1. Introduction

In 1954 R.H. Dicke has introduced the general phenomenon of the cooperative spontaneous emission in quantum systems with long dephasing time - often referred to as superradiance, subradiance or superfluorescence - depending on the initial state of the atomic ensemble [1]. In the same paper Dicke also presented a theoretical model aiming to describe the spontaneous emission of an ensemble of N identical two-level systems localized in a sub-wavelength volume element in vacuum. A rigorous theoretical analysis of this problem by Friedberg and Hartmann in 1974 [2] uncovered the basic problems of ensemble emission in free space and revealed that the original Dicke model is only applicable to a very special case of identical two-level systems resonantly coupled to a single damped cavity mode.

This discrepancy was not crucial for the interpretation of the first experiments on superfluo-rescence in macroscopic ensembles containing N ~ 10⁶ of identical Rydberg atoms [3]: these measurements were dominated by light propagation effects in elongated pencil-shaped atomic ensembles and the stochastic nature of superradiant emission. Being one of the most striking features of superradiance, the stochastic nature of the cooperative spontaneous emission manifests itself in fluctuations of delays times in the emission of superfluorescent pulses. The statistical distribution of delay times in superfluorescent emission could be successfully reproduced by incorporating the quantum-mechanical fluctuations of initial atomic polarization in the Dicke model [4, 5, 6].

Very recently, basic properties of the cooperative spontaneous emission both in atomic ensembles [7] and different solid-state based systems [8, 9] attracted a renewed fundamental interest. First studies are published that deal with few-number semiconductor quantum dot ensembles in microcavities with small, intermediate and high quality factors Q, i.e. covering a broad range in exciton-photon coupling constants. In most of previous studies, the cooperative modification of radiative lifetimes based on the original Dicke model is considered as the key signature of superradiance and subradiance. For example, to test whether quantum dots can interact with each other through their radiation field in free space, the decay rate of photoluminescence as a function of the number of interacting QDs and their respective separation has been investigated [10]. However, experiments addressing electron-hole recombination in semiconductor nanos-tructures are often performed in a way that not only two electronic levels are involved and correlations between carriers due to Coulomb interactions can result in non-exponential and fastened photoluminescence decay behavior [11]. Likewise, the present experimental status of radiative lifetime analysis in small ensembles illustrates the difficulties to distinguish between the cooperative variation in radiative lifetimes caused by the build up of coherent polarization (superradiance) and amplified spontaneous emission [12]. High-precision measurements with an isolated atom pair (N = 2) only revealed a marginal change of cooperative lifetime in free space [13].

Thus the modification of radiative lifetime in a mesoscopic (N < 10) ensemble of two-level systems does not appear to be a clear indication on superradiance. The identification of unique signatures for cooperative spontaneous emission, easily accessible in the experiment, represents a challenging topic from both theoretical and experimental point of view. A promising tool are investigations of photon statistics as e.g. conducted for resonance fluorescence in the strong coupling regime [14, 15, 16]. Spectrally resolved photon-statistics measurements proposed in Ref. [15] allow for unambiguous identification of the two-photon strong-coupling states in semiconductor quantum-dot microcavities. The limit of weak coupling and the extension to cooperative effects in ensembles of two-level emitters is another interesting and less-studied problem within the emerging concepts of second-order photon correlation spectroscopy.

Motivated by an explosive development in experiments measuring the second-order photon correlation function g ⁽²⁾(τ) leading to observation of photon bunching or antibunching) [17, 18, 19, 20, 21], we have investigated temporal fluctuations of superradiant light emission by two different theoretical methods: semiclassical Langevin equations and fully quantum Monte-Carlo simulations. We find a giant photon bunching, a natural generalization of the pulsed nature of superradiant emission at the single-photon level, and declare it as the most striking signature of superradiance in continuously pumped mesoscopic ensembles of N two-level systems. The results of our extended simulations show that exact matching of just two atomic transition frequencies out of a possibly much larger inhomogenously broadened ensemble should be sufficient to observe superradiant photon bunching in the experiment, a particularly important issue for cavity QED with semiconductor quantum dots [9]. This observation may trigger a development of photon bunching spectroscopy in the weak coupling regime.

2. Physical model and semiclassical description

Our physical model assumes a collection of N two-level systems incoherently pumped at the rate γ_pump coupled with strength g to a single damped cavity mode with photon life time (2κ)^-1. A weak coupling regime (bad cavity limit) is considered, i.e. g < κ, leading to adiabatic elimination of the cavity mode. The simplest theoretical description of this problem for a macroscopic ensemble (N ≫ 1), ignoring spontaneous emission into other modes and not taking into account any dephasing processes, is based on semiclassical equations for the coupled dynamics of cooperative atomic inversion Z and polarization P [22]:

 

Fig. 1. Emission intensity I(t) of superradiant pulse trains (a) for different numbers N ≫ 1 of incoherently pumped two-level systems and (b) corresponding trajectories in the population inversion vs. polarization diagram, dashed circles show the direction of motion (see text).

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Random Langevin noise terms F_p(t) and F_z(t) describe pump-induced fluctuations of cooperative polarization and inversion [22] and are crucial to explain the stochastic nature of superradiant emission [4, 5]. The numerical solution of eqations (1) is shown in Fig. 1, where we plot the emission intensity (I(t) = P ²) for different N keeping the pump rate constant, γ _pump = 0. 1g ²/κ. For N = 10000 and N = 1000 we observe the generation of superradiant pulse sequences. Whereas the number of excess inverted atoms (above zero population inversion) N_exc ~ Nγ_pump t grows linearly with time, the delay time of superradiant pulses ${\tau }_d~\frac{\kappa }{g^2}\frac{\ln N_{\mathrm{exc}}}{N_{\mathrm{exc}}}$ decreases with time. Once a critical number of excess atoms is achieved at t ~ τ_d [23] a super-radiant pulse is emitted, a process quasi-periodic in time domain. This behavior is visualized by circular trajectories on the population inversion-polarization diagram in Fig. 1(b) (analogous to superradiant dynamics on the Bloch sphere), where we clearly see the accumulation of population inversion followed by the build up of a macroscopic polarization and the emission of superradiant pulses. These pulsed dynamics of superradiance should not be mixed up with a quasi-periodic laser generation regime in a vicinity of laser threshold described by a full set of three Maxwell-Bloch equations [24]. As N decreases (N = 100 in Fig. 1(a)) the intensity traces become less regular with no signatures of superradiant pulse trains.

 

Fig. 2. Second-order photon correlation function g ⁽²⁾(τ) obtained from semiclassical intensity traces I(t) for different N. A curve for N=10 reveals a strong bunching maximum.

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It appears to be useful to calculate the second-order intensity autocorrelation function

using the semiclassical intensity traces I(t). For a large number of emitters (N=1000 in Fig. 2) the quasi-periodic superradiant generation results in damped oscillations of g ⁽²⁾(τ) around the central bunching maximum with a magnitude g ⁽²⁾(0) ~ 2. For lower number of emitters (N=10 in Fig. 2) a strong bunching maximum is observed with a magnitude g ⁽²⁾(0) substantially exceeding the value obtained for macroscopic superradiant ensembles.

3. Microscopic master-equation simulations

Being aware of the limitations and possible failure of our semiclassical approximation for small N, we have solved the exact dissipative master equation for the atomic density matrix. The full quantum-mechanical description for spontaneous emission of an ensemble of two-level systems into cavity mode is governed by dissipative evolution of joint atoms + cavity field density matrix ρ described by the following master equation:

 

Fig. 3. Second-order photon correlation function g ⁽²⁾(τ) for N incoherently pumped two-level systems (γ _pump = 0.01) for different values of dephasing rate γ _deph. For N=1 anti-bunching is observed, which does not depend on dephasing. For N > 1 a pronounced bunching is observed, which is extremely sensitive to γ _deph. Inset shows the dependence of bunching amplitude g ⁽²⁾(0) on N. All rates are in units of g ²/κ.

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The rate of spontaneous emission into the cavity mode is given by P ² = 〈J ₊ J _-〉 with the cooperative atomic polarization J _± (t) = ∑_n=1 ^N σ _± ⁽ⁿ⁾exp(±iδω_nt) [25], where σ _± ^(j) = σ_x ^(j) ± iσ_y ^(j) represent polarization operators for individual atoms (σ_x,y,z ^(j)-Pauli matrices). Except for possible inhomogeneous broadening, the polarization of the atoms can be additionally destroyed by an elastic dephasing process at rate γ _deph. It is formulated in terms of the projection operators P_e ^(j) and P_g ^(j) [26] and describes the random rotation of atomic polarization without changes in population inversion. Spontaneous emission into continuum modes at rate γ _sp is found to have similar impact on the photon statistics as elastic dephasing and is neglected in most simulations (γ_sp = 0). The master equation is solved numerically by stochastic Monte-Carlo wave-function approach [26]. The obtained sequences of random photon emission events (quantum jumps) with random time intervals τ_i in between are used to calculate the second-order photon (intensity) correlation function g ⁽²⁾ (τ), which represents the key physical variable in our simulations.

Figure 2 shows an example for g ⁽²⁾(τ) for different but now low numbers N of two-level systems and homogeneous dephasing rates. For N = 1 we observe the expected antibunching behaviour described by g ⁽²⁾(τ) = 1 -exp(-2g ² τ/κ), independent on the atomic dephasing rate γ _deph [17]. In contrast, for larger N > 1 we observe a strong bunching maximum with g ⁽²⁾ (0) ≫ 2, which is strongly suppressed in the presence of even small homogeneous dephasing rate γ _deph = 0.1g ²/κ. However, the magnitude of the bunching peak still exceeds the value g ⁽²⁾(0) = 2 characteristic for a thermal light source. The bunching peak is strongly suppressed not only in the presence of the homogeneous dephasing but for higher incoherent pump and spontaneous emission rates as well. All curves presented in Fig. 2 can be well approximated by a dependence g ⁽²⁾(τ) = 1 + Aexp(-4g ² τ/κ), where only the fit parameter A depends on N and the homogeneous dephasing rate. The dependence of the bunching amplitude g ⁽²⁾(0) on the number of emitters in Fig. 2 shows that the largest amplitude of the bunching peak is observed for N = 2.

4. Microscopic origin of bunching for N=2

In order to understand the microscopic mechanism of photon bunching we have analyzed the level scheme and the excitation/emission pathways for the simplest case of N = 2 shown in Fig. 3.

 

Fig. 4. Level structure for two two-level systems. The bunching pathway after simultaneous excitation of both atoms via a subradiant (dark) state leads to emission of photon pairs via a superradiant (bright) state. See text for details.

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The basis of collective atomic states in the angular momentum representation for two two-level systems with a ground state |g〉 and excited state |e〉 consists of a fully excited state |1,1〉 = |e,e), ground state |1,-1〉 = |g,g〉 and a doublet of half-excited bright state |1,0〉 = (|e,g〉 + |g,e〉)/ℚ2 and dark state |0,0〉 = (|e,g〉 - |g,e〉)/ℚ2. When starting from the ground state, the low incoherent pump results into two emission pathways with equal probability. The first path represents pumping from the ground state at the rate γ _pump to the bright state followed by the emission of a single photon at a rate 4g ²/κ. The time interval between two successive photon emission events is determined by the value of the incoherent pump rate γ _pump. This pathway resembles the antibunching path for N=1. The second (bunching) pathway evolves two consequent pump events via the dark state into a fully excited state. After being pumped into the dark state the system will be inevitably pumped into the bright state. After that, emission of a photon pair via the bright state occurs. The second photon in a pair is emitted at a double rate 4g ²/κ giving rise to and explaining the time scale of the strong bunching peak. Thus the physical origin of strong bunching for N = 2 is microscopically explained by emission of superradiant photon pairs. It is remarkable that the microscopic bunching pathway in Fig. 3 evolving generation of photon pairs is qualitatively similar to the classical evolution of superradiant emission on the P-Z diagram in Fig. 1 obtained for a macroscopic number of emitters.

Note that average populations of the ground and dark states are very close to 0.5 in case of weak pumping. Although both pathways have comparable probabilities, the amplitude of the bunching peak is very big just because the emission of the second photon in a superradiant photon pair occurs on a much faster time scale as compared to very rear pump and single-photon emission events via the antibunching path.

On the basis of Fig. 3 it can be easily understood why the increase of incoherent pump rate, elastic dephasing rate or spontaneous emission in to continuum leads to the suppression of the bunching peak. Indeed, if the elastic dephasing rate is comparable or larger than the incoherent pump rate, the dark state can be scattered into a bright state destroying the bunching loop. For higher incoherent pump rates, the average rate of photon generation increases and the relative contribution of the bunching peak due to photon pair generation decreases. The results of extensive simulations show that for γ _pump, γ _deph, γ _sp ≪ g ²/κ the amplitude of the bunching peak can be well approximated by

5. Photon bunching spectroscopy

So far we have analyzed a rather exotic situation of all atoms having the same transition frequency. In order to estimate the influence of the large inhomogeneous broadening intrinsic to low-dimensional semiconductor nanostructures (such as quantum dots) we have analyzed the influence of detuning on the bunching curves. In the simplest case of N = 2, the inhomogeneous broadening (detuning δω ₁) leads to the suppression of the bunching peak since the dark and bright states become coupled, similar to the case of elastic dephasing (Fig. 4(a)). The magnitude of the bunching peak is given by Eq. (4). The spectral width of the bunching maximum

is determined by the pump rate (γ _pump = 0.01g ²/κ) and is in our case one order of magnitude smaller that the homogeneous linewidth of individual atoms (= 2g ²/κ).

 

Fig. 5. (a) Dependence of bunching amplitude on the detuning δω ₁: (a) N = 2, (b) N = 4; the other three frequencies are kept constant: δω ₂ = -0.5g ²/κ, δω ₃ = δω ₄ = 0.5g ²/κ. The full width at half maximum of individual atomic resonances is 2g ²/κ.

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Keeping in mind the possibility to tune the excitonic resonances of quantum dots by external electric or magnetic field or temperature we have performed a simulation for four two-level systems, where the frequency ω ₁ (detuning δω ₁) of the first quantum dot is varied while keeping the remaining three transition frequencies constant: δω ₂ = -0.5g ²/κ, δω ₃ = δω ₄ = 0.5g ²/κ. The clearly resolved and sharp bunching peaks are apparent in Fig. 5(b) when the tuned frequency coincides with one of the fixed frequencies. Thus the results of Fig. 4(b) open a way to a new experimental technique - the photon bunching spectroscopy. Moreover, the observation of sharp photon bunching peaks may serve as a sensitive indicator on long-living coherence -an alternative technique to the sophisticated photon-echo (four wave mixing) experiments in inhomogeneously broadened ensembles of optical emitters. The proposed technique would be ultimatively sensitive to the long-living collective polarization states of any two optical transitions resonant to each other and weakly coupled to a common damped cavity mode, irrespective to the presence of other non-resonant emitters.

This physical picture based on two-particle entanglement favors experimental feasibility of the phenomenon. It is not necessary to have equal atom-field coupling constant g for two spectrally overlapping emitters. Indeed, for N = 2 the level structure with a doublet of bright and dark states required for photon pair generation persists for different coupling constants g ₁ ≠ g ₂ [27].

Generally speaking, the cooperative interaction of continuously pumped few-emitter ensembles interacting with a single damped cavity below the lasing threshold represent a new class of photon sources with non-trivial statistical properties, which can be tailored for particular type of applications. For example, a possibility to quickly bring the two resonant quantum dots in and out of resonance with a single cavity mode would allow to implement a deterministic source of indistinguishable (superradiant) photon pairs required for quantum information processing. The required picosecond switching time scale might be achieved by using ultrashort acoustic strain pulses to shift the microcavity resonance energy.

6. Conclusions

To summarize, the photon statistics, quantified in terms of second-order photon correlation function g ⁽²⁾(τ), is investigated theoretically for few-emitter systems in a weak coupling regime. The numerical simulations predict a giant photon bunching caused by generation of correlated photon pairs which should be observable in HBT experiments with atoms or quantum dots in low-Q cavities. Changing the emitter frequency in the photon correlations measurements, i.e. the resonance detuning in a new type of photon bunching spectroscopy, allows us to extract information about dephasing times and number of atoms in the cavity.

7. Acknowledgements

The financial support of Deutsche Forschungsgemeinschaft (TE770/1) and EU Networks of Excellence “PhOREMOST” and “SANDiE” is gratefully acknowledged.