probability: p( k) 2 1 2 = ±| | k,t| , with p( k):= [1 sin ✓ cos(4⇡k/N)] | k,t+1| p (t) ±| 2 ± ± Quantum mechanics code for the Bayesian rules. Collapse is « classical »: An echo of random Bayesian updating Generalisation and application: — The previous statement is general:

Iterating ad infinitum « system-probe interaction » plus « probe measurement » results in the collapse of the system probability distribution, provided that the interaction between the probes and the system preserves a specified basis of states, call « pointer » states.

These are identical phenomena.

— These are mesoscopic (quantum non-demolition) measurements (they become macroscopic only when an infinite number of probes have been involved) A Bayesian point of view.

— Many applications: Measurement continuously in time, (e.g. imaging « Bohr » virtual quantum jumps (experiments/theory)). Quantum control, etc.… Open quantum random walks (I)

— Back to the quantum parallelism on random walks but we add internal degree of freedom, « quantum gyroscope » to the walker.

2 * Hilbert space of states: c = C , o = CZ p=probes (or coins); H H c=color (or spin), internal d.o.f’s; 2 2 c o p⌦1 ⌦1 = C C o=position, orbital d.o.f.’s. H ⌦H ⌦H Hp ⌦ ⌦··· Coins are reset at each step.

— Interaction (without measurements):

On state c n o p in , the unitary evolution | i ⌦| i ⌦| i Hc ⌦Ho ⌦Hp

(B+ c n +1 o + p +(B c) n 1 o p | i⌦| i ⌦| i | i ⌦| i ⌦|i

Iterating interactions with different probes produce an «entangled» state, sum of states each indexed by a random walk (as before).

— The original definition of open QRW was the mean of this process (not keeping track of the measurements). Attal, Petruccione, Sabot, Sinayskiy, 2012. Open quantum random walks trajectories (II).

— Measurements and «quantum trajectories»:

(B+ c n +1 o + p +(B c) n 1 o p | i⌦| i ⌦| i | i ⌦| i ⌦|i Measuring the probes, one may find + or - with probabilities, and then «project» the state on |+> or |->:

(B c) n 1 o with probability c B† B c / ±| i ⌦| ± i h | ± ±| i The events, the output of the probe measurements, are in one-to-one correspondence with random walks.

— These are classical random processes with values on the line x the internal states.

ie. the system is not described by a vector but by a density — For mixed (internal) state: matrix (a positive hermitian normalised matrix).

If after n step, (⇢n,xn) 1 The updating is: (p (n) B ⇢nB† ,xn 1) For a pure state: ± ± ± ± ⇢n = n n with probability: p (n)=tr(B ⇢nB† ) | ih | ± ± ±

The matrices B are the moduli parameters of the walks. Events are random walks (output measurements), but with probabilities induced by the gyroscope motion. Numerical simulations:

— Unitarity or normalisation of probabilities imposes: B+† B+ + B† B = I 1 ur 1 vs A choice for the simulations: B+ = and B = r u sv with = pu2 +v2 + r2 + s2

A Brownian like regime. A ballistic like (but diffusive) regime. u =1.005, v =1.00 and r = s =0.00015 u =1.1, v =1.00 and r = s =0.00015

— Kramer’s like transition for a particle in a double well potential (... but not quite). Scaling limit: «Open quantum Brownian motion».

— In the scaling limit one gets a time continuous process (with continuous measurement) The scaling limit is ✏ 0, t = n✏ fixed, and dx2 dt. ! ⇠ — As for classical Brownian motion, scale time, distance and moduli simultaneously:

1 1 B = [I p✏N + ✏(iH M N †N)+o(✏)] ± p2 ± ± ± 2 with ✏ a small parameter and H , M hermitian but not N. ±

— In the scaling limit, the series of +/- of output probe measurements become a random process driven by a « classical » Brownian motion.

d⇢t = i[H,⇢t]+LN (⇢t) dt + DN (⇢t) dBt, A Brownian motion, coding for all probe dXt = UN (⇢t) dt + dBt, measurements. 1 with L (⇢):=N⇢N † (N †N⇢+ ⇢N †N) N 2 D (⇢):=N⇢+ ⇢N † ⇢U (⇢) and U (⇢ ):=tr(N⇢+ ⇢N †) N N N t — Both the density matrix and the position have stochastic evolution but the position drift is guided by the internal gyroscope. Transition between two regimes: 2 3 1,2,3 Take H := !0 and N = a with the usual Pauli matrices. 1 1 3 Describe ⇢t as a pure state: ⇢t = 2 (I+q1 +q3 )withq1 =sin✓, q3 = cos ✓.

Eqs. d⇢t = i[H,⇢t]+LN (⇢t) dt + DN (⇢t) dBt, becomes: d✓ = 2(! + a2 sin ✓ cos ✓ )dt 2a sin ✓ dB t 0 t t t t 2 a

✓t — Rabi like oscillations: 2 a

2 a >!0

✓ ✓t — Quantum jumps +⇤

2 a >!0 ✓⇤

— Two potential minima, with Kramer’s transition between them. « but not quite », because jumps arise for « strong noise ». Ballistically induced diffusion:

— Trajectories are ballistic, with seesaw profiles induced by gyroscope flips, and large mean free paths.

dXt = UN (⇢t) dt + dBt becomes:

dXt =2a cos ✓t dt + dBt

with 2a cos ✓⇤ 2a ± '⌥

The seesaw behavior is not due to collision but to the internal gyroscope behavior.

— But at very large time the position is Gaussian, with large effective diffusion constant.

2 4 2 E[Xt ]=De↵ t ,withDe↵ =1+4a /!0

Can we find similar phenomena (phenomenological description) Question: producing a large effective diffusion constant (at large time)? How to make the Open Quantum Brownian Motion conformally invariant ? In progress

— The 2D Brownian motion is conformally invariant (Levy), ¯ up to random time parametrization: dBt dBt =2dt

— A way to render a 2D quantum walker conformally invariant is to couple it to an internal conformal field theory (CFT). Which CFT? Which evolution equation for the internal state, The walker position will satisfy a SDE with a drift of the form: dXt = UN (⇢t) dt + dBt — To preserve conformal the drift has to be a (0,1)-form, because time scales under conformal transformation: dZt = ⌦¯ D(Zt, Z¯t) dt + dBt.

Hence, the CFT has to have a U(1) current and ⌦¯ D(Zt, Z¯t):=4p!D J¯(Z¯t) t h i — We then have to write the corresponding SDE for the evolution of the internal CFT state !D according to the rule of the OQBM. t h···i d!D A = p !D J(Z )A + AJ(Z ) 2 !D J(Z ) !D A dB +c.c., t h i t h t t i t h t i t h i t dZ =4p!⇥ D J¯(Z¯ ) dt + dB ⇤ t t h t i t

D The SDE for the internal state has an additional drift term, !t L⇤ (A) dt h [Zt] i but in general it doe snot contribute. Thank you. Open Quantum Brownian motion:

--- Quantum trajectories: Probes are measured (with random outputs), and the system density matrix evolves randomly (according to the measurement outputs). .... and this can be generalised with many packets more below..... and/or entangled states.

--- Quantum dynamical map: Probes (reservoir) are *not* measured but trace out, and the system density matrix evolves with Lindblad equation

1 @ ⇢¯ = [P, [P, ⇢¯ ]] + i N[P, ⇢¯ ]+[P, ⇢¯ ]N † + i[H, ⇢¯ ]+L (¯⇢ ) t t 2 t t t t N t for the density matrix ⇢¯ := dx ⇢(x, t) x x t ⌦| ioh | R --- Quantum Stochastic equation: Probes (reservoir) are *not* measured but *not* trace out, and the *total* state evolves with a quantum-SDE.

dAt = i[P + iN †,At] d⇠t + i[P iN, At] d⇠t† + L (At) dt ⇤ for observable A, and dual Linbladian L.,

and (quantum) noises: d⇠t d⇠t† = dt, d⇠t† d⇠t =0