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Quantum interference due to the spontaneous emission in nonlinear metallic photonic crystals
ARTICLE IN PRESS Microelectronics Journal 40 (2009) 854–856
Contents lists available at ScienceDirect
Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
Quantum interference due to the spontaneous emission in nonlinear metallic photonic crystals Mahi R. Singh , A. Hatef Department of Physics and Astronomy, University of Western Ontario, London, Canada N6A 3K7
a r t i c l e in f o
a b s t r a c t
Available online 4 February 2009
The time evolution of the spontaneous emission cancellation is studied in metallic photonic crystals doped with an ensemble of three-level particles. The particles are interacting with each other with the dipole–dipole interaction. We have considered that the two excited states spontaneously decay due to the interaction between the nanoparticles and the photonic crystal. It is found when the detuning parameter between electron resonance energy and probe laser field is zero the system reaches the steady states exponentially. However, when the detuning parameter is not zero the absorption coefficient and the populations of upper two levels oscillated with time and then reach the steady state. & 2008 Elsevier Ltd. All rights reserved.
PACS: 42.55.Tv 42.50.Nn 42.50.Hz 42.50.Ct Keywords: Metallic photonic crystals Absorption coefficient Nanoparticles Quantum interference Spontaneous emission
1. Introduction The aim of the paper is to study the time evolution of the spontaneous emission cancellation in metallic photonic crystals. This phenomenon occurs due to the quantum interference between the two emitted photons in the system. Metallic photonic crystals have a large energy gap which can be used to fabricate new type optoelectronic devices working at high temperature (above 1000 C) [1–5]. The energy band gap in the photonic spectrum in these crystals is due to the combination of the plasma screening and Brag scattering. A lower effective plasma frequency in these crystals has been observed compared to bulk metals below which electromagnetic wave is screened out [2]. It has also been found that electronic (plasmons) and photonic resonances occur in the same spectral range in these materials. Recently, we have studied that the phenomenon of spontaneous emission cancellation has been investigated in photonic crystals in the presence of dipole–dipole interaction (DDI) [6]. The photonic crystals are made from the dielectric materials such as silicon. Crystals have densely doped with an ensemble of threelevel nanoparticles. The mean field theory was used to calculate the effect of the DDI. The linear response theory was used to calculate the expressions for the real and imaginary susceptibilities in the steady state. Numerical simulations are performed for
Corresponding author.
E-mail address: [email protected] (M.R. Singh). 0026-2692/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2008.11.055
an isotropic photonic band gap material. It was found that the spontaneous emission cancellation can be controlled by moving the resonance energies between the energy band and energy gap. It was also predicted that the system can be switched between absorptive and nonabsorptive states by changing the strength of the DDI and the resonance energies in the energy band. The spontaneous emission cancellation has also been investigated in three-level atomic gases in the DDI by Calderon et al. [7]. They found that the system can be switched between absorption and gain by controlling the strength of the DDI and the relative phase difference between the probe and pump fields. In this paper, we have performed the numerical simulation on the time evolution of the spontaneous emission cancellation when the metallic crystal is doped with an ensemble of nanoparticles. Three electronic levels of nanoparticles participate in the absorption process. The crystal is made from metallic spheres and acting as reservoir for the nanoparticles. The numerical simulations for absorption coefficient are performed based on the theory of Singh [8]. We have considered that the two excited states spontaneously decay due to the interaction between the nanoparticles and the photonic crystal. A probe laser field is applied to measure the absorption coefficient. It is found when the detuning parameter between electron resonance energy and probe laser field is zero the system reaches to the steady states exponentially. However, when the detuning parameter is not zero the absorption coefficient and the populations of upper two levels oscillated with time and then reach the steady state.
ARTICLE IN PRESS M.R. Singh, A. Hatef / Microelectronics Journal 40 (2009) 854–856
2. Disunity matrix method An ensemble of three-level nanoparticles is doped densely in the material. The three levels of a nanoparticle are denoted by jai, jbi and jci as shown in Fig. 1. Note that the three levels jai, jbi and jci are in V configuration. The probe and pump lasers with Rabi frequencies Oab and Oac , respectively, are applied to monitor the time evolution of the absorption coefficient. The laser field induces transitions jai2jbi and jai2jci which are responsible for creating dipole moments in the nanoparticles. The time dependent absorption coefficients is written as
a¼
aab rab ðtÞ Oab
aac rac ðtÞ
þ
Oac
drba ¼ ½F ba þ iab ðrbb raa Þ þ ibrbc rba dt pffiffiffiffiffiffiffiffiffiffi p½1 þ i ab ac ðrcc raa Þrca (2)
drca ¼ ½F ca þ iac ðrcc raa Þ þ ibrcb rca dt pffiffiffiffiffiffiffiffiffiffi p½1 þ i ab ac ðrcc raa Þrba iOca ðrcc raa Þ iðOba þ ab rba Þrcb
(3)
drcb ¼ F cb rcb pðrcc þ rbb Þ þ iOca rab iOba rca dt þ iðac ab Þrca rab þ ibðjrba j2 jrca j2 Þ
(4)
drbb ¼ Gb rbb pðrcb þ rbc Þ þ iOba ðrab rba Þ dt þ ibðrab rca rba rac Þ
(5)
drcc ¼ Gc rcc pðrcb þ rbc Þ þ iOca ðrac rca Þ dt þ ibðrba rac rab rca Þ
(6)
F ca ¼
F ba ¼
F cb ¼
Gc 2
Gb 2
þ idc
þ idb
Gc þ Gb 2
þ iðdc db Þ
(7)
(8)
3. Numerical simulations We consider that the metallic crystal is made from metallic spheres which are arranged periodically in a background dielectric material. The lattice constant of the crystals is denoted as L and the diameter of a sphere is assigned the letter a. The refractive indices for a sphere is taken as !1=2 nðek Þ ¼
1
e2p e2k
(12)
where ep and ek are called the plasmon energy and photon energy, respectively. The background material for the metallic crystal is taken as air. Parameters used in the calculation of the absorption coefficient for the metallic photonic crystals are given in Table 1. The simplicity energies in the calculations are measured with respect to g0 which is taken as g0 ¼ 1:0 MeV [5]. Initially, it is considered that an electron is state jai. We have used the conservation of particles condition in the our simulations raa þ rbb þ rcc ¼ 1. The best way to understand the mechanism of the spontaneous emission in metallic photonic crystals is to study the temporal evolution of the absorption coefficient. We consider that the resonance energy lies away from the conduction band edge of the photonic crystals. The results are plotted in Figs. 2 and 3 which show the time evolution of absorption coefficient as a function of the normalized time t and plasmon frequency at zero detuning (i.e. dk ¼ 0) and p0 ¼ 1. Fig. 2 is plotted when the DDI is zero; whereas, Fig. 3 includes the effect of the DDI. In Fig. 2, the system evolves to the trapping state monotonically. However, in Fig. 3 the absorption exhibit an oscillatory behavior until reaching the steady state. During this time interval the system goes to the gain state with population inversion during part of the cycle before reaching the steady state. This type of behavior has also been observed by Calderon et al. [7] in atomic gases. It seems that there
(9)
G ¼ g0 Z 2 ðeÞ
(10)
p ¼ p0
(11)
pffiffiffiffiffiffiffiffiffiffiffi Gc Gc =2
where the function ZðeÞ is called the form factor which contains the information about the electron–photon interaction,t ¼ g0 t, Gc , Gb are the linewidths and Oba and Oca are Rabi energies for transitions jbi ! jai and jci ! jai, respectively. Detuning parameters for transitions jbi ! jai and jci ! jai are defined as db ¼ ðeba ep Þ=g0 and dc ¼ ðeca ep Þ=g0 . Resonance energies eca and eba are the energy differences between the states jci and jai and jbi and jai; respectively. Also ecb is the energy difference between the levels jci and jbi and can be obtained from ecb ¼ ðeca eba Þ=g0 . P0 is the strength of quantum interference. pffiffiffiffiffiffiffiffiffiffi ab , ac and b ¼ ac ab p are the local field parameters when the DDI is taken into account.
(1)
where aab ¼ ðNm2 eba =2e0 g0 _cÞ and aac ¼ ðNm2 eca =2e0 g0 _cÞ. rab ðtÞ and rac ðtÞ are the time dependent density matrices for the transitions jci2jai and jbi2jai; respectively. They are evaluated by using the Master equation method. Using the density matrix method the following expression for the density matrix elements is found as [8]
iOba ðrbb raa Þ iðOca þ ac rca Þrbc
855
Table 1 Numerical value of the parameters used in the calculation of the absorption coefficient for the metallic photonic crystal. Parameter
Description
Amount
L¼aþb a b
Lattice constant Spheres diameters Distance between spheres Plasmon energy Conduction band energy Valence band energy Local field parameter Local field parameter Strength of quantum interference Linewidth of level b Linewidth of level c Detuning parameter Rabi frequency
320 nm a=L ¼ 0:8 b=L ¼ 0:2 9.0 ev 11.41 ev 10.12 ev 0, 5, 10 0, 5, 10 1 2.05 1.99 0 0.01
p c v a2 ¼ a2 ¼ a pffiffiffiffiffiffiffiffiffiffiffi b ¼ a3 a2 p P0
Fig. 1. A schematic diagram of a three-level nanoparticle. The levels are denoted by jai, jbi and jci. A probe and a pump lasers couple the transitions jai ! jbi and jai ! jci, respectively.
Gb Gc dk ¼ ðdb þ dc Þ=g0 Oba and Oca ¼ O
ARTICLE IN PRESS 856
M.R. Singh, A. Hatef / Microelectronics Journal 40 (2009) 854–856
is a temporal retardation in the presence of the DDI in achieving the trapping condition. The origin of the retardation arises from the dynamical detuning introduced by the DDI. Thus, establishing a competition between the quantum interference and the DDI internal field [8]. It is found that as the DDI parameter increases the number of oscillations also increases.
Acknowledgment One of the authors (M.R.S) is thankful to NSERC of Canada for financial support in the form of a research grant. References
Fig. 2. The plot of the absorption coefficient versus normalized time when the DDI is zero.
Fig. 3. The plot of the absorption coefficient versus normalized time when the DDI is not zero for a ¼ 5 and 10.
[1] [2] [3] [4] [5] [6] [7] [8]
J.G. Fleming, et al., Nature 417 (2002) 52. A. Kaso, S. John, Phys. Rev. A 76 (2006) 5A. S.S. Wu, et al., J. Phys. C 61 (2007) 1261. M.M. Singalas, et al., Phys. Rev. B 52 (1995) 11744. S. Fan, et al., Phys. Rev. B 54 (1996) 11245. M.R. Singh, Phys. Rev. A 75 (2007) 033810. O.G. Calderon, et al., Eur. Phys. J. D 25 (2003) 77. M.R. Singh, Phys. Rev. A, 2009, in press.
Contents lists available at ScienceDirect
Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
Quantum interference due to the spontaneous emission in nonlinear metallic photonic crystals Mahi R. Singh , A. Hatef Department of Physics and Astronomy, University of Western Ontario, London, Canada N6A 3K7
a r t i c l e in f o
a b s t r a c t
Available online 4 February 2009
The time evolution of the spontaneous emission cancellation is studied in metallic photonic crystals doped with an ensemble of three-level particles. The particles are interacting with each other with the dipole–dipole interaction. We have considered that the two excited states spontaneously decay due to the interaction between the nanoparticles and the photonic crystal. It is found when the detuning parameter between electron resonance energy and probe laser field is zero the system reaches the steady states exponentially. However, when the detuning parameter is not zero the absorption coefficient and the populations of upper two levels oscillated with time and then reach the steady state. & 2008 Elsevier Ltd. All rights reserved.
PACS: 42.55.Tv 42.50.Nn 42.50.Hz 42.50.Ct Keywords: Metallic photonic crystals Absorption coefficient Nanoparticles Quantum interference Spontaneous emission
1. Introduction The aim of the paper is to study the time evolution of the spontaneous emission cancellation in metallic photonic crystals. This phenomenon occurs due to the quantum interference between the two emitted photons in the system. Metallic photonic crystals have a large energy gap which can be used to fabricate new type optoelectronic devices working at high temperature (above 1000 C) [1–5]. The energy band gap in the photonic spectrum in these crystals is due to the combination of the plasma screening and Brag scattering. A lower effective plasma frequency in these crystals has been observed compared to bulk metals below which electromagnetic wave is screened out [2]. It has also been found that electronic (plasmons) and photonic resonances occur in the same spectral range in these materials. Recently, we have studied that the phenomenon of spontaneous emission cancellation has been investigated in photonic crystals in the presence of dipole–dipole interaction (DDI) [6]. The photonic crystals are made from the dielectric materials such as silicon. Crystals have densely doped with an ensemble of threelevel nanoparticles. The mean field theory was used to calculate the effect of the DDI. The linear response theory was used to calculate the expressions for the real and imaginary susceptibilities in the steady state. Numerical simulations are performed for
Corresponding author.
E-mail address: [email protected] (M.R. Singh). 0026-2692/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2008.11.055
an isotropic photonic band gap material. It was found that the spontaneous emission cancellation can be controlled by moving the resonance energies between the energy band and energy gap. It was also predicted that the system can be switched between absorptive and nonabsorptive states by changing the strength of the DDI and the resonance energies in the energy band. The spontaneous emission cancellation has also been investigated in three-level atomic gases in the DDI by Calderon et al. [7]. They found that the system can be switched between absorption and gain by controlling the strength of the DDI and the relative phase difference between the probe and pump fields. In this paper, we have performed the numerical simulation on the time evolution of the spontaneous emission cancellation when the metallic crystal is doped with an ensemble of nanoparticles. Three electronic levels of nanoparticles participate in the absorption process. The crystal is made from metallic spheres and acting as reservoir for the nanoparticles. The numerical simulations for absorption coefficient are performed based on the theory of Singh [8]. We have considered that the two excited states spontaneously decay due to the interaction between the nanoparticles and the photonic crystal. A probe laser field is applied to measure the absorption coefficient. It is found when the detuning parameter between electron resonance energy and probe laser field is zero the system reaches to the steady states exponentially. However, when the detuning parameter is not zero the absorption coefficient and the populations of upper two levels oscillated with time and then reach the steady state.
ARTICLE IN PRESS M.R. Singh, A. Hatef / Microelectronics Journal 40 (2009) 854–856
2. Disunity matrix method An ensemble of three-level nanoparticles is doped densely in the material. The three levels of a nanoparticle are denoted by jai, jbi and jci as shown in Fig. 1. Note that the three levels jai, jbi and jci are in V configuration. The probe and pump lasers with Rabi frequencies Oab and Oac , respectively, are applied to monitor the time evolution of the absorption coefficient. The laser field induces transitions jai2jbi and jai2jci which are responsible for creating dipole moments in the nanoparticles. The time dependent absorption coefficients is written as
a¼
aab rab ðtÞ Oab
aac rac ðtÞ
þ
Oac
drba ¼ ½F ba þ iab ðrbb raa Þ þ ibrbc rba dt pffiffiffiffiffiffiffiffiffiffi p½1 þ i ab ac ðrcc raa Þrca (2)
drca ¼ ½F ca þ iac ðrcc raa Þ þ ibrcb rca dt pffiffiffiffiffiffiffiffiffiffi p½1 þ i ab ac ðrcc raa Þrba iOca ðrcc raa Þ iðOba þ ab rba Þrcb
(3)
drcb ¼ F cb rcb pðrcc þ rbb Þ þ iOca rab iOba rca dt þ iðac ab Þrca rab þ ibðjrba j2 jrca j2 Þ
(4)
drbb ¼ Gb rbb pðrcb þ rbc Þ þ iOba ðrab rba Þ dt þ ibðrab rca rba rac Þ
(5)
drcc ¼ Gc rcc pðrcb þ rbc Þ þ iOca ðrac rca Þ dt þ ibðrba rac rab rca Þ
(6)
F ca ¼
F ba ¼
F cb ¼
Gc 2
Gb 2
þ idc
þ idb
Gc þ Gb 2
þ iðdc db Þ
(7)
(8)
3. Numerical simulations We consider that the metallic crystal is made from metallic spheres which are arranged periodically in a background dielectric material. The lattice constant of the crystals is denoted as L and the diameter of a sphere is assigned the letter a. The refractive indices for a sphere is taken as !1=2 nðek Þ ¼
1
e2p e2k
(12)
where ep and ek are called the plasmon energy and photon energy, respectively. The background material for the metallic crystal is taken as air. Parameters used in the calculation of the absorption coefficient for the metallic photonic crystals are given in Table 1. The simplicity energies in the calculations are measured with respect to g0 which is taken as g0 ¼ 1:0 MeV [5]. Initially, it is considered that an electron is state jai. We have used the conservation of particles condition in the our simulations raa þ rbb þ rcc ¼ 1. The best way to understand the mechanism of the spontaneous emission in metallic photonic crystals is to study the temporal evolution of the absorption coefficient. We consider that the resonance energy lies away from the conduction band edge of the photonic crystals. The results are plotted in Figs. 2 and 3 which show the time evolution of absorption coefficient as a function of the normalized time t and plasmon frequency at zero detuning (i.e. dk ¼ 0) and p0 ¼ 1. Fig. 2 is plotted when the DDI is zero; whereas, Fig. 3 includes the effect of the DDI. In Fig. 2, the system evolves to the trapping state monotonically. However, in Fig. 3 the absorption exhibit an oscillatory behavior until reaching the steady state. During this time interval the system goes to the gain state with population inversion during part of the cycle before reaching the steady state. This type of behavior has also been observed by Calderon et al. [7] in atomic gases. It seems that there
(9)
G ¼ g0 Z 2 ðeÞ
(10)
p ¼ p0
(11)
pffiffiffiffiffiffiffiffiffiffiffi Gc Gc =2
where the function ZðeÞ is called the form factor which contains the information about the electron–photon interaction,t ¼ g0 t, Gc , Gb are the linewidths and Oba and Oca are Rabi energies for transitions jbi ! jai and jci ! jai, respectively. Detuning parameters for transitions jbi ! jai and jci ! jai are defined as db ¼ ðeba ep Þ=g0 and dc ¼ ðeca ep Þ=g0 . Resonance energies eca and eba are the energy differences between the states jci and jai and jbi and jai; respectively. Also ecb is the energy difference between the levels jci and jbi and can be obtained from ecb ¼ ðeca eba Þ=g0 . P0 is the strength of quantum interference. pffiffiffiffiffiffiffiffiffiffi ab , ac and b ¼ ac ab p are the local field parameters when the DDI is taken into account.
(1)
where aab ¼ ðNm2 eba =2e0 g0 _cÞ and aac ¼ ðNm2 eca =2e0 g0 _cÞ. rab ðtÞ and rac ðtÞ are the time dependent density matrices for the transitions jci2jai and jbi2jai; respectively. They are evaluated by using the Master equation method. Using the density matrix method the following expression for the density matrix elements is found as [8]
iOba ðrbb raa Þ iðOca þ ac rca Þrbc
855
Table 1 Numerical value of the parameters used in the calculation of the absorption coefficient for the metallic photonic crystal. Parameter
Description
Amount
L¼aþb a b
Lattice constant Spheres diameters Distance between spheres Plasmon energy Conduction band energy Valence band energy Local field parameter Local field parameter Strength of quantum interference Linewidth of level b Linewidth of level c Detuning parameter Rabi frequency
320 nm a=L ¼ 0:8 b=L ¼ 0:2 9.0 ev 11.41 ev 10.12 ev 0, 5, 10 0, 5, 10 1 2.05 1.99 0 0.01
p c v a2 ¼ a2 ¼ a pffiffiffiffiffiffiffiffiffiffiffi b ¼ a3 a2 p P0
Fig. 1. A schematic diagram of a three-level nanoparticle. The levels are denoted by jai, jbi and jci. A probe and a pump lasers couple the transitions jai ! jbi and jai ! jci, respectively.
Gb Gc dk ¼ ðdb þ dc Þ=g0 Oba and Oca ¼ O
ARTICLE IN PRESS 856
M.R. Singh, A. Hatef / Microelectronics Journal 40 (2009) 854–856
is a temporal retardation in the presence of the DDI in achieving the trapping condition. The origin of the retardation arises from the dynamical detuning introduced by the DDI. Thus, establishing a competition between the quantum interference and the DDI internal field [8]. It is found that as the DDI parameter increases the number of oscillations also increases.
Acknowledgment One of the authors (M.R.S) is thankful to NSERC of Canada for financial support in the form of a research grant. References
Fig. 2. The plot of the absorption coefficient versus normalized time when the DDI is zero.
Fig. 3. The plot of the absorption coefficient versus normalized time when the DDI is not zero for a ¼ 5 and 10.
[1] [2] [3] [4] [5] [6] [7] [8]
J.G. Fleming, et al., Nature 417 (2002) 52. A. Kaso, S. John, Phys. Rev. A 76 (2006) 5A. S.S. Wu, et al., J. Phys. C 61 (2007) 1261. M.M. Singalas, et al., Phys. Rev. B 52 (1995) 11744. S. Fan, et al., Phys. Rev. B 54 (1996) 11245. M.R. Singh, Phys. Rev. A 75 (2007) 033810. O.G. Calderon, et al., Eur. Phys. J. D 25 (2003) 77. M.R. Singh, Phys. Rev. A, 2009, in press.