Dynamics of nonlinear processes in ultracold fermionic gases ...wlazlowski.fizyka.pw.edu.pl/pdfs/2015-UJ-ZOA.pdfsolid framework (see for example: Bulgac, Forbes, Phys. Rev. C 87, 051301(R)

Dynamics of nonlinear Dynamics of nonlinear processes in processes in

ultracold fermionic gasesultracold fermionic gaseswithin within

Density Functional TheoryDensity Functional Theory

Gabriel WlazłowskiWarsaw University of Technology

University of Washington

Warsaw group: Piotr Magierski, Janina Grineviciute, Kazuyuki Sekizawa,

Seattle grup: Aurel Bulgac (UW), Michael McNeil Forbes (WSU, INT), Kenneth J. Roche (PNNL,UW)

Kraków, Poland, 12-10-2015

Supported bySupported by: ● Polish National Science Polish National Science Center (NCN) grant Center (NCN) grant under decision No. DEC-under decision No. DEC-2014/13/D/ST3/01940.2014/13/D/ST3/01940.

Present challenge for MBT: Unified description of static and dynamic properties of large Fermi systems

Methods:

QMC (static)

DFT (static and dynamic)

...(effective theories)...

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Cold atoms near a Feshbach Cold atoms near a Feshbach resonance = unitary Fermi gas resonance = unitary Fermi gas

System is dilute but... strongly interacting!

Unitary limit: no interaction length scale... Universal physics...

Cold atomic gasesNeutron matterHigh-Tc superconductors

Simple, but hard to calculate! (Bertsch Many Body X-challenge)

feed

We know what Eq. should be solved...The only problem: How to do it in practice?

Density Functional Theory – Idea

It can be shown that instead of wave function one may use a density distribution:

contains vastly more information contains vastly more information than the one neededthan the one needed

reduced object - sufficient to extractreduced object - sufficient to extractone body observables one body observables

`

In general:The energy is a functional of the density

“Universal” part External field

Fig. from: Prog.Part.Nucl.Phys.64:120-168,2010

Kohn-Sham method:

interacting system

non-interacting system

Both systemsdescribed by the same density

Formally rigorous way of approaching any interacting problem bymapping it exactly to a much easier-to-solve noninteracting system.

easy, if Energy Density Functional (EDF)is known...

More general:

InteractingSystem

System ofnon-interactingquasiparticles

Note: There are easy and difficult observables in DFT.In general: easy observables are one-body observables. They are easily extractedand reliable.

Energy density functional

How to derive EDF?

We build models of EDF (typically here we introduce approximation)

From orbitalswe build other densities(correlated with ρ)

Example:Local Density Approximation(only dependence on diagonalparts of densities )

Strategy

Propose EDF(exploit symmetries,

dimensional arguments...)

Use ab-initio results(or experimental)

to fix coupling constants

Validate

refine

Ultracold atoms are superfluid!N

orm

al

syst

em

BCS-like

pairing (anomalous) density

Note: diagonal part of pairing density is divergentRegularization required!

We use prescription given in:Bulgac, Yu, Phys. Rev. Lett. 88 (2002) 042504Bulgac, Phys. Rev. C65 (2002) 051305

Extension to time-dependent case

and

Runge & Gross theorem

Up to arbitrary function α(t)

… and consequently density functional exists...NOTE: in general this functional: - is initial state dependent- at time t depends on densities in previous times (memory effect)

Time evolution ofinteracting system

Time evolution ofnon-interacting system

van Leeuwen’s Theorem:Very little is known about the memory terms, but in principle it can be longranged (see eg. Dobson, Brunner, Gross, Phys. Rev. Lett. 79 (1997) 1905)

Memory effects are usually neglected = adiabatic approximation[Result: dissipation effects are not correctly taken into account exceptfor one-body dissipation]

...but our system is superfluid...

DFT: workhorse for electronic structure simulations

The Hohenberg-Kohn theorem assures that the theory can reproduce exactly the ground state energy if the “exact” Energy Density Functional (EDF) is provided

Often called as ab initio method

Extension to Time-Dependent DFT is straightforward

Can be extended to superfluid systems... (numerical cost increases dramatically)

Very successfulVery successful – DFT industry (commercial codes for quantum chemistry and solid-state physics)

1990

2012

EDF for UFG: [unpolarized case]Superfluid Local Density Approximation (SLDA)

Dimensional arguments, renormalizability, Galilean invariance, and symmetries (translational, rotational, gauge, parity) determine the functional (energy density)

Only local densities

unique combination of the kinetic and anomalous densities required by the renormalizability of the theory

Self-energy term - the only functionof the density alone allowed by dimensional arguments lowest gradient

correction- negligiblerequired by Galilean invariance

Review: A. Bulgac, M.M. Forbes, P. Magierski,

Lecture Notes in Physics, Vol. 836, Chap. 9, p.305-373 (2012)

Three dimensionless constants α, β, and γ determining the functional are extracted from QMC for homogeneous systems by fixing the total energy, the pairing gap and the effective mass.NOTE: there is no fit to experimental results

Forbes, Gandolfi, Gezerlis,PRL 106, 235303 (2011)

SLDA has been verified and validated against a large number of quantum Monte Carlo results for inhomogeneous systems and experimental data as well

So simple ...… so accurate!

Set to α=1

EDF for UFG: [polarized case] ASLDA

Polarization dependent parameters

Unpolarized case:QMC available

QMC for “polaron “ problem

QMC for system forced to be in normal state

Small polarizationsQMC doable (but hard)

Experimental data

Figure from: A. Bulgac, M.M. Forbes, P. Magierski,

Lecture Notes in Physics, Vol. 836, Chap. 9, p.305-373 (2012)

Table from: A. Bulgac, M.M. Forbes, P. Magierski,

Lecture Notes in Physics, Vol. 836, Chap. 9, p.305-373 (2012)

Solving time-dependent problem...

nonlinear coupled 3D

Partial Differential Equations

Supercomputing

We simulate fermionic systems consisting of

103 – 104 particles(cold atoms, neutron stars)

… also nuclear reactions(spin-orbit term required)

Solving...

The system is placed on a large 3D spatial latticeof size N

x×N

y×N

z

Discrete Variable Representation (DVR) - solid framework (see for example: Bulgac, Forbes,Phys. Rev. C 87, 051301(R) (2013))

Errors are well controlled – exponential convergence

No symmetry restrictions

Number of PDEs is of the order of the number of spatial lattice points

Typically (without spin-orbit term): 105 - 106

Solving...

Derivatives are computed with FFT

insures machine accuracy

very fast

Integration methods:

Adams-Bashforth-Milne fifth order predictor-corrector-modifier integrator – very accurate but memory intensive

Split-operator method that respects time-reversal invariance (third order) – very fast, but can work with simple EDF

It sets scaling (N-number of lattice points)

Number of wave-functions

FFT for large lattice

If non local densitiesN3!!!

(beyond our reach)

The spirit of SLDA is to exploit only local densities...

Suitable for efficient parallelization (MPI)

Excellent candidate for utilization multithreading computing units like GPUs

Lattice 643, 137,062 (2-component) wave functions, ABMCPU version running on 16x4096=65,536 coresGPU version running on 4096 GPUs

15 times 15 times Speed-up!!!Speed-up!!!

Operation(double complex)

50GFlops (laptop)

100TFlops (supercomputer)

10PFlops (leadership-

class supercomputer)

N^2 ~10min ~0.1sec ~0.001sec

N^2 logN ~1hour ~1sec ~0.01sec

N^3 ~25years ~100hours ~1hour

Example: Lattice 1003,

Time evolution

Ground state solver (initial state for real time evolution)

TDDFT – challenge for computational physics

6Li atoms near a Feshbach resonance (N≈106) cooled in harmonic trap

Step potential used to imprint a soliton (evolve to π phase shift)

Let system evolve...

Take picture (subtle imaging with

tomography)

Fig. from Nature 499, 426 (2013)

Validation against dynamical properties of the system

Recent MIT experiments: Nature 499, 426 (2013), PRL 113, 065301(2014)

Fig. from PRL 113, 065301(2014)

NOTE: We did tests against other dynamical properties

Experimental results

Yefsah et al., Nature 499, 426 (2013) PRL 113, 065301(2014)

RESULTS:

In the final state: Observe an oscillating vortex line with long period

Intertial mass 200 times larger than the free fermion mass

Precessional motion

...

Experimental results – Cascade of Solitary WavesFigures taken from: M. Zwierlein talk, (http://en.sif.it/activities/fermi_school/mmxiv)School of Physics E. Fermi – Quantum Matter at Ultralow Temperatures Varenna, July 9th , 2014

See also: Mark J.H. Ku, et al., arXiv:1507.01047

Experimental results – Cascade of Solitary WavesFigures taken from: M. Zwierlein talk, (http://en.sif.it/activities/fermi_school/mmxiv)School of Physics E. Fermi – Quantum Matter at Ultralow Temperatures Varenna, July 9th , 2014

Challenge for theory to describe all stages of the cascade!

Realistic simulation

Trapping potential:

The optical trapping potential in the x and y directionsis an axially symmetric gaussian

altered by gravity in thevertical direction y

anisotropy anharmonicity

NOTE: Unconstrained

calculations (3D)...

What do fully 3D simulations see?

Movie 2

Movie 3

Crossing and reconnection!

Phys. Rev. A 91, 031602 (2015)

Dynamics of solitons and vortices – GPE

Fermionic simulations – numerically expensive, cannot reach 106 particles...

Other solution: use a bosonic theory for the dimer/Cooper-pair wavefunction) with DFT...

Michael McNeil Forbes, Rishi Sharma, Phys. Rev. A 90, 043638 (2014)

Note similarity to GPE...

Accurate Equation of State state for a>0, speed of sound, phonon dispersion, static response, respects Galilean invariance

Ambiguous role played by the ‘’wave function,’’ as it describes at the same time both the number density and the order parameter.

Density depletion at vortex/soliton core exaggerated! Systematically underestimates time scales by a factor of close to 2

What about cascade?

Movie 2 - ETF

Movie 3 - ETF

Simulation vs experiment...Mark J.H. Ku, et al., arXiv:1507.01047

?

Cascad e

A. Munoz Mateo and J. Brand, PRL 113, 255302 (2014)

Subtle imaging needed:- needed expansion- must ramp to specific value of magnetic field

What do fully 3D simulations see? Phys. Rev. A 91, 031602 (2015) Fermionic simulation:

Crossing and reconnection!

No psi-soliton

Note: ETF (GPE-like approach) do not admit creation of psi-soliton...… but in these model also there is no crossings and reconnections In intermediate state (between vortex ring and vortex line)...

Our observation:It is very hard to force GPEto get crossing and reconnection...… by contrast to fermionic simulations!

Aurel Bulgac, Yongle Yu,J. Low Temp. Phys. 138, 741 (2005)

Depletion of density

Thus: possible of QT in cold atoms...

Towards quantum turbulence...

Tangle of many vortices!Phys. Rev. A 91, 031602 (2015)

How to generate QT state?Our proposition: “phase imprint” of a lattice of vortices

Movie 4

Thank you

CONCLUSIONS:

DFT – route for unified description of static and dynamic properties of large Fermi systems

We have EDF for UFG – validation of (TD)DFT in progress...

Correctly describes generation, dynamics, evolution, and eventual decay - large number of degrees of freedom in the SLDA permit many mechanisms for superfluid relaxation: various phonon processes, Cooper pair breaking, and Landau damping

Can be used to engineer interesting scenarios: colliding of vortices, QT, vortex interactions...

DFT with LDA idea results in numerically tractable tool

DFT com

Computational challenge: Finding initial (ground) state?

En

Diagonalization:

requires repeatedly diagonalizing the NxN single-particle Hamiltonian (an O(N3) operation) for the hundreds of iterations required to converge to the self-consistent ground state

We need full spectrum (eigenvalues and eigenstates)

only suitable for “small” problems or if symmetries can be used

Note: Imaginary time evolution is also prohibitively expensive(Non-unitary evolution: spoils orthogonality of wavefunctions, Re-orthogonalization of states at each time step is required)

Real time evolution scaling:

t=0

Self-consistent problem

Until convergence

Initial guess

The most time consuming part!

To put this in perspective:single diagonalization of nuclear problem(both proton and neutron single-particle Hamiltonians, HFB matrix size: 384,000)

on lattice 40 x 40 x 60 took essentially the entire (now retired) JaguarPF computer(217 800 of the 224 256 processor cores) about 6 hours of wall-time (about one million CPU hours) => about a month to determine just the initial state

Technical problems: Very hard for GPU acceleration Very hard to exploit matrix sparsity Memory demanding

(nuclear problem on 40x40x60 lattice: 2.15TB)

Quantum frictionEnergy density functional

Generalized density matrix

Single particle Hamiltonian

Equation of motion

Consider evolution with “external” potential:

Energy of the system

Quantum friction

Note:

Non-local potential equivalent to “complex time” evolution

Not suitable for fermionic problem

“Local” option:

current

dimensionless constant of order unity

removes any irrotational currents in the system, damping currents

by being repulsive where they are converging

Quantum frictionDoes not guarantee convergence to ground state

proceed with adiabatic state preparation(generally much faster than pure adiabatic state preparation)

Combine with a few diagonalization

Gain: computational scaling:

Additional “cooling” potentialcan be added in pairing channel

Movie 1

Bulgac, Forbes,Roche, and Wlazłowski, arXiv:1305.6891

Vortex in neutron matter pinned to impurity

Volume: 40 fm3

N=1500Z=40Size of HFB matrix: 128,000

Cloud of cold atoms (at Feshbach resonance)with vortex latticeN=1400Size of HFB matrix: 589,824

Example of configurations with vortices

PRELIMINARY

(ongoing project)