1. Introduction

Rare-earth-ion-doped crystals [1] have long coherence times for optical transitions up to 6.4 ms [2, 3] and for hyperfine transitions up to 1 s [4, 5]. Therefore, such crystals have been expected to be useful, e.g., for quantum information technologies [6–13].

So far, the following interesting experiments have been achieved with such crystals: the observation of electromagnetically induced transparency in solids [14–16], light storage or quantum memory for light [17–20], stimulated Raman adiabatic passage in solids [21–24], and quantum gate operations [22, 25].

In all the above experiments, the dopant ions were treated as an ensemble. This is partially because it is extremely difficult to treat a single dopant ion. If it is possible to manipulate and observe a single dopant ion, further new possibilities will be opened. An approach to realize this is based on cavity quantum electrodynamics (QED) [26]. Several cavity-QED schemes with rare-earth-ion-doped crystals have been theoretically proposed for quantum information technologies [6, 7, 11, 12, 27]. However, cavity-QED experiments with such crystals have been very few so far.

In the experiments of Refs. [28–30], a Fabry-Perot cavity inside which a Tm³⁺ :YAG thin crystal was placed was used as a superradiant system. From a cavity-QED point of view, however, this structure has a disadvantage that the intracavity loss may be high due to the reflection or scattering at the surfaces of the crystal. (In this paper, “intracavity loss” means the unwanted loss other than transmission losses at cavity mirrors. See Sec. 2 and Appendix for the details.)

On the other hand, in Ref. [31], a monolithic Fabry-Perot cavity made of Pr³⁺:Y₂SiO₅ (Pr:YSO) has been studied experimentally. Its monolithic structure that two multilayer dielectric mirrors are formed on two surfaces of the crystal is desirable to realize low intracavity loss. In Ref. [31], normal-mode splitting and optical bistability have been experimentally demonstrated. While these results have indicated a fairly strong interaction between the cavity mode and the ensemble of the dopant ions, the detailed characterization of such a monolithic cavity has not been done. In particular, the evaluation of the intracavity loss is important for future applications based on cavity QED, such as a single-photon source, quantum memory for light, single-ion detection, and entangling gate operations.

In this paper, we propose an experimental method to determine the intracavity loss of a monolithic Fabry-Perot cavity. This method also allows one to determine the individual cavity-mirror transmittances of the monolithic Fabry-Perot cavity and furthermore the coupling efficiency between the cavity mode and the incident light. And importantly, the modified version of this method can also be applied to whispering-gallery-mode cavities. Using this method, we measured the intracavity losses of monolithic Fabry-Perot cavities made of Pr:YSO at room temperature. The reason why we chose Pr:YSO is that Pr:YSO has very good properties (see Refs. [4, 5, 14, 15, 17–23, 25]) and therefore is one of the most promising materials. The measured intracavity loss of a sample with extremely low dopant concentration (2 × 10⁻⁵ at. %) is about 0.1%. We discuss how good this value is for cavity-QED applications. The experimental results also indicate that the loss may be mainly due to the bulk loss of YSO crystal.

This paper is organized as follows. In Sec. 2, we present our method to determine the intra-cavity loss of a monolithic Fabry-Perot cavity. In Sec. 3, the data on the samples used here are provided. In Sec. 4, the experimental setup and procedure are explained. In Sec. 5, the experimental results are presented, and we discuss them, especially from a cavity-QED point of view, in Sec. 6. The conclusion and outlook are presented in Sec. 7.

2. Method

Note that it is not easy to determine the intracavity loss of a monolithic cavity separately from the transmission losses. Here we propose a new method to achieve that. (Although a similar method has been proposed in Ref. [32], that is not for monolithic cavities and more importantly the way to determine the losses in Ref. [32] is completely different from the present one.)

The intracavity loss is mainly composed of two parts: bulk loss (absorption and scattering in propagating in the medium) and mirror loss (absorption and scattering by the cavity mirrors). By our method, we can determine the intracavity loss separately from the transmission losses, but cannot determine the bulk loss and the mirror loss separately (see Appendix). This method also enables to determine the individual cavity-mirror transmittances of the monolithic Fabry-Perot cavity and furthermore the coupling efficiency between the cavity mode and the incident light, as explained below.

A Fabry-Perot cavity is composed of two mirrors. Here we call them M1 and M2. First, the transmission and reflection spectra of the cavity are measured in the case where a laser is incident on M1. These spectra can be well fitted to the following functions (see Fig. 2):

 

Fig. 2 Measured transmission (a) and reflection (b) spectra of Sample 1. The smooth curves are the fitting ones. The polarization is parallel to the D1 axis. The wavelength is 606 nm. The laser is incident on the plain mirror.

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The free spectral range (FSR) ν_FSR is also measured. The finesse of the cavity F is given by [33]

Note that the above measurements are not sufficient to determine the intracavity loss. The following measurement is necessary to achieve that.

Next, the spectra are measured in the case where a laser is incident on M2. These spectra are fitted to the following functions:

Thus, we obtain the following five experimental data: F, P₁, D₁, P₂, and D₂. Using these, we can evaluate the intracavity loss L_in and the transmittances T₁ and T₂ of M1 and M2, respectively, with the following equations (see Appendix for the derivation):

Interestingly, the coupling efficiencies C₁ and C₂ in the cases of the incidences on M1 and M2, respectively, can also be determined with the data as follows:

Note that the five parameters L_in, T₁, T₂, C₁, and C₂ are determined with the five data F, P₁, D₁, P₂, and D₂. The above formula may be new.

The present method will be useful to evaluate various monolithic Fabry-Perot cavities. Furthermore, this method can be modified and applied to whispering-gallery-mode cavities (see Appendix). Therefore, the present method will also be useful to evaluate microspheres [34], toroid microcavities [35], and crystalline whispering-gallery-mode cavities [36, 37].

3. Samples

In the present work, we studied two monolithic Fabry-Perot cavities made of Pr:YSO. One of the two samples has extremely low dopant concentration (2 × 10⁻⁵ at. %), and the other has fairly high concentration (4 × 10⁻² at. %). In this paper, we call them Sample 1 and Sample 2, respectively.

The two samples have the same design except for concentration. The design is as follows. Two cavity mirrors are formed on two surfaces of a crystal, as depicted in Fig. 1. One of the two mirrors is plain, and the other is spherical with a curvature radius of about 9 mm. Both the mirrors have a diameter of 3 mm and a reflectance of about 99.95%. The cavity length is about 7.5 mm. The radius of the mode waist, which is at the plain mirror, is about 19 μm.

 

Fig. 1 Experimental setup. L: coupling lens. BS: beamsplitter. PM: plane mirror. SM: spherical mirror. DT: photodetector for transmission. DR: photodetector for reflection. DI: photodetector for input. HWP: half-wave plate.

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The detailed data on the samples are summarized in Table 1.

Table 1. Sample data. See the text for the sample design. “Radius” means curvature radius of spherical mirror. The reflectances are design values, not measured values

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4. Experimental

The experimental setup is depicted in Fig. 1.

The light source is a ring dye (Rhodamine 6G) laser (Coherent 699-29) pumped by an argon ion laser (Coherent INNOVA400). The frequency jitter of the laser is about 1.2 MHz. In the experiments with Sample 1, the frequency jitter was reduced to several kHz by locking to the resonance frequency of an external stable cavity using the Pound-Drever-Hall method [38]. For Sample 2, this frequency stabilization was not used because the cavity has a wide linewidth compared to the jitter.

In the experiments with Sample 1, the laser frequency was scanned by an acousto-optic modulator in the double-pass configuration (60 MHz/20 ms). In the case of Sample 2, the frequency scan was done with the laser controller (990 MHz/250 ms).

The wavelength was set to 606 nm or 609 nm. (The absorption by Pr³⁺ ions in YSO is maximal at 606 nm.)

The polarization of the incident laser was set to that parallel to one of two YSO optical axes D1 and D2 with a half-wave plate. (It is known that the absorption by Pr³⁺ ions in YSO is larger in the case of D2 than D1.) The laser direction and the cavity mode were parallel to the other axis (b axis) of the YSO crystal.

The incident power was set to 10–30 μW. To check the intensity dependence, the experiments with about 0.5 μW were also done, and it has been concluded that there is almost no intensity dependence. So, the incident power is not important in the present experiments.

The experimental procedure is as follows. First, the laser is incident on the plain mirror, and the transmission and reflection spectra are measured with the photodetectors DT and DR, respectively, as depicted in Fig. 1(a). A part of the incident power is also measured simultaneously with the photodetector DI. This is used to eliminate the effect of laser power fluctuation. After that, the FSR is also measured. (This frequency scan was done with the laser controller.) Next, the laser is incident on the spherical mirror, and the similar measurements are done, as depicted in Fig. 1(b). Finally, the measurements with DT and DI are done without the sample, as depicted in Fig. 1(c). This is used to obtain the ratio between the incident power measured by DT and the power measured by DI. Using these data, we can evaluate all the necessary quantities (F, P₁, D₁, P₂, and D₂).

All the present experiments were done at room temperature. This is because we can surely perform the experiments with the new method proposed here. (The experiments at cryogenic temperature where the cavity is placed in a cryostat are much more complicated and difficult to be surely performed.) In addition, this is also because it had been expected to obtain ultralow losses even at room temperature. (Previous experiments with whispering-gallery-mode crystalline cavities made of CaF₂ [36] and crystalline quartz [37] have also been done at room temperature and very high finesses have been obtained.) Experiments at cryogenic temperature will be done in the near future.

5. Results

The examples of the fittings to the spectra are shown in Fig. 2. The fittings are excellent. From such fittings, we obtained ν₀ (HWHM), P₁, D₁, P₂, and D₂.

The results for Sample 1 and Sample 2 are summarized in Tables 2 and 3, respectively.

Table 2. Results for Sample 1. WL: wavelength. Pol: polarization (“D1” means the polarization is parallel to the D1 axis of YSO). FWHM: full width at half maximum of transmission spectrum (2ν₀). F: finesse. L_in: intracavity loss (± denotes the statistical error estimated using the standard errors of ν₀, ν_FSR, P₁, D₁, P₂, and D₂ determined by measurements repeated three times). T₁ and T₂ are the transmittances of the plain and spherical mirrors, respectively. C₁ and C₂ are the coupling efficiencies in the cases of the incidences on the plain and spherical mirrors, respectively

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Table 3. Results for Sample 2. See the caption of Table 2 for details

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6. Discussion

We are most interested in the intracavity loss L_in in the case of Sample 1 (the sample with extremely low dopant concentration) because if this loss is very low, cavity-QED applications with such a cavity are very promising. The result is L_in ≃ 0.1%, as shown in Table 2.

How good is the cavity-QED system with this intracavity loss? To discuss this, the best figure of merit may be the critical atom number n_a [39], which is a fundamental dimensionless parameter in cavity QED defined as

Thus, a criterion for a useful cavity-QED system is given by n_a < 1.

In the case of Pr³⁺ ions in YSO, γ⁻¹ ≃ 100 μs [43].

κ is given by

g for a Gaussian standing-wave mode is given by [26, 31]

The w dependence of the smallest n_a is shown in Fig. 3. (In the present experiments, w = 19 μm.) The other parameters are set as follows: L_in = 10⁻³, γ = 10 kHz, n = 1.9, μ ≃ 3 × 10⁻³² C·m [44], and λ_c = 606 nm. (n was determined by the measurement results of FSR.) From Fig. 3, it turns out that n_a(w) becomes smaller than 1 if the mode radius becomes smaller than 1 μm. This is hard though this is possible in principle. So unfortunately, the measured intracavity loss, L_in = 0.1%, is fairly high for cavity-QED applications.

 

Fig. 3 n_a(w) on a log-log scale. See the text for the parameter setting.

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Is the intracavity loss due to Pr³⁺ dopants? The answer is probably no. The reason is as follows. First, the polarization and wavelength dependences are too small, compared to Sample 2. (Since the dopant concentration of Sample 2 is high, the data in Table 3 clearly show the polarization and wavelength dependences consistent with the properties of Pr³⁺ ions in YSO. See Sec. 4 for the properties.) Second, while the concentration of Sample 1 is three orders of magnitude lower than that of Sample 2, L_in of Sample 1 is only one order of magnitude smaller than that of Sample 2. This strongly indicates that L_in of Sample 1 is not due to Pr³⁺ dopants.

Additionally, the mirror loss other than the transmission loss is typically 0.01% (100 ppm), and therefore this is much lower than the measured intracavity loss. Thus, we conclude that the intracavity loss may be mainly due to the bulk loss of YSO crystal. It is estimated that the bulk loss of YSO crystal at room temperature is 7 × 10⁻⁴ cm⁻¹ (0.003 dB/cm) or lower.

7. Conclusion and outlook

We have proposed a method with which the intracavity loss and individual cavity-mirror transmittances of a monolithic Fabry-Perot cavity can be determined separately. The coupling efficiency between the cavity mode and the incident light can also be determined simultaneously by the method. This method can also be modified and applied to whispering-gallery-mode cavities. By using this method, we have experimentally determined the intracavity losses of two monolithic Fabry-Perot cavities made of Pr:YSO at room temperature. The measured intracavity loss for the sample with extremely low dopant concentration (2 × 10⁻⁵ at. %) is about 0.1%. By evaluating the critical atom number theoretically, we have concluded that this value is fairly high for cavity-QED applications.

From the experimental results, we have also concluded that the intracavity loss may be mainly due to the bulk loss of YSO crystal. We have estimated that the bulk loss of YSO crystal at room temperature is 7 × 10⁻⁴ cm⁻¹ (0.003 dB/cm) or lower.

Since long coherence times of Pr³⁺ ions are realized only at cryogenic temperature below 4 K, experiments at cryogenic temperature are important toward cavity-QED experiments. Such experiments will be done in the near future. It is very interesting and important whether the intracavity losses become lower at cryogenic temperature or not.

Appendix: Derivation of Eqs. (6)–(10)

The present model is depicted in Fig. 4. This figure corresponds to the case where a laser is incident on M1.

 

Fig. 4 Model for a monolithic Fabry-Perot cavity. M1 and M2 are the cavity mirrors. L: bulk loss on the path from M1 (M2) to M2 (M1). θ: phase shift from M1 (M2) to M2 (M1). T₁ and T₂ : transmittances of M1 and M2, respectively. R₁ and R₂ : reflectances of M1 and M2, respectively. L₁ and L₂ : mirror losses of M1 and M2, respectively. They satisfy L₁ = 1 – T₁ – R₁ and L₂ = 1 – T₂ – R₂ by definition. E denotes the electric field.

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From the boundary and propagation conditions, we obtain

(16)$$E_1=\sqrt{\frac{n_1}{n_2}}{t}_1{e}^{i{\varphi }_1}{E}_I-r_1{e}^{i{\varphi }_1}{E}_4,$$

(17)$$E_2=\sqrt{1-L}{e}^{i\theta }{E}_1,$$

(18)$$E_3=-r_2{e}^{i{\varphi }_2}{E}_2,$$

(19)$$E_4=\sqrt{1-L}{e}^{i\theta }{E}_3,$$

(20)$$E_T=\sqrt{\frac{n_2}{n_1}}{t}_2^{*}{e}^{i{\varphi }_2}{E}_2,$$

(21)$$E_R=\sqrt{\frac{n_2}{n_1}}{t}_1^{*}{e}^{i{\varphi }_1}{E}_4+r_1^{*}{e}^{i{\varphi }_1}{E}_I,$$

where n₁ and n₂ are the refractive indices of air and the medium (crystal), respectively, ϕ_j (j = 1, 2) is related to the phase shift at the jth mirror, r_j and t_j (j = 1, 2) are the reflection and transmission coefficients for the jth mirror, which satisfy R_j = |r_j|² and T_j = |t_j|². Note that the information of the mirror losses, L₁ and L₂, is included as the relations L₁ = 1 – T₁ – R₁ and L₂ = 1 – T₂ – R₂.

Solving Eqs. (16)–(21), we obtain the spectra as follows:

(22)$$S_{T1}(\theta )={\left|\frac{E_T(\theta )}{E_I}\right|}^2=\frac{T_1{T}_2(1-L)}{{|1-(1-L)\sqrt{R_1{R}_2}{e}^{2i\theta }|}^2},$$

(23)$$S_{R1}\left(\theta \right)={\left|\frac{E_R(\theta )}{E_I}\right|}^2=\frac{{\left|\sqrt{R_1}-(1-L_1)(1-L)\sqrt{R_2}{e}^{2i\theta }\right|}^2}{{\left|1-(1-L)\sqrt{R_1{R}_2}{e}^{2i\theta }\right|}^2},$$

where without losing generality unimportant phase factors have been determined so that the resonance corresponds to θ = 0.

Assuming that the transmittances and the losses are sufficiently low, we can approximate the above results around the resonance frequency (θ ≪ 1) as follows:

(24)$$S_{T1}(\theta )\simeq \frac{4T_1{T}_2}{{(L_{\mathit{in}}+T_1+T_2)}^2+{(4\theta )}^2},$$

(25)$$S_{R1}(\theta )\simeq \frac{{(L_{\mathit{in}}+T_2-T_1)}^2+{(4\theta )}^2}{{(L_{\mathit{in}}+T_1+T_2)}^2+{(4\theta )}^2},$$

where we have defined the intracavity loss as

(26)$$L_{\mathit{in}}\equiv 2L+L_1+L_2.$$

Note that we cannot determine L₁, L₂, and L separately. We can determine only L_in. And also note that in the case of a monolithic cavity, we cannot determine L₁ and L₂ by investigating only one of the mirrors because the mirrors cannot be detached from the surfaces of the crystal.

The above results hold under the condition that the coupling of the incident laser to the cavity mode is perfect. If the coupling efficiency is C₁ (0 ≤ C₁ ≤ 1), the spectra are modified as follows:

(27)$$S_{T1}(\theta )=C_1\frac{4T_1{T}_2}{{(L_{\mathit{in}}+T_1+T_2)}^2+{(4\theta )}^2},$$

(28)$$\begin{array}{lll}{S}_{R1}(\theta )\hfill & =\hfill & C_1\frac{{(L_{\mathit{in}}+T_2-T_1)}^2+{(4\theta )}^2}{{(L_{\mathit{in}}+T_1+T_2)}^2+{\left(4\theta \right)}^2}+(1-C_1)\hfill \\ \hfill & =\hfill & 1-C_1\frac{4T_1\left(L_{\mathit{in}}+T_2\right)}{{(L_{\mathit{in}}+T_1+T_2)}^2+{(4\theta )}^2}.\hfill \end{array}$$

Using the above results, we obtain

(29)$$F=\frac{2\pi }{L_{\mathit{in}}+T_1+T_2},$$

(30)$$P_1=C_1\frac{4T_1{T}_2}{{(L_{\mathit{in}}+T_1+T_2)}^2}=\frac{4C_1{T}_1{T}_2}{{(2\pi /F)}^2},$$

(31)$$D_1=C_1\frac{4T_1(L_{\mathit{in}}+T_2)}{{(L_{\mathit{in}}+T_1+T_2)}^2}=\frac{L_{\mathit{in}}+T_2}{T_2}{P}_1.$$

In a similar manner, we also obtain

(32)$$P_2=\frac{4C_2{T}_1{T}_2}{{(2\pi /F)}^2},$$

(33)$$D_2=\frac{L_{\mathit{in}}+T_2}{T_2}{P}_2.$$

Thus, we obtain the formula given by Eqs. (6)–(10).

Importantly, the modified version of the present method can be applied to whispering-gallery-mode cavities. The model for this is depicted in Fig. 5. This figure corresponds to the case where prisms are used for couplers. The method can also be applied to the case where tapered fibers are used as couplers instead of prisms.

 

Fig. 5 Model for a whispering-gallery-mode cavity. P1 and P2 are prism couplers. L: bulk loss on the path from P1 (P2) to P2 (P1). θ: phase shift from P1 (P2) to P2 (P1). T₁, R₁, and L₁ are the transmittance, reflectance, and loss between the cavity and P1, respectively. T₂, R₂, and L₂ are the transmittance, reflectance, and loss between the cavity and P2, respectively. By definition, L₁ = 1 – T₁ – R₁ and L₂ = 1 – T₂ – R₂. E denotes the electric field.

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The crucial point is that two couplers are used and two experiments are performed, where a laser is incident from one of the couplers in one of the experiments and from the other coupler in the other experiment. Note that we cannot apply the present method with only one coupler. This method has the great advantage that we do not need to decrease the “transmittances” for the measurement of the intracavity loss, unlike usual measurements. It is also notable that the coupling efficiency between the cavity mode and the incident light is not important for the present method.

The electric fields satisfy the same equations as Eqs. (16)–(21) if the coupling between the cavity mode and the incident light is perfect. (Rather, these equations are the definitions for the reflection and transmission coefficients.) In the present case, n₁ is the refractive index of the prism coupler. The imperfect coupling is treated by introducing the coupling efficiency, in a similar manner to the above. Thus, the above discussion on a monolithic Fabry-Perot cavity holds, and therefore we can evaluate the intracavity loss of the whispering-gallery-mode cavity in a similar manner.

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