Modeling and computation of Bose-Einstein condensates: stationary states, nucleation, dynamics, stochasticity.

*(English)*Zbl 1344.35114
Besse, Christophe (ed.) et al., Nonlinear optical and atomic systems. At the interface of physics and mathematics. Based on lecture notes given at the 2013 Painlevé-CEMPI-PhLAM thematic semester. Cham: Springer; Lille: Centre Européen pour les Mathématiques, la Physiques et leurs Interactions (CEMPI) (ISBN 978-3-319-19014-3/pbk; 978-3-319-19015-0/ebook). Lecture Notes in Mathematics 2146, 49-145 (2015).

The authors start with the description of the historical background of the discovery of Bose-Einstein condensates (BECs), beginning with the paper of S. N. Bose in 1924, who proposed for photons a new statistics which includes quantum effects in contrast to the Maxwell-Boltzmann statistics. The generalization of this idea to atoms by A. Einstein and his prediction of a new state of matter led to the now so-called Bose-Einstein condensates. Mentioned are also the most important attempts and progress of the experimentalists to approve the existence of the condensates before E. A. Cornell and C. E. Wiemar (using rubidium atoms) on the one hand, and W. Ketterle (using sodium atoms) otherwise have been awarded the Nobel 2001 price for their realization of Bose-Einstein condensates in 1995 in two independent experiments.

The authors choose the Gross-Pitaevskii equations (GPEs) under various possibilities to mathematically describe the BECs. Starting with the Euler-Lagrange equations and the corresponding Hamiltonian equations, which characterize the dynamics in classical mechanics, the authors adopt the Hamiltonian approach to quantum particles which are realized through a wave function. The wave function, associated to the particle, determines the probability that the particle is located in a given volume at a time \(t\). The particle is described by the de Broglie’s relations. The total energy is given via the Hamiltonian. Using these facts an evolution equation for the wave function with the Hamiltonian is deduced, that is the Schrödinger equation, which describes the dynamics of the wave function associated to the particles. The wave function is generalized to a system of \(N\) particles and the Hamiltonian is formulated in the first instance for an example of \(N\) noninteracting particles subject to an exterior potential.

The theory is applied to BECs, in which the set of condensed particles occupies the same ground state, that is the lowest level quantum energy state. The Hamiltonian of the system is deduced assuming that the condensate consists of \(N\) indistinguishable particles with the same wave function subject to an exterior potential and a force which depends on the interaction between the particles. The corresponding Schrödinger equation results in the so-called Gross-Pitaevskii equation using simplifications for the particle interaction force. Some classes of GPEs are deduced, such as for rotating BECs to describe superfluids, BECs without (e.g. alkali and hydrogen atoms) and including (e.g. chromium atoms) bipolar interactions, multi-components BECs, and BECs with stochastic effects.

Furthermore, some details are outlined, such as that the stationary states are the eigenfunctions of the Hamiltonian operator and the corresponding eigenvalues quantified energies. It is proved that the stationary states are critical points of the energy functional. Various approaches for the potential are discussed. Dimensionless forms of GPEs and a dimension reduction are treated.

The practical realization of a BEC and especially its imaging is a very difficult task. Thus, numerical simulations are required to compute the features of a BEC.

Stationary states correspond to stable or metastable states of BECs. The stationary states can be computed solving a nonlinear eigenvalue problem or minimizing the energy functional under constraint. The last one is a nonlinear optimization problem and is discussed in this paper using the so-called Conjugate Normalized Gradient Flow (CNGF) method (also known as imaginary time method), which generates a minimizing sequence of the energy functional. Several time and space discretizations of the corresponding partial differential equation are discussed. The authors consider a semi-implicit backward Euler scheme in time with the advantage, that a minimizing sequence is produced without a Courant-Friedrichs-Lewy condition, and compare it with Crank-Nicolson schemes. Two approaches are represented for the spatial discretization: a second-order finite difference scheme and a pseudo-spectral discretization technique based on a Fast Fourier Transform (FFT). The advantages and disadvantages of the methods are discussed in detail and validated for different BECs.

The next topic consists in the determination of an suitable initial guess to the nonlinear optimization problem for different situations and in the construction of simple approximations. Using the Thomas-Fermi approximation based on the neglecting of the kinetic energy in the strong interaction, simplified minimization problems are deduced for various potentials.

Furthermore, it is outlined that Krylov subspace iterative solvers, such as GMRES and BiCGStab, accelerated by preconditioning, are the most robust and effective algorithms for the solution of the linear systems, which have to be solved in each iteration step of the minimization problem using the semi-implicit backward Euler scheme for the FFT based pseudo-spectral discretization.

The presented numerical methods are implemented in a freely available Matlab toolbox named GPELab (Gross-Pitaevskii Equation Laboratory). The authors indicates that not only different kinds of Gross-Pitaevskii equations and systems can be solved but also nonlinear Schrödinger equations. The effectivity of the software is demonstrated by means of some examples.

The numerical solution of the dynamics of deterministic or stochastic GPEs is the next topic of the paper. After the formulation of the corresponding GPEs time-splitting pseudo-spectral schemes and relaxation schemes for rotating GPEs are treated and applied for various BECs. Also the essential properties of other schemes are outlined.

For the entire collection see [Zbl 1328.35002].

The authors choose the Gross-Pitaevskii equations (GPEs) under various possibilities to mathematically describe the BECs. Starting with the Euler-Lagrange equations and the corresponding Hamiltonian equations, which characterize the dynamics in classical mechanics, the authors adopt the Hamiltonian approach to quantum particles which are realized through a wave function. The wave function, associated to the particle, determines the probability that the particle is located in a given volume at a time \(t\). The particle is described by the de Broglie’s relations. The total energy is given via the Hamiltonian. Using these facts an evolution equation for the wave function with the Hamiltonian is deduced, that is the Schrödinger equation, which describes the dynamics of the wave function associated to the particles. The wave function is generalized to a system of \(N\) particles and the Hamiltonian is formulated in the first instance for an example of \(N\) noninteracting particles subject to an exterior potential.

The theory is applied to BECs, in which the set of condensed particles occupies the same ground state, that is the lowest level quantum energy state. The Hamiltonian of the system is deduced assuming that the condensate consists of \(N\) indistinguishable particles with the same wave function subject to an exterior potential and a force which depends on the interaction between the particles. The corresponding Schrödinger equation results in the so-called Gross-Pitaevskii equation using simplifications for the particle interaction force. Some classes of GPEs are deduced, such as for rotating BECs to describe superfluids, BECs without (e.g. alkali and hydrogen atoms) and including (e.g. chromium atoms) bipolar interactions, multi-components BECs, and BECs with stochastic effects.

Furthermore, some details are outlined, such as that the stationary states are the eigenfunctions of the Hamiltonian operator and the corresponding eigenvalues quantified energies. It is proved that the stationary states are critical points of the energy functional. Various approaches for the potential are discussed. Dimensionless forms of GPEs and a dimension reduction are treated.

The practical realization of a BEC and especially its imaging is a very difficult task. Thus, numerical simulations are required to compute the features of a BEC.

Stationary states correspond to stable or metastable states of BECs. The stationary states can be computed solving a nonlinear eigenvalue problem or minimizing the energy functional under constraint. The last one is a nonlinear optimization problem and is discussed in this paper using the so-called Conjugate Normalized Gradient Flow (CNGF) method (also known as imaginary time method), which generates a minimizing sequence of the energy functional. Several time and space discretizations of the corresponding partial differential equation are discussed. The authors consider a semi-implicit backward Euler scheme in time with the advantage, that a minimizing sequence is produced without a Courant-Friedrichs-Lewy condition, and compare it with Crank-Nicolson schemes. Two approaches are represented for the spatial discretization: a second-order finite difference scheme and a pseudo-spectral discretization technique based on a Fast Fourier Transform (FFT). The advantages and disadvantages of the methods are discussed in detail and validated for different BECs.

The next topic consists in the determination of an suitable initial guess to the nonlinear optimization problem for different situations and in the construction of simple approximations. Using the Thomas-Fermi approximation based on the neglecting of the kinetic energy in the strong interaction, simplified minimization problems are deduced for various potentials.

Furthermore, it is outlined that Krylov subspace iterative solvers, such as GMRES and BiCGStab, accelerated by preconditioning, are the most robust and effective algorithms for the solution of the linear systems, which have to be solved in each iteration step of the minimization problem using the semi-implicit backward Euler scheme for the FFT based pseudo-spectral discretization.

The presented numerical methods are implemented in a freely available Matlab toolbox named GPELab (Gross-Pitaevskii Equation Laboratory). The authors indicates that not only different kinds of Gross-Pitaevskii equations and systems can be solved but also nonlinear Schrödinger equations. The effectivity of the software is demonstrated by means of some examples.

The numerical solution of the dynamics of deterministic or stochastic GPEs is the next topic of the paper. After the formulation of the corresponding GPEs time-splitting pseudo-spectral schemes and relaxation schemes for rotating GPEs are treated and applied for various BECs. Also the essential properties of other schemes are outlined.

For the entire collection see [Zbl 1328.35002].

Reviewer: Georg Hebermehl (Berlin)

##### MSC:

35Q40 | PDEs in connection with quantum mechanics |

35Q55 | NLS equations (nonlinear Schrödinger equations) |

82B10 | Quantum equilibrium statistical mechanics (general) |

82B26 | Phase transitions (general) in equilibrium statistical mechanics |

82-08 | Computational methods (statistical mechanics) (MSC2010) |

82-03 | History of statistical mechanics |

01A60 | History of mathematics in the 20th century |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

65T50 | Numerical methods for discrete and fast Fourier transforms |