The coherent state representations of the group $G = W_1 \otimes G_0$ (where $G_0 = SU(2), SU(1,1)$) are used in computer simulation of the dynamics of single two-level atom $(G_0 = SU(2))$ interacting with a quantized photon cavity mode - the Jaynes - Cummings model (JCM) without the rotating wave approximation and, in general, nonlinear in photon variables). The second case (hyperbolic Jaynes - Cummings model (HJCM), $G_0 = SU(1,1))$ corresponds to the quantum dynamics of quadratic nonlinear coupled oscillators (the parametric resonance on double field frequency and a three - wave parametric processes of nonlinear optics). Quasiclassical dynamical equations for parameters of approximately factorizable coherent states for these models are derived and regimes of motion for "atom" and field variables are analyzed.
We experimentally demonstrate coherent control of a quantum system, whose dynamics is chaotic in the classical limit. Interaction of diatomic molecules with a periodic sequence of ultrashort laser pulses leads to the dynamical localization of the molecular angular momentum, a characteristic feature of the chaotic quantum kicked rotor. By changing the phases of the rotational states in the initially prepared coherent wave packet, we control the rotational distribution of the final localized state and its total energy. We demonstrate the anticipated sensitivity of control to the exact parameters…
We propose a method to decouple the nanomechanical resonator in optomechanical systems from the environmental noise by introducing a chaotic coherent feedback loop. We find that the chaotic controller in the feedback loop can modulate the dynamics of the controlled optomechanical system and induce a broadband response of the mechanical mode. This broadband response of the mechanical mode will cut off the coupling between the mechanical mode and the environment and thus suppress the environmental noise of the mechanical modes. As an application, we use the protected optomechanical system to act…
Extensive coherent control over quantum chaotic diffusion using the kicked rotor model is demonstrated and its origin in deviations from random matrix theory is identified. Further, the extent of control in the presence of external decoherence is established. The results are relevant to both areas of quantum chaos and coherent control.
We present a novel experimental technique that can differentiate unequivocally between chaotic light and coherent light with amplitude fluctuations, and thus permits to characterize unambiguously the output of a laser. This technique consists of measuring the second-order intensity cross-correlation at the outputs of an unbalanced Michelson interferometer. It is applied to a chaotic light source and to the output of a semiconductor nanolaser whose "standard" intensity correlation function above-threshold displays values compatible with a mixture of coherent and chaotic light. Our experimental …
The population decay due to a small opening in an otherwise closed cavity supporting chaotic classical dynamics displays a quantum correction on top of the classical exponential form, a pure manifestation of quantum coherence that acquires a universal form and can be explained by path interference. Being coherent, such enhancement is prone to decoherence effects due to the coupling of the system to an external environment. We study this interplay between incoherent and coherent quantum corrections to decay by evaluating, within a Caldeira-Legget scenario, off-diagonal contributions to the deco…
We calculate survival probability of a special state which couples randomly to a regular or chaotic environment. The environment is modelled by a suitably chosen random matrix ensemble. The exact results exhibit non--perturbative features as revival of probability and non--ergodicity. The role of background complexity and of coupling complexity is discussed as well.
We investigate a Bose-Einstein condensate held in a quasi-one-dimensional Wannier-Stark lattice which is a combination of linear potential with an accelerated optical lattice. It is demonstrated that the system can be reduced to a periodically driven Gross-Pitaevskii one, in which we find the first exact analytical solution and the regular and chaotic numerical solutions with accelerated atomic flow densities. The results suggest an experimental scheme for generating and controlling the accelerating regular and chaotic matter-waves.
We study the differences between the process of decoherence induced by chaotic and regular environments. For this we analyze a family of simple models wich contain both regular and chaotic environments. In all cases the system of interest is a "quantum walker", i.e. a quantum particle that can move on a lattice with a finite number of sites. The walker interacts with an environment wich has a D dimensional Hilbert space. The results we obtain suggest that regular and chaotic environments are not distinguishable from each other in a (short) timescale t*, wich scales with the dimensionality of t…
The quantum dynamics of initial coherent states is studied in the Dicke model and correlated with the dynamics, regular or chaotic, of their classical limit. Analytical expressions for the survival probability, i.e. the probability of finding the system in its initial state at time $t$, are provided in the regular regions of the model. The results for regular regimes are compared with those of the chaotic ones. It is found that initial coherent states in regular regions have a much longer equilibration time than those located in chaotic regions. The properties of the distributions for the init…
We review the fictitious integrable system approach which predicts dynamical tunneling rates from regular states to the chaotic region in systems with a mixed phase space. It is based on the introduction of a fictitious integrable system that resembles the regular dynamics within the regular island. We focus on the direct regular-to-chaotic tunneling process which dominates, if nonlinear resonances within the regular island are not relevant. For quantum maps, billiard systems, and optical microcavities we find excellent agreement with numerical rates for all regular states.
In generic Hamiltonian systems tori of regular motion are dynamically separated from regions of chaotic motion in phase space. Quantum mechanically these phase-space regions are coupled by dynamical tunneling. We introduce a semiclassical approach based on complex paths for the prediction of dynamical tunneling rates from regular tori to the chaotic region. This approach is demonstrated for the standard map giving excellent agreement with numerically determined tunneling rates.
The dynamics of a (quasi)one-dimensional interacting atomic Bose-Einstein condensate in a tilted optical lattice is studied in a discrete mean-field approximation, i.e., in terms of the discrete nonlinear Schrödinger equation. If the static field is varied the system shows a plethora of dynamical phenomena. In the strong field limit we demonstrate the existence of (almost) non-spreading states which remain localized on the lattice region populated initially and show coherent Bloch oscillations with fractional revivals in the momentum space (so called quantum carpets). With decreasing field, th…
We derive a prediction of dynamical tunneling rates from regular to chaotic phase-space regions combining the direct regular-to-chaotic tunneling mechanism in the quantum regime with an improved resonance-assisted tunneling theory in the semiclassical regime. We give a qualitative recipe for identifying the relevance of nonlinear resonances in a given $\hbar$-regime. For systems with one or multiple dominant resonances we find excellent agreement to numerics.
A new micro-irreversible 3D theory of quantum multichannel scattering in the three-body system is developed. The quantum approach is constructed on the generating trajectory tubes which allow taking into account influence of classical non-integrability on the dynamical quantum system. It was shown that when the volume of classical chaos in phase space is larger than quantum sell in the main object of quantum system the wavefunction generates chaos (quantum chaos). The probability of quantum transitions is constructed for this case. On the example of collinear collision of Li+(FH) -> (LiF)+H sy…
We derive a formula predicting dynamical tunneling rates from regular states to the chaotic sea in systems with a mixed phase space. Our approach is based on the introduction of a fictitious integrable system that resembles the regular dynamics within the island. For the standard map and other kicked systems we find agreement with numerical results for all regular states in a regime where resonance-assisted tunneling is not relevant.
We investigate the time evolution of wave packets in systems with a mixed phase space where regular islands and chaotic motion coexist. For wave packets started in the chaotic sea on average the weight on a quantized torus of the regular island increases due to dynamical tunneling. This flooding weight initially increases linearly and saturates to a value which varies from torus to torus. We demonstrate for the asymptotic flooding weight universal scaling with an effective tunneling coupling for quantum maps and the mushroom billiard. This universality is reproduced by a suitable random matrix…
Regular and chaotic dynamics of nonlinear optomechanical systems controlled by modulated light
Scientific-Practical Material Research Centre, Belarus National Academy of Sciences（白俄罗斯科学院科学实用材料研究中心）；Institute of Molecular Physics, Polish Academy of Sciences（波兰科学院分子物理研究所）
We present exact, explicit, convergent periodic-orbit expansions for individual energy levels of regular quantum graphs. One simple application is the energy levels of a particle in a piecewise constant potential. Since the classical ray trajectories (including ray splitting) in such systems are strongly chaotic, this result provides the first explicit quantization of a classically chaotic system.
In the spectrum of systems showing chaos-assisted tunneling, three-state crossings are formed when a chaotic singlet intersects a tunnel doublet. We study the dissipative quantum dynamics in the vicinity of such crossings. A harmonically driven double well coupled to a bath serves as a model. Markov and rotating-wave approximations are introduced with respect to the Floquet spectrum of the time-dependent central system. The resulting master equation is integrated numerically. We find various types of transient tunneling, determined by the relation of the level width to the inherent energy scal…
We study the quantum mechanics of a billiard (Robnik 1983) in the regime of mixed-type classical phase space (the shape parameter λ=0.15) at very high-lying eigenstates, starting at about 1.000.000th eigenstate and including the consecutive 587654 eigenstates. By calculating the normalized Poincaré Husimi functions of the eigenstates and comparing them with the classical phase space structure, we introduce the overlap criterion which enables us to separate with great accuracy and reliability the regular and chaotic eigenstates, and the corresponding energies. The chaotic eigenstates appear all…
Superposed quantum evolutions across chaotic and regular regimes
Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）；Department of Physics and Astronomy, University of Waterloo（滑铁卢大学物理与天文学系）；University of Vienna（维也纳大学）；Vienna Doctoral School in Physics（维也纳物理博士学院）；Vienna Center for Quantum Science and Technology (VCQ)（维也纳量子科学与技术中心）；Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences（奥地利科学院量子光学与量子信息研究所）；Institute for Theoretical Studies, ETH Zürich（苏黎世联邦理工学院理论研究所）；Perimeter Institute for Theoretical Physics（理论物理珀蒂默研究所）；Department of Physics and Computer Science, Wilfrid Laurier University（威尔弗里德·劳雷尔大学物理与计算机科学系）
Centre-of-mass atomic motion in an optical lattice near the resonance is shown to be a chaotic walking due to the interplay between coherent internal atomic dynamics and spontaneous emission. Statistical properties of chaotic atomic motion can be controlled by the single parameter, the detuning between the atomic transition frequency and the laser frequency. We derive a Fokker-Planck equation in the energetic space to describe the atomic transport near the resonance and demonstrate numerically how to manipulate the atomic motion varying the detuning.
Based on the connection between the spectral form factor and the probability to return, the origin of the $1/f^α$-noise in fully chaotic and fully integrable systems is traced to the quantum interference between invariant manifolds of the classical dynamics and the dimensionality of those invariant manifolds. This connection and the order-to-chaos transition are analyzed in terms of the statistics of Floquet's quasienergies of a classically chaotic driving non-linear system. An immediate prediction of the connection established here is that in the presence of decoherence, the spectral exponent…
Mean fidelity amplitude and parametric energy--energy correlations are calculated exactly for a regular system, which is subject to a chaotic random perturbation. It turns out that in this particular case under the average both quantities are identical. The result is compared with the susceptibility of chaotic systems against random perturbations. Regular systems are more susceptible against random perturbations than chaotic ones.
One of the principal goals of controlling classical chaotic dynamical systems is known as targeting, which is the very weakly perturbative process of using the system's extreme sensitivity to initial conditions in order to arrive at a predetermined target state. It is shown that a generalization to chaotic quantum systems is possible in the semiclassical regime, but requires tailored perturbations whose effects must undo the dynamical spreading of the evolving quantum state. The procedure described here is applied to initially minimum uncertainty wave packets in the quantum kicked rotor, a pre…
We study the classical dynamics in a generic first-order quantum phase transition between the U(5) and SU(3) limits of the interacting boson model. The dynamics is chaotic, of Hénon-Heiles type, in the spherical phase and is regular, yet sensitive to local degeneracies, in the deformed phase. Both types of dynamics persist in the coexistence region resulting in a divided phase space.
The effect of chaotic bath dynamics on the decoherence of a quantum system is examined for the vibrational degrees of freedom of a diatomic molecule in a realistic, constant temperature collisional bath. As an example, the specific case of I$_2$ in liquid xenon is examined as a function of temperature, and the results compared with an integrable xenon bath. A crossover in behavior is found: the integrable bath induces more decoherence at low bath temperatures than does the chaotic bath, whereas the opposite is the case at the higher bath temperatures. These results, verifying a conjecture due …
We investigate the influence of the photon statistics on the excitation dynamics of a single two level system. A single semiconductor quantum dot represents the two level system and is resonantly excited either with coherent laser light, or excited with chaotic light, with photon statistics corresponding to that of thermal radiation. Experimentally, we observe a reduced absorption cross-section under chaotic excitation in the steady-state. In the transient regime, the Rabi oscillations observable under coherent excitation disappear under chaotic excitation. Likewise, in the emission spectrum t…
We investigate the asymptotic state of time-periodic quantum systems with regular and chaotic Floquet states weakly coupled to a heat bath. The asymptotic occupation probabilities of these two types of states follow fundamentally different distributions. Among regular states the probability decreases from the state in the center of a regular island to the outermost state by orders of magnitude, while chaotic states have almost equal probabilities. We derive an analytical expression for the occupations of regular states of kicked systems, which depends on the winding numbers of the regular tori…
Classically integrable approximants are here constructed for a family of predominantly chaotic periodic systems by means of the Baker-Hausdorff-Campbell formula. We compare the evolving wave density for the corresponding exact quantum systems using semiclassical approximations based alternatively on the chaotic and on the integrable trajectories. It is found that the latter reproduce the quantum oscillations and provide superior approximations even when the initial coherent state is placed in a broad chaotic region. Time regimes are then accessed in which the propagation based on the system's …
We numerically investigate the properties of speckle patterns formed by nonlinear point scatterers. We show that, in the weak localization regime, dynamical instability appears, eventually leading to chaotic behavior of the system. Analysing the statistical properties of the instability thresholds for different values of the system size and disorder strength, a scaling law is emphasized. The later is also found to govern the smallest decay rate of the linear system, putting thus forward the crucial importance of interference effects. This is also underlined by the fact that coherent backscatte…
Just as a coherent state may be considered as a quantum point, its restriction to a factor space of the full Hilbert space can be interpreted as a quantum plane. The overlap of such a factor coherent state with a full pure state is akin to a quantum section. It defines a reduced pure state in the cofactor Hilbert space. The collection of all the Wigner functions corresponding to a full set of parallel quantum sections defines the Husimi-Wigner reresentation. It occupies an intermediate ground between drastic suppression of nonclassical features, characteristic of Husimi functions, and the daun…
The dynamics of chaotic billiards is significantly influenced by coexisting regions of regular motion. Here we investigate the prevalence of a different fundamental structure, which is formed by marginally unstable periodic orbits and stands apart from the regular regions. We show that these structures both {\it exist} and {\it strongly influence} the dynamics of locally perturbed billiards, which include a large class of widely studied systems. We demonstrate the impact of these structures in the quantum regime using microwave experiments in annular billiards.
Coherent control of chaotic molecular systems, using laser-assisted alignment of sulphur dioxide (SO$_2$) molecules in the presence of a static electric field as an example, is considered. Conditions for which the classical version of this system is chaotic are established, and the quantum and classical analogs are shown to be in very good correspondence. It is found that the chaos present in the classical system does not impede the alignment, neither in the classical nor the quantum system. Using the results of numerical calculations, we suggest that laser-assisted alignment is stable against…
Distributions of eigenmodes are widely concerned in both bounded and open systems. In the realm of chaos, counting resonances can characterize the underlying dynamics (regular vs. chaotic), and is often instrumental to identify classical-to-quantum correspondence. Here, we study, both theoretically and experimentally, the statistics of chaotic resonances in an optical microcavity with a mixed phase space of both regular and chaotic dynamics. Information on the number of chaotic modes is extracted by counting regular modes, which couple to the former via dynamical tunneling. The experimental da…
Dielectric resonators are open systems particularly interesting due to their wide range of applications in optics and photonics. In a recent paper [PRE, vol. 78, 056202 (2008)] the trace formula for both the smooth and the oscillating parts of the resonance density was proposed and checked for the circular cavity. The present paper deals with numerous shapes which would be integrable (square, rectangle, and ellipse), pseudo-integrable (pentagon) and chaotic (stadium), if the cavities were closed (billiard case). A good agreement is found between the theoretical predictions, the numerical simul…
We consider a simple model of lossless interaction between a two-level single atom and a standing-wave single-mode laser field which creates a one-dimensional optical lattice. Internal dynamics of the atom is governed by the laser field which is treated to be classical with a large number of photons. Center-of-mass classical atomic motion is governed by the optical potential and the internal atomic degree of freedom. The resulting Hamilton-Schrödinger equations of motion are a five-dimensional nonlinear dynamical system with two integrals of motion. The main focus of the paper is chaotic atomi…
We study Wigner function value statistics of classically chaotic quantum maps on compact 2D phase space. We show that the Wigner function statistics of a random state is a Gaussian, with the mean value becoming negligible compared to the width in the semi-classical limit. Using numerical example of quantized sawtooth map we demonstrate that the relaxation of time-dependent Wigner function statistics, starting from a coherent initial state, takes place on a logarithmically short log (hbar) time scale.
We derive Fourier's law for a completely coherent quasi one--dimensional chaotic quantum system coupled locally to two heat baths at different temperatures. We solve the master equation to first order in the temperature difference. We show that the heat conductance can be expressed as a thermodynamic equilibrium coefficient taken at some intermediate temperature. We use that expression to show that for temperatures large compared to the mean level spacing of the system, the heat conductance is inversely proportional to the level density and, thus, inversely proportional to the length of the sy…
We present a comprehensive analysis of the emerging order and chaos and enduring symmetries, accompanying a generic (high-barrier) first-order quantum phase transition (QPT). The interacting boson model Hamiltonian employed, describes a QPT between spherical and deformed shapes, associated with its U(5) and SU(3) dynamical symmetry limits. A classical analysis of the intrinsic dynamics reveals a rich but simply-divided phase space structure with a Hénon-Heiles type of chaotic dynamics ascribed to the spherical minimum and a robustly regular dynamics ascribed to the deformed minimum. The simple…
