Signatures of quantum mechanics in chaotic systems Kevin M Short1 and Matthew A Morena2 1 Integrated Applied Mathematics Program, University of New Hampshire, Durham, 7 NH 03824 1 2 Division of Mathematics and Science, Young Harris College, Young Harris, GA 0 30582 2 n E-mail: [email protected] a J Received 24 October 2016 3 ] Abstract. We consider the quantum-classical correspondence from a classical h perspective by discussing the potential for chaotic systems to support behaviors p - normally associated with quantum mechanical systems. Our main analytical tool is a t n chaotic system’s set of cupolets, which are essentially highly-accurate stabilizations of a its unstable periodic orbits. The discussion is motivated by the bound or entangled u states that we have recently detected between interacting chaotic systems, wherein q [ pairs of cupolets are induced into a state of mutually-sustaining stabilization that can be maintained without external intervention. This state is known as chaotic 1 v entanglement as it has been shown to exhibit several properties consistent with 7 quantum entanglement. For instance, should the interaction be disturbed, then the 7 chaotic entanglement would be broken. In this paper, we further describe chaotic 7 entanglement and go on to discuss the capacity for chaotic systems to exhibit other 0 0 characteristics that are conventionally associated with quantum mechanics, namely 1. analogs to wave function collapse, the measurement problem, the superposition of 0 states, and to quantum entropy definitions. In doing so, we argue that these 7 characteristics need not be regarded exclusively as quantum mechanical. 1 : v i X AMS classification scheme numbers: 81P40, 81Q50, 37C27, 34H10 r a Keywords: quantum entanglement, chaotic systems, cupolets, correspondence, unstable periodic orbits Submitted to: Nonlinearity (Some figures may appear in colour only in the online journal) 1. Introduction Chaotic behavior is generally attributed to a sensitive dependence on initial conditions and is characterized by a positive maximal Lyapunov exponent that causes nearby trajectories to diverge from each other exponentially fast. Despite its ubiquity in Signatures of quantum mechanics in chaotic systems 2 classical physics, chaos is yet to be rigorously established within quantum settings. One explanation for this disparity is that unlike chaotic or classical systems, whose states may be completely described by a set of dynamical variables, in quantum mechanics conjugate observables such as position and momentum cannot take on well- defined values at the same time. Particle dynamics are instead determined in part by the uncertainity principle and by the linearity of the Schr¨odinger equation, which preserves the overlap between quantum states. In other words, the nonlinearity required for chaotic dynamics and the exponential divergence of neighboring trajectories seem fundamentally incompatible with quantum mechanics in its present formulation. And yet, much effort has recently been devoted to detecting signatures of chaos in quantum systems [1, 2, 3, 4, 5]. One such signature is the sensitivity of some quantum systems to perturbation. This has been experimentally observed in the decay in the overlap between quantum states that are evolving under slightly different Hamiltonians and is thought to be associated with a positive Lypaunov exponent [1, 4]. In particular, the rate of overlap decay is known to transpire at different rates depending on whether the evolution begins from initial conditions that correspond classically to chaotic versus regular regimes [5]. A second signature is quantum scarring, which refers to the scenario in which a quantum system’s associated wave function is concentrated on paths that represent periodic orbits in the classical limit [6, 7]. This phenomenon has been experimentally observed in several recent studies [8, 9]. Entanglement in the purely quantum sense has also been observed to be a reliable indicator of classical chaos [10, 11, 12, 13]. For example, in Chaudhury et al.’s recent kicked top experiments of laser-cooled Cesium (133Cs) atoms, each atom’s initial state is followed for several periods of the “kicked” Hamiltonian, and the corresponding classical phase space reveals islands of regular motion surrounded by a sea of chaos [5]. When entropy is used to measure the entanglement, greater entanglement is detected in initial states that are prepared from chaotic regimes as opposed to weaker entanglement which is generated by those originating from regular regions. It is as if the quantum regime respects an underlying classical presence [13]. One feature of chaotic systems typically encountered in investigations is an infinite set of unstable periodic orbits (UPOs) that are found densely embedded in many attractors. These orbits collectively provide a rich source of qualitative information about the parent chaotic system and are a focus of numerous theoretical and practical applications [14, 15, 16]. As a result, several control schemes have been designed to detectandstabilizetheseorbits[17,18,19,20]. InSection2, wediscussanadaptationof oneparticularcontrolmethodthatveryefficientlystabilizestheUPOsofchaoticsystems onto cupolets (Chaotic, Unstable, Periodic, Orbit-LETS) [21, 22, 23, 24]. Cupolets are controlled and stabilized periodic orbits of a chaotic system that would normally be unstable without the presence of the control mechanism. These orbits represent a subset of the UPOs, but are distinguished because this stabilization supports a one- to-one correspondence between a given sequence of controls and a specific cupolet, with each cupolet able to be generated independently of initial condition. All of this allows Signatures of quantum mechanics in chaotic systems 3 for large collections of cupolets to be generated very efficiently, thereby making these orbits well suited for analyzing chaotic systems. In recent studies, we reported on the proclivity for chaotic systems to enter into bound or entangled states [25, 26, 24]. We demonstrated how pairs of interacting cupoletsmaybeinducedintoastateofmutually-sustainingstabilizationthatrequiresno external controls in order to be maintained. This state is known as chaotic entanglement and it is self-perpetuating within the cupolet-stabilizing control scheme, meaning that each cupolet of an entangled pair is effectively controlling the stability of its partner cupolet via their continued interaction. The controls used are all information-theoretic, so we stress that additional work is required to more rigorously relate this research to physical systems. However, since many of our simulated cupolet-to-cupolet interactions are based on the dynamics of physical systems, our findings indicate the potential of chaotic entanglement to be both physically realizable and naturally occurring. It is worth noting the sensitivity of chaotic entanglement to disturbance since any disruption to the stability of either cupolet of an entangled pair may be enough to destroy the entanglement, therefore supporting a reasonable analog to quantum entanglement. We are aware that entanglement is regarded as a quantum phenomenon and that there are characteristics of quantum entanglement that are not compatible with chaotic entanglement, such as nonlocality. We are also aware that chaotic entanglement has been previously examined in [27] and that a classical version of entanglement has been proposedin[28]. Inthefirststudy, linearandnonlinearsubsystemsarecoupledtogether to produce composite chaotic systems, a synthesis the authors refer to as chaotic entanglement. In the second study, a classical version of quantum entanglement is demonstrated via a beam of photons and is shown to be consistent with many features of quantum entanglement, apart from nonlocality. In contrast, the novelty of the chaotic entanglement that we have documented arises in how two interacting chaotic systems are induced into a state of mutual stabilization. First, the chaotic behavior of the two systems is collapsed onto unique periodic orbits (cupolets). Following the collapse, the ensuing periodicity of each chaotic system and the stability of each cupolet are maintained intrinsically by each system’s dynamical behavior and will persist until the interaction is disturbed. To our knowledge, this is the first documentation of chaotic systems interacting to such an extent. Our initial results are very promising since several hundred pairs of entangled cupolets have been identified from low-dimensional chaotic systems. When regarded as a parallel to quantum entanglement, chaotic entanglement is further intriguing as it not only signals a new correlation between classical and quantum mechanics, but it also demonstrates that chaotic systems are capable of exhibiting behavior that has conventionally been associated with quantum systems. We now discuss the potential for classically chaotic systems to support additional parallels with quantum mechanics, namely the measurement problem, notions of wave function collapse, superposition of states, and entropy definitions. Our discussion uses cupolets and chaotic entanglement as reference points and is Signatures of quantum mechanics in chaotic systems 4 organized as follows. In Section 2, we begin by providing a brief introduction to cupolets and how they are generated, and then we discuss a few of their interesting properties and applications. In Section 3, we describe chaotic entanglement and how it can be induced and detected between pairs of interacting cupolets. The main discussion of chaotic systems supporting quantum behavior is found in Section 4. Finally, we offer a few concluding remarks in Section 5. 2. Background on cupolets Broadly speaking, cupolets are a relatively new class of waveforms that were originally detected while controlling a chaotic system in a secure communication application. The theory behind these orbits and their applications have been well-documented [21, 22, 15, 16, 23, 29, 25, 30, 26, 24]. In this section, we summarize the control technique that is used to generate cupolets and then describe the applications of cupolets that have particular relevance to our chaotic entanglement research. More technical details of the control process can be found in [23, 29, 30, 26, 24]. The control scheme that is used to stabilize cupolets is adapted from the chaos control method designed by Hayes, Grebogi, and Ott (HGO) [19, 20]. In the HGO scheme, small perturbations are used to steer trajectories of the double scroll system, also known as Chua’s oscillator [31], around an attractor. The differential equations describing this system are given by: G(v −v )−g(v ) v˙ = C2 C1 C1 , C1 C 1 G(v −v )+i v˙ = C1 C2 L, (1) C2 C 2 v i˙ = − C2, L L where the piecewise linear function, g(v), is given by:  m v, if |v| ≤ B ,  1 p g(v) = m (v +B )−m B , if v ≤ −B , (2) 0 p 1 p p   m (v −B )+m B , if v ≥ B . 0 p 1 p p When C = 1, C = 1, L = 1, G = 0.7, m = −0.5, m = −0.8, and B = 1, the double 1 9 2 7 0 1 p scroll system is known to be chaotic and its attractor consists of two lobes that each surrounds an unstable fixed point [31]. Figure 1 shows a typical trajectory tracing out this attractor. Control of the double scroll system is first achieved by setting up two control planes on the attractor (via a Poincar´e surface of section) and then by partitioning each control plane into small control bins. Perturbations are applied only when a trajectory evolves through the control bins, otherwise the trajectory is allowed to freely evolve around the attractor. Figure 1 also shows the positions of these control planes which emanate outward from the center of each lobe. The control planes are assigned binary values Signatures of quantum mechanics in chaotic systems 5 2.5 1.5 0 0.5 iL −0.5 1 −1.5 −2.5 −2.5 −1.5 −0.5 0.5 1.5 2.5 VC1 Figure1. 2Dprojectionofthedoublescrollattractorshowingthecontrolsurfaces[31]. so that a binary symbolic sequence may be recorded whenever a trajectory intersects a control plane. This sequence is known as a visitation sequence. Parker and Short [21] later combined this control scheme with ideas from the study of impulsive differential equations [32] and discovered that when a repeating binary control sequence is used to define the controls, with a ‘1’ bit corresponding to a perturbation and a ‘0’ bit corresponding to no perturbation, then the double scroll system stabilizes onto a periodic orbit. These perturbations are defined via the HGO technique to be the smallest disturbance along a control plane that produces a change of lobe M loops downstream. For almost all repeating control sequences, the resulting periodic orbits are generated completely independently of the initial state of the system and a one-to-one relation exists between a given control sequence and a particular periodic orbit. These periodic orbits have been given the name cupolets, and this work has since been extended to chaotic maps and a variety of other continuous chaotic systems such as the Lorenz and R¨ossler systems. The examples of double scroll cupolets appearing in Figure 2 are generated by repeating the indicated sequences of control bits. To summarize, cupolets are highly-accurate approximations to the UPOs of chaotic systems that are generated by an adaptation of the HGO control technique [21, 23, 24]. Cupolets have the interesting properties of being stabilized independently of initial condition and also of being in one-to-one correspondence with the control sequences. These controls can be made arbitrarily small and thus do not significantly alter the topology of the orbits on the chaotic attractor. This suggests that cupolets are shadowingtrueperiodicorbitsandtheoremshavebeendevelopedtoestablishconditions under which this holds [33, 34, 35, 36, 23]. Furthermore, the effect of combining chaos control with impulsive perturbations has resulted not only in the ability to stabilize chaotic systems onto (approximate) periodic orbits, but has also simplified the search for periodic orbits since a simple program can be written to generate all possible N- bit control sequences and then feed them into the control scheme. What further Signatures of quantum mechanics in chaotic systems 6 0.5 1 0 0.5 0 −0.5 −0.5 −1 −1 −1.5 −1.5 −2 −2 −2.50 0.5 1 1.5 2 2.5 −2−.50.5 0 0.5 1 1.5 2 2.5 (a) (b) 2.5 2.5 1.5 1.5 0.5 0.5 −0.5 −0.5 −1.5 −1.5 −2−.52.5 −1.5 −0.5 0.5 1.5 2.5 −2−.52.5 −1.5 −0.5 0.5 1.5 2.5 (c) (d) Figure 2. Cupolets of various periods belonging to the double scroll system. The controlsequenceswhichmustbeperiodicallyappliedinordertostabilizetheirperiodic orbits are (a) ‘00’, (b) ‘11’, (c) ‘00001’, and (d) ‘001’ [26]. distinguishes cupolets from UPOs, which are traditionally stabilized via techniques such as Newton’s or first-return algorithms, is that large numbers of cupolets can be inexpensively generated by only a few bits of binary control information. For example, over8,800doublescrollcupoletscanbestabilizedfrom16-bitorfewercontrolsequences. 2.1. Application of Cupolets At a fundamental level, cupolets are very rich in structure and may be used to generate a variety of different waveforms ranging from a simple sine-like wave with a single dominant spectral peak to more involved waveforms consisting of many harmonics. Figure 3(a) illustrates the high diversity in spectral signatures found among four cupolets. ThedataistakenfromtheFFTofasingleperiodofoscillationofeachcupolet, andeachcupolet’scorrespondingtimedomainrepresentationisseeninFigure3(b). Itis clearthatthesimplestcupoletinthesefiguresisessentiallysinusoidal, whileincreasingly richer structure is evident in the other cupolets. This figure will be referenced later in Section 4. In addition to secure communication [21, 22], image processing [29], and data compression [15, 16], cupolets have also been found to provide particularly useful pathways when directing a chaotic system to a target state [30]. This is a recent application of targeting in dynamical systems and it relies heavily on the fact that Signatures of quantum mechanics in chaotic systems 7 0.2 Simple Cupolet Spectrum - axis scaled to 128 bins 1 Simple Cupolet Time Series 0.1 0 0 -1 0 20 40 60 80 100 120 0 100 200 300 400 500 600 700 800 900 1000 Intermediate Complexity Cupolet Spectrum Intermediate Complexity Cupolet Time Series 5 0.4 0 0.2 0 -5 0 20 40 60 80 100 120 0 100 200 300 400 500 600 700 800 900 1000 Complex Cupolet Spectrum - axis scaled to 256 bins Complex Cupolet Time Series 0.5 0.04 0 0.02 0 -0.5 0 50 100 150 200 250 0 100 200 300 400 500 600 700 800 900 1000 Highly Complex Cupolet Spectrum Highly Complex Cupolet Time Series 2 0.2 0 0 -2 0 50 100 150 200 250 0 100 200 300 400 500 600 700 800 900 1000 (a) (b) Figure 3. Cupolet diversity: in (a), spectral variation and in (b), time-domain variation among cupolets [15, 16]. The same cupolets are used to produce the corresponding graphs between the two sets of figures. cupolet stabilization occurs independently of the current state of the system. For a given cupolet to remain stabilized, all that is required is the repeated application of its control sequence to the system, and so applying different controls would induce the system to destabilize from the stabilized cupolet and to revert to chaotic behavior. If a second sequence of controls were to then be periodically applied, the chaotic system would eventually restabilize onto a second cupolet, possibly after some intermediary transient phase. Any transient is the result of the trajectory evolving while the chaotic system sifts through all possible states until it reaches one where the behavior of an UPO falls into synchrony with the control sequence, thus stabilizing the cupolet. Cupolet restabilization is guaranteed because of the injective relationship that exists between cupolets and the control sequences. This makes it possible to transition between cupolets, and thus between UPOs, simply by switching control sequences. It is further shown in [30] that this simple targeting method can be combined with algebraic graph theory and Dijkstra’s shortest path algorithm in order to achieve highly efficient targeting of desired cupolets. We shall refer to cupolet transitions throughout Section 3. 3. Chaotic entanglement In previous work [25, 26, 24], we document the surprising observation that pairs of chaotic systems may interact in such a way that they chaotically entangle. To do so, the two chaotic systems must first induce each other to collapse and stabilize onto a cupolet via the exchange of control information. Then, the stabilities of the two stabilized cupolets must become deterministically linked: disturbing one cupolet from itsperiodicorbitsubsequentlyaffectsthestabilityofthepartnercupolet, andviceversa. Hundreds of entangled cupolet pairs have been identified for the double scroll system, Signatures of quantum mechanics in chaotic systems 8 and it has been shown that chaotic entanglement evokes several connections to quantum entanglement as we discuss in Section 3.2, below. Cupolets from the two entangled chaotic systems are regarded as mutually stabilizing since their interaction essentially serves as a two-way coupling that is self- perpetuatating within the control scheme just described in Section 2. In particular, once entanglement has been established between two chaotic systems, no outside intervention or user-defined controls are needed to sustain the stabilities of their respective cupolets. Instead, the stability of each cupolet is maintained by the dynamics of the partner cupolet. In summary, not only has the original chaotic behavior of the two parent systemscollapsedontotheperiodicorbitsofthetwocupolets,buttheirperiodicbehavior will persist as long as their interaction is undisturbed. Chaotic entanglement is typically mediated by an exchange function that defines the interaction between the two chaotic systems and their cupolets. In [26], exchange functions are described more fully as catalysts for the entanglement and are taken to represent the environment or medium in which the chaotic systems are found. For instance, we have designed several types of exchange functions that simulate the interactions of various physical systems such as the integrate-and-fire dynamics of laser systems and networks of neurons. 3.1. Chaotic entanglement through cupolets InSection2, wedefinedacupolet’svisitationsequencetobethebinarysequenceoflobes that its orbit visits. Visitation sequences thus serve as a type of symbolic dynamics of chaoticsystems; i.e., dynamicinformationthatisgeneratedassolutionstothesesystems evolve over time. With this in mind, chaotic entanglement can be more technically characterizedasanexchangeofsymbolicinformationintheformofvisitationsequences. Consider a pair of cupolets, say C and C , that have been stabilized from two A B arbitrarybutinteractingchaoticsystems. AscupoletC evolvesaboutitsattractor, the A bits of its visitation sequence are passed to an exchange function which then performs a binary operation on the visitation sequence. The outputted sequence of bits is known as an emitted sequence and is taken as a control sequence and applied to cupolet C . B Concurrently, but in the reverse direction, the visitation sequence belonging to C B passes through the same exchange function and the resulting emitted sequence is used to control C . At this point, each cupolet is both receiving and transmitting control A information via the exchange function, but if the emitted sequence generated from the visitation sequence of C matches the control sequence needed to maintain the stability A of cupolet C —and vice versa—then the two cupolets, and the two parent chaotic B systems, become intertwined in a mutually-stabilizing feedback loop and are considered chaotically entangled. Any external controlling can be subsequently discontinued now that each cupolet’s visitation sequence is preserving the partner cupolet’s stabilization. As an example, we will demonstrate how the two cupolets shown in Figure 4 can becomechaoticallyentangled. ThisprocessisalsodepictedinFigure5asaseriesofstep- Signatures of quantum mechanics in chaotic systems 9 2.5 2.5 1.5 1.5 0.5 0.5 -0.5 -0.5 -1.5 -1.5 -2.5 -2.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 (a) (b) Figure 4. Entangled cupolets (a) C00000000011 and (b) C0000110011110011 with visitation sequences V0000011100011111000111 and V0000111111111111, respectively [26]. by-step illustrations. First, two double scroll systems, Systems I and II, are simulated without control. In order to stabilize one of these cupolets, say C00000000011, the controlsequence‘00000000011’mustbeappliedtoSystemIusingthecupolet-generating control technique described in Section 2. This step is illustrated in Figure 5(a), where the depiction of the control planes indicates that System I is being controlled via the (yellow) external control pump. Once C00000000011 completes one full period around the attractor, its visitation sequence, V0000011100011111000111, is realized. Figure 5(b) captures this stage of the entanglement process. This visitation sequence is then passed to an exchange function where it is modified according to a predefined binary operation and sent to System II as an emitted sequence. In this example, a ‘complement’ exchange function converts V0000011100011111000111 into the emitted sequence E0000110011110011 essentially by interchanging subsequences of ones and zeros for zeros and ones, respectively.‡ As the emitted sequence passes from the exchange function, it is applied to System II as instructions for controlling this system. In this case, System II stabilizes onto the cupolet C0000110011110011 since the emitted sequence actually is this second cupolet’s control sequence. These particular steps are visualized in Figure 5(c). The cupolets’ interaction now repeats in the reverse direction. The visitation sequence of cupolet C0000110011110011 is found to be V0000111111111111 (see Figure 5(c)), which is converted by the same exchange function into the emitted sequenceE00000000011. WhenappliedasacontrolsequencetoSystemI,E00000000011 preservesthestabilityofcupoletC00000000011becausetheemittedbits, ‘00000000011’, match the control information needed to maintain this cupolet’s stability. This two- way exchange of control information between Systems I and II defines the cupolets’ interaction which has been managed by the exchange function. Notice that both emitted sequences, E00000000011 and E0000110011110011, match the required control ‡ This particular type of exchange function is more thoroughly described in [26]. Signatures of quantum mechanics in chaotic systems 10 Table 1. (Color online) The following table summarizes the chaotic entanglement induced between two interacting cupolets, C00000000011 (of System I) and C0000110011110011 (of System II). The orbits of these cupolets are depicted in Figure 4, while the generation of the entanglement via a ‘complement’ exchange functionisillustratedinFigure5. Noticethatthecontrolsequencerequiredtosustain the stability of cupolet C00000000011 is contributed by cupolet C0000110011110011 via this cupolet’s emitted sequence, E00000000011. Similarly, the stability of C0000110011110011 is maintained by the repeated application of emitted sequence E0000110011110011, which is generated by C00000000011 via the same exchange function. The font colors in this table are intended to accentuate the correspondence between the cupolets’ control sequences and their emitted sequences. Details of the entanglement generation are found in the text. Cupolet Visitation Sequence Emitted Sequence System I C00000000011 V0000011100011111000111 E0000110011110011 System II C0000110011110011 V0000111111111111 E00000000011 sequences for cupolets C00000000011 and C0000110011110011, respectively. The (yellow) external control pump is thus rendered redundant and can be discarded now that the cupolets are dynamically generating the necessary control instructions themselves. Since the cupolets are effectively driving each other’s stability, they are consideredchaoticallyentangledandtheirstabilitiesareguaranteedsolongastheirtwo- wayinteractionisundisturbed. Figure5(d)illustratesthisfinalstepoftheentanglement andTable1summarizesthecorrespondencebetweenthecontrol, visitation, andemitted sequences of each cupolet. Strictly speaking, chaotic entanglement need not be associated exclusively with interacting cupolets because its generation extends naturally to pairs of interacting UPOs. CupoletsrepresenthighlyaccurateapproximationstotheseUPOs,butingeneral two interacting chaotic systems chaotically entangle once each system stablizes onto a particular UPO whose stability is then maintainted by the symbolic dynamics of the partner UPO. The visitation sequences of the UPOs would continue to provide an appropriate symbolic dynamics, but the advantage of inducing and detecting entanglement with cupolets is twofold. First, the control technique described in Section 2 is designed to stabilize cupolets. In doing so, the technique makes accessible the symbolic dynamics of chaotic systems while greatly simplifying how the interactions between the systems are simulated. For instance, perturbations are applied only when a cupolet intersects a control plane, which means that the cupolet’s remaining evolution is freely determined by the system’s governing equations. Therefore, when detecting entanglement between two systems, one only needs to monitor finitely-many intersections with the control planes, which allows one to simultaneously observe the visitation sequence of each evolving cupolet. Second, given that cupolets can be generated very efficiently, a great deal of useful information