Optical Frequency Comb-Based Direct Two-Photon Cooling for Cold Atom Clock
Optical Frequency Comb-Based Direct Two-Photon Cooling for Cold Atom Clock
Dan, Lin;Xu, Hao;Guo, Ping;Zhao, Jianye
2022-04-18 00:00:00
hv photonics Article Optical Frequency Comb-Based Direct Two-Photon Cooling for Cold Atom Clock Lin Dan , Hao Xu , Ping Guo and Jianye Zhao * Department of Electronics, Peking University, Beijing 100871, China; l_d@pku.edu.cn (L.D.); hall_xu@pku.edu.cn (H.X.); pingguo@pku.edu.cn (P.G.) * Correspondence: zhaojianye@pku.edu.cn; Tel.: +86-010-62754253 Abstract: The performance of the cold atom clock based on coherent population trapping (CPT) improved when the temperature decreased. In order to obtain a lower temperature in the cold atom clock, we proposed a cooling scheme in this paper that employs direct two-photon transition using optical frequency combs (OFCs). Two trains of time-delayed pulses from opposite directions were utilized to interact with atoms. It was found that the temperature of the cold atoms reached the minimum if the pulse area was and the time delay between the absorption pulse and the stimulated emission pulse was in the range from 0.7t to t. In this paper, it was confirmed that the proposed cooling process allowed for faster and more efficient momentum exchange between light and atoms, and the proposed cooling process could be applied to the atoms or molecules that could not be cooled to desired temperature through the single-photon cooling process. The Rb cooling, together with the CPT interrogating scheme using OFCs reduced the ratio value of linewidth/contrast, and the frequency stability of the cold atom clock hence improved by more than six times as per our calculation. Keywords: CPT; OFCs; direct two-photon transition; stimulated emission; cold atom clock Citation: Dan, L.; Xu, H.; Guo, P.; 1. Introduction Zhao, J. Optical Frequency The performance of the atomic frequency standard based on the coherent population Comb-Based Direct Two-Photon trapping (CPT) using vapour-cell alkali metal atoms has greatly improved in stability Cooling for Cold Atom Clock. and accuracy [1–3]. Several optimization schemes based on proposed CPT were aimed Photonics 2022, 9, 268. https:// at improving the contrast of the signal, thereby improving the frequency stability, such doi.org/10.3390/photonics9040268 as push-pull [4], - [5], lin//lin [6], and lin?lin [7]. Field applications of chip-scale Received: 23 February 2022 atomic clocks (CSACs) [8,9] based on CPT are growing rapidly because of their small size Accepted: 13 April 2022 and low power consumption. Typically, CSACs deliver short-term fractional frequency Published: 18 April 2022 stability of <1 10 [10], but the devices are substantively not accurate because of large systematic frequency shifts derived from high-pressure buffer gases and light shifts [11]. Publisher’s Note: MDPI stays neutral The temperature coefficient for the change in fractional frequency of high-performance with regard to jurisdictional claims in buffer-gas CPT clocks has been measured to be in the order of 10 /K [2,12,13]. published maps and institutional affil- The frequency shifts and associated instability derived from buffer gases can be iations. eliminated by performing CPT with cold atoms [14–16]. Using laser-cooled atoms can eliminate Doppler broadening, thus narrowing the atoms’ optical spectra and creating a clean system in which all cold atoms are uniformly interrogated and light shifts can be Copyright: © 2022 by the authors. precisely studied [15]. Licensee MDPI, Basel, Switzerland. A typical L system for CPT interrogation is shown in Figure 1. The atoms are evenly This article is an open access article distributed in the ground states j1i and j2i, and the bichromatic laser (with angular fre- distributed under the terms and quency ! and ! ) interacts with atoms. The frequency detuning between the ground states a b conditions of the Creative Commons j1i and j2i, and the excited states j3i is defined as d (= ! ! ) and d (= ! ! ), a 23 23 13 13 b Attribution (CC BY) license (https:// respectively, where ! and ! are the frequency splitting between the ground states 13 23 creativecommons.org/licenses/by/ j1i and j2i, and the excited states j3i, respectively, and the Raman detuning d is defined 4.0/). Photonics 2022, 9, 268. https://doi.org/10.3390/photonics9040268 https://www.mdpi.com/journal/photonics Photonics 2022, 9, x FOR PEER REVIEW 2 of 11 Photonics 2022, 9, 268 2 of 11 ω −ω ), respectively, where ω and ω are the frequency splitting between the ground states |1⟩ and |2⟩, and the excited states |3⟩, respectively, and the Raman detun- ing 𝛿 is defined as 𝛿= 𝛿 −𝛿 . The Rabi frequency Ω for the transition with electric as d = d d . The Rabi frequency W for the transition with electric dipole moment 23 13 ij dipole moment 𝜇 (𝑖 = 1,2 𝑎𝑛𝑑 𝑗 = 3) is defined as Ω =𝜀𝜇 /ℏ, where 𝜀 is the electric m (i = 1, 2 and j = 3) is defined as W = #m /}, where # is the electric field amplitude. G ij ij ij 3 field amplitude. Γ is the spontaneous decay rate of the excited state |3⟩, and γ is the re- is the spontaneous decay rate of the excited state j3i, and g is the relaxation rate of the laxation rate of the coherence between the hyperfine ground states. coherence between the hyperfine ground states. Figure 1. Scheme of three-level L system for CPT interrogating. Figure 1. Scheme of three-level Λ system for CPT interrogating. Nowadays, technology that uses magneto-optical trap (MOT) to cool and trap neutral Nowadays, technology that uses magneto-optical trap (MOT) to cool and trap neutral at- atoms is popular. To reach a lower temperature beyond the single-photon Doppler limit, oms is popular. To reach a lower temperature beyond the single-photon Doppler limit, two-photon transition (TPT) was also used for further cooling processes [17–22]. The two-photon transition (TPT) was also used for further cooling processes [17–22]. The op- optical frequency combs (OFCs) correspond to a train of ultrafast pulses generated by tical frequency combs (OFCs) correspond to a train of ultrafast pulses generated by mode- mode-locked (ML) laser in the time domain. This laser connects optical frequency with locked (ML) laser in the time domain. This laser connects optical frequency with micro- microwave frequency and could provide the purest microwave frequency source [23]. Due wave frequency and could provide the purest microwave frequency source [23]. Due to to its narrow linewidth and coherent frequencies, OFCs not only are suitable for frequency its narrow linewidth and coherent frequencies, OFCs not only are suitable for frequency standards based on CPT [24–26] but also can be used to cool atoms through the TPT standards based on CPT [24–26] but also can be used to cool atoms through the TPT pro- process [22]. There is a potential to combine these two functions of the combs together cess [22]. There is a potential to combine these two functions of the combs together in the in the cold atomic clock. The repetition rate of OFCs could be locked to the cooled Rb cold atomic clock. The repetition rate of OFCs could be locked to the cooled Rb TPT TPT resonance too and has served as a frequency standard, with the stability reaching resonance too and has served as a frequency standard, with the stability reaching 1.5 × 1.5 10 at an averaging time of 100 s [27]. This paper proposes a further cooling scheme −13 10 at an averaging time of 100 s [27]. This paper proposes a further cooling scheme using using OFCs. This cooling scheme utilizes the direct two-photon transitions (DTPT) cooling OFCs. This cooling scheme utilizes the direct two-photon transitions (DTPT) cooling method during the stimulated emission process to obtain a lower cooling temperature. In method during the stimulated emission process to obtain a lower cooling temperature. In the present paper, we study the applications of this kind of cooling scheme on the cold the present paper, we study the applications of this kind of cooling scheme on the cold atomic clocks to obtain better frequency stability in the future. In previous studies about atomic clocks to obtain better frequency stability in the future. In previous studies about atom cooling with OFCs, the cooling processes mainly involved single-photon interactions atom cooling with OFCs, the cooling processes mainly involved single-photon interac- with the atoms [28–37], which used only a small fraction of the laser ’s total power and tions with the atoms [28–37], which used only a small fraction of the laser’s total power output spectrum. On the contrary, the DTPT cooling method for the atom cooling involves and output spectrum. On the contrary, the DTPT cooling method for the atom cooling all of the comb teeth contributing together [38]. involves all of the comb teeth contributing together [38]. 2. Two-Photon Cooling Model by Pulses 2. Two-Photon Cooling Model by Pulses We introduced a stimulated emission scheme for the two-photon cooling method by We introduced a stimulated emission scheme for the two-photon cooling method by pulses, as shown in Figure 2. A train of pulses generated by ML laser propagating toward pulses, as shown in Figure 2. A train of pulses generated by ML laser propagating toward the right side interact with the atoms moving to the left side. In momentum–time space, the right side interact with the atoms moving to the left side. In momentum–time space, the atoms initially locate at (P , t ) with momentum p toward the left side, and after they 0 0 0 the atoms initially locate at (P0, t0) with momentum p0 toward the left side, and after they absorb TPs with wave vectors k and k coming from the left side, the atoms go to the final 1 2 astate bsorbj T fiPfr s wi omth the wa initial ve vecstate tors 𝒌jii and directly 𝒌 thr com ough ing from the pr th ocess e left si ofde, DTPT the whi atom les losing go to th the e 𝟏 𝟐 ⟩ ⟩ fin momentum al state |𝑓 of fr}om the (k + k in) it . ia Since l state the |𝑖 excited directatoms ly through do not the absorb process of D resonantly TPT while , their return losing to 1 2 the ground state without the spontaneous emission needed for further absorption can only the momentum of ℏ (𝒌 +𝒌 ). Since the excited atoms do not absorb resonantly, their 𝟏 𝟐 retu occur rn to bythe stimulated ground s emission tate witho [39 ut, th 40]. e s If pontaneo the stimulated us emiss emission ion needis ed fo caused r furt by her a ab counter sorp- - tion c propagating an only occur by pulse towar stim d ulated emission [39, the left side, the excited 40]. If the sti atoms m at ul( at Ped , temission ) would is c lose aused by another 1 1 momentum of } (k + k ); hence, the total momentum lost is } (k + k + k + k ). These a counter-propagating pulse toward the left side, the excited atoms at (P1, t1) would lose 3 4 1 2 3 4 stimulated processes repeat at the Rabi frequency W that is much larger than the decay another momentum of ℏ (𝒌 +𝒌 ); hence, the total momentum lost is ℏ (𝒌 +𝒌 +𝒌 + i f 𝟑 𝟒 𝟏 𝟐 𝟑 rate G . Because the excited atoms return to their ground states more frequently during 𝒌 ). These f stimulated processes repeat at the Rabi frequency Ω that is much larger than the stimulated emission process rather than during the spontaneous emission process, Photonics 2022, 9, x FOR PEER REVIEW 3 of 11 Photonics 2022, 9, 268 3 of 11 the decay rate Γ . Because the excited atoms return to their ground states more frequently during the stimulated emission process rather than during the spontaneous emission pro- cess, the pressure radiation force from photons will not be in a saturated state at a mo- the pressure radiation force from photons will not be in a saturated state at a momentum mentum exchange rate limited by Γ . exchange rate limited by G . Figure 2. Stimulated emission scheme in the momentum–time space, the atoms at (p , t ) lose a 0 0 Figure 2. Stimulated emission scheme in the momentum–time space, the atoms at (p0, t0) lose a mo- momentum of } (k + k ) during the TP absorption process (inset); after a time delay of less than the mentum of ℏ (𝒌 +𝒌 1 ) 2 during the TP absorption process (inset); after a time delay of less than the 𝟏 𝟐 lifetime , the excited atoms at (P , t ) lose another momentum of } (k + k ) during the TP stimulated lifetime τ, the excited atoms at (P1, t1) lose another momentum of ℏ (𝒌 +𝒌 ) during the TP stim- 1 1 3 4 𝟑 𝟒 ulated emission process, and the total momentum transfer or loss of the atoms is ℏ (𝒌 +𝒌 +𝒌 + emission process, and the total momentum transfer or loss of the atoms is } (k + k + k + k ) in 𝟑 𝟒 𝟏 3 4 1 2 𝒌 ) in one absorption-stimulated emission cycle. one 𝟐 absorption-stimulated emission cycle. By Byus using ing the the schem scheme e oof f ““absorption-stimulated absorption-stimulated em emission”, ission”, the the re recoil coil m momentum omentum ob ob- - ta tained ined by by a atoms toms or or molec molecules ules m mainly ainlycomes comes frfrom om the thfast e fas “absorption-stimulated t “absorption-stimulated emissi emi on” s- cycle. By choosing appropriate laser parameters, the rate of this transition cycle can be sion” cycle. By choosing appropriate laser parameters, the rate of this transition cycle can made much larger than the spontaneous radiation rate, so that the force on the atom or be made much larger than the spontaneous radiation rate, so that the force on the atom or the molecule obviously exceeds the spontaneous radiation force, thus ensuring a shorter the molecule obviously exceeds the spontaneous radiation force, thus ensuring a shorter deceleration distance and a larger deceleration efficiency. In addition, the experimental deceleration distance and a larger deceleration efficiency. In addition, the experimental setup required by this scheme is relatively simpler. The advantage of this scheme of stimu- setup required by this scheme is relatively simpler. The advantage of this scheme of stim- lated radiation deceleration has been confirmed in previous experiments [41] and was used to ulated radiation deceleration has been confirmed in previous experiments [41] and was cool [42] or accelerate [39] atoms/molecules, depending on the details of experiment designs. used to cool [42] or accelerate [39] atoms/molecules, depending on the details of experi- Assuming that the atoms are initially in the ground state, the density operator is ment designs. r ˆ (0) =jiihij, and n is used to represent the number of photons in the laser field. In the Assuming that the atoms are initially in the ground state, the density operator is form of dressed states, the atom-photon density operator is written as 𝜌 (0) =|𝑖 ⟩⟨𝑖| , and n is used to represent the number of photons in the laser field. In the form of dressed states, the atom-photon densi ty operator is written as i, n n, i 0 j ih j r ˆ(0) = (1) |𝑖, 𝑛 ⟩⟨𝑛, 𝑖| 0 0 0 𝜌 (0) = (1) We define the pulse area as We define the pulse area as +¥ q = W g((t) )dt (2) θ = Ω 𝑔 𝑡 𝑑𝑡 (2) i f where g(t) is the pulse envelope and Ω is the two-photon Rabi frequency. where g(t) is the pulse envelope and W is the two-photon Rabi frequency. i f The absorption pulse operator can be written as The absorption pulse operator can be written as 𝜃 𝜃 q q cos ji, nihn, ij i sin ji, nihn 2, fj cos |𝑖, 𝑛 ⟩⟨𝑛, 𝑖| −𝑖sin |𝑖, 𝑛 ⟩⟨𝑛− 2, 𝑓 | b 2 2 U = (3) 1 2 2 q q 𝑈 = (3) i sin j f , n 2ihn, ij cos j f , n 2ihn 2, fj 2 2 𝜃 𝜃 −𝑖sin |𝑓 ,𝑛 − 2 ⟩⟨𝑛, 𝑖| cos |𝑓 ,𝑛 − 2 ⟩⟨𝑛− 2, 𝑓 | 2 2 After the pulse is applied, the density operator of the atoms becomes After the pulse is applied, the density operator of the atoms becomes r ˆ = U r ˆ(0)U (4) 1 1 1 Photonics 2022, 9, x FOR PEER REVIEW 4 of 11 𝜌 =𝑈 𝜌 (0)𝑈 (4) Considering the fact that the atoms in the excited state undergo spontaneous emis- sion at the same time, the atoms in the excited state exponentially decay between the two trains of pulses by a factor of 𝑒 before the stimulated emission pulse is applied. The stimulated emission pulse operator is written as 𝜃 𝜃 Photonics 2022, 9, 268 4 of 11 ⟩⟨ ⟩⟨ cos |𝑖, 𝑛 + 2 𝑛+ 2, 𝑖| −𝑖sin |𝑖, 𝑛 + 2 𝑛, 𝑓 | 2 2 𝑈 = (5) 𝜃 𝜃 −𝑖sin |𝑓 ,𝑛 ⟩⟨𝑛 + 2, 𝑖| cos |𝑓 ,𝑛 ⟩⟨𝑛, 𝑓 | 2 2 Considering the fact that the atoms in the excited state undergo spontaneous emission The density operator of the atoms after the stimulated emission pulse becomes at the same time, the atoms in the excited state exponentially decay between the two trains t/t of pulses by a factor of e before the stimulated emission pulse is applied. (6) 𝜌 =𝑈 𝜌 𝑈 The stimulated emission pulse operator is written as By tracing the final density operator, the probability of the population distribution in q q the ground state (𝑃 ) and the excited state (𝑃 ) after the two trains of pulses can be obtained cos ji, n + 2ihn + 2, ij i sin ji, n + 2ihn, fj 2 2 U = (5) q q as shown in Equation (8) and Figure 3. i sin j f , nihn + 2, ij cos j f , nihn, fj 2 2 𝜃 𝜃 (7) The density operator 𝑃 of =t the r ( atoms ⟨𝑖|𝜌 |𝑖 after ⟩) = cos the stimulated −𝑒 sin emission cos𝜃 pulse becomes 2 2 b b𝜃 𝜃 𝜃 r ˆ = U r ˆ U (6) 2 2 1 (⟨ ⟩) (8) 𝑃 =tr 𝑓 |𝜌 |𝑓 =(1 + 𝑒 )sin cos +(1−𝑒 )sin 2 2 2 By tracing the final density operator, the probability of the population distribution in The repetition of the period of pulses is the time delay, which is usually less than the the ground state (P ) and the excited state (P ) after the two trains of pulses can be obtained g e lifetime 𝜏 of the atomic excited state. As apparent in Figure 3, the population distribution as shown in Equation (8) and Figure 3. of the excited state is simultaneously controlled by both the pulse area 𝜃 and the time delay t of stimulated emission pulse. When the pulse area 𝜃= π , the highest probability q q 2 2 P = tr hi r ˆ ii = cos e sin cos q (7) ( j j ) g n 2 is obtained as shown in Figure 3a. Additionally, in order to keep the high probability, the 2 2 stimulated emission pulse should be applied at a time delay between 0.7𝜏 and 𝜏 , as t t q q q 2 2 4 shown in Figure 3b. If the stimulated emission pulse is applied too early, the excitation t t P = tr (h fjr j fi) = 1 + e sin cos + (1 e ) sin (8) e n 2 2 2 2 probability will be low. (b) Stimulated 0.9 no stimulated 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time delay/ Figure 3. Figure 3.The The probability of probability of t th hee popu popula lation tion ddistribu istributition on in in ththe excited e excited sta sta te v te varie aries wsi with th the tp hu e pu lse als re ea area and the time delay of the stimulated emission pulse simultaneously. The curved surface graph and the time delay of the stimulated emission pulse simultaneously. The curved surface graph in (a) and in (a) and the red line in (b) indicate the case in which the stimulated emission pulse is applied, the red line in (b) indicate the case in which the stimulated emission pulse is applied, whereas the whereas the curved line graph in (a) and the blue line in (b) indicate that there is no stimulated curved line graph in (a) and the blue line in (b) indicate that there is no stimulated emission pulse. emission pulse. The repetition of the period of pulses is the time delay, which is usually less than the 3. Results lifetime t of the atomic excited state. As apparent in Figure 3, the population distribution of the To show the excited state cooling is simultaneously effect of the st contr imulolled ated emissio by bothn pulse on the pulse th are atoms more ea q and the time di- rectly delayand by t of stimulated employin emission g the concept o pulse. When f entropy the ,pulse we compared area q = th , the e temper highest atur pr e chang obability e is obtained as shown in Figure 3a. Additionally, in order to keep the high probability, the before applying the proposed cooling scheme with that after applying the proposed cool- stimulated emission pulse should be applied at a time delay between 0.7t and t, as shown ing scheme. in Figure 3b. If the stimulated emission pulse is applied too early, the excitation probability The relationship between entropy operator 𝜂 ̂ and density operator 𝜌 is will be low. 3. Results To show the cooling effect of the stimulated emission pulse on the atoms more directly and by employing the concept of entropy, we compared the temperature change before applying the proposed cooling scheme with that after applying the proposed cooling scheme. The relationship between entropy operator h ˆ and density operator r ˆ is h ˆ = ln r ˆ (9) Photonics 2022, 9, x FOR PEER REVIEW 5 of 11 Photonics 2022, 9, 268 5 of 11 𝜂 ̂ =−ln𝜌 (9) The ensemble of the entropy operator h ˆ is multiplied by Boltzmann’s constant k to obtain the corresponding macroscopic quantity information entropy S: The ensemble of the entropy operator 𝜂 ̂ is multiplied by Boltzmann’s constant 𝑘 to obtain the corresponding macroscopic quantity information entropy S: S = k hh ˆi = k tr(r ˆ ln r ˆ) = k (r ˆ ln r ˆ) (10) B B Bå nn 𝑆= 𝑘 〈𝜂 ̂ 〉 =−𝑘 tr(𝜌 ln𝜌 ) =−𝑘 (𝜌 ln𝜌 ) (10) For our atomic systems, For our atomic systems, S = k P ln P + P ln P (11) B g g e e 𝑆= −𝑘 (𝑃 ln𝑃 +𝑃 ln𝑃 ) (11) When N particles are confined to a volume V at a temperature T, we have [43,44] When N particles are confined to a volume V at a temperature T, we have [43,44] V 3 𝑉 3 S = Nk [ln + ln T + s ] (12) (12) 𝑆= 𝑁𝑘 B ln + ln𝑇 + 𝜎 0 N 2 𝑁 2 Which is based on assuming that T1 and S1, and T2 and S2 are the temperature and the Which is based on assuming that T and S , and T and S are the temperature 1 1 2 2 entropy of the atomic group before and after the stimulated emission pulse is applied. and the entropy of the atomic group before and after the stimulated emission pulse is According to Equations (11) and (12), the ratio (R) of the temperature obtained after ap- applied. According to Equations (11) and (12), the ratio (R) of the temperature obtained plying the proposed cooling scheme to the ratio obtained before applying the proposed after applying the proposed cooling scheme to the ratio obtained before applying the cooling scheme can be obtained as proposed cooling scheme can be obtained as 𝑇 2 T 2 𝑅= 2 = 𝑒 (13) S S 2 1 R = = e (13) 𝑇 3 T 3 Figure 4a shows that the temperature ratio varies with the time delay, and each line Figure 4a shows that the temperature ratio varies with the time delay, and each line has a corresponding pulse area. When the pulse area is small (π/10), the time delay has has a corresponding pulse area. When the pulse area is small (/10), the time delay has little effect on the result, and when the pulse area is π/2, the ratio increases with the time little effect on the result, and when the pulse area is /2, the ratio increases with the time delay. When the pulse area becomes larger, it can reduce the ratio to the minimum when delay. When the pulse area becomes larger, it can reduce the ratio to the minimum when faced with a proper time delay, such as for 2π/3 pulse area, and the ideal time delay is faced with a proper time delay, such as for 2/3 pulse area, and the ideal time delay is 0.4. 0.4. For π pulse area, the ratio is independent of the time delay. Figure 4b shows the var- For pulse area, the ratio is independent of the time delay. Figure 4b shows the variation iation of the temperature ratio with the pulse area, and each line has a corresponding time of the temperature ratio with the pulse area, and each line has a corresponding time delay. delay. When pulse area θπ/2 , the shorter the time delay, the lower the ratio will be. When pulse area q < /2, the shorter the time delay, the lower the ratio will be. When When pulse area 𝜃= π , the ratio does not depend on the time delay, and the ratio is the pulse area q = , the ratio does not depend on the time delay, and the ratio is the lowest, lowest, which means the lowest temperature could be obtained at pulse area 𝜃= π . which means the lowest temperature could be obtained at pulse area q = . Figure 4. The ratio R temperature obtained before and after the application of the stimulated radiation Figure 4. The ratio R temperature obtained before and after the application of the stimulated radia- tion pulse i pulse is shown s shown as a funct as a functionion of ( of (a) the a) the t timeime delay, whe delay, whereinrein the lin the lines each es ea have ch ha pulse ve pul ar seas e arof eas o /f 10 , π/10, π/2, 2π/3, and π and (b) the pulse area, wherein the lines each have time delays of 0.1 𝜏 , 0.3 /2, 2/3, and and (b) the pulse area, wherein the lines each have time delays of 0.1 t, 0.3 t, 0.6 t, 𝜏 , 0.6 𝜏 , and 0.9 𝜏 . and 0.9 t. Considering both the high transition probability and the low temperature obtained, it Considering both the high transition probability and the low temperature obtained, is best to apply the stimulated emission pulse with the pulse area of about , where the time it is best to apply the stimulated emission pulse with the pulse area of about π, where the delay is between 0.7t and t, such that the temperature of the atom can be decreased to the time delay is between 0.7𝜏 and 𝜏 , such that the temperature of the atom can be decreased minimum. As the ML-based DTPT cooling could reduce the temperature to a value much to the minimum. As the ML-based DTPT cooling could reduce the temperature to a value lower than that achieved by the single-photon Doppler [45] and the stability of cold atomic much lower than that achieved by the single-photon Doppler [45] and the stability of cold clock is thus improved at the lower temperature, it is predicted that the frequency stability Photonics 2022, 9, 268 6 of 11 of cold atom clock based on CPT is will reach a better performance than the previous or current level. The primary advantages of stimulated pulse together with a DTPT process for atom cooling arises from the use of stimulated emissions in place of spontaneous emissions to return atoms to their ground states. Therefore, the proposed cooling scheme allows for rapid and repeated momentum exchanges between the atom and the light field by restricting the atom–light interaction to a shorter time, compared to the traditional way in which a cycle of absorption is followed by natural decay. Because of the complex internal structure of some kinds of atoms and molecules, the cw lasers used during single-photon Doppler cooling are not suitable for them, especially in the deep ultraviolet (UV) range, as the single-photon laser cooling is not currently available for the most atoms [45]. For instance, single-photon laser cooling is not suitable for the 1S-2S transition of H atoms for the following two reasons, first, that there is no intermediate real en- ergy level between 1S and 2S of H atoms, and second, that there is no commercially available deep UV laser that will generate 243 nm or shorter wavelength to drive the transition. However, DTPT cooling with stimulated emission scheme proposed herein can be used to cool these kinds of atoms and molecules. The DTPT process played a main role in interrogating the 1S-2S transition of H atoms [46]. In contrast to cw lasers, OFCs generated by ML lasers can be efficiently frequency-multiplied to apply to the UV and can be used to control the motion of atoms and molecules in ways that are not possible in cw lasers. 4. Discussion and Application When the DTPT stimulated emission cooling method is used in the Rb atomic frequency standard, we can calculate the laser source characteristics required. ML laser 2 2 electric field strength can be written in the form #(r, t) = # g(t) exp r /w , where r is the radial distance from the central axis, ! is the beam waist radius, and # is the 0 0 peak amplitude of electric field at the central axis. Assuming that the driving pulse has a 2 2 Gaussian envelope g(t) = exp [ t /2T ], and T is the pulse width, it can be calculated p p 2 3 2 }w f w 0 i f that the time average laser power of the pulse condition is p = . For the 24G T c Gaussian pulses from the ML lasers, the beam waist radius is w = 20 m, the pulse width is T = 1 ps, the comb tooth repetition frequency is f = 100 MHz, G is the decay rate of the p r f upper state, and w is the resonance frequency between the upper and lower states. In a i f popular MOT, a 780 nm laser source is often used, the upper energy level is 5P , and the 3/2 laser wavelength that creates the CPT resonance is 795 nm with the upper state of 5P due 1/2 to which the hyperfine energy levels are easily distinguished. When the TPT process is used for cooling, the upper energy level can be 5D or 5D , and a total of 39 pathways and 5/2 3/2 14 transitions can be identified for the 5S -5D and 5S -5D two-photon resonances 1/2 3/2 1/2 5/2 ; the relative transition probabilities are shown in Figure 5. We compared the difference between circularly polarized photons and linearly polarized photons. As shown in Figure 5, the 5S -5D transition from F = 2 to F = 4 excited by circularly polarized photons has 1/2 5/2 the largest transition probability and can be used for DTPT stimulated emission cooling. We take the line width of the upper state 5D as 660 kHz and the equivalent DTPT resonance frequency as 7.7 10 Hz (5S -5D ). The time average power of the 1/2 5/2 Gaussian pulse generated by ML laser is approximately 420 mW and the peak amplitude of the pulse is about 5 10 V/m, such that the temperature obtained from the DTPT cooling scheme is supposed to reach one tenth of that of the single-photon Doppler cooling limit, which is 145 K for Rb [45]. Photonics 2022, 9, x FOR PEER REVIEW 7 of 11 Photonics 2022, 9, 268 7 of 11 Figure 5. Relative probabilities of 87Rb 5S-5D two-photon transitions, where blue represents excitation Figure 5. Relative probabilities of Rb 5S-5D two-photon transitions, where blue represents excita- with circularly polarized photons and red represents excitation with linearly polarized photons. tion with circularly polarized photons and red represents excitation with linearly polarized photons. Figure 6 illustrates the results of these pulses interacting with the same sample of We take the line width of the upper state 5D as 660 kHz and the equivalent DTPT cold Rb atoms to interrogate the CPT signals based on the configuration in Figure 1. For resonance frequency as 7.7 × 10 Hz (5S1/2-5D5/2). The time average power of the Gaussian both cases, the Doppler shifts d are assumed to be 0.01G . In the calculation, we used π pulse generated by ML laser is approximately 420 mW and the peak amplitude of the 500 Gaussian pulses to simulate the optical combs, the pulse–atom interaction is de- pulse is about 5×10 V/m, such that the temperature obtained from the DTPT cooling scribed by the Bloch equations as in Ref. [47], and the temporal evolution of the ele- scheme is supposed to reach one tenth of that of the single-photon Doppler cooling limit, ments r of the atomic density matrix can be numerically resolved by using the fourth- ij which is 145 μK for Rb [45]. order Runge–Kutta method. The Rabi frequency is defined as W (t) = #(t)m /}, for ML, ij ij Figure 6 illustrates the results of these pulses interacting with the same sample of #(t) = # g(t nT ), # = 5 10 V/m, the pulse width is 1 ps, and the pulse repetition 87 å r 0 n 0 cold Rb atoms to interrogate the CPT signals based on the configuration in Figure 1. For period is 10 ns. For cw, the Rabi frequency is W = W = G ; g = 10 G ; and for the 13 23 3 3 both cases, the Doppler shifts 𝛿 are assumed to be 0.01Γ . In the calculation, we used 500 Rb atom G = 37 MHz. We assumed r + r + r = 1 and r = r = 0.5 under the 3 33 22 11 22 11 Gaussian pulses to simulate the optical combs, the pulse–atom interaction is described by initial condition to obtain the results shown in Figure 6a. The red, green, and black lines in the Bloch equations as in Ref. [47], and the temporal evolution of the elements 𝜌 of the the figure represent the population of excited state, bright state, and dark state changing atomic density matrix can be numerically resolved by using the fourth-order Runge–Kutta with the number of pulses, respectively, where the bright state and dark state are defined as method. The Rabi frequency is defined as Ω (𝑡 ) =𝜀 (𝑡 )𝜇 /ℏ, for ML, 𝜀 (𝑡 ) =𝜀 ∑ 𝑔(𝑡 − r = (r + r )/2 + Re(r ), r = (r + r )/2 Im(r ), respectively. From Figure 6a, B 11 22 12 D 11 22 12 𝑛𝑇 ) ,𝜀 =5 × 10 𝑉/𝑚 , the pulse width is 1ps, and the pulse repetition period is 10 ns. we can see that the bright and dark states are at 50% each in the beginning. When the atoms For cw, the Rabi frequency is Ω =Ω =Γ ; 𝛾= 10 Γ ; and for the Rb atom Γ = interact with about 40–50 pulses, the bright state (green line) is slowly pumped to the dark 37 MHz. We assumed 𝜌 +𝜌 +𝜌 = 1 and 𝜌 =𝜌 =0.5 under the initial condition state (black line), while the atomic excited state (red line) population slowly becomes 0, the to obtain the results shown in Figure 6a. The red, green, and black lines in the figure rep- atoms have minimum fluorescence, which is in the dark state, and the CPT phenomenon resent the population of excited state, bright state, and dark state changing with the num- occurs. We denote the difference from the maximum value of r to the value at the end ber of pulses, respectively, where the bright state and dark state are defined as 𝜌 = of 50th pulse as Dr . The change in the population of excited states during this process (𝜌 +𝜌 )/2 + Re(𝜌 ), 𝜌 =(𝜌 +𝜌 )/2 − Im(𝜌 ), respectively. From Figure 6a, we corresponds to the process of the fluorescence changing from bright to dark. can see that the bright and dark states are at 50% each in the beginning. When the atoms interact with about 40–50 pulses, the bright state (green line) is slowly pumped to the dark state (black line), while the atomic excited state (red line) population slowly becomes 0, the atoms have minimum fluorescence, which is in the dark state, and the CPT phenome- non occurs. We denote the difference from the maximum value of 𝜌 to the value at the end of 50th pulse as Δ𝜌 . The change in the population of excited states during this pro- cess corresponds to the process of the fluorescence changing from bright to dark. Photonics Photonics 2022 2022 , 9, x FO , 9, 268 R PEER REVIEW 8 of 11 8 of 11 Figure 6. (a) The population of excited (red), bright (green) and dark (black) states change with the Figure 6. (a) The population of excited (red), bright (green) and dark (black) states change with the number of pulses. (b)The excited state population change (Δ𝜌 ) of the cold Rb 87 atoms varies with number of pulses. (b)The excited state population change (Dr ) of the cold Rb atoms varies with the Raman detuning (𝛿 ), and the Doppler shift is set to be 𝛿 = 0.01Γ . The interrogating laser fields the Raman detuning (d), and the Doppler shift is set to be d = 0.01G . The interrogating laser fields D 3 are cw (blue) and ML (red). are cw (blue) and ML (red). By scanning the Raman detuning d, we obtained the variation of Dr changing with By scanning the Raman detuning δ, we obtained the variation of Δ𝜌 changing with as shown in Figure 6b. The blue line is obtained by using cw, and the red line is obtained δ as shown in Figure 6b. The blue line is obtained by using cw, and the red line is obtained using the pulses discussed above. We obtained the shape and linewidth/contrast value using the pulses discussed above. We obtained the shape and linewidth/contrast value of of the pulse-excited CPT signal by numerical simulation and compared it with that from the pulse-excited CPT signal by numerical simulation and compared it with that from the the cw-excited signal. From Figure 6b, we calculated the linewidth/contrast values of 0.69 cw-excited signal. From Figure 6b, we calculated the linewidth/contrast values of 0.69 and and 0.11 from cw and ML excitations, respectively, as shown in Table 1. As the frequency 0.11 from cw and ML excitations, respectively, as shown in Table 1. As the frequency sta- stability based on CPT is proportional to the linewidth/contrast value, it can be seen bility based on CPT is proportional to the linewidth/contrast value, it can be seen that the that the frequency stability based on CPT87 in the cold Rb atoms has the potential to be frequency stability based on CPT in the cold Rb atoms has the potential to be improved improved by more than six times when using proposed pulses instead of cw. by more than six times when using proposed pulses instead of cw. Table 1. The linewidth/contrast values of CPT signals based on different schemes and Doppler shifts. Table 1. The linewidth/contrast values of CPT signals based on different schemes and Doppler shifts. Schemes d Linewidth/Contrast G 1.06 Schem cw e CPT s 𝜹 Linewidth/Contrast 0.1G 0.70 cw CPT cw CPT Γ 1.06 DTPT cooling + cw CPT 0.01G 0.69 cw CPT 0.1Γ 0.70 DTPT cooling + ML CPT 0.01G 0.11 DTPT cooling + cw CPT 0.01Γ 0.69 DTPT cooling + ML CPT 0.01Γ 0.11 As discussed above, the OFCs from ML lasers can be used to cool and interrogate atoms in the one cold atom clock system. Traditional cold atomic clocks usually require As discussed above, the OFCs from ML lasers can be used to cool and interrogate three semiconductor lasers for atom cooling, repumping, and CPT interrogation, and atoms in the one cold atom clock system. Traditional cold atomic clocks usually require each laser requires additional equipment for frequency locking. The entire optical system three semiconductor lasers for atom cooling, repumping, and CPT interrogation, and each is complex and bulky. As current mode-locked lasers can easily cover the wavelengths laser requires additional equipment for frequency locking. The entire optical system is required for Rb atomic cooling, repumping, and CPT detection and can also provide watt- complex and bulky. As current mode-locked lasers can easily cover the wavelengths re- level power output, a single mode-locked laser can be used for atomic cooling, repumping, quired for Rb atomic cooling, repumping, and CPT detection and can also provide watt- and CPT detection, which is beneficial to the integration of the optical system. The ML laser level power output, a single mode-locked laser can be used for atomic cooling, repump- parameters depend on the specified element and its corresponding transition energy levels ing, that and we CPT study de .tecRb, tion for , wexample, hich is bene during ficial the to cooling the intepr gra ocess, tion o the f th upper e optiener cal s gy ystlevel em. The can be ML laser parameters depend on the specified element and its corresponding transition 5D , and the lifetime in this situation is about t = 240 ns. According to the conclusions 5/2 energy levels that we study. Rb, for example, during the cooling process, the upper en- drawn in this paper, the time delay can be selected from 170 ns to 240 ns, and the specific ergy delay levemeasur l can bees 5D can 5/2, and refer the to l the ifetime method in thin is s ref. ituat [39 ion ]. isRb abou can t 𝜏 also = 2be 40 ns. Accordi used as the n cooling g to the concl element usistudied, ons drawn as in t selected his paper in the , the literatur time de ela [45 y ca ]. n be selec If Cs isted selected, from 170 n the upper s to 240 ns, ener gy level can be 6D , 6D , or 8S , and the corresponding lifetime t is different. As the and the specific delay measures can refer to the method in ref. [39]. Rb can also be used 3/2 5/2 1/2 optical comb frequency is determined by the initial frequency and repetition frequency, the as the cooling element studied, as selected in the literature [45]. If Cs is selected, the repetition frequency is set as 100 MHz in our calculation and the specific transition can also upper energy level can be 6D3/2, 6D5/2, or 8S1/2, and the corresponding lifetime 𝜏 is differ- meet the requirements by selecting the appropriate initial frequency parameters. ent. As the optical comb frequency is determined by the initial frequency and repetition frequency, the repetition frequency is set as 100 MHz in our calculation and the specific transition can also meet the requirements by selecting the appropriate initial frequency parameters. Photonics 2022, 9, 268 9 of 11 5. Conclusions In conclusion, as the ratio value of linewidth/contrast of CPT signal was reduced as the temperature of atoms decreased; hence, the performance of cold atom clock improved. We proposed a more efficient cooling scheme, which utilizes the DTPT process of the OFCs and stimulated emission pulse, to cool atoms for the cold atomic clock to a lower temperature. It was found that the temperature of the atomic sample could be reduced to the minimum when the pulse area was about p and the time delay was 0.7t~t, which was dependent on the specified element and its corresponding transition energy levels that we studied. We also calculated the pulse power required for the corresponding Rb cooling process. Additionally, this cooling scheme could be used to cool other elements that cannot be cooled to a desired temperature by traditional single-photon cooling methods. When compared with traditional single-photon cooling methods using cw, if the same optical combs were used to interrogate the CPT signal and when the proposed cooling scheme was utilized, the ratio value of the linewidth/contrast decreased by more than six times, and so did its frequency stability. The optical system of this OFC-based cold atomic clock was integrated and will have potential application in the future. Author Contributions: Conceptualization, L.D.; Data curation, P.G.; Formal analysis, L.D., H.X. and P.G.; Methodology, L.D. and H.X.; Project administration, J.Z.; Resources, J.Z.; Software, H.X.; Supervision, J.Z.; Writing—original draft, L.D.; Writing—review and editing, L.D. and P.G. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Science Foundation of China, grant numbers 91836301 and 61535001. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data that support the findings of this study are available upon reasonable request from the authors. Conflicts of Interest: The authors declare no conflict of interest. References 1. Vanier, J. Atomic clocks based on coherent population trapping: A review. Appl. Phys. B 2005, 81, 421–442. [CrossRef] 2. Vanier, J.; Levine, M.W.; Kendig, S.; Janssen, D.; Everson, C.; Delaney, M.J. 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