Relation between generalized Bogoliubov and Bogoliubov–de Gennes approaches including Nambu–Goldstone mode

Relation between generalized Bogoliubov and Bogoliubov–de Gennes approachesincluding Nambu–Goldstone modeM. Mine, M. Okumura, and Y. Yamanaka Citation: Journal of Mathematical Physics 46, 042307 (2005); doi: 10.1063/1.1865322 View online: http://dx.doi.org/10.1063/1.1865322 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/46/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A scattering view of the Bogoliubov-de Gennes equations AIP Conf. Proc. 1485, 312 (2012); 10.1063/1.4755832 Some mathematical aspects of the correspondence between the generalized nonlinear Schrödinger equationand the generalized Kortewegde Vries equation AIP Conf. Proc. 1188, 365 (2009); 10.1063/1.3266814 On the incompatibility between General Relativity and Quantum Theory AIP Conf. Proc. 977, 3 (2008); 10.1063/1.2902797 Triplet Vortex Lattice Solutions of the Bogoliubovde Gennes Equation in a Square Lattice AIP Conf. Proc. 850, 839 (2006); 10.1063/1.2354966 Non-perturbative canonical formulation of the finite temperature Nambu-Goldstone theorem AIP Conf. Proc. 519, 745 (2000); 10.1063/1.1291657

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Relation between generalized Bogoliubov andBogoliubov–de Gennes approaches includingNambu–Goldstone mode

M. Minea!

Department of Physics, Waseda University, Tokyo 169-8555, Japan

M. Okumurab!

Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan

Y. Yamanakac!

Department of Materials Science and Engineering, Waseda University, Tokyo 169-8555,Japan

sReceived 1 June 2004; accepted 13 December 2004; published online 25 March 2005d

The two approaches of consistent quantum field theory for systems of the trappedBose–Einstein condensates are known, one is the Bogoliubov–de Gennes approachand the other is the generalized Bogoliubov approach. In this paper, we investigatethe relation between the two approaches and show that they are formally equivalentto each other. To do this one must carefully treat the Nambu–Goldstone modewhich plays a crucial role in the condensation. It is emphasized that the choice ofvacuum is physically relevant. © 2005 American Institute ofPhysics.fDOI: 10.1063/1.1865322g

I. INTRODUCTION

The Bose–Einstein condensation is associated with the appearance of the Nambu–GoldstonesNGd mode1 which is gapless. This is because the phenomenon is a manifestation of a spontaneousbreakdown of a global phasesgauged symmetry in quantum field theorysQFTd and then theGoldstone theorem requires that there must be a gapless mode. The NG mode plays a crucial rolein creating and maintaining an ordered state such as the Bose–Einstein condensatesBECd.

For systems of the trapped BECs,2–4 there are two approaches in QFT involving the NG modeexplicitly. They are called the Bogoliubov–de GennessBdGd5,6 and the generalized BogoliubovsGBd7,8 approaches. Both seem to offer consistent formulations in the sense that the canonicalcommutation relations are respected and that the unperturbed Hamiltonian is diagonalized. Thepurpose of this paper is to study the relation between them. We can prove that they are formallyequivalent to each other, finding the explicit linear transformation in operators from one to theother. The treatment of the NG mode is subtle. We will clarify this point.

After the formal equivalence of the two approaches are established, we emphasize that thechoice of vacuum is relevant physically while the choice of approach is not.

The paper is organized as follows. In Sec. II, the model action and Hamiltonian are given. Anartificial breaking term is introduced there. Sections III and IV are devoted to reviews of the GBand BdG approaches, respectively. Section V is a central part of this paper, in which first theequivalence in the excitation modes is proved and then the correspondence between the NG modesin the two approaches is established. It will be found that the two sets of the NG modes are relatedthrough a squeezing transformation. In Sec. VI, we give a summary and discuss implications of

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our conclusion. Finally we comment that the two different types of vacua lead to different observ-able results. In the Appendix, we give expressions for free propagators when the unperturbedvacuum associated with quantum coordinates is chosen.

II. MODEL ACTION AND HAMILTONIAN

We start with the following action to describe the trapped BEC of neutral atoms:

S=E dt d3xFc†sxdsT − K − V + mdcsxd −g

2c†sxdc†sxdcsxdcsxdG , s1d

where

T = i]

]t, s2ad

K = −1

2m¹2, s2bd

V = 12mv2sx2 + y2 + z2d. s2cd

with the chemical potentialm and the coupling constantg. Here we assumed for simplicity theisotropic trapping potential with its frequencyv, but the essences of the present paper are not losteven in anisotropic trapping potentials. Throughout this paper" is set to unity. This action isinvariant under the global phase transformation,

csxd → eihcsxd and c†sxd → e−ihc†sxd, s3d

whereh is an arbitrary constant phase.When a continuous symmetry is broken spontaneously and the NG mode is present, one

usually needs an artificial breaking interaction to control an infrared behavior of the system. In thesystem of the BEC, we add

DS= «eE dt d3xfe−iuvsxdcsxd + eiuvsxdc†sxdg s4d

to the original actions1d,8

S« = S+ DS. s5d

Here« is an infinitesimal dimensionless parameter which is taken to be vanishing at the final stageof calculation, ande is a typical energy scale of the system given byv. As we will see below, sucha breaking term is necessary for the GB formalism, while it seems that the BdG formalism can beformulated without it.

Let us divide the original fieldcsxd into the classical and quantum parts as

csxd = eiuvsxd + eiuwsxd, s6d

where it is assumed that thec-number real functionvsxd, whose square is a distribution functionof condensed particle, is time independent, and thatu is real, time and space independent, corre-sponding to the situation without vortices. We factorizeeiu in wsxd so that there appears no phasefactor in the following formulations. Equations6d is substituted into Eq.s5d, which is rewritten interms ofvsxd andwsxd as follows:

S« = S0 + S1 + S2 + S3,4, s7d

where

042307-2 Mine, Okumura, and Yamanaka J. Math. Phys. 46, 042307 ~2005!


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S0 =E dt d3xFvsxds− K − V + m + 2«edvsxd −g

2v4sxdG , s8ad

S1 =E dt d3xhvsxdf− K − V + m − gv2sxd + «egwsxd + w†sxdf− K − V + m − gv2sxd + «egvsxdj,

s8bd

S2 =E dt d3xHw†sxdfT − K − V + mgwsxd −g

2v2sxdf4w†sxdwsxd + w2sxd + w†2sxdgJ , s8cd

S3,4=E dt d3xH− gvsxdfw†sxdw†sxdwsxd + w†sxdwsxdwsxdg −g

2w†sxdw†sxdwsxdwsxdJ . s8dd

At the tree level, thec-number functionvsxd satisfies

fK + V − m + gv2sxd − «egvsxd = 0, s9d

which, at the limit of vanishing«, is reduced to the Gross–PitaevskiisGPd equation:9

fK + V − m + gv2sxdgvsxd = 0. s10d

We rewritevsxd as

vsxd = ÎNcf«sxd, s11d

since the condensate particle numberNc is given by

Nc =E d3x v2sxd, s12d

and f«sxd can be normalized to unity,

E d3x f«2sxd = 1. s13d

The suffix « in f«sxd is put to remind us that it is«-dependent, which will be relevant in latersections.

The total Hamiltonian of the system is now written as

H = H0 + Hint, s14d

where

H0 =E d3xHw†sxdsK + V − mdwsxd +gNc

2f«2sxdf4w†sxdwsxd + w2sxd + w†2sxdgJ , s15d

Hint =E d3xHgÎNcf«sxdfw†sxdw†sxdwsxd + w†sxdwsxdwsxdg +g

2w†sxdw†sxdwsxdwsxdJ . s16d

It is emphasized that the canonical commutation relations

fwsx,td,w†sx8,tdg = d3sx − x8d s17d

and fwsx ,td ,wsx8 ,tdg=fw†sx ,td ,w†sx8 ,tdg=0, must hold for consistent QFT.

042307-3 Bogolubov-de Gennes approach J. Math. Phys. 46, 042307 ~2005!


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III. GENERALIZED BOGOLIUBOV APPROACH

First let us review the GB approach, in which the field operatorwsxd andw†sxd are expandedas

wsxd = on=0

`

anstdwns«dsxd, s18ad

w†sxd = on=0

`

an†stdwn

s«dsxd, s18bd

where the complete orthonormal set of real functionshwns«dsxdj,

E d3xwns«dsxdwm

s«dsxd = dnm, s19d

on=0

`

wns«dsxdwn

s«dsx8d = d3sx − x8d, s20d

is obtained from the following eigenequation:

sL − Mdwns«dsxd = sen + «edwn

s«dsxd. s21d

For later convenience, we have introduced the abbreviated notations of

L = K + V − m + 2gNcf«2sxd, s22ad

M = gNcf«2sxd, s22bd

and the indexs«d in hwns«dsxdj indicates that they are«-dependent, andhwn

s«dsxdj are expanded as

wns«dsxd = wn

s0dsxd + «wns1dsxd + ¯ . s23d

So in the limit «→0, the functionshwns«dsxdj reduce tohwn

s0dsxdj, which are the solutions of theeigenequations

sL − Mdwns0dsxd = enwn

s0dsxd. s24d

The eigenequations21d with n=0 is identical with thes«-modifiedd GP equations9d, andw0s«dsxd is

nothing but f«sxd defined in Eq.s11d. Hereafter the indexs«d in hws«dsxdj is omitted otherwisementioned.

For the operatorsanstd and an†std the following commutation relations are assumed:

fanstd,am† stdg = dnm s25d

andfanstd ,amstdg=fan†std ,am

† stdg=0. Combining these with the completeness condition in Eq.s20d,one can reproduce the canonical commutation relations of Eq.s17d. In the formulation under the“Bogoliubov” approximation, meaning that the terma0wnsxd is absent in the expansion of Eq.s18d, the canonical commutation relations are violated as

fwsx,td,w†sx8,tdg = d3sx − x8d − w0sxdw0sx8d. s26d

Substituting Eq.s18d into Eq. s15d, we rewriteH0 in terms of thea-operators,



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H0 = on=0

`

enan†an + o

n,m=0

`

f2an†Unmam + anUnmam + an

†Unmam† g, s27d

where

Unm=gNc

2E d3x f«

2sxdwnsxdwmsxd. s28d

This Hamiltonian can be diagonalized by introducing the generalized Bogoliubov transformation,

bn = om=0

`

sCnmam + Snmam† d, s29ad

bn† = o

m=0

`

sCnmam† + Snmamd, s29bd

and

H0 = on=0

`

fEG,nbn†bn + sc-numbersdg. s30d

Here the real matricesC andS in the matrix notation are given by

C = 12sEG

1/2Oe−1/2 + EG−1/2Oe1/2d, s31ad

S= 12sEG

1/2Oe−1/2 − EG−1/2Oe1/2d, s31bd

with

sEGdnm= EG,ndnm, s32ad

sednm= sen + «eddnm, s32bd

and the orthogonal matrixO diagonalizes the real symmetric matrixW with eigenvalues ofEG,n

2 sn=0,1,2, . . .d,

OWOT = EG2 , s33d

where

W= S4s«edU00 + Os«2d Î«eu8T + Os«3/2dÎ«eu8 + Os«3/2d W8 + Os«d

D , s34d

with

u8 = 14Îe1U10

4Îe2U20

]

2 , s35ad

Wnm8 = en2dnm+ 4ÎenUnm

Îem sn,m= 1,2, . . .d. s35bd

We use the following notations for matrices such asA and A8: A8 stands for a matrixAnm withn,m=1,2, . . .while A does for one withn,m=0,1,2, . . .. And wenotice that when we investigate



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the leading order behavior of «, it does not matter if we usehwn

s0dsxdj for Unm instead ofhwns«dsxdj fsee Eq.s28dg.

One can easily check the properties ofC andS from Eq. s31d,

om=0

`

sCnmCn8m − SnmSn8md = dnn8, s36ad

om=0

`

sCnmSn8m − SnmCn8md = 0. s36bd

The above expressions are expanded with respect to the infinitesimal parameter«. We note that thezeroth eigenvalue,

EG,0 = Î«eÎE0 + Os«3/2d, s37ad

E0 ; 4U00 − u8TW8−1u8, s37bd

approaches zero as it should be, but is kept nonvanishing due to«, which enables us to diagonalize

H0.The matrixO is obtained explicitly as

O = S1 − 12s«edu8TW8−2u8 + Os«2d − Î«eu8TW8−1O8T + Os«3/2d

Î«eO8W8−1u8 + Os«3/2d O8 + Os«dD , s38d

whereO8 is an orthogonal matrix diagonalizing the matrixW8,

O8W8O8T = EG82, s39d

with the diagonal matrix

sEG82dnm= EG,n

2 dnm+ Os«1/2d sn,m= 1,2, . . .d. s40d

Thus the mixing amongan sn=1,2, . . .d is regular with respect to«, only the mixing betweena0

and otheran’s gives rise to a singularity.

Thus the field operators in the GB approach are expanded in terms ofb-operators diagonal-izing the unperturbed Hamiltonian as

wsxd = on=0

`

fbnwCnsxd − bn†wSnsxdg, s41ad

w†sxd = on=0

`

fbn†wCnsxd − bnwSnsxdg s41bd

with

wCnsxd = om=0

`

Cnmwmsxd, s42ad

wSnsxd = om=0

`

Snmwmsxd. s42bd



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IV. BOGOLIUBOV–DE GENNES APPROACH

Next we review the BdG approach. The parameter« is set to be zero throughout this section,so all the quantities are those of«=0.

One sets up the coupled eigenequations,

Lunsxd − Mvnsxd = EB,nunsxd, s43ad

Lvnsxd − Munsxd = − EB,nvnsxd, s43bd

where the notations in Eq.s22d are used and we assume the functionsunsxd andvnsxd to be real.The orthonormal condition reads as

E d3xfunsxdumsxd − vnsxdvmsxdg = dnm, s44ad

E d3xfunsxdvmsxd − vnsxdumsxdg = 0. s44bd

Several authors have already made remarks on states with zero eigenvalue for Eq.s43d.10–12Itis easy to see thatu0sxd=v0sxd= fsxd fthe normalized solution of GP equation, see Eqs.s10d ands11dg are eigenfunctions withEB=0. However, the set offsxd and hunsxd ,vnsxdj with EB,nÞ0 isnot complete. One needs one more state denoted byhsxd for the completeness,12 which is asolution of

sL + Mdhsxd =1

Ifsxd, s45d

whereI is a positive constant. The orthonormal properties are, in addition to Eqs.s13d ands44d foreigenfunctions of nonzero eigenvalues,

0 =E d3xfhunsxd − vnsxdjfsxdg, s46ad

0 =E d3xfhunsxd + vnsxdjhsxdg, s46bd

12 =E d3x fsxdhsxd. s46cd

The last line fixes the value ofI. The completeness condition now reads as

on=1

`

funsxdunsx8d − vnsxdvnsx8dg + ffsxdhsx8d + hsxdfsx8dg = d3sx − x8d, s47ad

on=1

`

funsxdvnsx8d − vnsxdunsx8dg + ffsxdhsx8d − hsxdfsx8dg = 0. s47bd

Consider the field operators. We introduce the oscillator-operatorsanstd=ane−iEB,nt and an

†std=an

†eiEB,nt associated withunsxd and vnsxd, respectively, and also a set of canonical variables

hQstd ,Pstdj for fsxd andhsxd,



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fan,am† g = dnm, s48d

fQstd,Pstdg = i , s49d

and other vanishing commutation relations. Then the field operators are expanded as

wsxd = Pstdhsxd − iQstdfsxd + on=1

`

fanstdunsxd − an†stdvnsxdg, s50ad

w†sxd = Pstdhsxd + iQstdfsxd + on=1

`

fan†stdunsxd − anstdvnsxdg. s50bd

It is easy to check that the canonical commutation relations are derived from these expressions and

the completeness condition in Eq.s47d. Also the substitution of Eq.s50d into the H0 in Eq. s15dleads to

H0 =P2

2I+ o

n=1

`

fEB,nan†an + sc-numbersdg, s51d

where we have employed the orthonormal conditions,s13d, s44d, ands46d. Important observationshere are that the canonical commutation relations would be violated without introducinghsxd and

that we have the termP2/ s2Id in the Hamiltonian as a result of the existence of the NG mode.

V. RELATION BETWEEN GENERALIZED BOGOLIUBOV AND BOGOLIUBOV–DEGENNES APPROACHES

We have reviewed the GB and BdG approaches in the preceding two sections. Both of themseem to be consistent canonical theories in which the NG mode is taken account of properly.

On the other hand, the two approaches follow different procedures. For example, one sees in

the BdG approach the appearance of canonical variablesP andQ, while one does not in the GBapproach. Furthermore, in the BdG approach one apparently sees no problem of infrared diver-gence associated with the NG modes, while one encounters it in the GB approach.

Then arises a natural question whether the two approaches are equivalent. In this section, wefirst show that these two approaches are equivalent in the excitation modes. After that, we find acorrespondence in the zero-mode eigenspaces of the two approaches, investigating the functionsfsxd andhsxd in the BdG approach in terms ofwnsxd in the GB one. Then the limit of«→0 in theGB approach must be taken carefully.

First, let us deal with the following simultaneous equations for excited states withnù1,equivalent to Eq.s43d,13

sL − Mdu+,nsxd = EB,nu−,nsxd, s52ad

sL + Mdu−,nsxd = EB,nu+,nsxd, s52bd

where

u±,nsxd = unsxd ± vnsxd. s53d

Then we have

sL + MdsL − Mdu+,nsxd = EB,n2 u+,nsxd s54ad

sL − MdsL + Mdu−,nsxd = EB,n2 u−,nsxd. s54bd



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Suppose that the eigenfunctionsu±,nsxd sn=1,2, . . .d in the BdG approach can be expanded interms ofwn

s«dsxd sn=0,1, . . .d in the GB approach,

u±,nsxd = om=0

`

R±,nmwms«dsxd. s55d

Substitute Eq.s55d into Eq. s52d, apply Eq.s21d, multiply wn8s«dsxd from the left and integrate

over x, then we derive the matrix equations

R+e = EBR−, s56ad

R−se + 4Ud = EBR+, s56bd

where the matrix notations aresEBdnm=EB,nm andsUdnm=Unm in Eq. s28d, andsednm is defined inEq. s32d. Remark that the indices ofsR±dnm run fromn=1 andm=0. From these equations derivethe equations,

R+ese + 4Ud = EB2R+, s57ad

R−se + 4Ude = EB2R−. s57bd

One can easily find the solutions forR± by usingO satisfying Eq.s33d, that is,

R+ = H+Oe−1/2, s58ad

R− = H−Oe1/2, s58bd

whereH± are arbitrary matricessn=1,2, . . . ;m=0,1,2, . . .d whose off-diagonal elements are van-ishing. In fact we have from Eqs.s31d–s33d,

R+ese + 4Ud = EG2 R+, s59ad

R−se + 4Ude = EG2 R−, s59bd

wheresEG8 dnm=EG,ndnm sn,m=1,2, . . .d. The comparison of Eqs.s57d ands59d clearly shows thatthe energy spectraEB andEG are identical for excited states, so let us denote them simply byE.If we take

H+ = E1/2, s60ad

H− = E−1/2, s60bd

we can find in the limit of«→0 from arguments in Sec. IV that

unsxd = wCns0dsxd, s61ad

vnsxd = wSns0dsxd. s61bd

Thus it has been proved that the two approaches are essentially equivalent to each other in theexcited modes.

Next, in order to include the zero-mode eigenspace, we introduce a new functionh«sxd,corresponding to Eq.s45d, through the relation,



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sL + Mdh«sxd =1

I«

f«sxd. s62d

Here f«sxd is defined in Eq.s11d with Eq. s9d, and the constantI« is determined from the condition

1

2=E d3x f«sxdh«sxd. s63d

It is clear from their definitions that the functionsf«sxd and h«sxd and the constantI« reduce tofsxd, hsxd, andI, respectively, as« goes to zero.

We attempt to expandh«sxd in terms ofhwns«dj,

h«sxd = on=0

`

knwns«dsxd. s64d

Remark thatw0s«dsxd is included in this summation. Substitute this into Eq.s45d,

sL + Mdh«sxd = on=0

`

knfsen + «ed + 2Mgwns«dsxd =

1

I«

f«sxd, s65d

and multiply it byed3x wms«dsxd, one obtains in the matrix notation

fe + 4Ugk =1

I«

e0, s66d

with se0dn=d0n and skdn=kn. We manipulate this as

fe2 + 4e1/2Ue1/2ge−1/2k = We−1/2k =1

I«

e1/2e0, s67d

whereW is defined in Eq.s34d, and we have

k =1

I«

e1/2W−1e1/2e0. s68d

We remark thatsednm should be interpreted not asendnm but as sen+«eddnm. Since OW−1OT

=EG−2 from Eq. s33d andC−S=EG

−1/2Oe1/2 from Eq. s31d, Eq. s68d reduces to

k =1

I«

sC − SdTEG−1sC − Sde0. s69d

The behaviors of the matricesEG−1, sC−Sd, and sC+Sd=EG

1/2Oe−1/2 with respect to« can beestimated from Sec. III,

EG−1 = SOs«−1/2d 0

0 Os«0dD s70ad

C − S= SOs«1/4d Os«1/4dOs«d Os«0d

D s70bd

C + S= SOs«−1/4d Os«3/4dOs«0d Os«0d

D , s70cd

where the first rows and columns correspond ton=0, and the second ones ton=1,2, . . ..There-fore, up to the leading order of«, we have after some manipulations



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EG−1sC − Sde0 = E0

−3/4s«ed−1/4e0 + Os«d. s71d

Recall the definition of Eq.s37d for E0 which is independent of«. Finally k becomes

k =1

I«

E0−3/4s«ed−1/4sC − SdTe0 + Os«d, s72d

implying that

h«sxd =1

I«

sE0d−3/4s«ed−1/4swC0s«dsxd − wS0

s«dsxdd + Os«d

=1

I«

E0−1Fw0

s«dsxd − on=1

`

su8TW8−1dnen1/2wn

s«dsxdG + Os«1/2d. s73d

The constantI« can be evaluated from Eq.s73d, Eq. s63d, and f«sxd=w0s«dsxd,

I« =2

E0

+ Os«1/2d. s74d

Then we have an explicit form ofh«sxd as

h«sxd =1

2Fw0s0dsxd − o

n=1

`

su8TW8−1dnen1/2wn

s0dsxdG + Os«1/2d. s75d

We rewrite the functionf«sxd, which isw0s0dsxd in the limit of «→0, as

f«sxd = w0s«dsxd = sE0d−1/4s«ed1/4fswC0sxd + wS0sxdd + Os«1/2dg, s76d

since then the completeness condition becomes evidentfsee Eqs.s20d and s47d with Eq. s61dg,

d3sx − x8d = on=0

`

wns«dsxdwn

s«dsx8d

= on=1

`

fwCns«dsxdwCn

s«dsx8d − wSns«dsxdwSn

s«dsx8dg + ff«sxdh«sx8d + h«sxdf«sx8dg. s77d

This way it has been confirmed thatf«sxd andh«sxd thus defined in the GB approach correspondto fsxd andhsxd in the BdG approach in the limit of«→0.

As the next step, we rewrite the oscillatorlike operatorb0 by a new set of canonical operators

hQb,Pbj,

b0 =1Î2

sPb − iQbd, s78ad

b0† =

1Î2

sPb + iQbd, s78bd

with

fQb,Pbg = i s79d

and



129.12.234.185 On: Mon, 24 Nov 2014 02:15:10

fQb,Qbg = fPb,Pbg = 0. s80d

Using Eqs.s73d, s76d, ands75d one can express the field operator in the GB approach as

wsxd = b0stdwC0s«dsxd − b0

†stdwS0s«dsxd + wexcsxd = jPbstdh«sxd − ij−1Qbstdf«sxd + wexcsxd + Os«1/2d,

s81d

where

j = Î2E0−1/4s«ed1/4, s82d

wexcsxd = on=1

`

fbnstdwCns«dsxd − bn

†stdwSns«dsxdg. s83d

Recall Eq.s74d and thatwexcsxd is common, which has been proved in the first part of this section,and we see that Eqs.s81d and s50d become identical to each other in the limit of«→0, if weidentify

P = jPb, s84ad

Q = j−1Qb. s84bd

Namely, the field operators in the GB approach may be expressed as

wsxd = Pstdh«sxd − iQstdf«sxd + wexcsxd + Os«1/2d, s85ad

w†sxd = Pstdh«sxd + iQstdf«sxd + wexc† sxd + Os«1/2d. s85bd

The Hamiltonians30d in the GB approach is in terms ofP andQ,

H0 =1

2I«

j2Pb2 + s«edj−2Qb

2 + on=1

`

Enbn†bn + sc-numbersd

=1

2I«

P2 + s«edQ2 + on=1

`

Enbn†bn + sc-numbersd → 1

2IP2 + o

n=1

`

Enbn†bn + sc-numbersd,

s86d

which is the same as Eq.s51d.Thus Eqs.s78d, s82d, and s84d reveal the correspondence between the two sets of operators

representing the NG modes in the two approaches,hb0,b0†j and hP ,Qj, in the limit of «→0.

VI. SUMMARY AND DISCUSSIONS

It was shown explicitly in this paper that in the unperturbed representation for systems of thetrapped BECssid the operators of excitation modes in the GB and BdG approaches are identical,

and sii d the operatorsb0 and b0† of the NG mode in the GB approach are related to the quantum

coordinatesP andQ in the BdG approach through the linear relationss78d ands84d with Eq. s82d.The introduction of the breaking term in Eq.s4d and the parameter« in the GB is very importantto find the relations. The formal equivalence of the two approaches is established since theoperators turned out to be related to each other linearly.

We note that the scale transformationfsee Eq.s84dg,



129.12.234.185 On: Mon, 24 Nov 2014 02:15:10

Pb → P = jPb, s87ad

Qb → Q = j−1Qb, s87bd

which is canonical, corresponds to the squeezing, although it is singular asj−1 diverges in the limitof «→0.

Let us discuss on vacuum states. Because of the resultsid, the vacuum structures with respectto the excitation modes are identical in the two approaches, we simply require

bnuVexcl = 0 sn ù 1d. s88d

But there are two ways to determine the vacuum structure in the NG mode sector. One is aprescription, naturally adopted in the GB approach, to have the vacuum defined by

b0uV0l = 0. s89d

Here the operatorb0 is treated as a gapless mode. The other is to have no unique vacuum but a

quantum mechanical state associated with the quantum coordinateshP ,Qj, denoted byuQl. Al-though this state must satisfy

kQuQuQl = kQuPuQl = 0, s90d

due to the condition

kVuwsxduVl = 0, s91d

whereuVl is the total vacuum, it cannot be determined completely from general considerations. Asan example, one may take for it a Gaussian wave function, centered atQ=0. HereQ is a quantum

number associated with the operatorQ.In summary, we have the two types of total vacuum, depending on whether the NG mode is

treated as a gapless mode or as quantum coordinates,

uVl = HuV0l ^ uVexcl sas gapless moded,

uQl ^ uVexcl sas quantum coordinated.J s92d

One should not confuse the choice of approach and that of vacuum. We can develop in the GBapproach a consistent formulation with the vacuumuQl ^ uVexcl as quantum coordinates, guided bythe relationss78d ands84d with Eq. s82d. What matters physically is the choice of vacuum but notthat of approach. We are discussing the unperturbed representation in this paper. The choice ofunperturbed vacuum affects the unperturbedsfreed propagator and inevitably all of the higherorder effects. The corrections at one-loop level to the original GP equation have been evaluatedwhen the vacuumuV0l ^ uVexcl is chosen.8 We have reported the evaluation of the same correc-tions with the choice ofuQl ^ uVexcl in Ref. 14. The expressions for free propagators withuQl^ uVexcl are given in the Appendix, while those withuV0l ^ uVexcl are found in Ref. 8.

ACKNOWLEDGMENTS

The authors would like to thank Professors I. Ohba and H. Nakazato for helpful commentsand encouragements, and Dr. H. Iwasaki for useful discussions. The authors thank the YukawaInstitute for Theoretical Physics at Kyoto University for offering us the opportunity of discussionsduring the YITP workshop YITP-W-03-10 on “Thermal Quantum Field Theories and Their Ap-plications,” useful to complete this work. M.M. is supported partially by Fujukai. M.M. and M.O.are also supported partially by the Grant-in-Aid for The 21st Century COE ProgramsPhysics ofSelf-organization Systemsd at Waseda University and that for Priority Area BsNo. 763d, MEXT.Y.Y. wishes to express his thanks for Waseda University Grant for Special Research Projects.



129.12.234.185 On: Mon, 24 Nov 2014 02:15:10

APPENDIX: PROPAGATORS USING QUANTUM COORDINATES

In this Appendix, we construct unperturbedsfreed propagators withuQl ^ uVexcl. These arenecessary to calculate quantum and thermal corrections.

The time dependence ofQ and P are governed by the Hamiltonians86d,

Qstd = cossÎ«EtdQ +1

Î«EI«

sinsÎ«EtdP, sA1ad

Pstd = cossÎ«EtdP − sÎ«EI«dsinsÎ«EtdP, sA1bd

whereE=ÎeE0+Os«d.We substitute Eq.sA1d into the field operators in Eq.s85d,

wsxd = fcossÎ«EtdP − Î«EI« sinsÎ«EtdQgh«sxd − iFcossÎ«EtdQ +1

Î«EI«

sinsÎ«EtdPG f«sxd

+ wexcsxd + Os«d, sA2ad

w†sxd = fcossÎ«EtdP − Î«EI« sinsÎ«EtdQgh«sxd + iFcossÎ«EtdQ +1

Î«EI«

sinsÎ«EtdPG f«sxd

+ wexc† sxd + Os«d. sA2bd

When the limit«→0 is taken, they are

wsxd = Phsxd − iSQ +1

IPtD fsxd + wexcsxd, sA3ad

w†sxd = Phsxd + iSQ +1

IPtD fsxd + wexc

† sxd. sA3bd

These representations are the same as in Ref. 12.Next, let us construct the propagators with Eq.sA2d and the vacuumuQl ^ uVexcl. The

232-matrix propagator is derived as

GQsx,x8;t,t8d = SGQ,11sx,x8;t,t8d GQ,12sx,x8;t,t8dGQ,21sx,x8;t,t8d GQ,22sx,x8;t,t8d

D , sA4d

where

GQ,11sx,x8;t,t8d ; kVuTfwsxdw†sx8dguVl = FcossÎ«EtdcossÎ«Et8dh«sxdh«sx8d

+ S 1

Î«EI«

D2

sinsÎ«EtdsinsÎ«Et8df«sxdf«sx8d

+ i1

Î«EI«

cossÎ«EtdsinsÎ«Et8dh«sxdf«sx8d

− i1

Î«EI«

sinsÎ«EtdcossÎ«Et8df«sxdh«sx8dGkQuP2uQl



129.12.234.185 On: Mon, 24 Nov 2014 02:15:10

+ fcossÎ«EtdcossÎ«Et8df«sxdf«sx8d + sÎ«EI«d2sinsÎ«EtdsinsÎ«Et8dh«sxdh«sx8d

+ isÎ«EI«dcossÎ«EtdsinsÎ«Et8df«sxdh«sx8d

− isÎ«EI«dsinsÎ«EtdcossÎ«Et8dh«sxdf«sx8dgkQuQ2uQl

+ Hust − t8dF− sÎ«EI«dcossÎ«EtdsinsÎ«Et8dh«sxdh«sx8d

+1

Î«EI«

sinsÎ«EtdcossÎ«Et8df«sxdf«sx8d + i sinsÎ«EtdsinsÎ«Et8df«sxdh«sx8d

+ i cossÎ«EtdcossÎ«Et8dh«sxdf«sx8dG + ust8 − td

3F− sÎ«EI«dsinsÎ«EtdcossÎ«Et8dh«sxdh«sx8d

+1

Î«EI«

cossÎ«EtdsinsÎ«Et8df«sxdf«sx8d − i sinsÎ«EtdsinsÎ«Et8dh«sxdf«sx8d

− i cossÎ«EtdcossÎ«Et8df«sxdh«sx8dGJkQuPQuQl

+ Hust − t8dF− sÎ«EI«dsinsÎ«EtdcossÎ«Et8dh«sxdh«sx8d

+1

Î«EI«

cossÎ«EtdsinsÎ«Et8df«sxdf«sx8d − i cossÎ«EtdcossÎ«Et8df«sxdh«sx8d

− i sinsÎ«EtdsinsÎ«Et8dh«sxdf«sx8dG + ust8 − td

3F− sÎ«EI«dcossÎ«EtdsinsÎ«Et8dh«sxdh«sx8d

+1

Î«EI«

sinsÎ«EtdcossÎ«Et8df«sxdf«sx8d + i cossÎ«EtdcossÎ«Et8dh«sxdf«sx8d

+ i sinsÎ«EtdsinsÎ«Et8df«sxdh«sx8dGJkQuQPuQl sA5d

GQ,22sx,x8;t,t8d = GQ,11sx8,x;t8,td, sA6d

GQ,12sx,x8;t,t8d ; kVuTfwsxdwsx8dguVl = FcossÎ«EtdcossÎ«Et8dh«sxdh«sx8d

+ S 1

Î«EI«

D2

sinsÎ«EtdsinsÎ«Et8df«sxdf«sx8d



129.12.234.185 On: Mon, 24 Nov 2014 02:15:10

− i1

Î«EI«

cossÎ«EtdsinsÎ«Et8dh«sxdf«sx8d

− i1

Î«EI«

sinsÎ«EtdcossÎ«Et8df«sxdh«sx8dGkQuP2uQl

+ fcossÎ«EtdcossÎ«Et8df«sxdf«sx8d + sÎ«EI«d2sinsÎ«EtdsinsÎ«Et8dh«sxdh«sx8d

+ isÎ«EI«dcossÎ«EtdsinsÎ«Et8df«sxdh«sx8d

+ isÎ«EI«dsinsÎ«EtdcossÎ«Et8dh«sxdf«sx8dgkQuQ2uQl

+ Hust − t8dF− sÎ«EI«dcossÎ«EtdsinsÎ«Et8dh«sxdh«sx8d

+1

Î«EI«

sinsÎ«EtdcossÎ«Et8df«sxdf«sx8d + i sinsÎ«EtdsinsÎ«Et8df«sxdh«sx8d

− i cossÎ«EtdcossÎ«Et8dh«sxdf«sx8dG + ust8 − td

3F− sÎ«EI«dsinsÎ«EtdcossÎ«Et8dh«sxdh«sx8d

+1

Î«EI«

cossÎ«EtdsinsÎ«Et8df«sxdf«sx8d + i sinsÎ«EtdsinsÎ«Et8dh«sxdf«sx8d

− i cossÎ«EtdcossÎ«Et8df«sxdh«sx8dGJkQuPQuQl

+ Hust − t8dF− sÎ«EI«dsinsÎ«EtdcossÎ«Et8dh«sxdh«sx8d

+1

Î«EI«

cossÎ«EtdsinsÎ«Et8df«sxdf«sx8d − i cossÎ«EtdcossÎ«Et8df«sxdh«sx8d

+ i sinsÎ«EtdsinsÎ«Et8dh«sxdf«sx8dG + ust8 − td

3F− sÎ«EI«dcossÎ«EtdsinsÎ«Et8dh«sxdh«sx8d

+1

Î«EI«

sinsÎ«EtdcossÎ«Et8df«sxdf«sx8d − i cossÎ«EtdcossÎ«Et8dh«sxdf«sx8d

+ i sinsÎ«EtdsinsÎ«Et8df«sxdh«sx8dGJkQuQPuQl sA7d

GQ,21sx,x8;t,t8d = GQ,12sx8,x;t8,td. sA8d

In the above definitions, we use the notation for propagators without NG mode,

Gexc,11sx,x8;t − t8d ; kVexcuTfwexcsxdwexc† sx8dguVexcl, sA9ad



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Gexc,12sx,x8;t − t8d ; kVexcuTfwexcsxdwexcsx8dguVexcl, sA9bd

Gexc,21sx,x8;t − t8d ; kVexcuTfwexc† sxdwexc

† sx8dguVexcl, sA9cd

Gexc,22sx,x8;t − t8d ; kVexcuTfwexc† sxdwexcsx8dguVexcl. sA9dd

One can find their explicit forms in Ref. 8.In the limit of «→0, the propagators become

GQ,11sx,x8;t,t8d = Fhsxdhsx8d + fsxdfsx8dS t

IDS t8

ID + ihsxdfsx8dS t8

ID − i f sxdhsx8dS t

IDGkQuP2uQl

+ fsxdfsx8dkQuQ2uQl + Hust − t8dF fsxdfsx8dS t

ID + ihsxdfsx8dG + ust8 − td

3F fsxdfsx8dS t8

ID − i f sxdhsx8dGJkQuPQuQl + Hust − t8dF fsxdfsx8dS t8

ID

− i f sxdhsx8dG + ust8 − tdF fsxdfsx8dS t

ID + ihsxdfsx8dGJkQuQPuQl

+ Gexc,11sx,x8;t − t8d, sA10d

GQ,22sx,x8;t,t8d = GQ,11sx8,x;t8,td, sA11d

GQ,12sx,x8;t,t8d = Fhsxdhsx8d + fsxdfsx8dS t

IDS t8

ID − ihsxdfsx8dS t8

ID − i f sxdhsx8dS t

IDGkQuP2uQl

+ fsxdfsx8dkQuQ2uQl + Hust − t8dF fsxdfsx8dS t

ID − ihsxdfsx8dG + ust8 − td

3F fsxdfsx8dS t8

ID − i f sxdhsx8dGJkQuPQuQl + Hust − t8dF fsxdfsx8dS t8

ID

− i f sxdhsx8dG + ust8 − tdF fsxdfsx8dS t

ID − ihsxdfsx8dGJkQuQPuQl

+ Gexc,12sx,x8;t − t8d, sA12d

GQ,21sx,x8;t,t8d = GQ,12sx8,x;t8,td. sA13d

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2M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science269, 198 s1995d.3K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Keterle, Phys. Rev. Lett.

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6A. L. Fetter, Ann. Phys.sN.Y.d 70, 67 s1972d; Phys. Rev. A53, 4245s1996d.7H. Ezawa, K. Nakamura, K. Watanabe, and Y. Yamanaka, inMathematical Physics and Stochastic Analysis—Essays ofHonor of Ludvig Streit, edited by S. Albeverioet al. sWorld Scientific, Singapore, 2000d, p. 169; H. Ezawa, K. Watanabe,and K. Nakamura, inFoundations of Quantum Mechanics in the Light of New Technology ISQM–Tokyo ‘01, edited by Y.A. Ono and K. FujikawasWorld Scientific, Singapore, 2002d.

8M. Okumura and Y. Yamanaka, Phys. Rev. A68, 13609s2003d.9E. P. Gross, Nuovo Cimento20, 454s1961d; J. Math. Phys.4, 195s1963d; L. P. Pitaevskii, Zh. Eksp. Teor. Fiz.40, 646s1961d; Sov. Phys. JETP13, 451 s1961d.



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Relation between generalized Bogoliubov and Bogoliubov–de Gennes approaches including Nambu–Goldstone mode