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Semiconductor Lasers and Laser Dynamics

Two-particle quantum correlations in nano and macro single-mode lasers

F. Papoff, Giampaolo D'Alessandro, Gian Luca Lippi, Gian‐Luca Oppo

Published Jun 20, 2024• 0 citations
Publisher

Abstract

We compare a novel quantum mechanical model of lasers, which includes all two-particle correlations, with the Coherent-Incoherent Model (CIM), where truncation eliminates these effects. Numerical results for the simple case of identical single-electron quantum dots are presented for two cases: only the laser mode is coupled to the quantum dots; the coupling is extended to non-resonant modes. We find coexistence between non-lasing and lasing states, together with a minimum number of photons required to initiate the laser action. The correlations introduce a non-zero variance in the field, which is otherwise strictly zero in the absence of interparticle correlations.

DOI
10.1117/12.3024493

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We compare a novel quantum mechanical model of lasers, which includes all two-particle correlations, with the Coherent-Incoherent Model (CIM), where truncation eliminates these effects. Numerical results for the simple case of identical single-electron quantum dots are presented for two cases: only the laser mode is coupled to the quantum dots; the coupling is extended to non-resonant modes. We find coexistence between non-lasing and lasing states, together with a minimum number of photons required to initiate the laser action. The correlations introduce a non-zero variance in the field, which is otherwise strictly zero in the absence of interparticle correlations.
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We detail the derivation of nanolaser models that include coherent and incoherent variables and predict the existence of a laser threshold, irrespective of cavity size and emitter number, for both single- and multielectron systems. The growth in photon number in the lasing mode is driven by an increase in correlation between absorption and emission processes, leading to the onset of self-sustained stimulated emission (laser threshold), followed in turn by a correlation decrease and ending with the dominance of coherent emission. The first-order coherence g(1){g}^{(1)}g(1) steadily increases, as the pump grows towards the laser threshold value, and reaches unity at or beyond threshold. The transition toward coherent emission becomes increasingly sharp as the numbers of emitters and coupled electromagnetic cavity modes increase, continuously connecting, in the thermodynamic limit, the physics of nano- and macroscopic lasers at threshold. Our predictions are in remarkable agreement with experiments whose first-order coherence measurements have so far been explained only phenomenologically. A consistent evaluation of different threshold indicators provides a tool for a correct interpretation of experimental measurements at the onset of laser action.
Multi-photon lasing has been realized in systems with strong nonlinear interactions between emitters and cavity modes, where single-photon processes are suppressed. Coherence between the internal states of a quantum emitter, or among multiple emitters, plays a key role. Such continuous nonclassical sources of light can find applications in quantum computation, quantum sensing, quantum metrology, and quantum communication. This thesis explores the multi-photon lasing phenomena in various quantum dot-photonic crystal cavity quantum electrodynamic (QED) setups. Exciton-phonon interactions are inevitable in such systems and are incorporated using the polaron-transformed master equation. The Born-Markov approximation is employed to obtain the reduced density matrix rate equation. Using quantum laser theory, we derived the Scully-Lamb laser rate equations and evaluated the single- and multi-photon excess emission rates defined as the difference between emission and absorption rates into the cavity mode without mean-field approximations. We investigated cooperative two-photon lasing, correlated emission lasing, hyperradiant lasing, non-degenerate two-mode two-photon lasing, and continuous variable entanglement in open quantum systems with single or multiple semiconductor quantum dots (two-level, three-level, and four-level) driven coherently/incoherently and coupled to single/ bimodal cavities.
Entanglement and lasing of two quantum dots in a microcavity are investigated in the steady state achieved under a continuous incoherent pumping. Lasing is favored in the case of independent pumping of the dots, i.e., when an electron-hole pair which reaches one dot does not affect the other. In this case, having two dots allows a qualitative enhancement of the lasing emission as compared to the single dot case. On the other hand, when the two dots share the excitation bath, giving rise to a quantum uncertainty of the final state of the excitation, entanglement properties are enhanced in detriment of the photon emission. Slightly different couplings between the dots and the cavity induce good singlet-state population. This allows good entanglement in a system pumped continuously and incoherently.
Traditionally, the active material in a laser is modelled as independent emitters, but in recent years it has become increasingly clear that radiative coupling between emitters can significantly change the characteristics of small lasers. Collective effects in free space such as superradiance have been studied extensively [1,2], but the effects of inter-emitter correlation in micro- and nano-cavities need further examination to be put on firm theoretical ground. Several studies of collective effects in nano-cavities have been made [3-6], but the theoretical models employed are intricate, and numerical methods are needed both to generate the dynamic equations and to solve them. We propose a model where the complexity is strongly reduced, allowing analytical solutions [7]. We consider a collection of identical two-level emitters interacting with a single cavity mode. We start from Maxwell-Bloch equations, but instead of making the typical adiabatic elimination of the polarization, we allow the polarization decay rate to be of the same magnitude or smaller than other decay rates. Hence, the traditional laser rate equations for the photon number and the population inversion must be supplemented by equations for the emitter-field correlation and the emitter-emitter correlation. This gives us four generalized laser rate equations, which we solve analytically in steady state. Comparing with the steady state results obtained from the traditional laser rate equations we see that inclusion of collective effects leads to a reduction of the photon number for small pump rates, similarly to what is found in [4]. From the generalized laser rate equations, we derive a measure of the strength of collective effects in terms of laser parameters: This describes the difference between results with and without inter-emitter correlations, and it goes smoothly to zero as we approach parameter values where the traditional laser rate equations become valid. To gain insight into the photon statistics of the laser, we construct dynamic equations for higher order correlations of operators. We derive an analytical expression for the zero-delay photon auto-correlation function, and for low pump rates we find that the interaction of emitters results in super-thermal values of the auto-correlation. This feature is observed in experiments and numerical models [4-5], and with our analytical expressions, we are able to pinpoint the parameter combinations for which the collective effects have the largest impact. Considering the same model in terms of the Fourier components of the operators, we find results for the photon number that agree well with the previous approach, while allowing computation of the linewidth. Thus, we can examine how emitter-emitter correlation affects the line broadening of the laser.