TY - JOUR
AU - Frank, Rupert L
AU - Lieb, Élliott H
AU - Robert Seiringer
ID - 2379
IS - 4
JF - Journal of the American Mathematical Society
TI - Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators
VL - 21
ER -
TY - JOUR
AB - The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential.
AU - Hainzl, Christian
AU - Hamza, Eman
AU - Robert Seiringer
AU - Solovej, Jan P
ID - 2380
IS - 2
JF - Communications in Mathematical Physics
TI - The BCS functional for general pair interactions
VL - 281
ER -
TY - JOUR
AB - We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.
AU - Frank, Rupert L
AU - Robert Seiringer
ID - 2381
IS - 12
JF - Journal of Functional Analysis
TI - Non-linear ground state representations and sharp Hardy inequalities
VL - 255
ER -
TY - JOUR
AB - We show that the Lieb-Liniger model for one-dimensional bosons with repulsive δ-function interaction can be rigorously derived via a scaling limit from a dilute three-dimensional Bose gas with arbitrary repulsive interaction potential of finite scattering length. For this purpose, we prove bounds on both the eigenvalues and corresponding eigenfunctions of three-dimensional bosons in strongly elongated traps and relate them to the corresponding quantities in the Lieb-Liniger model. In particular, if both the scattering length a and the radius r of the cylindrical trap go to zero, the Lieb-Liniger model with coupling constant g ∼ a/r 2 is derived. Our bounds are uniform in g in the whole parameter range 0 ≤ g ≤ ∞, and apply to the Hamiltonian for three-dimensional bosons in a spectral window of size ∼ r -2 above the ground state energy.
AU - Robert Seiringer
AU - Yin, Jun
ID - 2382
IS - 2
JF - Communications in Mathematical Physics
TI - The Lieb-Liniger model as a limit of dilute bosons in three dimensions
VL - 284
ER -
TY - JOUR
AB - We study the relativistic electron-positron field at positive temperature in the Hartree-Fock approximation. We consider both the case with and without exchange terms, and investigate the existence and properties of minimizers. Our approach is non-perturbative in the sense that the relevant electron subspace is determined in a self-consistent way. The present work is an extension of previous work by Hainzl, Lewin, Séré and Solovej where the case of zero temperature was considered.
AU - Hainzl, Christian
AU - Lewin, Mathieu
AU - Robert Seiringer
ID - 2383
IS - 10
JF - Reviews in Mathematical Physics
TI - A nonlinear model for relativistic electrons at positive temperature
VL - 20
ER -
TY - CONF
AB - We review our proof that in a scaling limit, the time evolution of a quantum particle in a static random environment leads to a diffusion equation. In particular, we discuss the role of Feynman graph expansions and of renormalization.
AU - László Erdös
AU - Salmhofer, Manfred
AU - Yau, Horng-Tzer
ID - 2702
TI - Feynman graphs and renormalization in quantum diffusion
ER -
TY - JOUR
AB - The field of cavity quantum electrodynamics (QED), traditionally studied in atomic systems, has gained new momentum by recent reports of quantum optical experiments with solid-state semiconducting and superconducting systems. In cavity QED, the observation of the vacuum Rabi mode splitting is used to investigate the nature of matter-light interaction at a quantum-mechanical level. However, this effect can, at least in principle, be explained classically as the normal mode splitting of two coupled linear oscillators. It has been suggested that an observation of the scaling of the resonant atom-photon coupling strength in the Jaynes-Cummings energy ladder with the square root of photon number n is sufficient to prove that the system is quantum mechanical in nature. Here we report a direct spectroscopic observation of this characteristic quantum nonlinearity. Measuring the photonic degree of freedom of the coupled system, our measurements provide unambiguous spectroscopic evidence for the quantum nature of the resonant atom-field interaction in cavity QED. We explore atom-photon superposition states involving up to two photons, using a spectroscopic pump and probe technique. The experiments have been performed in a circuit QED set-up, in which very strong coupling is realized by the large dipole coupling strength and the long coherence time of a superconducting qubit embedded in a high-quality on-chip microwave cavity. Circuit QED systems also provide a natural quantum interface between flying qubits (photons) and stationary qubits for applications in quantum information processing and communication.
AU - Johannes Fink
AU - Göppl, M
AU - Baur, Matthias P
AU - Bianchetti, R
AU - Leek, Peter J
AU - Blais, Alexandre
AU - Wallraff, Andreas
ID - 1763
IS - 7202
JF - Nature
TI - Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system
VL - 454
ER -
TY - JOUR
AB - High quality on-chip microwave resonators have recently found prominent new applications in quantum optics and quantum information processing experiments with superconducting electronic circuits, a field now known as circuit quantum electrodynamics (QED). They are also used as single photon detectors and parametric amplifiers. Here we analyze the physical properties of coplanar waveguide resonators and their relation to the materials properties for use in circuit QED. We have designed and fabricated resonators with fundamental frequencies from 2 to 9 GHz and quality factors ranging from a few hundreds to a several hundred thousands controlled by appropriately designed input and output coupling capacitors. The microwave transmission spectra measured at temperatures of 20 mK are shown to be in good agreement with theoretical lumped element and distributed element transmission matrix models. In particular, the experimentally determined resonance frequencies, quality factors, and insertion losses are fully and consistently explained by the two models for all measured devices. The high level of control and flexibility in design renders these resonators ideal for storing and manipulating quantum electromagnetic fields in integrated superconducting electronic circuits.
AU - Göppl, M
AU - Fragner, A
AU - Baur, Matthias P
AU - Bianchetti, R
AU - Filipp, Stefan
AU - Johannes Fink
AU - Leek, Peter J
AU - Puebla, G
AU - Steffen, L. Kraig
AU - Wallraff, Andreas
ID - 1765
IS - 11
JF - Journal of Applied Physics
TI - Coplanar waveguide resonators for circuit quantum electrodynamics
VL - 104
ER -
TY - JOUR
AB - We consider the linear stochastic Cauchy problem dX (t) =AX (t) dt +B dWH (t), t≥ 0, where A generates a C0-semigroup on a Banach space E, WH is a cylindrical Brownian motion over a Hilbert space H, and B: H → E is a bounded operator. Assuming the existence of a unique minimal invariant measure μ∞, let Lp denote the realization of the Ornstein-Uhlenbeck operator associated with this problem in Lp (E, μ∞). Under suitable assumptions concerning the invariance of the range of B under the semigroup generated by A, we prove the following domain inclusions, valid for 1 < p ≤ 2: Image omitted. Here WHk, p (E, μinfin; denotes the kth order Sobolev space of functions with Fréchet derivatives up to order k in the direction of H. No symmetry assumptions are made on L p.
AU - Jan Maas
AU - van Neerven, Jan M
ID - 2120
IS - 4
JF - Infinite Dimensional Analysis, Quantum Probability and Related Topics
TI - On the domain of non-symmetric Ornstein-Uhlenbeck operators in banach spaces
VL - 11
ER -
TY - JOUR
AB - Let H be a separable real Hubert space and let double struck F sign = (ℱt)t∈[0,T] be the augmented filtration generated by an H-cylindrical Brownian motion (WH(t))t∈[0,T] on a probability space (Ω, ℱ ℙ). We prove that if E is a UMD Banach space, 1 ≤ p < ∞, and F ∈ double struck D sign1,p(Ω E) is ℱT-measurable, then F = double struck E sign(F) + ∫0T Pdouble struck F sign(DF) dW H, where D is the Malliavin derivative of F and P double struck F sign is the projection onto the F-adapted elements in a suitable Banach space of Lp-stochastically integrable ℒ(H, E)-valued processes.
AU - van Neerven, Jan M
AU - Jan Maas
ID - 2121
JF - Electronic Communications in Probability
TI - A Clark-Ocone formula in UMD Banach spaces
VL - 13
ER -
TY - JOUR
AB - We present an analytic model of thermal state-to-state rotationally inelastic collisions of polar molecules in electric fields. The model is based on the Fraunhofer scattering of matter waves and requires Legendre moments characterizing the “shape” of the target in the body-fixed frame as its input. The electric field orients the target in the space-fixed frame and thereby effects a striking alteration of the dynamical observables: both the phase and amplitude of the oscillations in the partial differential cross sections undergo characteristic field-dependent changes that transgress into the partial integral cross sections. As the cross sections can be evaluated for a field applied parallel or perpendicular to the relative velocity, the model also offers predictions about steric asymmetry. We exemplify the field-dependent quantum collision dynamics with the behavior of the Ne–OCS(Σ1) and Ar–NO(Π2) systems. A comparison with the close-coupling calculations available for the latter system [Chem. Phys. Lett.313, 491 (1999)] demonstrates the model’s ability to qualitatively explain the field dependence of all the scattering features observed.
AU - Mikhail Lemeshko
AU - Friedrich, Břetislav
ID - 2146
IS - 2
JF - Journal of Chemical Physics
TI - An analytic model of rotationally inelastic collisions of polar molecules in electric fields
VL - 129
ER -
TY - JOUR
AB - Homeostasis of internal carbon dioxide (CO2) and oxygen (O2) levels is fundamental to all animals. Here we examine the CO2 response of the nematode Caenorhabditis elegans. This species inhabits rotting material, which typically has a broad CO2 concentration range. We show that well fed C. elegans avoid CO2 levels above 0.5%. Animals can respond to both absolute CO2 concentrations and changes in CO2 levels within seconds. Responses to CO2 do not reflect avoidance of acid pH but appear to define a new sensory response. Sensation of CO2 is promoted by the cGMP-gated ion channel subunits TAX-2 and TAX-4, but other pathways are also important. Robust CO2 avoidance in well fed animals requires inhibition of the DAF-16 forkhead transcription factor by the insulin-like receptor DAF-2. Starvation, which activates DAF-16, strongly suppresses CO2 avoidance. Exposure to hypoxia (<1% O2) also suppresses CO2 avoidance via activation of the hypoxia-inducible transcription factor HIF-1. The npr-1 215V allele of the naturally polymorphic neuropeptide receptor npr-1, besides inhibiting avoidance of high ambient O2 in feeding C. elegans, also promotes avoidance of high CO2. C. elegans integrates competing O2 and CO2 sensory inputs so that one response dominates. Food and allelic variation at NPR-1 regulate which response prevails. Our results suggest that multiple sensory inputs are coordinated by C. elegans to generate different coherent foraging strategies.
AU - Bretscher, A. J.
AU - Busch, K. E.
AU - de Bono, Mario
ID - 6146
IS - 23
JF - Proceedings of the National Academy of Sciences
SN - 0027-8424
TI - A carbon dioxide avoidance behavior is integrated with responses to ambient oxygen and food in Caenorhabditis elegans
VL - 105
ER -
TY - JOUR
AB - We calculate the E-polynomials of certain twisted GL(n,ℂ)-character varieties Mn of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n, q) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,ℂ)-character variety. The calculation also leads to several conjectures about the cohomology of Mn: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n=2.
AU - Tamas Hausel
AU - Rodríguez Villegas, Fernando
ID - 1460
IS - 3
JF - Inventiones Mathematicae
TI - Mixed Hodge polynomials of character varieties: With an appendix by Nicholas M. Katz
VL - 174
ER -
TY - JOUR
AB - We report on the control of interaction-induced dephasing of Bloch oscillations for an atomic Bose-Einstein condensate in an optical lattice. We quantify the dephasing in terms of the width of the quasimomentum distribution and measure its dependence on time for different interaction strengths which we control by means of a Feshbach resonance. For minimal interaction, the dephasing time is increased from a few to more than 20 thousand Bloch oscillation periods, allowing us to realize a BEC-based atom interferometer in the noninteracting limit.
AU - Gustavsson, Mattias
AU - Haller, Elmar
AU - Mark, Manfred
AU - Danzl, Johann G
AU - Rojas Kopeinig, Gabriel
AU - Nägerl, Hanns
ID - 1036
IS - 8
JF - Physical Review Letters
TI - Control of interaction-induced dephasing of bloch oscillations
VL - 100
ER -
TY - JOUR
AB - We experimentally demonstrate Cs2 Feshbach molecules well above the dissociation threshold, which are stable against spontaneous decay on the time scale of 1s. An optically trapped sample of ultracold dimers is prepared in a high rotational state and magnetically tuned into a region with a negative binding energy. The metastable character of these molecules arises from the large centrifugal barrier in combination with negligible coupling to states with low rotational angular momentum. A sharp onset of dissociation with increasing magnetic field is mediated by a crossing with a lower rotational dimer state and facilitates dissociation on demand with a well-defined energy.
AU - Knoop, Steven
AU - Mark, Michael
AU - Ferlaino, Francesca
AU - Danzl, Johann G
AU - Kraemer, Tobias
AU - Nägerl, Hanns
AU - Grimm, Rudolf
ID - 1037
IS - 8
JF - Physical Review Letters
TI - Metastable feshbach molecules in high rotational states
VL - 100
ER -
TY - JOUR
AB - Molecular cooling techniques face the hurdle of dissipating translational as well as internal energy in the presence of a rich electronic, vibrational, and rotational energy spectrum. In our experiment, we create a translationally ultracold, dense quantum gas of molecules bound by more than 1000 wave numbers in the electronic ground state. Specifically, we stimulate with 80% efficiency, a two-photon transfer of molecules associated on a Feshbach resonance from a Bose-Einstein condensate of cesium atoms. In the process, the initial loose, long-range electrostatic bond of the Feshbach molecule is coherently transformed into a tight chemical bond. We demonstrate coherence of the transfer in a Ramsey-type experiment and show that the molecular sample is not heated during the transfer. Our results show that the preparation of a quantum gas of molecules in specific rovibrational states is possible and that the creation of a Bose-Einstein condensate of molecules in their rovibronic ground state is within reach.
AU - Danzl, Johann G
AU - Haller, Elmar
AU - Gustavsson, Mattias
AU - Mark, Manfred
AU - Hart, Russell
AU - Bouloufa, Nadia
AU - Dulieu, Olivier
AU - Ritsch, Helmut
AU - Nägerl, Hanns
ID - 1039
IS - 5892
JF - Science
TI - Quantum gas of deeply bound ground state molecules
VL - 321
ER -
TY - JOUR
AB - We give many examples of applying Bogoliubov's forest formula to iterative solutions of various nonlinear equations. The same formula describes an extremely wide class of objects, from an ordinary quadratic equation to renormalization in quantum field theory.
AU - Morozov, Alexei Y
AU - Maksym Serbyn
ID - 965
IS - 2
JF - Theoretical and Mathematical Physics
TI - Nonlinear algebra and Bogoliubov's recursion
VL - 154
ER -
TY - JOUR
AB - Gene expression levels fluctuate even under constant external conditions. Much emphasis has usually been placed on the components of this noise that are due to randomness in transcription and translation. Here we focus on the role of noise associated with the inputs to transcriptional regulation; in particular, we analyze the effects of random arrival times and binding of transcription factors to their target sites along the genome. This contribution to the total noise sets a fundamental physical limit to the reliability of genetic control, and has clear signatures, but we show that these are easily obscured by experimental limitations and even by conventional methods for plotting the variance vs. mean expression level. We argue that simple, universal models of noise dominated by transcription and translation are inconsistent with the embedding of gene expression in a network of regulatory interactions. Analysis of recent experiments on transcriptional control in the early Drosophila embryo shows that these results are quantitatively consistent with the predicted signatures of input noise, and we discuss the experiments needed to test the importance of input noise more generally.
AU - Gasper Tkacik
AU - Gregor, Thomas
AU - Bialek, William S
ID - 3734
IS - 7
JF - PLoS One
TI - The role of input noise in transcriptional regulation
VL - 3
ER -
TY - JOUR
AB - In the simplest view of transcriptional regulation, the expression of a gene is turned on or off by changes in the concentration of a transcription factor (TF). We use recent data on noise levels in gene expression to show that it should be possible to transmit much more than just one regulatory bit. Realizing this optimal information capacity would require that the dynamic range of TF concentrations used by the cell, the input/output relation of the regulatory module, and the noise in gene expression satisfy certain matching relations, which we derive. These results provide parameter-free, quantitative predictions connecting independently measurable quantities. Although we have considered only the simplified problem of a single gene responding to a single TF, we find that these predictions are in surprisingly good agreement with recent experiments on the Bicoid/Hunchback system in the early Drosophila embryo and that this system achieves approximately 90% of its theoretical maximum information transmission.
AU - Gasper Tkacik
AU - Callan,Curtis G
AU - Bialek, William S
ID - 3740
IS - 34
JF - PNAS
TI - Information flow and optimization in transcriptional regulation
VL - 105
ER -
TY - JOUR
AB - It is widely acknowledged that detailed timing of action potentials is used to encode information, for example, in auditory pathways; however, the computational tools required to analyze encoding through timing are still in their infancy. We present a simple example of encoding, based on a recent model of time-frequency analysis, in which units fire action potentials when a certain condition is met, but the timing of the action potential depends also on other features of the stimulus. We show that, as a result, spike-triggered averages are smoothed so much that they do not represent the true features of the encoding. Inspired by this example, we present a simple method, differential reverse correlations, that can separate an analysis of what causes a neuron to spike, and what controls its timing. We analyze with this method the leaky integrate-and-fire neuron and show the method accurately reconstructs the model's kernel.
AU - Gasper Tkacik
AU - Magnasco, Marcelo O
ID - 3744
IS - 1-2
JF - Biosystems
TI - Decoding spike timing: The differential reverse-correlation method
VL - 93
ER -