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Excitation probability of the hyperfine ground state of Doppler-broadened muonic hydrogen through dipole-forbidden transition
Optik - International Journal for Light and Electron Optics 200 (2020) 163380
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Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Excitation probability of the hyperfine ground state of Dopplerbroadened muonic hydrogen through dipole-forbidden transition
T
⁎
Rakesh Mohan Dasa, , Katsuhiko Ishidab, Masahiko Iwasakib, Takashi Nakajimaa a b
Institute of Advanced Energy, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan RIKEN (The Institute of Physical and Chemical Research), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
A R T IC LE I N F O
ABS TRA CT
Keywords: Muonic hydrogen Magnetic dipole Lineshape Spin-polarization
We study the magnetic dipole induced excitation probability of the hyperfine ground state of Doppler-broadened muonic hydrogen (pμ−) by a single nanosecond laser pulse with pulse durations of 20, 2, and 0.2 ns pulses with a Gaussian temporal shape such that the pulse bandwidth is, respectively, narrower than, comparable to, and broader than the Doppler width at 300 K. We numerically solve a set of density matrix equations to obtain the excitation probability at the various laser intensities and detunings. For the resonant 20 ns pulse, the excitation probability increases very slowly as the laser intensity increases. The width of the excitation probability lineshape mainly comes from the Doppler broadening and hardly increases with intensity. For the resonant 0.2 ns pulse a complete population inversion occurs at a certain laser intensity, and the excitation probability oscillates between 0 and unity as the intensity further increases. Owing to the inherent pulse bandwidth the width of the excitation probability lineshape is very broad even at the low intensity, and it increases further as the intensity increases. For the resonant 2 ns pulse at a moderate intensity (∼1011 W/cm2) the excitation probability is higher than those by the 20 and 0.2 ns pulses with a relatively narrow width. Our study can serve as a guideline for the development of the relevant laser source, depending on the purpose to excite muonic hydrogen, i.e., precise measurement of the ground state hyperfine splitting of muonic hydrogen or production of spin-polarized muonic hydrogen.
1. Introduction Efficient production and manipulation of exotic atoms has paved the way for various discoveries in the field of atomic and nuclear physics. Among them muon (μ ± ), is of great importance. Being a fundamental building block of the Standard model of elementary particles, muon is valuable in its own right to be studied, and also very useful for various applications such as μSR experiment, muoncatalized fusion, etc. [1–9]. Whether it has a positive or negative charge after ejected from the particle accelerator with very high kinetic energy (a few MeV), a common requirement for the application purpose of muon is to reduce its kinetic energy down to a few keV ∼ a few eV in some way. During this process, positive muons form muonium atoms by capturing electrons, while negative muons form muonic hydrogen atoms by being captured by protons. In both cases the degree of polarization, which is 100% upon birth, is significantly lost through the recombination processes. Subsequently, the muon beam after slowed down needs to be spin-repolarized. For the muonium there are several works to obtain slow positive muon beam [14–17]. We have reported a new scheme to polarize muoniums using a sequence of transform-limited short laser pulses [18], and compared the polarization and photoionization
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Corresponding author. E-mail address: [email protected] (R. Mohan Das).
https://doi.org/10.1016/j.ijleo.2019.163380 Received 7 June 2019; Accepted 7 September 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 200 (2020) 163380
R. Mohan Das, et al.
Fig. 1. Comparsion of the hydrogen and hydrogen-like systems involving positive and negative muons.
efficiency by the transform-limited picosecond and broadband nanosecond laser pulses [19,20]. More recently, we have studied the effects of chirped nanosecond laser pulses to polarize muoniums [21]. Being approximately 200 times heavier than electron, the atomic radius of muonic hydrogen is about 1/200 of hydrogen and muonium. As a result, the negative muon in muonic hydrogen stays much closer to the proton than electron in ordinary hydrogen, and accordingly, the hyperfine splitting of the former is much larger than that of the latter, and the ground state hyperfine splitting (1S(F = 0 − F = 1)) of muonic hydrogen is 44 THz while those of hydrogen and muonium are 1.4 GHz and 5.4 GHz, respectively. In Fig. 1, we summarize the relevant parameters of hydrogen, muonium, and muonic hydrogen. We note that muonic hydrogen is a very nice system to precisely determine the size of proton, since negative muon stays very close to the proton (Fig. 1). Recent study on the Lamb shift (2S1/2 − 2P1/2 energy difference) of muonic hydrogen at PSI [2] reports that the proton radius is about 4% smaller than the widely accepted value. An independent measurement for the hyperfine splitting (1S(F = 0 − F = 1)) of the ground state of muonic hydrogen can also serve as a pathway to accurately determine the proton size [10], since the hyperfine splitting is mathematically expressed as a sum of the pure QED correction and a term that describes the effects of proton structure [11,12]. In the experimental scheme originally proposed by Bakalov et al. [13], negative muons are slowed down and stopped in a pure hydrogen target between two parallel gold or aluminium plates to form muonic hydrogen. However, the measurement of the hyperfine splitting of the ground state of muonic hydrogen is not as easy as it sounds: Although the 1S(F = 0) − 1S(F = 1) transition of muonic hydrogen is large as mentioned above, and the transition wavelength is in the mid-infrared range (6.76 μm), this transition is dipole-forbidden, and we ought to employ the magnetic dipole transition which is several orders of magnitude weaker than the electric dipole transition. Indeed, by assuming a long laser pulse at 6.76 μm with 0.25 mJ pulse energy to induce the 1s(F = 0) − 1s (F = 1) transition of Doppler-broadened muonic hydrogen via magnetic dipole interaction, Adamczak et al. have found that the excitation probability is only 1.2 × 10−5, which is too low for the proposed experiment to be feasible [10]. They have also proposed a complicated scheme by introducing a multipass cavity to increase the excitation probability up to 12%. Undoubtedly, this is a very challenging task. In the present work, we study the excitation probability of Doppler-broadened muonic hydrogen by a single nanosecond laser pulse with three different pulse durations so that the pulse bandwidth is narrower than (similar to the situation in Ref. [10]), comparable to, and broader than the Doppler width at 300 K. The results we present in this work would be useful to develop the laser source for the specific purpose with muonic hydrogen. 2. System description We consider the hyperfine sublevels of the ground (1s) state of the muonic hydrogen. The temporal shape of the pulse we assume is Gaussian pulse in the mid-infrared range (6.7 μm), and it excites the muonic hydrogen from the hyperfine singlet state, 1s(F = 0), to the triplet state, 1s(F = 1), through magnetic dipole (M1) interaction. The magnetic field of the pulse with central frequency ω is written as
(t ) = B (t )exp[−iωt ] + c. c.,
(1)
where B(t) is the magnetic field amplitude of the pulse. The magnetic transition dipole moment of the muonic hydrogen atom 2
Optik - International Journal for Light and Electron Optics 200 (2020) 163380
R. Mohan Das, et al.
evaluated by applying the Wigner–Eckart theorem to the corresponding matrix element for the spherical component q(= 0) of the magnetic dipole operator M along the direction of the magnetic field, then uncoupling the nuclear spin angular momentum I, from the electronic angular momentum, is
M = 〈1s1/2 (F = 0, mF = 0)|Mq(1) = 0|1s1/2 (F ′ = 1, mF′ = 0) 〉 F 1 F′ ⎞ (−1) J + I + F ′+ 1 = (−1) F − mF ⎛ ⎝−mF 0 mF ′⎠ J I F × (2F + 1)(2F ′ + 1) F′ 1 J′ e (Lq = 0 + 2Sq = 0) × 〈n (ls) J∥ ∥n′ (l′s′) J ′〉 2mμ ⎜
⎟
{
}
(2)
where the term in the parenthesis is a Wigner 3−j symbol, and the term in braces is a 6−j symbol, e is the electronic charge and mμ is the reduced mass of the muonic hydrogen atom. We substitute the values of the quantum numbers, n = n′ = 1, l = l′ = 0, s = s′ = 1/ 2, J = J′ = 1/2, I = 1/2 and note that the orbital angular momentum L = 0 in Eq. (2), which gives us
M=
1 eℏ . 6 2mμ
(3)
Hence, the Rabi frequency Ω, corresponding to the magnetic dipole transition becomes
Ω = 〈 (F = 1)|M B(t )|1s (F = 0) 〉 1 eℏ = B (t ) 6 2mμ =
1 eℏ E (t ) , 6 2mμ c
(4)
where E(t) is the electric field amplitude of the pulse and c is the speed of light. Converting the above equation into S.I. unit by incorporating the appropriate conversion factors, we obtain
Ω(ns−1) = 1.77 × 10−6 I (t )
(5) 2
where I(t) is the laser intensity in units of W/cm . The semi-classical set of density matrix equations reads
σ˙ 00 = −iΩ(σ10 − σ01),
(6)
σ˙ 11 = iΩ(σ10 − σ01),
(7)
σ˙ 10 = iΔσ10 + iΩ(σ11 − σ00),
(8)
where the diagonal elements, σ00 and σ11, represent the population of state 1s(F = 0) and state 1s(F = 0), respectively, while the offdiagonal element, σ10, represents coherence between the two states. Δ is the detuning of the pulse between the laser frequency and transition frequency. We represent the peak intensity of the laser pulse by I0 and the pulse duration by τ. The Doppler bandwidth of the muonic hydrogen atoms at 300 K is 0.524 GHz. With the initial condition that all the atoms are in the singlet state, 1s(F = 0), we numerically calculate the Doppler-averaged excitation probability, P, to the ground hyperfine state, 1s(F = 1). 3. Results and discussions To start with, we study the variation of P as a function of peak intensity, I0, for the pulses with the durations of 20, 2, and 0.2 ns. The laser detunings, Δ, are 0, 0.25, and 0.5 GHz. The results are summarized in Fig. 2. For a moment, we focus on the case of Δ = 0 only, and investigate how the duration of the employed pulse influences the excitation probability, P, by recalling that the bandwidths of 20, 2, and 0.2 ns pulses are, respectively, narrower than, comparable to, and broader than the Doppler width at 300 K. The discussions for the cases of Δ ≠ 0 will be postponed until Section 3.4. For a fair comparison, we should look at the results by the 20, 2, and 0.2 ns pulses under the same pulse energies and beam diameter. This means that the peak intensities, I0's, of the 20, 2, and 0.2 ns pulses at which P's are compared should be in the ratio 1 : 10 : 102. Accordingly, the horizontal scale of Figs. 2(a)–(c) are chosen so that P's by the different pulse durations can be compared vertically. At first glance, we observe in Figs. 2(a)–(c) that P reaches its first maximum at around I0 = 109, 1011, and 1013 W/cm2 for the 20, 2, and 0.2 ns pulses, respectively. For all three pulse durations the value of P modulates as the intensity increases, and obviously this is comes from the increase of the pulse area. However, the way it modulates is different. For the 20 ns pulse, the value of P rapidly modulates without an increase of the maximum value, and the first maximum is only 0.04 at I0 = 109 W/cm2, For the 2 ns pulse, the value of P modulates with a gradual increase of the maximum value, and the first maximum is 0.37 at I0 = 1011 W/cm2, For the 0.2 ns pulse, the maximum value of P reaches almost unity at the first maximum, and it is 0.97 at I0 = 1013 W/cm2. The modulation is slowest in this case. To understand the results shown in Fig. 2, we consider the dynamics in the spectral domain. In Fig. 3 we illustrate the spectral intensity profiles of the employed resonant laser pulses and the Doppler-broadened transition line. The 20 ns pulse has a bandwidth of 3
Optik - International Journal for Light and Electron Optics 200 (2020) 163380
R. Mohan Das, et al.
Fig. 2. Doppler-averaged excitation probability, P, of muonic hydrogen as a function of peak laser intensity, I0, by the laser pulse with the durations of (a) 20, (b) 2, and (c) 0.2 ns. In each panel, black, red and green lines correspond to the detuning, Δ, of 0, 0.25, and 0.5 GHz, respectively. When the pulse energy and beam diameter are chosen to be the same for the pulses with different durations the respective peak intensities are different following the scaling of 1 : 10 : 102, and accordingly the horizontal scale of panel (a)–(c) are appropriately chosen.
Fig. 3. Illustration of the spectral intensity profiles of the employed resonant laser pulses (black solid line filled with yellow) and the Dopplerbroadened transition line of muonic hydrogen at 300 K (black dashed line). Panels (a), (c), and (e) ((b), (d), and (f)) correspond to the 20, 2, and 0.2 ns pulses, respectively, in the low (high) intensity regime. The peak heights of the spectral intensity profiles in panels (a), (c), and (e) are in the ratio of 102 : 10 : 1 (not to scale) as mentioned in the text, and the same is true for panels (b), (d), and (f). 4
Optik - International Journal for Light and Electron Optics 200 (2020) 163380
R. Mohan Das, et al.
0.022 GHz, which is quite narrower than the Doppler width (0.524 GHz) at 300 K. Qualitatively, this scenario resembles the case of monochromatic laser source in Ref. [10]. The 2 ns pulse has a bandwidth of 0.22 GHz, which is comparable to the Doppler width, and the 0.2 ns pulse has a bandwidth of 2.2 GHz, which is larger than the Doppler width. As mentioned above, if we assume, for the fair comparison, that the pulse energy and the beam diameter are the same for the three pulse durations, then the areas of the spectral intensity profiles for the three pulse durations are also the same. This fact can be easily confirmed by carrying out the Fourier transform of the pulses. Accordingly, in the spectral domain, the heights of the spectral intensity profiles for the 20, 2, and 0.2 ns pulses should be in the ratio 102 : 10 : 1. Note that in Fig. 3(a) the yellow-colored area under the spectral intensity profile is almost confined to the center, i.e., around the atoms at rest, while that in Fig. 3(c) is more spread over the moving atoms. In Fig. 3(e), the yellow-colored area is even more spread so that a significant part of the area lies beyond the Doppler distribution. The detailed discussions for Fig. 2 by referring to Fig. 3 will be given in Sections 3.1–3.3 where the laser pulse is still assumed to be on resonance, i.e., Δ = 0, for all pulse durations. 3.1. Low intensity regime In the low intensity regime the narrower the pulse bandwidth or equivalently the higher the spectral intensity is at resonance the more the atoms are excited. This can be easily understood by referring to Figs. 3(a), (c), and (e) by bearing in mind that the central frequency components of the respective pulses are most efficiently used to excite the atoms at the low intensity regime. Namely, the 20 ns pulse is most efficient and 0.2 ns pulse is least efficient. Indeed the numerical results of P for 20, 2 and 0.2 ns pulses at 107, 108, and 109 W/cm2, are 0.032, 0.0032, and 0.00032, respectively, which are practically in in the ratio of 102 : 10 : 1, and this ratio is practically equal to the ratio of the height of the spectral intensity profiles for the rest atoms. 3.2. Moderate intensity regime In the moderate intensity regime the spectral intensity of the 20 ns pulse at the central region is too much for the central area of atoms in the Doppler distribution, and some atoms in this region goes back to the lower state after passing through the excited state. This explains why the height of modulation in P for the 20 ns pulse is so low (Fig. 3(a)). For the 2 ns pulse the height of the spectral intensity at the central region is not yet too much (i.e., return of the excited atoms back to the lower state does not occur even for the rest atoms), and all the spectral components are effectively used to excite atoms all over the Doppler distribution. For the 0.2 ns pulse the spreading of the spectral intensity is too broad, and a lot of spectral components in the wing region of this pulse at this intensity region are not used to excite atoms, as a result of which the excitation probability by the 0.2 ns pulse is smaller than that by the 2 ns pulse. For example, at I0 = 1010 W/cm2 by the 20 ns pulse, P is no more than 0.05, while at I0 = 1011 W/cm2 by the 2 ns pulse, P = 0.37, and at I0 = 1012 W/cm2 by the 0.2 ns pulse, P = 0.25, and these results are well explained by the above interpretation. 3.3. High intensity regime In the high intensity regime the spectral intensities at the central region and also at the region a little more away from the center are too much for the 20 and 2 ns pulses (Figs. 3(b) and (d)), and this means that some of the atoms in these regions goes back to the lower state through the excited state. At the same time more and more spectral components toward the wings of the bandwidth become large to excite more atoms in the wings of Doppler distribution. As a result of those two factors to the opposite directions in terms of excitation probability the increase of the maximum value of P by the 20 ns pulse is so slow and no more than 0.1, while that by the 2 ns pulse increases more than 0.6 as the peak intensity further increases. In contrast, the spectral intensity profile of the 0.2 ns pulse is so flat that all the atoms are considered to be on resonance with the laser pulse (Fig. 3(f)), and accordingly, a deep modulation is observed between 0 and 1 as the peak intensity increases. For instance, at I0 = 1011 W/cm2 by the 0.2 ns pulse P is about 0.05, while at I0 = 1012 W/cm2 by the 2 ns pulse P is about 0.2, and at I0 = 1013 W/cm2 by the 0.2 ns pulse P exceeds 0.9. 3.4. Laser detuning We now study how the excitation probability varies with laser detuning. In Figs. 4(a)–(c), we plot P as a function of laser detuning, Δ, for the pulse durations of 20, 2, and 0.2 ns. All the chosen values of I0's in Figs. 4(a)–(c) correspond to the peak intensities at which the maxima of P appear in Figs. 2(a)–(c), respectively, and the same color in Figs. 4(a)–(c) means that the employed pulse energies (as well as the beam diameter) are nearly the same. At first glance we notice that, depending on the pulse duration the evolution of the lineshape, when the peak intensity is varied from the low to the high intensity regimes, is very different. For the case of 20 ns pulse the width of the lineshape in the low intensity regime is practically equal to that of the Doppler width (Fig. 4(a)). As the peak intensity increases, the height of the excitation spectrum slightly increases, but the increase of the width of the lineshape is negligibly small. For the case of 0.2 ns pulse (Fig. 4(c)) the width of the lineshape in the low intensity regime is practically equal to the bandwidth of the pulse. As the peak intensity increases the width of the lineshape drastically increases. For the case of 2 ns pulse (Fig. 4(b)) the width of the lineshape in the low intensity regime is equal to the Doppler width, which is just like the case of 20 ns pulse. But unlike the case of 20 ns pulse, the width as well as the height of the lineshape gradually increases as the peak intensity increases. To explain the above observations, we illustrate in Fig. 5 the spectral intensity profiles of the laser pulses with different pulse durations for the cases of Δ = 0, 0.25, and 0.5 GHz. In each of Figs. 5(a)–(f), the spectral intensity profile in the low (high) intensity 5
Optik - International Journal for Light and Electron Optics 200 (2020) 163380
R. Mohan Das, et al.
Fig. 4. Excitation probability of Doppler-broadened muonic hydrogen as a function of laser detuning, Δ, under the different peak intensities, I0. The employed pulse durations are (a) 20, (b) 2, and (c) 0.2 ns. Note that the lines in panels (a)–(c) with the same color correspond to same pulse energies and beam diameter.
regime is represented by the thin (thick) black solid line with the area filled with light (dark) yellow, and the Doppler distribution of the atoms is represented by the black dashed line. First, we discuss the case of 20 ns pulse. At Δ = 0 (Fig. 5(a)), the peak of the spectral intensity profile coincides with that of Doppler distribution, and the excitation process is most efficient. At Δ = 0.5 GHz (Fig. 5(b)), the number of available atoms within the Doppler distribution is much fewer than that at Δ = 0, and by recalling that the excitation probability is linearly proportional to the number of available atoms in the low intensity regime it is clear that the width of P by the 20 ns pulse is practically the same with that of Doppler distribution. An increase of I0 does not help so much to increase P, because in the high intensity regime the fall-off of the spectral intensity toward the wings is very fast, and accordingly, even at the value of I0 for which the maximum of P appears (Fig. 2(a)) the entire spectral intensity profile of the 20 ns pulse is much less likely to make a positive contribution to excite atoms. This results in the very limited increase of the width of P as I0 increases, as we see in Fig. 4(a). Figs. 5(a) and (b) also explain the results of Fig. 2(a) for Δ ≠ 0 (red and green lines in Fig. 2(a)). In the case of 0.2 ns pulse, the spectral intensity profile is much broader than the Doppler width, and even at Δ = 0.5 GHz the pulse bandwidth covers the entire Doppler distributions. Therefore, in the low intensity regime, the lineshape of P is practically equal to the pulse bandwidth as mentioned before. As the intensity increases, the peak value of P reaches unity, and consequently, the lineshape of P becomes flatter or equivalently the width of P increases when I0 further increases (Fig. 4(c)). From the comparison of Figs. 5(e) and (f) it is also obvious that the excitation probability for Δ = 0.5 GHz is smaller than that for Δ = 0, as we have seen in Fig. 2(c). Finally, in the case of 2 ns pulse, we recall that its bandwidth, 0.22 GHz, is about half of the Doppler width at 300 K, i.e., 0.524 GHz. Accordingly, in the low intensity regime, the width of P is nearly the same with that of Doppler width, and this situation is just similar to the case of the 20 ns pulse, as we can imagine by referring to Figs. 5(c) and (d). As the intensity increases, what happens is just between the cases described above for the 20 and 0.2 ns pulses: The width of P increases, but not too much, because the value of P does not reach unity, as we see in Fig. 4(b). The illustrations shown in Fig. 5(c) and (d) explain the qualitative behavior of the excitation process we have explained above. Before closing the discussions we emphasize that the 2 ns pulse is most efficient in the moderate intensity range, and the good thing is that the width of P does not so much increase as the intensity increases. Accordingly, the 2 ns pulse is most suitable for the measurement of hyperfine splitting of the ground state of muonic hydrogen. In contrast, the 0.2 ns pulse is most useful to polarize the muonic hydrogen, since the excitation probability reaches nearly unity and hence the degree of spin-polarization can be nearly 100% only if sufficient intensity is available. 6
Optik - International Journal for Light and Electron Optics 200 (2020) 163380
R. Mohan Das, et al.
Fig. 5. Illustration of the spectral intensity profiles of the employed laser pulses and the Doppler-broadened transition line of muonic hydrogen at 300 K (black dashed line). Panels (a), (c), and (e) ((b), (d), and (f)) correspond to the pulse durations of 20, 2, and 0.2 ns, respectively, under resonance (detuning of Δ = 0.5 GHz), and in each panel the thin (thick) black solid line filled with light (dark) yellow represents the case of low (high) intensity regime. The peak heights of the spectral intensity profiles in panels (a), (c), and (e) for the respective intensity regimes are in the ratio of 102 : 10 : 1 (not to scale) as mentioned in the text.
4. Conclusion In conclusion, we have studied the excitation probability between the hyperfine ground states of Doppler-broadened muonic hydrogen by a single nanosecond laser pulse with three different pulse durations, 20, 2, and 0.2 ns. This transition is dipole-forbidden, and it is the magnetic dipole interaction that induces the transition. The Doppler width of muonic hydrogen at 300 K is 0.524 GHz, and accordingly the bandwidths of the 20, 2, and 0.2 ns pulses are, respectively, narrower than, comparable to, and broader than the Doppler width. These relations result in the different excitation dynamics as the laser intensity and detuning are varied. At the moderate laser intensity, say, 1010, 1011, and 1012 W/cm2, respectively, by the 20, 2, and 0.2 ns pulses for a fair comparison (i.e., same pulse energy and beam diameter), we have found that the use of the resonant 2 ns pulse results in the highest excitation probability with a relatively narrow linewidth, and hence the 2 ns pulse is most suitable for the precise measurement of hyperfine splitting of the ground state of muonic hydrogen. In contrast, if the purpose is to spin-polarize the muonic hydrogen, the use of a 0.2 ns pulse is most efficient, as, for instance, the excitation probabilities at the intensities of 1011, 1012, and 1013 W/cm2, respectively, by the 20, 2, and 0.2 ns pulses are ∼0.05, 0.3, and 0.9. The results we have presented in this work would serve as a guideline for the development of the mid-infrared laser source to study the muonic hydrogen. Acknowledgement Part of this work was supported by the Grant-in-Aid for scientific research from the Ministry of Education, Culture, Sports, Science and Technology (Japan). References [1] [2] [3] [4] [5]
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Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Excitation probability of the hyperfine ground state of Dopplerbroadened muonic hydrogen through dipole-forbidden transition
T
⁎
Rakesh Mohan Dasa, , Katsuhiko Ishidab, Masahiko Iwasakib, Takashi Nakajimaa a b
Institute of Advanced Energy, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan RIKEN (The Institute of Physical and Chemical Research), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
A R T IC LE I N F O
ABS TRA CT
Keywords: Muonic hydrogen Magnetic dipole Lineshape Spin-polarization
We study the magnetic dipole induced excitation probability of the hyperfine ground state of Doppler-broadened muonic hydrogen (pμ−) by a single nanosecond laser pulse with pulse durations of 20, 2, and 0.2 ns pulses with a Gaussian temporal shape such that the pulse bandwidth is, respectively, narrower than, comparable to, and broader than the Doppler width at 300 K. We numerically solve a set of density matrix equations to obtain the excitation probability at the various laser intensities and detunings. For the resonant 20 ns pulse, the excitation probability increases very slowly as the laser intensity increases. The width of the excitation probability lineshape mainly comes from the Doppler broadening and hardly increases with intensity. For the resonant 0.2 ns pulse a complete population inversion occurs at a certain laser intensity, and the excitation probability oscillates between 0 and unity as the intensity further increases. Owing to the inherent pulse bandwidth the width of the excitation probability lineshape is very broad even at the low intensity, and it increases further as the intensity increases. For the resonant 2 ns pulse at a moderate intensity (∼1011 W/cm2) the excitation probability is higher than those by the 20 and 0.2 ns pulses with a relatively narrow width. Our study can serve as a guideline for the development of the relevant laser source, depending on the purpose to excite muonic hydrogen, i.e., precise measurement of the ground state hyperfine splitting of muonic hydrogen or production of spin-polarized muonic hydrogen.
1. Introduction Efficient production and manipulation of exotic atoms has paved the way for various discoveries in the field of atomic and nuclear physics. Among them muon (μ ± ), is of great importance. Being a fundamental building block of the Standard model of elementary particles, muon is valuable in its own right to be studied, and also very useful for various applications such as μSR experiment, muoncatalized fusion, etc. [1–9]. Whether it has a positive or negative charge after ejected from the particle accelerator with very high kinetic energy (a few MeV), a common requirement for the application purpose of muon is to reduce its kinetic energy down to a few keV ∼ a few eV in some way. During this process, positive muons form muonium atoms by capturing electrons, while negative muons form muonic hydrogen atoms by being captured by protons. In both cases the degree of polarization, which is 100% upon birth, is significantly lost through the recombination processes. Subsequently, the muon beam after slowed down needs to be spin-repolarized. For the muonium there are several works to obtain slow positive muon beam [14–17]. We have reported a new scheme to polarize muoniums using a sequence of transform-limited short laser pulses [18], and compared the polarization and photoionization
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Corresponding author. E-mail address: [email protected] (R. Mohan Das).
https://doi.org/10.1016/j.ijleo.2019.163380 Received 7 June 2019; Accepted 7 September 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 200 (2020) 163380
R. Mohan Das, et al.
Fig. 1. Comparsion of the hydrogen and hydrogen-like systems involving positive and negative muons.
efficiency by the transform-limited picosecond and broadband nanosecond laser pulses [19,20]. More recently, we have studied the effects of chirped nanosecond laser pulses to polarize muoniums [21]. Being approximately 200 times heavier than electron, the atomic radius of muonic hydrogen is about 1/200 of hydrogen and muonium. As a result, the negative muon in muonic hydrogen stays much closer to the proton than electron in ordinary hydrogen, and accordingly, the hyperfine splitting of the former is much larger than that of the latter, and the ground state hyperfine splitting (1S(F = 0 − F = 1)) of muonic hydrogen is 44 THz while those of hydrogen and muonium are 1.4 GHz and 5.4 GHz, respectively. In Fig. 1, we summarize the relevant parameters of hydrogen, muonium, and muonic hydrogen. We note that muonic hydrogen is a very nice system to precisely determine the size of proton, since negative muon stays very close to the proton (Fig. 1). Recent study on the Lamb shift (2S1/2 − 2P1/2 energy difference) of muonic hydrogen at PSI [2] reports that the proton radius is about 4% smaller than the widely accepted value. An independent measurement for the hyperfine splitting (1S(F = 0 − F = 1)) of the ground state of muonic hydrogen can also serve as a pathway to accurately determine the proton size [10], since the hyperfine splitting is mathematically expressed as a sum of the pure QED correction and a term that describes the effects of proton structure [11,12]. In the experimental scheme originally proposed by Bakalov et al. [13], negative muons are slowed down and stopped in a pure hydrogen target between two parallel gold or aluminium plates to form muonic hydrogen. However, the measurement of the hyperfine splitting of the ground state of muonic hydrogen is not as easy as it sounds: Although the 1S(F = 0) − 1S(F = 1) transition of muonic hydrogen is large as mentioned above, and the transition wavelength is in the mid-infrared range (6.76 μm), this transition is dipole-forbidden, and we ought to employ the magnetic dipole transition which is several orders of magnitude weaker than the electric dipole transition. Indeed, by assuming a long laser pulse at 6.76 μm with 0.25 mJ pulse energy to induce the 1s(F = 0) − 1s (F = 1) transition of Doppler-broadened muonic hydrogen via magnetic dipole interaction, Adamczak et al. have found that the excitation probability is only 1.2 × 10−5, which is too low for the proposed experiment to be feasible [10]. They have also proposed a complicated scheme by introducing a multipass cavity to increase the excitation probability up to 12%. Undoubtedly, this is a very challenging task. In the present work, we study the excitation probability of Doppler-broadened muonic hydrogen by a single nanosecond laser pulse with three different pulse durations so that the pulse bandwidth is narrower than (similar to the situation in Ref. [10]), comparable to, and broader than the Doppler width at 300 K. The results we present in this work would be useful to develop the laser source for the specific purpose with muonic hydrogen. 2. System description We consider the hyperfine sublevels of the ground (1s) state of the muonic hydrogen. The temporal shape of the pulse we assume is Gaussian pulse in the mid-infrared range (6.7 μm), and it excites the muonic hydrogen from the hyperfine singlet state, 1s(F = 0), to the triplet state, 1s(F = 1), through magnetic dipole (M1) interaction. The magnetic field of the pulse with central frequency ω is written as
(t ) = B (t )exp[−iωt ] + c. c.,
(1)
where B(t) is the magnetic field amplitude of the pulse. The magnetic transition dipole moment of the muonic hydrogen atom 2
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R. Mohan Das, et al.
evaluated by applying the Wigner–Eckart theorem to the corresponding matrix element for the spherical component q(= 0) of the magnetic dipole operator M along the direction of the magnetic field, then uncoupling the nuclear spin angular momentum I, from the electronic angular momentum, is
M = 〈1s1/2 (F = 0, mF = 0)|Mq(1) = 0|1s1/2 (F ′ = 1, mF′ = 0) 〉 F 1 F′ ⎞ (−1) J + I + F ′+ 1 = (−1) F − mF ⎛ ⎝−mF 0 mF ′⎠ J I F × (2F + 1)(2F ′ + 1) F′ 1 J′ e (Lq = 0 + 2Sq = 0) × 〈n (ls) J∥ ∥n′ (l′s′) J ′〉 2mμ ⎜
⎟
{
}
(2)
where the term in the parenthesis is a Wigner 3−j symbol, and the term in braces is a 6−j symbol, e is the electronic charge and mμ is the reduced mass of the muonic hydrogen atom. We substitute the values of the quantum numbers, n = n′ = 1, l = l′ = 0, s = s′ = 1/ 2, J = J′ = 1/2, I = 1/2 and note that the orbital angular momentum L = 0 in Eq. (2), which gives us
M=
1 eℏ . 6 2mμ
(3)
Hence, the Rabi frequency Ω, corresponding to the magnetic dipole transition becomes
Ω = 〈 (F = 1)|M B(t )|1s (F = 0) 〉 1 eℏ = B (t ) 6 2mμ =
1 eℏ E (t ) , 6 2mμ c
(4)
where E(t) is the electric field amplitude of the pulse and c is the speed of light. Converting the above equation into S.I. unit by incorporating the appropriate conversion factors, we obtain
Ω(ns−1) = 1.77 × 10−6 I (t )
(5) 2
where I(t) is the laser intensity in units of W/cm . The semi-classical set of density matrix equations reads
σ˙ 00 = −iΩ(σ10 − σ01),
(6)
σ˙ 11 = iΩ(σ10 − σ01),
(7)
σ˙ 10 = iΔσ10 + iΩ(σ11 − σ00),
(8)
where the diagonal elements, σ00 and σ11, represent the population of state 1s(F = 0) and state 1s(F = 0), respectively, while the offdiagonal element, σ10, represents coherence between the two states. Δ is the detuning of the pulse between the laser frequency and transition frequency. We represent the peak intensity of the laser pulse by I0 and the pulse duration by τ. The Doppler bandwidth of the muonic hydrogen atoms at 300 K is 0.524 GHz. With the initial condition that all the atoms are in the singlet state, 1s(F = 0), we numerically calculate the Doppler-averaged excitation probability, P, to the ground hyperfine state, 1s(F = 1). 3. Results and discussions To start with, we study the variation of P as a function of peak intensity, I0, for the pulses with the durations of 20, 2, and 0.2 ns. The laser detunings, Δ, are 0, 0.25, and 0.5 GHz. The results are summarized in Fig. 2. For a moment, we focus on the case of Δ = 0 only, and investigate how the duration of the employed pulse influences the excitation probability, P, by recalling that the bandwidths of 20, 2, and 0.2 ns pulses are, respectively, narrower than, comparable to, and broader than the Doppler width at 300 K. The discussions for the cases of Δ ≠ 0 will be postponed until Section 3.4. For a fair comparison, we should look at the results by the 20, 2, and 0.2 ns pulses under the same pulse energies and beam diameter. This means that the peak intensities, I0's, of the 20, 2, and 0.2 ns pulses at which P's are compared should be in the ratio 1 : 10 : 102. Accordingly, the horizontal scale of Figs. 2(a)–(c) are chosen so that P's by the different pulse durations can be compared vertically. At first glance, we observe in Figs. 2(a)–(c) that P reaches its first maximum at around I0 = 109, 1011, and 1013 W/cm2 for the 20, 2, and 0.2 ns pulses, respectively. For all three pulse durations the value of P modulates as the intensity increases, and obviously this is comes from the increase of the pulse area. However, the way it modulates is different. For the 20 ns pulse, the value of P rapidly modulates without an increase of the maximum value, and the first maximum is only 0.04 at I0 = 109 W/cm2, For the 2 ns pulse, the value of P modulates with a gradual increase of the maximum value, and the first maximum is 0.37 at I0 = 1011 W/cm2, For the 0.2 ns pulse, the maximum value of P reaches almost unity at the first maximum, and it is 0.97 at I0 = 1013 W/cm2. The modulation is slowest in this case. To understand the results shown in Fig. 2, we consider the dynamics in the spectral domain. In Fig. 3 we illustrate the spectral intensity profiles of the employed resonant laser pulses and the Doppler-broadened transition line. The 20 ns pulse has a bandwidth of 3
Optik - International Journal for Light and Electron Optics 200 (2020) 163380
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Fig. 2. Doppler-averaged excitation probability, P, of muonic hydrogen as a function of peak laser intensity, I0, by the laser pulse with the durations of (a) 20, (b) 2, and (c) 0.2 ns. In each panel, black, red and green lines correspond to the detuning, Δ, of 0, 0.25, and 0.5 GHz, respectively. When the pulse energy and beam diameter are chosen to be the same for the pulses with different durations the respective peak intensities are different following the scaling of 1 : 10 : 102, and accordingly the horizontal scale of panel (a)–(c) are appropriately chosen.
Fig. 3. Illustration of the spectral intensity profiles of the employed resonant laser pulses (black solid line filled with yellow) and the Dopplerbroadened transition line of muonic hydrogen at 300 K (black dashed line). Panels (a), (c), and (e) ((b), (d), and (f)) correspond to the 20, 2, and 0.2 ns pulses, respectively, in the low (high) intensity regime. The peak heights of the spectral intensity profiles in panels (a), (c), and (e) are in the ratio of 102 : 10 : 1 (not to scale) as mentioned in the text, and the same is true for panels (b), (d), and (f). 4
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0.022 GHz, which is quite narrower than the Doppler width (0.524 GHz) at 300 K. Qualitatively, this scenario resembles the case of monochromatic laser source in Ref. [10]. The 2 ns pulse has a bandwidth of 0.22 GHz, which is comparable to the Doppler width, and the 0.2 ns pulse has a bandwidth of 2.2 GHz, which is larger than the Doppler width. As mentioned above, if we assume, for the fair comparison, that the pulse energy and the beam diameter are the same for the three pulse durations, then the areas of the spectral intensity profiles for the three pulse durations are also the same. This fact can be easily confirmed by carrying out the Fourier transform of the pulses. Accordingly, in the spectral domain, the heights of the spectral intensity profiles for the 20, 2, and 0.2 ns pulses should be in the ratio 102 : 10 : 1. Note that in Fig. 3(a) the yellow-colored area under the spectral intensity profile is almost confined to the center, i.e., around the atoms at rest, while that in Fig. 3(c) is more spread over the moving atoms. In Fig. 3(e), the yellow-colored area is even more spread so that a significant part of the area lies beyond the Doppler distribution. The detailed discussions for Fig. 2 by referring to Fig. 3 will be given in Sections 3.1–3.3 where the laser pulse is still assumed to be on resonance, i.e., Δ = 0, for all pulse durations. 3.1. Low intensity regime In the low intensity regime the narrower the pulse bandwidth or equivalently the higher the spectral intensity is at resonance the more the atoms are excited. This can be easily understood by referring to Figs. 3(a), (c), and (e) by bearing in mind that the central frequency components of the respective pulses are most efficiently used to excite the atoms at the low intensity regime. Namely, the 20 ns pulse is most efficient and 0.2 ns pulse is least efficient. Indeed the numerical results of P for 20, 2 and 0.2 ns pulses at 107, 108, and 109 W/cm2, are 0.032, 0.0032, and 0.00032, respectively, which are practically in in the ratio of 102 : 10 : 1, and this ratio is practically equal to the ratio of the height of the spectral intensity profiles for the rest atoms. 3.2. Moderate intensity regime In the moderate intensity regime the spectral intensity of the 20 ns pulse at the central region is too much for the central area of atoms in the Doppler distribution, and some atoms in this region goes back to the lower state after passing through the excited state. This explains why the height of modulation in P for the 20 ns pulse is so low (Fig. 3(a)). For the 2 ns pulse the height of the spectral intensity at the central region is not yet too much (i.e., return of the excited atoms back to the lower state does not occur even for the rest atoms), and all the spectral components are effectively used to excite atoms all over the Doppler distribution. For the 0.2 ns pulse the spreading of the spectral intensity is too broad, and a lot of spectral components in the wing region of this pulse at this intensity region are not used to excite atoms, as a result of which the excitation probability by the 0.2 ns pulse is smaller than that by the 2 ns pulse. For example, at I0 = 1010 W/cm2 by the 20 ns pulse, P is no more than 0.05, while at I0 = 1011 W/cm2 by the 2 ns pulse, P = 0.37, and at I0 = 1012 W/cm2 by the 0.2 ns pulse, P = 0.25, and these results are well explained by the above interpretation. 3.3. High intensity regime In the high intensity regime the spectral intensities at the central region and also at the region a little more away from the center are too much for the 20 and 2 ns pulses (Figs. 3(b) and (d)), and this means that some of the atoms in these regions goes back to the lower state through the excited state. At the same time more and more spectral components toward the wings of the bandwidth become large to excite more atoms in the wings of Doppler distribution. As a result of those two factors to the opposite directions in terms of excitation probability the increase of the maximum value of P by the 20 ns pulse is so slow and no more than 0.1, while that by the 2 ns pulse increases more than 0.6 as the peak intensity further increases. In contrast, the spectral intensity profile of the 0.2 ns pulse is so flat that all the atoms are considered to be on resonance with the laser pulse (Fig. 3(f)), and accordingly, a deep modulation is observed between 0 and 1 as the peak intensity increases. For instance, at I0 = 1011 W/cm2 by the 0.2 ns pulse P is about 0.05, while at I0 = 1012 W/cm2 by the 2 ns pulse P is about 0.2, and at I0 = 1013 W/cm2 by the 0.2 ns pulse P exceeds 0.9. 3.4. Laser detuning We now study how the excitation probability varies with laser detuning. In Figs. 4(a)–(c), we plot P as a function of laser detuning, Δ, for the pulse durations of 20, 2, and 0.2 ns. All the chosen values of I0's in Figs. 4(a)–(c) correspond to the peak intensities at which the maxima of P appear in Figs. 2(a)–(c), respectively, and the same color in Figs. 4(a)–(c) means that the employed pulse energies (as well as the beam diameter) are nearly the same. At first glance we notice that, depending on the pulse duration the evolution of the lineshape, when the peak intensity is varied from the low to the high intensity regimes, is very different. For the case of 20 ns pulse the width of the lineshape in the low intensity regime is practically equal to that of the Doppler width (Fig. 4(a)). As the peak intensity increases, the height of the excitation spectrum slightly increases, but the increase of the width of the lineshape is negligibly small. For the case of 0.2 ns pulse (Fig. 4(c)) the width of the lineshape in the low intensity regime is practically equal to the bandwidth of the pulse. As the peak intensity increases the width of the lineshape drastically increases. For the case of 2 ns pulse (Fig. 4(b)) the width of the lineshape in the low intensity regime is equal to the Doppler width, which is just like the case of 20 ns pulse. But unlike the case of 20 ns pulse, the width as well as the height of the lineshape gradually increases as the peak intensity increases. To explain the above observations, we illustrate in Fig. 5 the spectral intensity profiles of the laser pulses with different pulse durations for the cases of Δ = 0, 0.25, and 0.5 GHz. In each of Figs. 5(a)–(f), the spectral intensity profile in the low (high) intensity 5
Optik - International Journal for Light and Electron Optics 200 (2020) 163380
R. Mohan Das, et al.
Fig. 4. Excitation probability of Doppler-broadened muonic hydrogen as a function of laser detuning, Δ, under the different peak intensities, I0. The employed pulse durations are (a) 20, (b) 2, and (c) 0.2 ns. Note that the lines in panels (a)–(c) with the same color correspond to same pulse energies and beam diameter.
regime is represented by the thin (thick) black solid line with the area filled with light (dark) yellow, and the Doppler distribution of the atoms is represented by the black dashed line. First, we discuss the case of 20 ns pulse. At Δ = 0 (Fig. 5(a)), the peak of the spectral intensity profile coincides with that of Doppler distribution, and the excitation process is most efficient. At Δ = 0.5 GHz (Fig. 5(b)), the number of available atoms within the Doppler distribution is much fewer than that at Δ = 0, and by recalling that the excitation probability is linearly proportional to the number of available atoms in the low intensity regime it is clear that the width of P by the 20 ns pulse is practically the same with that of Doppler distribution. An increase of I0 does not help so much to increase P, because in the high intensity regime the fall-off of the spectral intensity toward the wings is very fast, and accordingly, even at the value of I0 for which the maximum of P appears (Fig. 2(a)) the entire spectral intensity profile of the 20 ns pulse is much less likely to make a positive contribution to excite atoms. This results in the very limited increase of the width of P as I0 increases, as we see in Fig. 4(a). Figs. 5(a) and (b) also explain the results of Fig. 2(a) for Δ ≠ 0 (red and green lines in Fig. 2(a)). In the case of 0.2 ns pulse, the spectral intensity profile is much broader than the Doppler width, and even at Δ = 0.5 GHz the pulse bandwidth covers the entire Doppler distributions. Therefore, in the low intensity regime, the lineshape of P is practically equal to the pulse bandwidth as mentioned before. As the intensity increases, the peak value of P reaches unity, and consequently, the lineshape of P becomes flatter or equivalently the width of P increases when I0 further increases (Fig. 4(c)). From the comparison of Figs. 5(e) and (f) it is also obvious that the excitation probability for Δ = 0.5 GHz is smaller than that for Δ = 0, as we have seen in Fig. 2(c). Finally, in the case of 2 ns pulse, we recall that its bandwidth, 0.22 GHz, is about half of the Doppler width at 300 K, i.e., 0.524 GHz. Accordingly, in the low intensity regime, the width of P is nearly the same with that of Doppler width, and this situation is just similar to the case of the 20 ns pulse, as we can imagine by referring to Figs. 5(c) and (d). As the intensity increases, what happens is just between the cases described above for the 20 and 0.2 ns pulses: The width of P increases, but not too much, because the value of P does not reach unity, as we see in Fig. 4(b). The illustrations shown in Fig. 5(c) and (d) explain the qualitative behavior of the excitation process we have explained above. Before closing the discussions we emphasize that the 2 ns pulse is most efficient in the moderate intensity range, and the good thing is that the width of P does not so much increase as the intensity increases. Accordingly, the 2 ns pulse is most suitable for the measurement of hyperfine splitting of the ground state of muonic hydrogen. In contrast, the 0.2 ns pulse is most useful to polarize the muonic hydrogen, since the excitation probability reaches nearly unity and hence the degree of spin-polarization can be nearly 100% only if sufficient intensity is available. 6
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Fig. 5. Illustration of the spectral intensity profiles of the employed laser pulses and the Doppler-broadened transition line of muonic hydrogen at 300 K (black dashed line). Panels (a), (c), and (e) ((b), (d), and (f)) correspond to the pulse durations of 20, 2, and 0.2 ns, respectively, under resonance (detuning of Δ = 0.5 GHz), and in each panel the thin (thick) black solid line filled with light (dark) yellow represents the case of low (high) intensity regime. The peak heights of the spectral intensity profiles in panels (a), (c), and (e) for the respective intensity regimes are in the ratio of 102 : 10 : 1 (not to scale) as mentioned in the text.
4. Conclusion In conclusion, we have studied the excitation probability between the hyperfine ground states of Doppler-broadened muonic hydrogen by a single nanosecond laser pulse with three different pulse durations, 20, 2, and 0.2 ns. This transition is dipole-forbidden, and it is the magnetic dipole interaction that induces the transition. The Doppler width of muonic hydrogen at 300 K is 0.524 GHz, and accordingly the bandwidths of the 20, 2, and 0.2 ns pulses are, respectively, narrower than, comparable to, and broader than the Doppler width. These relations result in the different excitation dynamics as the laser intensity and detuning are varied. At the moderate laser intensity, say, 1010, 1011, and 1012 W/cm2, respectively, by the 20, 2, and 0.2 ns pulses for a fair comparison (i.e., same pulse energy and beam diameter), we have found that the use of the resonant 2 ns pulse results in the highest excitation probability with a relatively narrow linewidth, and hence the 2 ns pulse is most suitable for the precise measurement of hyperfine splitting of the ground state of muonic hydrogen. In contrast, if the purpose is to spin-polarize the muonic hydrogen, the use of a 0.2 ns pulse is most efficient, as, for instance, the excitation probabilities at the intensities of 1011, 1012, and 1013 W/cm2, respectively, by the 20, 2, and 0.2 ns pulses are ∼0.05, 0.3, and 0.9. The results we have presented in this work would serve as a guideline for the development of the mid-infrared laser source to study the muonic hydrogen. Acknowledgement Part of this work was supported by the Grant-in-Aid for scientific research from the Ministry of Education, Culture, Sports, Science and Technology (Japan). References [1] [2] [3] [4] [5]
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