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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 20 — Oct. 7, 2013
  • pp: 24332–24343
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Cavity-enhanced spectroscopy of a rare-earth-ion-doped crystal: Observation of a power law for inhomogeneous broadening

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


Optics Express, Vol. 21, Issue 20, pp. 24332-24343 (2013)
http://dx.doi.org/10.1364/OE.21.024332


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Abstract

We experimentally demonstrate cavity-enhanced spectroscopy of a rare-earth-ion-doped crystal (Pr3+:Y2SiO5). 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 OSA

1. Introduction

Rare-earth-ion-doped crystals [1

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

, 2

2. G. Liu and B. Jacquier, Spectroscopic Properties of Rare Earths in Optical Materials (Springer, 2005).

] have attractive features for quantum information technology, e.g., very long coherence times up to 30 s [3

3. E. Fraval, M. J. Sellars, and J. J. Longdell, “Method of extending hyperfine coherence times in Pr3+:Y2SiO5,” Phys. Rev. Lett. 92, 077601 (2004). [CrossRef]

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]

]. With such crystals, quantum memory for light [6

6. 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]

10

10. B. Lauritzen, J. Minář, H. de Riedmatten, M. Afzelius, N. Sangouard, C. Simon, and N. Gisin, “Telecommunication-wavelength solid-state memory at the single photon level,” Phys. Rev. Lett. 104, 080502 (2010). [CrossRef] [PubMed]

], stimulated Raman adiabatic passage in solids [11

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

14

14. 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 [12

12. 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]

, 15

15. 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]

] 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

16. 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]

20

20. 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]

]. The key technologies for these schemes are based on cavity quantum electrodynamics (cavity QED) [21

21. P. R. Berman, Cavity Quantum Electrodynamics(Academic, 1994).

]. 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 4f-5d transitions [22

22. R. Kolesov, K. Xia, R. Reuter, R. Stöhr, A. Zappe, J. Meijer, P. R. Hemmer, and J. Wrachtrup, “Optical detection of a single rare-earth ion in a crystal,” Nat. Commun. 3, 1029 (2012). [CrossRef] [PubMed]

] or a single-electron transistor [23

23. C. Yin, M. Rancic, G. G. de Boo, N. Stavrias, J. C. McCallum, M. J. Sellars, and S. Rogge, “Optical addressing of an individual erbium ion in silicon,” Nature 497, 91–94 (2013). [CrossRef] [PubMed]

]. 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

24. 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]

28

28. D. L. McAuslan, D. Korystov, and J. J. Longdell, “Coherent spectroscopy of rare-earth-metal-ion-doped whispering-gallery-mode resonators,” Phys. Rev. A 83, 063847 (2011). [CrossRef]

]. 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 Pr3+:Y2SiO5 (Pr:YSO) [29

29. H. Goto, S. Nakamura, and K. Ichimura, “Experimental determination of intracavity losses of monolithic Fabry-Perot cavities made of Pr3+:Y2SiO5,” Opt. Exp. 18, 23763 (2010). [CrossRef]

]. 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 |νν0|−2.4, where ν is the ionic transition frequency and ν0 is the center frequency of the broadening. Compared with a theoretical model [30

30. A. M. Stoneham, “Shapes of inhomogeneously broadened resonance lines in solids,” Rev. Mod. Phys. 41, 82–108 (1969). [CrossRef]

], 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 μm).

Fig. 1 Sample cavity. This is a monolithic Fabry-Perot cavity made of Pr:YSO, the dopant concentration of which is 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. The radius of the mode waist, which is on the planar mirror, is about 10 μm. The free spectral range 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 Pr3+ ions is about 0.15%.

The maximum coupling rate gmax between ions and the cavity mode is theoretically given by [21

21. P. R. Berman, Cavity Quantum Electrodynamics(Academic, 1994).

, 27

27. 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]

, 29

29. H. Goto, S. Nakamura, and K. Ichimura, “Experimental determination of intracavity losses of monolithic Fabry-Perot cavities made of Pr3+:Y2SiO5,” Opt. Exp. 18, 23763 (2010). [CrossRef]

]
gmax=μnωc2h¯ε0V=2μnwch¯ε0lcλc,
(1)
where μ is the transition dipole moment, n is the refractive index of the crystal, V is the mode volume, ε0 is the permittivity of vacuum, ωc and λc are the cavity resonance angular frequency and wavelength, respectively, lc is the cavity length, and w is the mode radius. (In general, the coupling rate depends on the ionic position [21

21. P. R. Berman, Cavity Quantum Electrodynamics(Academic, 1994).

, 27

27. 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]

]. The maximum value gmax 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 gmax/(2π) reaches 3 kHz and exceeds the decay rate, γ, of the ionic excited states [γ/(2π) ≃ 1 kHz [31

31. 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]

]].

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 Pr3+ ions is about 0.15%. The transmittances and intracavity loss of the cavity were determined by the method proposed in [29

29. H. Goto, S. Nakamura, and K. Ichimura, “Experimental determination of intracavity losses of monolithic Fabry-Perot cavities made of Pr3+:Y2SiO5,” Opt. Exp. 18, 23763 (2010). [CrossRef]

].

2.2. Experimental setup

The experimental setup is depicted in Fig. 2.

Fig. 2 Experimental setup. AOM: acousto-optic modulator. AWG: arbitrary waveform generator. SA: spectrum analyzer. Amp: amplifier. PBS: polarizing beamsplitter. HWP: half-wave plate. QWP: quarter-wave plate. HBS: half beamsplitter. CL: cylindrical lens. PD: photodetector. ND: variable neutral density filter. LO: local oscillator. PS: power splitter.

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

29. H. Goto, S. Nakamura, and K. Ichimura, “Experimental determination of intracavity losses of monolithic Fabry-Perot cavities made of Pr3+:Y2SiO5,” Opt. Exp. 18, 23763 (2010). [CrossRef]

, 32

32. 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]

].

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.)

Fig. 3 Laser frequency shift. All the frequencies of the three beams (probe, LO, and re-pump) are controlled simultaneously by the single AOM with AWG1 (see Fig. 2). The origin of this frequency shift is set at the starting point of the spectrum measurement. The lower figures show how the repumping is achieved for an ion. The arrows with (a), (b), and (c) in the upper figure indicate the points at which the situations in the lower figures (a), (b), and (c) occur, respectively. See Fig. 4 for the detailed energy structure of Pr3+ ions in YSO.

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 D2 axis of YSO crystal so that the absorption due to ions becomes maximum [2

2. G. Liu and B. Jacquier, Spectroscopic Properties of Rare Earths in Optical Materials (Springer, 2005).

]. 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/cm2. The polarization of the repump is also set to be parallel to the D2 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 103 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 Pr3+ 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 S0(ν) can be expressed as [29

29. H. Goto, S. Nakamura, and K. Ichimura, “Experimental determination of intracavity losses of monolithic Fabry-Perot cavities made of Pr3+:Y2SiO5,” Opt. Exp. 18, 23763 (2010). [CrossRef]

]
S0(ν)=C04TpTs(Lin+Tp+Ts)2+(4πν/νFSR)2,
(2)
where Tp and Ts are the transmittances of the planar and spherical mirrors, respectively, Lin is the intracavity loss per one round trip not including the absorption due to Pr3+ ions, νFSR is the FSR of the cavity, and C0 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 Pr3+ ions) is measured. (The repumping mechanism is presented in detail below.) This spectrum S(ν) can be expressed as
S(ν)=C4TpTs[Lin+2A(ν)+Tp+Ts]2+(4πν/νFSR)2,
(3)
where A(ν) is the single-pass absorption loss due to Pr3+ 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 S0(ν), we obtain
S¯(ν)S(ν)S0(ν)CC0(14(Lin+Tp+Ts)A(ν)(Lin+Tp+Ts)2+(4πν/νFSR)2),
(4)
where the first-order approximation with respect to 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/C0 and eliminate this factor from (ν) dividing (ν) by this estimated value. The renormalized (ν) is denoted by S¯(ν).

If |ν| is sufficiently smaller than the half width at half maximum of the cavity [(Lin + Tp + Ts)νFSR/(4π) ≃ 1.35 MHz], the following further approximation is valid:
S¯(ν)12πFA(ν),
(5)
where F is the finesse of the cavity given by F = 2π/(Lin + Tp + Ts).

Thus, it has been shown that the signal of the absorption becomes larger by the factor of the finesse (more correctly, 2F/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 Pr3+ ions in YSO is depicted in Fig. 4 [33

33. M. Lovrić, P. Glasenapp, and D. Suter, “Spin Hamiltonian characterization and refinement for Pr3+:YAlO3and Pr3+:Y2SiO5,” Phys. Rev. B 85, 014429 (2012). [CrossRef]

]. We chose the |± 5/2〉g−|± 5/2〉e transition to observe the absorption because this transition has the largest oscillator strength (electric dipole moment: μ = 2.7 × 10−32 C m) [34

34. 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]

]. The purpose of the repumping is to return the ionic population from |±1/2〉g 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〉e and |± 3/2〉g−|± 1/2〉e 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〉g−| ± 5/2〉e 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 Pr3+ 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/C0, 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.

Fig. 4 Energy-level structure of Pr3+ ions in YSO [33].

Fig. 5 Example of actual data. (a) Transmission spectra with and without the repump, S(ν) and S0(ν), respectively, measured by the balanced heterodyne detection. (The spectrum analyzer (SA) output of balanced heterodyne detection is proportional to the transmission power.) (b) (ν) (dB), which is obtained by subtracting S0(ν) (dBm) from S(ν) (dBm). (c) S¯(ν), which is obtained by renormalizing (ν) with the line fitted to (ν) in the range lower than 0.8 MHz (the dashed line in (b)).
Fig. 6 Experimental results. The cavity modes are numbered 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. Each result is the average of five measurements. The spectra used for each measurement are the ones obtained by 4 × 103 times accumulation for Modes 1 and −1 and 1 × 104 times accumulation for the other modes.

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 × 103 times accumulation for Modes 1 and −1 and 1 × 104 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|nn0|b, where d is the magnitude of the dip, n is the continuous mode number, n0 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 n0 are converted to frequencies ν and ν0 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.

Fig. 7 Power law for the inhomogeneous broadening. (a) The circles are the experimental data. The curve is the result of fitting the function d(n) = a|nn0|b to the data, where n is the continuous mode number, n0 is the value of n corresponding to the center of the inhomogeneous broadening, and a and b are constants. (b) Log-log plot of the same data as (a). n and n0 are converted to frequencies ν and ν0 with the FSR of the cavity (9.2 GHz). The circles and triangles correspond to the positive and negative mode numbers, respectively.

Finally, we briefly discuss the origin of the inhomogeneous broadening. The shape of inhomogeneous broadening has been theoretically discussed in detail in [30

30. A. M. Stoneham, “Shapes of inhomogeneously broadened resonance lines in solids,” Rev. Mod. Phys. 41, 82–108 (1969). [CrossRef]

]. 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 Pr3+:Y2SiO5 (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 Pr3+ 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.

References and links

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2.

G. Liu and B. Jacquier, Spectroscopic Properties of Rare Earths in Optical Materials (Springer, 2005).

3.

E. Fraval, M. J. Sellars, and J. J. Longdell, “Method of extending hyperfine coherence times in Pr3+:Y2SiO5,” Phys. Rev. Lett. 92, 077601 (2004). [CrossRef]

4.

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6.

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7.

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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]

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B. Lauritzen, J. Minář, H. de Riedmatten, M. Afzelius, N. Sangouard, C. Simon, and N. Gisin, “Telecommunication-wavelength solid-state memory at the single photon level,” Phys. Rev. Lett. 104, 080502 (2010). [CrossRef] [PubMed]

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

12.

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]

13.

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]

14.

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]

15.

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]

16.

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]

17.

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]

18.

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]

19.

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]

20.

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]

21.

P. R. Berman, Cavity Quantum Electrodynamics(Academic, 1994).

22.

R. Kolesov, K. Xia, R. Reuter, R. Stöhr, A. Zappe, J. Meijer, P. R. Hemmer, and J. Wrachtrup, “Optical detection of a single rare-earth ion in a crystal,” Nat. Commun. 3, 1029 (2012). [CrossRef] [PubMed]

23.

C. Yin, M. Rancic, G. G. de Boo, N. Stavrias, J. C. McCallum, M. J. Sellars, and S. Rogge, “Optical addressing of an individual erbium ion in silicon,” Nature 497, 91–94 (2013). [CrossRef] [PubMed]

24.

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]

25.

C. Greiner, T. Wang, T. Loftus, and T.W. Mossberg, “Instability and pulse area quantization in accelerated super-radiant atom-cavity systems,” Phys. Rev. Lett. 87, 253602 (2001). [CrossRef]

26.

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]

27.

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]

28.

D. L. McAuslan, D. Korystov, and J. J. Longdell, “Coherent spectroscopy of rare-earth-metal-ion-doped whispering-gallery-mode resonators,” Phys. Rev. A 83, 063847 (2011). [CrossRef]

29.

H. Goto, S. Nakamura, and K. Ichimura, “Experimental determination of intracavity losses of monolithic Fabry-Perot cavities made of Pr3+:Y2SiO5,” Opt. Exp. 18, 23763 (2010). [CrossRef]

30.

A. M. Stoneham, “Shapes of inhomogeneously broadened resonance lines in solids,” Rev. Mod. Phys. 41, 82–108 (1969). [CrossRef]

31.

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]

32.

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]

33.

M. Lovrić, P. Glasenapp, and D. Suter, “Spin Hamiltonian characterization and refinement for Pr3+:YAlO3and Pr3+:Y2SiO5,” Phys. Rev. B 85, 014429 (2012). [CrossRef]

34.

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:
Materials

History
Original Manuscript: September 19, 2013
Manuscript Accepted: September 24, 2013
Published: October 3, 2013

Citation
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," Opt. Express 21, 24332-24343 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-24332


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References

  1. A. A. Kaplyanskii and R. M. Macfarlane, Spectroscopy of Solids Containing Rare-Earth Ions (North-Holland, 1987).
  2. G. Liu and B. Jacquier, Spectroscopic Properties of Rare Earths in Optical Materials (Springer, 2005).
  3. E. Fraval, M. J. Sellars, and J. J. Longdell, “Method of extending hyperfine coherence times in Pr3+:Y2SiO5,” Phys. Rev. Lett.92, 077601 (2004). [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. A80, 032308 (2009). [CrossRef]
  6. 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]
  7. 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]
  8. M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, “Efficient quantum memory for light,” Nature465, 1052–1056 (2010). [CrossRef] [PubMed]
  9. 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]
  10. B. Lauritzen, J. Minář, H. de Riedmatten, M. Afzelius, N. Sangouard, C. Simon, and N. Gisin, “Telecommunication-wavelength solid-state memory at the single photon level,” Phys. Rev. Lett.104, 080502 (2010). [CrossRef] [PubMed]
  11. H. Goto and K. Ichimura, “Population transfer via stimulated Raman adiabatic passage in a solid,” Phys. Rev. A74, 053410 (2006). [CrossRef]
  12. 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. A75, 033404 (2007). [CrossRef]
  13. 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]
  14. A. L. Alexander, R. Lauro, A. Louchet, T. Chanelière, and J. L. Le Gouët, “Stimulated Raman adiabatic passage in Tm3+:YAG,” Phys. Rev. B78, 144407 (2008). [CrossRef]
  15. 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]
  16. 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]
  17. 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]
  18. 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. A70, 042314 (2004). [CrossRef]
  19. Y.-F. Xiao, Z.-F. Han, Y. Yang, and G.-C. Guo, “Quantum CPF gates between rare earth ions through measurement,” Phys. Lett. A330, 137–141 (2004). [CrossRef]
  20. 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. A80, 062307 (2009). [CrossRef]
  21. P. R. Berman, Cavity Quantum Electrodynamics(Academic, 1994).
  22. R. Kolesov, K. Xia, R. Reuter, R. Stöhr, A. Zappe, J. Meijer, P. R. Hemmer, and J. Wrachtrup, “Optical detection of a single rare-earth ion in a crystal,” Nat. Commun.3, 1029 (2012). [CrossRef] [PubMed]
  23. C. Yin, M. Rancic, G. G. de Boo, N. Stavrias, J. C. McCallum, M. J. Sellars, and S. Rogge, “Optical addressing of an individual erbium ion in silicon,” Nature497, 91–94 (2013). [CrossRef] [PubMed]
  24. 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]
  25. C. Greiner, T. Wang, T. Loftus, and T.W. Mossberg, “Instability and pulse area quantization in accelerated super-radiant atom-cavity systems,” Phys. Rev. Lett.87, 253602 (2001). [CrossRef]
  26. C. Greiner, B. Boggs, and T. W. Mossberg, “Frustrated pulse-area quantization in accelerated superradiant atom-cavity systems,” Phys. Rev. A67, 063811 (2003). [CrossRef]
  27. K. Ichimura and H. Goto, “Normal-mode coupling of rare-earth-metal ions in a crystal to a macroscopic optical cavity mode,” Phys. Rev. A74, 033818 (2006). [CrossRef]
  28. D. L. McAuslan, D. Korystov, and J. J. Longdell, “Coherent spectroscopy of rare-earth-metal-ion-doped whispering-gallery-mode resonators,” Phys. Rev. A83, 063847 (2011). [CrossRef]
  29. H. Goto, S. Nakamura, and K. Ichimura, “Experimental determination of intracavity losses of monolithic Fabry-Perot cavities made of Pr3+:Y2SiO5,” Opt. Exp.18, 23763 (2010). [CrossRef]
  30. A. M. Stoneham, “Shapes of inhomogeneously broadened resonance lines in solids,” Rev. Mod. Phys.41, 82–108 (1969). [CrossRef]
  31. R. W. Equall, R. L. Cone, and R. M. Macfarlane, “Homogeneous broadening and hyperfine structure of optical transitions in Pr3+:Y2SiO5,” Phys. Rev. B52, 3963–3969 (1995). [CrossRef]
  32. 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]
  33. M. Lovrić, P. Glasenapp, and D. Suter, “Spin Hamiltonian characterization and refinement for Pr3+:YAlO3and Pr3+:Y2SiO5,” Phys. Rev. B85, 014429 (2012). [CrossRef]
  34. 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. B70, 214116 (2004). [CrossRef]

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