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Decelerating a pulsed subsonic molecular beam by a quasi-cw optical lattice: 3D Monte-Carlo simulations
Optics Communications 287 (2013) 128–136
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Decelerating a pulsed subsonic molecular beam by a quasi-cw optical lattice: 3D Monte-Carlo simulations Xiang Ji, Yaling Yin, Qi Zhou, Zhenxia Wang, Jianping Yin n State Key Laboratory of Precise Spectroscopy, Department of Physics, East China Normal University, Shanghai 200062, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 February 2012 Received in revised form 6 May 2012 Accepted 18 June 2012 Available online 24 September 2012
We propose a practical scheme to realize the deceleration of a pulsed subsonic molecular beam by using a multistage optical Stark decelerator (i.e., a 1D quasi-cw, cavity-enhanced optical lattice), which is composed of two nearly counter-propagating, time-varying, red-detuned, light fields with an intensity of 108 W cm 2 in a folded ring resonator. The dependences of the molecular slowing effects on the synchronous phase angle, the deceleration-stage number and the initial central velocity of incident molecular beam as well as the cavity enhancement factor and its cavity length are investigated by the 3D Monte-Carlo method. Our study shows that the proposed decelerator cannot only be used to slow a pulsed subsonic beam from 240 m/s to standstill, but also to obtain a ultracold molecular packet with a temperature of a few mK due to the bunching effect of our multistage optical Stark decelerator, and the corresponding fraction of cold molecules is 10 4–10 6, which strongly depends on the synchronous phase angle, the cavity enhancement factor and the initial central velocity of incident molecular beam, etc.. & 2012 Elsevier B.V. All rights reserved.
Keywords: Atomic and molecular physics Laser cooling Laser trapping Stark effect Cavity enhancement
1. Introduction It is well known that cold or ultracold neutral molecules would offer many opportunities for the basic researches and have some important applications in the fields of high-resolution cold molecular spectroscopy and precision measurements [1,2], cold chemistry and cold collisions [3], quantum computing and quantum information processing [4], and so on. In recent years, some promising approaches, such as buffer gas cooling [5], electrostatic Stark decelerator for polar molecules [6], a magnetic Zeeman slower for paramagnetic molecules [7], and optical Stark decelerator for all kinds of neutral molecules [8], low-pass velocity filter [9–11], laser cooling [12] and so on, have been proposed and developed to prepare cold or ultracold molecules. In particular, a single-stage (or multistage) optical Stark decelerator using a far-offresonance, red-detuned, moving (or static), pulsed (or quasi-cw) optical lattice has attracted much attention of people [8,13–15]. In 2002, Barker and Shneider put forward a traveling optical lattice created by high-power pulsed lasers to decelerate molecules [16]. 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 22 K (i.e., a single-stage optical Stark decelerator) [8], and Ramirez-Serrano et al. experimentally
n
Corresponding author. Tel.: þ86 02162232650; fax: þ 86 02162232056. E-mail address: [email protected] (J. Yin).
0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.06.092
demonstrated a similar single-stage optical Stark deceleration for a supersonic H2 molecular beam with a static pulse optical lattice [13]. In 2009, Momose and Kuma proposed a multistage optical Stark decelerator scheme for a pre-cooling pulsed molecular beam with an initial velocity of 13–15 m/s and a temperature of 100– 500 mK by using a quasi-cw, near-resonant and cavity-enhanced IR optical lattice [14]. Also, our group proposed a useful and promising multistage optical Stark decelerator for a supersonic (or subsonic) molecular beam with an initial velocity of 240–400 m/s by using a far-off-resonance, red-detuned, quasi-cw optical lattice with a waist of 20 mm (which is equal to the experimental parameter in Ref [8]) and a power of 264 W, and studied its deceleration effects for a subsonic CH4 beam and a supersonic NO beam by using 1D MonteCarlo simulations [15]. It is clear that there are three problems in our proposed scheme [15]: (1) we did not consider the initial longitudinal and transverse spatial distributions of an incident pulsed molecular beam in our 1D simulations; (2) the laser waist (20 mm) is too small to obtain a longer lattice and more cold molecules because the waist (100–500 mm) of the supersonic beam is far larger than the laser waist (20 mm) and (3) we did not consider the transverse velocity distribution of the incident molecular beam and its transverse molecular loss in our 1D simulations [15], so the obtained fraction of cold molecules in our 1D simulations does not have any reference value. To solve these problems, in this paper, we propose a desirable multistage optical Stark decelerator for a subsonic (or supersonic) molecular beam by using a far-off-resonance, red-detuned, quasi-cw and cavity-enhanced
X. Ji et al. / Optics Communications 287 (2013) 128–136
optical lattice with a diameter of 200 mm and an incident power of 200 W, and demonstrate its feasibility and study its decelerating effects for a subsonic CH4 beam by using 3D Monte-Carlo simulations, and obtain some new and valuable results. This paper is organized as follows: In Section 2, we propose a practical scheme to slow a subsonic molecular beam and introduce its slowing principle. In Section 3, we briefly introduce our cavity-enhancing scheme and its theory. In Sections 4 and 5, we study the slowing effects of our optical Stark decelerator, and discuss the influence of cavity enhancement factor on the slowing results, including the dependences of the final central (i.e., the most-probable) velocity, the fraction of cold molecules and its temperature on the synchronous phase angle, the deceleration-stage number, the initial central velocity of incident pulsed molecular beam, and the length of the ring cavity employed in our scheme, and obtain some new and important results. In Section 6, we compare our scheme with the chirped optical lattice [16] by 3D Monte-Carlo simulations. Some main results and conclusions are summarized in the final section.
2. Decelerator 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. In our scheme, our proposed multistage Stark decelerator is composed of a red-detuned, quasi-cw optical lattice with a diameter of 200 mm in a folded ring cavity, and the optical lattice is created by two nearly counter-propagating, far-offresonance, single-frequency, linear-polarized infrared lasers with a same wavelength of l ¼1064 nm. In Fig. 1, b is the angle between these two laser beams and is about 1721 in our scheme [17], and the spatial period L of the optical lattice is nearly equal to half of the laser wavelength l. When molecules travel along the z axis, they will experience an optical dipole interaction potential in the red-detuned optical lattice field as follows: U ¼ ð2aI=e0 cÞcos2 ðkz=2Þ, 2
2
ð1Þ
129
According to Eq. (1), 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 larger than the transverse lattice potential will escape from the lattice. In the longitudinal direction (i.e., the z axis), if molecules enter the optical lattice from its antinode, 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 singlestage decelerating process, the slowed molecules will lose their kinetic energy, which is equal to the corresponding lattice potential energy. For convenient sake, the longitudinal position (z) of the slowed molecules in the optical lattice is defined as a synchronous phase angle f, where f ¼z/L 3601. The point of z ¼0 (i.e., f ¼01) is set at the antinode of the lattice (i.e., the minimum of the optical potential) due to its red-detuned. It is clear from Eq. (1) that the region of 01o f o1801 (the rear half of each lattice cell) is the decelerating phase-angle region for molecules in the red-detuned lattice, while the region of 1801 o f o01 is the accelerating one. When the synchronized molecule reaches the antinode (f ¼01) of the lattice, the lattice light field should be switched on in time. As soon as it arrives at the position of f ¼ f0 (where f0 is the synchronous phase angle of the synchronized molecule, usually we choose 01o f0 o1801), the lattice should be switched off rapidly. Then in the process of 01o f r f0, the synchronized molecules will be slowed effectively. For the red-detuned optical lattice, if this process can be repeated continuously for m times, the subsonic molecular beam will be efficiently slowed to a few 10 m/s, even to standstill. 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 supersonic (or subsonic) molecular beam by using a strict time-sequence synchronous control system and an electro-optic modulator (EOM).
)/w20]
is the intensity of a Gaussian laser where I¼I0exp[ 2(x þy beam, w0 is the waist (half-width at 1/e2 maximum intensity) of the laser beam, a is the averaged molecular polarizability, e0 is the permittivity in free space, c is the speed of light in vacuum, and k¼(4psin(b/2))/l is the wave number of this lattice. For CH4 molecule, we have a ¼2.9 10 40Cm2/V.
3. Cavity enhancement scheme and theory In order to product a time-varying lattice field, an electro-optic modulator (EOM) is used in our scheme. Meanwhile the EOM
Fig. 1. Schematic diagram of a multistage optical Stark decelerator, which is created by two nearly counter-propagating, far-off-resonance, red-detuned laser beams (originated by one incident laser) and located at the center of a folded ring cavity composed of four mirrors (M1, M2, M3 and M4). A pulsed subsonic molecular beam propagates and enters the quasi-cw lattice along the z axis. The inset figure is the enlarged lattice. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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should be sustained by a strict time-sequence synchronous and control signal so as to modulate the intensities of light fields precisely. When a lattice period is L ¼0.532 mm and an initial central velocity of incident molecular beam is v0 ¼240 m/s, the maximum initial modulation frequency of the quasi-cw lattice is about 451 MHz. In addition, a resonant cavity is applied to enhance the intensity of our lattice field and its optical potential. In our scheme, four mirrors (M1, M2, M3 and M4) with different reflectivity are used to form a folded ring resonator with a length of 1.5 cm, as shown in Fig. 1. In our scheme, the enhancement factor Ecav in a folded ring (i.e., traveling-wave) cavity can be expressed by [18] ffi pffiffiffi ð1R Þ2 ðpffiffiffiffiffi R0 R expðsdÞÞ 0 pffiffiffiffiffiffiffiffi ð2Þ Ecav , ð1 R0 R expðsdÞÞ5 where R0 is the power reflection coefficient of the mirror M1, R is the power reflection coefficient of other three mirrors (M2, M3 and M4), s is the absorption coefficient inside the cavity, and d is the optical length of this folded ring cavity. When R0 ¼0.6561 and R¼0.9905, the enhancement factor of the resonator is Ecav E80. This shows that the power of the light field in the cavity can reach 1.6 104 W as the input laser power is P0 ¼200 W. According to the theory of optical resonant cavities [19], the rising time (i.e., building time) of light field in the cavity is given by
tr ¼ nround
2Lcav , c
ð3Þ
and nround ¼ logR1 R2 ð1KÞ,
ð4Þ
where K is the percentage of the maximum value of light field energy in the cavity, R1 and R2 are mirror reflectivities, Lcav is the cavity length, and nround is the number of round trips needed for photons to establish this light field in the cavity. The photon lifetime (i.e., the falling time of the lattice field) in the cavity is
tf ¼
2Lcav =c , 1R1 R2
ð5Þ
where 2Lcav/c is the time required for photons to make one round trip in the cavity. So when we take tr ¼ 0.425 ns, the field energy inside the cavity will reach 60% of its maximum value, and the falling time of the lattice field (i.e., the photon lifetime) in the cavity is tf ¼0.516 ns after the light field is turned off. Assuming that the wavelength of incident laser field and the initial central velocity of a pulsed subsonic (supersonic) molecular beam are l ¼1.064 mm and v0 ¼240 m/s (or 360 m/s) respectively, the transit time for the synchronized molecule to pass a single cell of the lattice is about ttran ¼ 2.22 ns. In our scheme, the durations of the switch-on and switch-off signal of a single laser pulse are denoted as ton and toff respectively. Obviously, it is feasible for the cavity-enhanced method to be applied in our scheme, because ttran ( Eton þtoff) is much larger than tr ( ¼0.425 ns) and tf ( ¼0.516 ns). That means that there is enough time for the lattice field to finish all of processes, including the rising and falling of light field in the cavity as well as decelerating of molecules. However, the existence of tr and tf will impact on the deceleration effects of molecules exactly. In the next section, we will show the influence of them on the slowing effects. In Section 5, we will demonstrate and study the deceleration effects of our optical Stark decelerator for a subsonic molecular beam, including the final central velocity of cold molecular packet and its temperature Tl as well as the fraction n of cold molecules (n¼N/N0, where N and N0 are the molecular numbers in the slowed molecular packet and the incident molecular pulse, respectively), which will be affected by cavity enhancement factor Ecav. Additionally, in order to
know the influence of tr and tf on the slowing effects, we also compare the slowing results (see the Section 4) between the ideal (ignoring the influence of tr and tf) and real (considering the influence of tr and tf) situations.
4. Results of 3D Monte-Carlo simulations The slowing effect of our multistage optical-lattice decelerator was demonstrated by 1D Monte-Carlo simulations in 2009 [15]. However, the transverse loss of molecules escaping from the lattice field couldn’t be taken in account. In order to obtain some reliable (or valuable) slowing results, here we will carry out 3D Monte-Carlo simulations for 1D molecular slowing in the longitudinal direction and 2D trapping in the transverse one. In our simulations, the pulsed subsonic (or supersonic) methane (CH4) molecular beam with an initial central velocity of 240 m/s and a longitudinal temperature of 0.5 K is used and traveled along the z axis, as shown in Fig. 1. We assume that the transverse and longitudinal spatial distributions of incident pulsed molecular beam are Gaussian ones, and their spatial sizes are 0.1 mm and 1 mm, respectively. Also, the transverse and longitudinal velocity distributions of incident pulsed molecular beam are Gaussian ones, and their velocity half widths are 2.3 m/s and 40 m/s respectively, and the initial position and velocity of each simulated molecule are random. A single-frequency, singlemode, linear-polarized ytterbium-doped fiber laser with a power of 200 W and a wavelength of 1064 nm [20], as an input laser beam, is used to create our optical-lattice decelerator. In the slowing process, if the transverse position of a molecule is larger than the beam waist w0 of the optical lattice, it will be considered as a molecular loss in our 3D Monte-Carlo simulations. The final velocity distributions [21] of the slowed molecules are detected by a micro-channel plate (MCP), which is placed at the position of about 20 cm away from the start point of the lattice on the z axis. In addition, when the length of a folded ring cavity used in our scheme is about 1.5 cm, and the reflectivity of four mirrors are R1 ¼81% and R2 ¼R3 ¼R4 ¼99.525%, the corresponding enhancement factor is Ecav E80. First of all, we study the relationship between the deceleration effects and the deceleration stage number m, and the simulated results are shown in Fig. 2. The horizontal axis represents the longitudinal velocity of slowed molecules, and the vertical one represents the relative molecular number (i.e., the ratio of the molecular number at each longitudinal velocity to one at the initial most-probable velocity of incident molecular beam). Here one small peak on the longitudinal velocity distribution (i.e., on the horizontal axis) is pointed out by a red circle due to its small size, which is the slowed cold molecular packet and shows that the incident CH4 molecular beam has been decelerated successfully. To clearly show such a small slowed packet, we enlarge its velocity distribution (see the left enlarged peak). Fig. 2(a)– (c) shows the longitudinal velocity distributions of the slowed molecular packet for f0 ¼901 and the slowing stage number m¼3060, 4060 and 6060. It is clear from Fig. 2 that with the increase of the deceleration stages, the final central velocity of the slowed packet will be decreased gradually when the synchronous phase angle keeps invariant (f0 ¼ 901). In particular, when the slowing stage m is increased from 3060 to 6060, the final central velocity of the slowed packet will be efficiently decreased from 176.47 m/s to 78.67 m/s, and the corresponding temperature will be slightly increased from 56.85 mK to 73.68 mK, and the fraction n of cold molecules in the slowed packet will be slightly decreased from 5.13 10 4 to 4.81 10 4, which is far smaller than 2% in 1D Monte-Carlo simulations [15].
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Fig. 2. Simulated longitudinal velocity distributions of CH4 molecules at the outlet of the decelerator for the synchronous phase angle f0 ¼ 901 and the deceleration-stage number m¼3060, 4060 and 6060. Other simulation parameters are given in the plots and text.
Secondly, we investigate the relationship between the deceleration effects and the synchronous phase angles, and the simulated results are shown in Fig. 3. We can see from Fig. 3 that with the increase of the synchronous phase angle f0, the final central velocity of incident subsonic CH4 beam will be efficiently decreased, and the corresponding temperature of the slowed CH4 packet will be lowered to sub-mK even reach to a few mK. In particular, with the increase of f0 from 301 to 1501, the final central velocity of the slowed molecular packet will be decreased from 225.7 m/s to 11.55 m/s, and its temperature will be lowed from 0.5 K to about 1.6 mK. However, the velocity reduction of the slowed packet and the lowing of its temperature will accompany with the great loss of cold molecular number in the slowed packet, and when f0 ¼1501, the fraction of cold CH4 molecules is 1.0 10 6, which is far lower than 4.0 10 3 in the 1D Monte-Carlo simulation [15]. This shows that (1) the results from 1D Monte-Carlo simulations are not reliable due to without considering 2D transverse molecular loss; (2) 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 [22]. This is because the cold molecular number in the slowed packet is proportional to the area of the phase stability space; (3) 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. Also, we can find that when f0 ¼ 1501, it only needs 4060 deceleration stages to slow
CH4 molecules to 11.55 m/s, and the corresponding length of our optical lattice is only about 2.16 mm. Additionally, to know the influence of tr and tf on the decelerating effects, we study the dependences of the final central velocity vl of the slowed packet and its temperature as well as the fraction of cold molecules on the synchronous phase angle for the enhancement factor Ecav E80 and the deceleration-stage number m¼3680, and compare the simulated results between the ideal situation (ignoring the influence of tr, tf) and real one (considering influence of tr, tf), and the simulated results are shown in Fig. 4. We can find from Fig. 4(a) that with the increase of the synchronous phase angle f0, the final central velocity of the slowed packet will be reduced effectively, and the final central velocity in the real case becomes larger than one in the ideal one gradually. Additionally, we can see from Fig. 4(b) and (c) that with the increase of the synchronous phase angle f0, the fraction of cold molecules in the slowed packet and its temperature will be reduced due to the reduction of the phase-space area, and both the fraction of cold molecules and its temperature in the ideal case are always higher than ones in the real one. This is because the existence of tr and tf will reduce the effective potential-well depth of the optical lattice slightly, which will result in the decrease of the molecular number of the slowed packet and a lower temperature in the real case, as shown in Fig. 4(b) and (c). These investigations show that (1) a far-off-resonance, red-detuned, quasi-cw optical lattice, as a multistage optical Stark decelerator, can
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Fig. 3. Simulated longitudinal velocity distributions of CH4 molecules at the outlet of the decelerator for the deceleration-stage number m¼ 4060 and the synchronous phase angle f0 ¼ 301, 601, 901, 1201, and 1501. Other simulation parameters are given in the plots and text.
be used to efficiently slow a subsonic CH4 beam from 240 to 10 m/s, even to zero; (2) the temperature of the slowed cold molecular packet can be lowed to sub-mK, even reach to a few mK, but the fraction of cold molecules will be equal to about 1.0 10 6, which is far smaller than one (4.0 10 3) in 1D simulations. If the molecular number in an incident CH4 pulse is about 1012, the number of cold molecules in a slowed packet is about 106 and (3) the rising time (tr) and falling time (tf) of the lattice field in the cavity will markedly influence the slowing effects for molecules, including the final central velocity, the temperature and the fraction of cold molecules, etc. In final, we investigate the dependences of the final central velocity vl of slowed packet on both the initial central velocity v0
of incident molecular beam and the length Lcav of the folded ring resonator, and the results are shown in Fig. 5. We can find from Fig. 5(a) that with the increase of v0, the final central velocity vl of slowed packet will be increased. In particular, when f0 ¼901 and Ecav ¼ 80, we need the deceleration stages of m¼6850 to slow a pulsed molecular beam with an initial central velocity of v0 ¼240 m/s (or 360 m/s) to 7.34 m/s (or 261.54 m/s). While f0 ¼1201 and Ecav ¼ 80, we only need m¼4830 to slow the molecular beam from v0 ¼240 m/s (or 360 m/s) to 10.28 m/s (or 266.69 m/s). Also, it is clear from Fig. 5(b) that with the increase of the cavity length, the final central velocity vl of slowed packet will first drop and then rise, and for different synchronous phase angles such as f0 ¼ 901 and f0 ¼1201, the turning points appear at Lcav ¼3.5 cm and
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Fig. 4. Comparison of slowing effects between the ideal situation (ignoring the influence of tr and tf) and the real one (considering influence of tr and tf) for m¼3680 and Ecav ¼80. Other simulation parameters are given in the plots and text.
Fig. 5. Dependence of the final central velocity of slowed molecular packet on (a) the initial central velocity of incident pulsed CH4 beam and (b) the length of the folded ring cavity with Ecav ¼ 80. (c) Variation of optical dipole force experienced by molecules on the z axis in one spatial period of lattice. Other simulation parameters are given in the plots.
2.5 cm, respectively. In order to explain these results, we choose f0 ¼901 as an example [see Fig. 5(c)]. Fig. 5(c) shows the optical dipole force experienced by molecules on the z axis in one spatial period of the lattice. Here we assume that a group of molecules travel from point A to B within the time tr and from A0 to B0 within tf, as shown in Fig. 5(c). However, it is clear from Fig. 5(c) that the optical dipole force Fz experienced by the molecules in the motion of A0 -B0 is larger than that in A-B. That is, an appropriate tr (and tf) will
make the deceleration effect increased by tf be greater than the loss of slowing effect resulted from tr, so a pair of appropriate tr and tf will lead to a better slowing effect. According to Eqs.(3) and (5), we can find easily that a longer cavity length Lcav will lead to a longer tr and tf. When tr and tf are too long, the deceleration effect increased by tf will be equal to the loss of the slowing effect resulted from tr. Also, they will shallow the optical lattice potential significantly and even lead to the acceleration of molecules in the next
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accelerating phase angle region of 1801o f o01. Thus the slowing effect will be declined seriously. It can also be understood easily that the larger the synchronous phase angle f0 is, the shorter tr and tf (corresponding to Lcav) contributing to the slowing of molecules will be. So the turning point for f0 ¼1201 appears earlier than one for f0 ¼901 on the horizontal axis in Fig. 5(b). These results show that the slowing effects strongly depend on both the initial central velocity of incident molecular beam and the length of the ring cavity.
5. Influence of cavity enhancement factor Ecav In our scheme, it is clear that a larger or smaller (i.e., an improper) cavity enhancement factor Ecav will also influence the slowing effects for molecules. For this, we study the influences of the cavity-enhancement factor on the decelerating effects, and the simulated results are shown in Fig. 6. It can be seen from Fig. 6(a) that when Ecav rises from 60 to 140, the final central velocity of cold molecular packet slowed by the enhanced optical lattice in the cavity will be lower and lower. This is because a larger Ecav leads to a deeper potential well in the longitudinal direction so that molecules can be slowed to a lower velocity vl. Afterwards, we also explore the influence of Ecav on both the temperature of the slowed molecular packet and its fraction of cold molecules, and the results are shown in Fig. 6(b) and (c). Fig. 6(b) shows that for the same Ecav, we can get more slowed molecules when f0 ¼901 than that with f0 ¼1201, and with the increase of Ecav, n will first rise and then drop. As we know, a larger Ecav results in longer falling and rising times (tr and tf) of the optical lattice field in the folded ring cavity and higher lattice intensity. An optical lattice with higher intensity can capture more molecules in the transverse direction and slow more molecules in the longitudinal direction. And longer tr and tf will lead to the smaller efficient lattice-potential depth within the transit time ttran for synchronized molecule to pass a single cell of lattice, especially at
the beginning of deceleration when ttran is shorter, and then more molecules with larger transverse velocities will escape from the optical lattice in the radial direction. For f0 ¼901 and m¼3850, when Ecav rises from 60 to 100, the efficient lattice potential in the cavity also increases due to the relatively appropriate tr and tf, and then more molecules can be captured and slowed. However, when Ecav rises from 100 to 140, the loss of molecules will be increased gradually if even the maximum of optical lattice intensity is enhanced evidently, because much longer (i.e., improper) tr and tf will result in more shallow virtual lattice-potential depth in the cavity. So Ecav ¼100 becomes an optimal point for us to obtain the most cold molecules when f0 ¼901. But for f0 ¼1201, the turning point appears earlier at Ecav ¼70. It can be explained that the larger the synchronous phase angle is, the less the cold molecular number in the slowed packet is, and then the more shallow optical potential well is needed to decelerate molecules. Therefore the optimal point of Ecav for f0 ¼1201 will be lower. Fig. 6(c) shows that with the increase of Ecav, the longitudinal temperature Tl of the slowed molecular packet will be higher and higher. This is because that a higher Ecav, corresponding to a deeper longitudinal potential well, can decelerate more molecules in the z axis, which results in a broader longitudinal velocity distribution of the slowed molecular packet, and then the temperature of the slowed packet will be higher. According to the simulated results mentioned above, we think that the enhancement factor of Ecav ¼80 is more appropriate for our proposed Stark decelerating scheme.
6. Discussion In 2002, Barker and Shneider put forward a general scheme for creating stationary cold molecules by rapid deceleration of supersonically cooled molecules in a high-intensity pulsed optical lattice [16]. Next, we will compare the efficiency of their scheme
Fig. 6. Influence of the cavity enhancement factor on the slowing effects, including (a) the final central velocity, (b) the fraction of cold molecules, and (c) the temperature of the slowed molecular packet. Other simulation parameters are given in the plots.
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Fig. 7. Comparison between the deceleration effects of (a) our scheme and (b) a chirped optical lattice by 3D Monte-Carlo simulations. Other simulated parameters are shown in the plots and text.
using a chirped decelerating optical lattice with our proposed scheme by 3D Monte-Carlo simulations, and the results are shown in Fig. 7. In our simulations, we take molecule I2 as an example, and the pulsed I2 molecular beam has the same initial central velocity of 240 m/s and longitudinal temperature of 1.0 K. Fig. 7(a) shows the deceleration results of our scheme when I0 ¼ 1.02 108 W cm 2, f0 ¼901 and m¼23200. When the slowed packet is decelerated to a velocity of 77.1 m/s, the corresponding temperature and the fraction n of cold molecules are Tl ¼18.15 mK and n¼0.069%, respectively. Fig. 7(b) shows the deceleration results of a chirped optical lattice [16] when I0 ¼ 12 109 W cm 2, d ¼5.28 1015 rad/s (frequency chirp), q¼1.56 107/m (lattice wave number), and c ¼0.735 (dimensionless quantity, associated with d, I0, and q), which are the same as ones used in Ref [16]. When molecules are decelerated to the same velocity 78 m/s, the velocity width of the slowed packet is very broad, the corresponding Tl and n are 66.52 mK and 9.41%, respectively. It is obviously 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. That is because the optical intensity I0 employed in their scheme is much higher than that in ours, which can not only decelerate more molecules in the z axis but also capture more ones in the radial direction. But the temperature Tl of the slowed packet in our scheme is much lower than that in the chirped optical-lattice scheme by more than three orders of magnitude (by larger than 3.67 103 times) owing to the relatively shallow optical potential well and the bunching effect of our multistage optical Stark decelerator.
7. Conclusions In this paper, we have proposed and demonstrated a practical (or desirable) scheme to slow a pulsed subsonic molecular beam
by using a far-off-resonance, red-detuned, quasi-cw optical lattice. In order to enhance the lattice intensity, our multistage optical Stark decelerator is placed in a folded ring cavity with an enhancement factor of 80 and has an intensity of 108 W cm 2. By using 3D Monte-Carlo simulations, we have studied the dependences of the slowing results (including the final central velocity vl of the slowed packet and its temperature as well as the fraction of cold molecules) on the synchronous phase angle, the deceleration-stage number, the initial central velocity of an incident molecular beam and the cavity length, and discussed the influences of the cavity enhancement factor on the slowing results. Our research shows that a far-offresonance, red-detuned, cavity-enhanced, quasi-cw optical lattice, as a multistage optical Stark decelerator, can be used to efficiently slow a subsonic molecular beam from 240 to 10 m/s even to 0, and an ultracold molecular packet with a temperature of sub-mK even a few mK can be obtained, but the corresponding fraction of cold molecules will be 10 4–10 6, which strongly depends on the our choosing f0 and the initial central velocity of incident molecular beam, etc.. 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 106–108. We have also found that the slowing results strongly depend on the cavity enhancement factor Ecav, and we can obtain the better slowing results when Ecav ¼80. However, a more useful or practical slowing scheme should be as follows: we can first use a static (or moving) pulsed optical lattice (i.e., a single-stage optical Stark decelerator) [8,13] to beforehand slow a subsonic molecular beam from 240 m/s to 24 m/s, and then using our quasi-cw, cavity-enhanced optical lattice (i.e., a multistage optical Stark decelerator) to further decelerate a pre-slowed molecular beam from 24 m/s to 0. In this case, starting from an initial central velocity of 24 m/s, the diameter of our quasi-cw lattice can be increased from 0.2 mm to 1 mm, and the diameter of incident molecular beam can also be increased from
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0.2 mm to 1 mm, and then we can obtain more cold molecules. And under these conditions, the deceleration stage number we needed is only about 1270 if we choose f0 ¼1201, and the corresponding length of our multistage optical Stark decelerator (or quasi-cw optical lattice) is only 0.676 mm, and then the angle b (1721) between two laser beams (see Fig. 1) can be further reduced. Because our proposed multistage 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 (which is similar to a single-stage optical-lattice Stark decelerator [8,13]), but also to obtain an ultracold molecular packet with a temperature of a few mK and a number of 106–108, such a multistage optical Stark decelerator and its obtainable ultracold molecules have some important applications in the fields of cold molecular physics, molecule optics, cold molecular spectrum, cold molecular collisions and cold chemistry, precise measurement, quantum computing and its information processing, and cold molecular lithography, even can be used to realize an all-optical, chemically-stabled molecular BoseEinstein condensates (BEC) by using an optical-potential evaporative cooling [23], all-optical molecular Fermi quantum degeneration (FQNG), quantum and nonlinear molecule optics, and so on.
Acknowledgment This work is supported by the National Nature Science Foundation of China under Grant nos. 10674047, 10804031, 10904037, 10974055 and 11034002, the National Key Basic Research and Development Program of China under Grant nos. 2006CB921604 and 2011CB921602, and the Basic Key Program of Shanghai Municipality under Grant no. 07JC14017, and the Shanghai Leading Academic Discipline Project under Grant no. B408.
References [1] J.J. Hudson, B.E. Sauer, M.R. Tarbutt, E.A. Hinds, Physical Review Letters 89 (2002) 023003. [2] D. DeMille, F. Bay, S. Bickman, D. Kawall, D. Krause Jr., S.E. Maxwell, L.R. Hunter, Physical Review A 61 (2000) 052507-1–052507-8. [3] N. Balakrishnan, A. Dalgarno, Chemical Physics Letters 341 (2001) 652. [4] D. DeMille, Physical Review Letters 88 (2002) 067901-1–067901-4. [5] J.D. Weinstein, R. de Carvalho, T. Guillet, B. Friedrich, J.M. Doyle, Nature 395 (1998) 148. [6] M.R. Tarbutt, H.L. Bethlem, J.J. Hudson, V.L. Ryabov, V.A. Ryzhov, B.E. Sauer, G. Meijer, E.A. Hinds, Physical Review Letters 92 (2004) 173002. [7] N. Vanhaecke, U. Meier, M. Andrist, B.H. Meier, F. Merkt, Physical Review A 75 (2007) 031402-1–031402-4. [8] R. Fulton, A.I. Bishop, M.N. Shneider, P.F. Barker, Nature Physics 2 (2006) 465. [9] Yang Liu, Min Yun, Yong Xia, Lianzhong Deng, Jianping Yin, Physical Chemistry Chemical Physics 12 (2010) 745. [10] B. Ghaffari, J.M. Gerton, W.I. McAlexander, K.E. Strecker, D.M. Homan, R.G. Hulet, Physical Review A 60 (1999) 3878. [11] Runqin Liu, Qi Zhou, Yaling Yin, Jianping Yin, Journal of the Optical Society of America B 26 (2009) 1076. [12] E.S. Shuman, J.F. Barry, D. DeMille, Nature 467 (2010) 820. [13] J. Ramirez-Serrano, K.E. Strecker, D.W. Chandler, Physical Chemistry Chemical Physics 8 (2006) 2985. [14] Susumu Kuma, Takamasa Momose, New Journal of Physics 11 (2009) 055023. [15] Yaling Yin, Qi Zhou, Lianzhong Deng, Yong Xia, Jianping Yin, Optics Express 17 (2009) 10706. [16] P.F. Barker, M.N. Shneider, Physical Review A 66 (2002) 065402-1–065402-4. [17] R. Fulton, A.I. Bishop, M.N. Shneider, P.F. Barker, Journal of Physics B: Atomic, Molecular and Optical Physics 39 (2006) S1097. [18] Ma Longsheng, J.L. Hall, IEEE Journal of Quantum Electronics 26 (11) (1990). [19] Joseph ThomasVerdeyen, Laser Electronics, Prentice-Hall, Englewood Cliffs, N. J., 1981, p. 444. [20] Y. Jeong, J. Nilsson, J.K. Sahu, D.N. Payne, R. Horley, L.M.B. Hickey, P.W. Turner, IEEE Journal of Selected Topics in Quantum Electronics 13 (2007) 546. [21] D.C. Clary, Journal of Chemical Physics 84 (1986) 4288. [22] J.R. Bochinski, E.R. Hudson, H.J. Lewandowski, G. Meijer, J. Ye, Physical Review Letters 91 (2003) 243001-1–243001-4. [23] Jianping Yin, Physics Reports 430 (2006) 1.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Decelerating a pulsed subsonic molecular beam by a quasi-cw optical lattice: 3D Monte-Carlo simulations Xiang Ji, Yaling Yin, Qi Zhou, Zhenxia Wang, Jianping Yin n State Key Laboratory of Precise Spectroscopy, Department of Physics, East China Normal University, Shanghai 200062, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 February 2012 Received in revised form 6 May 2012 Accepted 18 June 2012 Available online 24 September 2012
We propose a practical scheme to realize the deceleration of a pulsed subsonic molecular beam by using a multistage optical Stark decelerator (i.e., a 1D quasi-cw, cavity-enhanced optical lattice), which is composed of two nearly counter-propagating, time-varying, red-detuned, light fields with an intensity of 108 W cm 2 in a folded ring resonator. The dependences of the molecular slowing effects on the synchronous phase angle, the deceleration-stage number and the initial central velocity of incident molecular beam as well as the cavity enhancement factor and its cavity length are investigated by the 3D Monte-Carlo method. Our study shows that the proposed decelerator cannot only be used to slow a pulsed subsonic beam from 240 m/s to standstill, but also to obtain a ultracold molecular packet with a temperature of a few mK due to the bunching effect of our multistage optical Stark decelerator, and the corresponding fraction of cold molecules is 10 4–10 6, which strongly depends on the synchronous phase angle, the cavity enhancement factor and the initial central velocity of incident molecular beam, etc.. & 2012 Elsevier B.V. All rights reserved.
Keywords: Atomic and molecular physics Laser cooling Laser trapping Stark effect Cavity enhancement
1. Introduction It is well known that cold or ultracold neutral molecules would offer many opportunities for the basic researches and have some important applications in the fields of high-resolution cold molecular spectroscopy and precision measurements [1,2], cold chemistry and cold collisions [3], quantum computing and quantum information processing [4], and so on. In recent years, some promising approaches, such as buffer gas cooling [5], electrostatic Stark decelerator for polar molecules [6], a magnetic Zeeman slower for paramagnetic molecules [7], and optical Stark decelerator for all kinds of neutral molecules [8], low-pass velocity filter [9–11], laser cooling [12] and so on, have been proposed and developed to prepare cold or ultracold molecules. In particular, a single-stage (or multistage) optical Stark decelerator using a far-offresonance, red-detuned, moving (or static), pulsed (or quasi-cw) optical lattice has attracted much attention of people [8,13–15]. In 2002, Barker and Shneider put forward a traveling optical lattice created by high-power pulsed lasers to decelerate molecules [16]. 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 22 K (i.e., a single-stage optical Stark decelerator) [8], and Ramirez-Serrano et al. experimentally
n
Corresponding author. Tel.: þ86 02162232650; fax: þ 86 02162232056. E-mail address: [email protected] (J. Yin).
0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.06.092
demonstrated a similar single-stage optical Stark deceleration for a supersonic H2 molecular beam with a static pulse optical lattice [13]. In 2009, Momose and Kuma proposed a multistage optical Stark decelerator scheme for a pre-cooling pulsed molecular beam with an initial velocity of 13–15 m/s and a temperature of 100– 500 mK by using a quasi-cw, near-resonant and cavity-enhanced IR optical lattice [14]. Also, our group proposed a useful and promising multistage optical Stark decelerator for a supersonic (or subsonic) molecular beam with an initial velocity of 240–400 m/s by using a far-off-resonance, red-detuned, quasi-cw optical lattice with a waist of 20 mm (which is equal to the experimental parameter in Ref [8]) and a power of 264 W, and studied its deceleration effects for a subsonic CH4 beam and a supersonic NO beam by using 1D MonteCarlo simulations [15]. It is clear that there are three problems in our proposed scheme [15]: (1) we did not consider the initial longitudinal and transverse spatial distributions of an incident pulsed molecular beam in our 1D simulations; (2) the laser waist (20 mm) is too small to obtain a longer lattice and more cold molecules because the waist (100–500 mm) of the supersonic beam is far larger than the laser waist (20 mm) and (3) we did not consider the transverse velocity distribution of the incident molecular beam and its transverse molecular loss in our 1D simulations [15], so the obtained fraction of cold molecules in our 1D simulations does not have any reference value. To solve these problems, in this paper, we propose a desirable multistage optical Stark decelerator for a subsonic (or supersonic) molecular beam by using a far-off-resonance, red-detuned, quasi-cw and cavity-enhanced
X. Ji et al. / Optics Communications 287 (2013) 128–136
optical lattice with a diameter of 200 mm and an incident power of 200 W, and demonstrate its feasibility and study its decelerating effects for a subsonic CH4 beam by using 3D Monte-Carlo simulations, and obtain some new and valuable results. This paper is organized as follows: In Section 2, we propose a practical scheme to slow a subsonic molecular beam and introduce its slowing principle. In Section 3, we briefly introduce our cavity-enhancing scheme and its theory. In Sections 4 and 5, we study the slowing effects of our optical Stark decelerator, and discuss the influence of cavity enhancement factor on the slowing results, including the dependences of the final central (i.e., the most-probable) velocity, the fraction of cold molecules and its temperature on the synchronous phase angle, the deceleration-stage number, the initial central velocity of incident pulsed molecular beam, and the length of the ring cavity employed in our scheme, and obtain some new and important results. In Section 6, we compare our scheme with the chirped optical lattice [16] by 3D Monte-Carlo simulations. Some main results and conclusions are summarized in the final section.
2. Decelerator 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. In our scheme, our proposed multistage Stark decelerator is composed of a red-detuned, quasi-cw optical lattice with a diameter of 200 mm in a folded ring cavity, and the optical lattice is created by two nearly counter-propagating, far-offresonance, single-frequency, linear-polarized infrared lasers with a same wavelength of l ¼1064 nm. In Fig. 1, b is the angle between these two laser beams and is about 1721 in our scheme [17], and the spatial period L of the optical lattice is nearly equal to half of the laser wavelength l. When molecules travel along the z axis, they will experience an optical dipole interaction potential in the red-detuned optical lattice field as follows: U ¼ ð2aI=e0 cÞcos2 ðkz=2Þ, 2
2
ð1Þ
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According to Eq. (1), 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 larger than the transverse lattice potential will escape from the lattice. In the longitudinal direction (i.e., the z axis), if molecules enter the optical lattice from its antinode, 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 singlestage decelerating process, the slowed molecules will lose their kinetic energy, which is equal to the corresponding lattice potential energy. For convenient sake, the longitudinal position (z) of the slowed molecules in the optical lattice is defined as a synchronous phase angle f, where f ¼z/L 3601. The point of z ¼0 (i.e., f ¼01) is set at the antinode of the lattice (i.e., the minimum of the optical potential) due to its red-detuned. It is clear from Eq. (1) that the region of 01o f o1801 (the rear half of each lattice cell) is the decelerating phase-angle region for molecules in the red-detuned lattice, while the region of 1801 o f o01 is the accelerating one. When the synchronized molecule reaches the antinode (f ¼01) of the lattice, the lattice light field should be switched on in time. As soon as it arrives at the position of f ¼ f0 (where f0 is the synchronous phase angle of the synchronized molecule, usually we choose 01o f0 o1801), the lattice should be switched off rapidly. Then in the process of 01o f r f0, the synchronized molecules will be slowed effectively. For the red-detuned optical lattice, if this process can be repeated continuously for m times, the subsonic molecular beam will be efficiently slowed to a few 10 m/s, even to standstill. 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 supersonic (or subsonic) molecular beam by using a strict time-sequence synchronous control system and an electro-optic modulator (EOM).
)/w20]
is the intensity of a Gaussian laser where I¼I0exp[ 2(x þy beam, w0 is the waist (half-width at 1/e2 maximum intensity) of the laser beam, a is the averaged molecular polarizability, e0 is the permittivity in free space, c is the speed of light in vacuum, and k¼(4psin(b/2))/l is the wave number of this lattice. For CH4 molecule, we have a ¼2.9 10 40Cm2/V.
3. Cavity enhancement scheme and theory In order to product a time-varying lattice field, an electro-optic modulator (EOM) is used in our scheme. Meanwhile the EOM
Fig. 1. Schematic diagram of a multistage optical Stark decelerator, which is created by two nearly counter-propagating, far-off-resonance, red-detuned laser beams (originated by one incident laser) and located at the center of a folded ring cavity composed of four mirrors (M1, M2, M3 and M4). A pulsed subsonic molecular beam propagates and enters the quasi-cw lattice along the z axis. The inset figure is the enlarged lattice. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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should be sustained by a strict time-sequence synchronous and control signal so as to modulate the intensities of light fields precisely. When a lattice period is L ¼0.532 mm and an initial central velocity of incident molecular beam is v0 ¼240 m/s, the maximum initial modulation frequency of the quasi-cw lattice is about 451 MHz. In addition, a resonant cavity is applied to enhance the intensity of our lattice field and its optical potential. In our scheme, four mirrors (M1, M2, M3 and M4) with different reflectivity are used to form a folded ring resonator with a length of 1.5 cm, as shown in Fig. 1. In our scheme, the enhancement factor Ecav in a folded ring (i.e., traveling-wave) cavity can be expressed by [18] ffi pffiffiffi ð1R Þ2 ðpffiffiffiffiffi R0 R expðsdÞÞ 0 pffiffiffiffiffiffiffiffi ð2Þ Ecav , ð1 R0 R expðsdÞÞ5 where R0 is the power reflection coefficient of the mirror M1, R is the power reflection coefficient of other three mirrors (M2, M3 and M4), s is the absorption coefficient inside the cavity, and d is the optical length of this folded ring cavity. When R0 ¼0.6561 and R¼0.9905, the enhancement factor of the resonator is Ecav E80. This shows that the power of the light field in the cavity can reach 1.6 104 W as the input laser power is P0 ¼200 W. According to the theory of optical resonant cavities [19], the rising time (i.e., building time) of light field in the cavity is given by
tr ¼ nround
2Lcav , c
ð3Þ
and nround ¼ logR1 R2 ð1KÞ,
ð4Þ
where K is the percentage of the maximum value of light field energy in the cavity, R1 and R2 are mirror reflectivities, Lcav is the cavity length, and nround is the number of round trips needed for photons to establish this light field in the cavity. The photon lifetime (i.e., the falling time of the lattice field) in the cavity is
tf ¼
2Lcav =c , 1R1 R2
ð5Þ
where 2Lcav/c is the time required for photons to make one round trip in the cavity. So when we take tr ¼ 0.425 ns, the field energy inside the cavity will reach 60% of its maximum value, and the falling time of the lattice field (i.e., the photon lifetime) in the cavity is tf ¼0.516 ns after the light field is turned off. Assuming that the wavelength of incident laser field and the initial central velocity of a pulsed subsonic (supersonic) molecular beam are l ¼1.064 mm and v0 ¼240 m/s (or 360 m/s) respectively, the transit time for the synchronized molecule to pass a single cell of the lattice is about ttran ¼ 2.22 ns. In our scheme, the durations of the switch-on and switch-off signal of a single laser pulse are denoted as ton and toff respectively. Obviously, it is feasible for the cavity-enhanced method to be applied in our scheme, because ttran ( Eton þtoff) is much larger than tr ( ¼0.425 ns) and tf ( ¼0.516 ns). That means that there is enough time for the lattice field to finish all of processes, including the rising and falling of light field in the cavity as well as decelerating of molecules. However, the existence of tr and tf will impact on the deceleration effects of molecules exactly. In the next section, we will show the influence of them on the slowing effects. In Section 5, we will demonstrate and study the deceleration effects of our optical Stark decelerator for a subsonic molecular beam, including the final central velocity of cold molecular packet and its temperature Tl as well as the fraction n of cold molecules (n¼N/N0, where N and N0 are the molecular numbers in the slowed molecular packet and the incident molecular pulse, respectively), which will be affected by cavity enhancement factor Ecav. Additionally, in order to
know the influence of tr and tf on the slowing effects, we also compare the slowing results (see the Section 4) between the ideal (ignoring the influence of tr and tf) and real (considering the influence of tr and tf) situations.
4. Results of 3D Monte-Carlo simulations The slowing effect of our multistage optical-lattice decelerator was demonstrated by 1D Monte-Carlo simulations in 2009 [15]. However, the transverse loss of molecules escaping from the lattice field couldn’t be taken in account. In order to obtain some reliable (or valuable) slowing results, here we will carry out 3D Monte-Carlo simulations for 1D molecular slowing in the longitudinal direction and 2D trapping in the transverse one. In our simulations, the pulsed subsonic (or supersonic) methane (CH4) molecular beam with an initial central velocity of 240 m/s and a longitudinal temperature of 0.5 K is used and traveled along the z axis, as shown in Fig. 1. We assume that the transverse and longitudinal spatial distributions of incident pulsed molecular beam are Gaussian ones, and their spatial sizes are 0.1 mm and 1 mm, respectively. Also, the transverse and longitudinal velocity distributions of incident pulsed molecular beam are Gaussian ones, and their velocity half widths are 2.3 m/s and 40 m/s respectively, and the initial position and velocity of each simulated molecule are random. A single-frequency, singlemode, linear-polarized ytterbium-doped fiber laser with a power of 200 W and a wavelength of 1064 nm [20], as an input laser beam, is used to create our optical-lattice decelerator. In the slowing process, if the transverse position of a molecule is larger than the beam waist w0 of the optical lattice, it will be considered as a molecular loss in our 3D Monte-Carlo simulations. The final velocity distributions [21] of the slowed molecules are detected by a micro-channel plate (MCP), which is placed at the position of about 20 cm away from the start point of the lattice on the z axis. In addition, when the length of a folded ring cavity used in our scheme is about 1.5 cm, and the reflectivity of four mirrors are R1 ¼81% and R2 ¼R3 ¼R4 ¼99.525%, the corresponding enhancement factor is Ecav E80. First of all, we study the relationship between the deceleration effects and the deceleration stage number m, and the simulated results are shown in Fig. 2. The horizontal axis represents the longitudinal velocity of slowed molecules, and the vertical one represents the relative molecular number (i.e., the ratio of the molecular number at each longitudinal velocity to one at the initial most-probable velocity of incident molecular beam). Here one small peak on the longitudinal velocity distribution (i.e., on the horizontal axis) is pointed out by a red circle due to its small size, which is the slowed cold molecular packet and shows that the incident CH4 molecular beam has been decelerated successfully. To clearly show such a small slowed packet, we enlarge its velocity distribution (see the left enlarged peak). Fig. 2(a)– (c) shows the longitudinal velocity distributions of the slowed molecular packet for f0 ¼901 and the slowing stage number m¼3060, 4060 and 6060. It is clear from Fig. 2 that with the increase of the deceleration stages, the final central velocity of the slowed packet will be decreased gradually when the synchronous phase angle keeps invariant (f0 ¼ 901). In particular, when the slowing stage m is increased from 3060 to 6060, the final central velocity of the slowed packet will be efficiently decreased from 176.47 m/s to 78.67 m/s, and the corresponding temperature will be slightly increased from 56.85 mK to 73.68 mK, and the fraction n of cold molecules in the slowed packet will be slightly decreased from 5.13 10 4 to 4.81 10 4, which is far smaller than 2% in 1D Monte-Carlo simulations [15].
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Fig. 2. Simulated longitudinal velocity distributions of CH4 molecules at the outlet of the decelerator for the synchronous phase angle f0 ¼ 901 and the deceleration-stage number m¼3060, 4060 and 6060. Other simulation parameters are given in the plots and text.
Secondly, we investigate the relationship between the deceleration effects and the synchronous phase angles, and the simulated results are shown in Fig. 3. We can see from Fig. 3 that with the increase of the synchronous phase angle f0, the final central velocity of incident subsonic CH4 beam will be efficiently decreased, and the corresponding temperature of the slowed CH4 packet will be lowered to sub-mK even reach to a few mK. In particular, with the increase of f0 from 301 to 1501, the final central velocity of the slowed molecular packet will be decreased from 225.7 m/s to 11.55 m/s, and its temperature will be lowed from 0.5 K to about 1.6 mK. However, the velocity reduction of the slowed packet and the lowing of its temperature will accompany with the great loss of cold molecular number in the slowed packet, and when f0 ¼1501, the fraction of cold CH4 molecules is 1.0 10 6, which is far lower than 4.0 10 3 in the 1D Monte-Carlo simulation [15]. This shows that (1) the results from 1D Monte-Carlo simulations are not reliable due to without considering 2D transverse molecular loss; (2) 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 [22]. This is because the cold molecular number in the slowed packet is proportional to the area of the phase stability space; (3) 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. Also, we can find that when f0 ¼ 1501, it only needs 4060 deceleration stages to slow
CH4 molecules to 11.55 m/s, and the corresponding length of our optical lattice is only about 2.16 mm. Additionally, to know the influence of tr and tf on the decelerating effects, we study the dependences of the final central velocity vl of the slowed packet and its temperature as well as the fraction of cold molecules on the synchronous phase angle for the enhancement factor Ecav E80 and the deceleration-stage number m¼3680, and compare the simulated results between the ideal situation (ignoring the influence of tr, tf) and real one (considering influence of tr, tf), and the simulated results are shown in Fig. 4. We can find from Fig. 4(a) that with the increase of the synchronous phase angle f0, the final central velocity of the slowed packet will be reduced effectively, and the final central velocity in the real case becomes larger than one in the ideal one gradually. Additionally, we can see from Fig. 4(b) and (c) that with the increase of the synchronous phase angle f0, the fraction of cold molecules in the slowed packet and its temperature will be reduced due to the reduction of the phase-space area, and both the fraction of cold molecules and its temperature in the ideal case are always higher than ones in the real one. This is because the existence of tr and tf will reduce the effective potential-well depth of the optical lattice slightly, which will result in the decrease of the molecular number of the slowed packet and a lower temperature in the real case, as shown in Fig. 4(b) and (c). These investigations show that (1) a far-off-resonance, red-detuned, quasi-cw optical lattice, as a multistage optical Stark decelerator, can
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Fig. 3. Simulated longitudinal velocity distributions of CH4 molecules at the outlet of the decelerator for the deceleration-stage number m¼ 4060 and the synchronous phase angle f0 ¼ 301, 601, 901, 1201, and 1501. Other simulation parameters are given in the plots and text.
be used to efficiently slow a subsonic CH4 beam from 240 to 10 m/s, even to zero; (2) the temperature of the slowed cold molecular packet can be lowed to sub-mK, even reach to a few mK, but the fraction of cold molecules will be equal to about 1.0 10 6, which is far smaller than one (4.0 10 3) in 1D simulations. If the molecular number in an incident CH4 pulse is about 1012, the number of cold molecules in a slowed packet is about 106 and (3) the rising time (tr) and falling time (tf) of the lattice field in the cavity will markedly influence the slowing effects for molecules, including the final central velocity, the temperature and the fraction of cold molecules, etc. In final, we investigate the dependences of the final central velocity vl of slowed packet on both the initial central velocity v0
of incident molecular beam and the length Lcav of the folded ring resonator, and the results are shown in Fig. 5. We can find from Fig. 5(a) that with the increase of v0, the final central velocity vl of slowed packet will be increased. In particular, when f0 ¼901 and Ecav ¼ 80, we need the deceleration stages of m¼6850 to slow a pulsed molecular beam with an initial central velocity of v0 ¼240 m/s (or 360 m/s) to 7.34 m/s (or 261.54 m/s). While f0 ¼1201 and Ecav ¼ 80, we only need m¼4830 to slow the molecular beam from v0 ¼240 m/s (or 360 m/s) to 10.28 m/s (or 266.69 m/s). Also, it is clear from Fig. 5(b) that with the increase of the cavity length, the final central velocity vl of slowed packet will first drop and then rise, and for different synchronous phase angles such as f0 ¼ 901 and f0 ¼1201, the turning points appear at Lcav ¼3.5 cm and
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Fig. 4. Comparison of slowing effects between the ideal situation (ignoring the influence of tr and tf) and the real one (considering influence of tr and tf) for m¼3680 and Ecav ¼80. Other simulation parameters are given in the plots and text.
Fig. 5. Dependence of the final central velocity of slowed molecular packet on (a) the initial central velocity of incident pulsed CH4 beam and (b) the length of the folded ring cavity with Ecav ¼ 80. (c) Variation of optical dipole force experienced by molecules on the z axis in one spatial period of lattice. Other simulation parameters are given in the plots.
2.5 cm, respectively. In order to explain these results, we choose f0 ¼901 as an example [see Fig. 5(c)]. Fig. 5(c) shows the optical dipole force experienced by molecules on the z axis in one spatial period of the lattice. Here we assume that a group of molecules travel from point A to B within the time tr and from A0 to B0 within tf, as shown in Fig. 5(c). However, it is clear from Fig. 5(c) that the optical dipole force Fz experienced by the molecules in the motion of A0 -B0 is larger than that in A-B. That is, an appropriate tr (and tf) will
make the deceleration effect increased by tf be greater than the loss of slowing effect resulted from tr, so a pair of appropriate tr and tf will lead to a better slowing effect. According to Eqs.(3) and (5), we can find easily that a longer cavity length Lcav will lead to a longer tr and tf. When tr and tf are too long, the deceleration effect increased by tf will be equal to the loss of the slowing effect resulted from tr. Also, they will shallow the optical lattice potential significantly and even lead to the acceleration of molecules in the next
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accelerating phase angle region of 1801o f o01. Thus the slowing effect will be declined seriously. It can also be understood easily that the larger the synchronous phase angle f0 is, the shorter tr and tf (corresponding to Lcav) contributing to the slowing of molecules will be. So the turning point for f0 ¼1201 appears earlier than one for f0 ¼901 on the horizontal axis in Fig. 5(b). These results show that the slowing effects strongly depend on both the initial central velocity of incident molecular beam and the length of the ring cavity.
5. Influence of cavity enhancement factor Ecav In our scheme, it is clear that a larger or smaller (i.e., an improper) cavity enhancement factor Ecav will also influence the slowing effects for molecules. For this, we study the influences of the cavity-enhancement factor on the decelerating effects, and the simulated results are shown in Fig. 6. It can be seen from Fig. 6(a) that when Ecav rises from 60 to 140, the final central velocity of cold molecular packet slowed by the enhanced optical lattice in the cavity will be lower and lower. This is because a larger Ecav leads to a deeper potential well in the longitudinal direction so that molecules can be slowed to a lower velocity vl. Afterwards, we also explore the influence of Ecav on both the temperature of the slowed molecular packet and its fraction of cold molecules, and the results are shown in Fig. 6(b) and (c). Fig. 6(b) shows that for the same Ecav, we can get more slowed molecules when f0 ¼901 than that with f0 ¼1201, and with the increase of Ecav, n will first rise and then drop. As we know, a larger Ecav results in longer falling and rising times (tr and tf) of the optical lattice field in the folded ring cavity and higher lattice intensity. An optical lattice with higher intensity can capture more molecules in the transverse direction and slow more molecules in the longitudinal direction. And longer tr and tf will lead to the smaller efficient lattice-potential depth within the transit time ttran for synchronized molecule to pass a single cell of lattice, especially at
the beginning of deceleration when ttran is shorter, and then more molecules with larger transverse velocities will escape from the optical lattice in the radial direction. For f0 ¼901 and m¼3850, when Ecav rises from 60 to 100, the efficient lattice potential in the cavity also increases due to the relatively appropriate tr and tf, and then more molecules can be captured and slowed. However, when Ecav rises from 100 to 140, the loss of molecules will be increased gradually if even the maximum of optical lattice intensity is enhanced evidently, because much longer (i.e., improper) tr and tf will result in more shallow virtual lattice-potential depth in the cavity. So Ecav ¼100 becomes an optimal point for us to obtain the most cold molecules when f0 ¼901. But for f0 ¼1201, the turning point appears earlier at Ecav ¼70. It can be explained that the larger the synchronous phase angle is, the less the cold molecular number in the slowed packet is, and then the more shallow optical potential well is needed to decelerate molecules. Therefore the optimal point of Ecav for f0 ¼1201 will be lower. Fig. 6(c) shows that with the increase of Ecav, the longitudinal temperature Tl of the slowed molecular packet will be higher and higher. This is because that a higher Ecav, corresponding to a deeper longitudinal potential well, can decelerate more molecules in the z axis, which results in a broader longitudinal velocity distribution of the slowed molecular packet, and then the temperature of the slowed packet will be higher. According to the simulated results mentioned above, we think that the enhancement factor of Ecav ¼80 is more appropriate for our proposed Stark decelerating scheme.
6. Discussion In 2002, Barker and Shneider put forward a general scheme for creating stationary cold molecules by rapid deceleration of supersonically cooled molecules in a high-intensity pulsed optical lattice [16]. Next, we will compare the efficiency of their scheme
Fig. 6. Influence of the cavity enhancement factor on the slowing effects, including (a) the final central velocity, (b) the fraction of cold molecules, and (c) the temperature of the slowed molecular packet. Other simulation parameters are given in the plots.
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Fig. 7. Comparison between the deceleration effects of (a) our scheme and (b) a chirped optical lattice by 3D Monte-Carlo simulations. Other simulated parameters are shown in the plots and text.
using a chirped decelerating optical lattice with our proposed scheme by 3D Monte-Carlo simulations, and the results are shown in Fig. 7. In our simulations, we take molecule I2 as an example, and the pulsed I2 molecular beam has the same initial central velocity of 240 m/s and longitudinal temperature of 1.0 K. Fig. 7(a) shows the deceleration results of our scheme when I0 ¼ 1.02 108 W cm 2, f0 ¼901 and m¼23200. When the slowed packet is decelerated to a velocity of 77.1 m/s, the corresponding temperature and the fraction n of cold molecules are Tl ¼18.15 mK and n¼0.069%, respectively. Fig. 7(b) shows the deceleration results of a chirped optical lattice [16] when I0 ¼ 12 109 W cm 2, d ¼5.28 1015 rad/s (frequency chirp), q¼1.56 107/m (lattice wave number), and c ¼0.735 (dimensionless quantity, associated with d, I0, and q), which are the same as ones used in Ref [16]. When molecules are decelerated to the same velocity 78 m/s, the velocity width of the slowed packet is very broad, the corresponding Tl and n are 66.52 mK and 9.41%, respectively. It is obviously 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. That is because the optical intensity I0 employed in their scheme is much higher than that in ours, which can not only decelerate more molecules in the z axis but also capture more ones in the radial direction. But the temperature Tl of the slowed packet in our scheme is much lower than that in the chirped optical-lattice scheme by more than three orders of magnitude (by larger than 3.67 103 times) owing to the relatively shallow optical potential well and the bunching effect of our multistage optical Stark decelerator.
7. Conclusions In this paper, we have proposed and demonstrated a practical (or desirable) scheme to slow a pulsed subsonic molecular beam
by using a far-off-resonance, red-detuned, quasi-cw optical lattice. In order to enhance the lattice intensity, our multistage optical Stark decelerator is placed in a folded ring cavity with an enhancement factor of 80 and has an intensity of 108 W cm 2. By using 3D Monte-Carlo simulations, we have studied the dependences of the slowing results (including the final central velocity vl of the slowed packet and its temperature as well as the fraction of cold molecules) on the synchronous phase angle, the deceleration-stage number, the initial central velocity of an incident molecular beam and the cavity length, and discussed the influences of the cavity enhancement factor on the slowing results. Our research shows that a far-offresonance, red-detuned, cavity-enhanced, quasi-cw optical lattice, as a multistage optical Stark decelerator, can be used to efficiently slow a subsonic molecular beam from 240 to 10 m/s even to 0, and an ultracold molecular packet with a temperature of sub-mK even a few mK can be obtained, but the corresponding fraction of cold molecules will be 10 4–10 6, which strongly depends on the our choosing f0 and the initial central velocity of incident molecular beam, etc.. 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 106–108. We have also found that the slowing results strongly depend on the cavity enhancement factor Ecav, and we can obtain the better slowing results when Ecav ¼80. However, a more useful or practical slowing scheme should be as follows: we can first use a static (or moving) pulsed optical lattice (i.e., a single-stage optical Stark decelerator) [8,13] to beforehand slow a subsonic molecular beam from 240 m/s to 24 m/s, and then using our quasi-cw, cavity-enhanced optical lattice (i.e., a multistage optical Stark decelerator) to further decelerate a pre-slowed molecular beam from 24 m/s to 0. In this case, starting from an initial central velocity of 24 m/s, the diameter of our quasi-cw lattice can be increased from 0.2 mm to 1 mm, and the diameter of incident molecular beam can also be increased from
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0.2 mm to 1 mm, and then we can obtain more cold molecules. And under these conditions, the deceleration stage number we needed is only about 1270 if we choose f0 ¼1201, and the corresponding length of our multistage optical Stark decelerator (or quasi-cw optical lattice) is only 0.676 mm, and then the angle b (1721) between two laser beams (see Fig. 1) can be further reduced. Because our proposed multistage 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 (which is similar to a single-stage optical-lattice Stark decelerator [8,13]), but also to obtain an ultracold molecular packet with a temperature of a few mK and a number of 106–108, such a multistage optical Stark decelerator and its obtainable ultracold molecules have some important applications in the fields of cold molecular physics, molecule optics, cold molecular spectrum, cold molecular collisions and cold chemistry, precise measurement, quantum computing and its information processing, and cold molecular lithography, even can be used to realize an all-optical, chemically-stabled molecular BoseEinstein condensates (BEC) by using an optical-potential evaporative cooling [23], all-optical molecular Fermi quantum degeneration (FQNG), quantum and nonlinear molecule optics, and so on.
Acknowledgment This work is supported by the National Nature Science Foundation of China under Grant nos. 10674047, 10804031, 10904037, 10974055 and 11034002, the National Key Basic Research and Development Program of China under Grant nos. 2006CB921604 and 2011CB921602, and the Basic Key Program of Shanghai Municipality under Grant no. 07JC14017, and the Shanghai Leading Academic Discipline Project under Grant no. B408.
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