Supplemental online material for:

Spin-polarized droplets in the unitary Fermi gas

 Piotr Magierski1,2, Buğra Tüzemen1, and Gabriel Wlazłowski1,2

1Faculty of Physics, Warsaw University of Technology
 2Department of Physics, University of Washington

When you use data from this webpage, please make credit to: P. Magierski, B. Tüzemen, G. Wlazłowski, Phys. Rev. A 100, 033613 (2019).

What is ferron?

It is localized excitation in the unitary Fermi gas. It looks like a droplet with a peculiar internal structure involving the abrupt change of the pairing phase at the surface of the droplet. It resembles the structure of the Josephson-π junction occurring when a slice of a ferromagnet is sandwiched between two superconductors. The polarized droplet is stabilized  by the mutual interplay between the polarization effects and the pairing field structure resulting in an exceptionally long-lived state.

The left and right parts of the figure present the polarization p(r) and the pairing field ∆(r) distributions respectively of the ferron. The characteristic feature is the presence of the nodal surface of the pairing field where the pairing changes its sign and it also corresponds to the surface where polarization reaches maximum.

Accompanying movies - highlights

Dynamical creation of the ferron (Movie 2)

Simulation demonstrating response of the unitary Fermi gas after application of the spherically symmetric polarizing potential. Maximal amplitude (A) of the potential and its width (σ) are displayed in top line of the movie. The movie present distribution of absolute value of the paring field |Δ(r)|, the phase difference of the paring filed with respect to the value at the boundary of the box Δφ, and the local polarization p(r) in plane crossing the impurity center. For this particular parameters the stable ferron is created!

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Non-central collision of two ferrons (Movie 9)

Movie demonstrating non-head collision of two ferrons. Colliding ferrons have been generated by applying two spin-selective potentials moving toward each other. The potentials were moving along x-axis with speed v ≈ 0.45vF , where vF is Fermi velocity. The amplitude is A = 2εF and width σ = 3.14ξ. The movie demonstrates that  ferrons behave like droplets: they fused and a new droplet of bigger size is created with the typical pairing nodal structure.

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Protocol for experimental creation of the ferron in a box-like trap (Movie 11)

Simulation demonstrating creation of the ferron in a box-like trap. It is created as a result of applications of two crossing beams, each of them has amplitude A = 1εF and width σ = 3.14ξ. In crossing region of beams strength of the potential is sufficiently large in order to polarize locally the system and create the stable ferron.

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Other movies

 Movie 1

Run demonstrating response of the UFG after application of the spherically symmetric polarizing potential. Lattice used in calculations was 403. Maximal amplitude of the potential is A = 2εF and width σ = 2.36ξ. The potential was turning on and off in time 25εF-1 and hold time Fwas set to 100εF-1. In this case the width of the potential is to narrow in order to produce the ferron.

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 Movie 3

The same as Movie 1 but with width σ = 5.5ξ.

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 Movie 4

The same as Movie 1 but with width σ = 6.28ξ. In this case the time-dependent part of the potential generates significant amount of the gas fluctuations. They impact structure of the ferron.

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 Movie 5

The same as Movie 1 but with width σ = 7.07ξ. In this case the hold time is not sufficient in order to allow for creation of the ferron.

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 Movie 6

The same as Movie 5 but hold on time was increased to 250εF-1, which allows for creation of short-lived ferron-like structure.

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 Movie 7

The same as Movie 5 but hold on time was increased to 400εF-1 , which allows for creation of stable ferron.

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 Movie 8

Movie demonstrating evolution of a deformed impurity. The simulation lattice was 603. Maximal amplitude of the potential is A = 2εF , while widths are different for each direction. Time dependency of the potential is the same as for Movie 1. The movie demonstrates that spherically symmetric shape of the ferron is energetically preferable.

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 Movie 10

The same as Movie 9, but for head-on collision.

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 Movie 12

The same as Movie 11, but with width of the beams σ = 4.71ξ.

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 Movie 13

Run demonstrating creation of the impurity with more complex internal structure. Lattice used in calculations was 643 . Maximal amplitude of the potential is A = 3.5εF and width σ = 11.78ξ. The potential was turning on and off in time 30εF-1 and hold time was set to 112εF-1

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 Movie 14

Movie demonstrating creation of a ferron in 1D Fermi, using the external potential of the gaussian type with amplitude A = 1.8εF . The lattice size is 200. The potential is turned off after t = 78.8εF-1. The expansion of polarized shell is clearly visible after switching off the potential. The movie demonstrates instability of ferrons in 1D.

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 Movie 15

Simulation demonstrating creation of the impurity within the BdG approach. Lattice size used in the calculation was set to 403 . The amplitude of the potential is A0 = 1.5εF and width σ = 4.71ξ. The potential has been turned on and off within time interval 25εF-1 and kept fixed at its maximal strength within 100εF-1.

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Related materials

Our talks & posters based on materials obtained within this work
  1. P. Magierski, Towards exascale simulations of quantum superfluids far from equilibrium,
    10th Symposium on Discovery, Fusion, Creation of New Knowledge by Multidisciplinary Computational Sciences, Tsukuba, Japan, Oct. 15-16, 2018
  2. B. Tüzemen, Stable, polarized impurity in the Unitary Fermi Gas,
    Quantum Simulations with Atoms and Light, Aarhus, Denmark, 13-17 Aug, 2018
  3. P. Magierski, Solitonic excitations in ultracold atoms and heavy-ion collisions,
    INT Program INT-19-1a Quantum Turbulence: Cold Atoms, Heavy Ions, and Neutron Stars, March 18 - April 19, 2019, Seattle, USA
  4. G. Wlazłowski, Dynamics of ultracold atomic clouds by means of time dependent density functional theory,
    Seminar at Nicolaus Copernicus University, Toruń, Poland, 07 May 2019
  5. G. Wlazłowski, Searching for new superfluid effects by means of numerical experiments,
    EuroHPC Summit Week 2019, Poznań, Poland, 13-17 May 2019
  6. B. Tüzemen, Stable, Polarized Impurity in the Unitary Fermi Gas,
    Compound (Atomic) Quantum Systems, Leiden, Netherlands, 20-24 May, 2019
  7. P. Magierski, Exotic features of superfluid dynamics and collisions: stable and unstable pairing nodal structures,
    Symposium: "Challenges in theory of heavy nuclei", July 17-20, York, UK

Acknowledgments

Faculty of Physics @ WUT

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