Open Access
Add to BibSonomyAbstract
We theoretically investigate the dynamic regime of coherent population trapping (CPT) in the presence of frequency modulation (FM). We have formulated the criteria for quasi-stationary (adiabatic) and dynamic (non-adiabatic) responses of atomic system driven by this FM. Using the density matrix formalism for Λ system, the error signal is exactly calculated and optimized. It is shown that the optimal FM parameters correspond to the dynamic regime of atomic-field interaction, which significantly differs from conventional description of CPT resonances in the frame of quasi-stationary approach (under small modulation frequency). Obtained theoretical results are in good qualitative agreement with different experiments. Also we have found CPT-analogue of Pound-Driver-Hall regime of frequency stabilization.
© 2017 Optical Society of America
1. Introduction
At the present time, atomic clocks are one of most important and valuable quantum devices, and have a wide spectrum of applications in different areas of science (high precision measurements, testing of fundamental theories) and technology (navigation and telecommunication systems, geodesy, gravimetry, etc.) [1]. The principle of operation of these instruments is based on the modern methods of laser physics and high-precision spectroscopy. Atomic clocks based on coherent population trapping (CPT) [2–5] are particularly significant. Main advantages of CPT clocks consist of their low-power and compactness in combination with relatively high metrological characteristics (stability and accuracy) [6–8]. Therefore, these devices are utterly attractive for large consumer segment.
In the context of clock operation, determination of optimal regime of frequency stabilization is one of main tasks for CPT clocks. Systematic investigations of this question were done in experimental papers [9–11]. However, detail theoretical consideration, where it is necessary to use the dynamic solution for atomic density matrix, was not still done. In this paper we make up this deficiency. For this purpose, we use recently developed method [12], which allows us to construct the exact periodic solution of density matrix equation omitting the Fourier analysis. We calculate and optimize the error signal formed by the use of frequency modulation, which is usually applied for frequency stabilization in atomic clocks. Obtained theoretical results are in good qualitative agreement with different experiments [9–11, 13, 14].
2. General theory
As a model, we will consider so-called dark resonance, which is formed in three-level Λ system under interaction with resonant bichromatic field
This resonance is observed when the difference between optical frequencies (ω1 ω2) is varied near the low-frequency transition between lower energy levels |1〉 and |2〉: ω1−ω2 ≈ Δ [see Fig. 1(a)]. For description we will use standard formalism of atomic density matrix: (where j, k =1,2,3). In this case, the dynamics of the Λ system in the rotating wave approximation is described by the differential equation system for density matrix components ρjk(t): Here δ1−ph is an effective one-photon detuning of frequency components ω1 and ω2 on the optical transitions with frequencies and [see Fig. 1(a)]. Two-photon (Raman) detuning δr = (ω1 − ω2 − Δ) is a main spectroscopy parameter for description of the narrow dark resonance (under |δr | ≪ γopt). Ω1=d31E1/ħ and Ω2=d32E2/ħ are Rabi frequencies for the transitions |1〉 ↔ |3〉 and |2〉 ↔ |3〉 (d31 and d32 are reduced matrix elements of dipole moment for these transitions). γ is spontaneous decay rate of upper level |3〉; γopt is rate of decoherence (spontaneous, collisional, etc.) of optical transitions |1〉 ↔|3〉 and |2〉 ↔|3〉; Γ0 is relatively slow (Γ0 ≪ γ, γopt) rate of relaxation to the equilibrium isotropic state:As a detected signal we consider the output laser power from atomic cell. In our case of Λ system and for optically thin medium, the main spectroscopic information (per one atom) is contained in slowly varying value:
In the steady-sate regime (∂t ρjk = 0), this value is directly proportional to the excited state population ρ33: Ast = (γ + Γ0)ρ33. The dependence Ast (δr) describes well-known lineshape of the CPT resonance as a function on two-photon detuning δr (under |δr| ≪ γopt), which has the form of generalised Lorentzian (as it was shown in Ref. [15]): Here B, C, D, and are some values, which depend on model parameters: {Ω1, Ω2, δ1−ph, γ, Γ0}. In the case of δ1−ph = 0, the symmetric lineshape takes place, when D = 0 and . Detail theory of the steady-state dark resonances for alkali-metal vapors was developed in Ref. [16]However, in atomic clocks the harmonic frequency modulation (FM) is usually applied, when the two-photon detuning depends on the time:
where δ(0) is some constant term of two-photon detuning, F is amplitude of modulation, fm is frequency of modulation, and M = F / fm is modulation index. In this case, the signal (3) becomes periodically depended on the time, A(t + T) = A(t), with period T = 2π/fm. Then, for frequency stabilization the lock-in detection is used, when the error signal is formed as a function on δ(0): where cos(fm t + ϕ) is the reference signal, and ϕ is the phase of the reference signal relative to the harmonic dependence (5). In experiments, the clock loop is stabilized on zero crossing point of the error signal. In the case of ϕ = 0 the error signal (6) can be named as “in-phase”, Sin−ph (δ(0)), and for ϕ = −π/2 we will name the error signal as “quadrature”, Squad (δ(0)): Then the expression (6) can be presented as following superposition At the qualitative level, the view of the error signal can be obtained by the use of stationary signal Ast(δr) [see Eq. (4)] in the integral (6), performing the formal replacement δr → δ(0) + F cos (fmt), i.e., A(t) ≈ Ast(δ(0) + F cos(fmt)). However, this quasi-stationary approach is correct only for small frequency of modulation fm. In the general case, it is necessary to use a dynamic periodic solution of the differential equation system (2).A typical view of the functional dependence Serr(δ(0)) is shown in Fig. 1(b), and it has a form of dispersion curve. In this case, a key role for clock stabilization plays the slope of the curve in the center of lineshape:
Indeed, the clock instability is proportional to the value N(fm) / |K|, where N(fm) is a noise spectral density at the frequency fm. Thus, there is an important task to investigate and maximize the slope |K|, which depends on several parameters {Ω1, Ω2, δ1−ph, F, fm, ϕ controlled in experiments. Because of Eq. (8), the slope K can be expressed as superposition where Kin−ph and Kquad are slopes of the error signals Sin−ph(δ(0)) and Squad(δ(0)), correspondingly.To conclude the general theory, let us derive the criteria for adiabatic and non-adiabatic responses of atomic system driven by the frequency modulation (5). If we assume that the half-width of the CPT resonance in Eq. (4) corresponds to the minimal damping rate in the Λ system, then we can formulate the criterion of quasi-stationary (adiabatic) regime as following:
Indeed, let us consider at first the condition . In this case, the typical scanning time of half-width in the time area can be approximately determined from the relationship: . Then, the quasi-stationary regime corresponds to the condition: , i.e. the scanning time should be much more than the maximal time of damping in the Λ system, . Using both these relationships , we obtain the inequality (11). If we will consider other condition , then quasi-stationary regime corresponds to the obvious relationship: . Using these relationships , we again obtain the same inequality (11). Thus, the formula (11) is quite universal criterion for adiabatic regime. At the same time, other inequality: can be used as criterion of dynamic (non-adiabatic) regime for atomic system driven by the frequency modulation (5).3. Computation algorithm
As we have mentioned in Introduction, pure experimental investigations of dependence of the coefficient |K| on frequency modulation parameters {F, fm, ϕ} are contained in Refs. [9–11]. However, detail theoretical consideration of this problem, using the exact dynamic solution of the equation system (2) for atomic density matrix , was not previously done. In the present paper, for calculations we have applied the method [12], which allows us to find the periodic solution of the equation system (2) without Fourier analysis. The essence of our approach is following. First of all, let us rewrite the differential equation for the density matrix [Eq. (2) in our case] in the vector form:
where the column-vector is formed by the matrix elements ρjk(t), the linear operator is constructed from the coefficients of dynamic equation [Eq. (2) in our case]. If for some instant of time t1 we have vector , then, in accordance with Eq. (13), for other instant of time t2 we can write: where the two-time evolution operator is determined by the matrix . In the case of periodicity condition, the following relationship takes place for arbitrary t1, t2. In Ref. [12], it was rigorously proven the existence of the periodic solution, , for arbitrary periodically driven system. Due to the relaxation processes this state is realized as an asymptotics (t→+ ∞) independently of initial conditions. Thus, the periodicity is the main attribute of dynamic steady-state, which satisfies the following equation: This equation can be used for construction of universal computation algorithm (without using either the Floquet or Fourier formalisms). Indeed, let us consider some selected time interval [t0, t0 + T], which can be divided into N small subintervals between points tn = t0 + nτ (n = 0, 1, …, N), where τ = T/N is duration of subintervals. The dependence we will approximate by step function, where the matrix has the constant value inside of subinterval (tn−1, tn]. In this case, the vector in initial t0 is determined by Eq. (16), where the evolution operator has the form of a chronologically ordered product of the matrix exponents: The vectors in other points of the interval [t0, t0 + T] are determined by the recurrence relation: Our approach automatically guarantees a full account of all frequency components and considerably simplifies numerical calculations regardless of periodic modulation character: from smoothly harmonic type to the ultrashort pulses (practically without significant change of computation time). In contrast, using the Fourier analysis to numerically solve Eq. (13) we should use the following decomposition: where the components satisfy certain recurrent relations, which can be very complicated, in the general case, and they can lead to a huge computational burden.4. Numerical calculations, comparison with experiments, and discussion of results
The results obtained by the use of above computation algorithm for Eq. (2) (with N > 1000) are shown in Figs. 2–5. In the Fig. 2, the time scanning of the signal A (t) [see Eq. (3)] is presented for different values of modulation frequency fm. As it is seen from the first Fig. 2(a), in the case of small frequency modulation fm the time scanning of the signal has a typical view of the dark resonance as a function on two-photon detuning in the stationary regime, i.e., A(t) ≈ Ast(δ(0) + F cos(fmt)). However, as the value fm increases the signal A(t) becomes strongly distorted (with significant asymmetry and some oscillations) [see Figs. 2(b)–(e)], and finally it takes the form of regular harmonic oscillation [see Fig. 2(f)], the phase of which significantly differs from the phase of initial harmonic law (5). Even the superficial glance on the obtained dependencies leads to the conclusion that the standard notion about dark resonance, which is described by the simple resonance lineshape (4), is not useful for the analysis of FM dynamic regime. Note also that the calculations in Fig. 2 confirm the criteria (11) and (12) for quasi-stationary and dynamic regimes. Indeed, for given parameters of the field the half-width of the steady-state dark resonance is approximately equal to the 2Γ0. Then for Fig. 2(a) we have , that good corresponds to the quasi-stationary criterion (11), while for Fig. 2(b) the value of this ratio corresponds already to the dynamic criterion (12).
Fig. 2 Time scanning of normalized signal at one period (0 ⩽ fmt ⩽ 2π) for two values of constant two-photon detuning: δ(0) = 0 (solid black line) and δ(0) = 10Γ0 (dashed red line), and for different values of modulation frequency: fm/Γ0 ={0.01; 0.2; 1; 5; 10; 20} [see (a)–(f)]. All graphs are obtained under following parameters: F = 20Γ0, γopt = 50γ, γ = 104Γ0, Ω1 = Ω2 = 500Γ0, δ1−ph = 0.
Fig. 3 (a) Dependencies of the slope Kin−ph(F, fm) and Kin−ph(M, fm) in the case of “in-phase” error signal (ϕ = 0). (b) Dependencies of the slope Kquad(F, fm) and Kquad(M, fm) in the case of “quadrature” error signal (ϕ = −π 2). All graphs are obtained under following parameters: γopt = 50γ, γ = 104Γ0, Ω1 = Ω2 = 500Γ0, δ1−ph = 0.
Fig. 4 Comparison of different error signals Serr[δ(0)]: (a) “In-phase” error signal (dashed blue line for ϕ = 0, F = 1:4Γ0, fm = 0:01Γ0) and “quadrature” error signal (solid red line for ϕ = −π/2, F = 3.6Γ0, fm = 3.4Γ0) with comparable maximal slopes. (b) Demonstration of sign inversion of the slope K in “quadrature” error signal Squad (ϕ = −π/2) for fixed modulation amplitude F = 7.0Γ0 and for different modulation frequencies: the curve with K > 0 (dashed red line) for fm = 2.4Γ0, the curve with K = 0 (solid black line) for fm = 2.965Γ0, the curve with K < 0 (dashed blue line) for fm = 6.3Γ0. All graphs are obtained under following parameters: γopt = 50γ, γ = 104Γ0, Ω1 = Ω2 = 500Γ0, δ1−ph = 0.
Fig. 5 Dependence of optimal slope |K|opt and optimal phase ϕopt (−π/2 < ϕopt < π/2). The position of the maximal slope |K|max = max {|K|opt} (for ϕopt ≈ −π/4) is tagged by the cross. (a) |K| in the plane of (F, fm), the position of |K|max corresponds to the point F/Γ0 ≈ 2.0 and fm/Γ0 ≈ 1.55; (b) |K|opt in the plane of (M, fm), the position of |K|max corresponds to the point M ≈ 1.3 and fm Γ0 ≈ 1.55; (c) ϕopt in the plane of (F, fm); (d) ϕopt in the plane of (M, fm). All graphs are obtained under following parameters: γopt = 50γ, = γ =104Γ0, Ω1 = Ω2 = 500Γ0, δ1−ph = 0.
Next, the dependencies of the slope Kin−ph (F, fm) and Kin−ph (M, fm), for “in-phase” error signal Sin−ph(δ(0)) [see Eq. (7)] is shown in Fig. 3(a). In this case, the maximal slope is observed near F ∼ 1.5Γ0 (for given parameters of the field) and for small modulation frequency fm, which corresponds to the quasi-stationary regime. The dependencies of the slope Kquad (F, fm) and Kquad(M, fm) for “quadrature” error signal Squad(δ(0)) [see Eq. (7)] are presented in Fig. 3(b). There, in contrast to the “in-phase” signal, the maximal slope is observed for high values fm and F, which correspond to the significantly dynamic regime [see criterion (12)].
In the Fig. 4(a), the“in-phase” and “quadrature” error signals are presented at the same panel for comparison. It is in good agreement with results [9], where the “quadrature” signal also has three cross-points with the horizontal axis (see Fig. 2(a) in Ref. [9]). Additional interesting feature of “quadrature” error signal is connected with existence of areas with opposite signs of the slope Kquad [see red and blue areas in Fig. 3(b)]. Demonstration of this effect is presented in Fig. 4(b), where the error signals with opposite signs of K are shown, as well as the error signal for transitional zone with horizontal slope (K=0).
The slope |K| can be maximized by the choice of optimal phase ϕ = ϕopt of the reference signal in (6), as it was done, for example, in Ref. [9–11]. Using Eq. (10), we find the optimal phase ϕopt and corresponding maximized slope |K|opt:
Calculations of this optimization are presented in Fig. 5. In Figs. 5(a),(b), the dependencies of maximized slope |K|opt are shown in the planes (F, fm) and (F, fm). Corresponding graphs for the optimal phase ϕopt are shown in Figs. 5(c),(d). Our calculations show that for high modulation frequency fm the optimization of slope |K| is achieved for the “quadrature” signal (ϕopt ≈ −π/2) [see main ridge along the line of M ≈ 1.1 in Fig. 5(b) and corresponding blue area in Fig. 5(c)]. However, the absolutely maximal value |K|max = max{|K|opt} corresponds to the phase ϕopt ≈ −π/4, which lies in the interjacent zone between “in-phase” and “quadrature” signals.The obtained theoretical results are in good agreement with experiments [9–11, 14]. For example, let us consider experimental results Ref. [9, 10] with the use of correcting multiplier 2 for modulation index M in these papers. It is necessary for comparison with our calculations, because in Ref. [9, 10] the semiconductor laser (VCSEL) has been modulated at the frequency 3.4 GHz, i.e., at the half of hyperfine splitting (Δhfs ≈ 6.8 GHz) in the ground state of 87Rb. In this case, two ±1-st sidebands with frequency difference of 6.8 GHz were used in the capacity of the resonant components. If we will compare our Fig. 5(b) with upper graph in Fig. 2 from Ref. [10], then we see practically identical pictures. Indeed, at the both graphs it is seen that the largest slope is obtained along clear ridge that converges to a modulation index of M ≈ 1.1 (M ≈ 0.55 in Fig. 2 from Ref. [10]). Moreover, there is the secondary ridge (less significant) along the line of M ≈ 3 (M ≈ 1.5 in the Fig. 2 from Ref. [10]). Note also that our Fig. 5(a) good conforms with Fig. 4 from Ref. [11].
High degree of agreement with different experiments is additionally confirmed if we compare the position of the maximal slope |K|max in our Fig. 5 and in the papers Ref. [10, 11]. Indeed, our calculations show [see Fig. 5(b)] that the maximal value |K|max to the modulation index M ≈ 1 3. For comparison: in the Ref. [10] the value of M ≈ 0.6 (M ≈ 1 2 in our parametrization) is presented, and the Ref. [11] contains the result of M ≈ 1.3 (for F/2π = 4 kHz and fm/2π = 3 kHz, see text and Fig. 4 in Ref [11]), which is practically equivalent to the our calculation.
However, as we said above, for optimization of the frequency stabilization we also need to take into account the noise spectral density N(fm) at the frequency fm, i.e., it is necessary to maximize the value |K|opt/N(fm). Because low-frequency noise usually becomes smaller with rising of fm, then the position of maximal ratio max {|K|/N (fm)} can be shifted to the area of more high frequencies fm in relation to the position of maximal slope |K|max, and optimal regime will correspond to the main ridge along the line of M ≈ 1.1 [see Fig. 5(b)]. Moreover, this shift can strongly differ for different experimental setups (due to difference of noise characteristics). In this context, note that along the main ridge with M ≈ 1.1 [see Fig. 5(b)] the value |K|opt becomes saturated at nonzero value (≈ 0.85|K|max for given field parameters) as fm grows till the megahertz level (when fm, F ∼ γ). We suppose that this high-frequency area can be considered as CPT-analogue of Pound-Driver-Hall regime of frequency stabilization [17, 18]. This regime requires additional investigations, because in CPT-clock experiments relatively slow-frequency stabilization (at the kilohertz level) is usually used.
In addition, we emphasize that in our calculation we have used very simple theoretical model of the Λ system with three relaxation constants {γopt, γ, Γ0}. Indeed, we have not taken into account many known factors: atomic motion (i.e., Doppler and time-of-flight effects), real hyperfine and Zeeman structure of energy levels, transfer of atomic velocities due to collisions with buffer gas or wall-coated cells, effects of spatial profile of laser beam intensity, an so on. In this context, it seems quite surprising that even simplest model demonstrates good qualitative agreement with results obtained in different experimental setups.
5. Conclusion
We theoretically have investigated the excitation of CPT resonances in the presence of harmonic FM. We have formulated the criteria of quasi-stationary (adiabatic) [see Eq. (11)] and dynamic (non-adiabatic) [see Eq. (12)] responses of atomic system driven by this frequency modulation. The study of the error signal slope |K| and its optimization on the FM parameters (modulation frequency fm, modulation amplitude F, phase of reference signal ϕ) has been done. The existence of the maximal slope |K |max under modulation index M = F fm 1.3 was theoretically shown. We have find that the dependence of optimal slope |K |opt in the plane of parameters (M, fm) has a relief form of alternating ridges along lines with modulation indexes of M ≈ 1.1, M ≈ 3.0, M ≈ 4.6, and so on (this fact can be considered as a subject for further theoretical and experimental study). Using our calculations and taking into account the typical peculiarity of the low-frequency noise (decreasing of the noise with the rise of frequency), we can estimate the area of FM parameters (fm,M), which contains the point of most optimal stabilization for atomic clock. This area is approximately determined as following: [where is half-width of the dark resonance for steady-state regime, see Eq. (4)], and 1.0 < M < 1.4. Note that this area corresponds to the criterion of dynamic regime (12). The position (on fm) of optimal regime can strongly differ for different experimental setups (due to difference of noise characteristics). Also we have found high-frequency area, which corresponds to the CPT Pound-Driver-Hall regime of stabilization, and which was not previously studied (as far as we know) nor theoretically nor experimentally.
Obtained theoretical results are in good qualitative agreement with different experiments. Proved in our calculations new theoretical method [12] has confirmed a high efficiency, and it will be used as a base for further detail investigations of different spectroscopic signals (not only CPT) in atomic systems driven by periodically modulated fields.
An assessment of the potential performance improvement as applied to atomic clocks would require a full analysis not just of the signal but also of the noise, which we have not considered here. Rather, this work extends previously understood analysis of CPT lineshapes at low modulation frequencies to moderate and high modulation frequencies. This extension enables us to understand these lineshapes from a firm theoretical foundation.
Funding
The work was supported by the Russian Science Foundation (RSF) (project No. 16-12-10147). M. Yu. Basalaev was supported by the Russian Foundation for Basic Research (RFBR) (projects No. 16-32-60050 mol_a_dk, No. 16-32-00127 mol_a).
Acknowledgments
We thank J. Kitching, S. M. Kobtsev, D. A. Radnatarov, and S. A. Khripunov for useful discussions and comments.
References and links
1. A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87(2), 637–701 (2015). [CrossRef]
2. G. Alzetta, A. Gozzini, M. Moi, and G. Orriols, “An experimental method for the observation of r.f. transitions and laser beat resonances in oriented Na vapour,” Il Nuovo Cimento B 36, 5 (1976). [CrossRef]
3. E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. 35, 257–354 (1996). [CrossRef]
4. J. Vanier, “Atomic clocks based on coherent population trapping: a review,” Appl. Phys. B 81(4), 421–442 (2005). [CrossRef]
5. V. Shah and J. Kitching, “Advances in coherent population trapping for atomic clocks,” Adv. At. Mol. Opt. Phys. 59, 21–74 (2010). [CrossRef]
6. S. Knappe, P. Schwindt, V. Shah, L. Hollberg, J. Kitching, L. Liew, and J. Moreland, “A chip-scale atomic clock based on 87Rb with improved frequency stability,” Opt. Express 13(4), 1249–1253 (2005). [CrossRef] [PubMed]
7. R. Lutwak, A. Rashed, M. Varghese, G. Tepolt, J. Leblanc, M. Mescher, D. K. Serkland, and G. M. Peake, “The miniature atomic clock – pre-production results,” Proc. IEEE International Frequency Control Symposium and European Frequency and Time ForumGeneva, Switzerland, 1327–1333 (2007).
8. Z. Wang, “Review of chip-scale atomic clocks based on coherent population trapping,” Chin. Phys. B 23(3), 030601 (2014). [CrossRef]
9. I. Ben-Aroya, M. Kahanov, and G. Eisenstein, “Optimization of FM spectroscopy parameters for a frequency locking loop in small scale CPT based atomic clocks,” Opt. Express 15(23), 15060–15065 (2007). [CrossRef] [PubMed]
10. M. Kahanov, I. Ben-Aroya, and G. Eisenstein, “Dependence of small-scale atomic clock performance on frequency modulation parameters used in the frequency control loop,” Opt. Lett. 33(9), 944–946 (2008). [CrossRef] [PubMed]
11. E. E. Mikhailov, T. Horrom, N. Belcher, and I. Novikova, “Performance of a prototype atomic clock based on lin||lin coherent population trapping resonances in Rb atomic vapor,” J. Opt. Soc. Am. B 27, (3)417–422 (2010). [CrossRef]
12. V. I. Yudin, A. V. Taichenachev, and M. Yu. Basalaev, “Dynamic steady state of periodically driven quantum systems,” Phys. Rev. A 93, 013820 (2016). [CrossRef]
13. S. Khripunov, D. Radnatarov, and S. Kobtsev, “Atomic clock based on a coherent population trapping resonance in 87Rb with improved high-frequency modulation parameters,” Proc. SPIE 9378, 93780A (2015). [CrossRef]
14. S. A. Khripunov, D. A. Radnatarov, S. M. Kobtsev, V. I. Yudin, A. V. Taichenachev, M. Y. Basalaev, and I. D. Popkov, “Transient processes under dynamic excitation of a coherent population trapping resonance,” Quantum Electronics 46(7), 668–671 (2016). [CrossRef]
15. S. Knappe, M. Stahler, C. Affolderbach, A. V. Taichenachev, V. I. Yudin, and R. Wynands, “Simple parameterization of dark-resonance line shapes,” Appl. Phys. B 76(1), 57–63 (2003). [CrossRef]
16. A. V. Taichenachev, V. I. Yudin, R. Wynands, M. Stahler, J. Kitching, and L. Hollberg, “Theory of dark resonances for alkali-metal vapors in a buffer-gas cell,” Phys. Rev. A 67(3), 033810 (2003). [CrossRef]
17. 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 31(2), 97–105 (1983). [CrossRef]
18. Y. Torii, H. Tashiro, N. Ohtsubo, and T. Aoki, “Laser-phase and frequency stabilization using atomic coherence,” Phys. Rev. A 86(3), 033805 (2012). [CrossRef]
