1. Introduction

For many years, excitons have played an important role in the description of optical properties of insulators and semiconductors. They consist of an electron and a hole which are bounded by their Coulomb attraction, closely resembling atoms with a series of energy levels similar to hydrogen. Cuprous oxide Cu${ }_2$O is a generic example in which four different excitonic states (yellow, green, blue and violet) have been observed. In 2014 an outstanding experiment realized by the Dortmund group [1] has reneved the interest in the field of excitons as it turned out that the yellow series could be followed up to high principal quantum number n=25. Those high-lying excitons in analogy to Rydberg atoms have been called Rydberg excitons (RE). A large spatial extent of RE reaching micrometers, significantly exceeding the length of the wave which has created them, large lifetimes which scale as n² and the energy spacing of neighboring states which decreases as $n^{-3}$ allow for an observation of RE in ranges of external parameters much different from other quantum situations. Their excitation energy of about 90 meV is lower by two orders of magnitude than the atomic Rydberg energy. This small binding energy of RE makes them sensitive to external electric or magnetic fields as compared with other systems. Those specific properties od RE in Cu${ }_2$O have motivated both theoretical and experimental interest in this field, with studies ranging from their spectroscopy, i.e., optical [2], electrooptical [3] and magnetooptical spectra [4] through non-atomic scaling laws [5] to quantum chaos [6]. Rydberg excitons can easily be excited and manipulated with laser light and they feature a quasicontinuum of narrow states with strong transition coupling to microwave and terahertz radiation. Those two properties endow RE with unusual potential for applications combining optical light with THz waves and microwaves in quantum sensing and information processing.

On the other hand, investigations of the dynamical properties of the systems with RE have recently attracted more interest. The dipole moments of RE, due to a large orbital radius, are exceptionally large for higher values of n. Therefore, the interaction between high lying Rydberg excitons is particularly strong and leads to the so-called Rydberg blockade that prevents the optical excitation of nearby excitons by shifting their corresponding levels out of the resonance with the exciting electromagnetic field. The appearance of Rydberg blockade clearly distinguishes linear and nonlinear regimes of dynamical phenomena regarding RE. In the nonlinear regime, the possibility of observing giant optical nonlinearities of RE in microcavities has been considered [7], while the paper [8] deals with strong interaction phenomena at very low densities of RE, which enables one to determine the contribution of the nonlinear optical response of the medium. In the case of a smaller exciton density it is possible to remain in the linear range for which a single-photon source has been proposed [9].

Recently a lot of effort has been devoted to develop a solid-state maser devices which are characterized by compact size, material stability and high optical pumping efficiency. While the generation of gain based on population inversion due to electromagnetically induced transparency in N-type four-level atomic systems has been dicussed in [10, 11] we focus our attention on RE in couprus oxide. Concerning RE it is worth mentioning that the wavelenghts corresponding to the transitions between higher states fall into sub- and milimeter region. Matching together their attractive properties, i.e., long lifetimes, long dephasing rates, huge dipole moments and their sensitivity to external fields, we have proposed to use an ensemble of optically pumped RE in cuprous oxide as the gain medium for a continuous-wave maser oscillators. The existing atomic gaseous masers demand high vacuum [12] and many solid-state masers require helium temperatures [13]. Moreover, once constructed, such a maser operates at a fixed wavelength. Continuous-wave room temperature maser has been first realized in 2012 in the organic material of p-terpenthenyl molecular crystal [14] and then room-temperature solid-state maser and microwave amplifiers using nitrogen-vacancy NV centere in diamond have been constructed [15, 16].

It seems that Rydberg excitons are promising candidates for the realization of the population inversion, which leads to a tunable maser [17] because the excitonic levels can be accessed separately by tuning the wavelength of the optical excitationvia an external electric field, which due to Stark shift, can displace the resonances (energy levels) [18]. By tuning specific transitions to the frequency of microwave cavity, one can establish population inversion, so that the Cu${ }_2$O crystal becomes a gain medium. Although RE were observed at helium temperatures, we claim that in principle a maser might operate up to nitrogen or even higher temperatures [18] and it can be tuned by an external electric field, generating a wide range of emission at different wavelengths with a significant output power. In this paper we theoretically analyse and discuss in detail the case of pulse maser established in a three or four-level system consisting of of Rydberg excitons in Cu${ }_2$O. Our numerical simulation demonstrates that masing and microwave amplification are feasible under accessible conditions and our devices can reach output power up to ${10}^{-3}$ W at the temperature of 100 K.

In the first section we recall the characteristic features of REs particularly useful for obtaining population inversion and stimulated microwave emission. Next, we propose a masing systems based on a ground state and three or two excitonic levels. The basic equations describing the population dynamics are presented and the influence of the microwave cavity geometry and Q factor on the performance is indicated. The following sections 4 and 5 are devoted to the discussion of numerical results obtained for three- and four- level maser systems. The general characteristics of setups based on all accessible state configurations are presented and selected, representative examples are discussed in detail. Finally, the conclusions are presented.

2. Rydberg exciton properties

The Rydberg excitons in Cu${ }_2$O feature several unique properties, making this material especially suitable as a gain medium in a solid-state maser. Importantly, they exhibit many similarities to Rydberg atoms, which have been successfully used for stimulated microwave emission in atomic vapor [19]. The key common properties and differences between those media are outlined below.

One of the striking characteristics of Rydberg atoms and Rydberg excitons are their large sizes. Due to the large distance between electron and hole, the P excitons with large n are characterized by enormous dipole moments. Assuming a hydrogen-like wavefunction, the radius of exciton with quantum number n is [1]

Keeping the assumption of hydrogen-like wavefunctions one can calculate from their overlap the $S\to P$ transition dipole moments d_ij [17, 20]. Our numerical results for the first 10 states are shown in the Table 2.

Table 1. Transition dipole moments $d_{n_1{n}_2}$ between $n_{1}S$ (rows) and $n_{2}P$ (columns) states given in units of $e{a}_B$.

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From the calculated values shown in the Table 1, several general observations follow; the largest moments are obtained for $nP\to nS$ transitions. The magnitude of those moments scales roughly as n². For $n_2\to n_1$ transition ($n_2>n_1$), the scaling is on the order of $4^{\left(n_1-n_2\right)}$.

The large excitonic radius is a limiting factor to the maximum density of excitons. For a sufficient concentration, the efficiency of exciton creation is diminished due to the Rydberg blockade mechanism [1]; as the mean distance between excitons decreases, their mutual dipole-dipole interaction causes a shift of energy levels which prevents further light absorption. In our calculations this mechanism is taken into account by multiplying the absorbed pump power by a factor $b=\left(1-\rho V_{blockade}\right)$ [9], where the $V_{blockade}$ is the volume occupied by the pumped excitonic state, which is given by

This estimation can be used to calculate the upper limit of the population; for example, for low n = 3 state, assuming that the whole available volume is occupied by excitons, one can achieve exciton density of up to $\rho =1.5\cdot {10}^{12}$ mm${ }^{-3}$. This value is much larger than the density of Rydberg atoms used in masers [19] and comparable with other solid-state devices such as the diamond maser [16], which is based on $N\sim {10}^{14}$ active NV centres.

As shown by Kazimierczuk et al. [1], by increasing the input power from $P=0.01$ mW/mm${ }^2$ to P = 1 mW/mm${ }^2$, one can expect a 90% reduction of absorption for $n=18$ state and 99% reduction for $n=22$. On the other hand, due to the fact that the blockade efficiency scales with the state number as $n^{10}$, it is dramatically smaller for low n states; for $n<10$ the absorption efficiency should remain high even for input power of up to 100 W/mm${ }^2$, where thermal considerations become the limiting factor.

Another characteristic aspect of Rydberg excitons, which will be usefull for obtaining population inversion, are their unusually long lifetime and the corresponding dissipation parameters γ. For the P exciton with quantum number n, one can use a fit [2] to the experimental data [1]

The second term of the r.h.s. of above formula represents exciton scattering on acoustic (AC) phonons, and the third one is due to interactions with longitudinal-optical (LO) phonons. The coefficients ${\gamma }_{AC}=22.9\cdot {10}^{-3}\text{ meV}/\mathrm{K}$ and ${\gamma }_{LO}=1633\text{ meV}$ represent the magnitude of those two interactions, respectively. The relation predicts almost a constant linewidth below $T=50$ K, as explained by Stolz et al [24]. This effect has been taken into account in other models [25] and has also been observed experimentally by Kitamura et al. [21]. For the non-radiative relaxation of S - excitons, one can use a general estimation ${\gamma }_S\sim 0.1{\gamma }_P$ [26]. Finally, since the transition probability between states $i,j$ scales as ${\omega }_{ij}^3$ [5, 27], one can estimate the inter-excitonic transition rates

The energy levels of the upper excitonic states are located closely together, so that the inter-excitonic transitions are located in the microwave regime. The exciton energy depends on the quantum number n as follows

3. System setup

As in our previous work [17], the system consists of a Cu${ }_2$O crystal placed inside a metal cavity (See Fig. 1, left panel). The cavity is characterized by a resonant frequency ${\omega }_{c2}$ and a quality factor of $Q_2={10}^5$, which is a typical of dielectric-loaded cavities [16, 30] and can reach values of up to ${10}^8$ for superconductor cavities [31]. The crystal has a form of a cylinder bounded by two parallel metallic mirrors (one semi-transparent), forming an inner cavity with frequency ${\omega }_{c1}$ and $Q_1={10}^4$. These two frequencies are tuned to the respective transitions in a 4-level system based on excitonic levels (Fig. 1, right panel). Due to the Purcell effect, which is an environment-induced enhancement of the rate of spontaneous emission [17, 32], the probability of the selected microwave transition $i\to j$ is muliplied by a factor proportional to the Q factor and a ratio ${\lambda }_{ij}^3/V$, where V is a value close to the geometric volume of the cavity [33], which is in our case of order of millimeters. The purpose of the outer cavity is to promote a fast transition between $n_{3}P$ and $n_{2}S$ state, leaving the upper, optically pumped level relatively empty. The n₂ level, which due to selections rules has to be S-exciton state, is relatively metastable; its population increases up to the point where the masing action starts based on $n_{2}S\to n_{1}P$ transition. To describe the dynamics of the system, we used a modified set of the rate equations [17, 34]

It should be stressed that in contrast to the continuous-wave system [17], the discussed system is characterized by a complex dynamics; the presence of a strong microwave field in the cavity causes a Stark shift, detuning the transitions from the respective cavity frequencies and changing the Purcell factors by up to 4 orders of magnitude and thus enabling or disabling the masing action. Moreover, the temperature of the system is not constant and has a nontrivial effect on the population dynamics. The optimal values of parameters such as geometric dimensions of the crystal, inner and outer cavity Q

factors, peak pump power and pumping pulse duration are highly dependent on the chosen excitonic levels.

Finally, we note that a masing action can be also obtained in a simpler, three level system which can be described by modifying Eq. (9), in particular omitting the N₃ populations and instead applying pump to the ensemble of $N_{2\pm i}$ states. However, this approach allows for less flexibility regarding the choice of the excitonic states and utilizes one tunable cavity as opposed to two in the 4-level scheme.

 

Fig. 1 Schematic depiction of the proposed system and excitonic energy level scheme.

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4. Three-level system

Let’s consider a 3 level system based on the modified setup shown on Fig. 1. The pump is tuned to the $n_{2}P$ state and masing occurs on the $n_2\to n_1$ transition. The system uses one microwave cavity tuned to the masing transition. By solving modified Eq. (9), one can obtain populations and emission power as a function of time.

 

Fig. 2 Emission power in 3-level system, as a function of wavelength for T=10K and T=100K. The principal numbers n₁, n₂ are given in the brackets.

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Figure 2 shows the peak maser emission power as a function of wavelength for cavity $Q={10}^4$, pump power $P=10$ W, pumping pulse duration τ = 20 ns and two selected temperatures. A significant number of state combinations (shown in brackets) exhibit an efficient microwave emission with maximum power of up to 10 mW. The wavelength depends mostly on the lower level n₁ of the $n_2\to n_1$ transition. The systems where $n_2=n_1+1$ are the most efficient due to the high overlap of wavefunctions, resulting in a significant transition dipole moment (e.g., for the system of [1, 2] or [2, 3] states). However, in some cases the transition dipole moment is sufficiently large to cause emission before any significant population inversion can occur, so that not all possible combinations of states are present on Fig. 2. For the higher temperature T=100 K, the configurations with $n_1>5$ become inaccessible due to a significant broadening of the spectral lines. Notably, in these conditions, the quantum number of the upper state n₂ can reach values exceeding 10; however, in such a case, the pump laser excites not only the chosen state n₂, but also a whole ensemble of nearby states, lowering the overall efficiency of the system. Nevertheless, there are quite a lot of accessible states that enable the maser action of various wavelengths.

 

Fig. 3 (a) State populations and blockade volume occupied by excitons in maser system based on [3, 5] levels. (b) Emission power and transition detuning from the cavity caused by Stark shift. (c) Total absorbed and emitted energy.

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On Fig. 3(a) one can see the populations of the maser states for the states’ pair [3,5] at T=10 K and the same conditions as in Fig. 2 (e.g., τ = 20 ns, P = 10 W). The chosen $5\to 3$ transition has a frequency of${\omega }_{53}=12.1$ THz. The time t = 0 marks the maximum of the pump laser power. As the power increases, the upper state reaches a significant population and the condition of population inversion is established. The $n_{2}P$ exciton population increases until the fraction of crystal taken by Rydberg blockade reaches unity, e. g. the point where the medium is saturated and no more excitons can be created. This is possible because the cavity is initially detuned from the $n_{2}P\to n_{1}S$ transition, as shown on Fig. 3(b). This can be achieved by applying a constant electric field to the system, inducing the Stark shift. At $t\approx -20$ ns, the inversion is sufficiently high to start masing despite the detuning. As the microwave field builds up in the cavity, the Stark shift changes the $n_{2}P\to n_{1}S$ transition frequency ω₂₁ so that it is closer to the cavity frequency ${\omega }_{c1}$. This, in turn, further enhances the emission. As a result, there is an exponential increase in the emitted field up to the moment when the inversion is destroyed. This mechanism, which can be seen as a form of passive Q-switching, is a key feature of our proposal; it is only possible due to the exceptionally high sensitivity of excitonic levels to the applied electric field. One can see on Fig. 3(a) that at $t\approx -20$ ns there is a sudden drop of the population N₂ and corresponding emission power peak on Fig. 3(b). Notably, the maximum power is in the range of 200 mW, much higher than in the same system on Fig. 2. This means that the introduction of an initial detuning allows for a much higher population inversion. One can see on Fig. 3(b) that the considered system has a second, wider emission peak starting at t = 0. At this point, the population N₁ is still relatively high, but significant pump power maintains a small population inversion, leading to a quasi - stationary regime where all populations remain relatively constant, up to the time $t\approx 30$ ns. The emission power is smaller than in the initial peak, but still considerable. Moreover, due to the much longer emission time, this peak represents the biggest contribution to the total emitted energy (Fig. 3(c)). Another notable feature is the second, narrow peak at $t\approx 40$ ns. This point corresponds to the condition where the masing transition is perfectly tuned to the cavity. The overall timescale of the whole processes ($\sim 400$ ns) is consistent with the experimental results with Rydberg atom masers [19]. The shortest emission peak has a duration of ${\tau }^{\prime }\sim 1$ ns which corresponds to the spectral width of $\delta \omega \sim 1$ GHz. During the emission peaks, the field is $E\sim {10}^2$ V/cm, which is significantly below the breakdown voltage of ${10}^5$ V/cm. From Fig. 3(c) one can conclude that the energetic efficiency of the system is on the order of ${10}^{-4}$. This is caused mainly by the fact that single optical frequency photon creates up to one exciton, which in turn emits one microwave frequency photon. The total emitted energy $E_{emit}\sim 10$ GeV corresponds to $1.3\cdot {10}^{12}$ microwave photons ($E_{53}\approx 8$ meV), which is comparable with the population of the upper state. Assuming that, apart from microwave emission, all the remaining energy is dissipated as heat, a single cycle increases the temperature of 1 mm${ }^3$ crystal by ${10}^{-1}$ K (the specific heat of 489 J kg${ }^{-1}$ K${ }^{-1}$ is assumed). In conclusion, even for high pumping intensity, the maximum power is not thermally limited.

 

Fig. 4 (a) State populations and blockade volume occupied by excitons in maser system based on [7, 8] levels. (b) Emission power and transition detuning from the cavity caused by Stark shift. (c) Total absorbed and emitted energy.

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To get a better insight into the dynamics of the maser system based on the upper excitonic levels, let’s consider $n_1=7$ and $n_2=8$ states with a much lower emission frequency ${\omega }_{78}=0.734$ THz. To avoid excessive bleaching [1], the pump power is reduced to $P=10$ mW but the pulse duration is increased to τ = 80 ns. Interestingly, one can see on Fig. 4(a) that even in these conditions, the system reaches saturation with almost 40% volume taken by excitons. This is caused by the fact that n = 8 excitons are much larger than the earlier considered n = 5 ones. One can see that the maximum population is of the order of 10¹¹. As mentioned earlier, the large transition dipole moment between upper excitonic states means that the emission is possible even in the condition of detuning from the cavity. As seen on Fig. 4 (b), ${\omega }_{87}\ne {\omega }_{c1}$ at all times, which facilitates the build-up of a significant population inversion. In this system, the emission occurs at $t\approx 50$ ns, after the peak of the pump power. Again, one can see that the sudden increase of the output power is caused by a reduction of the detuning caused by the Stark shift. The maximum power is $0.2$ mW, which is by 3 orders of magnitude lower than in the previous example, accordingly to the proportionally reduced pump power. In contrast to the previous system, the emission occurs after the pump has been switched off; as the population inversion is destroyed, the power decreases exponentially, which is marked on Fig 4 (b) by a straight line. The emission time ${\tau }^{\prime }\sim 10$ ns is considerably longer than in the [3, 5] system, which yields a narrower emission spectrum with full width at half maximum $\delta \omega \sim 100$ MHz. Finally, despite the much lower pump power, the total absorbed and emitted energy (Fig. 4(c)) are only smaller by a factor of 250 due to a longer pump pulse duration. The overall efficiency is still in the range of ${10}^{-4}$. To sum up, the masing systems based on higher excitonic levels are characterized by slower dynamics and much lower peak emission power. On the other hand, masing in these setups is feasible even with a very low pump power.

5. Four-level system

A four level system consists of the upper, optically pumped $n_{3}P$ state, intermediate, metastable $n_{2}S$ state and lower $n_{1}P$ state, as shown on the Fig. 1. The setup consists of two microwave cavities tuned to $n_3\to n_2$ and $n_2\to n_1$ transitions. Separating the pumped and metastable state and introducing two cavities allow for more flexibility in tuning the system. Specifically, one can utilize high frequency, high power lower level masing transition $n_2\to n_1$ in a system which is highly susceptible to the electric field due to the strong Stark shift of upper n₃ level altering its tuning to the outer cavity. This results in the maser’s operation in strong, passive Q-switching regime, leading to a very rich dynamics.

 

Fig. 5 Emission power in 4-level system, as a function of wavelength for T = 10K and T = 100K. The principal numbers n₁, n₂, n₃ are given in the brackets. For any set of n₁ and n₂, the combinations with lowest and highest n₃ are shown.

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Figure 5 shows the maser emission power as a function of wavelength for a 4-level system with inner cavity $Q_1={10}^4$ and outer cavity $Q_2={10}^5$. The pump power is $P=10$ W and the pulse duration is again τ = 20 ns. One can see that the results are grouped in columns corresponding to the common values of n₁, n₂ resulting in the same emission frequency ω₂₁. In comparison with Fig. 2, more systems based on upper states are present. This is caused by the introducing the third excitonic state and second cavity; the pumped level n₃ remains relatively empty at all times due to the Purcell effect, which increases the probability of $n_3\to n_2$ transition. This means that every pump photon has a high probability of creating $n_{3}P$ exciton and the effect of Rydberg blockade is not very significant as long as n₂ is relatively low. The important advantage over a 3-level system is that the middle state n₂ is an S-exciton state, which is characterized by a relatively longer lifetime [26], helping to achieve significant population inversion $n_2-n_1$. Additionally, the lower level $n_{1}P$ has a shorter lifetime than in 3-level system which is beneficial for keeping this level empty.

 

Fig. 6 (a) State populations and blockade volume occupied by excitons in maser system based on [2–4] levels. (b) Emission power and transition detuning from the cavity caused by Stark shift. (c) Total absorbed and emitted energy.

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Figure 6 shows a typical dynamics of 4-level maser with small initial detuning of the inner cavity ${\omega }_{c1}$ and no detuning of the outer cavity ${\omega }_{c2}$. The population of the relatively stable intermediate state $n_{2}S$ reaches a maximum value of 10¹¹, filling 1% of the available space (Fig. 6(a)). The emission starts early at $t=-18$ ns, with maximum power of 100 mW (Fig. 6(b)). Both $n_{1}P$ and $n_{3}P$ states are affected by the Stark shift. The higher frequency ω₂₁ is not as strongly affected as the ω₃₂ due to the fact that the higher levels are more shifted. Additionally, the higher Q factor of the outer cavity means that its operating frequency range is narrower. This means that the dynamics of the system is mainly governed by the detuning of ω₃₂ from the outer cavity frequency ${\omega }_{c2}$ (Fig. 6(b)). Specifically, after some time a significant microwave field builds up in the cavity and the following Stark shift prevents any further emission. As a result, masing occurs in the form of a series of bursts (${\tau }^{\prime }\sim 1$ ns). As mentioned above, this occurs for Rydberg blockade fraction as small as 0.01; this value can be increased by changing ${\omega }_{c2}$ and delaying the emission. However, the output power of the maser would still be limited by the Stark shift of n₃ level.

One can conclude that the system dynamics is highly complex and strongly dependent on the initial detuning of the cavities. The oscillation of populations on Fig. 6(a) reflects the emission power spectrum. Apart from these fast changes, one can also see that the populations of $n_{1}P$ and $n_{3}P$ states decrease exponentially for $t>0$ while that of $n_{2}S$ remains almost constant due to its longer lifetime. Also, the fast decay of $n_{1}P$ population allows for higher pulse repetition rate than in three level systems. Finally, the energetic efficiency is once again on the order of ${10}^{-4}$ (Fig. 6(c)).

6. Conclusions

We have discussed and numerically analysed a proposal for a high power, pulsed maser based on Rydberg exciton states. Our results indicate that the masing action is feasible in many three- and four level systems under a wide range of conditions, even at temperatures of over 100 K and with relatively low Q-factor cavities. A generation of nanosecond pulses with peak power of over 200 mW and overall device efficiency of ${10}^{-4}$ is reported. Moreover, we show that the dynamics of the system is much richer than that for the continuous-wave maser [17]; due to the Stark shift of energy levels, the conditions for population inversion are highly sensitive to the microwave field present in the cavity, making the system highly complex. This self-detuning affects the dynamics in a manner similar to passive Q-switching, enabling or disabling the masing action depending on the power stored in the cavity. The impact of the Rydberg blockade and temperature on the maser performance is also discussed; three level systems based on low n excitons offer highest emission power while the four level systems based on upper states provide the greatest tuning flexibility and pulse repetition rate.