## Abstract

We experimentally demonstrate cavity-enhanced spectroscopy of a rare-earth-ion-doped crystal (Pr^{3+}:Y_{2}SiO_{5}). We succeeded in observing very small absorption due to the ions appropriately prepared by optical pumping, which corresponds to the single-pass absorption of 4 × 10^{−6}. We also observed a power law for the inhomogeneous broadening of optical transitions of ions in the crystal. Compared with a theoretical model, the result of the power law indicates that the dominant origin of the inhomogeneous broadening may be some charged defects.

© 2013 Optical Society of America

## Corrections

Hayato Goto, Satoshi Nakamura, Mamiko Kujiraoka, and Kouichi Ichimura, "Cavity-enhanced spectroscopy of a rare-earth-ion-doped crystal: observation of a power law for inhomogeneous broadening: erratum," Opt. Express**24**, 4980-4980 (2016)

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-24-5-4980

## 1. Introduction

Rare-earth-ion-doped crystals [1, 2] have attractive features for quantum information technology, e.g., very long coherence times up to 30 s [3–5]. With such crystals, quantum memory for light [6–10], stimulated Raman adiabatic passage in solids [11–14], and quantum gate operations [12, 15] have been demonstrated.

While dopant ions were treated as an ensemble in all the above experiments, some schemes in which single dopant ions are used as quantum bits (qubits) have been theoretically proposed [16–20]. The key technologies for these schemes are based on cavity quantum electrodynamics (cavity QED) [21]. Entangling gate operations between two distant ions can be performed via cavity photons. Such cavity-QED schemes also allow one to observe single ions and read ionic qubits. (Very recently, the observation of single rare-earth ions in a crystal has been realized with 4*f*-5*d* transitions [22] or a single-electron transistor [23]. Although these techniques are useful for such observations, they cannot be applied to entangling gate operations, unlike cavity-QED approaches.)

Cavity-QED experiments with rare-earth-ion-doped crystals have been very few so far [24–28]. Towards the observation of a single ion in a crystal based on cavity QED, here we demonstrate cavity-enhanced spectroscopy of a rare-earth-ion-doped crystal. The sample is a monolithic Fabry-Perot cavity made of Pr^{3+}:Y_{2}SiO_{5} (Pr:YSO) [29]. The coupling rate between ions and the cavity mode is fairly high in the sense that the maximum coupling rate exceeds the decay rate of the ionic excited states. The ionic hyperfine state is initialized by optical pumping such that the absorption due to the ions occurs in a specific region in the cavity transmission spectrum. The smallest absorption observed in this work corresponds to the single-pass absorption of 4 × 10^{−6}.

We also found the fact that the inhomogeneous broadening in the region far from the center of the broadening obeys a power law such that the ionic density is proportional to |*ν* − *ν*_{0}|^{−2.4}, where *ν* is the ionic transition frequency and *ν*_{0} is the center frequency of the broadening. Compared with a theoretical model [30], this result indicates that the dominant origin of the inhomogeneous broadening may be some charged defects. To the best of our knowledge, this is the first report of such a suggestion for optical transitions of rare-earth ions in a crystal.

This paper is organized as follows. In Sec. 2, we explain our sample, experimental setup, and method. In Sec. 3, the experimental results are presented. In Sec. 4, we discuss the results. Finally, the conclusion is presented in Sec. 5.

## 2. Experimental

#### 2.1. Sample

In the present work, we studied a monolithic Fabry-Perot cavity made of Pr:YSO (see Fig. 1). The dopant concentration is fairly low (1 × 10^{−3} at. %). Two cavity mirrors are formed on two surfaces of a crystal, which are perpendicular to the *b* axis of YSO crystal. (The cavity mode is parallel to the *b* axis.) One of the two mirrors is planar, and the other is spherical with a radius of curvature of about 9 mm. Both the mirrors have a diameter of 3 mm. The cavity length is about 8.9 mm, which is designed to be close to the radius of curvature so that the radius of the mode waist, which is on the planar mirror, becomes small (about 10 *μ*m).

The maximum coupling rate *g _{max}* between ions and the cavity mode is theoretically given by [21, 27, 29]

*μ*is the transition dipole moment,

*n*is the refractive index of the crystal,

*V*is the mode volume,

*ε*

_{0}is the permittivity of vacuum,

*ω*and

_{c}*λ*are the cavity resonance angular frequency and wavelength, respectively,

_{c}*l*is the cavity length, and

_{c}*w*is the mode radius. (In general, the coupling rate depends on the ionic position [21, 27]. The maximum value

*g*corresponds to that for the ion at the position where the cavity field becomes maximum.) Because of the small radius of the mode waist, the coupling rate is fairly high in the sense that

_{max}*g*/(2

_{max}*π*) reaches 3 kHz and exceeds the decay rate,

*γ*, of the ionic excited states [

*γ*/(2

*π*) ≃ 1 kHz [31]].

The free spectral range (FSR) and full width at half maximum of the cavity are about 9.2 GHz and 2.7 MHz, respectively. (The finesse is about 3400.) The transmittances of the planar and spherical mirrors are about 0.014% and 0.018%, respectively. The intracavity loss not including the absorption loss due to Pr^{3+} ions is about 0.15%. The transmittances and intracavity loss of the cavity were determined by the method proposed in [29].

#### 2.2. Experimental setup

The experimental setup is depicted in Fig. 2.

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. This frequency jitter is reduced to several kHz on a time scale of 1 s by locking to the resonance frequency of a stable external cavity using the Pound-Drever-Hall method [29, 32].

The frequency of the frequency-stabilized laser is scanned with an acousto-optic modulator (AOM) in the double-pass configuration. The AOM is controlled by an arbitrary waveform generator (AWG1) (see below and Fig. 3 for details). (While the MHz-scale scan is done by the AOM, the frequency adjustment on a GHz scale is done by the controller of the dye laser.)

The laser is split into three beams by polarizing beamsplitters (PBSs) and half-wave plates (HWPs), which are used as a probe beam incident to the sample cavity, a local oscillator (LO) for balanced heterodyne detection, and a repump beam for the initialization of the ionic state. The frequency of each beam is shifted appropriately by an AOM in a single-pass configuration. These frequency shifts for the probe, LO, and repump are 104 MHz, 80 MHz, and 75 MHz, respectively. Note that the frequency diffferences are always constant and all the frequencies are shifted simultaneously by the single AOM with AWG1 explained above. It is important for successful initialization of the ionic state that the frequency difference between the probe and the repump is set to 29 MHz. The power of the repump beam is controlled by the AOM with an AWG (AWG2) so that the repump is not incident on the sample during the spectrum measurement [see Fig. 3(c)].

The probe power is reduced by two variable neutral density filters (NDs) to about 10 pW, which corresponds to cavity photons less than ten. Such a weak probe is necessary so that the ions in the cavity are not saturated. The polarization of the probe is set to be parallel to the *D*_{2} axis of YSO crystal so that the absorption due to ions becomes maximum [2]. The probe is incident to the sample cavity cooled at about 4.1 K in a cryostat. The probe transmitted through the cavity is combined with the LO (∼ 5 mW) at a half beamsplitter (HBS). The transmission power is measured by balanced heterodyne detection.

The repump beam is focused in the horizontal direction with a cylindrical lens to obtain higher intensity and is incident to the sample from the side. The highest intensity at the sample is about 0.6 W/cm^{2}. The polarization of the repump is also set to be parallel to the *D*_{2} axis of YSO crystal. The position of the repump is set near the planar mirror because the mode waist is on this mirror and here the coupling rate between ions and the cavity mode becomes maximum. The absorption to be measured is due to the ions at the crossing point of the probe and repump beams.

#### 2.3. Method

Our experimental method to observe small absorption due to ions is as follows.

First of all, the probe power is set to the value 10^{3} times higher (∼ 10 nW) to optically pump out unnecessary ions in the cavity mode. This is done without the repump beam. Thus, the absorption due to only the ions at the crossing point of the probe and repump beams can be measured. Then, the probe power is reset to the small value (∼ 10 pW) and the measurement is started.

First, the transmission spectrum without the repump beam (thus without the absorption due to Pr^{3+} ions) is measured. (Because of hole burning by the above optical pumping with the strong probe, there are no ions absorbing the probe as long as the repump beam is not incident.) This spectrum *S*_{0}(*ν*) can be expressed as [29]

*T*and

_{p}*T*are the transmittances of the planar and spherical mirrors, respectively,

_{s}*L*is the intracavity loss per one round trip not including the absorption due to Pr

_{in}^{3+}ions,

*ν*is the FSR of the cavity, and

_{FSR}*C*

_{0}is a proportionality constant. (The origin of frequency

*ν*has been taken at the peak of the transmission spectrum.)

Next, the transmission spectrum with the repump beam (thus with the absorption due to Pr^{3+} ions) is measured. (The repumping mechanism is presented in detail below.) This spectrum *S*(*ν*) can be expressed as

*A*(

*ν*) is the single-pass absorption loss due to Pr

^{3+}ions and

*C*is another proportionality constant. Here, we have assumed that the absorption loss depends on frequency. An appropriate frequency dependence is achieved by the repumping explained below.

Normalizing *S*(*ν*) with *S*_{0}(*ν*), we obtain

*A*(

*ν*) has been done assuming that

*A*(

*ν*) is sufficiently small.

Here we assume that *A*(*ν*) becomes zero in some frequency region. Using the data in this region, we can estimate *C/C*_{0} and eliminate this factor from *S̄*(*ν*) dividing *S̄*(*ν*) by this estimated value. The renormalized *S̄*(*ν*) is denoted by
$\overline{{S}^{\prime}}(\nu )$.

If |*ν*| is sufficiently smaller than the half width at half maximum of the cavity [(*L _{in}* +

*T*+

_{p}*T*)

_{s}*ν*/(4

_{FSR}*π*) ≃ 1.35 MHz], the following further approximation is valid:

*F*is the finesse of the cavity given by

*F*= 2

*π*/(

*L*+

_{in}*T*+

_{p}*T*).

_{s}Thus, it has been shown that the signal of the absorption becomes larger by the factor of the finesse (more correctly, 2*F/π*) compared to the single-pass case.

The detailed experimental method is as follows.

All the frequencies of the three beams are controlled simultaneously by the single AOM with AWG1 (see Fig. 2). (As mentioned above, the frequency differences of the three beams are constant.) Figure 3 shows the frequency shift in one period. (In Fig. 3, the origin of this frequency shift is taken at the starting point of the spectrum measurement.) The former part from 0 to 1.75 ms is used for the repumping. The latter part from 1.75 ms to 2.5 ms is used for the measurement of the cavity transmission spectrum. The spectrum measurement is performed by scanning the probe frequency by 1.5 MHz. The repump beam is not incident during the spectrum measurement not to induce some undesirable effects. This control is done by the AOM with AWG2 (see Fig. 2).

The repumping mechanism is as follows. The energy-level structure of Pr^{3+} ions in YSO is depicted in Fig. 4 [33]. We chose the |± 5/2〉* _{g}*−|± 5/2〉

*transition to observe the absorption because this transition has the largest oscillator strength (electric dipole moment:*

_{e}*μ*= 2.7 × 10

^{−32}C m) [34]. The purpose of the repumping is to return the ionic population from |±1/2〉

*and |± 3/2〉*

_{g}*to |± 5/2〉*

_{g}*. (The probe pumps out the ionic population from |± 5/2〉*

_{g}*to the others during the spectrum measurement.) This is achieved by the repump beam resonant with the |± 1/2〉*

_{g}*−|± 5/2〉*

_{g}*and |± 3/2〉*

_{e}*−|± 1/2〉*

_{g}*transitions. These repumpings for an ion are depicted in Figs. 3(a) and 3(b), respectively. The transition frequency of this ion is such that the |± 5/2〉*

_{e}*−| ± 5/2〉*

_{g}*transition resonates with the probe at the frequency shift of 1 MHz, as depicted in Fig. 3(c). The arrows with (a), (b), and (c) in the upper figure in Fig. 3 indicate the points at which the situations depicted in Figs. 3(a), 3(b), and 3(c) occur, respectively. Note that the probe does not affect the ion during the repumping because of large detuning. The frequency shift in Fig. 3 is carefully designed so that the absorption due to the ions occurs only in the range from 1 MHz to 1.5 MHz of the 1.5-MHz scan for the spectrum measurement. Here it is important that the frequency difference between the probe and the repump is set to 29 MHz. There is no absorption due to Pr*

_{e}^{3+}ions in the frequency region from 0 to 1 MHz, and therefore the data in this region can be used to estimate the factor of

*C/C*

_{0}, as explained above. The peak of the transmission spectrum is set at about 1.25 MHz so that the approximation in Eq. (5) is valid.

Finally, we show an example of actual data. Figure 5(a) shows the transmission spectra with and without the repump, *S*(*ν*) and *S*_{0}(*ν*), respectively, measured by the balanced heterodyne detection, where the peaks of the transmission spectra are at *ν* ≃ 1.25 MHz, unlike Eqs. (2)–(5). [The spectrum analyzer (SA) output of balanced heterodyne detection is proportional to the transmission power.] Normalizing *S*(*ν*) with *S*_{0}(*ν*), we obtain *S̄*(*ν*), which is shown in Fig. 5(b). [The unit “dB” means that *S̄*(*ν*) (dB)= 10log_{10}*S̄*(*ν*). This can be easily obtained by subtracting *S*_{0}(*ν*) (dBm) from *S*(*ν*) (dBm). This “dB” is also used in Fig. 6.] The next step is to estimate the factor *C/C*_{0} and renormalize *S̄*(*ν*) with the estimated value. However, *S̄*(*ν*) in the range lower than 0.8 MHz is not constant but a line with a small slope. This small tilt is due to a little difference between the peak frequencies of *S*(*ν*) and *S*_{0}(*ν*), which is inevitable by laser frequency drift between the measurements. Therefore, instead of estimating a constant factor, we fit a line to *S̄*(*ν*) in the range lower than 0.8 MHz. The fitted line is the dashed line in Fig. 5(b).
$\overline{{S}^{\prime}}(\nu )$ is obtained by renormalizing *S̄*(*ν*) with the line [subtracting the line (dB) from *S̄*(*ν*) (dB)], which is shown in Fig. 5(c). [The line includes the information on not only *C/C*_{0} but also the difference between the peak frequencies. Thus,
$\overline{{S}^{\prime}}(\nu )$ obtained here is a good approximation of
$\overline{{S}^{\prime}}(\nu )$ without the difference.]

The dip in Fig. 5(c) is the very signal to be obtained. The absorption due to ions occureed only in the range from 1 MHz to 1.5 MHz, as expected. This means that the initialization of the ionic state by the repumping was successfully performed. With this dip and Eq. (5), we can estimate the single-pass absorption loss *A*(*ν*).

## 3. Results

We number the cavity modes so that Mode 0 corresponds to the mode nearest to the center of the inhomogeneous broadening and higher numbers correspond to modes with higher frequencies.

We performed the measurement described in Sec. 2.3 for the modes from Mode −5 to Mode 4 except Mode 0. (The absorption for Mode 0 is too large to measure accurately.) For each mode, we repeat the measurement five times, where the spectra used for each measurement are the ones obtained by 4 × 10^{3} times accumulation for Modes 1 and −1 and 1 × 10^{4} times accumulation for the other modes. The results shown in Fig. 6 are the averages of the five measurements. (The noises in these data are mainly due to quantum noise of the power of the weak probe. Because of the quantum noise, we must take an average of many times.)

## 4. Discussion

First, we discuss the sensitivity of the present measurement. The smallest absorption observed is about −0.04 dB (0.9%) for Mode −5. From Eq. (5), the single-pass absorption *A* corresponding to this value is *A* ≃ 4 × 10^{−6} (*F* ≃ 3400). This absorption may be too small to observe with an ordinary bulk sample of Pr:YSO (without the cavity structure). (Because of the long excited-state lifetime and hole burning, the measurement of small absorption in Pr:YSO is difficult.) This sensitivity can be raised further by increasing the number of measurements for averaging. Our numerical simulation indicates that the signal for a single ion with the maximum coupling rate to the cavity mode will be about −0.004 dB (0.09%). Thus, we will be able to observe a single-ion signal by averaging 100 times more measurements than the case of Mode −5.

Next, we discuss the frequency dependence of the ionic density, i.e., the shape of the inhomogeneous broadening. Here we define the magnitude of the dip for each mode as the absolute value of the average from 1.2 MHz to 1.3 MHz in Fig. 6. [Since the peak of the transmission spectrum is at about 1.25 MHz, the approximation in Eq. (5) is valid in this range.] The dependence of the magnitude of the dip on the mode number is shown in Fig. 7(a). We found that the data are well fitted with the function *d*(*n*) = *a*|*n* − *n*_{0}|^{−}* ^{b}*, where

*d*is the magnitude of the dip,

*n*is the continuous mode number,

*n*

_{0}is the value of

*n*corresponding to the center of the inhomogeneous broadening, and

*a*and

*b*are constants. The curve in Fig. 7(a) is the fitting result. Figure 7(b) is a log-log plot of the same data as Fig. 7(a), where the continuous mode numbers

*n*and

*n*

_{0}are converted to frequencies

*ν*and

*ν*

_{0}with the FSR of the cavity (9.2 GHz). Assuming that the magnitude of the dip, which is proportional to the absorption from its definition, is proportional to the ionic density, this result means that the inhomogeneous broadening obeys a power law. The exponent

*b*is 2.39 ± 0.04.

Finally, we briefly discuss the origin of the inhomogeneous broadening. The shape of inhomogeneous broadening has been theoretically discussed in detail in [30]. According to this, inhomogeneous broadening in the frequency region far from the center obeys a power law. The exponent depends on the origin of the broadening: if the origin is the strain by dislocation, then the exponent is 3; if the origin is the strain by point defects, then the exponent is 2 (Lorentzian); if the origin is the electric field from charged point defects, then the exponent is 2.5. The present result that the exponent is about 2.4 strongly indicates that the inhomogeneous broadening of optical transitions of Pr:YSO is mainly caused by the electric field from charged point defects.

## 5. Conclusion

We have demonstrated cavity-enhanced spectroscopy of a rare-earth-ion-doped crystal with a monolithic Fabry-Perot cavity made of Pr^{3+}:Y_{2}SiO_{5} (Pr:YSO). We have successfully observed very small absorption due to the ions appropriately prepared by optical pumping. The smallest absorption measured corresponds to the single-pass absorption of 4 × 10^{−6}. We have also observed a power law for the inhomogeneous broadening of optical transitions of Pr^{3+} ions in YSO. The result indicates that the dominant origin of the inhomogeneous broadening may be some charged defects.

The present results are useful and make the present technique promising for single-ion observation based on cavity QED.

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