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Creation of entangled states of excitons in coupled quantum dots Emmanuel Paspalakis a,∗ , Andreas F. Terzis b a Materials Science Department, School of Natural Sciences, University of Patras, Patras 265 04, Greece b Physics Department, School of Natural Sciences, University of Patras, Patras 265 04, Greece

Received 8 February 2005; accepted 1 September 2005 Available online 27 October 2005 Communicated by J. Flouquet

Abstract We present two methods for the creation of two-particle entangled states of excitons in a coupled quantum dot system. The system contains two identical quantum dots that are coupled by an inter-dot hopping process. The manipulation of the system is succeeded by proper application of an external laser field. © 2005 Elsevier B.V. All rights reserved. PACS: 71.10.Li; 03.67.Mn; 73.20.Mf; 71.35.-y Keywords: Entanglement; Excitons; Quantum dot; Förster coupling; Adiabatic elimination

1. Introduction Semiconductor quantum dots have been one of the most important candidates for a solid state implementation of quantum computers [1]. In this area excitons in quantum dots have been proposed as potential candidates for qubits [2–27], and their manipulation is succeeded with the application of ultrashort laser pulses [28,29]. A particular scheme in this area [1] involves quantum dots that are coupled via an energy-transfer process, known as the Förster process [30]. Initially, Quiroga and Johnson [2] studied two and three identical and equidistant quantum dots coupled via this process and used a weak laser pulse for the creation of Bell states and Greenberger–Horne–Zeilinger states of spatially separated excitons. Their result was later extended beyond the weak coupling limit by Reina et al. [3]. A generalized method for the creation of any entangled state between the vacuum and the bi-exciton states is presented recently by Kis and Paspalakis [27]. In the same work an adiabatic passage method for the creation of a maximally entangled single-exciton state

* Corresponding author.

E-mail address: [email protected] (E. Paspalakis). 0375-9601/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.09.086

was also proposed. Other works in the same area also exist [4,8,15,20–22]. In this Letter we propose two methods for the creation of two-particle entangled states in a system of two identical quantum dots coupled by the Förster process and interacting with a classical laser field. Both methods are based on adiabatic elimination of an unwanted transition [31] and give rise to different types of entangled states. In the first case we work in the limit that we eliminate the single exciton entangled state. With this approach we create a two-particle entangled state between the vacuum and the bi-exciton states. In the second case we work in the limit that the bi-exciton state can be eliminated. Then, we can create a two-particle single-exciton entangled state. 2. Theoretical model and basic equations We study a system with two identical quantum dots that are coupled by the Förster process [2–4,8,15,30]. This process originates from the Coulomb interaction whereby an exciton can hop between the two dots [30]. The quantum dots contain no net charge and interact with a high frequency laser pulse. The formation of single excitons within the quantum dots, the interdot Förster hoping process and the interaction of the system of the quantum dots with the laser field, in the rotating wave ap-

E. Paspalakis, A.F. Terzis / Physics Letters A 350 (2006) 396–399

proximation [32], are described by the following Hamiltonian ε † Hˆ = eˆj eˆj − hˆ j hˆ †j 2 j =1,2

−

h¯ W † ˆ eˆj hk eˆk hˆ †j + hˆ j eˆk† hˆ †k eˆj 2

h¯ A(t)eˆj† hˆ †j + A∗ (t)hˆ j eˆj . 2

This approximation is of crucial importance as the level |1 is adiabatically eliminated and the system reduces to an effective two-level system, where only the states |0, |2 are involved. Therefore, Eqs. (4) and (6) read 1 Ω2 1 Ω 2 e−2iφ a0 (t) − a2 (t), 2Δ−W 2 Δ−W 1 Ω 2 e2iφ 1 Ω2 i a˙ 2 (t) = − a0 (t) + 2Δ − a2 (t). 2 Δ−W 2Δ−W

i a˙ 0 (t) = −

j,k=1,2

+

397

(1)

j =1,2

Here, eˆj (hˆ j ) is the electron (hole) annihilation operator and eˆj† (hˆ †j ) is the electron (hole) creation operator in the j th quantum dot. Also, ε is the band gap energy of the quantum dot and W is the inter-dot process hoping rate. In addition, A(t) = Ωe−iωt+iφ with h¯ Ω = μE, where μ is the coupling strength and E, φ denotes the laser pulse electric field amplitude and phase, respectively. Ω is the so-called Rabi frequency which characterizes the laser-quantum dot coupling and it is assumed real and positive in our study. In addition, ω is the angular frequency of the laser field. It is useful to write the Hamiltonian of Eq. (1) in the basis {|0, |1, |2}, where |0 ≡ |0, 0 is the vacuum state, √ |1 ≡ (|1, 0+|0, 1)/ 2 is the single-exciton, symmetric state, |2 ≡ |1, 1 is the bi-exciton state. Then, in the rotating wave picture [32] and after a unitary transformation the Hamiltonian of Eq. (1) is written as [3,15,27] ⎤ ⎡ √1 Ωe−iφ 0 0 2 ⎢ √1 Ωe−iφ ⎥ Δ−W Hˆ = h¯ ⎣ √12 Ωeiφ (2) ⎦. 2 1 iφ √ Ωe 0 2Δ 2

Here, h¯ Δ = ε − hω ¯ is the detuning of the laser pulse from exact resonance. Note, that the basis states {|0, |1, |2} form a closed subspace under the action of the Hamiltonian (1). The wavefunction of the system at any time t can be written as ψ(t) = a0 (t)|0 + a1 (t)|1 + a2 (t)|2. (3) Substituting the wavefunction in the time-dependent Schrödinger equation we obtain the evolution of the probability amplitudes as 1 i a˙ 0 (t) = √ Ωe−iφ a1 (t), (4) 2 1 1 i a˙ 1 (t) = √ Ωeiφ a0 (t) + (Δ − W )a1 (t) + √ Ωe−iφ a2 (t), 2 2 (5) 1 iφ i a˙ 2 (t) = √ Ωe a1 (t) + 2Δa2 (t). (6) 2

(8) (9)

We further make the change of variables b0,2 (t) = a0,2 (t) × exp (igt), with g = Ω 2 /[2(W − Δ)], and obtain i b˙0 (t) = ge−2iφ b2 (t), i b˙2 (t) = ge2iφ b0 (t) + 2Δb2 (t).

(10) (11)

The solutions of Eqs. (10) and (11) are given by b0 (t) = α(t)e−iΔt b0 (0) − β ∗ (t)e−iΔt b2 (0), b2 (t) = β(t)e

−iΔt

∗

b0 (0) + α (t)e

−iΔt

b2 (0),

(12) (13)

where ˜ +i α(t) = cos(Ωt)

Δ ˜ sin(Ωt), Ω˜

(14)

ge2iφ ˜ sin(Ωt), (15) Ω˜ with Ω˜ = g 2 + Δ2 . For the initial condition we assume that the system is initially in the vacuum state |ψ(0) = |0, or a0 (0) = 1. Then, Δ ˜ ˜ a0 (t) = cos(Ωt) + i sin(Ωt) e−i(g+Δ)t , (16) Ω˜ ge2iφ ˜ −i(g+Δ)t . a2 (t) = −i (17) sin(Ωt)e Ω˜ β(t) = −i

Choosing g = −Δ,

(18)

˜ then and the duration of the laser pulse as tp = π/(2Ω) 1 a0 (tp ) = i √ sign Δ, 2

1 a2 (tp ) = ie2iφ √ sign Δ. 2

(19)

Therefore, a maximally entangled state of the vacuum state and the bi-exciton state is created. The entangled state created in√ this case is, up to a global phase, of the form (|0 + e2iφ |2)/ 2. The condition of Eq. (18) gives π , tp = √ 2 2|Δ|

(20)

3. First case: Adiabatic elimination of state |1

and

√ In the first case we take Ω/ 2 to be much smaller than |Δ − W |. Then, we can approximate a1 (t) from Eq. (5) as [31]

1 W ± W 2 + 2Ω 2 . (21) 2 For√ the case of interest, |Δ − W | should be much larger than Ω/ 2, so the + sign in Eq. (21) does not fulfill the previous condition and should be omitted. For the − sign the condition

1 Ω a0 (t)eiφ + a2 (t)e−iφ . a1 (t) ≈ − √ 2Δ−W

(7)

Δ=

398

E. Paspalakis, A.F. Terzis / Physics Letters A 350 (2006) 396–399

sults are in very good agreement with the analytical results of Eqs. (16), (17) and (20). 4. Second case: Adiabatic elimination of state |2 In the second case we adapt the method proposed in Ref. [33] for√ the creation of the single-exciton entangled state. We take Ω/ 2 to be much smaller than 2|Δ| but not much smaller than |Δ − W |. Then, we can approximate a2 (t) from Eq. (6) as [31] eiφ Ω a1 (t). a2 (t) ≈ − √ (23) 2 2Δ In this case too the system reduces to an effective two-level system, where only the states |0, |1 are involved. Therefore, Eqs. (4) and (5) read e−iφ i a˙ 0 (t) = √ Ωa1 (t), 2 eiφ Ω2 a1 (t). i a˙ 1 (t) = √ Ωa0 (t) + Δ − W − 4Δ 2 In the case that Ω2 = 0, 4Δ which is approximated by

(24) (25)

Δ−W −

(26)

Δ ≈ W,

(27)

Eqs. (24) and (25) read

Fig. 1. The time evolution of the real part (solid curve) and imaginary part (dashed curve) of the probability amplitudes a0 (t) in (a), and a2 (t) in (b). In (c) we present the probabilities |an (t)|2 , with n = 0, 1, 2, in states |0 (solid curve), |1 (dashed curve) and |2 (dot–dashed curve). The parameters used in these figures are h¯ W = 0.1 eV, φ = 0, Ω = 0.008 fs−1 and Δ is chosen to fulfill Eq. (22).

of Eq. (21) can be approximated, in the case that Ω W , by Ω2 (22) . 2W We note that in addition to the creation of an entangled state with the present method one can induce a π phase to the state√|0. If the condition of Eq. (18) is fulfilled and tp = π/( 2|Δ|) then a0 (tp ) = −1 and a2 (tp ) = 0. An example of the creation of the two-particle maximally entangled state is shown in Fig. 1. The results of this figure are obtained by numerical solution of Eqs. (4)–(6) for parameters that fulfill the above conditions. We note that the numerical re-

Δ≈−

e−iφ i a˙ 0 (t) = √ Ωa1 (t), (28) 2 eiφ i a˙ 1 (t) = √ Ωa0 (t). (29) 2 The solution of Eqs. (28) and (29) are given by 1 1 −iφ a0 (t) = cos √ Ωt a0 (0) − ie sin √ Ωt a1 (0), (30) 2 2 1 1 iφ a1 (t) = −ie sin √ Ωt a0 (0) + cos √ Ωt a1 (0). (31) 2 2 We assume again that the system is initially in the vacuum state. Then, 1 a0 (t) = cos √ Ωt , (32) 2 1 a1 (t) = −ieiφ sin √ Ωt . (33) 2 √ Choosing the duration of the laser pulse as tp = π/( 2Ω) then a0 (tp ) = 0,

a1 (tp ) = −ieiφ .

(34)

Therefore, a symmetric maximally entangled single-exciton state is created. √ Furthermore, if the laser pulse duration is chosen as tp = 2π/Ω and also the condition of Eq. (26) is fulfilled, then Eqs. (32) and (33) show that a π phase is induced in state |0.

E. Paspalakis, A.F. Terzis / Physics Letters A 350 (2006) 396–399

399

5. Summary We have studied the creation of two-particle maximally entangled states in an interacting two quantum dot system. The proposed entanglement-creation processes are mediated by the inter-dot Förster process between the excitons residing in the quantum dots. We present two cases and show that lead to the creation of two different types of maximally entangled states of the system, an entangled state between the vacuum and the bi-exciton states and a single-exciton entangled state. Acknowledgements The authors acknowledge financial support by the “Archimedes-EPEAEK II Research Programme” co-funded by the European Social Fund and National Resources. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

Fig. 2. The time evolution of the real part (solid curve) and imaginary part (dashed curve) of the probability amplitudes a0 (t) in (a), and a1 (t) in (b). In (c) we present the probabilities |an (t)|2 , with n = 0, 1, 2, in states |0 (solid curve), |1 (dashed curve) and |2 (dot–dashed curve). The parameters used in these figures are h¯ W = 0.1 eV, φ = 0, Ω = 0.008 fs−1 and Δ = W in order to fulfill Eq. (27).

An example of the creation of the two-particle maximally entangled single-exciton state is shown in Fig. 2. The results of this figure are also obtained by numerical solution of Eqs. (4)– (6) for parameters that fulfill the conditions of this section. In this case, too, the numerical results are in very good agreement with the analytical results of Eqs. (32) and (33). We use the same parameters for W and Ω as in Fig. 1. We note that the entangled state in this case is created in significantly shorter times, thus it leaves less possibility for the development of decoherence effects that could be detrimental in the creation of the necessary quantum gates.

[17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

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