Non-exponential spontaneous emission dynamics for emitters in a time-dependent optical cavity

We have theoretically studied the effect of deterministic temporal control of spontaneous emission in a dynamic optical microcavity. We propose a new paradigm in light emission: we envision an ensemble of two-level emitters in an environment where the local density of optical states is modified on a time scale shorter than the decay time. A rate equation model is developed for the excited state population of two-level emitters in a time-dependent environment in the weak coupling regime in quantum electrodynamics. As a realistic experimental system, we consider emitters in a semiconductor microcavity that is switched by free-carrier excitation. We demonstrate that a short temporal increase of the radiative decay rate depletes the excited state and drastically increases the emission intensity during the switch time. The resulting time-dependent spontaneous emission shows a distribution of photon arrival times that strongly deviates from the usual exponential decay: A deterministic burst of photons is spontaneously emitted during the switch event.


Willem L. Vos
Complex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands (Dated: February 1, 2013) We have theoretically studied the deterministic switching in the time domain of the spontaneous emission of two-level emitters in an optical microcavity. We consider a system in the weak coupling regime, for which the refractive index of the cavity material is modified on a time scale shorter than the emitted lifetime, through the injection of free charge carriers. The equation of motion is introduced for the excited population of two-level emitters coupled to a time-dependent local density of optical states. We then derived the emitted intensity as a function of time for both continuous wave and pulsed excitation of the embedded emitters. We demonstrate that a short temporal increase of the radiative decay rate drastically increases the emission intensity during the switch time for both continuous wave excitation and pulsed excitation. For pulsed excitation, the resulting spontaneous emission shows a distribution of photon arrival times with a deterministic burst of photons during the switch event, that strongly deviates from the standard exponential law. Finally, we derive a figure of merit that quantifies the total switching action.

I. INTRODUCTION
Light is essential for myriad processes around us: in nature, to human life, and to technological applications and everyday appliances. It is known that an elementary light source such as an excited atom, quantum dot or molecule emits a photon either spontaneously or stimulated by an external field [1]. The rate of spontaneous emission is not an immutable property of a light source [2,3] but the rate strongly depends on its surroundings. For an emitter coupled to a continuum of electromagnetic modes, quantum mechanics predicts the rate of spontaneous emission of an excited two-level source to follow Fermi's golden rule [4]: the rate is determined by a product of atomic transition matrix elements of the dipole operator with the local radiative density of optical states (LDOS), that typifies the surroundings [5]. The LDOS counts the number of modes in which a photon can be emitted, and it can be interpreted as the density of vacuum fluctuations at the position of the source. A main feature of spontaneous emission is its dynamics: an emitted photon is detected at a random moment in time after the emitter has been excited. Both the distributions of emitted photons and of the excited-state population decay exponentially in time.
A well-known tool to modify the average spontaneous decay rate of a source in the frequency domain is a resonant cavity tuned to the source's emission frequency. It was first realized by Purcell that the decay rate of an atom can be increased [6], known as the Purcell effect. In the weak coupling regime the Purcell factor F p gauges the change of the radiative decay rate Γ cavity of an emitter in a cavity relative to the rate Γ 0 in a homogeneous medium: F p ≡ Γ cavity /Γ 0 . Following the pioneering work by Gérard et al. [7], many groups have demonstrated the Purcell effect with quantum dots embedded in solid-state microcavities [8][9][10]. To date, impressive progress has been achieved in controlling spontaneous emission in the frequency domain with nanophotonic structures [11][12][13], like microcavities and photonic crystals [14,15] and nano-antennas [16]. In all cases, however, the modification of the LDOS is stationary in time. Thus, the radiative decay rate is time independent and the distribution of photon emission times is single exponential in time and completely determined by this rate.
In this work, we theoretically propose a novel paradigm in light emission by modifying the environment of an ensemble of two-level sources in time during their lifetime. By utilizing fast optical switching of a microcavity, we can tune the cavity resonance and drastically change the LDOS at the emission frequency within the emission lifetime [17]. As a result, we anticipate bursts of dramatically enhanced emission, concentrated within short time intervals. The spontaneous emission process remains stochastic but results in a strongly non-exponential temporal distribution that is completely controlled by the experimentalist. Our approach offers a tool to dynamically control the light-matter coupling [18] and, in the long run, offers a novel tool to realize the breakdown of the weak-coupling approximation and realize non-Markovian dynamics in cavity quantum electrodynamics.
First we introduce the equation of motion that describes the population density of an ensemble of two-level sources with a time-dependent decay rate due to a time-dependent LDOS. From this equation, we derive the time-dependent emitted intensity for continuous wave and pulsed excitation of an ensemble of two-level sources such as quantum dots [7] or rare earth atoms [19] in a cavity. Since micropillar cavities are known to be a versatile class of microcavities we choose them as an example. The decay rate of the ensemble, determined by the LDOS, is switched by exciting free carriers, which is a well-known control mechanism in the time domain for nano-cavities [20][21][22][23][24][25].

A. Rate equations
We consider a two-level emitter in a medium with a strongly dispersive LDOS ρ(ω, r) in a photonic microcavity and investigate what happens if the LDOS changes in time, thereby modifying the radiative decay rate in time. To derive the rate equation of a two-level source we start with the equation of motion of the probability amplitude of the excited two-level source c a (t) [26] Here d and e d are the amplitude and orientation vector of the transition dipole moment, respectively, the reduced Planck's constant, ǫ 0 the dielectric constant of vacuum, r the emitter position, and ω d the emission frequency.
For convenience, we only write the time dependency of c a (t), but it should be kept in mind that the amplitude c a (t, r, e d , ω d ) also depends on r, e d and ω d [27]. In the following we limit ourselves to the weak coupling limit where the factor (ωρ(ω, e d , r, t ′ )) varies slowly over the linewidth of the emitter. This approximation is known as the Markov approximation [28] or the Wigner-Weisskopf approximation [1]. In the Markov approximation we can take ωρ(ω, e d , r, t ′ ) out of the frequency integral and Eq. (1) can be simplified to [29] dc a (t) The integral in Eq. (2) can be evaluated to yield [30] dc a (t) which can be written as with Γ rad the radiative rate Equation (5) is Fermi's golden rule [4] augmented with a time-depend LDOS. This shows that in the Markov limit the instantaneous radiative rate Γ rad (t) directly follows the time dependence of the LDOS. In case of a time-independent LDOS the rate Γ rad (t) = Γ rad is constant and Eq. (4) shows the well-known feature that the amplitude c a (t) decreases exponentially with the rate Γ rad 2 [1]. Similarly, the probability |c a (t)| 2 of the two-level system to be excited decreases exponentially according to For a time-dependent LDOS the rate in Eq. (6) is no longer constant and the excited state population decreases non-exponentially and thus deviates from the standard Markovian dynamics. With Eq. (4) we can write the equation of motion for the population density N 2 (t) for an ensemble of N identical (no inhomogeneous broadening) and non-interacting two-level sources. To complete the model we include a timedependent excitation term for the sources and a non-radiative decay rate Γ nrad . The equation of motion for the population density becomes The first term describes excitation and depends on the excitation power P exc (t), the excitation frequency ω exc and, the absorption efficiency of the excitation power that reaches the two-level source η abs . The second describes radiative decay and the third term non-radiative decay. For convenience, we write N 2 (t) only as function of time t in Eq. (7). For any sub-ensemble of the N two-level sources N 2 (t) also depends on r, e d and ω d , hence for an inhomogeneously broadened (sub-)ensemble we should average over r and e d . At any rate the general solution of Eq. (7) is The corresponding radiated emission intensity I em (t) is given by [31] which means that the total emitted light intensity is proportional to the instantaneous radiative decay rate and the population density. Equations (8) and (9) form the basis for our further discussion and they will be used to calculate the emission for an ensemble of emitters with a time-dependent radiative rate in the cases of continuous-wave and pulsed excitation.

B. Time dependent radiative decay rate
The central goal of this work is to describe the effects of a time-dependent radiative decay rate Γ rad (t) that is realized by dynamically changing the LDOS in time at the position and frequency of an emitter. In general, we can separate the time-dependent decay rate into a constant rate Γ 0 and a time-dependent change in the decay rate ∆Γ rad (t) where ∆ rad (t) is proportional to the change in the LDOS We assume that the time-depended part is the result of a short switching event that quickly changes the LDOS within a characteristic switching time τ sw . Two examples of normalized time-dependent decay rates are shown in Fig. 1 for emitters excited at t = 0 ps followed by a short switching event at t = t 0pu = 10 ps. In the examples in Fig. 1 the switching gives either an enhancement or inhibition by a factor of 5 that decay with a characteristic effective switching time of τ sw . In the following we choose for practical reasons a scheme where the emitter is embedded in a semiconductor cavity. Moreover, the LDOS is modified in time by controlling the refractive index by means of the free carrier density in the semiconductor, as excited by a short optical (or electrical) pump pulse at t = t pu . The induced change in the refractive index is proportional to the free carrier density [32] and the resulting change in the LDOS depends strongly on the kind of the microcavity [33]. The excited free carriers recombine exponentially with a characteristic recombination time τ sw , after which the refractive index is restored to its original value [24,32]. Here we use τ sw = 35 ps, characteristics for GaAs [24].
As an illustration, we consider in Fig. 2 the effect of switching the resonance frequency ω cav,0 of a microcavity with a Lorentzian LDOS with linewidth Γ cav in the spectral vicinity of the emission frequency ω d of a emitter. The decrease in the refractive index induced by the free carriers will lead to a positive frequency shift of the resonance frequency ω cav (t) as indicated in Fig. 2. The emitter is initially detuned from the cavity resonance and experiences a low radiative rate Γ 0rad . During the switch event the cavity peak is tuned into resonance with the emitter as shown as the dashed Lorentzian in Fig. 2. This change results in a rapid increase in the LDOS at the emitter frequency and greatly enhances the decay rate Γ rad (t) from the initial value Γ rad (0) = Γ 0 to its maximum value of Γ rad (∆t) = Γ 0 + ∆Γ rad and back to Γ 0 at a time ∆t. The effective switching time in this scenario is therefore given by A shorter effective switching time can thus be realized by either faster tuning of the cavity resonance in a time ∆ or by increasing the spectral tuning range relative to the cavity linewidth within the time ∆t. We note that this switching procedure is very flexible and we can effectively move along different trajectories on the cavity's LDOS by choosing the initial detuning and strength of the switching effect. For example, we can consider an (color online) Schematic graph of the switching process as experienced by a quantum emitter (green) emitting at a frequency ω d in the spectral vicinity of a cavity resonance whose frequency is switched in time. The cavity has a Lorentzian local density of states (solid line). Initially the emitter is detuned from the cavity resonance ωcav,0 by nearly one cavity linewidth, leading to an effective radiative rate Γ0. The switching process moves the cavity resonance up in frequency ωcav(t) (gray dashed). The cavity is then tuned into resonance with the emitter that thus experiences a decay rate strongly enhanced by ∆Γ rad . Within one cavity linewidth from the resonance, switching of the cavity resonance can be approximated by a linear shift of the decay rate versus frequency (red dashed line). alternative experiment where the emitter starts on resonance with a radiative rate that is already Purcell enhanced. The switch then detunes the cavity resonance away from the emitter's frequency and thus inhibits the spontaneous decay rate. The steep slope of the cavity LDOS gives a rapid change in the LDOS that can be used to either greatly enhance or inhibit the radiative decay rate, relative to the unswitched rate.
For initial detunings between the cavity and emitter frequency smaller than the cavity linewidth (ω d −ω cav,0 < Γ cav ) we can approximate the steep slope of the Lorentzian resonance as a linear trend shown as the red dashed line. We can therefore effectively approximate a linear relation between the excited free carrier density and the radiative decay rate. For a typical switching pulse with a Gaussian temporal width τ pu = 120 fs, much shorter than the carrier recombination time (35 ps, see Ref. [22][23][24]), we can separate the excitation and relaxation time scales of the free carriers. Using the linear relation between the carrier density and the decay rate discussed above, the time-dependent decay rate is then decomposed into an constant decay rate Γ 0 and a change induced by the switch that is turned on at time t 0pu . The magnitude of the switched term in Eq. (13) then decays exponentially with an effective switching time comparable to the free carrier relaxation time. The initialization of the switch is evaluated as an integral over the Gaussian switch pulse resulting in an error function that in the limit τ pu ≪ τ sw can be approximated with a Heaviside step function. The normalized time-dependent decay rate in Fig. 1 giving an enhancement thus depicts the situation where we quickly tune the cavity frequency, initially off-resonance, into resonance with the emitter as illustrated in Fig. 2. As a result the radiative rate is quickly and greatly increased at t = 10 ps before decreasing at a speed set by switching time. Similarly, the other curve illustrates the situation where the light source is initially on resonance and the cavity is then switched out of resonance. In the examples in Fig. 1 the switching gives either an enhancement or inhibition by a factor of 5, which is a realistic change observed on ensemble of quantum dots in micropillar cavities [7]. Note that a constant relative decay rate of Γ rad (t)/Γ 0 = 1 corresponds to the unswitched case, typical for all Purcell experiments performed to date [7][8][9][10]33]. Most striking is the fast dynamics in the decay rate: both the switch pulse duration τ pu and the exponential decrease with decay time τ sw are much faster than the intrinsic lifetime 1/Γ 0 = 1 ns typical for quantum dot emitters.
With the time-dependent decay rate given in Eq. (13) we can state that the assumptions for the population dynamics (from Eq. (1) to Eq. (2))) are valid if ω d τ pu ≫ 1. This is the case for a pump-pulse duration τ pu = 120 fs and ω d = 2 × 10 15 s −1 (corresponding to λ d = 950 nm). In this case the fastest variation of the LDOS in time has a characteristic time that is much longer than the width of the delta function resulting from the exponential in Eq. (1).
In the following sections we will focus on the effects of the time-dependent decay rate on the population density and emitted intensity of an ensemble of two-level sources. We will look at two examples: one where we switch an ensemble of sources that are continuously excited and one where the sources are excited by a short pulse (see Fig. 5) and the environment is subsequently switched.

C. Population and emission dynamics for continuous wave excitation
In the case of continuous wave (CW) excitation of the two-level sources the direct excitation is constant in time P exc (t) = P 0exc = ω exc /η abs . We account for the possibility that a small fraction of the free-carriers used in the switching process will excite the emitters as discussed in Appendix A. We therefore additionally include an excitation term that is proportional to the free carrier concentration. Substituting the time-dependent decay rate (Eq. (13)) and excitation power (Eq. (A1)) into Eq. (8) and integrating over time yields the population dynamics The first term N 0 is the initial population density at time zero set to the steady state population density without the switch. The two next terms represent the pumping of the emitters and the last is the relaxation term from spontaneous emission. Since Eq. (15) cannot be solved analytically, we have solved it numerically and the results are plotted in Fig. 3 for three examples. In the unswitched case with a continuous excitation and a time-independent radiative decay rate, both the emission intensity and excited state population are constant. This value is determined by the equilibrium between the spontaneous emission and excitation of the emitters. Figure 3b shows the dramatic influence on the emission dynamics for a switch pulse at t = t pu = 0 ps. The switching parameters are the same as in Fig. 1 with Γ 0 = 1 ns, ∆Γ rad = 4Γ 0 , η rec = 0 and τ sw = 35 ps. In the first 100 ps after the switch at t = 0, we see a strong burst of spontaneous emission with an intensity that is 6× higher than the steady state value. The switch therefore creates a short intense pulse above a weak background. The duration of the burst is much shorter than both the unswitched and switched radiative decay times of 1/Γ 0 = 1 ns and 1/(Γ 0 + ∆Γ rad ) = 200 ps, respectively. The temporal width is defined by the switching time of 35 ps. After the switch event is completed at t ≈ 200 ps, the emission intensity decreases below the unswitched emission level. This drop is caused by the rapid depletion of the population density during the emission burst shown in Fig. 3a. During this burst, the emission rate is indeed much larger than the (constant) pumping rate. After the switching event, the depleted population of excited emitters emits therefore less light than in the steady state regime. The recovery dynamics of the population of excited emitters and of the emitted signal follow an exponential law, governed by the radiative lifetime Γ 0 .
We next analyze the effect of a potential excitation of the emitters by the free carriers excited by the switching pulse. In Fig. 3 we show the population dynamics and emission intensity for Γ rad = Γ 0 and P pu η rec = 0.5P 0exc . For these parameters the switching pulse does not change the decay rate but only excites the emitters. We therefore see a small increase in the population density just after the switching pulse. This increase then decays back to the steady state level with a time constant given by the radiative decay time 1/Γ 0 . The corresponding emission dynamics shows an almost negligible increase in intensity. Even when the free carrier excitation rate equals the optical free carrier excitation rate (P pu η rec = P 0exc ) the emission intensity is only enhanced by 3%. The small increase is caused by the short duration of the pump. Therefore the difference in time scales between the carrier relaxation time τ sw and the radiative decay time 1/Γ rad (t) governing the intensity dynamics let us distinguish between true decay rate switching or emission increase due to residual free carrier excitation.
From Fig. 3 we conclude that switching the decay rate leads to a strong enhancement of the emission intensity compared to the unswitched case. The duration of the emission enhancement is determined by effective switching time, while the timing of the emission burst is determined by the arrival time of the switching pulse. By applying the switch pulse we are able to deterministically enhance an otherwise small and constant radiative decay rate and to create an emission burst, whose duration is only limited by the effective switching time.

D. Figure of merit for pulsed excitation
In this section we study the dynamics for the excited population of emitters after a pulsed excitation, when the environment is subsequently switched during their decay time. It will appear that the dynamics for pulsed excitation strongly differs from the dynamics under continuous wave excitation. We assume that a short excitation pulse with amplitude P 0exc initializes the system at t = t 0exc such that we have an initial population density N 2 (t = t 0exc ) = N 02 . At times after the excitation pulse the dynamics of the population density is governed only by the time-dependent decay rate and this gives a monotonous decrease in the population density. If we approximate the short excitation pulse by a Dirac delta pulse P exc (t) = δ(t − t 0exc )P 0exc in the rate equation (Eq. (7)) it can be solved analytically for times after the excitation (t > t 0exc ). In this case Eq. (7) simplifies to which can be integrated to yield Equation 17 describes the population density for any time-depended decay rate Γ rad (t) as a function of time t after the excitation process is over. Inserting the switched decay rate Eq. (13) into Eq. (17) and solving the integral over the constant part of the decay rate yields where we have defined a dimensionless time-dependent switch parameter ∆α rad (t) This parameter is a figure of merit that describes the change in population density due to the change in the decay rate. A negative ∆α rad (t) results in a population density that decays slower compared to the unswitched situation, while a positive ∆α rad (t) results in a faster decay. If we assume that the duration of the switch pulse τ pu is short compared to the effective switch time τ sw , the integral in Eq. (19) can be split into two parts -before and after the switch t = τ pu -and ∆α rad simplifies to Here Θ(t − t 0pu , τ pu ) is a step function from 0 to 1 that accounts for the fact that there is no change in the decay rate before the switching pulse arrives at t = t 0pu . In the limit of time t going to infinity ∆α rad (t) becomes Equation (21) shows that ∆α rad (t) is nonzero even in the long time limit and is given by a product of the magnitude ∆Γ rad of the switch and the effective switch duration. The switch therefore has an effect on the population dynamics even long after the switch event. The dimensionless switching parameter ∆α ∞ is therefore a useful figure of merit for the total switching effect on the excited state population. We can quantify the effect the switch has on the population at long times by using Eq. (21) and the limit in Eg. (18) to calculate the ratio between the switched population density N 2s and the unswitched population density N 2us in the limit of t going to infinity Equation (22) quantifies the long term effect of the switching on the population density as a result of a momentarily short change in the decay rate. Figure 4a displays the excited state population Eq. (18) for three cases: Without switch pulse the decay rate is stationary and as expected the population and intensity decay exponentially with the initial rates Γ 0 . Secondly the green long dashed curve shows a switch that tunes a cavity into resonance with an emitter and induces an enhanced decay rate by a factor of 5 (∆Γ rad = 4Γ 0 ) from Γ 0 = 1 ns −1 . Thirdly the red short dashed curve represents the opposite case where a cavity is tuned out of resonance by the switch and induces an inhibition in the decay rate by a factor of 5 starting from a high initial rate Γ 0 = 5 ns −1 . For the two switched examples (dashed lines) the population density clearly decays non-exponentially. In the enhanced case the population decays exponentially before the switching pulse with the same rate as in the unswitched case. As the switching pulse arrives at t = t pu = 150 ps the population decreases faster and thus deviates from exponential decay. During the effective switching time of 35 ps the population density continues to deviate from an exponential decay. A few recombination times after the switch event (t > 250 ps) the decay returns to its original rate but the absolute value of the populations is reduced compared to the unswitched case. Using Eq. (18) and the figure of merit (Eq. (21)) we see that the larger decay rate induced by the switch depletes the excited state population faster, thereby lowering the population density at long times. The population density does not converge back to the unswitched curve (unlike the continuous wave case) since there is no continuous reexcitation of the emitters. The situation is reversed for a switch that induces an inhibition of the spontaneous emission : the population experiences also a non-exponential decay after the switch; however, the population is now larger than its reference value (unsswitched case) at long times.

F. Emission dynamics for pulsed excitation
In this section we discuss the corresponding effects on the dynamics of the emitted intensity from emitters in a switched environment. According to Eq. (9) the emitted intensity I(t) is given by the product between the excited state population and the radiative rate that is also time-dependent. Modifications to the decay rate are therefore directly reflected in the total intensity emitted by the emitter. So, for a large dynamic change in the decay rate, we also expect large changes in the emitted intensity. One striking consequence is that for a time-dependent decay rate the population density and the emission intensity are no longer directly proportional, contrary to the steady state case [31]. Inserting the dynamic decay rate Eq. (13) and the population density Eq. (18) into Eq. (9) with η rec = 0 yields where ∆Γ rad (t) = ∆Γ rad e −(t−t 0pu ) τsw Θ(t − t 0pu , τ pu ) , and ∆α(t) is given by Eq. (20). The main difference between the population density dynamics and the emitted intensity is the presence of the decay rate prefactor (Γ 0 + ∆Γ rad (t)). Nevertheless, the intensity in Eq. (23) is still proportional to the population density so that the influence of the switching process will remain visible in the emission intensity even long after the switch event has passed as discussed in section II D. The relative intensity to the unswitched intensity at long times is thus given by lim t→∞ I(t)/I us (t) = e −∆Γ rad τsw as the exponent is the same as in Eq. (22) and the time-dependent decay rate ∆Γ rad (t) in the prefactor tends to zero for long times. Figure 4b shows the normalized time-resolved emission dynamics for the same two cases as for the population density in Fig. 4a: one where the radiative rate is quickly enhanced from an initial low rate of Γ 0 = 1 ns −1 and another where the radiative rate is inhibited from a high value of Γ 0 = 5 ns −1 . The emitters are first excited at t ex = 0 ps and the emission intensities then start to decay exponentially with the same rate as the population density as expected in the weak coupling limit. A switching pulse arrives at t = t pu = 150 ps whose effect is to either quickly enhance (green long dashed) or inhibit (red short dashed) the radiative decay rate from the initial rate by a factor of 5.
In the case where the switching pulse quickly enhances the decay rate we see in Fig. 4b a short and high intensity burst on top of the normally decaying signal; the intensity thus strongly deviates from an exponential decay. The relative magnitude of the enhancement is equal to the maximum Purcell enhancement and the temporal shape closely follows the modulation in the decay rate as expected from Eq. (23). In this example the temporal shape follows the exponential change in the decay rate and the temporal width is limited by the effective switching time of 35 ps. This time is much shorter than the minimum Purcell enhanced decay time of 200 ps. Let us note at this stage that the effective switching time could be further engineered to be as short as 2 − 3 ps by either decreasing the free carrier lifetime [34] or increasing the frequency shift of the cavity resonance. For times much longer than the switch time, we see a lower intensity relative to the unswitched case due to the depletion of the population density of the emitter discussed in Sec. II D.
In the second case in Fig. 4b the switching event inhibits the decay rate, which results in a short temporal drop in the emission intensity relative to the stationary case as seen in Fig. 4b. The temporal duration of the drop is again limited by the switch time. The drop in intensity exemplifies a highly unusual shape for a decay curve where the radius of curvature, during the free switching time, is negative during the decay. In the traditional paradigm of steady state spontaneous emission such a negative radius of curvature would be unphysical. An exponential decay with a stationary decay rate or even a sum of exponentials with stationary rates will always result in a positive radius of curvature in the resulting decay curve. This shows the flexibility of the switching mechanism to shape the temporal emission distribution of the emitted photons at will.

III. DISCUSSION
Spontaneous emission can be described as a stochastic process for a photon to be emitted from an excited light source [1]. It is thus impossible to predict the exact time when an excited source will emit a photon. On the other hand, the distribution of photon emission times, averaged over many excitation cycles, is usually well-known. In the weak coupling limit the distribution of photon arrival times is exponential and characterized by a single rate given by Einstein's A coefficient. Much interest has been devoted to controlling spontaneous emission by modifying this rate using the Purcell effect by embedding emitters in a nano-structured environment [7,8,10,33]. Several schemes have been implemented to tune the decay rate in time such as gas deposition, temperature and, electronic gating. However, the modification in the rate has in all cases remained constant in time during a single decay process. For this reason, the distribution of photon arrival times thus remains single exponential, and is simply described by one enhanced or inhibited single exponent. We have here introduced a new paradigm with dynamic control of the spontaneous decay rate in time using all optical switching. The spontaneous emission process remains stochastic in time but the dynamical change in the decay rate results in a strongly non-exponential temporal distribution of photon emission times. The active switching process allows us to deterministically control the photon distribution in time. We have shown (in Fig. 4) that photon arrival times can be bunched in short bursts where timing and duration of the burst can be fully controlled by the experimentalist. Naturally, within these short emission pulses, the individual photons still arrive at unpredictable moments in time.
On a more fundamental level spontaneous emission arises from the interaction between a single quantum emitter and fluctuations in the vacuum field at the emitter position. By dynamically modifying the environment of the emitter our approach gives direct temporal control of the local strength of the vacuum field on timescales much shorter than the excited state lifetime. As shown in Fig. 4 this allows to manipulate the excited state probability for a quantum emitter in time and subsequently control the time dependence of the single photon wave function of the emitted photon. Such control opens great prospects in quantum information processing and allows to shape the photon wave function emitted by single photon sources, for example for optimal mode matching of photons [35] and to enhance the absorption of single photons [36,37]. More generally, by dynamically tuning a cavity in the vicinity of a quantum emitter we can drastically modulate the light matter coupling between the emitter and the cavity mode. This offers interesting prospects where a system is modulated between the weak and strong coupling regime while emitting a single photon.
For very fast switching events the decay rate of the emitter can no longer adiabatically follow changes in the environment and the decay rate is not proportional to the instantaneous LDOS but depends also upon the past history of changes in the LDOS. So even a coupled cavity-emitter system, that in the steady state case would be weakly coupled, can by fast switching of the environment be brought into a regime where the weak-coupling approximation breaks down. Our method thus offers a novel tool to realize non-Markovian dynamics in cavity quantum electrodynamics, namely by very fast modulation.
For a large ensemble of emitters our approach offers a tool to implement a bright ultra-fast light source based on spontaneous emission with a low temporal coherence. This source has potentially much shorter pulse duration than electronically controlled LEDs. The photon statistics of such a source differs significantly from known laser action such as Q-switching or cavity dumping). An ultra fast low coherence source may find applications in speckle-free imaging which requires low coherence [38].

IV. CONCLUSION
We have studied the deterministic switching of the Purcell factor and thereby the decay rate for spontaneous emission of light sources in a cavity. We have introduced a model geared toward experimental validation using free carrier switching of micropillars cavities with embedded quantum dots. We have demonstrated that by dynamically controlling the radiative decay rate of emitters during the decay time the emission intensity can be drastically modified during the switch time for both continuous wave excitation and pulsed excitation. For pulsed excitation the dynamic decay rate reveals a strongly non-exponential distribution of photon arrival times. Finally, we have introduced a figure of merit of spontaneous emission switching, equal to the product between the radiative decay rate and the switch duration.

Appendix A: Direct excitation and free carrier excitation
In this appendix we elaborate on the excitation of the emitters, described by the term P exc (t) in Eq. (8). We consider that the carriers excited by the switching pulse may not be perfectly shielded from the quantum dot region. Thus, there are two main contributions to the excitation power. The first term results from the direct excitation pulse, which excites the ensemble of quantum dots sources. The time behaviour of the excitation pulse is modelled with a Gaussian function [39]. The second contribution results from the leakage of free carriers into the quantum dots created by the switch pulse. The contribution is therefore proportional to the free carrier density and has the same temporal behaviour as the free carriers. The total excitation power is thus given by P exc (t) = P 0exc e −2(t−t 0exc ) 2 τ 2 exc + P 0pu η rec e −(t−t 0pu ) τsw Θ(t − t 0pu , τ pu ).
Here P 0exc is the amplitude of the excitation pulse, t 0exc the time at which the excitation pulse excites the quantum dots, τ exc the duration of the excitation pulse and, P pu the amplitude of the pump pulse. The rest of the parameters are defined in relation to Eq. (13). An example of the time dependence excitation power from eq. (A1) is shown in Figure 5: first the short peak from the intentional excitation pulse at t 0exc = 5 ps used to excited the quantum dots followed a longer peak resulting from leakage of free carriers generated by the switching pulse arriving at t 0pu = 10 ps. The magnitude of the excitation due to free carriers is taken to be P 0pu η rec = 0.1P 0exc . Since the leakage rate is proportional to the free concentration this contribution has a sharp increase at t 0pu and then decreases exponentially with the free carrier recombination time taken to be τ sw = 35 ps. In a practical experiment it is desirable to reduce η rec by shielding the quantum dots from the free carriers generated by the switching pulse. One way is the use of a cavity where the optical mode for the switching cavity extends over two dielectric regions separated by an air gap. Switching the refractive index of one region of the cavity would change the combined cavity frequency whereas the quantum dots in the other region would be isolated from the excited free carriers.