Axel U. J. Lode


Department of Physics
University of Basel
Klingelbergstrasse 82
CH-4056 Basel, Switzerland

CV as .pdf

MCTDH-X Homepage:


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Research interests:

Development and Applications of the MCTDH-X package

MCTDH-X is a modern implementation of the MultiConfigurational Time-Dependent Hartree method for indistinguishable particles X . It is a program to solve the time-dependent many-body Schrödinger equation numerically exactly for Hamiltonians with generally time-dependent or time-independent one- and/or two-body potentials for bosons or fermions. A graphical user interface named Guantum facilitates the usage of the MCTDH-X package. For further reading and download, please click here.

Making of the X.

Study of 2D and 3D Interacting Ultracold Many-Body Systems

Since recent developments of especially MCTDH-X made computations in 2 and 3 dimensions feasible, I study bosonic and fermionic systems' properties in the crossover from weak to strong interactions as well as with and without angular momentum. The MCTDH-X idea can be adopted also for the case of optical lattices, i.e., Hubbard Hamiltonians. Contrary to other methods in the field, large 2D and 3D lattice systems can be treated on an accurate beyond-mean-field many-body level.

Tunneling Dynamics in Open Ultracold Many-Body Systems

One of my research interests is the tunneling process in systems of interacting indistinguishable particles which are not completely trapped, i.e., systems in which there is no/less than the particle number of bound states. So far it was found for bosons that the initial coherence will be destroyed throughout the dynamics, that the decay times are determined by the interactions, the particle number, and the potential threshold. The mechanism of the loss of coherence is driven by simultaneously happening single-particle processes. The pace of these processes depends on the chemical potentials of subsystems made of a different number of particles. In all the systems several natural orbitals are needed to qualitatively describe the dynamics. The time-dependent Gross-Pitaevskii approximation fails to cover the dynamics qualitatively and quantitatively. The devised model consideration was recently shown to also apply for fermionic particles with Coulomb interactions.

New Analysis Tools and Measures for Quantum Many-Body Systems and their Dynamics

One very general problem about quantum many-body physics is that even knowing the full solution to a problem might sometimes not be enough: The solution might be so complicated that one needs an appropriate tool to grasp the useful information contained in this solution. I try to tackle this question by developing new tools and measures to analyse and classify the plethora of phenomena in quantum many-body dynamics. Recently implemented and tested measures for quantum dynamics include several genuine many-body entropies, single shot simulations, and high-order correlation functions.

Useful Software

Picture Gallery

Video Gallery

MCTDH-X on Youtube

Phantom vortices playlist

Vortex reconnections playlist

MCTDHB introduction at the Annual Colloquium of the HGS MathComp.

MCTDHB introduction in a group seminar coinciding with the birthday of Lorenz Cederbaum.

MCTDHB introduction in the MCTDH group seminar.

MCTDHB introduction in a talk at the Technion in Haifa.

Movie depicting the first order momentum correlation function of the ejected bosons in the decay by tunneling to open space.

Movie illustrating the time-adaptivity of MCTDHB orbitals in the decay by tunneling to open space.

Movie showing a decomposition of the momentum distribution to a fraction corresponding to a harmonic oscillator and a peak structure for the decay by tunneling to open space.

Movie of the depletion of an expanding BEC of 10000 atoms visualized with the second order correlation function.

Movie of the vortex formation in a stirred 2D BEC.

Talks / Workshops / Conferences



Show all abstracts.

1.  Fragmented Superradiance of a Bose-Einstein Condensate in an Optical Cavity
Axel U. J. Lode and Christoph Bruder.
This paper shows how the interactions between laser-pumped superradiant atoms in an optical cavity lead to the emergence of fragmentation and the breakdown of the Dicke model description of the system.
Phys. Rev. Lett. 118, 013603 (2017); arXiv:1606.06058; .

The Dicke model and the superradiance of two-level systems in a radiation field have many applications. Recently, a Dicke quantum phase transition has been realized with a Bose-Einstein condensate in a cavity. We numerically solve the many-body Schrödinger equation and study correlations in the ground state of interacting bosons in a cavity as a function of the strength of a driving laser. Beyond a critical strength, the bosons occupy multiple modes macroscopically while remaining superradiant. This fragmented superradiance can be detected by analyzing the variance of single-shot measurements.

2.  Phantom vortices: hidden angular momentum in ultracold dilute Bose-Einstein condensates
Storm E. Weiner, Marios C. Tsatsos, Lorenz S. Cederbaum, and Axel U. J. Lode.
This paper discusses how the slow acquisition of angular momentum leads to the fragmentation of two-dimensional BECs and the occurence of phantom vortices, a new many-body mode of quantized vorticity.
Scientific Reports 7, 40122 (2017); arXiv:1409.7670 [cond-mat.quant-gas].

Vortices are essential to angular momentum in quantum systems such as ultracold atomic gases. The existence of quantized vorticity in bosonic systems stimulated the development of the Gross-Pitaevskii mean-field approximation. However, the true dynamics of angular momentum in finite, interacting many-body systems like trapped Bose-Einstein condensates is enriched by the emergence of quantum correlations whose description demands more elaborate methods. Herein we theoretically investigate the full many-body dynamics of the acquisition of angular momentum by a gas of ultracold bosons in two dimensions using a standard rotation procedure. We demonstrate the existence of a novel mode of quantized vorticity, which we term the phantom vortex. Contrary to the conventional mean-field vortex, can be detected as a topological defect of spatial coherence, but not of the density. We describe previously unknown many-body mechanisms of vortex nucleation and show that angular momentum is hidden in phantom vortices modes which so far seem to have evaded experimental detection. This phenomenon is likely important in the formation of the Abrikosov lattice and the onset of turbulence in superfluids.

3.  Dynamics of Hubbard Hamiltonians with the multiconfigurational time-dependent Hartree method for indistinguishable particles
Axel U. J. Lode and Christoph Bruder.
This paper applies the philosophy of the multiconfigurational time-dependent Hartree for indistinguishable particles to solve the time-dependent Schrödinger equation with Hubbard Hamiltonians
Phys. Rev. A 94, 013616 (2016);

We apply the multiconfigurational time-dependent Hartree method for indistinguishable particles (MCTDH-X) to systems of bosons or fermions in lattices described by Hubbard type Hamiltonians with long-range or short-range interparticle interactions. The wavefunction is expanded in a variationally optimized time-dependent many-body basis generated by a set of effective creation operators that are related to the original particle creation operators by a time-dependent unitary transform. We use the time-dependent variational principle for the coefficients of this transform as well as the expansion coefficients of the wavefunction in the time-dependent many-body basis as variational parameters to derive equations of motion. The convergence of MCTDH-X is shown by comparing its results to the exact diagonalization of one-, two-, and three-dimensional lattices filled with bosons with contact interactions. We use MCTDH-X to study the buildup of correlations in the long-time splitting dynamics of a Bose-Einstein condensate loaded into a large two-dimensional lattice subject to a barrier that is ramped up in the center. We find that the system is split into two parts with emergent time-dependent correlations that depend on the ramping time -- for most barrier-raising-times the system becomes two-fold fragmented, but for some of the very fast ramps, the system shows revivals of coherence.

4.  The multiconfigurational time-dependent Hartree method for bosons with internal degrees of freedom: Theory and composite fragmentation of multi-component Bose-Einstein condensates
Axel U. J. Lode
This paper applies the multiconfigurational time-dependent Hartree method for bosons with internal degrees of freedom and describes the composite fragmentation of harmonically trapped bosons as a function of the spatial separation between their two internal states.
Phys. Rev. A 93, 063601 (2016);

In this paper the multiconfigurational time-dependent Hartree for bosons method (MCTDHB) is derived for the case of N identical bosons with internal degrees of freedom. The theory for bosons with internal degrees of freedom constitutes a generalization of the MCTDHB method that substantially enriches the many-body physics that can be described. We demonstrate that the numerically exact solution of the time-dependent many-body Schr\"odinger equation for interacting bosonic particles with internal degrees of freedom is now feasible. We report on the MCTDHB equations of motion for bosons with internal degrees of freedom and their implementation for a general many-body Hamiltonian with one-body and two-body terms that, both, may depend on the internal states of the considered particles. To demonstrate the capabilities of the theory and its software implementation integrated in the MCTDH-X software, we apply MCTDHB to the emergence of fragmentation of parabolically trapped bosons with two internal states: we study the groundstate of N=100 parabolically confined bosons as a function of the splitting between the state-dependent minima of the two parabolic potentials. To quantify the coherence of the system we compute its normalized one-body correlation function. We find that the coherence within each internal state of the atoms is maintained, while it is lost between the different internal states. This is a hallmark of a new kind of fragmentation which is absent in bosons without internal structure. We term the emergent phenomenon "composite fragmentation".

5.  Multiconfigurational time-dependent Hartree method for fermions: Implementation, exactness, and few-fermion tunneling to open space
Elke Fasshauer and Axel U. J. Lode.
This paper introduces a new implementation of MCTDHF, shows its exactness and discusses the physics of charged fermions tunneling through a barrier to open space.
Phys. Rev. A 93, 033635 (2016); arXiv:1510.02984.

We report on an implementation of the multiconfigurational time-dependent Hartree method (MCTDH) for spin-polarized fermions (MCTDHF). Our approach is based on a mapping for operators in Fock space that allows a compact and efficient application of the Hamiltonian and solution of the MCTDHF equations of motion. Our implementation extends, builds on, and exploits the recursive implementation of MCTDH for bosons (r-mctdhb) package. Together with r-mctdhb, the present implementation of MCTDHF forms the mctdh-x package. We benchmark the accuracy of the algorithm with the harmonic interaction model and a time-dependent generalization thereof. These models consider parabolically trapped particles that interact through a harmonic interaction potential. We demonstrate that MCTDHF is capable of solving the time-dependent many-fermion Schrödinger equation to an arbitrary degree of precision and can hence yield numerically exact results even in the case of Hamiltonians with time-dependent one-body and two-body potentials. We study the problem of two initially parabolically confined and charged fermions tunneling through a barrier to open space. We demonstrate the validity of a model proposed previously for the many-body tunneling to open space of bosonic particles with contact interactions [Proc. Natl. Acad. Sci. USA 109, 13521 (2012)]. The many-fermion tunneling can be built up from sequentially happening single-fermion tunneling processes. The characteristic momenta of these processes are determined by the chemical potentials of trapped subsystems of smaller particle numbers: The escaped fermions convert the different chemical potentials into kinetic energy. Using the two-body correlation function, we present a detailed picture of the sequentiality of the process and are able to tell tunneling from over-the-barrier escape.

6.  Many-body entropies, correlations, and emergence of statistical relaxation in interaction quench dynamics of ultracold bosons
Axel U. J. Lode, Barnali Chakrabarti, and Venkata K. B. Kota.
This paper introduces several genuine many-body measures for entropy and discusses their time-evolution for interaction quench dynamics of ultracold bosons.
Phys. Rev. A 92, 033622 (2015); arXiv:1501.02611 [cond-mat.quant-gas].

We study the quantum many-body dynamics and the entropy production triggered by an interaction quench in a system of N=10 interacting identical bosons in an external one-dimensional harmonic trap. The multiconfigurational time-dependent Hartree method for bosons (MCTDHB) is used for solving the time-dependent Schrödinger equation at a high level of accuracy. We consider many-body entropy measures such as the Shannon information entropy, number of principal components, and occupation entropy that are computed from the time-dependent many-body basis set used in MCTDHB. These measures quantify relevant physical features such as irregular or chaotic dynamics, statistical relaxation, and thermalization. We monitor the entropy measures as a function of time and assess how they depend on the interaction strength. For larger interaction strength, the many-body information and occupation entropies approach the value predicted for the Gaussian orthogonal ensemble of random matrices. This implies statistical relaxation. The basis states of MCTDHB are explicitly time-dependent and optimized by the variational principle in a way that minimizes the number of significantly contributing ones. It is therefore a nontrivial fact that statistical relaxation prevails in MCTDHB computations. Moreover, we demonstrate a fundamental connection between the production of entropy, the buildup of correlations and loss of coherence in the system. Our findings imply that mean-field approaches such as the time-dependent Gross-Pitaevskii equation cannot capture statistical relaxation and thermalization because they neglect correlations. Since the coherence and correlations are experimentally accessible, their present connection to many-body entropies can be scrutinized to detect statistical relaxation. In this work we use the recent recursive software implementation of the MCTDHB (R-MCTDHB).

7.  Condensate fragmentation as a sensitive measure of the quantum many-body behavior of bosons with long-range interactions
Uwe R. Fischer, Axel U. J. Lode, and Budhaditya Chatterjee.
This manuscript investigates the degree of fragmentation in ultracold bosons in one and two spatial dimensions in dependence on the functional shape of the interparticle interaction potential.
Phys. Rev. A 91, 063621 (2015); arXiv:1502.04889 [cond-mat.quant-gas].

The occupation of more than one single-particle state, and hence the emergence of fragmentation, is a many-body phenomenon occurring for systems of spatially confined strongly interacting bosons. In the present study, we investigate the effect of the range of the interparticle interactions on the fragmentation degree of one- and two-dimensional systems in single wells. We solve the full many-body Schrödinger equation of the system using the recursive implementation of the multiconfigurational time-dependent Hartree for bosons method (R-MCTDHB). The dependence of the degree of fragmentation on dimensionality, particle number, areal or line density, and interaction strength is assessed. For contact interactions, it is found that the fragmentation is essentially density independent in two dimensions. However, fragmentation increasingly depends on density the more long ranged the interactions become. At fixed particle number N, the degree of fragmentation is increasing when the density is decreasing, as expected in one spatial dimension. We demonstrate that this, nontrivially, remains true also for long-range interactions in two spatial dimensions. We, finally, find that fragmentation in a single well is a mesoscopic phenomenon: Within our fully self-consistent approach, the degree of fragmentation, to a good approximation, decreases universally as N^(-0.5) when only N is varied.

8.  Vortex reconnections in anisotropic trapped three-dimensional Bose-Einstein condensates
Tomos Wells, Axel U. J. Lode, Vanderlei S. Bagnato, and Marios C. Tsatsos.
This paper investigates the reconnection dynamics of two coherent perpendicular vortices in a three-dimensional sample of ultracold bosons.
J. Low Temp. Phys 180, 133 (2015); arXiv:1410.2859 [cond-mat.quant-gas].

Quantum vortex reconnections can be considered as a fundamental unit of interaction in complex turbulent quantum gases. Understanding the dynamics of single vortex reconnections as elementary events is an essential precursor to the explanation of the emergent properties of turbulent quantum gases. It is thought that a lone pair of quantum vortex lines will inevitably interact given a sufficiently long time. This paper investigates aspects of reconnections of quantum vortex pairs imprinted in a Bose–Einstein condensate with 101 bosons held in an anisotropic three-dimensional trap using an exact many-body treatment. In particular, the impact of the interaction strength and the trap anisotropy in the reconnection time is studied. It is found that interaction strength has no effect on reconnection time over short time scales and that the trap anisotropy can cause the edge of the condensate to interfere with the reconnection process. It is also found that the initially coherent system fragments very slowly, even for a relatively large interaction strength, and therefore the system tends to stay condensed during the reconnections.

9.  Resonances and Dynamical Fragmentation in a Stirred Bose-Einstein Condensate
Marios C. Tsatsos and Axel U. J. Lode.
This paper investigates the resonant behavior of a two-dimensional stirred sample of ultracold bosons and establishes a connection of fragmentation and vortex nucleation in the system.
J. Low Temp. Phys. 181, 171 (2015); arXiv:1410.0414 [cond-mat.quant-gas].

Superfluids are distinguished from ordinary fluids by the quantized manner in which the rotation is manifested in them. Precisely, quantized vortices are known to appear in the bulk of a superfluid subject to external rotation. In this work we study a trapped ultracold Bose gas of N=101 atoms interacting with finite-range potential in two spatial dimensions that is stirred by a rotating beam. We use the multiconfigurational Hartree method for bosons, which goes beyond the mainstream mean-field theory, to calculate the dynamics of the gas in real time. As the gas is rotated, the wavefunction of the system changes symmetry and topology. We see a series of resonances, i.e., peaks in the total energy, as the stirring frequency is increased. Fragmentation and a change of the symmetry of the density of the gas accompany the appearance of these resonances. We conclude that fragmentation of the gas appears hand-in-hand with resonant absorption of energy and angular momentum from the external agent of rotation.

10.  Breaking the resilience of a two-dimensional Bose-Einstein condensate to fragmentation
Shachar Klaiman, Axel U. J. Lode, Alexej I. Streltsov, Lorenz S. Cederbaum, and Ofir E. Alon.
This paper discusses how to obtain a fragmented BEC by dynamically transforming radially symmetric single well to double well potentials.
Phys. Rev. A 90, 043620 (2014); arXiv:1409.0323 [cond-mat.quant-gas].

A two-dimensional Bose-Einstein condensate (BEC) split by a radial potential barrier is investigated. We determine on an accurate many-body level the system's ground-state phase diagram as well as a time-dependent phase diagram of the splitting process. Whereas the ground state is condensed for a wide range of parameters, the time-dependent splitting process leads to substantial fragmentation. We demonstrate the dynamical fragmentation of a BEC despite its ground state being condensed. The results are analyzed using a mean-field model and suggest that a large manifold of low-lying fragmented excited states can significantly impact the dynamics of trapped two-dimensional BECs.

11.  Controlling the Velocities and Number of Emitted Particles in the Tunneling to Open Space Dynamics
Axel U. J. Lode, Shachar Klaiman, Ofir E. Alon, Alexej I. Streltsov, and Lorenz S. Cederbaum.
This paper discusses how many-body tunneling to open space can be controlled by manipulating the potential threshold and the interparticle repulsion.
Phys. Rev. A 89, 053620 (2014); arXiv:1309.4253 [quant-ph].

A scheme to control the many-boson tunneling process to open space is derived and demonstrated. The number of ejected particles and their velocities can be controlled by two parameters, the threshold of the potential and the interparticle interaction. Since these parameters are fully under experimental control, this is also the case for the number of ejected particles and their emission spectrum. The process of tunneling to open space can hence be used, for example, for the quantum simulation of complicated tunneling ionization processes and atom lasers. To understand the many-body tunneling process, a generalization of the model introduced in [Proc. Natl. Acad. Sci. USA, 109, 13521 (2012)] for tunneling in the absence of a threshold is put forward and proven to apply for systems with a non-zero threshold value. It is demonstrated that the model is applicable for general interparticle interaction strengths, particle numbers and threshold values. The model constructs the many-body process from single-particle emission processes. The rates and emission momenta of the single-particle processes are determined by the chemical potentials and energy differences to the threshold value of the potential for systems with different particle numbers. The chemical potentials and these energy differences depend on the interparticle interaction. Both the number of confined particles and their rate of emission thus allow for a control by the manipulation of the interparticle interaction and the threshold. Numerically exact results for two, three and one hundred bosons are shown and discussed. The devised control scheme for the many-body tunneling process performs very well for the dynamics of the momentum density, the correlations, the coherence and of the final state, i.e., the number of particles that remain confined in the potential.

12.  How an interacting many-body system tunnels through a potential barrier to open space
Axel U. J. Lode, Alexej I. Streltsov, Kaspar Sakmann, Ofir E. Alon, and Lorenz S. Cederbaum.
This manuscript explains in detail how the tunneling of an initially coherent interacting ultracold atomic sample through a potential barrier to open space works.
Proc. Natl. Acad. Sci. USA, 109, 13521 (2012); arXiv:1202.3447 [cond-mat.quant-gas].

The tunneling process in a many-body system is a phenomenon which lies at the very heart of quantum mechanics. It appears in nature in the form of α-decay, fusion and fission in nuclear physics, and photoassociation and photodissociation in biology and chemistry. A detailed theoretical description of the decay process in these systems is a very cumbersome problem, either because of very complicated or even unknown interparticle interactions or due to a large number of constituent particles. In this work, we theoretically study the phenomenon of quantum many-body tunneling in a transparent and controllable physical system, an ultracold atomic gas. We analyze a full, numerically exact many-body solution of the Schrödinger equation of a one-dimensional system with repulsive interactions tunneling to open space. We show how the emitted particles dissociate or fragment from the trapped and coherent source of bosons: The overall many-particle decay process is a quantum interference of single-particle tunneling processes emerging from sources with different particle numbers taking place simultaneously. The close relation to atom lasers and ionization processes allows us to unveil the great relevance of many-body correlations between the emitted and trapped fractions of the wave function in the respective processes.

13.  Numerically exact quantum dynamics of bosons with time-dependent interactions of harmonic type
Axel U. J. Lode, Kaspar Sakmann, Ofir E. Alon, Lorenz S. Cederbaum, and Alexej I. Streltsov.
This manuscript assesses the capability of MCTDHB to provide numerically exact solutions of the full time-dependent many-boson Schrödinger equation
Phys. Rev. A 86, 063606 (2012); arXiv:1207.5128 [cond-mat.quant-gas].

The exactly solvable quantum many-particle model with harmonic one- and two-particle interaction terms is extended to include time dependency. We show that when the external trap potential and interparticle interaction have a time dependency, the numerically exact solutions of the corresponding time-dependent many-boson Schrödinger equation are still available. We use these exact solutions to benchmark the recently developed multiconfigurational time-dependent Hartree method for bosons (MCTDHB) [ Phys. Rev. Lett. 99 030402 (2007); Phys. Rev. A 77 033613 (2008)]. In particular, we benchmark the MCTDHB method for (i) the ground state; (ii) the breathing many-body dynamics activated by a quench scenario where the interparticle interaction strength is suddenly turned on to a finite value; (iii) the nonequilibrium dynamic for driven scenarios where both the trap- and interparticle-interaction potentials are time-dependent. Excellent convergence of the ground state and dynamics is demonstrated. The great relevance of the self-consistency and time adaptivity, which are the intrinsic features of the MCTDHB method, is demonstrated by contrasting the MCTDHB predictions and those obtained within the standard full configuration interaction method spanning a Fock space of the same size, but utilizing as one-particle basis set the fixed-shape eigenstates of the one-particle potential. Connections of the model's results to ultracold Bose-Einstein condensed systems are addressed.

14.  Numerically-Exact Schrödinger Dynamics of Closed and Open Many- Boson Systems with the MCTDHB Package
Axel U. J. Lode, Kaspar Sakmann, Rostislav A. Doganov, Julian Grond, Ofir E. Alon, Alexej I. Streltsov, and Lorenz S. Cederbaum.
This review article discusses the capabilities, various applications and the parallelization of the MCTDHB program package.
Springer, High Performance Computing in Science and Engineering '13, pp 81-92, Nagel, Wolfgang E.; Körner, Dietmar H.; Resch, Michael M.

This review article discusses the capabilities, various applications and the parallelization of the MCTDHB program package.

15.  Thesis: Tunneling Dynamics in Open Ultracold Bosonic Systems
Axel U. J. Lode
My thesis discusses the tunneling dynamics in open ultracold bosonic systems.
Link to Springer Theses website; Link.

This thesis explores the quantum many-body tunneling dynamics of open ultracold bosonic systems with the recently developed multiconfigurational time-dependent Hartree for bosons (MCTDHB) method. The capabilities of MCTDHB to provide solutions to the full time-dependent many-body problem are assessed in a benchmark using the analytically solvable harmonic interaction Hamiltonian and a generalization of it with time-dependent both one- and two-body potentials. In a comparison with numerically exact MCTDHB results, it is shown that e.g. lattice methods fail qualitatively to describe the tunneling dynamics. A model assembling the many-body physics of the process from basic simultaneously happening single-particle processes is derived and verified with a numerically exact MCTDHB description. The generality of the model is demonstrated even for strong interactions and large particle numbers. The ejection of the bosons from the source occurs with characteristic velocities. These velocities are defined by the chemical potentials of systems with different particle numbers which are converted to kinetic energy. The tunneling process is accompanied by fragmentation: the ejected bosons lose their coherence with the source and among each other. It is shown that the various aspects of the tunneling dynamics’ can be controlled well with the interaction and the potential threshold.

16.  Exact decay and tunnelling dynamics of interacting few-boson systems
Axel U.J. Lode, Alexej I. Streltsov, Ofir E. Alon, Hans-Dieter Meyer, and Lorenz S. Cederbaum.
This paper discusses the decay of a few-boson system modeled described numerically exactly with the multiconfigurational time-dependent Hartree method and absorbing boundary conditions.
J. Phys. B: At. Mol. Opt. Phys. 42, 044018 (2009)

The decay and tunnelling dynamics of repulsive few-boson systems through a one-dimensional potential barrier is studied from first principles. To this end, we solve the numerically exact time-dependent few-boson Schrödinger equation by utilizing the successful multiconfiguration time-dependent Hartree method. Benchmark results for a wide range of interactions are reported. Deviations from the time-dependent Gross–Pitaevskii approach are identified. Counterintuitively, the mean-field approach can overestimate the tunnelling times even for relatively weakly-interacting few-boson systems. Implications are discussed.

17.  Wave chaos as signature for depletion of a Bose-Einstein condensate
Iva Březinova, Axel U. J. Lode, Alexej I. Streltsov, Ofir E. Alon, Lorenz S. Cederbaum, and Joachim Burgdörfer.
This paper shows that the occurence of chaos in the Gross-Pitevskii equation is a sign of the occurence of depletion or fragmentation on the many-body level.
Phys. Rev. A 86, 013630 (2012); arXiv:1202.5869 [cond-mat.quant-gas].

We study the expansion of repulsively interacting Bose-Einstein condensates (BECs) in shallow one-dimensional potentials. We show for these systems that the onset of wave chaos in the Gross-Pitaevskii equation (GPE), that is, the onset of exponential separation in Hilbert space of two nearby condensate wave functions, can be used as an indication for the onset of depletion of the BEC and the occupation of excited modes within a many-body description. Comparison between the multiconfigurational time-dependent Hartree for bosons method and the GPE reveals a close correspondence between the many-body effect of depletion and the mean-field effect of wave chaos for a wide range of single-particle external potentials. In the regime of wave chaos the GPE fails to account for the fine-scale quantum fluctuations because many-body effects beyond the validity of the GPE are non-negligible. Surprisingly, despite the failure of the GPE to account for the depletion, coarse-grained expectation values of the single-particle density such as the overall width of the atomic cloud agree very well with the many-body simulations. The time-dependent depletion of the condensate could be investigated experimentally, for example, via decay of coherence of the expanding atom cloud.

18.  Elastic scattering of a Bose-Einstein condensate at a potential landscape
Iva Březinova, Axel U. J. Lode, Alexej I. Streltsov, Lorenz S. Cederbaum, Ofir E. Alon, Lee A. Collins, Barry I. Schneider, and Joachim Burgdörfer.
This paper demonstrates the importance of depletion and fragmentation in scattering a Bose-Einstein condensate from a shallow optical or disorder potential.
J. Phys. Conf. Ser. 488, 012032 (2014); arXiv:1310.0622 [cond-mat.quant-gas].

We investigate the elastic scattering of Bose-Einstein condensates at shallow periodic and disorder potentials. We show that the collective scattering of the macroscopic quantum object couples to internal degrees of freedom of the Bose-Einstein condensate such that the Bose-Einstein condensate gets depleted. As a precursor for the excitation of the Bose-Einstein condensate we observe wave chaos within a mean-field theory.

19.  Excitation spectra of many-body systems by linear response: General theory and applications to trapped condensates
Julian Grond, Alexej I. Streltsov, Axel U. J. Lode, Kaspar Sakmann, Lorenz S. Cederbaum, and Ofir E. Alon.
This paper introduces and tests a linear response theory on top of the multiconfigurational time-dependent Hartree method for bosons.
Phys. Rev. A 88, 023606 (2013); arXiv:1307.1667 [cond-mat.quant-gas].

We derive a general linear-response many-body theory capable of computing excitation spectra of trapped interacting bosonic systems, e.g., depleted and fragmented Bose-Einstein condensates (BECs). To obtain the linear-response equations we linearize the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method, which provides a self-consistent description of many-boson systems in terms of orbitals and a state vector (configurations), and is in principle numerically-exact. The derived linear-response many-body theory, which we term LR-MCTDHB, is applicable to systems with interaction potentials of general form. From the numerical implementation of the LR-MCTDHB equations and solution of the underlying eigenvalue problem, we obtain excitations beyond available theories of excitation spectra, such as the Bogoliubov-de Gennes (BdG) equations. The derived theory is first applied to study BECs in a one-dimensional harmonic potential. The LR-MCTDHB method contains the BdG excitations and, also, predicts a plethora of additional many-body excitations which are out of the realm of standard linear response. In particular, our theory describes the exact energy of the higher harmonic of the first (dipole) excitation not contained in the BdG theory. We next study a BEC in a very shallow one-dimensional double-well potential. We find with LR-MCTDHB low-lying excitations which are not accounted for by BdG, even though the BEC has only little fragmentation and, hence, the BdG theory is expected to be valid. The convergence of the LR-MCTDHB theory is assessed by systematically comparing the excitation spectra computed at several different levels of theory.

20.  Recursive formulation of the multiconfigurational time-dependent Hartree method for fermions, bosons and mixtures thereof in terms of one-body density operators
Ofir E. Alon, Alexej I. Streltsov, Kaspar Sakmann, Axel U. J. Lode, Julian Grond, and Lorenz S. Cederbaum.
This paper builds recursively from one-body operators the equations of motion of the multiconfigurational time-dependent Hartree method for three (different) kinds of indistinguishable particles.
Chemical Physics Volume 401, Pages 2-14 (2012); arXiv:1109.4429 [quant-ph].

The multiconfigurational time-dependent Hartree method (MCTDH) [H.-D. Meyer, U. Manthe, L.S. Cederbaum, Chem. Phys. Lett. 165, 73 (1990); U. Manthe, H.-D. Meyer, L.S. Cederbaum, J. Chem. Phys. 97, 3199 (1992)] is celebrating nowadays entering its third decade of tackling numerically-exactly a broad range of correlated multi-dimensional non-equilibrium quantum dynamical systems. Taking in recent years particles’ statistics explicitly into account, within the MCTDH for fermions (MCTDHF) and for bosons (MCTDHB), has opened up further opportunities to treat larger systems of interacting identical particles, primarily in laser-atom and cold-atom physics. With the increase of experimental capabilities to simultaneously trap mixtures of two, three, and possibly even multiple kinds of interacting composite identical particles together, we set up the stage in the present work and specify the MCTDH method for such cases. Explicitly, the MCTDH method for systems with three kinds of identical particles interacting via all combinations of two- and three-body forces is presented, and the resulting equations-of-motion are briefly discussed. All four possible mixtures (Fermi–Fermi–Fermi, Bose–Fermi–Fermi, Bose–Bose–Fermi and Bose–Bose–Bose) are presented in a unified manner. Particular attention is paid to represent the coefficients’ part of the equations-of-motion in a compact recursive form in terms of one-body density operators only. The recursion utilizes the recently proposed Combinadic-based mapping for fermionic and bosonic operators in Fock space [A.I. Streltsov, O.E. Alon, L.S. Cederbaum, Phys. Rev. A 81, 022124 (2010)], successfully applied and implemented within MCTDHB. Our work sheds new light on the representation of the coefficients’ part in MCTDHF and MCTDHB without resorting to the matrix elements of the many-body Hamiltonian with respect to the time-dependent configurations. It suggests a recipe for efficient implementation of the schemes derived here for mixtures which is suitable for parallelization.

21.  What to do with targeted IL-2.
H. N. Lode, R. Xiang, P. Perri, U. Pertl, A. Lode, S. D. Gillies, and R. A. Reisfeld.
This article assesses the usage of IL-2 immunocytokines for an immunotherapy of various sorts of cancer.
Drugs Today, 36(5): 321 (2000)

A common strategy in immunotherapy of cancer is the induction of an increased immunogenicity of syngeneic malignancies. A novel approach to achieve this goal is the targeting of cytokines into the tumor microenvironment with antibody-cytokine fusion proteins, called immunocytokines. This report summarizes therapeutic efficacy and immune mechanisms involved in targeting IL-2 to syngeneic tumors and describes their extended use as a synergistic treatment modality for cancer vaccines and antiangiogenesis. Treatment of established melanoma and colon carcinoma metastases with IL-2 immunocytokines resulted in eradication of disease, followed by a vaccination effect protecting mice from lethal challenges with wild-type tumor cells. In a syngeneic neuroblastoma model, targeted IL-2 elicited effective antitumor responses mediated by NK cells in the absence of a T-cell memory. Interestingly, targeted IL-2 was effective in amplification of memory immune responses previously induced by cancer vaccines. Furthermore, a synergistic effect achieved by combining targeted IL-2-immunotherapy with an antiangiogenic inhibitor of integrin alphavbeta3 extends the potential of this immunotherapeutic strategy in combination with antiangiogenesis as demonstrated in three syngeneic tumor models. Based on these findings, targeted IL-2 may provide an effective tool for the adjuvant treatment of cancer either applied as a single strategy or in combination with cancer vaccines and antiangiogenic strategies.