Detection of the nonlocal atomic entanglement assisted with single photons

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by  Yu-Bo Sheng

Detection of the nonlocal atomic entanglement assisted with single photons
Yu-Bo Sheng,1,2 and Lan Zhou2,3
∗
1 Institute of Signal Processing Transmission, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China
2 Key Lab of Broadband Wireless Communication and Sensor Network Technology,
Nanjing University of Posts and Telecommunications, Ministry of Education, Nanjing, 210003, China
3College of Mathematics & Physics, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China
Wepresentan efficient way formeasuring theentanglementof theatoms. Through theauxiliary
single photons input-output process in cavity quantum electrodynamics (QED), the concurrence
of the atomic entanglement can be obtained according to the success probability of picking up
the singlet states of the atoms. This protocol has three advantages: First, we do not require the
sophisticatedcontrolled-not(CNOT)gates. Second,thedistributedatomsarenotrequiredtointact
4 with each other. Third, the atomic entanglement can be distributed nonlocally, which provides its
1 important applications in distributed quantumcomputation.
0
2 PACSnumbers: 03.67.Mn,03.67.Hk,03.65.Lx

n
a
J Entanglement plays an important role in current can be described as C = 2α 1 α2. They require
1 quantum information processing (QIP)[1]. Quantum the hyperentanglement to co|m|pplet−e t|he| task. Recently,
1 teleportation[2], quantum key distribution [3], quantum the group of Cao also discussed the measurement of the
dense coding [4], quantum secure direct communication concurrence for two-photon polarization entangled pure

] [5, 6] and other quantum information protocols[7–9], all state with the help of cross-Kerrnonlinearity [18, 19].
h
p rely on the entanglement between distant parties. How- Ontheotherhand,the cavityquantumelectrodynam-
- ever,entanglementisdifficultto characterizeexperimen- ics (QED) has become an important platform to realize
nt tally. Bell inequalities [10], entanglement witnesses [11] the QIP [20]. Over past decades, both excellent the-
a cannot provide satisfactory results of the entanglement atrical and experiments have focused on QED for QIP,
u because they disclose the entanglement of some states such as the generation the single photons [21], the per-
q while fail for other states. Another indirect method for formance of the logic gate [22], and so on [23]. In the
[
measuring the entanglement is the quantum state tomo- early work of QED, they should require the high-quality

1 graphic reconstruction [12, 13]. By reading out the 15 cavity and the strong coupling between the solid qubit
v parameters,they canreconstructthe density matrix ofa and the cavity. Recently, the QIPs based on the low-Q
6 two-qubit state, but it is quite complicated. cavity have been widely discussed. For example, An et
2 In the early work of Bennett et al., they proposed the al. showed that with the help of Faraday rotation, the
5
way for measuring the quantify entanglement named en- differentpolarizedphotonscanobtainthedifferentphase
2
. tanglement of formation [14]. In their protocol, an arbi- shiftaftertheyinteractwiththeatomswhicharetrapped
1 trary two-qubit state with the density matrix ρ can be in a low-Q cavity [24]. Subsequently, a lot of excellent
0
described in terms of exactly calculable quantity, i. e. works for atoms were discussed, including the quantum
4
the concurrence (C). The concurrence of the two-qubit teleportation[25],controlledteleportation[26],swapping
1
: state can be defined as [15, 16] [27], entanglement concentration[28], and so on [29, 30].
v
Interestingly, inspired by the previous works of atoms
i C(ρ)=max 0,λ , λ , λ , λ , (1)
X { 1 − 2 − 3 − 4} and the measurement of the concurrence, we find that
r with the λ (i=1,2,3,4). λ are the non-negative eigen- the Faraday rotation can also be used to perform the
a values of tihe Hermitian maitrix R = √ρρ√ρ in de- measurement of the atomic entanglement. Actually, in
creasing order. Here ρ = (σ σ )ρ (σp σ ), and σ 2007, Romero et al. first discussed the direct measure-
y y ∗ y y y
is the Pauli operator, and ρ ⊗is the comp⊗lexeconjugate mentforconcurrenceofatomictwo-qubitpurestate[31].
∗
of ρ. If we consider ean arbitrary two-qubit pure state Intheirprotocol,theyillustratetheprotocolforRydberg
of the form φ = α0 0 + β 0 1 +γ 1 0 + δ 1 1 , atomscrossingthree-dimensionalmicrowavecavitiesand
with α2 + β| 2i+ γ 2|+i|δi2 =1|,it|hie conc|uir|reince c|ani|bie confined ions in a linear paul trap. They require the
descri|bed| as|C|(φ|)|= φ| σ| σ φ =2αδ βγ . controlled-not(CNOT)gateorthecontrolled-phasegate
y y ∗
In 2006, Wal|boirn e|th a|l. ⊗repo|rteid the|ir e−xper|iment betweentwoatomsto completethe task. In2008,Leeet
about the determination of entanglement with a single al. presented the concurrence of a two-qubit cavity sys-
measurement [17]. In their experiment, they demon- temwiththehelpofflyingatoms[32]. Themeasurement
stratedthemeasuringoftheconcurrenceforatwopolar- forconcurrencebasedontrappedionswerealsoproposed
ized state of the form α01 +β 10 and the concurrence [33].
| i | i
In this paper, we will propose the detection of the
remote atomic entanglement with the help of single
photons. This protocol is quite different from others.
Emailaddress: [email protected] First, it does not require the sophisticated controlled-
∗
2
not (CNOT) gates and the high-Q cavities, which will field is ω . The σ , σ , σ and σ are the lower-
c R R+ L L+
greatlyreleasethe experimentalcomplexity. Second, the ing and raising ope−rators of th−e transition R(L), respec-
distributed atoms are not required to intact with each tively. The ω is the atomic frequency and H means
0 R0
other. Third,theatomicentanglementcanbedistributed the Hamiltonian of the free reservoirs.
nonlocally, which provides its important applications in Briefly speaking, we consider a single photon pulse
distributed quantum computation. with frequency ω enters the optical cavity. Using the
P
This paper is organized as follows: In Sec. 2, we first adiabaticapproximation,wecanobtaintheinput-output
brieflydescribethekeyelementofthisprotocol,i. e. the relation of the cavity field in the form of [26, 27, 29]
basicprincipleoftheFaradayrotation. InSec. 3,wewill
[i(ω ω ) κ][i(ω ω )+ γ]+λ2
explainourprotocolwithasimpleexample. InSec. 4,we r(ω )= c− p − 2 0− p 2 . (4)
will make a discussion about the practical imperfection p [i(ωc−ωp)+ κ2][i(ω0−ωp)+ γ2]+λ2
in experiment. In Sec. 5, we will make a conclusion.
From Eq. (4), if λ = 0, above equation for an empty
cavity can be written as
I. BASIC THEORETICAL MODEL OF THE r (ω )= i(ωc−ωp)− κ2. (5)
PHOTONIC FARADAY ROTATION 0 p i(ω ω )+ κ
c− p 2
Here ω is the frequency of the photon. κ and γ are the
In this section, we will first explain the basic principle p
cavitydamping rateandatomicdecayrate. The λis the
of the photonic Faraday rotation.
atom-cavitycouplingstrength. Thetotalphysicalpicture
oftheprocesscanbedescribedasfollows: ifaleftcircular
polarization photon enters the cavity, and couples with
the atom in the state g , the photon will be reflected
L
| i
to the output mode but induce a phase shift, which can
be described as L g r(ω )L g eiθ L g .
L p L L
Here eiθ =r(ω ).|Init|ereist→ingly, if t|hei|rigiht≈-circul|ari|phoi-
p
ton enters the cavity and couples with the atom in the
state g , it only senses the empty cavity. After the
L
| i
photon reflecting from the cavity, the whole process can
be written as R g r (ω )R g eiθ0 R g .
L 0 p L L
| i| i → | i| i ≈ | i| i
Here eiθ0 = r (ω ). Therefore, if a single photon state
0 p
FIG. 1: The relevant atomic structure subjected to a low-Q 1 (L + R )entersthecavityandcoupleswiththe g
√2 | i | i | Li
cavity fielded. It has an excited state |ei and two degenerate atom, the whole system can be described as
ground states |gLi and|gRi. Itis also shown in Refs.[26,28].
1 1
(L + R )g (eiθ L +eiθ0 R )g . (6)
L L
√2 | i | i | i→ √2 | i | i | i
FromFig. 1,athree-levelatomistrappedinthecavity.
On the other hand, if 1 (L + R ) enters the cavity
It has an excited state e and two degenerate ground √2 | i | i
| i and couples with the g atom, the whole system can
states g and g . Thetransitionsbetweenthe e and R
L R | i
| i | i | i be described as
g , e and g are assisted with a left-circularly (L )
L R
| i | i | i | i
and right-circularly (R ) polarized photon, respectively. 1 1
We can write the Ham| ilitonian of the system as [26] √2(|Li+|Ri)|gRi→ √2(eiθ0|Li+eiθ|Ri)|gRi. (7)
H = j=XL,R[¯hω02σjz +¯hωca†jaj] Tλh=Iefwκ∆2,eΘwcFoen=csaidn|θeor−bthtθae0i|nciasθsec=saltlπhedaatntωhde0θ=F0a=ωracπ2d,a.ωypWr=eotcωaactni−ornκ2e.w,arnitde
the relationship between the photon and the atom as
+ ¯hλ (a†jσj +ajσj+)+HR, (2)
X −
j=L,R L g L g , R g iR g ,
L L L L
| i| i→−| i| i | i| i→ | i| i
L g iL g , R g R g . (8)
with | i| Ri→ | i| Ri | i| Ri→−| i| Ri
∞
HR = HR0+i¯h[Z dω α(ω)(b†j(ω)aj +bj(ω)a†j)]
X
−∞ j=L,R II. MEASURING THE CONCURRENCE OF
+ Z ∞ dω X α(ω)(c†jσj−+cjσj+). (3) THE ATOMIC ENTPAHNOGTLOEMNSENT WITH SINGLE
−∞ j=L,R
Here a and a are the creation and annihilation opera- Nowwestarttoexplainourprotocol. AsshowninFig.
†
tors of the cavity field and the frequency of the cavity 1, two pairs of atomic entanglement are shared by Alice
3
βγ
+ (0 1 1 0
a1 a2 b1 b2
2(αδ 2+ βγ 2) | i | i | i | i
| | | |
p
+ 1 0 0 1 ), (11)
a1 a2 b1 b2
| i | i | i | i
withthetotalsuccessprobabilityofP =2αδ 2+2βγ 2.
1
| | | |
After they obtain the state Φ in Eq. (11), they
a1a2b1b2
| i
perform the Hadamard operations on the atoms a1 and
a2, which makes the Φ become
a1a2b1b2
| i
αδ+βγ
Φ = (0 0 1 1 )
| i′a1a2b1b2 2 2(αδ 2+ βγ 2) | ia1| ia2−| ia1| ia2
| | | |
p
(0 1 + 1 0 )
b1 b2 b1 b2
⊗ | i | i | i | i
αδ βγ
+ − (0 1 1 0 )
a1 a2 a1 a2
2 2(αδ 2+ βγ 2) | i | i −| i | i
| | | |
p
(0 1 1 0 ). (12)
b1 b2 b1 b2
⊗ | i | i −| i | i
FIG. 2: The schematic drawing of the principle of our pro-
From Eq. (12), after performing the Hadamard opera-
tocol. Two input polarized photons are in the same state
1 (|Li+|Ri). The polarization beam splitter can transmit tion, the atoms a1 and a2 are entangled. Finally, they
√2 still prepare a single photon in the input1 mode of the
the photon in the |+i polarization and reflect the photon in
the |−i polarization. Here |±i= 1 (|Li±|Ri). D1, D2, D3 form √12(|Li+|Ri), and let it pass throughthe two cav-
√2 ities a1 and a2. If the photon’s polarization does not
and D4 are four single-photon detectors.
change, it will project the atoms a1 and a2 into the sin-
gletstate 0 1 1 0 . Therefore,ifthe photon
a1 a2 a1 a2
| i | i −| i | i
passes through the cavities and finally be detected by
andBob,respectively. Supposethattheyareinthesame thesingle-photondetectorD inasecondtime, thestate
1
arbitrary two-qubit states of the form Φ will collapse to
| i′a1a2b1b2
Φ =α0 0 +β 0 1 +γ 1 0 +δ 1 1 . (9) 1
a b a b a b a b
| i | i | i | i | i | i | i | i | i Φ = (0 1 1 0 )
| i′a′1a2b1b2 2 | ia1| ia2−| ia1| ia2
Here g 0 and g 1 . First, they prepare two
single|pLhiot≡on|siwith t|heRisa≡me| piolarization Φ1 = Φ2 = ⊗ (|0ib1|1ib2−|1ib1|0ib2). (13)
1 (L + R ). They let the two photons i|n tihe in|tpuit1 Thesuccessprobabilityforobtainingthestate Φ
a√n2Td|hineipfuout2|rpaitaosms tshcraonugbhetwheritctaevnitaiess, respectively. is TP2he=t2o(t|a|ααlδδ|−2s+uβc|γβc|γ2e|s2s).probability for obtaining| tih′ae′1as2tba1tb2e
Φ is P =P P = αδ βγ 2. Therefore, we can
Φ a1b1 Φ a2b2 | i′a′1a2b1b2 1 2 | − |
| i ⊗| i get
= (α0 0 +β 0 1 +γ 1 0 +δ 1 1 )
a1 b1 a1 b1 a1 b1 a1 b1
| i | i | i | i | i | i | i | i
(α0 0 +β 0 1 +γ 1 0 +δ 1 1 ) C(Φ)=2αδ βγ =2√P. (14)
a2 b2 a2 b2 a2 b2 a2 b2
⊗ | i | i | i | i | i | i | i | i | − |
= α2 0 0 0 0 +αβ 0 0 0 1
| ia1| ia2| ib1| ib2 | ia1| ia2| ib1| ib2 Interestingly, from Eq. (14), we have established the
+ αγ 0 a1 1 a2 0 b1 0 b2+αδ 0 a1 1 a2 0 b1 1 b2 relationship between the concurrence of the atomic en-
| i | i | i | i | i | i | i | i
+ αβ 0 0 1 0 +β2 0 0 1 1 tanglementandthesuccessprobabilityformeasuringthe
a1 a2 b1 b2 a1 a2 b1 b2
| i | i | i | i | i | i | i | i
photons. Inapracticalexperiment,weshouldrepeatthis
+ βγ 0 1 1 0 +βδ 0 1 1 1
a1 a2 b1 b2 a1 a2 b1 b2
| i | i | i | i | i | i | i | i protocolformanytimestoobtainthesuccessprobability
+ αγ 1 0 0 0 +βγ 1 0 0 1
| ia1| ia2| ib1| ib2 | ia1| ia2| ib1| ib2 P by calculating the ratio between the detected photon
+ γ2 1 a1 1 a2 0 b1 0 b2+γδ 1 a1 1 a2 0 b1 1 b2 number and the initial total photon number.
| i | i | i | i | i | i | i | i
+ αδ 1 0 1 0 +βδ 1 0 1 1
a1 a2 b1 b2 a1 a2 b1 b2
| i | i | i | i | i | i | i | i
+ γδ 1 1 1 0 +δ2 1 1 1 1 . (10)
| ia1| ia2| ib1| ib2 | ia1| ia2| ib1| ib2 III. DISCUSSION AND CONCLUSION
Therefore,afterthetwophotonspassingthroughthefour
cavities,if both photons do not changeand transmit the So far, we have briefly described our protocol with a
polarization beam splitters (PBSs) and finally are de- simpleexample. Itisshownthatwecancompletethede-
tected by single photondetectors D andD , the atomic tection of the concurrence of arbitrary two-qubit atomic
1 3
state will become purestate by input-output processingofphotons. Inthe
pioneer work of Romero et al., they also discussed the
αδ
Φ = (0 1 0 1 measurement for concurrence of atomic two-qubit pure
a1a2b1b2 a1 a2 b1 b2
| i 2(αδ 2+ βγ 2) | i | i | i | i
states. They require the atomic CNOT gate to com-
| | | |
p
+ 1 0 1 0 ) plete the task. However, the CNOT gate cannot be
a1 a2 b1 b2
| i | i | i | i
4
the photon change, which will has some probability to
be 1 (L + R ), and finally be detected by D or D .
√2 | i | i 1 3
In this way, it will contribute a successful case with the
probability of
1 1
P = ( L + R) (ei2θ L R )2
error
|√2 h | h | √2 | i−| i |
ei2θ 12
= | − | . (17)
4
In Fig. 3, we calculate the P altered with θ. From
error
Eq. (10), in an ideal case, the item 0 0 0 0
a1 a2 b1 b2
| i | i | i | i
has no contribution to the successful case, while it will
hastheprobabilityof α4P2 tomakeD andD both
| | error 1 3
fire. The second item 0 0 0 1 also has the
a1 a2 b1 b2
| i | i | i | i
FIG. 3: The probability of error is altered with θ. We let probability of αβ 2Perror to make D1 and D3 both fire.
| |
θ∈(0,2π). Therefore, if P 1, we can estimate the total suc-
error
≪
cess probability as
P αδ βγ 2(1+P )2. (18)
′ error
well performed in current experimental conditions. In ≈| − |
this protocol, we do not require the CNOT gate, and Here we omit the contribution of P2 for P2
we also do not require the atoms to interact with each P . Therefore, the concurrence weilrlrobre increaersreodrb≪e-
error
other. In our protocol, 87Rb atom trapped in a fiber- cause ofthe imperfection. Onthe other hand, we should
basedFabry-Perotcavityisagoodcandidatesystem[34]. also consider the photon loss. The photon loss is the
In Ref.[34], the two ground states gL and gR can be main obstacle in realistic experiment. The minor mis-
| i | i
the states of F = 2 ,mF = 1 of level 5S1/2. The alignment, cavity mirror absorption and scattering, and
| i ±
transition frequency between the excited state and the even the tiny piece of dust, can induce the photon loss.
ground state is ω0 = 2πc/λ at λ = 780nm. the cavity The photon loss will decrease the total success probabil-
length L = 38.6µm. The cavity decay rate κ = 2π 53 ity. Suppose the efficiency of a single-photon detector
×
MHz, and the finesse F = 37000. Certainly, in a practi- is η , the efficiency of coupling and transmission of the
d
calexperiment,wealsoshouldconsidertheexperimental photonfromfibertocavityisT ,andthetransmissionof
f
imperfection. For example, in this protocol, we require eachphotonthroughthe other opticalcomponents is T ,
o
ω0 = ωc, ωp = ωc − κ2, and λ = κ2. Actually, in practi- wecanrewritethetotalsuccessprobabilityinadetection
cal experiment, we cannot ensure to control θ =π to let round as
λ = κ. On the other hand, the photon loss and the im-
2
perfect detection will also affect the quality of the whole P =P T3 T3 η3. (19)
T ′× f × o × d
protocol. If θ = π, we should rewrite the relationship in
6
Eq. (8) as In practicalcondition, the efficiency ofthe single-photon
detector is η = 0.28. Suppose that T = 0.5, and T =
d f o
|Li|gLi→eiθ|Li|gLi,|Ri|gLi→i|Ri|gLi, 0.95. We can obtain that PT ≈ 2.4×10−3×P′. If the
|Li|gRi→i|Li|gRi,|Ri|gRi→eiθ|Ri|gRi. (15) generation rate of the single-photon is 1×10−5s−1, this
protocol can be completed in 0.01s.
We can also obtain that In conclusion, we have described the detection of the
1 1 nonlocal atomic entanglement assisted with single pho-
(L + R )0 0 (ei2θ L R )0 0 , tons. Throughthe auxiliarysinglephotons input-output
√2 | i | i | i| i → √2 | i−| i | i| i
process in cavity quantum electrodynamics (QED), the
1 1
(L + R )1 1 ( L +ei2θ R )1 1 , concurrenceofthe atomic entanglementcanbe obtained
√2 | i | i | i| i → √2 −| i | i | i| i according to the success probability of picking up the
1 1 singlet states of the atoms. We also discussed the prac-
(L + R )0 1 ieiθ(L + R )0 1 ,
√2 | i | i | i| i → √2 | i | i | i| i tical imperfection of this protocol. It is shown that the
1 1 success probability will be increased if κ = λ. On the
(L + R )1 0 ieiθ(L + R )1 0 . (16) 2 6
other hand, the photon loss will decrease the total suc-
√2 | i | i | i| i → √2 | i | i | i| i
cess probability. This protocol has several advantages:
From above description, it is shown that if the atomic First, we do not require the sophisticated CNOT gates.
state is 0 1 or 1 0 , the polarization state is not Second, the distributed atoms are not required to intact
| i| i | i| i
changed. That is to say, the imperfect operation does with each other. Third, the atomic entanglement can
notaffectthefidelityofthestate. However,iftheatomic be distributed nonlocally, which provides its important
state is 0 0 or 1 1 , it will make the polarization of applications in distributed quantum computation.
| i| i | i| i
5
ACKNOWLEDGEMENTS and 11347110. and the Project Funded by the Priority
Academic Program Development of Jiangsu Higher Ed-
This work is supported by the National Natural Sci- ucation Institutions.
ence Foundation of China under Grant Nos. 11104159
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