We introduce a non-commutative generalization of the Gross-Pitaevskii equation for one-dimensional quantum gasses and quantum liquids. This generalization is obtained by applying the time-dependent variational principle to the variational manifold of continuous matrix product states. This allows for a full quantum description of many body system —including entanglement and correlations— and thus extends significantly beyond the usual mean-field description of the Gross-Pitaevskii equation, which is known to fail for (quasi) one-dimensional systems. By linearizing around a stationary solution, we furthermore derive an associated generalization of the Bogoliubov – de Gennes equations. This framework is applied to compute the steady state response amplitude to a periodic perturbation of the potential.
We introduce a non-commutative generalization of the Gross-Pitaevskii equation for one-dimensional quantum gasses and quantum liquids. This generalization is obtained by applying the time-dependent variational principle to the variational manifold of continuous matrix product states. This allows for a full quantum description of many body system —including entanglement and correlations— and thus extends significantly beyond the usual mean-field description of the Gross-Pitaevskii equation, which is known to fail for (quasi) one-dimensional systems. By linearizing around a stationary solution, we furthermore derive an associated generalization of the Bogoliubov – de Gennes equations. This framework is applied to compute the steady state response amplitude to a periodic perturbation of the potential.
Abstract Using the linearized version of the time dependent Gross–Pitaevskii equation, we calculate the dynamic response of a Bose–Einstein condensed gas to periodic density and particle perturbations. The zero temperature limit of the fluctuation—dissipation theorem is used to evaluate the corresponding quantum fluctuations induced by the elementary excitations in the ground state. In uniform conditions the predictions of Bogoliubov theory, including the infrared divergency of the particle distribution function and the quantum depletion of the condensate, are exactly reproduced by Gross–Pitaevskii theory. Results are also given for the crossed particle-density response function and the extension of the formalism to nonuniform systems is discussed. The generalization of the Gross–Pitaevskii equation to include beyond mean field effects is finally considered and an explicit result for the chemical potential is found, in agreement with the prediction of Lee–Huang–Yang theory.
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.
We have derived some excitation states of the one-dimensional Gross-Pitaevskii equation coupled by the gravitational potential. The methods that we have used here are taken by pursuing the recent work of Kivshar et. al. by considering the equation as a macroscopic quantum oscillator. To obtain the states, we have made the appropriate transformation to reduce the three-dimensional Gross-Pitaevskii equation into the one-dimensional Gross-Pitaevskii equation and applying the time-independent perturbation theory in the general solution of the one-dimensional Gross-Pitaevskii equation as a linear superposition of the normalized eigenfunctions of the Schrödinger equation for the harmonic oscillator potential. Moreover, we also impose the condition by assuming that some terms in the equation should be so small in order to preserve the use of the perturbation method.
Systems of interest in physics are usually composed by a very large number of interacting particles. At equilibrium, these systems are described by stationary states of the many-body Hamiltonian (at zero temperature, by the ground state). The reaction to perturbations, for example to a change of the external fields, is governed by the time-dependent many-body Schroedinger equation. Since it is typically very difficult to extract useful information from the Schroedinger equation, one of the main goals of non-equilibrium statistical mechanics is the derivation of effective evolution equations which can be used to predict the macroscopic behavior of the system. In these notes, we are going to consider systems of interacting bosons in the so called Gross-Pitaevskii regime, and we are going to show how coherent states and Bogoliubov transformations can be used to approximate the many body dynamics.
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 present a microscopic derivation of the time-dependent Gross-Pitaevskii equation starting from an interacting N-particle system of Bosons. We prove convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the respective Gross-Pitaevskii equation. Our work extends a previous result by one of us (P.P.[44]) to interaction potentials which need not to be nonnegative, but may have a sufficiently small negative part. One key estimate in our proof is an operator inequality which was first proven by Jun Yin, see [49].