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Concentration scheme for partially entangled photon states via entanglement reflector and no Bell-state analysis
Optics Communications 284 (2011) 2402–2407
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Concentration scheme for partially entangled photon states via entanglement reflector and no Bell-state analysis Shao-Hua Xiang ⁎, Zhen-Gang Shi, Wei Wen, De-Hua Lu, Xi-Xiang Zhu, Ke-Hui Song Department of Physics and Information Engineering, Huaihua University, Huaihua 418008, PR China
a r t i c l e
i n f o
Article history: Received 7 September 2010 Received in revised form 13 November 2010 Accepted 30 December 2010 Available online 18 January 2011 Keywords: Entanglement concentration Spin–photon interface Entanglement reflector Spin detection
a b s t r a c t We propose a protocol to concentrate partially entangled states of photons using entanglement reflector, which consists of a single electron spin confined in a charged quantum dot inside a single-sided microcavity. The outstanding advantage of the proposed scheme is its experimental simplicity and feasibility since it only needs to perform a single local measurement on electronic spins rather than a joint Bell-state measurement on photons. We then extend this scheme to concentrate N-photon Greenberger–Horne–Zeilinger state. Finally, we analyze the influence of various imperfections on the scheme. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Quantum entanglement has become a useful physical resource for quantum information processing, such as quantum teleportation [1], quantum dense coding [2], quantum cryptography [3], and other nonclassical interference phenomena [4]. To accomplish these tasks successfully, one needs to distribute and store maximally entangled states of two or more qubits between the distant parties. In practical situations, however, a quantum system cannot completely isolate from its environment, so the degradation of entanglement is unavoidable during storing, processing and transmission processes. As entanglement cannot be increased by local operations and classical communication (LOCC) [5], entanglement purification will be essential for the best performances of quantum communication and quantum computation. Entanglement purification is a process that one can extract a certain number of almost perfectly entangled states from a large number of less entangled states using LOCC. For the pure nonmaximally entangled states case, this process will be called entanglement concentration. In 1996, Bennett et al. [6] proposed the first concentration protocol for identical partially entangled pairs of two-state particles where the two parties of quantum communication need to perform a collective and nondestructive measurement for the joint state of n pairs of qubits; unfortunately, this scheme is hard to implement in practice because so many particles are needed to measure simultaneously. Bose et al. [7]
⁎ Corresponding author. Tel.: +86 745 2851011; fax: +86 745 2851305. E-mail address: [email protected] (S.-H. Xiang). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.12.088
presented another concentration scheme of polarization entangled photons via entanglement swapping. After that, a lot of authors have been devoted to polarization entanglement concentration using by linear optical elements and collective Bell-state measurement [8–11] or by nonlinear optics and quantum nondemolition detector [12]. Experimental concentration of photon entangled state has been demonstrated [13,14]. Very recently, entanglement concentrations for unknown atomic entangled states [15] and electron-spin entanglement [16] have been proposed. As for the case of linear optical system, its major disadvantage is that it would be more difficult to perfectly distinguish four different Bell states from each other within the current experimental capabilities, while quantum nondemolition measurement is the requirement of strong cross-Kerr media. Such do not exist in readily available materials [17]. Currently, Hu et al. proposed a novel quantum device consisting of a singly charged quantum dot (QD) coupled to a microcavity, which can directly split an initial product state of photon and spin into the entangled state via transmission and reflection in a deterministic way and is immune to the fine-structure splitting in realistic semiconductor QDs [18]. Using this device, they presented a deterministic and scalable scheme for creating spin–spin [18], photon–photon [19,20], photon– spin entanglements [20] as well as quantum teleportation [21]. In this paper, we present a way for entanglement concentration of partial polarization entangled state of photons via a single QD spin in an optical microcavity based on giant circular birefringence. Unlike previous protocols, our scheme only requires the single-spin measurements of QDs instead of the routine Bell measurement on photons, which will make the scheme easier to implement experimentally. This paper is organized as follows. In Section 2, we describe the photon–spin entanglement reflector. In Section 3, entanglement
S.-H. Xiang et al. / Optics Communications 284 (2011) 2402–2407
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concentration of two partially entangled photon pairs is presented and then it is extended to the case of N-photon GHZ state. In Section 4, we discuss the influence of imperfect setup such as cavity photon loss and spin decoherence on the current scheme. Finally, we summarize the paper in Section 5.
uncoupled cavity and gets a phase shift of φ0(ω). This interaction is summarized as follow:
2. Photon–spin entanglement reflector
jRij↑i→r0 e
Before implementing our concentration scheme, we briefly recapitulate the photon–spin entanglement reflector introduced in Ref. [18], which consists of a singly charged quantum dot with only one excess electron placed at the antinode of an optical microcavity where the distributed Bragg reflectors and transverse index guiding provide the three-dimensional confinement of light [18–21]. Such nanostructures may be in fact the self-assembled In(Ga)As QD, GaAs interfacial QD, or semiconductor nanocrystal. As is well known, the optical properties of singly charged QDs are strongly determined by the optical transitions of the trion X− state (bond state of two electrons and hole). The single electron states have Jz = ± 1/2 spin (|↑〉, |↓〉) and the holes have Jz = ± 3/2 spin (|⇑〉, |⇓〉). Due to the Pauli blocking, the interaction between light and X− not only relies on the polarization of light but also on the initial spin state of the excess electron. That is: the left circularly polarized photon (labeled by L photon) only couples the electron in the spin state |↑〉 to X− in the spin state |↑ ↓⇑〉 with the two antiparallel electron spins, while the right circularly polarized photon (indicated by R photon) only couples the electron in the spin state |↓〉 to X− in the spin state |↑↓⇑〉. Here we assume the spin-quantization axis is along the normal direction of cavity. For a trion state, the two electrons form a single state and therefore have total spin zero, which preserves electron-spin interaction with the hole spin. In the weak-excitation approximation, the reflection coefficient of this X−-cavity system is given by [18] κ iðωX − −ωÞ + γ2 r ðωÞ = 1− iðωX − −ωÞ + γ2 iðωc −ωÞ + κ2 +
κs 2
+ g2
;
ð1Þ
where ω, ωc, and ωX− are the frequencies of the input photon, cavity mode, and X− transition, respectively. g is the X−-cavity coupling strength and γ/2 is the X− dipole decay rate. κ and κs/2 are the cavity decay rates into the input/output modes and the leaky modes (side leakage), respectively. The background absorption can be included in κs/2. Here we focus on the resonant interaction with ωc = ωX− = ω0 and the cavity is assumed to be single-side cavity with back mirror perfectly reflective and front mirror partially reflective. By setting g = 0, we obtain the reflection coefficient r0(ω) for a uncoupled cavity where the QD does not coupled to the cavity
r0 ðωÞ =
iðω0 −ωÞ + κ2s − κ2 : iðω0 −ωÞ + κ2s + κ2
ð2Þ
If the side leakage is much smaller than the main cavity decay, i.e., κs ≪ κ, then we can obtain near-unity reflectance r0(ω) ≃ 1 in this case, whereas for the coupled cavity where X− strongly couples to the cavity, i.e., g N (κ, γ), we obtain |rh(ω)| ≃ 1 and φh(ω) ≃ 0 within a frequency window |ω − ωc| ≪ g. It has been shown in Ref. [18] that the X−-cavity system shows large transmittance and reflectance difference between the uncoupled cavity with g = 0 and the coupled cavity with g ≠ 0. If the single excess electron lies in the spin state |↑〉, the L photon feels a coupled cavity and gets a phase shift of φh(ω) after reflection, while the R photon feels the uncoupled cavity and gets a phase shift of φ0(ω). Conversely, if the electron lies in the state |↓〉, the R photon feels a coupled cavity and gets a phase shift of φh(ω) after reflection, while the L photon feels the
iφh
jLij↑i;
ð3aÞ
iφ0
jRij↑i;
ð3bÞ
iφ0
jLi j↓i;
ð3cÞ
iφh
jRij↓i;
ð3dÞ
jLij↑i→rh e
jLij↓i→r0 e
jRij↓i→rh e
where r0 = |r0(ω)| and φ0 = arg[r0(ω)] are the reflection ratio and phase shift of the reflected light for the uncoupled cavity, and rh = |r(ω)| and φh = arg[r(ω)] for the coupled cavity, respectively. This phenomenon is called the giant circular birefringence (GCB), allowing us to construct an entanglement reflector of photon–spin system. In fact, the change can also be described by a reflection operator,
rˆðωÞ = jr0 ðωÞeiφ0 ð jRihRj⊗ j↑ih↑j + jLihLj⊗ j↓ih↓j Þ iφh
+ jrh ðωÞe
ð4Þ
ð jLihLj ⊗j↑ih↑j + jRihRj⊗j↓ih↓j Þ:
For |r0(ω)| ≃ 1 and |rh(ω)| ≃ 1 or for balanced reflectance |r0(ω)| = |rh(ω)|, the reflection operator can be simplified as rˆðωÞ = jr0 ðωÞj eiφ0Uˆ ðΔφÞ with Δφ = φh − φ0. Controlling the photon frequency ω and setting Δφ = π/2, we can obtain a two-qubit phase shift operator iπ ð jLihL j⊗j↑ih↑j + jRihRj⊗j↓ih↓ j Þ ; Uˆ ðπ = 2Þ = exp 2
ð5Þ
which means that it could create photon–spin entanglement. In the following, we exploit this interaction to concentrate partially entangled photon pairs between distant parties. 3. Concentration scheme for partially photon entangled states The principle of our concentration scheme is shown in Fig. 1. Initially two distant parties, Alice and Bob, share two photon pairs (1, 3) and pffiffiffi (2,ffiffiffiffiffiffiffiffiffiffi 4) in the nonmaximally entangled state j ϕiij = a jRii jRij + p 1−a jLii jLij with the Schmidt coefficient a being positive real number, where photons 1 and 2 belong to Alice and photons 3 and 4 to Bob. In order to concentrate the maximally entangled state from these less entangled states, they are asked for the help from the third partner, Cliff, who holds two identical QD-cavity systems described above, an optical switch (SW), an optical circulator (OC), a T transformation, and two electronic-spin detectors (Dj, j = 1, 2). Photons 2 and 3 are sent to Cliff, at whose situation an optical switch and an optical circulator successively direct these photons to two QD-cavity systems with the spin electron pffiffiffi prepared in a superposition state j + ij = j↑ij + j↓ij = 2 (j = 1, 2). It is important to note that a T operation is needed before these photons are reflected by the second QD-cavity unit. Finally, Cliff makes measurements on two single-electron spins, and thereby our scheme can complete with a high probability of success. The detailed analysis of our scheme is followed as. Two pairs of photons are prepared initially in the following state jϕi1234 = jϕi13 ⊗jϕi24 = a jRi1 jRi2 jRi3 jRi4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + að1−aÞ jRi1 jLi2 jRi3 jLi4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + að1−aÞ jLi1 jRi2 jLi3 j Ri4 + ð1−aÞjLi1 jLi2 jLi3 jLi4 :
ð6Þ
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a
After the reflection of photons 2 and 3 at the first QD-cavity system, the total state of four photons and two spins is transformed into j ϕir = a jRi1 jRi2 jRi3 jRi4 j−i1 j + i2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + i að1−aÞ jRi1 jLi2 jRi3 jLi4 j + i1 j + i2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + i að1−aÞ jLi1 jRi2 jLi3 jRi4 j + i1 j + i2
ð7Þ
−ð1−aÞjLi1 jLi2 jLi3 jLi4 j−i1 j + i2 ; iφh has been discarded and the state where the global phase pffiffiffi e j−ik = ð j↑ik −j↓ik Þ = 2. Next, Cliff performs a T transformation represented by
i jRij → pffiffiffi jRij′ + j Lij′ ; 2
ð8aÞ
1 jLij → pffiffiffi jRij′ −jLij′ 2
ð8bÞ
on photons 2 and 3. Such an operation can be realized via quantum circuit, as shown in Fig. 1(b). It is known that the action of the polarization beam splitter (PBS) is it transmits only horizontal polarization component and reflects vertical component, while the transformation of a quarter-wave plate (QWP) is expressed by jHij → p1ffiffi2 jHij′ + jVij′ and jVij → p1ffiffi2 jHij′ − jVij′ . Meanwhile, the phaseshifter is assumed to have phase θ = π/2and we define jRij = p1ffiffi2 jHij + ijVij and jLij = p1ffiffi2 jHij −ijVij , respectively. So, the T operation process can be described as follows: 1 i j Ria PBS 1 pffiffiffi ð jHic + ijVib Þ PS pffiffiffi ð jHid + jVib Þ → 2 → 2 i i PBS 2 pffiffiffi ð jHie + jVie Þ QWP pffiffiffi jRia′ + jLia′ ; → 2 → 2 1 i p ffiffiffi ð jHic −ij Vib Þ PS pffiffiffi ð jHid −jVib Þ jLia PBS 1 → 2 → 2 i 1 PBS 2 pffiffiffi ð jHie −jVie Þ QWP pffiffiffi jRia′ −jLia′ : → 2 → 2
ð9Þ
with certainty. For example, if both spins are detected in the state | + i 1| − i 2 or | + i 1| + i 2 the photons 1 and 4 are projected into one of the following states with success probability a(1 − a):
+ ½a jRi1 jRi4 −ð1−aÞjLi1 jLi4 j−i1 j + i2 jψiþ 2′ 3′ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + að1−aÞð jRi1 jLi4 + jLi1 jRi4 Þj + i1 j + i2 jϕi− 2′ 3′ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; + að1−aÞð jRi1 jLi4 − jLi1 jRi4 Þ j + i1 j + i2 jψi− 2′ 3′
ð10Þ
where the four Bell states are jRi2′ j Ri3′ jLi2′ jLi3′ 1 and jψi2′ 3′ = pffiffi2 jRi2′ jLi3′ jLi2′ jRi3′ . Subsequently, let photons 2′ and 3′ send into the second QD-cavity unit. When these photons are reflected by this unit, Eq. (10) is transformed into j ϕir2 = ½ajRi1 jRi4 + ð1−aÞjLi1 jLi4 j−i1 j−i2 jϕi− 2′ 3′
+ i½ajRi1 jRi4 −ð1−aÞ jLi1 jLi4 j−i1 j + i2 jψiþ 2′ 3′ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + að1−aÞð jRi1 jLi4 + jLi1 jRi4 Þj + i1 j−i2 jϕiþ 2′ 3′ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; + i að1−aÞð jRi1 jLi4 − jLi1 jRi4 Þj + i1 j + i2 jψi− 2′ 3′ 2iφ0
ð11Þ
where we have discarded the global phase factor e . It is interesting to see that our scheme is realized either by detecting the electronic spins or by implementing the Bell-state measurement of photons 2′ and 3′. However, it is impossible to completely distinguish four Bell states of photons with current technology. Therefore, Cliff can perform a single measurement on electronic spins 1 and 2 in such a way that photons 1 and 4 can be projected into the desired state
ð12Þ
Thus, Alice and Bob obtain the maximally entangled photon pair from two previously shared partially entangled photon pairs by LOCC. The concentration succeeds. But when the spins 1 and 2 are detected in the state | − i 1| + i 2 or | − i 1| − i 2, photons 1 and 4 will be projected into one of the following entangled states with probability (2a2 − 2a + 1)/2: jϕi14 = N½a jRi1 jRi4 j ð1−aÞjLi1 jLi4 ;
j ϕir1 = ½ajRi1 jRi4 + ð1−aÞjLi1 jLi4 j−i1 j + i2 jϕiþ 2′ 3′
= p1ffiffi2
Fig. 1. (a) Scheme of entanglement concentration. Black spheres with and without the arrow stand for quantum dots and photon modes, respectively. SW: optical switch, OC: optical circulator, Di: electron spin detectors, T: T operation. (b) T operation. PBS: polarizing beam splitter, PS: phase shifter, QWP: quarter-wave plate, M: plane mirror.
1 jψi14 = pffiffiffi ½ jRi1 jLi4 j jLi1 jRi4 : 2
After this transformation, Eq. (7) becomes
jϕi 2′ 3′
b
ð13Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where N = 1 = 2a2 −2a + 1. A question arises: which is more entangled—the state (13) or the originally non-maximally-entangled states? Using Wootters' formula [22], we calculate the ratio of the degree of entanglement of the concentrated state to that of the desired state as
r=
Ccen = Cdes
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi að1−aÞ : 2a2 −2a + 1
ð14Þ
It is clear to see that when a = 1/2, the concentrated state (13) has the same degree of entanglement as the initial shared state (i.e., r = 1). This process is in fact amount to entanglement swapping [23]. In other cases, we have r b 1, which implies that the concentration scheme fails. The above scheme can be directly generalized to the case of N-photon entangled state. Assume, for example, that Alice and Bob share the following partially entangled states: ⊗N
jϕi1⋯N = α jRi
⊗N
+ β jLi
;
jϕiab = α jRia jRib + β jLia jLib ;
ð15aÞ ð15bÞ
where |α|2 + |β|2 = 1 and the notation | i ⊗ N = | i 1| i 2 … | i N. Without loss of generality, assume that Alice owns k(k b N) photons of the first entangled pair and one photon (or, say, photon b) of the second pair, while all the remaining photons belong to Bob. To this aim, let Alice
S.-H. Xiang et al. / Optics Communications 284 (2011) 2402–2407
and Bob send photons N and a to Cliff. After the reflection at the first QD-cavity system, the total state of the system evolves into 2 ⊗N−1 2 ⊗N−1 j ψi = α j Ri j Rib j RiN jRia −β j Li jLib jLiN j Lia j −i1 j + i2 ð16Þ ⊗N−1 ⊗N−1 + iαβ j Ri j Lib j RiN jLia + j Li j Rib jLiN j Ria j + i1 j + i2 :
When photons a and N are reflected by the second QD-cavity unit, the state (16) becomes 2 ⊗N−1 2 ⊗N−1 − jϕi = α jRi jRib + β jLi jLib j −i1 j−i2 j ϕiN′ a′ 2 ⊗N−1 2 ⊗N−1 þ + i α jRi j Rib −β jLi jLib j −i1 j + i2 j ψiN′ a′ ⊗N−1 ⊗N−1 þ + αβ jRi jLib + j Li j Rib j + i1 j−i2 j ϕiN′ a′ ⊗N−1 ⊗N−1 − + iαβ jRi j Lib −j Li jRib j + i1 j + i2 jψiN′ a′ ;
where we have discarded the global phase factor ei4φ0. Cliff performs a single measurement on the electron spins 1 and 2, which is described ðiÞ ðiÞ ˆ = j+ i h + j; Π ˆ = j−i h−j . The conditional by a collection Π ii ii þ − state of photons 1 and 4 reads ρ14 =
h ð1Þ ð2Þ þ i 1 ˆ ðl 1Þ Π ˆl Π ˆm ˆ ðm2Þ ; ðl; m = Þ; Tr Π jϕiTT hϕj Π P Δ
ð17Þ
±
1 2 ⊗N−1 2 ⊗N−1 jϕi = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α jRi jRib + β jLi jLib ; 4 4 α +β
ð18aÞ
1 2 2 ⊗N−1 2 ⊗N−1 jϕi = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α jRi jRib −β jLi jLib ; 4 4 α +β
ð18bÞ
1 ⊗N−1 3 ⊗N−1 jϕi = pffiffiffi jRi jLib + jLi jRib ; 2
ð18cÞ
1 ⊗N−1 4 ⊗N−1 jϕi = pffiffiffi jRi jLib −jLi jRib : 2
ð18dÞ
It is to see that Alice and Bob can concentrate GHZ state of N photons with the total success probability 2α2β2 by aid of the third partner.
h ð1Þ ð2Þ þ i ˆl Π ˆm ˆ ðm2Þ : ˆ ðl 1Þ Π P = Tr Π jϕiTT hϕj Π
þ
i 1h 2 2 2 2 jϕiT = aj Ri1 jRi4 r0 j ↑i1 −rh j↓i1 + ð1−aÞ jLi1 jLi4 rh j↑i1 −r0 j ↓i1 4 h i 2 2 2 2 × r0 j ↑i2 −rh j ↓i2 jRi2′ jRi3′ − rh j ↑i2 −r0 j↓i2 j Li2′ j Li3′ r r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + 0phffiffiffi að1−aÞ j + 〉1 ½ j Ri1 jLi4 + jLi1 j Ri4 2 h 2 i 2 2 2 2 × r0 j↑i2 −rh j ↓i2 jRi2′ jRi3′ + rh j↑i2 −r0 j ↓i2 jLi2′ jLi3′ ir r h 2 2 + 0 h ajRi1 j Ri4 r0 j↑i1 −rh j ↓i1 2 i 2 2 þ −ð1−aÞ jLi1 jLi4 rh j↑i1 −r0 j↓i1 j + i2 j ψi2′ 3′ 2 2
ð19Þ
þ
over the states of photons 2′ and 3′, is given by 0 B B B B B B 4 B g ++ B ρ14 = B 8P B B B B B B @
2 2
a h g2
0
0
2h r0 rh að1−aÞ g4
0
2r0 rh að1−aÞ g4
2h r0 rh að1−aÞ g4
0
0
0
2 2 2
2 2
−
að1−aÞ g2
2 2
2r0 rh að1−aÞ g4 2 2 2
1 að1−aÞ C C g2 C C C C 0 C C C; C C 0 C C C 2 2 C ð1−aÞ h A g2
−
ð22Þ
pffiffiffi
pffiffiffi where g = r02 −rh2 = 2, h = r02 + rh2 = 2, ε = g2 − 2r20r2h and the
1 2 2 2 1 success probability P = g h 2a −2a + 1 + h2 r02 rh2 að1−aÞ. 8 2 Similarly, in the case where both spins are detected in the state | + i 1| − i 2, the state of photons 1 and 4 is easily calculated to be 0
In the above scheme, we have neglected the cavity side leakage and the electron-spin decoherence; i.e., we only consider the case where g ≫ (κ, γ), κs = 0, and the electron-spin coherence time T → ∞. This is an ideal case. In practical situations, however, our scheme can be affected by a number of noises, including photon loss and the electronic spin decoherence. In the following, we investigate the impact of these factors on our entanglement concentration scheme. We first consider cavity side leakage effect and take the twophoton concentration scheme as example. Assume that the parameters r0 and rh will be little deviation from unity. Using Eqs. (3a), (3b), (3c) and (3d) and repeating the same procedure as before, we can calculate the state of the system after photons 2′ and 3′ exiting through the second QD-cavity system as
ð21Þ
Next, we proceed to consider the case in which both spins are detected in the state | + i 1| + i 2 or | + i 1| − i 2. For the case of the ð1Þ ð2Þ ˆ ˆ Π , the state of photons 1 and 4, after tracing measurement Π
4. Feasibility of entanglement concentration scheme
ir r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − + p0 ffiffiffih að1−aÞ½ jRi1 j Li4 −j Li1 j Ri4 j + i1 j + i2 j ψi2′ 3′ ; 2
ð20Þ
where TrΔ stands for tracing over the states of both spins and photons 2′ and 3′ and the probability of finding this state is calculated as
where |ϕ i N ′a′ and |ψ i − N′a′ are four Bell states of photons N′ and a′, and we have discarded the global phase factor e2iφ0 in deriving the above Eqs. (16) and (17). Finally, Cliff operates a detection of the electron spins 1 and 2 in the {| + i i, | − i i} basis so that all the photons owned by Alice and Bob will collapse into one of the following states 1
2405
+−
ρ14
B B B B B B 4 2B g h B = B 8P B B B B B B @
2
a g2
0
0
0
2r02 rh2 að1−aÞ g4
2r02 rh2 að1−aÞ g4
0
2r0 rh að1−aÞ g4
2r0 rh að1−aÞ g4
0
0
2 2
−
að1−aÞ g2
2 2
1 að1−aÞ C g2 C C C C C 0 C C C: C C 0 C C C 2 C ð1−aÞ A g2 ð23Þ
−
Thus, we can compute the total success probability of concentrating maximally entangled states as Pt =
1 2 2 2 2 2 2 g h 2a −2a + 1 + h r0 rh að1−aÞ: 4
ð24Þ
We use the fidelity to measure how close the states (22) and (23) come to the desired states (12). It is very easy to calculate the corresponding fidelities as ++
F14
+−
=
F14 =
þ ++ 14 hψjρ14
þ
1 4 4 r r að1−aÞ; P 0 h
ð25aÞ
1 2 2 2 r r h að1−aÞ: 2P 0 h
ð25bÞ
jψi14 =
− +− − 14 hψjρ14 jψi14
=
It is not difficult to see that when the balanced reflection condition, r0 = rh ≠ 1, is fulfilled, one has the unity fidelity and the success
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probability 2r80a(1− a). It implies that the cavity side leakage (or called photon loss) has no influence on the fidelity of the concentrated entangled states and only decreases the success probability by a factor 2r80. To meet this condition, we will require the parameters of QD-cavity system as ω − ωc = κ/2, ωX = ωc, κs/κ = 0.01, γ/κ = 0.15, and g/κ = 2.83; moreover, the mirror of the cavity has the reflection coefficient 0.991, which is completely realistic with current experimental technologies. Recently, for micropillars with diameter around 1.5 μm, the coupling strength g = 80 μev and the microcavity Q factor of 104 ∼ 105 have been reported [24,25]. With these data, we can find g/κ = 2.4. Thus, all these parameters mentioned-above are achieved and then this condition can be met experimentally. In addition, it has been shown in Ref. [24,25] that γ is about several μev in this system. Therefore, although the complete reflection is impossible, our scheme would be feasible with high fidelity and high success probability using current techniques as long as the balance reflection condition holds. Another important limitation is the effect of the electron-spin decoherence and relaxation. It has been found that due to the hyperfine interaction between the electron spin and 104 ∼ 105 host nuclear spins, the spin coherence time in GaAs-based or InAs-based QD is extremely short and, for example, the spin-dephasing time is typically around 5∼ 10 ns [26,27], but can be increased by several orders of magnitude by spin echo techniques and manipulations of the nuclear spins [28,29], whereas the electron spin relaxation time is relatively long (∼ ms) due to the suppressed electron–photon and spin–orbit interactions in QDs, so the time required to complete the proposed scheme should be much shorter than the spin coherence time. On the other hand, the QD electron–spin superposition states (| ± i) can be prepared, for instance, by optical pumping and/or optical cooling [30,31] or by picosencond or femtosecond optical pulses [32,33]. From the spin basis state, there are two methods to obtain the spin superposition state: either via spin–flip Raman transitions [30] or by implementing single spin rotations using nanosecond microwave pulses [27]. Ultrafast spin manipulation through ac-Stack effect has been reported in semiconductor quantum wells on the picosecond time scales and in semiconductor QDs on the picosecond time scales [26], which are much shorter than the spin coherence time. Additionally, it is worthy to point out that quantum dot electron spin detection remains a challenging task, but in our case, the detection scheme can be implemented using a spin quantum nondemolition (QND) method. Here we only consider the two-photon concentration scheme; this method can be applied to other cases. The proposed detection scheme includes following three steps, as shown in Fig. 2. (1) Cliff applies a π/2 microwave pulse on each electron spin pffiffiffi(It is indeed equivalent pffiffiffito a Hadamard gate, i.e., j↑i→ð j↓i + j↑iÞ = 2 and j↓i→ð j↓i− j↑iÞ = 2) in such way that these spin superposition states can be rotated to the states | ↑ i 1| ↑ i 2, | ↑ i 1| ↓ i 2, | ↓ i 1| ↑ i 2, and | ↓ i 1| ↓ i 2. (2) Two photons A and B in pffiffiffi the state jRij + jLij = 2ð j = A; BÞ are imported into two QD-cavity units, respectively. Note that photons A and B have the same frequency as these photons above. After reflection, the total state of the system is given by W
jϕi2 = ½ajRi1 j Ri4 + ð1−aÞj Li1 jLi4 j + 45o iA j + 45o iB j↑i1 j↑i2 jϕi− 2′ 3′ o o þ −½ajRi1 jRi4 −ð1−aÞj Li1 jLi4 j + 45 iA j−45 iB j↑i1 j↓i2 jψi2′ 3′ ð26Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o o þ + i að1−aÞð jRi1 jLi4 + jLi1 jRi4 Þj−45 iA j + 45 iB j↓i1 j↑i2 jϕi2′ 3′ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; −i að1−aÞð jRi1 j Li4 −j Li1 jRi4 Þj−45o iA j−45o iB j ↓i1 j↓i2 jψi− 2′ 3′
pffiffiffi where j 45o i = ð jHi jViÞ = 2. (3) Finally, Cliff makes a measurement on each of the photons A and B in the orthogonal linear polarizations so that photons 1 and 4 held by Alice and Bob will be projected into one of Eqs. (12) and (13). For example, photons A and B are detected in the state | − 45o i A| − 45o i B, so both electron spins are definitely in the state | ↓ i 1| ↓ i 2 and Alice and Bob can obtain the
QD-cavity 1
QD-cavity 2 photon A
+45 -45
MW 1
photon B
+45 -45
MW 2
Fig. 2. Schematic of QND electron spin. The thick arrow indicates a π/2 microwave pulse.
maximally entangled state |ψ i − 14. We emphasize that the detection of photons in these bases has been experimentally realized [13,34]. It is interesting to note that Petta et al. [35] have recently demonstrated coherent control of electronic spin states in a double quantum dot (DQD) by sweeping an initially prepared spin-singlet through a single-triplet anticrossing in the energy-level spectrum where such an anticrossing serves as a beam splitter for the incoming quantum states, yielding the coherent quantum oscillations between the singlet state and a triplet state. Therefore, this type of photon–spin interface will open interesting perspectives for solid quantum information processing. 5. Conclusions In conclusion, we have proposed a practical scheme for entanglement concentration of unknown polarization entangled state based on the spin–photon interface in QD-cavity system. Compared with other concentration schemes, this protocol is simpler and convenient as it does not require the joint Bell state measurement on photons. Moreover, it does not need to know accurately the information about the less-entangled state in advance. We have extended this scheme to distill the case of N-photon GHZ state. In addition, we have studied the influence of numerous practical situations on the proposed scheme. Our results indicate that, when the balance reflection condition is satisfied, our scheme is immune to photon loss but greatly affected by spin decoherence of the electrons. With the development of cooling and fast coherent control of hole–spin states, we hope that the present scheme could be realized in the foreseeable future. Acknowledgments This work was partially supported by Hunan Provincial Natural Science Foundation of China (Grant no. 10JJ6010), the Key Project Foundation of Hunan Provincial Education Department, China (Grant no. 10A095), and the Youth Foundation from Huaihua University Grant no. HHUQ2009-09. References [1] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895. [2] C.H. Bennett, S.J. Wiesner, Phys. Rev. Lett. 69 (1992) 2881. [3] A.K. Ekert, Phys. Rev. Lett. 67 (1991) 661. [4] Y.H. Shih, A.V. Sergienko, Phys. Rev. A 50 (1994) 2564. [5] M. Plenio, V. Vedral, Contemp. Phys. 39 (1998) 431. [6] C.H. Bennett, H.J. Bernsten, S. Popescu, B. Schumacher, Phys. Rev. A 53 (1996) 2046. [7] S. Bose, V. Vedral, P.L. Knight, Phys. Rev. A 60 (1999) 194. [8] T. Yamamoto, M. Koashi, N. Imoto, Phys. Rev. A 64 (2001) 012304. [9] Z. Zhao, J.W. Pan, M.S. Zhan, Phys. Rev. A 64 (2001) 014301. [10] G.A. Durkin, C. Simon, D. Bouwmeester, Phys. Rev. Lett. 88 (2002) 187902. [11] M.I. Hwang, Y.H. Kim, Phys. Lett. A 369 (2007) 280. [12] Y.B. Sheng, F.G. Deng, H.Y. Zhou, Phys. Rev. A 77 (2008) 042308. [13] Z. Zhao, T. Yang, Y.A. Chen, A.N. Zhang, J.W. Pan, Phys. Rev. Lett. 90 (2003) 207901. [14] Y.H. Kim, S.P. Kulik, M.V. Chkhova, W.P. Grice, Y.H. Shih, Phys. Rev. A 67 (2003)8 010301(R). [15] M. Yang, Y. Zhao, W. Song, Z.L. Cao, Phys. Rev. A 71 (2005) 044302. [16] Z.L. Cao, P. Dong, Phys. B 404 (2009) 1917. [17] R.W. Boyd, Nonlinear Optics, 2nd edAcademic Press, New York, 20038 Chap. 4.
S.-H. Xiang et al. / Optics Communications 284 (2011) 2402–2407 [18] C.Y. Hu, A. Yong, J.L. O'Brien, W.J. Munro, J.G. Rarity, Phys. Rev. B 78 (2008) 085307. [19] C.Y. Hu, W.J. Munro, J.G. Rarity, Phys. Rev. B 78 (2008) 125318. [20] C.Y. Hu, W.J. Munro, J.L. O'Brien, J.G. Rarity, Phys. Rev. B 80 (2009) 205326. [21] C.Y. Hu, J.G. Rarity, e-print, arXiv:quant-ph/1005.5545v1. [22] W.K. Wootters, Phys. Rev. Lett. 80 (1998) 2245. [23] M. Zukowski, A. Zeilinger, M.A. Horne, A.K. Ekert, Phys. Rev. Lett. 71 (1993) 4287. [24] J.P. Reithmaier, G. Sek, A. Loeffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. Keldysh, V. Kulakovskii, T. Reinecke, A. Forchel, Nature (London) 432 (2004) 197. [25] S. Reitzenstein, C. Hofmann, A. Gorbunov, M. Strauβ, S.H. Kwon, C. Schneider, A. Loffler, S. Hofling, M. Kamp, A. Forchel, Appl. Phys. Lett. 90 (2007) 251109. [26] J. Berezovsky, M.H. Mikkelsen, N.G. Stoltz, L.A. Coldren, D.D. Awschalom, Science 320 (2008) 349. [27] J.R. Petta, A.C. Johnson, J.M. Taylor, E.A. Laird, A. Yacoby, M.D. Lukin, C.M. Marcus, M.P. Hanson, A.C. Gossard, Science 309 (2005) 2180. [28] S.M. Clark, Kai-Mei, C. Fu, Q. Zhang, T.D. Ladd, C. Stanley, Y. Yamamoto, Phys. Rev. Lett. 102 (2009) 247601;
2407
D. Press, T.D. Ladd, B.Y. Zhang, Y. Yamamoto, Nature 456 (2008) 218. [29] R. Oulton, A. Greilich, S.Yu. Verbin, R.V. Cherbunin, T. Auer, D.R. Yakovlev, M. Bayer, I.A. Merkulov, V. Stavarache, D. Reuter, A.D. Wieck, Phys. Rev. Lett. 98 (2007) 107401; D. Press, K. De Greve, P.L. McMahon, T.D. Ladd, B. Friess, C. Schneider, M. Kamp, S. Hoefling, A. Forchel, Y. Yamamoto, Nat. Photonics 4 (2010) 367. [30] M. Atature, J. Dreiser, A. Badolato, A. Hogele, K. Karrai, A. Imamoglu, Science 312 (2006) 551. [31] C. Emary, X.D. Xu, D.G. Steel, S. Saikin, L.J. Sham, Phys. Rev. Lett. 98 (2007) 047401. [32] J.A. Gupta, Science 292 (2005) 2458. [33] P.C. Chen, C. Piermarocchi, L.J. Sham, D. Gammon, D.G. Steel, Phys. Rev. B 69 (2004) 075320. [34] P. Walther, M. Aspelmeyer, K.J. Resch, A. Zeilinger, Phys. Rev. Lett. 95 (2005) 020403; A.M. Goebel, C. Wagenknecht, Q. Zhang, Y.A. Chen, K. Chen, J. Schmiedmayer, J.W. Pan, Phys. Rev. Lett. 101 (2008) 080403. [35] J.R. Petta, H. Lu, A.C. Gossard, Science 321 (2010) 669.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Concentration scheme for partially entangled photon states via entanglement reflector and no Bell-state analysis Shao-Hua Xiang ⁎, Zhen-Gang Shi, Wei Wen, De-Hua Lu, Xi-Xiang Zhu, Ke-Hui Song Department of Physics and Information Engineering, Huaihua University, Huaihua 418008, PR China
a r t i c l e
i n f o
Article history: Received 7 September 2010 Received in revised form 13 November 2010 Accepted 30 December 2010 Available online 18 January 2011 Keywords: Entanglement concentration Spin–photon interface Entanglement reflector Spin detection
a b s t r a c t We propose a protocol to concentrate partially entangled states of photons using entanglement reflector, which consists of a single electron spin confined in a charged quantum dot inside a single-sided microcavity. The outstanding advantage of the proposed scheme is its experimental simplicity and feasibility since it only needs to perform a single local measurement on electronic spins rather than a joint Bell-state measurement on photons. We then extend this scheme to concentrate N-photon Greenberger–Horne–Zeilinger state. Finally, we analyze the influence of various imperfections on the scheme. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Quantum entanglement has become a useful physical resource for quantum information processing, such as quantum teleportation [1], quantum dense coding [2], quantum cryptography [3], and other nonclassical interference phenomena [4]. To accomplish these tasks successfully, one needs to distribute and store maximally entangled states of two or more qubits between the distant parties. In practical situations, however, a quantum system cannot completely isolate from its environment, so the degradation of entanglement is unavoidable during storing, processing and transmission processes. As entanglement cannot be increased by local operations and classical communication (LOCC) [5], entanglement purification will be essential for the best performances of quantum communication and quantum computation. Entanglement purification is a process that one can extract a certain number of almost perfectly entangled states from a large number of less entangled states using LOCC. For the pure nonmaximally entangled states case, this process will be called entanglement concentration. In 1996, Bennett et al. [6] proposed the first concentration protocol for identical partially entangled pairs of two-state particles where the two parties of quantum communication need to perform a collective and nondestructive measurement for the joint state of n pairs of qubits; unfortunately, this scheme is hard to implement in practice because so many particles are needed to measure simultaneously. Bose et al. [7]
⁎ Corresponding author. Tel.: +86 745 2851011; fax: +86 745 2851305. E-mail address: [email protected] (S.-H. Xiang). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.12.088
presented another concentration scheme of polarization entangled photons via entanglement swapping. After that, a lot of authors have been devoted to polarization entanglement concentration using by linear optical elements and collective Bell-state measurement [8–11] or by nonlinear optics and quantum nondemolition detector [12]. Experimental concentration of photon entangled state has been demonstrated [13,14]. Very recently, entanglement concentrations for unknown atomic entangled states [15] and electron-spin entanglement [16] have been proposed. As for the case of linear optical system, its major disadvantage is that it would be more difficult to perfectly distinguish four different Bell states from each other within the current experimental capabilities, while quantum nondemolition measurement is the requirement of strong cross-Kerr media. Such do not exist in readily available materials [17]. Currently, Hu et al. proposed a novel quantum device consisting of a singly charged quantum dot (QD) coupled to a microcavity, which can directly split an initial product state of photon and spin into the entangled state via transmission and reflection in a deterministic way and is immune to the fine-structure splitting in realistic semiconductor QDs [18]. Using this device, they presented a deterministic and scalable scheme for creating spin–spin [18], photon–photon [19,20], photon– spin entanglements [20] as well as quantum teleportation [21]. In this paper, we present a way for entanglement concentration of partial polarization entangled state of photons via a single QD spin in an optical microcavity based on giant circular birefringence. Unlike previous protocols, our scheme only requires the single-spin measurements of QDs instead of the routine Bell measurement on photons, which will make the scheme easier to implement experimentally. This paper is organized as follows. In Section 2, we describe the photon–spin entanglement reflector. In Section 3, entanglement
S.-H. Xiang et al. / Optics Communications 284 (2011) 2402–2407
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concentration of two partially entangled photon pairs is presented and then it is extended to the case of N-photon GHZ state. In Section 4, we discuss the influence of imperfect setup such as cavity photon loss and spin decoherence on the current scheme. Finally, we summarize the paper in Section 5.
uncoupled cavity and gets a phase shift of φ0(ω). This interaction is summarized as follow:
2. Photon–spin entanglement reflector
jRij↑i→r0 e
Before implementing our concentration scheme, we briefly recapitulate the photon–spin entanglement reflector introduced in Ref. [18], which consists of a singly charged quantum dot with only one excess electron placed at the antinode of an optical microcavity where the distributed Bragg reflectors and transverse index guiding provide the three-dimensional confinement of light [18–21]. Such nanostructures may be in fact the self-assembled In(Ga)As QD, GaAs interfacial QD, or semiconductor nanocrystal. As is well known, the optical properties of singly charged QDs are strongly determined by the optical transitions of the trion X− state (bond state of two electrons and hole). The single electron states have Jz = ± 1/2 spin (|↑〉, |↓〉) and the holes have Jz = ± 3/2 spin (|⇑〉, |⇓〉). Due to the Pauli blocking, the interaction between light and X− not only relies on the polarization of light but also on the initial spin state of the excess electron. That is: the left circularly polarized photon (labeled by L photon) only couples the electron in the spin state |↑〉 to X− in the spin state |↑ ↓⇑〉 with the two antiparallel electron spins, while the right circularly polarized photon (indicated by R photon) only couples the electron in the spin state |↓〉 to X− in the spin state |↑↓⇑〉. Here we assume the spin-quantization axis is along the normal direction of cavity. For a trion state, the two electrons form a single state and therefore have total spin zero, which preserves electron-spin interaction with the hole spin. In the weak-excitation approximation, the reflection coefficient of this X−-cavity system is given by [18] κ iðωX − −ωÞ + γ2 r ðωÞ = 1− iðωX − −ωÞ + γ2 iðωc −ωÞ + κ2 +
κs 2
+ g2
;
ð1Þ
where ω, ωc, and ωX− are the frequencies of the input photon, cavity mode, and X− transition, respectively. g is the X−-cavity coupling strength and γ/2 is the X− dipole decay rate. κ and κs/2 are the cavity decay rates into the input/output modes and the leaky modes (side leakage), respectively. The background absorption can be included in κs/2. Here we focus on the resonant interaction with ωc = ωX− = ω0 and the cavity is assumed to be single-side cavity with back mirror perfectly reflective and front mirror partially reflective. By setting g = 0, we obtain the reflection coefficient r0(ω) for a uncoupled cavity where the QD does not coupled to the cavity
r0 ðωÞ =
iðω0 −ωÞ + κ2s − κ2 : iðω0 −ωÞ + κ2s + κ2
ð2Þ
If the side leakage is much smaller than the main cavity decay, i.e., κs ≪ κ, then we can obtain near-unity reflectance r0(ω) ≃ 1 in this case, whereas for the coupled cavity where X− strongly couples to the cavity, i.e., g N (κ, γ), we obtain |rh(ω)| ≃ 1 and φh(ω) ≃ 0 within a frequency window |ω − ωc| ≪ g. It has been shown in Ref. [18] that the X−-cavity system shows large transmittance and reflectance difference between the uncoupled cavity with g = 0 and the coupled cavity with g ≠ 0. If the single excess electron lies in the spin state |↑〉, the L photon feels a coupled cavity and gets a phase shift of φh(ω) after reflection, while the R photon feels the uncoupled cavity and gets a phase shift of φ0(ω). Conversely, if the electron lies in the state |↓〉, the R photon feels a coupled cavity and gets a phase shift of φh(ω) after reflection, while the L photon feels the
iφh
jLij↑i;
ð3aÞ
iφ0
jRij↑i;
ð3bÞ
iφ0
jLi j↓i;
ð3cÞ
iφh
jRij↓i;
ð3dÞ
jLij↑i→rh e
jLij↓i→r0 e
jRij↓i→rh e
where r0 = |r0(ω)| and φ0 = arg[r0(ω)] are the reflection ratio and phase shift of the reflected light for the uncoupled cavity, and rh = |r(ω)| and φh = arg[r(ω)] for the coupled cavity, respectively. This phenomenon is called the giant circular birefringence (GCB), allowing us to construct an entanglement reflector of photon–spin system. In fact, the change can also be described by a reflection operator,
rˆðωÞ = jr0 ðωÞeiφ0 ð jRihRj⊗ j↑ih↑j + jLihLj⊗ j↓ih↓j Þ iφh
+ jrh ðωÞe
ð4Þ
ð jLihLj ⊗j↑ih↑j + jRihRj⊗j↓ih↓j Þ:
For |r0(ω)| ≃ 1 and |rh(ω)| ≃ 1 or for balanced reflectance |r0(ω)| = |rh(ω)|, the reflection operator can be simplified as rˆðωÞ = jr0 ðωÞj eiφ0Uˆ ðΔφÞ with Δφ = φh − φ0. Controlling the photon frequency ω and setting Δφ = π/2, we can obtain a two-qubit phase shift operator iπ ð jLihL j⊗j↑ih↑j + jRihRj⊗j↓ih↓ j Þ ; Uˆ ðπ = 2Þ = exp 2
ð5Þ
which means that it could create photon–spin entanglement. In the following, we exploit this interaction to concentrate partially entangled photon pairs between distant parties. 3. Concentration scheme for partially photon entangled states The principle of our concentration scheme is shown in Fig. 1. Initially two distant parties, Alice and Bob, share two photon pairs (1, 3) and pffiffiffi (2,ffiffiffiffiffiffiffiffiffiffi 4) in the nonmaximally entangled state j ϕiij = a jRii jRij + p 1−a jLii jLij with the Schmidt coefficient a being positive real number, where photons 1 and 2 belong to Alice and photons 3 and 4 to Bob. In order to concentrate the maximally entangled state from these less entangled states, they are asked for the help from the third partner, Cliff, who holds two identical QD-cavity systems described above, an optical switch (SW), an optical circulator (OC), a T transformation, and two electronic-spin detectors (Dj, j = 1, 2). Photons 2 and 3 are sent to Cliff, at whose situation an optical switch and an optical circulator successively direct these photons to two QD-cavity systems with the spin electron pffiffiffi prepared in a superposition state j + ij = j↑ij + j↓ij = 2 (j = 1, 2). It is important to note that a T operation is needed before these photons are reflected by the second QD-cavity unit. Finally, Cliff makes measurements on two single-electron spins, and thereby our scheme can complete with a high probability of success. The detailed analysis of our scheme is followed as. Two pairs of photons are prepared initially in the following state jϕi1234 = jϕi13 ⊗jϕi24 = a jRi1 jRi2 jRi3 jRi4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + að1−aÞ jRi1 jLi2 jRi3 jLi4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + að1−aÞ jLi1 jRi2 jLi3 j Ri4 + ð1−aÞjLi1 jLi2 jLi3 jLi4 :
ð6Þ
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a
After the reflection of photons 2 and 3 at the first QD-cavity system, the total state of four photons and two spins is transformed into j ϕir = a jRi1 jRi2 jRi3 jRi4 j−i1 j + i2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + i að1−aÞ jRi1 jLi2 jRi3 jLi4 j + i1 j + i2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + i að1−aÞ jLi1 jRi2 jLi3 jRi4 j + i1 j + i2
ð7Þ
−ð1−aÞjLi1 jLi2 jLi3 jLi4 j−i1 j + i2 ; iφh has been discarded and the state where the global phase pffiffiffi e j−ik = ð j↑ik −j↓ik Þ = 2. Next, Cliff performs a T transformation represented by
i jRij → pffiffiffi jRij′ + j Lij′ ; 2
ð8aÞ
1 jLij → pffiffiffi jRij′ −jLij′ 2
ð8bÞ
on photons 2 and 3. Such an operation can be realized via quantum circuit, as shown in Fig. 1(b). It is known that the action of the polarization beam splitter (PBS) is it transmits only horizontal polarization component and reflects vertical component, while the transformation of a quarter-wave plate (QWP) is expressed by jHij → p1ffiffi2 jHij′ + jVij′ and jVij → p1ffiffi2 jHij′ − jVij′ . Meanwhile, the phaseshifter is assumed to have phase θ = π/2and we define jRij = p1ffiffi2 jHij + ijVij and jLij = p1ffiffi2 jHij −ijVij , respectively. So, the T operation process can be described as follows: 1 i j Ria PBS 1 pffiffiffi ð jHic + ijVib Þ PS pffiffiffi ð jHid + jVib Þ → 2 → 2 i i PBS 2 pffiffiffi ð jHie + jVie Þ QWP pffiffiffi jRia′ + jLia′ ; → 2 → 2 1 i p ffiffiffi ð jHic −ij Vib Þ PS pffiffiffi ð jHid −jVib Þ jLia PBS 1 → 2 → 2 i 1 PBS 2 pffiffiffi ð jHie −jVie Þ QWP pffiffiffi jRia′ −jLia′ : → 2 → 2
ð9Þ
with certainty. For example, if both spins are detected in the state | + i 1| − i 2 or | + i 1| + i 2 the photons 1 and 4 are projected into one of the following states with success probability a(1 − a):
+ ½a jRi1 jRi4 −ð1−aÞjLi1 jLi4 j−i1 j + i2 jψiþ 2′ 3′ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + að1−aÞð jRi1 jLi4 + jLi1 jRi4 Þj + i1 j + i2 jϕi− 2′ 3′ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; + að1−aÞð jRi1 jLi4 − jLi1 jRi4 Þ j + i1 j + i2 jψi− 2′ 3′
ð10Þ
where the four Bell states are jRi2′ j Ri3′ jLi2′ jLi3′ 1 and jψi2′ 3′ = pffiffi2 jRi2′ jLi3′ jLi2′ jRi3′ . Subsequently, let photons 2′ and 3′ send into the second QD-cavity unit. When these photons are reflected by this unit, Eq. (10) is transformed into j ϕir2 = ½ajRi1 jRi4 + ð1−aÞjLi1 jLi4 j−i1 j−i2 jϕi− 2′ 3′
+ i½ajRi1 jRi4 −ð1−aÞ jLi1 jLi4 j−i1 j + i2 jψiþ 2′ 3′ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + að1−aÞð jRi1 jLi4 + jLi1 jRi4 Þj + i1 j−i2 jϕiþ 2′ 3′ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; + i að1−aÞð jRi1 jLi4 − jLi1 jRi4 Þj + i1 j + i2 jψi− 2′ 3′ 2iφ0
ð11Þ
where we have discarded the global phase factor e . It is interesting to see that our scheme is realized either by detecting the electronic spins or by implementing the Bell-state measurement of photons 2′ and 3′. However, it is impossible to completely distinguish four Bell states of photons with current technology. Therefore, Cliff can perform a single measurement on electronic spins 1 and 2 in such a way that photons 1 and 4 can be projected into the desired state
ð12Þ
Thus, Alice and Bob obtain the maximally entangled photon pair from two previously shared partially entangled photon pairs by LOCC. The concentration succeeds. But when the spins 1 and 2 are detected in the state | − i 1| + i 2 or | − i 1| − i 2, photons 1 and 4 will be projected into one of the following entangled states with probability (2a2 − 2a + 1)/2: jϕi14 = N½a jRi1 jRi4 j ð1−aÞjLi1 jLi4 ;
j ϕir1 = ½ajRi1 jRi4 + ð1−aÞjLi1 jLi4 j−i1 j + i2 jϕiþ 2′ 3′
= p1ffiffi2
Fig. 1. (a) Scheme of entanglement concentration. Black spheres with and without the arrow stand for quantum dots and photon modes, respectively. SW: optical switch, OC: optical circulator, Di: electron spin detectors, T: T operation. (b) T operation. PBS: polarizing beam splitter, PS: phase shifter, QWP: quarter-wave plate, M: plane mirror.
1 jψi14 = pffiffiffi ½ jRi1 jLi4 j jLi1 jRi4 : 2
After this transformation, Eq. (7) becomes
jϕi 2′ 3′
b
ð13Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where N = 1 = 2a2 −2a + 1. A question arises: which is more entangled—the state (13) or the originally non-maximally-entangled states? Using Wootters' formula [22], we calculate the ratio of the degree of entanglement of the concentrated state to that of the desired state as
r=
Ccen = Cdes
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi að1−aÞ : 2a2 −2a + 1
ð14Þ
It is clear to see that when a = 1/2, the concentrated state (13) has the same degree of entanglement as the initial shared state (i.e., r = 1). This process is in fact amount to entanglement swapping [23]. In other cases, we have r b 1, which implies that the concentration scheme fails. The above scheme can be directly generalized to the case of N-photon entangled state. Assume, for example, that Alice and Bob share the following partially entangled states: ⊗N
jϕi1⋯N = α jRi
⊗N
+ β jLi
;
jϕiab = α jRia jRib + β jLia jLib ;
ð15aÞ ð15bÞ
where |α|2 + |β|2 = 1 and the notation | i ⊗ N = | i 1| i 2 … | i N. Without loss of generality, assume that Alice owns k(k b N) photons of the first entangled pair and one photon (or, say, photon b) of the second pair, while all the remaining photons belong to Bob. To this aim, let Alice
S.-H. Xiang et al. / Optics Communications 284 (2011) 2402–2407
and Bob send photons N and a to Cliff. After the reflection at the first QD-cavity system, the total state of the system evolves into 2 ⊗N−1 2 ⊗N−1 j ψi = α j Ri j Rib j RiN jRia −β j Li jLib jLiN j Lia j −i1 j + i2 ð16Þ ⊗N−1 ⊗N−1 + iαβ j Ri j Lib j RiN jLia + j Li j Rib jLiN j Ria j + i1 j + i2 :
When photons a and N are reflected by the second QD-cavity unit, the state (16) becomes 2 ⊗N−1 2 ⊗N−1 − jϕi = α jRi jRib + β jLi jLib j −i1 j−i2 j ϕiN′ a′ 2 ⊗N−1 2 ⊗N−1 þ + i α jRi j Rib −β jLi jLib j −i1 j + i2 j ψiN′ a′ ⊗N−1 ⊗N−1 þ + αβ jRi jLib + j Li j Rib j + i1 j−i2 j ϕiN′ a′ ⊗N−1 ⊗N−1 − + iαβ jRi j Lib −j Li jRib j + i1 j + i2 jψiN′ a′ ;
where we have discarded the global phase factor ei4φ0. Cliff performs a single measurement on the electron spins 1 and 2, which is described ðiÞ ðiÞ ˆ = j+ i h + j; Π ˆ = j−i h−j . The conditional by a collection Π ii ii þ − state of photons 1 and 4 reads ρ14 =
h ð1Þ ð2Þ þ i 1 ˆ ðl 1Þ Π ˆl Π ˆm ˆ ðm2Þ ; ðl; m = Þ; Tr Π jϕiTT hϕj Π P Δ
ð17Þ
±
1 2 ⊗N−1 2 ⊗N−1 jϕi = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α jRi jRib + β jLi jLib ; 4 4 α +β
ð18aÞ
1 2 2 ⊗N−1 2 ⊗N−1 jϕi = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α jRi jRib −β jLi jLib ; 4 4 α +β
ð18bÞ
1 ⊗N−1 3 ⊗N−1 jϕi = pffiffiffi jRi jLib + jLi jRib ; 2
ð18cÞ
1 ⊗N−1 4 ⊗N−1 jϕi = pffiffiffi jRi jLib −jLi jRib : 2
ð18dÞ
It is to see that Alice and Bob can concentrate GHZ state of N photons with the total success probability 2α2β2 by aid of the third partner.
h ð1Þ ð2Þ þ i ˆl Π ˆm ˆ ðm2Þ : ˆ ðl 1Þ Π P = Tr Π jϕiTT hϕj Π
þ
i 1h 2 2 2 2 jϕiT = aj Ri1 jRi4 r0 j ↑i1 −rh j↓i1 + ð1−aÞ jLi1 jLi4 rh j↑i1 −r0 j ↓i1 4 h i 2 2 2 2 × r0 j ↑i2 −rh j ↓i2 jRi2′ jRi3′ − rh j ↑i2 −r0 j↓i2 j Li2′ j Li3′ r r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + 0phffiffiffi að1−aÞ j + 〉1 ½ j Ri1 jLi4 + jLi1 j Ri4 2 h 2 i 2 2 2 2 × r0 j↑i2 −rh j ↓i2 jRi2′ jRi3′ + rh j↑i2 −r0 j ↓i2 jLi2′ jLi3′ ir r h 2 2 + 0 h ajRi1 j Ri4 r0 j↑i1 −rh j ↓i1 2 i 2 2 þ −ð1−aÞ jLi1 jLi4 rh j↑i1 −r0 j↓i1 j + i2 j ψi2′ 3′ 2 2
ð19Þ
þ
over the states of photons 2′ and 3′, is given by 0 B B B B B B 4 B g ++ B ρ14 = B 8P B B B B B B @
2 2
a h g2
0
0
2h r0 rh að1−aÞ g4
0
2r0 rh að1−aÞ g4
2h r0 rh að1−aÞ g4
0
0
0
2 2 2
2 2
−
að1−aÞ g2
2 2
2r0 rh að1−aÞ g4 2 2 2
1 að1−aÞ C C g2 C C C C 0 C C C; C C 0 C C C 2 2 C ð1−aÞ h A g2
−
ð22Þ
pffiffiffi
pffiffiffi where g = r02 −rh2 = 2, h = r02 + rh2 = 2, ε = g2 − 2r20r2h and the
1 2 2 2 1 success probability P = g h 2a −2a + 1 + h2 r02 rh2 að1−aÞ. 8 2 Similarly, in the case where both spins are detected in the state | + i 1| − i 2, the state of photons 1 and 4 is easily calculated to be 0
In the above scheme, we have neglected the cavity side leakage and the electron-spin decoherence; i.e., we only consider the case where g ≫ (κ, γ), κs = 0, and the electron-spin coherence time T → ∞. This is an ideal case. In practical situations, however, our scheme can be affected by a number of noises, including photon loss and the electronic spin decoherence. In the following, we investigate the impact of these factors on our entanglement concentration scheme. We first consider cavity side leakage effect and take the twophoton concentration scheme as example. Assume that the parameters r0 and rh will be little deviation from unity. Using Eqs. (3a), (3b), (3c) and (3d) and repeating the same procedure as before, we can calculate the state of the system after photons 2′ and 3′ exiting through the second QD-cavity system as
ð21Þ
Next, we proceed to consider the case in which both spins are detected in the state | + i 1| + i 2 or | + i 1| − i 2. For the case of the ð1Þ ð2Þ ˆ ˆ Π , the state of photons 1 and 4, after tracing measurement Π
4. Feasibility of entanglement concentration scheme
ir r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − + p0 ffiffiffih að1−aÞ½ jRi1 j Li4 −j Li1 j Ri4 j + i1 j + i2 j ψi2′ 3′ ; 2
ð20Þ
where TrΔ stands for tracing over the states of both spins and photons 2′ and 3′ and the probability of finding this state is calculated as
where |ϕ i N ′a′ and |ψ i − N′a′ are four Bell states of photons N′ and a′, and we have discarded the global phase factor e2iφ0 in deriving the above Eqs. (16) and (17). Finally, Cliff operates a detection of the electron spins 1 and 2 in the {| + i i, | − i i} basis so that all the photons owned by Alice and Bob will collapse into one of the following states 1
2405
+−
ρ14
B B B B B B 4 2B g h B = B 8P B B B B B B @
2
a g2
0
0
0
2r02 rh2 að1−aÞ g4
2r02 rh2 að1−aÞ g4
0
2r0 rh að1−aÞ g4
2r0 rh að1−aÞ g4
0
0
2 2
−
að1−aÞ g2
2 2
1 að1−aÞ C g2 C C C C C 0 C C C: C C 0 C C C 2 C ð1−aÞ A g2 ð23Þ
−
Thus, we can compute the total success probability of concentrating maximally entangled states as Pt =
1 2 2 2 2 2 2 g h 2a −2a + 1 + h r0 rh að1−aÞ: 4
ð24Þ
We use the fidelity to measure how close the states (22) and (23) come to the desired states (12). It is very easy to calculate the corresponding fidelities as ++
F14
+−
=
F14 =
þ ++ 14 hψjρ14
þ
1 4 4 r r að1−aÞ; P 0 h
ð25aÞ
1 2 2 2 r r h að1−aÞ: 2P 0 h
ð25bÞ
jψi14 =
− +− − 14 hψjρ14 jψi14
=
It is not difficult to see that when the balanced reflection condition, r0 = rh ≠ 1, is fulfilled, one has the unity fidelity and the success
2406
S.-H. Xiang et al. / Optics Communications 284 (2011) 2402–2407
probability 2r80a(1− a). It implies that the cavity side leakage (or called photon loss) has no influence on the fidelity of the concentrated entangled states and only decreases the success probability by a factor 2r80. To meet this condition, we will require the parameters of QD-cavity system as ω − ωc = κ/2, ωX = ωc, κs/κ = 0.01, γ/κ = 0.15, and g/κ = 2.83; moreover, the mirror of the cavity has the reflection coefficient 0.991, which is completely realistic with current experimental technologies. Recently, for micropillars with diameter around 1.5 μm, the coupling strength g = 80 μev and the microcavity Q factor of 104 ∼ 105 have been reported [24,25]. With these data, we can find g/κ = 2.4. Thus, all these parameters mentioned-above are achieved and then this condition can be met experimentally. In addition, it has been shown in Ref. [24,25] that γ is about several μev in this system. Therefore, although the complete reflection is impossible, our scheme would be feasible with high fidelity and high success probability using current techniques as long as the balance reflection condition holds. Another important limitation is the effect of the electron-spin decoherence and relaxation. It has been found that due to the hyperfine interaction between the electron spin and 104 ∼ 105 host nuclear spins, the spin coherence time in GaAs-based or InAs-based QD is extremely short and, for example, the spin-dephasing time is typically around 5∼ 10 ns [26,27], but can be increased by several orders of magnitude by spin echo techniques and manipulations of the nuclear spins [28,29], whereas the electron spin relaxation time is relatively long (∼ ms) due to the suppressed electron–photon and spin–orbit interactions in QDs, so the time required to complete the proposed scheme should be much shorter than the spin coherence time. On the other hand, the QD electron–spin superposition states (| ± i) can be prepared, for instance, by optical pumping and/or optical cooling [30,31] or by picosencond or femtosecond optical pulses [32,33]. From the spin basis state, there are two methods to obtain the spin superposition state: either via spin–flip Raman transitions [30] or by implementing single spin rotations using nanosecond microwave pulses [27]. Ultrafast spin manipulation through ac-Stack effect has been reported in semiconductor quantum wells on the picosecond time scales and in semiconductor QDs on the picosecond time scales [26], which are much shorter than the spin coherence time. Additionally, it is worthy to point out that quantum dot electron spin detection remains a challenging task, but in our case, the detection scheme can be implemented using a spin quantum nondemolition (QND) method. Here we only consider the two-photon concentration scheme; this method can be applied to other cases. The proposed detection scheme includes following three steps, as shown in Fig. 2. (1) Cliff applies a π/2 microwave pulse on each electron spin pffiffiffi(It is indeed equivalent pffiffiffito a Hadamard gate, i.e., j↑i→ð j↓i + j↑iÞ = 2 and j↓i→ð j↓i− j↑iÞ = 2) in such way that these spin superposition states can be rotated to the states | ↑ i 1| ↑ i 2, | ↑ i 1| ↓ i 2, | ↓ i 1| ↑ i 2, and | ↓ i 1| ↓ i 2. (2) Two photons A and B in pffiffiffi the state jRij + jLij = 2ð j = A; BÞ are imported into two QD-cavity units, respectively. Note that photons A and B have the same frequency as these photons above. After reflection, the total state of the system is given by W
jϕi2 = ½ajRi1 j Ri4 + ð1−aÞj Li1 jLi4 j + 45o iA j + 45o iB j↑i1 j↑i2 jϕi− 2′ 3′ o o þ −½ajRi1 jRi4 −ð1−aÞj Li1 jLi4 j + 45 iA j−45 iB j↑i1 j↓i2 jψi2′ 3′ ð26Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o o þ + i að1−aÞð jRi1 jLi4 + jLi1 jRi4 Þj−45 iA j + 45 iB j↓i1 j↑i2 jϕi2′ 3′ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; −i að1−aÞð jRi1 j Li4 −j Li1 jRi4 Þj−45o iA j−45o iB j ↓i1 j↓i2 jψi− 2′ 3′
pffiffiffi where j 45o i = ð jHi jViÞ = 2. (3) Finally, Cliff makes a measurement on each of the photons A and B in the orthogonal linear polarizations so that photons 1 and 4 held by Alice and Bob will be projected into one of Eqs. (12) and (13). For example, photons A and B are detected in the state | − 45o i A| − 45o i B, so both electron spins are definitely in the state | ↓ i 1| ↓ i 2 and Alice and Bob can obtain the
QD-cavity 1
QD-cavity 2 photon A
+45 -45
MW 1
photon B
+45 -45
MW 2
Fig. 2. Schematic of QND electron spin. The thick arrow indicates a π/2 microwave pulse.
maximally entangled state |ψ i − 14. We emphasize that the detection of photons in these bases has been experimentally realized [13,34]. It is interesting to note that Petta et al. [35] have recently demonstrated coherent control of electronic spin states in a double quantum dot (DQD) by sweeping an initially prepared spin-singlet through a single-triplet anticrossing in the energy-level spectrum where such an anticrossing serves as a beam splitter for the incoming quantum states, yielding the coherent quantum oscillations between the singlet state and a triplet state. Therefore, this type of photon–spin interface will open interesting perspectives for solid quantum information processing. 5. Conclusions In conclusion, we have proposed a practical scheme for entanglement concentration of unknown polarization entangled state based on the spin–photon interface in QD-cavity system. Compared with other concentration schemes, this protocol is simpler and convenient as it does not require the joint Bell state measurement on photons. Moreover, it does not need to know accurately the information about the less-entangled state in advance. We have extended this scheme to distill the case of N-photon GHZ state. In addition, we have studied the influence of numerous practical situations on the proposed scheme. Our results indicate that, when the balance reflection condition is satisfied, our scheme is immune to photon loss but greatly affected by spin decoherence of the electrons. With the development of cooling and fast coherent control of hole–spin states, we hope that the present scheme could be realized in the foreseeable future. Acknowledgments This work was partially supported by Hunan Provincial Natural Science Foundation of China (Grant no. 10JJ6010), the Key Project Foundation of Hunan Provincial Education Department, China (Grant no. 10A095), and the Youth Foundation from Huaihua University Grant no. HHUQ2009-09. References [1] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895. [2] C.H. Bennett, S.J. Wiesner, Phys. Rev. Lett. 69 (1992) 2881. [3] A.K. Ekert, Phys. Rev. Lett. 67 (1991) 661. [4] Y.H. Shih, A.V. Sergienko, Phys. Rev. A 50 (1994) 2564. [5] M. Plenio, V. Vedral, Contemp. Phys. 39 (1998) 431. [6] C.H. Bennett, H.J. Bernsten, S. Popescu, B. Schumacher, Phys. Rev. A 53 (1996) 2046. [7] S. Bose, V. Vedral, P.L. Knight, Phys. Rev. A 60 (1999) 194. [8] T. Yamamoto, M. Koashi, N. Imoto, Phys. Rev. A 64 (2001) 012304. [9] Z. Zhao, J.W. Pan, M.S. Zhan, Phys. Rev. A 64 (2001) 014301. [10] G.A. Durkin, C. Simon, D. Bouwmeester, Phys. Rev. Lett. 88 (2002) 187902. [11] M.I. Hwang, Y.H. Kim, Phys. Lett. A 369 (2007) 280. [12] Y.B. Sheng, F.G. Deng, H.Y. Zhou, Phys. Rev. A 77 (2008) 042308. [13] Z. Zhao, T. Yang, Y.A. Chen, A.N. Zhang, J.W. Pan, Phys. Rev. Lett. 90 (2003) 207901. [14] Y.H. Kim, S.P. Kulik, M.V. Chkhova, W.P. Grice, Y.H. Shih, Phys. Rev. A 67 (2003)8 010301(R). [15] M. Yang, Y. Zhao, W. Song, Z.L. Cao, Phys. Rev. A 71 (2005) 044302. [16] Z.L. Cao, P. Dong, Phys. B 404 (2009) 1917. [17] R.W. Boyd, Nonlinear Optics, 2nd edAcademic Press, New York, 20038 Chap. 4.
S.-H. Xiang et al. / Optics Communications 284 (2011) 2402–2407 [18] C.Y. Hu, A. Yong, J.L. O'Brien, W.J. Munro, J.G. Rarity, Phys. Rev. B 78 (2008) 085307. [19] C.Y. Hu, W.J. Munro, J.G. Rarity, Phys. Rev. B 78 (2008) 125318. [20] C.Y. Hu, W.J. Munro, J.L. O'Brien, J.G. Rarity, Phys. Rev. B 80 (2009) 205326. [21] C.Y. Hu, J.G. Rarity, e-print, arXiv:quant-ph/1005.5545v1. [22] W.K. Wootters, Phys. Rev. Lett. 80 (1998) 2245. [23] M. Zukowski, A. Zeilinger, M.A. Horne, A.K. Ekert, Phys. Rev. Lett. 71 (1993) 4287. [24] J.P. Reithmaier, G. Sek, A. Loeffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. Keldysh, V. Kulakovskii, T. Reinecke, A. Forchel, Nature (London) 432 (2004) 197. [25] S. Reitzenstein, C. Hofmann, A. Gorbunov, M. Strauβ, S.H. Kwon, C. Schneider, A. Loffler, S. Hofling, M. Kamp, A. Forchel, Appl. Phys. Lett. 90 (2007) 251109. [26] J. Berezovsky, M.H. Mikkelsen, N.G. Stoltz, L.A. Coldren, D.D. Awschalom, Science 320 (2008) 349. [27] J.R. Petta, A.C. Johnson, J.M. Taylor, E.A. Laird, A. Yacoby, M.D. Lukin, C.M. Marcus, M.P. Hanson, A.C. Gossard, Science 309 (2005) 2180. [28] S.M. Clark, Kai-Mei, C. Fu, Q. Zhang, T.D. Ladd, C. Stanley, Y. Yamamoto, Phys. Rev. Lett. 102 (2009) 247601;
2407
D. Press, T.D. Ladd, B.Y. Zhang, Y. Yamamoto, Nature 456 (2008) 218. [29] R. Oulton, A. Greilich, S.Yu. Verbin, R.V. Cherbunin, T. Auer, D.R. Yakovlev, M. Bayer, I.A. Merkulov, V. Stavarache, D. Reuter, A.D. Wieck, Phys. Rev. Lett. 98 (2007) 107401; D. Press, K. De Greve, P.L. McMahon, T.D. Ladd, B. Friess, C. Schneider, M. Kamp, S. Hoefling, A. Forchel, Y. Yamamoto, Nat. Photonics 4 (2010) 367. [30] M. Atature, J. Dreiser, A. Badolato, A. Hogele, K. Karrai, A. Imamoglu, Science 312 (2006) 551. [31] C. Emary, X.D. Xu, D.G. Steel, S. Saikin, L.J. Sham, Phys. Rev. Lett. 98 (2007) 047401. [32] J.A. Gupta, Science 292 (2005) 2458. [33] P.C. Chen, C. Piermarocchi, L.J. Sham, D. Gammon, D.G. Steel, Phys. Rev. B 69 (2004) 075320. [34] P. Walther, M. Aspelmeyer, K.J. Resch, A. Zeilinger, Phys. Rev. Lett. 95 (2005) 020403; A.M. Goebel, C. Wagenknecht, Q. Zhang, Y.A. Chen, K. Chen, J. Schmiedmayer, J.W. Pan, Phys. Rev. Lett. 101 (2008) 080403. [35] J.R. Petta, H. Lu, A.C. Gossard, Science 321 (2010) 669.