1. Introduction

In recent years, many promising approaches have been developed for obtaining cold or ultracold neutral molecules, such as photoassociation [1] and Feshbach resonances [2] for laser-cooled atoms, buffer gas cooling [3], electrostatic Stark decelerator for polar molecules [4], a magnetic Zeeman slower for paramagnetic molecules [5], and optical Stark decelerator for all kinds of neutral molecules [6], low-pass velocity filter [7–9], laser cooling [10] and so on. Recently, a single-stage (or multistage) optical Stark decelerator using a far-off-resonance, red-detuned, moving (or static), pulsed (or quasi-cw) optical lattice has attracted much attention of people [6, 11–15]. In 2001 and 2002, Barker’s group put forward a traveling periodic dipole potential of an optical lattice created by high-intensity short-pulse lasers to accelerate and decelerate atoms or molecules [11, 12]. In 2006, they also proposed and realized an effectively slowing of a supersonic NO molecular beam by using a traveling pulse optical lattice with a well-depth of 22K (i.e., a single-stage optical Stark decelerator) [6], and J. Ramirez-Serrano, et al experimentally demonstrated a similar single-stage optical Stark deceleration for a supersonic H₂ molecular beam with a static pulse optical lattice [13]. In 2009, Momose’s group proposed a multistage optical Stark decelerator scheme for a pre-cooling pulsed molecular beam with an initial velocity of 13-15m/s and a temperature of 100-500mK by using a quasi-cw, near-resonant and cavity-enhanced IR optical lattice [14]. Also, our group proposed a useful multistage optical Stark decelerator for a supersonic (or subsonic) molecular beam with an initial velocity of 240-400m/s by using a far-off-resonance, red-detuned, quasi-cw optical lattice with a waist of 20µm and a power of 264W in 2009, and studied its deceleration effects for a subsonic CH₄ beam and a supersonic NO beam by using 1D Monte-Carlo simulations [15]. But there are three problems in this scheme [15]: (1) The laser waist (20µm) is too small to obtain a longer lattice and more cold molecules; (2) We didn’t consider the transverse molecular loss in our 1D simulations; (3) It needs a longer optical lattice in order to slow molecular velocity to zero because the relative velocity between the initial velocity (240-400m/s) of supersonic (or subsonic) molecular beam and a static optical lattice is too large. To solve these problems, in this paper, we propose a more desirable multistage optical Stark decelerator for a subsonic (or supersonic) molecular beam by using a far-off-resonance, red-detuned, quasi-cw, traveling optical lattice with a diameter of 100µm and an intensity of ~10⁷Wcm⁻², and study its slowing results by 3D Monte-Carlo simulations. In our scheme, a shorter lattice and a shallower optical potential are only needed compared with the static lattice [15]. This paper is organized as follows: In section 2, our proposed new slowing scheme and its principle for molecules are briefly introduced. In section 3, the slowing effects (including the final most-probable velocity, the fraction of cold CH₄ molecules and its temperature) of our moving optical lattice with different synchronous phase angles are studied. Also, the dependences of the deceleration-stage numbers on the velocity reduced amount of the moving lattice, lattice power, and synchronous phase angle are investigated. In section 4, we study the influence of temporal shape of laser pulses on the slowing results. In section 5, we compare the slowing results of this moving lattice with the static one using the same parameters to show some advantages of our slowing scheme and also compare the slowing results of our scheme with the chirped optical lattice [12] by 1D Monte-Carlo simulations. Some main results and conclusions are summarized in the final section.

2. Slowing scheme and its principle

The schematic diagram of our multistage optical Stark decelerator for a pulsed subsonic (or supersonic) molecular beam is indicated in Fig. 1(a) . In our scheme, the multistage decelerator is composed of a red-detuned, quasi-cw moving optical lattice with a diameter of 100µm (i.e., a waist of w₀ = 50µm), which is created by two nearly counter-propagating, far-off-resonance, single-frequency, linear-polarized infrared lasers with an equal intensity of I₀ = 1.02 × 10⁷Wcm⁻² and a wavelength of λ = 1064nm. In Fig. 1(a), β is an angle between the two laser beams [16], and both the pulsed molecular beam and the moving optical lattice are traveled along the z axis. The velocity of the moving lattice is given by v_latt = (ω₁-ω₂)/|${\stackrel{\rightharpoonup }{k}}_1-{\stackrel{\rightharpoonup }{k}}_2$|, where ${\stackrel{\rightharpoonup }{k}}_1$ and ${\stackrel{\rightharpoonup }{k}}_2$ are the wave vectors of the two counter-propagating fields, and ω₁ and ω₂ are their laser frequencies.

 

Fig. 1 (a) Schematic diagram of a multistage optical Stark decelerator using a quasi-cw traveling optical lattice, which is created by two nearly counter-propagating, far-off-resonance, red-detuned laser beams with a fixed frequency difference between them, and the lattice moves along the z axis. A pulsed subsonic molecular beam propagates along the z axis and enters the quasi-cw moving lattice. The inset figure is the enlarged optical lattice. (b) The time-sequence control of moving lattice velocity for v_m0 = 240m/s, ϕ₀ = 90^° and 120^°.

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If the moving lattice is far red-detuned for the molecular resonant frequency, the optical lattice potential for molecules is given by [17]

When molecules enter a moving optical lattice along the z axis, they will experience an optical dipole interaction potential in the red-detuned optical lattice field as follows [6]:

According to Eq. (3), molecules in the red-detuned lattice will experience an optical dipole force in both the longitudinal and transverse directions. In the transverse (i.e., radial) direction, some cold and slow molecules will be confined by the oscillation motions, other molecules with a kinetic energy of larger than the transverse lattice potential will escape from the lattice. Because the optical lattice field has a constant velocity v_latt in the longitudinal direction (i.e., the z axis), it is useful to consider the motion of molecules in a reference frame of the moving lattice. In the relative reference frame, if molecules enter the optical lattice from its antinode in the longitudinal direction, they will undergo a dipole force, which is anti-parallel to their motion direction, and then gain an optical Stark energy due to ac Stark effect. The gain in potential energy will be compensated by a loss in the molecular kinetic energy due to the law of energy conservation. During this single-stage decelerating process, the slowed molecules will lose their kinetic energy, which is equal to the corresponding lattice potential energy.

In the relative reference frame, the relative longitudinal position (z_r) of the slowed molecules in the traveling optical lattice is defined as a synchronous phase angle ϕ, where ϕ = z_r/Λ × 360^° and Λ is the spatial period of moving lattice. The point of z_r = 0 (i.e., ϕ = 0^°) is set at the antinode of the lattice (i.e., the maximum of the light intensity) due to its red-detuned. It is clear from Eq. (3) that the region of 0⁰<ϕ<180^° (the rear half of each lattice cell) is the decelerating phase-angle region for molecules in the red-detuned lattice, while the region of −180^°<ϕ<0^° is the accelerating one.

When a synchronized molecule reaches the antinode (ϕ = 0^°) of the lattice, the lattice light field should be switched on in time. As soon as it arrives at the position of ϕ = ϕ₀ (where ϕ₀ is the synchronous phase angle of the synchronized molecule, usually we choose 0^°<ϕ₀≤180^°), the lattice should be switched off rapidly, and then in the region of 0^°<ϕ≤ϕ₀, the synchronized molecules will be slowed effectively. For the red-detuned optical lattice, if this slowing process can be repeated continuously for m times, the subsonic molecular beam will be efficiently slowed to standstill. In the deceleration process, when the relative velocity of the synchronized molecule is slowed to zero, the velocity of the reference frame (i.e., the velocity of the moving lattice v_latt) should be reduced to another constant (until it reaches zero) by modulating Δω, as shown in Fig. 1(b), and the velocity reduced amount of the lattice is denoted as Δv_latt. In this paper, the initial velocity of the moving lattice is set as v_latt0 = v_m0-Δv_latt for each Δv_latt, here v_m0 is the initial central velocity (240m/s) of incident molecular beam in our simulations. If Δv_latt is equal to 80m/s [see Fig. 1(b)], we will have v_latt0 = 160m/s. In the deceleration process, for ϕ₀ = 90^°, when the deceleration time t arrives at 101μs and later 204μs (i.e., as the relative velocity of the synchronized molecule is slowed to zero), the lattice velocity v_latt should be reduced to 80m/s and then to 0m/s. Also, for ϕ₀ = 120^°, v_latt should be reduced to 80m/s and then to 0m/s when t = 67μs and 137μs.

In order to form a multistage optical Stark decelerator, we also need a time-varying lattice field (so-called a quasi-cw optical lattice) to realize an efficient deceleration for a subsonic (or supersonic) molecular beam by using a strict time-sequence synchronous and control system and an electro-optic modulator (EOM).

3. Results of 3D Monte-Carlo simulation

The slowing effect of our multistage optical Stark decelerator using a traveling optical lattice is demonstrated by 3D Monte-Carlo simulations, and 1D molecular slowing in the longitudinal direction and 2D trapping in the transverse one are studied. In our simulations, the pulsed subsonic methane (CH₄) molecular beam with an initial central velocity of 240m/s and a longitudinal temperature of 0.5K is used and traveled along the z axis, as shown in Fig. 1(a). We assume that the transverse and longitudinal spatial distributions of incident pulsed molecular beam are Gaussian ones, and their spatial sizes are 0.1mm and 5mm, respectively. Also, the transverse and longitudinal velocity distributions of incident pulsed molecular beam are Gaussian ones, and their velocity half widths are 2.3m/s and 40m/s, respectively. Two nearly counter-propagating, far-off-resonance, single-frequency, linear-polarized, frequency tunable ytterbium-doped fiber lasers with a power of 400W and a wavelength of 1064nm [19], as input laser beams with a waist of w₀ = 50μm, are used to create our moving optical-lattice decelerator. A frequency difference of 944MHz or 1890MHz between two input lasers gives rise to a lattice velocity of approximately 80m/s or 160m/s, respectively. The longitudinal velocity distributions [20] of the slowed molecules are detected by a micro-channel plate (MCP), which is placed at the position of about 20cm away from the start point of the lattice on the z axis.

When the deceleration stage number is m = 23122, the synchronous phase angle is ϕ₀ = 90^°, the velocity reduced amount of the traveling lattice is Δv_latt = 80m/s, and the initial lattice velocity is v_latt0 = 160m/s (i.e., v_latt0 = 240m/s-Δv_latt), we first study the slowing results of our moving optical lattice by 3D Monte-Carlo method, and the simulated results are shown in Fig. 2(a) . The horizontal axis Fig. 2 represents the longitudinal velocity of molecules, and the vertical one represents the relative molecular number (i.e., the ratio of the molecular number at each longitudinal velocity to the one at the initial most-probable velocity of the incident molecular beam). It is clear from Fig. 2(a) that there are three slowed packets after deceleration. Here three small peaks on the horizontal axis are pointed out by red circles because they are very small even difficult to be seen. To clearly show such small slowed packets, we enlarge their velocity distributions (see the above enlarged peaks). Among them, the slowed molecular packet 3 is the slowest packet with a central velocity of ~0m/s, which is our hoped result to be obtained eventually, and its final central velocity, temperature, and fraction of cold molecules are v_l = 0.9m/s, T_l = 11.5μK, and n = 5.27 × 10⁻⁶ (n = N/N₀, where N and N₀ are the molecular numbers in the slowed molecular packet and the incident molecular pulse, respectively). The slowed molecular packet 1 and 2 are our obtained middle results due to two reductions of the lattice velocity during the deceleration process (i.e., the packet 1 is the first outcome when v_latt is reduced from 160m/s to 80m/s, and the packet 2 is the second outcome when v_latt is reduced from 80m/s to 0m/s), and their central velocities are v_l = 160.7m/s and v_l = 80.3m/s, respectively. From Fig. 2(a), we can find that the temperatures T_l and fractions n of cold molecules in the packet 1, 2 and 3 become lower one by one. Though the packet 3 seems higher than the packet 2, its half width is much narrower than one of the packet 2, which means both its T_l and n are lower than those of the packet 2. We also study the deceleration effects when ϕ₀ = 120^°, and obtain some similar results as ϕ₀ = 90^°, see Fig. 2(b). We can see from Fig. 2(b) that the moving lattice with only m = 15422 (which is far smaller than m = 23122 for the case of ϕ₀ = 90^°) can be used to decelerate CH₄ molecular beam to ~0m/s from 240m/s when the other parameters are the same as ones used in Fig. 2(a), and the final central (most-probable) velocity, temperature of the slowed packet 3 and the fraction of cold molecules are v_l = 0.925m/s, T_l = 4.2μK, and n = 6.4 × 10⁻⁷, respectively. Moreover, when m = 15422, the length of our moving optical Stark decelerator is only about 1.38cm.

 

Fig. 2 Simulated longitudinal velocity distributions of CH₄ molecules at the outlet of the decelerator for the synchronous phase angle (a) ϕ₀ = 90^° and (b) ϕ₀ = 120^°. Other simulated parameters are given in the plots and text.

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Secondly, we investigate the dependences of the final central velocity of the slowed packet (i.e., the slowed packet 3 in Fig. 2) and its temperature as well as the fraction of cold molecules on the deceleration stage number m, and the simulated results are shown in Fig. 3 . Figure 3(a) shows the relationship between the molecular velocity v_m and the stage number m when the initial velocity of the traveling lattice is v_latt0 = 160m/s and its velocity reduced amount is Δv_latt = 80m/s, the solid lines are the fitted curves. Figure 3(b) shows the relationship between the fraction n of cold molecules in the slowed packet and the stage number m. It can be found from Fig. 3(b) that with the increase of m, the fraction n of cold molecules will be decreased gradually, and the larger the synchronous phase angle ϕ₀ is, the smaller the fraction of cold molecules will be. This is because the larger the synchronous phase angle is, the smaller the area of the phase stability space is, and then the less the cold molecular number in the slowed packet is [21]. Figure 3(c) shows that with the increase of m, the temperature T_l of the slowed molecular packet will be reduced gradually, and the larger the synchronous phase angle ϕ₀ is, the lower the temperature T_l of the slowed molecular packet is. This is because the number of cold molecules in the slowed packet is proportional to the area of the phase stability space. That is, the larger the synchronous phase angle is, the less the cold molecular number in the slowed packet is, and then the cooler the temperature of the slowed packet is, which is similar to the results in the evaporative cooling due to the bunching effect.

 

Fig. 3 Dependences of (a) the final central velocity, (b) fraction and (c) temperature of the slowest packet (i.e., the slowed packet 3 in Fig. 2) on the deceleration stage number m. Other simulated parameters are given in the plots and text.

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Also, it is obvious from Fig. 3 that it needs much fewer deceleration stages for slowing molecules from 240m/s to ~0m/s with ϕ₀ = 120^° than that with ϕ₀ = 90^°, and the slowed effects (v_l, n and T_l) with ϕ₀ = 120^° are greatly better than that with ϕ₀ = 90^°.

In final, we study the dependences of the deceleration stage number m that need to slow a subsonic CH₄ molecular beam from 240m/s to zero (i.e., to obtain the slowest packet) on the velocity reduced amount Δv_latt of the traveling lattice, the synchronous phase angle ϕ₀ and the input power P₀ of lattice beams, and the simulated results are shown in Fig. 4 . We can see from Fig. 4(a) that with the increase of Δv_latt, the needed deceleration-stage number m will be increased gradually when P₀ = 400W, and the larger the synchronous phase angle is, the less the needed deceleration-stage number m will be. Also, we find that the fraction n of cold molecules in the slowest molecular packet is very low as ϕ₀ = 150^°. So in our simulations, ϕ₀ = 90^° and ϕ₀ = 120^° are employed to demonstrate our deceleration scheme and study its slowing results (see Figs. 2, 3, and later Fig. 5 ). It can be found from Fig. 4(b) that with the increase of ϕ₀, the needed deceleration-stage number m will be decreased gradually when P₀ = 400W, and when the synchronous phase angle ϕ₀ is given, the smaller the velocity reduced amount Δv_latt of the moving lattice is, the less the needed deceleration-stage number will be. In particular, when ϕ₀ = 150^° and Δv_latt = 40m/s, the needed deceleration-stage number is only m = 6500. However, we know from our simulated results that the larger the velocity reduced amount Δv_latt is, the more the obtained cold molecules in the slowest packet will be. So Δv_latt = 80m/s is used in our simulations (see Figs. 2, 3, and later Fig. 5). We can also find from Fig. 4(c) that the larger the laser power P₀ is, and the smaller the velocity reduced amount Δv_latt is, the less the needed deceleration-stage number m will be.

 

Fig. 4 Dependences of the deceleration stage number m that needs to slow a subsonic CH₄ molecular beam to zero on (a) the velocity reduced amount Δv_latt of the traveling lattice, (b) the synchronous phase angle ϕ₀, and (c) the input power P₀ of lattice beams. Other simulated parameters are given in the plots and text.

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Fig. 5 Influence of the rising and falling times of laser pulses (t_r and t_f) on (a) final most-probable velocity v_l (where lines are fitted curves and points are simulated results), (c) fraction n of cold molecules in slowed packet and temperature T_l of it. Variation of (b) turning point of t_r (and t_f) with the synchronous phase angle ϕ₀ and (d) optical dipole force experienced by molecules on the z axis in one spatial period of traveling lattice. Other simulated parameters are given in the plots and text.

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These results show that the deceleration-stage number m that needs to slow a subsonic molecular beam from 240/s to 0m/s is not only relative to the velocity reduced amount Δv_latt of the moving lattice and its power P₀, but also to the synchronous phase angle ϕ₀, and when the velocity reduced amount Δv_latt is smaller, and the synchronous phase angle ϕ₀ is larger, the needed deceleration-stage number m is less, but in this case, the obtainable cold molecules are less. So in order to obtain more and colder molecules, we should make a compromising consideration for Δv_latt and ϕ₀.

4. Influences induced by rising and falling times of laser pulses

As we known, in a real experiment, it is not easy to obtain a top-hat temporal shape of laser pulses by using the EOM. Usually, the laser pulses are time-dependent. So we should take the time-dependent shape of each laser pulse (formed by EOM) into account instead of the top-hat temporal one. Assuming that it needs a rising time t_r of laser lattice pulse to reach its maximum intensity and a falling time t_f of laser pulse to zero. We investigate the influence of t_r and t_f on the slowing results, including the final central velocity, the fraction and the temperature of slowed molecules provided that t_r is equal to t_f, and the simulated results are shown in Fig. 5. We can find from Fig. 5(a) that with the increase of t_r and t_f, the final velocity v_l of the slowed packet will first drop and then rise, and for different synchronous phase angles such as ϕ₀ = 60^°, ϕ₀ = 90^°, ϕ₀ = 120^° and ϕ₀ = 150^°, and the turning points appear at t_r = t_f = 4.0ns, 1.5ns, 0.8ns and 0.4ns respectively. In order to explain these results, we use ϕ₀ = 90^° as an example [see Fig. 5(d)]. Figure 5(d) shows the optical dipole force experienced by molecules on the z axis in one spatial period of traveling lattice. The region of 0^°<ϕ<180^° (the rear half of each lattice cell) is the decelerating phase-angle region for molecules in the red-detuned lattice. For ϕ₀ = 90^°, if the laser pulses have top-hat temporal shape, the lattice light can be switched on completely when the synchronized molecules arrive at ϕ = 0^° and can also be switched off 100% when they reach the position of ϕ = 90^°. However, for the laser pulses with a time-dependent intensity, the lattice field cannot reach its maximum intensity immediately due to the existence of t_r and can also not be reduced to zero in time when the synchronized molecules arrive at ϕ = 90^° owing to t_f. Here we assume that a group of molecules travel from point A to B within the time t_r and from A’ to B’ within t_f, as shown in Fig. 5(d). In this case, when the molecules move from A to B, the deceleration effect of the molecules owing to t_r will be poorer than that in the case of the rising time t_r = 0 (i.e., the top-hat temporal shape), while the molecules are in the motion of A’→B’, they will still be exposed to the interaction of the lattice field and still be decelerated due to the existence of t_f, and then the slowing effect due to t_f will be far better than that in the case of the falling time t_f = 0 (i.e., the top-hat temporal shape). However, it is clear from Fig. 5(d) that the optical dipole force F_z experienced by the molecules in the motion of A’→B’ is larger than that in A→B, That is, an appropriate t_r (and t_f) will make the deceleration effect increased by t_f be greater than the loss of the slowing effect resulted from t_r, so a pair of appropriate t_r and t_f will lead to a better slowing effect than that produced by the top-hat temporal laser pulses. However, when t_r and t_f are too long, the deceleration effect increased by t_f will be equal to the loss of the slowing effect resulted from t_r. Also, they will shallow the optical lattice potential significantly and even lead to the acceleration of molecules in the next accelerating phase angle region of −180^°<ϕ<0^°. Thus the slowing effect will be declined seriously.

It can also be understood easily that the larger the synchronous phase angle ϕ₀ is, the shorter the rising and falling times (t_r and t_f) of laser pulses contributing to the slowing of molecules will be. Thus the turning point with ϕ₀ = 150^° appears earlier than one with other synchronous phase angles such as ϕ₀ = 120^°, ϕ₀ = 90^° and ϕ₀ = 60^° on the horizontal axis. As shown in Fig. 5(b), with the increase of ϕ₀, the turning point of t_r and t_f will be lower and lower.

Figure 5(c) shows the influence of t_r and t_f on the temperature T_l of the slowed packet and fraction n of cold molecules when ϕ₀ = 90^° and the final most-probable velocity v_l of slowed molecules is below 1m/s. It is clear that with increase of t_r and t_f, both T_l and n decrease gradually. This is because t_r and t_f will result in more shallow optical potentials in the deceleration process, so that less cold molecules can be trapped and T_l of the slowed packet will also be lower.

5. Discussion

In 2009, our group proposed a multistage optical Stark decelerator using a 1D quasi-cw static optical lattice for a pulsed supersonic (or subsonic) molecular beam [15], and studied its deceleration effects for a subsonic CH₄ beam and a supersonic NO beam by using 1D Monte-Carlo simulations. In this section, we will compare the deceleration results of this static-lattice decelerator with that of our proposed traveling lattice one for the same subsonic CH₄ beam, and the simulated results are shown in Fig. 6 . In our simulations and comparison, we use the same lattice parameters and incident pulsed subsonic CH₄ beam, and for the quasi-cw traveling lattice, the velocity reduced amount of the traveling lattice is chosen as Δv_latt = 80m/s (i.e., its initial velocity is v_latt0 = 160m/s). It is clear from Fig. 6 that the velocity reduction of subsonic molecular beam produced by the traveling optical lattice is much larger than that by the static one. In particular, within the deceleration stage number of m = 0-16000, the traveling lattice can be used to slow a subsonic CH₄ beam from 240m/s to about 0.9m/s, but the static one can only be used to slow the CH₄ beam from 240m/s to ~195.5m/s.

 

Fig. 6 Simulated final central velocities of slowed CH₄ molecular packets decelerated by a static or traveling optical lattice, respectively. Other simulated parameters are given in the plots and text.

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That means it needs much more deceleration stages for slowing a subsonic molecular beam to standstill by using a quasi-cw static optical lattice than that by using a quasi-cw traveling one. This shows that our traveling optical lattice can be used to realize more efficient deceleration than the static one. This is because the relative velocity between the moving lattice and subsonic molecular beam in our proposed traveling-lattice decelerator is much lower than that in the static-lattice decelerator, while the interaction time of the slowed molecules with the traveling lattice is much longer. Then molecules can obtain better deceleration effects when the moving optical lattice is employed. Also, Fig. 6 shows that the traveling lattice with a length of only about 1.38cm (i.e., a deceleration stage of 16000) can be used to slow a subsonic molecular beam from 240m/s to ~0m/s, but for the static one, m = 40000 (i.e., a lattice length of about 2.13cm) is needed to slow a subsonic beam from 240m/s to ~0m/s.

In 2002, Barker’s group put forward a general scheme for creating stationary cold molecules by rapid deceleration of supersonically cooled molecules in a high-intensity pulsed optical lattice [12]. Next, we will compare the efficiency of their scheme using a chirped decelerating optical lattice with our proposal using a traveling lattice with a constant velocity (i.e., the frequency chirp δ = 0) by 1D Monte-Carlo simulation, and the results are shown in Fig. 7 . In the simulations, we take molecule I₂ as an example and the pulsed I₂ molecular beam has the same initial central velocity of 400m/s and longitudinal temperature of 1.0K. Figure 7(a) shows the results of our scheme when I₀ = 6.4 × 10⁷Wcm⁻², ϕ₀ = 90^° and Δv_latt = 100m/s.

 

Fig. 7 Comparison of the deceleration effects of (a) our traveling lattice with (b) a chirped optical lattice. Other simulated parameters are shown in the plots and text.

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When the slowed packet is decelerated to a velocity of 160m/s, the corresponding temperature and the fraction n of cold molecules are T_l = 6.08μK and n = 0.15%, respectively. Figure 7(b) shows the deceleration results of a chirped optical lattice when I₀ = 12 × 10⁹Wcm⁻², ψ = 0.735, δ = 5.28 × 10¹⁵rad/s and q = 1.56 × 10⁷/m which are the same as ones used in Ref. [12]. When molecules are decelerated to the same velocity 160m/s, the velocity width of the slowed packet is very broad, the corresponding T_l and n are 453.3mK and 11.49%, respectively. These show that the number of cold molecules in the chirped optical-lattice slowing scheme is almost two orders of magnitude higher than that of our slowing one, but the temperature T_l of the slowed packet in our scheme is much lower than that in the chirped optical-lattice scheme by about five orders of magnitude owing to the bunching effect of our multistage optical Stark decelerator.

6. Summary

In this paper, we have proposed and demonstrated a promising scheme to slow a pulsed subsonic molecular beam by using a far-off-resonance, red-detuned, quasi-cw traveling optical lattice with an intensity of ~10⁷Wcm⁻². By using 3D Monte-Carlo method, we have studied the slowing results (including the final central velocity v_l of the slowed packet and its temperature as well as the fraction of cold molecules) of our moving optical lattice for different synchronous phase angle. We have also investigated the dependences of the deceleration-stage number m needed to slow a subsonic molecular beam from 240m/s to zero on the velocity reduced amount Δv_latt of the traveling lattice and its power as well as the synchronous phase angle ϕ₀. Our study shows that a far-off-resonance, red-detuned, quasi-cw traveling optical lattice, as a multistage optical Stark decelerator with a length of only short than 1.4cm, can be used to efficiently slow a subsonic molecular beam from 240m/s to zero, and a cold molecular packet with a temperature of a few μK can be obtained, but the corresponding fraction of cold molecules will be 10⁻⁶~10⁻⁷, which strongly depends on our choosing ϕ₀. This shows that when the molecular number in an incident molecular pulse is equal to 10¹², our obtainable number of cold molecules in the slowed packet can reach 10⁵~10⁶. We have also found that the larger the synchronous phase angle ϕ₀ is, the lower the final central velocity of the slowed packet is, and the less the cold molecular number is, and then the lower the temperature of cold molecules is. In addition, a larger ϕ₀, a smaller Δv_latt, or a higher P₀ will result in a less deceleration stage number m that needs to slow a subsonic molecular beam to zero and a shorter quasi-cw, traveling optical lattice. Besides, we have taken the time-dependent intensity of laser pulses into account and investigated its influence on the slowing effects. Our study shows that (1) with the increase of the rising time t_r and the falling time t_f, the final velocity v_l of the slowed packet will first drop and then rise, that is, there is an optimal turning point for each synchronous phase angle. (2) The larger the synchronous phase angle is, the shorter the rising and falling times (t_r and t_f) of laser pulses contributing to the slowing of molecules will be. (3) A pair of appropriate t_r and t_f will lead to a better slowing effect than that produced by the top-hat temporal laser pulses. Moreover, we have compared the slowing results of our proposed traveling-lattice decelerator with the static-lattice one, and found that the traveling-lattice molecular decelerator is more efficient than the static-lattice one under the same slowing conditions. Finally, we have compared the slowing performances of our scheme with a chirped optical lattice proposed in Ref. [12], and found that our multistage decelerator can obtain a cooler molecular packet due to the bunching effect, but a chirped lattice scheme can get more cold molecules.

Our proposed traveling optical-lattice decelerator cannot only be used to efficiently slow all kinds of neutral molecules, including polar molecules, paramagnetic molecules and those molecules without a permanent electric or magnetic dipole moment, but also to obtain an ultracold molecular packet with a temperature of 4.2μK and a number of 10⁵~10⁶. So such a multistage optical Stark decelerator using a traveling lattice and its obtainable ultracold molecules have some important applications in the fields of cold molecular physics, cold molecular spectrum, cold molecular collisions and cold chemistry, precise measurement, quantum computing and its information processing, and cold molecular lithography, even it can be used to realize an all-optical, chemically-stabled molecular Bose-Einstein condensates (BEC) by using an optical-potential evaporative cooling [22], all-optical molecular Fermi quantum degeneration (FQNG), quantum and nonlinear molecule optics, and so on.