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arXiv:1301.2024 · Cold atoms

Time-Dependent Superfluid Local-Density Approximation

Aurel Bulgac, Michael McNeil Forbes

Published Mar 18, 2013• 2 citations
arXivHTMLPDF

Abstract

The time-dependent superfluid local density approximation (TDSLDA) is an extension of the Hohenberg-Kohn density functional theory (DFT) to time-dependent phenomena in superfluid fermionic systems. Unlike linear response theory, which is only valid for weak external fields, the (TDSLDA) approach allows one to study non-linear excitations in fermionic superfluids, including large amplitude collective modes, and the response to strong external probes. Even in the case of weak external fields, the (TDSLDA) approach is technically easier to implement. We will illustrate the implementation of the (TDSLDA) for the unitary Fermi gas, where dimensional arguments and Galilean invariance simplify the form of the functional, and ab initio input from (QMC) simulations fix the coefficients to quite high precision.

DOI
10.1142/9781848168121_0026

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The time-dependent superfluid local density approximation (TDSLDA) is an extension of the Hohenberg-Kohn density functional theory (DFT) to time-dependent phenomena in superfluid fermionic systems. Unlike linear response theory, which is only valid for weak external fields, the (TDSLDA) approach allows one to study non-linear excitations in fermionic superfluids, including large amplitude collective modes, and the response to strong external probes. Even in the case of weak external fields, the (TDSLDA) approach is technically easier to implement. We will illustrate the implementation of the (TDSLDA) for the unitary Fermi gas, where dimensional arguments and Galilean invariance simplify the form of the functional, and ab initio input from (QMC) simulations fix the coefficients to quite high precision.
The first detailed comparison between ab initio calculations of finite fermionic superfluid systems, performed recently by Chang and Bertsch [Phys. Rev. A 76, 021603(R) (2007)] and by von Stecher, Grange, and Blume [e-print arXiv:0705.0671v1] and the extension of the density-functional theory superfluid local-density approximation (SLDA) is presented. It is shown that SLDA reproduces the total energies, number density distributions in inhomogeneous systems along with the energy of the normal state in homogeneous systems. Unlike the Kohn-Sham LDA, in SLDA the effective fermion mass differs from the bare fermion mass and the spectrum of elementary excitations is also reproduced.
I describe the time-dependent superfluid local density approximation, which is an adiabatic extension of the density functional theory to superfluid Fermi systems and their real-time dynamics. This new theoretical framework has been used to describe several phenomena in cold atomic gases and nuclear collective motion: excitation of the Higgs modes in strongly interacting Fermi superfluids, generation of quantized vortices, crossing and reconnection of vortices, excitation of the superflow at velocities above the critical velocity, excitation of quantum shock waves and domain walls in the collisions of superfluid atomic clouds, and excitation of collective states in nuclei.
A density-functional theory is established for inhomogeneous superfluids at finite temperature, subject to time-dependent external fields in isothermal conditions. After outlining parallelisms between a neutral superfluid and a charged superconductor, Hohenberg-Kohn-Sham-type theorems are proved for gauge-invariant densities and a set of Bogolubov-Popov equations including exchange and correlation is set up. Earlier results applying in the linear response regime are recovered. 1 Experiments on confined Bose-condensed gases have revealed a rich variety of dynamical behaviours. These include elementary excitations of lowlying shape deformation modes [1], propagation of sound waves in both the condensate and the thermal cloud [2], Josephson-like oscillations in double condensates [3], creation of vortices [4], phase dynamics from various atomlaser configurations [5] and Bloch oscillations of a condensate in an optical
We describe the response of superfluid nuclei to any external time-dependent probe within an extension of the density functional theory, the time-dependent superfluid local density approximation (TD-SLDA). All quasi-particle wave functions (qpwfs) are represented on a three-dimensional spatial grid to allow for the breaking of all possible symmetries, and subsequently these functions are evolved in time using a high order multi-step algorithm. Mathematically this problem is a system of coupled, time-dependent, nonlinear partial differential equations, which when discretized becomes equivalent to a nonlinear classical mechanics problem with α N x 6 ≈ 10 12 degrees of freedom, where N x is number of spatial mesh points in one direction. The lattice representation of the qpwfs allows for fast and accurate evaluation of various spatial derivatives required for the construction of various physical quantities that appear in the theory. The total number of independent qpwfs is of the order of spatial mesh points, and the system is evolved for tens to hundreds of thousands of time steps. We describe how we implemented the TD-SLDA method and report some of the challenges encountered.
The dynamics of a confined fluid of Bose atoms is treated within the linear response regime, with a view to establishing a current-density functional formalism for an inhomogeneous superfluid state. After evaluating in full detail a simplified case of an external coupling to the density and phase of the condensate, the theory is extended to include the coupling to the total current density. The Kohn-Sham response functions of the condensate and all the exchange-correlation kernels for the superfluid are introduced from the microscopic equations of motion and are expressed in a physically transparent way through functional derivatives of correlation functions. A microscopic formula for the superfluid density is derived and used to introduce a generalized hydrodynamic approach for a weakly inhomogeneous two-fluid model in isothermal conditions. Local-density expressions are thereby derived for the velocities of first and second sound in the weakly inhomogeneous superfluid and for visco-elastic functions describing the transition from the hydrodynamic to the collisionless regime. Landau's hydrodynamic theory and known results in Green's functions language are recovered in the limiting case of a homogeneous superfluid.