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The virtual-photon effects in spontaneous emission from an atom embedded in a photonic band gap structure
20 December 1999
Physics Letters A 264 Ž1999. 137–141 www.elsevier.nlrlocaterphysleta
The virtual-photon effects in spontaneous emission from an atom embedded in a photonic band gap structure Zhengdong Liu a
a,1
, Yu Lin a , Shiyao Zhu b, Ke Shang a , Liang Zeng
c,)
Department of Physics, Zhejiang UniÕersity, Yuquan, Hangzhou 310027, China b Department of Physics, Hong Kong Baptist UniÕersity, Hong Kong, China c The UniÕersity of Texas-Pan American, Edinburg TX 78539, USA Received 14 September 1999; accepted 12 November 1999 Communicated by P.R. Holland
Abstract In a three-level atom embedded in a photonic band gap structure, the spontaneous emission is studied both with and without rotating-wave approximation. The virtual-photon-localized field is found, which causes a greater population of the atom trapped in its upper states. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 42.50 Gy; 42.70.Qs Keywords: Photonic band gap structure; Spontaneous emission; Virtual-photon processes; Virtual-photon-localized field
Spontaneous emission is a fundamental concept in atomic physics. The spectrum of spontaneous emission can display vacuum Rabi splitting in a resonant high-Q cavity w1,2x. Furthermore, to control spontaneous emission well is important for the control of an atom in terms of its stability. A new generation of experiments reveals that spontaneous radiation from excited atoms can be greatly suppressed or enhanced by placing the atoms between mirrors or in a cavity w3–5x. During recent years, considerable attention has been paid to the behavior of the spontaneous emission of an atom embedded in a photonic band gap structure ŽPBGS. w6–11x. In PBGS the prohibition of
) 1
Corresponding author. E-mail: [email protected] Fax: q86-571-7951358; e-mail: [email protected]
light wave transmission can be achieved for some frequency ranges in all directions. Furthermore, it was showed that there are three modes in the spontaneous emission of a three-level atom that could be partly trapped in the upper levels, a localized field, a propagating field and a decaying field w7x. It may be noted that the above work was carried out within the rotating-wave approximation ŽRWA.. However, virtual-photon processes, which are introduced without the RWA, may have significant influence on the RWA results with the problems such as the accurate treatment of the photon absorption w12,13x, the atomic level’s Lamb shift w14,15x, and the interference between virtual-photon and real-photon processes in a micromaser cavity field w16x. In this Letter, the evolution of a three-level atom embedded in a photonic band gap structure will be studied without the RWA ŽWRWA. and then com-
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 8 0 9 - 9
Z. Liu et al.r Physics Letters A 264 (1999) 137–141
138
pared with the RWA results. The two upper levels are coupled to the lower level via the same field continuum. Some notable quantum effects induced by the virtual photons will be discussed. Consider a three-level atom with two upper levels < a1 :, < a 2 : and a lower level < b :, as shown in Fig. 1. The dipole vectors between < a1 : and < b : and between < a 2 : and < b : are parallel to each other. The dispersion relationship of the band gap material near the band gap edge v c can be approximated by w6,17,18x: 2
v k s vc q A Ž k y k 0 . ,
A s vcrk 02 .
Ž 1.
The Hamiltonian for the system is:
In the interaction picture, the Hamiltonian of the system can be written as: HˆI s i"
k
a k < a1 : ² b <
qe iŽ v a 2 byv k .t a k < a 2 : ² b < < : ²b< qe iŽ v a1 bqv k .t aq k a1 < : ² b < . q H.C. qe iŽ v a 2 bq v k .t aq k a2
Ž 3.
Assume that the atom is initially in its upper states, < a1 :, and < a2 :, and the field is in the vacuum state. The wave function of the system then has the form: < c Ž t . : s AŽ1. Ž t . < a1 :<0: q AŽ2. Ž t . < a2 :<0: q Ý b 1, k < b :<1: k k `
Ý Ž g kŽ1. a k < a1 : ² b <
qÝ
k
Ý
< :< : aŽ1. n , k Ž t . a1 n k
k ns1
< : ²b< qg kŽ2. a k < a 2 : ² b < q g kŽ1. aq k a1 < : ² b < . q H.C. . qg kŽ2. aq k a2
a1 b y v k .t
k
Hˆ s " v a1 < a1 : ² a1 < q " v a 2 < a 2 : ² a 2 < q " v b < b : ² b < q Ý " v k aq k a k q i"
Ý g k Ž e iŽ v
qaŽ2. n,k
Ž t . < a2 :< n:k q bnq1, k < b :< n q 1:k Ž 4.
Ž 2.
Ž . Ž . Here aq k a k is the creation annihilation operator for the light field. The atom-field coupling constants g kŽ1. and g kŽ2. are related to the decay coefficients of each upper level. We assume that g kŽ1. s g kŽ2. s g k when v a1 b and v a 2 b satisfy: v a1 b y vc s vc y v a 2 b s D and < D < < vc , and that g k s r Ž v . g k 0 with r Ž v . having a negative exponential function as shown in Fig. 1.
Ž2. Ž . Ž . with AŽ1. Ž0., AŽ2. Ž0. / 0, and aŽ1. n, k 0 , a n, k 0 , Ž . < Ž .: bnq 1, k 0 s 0. And c t obeys the Schrodinger ¨ equation: d i" < c Ž t . : s HˆI < c Ž t . : . Ž 5. dt So the first-order equations set is deduced as: d Ž1. A Ž t . s Ý g k e iŽ v a1 by v k .t b 1, k Ž t . , dt k
d dt d dt
AŽ2. Ž t . s Ý g k e iŽ v a 2 by v k .t b 1, k Ž t . , k
b 1, k Ž t . s yg k eyi Ž v a1 by v k .tAŽ1. Ž t . q'2 eyi Ž v a1 bq v k .t aŽ1. 2, k Ž t . qeyi Ž v a 2 byv k .tAŽ2. Ž t . q'2 eyi Ž v a 2 bq v k .t aŽ2. 2, k Ž t . ,
d dt
aŽ1. nk Ž t . s gk
'n q 1 e iŽ v
a1 b y v k .t
PPP ,
bnq1, k Ž t .
q'n e iŽ v a1 bq v k .t bny 1, k Ž t . , d Fig. 1. A three-level atom in a photonic band gap structure. The two upper levels, < a1 : and < a2 :, are symmetrically placed from the band gap edge by D.
dt
aŽ2. nk Ž t . s gk
'n q 1 e iŽ v
a 2 b y v k .t
bnq1, k Ž t .
q'n e iŽ v a 2 bq v k .t bny 1, k Ž t . ,
Z. Liu et al.r Physics Letters A 264 (1999) 137–141
d dt
139
bnq 1, k Ž t . s yg k w 'n q 1 eyi Ž v a1 by v k .t aŽ1. n,k Ž t . q'n q 2 eyi Ž v a1 bq v k .t aŽ1. nq 2, k Ž t . q'n q 1 eyi Ž v a 2 by v k .t aŽ2. n,k Ž t . q'n q 2 eyi Ž v a 2 bq v k .t aŽ2. nq 2, k Ž t . .
Ž 6. It can be noted that all these equations have the atom-field coupling constant g k on their right-hand sides. When k is far from k 0 , g k becomes so small because of a sharp decrease of r Ž v . that those modes will approach zero. Without considering the influence of the virtual-photon processes, i.e., in the RWA, one finds that the equation set Ž6. reduces to: d dt d dt d dt
AŽ1. Ž t . s Ý g k e iŽ v a1 by v k .t b 1, k Ž t . , k
AŽ2. Ž t . s Ý g k e iŽ v a 2 by v k .t b 1, k Ž t . , k
b 1, k Ž t . s yg k eyi Ž v a1 by v k .tAŽ1. Ž t . qeyi Ž v a 2 by v k .tAŽ2. Ž t . ,
Ž 7.
which is similar to the equations in Ref. w6x. Comparing the equation sets Ž6. and Ž7., one can find that it is the virtual-photon effects that introduce the addiŽ2. Ž n s 1,2, . . . ,`., tional terms aŽ1. n, k , a n, k and bnq1, k which are respectively the coefficients of < a1 :< n: k , < a2 :< n: k and < b :< n q 1: k . The virtual-photon processes are non-energy-conserving processes, and the rapidly emitted and absorbed energy in the form of two-photon creates a virtual-photon cloud around the field source, so only the contributions of the virtualphoton equation terms around the source of real-photon processes will have a noticeable effect. With the numerical results of equation set Ž6., the population of all the states of the atom, P Ž a1 ., P Ž a 2 . and P Ž b . can be derived. Generally, the effects of virtual-photon processes are quite weak, for the virtual-photon processes are perturbation to the real-photon processes. However, there are times that the effects of virtual-photon processes are not negligible. When the system parameters are set at specific values Žas seen in Fig. 2., the virtual-photon processes may alter or completely destroy the rele-
Fig. 2. The population evolution of the states of the atom. P Ž a1 ., P Ž a 2 . and P Ž b . are the populations in level < a1 :, < a 2 : and < b :, respectively. Ža. is with the RWA, and Žb. and Žc. are without the RWA. The atom is initially in < c Ž0.: s Ž1r'2 .Ž< a1 :q < a 2 :.. Parameters: g k s1.0, vc s 4.8p , D s 0.48p .
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Z. Liu et al.r Physics Letters A 264 (1999) 137–141
vant evolution of the populations of the atomic states within the RWA. In Fig. 2 these non-negligible effects of virtualphoton processes are shown. Fig. 2a shows the evolution of the populations of the atomic states with the RWA, where the band edge v c s 4.8p with D s 0.48p , the atom-field coupling constant at the edge g k 0 s 1.0 and the two upper atomic states are initialized as AŽ1. Ž0. s AŽ2. Ž0.. The RWA evolution curves of the populations of all the atomic states are equalamplitude oscillations with constant periods. Moreover, the curves for the two upper states share the same amplitude, which is nearly 0.5. The evolution for the lower state has the same period as that of the upper two levels, but its amplitude is twice as large. The three curves indicate that: Ž1. The two upper states are thoroughly coherent, the phase difference between them is zero, and their population evolution curves descend and ascend together between the maximum of nearly 0.5 and the minimum of 0; Ž2. The phase difference between the curves for the lower state and the two upper states is p . There is no energy exchange between the two upper states, yet the two upper states commute energy with the lower state, respectively and synchronously. With the same parameters as in Fig. 2a, the evolutions of the populations of the two upper states without the RWA are shown in Fig. 2b. They are both quasioscillations under WRWA. Different from the constant amplitudes in the RWA curves, the amplitudes of the WRWA curves are time-dependent functions. In the beginning in a very brief time period, the WRWA curves have the similar disciplinarians to the RWA curves. But very soon the two WRWA evolution curves separate from each other and their amplitudes diminish sharply; one of their equilibrium points ascends and the other descends. Then both of the population quasioscillations augment their amplitudes again and eventually stabilize at different values respectively with comparatively smaller amplitudes. The envelopes of the two quasioscillations have slightly quasi-harmonic oscillations. In addition, a similar phenomenon can also be seen in Fig. 2c where the population evolution of the lower state is shown. All these indicate that a virtual-photon-localized field exists in the spontaneous emission process of this atom. Comparing the WRWA curves with the relevant RWA curves, the
virtual-photon processes destroy the coherence between the population evolutions of the two upper states. Moreover, their oscillations are no longer simply harmonious. Above all, the virtual-photon processes induce the virtual-photon-localized field, which traps a portion of the atom’s population in the upper levels. In Fig. 2a, the upper states’ populations approach zero when the curves reach the bottoms. However, the curves in WRWA can never reach zero, for instance, the bottoms of P Ž a2 . are always higher than 0.25; the lower state’s population evolution curve P Ž b . under WRWA in Fig. 2c is also a quasioscillation with an equilibrium point around 0.45 when it is stabilized. The amplitude is greater than 0.5 and the population cannot go beyond the magnitude of 0.7. Thus, a portion of population of the atom is trapped in the upper states under WRWA. The population trapping is similar to the phenomenon of a localized field under RWA in Refs. w6,7x However, the virtual-photon-localized field is quite different from that localized field. Firstly, the oscillation of the WRWA population curves cannot decay, which both of the two upper states oscillations’ amplitudes are greater than 0.2 when they are stabilized. The fluctuations of RWA population curves will dwindle sharply as time lapses and retain a certain value eventually. Secondly, the oscillation period in WRWA curves maintain as a constant while the periods in RWA curves will be greater and
Fig. 3. The population evolution of the upper state < a1 :, with the RWA Žsolid line. and without the RWA Ždotted line.. The atom is initially in < c Ž0.: s < a2 :. Parameters: g k s1.0, vc s6.5p , D s 0.65p .
Z. Liu et al.r Physics Letters A 264 (1999) 137–141
greater. Furthermore, the oscillation periods in WRWA curves are quite shorter than those RWA ones, namely the WRWA curves are rapid equalfrequency oscillations. Lastly, the ratio of the populations in the two upper levels, AŽ1. Ž`.rAŽ2. Ž`., is obviously greater in WRWA than that in RWA. The localized field mentioned in Ref. w7x is generated by the quantum interference effects of real-photon processes. But the virtual-photon-localized field is induced by the intensive interference between virtual photons and real photons. The virtual-photon processes are rapid oscillation processes, which result in rapid oscillation interference between virtualphoton processes and the real-photon processes. The virtual-photon-localized field is generally too weak to be found, because the rapidly emitted and absorbed virtual photons only cause a perturbation to the real-photon processes. For instance, when g k 0 s 1.0, the band edge v c s 100p and D s 10p , the atomic state is initially AŽ1. Ž0. s AŽ2. Ž0.; the evolution disciplinarians of atom states’ populations are the same as those described in the Ref. w7x. As the band edge vc decreases, the influence of the virtual-photon processes will increase, for example, in Fig. 3, in which vc s 6.5p and the atomic state is initialized as AŽ1. Ž0. s 0.0 and AŽ2. Ž0. s 1.0; the WRWA curves are slight modifications of the relevant RWA curves. However, the influence of the virtual-photon processes will not increase all the time with the decrease of the band edge. On the contrary, it will dwindle when the band edge gets quite weak, for instance, when vc s 1.5p the WRWA curves and the RWA curves nearly overlap each other. This indicates that the interference between the virtual-photon and the real-photon is maximized only under some specific parameter sets. In this Letter, we have studied the evolution behavior of the spontaneous emission of an atom that was embedded in a photon band gap structure, both with RWA and WRWA. Although the effects of the
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virtual-photon processes are generally perturbations to the evolution curve in RWA, the effect of the virtual-photon processes can be quite significant in certain cases. The virtual-photon-localized field was discussed in the spontaneous emission of an atom under certain series of system parameters, which do cause a greater population of the atom trapped in the upper states. Acknowledgements This work is supported by the National Natural Science Foundation of China, No. 19674048 & No.29676037, and the Zhejiang Province Natural Science Foundation, No. 198023. References w1x M.G. Raizen, R.J. Thompson, R.J. Brecha, H.J. Kimble, H.J. Carmichael, Phys. Rev. Lett. 63 Ž1989. 240. w2x R.J. Thompson, G. Rempe, H.J. Kimble, Phys. Rev. Lett. 68 Ž1990. 1132. w3x E.M. Purcell, Phys. Rev. 69 Ž1946. 681. w4x D. Kleppner, Phys. Rev. Lett. 47 Ž1981. 232. w5x S. Harochee, D. Kleppner, Phys. Today 42 Ž1989. 24, and references therein. w6x S. John, T. Quang, Phys. Rev. A 50 Ž1994. 1764. w7x S.Y. Zhu, H. Chen, H. Huang, Phys. Rev. Lett. 79 Ž1997. 205. w8x S. John, Phys. Rev. Lett. 53 Ž1984. 2169. w9x P.W. Anderson, Phil. Mag. B 52 Ž1985. 505. w10x E. Yablonovitch, Phys. Rev. Lett. 58 Ž1987. 2059. w11x S. John, Phys. Rev. Lett. 58 Ž1987. 2486. w12x G. Compagno, G.M. Palma, R. Passante, F. Perisco, Europhys. Lett. 9, 215. w13x G. Compagno, G.M. Palma, Phys. Rev. A 37 Ž1988. 2979. w14x X.Y. Huang, L.S. Peng, Phys. Scripta T 21 Ž1988. 100. w15x G. Compagno, R. Passante, F. Persico, Phys. Lett. A 98 Ž1983. 253. w16x L. Zeng, Z.D. Liu, Y. Lin, S.Y. Zhu, Phys. Lett. A 246 Ž1998. 43. w17x S. John, J. Wang, Phys. Rev. Lett. 64 Ž1990. 2418. w18x S. John, T. Quang, Phys. Rev. B 43 Ž1991. 12772.
Physics Letters A 264 Ž1999. 137–141 www.elsevier.nlrlocaterphysleta
The virtual-photon effects in spontaneous emission from an atom embedded in a photonic band gap structure Zhengdong Liu a
a,1
, Yu Lin a , Shiyao Zhu b, Ke Shang a , Liang Zeng
c,)
Department of Physics, Zhejiang UniÕersity, Yuquan, Hangzhou 310027, China b Department of Physics, Hong Kong Baptist UniÕersity, Hong Kong, China c The UniÕersity of Texas-Pan American, Edinburg TX 78539, USA Received 14 September 1999; accepted 12 November 1999 Communicated by P.R. Holland
Abstract In a three-level atom embedded in a photonic band gap structure, the spontaneous emission is studied both with and without rotating-wave approximation. The virtual-photon-localized field is found, which causes a greater population of the atom trapped in its upper states. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 42.50 Gy; 42.70.Qs Keywords: Photonic band gap structure; Spontaneous emission; Virtual-photon processes; Virtual-photon-localized field
Spontaneous emission is a fundamental concept in atomic physics. The spectrum of spontaneous emission can display vacuum Rabi splitting in a resonant high-Q cavity w1,2x. Furthermore, to control spontaneous emission well is important for the control of an atom in terms of its stability. A new generation of experiments reveals that spontaneous radiation from excited atoms can be greatly suppressed or enhanced by placing the atoms between mirrors or in a cavity w3–5x. During recent years, considerable attention has been paid to the behavior of the spontaneous emission of an atom embedded in a photonic band gap structure ŽPBGS. w6–11x. In PBGS the prohibition of
) 1
Corresponding author. E-mail: [email protected] Fax: q86-571-7951358; e-mail: [email protected]
light wave transmission can be achieved for some frequency ranges in all directions. Furthermore, it was showed that there are three modes in the spontaneous emission of a three-level atom that could be partly trapped in the upper levels, a localized field, a propagating field and a decaying field w7x. It may be noted that the above work was carried out within the rotating-wave approximation ŽRWA.. However, virtual-photon processes, which are introduced without the RWA, may have significant influence on the RWA results with the problems such as the accurate treatment of the photon absorption w12,13x, the atomic level’s Lamb shift w14,15x, and the interference between virtual-photon and real-photon processes in a micromaser cavity field w16x. In this Letter, the evolution of a three-level atom embedded in a photonic band gap structure will be studied without the RWA ŽWRWA. and then com-
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 8 0 9 - 9
Z. Liu et al.r Physics Letters A 264 (1999) 137–141
138
pared with the RWA results. The two upper levels are coupled to the lower level via the same field continuum. Some notable quantum effects induced by the virtual photons will be discussed. Consider a three-level atom with two upper levels < a1 :, < a 2 : and a lower level < b :, as shown in Fig. 1. The dipole vectors between < a1 : and < b : and between < a 2 : and < b : are parallel to each other. The dispersion relationship of the band gap material near the band gap edge v c can be approximated by w6,17,18x: 2
v k s vc q A Ž k y k 0 . ,
A s vcrk 02 .
Ž 1.
The Hamiltonian for the system is:
In the interaction picture, the Hamiltonian of the system can be written as: HˆI s i"
k
a k < a1 : ² b <
qe iŽ v a 2 byv k .t a k < a 2 : ² b < < : ²b< qe iŽ v a1 bqv k .t aq k a1 < : ² b < . q H.C. qe iŽ v a 2 bq v k .t aq k a2
Ž 3.
Assume that the atom is initially in its upper states, < a1 :, and < a2 :, and the field is in the vacuum state. The wave function of the system then has the form: < c Ž t . : s AŽ1. Ž t . < a1 :<0: q AŽ2. Ž t . < a2 :<0: q Ý b 1, k < b :<1: k k `
Ý Ž g kŽ1. a k < a1 : ² b <
qÝ
k
Ý
< :< : aŽ1. n , k Ž t . a1 n k
k ns1
< : ²b< qg kŽ2. a k < a 2 : ² b < q g kŽ1. aq k a1 < : ² b < . q H.C. . qg kŽ2. aq k a2
a1 b y v k .t
k
Hˆ s " v a1 < a1 : ² a1 < q " v a 2 < a 2 : ² a 2 < q " v b < b : ² b < q Ý " v k aq k a k q i"
Ý g k Ž e iŽ v
qaŽ2. n,k
Ž t . < a2 :< n:k q bnq1, k < b :< n q 1:k Ž 4.
Ž 2.
Ž . Ž . Here aq k a k is the creation annihilation operator for the light field. The atom-field coupling constants g kŽ1. and g kŽ2. are related to the decay coefficients of each upper level. We assume that g kŽ1. s g kŽ2. s g k when v a1 b and v a 2 b satisfy: v a1 b y vc s vc y v a 2 b s D and < D < < vc , and that g k s r Ž v . g k 0 with r Ž v . having a negative exponential function as shown in Fig. 1.
Ž2. Ž . Ž . with AŽ1. Ž0., AŽ2. Ž0. / 0, and aŽ1. n, k 0 , a n, k 0 , Ž . < Ž .: bnq 1, k 0 s 0. And c t obeys the Schrodinger ¨ equation: d i" < c Ž t . : s HˆI < c Ž t . : . Ž 5. dt So the first-order equations set is deduced as: d Ž1. A Ž t . s Ý g k e iŽ v a1 by v k .t b 1, k Ž t . , dt k
d dt d dt
AŽ2. Ž t . s Ý g k e iŽ v a 2 by v k .t b 1, k Ž t . , k
b 1, k Ž t . s yg k eyi Ž v a1 by v k .tAŽ1. Ž t . q'2 eyi Ž v a1 bq v k .t aŽ1. 2, k Ž t . qeyi Ž v a 2 byv k .tAŽ2. Ž t . q'2 eyi Ž v a 2 bq v k .t aŽ2. 2, k Ž t . ,
d dt
aŽ1. nk Ž t . s gk
'n q 1 e iŽ v
a1 b y v k .t
PPP ,
bnq1, k Ž t .
q'n e iŽ v a1 bq v k .t bny 1, k Ž t . , d Fig. 1. A three-level atom in a photonic band gap structure. The two upper levels, < a1 : and < a2 :, are symmetrically placed from the band gap edge by D.
dt
aŽ2. nk Ž t . s gk
'n q 1 e iŽ v
a 2 b y v k .t
bnq1, k Ž t .
q'n e iŽ v a 2 bq v k .t bny 1, k Ž t . ,
Z. Liu et al.r Physics Letters A 264 (1999) 137–141
d dt
139
bnq 1, k Ž t . s yg k w 'n q 1 eyi Ž v a1 by v k .t aŽ1. n,k Ž t . q'n q 2 eyi Ž v a1 bq v k .t aŽ1. nq 2, k Ž t . q'n q 1 eyi Ž v a 2 by v k .t aŽ2. n,k Ž t . q'n q 2 eyi Ž v a 2 bq v k .t aŽ2. nq 2, k Ž t . .
Ž 6. It can be noted that all these equations have the atom-field coupling constant g k on their right-hand sides. When k is far from k 0 , g k becomes so small because of a sharp decrease of r Ž v . that those modes will approach zero. Without considering the influence of the virtual-photon processes, i.e., in the RWA, one finds that the equation set Ž6. reduces to: d dt d dt d dt
AŽ1. Ž t . s Ý g k e iŽ v a1 by v k .t b 1, k Ž t . , k
AŽ2. Ž t . s Ý g k e iŽ v a 2 by v k .t b 1, k Ž t . , k
b 1, k Ž t . s yg k eyi Ž v a1 by v k .tAŽ1. Ž t . qeyi Ž v a 2 by v k .tAŽ2. Ž t . ,
Ž 7.
which is similar to the equations in Ref. w6x. Comparing the equation sets Ž6. and Ž7., one can find that it is the virtual-photon effects that introduce the addiŽ2. Ž n s 1,2, . . . ,`., tional terms aŽ1. n, k , a n, k and bnq1, k which are respectively the coefficients of < a1 :< n: k , < a2 :< n: k and < b :< n q 1: k . The virtual-photon processes are non-energy-conserving processes, and the rapidly emitted and absorbed energy in the form of two-photon creates a virtual-photon cloud around the field source, so only the contributions of the virtualphoton equation terms around the source of real-photon processes will have a noticeable effect. With the numerical results of equation set Ž6., the population of all the states of the atom, P Ž a1 ., P Ž a 2 . and P Ž b . can be derived. Generally, the effects of virtual-photon processes are quite weak, for the virtual-photon processes are perturbation to the real-photon processes. However, there are times that the effects of virtual-photon processes are not negligible. When the system parameters are set at specific values Žas seen in Fig. 2., the virtual-photon processes may alter or completely destroy the rele-
Fig. 2. The population evolution of the states of the atom. P Ž a1 ., P Ž a 2 . and P Ž b . are the populations in level < a1 :, < a 2 : and < b :, respectively. Ža. is with the RWA, and Žb. and Žc. are without the RWA. The atom is initially in < c Ž0.: s Ž1r'2 .Ž< a1 :q < a 2 :.. Parameters: g k s1.0, vc s 4.8p , D s 0.48p .
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Z. Liu et al.r Physics Letters A 264 (1999) 137–141
vant evolution of the populations of the atomic states within the RWA. In Fig. 2 these non-negligible effects of virtualphoton processes are shown. Fig. 2a shows the evolution of the populations of the atomic states with the RWA, where the band edge v c s 4.8p with D s 0.48p , the atom-field coupling constant at the edge g k 0 s 1.0 and the two upper atomic states are initialized as AŽ1. Ž0. s AŽ2. Ž0.. The RWA evolution curves of the populations of all the atomic states are equalamplitude oscillations with constant periods. Moreover, the curves for the two upper states share the same amplitude, which is nearly 0.5. The evolution for the lower state has the same period as that of the upper two levels, but its amplitude is twice as large. The three curves indicate that: Ž1. The two upper states are thoroughly coherent, the phase difference between them is zero, and their population evolution curves descend and ascend together between the maximum of nearly 0.5 and the minimum of 0; Ž2. The phase difference between the curves for the lower state and the two upper states is p . There is no energy exchange between the two upper states, yet the two upper states commute energy with the lower state, respectively and synchronously. With the same parameters as in Fig. 2a, the evolutions of the populations of the two upper states without the RWA are shown in Fig. 2b. They are both quasioscillations under WRWA. Different from the constant amplitudes in the RWA curves, the amplitudes of the WRWA curves are time-dependent functions. In the beginning in a very brief time period, the WRWA curves have the similar disciplinarians to the RWA curves. But very soon the two WRWA evolution curves separate from each other and their amplitudes diminish sharply; one of their equilibrium points ascends and the other descends. Then both of the population quasioscillations augment their amplitudes again and eventually stabilize at different values respectively with comparatively smaller amplitudes. The envelopes of the two quasioscillations have slightly quasi-harmonic oscillations. In addition, a similar phenomenon can also be seen in Fig. 2c where the population evolution of the lower state is shown. All these indicate that a virtual-photon-localized field exists in the spontaneous emission process of this atom. Comparing the WRWA curves with the relevant RWA curves, the
virtual-photon processes destroy the coherence between the population evolutions of the two upper states. Moreover, their oscillations are no longer simply harmonious. Above all, the virtual-photon processes induce the virtual-photon-localized field, which traps a portion of the atom’s population in the upper levels. In Fig. 2a, the upper states’ populations approach zero when the curves reach the bottoms. However, the curves in WRWA can never reach zero, for instance, the bottoms of P Ž a2 . are always higher than 0.25; the lower state’s population evolution curve P Ž b . under WRWA in Fig. 2c is also a quasioscillation with an equilibrium point around 0.45 when it is stabilized. The amplitude is greater than 0.5 and the population cannot go beyond the magnitude of 0.7. Thus, a portion of population of the atom is trapped in the upper states under WRWA. The population trapping is similar to the phenomenon of a localized field under RWA in Refs. w6,7x However, the virtual-photon-localized field is quite different from that localized field. Firstly, the oscillation of the WRWA population curves cannot decay, which both of the two upper states oscillations’ amplitudes are greater than 0.2 when they are stabilized. The fluctuations of RWA population curves will dwindle sharply as time lapses and retain a certain value eventually. Secondly, the oscillation period in WRWA curves maintain as a constant while the periods in RWA curves will be greater and
Fig. 3. The population evolution of the upper state < a1 :, with the RWA Žsolid line. and without the RWA Ždotted line.. The atom is initially in < c Ž0.: s < a2 :. Parameters: g k s1.0, vc s6.5p , D s 0.65p .
Z. Liu et al.r Physics Letters A 264 (1999) 137–141
greater. Furthermore, the oscillation periods in WRWA curves are quite shorter than those RWA ones, namely the WRWA curves are rapid equalfrequency oscillations. Lastly, the ratio of the populations in the two upper levels, AŽ1. Ž`.rAŽ2. Ž`., is obviously greater in WRWA than that in RWA. The localized field mentioned in Ref. w7x is generated by the quantum interference effects of real-photon processes. But the virtual-photon-localized field is induced by the intensive interference between virtual photons and real photons. The virtual-photon processes are rapid oscillation processes, which result in rapid oscillation interference between virtualphoton processes and the real-photon processes. The virtual-photon-localized field is generally too weak to be found, because the rapidly emitted and absorbed virtual photons only cause a perturbation to the real-photon processes. For instance, when g k 0 s 1.0, the band edge v c s 100p and D s 10p , the atomic state is initially AŽ1. Ž0. s AŽ2. Ž0.; the evolution disciplinarians of atom states’ populations are the same as those described in the Ref. w7x. As the band edge vc decreases, the influence of the virtual-photon processes will increase, for example, in Fig. 3, in which vc s 6.5p and the atomic state is initialized as AŽ1. Ž0. s 0.0 and AŽ2. Ž0. s 1.0; the WRWA curves are slight modifications of the relevant RWA curves. However, the influence of the virtual-photon processes will not increase all the time with the decrease of the band edge. On the contrary, it will dwindle when the band edge gets quite weak, for instance, when vc s 1.5p the WRWA curves and the RWA curves nearly overlap each other. This indicates that the interference between the virtual-photon and the real-photon is maximized only under some specific parameter sets. In this Letter, we have studied the evolution behavior of the spontaneous emission of an atom that was embedded in a photon band gap structure, both with RWA and WRWA. Although the effects of the
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virtual-photon processes are generally perturbations to the evolution curve in RWA, the effect of the virtual-photon processes can be quite significant in certain cases. The virtual-photon-localized field was discussed in the spontaneous emission of an atom under certain series of system parameters, which do cause a greater population of the atom trapped in the upper states. Acknowledgements This work is supported by the National Natural Science Foundation of China, No. 19674048 & No.29676037, and the Zhejiang Province Natural Science Foundation, No. 198023. References w1x M.G. Raizen, R.J. Thompson, R.J. Brecha, H.J. Kimble, H.J. Carmichael, Phys. Rev. Lett. 63 Ž1989. 240. w2x R.J. Thompson, G. Rempe, H.J. Kimble, Phys. Rev. Lett. 68 Ž1990. 1132. w3x E.M. Purcell, Phys. Rev. 69 Ž1946. 681. w4x D. Kleppner, Phys. Rev. Lett. 47 Ž1981. 232. w5x S. Harochee, D. Kleppner, Phys. Today 42 Ž1989. 24, and references therein. w6x S. John, T. Quang, Phys. Rev. A 50 Ž1994. 1764. w7x S.Y. Zhu, H. Chen, H. Huang, Phys. Rev. Lett. 79 Ž1997. 205. w8x S. John, Phys. Rev. Lett. 53 Ž1984. 2169. w9x P.W. Anderson, Phil. Mag. B 52 Ž1985. 505. w10x E. Yablonovitch, Phys. Rev. Lett. 58 Ž1987. 2059. w11x S. John, Phys. Rev. Lett. 58 Ž1987. 2486. w12x G. Compagno, G.M. Palma, R. Passante, F. Perisco, Europhys. Lett. 9, 215. w13x G. Compagno, G.M. Palma, Phys. Rev. A 37 Ž1988. 2979. w14x X.Y. Huang, L.S. Peng, Phys. Scripta T 21 Ž1988. 100. w15x G. Compagno, R. Passante, F. Persico, Phys. Lett. A 98 Ž1983. 253. w16x L. Zeng, Z.D. Liu, Y. Lin, S.Y. Zhu, Phys. Lett. A 246 Ž1998. 43. w17x S. John, J. Wang, Phys. Rev. Lett. 64 Ž1990. 2418. w18x S. John, T. Quang, Phys. Rev. B 43 Ž1991. 12772.