An electron gas in a one-dimensional periodic potential can be transported even in the absence of a voltage bias if the potential is slowly and periodically modulated in time. Remarkably, the transferred charge per cycle is sensitive only to the topology of the path in parameter space. Although this so-called Thouless charge pump was first proposed more than thirty years ago 1 , it has not yet been realized. Here we report the demonstration of topological Thouless pumping using ultracold fermionic atoms in a dynamically controlled optical superlattice. We observe a shift of the atomic cloud as a result of pumping, and extract the topological invariance of the pumping process from this shift. We demonstrate the topological nature of the Thouless pump by varying the topology of the pumping path and verify that the topological pump indeed works in the quantum regime by varying the speed and temperature.Topology manifests itself in physics in a variety of ways 2-4 , with the integer quantum Hall effect (IQHE) being one of the best-known examples in condensed matter systems. There, the Hall conductance of a two-dimensional electron gas is quantized very precisely in units of fundamental constants 5 . As discussed in the celebrated Thouless-Kohmoto-Nightingale-den Nijs paper 6 , this quantized value is given by a topological invariant, the sum of the Chern numbers of the occupied energy bands.In 1983, Thouless considered a seemingly different phenomenon of quantum transport of an electron gas in an infinite onedimensional periodic potential, driven in a periodic cycle 1 . This seems to be similar to the famous Archimedes screw 7 , which pumps water via a rotating spiral tube. However, whereas the Archimedes screw follows classical physics and the pumped amount of water can be changed continuously by tilting the screw, the charge pumped by the Thouless pump is a topological quantum number and not affected by a smooth change of parameters 1 . Interestingly, this quantization of pumped charge shares the same topological origin as the IQHE. The charge pumped per cycle can be expressed by the Chern number defined over a (1 + 1)-dimensional periodic Brillouin zone formed by quasimomentum k and time t. Although several single-electron pumping experiments have been implemented in nanoscale devices, such as quantum dots with modulated gate voltages 8-10 or surface acoustic waves to create a potential periodic in time 11 , the topological Thouless pump, which should have the spatial periodicity to define the Bloch wavefunction as well as the temporal periodicity, has not been realized in electron systems.In this Letter, we report a realization of Thouless' topological charge pump by exploiting the controllability of ultracold atoms in an optical superlattice. Differently from recent realizations of topological bands in two (spatial or synthetic) dimensions 12-17 , our experiment explores the topology of a (1 + 1)-dimensional adiabatic process, in which a dynamically controllable onedimensional optical superlattice is implemente...

We study measure perturbations of the Laplacian in L 2 (R 2 ) supported by an infinite curve Γ in the plane which is asymptotically straight in a suitable sense. We show that if Γ is not a straight line, such a "leaky quantum wire" has at least one bound state below the threshold of the essential spectrum.

We examined whether protein-carbohydrate (CHO) supplementation immediately after exercise each day during aerobic training facilitated plasma volume (PV) expansion and thermoregulatory and cardiovascular adaptations in older men. Fourteen moderately active older men [68 +/- 5 (SD) yr] were divided into two groups so as to have no significant differences in anthropometric measures, PV, and peak oxygen consumption rate (Vo(2peak)). Each group was provided with a mixture of protein and CHO (3.2 kcal, 0.18 g protein/kg body wt, Pro-CHO, n = 7) or a non-protein and low-calorie placebo (0.5 kcal, 0 g protein/kg body wt, CNT, n = 7) immediately after cycling exercise (60-75% Vo(2peak), 60 min/day, 3 days/wk) each day for 8 wk at approximately 19 degrees C ambient temperature (T(a)) and approximately 43% relative humidity (RH). Before and after training, we measured PV, cardiac stroke volume (SV), and esophageal temperature (T(es)) during 20-min exercise at 60% of pretraining Vo(2peak) at 30 degrees C T(a) and 50% RH. Moreover, we determined the sensitivity of the chest sweat rate (DeltaSR/DeltaT(es)) and forearm vascular conductance (DeltaFVC/DeltaT(es)) in response to increased T(es) during exercise. After training, PV increased by approximately 6% in Pro-CHO (P < 0.001), with an approximately 10% increase in SV during exercise (P < 0.001), but not in CNT (P > 0.07). DeltaFVC/DeltaT(es) increased by 80% and DeltaSR/DeltaT(es) by 18% in Pro-CHO (both P < 0.01) but not in CNT (P > 0.07). Moreover, we found a significant interactive effect of group x training on PV, SV, and DeltaFVC/DeltaT(es) (all P < 0.02) but with no significant effect of group (P > 0.4), suggesting that the supplement enhanced these responses to aerobic training. Thus postexercise protein-CHO supplementation during training caused PV expansion and facilitated thermoregulatory and cardiovascular adaptations, possibly providing a new training regimen for older men.

A rigorous path integral representation of the solution of the Cauchy problem for the pure-imaginary-time Schrodinger equation d t ψ (t,x} = -[_H -mc 2~] ψ(t,x) is established. H is the quantum Hamiltonian associated, via the Weyl correspondence, with the classical Hamiltonian \_(cp -eA(x)) 2 + w 2 c 4 ] 1/2 + eΦ(x) of a relativistic spinless particle in an electromagnetic field. The problem is connected with a time homogeneous Lέvy process.

We report the momentum-resolved measurement of Bloch bands in an optical Lieb lattice for a Bose-Einstein condensate (BEC). A BEC in the lattice is transported to a desired quasimomentum by applying a constant force. The energy dispersion of the lowest band is obtained by integrating measured group velocities. We also measure the gap from the lowest band to the higher bands with the same quasimomentum, which can be extracted from the oscillation of the sublattice populations after preparing a superposition of the band eigenstates. We show that the experimental results agree with a band calculation based on the Bogoliubov approximation. It is revealed that the second band, which should be flat in a single-particle description, is shifted and, in particular, distorted around the Brillouin zone edge as the interaction strength increases.

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