Quantum phase diffusions of a spinor condensate

5 Pages·0.15 MB·English

by  S. Yi

Quantum phase diffusions of a spinor condensate
S. Yi, O¨. E. Mu¨stecaplıog˘lu, and L. You
School of Physics, Georgia Institute of Technology, Atlanta, GA 30332-0430, USA
(Dated: February 2, 2008)
Wediscussthequantumphasesandtheirdiffusioninaspinor-1atomicBose-Einsteincondensate.
For ferromagnetic interactions, we obtain the exact ground state distribution of the phase fluctu-
ations corresponding to the total atom number (N), the magnetization (M), and the alignment
(or hypercharge) (Y) of the system. The mean field ground state is shown to be stable against
these fluctuations, which dynamically recover the two continuous symmetries associated with the
conservation of N and M as in current experiments.
3 SincetheobservationofBose-Einsteincondensationof weakmagneticfieldB(asalwaysexistsinanexperiment)
0
trapped atomic clouds [1, 2], the coherence properties of fixesthe quantizationz-axissuchthatquadraticZeeman
0
thecondensatehasbecomethefocusofmanytheoretical effect can be neglected. In the second-quantized form,
2
studies [3, 4, 5, 6, 7, 8, 9]. Within the mean-field theory, the Hamiltonian is [23, 24]

n it is commonly assumed that the condensate can be de-
a scribedbyaU(1)symmetrybreakingfield[10,11],equiv- ¯h2 2
J H = d~rψ†(~r) ∇ +V (~r) ¯hω F ψ (~r)
alenttoacoherentstateassumptionofthe groundstate. i − 2M ext − L z i
2 (see Refs. [10, 11] for discussions of U(1)-symmetric ap- Xi Z (cid:20) (cid:21)
1 c
1 porreotaicchaelsu).ndAerltshtaonudgihngqutiotemsaunccyesesxfupleriinmpenrotavlidoinbgsetrhvae-- + 20 i,j Z d~rψi†(~r)ψj†(~r)ψi(~r)ψj(~r)
v tions,suchacoherentstateassumptionisnotnecessarily X
c
8 consistent with real experimental situations, where the + 2 d~rψ†(~r)ψ†(~r)F~ F~ ψ (~r)ψ (~r), (1)
7 fluctuations of the atom numbers are difficult to control 2 i,j,k,lZ i j ik· jl l k
1 X
[12, 13]. A coherent state leads to a Poisson distribu-
1
0 tion of atoms. For a ground state with average of N where ψj(~r) (j = +,0,−) denotes the annihilation op-
atoms, the associated number fluctuations are of the or- erator for the j-th component of a spinor-1 field. The
3
0 der √N. As wasinitially pointedout in Refs. [3, 4], this trappingpotentialVext(~r)isassumedharmonicandspin-
/ number fluctuation of a coherent state condensate leads independent. The Larmor precessing frequency is ωL =
t
a to the “diffusion” (or spreading) of its initial phase. In BµB/¯h with µB the magnetic dipole moment for state
m a scalar condensate, this diffusion, a dynamic attempt F = 1,MF = 1 . The pseudo potential coefficients are
- to restore the U(1) symmetry of the interacting atomic c|0 = 4π¯h2(a0 +i2a2)/3M and c2 = 4π¯h2(a2 a0)/3M,
d system, can be studied in terms of a zero mode, or the with a0 (a2) the s-wave scattering length for−two spin-1
n
Goldstone mode of the condensate [3]. More physically atoms in the combined symmetric channel of total spin
o
meaningful discussions in terms of the relative phase of 0 (2). M is the mass of atom, and F~ is the spin 1 oper-
c
: twocondensateswerestudiedsoonafterwards[5,6,7,8]. ator [25]. Hamiltonian (1) is invariant under U(1) gauge
v
Experimentally, starting with the remarkable direct ob- transformationeiθ andSO(3)spinrotations (α,β,τ) =
Xi servation of the first order coherence in an interference e−iFzαe−iFyβe−iFzτ (for B = 0) [23]. A noUn-zero B or
r experiment [14], direct correlationsbetween number and theconservationofmagnetization =N+ N−reduces
a phase fluctuations were observed with a condensate in the SO(3) to its subgroup SO(2) gMenerated−by e−iFzα.
a periodic potential [15], and more recently, in the re- Following the Bogoluibov theory we assume there ex-
markable Mott insulating state [16] obtained by loading istthree‘large’condensatecomponentsφ aroundwhich
j
a superfluid condensate into an optical lattice [17]. westudythe smallquantumfluctuations(off-condensate
The emergence of spinor-1 condensates [18, 19] (of excitations) via
atoms with hyperfine quantum number F = 1) has cre-
ated new opportunities to understand quantum coher- ψ (~r)=√Nφ (~r)+δψ (~r), (2)
j j j
ence and the associated number/phase dynamics in a
three component condensate [20, 21]. In this paper, we where N d~r φ 2 = N is the number of condensed
j j
| |
investigate the quantum phase dynamics of a spinor-1 atomsincomponentj. Atnearzerotemperatures,higher
R
condensate due to atom number fluctuations. We will than quadratic terms of δψ (~r) in the Hamiltonian (1)
j
focus on ferromagnetic interactions, when the conden- are neglected. The assumed mean field ground state
sate wave functions for the three spin components share √Nφ (~r) breaks both continuous symmetries U(1) and
j
thesamespatialmode[22];inthiscase,thephasefluctu- SO(2), thus we expect to observe multiple zero energy
ations translate into fluctuations of the direction of the Goldstone modes [3].
macroscopic condensate spin. In the first order, we obtain the usual coupled Gross-
We consider a system of N spin-1 bosonic atoms in- Pitaevskii equation (GPE) for the condensate modes
teracting via only s-wave scattering [23, 24, 25, 26]. A φ (~r). The quantum fluctuations δψ obey the usual
j j
2
Bogoluibov-deGennesequations(BdGEs)[27]. Thecon- Similarly, the phase functions share the same spatial
densatenumberfluctuations,canbe studiedthroughthe mode and can be generally expressed as
number fluctuation operators [3]
θj(~r,t)=√njθ(~r)e−i(µ−jh¯ωL)t/h¯+iαj, (9)
Pj ≡ d~r φ∗j(~r)δψj(~r)+φj(~r)δψj†(~r) with θ(~r) governed by the following equation
Z h i
≈ √1N Z d~r[ψj†(~r)ψj(~r)−Nφ∗j(~r)φj(~r)]. (3) −¯h22M∇2 +Vext−µ+3c+Nφ2 θ(~r)=Nu˜φ(~r). (10)
(cid:20) (cid:21)
Using the GPEs and the BdGEs [27] their equations of
The normalization d~rφ(~r)θ(~r) = 1/2 determines the
motion are [3],
Goldstoneparameteru˜,whichintheThomas-Fermi(TF)
limitisu˜=[c /(4πaR3 /3)](a /15a )3/5N−3/5,witha
i¯hP˙ = i¯hP˙ /2, (4) + ho ho 2 ho
± − 0 the ground state size (h¯/Mωho)1/2 of a harmonic trap
with a frequency ω =(ω ω ω )1/3.
from which we find that both P = P +P +P and ho x y z
tot + 0 −
Equations (8) and (10) are identical to their counter-
P P are constants of motion, an obvious outcome
+ −
− parts for a scalar condensate [3]. In the ferromagnetic
since the Hamiltonian (1) commutes with operators of
ground state, individual atomic spins align along the
both the total number of atoms and magnetization.
same direction, thus they collide only in the symmetric
We define the phase operator
total spin F = 2 channel. To conserve the magnetiza-
tionwithrespecttotheB-fielddirection,thecondensate
Qj =i¯h d~r θj∗(~r)δψj(~r)−θj(~r)δψj†(~r) , (5) spin direction is simply tilted at an angle θ = cos−1m.
Z h i If this direction is taken as the quantization axis, then
with θ (~r) the associated phase mode functions. De- the spinor condensate behaves essentially as a scalar one
j
note Q = Q + Q + Q , the canonical quantiza- [22, 23, 30] when the B-field is zero.
tot + 0 −
tion condition [Q ,P ] = i¯h is satisfied if the con- With the mode functions for atom number and phase
tot tot
straint J + J + J 1 is enforced, where J fluctuations, we find the zero mode dynamics can be ex-
+ 0 − j
d~r θ∗(~r)φ (~r)+c.c. .≡As for a scalar or binary con≡- pressed in terms of an associated Hamiltonian. After
j j
densate [6] laborious calculation we arrive at [28]
R (cid:2) (cid:3)
dQtot =Nu˜P . (6) Hzero = p′T p′+q′T q′, (11)
dt tot N A B
We note that [Qj,Pk] = i¯hδjkJj, [δψj(~r),Pk] = with p′T = (p′+,p′0,p′−), q′T = (q+′ ,q0′,q−′ ), redefined
δδtojψkjdφ(ej~rfi()~rn≈)e,taθhnje(d~rc)o[Pδmψjp/jl(Je~rjt)e+,Qdφykjn](a=~rm)Q−icjie/¯hqiδ¯hujJkaθjtjio[(6~nr,)s.8fo]T,rhwtehhseiecnhlueahmdeblpteosr aQmsja/tcJraijnc.eosnAiicnav&lolylBvicnoagnrejountgwlayotepHdaervarammriietatibealrnessaopnf′jdth≡peossPyitjsitveaemn-dd.efiTqjn′hiut≡es
and phase fluctuations [28]. Hzero 0, the assumed ground state, a mean field sym-
≥
Including boththe initialandtime phases,the ground metry breaking state with coherent condensate ampli-
state wave functions can be generally expressed as tudes √Nφj(~r) is stable. The associated quantum fluc-
φj(~r)e−i(µ−jh¯ωL)t/h¯+iαj withacommonchemicalpoten- tuations of atom numbers can be studied with the lin-
t|ial µ,|a Lagrange multiplier enforcing the conservation earization approximation Eq. (2). Matrices &
of atom numbers. Minimization of the total energy Eq. are simplified with φθ d~rφ2(~r)θ2(~r) andAφφ B
O ≡ O ≡
i(n1)Rleeaf.ds[2t2o]αth+a+ttαh−e−st2eαad0y=st0a[t2e9s].olIuttwioanstfaukrethsetrhpersoavmene 2 d~dr~φrθ4((~r~r)). Weµa+lsoc fiNndφ2u˜(~rR)=θε(~r+)a4cn+oOn-φnθegwatitivheNquεan=-
R L− +
spatial mode for each of its three spin components (for tity.
R (cid:2) (cid:3)
ferromagnetic interactions), i.e. We note that matrix is diagonalized by an orthog-
onal transformation q B= (q′ + q′ + q′ )/√3, q =
N + 0 − M
φj(~r)=√njφ(~r)eiαj, (7) (q+′ −q−′ )/√2, and qY = (q+′ −2q0′ +q−′ )/√6 [31]. The
corresponding p , p , and p are respectively fluctua-
N M Y
where the real-valued mode function φ(~r) is normalized
tions of the total number of atoms, the magnetization,
to unity, and governedby an equivalent scalar GPE
and the alignment. With these new collective operators,
the zero mode Hamiltonian becomes
¯h2 2
∇ +V (~r)+c Nφ2(~r) φ(~r)=µφ(~r), (8)
(cid:20)− 2M ext + (cid:21) HNzero = ap2N +bp2M +cp2Y
ofascatteringlengtha2 (notec+ =c0+c2 ∝a2). Define +αpNpY +βpMpY +γpNpM +ηqY2,(12)
n =N /N andm= /N astherelativeatomnumbers
j j
and relative magnetizMation, n = (1 m)2/4 and n = where all the coefficients are listed in the Appendix. Re-
± 0
(1 m2)/2 in the ferromagnetic grou±nd state [22, 25]. placing p by i¯h∂/∂q , Hamiltonian (12) leads to the
j j
− −
3
2 (12) for xT(t) = [pN(t),pM(t),pY(t),qN(t),qM(t),qY(t)]
can be used. This solution has a simple structure [28];
in addition to oscillating terms of the forms cosΩt and
)
⊥
ω 1 sinΩt, the phase fluctuations of qN and qM also contain
(
Ω diffusion terms proportional to Nt, which indicates the
linearizationapproximationof Eq. (2) is valid only for a
finite duration.
0
Finally, let’s consider the diffusion of the direction of
2 thecondensatespinf~(~r,t) ψ†(~r,t)F~ ψ (~r,t). Be-
≡ ij i ij j
cause all three spin components share the same spa-
2Y0 tial mode function [22], indiPvidual spins of Bose con-
∆ 1
densed atoms are parallelfor ferromagnetic interactions,
20
n
i.e. they act as a macroscopic magnetic dipole pointing
alongthesamedirection(independentofthespatialcoor-
1001 102 103 104 105 106 dinates): f~0(~r,t) ∝ (√1−m2cosΘ,−√1−m2sinΘ,m)
while undergoing precessing with respect to the z-axis.
N
Θ = α α +ω t. As the phase dynamics attempts
+ 0 L
−
to restorethe U(1)andSO(2)symmetries ofthe system,
FIG. 1: The N-dependence of Ω and 3n20∆2Y0 for a 87Rb theirrespectiveinitialvaluesbecomeirrelevant. Thefluc-
condensateinacylindrically symmetrictrap withωx =ωy = tuation becomes δf~(t) = d~r[f~(~r,t) f~(~r,t)] with a
ωλ⊥==1.0(2(dπa)s1h0e0d(lHinze)),aannddωλz==10λ.0ω⊥(d.asλh-=dot0t.e1d(lsinolei)d. line), zeroaverage. UsingσN2 ,σM2R,andσY2 to−den0otetheinitial
variances of N, , and Y, we find in spherical coordi-
nates (rˆ,θˆ,φˆ) (nMote sinθ = √1 m2), δf (t) δf (0)
r r
− ≡
and δf (t) δf (0). Both are fixed constants
ground state distribution of fluctuations θ θ
≡
1 q2 [δfr(t)]2 = 3σN2 , (14)
ϕ (q ,q ,q )= exp Y , (13)
0 N M Y √π¯h∆Y0 −2h¯2∆2Y0! D[δf (t)]2E = 1 3m2σ2 +2σ2 , (15)
θ 1 m2 N M
with a width p D E − (cid:0) (cid:1)
(duetoconservationsofN and )whenrespectivefluc-
M
1/2 tuations in N, , and Y are uncorrelated. In the az-
1 ε 4c
∆2Y0 = 3n2 −c 2Oφθ , imuthal φˆ directMion a simple phase diffusion results
0 (cid:18) − 2Oφφ (cid:19)
δf (t)=δf (0) R t, (16)
and the zero point energy E0 ¯hΩ/2 = N[ c2 φφ(ε φ φ − d
≡ − O −
4t−ac(k2cOi2n/φgπθt)ah]3h1eo/2)s(.mahaoWl/laeε2)fi3n/50dN).n2F/205∆ig2Yuinr0et=1heshpTowF7/s1lsi0me/lei3tcta(enadndrdeΩsaullst∝os aδwfnitdh(0a)δ,df(iφwff(uh0si)ciohn=carant√bee1Re−dxpm=la22i(cid:2)nε√eNd2δqifnMθt((e00r))m/+¯hs,√opf6rtmohpeqoYsrit(ni0go)ln(cid:3)ea/al2xth¯ios,
for the N-dependen→ce of Ω and ∆2 . While Ω increases twθisting of T2 in the isospin subspace of a spior-1 con-
Y0 z
monotonically with N, the ground state width ∆2 for densate [31]). The N- and λ-dependence of the diffusion
Y0
qY peaks at some intermediate values of atom numbers parameterNεareshowninFig. 2fora87Rbcondensate.
andeventuallysaturatesinthelimitoflargeN (orstrong In the TF limit and applying the results of Ref. [32], we
interactions). findthatε=(4h¯2/5MR2)ln(R/1.3a ) N−2/5,gener-
ho
The above ground state distribution of Hzero can eas- ally a very small quantity. R=aho(15a2∝/aho)1/5N1/5 is
ily be understood. For a condensate with fixed total theTFradius. Typically,thedephasingrateisafraction
number of atoms and magnetization, the conservations ofω for acondensateof 105 atomsandλ 1,i.e. its
⊥
of N and require p2 = p2 = 0, which lead to macroscopic spin direction∼is lost in a few cy∼cles of trap
q2 = q2M= , i.e.h, cNoimplehteMlyidiffused phases; The oscillation. Figure2alsoindicatesthatthefasteraferro-
h Ni h Mi ∞
ferromagnetic interaction, nevertheless, prepares a cor- magnetic condensatediffuses its pointing directionalong
related ground state such that both distributions for pY theprecessiondirection,thelargerandmoretightlycon-
and qY take a Gaussian form with hp2Yi = 1/2∆2Y0 and fined a condensate is.
q2 = h¯2∆2 /2. Such a distribution in p and q is WhenB =0,zeromodedynamicsbecomesmuchsim-
h Yi Y0 Y Y
in fact the minimal uncertainty coherent state consis- pler if the condensate spin direction is taken as z-axis.
tent with the symmetry breaking ground state. How- AlthoughtherestillexistmorethanoneGoldstonemode
ever, it is experimentally difficult to produce such a in this case, the mean field ground state corresponds to
condensate having fixed atom numbers and phase fluc- all atoms in state + [23, 30]. Thus to leading order,
| i
tuations. For any initial state, the dynamic solution the only relevant fluctuation is that of the total num-
x(t) = (t)x(0) as governed by the Hamiltonian Eq. berofatoms(correspondingtothe U(1)symmetry),and
T
4
3 coefficient for the pointing direction of the condensate
λ=0.1 spin and recovered its small-time quadratic t-dependent
λ=1 spreading.
)⊥ 2 λ=10 This work is supported by the NSF grant No. PHYS-
ω 0140073and by a grantfrom NSA, ARDA, and DARPA
h
( under ARO Contract No. DAAD19-01-1-0667.
ε
N 1
APPENDIX A: COEFFICIENTS IN EQ. (12)
0
1 2 3 4 5 6
10 10 10 10 10 10
ThecoefficientsinthezeromodeHamiltonian(12)are
FIG. 2: The same as in Fig. 1 but for Nε.
1 ε
a = (5+3m2)+2 3c (1 m2)2
4n2 3 Oφθ 0 −
0
the phase diffusion dynamics becomes the same as in a h+8c (1 3m2) , (cid:2)
2
scalar condensate [3]. Our theory as developed in this 3 −
paper does reduce to this simple limit. 1 (cid:3)i
b = ε(1+m2) 8c m2
Inconclusion,wehavestudiedindetailquantumphase 2n2 − 2Oφθ
0
diffusionsofaspinor-1condensatewithferromagneticin- 1 (cid:2) (cid:3)
teractions. The condensate ground state is very simple, c = 3n2(ε−4c2Oφθ),
all three spin components have the same spatial mode 0
function and their associated phase functions are also α = √2(1+3m2)(ε 4c ),
identical. We have constructed the zero mode Hamilto- 6n2 − 2Oφθ
0
nian for the condensate number and phase fluctuations, 2m
β = (ε 4c ),
andsolvedforthegroundstatedistributionofthesefluc- −√3n2 − 2Oφθ
tuations when both N and are conserved. Further- 2m 02
more, we have obtained anaMlytically the dynamic num- γ = −n2 3 ε−c2Oφθ(1+3m2) ,
ber andphase fluctuations relating to both the quantum 0 r
phase diffusion and the initial distribution of these fluc- η = −3c2n20Oφ(cid:2)φ/¯h2. (cid:3)
tuations. We have identified a quantum phase diffusion
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−4
x 10
6
λ
=0.1
λ
=1
4
λ
=10
)
⊥
ω
h
(

ε
2
0
2 4 6
10 10 10
N