Classical Separatrix Research Articles

Bifurcation Oscillator as an Advanced Sensor for Quantum State Control.

We study bifurcation behavior of a high-quality (high-Q) Josephson oscillator coupled to a superconducting qubit. It is shown that the probability of capture into the state of dynamic equilibrium is sensitive to qubit states. On this basis we present a new measurement method for the superposition state of a qubit due to its influence on transition probabilities between oscillator levels located in the energy region near the classical separatrix. The quantum-mechanical behavior of a bifurcation oscillator is also studied, which makes it possible to understand the mechanism of "entanglement" of oscillator and qubit states during the measurement process. The optimal parameters of the driving current and the state of the oscillator are found for performing one-qubit gates with the required precision, when the influence on the qubit from measurement back-action is minimal. A measurement protocol for state populations of the qubit entangled with the oscillator is presented.

Floquet systems with Hall effect: Topological properties and phase transitions

We study the quantum topological properties of Floquet (time-periodic) systems exhibiting Hall effects due to perpendicular magnetic and electric fields. The systems are charged particles periodically kicked by a one-dimensional cosine potential in the presence of such fields, where the magnetic field is perpendicular also to the kicking direction. We consider parameter values including small kicking strength, which enables the quantum evolution to be described by effective Floquet Hamiltonians. We also assume the semiclassical regime of a small scaled Planck constant. In the case of an electric field parallel to the kicking direction, one observes, as the electric-field strength is varied, a series of topological phase transitions, causing changes in the Chern numbers of quasienergy (QE) bands. These transitions are due to band degeneracies which, in an initial wide range of electric-field strengths, are shown to occur when the energy of the classical separatrix is approximately equal to the average QE of the degenerating bands. For larger electric-field strengths, the degeneracies reflect changes in the classical phase-space structure. In the case of an electric field not parallel to the kicking direction and satisfying some resonance conditions, we show that the topological properties of the QE bands are characterized by universal (electric-field independent) Chern numbers. These same numbers also characterize the QE band spectrum in the first case, at the end of the basic electric-field interval where the topological phase transitions take place. The two cases above are known to exhibit significantly different dynamical rates, both classical and quantum. This is well reflected by the different topological properties in the two cases.

Connection between a Classical Separatrix and Quantum Inflection Point.

Connection between a Classical Separatrix and Quantum Inflection Point.

Topological phase transitions from Harper to Fibonacci crystals

Topological properties of Harper and generalized Fibonacci chains are studied in crystalline cases, i.e., for rational values of the modulation frequency. The Harper and Fibonacci crystals at fixed frequency are connected by an interpolating one-parameter Hamiltonian. As the parameter is varied, one observes topological phase transitions, i.e., changes in the Chern integers of two bands due to the degeneracy of these bands at some parameter value. For small frequency, corresponding to a semiclassical regime, the degeneracies are shown to occur when the average energy of the two bands is approximately equal to the energy of the classical separatrix. Spectral and topological features of the Fibonacci crystal for small frequency leave a clear imprint on the corresponding Hofstadter butterfly for arbitrary frequency.

Resonances near the Classical Separatrix of a Weakly Deformed Circular Microcavity

We study localized resonance patterns of a weakly deformed circular dielectric microcavity. The dominant pattern among them is detected when the classical motion of the light is on the separatrix of the effective potential. In this case, the topological shape of the pattern is automatically determined by the refractive index of the microcavity n. Also, the pattern can be interpreted as a linear combination of nearly degenerate eigenstates in the circular dielectric microcavity. Applying this analysis to a spiral-shaped microcavity, we generate quasiscarred patterns for n=2 and n=3.

Entanglement and classical instabilities: Fingerprints of electron-hole-to-exciton phase transition in a simple model

We propose a schematic model to study the formation of excitons in bilayer electron systems. The phase transition is signalized both in the quantum and classical versions of the model. In the present contribution we show that not only the quantum ground state but also higher energy states, up to the energy of the corresponding classical separatrix orbit, "sense" the transition. We also show two types of one-to-one correspondences in this system: On the one hand, between the changes in the degree of entanglement for these low-lying quantum states and the changes in the density of energy levels; on the other hand, between the variation in the expected number of excitons for a given quantum state and the behavior of the corresponding classical orbit.

INVESTIGATION OF THE QUANTUM CHAOS OF INTERNAL ROTATIONAL MOTION IN POLYATOMIC MOLECULES

Polyatomic molecules can perform internal rotational motion of two types: torsional oscillation and free rotation of one part of the molecule with respect to the other part. On the phase plane, these two types of motion are separated by the separatrix. Phase trajectories, originated as a result of periodical external force action on the system, have stochastic nature. For quantum consideration, regarding the motion near to the classical separatrix, transition from the pure quantum-mechanical state to the mixed one takes place. Originating at that mixed state, this must be considered as the quantum analog of the classical dynamic stochasticity and is named as the quantum chaos. This work is devoted to the investigation of the quantum chaos manifestation in the polyatomic molecules, which have a property that performs internal rotation. For the molecule of ethane C 2 H 6, the emergence of quantum chaos and possible ways of its experimental observation has been studied. It is shown that radio-frequency field can produce the non-direct transitions between rotational and oscillatory states. These transitions, being the sign of the existence of quantum chaos, are able to change population levels sizeably. Due to this phenomenon, experimental observation of the infrared absorption is possible.

Entanglement and decoherence in a quantum dimer

The dynamical properties of quantum entanglement in an integrable quantum dimer are studied in terms of the reduced-density linear entropy with various coupling parameters and total boson numbers. The characteristic time of decoherence process in the early-time evolution of the linear entropy is obtained, indicating that the characteristic time and the corresponding entropy exhibit a maximum near the position of the corresponding classical separatrix energy.

Entanglement and bifurcation in the integrable dimer

In this Brief Report the properties of both dynamical and static entanglement in the integrable quantum dimer are studied in terms of the reduced-density linear entropy and von Neumann entropy with various coupling parameters, total boson numbers, and initial states. The mean entanglement, which is defined to be averaged over time, is used to describe the influence of the classical separatrix on the behavior of entanglement. It is shown that the mean entanglement exhibits a maximum near the position of the corresponding classical separatrix energy and that the static entanglement of the state with the largest eigenvalue of the quantum spectrum displays a maximum near the bifurcation point. For weak coupling and larger total boson number the maximum entanglement state is exactly at the position of the classical separatrix and bifurcation. In strong coupling all initial states have nearly the same mean entanglement.

Quantum-mechanical research on nonlinear resonance and the problem of quantum chaos.

The quantum-mechanical investigation of nonlinear resonance in terms of approximation to moderate nonlinearity is reduced to the investigation of eigenfunctions and eigenvalues of the Mathieu-Schrodinger equation. The eigenstates of the Mathieu-Schrodinger equation are nondegenerate in a certain area of pumping amplitude values in the neighborhood of the classical separatrix. Outside this area, the system finds itself in a degenerate state for both small and large pumping amplitude values. Degenerate energy terms arise as a result of merging and branching of pairs of nondegenerate energy terms. Equations are obtained for finding the merging points of energy terms. These equations are solved by numerical methods. The main objective of this paper is to establish a quantum analog of the classical stochastic layer formed in the separatrix area. With this end in view, we consider a nonstationary quantum-mechanical problem of perturbation of the state of the Mathieu-Schrodinger equation. It is shown that in passing through the branching point the system may pass from the pure state to the mixed one. At multiple passages through branching points there develops the irreversible process of "creeping" of the system to quantum states. In that case, the observed population of a certain number of levels can be considered, in our opinion, to be a quantum analog of the stochastic layer. The number of populated levels is defined by a perturbation amplitude.

Resonance Zones in Action Space

Abstract The classical and quantum mechanics of isolated, nonlinear resonances in integrable systems with N ≥ 2 degrees of freedom is discussed in terms of geometry in the space of action vari­ ables. Energy surfaces and frequencies are calculated and graphically presented for invariant tori inside and outside the resonance zone. The quantum mechanical eigenvalues, computed in the sem iclassical WKB approximation, show a regular pattern when transformed into the action space of the associated symmetry reduced system: eigenvalues inside the resonance zone are arranged on iV-dimensional cubic lattices, whereas those outside are, in general, non-periodically distributed. However, TV-dimensional triclinic (skewed) lattices exist locally. Both kinds of lattices are joined smoothly across the classical separatrix surface. The statements are illustrated with the help of two and three coupled rotors. The energy-level statistics of this system are found numerically to be in very good agreement with the Poisson distribution, despite of the complex lattice structure. PACS: 03.65.Sq, 05.45.-a

Partitioning dynamic and thermal factors in quantum rate calculations: a coherent state approach

At the heart of the search for a quantum transition state theory is the partitioning of dynamic from thermal factors in the quantum rate expression. We explore the possibility of achieving an approximate partitioning by using the coherent state basis. The coherent states provide a tetradic representation of both the dynamic and thermal factors; the degree to which these factors partition is tied to the degree to which one or both of the tetradics is diagonal, and hence phase space localized. We find that for the dynamical factor the off-diagonal contributions are small, except for matrix elements between coherent states positioned anywhere along the stable branch of the classical separatrix. The thermal factor is nearly diagonal at high temperatures, but has significant off-diagonal contributions at low temperatures. As a result, at high temperatures the thermal factor cuts off long range correlation, leading to the classical limit. At low temperatures, there is a subtle interplay of the thermal and dynamical factors, with the long range off-diagonal portions of the thermal factor combining with the long range off-diagonal portions of the dynamical factor. This phase space picture sheds light on the physical assumptions underlying several commonly applied approximations for calculating thermal reaction rates. In particular, by elucidating the subtlety of the contributions to the low temperature rate it becomes clear why a simple, yet accurate, estimate of the rate in this regime is elusive, if not impossible.

Wavenumber transport: scattering of small-scale internal waves by large-scale wavepackets

In this work, we treat the problem of small-scale, small-amplitude, internal waves interacting nonlinearly with a vigorous, large-scale, undulating shear. The amplitude of the background shear can be arbitrarily large, with a general profile, but our analysis requires that the amplitude vary on a length scale longer than the wavelength of the undulations. In order to illustrate the method, we consider the ray-theoretic model due to Broutman & Young (1986) of high-frequency oceanic internal waves that trap and detrap in a near-inertial wavepacket as a prototype problem. The near-inertial wavepacket tends to transport the high-frequency test waves from larger to smaller wavenumber, and hence to higher frequency. We identify the essential physical mechanisms of this wavenumber transport, and we quantify it. We also show that, for an initial ensemble of test waves with frequencies between the inertial and buoyancy frequencies and in which the number of test waves per frequency interval is proportional to the inverse square of the frequency, a single nonlinear wave–wave interaction fundamentally alters this initial distribution. After the interaction, the slope on a log-log plot is nearly flat, whereas initially it was -2. Our analysis captures this change in slope. The main techniques employed are classical adiabatic invariance theory and adiabatic separatrix crossing theory.

Quantum dynamics near a classical separatrix.

Quantum-mechanical evolution in a slowly varying double-well potential is analyzed in the semiclassical limit to determine the transition probability due to loss of adiabatic invariance of states having energy close to that of a separatrix in the equivalent classical system. Transformation to the rotating-axis basis reduces the problem to the integration of a set of coupled differential equations. In the adiabatic limit the integration is carried out by the method of successive approximations. For adiabatic parameter \ensuremath{\epsilon} of the order of the inverse, 1/N, of the number of quantum states with energy below the separatrix, the spread of the quantum adiabatic invariant is exponentially small in the inverse of the adiabaticity parameter. Surprisingly, the transition to this quantum adiabatic behavior occurs far below the minimum transition frequency of the system.

Semiclassical mechanics of the quadratic Zeeman effect.

A comprehensive approach to semiclassical quantization of the quadratic Zeeman effect in the hydrogen atom (QZE) is presented which utilizes the connection between the Kepler system and the four-dimensional harmonic oscillator. Past attempts to quantize the QZE have encountered problems associated with singularities arising because of a classical separatrix. Here, a uniform semiclassical quantization scheme that passes smoothly through the separatrix is developed using classical perturbation theory. This method accounts for the topologically distinct kinds of dynamics that arise in the QZE, and provides good estimates of tunneling splittings. Extension to the quantization of arbitrarily-high-order classical perturbation expansions is straightforward.

WKB analysis of energy band structure of modulated systems

A discrete WKB approximation is applied to Harper's equation describing the motion of an electron in a square lattice subject to a magnetic field. The treatment is valid when the modulation parameter (proportional to the magnetic field) is a rational number M/N, in the limit of large N. A uniform asymptotic expression is derived for the Floquet wavenumber which characterizes the dynamics and spectral properties. The case of energies near the classical separatrix, crucial to the properties of the spectrum near the transition between localized and extended states is included. An application is a detailed calculation of the total width of the bands in the critical region, for M=1 and 2. The results are consistent with Thouless' conjecture that the bandwidth obeys a universal scaling form, independent of M.

Uniform quantization conditions in the presence of symmetry: The rotational spectrum of SF6.

Uniform semiclassical quantization conditions are obtained for a one-dimensional Hamiltonian possessing octahedral symmetry. The Hamiltonian describes the rotational dynamics of ${\mathrm{SF}}_{6}$, a system in which tunneling plays an important role. Quantization conditions are obtained for each symmetry class. These are shown to agree with previously obtained primitive quantization conditions in the small tunneling limit and to reproduce a characteristic periodicity in the symmetries labels. Quantum and semiclassical eigenvalues are computed numerically. Near the classical separatrix the uniform quantization provides orders of magnitude improvement in accuracy over primitive quantization. Our calculation is based on periodic orbit theory, modified here to include classically forbidden reflections and transmissions, and completes a study undertaken in a previous paper. The methods used may be generalized to other Hamiltonians and symmetry groups.

Clustering of levels in a solvable many-body problem with a degenerate ground state.

The level spacing of the Bassichis-Foldy model of an interacting boson system is studied as a function of the coupling F between the two levels of opposite momenta. For a specific value of this coupling F=${F}_{T}$, the ground state of the system becomes degenerate, and the histogram of the energy differences between neighboring levels shows a sharp peak near the origin. The dependence of the energy levels corresponding to the minimum spacing on F follows the classical separatrix energy for a wide range of the parameters of the system, except when F approaches ${F}_{T}$. Around ${F}_{T}$ the ground-state energy changes very rapidly as a function of F unlike the slow variation of the classical separatrix energy. By studying the classical description of the motion in terms of the action-angle variables, one can show an additional symmetry of the Hamiltonian for F very close to ${F}_{T}$. Thus, in this model the nonlinear classical Hamiltonian can be used as an indicator of the degeneracy of the system, even though the quantum-mechanical system does not have the symmetry of the classical problem.