Bell test for the free motion of material particles

0.18 MB·

by  Clemens Gneiting

published in: Phys.Rev.Lett. 101, 260503 (2008)
Bell test for the free motion of material particles
Clemens Gneiting and Klaus Hornberger
Arnold Sommerfeld Center for Theoretical Physics,
Ludwig-Maximilians-Universit¨at Mu¨nchen, Theresienstraße 37, 80333 Munich, Germany
We present a scheme to establish non-classical correlations in the motion of two macroscopically
separated massive particles without resorting to entanglement in their internal degrees of freedom.
It is based on the dissociation of a diatomic molecule with two temporally separated Feshbach
pulses generating a motional state of two counter-propagating atoms that is capable of violating
a Bell inequality by means of correlated single particle interferometry. We evaluate the influence
of dispersion on the Bell correlation, showing it to be important but manageable in a proposed
experimental setup. The latter employs a molecular BEC of fermionic Lithium atoms, uses laser-
9 guided atom interferometry, and seems to be within thereach of present-day technology.
0
0
PACSnumbers: 03.67.Bg,37.25.+k,03.65.Ud,03.75.Gg
2

n
a Introduction.—The possibility of entangling macro-
J scopically separate, non-interacting particles challenges

3 our classical view of the world by putting into question
1 the concepts of realism and locality [1]. Nowadays en-

tangledstates areroutinely establishedwithphotons [2].
]
h Also the internal states of material particles have been
p entangled, e.g., using non-classical light as a carrier for
-
t thequantumcorrelations[3,4,5,6,7],ortheirCoulomb
n
interaction in an ion trap [8, 9]. However, the origi-
a
nal discussion of entanglement focused on the motional
u
q state of massive particles, whose spatial separation is a
[ dynamicfeatureoftheentangledtwo-particlewavefunc-

tion[10,11]. Thelatterisspatiallyextended,unlikewith
2
v internal entanglement, where the positions only play the FIG.1: (a)ThetwoparticlesinaDTEstateareeachcharac-
8 passiveroleofseparatingthe parties. Sincethepositions terized byan early andalatewavepacketcomponentresult-
4 andmomentaarequantumobservableswithadirectclas- ing from two different dissociation times. (b) Each particle
4 sical analog, an observation of non-classical correlations is passed through an unbalanced Mach-Zehnder interferom-
1 eter with a switch deflecting the early wave component into
arisingfrommacroscopicallydistinctphasespaceregions
. the long arm, while conducting the late component through
7 wouldthereforebeastrikingdemonstrationofthefailure
0 the short arm. (c) The path difference is chosen such that
of classical mechanics.
8 the early and the late wave packets overlap. Detecting the
0 A convincing demonstration of non-classical correla- particle in an output port, at a given phase ϕ and splitting
: tions between two parties requires the experimental vi- ratioϑ,amountstoadichotomicmeasurementanalogoustoa
v
olation of a Bell-type inequality [12]. In the simplest spin 1/2 detection in an arbitrary orientation. The existence
i
X case, it involves detecting dichotomic properties on each ofDTEisestablished,implyingamacroscopicallydelocalized
two-particlestate,ifthecorrelations atthesingle-particle in-
r side, such as the polarization of spin-1/2 systems, where
a terferometers violate a Bell inequality.
a maximal violation is obtained if the spins are in a Bell
state, say Φ = ( +eiφ )/√2. Such a
1 2 1 2
| i |↑i |↑i |↓i |↓i
dichotomic property is not readily available in the mo- temporallyseparateddissociationpulses. Eachofthetwo
tional state of free, structureless particles. One possi- counter-propagatingdissociatedatomsthen has anearly
bility is to consider observables which have no classical and a late wave packet component corresponding to the
analogue,such as the displaced parity or pseudospin op- twopossibledissociationtimes,andtheircorrelationmay
eratorsusedtodiscussentanglementinthe‘originalEPR be called “dissociation time entanglement” (DTE), see
state’ [13, 14]. While they are expedient for character- Fig. 1 (a). If one regardsthese components as spin state
izing the quadratures of light fields, their experimental analogues in the motion (say, early corresponds to spin
implementationseems exceedingly difficult inthe caseof up, late to spin down), the DTE state shows the same
free, macroscopicallyseparatedmaterialparticles,where structure as the Bell state Φ . Switched, unbalanced
| i
only position measurements are easily realized. Mach-Zehnder interferometers on each side then serve
Hereweaiminsteadatproducingamotionalanalogue to mimic arbitrary spin rotations, and the detection of
of the Bell state Φ and implementing a Bell test which each particle in one of the interferometer output ports
| i
requires a simple position measurement in the end. The completes the Bell test, see Fig. 1 (b,c).
ideaistoexposeadiatomicmoleculetoasequenceoftwo The DTE state is a variantof the “energy-timeentan-
2
gled” state introduced in [15], and it is closely related creteBelltest,arobustquantitycharacterizingentangle-
to the “time-bin entanglement” of photons [2, 16, 17], ment is obtained by integrating the port-specific proba-
which has been used e.g. for establishing non-local cor- bilities over all times, P = pr(σ ,σ ;t ,t )dt dt .
σ1,σ2 1 2 1 2 1 2
relations over fiber distances of more than 50km [18] in For any reasonable model of the time-of-arrival detec-
R
a similar setup as in Fig. 1. As a main difference, the tion, the correlation function can be equally evaluated
DTE state is not composed of two single-particle prod- by means of the projections Πˆ Πˆ Πˆσ1 onto the
σ1 ⊗ σ2 ≡ σ2
uct states, but it superposes their relative coordinate, (unbounded) spatialregionsbehind the respective ports,
while the center-of-mass state remains separable during Pσ1,σ2 =limt→∞||Πˆσσ12Uˆz,t|ΨDTEi||2.
the two-pulse dissociation process. Since we are only interested in the interferometer out-
Our use ofDTE reflects the necessity to comeup with put,theeffectofdispersionisbestincorporatedbytime-
a state generationscheme appropriate to material parti- dependentscatteringtheory,whichseparatesthe‘rawac-
cles. Their finite mass and internal structure entail sub- tion’ of the interferometers, described by the S-matrices
stantial complications which require a careful investiga- Sˆ ,fromthefreedispersivetimeevolutionUˆ(0). Thepro-
j z,t
tion of whether a Bell violation can be expected at all. jection of the DTE state component ψc.m. ψrel onto a
Mostprominently,one mustaccountforthe unavoidable particularoutput-portcombinationσ|,σ0 tih|en0tiakesthe
1 2
wave packet dispersion, and only the recent progress in form
manipulatingultra-coldmolecules(suchastheirconden-
sation [19] and controlled dissociation [20]) suggests the Πˆσσ12Uˆ(zo,tn)|ψ0c.m.i|ψ0reli=Uˆ(z0,t)[Sˆ(σo1n)⊗Sˆ(σo2n)]|ψ0c.m.i|ψ0reli.
possibility to generate motional states that allow one to (1)
keep the detrimental effect of dispersion under control. Here we assume t to be sufficiently large, so that the
We note that other ways to demonstrate nonlocal cor- wave packets have passed the interferometers, and we
relations of molecular dissociation products have been take the entrance switches to be in the “on” configura-
proposed in [21, 22]. tion, i.e., routing towards the long arms. Implementing
We will show below that, by appropriately choosing a the phaseshifts bythe arm-lengthvariationsℓj,the pro-
sequence of magnetic field pulses, the Feshbach-induced jectedS-matricesSˆ(on) =Πˆ Sˆ(on) aregivenbySˆ(on) =
σj σj j σj=+1
dissociation of an ultracold 6Li2 molecule within a guid- exp(ipˆ ℓ /~)cosϑ and Sˆ(on) = exp(ipˆ ℓ /~)sinϑ .
ing laser beam can generate a motional state of macro- j j j σj=−1 j j j
For the “off” configuration we have correspondingly
scopically entangled atoms that is capable of violating Sˆ(off) =sinϑ and Sˆ(off) = cosϑ . Analogous rela-
a Bell inequality. But first, to clarify the implications σj=+1 j σj=−1 − j
tions hold for ψc.m. Pˆ ψrel .
of dispersion, we determine the correlation function for | 0 i | 0 i
The setup requires the dispersion-induced broadening
a generic DTE state subject to correlated single-particle
of the wave packets to remain much smaller than the
interferometry.
The general DTE Bell test.—It is natural to take the separation between the early and the late components,
so that the switches can be changed in between. In this
bound molecular two-particle state to be separable in
case the correlation function P can be evaluated by
the center-of-mass (c.m.) and the relative (rel) coordi- σ1,σ2
usingEq.(1)anditsvariantsevenfornon-pureandnon-
nate. Denoting by τ the period between the two dis-
separableinitial two-particlestates, Ψ Ψ ̺ . One
sociation pulses and assuming the transverse motion to 0 0 0
| ih |→
obtains
be frozen in the ground state of the guiding laser beam,
the longitudinal part of a pure DTE state then takes 1
P (ℓ ,ℓ )= 1+σ σ Re e−iφτ dp dp
the form ΨDTE = (Uˆ(z0,τ) Ψ0 + eiφτ Ψ0 )/√2, where σ1,σ2 1 2 4 1 2 1 2
t|Ψim0ei-=evo|ψlu0c|t.mio.ni(o|ψpi0reerlait+orPˆUˆ|ψ(z0,0rt|)e,lit)hi/e√p2a.riItty|ionpvioerlvaetsorthPˆe, afrnede ×nexp ip~~·~ℓ −hi2p~m2τ~ Zpr(p1,Zp2) , (2)
a relativephase φ determinedby the details of the two- (cid:16) (cid:17) io
τ with p~ = (p ,p )T, ~ℓ = (ℓ ,ℓ )T, and pr(p ,p ) =
pulse dissociation process. The center-of-mass state of 1 2 1 2 1 2
the original molecule ψc.m. is taken to rest in a wave p1,p2 ̺0 p1,p2 the momentum distribution function.
guide, while ψrel prop|a0gateis into positive direction. Fhorsim|pl|icitywietakeherethebeamsplitterstobesym-
| 0 i metric (ϑ = π/4) and the particles to have equal mass
Indicating the output ports of the two interferome- j
m. Note that (2) is independent of the total time of
ters j = 1,2 by σ = 1, the immediate experimen-
j
± flight and invariantunder momentum phase transforma-
tal outcome is characterized by the two-time probabil-
tions p ,p Ψ exp[iξ(p ,p )] p ,p Ψ , which in-
ity density pr(σ ,σ ;t ,t ) for detecting particles be- 1 2 0 1 2 1 2 0
1 2 1 2 h | i → h | i
cludesspatialtranslations,thusrenderingthecorrelation
hind the respective ports at the times t ,t . It de-
1 2
P (ℓ ,ℓ ) a robust signal.
pends on the phase settings ϕj and mirror transmis- σ1,σ2 1 2
For generic Gaussian states in the center-of-mass and
sion angles ϑ of the interferometers, and the disper-
j
relative motion Eq. (2) can be evaluated in closed form.
sive evolution of the DTE state leads to a complicated
The variances σ2 and σ2 then determine charac-
fringe pattern as a function of t1, t2, and τ. How- p,c.m. p,rel
ever, as suggested by the above analogy with a dis- teristic dispersion times, Tc.m. = 2m~/σp2,c.m. and Trel =
3
m~/2σ2 , indicating the transition to a dispersion- mv /2 defines the reduced wave length λ = ~/p ,
p,rel rel rel 0,rel
dominated spatial extension of the wave packets. The which sets the scale for the non-localinterference fringes
expectation value of the relative momentum p = in the explicit correlationfunction,
0,rel
1 1+τ2/T2 −1/4 T (ℓ ℓ τv )2 T (ℓ +ℓ )2
P (ℓ ,ℓ )= 1+σ σ c.m. exp rel 1− 2− rel c.m. 1 2
σ1,σ2 1 2 4( 1 2(cid:0) 1+τ2/Tr2el(cid:1)1/4 (cid:20)− Tr2el+τ2 2vrelλrel − Tc2.m.+τ2 2vrelλrel (cid:21)
ℓ1(cid:0) ℓ2 τ(cid:1) (ℓ1 ℓ2 τvrel)2 τ (ℓ1+ℓ2)2 ϕ0
cos − + − − + , (3)
× (cid:20) λrel Tr2el+τ2 2vrelλrel Tc2.m.+τ2 2vrelλrel − 2 (cid:21))
withϕ =τv /λ +arctan(τ/T )+arctan(τ/T )+ U /k = 100nK, ω /2π = 0.5Hz) and only a small
0 rel rel c.m. rel T B T
2φ . number of molecules (on the order of 102) remains in
τ
These DTE correlations can violate a Bell inequal- the BEC. These can be taken to be non-interacting, so
ity. This is seen from the structural similarity of (3) to that the initial longitudinal center of mass state ψ of
T
| i
the correlation function of the standard spin-1/2-based the molecules is straightforwardly defined by the trap
setup, Pspin (ϕ ,ϕ ) = 1+σ σ cos(ϕ ϕ ) /4, parameters.
σ1,σ2 1 2 { 1 2 1− 2 }
where the σ = 1 denote the spin measurement out- Each interferometer (with path length difference
j
±
comes for analyzers tilted by the angles ϕ with respect τv /2 5mm) is implemented by two more red-
j rel
≃
to the Bell state quantization axis. It follows from this detuned laser beams crossing the guide in a triangu-
analogythatanunambiguousdemonstrationofentangle- lar arrangement at small angles. While the crossings
mentrequiresthefringevisibilityofthecorrelationsignal act as beam splitters, the required atom mirror may
to exceed 1/√2 over at least a few periods. be realized using an evanescent light field or a blue-
Thedispersivesuppressionofthisfringevisibilityisde- detuned laser beam perpendicular to the interferometer
scribedbythosetermsin(3)whichdependonthecharac- plane [25, 26]. The time controlled appliance of such
teristic times T and T . Specifically, the dispersion- perpendicular blocking beams could also implement the
c.m. rel
induced distortion between the early and the late wave switch. However, a simplified setup could replace the
packet components is reflected in the Lorentzian reduc- switch by an ordinary beam splitter, at the cost of 50%
tionfactorandinthequadraticcompressionofthefringe post-selection.
pattern, while their envelopemismatchcausesthe Gaus- The fluorescence detection of the slow, strongly con-
siansuppression. Basedonthisresultonefindsthatnon- fined atoms at the two output guides can be done
local correlations can be observed, for λ /(τv ) 1, with near unit efficiency and with single particle reso-
rel rel
≪
provided(1+τ2/T2 )(1+τ2/T2 )<4. Inthefollowing, lution, since no particular spatial or temporal accuracy
c.m. rel
we present a conceivable scenario for the generation of a is needed. This single-particle resolution is crucial since
DTE state, which meets these conditions. events where more than one molecule gets dissociated
Experimental scenario for a DTE Bell test.—We sug- are disregarded in the present scenario, relying on post-
gest to use a dilute molecular Bose-Einstein condensate selection. In a more refined setup, it is conceivable to
(BEC) produced from a 50:50 spin mixture of fermionic use a specially preparedoptical lattice where eachsite is
6Li as a starting point. It can be prepared efficiently occupied by at most one molecule [27].
andwithnear-perfectpurity,displayingahugelifetimeof All this implies that the molecular dissociation in
morethan10sduetothePauliblockingofdetrimental3- presence of the wave guide must meet a number of
bodycollisions[23]. Atrulymacroscopictimeseparation rather restrictive criteria to render a demonstration of
between the two pulses, say τ = 1s, is thus conceivable, macroscopic entanglement possible. Only the transverse
andforarealisticdissociationvelocityofv =1cm/sthe groundstateoftheguidemaybepopulatedtoadmitthe
rel
de Broglie wave length λ =13.3µm is large enough to above quasi-one-dimensional description of the interfer-
rel
poseviablestabilityrequirementsfortheinterferometers. ometers, while both the momentum spread of the wave
TheBECispreparedinared-detuned,faroff-resonant packets and the dissociation probability must be suffi-
laser beam (transverse trap frequency ω /2π =300Hz), ciently small. In order to judge the feasibility of our
G
strong enough to guide the dissociated atoms towards experimental scenario, we now discuss how the dissoci-
the singleparticle interferometers. The intersectionwith atedpartofthe state, Φ (t) , depends onthe magnetic
bg
| i
a second, weak laser beam creates an elongated dipole field pulse sequence and the resonance parameters.
trap for the BEC within the laser guide [24]. At the A Green function analysis within the two-channel
end of the preparation steps it is very shallow (depth single-resonance approach shows that after an arbitrary
4
magnetic field pulse sequence (close to an isolated reso- resonant photons or backgroundgas particles.
nance) Φ (t) isdescribed,forlowenergies,atpositions
bg
| i Agreatadvantageofthissetupisthatnointerferomet-
far from the dissociation center, and at large times, by
theasymptoticform Φ (t) C ϕc.m. ϕrel Uˆ(0) Ψ , ric stability is required between the two interferometers,
| bg i∼ bg| 0,0 i| 0,0i z,t| zi sothattrulymacroscopicseparationsarefeasible. More-
where Uˆ(0) is the free propagator in the longitudinal
z,t over, the DTE state reveals its entanglement robustly
direction and where the transverse motion is frozen in
sinceneitheraspatialnortemporalresolutionisrequired
the harmonic ground state, ϕc.m. and ϕrel , respec-
| 0,0 i | 0,0i in the detection.
tively, of the guiding laser beam. The longitudinal
state is determined by p ,p Ψ = C˜(p2 /4m~+ This work was supported by the DFG Emmy Noether
p2 /m~+2ω ) p ψ h/c.mC˜. ,rwelh|erzeiC˜(ω)isct.mhe. Fourier program.
rel G h c.m.| Ti || ||
transform of the closed channel probability amplitude
C(t) and C˜ 2 = dp dp C˜(p2 /4m~+p2 /m~+
|| || c.m. rel| c.m. rel
2ω )2 p ψ 2. C(t), in turn, is determined by the
G c.m. T
| |h | i|R
coupled channel dynamics as induced by the externally
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