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Physics Letters A www.elsevier.com/locate/pla

Creating W-type entangled states in quantum dot systems Jian-Qi Zhang, Lin-Lin Xu, Zhi-Ming Zhang ∗ Laboratory of Photonic Information Technology, SIPSE and LQIT, South China Normal University, Guangzhou 510006, China

a r t i c l e

i n f o

Article history: Received 27 January 2010 Received in revised form 8 May 2010 Accepted 19 July 2010 Available online 30 July 2010 Communicated by P.R. Holland

a b s t r a c t We show that three initially nonresonant quantum dots (QDs) could be used to generate W-type entangled states with the application of a single detuned light via the ac Stark effect. This gives a way to prepare the highly entangled excitonic states in the picosecond time scale by controlling the interactions between QDs. Published by Elsevier B.V.

Keywords: Entanglement Decoherence Quantum dots Exciton Stark shift

1. Introduction Entanglement plays an essential role not only in understanding quantum mechanics [1], but also in the applications in quantum computation and quantum information processing [2,3]. Now the bipartite entanglement has been well understood, but it is still diﬃcult to characterize a general multipartite entanglement. The Greenberger–Horne–Zeilinger-type (GHZ-type) [4] states and the W-type [5] states are two kinds of well-understood multipartite entangled states which, as proved in Refs. [5,6], cannot be converted to each other by local operation and classical communication (LOCC). As it is well known, tripartite W-type entanglement has both bipartite and tripartite quantum entanglement simultaneously, thus it is robust to the loss of one qubit [5] and can be used in some quantum information processing such as the perfect teleportation [7]. So it is of practical signiﬁcance to prepare the W-type entangled states. On the other hand, with the development in semiconductor technology in the past decades, semiconductor quantum dots (QDs) have been used to replace atoms in some applications [8]. Because of their low-dimensional character, QDs can conﬁne the electrons and holes along all three dimensions which lead to a discrete energy structure with sharp optical absorption lines [9], therefore the physics of QDs shows many parallels with the behavior of naturally occurring quantum systems in atomic and nuclear physics [10]. These important physical properties facilitate the QDs to be one

*

Corresponding author. Tel.: +86 20 39310154; fax: +86 20 39310309. E-mail address: [email protected] (Z.-M. Zhang).

0375-9601/$ – see front matter Published by Elsevier B.V. doi:10.1016/j.physleta.2010.07.042

of the candidates for entanglement generation and applications in quantum information processing. Based on the idea of Lloyd [11], Loss et al. presented the ﬁrst scheme to generate Bell states of two QDs [12]. Afterwards, many schemes for the preparation of entangled states of QDs were proposed. Hanson et al. proposed to prepare entangled states of electrons in QDs [13]. Wang et al. presented a scheme to generate multipartite entangled states of electrons in QDs which are conﬁned in a cavity [14]. Yuan et al. showed a scheme which generates the W-type states with QDs conﬁned in a single-mode microcavity [8]. And the literaturies [15–18] studied how to generate the entangled states of two excitons by using the coupling between QDs and the external classical light ﬁelds. However, the preparation of the Wtype states has not been taken into account in this model until now. In Ref. [17], Nazir et al. proposed a scheme to prepare the highly entangled excitonic states of two QDs by resorting to the so-called Förster effect [15]. They showed that two initially nonresonant QDs could be brought into resonance by applying a detuned light ﬁeld. The essence of this phenomenon is due to the optical Stark shift induced by the light ﬁeld. In such a situation the energy can be transferred to each other between the two neighboring QDs, one as an emitter QD and the other as an acceptor QD. This phenomenon is called Förster effect. To the best of our knowledge, the generation of the W-type entangled states of three QDs by using the Förster effect and the Stark shift has not been studied until now. In this Letter, we consider three QDs in a chain that are driven by the external light ﬁeld under different conditions, and present one-step and two-step schemes to generate W-type states. The advantages of our scheme are as follows. Firstly, this model could be controlled by an external light ﬁeld. Secondly, it is possi-

J.-Q. Zhang et al. / Physics Letters A 374 (2010) 3818–3822

ble to generate and maintain long-lived entangled excitonic states. Thirdly, this scheme considers nonidentical QDs and this is more practical. The organization of this Letter is as follows: in Section 2 we derive the effective Hamiltonian of our model; in Section 3 we study how to prepare the W-type states of three QDs in different situations; the discussion about the realizability is given in Section 4; a conclusion is presented in Section 5. 2. Theoretical model and the effective Hamiltonian We consider a linear chain of three coupled self-assembled QDs, each of QDs has two-level, and they are driven by an external classical light ﬁeld. The total Hamiltonian of this system takes the form (h¯ = 1):

ˆ QD = 1 ω1 σ z + 1 ω2 σ z + 1 ω3 σ z , H 1 2 3 2 2 2 ˆH QD–QD = V F σ + σ − + σ + σ − + σ − σ + + σ − σ + 1 2 2 3 1 2 2 3 + V C σ111 σ211 + σ211 σ311 , ˆ light–QD = Ω1 cos(ωl t ) σ + + σ − + Ω2 cos(ωl t ) σ + + σ − H 1 1 2 2 (1) + Ω3 cos(ωl t ) σ3+ + σ3− , ˆ QD is the free Hamiltonian of the QDs, Hˆ QD–QD is the interwhere H ˆ light–QD is the Hamiltoaction Hamiltonian between the QDs, and H nian describing the interactions between the QDs and the classical light ﬁeld. ω p ( p = 1, 2, 3) is the exciton creation energy for the QD p, ωl is the frequency for the classical light. The operators for QDs are deﬁned as σ p+ = |1 p 0|, σ p− = |0 p 1|, σ pz = σ p11 − σ p00 ,

σ p11 = |1 p 1|, σ p00 = |0 p 0|, with |0 p and |1 p being the ground state and the low-lying excited state of the QD p; V F is the Förster coupling between the two nearest neighbor QDs [15]; V C is the direct Coulomb binding energy between two excitons, which are located in two nearest neighbor QDs. We assume that each QD could couple to the classical light ﬁeld but with a different strength, denoted by the corresponding Rabi frequency Ω p ≡ −2d p · E(r, t ). Here, d p is the dipole moment for QD p, and E(r, t ) is the amplitude of the external classical light ﬁeld at time t and position r. Therefore, the Rabi frequency can be controlled by the amplitude of the external classical light ﬁeld. Under the rotating-wave approximation (RWA) and in the frame rotating with the laser frequency ωl , Eq. (1) can be transformed into:

ˆ r = δ1 σ111 + δ2 σ211 + δ3 σ311 H

+ V F σ1+ σ2− + σ2+ σ3− + σ1− σ2+ + σ2− σ3+ Ω1 + σ1 + σ1− + V C σ111 σ211 + σ211 σ311 + +

2

−

σ2+ + σ2 +

Ω3 2

is very small, Eq. (3) ensures that the eigenstates can be approximately treated degenerately and the interactions for QD–QD and light–QD can be treated as perturbations. The second condition is

E mnk ≈ δ1 δm,1 + δ2 δn,1 + δ3 δk,1 + V C (δm,1 δn,1 + δn,1 δk,1 ),

2

−

σ3+ + σ3 ,

σ1+ σ2− + σ1− σ2+

+ V 13 σ1+ σ3− + σ1− σ3+ + V 23 σ2+ σ3− + σ2− σ3+

(5)

with

1 = δ1 +

1 4

EQ

Ω32 Ω22 , + + E Q − E 011 E Q − E 101

Ω32 Ω12 Ω22 , + + 4 E Q − E 011 EQ − 0 E Q − E 110 Ω2 Ω12 Ω22 1 3 = δ3 + + + 3 , 4 E Q − E 101 E Q − E 011 EQ Ω1 Ω2 1 Ω 1 Ω2 , V 12 = V F + + 4 EQ E Q − E 011 Ω1 Ω3 1 Ω 1 Ω3 , V 13 = + 4 EQ E Q − E 101 Ω2 Ω3 1 Ω 2 Ω3 . V 23 = V F + + 4 EQ E Q − E 110 2 = δ2 +

1

Ω12

(6)

The eigenenergies of Eq. (5) with different parameters are plotted in Fig. 1, where each curve of the eigenenergies corresponds to a dressed state [17,18]. This ﬁgure shows clearly that there are the anticrossings of the eigenenergies. This phenomenon is resulted from the exciton Rabi splitting which has been observed in the case of single QD in 2008 [19]. Because the general analytical solution for the Hamiltonian (5) is lengthy, we do not list it in this Letter. Instead, we will give some typical analytical solutions, the others will be given by ﬁgures. In addition, we should point out that the values of the parameters in this Letter are quoted from Ref. [17]. 3. Preparing the W-type entangled states

(2)

where δ p = ω p − ωl is the detuning of the light ﬁeld and the QD p. Note that in the derivation of the above equation we have chosen 3ωl − (ω1 + ω2 + ω3 ) = 0. For further processing we assume the following two conditions. The ﬁrst one is

|δ1 − δ2 |, |δ2 − δ3 |, |δ1 − δ3 |, | V F |, |Ω p /2| min | E Q |, | E Q − E mnk | ,

(4)

where δ j ,1 ( j = m, n, k) is the Kronecker delta function. As the interactions for QD–QD and light–QD are weak and can be treated as the perturbations, the excited energy for Eq. (4) can be approximately replaced by the eigenenergy for Eq. (2) with Ω2 = 0 and V F = 0. Under above conditions, using the degenerate perturbation theory, one can obtain the following effective Hamiltonian for the subspaces {|001, |010, |100} with an abbreviation |mnk = |m3 |n2 |k1 :

ˆ eff = 1 σ111 + 2 σ211 + 3 σ311 + V 12 H

ˆ = Hˆ QD + Hˆ QD–QD + Hˆ light–QD , H

Ω2

3819

(3)

where m, n, k = 0, 1, respectively, E Q = (δ1 +δ2 +δ3 )/3, and E mnk is the eigenenergy of Eq. (2) with Ωi = 0. As min(| E Q |, | E Q − E mnk |)

In this section we shall discuss how to generate the W-type entangled states of three QDs. Here two different situations are taken into account. In the ﬁrst situation, we propose a one-step scheme to entangle the three QDs where QD 1 and QD 3 are assumed to be the same but different from the middle one. In the second situation, we entangle the three different QDs in two steps (two-step scheme) in which two different light ﬁelds act successively on the QDs 1, 2 and then on QDs 2, 3, respectively. Now let us ﬁrst discuss the one-step scheme. In this case, the QDs 1 and 3 are assumed of the same structure, that is, ω1 = ω3 , δ1 = δ2 , Ω1 = Ω3 , and V 12 = V 23 , while the transition frequency of the QD 2 is different and satisﬁes the relation |ω2 − ω1,3 | V F . Then the effective Hamiltonian (5) reduces to:

ˆ one-step = 1 H

σ111 + σ311 + 2 σ211

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J.-Q. Zhang et al. / Physics Letters A 374 (2010) 3818–3822

Fig. 1. (Color online.) The anticrossing induced by the classical light for ﬁxed ratio k = Ω1 /Ω2 = Ω3 /Ω2 = 0.55, and V F = 0.088 meV. Eigenvalues are calculated from Eq. (2). (a) δ1 = 192.59 meV, δ2 = 190.59 meV and δ3 = 192.59 meV; (b) δ1 = 192.54 meV, δ2 = 190.59 meV and δ3 = 192.59 meV; (c) δ1 = 192.54 meV, δ2 = 190.59 meV and δ3 = 191.59 meV. [Description: Numerical simulation of eigenvalues varies with the external classical light with the different parameters.]

+ V 12 σ1+ σ2− + σ1− σ2+ + σ2+ σ3− + σ2− σ3+ + V 13 σ1+ σ3− + σ1− σ3+ , here

3 Ω12 1 = δ1 + 4

+

1 2 δ1 + δ 2

3

(7)

Ω22 − , δ1 + 2δ2

Ω22 , 4 δ1 + 2δ2 2 δ1 + δ 2 3 1 1 Ω1 Ω 2 , V 12 = V F + − 4 2 δ1 + δ 2 δ1 + 2δ2

2 = δ2 −

V 13 =

3

1

δ2 − 4δ1

W-type entangled state | W = √1 [i (|001 + |100) + e i γ0 |010],

2Ω12

−

3(δ1 − δ2 )Ω12 16δ12 + 4δ1 δ2 − 2δ22

(8)

.

Suppose initially the system of the three QDs is in the state

|ψ(0) = |010. It is not diﬃcult to prove that at time t the system will evolve to the following state under the action of the Hamiltonian (7),

ψ(t ) = − i V 12 sin(γ t ) |001 + |100

γ

+ cos(γ t ) +

i K 1 sin(γ t )

|010,

2γ

(9)

where K 1 = δ1 − δ2 + V 13 ,

γ =

2 8V 12 + K 12

2

. Here a global phase

factor e i K 2 t /2 with K 2 = δ1 + δ2 + V 13 has been discarded. When γ t = [2kπ ± arcsin( √ )]/γ , the state |ψ(t ) turns out to be a 3V 12

where

γ0 = arctan

K1 2 V 12 − K 12

. In other words, the system of inter-

est evolves from a separable state |010 into a W-type entangled state. It is worthy to point out that this process can be controlled by an external light ﬁeld, this is an advantage of this model. When the external light is tuned off, for δ > 2V , the difference of energy-levels between the two QDs is much larger than the effective interaction strength, there will be no energy exchange between the QDs. On the contrary, with the application of a single detuned laser which satisﬁes 1 = 2 , the three initially nonresonant QDs could be brought into resonant by the Stark shift. Here, we should point out that the Förster effect will give rise to the small oscillations in the populations of the ﬁnal state. One could suppress such oscillations by applying an electric ﬁeld [15]. The phenomena mentioned above are shown in Fig. 2. In practice, it is hard to grow two identical self-assembled QDs, so the QDs 1 and 3 may differ from each other in the transition frequencies, that is, ω1 = ω3 and δ1 = δ3 . In the following, we consider the inﬂuence of such differences. Under this situation, the effective Hamiltonian cannot be simpliﬁed to Eq. (7), we should adopt the effective Hamiltonian equation (5) to numerically simulate this problem. The numerical results are presented in Fig. 3. Fig. 3 shows the populations of the states varying as a function of time under the application of a detuned light. At ﬁrst, the light is on, the coherent oscillations take place among the states |001, |100 and |010. In this case, with the decrease of the population in state |010, the evolution curves for the populations of states |001 and |100 increase. Secondly, at the time

J.-Q. Zhang et al. / Physics Letters A 374 (2010) 3818–3822

Fig. 2. (Color online.) Populations of the states |001, |010 and |100 with ω1 = ω3 . They are calculated from Eq. (2) with the initial state |010, V F = 0.088 meV, k = Ω2 /Ω1,3 = 0.55, ω1 = ω3 = 1792.54 meV, and ω2 = 1790.59 meV. It describes the light with Ω2 = 32.35 meV acting for t = 7.5 ps and then turned off. [Description: When the light is on, with the population of state |010 decreasing and the states of |010 and |010 increasing, this system can evolve into the W-type state; afterward, with the light turned off, this W-type state can be kept. But there are oscillations in the ﬁnal state for the Förster effect.]

3821

Fig. 4. (Color online.) Populations of the states |001, |010 and |100 with |ω p =1,2,3 − ωq=1,2,3 ( p =q) | 0. They are calculated from Eq. (2) with the initial state |001, ω1 = 1792.54 meV, ω2 = 1790.59 meV, ω3 = 1791.59 meV, k = Ω2 /Ω1,3 = 0.55, and V F = 0.088 meV. When 0 ps t 10 ps, the light is turn on with Ω2 = 31.85 meV; when 10 ps t 18.54 ps, the classical light is turn on with Ω2 = 23 meV; when t > 18.54 ps, there is no light acting on. [Description: Generation the W-type states by two steps. When 0 ps t 10 ps, the population of state |001 is decreasing and the state of |010 is increasing. When 10 ps t 18.54 ps, the population of state |010 is decreasing and the state of |100 is increasing. And the population of state |010 centers on the population with 13 to oscillate. Afterward, the W-type state can be kept.]

acts on the QDs 2 and 3 for a time duration T 2 − T 1 to generate the target entangled state. In this case, the effective Hamiltonian can be written as,

ˆ two-step = H

Fig. 3. (Color online.) Populations of the states |001, |010 and |100 with ω1 ≈ ω3 . They are calculated from Eq. (2) with the initial state |010, V F = 0.088 meV, k = Ω2 /Ω1,3 = 0.55, ω1 = ω3 = 1792.54 meV, and ω2 = 1790.59 meV. It also describes the light with Ω2 = 32.35 meV acting for t = 7.5 ps and then turned off. (a) The time of evolution lasting for 50 ps. (b) The time of evolution lasting for 150 ps. [Description: When the light is on, with the population of state |010 decreasing and the states of |010 and |010 increasing, this system can evolve into the W-type state; afterward, with the light turned off, this W-type state can be kept. Besides the small oscillations in the ﬁnal state for the Förster effect, there are the large oscillations that, the population-curves for states |001 and |100 separate at ﬁrst, and then converge, which is caused by the difference of the energy structure in QD1 and QD3 .]

t = arcsin( √

γ 3V 12

)/γ , we turn off the light, the QDs can be decou-

pled with each other, and the system will be prepared in a W-type state. Comparing with Fig. 2, an essential difference in Fig. 3 is, when the W-type states have been prepared, the population-curves for states |001 and |100 separate at ﬁrst, and then converge, which is caused by the difference of the energy structure in QD1 and QD3 . As mentioned above, in the one-step scheme for generating the entangled W-type states, the difference between the QDs 1 and 3 may reduce the entanglement degree of the target W-type states. In order to avoid such a problem, we propose a two-step scheme to prepare the W-type entangled states for three different QDs. In the ﬁrst step, the light ﬁeld acts on the QDs 1 and 2 for a time T 1 so as to entangle the QDs 1 and 2. In the second step, the light ﬁeld

ˆ eff (Ω3 = 0), H

0 t T 1; ˆ H eff (Ω1 = 0), T 1 t T 2 ,

(10)

ˆ eff is given by Eq. (5). where H In principle, we can achieve any desired target W-type entangled states by controlling the Rabi frequencies and the interaction time. Because the analytical solution is tedious we shall not list it here for simplicity. Instead, we would like to give simulation results as plotted in Fig. 4. At the ﬁrst step to entangle the QDs 1 and 2, we choose the light in such a way that its intensity is adjusted so that energy level shifts for the QDs 1 and 2 satisfy 1 = 2 and are largely different from that of the QD 3, i.e., 3 . In this situation, |3 − 1,2 | V F , thus the interaction between the QD 3 and QD 2 (QD 1) can be neglected. As a result, in the ﬁrst step, the energy transfer mainly takes place between the QDs 2 and 3. Our numerical result shows that in this step the light ﬁeld needs to act for a time T 1 = π /(3V 12 ). From Fig. 4 we can see the population of the state |001 decreases while that for the state |010 increases in this situation. Similarly, in the second step, the light intensity is chosen so that energy level shifts for the QDs 2 and 3 satisfy 2 = 3 and are largely different from that of the QD 1, i.e., 1 . In this situation, |1 − 2,3 | V F , thus the interaction between the QD 1 and QD 2 (QD 3) is neglectible. Therefore, in the second step, the energy transfer mainly takes place between the QDs 2 and 3. With the light ﬁeld acting for a time T 2 − T 1 = π /(4V 23 ), we can ﬁnally obtain the W-type states. Similar to the one-step scheme, due to the Förster effect, there are oscillations in the population-curves of the states |001, |100 and |010. 4. Discussion In this section, we will give a brief discussion on the realizability. In the self-assembled AlAs/GaAs QD system, the optical Stark shifts of excitons have been observed, and the Rabi frequencies

3822

J.-Q. Zhang et al. / Physics Letters A 374 (2010) 3818–3822

all generated states are higher than 0.97. On the other hand, with the decoherence, the corresponding states can keep in the W state with the ﬁdelity F > 0.92 for about 30 ps. Moreover, as shown in Fig. 5, by comparing the time scale for the generation of the W-type states with the one for decoherence, we ﬁnd that those entangled states can be thought to be long-lived entangled states. This is also an advantage of the present system. Therefore, this scheme is realizability. 5. Conclusion

Fig. 5. (Color online.) Numerical simulation of the ﬁdelity about the generated states, they are calculated from Eqs. (11) and (12). The solid lines denote the states without decoherence, the dash ones represent the states with the decoherence. Here, the red line, green line and blue line are corresponding to the cases for Figs. 2, 3 and 4. And the decay rates are (Γ1 = Γ3 = 1.65 ns, Γ2 = 0.5 ns), (Γ1 = 1.84 ns, Γ2 = 0.5 ns, Γ3 = 1.65 ns), and (Γ1 = 0.826 ns, Γ2 = 0.5 ns, Γ3 = 1.65 ns), respectively. [Description: Without the decoherence, the ﬁdelities of those states are increasing at ﬁrst, and then they will center on a ﬁxed value to oscillate. All the ﬁdelities for all generated states are higher than 0.97. On the other hand, with the decoherence, the ﬁdelities of those states are increasing at ﬁrst, and then they will decrease. But those generated states can keep in the W state with the ﬁdelity F > 0.92 for about 30 ps.]

for different QDs are different [21]. Although the highest coupling strength between the exciton and light ﬁeld in the AlAs/GaAs QD system hasn’t been reported in the experiment, the one in the ZnO QD system with the value of 51 meV has been observed at 10 K [22]. Moreover, Eqs. (3) and (4) set upper limit on the energy selectivity, and the square pulses with the different laser amplitudes can be used as the controlling pulses. As a result, this scheme can adopt the AlAs/GaAs or ZnO QD system to realize. Now, we will take the AlAs/GaAs QD system as an example to discuss the stability of the entangled states mentioned above [17]. In order to do that, we should take the decoherence into account and use the ﬁdelity to measure those entangled states. Considering the ﬁnite exciton lifetime and spontaneous emission, we can write the master equation for our system as follows:

ρ˙ = −i [ H , ρ ] +

1 2

Γ p 2σ p− ρσ p+ − σ p+ σ p− ρ − ρσ p+ σ p−

Acknowledgements The authors thank Prof. Xun-Li Feng for helpful discussions. This work was supported by the National Natural Science Foundation of China under Grant No. 60978009, and the National “973 Program” under Grant Nos. 2009CB929604 and 2007CB925204. References [1] [2] [3] [4]

[5] [6] [7] [8] [9]

p =1,2,3

(11) where ρ is the density operator of the system, and Γ p are decoherence rates. The ﬁdelity of the state can be expressed as [20]:

F = Ψ |ρ |Ψ

In conclusion, we have shown that three initially nonresonant QDs can be brought into resonance by adjusting the amplitude of the external classical light ﬁeld which induces different Stark shift for each QD. That gives a way to control over the interaction between the QDs and to prepare the W-type states. During the process, the Förster coupling sets the interaction time of the picoseconds’ scale. Moreover, the advantages of this system are as follows: ﬁrst, it is controllable; secondly, it can be used to generate long-lived entangled excitonic states; ﬁnally, since the QDs we consider here can be nonidentical, it is more practical. In general, the idea to bring different QDs into resonance with the Stark effect can be used in other similar systems, such as, atom-cavity system. Therefore, this idea has a universal signiﬁcance in quantum optics and quantum information processing.

(12)

where |Ψ is a target state. Here we choose the W state (|Ψ = | W ) as a scale to measure those generated states. According to Eq. (11), ρ is a function of time. Therefore, the ﬁdelities for the generated states are also the functions of time. Here, ﬁdelities can clearly demonstrate how entangled states are generated over the course of the process. The ﬁdelities of those states with various values of ω p and Γ p [17] are shown in Fig. 5. Plots show that all the schemes have the analogous trends. Without the decoherence, the ﬁdelities for

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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