A Reply to the Comment by Wm. G. Hoover, et al. and Peter Reimann

We devise a new and highly accurate quantization procedure for the inner product representation, both in configuration and momentum space. Utilizing the representation $\Psi \left(\xi \right)=\Sigma i^{}{a}_i\left[E\right]{\xi }^i{R}_{\beta }\left(\xi \right)$, for an appropriate reference function, $R_{\beta }\left(\xi \right)$, we demonstrate that the (convergent) zeros of the coefficient functions, $a_i\left[E\right]=0\mathrm{}$, approximate the exact bound state energies with increasing accuracy as $i\to \infty$. The validity of the approach is shown to be based on an approximation to the Hill determinant quantization procedure. Our method has been applied, with remarkable success, to various quantum mechanical problems in one and two space dimensions.

We find exact coherent states for a one-dimensional quantum fluid interacting by an inverse-square pair potential, contained by a time-dependent harmonic trapping potential. These states are those that would evolve from the ground state of the time-independent problem. Correlations are determined, and a hydrodynamic description is shown to be exact. We treat the case of a nondissipative “sloshing” mode characteristic of superfluidity, free expansion where time-of-flight measurements do not give the momentum distribution, and a periodically varying trapping potential exhibiting alternating regions of stable and unstable behavior.

In the interaction picture, population transfer among coupled degrees of freedom is greatly enhanced by resonances. We show that statistically the number of resonances increases rapidly with degrees of freedom, changing the characteristic of population transfer from being bounded to diffusive. From the diffusion rate we derive simple expressions for the time scales of energy relaxation and equipartition. These expressions are supported by a wide range of experimental data. The analysis elucidates quantitatively the dependence of equipartition on resonances.

The dynamical structure of an atomic Bose-Einstein condensate limits the efficiency of the condensate in cooling slow impurity atoms. To illustrate the point, we show that an impurity atom moving in a homogeneous zero-temperature condensate is not scattered incoherently if its velocity is lower than the condensate sound velocity $c$, limiting cooling to velocities $v\ge c$. This striking effect is an expression of superfluidity and provides a direct means to detect the fundamental property of superfluidity in atomic condensates. Furthermore, we show that the fermionic lithium isotope, ${}^6\mathrm{Li}$, is a reasonable candidate for sympathetic cooling by a ${}^{23}\mathrm{Na}$ condensate.

The characteristic features of Bose-Einstein quantum statistics are derived resorting to the notion of coproduct associated with any Lie-Hopf structure and to generalized coherent states. It is shown that, while Boltzmann particles are well described by coherent states of $h\left(1\right)$, orthodox bosons require the $D_{1/2}^{\left(+\right)}$ representation of SU(1,1). A whole class of inequivalent generalized bosons is then defined by multimode coherent states of $\widetilde{\mathrm{SU}}\left(1,1\right)$. Known experimental results are in agreement with our generalizations—as well as with the Bose hypothesis—but experiments with discrete spectra, such as the recent ones on Bose condensation, could discriminate among all these possibilities.

Mechanically induced degradation of diamond, as occurs during polishing, is studied using total-energy pseudopotential calculations. The strong asymmetry in the rate of polishing between different directions on the diamond (110) surface is explained in terms of an atomistic mechanism for nanogroove formation. The postpolishing surface morphology and the nature of the polishing residue predicted by this mechanism are consistent with experimental evidence.

We have carried out a neutron interferometry experiment using longitudinally polarized neutrons to observe the scalar Aharonov-Bohm effect. The neutrons inside the interferometer are polarized parallel to an applied pulsed magnetic field B(t). The pulsed B field is spatially uniform so it exerts no force on the neutrons. Its direction also precludes the presence of any classical torque to change the neutron polarization.

We demonstrate a contradiction of quantum mechanics with local hidden variable theories for continuous quadrature phase amplitude (“position” and “momentum”) measurements. For any quantum state, this contradiction is lost for situations where the quadrature phase amplitude results are always macroscopically distinct. We show that for optical realizations of this experiment, where one uses homodyne detection techniques to perform the quadrature phase amplitude measurement, one has an amplification prior to detection, so that macroscopic fields are incident on photodiode detectors. The high efficiencies of such detectors may open a way for a loophole-free test of local hidden variable theories.

We report on the sub-kilohertz optical spectroscopy on the ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$- ${}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$ intercombination transition in magnesium at 457 nm. The spectroscopic signal is probed by a time domain atom interferometer. The realization of this time domain atom interferometer with laser cooled and trapped atoms allows extremely long interaction times and leads to resolutions down to 491 Hz (FWHM). This corresponds to a high line $Q$ factor of $1.3\times {10}^{12}$. Because of the high accuracy in the determination of the line center, applications with respect to an optical frequency standard are possible.

Four-particle correlations in fermion systems at finite temperatures are investigated with special attention to the formation of a condensate. Instead of the instability of the normal state with respect to the onset of pairing described by the Gorkov equation, a new equation is obtained which describes the onset of “quartetting.” Applying to symmetric nuclear matter, below a critical density, the four-particle condensation ( $\alpha$-like quartetting) is found to be favored over deuteron condensation (triplet pairing). This pairing-quartetting competition is expected to be a general feature of interacting fermion systems, such as the exciton-biexciton system in excited semiconductors. Possible experimental consequences are pointed out.

Oscillations between two quantum states may be a sensitive probe for a loss of quantum coherence due to possible violations of quantum mechanics, e.g., caused by gravitation. We show, for the strangeness oscillations in the neutral kaon system, that there exist four experiments which uniquely determine all nine non-quantum-mechanical parameters that occur in a density matrix description using the Pauli matrix basis.

We propose and analyze a scheme to create dark solitons and vortices in Bose-Einstein condensates. This is achieved starting from a condensate in the internal state $|a〉$ and transferring the atoms to the internal state $|b〉$ via a Raman transition induced by laser light. By scanning adiabatically the Raman detuning, dark solitons and vortices are created.

Quantum scattering systems described by Hamiltonians which are constructed from the Casimir operators of certain noncompact groups $G$ are considered. We obtain the following result: If $U^{\chi }$ and $U^{\tilde{\chi }}$ are the Weyl-equivalent representations of the symmetry group $G$ of the dynamical system, the corresponding $S$ matrices are constrained to satisfy ${\mathrm{SU}}^{\chi }\left(g\right){=U}^{\tilde{\chi }}\left(g\right)S$, for all $g\in G$. This relation enables one to derive $S$. As applications, the $S$ matrices corresponding to the dynamical groups ${\mathrm{SO}}_0(p,q)$ are derived.

We investigate the low-lying spectra of many-body systems with random two-body interactions, specifying that the ensemble be invariant under particle-hole conjugation. Surprisingly we find patterns reminiscent of more orderly interactions, such as a predominance of $J=0\mathrm{}$ ground states separated by a gap from the excited states, and evidence of phonon vibrations in the low-lying spectra.

Both the level of broadband microturbulence and the behavior of driven waves can be strongly affected by local coincidences that produce nonlinear levels of the total fluctuation amplitude. We evaluate the rate at which such coincidences occur, for moderately damped modes which frequently reset their phases. They can produce significant nonlinear effects and can limit the amplitude of individual modes to low levels. These effects, which involve very many modes, are inherently difficult to observe using standard Fourier space or particle simulation techniques.

Now that the properties of low temperature Bose gases at low density, $\rho$, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors four to five decades ago. One of these is that the leading term in the energy/particle is $2\pi {ħ}^2\rho a/m$, where $a$ is the scattering length. Owing to the delicate and peculiar nature of bosonic correlations, four decades of research have failed to establish this plausible formula rigorously. The only known lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct bound is proved here.