Counterfactual entanglement distribution without transmitting any particles Qi Guo,1 Liu-Yong Cheng,1 Li Chen,1 Hong-Fu Wang,2 and Shou Zhang,1,2∗ 1 Center

for the Condensed-Matter Science and Technology, Department of Physics, Harbin Institute of Technology, Harbin 150001, China 2 Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, China ∗

[email protected]

Abstract: To date, all schemes for entanglement distribution needed to send entangled particles or a separable mediating particle among distant participants. Here, we propose a counterfactual protocol for entanglement distribution against the traditional forms, that is, two distant particles can be entangled with no physical particles travel between the two remote participants. We also present an alternative scheme for realizing the counterfactual photonic entangled state distribution using Michelson-type interferometer and self-assembled GaAs/InAs quantum dot embedded in a optical microcavity. The numerical analysis about the effect of experimental imperfections on the performance of the scheme shows that the entanglement distribution may be implementable with high fidelity. © 2014 Optical Society of America OCIS codes: (230.5750) Optical devices: Resonators; (270.5565) Quantum communications; (270.5585) Quantum information and processing.

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Received 28 Jan 2014; revised 10 Mar 2014; accepted 12 Mar 2014; published 7 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.008970 | OPTICS EXPRESS 8970

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Received 28 Jan 2014; revised 10 Mar 2014; accepted 12 Mar 2014; published 7 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.008970 | OPTICS EXPRESS 8971

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1.

Introduction

Quantum physics has never stopped surprising us since its establishment in the early 20th century. Many novel counterintuitive effects predicted by quantum mechanics have been confirmed by experiments. Specially, some quantum properties can be applied to information processing, and an interdisciplinary field has emerged in recent decades, i.e. quantum information science [1], which can achieve lots of information processing tasks that appear unimaginable in the classical domain. In the field of quantum information science, entanglement is considered as an important physical resource, because it can provide anomalously strong correlations between communicating participants. Therefore, the remote establishment of entanglement, i.e. entanglement distribution, is a key procedure for most remarkable quantum processing tasks, such as quantum teleportation [2], dense coding [3], quantum cryptography [4], and so on. Besides, entanglement distribution also enables fundamental tests of the laws of quantum mechanics [5,6]. In recent years, entanglement distribution has been extensively researched both theoretically and experimentally [7–12]. Normally, in order to achieve entanglement distribution, entangled states should be created from an entanglement source, then the entangled particles are sent sepa#205632 - $15.00 USD (C) 2014 OSA

Received 28 Jan 2014; revised 10 Mar 2014; accepted 12 Mar 2014; published 7 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.008970 | OPTICS EXPRESS 8972

rately to distant parties. It is clear that the traditional entanglement distribution requires sending entangled particles among remote quantum nodes. However, in 2003, Cubitt et al. [13] showed that entanglement can be distributed by using a separable mediating particle, which means that two distant particles can be entangled by sending a third particle that is never entangled with the other two. The above counterintuitive entanglement distribution is actually achieved by using a form of nonclassical correlations that can be called quantum discord, which has been proved to be an important form of currency in quantum information sharing [14, 15]. Very recently, three research groups have independently implemented entanglement distribution by sending a separable carrier in experiment [16–18]. Another more surprising phenomenon that results from quantum mechanics is counterfactual quantum communication [19, 20], which indicates communication tasks can be achieved without any particles travelling the channel between participants. The researches in this aspect originate from the interaction-free measurements proposed by Elitzur and Vaidman in 1993 [21], whose basic idea is that an absorptive object in one of the arms of the Mach-Zehnder interferometer can destroy the interference even if no photon is absorbed by the object. Thus, one can ascertain the existence of the object in the given arm of the interferometer with the maximum attainable efficiency 50%, though no photon “touched” this object. In 1995, Kwiat et al. improved the interaction-free measurements [22] and the probability of an interaction-free measurement could be made arbitrarily close to 100% by applying a discrete form of the quantum Zeno effect [23]. If let the absorptive object be in a quantum superposition state of presence and absence, entanglement can be created by the interaction-free measurement interferometer [24,25]. Using a novel “chained” version of the quantum Zeno effect, Hosten et al. demonstrated counterfactual quantum computation and implemented Grover’s search algorithm [26] with boosting the counterfactual inference probability to unity. Subsequently, counterfactual quantum key distribution (CQKD) scheme with no particle transmitted through the quantum channel based on quantum interrogation was studied deeply [19, 27–31]. Recently, Salih et al. presented a direct counterfactual quantum communication protocol [20], which allowed a classical bit to be transferred from the sender to the receiver without any particles travelling between them by using the “chained” quantum Zeno effect. And this work maybe open the way for optical communication without photons [32]. Here, we will propose a more counterfactual protocol for entanglement distribution by constructing a chained Mach-Zehnder type interferometer, which is composed of the tandem repeat sequence of Kwiat et al.’s interferometer for interaction-free measurement [22]. Inspired by the basic idea of the counterfactual quantum communication in [20], we arrange the presence and absence of the absorptive object is controlled by a quantum state rather than classical means, and the quantum absorptive object locates in a third party. We are surprised to find that the two photons at two distant participants can be entangled with each other, but no particle passes through the transmission channels between participants during the entanglement creation process. In order to demonstrate the feasibility of the present protocol, we design an implementation scheme by using flying photon qubits and stationary electron-spin qubits assisted by quantum dot (QD) inside a double-sided optical microcavity, which has been considered as one of the most promising candidates for solid-state quantum computation and communication [33–36]. In the scheme, the absorption or transmission of the photon in one arm of a Michelson-type interferometer is controlled by the state of the electron spin in the QD. We also analyze the influences of experimental imperfections on the present counterfactual scheme, and the numerical analysis results show that the scheme can be achieved successfully in experiment.

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Received 28 Jan 2014; revised 10 Mar 2014; accepted 12 Mar 2014; published 7 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.008970 | OPTICS EXPRESS 8973

2.

Protocol for counterfactual entanglement distribution

First of all, we give a short review of the entanglement generation using Kwiat et al.’s interaction-free measurement interferometer by replacing the classical control of the absorption object with the quantum control. As shown in Fig. 1(a), the interferometer consists of N unbalanced beam splitters (BS) (or repeatedly using a BS N times), whose reflectivity and transmissivity are cos2 θ and sin2 θ (θ = π /2N), respectively. The two pathes of the interferometer are described as a and b. In principle, if the absorption object is absent, the photon injected from the input port path b will exit from path a; if the absorption object is present, the photon will exit from path b with high probability for large N. Now, let the absorption object be in a quantum superposition state of absence and presence, and define the absence and presence as quantum states |0i and |1i, respectively. We describe a state with a photon in path a and no photon in path b as |0i, and a state with a photon in path b and no photon in path a as |1i, i.e. |0i ≡ |1ia |0ib and |1i ≡ |0ia |1ib . Let a photon enter the interferometer from path a, and the initial joint state of the photon and the absorption object is |ϕ i0 = √12 |0i p (|0io + |1io ), where the subscripts p and o indicate photon and object, respectively. After passing through the first BS that performs the transformations {|0i p → cos θ |0i p + sin θ |1i p , |1i p → cos θ |1i p − sin θ |0i p }, the state will become 1 |ϕ i0 → √ (cos θ |0i p + sin θ |1i p ) (|0io + |1io). 2

(1)

The component |1i p |1io in above equation indicates that the photon is in path b and the absorption object is presence, so the photon will be absorbed and this component can be ignored. Therefore, after the first cycle of the photon travelling in the interferometer, i.e. before the photon arrives the second BS, if the photon isn’t absorbed (with the probability of 1 − 12 sin2 θ ), the system will be in a non-normalized state 1 |ϕ i1 = √ [(cos θ |0i p |1io + (cos θ |0i p + sin θ |1i p )|0io )] . 2

(2)

In the same way, after N cycles, if the photon isn’t absorbed yet, the system state becomes |ϕ iN

= =

 1  √ (cosN θ |0i p |1io + (cosN θ |0i p + sin N θ |1i p )|0io ) 2
1 √ cosN θ |0i p |1io + |1i p|0io , 2

(3)

where θ = π /2N. Obviously, for the large cycles N, the state will be approximatively normalized, i.e. |ϕ iN ∼ √12 (|0i p |1io + |1i p |0io ), which is a normal Bell state. Now, we begin to construct a chained nested Mach-Zehnder type interferometer using the interaction-free measurement interferometer in Fig. 1(a) to achieve the counterfactual entanglement distribution. Suppose that we want to distribute entanglement between two distant parties Alice and Bob. As shown in Fig. 1(b), we put the absorption object at a third party Charlie and construct interaction-free measurement interferometer of Fig. 1(a) between Charlie and Alice and between Charlie and Bob, respectively, and the absorption object is shared by the two interferometer. Then, the interaction-free measurement interferometers, as inner interferometers, are respectively inserted in one of the arms of outer Mach-Zehnder interferometers. The three pathes of the nested Mach-Zehnder type interferometer between Alice and Charlie are described as a1 , a2 , and a3 . And the three pathes of the interferometer between Bob and Charlie are described as b1 , b2 , and b3 . Every outer interferometer consists of M unbalanced

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Received 28 Jan 2014; revised 10 Mar 2014; accepted 12 Mar 2014; published 7 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.008970 | OPTICS EXPRESS 8974

(a)

(b)

a

b

Alice a1

a3 b BS o

b

b1

2

Bob

3

BS i

BS i

AO

BS

Charlie

a2

BS o

M

M

.

.

M

..

...

..

Db Da

T1

T2

... a1

Db Da

.

. a2

b1

b

2

Fig. 1. (a) The quantum version of Kwiat et al.’s interaction-free measurement interferometer. (b) The chained nested Mach-Zehnder type interferometer for counterfactual entanglement distribution. The interferometer in (a) is inserted one arm of a outer interferometer. Alice, Bob, and Charlie locate different places, and the wathet strip T1(2) denotes the transmission channel between Charlie and Alice (Bob). The red (blue) lines is the optical pathes of the nested interferometer between Charlie and Alice (Bob). BSo(i) : the beam splitter in outer (inner) interferometer. M: normal mirror. AO: absorption object in a quantum superposition state of absence and presence. Da(b) : conventional photon detector. ai and bi (i = 1, 2, 3) describe the different spatial pathes of the two nested interferometers, respectively.

beam splitters (BSo ) (or repeatedly using a BSo M times) with reflectivity cos2 ϑ and transmissivity sin2 ϑ (ϑ = π /2M). Initially, the absorption object is in a quantum superposition state of absence and presence, i.e. √12 (|0io + |1io). The first step of the entanglement distribution is that Alice sends a photon in the interferometer from the input port of path a1 , that is, the initial state of Alice’s photon is |1ia1 |0ia2 |0ia3 . The BSo performs the transformations {|1ia1 |0ia2 → cos ϑ |1ia1 |0ia2 + sin ϑ |0ia1 |1ia2 , |0ia1 |1ia2 → cos ϑ |0ia1 |1ia2 − sin ϑ |1ia1 |0ia2 }, so after the photon passing through the first BSo , the state of the photon and the absorption object becomes 1 |φ i0 → √ (cos ϑ |1ia1 |0ia2 |0ia3 + sin ϑ |0ia1 |1ia2 |0ia3 )(|0io + |1io ). 2

(4)

Then for the component |1ia1 |0ia2 |0ia3 , i.e. the photon is in path a1 , it will direct towards the second BSo ; but for the component |0ia1 |1ia2 |0ia3 , it will enter the inner interferometer. From the Eqs. (1)-(3), we know if the photon is not absorbed in the inner interferometer, the system state is given by |φ i0 #205632 - $15.00 USD (C) 2014 OSA

→

1 √ [cos ϑ |1ia1 |0ia2 |0ia3 (|0io + |1io ) 2

Received 28 Jan 2014; revised 10 Mar 2014; accepted 12 Mar 2014; published 7 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.008970 | OPTICS EXPRESS 8975

+ sin ϑ |0ia1 (cosN θ |1ia2 |0ia3 |1io + |0ia2 |1ia3 |0io )].

(5)

After passing through the inner interferometer, if the photon is in path a3 , it will be detected by the detector Da and the component |0ia1 |0ia2 |1ia3 will be ignored. While, other components will arrive the second BSo and begin the next outer cycle. So at the end of the first cycle in the outer interferometer, the state can be written as 1 |φ i1 = √ [cos ϑ |1ia1 |0ia2 |0ia3 (|0io + |1io ) + sin ϑ cosN θ |0ia1 |1ia2 |0ia3 |1io ]. 2

(6)

Obviously, Eqs. (5) and (6) are not normalized, because the components that absorbed by the object or Da in Charlie’s site can’t reach the next BSo and are ignored. Note that after every outer cycle, only the terms that include the component |0ia1 |1ia2 |0ia3 enter the inner interferometer. Using the same derivation process, we can calculate the system state after the photon finishing the mth (2 ≤ m ≤ M) outer cycle, which can be written as 1 |φ im = √ (xm |1ia1 |0ia2 |0ia3 |0io + ym |1ia1 |0ia2 |0ia3 |1io + zm |0ia1 |1ia2 |0ia3 |1io ), 2

(7)

where the parameters xm , ym , and zm denote the probability amplitudes of |1ia1 |0ia2 |0ia3 |0io , |1ia1 |0ia2 |0ia3 |1io , and |0ia1 |1ia2 |0ia3 |1io , respectively, and satisfy the recursion relations xm = xm−1 cos ϑ , ym = ym−1 cos ϑ − zm−1 sin ϑ , zm = (ym−1 sin ϑ + zm−1 cos ϑ ) cosN θ ,

(8)

with ϑ = π /2M, θ = π /2N, x1 = y1 = cos ϑ and z1 = sin ϑ cosN θ . It should be noticed that the parameters xm , ym , and zm are useful, because all the probability amplitudes of the quantum states in the following are composed of them. Note that, after all the outer cycles, the photon appears on the output pot of path either a1 or a2 , which means that the photon never passed through the transmission channel between Alice and Charlie. Therefore, we can describe the photon state in the Hilbert space spanned by |1ia1 |0ia2 and |0ia1 |1ia2 , and define the basis vectors as |0ia ≡ |1ia1 |0ia2 and |1ia ≡ |0ia1 |1ia2 . Hence, when the photon finishes all the M outer cycles, the state can be expressed as 1 |φ iM = √ (xM |0ia |0io + yM |0ia |1io + zM |1ia |1io ). 2

(9)

Now, we begin the second step of the entanglement distribution, i.e., Bob sends a photon in the interferometer from the input port of path b1 , so the initial state of Bob’s photon is |1ib1 |0ib2 |0ib3 . For convenience, we provisionally set |α ia ≡ xM |0ia and |β ia ≡ yM |0ia + zM |1ia during the derivation process. Therefore, the second step starts with the state 1 |ψ i0 = √ (|α ia |0io + |β ia |1io )|1ib1 |0ib2 |0ib3 . 2

(10)

Similar to the first step, when the Bob’s photon finishes the mth outer cycle and isn’t absorbed yet, the system state will become 1 |ψ im = √ (xm |α ia |0ib |0io + ym |β ia |0ib |1io + zm |β ia |1ib |1io ), 2

(11)

where the Bob’s photon has been expressed in the Hilbert space {|0ib ≡ |1ib1 |0ib2 , |1ib ≡ |0ib1 |1ib2 }, and the parameters xm , ym , and zm also satisfy the recursion relations in Eq. (8). #205632 - $15.00 USD (C) 2014 OSA

Received 28 Jan 2014; revised 10 Mar 2014; accepted 12 Mar 2014; published 7 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.008970 | OPTICS EXPRESS 8976

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When Bob’s photon finishes the Mth outer cycle and is not absorbed by the object or the detector Db , the state is given by |ψ iM

=

1 √ (x2M |0ia |0ib |0io + y2M |0ia |0ib |1io 2 +yM zM |1ia |0ib |1io + yM zM |0ia |1ib |1io + z2M |1ia |1ib |1io ).

(12)

In order to obtain maximal entangled state, we should tune the coefficients (probability amplitudes) x2M , y2M , yM zM , and z2M by choosing suitable M and N. We plot the variation trend of the coefficients with the values of N and M, as shown in Fig. 2. It’s obvious that x2M → 1, y2M → 0, yM zM → 0, and z2M → 1 for large values of N and M, for example, x2M = 0.9210, y2M = 0.0001, yM zM = 0.0114, and z2M = 0.9626 for M = 30 and N = 1000, which means that after the Mth outer cycle, the state in Eq. (12) can be approximatively expressed as |ψ iM ≃ √12 (|0ia |0ib |0io + |1ia|1ib |1io ), which is a maximal entangled state for the absorption object and the photons in Alice’s and Bob’s sites. In order to achieve the two-photon entangled state distribution between √ Alice and Bob, Charlie√needs to perform a Hadamard transformation {|0io → (|0io + |1io )/ 2, |1io → (|0io − |1io )/ 2} on the absorption object, the system state will become 1 |ψ iM → √ [(|0ia |0ib + |1ia|1ib )|0io + (|0ia |0ib − |1ia |1ib )|1io ]. 2 #205632 - $15.00 USD (C) 2014 OSA

(13)

Received 28 Jan 2014; revised 10 Mar 2014; accepted 12 Mar 2014; published 7 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.008970 | OPTICS EXPRESS 8977

Then Charlie measures the object in the basis {|0io , |1io }, i.e. observe the presence or absence of the object. Obviously, if Charlie doesn’t broadcast his measurement result, Alice and Bob will know which entangled state they shared with probability of 50%; if Alice and Bob receive Charlie’s measurement result, they will succeed in sharing entanglement deterministically. So far, the entanglement distribution between two distant parties has been achieved in principle, and the specific experimental implementation will be introduced in next section. During the whole process, once Alice’s or Bob’s photon passes through the transmission channel, i.e. enters path a3 or b3 , it will be absorbed by either the absorption object or the detector Da(b) . Choosing suitable values N and M, the probability of finding a photon in the transmission channel is virtually zero, so the present protocol is a counterfactual entanglement distribution scheme. 3.

Physical implementation for the counterfactual entanglement distribution

The counterfactual entanglement distribution scheme has been presented in above section, in which the difficulty is the quantum control and manipulate of the presence of absence of the absorption object. Here, we discuss the implementation for the counterfactual entanglement distribution using a quantum-dot-microcavity coupled system. Consider a singly charged selfassembled GaAs/InAs QD being embedded in an optical resonant double-sided microcavity, which proposed by Hu et al. [33] and recognized by Bonato et al. [34] recently. The four relevant electronic levels are shown in Fig. 3(a), where the symbols ⇑ (⇓) and ↑ (↓) represent a heavy hole and an electron with z-direction spin projections + 32 (− 32 ) and + 12 (− 12 ), respectively. The charged exciton X − , produced by the optical excitation of the system, consists of two electrons bound in one hole. The two electrons in the exciton are in a singlet state, which means the two electrons have total spin zero, so the electron-spin interactions with the heavy hole spin are avoided. According to the optical selection rules and the transmission and reflection rules of the cavity for an incident circular polarization photon with sz = ±1 conditioned on the QD-spin state, the interaction between photons and electrons in the QD-microcavity coupled system can be described as follows [35]: |R↑ , ↑i → |L↓ , ↑i, |L↑ , ↑i → −|L↑ , ↑i, |R↓ , ↑i → −|R↓ , ↑i, |L↓ , ↑i → |R↑ , ↑i, |R↑ , ↓i → −|R↑ , ↓i, |L↑ , ↓i → |R↓ , ↓i, |R↓ , ↓i → |L↑ , ↓i, |L↓ , ↓i → −|L↓ , ↓i, (14) where |Ri and |Li denote the right-circularly polarized photon state and the left-circularly polarized photon state, respectively, and the superscript uparrow (downarrow) denotes the propagating direction of polarized photon along (against) the z axis. The formulas above show that, for an incident photon with spin sz = +1 (|R↑ i or |L↓ i), if the electron is in the state | ↑i, the photon will couple with the electron and be reflected by the cavity. Then the photon state is transformed into the state |L↓ i or |R↑ i, respectively. On the other hand, if the electron is in the state | ↓i, there is no dipole interaction and the photon is transmitted through the cavity and acquire a π mod2π phase shift relative to a reflected photon. Similarly, a photon with spin sz = −1 (|R↓ i or |L↑ i) will be reflected when the electron-spin state is | ↓i and will be transmitted through the cavity when the electron-spin state is | ↑i. From the introduction above, we can see that the photon-QD-microcavity system is very suitable to serve as the quantum version of the absorption object in the above section. Construct a quantum device as shown in Fig. 3(b). Let a right-circularly polarized photon |Ri first pass through a polarizing beam splitter in the circular basis (c-PBS), which transmits the right circularly polarized photon and reflects the left circularly polarized photon. Then the photon will inject the cavity along the z axis. According to Eq. (14), for the electron state | ↑i, the photon couples with the QD and both polarization and the propagation direction will be flipped, then it is reflected by c-PBS and absorbed by the detector D. While, for the electron state | ↓i, the #205632 - $15.00 USD (C) 2014 OSA

Received 28 Jan 2014; revised 10 Mar 2014; accepted 12 Mar 2014; published 7 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.008970 | OPTICS EXPRESS 8978

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photon passes through the cavity. Therefore, using the device in Fig. 3(b), the absorbing or not of a right-circularly polarized photon is controlled by the quantum state of the electron spin in QD. Now, Alice and Bob connect with Charlie by a nested Michelson interferometer, respectively, as shown in Fig. 4. Alice and Bob have identical device. The light sources of Alice and Bob emit right circularly polarized photons. Switchable polarization rotators (SPRs) and switchable mirrors (SMs) can be switched on and off by external means. Detectors (Ds) are used to absorb photons. The grey part in Alice’s (Bob’s) site and the device in Charlie’s site together form a Michelson-type interferometer, which, as an inner interferometer, is inserted in one of the arms of an outer Michelson interferometer. That is, between Alice (Bob) and Charlie, the two optical paths SMo → M1(6) and SMo → M3 form the outer interferometer, and the two optical paths SMi → M2(7) and SMi → M3 form the inner interferometer. The SMo(i) in the outer (inner) interferometer is switched off initially (transmits photons) and then remains on (reflects the photon) once the photon enters the outer (inner) interferometer until the photon completes the M(N)th cycle in the outer (inner) interferometer. SPRo and SPRi perform the transformations {|Li → cos ϑ |Li + sin ϑ |Ri, |Ri → cos ϑ |Ri − sin ϑ |Li} and {|Li → cos θ |Li + sin θ |Ri, |Ri → cos θ |Ri − sin θ |Li}, respectively, where ϑ = π /2M and θ = π /2N. Experimentally, the SMo(i) can be controlled accurately by a computer via the switching-time sequences [36–38]. On the other hand, a suitable ultrafast optical switch with minimal loss and without disturbing the photon’s quantum state has been demonstrated, and the switching window can achieve 10 ps [39, 40]. Because the quarter-wave plate can achieve the photon’s transformation between linear and circular polarization |Li(|Ri) ↔ |Hi(|V i), and the half-wave plate can rotate the linearly polarized photon by arbitrary angle α {|Hi → cos α |Hi + sin α |V i, |V i → cos α |V i − sin α |Hi}. Therefore, for given values of M and N, the SPRo(i) can be implemented by the combination of quarter-wave plate and half-wave plate oriented at specified angles. Here, the photonic qubit is encoded in polarization degree of freedom, which corresponds to the spatial-path encoding in the above section. The combination of SPR and c-PBS is equivalent

#205632 - $15.00 USD (C) 2014 OSA

Received 28 Jan 2014; revised 10 Mar 2014; accepted 12 Mar 2014; published 7 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.008970 | OPTICS EXPRESS 8979

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in function to the BS in Fig. 1. The inner interferometer serves as the interaction-free measurement interferometer, and the outer interferometer correspond to the outer cycle of Fig. 1. The QD-cavity unit at Charlie’s site is placed in one arm of the interferometer, and the electron in the QD is initially in the state √12 (| ↑i + | ↓i). At original time t0 , the optical switch K in Charlie’s site connects port a and Alice sends a photon in the interferometer. When Alice’s photon completes the M outer cycles in the interferometer and is not absorbed yet, i.e. at time t = t0 + L/c, where L is the overall optical path length of M outer cycles and c light velocity, Charlie switches K to port b and Bob sends a photon in the interferometer. With the similar calculation process as introduced in above section, when Bob’s photon completes the M outer cycles, the system state can be expressed as |ΨiM

=

1 √ (x2M |Ria |Rib | ↓ic + y2M |Ria |Rib | ↑ic 2 −yM zM |Lia |Rib | ↑ic − yM zM |Ria |Lib | ↑ic + z2M |Lia |Lib | ↑ic ),

(15)

where the subscripts a, b, and c indicate the qubit in Alice’s, Bob’s, and Charlie’s site, respectively. The parameters xM , yM , and zM also satisfy the recursion relations in Eq. (8). Choosing suitable values of M and N as shown in Fig. 2, for example, M = 30 and N = 1000, then x2M = 0.9210, y2M = 0.0001, yM zM = 0.0114, and z2M = 0.9626, so we can approximately obtain the maximally entangled state |ΨiM ∼ √12 [|Ria |Rib | ↓ic + |Lia |Lib | ↑ic ]. Then Charlie performs √ √ a Hadamard transformation {| ↑i → (| ↑i + | ↓i)/ 2, | ↓i → (| ↑i − | ↓i)/ 2} on the electron state, which can be implemented by using a π /2 microwave or optical pulse [41–43]. The counterfactual photonic entanglement distribution can be achieved after Charlie’s detecting the electron state in the basis {| ↑i, | ↓i}. Now we surprisingly note that the two distant photons have never passed through transmission channels during the process from separable to entangled state, that is the perfect counterfactual entanglement distribution is possible in experiment.

#205632 - $15.00 USD (C) 2014 OSA

Received 28 Jan 2014; revised 10 Mar 2014; accepted 12 Mar 2014; published 7 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.008970 | OPTICS EXPRESS 8980

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4.

Analysis and discussion

Now we begin to analyze the effect of the imperfections in experiment on the performance of the counterfactual entanglement distribution. First of all, an obvious influence factor is the number of the outer (inner) cycle M(N). The perfect entanglement distribution requires that the state of photon-electron system becomes a maximal GHZ-type entangled state after the M outer cycles, i.e. |Ψ′ iM = √12 [|Ria |Rib | ↓ic + |Lia |Lib | ↑ic ]. Hence, the fidelity of the Eq. (15) can be obtained

as F = |M hΨ′ |ΨiM |2 = (x2M +z2M )2 /4. We numerically evaluate the fidelity change with M and N as shown in Fig. 5, which shows that the fidelity approach perfect for large M and N. However, as the values of M and N increase, we should consider the photon loss caused by linear optical elements, which may reduce the performance of the scheme. Fortunately, the present scheme works in repeat-until-success fashion, that is, the scheme is successfully completed if and only if photons appear at the output ports of Alice and Bob, which means the photons didn’t lose during the whole process. Therefore, the photon loss does not affect the fidelity, but affects the efficiency of the scheme. Suppose that the photon loss rate in every outer cycle is ε , the success rate of the scheme will be (1 − ε )2M , i.e. the success rate will be reduced exponentially with the cycle number. Nonetheless, if the light source can continuously generate a lot of single photons per second as in [44], the entanglement distribution can be accomplished with high fidelity within a short time. We also note that the present scheme requires high-precision switchable polarization rotators SPRo and SPRi to rotate a single circularly polarized photon by angles ϑ = π /(2M) and θ = π /(2N) respectively. However, it is bound to introduce a slight error in the practical experiment. The error coefficient of the SPRo(i) can be described by so(i) , which means the photon state is rotated with an additional angle ∆ϑ = so ϑ /M (∆θ = si θ /N) after each outer (inner) cycle [20]. So the angle ϑ (θ ) in the parameter recursion relations of Eq. (8) should be replaced with ϑ + ∆ϑ (θ + ∆θ ). In this way, the fidelity of Eq. (15) can be estimated numerically as shown in Fig. 6, in which we have set the error coefficients so = si = s and plotted the fidelity change curves versus s for different values of M and N. Other crucial influence factors mainly come from the performance of the spin-cavity unit, which seemingly does not affect the entanglement distribution since no photon passes through

#205632 - $15.00 USD (C) 2014 OSA

Received 28 Jan 2014; revised 10 Mar 2014; accepted 12 Mar 2014; published 7 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.008970 | OPTICS EXPRESS 8981

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transmission channels to interact with the quantum dot during the whole process. However, we need to note that the rules of optical transitions in Eq. (14) were obtained in ideal case, i.e. without regard to the side leakage and cavity loss. When the side leakage and the cavity loss are not negligible, the reflection and transmission coefficients of the coupled and the uncoupled cavities are generally different, and the rules of optical transitions in Eq. (14) need to be corrected. Therefore, the performance of spin-cavity system certainly will influence the fidelity of the entanglement distribution. The reflection and transmission coefficients of a double-sided optical microcavity for weak excitation limit can be described by [33, 35] [i(ωX − − ω ) + 2γ ][i(ωc − ω ) + κ2s ] + g2 , [i(ωX − − ω ) + 2γ ][i(ωc − ω ) + κ + κ2s ] + g2 −κ [i(ωX − − ω ) + 2γ ] t(ω ) = , [i(ωX − − ω ) + γ2 ][i(ωc − ω ) + κ + κ2s ] + g2 r(ω ) =

(16)

where g is the coupling strength; ω , ωc , and ωX − are respectively the frequencies of the input photon, the cavity field, and the X − transition; κ and γ /2 are the decay rate of cavity field and the X − dipole decay rate, respectively; κs /2 describes the side leakage rate. By setting g = 0 and under the resonant interaction condition ωc = ωX − = ω0 , we can obtain the reflection and transmission coefficients for the case that the QD does not interact with the input photon as follow r0 (ω ) =

[i(ω0 − ω ) + κ2s ] −κ . κs , t0 (ω ) = [i(ω0 − ω ) + κ + 2 ] [[i(ωc − ω ) + κ + κ2s ]

(17)

From the introduction above, the dynamics of the interaction between photons and electrons in the QD-microcavity can be rewritten as the form contained the side leakage and the cavity loss. In the case of photon state |R↑ i used in the present scheme, the interaction can be described by |R↑ , ↑i → |r(ω )||L↓ , ↑i + |t(ω )||R↑, ↑i, |R↑ , ↓i → −|t0 (ω )||R↑ , ↓i − |r0 (ω )||L↓ , ↓i.

(18)

Therefore, we can derive the realistic evolution of the system by replacing |R↑ , ↑i → |L↓ , ↑i and |R↑ , ↓i → −|R↑ , ↑i with Eq. (18), which will reduce the fidelity of the final state in Eq. (15). In #205632 - $15.00 USD (C) 2014 OSA

Received 28 Jan 2014; revised 10 Mar 2014; accepted 12 Mar 2014; published 7 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.008970 | OPTICS EXPRESS 8982

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order to quantitatively analyze the effect of the side leakage and the cavity loss, we numerically simulate the varying of the fidelity of the final photon-spin state versus κs /κ and g/κ for N = 500 and M = 60 as shown in Fig. 7. It’s obvious that the scheme has higher fidelity for κs ≪ κ and the effect of the coupling strength is negligible, for example, the fidelity equals 99.81% for κs = 0.01κ and g = 3κ . Although the cavity with high quality is still a challenge in experiments [45], the side leakage of the cavity can be suppressed with the improvement of fabrication techniques [46]. The electron spin qubit used in the present scheme and its fast initialization have also been demonstrated [47–49]. Strong coupling in the QD-microcavity system has been demonstrated. Reitzenstein et al. [46] reported g/(κ + κs ) ≃ 0.5 for a micropillar cavity with diameter d = 1.5µ m and quality factor Q = 4 × 8800. Young et al. [50] also confirmed g > (κ + κs + γ )/4 in a pillar microcavity, and they showed the radio of κs /κ can be reduced in the micropillar structures by increasing the coupling efficiency. One can see the fidelity of single photon input-output process is not ideal in current experiment conditions, but the prospect of a high-fidelity spin-photon interface can be seen from [50]. Besides the side leakage and cavity loss, the electron-spin decoherence will also reduce the fidelity of the scheme. Describe the electron-spin relaxation time and coherence time by T1e and T2e , the spin decoherence will decrease the fidelity by a factor [33, 35] F ′ = [1 + exp(−∆t/T2e )]/2, where ∆t is the time interval between two inputs of a photon. During the present entanglement distribution process, a photon needs to enter the cavity thousands of times, and the scheme should be completed within the electron-spin coherence time. That means the spin coherence time is at least three orders of magnitude longer than the cavity photon time interval, so the electron-spin decoherence can reduce the fidelities by few presents. However, on the other hand, it’s obvious that the scheme requires electron spin to be shielded from the environment long enough or have long electron-spin coherence time, because Alice’s photon and Bob’s photon must successively go through a large number of cycles. Suppose the distance between Alice (Bob) and Charlie is L, the required time to accomplish the scheme is t ≈ 2NML/c (c is the speed of light), so the electron-spin coherence time should satisfy T2e ≥ 2NML/c. Hence, for given T2e , the distance of the entanglement distribution is limited. However, due to the hyperfine interaction between the electron spin and 104 − 105 host nuclear spins, the coher#205632 - $15.00 USD (C) 2014 OSA

Received 28 Jan 2014; revised 10 Mar 2014; accepted 12 Mar 2014; published 7 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.008970 | OPTICS EXPRESS 8983

ence time of electron spin in QD is limited to hundreds of microseconds or less [51]. If nuclear spin fluctuations are suppressed [52], the coherence time can be greatly prolonged. Fortunately, more and more works have demonstrated that spin echo techniques can suppress the nuclear spin fluctuations effectively and prolong the electron spin coherence [53–57], which suggest strong promise for long electron-spin coherence time. For the coherence time T2e ∼ 100ms, M = 30, and N = 1000, the entanglement distribution distance will only close to 200m, which means the present entanglement distribution protocol may not promise a practical advantage over existing ones. However, the counterfactual protocol demonstrates at least the nonlocal entanglement can be created without exchanging any particles between remote nodes. The realization of the present protocol relies on single-photon input-output process via quantum dot and cavity coupling, and the single photons can be created via the well-known spontaneous parametric down-conversion of the ultraviolet laser beam in a β -barium borate crystal [58, 59]. However, the background noise will be produced during the process of photons’ generation, transmission, and detection, and the background noise is larger than the single photon signal. In order to accomplish the scheme, the experiment should be performed under the condition with high signal-noise ratio. Some recent works [60, 61] have demonstrated the unwanted background photons can be efficiently reduced in the parametric down-conversion process by using the optical shutter controlled by a simple field programable gate array. The background noise can also be suppressed by optimizing the detuning between the frequencies of the pump and photon pairs and cooling the nonlinear fiber [61]. We note that the detectors in the present scheme are only used to absorb the photon passed through the transmission channels, so the sensitivity and dark counts of the detectors does not influence the efficiency of the scheme. Hence the detectors can even be replaced with other absorption objects. From the analysis above, the present scheme can be realized in principle, but its implementation in practice may be a challenge for current experiment conditions. Nevertheless, compared with the experimental difficulties, maybe the counterfactual fact that entanglement distribution can be completed without transmitting any particles is more interesting. 5.

Conclusion

In conclusion, we have proposed a counterfactual scheme for entanglement distribution, which dose not require any particles to travel the transmission channels between participants. The scheme has shown that two distant photons can be surprisingly entangled with each other without interaction or entanglement swapping. In order to demonstrate the feasibility of the present scheme, we also constructed a quantum device for entanglement distribution by using QD spins in optical microcavities based on spin-selective photon reflection from the cavity. The results of numerical analyses showed that the scheme can be effectively implemented in the ideal asymptotic limit. On one hand, perhaps the present entanglement distribution scheme can be used for quantum information processing in the future; on the other hand, some deeper physical mechanisms may be implied behind the mind-boggling and highly counter-intuitive fact of the counterfactual entanglement distribution. Acknowledgment This work is supported by the National Natural Science Foundation of China under Grant Nos. 61068001 and 11264042; China Postdoctoral Science Foundation under Grant No. 2012M520612; the Program for Chun Miao Excellent Talents of Jilin Provincial Department of Education under Grant No. 201316; and the Talent Program of Yanbian University of China under Grant No. 950010001.

#205632 - $15.00 USD (C) 2014 OSA

Received 28 Jan 2014; revised 10 Mar 2014; accepted 12 Mar 2014; published 7 Apr 2014 21 April 2014 | Vol. 22, No. 8 | DOI:10.1364/OE.22.008970 | OPTICS EXPRESS 8984