728x90
반응형
by Kang-Soo Lee and Uwe R. Fischer
I apologize for not finishing this paper. I just open this in the internet. But this document is flawed as it is originally written in LaTeX and not translated to the docuK format perfectly yet. This paper is published in Int. J. Mod. Phys. B 84, 1550021 (2014) and arXiv>cond-mat>arXiv:1301.2199.
##[.no-sec-N] Abstract
We introduce a scheme to describe the evolution of an interacting system of bosons, for which the field operator expansion is truncated after a finite number of modes, in a rigorously controlled manner. Using McLachlan's principle of least error , we find a set of equations for the many-body state. As a particular benefit, and in distinction to previously proposed approaches , our approach allows for the dynamical increase of the number of orbitals during the temporal evolution, because we can rigorously monitor the error made by increasing the truncation dimension. The additional orbitals, determined by the condition of least error of the truncated evolution relative to the exact one, are obtained from an initial trial state by a method we call steepest constrained descent.
## PH
- 2022-07-06: Update (images, bold, italic and so on.)
- 2014-12-17: To the docuK.
- 2014-06-23: First posting?
Schematic potential we used. It is an infinite potential with width $L$ and a finite bump with width $a$ and height $h(t)$ on center of the infinite potential. The ratio $a/L$ is chosen to be 0.1 in the following implementations.
Natural occupation number of the ground state on the first symmetric orbital according to the height $h$ of the bump on the center of the infinite potential. Here we set $L=10 \mu m$, $g=0.01$, $N=100$. As $h$ increases, particles get fragmented and gradually condensed from the symmetric orbital into the antisymmetric orbital. The details are discussed in the script.
Ground state without any bump. Therefore $h=0$, $\hbar = m = 1$ unit, and $N=100,$ $L=10 \mu m$, $g=0.01$. Red dot indicates real value and blue squre indicates imaginary value of complex number. Green x in orbital graphs indicates $|phi (x)|^2$, i.e. square norm of the complex orbital at that position. The graph shows that the bosonic particles in the ground state are nearly condensated on the first symmetric orbital in (a). Since there is unitary transformation freedom or arbitrariness, the state can be represented in several different ways with different orbital bases. Consequently (b) is unitarilly transformed one from (a) and therefore these two states are exactly the same many-body state.
Ground state with a bump of height $h=1000$. Although (b) representation is more physically understandable, (a) represents the natural orbitals of eigenvalues of single particle density matrix. Here we used $\hbar = m = 1$ unit, and $N=100, L=10 \mu m, g=0.01$.
Ground state with a bump of height $h=7000$. Although (b) representation is more physically understandable, (a) represents the natural orbitals of eigenvalues of single particle density matrix. Here we used $\hbar = m = 1$ unit, and $N=100$, $L=10 \mu m$, $g=0.01$.
Ground state with a bump of height $h=11000$. Here we used 1hbar = m = 1$ unit, and $N=100, L=10 \mu m, g=0.01$.
(a) the instantaneous error Eq. of $M=1$ simulation. (b,c) particle occupation along the time in different y-scale.
(Graph will be put.) The error divided by double derivative of the particle conservation.
\label{error vs dt2}
-
Science 269, 198 (1995);
Observation of BEC in a Dilute Atomic Vapor;
by M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell;
Phys. Rev. Lett. 75, 3969 (1995); BEC in a Gas of Sodium Atoms; by K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle;
Phys. Rev. Lett. 78, 985 (1997); Bose-Einstein Condensation of Lithium - Observation of Limited Condensate Number; by C. C. Bradley, C. A. Sackett, and R. G. Hulet; - For an overview, see Ultracold Bosonic and Fermionic Gases, Eds. K. Levin, A. L. Fetter and D. M. Stamper-Kurn, Contemporary Concepts of Condensed Matter Science Vol. 5, Elsevier, 2012.
- Rev. Mod. Phys. 71, 463 (1999); Theory of trapped Bose-condensed gases; by F. Dalfovo, S. Giorgini, L. P. Pitaevski$\check{i}$, and S. Stringari;
- Rev. Mod. Phys. 80, 885 (2008); Many-Body Physics with Ultracold Gases; by I. Bloch, J. Dalibard, and W. Zwerger,
- Proc. Cambridge Philos. Soc. 26, 376 (1930); by P. A. M. Dirac;
- J. Frenkel, Wave Mechanics, Clarendon Press, Oxford, 1934.
- Mol. Phys. 8, 39 (2006); A variational solution of the time-dependent Schrodinger equation; by A. D. McLachlan;
- Chem. Phys. Lett. 149, 547 (1988); On the equivalence of time-dependent variational principle; by J. Broeckhove, L. Lathouwers, E. Kesteloot, and P. Van Leuven;
- Phys. Rev. A. 77, 033613 (2008); Multiconfigurational time-dependent Hartree method for bosons: Many-body dynamics of bosonic systems; by O. E. Alon, A. I. Streltsov, and L. S. Cederbaum;
- A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Phys. Rev. Lett. 100, 130401 (2008). %Formation and dynamics of many-boson fragmented states in one-dimensional attractive ultracold gases
- P. Bader and U. R. Fischer, Phys. Rev. Lett. 103, 060402 (2009). %Fragmented Many-Body Ground States for Scalar Bosons in a Single Trap
- K. Sakmann, A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Phys. Rev. Lett. 103, 220601 (2009). %Exact Quantum Dynamics of a Bosonic Josephson Junction
- A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Phys. Rev. Lett. 106, 240401 (2011). %Swift Loss of Coherence of Soliton Trains in Attractive Bose-Einstein Condensates
- A. Raab, Chem. Phys. Lett. 319, 674 (2000). %On the Dirac–Frenkel/McLachlan variational principle.
- H.-J. Kull and D. Pfirsch, Phys. Rev. E 61, 5940 (2000). %Generalized variational principle for the time-dependent Hartree-Fock equations for a Slater determinant
- Z. Baci\'c and J. C. Light, J. Chem. Phys. 85, 4594 (1986). %Highly excited vibrational levels of ``floppy'' triatomic molecules: A discrete variable representation — Distributed Gaussian basis approach.
- We illustrate this statement as follows. The reason why we can use $df = \frac{\partial f}{\partial z}dz + \frac{\partial f}{\partial z^*}dz^*$ is not because $dz$ and $dz^*$ are linearly independent differentials, but because $dz$ and $dz^*$ are differentials, that is infinitesimal quantities; $dz^*$ is in fact 100\% dependent on $dz$ since $dz^*$ is simply the complex conjugate of $dz$. Therefore we cannot simply argue that because $dz$ and $dz^*$ are independent variations, the partial derivatives $\frac{\partial f}{\partial z}$ and $\frac{\partial f}{\partial z^*}$ must be zero for the stationarity of $f$. There is a rather widespread misunderstanding of this mathematical fact found in the literature.
- When the variables and constraints are given in complex form, the method of Lagrange multipliers is expressed by $\frac{\partial f} {\partial z_k^*} = \sum_l \bigg[ \lambda_l \frac{\partial g_l} {\partial z_k^*} + \lambda_l^* \frac{\partial g_l^*} {\partial z_k^*} \bigg]$. Though $g_l (z_k, z_k^*)=c$ and $g_l^* (z_k, z_k^*)=c^*$ are the same constraint, we have to add the complex conjugate of it in the Lagrange equation. Be careful to note that $(\frac{\partial g_l}{\partial z_k^*})^*$ is not equal to $\frac{\partial g_l^*}{\partial z_k^*}$. The above equation gives the condition for the stationarity of the real-valued function $f$.
- $\vec{m}_k^l$ indicates the short-handed notation of $\hat{a}_{l}^{\dagger} \hat{a}_{k} | \vec{m} \rangle = \sqrt{(m_{l}+1)m_{k}} | \vec{m}_k^l \rangle$ which means that, from the given configuration $\vec{m}$, one particle is removed from $k$-orbital and one particle is added to $l$-orbital and the state $| \vec{m}_k^l \rangle$ is properly normalized.
- A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Phys. Rev. A. 73, 063626 (2006). %General variational many-body theory with complete self-consistency for trapped bosonic systems.
- J. J. McKeown, D. Meegan, and D. Sprevak, An Introduction to Unconstrained Optimization, IOP Publishing, 1990.
- The steepest descent direction in terms of real variables is given by the gradient: $d\vec{x} = - \vec{\nabla} f$. On the other hand, the steepest descent direction in terms of complex variables is given by the differential with respect to the complex conjugate of the variables: $dz_k = - \frac{\partial f}{\partial z_k^*}$. Then the steepest constrained descent is given by $\frac{d z_k} {d \tau} = -\Delta(\tau) \bigg[ \frac{\partial f} {\partial z_k^*} - \sum_l \Big[\lambda_l \frac{\partial g_l} {\partial z_k^*} + \lambda_l^* \frac{\partial g_l^*} {\partial z_k^*} \Big] \bigg]$ where $\Delta(\tau)$ can be any arbitrary positive definite function.
- A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Phys. Rev. Lett. 99, 030402 (2007). %Role of Excited States in the Splitting of a Trapped Interacting Bose-Einstein Condensate by a Time-Dependent Barrier
- kipid's blog - Method of Lagrange multipliers
728x90
반응형
'[Physics/Math] > Physics' 카테고리의 다른 글
엔트로피 (Entropy) 개념에 대한 이해 (1) | 2023.05.30 |
---|---|
Tensor and Relativity - 0. What is Tensor? (0) | 2023.03.11 |
텐서와 상대론 (Tensor and Relativity) - 0. 텐서 (Tensor) 란? (38) | 2023.03.10 |
물리학자들의 명언(名言)들 (Physicist Quotations) (0) | 2023.03.08 |
Time evolution (Hamiltonian/Energy) vs Space/Position Translation (Momentum) (0) | 2023.03.08 |
SI, cgs 단위계(unit) 및 물리상수(physical constants) (2) | 2023.01.30 |
과학적으로 효율적으로 난방하기 (에너지는 아끼면서 따뜻하게, 피부보호) (0) | 2022.12.13 |