Atomic if we slow down these atoms

   AtomicPhysicsStudent:Ranim TomanReg. #:21410578Instructor:Salman M SalmanTopic:Lasercooling & trappingIn science fictionmovies, LASER means hot! With uses ranging from LASER guns, LASER shooting outof eyes to burn stuff, cut through things and many other hot         – and somewhat fictional- uses.However it turned out that LASERs have a very unusual, but very remarkableimplementation…Cooling! It might seemcounterintuitive, but LASERs are actually used to cool atoms down totemperatures near absolute zero! The basic idea of LASER cooling is that whenan atom absorbs and re-emits a photon, its momentum changes, and for anensemble of atoms, their temperature is proportional to the variance in theirvelocity, so if we slow down these atoms and lead them towards more homogeneousvelocities, they will cool down.

But why Lasers?  LASERsThe acronym LASER stands for:Light Amplification by Stimulated Emission of Radiation.A LASER is a device that mainly consists of the following components:-         A mirror with a reflectivity of %100 is installed atone end of the device.-         A gas, a liquid or a solid that serves us the gainmedium and creates identical photons.

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–         Another mirror with a reflectivity of about %98(<%100) must be installed at the other end of the gain medium.-         An energy source, in order to give out photons toexcite electrons (optical pumping)Now, the gain medium is where the magic happens, the electrons inside theatom are in their ground state (E0). Now, optical pumping, which isemitting energy in the form of photons, is used to raise the electron into ahigher energy level (E2), after a short period of time, the electronjumps into a lower energy level (E1) without emitting a photon,consequently, the process of optical pumping cannot cause stimulated emission,because there are two amounts of energy present. The electron will maintain itsenergy level (E1) for some time, then, the atom will spontaneouslyemit a photon in a random direction as it relaxes to a lower electronic state-the ground state-, this photon might interact with an excited atom that has notemitted a photon yet, so, the photon stimulates the excited atom and causes astimulated emission, because the photon has the exact amount of energy to doso. Only photons emitted in a direction perpendicular to the mirrors will bereflected, the reflected photons will initiate a chain reaction, to producemore and more photons of the same kind. Furthermore, only photons with the sameamount of energy and momentum will be part of the chain reaction, this is thereason why LASER beams are strongly coherent and monochromatic.

The very important aspect is the fact that many atoms must be in an excitedstate. For LASER to work, more atoms must be in an excited state than in lower energystates, this is called “population inversion”. Now, the semi-transparentmirror allows some of the LASER energy to be emitted, while reflecting most ofit back through the LASER.So in a nutshell, when an atom is excited by electricity, a chemicalreaction, or light, and it gets stimulated by a photon of a particular energy,the atom will simultaneously emit a daughter photon with exactly the sameenergy, direction and momentum of the other photon (stimulated emission), if weamplify this signal, we get LASER and end up with the followingcharacteristics:1-      Mono-chromaticity(same wavelength and frequency).

2-      Coherence (samephase).3-      High intensityand directionality (narrow beam).Which is in contrary to other ordinary sources of light, which emit wavesin all directions (highly divergent) and cancel out each other after shortdistances due to the incoherence of these waves, and somewhat broad range ofwavelengths.Now, we know why LASERs are best suited for cooling atoms. If it weren’tfor coherence, mono-chromaticity and directionality it would be a mess, and wewouldn’t be able to control the process because we don’t know what would behappening with ordinary light!We know why we use LASERs to cool atoms down, but what does it mean to becool in the first place?                                                           Cooling & TrappingFirst of all, hot and cold are related to thermal energy, which is related tokinetic energy. Hot is simply a measure of how much energy something has, andthis energy takes translational, rotational and vibrational forms. Cold meansthe absolute opposite, when something gets cold it means it’s getting stillerand stiller, first of all, things lose their vibration, meaning they get lockedinto their basic shape, then the rotation is lost; things stop rotating, andfinally things lose translation and become very still. Now of course when theyare stationary, we mean (?X=0)  and wecannot do that due to Heisenberg’s uncertainty principle, that says we cannotsimultaneously know the position and momentum of a particle, so we’d never getthe atoms to a full stop, they will always have zero-point motion.

So, if we slow atoms down and make them more still, we are actually coolingthem down, and of course gases have atoms and molecules in constant, randommotion, so in order to slow them down we have to push them in the oppositedirection of that motion, and to do that we need to get down to their level,using light (LASER). Light has momentum in addition to energy (even though it’smassless), so if an atom moving in a certain direction absorbs a photon movingin the opposite direction, that momentum of the photon gets transferred intothe atom and that will slow it down, and thus cool it. However, gasses are inconstant and random motion, so not all of the atoms are moving in the oppositedirection to the LASER beam, so we’ll need 6 LASERs or 3 LASERs with mirrors tomake sure we slow down atoms moving in any combination of the six directions.In a nutshell, we want the thermal motion of the atoms to reach itsminimum, but never zero, because we cannot get to absolute zero due to atomshaving a finite zero-point energy in quantum mechanical description. But wemanaged to get the atoms down to micro kelvins (?K).  Laser cooling techniquesThere are multiple techniques used to achieve LASER cooling, the reasonthere is more than one, is that we started with one way, but we were faced withdeficiencies conducting the experiments, and so the new techniques were merelyadditions to the original one to compensate for its insufficiency of adequatecooling.  This original technique isstill being broadly used for cooling atoms, and it is called Doppler cooling,or in its simplest form (optical molasses). Doppler Cooling Doppler cooling was the first technique used to cool down atoms, its principalrelies on the fact that if an atom is moving in a direction and is bombarded bya photon that has its resonant frequency, then the atom will absorb the photon,and thus its momentum will change (loss of momentum) and it will slow down dueto the conservation of momentum.

Now, since the atom has absorbed the photon,it is going to be in an excited state, but not for long, because as the atomrelaxes to its ground state, it is going to emit a photon in a random direction,but with the same momentum of the original photon (spontaneous emission), whatthis does is give the atom a sideway thrust, and after numerous absorptions andemissions (tens of thousands) the atom will come to rest. You see, photons areemitted in random directions, but with a symmetric average distribution, sotheir contribution to the atoms’ momenta will average to zero. Now, thetechnique is called “Doppler cooling”, the reason that is, is becauseof the Doppler effect which is the essence of this method. See, the frequencyof the photons in the LASER beam has to be tuned just below the resonantfrequency of the atoms, so that when the atoms move in the opposite directionto the photons, the frequency of the photons would be ” blue shifted”and the atoms will be able to see the photons and interact with them, i.e.absorb them, and lose a part of their momentum equal to the momentum of thatphoton. But wait, momentum has do to with mass (p=mv) and photons are massless,how do they possess a momentum?  Well, inthe theory of special relativity (E=mc2) is not the full story, thisnotation is only an expression of the energy-mass equilibrium for particlesthat have mass, but in the case of massless photons, we need the wholeequation:E2 = (pc)2 + (mc2)2 ………….

.m=0E = pcp = E/c = h?/cSo photons actually have momentum, which is equivalent to h?/c, what this tells us is that ifwe increase the frequency of the photons, their energy increases, and thustheir momentum. So, tune the frequency of these photons to fit the resonancefrequency of the atoms’ -after taking into account the Doppler effect-, andyou’ll have yourself a cool atom…or will you?Optical pumpingTo elucidate, take a beam of sodium atoms coming out of an oven, movingat a velocity of 105 cm/sec, and we want to slow them down usingyellow light LASER, now the photon interaction with the sodium atom would resultin a recoil velocity Vrec = 3 cm/sec. So, if we want to bring theseatoms to rest, the absorption-emission process will have to occur about 30,000times. However, it is not as simple as that, sodium is not a two-level atom,but it does have two hyperfine levels (F=1 and F=2).

So, as the atom leaves thesource, it starts to absorb the photons, but it might relax to the ground state(F=1) instead of (F=2), and when it does, it will no longer see the photonsthat have a resonant frequency fit for the transition from (F=2) to (F=3), andbecause of this it stops absorbing the photons, this is called “opticalpumping”. To solve the problem, another LASER called a “re-pumper”was used to emit photons that have the energy sufficient to excite an electronout of the unwanted state (F=1), so that the atom relaxes to (F=2) and itcontinues to cool. Hence, the optical pumping problem was solved using a re-pumper. The Doppler shiftAfter dealing with the optical pumping problem, another problem emerged.After a couple hundred absorptions, the atoms stopped absorbing the photons,this is due to the Doppler shift; As the atom repeatedly absorbs and re-emits photons, it slows down to a certainlevel, as it slows down, it goes out of resonance with the photons because theDoppler shift has changed. Since we have tuned the LASER frequency just belowresonance, the frequency of the photons had to be (kv) below the resonantfrequency of the atom at rest. As v (velocity of the atom) decreases, theDoppler shift changes and there is nothing to compensate for that change. Soagain, after about 200 absorptions, the sodium atoms can no longer see the photons,only atoms with the proper velocity to be resonant with the LASER are sloweddown.

So far we have only reached 200 out of the 30,000 absorptions needed tocool and stop the atoms!To solve this problem, two ingenious solutions were offered, one of themwas   “chirp cooling”, and theother was the rather elegant “Zeeman cooling”.   Chirp coolingThe idea of chirp cooling is to keep changing the frequency of the LASERas the Doppler shift changes due to the slowing atoms. Meaning resonance wouldkeep occurring until the atoms are cooled and the spread of their velocitiespeaks. Zeeman cooling This technique suggests the same idea of sustaining the resonance withthe LASER beam, but instead of changing the frequency of the LASER beam, whatthe Zeeman slower offers is the use of a tapered solenoid; a magnet with whichalong its axis, the atomic beam would be directed.

Now, the solenoid has more windings at the entrance than exit, which means thatthe magnetic field would be highest at the entrance (near the atomic beamsource), and it will decrease along the solenoid. Due to this varying magneticfield with which the atoms with varied velocities are moving through, theZeeman effect occurs, and the atomic energy levels would be perturbed,resulting in a resonance that matches the fixed LASER frequency(of course aftertaking into account the Doppler shift into account). So, as an atom with avelocity V0 enters the solenoid -where the magnetic field ismaximum- , it would be in resonance with the fixed LASER, as the atom absorbsand emits photons, it will slow down to the point where the Doppler shiftoccurs, but when that happens, the atom would have already entered a region ofa lower magnetic field, which will change the energy levels’ splits and create adifferent resonance that will match the frequency of the LASER and compensatefor the Doppler shift. At the same time, when an atom with a velocity slightlylower than V0 enters the solenoid, it will not interact with thephotons, but when it reaches a lower magnetic field it will begin to resonatewith the photons and begin to slow down, and similar to the other atoms, thefield-induced Zeeman shift will compensate for the velocity-induced Dopplershift, and resonance will be sustained until the atoms reach the exit of thesolenoid, where they will be  cooled tothe desired temperature, and possess just enough velocity for them to exit thesolenoid and enter the detection region where they will come to rest and havetheir temperature measured.With that being said, the Zeeman slower compensates for the problems of theDoppler shift and the optical pumping.

In that, it is truly elegant.As the atoms entered the observation region cool and slow, they weredetected to have a velocity of 40 m/sec with a spread of 10 m/sec,corresponding to a temperature (in the atoms’ rest frame) of 70 mk(milli-kelvin). So far, we have only decelerated the atoms, but we have not trapped them.Magnetic TrappingThe idea of magnetic trapping is the utilization of the magnetic moments ofneutral atoms. When put in an external magnetic field, atoms with quantizedmagnetic moments will align with the external magnetic field due to the torqueimposed on them by it. Now, if the magnetic moment of the atom is already inthe direction of the magnetic field lines, the energy of the Zeeman states inthis atom will decrease as the magnetic field increases, and so these arecalled “high-field seekers”, because everything seeks minimum energyin the universe.

As for atoms with magnetic moments oriented opposite or in anyother different direction from the magnetic field lines, then the imposedtorque of the magnetic field on the atom will increase the energy of its Zeemanstates, meaning the energy of the atom increases as the magnetic field strengthincreases, and these are called “low-field seekers”. The trick hereis that in free space –vacuum- it is impossible to produce a local maximum ofthe magnetic field magnitude, but a local minimum can be easily produced, sothat only the low-field seekers can be trapped in this minimum. The potentialwell produced in the local minimum is shallow, so the atoms we are trappingmust have low kinetic energy and temperatures of a fraction of a kelvin, whichhas been already achieved using the Zeeman slower.

Also, to make sure that thelow-field seekers are trapped, their magnetic moments’ orientation must remainthe same with respect to the magnetic field lines to make sure that anadiabatic process is sustained and the atoms are successfully trapped.The magnetic trap is made up of two loops (spherical quadrupole) thatwill produce equal field magnitudes or equipotentials that represent theminimum magnitude of the magnetic field and hence, the trap. The process of trapping must beginwith the Zeeman slower, which decelerates the atoms to about 100m/s in thesolenoid. Then the cooling LASER is turned off for about 4ms to allow thecooled and slow atoms to proceed to the magnetic trap.

At this exact moment, currentis allowed to circulate in one coil, and the cooling LASER is simultaneouslyturned back on for 400ms. Because of the effect of Doppler and Zeeman cooling techniquesagain in this region, the atoms are brought to rest, when they are brought torest (stopped) the other coil is energized, producing –along with the othercoil- the equipotentials in the figure and thus the atoms are trapped.However, if the vacuum tube is imperfect then the atoms might be knockedout of the trap by the room-temperature background molecules seeping into thetube. So the vacuum tube must be of high efficiency.

After the atoms have been trapped, the magnetic field is turned off, and aprobe LASER is turned on to determine the velocity distribution of the trappedatoms, by knocking them with varying-frequency photons and observing theirDoppler shifts. We end up with temperatures around300?k.    Optical molassesWe have said that atoms in a gas move very randomly and rapidly, and we wouldneed 6 LASERs in 6 directions to cool atoms moving in any combination of thosedirections (3 degrees of freedom). Optical molasses technique offers just that;it consists of 3 pairs of counter-propagating circularly polarized LASER beamsthat cause 3 orthogonal standing waves intersecting in the region where theatoms have already been decelerated by the other cooling techniques. The basic idea of optical molasses is”polarization gradient cooling”, light coming out of the LASER islinearly polarized, but it rotates around the direction of the LASER beam atvery high rates. Atoms moving in this polarization gradient are more likely toabsorb photons moving in the direction opposite to their motion, than photonsmoving in the same direction, of course this is due to the Doppler effect discussedearlier; the LASER beams are tuned just below the resonant frequency of theatoms, and so the atoms are more likely to absorb photons hitting them head-oninstead of from behind.

This velocity dependent Doppler effect causes animbalance of the radiation forces of the LASER beams where Fmolasses =-?v. This means the light exerts africtional or damping force on the atoms just like a particle being submergedin a viscous fluid, which is why it is called “optical molasses”. The cooled atoms (say sodium)  will havea mean free path ( the mean distance an atom moves before its initialvelocity is damped out and the atom starts to move with a different and randomvelocity) of only 20?m, while the size of the LASER beams causing the coolingis usually around 1cm. So the atom undergoes a Brownian-like motion and getsstuck in the LASER beam. However, optical molasses should not be thought of asa trapping technique, for there is no restoring force keeping the atoms in themolasses, they are just stuck there. Also, optical molasses technique can coolatoms down to 40?k.This technique along with the magnetic trapping technique, result in amagneto-optical trap (MOT).

In the analogy made here, the cooling (damping) effect of the forces wastaken into account. However, this is not how it actually goes; the changingvelocities of the damped atoms will cause force fluctuations that will resultin heating in the system, which sets a limit on how cool we can get throughoptical molasses.  The Doppler cooling limitThe average of the force of the absorption of photons is the scattering forceand the random kick force resulting from spontaneous emissions.

However, wehave not taken into account the effect of the fluctuations in these two processes.The random nature of both the absorption and emission of photons results aheating process, this heating process will be somewhat compensated for by theLASER cooling process (equilibrium between the two) and this is how we get thefinal temperature resulting from LASER cooling. The random addition to theaverage momentum transfer produces a random walk of the atomic momentum and anincrease in the mean square atomic momentum. This heating is countered by thecooling force opposing the atomic motion. The force is proportional to thevelocity of the atoms. The rate at which energy is removed by cooling is F.vwhich is proportional to v2, so the cooling rate is proportional tothe kinetic energy of the atoms.

While the heating rate, proportional to thetotal photon scattering rate, is independent of atomic kinetic energy at lowvelocities. As a result, the heating and cooling come to equilibrium at acertain value of the average kinetic energy. This defines the Doppler coolinglimit to be:TDopp= ??/2KB? is the rate of spontaneous emission of the excited state.?-1 is the lifetime of the excited state.KB is the Boltzmann constant.? is the reduced Planck constant.

For sodium TDopp= 240?k.The atoms were found to be much cooler than the Doppler limit (about 40?k). SummaryLASER cooling and trapping refers to a number of techniques used to cool atomsdown by decelerating them using light, and bringing them to a stop usingmagnetic fields. The reason we want to cool these neutral atoms, is to help developour understanding of the atomic spectra, improve atomic clocks, createultra-cold atoms which exhibit quantum mechanical behavior, so these cold atomscan help us expand our understanding of the quantum world, and if we dig deeperand cool them even more, we create a new state of matter, the Bose-Einsteincondensate which is a remarkable discovery and could help us understand manyphenomena in the universe. Not to mention the use of trapped ions in quantumcomputing -even though my paper is on the cooling of neutral atoms, but it isworth mentioning-.

Reference William D. Phillips, Nobel lecture, 1997. 


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