Two-photon interference from a quantum dot–microcavity: Persistent pure-dephasing and suppression of time-jitter

Two-photon interference from a quantum dot–microcavity: Persistent pure-dephasing and suppression of time-jitter

Abstract

We demonstrate the emission of highly indistinguishable photons from a quasi-resonantly pumped coupled quantum dot–microcavity system operating in the regime of cavity quantum electrodynamics. Changing the sample temperature allows us to vary the quantum dot–cavity detuning, and on spectral resonance we observe a three-fold improvement in the Hong–Ou–Mandel interference visibility, reaching values in excess of 80%. Our measurements off-resonance allow us to investigate varying Purcell enhancements, and to probe the dephasing environment at different temperatures and energy scales. By comparison with our microscopic model, we are able to identify pure-dephasing and not time-jitter as the dominating source of imperfections in our system.

I Introduction

Single indistinguishable photons are key to applications in quantum networks Pan et al. (2012), linear optical quantum computing Kok et al. (2007); O’Brien (2007) and quantum teleportation Nilsson et al. (2013); Gao et al. (2013). One of the most promising platforms for single photon sources are solid-state quantum dots (QDs) Michler et al. (2000); Santori et al. (2002); Flagg et al. (2010); Gold et al. (2014); Müller et al. (2014). Compared to alternative platforms, such as cold atoms or trapped ions, single QDs offer several advantages: they can be driven electrically, which is of crucial importance for compact future applications Yuan et al. (2002); Heindel et al. (2010); Ellis et al. (2008), and in principle can be integrated in complex photonic environments and architectures, such as on-chip quantum optical networks Yao et al. (2010); Hoang et al. (2012). When embedded in a bulk semiconductor, however, QDs suffer from poor photon extraction efficiencies, since only a minor fraction of the photons can leave the high refractive index material. This problem can be mitigated by integrating QDs into optical microcavities Heindel et al. (2010); Reitzenstein and Forchel (2010); Gazzano et al. (2013); Gérard et al. (1998) or photonic waveguides Claudon et al. (2010); Reimer et al. (2012); Arcari et al. (2014), which can enhance extraction efficiencies to values beyond 50%.

In addition to increased extraction efficiencies, exploiting cavity quantum electrodynamics (cQED) effects in QD-based sources can have a positive effect on the interference properties (and hence the indistinguishability) of the emitted photon wave packets. Ideally, the wave packets emitted by an indistinguishable photon source are Fourier-limited, with a recombination time , and temporal extension of the wave packet given by He et al. (2013). If additional dephasing channels with a characteristic time exist, such as coupling to phonons or spectral diffusion, the coherence time is reduced according to , which consequently leads to a reduction of the two photon interference visibility. It was theoretically shown that pure dephasing strongly affects the detuning dependence of the relative strength of the cavity and QD-emission peaks Naesby et al. (2008); Auffeves et al. (2009). In the regime of cQED, the lifetime of the QD excitons can be manipulated via the photonic density of states in the cavity (the Purcell effect). If the timing of emission events is precisely known, and is constant, shortening of the emitter lifetime via the Purcell effect leads to an improved interference visibility as the condition can be approximately restored Santori et al. (2002); Gazzano et al. (2013); Varoutsis et al. (2005). This simple picture, however, is known to breakdown if there are uncertainties in the timing of emission events (time-jitters) Kiraz et al. (2004); Troiana et al. (2006); Kaer et al. (2013a), or if the dephasing environment gives rise to more than a simple constant pure-dephasing rate, as is known to be the case for phonons Kaer et al. (2013a, b); Kaer and Mørk (2014); McCutcheon and Nazir (2013); Wei et al. (2014). As such, with the aim of designing improved single indistinguishable photon sources, it is crucially important to first establish the magnitude of time-jitters and the nature of any dephasing environments.

In this work, we exploit a microcavity with a high Purcell factor and weak non-resonant contributions of spectator QDs to probe the interference properties of photons emitted from a single QD as a function of the QD–cavity detuning. In contrast to previous studies, where non-resonant coupling to spectator QDs Weiler et al. (2011) or strong temperature induced dephasing Varoutsis et al. (2005) dominated the experiments, we observe a strong improvement of the two-photon visibility on resonance, which exceeds a factor of 3 compared to the off-resonant case. We extend the theoretical model of Ref. Kaer et al. (2013a) to derive an expression for the Hong–Ou–Mandel dip including the effects of both time-jitter and pure-dephasing on- and off-resonance. This allows us to reject timing-jitter, and definitively attribute sources of pure-dephasing as the dominant factor limiting the indistinguishability of our photons. Furthermore, we show that the degree of symmetry we observe for positive and negative detuning suggests pure-dephasing caused by both phonon coupling and spectral diffusion.

Ii Quantum dot–cavity system

The device under investigation comprises a QD embedded in a micropillar cavity with a quality factor of . The layer structure consists of 25 (30) alternating -thick GaAs/AlAs mirror pairs which form the upper (lower) distributed Bragg reflector (DBR). The cavity region is composed of six alternating GaAs/AlAs layers with decreasing (lower part) and increasing (upper part) thickness. A single layer of partially capped and annealed InAs QDs is integrated in the central layer of the tapered segment, i.e. in the vertical maximum of the optical field Lermer et al. (2012). Micropillars with varying diameters were etched into the wafer (the pillar under investigation has a diameter of ) to provide zero dimensional mode confinement. As a result of the Bloch mode engineering Lermer et al. (2012), our micropillars support optical resonances with comparably large quality factors down to the sub-micron diameter range, which yields the possibility to significantly increase the Purcell factor in such microcavities compared to conventional DBR resonators based on -thick cavity spacers. The sample was placed inside an optical cryostat, and the QD was excited via a picosecond-pulsed Ti:sapphire laser with a repetition frequency of MHz (pulse separation ). The laser beam was coupled into the optical path via a polarizing beam splitter, which also suppresses the scattered laser light from the detection path of the setup. Further filtering was implemented by a long-pass filter in front of the monochromator. After spectral filtering, the emitted photons were coupled into a polarization maintaining single mode fibre followed by a fibre coupled Mach-Zehnder-Interferometer (MZI) with a variable fibre-coupled time delay in one arm to measure the two-photon-interference in a Hong–Ou–Mandel (HOM) setup. The second beamsplitter of the MZI can be removed to directly measure the autocorrelation function of the signal.

Figure 1: a) Temperature dependent intensity map of the QD–cavity system. The QD reaches spectral resonance with the cavity mode at K. b) QD lifetime as a function of QD–cavity detuning. The fit is a Lorentzian profile where the linewidth is fixed to the cavity linewidth. A Purcell enhancement of is extracted.

Fig. 1 (a) shows the temperature dependent micro-photoluminescence (-PL) map of the investigated QD–cavity system, which was recorded under non-resonant excitation conditions. The QD emission line, which we attribute to the neutral exciton, can be tuned through the cavity mode by changing the sample temperature. Spectral resonance with the fundamental cavity mode is achieved at  K. Due to the Purcell enhancement, the integrated intensity of the QD increases by a factor of more than three when the QD and cavity are tuned into resonance. In order to directly and accurately extract the Purcell factor of our coupled system, we measured the exciton lifetime via time-resolved -PL as a function of the QD-cavity detuning (see Appendix A.2). As seen in Fig. 1 (b), we observe a strong decrease of the lifetime when the QD is tuned into resonance as a result of the Purcell effect. The Purcell factor  Munsch et al. (2009) is extracted by fitting the data with a Lorentzian profile (the width being fixed to the cavity linewidth ), and we find a value as high as as a result of the small mode volume of our microcavity.

Figure 2: a) Autocorrelation histogram on QD–cavity resonance, from which we extract . b) Autocorrelation function as a function of QD–cavity detuning (the inset shows a zoom-in of the y-axis).

We now study the single photon emission properties of our system, which is particularly important on-resonance, where the single photon characteristics can be deteriorated by non-resonant contributions to the cavity from spectator QDs, or luminescence from the background continuum funnelled into the cavity mode Hennessy et al. (2007). The second-order photon-autocorrelation was probed under quasi-resonant excitation conditions, with a laser tuned to the high energy side of the single exciton emission feature, with a (below saturation) power of . The on-resonance ( K) autocorrelation histogram is shown in Fig. 2 (a). The strongly suppressed peak around is a clear signature of single photon emission. We extract the value by dividing the area of the central peak by the average area of all the side peaks, leading to , reflecting the high purity of our cavity enhanced single photon source. Off-resonance we find a minimum value of at meV ( K). For increasing temperatures, we note a modest increase up to for meV ( K). This value is still close to perfect single photon emission, and we attribute the slight rise to a lowered signal to background ratio between QD and cavity emission. We note that no deterioration of the value can be observed on spectral resonance, which suggests only very weak contributions from spectator QDs to the cavity signal.

Figure 3: a) Histogram of the two-photon-interference for zero time delay between the two arms of the interferometer, from which we extract . b) Measured two-photon-interference versus the time delay. The measurement shows the clear HOM-dip (black dots). Due to a non-vanishing two-photon probability, the data points go slightly above for large . These are corrected by subtracting half the measured value on resonance shown in Fig. 2 (red markers). These corrected data were fitted to extract a visibility of %.

Iii Photon Indistinguishability

We now assess the indistinguishable nature of the emitted photons, which we probe in the HOM interferometer under the same pulsed quasi-resonant excitation conditions. The second order correlation histogram for zero time delay in the MZI for the resonant case is shown in Fig. 3 (a). Strong suppression of the central correlation peak directly reflects a strong degree of photon indistinguishability. The black markers in Fig. 3 (b) are obtained by dividing the area of the peak centred around by that centred around for various time delays , and we observe a clear HOM-dip. For large time delays the correlation values slightly exceed as a result of the finite two photon emission probability, as seen in Fig. 2. We correct the interference data by subtracting half the corresponding experimentally extracted on-resonance value of [red markers] (see Appendix B.2). We then fit our data to the function , where we set (see Fig. 1 (b)), and we find a visibility of . This high value is a direct consequence of the large Purcell factor in our high quality QD–cavity system.

To further analyse our experimental data, and in particular, to determine the relative influences of time-jitter and pure-dephasing on the indistinguishability of the emitted photons, we extend the theory of Ref. [Kaer et al., 2013a] to derive an expression for the TPI as a function of both time delay and detuning. Dephasing caused by coupling to phonons is known to affect the two-photon interference (TPI) properties of the emission from a QD–cavity system in a highly non-trivial way, giving rise, for example, to pronounced asymmetries for positive and negative QD–cavity detunings Kaer et al. (2013a, b); Kaer and Mørk (2014). We find, however, that nearly all features seen in our data can be well reproduced by a model assuming a simple constant pure-dephasing rate. We present this simplified model first, and then go on to show that by including phonons in a rigorous manner at a Hamiltonian level, the behaviour off-resonance allows us to approximately determine the relative influence of phonons as compared to other sources of dephasing.

We model the QD as a three-level-system, and consider the vacuum and single photon Fock states of the cavity. Provided the QD–cavity coupling strength is sufficiently weak, and/or the cavity decay rate is sufficiently large, the cavity degrees of freedom can be adiabatically eliminated from the equations of motion for the QD–cavity system Kaer et al. (2013a). The result is a master equation of the form (see Appendix B.3)

(1)

where the states , represent the QD in ground (g), single exciton state (e), or pump-level (p), with the cavity containing zero or one excitations. The QD–cavity detuning is , while is the pure-dephasing rate, and is the rate at which the pump-level decays into the single exciton state, with determining the magnitude of the time-jitter (i.e. represents the ideal case in which there is no time-jitter). The Purcell enhanced spontaneous emission rate is

(2)

with the background decay rate, the QD–cavity coupling strength, and with the cavity decay rate. The validity of Eq. (1) relies on the condition , which is satisfied in all our experiments.

Eq. (1) can then be used to derive an expression for the normalised coincidence events in the TPI measurements (for details see Appendix B.3). The second order correlation function for the HOM interference measurements is found to read

(3)

where the detuning dependence enters through [see Eq. (2)], and is the visibility. We note that while the expression for has been derived before Kaer et al. (2013a), to our knowledge Eq. (3) represents the first time the full behaviour of the HOM-dip for nonzero values of including time-jitter and pure-dephasing has been presented. This model provides us with simple analytical expressions with which we can fit the experimental TPI data. Crucially, it allows us to explore how a given set of parameters simultaneously affects the HOM-dip and the TPI visibility as the QD and cavity are moved off resonance.

Figure 4: a) HOM-dip and HOM-dip depth as a function of detuning (b). The dashed blue curves correspond to a parameter set for which time-jitter dominates, which we find to be inconsistent with the data off resonance shown in (b). The dotted green curves correspond to a parameters set for which pure-dephasing dominates, which is able to consistently reproduce all data. The solid orange curves again correspond to a parameter set dominated by pure-dephasing, but where of the dephasing on-resonance is caused by coupling to phonons.

In Fig. 4 (a) we show again the HOM-dip, while in Fig. 4 (b) we show the depth of HOM-dip as a function of detuning. We see a pronounced rise of the HOM-dip as the QD is brought out of resonance, corresponding to visibilities on- and off-resonance which differ by more than a factor of . The dashed blue curves in Fig. 4 show a fit to Eq. (3) where only the data in Fig. 4 (a) (the HOM-dip) is considered. Fitting parameters and are found, corresponding to a regime where time-jitter dominates. These parameters are able to reproduce the HOM-dip well, but fail to describe the data off resonance. Indeed, we find that if we simultaneously fit all the data shown in Fig. 4, we find parameters and , corresponding to a regime for which pure-dephasing dominates. These parameters are shown by the dotted green curves, and much better agreement is found. These two fits show that while both time-jitter and pure-dephasing affect the shape of the HOM dip in a similar way, the reduction in the visibility seen off-resonance can only be explained by a system in which pure-dephasing dominates. As the QD and cavity are moved off resonance, the Purcell effect weakens (see Fig. 1) and decreases. For a source dominated by pure-dephasing, the visibility is given by , and a reduction in causes a reduction in . For a source dominated by time-jitter, the visibility instead follows , and a reduction in increases . We stress that which of two regimes is relevant for a particular system has important consequences for how experimental improvements will translate to improvements in photon indistinguishabilities. In the present case, since pure-dephasing dominates, a complete elimination of time-jitters (achieved, for example, via strictly resonant excitation conditions), will lead to only a modest increase in the visibility, while an elimination of sources of pure-dephasing will lead to an increase of up to .

Iv Discussion

The low value of implies that our quasi-resonant excitation scheme leads to a very fast relaxation to the desired single exciton state. This is also supported by the laser detuning we use (), which corresponds to the energy of a longitudinal optical phonon, known to relax on this timescale Grange et al. (2007). We attribute pure-dephasing in our sample as caused by exciton–phonon coupling and spectral fluctuation of the QD energy levels on a timescale shorter than the pulse separation of . The constant pure-dephasing rate used in our theory is expected to well approximate the spectral fluctuations, but the influence of phonons is known to give rise to more complicated behavior Kaer et al. (2013a, b); Kaer and Mørk (2014); McCutcheon and Nazir (2013). In particular, differing phonon absorption and emission rates at low temperatures are expected to lead to asymmetries for positive and negative detuning Kaer et al. (2013b). By including phonons using a time-convolutionless master equation technique (see e.g. Ref. Kaer et al. (2013a) or Appendix B.5), we find that these asymmetries can improve our fits. The solid orange curves in Fig. 4 show the predictions of a parameter set similar to that of the dotted green curve, but where we have included phonons with a strength corresponding to approximately of the total pure-dephasing on-resonance 1, and it can be seen that the phonon contribution improves the fits to the data. We note, however, that when increasing the phonon contribution yet further, the fits become worse as the asymmetry becomes too strong. The relatively strong symmetry seen in Fig. 4 (b) therefore leads us to conclude that both phonons, and additional sources of constant pure-dephasing (such as a spectral diffusion) are present in our system.

In conclusion, we have demonstrated the feasibility of our novel cavity design to enhance the emission of indistinguishable single photons generated in epitaxially grown InAs-QDs by a quasi-resonant excitation scheme with a TPI-visibility as high as %, and a two-photon emission probability as low as . We studied the influence of the QD–cavity detuning on both the two-photon-probability and the degree of indistinguishability of the emitted photons. The TPI measurements are explained by our new theory which takes the QD-cavity-detuning, time-jitter and pure-dephasing into account, and which identifies sources of pure-dephasing as the ultimate factor limiting the indistinguishably of emitted photons.

Acknowledgements

The authors would like to thank M. Emmerling and A. Wolf for sample preparation. We acknowledge financial support by the State of Bavaria and the German Ministry of Education and Research (BMBF) within the projects Q.com-H, the Chist-era project SSQN, as well as the Villum Fonden via the NATEC Centre of Excellence. This work was additionally funded by project SIQUTE (contract EXL02) of the European Metrology Research Programme (EMRP). The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. S.H. gratefully acknowledges support by the Royal Society and the Wolfson Foundation.

Appendix A Experimental Methods

a.1 Coherence Measurements

In addition to the Hong–Ou–Mandel (HOM) interference measurements, the coherence of the emitted photons was measured using a free-beam unbalanced Michelson interferometer. One mirror is mounted on a long linear stage which defines the path length difference between both optical arms, and using an additional implemented piezo crystal at one mirror, the contrast of the emitted photons is measured as a function of the path length difference. The measurements (black data points) are shown in Fig. A.1 (a) for a QD in spectral resonance with the cavity mode. Fitting these data points to a Gaussian function of the form we extract a coherence time of . This value is slightly lower than the coherence time extracted from the HOM-dip in Fig. 3, for which . We attribute this slight discrepancy to a long term spectral jitter which affects the QD emission energy on timescales which are longer than the pulse separation. In the HOM-measurement, only subsequently emitted photons separated by (the laser pulse separation) contribute to the measured indistinguishability, and hence the inferred coherence time of . The HOM measurements therefore include an effective time filter. In contrast, the measurements made using the Michelson interferometer are time-integrated, and as such long-term drifts and spectral diffusion result in a deterioration of the extracted value Santori et al. (2002); Kuhlmann et al. (2013); Gold et al. (2014).

a.2 Lifetime Measurements

In order to measure the lifetime of the QD emission we couple the spectrally filtered photons into a single mode fibre attached to an avalanche photo diode (APD) with resolution . Fig. A.1 (b) shows two representative time-resolved measurements of the QD emission under quasi-resonant excitation. The blue round data points correspond to spectral resonance between QD and fundamental cavity mode (), while the red square data points correspond to a detuning of . The measurements (time window  ns) each contain six complete decay curves similar to those shown in Fig. A.1 (b), which we fit to a biexponential decay function. The shorter time constant represents the lifetime of the bright-exciton, while the longer originates from a dark exciton effect. For the decay curves in Fig. A.1 (b) we find on resonance and for .

Figure A.1: a) Contrast of the emission measured using a free-beam unbalanced Michelson interferometer as a function of path length difference of both arms. Fitting the data points with a Gaussian distribution (red solid curve) we extract a coherence time of ps. b) Time-resolved -PL measurements. The blue dotted data points correspond to spectral resonance between QD and fundamental cavity mode ( meV), while the red square points were taken for a detuning of  meV. The solid curves correspond to fits to a biexponential function, from which we extract on resonance and off resonance.

Appendix B Two-Photon Interference Theory

Here we provide the necessary background for the theoretical analysis of the data presented in the main text.

b.1 Hanbury Brown and Twiss Measurements

We first consider the Hanbury Brown and Twiss (HBT) experimental setup used to measure the two-photon emission probability of our source. Emission from the source is incident upon a 50/50 beam splitter, and two detectors are placed equidistantly on the two output arms. We label the time of the detection event at detector 1, and that of detector 2. The probability of detecting a photon at detector 1 at , and at detector 2 at is proportional to the second order field correlation function

(4)

where is the creation operator for the mode propagating to detector 1 in the Heisenberg picture, and similarly for . We relate these modes to those on the input arms, described by creation operators and , using the unitary mode transformation Kiraz et al. (2004)

(5)

where is the delay introduced between arrival times at the beam-splitter. For the HBT measurement, there is no input in arm 1, and we simply have with

(6)

where the subscripts on the operators have been dropped since they are all equal.

To measure the two-photon emission probability, , we integrate Eq. (6) over all and , and divide this area by an adjacent peak. The adjacent peaks correspond to Eq. (6), but where and differ sufficiently that mode operators at these times are completely uncorrelated. This gives the uncorrelated coincidence probability in the HBT measurement with

(7)

The normalised autocorrelation function is then defined as

(8)

which is equal to zero for .

b.2 Hong-Ou-Mandel Experiment

We now consider the Hong–Ou–Mandel (HOM) experimental setup used to measure the indistinguishable nature of the emitted photons. Two emission events are incident on a 50/50 beam-splitter, with a delay introduced into input arm one. The unnormalised probability of a coincidence event is again given by Eq. (4), and the beam-splitter is described by Eq. (5). Upon combining these equations we find 16 terms. These can be simplified by assuming that modes 1 and 2 are identical but statistically independent, which allows us to write , where is any product of mode operators pertaining to mode 1, and similarly for . We then find eight terms linear in and . For an electromagnetic field state of the form , with a Fock state, expectation values linear in the ladder operators are zero, and we neglect these terms. This leaves second and fourth order terms. The second order terms involve expectation values of the form , which also give zero for electromagnetic fields as discussed above. The remaining terms give

(9)

where is the unnormalised first order correlation function.

To normalise this quantity we again consider the scenario in which and are sufficiently separated that mode operators evaluated at these two times are uncorrelated. In doing so we find the uncorrelated coincidence probability for the HOM setup . Since we integrate over all and , the appearances of can be neglected, i.e. we have

(10)

by a simple change of variables. An identical argument can be made for the appearing in in Eq. (9). In the HOM setup we therefore measure the normalised quantity

(11)

where we have defined the strictly-two-photon coalescence probability

(12)

which for becomes the visibility .

b.3 Quantum dot–cavity system

We now develop a master equation which will allow us to derive an analytic expression for in the presence of time-jitter and pure-dephasing. We follow Ref. Kaer et al. (2013a) and model the quantum dot (QD) as a three-level-system, with crystal ground state , single exciton state , and pump level , having energies , and respectively. The cavity mode is described by creation and annihilation operators and , and has frequency . The system is depicted in Fig. (B.2). In a rotating frame the QD–cavity system is described by the Jaynes-Cummings Hamiltonian

(13)

where is the detuning of the QD transition from the cavity mode, and is the QD–cavity coupling strength. Relaxation processes are added using the Lindblad formalism Breuer and Petruccione (2002), and the master equation describing the QD–cavity degrees of freedom becomes

(14)

where the Lindblad operators satisfy , with and the decay rates of the pump-level and cavity respectively. The background spontaneous emission rate of the QD is , and the rate describes pure-dephasing of the QD excited state level.

Figure B.2: Schematic diagram of the system under consideration; a quantum dot with pump-level , excited state , and ground state couples to a cavity mode with strength and detuning . We consider only the zero photon and one photon manifolds of the cavity mode, which decays with rate . The transition has rate , while the spontaneous emission process of has background rate . The excited state undergoes pure-dephasing with rate . Adiabatic cavity elimination results in an effective three-level system with a modified spontaneous emission rate .

In the limit of weak QD–cavity coupling and/or strong cavity decay, the cavity can be adiabatically eliminated from equations of motion describing our system. Formally, we require with , and provided we consider the initial state with the vacuum state of the cavity mode, the dynamics can be well approximated by the master equation Kaer et al. (2013a)

(15)

which is Eq. (1) in the main text.

b.4 Photon Indistinguishability for the QD–Cavity System

We now use Eq. (15) to calculate the two-photon interference probability, Eq. (11). To proceed, we note that in the far field we can make the replacement with  Kiraz et al. (2004) in Eq. (11). Then, to calculate the second order correlation function , we make use of quantum regression theorem to write Carmichael (1998)

(16)

For we find since , and as such and we can set in Eq. (11). This reflects that for the theory presented here we have strictly one (or less) excitation in the system at any time.

We now calculate the two-photon coalescence probability expressed in Eq. (12). To begin we consider the uncorrelated probability . The quantity is just the excited state population at time , and from Eq. (15) we have

(17)

where the Heaviside theta function ( for and for ) has been introduced to ensure no excitations are present before emission events. From the quantum regression theorem the first order correlation function obeys the equation of motion

(18)

with initial condition , which gives

(19)

Finally, performing the integrals in Eq. (11) we arrive at Eq. (3) in the main text.

b.5 Exciton–phonon coupling

To explore the influence of phonons seen in our data, a weak exciton–phonon coupling time convolutionless master equation technique is used Kaer et al. (2013a). To second order in the exciton–phonon coupling strength, and within the Born-Markov approximation, the master equation for the complete QD–cavity system (i.e. before adiabatic elimination) becomes

(20)

where the new phonon-induced dissipator is given by

(21)

where denotes a trace over the phonon modes. The interaction Hamiltonian is written

(22)

where is a creation operator for a phonon mode with wave-vector , and describes its coupling strength to the QD exciton. The interaction picture interaction Hamiltonian is defined by , where , with phonon Hamiltonian and the frequency of mode . Finally, we assume a thermal state for the phonon density operator: , with and the sample temperature.

The strength of the QD–phonon coupling is characterised by the spectral density, defined as , and which for excitons in QDs has been shown to be adequately described by the function

(23)

where captures the overall strength of the interaction determined by material parameters, and is the photon cut-off frequency Ramsay et al. (2010). The behaviour of the phonon dissipator in Eq. (21) in different parameter regimes has been discussed in detail elsewhere Kaer et al. (2013a, b); Kaer and Mørk (2014). The parameters used to obtain improved fits to the data in the main text (the solid orange curves in Fig. 4) are and , while the constant pure-dephasing rate was reduced to . These parameters correspond to phonons contributing approximately of the dephasing on-resonance. We note that the other parameters in the model were adjusted to and in order that the times as a function of detuning were well reproduced.

The density operator entering Eq. (20) contains both QD and cavity degrees of freedom. When relating the field operator to the QD–cavity system, we have a choice to consider QD emission or cavity emission, making respectively the replacements or in the field correlation functions. Our data was better described by cavity emission, which we attribute to the high Purcell factor of our QD–cavity system.

Footnotes

  1. We have also adjusted g and so that the data are well reproduced off resonance

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