Threshold for formation of atom-photon bound states in a coherent photonic band-gap reservoir

Optics Communications 366 (2016) 431–441

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Invited Paper

Threshold for formation of atom-photon bound states in a coherent photonic band-gap reservoir Yunan Wu b, Jing Wang a,n, Hanzhuang Zhang b a b

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

art ic l e i nf o

a b s t r a c t

Article history: Received 29 July 2015 Received in revised form 20 December 2015 Accepted 21 December 2015 Available online 31 December 2015

We study the threshold for the formation of atom-photon bound (APB) states from a two-level atom embedded in a coherent photonic band-gap (PBG) reservoir. It is shown that the embedded position of the atom plays an important role in the threshold. By varying the atomic embedded position, a part of formation range of APB states can be moved from inside to outside the band gap. The direct link between the steady-state entanglement and APB states is also investigated. We show that the values of entanglement between reservoir modes reflect the amount of bounded energy caused by APB states. The feasible experimental systems for verifying the above phenomena are discussed. Our results provide a clear clue on how to form and control APB states in PBG materials. & 2015 Elsevier B.V. All rights reserved.

Keywords: Non-Markovianity Threshold PBG Atom-photon bound state

1. Introduction Photonic band-gap materials are structures where the photonic mode density is zero [1]. For an atom with transitions inside a PBG, a photon emitted by the atom will only penetrate a finite length scale, forming the APB state [2–4]. This Non-Markovian atom-field interaction leads to many remarkable phenomena, including suppression of spontaneous emission [5–8], fractionalized singleatom inversion [9,10], photon hopping conduction [11], and population trapping [12]. Additionally, the formation of APB states has broad applications in quantum information processing. Theoretical studies [13–15] have shown that the existence of APB states can lead to quantum entanglement preservation between atoms as well as permanent effective coupling between reservoir modes. Most notably, the photonic component of the APB state can be thought of as an atom-induced cavity mode, which can be used to realize tunable long-range interaction between atoms [16]. Experimentally, the Non-Markovian atom-field interaction in PBG reservoirs has been directly observed [17–20]. In view of these applications, it is necessary to further study the dynamical control of APB states in PBG reservoirs. Studies have shown that APB states can be controlled by a classical driving field [21,22]. An and her co-workers [23–25] have studied the condition for formation of APB states and its direct consequence on the population trapping of atoms embedded in a single-band PBG n

Corresponding author. E-mail address: [email protected] (J. Wang).

http://dx.doi.org/10.1016/j.optcom.2015.12.055 0030-4018/& 2015 Elsevier B.V. All rights reserved.

reservoir. The criteria for population trapping in thermalization processes have been analyzed in Ref. [26]. It leads us to pose the next questions: (1) How to determine the threshold of APB states formation reflected by the population trapping? (2) What is the effect of the relative position of the atom embedded in photonic crystals on the formation of APB states? In this paper we focus on these questions and elucidate the physical nature of population trapping in PBG materials. We consider a two-level atom coupled to a coherent two-band PBG reservoir, where the atom-coupling fields from the two-band reservoir are two coherent waves and depend on the atomic position. While the spontaneous emission of a three-level atom embedded in a coherent PBG reservoir has been mentioned by Cheng [27], the discussions are limited to long-time spontaneous emission spectra. In contrast, we focus in this work on the threshold for formation of APB states. It is shown that the condition for formation of the APB state is just the criteria for population trapping. By means of this criteria, we find that the embedded position of the atom plays a key role in manipulating the formation of APB states. With the variation of the atomic embedded position, a part of the formation range of APB states can be moved from inside to far outside the band gap, and the threshold coupling strength βc, above which the APB state occurs, can be changed. The direct link between APB states and the population trapping as well as the steady-state entanglement is also investigated. These results would be useful for experimental exploration of Non-Markovian features in quantum systems composed of quantum dots or Rydberg atoms in PBG materials.

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Y. Wu et al. / Optics Communications 366 (2016) 431–441

This paper is organized as follows. The theoretical model is given in Section 2. Section 3 is devoted to presenting population trapping caused by the formation of APB states. In Section 4, we derive the threshold for formation of APB states and discuss the effect of the atomic position on the threshold. In Section 5, the steady-state entanglement between the atom and its reservoir modes and also between different reservoir modes caused by APB states are discussed. We summarize our results in Section 6

material. 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˜ u and 0˜ ι , respectively. The state vector is therefore

= a (t ) e−iω 0 t 1, 0˜ u , 0˜ ι

φ (t )

+

∑ cu (r0, t ) e−iωut

0, 1˜ u , 0˜ ι

u

∑ cι (r0, t ) e−iωιt

+

0, 0˜ u , 1˜ ι

, (9)

ι

2. Physical model

where the radiation state 0, 1˜u , 0˜ ι ( 0, 0˜ u , 1˜ι ) describes the mode We consider a two-level atom situated at location r0 in a double-band isotropic photonic crystal. The Hamiltonian, in the rotating-wave approximation, for this system is ( = = 1)

(1)

H = H0 + HI

of upper (lower) band reservoir with frequency ωu (ι) having one excitation. Using the Schrödinger equation, the expansion coefficients can be expressed as a set of coupled equations:

∑ gu (r0) cu (r0, t ) e−i ( ωu− ω0 )t

iȧ (t ) =

with

H0 = ω 0 σ+σ− +

∑ ωu a u† au + ∑ ωι bι† bι u

ι

u

+

(2)

∑ gι (r0) cι (r0, t ) e−i ( ωι− ω0 )t ,

(10)

ι

and

HI = i ∑ (gu (r0) a u+σ− − H . c . ) + i ∑ (gι (r0) bι+σ− − H . c . ). u

ι

icι̇ (r0, t ) = gι (r0) a (t ) ei ( ωι − ω 0 ) t .

(12)

Formally integrating Eqs. (11) and (12) and substituting the solution into Eq. (10), we can obtain a ̇ (t ) = −

∫0

t

( )

( )

dτa (τ ){Γu (t − τ ) cos2 θ (r0)[ cos2 θ r0 + e iΔc t sin2 θ r0 ] 2

2

( )

( )

+ Γι (t − τ ) sin θ (r0)[sin θ r0 + e −iΔc t cos2 θ r0 ]} ,

(4)

Here V is the quantization volume, d0 and u d are the magnitude and the unit vector of the atomic dipole moment, respectively, and E*u (ι) (r0) is the atom-coupling electric field from the upper-band (lower-band) reservoir. We note that the eigenmodes in photonic crystals can be characterized by Bloch modes, which are different from that in free space. As a result, the electric field varies from point to point within a unit cell of the crystal. Here, we assume that the distributions of electric fields from the double-band reservoir can be written as [27]

E*u (r0) = Ek cos θ (r0) e,

(11)

(3)

where ω0 is the atomic transition frequency, ωu (ι) is the photonic eigenmode frequency, σ _ = 0 1 and σ+ = 1 0 are the atomic transition operators, au† (au) and bι† ( bι ) are the creation (annihilation) operators for the upper and lower band reservoirs, respectively. The spatial dependence atom-mode coupling strength can be given by [28]

⎛ ⎞1/2 1 ⎟ u d ·E*u (ι) (r0). gu (ι) (r0) = ω 0 d0 ⎜ ⎝ 2ε0 ω u (ι) V ⎠

icu̇ (r0, t ) = gu (r0) a (t ) ei ( ω u − ω 0 ) t ,

(5)

(13)

where we have approximated cu (r0, t ) = ck (t ) cos θ (r0) and cι (r0, t ) = ck (t ) sin θ (r0), and Δc = ωu − ωι ≅ ωc1 − ωc2. The memory kernels from the two-band reservoir read

∑ gk2 e−i ( ωu− ω0 )( t − τ),

Γu (t − τ ) =

(14)

k

Γι (t − τ ) =

∑ gk2 e−i ( ωι− ω0 )( t − τ).

(15)

k

The Laplace transform of a (t ) can be given by

( )

( )

1 − sin2 θ r0 cos2 θ r0 E*ι (r0) = Ek sin θ (r0) e,

(6)

where Ek and e are the amplitude and the unit vector of the electric field with wave vector k , respectively, θ (r0) is the angle parameter seen by the atom located at r0 . Thus, the fields of the two-band reservoir are two coherent modes with phase difference π/2. The coupling constants can be assumed to be gu (r0) ≅ gk cos θ (r0) and gι (r0) ≅ gk sin θ (r0) with real constant

gk = ω0 d0 ( 2ε

1 )1/2Ek (u d·e). 0 ωk V

a˜ (s ) =

⎡⎣ a ( s + iΔc ) Γι ( s + iΔc ) + a ( s − iΔc ) Γu ( s − iΔc ) ⎤⎦

()

( )

( )

( )

()

ω1 = ω c1 + A1 (k − k 0 )2

ω1 ≥ ω c1,

(7)

ω2 = ω c 2 − A2 (k − k 0 )2

ω 2 ≤ ω c 2,

(8)

here Am = ωcm/k 02 (m = 1, 2), ωc1(c2 ) is the upper (lower) band edge frequency and k0 is a constant characteristic of the dielectric

≃ a˜ (0) (s )

{1 − sin2 θ r0 cos2 θ r0

⎡ a(0) s + iΔ Γ s + iΔ ⎤⎦ + a(0) s − iΔ Γ s − iΔ c ι c c u c ⎣

(

) (

(

) (

))}

(16)

with a˜ (0) (s ) = [s + Γu (s ) cos4 θ (r0) + Γι (s ) sin4 θ (r0)]−1. The Laplace transforms of Γu (t − τ ) and Γ(ι) (t − τ ) are

Near the two band edges, the dispersion relation has the form of

( )

s + Γu s cos4 θ r0 + Γι s sin4 θ r0

∞

Γu (s ) = β13/2

∫−∞

Γι (s ) = β23/2

∫−∞

∞

ρ1 ( ω) dω = s + i (ω − ω 0 )

ρ2 ( ω ) dω = s + i (ω − ω 0 )

−iβ13/2 ε +

−is − δ1

iβ23/2 ε +

is + δ2

,

,

where ρ (ω) is the density of modes, which has the form of

(17)

(18)

Y. Wu et al. / Optics Communications 366 (2016) 431–441

ρ1 (ω) =

1 ω − ω c1 Θ (ω − ω c1 ) ω ≥ ω c1, π ε + ω − ω c1

(19)

ρ2 (ω) =

1 ωc 2 − ω Θ (ω c 2 − ω) ω ≤ ω c 2, π ε + ωc 2 − ω

(20)

influence of the coherent PBG reservoir on the excited population dynamics of the atom. When we have one and only one pure imaginary root x (1) = ib(1), the dressed state caused by strong atom-reservoir coupling occurs at frequency ω0 − b(1) . According to Appendix Appendix B, the dressed state lies in the band gap region. The photon emitted by the atom will tunnel for a localization length and then be reflected back to the atom, forming the APB state. The APB state leads to population trapping and depends on the position of the embedded atom. This can be easily seen from Fig. 1, which gives the evolution of the atomic population P = |a (t )|2 with different values of angle parameter θ (r0). When the atomic transition frequency is located at the lower-band edge with δ2 = 0, the excited atomic population is mainly controlled by the influence of the lower-band reservoir. As the values of θ (r0) are increasing, the coherent coupling strength between atom and the lower-band reservoir becomes stronger. When the coupling strength is strong enough to generate APB state, the population trapping occurs. In Fig. 1(a), we see as expected that a fractionalized population is trapped in the upper level for θ (r0) = π /4 , and a greater fraction of the light is localized when θ (r0) increases to π/3. While, in the case of θ (r0) = π /6, the APB state is absent. As a result, the upper-level population decays in a power-law manner. On the contrary, the coupling strength between atom and the upper-band reservoir will decrease with increase of θ (r0). Thus we see as expected that the effect of θ (r0) on the population dynamics for δ1 = 0 is in contrast to that in the case where δ2 = 0 (see Fig. 1

with the Heaviside step function Θ and the smoothing parameter ε. Such a density of modes is more general and has been used in the studies of atoms embedded in PBG materials [29]. β1 (2) denotes the coupling constant of the atom to the upper (lower) band re3/2 3 2 servoir with β13/2 (2) = [(ω0 d0 ) /6πε0 ] k 0 / ωc1 (2) . δ1 (2) = ω0 − ωc1 (2) and the phase angle is defined as

433

−π /2 < arg −is − δ1 < π /2 and

−π /2 < arg is + δ2 < π /2. For simplicity sake, we can assume β13/2 = β23/2 = β 3/2. With the help of the inverse Laplace transform [30] and the residue theorem, the amplitude a (t ) can be rewritten as Eq.(A.8) (see Appendix A).

3. Population dynamics From the expression of the amplitude a (t ), we can see that the characteristics of roots xj, which are directly related to the relative position parameter θ (r0), are important in the study of the dynamical properties of the atom. In this section, we study the

(a)

(b)

1

1

θ(r )= π /3

θ(r 0)= π /3

0

θ(r )= π /4

θ(r 0)= π /4

0.8

0

0.8

θ(r 0)= π /6

θ(r 0)= π /6

0.6

P

P

0.6

0.4

0.4

0.2

0.2

0 0

5

10

15

0 0

20

5

10

εt

15

20

εt (c)

1

θ(r0)=π/4 θ(r0)=π/3

0.9

θ(r )=π/6 0

P

0.8

0.7

0.6

0.5 0

5

10 εt

15

20

Fig. 1. The evolution of excited-state population as a function of εt for different values of θ (r0). (a) ω0 = 99ε , (b) ω0 = 101ε and (c) ω0 = 100ϵ . The other parameters are ωc1 = 101ε , ωc2 = 99ε and β = 0.7ε .

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Y. Wu et al. / Optics Communications 366 (2016) 431–441

(b)

(a)

0.8

1 0.9

0.7

0.8

0.6 0.7

0.5 P0

P

0

0.6 0.5

Δ =2ε

0.4

0.3

c

Δ =4ε

0.3

c

Δ =6ε

0.2

c

0.2

0.1

0.1 0 96

0.4

97

98

99

100 ω /ε

101

102

103

104

0 98

98.5

99

99.5

0

100 ω /ε

100.5

101

101.5

102

0

Fig. 2. The steady-state population P0 as a function of ω0/ε . (a) θ (r0) = π /4 and β = 0.7ε . (b) β = 0.8ε , ωc1 = 101ε , and ωc2 = 99ε , θ (r0 ) = π /3 (solid line), θ (r0 ) = π /6 (dashed line). The shaded area illustrates the photonic band gap region.

(b)). For the symmetric case, in which the atomic transition frequency is in the middle of the band gap, the change of the θ (r0) value hardly has any influence on the evolution of the population. This is because that the APB state always exists for the symmetric case and the whole coupling strength hardly changes with different values of θ (r0). In Fig. 2, we plot the steady-state atomic population P0 on the excited state as a function of ω0/ε . Here, the steady-state population a (∞) 2 can be obtained from the first term of Eq. (A.8) [31]. We can easily see from Fig. 2(a) that the values of population trapping increase with the width of the band gap Δc. For the case of θ (r0) = π /4 , population trapping region lies inside the band gap ( ωc2 < ω0/ε < ωc1) and slightly outside the band edge, which is the same as the former results in Ref. [31]. However, a remarkable phenomenon appears when θ (r0) = π /3 and θ (r0) = π /6, e.g., in Fig. 2(b), the population trapping occurs even when ω0 is far outside the band gap, and does not necessarily occur when ω0 is inside the band gap. That is to say, due to the effect of the atomic embedded position, it is not always the truth that population trapping occurs inside the band gap and disappears far outside the band edge. So we therefore reach the interesting results that a part of steady-state population can be moved from inside to outside the band gap by manipulating the atomic position. This evident phenomenon can be observed in the experimental system of InAs quantum dots embedded in a GaAs photonic crystal membrane [32]. By using the positioning technique [33], the strong coupling between only one quantum dot and a photonic crystal membrane can be realized. Beyond that, the coherent manipulations of the exciton wave function and energy levels in quantum dots are experimentally achievable. Suppose that the optical transition dipole moment and the typical transition frequency of the InAs quantum dot are observed to be d0 ∼ 3.3 × 10−28 cm and ω0 ∼ 1.3 PHz [34], respectively, which in turn gives β ∼ 160 MHz . By Stark-shifting the quantum dot transition frequency with typical shifts Δ ∼ 1 GHz, the dimensionless shifts is Δ/β ∼ 6, which is large enough to move the atomic transition frequency from a ratio δ1/β ∼ − 3 inside the gap to the ratio δ1/β ∼ 2 outside the gap (see Fig. 2(b)). The phenomenon of population trapping can also been observed in experimental system composed of Rydberg atoms in photonic crystals. In the Rydberg-atom context, by using interference effect, a single Rydberg atom trapped in photonic crystals should be achieved, and one can tune the atom's phase by a single

photon [35]. Typical values of transition frequency and dipole moment for Rb Rydberg atoms are observed to be ω0 ∼ 35 GHz [36] and d0 ∼ 2 × 10−26 C m , respectively [37]. This in turn gives β ∼ 16 kHz, while the Stark shifts are Δ ∼ 160 kHz [38]. Therefore, Δ/β ∼ 10 and the conditions for atom coherent manipulation can be accomplished. Thus, quantum dots and Rydberg atoms are good candidates for observing the phenomenon of population trapping in photonic crystals. How can we explain the above phenomena? To answer this question, in the following section we will study the criteria for population trapping of this model.

4. Criteria for population trapping In order to explain the phenomenon in the previous section, we study the criteria for population trapping in this section. As mentioned in the previous section, population trapping signifies the formation of APB states. This is because that the bound state is actually the stationary state of the whole system. The eigenvalue equation for the bound atomic frequency(ies) ϖ can be given by the equation of the real part of the poles of a˜ (s → − iϖ ) shown in Eq. (16) [28]; the imaginary part is responsible for atomic decay. The criteria for population trapping can be obtained by making the imaginary part go to zero. By means of the Eq. (16), the criteria equation for population trapping has the form of

ω0 − ϖ =

( )

β 3/2 cos4 θ r0 ε +

ω c1 − ϖ

−

( ).

β 3/2 sin4 θ r0 ε +

ϖ − ωc 2

(21)

As long as a solution ϖ satisfying the Eq. (21) exists, population trapping occurs. In the band gap region ϖ ∈ (ωc2, ωc1), the righthand side of the Eq. (21) is monotonically increasing, while the left-hand side is a monotonous decreasing function of ω0. Thus, we find the conditions for population trapping:

ω 0 − ω c 2 ≥ β 3/2J2

(22)

and

ω 0 − ωc1 ≤ β 3/2J1 with

(23)

Y. Wu et al. / Optics Communications 366 (2016) 431–441

435

0.25

0.2

J12/ε

0.15

0.1

0.05

0 0

1

2

3 Δ /ε

4

5

6

c

Fig. 3. The difference J12 as a function of band gap width Δc with β = 0.7ε and θ (r0) = π /4 .

J1 =

( ) − sin4 θ ( r0)

cos4 θ r0 ε

ε +

(24)

Δc

and

J2 =

( ) − sin4 θ ( r0) .

cos4 θ r0 ε +

(25)

ε

Δc

It is obvious that population trapping occurs when

ωc2 + β 3/2J2 < ω 0 < ωc1 + β 3/2J1 .

(26)

Eq. (26) provides us with a route to explore the effects of atomic position on the population trapping. For θ (r0) = π /4 , we can obtain J2 < 0 < J1 and J1 = − J2. The population trapping in this case occurs even when ω0 is outside the band gap (see Fig. 2(a)). Moreover, the width of ω0 outside the gap is determined by the difference J12 = J1 − J2 with fixed β. It is easy to find that, the J12 increases with the width of PBG, as seen from Fig. 3. Therefore, the population trapping region outside the band gap is enlarged with the increase of the width of PBG. However, for the cases of θ (r0) = π /6 and θ (r0) = π /3, it is easy to check that the values of Ji (i = 1, 2) are all positive and negative,

respectively. When Ji (i = 1, 2) are all positive ( θ (r0) = π /6), the phenomenon of population trapping will disappear in the band gap region from ωc2 to ωc2 + β 3/2J2. On the other hand, the value of J1 is increased comparing with that of θ (r0) = π /4 . Thus, the population trapping region in the upper-band is expanded as seen from Fig. 2(b). Similarly, in the case of θ (r0) = π /3, the population trapped in the lower band region will increase. Again, we see that the relative position of the embedded atom plays an important role in population trapping. Next, we shall consider the threshold coupling strength above which population trapping occurs. Here, we focus on the situation of θ (r0) = π /4 , where J2 < 0 < J1. In this situation, the conditions (22) and (23) become

β 3/2 ≥ βc3/2 ,

(27)

where the threshold coupling strength is

⎧ ω0 − ωc ω0 − ωc ⎫ 2 1 ⎬. βc3/2 = max ⎨ , J2 J1 ⎩ ⎭

Here we consider three different cases of (a) ω0 < ωc2, (b) ω0 > ωc1 and (c) ωc2 < ω0 < ωc1. It is easy to find that βc is always

(a)

(b)

25 Δ =2ε

1

20

Δ =4ε

0.9

15

Δ =6ε

0.8

β=0 β=ε β=2ε β=7.5ε

c c c

0.7

10

0.6

5 p0

β c /ε

(28)

0

0.5 0.4

−5

0.3

−10

0.2

−15 −20 96

0.1 97

98

99

100 ω /ε 0

101

102

103

104

0 98

98.5

99

99.5

100 ω /ε

100.5

101

101.5

102

0

Fig. 4. (a) The threshold coupling βc as a function of atomic transition frequency for virous values of Δc with β = 0.7ε and θ (r0) = π /4 . (b) The steady-state population as a function of atomic transition frequency for various values of coupling strength with ωc1 = 101ε , ωc2 = 99ε and θ (r0) = π /4 .

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Y. Wu et al. / Optics Communications 366 (2016) 431–441

(a)

(b)

0.9

0.7 C

0.8

ar

P

0.6

0

0.7 0.5 0.6 0.4 Car

0.5 0.4

0.3

0.3 0.2

θ(r )=π/4 0

0.2

θ(r )=π/3 0

0.1

0.1

θ(r )=π/6 0

0 98

98.5

99

99.5

100 ω /ε

100.5

101

101.5

0 98

102

98.5

99

99.5

0

100 ω /ε

100.5

101

101.5

102

0

Fig. 5. (a) The steady-state population (solid line) and entanglement Car (dashed line) as a function of atomic transition frequency with θ (r0) = π /4 . (b) The steady-state entanglement Car as a function of atomic transition frequency for various values of θ (r0). The other parameters are ωc1 = 101ε , ωc2 = 99ε and β = 0.8ε .

(a)

(b)

29

−26

x 10

x 10

5

1.5

ξ (ω ,ω ,∞)

1

m

n

3

R

2

R

m

n

ξ (ω ,ω ,∞)

4

0.5

1 0 5

0 5 5 0 δ

5

0 −5

n

−5

0

δ

0

n

−5

δ

δ

−5

m

m

(c)

(d)

1.5

3

,ω n ,∞ ) m

1

ξ R( ω

m

ξ (ω ,ω ,∞) R n

2.5

0.5

2 1.5 1 0.5

0 5

0 5 5

5 δ

0

0

0

n

−5

−5

δ

m

0

δ

n

−5

−5

δ

m

Fig. 6. The steady-state entanglement ξR (ωm, ωn, ∞) for (a) ω0 = 100ε , (b) ω0 = 98.5ε , (c) ω0 = 100.4ε and (d) ω0 = 99.8ε . The other parameters are ωc1 = 101ε , ωc2 = 99ε , θ (r0) = π /4 and β = 0.7ε .

greater than zero in cases (a) and (b). For the third case, by contrast, the threshold coupling strength βc goes negative. That is, when the position of ω0 is in the band gap region, no matter how

weak the interaction strength is, population trapping always exists. Once the atomic transition frequency is outside the band gap, fractionalized steady-state atomic population occurs if and only if

Y. Wu et al. / Optics Communications 366 (2016) 431–441

the coupling strength exceeds the threshold βc. It is evident that the more the atomic transition frequency is far from the band edge, the larger is the threshold strength βc. This can be seen from Fig. 4(a), which shows the threshold strength as a function of ω0/ε for various values of Δc. We also find that the range of ω0 in which population trapping occurs will become larger with the increase of the β value, but at the cost of decreasing the values of population trapping, as shown in Fig. 4(b). Thus, in the PBG region, the weaker the coupling strength, the greater the values of population trapping. Interestingly, the criteria for population trapping, Eqs. (22) and (23), are just the conditions for the existence of one and only one purely imaginary root for the equation G (x ) = 0 (see Appendix B). This again reflects the direct relationship between the APB state and the population trapping. In general, whether the APB state exists or not plays a key role in population trapping. and provides a dynamical way to preserve the quantum entanglement. In what follows, we will explore the steady-state entanglement caused by the APB state.

5. Steady-state entanglement In order to consider the entanglement between atom and its reservoir modes and also between different reservoir modes, we use the method of density of entanglement [39], which can be derived from the reservoir spectrum S (ω, t ). The entanglement density between atom and the reservoir modes with frequencies in the region (ω, ω + dω) is

ξ A (ω, t ) = 4 a (t ) 2 S (ω, t )

(29)

and the density of entanglement between reservoir modes ωm and ωn is

ξR (ω m, ω n, t ) = 2S (ω m, t ) S (ω n, t ).

(30)

In the longtime limit t → ∞, the reservoir spectrum S (ω, ∞), with the help of the final value theory, reduces to

⎡ ρ (ω) cos2 θ (r0) ⎤ 1 ⎥ a˜ (s )s →−i (ω − ω 0 ) 2 . S (ω, ∞) = β 3/2 ⎢ ⎢⎣ + ρ2 (ω) sin2 θ (r0)⎥⎦

(31)

In terms of the distribution (29) and the reservoir spectrum (31), the steady-state entanglement between atom and the reservoir modes can be given by

Car =

+∞

∫−∞

ξ A (ω, ∞) dω.

(32)

Fig. 5(a) shows Car as a function of ω0/ε . It is easy to find that the formation of the APB state leads to the generation of the steady-state entanglement Car. When the APB state is formed, population is trapped, and thus steady-state entanglement occurs. When the APB state is absent, no entanglement is present. Thus, the APB states play a key role in generating entanglement in PBG reservoir. Moreover, the values of the preserved entanglement depend on the energy leaving the atom when population trapping occurs. The smaller the values of population trapping, the higher is the preserved atom-reservoir entanglement. We also find that the variation of atomic position can lead to the change of the entanglement distribution (see Fig. 5(b)). The strong coupling between atom and the reservoir will also induce an indirect entanglement between different reservoir

437

modes. Fig. 6 displays the mode–mode density of entanglement in different parameter regimes obtained from Eq. (30). When the APB state is available and the population trapping is nearer to its maximum value, i.e., (ωc1, ωc2, ω0 ) = (101ϵ, 99ϵ, 100ϵ), we see four sharp peaks as shown in Fig. 6(a). The formation of these peaks means the existence of the Bell-like state between two symmetric reservoir modes. When the APB state is absent, i.e., (ωc1, ωc2, ω0 ) = (101ϵ, 99ϵ, 98.5ϵ), the entanglement between reservoir modes disappears (see Fig. 6(b)). That is to say, the APB state provides the ability to generate entanglement in PBG reservoir. Unlike the atom-reservoir entanglement, the entanglement between reservoir modes depends on the energy localized around the atom. As we can see from Fig. 6(c) and (d), when the position of ω0 is slightly off the PBG center, the four sharp peaks become asymmetry. This asymmetry signifies that the entanglement between reservoir modes does not achieve its maximum as a result of the decrease of the values of population trapping. The transition of entanglement density ξR from Bell-like states to asymmetry structures and then to a single peaked structure is significant in the sense that it reflects the change of the values of population trapping.

6. Conclusion Summarizing, in this paper we have studied the threshold for formation of APB states in PBG reservoirs, and have presented the effect of the atomic position on the threshold. For the situation of θ (r0) = π /4 , the APB state can exist when atomic transition frequency ω0 is outside the band gap, but it needs the coupling strength exceeding the threshold βc; when ω0 is in the PBG region, no matter how weak the interaction strength is, the APB state is always available, and thus the reservoir is permanent Non-Markovian. By manipulating the atomic embedded position, a part of steady-state population can be moved from inside to far outside the band gap. The result here gives us an active way on how to form a bound state in PBG reservoirs. This is quiet significant in suppressing decoherence in nanostructured materials. The steady-state entanglement caused by the formation of APB state has also been presented. It is shown that the APB state provides the ability to generate entanglement in the long time limit. When the APB state is available, the entanglement between atom and the reservoir modes depends on the energy leaving the atom. On the contrary, the entanglement between different reservoir modes depends on the energy bounded around the atom. As for application, the density of entanglement between reservoir modes could be tested with a measure of how much information had been preserved by the PBG reservoir, i.e., Non-Markovianity.

Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant numbers 11447157, 11405073, 11447232), the Shandong Young Scientists Award Fund (Grant no. BS2013SF021) and Doctoral Foundation of University of Jinan (Grant no. XBS1325).

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Y. Wu et al. / Optics Communications 366 (2016) 431–441

Appendix A. Calculation of a (t ) In this appendix we will derive the expression for the amplitude a (t ) used in the text. We first define some functions as follows:

⎤ ⎥ ⎥, 3/2 ⎥ −iβ ε + −ix − δ 2 ⎥ ⎦

⎡ ⎢ G−1 (x + iΔc ) f1 (x) = 1 − sin2 θ (r0) cos2 θ (r0) ⎢ ⎢ −1 ⎢ + G (x − iΔc ) ⎣

iβ 3/2

ε + ix + δ1

⎡ ⎤ iβ 3/2 ⎢ H−1 (x + iΔc ) ⎥ ε i ix δ + − − 1 ⎥, f2 (x) = 1 − sin2 θ (r0) cos2 θ (r0) ⎢ ⎢ ⎥ −iβ 3/2 −1 (x − iΔ ) H + c ⎢ ε + −ix − δ 2 ⎥ ⎣ ⎦ ⎡ ⎢ K−1 (x + iΔc ) 2 2 f3 (x) = 1 − sin θ (r0) cos θ (r0) ⎢ ⎢ −1 ⎢ + K (x − iΔc ) ⎣ −iβ 3/2

G (x) = x + H (x) = x + K (x) = x +

−iβ 3/2 −iβ 3/2

ε + − i ix + δ1

iβ 3/2

M (x) = − x + iδ1 +

−ix + Δc

ε +

ε + i −ix − δ2 iβ 3/2 ix + δ2

ε +

sin4 θ (r0), sin4 θ (r0),

cos4 θ (r0),

ix + Δc

ε +

iβ 3/2

cos4 θ (r0) +

−iβ2/3

N (x) = − x + iδ2 +

sin4 θ (r0),

ix + δ2

ε +

cos4 θ (r0) +

−ix − δ1

ε +

iβ 3/2

cos4 θ (r0) +

−ix − δ1

ε +

⎤ ⎥ ⎥, ⎥ −iβ 3/2 ε − i ix + δ 2 ⎥ ⎦ iβ 3/2

ε + ix + δ1

sin4 θ (r0),

f3 (−x + iδ1) f1 (−x + iδ1) , − −iβ2/3 −iβ2/3 cos4 θ (r0) cos4 θ (r0) M (x) + M (x) + ε − i −ix ε + ix f1 (−x + iδ2 ) f2 (−x + iδ2 ) − . η2 (x) = iβ2/3 iβ2/3 N (x) + sin4 θ (r0) N (x) + sin4 θ (r0) ε + −ix ε + i ix η1 (x) =

The amplitude a(t) can be found from the inverse Laplace transform of a˜ (s ) given by Eq. (16) through the complex inversion formula

a (t ) =

1 2π i

σ+i∞

∫σ− i ∞

a˜ (s ) e st ds .

(A.1)

Here the real number s is chosen so that s = σ lies to the right of all the singularities of the function a˜ (s ). It is apparent from Eq. (16) that s = iδ1 (2) are branch points of a˜ (s ). In order to evaluate Eq. (A.1), we consider the integration contour shown in Fig. A.1. According to the residue theorem and Jordan's Lemma, we obtain

a (t ) =

∑ j

f1 (x (j1) ) G′(x (j1) )

(1 )

ex j

t

−

⎡ 1 ⎢ 2πi ⎢⎣

∫c

+ 3

∫c

∫c

+ 4

+ 6

∫c

⎤ ⎥ f1 (x) e xt dx, 7⎥ ⎦ G (x)

(A.2)

where x (j1) are the roots of the equation G (x ) = 0 in region [Re (x ) > 0] or [δ1 < Im (x ) < δ2 ]. The integral along c3 on the right-hand side of Eq. (A.2) can be obtained by the contour shown in Fig. A.2. Thus

1 2π i

∫c3

f1 (x) xt 1 e dx = G (x) 2π i

iδ 2+ 0

∫i ∞+ 0

= −

∑ j

f1 (x) xt 1 e dx = G (x) 2π i

f2 (x (j2) ) H′(x (j2) )

(2 )

ex j

1− −

eiδ 2 t 2π i

∫0

t

−

sin2

f2 (x) xt e dx H (x)

−∞

f2 (x + iδ2 ) xt e dx = − H (x + iδ2 )

eiδ 2 t 2π i

∫0

θ (r0

)cos2

−∞

x + iδ2 +

iδ 2

∫i ∞

ε +

−ix + Δc

cos4 θ (r0) +

H′(x (j2) )

j

⎡ −1 ⎢ H (x + i (Δc + δ2 )) θ (r0) ⎢ ⎢ + H−1 (x + iδ1) ⎣

−iβ2/3

f2 (x (j2) )

∑

(2 )

ex j

iβ2/3 ε + i −ix + Δc −iβ2/3 ε + i −ix

iβ2/3 ε + i −ix

t

⎤ ⎥ ⎥ ⎥ ⎦

sin4 θ (r0)

e xt dx, (A.3)

Y. Wu et al. / Optics Communications 366 (2016) 431–441

Fig. A.1. The integrating contours used in the formula (A.2).

439

Fig. A.3. The integrating contours used in the formula (A.6).

Fig. A.2. The integrating contours used in the formula (A.3).

where x (j2) are the roots of the equation H (x ) = 0 in region [Re (x ) < 0 and Im (x ) > δ2 ]. The integral along c4 on the right-hand side of Eq. (A.2) is

1 2π i

∫c4

f1 (x) xt eiδ 2 t e dx = G (x) 2π i

∫0

−∞

f1 (x + iδ2 ) xt eiδ 2 t e dx = G (x + iδ2 ) 2π i

∫0

⎡ −1 ⎢ G (x + i (Δc + δ2 )) 1 − sin2 θ (r0)cos2 θ (r0) ⎢ ⎢ + G−1 (x + iδ1) ⎣

−∞

−iβ2/3

x + iδ2 +

ε +

−ix + Δc

cos4

iβ2/3 ϵ + ix − Δc −iβ2/3 ε + −ix

iβ2/3

θ (r0) +

ε +

ix

sin4

⎤ ⎥ ⎥ ⎥ ⎦

e xt dx.

θ (r0)

(A.4)

The integral along c6 on the right-hand side of Eq. (A.2) is

1− 1 2π i

∫c

eiδ1t

6

f1 (x) xt e dx = G (x) 2π i

0

∫−∞

eiδ1t

f1 (x + iδ1) xt e dx = G (x + iδ1) 2π i

sin2

θ (r0)

cos2

0

∫−∞

x + iδ1 +

⎤ ⎡ −iβ 3/2 G−1 (x + iδ2 ) ⎥ ⎢ ε + −ix ⎥ θ (r0) ⎢ 2/3 i − β ⎥ ⎢ + G−1 (x + i (δ − Δ )) 1 c ix ε + − − Δ ⎣ c ⎦

−iβ2/3 ε +

−ix

cos4

θ (r0) +

iβ 3/2 ϵ +

ix + Δc

sin4

θ (r0)

e xt dx. (A.5)

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Y. Wu et al. / Optics Communications 366 (2016) 431–441

The integral along c7 on the right-hand side of Eq. (A.2) can be obtained by the contour shown in Fig. A.3. We obtain −∞ i + 0 f (x ) 1 1 e xt dx = 2πi G (x ) 1

f (x ) 1 ∫ 1 e xt dx = 21πi 2πi c 7 G (x )

∫iδ +0

f3 (x (j3) ) x (3) t eiδ1t e j − (3 ) 2πi j K ′ (x j )

−∞ i f (x ) 3 e xt dx = − K (x ) 1

∫iδ

∑

⎡ ⎢

In Eq. (A.6),

x (j3)

f3 (x (j3) ) x (3) t eiδ1t e j − (3 ) 2πi j K ′ (x j )

0

∫−∞

∑

x + iδ1 +

f3 (x + iδ1) xt e dx

K (x + iδ1)

⎤ ⎥ ⎥ ⎥ −iβ 2/3 ⎢ + K −1 (x + i (δ1 − Δc ) ) ⎥ ⎢⎣ ε − i ix + Δc ⎥ ⎦ xt e dx . 2/3 3/2 iβ − iβ 4 4 cos θ (r 0) + sin θ (r 0) ε − i ix ε + ix + Δc K −1 (x + iδ 2 )

⎢ 1 − sin2 θ (r 0)cos2 θ (r 0) ⎢ = −

0

∫−∞

iβ 3/2

ε + ix

(A.6)

are the roots of the equation K (x ) = 0 in region [Re (x ) < 0 and Im (x ) < δ1].

The amplitude a (t ) can be finally obtained by substituting Eqs. (A.3)– (A.6) into Eq. (A.2). That is

∑

a (t ) =

j

f1 (x (j1) ) G′(x (j1) )

e iδ 2 t + 2πi

∫0

∞

x (1 ) t e j

+

∑ j

f2 (x (j2) ) H′(x (j2) )

x (2 ) t e j

+

∑ j

f3 (x (j3) ) K ′(x (j3) )

x (3 ) t e j

e iδ1t + 2πi

∫0

∞

⎡ ⎢ ⎢ ⎢ ⎢ M (x ) + ⎣

f3 (−x + iδ1) −iβ 2/3 ε − i −ix

cos4 θ (r0)

⎤ ⎥ ⎥ −xt ⎥ e dx 2/3 − iβ 4 cos θ (r0) ⎥ ⎦ ε + ix

f1 (−x + iδ1)

− M (x ) +

⎡ ⎤ ⎢ ⎥ f1 (−x + iδ 2 ) f2 (−x + iδ 2 ) ⎢ ⎥ −xt − ⎢ ⎥ e dx. iβ 2/3 iβ 2/3 4 4 ⎢ N (x ) + sin θ (r0) sin θ (r0) ⎥ N (x ) + ⎣ ⎦ ε + −ix ε + i ix

(A.7)

Finally, we obtain

a (t ) =

∑ j

f1 (x (j1) ) G′(x (j1) )

(1 )

ex j

t

+

∑ j

f2 (x (j2) ) H′(x (j2) )

(2 )

ex j

t

+

∑ j

f3 (x (j3) ) K′(x (j3) )

(3 )

ex j

t

+

1 2π i

∫0

∞

[η1 (x) eiδ1t + η2 (x) eiδ2 t ] e−xt dx.

(A.8)

Appendix B. Purely imaginary roots for G (x ) = 0 in the region δ1 < Im (x ) < δ2 Here we only discuss the purely imaginary roots for

G (x) = x +

−iβ 3/2 −ix − δ1

ε +

cos4 θ (r0) +

iβ 3/2 ε +

ix + δ2

sin4 θ (r0) = 0.

(B.1)

If we set x¼ iy (y is a real number) the above equation becomes

f (y) = y +

−β 3/2 cos4 θ (r0) + ε + y − δ1

β 3/2 sin4 θ (r0) = 0. ε + −y + δ 2

(B.2)

Obviously, the left side of the above equation is monotonically increasing. For the limits y → δ2 and y → δ1, we obtain

f (y)|y→ δ2 = δ1 +

−β 3/2 cos4 θ (r0) + ε

f (y)|y→ δ2 = δ2 +

β 3/2 sin4 θ (r0), ε + Δc

−β 3/2 β 3/2 cos4 θ (r0) + sin4 θ (r0). ε ε + Δc

(B.3)

(B.4)

Thus, the conditions for existence only one purely imaginary root are

ω 0 − ω c 2 ≥ β 3/2J2

(B.5)

and

ω 0 − ωc1 ≤ β 3/2J1 , where Ji (i = 1, 2) are the expressions given by Eqs. (24) and (25).

(B.6)

Y. Wu et al. / Optics Communications 366 (2016) 431–441

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