The problem with the construction of the Gross-Pitaevskii (GP) equation and related wave functions in a medium is associated with the necessity to begin with a description of the configuration space and proceed to a description of the physical space. We show that the balance equations of the number of particles and momentum immediately follow from the multiparticle Schr\"odinger equation. From the obtained set of balance equations, the equation for the wave function in the medium coincides in form with the GP equation, where we can restrict ourselves only by the first-order terms of the interaction radius of the stress tensor of bosons, for dilute gases. As a generalization of the GP equation, we made allowance for the contribution of the third-order terms of the interaction radius to the stress tensor of bosons. For a system of particles that comprises an ultracold mixture of bosons and fermions, a two-kind quantum hydrodynamics is constructed for the third-order terms of the interaction radius. The spectrum of eigenmodes involves additional information on the interparticle interaction as a correction to the Bogoliubov spectrum.
The problem with the construction of the Gross-Pitaevskii (GP) equation and related wave functions in a medium is associated with the necessity to begin with a description of the configuration space and proceed to a description of the physical space. We show that the balance equations of the number of particles and momentum immediately follow from the multiparticle Schr"odinger equation. From the obtained set of balance equations, the equation for the wave function in the medium coincides in form with the GP equation, where we can restrict ourselves only by the first-order terms of the interaction radius of the stress tensor of bosons, for dilute gases. As a generalization of the GP equation, we made allowance for the contribution of the third-order terms of the interaction radius to the stress tensor of bosons. For a system of particles that comprises an ultracold mixture of bosons and fermions, a two-kind quantum hydrodynamics is constructed for the third-order terms of the interaction radius. The spectrum of eigenmodes involves additional information on the interparticle interaction as a correction to the Bogoliubov spectrum.
We describe recent development of quantum hydrodynamics for ultracold Bose particle studying and consider different kinds of interactions. The method of derivation of equations describing the evolution of the neutral Bose particle system at low temperatures is described. Despite the fact that we consider the neutral particles we account the short-range interaction between particles. We consider the particles in the Bose-Einstein condensate (BEC) state. This method is called the method of quantum hydrodynamics, because natural for of the quantum mechanics rewritten in terms of material fields of observable quantities in three dimensional space is the set of equations, which look like the hydrodynamics equations. It can be shown that from the quantum hydrodynamics equations can be derived macroscopic non-linear Schrodinger equation. Most famous non-linear Schrodinger equation is the Gross-Pitaevskii (GP) equation, which contains nonlinearity of the third degree. There are generalizations of the GP equation. New term appears in the GP equation at account of the three-particle interaction. This term contains nonlinearity of the fifth degree. At more detailed account of the two particle interaction we come to the nonlocal non-linear Schrodinger equation. This equation contains spatial derivatives of the order parameter in the non-linear terms caused by the interaction. In this terminology the GP equation corresponds to the first order by the interaction radius. For the BEC of the neutral particles with anisotropic long-range dipole-dipole interaction the generalization of the GP equation was also suggested. Detailed analyses of the applicability conditions shows that this equation valid for the system of dipoles parallel to each other, which do not change their direction, and where the dipole-dipole interaction interferences translational motion of particles.
A simple derivation of the static Gross-Pitaevskii (GP) equation is given from an energy variational principle. The result is then generalized heuristically to the time-dependent GP form. With this as background, a number of different experimental areas explored very recently are reviewed, in each case contact being established between the measurements and the predictions of the GP equations. The various limitations of these equations as used on dilute inhomogeneous condensed Boson atomic gases are then summarized, reference also being made to the fact that there is no many-body wave function underlying the GP formulation. This then leads into a discussion of a recently proposed integral equation, derived by taking the Bogoliubov-de Gennes equation as starting point. Some limitations of the static GP differential equation are thereby removed, though it is a matter of further study to determine whether a correlated wave function exists as underpinning for the integral equation formulation.
The Bogoliubov method for the excitation spectrum of a Bose-condensed gas is generalized to apply to a gas with an exact large number N of particles. This generalization yields a description of the Schr\"odinger picture field operators as the product of an annihilation operator A for the total number of particles and the sum of a ``condensate wave function'' $\ensuremath{\xi}(\mathbf{x})$ and a phonon field operator $\ensuremath{\chi}(\mathbf{x})$ in the form $\ensuremath{\psi}(\mathbf{x})\ensuremath{\approx}A{\ensuremath{\xi}(\mathbf{x})+\ensuremath{\chi}(\mathbf{x})/\sqrt{N}}$ when the field operator acts on the N particle subspace. It is then possible to expand the Hamiltonian in decreasing powers of N, and thus obtain solutions for eigenvalues and eigenstates as an asymptotic expansion of the same kind. It is also possible to compute all matrix elements of field operators between states of different N. The excitation spectrum can be obtained by essentially the same method as Bogoliubov only if $\ensuremath{\xi}(\mathbf{x})$ is a solution of the time-independent Gross-Pitaevskii equation for N particles and any chemical potential $\ensuremath{\mu}$, which yields a valid and stable solution of the Gross-Pitaevskii equation. The treatment within a subspace of fixed N is identical in form to that usually used, but the interpretation of the operators is slightly different. A time-dependent generalization is then made, yielding an asymptotic expansion in decreasing powers of N for the equations of motion. In this expansion the condensate wave function has the time-dependent form $\ensuremath{\xi}(\mathbf{x},t)$, and the condition for the validity of the expansion is that $\ensuremath{\xi}(\mathbf{x},t)$ satisfies the time-dependent Gross-Pitaevskii equation $\ensuremath{\partial}\ensuremath{\xi}/\ensuremath{\partial}t=\ensuremath{-}({\ensuremath{\Elzxh}}^{2}/2m){\ensuremath{\nabla}}^{2}\ensuremath{\xi}+V\ensuremath{\xi}+Nu|\ensuremath{\xi}{|}^{2}\ensuremath{\xi}$. The physics is then described in a kind of interaction picture, called the condensate picture, in which the phonon operator can be expressed as $\ensuremath{\chi}(\mathbf{x},t)={\ensuremath{\sum}}_{k}{\ensuremath{\xi}}_{k}(\mathbf{x},t){\ensuremath{\alpha}}_{k},$ where the operators ${\ensuremath{\alpha}}_{k}$ are time-independent annihilation operators, and the state vector has a time evolution described by a Schr\"odinger equation in which the Hamiltonian is a time-dependent quadratic form in the phonon creation and annihilation operators, whose coefficients are explicitly determined in terms of the time-dependent condensate wave function $\ensuremath{\xi}(\mathbf{x},t)$.
We establish a complete set of hydrodynamic equations for a spin-1 Bose-Einstein condensate, which are equivalent to the multicomponent Gross-Pitaevskii equations and expressed in terms of only observable physical quantities: the spin and the nematic (or quadrupolar) tensor in addition to the density of particles and the mass current that appear in the hydrodynamic description of a scalar condensate. The obtained hydrodynamic equations involve a generalized Mermin-Ho relation that is valid regardless of the spatiotemporal dependence of the spin polarization. Low-lying collective modes for phonons and magnons are reproduced by linearizing the hydrodynamic equations. We also apply the single-mode approximation to the hydrodynamic equations and find a complete set of analytic solutions.
We show that the Josephson plasma frequency for a condensate in a double-well potential, whose dynamics is described by the Gross-Pitaevskii (GP) equation, can be obtained with great precision by means of the usual Bogoliubov approach, whereas the two-mode model---commonly constructed by means of a linear combination of the low-lying states of the GP equation---generally provides accurate results only for weak interactions. A proper two-mode model in terms of the Bogoliubov functions is also discussed, revealing that in general a two-mode approach is formally justified only for not too large interactions, even in the limit of very small amplitude oscillations. Here we consider specifically the case of a one-dimensional system, but the results are expected to be valid in arbitrary dimensions.
It has been proven theoretically for bosons with two-body repulsive interaction potentials in the dilute limit that the Gross-Pitaevskii equation provides the exact energy and density per particle as does the basic many-particle Schr\"odinger equation [E. H. Lieb and R. Seiringer, Phys. Rev. Lett. 88, 170409 (2002)]. Here, we investigate the overlap of the Gross-Pitaevskii and exact ground-state wave functions. It is found that this overlap is always smaller than unity and may even vanish despite the fact that both wave functions provide the same energy and density per particle. Consequences are discussed.