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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 22 — Oct. 27, 2008
  • pp: 18067–18081
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Photon emission by nanocavity-enhanced quantum anti-Zeno effect in solid-state cavity quantum-electrodynamics

Makoto Yamaguchi, Takashi Asano, and Susumu Noda  »View Author Affiliations


Optics Express, Vol. 16, Issue 22, pp. 18067-18081 (2008)
http://dx.doi.org/10.1364/OE.16.018067


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Abstract

Solid-state cavity quantum-electrodynamics (QED) has great potential owing to advances such as coupled systems combining a nanocavity and a quantum dot (QD). These systems involve two photon-emission mechanisms: the Purcell effect in the weak coupling regime and vacuum Rabi-splitting in the strong coupling regime. In this paper, we describe a third emission mechanism based on the quantum anti-Zeno effect (AZE) induced by the pure-dephasing in a QD. This is significantly enhanced by the inherent characteristics of the nanocavity. This mechanism explains the origin of strong photon emission at a cavity mode largely detuned from a QD, previously considered a counterintuitive, prima facie non-energy-conserving, light-emission phenomenon. These findings could help in controlling the decay and emission characteristics of solid-state cavity QED, and developing solid-state quantum devices.

© 2008 Optical Society of America

1. Introduction

According to quantum mechanics, the state of a microscopic system is modified when it is measured [20

20. A. G. Kofman and G. Kurizki, “Acceleration of quantum decay processes by frequent observations,” Nature 405, 546–550 (2000). [CrossRef] [PubMed]

-22

22. A. G. Kofman and G. Kurizki, “Universal dynamical control of quantum mechanical decay: modulation of the coupling to the continuum,” Phys. Rev. Lett. 87, 270405 (2001). [CrossRef]

]. In this context, the quantum AZE and the quantum Zeno effect (QZE) are respectively known as the acceleration and deceleration of a quantum decay rate when subjected to frequent measurements [20

20. A. G. Kofman and G. Kurizki, “Acceleration of quantum decay processes by frequent observations,” Nature 405, 546–550 (2000). [CrossRef] [PubMed]

, 21

21. G. S. Agarwal, M. O. Scully, and H. Walther, “Accelerating decay by multiple 2π pulses,” Phys. Rev. A 63, 044101 (2001). [CrossRef]

] (including non-referred ones) or dephasing [22

22. A. G. Kofman and G. Kurizki, “Universal dynamical control of quantum mechanical decay: modulation of the coupling to the continuum,” Phys. Rev. Lett. 87, 270405 (2001). [CrossRef]

]. This modified decay rate W is in general expressed as follows [20

20. A. G. Kofman and G. Kurizki, “Acceleration of quantum decay processes by frequent observations,” Nature 405, 546–550 (2000). [CrossRef] [PubMed]

22

22. A. G. Kofman and G. Kurizki, “Universal dynamical control of quantum mechanical decay: modulation of the coupling to the continuum,” Phys. Rev. Lett. 87, 270405 (2001). [CrossRef]

]:

W=2π0dωξ(ω)2D(ω)F(ω).
(1)

In a case of a radiative decay of a QD combined with a nanocavity, the general expression of AZE and QZE can be explained as follows. Importantly, an elastic interaction between the QD and electrons in the surrounding bulk region is considered to induce the pure-dephasing effect in the QD, which corresponds to (non-referred) measurements. First, consider the case in which a QD is placed in free space instead of a nanocavity. |ξ(ω)|2 D(ω) in equation (1) is flat with respect to frequency ω, and although F(ω) is broadened by the puredephasing, the magnitude of the overlap integral in equation (1) is constant. Therefore, neither QZE nor AZE occurs. By contrast, when a QD is introduced into a nanocavity, it begins to couple to the nanocavity mode, and the additional radiative decay path through the nanocavity is generated. When we consider the case of a weak coupling regime, the reservoir spectrum seen from the QD looks like a strong peak at the cavity mode over the flat background. As described in the general expression, the spectrum overlap between |ξ(ω)|2 D(ω) and F(ω) will be modified when F(ω) is broadened by the pure-dephasing effect; thus, AZE and QZE are expected to occur.

In reality, the situation is complicated because the coupling between the QD and the nanocavity can range from the weak to strong coupling regime, and various detuning between the cavity mode and the QD can occur. To treat the QZE near the on-resonant condition in the strong coupling regime, equation (1) is not appropriate and consideration in the time domain is required.

Fig. 1. Schematic illustration of the system investigated. (a) Example of a solid-state cavity QED system, where a single QD is embedded in a two-dimensional photonic crystal nanocavity. (b) Schematic illustration of our analysis model, a TLS interacting with a single-mode cavity. Under the initial condition of an excited TLS, photons can be emitted to free space by two pathways: a direct pathway to free space or an indirect pathway through the cavity mode. (c) Schematic illustration of elastic interactions between the TLS and electrons in the cladding layer.

2. Theoretical analysis

2.1 Analysis model

Our analysis model is shown in Fig. 1(b). As QDs are known to possess discrete energy levels [13

13. Y. Arakawa and H. Sakai, “Multidimensional quantum well laser and temperature dependence of its threshold current,” Appl. Phys. Lett. 40, 939–941 (1982). [CrossRef]

16

16. M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B 65, 041308 (2002). [CrossRef]

] due to their three-dimensional confinement of electrons, we can approximate a single QD as a two-level system (TLS) in a solid. This TLS (with transition energy ħω TLS) can interact with a single-mode electromagnetic field formed in a cavity (with resonant energy ħω cav) with a coupling constant g. The TLS and cavity mode can also interact with vacuum photon modes in free space. The excited state composed of the TLS and the cavity mode can spontaneously emit photons to these vacuum modes. The critical feature of this model is that the TLS can interact elastically with electrons in the surroundings (Fig. 1(c)).

We analyze the quantum dynamics in this model using the quantum master equation [7

7. K. Srinivasan and O. Painter, “Mode coupling and cavity-quantum-dot interactions in a fiber-coupled microdisk cavity,” Phys. Rev. A 75, 023814 (2007). [CrossRef]

, 23

23. H. J. Carmichael, An Open Systems Approach to Quantum Optics (Springer-Verlag, Berlin, 1993).

25

25. H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, “Subnatural linewidth averaging for coupled atomic and cavity-mode oscillators,” Phys. Rev. A 40, 5516–5519 (1989). [CrossRef] [PubMed]

], focusing on the system of interest consisting of the TLS and the cavity mode, and treating its surroundings (the vacuum photon modes in free space and the surrounding electrons) as heat baths (see also Appendix A and B). The quantum master equation is then expressed as

dρSdt=i1[HS,ρS]+ΣmLmρm,
(2)

where ρ S is a reduced density operator, H S represents the interaction Hamiltonian between the TLS and the cavity mode (non-Markov process), and the L m terms (Liouvillians) represent the irreversible damping that arises from the interactions between the system and its surroundings (Markov process). The vacuum fields in free space cause irreversible energy damping to the TLS and cavity mode, whose original damping rates (without taking into account any mutual interaction) are represented by 2Γ spon and 2Γ cav (≡ω cav/Q), respectively. The elastic Coulomb interactions between the TLS and the surrounding electrons cause pure-dephasing of the TLS (with a rate denoted by γ phase) but do not induce any energy damping [7

7. K. Srinivasan and O. Painter, “Mode coupling and cavity-quantum-dot interactions in a fiber-coupled microdisk cavity,” Phys. Rev. A 75, 023814 (2007). [CrossRef]

, 23

23. H. J. Carmichael, An Open Systems Approach to Quantum Optics (Springer-Verlag, Berlin, 1993).

] (see also Appendix B). An analogy between the effect of elastic Coulomb interactions and superselection theories [26

26. W. H. Zurek, “Pointer basis of quantum apparatus,” Phys. Rev. D 24, 1516–1525 (1981). [CrossRef]

] can also be made. Unless otherwise stated, we used a standard set of parameters in the numerical calculations, 2ħΓ spon=0.044 µeV (1/2Γ spon=15 ns, corresponding to the decay time of a QD in the photonic band gap [4

4. K. Kounoike, M. Yamaguchi, M. Fujita, T. Asano, J. Nakanishi, and S. Noda, “Investigation of spontaneous emission from quantum dots embedded in a two-dimensional photonic-crystal slab,” Electron. Lett. 41, 1402–1403 (2005). [CrossRef]

]) and 2ħΓ cav=100 µeV (Q=1.1×104) [1

1. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007). [CrossRef] [PubMed]

, 2

2. T. Yoshie, A. Scherer, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004). [CrossRef] [PubMed]

]. For the coupling constant g, we used a value of ħg=76 µeV (2π/2g=27.0 ps), which is reasonable for a QD that is aligned to the centre of the cavity used in the experiments [1

1. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007). [CrossRef] [PubMed]

] mentioned above. For ħγ phase, it is known to range from ~20 to ~70 µeV at cryogenic temperatures (< 50 K) [15

15. D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer, and D. Park, “Fine structure splitting in the optical spectra of single GaAs quantum dots,” Phys. Rev. Lett. 76, 3005–3008 (1996). [CrossRef] [PubMed]

, 16

16. M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B 65, 041308 (2002). [CrossRef]

] and when the temperature or density of surrounding electrons is increased, ħγ phase increases larger than 100 µeV [16

16. M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B 65, 041308 (2002). [CrossRef]

]. In our analysis, in order to investigate the effect of pure-dephasing systematically, we used values from 0 to 350 µeV.

2.2 Decay rate

Initially, we analyze the decay rate W of the TLS using equation (2), setting the initial condition as an excited TLS. At the beginning of the time evolution for the parameters listed above (Appendix C), W can be expressed as

WΓcav+Γspon+R(ΓcavΓspon)2+R2,
(3)

where

R2g2ΓtotalδωTLS,cav2+Γtotal2.
(4)

Here, Γ totalΓ cav+Γ spon+γ phase (Γ total is the total dephasing rate for the system composed of the TLS and the cavity mode), and δω TLS,cavω TLS-ω cav (δω TLS,cav is the detuning between the resonant angular frequencies of the TLS and the cavity mode). The decay rate of the excited-state TLS is shown in Fig. 2(a) for ħγ phase values of 0 µeV, 35 µeV, 70 µeV, and 350 µeV. Pure-dephasing significantly modifies the decay rate of the TLS. This is consistent with the analysis [22

22. A. G. Kofman and G. Kurizki, “Universal dynamical control of quantum mechanical decay: modulation of the coupling to the continuum,” Phys. Rev. Lett. 87, 270405 (2001). [CrossRef]

] by Kofman et al. showing that stochastic phase changes (or dephasing) induce changes in the decay rate. Under off-resonant conditions in Fig. 2(a), W increases as ħγ phase increases from 0 µeV to 350 µeV. Hence, pure-dephasing induces the AZE under off-resonant conditions (Fig. 2(b)). Conversely, W decreases with ħγ phase under on-resonant conditions (gray region in Figs. 2(a) and 2(b)), so pure-dephasing induces the QZE under on-resonant conditions. When we consider ħγ phase is between ~20 and ~70 µeV at cryogenic temperatures, and increases at higher temperatures or when the density of surrounding electrons increases, AZE and QZE indeed occur in the solid-state nanocavity QED systems.

Fig. 2. Decay rate of the excited TLS. (a) Decay rate for pure-dephasing rates ħγ phase of 0 µeV (blue line), 35 µeV (red line), 70 µeV (green line) and 350 µeV (black line). (b) Decay rate for ħγ phase=35 µeV (red line), 70 µeV (green line) and 350 µeV (black line) normalized by the decay rate for ħγ phase=0 µeV in Fig. 2(a). The inset shows the magnified image under the onresonant condition. The gray dashed line denotes that the value along the longitudinal axis equals to one.

2.3 Emitted photon spectrum

The decay rate W does not provide information on the emitted photon spectrum, which is crucial for determining the optical performance of cavity QED systems. Therefore, we now analyze the emitted photon spectra from the correlation function of the electric fields in free space. The emitted photon spectrum S(ω) is the sum of the emission spectra for light escaping the TLS directly to free space and indirectly passing through the cavity (Fig. 1(b)). An analytical expression for S(ω) can be obtained using equation (2), which is applicable regardless of the weak or strong coupling regime between the TLS and the cavity:

S(ω)2πΛ+ΛRe[1γ+γ{f(γ+)iω+γ+f(γ)iω+γ}].
(5)

Here, we define

2γ±[Γtotal+i(ωTLS+ωcav)]±(ΓcavΓsponγphaseiδωTLS,cav)24g2.
(6)

Exact expressions for the coefficients f(γ±) and Λ± are given in Appendix C and D. The emission spectra S(ω) for various values of detuning are shown in Fig. 3, where the pure-dephasing rates are colour coded as in Fig. 2(a). The vertical axes represent the number of emitted photons per unit angular frequency.

Fig. 3. Emission spectra for various values of detuning. (a) No pure-dephasing. (b) With puredephasing rate ħγ phase=35 µeV. (c) With pure-dephasing rate ħγ phase=70 µeV. (d) With pure-dephasing rate ħγ phase=350 µeV. The labels TLS and Cav denote the peaks at the transition energy of the TLS and at the cavity resonance energy, respectively.

For the case with no pure-dephasing in Fig. 3(a), the emission spectra show two peaks when detuned < 0.4 meV, which is the typical Rabi-splitting spectra [19

19. G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, “Vacuum Rabi splitting in semiconductors,” Nature Phys. 2, 81–90 (2006). [CrossRef]

, 18

18. I. I. Rabi, “Space quantization in a gyrating magnetic field,” Phys. Rev. 51, 652 (1937). [CrossRef]

] in the strong coupling regime. For larger detuning, only one emission peak at the transition energy of the TLS is seen. However, when a small degree of pure-dephasing (ħγ phase=35 µeV) is introduced, two clear emission peaks can be seen even for large detuning as in Fig. 3(b). As ħγ phase is further increased to 70 µeV, a large peak at the cavity resonance energy develops (Fig. 3(c)). Finally, when ħγ phase is sufficiently large (350 µeV), a single emission peak at the cavity resonance energy is observed (Fig. 3(d)). The presence of a peak at the cavity resonance energy for large detuning has been experimentally observed, but the origin has been unresolved. Our analysis suggests that the pure-dephasing effect is the origin of the emission. In addition, the Rabisplitting under the on-resonant condition is observed when the pure-dephasing rate is relatively small (< 70 µeV), which indicates that the Rabi splitting and QZE are not exclusive when considering the result of Fig. 2 simultaneously.

2.4 Individual emission rate

Now we consider the phenomena under the off-resonant condition, where two peaks at the TLS transition energy and the cavity resonant energy are observed. To obtain the individual photon emission rates W TLS and W cav at the TLS transition energy and at the cavity resonance energy, we integrated the emission spectra of individual peaks with respect to the emission energy and multiplied them with the total decay rate W, which can then be written as W=W TLS+W cav. The obtained photon emission rates W TLS and W cav are shown as a function of the pure-dephasing rate in Fig. 4(a), where a large detuning of 4 meV is assumed. Notably, the value of detuning is much larger than the total spectral broadening of 2ħΓ total=2ħ(γ phase+Γ cav+Γ spon) (<~500 µeV). Nevertheless, Fig. 4(a) clearly indicates that the photon emission rate at the cavity resonance energy increases linearly with increasing ħγ phase, while the photon emission rate at the TLS transition energy is independent of ħγ phase. We conclude that the increase of the cavity mode emission rate is due to the AZE that is induced by the pure dephasing effect by simultaneously considering the results shown in Fig. 2. Figure 4(b) illustrates the ratio of the cavity mode emission rate to the total emitted photon rate, defined as a factor F,

FWcav(WTLS+Wcav),
(7)

where F increases with increasing ħγ phase. Consequently, the photon emission at the cavity resonance dominates the emission spectra for higher ħγ phase. These results explain the photon emission mechanism: despite large detuning, the AZE induced by the pure dephasing effect generates cavity-mode photons.

When the detuning is large enough, (|δω TLS,cav| ≫ g, Γ total), the photon emission rates W TLS and W cav shown in Fig. 4(a) can be analytically expressed as:

WTLS2Γspon+2g2ΓcavδωTLS,cav2,
(8)
Wcav2g2(Γspon+γphase)δωTLS,cav2,
(9)

where the terms of W TLS indicate the direct emission rate to free space and an approximated expression for the Purcell effect [17

17. E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

], respectively. Note that the Purcell effect occurs at the energy of TLS instead of the cavity resonant energy under off-resonant conditions, where the cavity mode is forced to oscillate at the TLS transition energy. On the other hand, W cav is proportional to the line width of the TLS expressed as 2(Γ spon+γ phase), which is multiplied by g 2/δω 2 TLS,cav. Thus, F can be expressed as

FΓspon+γphaseΓspon(δωTLS,cavg)2+Γspon+(Γspon+γphase),
(10)

Fig. 4. Pure-dephasing dependence of the photon emission rates for individual peaks in the emission spectrum. (a) Photon emission rates for detuning of 4 meV. The labels W TLS and W cav denote the photon emission rates from the peaks at the transition energy of the TLS and at the cavity resonance energy, respectively. (b) Factor of F is defined as W cav/(W TLS+W cav), which represents the ratio of the cavity mode emission rate to the total emitted photon rate. (c) Corresponding emission spectrum calculated for ħγ phase=35 µeV, where the percentage represents the ratio of each peak to the total integral value.

3. Discussions

We have described a case in which the number of QD is single and a nanocavity is based on photonic crystals. Here, we will discuss more general cases. First, we consider a nanocavity containing tens to hundreds of QDs variously detuned by 10–20 meV [2

2. T. Yoshie, A. Scherer, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004). [CrossRef] [PubMed]

4

4. K. Kounoike, M. Yamaguchi, M. Fujita, T. Asano, J. Nakanishi, and S. Noda, “Investigation of spontaneous emission from quantum dots embedded in a two-dimensional photonic-crystal slab,” Electron. Lett. 41, 1402–1403 (2005). [CrossRef]

]. Second, we discuss the case for various cavity geometries [6

6. K. Srinivasan and O. Painter, “Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system,” Nature 450, 862–865 (2007). [CrossRef] [PubMed]

10

10. J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature 432, 197–200 (2004). [CrossRef] [PubMed]

].

4. Conclusion

We have brought the concept of AZE (and QZE) to solid-state quantum systems, and found that the AZE enhanced by the inherent characteristics of the nanocavity produces a third emission mechanism in solid-state nanocavity QED systems. This emission mechanism based on the nanocavity-enhanced AZE explains the counterintuitive photon emission of a cavity mode largely detuned from QD peaks. We believe that our findings not only provide a basis that could lead to methods for controlling the decay and emission characteristics for solidstate cavity QED, but will also accelerate the development of various types of solid-state quantum devices.

Appendix A:

In this section we give a simple explanation for the AZE-induced emission from a quantitative point of view, where the emission channels from the excited two-level system (TLS) to free space are discussed in terms of the limited bases of the density operator. In general, a density operator ρ T of a quantum system (a system of interest and a reservoir) follows the Liouvillevon Neumann form, which is directly derived from the Schrödinger equation

dρTdt=i1[HT,ρT],
(11)

As can be seen in the figure, the direct emission from the TLS to free space is composed of all pathways that reach (C) via (F), whereas the indirect emission through the cavity mode is composed of all pathways that reach (C) via (E). Therefore, the lowest-order channel for direct emission is (A)-(F)-(C). This is intuitive because the channel (A)-(F)-(C) represents the original emission channel (without the cavity mode) from the excited TLS, where state (A) gives rise to an oscillating electric polarization (F) and the radiation of a photon to state (C) in free space. It is clear that the emission probability spectrum for the channel (A)-(F)-(C) has only one peak, located at the transition energy of the TLS. In contrast, there are two lowest-order channels for indirect emission through the cavity: channel 1 is along the pathway (A)-(D)-(B)-(E)-(C) and channel 2 is along the pathway (A)-(D)-(F)-(E)-(C). We note that channel 1 goes through only one correlation state (E), whereas channel 2 goes through two correlation states (F) and (E). As mentioned above, the states (E) and (F) give rise to spectral peaks when the energy of the free-space modes is at the cavity resonant energy and the transition energy of the TLS, respectively. Therefore, channel 1 gives rise to only one peak at the cavity resonant energy, whereas channel 2 results in peaks at both the cavity resonant energy and the transition energy of the TLS. Therefore, indirect emission can in principle give rise to peaks both at the transition energy of the TLS and at the cavity resonant energy. However, in the absence of pure dephasing, channels 1 and 2 can interfere at state (E), which leads to cancellation of the peak at the cavity resonant energy. The probability of emitting a photon to free space at the cavity resonant energy will then be zero; thus, there is only one peak at the transition energy of the TLS in the absence of pure dephasing. This process can be interpreted as the forced oscillation (or virtual excitation) of state (E) by the transition energy of the TLS. In contrast, in the presence of pure dephasing the coherence of the dipoles vanishes. Channels 1 and 2 are then unable to interfere completely. As a result, a peak at the cavity resonant energy can be obtained. These scenarios clearly explain the mechanism of emission from the spectrally detuned TLS at the cavity resonant energy.

Fig. 5. Emission channels through density operator states. (A)–(C) represent diagonal states and (D)–(F) represent off-diagonal (correlation) states. Gray arrows denote couplings between states within first-order perturbation. For simplicity, the Hermitian conjugate states and their related couplings are not shown.

Appendix B:

In this section we discuss the Liouvillians in the quantum master equation. The total Hamiltonian of H T can be decomposed as

HT=HS+HR+Hint,
(12)

where the constituent Hamiltonians represent the system (H S), the environment of the heat bath (H R), and their interaction (H int). Under the Born and Markov approximation, an integro-differential equation for the quantum master equation [23

23. H. J. Carmichael, An Open Systems Approach to Quantum Optics (Springer-Verlag, Berlin, 1993).

, 24

24. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1990).

] in the interaction picture can be obtained from equations (11) and (12) under the assumption that ρ T(t) ≈ ρ S(t)⊗ρR(0):

dρS(t)dt=i1[HS(t),ρS(t)]i1τ1tt+τdt1TrR[Hint(t1),ρS(t)ρR(0)]
2τ1tt+τdt1tt1dt2TrR[Hint(t1),[Hint(t2),ρS(t)ρR(0)]],
(13)

where ρ S(t)≡Tr R[ρ T(t)] represents the reduced density operator, and ρ S and ρ R represent individual density operators for a system of interest and its reservoirs. In addition, S(t)/dt≡{ρS(t+τ)-ρ S(t)}/τ expresses a coarse temporal differentiation. The first term represents the coherent time evolution within the system, whereas the second and third terms represent irreversible damping. Therefore, each Liouvillian is formed by the second and third terms in equation (13).

We now derive the Liouvillian L i,phase for the Coulomb interactions in the second quantization framework. The Coulomb interaction Hamiltonian Ĥ Coulomb in the solid can be expressed as

HCoulomb=12d3rd3rψ̂(r)ψ̂(r)e2rrψ̂(r)ψ̂(r),
(14)

where ψ̂(r) represents the electron field operator. ψ̂(r) can be decomposed as

ψ̂(r)=ĉ1φ1(r)+ĉ2φ2(r)+Σmĉmφm(r),
(15)

where φ j(r) (j=1, 2, m) represents the j-th electron state wave function, and ĉ 1 and ĉ 2 are the annihilation operators of the first (lower) and second (upper) levels in the TLS, respectively. In addition, ĉ m is the annihilation operator of the m-th electron state in the solid. These are Fermi operators satisfying anti-commutation relations. By substituting equation (15) into (14) and ignoring the Auger processes [28

28. J. L. Pan, “Reduction of the Auger rate in semiconductor quantum dots,” Phys. Rev. B 46, 3977–3998 (1992). [CrossRef]

] in the first approximation, we obtain

HCoulombH12,El+H11,El+H22,El,
(16)

where H 12,El represents the Coulomb interaction between the first and second levels in the TLS, and H 11,El (H 22,El) represents the interactions between the first (second) level in the TLS and the other electrons in the solid. The two Hamiltonians can be expressed as

Hii,El=ĉiĉiΣmĉmĉmG(i,i,m,m)+ĉiĉiΣn,kĉnĉkG(i,i,n,k),(nk,i=1or2)
(17)
H12,El=ĉ1ĉ1ĉ2ĉ2G(1,1,2,2),
(18)

where we define

G(i,j,k,l)d3rd3re2rr[φi*(r)φj(r)φk*(r)φl(r)φi(r)φj*(r)φl(r)φk*(r)].
(19)

Here, the first term of equation (17) describes the Coulomb interactions between the TLS and the other fixed states in the solid, whereas the second term represents the elastic Coulomb scattering of the electrons in the solid. Because H ii,El(i=1 or 2) represents the interaction between the system (the TLS and the cavity mode) and its environment (the electron heat bath), it is treated as H int in equation (13). The Hamiltonian H 12,El describes the interaction within the system. Therefore, H 12,El is treated as H S. It should be noted that H 11,El and H 22,El satisfy the commutation relations [ĉi ĉ i,H 11,El]=[ĉi ĉ i,H 22,El]=0, which suggests that these Hamiltonians do not change the temporal differentiation of <ĉd ĉ i(t)> [26

26. W. H. Zurek, “Pointer basis of quantum apparatus,” Phys. Rev. D 24, 1516–1525 (1981). [CrossRef]

, 29

29. V. B. Braginsky and F. Y. Khalili, Quantum Measurement (Cambridge University Press, Cambridge, 1992). [CrossRef]

]. As a result, the Liouvillians derived from these Hamiltonians do not contribute directly to the temporal change of the expectation value of < ĉi ĉ i(t)> either, leading to pure dephasing. After transforming equation (17) to the interaction picture and substituting into (13), standard procedures to obtain the Liouvillian [24

24. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1990).

] yield

Li,phaseX=iΔi[ĉiĉi,X]γphase,1(ĉiĉiX+Xĉiĉi2ĉiĉiXĉiĉi),(i=1,or2)
(20)

where Δi denotes the angular frequency shifts of the i-th level in the TLS due to the Coulomb interactions, and γ phase, i is the pure dephasing rate of the i-th level in the TLS. We have ignored the Δi in the analytical expressions and have defined the total pure dephasing rate as γ phaseγ phase,1+γ phase,2. In a similar manner, we can also obtain the Liouvillians L cav and L spon, representing the individual energy damping from the cavity mode and the TLS to the vacuum fields in free space. The Liouvillians are expressed as [24

24. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1990).

]

LcavX=Γcav(âcavâcavX+Xâcavâcav2âcavXâcav),
(21)
LsponX=Γspon(ĉ2ĉ1ĉ1ĉ2X+Xĉ2ĉ1ĉ1ĉ22ĉ1ĉ2Xĉ2ĉ1),
(22)

where â cav (â cav) is the annihilation (creation) operator of the cavity mode. In addition, the Hamiltonian describing the interaction within the system can be expressed as

HS(t)=(ĉ2ĉ1âcavgexp(iδωTLS,cavt)+H.c.)+H12,El,
(23)

where the first terms represent the interaction between the cavity mode and the TLS, and δω TLS,cavω TLS-ω cav is the detuning. Using equation (13), the dynamical time evolution of the system (the cavity mode and the TLS) can be expressed as

dρS(t)dt=LρS(t),
(24)

with the total Liouvillian

LLS+Lcav+Lspon+L1,phase+L2,phase,
(25)

where we defined

LSXi1[HS,X].
(26)

Appendix C:

In this section we analyze the time evolution of the expectation values <ĉ2 ĉ 2(t)> and <â cav â cav(t)> in order to find an electron in the upper level and a photon in the cavity, respectively. The coupled equations for these expectation values are obtained from equation (24) as

d<ĉ2ĉ2(t)>dt=Re[ig<ĉ2ĉ1âcav(t)>]2Γspon[<ĉ2ĉ2(t)><ĉ2ĉ1ĉ1ĉ2(t)>],
(27)
d<âcavâcav(t)>dt=Re[ig<ĉ2ĉ1âcav(t)>]2Γcav<âcavâcav(t)>,
(28)
d<ĉ2ĉ1âcav(t)>dt=ig*<ĉ2ĉ2(t)>ig*<âcavâcav(t)>+(iδωTLS,cavΓtotal)<ĉ2ĉ1âcav(t)>,
(29)
d<ĉ2ĉ1ĉ1ĉ2(t)>dt=0,
(30)

where the total dephasing rate is Γ totalΓ cav+Γ spon+γ phase. In these coupled equations, <ĉ 2 ĉ 1 â cav(t)> represents the expectation value of the electronic polarization for the cavity mode (state (D) in Fig. 5), and <ĉ 2 ĉ 1 ĉ 1 ĉ 2(t)> represents the probability that both levels of the TLS are occupied by electrons. Because we have ignored the Auger processes, <ĉ 2 ĉ 1 ĉ 1 ĉ 2(t)> remains the initial value, as described by equation (30). The analytical solution of these coupled equations is difficult. Therefore, we make the approximation that d/dt ≈ 0 in equation (29), which is similar to an adiabatic elimination of < ĉ 2 ĉ 1 â cav(t)>. It follows that this approximation is always valid for g << Γ total or g << δω TLS,cav. When these conditions are not satisfied, the solutions obtained from the remaining coupled equations (27), (28), and (30) become inaccurate on the time-scale g -1, and only temporally averaged solutions are obtained. In this limit, we can obtain simplified coupled equations for (27)–(30) using the initial condition <ĉ 2 ĉ 1 ĉ 1 ĉ 2(0)>=0.

d<ĉ2ĉ2(t)>dt=R<ĉ2ĉ2(t)>+R<âcavâcav(t)>2Γspon<ĉ2ĉ2(t)>,
(31)
d<âcavâcav(t)>dt=R<ĉ2ĉ2(t)>R<âcavâcav(t)>2Γcav<âcavâcav(t)>,
(32)

where we have defined the following:

R+iK2g21iδωTLS,cav+Γtotal,
(33)

Here, equations (31) and (32) are classical coupled-rate equations and can be solved analytically using two eigenfrequencies of Λ ± expressed as

Λ±(Γcav+Γspon+R)±(ΓcavΓspon)2R2.
(34)

These eigenfrequencies characterize the decay rate W of <ĉ 2 ĉ 2(t)>. Particularly for Γ cav > Γ spon, we can make the approximation that W ≈ -Λ+, which immediately leads to equation (3). We now discuss the validity of this exppression. As mentioned above, the approximation W ≈-Λ+(for Γ cav > Γ spon) is evidently valid under the condition that g << Γ total or g << δω TLS,cav. Furthermore, if we ignore the oscillation component (Rabi oscillation) that changes over the time-scale of g -1 and focus on the time evolution over a larger time-scale, the coupled equations (31) and (32) are also useful even when the conditions g << Γ total or g << δω TLS,cav are not satisfied. In general, it should be possible to estimate the decay rate W on this larger time-scale, thus the approximation W ≈ -Λ+ (for Γ cav > Γ spon) is also valid even under the conditions g >> Γ total and g >> δω TLS,cav.

Appendix D:

In this section we describe the total emission spectrum S(ω). Following the Wiener- Khintchine theorem, the total emission spectrum S(ω) can be expressed as

S(ω)=2ε0c0(2π)10dτ0dt<E(t+τ)·E+(t)>exp(iωτ),
(35)

where E +(t) is the electric field operator in free space in the Heisenberg picture, ε 0 is the dielectric constant, and c 0 is the velocity of light in free space. Using the Heisenberg equations of motion, E +(t) can be expressed in the form E +(t)=G cav aâcav(t)+G spon ĉ 1 ĉ 2(t), where we ignore the solution of the homogeneous or free-field wave equation. The factors G cav and G spon represent the coupling between the system (the cavity mode and the TLS) and the free-space photon modes. As a result, further steps yield the relation

2ε0c0<E(t+τ)·E+(t)>2Γcav<âcav(t+τ)âcav(t)>+2Γspon<ĉ2ĉ1(t+τ)ĉ1ĉ2(t)>,
(36)

in which we have ignored the cross-terms for simplicity; these are not essential for our main conclusion. In the case of τ=0, the first term 2Γ cav< â cav â cav(t)> represents the photon emission through the cavity (pathways (A)-(D)-(B)-(E)-(C) and (A)-(D)-(F)-(E)-(C) in Fig. 5). The second term 2Γ spon< ĉ 2 ĉ 1 ĉ 1 ĉ 2(t)> represents the direct photon emission from the TLS (pathway (A)-(F)-(C) in Fig. 5). By substituting equation (36) into (35), we can express the total emission spectrum S(ω) as

S(ω)Sspon(ω)+Scav(ω),
(37)

where

Scav(ω)=ΓcavπRe[0dτ0dt<âcav(t)âcav(t+τ)>exp(iωτ)],
(38)
Sspon(ω)=ΓsponπRe[0dτ0dt<ĉ2(t)ĉ1(t)ĉ1(t+τ)ĉ2(t+τ)>exp(iωτ)].
(39)

In order to calculate these emission spectra, the expectation values for the two-time correlation functions < â cav(t)aĉcav(t+τ)> and <ĉ 2(t)ĉ 1(t) ĉ 1(t+τ)ĉ 2(t+τ)> are required; they can be obtained using the quantum master equation derived above together with the quantum regression theorem [23

23. H. J. Carmichael, An Open Systems Approach to Quantum Optics (Springer-Verlag, Berlin, 1993).

25

25. H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, “Subnatural linewidth averaging for coupled atomic and cavity-mode oscillators,” Phys. Rev. A 40, 5516–5519 (1989). [CrossRef] [PubMed]

]. In addition, the time integral values <ĉ 2 ĉ 2(t)> and <â cav aĉcav(t)> discussed above are also required so that the analytical solutions of the simplified equations (31) and (32) can be used. As a result, each emission spectrum can be expressed analytically in a unified form:

Sα(ω)2πΛ+ΛRe[1γ+γ{fα(γ+)iω+γ+fα(γ)iω+γ}],(α=spon,orcav)
(40)

where we define the coefficients γ ± and f α(γ ±) (α=spon, or cav) as

2γ±[Γtotal+i(ωTLS+ωcav)]±(ΓcavΓsponγphaseiδωTLS,cav)24g2,
(41)
fcav(γ±)Γcav[R(γ±+iωTLS+Γtotal)+iKΓcav],
(42)
fspon(γ±)Γspon[Γcav(iK2Γcav)(γ±+iωcav)(R+2Γcav)].
(43)

Using equation (37), these expressions result in Eq. (5), where f(γ ±)≡f cav(γ ±)+f spon(γ ±).

Appendix E:

As described in the main text, we have demonstrated that the experimental results reported in reference [1

1. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007). [CrossRef] [PubMed]

] of main text, especially those concerning the intensity of the cavity mode emission under conditions where detuning from the exciton emission peak is large, can be explained very well by our theory. However, the reference [1

1. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007). [CrossRef] [PubMed]

] also reported very strange intensity correlations involving the cavity and exciton emission peaks. The cavity emission peak (when filtered) showed Poissonian-like intensity correlations, whereas the exciton emission peak (also filtered) showed clear anti-bunched intensity correlations. The cross-correlation of the cavity emission intensity and the exciton emission intensity, however, showed anti-bunched correlations. In the following discussion, we consider why such strange correlation features were observed.

We first point out that similar experiments to those in reference 1 were also performed in reference [9

9. D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett. 98, 117402 (2007). [CrossRef] [PubMed]

] of main text using a system composed of a micropillar-type nanocavity and a QD, where it was demonstrated that not only cross-correlation measurements between the bare-cavity mode and exciton emissions, but also auto-correlation measurements for the bare-cavity emission, showed anti-bunching. In this context, the experimental results reported in references [1

1. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007). [CrossRef] [PubMed]

] and [9

9. D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett. 98, 117402 (2007). [CrossRef] [PubMed]

] are quite different: all of the correlation measurements showed anti-bunching in reference [9

9. D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett. 98, 117402 (2007). [CrossRef] [PubMed]

], whereas the bare-cavity auto-correlation measurement showed Poissonian-like intensity correlations in reference [1

1. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007). [CrossRef] [PubMed]

].

Here, we also point out that the experimental conditions employed were different in references [1

1. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007). [CrossRef] [PubMed]

] and [9

9. D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett. 98, 117402 (2007). [CrossRef] [PubMed]

]. In reference [9

9. D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett. 98, 117402 (2007). [CrossRef] [PubMed]

] a resonant excitation method was utilized, where the excitation laser was tuned to resonantly pump the first excited state (p-shell) exciton in a selected QD. This exciton thermalizes to the QD ground state (s-shell), which implies that only one two-level system can interact with the cavity mode. These experimental conditions are identical to our theoretical model, and the experimental results in reference [9

9. D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett. 98, 117402 (2007). [CrossRef] [PubMed]

] agree well with our calculations.

In contrast, the study in reference [1

1. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007). [CrossRef] [PubMed]

] employed a higher energy pumping method (above-band pumping). Furthermore, in reference [1

1. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007). [CrossRef] [PubMed]

] the experimental conditions were different for the cross-correlation measurements between the bare-cavity and the exciton emission (CW excitation) and for the auto-correlation measurements of the cavity mode emission (pulsed excitation). It is likely that these different excitation conditions would affect the characteristics of the intensity correlation measurements. In particular, when the number of excited carriers increases (or under conditions of strong excitation), higher states in the QD will be simultaneously occupied. From this point of view, one possible explanation for the results reported in reference [1

1. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007). [CrossRef] [PubMed]

] could be as follows.

(1) For the cross-correlation measurements under CW excitation and off-resonant conditions, the number of carriers created might be relatively small, and the QD can be treated as a single two-level system. Consequently, the cavity emission can originate from the two-level system and shows anti-bunching in the cross-correlation measurements.

(2) For the auto-correlation measurements under pulsed excitation and off-resonant conditions, the number of instantaneously created carriers could be much greater, so that higher states in the QD could be simultaneously occupied by excess excited carriers, and these filled higher states may act as multiple two-level systems. For the case of large detuning, the cavity mode would simultaneously receive multiple photons from these multiple two-level systems. In this case, the auto-correlation measurements of the bare-cavity emission do not need to show anti-bunching.

Acknowledgment

This investigation was supported by Research Programs (Grant-in-Aid, Centre of Excellence and Special Coordination Fund) for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

References and links

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K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007). [CrossRef] [PubMed]

2.

T. Yoshie, A. Scherer, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004). [CrossRef] [PubMed]

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S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, “Self-tuned quantum dot gain in photonic crystal lasers,” Phys. Rev. Lett. 96, 127404 (2006). [CrossRef] [PubMed]

4.

K. Kounoike, M. Yamaguchi, M. Fujita, T. Asano, J. Nakanishi, and S. Noda, “Investigation of spontaneous emission from quantum dots embedded in a two-dimensional photonic-crystal slab,” Electron. Lett. 41, 1402–1403 (2005). [CrossRef]

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M. Yamaguchi, T. Asano, and S. Noda, “Origin of unexpected light emission in a coupled system of a photonic-crystal nanocavity and a quantum dot,” presented at the 8th International Conference on Physics of Light-Matter Coupling in Nanostructures, Tokyo, Japan , 7–11 April 2008. [PubMed]

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K. Srinivasan and O. Painter, “Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system,” Nature 450, 862–865 (2007). [CrossRef] [PubMed]

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K. Srinivasan and O. Painter, “Mode coupling and cavity-quantum-dot interactions in a fiber-coupled microdisk cavity,” Phys. Rev. A 75, 023814 (2007). [CrossRef]

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P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, and A. Imamoğlu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000). [CrossRef] [PubMed]

9.

D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett. 98, 117402 (2007). [CrossRef] [PubMed]

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J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature 432, 197–200 (2004). [CrossRef] [PubMed]

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B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005). [CrossRef]

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M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B 65, 041308 (2002). [CrossRef]

17.

E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

18.

I. I. Rabi, “Space quantization in a gyrating magnetic field,” Phys. Rev. 51, 652 (1937). [CrossRef]

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G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, “Vacuum Rabi splitting in semiconductors,” Nature Phys. 2, 81–90 (2006). [CrossRef]

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H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, “Subnatural linewidth averaging for coupled atomic and cavity-mode oscillators,” Phys. Rev. A 40, 5516–5519 (1989). [CrossRef] [PubMed]

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M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science 308, 1296–1298 (2005). [CrossRef] [PubMed]

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OCIS Codes
(270.0270) Quantum optics : Quantum optics
(270.5580) Quantum optics : Quantum electrodynamics
(300.6470) Spectroscopy : Spectroscopy, semiconductors
(350.4238) Other areas of optics : Nanophotonics and photonic crystals

ToC Category:
Quantum Optics

History
Original Manuscript: August 28, 2008
Revised Manuscript: October 16, 2008
Manuscript Accepted: October 16, 2008
Published: October 21, 2008

Citation
Makoto Yamaguchi, Takashi Asano, and Susumu Noda, "Photon emission by nanocavity-enhanced quantum anti-Zeno effect in solid-state cavity quantum-electrodynamics," Opt. Express 16, 18067-18081 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-18067


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References

  1. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, "Quantum nature of a strongly coupled single quantum dot-cavity system," Nature 445, 896−899 (2007). [CrossRef] [PubMed]
  2. T. Yoshie, A. Scherer, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, "Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity," Nature 432, 200−203 (2004). [CrossRef] [PubMed]
  3. S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, "Self-tuned quantum dot gain in photonic crystal lasers," Phys. Rev. Lett. 96, 127404 (2006). [CrossRef] [PubMed]
  4. K. Kounoike, M. Yamaguchi, M. Fujita, T. Asano, J. Nakanishi, and S. Noda, "Investigation of spontaneous emission from quantum dots embedded in a two-dimensional photonic-crystal slab," Electron. Lett. 41, 1402−1403 (2005). [CrossRef]
  5. M. Yamaguchi, T. Asano, and S. Noda, "Origin of unexpected light emission in a coupled system of a photonic-crystal nanocavity and a quantum dot," presented at the 8th International Conference on Physics of Light-Matter Coupling in Nanostructures, Tokyo, Japan, 7-11 April 2008. [PubMed]
  6. K. Srinivasan and O. Painter, "Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system," Nature 450, 862−865 (2007). [CrossRef] [PubMed]
  7. K. Srinivasan and O. Painter, "Mode coupling and cavity-quantum-dot interactions in a fiber-coupled microdisk cavity," Phys. Rev. A 75, 023814 (2007). [CrossRef]
  8. P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, and A. Imamoğlu, "A quantum dot single-photon turnstile device," Science 290, 2282−2285 (2000). [CrossRef] [PubMed]
  9. D. Press, S. Götzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, "Photon antibunching from a single quantum-dot-microcavity system in the strong coupling regime," Phys. Rev. Lett. 98, 117402 (2007). [CrossRef] [PubMed]
  10. J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, "Strong coupling in a single quantum dot-semiconductor microcavity system," Nature 432, 197−200 (2004). [CrossRef] [PubMed]
  11. Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944−947 (2003). [CrossRef] [PubMed]
  12. B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207−210 (2005). [CrossRef]
  13. Y. Arakawa and H. Sakai, "Multidimensional quantum well laser and temperature dependence of its threshold current," Appl. Phys. Lett. 40, 939−941 (1982). [CrossRef]
  14. M. Tabuchi, S. Noda, and A. Sasaki, "Mesoscopic structure in lattice-mismatched heteroepitaxial interface layers," in Science and Technology of Mesoscopic Structures (eds Namba, S., Hamaguchi, C. & Ando, T.) 379−384 (Springer Verlag, Tokyo, 1992).
  15. D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer, and D. Park, "Fine structure splitting in the optical spectra of single GaAs quantum dots," Phys. Rev. Lett. 76, 3005−3008 (1996). [CrossRef] [PubMed]
  16. M. Bayer and A. Forchel, "Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots," Phys. Rev. B 65, 041308 (2002). [CrossRef]
  17. E. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).
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