arXiv subjects

Roman Schubert

Publications and source records attributed to Roman Schubert.

At least 19 recordsLinked to original sources

arXiv2023

Decoherence Time Scales and the H\"ormander condition

We consider an open quantum system described by the GKLS equation and we are interested in the onset of decoherence. We are in particulary interested in situations where only some degrees of freedom of the system are coupled to the environment, and we want to understand if, and how fast, the noise travels through the system and eventually affects all degrees of freedom. We find that this can be understood in terms of the H\"ormander condition, a condition on the commutators of the Hamiltonian vectorfields of the Lindblad operators and the internal Hamiltonian, which is a condition for hypoellipticity known from the theory of PDE's. We show that for Gaussian quantum channels this condition leads to a delay in the onset of decoherence and can as well be used to detect decoherence free subsystems.

quant-ph↗

arXiv2023

Beyond Gaussian Quantum Channels: A model case

Gaussian quantum channels are well understood and have many applications, e.g., in Quantum Information Theory and in Quantum Optics. For more general quantum channels one can in general use semiclassical approximations or perturbation theory, but it is not easy to judge the accuracy of such methods. We study a relatively simple model case, where the quantum channel is generated by a Lindblad equation where one of the Lindblad operators is a multiple of the internal Hamiltonian, and therefore the channel is not Gaussian. For this model we can compute the characteristic function of the action of the channel on a Gaussian state explicitly and we can as well derive a representation of the propagator in an integral form. This allows us to compare the exact results with semiclassical approximations and perturbation theory and evaluate their accuracy. We finally apply these results to the study of the evolution of the von Neumann entropy of a state.

quant-ph↗

arXiv2022

Quantum-jump vs stochastic Schr\"{o}dinger dynamics for Gaussian states with quadratic Hamiltonians and linear Lindbladians

The dynamics of Gaussian states for open quantum systems described by Lindblad equations can be solved analytically for systems with quadratic Hamiltonians and linear Lindbladians, showing the familiar phenomena of dissipation and decoherence. It is well known that the Lindblad dynamics can be expressed as an ensemble average over stochastic pure-state dynamics, which can be interpreted as individual experimental implementations, where the form of the stochastic dynamics depends on the measurement setup. Here we consider quantum-jump and stochastic Schr\"odinger dynamics for initially Gaussian states. While both unravellings converge to the same Lindblad dynamics when averaged, the individual dynamics can differ qualitatively. For the stochastic Schr\"odinger equation, Gaussian states remain Gaussian during the evolution, with stochastic differential equations governing the evolution of the phase-space centre and a deterministic evolution of the covariance matrix. In contrast to this, individual pure-state dynamics arising from the quantum-jump evolution do not remain Gaussian in general. Applying results developed in the non-Hermitian context for Hagedorn wavepackets, we formulate a method to generate quantum-jump trajectories that is described entirely in terms of the evolution of an underlying Gaussian state. To illustrate the behaviours of the different unravellings in comparison to the Lindblad dynamics, we consider two examples in detail, which can be largely treated analytically, a harmonic oscillator subject to position measurement and a damped harmonic oscillator. In both cases, we highlight the differences as well as the similarities of the stochastic Schr\"odinger and the quantum-jump dynamics.

quant-ph↗

arXiv2016

Propagation of Gaussian beams in the presence of gain and loss

We consider the propagation of Gaussian beams in a waveguide with gain and loss in the paraxial approximation governed by the Schrödinger equation. We derive equations of motion for the beam in the semiclassical limit that are valid when the waveguide profile is locally well approximated by quadratic functions. For Hermitian systems, without any loss or gain, these dynamics are given by Hamilton's equations for the center of the beam and its conjugate momentum. Adding gain and/or loss to the waveguide introduces a non-Hermitian component, causing the width of the Gaussian beam to play an important role in its propagation. Here we show how the width affects the motion of the beam and how this may be used to filter Gaussian beams located at the same initial position based on their width.

quant-ph↗

arXiv2015

Non-Hermitian propagation of Hagedorn wavepackets

We investigate the time evolution of Hagedorn wavepackets by non-Hermitian quadratic Hamiltonians. We state a direct connection between coherent states and Lagrangian frames. For the time evolution a multivariate polynomial recursion is derived that describes the activation of lower lying excited states, a phenomenon unprecedented for Hermitian propagation. Finally we apply the propagation of excited states to the Davies--Swanson oscillator.

math-ph↗

arXiv2014

Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator

The non-Hermitian quadratic oscillator studied by Swanson is one of the popular $PT$-symmetric model systems. Here a full classical description of its dynamics is derived using recently developed metriplectic flow equations, which combine the classical symplectic flow for Hermitian systems with a dissipative metric flow for the anti-Hermitian part. Closed form expressions for the metric and phase-space trajectories are presented which are found to be periodic in time. Since the Hamiltonian is only quadratic the classical dynamics exactly describes the quantum dynamics of Gaussian wave packets. It is shown that the classical metric and trajectories as well as the quantum wave functions can diverge in finite time even though the $PT$-symmetry is unbroken, i.e., the eigenvalues are purely real.

quant-ph↗

arXiv2014

Entropy of eigenfunctions on quantum graphs

We consider families of finite quantum graphs of increasing size and we are interested in how eigenfunctions are distributed over the graph. As a measure for the distribution of an eigenfunction on a graph we introduce the entropy, it has the property that a large value of the entropy of an eigenfunction implies that it cannot be localised on a small set on the graph. We then derive lower bounds for the entropy of eigenfunctions which depend on the topology of the graph and the boundary conditions at the vertices. The optimal bounds are obtained for expanders with large girth, the bounds are similar to the ones obtained by Anantharaman et.al. for eigenfunctions on manifolds of negative curvature, and are based on the entropic uncertainty principle. For comparison we compute as well the average behaviour of entropies on Neumann star graphs, where the entropies are much smaller. Finally we compare our lower bounds with numerical results for regular graphs and star graphs with different boundary conditions.

math-ph↗

arXiv2014

Negative Conditional Entropy of Post-Selected States

We define a quantum entropy conditioned on post-selection which has the von Neumann entropy of pure states as a special case. This conditional entropy can take negative values which is consistent with part of a quantum system containing less information than the whole which can be in a pure state. The definition is based on generalised density operators for postselected ensembles. The corresponding density operators are consistent with the quantum generalisation of classical conditional probabilities following Dirac s formalism of quasiprobability distributions.

quant-ph↗

arXiv2012

Complexified coherent states and quantum evolution with non-Hermitian Hamiltonians

The complex geometry underlying the Schrödinger dynamics of coherent states for non-Hermitian Hamiltonians is investigated. In particular two seemingly contradictory approaches are compared: (i) a complex WKB formalism, for which the centres of coherent states naturally evolve along complex trajectories, which leads to a class of complexified coherent states; (ii) the investigation of the dynamical equations for the real expectation values of position and momentum, for which an Ehrenfest theorem has been derived in a previous paper, yielding real but non-Hamiltonian classical dynamics on phase space for the real centres of coherent states. Both approaches become exact for quadratic Hamiltonians. The apparent contradiction is resolved building on an observation by Huber, Heller and Littlejohn, that complexified coherent states are equivalent if their centres lie on a specific complex Lagrangian manifold. A rich underlying complex symplectic geometry is unravelled. In particular a natural complex structure is identified that defines a projection from complex to real phase space, mapping complexified coherent states to their real equivalents.

math-ph↗

arXiv2011

Semiclassical wave propagation for large times

We study solutions of the time dependent Schrödinger equation on Riemannian manifolds with oscillatory initial conditions given by Lagrangian states. Semiclassical approximations describe these solutions for small h (where h is the semiclassical parameter), but their accuracy for large times is in general only understood up to the Ehrenfest time T ~ ln(1/h), and the most difficult case is the one where the underlying classical system is chaotic. We show that on surfaces of constant negative curvature semiclassical approximations remain accurate for times at least up to T ~ h^(-1/2) in the case that the Lagrangian state is associated with an unstable manifold of the geodesic flow.

math-ph↗

arXiv2011

How do wave packets spread? Time evolution on Ehrenfest time scales

We derive an extension of the standard time dependent WKB theory which can be applied to propagate coherent states and other strongly localised states for long times. It allows in particular to give a uniform description of the transformation from a localised coherent state to a delocalised Lagrangian state which takes place at the Ehrenfest time. The main new ingredient is a metaplectic operator which is used to modify the initial state in a way that standard time dependent WKB can then be applied for the propagation. We give a detailed analysis of the phase space geometry underlying this construction and use this to determine the range of validity of the new method. Several examples are used to illustrate and test the scheme and two applications are discussed: (i) For scattering of a wave packet on a barrier near the critical energy we can derive uniform approximations for the transition from reflection to transmission. (ii) A wave packet propagated along a hyperbolic trajectory becomes a Lagrangian state associated with the unstable manifold at the Ehrenfest time, this is illustrated with the kicked harmonic oscillator.

math-ph↗

arXiv2011

Wave packet evolution in non-Hermitian quantum systems

The quantum evolution of the Wigner function for Gaussian wave packets generated by a non-Hermitian Hamiltonian is investigated. In the semiclassical limit $\hbar\to 0$ this yields the non-Hermitian analog of the Ehrenfest theorem for the dynamics of observable expectation values. The lack of Hermiticity reveals the importance of the complex structure on the classical phase space: The resulting equations of motion are coupled to an equation of motion for the phase space metric---a phenomenon having no analog in Hermitian theories.

quant-ph↗

arXiv2010

The Flux-Flux Correlation Function for Anharmonic Barriers

The flux-flux correlation function formalism is a standard and widely used approach for the computation of reaction rates. In this paper we introduce a method to compute the classical and quantum flux-flux correlation functions for anharmonic barriers essentially analytically through the use of the classical and quantum normal forms. In the quantum case we show that the quantum normal form reduces the computation of the flux-flux correlation function to that of an effective one dimensional anharmonic barrier. The example of the computation of the quantum flux-flux correlation function for a fourth order anharmonic barrier is worked out in detail, and we present an analytical expression for the quantum mechanical microcanonical flux-flux correlation function. We then give a discussion of the short-time and harmonic limits.

physics.chem-ph↗

arXiv2010

Quantum Theory of Reactive Scattering in Phase Space

We review recent results on quantum reactive scattering from a phase space perspective. The approach uses classical and quantum versions of normal form theory and the perspective of dynamical systems theory. Over the past ten years the classical normal form theory has provided a method for realizing the phase space structures that are responsible for determining reactions in high dimensional Hamiltonian systems. This has led to the understanding that a new (to reaction dynamics) type of phase space structure, a {\em normally hyperbolic invariant manifold} (or, NHIM) is the "anchor" on which the phase space structures governing reaction dynamics are built. The quantum normal form theory provides a method for quantizing these phase space structures through the use of the Weyl quantization procedure. We show that this approach provides a solution of the time-independent Schrödinger equation leading to a (local) S-matrix in a neighborhood of the saddle point governing the reaction. It follows easily that the quantization of the directional flux through the dividing surface with the properties noted above is a flux operator that can be expressed in a "closed form". Moreover, from the local S-matrix we easily obtain an expression for the cumulative reactio probability (CRP). Significantly, the expression for the CRP can be evaluated without the need to compute classical trajectories. The quantization of the NHIM is shown to lead to the activated complex, and the lifetimes of quantum states initialized on the NHIM correspond to the Gamov-Siegert resonances. We apply these results to the collinear nitrogen exchange reaction and a three degree-of-freedom system corresponding to an Eckart barrier coupled to two Morse oscillators.

quant-ph↗

arXiv2010

A periodic orbit formula for quantum reactions through transition states

Transition State Theory forms the basis of computing reaction rates in chemical and other systems. Recently it has been shown how transition state theory can rigorously be realized in phase space using an explicit algorithm. The quantization has been demonstrated to lead to an efficient procedure to compute cumulative reaction probabilities and the associated Gamov-Siegert resonances. In this letter these results are used to express the cumulative reaction probability as an absolutely convergent sum over periodic orbits contained in the transition state.

nlin.CD↗

arXiv2009

The Quantum Normal Form Approach to Reactive Scattering: The Cumulative Reaction Probability for Collinear Exchange Reactions

The quantum normal form approach to quantum transition state theory is used to compute the cumulative reaction probability for collinear exchange reactions. It is shown that for heavy atom systems like the nitrogen exchange reaction the quantum normal form approach gives excellent results and has major computational benefits over full reactive scattering approaches. For light atom systems like the hydrogen exchange reaction however the quantum normal approach is shown to give only poor results. This failure is attributed to the importance of tunnelling trajectories in light atom reactions that are not captured by the quantum normal form as indicated by the only very slow convergence of the quantum normal form for such systems.

physics.chem-ph↗

arXiv2007

Wigner's Dynamical Transition State Theory in Phase Space: Classical and Quantum

A quantum version of transition state theory based on a quantum normal form (QNF) expansion about a saddle-centre-...-centre equilibrium point is presented. A general algorithm is provided which allows one to explictly compute QNF to any desired order. This leads to an efficient procedure to compute quantum reaction rates and the associated Gamov-Siegert resonances. In the classical limit the QNF reduces to the classical normal form which leads to the recently developed phase space realisation of Wigner's transition state theory. It is shown that the phase space structures that govern the classical reaction d ynamicsform a skeleton for the quantum scattering and resonance wavefunctions which can also be computed from the QNF. Several examples are worked out explicitly to illustrate the efficiency of the procedure presented.

nlin.CD↗

arXiv2007

Quantisations of piecewise affine maps on the torus and their quantum limits

For general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying classical system is quite difficult to understand. The Wignerfunctions of eigenstates converge weakly to invariant measures of the classical system, the so called quantum limits, and one would like to understand which invariant measures can occur that way, thereby classifying the semiclassical behaviour of eigenfunctions. We introduce a class of maps on the torus for whose quantisations we can understand the set of quantum limits in great detail. In particular we can construct examples of ergodic maps which have singular ergodic measures as quantum limits, and examples of non-ergodic maps where arbitrary convex combinations of absolutely continuous ergodic measures can occur as quantum limits. The maps we quantise are obtained by cutting and stacking.

math.DS↗