annals of physics, 328 158

Annals of Physics 328 (2013) 158219

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Beliaev theory of spinor BoseEinstein condensates

Nguyen Thanh Phuc a,, Yuki Kawaguchi b, Masahito Ueda aa Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japanb Department of Applied Physics and Quantum-Phase Electronics Center, University of Tokyo, 2-11-6 Yayoi, Bunkyo-ku,Tokyo 113-0032, Japan

a r t i c l e i n f o

Article history:Received 8 May 2012Accepted 3 October 2012Available online 16 October 2012

Keywords:Spinor BoseEinstein condensate (BEC)Beliaev theoryEnergy spectrumSpin waveBeliaev damping

a b s t r a c t

By generalizing the Greens function approach developed byBeliaev [S.T. Beliaev, Sov. Phys. JETP 7 (1958) 299; S.T. Beliaev, Sov.Phys. JETP 7 (1958) 289], we study effects of quantum fluctuationson the energy spectra of spin-1 spinor BoseEinstein condensates,in particular, of a 87Rb condensate in the presence of an externalmagnetic field. We find that due to quantum fluctuations, theeffective mass of magnons, which characterizes the quadraticdispersion relation of spin-wave excitations, increases comparedwith its mean-field value. The enhancement factor turns out to bethe same for two distinct quantum phases: the ferromagnetic andpolar phases, and it is a function of only the gas parameter. Thelifetime of magnons in a spin-1 87Rb spinor condensate is shownto be much longer than that of phonons due to the differencein their dispersion relations. We propose a scheme to measurethe effective mass of magnons in a spinor Bose gas by utilizingthe effect of magnons nonlinear dispersion relation on the timeevolution of the distribution of transverse magnetization. Thistype of measurement can be applied, for example, to precisionmagnetometry.

2012 Elsevier Inc. All rights reserved.

1. Introduction

Since the first experimental realization of atomic BoseEinstein condensates (BECs) [13], theBogoliubov theory of a weakly interacting dilute Bose gas has been successfully applied to describea variety of physical phenomena in atomic systems [46]. The Bogoliubov theory was originally

Corresponding author.E-mail address: [email protected] (N.T. Phuc).

0003-4916/$ see front matter 2012 Elsevier Inc. All rights reserved.doi:10.1016/j.aop.2012.10.004

N.T. Phuc et al. / Annals of Physics 328 (2013) 158219 159

proposed to describe properties of bosonic systems at absolute zero temperature [7,8]. It givesthe mean-field value of the energy spectrum, which is the leading term in the expansion of thespectrum with respect to the small gas parameter na3. Here, n and a are the atomic density andthe s-wave scattering length, respectively. The higher-order corrections to the mean-field value,which result from quantum fluctuations, are usually relatively small for dilute Bose gases [9,10]. Theleading correction to the mean-field energy spectrum was given by Beliaev [11], who developed adiagrammatic Greens function approach to describe the energy spectrum of elementary excitationsat absolute zero [12]. The Beliaev technique has also been generalized to finite-temperature [1316]. Recently, with the rapid development in precision measurement, the effects of quantumfluctuations on atomic BECs have been measured experimentally. Especially, by using a Feshbachresonance or by loading atoms into an optical lattice, the effective interatomic interaction canbe manipulated to cover a wide range from weakly to moderately interacting systems [1720].Recent experiments on a scalar (spin-polarized) BEC by the ENS group [20] have shown that theLeeHuangYang correction [9] can explain the deviation from the mean-field result as large as20%. Consequently, the leading correction to the mean-field value of the energy spectrum shouldbe sufficient to describe the effects of quantum fluctuations in atomic gases. Furthermore, theBeliaev theory for scalar BECs also predicts the so-called Beliaev damping which quantitativelycharacterizes the finite lifetime of phonons due to their collisions with condensate particles. TheBeliaev damping of quasiparticles under various conditions has been a subject of active study[2125].

Recently, BoseEinstein condensates with spin degrees of freedom (spinor BECs) have beenextensively studied (see, for example, [26]). These atomic systems simultaneously exhibitsuperfluidity and magnetism, and the combination of atomic motional and spin degrees of freedomgives rise to various interesting phenomena in the study of thermodynamic properties and quantumdynamics. In comparisonwith scalar BECs, there exist spin-wave excitations in spinor BECs in additionto conventional density-wave excitations. These are excitations of atoms out of the condensateaccompanied by a change in their spin state, resulting in spatial modulations of the distributionof spin density. The corresponding magnons have a quadratic dispersion relation at low momentaas opposed to the linear dispersion relation of phonons. Furthermore, in spinor Bose gases, thecollisions of atoms in different spin channels give rise to spin-independent and spin-dependentinteractions. Particularly, in some atomic species such as 87Rb, the ratio of the spin-independentto spin-dependent interactions is so large that a small number of quantum-fluctuation-inducednoncondensed atoms can, in principle, have a significant influence on the magnetic properties ofthe condensate. A remarkable example is a shift of the phase boundary between two quantumphases [27].

In this study, we apply the Beliaev theory [11,12] to a spin-1 87Rb Bose gas at zero temperatureto investigate the effects of quantum fluctuations on the energy spectra of elementary excitations.Our study is related to the work in Ref. [28] though there are mistakes in the calculation of second-order self-energies therein. In the presence of an external magnetic field, due to the competitionbetween the spin-dependent interatomic interaction and the quadratic Zeeman effect, the systemcan exist in various quantum phases with different spinor order parameters [2931]. In contrast tothe study in Ref. [27], we do not consider phase transitions. Instead, we assume that the magnitudeof the external magnetic field is appropriately chosen so that the system is stable in a certainquantum phase. We consider two characteristic phases of spin-1 spinor Bose gases: the fully spin-polarized ferromagnetic phase and the unmagnetized polar phase. Here, in the calculation of theleading corrections to the energy spectra of the 87Rb BEC, the spin-dependent interaction can beneglected as a good approximation since it is about 200 times smaller than the spin-independentinteraction. We find that quantum fluctuations increase the effective mass of magnons, while theenergy gap is not modified up to the leading order. This implies that the motion of magnons ishindered by their interaction with atoms excited by quantum fluctuations. Although the magnetic-field dependence of the magnons effective mass varies from the ferromagnetic phase to the polarphase, it turns out that the effective mass is enhanced by the same factor which is a function of onlythe gas parameter na3. This implies that the effect of density fluctuations is decoupled from that ofmagnetic interactions. In the case of a spin-1 87Rb BEC, we also find that the lifetime of magnons is

160 N.T. Phuc et al. / Annals of Physics 328 (2013) 158219

much longer than that of phonons. We show that this agrees with the mechanism of Beliaev dampingvia collisions between quasiparticles and condensate particles. This phenomenon is related to the caseof type-II Goldstone bosons, which are elementary excitations with a quadratic dispersion relation,in the model for kaon condensation in nuclear physics [32]. Finally, to probe the effect of quantumfluctuations in spinor BECs, we propose an experimental scheme to measure the effective mass ofmagnons based on the effect of magnons nonlinear dispersion relation on the time evolution of awave packet with transverse magnetization. This type of measurement can be applied to identifyspinor quantum phases, or to precision magnetometry in a way different from the method given inRef. [33].

This paper is organized as follows. Section 2 formulates the diagrammatic Greens functionapproach for spin-1 spinor BECs, which is a generalization of the Beliaev theory [11,12] to systemswith spin degrees of freedom. The explicit forms of thematrices of Greens functions and self-energiesfor the ferromagnetic and polar phases are given. The T -matrix which plays the role of an effectiveinteraction potential in dilute Bose gases is also introduced in this section. Section 3 reproduces themean-field (Bogoliubov) energy spectra by using the Greens function approach [34]. Section 4 dealswith the self-energies at the second order in the interaction and gives the leading corrections tothe mean-field energy spectra due to quantum fluctuations. Section 5 shows that the elementaryexcitations with quadratic dispersion relations should be physically interpreted as spin waves. Anexperimental scheme using a spinor wave packet is proposed to measure the effective mass ofmagnons. An order-of-magnitude estimation of the time evolution of the wave packet is also given.Section 6 concludes the paper by discussing the application of the measurement to some practicalpurposes. The detailed calculations are given in the Appendices to avoid digressing from the mainsubject.

2. Greens function formalism for a spinor BoseEinstein condensate

2.1. Hamiltonian

We consider a homogeneous system of identical bosons with mass M in the F = 1 hyperfinespin manifold that is subject to a magnetic field in the z-direction. The single-particle part of theHamiltonian is given in the matrix form by

(h0)jj = h

2 22M

+ qBj2jj , (1)

where the subscripts j, j = 0,1 refer to the magnetic sublevels, and qB is the coefficientof the quadratic Zeeman energy. Because of the conservation of the systems total longitudinalmagnetization, the linear Zeeman term vanishes. The total Hamiltonian of the F = 1 spinor Bosegas is then given in the second-quantized form by

H =

drjj

j (r)(h0)jjj(r)+ V, (2)

where j(r) is the field operator that annihilates an atom in magnetic sublevel mF = j at position r,and the interaction energy V is given by

V = 12

dr

dr

j,j,m,m

j (r)

m(r

)Vjm,jm(r r)m(r)j(r). (3)

Here, the matrix element Vjm,jm(r r) can be written as a sum of interactions in two spin channelsF = 0 and 2 (F denotes the total spin of two colliding atoms):


Vjm,jm(r r) = j,m|F = 0F = 0|j,mV0(r r)+j,m|F = 2F = 2|j,mV2(r r), (4)

where quantum statistics prohibits bosons from interacting via the spin channel F = 1.In the presence of a condensate, the field operator j(r) is decomposed into the condensate part,

which can be replaced by a classical fieldn0j, and the noncondensate part j(r):

j(r) = n0j + j(r). (5)For a homogeneous system, the condensate is characterized by the condensate number density n0 andthe spinor order parameter j(j = 0,1), which is normalized to unity:

j

|j|2 = 1. (6)

Substituting Eq. (5) into Eq. (3), we can decompose the interaction energy as

V = E0 +7

n=1Vn, (7)

where

E0 = 12n20

dr

dr j mVjm,jm(r r)mj , (8a)

V1 = 12n0

dr

dr j mVjm,jm(r r)m(r)j(r), (8b)

V2 = 12n0

dr

drj (r)m(r

)Vjm,jm(r r)mj , (8c)

V3 = 212n0

dr

dr j m(r

)Vjm,jm(r r)m j(r), (8d)

V4 = 212n0

dr

drj (r)mVjm,jm(r r)m j(r), (8e)

V5 = 212n1/20

dr

drj (r)m(r

)Vjm,jm(r r)m j(r), (8f)

V6 = 212n1/20

dr

drj (r)mVjm,jm(r r)m(r)j(r), (8g)

V7 = 12

dr

drj (r)m(r

)Vjm,jm(r r)m(r)j(r). (8h)These interactions are schematically illustrated by the Feynman diagrams in Fig. 1.

We consider a grand canonical ensemble of the above atomic system, and introduce the operator

K H N , (9)where denotes the chemical potential and N is the total number operator:

N =

drj

j (r)j(r). (10)

Using Eqs. (2), (5), (7) and (9), we have

K = E0 +

j

qBj2|j|2 N0 + K , (11)


Fig. 1. Feynman diagrams representing the eight terms (8a)(8h) in the interaction Hamiltonian. The dashed, solid, and wavylines represent a condensate atom, a noncondensate atom, and the interaction, respectively.

where E0, given in Eq. (8a), is the interaction energy between condensed atoms, N0 = Vn0 is the totalnumber of condensed atoms with V being the volume of the system, and

K K0 + K1 (12)is the corresponding operator for the noncondensate part with

K0 k=0,j

(0k + qBj2)aj,kaj,k, (13)

K1 7

n=1Vn. (14)

Here, 0k = h2 k2/(2M) is the kinetic energy of a particle with momentum hk, and aj,k is the Fouriertransform of the noncondensate field operator j(r):

aj,k = 1V

dreikrj(r). (15)

In the following sections, K0 and K1 are referred to as the noninteracting and interacting parts of K in Eq. (12), respectively. For a weakly interacting system, K1 can be treated as a perturbation to K0.

2.2. Greens functions

In the presence of the condensate, Greens function is given by [12,35]

iGtotaljj (x, y) = n0j j + iGjj(x, y), (16)where j, j = 0,1 indicate the spin components, and x = (r, t), y = (r, t ) are four-vectors of thespacetime coordinates. The noncondensate part of Greens function is defined as

iGjj(x, y) O|T j,H(x)j,H(y)|O

O|O . (17)


Here, |O is the ground state of the interacting system, and T andH denote the time-ordering operatorand the Heisenberg representation, respectively.

In the presence of the condensate, we must take into account the processes in which twononcondensed atoms collide into or out of the condensate. For this purpose, in addition to the normalGreens functions Gjj(x, y) defined in Eq. (17), it is necessary to introduce the so-called anomalousGreens functions defined as

iG12jj (x, y) O|T j,H(x)j,H(y)|O

O|O , (18)

iG21jj (x, y) O|T j,H(x)j,H(y)|O

O|O . (19)

In energymomentum space, the Dysons equations for the noncondensate Greens functions aregiven by

Gjj (p) = (G0)jj (p)+ (G0)jm (p) mm(p)Gmj(p), (20)where hp h(p0, p) is the four-momentum, and , , , = 1, 2 are used to label the normal andanomalous Greens functions as matrix elements of a 6 6 matrix:

G111,1(p) G111,0(p) G

111,1(p) G

121,1(p) G

121,0(p) G

121,1(p)

G110,1(p) G110,0(p) G

110,1(p) G

120,1(p) G

120,0(p) G

120,1(p)

G111,1(p) G111,0(p) G

111,1(p) G

121,1(p) G

121,0(p) G

121,1(p)

G211,1(p) G211,0(p) G

211,1(p) G

221,1(p) G

221,0(p) G

221,1(p)

G210,1(p) G210,0(p) G

210,1(p) G

220,1(p) G

2200(p) G

220,1(p)

G211,1(p) G211,0(p) G

211,1(p) G

221,1(p) G

221,0(p) G

221,1(p)

, (21)

where

G11jj (p) Gjj(p), G22jj (p) Gjj(p). (22)Eq. (20), which is illustrated in Fig. 2, can be written as a matrix equation of 6 6 matrices:

G(p) = G0(p)+ G0(p)(p)G(p), (23)where G, G0, and denote the 6 6 matrices of interacting, non-interacting Greens functions, andproper self-energy, respectively. The normal and anomalous self-energies are labeled in the samewayas Greens functions (i.e., superscripts 11 and 22 for normal self-energies, and 12 and 21 for anomalousself-energies). The solution to Eq. (23) can be written formally as

G(p) =1 G0(p)(p)

1G0(p). (24)

The non-interacting Greens function is defined as

iG0jj(x y) 0|T j,H0(x)j,H0(y)|0

0|0 , (25)

where |0 is the non-interacting ground state, and H0 indicates the free time evolution in theHeisenberg representation under the non-interacting Hamiltonian K0 given by Eq. (13). Here, |0


Fig. 2. Dysons equations for the normal and anomalous Greens functions. The thick line, thin line, and oval represent theinteracting, non-interacting Greens functions, and the proper self-energy, respectively.

is the vacuum state with respect to noncondensed particles; that is, ak,j|0 = 0 for all k = 0 andj = 1, 0. Substituting Eq. (13) into Eq. (25), we obtain the Fourier transform of G0jj(x y) as

G0jj(p) =

d4x eipxG0jj(x)

= jj 1p0 0p/h+ /h qBj2/h+ i jjG0j (p), (26)


Fig. 3. Normal (left) and anomalous (right) proper self-energies of noncondensed particles, where hp is the four-momentum,j, j label the spin components, and the dashed lines represent condensate particles.

where is an infinitesimal positive number. Note that the anomalous Greens functions in a non-interacting system are always zero, and thus, thematrix G0(p) is diagonal withmatrix elements givenby Eq. (26).

Now we consider two cases in which the mean-field ground state is in the ferromagnetic phaseand in the polar phase.

2.2.1. Ferromagnetic phaseIf the ground state is ferromagnetic, the condensates spinor is given by

(1, 0, 1) = (1, 0, 0), (27)i.e., all condensate atoms reside in themF = 1 magnetic sublevel. Then, the nonzero matrix elementsof (p) are

111,1(p) 0 0 121,1(p) 0 0

0 110,0(p) 0 0 0 00 0 111,1(p) 0 0 0

211,1(p) 0 0 111,1(p) 0 0

0 0 0 0 110,0(p) 00 0 0 0 0 111,1(p)

. (28)This can be understood by considering the spin conservation in normal and anomalous self-energies,which are illustrated in Fig. 3. For normal self-energies11jj (p), the conservation of the total projectedspin allows only j = j, i.e., diagonal elements. In contrast, for anomalous self-energies 12jj (p), onlythe j = j = 1 element is nonvanishing because the condensate atoms are all in themF = 1 magneticsublevel.

By substituting Eq. (28) into Eq. (24) and using the fact that G0(p) is a diagonalmatrix (see Eq. (26)),we find that the matrix G(p) of interacting Greens functions has the same form as (p):

G1,1(p) 0 0 G121,1(p) 0 00 G0,0(p) 0 0 0 00 0 G1,1(p) 0 0 0

G211,1(p) 0 0 G1,1(p) 0 00 0 0 0 G0,0(p) 00 0 0 0 0 G1,1(p)

. (29)


Both G(p) and (p) are block-diagonal matrices composed of one 2 2 and four 1 1submatrices.

The normal and anomalous Greens functions given by Eq. (24) can then be expressed in terms ofthe self-energies as

G1,1(p) =[G01(p)]1 +111,1(p)

D1= p0 +

0p/h+ qB/h+111,1(p) /h

D1, (30a)

G0,0(p) = 1[G00(p)]1 110,0(p)+ i, G1,1(p) = 1[G01(p)]1 111,1(p)+ i

, (30b)

G121,1(p) =121,1(p)

D1, G211,1(p) =

211,1(p)D1

, (30c)

where

D1 = [G01(p)]1[G01(p)]1 +111,1(p)[G01(p)]1 +111,1(p)[G01(p)]1111,1(p)111,1(p)+211,1(p)121,1(p)+ i

= p20 111,1(p)111,1(p)

p0 +211,1(p)121,1(p)

0p/h /h+ qB/h+

111,1(p)+111,1(p)2

2

+111,1(p)111,1(p)

2

2+ i. (31)

From Eqs. (30) and (31), we obtain the modified version of the HugenholtzPines condition [36]for an F = 1 spinor BEC in the ferromagnetic phase, that is, for the three elementary excitations to begapless, the following condition must be met:

11j,j (p0 = 0, p = 0)12j,j (p0 = 0, p = 0) = ( qBj2)/h. (32)Here, the excitation modes with j = 0 and 1 are single-particle like, and thus, the correspondinganomalous self-energies and Greens functions vanish. The energy shift of qB from the chemicalpotential on the right-hand side of Eq. (32) results from the difference in quadratic Zeeman energybetween magnetic sublevels mF = 1 and mF = 0 [37]. For the ferromagnetic phase, theHugenholtzPines condition (32) is met only for j = 1 in the presence of the quadratic Zeeman effect;therefore, only the corresponding phononmode (j = 1) is gapless. When qB = 0, the spin-wavemode(j = 0) also becomes gapless with a quadratic dispersion relation.

2.2.2. Polar phaseIf the ground state is polar, the condensates spinor is given by

(1, 0, 1) = (0, 1, 0), (33)i.e., all condensate atoms occupy themF = 0magnetic sublevel. Following an argument similar to theferromagnetic phase, we find that the nonzero matrix elements of (p) and G(p) are

111,1(p) 0 0 0 0 121,1(p)

0 110,0(p) 0 0 120,0(p) 0

0 0 111,1(p) 121,1(p) 0 0

0 0 211,1(p) 111,1(p) 0 0

0 210,0(p) 0 0 110,0(p) 0

211,1(p) 0 0 0 0 111,1(p)

, (34)


G1,1(p) 0 0 0 0 G121,1(p)0 G0,0(p) 0 0 G120,0(p) 00 0 G1,1(p) G121,1(p) 0 00 0 G211,1(p) G11(p) 0 00 G210,0(p) 0 0 G0,0(p) 0

G211,1(p) 0 0 0 0 G1,1(p)

. (35)

Both of these matrices are block-diagonal matrices composed of three 2 2 sub-matrices. Here,121,1(p) and G

121,1(p) are nonzero due to the projected-spin-conserved scattering process in which

two condensate atoms in the magnetic sublevel mF = 0 collide with each other to produce twononcondensed atoms with spin componentsmF = +1 and 1 (see Fig. 3).

The normal and anomalous Greens functions given by Eq. (24) can then be expressed in terms ofthe self-energies as

G1,1(p) =[G01(p)]1 +111,1(p)

D1= p0 +

0p/h+ qB/h+111,1(p) /h

D1, (36a)

G0,0(p) =[G00(p)]1 +110,0(p)

D0= p0 +

0p/h+110,0(p) /h

D0, (36b)

G1,1(p) =[G01(p)]1 +111,1(p)

D1= p0 +

0p/h+ qB/h+111,1(p) /h

D1, (36c)

G121,1(p) =121,1(p)

D1, G211,1(p) =

211,1(p)D1

, (36d)

G120,0(p) =120,0(p)

D0, G210,0(p) =

210,0(p)D0

, (36e)

G121,1(p) =121,1(p)

D1, G211,1(p) =

211,1(p)D1

, (36f)

where


= p20 111,1(p)111,1(p)

p0 +211,1(p)121,1(p)

0p + qB

h+

111,1(p)+111,1(p)

2

2

+111,1(p)111,1(p)

2

2+ i, (37a)


= p20 110,0(p)110,0(p)

p0 +210,0(p)120,0(p)

0p

h+

110,0(p)+110,0(p)

2

2+110,0(p)110,0(p)

2

2+ i, (37b)


D1 = [G01(p)]1[G01(p)]1 +1,11,1 (p)[G01(p)]1 +111,1(p)[G01(p)]1111,1(p)111,1(p)+211,1(p)121,1(p)+ i

= p20 111,1(p)111,1(p)

p0 +211,1(p)121,1(p)

0p + qB

h+

111,1(p)+111,1(p)

2

2

+111,1(p)111,1(p)

2

2+ i. (37c)

From Eqs. (36) and (37), we obtain a modified version of the HugenholtzPines condition for anF = 1 spinor BEC in the polar phase, that is, for the elementary excitations to be gapless, the followingcondition must be met:

11j,j (p0 = 0, p = 0)12j,j(p0 = 0, p = 0) = ( qBj2)/h. (38)For the polar phase, the HugenholtzPines condition (38) holds only for j = 0 in the presence of thequadratic Zeeman effect (qB = 0); therefore, only the corresponding phonon mode (j = 0) is gapless.

2.3. T -matrix

For a weakly interacting dilute Bose gas, the contributions from all ladder-type diagrams to theself-energies are shown to be of the same order of magnitude [11,35], and, therefore, all of thesecontributions must be taken into account. The T -matrix is defined as the sum of an infinite number ofladder-type diagrams as illustrated in Fig. 4. It is written as

jm,jm(p1, p2; p3, p4) = Vjm,jm(p1 p3)+ ihj,m

d4q(2)4

G0j(p1 q)G0m(p2 + q)

Vjm,jm(q)Vjm,jm(p1 q p3)+ = Vjm,jm(p1 p3)+

j,m

d3q(2)3

1h(p1)0 + h(p2)0 0p1q 0p2+q + 2 qB(j2 +m2)+ i

Vjm,jm(q)Vjm,jm(p1 q p3)+ = Vjm,jm(p1 p3)+

j,m

d3q(2)3

1h(p1)0 + h(p2)0 0p1q 0p2+q + 2 qB(j2 +m2)+ i

j,m|F = 0F = 0|j,mj,m|F = 0F = 0|j,m

V0(q)V0(p1 q p3)+j,m|F = 0F = 0|j,mj,m|F = 2F = 2|j,m V0(q)V2(p1 q p3)+j,m|F = 2F = 2|j,mj,m|F = 0F = 0|j,m V2(q)V0(p1 q p3)+j,m|F = 2F = 2|j,mj,m|F = 2F = 2|j,m V2(q)V2(p1 q p3)

+ . (39)


Fig. 4. T -matrix of a two-body scattering. Two atoms with momenta hp3, hp4 and magnetic quantum numbers j,m collideto produce two atoms with momenta hp1, hp2 and magnetic quantum numbers j,m. The T -matrix is defined as the sum of aninfinite number of ladder-type diagrams which describe virtual multiple-scattering processes (see Eq. (39)).

Here, the second identity in Eq. (39) is obtained by using Eq. (26) for G0j (p) in the integration ofG0j(p1 q)G0m(p2 + q)with respect to q0.

In spinor BECs, the stable quantum phase of the ground state is determined as the competitionbetween the spin-dependent interatomic interaction and the coupling of atoms to an externalmagnetic field via the quadratic Zeeman energy, and thus, the quadratic Zeeman energy usually hasthe same order of magnitude as the spin-dependent interaction: qB |c1|n c0n. It can be shownthat for such an external magnetic field, the spin dependence of intermediate states via the quadraticZeeman energies qB(j2 + m2) in the denominator of the right-hand side of Eq. (39) only gives asmall difference that is negligible up to the order of magnitude we are considering in this paper (seeAppendix A). Consequently, as a good approximation we can take the sum

g,h

|ghgh| = 1 (40)

out of the integral. Inside the integral, by using the fact that the F = 0 and F = 2 spin channels areorthogonal to each other (F = 0|F = 2 = 0), the T -matrix can be rewritten as

jm,jm(p1, p2; p3, p4) = j,m|F = 0F = 0|j,m0(p1, p2; p3, p4)+j,m|F = 2F = 2|j,m2(p1, p2; p3, p4), (41)

where F (p1, p2; p3, p4) is the T -matrix in the total spin F channel given byF (p1, p2; p3, p4) = VF (p1 p3)+ ih

d4q(2)4

G0(p1 q)G0(p2 + q)VF (q) VF (p1 q p3)+ . (42)

Here, G0(p) = 1/(p0 0p + + i) is the spinless non-interacting Greens function.The T -matrix F (p1, p2; p3, p4) can be expressed in terms of the vacuum scattering amplitude for

the spin channel F = 0, 2 as follows (see Appendix A) [11,35]:F (p1, p2; p3, p4) = F (p, p, P)

= fF (p, p)+

d3q(2)3

fF (p, q)

1

hP0 h2 P24M + 2 h2 q2M + i

+ 1h2 q2M h

2 p2M i

f F (p

, q), (43)

where MfF (p, p)/(4 h2) is the vacuum scattering amplitude of the two-body collision inwhich the relative momentum changes from hp to hp. As seen in Eq. (43), it can be shown that


F (p1, p2; p3, p4) depends only on the four-vector total momentum hP hp1 + hp2 = hp3 + hp4and the relative momenta hp (hp1 hp2)/2, hp (hp3 hp4)/2, but neither on p0 [(p1)0 (p2)0] /2 nor on p0 [(p3)0 (p4)0] /2 (see Appendix A).

3. First-order approximationBogoliubov theory

In the approximation to first order in the interatomic interaction, we can neglect the q-integralin Eq. (43) because it gives a contribution to second order. Indeed, its contribution is smaller in

magnitude than the first-order contribution by a factor of the diluteness parameterna3F 1,where

aF is the s-wave scattering length in spin channelF (=0, 2) (see Section 4). On the other hand, in thelow-energy regime |p| 1/aF , the momentum dependence of the vacuum scattering amplitudes isnegligible, and fF (p, q) reduces to fF 4 h2 aF /M in the limit of zero momentum: p, q 0. TheT -matrix then becomes

jm,jm(p, p, P) j,m|F = 0F = 0|j,mf0 + j,m|F = 2F = 2|j,mf2. (44)By using the relations

|F = 0F = 0| + |F = 2F = 2| = 1, (45)2|F = 0F = 0| + |F = 2F = 2| = F F, (46)

the T -matrix can be rewritten as

jm,jm(p, p, P) c0 jjmm + c1

(F)jj(F)mm , (47)

where (F) ( = x, y, z) are the components of the spin-1 matrix vector

Fx = 12

0 1 01 0 10 1 0

, Fy = i

2

0 1 01 0 10 1 0

, Fz =

1 0 00 0 00 0 1

, (48)

and c0 and c1 are the coefficients of the spin-independent and spin-dependent interactions,respectively, which are related to the s-wave scattering lengths by

c0 f0 + 2f23 =4 h2

Ma0 + 2a2

3, (49)

c1 f2 f03 =4 h2

Ma2 a0

3. (50)

For convenience,we define a characteristic length scale a (a) that corresponds to the spin-independentinteraction:

a a4

a0 + 2a23

, (51)

from which we have c0 = 4 h2 a/M = h2 a/M .Now, we consider two cases in which the ground state is ferromagnetic and polar.

3.1. Ferromagnetic phase

In the ferromagnetic phase, all condensate particles occupy themF = 1magnetic sublevel, and thecondensates spinor is

(1, 0, 1) = (1, 0, 0). (52)The proper self-energies and chemical potential in the first-order approximation are illustrated inFig. 5 and given by


c

a b

d

Fig. 5. First-order contributions to the proper self-energies (a)(c) and the chemical potential (d). Here, the squares representthe T -matrices, while condensate particles are not explicitly shown. In fact, in (a), there are one condensate particle moving inand another moving out; in (b) and (c), there are two condensate particles moving in and two moving out, respectively; in (d),all four particles belong to the condensate. This convention helps simplify the second-order diagrams in Section 4.

h11jj (p) = j1,j1(p/2, p/2, p)+ 1j,j1(p/2,p/2, p) c0n0(jj + j,1j,1)+ c1n0

(F)jj(F)11 + (F)j,1(f)1,j

= c0n0(jj + j,1j,1)+ c1n0(jjj + j,1j,1 + j,0j,0), (53a)

h12jj (p) = h21jj (p) = jj,11(p, 0, 0) c0n0j,1j,1 + c1n0

(F)j,1(F)j,1

= c0n0j,1j,1 + c1n0j,1j,1, (53b) = 11,11(0, 0, 0)+ qB c0n0 + c1n0

(F)11(F)11 + qB= (c0 + c1)n0 + qB. (53c)

Here, the quadratic Zeeman energy qB is added to the right-hand side of Eq. (53c) for the chemicalpotential to account for the fact that all condensate particles occupy the magnetic sublevel mF = 1,whose energy is raised by qB due to the quadratic Zeeman effect. The matrix elements of (p) in Eq.(28) are then given by

h1111 (p) = 2(c0 + c1)n0, (54a)h1100 (p) = (c0 + c1)n0, (54b)h111,1(p) = (c0 c1)n0, (54c)h1211 (p) = h2111 (p) = (c0 + c1)n0, (54d)others = 0. (54e)

By substituting Eqs. (53c) and (54) into Eq. (30), we obtain the first-order Greens functions:

G11(p) =p0 + 0p/h+ (c0 + c1)n0/h

p20 21,p + i, (55a)


G00(p) = 1p0 0,p + i , (55b)

G1,1(p) = 1p0 1,p + i , (55c)

G1211(p) = G2111 = (c0 + c1)n0/hp20 21,p + i

, (55d)

others = 0. (55e)The energy spectra of the elementary excitations, which are given by the poles of Greens functions,

are

h1,p =0p[0p + 2(c0 + c1)n0], (56a)

h0,p = 0p qB, (56b)h1,p = 0p 2c1n0. (56c)

Thus, the Greens function approach gives the Bogoliubov energy spectra of elementary excitations asthe first-order results [34].

It will be useful for the second-order calculation in Section 4 to rewrite the first-order Greensfunctions in Eq. (55) as follows:

G11(p) = A1,pp0 1,p + i B1,p

p0 + 1,p i , (57a)

G1211(p) = G2111 = C1,p

1p0 1,p + i

1p0 + 1,p i

, (57b)

where

A1,p =h1,p + 0p + (c0 + c1)n0

2h1,p, B1,p =

h1,p + 0p + (c0 + c1)n02h1,p

, (58)

C1,p = (c0 + c1)n02h1,p . (59)

3.2. Polar phase

In the polar phase, all condensate particles occupy the mF = 0 magnetic sublevel, and thecondensates spinor is

(1, 0, 1) = (0, 1, 0). (60)From Fig. 5, the first-order proper self-energies and chemical potential are given by

h11jj (p) = j0,j0(p/2, p/2, p)+ 0j,j0(p/2,p/2, p) c0n0(j,j + j,0j,0)+ c1n0

(F)j,j(F)0,0 + (F)j,0(F)0,j

= c0n0(j,j + j,0j,0)+ c1n0(j,1j,1 + j,1j,1), (61a)

h12jj (p) = h21jj (p) = jj,00(p, 0, 0) c0n0j,0j,0 + c1n0

(F)j,0(F)j,0

= c0n0j,0j,0 + c1n0(j,1j,1 + j,1j,1), (61b) = 00,00(0, 0, 0) c0n0, (61c)


The matrix elements of (p) in Eq. (34) are then given by

h1111 (p) = h111,1(p) = (c0 + c1)n0, (62a)h1100 (p) = 2c0n0, (62b)h121,1(p) = h121,1(p) = h211,1(p) = h211,1(p) = c1n0, (62c)h1200 (p) = h2100 (p) = c0n0, (62d)others = 0. (62e)

Substituting Eqs. (61c) and (62) into Eq. (36), we obtain the first-order Greens functions as follows:

G11(p) = G1,1(p) =p0 +

0p + c1n0 + qB

/h

p20 21,p + i, (63a)

G00(p) =p0 +

0p + c0n0

/h

p20 20,p + i, (63b)

G121,1(p) = G121,1(p) = G211,1(p) = G211,1(p) = c1n0/h

p20 21,p + i, (63c)

G1200(p) = G2100(p) = c0n0/h

p20 20,p + i, (63d)

others = 0. (63e)The energy spectra of the elementary excitations, which are given by the poles of Greens functions,are

h1,p = h1,p =(0p + qB)(0p + qB + 2c1n0), (64a)

h0,p =0p(

0p + 2c0n0). (64b)

Here, there is a two-fold degeneracy in the energy spectra:1,p = 1,p due to the symmetry betweentwomagnetic sublevelsmF = 1. Similarly to the ferromagnetic phase, by using the Greens functionapproach, we have obtained the Bogoliubov energy spectra of elementary excitations for the polarphase as the first-order results [34].

The first-order Greens functions given by Eq. (63) can be rewritten in the following form:

G11(p) = G1,1(p) = A1,pp0 1,p + i B1,p

p0 + 1,p i , (65a)

G00(p) = A1,pp0 0,p + i B0,p

p0 + 0,p i , (65b)

G121,1(p) = G121,1(p) = G211,1(p) = G211,1(p)= C1,p

1

p0 1,p + i 1

p0 + 1,p i, (65c)

G1200(p) = G2100(p) = C0,p

1p0 0,p + i

1p0 + 0,p i

, (65d)

where

A1,p =h1,p + 0p + c1n0 + qB

2h1,p, B1,p =

h1,p + 0p + c1n0 + qB2h1,p

, (66)


A0,p =h0,p + 0p + c0n0

2h0,p, B0,p =

h0,p + 0p + c0n02h0,p

, (67)

C1,p = c1n02h1,p , C0,p =c0n02h0,p

. (68)

These expressions will be used in the following sections.

4. Second-order approximationBeliaev theory

We now investigate how quantum fluctuations at absolute zero alter the energy spectra ofelementary excitations in a spin-1 spinor condensate of 87Rb by calculating the energy spectra upto the second order in the interaction. The spin-dependent interaction for 87Rb atoms is known to beferromagnetic (c1 < 0). Here, we only consider the case of qB < 0 and qB > 2|c1|n for the respectiveferromagnetic and polar phases, where the corresponding first-order energy spectra of elementaryexcitations show that the system is dynamically stable (see Eqs. (56) and (64)). On the other hand,when considering the second-order corrections to the first-order results, we only need to take intoaccount the spin-independent interaction since the spin-dependent interaction would make a muchsmaller contribution to the already small second-order quantities. This is due to the large ratio ofspin-independent to spin-dependent interactions of 87Rb atoms: c0/|c1| 200. However, for a typicalatomic density in experiments of ultracold atoms, the second-order contribution to the proper self-energies from the spin-independent interaction is of the order of c0n

na3 0.01c0n, which is of the

same order of magnitude as the first-order contribution from the spin-dependent interaction|c1|n.We may thus expect an interplay between quantum fluctuations and spinor physics.

4.1. Second-order proper self-energies and chemical potential

The second-order correction of the proper self-energies and chemical potential involves two terms.One is the second-order correction toF=0,2(p1, p2; p3, p4) in the first-order diagrams (see Fig. 5), thatis, the q-integrals and the imaginary part of fF=0,2(p, p) in Eq. (43). The other is the contribution fromthe second-order diagrams given in Figs. 69.

4.1.1. Ferromagnetic phaseFirst, we consider the second-order corrections to the self-energies and chemical potential

that result from the correction to the T -matrix in the first-order diagrams. They are obtained bysubstituting the q-integrals and the imaginary part of fF (p, p) in Eq. (43) into the first lines of Eqs.(53a)(53c) (for more details, see Appendix B):

h11jj (p) : i Im{c0(p/2, p/2)}n0jj + i Im{c0(p/2,p/2)}n0j,1j,1

+n0f 20 + 2f 22

3

d3q(2)3

1

hp0 + 2[(c0 + c1)n0 + qB] 0q 0k + i

10p 0q 0k + i

(jj + j,1j,1), (69a)

h12,21jj (p) : n0f 20 + 2f 22

3

d3q(2)3

1

2[(c0 + c1)n0 + qB] 20q + i+ 1

20q

j,1j,1, (69b)


Fig. 6. Second-order diagrams for the proper self-energy 11jj (p). The intermediate propagators are classified into threedifferent categories, depending on the number of noncondensed atoms moving into and out of the condensate. They arerepresented by curves with one arrow (), two out-arrows (), and two in-arrows (), and are described by thefirst-order normal Greens function Gjj (p) and two anomalous Greens functions G12jj (p) and G

21jj (p), respectively. Here, the two

horizontal dashes in diagrams (e1) and (e2) indicate the fact that we need to subtract from these diagrams the terms of non-interacting Greens functions to avoid a double counting of the contributions that have already been taken into account inthe T -matrix and the first-order diagrams (see Eqs. (C.18) and (C.19)). As in Fig. 5, we use the convention that the condensateparticles in diagrams (a1)(e2) are not explicitly shown.

: n0f 20 + 2f 22

3

d3q(2)3

1

2[(c0 + c1)n0 + qB] 20q + i+ 1

20q

, (69c)


Fig. 7. Second-order diagrams for the proper self-energy 12jj (p). As in diagrams (e1) and (e2) of Fig. 6, the vertical dash indiagram (e) indicates the fact thatwe need to subtract from this diagram a term containing non-interacting Greens functions toavoid double counting of the contribution that has already been taken into account in the T -matrix and the first-order diagrams(see Eq. (C.40)).

where Im denotes the imaginary part of a complex number, k q p, and

c0(p/2,p/2) f0(p/2,p/2)+ 2f2(p/2,p/2)3 . (70)


Fig. 8. Second-order diagrams for the proper self-energy21jj (p).

Fig. 9. Second-order diagrams for the chemical potential .


Using the optical theorem for scattering, the imaginary part of an on-shell vacuum scatteringamplitude fF (p, p)with |p| = |p| is given by [35]

Im{fF (p, p)} = Mh2

d3q(2)3

fF (p, q)f F (p, q)(p2 q2)

= |p|M162 h2

dq fF (p, q)f F (p

, q), (71)

whereq denotes the solid angle of the on-shell momentum q : |q| = |p| = |p|. Consequently, theimaginary parts of fF (p/2,p/2) and c0(p/2,p/2) in Eqs. (69) and (70) are given in the second-order approximation as follows:

Im{fF (p/2,p/2)} = |p|M8 h2

f 2F , (72)

Im{c0(p/2,p/2)} = |p|M8 h2

f 20 + 2f 22

3

, (73)

where we have replaced fF (p, p) on the right-hand side of Eq. (71) by its zero-momentum limit fF .Next, we calculate the second-order contributions to the proper self-energies and chemical

potential from the second-order diagrams illustrated in Figs. 69 by using the first-order Greensfunctions given in Eq. (57) (for more details, see Appendix C.1). By summing up the second-ordercorrections that arise from the correction to the T -matrix (Eq. (69)) and the contributions from thesecond-order diagrams (Eqs. (C.2)(C.40)), we obtain the second-order self-energies and chemicalpotential as follows:

h11(2)11 (p) =i|p|Mn04 h2

f 20 + 2f 22

3

+ 2n0

f 20 + 2f 22

3

d3q(2)3

1hp0 + 2[(c0 + c1)n0 + qB] 0q 0k + i

10p 0q 0k + i

+ n0c20

d3q(2)3

2A1,q, B1,k

+ 4C1,qC1,k 4 A1,q, C1,k+ 2A1,qA1,khp0 1,q 1,k

+ i 2

A1,q, B1,k

+ 4C1,qC1,k 4 B1,q, C1,k+ 2B1,qB1,khp0 + 1,q + 1,k

i 2

hp0 0q 0k + 2(c0 + c1)n0 + i

+ 2c0

d3q(2)3

B1,q, (74)

h11(2)00 (p) =i|p|Mn08 h2

f 20 + 2f 22

3

+ n0

f 20 + 2f 22

3

d3q(2)3

1hp0 + 2[(c0 + c1)n0 + qB] 0q 0k + i

10p 0q 0k + i

+ n0c20

d3q(2)3

A1,k + B1,k 2C1,k

hp0 0,q 1,k

+ i 1

hp0 0q 0k + 2(c0 + c1)n0 + qB + i

+ c0

d3q(2)3

B1,q, (75)


h11(2)1,1(p) =i|p|Mn08 h2

f 20 + 2f 22

3

+ n0

f 20 + 2f 22

3

d3q(2)3

1hp0 + 2[(c0 + c1)n0 + qB] 0q 0k + i

10p 0q 0k + i

+ n0c20

d3q(2)3

A1,k + B1,k 2C1,k

hp0 1,q 1,k

+ i 1

hp0 0q 0k + 2(c0 + c1)n0 + i

+ c0

d3q(2)3

B1,q, (76)

h12(2)11 (p) = h21(2)11 (p)= n0

f 20 + 2f 22

3

d3q(2)3

1

2[(c0 + c1)n0 + qB] 20q + i+ 1

20q

+ n0c20

d3q(2)3

2A1,q, B1,k

+ 6C1,qC1,k 2 A1,q + B1,q, C1,k

1hp0 1,q 1,k

+ i 1h p0 + 1,q + 1,k i

+ c0

d3q(2)3

C1,q + c0n020q 2(c0 + c1)n0 i

, (77)

(2) = n0f 20 + 2f 22

3

d3q(2)3

1

2[(c0 + c1)n0 + qB] 20q + i+ 1

20q

+ 2c0

d3q(2)3

B1,q + c0

d3q(2)3

C1,q + c0n020q 2(c0 + c1)n0 i

, (78)

where k q p and A1,q, B1,k A1,qB1,k + A1,kB1,q.Here we consider only the case in which the external magnetic field satisfies qB |c1|n c0n

(see Section 2.3), and we ignore terms of the order smaller than c0nna3, which is of the order of

magnitude of the second-order approximation under consideration. Then, Eqs. (74), (77) and (78)for 11(2)11 (p),

12(2)11 (p), and

(2), respectively, are the same as those for a spinless BoseEinsteincondensate [11]. This is because the condensate is in the mF = 1 sublevel, and the elementaryexcitation given by11;12(2)11 (p) is the density-wave excitation as in a spinless system. Consequently,it has a phonon-like second-order energy spectrum in the low-momentum regime (0p c0n):

hp0 =1+ 7

62n0a3

2n0(c0 + c1)

0p i

3640

n0c0n0a3

0p5/2

(n0c0)5/2

=1+ 28

3

n0a3

2n0(c0 + c1)

0p i

3

80n0c0

n0a3

0p5/2

(n0c0)5/2

=1+ 8

na3

2n(c0 + c1)0p i

3

80nc0

na3

0p5/2

(nc0)5/2, (79)

where a and a are defined in Eq. (51). Here, in the derivation of the last line of Eq. (79), we used theexpression for the condensate fraction in a homogeneous system [4,35]:


n0 = n1 8

3

na3. (80)

The first term (real part) on the right-hand side of Eq. (79) shows an increase in the sound velocitydue to quantum fluctuations, while the second term (imaginary part) is the so-called Beliaev damping,which indicates a finite lifetime of phonons due to their collisions with the condensate. The second-order contribution to the chemical potential is given by [11]

(2) = 532

n0c0n0a3. (81)

Now, to evaluate 11(2)00 (p) we expand it around p0 = 0,p, where h0,p is the first-order energyspectrum given by Eq. (56b):

11(2)00 (p) = 11(2)00 (p0 = 0,p)+

11(2)00 (p)p0

p0=0,p

p0 0,p

+O

p0 0,p

2+ . (82)We can stop at the linear term in this Taylor expansion, provided that the difference between thesecond-order energy spectrum and the first-order one is small:

|p0 0,p| 11(2)00 (p0 = 0,p)

11(2)00 /p0

(p0 = 0,p)

c0n0h. (83)

This condition is justified by the fact that the system is a dilute weakly interacting Bose gas, and willbe confirmed later by the final result of second-order energy spectrum obtained in Eq. (119). This willbe discussed in more detail at the end of Section 4.2.1.

It can be shown that the imaginary part of11(2)00 (p) vanishes for any finite momentum p = 0 (seeAppendix D.1):

Im11(2)00 (p) = 0. (84)This result implies that there is no damping for the elementary excitation given by 1100 (p) up to theorder of magnitude under consideration.

For the real parts of 11(2)00 (p0 = 0,p) and

11(2)00 /p0

(p0 = 0,p), we can make their Taylor

expansions around p = 0 in the low-momentum regime 0p c0n0:

Re11(2)00 (p0 = 0,p) = Re11(2)00 (p0 = 0,p)|p=0 +Re11(2)00 (p0 = 0,p)

1p

p=0

1,p

+ 122Re11(2)00 (p0 = 0,p)

(1,p)2

p=0

21,p + , (85a)

Re11(2)00 (p)p0

p0=0,p

= Re11(2)00 (p)p0

p0=0,p,p=0

+ 1,p

Re11(2)00 (p)

p0

p0=0,p

p=0

1,p + 122

(1,p)2

Re11(2)00 (p)p0

p0=0,p

p=0

21,p+ , (85b)


where1,p is given in Eq. (56a). Note that p = 0 is equivalent to1,p = 0, and 0p c0n0 is equivalentto h1,p c0n0.

With straightforward calculations, we obtain (see Appendix E.1 for details):

hRe11(2)00 (p0 = 0,p)|p=0 =5

32n0c0

n0a3, (86)

Re11(2)00 (p0 = 0,p)1,p

p=0

= 0, (87)

1h2Re11(2)00 (p0 = 0,p)

(1,p)2

p=0

= 493602

n0a3

1n0c0

, (88)

Re11(2)00 (p)p0

p0=0,p,p=0

= 132

n0a3, (89)

1,p

Re11(2)00 (p)p0

p0=0,p

p=0

= 0, (90)

1h2

2

(1,p)2

Re11(2)00 (p)p0

p0=0,p

p=0

= 13602

n0a3

1(n0c0)2

. (91)

Hence, the value of second-order self-energy11(2)00 (p) is

hRe11(2)00 (p) =5

32n0a3

1 49

1200

h1,pn0c0

2n0c0

132

n0a3

1+ 13

40

h1,pn0c0

2hp0 0,p

+ O (p0 0,p)2 . (92)Eq. (92) shows the modification of the self-energy1100 (p) due to quantum fluctuations. The first termin the first line is the value for p = 0, p0 = 0, the second term in the first line is the correction for anonzero momentum, while the second line is the correction for a nonzero energy. It can be seen thatthe self-energy1100 (p), which describe the effect of interactionwith other particles on the propagationof a quasiparticle, decreases with increasing momentum or frequency.

4.1.2. Polar phaseFollowing a procedure similar to the ferromagnetic case, the second-order corrections to the self-

energies and chemical potential that result from the correction to the T -matrix in the first-orderdiagrams are given by

h11jj (p) : i Im{c0(p/2, p/2)}n0jj + i Im{c0(p/2,p/2)}n0j,0j,0

+ n0f 20 + 2f 22

3

d3q(2)3

1

hp0 + 2c0n0 0q 0k + i

10p 0q 0k + i

(jj + j,0j,0), (93a)


h12,21jj (p) : n0f 20 + 2f 22

3

d3q(2)3

1

2c0n0 20q + i+ 1

20q

j,0j,0, (93b)

: n0f 20 + 2f 22

3

d3q(2)3

1

2c0n0 20q + i+ 1

20q

. (93c)

Summing up the second-order corrections that arise from the correction to the T -matrix (Eq. (93)) andthe contributions from the second-order diagrams (see Appendix C.2), we obtain the second-orderself-energies and chemical potential as follows:

h11(2)11 (p) = h11(2)1,1(p)= i|p|Mn0

8 h2

f 20 + 2f 22

3

+ n0

f 20 + 2f 22

3

d3q(2)3

1hp0 + 2c0n0 0q 0k + i

10p 0q 0k + i

+ n0c20

d3q(2)3

A0,k + B0,k 2C0,k A1,q

hp0 1,q 0,k

+ i B1,q

hp0 + 1,q + 0,k

i 1

hp0 0q 0k + 2c0n0 qB + i

+ c0

d3q(2)3

3B1,q + B0,q

, (94)

h11(2)00 (p) =i|p|Mn04 h2

f 20 + 2f 22

3

+ 2n0

f 20 + 2f 22

3

d3q(2)3

1hp0 + 2c0n0 0q 0k + i

10p 0q 0k + i

+ n0c20

d3q(2)3

2A0,q, B0,k

+ 4C0,qC0,k 4 A0,q, C0,k+ 2A0,qA0,khp0 0,q 0,k

+ i 2

A0,q, B0,k

+ 4C0,qC0,k 4 B0,q, C0,k+ 2B0,qB0,khp0 + 0,q + 0,k

i 2

hp0 0q 0k + 2c0n0 + i+

A1,q, B1,k+ 2C1,qC1,k

1hp0 1,q 1,k

+ i 1h p0 + 1,q + 1,k i

+ c0

d3q(2)3

2B0,q + 2B1,q

, (95)

h12(2)1,1 (p) = h12(2)1,1 (p) = h21(2)1,1 (p) = h21(2)1,1 (p)= n0c20

d3q(2)3

C1,q2C0,k A0,k B0,k

1hp0 1,q 0,k

+ i 1h p0 + 1,q + 0,k i

c0

d3q(2)3

C1,q c1n020q 2c0n0 + 2qB i

, (96)


h12(2)00 (p) = n0f 20 + 2f 22

3

d3q(2)3

1

2c0n0 20q + i+ 1

20q

+ n0c20

d3q(2)3

2A0,q, B0,k

+ 6C0,qC0,k 2 A0,q + B0,q, C0,k

1hp0 0,q 0,k

+ i 1h p0 + 0,q + 0,k i

+

A1,q, B1,k+ 2C1,qC1,k 1

hp0 1,q 1,k

+ i 1

hp0 + 1,q + 1,k

i

+ c0

d3q(2)3

C0,q + c0n020q 2c0n0 i

, (97)

(2) = n0f 20 + 2f 22

3

d3q(2)3

1

2c0n0 20q + i+ 1

20q

+ 2c0

d3q(2)3

B0,q + B1,q

+ c0

d3q(2)3

C0,q c0n02c0n0 20q + i

. (98)

It can be seen that the integrand of the q-integral in each of Eqs. (95), (97) and (98) is asum of a term that contains only A0,q;k, B0,q;k, C0,q;k, 0,q;k, c0n0 and a term that contains onlyA1,q;k, B1,q;k, C1,q;k, 1,q;k, c1n. By rewriting the corresponding q-integrals using dimensionlessvariables 0q/(c0n0) and

0q/(|c1|n0), we find that the value of the latter integral is smaller than that of

the former one by a factor of|c1|/c0 1, and thus, the latter integral can be ignored. Here, we used

hp0 h0,p |c1|n0c0n0 c0n0 for the low-momentum region 0p |c1|n under considerationfor the case of the polar phase. Consequently,11,12(2)00 (p) and

(2) are the same as the second-orderself-energies and chemical potential of a spinless BoseEinstein condensate [11]. Namely, the second-order contribution to the chemical potential is given by

(2) = 532

n0c0n0a3. (99)

Here, the elementary excitation given by 11;12(2)00 (p) is a density-wave excitation as in a spinlesssystem. It, therefore, has a phonon-like second-order energy spectrum in the low-momentum regime:

hp0 =1+ 8

na3

2nc00p i

3

80nc0

na3

0p5/2

(nc0)5/2. (100)

On the other hand, in Eq. (96) for 12(2)1,1 , the factor c1n0, which arises from C1,q, can be taken outof the integral, and thus,12(2)1,1 is negligibly small:

12(2)1,1 (p) = 12(2)1,1 (p) = 21(2)1,1 (p) = 21(2)1,1 (p)

= 0+ O|c1|n0

n0a3

. (101)


Now, to evaluate 11(2)11 (p) we expand it around p0 = 1,p, where h1,p is the first-order energyspectrum given by Eq. (64a):

11(2)11 (p) = 11(2)11 (p0 = 1,p)+

11(2)11 (p)p0

p0=1,p

p0 1,p

+O

p0 1,p

2+ . (102)We can stop at the linear term in this Taylor expansion, provided that the difference between thesecond-order energy spectrum and the first-order one is small:

p0 1,p 11(2)11 (p0 = 1,p)

11(2)11 /p0

(p0 = 1,p)

c0n0h. (103)

This condition is justified by the fact that the system is a dilute weakly interacting Bose gas, and willbe confirmed later by the final result of second-order energy spectrum obtained in Eq. (140).

It can be shown that the imaginary part of11(2)11 (p) vanishes for any finite momentum p = 0 (seeAppendix D.2):

Im11(2)11 (p) = 0. (104)Physically, this implies that there is no damping for this elementary excitation up to the order ofmagnitude under consideration.

For the real parts of 11(2)11 (p0 = 1,p) and

11(2)11 /p0

(p0 = 1,p), we can make their Taylor

expansions around p = 0 in the low-momentum regime 0p |c1|n0 c0n0:

Re11(2)11 (p)|p0=1,p = Re11(2)11 (p0 = 1,p)|p=0 +Re11(2)11 (p0 = 1,p)

0,p

p=0

0,p

+ 122Re11(2)11 (p0 = 1,p)

(0,p)2

p=0

20,p + , (105a)

Re11(2)11 (p)p0

p0=1,p

= Re11(2)11 (p)p0

p0=1,p,p=0

+ 0,p

Re11(2)11 (p)

p0

p0=1,p

p=0

0,p + 122

(0,p)2

Re11(2)11 (p)

p0

p0=1,p

p=0

20,p+ , (105b)

where 0,p is given in Eq. (64b). Note that 0,p = 0 at p = 0, and h0,p c0n0 implies 0p c0n0.With straightforward calculations, we obtain (see Appendix E.2 for details):

hRe11(2)11 (p0 = 1,p)|p=0 =5

32n0c0

n0a3, (106)

Re11(2)11 (p0 = 1,p)0,p

p=0

= 0, (107)


1h2Re11(2)11 (p0 = 1,p)

(0,p)2

p=0

= 132

qB + c1n0qB(qB + 2c1n0) +

713602

n0a3

1n0c0

, (108)

Re11(2)11 (p)p0

p0=1,p,p=0

= 132

n0a3, (109)

0,p

Re11(2)11 (p)

p0

p0=1,p

p=0

= 0, (110)

1h2

2

(0,p)2

Re11(2)11 (p)

p0

p0=1,p

p=0

= 132


7602

n0a3

1(n0c0)2

. (111)

Therefore, we have

hRe11(2)11 (p) =5

32n0c0

n0a3

1+

110


711200

h0,pn0c0

2

132

n0a3

1+

12

qB + c1n0qB(qB + 2c1n0)

740

h0,pn0c0

2 h p0 1,p+ O (p0 1,p)2 . (112)

Eq. (112) shows themodification of the self-energy1111 (p) due to quantum fluctuations. The first termin the first line is the value for p = 0, p0 = 0, the second term in the first line is the correction for anonzero momentum, while the second line is the correction for a nonzero frequency. It can be seenthat because (qB + c1n0)/qB(qB + 2c1n0) > 1 for any qB > 2|c1|n0, the self-energy 1111 (p), whichdescribes the effect of interactionwith other particles on the propagation of a quasiparticle, decreasesfor increasing momentum and frequency, regardless of the strength of the external magnetic field.Similarly, we have

hRe22(2)11 (p) hRe11(2)11 (p)

= 532

n0c0n0a3

1+

110


711200

h0,pn0c0

2

+ 132

n0a3

1+

12


740

h0,pn0c0

2 h p0 1,p+ O (p0 1,p)2 . (113)

4.2. Second-order energy spectra of elementary excitations

With the second-order self-energies and chemical potential obtained in Section 4.1, we are nowin a position to evaluate the second-order energy spectra of elementary excitations, which can beobtained from the poles of the second-order Greens functions. As shown in Section 4.1, there is alwaysone density-wave elementary excitation, which is described by G11;1211 and G

11;1200 for the ferromagnetic

and polar phases, respectively. It has a linear dispersion relation as the phononmode in spinless BECs(see Eqs. (79) and (100)). Due to quantum fluctuations, the sound velocity increases by a universalfactor of 1 + (8/)na3, while there appears the so-called Beliaev damping due to the collisions


between quasiparticles in the elementary excitation and the condensate. The second-order energyspectra of the other elementary excitations will be discussed in the following.

4.2.1. Ferromagnetic phaseThe poles of Greens functions G00(p) and G1,1(p) in Eqs. (30) and (31) are given by the solutions

of the following equations:

G00(p) : hp0 = 0p + h1100 (p)= 0p qB +

h11(2)00 (p) (2)

= h0,p +

h11(2)00 (p) (2)

, (114a)

G1,1(p) : hp0 = 0p + qB + h111,1(p)= 0p 2c1n0 +

h11(2)1,1(p) (2)

= h1,p +

h11(2)1,1(p) (2)

. (114b)

Here,weused the first-order self-energies and chemical potential in Eq. (54) and the first-order energyspectra h0;1,p in Eq. (56). Note that on the right-hand sides of Eqs. (114a) and (114b), the self-energies are functions of both p0 and p.

By substituting Eqs. (84) and (92) for11(2)00 (p) and Eq. (81) for the chemical potential(2) into Eq.

(114a), the equation for the pole of G00(p) becomes

p0 = 0,p + p + pp0 0,p

, (115)

where p, p are the lowest-order coefficients in the Taylor expansion around p0 = 0,p and givenby

hp = 497202 n0c0n0a3

h1,pn0c0

2, (116)

p = 132n0a3 131202

n0a3

h1,pn0c0

2. (117)

Using the fact that p n0a3 1, the solution to Eq. (115) is given by

p0 = 0,p + p1 p 0,p + p

= 0,p 1h49

7202n0c0

n0a3

h1,pn0c0

2. (118)

In the low-momentum region 0p c0n0, h1,p in Eq. (56a) reduces to h1,p 2(c0 + c1)n00p .

Substituting this and Eq. (56b) into Eq. (118), and neglecting all the terms that are small in the orderof magnitude, we obtain the energy spectrum:

hp0 = 0p qB 49

7202

n0a3

n0c0 2c0n00p

=1 49

3602n0a3

0p qB

=1 49

45

n0a3

0p qB

1 49

45

na30p qB. (119)


Fig. 10. Collision between a magnon in spin statemF = 0 and a condensate atom in spin statemF = 1. Because the dominantinteraction is the spin-independent one, which conserves the spin of particles, another magnon with mF = 0 and a phononwithmF = 1 are produced. The condensate atom is represented by a dashed line.

Here, we used Eqs. (51) and (80) in deriving the last two equalities. Eq. (119) shows that the energyspectrum of the elementary excitation with spin state mF = 0 has a quadratic dispersion relation atlow momenta, which can be expressed using an effective massMeff as

hp0 = h2 p2

2Meff qB, (120)

where

Meff = M1 4945

na3

. (121)

Compared with the first-order energy spectrum h0,p (see Eq. (56b)), the energy gap remainsunchanged, while the effective mass Meff of the corresponding quasiparticles increases by a factorof 1/[1 49/(45)na3]. From Eq. (80), it can be seen that enhancement factor of the effectivemass is proportional to the number of quantum depleted atoms, which is of the order of

na3. This

can be understood as the effect of the interaction between a quasiparticle and the quantum depletedatoms, which hinders the motion of the quasiparticle.

Furthermore, because the imaginary part of 11(2)00 vanishes up to the order of n0c0n0a3 (see

Eq. (84)), the damping of this spin-wave elementary excitation is much smaller than that of thedensity-wave excitation mode (the mode with spin state mF = 1). In other words, the lifetimeof the corresponding magnons is much longer than that of phonons. (The fact that the excitationwith spin state mF = 0 and its quasiparticles can be identified as a spin wave and magnons willbe discussed in Section 5.) This agrees with the mechanism of the Beliaev damping via collisionsbetween quasiparticles and condensate particles. Physically, the Beliaev damping can be understoodby considering the conservation of momentum and energy in the collisional process between amagnon and a condensate particle. Because c0/|c1| 200 1, the interaction between 87Rb atomsin a scattering process is dominated by the spin-independent interaction, which conserves the spinof particles in the collision. Consequently, the collision between a magnon (spin state mF = 0) and acondensate particle (spin state mF = 1) would produce another magnon (spin state mF = 0) and aphonon (spin statemF = 1). This is illustrated in Fig. 10.

The conservation of the total momentum and energy in the collision process requires that thefollowing simultaneous equations be satisfied:

p1 = p3 + p4, (122)

qB + h2 p21

2Meff= qB + h

2 p232Meff

+ hvs|p4|. (123)Here, we used the following energy spectra of magnons and phonons:

Emagp = qB +h2 p2

2Meff, (124)

Ephop = hvs|p|, (125)


where the effective massMeff and the sound velocity vs are given by (see Eqs. (119) and (79))

Meff = M1 4945

na3

, (126)

vs =1+ 8

na3

(c0 + c1)nM

. (127)

From Eqs. (122) and (123), we obtain

h|p1 + p3| cos 2Meff

= vs, (128)where is the angle between p1+p3 and p1p3 = p4. By using Eqs. (126) and (127) for the effectivemass and sound velocity, we can evaluate the ratio of the left-hand side to the right-hand side ofEq. (128):

h|p1 + p3|| cos |2Meffvs

|p|, we have1,q > 1,p, and, in turn, p01,q0,k < 1,p1,q0,k < 0.In contrast, for |q| |p|, we have 0q 0p |c1|n0, and thus, 1,q can be expanded at low momentaas

hp0 1,q 0,k

1 4945

na3


0p


0p+k

0k[0k + 2(c0 + c1)n0]

= 0k

qB + c1n0

qB(qB + 2c1n0)

0k + 20p cos

+0k + 2c0n0

49

45

na3


0p . (D.9)

Because 0p c0n0, the expression in the square bracket of the last line of Eq. (D.9) is always positivefor any value of (0, ). Therefore, for finite momentum p = 0, the argument of the Diracs deltafunction is always upper limited by a negative value, leading to the vanishing of the integral in Eq.(D.8). We thus obtain

Im11(2)11 (p) = 0. (D.10)

Appendix E. Real parts of self-energies

E.1. Ferromagnetic phase:11(2)00 (p)

By restricting our consideration to a small external magnetic field qB |c1|n c0n, andignoring any difference of the order smaller than |c0|n

na3, which is justified at the second-order

approximation, the real part of11(2)00 (p) given by Eq. (75) reduces to

Re11(2)00 (p) =n0c20h

d3q(2)3

P 1

0p 0q 0k+ A1,k + B1,k 2C1,k

1hp0 0,q 1,k

+ c0h

d3q(2)3

B1,q

= n0c20

h

d3q(2)3

P

10p 0q 0k

+ 14

1

h1,q+ 1

h1,k

+ n0c20

h

d3q(2)3

0k

h1,k

1hp0 0,q 1,k


+ 14

1

h1,q+ 1

h1,k

+ 1

32n0c0h

n0a3, (E.1)

where P denotes the principle value, and a is defined by Eq. (51). Note that p0 0,q 1,k < 0as shown in Appendix D.1, and thus, there is no principle value with respect to the correspondingintegral in Eq. (E.1). Here, we used A1,k + B1,k 2C1,k = 0k/(h1,k) and

c0h

d3q(2)3

B1,q = c022h

(c0 + c1)n0M

h2

3/2 0

dxx+ 1x(x+ 2)

2x+ 2

= c022h

(c0 + c1)n0M

h2

3/2 23

132

c0n0h

n0a3, (E.2)

where we have ignored terms that contain the factor |c1|/c0 1.First, we calculate the third line of Eq. (E.1). Putting q p/2+ q, we have

k = q p = q p/2, (E.3a)0p 0q 0k = 2(0p/2 0q), (E.3b)

d3q(2)3

1h1,k

=

d3k(2)3

1h1,k

=

d3q(2)3

1h1,q

, (E.3c)d3q(2)3

P1

0p 0q 0k= 1

2

d3q(2)3

P1

0p/2 0q= 1

2

d3q

(2)3P

10p/2 0q

= 12

d3q(2)3

P1

0p/2 0q. (E.3d)

The third line of Eq. (E.1) then becomes

n0c20

h

d3q(2)3

P

10p 0q 0k

+ 14

1

h1,q+ 1

h1,k

= n0c20

2h

d3q(2)3

P

10p/2 0q

+ 1h1,q

. (E.4)

By taking a transformation of variables |q| 0q , and using the indefinite integrationdx

x

a x = 2xa ln

xax+a , (E.5)

together with the definition of the principle valuedxP

x

a x = lim0 a

+a+

dxx

a x, (E.6)

we obtain

n0c20

2h

d3q(2)3

P

10p/2 0q

+ 1h1,q

= 12

n0c0h

n0a3. (E.7)


For the remaining term in Eq. (E.1), its value in the low-momentum region 0p c0n is obtainedanalytically by making Taylor expansions around p0 = 0,p and p = 0 as described in Eqs. (82) and(85). The expansion coefficients are then calculated as follows:

Re11(2)00 (p0 = 0,p) =4

32n0c0h

n0a3 + n0c

20

h2

d3q(2)3

0k

h1,k

10,p 0,q 1,k

+ 14

11,q

+ 11,k

, (E.8)

Re11(2)00 (p0 = 0,p)|p=0 =4

32n0c0h

n0a3 + n0c

20

h

d3q(2)3

0q

h1,q

1(0q h1,q)

+ 12h1,q

= 4

32n0c0h

n0a3 + n0c

20M

3/2

22 h4

0

d0q0q

0qh1,q(0q + h1,q)

+ 12h1,q

= 4

32n0c0h

n0a3 + n0c

20

(c0 + c1)n0M3/222 h4

0

dxx x

x(x+ 2)(x+x(x+ 2)) +1

2x(x+ 2)

= 4

32n0c0h

n0a3 + n0c

20

(c0 + c1)n0M3/222 h4

23

532

n0c0h

n0a3, (E.9)

Re11(2)00 (p0 = 0,p)1,p

p=0

= 0, (E.10)

2Re11(2)00 (p0 = 0,p)(1,p)2

p=0

= 49n0c20M

3/2

3602[(c0 + c1)n0]3/2 h2

493602

n0a3

hn0c0

, (E.11)

Re11(2)00 (p)p0

p0=0,p

= n0c20

h2

d3q(2)3

(0k)h1,k

1(0,p 0,q 1,k)2 , (E.12)

Re11(2)00 (p)p0

p0=0,p,p=0

= n0c20

d3q(2)3

(0q)h1,q

1(0q h1,q)2

= n0c20M

3/2

22 h3

0

d0q0q(0q)h1,q

1(0q h1,q)2

= 132

n0c20M3/2

(c0 + c1)n0 h3

= 132

n0a3, (E.13)


1,p

Re11(2)00 (p)

p0

p0=0,p

p=0

= 0, (E.14)

2

(1,p)2

Re11(2)00 (p)p0

p0=0,p

p=0

= 13n0c20M

3/2

602[(c0 + c1)n0]5/2h

13602

n0a3

h2

(n0c0)2. (E.15)

E.2. Polar phase:11(2)11 (p)

Neglecting terms of the order smaller than c0n0n0a3, which is justified at the second-order

approximation, the real part of11(2)11 (p) then reduces to

Re11(2)11 (p) =n0c20h

d3q(2)3

P 1

0p 0q 0k+ (A0,k + B0,k 2C0,k)

A1,qhp0 1,q 0,k

B1,qhp0 + 1,q + 0,k

+ c0h

d3q(2)3

3B1,q + B0,q

= n0c

20

h

d3q(2)3

P

10p 0q 0k

+ 14

1

h1,q+ 1

h0,k

+ n0c20

h

d3q(2)3

(A0,k + B0,k 2C0,k)

A1,q

hp0 1,q 0,k

B1,q

hp0 + 1,q + 0,k

+ 14

1

h1,q+ 1

h0,k

+ 132

c0n0h

n0a3. (E.16)

Here, because p00,q1,k < 0 as shown in Appendix D.2, there is no principle value with respectto the corresponding integral in Eq. (E.16). We also used

c0h

d3q(2)3

B0,q = c022

(c0n0)3/2M3/2

h4

0

xdx

x+ 1x(x+ 2)2x+ 2

= c022

(c0n0)3/2M3/2

h4

23

= 132

c0n0h

n0a3, (E.17)


c0h

d3q(2)3

B1,q M3/2

h4

0

0qd

0qE1q + 0q + c1n0 + qB

2E1q

c0(|c1|n0)3/2M3/2

h4

c0n0h

n0a3. (E.18)

First, we consider the first term in Eq. (E.16):

n0c20

h

d3q(2)3

P

10p 0q 0k

+ 14

1

h1,q+ 1

h0,k

= n0c20

2h

d3q(2)3

P

10p 0q 0k

+ 12h1,q

+P

10p 0q 0k

+ 12h0,q

= n0c20

4h

d3q(2)3

P

10p/2 0q

+ 1h1,q

+

d3q(2)3

P

10p/2 0q

+ 1h0,q

n0c20M

3/2

422 h4

d0q

0q

P

10p/2 0q

+ 1E0q

= 122

n0c0h

n0a3. (E.19)

Here, in obtaining the second equality in Eq. (E.19)weused a transformation of variables (see Eq. (E.3)).Furthermore, in the third line the main contributions to the first and the second integrals arise from0q |c1|n and 0q c0n, respectively, which results in the fact that the first integral is smaller thanthe second one by a factor of the order of

|c1|/c0 1, and thus, the second integral was neglected.The integral in the next to the last line was directly calculated by using

0

xdx

P

1a x +

1x(x+ b)

= 2x +

a ln

a+ aa+ +a

+ 2a+ 2aa ln

a aa +a

+ 2x + b 2b= 2b, (E.20)

where x limx, a a ( +0).For the remaining term in Eq. (E.16), we can obtain an analytic result for the low-momentum

region 0p |c1|n c0n by making Taylor expansions around p0 = 1,p and p = 0 as described inEqs. (102) and (105). The coefficients of the expansions are then calculated as follows:

Re11(2)11 (p)|p0=1,p =5

62n0c0h

n0a3 + n0c

20

h2

d3q(2)3

0k

h0,k

A1,q

1,p 1,q 0,k B1,q1,p + 1,q + 0,k

+ 1

4

11,q

+ 10,k

, (E.21)


Re11(2)11 (p0 = 1,p)|p=0 =5

62n0c0h

n0a3 + n0c

20

h2

d3q(2)3

0q

h0,q

A1,q

1,p=0 1,q 0,q B1,q1,p=0 + 1,q + 0,q

+ 1

4

11,q

+ 10,k

562

n0c0h

n0a3 + n0c

20

h2

d3q(2)3

0q

0,q(0q + h0,q)

+ 14

11,q

+ 10,k

= 5

62n0c0h

n0a3 + n0c

20M

3/2

22 h5

0

0qd

0q

0q

0,q(0q + h0,q)+ 1

4

11,q

+ 10,k

= 5

62n0c0h

n0a3 + n0c

20 (n0c0)

1/2M3/2

22 h4

0

xdx

x

x(x+ 2)(x+x(x+ 2)) +14

1x+ 1

x(x+ 2)

= 532

n0c0h

n0a3. (E.22)

Here, we used the fact that the main contribution to the integral in the second line of Eq. (E.22) comesfrom 0q c0n0 c1n0, and we can approximate h1,q 0q, h1,p=0 0, A1,q 1, B1,q 0. This isbecause the integral converges at both the upper limit 0q c0n0, and the lower limit 0q 0. Next,we have

Re11(2)11 (p0 = 1,p)0,p

p=0

= 0, (E.23)

2Re11(2)11 (p0 = 1,p)(0,p)2

p=0

= 132


713602

n0a3

hn0c0

, (E.24)

where we used the result of the following integral: 0

dx13x2 + 11x3 x3(2+ x)+ 29x5(2+ x)+ 8x7(2+ x)

3(2+ x)5/2 x+x(2+ x)3 = 71902 . (E.25)Therefore, the real part of the self-energy11(2)11 (p0 = 1,p) can be written as

Re11(2)11 (p0 = 1,p) 5

32n0c0h

n0a3 +

162


717202

n0a3

h20,pn0c0

. (E.26)

In a similar manner, the real part of the self-energy11(2)11 (p)|p0=1,p can be calculated by replacingk q pwith k q+ p, and we obtain

Re11(2)11 (p)|p0=1,p 5

32n0c0h

n0a3 +

1

62qB + c1n0

qB(qB + 2c1n0) +71

7202


n0a3

h20,pn0c0

. (E.27)

Next, we have

Re11(2)11 (p)p0

p0=1,p,p=0

= n0c20

h2

d3q(2)3

0q

h0,q

A1,q

(1,p=0 1,q 0,q)2

B1,q(1,p=0 + 1,q + 0,q)2

. (E.28)

As above, the main contribution to the integral in Eq. (E.28) arises from 0q c0n0 c1n0. We canthen approximate h1,q 0q, h1,p=0 0, A1,q 1, B1,q 0, and the integral can be evaluatedstraightforwardly as

Re11(2)11 (p)p0

p0=1,p,p=0

n0c20M

3/2

22 h4

0

0q d

0q

0q

0,q(0q + h0,q)2

= n0c20M

3/2

22 h3

(n0c0)1/2

0

xdx

xx(x+ 2)[x+x(x+ 2)]2

= 132

n0a3. (E.29)

Similarly, we have

Re11(2)11 (p)p0

p0=1,p,p=0

= 132

n0a3. (E.30)

For the derivatives with respect to 0,p, we obtain

0,p

Re11(2)11 (p)

p0

p0=1,p

p=0

= 0, (E.31)

0,p

Re11(2)11 (p)

p0

p0=1,p

p=0

= 0, (E.32)

2

(0,p)2

Re11(2)11 (p)

p0

p0=1,p

p=0

= 132


7602

n0a3

h2

(n0c0)2, (E.33)

2

(0,p)2

Re11(2)11 (p)

p0

p0=1,p

p=0

=

132


7602

n0a3

h2

(n0c0)2. (E.34)

We thus obtain the real parts of the self-energies11(2)11 (p) as given in Eqs. (112) and (113).


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Beliaev theory of spinor Bose--Einstein condensatesIntroductionGreen's function formalism for a spinor Bose--Einstein condensateHamiltonianGreen's functionsFerromagnetic phasePolar phase

T -matrix

First-order approximation---Bogoliubov theoryFerromagnetic phasePolar phase

Second-order approximation---Beliaev theorySecond-order proper self-energies and chemical potentialFerromagnetic phasePolar phase

Second-order energy spectra of elementary excitationsFerromagnetic phasePolar phase

Transverse magnetization and spin waveFerromagnetic phasePolar phase

ConclusionAcknowledgmentsRelation between T -matrix and vacuum scattering amplitudeDerivation of Eq. (69)Contributions to the self-energies and chemical potential from the second-order diagramsFerromagnetic phasePolar phase

Imaginary parts of self-energiesFerromagnetic phase: 0011(2) (p) Polar phase: 1111(2) (p)

Real parts of self-energiesFerromagnetic phase: 0011(2) (p) Polar phase: 1111(2) (p)

References

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