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Journal of Physics Communications

Application of Grassmann phase space theory to Cooper pair model

NADER KIDWANI, Bryan J. Dalton

Published Nov 5, 2019• 1 citation
Publisher

Abstract

Abstract This paper concerns the application of Grassmann phase space theory (GSPT) to treat the dynamical evolution of systems of identical fermions, such as ultracold gases of fermionic atoms. Phase space theory (which originated from quantum optics) is increasing in importance since it overcomes certain issues associated with other theoretical methods, such as Greens functions, variational methods, quantum-Monte-Carlo equations, etc. In phase-space theory quantum states are represented by quasi-probability distribution functions of phase space variables associated with canonical system operators—such as annihilation, creation operators. Evolution is described via a Fokker-Planck equation for the distribution function, which is equivalent to Ito stochastic equations for (time dependent) stochastic phase space variables. Quantum correlation functions given as averages of products of phase space variables over the quasi-probability distributions then become stochastic averages of products of stochastic phase space variables. In GSPT, the phase space variables are Grassmann numbers, but as computer representation of g-numbers is difficult, Grassmann phase space methods were regarded as being computationally inaccessible. However, previous work using the un-normalised B distribution shows that computer representation of Grassmann variables is unnecessary. Stochastic averages of products for quantum correlation functions at later times are related linearly to stochastic averages at earlier times via stochastic matrices only involving c-numbers. Thus, GSPT calculations of quantum correlation functions now only involve c-number computations. This paper presents the first correct numerical calculation of a quantum correlation function for a fermionic system using stochastic methods based on Grassmann phase space theory, namely the time dependence of the coherence between two Cooper pair states in a four-mode fermion system, where the short and finite time solutions can be compared to known exact results. Good agreement between the stochastic and exact results is found, showing that GPST is a valid approach for treating fermionic systems. The treatment of time evolution involves a novel use of the eigenvalues and biorthogonal column eigenvectors of a stochastically determined c-number matrix M and its transpose. Other topics of interest in ultra-cold fermi gases for which the GSPT could be applied are highlighted, such as the strong interaction regime for the BEC/BCS crossover achieved using magnetically tuned Feshbach resonance techniques.

DOI
10.1088/2399-6528/ab542d

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Abstract This paper concerns the application of Grassmann phase space theory (GSPT) to treat the dynamical evolution of systems of identical fermions, such as ultracold gases of fermionic atoms. Phase space theory (which originated from quantum optics) is increasing in importance since it overcomes certain issues associated with other theoretical methods, such as Greens functions, variational methods, quantum-Monte-Carlo equations, etc. In phase-space theory quantum states are represented by quasi-probability distribution functions of phase space variables associated with canonical system operators—such as annihilation, creation operators. Evolution is described via a Fokker-Planck equation for the distribution function, which is equivalent to Ito stochastic equations for (time dependent) stochastic phase space variables. Quantum correlation functions given as averages of products of phase space variables over the quasi-probability distributions then become stochastic averages of products of stochastic phase space variables. In GSPT, the phase space variables are Grassmann numbers, but as computer representation of g-numbers is difficult, Grassmann phase space methods were regarded as being computationally inaccessible. However, previous work using the un-normalised B distribution shows that computer representation of Grassmann variables is unnecessary. Stochastic averages of products for quantum correlation functions at later times are related linearly to stochastic averages at earlier times via stochastic matrices only involving c-numbers. Thus, GSPT calculations of quantum correlation functions now only involve c-number computations. This paper presents the first correct numerical calculation of a quantum correlation function for a fermionic system using stochastic methods based on Grassmann phase space theory, namely the time dependence of the coherence between two Cooper pair states in a four-mode fermion system, where the short and finite time solutions can be compared to known exact results. Good agreement between the stochastic and exact results is found, showing that GPST is a valid approach for treating fermionic systems. The treatment of time evolution involves a novel use of the eigenvalues and biorthogonal column eigenvectors of a stochastically determined c-number matrix M and its transpose. Other topics of interest in ultra-cold fermi gases for which the GSPT could be applied are highlighted, such as the strong interaction regime for the BEC/BCS crossover achieved using magnetically tuned Feshbach resonance techniques.
Grassmann Phase Space Theory (GSPT) is applied to the BEC/BCS crossover in cold fermionic atomic gases and used to determine the evolution (over either time or temperature) of the Quantum Correlation Functions (QCF) that specify: (a) the positions of the spin up and spin down fermionic atoms in a single Cooper pair and (b) the positions of the two spin up and two spin down fermionic atoms in two Cooper pairs The first of these QCF is relevant to describing the change in size of a Cooper pair, as the fermion-fermion coupling constant is changed via Feshbach resonance methods through the crossover from a small Cooper pair on the BEC side to a large Cooper pair on the BCS side. The second of these QCF is important for describing the correlations between the positions of the fermionic atoms in two Cooper pairs, which is expected to be small at the BEC or BCS sides of the crossover, but is expected to be significant in the strong interaction unitary regime, where the size of a Cooper pair is comparable to the separation between Cooper pairs. In GPST the QCF are ultimately given via the stochastic average of products of Grassmann stochastic momentum fields, and GPST shows that the stochastic average of the products of Grassmann stochastic momentum fields at a later time (or lower temperature) is related linearly to the stochastic average of the products of Grassmann stochastic momentum fields at an earlier time (or higher temperature), and that the matrix elements involved in the linear relations are all c-numbers. Expressions for these matrix elements corresponding to a small time or temperature increment have been obtained analytically, providing the formulae needed for numerical studies of the evolution that are planned for a future publication. Various initial conditions are considered, including those for a non-interacting fermionic gas at zero temperature and a high temperature gas.
A phase space theory for fermions has been developed using Grassmann phase space variables which can be used in numerical calculations for cold Fermi gases and for large fermion numbers. Numerical calculations are feasible because Grassmann stochastic variables at later times are related linearly to such variables at earlier times via c‐number stochastic quantities. A Grassmann field version has been developed making large fermion number applications possible. Applications are shown for few mode and field theory cases.
Abstract Recent experimental progress has enabled cold atomic gases to be studied at nanokelvin temperatures, creating new states of matter where quantum degeneracy occurs—Bose–Einstein condensates and degenerate Fermi gases. Such quantum states are of macroscopic dimensions. As its title suggests, this book presents the phase space theory approach to treating the physics of degenerate quantum gases, an approach already widely used in quantum optics. However, degenerate quantum gases involve massive bosonic and fermionic atoms, not massless photons. The book begins with a review of Fock states for systems of identical atoms, where large numbers of atoms occupy the various single-particle states or modes. First, separate modes are considered, and here the quantum density operator is represented by a phase space distribution function of phase space variables which replace mode annihilation and creation operators, the dynamical equation for the density operator determines a Fokker–Planck equation for the distribution function, and measurable quantities such as quantum correlation functions are given as phase space integrals. Finally, the phase space variables are replaced by time-dependent stochastic variables satisfying Langevin stochastic equations obtained from the Fokker–Planck equation, with stochastic averages giving the measurable quantities. Second, a quantum field approach is then treated, the density operator being represented by a distribution functional of field functions which replace field annihilation and creation operators, the distribution functional satisfying a functional Fokker–Planck equation, etc. A novel feature of this book is that the phase space variables for fermions are Grassmann variables, not c-numbers. However, the book shows that Grassmann distribution functions and functionals still provide equations for obtaining both analytic and numerical solutions.
Phase space distribution functional methods utilising c-number and Grassmann fields for bosons and fermions respectively are being developed for theoretical treatments of degenerate quantum gases with large fermionic and bosonic atom numbers. The Jaynes-Cummings model where a single mode bosonic system and a two mode fermionic system are coupled provides a simple testbed for phase space methods involving Grassmann variables. The Fokker-Planck equation for the distribution function is shown to be equivalent to six coupled equations for c-number bosonic functions. Numerical solutions enable well-known effects such as collapses/revivals of the atomic population difference to be demonstrated.
We discuss the numerical implementation of two related representations of fermionic density matrices which have been introduced in Dalton et al. (2016). In both of them, the density matrix is expanded in a basis of Bargmann coherent states with weights given by the two phase space distributions. We derive the equations of motion for the distributions when imaginary time evolution is generated by the Hubbard Hamiltonian. One of them is a Grassmann Fokker–Planck equation that can be re-cast into a remarkably simple Itô form involving solely complex variables. In spite of this simple form, we demonstrate that complications arise in numerically computing the expectation value of any observable. These are due to exponential growth in the matrix elements of the stochastic propagator, delicate numerical sensitivity in performing primitive linear algebra operations, and the re-appearance of a sign problem.
We discuss the numerical implementation of two related representations of fermionic density matrices which have been introduced in Annals of Physics 370, 12 (2016). In both of them, the density matrix is expanded in a basis of Bargmann coherent states with weights given by the two phase space distributions. We derive the equations of motion for the distributions when imaginary time evolution is generated by the Hubbard Hamiltonian. One of them is a Grassmann Fokker-Planck equation that can be re-cast into a remarkably simple Itô form involving solely complex variables. In spite of this simple form, we demonstrate that complications arise in numerically computing the expectation value of any observable. These are due to exponential growth in the matrix elements of the stochastic propagator, delicate numerical sensitivity in performing primitive linear algebra operations, and the re-appearance of a sign problem.