Entanglement manipulation by atomic position in photonic crystals

Optics Communications 356 (2015) 74–78

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Entanglement manipulation by atomic position in photonic crystals Yunan Wu a, Jing Wang b,n, Minglun Mo b, Hanzhuang Zhang a,nn a b

College of Physics, Jilin University, Changchun 130023, China School of Physics and Technology, University of Jinan, Jinan 250022, China

art ic l e i nf o

a b s t r a c t

Article history: Received 18 June 2015 Received in revised form 20 July 2015 Accepted 21 July 2015

We consider two entangled atoms, each of which is embedded in a coherent photonic-band-gap (PBG) reservoir. The effect of the atomic embedded position on the entanglement of the two-atom system is studied. We find that the embedded position of the atom plays an important role in the dynamics of entanglement. The variation of the atomic position can lead to the shift between entanglement sudden death and the entanglement trapping. We also consider the entanglement transfer between different subsystems. Our results could be applied to manipulation of entanglement in nanostructured materials. & 2015 Elsevier B.V. All rights reserved.

Keywords: Non-Markovianity Entanglement PBG

1. Introduction Photonic crystals [1,2] are periodic dielectric structures which can exhibit full PBGs. When atoms are embedded in photonic crystals, it is possible to realize key solid-state quantum information processing (QIP) tasks, such as entangling quantum systems in a controlled fashion [3]. Previous studies have shown that entanglement trapping for two atoms coupled to a PBG reservoir can be realized [4–6], and the entanglement can be controlled by the position of the atomic upper level [7] and the dipole–dipole interaction between atoms [8]. Moreover, the non-Markovian atom– field interaction in PBG reservoirs can lead to controlled entanglement between the atom and the reservoir modes [9]. In this paper we propose a different point of view on controlling the entanglement through changing the relative position of the embedded atom. It originates from the fact that the variation of the atomic position would lead to the change of photon–atom coupling strengths [10], which effects the spontaneous emission spectrum and optical properties of atoms [11]. We consider two entangled atoms, each coupled to a coherent two-band PBG reservoir, which depends on the embedded position of the atom. We highlight the effect of atomic position on the entanglement dynamics of the two atoms. A detailed asymptotic analysis shows that the variation of the atomic position would lead to significant changes of entanglement distribution. When the atomic transition

frequency is located at the band edge, the entanglement could change from entanglement sudden death (ESD) to entanglement trapping with little variation of the atomic embedded position. The accompanied dynamics of entanglement among other bipartite subsystems is also studied. Our results would be useful for experimental exploration of controlled entanglement in quantum systems composed of quantum dots or Rydberg atoms in PBG materials. This paper is organized as follows. The physical model is given in Section 2. In Section 3, the effects of the atomic position on the entanglement dynamics of the two-atom system are studied. In Section 4, we study the entanglement transfer between different subsystems. We summarize our results in Section 5.

2. Physical model We consider two entangled qubits A and B embedded, respectively, in two uncorrelated double-band photonic crystals a and b. Additionally, we assume that the subsystems Aa and Bb are identical. The qubit can be assumed to be a two-level atom with the ground state 0 and the excited state 1 . The Hamiltonian, in the rotating-wave approximation, for each local subsystem is (= = 1)

H = ω 0 |1〉〈1| + n

Corresponding author. nn Principal corresponding author. E-mail addresses: [email protected] (J. Wang), [email protected] (H. Zhang). http://dx.doi.org/10.1016/j.optcom.2015.07.054 0030-4018/& 2015 Elsevier B.V. All rights reserved.

∑ ωl bl† bl + ∑ ωμ aμ† aμ l

μ

+ i ∑ (gl (r) bl+|0〉〈1| − H . c . ) + i ∑ (gμ (r) aμ+|0〉〈1| − H . c . ), l

μ

(1)

Y. Wu et al. / Optics Communications 356 (2015) 74–78

75

where ω0 is the atomic transition frequency, r is the location of the embedded atom, aμ ( aμ† ) and bl (bl† ) are the annihilation (creation) operators for the upper and lower band reservoirs, respectively. The spatial dependence coupling constant can be given by [14,15]

modes can be obtained by

ω1i + 1 = ω1i + Δ1 ωi ,

(10)

⎛ ⎞1/2 ⁎ 1 ⎟ u d ·E μ (l) (r), gμ (l) (r) = ω 0 d0 ⎜ ⎝ 2ε0 ωμ (l) V0 ⎠

ω2j + 1 = ω2j + Δ2 ωj ,

(11)

(2)

where V0 is the quantization volume, u d and d0 are the unit vector and the magnitude of atomic dipole moment. The eigenmodes Eμ (l ) (r) can be characterized by Bloch modes, which varies from point to point within a unit cell of the photonic crystal. Here, we assume that the eigenmodes are [11] ⁎ Eu

⁎

(r) = Ek cos θ (r) e,

for upper and lower band reservoir, respectively. The coupling constant to the discrete modes of the two-band reservoir can be found by integration of Eqs. (8) and (9),

∑ gm2 (r)ΔN1 ≈ ∫ ω

()

El r = Ek sin θ r e,

(4)

where θ (r) is the angle parameter seen by the atom located at r , e and Ek are the unit vector and the amplitude of the electric field with wave vector k. From the above equation, we can find that the fields of the double-band reservoir are two coherent modes with phase difference π /2. Thus, the coupling constants can be rewritten as gu (r) ≅ gk cos θ (r) and gl (r) ≅ gk sin θ (r) with real constant gk = ω0 d0

(

c1

2

gu (r) ρ1 (ω) dω,

(12)

(3)

∑ g 2j (r)ΔN2 ≈ ∫ ω

()

ω v1

1/2 1 Ek (u d·e). 2ε0 ωV0

)

ωc 2 v2

2

gl (r) ρ2 (ω) dω,

(13)

where ω v1 (2) is the upper (lower) limit of the discretized part of the density

of

2

states.

()

gu (r) ρ1 ω = (β /π )[1/ ω − ωc1 ]

and

gl (r) ρ2 (ω) = (β /π )[1/ ωc2 − ω ] [11], where β is the effective coupling between the atom and the PBG reservoir. The detailed evolution of β is shown in Ref. [20]. We thus find 2

2β ω v1 − ω c1 cos θ (r), Nπ

gm (r) ≈

(14)

Near the two band edge frequencies, the dispersion relationship is 2 ⎧ ω ≥ ω c1 ⎪ ω c + A1 (k − k 0 ) , 1 ω=⎨ ⎪ 2 ⎩ ω c 2 − A2 (k − k 0 ) , ω ≤ ω c 2,

(5)

where k0 is wave number corresponding to the band edge, ωc1(c2 ) is the upper (lower) band edge frequency and Aj = ωcj /k 02 (j = 1, 2). The corresponding band-edge density of states takes the form [16,17]

k 02 Θ (ω − ω c1 ) V0 , ρ1 (ω) = 3 (2π ) 2 A1 ω − ω c1

k 02 Θ (ω c 2 − ω) V0 , ρ2 (ω) = (2π )3 2 A2 ω − ω c 2

where N is the number of discrete modes. We assume that at time t¼0, the atom is in the excited state 1 and the two reservoir modes are in the vacuum states 0˜ m and 0˜ j , respectively. The state vector of the system is therefore

ω ≥ ω c1,

(6)

+

∑ cm (r, t ) e−iωmt |0, 1˜m , 0˜ j 〉

+

∑ cj (r, t ) e−iωj t |0, 0˜ m , 1˜ j 〉,

m

(16)

j

ω ≤ ω c 2,

(7)

where the radiation state |0, 1˜m , 0˜ j 〉( 0, 0˜ m , 1˜ j ) accounts for the mode of upper (lower) band reservoir with frequency ωm (j ) having one excitation. The equations for the amplitudes are governed by the Schrödinger equation, and after eliminating [18] the off-resonant modes with frequency ω > ω v1 and ω < ω v2 , we obtain

∑

a ̇ (t ) = i

ωμ > ω v1

()

gμ r

2

ωμ − ω 0

a (t ) + i

∑ ωl < ω v 2

gl (r)

∑

a (t )

N

gm (r) cm (r , t ) e −i (ω m − ω 0 ) t − i

m=1

( )

2

ωl − ω 0

N

−i

∑ gj (r) cj (r, t ) e−i (ωj − ω0 ) t , j =1

() ()

̇ r, t = − igm r a t ei (ω m − ω 0 ) t , cm

(17)

(18)

(8)

( )

() ()

cj̇ r, t = − igj r a t ei (ω j − ω 0 ) t .

ΔN2 = ρ2 (ω) Δ2 ω.

(15)

|φ (t )〉 = a (t ) e−iω 0 t |1, 0˜ m , 0˜ j 〉

with the Heaviside step function Θ (x ). From the above equations, we can find that the density of states diverges at the edge frequency. Thus, the atom–reservoir interaction within PBG materials is highly non-Markovian [18]. In order to solve the problem of the non-Markovian dynamics, we use the discretization method [19]. The core of this method is to divide the density of modes into two parts: the discrete part which is near the band edge frequencies and the perturbance part which is far from the band edge. More specifically, the density of modes near the band edge is replaced by a finite (but large) number of discrete harmonic oscillators, while the rest of the mode density can be treated perturbatively. For the discrete part, we should obtain the frequencies and the atom-field coupling constants of the discrete oscillators. The differential forms of Eqs. (6) and (7) are

ΔN1 = ρ1 (ω) Δ1 ω,

2β ω c 2 − ω v 2 sin θ (r), Nπ

gj (r) ≈

(9)

For ΔNm = 1 (m = 1, 2) , we find Δ1 ωi = 1/ρ1 (ωi ) and Δ2 ωj = 1/ρ2 (ωj ) with discrete index i and j. Thus, the frequencies of the discrete

(19)

By numerically solving the above set of equations, we shall analyze the population and entanglement dynamics of the two-qubit system.

76

Y. Wu et al. / Optics Communications 356 (2015) 74–78

3. Entanglement dynamics

1

In this section, we research the influence of the coherent PBG reservoir on the entanglement dynamics of the two-qubit system. We choose the concurrence [21] to measure the entanglement of the two qubits. The concurrence is defined as

{

λ1 −

λ2 −

λ3 −

}

λ4 , 0 ,

(20)

where {λ i } are the eigenvalues, in the descending order of value, of ⁎ the matrix R = ρAB (σyA ⊗ σyB ) ρAB (σyA ⊗ σyB ). We assume that the two-qubit system is initially in a pure Belllike state Ψ (0) AB = η 11 AB + γ 00 AB , and the two-band reservoir is in the vacuum state. Exploiting Eq. (16), the reduced density matrix for the two-qubit system can be calculated as (in the basis [ 1 = 11 , 2 = 10 , 3 = 01 , 4 = 00 ])

⎛ ρ (t ) ρ14 (t )⎞ 0 0 ⎜ 11 ⎟ ⎜ 0 ρ22 (t ) 0 0 ⎟ ⎟, ρAB (t ) = ⎜ ⎜ 0 ρ33 (t ) 0 0 ⎟ ⎜⎜ ⎟ ρ44 (t ) ⎟⎠ 0 0 ⎝ ρ41 (t )

0.8 0.7 concurrence

C = max

()

ρ11 t =

a (t )

0.6 0.5 0.4 0.3 0.2 0.1 0

0

5

10

15

(21) 1

θ=π/4 θ=π/6 θ=π/3

0.9

4,

0.8

()

0.7

()

concurrence

⁎ ρ14 t = ρ14 t = γ ⁎η a (t ) 2 e−2iω 0 t ,

ρ22 (t ) = ρ33 (t ) = η2 a (t ) 2 (1 − a (t ) 2 ),

ρ44 (t ) = 1 − ρ11 (t ) − ρ22 (t ) − ρ33 (t ).

ρ22 (t ) ρ33 (t ) ).

0.4

0.1 0

(23)

The evolution of entanglement as a function of βt for different values of θ (r) is plotted in Fig. 1. We can find that the embedded position of the atom plays an important role in the entanglement while the position of the atomic upper level also affects the entanglement. When the atomic transition frequency is located at the lower-band edge with δ2 = 0, the concurrence of the two-qubit system decreases with the decrease of the θ (r). On the contrary, the effect of θ (r) on the entanglement dynamics for δ1 = 0 is in contrast to that in the case where δ2 = 0, as seen in Fig. 1(b). While for the atomic transition frequency is in the middle of the band gap, the change of the θ (r) value hardly has any influence on the evolution of the entanglement. The reason for the entanglement changes can be understood as follows. For the case of δ2 = 0, the atomic population and entanglement dynamics are mainly controlled by the lower-band reservoir. As increasing the values of θ (r), the coupling strength to the lower-band reservoir becomes stronger. When the coupling strength is strong enough, a photon emitted by the atom can only penetrate a distance given by the localization length, forming the atom-photon bound state [23]. This can lead to the population trapping. We see as expected that a fractionalized population is trapped for θ (r) = π /4 , and a greater fraction of the light is localized when θ (r) increases to π /3 (see Fig. 2(a)). Due to the existence of the atom-photon bound state, the initial entanglement between the two qubits can partially be preserved for θ (r) = π /3 and θ (r) = π /4 , as seen from Fig. 1(a). However, in the case of

0.5

0.2

0

5

10

15

20

βt 1

θ(r)= π/4 θ(r)= π/3 θ(r)= π/7

0.9 0.8 0.7 concurrence

()

0.6

0.3

(22)

With the density matrix (21), we can calculate the concurrence [22] of the two-qubit system. From Eq. (20) we have

CAB t = 2 ( ρ14 (t ) −

20

βt

with the density matrix elements evolving as

η2

θ(r)= π/4 θ(r)= π/6 θ(r)= π/3

0.9

0.6 0.5 0.4 0.3 0.2 0.1 0

0

5

10 βt

15

20

Fig. 1. The evolution of entanglement as a function of βt with different values of θ (r). (a) ω0 = 99β , (b) ω0 = 101β and (c) ω0 = 100β . The other parameters are ωc1 = 101β , ωc2 = 99β and η2 = 0.4 .

Y. Wu et al. / Optics Communications 356 (2015) 74–78

1

θ(r)= π/4 θ(r)= π/6 θ(r)= π/3

0.9 0.8

population

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

5

10

15

20

βt

1

θ(r)= π/4 θ(r)= π/6 θ(r)= π/3

0.9 0.8

population

In this section, we further study the effect of the embedded position of the atom on the distribution dynamics of entanglement among other bipartite subsystems Aa, Ab and ab. Using Eqs. (16) and (23), the concurrence can be expressed in the following forms:

0.6 0.5 0.4

Cab (t ) = 2 γ ⁎β (1 − a (t ) 2 ) − 2β2 a (t ) 2 (1 − a (t ) 2 ),

0.3 0.2

()

(24)

(25)

and 0

5

10

15

20

βt 1

θ(r)= π/4 θ(r)= π/3 θ(r)= π/7

0.9 0.8

population

()

CAa t = CBb t = 2β2 a (t ) 3 1 − a (t ) 2

0.1

0.7 0.6 0.5 0.4 0.3 0.2

θ (r) = π /6, the atom-photon state is absent. As a result, the initially entangled atoms spontaneously emit photons into the reservoir, which results in the ESD [12,13] between the two qubits (see Fig. 1 (a)). When the atomic transition frequency is located at the upperband edge (δ1 = 0), the coupling strength to the upper-band reservoir decreases with increase of θ (r). Thus, the effect of the position of the embedded atom on the entanglement for δ1 = 0 is in contrast to that in the case where δ2 = 0 (see Fig. 1(b)). For the atomic transition frequency is in the middle of the band gap, the atom-photon bound state always exists and the coupling strength between atom and the reservoir hardly changes with different values of θ (r). Thus, we see as expected that the change of the θ (r) value hardly has any influence on the evolution of the entanglement. In concluding this section, we would like to emphasize that the embedded position of the atom plays an important role in the process of entanglement preservation. When the atomic transition frequency is located at the band edge, the variation of the atomic position can lead to the shift between ESD and the entanglement trapping. If it is hard to manipulate the atomic transition frequency, one can simply control the position of the embedded atom to entangle atoms on demand.

4. Entanglement transfer

0.7

0

77

0

5

10

15

20

()

CAb t = 2 γ ⁎β a (t )

1 − a (t ) 2 − 2β2 a (t ) 2 (1 − a (t ) 2 )

(26)

for subsystems ab, Aa and Ab, respectively. In Fig. 3 we sketch the behavior of entanglement transfer. We find that, before the ESD between qubits, the qubit–qubit entanglement is mainly transferred to the qubit–reservoir entanglement, i.e., CAa (t ) and CAb (t ). However, the qubit–qubit entanglement after the ESD is connected with the growth of reservoir–reservoir entanglement Cab (t ). To understand why that happens we plot in Fig. 4 the population in the upper atomic state and the reservoir modes. From the figure, we find that there is an exchange of energy between the qubit and the reservoir. This energy exchange induce the generation of entanglement between qubit and reservoir. However, since both atomic and reservoir modes populations are needed for qubit–reservoir entanglement, the qubit–reservoir entanglement only maintains for a short while and then decays with the decrease of atomic population. After the ESD, all the energy is transferred to the reservoir, so the entanglement between reservoirs grows and maintain a steady-state value in the long time limit.

βt Fig. 2. The evolution of excited-state population as a function of βt with different values of θ (r). (a) ω0 = 99β , (b) ω0 = 101β and (c) ω0 = 100β . The other parameters are ωc1 = 101β , ωc2 = 99β and η2 = 0.4 .

5. Conclusion In conclusion, we have analyzed the entanglement dynamics of a system consisting in a pair of neutral two-level atoms A and B embedded in two uncorrelated photonic crystals a and b,

78

Y. Wu et al. / Optics Communications 356 (2015) 74–78

1

C

0.9

C

0.8

C Aa C

concurrence

0.7

entanglement. We find that the atom–atom entanglement is strongly related to the position of the embedded atom. With the variation of atomic position [θ (r)], we could observe the obvious changes of entanglement between atoms. By controlling the position of the embedded atom, prevention of ESD and entanglement trapping could occur. We have considered also the dynamics of entanglement among other bipartite subsystems. We show that the growth of atom–reservoir entanglement is correlated with the decrease of atom–atom entanglement before the ESD, while after the ESD, the entanglement between atoms is mainly transferred to the reservoir–reservoir entanglement.

AB ab

Ab

0.6 0.5 0.4 0.3 0.2

Acknowledgments

0.1 0

0

5

10

15

20

βt Fig. 3. The concurrences CAB (t ), CAb (t ) , CAa (t ) and CAb (t ) as a function of βt with ωc1 = 101β , ωc2 = 99β , ω0 = 99β , θ (r) = π /4 and η2 = 0.4 .

1

This work was supported by the National Natural Science Foundation of China (Grant numbers 11447157, 11405073, 11274142), the Shandong Young Scientists Award Fund (Grant no. BS2013SF021) and Doctoral Foundation of University of Jinan (Grant no. XBS1325).

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0.9 [1] [2] [3] [4]

0.8

popolation

0.7 0.6

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0.2 0.1 0 0

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20

βt Fig. 4. The population of the system as a function of βt . The solid line is the population in the excited atomic state. The long-dashed line is the population in the upper-reservoir modes. The dot-dashed line is the population in the lower-reservoir modes.

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