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BTELab, a MATLAB Programme Code for solving Quantum Boltzmann Transport Equation with all Collision Terms within Trapping Potential

Sunipa Som, Ravindra Dhar, Ram Kripal, Heinrich Stolz

Published Oct 9, 2026• 0 citations
Publisher

Abstract

Abstract This paper presents Quantum Boltzmann Transport Equation Laboratory (QBTELab), an advanced easy-to-use toolbox to solve many complex problems related to Bose-Einstein Condensate (BEC). The model equation that BTELab solves is the Quantum Boltzmann Transport Equation (QBTE) with all collision terms within trapping potential. Inelastic scattering (Exciton-phonon collision), elastic scattering (exciton-exciton collision), Auger decay and radiative non-radiative decay are included here as the collision terms. The programme code has been written in MATLAB. From the results it can be said that the programme is working properly and it has enough numerical stability. The aim of this paper is to provide programme code to solve the full QBTE. QBTE is a powerful tool for investigating the transport phenomena in various systems like semiconductors, plasmas, nuclear reactors, heat transfer process, fluid dynamics etc and specially different problems related to BEC.

DOI
10.21203/rs.3.rs-11150894/v1

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Abstract This paper presents Quantum Boltzmann Transport Equation Laboratory (QBTELab), an advanced easy-to-use toolbox to solve many complex problems related to Bose-Einstein Condensate (BEC). The model equation that BTELab solves is the Quantum Boltzmann Transport Equation (QBTE) with all collision terms within trapping potential. Inelastic scattering (Exciton-phonon collision), elastic scattering (exciton-exciton collision), Auger decay and radiative non-radiative decay are included here as the collision terms. The programme code has been written in MATLAB. From the results it can be said that the programme is working properly and it has enough numerical stability. The aim of this paper is to provide programme code to solve the full QBTE. QBTE is a powerful tool for investigating the transport phenomena in various systems like semiconductors, plasmas, nuclear reactors, heat transfer process, fluid dynamics etc and specially different problems related to BEC.
The Quantum Lattice Boltzmann Method (QLBM) has emerged as one of the most promising quantum computing approaches for the numerical simulation of problems in computational fluid dynamics (CFD). The dynamics is formulated in terms of mesoscopic particle distribution functions governed by a discrete Boltzmann transport equation, comprising local streaming and collision operations. In this work, the resulting macroscopic behavior corresponds to the advection-diffusion equation, which we adopt as a canonical model problem for transport phenomena. Building upon recent progress in QLBM implementations, we advance towards more realistic problem settings that better reflect conventional CFD requirements. We address, for the first time, transport under the action of non uniform velocity fields on quantum hardware. We implement our demonstration using IonQ's trapped-ion systems including Forte generation systems and a 64-qubit Barium development system similar to the forthcoming IonQ Tempo line. We identify the density readout and subsequent reloading of the fluid density as a potential bottleneck of the current algorithm and discuss several approaches to mitigate this bottleneck. We identify the use of MPS shadow tomography as a promising method to efficiently scale the readout to large system with complex density distributions. Lastly, we introduce and simulate a novel method to implement wall boundaries for advection-diffusion in QLBM, and discuss the prospects of scaling to higher-complexity problems.
The non-Markoffian transport equations for the systems of cold Bose atoms confined by a external potential both without and with a Bose-Einstein condensate are derived in the framework of nonequilibrium thermal filed theory (Thermo Field Dynamics). Our key elements are an explicit particle representation and a self-consistent renormalization condition which are essential in thermal field theory. The non-Markoffian transport equation for the non-condensed system, derived at the two-loop level, is reduced in the Markoffian limit to the ordinary quantum Boltzmann equation derived in the other methods. For the condensed system, we derive a new transport equation with an additional collision term which becomes important in the Landau instability.
The transport of quantum particles in non ideal material media (eg the conduction of electrons in an imperfect crystal) is strongly affected by scattering from impurities of the medium. Even for a weak disorder, semi-classical theories, such as those based on the Boltzmann equation for matter-waves scattering from the impurities, often fail to describe transport properties and full quantum approaches are necessary. The properties of the quantum systems are of fundamental interest as they show intriguing and non-intuitive phenomena that are not yet fully understood such as Anderson localization, percolation, disorder-driven quantum phase transitions and the corresponding Bose-glass or spin-glass phases. Understanding quantum transport in amorphous solids is one of the main issues in this context, related to electric and thermal conductivities. Ultracold atomic gases can now be considered to revisit the problem of quantum conductivity and quantum transport under unique control possibilities. Dilute atomic Bose-Einstein condensates (BEC) and degenerate Fermi gases (DFG) are produced routinely taking advantage of the recent progress in cooling and trapping of neutral atoms. Transport has been widely investigated in controlled potentials with no defects, for instance periodic potentials (optical lattices). Controlled disordered potentials can also be produced with various techniques such as the use of magnetic traps designed on atomic chips with rough wires, the use of localized impurity atoms, the use of radio-frequency fields or the use of optical potentials. This recently lead to the observation of the Anderson Localisation of a BEC and the measurement of the diffusion coefficient of a 2D gaz in the presence of disorder.
Transport phenomena still stand as one of the most challenging problems in computational physics. By exploiting the analogies between Dirac and lattice Boltzmann equations, we develop a quantum simulator based on pseudospin-boson quantum systems, which is suitable for encoding fluid dynamics transport phenomena within a lattice kinetic formalism. It is shown that both the streaming and collision processes of lattice Boltzmann dynamics can be implemented with controlled quantum operations, using a heralded quantum protocol to encode non-unitary scattering processes. The proposed simulator is amenable to realization in controlled quantum platforms, such as ion-trap quantum computers or circuit quantum electrodynamics processors.
We construct a nonequilibrium theory for the dynamics of two interacting finite-temperature atomic Bose-Einstein condensates and use it to numerically estimate the relative rates of the arising collisional processes near equilbrium. The condensates are described by dissipative Gross-Pitaevskii equations, coupled to quantum Boltzmann equations for the thermal atoms. The density-density interactions between atoms in different components facilitate a number of transport processes of relevance to sympathetic cooling: in particular, considering realistic miscible and immiscible trapped atomic $^{87}\mathrm{Rb}\text{\ensuremath{-}}^{41}\mathrm{K}$ and $^{87}\mathrm{Rb}\text{\ensuremath{-}}^{85}\mathrm{Rb}$ condensate mixtures, we highlight the dominance of an intercomponent scattering process associated with collisional ``exchange'' of condensed and thermal atoms between the components close to equilibrium.