Salman M Salman
cooling & trapping
In science fiction
movies, LASER means hot! With uses ranging from LASER guns, LASER shooting out
of 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 remarkable
It might seem
counterintuitive, but LASERs are actually used to cool atoms down to
temperatures near absolute zero! The basic idea of LASER cooling is that when
an atom absorbs and re-emits a photon, its momentum changes, and for an
ensemble of atoms, their temperature is proportional to the variance in their
velocity, so if we slow down these atoms and lead them towards more homogeneous
velocities, they will cool down.
But why Lasers?
The 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 at
one end of the device.
A gas, a liquid or a solid that serves us the gain
medium and creates identical photons.
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 to excite electrons (optical pumping) Now, the gain medium is where the magic happens, the electrons inside the atom are in their ground state (E0). Now, optical pumping, which is emitting energy in the form of photons, is used to raise the electron into a higher energy level (E2), after a short period of time, the electron jumps 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 its energy level (E1) for some time, then, the atom will spontaneously emit 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 not emitted a photon yet, so, the photon stimulates the excited atom and causes a stimulated emission, because the photon has the exact amount of energy to do so. Only photons emitted in a direction perpendicular to the mirrors will be reflected, the reflected photons will initiate a chain reaction, to produce more and more photons of the same kind. Furthermore, only photons with the same amount of energy and momentum will be part of the chain reaction, this is the reason why LASER beams are strongly coherent and monochromatic. The very important aspect is the fact that many atoms must be in an excited state. For LASER to work, more atoms must be in an excited state than in lower energy states, this is called "population inversion". Now, the semi-transparent mirror allows some of the LASER energy to be emitted, while reflecting most of it back through the LASER. So in a nutshell, when an atom is excited by electricity, a chemical reaction, or light, and it gets stimulated by a photon of a particular energy, the atom will simultaneously emit a daughter photon with exactly the same energy, direction and momentum of the other photon (stimulated emission), if we amplify this signal, we get LASER and end up with the following characteristics: 1- Mono-chromaticity (same wavelength and frequency). 2- Coherence (same phase). 3- High intensity and directionality (narrow beam). Which is in contrary to other ordinary sources of light, which emit waves in all directions (highly divergent) and cancel out each other after short distances due to the incoherence of these waves, and somewhat broad range of wavelengths. Now, we know why LASERs are best suited for cooling atoms. If it weren't for coherence, mono-chromaticity and directionality it would be a mess, and we wouldn't be able to control the process because we don't know what would be happening with ordinary light! We know why we use LASERs to cool atoms down, but what does it mean to be cool in the first place? Cooling & Trapping First of all, hot and cold are related to thermal energy, which is related to kinetic energy. Hot is simply a measure of how much energy something has, and this energy takes translational, rotational and vibrational forms. Cold means the absolute opposite, when something gets cold it means it's getting stiller and stiller, first of all, things lose their vibration, meaning they get locked into their basic shape, then the rotation is lost; things stop rotating, and finally things lose translation and become very still. Now of course when they are stationary, we mean (?X=0) and we cannot do that due to Heisenberg's uncertainty principle, that says we cannot simultaneously know the position and momentum of a particle, so we'd never get the 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 cooling them down, and of course gases have atoms and molecules in constant, random motion, so in order to slow them down we have to push them in the opposite direction 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's massless), so if an atom moving in a certain direction absorbs a photon moving in the opposite direction, that momentum of the photon gets transferred into the atom and that will slow it down, and thus cool it. However, gasses are in constant and random motion, so not all of the atoms are moving in the opposite direction to the LASER beam, so we'll need 6 LASERs or 3 LASERs with mirrors to make 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 its minimum, but never zero, because we cannot get to absolute zero due to atoms having a finite zero-point energy in quantum mechanical description. But we managed to get the atoms down to micro kelvins (?K). Laser cooling techniques There are multiple techniques used to achieve LASER cooling, the reason there is more than one, is that we started with one way, but we were faced with deficiencies conducting the experiments, and so the new techniques were merely additions to the original one to compensate for its insufficiency of adequate cooling. This original technique is still 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 principal relies on the fact that if an atom is moving in a direction and is bombarded by a 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 due to 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 atom relaxes 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), what this does is give the atom a sideway thrust, and after numerous absorptions and emissions (tens of thousands) the atom will come to rest. You see, photons are emitted in random directions, but with a symmetric average distribution, so their contribution to the atoms' momenta will average to zero. Now, the technique is called "Doppler cooling", the reason that is, is because of the Doppler effect which is the essence of this method. See, the frequency of the photons in the LASER beam has to be tuned just below the resonant frequency of the atoms, so that when the atoms move in the opposite direction to 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 that photon. But wait, momentum has do to with mass (p=mv) and photons are massless, how do they possess a momentum? Well, in the theory of special relativity (E=mc2) is not the full story, this notation is only an expression of the energy-mass equilibrium for particles that have mass, but in the case of massless photons, we need the whole equation: E2 = (pc)2 + (mc2)2 …………..m=0 E = pc p = E/c = h?/c So photons actually have momentum, which is equivalent to h?/c, what this tells us is that if we increase the frequency of the photons, their energy increases, and thus their momentum. So, tune the frequency of these photons to fit the resonance frequency of the atoms' -after taking into account the Doppler effect-, and you'll have yourself a cool atom…or will you? Optical pumping To elucidate, take a beam of sodium atoms coming out of an oven, moving at a velocity of 105 cm/sec, and we want to slow them down using yellow light LASER, now the photon interaction with the sodium atom would result in a recoil velocity Vrec = 3 cm/sec. So, if we want to bring these atoms to rest, the absorption-emission process will have to occur about 30,000 times. 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 the source, 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 photons that have a resonant frequency fit for the transition from (F=2) to (F=3), and because of this it stops absorbing the photons, this is called "optical pumping". To solve the problem, another LASER called a "re-pumper" was used to emit photons that have the energy sufficient to excite an electron out of the unwanted state (F=1), so that the atom relaxes to (F=2) and it continues to cool. Hence, the optical pumping problem was solved using a re-pumper. The Doppler shift After 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 certain level, as it slows down, it goes out of resonance with the photons because the Doppler shift has changed. Since we have tuned the LASER frequency just below resonance, the frequency of the photons had to be (kv) below the resonant frequency of the atom at rest. As v (velocity of the atom) decreases, the Doppler shift changes and there is nothing to compensate for that change. So again, 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 slowed down. So far we have only reached 200 out of the 30,000 absorptions needed to cool and stop the atoms! To solve this problem, two ingenious solutions were offered, one of them was "chirp cooling", and the other was the rather elegant "Zeeman cooling". Chirp cooling The idea of chirp cooling is to keep changing the frequency of the LASER as the Doppler shift changes due to the slowing atoms. Meaning resonance would keep occurring until the atoms are cooled and the spread of their velocities peaks. Zeeman cooling This technique suggests the same idea of sustaining the resonance with the LASER beam, but instead of changing the frequency of the LASER beam, what the Zeeman slower offers is the use of a tapered solenoid; a magnet with which along its axis, the atomic beam would be directed. Now, the solenoid has more windings at the entrance than exit, which means that the magnetic field would be highest at the entrance (near the atomic beam source), and it will decrease along the solenoid. Due to this varying magnetic field with which the atoms with varied velocities are moving through, the Zeeman effect occurs, and the atomic energy levels would be perturbed, resulting in a resonance that matches the fixed LASER frequency(of course after taking into account the Doppler shift into account). So, as an atom with a velocity V0 enters the solenoid -where the magnetic field is maximum- , it would be in resonance with the fixed LASER, as the atom absorbs and emits photons, it will slow down to the point where the Doppler shift occurs, but when that happens, the atom would have already entered a region of a lower magnetic field, which will change the energy levels' splits and create a different resonance that will match the frequency of the LASER and compensate for the Doppler shift. At the same time, when an atom with a velocity slightly lower than V0 enters the solenoid, it will not interact with the photons, but when it reaches a lower magnetic field it will begin to resonate with the photons and begin to slow down, and similar to the other atoms, the field-induced Zeeman shift will compensate for the velocity-induced Doppler shift, and resonance will be sustained until the atoms reach the exit of the solenoid, where they will be cooled to the desired temperature, and possess just enough velocity for them to exit the solenoid and enter the detection region where they will come to rest and have their temperature measured. With that being said, the Zeeman slower compensates for the problems of the Doppler shift and the optical pumping. In that, it is truly elegant. As the atoms entered the observation region cool and slow, they were detected 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 Trapping The idea of magnetic trapping is the utilization of the magnetic moments of neutral atoms. When put in an external magnetic field, atoms with quantized magnetic moments will align with the external magnetic field due to the torque imposed on them by it. Now, if the magnetic moment of the atom is already in the direction of the magnetic field lines, the energy of the Zeeman states in this atom will decrease as the magnetic field increases, and so these are called "high-field seekers", because everything seeks minimum energy in the universe. As for atoms with magnetic moments oriented opposite or in any other different direction from the magnetic field lines, then the imposed torque of the magnetic field on the atom will increase the energy of its Zeeman states, meaning the energy of the atom increases as the magnetic field strength increases, and these are called "low-field seekers". The trick here is that in free space –vacuum- it is impossible to produce a local maximum of the magnetic field magnitude, but a local minimum can be easily produced, so that only the low-field seekers can be trapped in this minimum. The potential well produced in the local minimum is shallow, so the atoms we are trapping must have low kinetic energy and temperatures of a fraction of a kelvin, which has been already achieved using the Zeeman slower. Also, to make sure that the low-field seekers are trapped, their magnetic moments' orientation must remain the same with respect to the magnetic field lines to make sure that an adiabatic process is sustained and the atoms are successfully trapped. The magnetic trap is made up of two loops (spherical quadrupole) that will produce equal field magnitudes or equipotentials that represent the minimum magnitude of the magnetic field and hence, the trap. The process of trapping must begin with the Zeeman slower, which decelerates the atoms to about 100m/s in the solenoid. Then the cooling LASER is turned off for about 4ms to allow the cooled and slow atoms to proceed to the magnetic trap. At this exact moment, current is allowed to circulate in one coil, and the cooling LASER is simultaneously turned back on for 400ms. Because of the effect of Doppler and Zeeman cooling techniques again in this region, the atoms are brought to rest, when they are brought to rest (stopped) the other coil is energized, producing –along with the other coil- the equipotentials in the figure and thus the atoms are trapped. However, if the vacuum tube is imperfect then the atoms might be knocked out of the trap by the room-temperature background molecules seeping into the tube. So the vacuum tube must be of high efficiency. After the atoms have been trapped, the magnetic field is turned off, and a probe LASER is turned on to determine the velocity distribution of the trapped atoms, by knocking them with varying-frequency photons and observing their Doppler shifts. We end up with temperatures around 300?k. Optical molasses We have said that atoms in a gas move very randomly and rapidly, and we would need 6 LASERs in 6 directions to cool atoms moving in any combination of those directions (3 degrees of freedom). Optical molasses technique offers just that; it consists of 3 pairs of counter-propagating circularly polarized LASER beams that cause 3 orthogonal standing waves intersecting in the region where the atoms 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 is linearly polarized, but it rotates around the direction of the LASER beam at very high rates. Atoms moving in this polarization gradient are more likely to absorb photons moving in the direction opposite to their motion, than photons moving in the same direction, of course this is due to the Doppler effect discussed earlier; the LASER beams are tuned just below the resonant frequency of the atoms, and so the atoms are more likely to absorb photons hitting them head-on instead of from behind. This velocity dependent Doppler effect causes an imbalance of the radiation forces of the LASER beams where Fmolasses = -?v. This means the light exerts a frictional or damping force on the atoms just like a particle being submerged in a viscous fluid, which is why it is called "optical molasses". The cooled atoms (say sodium) will have a mean free path ( the mean distance an atom moves before its initial velocity is damped out and the atom starts to move with a different and random velocity) of only 20?m, while the size of the LASER beams causing the cooling is usually around 1cm. So the atom undergoes a Brownian-like motion and gets stuck in the LASER beam. However, optical molasses should not be thought of as a trapping technique, for there is no restoring force keeping the atoms in the molasses, they are just stuck there. Also, optical molasses technique can cool atoms down to 40?k. This technique along with the magnetic trapping technique, result in a magneto-optical trap (MOT). In the analogy made here, the cooling (damping) effect of the forces was taken into account. However, this is not how it actually goes; the changing velocities of the damped atoms will cause force fluctuations that will result in heating in the system, which sets a limit on how cool we can get through optical molasses. The Doppler cooling limit The average of the force of the absorption of photons is the scattering force and the random kick force resulting from spontaneous emissions. However, we have 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 a heating process, this heating process will be somewhat compensated for by the LASER cooling process (equilibrium between the two) and this is how we get the final temperature resulting from LASER cooling. The random addition to the average momentum transfer produces a random walk of the atomic momentum and an increase in the mean square atomic momentum. This heating is countered by the cooling force opposing the atomic motion. The force is proportional to the velocity of the atoms. The rate at which energy is removed by cooling is F.v which is proportional to v2, so the cooling rate is proportional to the kinetic energy of the atoms. While the heating rate, proportional to the total photon scattering rate, is independent of atomic kinetic energy at low velocities. As a result, the heating and cooling come to equilibrium at a certain value of the average kinetic energy. This defines the Doppler cooling limit 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). Summary LASER cooling and trapping refers to a number of techniques used to cool atoms down by decelerating them using light, and bringing them to a stop using magnetic fields. The reason we want to cool these neutral atoms, is to help develop our understanding of the atomic spectra, improve atomic clocks, create ultra-cold atoms which exhibit quantum mechanical behavior, so these cold atoms can help us expand our understanding of the quantum world, and if we dig deeper and cool them even more, we create a new state of matter, the Bose-Einstein condensate which is a remarkable discovery and could help us understand many phenomena in the universe. Not to mention the use of trapped ions in quantum computing -even though my paper is on the cooling of neutral atoms, but it is worth mentioning-. Reference William D. Phillips, Nobel lecture, 1997. https://www.nobelprize.org/nobel_prizes/physics/laureates/1997/phillips-lecture.pdf