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

Optics Express, Vol. 21, Issue 20, pp. 23130-23144 (2013)

http://dx.doi.org/10.1364/OE.21.023130

Acrobat PDF (966 KB)

### Abstract

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.

© 2013 OSA

## 1. Introduction

1. S. Haroche and D. Kleppner, “Cavity quantum electrodynamics,” Phys. Today **42**, 24–30 (1989). [CrossRef]

6. D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. **47**, 233–236 (1981). [CrossRef]

7. M. D. Leistikow, A. P. Mosk, E. Yeganegi, S. R. Huisman, A. Lagendijk, and W. L. Vos, “Inhibited spontaneous emission of quantum dots observed in a 3d photonic band gap,” Phys. Rev. Lett. **107**, 193903 (2011). [CrossRef] [PubMed]

8. Q. Wang, S. Stobbe, and P. Lodahl, “Mapping the local density of optical states of a photonic crystal with single quantum dots,” Phys. Rev. Lett. **107**, 167404 (2011). [CrossRef] [PubMed]

9. T. Lund-Hansen, S. Stobbe, B. Julsgaard, H. Thyrrestrup, T. Sünner, M. Kamp, A. Forchel, and P. Lodahl, “Experimental realization of highly efficient broadband coupling of single quantum dots to a photonic crystal waveguide,” Phys. Rev. Lett. **101**, 113903 (2008). [CrossRef] [PubMed]

11. L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. García, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes.” Science **327**, 1352–1355 (2010). [CrossRef] [PubMed]

12. L. Novotny and N. van Hulst, “Antennas for light,” Nature Photon. **5**, 83–90 (2011). [CrossRef]

1. S. Haroche and D. Kleppner, “Cavity quantum electrodynamics,” Phys. Today **42**, 24–30 (1989). [CrossRef]

6. D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. **47**, 233–236 (1981). [CrossRef]

13. R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. **35**, 265–270 (1996). [CrossRef]

14. J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. **81**, 1110–1113 (1998). [CrossRef]

11. L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. García, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes.” Science **327**, 1352–1355 (2010). [CrossRef] [PubMed]

15. M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald, and A. Forchel, “Inhibition and enhancement of the spontaneous emission of quantum dots in structured microresonators,” Phys. Rev. Lett. **86**, 3168–3171 (2001). [CrossRef] [PubMed]

16. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoglu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature **445**, 896–899 (2007). [CrossRef] [PubMed]

*stationary in time*. Thus, the radiative decay rate is time independent and the distribution of photon emission times decays exponentially in time and is completely determined by this rate.

*in time during their lifetime*, as mediated by a time-dependent LDOS. This results in non-exponential time evolution of the internal dynamics of the emitters and the emitted intensity. By utilizing fast optical modulation of a microcavity, we can tune the cavity resonance and drastically change the LDOS at the emission frequency within the emission lifetime. 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 of detected photons that is completely controlled by the experimentalist. Our approach thus offers a tool to dynamically control the light-matter coupling [17

17. A. Majumdar, D. Englund, M. Bajcsy, and J. Vučković, “Nonlinear temporal dynamics of a strongly coupled quantum-dot cavity system,” Phys. Rev. A **85**, 033802 (2012). [CrossRef]

18. A. Lagendijk, “Vibrational relaxation studied with light,” in Ultrashort Processes in Condensed Matter , vol. 314 (1993), pp. 197–236. [CrossRef]

14. J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. **81**, 1110–1113 (1998). [CrossRef]

19. A. M. Vredenberg, N. E. J. Hunt, E. F. Schubert, D. C. Jacobson, J. M. Poate, and G. J. Zydzik, “Controlled atomic spontaneous emission from Er^{3+}in a transparent Si/SiO_{2}microcavity,” Phys. Rev. Lett. **71**, 517–520 (1993). [CrossRef] [PubMed]

20. J. L. Jewell, S. L. McCall, A. Scherer, H. H. Houh, N. A. Whitaker, A. C. Gossard, and J. H. English, “Transverse modes, waveguide dispersion, and 30 ps recovery in submicron GaAs/AlAs microresonators,” Appl. Phys. Lett. **55**, 22–24 (1989). [CrossRef]

25. X. Hu, P. Jiang, C. Ding, H. Yang, and Q. Gong, “Picosecond and low-power all-optical switching based on an organic photonicbandgap microcavity,” Nat. Phot. **2**, 185–189 (2008). [CrossRef]

## 2. Emission dynamics in a time-dependent environment

### 2.1. Rate equations

*ρ*(

*ω*,

**) in a photonic microcavity and we investigate the effect of a time-dependent LDOS, that modifies 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 emitter**

*r**c*(

_{a}*t*) [26

26. N. Vats, S. John, and K. Busch, “Theory of fluorescence in photonic crystals,” Phys. Rev. A **65**, 043808 (2002). [CrossRef]

*ρ*(

*ω*,

**e**

*,*

_{d}**,**

*r**t′*) that depends on time

*t′*Here

*d*and

**e**

*are the amplitude and orientation vector of the transition dipole moment, respectively,*

_{d}*h̄*the reduced Planck’s constant,

*ε*

_{0}the dielectric constant of vacuum,

**r**the emitter position, and

*ω*the emission frequency. For convenience, we only write the time dependency of

_{d}*c*(

_{a}*t*), but it should be kept in mind that the amplitude

*c*(

_{a}*t*,

**,**

*r*

*e**,*

_{d}*ω*) also depends on

_{d}**,**

*r*

*e**and*

_{d}*ω*[27

_{d}27. W. L. Vos, A. F. Koenderink, and I. S. Nikolaev, “Orientation-dependent spontaneous emission rates of a two-level quantum emitter in any nanophotonic environment,” Phys. Rev. A **80**, 053802 (2009). [CrossRef]

*ωρ*(

*ω*,

*e**,*

_{d}**,**

*r**t′*)). This approximation is known as the

*Markov approximation*[18

18. A. Lagendijk, “Vibrational relaxation studied with light,” in Ultrashort Processes in Condensed Matter , vol. 314 (1993), pp. 197–236. [CrossRef]

*Wigner-Weisskopf approximation*[28]. We thus neglect coherent interactions between the emitter and the environment where a full quantum mechanical description is necessary. 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 The integral in Eq. (2) can be evaluated to yield [29

29. I. S. Nikolaev, Spontaneous-Emission Rates of Quantum Dots and Dyes Controlled with Photonic Crystals, available online: http://www.photonicbandgaps.com, Ph.D. thesis, Universiteit of Twente (2006).

_{rad}(

*t*) the radiative rate Equation (5) is Fermi’s golden rule [30

30. E. Fermi, “Quantum theory of radiation,” Rev. Mod. Phys. **4**, 87–132 (1932). [CrossRef]

_{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 in time and Eq. (4) shows the well-known feature that the amplitude

*c*(

_{a}*t*) decreases exponentially with the rate

*c*(

_{a}*t*)|

^{2}of the two-level emitter 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.

*N*

_{2}(

*t*) for an ensemble of

*N*identical non-interacting two-level sources. To complete the model we include a time-dependent 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 the excitation and depends on the excitation power

*P*

_{exc}(

*t*) per emitter, the excitation frequency

*ω*, and the absorption efficiency of the excitation power that reaches the two-level source

_{exc}*η*

_{abs}. The second term describes the radiative decay and the third term the non-radiative decay. For convenience, we write

*N*

_{2}(

*t*) only as a function of time in Eq. (7), although for an inhomogeneous ensemble of two-level sources

*N*

_{2}(

*t*) also depends on

**,**

*r*

*e**and*

_{d}*ω*. The general solution of Eq. (7) is The corresponding radiated emission intensity

_{d}*I*(

*t*) is given by [31

31. A. F. van Driel, I. S. Nikolaev, P. Vergeer, P. Lodahl, D. Vanmaekelbergh, and W. L. Vos, “Statistical analysis of time-resolved emission from ensembles of semiconductor quantum dots: Interpretation of exponential decay models,” Phys. Rev. B **75**, 035329 (2007). [CrossRef]

*ω*we should average Eq. (9) over

_{d}**and**

*r*

*e**. Equation (8) and (9) are generally valid for any set of two-level emitters in environment with a time-dependent LDOS. Equations (8) and (9) form the basis for our further discussion and they will be used to calculate the emission of an ensemble of emitters that experience a time-dependent LDOS.*

_{d}### 2.2. Time dependent radiative decay rate in a microcavity

_{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 ΔΓ

_{rad}(

*t*) where ΔΓ

_{rad}(

*t*) is proportional to the change in the LDOS Δ

*ρ*(

*t*) 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}.

*t*=

*t*

_{pu}. The induced change in the refractive index is proportional to the free carrier density [32

32. T. G. Euser, A. J. Molenaar, J. G. Fleming, B. Gralak, A. Polman, and W. L. Vos, “All-optical octave-broad ultrafast switching of Si woodpile photonic band gap crystals,” Phys. Rev. B **77**, 115214 (2008). [CrossRef]

2. K. J. Vahala, “Optical microcavities,” Nature **424**, 839–846 (2003). [CrossRef] [PubMed]

*τ*

_{sw}, after which the refractive index is restored to its original value [24

24. P. J. Harding, T. G. Euser, Y.-R. Nowicki-Bringuier, J.-M. Gérard, and W. L. Vos, “Ultrafast optical switching of planar GaAs/AlAs photonic microcavities,” Appl. Phys. Lett. **91**, 111103 (2007). [CrossRef]

32. T. G. Euser, A. J. Molenaar, J. G. Fleming, B. Gralak, A. Polman, and W. L. Vos, “All-optical octave-broad ultrafast switching of Si woodpile photonic band gap crystals,” Phys. Rev. B **77**, 115214 (2008). [CrossRef]

*τ*

_{sw}= 35 ps, characteristic for GaAs [24

24. P. J. Harding, T. G. Euser, Y.-R. Nowicki-Bringuier, J.-M. Gérard, and W. L. Vos, “Ultrafast optical switching of planar GaAs/AlAs photonic microcavities,” Appl. Phys. Lett. **91**, 111103 (2007). [CrossRef]

33. P. M. Johnson, A. F. Koenderink, and W. L. Vos, “Ultrafast switching of photonic density of states in photonic crystals,” Phys. Rev. B **66**, 081102 (2002). [CrossRef]

*ω*

_{cav,0}of a microcavity with a Lorentzian LDOS with linewidth

*γ*

_{cav}in the spectral vicinity of an emitter with emission frequency

*ω*. The single emitter homogeneous linewidth is taken to be narrower than the cavity linewidth (

_{d}*γ*

_{em}< cav), to fulfill the Markov approximation. This criterion can easily be obtained with semiconductor quantum dots at low temperatures. The large inhomogeneous spectral broadening of semiconductor quantum dots further ensures that only a small sub-ensemble interacts with the cavity and the dots can be treated as non-interacting single emitters. Non-exponential modifications of the emission decay curve arising from non-local effects is therefore negligible [34

34. A. A. Svidzinsky, “Nonlocal effects in single-photon superradiance,” Phys. Rev. A **85**, 013821 (2012). [CrossRef]

35. M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B **65**, 041308 (2002). [CrossRef]

*ω*

_{cav}(

*t*) as indicated in Fig. 1[36

36. The resonant index change contribution from the excited emitters themselves can be neglected due to the low emitter density. Likewise the emission frequency shift caused by the refractive index change of the of the surrendering material is small compared to the cavity resonance shift and has been neglected.

_{0rad}. During the switch event the cavity peak is tuned into resonance with the emitter as shown as the dashed Lorentzian in Fig. 1. 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}(Δ

*t*) = Γ

_{0}+ ΔΓ

_{rad}and back to Γ

_{0}within a time Δ

*t*. The effective switching time in this scenario is therefore given by A shorter effective switching time can thus be realized by either a faster tuning of the cavity resonance in time Δ

*t*or by increasing the spectral tuning range relative to the cavity linewidth

*γ*

_{cav}within the time Δ

*t*.

*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 in Fig. 1. We can therefore effectively make a linear approximation 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, that is much shorter than the carrier recombination time (35 ps, see [22

22. I. Fushman, E. Waks, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Ultrafast nonlinear optical tuning of photonic crystal cavities,” Appl. Phys. Lett. **90**, 091118 (2007). [CrossRef]

24. P. J. Harding, T. G. Euser, Y.-R. Nowicki-Bringuier, J.-M. Gérard, and W. L. Vos, “Ultrafast optical switching of planar GaAs/AlAs photonic microcavities,” Appl. Phys. Lett. **91**, 111103 (2007). [CrossRef]

_{0}, and a change induced by the switch that is turned on at time

*t*

_{0pu}. The change is initiated by a Heaviside step function and the magnitude of the switched term in Eq. (13) then decays exponentially with an effective switching time comparable to the free carrier relaxation time.

*t*= 10 ps before decreasing again at a rate set by the inverse switching time. Similarly, the lower curve (short dashed) illustrates the situation where the emitter is initially on resonance and the cavity is switched out of resonance. In the examples in Fig. 2 we use either an enhancement or inhibition by a factor of 5, which is a realistic change observed on ensemble of quantum dots in micropillar cavities [14

14. J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. **81**, 1110–1113 (1998). [CrossRef]

_{rad}(

*t*)/Γ

_{0}= 1 corresponds to the unswitched case, typical for all Purcell experiments performed to date [11

11. L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. García, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes.” Science **327**, 1352–1355 (2010). [CrossRef] [PubMed]

**81**, 1110–1113 (1998). [CrossRef]

16. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoglu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature **445**, 896–899 (2007). [CrossRef] [PubMed]

*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.

### 2.3. Figure of merit for pulsed excitation

*P*

_{0exc}initializes the system at

*t*=

*t*

_{0exc}such that we have an initial population density

*N*

_{2}(

*t*=

*t*

_{0exc}) =

*N*

_{02}. 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 excitation (

*t*>

*t*

_{0exc}). In this case Eq. (7) simplifies to which can be integrated to yield Equation 15 describes the population density for any time-dependent decay rate Γ

_{rad}(

*t*) as a function of time

*t*after the excitation process is over. Despite the time-integral in Eq. (15) the equation does not describe non-Markovian dynamics, since the dynamics only depends on the present time (Eq. (14)) and only accumulate changes from the modification in the LDOS and not the light-matter dynamics [37

37. D. Chruściński and A. Kossakowski, “Non-Markovian Quantum Dynamics: Local versus Nonlocal”, Phys. Rev. Lett. **104**, 070406 (2010). [CrossRef]

*α*

_{rad}(

*t*) This parameter is a figure of merit that describes the relative change in the population density due to the change in the decay rate. A negative Δ

*α*

_{rad}(

*t*) results in a population density that decays slower compared to the unswitched situation, while a positive Δ

*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. (17) can be split into two parts - before and after the switch

*t*=

*τ*

_{pu}- and Δ

*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 Δ

*α*

_{rad}(

*t*) becomes Equation (19) shows that Δ

*α*

_{rad}(

*t*) is nonzero even in the long-time limit and is given by a product of the switch magnitude ΔΓ

_{rad}the effective switch duration. The switch therefore has an effect on the population dynamics even long after the switch event. The dimensionless switching parameter Δ

*α*

_{∞}is therefore a useful figure of merit for the total switching effect on the excited state population.

*N*

_{2}and the unswitched population density

*N*

_{2us}in the limit of

*t*tend to infinity Equation (20) quantifies the long term effect of the switching on the population density as a result of a momentarily short change in the decay rate.

### 2.4. Population dynamics for pulsed excitation

_{0}= 1 ns

^{−1}and Γ

_{0}= 5 ns

^{−1}) and two with switching pulses (solid lines) resulting in the time-dependent decay rates shown in Fig. 2. In the two stationary cases, as expected, the population decay exponentially with their initial rates Γ

_{0}. The green long dashed curve shows the case where a switch tunes the cavity into resonance with the emitter and induces an enhanced decay rate by a factor of 5 (ΔΓ

_{rad}= 4Γ

_{0}) from Γ

_{0}= 1 ns

^{−1}. 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 the population density clearly decays non-exponentially.

*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 switching times later (

*t*> 250 ps) the decay rate returns to its original value but the absolute value of the populations is reduced compared to the unswitched case. Using Eq. (16) and the figure of merit (Eq. (19)) 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 situation is reversed for a switch that induces an inhibition of the spontaneous emission: the population also experiences a non-exponential decay after the switch; however, the population is now larger than its reference value (unswitched case) at long times.

### 2.5. Emission dynamics for pulsed excitation

*I*(

*t*) is the product of 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 emitted intensity. For large dynamic changes in the decay rate, we therefore expect correspondingly 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 results in the steady-state case [31

31. A. F. van Driel, I. S. Nikolaev, P. Vergeer, P. Lodahl, D. Vanmaekelbergh, and W. L. Vos, “Statistical analysis of time-resolved emission from ensembles of semiconductor quantum dots: Interpretation of exponential decay models,” Phys. Rev. B **75**, 035329 (2007). [CrossRef]

*α*(

*t*) is given by Eq. (18). The main difference between the population density dynamics Eq. (16) and the emitted intensity is the presence of the decay rate prefactor (Γ

_{0}+ ΔΓ

_{rad}(

*t*)). In addition, the intensity in Eq. (21) is still proportional to the population density so that the influence of the switching process remains visible in the emission intensity even long after the switch event has passed as discussed in section 2.3. The relative intensity to the unswitched intensity at long times is thus given by lim

_{t}_{→∞}

*I*(

*t*)/

*I*

_{us}(

*t*) =

*e*

^{−ΔΓradτsw}as the exponent in Eq. (21) is the same as in Eq. (20) and the time-dependent decay rate ΔΓ

_{rad}(

*t*) in the prefactor tends to zero a long times.

_{0}= 1 ns

^{−1}and another where the radiative rate is inhibited from a high value of Γ

_{0}= 5 ns

^{−1}. The emitter is excited at

*t*= 0 ps, followed by an exponential decay of the emission intensities with the same rate as the population density as expected in the weak coupling limit. A switching pulse arrives at

_{ex}*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.

_{0}= 200 ps. Let us note in pasing that the effective switching time could be engineered to be as short as 2 to 3 ps by either decreasing the free carrier lifetime [38

38. H. Nemec, A. Pashkin, P. Kuzel, M. Khazan, S. Schnüll, and I. Wilke, “Carrier dynamics in low-temperature grown GaAs studied by terahertz emission spectroscopy,” J. Appl. Phys. **90**, 1303–1306 (2001). [CrossRef]

*N*≃ 10

^{18}cm

^{−3}is sufficient to switch several linewidths, even lower reductions are expected.

## 3. Discussion

2. K. J. Vahala, “Optical microcavities,” Nature **424**, 839–846 (2003). [CrossRef] [PubMed]

**327**, 1352–1355 (2010). [CrossRef] [PubMed]

**81**, 1110–1113 (1998). [CrossRef]

15. M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald, and A. Forchel, “Inhibition and enhancement of the spontaneous emission of quantum dots in structured microresonators,” Phys. Rev. Lett. **86**, 3168–3171 (2001). [CrossRef] [PubMed]

39. M. O. Scully, V. V. Kocharovsky, A. Belyanin, E. Fry, and F. Capasso, “Enhancing Acceleration Radiation from Ground-State Atoms via Cavity Quantum Electrodynamics,” Phys. Rev. Lett. **91**, 243004 (2003). [CrossRef] [PubMed]

1. S. Haroche and D. Kleppner, “Cavity quantum electrodynamics,” Phys. Today **42**, 24–30 (1989). [CrossRef]

40. P. P. Rohde, T. C. Ralph, and M. A. Nielsen, “Optimal photons for quantum-information processing,” Phys. Rev. A **72**, 052332 (2005). [CrossRef]

41. R. Johne and A. Fiore, “Single-photon absorption and dynamic control of the exciton energy in a coupled quantum-dot-cavity system,” Phys. Rev. A **84**, 053850 (2011). [CrossRef]

42. J. Dilley, P. Nisbet-Jones, B. W. Shore, and A. Kuhn, “Single-photon absorption in coupled atom-cavity systems,” Phys. Rev. A **85**, 023834 (2012). [CrossRef]

43. M. Fernée, H. Rubinsztein-Dunlop, and G. Milburn, “Improving single-photon sources with Stark tuning,” Phys. Rev. A **75**, 043815 (2007). [CrossRef]

44. C.-H. Su, A. D. Greentree, W. J. Munro, K. Nemoto, and L. C. L. Hollenberg, “Pulse shaping by coupled cavities: Single photons and qudits,” Phys. Rev. A **80**, 033811 (2009). [CrossRef]

18. A. Lagendijk, “Vibrational relaxation studied with light,” in Ultrashort Processes in Condensed Matter , vol. 314 (1993), pp. 197–236. [CrossRef]

45. K. E. Dorfman and S. Mukamel, “Nonlinear spectroscopy with time- and frequency-gated photon counting: A superoperator diagrammatic approach,” Phys. Rev. A **86**, 013810 (2012). [CrossRef]

## 4. Conclusion

## A. Effect of free carrier absorption on the dynamic emission intensity

*n*and the imaginary part

*n″*of the complex refractive index

*ñ*whose components are linked through the Kramers-Kronig relations. Thus, the free carriers induce absorption of the light in the cavity. The absorption manifests itself as a broadening of the cavity linewidth during the switch event. For applications where the interest is in the photons emitted from the cavity, such losses are an unwanted effect. A side effect of the loss mechanism, and the associated linewidth broadening, is a decrease of the local density of states experienced by the emitter, which can be exploited as an additional switching mechanism.

*γ*(

*t*) into a sum of the intrinsic linewidth of the unswitched cavity

*γ*and a loss rate due to free carrier absorption

_{i}*γ*(

_{a}*t*). For GaAs, the Drude model is a good approximation for relatively low carrier concentration

*N*< 10

^{19}cm

^{−3}after thermalization of the free carriers at

*t*> 6 ps. Within this approximation, the imaginary part of the refraction index and therefore the loss rate

*γ*(

_{a}*t*) is proportional to the free carrier concentration

*N*. Similarly, the change in the real part is proportional to

*N*. To first order we can therefore assume a linear relation between the shift of the cavity resonance frequency Δ

*ω*(

*t*) and the loss rate for small frequency shifts. Defining the relative shift as the relative linewidth can be written as where

*a*is a phenomenological constant. Equation (24) directly relates the relative broadening of the cavity linewidth with the switching magnitude. There are not many sources for the effective losses caused by free carrier absorption. Nevertheless, we can extract

*a*from previous published data on switched GaAs planar cavities [48

48. P. J. Harding, H. J. Bakker, A. Hartsuiker, J. Claudon, A. P. Mosk, J.-M. Gérard, and W. L. Vos, “Observation of a stronger-than-adiabatic change of light trapped in an ultrafast switched GaAs-AlAs microcavity,” J. Opt. Soc. Am. B **29**, A1–A5 (2012). [CrossRef]

47. P. J. Harding, Photonic crystals modified by optically resonant systems, available online: http://www.photonicbandgaps.com, Ph.D. thesis, Universiteit of Twente (2008).

*a*≃ 0.083 fits the data remarkably well for shifts smaller than 4 linewidths, which yields an increase in the linewidth by 25% for a 3 linewidth switch.

_{rad}(

*t*) is proportional to Q and the radiative rate Γ

_{rad}(

*t*) must be scaled by

*γ*(

_{i}/γ*t*). Secondly, a fraction [1 −

*γ*(

_{i}/γ*t*)] of the photons emitted into the cavity is absorbed, and only the remaining fraction

*γ*(

_{i}/γ*t*) leaves the cavity to be detected. The modified time-dependent intensity is therefore where Γ

_{rad}(

*t*) is given by Eq. (13) and

*γ*(

_{i}/γ*t*) is the inverse of Eq. (24). Similarly to the description in Sec. 2.2, we assume a linear relation between the switching magnitude

*S*(

*t*) and the free carrier concentration. Thus,

*S*(

*t*) has the form where

*S*

_{0}is the maximum frequency shift of the cavity resonance relative to the initial cavity linewidth.

*t*= 150 ps. When including free carrier absorption (solid red line) with

*a*= 0.083 and a switching magnitude of one linewidth

*S*

_{0}= 1, the shape of the peak is barely modified.. We only observe a small reduction by 15% of the original peak and a gentler slope back down to its unswitched dynamics compared to the direct change in the free carrier concentration. The intensity dynamics is therefore hardly affected by free carrier absorption.

## Acknowledgments

## References and links

1. | S. Haroche and D. Kleppner, “Cavity quantum electrodynamics,” Phys. Today |

2. | K. J. Vahala, “Optical microcavities,” Nature |

3. | J. M. Gérard, “Solid-state cavity-quantum electrodynamics with self-assembled quantum dots,” Topics Appl. Phys. |

4. | J. P. Reithmaier, “Strong exciton-photon coupling in semiconductor quantum dot systems,” Semicond. Sci. Technol. |

5. | S. Buckley, K. Rivoire, and J. Vučković, “Engineered quantum dot single-photon sources,” Rep. Prog. Phys. |

6. | D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. |

7. | M. D. Leistikow, A. P. Mosk, E. Yeganegi, S. R. Huisman, A. Lagendijk, and W. L. Vos, “Inhibited spontaneous emission of quantum dots observed in a 3d photonic band gap,” Phys. Rev. Lett. |

8. | Q. Wang, S. Stobbe, and P. Lodahl, “Mapping the local density of optical states of a photonic crystal with single quantum dots,” Phys. Rev. Lett. |

9. | T. Lund-Hansen, S. Stobbe, B. Julsgaard, H. Thyrrestrup, T. Sünner, M. Kamp, A. Forchel, and P. Lodahl, “Experimental realization of highly efficient broadband coupling of single quantum dots to a photonic crystal waveguide,” Phys. Rev. Lett. |

10. | H. Thyrrestrup, L. Sapienza, and P. Lodahl, “Extraction of the beta-factor for single quantum dots coupled to a photonic crystal waveguide,” Appl. Phys. Lett. |

11. | L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. García, S. Smolka, and P. Lodahl, “Cavity quantum electrodynamics with Anderson-localized modes.” Science |

12. | L. Novotny and N. van Hulst, “Antennas for light,” Nature Photon. |

13. | R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. |

14. | J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. |

15. | M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald, and A. Forchel, “Inhibition and enhancement of the spontaneous emission of quantum dots in structured microresonators,” Phys. Rev. Lett. |

16. | K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoglu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature |

17. | A. Majumdar, D. Englund, M. Bajcsy, and J. Vučković, “Nonlinear temporal dynamics of a strongly coupled quantum-dot cavity system,” Phys. Rev. A |

18. | A. Lagendijk, “Vibrational relaxation studied with light,” in Ultrashort Processes in Condensed Matter , vol. 314 (1993), pp. 197–236. [CrossRef] |

19. | A. M. Vredenberg, N. E. J. Hunt, E. F. Schubert, D. C. Jacobson, J. M. Poate, and G. J. Zydzik, “Controlled atomic spontaneous emission from Er |

20. | J. L. Jewell, S. L. McCall, A. Scherer, H. H. Houh, N. A. Whitaker, A. C. Gossard, and J. H. English, “Transverse modes, waveguide dispersion, and 30 ps recovery in submicron GaAs/AlAs microresonators,” Appl. Phys. Lett. |

21. | T. Rivera, F. R. Ladan, A. Izrael, R. Azoulay, R. Kuszelewicz, and J. L. Oudar, “Reduced threshold all-optical bistability in etched quantum well microresonators,” Appl. Phys. Lett. |

22. | I. Fushman, E. Waks, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Ultrafast nonlinear optical tuning of photonic crystal cavities,” Appl. Phys. Lett. |

23. | M. W. McCutcheon, A. G. Pattantyus-Abraham, G. W. Rieger, and J. F. Young, “Emission spectrum of electromagnetic energy stored in a dynamically perturbed optical microcavity,” Opt. Express |

24. | P. J. Harding, T. G. Euser, Y.-R. Nowicki-Bringuier, J.-M. Gérard, and W. L. Vos, “Ultrafast optical switching of planar GaAs/AlAs photonic microcavities,” Appl. Phys. Lett. |

25. | X. Hu, P. Jiang, C. Ding, H. Yang, and Q. Gong, “Picosecond and low-power all-optical switching based on an organic photonicbandgap microcavity,” Nat. Phot. |

26. | N. Vats, S. John, and K. Busch, “Theory of fluorescence in photonic crystals,” Phys. Rev. A |

27. | W. L. Vos, A. F. Koenderink, and I. S. Nikolaev, “Orientation-dependent spontaneous emission rates of a two-level quantum emitter in any nanophotonic environment,” Phys. Rev. A |

28. | R. Loudon, |

29. | I. S. Nikolaev, Spontaneous-Emission Rates of Quantum Dots and Dyes Controlled with Photonic Crystals, available online: http://www.photonicbandgaps.com, Ph.D. thesis, Universiteit of Twente (2006). |

30. | E. Fermi, “Quantum theory of radiation,” Rev. Mod. Phys. |

31. | A. F. van Driel, I. S. Nikolaev, P. Vergeer, P. Lodahl, D. Vanmaekelbergh, and W. L. Vos, “Statistical analysis of time-resolved emission from ensembles of semiconductor quantum dots: Interpretation of exponential decay models,” Phys. Rev. B |

32. | T. G. Euser, A. J. Molenaar, J. G. Fleming, B. Gralak, A. Polman, and W. L. Vos, “All-optical octave-broad ultrafast switching of Si woodpile photonic band gap crystals,” Phys. Rev. B |

33. | P. M. Johnson, A. F. Koenderink, and W. L. Vos, “Ultrafast switching of photonic density of states in photonic crystals,” Phys. Rev. B |

34. | A. A. Svidzinsky, “Nonlocal effects in single-photon superradiance,” Phys. Rev. A |

35. | M. Bayer and A. Forchel, “Temperature dependence of the exciton homogeneous linewidth in In0.60Ga0.40As/GaAs self-assembled quantum dots,” Phys. Rev. B |

36. | The resonant index change contribution from the excited emitters themselves can be neglected due to the low emitter density. Likewise the emission frequency shift caused by the refractive index change of the of the surrendering material is small compared to the cavity resonance shift and has been neglected. |

37. | D. Chruściński and A. Kossakowski, “Non-Markovian Quantum Dynamics: Local versus Nonlocal”, Phys. Rev. Lett. |

38. | H. Nemec, A. Pashkin, P. Kuzel, M. Khazan, S. Schnüll, and I. Wilke, “Carrier dynamics in low-temperature grown GaAs studied by terahertz emission spectroscopy,” J. Appl. Phys. |

39. | M. O. Scully, V. V. Kocharovsky, A. Belyanin, E. Fry, and F. Capasso, “Enhancing Acceleration Radiation from Ground-State Atoms via Cavity Quantum Electrodynamics,” Phys. Rev. Lett. |

40. | P. P. Rohde, T. C. Ralph, and M. A. Nielsen, “Optimal photons for quantum-information processing,” Phys. Rev. A |

41. | R. Johne and A. Fiore, “Single-photon absorption and dynamic control of the exciton energy in a coupled quantum-dot-cavity system,” Phys. Rev. A |

42. | J. Dilley, P. Nisbet-Jones, B. W. Shore, and A. Kuhn, “Single-photon absorption in coupled atom-cavity systems,” Phys. Rev. A |

43. | M. Fernée, H. Rubinsztein-Dunlop, and G. Milburn, “Improving single-photon sources with Stark tuning,” Phys. Rev. A |

44. | C.-H. Su, A. D. Greentree, W. J. Munro, K. Nemoto, and L. C. L. Hollenberg, “Pulse shaping by coupled cavities: Single photons and qudits,” Phys. Rev. A |

45. | K. E. Dorfman and S. Mukamel, “Nonlinear spectroscopy with time- and frequency-gated photon counting: A superoperator diagrammatic approach,” Phys. Rev. A |

46. | A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Phys. |

47. | P. J. Harding, Photonic crystals modified by optically resonant systems, available online: http://www.photonicbandgaps.com, Ph.D. thesis, Universiteit of Twente (2008). |

48. | P. J. Harding, H. J. Bakker, A. Hartsuiker, J. Claudon, A. P. Mosk, J.-M. Gérard, and W. L. Vos, “Observation of a stronger-than-adiabatic change of light trapped in an ultrafast switched GaAs-AlAs microcavity,” J. Opt. Soc. Am. B |

49. | P. J. Harding, A. P. Mosk, A. Hartsuiker, Y.-R. Nowicki-Bringuier, J.-M. Gérard, and W. L. Vos, “Time-resolved resonance and linewidth of an ultrafast switched GaAs/AlAs microcavity,” arXiv:0901.3855 [physics.optics] (2009). |

**OCIS Codes**

(270.5580) Quantum optics : Quantum electrodynamics

(320.5540) Ultrafast optics : Pulse shaping

(320.7130) Ultrafast optics : Ultrafast processes in condensed matter, including semiconductors

(130.4815) Integrated optics : Optical switching devices

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: June 12, 2013

Revised Manuscript: August 12, 2013

Manuscript Accepted: August 19, 2013

Published: September 24, 2013

**Citation**

Henri Thyrrestrup, Alex Hartsuiker, Jean-Michel Gérard, and Willem L. Vos, "Non-exponential spontaneous emission dynamics for emitters in a time-dependent optical cavity," Opt. Express **21**, 23130-23144 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-23130

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### References

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- R. Loudon, The Quantum Theory of Light (Oxford University, 2000), 3rd ed.
- I. S. Nikolaev, Spontaneous-Emission Rates of Quantum Dots and Dyes Controlled with Photonic Crystals, available online: http://www.photonicbandgaps.com , Ph.D. thesis, Universiteit of Twente (2006).
- E. Fermi, “Quantum theory of radiation,” Rev. Mod. Phys.4, 87–132 (1932). [CrossRef]
- A. F. van Driel, I. S. Nikolaev, P. Vergeer, P. Lodahl, D. Vanmaekelbergh, and W. L. Vos, “Statistical analysis of time-resolved emission from ensembles of semiconductor quantum dots: Interpretation of exponential decay models,” Phys. Rev. B75, 035329 (2007). [CrossRef]
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- P. M. Johnson, A. F. Koenderink, and W. L. Vos, “Ultrafast switching of photonic density of states in photonic crystals,” Phys. Rev. B66, 081102 (2002). [CrossRef]
- A. A. Svidzinsky, “Nonlocal effects in single-photon superradiance,” Phys. Rev. A85, 013821 (2012). [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]
- The resonant index change contribution from the excited emitters themselves can be neglected due to the low emitter density. Likewise the emission frequency shift caused by the refractive index change of the of the surrendering material is small compared to the cavity resonance shift and has been neglected.
- D. Chruściński and A. Kossakowski, “Non-Markovian Quantum Dynamics: Local versus Nonlocal”, Phys. Rev. Lett.104, 070406 (2010). [CrossRef]
- H. Nemec, A. Pashkin, P. Kuzel, M. Khazan, S. Schnüll, and I. Wilke, “Carrier dynamics in low-temperature grown GaAs studied by terahertz emission spectroscopy,” J. Appl. Phys.90, 1303–1306 (2001). [CrossRef]
- M. O. Scully, V. V. Kocharovsky, A. Belyanin, E. Fry, and F. Capasso, “Enhancing Acceleration Radiation from Ground-State Atoms via Cavity Quantum Electrodynamics,” Phys. Rev. Lett.91, 243004 (2003). [CrossRef] [PubMed]
- P. P. Rohde, T. C. Ralph, and M. A. Nielsen, “Optimal photons for quantum-information processing,” Phys. Rev. A72, 052332 (2005). [CrossRef]
- R. Johne and A. Fiore, “Single-photon absorption and dynamic control of the exciton energy in a coupled quantum-dot-cavity system,” Phys. Rev. A84, 053850 (2011). [CrossRef]
- J. Dilley, P. Nisbet-Jones, B. W. Shore, and A. Kuhn, “Single-photon absorption in coupled atom-cavity systems,” Phys. Rev. A85, 023834 (2012). [CrossRef]
- M. Fernée, H. Rubinsztein-Dunlop, and G. Milburn, “Improving single-photon sources with Stark tuning,” Phys. Rev. A75, 043815 (2007). [CrossRef]
- C.-H. Su, A. D. Greentree, W. J. Munro, K. Nemoto, and L. C. L. Hollenberg, “Pulse shaping by coupled cavities: Single photons and qudits,” Phys. Rev. A80, 033811 (2009). [CrossRef]
- K. E. Dorfman and S. Mukamel, “Nonlinear spectroscopy with time- and frequency-gated photon counting: A superoperator diagrammatic approach,” Phys. Rev. A86, 013810 (2012). [CrossRef]
- A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Phys.6, 283–292 (2012).
- P. J. Harding, Photonic crystals modified by optically resonant systems, available online: http://www.photonicbandgaps.com , Ph.D. thesis, Universiteit of Twente (2008).
- P. J. Harding, H. J. Bakker, A. Hartsuiker, J. Claudon, A. P. Mosk, J.-M. Gérard, and W. L. Vos, “Observation of a stronger-than-adiabatic change of light trapped in an ultrafast switched GaAs-AlAs microcavity,” J. Opt. Soc. Am. B29, A1–A5 (2012). [CrossRef]
- P. J. Harding, A. P. Mosk, A. Hartsuiker, Y.-R. Nowicki-Bringuier, J.-M. Gérard, and W. L. Vos, “Time-resolved resonance and linewidth of an ultrafast switched GaAs/AlAs microcavity,” arXiv:0901.3855 [physics.optics] (2009).

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