Positive P simulations of spin squeezing in a two-component Bose condensate

5 Pages·0.16 MB·English

by  Uffe V. Poulsen

Positive P simulations of spin squeezing in a two-component Bose condensate
Uffe V. Poulsen∗ and Klaus Mølmer
Institute of Physics and Astronomy, University of Aarhus, DK-8000 ˚Arhus C, Denmark
The collisional interaction in a Bose condensate represents anon-linearity which in analogy with
non-linear optics gives rise to unique quantum features. In this paper we apply a Monte Carlo
method based on the positive P pseudo-probability distribution from quantum optics to analyze
theefficiency of spin squeezingby collisions in a two-component condensate. Thesqueezing can be
controlled by choosing appropiate collision parameters or by manipulating the motional states of
thetwo components.
PACSnumbers: 03.75.Fi,05.30.Je
1
0 I. INTRODUCTION exactlyintoaccount. Weconfirmthevalidityofresultsof
0 a recentstudy [5], andwe proposean alternativescheme
2 for spin squeezing which relies on a spatial separationof

n In quantum optics it was realized some time ago atoms in different internal states.
a thatso-called’non-classical’statesoflight,inparticular, We consider in this paper a Bose-Einstein condensate
J
the so-called squeezed states, may outperform classical of two-level atoms in a trap. The dynamics of such a

8 field states in high precision experiments [1]. Moder- systemisinthesecondquantizedformalismwithcreation
1
atelysqueezedlighthasbeenproduced,andseveralnon- andannihilation operatorsof atoms in the state i=a or

classical properties have been demonstrated, but classi- b, ψˆ† , ψˆ , controlled by the Hamiltonian:
1 i i
cal technology has so far left enough space for improve-
v
9 ments that a real practical high precision application of Hˆ =
8 squeezed light remains to be seen. Atomic squeezing is
g
0 definedinawaysimilartosqueezingoflight,inthesense d3r ψˆ†(~r)hˆ ψˆ(~r)+ iiψˆ†(~r)ψˆ†(~r)ψˆ(~r)ψˆ(~r)
1 that the quantum mechanical uncertainty of a physical Z niX=a,bh i i 2 i i i i i
0
variable, here a component of the collective spin, is re-
1 +g ψˆ†(~r)ψˆ†(~r)ψˆ (~r)ψˆ (~r) . (1)
duced by a suitable manipulation of the quantum state ab a b b a
0 o
of the system.
/
h Due to quite strong interactions between atoms and be- Here, hi is the single particle Hamiltonian for atoms in
p internal state i and g is the effective two-body interac-
tween atoms and light, and due to the long storage and ij
- tionstrengthbetweenanatominstateiandoneinstate
t interactiontimesforatoms,thedegreeofcorrelationsand
n j. In terms of the corresponding scattering lengths they
squeezing obtainable in atomic systems exceeds the one
a are given by g =4π~2a /m.
u in light by orders of magnitude. Also, there is a real po- ij ij
Attemperaturessufficientlybelowthe criticaltemper-
q tential for practical application of squeezed atoms, since
ature for Bose-Einstein condensation almost all atoms
: current atom interferometers and atomic clocks operate
v occupy the same single particle wavefunction which to a
at a limit of precision which can only be improved by
i
X imposing quantum correlations between the atoms. very good approximation can be obtained from a two-
component Gross-Pitaevskii Equation. There are, how-
r Recently there has been a number of proposals for prac-
a ever,important effects about which the Gross-Pitaevskii
tical spin squeezing: absorption of squeezed light [2],
Equation gives no information: the population statistics
quantum non-demolition atomic detection [3], collisional
ofthecondensateisassumedtobepoissonian(oranum-
interactionsinclassicalordegenerategasses[4,5],photo-
ber state [8]), it is not treatedas a variable which has to
dissociation of molecular condensates [6], and controlled
be determined, and which can be manipulated by phys-
dynamics in quantum computers with ions or atoms [7].
ical processes. In multi-component condensates, the rel-
Since quantum mechanical squeezing implies quantum
ative populations and coherences of different states are
mechanical uncertainties below the level in ’natural’
importantdegreesoffreedomwhichrequirea moreelab-
states of the system, e.g., the ground state of the sys-
orate treatment.
tems, the analysis of squeezing has to be very precise,
and in particular one should avoid use of classical ap-
proximations or assumptions. In the present paper we
II. TWO-MODE MODEL
shallinvestigatethe possibilities for squeezing in a Bose-
Einstein condensate, using a method which takes the in-
teractions and the multi-mode character of the problem In a firstattempt to model the population statistics it
isconvenientforsimplicity toassumeaseparationofthe
spatial and internal degrees of freedom
∗Electronicaddress: [email protected] ψˆa(~r)=aˆφa(~r) ψˆb(~r)=ˆbφb(~r). (2)
2
The separation (2) is difficult to justify in general but it tonian (4) can be written
is certainly reasonable in the initial state that we have
in mind: A very pure (T=0) single component conden- Hˆ =e(t)Nˆ +k(t)Sˆ
two-mode z
sate is prepared. We then apply a π/2-pulse, coherently
transfering all atoms to an equal superposition of a and +E(t)Nˆ2+D(t)NˆSˆz +χ(t)Sˆz2 (6)
b. Then Eq.(2) is fulfilled to a good approximation with
where the time dependent coefficients are given by inte-
φ =φ .
a b
gralsinvolvingthetimedependentmodefunctionsfound
In the subsequent dynamics, the wavefunctions φ (x)
a
from Eq.(3). All the terms in the Hamiltonian (6) com-
and φ (x) may evolve with time, and we describe this
b
mute which is an important simplification as the coeffi-
evolution with Gross-PitaevskiiEquations
cients are time-dependent. The term proportional to Sˆ
z
will result in a rotation of the spin around the z-axis.
N N
i~∂ φ = hˆ +g φ 2+g φ 2 φ (3) The Nˆ and Nˆ2 terms give different dynamical overall
t i i ii i ii i i
(cid:18) 2 | | 2 (cid:19)
(cid:12) (cid:12) phases to states with different total numbers of atoms.
(cid:12) (cid:12)
Suchphasesareimmaterialifwehavenoexternalphase-
where a b and vice versa.
≡ standardtocomparewithandweneglectthesetermsfor
The dynamics associated with distribution of atoms the purpose of the present work. The NˆSˆ -term adds to
z
amongthe twomodesisnowstudiedby rewritingEq.(1)
the spin rotation with an angle linear in N. It cannot
in terms of the aˆ andˆb operators: be neglected as the direction of the spin (the phase be-
tween a- and b-components) can be probed by a second
N π/2 pulse phase locked to the first π/2 pulse. The term
Hˆtwo-mode =gbb(cid:18)ˆb†ˆb†ˆbˆb− 2ˆb†ˆb(cid:19)Z dr3|φb|4 of interest to us is Sˆz2, the effect of which is well known
from the work of Kitagawa and Ueda [10]. It produces
N
+g aˆ†aˆ†aˆaˆ aˆ†aˆ dr3 φ 4 spin-squeezing,i.e. itentanglestheindividualatomsina
aa a
(cid:18) − 2 (cid:19)Z | |
waythatreducesthe fluctuationsofthe totalspininone
+g ˆb†aˆ†aˆˆb Nˆb†ˆb Naˆ†aˆ dr3 φ 2 φ 2. (4) of the directions perpendicular to the average spin. The
ab(cid:18) − 2 − 2 (cid:19)Z | b| | a| strengthparameterof the squeezing operatoris givenby
1
Allterms inthis Hamiltoniancommutewith aˆ†aˆ andˆb†ˆb χ(t)= dr3 g φ 4+g φ 4 2g φ 2 φ 2 .
bb b aa a ab a b
and so the interesting dynamics takes place in the co- Z 2(cid:16) | | | | − | | | | (cid:17)
(7)
herencesbetweenthe twointernalstates. To studythese
coherencesitisconvenienttodefineeffectiveinternalspin
Giventhetime integralµ=2 χ(t′)dt′ ofthis parameter
operators
KitagawaandUedaprovidethReanalyticalexpressionsfor
the variance of the squeezed spin component
1
Sˆ ˆb†aˆ Sˆ aˆ†ˆb Sˆ ˆb†ˆb aˆ†aˆ . (5)
+ − z
≡ ≡ ≡ 2(cid:16) − (cid:17)
∆S2 =
h θi
Sˆ represents the difference of total population of states S 1 1 1 1
z [1+ (S )A] (S ) A2+B2] (8)
aandb,aquantitywhichcanbe measured,e.g., bylaser 2 (cid:26) 2 − 2 − 2 − 2 (cid:27)
induced fluoresence in an experiment. Sˆ and Sˆ or, p
+ −
equivalently, Sˆx (Sˆ++Sˆ−)/2 and Sˆy (Sˆ+ Sˆ−)/2i where A= 1 cos2S−1µ and B = 4sinµ/2cos2S−2µ/2,
≡ ≡ − −
represent the coherences between a and b. By suitable and they specify the direction of the squeezed spin com-
couplings they can also be turned into population dif- ponent Sˆ =cosθSˆ +sinθSˆ :
θ y z
ferences and are therefore interesting observables of the
system. Infactitturns outthatthe quantitythatdeter- π 1 B
θ = + arctan . (9)
mines the accuracy of a given spectroscopic experiment 2 2 A
is the ratio of the measured spin-component (the signal)
to the fluctuations in a component perpendicular to it It is of some interest to use simple analytical approxi-
(the noise) [9]. For the initial state prepared by a π/2 mations for χ(t) and we will do so in the specific cases
pulsewehaveN independentatomsallinthesameequal studied below. Another approach would of course be to
superposition of a and b. Then the mean spin is in the obtain χ(t) by a numerical solution of Eq. (3).
xy-planeandbydefinitionwe cantakeittobe alongthe
x-axis. The maximal signal we can obtain is the length
of the spin S = N/2 and the perpendicular fluctuations III. FULL MULTI-MODE DESCRIPTION
arethenS/2. Toimprovethis“standardquantumlimit”
signal/noise ratio we can introduce correlations among Equations (2)-(7) are based on a simplifying assump-
the atoms i.e. we can introduce spin squeezing. tion. Theactualobservablesofthesystemaremorecom-
With the internalspinoperatorsof Eq.(5) the Hamil- plicatedtodealwith,butwecandefineasetofoperators
3
obeying angular momentum commutation relations by On the computer we can simulate the Langevin equa-
tions to obtain an ensemble of realizations of ψ. This
1
Jˆx ≡ 2Z dr3(cid:16)ψˆb†ψˆa+ψˆa†ψˆb(cid:17) (10) uensesdemtbolecailscualafitneiteexspaemctpaltiinognovfaPlu+esavnidactahne tphreersecfroipretiobne
1
Jˆ dr3 ψˆ†ψˆ ψˆ†ψˆ (11) (14) and with a precision limited only by the number of
y ≡ 2iZ (cid:16) b a− a b(cid:17) realizations in the ensemble.
Jˆ 1 dr3 ψˆ†ψˆ ψˆ†ψˆ . (12) Three limitations of the method should be noted: (i)
z ≡ 2Z (cid:16) b b− a a(cid:17) The initial state of the system has to be expressedas an
initial P . This is trivial for a single coherent state and
ThetotalnumberoperatorNˆ dr3(ψˆ†ψˆ +ψˆ†ψˆ )com- +
≡ a a b b for any mixture of coherent states but it can be compli-
muteswiththesethreeoperatorsRandwhenthetwo-mode catedforotherinitialconditions. (ii)TheP methodhas
+
approximation applies well, the two-mode and multi-
a notorious divergence problem at large non-linearities.
mode operators are comparable by the replacement
This problem sets in after a certain time and is clearly
Jˆ =ρeiνSˆ Jˆ =ρe−iνSˆ Jˆ =Sˆ (13) noticable in the simulations. We can therefore easily tell
+ + − − z z
how long we can trust the results of the method. (iii)
where ρeiν dr3φ∗(~r)φ (~r). The factor ρ takes into
≡ b a Although we have so far written all equations in 3D it
account that RJˆx and Jˆy vanish unless the atoms in wouldbe computationallyveryheavytosimulateasuffi-
h i h i
state a and b occupy the same regionin phase space and cient number of realizations of Eq. (16). In what follows
ν is a dynamical phase from the spatial dynamics. we will thus restrict ourselves to 1D. It is reasonable to
Thetwo-modeapproximationprovidesanintuitivepic- assume that this may alter the quantitative results sig-
ture of the evolution of the system. It is however not nificantlybuta1Dcalculationcanbeusedtoinvestigate
easy to justify the factorization (2) and we shall there- the validity of the two-mode model which may hereafter
fore apply an exact method to determine more precisely be applied in 3D with more confidence.
whathappenstothemeanvaluesandthevariancesofthe
components of Jˆ~. The positive P function (P ) [11] is
+
a pseudo-probabilitydistribution giving expectation val- IV. SPIN SQUEEZING WITH CONTROLLED
ues of normally ordered operator products as c-number COLLISION STRENGTHS
averages. In our case with two internal states one has
Let us first focus on a situation with simple spatial
h:f[ψˆ(t)]:i=Z d[ψ]f[ψ]P+[ψ,t] (14) dynamics. If we set Va = Vb = mω2x2/2 (1D model)
and we assume that the values of the collision strengths
where can be controlled so that, e.g., g = g = 2g g the
aa bb ab
≡
spatial dynamics is limited to a slight breathing. The
ψˆa(t) ψa1 spin-dynamics is almost a pure squeezing, that is, the
ψˆ(t)≡ ψψψˆˆˆab†b†(((ttt))) and ψ ≡ ψψψabb212 . (15) ot0mhf)eeta=hsnteφastsbpit(oirtnne=nasrgtyat0hy)ssoaplisuanrttaithomheneeTotxefh-rtdohχimreeaoGcsft-PiFoEEenqr.m.w(Tiit7oha)pgawpellretoaxfiatninommdeassφttiiaionm(nttahtt=oee
Thedistributionisnotdetermineduniquelybutonepar- a-state. We then have:
ticular choiceobeys a functional Fokker-Planckequation
which is of course immensely difficult to solve. For nu- χ= 1 g2/3 m1/3ω2/3. (19)
merical purposes it is much better to translate it to cou- ∼ 22/332/35N1/3
pledLangevinequationsforthe4c-numberfields(“wave
Choosing g = 0.005~ωa and N = 2000 we get χ =
functions”). Theseequationsare“noisyGross-Pitaevskii 0
6.1 10−4~ωwhichshouldgiveasizablesqueezingwithin
equations” ×
~
a quarter of a trapping periode. a is the har-
dψiµ =(−1)µi~dt(cid:16)hˆi+giiψi2ψi1+gabψi2ψi1(cid:17)ψiµ+dWiµ monic oscillator length in the trap0a≡ndqmmiωs the atomic
(16) mass.
InFig.1weshowresultsofboththetwo-modeapprox-
where a b and vice versa. In order to treat the inter-
≡ imation (8) and of the P+ simulation for the parameters
actions exactly (within the approximation given by the
mentioned above. χ was assumed to be constant and of
form of Eq.(1)) the noise terms have to be gaussian and the value determined by Eq. (19). Jˆ =cosθJˆ +sinθJˆ
θ y z
to fulfill:
refers to the squeezed component of the spin. The di-
dW (~r,t) =0 (17) rection θ in the yz-plane is determined in the two mode
iµ
h i
model, i.e., from Eq. (9) and it is not independently op-
dW (~r,t)dW (~r′,t′) =δ(3)(~r ~r′)δ(t t′)δ
h iµ jν i − − µν timized for the full P+ results. The agreementis seen to
( 1)µidtgijψ (~r,t)ψ (~r,t). (18) be surprisingly goodconsidering the crudeness of the es-
× − ~ iµ jµ timateoftheparametersinthetwomodemodel. Within
4
1000 1000
hJ(cid:23)i
800 800
2
h(cid:1)J i
600 600 (cid:18)
400 400
2
h(cid:1)J i
(cid:18)
200 200
hJ(cid:23)i
0 0
-200 -200
0 1 2 3 4 5 0 1 2 3 4 5 6
!t !t
FIG. 1: Squeezing of the collective spin for favorable in- FIG. 2: Squeezing dueto spatial separation of thetwo inter-
teraction parameters: N=2000 atoms and (gaa,gab,gbb) = nal state potentials by x0 =3a0. The total numberof atoms
(1.0,0.5,1.0)×5×10−3~ωa0 (1Dmodel). +(witherrorbars) is 2000 and the interaction strengths are (gaa,gab,gbb) =
show P+ results, lines show results of thetwo-mode model. (1.03,1,0.97)×5×10−3~ωa0 (1D model).
the uncertaintyofthe positiveP results(the ∆Jˆ2 data certainamountx fromV . Inthismodelbothpotentials
h θi 0 a
are presented with errorbars in Fig. 1) the noise sup- are still harmonic and of the same strength. The spa-
pression is seen to be almost perfect. These calculations tial dynamics is now more complicated but it can be ap-
thus confirm the results of Sørensen et al. [5] where the proximated by the solutions to coupled Gross-Pitaevskii
simpletwo-modeapproachwassupplementedbyanother equations. To get a rough idea of the evolution we can
approximate method. use the well know evolution of a displaced ground state
wavefunction in a harmonic trap. We then get
V. SPIN SQUEEZING WITH CONTROLLED ρeiν =e−x20(1−cos(ωt))/2eix20sin(ωt)/2 (20)
MODE FUNCTIONS OVERLAPS
and
FI=n 2th,me fex=pe1rimsteanttessionn87tRheb[|1F2]=the1s,cmatfte=ring−l1eingatnhds Z |φb|2|φa|2dx= √12π e−x20(1−cos(ωt))/4. (21)
| i
andthustheinteractionstrengthsareactuallyinpropor-
tion g :g :g =1.03:1:0.97. This is far from ideal Ifwechoose(g ,g ,g )=(1.03,1,0.97) 5 10−3~ωa
aa ab bb aa ab bb × × 0
conditionsforsqueezingaswecanseefromEq.(7): χ=0 and x0 = 3a0 this model gives an order of magni-
when in addition φ =φ as is the case initially. To pro- tude estimate for the integrated strength parameter of
a b
duce a sizable squeezing effect we propose to make the 2π/ωχ(t)dt = 10−2~ω when the two components are
0 ∼
twomodefunctionsdiffer(seealso[13]). Thisisachieved Ragain overlapped. For 2000 atoms this corresponds to a
byapplyingdifferentpotentialstothetwointernalstates reductionoftheuncertaintyinthe squeezedspincompo-
which has actually already been done for magnetically nent by roughly a factor of 10 according to Eq. 8.
trappedRbmakinguseofgravityanddifferentmagnetic InFig.2weusethese estimatesto analyzethe P sim-
+
moments of the two internal states [12]. If Vb is dis- ulations, i.e., we plot the expectation value of the spin
placed from Va the b-component created by the initial componentpredictedto be maximal, Jˆν , andthe vari-
π/2-pulsewill moveawayfromthe a componentthereby ance of the perpendicular componenht piredicted to be
reducing the overlap dx φ 2 φ 2 and increasing χ. It squeezed, ∆Jˆ2 . As can be seen, after a fast drop a
| a| | b| h θi
is remarkable that thRe squeezing then takes place while large fraction of the original mean spin is recovered in
the two components are away from each other and are Jˆ when the two wave packets are again overlapped.
ν
h i
therefore not interacting. This confirms our prediction of ν and it implies that a
Todemonstratetheaccomplishmentsoftheschemede- sizable signal can be obtained in an experiment. At the
scribed above we have chosen simply to displace V by a same time the variance in the predicted perpendicular
b
5
componentis stronglysuppressedconfirming our predic- temporal dynamics and the population dynamics couple
tion for θ and implying that the noise in the experiment the way they should to produce spin squeezing but the
canbesignificantlyreducedbelowthestandardquantum resulting entanglement between spatial and internal de-
limit. greesoffreedomissmallenoughthatpurelyinternalstate
observablesshowstrongsqueezingandmulti-particleen-
tanglement. In particular the calculations of Fig. 2 re-
VI. DISCUSSION vealthatquite significantdistortionsofthe distributions
whenthecomponentsseparateandmergedonotprevent
We have presented the first exact numerical studies of sizable spin squeezing.
spin squeezingin two-componentcondensates. As in our Although we claim quantitative agreement above it
previous work on squezing of a single component atomic should of course be realized that this is within the 1D
field operator [6] we have applied the positive P distri- model. To use the two-mode model to describe a real
bution which is well suited for studies of transient, short experiment the spatial dynamics which enters the inter-
time behavior of interacting many-body systems. This naldynamics via Eqs.(7) and(13) mustbe treated with
formulation involves simulations, and due to the formal some accuracy. Apart from the 3D aspects it should be
separation of c-number representations of creation and noted that e.g. in some experiments on two-component
annihilation operators, ’non-physical’ results may occur condensates [14] the displacement of the potentials are
such as negative/complex atomic densities, and few in- accompanied (and in fact due to) different strenghts of
stants with negative variances as depicted in Figs. 1 and the potentials. This adds to the complexity of the dy-
2. For long times we cannot exclude that slight impreci- namics and will of course be important if the aim is pre-
sion due to discretization of space and time contributes cise quantitative predictions and not a proof-of-principle
to these results. The Monte Carlo nature of the method analysis as offered in this paper.
makes it somewhat tedious to check rigorously for dis- More tests need to be carried out for other kinds of
cretization errors but for resonable numbers of realiza- processes, but the present study suggests that the sim-
tions inthe ensemble samplingerrorswill dominate any- ple two-mode description provides good predictions for
how. the many-bodydynamics ofspin squeezing. Hence other
The conclusion of this paper is that the predictions of ideas,e.g.,forreducingthefluctuationsinthetotalnum-
thesimpletwo-modemodelreproducethefullmultimode berofatomsinacondensateorinout-coupledatomlaser
result also quantitatively. This means that the spatio- beams, may be reliably based on the terms of Eq. (6).
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