Amplitude modulation (AM) spectroscopy represents one of the widely exploited modulation techniques in the rapidly advancing field of microwave photonics, which merges the worlds of radio-frequency (RF) electronics and optoelectronics [1]. Modulation methods are used to convert RF signals into the optical domain and vice versa to characterize optical components such as high-Q optical filters [2,3], to perform signal processing, and to filter out undesired sidebands [4]. The basic idea of the implemented technique is to use a RF vector network analyzer to modulate the optical signal with its own RF signal and detect its magnitude and phase delay by performing transmission measurement in a closed optoelectronic loop. In this context, generation of optical sidebands by precisely controlled RF/microwave signals opens up a new possibility for quantum information processing with rare-earth-(RE)-doped solids. In particular, one can implement high-resolution spectroscopy, optical hole burning [5], electromagnetically induced transparency, state manipulation, and Raman echoes by using very powerful methods of RF vector network analysis in a continuous as well as a pulsed regime, such as those that have been developed for superconducting quantum circuits [6,7].

In the present work we perform a crucial step toward the realization of quantum control on solid-state atomic ensembles in a closed optoelectronic loop. We demonstrate optical vector network analysis (OVNA) by using normal AM spectroscopy of ultranarrow ${}^4{I}_{15/2}\leftrightarrow {{}^{4}I}_{13/2}$ optical transition of an isotopically purified ${{}^{166}\mathrm{Er}}^{3+}\text{:}{}^7{\mathrm{LiYF}}_4$ (Er:LYF) crystal. We also present a theory that accurately describes the observed modulation response, enabling the extraction of the absorption coefficient and change of refractive index in the vicinity of an atomic resonance.

Isotopically purified, low-strain crystals doped with specific isotopes of RE ions represent rather interesting materials for the implementation of optical quantum memory. Owing to the ordered crystal structure, the inhomogeneous broadening of optical transitions of RE ions is mainly limited by nuclear spin fields of a host matrix [8,9]. For instance, previous spectroscopic studies of ${}^7{\mathrm{LiYF}}_4$ crystals doped with ${\mathrm{Er}}^{3+}$ and ${\mathrm{Nd}}^{3+}$ ions demonstrated optical FWHM inhomogeneous broadening in the order of $\mathrm{\Gamma }/2\pi \sim 10–100\mathrm{MHz}$ [10,11]. The presence of optical transitions of ${\mathrm{Er}}^{3+}$ within the fiber transparency windows makes such solid-state media very attractive for realization of quantum memory protocols based on off-resonant Raman echoes, promising the highest quantum efficiency. The basic idea of such a protocol relies on using long-lived hyperfine transitions for the storage of optical photons. Isotopically purified crystals are the most suitable candidates for realization of off-resonant Raman memories because they offer direct optical addressing of their hyperfine states and large optical depth of $\alpha L>10$. However, the accurate measurement of the high optical depth with conventional transmission spectroscopy is often very challenging due to finite resolution of oscilloscopes and difficulty of precise offset calibrations. Certainly, OVNA based on a single-sideband (SSB) detection method, s.f. [12,13], would represent the best option for the characterization of materials for quantum optical storage. However, the implementation of SSB technique still requires quite expensive and/or technically challenging solutions, such as dual-parallel Mach–Zehnder modulators [14,15] or narrow-band tunable optical filters [16]. Contrary to that, in our work we show that conventional double-sideband OVNA accompanied with the proper theory is sufficient for the spectroscopy of a highly dispersive RE-doped solids.

The optical vector network analysis setup is sketched in Fig. 1. The erbium-doped tunable fiber laser (NKT Photonics Adjustik E15) emits a continuous signal at the wavelength of $\lambda =1530.37\mathrm{nm}$ (vac) in the vicinity of the optical transitions of Er:LYF. The laser is tuned to ${\omega }_0/2\pi \approx 195888.14\mathrm{GHz}$, which is 6 GHz away from the atomic transition at zero field, and its frequency is stabilized with the help of an optical wavemeter (High-Finess WS6-200). The optical carrier signal is modulated by applying a weak RF signal (0 dBm) from RF-VNA (Keysight PNA-L) to the Mach–Zehnder intensity modulator (MZ-IM), which is locked to its quad operating point by using a bias controller. The spectral analysis of the generated optical signal shows that there is less than 1% of the total light power contained in each of the generated optical sidebands. The laser light is focused into the sample, and the total light power at the sample is about 100 μW.

 

Fig. 1. Sketch of the experimental setup. MZ-IM stands for the Mach–Zehnder intensity modulator. PD is the high-speed InGaAs photodetector. RF-VNA is the radio-frequency vector network analyzer.

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Our isotopically purified Er:LYF crystal has 0.0013% doping concentration and a length of $L=0.5\mathrm{cm}$. To attain minimal optical transition linewidth, the sample is cooled to 3.8 K [17]. The external magnetic field is applied perpendicular to the crystal axis, and the polarization of the laser light is along the crystal axis $c$. The transmitted light is detected by a high-speed photodetector (Newport 1414A), which has a bandwidth of 25 GHz. The resulting microwave signal is amplified by a broadband 0.1–18 GHz, $+26\mathrm{dB}$ microwave amplifier (Minicircuits ZVA-183S+) and is detected by the RF-VNA. Further details regarding optical and cryogenic setup as well as fabrication of the sample are given in Ref. [18].

The measured microwave response ($|S_{21}|$, $\arg (S_{21})$) of the crystal at the magnetic field of $B=0.2\mathrm{T}$ is shown in Fig. 2. The intermediate frequency bandwidth (IFBW) filter is set to 10 kHz. The magnitude of the transmitted signal $|S_{21}(\mathrm{\Omega })|$ consists of two narrowband absorption components, namely 1 and 2, which are centered around ${\mathrm{\Omega }}_1/2\pi =7.6\mathrm{GHz}$ and ${\mathrm{\Omega }}_2/2\pi =5.2\mathrm{GHz}$, respectively. These components are associated with the optical transitions between Zeeman states with the same magnetic spin number $m_S$. Each spectral component reveals a double-peak feature. A similar feature is clearly recognizable in the phase response and is associated with a large phase delay acquired by the interacting optical sideband while it is swept across an atomic resonance. This auxiliary effect disappears for transition 2 at higher fields, when the Zeeman level of optical ground state ${}^4{I}_{15/2}$ becomes depopulated, and optical density becomes small. Below, we outline our simple theoretical model, which properly describes the observed effect.

 

Fig. 2. Measured (a) microwave transmission $|S_{21}(\mathrm{\Omega })|$ and (b) phase delay arg $(S_{21}(\mathrm{\Omega }))$ through the Er:LYF crystal. The inset shows addressed optical levels of Er:YLF crystal and transitions corresponding to the absorption components 1 and 2.

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The response of any dispersive medium is characterized by the complex susceptibility $\chi (\omega )={\chi }^{\prime }(\omega )+i{\chi }^{\prime \prime }(\omega )$, where its real and imaginary parts are connected via well-known Kramers–Kronig relations [19]. In the case of weak susceptibility, i.e., $|{\chi }^{\prime }(\omega )|$, $|{\chi }^{\prime \prime }(\omega )|\ll 1$, the corresponding change of the refractive index and the absorption coefficient can be expressed via real and imaginary parts of $\chi (\omega )$ as $\mathrm{\Delta }n(\omega )\approx {\chi }^{\prime }(\omega )/2$ and $\alpha (\omega )\approx \frac{\omega }{c}{\chi }^{\prime \prime }(\omega )$. By assuming a Lorentzian absorption profile, which is the case for ultranarrow optical transitions in Er:LYF crystal [18], we define the absorption coefficient as

When a nearly resonant laser wave propagates through an atomic ensemble it experiences both: absorption and phase delay (dispersion). In our experiment and thus in our model, only the blueshifted optical sideband at ${\omega }_{+}={\omega }_0+\mathrm{\Omega }$ is assumed to interact with the ions, while the carrier at ${\omega }_0$ and the redshifted sideband at ${\omega }_0-\mathrm{\Omega }$ remain unperturbed. The beat between the modulated optical sidebands and the carrier is detected by the photodetector and results in the following RF signal at the modulation frequency of $\mathrm{\Omega }$:

Two components of the RF signal are attributed to the beating of the carrier with redshifted and blueshifted sidebands. The detection of the RF signal by VNA yields the microwave transmission magnitude and phase response, which reads as

The fit of the measured signal by the equations above is shown in Fig. 2 with solid and dotted curves. At first, the magnitude response is fit by Eq. (3), and the phase response is fit by Eq. (4). Then, the parameters extracted from the fit of the magnitude ($\alpha L$, ${\omega }_r$ and $\mathrm{\Gamma }$) are used to simulate the phase response and vice versa. As it is seen from the graph, the measured data are in a very good agreement with the above presented theoretical model. Double peak structure appears in the absorption profile when the VNA response exceeds $-6\mathrm{dB}$.

By using our theoretical model, we reconstruct the optical depth of the studied optical transitions and the corresponding change of the refractive index of a crystal around atomic resonance. The result of calculations is shown in Fig. 3. Transitions 1 and 2 of Er:LYF are detected at RF of 7.601 GHz and 5.191 GHz. The maximal optical depth for transitions 1 and 2 is equal to 11.4 and 3.6 with the inhomogeneous broadening of ${\mathrm{\Gamma }}_1/2\pi =13\mathrm{MHz}$ and ${\mathrm{\Gamma }}_2/2\pi =22\mathrm{MHz}$, respectively. The optical absorption coefficient for transitions 1 and 2 is equal to $22{\mathrm{cm}}^{-1}$ and $7{\mathrm{cm}}^{-1}$, respectively. In a previous work optical depth as high as $\alpha \simeq 40{\mathrm{cm}}^{-1}$ at $\mathrm{\Gamma }/2\pi =16\mathrm{MHz}$ was reported for a similar Er:LYF system at 0.005% doping concentration [21]. We note here that the deep freezing of the LYF crystal to sub-Kelvin temperatures pumps the population into the ground Zeeman state and results in similar values of $\alpha$ for the line 1 at magnetic field above 100 mT [18].

 

Fig. 3. Recovered (a) optical depth $\alpha L$ and (b) the change of the refractive index $\mathrm{\Delta }n$ in the vicinity of atomic resonances measured by using OVNA.

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The dynamic range of the proposed OVNA is mainly determined by the noise floor that arises from shot-noise of the photodetector and thermal noise of the amplifier. For the current experimental parameters, it is about 35 dB. The present dynamical range would potentially allow us to measure optical depths up to $\alpha L\simeq 200$. Other detrimental contributions originating from the higher-order positive and negative RF sidebands, s.f. Ref. [22], are negligible in our case because of the small modulation index.

It is interesting to compare the data measured with the presented method to a standard transmission spectroscopy [11,17]. For this purpose, the modulation is turned off, and the laser frequency is swept by applying a triangular signal the piezo-transducer (PZT). The change of the laser wavelength is measured by an optical wavemeter. The transmitted optical signal is detected by an amplified, slow InGaAs photodetector, and the detected signal is recorded by a digital oscilloscope. The measured optical depth $\alpha L$ is shown in Fig. 4 with a solid line. The absorption signal measured for line 1 is clipped at $\alpha L\sim 5.5$ due to the finite vertical resolution of the oscilloscope. The dotted line shows the fit of the data with a Lorentzian spectral shape excluding the central truncated region, which yields $\alpha L$ to be equal to 14 and 4.9 for line 1 and line 2, correspondingly.

 

Fig. 4. Comparison between the optical depth measured by using the conventional transmission spectroscopy and OVNA. Solid line shows the conventional transmission spectrum and dotted line is the fit to Lorentzians. Dashed line shows the recovered optical depth measured by OVNA, shown also in Fig. 3.

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The dashed line in Fig. 4 shows the absorption reconstructed from AM data. The discrepancy in $\alpha L$ between the two data sets can be explained by the difficulty of a precise no-light-signal calibration in conventional spectroscopy. We found that it can vary by a few percent during the experiment. However, in the case of strong optical absorption, even such little offset variations can introduce a noticeable change into the $\alpha L$. The difference of the measured resonance frequency of about 20 MHz between the sets is due to the nonlinear response of the PZT with respect to the applied voltage. In this experiment, the laser frequency is recorded while being swept. Then, the recorded signal is approximated by a 5-order polynomial function. The described procedure can result in $\sim 10\mathrm{MHz}$ errors in the laser frequency determination. This discrepancy of the frequency scale also influences the linewidths of the transitions, found to be slightly higher for conventional spectroscopy in comparison to the AM spectroscopy. Linewidths $\mathrm{\Gamma }/2\pi$ measured by conventional spectroscopy are 14 MHz and 30 MHz, whereas by AM spectroscopy they are 13 MHz and 22 MHz for lines 1 and 2, respectively.

The AM spectroscopy technique yields a nonlinear response for large optical depths. To illustrate the limitations of the method we plot the absorption peak depth, measured by RF VNA at the atomic transition frequency ${\omega }_0+\mathrm{\Omega }={\omega }_r$, as a function of actual $\alpha L$ in Fig. 5. For a medium with $\alpha L\ll 1$ the quadratic magnitude term and phase shift in Eq. (3) can be neglected, and then it simplifies to ${|S_{21}|}^2(\mathrm{\Omega })\propto 1-\alpha ({\omega }_0+\mathrm{\Omega })L$. For the optical depth below 3, i.e., before the double-peak structure appears, it would be possible to convert the dB signal directly into $\alpha L$ with a relatively simple polynomial function, which can be extracted from the curve on Fig. 5. We also note here that double peak structure appears already at $\alpha L\simeq 4$. With increasing optical depth, the signal starts to significantly deviate from linear behavior and saturates at 6 dB, corresponding to the complete removal of one of the sidebands in the RF signal. At such high optical depth, a fit with our model is required in order to extract the real parameters.

 

Fig. 5. Solid line: the measured depth of the absorption peak at the resonance frequency of the atomic transition ${\omega }_0+\mathrm{\Omega }={\omega }_r$ as a function of the optical depth of the sample $\alpha L$. Dotted line represents linear response of OVNA at $\alpha L\ll 1$.

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In conclusion, we presented OVNA of record narrow (13 MHz) solid-state atomic ensembles with large optical density at telecom-C band. The auxiliary features observed on the AM response are attributed to the rapid phase change when one of the sidebands passes through the atomic resonance. We developed a simple theoretical model that accurately describes the measured modulation response of the medium. The model allows us to reconstruct the absorption profile and the refractive index change. We also compared the OVNA spectroscopy to conventional transmission spectroscopy and found reasonable agreement between the extracted parameters by using both methods. We also emphasize here that using the OVNA can potentially allow for a more precise determination of Hamiltonian parameters of the ground and excited states of isotopically purified crystals with optically resolved hyperfine states. A pair of ultranarrow resonant transitions with a tunable splitting allows implementing controllable optical delays using slow light [23] and potentially may be useful for implementing quantum memory schemes involving refractive index control [24,25]. In addition, ultranarrow solid-state atomic ensembles are very attractive for the resonant filtering and conversion of optical signals [26]. Overall, the presented experiment lays the foundation for the further integration of microwave and optical quantum systems that is required for the development of the quantum internet.