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Influences of pure dephasing and incoherent pumping in a coupled quantum dot–cavity system and its application
Physica B 406 (2011) 3805–3809
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Influences of pure dephasing and incoherent pumping in a coupled quantum dot–cavity system and its application Gui Cao, Zhongyuan Yu n, Yumin Liu, Wenjie Yao, Xia Xin State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), P.O. Box 72, Beijing 100876, China
a r t i c l e i n f o
abstract
Article history: Received 2 March 2011 Received in revised form 22 June 2011 Accepted 30 June 2011 Available online 6 July 2011
The influences of pure dephasing and incoherent pumping on the spectra from a coupled target quantum dot–cavity system are systematically investigated. We find that incoherent pumping has a similar effect on the spectra as pure dephasing does. The results of off-resonance study indicate that a frequency-stabilized single photon source can be achieved by controlling either pure dephasing rate or incoherent pumping rate. Furthermore, we investigate the strategies to optimize the efficiency of a single photon source. The obtained results offer appealing perspectives in developing an efficient frequency-stabilized solid-state single photon source. & 2011 Elsevier B.V. All rights reserved.
Keywords: Pure dephasing Incoherent pumping Single photon source
1. Introduction Motivated by applications [1–4] of cavity quantum electrodynamics (CQED) in quantum information science, the interaction of solid-state emitters with cavity has been studied experimentally [5–11] and theoretically [12–17] in recent years. For example, Cui and Raymer [12] have pointed out that the emission spectra peaks were broadened by increasing pure dephasing rate in quantum dot–cavity systems, and an intensity shift of emission spectra was observed in several experiments [6–8], which were perfectly explained in theory [14]. Using a quantum master equation (QME), which considers only pure dephasing process, the influence of pure dephasing on the spontaneously-emitted spectra was analyzed and computed in Refs. [16,18]. In the meantime, an effective coupling rate [18] has been derived to investigate the dynamics of atom–cavity system. On the other hand, two typical experiments investigate the pump-induced ¨ spectra, and have been reported by Munch et al. [19] and Laucht et al. [20]. Then Yao et al. [17] presented a useful QME theory, taking into account of pure dephasing, incoherent pumping and stimulated emission. However, in Ref. [17], the influence of incoherent pumping on the spectra was only analyzed in onresonance situation. In addition, there are still some interesting questions that have not been treated, such as the influence of incoherent pumping on the spectra in the case of off-resonance, and the role of pure dephasing on the pump-induced spectra.
n
Corresponding author. Tel.: þ86 010 62281490. E-mail addresses: [email protected] (G. Cao), [email protected] (Z. Yu). 0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.06.072
In this paper, we study the coupling between a target quantum dot (QD) and a leaky cavity based on a QME model provided by Yao et al. [17], and the parameters used for simulations come from Refs. [6,8,21]. Firstly, we investigate the roles of pure dephasing and incoherent pumping on the spectra. Then the influence of pure dephasing on the pump-induced spectra is analyzed. In addition, we generalize the effective coupling rate reported in Refs. [16,18], where the coupled system undergoes no incoherent pumping. Finally, the efficiency of emission process in a single photon source is obtained.
2. Influence of pure dephasing and incoherent pumping 2.1. QD–cavity system model To investigate the spectra from a coupled target QD–cavity system, we adopt the model of Yao et al. [17], represented in Fig. 1(a), where the coupling strength between a target QD and a leaky cavity is g; ox is resonance frequency of the target QD and oc is eigenfrequency of the leaky cavity, the detuning of QD–cavity is denoted by Dcx ¼ oc ox; the leaky cavity’s decay rate is Gc, the radiative decay rate and pure dephasing rate of the target QD exciton are Gx and Gnx ; Px and Pc are the driving incoherent pumping rates for target exciton and cavity, respectively, and Pc comes from other QDs coupled to the cavity offresonantly [21]. Taking into account of pure dephasing, incoherent pumping and stimulated emission, the evolution of this coupled QD–cavity system can be described by a quantum dissipative master equation as follows, which employs the thermal bath approximation and
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G. Cao et al. / Physica B 406 (2011) 3805–3809
Born–Markov approximation. i r_ ¼ H, r þ LðrÞ _
ð1Þ
where H ¼ _ox s þ s þ _oa ay a þi_gðay s s þ aÞ is the total Hamiltonian [22] of the coupled system, and the superoperator L(r) is given by the Lindblad operator [22] as LðrÞ ¼
Px Gx þ Px ð2s þ rs s s þ rrs s þ Þ þ ð2s rs þ s þ s 2 2 Pc rrs þ s Þ þ ð2ay raaay rraay Þ 2 Gc þ Pc Gn y ð2ara ay array aÞ þ x ðsz rsz rÞ þ ð2Þ 2 4
Here, ay and a are the creation and annihilation operators in cavitymode, s þ and s are the Pauli operators of target exciton. The spectra of Scav(o) and Srad(o) are emitted by cavity-mode and radiation-mode, respectively. According to the Wiener– Khintchine theorem [23], they can be written as Z 1 Gc limt-1 Re /ay ðtÞaðt þ tÞSeiot dt Scav ðoÞ ¼ p Z0 1 Gx limt-1 Re /s þ ðtÞs ðt þ tÞSeiot dt ð3Þ Srad ðoÞ ¼
p
0
The exact spectra formalism was obtained in Ref. [24] as follows, applying one photon-correlation approximation/szaS¼ /aS and fermion statistics [s ,s þ ] þ ¼1. Gc i/ay aSss DðoÞ ig/ay s Sss Scav ðoÞ ¼ Re þ p CðoÞDðoÞg 2 CðoÞDðoÞg 2 Gx i/ay aSss CðoÞ ig/s þ aSss Re þ ð4Þ Srad ðoÞ ¼ p CðoÞDðoÞg 2 CðoÞDðoÞg 2 where CðoÞ ¼ ooc þ 2i Gc and DðoÞ ¼ oox þ 2i ð2Px þ Gx þ Gnx Þ. The total spectrum of the QD–cavity system can be expressed as Stot ðr, oÞ ¼ Frad ðrÞSrad ðoÞ þFcav ðrÞScav ðoÞ
ð5Þ
As is shown in Refs. [13–15], cavity-mode emission is the main contribution to total spectrum. Thus here, we adopt Fcav ðrÞ=Frad ðrÞ ¼ 2=1. From Eq. (1), the populations’ steady-state solutions are derived to be h 2 i g 2 GðPx þ Pc Þ þ Pc ð2Px þ Gx Þ G4 þ D2cx h 2 i /ay aSss ¼ ð6Þ g 2 Gð2Px þ Gx þ Gc Þ þ Gc ð2Px þ Gx Þ G4 þ D2cx /ay s Sss ¼
g
/s þ s Sss ¼
G 2
iDcx ðGc þ Gx þ 2Px Þ/ay aSss þPc þPx 2 G þ D2 þ ðG þ 2P Þ x x cx 4
Gc /ay aSss þ Pc þ Px Gx þ 2Px
ð7Þ
ð8Þ
where G ¼ 2Px þ Gx þ Gnx þ Gc . Thus, the influences of pure dephasing and incoherent pumping on the spectra can be investigated numerically. Fig. 1. (a) Coupled target QD–cavity system, undergoing pure dephasing, incoherent pumping and stimulated emission. (b) The model for quantum energy exchanging between target QD and cavity. (c) The decay from a coupled QD–cavity system in QD point of view.
Table 1 The parameters used for simulations. g
Cx
Cc
Cnx
Px
Pc
0.045 meV
0.005 meV
0.085 meV
0.001 meV
0.01g
0.35g
2.2. Spectra simulations In this paper, we investigate the spectra in situations of onresonance (Dcx ¼ 0) and off-resonance (Dcx ¼1 meV). The parameters used for simulations are listed in Table 1 [6,8,21]. To highlight the role of pure dephasing on the spectra, we fix all other parameters, which are listed in Table 1, and change pure dephasing rate Gnx only, the value of Gnx ranges from 0.001 to 0.2 meV. Firstly, we consider the situation of on-resonance, shown in Fig. 2(a), a vacuum Rabi doublet can be observed when the value
Fig. 2. The influence of pure dephasing on the spectra: (a) on-resonance and (b) off-resonance.
G. Cao et al. / Physica B 406 (2011) 3805–3809
of Gnx is small, and with the value of Gnx increasing, there is an obvious trend that the peaks are broadened and the double peaks are gradually becoming a single peak. On the contrary, we consider the off-resonance situation. As shown in Fig. 2(b), when Gnx ¼ 0:001meV, the peaks of spectra emitted by cavity-mode and radiation-mode are separate. As a result, photons emitted by the coupled QD–cavity system have two different frequencies. However, as the value of Gnx increases, the spectrum of coupled QD– cavity main emitted at the frequency of cavity. When Gnx ¼ 0:2 meV, the spectrum from coupled QD–cavity emits at the cavity frequency. All this phenomenon of emission spectra by changing pure dephasing rate has already been shown in Ref. [16]. Correspondingly, to highlight the role of incoherent pumping on the spectra, all other parameters are invariable and listed in Table 1, while the value of incoherent pumping Px is changed from 0.01g to 1.36g. As show in Fig. 3(a) and (b), with Px increasing, similar trends of spectra are obtained as in Fig. 2(a) and (b). The phenomenon of on-resonance situation has already been shown in Ref. [17], which is perfectly corresponding to the experimental
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data [19]. As show in Fig. 3(b), we consider the situation of offresonance, when the value of incoherent pumping Px is small, there are two separate peaks of spectra emitted by cavity-mode and radiation-mode. However, the spectrum of coupled QD– cavity main emitted at cavity frequency with the value of Px increasing. When Px ¼1.36g, the spectrum from coupled QD– cavity emits at the frequency of cavity totally. Comparing Figs. 2 and 3, we can easily come to a conclusion that pure dephasing and incoherent pumping have similar influence on the spectra. In the situation of on-resonance, the splitting of the peaks is blurred and the peaks are broadened with Gnx or Px increasing, while in the off-resonance situation, the spectrum from coupled QD–cavity can emit at the frequency of cavity by changing Gnx or Px. Now, we investigate the influence of pure dephasing on the pump-induced spectra. Here, both pure dephasing and incoherent pumping are changed, while the other parameters are invariable and given in Table 1. As presented in Fig. 4, similar trends of spectra are observed as those plotted in Figs. 2 and 3. More
Fig. 3. The influence of incoherent pumping on the spectra: (a) on-resonance and (b) off-resonance.
Fig. 4. The influence of pure dephasing on the pump-induced spectra: (a) on-resonance and (b) off-resonance.
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G. Cao et al. / Physica B 406 (2011) 3805–3809
interestingly, it is obvious that the influence on spectra by changing both pure dephasing and incoherent pumping is greater than that by changing any of them independently. In addition, Fig. 4(b) indicates that when Px ¼1.0g, Gnx ¼ 0:1 meV, the spectrum from coupled QD–cavity emits at the frequency of cavity perfectly.
d/s þ s S ¼ ðGx þ2Px Þ/s þ s SR/s þ s S þR/ay aS þPx dt ð12Þ We derive the effective coupling rate R as R¼
Gg 2 2
ðG
=4 þ D2cx Þ
¼
4g 2 1 2Px þ Gx þ Gnx þ Gc 1 þ ð2Dcx =2Px þ Gx þ Gnx þ Gc Þ2
3. Application of QD–cavity system
ð13Þ
Based on above analysis and discussion, the phenomenon of off-resonance situation is appealing, because it provides a potential and easy way to realize a solid-state single photon source with photons emitted at the cavity frequency. As we know, efficiency plays an important role in a photonic device. Next, we pay more attention to the efficiency of emission process in a single photon source. 3.1. Efficiency model
Pcav ¼
RGc =ðR þ Gc Þ Gc ¼ Gx þ RGc =ðR þ Gc Þ ðGx þ Gc Þ þ GxRGc
ð14Þ
3.2. Efficiency simulations
_ ¼ TrðroÞ _ ¼ From Eq. (1), and the operational rule of /oS Trðr_ oÞ ¼ Trðor_ Þ, a set of equations are derived, which describe the evolution of populations d/ay aS ¼ Gc /ay aS þ gð/s þ aS þ/ay s SÞ þ Pc dt
The cavity seems as a further energy loss channel at rate of RGc =ðR þ Gc Þ, which is represented in Fig. 1(c). As represented in Ref. [18], we can easily derive the efficiency of emission process in a single photon source, which is given by
ð9Þ
d/s þ s S ¼ Gx /s þ s S2Px /s þ s Sx gð/s þ aS þ/ay s SÞ þ Px dt ð10Þ d/s þ aS 2Px þ Gx þ Gc þ Gnx ¼ iDcx /s þ aS /s þ aS þ gð/s þ s S/ay aSÞ dt 2
ð11Þ As stated in Ref. [18], an effective coupling rate R has been presented in atom–cavity system, undergoing pure dephasing only. Here, we generalize the coupling rate R in QD–cavity system, taking incoherent pumping and stimulated emission into additional account. As shown in Fig. 1(b), the system is modeled by a QD box and a cavity box, which undergoes a process of quantum energy escapes into environment at rates of Gx and Gc, respectively. At the same time, the exchanging rate of quantum energy between the two boxes is R. The coupled dynamical Eqs. (9) and (10) can be described as follows:
The efficiency Pcav is illustrated as a function of pure dephasing rate in Fig. 5(a) and (b). Here, the parameters used for simulations are listed in Table 2. In the case of on-resonance (Dcx ¼0), the efficiency is larger and at a trend of dropping. It drives from that that most spectra of QD and cavity are overlapping, and the overlapping decreases with increasing pure dephasing rate. On the contrary, in offresonance (Dcx ¼1 meV) case, the efficiency firstly rises and then drops with enhancing the pure dephasing rate, the maximal value is reached for 2Px þ Gx þ Gnx þ Gc ¼ 2Dcx , and the effective coupling rate R ¼ g 2 =Dcx . In Fig. 5(a), all other parameters are invariable and listed in Table 2, the roles of g and Px on the efficiency Pcav are illustrated by changing the values of g and Px, respectively. As shown in the figure, the influence of changing incoherent pumping rate on the efficiency Pcav is negligible. However, the efficiency Pcav can be greatly improved, by choosing a bigger coupling strength g. The roles of decay rates (both Gc and Gx) on the efficiency Pcav are shown in Fig. 5(b). In the similar method, all other parameters are Table 2 The parameters used for efficiency simulations. g
Cx
Cc
Cnx
Px
Pc
0.045 meV
0.005 meV
0.085 meV
0.1 meV
1.0g
0.35g
y
d/a aS ¼ Gc /ay aSR/ay aS þR/s þ s S þPc dt
Fig. 5. Efficiency Pcav as a function of pure dephasing rate. Dashed line: on-resonance; solid line: off-resonance. Red plots: all parameters are given in Table 2. (a) Blue plots: Px ¼ 1.36g. Green plots: g¼ 0.085 meV. (b) Blue plots: Gc ¼0.105 meV. Green plots: Gx ¼ 0.001 meV. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
G. Cao et al. / Physica B 406 (2011) 3805–3809
fixed and given in Table 2, while Gx and Gc are changed. As appeared in the figure, the changing of cavity decay rate has little effect on the efficiency Pcav, while the radiative decay rate plays an important role in enhancing it, and a higher value for Pcav could be achieved by a lower radiative decay rate.
4. Conclusions In this paper, we investigate the spectra of a coupled target QD–cavity system through a quantum master equation, which considers pure dephasing, incoherent pumping and stimulated emission. We find that pure dephasing and incoherent pumping have similar effect on the spectra. The results indicate that a frequency-stabilized solid-state single photon source can be got by controlling either pure dephasing rate or incoherent pumping rate. Then we investigate the strategies to enhance the efficiency of a single photon source, as a consequence, a higher value for the efficiency could be achieved by a bigger coupling strength and a lower radiative decay rate, these results are useful for developing an efficient solid-state single photon source. In addition, to obtain an ‘‘on-demand’’ single photon source, we should further consider the indistinguishabilities of photons and mulitiphoton probability in the QD–cavity system. Also, the inhomogeneous broadening of quantum dot should be taken into account in the future.
Acknowledgments This work is supported by the National High Technology Research and Development Program of China (Grant no. 2009AA03Z405) and the National Natural Science Foundation of China (Grant nos. 60908028, 60971068 and 10979065).
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References [1] M.O. Scully, M.S. Zubairy, Quantum Optics, first ed., Cambridge University Press, London, 1997, p. 220. [2] A. Boca, A.D. Boozer, J.R. Buck, H.J. Kimble, Nature 425 (2003) 268. [3] J. McKeever, A. Boca, A.D. Boozer, R. Miller, J.R. Buck, A. Kuzmich, H.J. Kimble, Science 303 (2004) 1992. [4] O. Astafiev, K. Inomata, A.O. Niskanen, T. Yamamoto, Yu.A. Pashkin, Y. Nakamura, J.S. Tsai, Nature 449 (2007) 588. [5] E. Moreau, I. Robert, J.M. Ge´rard, I. Abram, L. Manin, V. Thierry-Mieg, Appl. Phys. Lett. 79 (2001) 2865. [6] J.P. Reithmaier, G. Sek, A. Loffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L.V. Keldysh, V.D. Kulakovskii, T.L. Reinecke, A. Forchel, Nature 432 (2004) 197. [7] T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H.M. Gibbs, G. Rupper, C. Ell, O.B. Shchekin, D.G. Deppe, Nature 432 (2004) 200. [8] K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atature, S. Gilde, S. Falt, E.L. Hu, A. Lmamoglu, Nature 445 (2007) 896. [9] D. Press, S. Gotzinger, S. Reitzenstein, C. Hofmann, A. Loffler, M. Kamp, A. Forchel, Y. Yamamoto, Phys. Rev. Lett. 98 (2007) 117402. [10] J.M. Fink, M. Goppl, M. Baur, R. Bianchetti, P.J. Leek, A. Blais, A. Wallraff, Nature 454 (2008) 315. [11] D. Dalacu, K. Mnaymneh, V. Sazonova, P.J. Poole, G.C. Aers, J. Lapointe, R. Cheriton, A.J. SpringThorpe, R. Williams, Phys. Rev. B 82 (2010) 033301. [12] G. Cui, M.G. Raymer, Phys. Rev. A 73 (2006) 053807. [13] A. Auffeves, B. Besga, J.M. Gerard, J.P. Poizat, Phys. Rev. A 77 (2008) 063833. [14] A. Naesby, T. Suhr, P.T. Kristensen, J. Mørk, Phys. Rev. A 78 (2008) 045802. [15] M. Yamaguchi, T. Asano, S. Noda, Opt. Express 16 (2008) 18067. [16] A. Auffeves, Jean-Michel Gerard, Jean-Philippe Poizat, Phys. Rev. A 79 (2009) 053838. [17] P. Yao, P.K. Pathak, E. Illes, S. Hughes, Phys. Rev. B 81 (2010) 033309. [18] A. Auffeves, D. Gerace, J.M. Gerard, M.F. Santos, L.C. Andreani, J.P. Poizat, Phys. Rev. B 81 (2010) 245419. ¨ ¨ [19] S. Munch, S. Reitzenstein, P. Franeck, A. Loffler, T. Heindel, S. Hofling, L. Worschech, A. Forchel, Opt. Express 17 (2009) 12821. ¨ [20] A. Laucht, N. Hauke, J.M. Villas-Boˆ as, F. Hofbauer, M. Kaniber, G. Bohm, J.J. Finley, Phys. Rev. Lett. 103 (2009) 087405. [21] F.P. Laussy, E.D. Valle, C. Tejedor, Phys. Rev. Lett. 101 (2008) 083601. [22] J.C. Garrison, R.Y. Chiao, Quantum Optics, first ed., Oxford University Press, New York, 2008, p. 420. [23] M.O Scully, M.S. Zubairy, Quantum Optics, first ed., Cambridge University Press, London, 1997, p. 291. [24] T. Tawara, H. Kamada, T. Tanabe, T. Sogawa, H. Okamoto, P. Yao, P.K. Pathak, S. Hughes, Opt. Express 18 (2010) 2719.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Influences of pure dephasing and incoherent pumping in a coupled quantum dot–cavity system and its application Gui Cao, Zhongyuan Yu n, Yumin Liu, Wenjie Yao, Xia Xin State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), P.O. Box 72, Beijing 100876, China
a r t i c l e i n f o
abstract
Article history: Received 2 March 2011 Received in revised form 22 June 2011 Accepted 30 June 2011 Available online 6 July 2011
The influences of pure dephasing and incoherent pumping on the spectra from a coupled target quantum dot–cavity system are systematically investigated. We find that incoherent pumping has a similar effect on the spectra as pure dephasing does. The results of off-resonance study indicate that a frequency-stabilized single photon source can be achieved by controlling either pure dephasing rate or incoherent pumping rate. Furthermore, we investigate the strategies to optimize the efficiency of a single photon source. The obtained results offer appealing perspectives in developing an efficient frequency-stabilized solid-state single photon source. & 2011 Elsevier B.V. All rights reserved.
Keywords: Pure dephasing Incoherent pumping Single photon source
1. Introduction Motivated by applications [1–4] of cavity quantum electrodynamics (CQED) in quantum information science, the interaction of solid-state emitters with cavity has been studied experimentally [5–11] and theoretically [12–17] in recent years. For example, Cui and Raymer [12] have pointed out that the emission spectra peaks were broadened by increasing pure dephasing rate in quantum dot–cavity systems, and an intensity shift of emission spectra was observed in several experiments [6–8], which were perfectly explained in theory [14]. Using a quantum master equation (QME), which considers only pure dephasing process, the influence of pure dephasing on the spontaneously-emitted spectra was analyzed and computed in Refs. [16,18]. In the meantime, an effective coupling rate [18] has been derived to investigate the dynamics of atom–cavity system. On the other hand, two typical experiments investigate the pump-induced ¨ spectra, and have been reported by Munch et al. [19] and Laucht et al. [20]. Then Yao et al. [17] presented a useful QME theory, taking into account of pure dephasing, incoherent pumping and stimulated emission. However, in Ref. [17], the influence of incoherent pumping on the spectra was only analyzed in onresonance situation. In addition, there are still some interesting questions that have not been treated, such as the influence of incoherent pumping on the spectra in the case of off-resonance, and the role of pure dephasing on the pump-induced spectra.
n
Corresponding author. Tel.: þ86 010 62281490. E-mail addresses: [email protected] (G. Cao), [email protected] (Z. Yu). 0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.06.072
In this paper, we study the coupling between a target quantum dot (QD) and a leaky cavity based on a QME model provided by Yao et al. [17], and the parameters used for simulations come from Refs. [6,8,21]. Firstly, we investigate the roles of pure dephasing and incoherent pumping on the spectra. Then the influence of pure dephasing on the pump-induced spectra is analyzed. In addition, we generalize the effective coupling rate reported in Refs. [16,18], where the coupled system undergoes no incoherent pumping. Finally, the efficiency of emission process in a single photon source is obtained.
2. Influence of pure dephasing and incoherent pumping 2.1. QD–cavity system model To investigate the spectra from a coupled target QD–cavity system, we adopt the model of Yao et al. [17], represented in Fig. 1(a), where the coupling strength between a target QD and a leaky cavity is g; ox is resonance frequency of the target QD and oc is eigenfrequency of the leaky cavity, the detuning of QD–cavity is denoted by Dcx ¼ oc ox; the leaky cavity’s decay rate is Gc, the radiative decay rate and pure dephasing rate of the target QD exciton are Gx and Gnx ; Px and Pc are the driving incoherent pumping rates for target exciton and cavity, respectively, and Pc comes from other QDs coupled to the cavity offresonantly [21]. Taking into account of pure dephasing, incoherent pumping and stimulated emission, the evolution of this coupled QD–cavity system can be described by a quantum dissipative master equation as follows, which employs the thermal bath approximation and
3806
G. Cao et al. / Physica B 406 (2011) 3805–3809
Born–Markov approximation. i r_ ¼ H, r þ LðrÞ _
ð1Þ
where H ¼ _ox s þ s þ _oa ay a þi_gðay s s þ aÞ is the total Hamiltonian [22] of the coupled system, and the superoperator L(r) is given by the Lindblad operator [22] as LðrÞ ¼
Px Gx þ Px ð2s þ rs s s þ rrs s þ Þ þ ð2s rs þ s þ s 2 2 Pc rrs þ s Þ þ ð2ay raaay rraay Þ 2 Gc þ Pc Gn y ð2ara ay array aÞ þ x ðsz rsz rÞ þ ð2Þ 2 4
Here, ay and a are the creation and annihilation operators in cavitymode, s þ and s are the Pauli operators of target exciton. The spectra of Scav(o) and Srad(o) are emitted by cavity-mode and radiation-mode, respectively. According to the Wiener– Khintchine theorem [23], they can be written as Z 1 Gc limt-1 Re /ay ðtÞaðt þ tÞSeiot dt Scav ðoÞ ¼ p Z0 1 Gx limt-1 Re /s þ ðtÞs ðt þ tÞSeiot dt ð3Þ Srad ðoÞ ¼
p
0
The exact spectra formalism was obtained in Ref. [24] as follows, applying one photon-correlation approximation/szaS¼ /aS and fermion statistics [s ,s þ ] þ ¼1. Gc i/ay aSss DðoÞ ig/ay s Sss Scav ðoÞ ¼ Re þ p CðoÞDðoÞg 2 CðoÞDðoÞg 2 Gx i/ay aSss CðoÞ ig/s þ aSss Re þ ð4Þ Srad ðoÞ ¼ p CðoÞDðoÞg 2 CðoÞDðoÞg 2 where CðoÞ ¼ ooc þ 2i Gc and DðoÞ ¼ oox þ 2i ð2Px þ Gx þ Gnx Þ. The total spectrum of the QD–cavity system can be expressed as Stot ðr, oÞ ¼ Frad ðrÞSrad ðoÞ þFcav ðrÞScav ðoÞ
ð5Þ
As is shown in Refs. [13–15], cavity-mode emission is the main contribution to total spectrum. Thus here, we adopt Fcav ðrÞ=Frad ðrÞ ¼ 2=1. From Eq. (1), the populations’ steady-state solutions are derived to be h 2 i g 2 GðPx þ Pc Þ þ Pc ð2Px þ Gx Þ G4 þ D2cx h 2 i /ay aSss ¼ ð6Þ g 2 Gð2Px þ Gx þ Gc Þ þ Gc ð2Px þ Gx Þ G4 þ D2cx /ay s Sss ¼
g
/s þ s Sss ¼
G 2
iDcx ðGc þ Gx þ 2Px Þ/ay aSss þPc þPx 2 G þ D2 þ ðG þ 2P Þ x x cx 4
Gc /ay aSss þ Pc þ Px Gx þ 2Px
ð7Þ
ð8Þ
where G ¼ 2Px þ Gx þ Gnx þ Gc . Thus, the influences of pure dephasing and incoherent pumping on the spectra can be investigated numerically. Fig. 1. (a) Coupled target QD–cavity system, undergoing pure dephasing, incoherent pumping and stimulated emission. (b) The model for quantum energy exchanging between target QD and cavity. (c) The decay from a coupled QD–cavity system in QD point of view.
Table 1 The parameters used for simulations. g
Cx
Cc
Cnx
Px
Pc
0.045 meV
0.005 meV
0.085 meV
0.001 meV
0.01g
0.35g
2.2. Spectra simulations In this paper, we investigate the spectra in situations of onresonance (Dcx ¼ 0) and off-resonance (Dcx ¼1 meV). The parameters used for simulations are listed in Table 1 [6,8,21]. To highlight the role of pure dephasing on the spectra, we fix all other parameters, which are listed in Table 1, and change pure dephasing rate Gnx only, the value of Gnx ranges from 0.001 to 0.2 meV. Firstly, we consider the situation of on-resonance, shown in Fig. 2(a), a vacuum Rabi doublet can be observed when the value
Fig. 2. The influence of pure dephasing on the spectra: (a) on-resonance and (b) off-resonance.
G. Cao et al. / Physica B 406 (2011) 3805–3809
of Gnx is small, and with the value of Gnx increasing, there is an obvious trend that the peaks are broadened and the double peaks are gradually becoming a single peak. On the contrary, we consider the off-resonance situation. As shown in Fig. 2(b), when Gnx ¼ 0:001meV, the peaks of spectra emitted by cavity-mode and radiation-mode are separate. As a result, photons emitted by the coupled QD–cavity system have two different frequencies. However, as the value of Gnx increases, the spectrum of coupled QD– cavity main emitted at the frequency of cavity. When Gnx ¼ 0:2 meV, the spectrum from coupled QD–cavity emits at the cavity frequency. All this phenomenon of emission spectra by changing pure dephasing rate has already been shown in Ref. [16]. Correspondingly, to highlight the role of incoherent pumping on the spectra, all other parameters are invariable and listed in Table 1, while the value of incoherent pumping Px is changed from 0.01g to 1.36g. As show in Fig. 3(a) and (b), with Px increasing, similar trends of spectra are obtained as in Fig. 2(a) and (b). The phenomenon of on-resonance situation has already been shown in Ref. [17], which is perfectly corresponding to the experimental
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data [19]. As show in Fig. 3(b), we consider the situation of offresonance, when the value of incoherent pumping Px is small, there are two separate peaks of spectra emitted by cavity-mode and radiation-mode. However, the spectrum of coupled QD– cavity main emitted at cavity frequency with the value of Px increasing. When Px ¼1.36g, the spectrum from coupled QD– cavity emits at the frequency of cavity totally. Comparing Figs. 2 and 3, we can easily come to a conclusion that pure dephasing and incoherent pumping have similar influence on the spectra. In the situation of on-resonance, the splitting of the peaks is blurred and the peaks are broadened with Gnx or Px increasing, while in the off-resonance situation, the spectrum from coupled QD–cavity can emit at the frequency of cavity by changing Gnx or Px. Now, we investigate the influence of pure dephasing on the pump-induced spectra. Here, both pure dephasing and incoherent pumping are changed, while the other parameters are invariable and given in Table 1. As presented in Fig. 4, similar trends of spectra are observed as those plotted in Figs. 2 and 3. More
Fig. 3. The influence of incoherent pumping on the spectra: (a) on-resonance and (b) off-resonance.
Fig. 4. The influence of pure dephasing on the pump-induced spectra: (a) on-resonance and (b) off-resonance.
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G. Cao et al. / Physica B 406 (2011) 3805–3809
interestingly, it is obvious that the influence on spectra by changing both pure dephasing and incoherent pumping is greater than that by changing any of them independently. In addition, Fig. 4(b) indicates that when Px ¼1.0g, Gnx ¼ 0:1 meV, the spectrum from coupled QD–cavity emits at the frequency of cavity perfectly.
d/s þ s S ¼ ðGx þ2Px Þ/s þ s SR/s þ s S þR/ay aS þPx dt ð12Þ We derive the effective coupling rate R as R¼
Gg 2 2
ðG
=4 þ D2cx Þ
¼
4g 2 1 2Px þ Gx þ Gnx þ Gc 1 þ ð2Dcx =2Px þ Gx þ Gnx þ Gc Þ2
3. Application of QD–cavity system
ð13Þ
Based on above analysis and discussion, the phenomenon of off-resonance situation is appealing, because it provides a potential and easy way to realize a solid-state single photon source with photons emitted at the cavity frequency. As we know, efficiency plays an important role in a photonic device. Next, we pay more attention to the efficiency of emission process in a single photon source. 3.1. Efficiency model
Pcav ¼
RGc =ðR þ Gc Þ Gc ¼ Gx þ RGc =ðR þ Gc Þ ðGx þ Gc Þ þ GxRGc
ð14Þ
3.2. Efficiency simulations
_ ¼ TrðroÞ _ ¼ From Eq. (1), and the operational rule of /oS Trðr_ oÞ ¼ Trðor_ Þ, a set of equations are derived, which describe the evolution of populations d/ay aS ¼ Gc /ay aS þ gð/s þ aS þ/ay s SÞ þ Pc dt
The cavity seems as a further energy loss channel at rate of RGc =ðR þ Gc Þ, which is represented in Fig. 1(c). As represented in Ref. [18], we can easily derive the efficiency of emission process in a single photon source, which is given by
ð9Þ
d/s þ s S ¼ Gx /s þ s S2Px /s þ s Sx gð/s þ aS þ/ay s SÞ þ Px dt ð10Þ d/s þ aS 2Px þ Gx þ Gc þ Gnx ¼ iDcx /s þ aS /s þ aS þ gð/s þ s S/ay aSÞ dt 2
ð11Þ As stated in Ref. [18], an effective coupling rate R has been presented in atom–cavity system, undergoing pure dephasing only. Here, we generalize the coupling rate R in QD–cavity system, taking incoherent pumping and stimulated emission into additional account. As shown in Fig. 1(b), the system is modeled by a QD box and a cavity box, which undergoes a process of quantum energy escapes into environment at rates of Gx and Gc, respectively. At the same time, the exchanging rate of quantum energy between the two boxes is R. The coupled dynamical Eqs. (9) and (10) can be described as follows:
The efficiency Pcav is illustrated as a function of pure dephasing rate in Fig. 5(a) and (b). Here, the parameters used for simulations are listed in Table 2. In the case of on-resonance (Dcx ¼0), the efficiency is larger and at a trend of dropping. It drives from that that most spectra of QD and cavity are overlapping, and the overlapping decreases with increasing pure dephasing rate. On the contrary, in offresonance (Dcx ¼1 meV) case, the efficiency firstly rises and then drops with enhancing the pure dephasing rate, the maximal value is reached for 2Px þ Gx þ Gnx þ Gc ¼ 2Dcx , and the effective coupling rate R ¼ g 2 =Dcx . In Fig. 5(a), all other parameters are invariable and listed in Table 2, the roles of g and Px on the efficiency Pcav are illustrated by changing the values of g and Px, respectively. As shown in the figure, the influence of changing incoherent pumping rate on the efficiency Pcav is negligible. However, the efficiency Pcav can be greatly improved, by choosing a bigger coupling strength g. The roles of decay rates (both Gc and Gx) on the efficiency Pcav are shown in Fig. 5(b). In the similar method, all other parameters are Table 2 The parameters used for efficiency simulations. g
Cx
Cc
Cnx
Px
Pc
0.045 meV
0.005 meV
0.085 meV
0.1 meV
1.0g
0.35g
y
d/a aS ¼ Gc /ay aSR/ay aS þR/s þ s S þPc dt
Fig. 5. Efficiency Pcav as a function of pure dephasing rate. Dashed line: on-resonance; solid line: off-resonance. Red plots: all parameters are given in Table 2. (a) Blue plots: Px ¼ 1.36g. Green plots: g¼ 0.085 meV. (b) Blue plots: Gc ¼0.105 meV. Green plots: Gx ¼ 0.001 meV. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
G. Cao et al. / Physica B 406 (2011) 3805–3809
fixed and given in Table 2, while Gx and Gc are changed. As appeared in the figure, the changing of cavity decay rate has little effect on the efficiency Pcav, while the radiative decay rate plays an important role in enhancing it, and a higher value for Pcav could be achieved by a lower radiative decay rate.
4. Conclusions In this paper, we investigate the spectra of a coupled target QD–cavity system through a quantum master equation, which considers pure dephasing, incoherent pumping and stimulated emission. We find that pure dephasing and incoherent pumping have similar effect on the spectra. The results indicate that a frequency-stabilized solid-state single photon source can be got by controlling either pure dephasing rate or incoherent pumping rate. Then we investigate the strategies to enhance the efficiency of a single photon source, as a consequence, a higher value for the efficiency could be achieved by a bigger coupling strength and a lower radiative decay rate, these results are useful for developing an efficient solid-state single photon source. In addition, to obtain an ‘‘on-demand’’ single photon source, we should further consider the indistinguishabilities of photons and mulitiphoton probability in the QD–cavity system. Also, the inhomogeneous broadening of quantum dot should be taken into account in the future.
Acknowledgments This work is supported by the National High Technology Research and Development Program of China (Grant no. 2009AA03Z405) and the National Natural Science Foundation of China (Grant nos. 60908028, 60971068 and 10979065).
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