Controlling chaos in the quantum regime using adaptive measurements
Jessica K. Eastman, Stuart S. Szigeti, Joseph J. Hope, André R. R. Carvalho
CControlling chaos in the quantum regime using adaptive measurements
Jessica K. Eastman,
1, 2, 3, ∗ Stuart S. Szigeti, Joseph J. Hope, and Andr´e R. R. Carvalho Department of Quantum Science, Research School of Physics and Engineering,The Australian National University, Canberra, ACT 2601 Australia Centre for Quantum Computation and Communication Technology (Australian Research Council),Griffith University, Brisbane, Queensland 4111, Australia Centre for Quantum Dynamics, Griffith University, Brisbane, Queensland 4111, Australia (Dated: January 14, 2019)The continuous monitoring of a quantum system strongly influences the emergence of chaoticdynamics near the transition from the quantum regime to the classical regime. Here we present afeedback control scheme that uses adaptive measurement techniques to control the degree of chaosin the driven-damped quantum Duffing oscillator. This control relies purely on the measurementbackaction on the system, making it a uniquely quantum control, and is only possible due to thesensitivity of chaos to measurement. We quantify the effectiveness of our control by numericallycomputing the quantum Lyapunov exponent over a wide range of parameters. We demonstratethat adaptive measurement techniques can control the onset of chaos in the system, pushing thequantum-classical boundary further into the quantum regime.
Quantum systems possess uniquely nonclassical prop-erties, such as coherence and entanglement, which can bemanipulated for applications including quantum compu-tation [1, 2], quantum communication [3, 4], and quan-tum sensing [5, 6]. Designing controls that do this is adiverse and productive area of ongoing research [7–19].However, these nonclassical properties also considerablymodify the kinds of control strategies and mechanismsavailable to quantum systems.One key example of the differences is the role of mea-surement. It is a given in classical control that one canmeasure the system and act upon it based on the in-formation extracted about the system. However, for aquantum system measurement itself changes the state ofthe system and this has to be carefully accounted forin the design of many closed-loop control protocols [20–26]. Although measurement backaction is usually consid-ered undesirable—an unwanted effect to be minimized—from another perspective measurement is an extra “con-trol knob” unavailable in the classical context, whichcan be used to develop new control strategies for quan-tum dynamical systems [27, 28]. In particular, adaptivemeasurements have been used to improve phase estima-tion [29], in quantum state preparation [30], and to en-hance the precision of quantum measurements [31].In this paper, we explore how this uniquely quantumknob can be used to control the dynamics of a chaoticsystem. Classically, controlling these systems is both asignificant and nontrivial problem. In some situations itis desirable to induce chaotic dynamics, as in the case ofembedding data into chaotic signals for secure transmis-sion of information [32]. However, in other cases the taskis to lock the system to stable orbits, as when aimingto regularize the behavior of cardiac rhythms [33] or im-prove energy harvesting in cantilever devices [34, 35]. In ∗ [email protected] many of these stabilization problems, feedback methodsare used to turn an originally unstable orbit embedded inthe chaotic attractor into a regular one [36, 37]. In thiswork, we show that transitioning at will from chaos toregularity is possible by using a real-time adaptive mea-surement protocol. In particular, our protocol combinesthe tunability of quantum measurement backaction onthe quantum state with the underlying geometry of theclassical dynamical system. This opens up regimes ofcontrol not available to open-loop control schemes.This quantum control strategy cannot be borrowedstraightforwardly from an analogous classical problem,not only because of the aforementioned peculiarities ofquantum measurement, but also due to subtleties associ-ated with identifying emergent quantum chaotic orbits.In a closed quantum system, coherent interference ef-fects cause a breakdown in the correspondence princi-ple such that chaotic classical dynamics do not emergewhen the underlying quantum model is taken to themacroscopic limit [38]. However, in open quantum sys-tems, decoherence destroys such quantum interference ef-fects [39], allowing emergent chaotic dynamics in the clas-sical limit [39–46]. In particular, by considering stochas-tic unravelings of an open quantum system, which arephysically associated with making particular continuousmeasurements on the system [47–49], we can observechaos in the conditional system dynamics [42, 49]. Thestochastic unravelings allow chaos to be identified andquantified with the quantum Lyapunov exponent [50–54]and also provide the necessary ingredient for a closed-loop feedback control scheme.Previously, we showed that the behavior of the systemcan be chaotic or not depending on the initial (and fixed)choice of measurement, due to the interplay between theinterference effects induced by the nonlinear dynamicsand the effectiveness of the measurement in destroyingthem [55]. This sensitivity to measurement choice wasshown to be absent both in the macroscopic limit, wherethe effects of quantum measurement are naturally ex- a r X i v : . [ qu a n t - ph ] J a n pected to disappear, and in a highly-quantum regime,where noise dominates and measurement choice becomesirrelevant. Although the system behaves chaotically inthe former case, as in the classical analog, in the latter,chaos is suppressed by quantum effects. As the mainoutcome of the control protocol presented here, we areable to show that a judicious real-time choice of measure-ment can induce chaotic behavior deeper in the quantumregime, effectively pushing the quantum-classical bound-ary further towards the microscopic domain. I. QUANTUM DUFFING OSCILLATOR
To illustrate our adaptive protocol, we consider adriven-damped Duffing oscillator [56], a model that hasbeen extensively used in the investigation of chaotic dy-namics in open quantum systems [51, 54, 55, 57–59]. Themodel consists of a particle that oscillates in a double-well potential that is periodically tilted by an externaldriving force with amplitude g and frequency Ω. The di-mensionless quantum Hamiltonian describing this modelis given byˆ H = 12 ˆ P + β Q −
12 ˆ Q + Γ2 ( ˆ Q ˆ P + ˆ P ˆ Q ) − gβ ˆ Q cos (Ω t ) , (1)where time is in units of the trap period 2 π/ω andˆ Q = ˆ x/ (cid:112) (cid:126) / ( mω ) and ˆ P = ˆ p/ √ (cid:126) mω are, respec-tively, the dimensionless position and momentum opera-tors for a single particle of mass m . The first term in theHamiltonian describes the kinetic energy, the quartic andquadratic terms in ˆ Q describe the double-well potential,and the last term describes the periodic driving of theparticle. The dimensionless parameter β = (cid:126) / ( ml ω )defines the scale of the phase space relative to Planck’sconstant [51, 53, 58] (where l characterizes the size of thesystem). A larger β is therefore associated with a regimewhere quantum fluctuations have a larger effect on theoscillator dynamics. Thus, by tuning β we can studythe transition from the quantum regime to the classicalregime ( β → ρ = − i [ ˆ H, ρ ] + (cid:18) ˆ Lρ ˆ L † − { ˆ L † ˆ L, ρ } (cid:19) , (2)where dissipation effects arise from choosing the system-environment coupling, ˆ L = √ Γ( ˆ Q + i ˆ P ) = √ a , to beproportional to the annihilation operator of the harmonicoscillator.In the classical limit ( β → (cid:104) ˆ Q (cid:105) → x cl and (cid:104) ˆ P (cid:105) → p cl such that the equationsof motion for (cid:104) ˆ Q (cid:105) and (cid:104) ˆ P (cid:105) correspond to the dimension-less classical dynamics given by [51, 53, 55, 58]¨ x cl + 2Γ ˙ x cl + β x cl − x cl = gβ cos (Ω t ) . (3) Although the scaling factor β is crucial in determining therole of quantum effects in the dynamics, classically it is atrivial scaling factor due to the definition of x cl and p cl .Indeed, for rescaling X ≡ βx cl , the classical equation ofmotion is independent of β . Note also that the quantumdissipation, given in terms of ˆ L , is symmetric with respectto position and momentum. The extra term proportionalto the damping rate Γ in the Hamiltonian (1), breaksthis symmetry in such a way that the dissipative force isproportional to the velocity, exactly as expected in theclassical limit.Depending on the parameters, the classical model de-scribed by Eq. (3) exhibits chaotic dynamics as illus-trated by the strange attractor in phase space shown bythe black dots in Fig. 1. The steady state of the Wignerfunction, obtained by numerically solving Eq. (2), is alsoshown in Fig. 1 for the same set of parameters. This illus-trates that the Wigner function of the ensemble-averagedquantum state broadly matches the strange attractor,which is a signature of chaotic dynamics. However, thedegree of chaos cannot be quantified via the uncondi-tional dynamics of Eq. (2), since any two initial statesevolve to the same asymptotic state, giving a negativeLyapunov exponent. This does not mean that chaos isnot present; indeed, the same problem would arise in clas-sical chaos if one decided to calculate classical Lyapunovexponents by using the separation of average trajecto-ries over a classical ensemble, rather than the separa-tion of two classical trajectories. To define the degreeof chaos via a quantum Lyapunov exponent, we needto use a conditional quantum trajectory approach thathas a direct comparison with the classical trajectory ap-proach [42, 59–61].
II. CONTINUOUS MEASUREMENT OF ANOPEN QUANTUM SYSTEM
The master equation (2), describes the ensemble-averaged evolution of the open quantum system. How-ever, implementing a closed-loop control scheme that de-pends on the monitored real-time dynamics requires adescription of a single experimental realization (or tra-jectory). This is provided by stochastic unravelings ofthe master equation, which correspond to the evolutionof the quantum state conditioned on a continuous mea-surement record [47, 62–64].Here we consider the class of diffusive quantum trajec-tories which, in its most general form, is described by theIto stochastic Schr¨odinger equation (SSE) [47, 64]:d | ψ (cid:105) = (cid:32) − i ˆ H − ˆ L † ˆ L (cid:104) ˆ L † (cid:105) ˆ L − (cid:104) ˆ L † (cid:105)(cid:104) ˆ L (cid:105) (cid:33) | ψ (cid:105) d t + (cid:16) ˆ L − (cid:104) ˆ L (cid:105) (cid:17) | ψ (cid:105) d ξ, (4)where the noise term d ξ is a complex Wiener process with FIG. 1. Wigner function for the steady state of the uncon-ditional dynamics given by the master equation (2), for thedimensionless parameters Γ = 0 . g = 0 .
3, and Ω = 1.The Poincar´e section of the classical Duffing oscillator is alsooverlaid for these parameters (black dots). The system ex-hibits chaos for these parameters, as seen by the emergenceof the strange attractor and the positive Lyapunov exponent λ cl = 0 .
16. Here the Wigner function for the unconditionalstate also follows the shape of the strange attractor in phasespace, which is a signature of chaotic dynamics. The scalingparameter β = 0 . zero mean ( E [ dξ ] = 0) and correlationsd ξ d ξ ∗ = dt and d ξ d ξ = u dt , (5)with u being a complex number satisfying | u | ≤ u = exp ( − iφ ) so thatd ξ = exp ( − iφ ) d W , where dW is a real noise of zeromean and dW = dt . Physically, this choice correspondsto a continuous measurement of the quadrature operatorˆ X φ = [exp ( − iφ )ˆ a +exp ( iφ )ˆ a † ] / √
2. Experimentally, thiscould be achieved by performing a standard balanced ho-modyne detection on the output of the system, as shownin Fig. 2. The output channel ˆ L = √ a is combinedwith a local oscillator (LO) of phase φ at a beam split-ter, while the readings at the detectors are subtracted toyield a measurement signal I d t = √ Γ (cid:104) ˆ X φ (cid:105) + d W [47].The phase φ of the LO is a controllable parameter thatdetermines the quadrature to be measured. For instance, φ = 0 results in a measurement of ˆ Q = ˆ X φ =0 , whereas φ = π/ P = ˆ X φ = π/ .Within the context of quantum chaos, this quantumtrajectory approach has proven useful in the investigationof the quantum-classical transition [42, 49, 58, 64, 65].Furthermore, it offers a way to calculate quantum Lya-punov exponents, thereby unambiguously quantifyingthe degree of chaos within the system [50–55]. Similarto the classical protocol [66], this is done by followingthe separation of two initially close wave-packet centroids in phase space ( (cid:104) ˆ Q (cid:105) , (cid:104) ˆ P (cid:105) ) evolving according to Eq. (4)under the same noise realization [54, 55].Specifically, the quantum Lyapunov exponent is de-fined as λ = lim t →∞ lim d → ln ( d t /d ) t , (6)where d t = [∆ Q ( t ) + ∆ P ( t ) ] / is the dimension-less phase-space distance between two quantum trajec-tories with differences in the average position and aver-age momentum of the two trajectories given by ∆ Q ( t ) = (cid:104) ˆ Q (cid:105) − (cid:104) ˆ Q (cid:105) and ∆ P ( t ) = (cid:104) ˆ P (cid:105) − (cid:104) ˆ P (cid:105) , respectively. Thetwo quantum trajectories are initially prepared in coher-ent states displaced (in phase space) from each otherby a small distance d = d t =0 (i.e., | α (cid:105) = | α (cid:105) and | α (cid:105) = | α + d (cid:105) ), and then evolved stochastically viaEq. (4) under the same noise realization, which corre-sponds to the same measurement record. Using thisapproach, it was shown in Ref. [55] that the choice ofmeasurement angle φ has a direct effect on the quantumLyapunov exponent and, therefore, on the emergence ofchaos in quantum systems. III. ADAPTIVE MEASUREMENT PROTOCOLFOR CONTROLLING CHAOS
The continuous measurement approach described inSec. II naturally sets the scene for our main result: thedesign of a protocol to control chaos by using a tunable,and experimentally accessible, parameter. The parame-ter in question, the LO phase φ , is intrinsically linked tothe measurement backaction, making our control mecha-nism fundamentally quantum in nature.The scheme we consider is shown in Fig. 2. The con-tinuous monitoring of the system gives a measurementsignal, I ( t ), that allows for a real-time estimate of thequantum state. In possession of this information, onecan then design a feedback action to influence the sys-tem. Motivated by the effect that measurement has onthe system dynamics [55], here we propose to adaptivelychange the phase φ in real time, with the intent to controlthe Lyapunov exponent of the system.The design of an effective control strategy relies onfirst understanding how the feedback action affects thesystem. For that, we recall a fact observed in Ref. [55]:The stretches and foldings induced by the chaotic dynam-ics generate interference fringes in the Wigner functionof the system (see top panel of Fig. 3), and these lead tothe suppression of chaos in the quantum regime. Sincethese interference fringes are associated with quantumcoherence, destroying them shifts the dynamics towardsthe classical chaotic behavior. Therefore, in order to en-hance (suppress) chaos, our state-dependent controllerchooses the LO phase φ such that the measurement de-stroys the interference fringes in the state’s Wigner func-tion at the fastest (slowest) possible rate. More precisely, ! I LO System | ψ ! Control
FIG. 2. Adaptive measurement scheme in a quantum opticssetup. The state-dependent controller chooses the LO phase φ at each time step in order to change the measurement back-action applied to the system, which changes the evolution asdesired. this rate of fringe destruction is determined by the direc-tion of the interference fringes in phase space ( θ f ) rela-tive to the axis of measurement (determined solely by φ ),with fast destruction rates occurring when these axes arealigned. Our control protocol then consists of estimatingthe fringe structure in real time and picking a φ ( t ) thatwould maximize the control objective.Automating the process of determining the directionof the interference fringes in the Wigner function canbe done by examining the probability distributions fordifferent quadrature measurements: P X θ = |(cid:104) X θ | ψ (cid:105)| , (7)where | X θ (cid:105) is an eigenstate of the quadrature operatorˆ X θ . To understand how this can be used to estimatethe fringe structure, let us look at the particular case ofthe Schr¨odinger cat state | ψ cat (cid:105) ∝ | α (cid:105) + | − α (cid:105) shown inFig. 3(a). Projection onto the ˆ X quadrature is given bythe top red plot in Fig. 3(a). Here, a measurement of ˆ X distinguishes between the two coherent states, resultingin two peaks. In contrast, the projection onto the ˆ X π/ quadrature (the red plot to the left of the Wigner func-tion plot) reveals the overlap of the two coherent states,resulting in interference fringes and a large number ofpeaks. As shown directly below the Wigner function plot,looking at the number of peaks as a function of projec-tion angle θ reveals that the peak distribution is narrowlycentered around θ = π/ (cid:104) ˆ P (cid:105) axis], which is perpen-dicular to the interference fringe axis. This shows thatthe angle that maximizes the number of peaks ( θ max ) isa good indicator of the direction that is perpendicular tothe fringes in the Wigner function.In the actual quantum Duffing oscillator, the nonlineardynamics lead to interference fringe patterns with morecomplexity than those of a Schr¨odinger cat state. Ex-amples of the Wigner functions for typical evolved statesthat arise during this evolution are plotted in Figs. 3(b)- 3(d). Although more complicated, these Wigner func-tions still present a reasonably-well-defined direction inthe fringe structure, which can be determined by findingthe angle that leads to the maximum number of peaks in P X θ , as explained above.In summary, our protocol consists of the followingsteps:(i) Starting from a given | ψ ( t ) (cid:105) , calculate P X θ for var-ious θ ;(ii) Count the number of peaks for each P X θ and find θ max ;(iii) To maximize (minimize) the Lyapunov exponent,choose φ ( t ) = θ f = θ max − π/ φ ( t ) = θ max );(iv) Use the value of φ ( t ) from (iii) in Eq. (4) to calculatethe new state | ψ ( t + dt ) (cid:105) ;(v) Repeat steps (i) to (iv).Full details of the numerical implementation of thesesteps are given in the appendix. IV. RESULTS
We implemented the adaptive measurement schemedescribed in Sec. III for a range of scaling parameters β (spanning the transition from the quantum regime tothe classical regime) and two distinguishable strategies:maximization and minimization of the Lyapunov expo-nent ( λ ). The results are shown in Fig. 4 for both cases,specifically, where the LO phase is set to always measurealong an axis parallel ( φ = θ f , blue line, square points)or perpendicular ( φ = θ f + π/
2, green line, crosses) tothe interference fringes. To assess the effectiveness of ouradaptive protocol, we compare with the best nonadaptivestrategy by displaying the curves that maximize (blackline, triangles) and minimize (red line, circles) λ for a fixed LO phase.The adaptive maximization strategy leads to Lyapunovexponents that are always larger than the best fixed-anglescenario ( φ = 0). By destroying coherent interference ef-fects and localising the state faster, the adaptive caseallows the quantum system to track the classical chaoticdynamics more closely, increasing λ . Further evidence ofthis is provided by looking at the dynamical evolution ofthe Wigner function (see Fig. 5, top), showing states thatare more localized and possess less interference, and aretherefore more classical in nature. The opposite adaptivestrategy, the one designed to suppress chaos, also workseffectively, giving negative Lyapunov exponents for allvalues of β . In this case, the adaptive choice of moni-toring angle leads to the preservation of quantum inter-ference effects and therefore to highly nonclassical stateswith a large spread in phase space, as seen in the Wignerfunctions of Fig. 5 (bottom). ( d )( c )( b )( a ) ! ˆ Q " ! ˆ Q "! ˆ Q "! ˆ Q " N o . o f p e a k s ! ˆ P " θ θ θθ FIG. 3. Wigner functions and corresponding phase quadrature projections X θ =0 and X θ = π/ for (a) a Schr¨odinger cat state | ψ cat (cid:105) ∝ | α (cid:105) + | − α (cid:105) and (b)-(d) three snap shots typically seen in the evolution of the quantum Duffing oscillator. Here θ = 0(top) and θ = π/ θ for 32 different angles. The maximum in the number of peaks corresponds to the directionperpendicular to the interference fringes ( θ max − θ f = π/ λ ) as a functionof β for adaptive measurements ( φ = θ f , blue squares, and φ = θ f + π/
2, green crosses) and fixed LO measurements( φ = 0, black triangles, and φ = π/
2, red circles). Here Γ =0 . g = 0 . λ cl = 0 .
16. Each point is averaged over 10 differentnoise realizations and the shaded area within the dashed linessignifies twice the standard error.
Interestingly, the adaptive λ -maximization schemegives positive Lyapunov exponents for much larger val-ues of β (up to 0 . β , and so one would think that thechoice of measurement is irrelevant. This is clearly thecase for the fixed measurement (see Fig. 4), where thequantum Lyapunov exponent for all monitoring schemesother than λ -maximization converge to roughly the samenegative value, indicating regular dynamics. In starkcontrast, our λ -maximization protocol is able to sustainchaotic dynamics even at this scale.Although our adaptive λ -maximization scheme can sig-nificantly enhance chaos, the adaptive λ -minimizationscheme does not provide significantly enhanced regular-ity over the fixed measurement. This is a consequenceof using metric (7) to choose the measurement quadra-ture angle φ at each time point. The aim is to find thedirection of interference fringes in the Wigner function,and choose a measurement angle parallel (perpendicular)to this direction in order to enhance (suppress) chaos.However, the metric (7) becomes less effective when thestate is highly nonclassical and delocalized. This is shownclearly in the Wigner function plots of Fig. 5(b), in par-ticular at time Ω t = 70. In this case, the large degreeof delocalization means that there is no well-defined sin-gle direction of interference fringes. Consequently, in thisregime the adaptive control does not provide a substan-tially improved performance over a fixed-angle measure-ment. When trying to suppress chaos by picking a mea-surement that has the least deleterious effect on quantuminterferences, it is exactly this highly delocalized regime FIG. 5. Snap shots of the Wigner function for the first 100 cycles of the driving for both adaptive measurements [(a) φ = θ , and(b) φ = θ + π/ λ = 0 . ± .
001 and (b) λ = − . ± . that is encouraged. Therefore, it is unsurprising thatour adaptive measurement protocol provides little bene-fit over a fixed measurement angle, if the goal is to sup-press chaos. In contrast, our metric is more effectivewhen the Wigner function is localized and the fringe di-rection better defined [see Fig. 5(a) for Ω t = 70]. This isthe scenario arising from our strategy to enhance chaos:choosing measurements that destroy coherence and keepthe state localized. V. DISCUSSION
We briefly discuss the experimental prospects of re-alizing both the driven-damped quantum Duffing oscil-lator and our adaptive measurement protocol. Super-conducting circuits are excellent candidate systems, dueto their flexible architecture, wide range of experimentalparameters, and the existence of demonstrated contin-uous probing [67]. Specifically, superconducting circuitsin a parallel circuit configuration (i.e., a rf-SQUID) couldbe used to experimentally realize a quantum Duffing os-cillator [57, 68]. For the scheme proposed in Ref. [57], β = e / [3 (cid:126) ωC p (1 − L p /L J )], where ω = 1 / (cid:112) C p L p , C p is the capacitance of the Josephson junction in the cir-cuit, L − p = L − J − L − p is the parallel inductance formedfrom the Josephson inductance L J and the geometric in-ductance L pe , and e is the charge of an electron. Usingtypical experimental parameters from Ref. [69], we es-timate that β ∼ . V exp = 12 mω ˆ x + Ae − ˆ x / σ ≈ (cid:126) ω (cid:20) (cid:18) − Amω σ (cid:19) ˆ Q + 14 (cid:18) (cid:126) A m ω σ (cid:19) ˆ Q (cid:21) . (8)The choice of barrier height A = 2 mω σ realizes theneeded potential [see Eq. (1)] with β = (cid:126) / ( mω σ ).There are a number of techniques for creating this poten-tial, including via an optical lattice [76] or spatial lightmodulation [77]. For the 780 nm transition of Rb, abarrier waist of σ ∼ µ m is easily achievable. For typi-cal trapping frequencies ω ∈ π × [5 , β ∼ . − . single-particle system. Many-body quantum chaos is a growing research field,due to its potential connections to random unitaries [82],information scrambling and holographic duality [83–85],nonequilibrium thermodynamics [86], and even quantumsensing [87]. Whether measurement can be used to mean-ingfully control chaos in many-body quantum systems isan intriguing question that warrants further investiga-tion.
VI. CONCLUSION
In this work we have shown that the degree of chaos ina quantum Duffing oscillator can be controlled by apply- ing real-time state-dependent feedback via an adaptivemeasurement technique. The underlying mechanism forthis control is the rate at which the measurement back-action destroys interference fringes in the state’s Wignerfunction. By adaptively choosing measurements that aremore (less) destructive, the dynamics more (less) closelyresemble the corresponding classical trajectory, therebyenhancing (suppressing) chaos. Using this adaptive mea-surement technique, we have shown that the presence ofchaos can be pushed further into the quantum regime.This regime is more easily accessible for certain experi-mental setups, potentially enabling new, detailed studiesinto the emergence of chaos in quantum systems.
ACKNOWLEDGEMENTS
The authors would like to thank A. Pattanayak andS. Greenfield for thoughtful discussions. The authorswould also like to thank P.J. Everitt for discussions onthe experimental details and parameters.J.K.E. acknowledges the support of an Australian Gov-ernment Research Training Program (RTP) Scholarshipand the hospitality of the Centre for Quantum Dynam-ics at Griffith University, where part of this work wascompleted. J.K.E. and A.R.R.C acknowledge support bythe Australian Research Council (ARC) Centre of Ex-cellence for Quantum Computation and CommunicationTechnology (project CE110001027). S.S.S. received fund-ing from ARC projects DP160104965 and DP150100356.This research was undertaken with the assistance of re-sources and services from the National Computational In-frastructure (NCI), which is supported by the AustralianGovernment.
Appendix A: Numerical simulation
We numerically simulated the SSE (4) on a finite sub-space of N energy eigenstates of the harmonic oscilla-tor by using the software package XMDS2 [88]. That is,we write the conditional state as | ψ (cid:105) = (cid:80) N − n =0 C n ( t ) | n (cid:105) and numerically solve for the dynamics of the coefficients C n ( t ), governed by the set of Stratonovich stochastic dif-ferential equationsd C n = − i (cid:104) β (cid:112) ( n + 1)( n + 2)( n + 3)( n + 4) C n +4 + (cid:112) ( n + 1)( n + 2)( β (4 n + 6) − (1 + i Γ)) C n +2 − g √ β cos (Ω t ) √ n + 1 C n +1 + β (6 n + 6 n + 3) C n − g √ β cos (Ω t ) √ nC n − + (cid:112) n ( n − β (4 n − − (1 − i Γ)) C n − + β (cid:112) n ( n − n − n − C n − (cid:105) d t − n Γ C n d t − e iφ Γ (cid:112) ( n + 1)( n + 1) C n +2 d t + 2Γ (cid:0) (cid:104) ˆ a † (cid:105) + (cid:104) ˆ a (cid:105) e iφ (cid:1) √ n + 1 C n +1 d t + √ √ n + 1 C n +1 e iφ ◦ d W, (A1)where (cid:104) ˆ a (cid:105) = (cid:80) N − n =0 √ n + 1 C ∗ n C n +1 and C n = 0 for all n ≥ N . For our simulations, we use N = 64 basis states,a large enough number such that | C N − | + | C N − | + | C N − | + | C N − | < − at all times, while still smallenough to be numerically tractable.For the adaptive protocol, we calculate the probabilitydistribution for a number of quadratures, this is given by P X φ = |(cid:104) X φ | ψ (cid:105)| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) n C n ψ n ( x ) e − inφ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (A2)where ψ n ( x ) are the Hermite-Gauss functions: ψ n ( x ) = (2 n n ! √ π ) − / e − x / H n ( x ) , (A3)and H n ( x ) are Hermite polynomials.We use a grid-based search algorithm to determine theoptimum measurement phase for each time step. Todo this, we use a finite-difference method to calculatethe derivative of the probability distribution (7) for anequidistant grid of LO angles φ ∈ [0 , π ], allowing thenumber of peaks in the distribution to be calculated.The angle θ max corresponding to the maximum numberof peaks gives an axis perpendicular to the interferencefringes ( θ f + π/ φ = θ f (parallel to fringes), whereas to suppress chaos we choose φ = θ f + π/ φ = θ f + π/ φ = θ f ), a coarser grid of 8 angles was sufficient.We quantify the degree of chaos in our system by com-puting the quantum Lyapunov exponent as in Ref. [55],which is based on an adaptation of the usual classicalprocedure [66]. For our numerical calculations, one ofthe trajectories is periodically reset towards the otherone to remain within the linear regime and log ( d t /d ),calculated before every reset, is averaged over time. Theperturbed trajectory after the reset is a displaced ver-sion of the trajectory of interest. The displacement isgiven by the initial distance d in phase space, in the di-rection of expansion. The perturbed trajectory becomes | ψ (cid:105) = D ( α ) | ψ (cid:105) , where D ( α ) is the displacement opera-tor and α = d [( (cid:104) ˆ Q (cid:105) + i (cid:104) ˆ P (cid:105) ) − ( (cid:104) ˆ Q (cid:105) + i (cid:104) ˆ P (cid:105) )] / ( d t β ) isthe displacement in the direction of expansion.The simulations are run over 10 ,
000 cycles of the driv-ing term ( t = 10 / Ω) for both the adaptive- and thefixed -LO cases, and the final Lyapunov exponent is aver-aged over multiple realizations (10 runs) of the stochasticnoise. [1] M. A. Nielsen and I. L. Chuang,
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