The center-of-mass (c.m.) oscillation of a many-body system in a harmonic trap is known to be independent of the interparticle interaction. However, this is not necessarily the case if the interactions are treated approximately. Here, we prove a simple general criterion for preservation of the c.m. mode: the approximation has to preserve density and momentum. The result equally applies to zero and finite temperatures, as well as to nonequilibrium situations, and to the linear and nonlinear response regimes.
The center-of-mass (c.m.) oscillation of a many-body system in a harmonic trap is known to be independent of the interparticle interaction. However, this is not necessarily the case if the interactions are treated approximately. Here, we prove a simple general criterion for preservation of the c.m. mode: the approximation has to preserve density and momentum. The result equally applies to zero and finite temperatures, as well as to nonequilibrium situations, and to the linear and nonlinear response regimes.
The presence of undamped harmonic center-of-mass oscillations of a weakly interacting Bose gas in a harmonic trap is demonstrated within the dielectric formalism for a previously introduced finite-temperature approximation, including exchange. The consistency of the approximation with the Kohn theorem is thereby demonstrated. The Kohn modes are found explicitly, generalizing an earlier zero-temperature result found in the literature. It is shown how the Kohn mode disappears from the single-particle spectrum, while remaining in the density oscillation spectrum, when the temperature increases from below to above the condensation temperature.
The presence of (approximate) conservation laws can prohibit the fast relaxation of interacting many-particle quantum systems. We investigate this physics by studying the center-of-mass oscillations of two species of fermionic ultracold atoms in a harmonic trap. If their trap frequencies are equal, a dynamical symmetry (spectrum-generating algebra), closely related to Kohn's theorem, prohibits the relaxation of center-of-mass oscillations. A small detuning $\ensuremath{\delta}\ensuremath{\omega}$ of the trap frequencies for the two species breaks the dynamical symmetry and ultimately leads to a damping of dipole oscillations driven by interspecies interactions. Using memory-matrix methods, we calculate the relaxation as a function of frequency difference, particle number, temperature, and strength of interspecies interactions. When interactions dominate, there is almost perfect drag between the two species and the dynamical symmetry is approximately restored. The drag can either arise from Hartree potentials or from friction. In the latter case (hydrodynamic limit), the center-of-mass oscillations decay with a tiny rate, $1/\ensuremath{\tau}\ensuremath{\propto}{(\ensuremath{\delta}\ensuremath{\omega})}^{2}/\mathrm{\ensuremath{\Gamma}}$, where $\mathrm{\ensuremath{\Gamma}}$ is a single-particle scattering rate.
Single-component quantum gas confined in a harmonic potential, but otherwise isolated, is considered. From the invariance of the system of the gas under a displacement-type transformation, it is shown that the center of mass oscillates along a classical trajectory of a harmonic oscillator. It is also shown that this harmonic motion of the center has, in fact, been implied by Kohn's theorem. If there is no interaction between the atoms of the gas, the system in a time-independent isotropic potential of frequency ${\ensuremath{\nu}}_{c}$ is invariant under a squeeze-type unitary transformation, which gives collective radial breathing motion of frequency $2{\ensuremath{\nu}}_{c}$ to the gas. The amplitudes of the oscillating and breathing motions from the exact invariances could be arbitrarily large. For a Fermi system, appearance of $2{\ensuremath{\nu}}_{c}$ mode of the large breathing motion indicates that there is no interaction between the atoms, except for a possible long-range interaction through the inverse-square-type potential.
Two methods to calculate the density matrix of one• body or more•than•one•body which does not include the spurious center•of•mass excitation are presented.One is the method to calculate the density matrix for the many-body wave function of general type and the other is applicable to the case in which the center~of-mass motion separates from the internal motion and is in the (Os) orbit of harmonic oscillator.The applications of these methods are illustrated for some many-body wave function.§ 1. IntroductionTo represent the characteristic features of the many-body wave function of nuclei, the density matrices are often utilized.Experimentally, for example, in the electron scattering from nuclei, the nuclear structure is analyzed by the terms of the nuclear density and the current density, which are the special forms of the one-body density matrix.Theoretically, the one-body density matrix plays a crucial role in the current nuclear theories based on the mean field approximation, such as Hartree-Fock theory.And, in recent nuclear studies, interest is taken in not only the nuclear density which is the diagonal part of the one-body density matrix, but the non-local part of it.The non-local part of the one-body density matrix is often visualized with the use of the Wigner transform of it. 1),2)Especially, in the time-dependent Hartree-Fock theory, the Wigner transform of the one-body density matrix is interpreted as the classical distribution function and obeys the Vlasov equation approximately.3)These works clarify the importance of the non-local part of the one-body density matrix in the dynamics.Usually the spatial coordinate of the density matrix is measured from some spatially fixed origin.Due to the finiteness of the nuclear system, however, we need to measure the coordinate from the center of mass of the system; otherwise the spurious center-of-mass motion affects the density matrix.This effect of center-of-mass motion becomes important for light systems.For the diagonal part of the one-body transition density matrix, the effect of center-of-mass motion was discussed by Tassie and Barker for the case of harmonic oscillator shell mode1. 4) And for the diagonal part of the one-body density matrix in momentum representation, that is the nuclear momentum distribution, the effect of spurious center-of-mass motion has been discussed. 5) For the general non-local density matrix, however, there seems not to be presented any method to treat the spurious center-of-mass effect.In this paper, we present two methods to get the center-of-mass free density matrix.One is the method, applicable to the general wave function, which is especially suitable for the case where the wave function is a (linear superposition of) Slater determinant.The other is applicable to the cases in which the center-of-mass motion separates from other internal motions and is in the (Os) orbit of harmonic oscillator.For such wave functions, the latter method is more easy to calculate though the former method is also applicable.
We consider the criterion for Bose-Einstein condensation for particles in a harmonic trap. For a fixed angular momentum, the lowest energy state for a cloud of bosons with attractive interactions is the ground state of the cloud with all the angular momentum in the center-of-mass motion, and the one-particle reduced density matrix generally does not have a single large eigenvalue, but a number of them, suggesting that the state is an example of a fragmented condensate [N. K. Wilkin, J. M. F. Gunn, and R. A. Smith, Phys. Rev. Lett. 80, 2265 (1998)]. We show that a convenient way to describe correlations in the system is by defining an internal one-particle reduced density matrix, in which the center-of-mass motion is eliminated, and that this has a single eigenvalue equal to the number of particles for the problem considered here. Our considerations indicate that care is necessary in formulating a criterion for Bose-Einstein condensation.
A field theoretical formulation of the rather easy problem of N identical, one-dimensional bosons, interacting pairwise via harmonic oscillator potentials of equal force constants is given, in a way similar to the way the nonlinear Schrodinger model provides a field theoretical formulation of a one-dimensional, delta -function Bose gas. The nonlinear integro-differential equation obtained for the complex scalar field of the model is treated classically, and solved in the semiclassical limit of a large number of particles. Working in the centre of mass frame of the field, a sequence of operators that would raise or lower the energy of a state by definite amounts without changing its particle number are constructed from certain bilinear expressions in the field. These operators have, in the limit of a large number of particles, commutation relations similar to those of the Fourier modes of a compact free field, except that the first mode is missing as a result of the fact that the field in the centre of mass frame is subject to the obvious constraint that its centre of mass always coincides with the coordinate origin. The quantum states can be labelled by the total centre of mass momentum, and the quasi-particle numbers in each of the independent oscillator modes. The energy spectrum of the system is calculated, giving the same results as the exact, quantum-mechanical N-body analysis.