## Photon emission by nanocavity-enhanced quantum anti-Zeno effect in solid-state cavity quantum-electrodynamics

Optics Express, Vol. 16, Issue 22, pp. 18067-18081 (2008)

http://dx.doi.org/10.1364/OE.16.018067

Acrobat PDF (335 KB)

### 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

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]

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]

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]

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]

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,” Nature Mater. **4**, 207–210 (2005). [CrossRef]

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]

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]

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. M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In_{0.60}Ga_{0.40}As/GaAs self-assembled quantum dots,” Phys. Rev. B **65**, 041308 (2002). [CrossRef]

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

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]

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

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]

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]

*Q*factor, and control of spontaneous emission. This emission mechanism successfully explains the origin of strong photon emission at a cavity mode largely detuned from a QD, which has so far been considered as a counterintuitive, prima facie non-energy-conserving, light emission phenomenon in these systems [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]

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]

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

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]

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

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

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]

**405**, 546–550 (2000). [CrossRef] [PubMed]

**87**, 270405 (2001). [CrossRef]

*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.

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]

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]

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]

*Q*factor and control of spontaneous emission.

## 2. Theoretical analysis

### 2.1 Analysis model

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. M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In_{0.60}Ga_{0.40}As/GaAs self-assembled quantum dots,” Phys. Rev. B **65**, 041308 (2002). [CrossRef]

*ħω*

*) can interact with a single-mode electromagnetic field formed in a cavity (with resonant energy*

_{TLS}*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)).

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]

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]

*ρ*

*is a reduced density operator,*

_{S}*H*

*represents the interaction Hamiltonian between the TLS and the cavity mode (non-Markov process), and the*

_{S}*L*

*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 2*

_{m}*Γ*

_{spon}and 2

*Γ*

_{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]

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

*ħΓ*

_{spon}=0.044 µeV (1/2

*Γ*

_{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]

*ħΓ*

_{cav}=100 µeV (

*Q*=1.1×10

^{4}) [1

**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]

*g*, we used a value of

*ħg*=76 µeV (2

*π*/2

*g*=27.0 ps), which is reasonable for a QD that is aligned to the centre of the cavity used in the experiments [1

**445**, 896–899 (2007). [CrossRef] [PubMed]

*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. M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In_{0.60}Ga_{0.40}As/GaAs self-assembled quantum dots,” Phys. Rev. B **65**, 041308 (2002). [CrossRef]

*ħγ*

_{phase}increases larger than 100 µeV [16

_{0.60}Ga_{0.40}As/GaAs self-assembled quantum dots,” Phys. Rev. B **65**, 041308 (2002). [CrossRef]

## 2.2 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

*Γ*

_{total}≡

*Γ*

_{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},

_{cav}≡

*ω*

_{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

**87**, 270405 (2001). [CrossRef]

*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.

## 2.3 Emitted photon spectrum

*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:

*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.

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

*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

*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 2

*ħΓ*

_{total}=2

*ħ*(

*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*,

*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.

*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:

*W*

_{TLS}indicate the direct emission rate to free space and an approximated expression for the Purcell effect [17], 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

*δω*

_{TLS},

_{cav}| ≫

*g*,

*Γ*

_{total}. From this expression, a QD combined with a nanocavity gives rise to a highly advantageous situation for the enhancement of

*F*. This is due to the low photonic damping rate of

*Γ*

_{cav}(owing to the high quality factor) and a large coupling constant

*g*(owing to the small modal volume). Hence, the strong photon emission [1

**445**, 896–899 (2007). [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]

**445**, 896–899 (2007). [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]

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,” Nature Mater. **4**, 207–210 (2005). [CrossRef]

*F*increases further, because the

*Γ*

_{spon}is suppressed by the photonic bandgap effect [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]

27. 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]

*F*≈0.24 can be obtained for |

*δω*

_{TLS},

_{cav}| ~4 meV with

*ħγ*

_{phase}=35 µeV, which suggests that 24% of the total emitted photons are detected as cavity mode emission even though the detuning is large (Figs. 4(b) and 4(c)). This agrees with the experimental results (see also Appendix E).

## 3. Discussions

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. 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]

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]

**432**, 197–200 (2004). [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]

**432**, 197–200 (2004). [CrossRef] [PubMed]

*ħg*~30 µeV with (2

*ħΓ*

_{spon}, 2

*ħΓ*

_{cav},

*γ*

_{phase})=(0.044 µeV, 100 µeV, 35 µeV), the factor

*F*from each QD is estimated to be ~1.4% for 10 meV detuning and ~0.4% for 20 meV detuning. The combined effect of tens to hundreds of QDs results in a contribution to the cavity mode comparable to, or even exceeding, the individual QD exciton peaks. This estimation is reasonable compared with experimental results. Next, we consider the case for various cavity geometries. While the nanocavity we discussed above was photonic-crystal based, the nanocavity-enhanced AZE emission is also important for the other cavity geometries. In the case of a microdisk [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]

*g*), the total number of QDs is expected to increase due to the larger modal volume. Therefore, the total contribution of all the QDs can result in the strong cavity mode emission reaching or exceeding the individual QD emissions. In the case of micropillars or microdisks for photon turnstile devices [8

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]

*g*

^{(2)}(

*τ*=0) does not become zero when the QD is on resonance with the cavity mode. This unexpected second-order correlation in the photon-turnstile device can also be explained as a consequence of the nanocavity-enhanced AZE emission from the off-resonant QDs. The reference points out that using higher pump powers increases the degree of unexpected correlation. This is interpreted as the AZE emission when we consider that higher pump powers cause many carriers and higher rates of pure-dephasing.

## 4. Conclusion

## Appendix A:

*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

## Appendix B:

*H*

_{T}can be decomposed as

*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, 24] in the interaction picture can be obtained from equations (11) and (12) under the assumption that

*ρ*

_{T}(

*t*) ≈

*ρ*

_{S}(

*t*)⊗

*ρR*(0):

*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,

*dρ*

_{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).

*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

*ψ*̂(

*r*) represents the electron field operator.

*ψ*̂(

*r*) can be decomposed as

*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*

*is the annihilation operator of the*

_{m}*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]

*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

*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 [

*ĉ*†

_{i}*ĉ*

*,*

_{i}*H*

_{11},

_{El}]=[

*ĉ*†

_{i}*ĉ*

*,*

_{i}*H*

_{22},

_{El}]=0, which suggests that these Hamiltonians do not change the temporal differentiation of <

*ĉ*†

_{d}*ĉ*

*(*

_{i}*t*)> [26

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

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

*ĉ*†

_{i}*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] yield

*denotes the angular frequency shifts of the*

_{i}*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 Δ

_{i}in the analytical expressions and have defined the total pure dephasing rate as

*γ*

_{phase}≡

*γ*

_{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]

*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

*δω*

_{TLS},

_{cav}≡

*ω*

_{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

## Appendix C:

*ĉ*†

_{2}

*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

*Γ*

_{total}≡

*Γ*

_{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.

*Λ*

_{±}expressed as

*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:

*S*(

*ω*). Following the Wiener- Khintchine theorem, the total emission spectrum

*S*(

*ω*) can be expressed as

*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}

*a*â

_{cav}(

*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

*τ*=0, the first term 2

*Γ*

_{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 2

*Γ*

_{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

*â*

^{†}

_{cav}(

*t*)

*a*ĉ

_{cav}(

*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 [23–25

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]

*ĉ*

^{†}

_{2}

*ĉ*

_{2}(

*t*)> and <

*â*

^{†}

_{cav}

*a*ĉ

_{cav}(

*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:

*γ*

_{±}and

*f*

_{α}(

*γ*

_{±}) (

*α*=spon, or cav) as

*f*(

*γ*

_{±})≡

*f*

_{cav}(

*γ*

_{±})+

*f*

_{spon}(

*γ*

_{±}).

## Appendix E:

**445**, 896–899 (2007). [CrossRef] [PubMed]

**445**, 896–899 (2007). [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]

**445**, 896–899 (2007). [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]

**98**, 117402 (2007). [CrossRef] [PubMed]

**445**, 896–899 (2007). [CrossRef] [PubMed]

**445**, 896–899 (2007). [CrossRef] [PubMed]

**98**, 117402 (2007). [CrossRef] [PubMed]

**98**, 117402 (2007). [CrossRef] [PubMed]

**98**, 117402 (2007). [CrossRef] [PubMed]

**445**, 896–899 (2007). [CrossRef] [PubMed]

**445**, 896–899 (2007). [CrossRef] [PubMed]

**445**, 896–899 (2007). [CrossRef] [PubMed]

## Acknowledgment

## References and links

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 |

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 |

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. |

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. |

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 |

7. | K. Srinivasan and O. Painter, “Mode coupling and cavity-quantum-dot interactions in a fiber-coupled microdisk cavity,” Phys. Rev. A |

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 |

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. |

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 |

11. | Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature |

12. | B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Mater. |

13. | Y. Arakawa and H. Sakai, “Multidimensional quantum well laser and temperature dependence of its threshold current,” Appl. Phys. Lett. |

14. | M. Tabuchi, S. Noda, and A. Sasaki, “Mesoscopic structure in lattice-mismatched heteroepitaxial interface layers,” in |

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. |

16. | M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In |

17. | E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. |

18. | I. I. Rabi, “Space quantization in a gyrating magnetic field,” Phys. Rev. |

19. | G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, “Vacuum Rabi splitting in semiconductors,” Nature Phys. |

20. | A. G. Kofman and G. Kurizki, “Acceleration of quantum decay processes by frequent observations,” Nature |

21. | G. S. Agarwal, M. O. Scully, and H. Walther, “Accelerating decay by multiple 2 |

22. | A. G. Kofman and G. Kurizki, “Universal dynamical control of quantum mechanical decay: modulation of the coupling to the continuum,” Phys. Rev. Lett. |

23. | H. J. Carmichael, |

24. | W. H. Louisell, |

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 |

26. | W. H. Zurek, “Pointer basis of quantum apparatus,” Phys. Rev. D |

27. | M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S. Noda, “Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals,” Science |

28. | J. L. Pan, “Reduction of the Auger rate in semiconductor quantum dots,” Phys. Rev. B |

29. | V. B. Braginsky and F. Y. Khalili, |

**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

Sort: Year | Journal | Reset

### References

- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207−210 (2005). [CrossRef]
- Y. Arakawa and H. Sakai, "Multidimensional quantum well laser and temperature dependence of its threshold current," Appl. Phys. Lett. 40, 939−941 (1982). [CrossRef]
- 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).
- 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]
- 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]
- E. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).
- I. I. Rabi, "Space quantization in a gyrating magnetic field," Phys. Rev. 51, 652 (1937). [CrossRef]
- G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, and A. Scherer, "Vacuum Rabi splitting in semiconductors," Nat. Phys. 2, 81−90 (2006). [CrossRef]
- A. G. Kofman and G. Kurizki, "Acceleration of quantum decay processes by frequent observations," Nature 405, 546−550 (2000). [CrossRef] [PubMed]
- G. S. Agarwal, M. O. Scully, and H. Walther, "Accelerating decay by multiple 2π pulses," Phys. Rev. A 63, 044101 (2001). [CrossRef]
- 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]
- H. J. Carmichael, An Open Systems Approach to Quantum Optics (Springer-Verlag, Berlin, 1993).
- W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1990).
- 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]
- W. H. Zurek, "Pointer basis of quantum apparatus," Phys. Rev. D 24, 1516−1525 (1981). [CrossRef]
- 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]
- J. L. Pan, "Reduction of the Auger rate in semiconductor quantum dots," Phys. Rev. B 46, 3977-3998 (1992). [CrossRef]
- V. B. Braginsky and F. Y. Khalili, Quantum Measurement (Cambridge University Press, Cambridge, 1992). [CrossRef]

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