Ferron dynamics in ultracold atomic gas
FFerron dynamics in ultracold atomic gas
Piotr Magierski,
1, 2, ∗ Bu˘gra T¨uzemen, † and Gabriel Wlaz lowski
1, 2, ‡ Faculty of Physics, Warsaw University of Technology, Ulica Koszykowa 75, 00-662 Warsaw, Poland Department of Physics, University of Washington, Seattle, Washington 98195–1560, USA
We show that the motion of spin-polarized impurity (ferron) in ultracold atomic gas is character-ized by a certain critical velocity which can be traced back to the amount of spin imbalance insidethe impurity. We have calculated the effective mass of ferron in 2D. We show that the effectivemass scales with the surface of the ferron and in general it scales as M eff ∝ R D − , where D is thedimensionality of the system. We discuss the impact of these findings, in particular we demonstratethat ferrons become unstable in the vicinity of a vortex. Introduction — The ultracold atomic gases withnonzero spin polarization offer the possibility to inves-tigate the existence of metastable structures that mayspontaneously occur in such systems. These includerealizations of Fulde-Ferrell-Larkin-Ovchinnkov phase(FFLO) [1, 2] leading to the possible formation of liq-uid crystals [3], supersolids [4], which also include po-larized vortex cores [5–7] and Sarma phase [8–10]. Al-though the experimental confirmation of these phases isstill lacking, the progress in experimental techniques al-lows to treat spin imbalance as a controllable experimen-tal ”knob” and thus offers the possibility to investigatethe superfluid gas as a function of spin polarization [11–14]. In particular the evolution of spin imbalanced sys-tems from deep BCS regime through the unitary limitto the BEC side is predicted to generate various exoticphases [15–17]. Although the phase diagram as a func-tion of spin-polarization remains still merely a theoreti-cal prediction, yet another question may be posed: doesthe ultracold atomic gas with nonzero spin-polarizationadmit the presence of metastable structures inside thesuperfluid, where the polarization could be effectivelystored? One such structure in the form of ferron , re-sembling Larkin-Ovchinnikov droplet has be recently in-vestigated in Ref. [18, 19]. In this case it was found thatone can generate dynamically the local spin imbalance inthe form of droplet in otherwise unpolarized medium cor-responding to the unitary Fermi gas (UFG). Due to theparticular nodal structure of the pairing field, the ferronappears as an excitation mode of a metastable charac-ter. On the other hand, one may expect that under thecondition of nonzero spin imbalance spatially separatedferrons may appear spontaneously in the cooling process.This situation may occur in the limit when the spin im-balance is too small to generate the FFLO phase in thebulk.The structure of ferrons is stabilized by the existenceof Andreev states, induced by the spatial variations ofthe pairing field ∆, where the majority spin particles arestored, see Fig 1. It was shown that at the unitaritythe structure of the droplet remains preserved even un-der dynamic evolution including stretching and collisionswith other droplets [18]. In the case of a single ferron the lowest energy condition guarantees that the shape ofthe ferron remain spherical and its radius is a functionof polarization. The relation between radius and polar-ization reflects the fact that spin excess can be stored inAndreev states and their number scales with the radius.Therefore it is easy to realize that in the case of spher-ical 3D ferron the size (radius) R scales as | N ↑ − N ↓ | ,whereas in the case of 2D system (or cylindrical ferron)the relation is linear [19]. It is also possible to create aspherical ferron with multiple concentric nodal surfaces.Recently, the ferron-like structures have been generatedwithin an extension of Ginzburg-Landau (GL) approachwhich allows for consideration of spin-imbalanced system[20]. Within certain parameter range of GL model stable,circular solutions in 2D have been found correspondingto circular ferrons with single or multiple nodal lines.They have been described as ring solitons although theirstructure coincides with that of ferrons. The interactionbetween ferrons mediated by the superfluid have beendetermined [21]. FIG. 1. Schematic presentation of the ferron solution in 2D.It is characterized by a nodal line where the order parameter∆ changes phase by π . The localized Andreev states residearound the nodal line (red area) and they accumulate major-ity (spin-up) particles. These states are almost degenerate,with excitation energies about µ ↑ − µ ↓ . a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b In this letter we investigate dynamic properties of aferron from BCS regime towards the unitary point. Weshow that the ferron possess a certain effective masswhich scales with its surface. It is also characterized bya critical velocity which cannot be exceeded while mov-ing through the superfluid environment. We discuss theimplications of these findings.
Effective mass — In the case of nonzero polarizationthe nodal line (surface) of the pairing field may acquirestability as soon as Andreev states become populated.Clearly the nodal line shares the property of the vortexline (the phase changes abruptly by π ) which cannot endinside a superfluid. It may either form the closed struc-ture (eg. sphere in 3D) or may end at the boundary,where the density drops to zero. Similarly as in the caseof a vortex, one may ask the question: what are the lawsof dynamics governing the motion of such nodal struc-tures travelling through the superfluid? In the case ofvortices the answer to these questions gave rise to theformulation of the filament model which accurately pre-dicts dynamics of vortices and can be applied to describeturbulence phenomenon [22]. In order to be able to for-mulate an effective theory one needs to extract the in-ertia of the object and determine the conservative anddissipative forces present when moving in superfluid en-vironment. In this letter we focus on the effective mass ofthe ferron. Although in general the determination of themass of an impurity immersed in fermionic environmentis a challenging problem [23], due to the presence of thepairing gap the problem facilitates considerably.We determine the mass as the response of the systemwith ferron, being exposed to the superflow characterizedby the wave vector 2 q . Namely, we consider the pairingfield ∆( r ) which in the limit of large distance R from theferron behaves as: lim R →∞ ∆( r ) = | ∆ | exp( i q · r + iφ ),with φ being an arbitrary overall phase. However insteadof considering the superflow we change the referenceframe to the one moving with velocity q (we use units: ~ = m = 1). In this case it is sufficient to apply the trans-formation: u n, ↑↓ ( r ) → exp( i q · r ) u n, ↑↓ ( r ) , v n, ↑↓ ( r ) → exp( − i q · r ) v n, ↑↓ ( r ) to transform the intial BdG equa-tions (for u n, ↑ , v n, ↓ components) with superflow to: H ( q ) (cid:18) u n, ↑ ( r ) v n, ↓ ( r ) (cid:19) = E n (cid:18) u n, ↑ ( r ) v n, ↓ ( r ) (cid:19) (1)with Hamiltonian H ( q ) = (cid:18) − ( ∇ + i q ) − µ ↑ ∆( r )∆ ∗ ( r ) ( ∇ − i q ) + µ ↓ , (cid:19) (2)where µ ↑ , ↓ are chemical potentials for two spin compo- nents. The quasi-particle wave functions define densities: n ↑ ( r ) = X E n < | u n, ↑ ( r ) | , (3) n ↓ ( r ) = X E n > | v n, ↓ ( r ) | , (4) ν ( r ) = X E n < u n, ↑ ( r ) v ∗ n, ↓ ( r ) , (5) j ↑ ( r ) = − X E n < Im[ u n, ↑ ( r ) ∇ u ∗ n, ↑ ( r )] , (6) j ↓ ( r ) = X E n > Im[ v n, ↓ ( r ) ∇ v ∗ n, ↓ ( r )] . (7)The pairing field, ∆( r ) is calculated self-consistently:∆( r ) = − g eff ν ( r ) , (8)where g eff is the coupling constant which is tuned to ob-tain the required strength of the pairing field.The transformation of u n, ↑ , v n, ↓ amplitudes to themoving frame induces the transformation of the currents: j ↑↓ ( r ) → j ↑↓ ( r ) − q n ↑↓ ( r ), where the term q n ↑↓ ( r ) cor-responds to the uniform motion of spin-up ( n ↑ ) and spin-down ( n ↓ ) component, respectively. Consequently, in thisreference frame the resulting currents represent perturba-tion to the superflow induced by the presence of impurity.As a result we may define the effective mass of the ferronas a static response R : M eff = lim q → R ( q ) = lim q → | R d r ( j ↑ + j ↓ ) || q | . (9)The effective mass contains two components. The firstone is related to the number of majority spin particlesthat are accumulated inside the ferron and are draggedthrough the superfluid by the nodal structure. The sec-ond component comes from the modification of the sur-rounding environment by moving impurity: M eff = M pol + δM = ( N ↑ − N ↓ ) + δM (10)Although δM is expected to represent a small correc-tion to M pol it is not a priori obvious that it is indeednegligible. For example if one considers the problem inhydrodynamic limit the component of the effective massrelated to δM is generated by flow induced by movingobstacle and therefore it scales with the volume of theimpurity. On the other hand M pol scales with the surfacesince M pol ∝ R ( ∝ R in 2D) due to the relation betweenpolarization and the ferron radius (see Ref. [19]). It sug-gests that the hydrodynamic contribution will eventuallydominate the effective mass as the size of the ferron isincreasing, ie. will be sensitive to the spin-imbalance inthe system. One may also expect that the effective massis modified with increasing pairing strength (∆ /ε F ), asit corresponds to moving towards the irrotational hydro-dynamic limit.In order to resolve these apparently non obvious issueswe have performed a series of calculations in 2D and de-termined the response function (9). We have appliedBdG approach varying the value of ∆ /ε F from 0 .
36 to0 .
55. Subsequently we have determined numerically thelimit | q | = q → with Fermi momenta for k F ↑↓ ≈
1. W-SLDA Toolkit has been used for the calculations [24–26].We have evaluated the total current for a series of ve-locities q/v F = 0 . , . , . ... until the ferronic con-figuration is destroyed by the currents. For velocities q . .
04 we found that linear relation between the cur-rent and velocity holds with a very good accuracy. Wehave also analyzed the stability of the result with re-spect to the size of the box, by evaluating effective massin a box with lattice size 100 and found an agreementwith accuracy better than 1%. Finally, we have checkedthat extracted effective mass when plugged into equation E ( q ) − E (0) = M eff q reproduces reasonably well thebehavior of E ( q ) obtained from calculations, for q -valueswhere the linear relation j ∼ q holds (see [27] for details).The results shown in the Fig. 2 indicate that the con-tribution coming from the flow induced in the superfluidmedium δM is a correction to the dominating term M pol , FIG. 2. The effective mass M eff as a function of the magnitudeof the pairing field | ∆ | /ε F and the spin imbalance δN = N ↑ − N ↓ . In all cases the total number of particles in the simulationbox is N = N ↑ + N ↓ = 770 and the Fermi momentum k F = √ ε F ≈
1. The values of δN = 21 ,
41 correspond to the ferronradii R ≈ . ξ and R ≈ . ξ , respectively, where ξ = k F ε F | ∆ | . except for the small ferron size of the order of coherencelength. In order to understand this result one may no-tice that in pure irrotational hydrodynamics in 2D thecontribution to δM ∝ S n out − n in n out + n in , (where n in , n out corre-spond to superfluid density inside and outside impurity,respectively and S is its area) and thus it vanishes if n in → n out [27]. Clearly the largest discrepancy betweenmagnitude of pairing field inside from its bulk value oc-curs for small ferrons (and weak pairing). In that casedue to the fact that coherence length is of the order ofthe ferron size (ie. ξ ≈ R ) the value of the pairing gapis smaller than outside. It implies also that the polariza-tion inside the ferron does not vanish completely. As aconsequence there is larger contribution coming from theflow than in the case of large ferron. In the latter casethe magnitude of the pairing field inside the ferron is thesame as outside and therefore the perturbation related tothe flow occurs effectively around the pairing nodal area. Critical velocity — While moving through a superfluidstructure of ferron is affected. Spherical ferron has acharacteristic spectrum of Andreev states. It consistsof almost degenerate states at E ± ( l ) ≈ ∆ µ ± δE ( l ),where δµ = µ ↑ − µ ↓ and l denotes orbital quantum num-ber (in 2D case l = m and denotes projection of an-gular momentum on the ferron symmetry axis L z ). Inthe case of a small ferron the degeneracy is lifted dueto the tunneling effect through the interior of the ferron δE ( l ), which however decrease exponentially with the fer-ron size [28]. The spectrum of these states consists ofangular momenta between ± k F R , where R is the ferronradius. The 3D ferron possesses the same structure ofAndreev states with an additional 2 l + 1 degeneracy ofeach state. Apart from these degenerate states which ac-cumulate the spin polarization, there is a small fractionof states with L z ≈ ± k F R , which energy raises steeplywith angular momentum. These states can be interpretedas related to periodic orbits located in the nodal regionwhich represent trajectories between pairing potential ofthe same phase [28].It is important to realize that the stability of the fer-ron is exclusively related to the structure of Andreevstates. When ferron is moving through superfluid or,equivalently, when it is exposed to the superflow, thestructure and energies of these states are modified. Theperturbation is induced by the pairing field which is af-fected by the superflow. In particular the phase of thepairing field is modified on both sides of nodal line, de-pending on its orientation with respect to the directionof superflow. Namely, the spherically symmetric pairingfield becomes perturbed by the superflow in he followingway: ∆ ( r ) → ∆( r ) = ˜∆ ( r ) exp(2 i q · r ). Neglectingin the first approximation the modification of the mag-nitude of initial pairing field associated with ferron, ie.˜∆ ( r ) ≈ ∆ ( r ), it is easy to show that energies of An-dreev states will be shifted in energy proportionally to q . The modification of the spectrum of states inside the FIG. 3. Structure of the spectrum of Andreev states exposedto different strengths of superflow.
Left subfigure: mag-netic quantum numbers m ( m = h L z i / | v | , where | v | de-notes occupation probability of the state) corresponding toAndreev states are shown for two velocities of the superflow: q/v F = 0 .
01 (filled circles) and q/v F = 0 .
05 (empty dia-monds), where v F denotes Fermi velocity. Right subfigure: the expectation value of the momentum operator component,parallel to the direction of the superflow is shown for Andreevstates. The quasiparticle energies have been shifted by δµ and therefore the plots on both subfigures possess symmetrywith respect to E = 0. The spin imbalance corresponds to δN = 31 ( R ≈ . ξ ) and the strength of the pairing field | ∆ | /ε F = 0 . ferron can be seen in Fig. 3.All Andreev states inside the ferron at rest have van-ishing expectation value of linear momentum. When theferron is moving they acquire non zero component of mo-mentum in the direction of the flow. The most affectedstates are those for small angular momenta. As the ve-locity increases more states are affected and finally theycease to stabilize ferron structure leading to its decay.Consequently one may attribute to each ferron a certaincritical velocity v crit which constitute its maximum veloc-ity when moving through the uniform superfluid. In theFig. 4 the critical velocity in units of Fermi velocity hasbeen shown as a function of pairing gap and ferron size.It is of no surprise that the larger sizes of ferrons admitslarger velocities. Clearly, it is related to the fact thatstates with large momenta are less affected by superflow.As a consequence ferrons with larger polarization canmove with higher velocities through the medium. Therelation between critical velocity and polarization whichturn out to be approximately linear in 2D (apart from de-viations induced by deformation changes at the vicinityof critical velocities) represent an interesting manifesta-tion of relation between spatial pairing field modulationand its dynamic properties. In 3D all the argumentsremain valid, however one may expect that due to ad-ditional degeneracy the relation between critical velocityand polarization will read v crit ∝ √ δN . The deviationswhich are visible in Fig. 4 are attributed to the shell ef- FIG. 4. The ferron critical velocity as a function of the mag-nitude of the pairing field | ∆ | /ε F and the spin imbalance δN .The simulation settings are the same as for Fig. 2. Insetshows an example of two different sizes of ferron having thesame critical velocity where the smaller ferron is deformed. fects related to Andreev states. Namely, for velocitiesclose the v crit , some ferrons become deformed, which canbe seen in the inset in Fig. 4 Induced motion of ferron and interaction with a vor-tex — The results presented in previous section can bealso looked at from another perspective. Namely, assumethat one creates a ferron as an excited configuration inunpolarized superfluid medium. This can be achieved byapplying dynamically a spin selective potential, whichwill locally break Cooper pairs. If the potential is ap-plied for a sufficiently long time it allows pairing field toadjust by developing nodal surface. It was shown in theRef. [18] that such configuration is stable despite of thefact that the ferron is surrounded by phonon excitations.Taking into account results from the previous section, onemay ask the following question: What is going to happenif one attempts to accelerate ferron beyond the criticalvelocity. In order to investigate this issue we have per-formed the following time-dependent simulations in 3D.We have applied a spin selective potential in a form ofthe Gaussian (see [27] for details). When the ferron iscreated, we have accelerated the potential, which wasdragging the ferron through the superfluid with velocity v drag . Subsequently we have removed the potential allow-ing the ferron to move freely. It has been found that theferron, after switching off the potential, continues its mo-tion although it always slows down to the velocity v final (see Fig. 5). Still for velocities v drag (cid:28) v crit the relationbetween v drag and v final is approximately linear. Howeverwhen v drag becomes large enough the velocity v final satu-rates and attempts to increase the ferron velocity beyondcertain value fail. Note that the results shown in Fig. 5for different sizes of ferron are consistent with the static FIG. 5. Velocity of the ferron in the final state as a functionof the dragging velocity. The time-dependent spin-selectivepotential is dragged along the x-axis during its application.The horizontal dashed lines shows the ”plateau” of the finalvelocity for various sizes of ferrons corresponding to: σk F = 8, σk F = 6, σk F = 4 from top to bottom, where σ is the width ofthe Gaussian potential. Inset shows the absolute value of thepairing field in the left column, while in the right column thephase of the pairing field is shown. The images are taken afterthe external potential is turned-off while the ferron is moving.All three configurations correspond to v drag /v F = 0 .
06. Inthe simulations we used box of size 53 ξ × ξ × ξ in x,y,zdimensions, respectively. Fermi momentum k F ≈
1. For fullmovies see Supplemental Material [27]. results; the critical velocity increases with the size of theferron.The existence of the critical velocity has yet anotherimportant consequence, when it comes to the possibilityof creating vortices in the system with ferrons. Namely,it is possible to have a coexistence of a vortex and aspherical ferron as long as the distance between vortexcore and a ferron is large enough. In this case the super-flow generated by a vortex is weak enough to support theexistence of the ferron solution. On the other hand anattempt to create a ferron in the vicinity of the vortexcore fails which is shown in the Fig. 6. The snapshotsindicate stages of the ferron decay which was generatedby spin selective Gaussian potential. One expects thatlarge ferrons which are characterized by higher criticalvelocity may be created closer to the vortex core. How-ever in this case effects related to non-uniformity of thesuperflow withing the volume of the ferron may becomeimportant.
Summary — We have investigated dynamical proper-ties of ferrons related to its motion through the super-fluid. We have extracted the effective mass of this objectwhich turned out to be related mainly to spin imbalancewith a small correction coming from induced superfluidflow. Only for small ferrons (of sizes of the order of fewcoherence lengths) the latter contribution becomes im-portant. It implies that the effective mass scale ratherwith the surface than the volume of impurity. We havealso shown that each ferron is characterized by a certain
FIG. 6. Snapshots showing the attempt to create a stableferron solution in the presence of the vortex. The time-dependent potential to generate the ferron is turned off at tε F = 150. The vortex, with core located in the center, cre-ates currents rotating counter-clockwise. It is visible that fer-ron is destroyed because of these currents. The polarizationinside the ferron is pushed to the boundary of the system. Fordetailed info and full movies see [27]. critical velocity which cannot be exceeded while movingthrough the superfluid environment. The critical veloc-ity increases with the ferron size. It was demonstratedthat it is not possible to accelerate ferron dynamically bydragging it beyond certain velocity. For the same reasonit is not possible to create a stable configuration of ferronin the vicinity of the vortex core. Acknowledgments — One of the authors (PM) wouldlike to thank Centre for Computational Sciences at Uni-versity of Tsukuba, where part of this work has beendone for hospitality. This work was supported by thePolish National Science Center (NCN) under ContractsNo. UMO-2016/23/B/ST2/01789 (PM,BT) and UMO-2017/26/E/ST3/00428 (GW). We acknowledge PRACEfor awarding us access to resource Piz Daint based inSwitzerland at Swiss National Supercomputing Centre(CSCS), decision No. 2019215113. We also acknowl-edge Global Scientific Information and Computing Cen-ter, Tokyo Institute of Technology for resources at TSUB-AME3.0 (Project ID: hp200115) and InterdisciplinaryCentre for Mathematical and Computational Modelling(ICM) of Warsaw University for computing resources atOkeanos (grant No. GA83-9). The contribution of eachof the authors has been significant and the order of thenames is alphabetical. ∗ [email protected] † [email protected] ‡ [email protected][1] P. Fulde and R.A. 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B Quantum-coherent transport of a heavy particlein a fermionic bath , (Shaker-Verlag, Aachen, 1997)[24] G. Wlaz lowski, K. Sekizawa, M. Marchwiany, andP. Magierski, Suppressed Solitonic Cascade in Spin-Imbalanced Superfluid Fermi Gas, Phys. Rev. Lett. ,253002 (2018).[25] A. Bulgac, M.M. Forbes, M.M. Kelley, K.J. Roche, G.Wlaz lowski, Quantized Superfluid Vortex Rings in theUnitary Fermi Gas, Phys. Rev. Lett. , 025301 (2014).[26] W-SLDA Toolkit webpage: https://wslda.fizyka.pw.edu.pl [27] See supplemental online material at { URL will be pro-vided by the publisher } for details related to technicalaspects of numerical calculations, implementation of thespin-polarizing potential, extraction of the effective mass.The supplement includes also movies visualizing dynam-ics of ferrons.[28] P. Magierski, B. T¨uzemen, G. Wlaz lowski, (in prepara-tion). upplemental Material for:Ferron dynamics in ultracold atomic gas Piotr Magierski , , ∗ Bu˘gra T¨uzemen , † and Gabriel Wlaz lowski , ‡ Faculty of Physics, Warsaw University of Technology,Ulica Koszykowa 75, 00-662 Warsaw, Poland and Department of Physics, University of Washington, Seattle, Washington 98195–1560, USA
Details of calculations are presented. The prescription for effective mass extraction is discussed.The analytic formulas for effective masses of circular impurity in 2D irrotational hydrodynamics arederived.
PACS numbers: 67.85.De, 67.85.Lm, 74.40.Gh, 74.45.+c
Static BdG calculations in 2D systems
The total energy density of the system in BdG ap-proach is expressed through kinetic and anomalous den-sities: E BdG = τ ↑ + τ ↓ g eff ν † ν. (11)We obtain the stationary configuration by minimizing thefollowing functional: F = E − X s = {↑ , ↓} µ s N s − X s = {↑ , ↓} Z q · j s ( r ) d r , (12)where N s = R n s ( r ) d r denotes particle number of spin s component, µ s are corresponding chemical potentials and E = R E BdG ( r ) d r is the energy. The last term generatesthe flow in directions given by q . Minimization of the F functional provides Eqs. (1) and (2) from the mainpaper. In calculations we used velocity q directed along x direction.The ferronic solution corresponds to a particular choiceof pairing field ∆( r ) which involves a closed nodal line.To capture the ferron geometry we imposed the con-straint on the pairing potential to have the form:∆( r ) = (cid:26) − ∆ , r < R in , ∆ , r > R out , (13)To get the ground state for ferron we applied the aboveconstraint to the system for a couple of iterations dur-ing the energy minimization and subsequently released it.Values of R in and R out are selected in such a way that af-ter convergence the radius of the ferron is between thesevalues. Consequently, the initially imprinted pairing po-tential captures the main features of the ferron, whichconsist of outer and inner areas where the phase of thepairing field varies by π and the nodal region of the sizeof the coherence length where the pairing field vanishes.The Andreev states inside the circular ferron (at q =0) can be labeled by eigenvalues of angular momentumoperator component perpendicular to its area (which we denote by ˆ L z ). However, due to degeneracy of statescorresponding to positive and negative eigenvalues of ˆ L z these states are mixed in numerical calculations and donot have well defined L z value. Therefore in order toremove this degeneracy we add a small perturbation tothe system of equations (2) in the form: − ( ∇ + i q ) − ˜ µ − ωL z where ω is the radial frequency. We typically set thisvalue to ω ≈ . ε F . The perturbation is added only toextract and visualize the Andreev states (see Fig. 3 in themanuscript) and is not applied to get the self-consistentsolution.For 2D static calculations we use a simulation box witha lattice size of 70 k − in x and y directions. We set theFermi momentum k F = p π ( n ↑ + n ↓ ) ≈
1. The finitetemperature T has been used for numerical conveniencewith T /T c ≈ − , where the critical temperature is cal-culated from well-known BCS result ∆ /T c = 1 .
76. Whenthe finite temperature is applied the Eqs. (3-7) are mod-ified to the following form: n s ( r ) = X n | v n,s ( r ) | f β ( − E n ) , (14) τ s ( r ) = X n |∇ v n,s ( r ) | f β ( − E n ) , (15) ν ( r ) = X n v ∗ n, ↓ ( r ) u n, ↑ ( r ) f β ( − E n ) − f β ( E n )2 , (16) j s ( r ) = X n Im[ v n,s ( r ) ∇ v ∗ n,s ( r )] f β ( − E n ) , (17)where f β ( E n ) = 1 / ( e E n /T + 1) is the Fermi-Dirac distri-bution. Time-dependent calculations in 3D systems
Time-dependent simulations in 3D were executed inthe same manner as calculations presented in Ref. [1].Below we provide details of methodology of dynamicalcreation of ferrons. a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Simulations of the moving ferron
We start from the initial solution for unpolarized, uni-tary Fermi gas. Subsequently we apply the spin-selective,time-dependent external Gaussian potential to create lo-cal polarization: V s ( r , t ) = λ s A ( t ) exp (cid:20) − ( x + vt ) + y + z σ (cid:21) . (18)This potential is repulsive for spin-up components, λ ↑ =+1 and attractive for spin-down components, λ ↓ = − σ sets the size of theferron. Calculations are executed on spatial lattice of size68 k − × k − × k − in xyz-directions with periodicboundary conditions, and k F = (3 π ( n ↑ + n ↓ )) / ≈ A ( t ) is the time-dependent amplitude of the potentialand has the following form: A ( t ) = A s ( t, t on ) , t < t on ,A , t on t < t hold ,A [1 − s ( t − t hold , t off − t hold )] , t hold t < t off , , t > t off , (19)where s ( t, w ) denotes the function which smoothly variesfrom 0 to 1 within time interval [0 , w ]: s ( t, w ) = 12 + 12 tanh (cid:20) tan (cid:18) πtw − π (cid:19)(cid:21) . (20) A denotes the amplitude of the potential, which we setto be about A ≈ ε F .To drag the ferron we set the potential in motion byusing v = 0 in Eq. (18), where x is the initial positionof the center of the Gaussian potential along x -axis. Weextract the velocity of the potential ( v drag ) and the ve-locity with which the ferron travels on its own ( v final )by following the position of the center of the polarizedsphere. In Fig. 7 we provide an example for a potentialwidth σk F = 6. During the switching on of the potential,the polarized sphere experiences an acceleration and thepotential creates a force responsible for breaking of theCooper pairs. After the potential reaches its maximumamplitude, it is kept on until the nodal sphere is formed.We then turn the potential off and observe the movingimpurity.As v drag increases, v final eventually reaches a criticalvalue beyond which ferron can not be accelerated fur-ther (Fig. 5). If v drag is increased even more, we observethat the ferron is destroyed during its movement. Thereare two effects responsible of this: When the final veloc-ity gets closer to the critical value, the ferron undergoesdeformation and finally ceases to exist. Moreover, dur-ing the acceleration of the ferron, the external potentialexcites phonons in the system. These phonons scatter FIG. 7: The position of the moving ferron inside a box cor-responding to lattice size 68 × ×
40 which corresponds to53 ξ × ξ × ξ where ξ is the coherence length. The widthof the polarizing potential is σk F = 6 and its amplitude is A = 2 ε F . The potential is switched on at tε F = 50 andcompletely removed at tε F = 150. Different data sets corre-sponds to different dragging velocities. inside the simulation box and interact with the ferron.While for low dragging velocities ferron is stable againstthese perturbations, for high velocities the strength ofthe perturbation increases with the number of excitedphonons and eventually ferron loses its stability. Thiseffect hastens the destruction of the ferron. Ferron in presence of the quantum vortex
The numerical simulations with the presence of a vor-tex are conducted at the unitary limit. For these cal-culations we have used a box with the lattice size of80 × ×
32 which corresponds to 62 ξ × ξ × ξ with k F ≈
1. A straight vortex line along z -direction is obtained byimposing on the static solution the following structure ofthe pairing field: ∆( x, y ) = | ∆( x, y ) | e ( i tan − ( y/x )) . Next,the ferron is generated dynamically, by applying the spinselective potential (18), with v = 0 and x controls dis-tance of the ferron from the vortex core.Additional to the results presented in the main article,we present in Movie 5 the dynamics of ferron placed atthe center of the vortex. In the movie it can be seen thatthe polarization that forms the ferron is absorbed intothe vortex. Extraction of the effective mass
We introduce a superflow corresponding to the veloc-ity of q/v F = 0 . , . , .
03 and 0 .
04 and calculate themomentum to velocity ratio by executing the formula (9)from the main text. Next, we extrapolate results to the q → FIG. 8: The response function ( see eq. (9) in the main text)as a function of the superflow velocity q . Symbols correspondto numerical calculations. Lines are obtained as a result ofinterpolation. The value of effective mass is extracted in thelimit of q → /ε F = 0 .
365 (panel a)and for ∆ /ε F = 0 .
552 (panel b). The lattice size is (70 k − ) where k F ≈ in panel a) of Fig. 8 the response function exhibit morepronounced dependence on q than in the strong pairinglimit shown in panel b). This is due to the fact that inthe former case the critical velocity is lower and the shapeof ferron becomes affected already at relatively small q values.To demonstrate the agreement between the effectivemasses extracted from the behavior of the momentumand the energy as a function of q , we calculated the to-tal energy using the BdG density functional given by thevolume integral of Eq. (11). Subsequently we normalizedthese energies as E ex ( q ) = E ( q ) − E (0). Therefore E ex ( q )gives the contribution to the energy coming from the fer-ron’s response to the superflow. In Fig. 9 we comparethese values to the kinetic energy of the ferron moving ina superfluid environment, E f ( q ) = M eff q , where M eff is extracted as described in the previous paragraph. Mass of circular impurity in 2D irrotationalhydrodynamics
In this section we present a derivation of the effectivemass of circular impurity that can be obtained in irrota-tional hydrodynamics.Let us consider an impurity of radius R moving withvelocity v through the superfluid characterized by the FIG. 9: The excitation energy of the system E ex ( q ) as a func-tion of velocity q obtained in BdG calculations (lines). Thekinetic energy of the ferron E f ( q ) obtained using the extractedeffective mass (points). Both energies are shown in units ofnoninteracting Fermi gas E ffg . The spin-imbalance in the sys-tem is δN = 41. The lattice size is (70 k − ) where k F ≈ velocity potential Φ: ∇ Φ( r ) = (cid:18) ∂ ∂x + ∂ ∂y (cid:19) Φ( r ) = 0 . (21)Inside the impurity the density is denoted by n in whereasoutside - by n out . Conditions for velocity potential atinfinity and at the boundary of impurity lead to:lim r →∞ Φ( r ) = 0Φ( r ) | r = R − = Φ( r ) | r = R + (22) n in ∂ Φ ∂r | r = R − − n out ∂ Φ ∂r | r = R + = ( n in − n out ) v · n , where the last equation is a consequence of continuityrelation for the fluid and n denotes unit vector, normal(outward) to the boundary. The solutions of eq. (21)reads: Φ in ( r ) = n in − n out n in + n out v · r (23)Φ out ( r ) = n in − n out n in + n out R r v · r . (24)One can now evaluate the energy of the system, which isstored in the flow: E = 12 Z r
Below we present the list of movies. All movies are alsoaccessible on YouTube. The movies present the distribu-tion of absolute value of the paring field | ∆( r ) | , the phasedifference of the paring field with respect to the value atthe boundary of the box ∆ ϕ , and the local polarization p ( r ) = n ↑ ( r ) − n ↓ ( r ) n ↑ ( r )+ n ↓ ( r ) , in plane crossing the impurity center. Movie 1:
Simulation demonstrating the dynamic cre-ation of a ferron moving in x -direction. The gener-ating Gaussian potential has the amplitude A = 2 ε F and the width σk F = 4. The potential isdragged with the velocity of v drag = 0 . v F andit has been turned on and off within time interval25 ε − F and kept fixed at its maximum strength for100 ε − F .File: Movie-1.mp4
YouTube: https://youtu.be/0GjYPYKd_sc
Movie 2:
The same configuration as
Movie 1 but withthe width σk F = 6.File: Movie-2.mp4
YouTube: https://youtu.be/qSABK70YUBw
Movie 3:
The same configuration as
Movie 1 but withthe amplitude A = 2 . ε F and the width σk F = 8.File: Movie-3.mp4
YouTube: https://youtu.be/Om9cTbAWmjU
Movie 4:
Simulation demonstrating a ferron in thevicinity of a vortex line. The distance ( d ) betweenthe ferron and the vortex is dk F = 24. The generat-ing Gaussian potential has the amplitude A = 2 ε F and the width σk F = 6. The potential has beenturned on and off within time interval 25 ε − F andkept fixed at its maximum strength for 100 ε − F .File: Movie-4.mp4
YouTube: https://youtu.be/aYcG6mvrGxk
Movie 5:
The same configuration as
Movie 4 exceptthe ferron is located at the vortex core.File:
Movie-5.mp4
YouTube: https://youtu.be/gjAUg2QJG2g ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected][1] P. Magierski, B. T¨uzemen, G. Wlaz lowski, ”Spin-polarizeddroplets in the unitary fermi gas”, Phys. Rev. A100