- Email: [email protected]
Spontaneous radiation from a two-level atom in photonic crystals with three-dimensional dispersion relation
Physica B 279 (2000) 155}158
Spontaneous radiation from a two-level atom in photonic crystals with three-dimensional dispersion relation Yaping Yang!,", Shi-Yao Zhu",*, Hong Chen!,", Hang Zheng",# !Department of Physics, Tongji University, Shanghai 200092, China "Department of Physics, Hong Kong Baptist University, 224 Waterloo Road, Kowloon Tong, Hong Kong #Department of Physics, Jiaotong University, Shanghai 200030, China
Abstract The spontaneous radiation from a two-level atom embedded in photonic crystals with anisotropic three-dimensional dispersion relation is studied. We "nd the properties of emitted "eld depend on the atomic transition frequency and the band edge. The coexistence of a localized "eld and a propagating "eld is removed, because of no singularity in DOS. A di!usion "eld (incoherent propagating "eld) can be formed with all the initial energy of the atom under certain conditions. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 42.50.Dv; 32.80.Bx Keywords: Spontaneous emission; Photonic crystals
There has been growing interest on periodic dielectric structures (photonic crystals) [1}21], especially after much progress has been made recently in preparing photonic crystals working in visible light. The inhibition of light wave propagation within the gaps in photonic crystals provides a way to control spontaneous emission [7], which has attracted a lot of theoretical and experimental investigations because of many important applications [8]. The previous studies show that the gap edge has novel in#uences on optical behavior of an atom embedded in a photonic crystal, such as appearing of photon-atom bound states [9}19], spectral splitting [20], enhanced quantum interference e!ects [18,19], coherent control of spontaneous emission [21] etc. In these studies, the photonic crystal can be well represented by a band-gap structure, i.e. one band edge frequency u and a dispersion relation. For a real photonic crystal, c
* Corresponding author. Fax: #852-2339-5813. E-mail address: [email protected] (S.-Y. Zhu)
the dispersion relation is very complicated. A quadratic dispersion relation is quite a good approximation, especially for frequencies near the band edge. In most of the previous studies, the photon dispersion relation near the band edge was assumed to be isotropic in the momentum space (an one-dimensional dispersion relation). This assumption neglects the anisotropic structures in the momentum space (a three-dimensional dispersion relation) for a real photonic crystal. Furthermore, the density of states (DOS) in a three-dimensional case is proportional to (u !u )1@2, while in the one-dimensional case, it is k c proportional to (u !u )~1@2 leading to a singularity. In k c the one-dimensional approximation, the density of states is overestimated, and consequently its in#uence is also overestimated as we know that it plays a great role in the interaction between light and materials. Therefore, it is crucial to know how the spontaneous emission behaves if the three-dimensional dispersion relation is taken into account. We "nd, in the present paper, the spontaneous emission radiation is di!erent from that in the onedimensional dispersion relation approximation: no coexistence of a localized "eld with a propagating "eld and a very strong di!usion "eld.
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 7 0 6 - 1
156
Y. Yang et al. / Physica B 279 (2000) 155}158
Consider the spontaneous emission from a two-level atom embedded in a photonic crystal. The atom has an upper level DaT with frequency u and a ground state DbT with its energy set to zero. The upper level is coupled by electromagnetic modes to the ground level. In a photonic crystal, the vacuum dispersion relation is modi"ed strongly by the periodic dielectric structures, and anisotropic band-gap structure is formed on the surface of the "rst Brillouin zone in the reciprocal lattice space. Numerical calculations illustrate the band edge is associated, in general, with a "nite collection of symmetry related points ki , e.g., the eight ¸ points on the surface of the "rst 0 Brillouin zone of a diamond photonic crystal [1}5], leading to a three-dimensional band structure. In the present study, the atomic transition frequency u is assumed to be near the band edge u . The dispersion c relation for these k whose directions are near one of ki (i"1, 2,2, 8) could be expressed approximately by 0 uk "u #ADk!ki D2, where A is a model-dependent c 0 constant. The Hamiltonian of the atom and the electromagnetic modes is HK "+uDaTSaD#+ +u bsb k k k k # i++ [g bsDbTSaD!HC]. k k k
(1)
Here b (bs) is the annihilation (creation) operator for the k k kth electromagnetic mode with frequency u , and g is the k k coupling constant between the kth electromagnetic mode and the atomic transition and is assumed to be real. k represents both the momentum and polarization of the modes. The state vector of the system at arbitrary time t is given by Dt(t)T"A(t)e~*utDaTD0T #+ B (t)e~*uk tDbTD1 T f k k f k
P
]
=
b3@2i1@2Jx(u !ix)e~xt c dx, [(du#ix)(u !ix)!Ju b3@2]2!ib3x 0 c c (4)
where du"u!u , and b"[((ud )2/8pe +A3@2) c 1 0 ](+ sin2h )]2@3. In deriving Eq. (4) the summation over i i k was replaced by an integation over k. Due to the anisotropy, the integration over k has to be carried out around the directions of each ki , separately. Here h 0 i is the angle between the dipole vector of the atom and ki . 0 Functions F(x) and G(x) are de"ned as F(x)"1!x2/(2b3@2J!ix!du) and G(x)"1!ix2/ (2b3@2Jix#du). x is the root of x!ib3@2/(Ju # 1 c J!ix!du)"0 in the region (Re(x)'0 or Im(x)'du), and x is the root of x!ib3@2/ 2 (Ju !iJix#du)"0 in the region (Re(x)(0 and c Im(x)(du). The existence of x and x depends on the 1 2 relative positions between the atomic frequency u and the frequency of the band edge u . If x (or x ) does not c 1 2 exist, the "rst (or second) term in Eq. (4) will be replaced by zero. There are three regions separated by u and u 1 2 (see Fig. 1) where we have di!erent roots: u "u #b3@2/u1@2, 1 c c u "2u !b3@2/(2u1@2/3#(q !q )1@3!(q #q )1@3) 2 c c 1 2 1 2 with q "((4u3!20u3@2b3@2)/27#b3)1@2, and q " 1 c c 2 10u3@2/27!b3@2. In region I (u(u ), we have x only c 1 1 (no x ), in region II (u )u)u ), we have neither 2 1 2 x nor x , and in region III (u'u ), we have x only. 1 2 2 2 The emitted radiation E(r, t) can be obtained from A(t) in a standard way [18,22]. The three terms in Eq. (4) yield three di!erent emission "elds. Because x is a pure 1
(2)
with the atom initially in the upper level, A(0)"1 and B (0)"0. The state vector D0T describes no photons k f existing in any modes, and the state vector D1 T represk f ents one photon in the kth mode. From the SchroK dinger equation, we can obtain the following equations for the amplitudes A(t) and B (t): k R A(t)"!+ g B (t)e~*(uk ~u)t, k k Rt k
(3a)
R B (t)"g A(t)e*(uk ~u)t. k Rt k
(3b)
Eqs. (3) can be solved by the Laplace transform method, the resulting expression for the amplitude being ex1 t ex2 t e*dut A(t)" # ! F(x ) G(x ) p 1 2
Fig. 1. The amplitude square (arbitary unit) of the localized mode and the propagating mode as a function of detuning of resonant frequency u from photonic band edge; the dotted curve represents the localized mode, the solid curve the propagating mode, and u "100b. c
Y. Yang et al. / Physica B 279 (2000) 155}158
157
imaginary root [18}20], the "rst "eld is a localized one, which localized near the atom. As x is a complex root, 2 the second "eld is a propagating "eld, which propagates out as a pulse with the energy velocity JA Re(x )/ 2 Im(Jix #du). The third "eld comes from the integra2 tion along the cut-o! the single valued branches and represents a di!usion "eld. For the localized and propagating "elds, we can "nd a well-de"ned frequencies which are u!Im(x ). For the di!usion "eld, we could 1,2 not "nd a de"ned frequency, although its energy propagates out like the propagating "eld. When the upper level is below the band edge or in the band but still in region I (u(u ), the emitted radiation 1 is composed of a localized and a di!usion "eld. When the upper level gets into region II (in the band), the emitted radiation consists of only a di!usion "eld. When the upper level is in region III, the emitted radiation has a propagating "eld and a di!usion "eld. The di!usion "elds in regions I and III are extremely small and can be neglected. However, it is very important in region II (see Fig. 2) (almost several hundred times stronger than in regions I and III), because all the energy is in this "eld. From the above discussion we get the following picture. The property of the emitted radiation depends on the relative position of the upper level to the band edge. As the upper level moves from within the gap to deep inside the band, the emission radiation changes from mainly a localized "eld with a well-de"ned frequency less than u to a mainly a propagating "eld of a pulse (like the c atom in a vacuum) with a frequency larger than u . c During this process, the di!usion "eld "rst increases while the localized "eld decreases. There is energy transfer between the di!usion "eld and the localized "eld. Then the di!usion "eld decreases, while the propagating "eld increases. The energy in the di!usion "eld gives to the propagating "eld.
In region I (u(u ), the localized "eld can be ex1 pressed as
Fig. 2. The amplitude square (arbitary unit) of the di!usion "eld as a function of the transition frequency u with u " c 100b, rJb/A"1 and bt"3.
Fig. 3. The evolution of the di!usion "eld (arbitary unit) with time t and with the distance r from the atom. u " c 100b, u"100.1b.
1 E (r, t)"E (0) e(x1 ~*u)t~r@l, l l r
(5)
where E (0) is a u dependent constant, the size of the l localized "eld being determined by the localization length, l"[(!ix(1)#u !u)/A]~1@2. The localized c "eld results in fractionalized steady-state upper level population P&DE (0)D2 [7]. The upper level population l decreases to zero as u approaches u . It can be proved 1 that the localization length tends to in"nity, and the amplitude of the localized "eld tends to zero as u goes to u . In region II (u (u(u , see inset in Fig. 1), there 1 1 2 are no localized "elds and no propagating "elds, the spontaneous emission being a typical di!usion "eld. 1 E (r, t)"E (0) e~*uc t`*r2@(4At)~*p@4 $ $ r
P P
]
]
=
b3@2Jx(u !ix) dx c [(du#ix)(u !ix)!Ju b3@2]2!ib3x 0 c c =
[ye3p*@4#r/(2JAt)]e~y2 dy
.
(6)
~= !xt#i[ye3p*@4#r/(2JAt)]2
The evolution of E (r, t), with time t and with the $ distance r from the atom, is shown in Fig. 3. As time t increases, E (r, t) at any space point increases "rstly $ from zero to a maximum value, and then decreases to zero again. Similar behavior can be seen considering E (r, t) as a function of the distance r from the atom at $ a given time. The position of the maximum E (r, t) goes $ away from the atom with time (see Fig. 3). At the long time limit we have E (r, t)&t~a, where a is a constant $ decreasing with increasing u. This region, in which there is only a di!usion "eld, is caused by the anisotropic
158
Y. Yang et al. / Physica B 279 (2000) 155}158
is not large enough. Further movement into the band will result in the propagating "eld with "xed phase. In conclusion, we "nd that the properties of the spontaneous radiation from a two-level atom in a photonic crystal heavily depend on the position of the atomic transition frequency u relative to the band edge u . The c anisotropic dispersion relation results in non-coexistence of the localized "eld and the propagating "eld, and results in a di!usion "eld, which carries all the initial energy of the atom. Acknowledgements Fig. 4. The width of the frequency region for u, in which only di!usion "eld exists, as a function of the band edge u . c
dispersion relation and does not exist in the case of the isotropic dispersion relation. In the case of isotropic dispersion relation, the di!usion "eld is negligible. The degree of the anisotropy is determined by the coe$cient A/u and decreases as u increases. Therefore, the width c c of region II (D"u !u ) decreases as u increases (see 2 1 c Fig. 4). In region III, u'u , the dominant component 2 of the emitted "eld is a propagating "eld . It has the form 1 E (r, t)"E (0) e(x2 ~*u)t`*qr 1 1 r
This reseach was supported by FRG from the Hong Kong Baptist University and UGC from Hong Kong Government and the Chinese National Science Foundation.
References [1] [2] [3] [4] [5] [6]
(7)
with q"[(ix #u!u )/A]1@2. The behavior of DE (0)D2 2 c 1 is given in Fig. 1, indicating a pronounced switch-on e!ect for the propagating "eld when u changes from being smaller than u to larger than u . 2 2 In the photonic crystal with the isotropic dispersion relation, a localized "eld and a propagating "eld can coexist, but this is not the case with anisotropic dispersion relation. This di!erence comes from the di!erence of DOS near the band edge in the two cases. There is a singularity in the isotropic case, but none in the anisotropic case. For the isotropic case, any potential from an impurity (no matter how small) will lead to localization [23] because of the singularity in DOS. This is why we always have the localized "eld in the isotropic case. When the upper level moves up into to the band from the gap, the frequency of the localized "eld approaches u . In c the isotropic case, the frequency cannot be u , because of c the in"nite DOS at u , which means in"nite energy c requirement for the localized "eld. In the anisotropic case, it can be u . When it is u , the localized "eld c c disappears, and the di!usion "eld (an incoherent propagating "eld) appears, because the number of electromagnetic modes near the atomic transition frequency
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
S. John, Phys. Rev. Lett. 53 (1984) 2169. S. John, Phys. Rev. Lett. 58 (1987) 2486. E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. K.M. Ho, T.J. Chan, C.M. Soukoulis, Phys. Rev. Lett. 65 (1990) 3152. J.E.G.J. Wijnhoven, W.L. Vos, Science 281 (1998) 802. Y.S. Chan, C.T. Chan, Z.Y. Liu, Phys. Rev. Lett. 80 (1988) 956. E. Yablonovitch, J. Opt. Soc. Am. B 10 (1993) 283. J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals, Princeton, New York, 1995. S. John, J. Wang, Phys. Rev. Lett. 64 (1990) 2418. S. John, J. Wang, Phys. Rev. B 43 (1991) 12 772. S. John, T. Quang, Phys. Rev. Lett. 74 (1995) 3419. S. John, T. Quang, Phys. Rev. Lett. 76 (1996) 1320. S. John, T. Quang, Phys. Rev. Lett. 78 (1997) 1888. S. Bay, P. Lambropoulos, K. Molmer, Phys. Rev. Lett. 79 (1997) 2654. S. Bay, P. Lambropoulos, K. Molmer, Opt. Commun. 132 (1996) 237. S. Bay, P. Lambropoulos, K. Molmer, Opt. Commun. 146 (1998) 130. S. Bay, P. Lambropoulos, K. Molmer, Phys. Rev. A 55 (1997) 1485. S.-Y. Zhu, H. Chen, H. Huang, Phys. Rev. Lett. 79 (1997) 205. H. Huang, X.H. Lu, S.-Y. Zhu, Phys. Rev. A 57 (1998) 4945. S. John, T. Quang, Phys. Rev. A 50 (1994) 1764. T. Quang, M. Woldeyohannes, S. John, G.S. Agarwal, Phys. Rev. Lett. 79 (1997) 5238. M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, England, 1996 (Chapter 6). J. Callaway, Quantum Theory of the Solid State, Academic Press, New York, 1976 (Chapter 5).
Spontaneous radiation from a two-level atom in photonic crystals with three-dimensional dispersion relation Yaping Yang!,", Shi-Yao Zhu",*, Hong Chen!,", Hang Zheng",# !Department of Physics, Tongji University, Shanghai 200092, China "Department of Physics, Hong Kong Baptist University, 224 Waterloo Road, Kowloon Tong, Hong Kong #Department of Physics, Jiaotong University, Shanghai 200030, China
Abstract The spontaneous radiation from a two-level atom embedded in photonic crystals with anisotropic three-dimensional dispersion relation is studied. We "nd the properties of emitted "eld depend on the atomic transition frequency and the band edge. The coexistence of a localized "eld and a propagating "eld is removed, because of no singularity in DOS. A di!usion "eld (incoherent propagating "eld) can be formed with all the initial energy of the atom under certain conditions. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 42.50.Dv; 32.80.Bx Keywords: Spontaneous emission; Photonic crystals
There has been growing interest on periodic dielectric structures (photonic crystals) [1}21], especially after much progress has been made recently in preparing photonic crystals working in visible light. The inhibition of light wave propagation within the gaps in photonic crystals provides a way to control spontaneous emission [7], which has attracted a lot of theoretical and experimental investigations because of many important applications [8]. The previous studies show that the gap edge has novel in#uences on optical behavior of an atom embedded in a photonic crystal, such as appearing of photon-atom bound states [9}19], spectral splitting [20], enhanced quantum interference e!ects [18,19], coherent control of spontaneous emission [21] etc. In these studies, the photonic crystal can be well represented by a band-gap structure, i.e. one band edge frequency u and a dispersion relation. For a real photonic crystal, c
* Corresponding author. Fax: #852-2339-5813. E-mail address: [email protected] (S.-Y. Zhu)
the dispersion relation is very complicated. A quadratic dispersion relation is quite a good approximation, especially for frequencies near the band edge. In most of the previous studies, the photon dispersion relation near the band edge was assumed to be isotropic in the momentum space (an one-dimensional dispersion relation). This assumption neglects the anisotropic structures in the momentum space (a three-dimensional dispersion relation) for a real photonic crystal. Furthermore, the density of states (DOS) in a three-dimensional case is proportional to (u !u )1@2, while in the one-dimensional case, it is k c proportional to (u !u )~1@2 leading to a singularity. In k c the one-dimensional approximation, the density of states is overestimated, and consequently its in#uence is also overestimated as we know that it plays a great role in the interaction between light and materials. Therefore, it is crucial to know how the spontaneous emission behaves if the three-dimensional dispersion relation is taken into account. We "nd, in the present paper, the spontaneous emission radiation is di!erent from that in the onedimensional dispersion relation approximation: no coexistence of a localized "eld with a propagating "eld and a very strong di!usion "eld.
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 7 0 6 - 1
156
Y. Yang et al. / Physica B 279 (2000) 155}158
Consider the spontaneous emission from a two-level atom embedded in a photonic crystal. The atom has an upper level DaT with frequency u and a ground state DbT with its energy set to zero. The upper level is coupled by electromagnetic modes to the ground level. In a photonic crystal, the vacuum dispersion relation is modi"ed strongly by the periodic dielectric structures, and anisotropic band-gap structure is formed on the surface of the "rst Brillouin zone in the reciprocal lattice space. Numerical calculations illustrate the band edge is associated, in general, with a "nite collection of symmetry related points ki , e.g., the eight ¸ points on the surface of the "rst 0 Brillouin zone of a diamond photonic crystal [1}5], leading to a three-dimensional band structure. In the present study, the atomic transition frequency u is assumed to be near the band edge u . The dispersion c relation for these k whose directions are near one of ki (i"1, 2,2, 8) could be expressed approximately by 0 uk "u #ADk!ki D2, where A is a model-dependent c 0 constant. The Hamiltonian of the atom and the electromagnetic modes is HK "+uDaTSaD#+ +u bsb k k k k # i++ [g bsDbTSaD!HC]. k k k
(1)
Here b (bs) is the annihilation (creation) operator for the k k kth electromagnetic mode with frequency u , and g is the k k coupling constant between the kth electromagnetic mode and the atomic transition and is assumed to be real. k represents both the momentum and polarization of the modes. The state vector of the system at arbitrary time t is given by Dt(t)T"A(t)e~*utDaTD0T #+ B (t)e~*uk tDbTD1 T f k k f k
P
]
=
b3@2i1@2Jx(u !ix)e~xt c dx, [(du#ix)(u !ix)!Ju b3@2]2!ib3x 0 c c (4)
where du"u!u , and b"[((ud )2/8pe +A3@2) c 1 0 ](+ sin2h )]2@3. In deriving Eq. (4) the summation over i i k was replaced by an integation over k. Due to the anisotropy, the integration over k has to be carried out around the directions of each ki , separately. Here h 0 i is the angle between the dipole vector of the atom and ki . 0 Functions F(x) and G(x) are de"ned as F(x)"1!x2/(2b3@2J!ix!du) and G(x)"1!ix2/ (2b3@2Jix#du). x is the root of x!ib3@2/(Ju # 1 c J!ix!du)"0 in the region (Re(x)'0 or Im(x)'du), and x is the root of x!ib3@2/ 2 (Ju !iJix#du)"0 in the region (Re(x)(0 and c Im(x)(du). The existence of x and x depends on the 1 2 relative positions between the atomic frequency u and the frequency of the band edge u . If x (or x ) does not c 1 2 exist, the "rst (or second) term in Eq. (4) will be replaced by zero. There are three regions separated by u and u 1 2 (see Fig. 1) where we have di!erent roots: u "u #b3@2/u1@2, 1 c c u "2u !b3@2/(2u1@2/3#(q !q )1@3!(q #q )1@3) 2 c c 1 2 1 2 with q "((4u3!20u3@2b3@2)/27#b3)1@2, and q " 1 c c 2 10u3@2/27!b3@2. In region I (u(u ), we have x only c 1 1 (no x ), in region II (u )u)u ), we have neither 2 1 2 x nor x , and in region III (u'u ), we have x only. 1 2 2 2 The emitted radiation E(r, t) can be obtained from A(t) in a standard way [18,22]. The three terms in Eq. (4) yield three di!erent emission "elds. Because x is a pure 1
(2)
with the atom initially in the upper level, A(0)"1 and B (0)"0. The state vector D0T describes no photons k f existing in any modes, and the state vector D1 T represk f ents one photon in the kth mode. From the SchroK dinger equation, we can obtain the following equations for the amplitudes A(t) and B (t): k R A(t)"!+ g B (t)e~*(uk ~u)t, k k Rt k
(3a)
R B (t)"g A(t)e*(uk ~u)t. k Rt k
(3b)
Eqs. (3) can be solved by the Laplace transform method, the resulting expression for the amplitude being ex1 t ex2 t e*dut A(t)" # ! F(x ) G(x ) p 1 2
Fig. 1. The amplitude square (arbitary unit) of the localized mode and the propagating mode as a function of detuning of resonant frequency u from photonic band edge; the dotted curve represents the localized mode, the solid curve the propagating mode, and u "100b. c
Y. Yang et al. / Physica B 279 (2000) 155}158
157
imaginary root [18}20], the "rst "eld is a localized one, which localized near the atom. As x is a complex root, 2 the second "eld is a propagating "eld, which propagates out as a pulse with the energy velocity JA Re(x )/ 2 Im(Jix #du). The third "eld comes from the integra2 tion along the cut-o! the single valued branches and represents a di!usion "eld. For the localized and propagating "elds, we can "nd a well-de"ned frequencies which are u!Im(x ). For the di!usion "eld, we could 1,2 not "nd a de"ned frequency, although its energy propagates out like the propagating "eld. When the upper level is below the band edge or in the band but still in region I (u(u ), the emitted radiation 1 is composed of a localized and a di!usion "eld. When the upper level gets into region II (in the band), the emitted radiation consists of only a di!usion "eld. When the upper level is in region III, the emitted radiation has a propagating "eld and a di!usion "eld. The di!usion "elds in regions I and III are extremely small and can be neglected. However, it is very important in region II (see Fig. 2) (almost several hundred times stronger than in regions I and III), because all the energy is in this "eld. From the above discussion we get the following picture. The property of the emitted radiation depends on the relative position of the upper level to the band edge. As the upper level moves from within the gap to deep inside the band, the emission radiation changes from mainly a localized "eld with a well-de"ned frequency less than u to a mainly a propagating "eld of a pulse (like the c atom in a vacuum) with a frequency larger than u . c During this process, the di!usion "eld "rst increases while the localized "eld decreases. There is energy transfer between the di!usion "eld and the localized "eld. Then the di!usion "eld decreases, while the propagating "eld increases. The energy in the di!usion "eld gives to the propagating "eld.
In region I (u(u ), the localized "eld can be ex1 pressed as
Fig. 2. The amplitude square (arbitary unit) of the di!usion "eld as a function of the transition frequency u with u " c 100b, rJb/A"1 and bt"3.
Fig. 3. The evolution of the di!usion "eld (arbitary unit) with time t and with the distance r from the atom. u " c 100b, u"100.1b.
1 E (r, t)"E (0) e(x1 ~*u)t~r@l, l l r
(5)
where E (0) is a u dependent constant, the size of the l localized "eld being determined by the localization length, l"[(!ix(1)#u !u)/A]~1@2. The localized c "eld results in fractionalized steady-state upper level population P&DE (0)D2 [7]. The upper level population l decreases to zero as u approaches u . It can be proved 1 that the localization length tends to in"nity, and the amplitude of the localized "eld tends to zero as u goes to u . In region II (u (u(u , see inset in Fig. 1), there 1 1 2 are no localized "elds and no propagating "elds, the spontaneous emission being a typical di!usion "eld. 1 E (r, t)"E (0) e~*uc t`*r2@(4At)~*p@4 $ $ r
P P
]
]
=
b3@2Jx(u !ix) dx c [(du#ix)(u !ix)!Ju b3@2]2!ib3x 0 c c =
[ye3p*@4#r/(2JAt)]e~y2 dy
.
(6)
~= !xt#i[ye3p*@4#r/(2JAt)]2
The evolution of E (r, t), with time t and with the $ distance r from the atom, is shown in Fig. 3. As time t increases, E (r, t) at any space point increases "rstly $ from zero to a maximum value, and then decreases to zero again. Similar behavior can be seen considering E (r, t) as a function of the distance r from the atom at $ a given time. The position of the maximum E (r, t) goes $ away from the atom with time (see Fig. 3). At the long time limit we have E (r, t)&t~a, where a is a constant $ decreasing with increasing u. This region, in which there is only a di!usion "eld, is caused by the anisotropic
158
Y. Yang et al. / Physica B 279 (2000) 155}158
is not large enough. Further movement into the band will result in the propagating "eld with "xed phase. In conclusion, we "nd that the properties of the spontaneous radiation from a two-level atom in a photonic crystal heavily depend on the position of the atomic transition frequency u relative to the band edge u . The c anisotropic dispersion relation results in non-coexistence of the localized "eld and the propagating "eld, and results in a di!usion "eld, which carries all the initial energy of the atom. Acknowledgements Fig. 4. The width of the frequency region for u, in which only di!usion "eld exists, as a function of the band edge u . c
dispersion relation and does not exist in the case of the isotropic dispersion relation. In the case of isotropic dispersion relation, the di!usion "eld is negligible. The degree of the anisotropy is determined by the coe$cient A/u and decreases as u increases. Therefore, the width c c of region II (D"u !u ) decreases as u increases (see 2 1 c Fig. 4). In region III, u'u , the dominant component 2 of the emitted "eld is a propagating "eld . It has the form 1 E (r, t)"E (0) e(x2 ~*u)t`*qr 1 1 r
This reseach was supported by FRG from the Hong Kong Baptist University and UGC from Hong Kong Government and the Chinese National Science Foundation.
References [1] [2] [3] [4] [5] [6]
(7)
with q"[(ix #u!u )/A]1@2. The behavior of DE (0)D2 2 c 1 is given in Fig. 1, indicating a pronounced switch-on e!ect for the propagating "eld when u changes from being smaller than u to larger than u . 2 2 In the photonic crystal with the isotropic dispersion relation, a localized "eld and a propagating "eld can coexist, but this is not the case with anisotropic dispersion relation. This di!erence comes from the di!erence of DOS near the band edge in the two cases. There is a singularity in the isotropic case, but none in the anisotropic case. For the isotropic case, any potential from an impurity (no matter how small) will lead to localization [23] because of the singularity in DOS. This is why we always have the localized "eld in the isotropic case. When the upper level moves up into to the band from the gap, the frequency of the localized "eld approaches u . In c the isotropic case, the frequency cannot be u , because of c the in"nite DOS at u , which means in"nite energy c requirement for the localized "eld. In the anisotropic case, it can be u . When it is u , the localized "eld c c disappears, and the di!usion "eld (an incoherent propagating "eld) appears, because the number of electromagnetic modes near the atomic transition frequency
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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