Quantum dots in semiconductor photonic reservoirs are important systems for studying and exploiting quantum optics on a chip, and it is essential to understand fundamental concepts such as spontaneous emission. According to Fermi’s golden rule, the spontaneous emission rate of a quantum emitter weakly coupled to a structured photonic reservoir is proportional to the local density of photon states (LDOS) at the emitter’s position and frequency. Coupling to lattice vibrations or phonons, however, significantly modifies the emission properties of a quantum dot (QD) compared to an isolated emitter (e.g., an atom). In the regime of phonon-dressed reservoir coupling, we demonstrate why and how the broadband frequency dependence of the LDOS determines the spontaneous emission rate of a QD, manifesting in a dramatic breakdown of Fermi’s golden rule. We analyze this problem using a polaron transformed master equation and consider specific examples of a semiconductor microcavity and a coupled-cavity waveguide. For a leaky single cavity resonance, we generalize Purcell’s formula to include the effects of electron–phonon coupling, and for a waveguide, we show a suppression and a 200-fold enhancement of the photon emission rate. These results have important consequences for modeling and understanding emerging QD experiments in a wide range of photonic reservoir systems.
© 2015 Optical Society of America
Recent developments in chip-scale quantum optical technologies  have generated substantial interest in quantum dots (QDs), which act as “artificial atoms” in solid-state media. However, electron–phonon coupling in solid-state media has been shown to significantly modify the emission properties of a QD as compared to an isolated atom . Studying phonon interactions in governing the emission properties of QDs has been an intense area of research, leading to a number of effects beyond a simple pure-dephasing model . For driven QD excitons yielding Rabi oscillations, phonon coupling manifests in damping and frequency shifts [4 –6]. In QD-cavity systems, phonons cause intensity-dependent broadening of Mollow side-bands , off-resonant cavity feeding , and asymmetric vacuum Rabi doublets [9,10].
Semiconductor QDs coupled to structured photonic reservoirs provide a promising platform for tailoring light–matter interaction in a solid-state environment. One of the primary interests in coupling QDs to structured reservoirs is for modifying the spontaneous emission rate (SE), , via the Purcell effect [11,12]. Photonic crystals are a paradigm example of a structured photonic reservoir, and both photonic crystal cavities [Fig. 1(a)] and coupled-cavity optical waveguide (CROW) [Fig. 1(b)] structures have been investigated for modifying QD SE rates [12,13]. For an unstructured reservoir, remains unchanged in the presence of phonons . For structured reservoirs, previous theories have assumed phonon processes to be much faster than all relevant system dynamics [15,16], thus restricting them to structures with sharp variations of photon local density of states (LDOS) (e.g., high- cavity or photonic band edge). A primary example of a structured reservoir is a microcavity, and existing theories [15,17,18] treat the cavity mode as a system operator and find that phonons modify the QD-cavity coupling rate, through [17,19], where is the thermal average of the coherent phonon bath displacement operators . Hence the Purcell factor is believed to scale as . However, such theories do not apply to large cavities, where is the cavity decay rate, and one would expect to recover the result that —and thus —are not affected by phonons. Moreover, for an arbitrary photonic bath medium, it is not known how phonons affect the SE rates, yet clearly such an effect is of significant fundamental interest and is also important for understanding emerging QD experiments.
In this Letter we introduce a self-consistent master equation (ME) approach with both phonon and photon reservoirs included, and we explore in detail the influence of a photon reservoir on the phonon-modified SE rate. When the relaxation times of the photon and phonon baths compare, the frequency dependence of the LDOS is found to dictate how phonons modify the SE rates, causing a clear breakdown of Fermi’s golden rule. Such an effect arises due to dressing of QD excitations by phonons in a solid-state medium. Importantly, our theory can be applied to any general LDOS medium, and as specific examples we consider a semiconductor microcavity and a slow-light coupled-cavity waveguide.
We model the QD as a two-level system interacting with an inhomogeneous semiconductor-based photonic reservoir and an acoustic phonon bath  (see Fig. 1(c)). Assuming the QD of dipole moment at spatial position , the total Hamiltonian of the system in a frame rotating at the QD exciton frequency, , is 20]. To include phonon interactions to all orders, we perform the polaron transform on the Hamiltonian given by , where . Assuming weak-to-intermediate coupling with the photon bath, we derive a time-local polaron ME for the QD reduced density operator , using the Born approximation. The usual incoherent terms from the photon reservoir can be written as . Subsequently, the phonon-modified SE decay rate can be obtained from , yielding the familiar Lindblad superoperator, , where the SE decay rate is derived to be (see Supplement 1, Section 2) 21]. The photon bath correlation function can be expressed in terms of the photon-reservoir spectral function , with . In the Markov limit (), Eq. (2) generalizes Fermi’s golden rule for QD SE, since the LDOS at various frequencies can now contribute to the phonon-modified SE rate, . Similar expressions for the SE rate in the frequency domain have been used to explain mode pulling effects in QD cavities . In the absence of phonon coupling, the SE decay rate of the QD in a structured photon reservoir reduces to , where . McCutcheon and Nazir  use a similar approach to show that for a free-space bath function.
To appreciate how phonons modify the SE rate in a structured photonic reservoir, we first consider a simple Lorentzian cavity [cf. Fig. 1(a)]. For a single cavity mode, in a dielectric with a dielectric constant ,2) and (3) we obtain 19] and . We can now generalize the Purcell factor (PF) for the enhanced SE rate of a QD in a semiconductor cavity:
To obtain a mean-field approximation in the high- limit, i.e., in the limit (where is the phonon relaxation time), then , where the phonon-mediated cavity scattering rate . This is exactly the expression for the SE rate [15,21], which is derived (see Supplement 1, Section 3) with a polaron ME  when treating the cavity mode as a system operator, with phenomenological damping and using the bad cavity limit (). Our theory thus not only recovers previous (polaronic) cavity-QED results in the appropriate limit, but also reveals a fundamental limitation of these approaches for sufficiently large (low-) cavities. Specifically, when the cavity relaxation time becomes comparable to , or smaller, these formalisms break down.
To test this hypothesis, consider the example of three different cavity decay rates, , and 2.4 meV, corresponding to a factor of around 23000, 2300, and 600, respectively (with ). In Fig. 2 we plot the PF (left panels) and phonon-modified SE factor (right panels). For each cavity, we investigate two different bath temperatures (4 and 40 K), and the dashed lines on the left panels represent PFs without phonon modification. The clear asymmetry in about the LDOS peak arises since phonon emission is more probable than absorption  at low temperatures. The main results can be explained by writing , where is the coherently renormalized bare SE rate and arises due to local () sampling of photonic LDOS, while accounts for the nonlocal contribution (i.e., frequencies that would not contribute to a Fermi’s golden rule expression). Note that when is small, . Due to the nonlocal component, the reduction of the SE rate is always , at zero detuning. At large detunings, the nonlocal component dominates, leading to an overall enhancement of the SE rate. Figures 2(b), 2(d), and 2(f) show that varies significantly over several meV. The dashed lines on the right panels represent , which evidently differs from our full calculations in the limit [Figs. 2(d) and 2(f)]. This is because the reservoir structure of the photon bath is properly accounted in the present calculations and is not approximated as a high- cavity (also see Supplement 1, Section 1). It is also important to note that low- (several hundred) cavities are commonly employed for measuring the vertical emission from QDs in planar cavities  and for modifying the SE rates in simple photonic crystal cavities , while intermediate () cavities are used for all-optical switching  and enabling single photon sources .
We now depart from the simple cavity, and consider the richer case of a photonic crystal CROW [cf. Fig. 1(b)]. Photonic crystal waveguides [Fig. 1(b)] can be used for realizing slow-light propagation  and for manipulating the emission properties of embedded QDs for on-chip single photon emission [26,27], with a number of recent experiments emerging. Current theories in this regime either ignore phonon coupling or assume a coherent renormalization factor (), consistent with Fermi’s golden rule (modified by a mean-field reduction factor). For the photon reservoir function, we adopt a model LDOS for a CROW , and use an analytical tight-binding technique to calculate the photonic band structure ; the photon reservoir spectral function is obtained analytically, as29], represents the mode-edge frequencies of the waveguide [see Fig. 3(a)], are effective damping rates, and is the mode volume of a single cavity. The photonic LDOS has a rich nontrivial spectral structure compared to a smooth Lorentzian cavity, especially within the bandwidth of the phonon bath (which spans about 5–10 meV; see Supplement 1, Section 1). For our calculations, we use parameters that closely represent a CROW made up of a local width modulation of a line-defect photonic crystal cavity , which yields a band structure  consistent with experiments . In Fig. 3(a), we show the calculated PF with (solid) and without (dashed) phonons, using a bath temperature of 40 K. We see that in contrast to current theories, phonons significantly influence the spectral shape of the SE rates, causing a reduction at the mode edges and a significant enhancement inside and outside the waveguide band. Figure 3(b) shows a slight asymmetry (which increases with decreasing temperatures) in , which is again due to unequal phonon emission and absorption rates. The spectral dependence of in the waveguide can be qualitatively understood by treating approximately as a sum of two Lorentzians located at the mode edges (). The corresponding is a sum of two exponentially damped oscillatory functions, where is the bandwidth of the waveguide mode-edge LDOS.
At a sharp mode edge of the LDOS, the contribution from local () photonic LDOS dominates and . Away from the mode edge [Figs. 3(a) and 3(b)], the SE rate is enhanced due to nonlocal effects. This simple model discussion is, however, approximate as in reality the mode-edge LDOS is non-Lorentzian. For a symmetric Lorentzian with the same bandwidth, , damps much faster than initially and damps very slowly thereafter [Fig. 3(a), inset]. The long time decay rate is set by the linewidth of the sharper side of the mode-edge LDOS (). This non-Lorentzian mode edge in turn leads to a very strong enhancement of the PF () outside the waveguide band [Fig. 3(b)], compared to a symmetric Lorentzian lineshape [see Fig. 2(b)].
In conclusion, we have demonstrated how the frequency dependence of the LDOS of a photonic reservoir determines the extent to which phonons modify the SE of a coupled QD. The relative dynamics between the phonon and the photon bath correlation functions is found to play a fundamentally important role; specifically, when the relaxation times are comparable, phonons strongly modify the emission spectra leading to non-Lorentzian cavity lineshapes and even enhanced SE. These effects are not obtained using the usual Fermi’s golden rule. Our formalism is important for understanding related experiments with QD-cavity systems, such as with photoluminescence intensity measurements with a coherent drive , and is broadly applicable to various photonic reservoir systems.
Natural Sciences and Engineering Research Council of Canada (NSERC).
We thank D. P. S. McCutcheon and A. Nazir for useful discussions and for sharing their results of  prior to publication.
See Supplement 1 for supporting content.
1. H. Kim, R. Bose, T. Shen, G. Solomon, and E. Waks, Nat. Photonics 7, 373 (2013). [CrossRef]
2. S. Weiler, A. Ulhaq, S. M. Ulrich, D. Richter, M. Jetter, P. Michler, C. Roy, and S. Hughes, Phys. Rev. B 86, 241304(R) (2012).
3. B. Krummheuer, V. M. Axt, and T. Kuhn, Phys. Rev. B 65, 195313 (2002).
4. J. Förstner, C. Weber, J. Danckwerts, and A. Knorr, Phys. Rev. Lett. 91, 127401 (2003). [CrossRef]
5. A. J. Ramsay, T. M. Godden, S. J. Boyle, E. M. Gauger, A. Nazir, B. W. Lovett, A. M. Fox, and M. S. Skolnick, Phys. Rev. Lett. 105, 177402 (2010). [CrossRef]
6. L. Monniello, C. Tonin, R. Hostein, A. Lemaitre, A. Martinez, V. Voliotis, and R. Grousson, Phys. Rev. Lett. 111, 026403 (2013). [CrossRef]
7. S. M. Ulrich, S. Ates, S. Reitzenstein, A. Löffler, A. Forchel, and P. Michler, Phys. Rev. Lett. 106, 247402 (2011). [CrossRef]
8. A. Majumdar, E. D. Kim, Y. Gong, M. Bajcsy, and J. Vučković, Phys. Rev. B 84, 085309 (2011).
9. F. Milde, A. Knorr, and S. Hughes, Phys. Rev. B 78, 035330 (2008).
10. Y. Ota, S. Iwamoto, N. Kumagai, and Y. Arakawa, “Impact of electron-phonon interactions on quantum-dot cavity quantum electrodynamics,” arXiv:0908.0788 (2009).
11. E. M. Purcell, Phys. Rev. 69, 37 (1946). [CrossRef]
12. D. Englund, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vučković, Phys. Rev. Lett. 95, 013904 (2005). [CrossRef]
13. T. Lund-Hansen, S. Stobbe, B. Julsgaard, H. Thyrrestrup, T. Sünner, M. Kamp, A. Forchel, and P. Lodahl, Phys. Rev. Lett. 101, 113903 (2008). [CrossRef]
14. D. P. S. McCutcheon and A. Nazir, Phys. Rev. Lett. 110, 217401 (2013); see also the Supplementary Material. [CrossRef]
15. P. Kaer, T. R. Nielsen, P. Lodahl, A.-P. Jauho, and J. Mørk, Phys. Rev. B 86, 085302 (2012).
16. C. Roy and S. John, Phys. Rev. A 81, 023817 (2010). [CrossRef]
17. C. Roy and S. Hughes, Phys. Rev. Lett. 106, 247403 (2011). [CrossRef]
18. U. Hohenester, Phys. Rev. B 81, 155303 (2010).
19. I. Wilson-Rae and A. Imamoğlu, Phys. Rev. B 65, 235311 (2002).
20. S. Scheel, L. Knöll, and D.-G. Welsch, Phys. Rev. A 60, 4094 (1999). [CrossRef]
21. C. Roy and S. Hughes, Phys. Rev. X 1, 021009 (2011). [CrossRef]
22. D. Valente, J. Suffczyński, T. Jakubczyk, A. Dousse, A. Lemaître, I. Sagnes, L. Lanco, P. Voisin, A. Auffèves, and P. Senellart, Phys. Rev. B 89, 041302(R) (2014).
23. E. Waks and J. Vučković, Phys. Rev. Lett. 96, 153601 (2006). [CrossRef]
24. C. A. Foell, E. Schelew, H. Qiao, K. A. Abel, S. Hughes, F. C. J. M. van Veggel, and J. F. Young, Opt. Express 20, 10453 (2012). [CrossRef]
25. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, Phys. Rev. Lett. 87, 253902 (2001). [CrossRef]
26. V. S. C. Manga Rao and S. Hughes, Phys. Rev. B 75, 205437 (2007).
27. A. Schwagmann, S. Kalliakos, I. Farrer, J. P. Griffiths, G. A. C. Jones, D. A. Ritchie, and A. J. Shields, Appl. Phys. Lett. 99, 261108 (2011). [CrossRef]
28. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, Opt. Lett. 24, 711 (1999). [CrossRef]
29. D. P. Fussell and M. M. Dignam, Phys. Rev. A 76, 053801 (2007). [CrossRef]
30. E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, Appl. Phys. Lett. 88, 041112 (2006). [CrossRef]
31. D. P. Fussell, S. Hughes, and M. M. Dignam, Phys. Rev. B 78, 144201 (2008).
32. M. Notomi, E. Kuramochi, and T. Tanabe, Nat. Photonics 2, 741 (2008). [CrossRef]
33. K. Roy-Choudhury and S. Hughes, Opt. Lett. 40, 1838 (2015).