Dissipative quantum chaos: transition from wave packet collapse to explosion

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 nonlinear dynamics the explosion practically disappears leaving place to collapse.     

Quantum geometry and quantum mechanics of integrable systems

Quantum integrable systems and their classical counterparts are considered. We show that the symplectic structure and invariant tori of the classical system can be deformed by a quantization parameter ħ to produce a new (classical) integrable system. The new tori selected by the ħ-equidistance rule represent the spectrum of the quantum system up to O(ħ ^∞) and are invariant under quantum dynamics in the long-time range O(ħ ^−∞). The quantum diffusion over the deformed tori is described. The analytic apparatus uses quantum action-angle coordinates explicitly constructed by an ħ-deformation of the classical action-angles.     

On Algebraic Integrability of the Deformed Elliptic Calogero-Moser Problem

Abstract

We discuss stationary solutions of the discrete nonlinear Schrödinger equation (DNSE) with a potential of the ? ⁴ type which is generically applicable to several quantum spin, electron and classical lattice systems. We show that there may arise chaotic spatial structures in the form of incommensurate or irregular quantum states. As a first (typical) example we consider a single electron which is strongly coupled with phonons on a 1D chain of atoms — the (Rashba)–Holstein polaron model. In the adiabatic approximation this system is conventionally described by the DNSE. Another relevant example is that of superconducting states in layered superconductors described by the same DNSE. Amongst many other applications the typical example for a classical lattice is a system of coupled nonlinear oscillators. We present the exact energy spectrum of this model in the strong coupling limit and the corresponding wave function. Using this as a starting point we go on to calculate the wave function for moderate coupling and find that the energy eigenvalue of these structures of the wave function is in exquisite agreement with the exact strong coupling result. This procedure allows us to obtain (numerically) exact solutions of the DNSE directly. When applied to our typical example we find that the wave function of an electron on a deformable lattice (and other quantum or classical discrete systems) may exhibit incommensurate and irregular structures. These states are analogous to the periodic, quasiperiodic and chaotic structures found in classical chaotic dynamics.     

Diagram Expansions in Nonequilibrium Statistical Mechanics Classical Limit Considerations

A diagram expansion is proposed for calculating traces of the kind Tr{Ae^?itLB} which are of interest for calculating time correlation functions and expectation values in nonequilibrium statistical mechanics. Arbitrary initial conditions are considered. In the classical limit the diagram expansion of FUJITA is obtained. A systematic method for obtaining quantum corrections including exchange and symmetry effects is proposed.     

Semiclassical Behaviour of Expectation Values in Time Evolved Lagrangian States for Large Times

We study the behaviour of time evolved quantum mechanical expectation values in Lagrangian states in the limit 0 and t. We show that it depends strongly on the dynamical properties of the corresponding classical system. If the classical system is strongly chaotic, i.e. Anosov, then the expectation values tend to a universal limit. This can be viewed as an analogue of mixing in the classical system. If the classical system is integrable, then the expectation values need not converge, and if they converge their limit depends on the initial state. An additional difference occurs in the timescales for which we can prove this behaviour; in the chaotic case we get up to Ehrenfest time, t ln (1/), whereas for integrable system we have a much larger time range.     

Growing Classical and Quantum Entropies in the Early Universe

Following the idea that the global and local arrow of time has a cosmological origin, we define an entropy in the classical and in the quantum periods of the universe evolution. For the quantum period a semi-classical approach is adopted, modelling the universe with Wheeler-De Witt equation and using WKB. By applying the self-induced decoherence to the state of the universe it is proved that the quantum universe becomes a classical one. This allows us to define a conditional entropy which, in our simplified model, is proportional to e ^{2γ t} where γ is the dumping factor associated with the interaction potential of the scalar fields. Finally we find both Gibbs and thermodynamical entropy of the universe based in the conditional entropy.     

Quantum Instantons with Classical Moduli Spaces

 We introduce a quantum Minkowski space-time based on the quantum group SU(2) _q extended by a degree operator and formulate a quantum version of the anti-self-dual Yang-Mills equation. We construct solutions of the quantum equations using the classical ADHM linear data, and conjecture that, up to gauge transformations, our construction yields all the solutions. We also find a deformation of Penrose's twistor diagram, giving a correspondence between the quantum Minkowski space-time and the classical projective space ℙ³. Received: 10 May 2002 / Accepted: 10 January 2003 Published online: 5 May 2003 Communicated by L. Takhtajan     

Quantum Integrable Models and Discrete Classical Hirota Equations

The standard objects of quantum integrable systems are identified with elements of classical nonlinear integrable difference equations. The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's Q-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to the Bethe ansatz are studied. The nested Bethe ansatz equations for A _k-1 -type models appear as discrete time equations of motions for zeros of classical τ-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained. Difference equations for eigenvalues of the Q-operators which generalize Baxter's three-term T−Q-relation are derived. Received: 15 May 1996 / Accepted: 25 November 1996     

Numerical on-the-fly implementation of the action of the kinetic energy operator on a vibrational wave function: application to methanol

ABSTRACT

In quantum dynamics, physically well-adapted curvilinear coordinates are coordinates that lead to a Hamiltonian operator as separable as possible, in order to simplify the resolution of the corresponding time-independent or time-dependent Schrödinger equations. Various equivalent curvilinear expressions of the kinetic energy operator (KEO) are well known. They can be used in either an analytical or a numerical approach. The latter has the feature of allowing to straightforwardly compute the KEO in terms of sophisticated (yet easy to define) physically well-adapted curvilinear coordinates. Nevertheless, the number of terms to be computed on a full grid, scales as n²/2 (n being the number of degrees of freedom), so that, for systems with n?>?10, the memory storage of the KEO's becomes extremely demanding and therefore often unrealistic. We show here that it is possible, starting from the basic quantum expression of the KEO as a curvilinear Laplacian operator, to reduce the memory storage bottleneck by numerically computing the KEO on-the-fly, i.e. each time it is required, without computing the extrapotential term. This new approach opens the way to rigorous quantum studies of systems with many degrees of freedom. The comparison of torsional levels of methanol obtained by the present on-the-fly method with our previous results shows excellent agreement.     

Wave packet revivals and the energy eigenvalue spectrum of the quantum pendulum

The rigid pendulum, both as a classical and as a quantum problem, is an interesting system as it has the exactly soluble harmonic oscillator and the rigid rotor systems as limiting cases in the low- and high-energy limits, respectively. The energy variation of the classical periodicity (τ) is also dramatic, having the special limiting case of τ→∞ at the ‘top’ of the classical motion (i.e., the separatrix.) We study the time-dependence of the quantum pendulum problem, focusing on the behavior of both the (approximate) classical periodicity and especially the quantum revival and superrevival times, as encoded in the energy eigenvalue spectrum of the system. We provide approximate expressions for the energy eigenvalues in both the small and large quantum number limits, up to fourth order in perturbation theory, comparing these to existing handbook expansions for the characteristic values of the related Mathieu equation, obtained by other methods. We then use these approximations to probe the classical periodicity, as well as to extract information on the quantum revival and superrevival times. We find that while both the classical and quantum periodicities increase monotonically as one approaches the ‘top’ in energy, from either above or below, the revival times decrease from their low- and high-energy values until very near the separatrix where they increase to a large, but finite value.     

Competing Species: Integrability and Stability

Abstract

We examine the classical model of two competing species for integrability in terms of analytic functions by means of the Painlevé analysis. We find that the governing equations are integrable for certain values of the essential parameters of the system. We find that, for all integrable cases with the nontrivial equilibrium point in the physically acceptable region, the nontrivial equilibrium point is stable.     

Quantum and classical correlations for a two-qubit X structure density matrix

We derive explicit expressions for quantum discord and classical correlation for an X structure density matrix. Based on the characteristics of the expressions, the quantum discord and the classical correlation are easily obtained and compared under different initial conditions using a novel analytical method. We explain the relationships among quantum discord, classical correlation, and entanglement, and further find that the quantum discord is not always larger than the entanglement measured by concurrence in a general two-qubit X state. The new method, which is different from previous approaches, has certain guiding significance for analysing quantum discord and classical correlation of a two-qubit X state, such as a mixed state.     

Quantum breaking time near classical equilibrium points

In the evolution of distributions localized around classical equilibrium points, the quantum-classical correspondence breaks down at a time, the so-called quantum breaking, or Ehrenfest time, which is related to the minimal separation of the quantum levels in proximity of the classical equilibrium energy. By studying one-dimensional systems with single- and double-well polynomial potentials, we find that the Ehrenfest time diverges logarithmically with the inverse of the Planck constant whenever the equilibrium point is exponentially unstable. In all the other cases, we have a power law divergence with the exponent determined by the degree of the potential near the equilibrium point.     

The classical limit for quantum mechanical correlation functions

For quantum systems of finitely many particles as well as for boson quantum field theories, the classical limit of the expectation values of products of Weyl operators, translated in time by the quantum mechanical Hamiltonian and taken in coherent states centered inx- andp-space around? ^?1/2 (coordinates of a point in classical phase space) are shown to become the exponentials of coordinate functions of the classical orbit in phase space. In the same sense,? ^?1/2 [(quantum operator) (t) — (classical function) (t)] converges to the solution of the linear quantum mechanical system, which is obtained by linearizing the non-linear Heisenberg equations of motion around the classical orbit.     

A Remark on the Notion of Independence of Quantum Integrals of Motion in the Thermodynamic Limit

Studies of integrable quantum many-body systems have a long history with an impressive record of success. However, surprisingly enough, an unambiguous definition of quantum integrability remains a matter of an ongoing debate. We contribute to this debate by dwelling upon an important aspect of quantum integrability—the notion of independence of quantum integrals of motion (QIMs). We point out that a widely accepted definition of functional independence of QIMs is flawed, and suggest a new definition. Our study is motivated by the PXP model—a model of N spins 1/2 possessing an extensive number of binary QIMs. The number of QIMs which are independent according to the common definition turns out to be equal to the number of spins, N. A common wisdom would then suggest that the system is completely integrable, which is not the case. We discuss the origin of this conundrum and demonstrate how it is resolved when a new definition of independence of QIMs is employed.

     

Masking Quantum Information Encoded in Pure and Mixed States

Masking of quantum information means that information is hidden from a subsystem and spread over a composite system. Modi et al. proved in [Phys. Rev. Lett. 120, 230501 (2018)] that this is true for some restricted sets of nonorthogonal quantum states and it is not possible for arbitrary quantum states. In this paper, we discuss the problem of masking quantum information encoded in pure and mixed states, respectively. Based on an established necessary and sufficient condition for a set of pure states to be masked by an operator, we find that there exists a set of four states that can not be masked, which implies that to mask unknown pure states is impossible. We construct a masker S^? and obtain its maximal maskable set, leading to an affirmative answer to a conjecture proposed in Modi’s paper mentioned above. We also prove that an orthogonal (resp. linearly independent) subset of pure states can be masked by an isometry (resp. injection). Generalizing the case of pure states, we introduce the maskability of a set of mixed states and prove that a commuting subset of mixed states can be masked by an isometry S^◇ while it is impossible to mask all of mixed states by any operator. We also find the maximal maskable sets of mixed states of the isometries S^? and S^◇, respectively.

     

Interface D − complexes in a two-dimensional electron system

The excitation spectrum of a two-dimensional electron system in high-quality AlGaAs/GaAs quantum wells has been studied by Raman scattering. New Raman lines due to the excitation of interface D ^? complexes in which two electrons localized in a quantum well are coupled to a charged impurity at the quantum well interface have been identified. The ground state of the interface D ^? complexes has been found to change in the transverse magnetic field from spin-singlet to spin-triplet, similar to a change in the ground state of the system of two electrons localized in a harmonic potential.

     

Diagrammar in classical scalar field theory

In this paper we analyze perturbatively a g?⁴classical field theory with and without temperature. In order to do that, we make use of a path-integral approach developed some time ago for classical theories. It turns out that the diagrams appearing at the classical level are many more than at the quantum level due to the presence of extra auxiliary fields in the classical formalism. We shall show that a universal supersymmetry present in the classical path-integral mentioned above is responsible for the cancelation of various diagrams. The same supersymmetry allows the introduction of super-fields and super-diagrams which considerably simplify the calculations and make the classical perturbative calculations almost “identical” formally to the quantum ones. Using the super-diagrams technique, we develop the classical perturbation theory up to third order. We conclude the paper with a perturbative check of the fluctuation-dissipation theorem.     

Dynamics of ‘quantumness’ measures in the decohering harmonic oscillator

We studied the behaviour under decoherence of four different measures of the distance between quantum states and classical states for the harmonic oscillator coupled to a linear Markovian bath. Three of these are relative measures, using different definitions of the distance between the given quantum states and the set of all classical states. The fourth measure is an absolute one, the negative volume of the Wigner function of the state. All four measures are found to agree, in general, with each other. When applied to the eigenstates |n〉, all four measures behave non-trivially as a function of time during dynamical decoherence. First, we find that the first set of classical states to which the set of eigenstate evolves is (by all measures used) closest to the initial set. That is, all the states decohere to classicality along the ‘shortest path’. Finding this closest classical set of states helps improve the behaviour of all the relative distance measures. Second, at each point in time before becoming classical, all measures have a state n^? with maximal quantum-classical distance; the value n^? decreases as a function of time. Finally, we explore the dynamics of these non-classicality measures for more general states.