arXiv subjects

Heinz-Jürgen Schmidt

Publications and source records attributed to Heinz-Jürgen Schmidt.

At least 37 records · Page 2Linked to original sources

arXiv2021

The classical spin triangle as an integrable system

The classical spin system consisting of three spins with Heisenberg interaction is an example of a completely integrable mechanical system. In this paper we explicitly calculate its time evolution and the corresponding action-angle variables. This calculation is facilitated by splitting the six degrees of freedom into three internal and three external variables, such that the internal variables evolve autonomously. Their oscillations can be explicitly calculated in terms of the Weierstrass elliptic function. We test our results by means of an example and comparison with direct numerical integration. A couple of special cases is analyzed where the general theory does not apply, including the aperiodic limit case for special initial conditions. The extension to systems with a time-depending magnetic field in a constant direction is straightforward.

math-ph↗

arXiv2021

Stochastic thermodynamics of a finite quantum system coupled to two heat baths

We consider a situation where an $N$-level system (NLS) is coupled successively to two heat baths with different temperatures without being necessarily thermalized and approaches a steady state. For this situation we apply a general Jarzinski-type equation and conclude that heat and entropy is flowing from the hot bath to the cold one. The Clausius relation between increase of entropy and transfer of heat divided by a suitable temperature assumes the form of two inequalities. Our approach is illustrated by an analytical example. For the linear regime, i.~e., for small temperature differences between the two heat baths we derive an expression for the heat conduction coefficient.

cond-mat.stat-mech↗

arXiv2021

Dilation of stochastic matrices by coarse graining

We consider two different ways of representing stochastic matrices by bi-stochastic ones acting on a larger probability space, referred to as ``dilation by uniform coarse graining" and ``environmental dilation". The latter is motivated by analogy to the dilation of operations in quantum theory. Both types of dilation can be viewed as special cases of a general ``dilation by coarse graining". We also discuss the entropy balance and illustrate our results, among others, by an example of a stochastic $4\times 4$-matrix, which serves as a simplified model of the conditional action of Maxwell's demon.

math-ph↗

arXiv2021

Stochastic thermodynamics of a finite quantum system coupled to a heat bath

We consider a situation where an $N$-level system (NLS) is coupled to a heat bath without being necessarily thermalized. For this situation we derive general Jarzinski-type equations and conclude that heat and entropy is flowing from the hot bath to the cold NLS and, vice versa, from the hot NLS to the cold bath. The Clausius relation between increase of entropy and transfer of heat divided by a suitable temperature assumes the form of two inequalities which have already been considered in the literature. Our approach is illustrated by an analytical example.

quant-ph↗

arXiv2021

Conditional Action and imperfect Erasure of Qubits

We consider state changes in quantum theory due to "conditional action" and relate these to the discussion of entropy decrease due to interventions of "intelligent beings" and the principles of Szilard and Landauer/Bennett. The mathematical theory of conditional actions is a special case of the theory of "instruments" which describes changes of state due to general measurements and will therefore be briefly outlined in the present paper. As a detailed example we consider the imperfect erasure of a qubit that can also be viewed as a conditional action and will be realized by the coupling of a spin to another small spin system in its ground state.

quant-ph↗

arXiv2020

Conditional action and quantum versions of Maxwell's demon

We propose a new way of looking at the quantum Maxwell's demon problem in terms of conditional action. A "conditional action" on a system is a unitary time evolution, selected according to the result of a previous measurement, which can reduce the entropy of the system. However, any conditional action can be realized by an (unconditional) unitary time evolution of a larger system and a subsequent Lüders measurement, whereby the entropy of the entire system is either increased or remains constant. We give some examples that illustrate and confirm our proposal, including the erasure of $N$ qubits and the Szilard engine, thus relating the present approach to the Szilard principle and the Landauer principle that have been discussed as possible solutions of the Maxwell's demon problem.

quant-ph↗

arXiv2020

The Rabi problem with elliptic polarization

We consider the solution of the equation of motion of a classical/quantum spin subject to a monochromatical, elliptically polarized external field. The classical Rabi problem can be reduced to third order differential equations with polynomial coefficients and hence solved in terms of power series in close analogy to the confluent Heun equation occurring for linear polarization. Application of Floquet theory yields the physically interesting quasienergy as a function of the parameters of the problem and expressions for the Bloch-Siegert shift of resonance frequencies. Various limit cases cases have been thoroughly investigated.

quant-ph↗

arXiv2020

Environment-controlled Floquet-state paramagnetism

We study the response of ideal spin systems which are interacting with both a strong oscillating magnetic field, and a thermal environment, to a weak probing magnetic field. We demonstrate that even the sign of the resulting mean magnetization depends on the amplitude of the driving field, and that its absolute value can be significantly larger than the equilibrium magnetization in the absence of time-periodic forcing. Since the underlying Floquet-state occupation probabilities are determined by the precise form of the system-bath coupling, future measurements of such effects have the potential to establish a particularly innovative line of research, providing information on nonequilibrium thermodynamics, and giving access to quantities which usually remain hidden when probing equilibrium systems.

quant-ph↗

arXiv2020

Theory of ground states for classical Heisenberg spin systems VI

We formulate part VI of a rigorous theory of ground states for classical, finite, Heisenberg spin systems. After recapitulating the central results of the parts I - V previously published we consider a magnetic field and analytically calculate the susceptibility at the saturation point. To this end we have to distinguish between parabolic and non-parabolic systems, and for the latter ones between two- and three-dimensional ground states. These results are checked for a couple of examples.

cond-mat.stat-mech↗

arXiv2020

Theory of ground states for classical Heisenberg spin systems V

We formulate part V of a rigorous theory of ground states for classical, finite, Heisenberg spin systems. After recapitulating the central results of the parts I - IV previously published we extend the theory to the case where an involutary symmetry is present and the ground states can be distinguished according to their degree of mixing components of different parity. The theory is illustrated by a couple of examples of increasing complexity.

cond-mat.str-el↗

arXiv2020

Periodic thermodynamics of a two spin Rabi model

We consider two $s=1/2$ spins with Heisenberg coupling and a monochromatic, circularly polarized magnetic field acting only onto one of the two spins. This system turns out to be analytically solvable. Also the statistical distribution of the work performed by the driving forces during one period can be obtained in closed form and the Jarzynski equation can be checked. The mean value of this work, viewed as a function of the physical parameters, exhibits features that can be related to some kind of Rabi oscillation. Moreover, when coupled to a heat bath the two spin system will approach a non-equilibrium steady state (NESS) that can be calculated in the golden rule approximation. The occupation probabilities of the NESS are shown not to be of Boltzmann type, with the exception of a single phase with infinite quasitemperature. The parameter space of the two spin Rabi model can be decomposed into eight phase domains such that the NESS probabilities possess discontinuous derivatives at the phase boundaries. The latter property is shown to hold also for more general periodically driven $N$-level systems.

quant-ph↗

arXiv2020

Sequential measurements and entropy

We sketch applications of the so-called J-equation to quantum information theory concerning fundamental properties of the von Neumann entropy. The J-equation has recently be proposed as a sort of progenitor of the various versions of the Jarzynski equation. It has been derived within a general framework of sequential measurements that is slightly generalised here.

quant-ph↗

arXiv2019

Periodic thermodynamics of the Rabi model with circular polarization for arbitrary spin quantum numbers

We consider a spin $s$ subjected to both a static and an orthogonally applied oscillating, circularly polarized magnetic field while being coupled to a heat bath, and analytically determine the quasi\-stationary distribution of its Floquet-state occupation probabilities for arbitrarily strong driving. This distribution is shown to be Boltzmannian with a quasitemperature which is different from the temperature of the bath, and independent of the spin quantum number. We discover a remarkable formal analogy between the quasithermal magnetism of the nonequilibrium steady state of a driven ideal paramagnetic material, and the usual thermal paramagnetism. Nonetheless, the response of such a material to the combined fields is predicted to show several unexpected features, even allowing one to turn a paramagnet into a diamagnet under strong driving. Thus, we argue that experimental measurements of this response may provide key paradigms for the emerging field of periodic thermodynamics.

quant-ph↗

arXiv2019

On the geometry of the classical Rabi problem

We investigate the motion of a classical spin precessing around a periodic magnetic field using Floquet theory as well as elementary differential geometry and considering a couple of examples. Under certain conditions the rôle of spin and magnetic field can be interchanged, leading to the notion of "duality of loops" on the Bloch sphere.

physics.class-ph↗

arXiv2019

A framework for sequential measurements and general Jarzynski equations

We formulate a statistical model of two sequential measurements and prove a so-called J-equation that leads to various diversifications of the well-known Jarzynski equation including the Crooks dissipation theorem. Moreover, the J-equation entails formulations of the Second Law going back to Wolfgang Pauli. We illustrate this by an analytically solvable example of sequential discrete position-momentum measurements accompanied with the increase of Shannon entropy. The standard form of the J-equation extends the domain of applications of the quantum Jarzynski equation in two respects: It includes systems that are initially only in local equilibrium and it extends this equation to the cases where the local equilibrium is described by microcanononical, canonical or grand canonical ensembles. Moreover, the case of a periodically driven quantum system in thermal contact with a heat bath is shown to be covered by the theory presented here. Finally, we shortly consider the generalized Jarzynski equation in classical statistical mechanics.

quant-ph↗

arXiv2019

Reduction and extremality of finite observables

We investigate the theory of finite observables, i.e., resolutions of the finite-dimensional identity by means of positive operators, that have a physical interpretation in terms of measurement schemes. We focus on extremal and rank-one observables and consider various constructions that reduce observables to simpler ones. However, these constructions do not suffice to generate all finite extremal observables, as we show by means of counter-examples.

quant-ph↗

arXiv2018

Floquet theory of the analytical solution of a periodically driven two-level system

We investigate the analytical solution of a two-level system subject to a monochromatical, linearly polarized external field that was published a couple of years ago. In particular, we derive an explicit expression for the quasienergy. Moreover, we calculate the time evolution of a typical two-level system over a full period by evaluating series solutions of the confluent Heun equation. This is possible without invoking the connection problem of this equation since the complete time evolution of the system under consideration can be reduced to that of the first quarter-period.

quant-ph↗

arXiv2018

Snellius meets Schwarzschild - Refraction of brachistochrones and time-like geodesics

The brachistochrone problem can be solved either by variational calculus or by a skillful application of the Snellius' law of refraction. This suggests the question whether also other variational problems can be solved by an analogue of the refraction law. In this paper we investigate the physically interesting case of free fall in General Relativity that can be formulated as a variational problem w. r. t. proper time. We state and discuss the corresponding refraction law for a special class of spacetime metrics including the Schwarzschild metric.

physics.pop-ph↗