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arXiv:2002.03254 · Laser & Photonics Review

Exceptional Points as Lasing Prethresholds

A. A. Zyablovsky, Ilya V. Doronin, Eugeny S. Andrianov, Alexander A. Pukhov, Yu. E. Lozovik, Alexey P. Vinogradov, Alexander A. Lisyansky

Published Jan 15, 2021• 26 citations
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Abstract

Abstract The genesis of lasing, as an evolution of the laser hybrid eigenstates comprised of electromagnetic modes and atomic polarization, is considered. It is shown that the start of coherent generation at the laser threshold is preceded by the formation of a special hybrid state at the lasing prethreshold. This special state is characterized by an enhanced coupling among excited atoms and electromagnetic modes. This leads to an increase in the rate of stimulated emission in the special state and, ultimately, to lasing. At the lasing prethreshold, the transformation of hybrid eigenstates has the features of an exceptional point (EP) observed in non‐Hermitian systems. The special state is formed when eigenfrequencies of two hybrid states coalesce or come close to each other. Below the prethreshold, lifetimes of all hybrid states grow with increasing pump rate. When the pump rate crosses the prethreshold, resonance trapping occurs with the lifetime of the special state continuing to increase while the lifetimes of all other eigenstates begin to decrease. Consequently, the latter eigenstates do not participate in the lasing. Thus, above the prethreshold, a laser transitions into the single‐mode regime.

DOI
10.1002/lpor.202000450

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Abstract The genesis of lasing, as an evolution of the laser hybrid eigenstates comprised of electromagnetic modes and atomic polarization, is considered. It is shown that the start of coherent generation at the laser threshold is preceded by the formation of a special hybrid state at the lasing prethreshold. This special state is characterized by an enhanced coupling among excited atoms and electromagnetic modes. This leads to an increase in the rate of stimulated emission in the special state and, ultimately, to lasing. At the lasing prethreshold, the transformation of hybrid eigenstates has the features of an exceptional point (EP) observed in non‐Hermitian systems. The special state is formed when eigenfrequencies of two hybrid states coalesce or come close to each other. Below the prethreshold, lifetimes of all hybrid states grow with increasing pump rate. When the pump rate crosses the prethreshold, resonance trapping occurs with the lifetime of the special state continuing to increase while the lifetimes of all other eigenstates begin to decrease. Consequently, the latter eigenstates do not participate in the lasing. Thus, above the prethreshold, a laser transitions into the single‐mode regime.
The emission linewidth in active medium emerges due to homogeneous and inhomogeneous broadening. We demonstrate that in lasers with inhomogeneous broadening there is a critical pump rate, above which the special mode forms. This mode consists of locked-in oscillations of cavity mode and of the active atoms with different transition frequencies. Below the critical value of the pump rate, the radiation spectrum of the laser has a Gaussian profile, provided that inhomogeneous broadening is dominant. Above the critical value of pump rate, the special mode mostly determines the laser radiation spectrum. As a result, the spectrum attains Lorentz shape characteristic for homogeneous broadening. The formation of the special mode precedes lasing and that the critical pump rate plays the role of lasing prethreshold. We suggest a system where such a transition could be experimentally observed separately from lasing. We obtain expressions for the threshold and generation frequency of a single-mode laser where both homogeneous and inhomogeneous broadening are present.
We propose an approach to obtain lasing without inversion, which does not require preparation of the quantum superposition of states in the active medium. We demonstrate that with periodically variable pumping, lasing is achievable even in the case when the population inversion of the active medium is always negative. This is possible due to the fact that laser in strong-coupling regime is a non-Hermitian system with an exceptional point existing at negative value of population inversion. We show that due to nonorthogonality of eigenstates near the exceptional point, the time variation of pump power can result in parametric instability in the laser and excitation of the electromagnetic field self-oscillations. These oscillations take place even when the population inversion of the active medium is always negative. Our results help to pave the way for the creation of a new class of lasers operating at negative value of population inversion.
We develop a theory of lasing of a collection of pumped active atoms without a resonator (either regular or random). Due to spontaneous emission into free space, phases of free space electromagnetic modes fluctuate. These phase fluctuations can be reduced to frequency fluctuations. The closer the frequency of fluctuation to the transition frequency of the active atoms, the higher lifetime of the fluctuation. We show that because of this, the average frequency of modes pulls toward the transition frequency. This leads to a maximum in the density of states of the electromagnetic field and a decrease of the mode group velocity. Consequently, the coupling of modes with atoms as well as the lifetime of fluctuations increase. Thus, mode pulling provides positive feedback. When the pump rate exceeds a certain threshold, the lifetime of one of the realized fluctuations diverges, and radiation becomes coherent.
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.
We review the physics underlying the process of spontaneous emission, with a special focus on spontaneous emission into a resonator mode. We define the mode volume, verify the fundamental modal dimensions, present the spectral mode profile, the coherence time, the Q-factor, the Füchtbauer-Ladenburg equation, and the Purcell factor, and discuss their influence on different types of lasers. We obtain the relation between peak emission cross section, radiative lifetime, and emission linewidth. By interpreting spontaneous emission as stimulated emission driven by vacuum fluctuations, we derive the spontaneous-emission rate into a resonator mode and establish physical expressions for the fractions of spontaneous emission and total decay from the upper laser level into this mode. Furthermore, we discuss coupling of the atomic system with the coherent field inside a lasing resonator mode, resulting in the formation of a Mollow triplet, and demonstrate that it leads to a reduction of the spontaneous-emission rate by a factor of 2.
It is commonly accepted that a collection of pumped atoms without a resonator, which provides feedback, cannot lase. We show that intermodal coupling via active atoms pulls the frequencies of the free-space modes towards the transition frequency of the atoms. Although at a low pump rate mode phases randomly fluctuate, phase realizations at which interference of pulled modes is constructive emerge. This results in an increase of stimulated emission into such realizations and makes their lifetime longer. Thus, mode pulling provides positive feedback. When the pump rate exceeds a certain threshold, the lifetime of one of the realizations diverges, and radiation becomes coherent.