A new class of solutions for Bose crystals with a simple cubic lattice consisting of N atoms is found. The wave function (WF) of the ground state takes the form Ψ_0=e^{S_{w}^{l}+S_{b}}*\prod_j [\sin{k_{l_x}x_{j}}\sin{k_{l_y}y_{j}}\sin{k_{l_z}z_{j}}], where e^{S_{b}} is the ground-state WF of a fluid, and \textbf{k}_l=(π/a_l, π/a_l, π/a_l) (a_l is the lattice constant). The state with a single longitudinal acoustic phonon is described by the WF Ψ_k=[ρ_{-k}+corrections + 7 permutations]Ψ_0, where the permutations give the terms with different signs of components of vector k. The structure of Ψ_k is such that the excitation corresponds, in fact, to the replacement of \textbf{k}_l in some triple of sines from Ψ_0 by \textbf{k}. Such a structure of Ψ_0 and Ψ_k means that the crystal is created by sound: the ground state of a cubic crystal is formed by N identical three-dimensional standing waves similar to a longitudinal sound. It is also shown that the crystal in the ground state has a condensate of atoms with \textbf{k}=\textbf{k}_l. The nonclassical inertia moment observed in crystals He-4 can be related to the synchronous tunneling of condensate atoms.
A new class of solutions for Bose crystals with a simple cubic lattice consisting of N atoms is found. The wave function (WF) of the ground state takes the form Ψ_0=e^{S_{w}^{l}+S_{b}}*\prod_j [\sin{k_{l_x}x_{j}}\sin{k_{l_y}y_{j}}\sin{k_{l_z}z_{j}}], where e^{S_{b}} is the ground-state WF of a fluid, and \textbf{k}l=(π/a_l, π/a_l, π/a_l) (a_l is the lattice constant). The state with a single longitudinal acoustic phonon is described by the WF Ψ_k=[ρ{-k}+corrections + 7 permutations]Ψ_0, where the permutations give the terms with different signs of components of vector k. The structure of Ψ_k is such that the excitation corresponds, in fact, to the replacement of \textbf{k}_l in some triple of sines from Ψ_0 by \textbf{k}. Such a structure of Ψ_0 and Ψ_k means that the crystal is created by sound: the ground state of a cubic crystal is formed by N identical three-dimensional standing waves similar to a longitudinal sound. It is also shown that the crystal in the ground state has a condensate of atoms with \textbf{k}=\textbf{k}_l. The nonclassical inertia moment observed in crystals He-4 can be related to the synchronous tunneling of condensate atoms.
At the present time, it is considered that Bose crystals are formed at the cooling of a fluid, because the state of crystal is more favorable by energy. It is also believed [1,2] that no ordering factor forming a crystal is present, except for the interatomic interaction. However, the available solutions [1,2,3] for the wave functions (WFs) of the ground and excited states of a crystal are approximate and are obtained for cyclic boundary conditions, which are not realized in the Nature. Here, we present the exact solutions for the WFs of a Bose crystal with rectangular lattice under natural zero boundary conditions. The structure of WFs implies that 1) a crystal is formed by a standing wave in the probability field; 2) a crystal in the ground state contains a condensate of atoms with the wave vector \textbf{k}_l=(π/\bar{R}_x, π/\bar{R}_y, π/\bar{R}_z) (\bar{R}_x, \bar{R}_y, \bar{R}_z are the periods of the lattice) that is equal to a half of the vector of the reciprocal lattice. These solutions indicate that the ordering factor forming a crystal is an intense standing wave similar to a sound one. Thus, the periodicity of a lattice is caused by that of a sound wave, but not only by the energy minimum principle. Apparently, the crystals of other types and with different lattices have the wave nature as well. The condensate opens a possibility to explain the nonclassical inertia moment discovered by Kim and Chan [4,5] in solid He-4, which testifies, probably, to the presence of a superfluid subsystem in the crystal.
In this paper we suggest a new ground-state wave function and low-temperature density matrix for a strongly interacting system of bosons. Our basic assumption is that the system can support long-wavelength phonons and that these can propagate independently of any other mode of motion. We therefore write the ground-state function as the product of two factors. One factor arises from the zero-point motion of the phonons, and we show that it has the form ${\ensuremath{\Pi}}_{i<j}f({r}_{\mathrm{ij}})$, where lnf(r) has an infinite range. The other factor is assumed to have this same form but with lnf(r) of finite range; it takes into account the short-range correlations arising from the strong repulsive part of the interparticle potential. The function we have chosen to represent the short-range correlations is not new; functions of this kind were first introduced by Bijl and later by Jastrow. At finite temperatures we use a density matrix for an ensemble of excited phonon states. We find that for small wave vectors k, the structure factor S(k) is equal to $\frac{\ensuremath{\hbar}k}{2mc}$, a result that was first derived by Feynman. At a finite temperature T, S(k) tends to the constant value mc2kBT, where m is the mass of the particles and c the velocity of propagation of the phonons. The momentum distribution nk has a ${k}^{\ensuremath{-}1}$ singularity at absolute zero and a stronger, ${k}^{\ensuremath{-}2}$ singularity at finite temperatures. The small-k behavior of both S(k) and nk is completely controlled by the correlations introduced by the phonons. Both the ground-state function and the density matrix imply that there is a finite fraction of particles in the zero-momentum state in three dimensions; this fraction does not seem to be appreciably affected by the infinite-range correlations introduced by the phonons. We find, however, that these correlations imply that a one-dimensional Bose system does not exhibit Bose-Einstein condensation at any temperature, while a two-dimensional system exhibits Bose-Einstein condensation only at absolute zero.
It is usually assumed that the Bose crystal at T=0 corresponds to the genuine ground state of a Bose system, i.e., this state is non-degenerate and is described by the wave function without nodes. By means of symmetry analysis we show that the ground state of a Bose system of any density should correspond to a liquid or gas, but not to a crystal. The main point is that any anisotropic state of a system of spinless bosons is degenerate. We prove this for an infinite three-dimensional (3D) system and a finite ball-shaped 3D system. One can expect that it is true also for a finite system of any form. Therefore, the anisotropic state cannot be the genuine ground state. Hence, a zero-temperature natural 3D crystal should correspond to an excited state of a Bose system. The wave function Ψ0c of a zero-temperature 3D Bose crystal is proposed for zero boundary conditions. Apparently, such Ψ0c corresponds to a local minimum of energy (absolute minimum corresponds to a liquid). Those properties yield the possibility of existence of superfluid liquid H2, Ne, Ar, and other inert elements. We propose several possible experimental ways of obtaining them.
We study two different subjects related to ultra-cold Bose gases. First we study the influence of the trap and the background pressure on particle loss and temperature change for trapped atom clouds. We find the number of particles to exponentially decrease over time due to the imperfect vacuum, while the rate of temperature change differs between a magnetic trap and a far off resonance optical trap. Secondly, we study the properties of a standing sound wave induced in a Bose-Einstein condensate. Phase contrast imaging is used to image the condensate in order to observe the density profile as a function of time. We find that there is coupling between the standing wave and other collective modes. There is a clear relation between the frequencies of these collective modes, and also between the frequencies of these modes and the frequency used to induce the modes. We also find that the amplitude of the standing wave decays over time.
We experimentally and theoretically investigate the anisotropic speed of sound of an atomic superfluid (SF) Bose-Einstein condensate in a 1D optical lattice. Because the speed of sound derives from the SF density, this implies that the SF density is itself anisotropic. We find that the speed of sound is decreased by the optical lattice, and the SF density is concomitantly reduced. This reduction is accompanied by the appearance of a zero entropy normal fluid in the purely Bose condensed phase. The reduction in SF density-first predicted [A. J. Leggett, Phys. Rev. Lett. 25, 1543 (1970).PRLTAO0031-900710.1103/PhysRevLett.25.1543] in the context of supersolidity-results from the coexistence of superfluidity and density modulations, but is agnostic about the origin of the modulations. We additionally measure the moment of inertia of the system in a scissors mode experiment, demonstrating the existence of rotational flow. As such we shed light on some supersolid properties using imposed, rather than spontaneously formed, density order.
The cubic nonlinear Schrödinger equation with repulsive nonlinearity and an elliptic function potential models a quasi-one-dimensional repulsive dilute gas Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schrödinger equation nor in the integrable nonlinear Schrödinger equation. Their stability is examined using analytical and numerical methods. All trivial-phase stable solutions are deformations of the ground state of the linear Schrödinger equation. Our results show that a large number of condensed atoms is sufficient to form a stable, periodic condensate. Physically, this implies stability of states near the Thomas-Fermi limit.