Recent advances in the active field of cavity quantum electrodynamics have opened the doors to a fascinating regime of physics, where a single emitter strongly interacts with the electromagnetic field of a resonator. Experimental achievements in this field date back to the micro-maser with individual atoms traversing a microwave cavity [1] and its optical analog [2], and the emission of a train of individual photons from a single atom [3]. Single-atom laser emission is connected with the ultimate quantum limit where one photon is sufficient to saturate the atomic transition and novel properties unknown from conventional lasers are seen, which include nonclassical and more orderly light emission [4].

Single quantum dot (QD) lasers have been proposed as the solid-state analog to the single-atom laser [5]. In the past decade, considerable progress has been made in the fabrication of semiconductor microcavities with quasi-three-dimensional photon confinement and single QD emitters. These systems have demonstrated single-photon generation on demand [6, 7] and strong coupling [8, 9]. The laser emission has been studied with a small number of QD emitters [10, 11, 12, 13] and the goal of current investigations is the realization of a single-QD laser. In recent experiments [14, 15, 16] the dominance of a single QD in the laser emission was demonstrated.

The discrete level structure of QDs and the resulting similarity to atomic systems has been widely used throughout the literature by invoking atomic models to describe QD-based systems. In the limit of a single-emitter laser, the question what differences to expect between atomic and QD systems becomes even more prominent than for the many-emitter case where inhomogeneous broadening due to QD size and composition fluctuations masks experimental observations. Present laser experiments with dominating contributions from single QDs [14, 15, 16] are considered to show stimulated emission in the weak-coupling regime. In this paper, we analyze these experimental situations and predict results for further cavity quality improvements. A central point of our investigations are conceptual differences in the physics of atomic and QD emitters.

In an atomic system one usually assumes optical transition between two electronic configurations (the “laser levels”) that are resonantly coupled to a high-quality cavity mode. In the corresponding theoretical models, this coupling is described by the Jaynes-Cummings Hamiltonian. Additional configurations are used for the pump process. The theoretical description of the single-atom laser was introduced in [17]. On the other hand, a microscopic description of semiconductor systems usually starts from the electronic single-particle states. The occupation of these states with electrons and holes in the conduction and valence band states, respectively, leads even for a single QD to various system configurations with energies depending on the Coulomb interaction of the excited carriers. This is very much like in a multi-electron atom and it is what has earned QDs the name of artificial atoms.

Yet, the analogy breaks down in several important respects, related to the strength of the configuration (Coulomb) interaction and its consequences. In atoms the interaction induces a large energetic separation between configurations, allowing in atomic laser models to consider only a limited number of configurations and neglect those which lie outside the energetic window of interest. For instance, just two configurations are supposed to interact with the cavity mode. We argue below that in QDs such a restriction is usually not possible. The involvement of multiple configurations in the same interaction process changes the physics of the QD laser and requires a more flexible model.

To begin with, QDs are much larger than atoms, which leads to proportionally reduced Coulomb interaction of carriers in various states. This considerably weakens the strength of the configuration interaction. Secondly, in an atom a change in the configuration when an electron changes its orbital amounts to an important modification in the charge distribution, and the configuration energy is substantially shifted by the direct (Hartree) Coulomb interaction. On the contrary, in QDs, to a good accuracy [18] the conduction and valence band states have pair-wise equal wave functions and a carrier jump between such states does not modify the charge distribution in the dot.

As a typical example we consider self-assembled QDs that currently have many technological applications. The confinement potential gives rise to localized states, both for electrons and holes, below a quasi-continuum of wetting layer (WL) and bulk states [18]. For a QD laser, the lowest confined QD states of the electrons and holes provide the laser levels when tuned to the cavity mode. Additional levels are necessary for the pump process. Here one has to distinguish between different situations. In the limit of a shallow QD with only one confined state for electrons or holes, the quasi-continuum of delocalized WL states at higher energies would need to be used for the pump process. Efficient carrier relaxation between the WL and QD states due to carrier-carrier and carrier-phonon interaction easily facilitates the laser action [19]. Alternatively, deeper confinement potentials provide additional localized excited QD states in each band, which can be used directly for an optical pump process or as intermediate levels for carrier relaxation from the WL to the laser levels. Various system configurations arise from different occupations of these QD single-particle states with carriers¹.

In the following we investigate the simplest possible QD system suitable for laser action under optical pumping². Two confined QD states for holes (∣1〉, ∣2〉) and electrons (∣3〉, ∣4〉) are considered, as shown in Fig. 1. The pump process and the carrier relaxation are indicated by blue arrows (solid lines). Carrier transitions due to photon emission into and reabsorption from the laser mode are represented by the red arrow (dashed line) and optical processes coupled to non-lasing modes are shown in magenta (dotted lines). (Note that the conduction-valence-band picture and not the electron-hole picture is used.) In the ground state system configuration, the valence-band states are occupied with electrons. All states are spin-degenerate, however, optical transitions between carriers from the respective spin-subsystems couple to different light polarizations. The two spin-subsystems can be treated separately provided that the Coulomb interaction between configurations with opposite spin carriers is smaller than the line broadening. Also, to keep the discussion and the model as simple as possible, we do not consider the option of an additional degeneracy of the excited states.

 

Fig. 1. Localized single-particle states ∣1〉 - ∣4〉 in the QD confinement potential and carrier transitions for an optically pumped QD laser. The pump process P excites holes and electrons in the states ∣1〉 and ∣4〉 while carrier relaxation with rates γ ₃₄ and γ ₁₂ couples the pump levels to the laser levels; γ ₂₃, γ ₁₄ are the laser- and pump-level spontaneous emission rate into non-lasing modes.

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For one spin-subsystem, with two carriers in the ground-state we obtain the 6 possible system configurations of Fig. 2 which are connected by the transition processes depicted in Fig. 1. As we outline subsequently, two distinct configurations allow for the emission of a photon into the laser mode. Fig. 3a shows the laser transition in the presence of a valence electron in the state ∣1〉 or, in other words, the recombination of an s-exciton 1X_S → 0X (the exciton laser transition). Similarly the transition with the second electron in ∣4〉 shown in Fig. 3b amounts to a transition from an sp-biexciton to a p-exciton 2X_sp → 1X_p (the biexciton laser transition). Since the pair of s-states have similar wave functions in good approximation, the s-exciton is not only globally but also locally neutral. The same is true for the p-exciton and therefore there is no direct Coulomb interaction between them. This shows that the Hartree interaction is not lifting the degeneracy between the exciton and biexciton transitions in Fig. 3 and, in the absence of other effects, they would both be resonant with the cavity mode.

 

Fig. 2. System configurations corresponding to two electrons in the four confined QD states of Fig. 1. The top row shows the state occupation for electrons in the conduction and valence-band picture while the bottom row corresponds to electrons (filled circles) and holes (open circles) in the electron-hole picture. The ground states (0X) is given by two valence electrons in the states ∣1〉, ∣2〉. Single excitons with electrons and holes in the lowest QD state (s-state) and excited QD state (p-state) are represented by 1X_S and 1X_p, respectively. The sp-biexciton is labeled as 2X_sp. Selection rules typically identify 0X_S and 0X_P as optically dark configurations.

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Fig. 3. Transitions coupled to the laser mode: (a) exciton and (b) biexciton recombination.

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On the other hand, for same spin carriers, one has to take into account the exchange Coulomb interaction as well. Indeed, the transition of Fig. 3b involves the loss of the s-p exchange energy in both bands, while the one in Fig. 3a does not. This would take in principle the 2X_sp → 1X_p transition out of resonance with the laser mode (tuned to 1X_S → 0X). In practice though the resonance condition is not sharp, but is a result of the interplay between the size of the detuning due to exchange (typically a few meVs [20, 21]) and the finite linewidths of the levels involved. Recent investigations showed that detuned QD transitions can still couple to high-quality microcavity modes [22].

Regarding the exchange Coulomb interaction of QD carriers as well as the finite linewidth of optical transitions, an important role is played by excited carriers in the WL. When in sufficient number, they efficiently screen the Coulomb interaction and thus reduce the strength of the exchange. Also, scattering processes involving these WL carriers (in addition to the carrier-phonon interaction) lead to finite lifetimes for the QD states. The WL screening and dephasing effects require the presence of an energetically close continuum of states, into which a population of carriers can be created, e.g. by direct WL pumping. This is not the case in atomic systems, but is quite common in QDs. One encounters here a third major difference between the two types of emitters, QDs and atoms.

With these facts in mind one should not exclude the biexcitonic configuration from the model, leaving the QD properties and the excitation condition to decide whether only one transition of Fig. 3 or, due to the excitation-induced screening and dephasing, both of them are efficiently coupled to the cavity mode. But this is not only the case for the Jaynes-Cummings coupling to the photon subsystem, each of the other processes considered (pumping, relaxation) may act upon several pairs of configurations. For instance the pumping gives rise to both the 0X → 1X_p and the 1X_s → 2X_sp transition. In a recent publication [23] coupling of several configurations by the same processes was also discussed. In the single-QD laser model which we define below, such possibilities are naturally taken into account.

We consider the four-level, single spin QD system described above. The formalism can be generalized in a straightforward manner to include both spins, p-level degeneracy, higher QD states a.s.o. The fermionic creation (annihilation) operators corresponding to the four states are denoted a_i ^† (a_i), i = 1…4. The cavity mode, assumed resonant with the 1X_s → 0X transition, is described by the bosonic operators b ^† and b. The equation of motion for the QD-photon density matrix ρ(t) in the interaction picture reads (h̄ = 1)

The first line in Eq. (1) is the von Neumann evolution generator corresponding to the Jaynes-Cummings and the Coulomb interaction Hamiltonian. The former is given by

where g is the coupling constant between the laser mode and the s-levels. The Coulomb Hamiltonian for this QD system [21] contains direct and exchange terms. For reasons discussed above only the exchange part

contributes to the dynamics of the density matrix (see Appendix I), and gives rise to a detuning of 2E^X_sp in the biexciton transition. E^X_sp is the sp-exchange energy and n_i = a_i ^† a_i are the occupation number operators.

The second line in Eq. (1) describes dissipation processes derived from the interaction with different reservoirs, treated in the Born-Markov limit [24]. The sum runs over the following pairs of indices (i,j) = (1,2), (3,4), (2,3), (1,4) and the coefficients γ_ij are each the relaxation rates for the ∣j〉 → ∣i〉 process. Additionally, in the sum we include the pumping term as an “inverse relaxation” ∣1〉 → ∣4〉, with the rate γ ₄₁ = P. This simulates the situation where carriers are initially generated by optical excitation in the WL, and then are captured in the QD. With Coulomb scattering processes as the main capture mechanism, the pumping rate P is a function of the incoherent WL population. The discussion about the relationship between pump rate, WL population and resulting excitation-induced dephasing are found in Sections 2 and 3 of the appendix. Finally, the third line of Eq. (1) describes cavity losses at a rate κ in a similar way.

The Lindblad terms in Eq. (1) look similar to those used in atomic models [17, 25, 26, 27], with the important difference that here they are formulated using the fermionic operators a_i ^† a_j, instead of the operators σ_ij = ∣i〉〈j∣ acting on configurations. This leads to a different structure of the density-matrix evolution operator, as seen by comparing σ_jiσ_ij = σ_jj of the atomic models with a_j ^† a_ia_i ^† = a_j ^† a_j - a_j ^† a_i ^† a_i a_j , which is a_j ^† a_j only in the case of a system with a single electron (a_i a_j = 0). With more than one electron, the difference amounts to the fact, described above, that a given physical process (relaxation, Jaynes-Cummings interaction) is involved in transitions between several pairs of configurations. In this sense the atomic models are formally equivalent to the present one with only one carrier [17], with the caveat that the levels considered there are configurations and not one-particle states.

To illustrate the implications of the discussed interplay of multiple configurations in single-QD lasers, the Liouville-von Neumann equation for the discussed system is solved numerically without further approximations. We present the input/output characteristics along with the photon autocorrelation function for different physical situations by considering typical QD parameters. To include the influence of a WL continuum, we account for the excitation-dependent broadening of the transition lines and screening of the Coulomb interaction, which are determined as functions of the pump rate as discussed in Appendix III. Additionally, we also consider the case without the impact of the WL. This more closely resembles the situation in an atomic system, where the energetic positions of the transitions are not excitation-density dependent. In the following examples, the equation of motion for the density matrix is solved numerically in time until a steady state is reached.

By tracing over the electronic degrees of freedom, our calculations provide the density matrix for the laser mode that contains the information about the statistical properties of the emitted photons. In experiments, frequently the photon autocorrelation function at zero delay time g ⁽²⁾(0) = (〈n ²〉-〈n〉)/〈n〉² with the photon number operator n = b ^† b is used to characterize statistical properties. Single-photon sources are associated with anti-bunching behavior and g ⁽²⁾(0) < 1, typical for a sub-poissonian photon statistics, indicates that immediately after a first photon emission, the likelihood for a second one is reduced. This is expected to be a common feature for single emitters, which need a certain re-excitation time after an emission event. Thermal emission with bunching behavior is characterized by g ⁽²⁾(0) = 2, while a Poisson distribution of photons corresponds to g ⁽²⁾(0) = 1.

In Fig. 4a,b the steady-state mean photon number 〈n〉 and the second-order photon correlation function g ⁽²⁾(0) are shown versus pump rate P for parameters similar to the experiments of [16]. For P > 5 × 10^-2/ps our results are reasonably close to the experimental findings. The s-shaped photon output curve in this pumping range resembles the behavior of solutions of the conventional rate equation for the multi-emitter laser. The characteristic kink appearing at the onset of the stimulated emission is related in these solutions to the value of the spontaneous emission coupling factor [28]. This similarity is surprising in the present context because, for single-emitter lasers, one expects saturation effects above the threshold region. The simplest argument is that for a fully inverted laser, a further increased pump rate does not enlarge the photon production. Moreover, in [17, 5] a quenching of the laser output has been discussed due to pump-induced dephasing in a situation where the pump process directly involves one of the laser levels.

Here the appearance of a kink has a different interpretation. In highly excited QDs, one expects excitation-dependent scattering rates into the laser levels. This effect introduces excitation-dependent dephasing which on the one hand lowers the rates for optical transitions and on the other hand relaxes the resonance condition for these transitions by line-broadening. In our model, these features are included via excitation-dependent rates γ ₁₂ and γ ₃₄. Additionally, in our multi-configuration model, with increasing pump rate first the probability for the 1X_S configuration increases, but at elevated pump rates its probability diminishes at the expense of the configuration 2X_sp. This can be seen from the contributions of the exciton and biexciton transitions to the total photon production, which can be separated as described in Appendix IV and are shown in Fig. 4a. A kink in the input/output curve is obtained when the biexciton transition takes over.

 

Fig. 4. (a) Steady-state mean photon number 〈n〉 vs. pump rate P (solid line) together with its contributions from the exciton (dashed line) and biexciton (dotted line) transitions according to Fig. 3. (b) Second-order photon correlation function g ⁽²⁾(0) versus pump rate P. (c) Excitation-induced dephasing (dashed line) and reduction of the Coulomb exchange energy E^X_sp due to WL carrier screening (solid line). (d) Photon emission spectrum for selected pump rates. Lines are vertically displaced. For all curves the light-matter coupling constant g = 0.9 ps^-1, cavity losses at rate κ = 0.1 ps^-1 and spontaneous emission rates γ ₂₃ = γ ₁₄ = 0.01 ps^-1 are used.

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For weak pump rates P < 10^-1/ps, when the excitonic recombination dominates the emission, we obtain the expected anti-bunching behavior g ⁽²⁾(0) < 1 in Fig. 4b. In the transition region between excitonic and biexcitonic emission, a clear photon-bunching behavior with 1 < g ⁽²⁾ (0) < 2 is observed. This might seem surprising for a single-emitter laser, and was tentatively attributed in the literature to background contributions [16]. However, within the present model, with the increasing contribution of the biexcitonic configuration, shown by the dotted line in Fig. 4a, successive photon-pair emission becomes possible without a repumping of the system. Note that in our system the subsequent emission by the sp-biexciton and the s-exciton requires a carrier relaxation from p- to s-shells. Even though the relaxation times are not constant, but increase with the excitation density, see Fig. 4a, they are short compared with the cavity lifetime of 10ps, enabling storage of the emitted photons and the development of cavity photon bunching. The contribution of the biexcitonic transition to photon emission into the laser mode is facilitated for larger pump rates by the increased dephasing of the optical transitions that broadens the emission lines, and the reduction of the exchange splitting of the exciton and biexciton lines, as depicted in Fig. 4c. Also our calculations exhibit a transition from the strong to weak-coupling regime. For a weak pump rate P = 10^-2/ps the emission spectrum in Fig. 4d shows two main peaks due to Rabi-splitting of the degenerate cavity and exciton line, weak side peaks due to the next step of the Jaynes-Cummings ladder and additional peaks from the sp-biexciton transitions. The latter ones are positioned at twice the Coulomb exchange energy E^X_sp Excitation-induced dephasing leads to a transition to the weak-coupling regime. We note that for pump rates P > 5 × 10^-2/ps, where our input-output curve shows similar behavior as the results in [16], the strong coupling regime gradually fades out, in an experiment of finite resolution possibly even earlier.

 

Fig. 5. (a) Steady-state mean photon number 〈n〉 vs. pump rate P. From bottom to top, the cavity losses are reduced using κ = 0.1 ps^-1 , 0.05 ps^-1, 0.025 ps^-1, 0.0125 ps^-1. For a cavity mode wavelength of 915 nm this amounts to a cavity quality Q ~ 21 × 10³,42 × 10³, 84 × 10³, 168 ×10³. Other parameters are the same as in Fig. 4. (b) Corresponding results for the second-order photon correlation function g ⁽²⁾(0) in the stationary regime vs. pump rate P.

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In the following, we discuss the impact of modified system parameters and properties in order to obtain a more general picture. Starting from the above example, the influence of the cavity quality and the WL-influence is analyzed. Additionally, we compare our QD results to the limiting case of an “atomic model”.

Fig. 5 reveals, that for reduced cavity losses (increasing cavity-quality factor Q = ω/κ defined with the cavity mode frequency ω) the mean photon number increases and the kink in the transition region is smoother. More remarkably, anti-bunching (for small P) and bunching (for intermediate P) are reduced. The steady state photon statistics depends not only on the QD emission rate, but also on the photon storage properties of the cavity. Therefore, if antibunching is desired, a compromise with respect to the cavity quality has to be made. For high pump rates, in all cases the lasing-regime with stimulated emission is reached, as reflected in g ⁽²⁾(0) = 1.

A different situation is found when the excitation-induced effects of the WL carriers are not considered. This would require a very large energy separation of the QD states from delocalized states, or other QD systems like semiconductor microcrystalites. We describe this situation by using pump-independent values for the scattering/dephasing rates γ ₁₂ and γ ₃₄ and for the exciton-biexciton Coulomb exchange splitting 2E^X_sp. The corresponding red lines in Fig. 6 exhibit a strong quenching of the mean photon number. In contrast to the systems of Refs. [17, 5], this is not induced by dephasing due to the pump process (as a result of the pump process acting directly on one of the system configurations contributing to the laser process) but rather by bringing the system into the biexcitonic configuration that inefficiently couples to the cavity mode (when the detuning is too large and line broadening is weak). From the red to the blue lines in Fig. 6 the Coulomb exchange energy E^X_sp is increased by a factor of two. This further reduces the biexciton contribution to the recombination and the quenching of the mean photon number is more pronounced. The decoupling of the QD from the WL induces a stronger antibunching, and the transition into the bunching-regime occurs at higher pump rates.

 

Fig. 6. (a) Steady-state mean photon number 〈n〉, (b) second-order photon correlation function g ⁽²⁾(0), (c) exciton (solid line) and biexciton (dashed line) contribution to the mean photon number, versus pump rate P. The black lines correspond to the results of Fig. 4. For the red lines the Coulomb energy and the dephasing are kept constant: E^X_sp = 2.68 meV, γ ₁₂ = γ ₃₄ = 0.05 meV. In comparison to the red curves, E^X_sp = 5.36 meV is used for the blue lines. The green curves correspond to the “atomic model”, using σ_ij instead of a_i ^† a_j. Apart from the Coulomb interaction, which is assumed to be already included in the atomic con-figuration energies (therefore E^X_sp = 0), the parameters are the same as for the red curves.

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As the true atomic limit we refer to the situation where off-resonant transitions are completely removed from the model and the remaining ones (between a reduced number of configurations) are described by σ_ij, operators. Replacing a_i ^† a_j by σ_ij, in our case leads to a four-state atomic model that was considered, e.g. in [17]. The results are shown in the green curve of Fig. 6. In the low-pumping regime, the mean photon number coincides with the QD-models. Differences occur with rising importance of the biexciton contribution (dashed lines in Fig. 6c), which is not considered in the atomic model. A striking difference to the QD-model is the pronounced antibunching behavior throughout the whole range of pump rates shown in Fig. 6b.

Returning to our starting point, we conclude that a single QD is truly a semiconductor system and in its nature subject to many-particle effects. A static configuration picture, which can be used in the description of atomic systems, is in general not justified for QDs in the presence of carriers in a WL. We find that the demonstrated transition from photon anti-bunching to bunching with increasing pump rate as well as the characteristic shape of the input/output curve depend on interaction-induced dephasing and screening effects. For parameters corresponding to present experiments the transition from strong to weak coupling is identified in a regime where spontaneous emission still dominates over stimulated emission and the average cavity photon number is below one.