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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 23 — Nov. 8, 2010
  • pp: 23763–23775
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Experimental determination of intracavity losses of monolithic Fabry-Perot cavities made of Pr3+:Y2SiO5

Hayato Goto, Satoshi Nakamura, and Kouichi Ichimura  »View Author Affiliations


Optics Express, Vol. 18, Issue 23, pp. 23763-23775 (2010)
http://dx.doi.org/10.1364/OE.18.023763


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Abstract

We propose an experimental method with which all the following quantities can be determined separately: the intracavity loss and individual cavity-mirror transmittances of a monolithic Fabry-Perot cavity and furthermore the coupling efficiency between the cavity mode and the incident light. It is notable that 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 Pr3+:Y2SiO5 at room temperature. The knowledge of the intracavity losses is very important for applications of such cavities, e.g., to quantum information technologies. It turns out that fairly high losses (about 0.1%) exist even for a sample with extremely low dopant concentration (2 × 10−5 at. %). The experimental results also indicate that the loss may be mainly due to the bulk loss of Y2SiO5 crystal. The bulk loss is estimated to be 7 × 10−4 cm−1 (0.003 dB/cm) or lower.

© 2010 Optical Society of America

1. Introduction

Rare-earth-ion-doped crystals [1

1. Spectroscopy of Solids Containing Rare-Earth Ions, edited by A. A. Kaplyanskii and R. M. Macfarlane (North-Holland, Amsterdam, 1987).

] have long coherence times for optical transitions up to 6.4 ms [2

2. Y. Sun, C. W. Thiel, R. L. Cone, R. W. Equall, and R. L. Hutcheson, “Recent progress in developing new rare earth materials for hole burning and coherent transient applications,” J. Lumin. 98, 281–287 (2002). [CrossRef]

, 3

3. R. M. Macfarlane, “High-resolution laser spectroscopy of rare-earth doped insulators: a personal perspective,” J. Lumin. 100, 1–20 (2002), and references therein. [CrossRef]

] and for hyperfine transitions up to 1 s [4

4. E. Fraval, M. J. Sellars, and J. J. Longdell, “Dynamic Decoherence Control of a Solid-State Nuclear-Quadrupole Qubit,” Phys. Rev. Lett. 95, 030506 (2005). [CrossRef] [PubMed]

, 5

5. S. E. Beavan, E. Fraval, M. J. Sellars, and J. J. Longdell, “Demonstration of the reduction of decoherent errors in a solid-state qubit using dynamic decoupling techniques,” Phys. Rev. A 80, 032308 (2009). [CrossRef]

]. Therefore, such crystals have been expected to be useful, e.g., for quantum information technologies [6

6. K. Ichimura, “A simple frequency-domain quantum computer with ions in a crystal coupled to a cavity mode,” Opt. Commun. 196, 119–125 (2001). [CrossRef]

13

13. J. H. Wesenberg, K. Mølmer, L. Rippe, and S. Kröll, “Scalable designs for quantum computing with rare-earth-ion-doped crystals,” Phys. Rev. A 75, 012304 (2007). [CrossRef]

].

So far, the following interesting experiments have been achieved with such crystals: the observation of electromagnetically induced transparency in solids [14

14. B. S. Ham, M. S. Shahriar, and P. R. Hemmer, “Enhanced nondegenerate four-wave mixing owing to electromagnetically induced transparency in a spectral hole-burning crystal,” Opt. Lett. 22, 1138–1140 (1997). [CrossRef] [PubMed]

16

16. E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Bretenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, “Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency,” Phys. Rev. B 81, 144303 (2010). [CrossRef]

], light storage or quantum memory for light [17

17. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of Ultraslow and Stored Light Pulses in a Solid,” Phys. Rev. Lett. 88, 023602 (2001). [CrossRef]

20

20. M. Sabooni, F. Beaudoin, A. Walther, N. Lin, A. Amari, M. Huang, and S. Kröll, “Storage and Recall of Weak Coherent Optical Pulses with an Efficiency of 25%,” Phys. Rev. Lett. 105, 060501 (2010). [CrossRef] [PubMed]

], stimulated Raman adiabatic passage in solids [21

21. H. Goto and K. Ichimura, “Population transfer via stimulated Raman adiabatic passage in a solid,” Phys. Rev. A 74, 053410 (2006). [CrossRef]

24

24. A. L. Alexander, R. Lauro, A. Louchet, T. Chanelière, and J. L. Le Gouët, “Stimulated Raman adiabatic passage in Tm3+ :YAG,” Phys. Rev. B 78, 144407 (2008). [CrossRef]

], and quantum gate operations [22

22. H. Goto and K. Ichimura, “Observation of coherent population transfer in a four-level tripod system with a rare-earth-metal-ion-doped crystal,” Phys. Rev. A 75, 033404 (2007). [CrossRef]

, 25

25. J. J. Longdell, M. J. Sellars, and N. B. Manson, “Demonstration of conditional quantum phase shift between ions in a solid,” Phys. Rev. Lett. 93, 130503 (2004). [CrossRef] [PubMed]

].

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

26. Cavity Quantum Electrodynamics, edited by P. R. Berman, Adv. At. Mol. Opt. Phys. Suppl.2 (Academic Press, San Diego, 1994).

]. Several cavity-QED schemes with rare-earth-ion-doped crystals have been theoretically proposed for quantum information technologies [6

6. K. Ichimura, “A simple frequency-domain quantum computer with ions in a crystal coupled to a cavity mode,” Opt. Commun. 196, 119–125 (2001). [CrossRef]

, 7

7. M. S. Shahriar, J. A. Bowers, B. Demsky, P. S. Bhatia, S. Lloyd, P. R. Hemmer, and A. E. Craig, “Cavity dark states for quantum computing,” Opt. Commun. 195, 411–417 (2001). [CrossRef]

, 11

11. Y.-F. Xiao, X.-M. Lin, J. Gao, Y. Yang, Z.-F. Han, and G.-C. Guo, “Realizing Quantum Controlled Phase Flip through Cavity-QED,” Phys. Rev. A 70, 042314 (2004). [CrossRef]

, 12

12. Y.-F. Xiao, Z.-F. Han, Y. Yang, and G.-C. Guo, “Quantum CPF gates between rare earth ions through measurement,” Phys. Lett. A 330, 137–141 (2004). [CrossRef]

, 27

27. D. L. McAuslan, J. J. Longdell, and M. J. Sellars, “Strong-coupling cavity QED using rare-earth-metal-ion dopants in monolithic cavities: What you can do with a weak oscillator,” Phys. Rev. A 80, 062307 (2009). [CrossRef]

]. However, cavity-QED experiments with such crystals have been very few so far.

In the experiments of Refs. [28

28. C. Greiner, B. Boggs, and T.W. Mossberg, “Superradiant Emission Dynamics of an Optically Thin Material Sample in a Short-Decay-Time Optical Cavity,” Phys. Rev. Lett. 85, 3793–3796 (2000). [CrossRef] [PubMed]

30

30. C. Greiner, B. Boggs, and T. W. Mossberg, “Frustrated pulse-area quantization in accelerated superradiant atom-cavity systems,” Phys. Rev. A 67, 063811 (2003). [CrossRef]

], 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

31. K. Ichimura and H. Goto, “Normal-mode coupling of rare-earth-metal ions in a crystal to a macroscopic optical cavity mode,” Phys. Rev. A 74, 033818 (2006). [CrossRef]

], 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

31. K. Ichimura and H. Goto, “Normal-mode coupling of rare-earth-metal ions in a crystal to a macroscopic optical cavity mode,” Phys. Rev. A 74, 033818 (2006). [CrossRef]

], 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.

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

32. G. Li, Y. Zhang, Y. Li, X. Wang, J. Zhang, J. Wang, and T. Zhang, “Precision measurement of ultralow losses of an asymmetric optical microcavity,” Appl. Opt. 45, 7628–7631 (2006). [CrossRef] [PubMed]

], that is not for monolithic cavities and more importantly the way to determine the losses in Ref. [32

32. G. Li, Y. Zhang, Y. Li, X. Wang, J. Zhang, J. Wang, and T. Zhang, “Precision measurement of ultralow losses of an asymmetric optical microcavity,” Appl. Opt. 45, 7628–7631 (2006). [CrossRef] [PubMed]

] 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):
Transmissionspectrum:ST1(ν)=P1ν02ν2+ν02,
(1)
Reflectionspectrum:SR1(ν)=1D1ν02ν2+ν02,
(2)
where ν is the laser detuning from the cavity resonance, ν0 is the half width at half maximum of the transmission spectrum, P1 is the maximum of the transmittance, and (1 – D1) is the minimum of the reflectance.

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.

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

33. A. Yariv, Optical Electronics in Modern Communications, Fifth Edition (Oxford University Press, New York, 1997).

]
F=νFSR2ν0.
(3)

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:
Transmissionspectrum:ST2(ν)=P2ν02ν2+ν02,
(4)
Reflectionspectrum:SR2(ν)=1D2ν02ν2+ν02.
(5)
Note that the half width at half maximum is the same as the above case. (In analysis of experimental data, ν0 is regarded as a fitting parameter in both the cases and the average of them are taken.)

Thus, we obtain the following five experimental data: F, P1, D1, P2, and D2. Using these, we can evaluate the intracavity loss Lin and the transmittances T1 and T2 of M1 and M2, respectively, with the following equations (see Appendix for the derivation):
Lin+T1+T2=2πF,
(6)
P1Lin+(P1D1)T2=0,
(7)
P2Lin+(P2D2)T1=0.
(8)
Here, Lin is defined as Lin = 2L + L1 + L2, where L is the bulk loss on the path from M1 (M2) to M2 (M1), L1 and L2 are the mirror losses of M1 and M2, respectively. (See Appendix for the details.)

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:
C1=P1(2π/F)24T1T2,
(9)
C2=P2(2π/F)24T1T2.
(10)
(In Ref. [32

32. G. Li, Y. Zhang, Y. Li, X. Wang, J. Zhang, J. Wang, and T. Zhang, “Precision measurement of ultralow losses of an asymmetric optical microcavity,” Appl. Opt. 45, 7628–7631 (2006). [CrossRef] [PubMed]

], coupling efficiencies were determined by additional experiments. This fact clearly shows that the method in Ref. [32

32. G. Li, Y. Zhang, Y. Li, X. Wang, J. Zhang, J. Wang, and T. Zhang, “Precision measurement of ultralow losses of an asymmetric optical microcavity,” Appl. Opt. 45, 7628–7631 (2006). [CrossRef] [PubMed]

] and the present one are different.)

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

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.

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

38. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. B31, 97–105 (1983). [CrossRef]

]. 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).

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. Lin: intracavity loss (± denotes the statistical error estimated using the standard errors of ν0, νFSR, 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 Lin 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 Lin ≃ 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 na [39

39. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microcavities for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005). [CrossRef]

], which is a fundamental dimensionless parameter in cavity QED defined as
na=κγg2,
(11)
where κ and γ are the decay rates for the cavity-mode field and the atomic excited state, respectively, and g is the coupling strength between the cavity mode and the atom. The other two fundamental parameter, the Purcell factor Fp and the single-atom cooperativity parameter C1, have the following relations to na:
Fp=4C1=4na.
(12)
In terms of these parameters, the followings are known:
  1. The success probabilities, Ps, of entangling gates in cavity-QED schemes are limited by na, e.g., as Pse4na [40

    40. H. Goto and K. Ichimura, “Upper bound for the success probability of cavity-mediated adiabatic transfer in the presence of dissipation,” Phys. Rev. A 77, 013816 (2008). [CrossRef]

    , 41

    41. H. Goto and K. Ichimura, “Condition for fault-tolerant quantum computation with a cavity-QED scheme,” Phys. Rev. A 82, 032311 (2010). [CrossRef]

    ].
  2. The enhancement of spontaneous emission to the cavity mode, which is useful for a single-photon source, is realized if Fp is larger than 1 (Purcell effect).
  3. Nonclassical photon statistics in cavity QED can be observed if C1 is comparable to or larger than 1 [42

    42. H. Goto and K. Ichimura, “Expectation-value approach to photon statistics in cavity QED,” Pnys. Rev. A 70, 023815 (2004), and references therein. [CrossRef]

    ].

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

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

43. R. W. Equall, R. L. Cone, and R. M. Macfarlane, “Homogeneous broadening and hyperfine structure of optical transitions in Pr3+ :Y2SiO5,” Phys. Rev. B 52, 3963–3969 (1995). [CrossRef]

].

κ is given by
κ=πνFSRF=c(Lin+T1+T2)4nlc>cLin4nlc,
(13)
where c is the speed of light in vacuum, lc is the cavity length, and n is the refractive index of YSO. Thus, the intracavity loss limits κ. (T1 and T2 can be reduced easily by increasing the number of dielectric layers of mirrors.)

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

26. Cavity Quantum Electrodynamics, edited by P. R. Berman, Adv. At. Mol. Opt. Phys. Suppl.2 (Academic Press, San Diego, 1994).

, 31

31. K. Ichimura and H. Goto, “Normal-mode coupling of rare-earth-metal ions in a crystal to a macroscopic optical cavity mode,” Phys. Rev. A 74, 033818 (2006). [CrossRef]

]
g=μnωc2h¯ɛ0V=2μnwch¯ɛ0lcλc,
(14)
where μ is the transition dipole moment, V is the mode volume, ɛ0 is the permittivity of vacuum, is Planck’s constant divided by 2π, ωc and λc are the cavity resonance angular frequency and wavelength, respectively, and w is the mode radius. Thus, the smallest na (T1 = T2 = 0) becomes
na(w)=h¯ɛ0nγλcLin16μ2w2.
(15)
Note that na(w) does not depend on lc explicitly.

The w dependence of the smallest na is shown in Fig. 3. (In the present experiments, w = 19 μm.) The other parameters are set as follows: Lin = 10−3, γ = 10 kHz, n = 1.9, μ ≃ 3 × 10−32 C·m [44

44. M. Nilsson, L. Rippe, S. Kröll, R. Klieber, and D. Suter, “Hole-burning techniques for isolation and study of individual hyperfine transitions in inhomogeneously broadened solids demonstrated in Pr3+:Y2SiO5,” Phys. Rev. B 70, 214116 (2004). [CrossRef]

], and λc = 606 nm. (n was determined by the measurement results of FSR.) From Fig. 3, it turns out that na(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, Lin = 0.1%, is fairly high for cavity-QED applications.

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

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, Lin of Sample 1 is only one order of magnitude smaller than that of Sample 2. This strongly indicates that Lin 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

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 – T1R1 and L2 = 1 – T2R2 by definition. E denotes the electric field.

Solving Eqs. (16)(21), we obtain the spectra as follows:
ST1(θ)=|ET(θ)EI|2=T1T2(1L)|1(1L)R1R2e2iθ|2,
(22)
SR1(θ)=|ER(θ)EI|2=|R1(1L1)(1L)R2e2iθ|2|1(1L)R1R2e2iθ|2,
(23)
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:
ST1(θ)4T1T2(Lin+T1+T2)2+(4θ)2,
(24)
SR1(θ)(Lin+T2T1)2+(4θ)2(Lin+T1+T2)2+(4θ)2,
(25)
where we have defined the intracavity loss as
Lin2L+L1+L2.
(26)
Note that we cannot determine L1, L2, and L separately. We can determine only Lin. 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:
ST1(θ)=C14T1T2(Lin+T1+T2)2+(4θ)2,
(27)
SR1(θ)=C1(Lin+T2T1)2+(4θ)2(Lin+T1+T2)2+(4θ)2+(1C1)=1C14T1(Lin+T2)(Lin+T1+T2)2+(4θ)2.
(28)

Using the above results, we obtain
F=2πLin+T1+T2,
(29)
P1=C14T1T2(Lin+T1+T2)2=4C1T1T2(2π/F)2,
(30)
D1=C14T1(Lin+T2)(Lin+T1+T2)2=Lin+T2T2P1.
(31)
In a similar manner, we also obtain
P2=4C2T1T2(2π/F)2,
(32)
D2=Lin+T2T2P2.
(33)

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

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 – T1R1 and L2 = 1 – T2R2. E denotes the electric field.

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.

References and links

1.

Spectroscopy of Solids Containing Rare-Earth Ions, edited by A. A. Kaplyanskii and R. M. Macfarlane (North-Holland, Amsterdam, 1987).

2.

Y. Sun, C. W. Thiel, R. L. Cone, R. W. Equall, and R. L. Hutcheson, “Recent progress in developing new rare earth materials for hole burning and coherent transient applications,” J. Lumin. 98, 281–287 (2002). [CrossRef]

3.

R. M. Macfarlane, “High-resolution laser spectroscopy of rare-earth doped insulators: a personal perspective,” J. Lumin. 100, 1–20 (2002), and references therein. [CrossRef]

4.

E. Fraval, M. J. Sellars, and J. J. Longdell, “Dynamic Decoherence Control of a Solid-State Nuclear-Quadrupole Qubit,” Phys. Rev. Lett. 95, 030506 (2005). [CrossRef] [PubMed]

5.

S. E. Beavan, E. Fraval, M. J. Sellars, and J. J. Longdell, “Demonstration of the reduction of decoherent errors in a solid-state qubit using dynamic decoupling techniques,” Phys. Rev. A 80, 032308 (2009). [CrossRef]

6.

K. Ichimura, “A simple frequency-domain quantum computer with ions in a crystal coupled to a cavity mode,” Opt. Commun. 196, 119–125 (2001). [CrossRef]

7.

M. S. Shahriar, J. A. Bowers, B. Demsky, P. S. Bhatia, S. Lloyd, P. R. Hemmer, and A. E. Craig, “Cavity dark states for quantum computing,” Opt. Commun. 195, 411–417 (2001). [CrossRef]

8.

N. Ohlsson, R. K. Mohan, and S. Kröll, “Quantum computer hardware based on rare-earth-ion-doped inorganic crystals,” Opt. Commun. 201, 71–77 (2002). [CrossRef]

9.

J. Wesenberg and K. Mølmer, “Robust quantum gates and a bus architecture for quantum computing with rare-earth-ion-doped crystals,” Phys. Rev. A 68, 012320 (2003). [CrossRef]

10.

I. Roos and K. Mølmer, “Quantum computing with an inhomogeneously broadened ensemble of ions: Suppression of errors from detuning variations by specially adapted pulses and coherent population trapping,” Phys. Rev. A 69, 022321 (2004). [CrossRef]

11.

Y.-F. Xiao, X.-M. Lin, J. Gao, Y. Yang, Z.-F. Han, and G.-C. Guo, “Realizing Quantum Controlled Phase Flip through Cavity-QED,” Phys. Rev. A 70, 042314 (2004). [CrossRef]

12.

Y.-F. Xiao, Z.-F. Han, Y. Yang, and G.-C. Guo, “Quantum CPF gates between rare earth ions through measurement,” Phys. Lett. A 330, 137–141 (2004). [CrossRef]

13.

J. H. Wesenberg, K. Mølmer, L. Rippe, and S. Kröll, “Scalable designs for quantum computing with rare-earth-ion-doped crystals,” Phys. Rev. A 75, 012304 (2007). [CrossRef]

14.

B. S. Ham, M. S. Shahriar, and P. R. Hemmer, “Enhanced nondegenerate four-wave mixing owing to electromagnetically induced transparency in a spectral hole-burning crystal,” Opt. Lett. 22, 1138–1140 (1997). [CrossRef] [PubMed]

15.

K. Ichimura, K. Yamamoto, and N. Gemma, “Evidence for electromagnetically induced transparency in a solid medium,” Phys. Rev. A 58, 4116–4120 (1998). [CrossRef]

16.

E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Bretenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, “Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency,” Phys. Rev. B 81, 144303 (2010). [CrossRef]

17.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of Ultraslow and Stored Light Pulses in a Solid,” Phys. Rev. Lett. 88, 023602 (2001). [CrossRef]

18.

J. J. Longdell, E. Fraval, M. J. Sellars, and N. B. Manson, “Stopped Light with Storage Times Greater than One Second Using Electromagnetically Induced Transparency in a Solid,” Phys. Rev. Lett. 95, 063601 (2005). [CrossRef] [PubMed]

19.

M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, “Efficient quantum memory for light,” Nature 465, 1052–1056 (2010). [CrossRef] [PubMed]

20.

M. Sabooni, F. Beaudoin, A. Walther, N. Lin, A. Amari, M. Huang, and S. Kröll, “Storage and Recall of Weak Coherent Optical Pulses with an Efficiency of 25%,” Phys. Rev. Lett. 105, 060501 (2010). [CrossRef] [PubMed]

21.

H. Goto and K. Ichimura, “Population transfer via stimulated Raman adiabatic passage in a solid,” Phys. Rev. A 74, 053410 (2006). [CrossRef]

22.

H. Goto and K. Ichimura, “Observation of coherent population transfer in a four-level tripod system with a rare-earth-metal-ion-doped crystal,” Phys. Rev. A 75, 033404 (2007). [CrossRef]

23.

J. Klein, F. Beil, and T. Halfmann, “Robust Population Transfer by Stimulated Raman Adiabatic Passage in a Pr3+ :Y2SiO5,” Phys. Rev. Lett. 99, 113003 (2007). [CrossRef] [PubMed]

24.

A. L. Alexander, R. Lauro, A. Louchet, T. Chanelière, and J. L. Le Gouët, “Stimulated Raman adiabatic passage in Tm3+ :YAG,” Phys. Rev. B 78, 144407 (2008). [CrossRef]

25.

J. J. Longdell, M. J. Sellars, and N. B. Manson, “Demonstration of conditional quantum phase shift between ions in a solid,” Phys. Rev. Lett. 93, 130503 (2004). [CrossRef] [PubMed]

26.

Cavity Quantum Electrodynamics, edited by P. R. Berman, Adv. At. Mol. Opt. Phys. Suppl.2 (Academic Press, San Diego, 1994).

27.

D. L. McAuslan, J. J. Longdell, and M. J. Sellars, “Strong-coupling cavity QED using rare-earth-metal-ion dopants in monolithic cavities: What you can do with a weak oscillator,” Phys. Rev. A 80, 062307 (2009). [CrossRef]

28.

C. Greiner, B. Boggs, and T.W. Mossberg, “Superradiant Emission Dynamics of an Optically Thin Material Sample in a Short-Decay-Time Optical Cavity,” Phys. Rev. Lett. 85, 3793–3796 (2000). [CrossRef] [PubMed]

29.

C. Greiner, T. Wang, T. Loftus, and T.W. Mossberg, “Instability and Pulse Area Quantization in Accelerated Superradiant Atom-Cavity Systems,” Phys. Rev. Lett. 87, 253602 (2001). [CrossRef] [PubMed]

30.

C. Greiner, B. Boggs, and T. W. Mossberg, “Frustrated pulse-area quantization in accelerated superradiant atom-cavity systems,” Phys. Rev. A 67, 063811 (2003). [CrossRef]

31.

K. Ichimura and H. Goto, “Normal-mode coupling of rare-earth-metal ions in a crystal to a macroscopic optical cavity mode,” Phys. Rev. A 74, 033818 (2006). [CrossRef]

32.

G. Li, Y. Zhang, Y. Li, X. Wang, J. Zhang, J. Wang, and T. Zhang, “Precision measurement of ultralow losses of an asymmetric optical microcavity,” Appl. Opt. 45, 7628–7631 (2006). [CrossRef] [PubMed]

33.

A. Yariv, Optical Electronics in Modern Communications, Fifth Edition (Oxford University Press, New York, 1997).

34.

D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-silica microspheres in the near infrared,” Opt. Lett. 23, 247–249 (1998). [CrossRef]

35.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Demonstration of ultra-high-Q small mode volume toroid microcavities on a chip,” Appl. Phys. Lett. 85, 6113–6115 (2004), [CrossRef]

36.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical cavities with ten million finesse,” Opt. Express 15, 6768–6772 (2007). [CrossRef] [PubMed]

37.

V. S. Ilchenko, A. A. Savchenkov, J. Byrd, I. Solomatine, A. B Matsko, D. Seidel, and L. Maleki, “Crystal quartz optical whispering-gallery cavities,” Opt. Lett. 33, 1569–1571 (2008). [CrossRef] [PubMed]

38.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. B31, 97–105 (1983). [CrossRef]

39.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microcavities for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005). [CrossRef]

40.

H. Goto and K. Ichimura, “Upper bound for the success probability of cavity-mediated adiabatic transfer in the presence of dissipation,” Phys. Rev. A 77, 013816 (2008). [CrossRef]

41.

H. Goto and K. Ichimura, “Condition for fault-tolerant quantum computation with a cavity-QED scheme,” Phys. Rev. A 82, 032311 (2010). [CrossRef]

42.

H. Goto and K. Ichimura, “Expectation-value approach to photon statistics in cavity QED,” Pnys. Rev. A 70, 023815 (2004), and references therein. [CrossRef]

43.

R. W. Equall, R. L. Cone, and R. M. Macfarlane, “Homogeneous broadening and hyperfine structure of optical transitions in Pr3+ :Y2SiO5,” Phys. Rev. B 52, 3963–3969 (1995). [CrossRef]

44.

M. Nilsson, L. Rippe, S. Kröll, R. Klieber, and D. Suter, “Hole-burning techniques for isolation and study of individual hyperfine transitions in inhomogeneously broadened solids demonstrated in Pr3+:Y2SiO5,” Phys. Rev. B 70, 214116 (2004). [CrossRef]

OCIS Codes
(140.4780) Lasers and laser optics : Optical resonators
(160.5690) Materials : Rare-earth-doped materials
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: September 7, 2010
Revised Manuscript: October 8, 2010
Manuscript Accepted: October 17, 2010
Published: October 27, 2010

Citation
Hayato Goto, Satoshi Nakamura, and Kouichi Ichimura, "Experimental determination of intracavity losses of monolithic Fabry-Perot cavities made of Pr3+:Y2SiO5," Opt. Express 18, 23763-23775 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-23763


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References

  1. Spectroscopy of Solids Containing Rare-Earth Ions, edited by A. A. Kaplyanskii and R. M. Macfarlane (North-Holland, Amsterdam, 1987).
  2. Y. Sun, C. W. Thiel, R. L. Cone, R. W. Equall, and R. L. Hutcheson, "Recent progress in developing new rare earth materials for hole burning and coherent transient applications," J. Lumin. 98, 281-287 (2002). [CrossRef]
  3. R. M. Macfarlane, "High-resolution laser spectroscopy of rare-earth doped insulators: a personal perspective," J. Lumin. 100, 1-20 (2002) (and references therein). [CrossRef]
  4. E. Fraval, M. J. Sellars, and J. J. Longdell, "Dynamic Decoherence Control of a Solid-State Nuclear-Quadrupole Qubit," Phys. Rev. Lett. 95, 030506 (2005). [CrossRef] [PubMed]
  5. S. E. Beavan, E. Fraval, M. J. Sellars, and J. J. Longdell, "Demonstration of the reduction of decoherent errors in a solid-state qubit using dynamic decoupling techniques," Phys. Rev. A 80, 032308 (2009). [CrossRef]
  6. K. Ichimura, "A simple frequency-domain quantum computer with ions in a crystal coupled to a cavity mode," Opt. Commun. 196, 119-125 (2001). [CrossRef]
  7. M. S. Shahriar, J. A. Bowers, B. Demsky, P. S. Bhatia, S. Lloyd, P. R. Hemmer, and A. E. Craig, "Cavity dark states for quantum computing," Opt. Commun. 195, 411-417 (2001). [CrossRef]
  8. N. Ohlsson, R. K. Mohan, and S. Kröll, "Quantum computer hardware based on rare-earth-ion-doped inorganic crystals," Opt. Commun. 201, 71-77 (2002). [CrossRef]
  9. J. Wesenberg, and K. Mølmer, "Robust quantum gates and a bus architecture for quantum computing with rare-earth-ion-doped crystals," Phys. Rev. A 68, 012320 (2003). [CrossRef]
  10. I. Roos, and K. Mølmer, "Quantum computing with an inhomogeneously broadened ensemble of ions: Suppression of errors from detuning variations by specially adapted pulses and coherent population trapping," Phys. Rev. A 69, 022321 (2004). [CrossRef]
  11. Y.-F. Xiao, X.-M. Lin, J. Gao, Y. Yang, Z.-F. Han, and G.-C. Guo, "Realizing Quantum Controlled Phase Flip through Cavity-QED," Phys. Rev. A 70, 042314 (2004). [CrossRef]
  12. Y.-F. Xiao, Z.-F. Han, Y. Yang, and G.-C. Guo, "Quantum CPF gates between rare earth ions through measurement," Phys. Lett. A 330, 137-141 (2004). [CrossRef]
  13. J. H. Wesenberg, K. Mølmer, L. Rippe, and S. Kröll, "Scalable designs for quantum computing with rare-earth ion-doped crystals," Phys. Rev. A 75, 012304 (2007). [CrossRef]
  14. B. S. Ham, M. S. Shahriar, and P. R. Hemmer, "Enhanced nondegenerate four-wave mixing owing to electromagnetically induced transparency in a spectral hole-burning crystal," Opt. Lett. 22, 1138-1140 (1997). [CrossRef] [PubMed]
  15. K. Ichimura, K. Yamamoto, and N. Gemma, "Evidence for electromagnetically induced transparency in a solid medium," Phys. Rev. A 58, 4116-4120 (1998). [CrossRef]
  16. E. Baldit, K. Bencheikh, P. Monnier, S. Briaudeau, J. A. Levenson, V. Crozatier, I. Lorgeré, F. Bretenaker, J. L. Le Gouët, O. Guillot-Noël, and Ph. Goldner, "Identification of Λ-like systems in Er3+:Y2SiO5 and observation of electromagnetically induced transparency," Phys. Rev. B 81, 144303 (2010). [CrossRef]
  17. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, "Observation of Ultraslow and Stored Light Pulses in a Solid," Phys. Rev. Lett. 88, 023602 (2001). [CrossRef]
  18. J. J. Longdell, E. Fraval, M. J. Sellars, and N. B. Manson, "Stopped Light with Storage Times Greater than One Second Using Electromagnetically Induced Transparency in a Solid," Phys. Rev. Lett. 95, 063601 (2005). [CrossRef] [PubMed]
  19. M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, "Efficient quantum memory for light," Nature 465, 1052-1056 (2010). [CrossRef] [PubMed]
  20. M. Sabooni, F. Beaudoin, A. Walther, N. Lin, A. Amari, M. Huang, and S. Kröll, "Storage and Recall of Weak Coherent Optical Pulses with an Efficiency of 25%," Phys. Rev. Lett. 105, 060501 (2010). [CrossRef] [PubMed]
  21. H. Goto, and K. Ichimura, "Population transfer via stimulated Raman adiabatic passage in a solid," Phys. Rev. A 74, 053410 (2006). [CrossRef]
  22. H. Goto, and K. Ichimura, "Observation of coherent population transfer in a four-level tripod system with a rare-earth-metal-ion-doped crystal," Phys. Rev. A 75, 033404 (2007). [CrossRef]
  23. J. Klein, F. Beil, and T. Halfmann, "Robust Population Transfer by Stimulated Raman Adiabatic Passage in a Pr3+:Y2SiO5," Phys. Rev. Lett. 99, 113003 (2007). [CrossRef] [PubMed]
  24. A. L. Alexander, R. Lauro, A. Louchet, T. Chanelière, and J. L. Le Gouët, "Stimulated Raman adiabatic passage in Tm3+:YAG," Phys. Rev. B 78, 144407 (2008). [CrossRef]
  25. J. J. Longdell, M. J. Sellars, and N. B. Manson, "Demonstration of conditional quantum phase shift between ions in a solid," Phys. Rev. Lett. 93, 130503 (2004). [CrossRef] [PubMed]
  26. C. Q. Electrodynamics, edited by P. R. Berman, Adv. At. Mol. Opt. Phys. Suppl. 2 (Academic Press, San Diego, 1994).
  27. D. L. McAuslan, J. J. Longdell, and M. J. Sellars, "Strong-coupling cavity QED using rare-earth-metal-ion dopants in monolithic cavities: What you can do with a weak oscillator," Phys. Rev. A 80, 062307 (2009). [CrossRef]
  28. C. Greiner, B. Boggs, and T. W. Mossberg, "Superradiant Emission Dynamics of an Optically Thin Material Sample in a Short-Decay-Time Optical Cavity," Phys. Rev. Lett. 85, 3793-3796 (2000). [CrossRef] [PubMed]
  29. C. Greiner, T. Wang, T. Loftus, and T. W. Mossberg, "Instability and Pulse Area Quantization in Accelerated Superradiant Atom-Cavity Systems," Phys. Rev. Lett. 87, 253602 (2001). [CrossRef] [PubMed]
  30. C. Greiner, B. Boggs, and T. W. Mossberg, "Frustrated pulse-area quantization in accelerated superradiant atom cavity systems," Phys. Rev. A 67, 063811 (2003). [CrossRef]
  31. K. Ichimura, and H. Goto, "Normal-mode coupling of rare-earth-metal ions in a crystal to a macroscopic optical cavity mode," Phys. Rev. A 74, 033818 (2006). [CrossRef]
  32. G. Li, Y. Zhang, Y. Li, X. Wang, J. Zhang, J. Wang, and T. Zhang, "Precision measurement of ultralow losses of an asymmetric optical microcavity," Appl. Opt. 45, 7628-7631 (2006). [CrossRef] [PubMed]
  33. A. Yariv, Optical Electronics in Modern Communications, Fifth Edition (Oxford University Press, New York, 1997).
  34. D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, "High-Q measurements of fused silica microspheres in the near infrared," Opt. Lett. 23, 247-249 (1998). [CrossRef]
  35. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Demonstration of ultra-high-Q small mode volume toroid microcavities on a chip," Appl. Phys. Lett. 85, 6113-6115 (2004). [CrossRef]
  36. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, "Optical cavities with ten million finesse," Opt. Express 15, 6768-6772 (2007). [CrossRef] [PubMed]
  37. V. S. Ilchenko, A. A. Savchenkov, J. Byrd, I. Solomatine, A. B. Matsko, D. Seidel, and L. Maleki, "Crystal quartz optical whispering-gallery cavities," Opt. Lett. 33, 1569-1571 (2008). [CrossRef] [PubMed]
  38. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys., B Photophys. Laser Chem. B31, 97-105 (1983). [CrossRef]
  39. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, "Ultrahigh-Q toroidal microcavities for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005). [CrossRef]
  40. H. Goto, and K. Ichimura, "Upper bound for the success probability of cavity-mediated adiabatic transfer in the presence of dissipation," Phys. Rev. A 77, 013816 (2008). [CrossRef]
  41. H. Goto, and K. Ichimura, "Condition for fault-tolerant quantum computation with a cavity-QED scheme," Phys. Rev. A 82, 032311 (2010). [CrossRef]
  42. H. Goto, and K. Ichimura, "Expectation-value approach to photon statistics in cavity QED," Phys. Rev. A 70, 023815 (2004) (and references therein). [CrossRef]
  43. R. W. Equall, R. L. Cone, and R. M. Macfarlane, "Homogeneous broadening and hyperfine structure of optical transitions in Pr3+:Y2SiO5," Phys. Rev. B 52, 3963-3969 (1995). [CrossRef]
  44. M. Nilsson, L. Rippe, S. Kröll, R. Klieber, and D. Suter, "Hole-burning techniques for isolation and study of individual hyperfine transitions in inhomogeneously broadened solids demonstrated in Pr3+:Y2SiO5," Phys. Rev. B 70, 214116 (2004). [CrossRef]

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