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Conventional lasers reach their threshold when the rate at which photons are coherently fed into the cavity mode exceeds losses and stimulated emission dominates over spontaneous emission into the lasing mode2,3. The most prominent characteristic of this threshold is a steep increase of photon flux from the laser while increasing the strength of the external pump exciting the gain medium. At the same time, intensity fluctuations are reduced11,12,16,17. These characteristics are induced by the nonlinear process of photon-stimulated emission of photons into the lasing mode. At the single-atom level, characteristics that strongly deviate from standard laser properties have been predicted11,12,16,18,19, in particular regarding the question of the existence of a threshold. For strong atom–cavity coupling, no threshold exists. In this regime, photons are coherently scattered into the cavity mode through stimulation by the vacuum field8. More importantly, strong coupling introduces a photon blockade, which suppresses cavity-mode populations with more than one photon20. As a consequence, photon-stimulated emission enabling the threshold is absent, the laser threshold disappears and non-classical radiation is emitted8,9,12,21. It was pointed out theoretically that a one-atom laser can show a threshold11,12 in an intermediate-coupling regime. Here we explore a single-ion device in this regime that shows a laser threshold for strong external pumping for which photon-stimulated processes become important, owing to the non-zero population of cavity modes with more than one photon.

The schematic experimental set-up of our ion-trap cavity quantum electrodynamics system is shown in Fig. 1a. Using frequency- and intensity-stabilized lasers, we excite a single 40Ca+ ion that is confined in a linear Paul trap at the centre of a near-concentric optical cavity22. The cavity provides a coherent coupling rate between the ion and a selected optical mode of g/2π=1.3(1) MHz, limited by the localization of the ion along the cavity standing wave. With a cavity finesse of ≈70,000 at 866 nm, the field decay rate from the cavity is given by κ/2π=54(1) kHz. A Hanbury-Brown and Twiss set-up consisting of a beamsplitter and two avalanche photodiodes is used to analyse photons leaving the cavity. It provides us with the steady-state population of photons in the cavity22, nss, and allows us to compute the second-order correlation function, g(2)(τ) (see Supplementary Information).

Figure 1: Experimental set-up and excitation scheme.
figure 1

a, Schematic experimental set-up. The ion (blue ball) is surrounded by a high-finesse cavity and positioned in an anti-node of the standing-wave cavity field. The single-ion device is excited to a meta-stable state through a Raman transition (blue drive-laser beam and cavity). A red laser provides repumping to close the excitation cycle. Photons leaving the cavity are coupled into a multimode optical fibre, split by a non-polarizing beam splitter and detected by two avalanche photodiodes (APD) in a Hanbury-Brown and Twiss set-up. Photon arrival times are recorded with ≈300 ps temporal resolution. b, Relevant level scheme of the 40Ca+ ion. A quantization axis is defined by applying a weak magnetic field, B=2.8 G, perpendicular to the cavity axis. The drive laser (Ω1) is polarized parallel to the B field: it excites transitions conserving angular momentum (Δm=0). The cavity (g) supports two linear polarizations, perpendicular and parallel to the B field. The recycling laser (Ω2) is polarized perpendicular to the B field.

Figure 1b shows a reduced three-level scheme of the 40Ca+ ion’s electronic structure, which allows us to qualitatively describe the excitation process. Using an off-resonant drive laser, we achieve an effective coupling on a vacuum-stimulated Raman transition4 between the electronic ground state S1/2 and the metastable state D3/2. A transfer from S1/2 to D3/2 deposits a photon in the cavity. The effective coupling between these levels is geffg·s1, where we have defined the drive parameter s1≡Ω1/2|Δ1|, with Ω1 denoting the Rabi frequency of the 397 nm driving laser and Δ1 its fixed detuning from the S1/2P1/2 resonance. To close the excitation cycle, an infrared recycling laser at 866 nm, with a resonant Rabi frequency Ω2 and detuning Δ2, transfers population from the D3/2 to the P1/2 state. From there, spontaneous emission projects the ion back to the S1/2 ground state at a rate (2γ1/2π)=20 MHz. The corresponding recycling rate is given by 2γr=2γ1s22 with the recycling parameter . Here, we define the effective detuning , where 2γ=2(γ1+γ2) is the total decay rate from the P1/2 state and 2 γ2/2π=1.69 MHz is the decay rate back to the D3/2 state. The total cycling rate Γtot for external pumping, that is, the rate at which one photon is added to the cavity mode, is thus given by 1/Γtot1/2geff+1/2γr. The strength of external pumping is then controlled by the drive and recycling lasers; however, note that the drive parameter s1 is kept constant in our experiments where the external pumping strength is varied only by the recycling parameter. Furthermore, the drive and recycling lasers introduce the broadenings γeff(1)γ s12 and γeff(2)γ s22, of the S1/2 and D3/2 states, respectively. The total linewidth of the Raman transition is therefore given by 2γeff=2(γeff(1)+γeff(2))=2γ(s12+s22). In this situation, an effective strong coupling is achieved for geff>(γeff,κ). In the following experiments, this condition is fulfilled for weak external pumping at a fixed detuning of the drive laser (Δ1/2π≈ 350 MHz). Note that this laser also provides weak Doppler cooling to reduce the motion of the ion in the trap22.

We carried out numerical simulations using a master-equation approach to predict measured observables, that is, the intra-cavity mean photon number, nss, the Mandel Q parameter2 and the ratio R between coherent and incoherent emission rates on the lasing transition for continuous application of all lasers. The Q parameter, Q=nss(g(2)(0)−1), quantifies intensity fluctuations in the quantum (Q<0) and classical (Q≥0) domains. The R parameter is defined as R=κ·nss/(γ2·nP), where nP is the steady-state population of the P1/2 level, which controls the amplitude of incoherent photon scattering. Experimentally, R is accessed by monitoring the fluorescence rate on the transition, which leads to nP.

In the following, we show that the ion–cavity device acts as a thresholdless laser when operated at the onset of the strong-coupling regime, that is, for geffγeff(1). Raman resonance between selected S1/2 and D3/2 states is established at low recycling rates by adjusting the drive-laser detuning22. In this situation, the strength of external pumping is given by the recycling-laser parameter, s2, which is held at a fixed detuning Δ2/(2π)=−20 MHz.

Figure 2 shows the measured steady-state photon number nss and the Q and R parameters as a function of external pumping together with simulation results (see Supplementary Information) for continuous laser excitation of the trapped ion at the boundary of the strong-coupling regime (geffγeff(1)). The following three regions are identified. At low recycling parameters (region (I) of Fig. 2), we observe the signature of a thresholdless single-atom laser. It is marked by an increase of the intra-cavity mean photon number, nss, whereas intensity fluctuations, as indicated by the Q parameter, are reduced. Lasing occurs in the quantum domain (Q=−2.3(4)% for Ω2/(2π)≈4.6 MHz) because single photons are stimulated into the cavity mode through the vacuum field whereas spontaneous emission is reduced (R≈0.70.8). In region (II) of Fig. 2, the laser starts losing its quantum properties as it approaches the classical domain (Q≥0), whereas the intra-cavity mean photon number reaches a maximum. For even stronger pumping, the recycling laser introduces phase noise, leading to an increase of the Q parameter (region (III) of Fig. 2). At the same time, it imposes a Stark shift of the D3/2 levels, effectively reducing the atom–cavity coupling and therefore the mean photon number. This is a manifestation of the theoretically predicted self-quenching of single-atom lasers9,12.

Figure 2: Thresholdless single-atom laser.
figure 2

Statistical behaviour of the single-ion device at the boundary of the strong-coupling regime. Shown are the mean intra-cavity photon number, nss, the Mandel Q parameter and the ratio R between coherent and incoherent scattering. Region (I) shows thresholdless lasing of the single-ion device, marked by decreasing intensity fluctuations (Q) and increasing photon flux (nss) as a function of external pumping, whereas R is of the order of one. In regions (II) and (III), the laser turns off (self-quenching). The drive-laser parameters are (Ω1, Δ1)=2π·(88,−350) MHz, leading to geff/(2π)=165 kHzγeff(1)/(2π)=170 kHz. The recycling rate is varied by the Rabi frequency, Ω2, for a constant detuning Δ2/2π≈−20 MHz. Experimental data are obtained with the same trapped ion continuously probed over 12 h. The solid lines show the results of numerical simulations using independently calibrated experimental parameters. The error bars are derived from calibration uncertainties of the corresponding quantities (see Supplementary Information).

In Fig. 3a–c, we have investigated the pronounced change in the statistics of the emitted light observed in regions (I)–(III) of Fig. 2 in more detail for a slightly different (albeit compatible) set of experimental parameters. Figure 3a shows the measured and calculated normalized correlation functions g(2)(τ) for a small total cycling rate Γtot/2π100 kHz. The slow recycling process introduces a time gap between successive photon emissions and sub-Poissonian light (Q−1.2(1)%) is emitted11,23. Different timescales are apparent in Fig. 3a. The decrease of the correlation function for |τ|<1.5 μs reveals the finite storage time for cavity photons. For longer times τ, the g(2) function approaches unity on a timescale given by the total cycling rate Γtot. Finally, for long time intervals τ, a modulation of the correlation function is observed, which can be attributed to the beat note between the ion’s two quasi-degenerate radial modes of motion in the trap24. As we slightly increase the recycling parameter, the photon flux remains almost constant, but the statistical properties of the emitted photons change markedly: the correlation function becomes almost flat (Q=0.0(1)% in Fig. 3b), akin to a Poissonian source of light. At even higher recycling parameters, the output statistics resembles that of a thermal source (Q=+1.0(1)% in Fig. 3c). In general, the measured coincident photon correlations, g(2)(0), are governed by the population of the cavity Fock states with two photons, p2, so that g(2)(0)2p2/p12, where p1 is the population of states with only one photon. The population p2 is established through an interplay between the rate Γtot at which photons are added to the cavity and the rate 2κ at which the photon state leaves the cavity (see Supplementary Table S2). The tunability of this population p2 and thus the statistical properties of the emitted light is a feature of our system, which can be adjusted to operate in the strong- or weak-coupling regimes, as in the following where we demonstrate a single-atom laser showing threshold behaviour.

Figure 3: Tunable photon statistics.
figure 3

ac, Normalized second-order photon correlations, g(2)(τ), for different photon generation rates. Increasing the photon generation rate leads to a transition from sub-Poissonian (a) through almost Poissonian (b) to super-Poissonian, bunched (c) light emitted by the cavity. The peak observed around τ=0 μs is a feature of the intermediate-coupling regime and stems from the accumulation of photons inside the cavity. The drive-laser excitation is such that (Ω11)≈2π(95,−400) MHz and the recycling laser is set at (Ω22)≈2π(7,−20) MHz in a, at (Ω22)≈2π(12,−20) MHz in b and at (Ω22)≈2π(16,−20) MHz in c. In every measurement, the cavity detuning is Δc/2π≈−400 MHz, the time resolution is set at 500 ns and accidental correlations have been subtracted (see Supplementary Information). The solid lines show the results of numerical simulations using independently calibrated experimental parameters.

In an experiment with different parameters, we studied the single-ion device for an even larger number of photons in the cavity. This was achieved by increasing the drive-laser intensity to Ω1=2π·130 MHz. Experimental results are shown in Fig. 4, where we identify the following regimes. For low recycling rates, that is, in region (I), the photon number and Q parameter both rise, indicating classical non-lasing behaviour. Increasing the external pumping further (region (II)), nss still grows whereas the Q parameter has passed through a maximum and decreases. Indeed, we note a 17% increase of the intra-cavity photon number together with a reduction of 33% in intensity fluctuations. At the same time, the rate of coherent versus incoherent emission on the lasing transition indicated by the ratio R is of the order of unity. These are the signatures for the onset of a laser threshold11,12. As for a conventional laser, the threshold is reached when the population of photons in the cavity exceeds a critical value for which photon-stimulated emission compensates incoherent processes. Most notably, this critical value is less than one for our one-atom device. Furthermore, we note that the variation of nss at threshold is less pronounced than for classical lasers. This is a consequence of the low photon number in the lasing mode, as theoretically predicted9,11,12,16,17, caused by a saturation of the rate of external pumping (Γtot) in region (II). Furthermore, we observe in region (III) quenching of the laser output. Unlike the experiments shown in Fig. 2, this is not due to a level Stark shift because the Raman detuning is chosen such that resonant coupling is established at threshold. Rather, numerical simulations indicate that it is due to the formation of dark states by coherent coupling by means of Ω2, which disables the recycling process. More precisely, the induced dressed states, generally composed of two D3/2 magnetic sub-states and one P1/2 sub-level, lose their P1/2 contribution in region (III), where Ω2 is large. This renders the external pumping inefficient, hence yielding self-quenching, which is a striking difference compared with conventional lasers. In fact, it constitutes an intrinsic limitation of single-atom lasers. In Fig. 4, we also present theoretical simulations, which agree qualitatively with the experimental data. We attribute the discrepancy to the non-zero temperature of the ion, which modifies the excitation spectrum for the parameter range of the experiments shown in Fig. 4 and is not included in the simulations (see Supplementary Information).

Figure 4: Single-atom laser with threshold behaviour.
figure 4

a,b, Comparison between experimental observations (a) and theoretical predictions (b) for the statistical response of the single-ion device operating as a laser with threshold behaviour. Shown are the mean intra-cavity photon number, nss, the Mandel Q parameter and the ratio R between coherent and incoherent scattering. Intensity (nss) and intensity fluctuations (Q) rise as a function of external pumping in region (I), indicating classical behaviour. Signatures of a laser at threshold are observed in region (II) where Q has passed through a maximum whereas nss is still rising and R is of the order of one. Self-quenching turns the laser off in region (III). The drive-laser parameters are (Ω11)=2π(130,−350) MHz, resulting in a geff240 kHz. The recycling laser has a fixed detuning Δ2/2π≈−20 MHz. Experimental data are obtained with the same trapped ion continuously probed over 12 h. The dashed lines are a guide to the eye. The error bars are derived from calibration uncertainties of the corresponding quantities (see Supplementary Information).

We have demonstrated that a laser threshold can be achieved at the single-atom level, although much less pronounced compared with classical lasers9,11,12,16,17 as a consequence of the low photon number in the lasing mode. Another striking difference between one-atom and classical lasers is the ability to tune the photon emission statistics from quantum to classical. Such a light source offers perspectives to study new types of spectroscopy, based on narrow-band optical excitation with quantum/classical statistical distributions25. Our trapped-ion laser is ideally suited to further investigate the transition between quantum and classical lasers through controlled addition of more and more ions interacting with the light field.