Dark solitons in a superfluid Fermi gas

a r X i v :0706.0601v 2 [c o n d -m a t .s t a t -m e c h ] 12 O c t 2007

Dark solitons in a super?uid Fermi gas

Mauro Antezza,1Franco Dalfovo,1Lev P.Pitaevskii,1,2and Sandro Stringari 1

1

Dipartimento di Fisica,Universit`a di Trento and CNR-INFM BEC,Via Sommarive 14,I-38050Povo,Italy

2

Kapitza Institute for Physical Problems,ul.Kosygina 2,119334Moscow,Russia

(Dated:February 1,2008)

We investigate the behavior of dark solitons in a super?uid Fermi gas along the BCS-BEC crossover by solving the Bogoliubov -de Gennes equations and looking for real and odd solutions for the order parameter.We show that in the resonance unitary region,where the scattering length is large,the density pro?le of the soliton has a deep minimum,di?erently from what happens in the BCS regime.The super?uid gap is found to be signi?cantly quenched by the presence of the soliton due to the occurrence of Andreev fermionic bound states localized near the nodal plane of the order parameter.

PACS numbers:03.75.Lm,03.75.Ss,03.75.Kk

INTRODUCTION

The interplay between e?ects of coherence and inter-action is one of the most interesting features exhibited by super?uids.It shows up,in particular,through the occurrence of topological excitations such as vortices and solitons.These have been the object of systematic inves-tigations in the case of ultracold Bose gases,where coher-ence is the result of Bose-Einstein condensation (BEC)and is associated with long range order in the one-body density matrix.The situation is even more interesting in the case of Fermi gases where coherence originates from the interaction between particles which,at low tempera-ture,brings the system into a super?uid phase character-ized by long range order in the two-body density matrix.For this reason,the consequences of coherence on mea-surable quantities,like the density distribution of the gas,are more indirect and subtle than for bosons.

Fermi super?uid gases are now available experimen-tally in 40K and 6Li and various regimes along the BCS-BEC crossover can be explored by tuning the atomic s -wave scattering length a through the Feshbach reso-nances exhibited by these atoms.When a is small and negative the system is described by the BCS theory of super?uidity.Conversely when a is small and positive dimers of atoms of di?erent spin are formed and the sys-tem behaves like a BEC of molecules.Near resonance the scattering length is much larger than the average in-terparticle distance and the system enters the so called unitary regime.Quantized vortices have been recently observed along the BCS-BEC crossover [1]and have been the object of several theoretical papers [2,3,4,5,6].In this work we investigate another important class of nonlinear topological excitations:dark solitons.In three dimensions (3D)a dark soliton is characterized by a real order parameter which changes sign at a planar node (a point node in 1D).In the BEC regime the soli-ton is a solution of the Gross-Pitaevskii (GP)equation for the order parameter of the condensate with repulsive interaction [7].In a uniform 3D system dark solitons are known to be unstable via the snake instability,i.e.,

a sinusoidal transverse oscillation of the planar node.In trapped gases,however,the instability timescale can be very long,so that solitons can indeed be observed [8].While in the BEC case the node of the order parame-ter causes a notch in the density distribution,in a BCS super?uid the density is almost una?ected by the pres-ence of the node [9].The situation is similar to the one of vortices.As in that case,the behavior of the density along the BCS-BEC crossover is expected to be inter-esting and rather nontrivial,as a result of the delicate balance of coherence and nonlinear interactions.A ma-jor question concerns the behavior at unitary where no exact many-body theory is presently available.The prob-lem also shares useful analogies with the interference be-tween two expanding Fermi super?uid gases where the order parameter is expected to exhibit an oscillating be-havior with a change of sign,but no quantitative predic-tions are available concerning the behavior of the density.

BOGOLIUBOV -DE GENNES THEORY

We investigate the problem by using a mean-?eld the-ory for a 3D Fermi gas at zero temperature,based on the solution of the Bogoliubov -de Gennes (BdG)equations [10,11]:

?H ?(r )??(r )??H

u η(r )

v η(r )

=εη

u η(r )

v η(r )

,

(1)

where ?H

=? 2?2/2M +V ext (r )?μis the single-particle grand-canonical Hamiltonian.For given chem-ical potential,μ,and order parameter of the super-?uid phase,?(r ),these equations provide the spec-trum εηand quasiparticle amplitudes u η(r )and v η(r ),which are required to satisfy the normalization rela-tion d 3r u ?η′(r )u η(r )+v ?η′(r )v η(r ) =δη′η.The above equations must be solved together with the equations for

2 the order parameter and the density:

?(r)=?g ηuη(r)v?η(r),(2)

n(r)=2 η|vη(r)|2.(3)

This is done by means of an iterative procedure,which

starts from a trial function?(r)and converges to the self-

consistent solution of(1)-(3).All sums in the equations

are limited by an energy cuto?,0≤εη≤E c.This cuto?

is required in order to cure the ultraviolet divergences

and is accompanied by a regularization of the interaction

parameter g according to[11]:

1

gk3

F +

2E c

k F/L⊥and

vη(r)=v n,n

⊥(z)e i k⊥·r⊥

√

3?10%on the BEC side and?1%in the other cases.A

larger discrepancy is of course obtained by using a lower

cuto?energy,especially on the BEC side of the resonance

where the formation of molecules with energy～ 2/ma2

requires large values of E c to reach convergence.

Figure1b shows the occurrence of a deep depletion of

the density at unitarity(solid line)with a?80%con-

trast,comparable to the one of the BEC regime(dash-

dotted).On the BCS side(dashed),conversely,the con-

trast is only?30%at(k F a)?1=?1and becomes expo-

nentially small in the limit k F|a|?1.These results are

consistent with those obtained for the pro?le of a vortex

core[5,6].

In Fig.1a one notices that,as for the core of a vortex,

the order parameter of the soliton in the BCS regime

exhibits two length scales:a steep slope in a narrow re-

gion of the order of k?1F and a smoother slope in wider

region of sizeξBCS= v F/?0,whereξBCS is the coher-

ence length of the Cooper pairs and v F=

2ξBEC),with n B0=

√

n0/2=k3F/(6π2),ξBEC= /

?2z+g BF n B(z) Ψ(z)=εΨ(z),(5)

where g BF=2π 2a BF/m r is the dimer-atom coupling

constants,while m r=m B m F/(m B+m F)=2M/3is

the reduced mass.The exact values of the a BB and

a BF scattering lengths are equal to0.6a[14]and1.2a

[22],respectively,while the BCS mean?eld theory yields

the Born approximation values2a and8/3a[23].Equa-

tion(5)corresponds to the Schr¨o dinger equation for a

particle in a modi?ed P¨o schl-Teller potential.The so-

lutions are analytic[24]and include bound states.The

energy of these bound states can be used to calculate the

4

FIG.3:Dashed lines:density pro?le n B(z)/n B0and wave func-tion of the lowest fermionic bound state,Ψ0(z),in the BEC limit. Solid lines:density pro?le n(z)/n0and lowest eigenvector u0(z)of the BdG equations for(k F a)?1=+1.

minigap for fermionic excitations in the BEC regime.In

fact the breaking of a pair causes the unbound motion of two fermions which will occupy the single particle states Ψ0with lowest https://www.360docs.net/doc/9d1125922.html,ing the mean?eld values for

the scattering length,the energy of the lowest state turns out to be exactly one half of the bulk value g BF n0/2[25] so that

?gap??minigap=4π 2an0/M.(6) Equation(6)provides a good quantitative estimate for the gap even for(k F a)?1=+1where the above ex-

pression gives the value?gap??minigap=0.26?gap to be compared with our numerical result?0.27?gap.In

Fig.3the analytic curve for the density n B(z)and the bound state wave functionΨ0(z)(dashed lines)are com-pared with n(z)/2and u0(z)obtained from BdG equa-tions(solid lines).The agreement is rather good and the small di?erences are due to the fact that(k F a)?1=+1 is still relatively far from the true BEC limit and to the use of a?nite cuto?energy E c.

CONCLUSIONS

In conclusion,we have theoretically investigated the behavior of dark solitons in a Fermi super?uid in the BCS-BEC crossover,showing that the soliton at unitarity has a large density contrast.We have also discussed the existence of bound states.Fermionic bound states have already been the object of investigations in the context of 1D dark solitons in conducting polymers[26].They play an important role also in the physics of domain walls in the Fulde-Ferrell-Larkin-Ovchinnikov states(see,for in-stance,Ref.[19,20]and references therein)and in the case of Josephson currents through a potential barrier [27].In the opposite BEC limit,the binding of the un-paired Fermi atoms on the solitonic plane shares an inter-esting analogy with the Andreev state of3He atoms on the free surface of super?uid4He[28].In the geometry of the soliton,the fermionic bound state can give rise to a 2D Fermi gas embedded into a molecular Bose super?uid. This can be particularly interesting when the molecular BEC is made starting from a slightly imbalanced spin population.The residual gas of unpaired atoms can eas-ily?ll the available states in the soliton,thus forming a polarized2D Fermi gas.From the experimental view-point,it seems quite possible to observe solitons in the BCS-BEC crossover,for instance,by producing them in the molecular BEC phase with known techniques[8]and then tuning the scattering length across the Feshbach resonance,similarly to what has already been done with vortices.A natural extension of this work is the study of solitons characterized by a complex order parameter and moving in the super?uid(grey solitons).

ACKNOWLEDGMENTS

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5

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