arXiv subjects

Fritz Haake

Publications and source records attributed to Fritz Haake.

At least 37 records · Page 2Linked to original sources

arXiv2004

Quantum statistics of overlapping modes in open resonators

We study the quantum dynamics of optical fields in weakly confining resonators with overlapping modes. Employing a recently developed quantization scheme involving a discrete set of resonator modes and continua of external modes we derive Langevin equations and a master equation for the resonator modes. Langevin dynamics and the master equation are proved to be equivalent in the Markovian limit. Our open-resonator dynamics may be used as a starting point for a quantum theory of random lasers.

quant-ph↗

arXiv2004

Quantization of Classical Maps with tunable Ruelle-Pollicott Resonances

We investigate the correspondence between the decay of correlation in classical system, governed by Ruelle--Pollicott resonances, and the properties of the corresponding quantum system. For this purpose we construct classical systems with controllable resonances together with their quantum counterpart. As an application of such tailormade resonances we reveal the role of Ruelle--Pollicott resonances for the localization properties of quantum energy eigenstates.

nlin.CD↗

arXiv2003

Universal spectral form factor for chaotic dynamics

We consider the semiclassical limit of the spectral form factor $K(τ)$ of fully chaotic dynamics. Starting from the Gutzwiller type double sum over classical periodic orbits we set out to recover the universal behavior predicted by random-matrix theory, both for dynamics with and without time reversal invariance. For times smaller than half the Heisenberg time $T_H\propto \hbar^{-f+1}$, we extend the previously known $τ$-expansion to include the cubic term. Beyond confirming random-matrix behavior of individual spectra, the virtue of that extension is that the ``diagrammatic rules'' come in sight which determine the families of orbit pairs responsible for all orders of the $τ$-expansion.

nlin.CD↗

arXiv2002

Statistics of Self-Crossings and Avoided Crossings of Periodic Orbits in the Hadamard-Gutzwiller Model

Employing symbolic dynamics for geodesic motion on the tesselated pseudosphere, the so-called Hadamard-Gutzwiller model, we construct extremely long periodic orbits without compromising accuracy. We establish criteria for such long orbits to behave ergodically and to yield reliable statistics for self-crossings and avoided crossings. Self-encounters of periodic orbits are reflected in certain patterns within symbol sequences, and these allow for analytic treatment of the crossing statistics. In particular, the distributions of crossing angles and avoided-crossing widths thus come out as related by analytic continuation. Moreover, the action difference for Sieber-Richter pairs of orbits (one orbit has a self-crossing which the other narrowly avoids and otherwise the orbits look very nearly the same) results to all orders in the crossing angle. These findings may be helpful for extending the work of Sieber and Richter towards a fuller understanding of the classical basis of quantum spectral fluctuations.

nlin.CD↗

arXiv2002

Decoherence scenarios from micro- to macroscopic superpositions

Environment induced decoherence entails the absence of quantum interference phenomena from the macroworld. The loss of coherence between superposed wave packets depends on their separation. The precise temporal course depends on the relative size of the time scales for decoherence and other processes taking place in the open system and its environment. We use the exactly solvable model of an harmonic oscillator coupled to a bath of oscillators to illustrate various decoherence scenarios: These range from exponential golden-rule decay for microscopic superpositions, system-specific decay for larger separations in a crossover regime, and finally universal interaction-dominated decoherence for ever more macroscopic superpositions.

quant-ph↗

arXiv2002

Unversality of decoherence in the macroworld

We consider environment induced decoherence of quantum superpositions to mixtures in the limit in which that process is much faster than any competing one generated by the Hamiltonian $H_{\rm sys}$ of the isolated system. This interaction dominated decoherence limit has previously not found much attention even though it is of importance for the emergence of classical behavior in the macroworld, since it will always be the relevant regime for large enough separations between superposed wave packets. The usual golden-rule treatment then does not apply but we can employ a short-time expansion for the free motion while keeping the interaction $H_{\rm int}$ in full. We thus reveal decoherence as a universal short-time phenomenon largely independent of the character of the system as well as the bath. Simple analytical expressions for the decoherence time scales are obtained in the limit in which decoherence is even faster than any timescale emerging from the reservoir Hamiltonian $H_{\rm res}$.

quant-ph↗

arXiv2001

Level Dynamics and Universality of Spectral Fluctuations

The spectral fluctuations of quantum (or wave) systems with a chaotic classical (or ray) limit are mostly universal and faithful to random-matrix theory. Taking up ideas of Pechukas and Yukawa we show that equilibrium statistical mechanics for the fictitious gas of particles associated with the parametric motion of levels yields spectral fluctuations of the random-matrix type. Previously known clues to that goal are an appropriate equilibrium ensemble and a certain ergodicity of level dynamics. We here complete the reasoning by establishing a power law for the $\hbar$ dependence of the mean parametric separation of avoided level crossings. Due to that law universal spectral fluctuations emerge as average behavior of a family of quantum dynamics drawn from a control parameter interval which becomes vanishingly small in the classical limit; the family thus corresponds to a single classical system. We also argue that classically integrable dynamics cannot produce universal spectral fluctuations since their level dynamics resembles a nearly ideal Pechukas-Yukawa gas.

nlin.CD↗

arXiv2001

Classical versus Quantum Time Evolution of Densities at Limited Phase-Space Resolution

We study the interrelations between the classical (Frobenius-Perron) and the quantum (Husimi) propagator for phase-space (quasi-)probability densities in a Hamiltonian system displaying a mix of regular and chaotic behavior. We focus on common resonances of these operators which we determine by blurring phase-space resolution. We demonstrate that classical and quantum time evolution look alike if observed with a resolution much coarser than a Planck cell and explain how this similarity arises for the propagators as well as their spectra. The indistinguishability of blurred quantum and classical evolution implies that classical resonances can conveniently be determined from quantum mechanics and in turn become effective for decay rates of quantum correlations.

nlin.CD↗

arXiv2001

Resonances of the Frobenius-Perron Operator for a Hamiltonian Map with a Mixed Phase Space

Resonances of the (Frobenius-Perron) evolution operator P for phase-space densities have recently attracted considerable attention, in the context of interrelations between classical and quantum dynamics. We determine these resonances as well as eigenvalues of P for Hamiltonian systems with a mixed phase space, by truncating P to finite size in a Hilbert space of phase-space functions and then diagonalizing. The corresponding eigenfunctions are localized on unstable manifolds of hyperbolic periodic orbits for resonances and on islands of regular motion for eigenvalues. Using information drawn from the eigenfunctions we reproduce the resonances found by diagonalization through a variant of the cycle expansion of periodic-orbit theory and as rates of correlation decay.

nlin.CD↗

arXiv2001

Frobenius-Perron Resonances for Maps with a Mixed Phase Space

Resonances of the time evolution (Frobenius-Perron) operator P for phase space densities have recently been shown to play a key role for the interrelations of classical, semiclassical and quantum dynamics. Efficient methods to determine resonances are thus in demand, in particular for Hamiltonian systems displaying a mix of chaotic and regular behavior. We present a powerful method based on truncating P to a finite matrix which not only allows to identify resonances but also the associated phase space structures. It is demonstrated to work well for a prototypical dynamical system.

nlin.CD↗

arXiv2000

Universality of Decoherence

We consider environment induced decoherence of quantum superpositions to mixtures in the limit in which that process is much faster than any competing one generated by the Hamiltonian $H_{\rm sys}$ of the isolated system. While the golden rule then does not apply we can discard $H_{\rm sys}$. By allowing for simultaneous couplings to different reservoirs, we reveal decoherence as a universal short-time phenomenon independent of the character of the system as well as the bath and of the basis the superimposed states are taken from. We discuss consequences for the classical behavior of the macroworld and quantum measurement: For the decoherence of superpositions of macroscopically distinct states the system Hamiltonian is always negligible.

quant-ph↗

arXiv1999

Fluctuations and Ergodicity of the Form Factor of Quantum Propagators and Random Unitary Matrices

We consider the spectral form factor of random unitary matrices as well as of Floquet matrices of kicked tops. For a typical matrix the time dependence of the form factor looks erratic; only after a local time average over a suitably large time window does a systematic time dependence become manifest. For matrices drawn from the circular unitary ensemble we prove ergodicity: In the limits of large matrix dimension and large time window the local time average has vanishingly small ensemble fluctuations and may be identified with the ensemble average. By numerically diagonalizing Floquet matrices of kicked tops with a globally chaotic classical limit we find the same ergodicity. As a byproduct we find that the traces of random matrices from the circular ensembles behave very much like independent Gaussian random numbers. Again, Floquet matrices of chaotic tops share that universal behavior. It becomes clear that the form factor of chaotic dynamical systems can be fully faithful to random-matrix theory, not only in its locally time-averaged systematic time dependence but also in its fluctuations.

chao-dyn↗

arXiv1999

Slow Decoherence of Superpositions of Macroscopically Distinct States

Linear superpositions of macroscopically distinct quantum states (sometimes also called Schrödinger cat states) are usually almost immediately reduced to a statistical mixture if exposed to the dephasing influence of a dissipative environment. Couplings to the environment with a certain symmetry can lead to slow decoherence, however. We give specific examples of slowly decohering Schrödinger cat states in a realistic quantum optical system and discuss how they might be constructed experimentally.

quant-ph↗

arXiv1999

Long-lived Quantum Coherence between Macroscopically Distinct States in Superradiance

The dephasing influence of a dissipative environment reduces linear superpositions of macroscopically distinct quantum states (sometimes also called Schrödinger cat states) usually almost immediately to a statistical mixture. This process is called decoherence. Couplings to the environment with a certain symmetry can lead to slow decoherence. In this Letter we show that the collective coupling of a large number of two-level atoms to an electromagnetic field mode in a cavity that leads to the phenomena of superradiance has such a symmetry, at least approximately. We construct superpositions of macroscopically distinct quantum states decohering only on a classical time scale and propose an experiment in which the extraordinarily slow decoherence should be observable.

quant-ph↗

arXiv1999

Semiclassics for a Dissipative Quantum Map

We present a semiclassical analysis for a dissipative quantum map with an area-nonpreserving classical limit. We show that in the limit of Planck's constant to 0 the trace of an arbitrary natural power of the propagator is dominated by contributions from periodic orbits of the corresponding classical dissipative motion. We derive trace formulae of the Gutzwiller type for such quantum maps. In comparison to Tabor's formula for area-preserving maps, both classical action and stability prefactor are modified by the dissipation. We evaluate the traces explicitly in the case of a dissipative kicked top with integrable classical motion and find good agreement with numerical results.

chao-dyn↗

arXiv1998

Superradiant Laser: First-Order Phase Transition and Non-stationary Regime

We solve the superradiant laser model in two limiting cases. First the stationary low-pumping regime is considered where a first-order phase transition in the semiclassical solution occurs. This discontinuity is smeared out in the quantum regime. Second, we solve the model in the non-stationary regime where we find a temporally periodic solution. For a certain parameter range well separated pulses may occur.

quant-ph↗

arXiv1998

WKB for a damped spin

The master equation for a damped spin well known from the theory of superradiance, is written as a finite-difference equation and solved by a WKB-like method. The propagator thus obtained looks like the van Vleck propagator of a certain classical Hamiltonian system with one degree of freedom. A new interpretation is provided of the temporal broadening of initially sharp probability distributions as the analogue of the spreading of the quantum mechanical wave packet.

chao-dyn↗

arXiv1998

Semiclassical spin damping: Superradiance revisited

A well known description of superradiance from pointlike collections of many atoms involves the dissipative motion of a large spin. The pertinent ``superradiance master equation'' allows for a formally exact solution which we subject to a semiclassical evaluation. The clue is a saddle-point approximation for an inverse Laplace transform. All previous approximate treatments, disparate as they may appear, are encompassed in our systematic formulation. A byproduct is a hitherto unknown rigorous relation between coherences and probabilities. Our results allow for generalizations to spin dynamics with chaos in the classical limit.

physics.optics↗