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Abstract
Doppler-free spectroscopy of 40Ca+ on the transition 3D3/2 → 4P1/2 known as the frequency standard for repumping beam of Calcium ion trap was performed by means of optogalvanic detection. This reference signal was applied to measure the frequency stability of laser locked to the resonance of an ultra-low expansion (ULE) glass made cavity. Lamb dip spectrum fitting of this Calcium ion spectra revealed that the long-term drift of our laser system is below 2 MHz per hour. A simple setup for frequency locking of dual colour of lasers at 866 nm and 780 nm was also demonstrated. Consistencies of the frequency difference between these two lasers were measured less than 2 MHz in a hour after stabilizing both lasers to the cavity.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Laser-cooled and trapped single or many ions nowadays is a magnificent experimental platform for pioneering researches, such as quantum computer [1,2], quantum sensing [3], and a potential candidate for primary clock [4]. These remarkable progresses of ion trap have provided approaches for academy to span from fundamental to applied physics. All of the topics mentioned above, the Ca$^+$ ion traps are participated.
Laser cooling of trapped ions requires stable multiple-laser frequency technologies [5–12] spanned from UV to NIR. The lasers used for establishing an ion trap are usually stabilized their frequencies via optical feedback from a stable reference cavity. The reference cavity stabilization provides an excellent "relative" stability. However, the absolute frequency and the long-term stability can only be realized using a stable absolute frequency reference. In the case of the ion trap laser cooling, two solutions are widely implemented: the wavemeter with MHz precision or the Doppler-free spectroscopy of atoms, molecules or ions. The wavemeter has two major weaknesses: first, it needs to be calibrated the measured frequencies with an atomic transition or absolute frequency standard periodically [13]. Second, performance of the frequency locking could be limited by grades of the wavemeter which directly determine the absolute frequency accuracy and feedback speed [14].
In this paper, we developed a multiple-laser stabilization system for Ca$^+$ ion trap based on an optical cavity (short-term stability) and Doppler-free optogalvanic (OG) spectroscopy [15]. Nonclosed 3D$_{3/2}$ $\rightarrow$ 4P$_{1/2}$ transition of $^{40}$Ca$^+$, one of the transitions for Doppler cooling of Calcium ions [16–18], was used to be an absolute frequency reference for all the other lasers within various wavelengths in the cooling system. A low-cost home-built optical cavity with sufficient stability is then used for the short-term stability. To show the versatile and the various applications of this experimental scheme in Ca$^+$ ion trap, we then demonstrate the simultaneous stabilization of two-laser system and characterize their absolute and relative stability.
2. Doppler-free spectroscopy of Calcium ion in hollow-cathode lamp (HCL)
HCL is a handy approach to provide a gaseous sample of the low vapor pressure substances without a delicate and power consuming high temperature oven. Meanwhile, it also generates certain amount of ionic sample, despite with very low number density, and has been widely used as a frequency reference for various ionic spectroscopy and ion trap. However, due to the low number density and the complicate mixture, which contains neutral atoms, various ions, and buffer gases, most of HCL spectroscopies are Doppler broadened and have reference linewidths of GHz. It is then challenging to obtain a Doppler-free spectrum for precision calibration with a MHz accuracy.
To our knowledge, particularly for ions, there are few successful demonstrations of Doppler-free HCL spectroscopies on only two species: Yb+ [19], Ca+ [20]. Both of them are in the regime of blue or UV. A stable ion-based quantum information machine requires a long term and reliable frequency reference for laser stabilization. A stable laser system built upon the Doppler-free HCL spectroscopy is one of the best solutions. Our experiment demonstrated another ionic calcium Doppler-free reference line in the near infrared regime with the 3D$_{3/2}$ metastable state as the lower level.
In hot cells, atomic sources with low vapor pressure usually need the cell to be heated to hundreds or thousands degree Celsius. And for preparing the ionic samples, the photo-ionization or electron discharge are needed. Different species of HCLs can efficiently provide sufficient vapors of atomic and ionic sources, and which have widely applied to the spectroscopies of metal vapors with low vapor pressures at room temperature, such as Cd$^+$ [21], Sr [22], Yb [23], Fe [24], Tl [25], Al [26] etc. In past applications, high-resolution spectroscopies of Yb$^+$ ions in HCL were used to stabilize an ultraviolet diode laser at 369 nm via its Doppler-free spectroscopy [19]. Furthermore, OG profiles of Ca$^+$ ions in HCL were combined with a Fabry-Pérot interferometer to reduce the long-term drift of lasers at 379 nm and 866 nm, and used them for the single Calcium ion confinement [27]. In this paper, our group demonstrate the Doppler-free spectroscopy of Ca$^+$ ions in a Calcium-HCL. Similar to the Yb$^+$ ions in [19], this Doppler-free ion spectroscopy could also be an ideal absolute frequency reference for Ca$^+$ ion cooling and trapping experiment and should be capable of deriving the differential error signals for frequency locking of the lasers.
To achieve the Doppler-free saturation spectroscopy with a low ionic number density, several approaches have been employed. Firstly, the laser should be low power noise and low frequency noise. An ECDL based on a monolithic cavity was used. Secondly, we utilized the optogalvanic technique to enhance the sensitive. And, in order to reduce 1/f noise, we operated the modulation at a high frequency 30 kHz using an acoustic-optical modulator (AOM).
In our experiment, absolute frequency between 3D$_{3/2}$ and 4P$_{1/2}$ of $^{40}$Ca$^+$ [28] was obtained from a Doppler-free OG spectroscopy [29,30] in a see-through HCL (Photron, P809ST). OG signal is highly sensitive in detecting changes of atomic population induced by light absorption, thus could be applied for Doppler-free spectroscopy [15,31,32]. Additionally, in the HCL, the non-optical OG detection is a better technique for detecting very small levels of optical pumping signature than the fluorescence and transmission detection [33]. In the HCL, the cathode surface was coated with Calcium, and buffer gas of neon at pressure of 10 torr was filled in the lamp. The experiment setup of the Doppler-free OG spectroscopy is illustrated in Fig. 1(a). The OG signal was generated by the excess conductivity of the neon plasma as the frequency of laser beam was on resonance with the Ca$^+$. Subsequently, the voltage across the 10 kΩ ballast resistor will be increased slightly. Here, the laser at 866 nm was chopped by an AOM and so was the voltage drop across the resistor. The laser was a homemade Littrow configuration external cavity diode laser (ECDL) [34–37] installed a 860 nm laser diode (Eagleyard Photonics, EYP-RWE-0860-06010-1500-SOT02-0000). The design of this ECDL is similar to that described in [38,39] such that the output beam direction is fixed when tuning the wavelength. Center wavelength of the ECDL was set to 866 nm with beam diameter 0.98 mm and output power 25 mW in typical operation condition (temperature 18 °C and current 80 mA). The voltage signal after a 0.22 $\mu$F capacitor was then demodulated by a lock-in amplifier (Stanford Research System SR830) to acquire the OG spectrum. For observation of the saturation Lamb dip signal as shown in Fig. 1(c), a mirror is installed at the exit of the HCL window to reflect the radiation which has been absorbed by Ca$^+$ in the HCL back into the cell. This counter-propagating arrangement can yield a saturation dip at the center of the absorption spectrum since only those ions within zero velocity parallel to the beam can absorb both of the radiation.
Fig. 1. (a) Experimental setup for probing the Ca$^{+}$ in the HCL. (b) Simplified energy level scheme of Ca$^+$. (c) Blue: Doppler-free OG spectra at 866 nm. Lamb dip on top of the spectrum is resolved. Red: fitting curve.
Figure 1(c) depicts the Doppler-free spectra by sweeping the frequency of ECDL over 3D$_{3/2}$ $\rightarrow$ 4P$_{1/2}$ transition of Ca$^+$ ion at 866 nm. With this Doppler-free spectra, the third derivative locking methods could be applied for the frequency stability of the ECDL at 866 nm [40]. Full width at half maximum (FWHM) of the absorption spectra resulting from the fitting is about 0.96 GHz and the Lamb dip FWHM is about 240 MHz. The center frequency of this transition can be measured by fitting the curve of Lamb dip. According to this graph, although the signal to noise (S/N) ratio in the full data range (0 to 2) is above 200, the S/N ratio in the dip data range (1.96 to 2) is only about 4 to 5. Therefore, to reduce the uncertainty of fitting the Lamb dip at the peak of Doppler profile caused by the low S/N ratio and the broaden spectral line shape, we will perform the data binning method to acquire the center frequency of the the Lamb dip in the later section 5, and use it as a precise frequency reference.
3. ULE reference cavity and frequency locking with extended PDH method
This section describes the design of our Fabry-Pérot optical cavity. The critical part of our optical cavity is an ULE glass made (Corning 7972) hollow cylinder spacer with the dimensions: 103 mm long, 27 mm outside diameter, and 17 mm inside diameter. A hole of diameter 1.5 mm was drilled through the wall of spacer for pressure vent. A custom made plano-concave fused silica mirror (radius of curvature 1000 mm) and a plano mirror were stuck on both ends of the spacer. Both mirrors have high-reflective coatings with reflectivity 99.5$\pm$0.15$\%$ (740-860$\pm$8 nm) on the side toward the spacer and anti-reflective coating of reflectivity less than 0.25$\%$ (700-900 nm) on the other sides. The mirrors were glued on the sidewall of spacer by UV curable adhesives (Norland Optical Adhesive, NOA 61). The optical cavity was hold by two aluminium containers. The upper container was fixed to a vacuum flange sealed by a viewport as illustrated in Fig. 2. Three sustained spacers made of stainless steels were installed between the upper container and the the flange to prevent the container directly touching the flange so as to reduce heat transfer into the vacuum chamber. The bottom side of the chamber was sealed by a flange and a viewport. The left port of the chamber was connected to a turbo molecular pump (TMP) by a pinch-off tube (Solid Sealing Technology, FA206CFCFC030), and the port on the right was connected to to a 2 L/s miniature ion pump (Agilent, VacIon 2 L/s pumps). First, vacuum system was pumped to about $10^{-7}$ torr by the TMP and kept it running for 30 days. Next, the tube connecting the TMP and the vacuum chamber was pinched off for chamber sealing. We turned on the ion pump and kept the pressure of the chamber below $10^{-8}$ torr. This vacuum pressure was maintained for more than 10 months by just one ion pump, and which indicates that it is almost a permanent vacuum chamber.
Fig. 2. Drawings of vacuum system and ULE optical cavity. (a) Side view of the semi-permanent vacuum chamber with optical cavity inside. The screws for fixing flanges are omitted for clarity. The left vacuum chamber port is connected to a pinch-off tube and right port to an ion pump. (b) Assembly of optical cavity and the cavity container. The optical cavity is clipped by the container, and the whole set is fixed to the flange with viewport by screws.
The temperature of the optical cavity was controlled and stabilized with a strip of insulated resistance wire wounded around the vacuum chamber. A resistance temperature sensor, Pt100, was attached on the surface of chamber to monitor the temperature of chamber. A PID temperature controller (Fotek, MT-48V) with accuracy 0.1 °C was used to control the temperature of chamber to 27 °C. Furthermore, the whole set was wrapped in aluminium foil and several layers of foam to reduce the thermal coupling from environment for better temperature stability. Schematic diagram of the experimental apparatus used for frequency cavity locking experiment is shown in Fig. 3. A portion of the light source after isolator was picked off and phase modulated by a FEOM (iXBlue, NIR-MPX800-LN-10) to perform the dual sideband (DSB) locking scheme [41]. The FEOM was driven by a composite signal that modulated the beam at two distinct frequencies $f_0$ and $f_\text {PDH}$, where $f_0$ is adjustable. This phase modulation resulted in a carrier with frequency $L_\text {c}$, two sets of sidebands with frequencies $L_\text {c} \pm f_0$ and $L_\text {c} \pm f_\text {PDH}$, and sub-sidebands $L_\text {c} +f_0 \pm f_\text {PDH}$ and $L_\text {c} -f_0 \pm f_\text {PDH}$. Spectrum of the modulating beam transmitted the cavity was detected by PD$_\text {T}$ and shown in Fig. 4. The error signal was generated referring to the standard PDH method [42,43], which multiplied the reflection signal PD$_\text {R}$ from the cavity by the $f_\text {PDH}$ signal within a RF mixer (Mini-Circuits ZP-3H+, 0.15-400 MHz). The high frequency components after the mixer were then removed by a low pass filter (Mini-Circuits BLP 1.9+, DC-1.9 MHz) and subsequently demodulated the PDH error signal as shown in Fig. 4. The frequency was locked to the PDH error signal generated at $L_\text {c} + f_0$ , and tuning of the remaining frequency components could be accomplished by adjusting $f_{0}$, and so was the frequency of laser.
Fig. 3. Experimental Setup for the modified PDH method. PBS: polarization beam splitter; QWP: quarter wave plate; LPF: low pass filter; Amp.: amplifier; PD: photo detector.
Fig. 4. PDH error signals (green) and the spectrum of laser at 866 nm transmitted through the optical cavity (purple). $f_0=600$ MHz and $f_\text {PDH}=15$ MHz. A residual PDH error signal at $\sim 0.84$ GHz was caused by higher-order cavity modes.
If the frequency between carrier and sideband is fixed, the relative stability of laser frequency is identical to the sideband frequency, and which could be measured by tracing the slope of PDH error signal crossing the zero point. After frequency locking, the error signal was recorded over a time span of 30 seconds and converted to frequency units by the slope of the error signal versus frequency as shown in the inset of Fig. 5. The relative frequency fluctuations of the locked laser are about 400 kHz. The Allan deviation [44,45] was then calculated according to the frequency deviation data. It shows that the short-term ($\sim$ 1 ms) relative stability of the laser is about 100 kHz and the long-term ($\sim$ 1 s) stability is about 3 kHz. Therefore this laser is available for measuring the long-term (larger than 1 s) frequency drift of the resonance of cavity and the accuracy will be several kHz.
Fig. 5. The Allan deviation of laser frequency relative to the cavity resonance frequency. The small box is the relative frequency deviation of the 866 nm laser by offset locking during 30 seconds.
4. Offset sideband locking of dual wavelength stabilization
This section demonstrates the frequency stabilization of two lasers at different frequencies on the same ULE cavity after merging both lasers into a single FEOM. Configuration of the dual-laser frequency locking system is shown in Fig. 6, where lasers at 866 nm and 780 nm were combined together by a fiber polarization combiner (THORLABS, PFC780A). The FEOM was driven by a signal source that combined two sinusoidal signals of frequencies $f_{1,o}=93$ MHz and $f_{2,o}=142$ MHz, both of which were frequency modulated by sine waveform at $f_{1,m}=93$ kHz and $f_{2,m}=97$ kHz respectively. The merged laser after the FEOM then generated multiple sidebands relative to the carriers $L_{1}$ (866 nm) and $L_{2}$ (780 nm) as shown in the spectrum of Fig. 7 acquired by the transmitted signal of PD$_\text {T}$. PD$_\text {R}$ detected the reflection light from the cavity followed by divided the signals into two parts by a power splitter. Each of the signals were sent into a lock-in amplifier for extracting the error signals. The error signals having $f_{1,o}$ or $f_{2,o}$ components shown in subfigure of Fig. 7 were demodulated by $f_{1,m}$ (upper) and $f_{2,m}$ (lower) separately. Two arrows indicate in the subfigure marked with $L_{2}-f_{1,o}$ and $L_{1}+f_{2,o}$ are error signals applied for frequency locking in this experiment. Marker $L_{2}-f_{1,o}$ (Marker $L_{1}+f_{2,o}$) stands for the sideband that its frequency is $-f_{1,o}$ ($+f_{2,o}$) from the carrier $L_{2}$ ($L_{1}$). The detuning of $L_{1}$ and $L_{2}$ could be completed by setting the modulation frequencies $f_{2,o}$ and $f_{1,o}$ respectively.
Fig. 6. Schematic diagram of the dual laser frequency stabilization system. Symbols in the diagram, OI: optical isolator, FEOM: fiber based electro-optic modulator, AMP: amplifier, PBS: polarization beam splitter, PD$_\text {R}$ detector sensing the reflection light, and PD$_\text {T}$ detector sensing the transmission light. Lock-in: lock-in amplifier. PID: servo loop for feedback control of piezo tube in the ECDL.
Fig. 7. Transmission signals detected by PD$_\text {T}$ placed behind the ULE cavity. Vertical lines under the spectrum mark the detected signals and classified by the wavelength, red: 780 nm and blue: 866 nm. We list and rank them by frequencies (from left to right): $L_{2}-(f_{1,o}+f_{2,o})$, $L_{2}-f_{2,o}$, $L_{2}-f_{1,o}$, $L_{2}-(f_{2,o}-f_{1,o})$, $L_{2}$, $L_{2}+(f_{2,o}-f_{1,o})$, $L_{2}+f_{1,o}$, $L_{1}-(f_{1,o}+f_{2,o})$, $L_{2}+f_{2,o}$, $L_{1}-f_{2,o}$, $L_{2}+(f_{1,o}+f_{2,o})$, $L_{1}-f_{1,o}$, $L_{1}-(f_{2,o}-f_{1,o})$, $L_{1}$, $L_{1}+(f_{2,o}-f_{1,o})$, $L_{1}+f_{1,o}$, $L_{1}+f_{2,o}$, $L_{1}+(f_{1,o}+f_{2,o})$. Subfigure is the error signals demodulated from modulation frequencies: $f_{1,m}=93$ kHz (purple) and $f_{2,m}=97$ kHz (green). Arrows point out the two error signals that we used to lock $L_{2}$ (780 nm) and $L_{1}$ (866 nm) respectively in this experiment.
Both of the absolute frequencies of the stabilized lasers were measured and recorded by a wavelength meter (HighFinesse, WSU-30) for 2 hours. In order to observe the frequency difference between $L_{1}$ and $L_{2}$ under the influence of drift length of the cavity, we turn off the temperature controller of the cavity and let the cavity length to drift with time. In Fig. 8, both lasers reduced about 20 MHz in two hours due to the change of the cavity length. The inset figure shown that the frequency differences were less than 2 MHz limited by the resolution of the wavemeter. This result implies that the relative drift of the two laser frequencies, an important quantity for STIRAP, Rydberg excitation, or EIT experiments, is much smaller than the drift of the cavity resonance.
Fig. 8. Frequency drift of two lasers measured by a wavemeter in 2 hours. Inset: frequency difference between two lasers during the measurement, where the error bar is the relative deviation of the readout frequencies.
5. Results and discussion
In section 2, our Ca-HCL contained Neon as buffer gas at a pressure of 10 torr. From the spectra in Fig. 1(c), the Lamb dip FWHM is about 240 MHz. This result of linewidth is comparable with that in [15]. The result in [15] indicated that the effect of velocity-changing collisions broaden the Doppler-free spectra and limited the resolution, and also mentioned that the effect could be greatly reduced by the choice of buffer gas. In their case, buffer gas of Krypton at 0.6 torr shown better resolution than Argon at 2.5 torr. This result also gives us some prospects for future research of improving our Doppler-free signal. The buffer gas in our Ca-HCL system was filled with relatively light Neon at higher pressure of 10 torr, this might explain the broaden dip of our spectra in Fig. 1(c) and which could result in additional uncertainty to determine the center frequency of the signal. This problem was solved by using a Gaussian function to fit the spectrum data near the peak and get the center frequency of the dip. The function we used for finding the peak frequency is given by,
where $y$ is the measured spectrum and $y_0$, $A$, $B$ and $f_\text {c}$ are the parameters to be fitted. $f_\text {c}$ is the fitted center frequency of the peak. Figure 9 shows a typical OG spectra near the Lamb dip and the Gaussian curve that fits the data. The spectra was obtained by employing the laser to scan back and forth over the dip for three times in about 36 minutes by varying the sideband offset frequency $f_\text {0}$. The spectrum was recorded in resolution of 1 MHz and therefore there were six data points for each recorded frequency. In order to reduce the uncertainty of curve fitting parameters, data binning was performed by grouping spectra data in 4 frequencies into one bin and the peak frequency could be obtained by fitting the Gaussian binning curve in Fig. 9 with Eq. (1).Fig. 9. OG spectra near the Lamb dip and Gaussian curve fitted the spectra (green line). Blue points are the raw data recorded in resolution of 1 MHz and each recorded frequencies were measured six times. Red lines are the average and standard deviation of the binning data (group of 4 frequencies). These red lines are then fitted by the Gaussian curve (green line). Horizontal axis is the frequency offset sideband f$_\text {0}$. From the curve fitting, the peak of the dip is at frequency 602.46 (0.3027) MHz.
The OG spectra of Ca$^+$ ion was obtained by scanning the offset frequency sideband $f_\text {0}$ which already be stabilized to the cavity. Therefore, drift of the frequency locking could be estimated by measuring the fitted center frequency of the peak that extracted by fitting the Gaussian binning curve of the OG signal in Fig. 9. Figure 10 recorded the fitted center frequencies of the OG spectra over six hours, which indicates that the cavity resonance frequency drift was about $\pm$1 MHz per hour. The resonance frequency of the cavity varies with the fluctuation of the cavity length dominated by the temperature control of the cavity. It was measured that the resonance frequency of the cavity will drift about 160 MHz while we changed the vacuum chamber temperature for 1 °C. From the data in Fig. 10, the resonance frequency drifted about 6 MHz during the experiment, which could be estimated that the temperature inside the optical cavity changed about 0.0375 °C. This result is consistent with the uncertainty of our temperature controller, $\pm$1 °C. This performance could afford the initial experiment of laser cooling, but the stability was worse than our expected. If the mirror surface and the cavity end surface contacts perfectly, the resonance frequency deviation induced by thermal variation should be
where $\Delta f$ is the resonance frequency deviation, $f$ is the light frequency, $\Delta L$ is the cavity length variation and $L$ is the cavity length. Since the coefficient of thermal expansion (CTE) of ULE is below 30 ppb/K (ppb = $\times 10^{-9}$) [46] at our setting temperature, where $\text {CTE} = \frac {1}{L}(\frac {\Delta L}{\Delta T})$. The laser frequency $f=346000$ GHz, so the resonance frequency deviation $\Delta f$ should be below 10.4 MHz/K. However, from our measurements, the resonance frequency variation was about 160 MHz/K and it means the cavity length variation is about 46 nm/K, much larger than the largest possible ULE variation, that is 3 nm/K. We speculate that the excess cavity length variation was come from the glue used to adhere the mirrors to both ends of the ULE spacer. The CTE of the glue we choose (Norland NOA 61) is about 220 ppm/K [47], so the total thickness variation will be 44 nm/K if the total thickness of the glue on the surface of both ends of the ULE spacer is about 200 $\mu$m. This estimation can explain the larger variation of the resonance frequency when our chamber temperature was changed. The feasible methods for us to further reduce the resonance variation of the cavity could be done by reducing glue thickness and adopting the low CTE glue, such as EP42HT-3AO [48] or EPO-TEK 353ND [49]. The CTE of EP42HT-3AO is only 10 to 20 ppm/K therefore if this glue is applied and the total glue thickness is reduced to 50 $\mu$m, the thickness variation will be less than 1 nm/K, less than the variation of the ULE spacer. Furthermore, the resonance frequency variation can be greatly reduced to below 100 kHz/K if the temperature variation of vacuum chamber is reduced to 0.01 °C or even 0.001 °C by using a better temperature controller and better thermal shielding materials.Fig. 10. The cavity long-term stability measurements was characterized by the drift of the center of Lamb dip spectra $f_\text {c}$. There was a 3 hours interruption during the measurements.
We also demonstrated two methodologies for frequency stabilization of lasers locked on the optical cavity: modified Pound-Drever-Hall and sideband modulation. Sideband modulation method can realize frequency locking of dual-channel lasers by utilizing a single photo-detector and FEOM [12] even for light sources of two different colours (in this paper, 866 nm and 780 nm). This method shows a simple and cost-effective way to lock multiple lasers. The disadvantage of this method is to recognize each error signal from the complex demodulated signals. In the future, this method could be improved by adding dichroic mirrors for the reflection light from the cavity to detect the specific wavelength range before the demodulation. The PDH method on the other hand needs corresponding number of photo-detectors and FEOMs for stabilizing multiple colour of lasers to the cavity [6,7]. That means the more lasers we intend to use, the more optical components will be required. On the contrary, issue of recognizing the error signals as the sideband modulation method is negligible.
6. Conclusions
In conclusion, we have reported a Doppler-free spectra of $^{40}\text {Ca}^+$ ions by the OG signals of a Ca-HCL. This Doppler-free spectra, in the future, could be applied on the laser frequency stabilization by the third derivative locking method and will be compatible with experiments of trapping $\text {Ca}^+$ ions. A miniature ULE cavity was used to lock the frequency of our laser by modified PDH method followed by acquiring its stability of frequency locking. The resonance drifts of the cavity were measured by statistically fitting the center frequency of the Lamb dip at the peak off a $^{40}$Ca$^+$ spectra. We also demonstrated an application of our ULE cavity that dual colour of lasers could lock their frequencies on the cavity separately at the same time, which kept the frequency difference of both lasers less than 2 MHz during the measurement. The optical setup is economic benefits and quite simple, which doesn’t need multiple FEOMs and detectors for each laser system, as in single FEOM and single detector. This application is particularly important for experiments such as two photon Rydberg atom excitation or stimulated Raman adiabatic passage (STIRAP). These experiments are under the framework of energy level configuration of two-step transition or a Lambda type scheme which require two light to keep the same frequency difference at the same time.
Funding
Ministry of Science and Technology, Taiwan (109-2112-M-007-020-MY3, 110-2811-M-007-539-MY2).
Acknowledgments
The authors acknowledge Yao-Chin Huang for useful experimental discussions. This work was financially supported by the Center for Quantum Technology from the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) and the Ministry of Science and Technology (MOST), Taiwan.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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