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Controllable atom–photon entanglement near a 3D anisotropic photonic band
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Controllable atom–photon entanglement near a 3D anisotropic photonic band edge
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IOP PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 085503 (6pp) doi:10.1088/0953-4075/43/8/085503
Controllable atom–photon entanglement near a 3D anisotropic photonic band edge
S Roshan Entezar
Physics Department, University of Tabriz, Tabriz, Iran E-mail: s-roshan@tabrizu.ac.ir
Received 29 November 2009, in final form 18 February 2010 Published 23 March 2010
Online at stacks.iop.org/JPhysB/43/085503
Abstract
The coherent control of the entanglement between a three-level atom located within a three-dimensional (3D) anisotropic photonic band gap (PBG) material, with one transition frequency near the edge of the PBG, and its radiation field is demonstrated. As a result of quantum interference and photon localization, the entanglement can be decreased or increased depending on the relative phase between the control laser coupling, the two upper levels and the pump laser pulse used to create an excited state of the atom in the form of a coherent superposition of the two upper levels. Unlike the free-space case, the steady-state entanglement is strongly dependent on the externally prescribed initial conditions. This non-zero steady-state entanglement is achieved by virtue of the localization of light in the vicinity of the emitting atom.
1. Introduction
Entanglement lies at the heart of quantum information and computing science [1, 2]; it is responsible both for the mysteries of quantum cryptography [3] and teleportation [4]. A system consisting of two subsystems is said to be entangled if its quantum state cannot be described by a product of the quantum states of the two subsystems [5]. For a bicomponent system in a pure state, it has been shown that the reduced quantum entropy is a very accurate measure of the degree of the entanglement between two components [6]. The higher the reduced quantum entropy, the greater the entanglement. Some works on the quantum entropy and the entanglement in the bicomponent system have been reported [7]. It is important for quantum information processing to be able to create entangled states in a controllable way. In recent years, many novel methods have been proposed to generate controllable entangled states [8–11]. Some of them are based on quantum interference effects associated with electromagnetically induced transparency (EIT) [12], including atom–atom, atom–photon and photon– photon entangled states [13–16]. Also, entanglement can exhibit the nature of a nonlocal correlation between quantum systems that have no classical interpretation. However, real quantum systems will unavoidably be influenced by surrounding environments. It is well known that the spontaneous emission and absorption properties depend not
only on the energy structure of an atom but also on the nature of the surrounding environment, more specifically on the density of states (DOS) of the radiation field [17]. So, the entanglement of the atom and its spontaneous emission will be affected by the surrounding environment of the atom. From the point of view of the surrounding environment of atoms, PBG structures have been shown to have different DOS compared with a free space vacuum field [18]. PBG structures offer unique ways to tailor light and the propagation of electromagnetic waves and have given rise to growing interest in recent years because it offers the possibility of controlling and manipulating light within a given frequency range through PBG [19]. In this paper the combined effects of coherent control and photon localization on the entanglement between a three-level atom with one transition frequency near the edge of a PBG and its radiation field are investigated. The time evolution of atomic quantum entropy and the property of entanglement are shown in detail. In our system, a pump laser pulse is used to create an excited state of the atom with an atomic Bloch vector specified by the ‘area’ of the incident pulse. A control, cw, laser field with a specific phase relation to the pump laser pulse stimulates radiative transitions between the upper two excited states. It is shown that entanglement can be decreased or increased by changing the optical paths and, hence, the relative phase between the control laser field and the initial atomic Bloch vector determined by the
0953-4075/10/085503+06$30.00 1
© 2010 IOP Publishing Ltd Printed in the UK & the USA
J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 085503
S R Entezar
pump laser pulse. Unlike the free space case, the steady- state entanglement is strongly dependent on the externally prescribed initial conditions.
2. Atom–photon entanglement
Consider a single three-level atom with the excited-states |3⟩ and |2⟩ and a ground state |1⟩ shown in figure 1. Level |3⟩ is coupled by radiation modes to the ground level |1⟩, and the transition frequency ω31 is assumed to be near the band edge frequency ωc of a PBG [20]. The transition between the two upper levels |3⟩ and |2⟩ is driven by a resonant control laser field. The Hamiltonian describing the dynamics of this system in the interaction picture and the rotating wave approximation takes the form
ω32
|3〉 |2〉
|1〉
H = ih ̄
gk e−ik t |3⟩⟨1|bˆk + ih ̄ eiφc |2⟩⟨3| + H.C. (1)
+
k
ak (t )|1, {1k }⟩,
(2)
rapidly in the vicinity of the atomic transition frequency. In this
I
6 h ̄ c 3
function is proportional to the delta function. This is because
a ̇3(t) = −e−iφc a2(t) +
gk e−iktak(t). (6)
ω31
k
Here, bˆk is the annihilation operator for the kth vacuum mode with frequency ωk, is the resonant Rabi frequency, k = ωk −ω31 is the detuning of the radiation mode frequency ωk from the atomic resonant frequency ω31 and φc is the phase of the control laser beam which depends on its optical path. gk = (ω31d31/h ̄ )(h ̄ /2ε0ωkV0)1/2ek.ud is the coupling constant between the kth vacuum mode and the atomic transition from the upper level |3⟩ to the ground level |1⟩, where d31 and ud are the magnitude and unit vector of the atomic dipole moment of the transition |3⟩ → |1⟩ respectively, V0 is the quantization volume, ek is the transverse (polarization) unit vector and ε0 is the Coulomb constant. In this model the upper levels |2⟩ and |3⟩ are assumed to have the same symmetry, and the transition frequency ω21 to be far inside the gap. So, the spontaneous emission in the transition |3⟩ → |2⟩ and the spontaneous emission from level |2⟩ to the ground state |1⟩ are ignored [21]. Accordingly, the control laser field drives a two-photon transition (2ωL = ω32) between states |2⟩ and |3⟩. In this case, the Rabi frequency is given by
The state vector of the system at time t may be written as |ψ(t)⟩ = a3(t)|3, {0}⟩ + a2(t)|2, {0}⟩ + a1(t)|1, {0}⟩
Figure 1. The schematic diagram of a V-type three-level atom driven by a resonant control laser field with frequency ω32. Here, the transition frequency ω31 is near a photonic band gap and the transition frequency ω21 is far inside the photonic band gap.
a ̇k(t) = −gk eikta3(t), (3)
a1(t) = a1(0), (4)
a ̇2(t) = eiφc a3(t), (5)
k
By formal time integration of equation (3) and eliminating ak(t) from equation (6) the following equation is obtained:
The resulting Green function depends very strongly on the photon density of states of the relevant photon reservoir. For a broadband reservoir, such as free vacuum with the photon dispersion relation ωk = ck, one can use the Wigner– Weisskopf approximation [24] to obtain
G(t − t′) = γ δ(t − t′), (8) 2
4ω3 d2
31 31 . Thus, in the free space, the Green
the free vacuum is an infinitely broad photon reservoir (flat
spectrum) and, therefore, its response should be instantaneous.
The Wigner–Weisskopf approximation, however, is not
valid when the density of electromagnetic modes changes
t 0
a ̇3(t) = − e−iφc a2(t) −
whereG(t−t′) = g2 e−iδk(t−t′) isthedelayGreenfunction.
kk
G(t − t′)a3(t′) dt′, (7)
h ̄=(d.E)(d.E)/h ̄(ω−ω)[22].Here,the
with γ = 1 4 π ε 0
3I 0 I1 0 L I2
summation is over all intermediate states |I ⟩ of the atom and E0 is the electric field of the control laser.
where the state vectors |1, {0}⟩, |2, {0}⟩ and |3, {0}⟩ describe the atom in the states |1⟩, |2⟩ and |3⟩ respectively, with no photons present in any vacuum mode, and the state vector |1, {1k }⟩ describes the atom in its ground state and a single photon in the kth mode with frequency ωk. We assume that the atom is initially in the superposition state cosθ|3⟩+sinθeiφp|2⟩,whereθandφparedefinedbythe ‘pulse area’ and the phase of the pumping laser pulse used for preparing the atom in a coherent superposition of states [23].
By substituting the Hamiltonian (1) into the Schro ̈dinger equation and using the state vector (2) the following set of equations can be obtained:
case, we must perform the exact summation g2 e−iδk(t−t′). kk
2
For the purpose of discussion we assume that our three-level atom is embedded in a 3D PBG material. In a PBG one finds a modified dispersion relation for the photons in the radiation reservoir, with a gap(s) in the photon density of states. We begin by considering an isotropic effective mass approximation [20] for the photon dispersion relation in a PBGmaterial,ωk=ωc+A(k−k0)2.Hereωcistheupper band edge frequency, k is the modulus of the wavevector k, k0 ≡ π/L, L is the lattice constant of the photonic crystal and A ≈ ωc / k02 ≈ c2 /ωc (this corresponds to a density of states of the form ρ(ω) ∼ (ω − ωc)(ω − ωc)−1/2, with being the Heaviside step function [20]). This dispersion relation is isotropic since it depends only on the magnitude k of the
J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 085503
wavevector k. However, there is no physical PBG material
with an isotropic gap. In a real 3D PBG material with an allowed point-group symmetry, the gap is highly anisotropic. Thus, in a more realistic picture we take the photon dispersion relation ωk = ωc + A(k − k0)2 with A ≈ fc2/ωc where f is a dimensionless scaling factor whose value depends on the nature of the dispersion relation near the band edge ωc. This anisotropic dispersion relation leads to a photonic density of states at a band edge ωc which behaves as ρ(ω) ∼ (ω−ωc)1/2, for ω > ωc , characteristic of a 3D phase space [20]. This dispersion relation is valid for frequencies close to the upper photonic band edge. If the PBG is large, and if the relevant atomic transitions are near the upper band edge, it is a very good approximation to completely neglect the effects of the lower band. Using the anisotropic dispersion relation we can evaluate the corresponding Green function as [17]
g1(x) = [(−x + iδ) cos θ − eiφ sin θ]√x, g2(x) = (−x + iδ)g1(x),
Z(x) = [(−x + iδ)2 + 2]2 + iα2(−x + iδ)2,
S R Entezar
(17) (18) (19)
whereφ=φp−φcanduj(j=1,…,4)aretherootsofthe quartic equation
x4 +αx3 +2δx2 +αδx−(2 −δ2)=0, (20) √ iπ/4
found by substituting s = x e into the equation D(s) = 0. These roots are given by [27]
′ ei(δ(t−t′)+π/4) G(t−t)=−α ′ 3,
′ ωc(t−t)≫1, (9)
u=−α±A+Y±3A+Y±√B ,(21) j 4s42t42s2A+2Y
where the two ±s must have the same sign and the ±t is independent. Here,
3α2
A=− 8 +2δ, (22)
3
5A P
Y=−6 −Z+3Z, (24)
3Q Q2 P3
Z= 2± 4+27, (25)
A2
P =−12 −, (26)
A3 A B2 Q=−−−, (27)
4π(t − t )
whereα2 ≈ ωc γ312 andδ=ω −ω. Incontrastto
B=α, (23) 8
16f3ω31 31 c
the free space case, equation (9) demonstrates that there is a
contribution in the current dynamics at time t from previous states of the system at time t ′ . Thus, it describes long-time memory effects in the atom–photon interaction due to the presence of the PBG material, indicating that the atom–photon interaction within a PBG material is highly non-Markovian [25].
We continue by taking the Laplace transform of equations (5) and (7) to derive the explicit time dependence of the atom’s evolution
eiφp sinθ+eiφca ̃3(s+iδ)
a ̃2(s+iδ)= s+iδ , (10)
iφ a ̃3(s+iδ)= (s+iδ)cosθ−e
108 3 8 3α4 α2δ 2 2
sinθ, (11) 2iπ√2
=−256− 8 − +δ. (28)
D(s)
Numerical analysis shows that two roots are real (one is positive (u1 ) but the other one is negative (u3 )) whereas the other two roots are the complex conjugates of each other (one with negative real and imaginary parts (u2) and the other with a negative real part and a positive imaginary part (u4)).
The entropy of the atom can be defined through its respective reduced density operator by
Sa(t) = −Tr(ρa(t)lnρa(t)), (29)
where ρa (t ) is the reduced density operator of the atom. To obtain the reduced density operator of the atom we rewrite the
with D(s) = (s + iδ) + αe 4 (s + iδ) s + . The amplitude aj (t) is found by inverting a ̃j (s + iδ) using the complex inversion formula [26] which involves a contour integration in the complex s plane as
2 j=1
i(φc +π/4)
PjRj ei(u2j+δ)t + αe ∞ g1(x)e−(x−iδ)t
a2(t)=
× Z(x) dx,
π
a3(t)=
Here,
i(u2+δ)t
αeiπ/4 π
2uj
∞ g2(x)e−(x−iδ)t Z(x)
02
(12)
dx. (13)
state with
vector |ψ (t )⟩ (equation (2)) as
|ψ(t)⟩ = |Q⟩|3⟩ + |P ⟩|2⟩ + |S⟩|1⟩, (30)
|Q⟩ = a3(t)|{0}⟩, (31) |P⟩ = a2|{0}⟩, (32)
PjQj e
j
+
j=1
0
Pj=(u−u)(u−u)(u−u), (14) jljmjn
|S⟩ = a1|{0}⟩ +
The density operator of the atom field is given by
ρaf (t) = |ψ(t)⟩⟨ψ(t)|, (34)
(l,m,n = 1,…,4,j ̸= l ̸= m ̸= n), Qj =u2j +δcosθ+ieiφsinθ,
k
Rj =(u2j +αuj +δ)eiφp sinθ−ieiφc cosθ,
(15) (16)
3
ak(t)|{1k}⟩. (33)
J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 085503
122 1
S R Entezar
0.8 0.6 0.4 0.2
Ω/α =2, δ/α =1, θ=π/4 φ=−π/2
0.8 0.6 0.4 0.2
Ω/γ=2, θ=π/4
φ=0
φ=±π/2
Entanglement
Entanglement
φ=0
φ=π/2
00 02468 02468
α2 t
Figure 2. The evolution of the entanglement as a function of the scaled time α2t for /α2 = 2, δ/α2 = 1, θ = π/4, and for various values of the relative phase φ.
and the reduced density operator of the atom can be obtained
γ t
Figure 3. Same as figure 2 for the free space case.
and (13) when the transition frequency ω31 of the atom to
be near the edge of the PBG. Since u1 is real, and u2 is
complex (with negative real and imaginary parts), the first
terms on the right-hand side of equations (12) and (13) are
non-decayingoscillatoryterms,whereasthesecondtermsare
also oscillatory but decay exponentially to zero as t → ∞.
The last terms containing the integral represent the branch-
cut contribution (arising from the deformation of the contour
of integration around a branch point in the complex inversion
formula). These also decay to zero as t → ∞, albeit faster than
the second terms. These equations show that the upper levels
|2⟩ and |3⟩ are split into two dressed states, respectively. This
dressed-state splitting is the combined effect of the vacuum-
field Rabi splitting by the gap [28] and the Autler–Townes
as
ρa(t)=Trf{ρaf(t)}=⎣⟨Q|P⟩ ⟨P|P⟩ ⟨S|P⟩⎦,
where
3. Results and discussion
Since we assumed that the transition frequency ω21 is far
inside the gap, both dressed states lie deeply inside the gap
and correspond to the photon–atom bound dressed states with
no decay in time [20]. The state of a photon emitted by the
atom in such a dressed state will be exponentially decaying
away from the atom, since the frequency ωk of the emitted
photon lies within the classically forbidden energy gap of the
PBG material. In other words, the spontaneously emitted
photon will tunnel through the crystal for a short length, called
the localization length, before being Bragg reflected back to
the emitting atom to re-excite it. The result is a strongly
coupled eigenstate of the electronic degrees of freedom of the
atom and the electromagnetic modes of the dielectric. When
the transition frequency is deeply inside the gap, the photon
tunnelling distance is on the scale of a few optical wavelengths.
As the transition frequency approaches the band edge ωc, the
photon localization length ξloc grows larger and eventually
⎡⟨Q|Q⟩ ⟨P |Q⟩ ⟨S|Q⟩⎤
⟨Q|S⟩ ⟨P |S⟩ ⟨Q|Q⟩ = a3(t)|2,
k
⟨S|S⟩
(35)
(36) (37) (38)
(39) (40) (41)
⟨P |P ⟩ = a2(t)|2, 22
⟨S|S⟩ = a1(t)| +
⟨P |Q⟩ = ⟨Q|P ⟩∗ = a2∗(t)a3(t),
|ck(t)| , ⟨S|Q⟩ = ⟨Q|S⟩∗ = a1∗(t)a3(t),
splitting [29] by the external field. The dressed states of
∗∗
⟨S|P⟩ = ⟨P|S⟩ = a (t)a (t).
ω−ω−Imiu =ω−ω−uandω−δ+Imiu = c32 1c321212
In what follows, we use equation (29) to study the entanglement of a three-level atom embedded in a 3D PBG material to its radiation field. In figure 2 we plot the entanglement as a function of the scaled time α2t for various values of the relative phase φ. This figure shows that, all other conditions being equal, the steady-state entanglement is the maximum or the minimum when the relative phase is φ = −π/2 or π/2, respectively. This is in contrast to the case where the three-level atom is embedded in the free space. In the latter case, the atom insensitive to the relative phase φ is disentangled from its radiation field at the steady state (see figure 3). Figure 4 depicts the entanglement for various values of . From this figure we note that, as is increased, entanglement oscillates faster and reaches its steady-state value more quickly. Moreover, the steady-state value of the entanglement decreases with . In order to explain the behaviour of the entanglement between the atom and its spontaneous emission field, consider equations (12)
diverges near ωc. The dressed states of level |3⟩ occur at
12
ωc − ω32 − Imiu2 = ωc − ω32 + Reu2, respectively. 22
4
level |2⟩ occur at the frequencies ω21 − δ + Imiu2 = 2 2 12
frequencies ω31 − δ + Imiu2 = ωc − Imiu2 = ωc − u2 1 11
andω31 − δ+Im iu2 =ωc −Im iu2 =ωc +Re u2 , respectively. The dressed state at the frequency ωc − u21 lies inside the gap and corresponds to the photon–atom
J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 085503 11
0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2
00 02468 12345
S R Entezar
δ=0, θ=π/4
φ=0
φ=−π/2
φ=π/2
φ=−π/2, δ=0, θ=π/4 Ω=0.5α2
Ω=2α2 Ω=5α2
Ω=0
α2 t
Figure 4. The evolution of the entanglement as a function of scaled time α2t for φ = −π/2, δ = 0, θ = π/4, and for various values
of .
1 0.8 0.6 0.4 0.2 0
−2 −1 0 1 2 δ /α2
Figure 5. Steady-state atom–photon entanglement as a function of the detuning δ from the anisotropic 3D band edge for φ = −π/2, θ =π/4and/α2 =2.
Ω/α2
Figure 6. Steady-state atom–photon entanglement as a function of
detuning for δ = 0, θ = π/4 and for different values of φ.
laser field and the pumping laser pulse. When there is no driving field, the system can be viewed as a two-level system consisting of levels |3⟩ and |1⟩, with the transition frequency ω31 near the edge of a PBG. For such a two-level atom and the anisotropic dispersion relation, the steady-state population on the excited level |3⟩ vanishes when the level is at the band edge or outside the gap. As a consequence, atom disentangles from its radiation field at the steady state (see the dashed-dotted line in figure 4). In the long-time limit, only the first terms in equations (12) and (13) remain dominant, since u1 is real whereas u2 is complex with a negative real part. The steady- state populations on the upper levels |2⟩ and |3⟩ are thus given by |P1 R1 |2 and |P1 Q1 |2 , respectively. This phenomenon of population trapping is due to the presence of a PBG material and is absent in the free space. It is apparent from equations (14)–(28) that the steady-state upper level population, and hence the steady-state entanglement, depends strongly on the parameters θ, φ, δ and . Figure 5 shows the variation of the steady-state entanglement with respect to the detuning δ. We see that as δ increases from zero (that is, as level |3⟩ is pushed farther away from the band edge into the continuum) the steady-state entanglement initially increases and attains its maximum value before it begins to decrease to zero very rapidly. This behaviour is the result of the fractionalized steady-state atomic population on the excited states |2⟩ and |3⟩ even when the bare excitation frequency of the level |3⟩ lies outside of the PBG, but not far from the band edge. Figure 6 depicts the variation of steady-state entanglement with respect to the strength of the driving field for various values of the relative phase φ. This figure shows that the steady-state entanglement can be an increasing or decreasing function of depending on the value of the relative phase φ. The steady- state entanglement can be obtain by using the following steady- state density operator of the atom:
⎡ |P Q |2 |P |2Q R∗ 0 ⎤ ⎣11111 ⎦
ρas = |P1|2R1Q∗1 |P1R1|2 0 . 0 0 1 − |P1Q1|2 − |P1R1|2
(42)
For a driving laser field so strong that ≫ α2,δ, we have P1 ∼ 1/2, Q1 ∼ (cosθ + ieiφ sinθ) and
φ=−π/2, θ=π/4, Ω/α2=2
bound dressed state. The dressed state at the frequency ω + Reu2 lies outside the gap and decays at a rate of
c2
Im u2 . It results in highly non-Markovian decay of the
atomic population of the upper level |3⟩. As ω31 is detuned further into the gap (i.e. as δ becomes more negative), a greater fraction of the light is localized in the gap dressed state. Conversely, as ω31 is moved out of the gap, the total emission intensity from the decaying dressed state is increased [21, 25]. As a result of the interference between the three terms in equation (13), the spontaneous emission dynamics displays oscillatory behaviour [21], which is the origin of the oscillatory behaviour of the atom–photon entanglement (see figures 2 and 4). As can be seen from equations (12) and (13), the dynamics of spontaneous emission, and so the entanglement between the atom and its spontaneous emission, strongly depend on the detuning δ = ω31 − ωc of level |3⟩ from the upper band edge, the initial coherent superposition state as defined by the parameter θ, the intensity of the control laser driving the transition between the upper levels, and the relative phase φ = φp − φc between the cw control
5
Steady−state entanglement Entanglement
Steady−stat entanglement
J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 085503 R1 ∼ −ieiφc (cosθ + ieiφ sinθ) so,
S R Entezar
[5] Bennett C H, DiVincenzo D P, Smolin J A and Wootters W K 1996 Phys. Rev. A 54 3824
[6] Bennett C H, Bernstein H J, Popescu S and Schumacher B 1996 Phys. Rev. A 53 2046
Phoenix S J D and Knight P L 1988 Ann. Phys., NY 186 381
⎡ (1 − sin 2θ sin φ)/4 ρas ∼⎣−ieiφc (1−sin2θsinφ)/4
0
i e−iφc (1 − sin 2θ sin φ)/4 (1−sin2θsinφ)/4
0
0
0
(1 + sin 2θ sin φ)/2
⎤ ⎦.
Thus, for the case of a strong laser field, when θ = 0 (i.e. when the atom is initially on level |3⟩) or when θ = π/2 (i.e. when the atom is initially on level |2⟩), the steady-state entanglement is independent of the initial relative phase φ (if the system is not initially prepared as a coherent superposition of the upper states). However, if the atom is initially prepared in a coherent superposition of the two upper states |3⟩ and |2⟩, the steady- state entanglement will also depend on φ. For instance, when θ = π/4, the entanglement is strongly increased for φ = 0, whereas it is decreased for φ = π/2.
4. Conclusion
In this paper the entanglement between a three-level atom embedded in a PBG material and its radiation field was investigated. Specifically, the combined effects of coherent control by an external driving field and photon localization facilitated by a PBG on the atom–photon entanglement were investigated. It was shown that due to the quantum interference, as well as the coherent photon localization, the entanglement can be increased or decreased, depending on the relative phase of the initial atomic Bloch vector (determined by the pump laser pulse) and the control laser field.
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