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The spontaneous emission from a V-type three-level atom embedded in an isotropic photonic crystal with dynamic photonic band edge is studied. We consider the situation where the atom interacts with all possible radiation modes, and calculate numerically the evolution of atomic population without using Markov approximation. The calculation method can be used in related researches. In the present paper, we mainly discuss the effects of modulation parameters and the quantum interference on spontaneous emission when the band edge is modulated with step function or triangle function. We hope that the results can contribute to the applications in the dynamic photonic crystal environment in controlling the spontaneous emission via the quantum interference. The results show that in the step-modulated situation, the number of the photon-atom bound dressed states after the modulation has happened depends on atomic transition frequencies and the band edge frequency at that time, and is identical to the one in the unmodulated situation with the same parameters. The long-time evolution of the atomic population is affected by the time when the modulation happens. Depending on the system initial state, after the modulation has happened, the quantum interference can weaken the probability amplitude components corresponding to the photon-atom bound dressed states, and cause the upper-level population to decay quickly from a great value to a value near zero; or on the contrary, it can strengthen the bound dressed states, and make the upper levels retain a high population. In the modulated situation with trigonometric functions, after long enough time, the total upper-level population presents a decaying quasi-periodic oscillation behaviour. And the evolution of the total upper-level population tends to synchronize with the modulation, so the frequency of the quasi-periodic oscillation is approximately equal to the modulation frequency. But, the quantum interference can destroy the synchronization under some conditions. The decay rate of the total upper-level population is affected by the modulation frequency, and also by the initial state of the system and the angle between two dipole moment because of the quantum interference.
[1] Scully M O, Zubairy M S 1997 Quantum Optics（Cambridge:Cambridge University Press) Chapter 7
[2] Fleischhauer M, Imamoglu A, Marangos J 2005 Rev. Mod. Phys. 77 633
[3] Zhu S Y, Chan R C F, Lee C P 1995 Phys. Rev. A 52 710
[4] Paspalakis E, Kylstra N J, Knight P L 1999 Phys. Rev. A 60 R33
[5] Angelakis D G, Paspalakis E, Knight P L 2001 Phys. Rev. A 64 013801
[6] Du C G, Hu Z F, Hou C F, Li S Q 2002 Chin. Phys. Lett. 19 338
[7] Yang Y P, Zhu S Y, Zubairy M S 1999 Opt. Commun. 166 79
[8] Yang Y P, Zhu S Y 2000 Phys. Rev. A 61 043809
[9] Yang Y P, Zhu S Y 2000 J. Mod. Opt. 47 1513
[10] Zhu S Y, Chen H, Huang H 1997 Phys. Rev. Lett. 79 205
[11] Yablonovitch E 1987 Phys. Rev. Lett. 58 2059
[12] John S 1987 Phys. Rev. Lett. 58 2486
[13] John S, Wang J 1990 Phys. Rev. Lett. 64 2418
[14] John S, Wang J 1991 Phys. Rev. B 43 12772
[15] John S, Quang T 1994 Phys. Rev. A 50 1764
[16] Yang Y P, Lin Z X, Xie S Y, Feng W G, Wu X 1999 Acta Phys. Sin. 48 603 （in Chinese)[羊亚平, 林志新, 谢双媛, 冯伟国, 吴翔1999物理学报48 603]
[17] Xie S Y, Yang Y P, Wu X 2001 Eur. Phys. J. D 13 129
[18] Quang T, Woldeyohannes M, John S, Agarwal G S 1997 Phys. Rev. Lett. 79 5238
[19] Wang X H, Gu B Y 2005 Appl. Phys. Lett. 86 051103
[20] Zhang L F, Huang J P 2010 Chin. Phys. B 19 024213
[21] Liu S Y, Du J J, Lin Z F, Wu R X, Chui S T 2008 Phys. Rev. B 78 155101
[22] Manzanares-Martinez J, Ramos-Mendieta F, Halevi P 2005 Phys. Rev. B 72 035336
[23] Hu X Y, Zhang Q, Liu Y H, Cheng B Y, Zhang D Z 2003 Appl. Phys. Lett. 83 2518
[24] Leonard S W, van Driel H M, Schilling J, Wehrspohn R B 2002 Phys. Rev. B 66 161102
[25] Jia F, Xie S Y, Yang Y P 2006 Acta Phys. Sin. 55 5835 （in Chinese)[贾飞, 谢双媛, 羊亚平2006物理学报55 5835]
[26] Pisipati U, Almakrami I M, Joshi A, Serna J D 2012 Am. J. Phys. 80 612
[27] Yang Y P, Xu J P, Li G X, Chen H 2004 Phys. Rev. A 69 053406
[28] Law C K, Zhu S Y, Zubairy M S 1995 Phys. Rev. A 52 4095
[29] Linington I E, Garraway B M 2006 J. Phys. B:At. Mol. Opt. Phys. 39 3383
[30] Linington I E, Garraway B M 2008 Phys. Rev. A 77 033831
[31] Kofman A G, Kurizki G 2001 Phys. Rev. Lett. 87 270405
[32] Linz P 1985 Analytical and Numerical Methods for Volterra Equations（Philadelphia:Society for Industrial and Applied Mathematics) Chapter 7
[33] Xia H R, Ye C Y, Zhu S Y 1996 Phys. Rev. Lett. 77 1032
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[1] Scully M O, Zubairy M S 1997 Quantum Optics（Cambridge:Cambridge University Press) Chapter 7
[2] Fleischhauer M, Imamoglu A, Marangos J 2005 Rev. Mod. Phys. 77 633
[3] Zhu S Y, Chan R C F, Lee C P 1995 Phys. Rev. A 52 710
[4] Paspalakis E, Kylstra N J, Knight P L 1999 Phys. Rev. A 60 R33
[5] Angelakis D G, Paspalakis E, Knight P L 2001 Phys. Rev. A 64 013801
[6] Du C G, Hu Z F, Hou C F, Li S Q 2002 Chin. Phys. Lett. 19 338
[7] Yang Y P, Zhu S Y, Zubairy M S 1999 Opt. Commun. 166 79
[8] Yang Y P, Zhu S Y 2000 Phys. Rev. A 61 043809
[9] Yang Y P, Zhu S Y 2000 J. Mod. Opt. 47 1513
[10] Zhu S Y, Chen H, Huang H 1997 Phys. Rev. Lett. 79 205
[11] Yablonovitch E 1987 Phys. Rev. Lett. 58 2059
[12] John S 1987 Phys. Rev. Lett. 58 2486
[13] John S, Wang J 1990 Phys. Rev. Lett. 64 2418
[14] John S, Wang J 1991 Phys. Rev. B 43 12772
[15] John S, Quang T 1994 Phys. Rev. A 50 1764
[16] Yang Y P, Lin Z X, Xie S Y, Feng W G, Wu X 1999 Acta Phys. Sin. 48 603 （in Chinese)[羊亚平, 林志新, 谢双媛, 冯伟国, 吴翔1999物理学报48 603]
[17] Xie S Y, Yang Y P, Wu X 2001 Eur. Phys. J. D 13 129
[18] Quang T, Woldeyohannes M, John S, Agarwal G S 1997 Phys. Rev. Lett. 79 5238
[19] Wang X H, Gu B Y 2005 Appl. Phys. Lett. 86 051103
[20] Zhang L F, Huang J P 2010 Chin. Phys. B 19 024213
[21] Liu S Y, Du J J, Lin Z F, Wu R X, Chui S T 2008 Phys. Rev. B 78 155101
[22] Manzanares-Martinez J, Ramos-Mendieta F, Halevi P 2005 Phys. Rev. B 72 035336
[23] Hu X Y, Zhang Q, Liu Y H, Cheng B Y, Zhang D Z 2003 Appl. Phys. Lett. 83 2518
[24] Leonard S W, van Driel H M, Schilling J, Wehrspohn R B 2002 Phys. Rev. B 66 161102
[25] Jia F, Xie S Y, Yang Y P 2006 Acta Phys. Sin. 55 5835 （in Chinese)[贾飞, 谢双媛, 羊亚平2006物理学报55 5835]
[26] Pisipati U, Almakrami I M, Joshi A, Serna J D 2012 Am. J. Phys. 80 612
[27] Yang Y P, Xu J P, Li G X, Chen H 2004 Phys. Rev. A 69 053406
[28] Law C K, Zhu S Y, Zubairy M S 1995 Phys. Rev. A 52 4095
[29] Linington I E, Garraway B M 2006 J. Phys. B:At. Mol. Opt. Phys. 39 3383
[30] Linington I E, Garraway B M 2008 Phys. Rev. A 77 033831
[31] Kofman A G, Kurizki G 2001 Phys. Rev. Lett. 87 270405
[32] Linz P 1985 Analytical and Numerical Methods for Volterra Equations（Philadelphia:Society for Industrial and Applied Mathematics) Chapter 7
[33] Xia H R, Ye C Y, Zhu S Y 1996 Phys. Rev. Lett. 77 1032
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