Relationship between spin squeezing and single-particle coherence in two-component Bose-Einstein condensates with Josephson coupling

0.44 MB·English

by  G. R. Jin

Relationship between spin squeezing and single-particle coherence in two-component
Bose-Einstein condensates with Josephson coupling
G. R. Jin1,2 and C. K. Law1
1Department of Physics and Institute of Theoretical Physics,
The Chinese University of Hong Kong, Shatin, NT, Hong Kong SAR, China
2 Department of Physics, Beijing Jiaotong University, Beijing 100044, China
(Dated: January 8, 2009)
We investigate spin squeezing of a two-mode boson system with a Josephson coupling. An exact
relation between the squeezing and the single-particle coherence at the maximal-squeezing time is
9 discovered, which provides a more direct way to measure the squeezing by readout the coherence
0 in atomic interference experiments. We prove explicitly that the strongest squeezing is along the
0 Jz axis, indicating the appearance of atom number-squeezed state. Power laws of the strongest
2 squeezing and theoptimal coupling with particle numberN are obtained based upon a wide range
of numerical simulations.
n
a
PACSnumbers: 03.75.Mn,05.30.Jp,42.50.Lc
J

8
I. INTRODUCTION ibility of the interference fringe [8, 9, 10, 11, 12].
]
r Our paper is organized as follows. In Sec. II, we in-
e Spin squeezing is a nonclassical effect of collective troduce theoretical model and derive some formulas for
h
t spin systems [1, 2, 3, 4], showing reduced spin fluc- the single-particle coherence and the squeezing param-
o tuation in one certain spin component normal to the eter. In Sec. III, quantum dynamics of the coherence
t. mean spin. Kitagawa and Ueda proposed spin squeez- and the squeezing are investigated for the OAT and the
a ing generated by the self-interaction Hamiltonian H = JLC models, respectively. In Sec. IV, we present exact
m 1
2κJ2,duetothe so-calledone-axistwisting(OAT)effect relationbetweenthe coherenceandthespinsqueezingat
d- [1].zThe OAT-type spin squeezing could be realized in the maximal-squeezing time t0. In Sec. V, power rules
n weakly interacting Bose-Einstein Condensate (BEC) [5], of the optimal coupling and the strongest squeezing as
o or atomic ensemble in a dispersive regime [6]. The self- a function of particle number N are investigated based
c interaction H also leads to phase diffusion of the BEC upon a wide range of numerical simulations. Moreover,
1
[ [7], which indicates a decay of single-particle coherence we compare numerical result of t0 for the optimal cou-
2 [8, 9, 10, 11, 12, 13]. pling case with its analytic solution. Finally, a summary
v Beyond the OAT model, an Josephson-like coupling of our paper is presented.
7 (JLC) term ΩJ was added to the Hamiltonian H with
x 1
5
purposetocoherentlycontrolthephasediffusion[13]and
9
the spin squeezing [14, 15, 16]. It was shown that the II. THEORETICAL MODEL AND SOME
1
FORMULAS
. JLCmodel[seeEq.(1)]resultsinstrongreductionofspin
8
fluctuation along the z (i.e., J ) axis, provided that the
0 z
additional field is tunned optimally [15]. We found the To begin with, we consider a two-component weakly-
8
maximal-squeezing time t of the JLC model, and pro- interacting BEC consisting of 2j atoms in two hyperfine
0 0
: posed a simple scheme to store the strongest squeezing states 1 and 2 coupled by a radio-frequency (or mi-
v | i | i
alongthez axisforalongtime[16]. Sofar,thereremain crowave) field [18, 19]. A tightly-confined BEC can be
Xi certainquestionsunsolved: Isthereanyrelationbetween described by the JLC Hamiltonian (~=1) [20]
thesqueezingandthe single-particlecoherence? Inaddi-
r
a tion,towhatdegreecanthestrongestsqueezingreachin H2 =ΩJx+2κJz2, (1)
the JLC model? The first question is important because
it relates to measurement of the squeezing. where the angular momentum operators J+ = (J−)† =
In this paper, we present an exact relation between aˆ†2aˆ1, Jz = (aˆ†2aˆ2−aˆ†1aˆ1)/2 obey the SU(2) Lie algebra.
the squeezing and the coherence by solving the Heisen- ThetotalparticlenumberN =aˆ†aˆ +aˆ†aˆ isaconserved
1 1 2 2
berg equation. Our results show that local minima of quantity. In Eq.(1), we have neglected the term propor-
thesqueezingandthecoherenceoccursimultaneouslyfor tional to J by assuming equal intraspecies atom-atom
z
the coupling Ω larger than its optimal value Ω0. Un- interaction strengthes [5]. The Rabi frequency Ω can be
like the OAT scheme, where number variance ∆Jz is controlledby the strengthofthe externalfield. The self-
time-independent, we prove explicitly that the squeez- interaction term 2κJ2 leads to spin squeezing, which is
z
ing at time t0 is along the z axis in the JLC model quantified by a parameter [1]:
[17]. The strongest squeezing obeys the power law
ξ0 = ∆Jz(t0)/ j/2 ∝ N−1/3, which can be measured ξ = √2(∆Jn)min, (2)
by readoutthepsingle-particlecoherencethroughthe vis- j1/2
2
where j = N/2 and (∆Jn)min represents the minimal p(m+)(0) = p(m+)(t) = 0, i.e., c−m(t) = cm(t). Quantum
−
variance of a spin component Jn = J~ n normal to the dynamicsofthe oddN systemdepends onthe equations
mean spin J~ . The coherent spin sta·te (CSS), defined ofpm(−). The aboveprocesseshavecertainadvantagesto:
h i
formally as [21] (i)reducethetotalHilbertspacedimensionfrom2j+1to
j+1(evenN)orj+1/2(oddN);(ii)Sincec = c ,
−m m
±
θ,φ =e−iθ(Jxsinφ−Jycosφ) j, j (3) we obtain Jy = Jz =0 and Jx = J+ =0, i.e., the
| i | − i h i h i h i h i6
mean spin is always along the x axis. Actually J is a
+
h i
has the minimal variance (∆Jn)min = j/2 and ξ = 1. real function; (iii) The correlation function is simplified
Therefore,a stateis calledspinsqueezepdstate ifits vari- as g(1) = j−1 J , and the spin component normal to
ance is smaller than that of the CSS, i.e. ξ <1. Besides the 1m2ean spin|hisxiJ|n = J n = Jysinθ Jzcosθ. The
the squeezing, the self-interaction 2κJz2 also leads to the varianceofJn is(∆Jn)2 =·hJn2i−hJni2 =−(C−Acos2θ−
phase diffusion, which indicates a decay of the single- Bsin2θ)/2, where A = J2 J2 , B = J J +J J ,
h y − zi h z y y zi
pbyarotffic-ldeiacoghoneraelnecleem. Senutcshoaftkhinedsionfgcleo-hpearretniccleeidsemnseiatysumread- andC =hJy2+Jz2i. Fromtherelation ∂θ(∆Jn)2 θmin ≡0,
we get tan(2θ ) = B/A and the minimal(cid:12)variance
tdruixceρs(ij1t)he=fihras†ita-ojrid/Nerwteimthpio,rjal=co1r,r2e.laFtoiormnafullnyc,toinonei[n1t3r]o:- (∆Jn)2min = Cm−in√A2+B2 /2. (cid:12)
(cid:0) (cid:1)
ρ(1) III. QUANTUM DYNAMICS OF THE
g(1) = (cid:12) 12(cid:12) |hJ+i| , (4) COHERENCE AND THE SQUEEZING
12 (cid:12)(cid:12)ρ(1)ρ(cid:12)(cid:12)(1) ≡ j2 J 2
q 11 22 q −h zi
The OAT model H (i.e., Ω = 0) can be solved ex-
1
whichisobservableinexperimentsby extractingthe vis- actly in Heisenberg picture [1], with its analytic results:
ibilityoftheRamseyfringes[8,9,10,11,12]. Oneofthe A = (j/2)(j 1/2) 1 cos2j−2(4κt) , B = j(2j
− − − −
goals of this paper is thereby to present the relation be- 1)sin(2κt)cos2j−2(2κt(cid:2)), and C =j+A(cid:3)due to the time-
tween the squeezing ξ and the first-order coherence g(1). independent variance J2 = j/2. The strongest (op-
Let us first examine the exact numerical solutions12of timal) squeezing ξ =h zξ(it ) (4/3)1/6N−1/3 occurs
0 s
the time-dependent Schr¨odinger equation governed by at time t 61/6N−2/3/2. T≃he single-particle coher-
s
≃
the JLCHamiltonianH2. We considerthatthe spinsys- ence g(1)(t) = cosN−1(2κt) e−(t/td)2 with the phase-
ete−miπJsyt/a2rtjs,frjom, athpearlotiwcueslatreiCgSenSv,eEctqo.r(3o)f,Jwxi,th|j,θ−=jixπ/=2 diffusio1n2 time κtd = (2N)−≃1/2. Obviously, g1(12)(td) =
| − i e−1g(1)(0) = 1/e. The coherence g(1)(t) has been mea-
and φ = π. Such an experimentally realizable state can 12 12
sured in experiment by extracting the visibility of the
be prepared by applying a two-photon π/2 pulse to the
Ramsey fringe [12]. As shown in Fig. 1(a), the opti-
ground state j, j with all the atoms in the internal
| − i mal squeezing occurs within the coherence time due to
state 1 [5, 12, 18]. The spin state at arbitrary time t
cpalintubde|esiexcmpanodbeedy aics˙m: |=Ψiε=mcPm +mcXm−|mj,cmmi+,1a+ndXtmhecma−m1-, tts0 <=tπd/.(4Mκo),reaonvderr,etchoevecrohtoereunnciteyga1(12t)2dte0ca[nyosttoshzoewrno aint
where ε = 2κm2, X = Ω (j+m)(j m+1) with Fig. 1(a), see Refs.[7, 13]].
m m 2 − Exceptfor N =2,3 cases,the JLC model (Ω=0) can
Xc −(j0)== 0j.,mTjh,ejamp=litu(−d1ep)sj+omf t2hje 1in/2it.ialOCbvSiSousalrye, not besolvedexactly[15,16,23]. Numericalsim6 ulations
m h | − ix 2j j+m of the single-particle coherence g(1)(t) and the squeezing
c (0) = c (0) for even N and (cid:0)c ((cid:1)0) = c (0) for 12
−m m −m − m ξ(t) are presented in Fig. 1(b)-(d) for N = 40 and vari-
odd N, which gives the expectation value J (0) = 0
and the variance J2(0) = j/2. Note thathsozmeirefer- ous Ω. Similar with the OAT case, the coherence g1(12)(t)
ences adopt the Hhamziltoinian H = ΩJ +2κJ2 to in- collapsestoitslocalminimumatt0 thenrevivespartially
3 − x z atabout2t , andthe maximalsqueezingoccurs att for
vestigate the BEC in a double-well potential [20], which 0 s
a small coupling Ω = 2κ [Fig. 1(b)]. Two time scales t
corresponds to H for Ω<0 case. In this work, we con- 0
2
andt tendtomergewiththeincreaseofΩ. ForΩ Ω ,
sider only positive Ω and κ by assuming repulsive atom- s ≥ 0
(1)
atominteractions. IfeitherΩorκisnegative,ourresults both the coherence g12 and the squeezing ξ reach local
remain valid by using initial state j,j x [22]. minimaatthesametimet0[Fig.1(c)and(d)]. HereΩ0is
Since c (0) = c (0) and X| i= X , we in- the optimal coupling to produce the strongest squeezing
−m −m ±m ∓m+1
troduce linear com±binations of the amplitudes p(±) = in the JLC model, such as Ω0 =4.2405κ for N =40.
m
It should be mentioned that the spin state at t ex-
c c . For even N case, one can derive a closed 0
m ± −m hibits a very sharp probability distribution and a strong
set of equations for p(−). However, all p(−)(0) = 0
m m reduction of the number variance ∆J [16]. The loss of
lead to pm(−)(t) = 0, thus c−m(t) = cm(t). As a re- thecoherence(orvisibility)asanevideznceofthenumber
sult, dynamical evolution of the even N system is deter- squeezing at time t , as shownin Fig.1(b)-(d), has been
0
(+)
minedsolelybytheequationsoftheamplitudesp with observed by Orzel et al. [8]. Moreover, our results show
m
m = 0,1,...,j. Similarly, for the odd N case, we obtain thatthe phasediffusion is suppresseddue to the appear-
3
1.0 between g1(12) and ξ
0.5 gx (112) ξ2(t0)= 2(cid:10)Jz2j(t0)(cid:11) =1− Ωκ h1−g1(12)(t0)i, (6)
1/e
which is valid for arbitrary Ω. For instance Ω = 0,
(1)g120.0 ts td (a) ts t0 (b) g(1) = 0 and ξ = 1 at t = π/(4κ); while for Ω > Nκ,
x, 1.0 fr1o2m Eq. (6) we get g(10)(t ) ξ(t ) 1; two trivial
12 0 ≃ 0 ≃
results due to weak squeezing. Hereafter, we focus on
the coupling around its optimal value Ω , with which
0.5 0

the strongest squeezing ξ = ξ(t ,Ω ) can be obtained
0 0 0
at time t . According to Eq. (6), one can measure ξ by
0.0 t0 (c) t0 (d) readoutt0he coherence g(1) in atomic interference exp0eri-
0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 12
ments [8, 9, 10, 11, 12].
t (k -1) t (k -1)
FIG. 1: Time evolution of the squeezing parameter ξ (solid 20
thicklines)andthesingle-particlecoherenceg(1)(t)(thinlines (a) (e)
12 0
with open circles) for N = 40 and various Rabi frequencies:
(a)Ω=0,(b)Ω=2,(c)Ω=Ω0 =4.2405(optimalcoupling), -20
(d) Ω=8. Time t is in units of κ−1, and the unitsof Ω is κ. 20 (b) (f)
0

pppp -20
f 20
(c) (g)
ance of number-squeezed state at t , consistent with the
0 0
experimental observations [11]. In experiments so far,
however, the number squeezing were detected through -20
20
the observation of increased phase fluctuation ∆φ, or (d) (h)
through an increased phase-diffusion time [24]. In fact, 0
the numberfluctuationhasanontrivialrelationwiththe -20
phase fluctuation, complex even for a single-mode light 0 ffff1/p 2 0 ffff1/p 2
field [25]. Consequently, a more direct way to measure
FIG. 2: (color online) Husimi Q function in the phase space
the number fluctuation ∆J is necessary.
z (φ,p ) for various times: (a) t = 0, (b) t = 0.045, (c)
φ
t = t = 0.09064 (the optimal squeezing), (d) t = 0.135,
0
(e) t = 0.183127, (f) t = 0.225, (g) t = 0.26988, and (h)
t = 0.315, corresponding to the times indicated by the ar-
IV. EXACT RELATION BETWEEN THE
rowsinFig.1(c),respectively. OtherparametersareN =40,
COHERENCE AND THE SQUEEZING AT THE
Ω=Ω =4.2405κ, and time t is in unitsof κ−1.
MAXIMAL-SQUEEZING TIME 0
There exists an exact relation between the coher- The time scale t can be obtained based upon the
0
ence g(1) and the squeezing ξ at time t . To see phase model [16]. By replacing J p = i∂ and
12 0 z → φ − φ
th2isκ,(JleztJyus+eJxyaJmz)inaendtheJ˙zH=eiseΩnJbye.rgTehqeuafitriostnse:quJa˙xtio=n JasxH→φj=co2sκφp,2φth+eΩJjLcCosHφam[2i6lt],onwihanereHφ2 cisanreblaetirveewprihttaesne
−
gives the relation between the coherence g(1) and the between two bosonic modes. The phase model allows us
12 to treat the spin system as a simple classical pendulum
squeezing angle θ
min
oscillatingaroundtheminimumoftheMathieupotential
cosφ,i.e.,φ=π. IntheJosephsonregime1<Ω/κ N,
dg(1) =2κAj−1tan(2θ ). (5) thependulum rotatesinthephasespace(φ,p )wit≪hthe
dt 12 min φ
effective frequency ω = √2κΩN [16, 27]. To illustrate
eff
this motion, we calculate the Husimi Q function
As g(1) reachesits localminimumatt ,(dg(1)/dt) 0,
12 0 12 t0 ≡
thenthesqueezingangleθmin =0(orπ)providedA=0. Q(θ,φ)= θ,φΨ(t) 2 (7)
Combining the two Heisenberg equations, we obtain6fur- |h | i|
ther dJ2/dt=Ω(J J +J J )= (Ω/2κ)dJ /dt, which in the phase space (φ,p ), where p = jcos(θ) de-
yields zJ2 = λ Ωz Jy /(2yκz) with−the integrxal constant scribes the population imφbalance of tφhe tw−o modes, and
h zi − h xi
λ. FortheinitialCSS j, j ,wehaveλ=j(1 Ω/κ)/2. the polar angle φ represents the relative phase [28]. The
x
| − i −
At time t , B = 0 and A> 0, and the minimal variance CSS θ,φ is given in Eq.(3). As shownin Fig. 2(a), the
0
t(h∆eJzn)a2mxinis=[17(C]. −AsAa)/r2es≡ulhtJ,z2wie, io.be.t,atinheassqiumepezleinrgelaaltoionng QPofiussn|ocntiodinistisriabuctirioclneofofrtthheeniunmitibaelrCvSaSr,iawnhciech(∆reJpr)e2seanntds
z
4
the phase uncertainty (∆φ)2 [12]. As time increases, it
becomes an elliptic shape [Fig. 2(b)], rotating clockwise

in the phase space. After a duration t , the ellipse elon-
0 60
gates horizontally corresponding to the optimal squeez-
ing of (∆Jz)2 [Fig. 2(c)]. It seems reasonable to suppose 50
that the motion of the ellipse is consistent with that of

the pendulum. Infact,the trajectoryofthe pendulum is 40 60
justpassingthroughthe horizontalaxis(p =0)attime 50
φ
T/4, where T = 2π/ωeff is the period of the pendulum. 030 40
As a result, we get the maximal-squeezing time [16]
0
20 30 Num.
Linear Fit
1/3
T π κ 10 0=N
κt0 ≃κ4 = 2r2ΩN, (8) 20104 N 105
0
0 50000 100000 150000 200000
N
whichisvalidforlargeN ( 103). AsshowninFig.2(e)-
FIG. 3: (color online) Numerical simulation of the optimal
≥
(h), for t ≥ 2t0 (≃ T/2) the Q functions almost recover coupling Ω0 as a function of N (solid circles) in the normal
to originalshapes,due to partialrevivalofthe squeezing scale. The solid line is the fit to power law Ω = aNb with
0
ξ and the coherence g(1) [see Fig. 1(c)]. a=1.07827andb=0.32655. Theinset: comparisonlinearfit
12 (blueline)ofnumericalresults(opencircles)withΩ =N1/3
0
(red line) in thelog-log scale. Ω is in unitsof κ.
0
V. OPTIMAL COUPLING AND THE
STRONGEST SQUEEZING

0
10
0
To create the strongest reduction of (∆J )2 as shown 10
z Num.
in Fig. 2(c), we need to determine the optimal coupling -1 Eq. (10)
10
Ω as a function of particle number N. Note that the
0
optimal squeezing occurs at ts for the OAT and t0 for t010-2
the JLC, respectively. For large N, the latter time scale -1
should be comparable with the former one as Ω = Ω . 010 10-3
0
Such a non-rigorous comparisonenables us to suppose a -4
10
power law as Ω /κ N1/3. Numerical solution of Ω is 0 50000 100000 150000 200000
0 0
presentedinFig.3fo≃rN upto2 105. Wefitthedataas N
×
Ω /κ = aNb and find the power-exponent b = 0.32655,
0
very close to the expected value 1/3. From the inset of 10-2
0 50000 100000 150000 200000
Fig.3,wealsofindthatthe largernumberN isadopted,
N
the better fit is obtained.
FIG. 4: (color online) The optimal squeezing ξ for theOAT
0
In Fig. 4, we investigate the optimal squeezing ξ as a (open circles) and the JLC (solid circles) as a function of N.
0
function of N. The fitting result is ξ0 0.8578N−1/3, Thebluelineisgivenbyξ0 ≃(4/3)1/6N−1/3 ≃1.0491N−1/3;
which is slightly smaller than the OA≃T result ξ0 theredlineisafittingcurveasξ0 =0.8578N−1/3. Theinset:
(4/3)1/6N−1/3 1.0491N−1/3. Smalldifferenceofξ0 be≃- numerical simulations of t0 (open circles), and the analytic
tweentheOAT≃andtheJLCdoesnotdeterioratethead- result of Eq. (8) with Ω0/κ=N1/3 (solid line) as a function
vantagesofthelatterscheme. Infact,thereisnonumber of N for optimal squeezing. The time t0 is in unitsof κ−1.
squeezing in the OAT model due to [J2,H ]=0. In our
z 1
case, one can realize J2(t ) = ξ2 J2(0) with ξ < 1,
h z 0 i 0h z i 0
indicating the appearance of the number-squeezed state
VI. CONCLUSION
[17]. Such a kind of squeezed state has been observed
in optical lattices [8], optical trap [10], and atom chip
[11]. However, the observed squeezing ξ 0.1 for Insummary,wehaveinvestigatedoptimalspinsqueez-
0
N =4 105[11],weakerthanourresultξ 1.≃467 10−2 inginatwo-componentBose-Einsteincondensatewith a
0
for Ω ×= 58.05κ and N = 2 105. Finally≃, within×inset Josephsoncoupling. Weshowthat: (i)thesqueezingξat
0
×
of Fig. 4, we show the time scale t as a function of N. time t aligns along the z axis, which is equivalent with
0 0
InsertingΩ /κ=N1/3 intoEq.(8),wefindthatanalytic the number squeezing [8, 9, 10, 11, 17], and is desirable
0
expressionoft givesgoodagreementwiththe exactnu- for high-precisionatom interferometry[28, 29]; (ii) there
0
merical simulations. exists a simple relation between the squeezing ξ and the
5
single-particle coherence g(1) at t , Eq.(5) and Eq.(6), work is supported by the Research Grants Council of
12 0
from which it is possible to measure the number vari- Hong Kong, Special Administrative Region of China
ance ∆J by readout the coherence g(1) in the interfer- (Project No. 401406). GRJ acknowledge additional sup-
z 12
ence experiments; (iii) the strongest squeezing with the portbytheNSFC (ProjectNo.10804007),andResearch
power law ξ 0.8578N−1/3 is achievable by applying Funds of Beijing Jiaotong University (No. 2007RC030
0
the optimal co≃upling Ω /κ N1/3. We also discuss the and No. 2007XM049).
0
≃
maximal-squeezing time t via the phase model and the
0
HusimiQfunction,andfindthatanalyticresult,Eq.(8),
agrees with its numerical simulations.
Acknowledgments
We thank Professor C. P. Sun, Professor W. M. Liu,
and Professor S. W. Kim for helpful discussions. This
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