1. Introduction

Driving spin systems with specific time-dependent fields enables engineering of spin ensembles, observation of many interesting phenomena, and important applications. A fundamental condition for such developments is a sound understanding of relaxation and decoherence and the ability to control interactions with other fields or systems. Significant progress in these aspects has been achieved through coherent dark-state formation and coherent population trapping [1–10], the engineering of quantum states [11–16], and their environments [17–22].

An important role in such control is played by optically detected magnetic resonance (ODMR) conducted with single radio-frequency or microwave (MW) field, continuous-wave (CW) or pulsed [23,24,25]. A recent development in the CW-ODMR methodology is the simultaneous application of two MW fields, that is, the hole-burning technique [26–31]. Generally, there are two kinds of hole-burning experiments. In the first case, one of two MW fields may be tuned to one of the transitions sharing a common level, while the second field is scanned across the other transition. This enables recording of spectral structures (holes) with the homogeneous linewidth on a background of inhomogeneously broadened ODMR line [26]. Another possible situation is when both MW fields are close to resonance with the same transition, and the beating of two coherent fields creates population pulsations or coherent population oscillations (CPO) [27,32,33,34] that lead to novel resonances with non-standard features. In this article, we call these resonances composite resonances. Such resonances occurring in hole-burning experiment with a bichromatic driving of the spin resonance in nitrogen-vacancy (NV) color center in diamond have been described in our Ref. [27] within the first-order perturbation theory.

Our motivation for this work is to present a non-perturbative study of the composite resonances and a detailed analysis of their proerties. The general model developed that is applicable to any open two-level system is verified in hole-burning experiments with NV ensembles in diamond. The model and experiments revealed unexpected immunity of one of the composite resonance components to power broadening. To our knowledge, such behavior has not been analyzed before. Here, we interpret this effect as a general consequence of the openness of the system. We also show that it is mathematically related with the effect of coherent population trapping (CPT). Specifically, we introduce power-dependent linear combinations of spin-state populations which act as the dark and bright states, similarly to the symmetric and antisymmetric superpositions of wavefunctions in the standard description of CPT.

In Section 2, we present a detailed experimental study of the composite resonance properties of three samples of NV diamonds that differ significantly in their densities aiming at verification of the theoretical model. In Section 3, we introduce the applied model and discuss its consequences. Similarly to earlier studies in laser spectroscopy [35,36], we find that the existence of the composite resonance is intrinsically related to the openness of the system and discuss the analogy of the stabilization effect with CPT. Section 4 is devoted to comparison of our experimental results with the model and detailed discussion of our findings, and the paper is concluded in Section 5.

2. Experimental

2.1. Setup

Our experimental setup was similar to that described in Ref. [27]. The samples were excited using green laser light (532 nm) and driven by two MW fields with frequencies ω₁ and ω₂ close to the ODMR resonance frequency. Figure 1 depicts the structure of energy levels of NV ⁻ and the transitions relevant for the experiment. The MW fields were created by separate, phase-synchronized generators connected to a signal combiner before powering an antenna structure attached to the diamond sample. A permanent magnet created a static magnetic field B ≈ 30 - 40 G aligned with the (111) crystallographic direction. The magnetic field split the m_S=±1 spin states and allowed us to use a two-level approximation with MW fields tuned to one of the resolved transitions: the resonance frequency ω₀ corresponds thus to one of the resolved transitions, m_S=0 ↔ m_S= -1 or m_S=0 ↔ m_S=+1. Excitation and detection were performed with a standard home-made confocal microscope arrangement with a dichroic mirror that combined excitation and fluorescent light.

 

Fig. 1. Energy levels of NV ⁻ color center in diamond. The green arrow marks excitation by the 532 nm laser beam, the red arrow represents optical transitions responsible for red fluorescence, the broken arrows represent transitions responsible for spin polarization of the ground state ³A₂, and the double arrow indicates MW transitions. The zoomed figure on the right shows the bichromatic driving (frequencies ω₁ and ω₂) responsible for the composite resonance when both MWs are tuned to the m_S=0 ↔ m_S= -1 transition.

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The experiments were conducted with three samples of NV center ensembles in diamond exhibiting varying levels of inhomogeneous broadening due to a range of nitrogen and NV densities, from the moderate-defect-density chemical vapor deposition (CVD) diamond sample to dense high-pressure high-temperature (HPHT) crystals.

All samples were purchased from Element Six and had main facets polished in the (100) plane. Sample E8 (3 × 3 × 0.3 mm³ in size) was produced using the CVD technique with an initial nitrogen concentration [N] of about 1 ppm. It was irradiated by a 14 MeV e-beam with a fluence of 1 × 10¹⁶cm^-2 prior to 4 hour annealing in vacuum at 650 °C. The estimated final concentration of NV⁻ centers was approximately 0.1 ppm. Samples E1 and W7 were HPHT diamonds. The E1 sample (2.6 × 2.6 × 0.5 mm³) had an initial N concentration [N] ∼200 ppm and was similarly irradiated by a 14 MeV e-beam with a fluence of 1 × 10¹⁸cm^-2 prior to 4 hour annealing in vacuum at 650 °C, resulting in an estimated concentration of NV centers of about 10 ppm. The W7 sample (3 × 3 × 0.5 mm³) had an initial N concentration [N] ∼200 ppm and was irradiated with a 1.5 × 10¹⁸cm^-2 fluence by a 14 MeV electron beam. After the irradiation it was annealed in vacuum for 2.5 hours at 750 ° C, which resulted in an estimated concentration of NV⁻ centers of about 20 ppm. The relevant densities of the samples used and their spin decay rates 1/T₂ and 1/T₂* are summarized in Table 1. Relaxation times T₂ and T₂* were measured by the Hahn spin echo and Ramsey fringe technique, respectively, in a magnetic field of 30 G aligned along the [111] direction.

Table 1. Densities of the samples used in the experiment

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2.2. Results

2.2.1 General features of the composite resonance

In general, all observed resonances exhibit the same general features, mostly consistent with Ref. [27], so we only briefly summarize them here. The recorded composite resonance occurs within an inhomogeneously broadened ODMR profile and consists of two components of different widths w₀, w₁ and amplitudes A₀, A₁ superimposed on a wider hole as shown schematically in Fig. 2. (note that with our CW detection, the “hole” reflects an increase or a “bump” in the fluorescence level).

 

Fig. 2. Fluorescence signal with ODMR (black trace, width σ) at the resonance frequency ω₀ and composite resonance at ω₁ detuned by Δ from ω₀. The composite resonance has two narrow components of widths w₀, w₁ and amplitudes A₀, A₁ (red and blue traces, respectively) superimposed on a hole of width Γ (green trace).

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When ω₁ ≠ ω₀, the position of the composite resonance follows the ω₁ frequency, i.e. the hole is shifted from ω₀ by Δ = ω₁ – ω₀ such that its center occurs at ω₂ = ω₁. In Sec. III.3 we discuss in detail the relevant lineshapes.

The composite resonances are deeper and narrower than the regular holes burnt when each MW field acts upon a different transition as discussed in Ref. [26]. We have observed such resonances with all studied samples, although the signal/noise ratios of the signals from various samples differed strongly, depending on the sample density. The best signals were recorded with sample W7 which had the highest NV⁻ concentration. The high sample densities caused a large inhomogeneous broadening of the induced MW transitions such that their hyperfine structure was not visible.

2.2.2 Amplitudes and widths of resonance

Figure 3 depicts the MW power dependencies of the amplitudes A₀, A₁ and the widths w₀, w₁ of the resonance components observed with three different samples specified in Table 1. For increasing MW power, the amplitudes of both components first rise and then stabilize or fall slowly for high MW power. However, the dependences of the widths are very different for both components: w₁ exhibits regular power broadening, whereas w₀ after an initial increase stabilizes at low values, much below the natural linewidth Γ. To better visualize the stabilization effect, the lower panels (Figs. 3 (g),(h),(i)) are presented with an expanded vertical scale.

 

Fig. 3. Dependencies of composite resonance amplitudes A₀, A₁ (a), (b), (c) and widths w₀, w₁ (d), (e), (f) on MW power for three samples E1, E8 (b), (e), (h), and W7 (c), (f), (i). The panels (d), (e), and (f) illustrate the power broadening of w₁ and the stabilization of w₀. In (g), (h), (i) the effect of stabilization of w₀ is shown with expanded vertical scale. Amplitudes A₀, A₁ are expressed relative to the off-resonant fluorescence level. They cannot be directly compared between different columns because of different sample geometries.

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Since the higher density of the spin system is related to its stronger interaction with the environment, we regard the reported dependences as an illustration of various stages of the opening of the system. However, for the lower-density samples E1, E8, precise characterization of the composite resonance components becomes difficult. On the one hand, the resonance signals seen with these samples have lower contrast and are noisier (Figs. 3 (g), (h) and (i)). On the other hand, the relevant resonance components and hyperfine splitting become comparably broad, so that it is uneasy to assign them properly. Therefore, for the E1 and E8 samples, several important assumptions in our simple model do not hold and not all theoretical predictions are reflected by the recorded resonances. In particular, complete stabilization is reached only with the most dense sample W7 (Fig. 3(i)), while for samples E1 and E8 only the onset of stabilization shows up.

3. Theoretical interpretation

3.1. Model, assumptions, equations

The model we have developed for interpretation of the described stabilization effect is very simple and general. It applies to a wide class of multilevel spin systems that are subjected to optical pumping and driven by two nearly resonant MW fields. Optical pumping creates spin polarization of the ground state and permits for its optical detection [37]. Appropriate magnetic-field splitting and nearly resonant MW frequencies enable to reduce the dynamics to essentially two-level system and a reservoir, while the beating of MW frequencies with frequency δ = ω₁ − ω₂ drives the population oscillations (CPO) [27]. In this work we apply the general model to experiments with NV ensembles in three high-density diamond samples and obtain results which agree well with the main prediction of our theory.

The analysis is performed with the density matrix formalism and the general quantum master equation in the form

Here γ_0,1 = γ (1∓ ɛ) denote the rates with which populations n_k (k = 0,1) relax to equilibrium values n_k⁰ and are expressed by the average rate γ and asymmetry parameter ɛ, while Γ=1/T₂ is the relaxation rate of the coherences ρ₀₁ and ρ₁₀. The initial values of n_k⁰ are established mainly by the dynamics of optical pumping and depend on the light intensity. In this work, however, we do not consider the optical pumping dynamics and restrict it solely to the establishment of initial n_k° values.

Different values of γ₀ and γ₁ are intrinsically related with the issue of the system’s openness. The trace of the density matrix represents the time derivative of the total probability (population) stored in the system. For a closed system $Tr\; \dot{\rho } = 0$, and the total probability is conserved. On the contrary, the system with $Tr\; \dot{\rho } \ne 0$ is referred to as the open one since the probability flows into and out of the system from a reservoir of probability. In our modelling, when ɛ = 0, the total probability is conserved, the populations satisfy the normalization condition n₁ + n₀ = 1, and the system is closed. If, however, ɛ ≠ 0, the time derivative of the total probability is non-zero and the system is open.

The connection between probability conservation and openness of the system implies that in an open system populations n_k are coupled to a probability sink (Fig. 4(b)), which we call the reservoir $\mathrm{{\cal R}}$ that participates in the dynamics of the system with an additional population n_R := 1 − n₁ − n₀.

For comparison with experiments with NV diamond samples we model the reservoir as a specific third state that contains all levels, including the ground-state sublevel m_S = + 1, which participate in the overall population flow but are not directly coupled by MWs, (Fig. 4). The coupling of states ${|0\rangle}$, ${|1\rangle}$ with $\mathrm{{\cal R}}$ is provided by relaxation and optical pumping. Such modelling of the reservoir enables simplification of the coherent dynamics to an open two-level system and has been employed, e.g., in Refs. [35,36]. Here, it enables us to demonstrate in the next sections the intrinsic links between the system openness and the appearance of the stabilization of bichromatically coupled states. We note that in our modelling there is no need to make any specific assumptions on the nature of the coupling with the reservoir. The phenomenon of stabilization is controlled solely by ɛ, i.e., the asymmetry between γ₀ and γ₁.

 

Fig. 4. (a) Electronic structure of the NV ⁻ color center in diamond with the visualization of the relevant MW (red arrows), optical excitation (double green arrow), and population decay channels (single black arrows). The shading marks the part of the system which is treated as a reservoir in our analysis. (b) An effective model of an open spin system driven by two MW fields nearly resonant with transition ${|0\rangle}$ − ${|1\rangle}$ and coupled to reservoir $\mathrm{{\cal R}}$, as symbolized by a thick double arrow.

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3.2 Reduced master equation and system evolution

With bichromatic MW driving, the Hamiltonian defined in the frame rotating with frequency ω₁ is

Equation (3) may be expressed with the help of the time-independent $\mathrm{{\cal L}}$₀ and time-dependent $\mathrm{{\cal L}}$₁(t) operators acting on the population vector

In the beginning of the analysis we take into account only the time-independent term $\mathrm{{\cal L}}$₀ and postpone discussion of the $\mathrm{{\cal L}}$₁ term to the next subsection. The formal solution of the time-independent part of Eq. (4) can be written as

To identify independently evolving states and their lifetimes, we diagonalize $\mathrm{{\cal L}}$₀ using the rotation matrix by angle θ, which represents the mixing of the n₀ and n₁ populations with $\tan \theta = \sqrt {1 + {{\left({\frac{\varepsilon}{S}} \right)}^2}} - \frac{\varepsilon}{S}$. Diagonalization yields two eigenvalues

In the η₀, η₁ basis the time-dependent solution of $\mathrm{{\cal L}}$₀ takes the form

Relations (6) show that the dynamics of the system can be described in terms of two combinations of populations η₀ and η₁, respectively, strongly and weakly self-coupled through the MW driving, while (7) suggests the interpretation of η₁ and η₀ as two states evolving with coefficients γ·λ_k being their effective relaxation rates. Diagonalization and transformation (6) are the main steps in understanding the stabilization effect. From now on, we analyze the problem in the transformed (perturbed) representation (6), rather than the initial representation of n₁, n₀.

For strong MWs (S >> 1), λ₁ and λ₂ depend very differently on S: λ₁ increases with S like λ₁ ≈ 1 + 2S, but λ₀ ≈1 and becomes independent of S. Consequently, states η₀, η₁ are, respectively, strongly and weakly coupled to the MW field, or short and long living. The further consequence is that one decay constant is power dependent, while the other one becomes independent of power, that is, stabilizes; hence superposition η₀ is power broadened while η₁ is power stabilized.

We note that the derived properties of the η₀, η₁ states follow straightforwardly from the diagonalization of 2 × 2 matrix, and are formally analogous to many familiar problems in physics. Most often, such diagonalization is applied to the energy level structure and describes situations like energy-level anticrossing, or dressed-energy states. Here, however, we apply the standard mathematical approach to a different problem and analyze real-valued state populations, rather than coherent superpositions of quantum states. That procedure appears perfectly analogous to the CPT. In a standard three-level Λ-systems where CPT is observed, coherent interaction with laser fields creates two superpositions: the dark one which is not excited, and the bright one strongly absorbing [1,2]. In the discussed case, we have superpositions of populations η₀ and η₁ one of which is interacting with MW and becomes power broadened, and the other which is not interacting and becomes stabilized.

3.3 Resonance features

The above discussion of the effects of the $\mathrm{{\cal L}}$₀ term revealed the main mechanism of stabilization of one of the components of composite resonance, but could not explain the time-dependent dynamics which is responsible for its full lineshape. This may be described by solving the complete time-dependent Eq. (3) with the $\mathrm{{\cal L}}$₁(t) term, accounting for the CPO wave beating effects. The beating creates additional dynamics that competes with the relaxational inertia of the individual spin states and enables addressing the corresponding components of the composite resonance associated with different relaxation rates γ₁, γ₂.

To analyze the effects of the time-dependent contributions to Eq. (3) we expand the populations into a Fourier series and interpret its components as amplitudes of the harmonics of δ. Assuming low MW power, we focus on the first harmonic and get the time-averaged population difference $\overline {\Delta n} $ = <n₁ – n₀> which describes the change in fluorescence intensity I, detected in ODMR-like experiments [38],

To reproduce the experimentally observed spectra, one needs to account for the inhomogeneous broadening of resonance transitions and an off-resonance fluorescence background B. We reproduce the broadening by introducing an additional inhomogeneity variable υ such that ω₀(υ) = ω₀ - υ, and assume a normal distribution N(υ) = $\Delta {n^0}{e^{ - {{\left( {\frac{\upsilon }{\mathrm{\sigma }}} \right)}^2}}}$ of υ over frequency range on the order of σ related with the inhomogeneous width 1/T₂* of the transition and wider than the individual spin-state population decay rates γ₀, γ₁.

The fluorescence signal can then be calculated as the convolution of N(υ) with function $\overline {\Delta n} ({{\omega_1},{\omega_2},\; \upsilon } )$ obtained from expression (8) by substituting ω₀ with ω₀ - υ in the L_Γ Lorentz functions

When only one MW is applied, Eq. (4) includes only the $\mathrm{{\cal L}}$₀ part and the signal predicted by Eq. (10) describes a familiar inhomogeneously broadened Gaussian-shaped ODMR centered at ω₀ with the width σ on the order of (T₂*)⁻¹. With two MWs, Eq. (10) yields the product of two types of resonances narrower than the inhomogeneous width. One, described by the fraction with S(ω₁, ω₂) before the square bracket, represents a regular hole-burning similar to the Doppler-free Lamb-dip spectroscopy [29,40]. This resonance appears as a peak of the width ∼Γ centered at ω₁ on the inhomogeneously-broadened ODMR background. If there was no inhomogeneous broadening, or σ>>Γ as in the case of single color centers, then the single hole of the width ∼Γ could not be observed.

The second kind of yet narrower structures is represented in Eq. (10) by the bracket with the ${L_{{w_0}}}(\delta )$, ${L_{{w_1}}}(\delta )$ functions and describes the composite resonance centered at δ=0, i.e. at ω₂= ω₁ It stems from CPO and occurs only when both MWs interact with the same transition, when the CPO dynamics is sufficiently slow with respect to the population time constants [27]. Otherwise, the population oscillations are too fast and average out, which explains why the composite resonances are observed only when both MWs act on the same transition.

Figure 5 presents inhomogeneously broadened lineshapes calculated with all terms of Eq. (10) as the function of ω₂ with fixed ω₁ and the assumption σ>>Γ, γ, and compared with the experimental results. Figures 5(a) and 5(c) show the plots with several detunings Δ to demonstrate how the position of the composite resonance and ODMR depends on ω₁, while Figs. 5(b) and 5(d) refer to the case of ω₁= ω₀. The calculated spectra very well reconstruct the experimental observations.

 

Fig. 5. Fluorescence signals vs. δ= ω₂-ω₁ for the m_S= 0 – m_S= -1 transition at B = 28 G. (a) Predictions of Eq. (9) for ɛ=0.85 and σ=3.5, Ω=γ=0.1 (in Γ units). The colored curves correspond to Δ= ω₁-ω₀=0, 2, and 4 Γ, respectively from the bottom. The top black curve shows the regular ODMR with a single MW field scanned around ω₀. (b) Blow-out of the resonance centrum at ω₂ ≈ ω₁ (δ ≈ 0) with differently colored individual Lorentzian contributions ${L_\mathrm{\Gamma }}$, ${L_{{w_0}}},\; {L_{{w_1}}}$. (c) Experimental signals measured for different detunings of ω₁; (d) central part of the red curve in (c) for comparison with (b).

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Figure 6 depicts the widths w₀, w₁, and amplitudes A₀, A₁ of the composite resonance calculated with equations (5-7) as functions of S for various values of the asymmetry parameter ɛ representing the difference between γ₀ and γ₁ that controls the openness of the system in our analysis.

 

Fig. 6. Amplitudes A₀(S), A₁(S) (a), and widths w₀(S), w₁(S) (b) of the composite resonance components calculated for various values of the asymmetry parameter ɛ. Note that A₀(S) is zero for ɛ=0 and is very small for ɛ=0.1, hence the corresponding two plots in (a) become indiscernible from the horizontal axis.

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For a closed system (ɛ=0), the width dependences w₀(S) and w₁(S) are represented in Fig. 6(b) by two straight lines (black) that cross at S = 0 when w₀(0)= w₁(0), or λ₀(0) =λ₁(0). When S increases, w₀(S) remains constant equal to w₀(0)= w₁(0) = γ, whereas w₁(S) rises linearly. The corresponding amplitude dependences (black) show that A₁(S) increases, but A₀(S) is identically zero for all values of S. Thus, for the closed system when ɛ=0, the composite resonance structure simplifies to a single power-broadened shape (shown in Fig. 5(c)) with its width nearly proportional to S and there is no second component, hence no stabilization for the closed system.

For an open system with ɛ≠0, the width degeneracy at S = 0 is lifted and the graphs of w₀(S) and w₁(S) in Fig. 6(b) represent two repelling curves approaching γ₀ and γ₁ when S → 0. They resemble the anticrossing of the energy levels or potential curves widely studied in many branches of physics [42]. Based on this similarity, we see that the parameter ɛ, or the system’s openness, plays a role of a repelling force between the two curves. For ɛ ≠ 0, both components of the composite resonance exhibit non-zero amplitudes, therefore in open systems the composite resonance has two components of different widths and amplitudes. Their S dependences are drastically different: w₁(S) becomes strongly power-broadened, while w₀(S) does not broaden but stabilizes at the mean value γ for S>>1. The amplitudes of both components become comparable for S≈1. For S < 1, we have A₀> A₁ but for S > 1 the A₀ amplitude is lower than A₁ and gradually weakens with increasing S.

4. Discussion

The resonance features calculated with our model in many aspects agree with the observations, particularly for the most dense sample W7. In particular, our main finding, the stabilization of the narrow component w₀ is well confirmed and illustrates the adequacy of the developed model for describing the essential mechanisms. However, a more quantitative comparison of the other calculated features: w₁(S), A₀(S), and A₁(S) shows differences larger than the measurement accuracies. Furthermore, the modelling predicts a linear increase of w₁(S) whereas the measured dependences are flatter (Figs. 3,7).

 

Fig. 7. Measured properties of the composite resonance vs. MW power recorded with sample W7 and compared with theoretical predictions (solid lines) for ω₂ =ω₀, γ = 0.1 and ɛ = 0.85: (a) amplitudes A₀, A₁; (b) widths w₀, w₁; (c) w₀(S) zoomed in.

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We attribute these discrepancies to the simplicity of our two-level model. In particular, the important assumption, Γ>> γ, is violated when w₁(S) approaches the width of the burnt hole, w₁(S) ∼ Γ. In such a case, the model needs to be extended to more states. Accounting for more states would yield more eigenvalues and eigenvectors of the Liouvillian $\mathrm{{\cal L}}$, and correspondingly more resonance components and width curves w_k(S) than just w₀ and w₁ considered above. Consequently, based on the “anticrossing theorem” [41], one could expect that in a larger system, more anticrossings would appear at non-zero S values in addition to the already discussed crossing of w₀(S) and w₁(S) at S≈0. In an extended model, the additional anticrossings would bend down the w₁(S) curve for higher S while having practically no impact on the w₀(S) one. This intuitive reasoning strongly suggests that the calculated linear increase of w₁(S), visibly different from the flattened dependence seen in Fig. 2, is most likely an artefact of our simple model.

An important assumption of our model is the reduction of the role of optical pumping to the mere establishment of the initial spin polarization Δn° and ignoring its role in the system dynamics. To gain more insight into the effect of light intensity, we measured the widths of the resonance components vs. laser power. In the range 0 - 20 mW we observed that the specific features of the composite resonances, such as the widths, amplitudes, and signal/noise ratios, depended on the light intensity. In particular, we observed a light-induced narrowing of w₁ and attributed it to the optical pumping and intersystem crossing, similar as in [42]. However, we did not see any qualitative changes in the described MW stabilization of w₀ that could change the general predictions of our simple model.

The role of relaxation and optical pumping in the openness of the system is crucial. We have shown that it can be controlled by an asymmetry between population relaxation rates γ₀, γ₁ which implicitly depend on relaxation, optical pumping and other possible interactions within the sample, like dipole-dipole, etc. This is illustrated by the fact that most efficient stabilization is observed for dense samples. Appropriate values of relaxation rates γ < Γ < σ are also necessary for creation of narrow structures (hole burning) in an inhomogenously broadened profile.

5. Conclusions

In conclusion, we have presented an experimental and theoretical analysis of an intriguing phenomenon of field-induced stabilization of composite resonances in open systems. The effect exists in systems which are open to reservoir constituted by all relevant states coupled in any way (relaxation, optical excitation/ de-excitation, intersystem crossing, dipole-dipole interaction, etc.) to the two-level system that is coherently driven. The system openness appears to be controlled by the difference of population relaxation rates of spin states, γ₀ ≠ γ₁ and does not require bichromatic driving. The latter, however, is necessary for observation of the composite resonance and its two-component structure. Competition between the relaxation and simultaneous driving of two spin sublevels by a bichromatic MW field in an open system leads to the formation of two states responsible for the two-component shape of the composite resonance associated with combinations of their populations characterized by different decay rates. The properties of these combinations are governed by the MW power and initial relaxation rates, and one of them becomes power-independent, i.e., stabilized in open systems.

We have found a formal analogy between the width stabilization and the creation of dark and bright states in CPT as a mathematical consequence of a 2 × 2 matrix diagonalization. In our case, however, the role of the strongly and weakly coupled states (respectively, bright and dark) is played by the population superpositions η₁ and η₀, rather than superpositions of wave functions routinely analyzed in CPT. This stabilization is evidenced in the described experiments carried out with three different NV samples. The described stabilization appears to be an intrinsic feature of the system’s openness. In that way, it is related to a wide class of phenomena where the population is not circulating among closed-system states, but may escape to reservoir [35,36], either constituted by a single state like our $\mathrm{{\cal R}}$, or by a continuum of states like in the case of sensitized atomic fluorescence [43,44] or pressure-induced extra-resonances (PIER) observed, for example, in Refs. [45,46].

The theoretically and experimentally discovered width stabilization is not only a formal curiosity. The described reduction of power broadening and the resulting small resonance width may be useful for precision spectroscopy. For such applications, the investigated samples should have different population relaxation rates γ₀, γ₁, much smaller than the overall dephasing rate, γ₀< γ₁<< Γ which is often realized in open solid state systems. Although our experiment dealt with a special case of NV diamond, the presented analysis of the discovered phenomena is general and may be useful for the characterization of the spin dynamics of various open paramagnetic systems and the control of their interaction with external fields. Related resonances created by interaction with the bichromatic field have also been studied in other recent experiments. In addition to studies with NV⁻ [27,29,31,34], composite resonances have been observed in other systems: SiC [30] and Exciplex States in Organic Light-Emitting Diode [33], and Refs. [33 and 34] reported also the absence of power-broadening in their spectra.