Effective low-dimensional Hamiltonian for strongly interacting atoms in a transverse trap

We derive an effective low-dimensional Hamiltonian for strongly interacting ultracold atoms in a transverse trapping potential near a wide Feshbach resonance. The Hamiltonian includes crucial information about transverse excitations in an effective model with renormalized interaction between atoms and composite dressed molecules. We fix all the parameters in the Hamiltonian for both one- and two-dimensional cases.Comment: v2: 5 pages, 1 figure; expanded presentation of the formalis

Anharmonicity Induced Resonances for Ultracold Atoms and their Detection

When two atoms interact in the presence of an anharmonic potential, such as an optical lattice, the center of mass motion cannot be separated from the relative motion. In addition to generating a confinement-induced resonance (or shifting the position of an existing Feshbach resonance), the external potential changes the resonance picture qualitatively by introducing new resonances where molecular excited center of mass states cross the scattering threshold. We demonstrate the existence of these resonances, give their quantitative characterization in an optical superlattice, and propose an experimental scheme to detect them through controlled sweeping of the magnetic field.Comment: 6 pages, 5 figures; expanded presentatio

Level crossing in the three-body problem for strongly interacting fermions in a harmonic trap

We present a solution of the three-fermion problem in a harmonic potential across a Feshbach resonance. We compare the spectrum with that of the two-body problem and show that it is energetically unfavorable for the three fermions to occupy one lattice site rather than two. We also demonstrate the existence of an energy level crossing in the ground state with a symmetry change of its wave function, suggesting the possibility of a phase transition for the corresponding many-body case.Comment: 5 pages, 6 figures, typos corrected, references adde

Effective single-band models for strongly interacting fermions in an optical lattice

To test effective Hamiltonians for strongly interacting fermions in an optical lattice, we numerically find the energy spectrum for two fermions interacting across a Feshbach resonance in a double well potential. From the spectrum, we determine the range of detunings for which the system can be described by an effective lattice model, and how the model parameters are related to the experimental parameters. We find that for a range of strong interactions the system is well described by an effective $t-J$ model, and the effective superexchange term, $J$, can be smoothly tuned through zero on either side of unitarity. Right at and around unitarity, an effective one-band general Hubbard model is appropriate, with a finite and small on-site energy, due to a lattice-induced anharmonic coupling between atoms at the scattering threshold and a weakly bound Feshbach molecule in an excited center of mass state.Comment: 7 pages, 7 figures; minor typos correcte

Efficient reverse engineering of one-qubit filter functions with dynamical invariants

We derive an integral expression for the filter-transfer function of an arbitrary one-qubit gate through the use of dynamical invariant theory and Hamiltonian reverse engineering. We use this result to define a cost functional which can be efficiently optimized to produce one-qubit control pulses that are robust against specified frequency bands of the noise power spectral density. We demonstrate the utility of our result by generating optimal control pulses that are designed to suppress broadband detuning and pulse amplitude noise. We report an order of magnitude improvement in gate fidelity in comparison with known composite pulse sequences. More broadly, we also use the same theoretical framework to prove the robustness of nonadiabatic geometric quantum gates under specific error models and control constraints

Characterization by NMR of Reactants and Products of Hydrofluoroether Isomers, CF3(CF2)3OCH3 and (CF3)2C(F)CF2OCH3, Reacting with Isopropyl Alcohol

The 3M Company product Novec™ 71IPA DL, a mixture of methoxyperfluorobutane, methoxyperfluoroisobutane and 4.5 wt.% isopropyl alcohol, has been found to be very stable at ambient temperature, producing fluoride at the rate of ~1 ppm/year. Our earlier kinetic and theoretical studies have identified the reaction mechanism. This paper identifies the 1H and 19F NMR chemical shifts, multiplicities, and coupling constants of reactants and the major products that result from aging the mixture in sealed Pyrex NMR tubes for periods up to 1.8 years at temperatures from 26 °C to 102 °C. Chemical shifts and coupling constants of fluorine and hydrogen atoms on the hydrofluoroethers and isopropyl alcohol are traced through the reactions to their values in the products – esters, isopropylmethyl ether, and HF. These spectral positions, multiplicities, and coupling constants are presented in table format and as figures to clarify the transformations observed as the samples age

Optimal leverage from non-ergodicity

In modern portfolio theory, the balancing of expected returns on investments against uncertainties in those returns is aided by the use of utility functions. The Kelly criterion offers another approach, rooted in information theory, that always implies logarithmic utility. The two approaches seem incompatible, too loosely or too tightly constraining investors' risk preferences, from their respective perspectives. The conflict can be understood on the basis that the multiplicative models used in both approaches are non-ergodic which leads to ensemble-average returns differing from time-average returns in single realizations. The classic treatments, from the very beginning of probability theory, use ensemble-averages, whereas the Kelly-result is obtained by considering time-averages. Maximizing the time-average growth rates for an investment defines an optimal leverage, whereas growth rates derived from ensemble-average returns depend linearly on leverage. The latter measure can thus incentivize investors to maximize leverage, which is detrimental to time-average growth and overall market stability. The Sharpe ratio is insensitive to leverage. Its relation to optimal leverage is discussed. A better understanding of the significance of time-irreversibility and non-ergodicity and the resulting bounds on leverage may help policy makers in reshaping financial risk controls.Comment: 17 pages, 3 figures. Updated figures and extended discussion of ergodicit