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Applying the direct simulation Monte Carlo (DSMC) method developed for ultracold Bose-Fermi mixture gases research, we study the sympathetic cooling process of 6Li and 133Cs atoms in a crossed optical dipole trap. The obstacles to producing 6Li Fermi degenerate gas via direct sympathetic cooling with 133Cs are also analyzed, by which we find that the side-effect of the gravity is one of the main obstacles. Based on the dynamic nature of 6Li and 133Cs atoms, we suggest a two-stage cooling process with two pairs of crossed beams in microgravity environment. According to our simulations, the temperature of 6Li atoms can be cooled to T = 29.5 pK and T/TF = 0.59 with several thousand atoms, which propose a novel way to get ultracold fermion atoms with quantum degeneracy near pico-Kelvin.
© 2015 Optical Society of America
1. Introduction
Since the first sympathetic cooling has been demonstrated, this technique has been successfully extended to the ultracold atoms field, which serves as a unique tool that opens up many possibilities for researches such as ultracold molecules, Efimov trimers, many-body physics and the exploration of BEC-BCS crossover [1–3]. In particular, one can get the different statics by mixing bosonic and fermionic atomic species. When combining the mixture with optical lattices, the mixture provides new opportunities for exploring the quantum phases such as supersolids, insulator with fermionic domains and boson mediated superfluids [4–6].
Among the different possible heteronuclear mixtures, the Li-Cs mixtures can be considered as an excellent candidate. Firstly, the large mass ratio of mCs/mLi=22 leads to an advantageous universal scaling factor of 4.88 for Efimov resonances, compared with 22.7 for a homonuclear system, which is a smaller scaling constant that facilitates the text of the scaling law in Efimov physics [7, 8]. Secondly, the Cs atoms can be cooled down to nano-Kelvin and the Fermi transition temperature of Li atoms can be 1μK. Through effective sympathetic cooling process, the Li atoms can be prepared at a temperature of T/TF=0.01 theoretically. Finally, the ground-state Li-Cs molecules have the largest dipole moment of 1.8×10−29 C · m among the combinations of two stable alkaline-metal atoms [9, 10].
Although there are certain advantages of ultracold Li-Cs mixtures, little is known about the interaction properties of Li and Cs until now. Recently, Chin’s and Weidemüller’s group both succeeded in producing an ultracold mixture of fermionic 6Li and bosonic 133Cs atoms and observing interspecies Feshbach resonances [9, 11]. However, they both take the tragedy that cooling 6Li atoms and 133Cs atoms separately, then transferring and mixing them together to observe Feshbach spectroscopy. It’s a good arrangement for the observation of 6Li-133Cs Feshbach resonances, but due to the inevitably heating effects in the long distance transferring, the 6Li atoms can not achieve a much lower temperature [12, 13].
During the evaporation cooling process, the gravitational sag is one of the main obstacles to the mixture of 6Li and 133Cs. In one species situation, magnetic levitation may be a unique technique to counteract the effect of the gravitational force, which can improve the efficiency of the evaporation cooling [13, 14]. However, one can not compensate the gravitational force of two different species at one time, which means that even with magnetic levitation, the gravitational sag still exists as the sympathetic cooling goes on [15]. The microgravity environment can consequently be regarded as an indispensable condition to improve the efficiency of sympathetic cooling with 6Li atoms and 133Cs atoms [16].
Moreover, in the microgravity environment, owe to the stable fully mixing and interacting of the 6Li atoms and 133Cs mixture, further cooling process can be applied on the two spceies. In 2003, A. E. Leanhardt et. al employed an adiabatically decompression and subsequent evaporation cooling process of partially condensed atomic vapors in a very shallow gravitomagnetic trap to obtain a temperature of hundreds of pico-Kelvin [17]. In 2013, the situation in microgravity was studied by Lu Wang et. al, and they proposed a two-stage path for 133Cs to pico-Kelvin temperatures in microgravity with a multi-beam dipole trap [16]. But until now, all the two-stage cooling process are proposed for the evaporation cooling of single bosonic atom, which leads to the good result of pico-Kelvin. Thus, we think that a similar two-stage process might also be efficient for the 6Li-133Cs mixture’s sympathetic cooling, which could lead to lower temperature with acceptable quantum degeneracy in the microgravity environment.
In this paper, we first analyze the obstacles to cooling 6Li atoms via direct sympathetic cooling with 133Cs atoms. The obstacles to producing 6Li Fermi degenerate gas via direct sympathetic cooling with 133Cs are also analyzed, by which we find that the side-effect of the gravity is one of the main obstacles. Then based on the dynamic nature of 6Li atoms and 133Cs atoms, we suggest a two-stage cooling process with two pairs of crossed beams in microgravity environment. According to our simulations, the temperature of 6Li atoms can be cooled to T = 29.5 pK and T/TF = 0.59 with several thousand atoms. In the end, factors that may affect the final temperature of 6Li atoms are also analyzed.
2. The simulation
We employ the DSMC method described in [18]. In our simulation, the atoms are prepared in the lowest hyperfine ground state (|F = 3,mF = 3〉 for 133Cs, |F = 1/2,mF = 1/2〉 for 6Li). And the atoms are treated as semiclassical particles, solely for the purpose of modeling their collision processes in the framework of well-established classical collision dynamics, so that the hard-sphere model and energy and momentum reservation laws can be utilized. However, the quantum nature of particles is also taken into consideration. The elastic collisions between particles are mostly induced by s-wave elastic scattering. As the frequencies of an atom colliding with others depend on the s-wave scattering cross section σ, which is a key factor for the simulations [14]. Since 133Cs atoms are bosonic particles, we set σ as
where aCs is the s-wave scattering length and kCs is the de Broglie wave vector of the 133Cs atoms. As kCs depends on the temperature, the scattering cross section is related to the temperature asAt ultralow temperature (T→0), σ reduces to σ =8π , which is the case of weakly interacting identical bosons. For weakly interacting identical fermions, the s-wave scattering is forbidden, thus we neglect the elastic collisions between the 6Li atoms themselves.
The three-body recombination is the dominant inelastic collision and the loss due to the background scattering is also considered [16].
The crossed dipole trap is created by two orthogonally focused laser beams with equal waists in the horizontal plane. The dipole potential can be expressed as,
with the individual trap depths, U1 and U2: in which, P1(2) are the powers of the two beams, w is the waist of the beams, c is the speed of light in space, Γ is the scattering rate, ω0 and ω1 are the resonance frequency of atoms and the frequency of laser beams.The dipole potential near the center can be well approximated as a symmetric harmonic oscillator,
In our simulation, we consider that the atoms with higher kinetic energies move to the boundary of the trap and then escape [19].
3. The sympathetic cooling process
In order to study the sympathetic cooling process, we numerically simulate the 6Li-133Cs sympathetic cooling process of Mudrich in 2002 [20]. They simultaneously trap typically 4×104 6Li atoms with 105 133Cs atoms in a quasi-electrostatic trap (QUEST) which is produced by a focused 10.4 μm CO2 beam. The radial and axial oscillation frequencies of trapped Cs(Li) atoms are ωx,y/2π=0.85 kHz (2.4 kHz) and ωz/2π=18 Hz (50 Hz). The effective s-wave scattering lengths are |aLi−Cs| = 180a0, |aCs−Cs| = 140a0 (a0 is the Bohr radius). We set the initial simulation parameters according to the chart in [20] and get the evolution curve of the 6Li-133Cs atoms’ number and temperature. As a comparison, the origin experimental results by Mudrich et. al are also illustrated. From the comparison in Fig. 1, we can see that the simulations are consistent with the experiment.
Fig. 1 The simulated evolution of the Li-Cs atoms’ number and temperature (blue curves for the Cs atoms, red curves for the Li atoms). As a comparison, the experimental results by Mudrich et. al are also illustrated as black squares and dots, courtesy of professor Grimm and professor Weidemüller.
Based on the discussions above, we take several series of simulations to find out the relation between the atom numbers and the final degeneracy T/TF of 6Li atoms, and the relation between the initial temperature of 133Cs atoms and final degeneracy of 6Li atoms. Here,
in which is the reduced Planck constant, kB is the Boltzmann constant, is the mean trapping frequencies and N is the atom number of 6Li atoms.In our simulations, the dipole trap is still formed by crossing two focused beams in the horizontal plane at the wavelength of 1064 nm, waist of 60 μm. The initial atom numbers and temperature of 6Li are 2× 105 and 20 μK, which is accessible according to [9, 11]. We take the simulations under different initial conditions of 133Cs atoms. The initial atom number ranges from 1× 105 to 1× 108 and the initial temperature ranges from 1 μK to 90 μK. The beam power is nonlinearly ramped down as,
where P0 is the initial beam power, τ and β are parameters associated with the ramping curve. We set P0=10 W, and the minimum beam power can be 3 mW. We take different τ (0.001 to 100) to find the maximum degeneracy T/TF in each group of simulations. The results in Fig. 2(a) shows the relationship between the final degeneracy of 6Li atoms and the initial number of 133Cs atoms. And Fig. 2(b) shows the relationship between the final degeneracy of 6Li atoms and the initial temperature of 133Cs atoms. From the figure we can see that the compatible atom number of 133Cs lies approximately between 5× 105 and 1× 106, and that the lower the initial temperature of 133Cs atoms is, the better degeneracy of 6Li atoms can be. However, when the initial temperature of 133Cs atoms is below 2 μK, the condition is hard to keep because of the inevitable heating effects in the long distance transfering.Fig. 2 (a) Relationship between the final degeneracy of 6Li atoms and the initial number of 133Cs atoms. The black blocks, red dots and the blue triangles show that the initial temperature of 133Cs atoms is respectively 10 μK, 5 μK, and 2 μK. (b) Relationship between the final degeneracy of 6Li atoms and the initial temperature of 133Cs atoms. The black blocks, red dots and the blue triangles show that the initial number of 133Cs atoms is respectively 1×106, 2×105, and 1×105.
In order to find out the obstacles to producing degenerate fermionic 6Li gases, we simulate the cooling process and depict the diagram of the evaluation of atom number and temperature of the 6Li atoms during the cooling process. The process is illustrated in Fig. 3(a), in which the initial conditions are TCs= 3 μK, NCs = 1×106, TLi= 20 μK, NLi =2 × 105 with the ramping parameters set to τ = 25 and β = −44.5, which is the best set according to the group simulations in Fig. 2.
Fig. 3 Evolution of the total atom number NLi (black curve above), degeneracy T/TF (red curve) and the temperature of 6Li atoms (blue curve). (a) The sympathetic cooling process of 6Li atoms with gravity(g=9.81 m/s). The lowest degeneracy T/TF is 1.09. (b) The sympathetic cooling process of 6Li atoms in microgravity(10−3g). The lowest degeneracy T/TF is 0.52.
From Fig. 3(a), we can see that the degeneracy T/TF reaches the lowest value 1.09 at 3 s and then shifts upward. In fact, the dipole force of the red-detuned laser serves to lift the atoms against the gravity of the earth, so that the equilibrium position of 6Li atoms shifts downward as the beam power ramps down. At 3 s, ΔzLi = 83.8 μm is greater than the beam waist(60 μm). So the 6Li atoms begin to fall down and leak out from the dipole trap. Because of the fast loss of the atom, the degeneracy T/TF shifts upward.
The other factor that may affect the cooling efficiency is the gravitational sag between the two atomic species. In the direction of the gravitational force, the total gravitational sag can be expressed as . The characteristic resonant lengths of the two species in the direction of the gravitational force are , . The gravitational sag and the characteristic resonant lengths in the cooling process are shown in Table 1.
Table 1. Gravitational sag and characteristic resonant lengths in the cooling process.
From Table 1 we can see that the gravitational sag increases as the cooling process goes on. After 1.75 s, the gravitational sag can be compared with the characteristic resonant lengths. When the two atom clouds separate too far apart, the inter-species collisions are not effective. This is one of the main obstacles to the sympathetic cooling. Thus, a microgravity environment could be a better condition for the cooling process.
Under the same initial conditions, we also simulate this sympathetic cooling process in the microgravity environment(10−3g). The process is illustrated in Fig. 3(b). Without the side-effect of the gravity, the sympathetic cooling process lasts 5.5 s. The final degeneracy T/TF is 0.52, the final temperature is 7.36 nK and the final atom number is 2985. In order to study how the gravity acceleration affects the final temperature and degeneracy of 6Li atoms, we run several simulations under the same initial conditions and summarize the results in Table 2.
Table 2. Gravity acceleration, final temperature and degeneracy of 6Li atoms.
From Table 2, we can see that for the first stage cooling, the final temperature can be different if the microgravity level is bigger than 10−3g. However, the final temperature stays almost the same when the micro-gravity level is smaller than 10−3g.
4. The two-stage crossed beam cooling process in microgravity
From previous analysis, we can see the advantages of microgravity for sympathetic cooling. However, in order to get sufficiently high degeneracy and low temperature, it is necessary to find an efficient method for cooling. We study the two-stage cooling process for 133Cs atoms in microgravity suggested by Lu Wang, which cools down the atoms’ temperature to pico-Kelvin [16]. We think that this process might also be workable for the sympathetic cooling process of boson-fermion mixture. So we take this method to the simulation of our cooling process to check if it also works well for the Bose-fermi mixture.
Similar with the two-stage cooling of boson atoms, the whole cooling process is divided into two parts. The schematic drawing of the multi-optical dipole trap is illustrated in Fig. 4 and the gravity acceleration is set to 10−5g. In the first stage, atoms are loaded into a tight-confining dipole trap created by two crossed laser beam of 1064 nm, each with a waist of 60 μm and power of 10 W. The loaded atoms are 1×106 133Cs atoms with a temperature of 3 μK and 2×105 6 Li atoms with a temperature of 20 μK, which produce the initial degeneracy T/TF = 4.28. In this tight confining dipole trap, the initial trapping frequency of 6Li(133Cs) is ωz/2π=1137.65 Hz(ωz/2π=616.24 Hz). In the first 5.5 s, the beam power is ramped down to 3 mW by Eq. (7), with τ=25 and β =−44.5. During this process, the waist of the laser beam is remained. The condensate we get from this process is 2985 6Li atoms with a temperature T=7.36 nK and a degeneracy T/TF = 0.52. At the same time, there also leaves 1.8×104 133Cs atoms with a temperature of 6.8 nK. At the end of this stage, the trapping frequency of 6Li(133Cs) is ωz/2π=13.99 Hz(ωz/2π=6.58 Hz). The first stage is same as the situation describe in Fig. 3(b).
Fig. 4 The proposed setup of the multi-beam optical dipole trap, which is designed for the two-stage sympathetic cooling. The arrowheads illustrate the process of the two-stage sympathetic cooling. (a) Sympathetic cooling in a tight-confining crossed dipole trap. (b) Overlapping the trap with a wider and weaker one. (c) Adiabatically decompressing the combined trap.
Then, we come to the second stage. It is an adiabatically decompressing process where a much shallower and wider crossed-beam trap is overlapped to the trap. The result of the simulation is shown in Fig. 5. The waist and power of this new trap is 3 mm and 80 mW each. We ramp down the power of the narrow laser by parameters τ=0.03 and β =0.8683 with Eq. (7). With another 9.5 s, the narrow beam is nearly shut down and there only remains a weaker confinement, in which the trapping frequency of 6Li(133Cs) is ωz/2π=0.058 Hz(ωz/2π=0.027 Hz). Finally, we get 2684 6Li atoms at the temperature of 29.5 pK, degeneracy of T/TF = 0.59, and 15295 133Cs atoms with temperature of 26.1 pK. The whole process is under the condition of microgravity(10−5g), which allows for the persistent flatness of the very shallow trap and avoids the separation of the two different species. In the end, the fermions could reach tens of pico-Kelvin with degeneracy T/TF around 0.5.
Fig. 5 The time evolution of atom number NLi (black curve above), degeneracy T/TF (red curve) and the temperature of Li atoms(blue curve). The lowest temperature is 29.5 pK with the degeneracy of T/TF = 0.59.
We also make several simulations to test how gravity acceleration affects the final temperature of the atoms in two-stage cooling process. We set different gravity acceleration values and get the final temperature of 6Li atoms when the reach the same degeneracy 0.59. The final temperature (gravity acceleration) of 6Li atoms are 5.72×10−9 K (10−2g), 1.03×10−9 K (10−3g), 8.52×10−11 K (10−4g), 2.95×10−11 K (10−5g). From the simulation results, we can see that the gravity acceleration must be smaller than 10−5g in order to achieve the temperature of pico-kelvin regime. To realize this microgravity environment, we can use techniques such as species-specific dipole trap, or setting up microgravity platforms such as recoverable satellites (10−5g), drop towers (10−5g) and space station (10−6g) [21, 22].
To insure the validity of two-stage cooling, we also estimate how the power and frequency dithering of laser can affect the final temperature of the mixture. From Eq. (3) and Eq. (4), near the center of the crossed dipole trap, the potential can be approximated as,
here we use the rotating-wave approximation and assume the two beams of the trap are produced by one laser. In the ultracold regime, the temperature is a statistic data according to the average kinetic energy K and potential energy Udip of the atoms in three direction.So we can estimate the influence of power dithering to the final temperature through the partial differential of P in both sides of Eq. (9).
Taking the parameters of 6 Li atoms, Γ = 5.8724 MHz, ω0 = 2π×4.47×1014 rad/s, the conditions of the laser, ω1=2π×2.82×1014 rad/s, w=60 μm and the constants c=3×108 m/s, kB=1.38×10−23 J/K [23]. We can find out
So we get the final expression of the temperature influenced by the power dithering of the laser, which means that if we want to keep the atoms’ temperature stable at 10 pK scale, we must insure the power dithering of laser less than 2×10−2 mW. In the same way, we get the expression of the temperature influenced by the frequency dithering,
here, Δ = ω0 −ω1. Eq. (12) means that if the frequency of laser dithered 1 MHz, the temperature of atoms in the shallow dipole trap will change −0.447×10−2 pK. Through the discussion above, taking the current experiment condition into consideration, we can see that the frequency stability of the laser can meet the requirement of 10 pK scale while the power stability needs a further development. We think that the multilevel feedback circuit might be a good choice.5. Conclusion
We studied the sympathetic cooling process of 6Li and 133Cs atoms in crossed optical dipole trap with the direct simulation Monte Carlo(DSMC) method developed for ultracold Bose-Fermi mixture gases research. Applying two-stage cooling technique to sympathetic cooling process, we put forward a new cooling process with two pairs of crossed beams for 6Li-133Cs mixture, which leads to tens of pico-Kelvin with degeneracy T/TF around 0.5. Even though the two-stage process just provides a possible path to ultralow temperature with no more increasing on quantum degeneracy, there are still several advantages for reaching such a ultracold Bose-Fermi system. With a slower atoms’ velocity, we are able to study the physics phenomena that only occur on very low energy scale such as phase transitions and new forms of matter. It can also be benefical to the precision measurement by enabling ultra-precise atomic sensors which contribute to the test of basic physical quantities, general relativity and gravitational wave detection [16, 17, 24]. This new cooling process can also be useful to create other Bose-Fermi mixtures such as 133Cs-40K, 87Rb-40K and 87Rb-6Li, which may be a new method for exploring the ultracold atoms world.
Acknowledgments
The first two author contributed equally to this work. We thank Professor Cheng Chin and Professor Xiaoji Zhou for their helpful discussions and suggestions. This work is supported by the National Fundamental Research Program of China under Grant No. SQ2010CB511493 and No. 2011CB921501, the National Natural Science Foundation of China under Grant No. 61027016, No. 61078026, No. 10934010 and No. 91336103.
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