E2science

A magneto-optical trap, or MOT, is the most common class of device used for laser trapping/cooling. The development of the MOT was an important development in atomic physics; its development brought Stephen Chu, Claude Cohen-Tannoudji, and William D. Phillips the 1997 Nobel Prize in Physics. It was an advance over the earlier 'optical molasses' as it allowed for trapping as well as slowing of the atoms. The atoms thus confined are useful for atomic measurements, used for production of Bose-Einstein condensates (itself a Nobel-prizewinning development), and allow high-precision nuclear measurements when used to trap unstable isotopes.

Pushing Atoms with Lasers

The full MOT depends on a succession of methods which each solve one part of the cooling and trapping problem. The most basic is the ability to push atoms around using laser beams; if this were impossible it would be unlikely that a laser trap would be feasible.

One might expect that a laser beam, carrying a coherent flow of energy, would exert a pressure upon materials in its path. For macroscopic objects, the energy in a laser beam is small and rarely produces a measurable pressure, but free atoms, light as they are, are a different matter. However, atoms are not solid objects but rather have complex structure described by quantum mechanics. Similarly, the behaviour of light on this scale is quantum in nature. Thus, only a quantum mechanical treatment will show how the laser light is able to exert a force on the atom.

Consider an atom with only two internal states, differing in energy by Δ; this is a vast oversimplification, but this is theory so it's allowed. The essential features are the same as for real atoms. If we illuminate the atom with a laser whose frequency is Δ/h, where h is Planck's constant, each photon in the laser beam will also have an energy Δ. Thus, by absorbing one photon from the beam, an atom in the lower (ground) state can transition into the upper (excited) state. Photons carry momentum, so this absorption exerts a force on the atom. The frequency must be tuned, since the atom only interacts with photons that are able to change its state, and the state can only change from photons with just the right energy, since all of these things are quantized. The condition of the photons matching the atom energy is called being "on resonance".

Now the thing about atomic excited states is that they decay by emitting photons. So the atom will, after absorbing a photon, quickly return to the ground state by emitting a second photon with equal energy (and thus the same momentum) to the first. On the face of the matter, this would appear to defeat the goal of exerting a force on an atom with a laser. But, an absorbed photon is always moving in the same direction, while the emitted photon comes out in a random direction. Through many cycles of absorption and emission, the momenta of the emitted photons will cancel out as the momenta of the incident photons add up. Thus, the atom will experience a net force in the direction of the laser.

The Doppler Effect: Optical Molasses

This effect is not enough to cool down the atoms, which is equivalent to slowing them down. Even if we use two opposing laser beams, the forces will just cancel, leaving the atoms in essentially the same condition as before adding the lasers. To slow down the motion of the atoms, the force exerted by the laser beams must depend on their speed. Fortunately, we can do this using the Doppler effect.

Consider two laser beams, pointing in opposite directions, with a photon energy slightly less than the atomic transition energy Δ. A stationary atom situated in the beams will absorb no photons and thus experience no force, since the laser photons are not on resonance. We want this, since any force on a stationary atom would accelerate the atom, contrary to our purposes. The situation is different for a moving atom. An atom moving along the direction of the beams will observe a Doppler shift in the frequency of the light photons. This increases the frequency of the light from the laser it is moving toward, and decreases the frequency of the other laser light. Frequency is directly related to photon energy, so the one energy increases while the other decreases. If the atom is moving fast enough, the laser light moving in the opposite direction as the atom will shift to have a photon energy of Δ, at which point the photon is on resonance and the atom excites. Thus, the atom will experience a force opposing its motion.

In this case, then, the atoms will experience forces that are selective on their velocity along the direction of the laser beams. For a real system, both the energy of the laser beam and the energy required to excite an atomic transition are 'fuzzy', so the atom will experience a force for a range of velocities rather than a single very well-defined velocity. This arrangement is referred to as 1D optical molasses, slowing the atoms' motion along one axis.

It is a relatively simple matter to combine multiple pairs of lasers along different axes to make multi-dimensional optical molasses. A 2D combination of four laser beams cools atoms in a plane; this arrangement is often called an 'optical funnel'. Six beams can be used to cool atoms in all three directions, making true, 3D optical molasses. This system impedes the motion of atoms in the area covered by all three beam pairs regardless of its direction, acting on the atoms like a viscous fluid does on macroscopic bodies.

Optical molasses have a fatal flaw for trapping atoms; there is no dependence on the position of the atoms, which prevents their confinement to a particular volume. If an atom happens to work its way toward the edge of the molasses region, no force will prevent it from escaping the lasers' influence completely. A result called the Optical Earnshaw Theorem, which I will not prove here, prevents the optical system itself from providing a position-sensitive potential in the manner that is needed. We avoid this and trap the atoms anyways by modifying the structure of the atom itself via the Zeeman effect.

The Operation of the MOT

A MOT cannot be constructed using the two-level atoms used to describe the optical molasses. Consider instead an atom with one ground state and three excited states. These excited states are distinguished from each other by having different projections of the electron's angular momentum along the z-axis: 1, 0, and -1 (in the basic unit of angular momentum, h/2π). Ordinarily, these three states will have exactly the same energy relative to the ground state, Δ.

If we introduce a magnetic field along the z-axis, the energies of the state are modified by the interaction of the electron with the magnetic field. This change in energy is proportional to the product of the angular momentum projection, m, and the magnetic field strength, B, so the states with m=1 and m=-1 change in energy in opposite directions by the same amount which we'll call δ. The other state has m=0, and so does the ground state, so they will not shift in energy. This energy shift is called the Zeeman shift.

If we set up the magnetic field for our MOT such that it increases proportional to the distance from a centre, where the field is zero, the Zeeman shift will also increase proportionally with this distance. If the magnetic field switches direction at the centre, one state will be shifted down at one side of the minimum and the other will be shifted down at the other side. This allows us to have the desired position sensitivity.

For this to work, we need the beam from one side to only excite the m=1 state, and the beam from the other side to only excite the m=-1 state. This can be arranged by the use of the correct beam polarization. There exists a polarization state that only excites to the m=1 state, called σ⁺, and one that only excites to the m=-1 state, σ^-. If we put σ⁺ light in from the side where the m=1 state shifts down, and σ^- light in from the other side, and fix its frequency at less than Δ/h, then the atoms will experience a restoring force toward the centre, having essentially the same form as a simple harmonic oscillator.

Moreover, the laser setup remains an optical molasses, so this oscillator is damped. Thus the atoms oscillate around the centre from the Zeeman force, and have their energy drained away by the Doppler force. This very quickly produces a cloud of slow, trapped atoms around the centre of the magnetic field.

The Construction of the MOT

All of these careful preparations are spoiled if air molecules are present along with the atoms to bump our trapped atoms away from the centre of the trap. Thus, a MOT must be made within a high-vacuum chamber, which can be a glass bulb or a metal chamber with optical windows. A perfect vacuum would mean that there are no atoms available to trap, though, so the vacuum chamber must be filled with a diffuse vapour of the element to be trapped. The MOT is strong enough to capture some atoms directly from the vapour.

The magnetic field is most conveniently provided by a quadrupole arrangement, realized with a pair of coils situated at the top and bottom of the chamber. If the currents in the coils travel in opposite directions, then the magnetic field generated is as desired. (This arrangement is opposite that of Helmholtz coils.) The polarizations σ⁺ and σ^- are oppositely-oriented circular polarizations, making them relatively easy to realize. In fact, since the orientation of the circular polarization is defined relative to the direction of the light, the two beams can come from the same source and become oppositely oriented by virtue of being oppositely directed.

The Limitations of the MOT

The most significant limitation of the MOT technique is that it requires that the atoms to be trapped have a relatively simple atomic structure. Specifically, it requires that the atom have a 'cycling transition', where if the atom is excited there is a reasonable chance it will return to the state it was excited from. This is difficult to achieve if there are multiple valence electrons available to excite and de-excite, as they will tend to "get in each other's way" and prevent trapping on a cycling transition.

Thus, the most common atoms used in MOTs are alkali metal atoms; most commonly rubidium and caesium, but potassium MOTs are also common, and sodium and lithium MOTs not unheard of. Francium is fairly easy to trap, but is very radioactive and so is difficult to obtain. Hydrogen is, in theory, easy to trap, but the (far-ultraviolet) laser frequency required is so high that producing and manipulating a laser beam at that frequency is presently infeasible. (Most optical materials are opaque to far-ultraviolet light) Present research interest towards hydrogen traps is directed with an eye towards containing anti-hydrogen atoms.

This is not to say that alkali atoms are the only ones that can be trapped using MOTs. The alkaline earth metals are being worked on, and magnesium, calcium, and strontium traps have been achieved. Noble gases have effectively no valence electrons, but if one electron is excited to a higher state it remains there long enough to trap with. This has already been done for all noble gases except radon, and radon will be trapped someday as well.

Applications of the MOT

As mentioned above, the MOT is a valuable tool in many fields of atomic and subatomic physics. Its applications include precision spectroscopy, production of Bose-Einstein condensates, and precision decay measurements. All of these applications leverage the MOT as a source of slow, cold atoms. Spectroscopy is improved by the use of a MOT because the motion of atoms in a warm gas smears out the atomic spectrum through the Doppler effect. The MOT, along with other laser cooling methods such as Sisyphus cooling enabled the production of Bose-Einstein condensates, which can only form at very low temperatures.

In my own research, we use the MOT to study the beta decay of short-lived alkali isotopes produced with the ISAC radioactive beam facility at TRIUMF. The Standard Model of particle physics gives very precise predictions of the distribution of decay products from a nucleus undergoing beta decay. That distribution is difficult to measure because one of the decay products, the neutrino, is essentially undetectable. If the energy and momentum of the beta particle and the daughter nucleus can be measured, though, conservation of energy and momentum can be used to determine the neutrino momentum, if the initial momentum of the nucleus is known. This is where the MOT comes in; if the nucleus decays while in a MOT, its initial momentum will be known to be very small. In addition, the MOT is well-defined in position, so the detection of the decay products can use methods that depend on knowing the initial position of the atom. These advantages combine to produce a more precise measurement than other methods of studying beta decay.

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Sources include:
H. Metcalf and P. van der Straten's book Laser Cooling and Trapping, published 1999.
Lectures at the 2005 TRIUMF Summer Institute, notes available at http://triumf.ca/tsi/
Experience as a graduate student with the TRINAT group at TRIUMF over the past year.

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(CC)
This writeup is copyright 2005 D.G. Roberge and is released under the Creative Commons Attribution-NoDerivs-NonCommercial licence. Details can be found at http://creativecommons.org/licenses/by-nd-nc/2.0/ .

Chromium (III) Sulfate, Hydrate
Cr₂(SO₄)₃ - nH₂O
CAS No. 10101-53-8
NFPA Rating: Health 1, Flammability 0, Reactivity 0

Physical Properties

Appearance: Violet and green crystals
Odor: Odorless
Solubility: Appreciable in water (24%)
Specific Gravity: 1.8
Boiling Point: Decomposes at red heat
Melting Point: 90C (194F)

Stability and Reactivity

Chromium sulfate is stable under ordinary conditions. When heated, it may decompose to produce chromic oxides (such as chromic acid) and sulfur oxides. It is incompatible (read: don't mix it) with strong oxidizers.

Summary: It's pretty stable. It won't burn, but be prepared for toxic gas if you put it in a fire.

Health Effects and First Aid

Inhalation, ingestion, skin contact, and eye contact all result in irritation of the exposed organs and tissues. Treat by discontinuing exposure and appropriately cleaning any affected body parts: lungs should be given fresh air; skin should be washed with soap and water for at least 15 minutes; eyes should be flushed with water for at least 15 minutes; the stomach should be purged. Get medical attention immediately.

Chromium sulfate is not a carcinogen (trivalent chromium, or chromium (III), is generally far less dangerous than hexavalent chromium, or chromium (VI)). It is not significantly bioaccumulative.

Summary: It's not terribly nasty. Don't eat it. Don't breathe it. Wear proper gloves and goggles, and take basic laboratory safety precautions.

Uses

Chromium sulfate is primarily used in chromium tanning, a common industrial process for turning skins/hides into leather. Leather made with chromium tanning is soft and stretchy, and has a blue-green color (before dyeing). It is also comparatively durable, especially against heat and water.

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References:
Mallinckrodt Baker, Inc., formerly J T Baker, Inc. (jtbaker.com)
Cornell University Environmental Health and Safety, material safety data sheets database (msds.ehs.cornell.edu)
Physical and Theoretical Chemistry Laboratory at Oxford University (http://physchem.ox.ac.uk/index.html)

One common use of the Bremsstrahlung effect is in high energy physics research, to produce an intense and well-characterized photon beam. Following is a description of a Bremsstrahlung photon tagging system:

1. Obtain an accelerated electron beam. For modern nuclear photoproduction research you'll want at least several GeV for the beam energy.
2. Direct the beam onto a thin radiator made of a very heavy metal (such as gold, maybe deposited on carbon foil). Be careful not to overdo it with the radiator; a couple of atoms thick is plenty.
3. The electrons will scatter off the heavy nuclei. Do not be alarmed, this is normal. Lost kinetic energy will be radiated as Bremsstrahlung photons along the original beam line; ideally very little will be lost in the radiator.
4. Use a magnetic field to direct the braked electrons onto a bank of scintillation counters. They will naturally strike at different positions depending on their energies.
5. When a counter fires, you know that an electron of a particular energy just hit it, and that a photon just left the tagger carrying the energy difference to the initial beam.

This system has two main attractions. First, you can figure out (by correcting for time-of-flight) exactly when each photon hit your target and what energy it had. This information will be likely very useful in whatever calculations your research advisor will require of you. Second, it's relatively easy to pump energy into an electron beam. RF LINACs are fairly old technology by now, relatively simple to maintain, and nicely scalable as long as you keep them cold and feed in power.

There are several approaches to Quantum Mechanics, called pictures. All of them come up with the same results; they only differ in their internal organization. The most commonly taught elementary form is the Schrodinger Picture, devised by Erwin Schrodinger. It is so dominant that the others are not typically taught below the graduate level. But what's the difference?

First, let's consider the various things that go into an expression in quantum mechanics.

• You have a state of interest. This is represented by a vector.
• You have an observable quantity of interest. For example, the momentum. This is represented by a linear operator acting on the vector.
• You have time dependence.

The pictures differ in where the time dependence goes.

Looked at from a straightforward point of view, it is obviously the state of the particle which varies in time. The way of getting the momentum out of the state does not vary in time -- the definition of momentum hasn't changed in time, certainly! This is the Schrodinger Picture, and it is fine for many purposes.

But that's not the only possibility. In classical mechanics, most commonly you express the state simply as the set of functions of the observable characteristics as a function of time. For example, you can give the center of mass position in each dimension as a function of time, x(t), y(t), z(t). Other observables such as the momentum or angular momentum can be given their own functions, and there are relationships between them.

From this point of view, it is most natural to consider the observable characteristics as what vary in time. So you put the time dependence in the linear operators and use the state as an initial condition. This is the Heisenberg Picture.

But that's not all! In Quantum Mechanics, there is a lot of trivial time dependence. If the potential is constant in time (i.e. the system is not pushed on from outside) then the way to find out the behavior of the system is to identify all of the states with definite energy (called energy eigenstates*). If you use these states as the basis for your state, then letting time advance is very simple: just let the complex phase of each component advance at the rate of E/ℏ. This is very simple to execute once you have the solution.

Now suppose the potential is not constant in time, but rather has one component which stays the same and one component which varies. No longer is the time dependence just that simple spinning**. However, the simple spinning does comprise a large fraction of the time dependence, if the time varying part is small. The Interaction Picture separates the trivial time dependence from the nontrivial time dependence, by hiding the trivial time dependence in the observables, and putting the more complicated time dependence in the state.

Note that the Interaction Picture itself is not an approximation. It is technically applicable in every situation. However, it is most useful when the perturbation is small. In that case, the Interaction Picture simplifies the task of making the appropriate approximation. See Path Integral.

There are other more esoteric pictures of quantum mechanics. Those of you who know of them, feel free to add them!

One interesting side-note, though, is that when one attempts to incorporate relativity, thus yielding Quantum Field Theory, the Schrodinger picture becomes impractical, because does not treat time and space on the same footing.

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unperson notes: since evolution in quantum mechanics is unitary, you can just think of it as a rotation (or the generalization there of, anyway). The difference between the Schroedinger and Heisenberg pictures is nothing but the difference between an active transformation and a passive transformation.

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* see eigenvector for some clarification on why they would be called this. Keep in mind that the Energy is found by applying the Hamiltonian, a linear operator (a.k.a. matrix), to the state vector.

** This is not the same thing as spin, as in fermions or bosons.

Atomic Spectroscopy is the technique of observing the amount of light absorbed or emitted by a specific substance in order to determine the concentration of a species in the substance.

The Theory

Electrons can exist in one of two states:

• Ground State. In this state, the electron contains the least energy possible, orbiting as close as it can to the nucleus.
• Excited State. In this state, the electron contains more energy than in its ground state, orbiting further from the nucleus of the atom.

An electron in its excited state can "decay" to its ground state, giving off the excess energy in the form of a photon of light, the wavelength of which is determined by the equation:

λ = h / E

Similarly, a photon of light with this wavelength may be absorbed by the electron, exciting it. Due to quantum mechanics, only certain wavelengths of light may be absorbed or emitted. These wavelengths are (generally) unique to each element, and therefore we can use these wavelengths to identify whether or not an element exists in a substance and, if so, its concentration. Analysis by shining a light at the substance and observing the wavelength(s) absorbed is known as absorption spectroscopy, while energising the substance and observing the wavelength(s) emitted is known as emission spectroscopy.

From our measured values we can obtain the absorbance of a substance (for absorption spectroscopy at least), defined as:

A = log₁₀( P₀/ P_t)

where P₀ is the power of the light beam from the cathode lamp, and P_t is the power of the light beam once it has passed through the substance. We can also show this as:

A=ε c l

where ε is a constant, c is the concentration of the element in the substance and l is the length through the substance that the ray of light passes. Rather than manually measure the length (which will generally stay constant throughout the experiment), we usually write:

A = ε' c (where ε' = ε l)

Thus we can find out the concentration by the equation:

c = (log₁₀( P₀/ P_t))/ε'

Thankfully, if you need to do Atomic Spectroscopy, you probably have access to a machine that does all of this for you.

The Equipment

You will need the following:

• An atomiser.
• A monochromator.
• A photomultiplier.
• Some sort of electronic recorder.
• For absorption spectroscopy, a light source.

A commercial spectrograph will have all of these built in, making your job a lot easier.

The equipment is set up something like this:

 _______                     Atomiser          _______________  Monochromator
|_______| - - - - - - - - ____________ - - - - +              |
Lamp                      \          /         |              |    Photomultiplier
                           |________|          |______________+ - )---|
                                                                      |    ___
                                                                      |---|___| Recorder

The Method

NB: Depending on your equipment, some of the following steps will be automated. If you're lucky, it'll all be automated and you can read the concentration straight from the machine.

Set the monochromator to a wavelength that is absorbed or emitted by the element you are observing. If you are using absorption spectroscopy, start the lamp and make sure it is shining through the atomiser.

In order to calibrate the machine, get several samples of differing (but known) concentrations of the element you will be observing, and place them in the atomiser. Measure the intensity of light for each sample and plot c versus A in order to get a nice straight line with gradient equal to ε'.

Take your sample of unknown composition and place it in the atomiser. Take your measurement, and from that and the graph you will be able to find the concentration of your element.

Complications

Because of the narrowness of the bands of light absorbed or emitted and the size of the light spectrum, it is very rare for spectral interference (that is, when two elements absorb or emit light of the same frequency) to occur. A lot more common is chemical interference, in which another element interferes chemically. An example of this is the phosphate ion, PO₄^3-, which bonds with calcium to form calcium phosphate, Ca₃(PO₄)₂. If we were to test a solution of Ca₃(PO₄ )₂ for calcium, we would get a very low value for c despite the presence of calcium, as the bonded PO₄ interferes chemically. There are two ways to stop this happening:

1. Add a releasing agent which forms a more stable compound with the interfering species, thus freeing the element being analysed.
2. If we are analysing a metal, add a complexing agent which forms a compound with the metal, but which readily decomposes into its component atoms in the flame.

Another way to solve this problem is to increase the temperature of the flame until it is hot enough to decompose the compound. However if the flame is too hot it will ionise the element being analysed, changing its spectral "signature". A way of solving this problem is to add an ionisation suppressant, that is, an element that ionises more rapidly and thus suppresses the ionisation of the element.