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Annals of Physics 328 (2013) 158219

Contents lists available atSciVerse ScienceDirect

Annals of Physics

journal homepage:www.elsevier.com/locate/aop

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 2012

Accepted 3 October 2012

Available online 16 October 2012

Keywords:

Spinor BoseEinstein condensate (BEC)Beliaev theory

Energy spectrum

Spin wave

Beliaev 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 fluctuations

on the energy spectra of spin-1 spinor BoseEinstein condensates,

in particular, of a 87Rb condensate in the presence of an external

magnetic field. We find that due to quantum fluctuations, the

effective mass of magnons, which characterizes the quadraticdispersion relation of spin-wave excitations, increases compared

with its mean-field value. The enhancement factor turns out to be

the same for two distinct quantum phases: the ferromagnetic andpolar phases, and it is a function of only the gas parameter. The

lifetime of magnons in a spin-1 87Rb spinor condensate is shown

to be much longer than that of phonons due to the difference

in their dispersion relations. We propose a scheme to measure

the effective mass of magnons in a spinor Bose gas by utilizingthe effect of magnons nonlinear dispersion relation on the time

evolution of the distribution of transverse magnetization. This

type of measurement can be applied, for example, to precision

magnetometry.

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


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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]. The

leading 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 comparison with 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 momenta

as 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 competition

between 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 polar

phase, 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


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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 of

magnons 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 the matrices 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. Section3reproduces themean-field (Bogoliubov) energy spectra by using the Greens function approach [34]. Section4dealswith the self-energies at the second order in the interaction and gives the leading corrections to

the mean-field energy spectra due to quantum fluctuations. Section 5shows 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.Section6 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

+ qBj2

jj , (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 longitudinal

magnetization, the linear Zeeman term vanishes. The total Hamiltonian of the F= 1 spinor Bosegas is then given in the second-quantized form by

H=

dr

jj

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

wherej(r)is the field operator that annihilates an atom in magnetic sublevel mF= jat positionr,and the interaction energyVis 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 elementVjm,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):


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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 operatorj(r)is decomposed into the condensate part,

which can be replaced by a classical field n0j, and the noncondensate partj(r):j(r) =

n0j +j(r). (5)

For a homogeneous system, the condensate is characterized by the condensate number density n0andthe 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=1 Vn, (7)

where

E0= 1

2n20

dr

drj

mVjm,jm (r r)m j , (8a)

V1= 1

2n0

dr

drj

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

V2= 1

2n0

dr

drj(r)

m(r

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

V3= 212

n0

dr

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

V4= 2

1

2n0

dr

drj(r)

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

V5= 2

1

2n

1/2

0

dr

drj(r)

m(r

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

V6= 2

1

2n

1/2

0

dr

drj(r)

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

V7= 1

2

dr

dr

j(r)

m(r)Vjm,jm (r r)m (r)j (r). (8h)These interactions are schematically illustrated by the Feynman diagrams inFig. 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 Nis the total number operator:

N=

dr

j

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

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

K= E0 +

j

qBj2|j|2

N0 +K, (11)


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Fig. 1. Feynman diagrams representing the eight terms(8a)(8h)in the interaction Hamiltonian. The dashed, solid, and wavy

lines represent a condensate atom, a noncondensate atom, and the interaction, respectively.

whereE0, given in Eq.(8a), is the interaction energy between condensed atoms,N0= Vn0is the totalnumber of condensed atoms withVbeing 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)

K17

n=1Vn. (14)

Here, 0k=h2 k2/(2M)is the kinetic energy of a particle with momentumhk, andaj,kis the Fouriertransform of the noncondensate field operatorj(r):

aj,k= 1

V dreikrj(r). (15)

In the following sections, K0and K1are referred to as the noninteracting and interacting parts ofKin Eq.(12), respectively. For a weakly interacting system, K1can 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) = n0jj+ iGjj (x,y), (16)

wherej,j= 0, 1 indicate the spin components, andx= (r, t),y= (r, t)are four-vectors of thespacetime coordinates. The noncondensate part of Greens function is defined as

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

O|O . (17)


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Here, |O is the ground state of the interacting system, andT and H 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 anomalous

Greens functions defined as

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

O|O , (18)

iG21jj(x,y) O|Tj,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)

wherehph(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)

G21

1,1(p) G21

1,0(p) G21

1,1(p) G22

1,1(p) G22

1,0(p) G22

1,1(p)

, (21)

where

G11jj(p) Gjj (p), G22jj(p) Gjj (p). (22)

Eq.(20), which is illustrated inFig. 2, can be written as a matrix equation of 6 6 matrices:

G(p) =G0(p) +G0(p)(p)G(p), (23)

whereG,G0, anddenote the 6 6 matrices of interacting, non-interacting Greens functions, andproper self-energy, respectively. The normal and anomalous self-energies are labeled in the same wayas 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)1

G0(p). (24)

The non-interacting Greens function is defined as

iG0jj (x y) 0|Tj,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


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Fig. 2. Dysons equations for the normal and anomalous Greens functions. The thick line, thin line, and oval represent the

interacting, 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 ofG0

jj (x y) as

G0jj (p)=

d4x eipxG0jj (x)

= jj1

p0 0p /h + /h qBj2/h + i jj G0j(p), (26)


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Fig. 3. Normal (left) and anomalous (right) proper self-energies of noncondensed particles, wherehpis the four-momentum,j,jlabel 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, the matrixG0(p) is diagonal with matrix 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 phase

If 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 0

0 0 111,1(p) 0 0 0211,1(p) 0 0

111,1(p) 0 0

0 0 0 0 110,0(p) 00 0 0 0 0 11

1,

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-energies 11

jj (p), the conservation of the total projected

spin allows onlyj= j, i.e., diagonal elements. In contrast, for anomalous self-energies 12jj (p), only

thej = 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 thatG0(p) is a diagonal matrix (see Eq. (26)),we find that the matrixG(p) of interacting Greens functions has the same form as(p):

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

0 G0,0(p) 0 0 0 00 0 G

1,

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)


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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) = [G0

1(p)]1

+ 11

1,1(p)D1

= p0 + 0

p /h + qB/h + 11

1,1(p) /hD1

, (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)]1

111,1(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 anF

=1 spinor BEC in the ferromagnetic phase, that is, for the three elementary excitations to be

gapless, 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 withj= 0 and 1 are single-particle like, and thus, the correspondinganomalous self-energies and Greens functions vanish. The energy shift ofqB 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 forj = 1 in the presence of the quadratic Zeeman effect;therefore, only the corresponding phonon mode (j = 1) is gapless. When qB= 0, the spin-wave mode(j = 0) also becomes gapless with a quadratic dispersion relation.

2.2.2. Polar phase

If the ground state is polar, the condensates spinor is given by

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


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

0 G0,0(p) 0 0 G120,0(p) 0

0 0 G1,1(p) G121,1(p) 0 00 0 G211,1(p) G11(p) 0 00 G21

0,0(p

) 0 0 G

0,0(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)andG

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 (seeFig. 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

D1 = [G01(p)]1[G01(p)]1 + 111,1(p)[G01(p)]1 + 111,1(p)[G01(p)]1

111,1(p)

11

1,

1(

p)

+21

1,1(p)

121,

1(p)

+i

= p20

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

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

0p + qBh

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

2

2

+

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

2+ i, (37a)

D0 = [G00(p)]1[G00(p)]1 + 110,0(p)[G00(p)]1 + 110,0(p)[G00(p)]1 110,0(p)110,0(p) + 210,0(p)120,0(p) + i

= 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)


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D1 = [G01(p)]1[G01(p)]1 + 1,11,1 (p)[G01(p)]1 + 111,1(p)[G01(p)]1 111,1(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 +11

1,

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 the

quadratic 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 inFig. 4. It is written as

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

h j,m

d4q

(2 )4G0j(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)


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Fig. 4. T-matrix of a two-body scattering. Two atoms with momentahp3, hp4 and magnetic quantum numbers j, m collideto produce two atoms with momentahp1, hp2and magnetic quantum numbersj, m. TheT-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 G0

j

(p) in the integration of

G0j(p1 q)G0m(p2 + q) with respect toq0.

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 quadratic

Zeeman energies qB(j2 + m2) in the denominator of the right-hand side of Eq. (39)only gives a

small 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 |

gh

gh

| =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), theT-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 theT-matrix in the total spin Fchannel given by

F(p1,p2;p3,p4)= VF(p1 p3) + i

h

d4q

(2 )4G0(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.

TheT-matrix F(p1,p2;p3,p4) can be expressed in terms of the vacuum scattering amplitude forthe spin channel F= 0, 2 as follows (seeAppendix A) [11,35]:

F(p1,p2;p3,p4)= F(p, p, P)

=fF(p, p) +

d3q

(2 )3fF(p, q)

1

hP0 h2 P24M + 2 h2 q2

M + i

+ 1h2 q2

M h2 p2

M i

fF

(p, q), (43)

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


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F(p1,p2;p3,p4)depends only on the four-vector total momentumhPhp1+hp2=hp3+hp4and the relative momentahp (hp1hp2)/2, hp (hp3hp4)/2, but neither on p0 [(p1)0 (p2)0] /2 nor onp0 [(p3)0 (p4)0] /2 (seeAppendix 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 parameter

na3

F 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, andfF(p, q)reduces tofF 4h2 aF/Min 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)

theT-matrix can be rewritten as

jm,jm(p, p, P) c0 jj mm+ c1

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

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

Fx

= 1

2 0 1 01 0 1

0 1 0 , Fy=

i

2 0 1 01 0

1

0 1 0 , Fz=

1 0 00 0 0

0 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 + 2f2

3= 4h

2

M

a0 + 2a23

, (49)

c1 f2 f0

3= 4h

2

M

a2 a03

. (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 havec0= 4h2 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 the mF= 1 magnetic 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. 5and given by


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c

a b

d

Fig. 5. First-order contributions to the proper self-energies (a)(c) and the chemical potential (d). Here, the squares represent

theT-matrices, while condensate particles are not explicitly shown. In fact, in (a), there are one condensate particle moving in

and 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 qBdue 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)


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G00(p) = 1

p0 0,p + i, (55b)

G1,1(p) = 1

p0 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)

h

1,p

=0

p2c1n0. (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 4to rewrite the first-order Greensfunctions in Eq.(55)as follows:

G11(p) = A1,p

p0 1,p + i B1,p

p0 + 1,p i, (57a)

G1211(p) = G2111= C1,p

1

p0 1,p + i 1

p0 + 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)n0

2h1,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)FromFig. 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

= c

0n

0j,0j,0 +c

1n

0(j,1j,1 + j,1j,1), (61b)

= 00,00(0, 0, 0) c0n0, (61c)


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The matrix elements of(p) in Eq.(34)are then given byh1111 (p) =h111,1(p) = (c0 + c1)n0, (62a)h1100 (p) = 2c0n0, (62b)

h121,1(p) =h

121,1(p) =h

211,1(p) =h

211,1(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

=h

1,p

= (0p+ qB)(0p+ qB + 2c1n0), (64a)h0,p=

0p (

0p+ 2c0n0). (64b)

Here, there is a two-fold degeneracy in the energyspectra: 1,p= 1,pdue to the symmetry betweentwo magnetic sublevels mF= 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,p

p0 1,p + i B1,p

p0 + 1,p i, (65a)

G00(p) = A1,p

p0 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

1

p0 0,p + i 1

p0 + 0,p i

, (65d)

where

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

2h1,p, B1,p=

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

, (66)


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

2h0,p, B0,p=

h0,p + 0p+ c0n02h0,p

, (67)

C1,p= c1n0

2

h1,p

, C0,p= c0n0

2

h0,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 87 Rb atoms is known to beferromagnetic (c1 < 0). Here, we only consider the case ofqB < 0 andqB >2|c1|nfor 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 of87Rb 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 ofc0n

na3 0.01c0n, which is of thesame 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.Oneis thesecond-order correction to F=0,2(p1,p2;p3,p4) in the first-order diagrams (see Fig. 5), thatis, the q-integrals and the imaginary part offF=0,2(p, p) in Eq. (43). The other is the contribution fromthe second-order diagrams given inFigs. 69.


First, we consider the second-order corrections to the self-energies and chemical potentialthat result from the correction to the T-matrix in the first-order diagrams. They are obtained bysubstituting theq-integrals and the imaginary part offF(p, p

)in Eq.(43)into the first lines of Eqs.(53a)(53c)(for more details, seeAppendix B):

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

+n0

f20+ 2f223

d3q

(2 )3

1

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

10p 0q 0k+ i

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

h12,21

jj (p) : n0

f20+ 2f223

d3q

(2)3

1

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

20q

j,1j,1, (69b)


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Fig. 6. Second-order diagrams for the proper self-energy 11jj (p). The intermediate propagators are classified into three

different categories, depending on the number of noncondensed atoms moving into and out of the condensate. They are

represented by curves with one arrow (), two out-arrows (), and two in-arrows (), and are described by thefirst-order normal Greens functionGjj (p) and two anomalous Greens functionsG

12jj(p) andG

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 in

theT-matrix and the first-order diagrams (see Eqs.(C.18)and(C.19)). As inFig. 5, we use the convention that the condensate

particles in diagrams (a1)(e2) are not explicitly shown.

: n0

f20+ 2f223

d3q

(2 )3

1

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

20q

, (69c)


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Fig. 7. Second-order diagrams for the proper self-energy12jj (p). As in diagrams (e1) and (e2) ofFig. 6, the vertical dash in

diagram (e) indicates the fact that we need to subtract from this diagram a term containing non-interactingGreens functionsto

avoid double counting of the contribution that hasalready 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)


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Fig. 8. Second-order diagrams for the proper self-energy 21jj (p).

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


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Using the optical theorem for scattering, the imaginary part of an on-shell vacuum scattering

amplitude fF(p, p) with |p| = |p| is given by [35]

Im

{fF(p, p)

} =

M

h

2 d3q

(2 )3

fF(p, q)fF(p, q)(p2

q2)

= |p|M16 2 h2

dqfF(p, q)fF(p, q), (71)

whereqdenotes the solid angle of the on-shell momentumq: |q| = |p| = |p|. Consequently, theimaginary parts offF(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|M

8

h2

f2F

, (72)

Im{c0(p/2, p/2)} =|p|M8h2

f20+ 2f223

, (73)

where we have replaced fF(p, p) on the right-hand side of Eq.(71)by its zero-momentum limitfF.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, seeAppendix 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|Mn0

4h2

f20+ 2f223

+ 2n0

f20+ 2f223

d3q

(2 )3

1

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

0p 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,k

h p0 + 1,q + 1,k i

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

+ 2c0

d3q

(2 )3B1,q, (74)

h11(2)00 (p)=

i|p|Mn08h2

f20+ 2f22

3

+ n0

f20+ 2f223

d3q

(2 )3

1

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

0p 0q 0k+ i

+ n0c20 d3q

(2 )3

A1,k + B1,k 2C1,k

h p0 0,q 1,k+ i

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

+ c0

d3q

(2 )3 B1,q, (75)


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h11(2)1,1(p)=

i|p|Mn08h2

f20+ 2f22

3

+ n0

f20+ 2f223

d3q

(2 )3

1

hp0

+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

f20+ 2f223

d3q

(2)3

1

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

20q

+ n0c20 d3q(2 )3 2 A1,q, B1,k+ 6C1,qC1,k 2 A1,q + B1,q, C1,k

1

hp0 1,q 1,k

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

+ c0

d3q

(2 )3

C1,q +

c0n0

20q 2(c0 + c1)n0 i

, (77)

(2) = n0

f20+ 2f223

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 + c0n0

20q 2(c0 + c1)n0 i

, (78)

wherek q pand 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 Section2.3), and we ignore terms of the order smaller thanc0n

na3, which is of the order ofmagnitude 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 by

11;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 + 76 2

n0a

3

2n0(c0 + c1)

0p i

3

640n0c0

n0a

3

0p5/2

(n0c0)5/2

=

1 + 283

n0a

3

2n0(c0 + c1)

0p i

3

80n0c0

n0a

3

0p5/2

(n0c0)5/2

=

1 + 8

na3

2n(c0 + c1)

0p i3

80nc0

na3

0p5/2

(nc0)5/2

, (79)

wherea andaare 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]:


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n0= n

1 83

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) = 53 2

n0c0

n0a

3. (81)

Now, to evaluate11(2)

00 (p)we expand it aroundp0= 0,p, whereh0,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 Section4.2.1.

It can be shown that the imaginary part of11(2)00 (p) vanishes for any finite momentum p = 0 (see

Appendix 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 of11(2)00 (p0= 0,p)and

11(2)00 /p0

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

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

Re11(2)

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

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

1p

p=0

1,p

+ 12

2Re11(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

Re

11(2)

00 (p)

p0

p0=0,p

p=0

1,p +1

2

2

(1,p)2 Re11(2)00 (p)

p0p0=0,p

p=0

2

1,p

+ , (85b)


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where 1,pis given in Eq. (56a). Note that p = 0 is equivalent to 1,p= 0, and 0p c0n0is equivalenttoh1,p c0n0.

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

hRe

11(2)

00 (p

0=

0,p)|p=0=

5

3 2n

0c

0n

0 a3, (86)

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

1,p

p=0

= 0, (87)

1

h

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

p=0

= 49360 2

n0a

3 1

n0c0, (88)

Re11(2)00 (p)

p0

p0=0,p,p=0

= 13 2

n0a

3, (89)

1,p

Re11(2)00 (p)

p0

p0=0,p

p=0

= 0, (90)

1

h2 2

(1,p)2

Re11(2)00 (p)

p0

p0=0,p

p=0

= 1360 2

n0a

3 1

(n0c0)2. (91)

Hence, the value of second-order self-energy 11(2)00 (p) is

hRe11(2)00 (p)= 53 2

n0a3

1 491200

h1,pn0c0

2n0c0

13 2

n0a

3

1 + 13

40

h1,p

n0c0

2hp0 0,p

+ O (p0 0,p)2 . (92)Eq. (92) shows the modification of the self-energy 1100 (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-energy 1100 (p), whichdescribetheeffectof interaction with other particleson the propagationof a quasiparticle, decreases with increasing momentum or frequency.

4.1.2. Polar phase

Following 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

+ n0

f20+ 2f223

d3q

(2 )3

1

hp0 + 2c0n0 0q 0k+ i

10p 0q 0k+ i

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


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h12,21

jj (p) : n0

f20+ 2f223

d3q

(2 )3

1

2c0n0 20q+ i+ 1

20q

j,0j,0, (93b)

:n0 f

20+ 2f22

3 d

3q

(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 (seeAppendix C.2), we obtain the second-orderself-energies and chemical potential as follows:

h11(2)11 (p)=h11(2)1,1(p)

=i|p|Mn08h2

f20+ 2f22

3

+ n0

f20+ 2f223

d3q

(2 )3

1

hp0

+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

1hp0 0q 0k+ 2c0n0 qB + i

+ c0

d3q

(2 )3

3B1,q + B0,q

, (94)

h11(2)00 (p)=

i|p|Mn04

h2

f20+ 2f22

3

+ 2n0

f20+ 2f223

d3q

(2 )3

1

hp0 + 2c0n0 0q 0k+ i 1

0p 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 2A0,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

1

hp0 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,q

2C0,k A0,k B0,k

1

h p0 1,q 0,k+ i

1

h p0 + 1,q + 0,k i

c0

d3q

(2 )3

C1,q

c1n0

20q 2c0n0 + 2qB i

, (96)


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h12(2)00 (p)= n0

f20+ 2f22

3

d3q

(2)3

1

2c0n0 20q+ i+ 1

20q

+ n0c20 d3q

(2 )3 2 A0,q, B0,k+ 6C0,qC0,k 2 A0,q + B0,q, C0,k

1

hp0 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

+

c0n0

2

0

q2c

0n

0 i, (97)

(2) = n0

f20+ 2f223

d3q

(2 )3

1

2c0n0 20q+ i+ 1

20q

+ 2c0

d3q

(2 )3

B0,q + B1,q

+ c0

d3q

(2 )3

C0,q

c0n0

2c0n0 20q+ i

. (98)

It can be seen that the integrand of the q-integral in each of Eqs. (95), (97) and (98) is a

sum 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

hp0h0,p|c1|n0c0n0 c0n0for the low-momentum region0p |c1|nunder consideration

for 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) = 53 2

n0c0n0a

3. (99)

Here, the elementary excitation given by 11;12(2)00 (p) is a density-wave excitation as in a spinless

system. It, therefore, has a phonon-like second-order energy spectrum in the low-momentum regime:

hp0=

1 + 8

na3

2nc0

0p 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 fromC1,q, can be taken out

of 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)


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Now, to evaluate11(2)11 (p)we expand it aroundp0= 1,p, whereh1,p is the first-order energy

spectrum 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,p2+ . (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 (see

Appendix 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 of11(2)11 (p0= 1,p)and

11(2)11 /p0

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

expansions aroundp = 0in 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

+ 12

2Re11(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)

p0p0=1,p

p=0

0,p +1

2

2

(0,p)2

Re

11(2)11 (p)

p0

p0=1,p

p=0

20,p

+ , (105b)where 0,pis given in Eq.(64b). Note that 0,p= 0 atp = 0, andh0,p c0n0implies 0p c0n0.

With straightforward calculations, we obtain (seeAppendix E.2for details):

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

5

3 2n0c0n0a

3, (106)

Re11(2)

11 (p0= 1,p)0,p

p=0

= 0, (107)


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1

h

2Re11(2)

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

2

p=0

= 1

3 2

qB + c1n0qB(qB + 2c1n0)

+ 71360 2

n0a

3 1

n0c0, (108)

Re11(2)11 (p)

p0p0=1,p,p=0 =

1

3 2n0 a

3, (109)

0,p

Re

11(2)11 (p)

p0

p0=1,p

p=0

= 0, (110)

1

h22

(0,p)2

Re

11(2)11 (p)

p0

p0=1,p

p=0

=

13 2


+ 760 2

n0a3

1

(n0c0)2. (111)

Therefore, we have

hRe11(2)11 (p)=

5

3 2n0c0

n0a

3

1 +

110


+ 711200

h0,p

n0c0

2

13 2

n0a

3

1 +

1

2


740

h0,p

n0c0

2

h p0 1,p+ O (p0 1,p)2 . (112)Eq. (112) shows the modification of the self-energy 1111 (p) due to quantum fluctuations. The first term

in 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-energy1111 (p), which

describes the effect of interaction with 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)

= 53 2

n0c0

n0a

3

1 +

1

10


+ 711200

h0,p

n0c0

2

+ 1

3 2

n0a3

1 + 12 qB + c1n0qB(qB + 2c1n0) 7

40 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 Section4.1, 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 always

one 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 phonon mode in spinless BECs(see Eqs.(79)and (100)). Due to quantum fluctuations, the sound velocity increases by a universal

factor of 1 + (8/ )

na3, while there appears the so-called Beliaev damping due to the collisions


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


The poles of Greens functions G00(p) and G

1,

1(p) in Eqs. (30) and (31) are given by the solutions

of the following equations:

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

h

11(2)00 (p) (2)

=h0,p +

h

11(2)00 (p) (2)

, (114a)

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

h

11(2)1,1(p) (2)

=h1,p + h

11(2)1,1(p) (2) . (114b)

Here, we used the first-order self-energies and chemical potential in Eq. (54) and the first-order energyspectrah0;1,p in Eq. (56). Note that on the right-hand sides of Eqs. (114a)and (114b), the self-energies are functions of bothp0andp.

By substituting Eqs.(84)and(92)for 11(2)00 (p) and Eq.(81)for the chemical potential

(2) into Eq.(114a), the equation for the pole ofG00(p) becomes

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

, (115)

wherep, p are the lowest-order coefficients in the Taylor expansion aroundp0= 0,p and givenby

hp

=

49

7202

n0c0n0 a3 h1,p

n0c0

2

, (116)

p= 1

3 2

n0a

3 13120 2

n0a

3

h1,p

n0c0

2. (117)

Using the fact that p

n0a3 1, the solution to Eq.(115)is given by

p0= 0,p + p

1 p 0,p + p

= 0,p 1

h

49

720 2n0c0n0a

3

h1,p

n0c0 2

. (118)

In the low-momentum region 0p c0n0, h1,p in Eq.(56a)reduces toh1,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

720 2

n0a

3

n0c0 2c0n00p

=

1 49360 2

n0a

3

0p qB

= 1

49

45n0a

3 0pqB

1 4945

na3

0p qB. (119)


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Fig. 10. Collision between a magnon in spin statemF= 0 and a condensate atom in spin state mF= 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 mass Meffas

hp0=h2 p2

2Meff qB, (120)

where

Meff= M

1 4945

na3

. (121)

Compared with the first-order energy spectrumh0,p (see Eq. (56b)), the energy gap remainsunchanged, while the effective mass Meff of the corresponding quasiparticles increases by a factor

of 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 of11(2)00 vanishes up to the order ofn0c0

n0a3 (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 Section5.) 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. Becausec0/|c1| 2001, the interaction between 87 Rb 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 a

condensate particle (spin statemF= 1) would produce another magnon (spin state mF= 0) and aphonon (spin statemF= 1). This is illustrated inFig. 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 +

h2 p23

2Meff+hvs|p4|. (123)

Here, we used the following energy spectra of magnons and phonons:

Emagp = qB +

h2 p2

2Meff, (124)

Ephop =hvs|p|, (125)


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where the effective massMeffand the sound velocity vsare given by (see Eqs.(119)and(79))

Meff= M

1 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 + p3and p1 p3= 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

< h|p1|

Meffvs

2Mc0n

Meff

=1 49

45

na3

1 + 8

na3

2c0

c0 + c1 O(1). (129)

Here, we used |p3| < |p1|, which results from Eq.(123), and |p1|

2Mc0nfor the low-momentumregion 0p1 c0n where the obtained second-order energy spectrum (Eq. (119)) is valid. FromEq.(129), it is obvious that there is no collision between a magnon and a condensate particle in whichthe momentum and energy conservations (Eqs. (122) and (123), or equivalently, Eqs. (122) and (128))are satisfied. Consequently, the damping of magnons vanishes up to the order of magnitude underconsideration as shown by Eq.(84). In fact, in the case of spin-1 spinor BECs, the collision between amagnon and a condensate particle via the small spin-dependent interaction is also suppressed sinceit only exchanges the spin states of the colliding particles.

Similarly, the energy spectrum of the elementary excitation given by G1,1(p)at low momenta0p c0n0is given by

hp0h1,p 49

720 2n0c0

n0a

3

h1,p

n0c0

2

0p 2c1n0 49

720 2n

0 a3

n0c0 2(c0 + c1)n00p

=

1 49360 2

n0a

3

0p 2c1n0

=

1 4945

na3

0p 2c1n. (130)

It can be seen from Eq. (130)that the energy spectrum of the excitation with spin state mF= 1also has a quadratic dispersion relation at low momenta, and compared with the first-order energyspectrum, the energy gap remains unchanged while the effective mass increases by the same factor

of 1/

[1

49/(45

)

na3

]as the excitation with spin state mF

=0.

Now we are in a position to examine the validity of the a priori assumption that the differencebetween the second-order and first-order energy spectra is small (see Eq. (83)). This assumption hasbeen used in Section4.1as the self-energies were Taylor expanded aroundp0= 0,p (Eq.(82)), and


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the expansions were stopped at the linear term. The condition for the validity of the Taylor expansioncan be obtained from Eq.(115), that is,

pp0 0,p

p. (131)

By using Eq.(118)for the second-order energy spectrum, we find that the left-hand side of Eq.(131)

is almost equal to pp, and thus, Eq.(131)is satisfied, provided that

p

n0a3 1. (132)

Eq. (132) is nothing but the diluteness condition, and it is usually satisfied in experiments of ultracoldBose gases.

4.2.2. Polar phase

For the polar phase, there is a two-fold degeneracy in the energy spectra of elementary excitationsdue to the symmetry between the mF = 1 sublevels. The poles of Greens function G11(p) (orequivalently,G1,1(p)) given in Eqs.(36)and(37)are

p0= + 2

0p + qB

h+

+ + 2

2 121,1211,1

= (2)+ (2)

2

0p+ c1n0 + qBh

+

(2)+ + (2)

2

(2)

h

2

c1n0

h+ 12(2)1,1

21/2

(2)+

(2)

2 21,p +

(2)+ + (2) 2(2)/h

0p+ c1n0 + qB

h

2 c1n0h

12(2)1,1

1/2

(2)+ (2)

2

1,p +

(2)+ + (2) 2(2)/h

(0p+ c1n0 + qB)

2h1,p

c1n0

h1,p

12(2)1,1

= 1,p + 1,p , (133)where denote 1111 (p), and

1,p

0p+ c1n0 + qB2h1,p

(2)+ + (2) 2(2)/h

c1n0

h1,p

12(2)1,1

(2)+ (2)

2. (134)

Here, we used the first-order self-energies and chemical potential in Eq.(62), the first-order energy

spectrum in Eq.(64a), and the fact that 12(2)

1,1 and(2)+ + (2) 2(2)/h are much smaller than

|c1|n0/h qB/h1,pin the low-momentum region 0p |c1|n0(see Eqs.(99),(101),(104),(112)and(113)).

By substituting Eqs.(99),(101),(104),(112)and(113)into Eqs.(133)and(134), we obtain theequation that determines the pole ofG11(p):

p0= 1,p + p + pp0 1,p

, (135)


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where

hp=

71

720 2

0p+ qB + c1n0h1,p

16 2

(qB + c1n0)qB(qB + 2c1n0)

n0c0

n0a

3

h0,p

n0c0

2, (136)

p= 13 2

n0a

3 +

7120 2

16 2


n0a

3

h0,p

n0c0

2

. (137)

Using the fact that p

n0a3 1, the solution to Eq.(135)is given by

p0= 1,p + p

1 p 1,p + p

= 1,p +

71

720 2

0p+ qB + c1n0h1,p

16 2


n0c0

n0a

3

h0,p

n0c0

2. (138)

In the low-momentum region 0p |c1|n0 qB c0n0, 1,pand 0,pcan be approximated as

h1,p=

(0p+ qB)(0p+ qB + 2c1n0)

qB(qB + 2c1n0) + (qB + c1n0)

qB(qB + 2c1n0)0p , (139a)

h0,p=

0p (0p+ 2c0n0)

2c0n0

0p . (139b)

Substituting Eq. (139) into Eq. (138), we obtain the energy spectrum which is correct up to the secondorder:

hp0=

qB(qB + 2c1n0) + (qB + c1n0)

qB(qB

+2c1n0)

0p 49

360 2n0a

3 (qB + c1n0)

qB(qB

+2c1n0)

0p

=

qB(qB + 2c1n0) +

1 49360 2

n0a

3

(qB + c1n0)

qB(qB + 2c1n0)0p

=

qB(qB + 2c1n0) +

1 4945

n0a

3

(qB + c1n0)

qB(qB + 2c1n0)0p

=

qB(qB + 2c1n) +

1 4945

na3

(qB + c1n)

qB(qB + 2c1n)0p . (140)

Here, in deriving the last equality, we usedna3 1 and(qB + c1n0)

qB(qB + 2c1n0) (qB + c1n)qB(qB + 2c1n) 1

(c1n)2

(qB + c1n)2 8

3 2

na3 . (141)From Eq.(140), it can be seen that the energy spectrum of the elementary excitation given by G11(p)has a quadratic dispersion relation at low momenta, which can be expressed using the effective massMeffas

hp0=h2 p2

2Meff+

qB(qB + 2c1n). (142)The effective mass depends on the quadratic Zeeman energy qBas

Meff=

qB(qB + 2c1n)(qB

+c1n)

M

1

49

45

na3

. (143)

Compared with the first-order energy spectrum in Eq. (139a), the energy gap remains unchanged

while theeffectivemass increases by a factor of 1/[149/(45 )

na3] due to quantum fluctuations.


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It can be seen that the effectof quantum fluctuations on the effective mass is characterized by thesameenhancement factor, regardless of whether the system is ferromagnetic or polar, and independent of

external parameters of the system. Furthermore, since the imaginary part of11(2)11 vanishes up to

the order ofn0c0n0a

3, the damping of this spin-wave excitation (see Section 5) is much smallerthan that of the density-wave excitation (with spin state mF

=0). In other words, the lifetime of

the corresponding magnons is much longer than that of phonons. This agrees with the mechanism ofBeliaev damping and can be understood by considering the conservation of momentum and energyin the collision process between a quasiparticle and a condensate particle (see Section4.2.1).

5. Transverse magnetization and spin wave

As shown in Sections 3 and 4, the elementary excitations of a spinor BEC include both density-waveand spin-wave excitations with quasiparticles being phonons and magnons, respectively. The energyspectrum of phonons with a linear dispersion relation can be experimentally measured by using theneutron scattering or the Bragg spectroscopy. The former has been widely used in experiments ofthe superfluid helium [3842], while the later has been applied to measurements of ultracold atoms

[18,4345]. Similarly, the neutron scattering has been used to measure the dispersion relation ofmagnons in solid crystals [46,47], though its application to ultracold atomic systems is limited bya huge difference in energy scales between neutrons and atoms. The Bragg spectroscopy can alsobe generalized to measure the energy spectrum of magnons by using appropriately polarized laserbeams to make spin-selective transitions. In this section, we propose an alternative experimentalscheme to obtain information about the effective mass of magnons in a spinor BEC. In ultracold atomexperiments, atoms can be optically excited to produce an appreciable number of quasiparticles.Furthermore, an in situ, non-destructive measurement of magnetization can be made with a highresolution [48]. It has been applied in a wide range of spinor BECs experiments to investigate, forinstance, the formation of spin textures and topological excitations as well as their dynamics [4953].

First, we show that in a spinor BEC, the elementary excitations with quadratic dispersion relations(see Sections 3 and 4) can be interpreted as waves of transverse magnetization. The transverse

magnetization density operatorsFx(r, t) andFy(r, t) in the Heisenberg representation are defined byF+(r, t) Fx(r, t) + iFy(r, t) =

2

1 (r, t)0(r, t) +0 (r, t)1(r, t)

, (144a)

F(r, t) Fx(r, t) iFy(r, t) =

2

0 (r, t)1(r, t) +1(r, t)0(r, t)

. (144b)

The squared modulus of the transverse magnetization is written in terms ofF+(r, t) andF(r, t) asF2(r, t)F2x (r, t) +F2y (r, t)

= 12

F+(r, t)F(r, t) +F(r, t)F+(r, t). (145)

5.1. Ferromagnetic phase

The condensate wavefunction is given by

= n0(1, 0, 0). (146)The lowest-order transverse magnetizationF+(r, t) andF(r, t) are then given by

F+(r, t) =

2n00(r, t) =

2n0

k

eikra0,k(t), (147a)

F(r, t) = 2n0

0 (r, t) = 2n0k

eikra0,k(t), (147b)

where a0,kis the Fourier transform of the field operator0(r) =0(r) as defined in Eq. (15). Here, wereplaced operators1(r, t) and1 (r, t) in Eq.(144)by the condensate wavefunction 1=

n0.


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At the lowest order (first-order) approximation, by using the Bogoliubov transformation

b1,k= uka1,k vka1,k, (148)the Hamiltonian for the noncondensate part given by Eq.(12)can be effectively written as

H=k=0 h

1,kb

1,kb1,k +k

h0,ka

0,ka0,k +h1,ka

1,ka1,k

, (149)

whereh1;0,k are the lowest-order energy spectra of elementary excitations given by Eq.(56). Theground state is defined as the vacuum of annihilation operatorsb1,k, a0,kanda1,k:

b1,k|g = 0, (150)a0,k|g = 0, (151)a1,k|g = 0. (152)

If a spin-1 particle with momentum hk0and magnetic sublevel mF= 0 is created above the groundstate, the system is in an excited state given by

|e =a0,k0 |g. (153)We now calculate the equal-time spatial correlation of transverse magnetization in the system for thisplane-wave excited state. Using Eq.(147), we have

e|F+(r, t)F(r, t)|e = 2n0k,k

ei(krkr)e|a0,k(t)a0,k(t)|e

= 2n0k,k

ei(krkr)ei(0,k0,k )t

k,k+ k,k0 k,k0

= 2n0k eik(rr) + 2n0eik0(rr), (154)

where

a0,k(t) eihHt

a0,ke i

hHt = ei0,kta0,k, (155)

a

0,k(t) eihHt

a

0,ke i

hHt = ei0,kta0,k. (156)

Here, the first term in the last line of Eq. (154), which is proportional to(r r), describes the self-correlation, and it exists for all states including the ground state. Similarly, we have

e|F(r, t)F+(r, t)|e = 2n0k,k

ei(krkr)e|a0,k(t)a0,k(t)|e

= 2n0e

ik0(rr). (157)

Using Eqs. (154) and (157), we obtain the spatial correlation of transverse magnetization in thesystem:

e|F(r, t) F(r, t)|e g|F(r, t) F(r, t)|g

= 12

e|F+(r, t)F(r, t) +F(r, t)F+(r, t)|e

g|F+(r, t)F(r, t) +F(r, t)F+(r, t)|g

= 2n0cos k0 (r r) . (158)Here, we subtract the correlation in the ground state from that in the excited state to removethe self-correlation term in Eq. (154) which is not an observable. Eq. (158) shows that the


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elementary excitation given by Eq.(153)can be interpreted as a spatial modulation of the transversemagnetization, or a transverse spin wave.

Now let us assume that one particle is excited to create a superposition state of different momenta:

|esp

= d3kf(k)

a

0,k

|g

, (159)

wheref(k)is the weight of the superposition. The expectation value of the squared modulus of the

transverse magnetization densityF2(r, t) with respect to this excited state is given by

esp|F2(r, t)|esp g|F2(r, t)|g = 1

2

esp|F+(r, t)F(r, t) +F(r, t)F+(r, t)|esp

g|F+(r, t)F(r, t) +F(r, t)F+(r, t)|g

= n0k,k

ei(kk)rei(0,k0,k )tf(k)f(k) + c.c.

= 2n0

k

ei(kr0,kt)f(k)

2

. (160)

The expression inside the vertical bars in the last line of Eq.(160)is nothing but the time evolution ofa wave packet, which is initially constructed by a superposition of plane waves with a weight function

f(k). Although the results in this section are derived at the lowest-order approximation, as we move tothe second-order approximation, the physical properties of the elementary excitations do not change(see Sections3and4). Namely, the elementary excitation given by Eq.(153)always has a quadratic

dispersion relation and can be interpreted as a transverse spin wave. The only difference betweenthe first-order and second-order results is the enhancement factor for the effective mass of magnons(see, for example, Eqs.(56b)and(119)). Therefore, we can apply the time evolution of the transversemagnetization density for a spinor wave packet, which is given by Eq. (160), to the second-orderapproximation with just a replacement of the first-order energy spectrumh0,k= 0k qB by thesecond-order oneh(2)0,k=

1 49

na3/(45

)

0k qB.As an example, let us consider a Gaussian wave packet in one dimension, which is a superposition

state with spectral weightf(k) in momentum space given by

f(k) = ed2(kk0)2 . (161)

This Gaussian wave packet has a width of the order ofdin the coordinate space and the center ofmass moves with momentumhk0. Although a generalization to three dimensions is straightforward,a quasi-one-dimensional atomic system is relevant to the experiments of ultracold atoms whichare tightly confined in the radial direction. Now, we can see how a quadratic dispersion relation

h(2)0,k=h2 k2/(2Meff)withMeffgiven by Eq.(121)affects the propagation of a spinor wave packet.

Note that the energy gap qBin the energy spectrum has no influence on the squared modulus of thetransverse magnetization density because its contribution drops out upon taking the absolute valuein Eq.(160). We then have

k

ei(kx(2)

0,kt)

f(k)= V2

dk exp

i

kx hk

2

2Mefft

d2(k k0)2

= V2

22d2 + iht

Meff

exp

Meffx2 2id2k0(hk0t 2Meffx)4d2Meff + 2iht

, (162)


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a b

Fig. 11. Two-photon Raman transition used to transfer 87 Rb atoms between themF= 1 andmF= 0 magnetic sublevels intheF= 1 ground-state manifold 52S1/2. To produce a localized wave packet with transverse magnetization, atoms in a smallregion of the atomic cloud need to be transferred from the mF= 1 tomF= 0 spin state for the ferromagnetic phase (a), andfrom themF

=0 tomF

=1 spin state for the polar phase (b) (see Section5.2). Here, +() denotes the right (left) circularly

polarized light, and the linearly polarized light.

and thus,

k

ei(kx(2)

0,kt)

f(k)

2

22d2

1 + h2 t24d4M2

eff

exp

(x hk0t/Meff)2

2d2

1 + h2 t2

4d4M2eff

. (163)

From Eq. (163), it can be seen that due to the nonlinear dispersion relation, the transversemagnetization of a spinor wave packet expands in space during its propagation with a group velocityvg

=hk0/Meff. The time dependence of the width of the wave packet is also governed by the effective

massMeff:

d(t) = d

1 + h2 t2

4d4M2eff. (164)

Consequently, by measuring either the group velocity or the rate of expansion of the transversemagnetization of a spinor wave packet, we can find the effective mass of magnons.

To produce a spinor wave packet, which is a localized excited state as given in Eqs. (159) and (161),a small region of the atomic condensate can be exposed locally to a pair of laser beams which couplethe states in the ground-state manifold to the electronically excited states. Via a Raman transition afraction of the atoms in the mF= 1 sublevel are transferred locally into the mF= 0 state (see Fig. 11).As can be seen from Eqs. (85)and (92), the second-order energy spectra obtained in Section4 are

valid only in the low-momentum region 0p c0n0. Using the parameters of

87

Rb [54], the maximummomentumhpmaxis given by

pmax

8 an 107 m1. (165)Therefore, as a superposition of plane waves with momenta in the above region, the spinor wavepacket should have a width of the order of

x 1pmax

107 m. (166)The condition(166)turns out to be well satisfied with the parameters of laser beams used in theexperimental setup. The pair of laser beams is set to be perpendicular to the single laser beam used asthe trapping potential (see Fig. 12). The wavelength of the pair of laser beams that couple the ground-

state manifold to electronically excited states is of the order of 0 .5 m and their beam waist is acouple of the wavelength, i.e., of the order of a micrometer. Finally, using Eq. (164) and the parametersof 87Rb [54], we can estimate the time it takes for the spinor wave packet to expand to the entire


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Fig. 12. A schematic configuration of laser beams used to produce a wave packet with transverse magnetization. A pair of laser

beams along thex-axisare used to transfer atoms between the mF= 1and mF= 0 magnetic sublevels: one is linearly polarized( ) in the direction of the external magnetic field, which determines the quantization axis (z-axis), while the other is linearly

polarized in a perpendicular direction (e.g., the y-axis), which can be regarded as a superposition of two circular polarizations

(++ ). An additional single laser beam along the z-axis is used as a trapping potential.

atomic cloud. For an atomic cloud whose axial length is 100 m, the evolution time of the spinorwave packet is about 40 ms. It is well within the lifetime of the condensate, which is of the order of asecond. Furthermore, theenhancement of the effective mass of magnons as a consequence of quantumfluctuations is manifested by a change in the width of the spinor wave packet, which is of the orderof 1 m after 40 ms of propagation.

5.2. Polar phase

The condensate wavefunction for the polar phase is

= n0(0, 1, 0). (167)The lowest-order transverse magnetizationF+(r, t) andF(r, t) are then given by

F+(r, t)=

2n0

1 (r, t) +1(r, t)

=

2n0

k

eikra1,k(t) + eikra1,k(t)

= 2n0k eikr a

1,k(t)

+a

1,

k(t) , (168a)

F(r, t)=

2n0

1(r, t) +1(r, t)

=

2n0

k

eikra1,k(t) + eikra1,k(t)

=

2n0

k

eikr

a

1,k(t) +a1,k(t)

. (168b)

At the lowest-order approximation, by using the Bogoliubov transformations

a0,k= u0,kb0,k + v0,kb0,k, (169a)a1,k= u1,kb1,k + v1,kb1,k, (169b)a1,k= u1,kb1,k + v1,kb1,k, (169c)


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the effective Hamiltonian for the noncondensate part can be written as

H=k=0

h0,kb

0,kb0,k +

k

h1,kb

1,kb1,k +h1,kb1,kb1,k

, (170)

where b0;1,kare the annihilation operators of quasiparticles. As seen in Eq.(64a), there is a two-folddegeneracy in energyh1,k=h1,k due to the symmetry between the mF= 1 sublevels. Theground state is defined as th

annals of physics, 328 158 (2013)

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