The linear response to a space- and time-dependent external disturbance of a
system of dilute condensed composite bosons at zero temperature, as obtained
from the linearized version of the time-dependent Gross-Pitaevskii equation, is
shown to result also from the strong-coupling limit of the time-dependent BCS
(or broken-symmetry RPA) approximation for the constituent fermions subject to
the same external disturbance. In this way, it is possible to connect
excited-state properties of the bosonic and fermionic systems by placing the
Gross-Pitaevskii equation in perspective with the corresponding fermionic
approximationsComment: 4 pages, 1 figur

A.J. Leggett

C.A.R. Sá de Melo

D.R. Anglin

E.A. Cornell

E.P. Gross

F. Dalfovo

G. C. Strinati

L.P. Pitaevskii

M. Greiner

M.W. Zwierlein

P. Pieri

S. Jochim

English

arXiv

The linear response to a space- and time-dependent external disturbance of a system of dilute condensed
composite bosons at zero temperature, as obtained from the linearized version of the time-dependent Gross-
Pitaevskii equation, is shown to result also from the strong-coupling limit of the time-dependent BCS ~or
broken-symmetry random-phase! approximation for the constituent fermions subject to the same external
disturbance. In this way, it is possible to connect excited-state properties of the bosonic and fermionic systems
by placing the Gross-Pitaevskii equation in perspective with the corresponding fermionic approximations

STRINATI CALVANESE, Giancarlo

PIERI, Pierbiagio

Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna

Time-dependent Gross-Pitaevskii equation for composite bosons as the  strong coupling limit of the fermionic broken-symmetry random-phase approximation

The linear response to a space- and time-dependent external disturbance of a system of dilute condensed
composite bosons at zero temperature, as obtained from the linearized version of the time-dependent Gross-
Pitaevskii equation, is shown to result also from the strong-coupling limit of the time-dependent BCS ~or
broken-symmetry random-phase! approximation for the constituent fermions subject to the same external
disturbance. In this way, it is possible to connect excited-state properties of the bosonic and fermionic systems
by placing the Gross-Pitaevskii equation in perspective with the corresponding fermionic approximations

Archivio istituzionale della ricerca - Università di Camerino

RAPID COMMUNICATIONSPHYSICAL REVIEW A 69, 011601~R! ~2004!Time-dependent Gross-Pitaevskii equation for composite bosons as the strong-coupling limitof the fermionic broken-symmetry random-phase approximationG. C. Strinati and P. PieriDipartimento di Fisica, UdR INFM, Universita` di Camerino, I-62032 Camerino, Italy~Received 10 September 2003; published 29 January 2004!The linear response to a space- and time-dependent external disturbance of a system of dilute condensedcomposite bosons at zero temperature, as obtained from the linearized version of the time-dependent Gross-Pitaevskii equation, is shown to result also from the strong-coupling limit of the time-dependent BCS~orbroken-symmetry random-phase! approximation for the constituent fermions subject to the same externaldisturbance. In this way, it is possible to connect excited-state properties of the bosonic and fermionic systemsby placing the Gross-Pitaevskii equation in perspective with the corresponding fermionic approximations.DOI: 10.1103/PhysRevA.69.011601 PACS number~s!: 03.75.Ss, 03.75.Hh, 05.30.Jpnopdiseungerton.pxterhei-dinheuca-uownn,se--froebothOne-tionme-stsanc-sonsedGPonInre-e-it.in-tstUltracold atomic Fermi gases are currently being intesively studied experimentally, as their low-temperature prerties can shed light on fundamental questions regardinggenerate Fermi systems. Producing a paired superfluidparticular challenge, which is being achieved at the prestime @1#. Since the attractive interaction responsible for sperfluidity in a Fermi gas can be tuned to reach the strocoupling condition, it also appears possible to study expmentally the evolution from superfluid fermionscondensed composite bosons.~These form as bound-fermiopairs in the strong-coupling limit of the fermionic attraction!This would permit for the first time a controlledcrossoverbetween two basic quantum systems sharing the same staneous broken-symmetry~superfluid! behavior.Dilute condensed bosons have already been studied esively both experimentally@2# and theoretically@3#. The di-luteness condition is matched in almost all current expments. Correspondingly, the Gross-Pitaevskii~GP! equations@4,5# have proven sufficient to describe the observed pnomena. This is true both for ground-state properties~via thetime-independent GP equation! and for the dynamical re-sponse of the condensate to an external disturbance~via thetime-dependent GP equation in its linearized form@6#!.With the aim of connecting properties of superfluid fermons and condensed composite bosons, the time-indepenGP equation for the condensate wave function was obtain a previous paper@7# as the strong-coupling limit of theBogoliubov–de Gennes equations@8# for superfluid fermionsat zero temperature.In this paper, a corresponding connection is establisbetween the linearized form of the time-dependent GP eqtion ~from which the dynamics of the bosonic condensatebe obtained! and the strong-coupling limit of the linear response for superfluid fermions with a BCS ground state, sject to the same external disturbance. We will explicitly shthat the fermionic density and current correlation functiomap, in the strong-coupling limit of the fermionic attractioonto the linear-response results for condensed compobosons. The fermionic correlation functions are obtainfrom the time-dependent BCS@or broken-symmetry randomphase approximation~RPA!# approximation, while the linearresponse results for the composite bosons are obtainedthe time-dependent GP equation. Thus a connection betw1050-2947/2004/69~1!/011601~4!/$22.50 69 0116--e-ant--i-on-en-i--entedda-nb-sitedmenground- and excited-state properties of the condensatein the weak- and strong-coupling limits is established.physical grounds the connection between the timindependent and time-dependent versions of the GP equafor condensed composite bosons and the BCS and tidependent BCS approximations for superfluid fermions, reon the zero-point motion in a quantum theory implyingoscillatory spectrum. The imprint of the quasiparticle spetrum should be found in the ground-state wave function@4#.The response functions of a system of condensed bodescribed by the wave functionF(r ,t) can be obtained fromthe time-dependent GP equation@4,5#:~1/2mB! @ i“1 ~qB/c! Aext~r ,t !#2F~r ,t !1Vext~r ,t !F~r ,t !1U0uF~r ,t !u2F~r ,t !5 i $@]F~r ,t !#/]t% ~1!in the presence of space- and time-dependent vectorAext andscalarVext external potentials. Here,c is the velocity of light,mB and qB are the mass and charge of the bosons, andU054paB /mB is the short-range repulsive potential expressin terms of the bosonic scattering lengthaB (\51 through-out!. Note that we have generalized the time-dependentequation to include a vector potential, as a mereformal de-vice to obtain, in addition to the density-density correlatifunction already considered for neutral bosons@6#, thecurrent-density and current-current correlation functions.this way we can compare these functions with the corsponding quantities for fermions obtained within the timdependent BCS approximation in the strong-coupling limAccordingly, no Coulomb repulsion between bosons iscluded.The linear-response solution of Eq.~1! for Aext50 is ob-tained @6# by taking Vext(r ,t)5Vqei (q•r2vt)1Vq* e2 i (q•r2vt)with wave vectorq and frequencyv, and by setting to linearorderF~r ,t !5AnB@11ūv~r !e2 ivt2 v̄v~r !* eivt#e2 imBt. ~2!nB is the ground-state bosonic density,mB5U0nB thebosonic chemical potential, and the ‘‘small’’ componenūv(r ) and v̄v(r ) are correspondingly taken of the formūv(r )5ūq,veiq•r and v̄v(r )5 v̄q,veiq•r. The time-dependen©2004 The American Physical Society01-1n.dy-enniontondis-.c--etate-ultsns.therebehela-rinc-a--For-RAPID COMMUNICATIONSG. C. STRINATI AND P. PIERI PHYSICAL REVIEW A69, 011601~R! ~2004!GP equation~1! then reduces to a linear system for the uknownsūq,v and v̄q,v . The solution isūq,v5v1EB0~q!v22EB~q!2Vq , v̄q,v5v2EB0~q!v22EB~q!2Vq . ~3!EB0(q)5q2/(2mB) is the free-boson energy andEB(q)5AEB0(q)212mBEB0(q) the Bogoliubov dispersion relationThe induced particle density and current are then obtaineexpanding the expressionsn(r ,t)5uF(r ,t)u2 and j (r ,t)5Im@F(r ,t)*“F(r ,t)/mB# to linear order inVext. One ob-tainsdn~r ,t !5nBmBq2v22EB~q!2Vext~r ,t !, ~4!j ~r ,t !5nBmBqvv22EB~q!2Vext~r ,t !. ~5!These satisfy the continuity equation]dn(r ,t)/]t1“• j (r ,t)50. By linear-response theory, the densitdensity and current-density correlation functions can be idtified asxnn~q,v!5 ~nB/mB!$q2/@v22EB~q!2#% , ~6!x jn~q,v!5 ~nB/mB!$qv/@v22EB~q!2#% , ~7!with the typical Bogoliubov form.Similarly, the linear-response solution of Eq.~1! whenVext50 is obtained by taking Aext(r ,t)5Aqei (q•r2vt)1Aq* e2 i (q•r2vt) and using the same form~2! for the wavefunction. Working in a gauge where“•Aext(r ,t) is nonzero~such thatq•AqÞ0) and retracing the above arguments, oobtainsūq,v5EB0~q!12mB1vEB~q!22v2qB2mBcq•Aq , ~8!v̄q,v5AnBEB0~q!12mB2vEB~q!22v2qB2mBcq•Aq ~9!in place of Eq.~3!. The ensuing induced density and currenow take the form:dn~r ,t !5nBmBvqv22EB~q!2•S 2 qBc Aext~r ,t ! D , ~10!j ~r ,t !52nB~2mB!2qEB0~q!12mBv22EB~q!2q•S 2 qBc Aext~r ,t ! D .~11!These replace Eqs.~4! and ~5!, with the continuity equationbeing satisfied once the diamagnetic contribut2(qBnB)/(mBc)Aext(r ,t) is added to the induced curren01160-byn-et~11!. The density-current and current-current correlatifunctions also acquire the typical Bogoliubov form:xnj~q,v!5 ~nB/mB!$vq/@v22EB~q!2#% , ~12!x jj ~q,v!5~nB/mB2 ! q$@mB1EB0~q!/2#/@v22EB~q!2#% q.~13!The correlation functions~6!, ~7! and~12!, ~13! describe thelinear response of the bosonic condensate to an externalturbance in terms of Bogoliubov quasiparticle excitationsWe now show that the same form of the correlation funtions is obtained from the fermionic time-dependent BCS~orbroken-symmetry RPA! approximation in the strongcoupling limit of the fermionic attraction. We describe thfermionic attraction by a short-range~contact! potential regu-larized in terms of the two-body scattering lengthaF @9,10#.By increasing the strength of the attraction, a bound sappears in the two-body problem andaF becomes positive.For strong attraction, the fermionic chemical potentialm ap-proaches2«0/2, where«05(maF2)21 is the binding energyof the composite bosons which form in this limit. The composite bosons have massmB twice the fermionic massm andpositivescattering lengthaB52aF . The positive scatteringlength ensures the stability of the bosonic system and resfrom the Pauli repulsion between the constituent fermio~It was first pointed out in Ref.@10# that the relationaB52aF is actually an approximate one, being the result ofBCS mean-field treatment of the fermionic attraction. Morefined treatments yield a reduction ofaB by about a factorof 3 @10,11#. This reduction, however, is not expected torelevant to the main results of this paper.!We first express the fermionic correlation functions in tbroken-symmetry phase in terms of the two-particle corretion functionL~1,2;18,28!5^Tt@c~1!c~2!c†~28!c†~18!#&2G~1,18!G~2,28!. ~14!Here,^•••& is a thermal average,Tt the time-order operatofor imaginary time t, 15(r1 ,t1 ,,1) with the index ,5(I,II) identifying the components of the field operatorthe Nambu representation@c I(r )5c↑(r ) and c II(r )5c↓†(r )], and G(1,2)52^Tt@c(1)c†(2)#& is the single-particle Green’s function. The two-particle correlation funtion ~14! can be expressed in terms of the many-particleTmatrix @12#2L~1,2;18,28!5G~1,28!G~2,18!1E d3456G~1,3!G~6,18!3T~3,5;6,4!G~4,28!G~2,5!. ~15!An explicit form for T results from the Bethe-Salpeter eqution once a choice for its kernel is made.Within the ~off-diagonal! BCS approximation we are considering, the Bethe-Salpeter equation forT can be explicitlysolved in the frequency and wave-vector representation.a short-range~contact! potential, the only nonvanishing elements of the many-particleT matrix correspond to the1-2leinyiath-neatistoofra-m-rre-ioningon-nc-honRAPID COMMUNICATIONSTIME-DEPENDENT GROSS-PITAEVSKII EQUATION . . . PHYSICAL REVIEW A69, 011601~R! ~2004!Nambu indices reported in Fig. 1~a!. In the strong-couplinglimit the binding energy«0 becomes the largest energy scathus yieldingbm→2` and D!umu (b being the inversetemperature andD the BCS gap function!. In this limit, onethus obtains the following expressions for the nonvanishelements of the many-particleT matrixT11~q,Vn!5T22~q,2Vn!.8pm2aFiVn1q2/~2mB!1mB~ iVn!22EB~q!2,~16!T12~q,Vn!5T21~q,Vn!.28pm2aFmB~ iVn!22EB~q!2.~17!Vn52np/b (n integer! is a bosonic Matsubara frequenc@13#, mB52m1«0 is the same bosonic chemical potentintroduced after Eq.~2!, andEB(q) coincides formally withthe Bogoliubov dispersion of Eq.~3!. The fermionic single-particle Green’s functions are correspondingly given byBCS expressionsG11~k,vn!52G22~k,2vn!52j~k!1 ivnE~k!21vn2, ~18!G21~k,vn!5G12~k,vn!5DE~k!21vn2. ~19!vn5(2n11)p/b (n integer! is a fermionic Matsubara frequency @13#, j(k)5k2/(2m)2m, and E(k)5Aj(k)21D2for an (s-wave! isotropic gap functionD.The fermionic correlation functions for the number desity and current can be quite generally obtained from thdefinitions as thermal averages of the time-order operacting on the number density and current operators. Thdone by expressing the number density and current operain terms of Nambu field operators and introducing the twFIG. 1. ~a! Nambu indices associated with the four nonvanising elements of the many-particleT matrix for a fermionic contactpotential;~b! Relation between the two-particle correlation functiL and the many-particleT matrix.01160,gle-irorisrs-particle correlation function~14! accordingly. One ends upwith the following expressions:xnn~q!52 (,,,8,,9,,-t,,,83 t,9,,-3 E dkdk8~2p!61b23 (n,n8L,,,-,8,,9~k,k8;q!, ~20!xnj~q!5212m (,,,8,,9,,-t,,,83 t,9,,-0 E dkdk8~2p!63~2k81q!1b2(n,n8L,,,-,8,,9~k,k8;q!, ~21!x jj ~q!521~2m!2(,,,8,,9,,-t,,,80 t,9,,-0 E dkdk8~2p!63~2k1q!~2k81q!1b2(n,n8L,,,-,8,,9~k,k8;q!.~22!The notation is q5(q,Vn), k5(k,vn), k85(k8,vn8),wheret0 andt3 are Pauli matrices, and the labels forL arespecified in Fig. 1~b!. The first term on the right-hand side othe diagrammatic representation of Fig. 1~b! corresponds tothe standard BCS contribution to the response functions@13#.This term vanishes in the strong-coupling limit for tempetures much below the dissociation temperature of the coposite bosons. The second term, on the other hand, cosponds to the broken-symmetry BCS-RPA approximatand is required to maintain full gauge invariance@14#. In thepresent context this term has a nontrivial strong-coupllimit and yields the response of the composite bosons cdensate. From this term one obtains for the correlation futions ~20!–~22!xnn~q!5 (,,,8,,9,,-D,,,8~q!T,,,-,8,,9~q!D,-,,9~q!, ~23!xnj~q!5 (,,,8,,9,,-D,,,8~q!T,,,-,8,,9~q!V,-,,9~q!, ~24!x jj ~q!5 (,,,8,,9,,-V,,,8~q!T,,,-,8,,9~q!V,-,,9~q!, ~25!withD,,,8~q![ (,9,,-t,9,,-3 E dk~2p!31b (n G,,,9~k!3G,-,,8~q1k!, ~26!-1-3heths-lainofentforcon-t GPap-ticGPenttab-eei-theyture.onlyan-ofim-iteera-orntndre-ur-ingc-rmsiteen-i-RAPID COMMUNICATIONSG. C. STRINATI AND P. PIERI PHYSICAL REVIEW A69, 011601~R! ~2004!V,,,8~q![12m (,9,,-t,9,,-0 E dk~2p!31b (n ~2k1q!3G,,,9~k!G,-,,8~q1k!. ~27!Recalling that, for a fermionic contact potential, only tfour elements~16! and~17! of the many-particleT matrix arenonvanishing, and exploiting the symmetry properties ofquantities~26! and ~27!, one arrives atxnn~q!5„D~q!,D~2q!…S T11~q! T12~q!T21~q! T22~q! D S D~q!D~2q! D ,xnj~q!5„D~q!,D~2q!…S T11~q! T12~q!T21~q! T22~q! D S V~q!V~2q! D ,x jj ~q!5„V~q!,V~2q!…S T11~q! T12~q!T21~q! T22~q! D S V~q!V~2q! D ,where D(q).24mD(maF)/(16p) and V(q).2qD(maF)/(16p). Using the approximate expression~16! and ~17!, one obtains finally for the fermionic correlation functions in the strong-coupling limitxnn~q!.D2~q50!@T11~q!1T11~2q!12T12~q!#54~nB/mB!$q2/@~ iVn!22EB~q!2#% , ~28!xnj~q!.D~q50!V~q!@T11~q!2T11~2q!#54~nB/mB!$ iVnq/@~ iVn!22EB~q!2#%, ~29!x jj ~q!.V~q!V~q!@T11~q!1T11~2q!22T12~q!#54nBmB2qmB1q2/~4mB!~ iVn!22EB~q!2q. ~30!The analytic continuationiVn→v1 id (d501) to the realfrequencyv is now required to obtain the retarded corretion functions. In this way the bosonic expressions~6!,~12!,and ~13! are recovered from their fermionic counterpartsodi-01160e-the strong-coupling limit. There is an additional factorfour, reflecting the fact that the bosonic number and currdensities are half the fermionic values.The present mapping has been explicitly establishedfermions ~and composite bosons! in the absence of a statiexternal potential. On physical grounds we expect the cnection we have established between the time-dependenequation and the fermionic broken-symmetry BCS-RPAproximation to remain valid also in the presence of a staconfining potential.A similar connection between the time-dependentequation in strong coupling and the time-dependGinzburg-Landau equation in weak coupling has been eslished in Ref.@9# at the level of the functional integral in thabsence of a static external potential.The above forms of the correlation functions obtainedther from the time-dependent GP equation or fromstrong-coupling limit of the fermionic broken-symmetrBCS-RPA approximation, containonly the contribution ofthe condensate and thus hold at or near zero temperaSince the current response of the condensate can belongitudinal, the transverse current correlation function cnot be obtained in this way since it requires inclusionexcitations out of the condensate. These excitations areportant for determining the behavior of the system at fintemperature and close to the superfluid transition tempture. Corrections to the time-dependent GP equation,equivalently additional fermionic contributions to the currecorrelation function in the broken-symmetry phase over aabove the BCS-RPA approximation, are thus required tocover, for instance, the Bogoliubov approximation to the crent correlation function for bosons@15#. This contains atransverse in addition to a longitudinal part.In conclusion, we have shown that in the strong-coupllimit of the mutual fermionic attraction the response funtions obtained at zero temperature from the linearized foof the time-dependent GP equation for a system of compobosons are equivalent to those obtained within the broksymmetry BCS-RPA approximation for the original fermonic system.@1# M. Greiner, C.A. Regal, and D.S. Jin, Nature~London! 426,537 ~2003!; S. Jochimet al., Science302, 2101~2003!; M.W.Zwierlein et al., Phys. Rev. Lett.91, 250401~2003!.@2# For reviews, see E.A. Cornell and C.E. Wieman, Rev. MPhys.74, 875 ~2002!; D.R. Anglin and W. Ketterle,ibid. 74,1131 ~2002!.@3# For reviews, see F. Dalfovoet al., Rev. Mod. Phys.71, 463~1999!; A.J. Leggett,ibid. 73, 307 ~2001!.@4# E.P. Gross, Nuovo Cimento20, 1766~1961!; J. Math. Phys.4,195 ~1963!.@5# L.P. Pitaevskii, Sov. Phys. JETP13, 451 ~1961!.@6# C.J. Pethick and H. Smith,Bose-Einstein Condensation in Dlute Gases~Cambridge University Press, Cambridge, 2002!.@7# P. Pieri and G.C. Strinati, Phys. Rev. Lett.91, 030401~2003!..@8# P.G. De Gennes,Superconductivity of Metals and Alloys~Ben-jamin, New York, 1966!.@9# C.A.R. Sáde Meloet al., Phys. Rev. Lett.71, 3202~1993!.@10# P. Pieri and G.C. Strinati, Phys. Rev. B61, 15 370~2000!.@11# D.S. Petrovet al., e-print cond-mat/0309010.@12# See, e.g., G.C. Strinati, Riv. Nuovo Cimento11, No. 12~1988!.@13# See, e.g., A.L. Fetter and J. D. Walecka,Quantum Theory ofMany-Particle Systems~McGraw-Hill, New York, 1971!.@14# J.R. Schrieffer,Theory of Superconductivity~Benjamin, NewYork, 1964!.@15# G.C. Strinati, inElectrons and Photons in Solids~Scuola Nor-male Superiore, Pisa, 2001!, p. 403.1-4

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Time-dependent Gross-Pitaevskii equation for composite bosons as the strong-coupling limit of the fermionic broken-symmetry random-phase approximation