International Journal of Modern Physics B Vol. 26, No. 32 (2012) 1250194 (11 pages) c World Scientific Publishing Company DOI: 10.1142/S0217979212501949 CONTROLLING SINGLE-PHOTON TRANSPORT IN A ONE-DIMENSIONAL RESONATOR WAVEGUIDE BY INTERATOMIC DIPOLE DIPOLE INTERACTION WEI TIAN and BIN CHEN * University of Shanghai for Science and Technology, Shanghai 200093, P. R. China * cb [email protected]GUANG TONG School of Automotive Studies, Tongji University, Shanghai 200092, P. R. China Received 10 June 2012 Accepted 30 September 2012 Published 8 November 2012 Controlling single-photon transport in a one-dimensional resonator waveguide can be realized by the interatomic dipole–dipole interaction (DDI). Our numerical results show that the effects of the DDI act as that of a positive detuning. Because of the DDI, the period of the transmission spectrum changes, and its amplitude increases for stronger DDI intensity. We also discuss the influences of the DDI on the transport in low-energy and high-energy regimes. Besides, a cooperation dissipation, induced by the two atoms coupling to a common reservoir, can lead to the increase of the atomic total decay rates and the decrease of the reflection amplitude of the incident photon. Keywords : Single-photon transport; one-dimensional resonator waveguide; dipole–dipole interaction. 1. Introduction In recent years, many efforts have been devoted to the coherent controlling of the single-photon transport in one-dimensional waveguides due to their wide applica- tions, such as in optical quantum computers 1–4 and all-optical quantum information processing. 5–8 Such a configuration can be realized by a resonator with an atom or a semiconductor quantum dot embedded in waveguides 9–13 or side-coupled to them. 14–18 By regulating the coupling intensity and the detuning between the res- onator and the quantum dot (or atom), the single-photon transport can be well controlled. Besides, by tuning the frequency of either one or two cavities in a coupled-cavity array 19 or using different interacting intensities of the two cavities nearest-neighbor coupled to one containing an atom, 20 the transmission probability of the resonantly incident photons can also be precisely manipulated from 0 to 1. 1250194-1 Int. J. Mod. Phys. B 2012.26. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA on 09/10/13. For personal use only.
Transcript
November 1, 2012 11:3 WSPC/Guidelines-IJMPB S0217979212501949
University of Shanghai for Science and Technology, Shanghai 200093, P. R. China∗cb [email protected]
GUANG TONG
School of Automotive Studies, Tongji University, Shanghai 200092, P. R. China
Received 10 June 2012Accepted 30 September 2012Published 8 November 2012
Controlling single-photon transport in a one-dimensional resonator waveguide can berealized by the interatomic dipole–dipole interaction (DDI). Our numerical results showthat the effects of the DDI act as that of a positive detuning. Because of the DDI, theperiod of the transmission spectrum changes, and its amplitude increases for strongerDDI intensity. We also discuss the influences of the DDI on the transport in low-energyand high-energy regimes. Besides, a cooperation dissipation, induced by the two atomscoupling to a common reservoir, can lead to the increase of the atomic total decay ratesand the decrease of the reflection amplitude of the incident photon.
November 1, 2012 11:3 WSPC/Guidelines-IJMPB S0217979212501949
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Recent experiments demonstrated that the one-dimensional coupled-resonator
waveguides under the tight-binding approximation can be made by the defect
resonators in photonic crystals,21,22 and coupled superconducting transmission line
resonators.23 Using a discrete-coordinate approach, the scattering process of the
single-photon transport controlled by a two-level or three-level atom in one of the
resonators is investigated theoretically.14 The results show that the complete re-
flection of the photon occurs when its frequency equals that of the atom, but the
photon undergoes the complete transmission at a large atom-cavity detuning. In
this paper, we study the system of a one-dimensional (1D) coupled-resonator24
waveguide (CRW) with two two-level atoms in one of the resonators. When the
two atoms are confined in a small distance, such as those produced by optical lat-
tices,25 or magnetic traps,26 the DDI between them cannot be neglected. The DDI
can profoundly affect the light absorption and lead to the shift of the atomic energy
levels.27 So its effects on the single-photon transport are also worth exploring.
The paper is organized as follows: Sec. 2 presents the theoretical model. In
Sec. 3, we study the dynamical behavior of the single-photon scattering in different
conditions and discuss the influences of the DDI on the single-photon transmission
spectra for low-energy and high-energy regimes. In Sec. 4, the effects of the atomic
decay induced by DDI on the single-photon transport is investigated. Finally, we
present our conclusions in Sec. 5.
2. Model
We study the system of a 1D CRW with two two-level atoms located in one of the
resonators, as shown in Fig. 1. The DDI between the two atoms is considered. The
Hamiltonian is described by a typical tight-binding boson model (~ = 1),14
Hc = ωc
∑
j
a†jaj − ξ∑
j
(a†jaj+1 + aja†j+1) (1)
with the dispersion relation Ωk = ωc − 2ξ cos k. Each resonator is modeled as a
harmonic oscillator with the frequency ωc. a†j and aj denotes the photon creation
and annihilation operators for the jth resonator. Neighboring resonators are coupled
with the strength ξ, so photons can hop among them.
Assuming that the frequencies of the two dipole-coupled two-level atoms are
equal and located in the 0th cavity. Under rotating wave approximation, the Hamil-
Fig. 1. Schematic illustration of the model, where two dipole-coupled two-level atoms are trappedin one of the resonators in a 1D coupled resonator waveguide.
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Interatomic Dipole–Dipole Interaction
tonian of the atom-cavity system is given by28
Ha = HA +HI +HDD
=
2∑
N=1
[ωaσ†NσN + g(a†σN + aσ†
N )] + J(σ†1σ2 + σ†
2σ1) , (2)
where HA is atomic Hamiltonian and HI describes the interaction of the atoms
and the cavity. HDD denotes the DDI Hamiltonian between the two atoms. σ†N
and σN represent the raising and lowering operators of the Nth atom (N = 1, 2),
respectively. g and J are the coupling intensities of the atom-cavity and the atom–
atom, respectively.
For two identical two-level atoms located inside a resonant single-mode cavity,
J is defined in the form28
J =3
4(Γ0c
3/ω3ad
3)(1 − 3 cos2 ϕ) , (3)
where d is the distance between the two atoms and ϕ is the angle between d and the
in free space. We assume that the dipole moments of the two atoms are parallel to
each other and are polarized in the direction perpendicular to the interatomic axis.
Then, cosϕ = 0, DDI intensity only depends on the positions of the two atoms.
When a single-photon with eigenenergy Ωk enters into the 1D CRW, the sta-
tionary eigenstate for the total Hamiltonian H = Ha +Hc has the form14
|Ψ〉 =∑
j
uk(j)|1j〉|g1g2〉+ v1|0〉|e1g2〉+ v2|0〉|g1e2〉 , (4)
where k denotes the momentum of the incident photons. |1j〉|g1g2〉 presents the
state that the jth cavity is occupied with one excitation, while other cavities and
the two atoms are not excited. |0〉|e1g2〉 and |0〉|g1e2〉 indicate that one of the two
atoms is excited and there is no photon in the coupled-cavity array. uk(j), v1 and
v1 are the probability amplitudes of the corresponding states.
Based on the equation (Ha+Hc)|Ψ〉 = Ωk|Ψ〉, we can get the discrete scattering
equation14
(
2ξ cos k +2g2δj0
Ωk − ωa − J
)
uk(j) = ξ[uk(j + 1) + uk(j − 1)] . (5)
We assume that a single-photon is injected from the left side of the coupled-
resonator waveguide, then a usual solution for the scattering equation is
uk(j) =
eikj + re−ikj , j < 0 ,
teikj , j > 0 ,(6)
where t and r are the transmission and reflection amplitudes, respectively. Using
the continuity condition at j = 0, uk(0+) = uk(0
−), we can obtain
r =g2
iξ sin k(Ωk − ωa − J)− g2(7)
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with the constraints 1 + r = t and |t|2 + |r|2 = 1.
So the reflection and transmission coefficients R, T are
R = |r|2 =
∣
∣
∣
∣
g2
iξ sink(Ωk − ωa − J)− g2
∣
∣
∣
∣
2
, (8)
T = |t|2 =
∣
∣
∣
∣
iξ sink(Ωk − ωa − J)
iξ sin k(Ωk − ωa − J)− g2
∣
∣
∣
∣
2
. (9)
By defining ∆ = ωa − ωc, from Eq. (8), we find that R = 1 at ∆ + 2ξ cos k +
J = 0. It indicates that ∆ + J = 0 is the essential condition for the reflection
coefficient R = 1. The result is different from the case that a single atom couples to
a one-dimensional coupled-resonator waveguide, for which the atom-cavity couples
as a perfect mirror of the resonant photon and thus the photon can be reflected
completely at ∆ = 0.
3. Single-Photon Transmission Spectrum without Dissipation
In Eq. (9), it is clearly shown that the dynamics of the single-photon transport
depends on the cavity–cavity coupling intensity ξ, the atom-cavity detuning ∆,
the momentum k of the incident photons, DDI intensity J , and the atom-cavity
coupling strength g. In this section, we will focus on the effects of these parameters
on the transmission coefficients T .
3.1. The behavior of single-photon transport in 1D CRW
controlled by the DDI between atoms
Figure 2 displays the transmission coefficients T as a function of cos k for different
values of ∆ and J . In Fig. 2(a), when J = 0, the single photon is reflected completely
at ∆ = 0, cos k = 0, which is similar to the case of a single atom coupling to a
1D CRW. For J 6= 0, from Figs. 2(b) and 2(c), we find that the effects of the DDI
are similar to that of a positive detuning. As a consequence, the DDI can enhance
the effects of a positive detuning but weaken that of a negative detuning. This
result can be confirmed from the effective Hamiltonian of the atom-cavity system.
For there is only a single photon in the total system, we can adopt the method in
Ref. 28 and introduce the unitary operator U , which is defined as
U = exp[
−π
4(σ†
1σ2 − σ†2σ1)
]
, (10)
then the Hamiltonian Ha can be transformed into
H ′a = U †HaU = (ωa + J)σ†
1σ1 + (ωa − J)σ†2σ2 +
√2g(a†σ1 + aσ†
1) . (11)
In the transformed Hamiltonian H ′a, there is only one of the two atoms inter-
acting with the cavity field with an effective frequency ωa+J , while the other atom
decouples to the cavity filed. The coupling intensity between atom and cavity field
also changes from g to√2g. Therefore, the atomic frequency ωa is renormalized to
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Interatomic Dipole–Dipole Interaction
(a) (b)
(c)
Fig. 2. The single-photon transmission spectrum as a function of cos k and for different valuesof δ and J are plotted. ξ = 2, (a) δ = 0, (b) δ = 1, (c) δ = −1. Parameters are in units of g.
ωa + J due to DDI. Then an effective detuning between atom and cavity is ∆ + J
instead of ∆. For example, in Fig. 2(c) when ∆ = −1 and J = 1, the transmission
spectrum is the same as that shown in Fig. 2(a) when ∆ = J = 0.
The behavior of the transmission coefficient T under a wide range of ∆ and J
is shown in Fig. 3. It also reflects the nature that DDI acts as a positive atom-
cavity detuning. However, when (∆ + J)/g ≫ 1, the atom and cavity are almost
decoupling, then the photon can be transported completely. For simplicity, we will
pay our attention to the case of ∆ = 0 in the following discussion.
Figure 4 exhibits the relation between the transmission probability of the inci-
dent photon and its momentum k for some different DDI intensities. Clearly, the
transmission coefficient T is a periodic function of k. For J = 0, the period is π/2,
while for J 6= 0, it changes to 2π. This result can also be obtained from Eq. (9).
Meanwhile, it is found that as the DDI intensity increases the amplitude of T also
increases. Particularly, when J/g ≫ 1, the value of T nearly reaches 1.
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Fig. 3. The single-photon transmission spectrum as a function of ∆ and J . ξ = 2, k = π/8.Parameters are in units of g.
(a) (b)
(c) (d)
Fig. 4. The single-photon transmission spectra as a function of k are plotted for different valuesof J . (a) J = 0; (b) J = 1; (c) J = 2; (d) J = 10. ξ = 1, ∆ = 0. Parameters are in units of g.
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Interatomic Dipole–Dipole Interaction
Fig. 5. The single-photon transmission spectra as a function of g are plotted for different valueof J . ∆ = 0, k = π/8. Parameters are in units of ξ.
Figure 5 explores T versus g for different DDI intensities. Obviously, T is a
monotonic decreasing function of g and quickly decays to zero. We can also find
that a larger J can slow down this decay process. The stronger the intensity of J , the
slower the process becomes. The results can be understood easily. In Eq. (9), when
∆ = 0, the expression reduces to T = |1/[1− g2/(iξ2 sin 2k+ iJξ sin k)]|2. It clearlyshows that T decreases with the increase of g, but increases for larger J . On the
other hand, the effect of DDI equals to a positive detuning, which has a competition
with that of atom-cavity coupling. If atoms decouple from the cavity completely,
then the photon can pass through the 1D CRW unimpededly. Conversely, good
atomic “reflection effect” relies on the strong atom-cavity coupling.
3.2. Single-photon transmission spectrum for low-energy and
high-energy regimes
In low-energy regime λ ≫ l, k → 0, so that cos k ≃ 1 − k2/2 and sin k ≃ k. This
regime leads to the quadratic form dispersion relation Ωk ≃ ωc − 2ξ + ξk2. Then
the expressions of reflection coefficient R and transmission coefficient T become
R =
∣
∣
∣
∣
g2
iξk(−δ − 2ξ + ξk2 − J)− g2
∣
∣
∣
∣
2
, (12)
T =
∣
∣
∣
∣
iξk(−δ − 2ξ + ξk2 − J)
iξk(−δ − 2ξ + ξk2 − J)− g2
∣
∣
∣
∣
2
. (13)
In high-energy regime, k → (π/2), the short-wavelength approximation leads to
a photon linear spectrum Ωk ≃ ωc − 2ξ ± 2ξk. So the reflection coefficient R and
transmission coefficient T can be written as
R =
∣
∣
∣
∣
g2
iξ(−∆− πξ ± 2ξk − J)− g2
∣
∣
∣
∣
2
, (14)
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Fig. 6. The single-photon transmission spectra versus the DDI intensity J . Three different casesof low-energy regime (k = π/30) (solid), sinusoidal form (k = π/8) (dashed) and high-energyregime (k = π/2) (dotted) are shown. ∆ = 0, ξ = 2. Parameters are in units of g.
T =
∣
∣
∣
∣
iξ(−∆− πξ ± 2ξk − J)
iξ(−∆− πξ ± 2ξk − J)− g2
∣
∣
∣
∣
2
. (15)
Figure 6 exhibits the transmission coefficient T as a function of J for the si-
nusoidal form, the low-energy and high-energy regimes. In high-energy regime, T
increases rapidly with the increasing of J and finally approaches to 1. While in
low-energy regime it increases at a more slower speed. For the sinusoidal form, the
influence of the DDI on the transmission coefficient is the smallest compared to the
former two cases. It also indicates that the effect of the DDI on the atom-cavity
coupling is the most obvious in high-energy regime. However, when J ≫ g the atom
and cavity almost decouples. So no matter what the value of k is, the photon nearly
can be transmitted completely.
4. The Effects of the Atomic Dissipation Induced by DDI on the
Single-Photon Transport Properties in 1D CRW
The impacts of cavity dissipation and atomic decay on the transport properties
of the single-photon along 1D CRW have been discussed by Lu et al. in Ref. 18.
Their studies revealed that when the atomic decay rate and the leakage rate of
the cavities are unequal, the total dissipation lowers the peak of the resonance and
broadens the width of the line shape of the reflection spectrum. However, when the
two decay rates are equal, their effects can be eliminated, and the incident single
photon would be reflected completely. Therefore the dissipation from cavities not
only influences the free propagation of the single photon, but also gives rise to its
inelastic scattering.
Our model is similar to that used in Ref. 18, excepting for there are two dipole-
coupled atoms contained in one of the resonances. At the present work, we only
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Interatomic Dipole–Dipole Interaction
focus on the effects of the atomic decay rate, particularly that of the decay rate
induced by the DDI on the single-photon transport.
Taking the atomic dissipation into account, the atomic and the DDI Hamiltonian
can be written as29 (~ = 1).
HA = (ωa − iΓ)(σ†1σ1 + σ†
2σ2) ,
HDD = (J − iγ)(σ†1σ2 + σ†
2σ1) ,(16)
where Γ is the atomic decay rate, and γ denotes the atom–atom cooperation induced
by their coupling with a common reservoir.28–30 It is important only when the
atomic distance are small relative to the radiation wavelength. In general, γ ≤ Γ,
only when the two atoms inside the cavity are closely located, the cooperation
dissipation between the two atoms is maximum,28 that is γ = Γ.
By substituting Eq. (14) into Eq. (2), the reflection and transmission amplitudes
are obtained
r =g2
iξ sin k(Ωk − ωa − J)− ξ sin k(γ + Γ)− g2, (17)
t =iξ sin k(Ωk − ωa − J)− ξ sink(γ + Γ)
iξ sin k(Ωk − ωa − J)− ξ sin k(γ + Γ)− g2. (18)
In general, both the values of γ and J depend on the atomic distance. However,
the intensity of J is usually much larger than the value of γ. So for simplicity, we
assume that γ and J are irrelevant. Equations (15) and (16) indicate that the atomic
total dissipation changes from Γ to Γ+γ due to the DDI. The influence of γ on the
transmission coefficient T is shown in Fig. 7. One can find that the photon cannot
be reflected completely because of the atomic dissipation, as depicted by a solid
line. On the other hand, the DDI leads to the shift of the momentum k for which
the photon is reflected maximally. More importantly, DDI induces a cooperation
Fig. 7. The single-photon transmission spectra as a function of cos k for different value of γ areshown. Other parameters ∆ = 0, ξ = 2, Γ = 0.1, which are in units of g.
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decay rate γ, which increases the total decay rates. With the increasing of γ, the
reflection amplitude reduces gradually.
5. Conclusion
In this paper, controlling single-photon transport in 1D CRW has been realized by
regulating DDI between two atoms. Our results show that the effects of the DDI
are similar to a positive detuning, which result in the photon reflection depending
on the total value of ∆ + J instead of ∆. The DDI can change the period and
amplitude of the transmission coefficient T evolving with momentum k. In addition,
there is a competition between DDI and the atom-cavity coupling. Because good
atomic “reflection effect” relies on the strong atom-cavity coupling, for a stronger
atom-cavity coupling, T decreases, but DDI can make this decay process become
slower. The influences of the DDI on the transmission spectrum for both the low-
energy and the high-energy regimes have also been studied. In high-energy regime,
it firstly increases rapidly and approaches to one and then becomes gentle with the
increase of DDI. While for the low-energy regime and sinusoidal form dispersion
relations, the transmission spectrum shows a slow increase tendency. Besides, there
is a cooperation dissipation induced by DDI, which increases the atomic total decay
rates and makes the reflection amplitude decrease.
Acknowledgments
This work was supported by National Natural Science Foundation of China Under
Grant No. 10774053.
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