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

doi:10.1364/OE.21.004382

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

Ata Ulhaq, Stefanie Weiler, Chiranjeeb Roy, Sven Marcus Ulrich, Michael Jetter, Stephen Hughes, and Peter Michler »View Author Affiliations

Optics Express, Vol. 21, Issue 4, pp. 4382-4395 (2013)
http://dx.doi.org/10.1364/OE.21.004382

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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.

1. Introduction

Resonant excitation of single QDs has recently gained significant interest [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]

], in part because this type of coherent excitation is promising for the generation of single photons with excellent coherence properties [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]

]. The techniques developed for effective laser stray light suppression have enabled the collection of resonance fluorescence from a single QD with high signal-to-noise ratio. Resonance fluorescence (RF) emission below saturation of the quantum emitter has revealed close-to-Fourier transform limited single photons with record-high emission coherence and two-photon interference visibility [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]

]. Recent experiments have even been able to beat the Fourier limit for single-photon emission coherence in the Heitler regime, e.g., with excitation strengths well below the saturation of the quantum emitter [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]

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

]. Another major achievement with respect to RF is the demonstration of single- and cascaded photon emission between the Mollow sidebands above saturation of the QD [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]

Recent investigations of single QD RF have revealed distinct differences of their emission coherence properties [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]

] which need to be theoretically treated beyond a simple two-level description usually used for atoms. One of the main consequences of the solid-state character of these quantum emitters is the consideration of specific dephasing channels primarily caused by carrier-phonon scattering. Dephasing of a resonantly driven QD system has been theoretically studied in detail with respect to effects of electron-phonon interaction on the dynamics of an optically driven system [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]

]. These studies anticipated excitation-induced dephasing (EID) for moderate Rabi frequencies. However, non-monotonic behavior was predicted for Rabi frequencies larger than a cut-off frequency defined by the material parameters and the QD size [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]

]. Non-monotonic behavior is also predicted for cavity structures with suitable microcavity coupling [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]

]. Experimental evidence of EID effects has recently been observed as oscillation damping in pulsed photocurrent measurements on a resonantly driven QD [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]

]. This damping was found to have a clear quadratic dependence on the effective Rabi frequency Ω. The effect of EID has also been observed under strictly resonant continuous wave excitation of a QD in a microcavity in terms of Mollow-triplet sideband broadening [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]

]. These experiments reveal good agreement with a theoretical description based on a polaron master equation approach to multi-phonon and multi-photon effects in a cavity-QED system [15

]. In the work of Ulrich et al.[18

], the phenomenon of spectral Mollow sideband narrowing in dependence of laser-excitation detuning from the bare emitter resonance had to be left open for further in-depth theoretical analysis.

Motivated by these findings and with the aim of fundamental interpretation, the focus of the current work lies on a detailed study of the detuning-dependent dressed state emission of a single QD without cavity coupling. Our theory is based on a polaron master equation approach from which we develop a fully analytical description of the emission spectrum. Consequently, we are able to clearly distinguish between different regimes of spectral broadening or narrowing of the Mollow sidebands, under strong influence by pure dephasing and phonon-induced scattering. The comparison of detuning-dependent resonance fluorescence data reveals very good agreement with the theoretical model.

2. Sample structure and experimental procedure

The planar sample employed for the measurements in this work is grown by metal-organic vapor epitaxy (MOVPE). The self-assembled In(Ga)As QDs are embedded in a GaAs 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

Δ E_{res}^{HRPL}

< 1μeV as described earlier [5

, 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

3. Experimental results: Detuning-dependent resonance fluorescence

In our experiments we apply pump powers well above the saturation of the quantum emitters. In this high-field regime, the excitation-induced Rabi rotation of the two-level emitter system becomes much faster than the spontaneous decay rate γ. The incoherent spectrum of the resulting dressed state is the characteristic Mollow triplet [19

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

]. Under strictly resonant excitation, i.e. for a laser detuning Δ = ω_L – ω_x = 0 from the QD exciton resonance (see Fig. 1(a), green center trace), the spectrum is composed of the central Rayleigh line “R” at the bare emitter energy ω₀ and two symmetric satellite peaks, i.e. the Three-Photon Line “T” and the Fluorescence Line “F ” at ω₀ ± Ω_r, respectively. Ω_r denotes the effective Rabi frequency including renormalization effects from the phonon bath as discussed in the theory section below.

Fig. 1: (a) HRPL of the QD resonance fluorescence under systematic variation of the laser-excitation detuning Δ = ω_L − ω_x, taken at a fixed power of P₀ = 500 μW. The parameter δ denotes the spectral emission detuning from the bare emitter resonance. Green center trace: Mollow triplet under strictly resonant excitation Δ = 0. (b) Spectral evolution of the Mollow sidebands with laser-detuning, Δ, extracted from (a).

Laser detuning (Δ)-dependent Mollow triplet spectra taken at a constant excitation strength of P₀ = 500 μW (Ω_r ∝ (P₀)^1/2 = const.) are depicted in Fig. 1(a). According to theory (see, e.g., Ref. [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]

]), the laser-detuning between the driving field and the bare emitter resonance ω₀ modifies the dressed emission. Besides the center transition at ω₀ + Δ the two sideband frequencies become ω₀ + Δ ± Ω, where

Ω = \sqrt{Ω_{r}^{2} + Δ^{2}}

denotes the generalized Rabi frequency at a given excitation strength. The extracted spectral positions for the red- and blue-shifted Mollow sidebands and the central Rayleigh line are depicted in Fig. 1(b) with a corresponding fit to the data. In addition, observations on the detuning-dependent Mollow triplet series depicted in Fig. 1(a) reveal distinct broadening of the Mollow sidebands with increasing Δ, accompanied by a change in the relative sideband intensities. In order to explain these observations, we develop a theoretical description in terms of a polaron master equation formalism and derive an analytical expression for the incoherent spectrum. The main theoretical findings used to interpret our experimental observations are described in the following section.

4. Theory

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

We model the QD as an effective two-level system interacting with a coherent pump field and an acoustic phonon reservoir. In a frame rotating with respect to the laser pump frequency ω_L, the model Hamiltonian (excluding QD zero-phonon line decay mechanisms) is

H = - \bar{h} Δ σ^{+} σ^{-} + \bar{h} η_{x} (σ^{+} + σ^{-}) + σ^{+} σ^{-} \sum_{q} \bar{h} λ_{q} (b_{q} + b_{q}^{†}) + \sum_{q} \bar{h} ω_{q} b_{q}^{†} b_{q},

(1)

where

b_{q} (b_{q}^{†})

are the annihilation and creation operators of the phonon reservoir, σ^+/− (σ^z = σ⁺σ⁻ − σ⁻σ⁺) are the Pauli operators of the exciton; η_x is the exciton pump rate, and λ_q (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

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

, 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]

] which is particularly well suited for studying quantum optical phenomena such as resonance fluorescence spectra. In the following we will closely follow (and extend where necessary) the theoretical formalisms described in Refs. [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

], except we can safely neglect cavity coupling terms.

Defining

P = σ^{+} σ^{-} \sum_{q} \frac{λ_{q}}{ω_{q}} (b_{q}^{†} - b_{q})

, the polaron transformed Hamiltonian, H′ → e^PHe⁻^P[22

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

], consists of a system part, reservoir part, and an interaction part, respectively:

H_{S}^{'} = \bar{h} (- Δ - Δ_{P}) σ^{+} σ^{-} + 〈 B 〉 X_{g}, H_{R}^{'} = \sum_{q} \bar{h} ω_{q} b_{q}^{†} b_{q}, H_{I}^{'} = X_{g} ζ_{g} + X_{u} ζ_{u},

(2)

with the coherent displacement operators

B_{\pm} = exp [\pm \sum_{q} \frac{λ_{q}}{ω_{q}} (b_{q} - b_{q}^{†})]

, and

ζ_{g} = \frac{1}{2} (B_{+} + B_{-} - 2 〈 B 〉)

and

ζ_{u} = \frac{1}{2 i} (B_{+} - B_{-})

. The polaron shift is

Δ_{P} = \int_{0}^{\infty} d ω \frac{J (ω)}{ω}

, where

J (ω) = α_{p} ω^{3} exp (- \frac{ω^{2}}{2 ω_{b}^{2}})

denotes the characteristic phonon spectral function that describes the LA-phonon interaction resulting from deformation potential coupling. The thermally-averaged bath displacement operator is defined through [22

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

]

〈 B 〉 = exp [- \frac{1}{2} \int_{0}^{\infty} d ω \frac{J (ω)}{ω^{2}} coth (β \bar{h} ω / 2)]

, with 〈B〉 = 〈B₊〉 = 〈B₋〉, at a bath temperature T = 1/k_bβ. For convenience, we will assume that the polaron shift is implicitly included in our definition of ω_x below. The operators X_g and X_u are defined through X_g = h̄η_x(σ⁻ + σ⁺) and X_u = ih̄η_x(σ⁺ − σ⁻).

We next present the time-local (or time-convolutionless) ME for the reduced density operator 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′_I in the Born approximation and trace over the phonon degrees of freedom. The full polaron ME takes the following form [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

, 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]

\begin{array}{l} \frac{\partial ρ}{\partial t} & = & \frac{1}{i \bar{h}} [H_{S}^{'}, ρ (t)] + \frac{γ}{2} ℒ [σ^{-}] + \frac{γ^{'}}{2} ℒ [σ_{11}] \\ - & \frac{1}{{\bar{h}}^{2}} \int_{0}^{t} d τ \sum_{m = g, u} (G_{m} (τ) [X_{m}, e^{- i H_{S}^{'} τ / \bar{h}} X_{m} e^{i H_{S}^{'} τ / \bar{h}} ρ (t)] + H . c .), \end{array}

(3)

where σ₁₁ = 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

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]

], with the phonon correlation function

ϕ (t) = \int_{0}^{\infty} d ω \frac{J (ω)}{ω^{2}} [coth (β \bar{h} ω / 2) cos (ω t) - i sin (ω t)]

. The Lindblad operators 𝒧[O] = 2OρO^† − O^†Oρ − ρO^†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

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]

]. Without the coherent pump term and the residual ZPL broadenings, this ME formally recovers the independent boson model [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]

For continuous wave (cw) excitation, the integration appearing in Eq. (3) can have the upper time limit t → ∞, resulting in a Markovian ME where the scattering rates are computed as a function of H′_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]

]. Such an approach is valid since the acoustic phonon lifetimes are very fast, i.e., on a few ps timescale and much faster than the characteristic time scales of the system dynamics. As was shown elsewhere [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]

], for the pump strengths we consider in this work, one can neglect the pump-dependence of H′_S appearing in the exponential phase terms above (which we will further justify below) to derive an effective phonon ME as follows:

\begin{array}{l} \frac{\partial ρ}{\partial t} & = & \frac{1}{i \bar{h}} [H_{S}^{'}, ρ (t)] + \frac{γ}{2} ℒ [σ^{-}] + \frac{γ^{'}}{2} ℒ [σ_{11}] + \frac{Γ_{ph}^{σ^{+}}}{2} ℒ [σ^{+}] + \\ - & Γ_{ph}^{cd} (σ^{+} ρ σ^{+} + σ^{-} ρ σ^{-}) . \end{array} \frac{Γ_{ph}^{σ^{-}}}{2} ℒ [σ^{-}]

(4)

Here the pump-driven incoherent scattering processes, mediated by the phonon bath, are obtained from

\begin{array}{l} Γ_{ph}^{σ^{+} / σ^{-}} & = & \frac{Ω_{r}^{2}}{2} Re [\int_{0}^{\infty} d τ e^{\pm i Δ τ} (e^{ϕ (τ)} - 1)], \end{array}

(5)

\begin{array}{l} Γ_{ph}^{cd} & = & \frac{Ω_{r}^{2}}{2} Re [\int_{0}^{\infty} d τ cos (Δ τ) (1 - e^{- ϕ (τ)})] . \end{array}

(6)

The classical Rabi frequency of the exciton pump, including renormalization effects from the phonon bath, is given by Ω_r = 2η_x〈B〉 (cf. the bare Rabi frequency Ω₀ = 2η_x. The scattering term

Γ_{ph}^{cd}

is a cross-dephasing rate that only affects the off-diagonal components of the resulting optical Bloch equations. Similar terms appear when a system is driven by a broadband squeezed light reservoir [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]

]. As might be expected, the excitation-dependent rates depend upon the phonon correlation function, the coherent pump rate, and the laser-exciton detuning. The

Γ_{ph}^{σ^{-}}

process corresponds to an enhanced radiative decay, while the

Γ_{ph}^{σ +}

process represents an incoherent excitation process [15

]. We stress that these mechanisms are quite different to simple pure dephasing models, which are frequently used to describe weak (i.e., perturbative) electron-phonon scattering [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]

]. The phonon-mediated incoherent excitation process has recently been confirmed experimentally [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]

]. Note that Ω_r can be significantly smaller than Ω₀ (for suitably large electron-phonon coupling), even at low temperatures. For example, using InAs QD parameters that closely represent our experimental samples [31

In order to derive the phonon scattering rates, we use parameters for InAs/GaAs QDs, which are ω_b = 1 meV and α_p/(2π)² = 0.15 ps², where ω_b is the high frequency cutoff proportional to the inverse of the typical electronic localization length in the QD and α_p is a material parameter (extracted from our experiments) that accounts for the difference between the deformation potential constants between electrons and holes.

] and a phonon bath temperature T ∼ 6 K, then 〈B〉 ≈ 0.75, and this value decreases (increases) with increasing (decreasing) temperature.

For a pump field strength of Ω_r = 50 μeV, example phonon scattering rates are shown in Fig. 2. Within the zoomed region of laser detunings |Δ| < 100 μeV, 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).

Fig. 2: Phonon-mediated scattering rates

Γ_{ph}^{σ^{+ / -}}

(blue solid and dashed lines, respectively), together with their average sum (black dashed line) and

Γ_{ph}^{cd}

(red solid line) for a phonon bath temperature T = 6.0 K. Only for large detunings Δ = ω_L − ω_x ≥ 0.5 meV these rates change appreciably. The overall magnitude of the phonon scattering rates is proportional to

Ω_{r}^{2}

. Here we assume Ω_r = 50 μeV. As is shown in the inset figure, the sum of rates

Γ_{ph}^{σ^{+}} + Γ_{ph}^{σ^{-}} = γ_{ph}

as well as

Γ_{ph}^{cd}

are constant within laser-exciton detunings over hundreds of μeV.

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

To gain better insight into the underlying physics of a QD driven Mollow triplet with a finite detuning, it is desirable to derive an analytical form for the Mollow triplet spectrum which we can derive from the effective phonon ME. Thus we will first investigate how good the approximation is to replace the phonon scattering terms in the 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 μeV, and high field strengths of Ω₀ = 50 μeV. 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.

Fig. 3: Numerically calculated Mollow triplet with Ω₀ = 50 μeV, with γ = γ′ = 1 μeV and a phonon bath temperature of T = 6.0 K. The spectra are obtained for (a) Δ = 0 μeV and (b) Δ = 30 μeV, plotted for the case of no phonon scattering (black solid line), and with phonon scattering using the full polaron ME (red solid line) and the effective ME (blue dashed line). The latter two give almost identical spectra, which justifies the accuracy of the simpler effective ME. Additionally, the effect of renormalization of the Rabi frequency can be seen in the different Mollow triplet center-to-sideband splittings for the case of no phonon scattering (black line) in comparison to the red/blue spectra.

We highlight that a cw Rabi field of Ω_r = 50 μeV 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_r becomes comparable to (or greater than) the phonon cut-off frequency. In this case, other approaches exists such as a variational ME approach [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 84, 081305(R) (2011).

] and path integral techniques [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]

]. Since our maximum Rabi field strengths are much less than ω_b (the characteristic phonon cut off frequency), as shown by McCutcheon 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

One of our main theoretical goals is to derive a useful analytical expression that will allow one to fit the experimental Mollow triplets over a wide range of parameters, explicitly including the laser-exciton detuning. From the effective phonon ME [Eq. (4)] and 〈Ȯ〉 = tr[ρ̇O] [37

e.g., see H. J. Carmichael, Statistical methods in quantum optics 1: Master equations and Fokker-Planck equations (Springer, 2003).

], we obtain the following optical Bloch equations:

\frac{d 〈 σ^{-} 〉}{d t} = - (γ_{pol} - i Δ) 〈 σ^{-} 〉 - γ_{cd} 〈 σ^{+} 〉 + i \frac{Ω_{r}}{2} 〈 σ^{z} 〉,

(7a)

\frac{d 〈 σ^{+} 〉}{d t} = - (γ_{pol} - i Δ) 〈 σ^{+} 〉 - γ_{cd} 〈 σ^{-} 〉 - i \frac{Ω_{r}}{2} 〈 σ^{z} 〉,

(7b)

\frac{d 〈 σ^{z} 〉}{d t} = i Ω_{r} 〈 σ^{-} 〉 - i Ω_{r} 〈 σ^{+} 〉 - γ_{pop} 〈 σ^{z} 〉 - γ_{pop}^{'},

(7c)

where we define the polarization decay

γ_{pol} = \frac{1}{2} (Γ_{ph}^{σ^{+}} + Γ_{ph}^{σ^{-}} + γ + γ^{'})

and the population decay

γ_{pop} = (Γ_{ph}^{σ^{+}} + Γ_{ph}^{σ^{-}} + γ)

, as well as

γ_{pop}^{'} = γ_{pop} - 2 Γ_{ph}^{σ^{+}}

. For notational convenience, we have also defined

γ_{cd} \equiv Γ_{ph}^{cd}

. The incoherent spectrum can be computed from an integration of the appropriate two-time correlation function [37

e.g., see H. J. Carmichael, Statistical methods in quantum optics 1: Master equations and Fokker-Planck equations (Springer, 2003).

S (r, ω) \equiv F (r) S (ω) \propto {lim}_{t \to \infty} Re {\int_{0}^{\infty} d τ 〈 δ σ^{+} (t) δ σ^{-} (t + τ) 〉 e^{i (ω - ω_{L}) τ}},

(8)

where 〈δO〉 = 〈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

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]

], as the emitted spectrum is detected via a weakly-coupled planar cavity mode, in which case 〈δa^†(t)δa(t + τ)〉 ∝ 〈δσ⁺(t)δσ⁻(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

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]

]. The coherent spectrum, if needed, can be derived in a similar way. By exploiting the quantum regression theorem and Eqs. (7a)–(7c), it is possible to derive the spectrum analytically, e.g., using Laplace transform techniques. We first define the steady-state expectation values f(0) ≡ 〈δσ⁺δσ⁻〉_ss, g(0) ≡ 〈δσ⁺δσ⁺〉_ss, and h(0) ≡ 〈δσ⁺δσ^z〉_ss, and keep the explicit laser-exciton detuning dependence in the solution. Using the frequency detuning δω = ω −ω_L, we can obtain the spectrum lineshape,

S (ω) \equiv Re {\frac{- f (0) D (ω) + i h (0) C (ω) D (ω) - [γ_{cd} + Ω_{r} C (ω)] [g (0) + i h (0) C (ω)]}{(D (ω) + i 2 Δ) D (ω) - {[γ_{cd} + Ω_{r} C (ω)]}^{2}}},

(9)

where

C (ω) = \frac{Ω_{r}}{2 (i δ ω - γ_{pop})}

and

D (ω) = i δ ω - γ_{pol} - i Δ + \frac{Ω_{r}^{2}}{2 (i δ ω - γ_{pop})}

The corresponding steady-state inversion and polarization components are calculated to be

{〈 σ^{z} 〉}_{s s} = - \frac{γ_{pop}^{'}}{γ_{pop} + \frac{Ω_{r}^{2} (γ_{pol} + γ_{cd})}{(γ_{pol}^{2} + Δ^{2} - γ_{cd}^{2})}}, {〈 σ^{-} 〉}_{s s} = \frac{i Ω_{r} (γ_{pol} + i Δ + γ_{cd})}{2 (γ_{pol}^{2} + Δ^{2} - γ_{cd}^{2})} {〈 σ^{z} 〉}_{ss} = {〈 σ^{+} 〉}_{s s}^{*},

(10)

from which we can obtain the following steady-state values for f, g, and h:

f (0) = \frac{1}{2} (1 + {〈 σ^{z} 〉}_{ss} - 2 {〈 σ^{+} 〉}_{ss} {〈 σ^{-} 〉}_{ss}),

(11a)

g (0) = - {〈 σ^{+} 〉}_{ss}^{2},

(11b)

h (0) = - {〈 σ^{+} 〉}_{ss} (1 + {〈 σ^{z} 〉}_{ss}) .

(11c)

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_{side} \approx \frac{3}{2} (γ_{0} + γ_{ph}) + \frac{1}{2} γ^{'} - γ_{cd}

and γ_center ≈ γ₀ + 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.

Fig. 4: (a) Analytically computed spectrum as a function of detuning for three different values of r. The solid red curve shows the analytical solution [Eq. (9)] and the blue dashed curve shows the three-Lorentzian fit. Positive and negative detunings |Δ| reveal simply a mirror image of each other. The phonon parameters are taken from Fig. 2, with γ_cd ≈ 0.6 μeV and γ_ph ≈ 1.6 μeV for the chosen Rabi field. Here we adjust γ and γ′ to maintain the same on-resonance FWHM value of

γ_{side} (Δ = 0) \approx \frac{3}{2} (γ + γ_{ph}) + \frac{1}{2} γ^{'} - γ_{cd} = 5.6

μeV: for r = 0.7,1.0,1.81, we use γ′ (γ) as 0.4(2.4),2.2(1.8), and 5.1(0.8) μeV, respectively. (b) Extracted FWHM of the lower (blue crosses) and higher energy sideband (red inverted triangles) as a function of detuning Δ. One clearly sees a trend of either increasing or decreasing sideband line width as a function of laser detuning, depending on the value of r, where r ≈ 1 denotes the crossover. The center Mollow line exhibits the opposite trend (now shown), e.g., it decreases if the sidebands increase.

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

From the analytical spectra above, we can discern when the Mollow sidebands will become asymmetric and whether the detuning dependence will exhibit broadening or narrowing of the sideband resonances. For convenience, we introduce the following ratio:

r = \frac{γ_{pol} + γ_{cd}}{γ_{pop}} = \frac{1}{2} [1 + \frac{γ^{'} + 2 γ_{cd}}{γ + γ_{ph}}],

(12)

where

γ_{ph} = Γ_{ph}^{σ^{+}} + Γ_{ph}^{σ^{-}}

. Worth to note, for off-resonant driving and 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

In the following, we show a detailed modeling of experimentally derived results from detuning-dependent high-resolution PL measurements using the above presented theory. The measurements are performed on a QD in a planar sample structure with negligible cavity coupling, though the planar cavity acts to separate out the QD emission from the pump field.

To reproduce the experimental results with the theoretical model we derive the characteristic values for the calculation via independent measurements or analysis. In the experiments, the sample temperature has been measured to be T = 6.0 ± 0.5 K. The QDs in the planar sample are found to have rather similar radiative lifetimes due to no Purcell-like enhancement. The radiative decay rate γ is extracted from time-correlated photon counting measurements that reveal a typical radiative lifetime of (750–850 ps), yielding γ ≈ (0.77 − 0.88)μeV. The Rabi field Ω_r = 22.7μeV is derived from the Mollow center-to-sideband splitting at zero laser-detuning Δ = 0. The cut-off frequency is set to ω_b = 1 meV.

To carefully identify the pure dephasing rate γ′, and electron-phonon coupling strength α_p, the spectra of a power-dependent Mollow triplet series of the QD under investigation at Δ = 0 have been modeled with γ′ and α_p as free parameters. The extracted FWHM can be well reproduced with a pure dephasing rate of γ′ = 4.08 γ = 3.43μeV (equivalent to a pure dephasing time of 192 ps) and α_p/(2π)² = 0.15 ± 0.01 ps². 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 (μeV)⁻¹] in the FWHM with

Ω_{r}^{2}

shows very good agreement with the experiment [slope: 9.8 × 10⁻⁴ (μeV)⁻¹].

Fig. 5: Determination of the pure dephasing rate γ′ and electron-phonon coupling strength α_p. The graph shows the expected linear increase of the FWHM of the Mollow sidebands versus

Ω_{r}^{2}

as extracted from a power-dependent Mollow series under strictly resonant excitation (Δ = 0). A comparison between (a) the theoretical predictions (revealing a slope of 9.3×10⁻⁴ (μeV)⁻¹) and (b) the experimental data (giving 9.8×10⁻⁴(μeV)⁻¹) reveals best consistency for a dephasing rate of γ′ = 4.08γ = 3.43μeV and α_p/(2π)² = 0.15 ± 0.01 ps². All other parameters are fixed according to T = 6.0 ± 0.5K, γ = 0.84 ± 0.04 μeV (784 ± 10 ps), γ_cd = 0.13 ± 0.01 μeV and γ_ph = 0.34 ± 0.02 μeV (at Ω_r = 22.7 μeV); the latter two rates scale with

Ω_{r}^{2}

and are for evaluated at Ω_r = 22.7 μeV.

The deformation potential constant is somewhat higher compared to the value used in Refs. [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]

]. However, the value for α_p and the dimensionless Huang-Rhys parameter

S_{HR} = α_{p} / {(2 π)}^{2} c_{l}^{2} / l_{e / h}^{2}

(with c_l the speed of sound and 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

The sum of the main phonon scattering rates

γ_{ph} = Γ_{ph}^{σ^{+}} + Γ_{ph}^{σ^{-}} = 0.34

μeV, which is constant in the detuning range accessible in our measurements, is determined by the derived value for α_p and and has been calculated according to Fig. 2. The cross-dephasing term has been extracted from the same graph as γ_cd = 0.13 μeV.

With all parameters at hand, the experimentally measured detuning-dependent Mollow triplet series is compared with the theoretical expectations in terms of sideband broadening and the change in the relative sideband areas 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

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

]. The higher magnitude of pure dephasing results in broadening of Mollow sidebands.

Fig. 6: Detuning-dependent Mollow triplet series at P = 500 μeV, showing theoretical predictions versus experimental results for a system with r = 2.01. (a) Mollow triplet spectra for increasing negative detunings, Δ, the spectra are plotted with respect to the energetic laser position set to zero. (b) FWHM of the blue and red Mollow sideband reveal distinct sideband broadening with increasing laser-detuning. (c) Change of the relative Mollow sideband area with Δ. The theoretically expected trend can be seen.

The regime of distinct sideband narrowing has also been experimentally observed by Ulrich et al.[18

] (see Fig. 3(d) of their paper) on a QD embedded in a micropillar cavity structure, grown by MBE. Even though the effect can be qualitatively understood from the theoretical model discussed above, a direct comparison to theory requires the inclusion of QED-cavity coupling into the polaron ME approach. Previous numerical studies (using a polaron ME approach with cavity coupling) were performed in this regime [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]

], but did not obtain this behaviour, suggesting that further work is likely required to explain the significant narrowing effects that are observed in the experiment.

6. Conclusion

In conclusion, we have presented a combined theoretical-experimental study on the impact of pure dephasing and phonon-induced scattering on the excitation detuning-dependence of Mollow triplet sidebands. Based on a polaron ME approach, supplemented by a useful analytical solution for the Mollow triplet spectrum, it is possible to distinguish different regimes of spectral broadening or narrowing, which depend on the ratio of different phonon-mediated scattering rates. Regimes of spectral squeezing and anti-squeezing have also been identified. For the case of experimentally observed distinct sideband broadening, we have found excellent agreement with the predictions of theory.

Acknowledgments

During the final preparation of this work we became aware of similar results obtained independently for off-resonant Mollow sideband narrowing [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

]. We would like to thank Dara McCutcheon and Ahsan Nazir for bringing these to our attention and for useful discussions. S. Weiler acknowledges financial support by the Carl-Zeiss-Stiftung. S. Hughes thanks Rong-Chun Ge for useful comments and the National Sciences and Engineering Research Council of Canada for research funding. We thank the German Research Foundation (DFG) for financial support via grant DFG MI 500/23-1. This work was also supported by the DFG within the funding program Open Access Publishing.

References and links

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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]
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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]
31.	In order to derive the phonon scattering rates, we use parameters for InAs/GaAs QDs, which are ω_b = 1 meV and α_p/(2π)² = 0.15 ps², where ω_b is the high frequency cutoff proportional to the inverse of the typical electronic localization length in the QD and α_p is a material parameter (extracted from our experiments) that accounts for the difference between the deformation potential constants between electrons and holes.
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37.	e.g., see H. J. Carmichael, Statistical methods in quantum optics 1: Master equations and Fokker-Planck equations (Springer, 2003).
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]
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
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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

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]
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.5203–207 (2009). [CrossRef]
A. Nick Vamivakas, Yong Zhao, Chao-Yang Lu, and Mete Atatüre, “Spin-resolved quantum-dot resonance fluorescence,” Nat. Physics5, 198–202 (2009). [CrossRef]
A. Kiraz, M. Atatüre, and A Imamoğlu, “Quantum-dot single-photon sources: Prospects for applications in linear optics quantum-information processing,” Phys. Rev. A69, 032305 (2004). [CrossRef]
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]
C. Matthiesen, A. N. Vamivakas, and M. Atatüre, “Subnatural linwidth single photons from a quantum dot,” Phys. Rev. Lett.108, 093602 (2012). [CrossRef] [PubMed]
H. S. Nguyen, C. Voisin, P. Roussignol, C. Diedrichs, and G. Cassabois, “Ultra-coherent single photon source,” App. Phys. Lett.99, 261904 (2011). [CrossRef]
A. Ulhaq, S. Weiler, S. M. Ulrich, R. Roßbach, M. Jetter, and P. Michler, “Cascaded single-photon emission from resonantly excited quantum dots,” Nat. Photonics6, 238 (2012). [CrossRef]
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. X1, 021009 (2011). [CrossRef]
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]
P. Machnikowski and L. Jacak, “Resonant nature of phonon-induced damping of Rabi oscillations in quantum dots,” Phys. Rev. B69, 193302 (2004). [CrossRef]
K. J. Ahn, J. Förstner, and A. Knorr, “Resonance fluorescence of semiconductor quantum dots: Signatures of the electron-phonon interaction,” Phys. Rev. B71, 153309 (2005). [CrossRef]
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]
A. Nazir, “Photon statistics from a resonantly driven quantum dot,” Phys. Rev. B78, 153309, (2008). [CrossRef]
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]
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]
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]
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]
B. R. Mollow, “Power spectrum of light scattered by two-level systems,” Phys. Rev.188, 169–175 (1969). [CrossRef]
I. Wilson-Rae and A. Imamoğlu, “Quantum dot cavity-QED in the presence of strong electron-phonon interactions,” Phys. Rev. B65, 235311 (2002). [CrossRef]
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]
G. D. Mahan, Many-Particle Physics (Plenum, New York, 1990). [CrossRef]
L. Besombes, K. Kheng, L. Marsal, and H. Mariette, “Acoustic phonon broadening mechanism in single quantum dot emission,” Phys. Rev. B63, 155307 (2001). [CrossRef]
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]

Aers, G. C.
Agarwal, G. S.
Ahn, K. J.
Atatüre, M.
Atatüre, Mete
Ates, S.
Axt, V. M.
Baldassarri Höger von Högersthal, G.
Bayer, M.
Besombes, L.
Bhatti, A. S.
Bianucci, P.
Bimberg, D.
Bissiri, M.
Borri, P.
Capizzi, M.
Carmichael, H. J.
Cassabois, G.
Cheriton, R.
Croitoru, M. D.
Dalacu, D.
Danckwerts, J.
Dara, P.
Dattani, N. S.
Deppe, D. G.
Diedrichs, C.
Fafard, S.
Figeri, P.
Flagg, E. B.
Forchel, A.
Förstner, C.
Förstner, J.
Founta, S.
Fox, A. M.
Franchi, S.
Frova, A.
Gauger, E. M.
Glässl, M.
Gopal, A. V.
Hughes, S.
Imamoglu, A
Imamoglu, A.
Jacak, L.
Jetter, M.
Kaer, Per
Kheng, K.
Kim, H.
Kiraz, A.
Knorr, A.
Krummheuer, B.
Kuech, T. F.
Kuhn, T.
Langbein, W.
Lapointe, J.
Löffler, A.
Lorke, Michael
Lovett, B. W.
Lu, Chao-Yang
Lüker, S.
Ma, W.
Machnikowski, P.
Mahan, G. D.
Mariette, H.
Marsal, L.
Matthiesen, C.
McCutcheon, D. P. S.
McCutcheon, S.
Michler, P.
Milde, F.
Mnaymneh, M.
Moelbjerg, Anders
Mollow, B. R.
Mørk, Jesper
Muljarov, E. A.
Muller, A.
Nazir, A.
Nazir, Ahsan
Nguyen, H. S.
Nick Vamivakas, A.
Ortner, G.
Ouyang, D.
Peeters, F. M.
Poole, P. J.
Puri, R. R.
Ramsay, A. J.
Reinecke, T. L.
Reiter, D. E.
Reitzenstein, S.
Richter, D.
Robertson, J. W.
Roßbach, R.
Roussignol, P.
Roy, C.
Rudin, S.
Salamo, G. J.
Sazonova, V.
Schneider, S.
Sellin, R. L.
Shih, C. K.
Skolnick, M. S.
Ulhaq, A.
Ulrich, S. M.
Vagov, A.
Vamivakas, A. N.
Voisin, C.
Waks, E.
Wang, X. Y.
Wasilewski, Z.
Weber, C.
Weiler, S.
Williams, R. L.
Wilson-Rae, I.
Woggon, U.
Xiao, M.
Yakovlev, D. R .
Yao, P.
Zhang, J.
Zhao, Yong
Zimmermann, R.

App. Phys. Lett.

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

Material Science Reports

T. F. Kuech, “Metal-organic vapor phase epitaxy of compound semiconductors,” Material Science Reports2, 1–50 (1987). [CrossRef]

Nat. Photonics

A. Ulhaq, S. Weiler, S. M. Ulrich, R. Roßbach, M. Jetter, and P. Michler, “Cascaded single-photon emission from resonantly excited quantum dots,” Nat. Photonics6, 238 (2012). [CrossRef]

Nat. Phys.

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.5203–207 (2009). [CrossRef]

Nat. Physics

A. Nick Vamivakas, Yong Zhao, Chao-Yang Lu, and Mete Atatüre, “Spin-resolved quantum-dot resonance fluorescence,” Nat. Physics5, 198–202 (2009). [CrossRef]

New J. Phys.

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]

Photon Nanostruct: Fundam. Appl.

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]

Phys Rev B

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]

Phys. Rev.

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

Phys. Rev. A

A. Kiraz, M. Atatüre, and A Imamoğlu, “Quantum-dot single-photon sources: Prospects for applications in linear optics quantum-information processing,” Phys. Rev. A69, 032305 (2004). [CrossRef]
G. S. Agarwal and R. R. Puri, “Cooperative behavior of atoms irradiated by broadband squeezed light,” Phys. Rev. A41, 3782 (1990). [CrossRef] [PubMed]

Phys. Rev. B

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]
L. Besombes, K. Kheng, L. Marsal, and H. Mariette, “Acoustic phonon broadening mechanism in single quantum dot emission,” Phys. Rev. B63, 155307 (2001). [CrossRef]
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]
I. Wilson-Rae and A. Imamoğlu, “Quantum dot cavity-QED in the presence of strong electron-phonon interactions,” Phys. Rev. B65, 235311 (2002). [CrossRef]
P. Machnikowski and L. Jacak, “Resonant nature of phonon-induced damping of Rabi oscillations in quantum dots,” Phys. Rev. B69, 193302 (2004). [CrossRef]
K. J. Ahn, J. Förstner, and A. Knorr, “Resonance fluorescence of semiconductor quantum dots: Signatures of the electron-phonon interaction,” Phys. Rev. B71, 153309 (2005). [CrossRef]
A. Nazir, “Photon statistics from a resonantly driven quantum dot,” Phys. Rev. B78, 153309, (2008). [CrossRef]
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]

Phys. Rev. Lett.

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

Phys. Rev. X

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. X1, 021009 (2011). [CrossRef]

Other

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.
e.g., see H. J. Carmichael, Statistical methods in quantum optics 1: Master equations and Fokker-Planck equations (Springer, 2003).
G. D. Mahan, Many-Particle Physics (Plenum, New York, 1990). [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

2012, Matthiesen, Phys. Rev. Lett.
2012, Ulhaq, Nat. Photonics
2012, Roy, Photon Nanostruct: Fundam. Appl.
2012, Moelbjerg, Phys. Rev. Lett.
2012, Weiler, Phys. Rev. B
2012, Roy, Phys Rev B
2011, Hughes, Phys. Rev. B
2011, McCutcheon, Phys. Rev. B
2011, Glässl, Phys. Rev. B
2011, Roy, Phys. Rev. Lett.
2011, Ulrich, Phys. Rev. Lett.
2011, Roy, Phys. Rev. X
2011, Nguyen, App. Phys. Lett.
2010, Ramsay, Phys. Rev. Lett.
2010, McCutcheon, New J. Phys.
2009, Flagg, Nat. Phys.
2009, Nick Vamivakas, Nat. Physics
2009, Ates, Phys. Rev. Lett.
2008, Nazir, Phys. Rev. B
2007, Vagov, Phys. Rev. Lett.
2007, Muller, Phys. Rev. Lett.
2005, Ahn, Phys. Rev. B
2004, Machnikowski, Phys. Rev. B
2004, Kiraz, Phys. Rev. A
2004, Muljarov, Phys. Rev. Lett.
2004, Ortner, Phys. Rev. B
2003, Förstner, Phys. Rev. Lett.
2003, Förstner, Phys. Rev. Lett.
2002, Krummheuer, Phys. Rev. B
2002, Bayer, Phys. Rev. B
2002, Wilson-Rae, Phys. Rev. B
2001, Besombes, Phys. Rev. B
2001, Borri, Phys. Rev. Lett.
2000, Bissiri, Phys. Rev. B
1990, Agarwal, Phys. Rev. A
1987, Kuech, Material Science Reports
1969, Mollow, Phys. Rev.

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