1. Introduction

Nonlinear frequency conversions are widely used techniques to obtain light at the wavelengths which are not within the direct reach by laser sources. Among them four-wave mixing (FWM) is commonly used to generate short-wavelength radiation with a gaseous medium.

Because of its broad applications for the plasma diagnosis, precision spectroscopy, and in recent years for the advanced lithography [1] and laser cooling [2, 3], etc., a Lyman-α light source has been recognized to be very important since early years. Although the study for the generation of Lyman-α radiation in nanosecond pulses or continuous waves (CW) started long time ago utilizing the FWM such as frequency tripling [4], sum-frequency mixing [5], and difference-frequency mixing [6, 7], the conversion efficiency does not exceed 10⁻⁶ ∼ 10⁻⁴. This is because the use of a gaseous medium is unavoidable to generate the vacuum-ultraviolet (VUV) radiation.

In order to overcome the conversion efficiency problem, a few different techniques have been invented. They include the introduction of the buffer gas to improve the phase matching condition [7] and the use of electromagnetically induced transparency (EIT) to suppress the reabsorption of the generated field [8]. Those techniques, however, have some limitations. Although the use of a buffer gas offers us some flexibility to tune the wavelength of the generated field, the conversion efficiency is still no better than 10⁻⁵ ∼ 10⁻⁴. As for the EIT-based scheme it does not allow us to tune the wavelength of the generated field, since a strong resonant coupling laser is needed to realize EIT. In any case the essential problem of the above-mentioned techniques is that the third order susceptibility, whose magnitude plays an important role for the conversion efficiency of FWM processes, generally remains very small.

With an aim to improve the third order susceptibility, a novel technique has been proposed and experimentally demonstrated by Harris and his coworkers [9,10]. They have demonstrated that the preparation of maximal coherence in a double-Λ system with a Pb vapor by the pump and Stokes pulses results in the extremely high conversion efficiency for the FWM processes within the coherence length when the probe pulse is subsequently injected to the coherently prepared medium. It is very important to note that the phase matching is not required in this case, since the FWM processes take place within the coherence length. Following the work by Harris et al., nonlinear frequency-conversion processes utilizing high atomic coherence in one way or the other have been studied by many groups. For instance, Stark-chirped rapid adiabatic passage (SCRAP) [11–13] and coherent population return (CPR) [14] schemes have been considered to generate maximal coherence for the enhancement of the FWM signals.

Although all these methods can result in high conversion efficiency, some limitations still exist. The double-Λ scheme [9, 10] cannot be directly applied for the generation of VUV radiation, since the use of double-Λ scheme implies that the generated wavelength would not be much shorter than the incident three fields. As for the CPR and SCRAP schemes, the wavelength of the generated field can be very short, since they are based on the ladder system with two-photon pumping. The common disadvantage of both schemes is that the third (probe) pulse must interact with the coherently prepared medium at the moment when the transient coherence is maximum. This means that the accurate control of the pulse timing is essential to realize the high conversion efficiency.

Recently Vitanov and Shore [15] proposed an interesting variant of stimulated Raman adiabatic passage (STIRAP) process in a two-level system. The idea in there is that the Bloch equation for a two-level system is mathematically similar to the Schrödinger equation for a Λ system. This implies that the counter-intuitive time sequence of the Stokes and pump pulses for the STIRAP in a Λ system can be replaced by the counter-intuitive time sequence of the time-dependent detuning and pump pulse in the two-level system. Accordingly the complete population transfer that is to be realized by the conventional STIRAP process in the Λ system corresponds to the stationary and perfect coherence by the detuning-induced STIRAP (D-STIRAP) process in the two-level system. In Fig. 1 we illustrate the D-STIRAP process. The adiabatic condition for the D-STIRAP process is ${\int }_{-\infty }^{\infty }\sqrt{{\mathrm{\Delta }}^2+{\mathrm{\Omega }}^2}\text{d}t\gg \pi /2$, where Δ is the detuning and Ω is the Rabi frequency of the pump pulse. Similar to the STIRAP process, the D-STIRAP process is robust against the choice of the laser intensity and the magnitude and even sign of the detuning. The D-STIRAP has been experimentally demonstrated with a single Ca ion using a linearly chirped pulse produced from the CW laser [16].

 

Fig. 1 (a) Proto-type scheme for the D-STIRAP process in a two-level system. (b) Time sequence of the detuning, Δ, and Rabi frequency, Ω, and the resulting time-evolution of atomic coherence to the maximum value, 0.5. As the STIRAP process realizes complete population transfer from the initial to the final state in a three level Λ-system, the D-STIRAP process realizes perfect coherence in the two-level system.

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At first glance the use of the D-STIRAP process seems more favorable than the CPR and SCRAP to generate the VUV radiation because of the existence of stationary and perfect coherence. But it has to be theoretically verified under the realistic context by including not only the Doppler broadening but also the ionization loss and Stark shifts by various lasers as well as spontaneous decay from the upper states, which are often neglected in the theoretical studies.

In this paper we theoretically study the generation of VUV radiation by difference-frequency mixing with a Doppler broadened gas utilizing high atomic coherence prepared by the D-STIRAP process. Unlike the schemes which make use of the CPR and SCRAP processes, our scheme is not very sensitive to the temporal jitter between the pump, Stark, and probe pulses. Our scheme is not sensitive to the exact value and even sign of the detuning induced by the Stark pulse, either, hence it is practically insensitive to Doppler broadening. To be most realistic, we consider the generation of picosecond Lyman-α (122 nm) pulses with a Kr gas. The specific choice to generate the picosecond Lyman-α radiation is motivated by our recent work to polarize muonium by the sequence of picosecond Lyman-α pulses [17]. Photoionization of polarized muoniums result in polarized muons, which are of great use for the muon facility at J-PARC [18].

2. Model

The scheme we consider in this paper is shown in Fig. 2. For the generation of Lyman-α pulses, the Kr gas is one of the good candidates as an efficient nonlinear medium due to its level structure, and we choose it as a specific example. Accordingly the states |1〉–|4〉 in Fig. 2 are labeled by 4p^{6 1}S₀, 5s [3/2]₁, 5s′ [1/2]₁, and 5p [1/2]₀, respectively. The pump pulse at the wavelength around 212.6 nm couples states |1〉 and |4〉 by two-photon transition under the presence of the auxiliary (Stark) pulse to induce Stark shifts. The purpose of the pump and Stark pulses is to prepare high atomic coherence in the medium through the D-STIRAP process, and hence nanosecond pulses can be efficiently used. After the preparation of high atomic coherence, a picosecond probe pulse at the wavelength of 843.5 nm is turned on to generate the picosecond Lyman-α pulse through the difference-frequency mixing, ω_Lyman = 2ω_pump − ω_probe. If we wish to generate the nanosecond Lyman-α pulse instead, a nanosecond probe pulse at the same wavelength can be alternatively used.

 

Fig. 2 Schematic diagram of the FWM process in Kr atoms. Δ₁₄ is a dynamic detuning due to the Stark shifts, Γ₄ is an ionization rate from state |4〉, γ₄₂ and γ₄₃ are spontaneous decay rates from state |4〉 to states |2〉 and |〉, respectively.

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In this work the wavelength of the Stark pulse is chosen to be 1064 nm, which is far off-resonant from any states. For the purpose of realizing high atomic coherence through the D-STIRAP process the use of two independent pulses, i.e., pump and Stark pulses, offers us more flexibility than the use of the linearly chirped pump pulse alone. There are two advantages. The first advantage is that the use of the Stark pulse enables us to easily control the detuning through its intensity. In contrast, control of the time-dependent detuning by the pump pulse alone through the chirp is not very easy, since there is an upper-limit for the chirp rate if the conventional linear optics is used to induce the chirp [19]. This makes a big difference in terms of the intensity requirement for the pump pulse. The adiabatic condition for the D-STIRAP process at hand is rewritten as, ${\int }_{-\infty }^{\infty }\sqrt{{\left(\delta +{\mathrm{\Delta }}_{14}\right)}^2+{\left[{\mathrm{\Omega }}_{\text{pump}}^{(2)}\right]}^2}\text{d}t\gg \pi /2$, where ${\mathrm{\Omega }}_{\text{pump}}^{(2)}$ stands for the two-photon Rabi frequency, δ is the static detuning of the pump laser while Δ₁₄ is the dynamic detuning as a result of the Stark shifts by the pump and Stark pulses. This adiabatic condition tells us that the intensity of the pump pulse does not have to be strong if the detuning is large, which can be easily realized by increasing the intensity of the Stark pulse. The second advantage is that we can realize the ideal counter-intuitive pulse sequence for the D-STIRAP process (Fig. 1(b)) by the use of the two pulses. If the linearly chirped pump pulse is used instead, the latter half of the pump pulse have to be cut out [16].

Before we move on to the next section, we clarify the difference between our scheme (the D-STIRAP scheme) and the SCRAP scheme [11–13] which apparently look similar. The essential difference between them is that, in the former the Stark pulse should be turned on before the pump pulse and the pump pulse should be resonant or near-resonant with the two-photon transition, while in the latter the Stark and pump pulse are simultaneously turned on, and the pump pulse must have a quite large detuning to ensure that it is much larger than the two-photon Rabi frequency during the rise of the pulses. The induced Stark shift by the Stark pulse then changes the sign of the detuning from a large negative value to a large positive value (or vice versa). As a result the maximum coherence appears in a transient manner [20]. In the D-STIRAP scheme the maximum coherence appears in a stationary manner.

3. Preparation of high atomic coherence using D-STIRAP

In this section, we investigate the dynamics for the preparation of coherence between the ground and excited states through the D-STIRAP process under the presence of the pump and Stark pulses only. Clearly the depletion/distortion of these pulses would be negligibly small because there are no single-photon resonant states involved. This means that the preparation process of coherence can be well-described by the single-atom response. The time evolution of the two states, |1〉 and |4〉, can be described by the density matrix equations in the following form:

We assume that the pump and Stark pulses have Gaussian temporal profiles, and they read

Because of the non-negligible contribution of the pump pulse on Δ₁₄ as we see in Eq. (7), the time sequence of δ + Δ₁₄ and ${\mathrm{\Omega }}_{\text{pump}}^{(2)}$ cannot be made ideal as shown in Fig. 1(b) if the pump pulse is two-photon resonant with the transition |1〉–|4〉, i.e., δ = 0. That is, if δ = 0, δ + Δ₁₄ disappears when ${\mathrm{\Omega }}_{\text{pump}}^{(2)}$ disappears, as illustrated by the red solid line in Fig. 3(a). Accordingly we cannot expect the perfect coherence if δ = 0. Recall that only the partial population transfer takes place for the STIRAP process [22] under the similar pulse sequence for the Stokes and pump pulses, and the same is true for the D-STIRAP process we utilize.

 

Fig. 3 (a) Illustration of the pulse sequence under the present study with Kr atoms for the case of δ = 0 (red solid line), which is to be compared with the prototype of the D-STIRAP process shown in Fig. 1. The case of finite positive/negative laser detuning (red dashed/dotted line) is also shown in this graph. Note that the detuning denoted by Δ₁₄ includes contributions from both pump and Stark pulses, while the Rabi frequency pulse denoted by ${\mathrm{\Omega }}_{\text{pump}}^{(2)}$ is solely from the pump pulse. (b) Color-coded plot of final coherence, |ρ₁₄|, by the D-STIRAP scheme as functions of the intensities of the pump and Stark pulses without Doppler broadening for the case of t_Stark = −0.9 ns and zero laser detuning, δ = 0. (c) Similar plot to graph (b) but with a finite laser detuning, δ = −3.2 GHz.

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The above qualitative argument has been confirmed by the numerical calculations. In order to highlight the importance of the pulse sequence for the pump and Stark pulses, we include neither the ionization and spontaneous decays nor the Doppler broadening for a moment, and solve Eqs. (1)–(3) with Eqs. (4), (7), and (8). The final coherence as functions of the intensities of the pump and Stark pulses is presented in Figs. 3(b) and 3(c) for the case of the zero and finite laser detunings, i.e., δ = 0 and −3.2 GHz, with the incident timing of the Stark pulse taken as t_Stark = −0.9 ns. We first look at Fig. 3(b). It clearly shows that the D-STIRAP process is indeed robust, since coherence does not vary significantly beyond the certain values of I_pump and I_Stark (and hence δ + Δ₁₄) when the adiabatic condition for the D-STIRAP process mentioned before is satisfied. Nevertheless, only partial coherence around |ρ₁₄| = 0.3 is achieved because of the simultaneous disappearance of Δ₁₄ and ${\mathrm{\Omega }}_{\text{pump}}^{(2)}$ since δ = 0. This result shows that, although the Stark shift induced by the pump pulse is much smaller than that by the Stark pulse (see Eq. (7)), its influence on the generation of high atomic coherence is non-negligible. In order to obtain the maximum coherence, δ + Δ₁₄ and ${\mathrm{\Omega }}_{\text{pump}}^{(2)}$ should not approach zero simultaneously. Now we introduce a non-zero laser detuning, δ, to realize such situation. Depending on the value and sign of δ, the total detuning, δ + Δ₁₄, in Fig. 3(a) will shift upward/downward, as shown by the red dashed/dotted line. A representative result is shown in Fig. 3(c) for δ = −3.2 GHz. Indeed nearly maximum coherence has been obtained by the introduction of an appropriate laser detuning at the expense of some loss of robustness against intensities. Notice that, although we see larger coherence in Fig. 3(c), there now exist certain intensity ranges for the pump and Stark pulses.

Having understood the basic behavior of the system towards the generation of high atomic coherence using the D-STIRAP process, we now consider the influence of the Doppler broadening of the gaseous medium. For an atom with a velocity v and the wave vector of the pump pulse, k, the Doppler shift of state |4〉 relative to state |1〉 is −2kv because it is a two-photon transition. Under the thermal equilibrium which is our case, we may assume that the velocity of Kr atoms obeys the Maxwell distribution, $\text{exp}\left(-v^2/v_{\text{p}}^2\right)/\left(\sqrt{\pi }{v}_{\text{p}}\right)$, where $v_{\text{p}}=\sqrt{2k_{\text{B}}T/m}$ is the mean velocity of Kr atoms at a given temperature, T, with m and k_B being the mass of the Kr atom and Boltzmann constant. We find that the Doppler broadening of the Kr gas at room temperature, T = 300 K, is about 3.8 GHz.

Now we carry out the similar calculations of atomic coherence with Doppler broadening of Kr atoms at room temperature. The results are shown in Fig. 4 without the ionization and spontaneous decays as in Fig. 3. From the comparison of Fig. 4(a) with Fig. 3(b), it is clear that the D-STIRAP scheme is almost insensitive to Doppler broadening. Note that the intensities of neither pump nor Stark pulses have to be strong. Figure 4(a) shows that maximum coherence can be obtained at the modest intensity of sub-GW/cm² for both pump and Stark pulses. This is encouraging for experimentalists. The variation of final coherence, |ρ₁₄|, as functions of the incident timing of Stark pulse, t_Stark, and laser detuning, δ, is also shown in Fig. 4(b), where it is apparent that the scheme is also robust against these two parameters.

 

Fig. 4 (a) Color-coded plot of final coherence, |ρ₁₄|, by the D-STIRAP scheme as functions of the intensities of the Stark and pump pulses with Doppler broadening. The laser detuning and incident timing of Stark pulse are chosen to be δ = −3.2 GHz and t_Stark = −0.9 ns. (b) Color-coded plot of final coherence, |ρ₁₄|, as functions the laser detuning and incident timing of Stark pulse under Doppler broadening. The intensities of the pump and Stark pulses are chosen to be I_Stark = 0.5 GW/cm² and I_pump = 0.4 GW/cm².

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We can easily understand the insensitive property of the D-STIRAP process to Doppler broadening if we plot the final coherence of a single atom with different velocities within the Maxwell distribution. Such a graph is presented in Fig. 5, where the intensities of the pump and Stark pulses as well as the laser detuning and incident timing of Stark pulse are fixed at I_pump = 0.4 GW/cm², I_Stark = 0.5 GW/cm², δ = −3.2 GHz, and t_Stark = −0.9 ns. It is obvious from this figure that, although the velocity distribution of Kr atoms ranges between ±600 m/s, almost all of them can contribute to produce nearly maximum coherence. Recalling the adiabatic condition for the D-STIRAP process, we notice that this is due to its robustness against the detuning. Indeed the insensitive property of the D-STIRAP process to Doppler broadening was already mentioned before in terms of the self-induced transparency in a two-level system [23]. The difference between their case and our case is that, in the former an ideal time sequence is assumed for the detuning and Rabi frequency (as in Fig. 1(b)), while in the latter the assumptions are more realistic (as in Fig. 3(a)). Another important remark for the insensitive property of the D-STIRAP process to Doppler broadening is that the result is robust against the change of the sign of total detuning, which somehow can be expected from the adiabatic condition.

 

Fig. 5 Prepared coherence by the D-STIRAP scheme in Kr atoms as a function of atomic velocity at I_pump = 0.4 GW/cm², I_Stark = 0.5 GW/cm², δ = −3.2 GHz, and t_Stark = −0.9 ns. The Maxwell distribution is also shown.

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The insensitive property of the D-STIRAP process to Doppler broadening implies that the coherence loss is only induced by the ionization and spontaneous decays. How much coherence is preserved at a certain time after the interaction can be determined by including not only the Doppler broadening but also both decays. The result of similar calculation to Fig. 3(b) and 4(a) at the time of 4 ns is now shown in Fig. 6(a). The lifetime of state |4〉 is known to be about 20 ns, i.e., γ₄ = 5×10⁷ (s⁻¹). Since the ionization cross section from |4〉 at this pump pulse wavelength is neither experimentally nor theoretically known,we assume that it is 3 Mb. Therefore,

 

Fig. 6 (a) Color-coded plot of coherence, |ρ₁₄|, by the D-STIRAP scheme at the time of 4 ns as functions of the intensities of the Stark and pump pulses under Doppler broadening with ionization and spontaneous decays. The laser detuning and incident timing of the Stark pulse are chosen to be δ = −3.2 GHz and t_Stark = −0.9 ns. (b) Comparison of our D-STIRAP scheme (black dashed line) with the two-photon resonant excitation scheme (black solid line) assuming the ionization cross section of 3 Mb from state |4〉 at 212.6 nm. For the D-STIRAP scheme I_pump = 0.4 GW/cm², I_Stark = 0.5 GW/cm², δ = −3.2 GHz, and t_Stark = −0.9 ns, while for the two-photon resonant excitation scheme I_pump = 2 GW/cm² and δ = −1.3 GHz to maximize the peak value of atomic coherence. Besides, we also show the results with ±20% different pump intensities by the red and green lines for each scheme, respectively. The results we obtain by assuming the ionization cross section of 30 Mb are also shown by blue lines.

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In order to further highlight the advantage of the D-STIRAP scheme, we make a comparison of the D-STIRAP and commonly used two-photon resonant excitation schemes. The relevant parameters for the D-STIRAP scheme have been chosen to be the same with those for Fig. 5. As for the two-photon resonant excitation scheme, we also take into account the Stark shift by the pump pulse, and hence the value of the obtained coherence depends on the laser detuning. Our calculations show that the transient coherence cannot be more than the value we can obtain at δ = −1.3 GHz and I_pump = 2 GW/cm², and we use those parameters. The results are shown in Fig. 6(b) together with the corresponding results by the D-STIRAP scheme. We notice that, although coherence takes similar maximum values for both schemes, the temporal behavior of coherence is very different. Coherence produced by the two-photon resonant excitation scheme dies away very rapidly, while that by the D-STIRAP scheme experiences a gradual decay. The rapid decay for the former is due to fast ionization because of the unavoidable choice of high intensity for the pump pulse and dephasing of coherence produced by different velocity components of Kr atoms. The gradual decay for the latter comes from the spontaneous decay of state |4〉 and there is no dephasing of coherence, because the D-STIRAP scheme is almost insensitive to Doppler broadening. Figure 6(b) clearly demonstrates the advantage of the D-STIRAP scheme to produce high coherence for the FWM process which is to be discussed in the next section, since it shows that timing of the probe pulse does not have to be accurately controlled. Finally we can see from Fig. 6(b) that the fluctuation of the pump pulse intensity has little influence on the evolution of coherence. We can also see that even if the ionization cross section is assumed to be as large as 30 Mb, coherence as much as 0.22 at the maximum is still obtained for both schemes. Besides, the advantage of the D-STIRAP scheme that the temporal decay of coherence is much slower than that of the two-photon resonant scheme is still seen regardless of the values of the ionization cross sections, 3 Mb or 30 Mb.

4. Generation of Lyman-α pulse

Once high atomic coherence has been prepared utilizing the D-STIRAP scheme, we can make use of it to efficiently generate the FWM signals by injecting the probe pulse at a certain timing. In this section we discuss the generation of picosecond Lyman-α pulse using a picosecond probe pulse. The probe pulse is also assumed to have the Gaussian temporal function at the wavelength of 843.5 nm, which is far-off resonant (39.6 THz and −108 THz, respectively) from the two intermediate states |2〉 = 5s′ [1/2]₁ and |3〉 = 5s [3/2]₁ (see Fig. 2). Strictly speaking, we must take into account the dynamics of states |2〉 and |3〉 during the interaction with the probe pulse. However, a significant simplification can be made for the present case as long as the probe pulse intensity is not very high. The condition for such a simplification is that the Rabi frequency induced by the probe pulse is much smaller than the corresponding detunings so that there is no notable populations in states |2〉 and |3〉 during the propagation of the probe pulse. Our estimation shows that the Rabi frequency by the probe pulse becomes comparable to the corresponding detunings at the probe pulse intensity of > 100 GW/cm². Therefore, if I_probe ≪ 100 GW/cm² at any time and space in the Kr gas, we may consider the interaction of the system with the probe pulse by adiabatically eliminating states |2〉 and |3〉. A similar argument holds for the generated Lyman-α pulse. Moreover, the Doppler broadening of the Kr gas at room temperature, 3.8 GHz, can be neglected because it is much smaller compared with the bandwidths of the 1 ps probe and generated Lyman-α pulses, which are about 440 GHz. As mentioned before, the depletion of the pump pulse is negligible since it is a two-photon excitation without any near-resonant intermediate states. This means that the pump pulse may be considered as constant during the FWM processes. After these considerations, we notice that we may use the propagation equations given by Harris [24]. The propagation equations of the probe and Lyman-α pulses under the slowly-varying envelope approximation read [24]

We solve the propagation equations, Eqs. (10) and (11), together with the density matrix equations, Eqs. (1)–(3), for states |1〉 and |4〉 to study the spatio-temporal variation of the probe and the generated Lyman-α pulses. We also include the Stark shifts for states |1〉 and |4〉 and ionization decays from |4〉 by the probe and generated Lyman-α pulses. After this modification Δ₁₄ and Γ₄ to be substituted in Eqs. (1)–(3) now read,

The coarse or fine temporal step sizes are chosen, depending on the duration of the interacting pulse at that moment. Namely, although we generally use the coarse temporal step size, 4 ps, we switch to the fine temporal step size, 4 fs, when the probe pulse is on. This is because the pump and Stark pulses have 1 ns duration, while the probe pulse has a 1 ps duration. The parameters we employ to perform the calculations are I_pump = 0.4 GW/cm², I_Stark = 0.5 GW/cm², δ = −3.2 GHz, and t_Stark = −0.9 ns. The product of the number density of Kr atoms, N, and the interaction length, L, is chosen to be NL = 10¹⁷ cm⁻².

Before proceeding we have ensured that the increment of the Lyman-α intensity is approximately 7 times larger than that of the probe pulse, as it should be. Recall that when the photon at the probe pulse wavelength is created the photon at the Lyman-α pulse wavelength is also created simultaneously, and hence the ratio of the increment is ΔI_Lyman/ΔI_probe = λ_probe/λ_Lyman = 843.5 nm/121.6 nm ≈ 7. Now we investigate the spatial evolution of the probe and Lyman-α pulses for the D-STIRAP scheme. First we show the results without Doppler broadening in Figs. 7(a) and 7(b) for different intensities but at the same incident timing of the probe pulse. From the result shown in Fig. 7(a) for I_probe = 0.05 GW/cm² and t_probe = 0.3 ns at which the prepared coherence is maximum for the D-STIRAP scheme (see Fig. 6(b)), it is clear that the coherence length is rather limited. In other words there is a notable phase mismatch. In Ref. [24] Harris suggested to introduce a small detuning to cancel the phase mismatch. Obviously the laser detuning we have at hand for the |1〉–|4〉 transition is far from the optimized detuning. Now we increase the probe pulse intensity to I_probe = 0.4 GW/cm² but with the same incident timing for the probe pulse. The result is shown in Fig. 7(b). We notice that the intensity of the Lyman-α pulse significantly increases. More interestingly the coherence length becomes much longer, because the larger laser detuning induced by the larger intensity of the probe pulse (see Eq. (16)) results in the better phase matching condition. Now we perform the calculation with Doppler broadening while all the other parameters are kept to be the same with those for Fig. 7(a). The result is shown in Fig. 7(c). We notice that the inclusion of Doppler broadening does not spoil the Lyman-α generation. There are two reasons for this. The first reason is that, the D-STIRAP process is nearly insensitive to Doppler broadening to prepare high atomic coherence as mentioned before (see Fig. 5). The second reason is that the Doppler broadening serves as an effective detuning for the atomic ensemble, which helps to improve the phase matching to generate the FWM signal as explained in Ref. [24]: To understand why the Doppler broadening serves as an effective detuning for the atomic ensemble, we refer to the plot of coherence as a function of atomic velocity, i.e., Fig. 5, which has been calculated without ionization and spontaneous decays. Since we have found that its shape does not differ so much with and without decays, we can utilize Fig. 5 to qualitatively understand the role of Doppler broadening. Figure 5 obviously shows that coherence is not symmetric with respect to the atomic velocity, and atoms with positive velocities make more contribution to coherence. This means that, if we have Doppler broadened atoms the effective velocity of the atomic ensemble can be considered to be non-zero, and hence the effective detuning is also non-zero. Then, following the argument in Ref. [24], we can conclude that the effective non-zero detuning of the Doppler broadened atoms helps to improve the phase matching condition. An interesting question is how much the incident timing of the probe pulse influences the generation efficiency of the Lyman-α pulse. To answer this question we have compared the results in Fig. 7(d) for t_probe = 0.3 ns and 4 ns with a probe pulse intensity of I_probe = 0.05 GW/cm². As we could expect from Fig. 6(b), the generation efficiency is rather insensitive to the incident timing, which is very favorable from the experimental viewpoint. The small reduction of generation efficiency at t_probe = 4 ns is mainly due to the loss of prepared coherence by the spontaneous decay from state |4〉.

 

Fig. 7 Spatial evolution of the probe and Lyman-α pulses by the D-STIRAP scheme for (a) I_probe = 0.05 GW/cm² and t_probe = 0.3 ns without Doppler broadening, (b) I_probe = 0.4 GW/cm² and t_probe = 0.3 ns without Doppler broadening, and (c) I_probe = 0.05 GW/cm² and t_probe = 0.3 ns with Doppler broadening. (d) Spatial evolution of the Lyman-α pulse by the D-STIRAP scheme for I_probe = 0.05 GW/cm² with Doppler broadening at two different timings of the probe pulse, t_probe = 0.3 and 4 ns. Ionization and spontaneous decays from state |4〉 are included for all cases.

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Having understood how Doppler broadening as well as intensity and temporal jitter of the probe pulse influence the nonlinear frequency conversion process, we now estimate the conversion efficiency for the D-STIRAP scheme under the parameters we have employed. By defining the conversion efficiency as

Note that the various parameters employed in this paper are not optimized. Therefore, it is interesting to consider how the choice of the intensities of the pump and probe pulses influences the conversion efficiency. If the pump pulse intensity is too high or too low, we can see from Fig. 6(a) that prepared coherence would become lower, which results in the lower conversion efficiency. In particular if the pump pulse intensity is too high, the conversion efficiency would rapidly go down due to the increased pump pulse energy and reduced output of the Lyman-α pulse energy. Hence the pump pulse intensity should be chosen with care. As for the probe pulse intensity, recall that preparation of coherence is already completed when the probe pulse is turned on. Therefore the probe pulse intensity does not influence the degree of prepared coherence. Due to the Stark shift, different probe pulse intensities result in the slight shifts of the generated VUV wavelength. Of course if we could use higher probe pulse intensity the conversion efficiency can be made higher, since the generation of Lyman-α pulses is a nonlinear frequency conversion process.

We would like to make an additional remark on the conversion efficiency. Here we assume one-dimensional propagation of the pulses to simplify the analysis and also highlight the essence of the work. In reality, however, the radial intensity distribution of the pulses and the focusing could influence the conversion efficiency. Therefore, the conversion efficiencies we estimated above should be considered as a rough guide for experiments to compare the performance of the D-STIRAP scheme with existing schemes.

It is interesting to consider how the conventional two-photon resonant scheme is compared with our D-STIRAP scheme. The results are summarized in Fig. 8. In Figs. 8(a)–8(c), the incident timing of the probe pulse is chosen to be t_probe = −0.65 ns so that the prepared coherence is maximum (see Fig. 6(b)). As we see in Fig. 8(a), the coherence length is limited for the case of a weak probe intensity, I_probe = 0.05 GW/cm², without the Doppler broadening. If the probe intensity is increased to, say, I_probe = 0.4 GW/cm², the coherence length becomes a bit longer. But it is not as significant as it is for the D-STIRAP scheme we have seen in Fig. 7(b). This is because the preparation of coherence by the two-photon resonance is very sensitive to the exact value of the detuning for the |1〉–|4〉 transition. When we include the Doppler broadening, the coherence length becomes much longer. This can be understood from Fig. 9 where the prepared coherence is plotted for the different velocity components of Kr atoms at room temperature. Since the Rabi frequency at I_pump = 2 GW/cm² is ${\mathrm{\Omega }}_{\text{pump}}^{(2)}=3.2\times {10}^{10}\text{rad}/\text{s}$ which is comparable to the Doppler width, 3.8 GHz, the curve for coherence in Fig. 9 shows a rather flat top and similar width with that of Doppler broadening. The small laser detuning, δ = −1.3GHz, slightly shifts the curve to the left, which helps to improve the phase matching condition as we see in Fig. 8(c). Finally Fig. 8(d) compares the spatial evolution of the Lyman-α pulse at two different incident timings, t_probe = −0.65 and 4 ns. If the incident timing is different from the optimized timing, we observe the significant reduction of the generation efficiency. Similar estimations of the conversion efficiency based on Fig. 8(d) for the two-photon resonant scheme show that η (t_probe = −0.65ns) ≈ 3.2×10⁻⁴ and η (t_probe = 4ns) ≈ 1.4×10⁻⁵. The comparison of Fig. 7 and Fig. 8 clearly demonstrates the superiority of the D-STIRAP scheme to the conventional two-photon resonant scheme if we wish to generate the VUV radiation by the FWM processes. This is so, partly because the pump pulse intensity required to prepare coherence by the D-STIRAP scheme is lower than that by the two-photon resonant scheme, and partly because the D-STIRAP scheme does not suffer from the temporal jitter between the pump and probe pulses.

 

Fig. 8 Spatial evolution of the probe and Lyman-α pulses by the two-photon resonant scheme for (a) I_probe = 0.05 GW/cm² and t_probe = −0.65 ns without Doppler broadening, (b) I_probe = 0.4 GW/cm² and t_probe = −0.65 ns without Doppler broadening, and (c) I_probe = 0.05 GW/cm² and t_probe = −0.65 ns with Doppler broadening. (d) Spatial evolution of the Lyman-α pulse by the two-photon resonant scheme for I_probe = 0.05 GW/cm² with Doppler broadening at two different timings of the probe pulse, t_probe = −0.65 and 4 ns. Ionization and spontaneous decays from state |4〉 are included for all cases.

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Fig. 9 Prepared coherence at t = −0.65 ns (see Fig. 6(b)) by the two-photon resonant scheme in Kr atoms as a function of atomic velocity at I_pump = 2 GW/cm² and δ = −1.3 GHz. The Maxwell distribution is also shown. Note that ionization and spontaneous decays are included.

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5. Conclusion

In conclusion, we have theoretically investigated the generation of VUV pulses with a Doppler-broadened gas utilizing the D-STIRAP process to prepare high atomic coherence. Specifically we have presented the results for the generation of picosecond Lyman-α pulses in the Kr gas.

Our D-STIRAP scheme makes use of the two-photon near-resonant excitation by the pump pulse assisted by the auxiliary (Stark) pulse. The role of the Stark pulse is to induce the Stark shifts and control the effective detuning between the two states. As the adiabatic condition of the D-STIRAP process tells us, the pump pulse intensity may be kept rather low if a relatively large detuning, which is a sum of the laser detuning and the Stark shifts, is chosen. The big advantage of the D-STIRAP scheme is that high atomic coherence can be prepared even if a Doppler broadened gas is employed under the not-very-accurately-controlled intensities of the pump and Stark pulses as well as the timing between them. This implies that our scheme is nearly insensitive to Doppler broadening to generate the VUV pulses. Moreover, the presence of Doppler broadening is rather helpful to improve the generation efficiency. The underlying mechanism for this improvement is that the overall phase matching condition of the gas becomes better because of the presence of Doppler shifts.