Quantum control tasks are classified either as classical-like or as quantum requiring interference of pathways. We study the generation of interference pathways and relate them to the fidelity of the control target at a fixed time for various tasks. The model drift Hamiltonian studied is the two-dimensional Henon-Heiles (HH) potential. This system shows regular classical dynamics for low energies and chaotic dynamics for higher energies. A control operator supported by the whole momentum space and therefore connecting the entire Hilbert phase space is a random spiky potential. The other extreme is a smooth control potential. Intermediate cases are obtained by filtering the random spiky potential in momentum space. The fidelity of achieving a control task was related to the connectivity in phase space of the control operators. Typical quantum tasks such as generating a superposition of generalized coherent states rely on interfering pathways. For these cases the nonlinearity in the drift or control Hamiltonian is a necessary requirement for creating interferences. Control over rapidly diverging components of the wave function is achieved by the use of highly nonlocal control operators. Quantum control under chaotic drift was found to give a better yield than control under regular dynamics for such cases. For classical tasks we study the transformation of an initial generalized coherent state to another one. The best fidelity is obtained for regular or harmonic regions of the potential and smooth control operators. The approach to the classical limit is checked by decreasing the effective value of $\ensuremath{\hbar}$. Control under both quantum and classical tasks suffered from the decrease of $\ensuremath{\hbar}$ and the approach to classical proximity. Classical control tasks which rely heavily on maintaining a generalized coherent state throughout the evolution were found to be dysfunctional and lead to a completely uncontrolled situation once the classical chaos starts to appear.
Quantum control tasks are classified either as classical-like or as quantum requiring interference of pathways. We study the generation of interference pathways and relate them to the fidelity of the control target at a fixed time for various tasks. The model drift Hamiltonian studied is the two-dimensional Henon-Heiles (HH) potential. This system shows regular classical dynamics for low energies and chaotic dynamics for higher energies. A control operator supported by the whole momentum space and therefore connecting the entire Hilbert phase space is a random spiky potential. The other extreme is a smooth control potential. Intermediate cases are obtained by filtering the random spiky potential in momentum space. The fidelity of achieving a control task was related to the connectivity in phase space of the control operators. Typical quantum tasks such as generating a superposition of generalized coherent states rely on interfering pathways. For these cases the nonlinearity in the drift or control Hamiltonian is a necessary requirement for creating interferences. Control over rapidly diverging components of the wave function is achieved by the use of highly nonlocal control operators. Quantum control under chaotic drift was found to give a better yield than control under regular dynamics for such cases. For classical tasks we study the transformation of an initial generalized coherent state to another one. The best fidelity is obtained for regular or harmonic regions of the potential and smooth control operators. The approach to the classical limit is checked by decreasing the effective value of \ensuremathℏ. Control under both quantum and classical tasks suffered from the decrease of \ensuremathℏ and the approach to classical proximity. Classical control tasks which rely heavily on maintaining a generalized coherent state throughout the evolution were found to be dysfunctional and lead to a completely uncontrolled situation once the classical chaos starts to appear.
The evolution of wave packets under the influence of a Hénon–Heiles potential has been investigated by direct numerical solution of the time-dependent Schrödinger equation. Coherent state Gaussians with a variety of mean positions and momenta were selected as initial wave functions. Three types of diagnostics were used to identify chaotic behavior, namely, phase space trajectories computed from the expected values of coordinates and momenta, the correlation function P(t)=‖〈ψ(0)‖ψ(t)〉‖2, and the uncertainty product or phase space volume V(t)=ΔxΔyΔpxΔpy. The three approaches lead to a consistent interpretation of the system’s behavior, which tends to become more chaotic as the energy expectation value of the wave packet increases. The behavior of the corresponding classical system, however, is not a reliable guide to regular or chaotic behavior in the quantum mechanical system.
Quantum entanglement in the Hénon–Heiles system is analyzed using the squeezed coherent state. Enhancement of quantum entanglement via squeezing is explored in connection with chaotic and regular dynamics of the system. It is found that the entanglement enhancement via squeezing is implicitly linked to the local structure of the classical phase-space and it shows a clear quantum-classical correspondence. In particular, the entanglement enhancement via squeezing is found to be negligible for a highly chaotic orbit compared to the regular and weakly chaotic orbits, and shows a clear correspondence to the degree of chaos present in the classical initial condition. We believe that these results might be useful to develop efficient strategies to enhance entanglement in quantum systems.
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.
In this paper we consider some of the consequences of classical stochasticity phenomena in the quantum mechanical Henon–Heiles and Barbanis systems. To explore classical–quantum analogies we have introduced a quantum mechanical phase space based on the coherent state representation. Quantum Poincaré maps (QPMs) were constructed from contour plots of the stationary phase-space densities. We have observed a correlation between the topological features of a QPM at an eigenenergy E and the sensitivity ‖dE/dε‖ of that energy to the strength ε of the nonlinear coupling. States with extreme values of ‖dE/dε‖, corresponding to both high and to low values of ‖dE/dε‖, show regular regions in the quantum phase space, while states with intermediate values of ‖dE/dε‖ span large parts of the quantum phase-space regions. These general features of the QPMs apply for almost each energy multiplet, and no qualitative change in the character of the QPMs was observed above the classical critical energy Ec for the stochastic transition. To investigate the quantum analog of classical trajectories, we have studied the dynamics of initially coherent Gaussian wave packets. We have discovered two limiting types of time evolution of intially coherent wave packets, which exhibit quasiperiodic time evolution or rapid dephasing. Quasiperiodic quantum dynamics is analogous to quasiperiodic classical trajectory dynamics, while rapid dephasing of the wave packet reflects the effects of spreading of the wave packet or/and stochastic classical dynamics. We were able to establish a correspondence between the topology of the quantum phase-space density and the wave packet dynamics. Quasiperiodic time evolution and rapid dephasing are exhibited by wave packets initially located in the regular and in the irregular regions of the quantum phase space, respectively.
While the superposition of quantum evolutions is known to produce interference effects, the interference between evolutions with regular and chaotic classical limits remains largely unexplored. Here, we use a Mach-Zehnder interferometer to investigate the superposition of two quantum evolutions, implemented via post-selection, and to compare it with the corresponding classical mixture. The quantum kicked top provides a natural platform for this study, as its classical dynamics ranges from regular to mixed to fully chaotic depending on the Hamiltonian parameters. We show that when a regular evolution is superposed with a chaotic one, the resulting subsystem entropy can exceed that of the classical mixture, provided the contribution of the chaotic branch dominates in the superposed quantum evolution. We further demonstrate that entropy production in such superpositions is strongly influenced by the structure of the underlying classical phase space. We further show that increased entropy generation can occur for purely regular dynamics at small values of the chaos parameter, given an appropriate choice of post-selection. These results reveal a nontrivial interplay between classical chaos and quantum interference in superposed quantum dynamics
I compare the quantum and classical dynamics of a particle moving in a cosine potential while subject to a time-dependent force. I concentrate here on the behavior of an initially well-localized wave packet at times before the classically chaotic motion is fully developed. I find that the quantum and classical dynamics are indistinguishable well beyond the Ehrenfest time where the wave packet delocalizes. The quantum and classical descriptions first differ precisely when the classical probability density is folded in the vicinity of a hyperbolic fixed point. At this point, the wave function acquires a nodal structure which I show to be the result of a simple beating phenomenon between paths in the semiclassical propagator. When the interaction of the classical manifold with the hyperbolic fixed point leads to escape from the remnant separatrix, representing the onset of classical chaos, the interference associated with the tendril leads to a long-lived accumulation of quantum amplitude on top of the cosine barrier. This effect also has a semiclassical interpretation, meaning that it arises from the interference between classical paths, but one must expand to second order in Planck's constant to describe the behavior correctly. Both the nodal structure and this pinning of amplitude on the barrier are dynamic mechanisms for the quantum inhibition of mixing. I then couple the system to a bath of harmonic oscillators in order to study the effect of the environment on these mechanisms. Although an oscillator bath environment brings dissipation as well as noise to the problem, the noise effect dominates for the high temperature, weak coupling regime I study. When there is sufficient noise to render the interfering classical paths indistinguishable, I find that the quantum interference gets erased. This dephasing occurs at very early times, long before there is appreciable dissipation of energy to the environment. Consequently, one can argue that the presence of an environment, even if its effect would be negligible in a nonchaotic setting, allows for the possibility of quantum mixing.