Continuous slowing of a gadolinium atomic beam
UUsing a solid-state Quartz Crystal µ -balance as akinetic energy sensor in laser cooling experiments A. Chavarr´ıa-Sibaja , Escuela de F´ısica, Universidad de Costa Rica, 2060 San Pedro, San Jos´e, Costa Rica Centro de Investigaci´on en Ciencia e Ingenier´ıa de Materiales, Universidad de CostaRica, 2060 San Pedro, San Jos´e, Costa RicaE-mail: [email protected]
A. Araya-Olmedo , Escuela de F´ısica, Universidad de Costa Rica, 2060 San Pedro, San Jos´e, Costa Rica Centro de Investigaci´on en Ciencia e Ingenier´ıa de Materiales, Universidad de CostaRica, 2060 San Pedro, San Jos´e, Costa RicaE-mail: [email protected]
O.A. Herrera-Sancho , , Escuela de F´ısica, Universidad de Costa Rica, 2060 San Pedro, San Jos´e, Costa Rica Centro de Investigaci´on en Ciencia e Ingenier´ıa de Materiales, Universidad de CostaRica, 2060 San Pedro, San Jos´e, Costa Rica Centro de Investigaci´on en Ciencias At´omicas Nucleares y Moleculares, Universidadde Costa Rica, San Jos´e 2060, Costa RicaE-mail: [email protected]
Abstract.
We present here the development of a new and innovative experimental method tofully characterize a solenoidal ”Spin-Flip” Zeeman slower (ZS) using a Quartz Crystal µ -balance (QCM) as a kinetic energy sensor. In this experiment, we focus a 447.1 nmlaser into a counter-propagating beam of Gd(I) atoms in order to drive the dipoletransition between ground D state and D exited stated. Also, during the process,we continuously measure the change of the oscillation frequency signal of a QCM due tothe deposition of the Gd(I) atoms in its surface. We use these measurements to studythe time-evolution of the velocity distribution of the Gd(I) atom beam during a coolingprocess. We obtain a maximum atom average velocity reduction of (43.45 ± τ e = 8.1988 ns and compare it with reported lifetime for 443.06 nm and 451.96 nmelectronic transitions of Gd(I). These results confirm that the QCM offers an accessibleand simple solution to have in direct overview for laser cooling experiment. Therefore,a new field of research in the borderland between solid state physics and ultra-coldatoms physics will enable highly sensitive test of postulates from fundamental physics. a r X i v : . [ phy s i c s . a t o m - ph ] S e p . Chavarr´ıa-Sibaja et al Keywords : Laser cooling, Fokker-Planck equation, Bose-Einstein condesate, Zeemanslower, dipole transition, momentum exchange, transition lifetime.
1. Introduction
Diminishing the kinetic energy (KE), or velocity, of atoms has been an important feat inatomic physics by producing innovations in experimental and fundamental theoreticalfields. These advances have allowed the investigation of nearly ideal quantum systemssuch as one-component plasmas and Bose-Einstein condensates of dilute gases [1].Techniques including mechanical diffractive structures—based on atom optics— andtraps, like Paul’s trap, have been used to control the atoms’ position and velocity inorder to obtain such systems; however, radiation pressure has proven to be one of thebest techniques to cool atoms and reduce the KE to obtain quantum-like environmentsby its wide range of methods like evaporative cooling and sub-Doppler laser cooling.Radiation pressure is based on the transfer of momentum between photons and atomsand itself. Recently, several successful attempts have been made in order to control andmanipulate the external and internal degrees of freedom for atoms [2] and ions [3, 4] viadiminishing the KE of the particles. Nevertheless, in its most basic form this methodpresents a number of challenges that limit the efficiency of KE reduction. For example,the detuning between atoms and photons [5], the existence of open transitions andhyperfine structures [6] in atoms and others.Some of these limitations —for example, sensibility to hyperfine structures [7]—have been overcomed in the last 30 years with great advances in both theoreticaland experimental areas, such as the perfectioning of radiation based atom slowers [8].The advancements by the three 1997 Nobel price winners S. Chu [9], C.Cohen-Tannoudji [10], and W.D. Phillips [11] are some of the most important,providing fundamental tools for the first experimental elaboration of a Bose-Einsteincondensate [12]. Furthermore, the influence that these contributions have had in solidstate physics have resulted in a wide range of applications, from the study of surfacescience, to many-body physics, including preparation and characterization of relatedphenomena, eg. optical lattices [13–17]. Phillips’ research resulted in the Zeeman slower(ZS) for laser cooling which solved the detuning limitation for slowing atoms. This isthe main tool in our laser cooling investigation.The principle of the ZS relies on a laser, resonant with the atoms’ energy leveltransition frequency, that interacts with the incoming evaporated particles, recoilingthem, hence, reducing its velocity. However, given the atoms’ recoil, a Doppler shiftoccurs, thus, making the atoms observe a laser frequency different from their own. Tocorrect this, a magnetic field is applied so that the energy levels of the atom split (theZeeman effect) and are again in resonance with the laser. For a comprehensive review,see [5, 18–20]. We used a variation of the ZS in which the magnetic field changes its . Chavarr´ıa-Sibaja et al µ -balance (QCM) to quantify the atoms’ velocity. This technique wasinspired by considering that through the measurement of frequency one can indirectlyobtain other quantities and describe, for example, relativistic effects and electromagneticfields [22,23]; thus, from the deposition of atoms at the balance, the frequency variationsof the QCM can be translated to momentum exchange from the incoming atoms [24].This rate of exchange can then be converted into a variation of velocities and, then, asa percentage of reduction from the initial velocity of the atoms. The verification of thisdeceleration is confirmed by a computational model based on the Fokker-Planck equation(FPE). The FPE has mathematical properties which ease the study dynamic variablesand has been applied in atom cooling as in [25, 26]. With the QCM implementation andthe use of a numerical model, a different approach to the atom’s velocity detection isachieved, dispensing the use of high-technology equipment to report these velocities.First, in section 2, we present the numerical model based on the FPE, and abrief explanation of its operation, prioritizing on how it translates to the experimentalobservations of atom deceleration. In section 3, the experimental setup is described,and, in section 4 and 5, we explain the experimental methods carried during the trialsand the analysis of the obtaining data. Finally, in section 6, the results are discussedand compared with those of the numerical model and, in section 7, the conclusions andoutlook of the experiment are set.
2. Numerical Model
As mentioned before, given that we used a QCM to measure the variation of velocities, anumerical model is implemented so that a simulation of the deceleration of the atoms canbe obtained and, as byproduct, verify the experimental results. This method is based ona stochastic partial differential equation, which in our case is called the Fokker-Planckequation [27]. In this case, the FPE translates the interactions between the atoms andphotons to velocity distributions that describe the atom cooling process.In order to associate the FPE with the physical phenomenom of atom cooling asemiclassical approach of laser cooling, based on a quantum mechanical treatment ofthe interaction between the light field and the atoms and a classical treatment of the . Chavarr´ıa-Sibaja et al Figure 1.
Time evolution of the particles’ distribution. i) The atoms enter our ZeemanSlower (ZS) with a Gaussian distribution given by the spread of the Gd evaporation. ii)During the passage through the first stage, the gaussian distribution gets flattened dueto the action of the laser cooling. iii) The process continues after the spin flip sectionand the bell continues to flatten in a more pronounced way due to a mayor magneticfield gradient. iv) The velocity distribution reaches its maximum flattening at the endof the ZS. The flattening of the center of the bell is an expression of slowdown processof the atoms since the interactions happen on the center portion of the distribution;hence, slowing the particles in this region first and consequently flattening the surface,allocating atoms in a more uniform distribution because of their reduced velocity. . Chavarr´ıa-Sibaja et al v r = (cid:126) k/M , where (cid:126) is Planck’sconstant, h , divided by 2 π , k is the wave number, and M is the atomic mass. Therefore,the velocity distribution, W(v) , of the atoms is described by the Fokker-Planck equation: ∂∂t W ( v ) = − M ∂∂v F ( v ) W ( v ) + 1 M ∂ ∂v∂v D ( v ) W ( v ) , (1)where F(v) is the velocity dependant force and
D(v) is a diffusion coefficient. Here,the first term of the equation is negative given that the force (radiation pressure) uponthe incoming atoms slows the atoms, while the diffusion coefficient adds variability tothe spread of the incoming atoms, For the extent of this paper, the diffusion coefficientis taken as Gaussian noise given that the atoms’ movement is considered as Brownian.This assumption is valid because the characteristics of the experimental setup generatea mean free path greater than the size of the space of interaction [28]. The velocitydependant force does not evolve directly with the time solution of the FPE. Rather,it is calculated by propagating the time solution through the experimental setup, suchthat the possible atom-photon interactions, Zeeman effect and the velocity itself aretaken into account. These last phenomena are a product of the Doppler effect given aninteraction between an atom and a photon, where the Zeeman effect is a compensationfor the possible detuning of this event.To solve the FPE that describes the previous behaviour, the matrix numericalmethod proposed in [29] is employed. Here, the plane in which the atoms travel isdiscretized such that, given the transition probability between adjacent points on thelattice, a transition rate is found. Then, the time evolution of the velocity distributionis obtained by a time-ordered exponential that sums over all the paths possible in thediscretized space. An extensive explanation of the procedure is found in [29].Note that the previous algorithm finds the evolution of the FPE as if it is stationary,e.g. not moving through the experimental setup. Therefore, the solution is propagatedthrough the setup such that the velocity through any given point in space can bedetermined. Thus, it is possible to obtain the velocity of the atoms depositing in theQCM. Figure 1 shows the results of the numerical model, where a flattening of the atoms’distribution occurs as their velocity is reduced. Here, the subfigure (i) corresponds tothe particles’ distribution at the entrance of the Zeeman Slower, (ii) and (iii) show theflattening of the curve before the spin-flip and after the spin-flip, respectively, and (iv)shows the final distributions when atoms are at the µ -balance. The flattening of thesurface indicates the slowing of the atoms’ by spreading the distribution through time. . Chavarr´ıa-Sibaja et al
751 m/s 326 m/s
Atom velocity
Figure 2.
Artistic visualization of our experimental setup using for the measurementof the slowdown effect of the Gd (I) atoms using the variation of frequency QCM.The technical details of our ZS is exposed in the reference [20]. The oven (left side ofthe figure) produces an atom beam by means of a hot filament. Initially, the velocityof the beam is (738.62, +66.66, − ±
3. Experimental Setup
Figure 2 shows a scheme of our experimental setup for the cooling of Gd atoms. Theexperiment was done in a high vacuum system with total internal volume ∼ × − m . The main part of the volume is used for the application of the magnetic field whilethe rest is used by the source of atoms and the QCM, both placed inside the apparatus.The internal base pressure is maintained at 10 − Torr by one ion pump model VarianVacion Plus 25 Triode of 20 L/s capacity and is constantly monitored by a pressuregauge model Bayard-Alpert gauge during all of the measurements reported below.For our experiment, neutral Gadolinium Gd (I) of 99.9 % purity is used. The Gd(I) has Z = 64 and it most abundance isotopes are five bosonic isotops:
Gd (0.2 %),
Gd (2.15 %),
Gd (20.47 %),
Gd (24.87 %) and
Gd (21.90 %), and anotherthree fermionic isotops:
Gd (14.73 %) and
Gd (15.68 %) [30–32]. Also, it belongsto the Lanthanide group that has partially filled 4 f and 5 d inner shells of electrons [33].This partially filled shield generates a rich very spectra produced by the overlappingelectronics configuration present making difficult to resolve the the individual lines ofthe different electronic states of the Gd(I). Also, the Gd (I) has a mass m Gd = 157.25 . Chavarr´ıa-Sibaja et al − N ∼ evaporated particles per second, causingan increasing of pressure of the system up to the 10 Torr range.Furthermore, we use spin-flip Zeeman slower (ZS), the technical details of which canbe found in the reference [20]. Our ZS generated magnetic field values in a range from53.5 G to 374.8 G for a current from 1 A to 7 A, respectively. During the experiment,we measured the current values applied to the ZS coils due to the impossibility of adirect measure of the magnetic field inside the apparatus. Also, the range of currentsused was determined during the development of the experiments in order to map theefficiency of our system.Moreover, in addition to our ZS, we use a blue laser model Laserglow Polaris-100.As shown in the 2, we align the laser slightly offset from the central axis of the Zeemanslower due to the presence of the QCM as shown in the Figure (1), so that both theQCM, the oven and the laser beam are in the same plane. The above promotes agreater number of interactions between the laser and the atom beam when the magneticfield is on. Also, the laser has a Gaussian distribution emission centered at 447.10 nm(6.71 × MHz), a FWHM of 2.13 nm (1.40 × MHz) with an initial power of P L = 100 mW and a beam diameter of 4 mm. However, P L is reduced to P o = 25.88mW inside our Zeeman slower generating a intensity I o = 2.085 × mW/cm . Thus,we use the laser to interact with two different dipole transitions of the Gd(I). Thefirst transition is located at 443.063 nm and corresponds to the transition between theground state with total angular momentum J = 2 and the excited state with J = 3,both with electronic spin number S = 4. These states have spectroscopic terms D and D respectively [35]. On the other hand, the second transition is located at λ = 451.965nm and corresponds to an intermediate transition between a state with J = 3 and astate with J = 2, with D while the second is not defined [35]. The two transitionshave measured lifetimes of τ . = 13.7 ns and τ . = 10.8 ns [36] and saturationintensity I . = 1.06 × mW/cm and I . = 1.35 × mW/cm . This intensityvalues generate ratios of I . /I sat = 1.544 and I . /I sat = 2.7456 respectively. Alsothe transitions suffer a Doppler-broadening of 15.43 GHz and 15.15 GHz respectively . Chavarr´ıa-Sibaja et al ν . = 9.38 × GHz for the D - D transition and∆ ν . = 6.29 × GHz for the transition. The transitions detuning are adjustedby the Zeeman effect according to theory [5, 20, 37].After the ZS section, we measure the momentum exchange of the Gd atoms witha QCM that has a surface detection area of 5.03 × − cm . The QCM is coupledto a thickness monitor model Maxtek TM-200 and a frequency meter model HewlettPackard 53131A universal counter. The QCM is a device normally used in thin filmdeposition processes and its operating principle is based on monitoring the change ofnatural frequency oscillation values of a quartz crystal, which are modified due to thedeposited mass in the surface of the crystal. This system allows us to measure anyfrequency variation for the QCM with a resolution of ±
4. Use of the QCM to measure the deceleration of Gd(I) atoms generatedfor a Zeeman slower
To begin with the experiment, we focus our attention on the measurement of theresonance frequency of the QCM. In this regard, we can first assume that any frequencyperturbations are only produced by two factors: the number of particles hitting persecond and the heat transfer to the QCM from the oven. The first factor generatesa momentum exchange that we want to measure with the QCM while the second is asource of undesirable noise in the measurement that is necessary to reduce.Therefore, in preparation for our experiment we carry out heating of our vacuumsetup from room temperature to ∼ C including an annealing of the oven, applyinga current of 1 A during eight hours. This heating reduces the impurities making thesystem reach a base pressure of 10 − Torr when the oven is off and increasing the meanfree path of the atoms inside it. The above allows us to consider a continuous flowof atoms between the oven and the QCM that undergoes without significant change.Thus, we can assume that the Gd (I) atoms move out from oven to the QCM withoutinteraction with other particles in their trajectory and only the desired effects can beconsidered. These results lets us ignore any difference between the number of atoms thatare emitted by our source and the number that reaches the QCM. Hence propitiatingthat any resonance frequency perturbation of the QCM will be created mostly by thearrival of the atoms to the surface of the crystal at a constant rate. . Chavarr´ıa-Sibaja et al Time (s) F r e qu e n c y s h i f t ( H z ) B Figure 3.
Time evolution of the frequency drop during our experiment (purple dots).We have three stages of five minutes each. In the white areas, we have the periodswhen our apparatus is Off and there no disturbances affecting the QCM frequencysignal. In the shaded purple area, a perturbation of the frequency signal is introducedby action of our Zeeman slower (ZS). The change in the slopes is more evident withthe linear fits done (violet line). Also, we have that due to the shape of the laser beam,when the ZS is On, not all atoms are slowed down.
Similarly, from the point of view of the QCM, the direct exposure to the heatradiated by the oven and the joule effect of the ZS can increase the measured frequencyvalues. The above creates a shielding effect that makes it impossible to determinethe frequency variation due to the effects of our interest. We reduce the shieldingeffect by slowly increasing the electric power up to 1000 W in the oven. Therefore,the temperature of the oven increases what is necessary to produce a constant flux ofatoms without creating an excessive and abrupt increase in the frequency of the QCM.Additionally, we use a continuous flow of water to extract the excess of heat from theQCM holder to stabilize the device’s temperature in a short time and to cool the ZScoils. The flow of water not only reduces the temperature of the QCM but also reducesthe noise of the frequency signal to an average value of (0.0451 ± ± . Chavarr´ıa-Sibaja et al s ) is taken for comparison with subsequentmeasurements.(iv) While the flux of atoms continues to the QCM, we turn On the ZS and the laserat the same time and maintain these conditions for five minutes. We maintainthe intensity of the laser at a stable value of 91.17 mW while the ZS is mon-itored to maintain a constant operating current value. If any slowdown pro-cess is generated a variation in the rate of incoming atoms to the QCM will beproduced. Therefore, a new slope ( s ) in the frequency curve is obtained, co-inciding with what was observed in the central shaded region of the Figure 3.(v) We turn Off the ZS and the laser at the same time and wait for other five minutesin order to see an approximate recovery in the fall the frequency signal. We observea downward slope ( s ) similar to the one in the right white section of the Figure 3with a trend close to that observed for the curve in the step (iii).(vi) We repeat this cycle until a minimum of six tests with favorable results for differentvalues of current applied to the ZS are attained. . Chavarr´ıa-Sibaja et al
5. Characterization of the momentum exchange produced by Zeemanslower
As mentioned in the reference [24], due to the momentum exchange with the atoms andthe deposition process in the surface of the QCM, it is possible to relate the measuredfrequency change with the momentum of the atoms. Therefore, in our experiment, theatoms exchange momentum with the QCM when deposited on it surface. The processcauses a constant rate of fall of natural oscillation frequency value of the QCM thatonly depends of the number of atoms arriving at the QCM surface. Consequently, if thepower of the oven is maintained constant, the number of atoms arriving and interactingwith the QCM will be approximately constant. Also the rate of fall of the frequencymeasured by the QCM will be nearly constant too. Thus, any perturbation observedover the rate of fall of the frequency signal will produced by the action of the laser andthe Zeeman slower (ZS) over the atoms.Figure 3 shows the effect on the slope in the three steps of our experimental cycle.The step (iii) and (v) corresponds with the white areas representing a free change of thefrequency of the QCM. In this white areas, there is no deceleration of the atoms andthey remain unperturbed during the trajectory from the oven to the QCM. Instead, theshaded area that corresponds with step (ii) where the slowdown process is present onthe atoms, which generates a positive increase in the slope value of the frequency. Asa result, we can estimate the variation of the momentum exchange between the atomsand the QCM in each experimental cycle and find the maximum efficiency current valueof our ZS.Consequently, we classify the data generated during each experimental cycleaccording to the current value used in the ZS coils. Also, we separate each data series inthe three steps mentioned and perform a linear fit on each one to obtain the value of s , s and s . Hence, we impose a selection criteria based on our observations realized forthe variation of the slopes and establish two conditions for a successful test. Therefore, ifwe get that s < s and s ∼ s the test can be classified as successful. Then, we estimate∆ s the percentage of variation between s and s and ∆ s the between s and s foreach experimental cycle. As a result, when the laser is getting closer to resonance with . Chavarr´ıa-Sibaja et al s is expected. The tendency continues until amaximum efficiency value for ∆ s is reached. After, the laser comes out of resonancewith the electronic transition again because the excessive Zeeman effect and ∆ s dropsagain. On the other hand, we only expect a little variation of the values of ∆ s becauseduring step one and step two there is not a slowdown process and therefore s and s must have similar values.To perform the analysis for the large number of test carried out withdifferent current values, a computer based program was developed using thePython programming language. The program uses PyWavelets [38], Pandas,Numpy [39] and Scipy [40] libraries for cleaning and processing data andreduce the signal noise to average value of (0.0267 ± ± ± ± s , s and s are calculated using a linear fit.Afterwards, the program compares s against s and s , and selects only the tests thatmeet the selection criteria previously mentioned. The analysis is performed by matching s to the 100% of the undisturbed fall of frequency and also representing a maximummomentum exchange between the atoms and the QCM. Following, the s is calculatedand compared with s to obtain the percentage of variation of s and the reduction inthe momentum exchange due to the effect of our ZS. The program can be accessed at https : //github.com/Rocketman /Zeeman P roject.git , and is open to download,use, modify and improve for free.Figure 4 presents the resulting efficiency curve for our experiment beginning with acurrent value of 0 A where ∆ s is initially 0 %. The above corresponds with no changein the the momentum exchange between the atoms and the QCM where no currentis used in the ZS and, therefore, no magnetic field is present. After, the value ∆ s begins to increase as more current is applied to the ZS, obtaining frequency curves likethe showed in the Figure 3 and indicating a reduction of momentum exchange. Thetendency continues until a maximum efficiency current of 3.8 A is reached, producing areduction of the (43.45 ± s reaches itsmaximum value.Next, once the maximum efficiency current value is exceeded, the efficiency of our . Chavarr´ıa-Sibaja et al M o m e n t u m e x c h a ng e ( a r b . un it s ) Current (A)
Figure 4.
Observed percent of momentum exchange with respect to theapplied currents values. The experimental values (blue dots) are obtainedcalculating the average change between the slope obtained when our ZS isOff and when is On for any current value applied. Hence, we can get anefficiency percentage and, through the application of the Gaussian fit (greenline), we can obtain the location of the maximum of efficiency. For ourexperiment, the maximum of efficiency of our ZS is at (43.45 ± ZS begins to decrease as expected. As a consequence, the momentum exchange increaseand ∆ s are reduced until a limit of detection for the QCM is reached. The aboveis because, although there may be changes in the value of the momentum exchangedbetween the QCM and the atoms, the magnitude of the natural noise of the QCM and∆ s begins to be comparable. For our experiment, the natural noise of the QCM isof 0.0451Hz and begins to be meaningful after 7 A of current applied to our ZS andany measurement after this point is not possible with the actual configuration of ourexperimental setup.
6. Determination of the final velocity of Gd(I) atoms and lifetimes for443.06 nm and 451.96 nm transitions from the mechanic perturbation ofthe QCM frequency
The atoms in a gas do not have a uniform velocity. Thus, we need a good parameter tocompare the changes produced by the cooling process taking into account the aleatoryvelocities of the atoms inside the vacuum system for better modeling. Therefore,the Maxwell-Boltzmann distribution (MBD) and its average velocity at a certaintemperature T is chosen to describe the behaviour of the atoms in a beam [5, 41].Moreover, the evolution of the MBA through the Jaymes-Cuming Model (JCM) and . Chavarr´ıa-Sibaja et al τ . In our case, we use two different values of τ measured using time-resolved laser-induced fluorescence (TRLIF) on a slow atomic beam and reported byE. A. Den Hartog et al in reference [36]. The values reported for each transition are τ . = 13.7 ns for the 443.06 nm transition and τ . = 10.8 ns for the 451.96 nm andwe take it as a reference for our numerical model. Thus, we compared the reported valuesof τ . and τ . with our τ values ( τ e ) obtained by our model and the experiment.There is an estimate of 2.00 × Gd(I) atoms hitting the surface areaof the QCM per second, therefore, we use a statistic view for the process.For the first approach, the atoms have an average initial experimental velocity v ie = (751.62, -53.38, +66.66)m/s according to the MBD when they leave out the oven atT = (4051.90, -53.3879, +66.66)K. Hence, when we take into a count at 443.06 nm, wehave a theoretical transition linewidth Γ . = 1/ τ . = 7.30 × MHz. Thus, theatoms fills a maximum theoretical deceleration a max = (cid:126) k Γ . / m Gd = 3.33 × m/s when our ZS is working at maximum efficiency according to the JCM [5, 42]. Therefore,we use our numerical model to study how the evolution of the MBD is affected bya tmax during the cooling process as showed in Figure 1. As a result, we obtain afinal average velocity of the model of v fm = 529.81 m/s corresponding with a finalmodel temperature T fm = 2084.76 K reached by the Gd (I) in our experiment. Theresult described indicates a reduction of 29.51% of the atoms velocity according toour numerical model. At the same time, the reduction percent directly measured bythe QCM was (43.45 ± v fe = (326.57, +29.03, -23.12) m/s. As a result, the final experimental temperatureobtained for v fe was T fe = (767.49, +142.53, -104.84)K. The latter represents adifference of 32.45% between the reduction value obtained by our numerical model incompassion with the experimental value obtained by the QCM. The results indicate aclear probability that we are cooling Gd (I) atoms based in the D - D electronictransition and we are able to measures the effect of the cooling process with theQCM. In a similar way, for the 451,96 nm transition using the first approach, we havea theoretical transition linewidth Γ . = 1/ τ . = 9.26 × MHz that producea max = 4.14 × m/s . This deceleration generates a v fm = 457.80 m/s according withthe study evolution of the velocity distribution with the JCM obtaining a T fm = 1556.57K. Therefore, for the transition of 451.96 nm, we obtain a reduction of 39.09% of theatom velocity according with our numerical model. The obtained percent presents adifference of 10.03% between the results obtained by our numerical model for 451.96 nmtransition in compassion with the resulting measure by the QCM. . Chavarr´ıa-Sibaja et al τ e from the experimental data obtained. The method takes into account the value of v ie and with the reduction percent obtained by the QCM in our experiment, we estimate v ef .But in this second approach, we impose the condition v im = v ie and v fm = v fe as initialand final average velocity values respectively for our numerical model. Thus, taking intothe reduction obtained of account the reduction of (43.45 ± τ e = 8.1988 ns. Then, we compare τ e with τ m obtaining a difference of 40.15% and 24.08 % between both values for the transitionof 443.06 nm and 451.96 nm respectively. Despite the difference between τ e and τ m , weconsider that that τ e is consistent with the reported value in [36]. The above due to thatthe operating principle of the QCM does not have the necessary sensitivity required fora better measurement of τ e by the actual use of our method. In addition, it may benecessary to add some modifications to our experimental setup like a collimator for abetter control of the spread and evolution of the atom beam as example. Hence, weconsider the necessary improvement and, therefore, the modification of our experimentalsetup to achieve more accuracy for our measurements in future experiments.However, we think that our technique is a new and innovative way to measure thelifetime of the electronic transitions based in the mechanical operating principles of theQCM. We can compare our method with TRLIF that can be carry out by two methods:pulse method and phase modulation-method, where the first is in time domain and thesecond in frequency domain [43]. Therefore, it can identify spectrally and temporallythe different frequencies and lifetime of the electronic transition of the atoms [44]. Hencethat the TRLIF is a predominant method of measure in laser cooling experiment. But itmay need different equipment like photo-multipliers, charge-coupled device (CCD). Asexample, others instrument used to carry out TRLIF are mentioned in references [43–45]as a part of different setups for independent purposes. On the other hand, the use ofthe QCM requires simple equipment that is more accessible for many not specialisedlaboratories. QCM is a pluck and play well known instrument that is normally usedin experimental solid state physics application. Thus, its use is simple and easy toimplement for the measurement of the speed change of atoms by exchanging momentumatoms-QCM. Due to the mechanical operating principle of the QCM, we cannot be usedour technique to determinate the excited population. But we think that the ability tomeasure the momentum exchange with the QCM in a laser cooling experiment allows usto obtain enough data to characterize the slowdown effect generated by a ZS. Also, ourmethod can make available to measure the lifetime of different electronic transition bya non optical technique as a first approach when specialized equipment is not available.As a result, we consider the possible future exploration of our technique as a new way ofmixing proprietary elements used in experimental solid state physics in a field such asultra-cold atom physics. We think that this implementation can be generate new fieldsof study and development of applications in experimental physics in future projects. . Chavarr´ıa-Sibaja et al
7. Conclusions
In conclusion, we demonstrated how a QCM can be implemented as an energy transfersensor to detect the change in the velocity of a beam of atoms produced by a Zeemanslower (ZS) in laser cooling experiments. By the use of the technique of the reference [24],our experiment is based in continuous monitoring of the perturbation of the oscillationfrequency of the QCM. We also improve the analysis of the measurements of frequencysignal of the QCM by the use of a post-processing software and estimate the change inthe dynamic between the hitting atoms and a QCM in time. From these measurements,we determine the maximum efficiency working parameter of our ZS. Moreover, we createa numerical model to study the particle dynamics of a beam with a Maxwell-Boltzmannvelocity distribution (MBD) during a laser cooling experiment. Thus, from this modelwe obtain a theoretical prediction of the final average velocity for MBD of a Gd (I)atoms beam that interact with a laser beam of 447.10 nm and a magnetic field of aZS. Furthermore, with the careful comparison between our numerical model and theexperimental data we also estimate the lifetime values of the 443.06 nm and 451.96nm transition of the Gd(I). We compare our results with the values reported in [36]as reference obtaining 40.15% . Furthermore, we determine the viability of the QCMas a mechanical sensor to measure quantum properties that are normally measured byrather complex optic systems in laser cooling experiments.With this set of results, we propose a simple and easy way to implement a pluckand play instrument like QCM in a laser cooling experiments to study the change in thedynamics of the particles during the cooling process. QCM is a well known device usedin solid state physics and its main advantage is its easy operation. Additionally, dueto its mechanical operating principle, the QCM is a good sensor for the study of thekinetic evolution of a particles distribution. Therefore, the implementation of the QCMin the laser cooling experiments can contribute in the development of new applicationof experimental method between solid state and ultra-cold atom physics. Our resultsconfirm that the use of a QCM can make available an innovative mechanical method tostudy the kinetic evolution of the atoms inside a ZS to obtain a deeper understandingof the cooling process.
8. Acknowledgments
We want to acknowledge Milena Guevara Bertsch for her work in the construction ofour Zeeman slower and the contribution on the first steps that were the base of ourexperiment. We also want to thank Alejandro God´ınez Sand´ı for his effort on the initialsteps of the experimental executions. The authors are very grateful for the supportgiven by the Vicerrector´ıa de Investigaci´on of the Universidad de Costa Rica to carryout this research work. . Chavarr´ıa-Sibaja et al References [1] C.E. Wieman, D.E. Pritchard, and D.J. Wineland. Atom cooling, trapping, and quantummanipulation.
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