Experimental realization and robustness of lasing without population inversion in an Er3+-doped YAG crystal

Sahrai, Asadpour, Eslami-Majd and Sadighi-Bonabi (Journal of Modern Optics 59, 446 (2012)) proposed two four-level density-matrix schemes for an $\mathrm{Er}^{3+}$:YAG crystal and numerically identified parameter windows - an enhanced spontaneous-emission branch $\gamma_{32}$ of order $10\gamma$, an incoherent pump rate $r$ of about $2\gamma$ ($\gamma=\gamma_{21}$, the reference decay rate), and a moderate coherent coupling $\Omega_c$ - in which a probe field is amplified while the populations on the probe transition remain uninverted. Whether these windows can be engineered and stabilized in a real crystal - where spontaneous-emission branching is fixed by the host, dephasing and inhomogeneous broadening are large, and the source model does no ensemble averaging - was left open and remains open. The problem asks for the experimental realization (or quantitative refutation) of inversionless gain in $\mathrm{Er}^{3+}$:YAG: engineering the required decay branches, dephasing rates and incoherent-pump coupling; mapping how inhomogeneous broadening and realistic solid-state noise shift the boundary between inversion-based and inversionless gain; and identifying the cavity or reservoir engineering needed to stabilize the lasing-without-inversion regime.

Background

In an ordinary laser, gain requires population inversion: more ions in the upper than in the lower level of the lasing transition, so that stimulated emission outpaces absorption. Lasing without population inversion (LWI) is a scheme in which an auxiliary coherent 'coupling' field (whose strength is expressed as the Rabi frequency $\Omega_c$, the rate at which the field coherently drives its transition) and usually an incoherent pump (an incoherent, rate-like transfer of population between two specified levels, here with rate $r$) prepare quantum coherence between atomic states so that the probability amplitudes of the pathways for absorbing the probe light interfere destructively. Absorption is then suppressed while emission is not, and net gain - a negative imaginary part $\chi''$ of the probe susceptibility - can occur even though the upper level of the probe transition is less populated than the lower one. The effect is a close relative of electromagnetically induced transparency (EIT), the coherent cancellation of absorption by a driving field. The theory of amplification and lasing without inversion was developed around 1990 (reviewed by Kocharovskaya, 1992), motivated in part by the prospect of lasers at wavelengths, such as the vacuum ultraviolet, where population inversion is prohibitively hard to build. Laser oscillation without inversion was demonstrated in rubidium vapor (Zibrov et al., 1995) and in a sodium atomic beam (Padmabandu et al., 1996), and light amplification without inversion had been observed in sodium vapor in 1992 (Gao et al., 1992). Kocharovskaya, Rostovtsev and Imamoglu (1998) later emphasized that a three-level 'inversionless' amplifier can hide a microscopic population inversion in a dressed-state basis (the basis of states diagonalizing the atom-plus-coupling-field system), so any new LWI claim must state which populations are being compared.

All quantum-coherence demonstrations of lasing without inversion to date are in gases or vapors; rare-earth-doped crystals remain untested ground. Rare-earth ions substituting in transparent crystals have optical 4f-shell transitions (transitions of the lanthanide 4f inner shell, shielded by outer shells and therefore sharp) with narrow lines and long coherence times, especially at cryogenic temperature. EIT has been demonstrated in a rare-earth-doped crystal ($\mathrm{Pr}^{3+}$:$\mathrm{Y}_2\mathrm{SiO}_5$; Ham, Hemmer and Shahriar, 1997), and coherence times longer than a second have since been reached in $\mathrm{Er}^{3+}$:$\mathrm{Y}_2\mathrm{SiO}_5$ (Rancic et al., 2018), so the coherence side-conditions of a solid-state coherence experiment are increasingly accessible. YAG (yttrium aluminium garnet, $\mathrm{Y}_3\mathrm{Al}_5\mathrm{O}_{12}$) is a standard laser host and $\mathrm{Er}^{3+}$:YAG is a workhorse of conventional inversion-based lasers. But a solid host brings obstacles absent from vapor experiments: inhomogeneous broadening, in which each ion sits in a slightly different static environment so its transition frequency is shifted and the ensemble line becomes orders of magnitude wider than the single-ion (homogeneous) width; phonon- and host-induced dephasing, the decay of exactly the coherences LWI relies on; and ion-ion interactions at the concentrations needed for gain.

Sahrai, Asadpour, Eslami-Majd and Sadighi-Bonabi (2012) proposed two four-level configurations of $\mathrm{Er}^{3+}$ crystal-field levels in YAG, each written as a closed set of density-matrix equations in the rotating-wave approximation, with a weak probe field, a coherent coupling field (Rabi frequency $\Omega_c$), an incoherent pump of rate $r$ between specified levels, spontaneous decay rates $\gamma_{ij}$ between the levels, and phenomenological coherence-dephasing rates $\gamma_{ij}^{\mathrm{dph}}$. Gain or loss for the probe is read from the computed susceptibility ($\chi''<0$ means amplification). For their first configuration the numerics show probe amplification only when the probe-transition populations are inverted; for the second they find parameter windows - for example with the spontaneous decay branch $\gamma_{32}$ (the rate from level $|3\rangle$ to level $|2\rangle$) set to about $10\gamma$, an incoherent pump $r$ of about $2\gamma$ ($\gamma=\gamma_{21}$, the reference decay rate), and a moderate coupling $\Omega_c$ - in which the probe is amplified while the probe-transition population difference remains negative. The mechanism is destructive interference between probe-absorption pathways. Quantum-coherence modeling of $\mathrm{Er}^{3+}$:YAG had begun earlier with EIT studies (Xu, Dai and Jiang, 2002).

The model, however, describes a single ion with one set of transition frequencies: it does not average over an inhomogeneously broadened ensemble, its dephasing rates are phenomenological input parameters, and the required enhancement of the decay branching $\gamma_{32}$ is imposed rather than derived from any real sample. Established ideas for modifying spontaneous emission - frequency-dependent reservoirs, microwave cavities, photonic-band-gap environments - are known from the broader LWI literature and were later pursued theoretically for rare-earth ions in a one-dimensional photonic crystal (Kamkar et al., 2019), but no implementation pathway for $\mathrm{Er}^{3+}$:YAG is established and no sensitivity analysis of the predicted windows against imperfections exists. Whether the predicted inversionless-gain regimes can be engineered in, and survive the realistic conditions of, an actual $\mathrm{Er}^{3+}$:YAG crystal is therefore an open experimental question, and it is the subject of this problem.

Problem Statement

Determine whether the inversionless-gain regime predicted by the four-level $\mathrm{Er}^{3+}$:YAG density-matrix models of Sahrai et al. (2012) can be realized and stabilized in an actual $\mathrm{Er}^{3+}$:YAG crystal. The question has three linked parts.

(i) Parameter engineering: can the spontaneous-emission branching (in particular the enhanced branch $\gamma_{32}$ relative to the reference decay $\gamma=\gamma_{21}$), the coherence-dephasing rates $\gamma_{ij}^{\mathrm{dph}}$, and the incoherent-pump coupling $r$ that the source models' inversionless-gain windows require be engineered in a real $\mathrm{Er}^{3+}$:YAG sample - via, e.g., frequency-dependent reservoirs, microcavities, photonic-band-gap environments, or other means (these routes are suggested examples, not mandatory) - given that in a bulk crystal these rates are set by the host and the 4f-shell dynamics rather than chosen freely?

(ii) Robustness of the inversion / inversionless boundary: quantitatively, how do inhomogeneous broadening (ensemble averaging over the ion transition-frequency distribution) and realistic crystal-host dephasing and noise shift the boundaries, found in the single-ion numerics of the source model, between inversion-based lasing and inversionless gain across the ($r$, $\gamma_{32}$, $\Omega_c$, dephasing) parameter space?

(iii) Stabilization: identify the cavity or reservoir engineering necessary to stabilize the predicted lasing-without-inversion regime in a real sample.

The problem is completely answered either by an experimental realization - inversionless probe amplification or laser oscillation in $\mathrm{Er}^{3+}$:YAG under the proposed four-level scheme, with measured probe-transition populations establishing the absence of inversion (stating the basis in which inversion is assessed), together with the parameter map under realistic operating conditions - or by a quantitative feasibility analysis using measured $\mathrm{Er}^{3+}$:YAG spectroscopic parameters, ensemble-averaged over the inhomogeneous distribution with realistic dephasing, that identifies which inversionless-gain windows survive or establishes that none do. A null result - a feasibility analysis establishing that no inversionless-gain window survives ensemble averaging - resolves part (ii) and the problem as a whole; parts (i) and (iii), which concern engineering the required rates and stabilizing a surviving regime, are required only if at least one window survives.

References

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