1. Introduction

Semiconductor lasers have become an invaluable tool for the preparation, coherent manipulation, and spectroscopy of trapped atoms and ions [1–8], with applications in quantum computing [9,10] and quantum simulation [11–13] as well as in quantum sensing and metrology [14–17]. Their ever-expanding range of directly accessible wavelengths along with their overall ease of use (setup, maintenance, spatial stability of output, compactness, energy efficiency, and small footprint) have firmly established their use in atomic physics experiments. For these lasers, however, the spectral noise properties determine the quality of coherent control of qubits [18]. Commercial distributed Bragg reflector (DBR) and distributed feedback (DFB) lasers exhibit linewidths of a few MHz [6], while extended-cavity diode lasers (ECDLs) achieve linewidths of below $100\,$kHz [19]. Laser linewidths below $1\,$Hz have been achieved by stabilizing the frequency of diode lasers to high-finesse cavities [5,20–26].

One of the major drawbacks of such semiconductor lasers is their significant noise at Fourier frequencies above a few $100\,$kHz [27]. Broadband amplified spontaneous emission (ASE) can be a limit in Doppler cooling [28]. It causes shifts in optical lattice clocks [29] by off-resonantly coupling ground and excited clock states to other levels and it deteriorates the performance of Ramsey-Bordé atom interferometers [30]. Phase noise at frequencies that match the motional frequencies of trapped atoms (100s of kHz up to several MHz) or nanoparticles [31] or the mechanical resonances of optomechanical oscillators [32] limit the sideband cooling performance. The fidelity of laser-based entangling gates on trapped ions is similarly limited by the laser’s noise spectrum incoherently coupling to the atomic qubit [18,33–36]. It is this specific issue that we will focus on here.

One approach to overcoming this limitation is to use solid-state ring lasers (e.g., titanium-doped-sapphire, Ti:Sa) with intrinsically lower phase noise at these Fourier frequencies and practically no ASE [29] due to their lower round trip gain and the longer excited-state lifetime of the gain medium. These advantages typically come at a greater system cost and footprint in addition to more frequent maintenance. A performance of similar quality can be achieved using fiber lasers, which, however, are only available at comparatively few wavelengths such as around 1.064 $\mu$m and 1.5 $\mu$m.

Cavity-stabilized diode lasers exhibit so-called "servo bumps" around the unity gain bandwidth of the servo loop. In some cases feedback bandwidths of several megahertz have been achieved, thus pushing the servo bumps well above 1 MHz [5,37], where they still limit applications that are sensitive in this frequency regime.

Linewidth narrowing down to a few kHz has been achieved by employing feedback from an optical cavity [38–44]. Phase noise reduction of up to 20 dB for Fourier frequencies up to 20 MHz compared to a laser in Littrow configuration has been demonstrated by embedding a semiconductor gain medium in a ring cavity of 2 m length [45]. Optical feedback from an external fiber cavity to a DBR laser has been shown to reduce the linewidth down to 300 Hz with a noise suppression of 33 dB for frequencies $>100$ kHz [46].

High-finesse optical filter cavities with subsequent amplification of the transmitted laser radiation have been employed to further reduce high-frequency noise [30,33,42,47,48]. In an alternative approach, the beat signal between the cavity-filtered light and the cavity-stabilized laser is employed to reduce the servo bumps through a high-speed electro-optical feedforward loop [49]. Often, the cavity serves to simultaneously stabilize the frequency of the laser. Amplification of the usually weak ($<1$ mW) transmitted beam is accomplished in a two-stage process by first injecting a laser diode, which then seeds a tapered amplifier.

In this work, we demonstrate self-injection locking [50–56] of a Fabry-Pérot diode to the low-pass filtered transmission of a linear high-finesse cavity. This self-injection locked laser (SILL) inherits the frequency stability of the cavity for Fourier frequencies below the low-pass corner frequency of the optical cavity, while above it the system exhibits the same phase noise as a laser injection locked by the low-pass filtered output of the cavity. In this way, servo bumps in the critical frequency regime starting at around 10 kHz are avoided and low phase noise is achieved all the way from dc to high Fourier frequencies. There, the performance is limited by the inherent phase noise of the source diode. The system can be conveniently upgraded from an existing semiconductor laser that is locked to a linear cavity. In section 2. we describe the laser and the apparatus, while in section 3. we describe how to translate the ion excitation directly into phase noise. Section 4. presents the excitation spectra obtained by the laser along with the resulting phase noise estimates.

2. Experimental setup

2.1 Self-injection locked laser system

The injection setup (see Fig. 1) consists of a Fabry-Pérot (Thorlabs HL7302MG) laser diode (FPLD) emitting at 729 nm, which is seeded through the rejection port of a Faraday isolator by the beam transmitted through a high-finesse cavity. The fundamental feedback mechanism is optical injection of light transmitted by the high-finesse cavity. Temperature and current are held constant and can be adjusted to achieve self-injection lasing at different external cavity longitudinal modes. The frequency is not freely tunable but is set by these modes. The self-injection process is self-starting through the large linewidth of the free-running FPLD.

 

Fig. 1. Laser setup for self-injection locking. The light for the laser is emitted from a Fabry-Perot laser diode (FPLD) and sent via a path length modulation piezo mirror, a grating filter, a fiber, and an electro-optic modulator (EOM) for Pound-Drever-Hall (PDH) locking to the high-finesse optical cavity. The cavity’s transmission is sent back to the FPLD and injects the laser via the rejection port of the laser’s Faraday-isolator (FI). A fraction of the laser output is split off for seeding an antireflection coated laser diode (ARLD)/tapered amplifier (TA) combination after frequency shifting by a double-pass acousto-optic modulator (AOM). The unshifted part of the output is sent to a wavemeter for monitoring. Further abbreviations used here: PD for photodiode, PBS for polarizing beamsplitter, and $\lambda /4$ and $\lambda /2$ for waveplates. The component library [57] was used to create this graphic.

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Since the cavity and the laser are located on different breadboards, we use an optical fiber to guide the light between the input and output of the cavity and the FPLD. The use of fiber is not essential to the setup, but rather adds flexibility and convenience. A piezo mirror between the laser diode and the cavity entrance is used to stabilize the path length to the cavity. Since the SILL-frequency is fixed, this is necessary to allow constructive interference between light inside the high-finesse cavity and the FPLD-output, and thus ensures that the laser remains locked rather than drifting in and out of lock. The lock is implemented as a PDH-type-stabilization [58–60]. The output of the laser diode is spectrally filtered by a ruled grating (Thorlabs GR13-1208). The fiber after the grating to the cavity input is used as a spatial filter. Fiber noise between cavity output and FPLD seed input gets imprinted directly onto the laser output as additional excess phase noise but does not interfere with self injection locking as it is also canceled by the piezo stabilization. This small and low-frequency additional noise is not relevant for our application and can be mitigated by stabilization techniques [61].

Note that the ruled grating does not ensure or improve the operation of the self-injection lock, but rather helps to select the correct cavity mode for the laser to operate on. In principle any mode of the cavity inside the FPLD gain profile can begin to oscillate. With a gain medium spectral width of >1 THz, several of the internal cavity modes of the FPLD (spacing ca. 200 GHz) can start to lase. The grating filter with 170 GHz bandwidth (FWHM) selects one of these modes, which can be brought into resonance with the desired longitudinal mode of the high-finesse cavity. Similar to the situation in an ECDL, this combination of frequency-selective elements selects all but one mode and suppresses mode jumps to neighboring FPLD cavity modes caused, for example, by large external disturbances or the length stabilization piezo (see Fig. 1) reaching the end of its travel and thus having to reset to the middle of its stroke.

The employed cavity has a length of $10~$cm and a finesse of $\mathcal {F}=11013(2)$, resulting in a full width half maximum linewidth of $\delta \nu \approx 140~$kHz. It transmits approximately $10~\mu$W at an input power of $800~\mu$W. Just before the cavity we introduce a phase modulation using an EOM (QUBIG, $12.7~$MHz).

In order to produce enough optical power for the subsequent use of ion excitation, the SILL-output is sent to injection seed an anti-reflection coated laser diode (ARLD), and subsequently amplified in a tapered amplifier module.

A comparison laser system is an external cavity diode laser (ECDL, Toptica TA-PRO), which is locked via conventional PDH-lock to the high-finesse cavity also used above. We modulate the laser diode current (ECDL piezo) for fast (slow) feedback to the laser frequency. For details we refer the reader to [62,63].

2.2 Ion trap apparatus

We use the electronic excitation of a single trapped ion to determine an upper limit for laser phase noise as a function of laser detuning from the narrow $^2\mathrm {S}_{1/2}$-$^2\mathrm {D}_{5/2}$ resonance in $\textrm {Ca}^+$ [36]. More detailed information about the spectral noise, especially at low Fourier frequencies, can be gained by employing a dynamical decoupling sequence [64]. The ion trap setup is described in detail in [62,63]. In brief, we trap ions in a linear segmented Paul trap [65] assembled from a stack of four laser-structured wafers made from Rogers$^{\mathrm {TM}}~$4350 material with electrodes comprised of a gold coating.

The trapping frequencies along the three principal axes of motion in the trap are $(\omega _{x},\, \omega _{y},\, \omega _{z}) = 2\pi \times (1.34(5),\, 1.10(5),\, 1.26(2))~$MHz, utilizing a drive frequency of $\Omega _{r\!f} = 2\pi \times 33.0(1)~$MHz. Doppler cooling followed by electromagnetically induced transparency (EIT) cooling [66,67] is used to attain an average phonon occupation of $(\bar {n}_x, \bar {n}_y, \bar {n}_z) \approx (0.5,\,0.5,\,0.5)$.

A single ion is irradiated with light from the laser system either in regular ECDL/PDH configuration (see [62]) or in self-injection configuration for $200~\mu$s $\dots \,50~$ms. With these long probe times, fast laser noise can induce incoherent spin flips. The laser beam has an optical power of $\approx 5~$mW at the ion. The excitation of the ion is read out via standard fluorescence detection using a detection time of $300~\mu$s. We scan the laser frequency from $-5\dots 0.2~$MHz relative to the resonance of the 4$^2\mathrm {S}_{1/2}\;m_J = -1/2 \rightarrow$ 3$^2\mathrm {D}_{5/2}\;m_J = -1/2$ transition in Ca$^+$ to determine the phase noise spectrum of the laser. Typical on-resonance Rabi frequencies for this transition given our system parameters range from $180\dots 300~$kHz.

3. Phase noise characterization

In the following we show how the atomic excitation can be converted to laser phase noise. Following [68–70], the time evolution of a two-level atom subject to a noisy field can be described as

Given that the phase noise of the laser is not white but rather has a slowly varying frequency dependence, we will make the approximation that the noise is approximately white where it is near-resonant with the carrier, i.e., around the detuning $\Delta$ (only here we have slowly rotating terms that have a significant effect), by replacing $N_0/P_0$ with $10^{\frac {\mathcal {L}(\Delta )}{10~\mathrm {dBc}}\mathrm {Hz}}/\mathrm {Hz}$ in Eq. (1), where $\mathcal {L}(\Delta )$ is the single sideband phase noise in units of dBc/Hz. We have verified by numerical integration that, to a good approximation, a fully polarized input state (ground state population probability $p_g(0)=1$, excited state population probability $p_e(0)=0$) yields the following evolution of $p_e$ over total probe time $t$ of Eq. (1):

The fundamental detection noise floor of this method is given by the probability of falsely detecting an excited state, when the laser noise would not excite the atom. The detection noise floor can thus be adapted to the quality of the laser under investigation.

4. Experimental results

Our first step was to verify the model developed in the previous section for deriving the noise spectral density from atomic excitation. For this we performed a beat measurement of the direct ECDL output and its low-pass filtered ($\approx 68$ kHz corner frequency) transmission output from the cavity [71] after amplification through injection locking of another laser diode. The results are shown in Fig. 2.

 

Fig. 2. Comparison between ECDL phase noise derived from atomic excitation and a beat measurement. The ECDL is locked via the PDH technique to the high-finesse cavity in both cases. Phase noise derived from atomic excitation is shown by the gray dots, while the solid blue line indicates the beat signal between the output of the ECDL, with the transmission of the cavity amplified by injection locking another laser diode. A two-point moving average filter was used to reduced measurement uncertainty. One datapoint (noted by a grey star yielded zero excitation and can not be displayed on this graph.)

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Beyond Fourier frequencies of 200 kHz, the phase noise of the transmitted light through the cavity is suppressed by more than 10 dB and therefore has negligible effect on the beat measurement. The injection locking is expected to limit noise performance at a broadband white-phase noise floor of approximately −115(2)dBc/Hz (see Fig. 4), which is extracted from the high-frequency performance of the SILL. Figure 2 shows good agreement between the inferred noise spectral density of the ECDL locked to the cavity from the beat signal and the result from ion excitation, thus confirming the noise excitation model.

In the next step, the regular ECDL/TA laser was compared with the SILL setup by measuring the respective excitation spectrum from a single trapped ion. The result, shown in Fig. 3, reveals a significant improvement of laser phase noise when employing the self-injection locking, especially considering that the effective probe time for the SILL is ten times longer than that of the ECDL/TA system and that the SILL’s Rabi frequency of $254\,$kHz is greater than the $190\,$kHz seen for the ECDL/TA system.

 

Fig. 3. Comparative excitation spectra for ECDL (gray, dotted) and SILL (blue, solid). The ECDL is PDH-locked to a high-finesse cavity, while for the SILL only the path length to the cavity is stabilized. The additional off-resonant excitation for the ECDL is clearly visible, with the shape of the noise being a consequence of the servo noise and locking bandwidth. The vertical dashed lines indicate motional sideband peaks of the ion that are accessible due to imperfect laser cooling and to neighboring transitions. The total probe time for the SILL case (and thus the measurement sensitivity) was increase in order to have a non-vanishing signal. The resulting power spectral densities derived from the signals are shown in Fig. 4.

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In Fig. 4 we compare the noise spectral density of the ECDL with the SILL for different incoherent probe times. We observe a maximum noise suppression of the SILL compared to the ECDL of up to 30 dB at a Fourier frequency of $1.5$ MHz and of more than 20 dB over the entire range from 0.1 to 6 MHz. We used probe times of $50\,$ms and $200\,\mu$s for the SILL and ECDL, respectively. The observed high-frequency noise floor is at the lower limit of what we estimate to be achievable with the Fabry-Perot laser diode we used to build the SILL [72]. The estimated fundamental noise floor of the data shown in Fig. 4, as discussed above in section 3, is calculated to be $-129~$dBc for the ECDL data and $-144~$dBc for the SILL data, by using an independent estimation of the respective false-excited state detection probabilities of $1(1)\times 10^{-5}$ (ECDL) and $1.2(3)\times 10^{-4}$ (SILL) respectively.

 

Fig. 4. Phase noise calculated from incoherent excitation for the SILL (blue crosses) and the ECDL (orange dots). The data is the identical to Fig. 2, where the star again marks the singular zero-excitation event. The dashed vertical lines indicate motional sideband peaks of the ion that are excited due to imperfect laser cooling, and due to neighboring transitions. The green solid line shows the estimated performance of injection locking a slave laser diode (ILL) to the light transmitted through the cavity. The line is calculated from the product of the measured ECDL noise and the noise-low pass filtering of the high-finesse resonator at $68~$kHz corner frequency, and offset by a white phase noise plateau discussed in section 5. The white phase noise plateau is estimated by the flat portion of the SILL spectrum between 5 MHz and 6 MHz. The shading indicates the uncertainty of the noise-plateau estimation.

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5. Discussion and summary

In the following we discuss the experimental results in the context of theoretical models for optical injection locking. The treatment will focus on the comparison case, where a primary master laser is PDH-locked to a high-finesse cavity, and the transmission of this resonator is then used to injection lock a secondary laser diode (injection locked laser, ILL) [30,33,47,48].

The noise suppression factor $\eta ^2$ is the ratio between the frequency noise of the free-running FPLD and a SILL. It qualitatively takes the form [72,73]:

The frequency-dependent noise floor of an ILL is given by approximately by [74,75]:

For identical feedback parameters, the resulting white phase noise plateau is identical for SILL and ILL, as it stems from fundamentally the same process. The important point of comparing SILL and ILL is thus, at which Fourier frequency this plateau is reached.

For the SILL it is simply the corner frequency of the high-finesse cavity. For the ILL it depends on the details of the feedback loop of the PDH locking (emergence of so-called servo-bumps that can be seen in Fig. 3) as well as its relationship to the high-finesse cavities linewidth. To provide a comparison between SILL and ILL, we have in Fig. 4 taken the known noise of our laser locked via the PDH-method, multiplied it by the transmission function of the cavity, and added the noise which the injection lock process would yield, according to Eq. (4). The value of the white phase-noise plateau was estimated using the data of the SILL. The result is the green line in the graph, and it can be seen that for our case, the SILL is predicted to have superior performance from $\sim 100~$kHz until $\sim 2~$MHz. Resonators with a smaller linewidth, or laser setups which achieve much faster servo-speeds will exhibit a different performance.

For very low Fourier frequencies, the cavity instability itself as well as other noise sources will limit the achievable performance for both cases.

A second interesting trade-off is the consideration that the ultimate level of white phase noise is a function of power transmitted through the cavity (see Eq. (3)). The more power that is transmitted through the cavity, the lower will be the white phase noise plateau, while the achievable linewidth in a laser lock to the cavity is inversely proportional to its finesse. Since finesse and transmission are typically anti-correlated, there is a trade-off between high-frequency noise suppression and achievable laser linewidth. In our specific system, the linewidth of the SILL is further narrowed by transfer locking [76] it to a narrow-linewidth reference laser [77]. We have recently set up a new cavity with higher transmission, which will enable more stable injection locking and further noise suppression [78].

In summary, we have developed and characterized a SILL using the transmission of a high-finesse cavity. A theoretical framework for converting phase noise into incoherent electronic excitation has been developed that can be used to judge the suitability of a laser system for sideband operations. We observe a suppression of laser phase noise by more than 30 dB compared to a conventionally cavity-stabilized ECDL, approaching the theoretical limit set by the white frequency noise of the utilized laser diode at −115(2) dBc/Hz. Compared to an ILL, the noise suppression may be stronger for Fourier frequencies below the cavity linewidth due to the strong optical feedback. It is noteworthy that the demonstrated level of phase noise enables two-qubit operations with an error of below $10^{-4}$ in trapped ion quantum computing using qubits based on a narrow optical transition [36].