A possibility of polarization of the gas of hydrogen-like atoms at zero temperature in the Bose–Einstein condensate is studied at equilibrium and nonequilibrium conditions. The Gross–Pitaevskii equation is generalized for a gas of atoms considering their electronic structure. It is shown that the wave function of the system of interacting bosons in a single-particle condensate at zero temperature looks like the Glauber coherent state. With the help of the generalized Gross–Pitaevskii equation the polarization properties of atomic gas in the Bose–Einstein condensation state are investigated. It is shown that the electron stationary states in atoms of a gas in the coherent state differ from the electron stationary states in free atoms. Because of this the superfluid system can be spontaneously polarized, even if the free atoms in the stationary states have no intrinsic electric dipole moment. The polarization properties of a system of double-level hydrogen-like atoms in the Bose–Einstein condensate are studied more comprehensively. The propagation of both sound and high-frequency small disturbances in the coherent system of polarized atoms is accompanied by oscillations of polarization. The applicability of the developed idea to the analysis of electrical phenomena in superfluid helium is discussed.
A possibility of polarization of the gas of hydrogen-like atoms at zero temperature in the Bose–Einstein condensate is studied at equilibrium and nonequilibrium conditions. The Gross–Pitaevskii equation is generalized for a gas of atoms considering their electronic structure. It is shown that the wave function of the system of interacting bosons in a single-particle condensate at zero temperature looks like the Glauber coherent state. With the help of the generalized Gross–Pitaevskii equation the polarization properties of atomic gas in the Bose–Einstein condensation state are investigated. It is shown that the electron stationary states in atoms of a gas in the coherent state differ from the electron stationary states in free atoms. Because of this the superfluid system can be spontaneously polarized, even if the free atoms in the stationary states have no intrinsic electric dipole moment. The polarization properties of a system of double-level hydrogen-like atoms in the Bose–Einstein condensate are studied more comprehensively. The propagation of both sound and high-frequency small disturbances in the coherent system of polarized atoms is accompanied by oscillations of polarization. The applicability of the developed idea to the analysis of electrical phenomena in superfluid helium is discussed.
It is shown that the condensate of a degenerated Bose gas consisting of neutral atoms possesses electrical properties which differ from a trivial polarization of the atoms in the electric field.A notion of an isotropic quadrupole moment (IQM) of a neutral atom is introduced.A distribution of IQM reflects a specific spatial ordering in the condensate and produces a distribution of the electric potential.Small vibrations of the Bose gas are considered and a correction to the Bogoliubov spectrum of elementary excitations in the degenerated Bose gas is obtained.An additional term in the Gross-Pitaevskii equation which is responsible for such a correction is found and a new type of the nonlinear Schr ödinger equation (NSE) is constructed.Since the Bose condensate is akin to the superfluid component in He II, a manifestation of its electrical activity could have a relation to the electrical activity of the superfluid liquid observed experimentally.
Spin-polarized atomic hydrogen continues to be one of the most promising candidates for Bose condensation of an atomic system. In contrast to liquid helium, hydrogen is gaseous and therefore its density can be changed to vary its behavior from the weakly to the strongly interacting Bose gas. Until now, efforts to Bose condense hydrogen have been thwarted by recombination and relaxation phenomena. After a long introduction to the subject, two promising approaches to observe quantum degenerate behavior are discussed: the microwave trap and a two-dimensional gas of hydrogen. Introduction Since the stabilization of atomic hydrogen as a spin-polarized gas (H↓), reported in 1980 [1], there has been a continuous effort to observe Bose–Einstein condensation (BEC) or other effects of quantum degeneracy in this Bose gas. This challenge has not yet been realized due to the instability of H↓ towards recombination to H 2 or relaxation among hyperfine states as the conditions for BEC are approached. Although this difficulty is formidable, hydrogen presents a unique opportunity to study BEC and the related superfluidity, as it remains a gas to T = 0 [2]. By comparison, 4 He, the only experimentally observed boson fluid, is a strongly interacting superfluid Bose liquid, with little flexibility for varying its density. As a result of the gaseous nature of hydrogen, its density, n , can be varied over several orders of magnitude with the possibility of studying BEC and its relationship to superfluidity, ranging from the weakly to the strongly interacting boson gas.
In the present chapter we consider the structure of the Bose–Einstein condensed state in the presence of interactions. Our discussion is based on the Gross–Pitaevskii equation, which describes the zero-temperature properties of the non-uniform Bose gas when the scattering length a is much less than the mean interparticle spacing. We shall first derive the Gross–Pitaevskii equation at zero temperature by treating the interaction between particles in a mean-field approximation (Sec. 6.1). Following that, in Sec. 6.2 we discuss the ground state of atomic clouds in a harmonic-oscillator potential. We compare results obtained by variational methods with those derived in the Thomas–Fermi approximation, in which the kinetic energy operator is neglected in the Gross–Pitaevskii equation. The Thomas–Fermi approximation fails near the surface of a cloud, and in Sec. 6.3 we calculate the surface structure using the Gross–Pitaevskii equation. In Sec. 6.4 we determine how the condensate wave function ‘heals’ when subjected to a localized disturbance. Finally, in Sec. 6.5 we show how the magnetic dipole–dipole interaction, which is long-ranged and anisotropic, may be included in the Gross–Pitaevskii equation and determine within the Thomas–Fermi approximation its effect on the density distribution. The Gross–Pitaevskii equation In the previous chapter we have shown that the effective interaction between two particles at low energies is a constant in the momentum representation, U 0 = 4π ħ 2 a/m .
Spin-polarized, atomic hydrogen at temperatures below 1K shows promise of becoming the first physically realizable example of a degenerate, weakly interacting Bose gas. Because of the hyperfine structure of the hydrogen atom the Bose condensed state is expected to exhibit novel magnetic properties such as spin wave excitations of the superfluid and possibly a coherent precessing, spontaneous magnetization. This paper discusses a variety of theoretical problems which arise in the study of atomic hydrogen gas. Theories are presented for spin relaxation of the hyperfine levels and for the recombination of H-atoms into H2 molecules. The problem of the binding of H atoms to a surface of superfluid 4He is reviewed together with the theory of surface superfluidity in the hydrogen film proposed by Edwards and Mantz. Finally we consider the problem of superfluidity in the bulk atomic hydrogen gas. We describe the theory of Siggia and Ruckenstein of double Bose condensation into the two lowest hyperfine states and speculate about the possible observable consequences of this phenomenon.
Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in L2-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others. Full-text embargoed until: 2026-01-01
The issue of adequate describing a gas of complex particles, composed of an even number of fermions, as point bosons is studied in alkali metals. In the low-density approximation we obtain the equation for the complex order parameter Φ(r1, r2) representing the wave function of atoms with taking into account the internal motion of the valence electrons. For superfluid systems formed by particles with internal degrees of freedom, this equation replaces the Gross-Pitaevskii equation. It is shown that, in general, exchange effects should be considered in the same approximation as effects of the direct interaction of atoms with each other. In particular, in the case of only the Coulomb interaction the neglect of exchange effects leads to qualitatively incorrect results. The problem of spontaneous polarization in superfluid systems is studied. The expression for the electric polarization of the inhomogeneous superfluid system is obtained.