Sensitivity of the time evolution to small changes in the state is a characteristic feature of classical chaos. It has been believed that state sensitivity could not exist in quantum mechanics because of the unitary invariance of the Hilbert-space overlap of states. We argue that this Hilbert-space criterion is irrelevant and show that both quantum states and classical statistical states exhibit a similar kind of state sensitivity. This is demonstrated by the degree to which the initial state can be recovered in computational motion reversal: forward evolution for a time T, perturbation of the state, and backward time evolution. Some differences between classical and quantum state sensitivity remain, and these seem to be insensitive to decoherence. \textcopyright{} 1996 The American Physical Society.
Sensitivity of the time evolution to small changes in the state is a characteristic feature of classical chaos. It has been believed that state sensitivity could not exist in quantum mechanics because of the unitary invariance of the Hilbert-space overlap of states. We argue that this Hilbert-space criterion is irrelevant and show that both quantum states and classical statistical states exhibit a similar kind of state sensitivity. This is demonstrated by the degree to which the initial state can be recovered in computational motion reversal: forward evolution for a time T, perturbation of the state, and backward time evolution. Some differences between classical and quantum state sensitivity remain, and these seem to be insensitive to decoherence. \textcopyright{} 1996 The American Physical Society.
Classical chaotic dynamics is characterized by exponential sensitivity to initial conditions. Quantum mechanics, however, does not show this feature. We consider instead the sensitivity of quantum evolution to perturbations in the Hamiltonian. This is observed as an attenuation of the Loschmidt echo M(t), i.e., the amount of the original state (wave packet of width sigma) which is recovered after a time reversed evolution, in the presence of a classically weak perturbation. By considering a Lorentz gas of size L, which for large L is a model for an unbounded classically chaotic system, we find numerical evidence that, if the perturbation is within a certain range, M(t) decays exponentially with a rate 1/tau(phi) determined by the Lyapunov exponent lambda of the corresponding classical dynamics. This exponential decay extends much beyond the Eherenfest time t(E) and saturates at a time t(s) approximately equal to lambda(-1)ln[N], where N approximately (L/sigma)(2) is the effective dimensionality of the Hilbert space. Since tau(phi) quantifies the increasing uncontrollability of the quantum phase (decoherence) its characterization and control has fundamental interest.
Numerical results show that, for quantum autonomous chaotic system, the evolution of initially coherent states are sensitive to perturbation. The overlap of a perturbed state with the unperturbed one decays exponentially, which is followed by fluctuation around N-1, N being the dimension of the Hilbert space. The matrix elements of the evolution operator in interaction picture tend to be a random distribution after sufficiently long time, where the interaction is the perturbation, even when the perturbation is very weak. The difference between a regular system and the chaotic one is shown clearly. In a regular system, the overlap shows strong revival. The distribution of the evolution matrix has only a few dominant terms.
Using the key properties of chaos, i.e. ergodicity and exponential instability, as a resource to control classical dynamics has a long and considerable history. However, in the context of controlling "chaotic" quantum unitary dynamics, the situation is far more tenuous. The classical concepts of exponential sensitivity to trajectory initial conditions and ergodicity do not directly translate into quantum unitary evolution. Nevertheless properties inherent to quantum chaos can take on those roles: i) the dynamical sensitivity to weak perturbations, measured by the fidelity decay, serves a similar purpose as the classical sensitivity to initial conditions; and ii) paired with the fact that quantum chaotic systems are conjectured to be statistically described by random matrix theory, implies a method to translate the ergodic feature into the control of quantum dynamics. With those two properties, it can be argued that quantum chaotic dynamical systems, in principle, allow for full controllability beyond a characteristic time that scales only logarithmically with system size and ℏ−1. In the spirit of classical targeting, it implies that it is possible to fine tune the immense quantum interference with weak perturbations and steer the system from any initial state into any desired target state, subject to constraints imposed by conserved quantities. In contrast, integrable dynamics possess neither ergodicity nor exponential instability, and thus the weak perturbations apparently must break the integrability for control purposes. The main ideas are illustrated with the quantum kicked rotor. The production of revivals, cat-like entangled states, and the transition from any random state to any other random state is possible as demonstrated.
Unless you are a trained juggler, it is very hard to balance a sharpened pencil on its tip.We expect it will fall, at some time, in some unpredictable direction.In contrast, in the ammonia maser, a simple molecule can be poised in a state between its isoenergetic inversion isomers, and ammonia will predictably oscillate between the two isomers.Indeed, the oscillation is so predictable that it revolutionized precision timekeeping (1).The lesson is that while classical motions can be very sensitive to perturbations, quantum motions can be remarkably stable and insensitive to perturbation (Fig. 1A).In this issue of PNAS, Sadhasivam, Meuser, Reichman, and Althorpe show that a limit to quantum sensitivity previously proposed for black hole dynamics holds for a molecular barrier model as well (2).In the 19th century, the sensitivity of classical motion to perturbation became an intellectual starting point for statistical mechanics.Aleksandr Lyapunov suggested a way to quantify this sensitivity through the eponymous Lyapunov exponent λ L , which measures how rapidly slightly perturbed systems will diverge from each other.In a dilute gas, the Lyapunov exponent is inversely related to the mean free time between collisions.Since the world is quantum mechanical beneath its classical surface, how can we reconcile classical sensitivity with the clock-like behavior of quantum systems?One possibility is mere physical size, but quantum superposition has been observed in cold macroscopic objects.The key seems then to be not the physical size of a system, but rather some combination of the size of its accessible quantum state space (classically: phase space), its physical size, and how easy it is to hop from one part of the system's state space to another.For a sufficiently connected large state space, quantum motion becomes entangled or "scrambled" and a statistical and sensitive behavior emerges even from quantum mechanics.Classically there is no upper bound on the size of λ L because classical phase space has infinite room in it, so the proverbial "Butterfly Effect" of the weather changing at Columbia University depending on how a butterfly beats its wings over the Andaman Islands can be very efficient.In contrast, quantum chaos has turned out to be a subtle problem because the Schrödinger equation is a linear equation.Time evolution is just a linear superposition of eigenstates with different phases, and so it's all in the nature of those eigenstates-do they look "regular" or do they look scrambled?The puzzles of quantum scrambling have a history even older than modern quantum mechanics itself: in 1919, using Old Quantum Theory, Herzfeld proposed that bimolecular reactive collisions in gases could scramble atoms at a maximal rate of k B T/h (3), which would be slowed down only by a Boltzmann factor exp[-E † /k B T ], if an energy E † , the activation energy, was needed to rearrange the atoms.It was only in 1969 that out-of-time-order correlators, or OTOCs, were introduced by Larkin and Ovchinnikov to describe the
We present a quantitative analysis of the reversibility properties of classically chaotic quantum motion. We analyze the connection between reversibility and the rate at which a quantum state acquires a more and more complicated structure in its time evolution. This complexity is characterized by the number M(t) of harmonics of the [initially isotropic, i.e., M(0)=0 ] Wigner function, which are generated during quantum evolution for the time t . We show that, in contrast to the classical exponential increase, this number can grow not faster than linearly and then relate this fact with the degree of reversibility of the quantum motion. To explore the reversibility we reverse the quantum evolution at some moment T immediately after applying at this moment an instant perturbation governed by a strength parameter xi . It follows that there exists a critical perturbation strength xic approximately sqrt 2/M(T) below which the initial state is well recovered, whereas reversibility disappears when xi > or approximately xic(T) . In the classical limit the number of harmonics proliferates exponentially with time and the motion becomes practically irreversible. The above results are illustrated in the example of the kicked quartic oscillator model.
The unitarity of quantum evolutions implies that an overlap between two initial states does not change in time. This property is commonly believed to explain the apparent lack of state sensitivity in quantum theory, a feature that is prevailing in classical chaotic systems. However, classical state sensitivity is based on a distance between two trajectories in phase space which is a completely different mathematical concept than an overlap between two vectors in Hilbert space. It is possible that state sensitivity in quantum theory can be detected with the help of some special metric. Here we show that the recently introduced Weighted Bures length achieves this task. We numerically investigate a unitary cellular automaton of N interacting qubits and analyze how a single-qubit perturbation affects the evolution of WBL between the unperturbed and perturbed states. We observe a linear growth of WBL if the qubits are arranged into a cyclic graph and an exponential growth if they are arranged into a random bipartite graph.