Dirk Witthaut - Academia.edu (original) (raw)
Papers by Dirk Witthaut
Physical review. E, 2017
We introduce the concept of network susceptibilities quantifying the response of the collective d... more We introduce the concept of network susceptibilities quantifying the response of the collective dynamics of a network to small parameter changes. We distinguish two types of susceptibilities: vertex susceptibilities and edge susceptibilities, measuring the responses due to changes in the properties of units and their interactions, respectively. We derive explicit forms of network susceptibilities for oscillator networks close to steady states and offer example applications for Kuramoto-type phase-oscillator models, power grid models, and generic flow models. Focusing on the role of the network topology implies that these ideas can be easily generalized to other types of networks, in particular those characterizing flow, transport, or spreading phenomena. The concept of network susceptibilities is broadly applicable and may straightforwardly be transferred to all settings where networks responses of the collective dynamics to topological changes are essential.
Scientific Reports, 2016
Synchronization constitutes one of the most fundamental collective dynamics across networked syst... more Synchronization constitutes one of the most fundamental collective dynamics across networked systems and often underlies their function. Whether a system may synchronize depends on the internal unit dynamics as well as the topology and strength of their interactions. For chaotic units with certain interaction topologies synchronization might be impossible across all interaction strengths, meaning that these networks are non-synchronizable. Here we propose the concept of interaction control, generalizing transient uncoupling, to induce desired collective dynamics in complex networks and apply it to synchronize even such non-synchronizable systems. After highlighting that non-synchronizability prevails for a wide range of networks of arbitrary size, we explain how a simple binary control may localize interactions in state space and thereby synchronize networks. Intriguingly, localizing interactions by a fixed control scheme enables stable synchronization across all connected networks regardless of topological constraints. Interaction control may thus ease the design of desired collective dynamics even without knowledge of the networks' exact interaction topology and consequently have implications for biological and self-organizing technical systems.
The European Physical Journal Special Topics, 2016
ABSTRACT
A useful semiclassical method to calculate eigenfunctions of the Schrodinger equation is the mapp... more A useful semiclassical method to calculate eigenfunctions of the Schrodinger equation is the mapping to a well-known ordinary differ- ential equation, as for example Airy's equation. In this paper we gen- eralize the mapping procedure to the nonlinear Schrodinger equation or Gross-Pitaevskii equation describing the macroscopic wave function of a Bose-Einstein condensate. The nonlinear Schrodinger equation is mapped to the second Painleve equation (PII), which is one of the best-known differential equations with a cubic nonlinearity. A quan- tization condition is derived from the connection formulae of these functions. Comparison with numerically exact results for a harmonic trap demonstrates the benefit of the mapping method. Finally we discuss the influence of a shallow periodic potential on bright soliton solutions by a mapping to a constant potential.
Physical Review Letters, Dec 1, 2008
We discuss the dynamics of a Bose-Einstein condensate in a double-well trap subject to phase nois... more We discuss the dynamics of a Bose-Einstein condensate in a double-well trap subject to phase noise and particle loss. The phase coherence of a weakly interacting condensate as well as the response to an external driving show a pronounced stochastic resonance effect: Both quantities become maximal for a finite value of the dissipation rate matching the intrinsic time scales of the system. Even stronger effects are observed when dissipation acts in concurrence with strong interparticle interactions, restoring the purity of the condensate almost completely and increasing the phase coherence significantly.
J Phys a Math Gen, 2006
A useful semiclassical method to calculate eigenfunctions of the Schrödinger equation is the mapp... more A useful semiclassical method to calculate eigenfunctions of the Schrödinger equation is the mapping to a well-known ordinary differential equation, such as for example Airy's equation. In this paper, we generalize the mapping procedure to the nonlinear Schrödinger equation or Gross-Pitaevskii equation describing the macroscopic wavefunction of a Bose-Einstein condensate. The nonlinear Schrödinger equation is mapped to the second Painlevé equation (PII), which is one of the best-known differential equations with a cubic nonlinearity. A quantization condition is derived from the connection formulae of these functions. Comparison with numerically exact results for a harmonic trap demonstrates the benefit of the mapping method. Finally we discuss the influence of a shallow periodic potential on bright soliton solutions by a mapping to a constant potential.
Starting from a mean-field model of the Bose-Einstein condensate dimer, we reintroduce classicall... more Starting from a mean-field model of the Bose-Einstein condensate dimer, we reintroduce classically forbidden tunneling through a Bohr-Sommerfeld quantization approach. We find closed-form approximations to the tunneling frequency more accurate than those previously obtained using different techniques. We discuss the central role that tunneling in the self-trapped regime plays in a quantitatively accurate model of a dissipative dimer leaking atoms to the environment. Finally, we describe the prospects of experimental observation of tunneling in the self-trapped regime, both with and without dissipation.
Physical review letters, 2016
Link failures repeatedly induce large-scale outages in power grids and other supply networks. Yet... more Link failures repeatedly induce large-scale outages in power grids and other supply networks. Yet, it is still not well understood which links are particularly prone to inducing such outages. Here we analyze how the nature and location of each link impact the network's capability to maintain a stable supply. We propose two criteria to identify critical links on the basis of the topology and the load distribution of the network prior to link failure. They are determined via a link's redundant capacity and a renormalized linear response theory we derive. These criteria outperform the critical link prediction based on local measures such as loads. The results not only further our understanding of the physics of supply networks in general. As both criteria are available before any outage from the state of normal operation, they may also help real-time monitoring of grid operation, employing countermeasures and support network planning and design.
Quantum Information Computation, Jul 1, 2003
We discuss the relation between discrete and continuous linear teleportation, i.e. teleportation ... more We discuss the relation between discrete and continuous linear teleportation, i.e. teleportation schemes that use only linear optical elements and photodetectors. For this the existing qubit protocols are generalized to qudits with a discrete and finite spectrum but with an arbitrary number of states or alternatively to continuous variables. Correspondingly a generalization of linear optical operations and detection is made. It is shown that linear teleportation is only possible in a probabilistic sense. A general expression for the success probability is derived which is shown to depend only on the dimensions of the input and ancilla Hilbert spaces. From this the known results p=1/2p=1/2p=1/2 and p=1p=1p=1 for the discrete and continuous cases can be recovered. We also discuss the probabilistic teleportation scheme of Knill, Laflamme and Milburn and argue that it does not make optimum use of resources.
J Phys a Math Gen, 2005
We study the stationary nonlinear Schrödinger equation, or Gross-Pitaevskii equation, for a singl... more We study the stationary nonlinear Schrödinger equation, or Gross-Pitaevskii equation, for a single delta potential and a delta-shell potential. These model systems allow analytical solutions, and thus provide useful insight into the features of stationary bound, scattering and resonance states of the nonlinear Schrödinger equation. For the single delta potential, the influence of the potential strength and the nonlinearity is studied as well as the transition from bound to scattering states. Furthermore, the properties of resonance states in a repulsive delta-shell potential are discussed.
Annalen der Physik, 2015
ABSTRACT Open many-body quantum systems have recently gained renewed interest in the context of q... more ABSTRACT Open many-body quantum systems have recently gained renewed interest in the context of quantum information science and quantum transport with biological clusters and ultracold atomic gases. A series of results in diverse setups is presented, based on a Master equation approach to describe the dissipative dynamics of ultracold bosons in a one-dimensional lattice. The creation of mesoscopic stable many-body structures in the lattice is predicted and the non-equilibrium transport of neutral atoms in the regime of strong and weak interactions is studied.
New Journal of Physics, 2015
Phys Rev a, Feb 20, 2006
We consider the Landau-Zener problem for a Bose-Einstein condensate in a linearly varying two-lev... more We consider the Landau-Zener problem for a Bose-Einstein condensate in a linearly varying two-level system, for the full many-particle system as well and in the mean-field approximation. The many-particle problem can be solved approximately within an independent crossings approximation, which yields an explicit Landau-Zener formula.
Aps March Meeting Abstracts, Feb 1, 2012
Recent experiments studying a Bose Einstein Condensate (BEC) in a two-mode system, equivalent to ... more Recent experiments studying a Bose Einstein Condensate (BEC) in a two-mode system, equivalent to a ``dimer,'' have shown that many qualitative dynamical features of the BEC can be understood from motions in the underlying classical (two-dimensional) phase space (phi, z). Using a Bose-Hubbard model for the dimer, we focus on quantum deviations from motions in the classical phase space. We introduce a ``quantum'' phase space (QPS), which we define as the minimum condensate fraction c(tau;phi,z) of initial coherent states (phi,z) in the time interval [0,tau]. We find that lines of equal condensate fraction in the QPS do mimic the classical trajectories of constant energy in many respects, such that the QPS clearly reflects Josephson oscillations and self-trapping. However, novel quantum features beyond the classical description appear at finite time tau. These include symmetry breaking and enhanced c(tau; phi, z) near the classical hyperbolic fixed point and along a ridge near the classical separatrix. These features of the QPS can be readily studied in current experiments.
Physical review. E, Statistical, nonlinear, and soft matter physics, 2015
We study the propagation of cascading failures in complex supply networks with a focus on nonloca... more We study the propagation of cascading failures in complex supply networks with a focus on nonlocal effects occurring far away from the initial failure. It is shown that a high clustering and a small average path length of a network generally suppress nonlocal overloads. These properties are typical for many real-world networks, often called small-world networks, such that cascades propagate mostly locally in these networks. Furthermore, we analyze the spatial aspects of countermeasures based on the intentional removal of additional edges. Nonlocal actions are generally required in networks that have a low redundancy and are thus especially vulnerable to cascades.
Physical Review A, 2014
ABSTRACT Starting from a mean-field model of the Bose-Einstein condensate dimer, we reintroduce c... more ABSTRACT Starting from a mean-field model of the Bose-Einstein condensate dimer, we reintroduce classically forbidden tunneling through a Bohr-Sommerfeld quantization approach. We find closed-form approximations to the tunneling frequency more accurate than those previously obtained using different techniques. We discuss the central role that tunneling in the self-trapped regime plays in a quantitatively accurate model of a dissipative dimer leaking atoms to the environment. Finally, we describe the prospects of experimental observation of tunneling in the self-trapped regime, both with and without dissipation.
We analyze the correspondence of many-particle and mean-field dynamics for a Bose-Einstein conden... more We analyze the correspondence of many-particle and mean-field dynamics for a Bose-Einstein condensate in an optical lattice. Representing many-particle quantum states by a classical phase space ensemble instead of one single mean-field trajectory and taking into account the quantization of the density by a modified integer Gross-Pitaevskii equation, it is possible to simulate the superfluid to Mott insulator transition and other phenomena purely classically. This approach can be easily extended to higher particle numbers and multidimensional lattices. Moreover it provides an excellent tool to classify true quantum features and to analyze the mean-field -- many particle correspondence.
Physical Review A
The number-conserving quantum phase space description of the Bose-Hubbard model is discussed for ... more The number-conserving quantum phase space description of the Bose-Hubbard model is discussed for the illustrative case of two and three modes, as well as the generalization of the two-mode case to an open quantum system. The phase-space description based on generalized SU(M) coherent states yields a Liouvillian flow in the macroscopic limit, which can be efficiently simulated using Monte Carlo methods even for large systems. We show that this description clearly goes beyond the common mean-field limit. In particular it resolves well-known problems where the common mean-field approach fails, such as the description of dynamical instabilities and chaotic dynamics. Moreover, it provides a valuable tool for a semiclassical approximation of many interesting quantities, which depend on higher moments of the quantum state and are therefore not accessible within the common approach. As a prominent example, we analyze the depletion and heating of the condensate. A comparison to methods ignor...
New Journal of Physics, 2015
Physical review. E, 2017
We introduce the concept of network susceptibilities quantifying the response of the collective d... more We introduce the concept of network susceptibilities quantifying the response of the collective dynamics of a network to small parameter changes. We distinguish two types of susceptibilities: vertex susceptibilities and edge susceptibilities, measuring the responses due to changes in the properties of units and their interactions, respectively. We derive explicit forms of network susceptibilities for oscillator networks close to steady states and offer example applications for Kuramoto-type phase-oscillator models, power grid models, and generic flow models. Focusing on the role of the network topology implies that these ideas can be easily generalized to other types of networks, in particular those characterizing flow, transport, or spreading phenomena. The concept of network susceptibilities is broadly applicable and may straightforwardly be transferred to all settings where networks responses of the collective dynamics to topological changes are essential.
Scientific Reports, 2016
Synchronization constitutes one of the most fundamental collective dynamics across networked syst... more Synchronization constitutes one of the most fundamental collective dynamics across networked systems and often underlies their function. Whether a system may synchronize depends on the internal unit dynamics as well as the topology and strength of their interactions. For chaotic units with certain interaction topologies synchronization might be impossible across all interaction strengths, meaning that these networks are non-synchronizable. Here we propose the concept of interaction control, generalizing transient uncoupling, to induce desired collective dynamics in complex networks and apply it to synchronize even such non-synchronizable systems. After highlighting that non-synchronizability prevails for a wide range of networks of arbitrary size, we explain how a simple binary control may localize interactions in state space and thereby synchronize networks. Intriguingly, localizing interactions by a fixed control scheme enables stable synchronization across all connected networks regardless of topological constraints. Interaction control may thus ease the design of desired collective dynamics even without knowledge of the networks' exact interaction topology and consequently have implications for biological and self-organizing technical systems.
The European Physical Journal Special Topics, 2016
ABSTRACT
A useful semiclassical method to calculate eigenfunctions of the Schrodinger equation is the mapp... more A useful semiclassical method to calculate eigenfunctions of the Schrodinger equation is the mapping to a well-known ordinary differ- ential equation, as for example Airy's equation. In this paper we gen- eralize the mapping procedure to the nonlinear Schrodinger equation or Gross-Pitaevskii equation describing the macroscopic wave function of a Bose-Einstein condensate. The nonlinear Schrodinger equation is mapped to the second Painleve equation (PII), which is one of the best-known differential equations with a cubic nonlinearity. A quan- tization condition is derived from the connection formulae of these functions. Comparison with numerically exact results for a harmonic trap demonstrates the benefit of the mapping method. Finally we discuss the influence of a shallow periodic potential on bright soliton solutions by a mapping to a constant potential.
Physical Review Letters, Dec 1, 2008
We discuss the dynamics of a Bose-Einstein condensate in a double-well trap subject to phase nois... more We discuss the dynamics of a Bose-Einstein condensate in a double-well trap subject to phase noise and particle loss. The phase coherence of a weakly interacting condensate as well as the response to an external driving show a pronounced stochastic resonance effect: Both quantities become maximal for a finite value of the dissipation rate matching the intrinsic time scales of the system. Even stronger effects are observed when dissipation acts in concurrence with strong interparticle interactions, restoring the purity of the condensate almost completely and increasing the phase coherence significantly.
J Phys a Math Gen, 2006
A useful semiclassical method to calculate eigenfunctions of the Schrödinger equation is the mapp... more A useful semiclassical method to calculate eigenfunctions of the Schrödinger equation is the mapping to a well-known ordinary differential equation, such as for example Airy's equation. In this paper, we generalize the mapping procedure to the nonlinear Schrödinger equation or Gross-Pitaevskii equation describing the macroscopic wavefunction of a Bose-Einstein condensate. The nonlinear Schrödinger equation is mapped to the second Painlevé equation (PII), which is one of the best-known differential equations with a cubic nonlinearity. A quantization condition is derived from the connection formulae of these functions. Comparison with numerically exact results for a harmonic trap demonstrates the benefit of the mapping method. Finally we discuss the influence of a shallow periodic potential on bright soliton solutions by a mapping to a constant potential.
Starting from a mean-field model of the Bose-Einstein condensate dimer, we reintroduce classicall... more Starting from a mean-field model of the Bose-Einstein condensate dimer, we reintroduce classically forbidden tunneling through a Bohr-Sommerfeld quantization approach. We find closed-form approximations to the tunneling frequency more accurate than those previously obtained using different techniques. We discuss the central role that tunneling in the self-trapped regime plays in a quantitatively accurate model of a dissipative dimer leaking atoms to the environment. Finally, we describe the prospects of experimental observation of tunneling in the self-trapped regime, both with and without dissipation.
Physical review letters, 2016
Link failures repeatedly induce large-scale outages in power grids and other supply networks. Yet... more Link failures repeatedly induce large-scale outages in power grids and other supply networks. Yet, it is still not well understood which links are particularly prone to inducing such outages. Here we analyze how the nature and location of each link impact the network's capability to maintain a stable supply. We propose two criteria to identify critical links on the basis of the topology and the load distribution of the network prior to link failure. They are determined via a link's redundant capacity and a renormalized linear response theory we derive. These criteria outperform the critical link prediction based on local measures such as loads. The results not only further our understanding of the physics of supply networks in general. As both criteria are available before any outage from the state of normal operation, they may also help real-time monitoring of grid operation, employing countermeasures and support network planning and design.
Quantum Information Computation, Jul 1, 2003
We discuss the relation between discrete and continuous linear teleportation, i.e. teleportation ... more We discuss the relation between discrete and continuous linear teleportation, i.e. teleportation schemes that use only linear optical elements and photodetectors. For this the existing qubit protocols are generalized to qudits with a discrete and finite spectrum but with an arbitrary number of states or alternatively to continuous variables. Correspondingly a generalization of linear optical operations and detection is made. It is shown that linear teleportation is only possible in a probabilistic sense. A general expression for the success probability is derived which is shown to depend only on the dimensions of the input and ancilla Hilbert spaces. From this the known results p=1/2p=1/2p=1/2 and p=1p=1p=1 for the discrete and continuous cases can be recovered. We also discuss the probabilistic teleportation scheme of Knill, Laflamme and Milburn and argue that it does not make optimum use of resources.
J Phys a Math Gen, 2005
We study the stationary nonlinear Schrödinger equation, or Gross-Pitaevskii equation, for a singl... more We study the stationary nonlinear Schrödinger equation, or Gross-Pitaevskii equation, for a single delta potential and a delta-shell potential. These model systems allow analytical solutions, and thus provide useful insight into the features of stationary bound, scattering and resonance states of the nonlinear Schrödinger equation. For the single delta potential, the influence of the potential strength and the nonlinearity is studied as well as the transition from bound to scattering states. Furthermore, the properties of resonance states in a repulsive delta-shell potential are discussed.
Annalen der Physik, 2015
ABSTRACT Open many-body quantum systems have recently gained renewed interest in the context of q... more ABSTRACT Open many-body quantum systems have recently gained renewed interest in the context of quantum information science and quantum transport with biological clusters and ultracold atomic gases. A series of results in diverse setups is presented, based on a Master equation approach to describe the dissipative dynamics of ultracold bosons in a one-dimensional lattice. The creation of mesoscopic stable many-body structures in the lattice is predicted and the non-equilibrium transport of neutral atoms in the regime of strong and weak interactions is studied.
New Journal of Physics, 2015
Phys Rev a, Feb 20, 2006
We consider the Landau-Zener problem for a Bose-Einstein condensate in a linearly varying two-lev... more We consider the Landau-Zener problem for a Bose-Einstein condensate in a linearly varying two-level system, for the full many-particle system as well and in the mean-field approximation. The many-particle problem can be solved approximately within an independent crossings approximation, which yields an explicit Landau-Zener formula.
Aps March Meeting Abstracts, Feb 1, 2012
Recent experiments studying a Bose Einstein Condensate (BEC) in a two-mode system, equivalent to ... more Recent experiments studying a Bose Einstein Condensate (BEC) in a two-mode system, equivalent to a ``dimer,'' have shown that many qualitative dynamical features of the BEC can be understood from motions in the underlying classical (two-dimensional) phase space (phi, z). Using a Bose-Hubbard model for the dimer, we focus on quantum deviations from motions in the classical phase space. We introduce a ``quantum'' phase space (QPS), which we define as the minimum condensate fraction c(tau;phi,z) of initial coherent states (phi,z) in the time interval [0,tau]. We find that lines of equal condensate fraction in the QPS do mimic the classical trajectories of constant energy in many respects, such that the QPS clearly reflects Josephson oscillations and self-trapping. However, novel quantum features beyond the classical description appear at finite time tau. These include symmetry breaking and enhanced c(tau; phi, z) near the classical hyperbolic fixed point and along a ridge near the classical separatrix. These features of the QPS can be readily studied in current experiments.
Physical review. E, Statistical, nonlinear, and soft matter physics, 2015
We study the propagation of cascading failures in complex supply networks with a focus on nonloca... more We study the propagation of cascading failures in complex supply networks with a focus on nonlocal effects occurring far away from the initial failure. It is shown that a high clustering and a small average path length of a network generally suppress nonlocal overloads. These properties are typical for many real-world networks, often called small-world networks, such that cascades propagate mostly locally in these networks. Furthermore, we analyze the spatial aspects of countermeasures based on the intentional removal of additional edges. Nonlocal actions are generally required in networks that have a low redundancy and are thus especially vulnerable to cascades.
Physical Review A, 2014
ABSTRACT Starting from a mean-field model of the Bose-Einstein condensate dimer, we reintroduce c... more ABSTRACT Starting from a mean-field model of the Bose-Einstein condensate dimer, we reintroduce classically forbidden tunneling through a Bohr-Sommerfeld quantization approach. We find closed-form approximations to the tunneling frequency more accurate than those previously obtained using different techniques. We discuss the central role that tunneling in the self-trapped regime plays in a quantitatively accurate model of a dissipative dimer leaking atoms to the environment. Finally, we describe the prospects of experimental observation of tunneling in the self-trapped regime, both with and without dissipation.
We analyze the correspondence of many-particle and mean-field dynamics for a Bose-Einstein conden... more We analyze the correspondence of many-particle and mean-field dynamics for a Bose-Einstein condensate in an optical lattice. Representing many-particle quantum states by a classical phase space ensemble instead of one single mean-field trajectory and taking into account the quantization of the density by a modified integer Gross-Pitaevskii equation, it is possible to simulate the superfluid to Mott insulator transition and other phenomena purely classically. This approach can be easily extended to higher particle numbers and multidimensional lattices. Moreover it provides an excellent tool to classify true quantum features and to analyze the mean-field -- many particle correspondence.
Physical Review A
The number-conserving quantum phase space description of the Bose-Hubbard model is discussed for ... more The number-conserving quantum phase space description of the Bose-Hubbard model is discussed for the illustrative case of two and three modes, as well as the generalization of the two-mode case to an open quantum system. The phase-space description based on generalized SU(M) coherent states yields a Liouvillian flow in the macroscopic limit, which can be efficiently simulated using Monte Carlo methods even for large systems. We show that this description clearly goes beyond the common mean-field limit. In particular it resolves well-known problems where the common mean-field approach fails, such as the description of dynamical instabilities and chaotic dynamics. Moreover, it provides a valuable tool for a semiclassical approximation of many interesting quantities, which depend on higher moments of the quantum state and are therefore not accessible within the common approach. As a prominent example, we analyze the depletion and heating of the condensate. A comparison to methods ignor...
New Journal of Physics, 2015