High momentum entanglement of cold atoms generated by a single photon scattering

0.21 MB·English

by  Rui Guo

.
High momentum entanglement of cold atoms generated by a single photon scattering
Rui Guo, Xiaoji Zhou, and Xuzong Chen
∗
School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China
(Dated: February 2, 2008)
With the mechanism of pairwise scattering of photons between two atoms, we propose a novel
scheme to highly entangle the motional state between two ultracold neutral atoms by a single
photonscattering anddetection. Undercertain conditions, it isshown thatan arbitrary amount of
entanglement can be obtained with this scheme.
8 PACSnumbers: 03.65.Ud,42.50.Vk,32.80.Lg
0
0
2 Introduction.— Quantum entanglement, as one of the more feasible and efficient for realistic implementations
central topics in quantum mechanics, has been exten- [7]. The basic physics for this scheme is that, for a cou-
n
a sivelystudiedontheHilbertspaceswithcontinuousvari- ple of cold atoms with the same electric dipoles, photon
J ables (CV) in recent years [1, 2, 3, 4, 5, 6, 7, 8]. Com- scattered from one atom can be efficiently re-absorbed

5 pared to the finite dimensional entanglement on pho- and re-scattered by the other, this mechanism of “pair-
1 tonicpolarizationsoratomicinternalstates,the CVsys- wise scattering” [9, 10] of photons entangles the atomic

tem provides entanglement in a rich diversity of forms wavepackets under the law of momentum conservation.
]
h [2, 3, 4, 5, 6], which provide unique roles in quan- Evaluated by the Schmidt number K [4, 5], it is shown
p tum information processing [1, 7] and in fundamental that an arbitrary amount of entanglement can be pro-
- test for quantum physics [2]. It is only in CV systems duced for ultracold atoms with proper physical control
t
n that the unbounded high degree of entanglement be- parameters.
a
comes possible, which has been studied recently for op- Theoretical analysis.— To describe the physical pro-
u
q tical squeezed state [1], optical parametric down conver- cess, as shown in Fig. 1, it is assumed that a pair of
[ sion [3, 4], atom–photon scattering [5], and atom–atom identical cold atoms “a” and “b” are coupled to a sin-

motional state [6], etc.. gle photon in a cavity along y-axis, with nearly paral-
1
v For motional entanglement between neutral atoms lel atomic electric dipoles d~a,b, i.e., d~a = d~b =d and
| | | |
0 without interactions, it is verified that [6] one can not their relative angle ϕ 0. We assume the coupling pho-
≈
4 produce entanglement beyond 1 ebit with single-photon ton is scattered and therefore recoils the atoms to the
2 emissionanddetection,therefore,itunavoidablyrequires x–direction where the photon detector is placed. Con-
2
nonlinear bi-photon or multi-modes detections for high sidering the momentum exchange along x–direction, the
.
1 entanglement [6]. By introducing effective interactions Hamiltonian can be written as Hˆtotal =Hˆ0+HˆI, where
0
between atoms in this letter, however, we find that it
8 (Pˆa)2 (Pˆb)2
v:0 iosnlpyosasisbilnegtleopphrootdouncescaartbtietrrianrgilyanhdigdheteencttainongl,emwhenicthbiys Hˆ0 = 2xm + 2xm +Xk,s ~ωka†k,sak,s
Xi + ~ω21(σ2a2+σ2b2)+~ωca†a,
ar HˆI = ~ ga(k,s)e−ikxaσ1a2a†k,s
Xk,s (cid:2)
+ gb(k,s)e−ikxbσ1b2a†k,s+H.C.
+ ~ g σa a +g σb a +H.C.(cid:3). (1)
c 12 † c 12 †
(cid:0) (cid:1)
Pˆa(b) is the x-dimensional momentum operator for the
x
two-levelatom“a”(“b”)withmassmandtransitionfre-
quencyω ,andσ i j . a (a)isthecreation(anni-
21 ij †
≡| ih |
hilation) operator for the coupling single mode with fre-
quency ωc and coupling strength gc; whereas a†k,s (ak,s)
is for the scattering modes along x-axis with frequency
FIG. 1: (Color online) Schematic plot of the physical model.
ω =ck andpolarizations andthe couplingstrengthis
k
Thecouplingphotonisinjectedinasingle–modecavityalong | |
ttheceteyd-axalios,nwghxe-raexaiss.thTehsecartetceoriilngofpahtootmonsiisscroelsltercitcetdedanindtdhee- ga(Wb)(eku,tsi)li=zeqth2eεc0|Dk~|Vira·cd~ak(ebt)·toǫˆk,dse.note the physical state,
x–direction. The inset shows the atomic configuration. e.g., 1 ,q ;1 ,q ;k represents that the atom “a” (“b”)
a a b b
| i
2
the Schr¨odingerequation,the dynamicalequations read:
dM(q)
dt = −i∆M(q)−2igc∗C(q)−γM(q)
t
eiEmt/~ dk ds g (k)g (k)
a b
− Z Z ×
0 (cid:2)
ei(ωc−c|k|)(t−s)M(qa+k,qb k,s) , (3)
−
dC(q) (cid:3)
= ig M(q), (4)
c
dt −
FIG. 2: (Color online) Spatial probability distribution of the dD(q,k)
= iexp[ i(ω ck )t] (5)
steady bipartite wavefunction D(x,k=kc), with kc=1, σ= dt − − c− | | ×
0.2,δ=0.1andEm=0[11]. Thesmallfigureshowstheinitial [g (k)A(q ,q +k)+g (k)B(q +k,q )],
b a b a a b
Gaussian distribution.
where the detuning ∆ = ω ω and γ =
21 c
−
(a) (b) 2π dk g (k)2δ(ω ω ) is the atomic natural
a(b) k c
| | −
lineRwidth, and Γ 2gc 2γ/∆2 denotes the scattering
3 rate. E =~2k2/≡m re|pr|esents the kinetic energy mis-
m c
2 match generated when the coupling photon is scattered
1 fromoneatomtoanotherandbere-scatteredbacktothe
qb0 couplingmode[9,10]. Whenthisenergydeficitissignifi-
−1
cant,thepairwisescatteringhappensonlyinatimescale
−2 τ < ~/E and further cascaded processes will be elimi-
m
−3
nated [9]. In this work, we assume E is small enough
m
−3−2−1 0 1 2 3 −3−2−1 0 1 2 3 to allow needed cascaded pairwise scattering, and treat
qa qa it as a constant [11].
The fourth term at the r.h.s. of Eq. (3) represents
FIG. 3: (Color online) Density plot of momentum distribu- thepairwisescatteringbetweentwoatoms. Iftheatomic
tionsforthesteadybipartitewavefunctionsD(q,k=kc)with dipoles are perpendicular (ϕ=π/2), this term vanishes
kc = 1, δ = 0.1, and Em = 0 [11]. (a) Different orders of
and the pairwise scattering will be forbidden in the Eqs.
scattered wavepackets are well separated with σ=0.3. (b)
(3–5), which therefore reduce to the model of a single
Wavepacketsbegin to overlap and interfere with σ=1.
atomscattering[5]. Whenthedipolesareparallel(ϕ 0),
≈
however,photonscatteredbyoneatomcanbeefficiently
re-scatteredby the other,therefore producesmomentum
has the internal state 1 and momentum ~q , and
the photon is scatter|ediaw(b)ith momentum ~k.a(bI)n the correlationbetween them.
Eqs. (3–5)canbeanalyticallydecoupledintheatomic
slowly–varyingframe, therefore, the system state can be
position coordinates. We use the argument x (x ,x )
expanded as: ≡ a b
for wavefunctions to indicate their fourier counterparts,
e.g., M(x) dq e ixqM(q), where x q x q +x q .
− · a a b b
|Ψi=Zdq e−i(E/~+ωc)t(cid:20)A(q)|1a,qa;2b,qb;0i EWit/h~ weaΓk,≡ltyhRecoaudpialibnagticcosonlduittiioonnso:f Egqc·,.γ(5≡)≪yiel∆ds,: and
m
≪
+B(q)2 ,q ;1 ,q ;0 +C(q)1 ,q ;1 ,q ;0
a a b b a a b b
| i | i(cid:21) D(q,k,t) = N dx eix·qG(x)(eixak+eixbk)
Z ×
+Zdqdk e−i(E/~+ωk)tD(q,k)|1a,qa;1b,qb;ki, (2) [1 e−t·Π(x,k,t)]/Π(x,k,t), (6)
−
Π(x,k,t) = i(ω 2g 2/∆ ck )
c c
− | | − | |
wheretheboldsymbolqsimplifiesthepairwisevariables +Γ[1+sinc(E t/~)cos(ϕ)cos(k x k x )],
m c a c b
(q ,q ), and the summation over the polarization s is −
a b
implied in the integration over k. The kinetic energy is where G(x) is the fourier counterpart of the ini-
E≡~2(qa2+qb2)/2m. tial atomic momentum wavefunction G(q) =
In Eq. (2), D(q,k) is the momentum wavefunction √2exp[ (q2+q2)/σ2]/√πσ which is generally specified
− a b
for the atoms and the scattered photon, which is our as an unentangled Gaussian with momentum width
main concern for its induced atom–atom entanglement σ. For atoms with nearly parallel dipoles (ϕ 0),
≈
in the steady state. A(q) [B(q)] is the instantaneous we introduce cos(ϕ) 1 δ2 and use δ ( 1) as a
≡ − ≪
wavefunction for the atom “b” (“a”) is excited, and we dipole parallelity parameter in the following. N is the
introduce M(q) A(q)+B(q) for the symmetry. From normalized factor and sinc(x) sin(x)/x.
≡ ≡
3
Generation of atom-atom entanglement with single- position [8] of D(q,k=k ):
c
photondetection—Eq. (6)isthetimeevolutionofcollec-
tiveatoms–photonwavefunction,whichwillbeprojected ∞
D(q,k=k )= λ S (q), (10)
to an atom–atom entangled state [6] after the detection c n n
n=Xp
of the scattered photon. In the following, we neglect the −∞
light shift (2|gc|2/∆) and assume the photon clicks on a where Sn(q) is the momentum Schmidt basis which is
narrow band detector with wavevector k =+kc. orthonormalandseparable,with nλn =1. Thedegree
When Em=0, from Eq. (6), the steady atomic mo- of entanglement is then defined aPs the Schmidt number
mentum wave6 function reduces to a Bell-like state: [4, 5]: K 1/ λ2.
≡ n n
For the singlPe–atom–scattering state in Eq. (7), com-
D(q,k=k ,t ) G(q +k ,q )+G(q ,q +k ), (7) pared with Eq. (10), one has at most K = 2 which is
c a c b a b c
→∞ ∝ only 1 ebit [6]. For the pairwise–scattering–state in Eq.
since for t ~/E the energy–conserved single–atom– (8), however, from the numerical results of K in Fig. 4
m
≫ in dependence of δ and σ, one sees that it may highly
scatteringprocessprevails[9],whichcanproduceatmost
exceed the 1 ebit limit [6] for cold atoms (σ<k ) with
1 ebit by a single–photon detection [6]. c
parallel dipoles (δ 0).
For Em = 0 [11], however, the atom–atom pairwise For atoms coole≈d well below the recoil temperature,
scatteringintroduceseffectiveinteractionsbetweenthem,
i.e., σ k , the Fourier series in Eq. (9) makes the rel-
which produces correlated wavepacketas: ≪ c
ative momentum highly localized in each single period,
therefore, the summation over spatial frequency can be
D(q,k=kc,t→∞)∝D′(qa+kc,qb)+D′(qa,qb+kc),(8) wellapproximatedbyasummationoverdiscretelocalized
D′(q)=exp[ (qa+qb)2]exp( |qa−qb|δ) (9) momentum modes. Along with Eqs. (8–10), it eventu-
− 2σ2 − √2k × allyyieldstheSchmidtdecompositionasasummationof
c
∞ σ2π2 π different orders of scattered wavepackets:
exp[ (2n+1)2]cos[ (2n+1)(q q )].
nX=0 − 8kc2 2kc a− b Sn(q)=( 1)|n|+1Sgn(n)G(qa nkc,qb+nkc+kc),(11)
− −
2
λ =N e √2δn e √2δn+1 , (12)
Eq. (9) characterizes the essences for the pairwise scat- n ′ − | | − | |
(cid:16) − (cid:17)
tering: the first Gaussian factor on the r.h.s. represents
the bipartite total momentum which is provided by the where Sgn(n) is the signum function with value +1 for
initial momentum width σ; the Fourier serieson the sec- n 0and 1forn<0,andN′isanormalizationfactor.
≥ −
ond line, which gives a periodic structure of the rela- With Eq. (12), the degree of entanglement is obtained:
tivemomentum,characterizesdifferentordersofpairwise
K 1/ λ2 =√2/δ, (δ 1). (13)
scattering, with the total number of scattering orders ≡ n ≪
controlled by the dipole parallelity parameter δ through Xn
the second Gaussian factor on the first line.
Eq. (13) fits well with the numerical results when
Thespatialandmomentumdistributionsoftheatomic σ<k /2 as shown in Fig. 4. It indicates that, by uti-
c
wavefunction of Eq. (8) are plotted in Fig. 2 and Fig. lizing ultracold atom pair, such as in BEC, where both
3, respectively. In Fig. 2, one sees that, the scatter- conditions δ 0 and σ k are well fulfilled, arbitrarily
c
≈ ≪
ing of photon with wavevector k = kc correlates the highatom–atomentanglementmaybeproducedbyasin-
atomic wavepacketswith resonantspatialperiodλc, and gle photon scattering and detection, once the mismatch
produces grating-like wavefunction with highly localized energy E is well compensated [11].
m
atomic relative positions. Meanwhile, as in Fig. 3, the For hotter atoms with temperature approaches or ex-
atomic momentum is correlatedalong qa+qb+kc=0 due ceeds the recoil temperature, or equivalently, σ & kc,
to the momentumconservation. When the initialatomic the scatteredmomentumwavepacketsS (q)willoverlap
n
momentum width σ is well below the recoil momentum eachother and exhibit quantum interferences. From Eq.
(σ kc) as in Fig. 3 (a), different orders of scattered (11), one sees that different orders of recoiled wavepack-
≪
wavepackets are clearly separated, which corresponds to ets take the same shape as the initial wavepacket G(q),
a well entangled bipartite state; when σ tends towards together with an extra overall phase: ( 1)n+1Sgn(n).
| |
−
kc, however,as in Fig. 3 (b), recoiled wavepacketsbegin Explicitly, the first two orders of scattered wavepackets
to overlapandexhibitdestructive quantuminterferences S (q)andS (q)takethesameoverallphase“ 1”,and
0 1
for the high–order scattering, which will eventually de- therefore ex−hibit constructive interference; for−higher–
stroythemomentumentanglementthatcanbeseenmore orderwavepackets,however,S (q)’ takeoppositephases
n
clearly in the following. “ 1”interchangeably,whichwillinducedestructivelyin-
±
TheentanglementencodedinthewavefunctionEq. (8) terferences as in Fig. 3 (b). Due to this quantum inter-
can be quantitatively evaluated by the Schmidt decom- ference, the entanglement will be significantly decreased
4
K)120 scattering in a couple of ultracold atoms may be used
er 2−1/2/δ to produce superhigh atom–atom entanglement. When
mb100 σ=0.8kc theatompairiscooledwellbelowtherecoiltemperature
u
midt N 80 σσ==k0c.2kc waribthitrpaarryaallmeloeulencttorifcednitpaonlgelse,mitenistpbyosasisbilnegtleopahchoiteovnedaen-
ch 60 σ=0.4k tectiononcethescattering–inducedenergymismatchE
S c m
nt ( iscompensated[11]. Thisschemecanbeusedtoproduce
e 40
m entangledatomresourcesforthetestofEPR–nonlocality
e
ngl 20 [2] and for quantum information processing [1, 7].
a
nt For experimental tests of this scheme, the spatially
E
0 overlappingultracoldatompairscanbeproducedbycou-
0 20 40 60 80
1/δ pling weak atom laser beams from a BEC, and then be
loaded into a cavity for detections of the scattered pho-
FIG. 4: (Color online) The Schmidt number K for the en-
ton [12]. Moreover, as in recent report [10] of BEC su-
tangled steady state of D(q,k = kc) is plotted against the
perradiant [9], the generation of pairwise scattering by
reciprocaldipoleparallelity parameter1/δ,withdifferentini-
tial momentum widthes σ as shown in theinset. atom pairs is preferred over single atom scattering when
the energymismatchE is compensatedbythe incident
m
K)15 coupling light [10], therefore, it is most probable to an-
er alyze the quantum correlation in the pairwise scattering
b
m process with this proposed model.
u
N
dt 10 It is our pleasure to acknowledge the helpful com-
mi
h ments on this manuscript from Dr. Yu Ting. This
c
S
ment ( 5 wgroarmk ifsorsuBpapsoicrteRdesbeyartchheosftaCtheinKaey(NDoe.v2e0lo0p5mCeBn7t24P5r0o3-
e
gl and 2006CB921401, 921402), NSFC(No. 60490280 and
n
nta 10574005).
E 0
0 50 100 150 200 250 300
t/Γ
FIG. 5: (Color online) Time evolution of the produced en-
tanglement. The dashed red line indicates Em/~ = 0 [11], ∗ E-mail: [email protected].
whereas the solid lines from top to bottom are plotted with [1] S. L. Braunstein and P. V. Loock, Rev. Mod. Phys. 77,
Em/~=0.1×10−3 (blue),Em/~=0.5×10−3 (green), Em/~= 513 (2005).
2×10−3(black), respectively. Γ=1,σ=0.2kc, δ=0.1. [2] A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777
(1935); John C. Howell, RyanS. Bennink,Sean J. Bent-
ley, R.W. Boyd,Phys. Rev.Lett. 92, 210403 (2004).
as in Fig. 4, therefore, one can hardly produce high en- [3] M.V.Fedorovet.al.,Phys.Rev.Lett.99,063901(2007).
tanglement for hot atoms with σ k , even when the [4] C.K.LawandJ.H.Eberly,Phys.Rev.Lett.92,127903
c
≫
conditions δ 0 and E =0 [11] are fulfilled. (2004).
m
The time≈evolution of the produced atomic entan- [5] K. W. Chan et. al., Phys. Rev. A 68, 022110 (2003); R.
Guo and H. Guo, ibid. 73, 012103 (2006); R. Guo and
glement can be obtained from the general solution in
H. Guo, ibid. 76, 012112 (2007).
Eq. (6). As in Fig. 5, the steady entangled state is
[6] L.Lamata,J.J.Garc´ıa-Ripoll,andJ.I.Cirac,Phys.Rev.
established with a time τ which can be estimated as Lett. 98, 010502 (2007).
τ 1/(δ2Γ). For ~/Em τ [11], the maximal entangle- [7] M. Paternostro, M. S. Kim, G. M. Palma, Phys. Rev.
∼ ≫
ment can be obtained by a single photon detection with Lett. 98, 140504 (2007).
the coupling time T satisfying τ T ~/E . For the [8] S. Parker et al.,Phys. Rev.A 61, 032305 (2000).
m
general case of E = 0, the produ≪ced≪entanglement will [9] D. Schnebleet. al.,Science 300, 475 (2003).
m
6 [10] K.M.R. van derStam et. al., arxiv: 0707.1465v1.
first increase due to the pairwise scattering, and then
decrease to a Bell–like state (K 2) when T ~/E , [11] Em canbemadesmallerforheavieratoms;furthermore,
≈ ≫ m itcanbeefficientlycompensatedbyextendingtheenergy
since the energy–conserved single–atom scattering pro- spectrum of the incident light, see ref. [10]. When ~/Em
cess dominates in this time scale [9]. is much larger than all other time scales in this physical
Conclusion.— From the first principle we have process, we simply set Em =0.
demonstrated that, the mechanism of pairwise photon– [12] T. Bourdel et. al.,Phys. Rev.A 73, 043602 (2006).