## Detuning-dependent Mollow triplet of a coherently-driven single quantum dot |

Optics Express, Vol. 21, Issue 4, pp. 4382-4395 (2013)

http://dx.doi.org/10.1364/OE.21.004382

Acrobat PDF (1222 KB)

### Abstract

We present both experimental and theoretical investigations of a laser-driven quantum dot (QD) in the dressed-state regime of resonance fluorescence. We explore the role of phonon scattering and pure dephasing on the detuning-dependence of the Mollow triplet and show that the triplet sidebands may spectrally broaden or narrow with increasing detuning. Based on a polaron master equation approach, which includes electron-phonon interaction nonperturbatively, we derive a fully analytical expression for the spectrum. With respect to detuning dependence, we identify a crossover between the regimes of spectral sideband narrowing or broadening. We also predict regimes of phonon-induced squeezing and anti-squeezing of the spectral resonances. A comparison of the theoretical predictions to detailed experimental studies on the laser detuning-dependence of Mollow triplet resonance emission from single In(Ga)As QDs reveals excellent agreement.

© 2013 OSA

## 1. Introduction

1. A. Muller, E. B. Flagg, P. Bianucci, X. Y. Wang, D. G. Deppe, W. Ma, J. Zhang, G. J. Salamo, M. Xiao, and C. K. Shih, “Resonance fluorescence from a coherently driven semiconductor quantum dot in a cavity,” Phys. Rev. Lett. **99**, 187402, (2007). [CrossRef] [PubMed]

3. A. Nick Vamivakas, Yong Zhao, Chao-Yang Lu, and Mete Atatüre, “Spin-resolved quantum-dot resonance fluorescence,” Nat. Physics **5**, 198–202 (2009). [CrossRef]

4. A. Kiraz, M. Atatüre, and A Imamoğlu, “Quantum-dot single-photon sources: Prospects for applications in linear optics quantum-information processing,” Phys. Rev. A **69**, 032305 (2004). [CrossRef]

5. S. Ates, S. M. Ulrich, S. Reitzenstein, A. Löffler, A. Forchel, and P. Michler, “Post-selected indistinguishable photons from the resonance fluorescence of a single quantum dot in a microcavity,” Phys. Rev. Lett. **103**, 167402 (2009). [CrossRef] [PubMed]

6. C. Matthiesen, A. N. Vamivakas, and M. Atatüre, “Subnatural linwidth single photons from a quantum dot,” Phys. Rev. Lett. **108**, 093602 (2012). [CrossRef] [PubMed]

7. H. S. Nguyen, C. Voisin, P. Roussignol, C. Diedrichs, and G. Cassabois, “Ultra-coherent single photon source,” App. Phys. Lett. **99**, 261904 (2011). [CrossRef]

8. A. Ulhaq, S. Weiler, S. M. Ulrich, R. Roßbach, M. Jetter, and P. Michler, “Cascaded single-photon emission from resonantly excited quantum dots,” Nat. Photonics **6**, 238 (2012). [CrossRef]

9. C. Roy and S. Hughes, “Influence of electron-acoustic-phonon scattering on intensity power broadening in a coherently driven quantum-dot-cavity system,” Phys. Rev. X **1**, 021009 (2011). [CrossRef]

10. C. Förstner, C. Weber, J. Danckwerts, and A. Knorr, “Phonon-assisted damping of Rabi oscillations in semiconductor quantum dots,” Phys. Rev. Lett. **91**, 127401 (2003). [CrossRef] [PubMed]

12. K. J. Ahn, J. Förstner, and A. Knorr, “Resonance fluorescence of semiconductor quantum dots: Signatures of the electron-phonon interaction,” Phys. Rev. B **71**, 153309 (2005). [CrossRef]

13. A. Vagov, M. D. Croitoru, V. M. Axt, T. Kuhn, and F. M. Peeters, “Nonmonotonous field dependence of damping and reappearance of rabi oscillations in quantum dots,” Phys. Rev. Lett. **98**, 227403 (2007). [CrossRef] [PubMed]

14. A. Nazir, “Photon statistics from a resonantly driven quantum dot,” Phys. Rev. B **78**, 153309, (2008). [CrossRef]

15. C. Roy and S. Hughes, “Phonon-dressed Mollow triplet in the regime of cavity quantum electrodynamics: Excitation-induced dephasing and nonperturbative cavity feeding effects,” Phys. Rev. Lett. **106**, 247403 (2011). [CrossRef] [PubMed]

16. C. Roy, H. Kim, E. Waks, and S. Hughes, “Anomalous phonon-mediated damping of a driven quantum dot embedded in a high-Q microcavity,” Photon Nanostruct: Fundam. Appl. **10**, 359 (2012). [CrossRef]

17. A. J. Ramsay, A. V. Gopal, E. M. Gauger, A. Nazir, B. W. Lovett, A. M. Fox, and M. S. Skolnick, “Damping of exciton rabi rotations by acoustic phonons in optically excited InGaAs/GaAs quantum dots,” Phys. Rev. Lett. **104**, 017402 (2010). [CrossRef] [PubMed]

18. S. M. Ulrich, S. Ates, S. Reitzenstein, A. Löffler, A. Forchel, and P. Michler, “Dephasing of Mollow triplet sideband emission of a resonantly driven quantum dot in a microcavity,” Phys. Rev. Lett. **106**, 247402, (2011). [CrossRef] [PubMed]

15. C. Roy and S. Hughes, “Phonon-dressed Mollow triplet in the regime of cavity quantum electrodynamics: Excitation-induced dephasing and nonperturbative cavity feeding effects,” Phys. Rev. Lett. **106**, 247403 (2011). [CrossRef] [PubMed]

*et al.*[18

18. S. M. Ulrich, S. Ates, S. Reitzenstein, A. Löffler, A. Forchel, and P. Michler, “Dephasing of Mollow triplet sideband emission of a resonantly driven quantum dot in a microcavity,” Phys. Rev. Lett. **106**, 247402, (2011). [CrossRef] [PubMed]

*narrowing*in dependence of laser-excitation detuning from the bare emitter resonance had to be left open for further in-depth theoretical analysis.

## 2. Sample structure and experimental procedure

*λ*-cavity, sandwiched between 29 (4) periods of

*λ*/4-thick AlAs/GaAs layers as the bottom (top) distributed Bragg reflectors (DBRs). For our experimental investigations, the sample is kept in a Helium flow cryostat providing highly stable temperature of

*T*= 6.0 ± 0.5 K. Suppression of parasitic laser stray-light is achieved by use of an orthogonal geometry between QD excitation and emission detection. In addition, polarization suppression and spatial filtering via a pinhole is applied in the detection path. Resonant (tunable) QD excitation is achieved by a narrow-band (≈ 500 kHz) continuous-wave (cw) Ti:Sapphire ring laser. For high-resolution spectroscopy (HRPL) of micro-photoluminescence (

*μ*-PL) we employ a scanning Fabry-Pérot interferometer with

*μ*eV as described earlier [5

5. S. Ates, S. M. Ulrich, S. Reitzenstein, A. Löffler, A. Forchel, and P. Michler, “Post-selected indistinguishable photons from the resonance fluorescence of a single quantum dot in a microcavity,” Phys. Rev. Lett. **103**, 167402 (2009). [CrossRef] [PubMed]

8. A. Ulhaq, S. Weiler, S. M. Ulrich, R. Roßbach, M. Jetter, and P. Michler, “Cascaded single-photon emission from resonantly excited quantum dots,” Nat. Photonics **6**, 238 (2012). [CrossRef]

18. S. M. Ulrich, S. Ates, S. Reitzenstein, A. Löffler, A. Forchel, and P. Michler, “Dephasing of Mollow triplet sideband emission of a resonantly driven quantum dot in a microcavity,” Phys. Rev. Lett. **106**, 247402, (2011). [CrossRef] [PubMed]

## 3. Experimental results: Detuning-dependent resonance fluorescence

*γ*. The incoherent spectrum of the resulting dressed state is the characteristic Mollow triplet [19

19. B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. **188**, 169–175 (1969). [CrossRef]

*ω*

*–*

_{L}*ω*

*= 0 from the QD exciton resonance (see Fig. 1(a), green center trace), the spectrum is composed of the central*

_{x}*Rayleigh line*“R” at the bare emitter energy

*ω*

_{0}and two symmetric satellite peaks, i.e. the

*Three-Photon Line*“T” and the

*Fluorescence Line*“F ” at

*ω*

_{0}± Ω

*, respectively. Ω*

_{r}*denotes the effective Rabi frequency including renormalization effects from the phonon bath as discussed in the theory section below.*

_{r}*P*

_{0}= 500

*μ*W (Ω

*∝ (*

_{r}*P*

_{0})

^{1/2}= const.) are depicted in Fig. 1(a). According to theory (see, e.g., Ref. [3

3. A. Nick Vamivakas, Yong Zhao, Chao-Yang Lu, and Mete Atatüre, “Spin-resolved quantum-dot resonance fluorescence,” Nat. Physics **5**, 198–202 (2009). [CrossRef]

*ω*

_{0}modifies the dressed emission. Besides the center transition at

*ω*

_{0}+ Δ the two sideband frequencies become

*ω*

_{0}+ Δ ± Ω, where

## 4. Theory

### 4.1. Hamiltonian, polaron master equation and phonon-induced scattering rates

*ω*

*, the model Hamiltonian (excluding QD zero-phonon line decay mechanisms) is where*

_{L}*σ*

^{+/−}(

*σ*

*=*

^{z}*σ*

^{+}

*σ*

^{−}−

*σ*

^{−}

*σ*

^{+}) are the Pauli operators of the exciton;

*η*

*is the exciton pump rate, and*

_{x}*λ*(assumed real) is the coupling strength of the electron-phonon interaction. We assume that only one exciton will be coherently excited in the spectral region of interest. In order to include electron-phonon scattering nonperturbatively, we transform the above Hamiltonian to the polaron frame. Consequently, we derive a polaron master equation (ME) [9

_{q}9. C. Roy and S. Hughes, “Influence of electron-acoustic-phonon scattering on intensity power broadening in a coherently driven quantum-dot-cavity system,” Phys. Rev. X **1**, 021009 (2011). [CrossRef]

15. C. Roy and S. Hughes, “Phonon-dressed Mollow triplet in the regime of cavity quantum electrodynamics: Excitation-induced dephasing and nonperturbative cavity feeding effects,” Phys. Rev. Lett. **106**, 247403 (2011). [CrossRef] [PubMed]

20. I. Wilson-Rae and A. Imamoğlu, “Quantum dot cavity-QED in the presence of strong electron-phonon interactions,” Phys. Rev. B **65**, 235311 (2002). [CrossRef]

21. D. P. S. McCutcheon and A. Nazir, “Quantum dot Rabi rotations beyond the weak exciton-phonon coupling regime,” New J. Phys. **12**, 113042 (2010). [CrossRef]

9. C. Roy and S. Hughes, “Influence of electron-acoustic-phonon scattering on intensity power broadening in a coherently driven quantum-dot-cavity system,” Phys. Rev. X **1**, 021009 (2011). [CrossRef]

**106**, 247403 (2011). [CrossRef] [PubMed]

*H*′ →

*e*

^{P}He^{−}

*[22*

^{P}22. G. D. Mahan, *Many-Particle Physics* (Plenum, New York, 1990). [CrossRef]

22. G. D. Mahan, *Many-Particle Physics* (Plenum, New York, 1990). [CrossRef]

*B*〉 = 〈

*B*

_{+}〉 = 〈

*B*

_{−}〉, at a bath temperature

*T*= 1/

*k*

_{b}*β*. For convenience, we will assume that the polaron shift is implicitly included in our definition of

*ω*

*below. The operators*

_{x}*X*and

_{g}*X*are defined through

_{u}*X*=

_{g}*h̄*

*η*

*(*

_{x}*σ*

^{−}+

*σ*

^{+}) and

*X*=

_{u}*ih̄*

*η*

*(*

_{x}*σ*

^{+}−

*σ*

^{−}).

*ρ*(

*t*) of the QD-bath system in the second-order Born approximation of the system-reservoir coupling. In the interaction picture, we consider the exciton-photon-phonon coupling

*H*′

*in the Born approximation and trace over the phonon degrees of freedom. The full polaron ME takes the following form [9*

_{I}**1**, 021009 (2011). [CrossRef]

**106**, 247403 (2011). [CrossRef] [PubMed]

20. I. Wilson-Rae and A. Imamoğlu, “Quantum dot cavity-QED in the presence of strong electron-phonon interactions,” Phys. Rev. B **65**, 235311 (2002). [CrossRef]

21. D. P. S. McCutcheon and A. Nazir, “Quantum dot Rabi rotations beyond the weak exciton-phonon coupling regime,” New J. Phys. **12**, 113042 (2010). [CrossRef]

*σ*

_{11}=

*σ*

^{+}

*σ*

^{−}and the time-dependent functions

*G*

_{α}(

*t*) ≡ 〈

*ζ*

_{α}(

*t*)

*ζ*

_{α}(0)〉 are given by

*G*(

_{g}*t*) = 〈

*B*〉

^{2}(cosh[

*ϕ*(

*t*)] − 1) and

*G*(

_{u}*t*) = 〈

*B*〉

^{2}sinh[

*ϕ*(

*t*)] [20

20. I. Wilson-Rae and A. Imamoğlu, “Quantum dot cavity-QED in the presence of strong electron-phonon interactions,” Phys. Rev. B **65**, 235311 (2002). [CrossRef]

22. G. D. Mahan, *Many-Particle Physics* (Plenum, New York, 1990). [CrossRef]

*O*] = 2

*O*

*ρ*

*O*

^{†}−

*O*

^{†}

*O*

*ρ*−

*ρ*

*O*

^{†}

*O*describe dissipation through zero-phonon line (ZPL), radiative decay (

*γ*) and ZPL pure dephasing (

*γ*′), where the latter process is known to increase as a function of temperature [23

23. L. Besombes, K. Kheng, L. Marsal, and H. Mariette, “Acoustic phonon broadening mechanism in single quantum dot emission,” Phys. Rev. B **63**, 155307 (2001). [CrossRef]

27. G. Ortner, D. R . Yakovlev, M. Bayer, S. Rudin, T. L. Reinecke, S. Fafard, Z. Wasilewski, and A. Forchel, “Temperature dependence of the zero-phonon linewidth in InAsGaAs quantum dots,” Phys. Rev. B **70**, 201301(R) (2004). [CrossRef]

22. G. D. Mahan, *Many-Particle Physics* (Plenum, New York, 1990). [CrossRef]

28. B. Krummheuer, V. M. Axt, and T. Kuhn, “Theory of pure dephasing and the resulting absorption line shape in semiconductor quantum dots,” Phys. Rev. B **65**, 195313 (2002). [CrossRef]

29. J. Förstner, C. Weber, J. Danckwerts, and A. Knorr, “Phonon-assisted damping of Rabi oscillations in semiconductor quantum dots,” Phys. Rev. Lett. **91**, 127401 (2003). [CrossRef] [PubMed]

*t*→ ∞, resulting in a Markovian ME where the scattering rates are computed as a function of

*H*′

*[30*

_{S}30. C. Roy and S. Hughes, “Polaron master equation theory of the quantum-dot Mollow triplet in a semiconductor cavity-QED system,” Phys Rev B **85**, 115309 (2012). [CrossRef]

**1**, 021009 (2011). [CrossRef]

*H*′

*appearing in the exponential phase terms above (which we will further justify below) to derive an*

_{S}*effective phonon ME*as follows: Here the pump-driven incoherent scattering processes, mediated by the phonon bath, are obtained from The classical Rabi frequency of the exciton pump, including renormalization effects from the phonon bath, is given by Ω

*= 2*

_{r}*η*

*〈*

_{x}*B*〉 (cf. the bare Rabi frequency Ω

_{0}= 2

*η*

*. The scattering term*

_{x}32. G. S. Agarwal and R. R. Puri, “Cooperative behavior of atoms irradiated by broadband squeezed light,” Phys. Rev. A **41**, 3782 (1990). [CrossRef] [PubMed]

*enhanced radiative decay*, while the

*incoherent excitation*process [15

**106**, 247403 (2011). [CrossRef] [PubMed]

33. Anders Moelbjerg, Per Kaer, Michael Lorke, and Jesper Mørk, “Resonance fluorescence from semiconductor quantum dots: Beyond the Mollow triplet,” Phys. Rev. Lett. **108**, 017401 (2012). [CrossRef] [PubMed]

34. S. Weiler, A. Ulhaq, S. M. Ulrich, D. Richter, M. Jetter, P. Michler, C. Roy, and S. Hughes, “Phonon-Assisted Incoherent Excitation of a Quantum Dot and its Emission Properties,” Phys. Rev. B **86**, 241304(R) (2012). [CrossRef]

*can be significantly smaller than Ω*

_{r}_{0}(for suitably large electron-phonon coupling), even at low temperatures. For example, using InAs QD parameters that closely represent our experimental samples [31

31. In order to derive the phonon scattering rates, we use parameters for InAs/GaAs QDs, which are *ω** _{b}* = 1 meV and

*α*

*/(2*

_{p}*π*)

^{2}= 0.15 ps

^{2}, where

*ω*

*is the high frequency cutoff proportional to the inverse of the typical electronic localization length in the QD and*

_{b}*α*

*is a material parameter (extracted from our experiments) that accounts for the difference between the deformation potential constants between electrons and holes.*

_{p}*T*∼ 6 K, then 〈

*B*〉 ≈ 0.75, and this value decreases (increases) with increasing (decreasing) temperature.

*= 50*

_{r}*μ*eV, example phonon scattering rates are shown in Fig. 2. Within the zoomed region of laser detunings |Δ| < 100

*μ*eV, the relevant phonon scattering rates can clearly be assumed to be constant. Therefore, these values will be treated as constant in the following to compute the analytical Mollow triplets—though this is not a model requirement).

### 4.2. Mollow triplet simulations: Full polaron versus effective phonon ME

*full polaron ME*[Eq. (3)] by the ones appearing in the

*effective phonon ME*[Eq. (4)]. In Fig. 3, a direct comparison between the numerically calculated Mollow triplet based on the full polaron and the effective phonon ME is shown, revealing excellent agreement even for large detunings, Δ = 30

*μ*eV, and high field strengths of Ω

_{0}= 50

*μ*eV. The main reason that one can neglect the pump-dependence of the phase terms in Eq. (3) is that—for the pump values we consider—phonon correlation times are much faster than the inverse Rabi oscillation.

*= 50*

_{r}*μ*eV is already close to the highest achievable experiments to date, and for our purposes can be considered the high-field regime. However, we note that the polaron approach, although nonperturbative, can break down if extremely high field strengths are used such that Ω

*becomes comparable to (or greater than) the phonon cut-off frequency. In this case, other approaches exists such as a variational ME approach [35] and path integral techniques [36*

_{r}36. e.g., see M. Glässl, A. Vagov, S. Lüker, D. E. Reiter, M. D. Croitoru, P. Machnikowski, V. M. Axt, and T. Kuhn, “Long-time dynamics and stationary nonequilibrium of an optically driven strongly confined quantum dot coupled to phonons,” Phys. Rev. B **84**, 195311 (2011). [CrossRef]

*ω*

*(the characteristic phonon cut off frequency), as shown by McCutcheon*

_{b}*et al.*[35], the polaron ME should be rigorously valid for the field strengths that we model.

### 4.3. Optical Bloch equations and analytical fluorescence spectrum

*Ȯ*〉 = tr[

*ρ*̇

*O*] [37], we obtain the following optical Bloch equations: where we define the polarization decay

*δ*

*O*〉 = 〈

*O*〉 −

*O*and

*F*(

**r**) is a geometrical factor. Note that we do not need to add in the phonon correlation phase (

*e*

^{−iϕ(τ)}) when computing the two-time correlation function [30

30. C. Roy and S. Hughes, “Polaron master equation theory of the quantum-dot Mollow triplet in a semiconductor cavity-QED system,” Phys Rev B **85**, 115309 (2012). [CrossRef]

*weakly-coupled*planar cavity mode, in which case 〈

*δ*

*a*

^{†}(

*t*)

*δ*

*a*(

*t*+

*τ*)〉 ∝ 〈

*δ*

*σ*

^{+}(

*t*)

*δ*

*σ*

^{−}(

*t*+

*τ*)〉; so we are actually obtaining the cavity emission which requires no change in the aforementioned correlation functions when coming out of the polaron frame [30

30. C. Roy and S. Hughes, “Polaron master equation theory of the quantum-dot Mollow triplet in a semiconductor cavity-QED system,” Phys Rev B **85**, 115309 (2012). [CrossRef]

*f*(0) ≡ 〈

*δ*

*σ*

^{+}

*δ*

*σ*

^{−}〉

_{ss},

*g*(0) ≡ 〈

*δ*

*σ*

^{+}

*δ*

*σ*

^{+}〉

_{ss}, and

*h*(0) ≡ 〈

*δ*

*σ*

^{+}

*δ*

*σ*

*〉*

^{z}_{ss}, and keep the explicit laser-exciton detuning dependence in the solution. Using the frequency detuning

*δ*

*ω*=

*ω*−

*ω*

*, we can obtain the spectrum lineshape, where*

_{L}*f*,

*g*, and

*h*: These equations are used to obtain

*S*(

*ω*). We stress that the resulting spectrum is an

*exact solution*of our effective phonon ME [Eq. (4)]. The full-width at half-maximum (FWHM) of spectral resonances can be obtained from Eq. (9), though these are too complicated to write down analytically. However, as we have verified, one can simply fit the analytical spectrum to a sum of Lorentzian line shapes (see discussion of Fig. 4) and easily extract the broadening parameters. In the high-field limit, the on-resonance (Δ = 0) FWHM values are

*γ*

_{center}≈

*γ*

_{0}+

*γ*

_{ph}+

*γ*′ +

*γ*

_{cd}for the sideband and center resonances, respectively; these show that the cross dephasing term acts to

*squeeze*the sidebands while broadening the center line or vice versa.

### 4.4. Off-resonant Mollow triplet: Regimes of spectral sideband broadening and narrowing

*γ*′ = 0, a completely symmetric Mollow triplet is expected

*only*if all phonon terms are neglected. Thus, phonon coupling causes an asymmetry for off-resonant driving. Under systematic increase of the excitation-detuning Δ, sideband spectral broadening or narrowing can be achieved depending upon the value of

*r*. In Figs. 4(a) and 4(b) we plot the Mollow triplet as a function of Δ, and extract the FWHM of the sidebands for three values of

*r*. As can be seen,

*r*< 1 (for a suitably small

*γ*′) leads to spectral sideband narrowing, whereas for

*r*> 1 the effect of spectral sideband broadening occurs. Interestingly, the reverse trend occurs for the center resonance (not shown), namely when the sidebands broaden (narrow) then the center line narrows (broadens); so depending on the

*r*value, one can observe squeezing or anti-squeezing of the spectral resonances with increasing Δ (in addition to the squeezing that already occurs from a finite

*γ*

_{cd}).

## 5. Comparison between experiment and theory

*γ*′, and electron-phonon coupling strength

*α*

*, the spectra of a power-dependent Mollow triplet series of the QD under investigation at Δ = 0 have been modeled with*

_{p}*γ*′ and

*α*

*as free parameters. The extracted FWHM can be well reproduced with a pure dephasing rate of*

_{p}*γ*′ = 4.08

*γ*= 3.43

*μ*eV (equivalent to a pure dephasing time of 192 ps) and

*α*

*/(2*

_{p}*π*)

^{2}= 0.15 ± 0.01 ps

^{2}. A direct comparison between the extracted FWHM of the experimental data and the theoretical model is shown in Fig. 5: The expected linear increase [slope: 9.3 × 10

^{−4}(

*μ*eV)

^{−1}] in the FWHM with

^{−4}(

*μ*eV)

^{−1}].

38. M. Bissiri, G. Baldassarri Höger von Högersthal, A. S. Bhatti, M. Capizzi, A. Frova, P. Figeri, and S. Franchi, “Optical evidence of polaron interaction in InAs/GaAs quantum dots,” Phys. Rev. B **62**, 4642 (2000). [CrossRef]

39. S. Hughes, P. Yao, F. Milde, A. Knorr, D. Dalacu, M. Mnaymneh, V. Sazonova, P. J. Poole, G. C. Aers, J. Lapointe, R. Cheriton, and R. L. Williams, “Influence of electron-acoustic phonon scattering on off-resonant cavity feeding within a strongly coupled quantum-dot cavity system,” Phys. Rev. B **83**, 165313 (2011). [CrossRef]

*α*

*and the dimensionless Huang-Rhys parameter*

_{p}*c*the speed of sound and

_{l}*l*

_{e/h}the electron/hole confinement length) reported in the literature (i.e.

*S*

_{HR}= 0.01 − 0.5) covers a large range and there are no well-accepted numbers to date. Additionally,

*S*

_{HR}has been shown to be enhanced in zero-dimensional QDs compared to bulk material, for which different explanations are proposed, e.g., in terms of non-adiabatic effects or the influence of defects [38

38. M. Bissiri, G. Baldassarri Höger von Högersthal, A. S. Bhatti, M. Capizzi, A. Frova, P. Figeri, and S. Franchi, “Optical evidence of polaron interaction in InAs/GaAs quantum dots,” Phys. Rev. B **62**, 4642 (2000). [CrossRef]

*μ*eV, which is constant in the detuning range accessible in our measurements, is determined by the derived value for

*α*

*and and has been calculated according to Fig. 2. The cross-dephasing term has been extracted from the same graph as*

_{p}*γ*

_{cd}= 0.13

*μ*eV.

*A*

_{red/blue}=

*I*

_{red/blue}/(

*I*

_{red}+

*I*

_{blue}). Figure 6(a) shows a direct comparison of the Mollow triplet spectra for increasing negative detuning Δ < 0, from which the FWHM and relative intensities are extracted. The discrepancy between the expected and measured central Mollow line intensity results from contributions of scattered laser stray-light to the true QD emission that can experimentally not be differentiated due to the equal emission frequency. For the detuning Δ ≠ 0 the spectral resolution of the high-resolution spectroscopy is not sufficient to distinguish between laser-excitation and QD Rayleigh line emission. The gray shaded peaks in Fig. 6(a) (lower panel) belong to a higher order interference of the

*Fabry Pérot*interferometer. The extracted FWHM values are depicted in Fig. 6(b). For the system under investigation,

*r*is calculated to be around 2.01, and therefore an increase in the sidebands’ width is expected according to the theoretical model. Indeed, we observe a systematic increase with increasing negative detuning Δ < 0. Moreover, we observe spectral narrowing (squeezing) of the center line though we do not attempt to fit this resonance as it has a large contribution from coherent scattering. Additionally, the relative sideband areas

*A*

_{red/blue}in dependence on Δ are plotted in Fig. 6(c). As becomes already visible from the Mollow spectra, for positive detunings the blue sideband gains intensity whereas the red sideband area decreases, and vice versa. The crossing between relative intensities is expected to occur at Δ = 0. Interestingly, we observe crossings at moderate negative laser-detuning values for all different QDs under study. A detailed interpretation of the physics behind this effect has to be left for further on-going analysis and may involve the inclusion of more excitons. The high value of pure dephasing which causes

*r*> 1 in our sample comes from the fact that the samples were manufactured using metal organic chemical vapor depositions, which are supposed to incorporate more impurities compared to sample grown by molecular beam epitaxy (MBE) [41

41. T. F. Kuech, “Metal-organic vapor phase epitaxy of compound semiconductors,” Material Science Reports **2**, 1–50 (1987). [CrossRef]

*et al.*[18

**106**, 247402, (2011). [CrossRef] [PubMed]

**85**, 115309 (2012). [CrossRef]

## 6. Conclusion

## Acknowledgments

## References and links

1. | A. Muller, E. B. Flagg, P. Bianucci, X. Y. Wang, D. G. Deppe, W. Ma, J. Zhang, G. J. Salamo, M. Xiao, and C. K. Shih, “Resonance fluorescence from a coherently driven semiconductor quantum dot in a cavity,” Phys. Rev. Lett. |

2. | E. B. Flagg, A. Muller, J. W. Robertson, S. Founta, D. G. Deppe, M. Xiao, W. Ma, G. J. Salamo, and C. K. Shih, “Resonantly driven coherent oscillations in a solid-state quantum emitter,” Nat. Phys. |

3. | A. Nick Vamivakas, Yong Zhao, Chao-Yang Lu, and Mete Atatüre, “Spin-resolved quantum-dot resonance fluorescence,” Nat. Physics |

4. | A. Kiraz, M. Atatüre, and A Imamoğlu, “Quantum-dot single-photon sources: Prospects for applications in linear optics quantum-information processing,” Phys. Rev. A |

5. | S. Ates, S. M. Ulrich, S. Reitzenstein, A. Löffler, A. Forchel, and P. Michler, “Post-selected indistinguishable photons from the resonance fluorescence of a single quantum dot in a microcavity,” Phys. Rev. Lett. |

6. | C. Matthiesen, A. N. Vamivakas, and M. Atatüre, “Subnatural linwidth single photons from a quantum dot,” Phys. Rev. Lett. |

7. | H. S. Nguyen, C. Voisin, P. Roussignol, C. Diedrichs, and G. Cassabois, “Ultra-coherent single photon source,” App. Phys. Lett. |

8. | A. Ulhaq, S. Weiler, S. M. Ulrich, R. Roßbach, M. Jetter, and P. Michler, “Cascaded single-photon emission from resonantly excited quantum dots,” Nat. Photonics |

9. | C. Roy and S. Hughes, “Influence of electron-acoustic-phonon scattering on intensity power broadening in a coherently driven quantum-dot-cavity system,” Phys. Rev. X |

10. | C. Förstner, C. Weber, J. Danckwerts, and A. Knorr, “Phonon-assisted damping of Rabi oscillations in semiconductor quantum dots,” Phys. Rev. Lett. |

11. | P. Machnikowski and L. Jacak, “Resonant nature of phonon-induced damping of Rabi oscillations in quantum dots,” Phys. Rev. B |

12. | K. J. Ahn, J. Förstner, and A. Knorr, “Resonance fluorescence of semiconductor quantum dots: Signatures of the electron-phonon interaction,” Phys. Rev. B |

13. | A. Vagov, M. D. Croitoru, V. M. Axt, T. Kuhn, and F. M. Peeters, “Nonmonotonous field dependence of damping and reappearance of rabi oscillations in quantum dots,” Phys. Rev. Lett. |

14. | A. Nazir, “Photon statistics from a resonantly driven quantum dot,” Phys. Rev. B |

15. | C. Roy and S. Hughes, “Phonon-dressed Mollow triplet in the regime of cavity quantum electrodynamics: Excitation-induced dephasing and nonperturbative cavity feeding effects,” Phys. Rev. Lett. |

16. | C. Roy, H. Kim, E. Waks, and S. Hughes, “Anomalous phonon-mediated damping of a driven quantum dot embedded in a high-Q microcavity,” Photon Nanostruct: Fundam. Appl. |

17. | A. J. Ramsay, A. V. Gopal, E. M. Gauger, A. Nazir, B. W. Lovett, A. M. Fox, and M. S. Skolnick, “Damping of exciton rabi rotations by acoustic phonons in optically excited InGaAs/GaAs quantum dots,” Phys. Rev. Lett. |

18. | S. M. Ulrich, S. Ates, S. Reitzenstein, A. Löffler, A. Forchel, and P. Michler, “Dephasing of Mollow triplet sideband emission of a resonantly driven quantum dot in a microcavity,” Phys. Rev. Lett. |

19. | B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev. |

20. | I. Wilson-Rae and A. Imamoğlu, “Quantum dot cavity-QED in the presence of strong electron-phonon interactions,” Phys. Rev. B |

21. | D. P. S. McCutcheon and A. Nazir, “Quantum dot Rabi rotations beyond the weak exciton-phonon coupling regime,” New J. Phys. |

22. | G. D. Mahan, |

23. | L. Besombes, K. Kheng, L. Marsal, and H. Mariette, “Acoustic phonon broadening mechanism in single quantum dot emission,” Phys. Rev. B |

24. | P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in InGaAs quantum dots,” Phys. Rev. Lett. |

25. | E. A. Muljarov and R. Zimmermann, “Dephasing in quantum dots: quadratic coupling to acoustic phonons,” Phys. Rev. Lett. |

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

27. | G. Ortner, D. R . Yakovlev, M. Bayer, S. Rudin, T. L. Reinecke, S. Fafard, Z. Wasilewski, and A. Forchel, “Temperature dependence of the zero-phonon linewidth in InAsGaAs quantum dots,” Phys. Rev. B |

28. | B. Krummheuer, V. M. Axt, and T. Kuhn, “Theory of pure dephasing and the resulting absorption line shape in semiconductor quantum dots,” Phys. Rev. B |

29. | J. Förstner, C. Weber, J. Danckwerts, and A. Knorr, “Phonon-assisted damping of Rabi oscillations in semiconductor quantum dots,” Phys. Rev. Lett. |

30. | C. Roy and S. Hughes, “Polaron master equation theory of the quantum-dot Mollow triplet in a semiconductor cavity-QED system,” Phys Rev B |

31. | In order to derive the phonon scattering rates, we use parameters for InAs/GaAs QDs, which are α/(2_{p}π)^{2} = 0.15 ps^{2}, where ω is the high frequency cutoff proportional to the inverse of the typical electronic localization length in the QD and _{b}α is a material parameter (extracted from our experiments) that accounts for the difference between the deformation potential constants between electrons and holes._{p} |

32. | G. S. Agarwal and R. R. Puri, “Cooperative behavior of atoms irradiated by broadband squeezed light,” Phys. Rev. A |

33. | Anders Moelbjerg, Per Kaer, Michael Lorke, and Jesper Mørk, “Resonance fluorescence from semiconductor quantum dots: Beyond the Mollow triplet,” Phys. Rev. Lett. |

34. | S. Weiler, A. Ulhaq, S. M. Ulrich, D. Richter, M. Jetter, P. Michler, C. Roy, and S. Hughes, “Phonon-Assisted Incoherent Excitation of a Quantum Dot and its Emission Properties,” Phys. Rev. B |

35. | D. P. S. McCutcheon, N. S. Dattani, E. M. Gauger, B. W. Lovett, and A. Nazir, “A general approach to quantum dynamics using a variational master equation: Application to phonon-damped Rabi rotations in quantum dots,” Phys. Rev. B |

36. | e.g., see M. Glässl, A. Vagov, S. Lüker, D. E. Reiter, M. D. Croitoru, P. Machnikowski, V. M. Axt, and T. Kuhn, “Long-time dynamics and stationary nonequilibrium of an optically driven strongly confined quantum dot coupled to phonons,” Phys. Rev. B |

37. | e.g., see H. J. Carmichael, |

38. | M. Bissiri, G. Baldassarri Höger von Högersthal, A. S. Bhatti, M. Capizzi, A. Frova, P. Figeri, and S. Franchi, “Optical evidence of polaron interaction in InAs/GaAs quantum dots,” Phys. Rev. B |

39. | S. Hughes, P. Yao, F. Milde, A. Knorr, D. Dalacu, M. Mnaymneh, V. Sazonova, P. J. Poole, G. C. Aers, J. Lapointe, R. Cheriton, and R. L. Williams, “Influence of electron-acoustic phonon scattering on off-resonant cavity feeding within a strongly coupled quantum-dot cavity system,” Phys. Rev. B |

40. | P. Dara, S. McCutcheon, and Ahsan Nazir, “Emission properties of a driven artificial atom: increased coherent scattering and off-resonant sideband narrowing,” arXiv:1208.4620v1 |

41. | T. F. Kuech, “Metal-organic vapor phase epitaxy of compound semiconductors,” Material Science Reports |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(300.6320) Spectroscopy : Spectroscopy, high-resolution

(300.6470) Spectroscopy : Spectroscopy, semiconductors

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: October 10, 2012

Revised Manuscript: January 11, 2013

Manuscript Accepted: January 11, 2013

Published: February 13, 2013

**Citation**

Ata Ulhaq, Stefanie Weiler, Chiranjeeb Roy, Sven Marcus Ulrich, Michael Jetter, Stephen Hughes, and Peter Michler, "Detuning-dependent Mollow triplet of a coherently-driven single quantum dot," Opt. Express **21**, 4382-4395 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-4382

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### References

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- P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L. Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephasing time in InGaAs quantum dots,” Phys. Rev. Lett.87, 157401 (2001). [CrossRef] [PubMed]
- E. A. Muljarov and R. Zimmermann, “Dephasing in quantum dots: quadratic coupling to acoustic phonons,” Phys. Rev. Lett.93, 237401 (2004). [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. B65, 041308 (2002). [CrossRef]
- G. Ortner, D. R . Yakovlev, M. Bayer, S. Rudin, T. L. Reinecke, S. Fafard, Z. Wasilewski, and A. Forchel, “Temperature dependence of the zero-phonon linewidth in InAsGaAs quantum dots,” Phys. Rev. B70, 201301(R) (2004). [CrossRef]
- B. Krummheuer, V. M. Axt, and T. Kuhn, “Theory of pure dephasing and the resulting absorption line shape in semiconductor quantum dots,” Phys. Rev. B65, 195313 (2002). [CrossRef]
- J. Förstner, C. Weber, J. Danckwerts, and A. Knorr, “Phonon-assisted damping of Rabi oscillations in semiconductor quantum dots,” Phys. Rev. Lett.91, 127401 (2003). [CrossRef] [PubMed]
- C. Roy and S. Hughes, “Polaron master equation theory of the quantum-dot Mollow triplet in a semiconductor cavity-QED system,” Phys Rev B85, 115309 (2012). [CrossRef]
- In order to derive the phonon scattering rates, we use parameters for InAs/GaAs QDs, which are ωb = 1 meV and αp/(2π)2 = 0.15 ps2, where ωb is the high frequency cutoff proportional to the inverse of the typical electronic localization length in the QD and αp is a material parameter (extracted from our experiments) that accounts for the difference between the deformation potential constants between electrons and holes.
- G. S. Agarwal and R. R. Puri, “Cooperative behavior of atoms irradiated by broadband squeezed light,” Phys. Rev. A41, 3782 (1990). [CrossRef] [PubMed]
- Anders Moelbjerg, Per Kaer, Michael Lorke, and Jesper Mørk, “Resonance fluorescence from semiconductor quantum dots: Beyond the Mollow triplet,” Phys. Rev. Lett.108, 017401 (2012). [CrossRef] [PubMed]
- S. Weiler, A. Ulhaq, S. M. Ulrich, D. Richter, M. Jetter, P. Michler, C. Roy, and S. Hughes, “Phonon-Assisted Incoherent Excitation of a Quantum Dot and its Emission Properties,” Phys. Rev. B86, 241304(R) (2012). [CrossRef]
- D. P. S. McCutcheon, N. S. Dattani, E. M. Gauger, B. W. Lovett, and A. Nazir, “A general approach to quantum dynamics using a variational master equation: Application to phonon-damped Rabi rotations in quantum dots,” Phys. Rev. B84, 081305(R) (2011).
- e.g., see M. Glässl, A. Vagov, S. Lüker, D. E. Reiter, M. D. Croitoru, P. Machnikowski, V. M. Axt, and T. Kuhn, “Long-time dynamics and stationary nonequilibrium of an optically driven strongly confined quantum dot coupled to phonons,” Phys. Rev. B84, 195311 (2011). [CrossRef]
- e.g., see H. J. Carmichael, Statistical methods in quantum optics 1: Master equations and Fokker-Planck equations (Springer, 2003).
- M. Bissiri, G. Baldassarri Höger von Högersthal, A. S. Bhatti, M. Capizzi, A. Frova, P. Figeri, and S. Franchi, “Optical evidence of polaron interaction in InAs/GaAs quantum dots,” Phys. Rev. B62, 4642 (2000). [CrossRef]
- S. Hughes, P. Yao, F. Milde, A. Knorr, D. Dalacu, M. Mnaymneh, V. Sazonova, P. J. Poole, G. C. Aers, J. Lapointe, R. Cheriton, and R. L. Williams, “Influence of electron-acoustic phonon scattering on off-resonant cavity feeding within a strongly coupled quantum-dot cavity system,” Phys. Rev. B83, 165313 (2011). [CrossRef]
- P. Dara, S. McCutcheon, and Ahsan Nazir, “Emission properties of a driven artificial atom: increased coherent scattering and off-resonant sideband narrowing,” arXiv:1208.4620v1
- T. F. Kuech, “Metal-organic vapor phase epitaxy of compound semiconductors,” Material Science Reports2, 1–50 (1987). [CrossRef]

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