Mathematical Horizons for Quantum Physics 2 - IMS

Mathematical Horizons for Quantum Physics 2
(12 Aug - 11 Oct 2013)

~ Abstracts ~

Modeling, analysis and simulation for degenerate dipolar quantum gases
Weizhu Bao, National University of Singapore

In this talk, I will present our recent work on mathematical models, asymptotic analysis and numerical simulation for degenerate dipolar quantum gases. As preparatory steps, I begin with the three-dimensional Gross-Pitaevskii equation with a long-range dipolar interaction potential which is used to model the degenerate dipolar quantum gas and reformulate it as a Gross-Pitaevskii-Poisson type system by decoupling the two-body dipolar interaction potential which is highly singular into short-range (or local) and long-range interactions (or repulsive and attractive interactions). Based on this new mathematical formulation, we prove rigorously existence and uniqueness as well as nonexistence of the ground states, and discuss the existence of global weak solution and finite time blowup of the dynamics in different parameter regimes of dipolar quantum gas. In addition, a backward Euler sine pseudospectral method is presented for computing the ground states and a time-splitting sine pseudospectral method is proposed for computing the dynamics of dipolar BECs. Due to the adoption of new mathematical formulation, our new numerical methods avoid evaluating integrals with high singularity and thus they are more efficient and accurate than those numerical methods currently used in the literatures for solving the problem. In addition, new mathematical formulations in two-dimensions and one dimension for dipolar quantum gas are obtained when the external trapping potential is highly confined in one or two directions. Numerical results are presented to confirm our analytical results and demonstrate the efficiency and accuracy of our numerical methods. Some interesting physical phenomena are discussed too.

[1] W. Bao, Y. Cai and H. Wang, Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates, J. Comput. Phys., 229 (2010), pp. 7874-7892.

[2] Y. Cai, M. Rosenkranz, Z. Lei and W. Bao, Mean-field regime of trapped dipolar Bose-Einstein condensates in one and two dimensions, Phys. Rev. A, 82 (2010), article 043623.

[3] W. Bao, N. Ben Abdallah and Y. Cai, Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement, SIAM J. Math. Anal, 44 (2012), pp. 1713-1741.

Localization in the ground state of mean field models with random potentials
Michael Bishop, University of Arizona, USA

The recent experimental realization of Bose-Einstein condensate and the development of techniques in cold atom experiments provide new methods for investigating quantum phenomena and the models that describe them. The Gross-Pitaevskii mean-field approximation is a popular model for describing these interacting boson systems. In this approximation, each particle in the many-particle state is assumed to have the same one-particle state, substituting a linear operator on a large tensor space with a nonlinear operator on a smaller function space. I will discuss research done with J. Wehr on the ground state of Gross-Pitaevskii mean-field model with local 'soft core' interactions and random potentials. The interplay of interactions and random potentials is unclear: particles localize in systems with random potentials, but repulsive interactions cause states to spread because localization of the entire multi-particle state is energetically expensive. The main result is a criterion for the minimal localization of a mean-field state given its per particle energy and the interaction strength. To help understand this theorem, it will be applied to the model in one discrete dimension with Bernoulli distributed potential in the thermodynamic limit.

Entanglement of multipartite and infinite systems
(ENTANGLEMENT)
Géza Giedke, Max-Planck-Institut für Quantenoptik, Germany

Clearly, entanglement is a central notion in quantum information, and the development of theory on the many particle case remains a challenge. One of the main problems still is to introduce suitable measures for entanglement for mixed states and on multipartite systems. It had already proved useful to study entanglement with the eyes of an operator algbraist so that non-trivial results on completely positive maps found their way into quantum information. The maximal tensor norm, introduced by Rudolph and Arveson as an ideal measure of entanglement, constituted a connection between abstract Banach space theory and quantum information. To find good estimates of this and similar norms for general n-particle states remains a major challenge. Restriction to particular classes, such as the completele symmetric or 'generalized Werner states' facilitate treatment of the problem. The characterization of separability for such states is connected with Lieb's 1966 permanental dominance conjecture. The quantum marginal problem asks, given a set of reduced states of a multipartite system, whether there exists a joint quantum state consistent with them. This problem is known to be hard to solve in general as it is a variant of the N-representability problem. Another subject gaining increasing importance is entanglement in infinite systems, such as spin chains and, in particular, systems composed of harmonic oscillators, i.e., modes of the quantized electromagnetic field. Clearly, treating states on infinite systems involves the language of operator algebras. Until now, the trace plays a major role in investigating entanglement. Many infinite systems, however, don't have traces, hence new concepts have to be developed for their treatment.

Spaces of operators, not necessarily operator algebras
(OPERATOR SPACES)
Marius Junge, University of Illinois at Urbana-Champaign, USA

Both quantum information theory and studies of operator spaces are active fields which have begun to influence each other strongly. Operator spaces are Banach subspaces of the bounded operators on a Hilbert space. David Perez Garcia, Michael Wolf, and Marius Junge used these operator spaces to solve a long-standing question about violations of Bell inequalities for tripartite physical systems. A major open problem in quantum information is the 'NPT-problem', asking which bipartite states possess entanglement that can be 'distilled' using local operations with classical communication, a notion entirely in the realm of operator spaces. Another open problem is the quest for a quantum analogue of Shannon's noisy coding theorem, or at least good bounds on quantum channel capacities. The use of operator spaces for these questions seems mandatory.

The thermodynamic limit for interacting quantum fermions in a random environment: the random pîeces model
Frederic Klopp, Université Pierre et Marie Curie, France

In this talk, we will present a simple model of one dimensional interacting electrons in a disordered environment and describe its thermodynamic limit. We shall describe both the ground state energy per particle and the ground state itself. Our main parameter to control the system is the density of particles; we work in the small density limit. The results were obtained in collaboration with N. Veniaminov.

Gapped ground state phases of quantum lattice systems
Bruno Nachtergaele, University of California Davis, USA

If the spectral gap above the ground state(s) of a family of quantum lattice models with short-range interactions depending on a continuous parameter $\lambda\in I$, has a positive lower bound independent of $\lambda$ and the size of the system, we say that the ground states belong to the same gapped quantum phase. We present new results characterizing gapped quantum phases. In particular, we discuss the role of edge excitations,the "protection" effect of an unbroken local symmetry, and the particle nature of elementary excitations. This talk is based on joint work with Sven Bachmann, Jutho Haegeman, Eman Hamza, Spirydon Michalakis, Tobias Osborne, Norbert Schuch, Robert Sims, Frank Verstraete, and Amanda Young.

Bogoliubov dymamics of mean-field Bose gases
Phan Thanh Nam, Université de Cergy Pontoise, France

We consider the dynamics of a large system of N interacting bosons in the mean-field limit where the interaction is of order 1/N. To the leading order, it is well-known that the system has the ability to undergo the Bose-Einstein condensation where most of the particles live in a common quantum state governed by the nonlinear Hartree equation. In this talk, I will discuss the fluctuations around the Hartree states, which can be described by an effective evolution derived from Bogoliubov's approximation. While the usual approach is based on coherent states in Fock space, I will focus on a direct method in the N-particle space. This is joint work with Mathieu Lewin and Benjamin Schlein.

Structure of large bosonic systems: the mean-field approximation and the quantum de Finetti theorem
Nicolas Rougerie, Université Grenoble and CNRS, LPMMC, France

The mean-field approximation is one of the cornerstones of many-body physics. For large bosonic systems, such as the dilute alkali vapors used to study Bose-Einstein condensation, it roughly amounts to assuming that all particles behave independently of one another. The fact that this approximation becomes sensible in the limit of many particles, for a great variety of systems, forms the basis of much of our understanding of quantum bosonic systems. In this talk I will argue that the validity of the mean-field approximation can be seen as a consequence of the special structure of the set of bosonic states, thereby explaining that it holds for systems described by very different Hamiltonians. The main relevant structural property is the quantum de Finetti theorem that describes the structure of bosonic density matrices in the limit of large particle numbers. This theorem was originally proved by Størmer and Hudson-Moody in the 1970's, and I shall discuss the original theorem along with recent variants. I will then outline a general strategy to derive Hartree's theory for the ground state of interacting many-bosons systems with mean-field scaling, based on the quantum de Finetti theorem. The approach is general and applies for example to famous examples such as the trapped Bose gas, bosonic atoms and boson stars. This is joint work with Mathieu Lewin and Phan Thành Nam.

Scales and phases of correlated fermion systems
Manfred Salmhofer, Universität Heidelberg, Germany

I will discuss some phenomena of current interest in solid state physics, as well as theoretical models used in the attempt to understand them. A key feature of these systems are competing ordering tendencies, which produce phases with strongly differing types of order at different scales. Needless to say, very little of this is understood rigorously. I will summarize some results and discuss open problems that may be in reach.

Bose gases, Bose-Einstein condensation, and the Bogoliubov approximation
Robert Seiringer, Institute of Science and Technology Austria (IST Austria), Austria

We present an overview of rigorous results on the low temperature properties of dilute quantum gases, which have been obtained in the past few years. The presentation includes a discussion of Bose-Einstein condensation, the excitation spectrum for trapped gases and its relation to superfluidity, as well as the appearance of quantized vortices in rotating systems. All these properties are intensely being studied in current experiments on cold atomic gases. We will give a description of the mathematics involved in understanding these phenomena, starting from the underlying many-body Schrödinger equation.

Localization for disordered quantum harmonic oscillators
Robert Sims, University of Arizona, USA

We consider quantum harmonic oscillator lattices with random coefficients. Our goal is to study localization properties in many-body systems. Under certain verifiable conditions on the effective one-particle Hamiltonian, we prove:
- dynamical localization, expressed in terms of a zero-velocity Lieb-Robinson bound;
- exponential decay of static and dynamic correlation functions at both zero and positive temperature;
- an area law for the bipartite entanglement of both the ground state and thermal states, as measured by the logarithmic negativity.
Our conditions are satisfied for some standard models that are almost surely gapless in the thermodynamic limit. The talk will provide a general overview of these results.

The classical entropy of quantum states
Jan Philip Solovej, University of Cophenhagen, Denmark

To quantify the inherent uncertainty of quantum states Wehrl ('79) suggested a definition of their \emph{classical entropy} based on the coherent state transform. He conjectured that this classical entropy is by states that also minimize the Heisenberg uncertainty inequality, i.e., Gaussian coherent states. Lieb ('78) proved this conjecture and conjectured that the same holds when Euclidean Glauber coherent states are replaced by SU(2) Bloch coherent states. This generalized Wehrl conjecture has been open for almost 35 years. This conjecture was settled last year in joint work with Elliott Lieb. Recently we simplified the proof and generalized it to SU(N) for general N. I will present this here.

Quantum quenches from a kinetic perspective
Herbert Spohn, Technische Universität München, Germany

There has been a lot of interest in homogeneous quenches for integrable quantum spin chains. The computations suggest that the long time limit has the form of a generalized Gibbs ensemble. In our talk we discuss the kinetic equation for the Fermi-Hubbard chain. For the integrable case we obtain convergence to GGE while for the non-integrable case convergence to the conventional Gibbs state. This
is joint work with M. Fuerst and C. Mendl.

Cones of positive maps on operator algebras
(POSITIVE CONES)
Erling Størmer, University of Oslo, Norway

The use of positive, but not completely positive, maps as entanglement witnesses has generated interest in cones of k-positive maps and their duals. The existence of non-decomposable positive maps on the 3-by-3 matrices is directly related to the failure of the PPT-criterion for entanglement in dimensions 3 and upward, and leads to the phenomenon of undistillable entanglement already mentioned in theme 2 above. Some good properties of separable states arise from the fact that the cone of positive maps is closed under composition. It is hence quite plausible that the problem of distillability is also to be attacked from the side of positive cones. The main question in quantum statistics, whether joint measurement on many particles charts quantum states faster than separate measurements in individual particles, arises in state discrimination and hypothesis testing. It is our aim to help it forward by studying positive cones; non-commutative versions of such classical quantities as Chernoff distance and Hoeffding bounds have already been obtained using cones of positive maps on operator algebras.

Thermodynamic limit for interacting quantum particles in a random environment. General model
Nikolaj Veniaminov, Paris Dauphine, France

In this talk, we will study the thermodynamic limit for a system of interacting quantum particles in disordered environment. Both fermions and bosons will be considered. We will prove that, under rather general assumptions on interactions and underlying one-particle random model, the ground state energy of the system per particle admits thermodynamic limit. This result is mainly based on subadditive type arguments. Next, a particular case of non interacting fermions will be discussed in greater details. Finally, basing on the non interacting case, some intuition behind the perturbative approach for the case of interacting fermions will be given.

Cones of positive maps on operator algebras
(POSITIVE CONES)
Michael Wolf, Technische Universität München, Germany

The use of positive, but not completely positive, maps as entanglement witnesses has generated interest in cones of k-positive maps and their duals. The existence of non-decomposable positive maps on the 3-by-3 matrices is directly related to the failure of the PPT-criterion for entanglement in dimensions 3 and upward, and leads to the phenomenon of undistillable entanglement already mentioned in theme 2 above. Some good properties of separable states arise from the fact that the cone of positive maps is closed under composition. It is hence quite plausible that the problem of distillability is also to be attacked from the side of positive cones. The main question in quantum statistics, whether joint measurement on many particles charts quantum states faster than separate measurements in individual particles, arises in state discrimination and hypothesis testing. It is our aim to help it forward by studying positive cones; non-commutative versions of such classical quantities as Chernoff distance and Hoeffding bounds have already been obtained using cones of positive maps on operator algebras.

Interactions and disorder. A brief introduction
Jakob Yngvason, Universität Wien, Austria

The talk is intended as a introduction to the topic of the first week of the session: Many-body systems in random potentials. An overview of some ideas in the physics and mathematics literature will be given and some rigorous results will be discussed, in particular for a one-dimensional model of interacting Bosons in a potential of randomly distributed point scatterers.

The Bogoliubov C-number approximation for random boson systems
Valentin Zagrebnov, Université d'Aix-Marseille, France

We discuss validity of the Bogoliubov c-number approximation for interacting Bose-gas in a homogeneous external random potential. We generalise the c-number substitution procedure to take into account a possible occurrence of the Bose-Einstein condensation in an infinitesimal band of low kinetic-energy modes without macroscopic occupation of any of them.

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