We discuss some features of the dissipative quantum model of brain in the frame of the formalism of quantum dissipation. Such a formalism is based on the doubling of the system degrees of freedom. We show that the doubled modes account for the quantum noise in the fluctuating random force in the system-environment coupling. Remarkably, such a noise manifests itself through the coherent structure of the system ground state. The entanglement of the system modes with the doubled modes is shown to be permanent in the infinite volume limit. In such a limit the trajectories in the memory space are classical chaotic trajectories.
I review the dissipative quantum model of brain and discuss its recent developments related with the role of entanglement, quantum noise and chaos. Some comments on consciousness in the frame of the dissipative model are also presented. Dissipation seems to account for the medial character of consciousness, for its being in the present (the Now), its un-dividable unity, its intrinsic subjectivity (autonomy). Finally, essential features of a conscious artificial device, if ever one can construct it, are briefly commented upon, also in relation to a device able to exhibit mistakes in its behavio…
We review the dissipative quantum model of brain and present recent developments related with the rôle of entanglement, quantum noise and chaos in the model.
A quantum manifestation of chaotic classical dynamics is found in the framework of oscillatory numbers statistics for the model of nonlinear dissipative oscillator. It is shown by numerical simulation of an ensemble of quantum trajectories that the probability distributions and variances of oscillatory number states are strongly transformed in the order-to-chaos transition. The nonclassical, sub-Poissonian statistics of oscillatory excitation numbers is established for chaotic dissipative dynamics in the framework of Fano factor and Wigner functions. These results are proposed for testing and …
We discuss some problems of dissipative chaos for open quantum systems in the framework of semiclassical and quantum distributions. For this goal, we propose a driven nonlinear oscillator with time-dependent coefficients, i.e. with time-dependent Kerr-nonlinearity and time-modulated driving field. This model showing both regular and chaotic dynamics in the classical limit is realized in several experimental schemes. Quantum dissipative chaos is analyzed on the base of numerical method of quantum trajectories. Three quantities are studied: the Wigner function of oscillatory mode from the point …
We propose an anharmonic oscillator driven by two periodic forces of different frequencies as a new time-dependent model for investigating quantum dissipative chaos. Our analysis is done in the frame of statistical ensemble of quantum trajectories in quantum state diffusion approach. Quantum dynamical manifestation of chaotic behavior, including the emergence of chaos, properties of strange attractors, and quantum entanglement are studied by numerical simulation of ensemble averaged Wigner function and von Neumann entropy.
The agenda of Dissipative Quantum Chaos is to create a toolbox which would allow us to categorize open quantum systems into "chaotic" and "regular" ones. Two approaches to this categorization have been proposed recently. One of them is based on spectral properties of generators of open quantum evolution. The other one utilizes the concept of Lyapunov exponents to analyze quantum trajectories obtained by unraveling this evolution. By using two quantum models, we relate the two approaches and try to understand whether there is an agreement between the corresponding categorizations. Our answer is…
Some recent developments of the dissipative quantum model of brain are reported. In particular, the time-dependent frequency case is considered with its implications on the different life-times of the collective modes.
We study the stability of classical structures in chaotic systems when a dissipative quantum evolution takes place. We consider a paradigmatic model, the quantum baker map in contact with a heat bath at finite temperature. We analyze the behavior of the purity, fidelity and Husimi distributions corresponding to initial states localized on short periodic orbits (scar functions) and map eigenstates. Scar functions, that have a fundamental role in the semiclassical description of chaotic systems, emerge as very robust against environmental perturbations. This is confirmed by the study of other st…
In this paper, the purity of quantum states is applied to probe chaotic dissipative dynamics. To achieve this goal, a comparative analysis of regular and chaotic regimes of nonlinear dissipative oscillator (NDO) are performed on the base of excitation number and the purity of oscillatory states. While the chaotic regime is identified in our semiclassical approach by means of strange attractors in Poincaré section and with the Lyapunov exponent, the state in the quantum regime is treated via the Wigner function. Specifically, interesting quantum purity effects that accompany the chaotic dynamic…
We revisit the problem of quantum bi- and multi-stability by considering the dissipative Double Resonance Model. For a large driving frequency, this system has a simpler phase structure than the driven dissipative nonlinear oscillator -- the paradigm model for classical and quantum bistability. This allows us to obtain an analytical estimate for the lifetime of quantum limit cycles. On the other hand, for a small driving frequency, the system is much reacher than the nonlinear oscillator. This allows us to address a novel phenomenon of dissipation- and chaos-assisted tunneling between quantum …
The mechanism of memory localization in extended domains is described in the framework of the parametric dissipative quantum model of brain. The size of the domains and the capability in memorizing depend on the number of links the system is able to establish with the external world.
The effects of quantum and thermal corrections on the dynamics of a damped nonlinearly kicked harmonic oscillator are studied. This is done via the Quantum Langevin Equation formalism working on a truncated moment expansion of the density matrix of the system. We find that the type of bifurcations present in the system change upon quantization and that chaotic behavior appears for values of the nonlinear parameter that are far below the chaotic threshold for the classical model. Upon increase of temperature or Planck's constant, bifurcation points and chaotic thresholds are shifted towards low…
A dissipative quantum system is treated here by coupling it with a heat bath of harmonic oscillators. Through quantum Langevin equations and Ehrenfest's theorem, we establish explicitly the quantum Duffing equations with a double-well potential chosen. A quantum noise term appears the only driving force in dynamics. Numerical studies show that the chaotic attractor exists in this system while chaos is certainly forbidden in the classical counterpart.
Using the methods of quantum trajectories we investigate the effects of dissipative decoherence in a quantum computer algorithm simulating dynamics in various regimes of quantum chaos including dynamical localization, quantum ergodic regime and quasi-integrable motion. As an example we use the quantum sawtooth algorithm which can be implemented in a polynomial number of quantum gates. It is shown that the fidelity of quantum computation decays exponentially with time and that the decay rate is proportional to the number of qubits, number of quantum gates and per gate dissipation rate induced b…
Using the method of quantum trajectories we study a quantum chaotic dissipative ratchet appearing for particles in a pulsed asymmetric potential in the presence of a dissipative environment. The system is characterized by directed transport emerging from a quantum strange attractor. This model exhibits, in the limit of small effective Planck constant, a transition from quantum to classical behavior, in agreement with the correspondence principle. We also discuss parameter values suitable for implementation of the quantum ratchet effect with cold atoms in optical lattices.
Using the quantum trajectories approach we study the quantum dynamics of a dissipative chaotic system described by the Zaslavsky map. For strong dissipation the quantum wave function in the phase space collapses onto a compact packet which follows classical chaotic dynamics and whose area is proportional to the Planck constant. At weak dissipation the exponential instability of quantum dynamics on the Ehrenfest time scale dominates and leads to wave packet explosion. The transition from collapse to explosion takes place when the dissipation time scale exceeds the Ehrenfest time. For integrable…
We study the ensemble of complex symmetric matrices. The ensemble is useful in the study of effect of dissipation on systems with time reversal invariance. We consider the nearest neighbor spacing distribution and spacing ratio to investigate the fluctuation statistics and show that these statistics are similar to that of dissipative chaotic systems with time reversal invariance. We show that, unlike cubic repulsion in eigenvalues of Ginibre matrices, these ensemble exhibits a weaker repulsion. The nearest neighbor spacing distribution exhibits $P(s) \propto -s^3 \log s$ for small spacings. We…
We show that in the dissipative quantum model of brain the time-dependence of the frequencies of the electrical dipole wave quanta leads to the dynamical organization of the memories in space (i.e. to their localization in more or less diffused regions of the brain) and in time (i.e. to their longer or shorter life-time). The life-time and the localization in domains of the memory states also depend on internal parameters and on the number of links that the brain establishes with the external world. These results agree with the physiological observations of the dynamic formation of neural circ…
Quantum chaos---the study of quantized nonintegrable Hamiltonian systems---is an extremely well-developed and sophisticated field. By contrast, very little work has been done in looking at quantum versions of systems which classically exhibit {\it dissipative} chaos. Using the decoherence formalism of Gell-Mann and Hartle, I find a quantum mechanical analog of one such system, the forced damped Duffing oscillator. I demonstrate the classical limit of the system, and discuss its decoherent histories. I show that using decoherent histories, one can define not only the quantum map of an entire de…
We investigate the dissipative dynamics of quantum discord in a decoherence model with two initially entangled qubits in addition to a quantum kicked top. The two qubits are uncoupled during the period of our study and one of them interacts with the quantum kicked top. We find that the long time behavior of quantum discord could be well described by the fidelity decay of the quantum kicked top; for short time behavior, however, the phase of the amplitude of the fidelity decay is necessary to provide more specific information about the system. We have made comparison between the quantum kicked …
We study a dissipative, mechanically coupled optomechanical system that accommodates gain and loss. The gain (loss) is engineered by driven a purely dispersive optomechanical cavity with a blue-detuned (red-detuned) electromagnetic field. By taking into account the dissipative coupling, the Exceptional Point (EP), which is the $\mathcal{PT}-$symmetry phase transition, occurs at low threshold driving strength compared to the purely dispersive system. In the linear regime, the $\mathcal{PT}-$symmetry is unbroken and the dissipative coupling induces strong coupling between the mechanical resonato…
Optomechanical systems attract a lot of attention because they provide a novel platform for quantum measurements, transduction, hybrid systems, and fundamental studies of quantum physics. Their classical nonlinear dynamics is surprisingly rich and so far remains underexplored. Works devoted to this subject have typically focussed on dissipation constants which are substantially larger than those encountered in current experiments, such that the nonlinear dynamics of weakly dissipative optomechanical systems is almost uncharted waters. In this work, we fill this gap and investigate the regular …
Open quantum systems can exhibit complex states, which classification and quantification is still not well resolved. The Kerr-nonlinear cavity, periodically modulated in time by coherent pumping of the intra-cavity photonic mode, is one of the examples. Unraveling the corresponding Markovian master equation into an ensemble of quantum trajectories and employing the recently proposed calculation of quantum Lyapunov exponents [I.I. Yusipov {\it et al.}, Chaos {\bf 29}, 063130 (2019)], we identify `chaotic' and `regular' regimes there. In particular, we show that chaotic regimes manifest an inter…
The complexity and the chaos degree can be used to examine the chaotic aspects of not only several nonlinear classical and quantum physical physics but also life sciences. We will construct a model describing the function of brain in the context of Quantum Information Dynamics.
We compare the predictions of the dissipative quantum model of brain with neurophysiological data collected from electroencephalograms resulting from high-density arrays fixed on the surfaces of primary sensory and limbic areas of trained rabbits and cats. Functional brain imaging in relation to behavior reveals the formation of coherent domains of synchronized neuronal oscillatory activity and phase transitions predicted by the dissipative model.
In the context of dissipative systems, we show that for any quantum chaotic attractor a corre- sponding classical chaotic attractor can always be found. We provide with a general way to locate them, rooted in the structure of the parameter space (which is typically bidimensional, accounting for the forcing strength and dissipation parameters). In the cases where an approximate point like quantum distribution is found, it can be associated to exceptionally large regular structures. Moreover, supposedly anomalous quantum chaotic behaviour can be very well reproduced by the classical dynamics plu…
We analyze the origin and properties of the chaotic dynamics of two atomic ensembles in a driven-dissipative experimental setup, where they are collectively damped by a bad cavity mode and incoherently pumped by a Raman laser. Starting from the mean-field equations, we explain the emergence of chaos by way of quasiperiodicity -- presence of two or more incommensurate frequencies. This is known as the Ruelle-Takens-Newhouse route to chaos. The equations of motion have a $\mathbb{Z}_{2}$-symmetry with respect to the interchange of the two ensembles. However, some of the attractors of these equat…
We propose a measure, which we call the dissipative spectral form factor (DSFF), to characterize the spectral statistics of non-Hermitian (and non-Unitary) matrices. We show that DSFF successfully diagnoses dissipative quantum chaos, and reveals correlations between real and imaginary parts of the complex eigenvalues up to arbitrary energy (and time) scale. Specifically, we provide the exact solution of DSFF for the GinUE and for a Poissonian random spectrum (Poisson) as minimal models of dissipative quantum chaotic and integrable systems respectively. For dissipative quantum chaotic systems, …
We introduce a complex-plane generalization of the consecutive level-spacing distribution, used to distinguish regular from chaotic quantum spectra. Our approach features the distribution of complex-valued ratios between nearest- and next-to-nearest neighbor spacings. We show that this quantity can successfully detect the chaotic or regular nature of complex-valued spectra. This is done in two steps. First, we show that, if eigenvalues are uncorrelated, the distribution of complex spacing ratios is flat within the unit circle, whereas random matrices show a strong angular dependence in additio…
We study the semiclassical limit of the anisotropic two-photon Dicke model with a dissipative bosonic field and describe its rich nonlinear dynamics. Besides normal and 'superradiant'-like phases, the presence of localized fixed points reflects the spectral collapse of the closed-system Hamiltonian. Through Hopf bifurcations of superradiant and normal fixed points, limit cycles are formed in certain regions of parameters. We also identify a pole-flip transition induced by anisotropy and a region of chaotic dynamics, which appears from a cascade of period-doubling bifurcations. In the chaotic r…
The dynamics of chaotic systems are, by definition, exponentially sensitive to initial conditions and may appear rather random. In this work, we explore relations between the chaotic dynamics of an observable and the dynamics of information (entropy) contained in this observable, focussing on a disordered metal coupled to a dissipative, e.g. phononic, bath. The chaotic dynamics is characterised by Lyapunov exponents $λ$, the rates of growth of out-of-time order correlators (OTOCs), quantities of the form $\langle[\hat A(t),\hat B(0)]^{2}\rangle\propto\exp(2λt)$, where $\hat A$ and $\hat B$ are…
Using the way of deriving infinitive sum representation of density operator as a solution to the master equation describing the amplitude dissipative channel by virtue of the entangled state representation, we show manifestly how the initial density operator of a single-mode squeezed vacuum state evolves into a definite mixed state which turns out to be a squeezed chaotic state with decreasing-squeezing. We investigate average photon number, photon statistics distributions for this state.
A measure describing the chaos of a dynamics was introduced by two complexities in information dynamics, and it is called the chaos degree. In particular, the entropic chaos degree has been used to characterized several dynamical maps such that logistis, Baker's, Tinckerbel's in classical or quantum systems. In this paper, we give a new treatment of quantum chaos by defining the entropic chaos degree for quantum transition dynamics, and we prove that every non-chaotic quantum dynamics, e.g., dissipative dynamics, has zero chaos degree. A quantum spin 1/2 system is studied by our chaos degree, …
