arXiv subjects

Holger Cartarius

Publications and source records attributed to Holger Cartarius.

At least 19 recordsLinked to original sources

arXiv2026

Roller coaster dynamics -- from point particles to a continuum model using Lagrange density

Analyzing the motion of a roller coaster allows for an instructive introduction of various theoretical concepts in a concrete and enjoyable context. We start by modeling the roller coaster train as a point particle. We then develop more realistic models for the train and finally we show how to introduce a continuum limit in a simple way. These studies instructively illustrate the relationships between different formalisms (Newtonian mechanics, Lagrangian mechanics of the first and second kind, as well as continuous Lagrangian mechanics using a Lagrangian density). We derive the equations of motion in all considered models and calculate the forces acting on the track and on the passengers. Numerical results are also provided and discussed.

physics.ed-ph↗

arXiv2023

Why and How to Implement Worked Examples in Upper Division Theoretical Physics

Studying worked examples has been shown by extensive research to be an effective method for learning to solve well-structured problems in physics and mathematics. The effectiveness of learning with worked examples has been demonstrated and documented in many research projects. In this work, we propose a new four-step approach for teaching with worked examples that includes writing explanations and finding and correcting errors. This teaching method can even be implemented in courses in which homework performance constitutes part of the grading system. This four-step approach is illustrated in the context of Lagrangian mechanics, which is ideal for the application of worked examples due to its universal approach to solve problems.

physics.ed-ph↗

arXiv2020

Balancing gain and loss in symmetrised multi-well potentials

Balanced gain and loss leads to stationary dynamics in open systems. This occurs naturally in PT-symmetric systems, where the imaginary part of the potential describing gain and loss is perfectly antisymmetric. While this case seems intuitive, stationary dynamics are also possible in asymmetric open systems. Open multi-well quantum systems can possess completely or partly real spectra if their Hamiltonian is symmetrised or semi-symmetrised, respectively. In contrast to similar concepts, symmetrisation allows for the description of physical multi-well potentials with gain and loss. A simple matrix model for the description of two and three-mode systems is used as an example, for which analytical symmetrised solutions are derived. It is explicitly shown how symmetrisation can be used to systematically find two-mode systems with a stable, stationary ground state and why only PT-symmetric two-mode systems can have stationary excited states.

quant-ph↗

arXiv2019

Hybrid Breathers in Nonlinear $\mathcal{PT}$-Symmetric Metamaterials

On a two-dimensional planar parity-time-($\mathcal{PT}$-)symmetric nonlinear magnetic metamaterial, consisting of split-ring dimers with balanced gain and loss, discrete breather solutions can be found. We extend these studies and by numerical calculations reveal the existence of further stable, long-lived oscillations, with certain frequencies, in the breather spectrum. We describe these oscillations in terms of an analytical breather theory, and show that they can be interpreted as superpositions of a breather oscillation and a plane wave. We coin the term `hybrid breather' solutions for these solutions.

nlin.PS↗

arXiv2019

Purity oscillations in coupled Bose-Einstein condensates

It is shown that the distinct oscillations of the purity of the single-particle density matrix for many-body open quantum systems with balanced gain and loss reported by Dast et al. [Phys. Rev. A 93, 033617 (2016)] can also be found in closed quantum systems of which subsystems experience a gain and loss of particles. This is demonstrated with two different lattice setups for cold atoms, viz. a ring of six lattice sites with periodic boundary conditions and a linear chain of four lattice wells. In both cases pronounced purity oscillations are found, and it is shown that they can be made experimentally accessible via the average contrast in interference experiments.

quant-ph↗

arXiv2018

Topological edge states in the Su-Schrieffer-Heeger model subject to balanced particle gain and loss

We investigate the Su-Schrieffer-Heeger model in presence of an injection and removal of particles, introduced via a master equation in Lindblad form. It is shown that the dynamics of the density matrix follows the predictions of calculations in which the gain and loss are modeled by complex $\mathcal{PT}$-symmetric potentials. In particular it is found that there is a clear distinction in the dynamics between the topologically different cases known from the stationary eigenstates.

quant-ph↗

arXiv2018

Topological invariants in dissipative extensions of the Su-Schrieffer-Heeger model

We investigate dissipative extensions of the Su-Schrieffer-Heeger model with regard to different approaches of modeling dissipation. In doing so, we use two distinct frameworks to describe the gain and loss of particles, one uses Lindblad operators within the scope of Lindblad master equations, the other uses complex potentials as an effective description of dissipation. The reservoirs are chosen in such a way that the non-Hermitian complex potentials are $\mathcal{PT}$-symmetric. From the effective theory we extract a state which has similar properties as the non-equilibrium steady state following from Lindblad master equations with respect to lattice site occupation. We find considerable similarities in the spectra of the effective Hamiltonian and the corresponding Liouvillean. Further, we generalize the concept of the Zak phase to the dissipative scenario in terms of the Lindblad description and relate it to the topological phases of the underlying Hermitian Hamiltonian.

quant-ph↗

arXiv2018

Using mixed many-body particle states to generate exact $\mathcal{PT}$-symmetry in a time-dependent four-well system

Bose-Einstein condensates with balanced gain and loss in a double-well potential have been shown to exhibit PT-symmetric states. As proposed by Kreibich et al [Phys. Rev. A 87, 051601(R) (2013)], in the mean-field limit the dynamical behaviour of this system, especially that of the PT-symmetric states, can be simulated by embedding it into a Hermitian four-well system with time-dependent parameters. In this paper we go beyond the mean-field approximation and investigate many-body effects in this system, which are in lowest order described by the single-particle density matrix. The conditions for PT symmetry in the single-particle density matrix cannot be completely fulfilled by using pure initial states. Here we show that it is mathematically possible to achieve exact PT symmetry in the four-well many-body system in the sense of the dynamical behaviour of the single-particle density matrix. In contrast to previous work, for this purpose, we use mixed initial states fulfilling certain constraints and use them to calculate the dynamics.

quant-ph↗

arXiv2018

Computing eigenfunctions and eigenvalues of boundary value problems with the orthogonal spectral renormalization method

The spectral renormalization method was introduced in 2005 as an effective way to compute ground states of nonlinear Schrödinger and Gross-Pitaevskii type equations. In this paper, we introduce an orthogonal spectral renormalization (OSR) method to compute ground and excited states (and their respective eigenvalues) of linear and nonlinear eigenvalue problems. The implementation of the algorithm follows four simple steps: (i) reformulate the underlying eigenvalue problem as a fixed point equation, (ii) introduce a renormalization factor that controls the convergence properties of the iteration, (iii) perform a Gram-Schmidt orthogonalization process in order to prevent the iteration from converging to an unwanted mode; and (iv) compute the solution sought using a fixed-point iteration. The advantages of the OSR scheme over other known methods (such as Newton's and self-consistency) are: (i) it allows the flexibility to choose large varieties of initial guesses without diverging, (ii) easy to implement especially at higher dimensions and (iii) it can easily handle problems with complex and random potentials. The OSR method is implemented on benchmark Hermitian linear and nonlinear eigenvalue problems as well as linear and nonlinear non-Hermitian $\mathcal{PT}$-symmetric models.

physics.comp-ph↗

arXiv2018

Numerical calculation of the complex berry phase in non-Hermitian systems

We numerically investigate topological phases of periodic lattice systems in tight-binding description under the influence of dissipation. The effects of dissipation are effectively described by $\mathcal{PT}$-symmetric potentials. In this framework we develop a general numerical gauge smoothing procedure to calculate complex Berry phases from the biorthogonal basis of the system's non-Hermitian Hamiltonian. Further, we apply this method to a one-dimensional $\mathcal{PT}$-symmetric lattice system and verify our numerical results by an analytical calculation.

quant-ph↗

arXiv2018

Realization of balanced gain and loss in a time-dependent four-mode Bose-Hubbard model

A quantum system exhibiting $\mathcal{PT}$ symmetry is a Bose-Einstein condensate in a double-well potential with balanced particle gain and loss, which is described in the mean-field limit by a Gross-Pitaevskii equation with a complex potential. A possible experimental realization of such a system by embedding it into a Hermitian time-dependent four-mode optical lattice was proposed by Kreibich et al. [Phys. Rev. A 87, 051601(R) (2013)], where additional potential wells act as reservoirs and particle exchange happens via tunneling. Since particle influx and outflux have to be controlled explicitly, a set of conditions on the potential parameters had to be derived. In contrast to previous work, our focus lies on a full many-body description beyond the mean-field approximation using a Bose-Hubbard model with time-dependent potentials. This gives rise to novel quantum effects, such that the differences between mean-field and many-body dynamics are of special interest. We further present stationary analytical solutions for the embedded wells in the mean-field limit, different approaches for the embedding into a many-body system, and a very efficient method for the evaluation of hopping terms to calculate exact Bose-Hubbard dynamics.

quant-ph↗

arXiv2017

Balanced gain and loss in Bose-Einstein condensates without PT symmetry

Balanced gain and loss renders the mean-field description of Bose-Einstein condensates PT symmetric. However, any experimental realization has to deal with unbalancing in the gain and loss contributions breaking the PT symmetry. We will show that such an asymmetry does not necessarily lead to a system without a stable mean-field ground state. Indeed, by exploiting the nonlinear properties of the condensate, a small asymmetry can stabilize the system even further due to a self-regulation of the particle number.

quant-ph↗

arXiv2017

PT-symmetric gain and loss in a rotating Bose-Einstein condensate

PT-symmetric quantum mechanics allows finding stationary states in mean-field systems with balanced gain and loss of particles. In this work we apply this method to rotating Bose-Einstein condensates with contact interaction which are known to support ground states with vortices. Due to the particle exchange with the environment transport phenomena through ultracold gases with vortices can be studied. We find that even strongly interacting rotating systems support stable PT-symmetric ground states, sustaining a current parallel and perpendicular to the vortex cores. The vortices move through the non-uniform particle density and leave or enter the condensate through its borders creating the required net current.

quant-ph↗

arXiv2017

Stationary states in the many-particle description of Bose-Einstein condensates with balanced gain and loss

Bose-Einstein condensates with balanced gain and loss can support stationary states despite the exchange of particles with the environment. In the mean-field approximation this is described by the PT-symmetric Gross-Pitaevskii equation with real eigenvalues. In this work we study the role of stationary states in the appropriate many-particle description. It is shown that without particle interaction there exist two non-oscillating trajectories which can be interpreted as the many-particle equivalent of the stationary PT-symmetric mean-field states. Furthermore the system has a non-equilibrium steady state which acts as an attractor in the oscillating regime. This steady state is a pure condensate for strong gain and loss contributions if the interaction between the particles is sufficiently weak.

quant-ph↗

arXiv2017

Simple models of three coupled $\mathcal{PT}$-symmetric wave guides allowing for third-order exceptional points

We study theoretical models of three coupled wave guides with a $\mathcal{PT}$-symmetric distribution of gain and loss. A realistic matrix model is developed in terms of a three-mode expansion. By comparing with a previously postulated matrix model it is shown how parameter ranges with good prospects of finding a third-order exceptional point (EP3) in an experimentally feasible arrangement of semiconductors can be determined. In addition it is demonstrated that continuous distributions of exceptional points, which render the discovery of the EP3 difficult, are not only a feature of extended wave guides but appear also in an idealised model of infinitely thin guides shaped by delta functions.

physics.optics↗

arXiv2017

$\mathcal{PT}$-symmetric wave guide system with evidence of a third-order exceptional point

An experimental setup of three coupled $\mathcal{PT}$-symmetric wave guides showing the characteristics of a third-order exceptional point (EP3) has been investigated in an idealized model of three delta-functions wave guides in W.~D. Heiss and G.~Wunner, J. Phys. A 49, 495303 (2016). Here we extend these investigations to realistic, extended wave guide systems. We place major focus on the strong parameter sensitivity rendering the discovery of an EP3 a challenging task. We also investigate the vicinity of the EP3 for further branch points of either cubic or square root type behavior.

physics.optics↗

arXiv2017

Relation between $\mathcal{PT}$-symmetry breaking and topologically nontrivial phases in the SSH and Kitaev models

Non-Hermitian systems with $\mathcal{PT}$ symmetry can possess purely real eigenvalue spectra. In this work two one-dimensional systems with two different topological phases, the topological nontrivial Phase (TNP) and the topological trivial phase (TTP) combined with $\mathcal{PT}$-symmetric non-Hermitian potentials are investigated. The models of choice are the Su-Schrieffer-Heeger (SSH) model and the Kitaev chain. The interplay of a spontaneous $\mathcal{PT}$-symmetry breaking due to gain and loss with the topological phase is different for the two models. The SSH model undergoes a $\mathcal{PT}$-symmetry breaking transition in the TNP immediately with the presence of a non-vanishing gain and loss strength $γ$, whereas the TTP exhibits a parameter regime in which a purely real eigenvalue spectrum exists. For the Kitaev chain the $\mathcal{PT}$-symmetry breaking is independent of the topological phase. We show that the topological interesting states -- the edge states -- are the reason for the different behaviors of the two models and that the intrinsic particle-hole symmetry of the edge states in the Kitaev chain is responsible for a conservation of $\mathcal{PT}$ symmetry in the TNP.

quant-ph↗

arXiv2017

Vortices in Bose-Einstein condensates with $\mathcal{PT}$-symmetric gain and loss

We investigate vortex excitations in dilute Bose-Einstein condensates in the presence of complex $\mathcal{PT}$-symmetric potentials. These complex potentials are used to describe a balanced gain and loss of particles and allow for an easier calculation of stationary states in open systems than in a full dynamical calculation including the whole environment. We examine the conditions under which stationary vortex states can exist and consider transitions from vortex to non-vortex states. In addition, we study the influences of $\mathcal{PT}$ symmetry on the dynamics of non-stationary vortex states placed at off-center positions.

quant-ph↗