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VNU Journal of Science Mathematics - Physics

Phenomenological Analogy between Gross-Pitaevskii Theory for Bose-Einstein Condensate and Newton Equation for Classical Mechanics

Nguyễn Văn Thụ, Pham Duy Thanh

Published Jun 25, 2024• 0 citations
Publisher

Abstract

In this work we found an interesting analogy between Gross-Pitaevskii theory for Bose-Einstein condensate at zero temperature and Newton equation of classical particle in the classical physics. Although this analogy is a pure phenomenology, it brings a new perspective to physics in general, in particular to quantum physics, as well as to many-body physics.

DOI
10.25073/2588-1124/vnumap.4922

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In this work we found an interesting analogy between Gross-Pitaevskii theory for Bose-Einstein condensate at zero temperature and Newton equation of classical particle in the classical physics. Although this analogy is a pure phenomenology, it brings a new perspective to physics in general, in particular to quantum physics, as well as to many-body physics.
Based on an analogy between the Gross-Pitaevskii (GP) equations for binary mixtures of Bose-Einstein condensates (BECs) at zero temperature and Newton’s equations of motion for a particle in a conservative field, we derive exact analytical solutions for the coupled GP equations in several phase-segregated BEC scenarios. In a straightforward manner, exact analytical solutions are obtained for both the symmetric and anti-symmetric cases. Furthermore, our approach enables us to propose approximate analytical solutions as an alternative to those derived from the widely used double-parabola approximation (DPA). Numerical computations demonstrate that our approximate solutions closely match the exact solutions within the GP framework. This finding is of particular significance for determining the wavefunctions of the condensates, a fundamental task in investigating the static properties of BECs.
The Gross–Pitaevskii equation (GPE) is discussed at the level of an advanced course on statistical physics. In the standard literature the GPE is usually obtained in the framework of the second quantization formalism, which in many cases goes beyond the material covered in many advanced undergraduate courses. In this paper, we motivate the derivation of the GPE in relationship to concepts from statistical physics, highlighting a number of applications from the dynamics of a Bose–Einstein condensate to the excitations of the gas cloud. This paper may be helpful for encouraging the discussion of modern developments in a statistical mechanics course, and can also be of use in other contexts such as mathematical physics and modelling. The paper is suitable for undergraduate and graduate students, as well as for general physicists.
An atomic Bose-Einstein condensate (BEC) is often described as a macroscopic object which can be approximated by a coherent state. This, on the surface, would appear to indicate that its behavior should be close to being classical. In this paper, we clarify the extent of how "classical" a BEC is by exploring the semiclassical equations for BECs under the mean field Gaussian approximation. Such equations describe the dynamics of a condensate in the classical limit in terms of the variables < x > and < p > as well as their respective variances. We compare the semiclassical solution with the full quantum solution based on the Gross-Pitaevskii Equation (GPE) and find that the interatomic interactions which generate nonlinearity make the system less "classical." On the other hand, many qualitative features are captured by the semiclassical equations, and the equations to be solved are far less computationally intensive than solving the GPE which make them ideal for providing quick diagnostics, and for obtaining new intuitive insight.
We show how to adapt the ideas of local energy and momentum conservation in order to derive modifications to the Gross-Pitaevskii equation which can be used phenomenologically to describe irreversible effects in a Bose-Einstein condensate. Our approach involves the derivation of a simplified quantum kinetic theory, in which all processes are treated locally. It is shown that this kinetic theory can then be transformed into a number of phase-space representations, of which the Wigner function description, although approximate, is shown to be the most advantageous. In this description, the quantum kinetic master equation takes the form of a Gross-Pitaevskii equation with noise and damping added according to a well defined prescription - an equation we call the stochastic Gross-Pitaevskii equation . From this, a very simplified description we call the phenomenological growth equation can be derived. We use this equation to study (i) the nucleation and growth of vortex lattices, and (ii) nonlinear losses in a hydrogen condensate, which it is shown can lead to a curious instability phenomenon.
The study of nonequilibrium dynamics in Bose-Einstein condensed gases has been dominated by the zero-temperature mean field Gross-Pitaevskii formalism. Motivated by recent experiments on the reflection of condensates from silicon surfaces, we revisit the so called classical field description of condensate dynamics, which incorporates the effects of quantum noise and can also be generalized to include thermal effects. The noise is included in a stochastic manner through the initial conditions. We show that the inclusion of such noise is important in the quantitative description of the recent reflection experiments.