![MOT setup](https://upload.wikimedia.org/wikipedia/commons/5/51/MOT_setup.png)
A magneto-optical trap (MOT) is an apparatus which uses
laser cooling
Laser cooling includes a number of techniques in which atoms, molecules, and small mechanical systems are cooled, often approaching temperatures near absolute zero. Laser cooling techniques rely on the fact that when an object (usually an atom) ...
and a spatially-varying magnetic field to create a trap which can produce samples of cold, trapped, neutral atoms. Temperatures achieved in a MOT can be as low as several
microkelvin
List of orders of magnitude for temperature
Detailed list for 100 K to 1000 K
Most ordinary human activity takes place at temperatures of this order of magnitude. Circumstances where water naturally occurs in liquid form are shown in light gre ...
, depending on the atomic species, which is two or three times below the photon recoil limit. However, for atoms with an unresolved hyperfine structure, such as
, the temperature achieved in a MOT will be higher than the Doppler cooling limit.
A MOT is formed from the intersection of a weak quadrupolar spatially-varying magnetic field and six circularly-polarized red-detuned optical molasses beams. As atoms travel away from the field zero at the center of the trap (halfway between the coils), the spatially-varying
Zeeman shift brings an atomic transition into resonance which gives rise to a scattering force that pushes the atoms back towards the center of the trap. This is why a MOT traps atoms, and because this force arises from photon scattering in which atoms receive momentum "kicks" in the direction opposite their motion, it also slows the atoms (i.e. cools them), on average, over repeated absorption and
spontaneous emission
Spontaneous emission is the process in which a quantum mechanical system (such as a molecule, an atom or a subatomic particle) transits from an excited energy state to a lower energy state (e.g., its ground state) and emits a quantized amount of ...
cycles. In this way, a MOT is able to trap and cool atoms with initial velocities of hundreds of meters per second down to tens of centimeters per second (again, depending upon the atomic species).
Although charged particles can be trapped using a
Penning trap
A Penning trap is a device for the storage of charged particles using a homogeneous axial magnetic field and an inhomogeneous quadrupole electric field. This kind of trap is particularly well suited to precision measurements of properties of io ...
or a
Paul trap
A quadrupole ion trap or paul trap is a type of ion trap that uses dynamic electric fields to trap charged particles. They are also called radio frequency (RF) traps or Paul traps in honor of Wolfgang Paul, who invented the device and shared the N ...
using a combination of electric and magnetic fields, those traps are ineffective for neutral atoms.
Theoretical description of a MOT
Two coils in an anti-Helmholtz configuration are used to generate a weak quadrupolar magnetic field; here, we will consider the coils as being separated along the
-axis. In the proximity of the field zero, located halfway between the two coils along the
-direction, the field gradient is uniform and the field itself varies linearly with position. For this discussion, consider an atom with ground and excited states with
and
, respectively, where
is the magnitude of the total angular momentum vector. Due to the
Zeeman effect
The Zeeman effect (; ) is the effect of splitting of a spectral line into several components in the presence of a static magnetic field. It is named after the Dutch physicist Pieter Zeeman, who discovered it in 1896 and received a Nobel prize ...
, these states will each be split into
sublevels with associated values of
, denoted by
(note that the Zeeman shift for the ground state is zero and that it will not be split into sublevels by the field). This results in spatially-dependent energy shifts of the excited-state sublevels, as the Zeeman shift is proportional to the field strength and in this configuration the field strength is linear in position. As a note, the
Maxwell equation implies that the field gradient is twice as strong along the
-direction than in the
and
-directions, and thus the trapping force along the
-direction is twice as strong.
In combination with the magnetic field, pairs of counter-propagating circularly-polarized laser beams are sent in along three orthogonal axes, for a total of six MOT beams (there are exceptions to this, but a minimum of five beams is required to make a 3D MOT). The beams are red-detuned from the
transition by an amount
such that
, or equivalently,
, where
is the frequency of the laser beams and
is the frequency of the transition. The beams must be circularly polarized to ensure that photon absorption can only occur for certain transitions between the ground state
and the sublevels of the excited state
, where
. In other words, the circularly-polarized beams enforce selection rules on the allowed electric dipole transitions between states.
At the center of the trap, the magnetic field is zero and atoms are "dark" to incident red-detuned photons. That is, at the center of the trap, the Zeeman shift is zero for all states and so the transition frequency
from
remains unchanged. The detuning of the photons from this frequency means that there will not be an appreciable amount of absorption (and therefore emission) by atoms in the center of the trap, hence the term "dark". Thus, the coldest, slowest moving atoms accumulate in the center of the MOT where they scatter very few photons.
Now consider an atom which is moving in the
-direction. The Zeeman effect shifts the energy of the
state lower in energy, decreasing the energy gap between it and the
state; that is, the frequency associated with the transition decreases. Red-detuned
photons, which only drive
transitions, propagating in the
-direction thus become closer to resonance as the atom travels further from the center of the trap, increasing the scattering rate and scattering force. When an atom absorbs a
photon, it is excited to the
state and gets a "kick" of one photon recoil momentum,
, in the direction opposite to its motion, where
. The atom, now in an excited state, will then spontaneously emit a photon in a random direction and after many absorption-spontaneous emission events, the atom will have, on average, been "pushed" back towards the field-zero of the trap. This trapping process will also occur for an atom moving in the
-direction if
photons are traveling in the
-direction, the only difference being that the excitation will be from
to
since the magnetic field is negative for
. Since the magnetic field gradient near the trap center is uniform, the same phenomenon of trapping and cooling occurs along the
and
-directions as well.
Mathematically, the radiation pressure force that atoms experience in a MOT is given by:
where
is the damping coefficient,
is the
Landé g-factor
In physics, the Landé ''g''-factor is a particular example of a ''g''-factor, namely for an electron with both spin and orbital angular momenta. It is named after Alfred Landé, who first described it in 1921.
In atomic physics, the Landé '' ...
and
is the Bohr magneton.
Doppler cooling
Photons have a momentum given by
(where
is the
reduced Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
and
the photon
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
), which is conserved in all atom-photon interactions. Thus, when an atom absorbs a photon, it is given a momentum kick in the direction of the photon before absorption. By
detuning a laser beam to a frequency less than the resonant frequency (also known as red detuning), laser light is only absorbed if the light is frequency up-shifted by the
Doppler effect
The Doppler effect or Doppler shift (or simply Doppler, when in context) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who d ...
, which occurs whenever the atom is moving towards the laser source. This applies a friction force to the atom whenever it moves towards a laser source.
For cooling to occur along all directions, the atom must see this friction force along all three Cartesian axes; this is most easily achieved by illuminating the atom with three orthogonal laser beams, which are then reflected back along the same direction.
Magnetic trapping
Magnetic trapping is created by adding a spatially varying magnetic quadrupole field to the red detuned optical field needed for laser cooling. This causes a
Zeeman shift in the magnetic-sensitive m
f levels, which increases with the radial distance from the center of the trap. Because of this, as an atom moves away from the center of the trap, the atomic resonance is shifted closer to the frequency of the laser light, and the atom becomes more likely to get a photon kick towards the center of the trap.
The direction of the kick is given by the polarization of the light, which is either left or right handed circular, giving different interactions with the different m
f levels. The correct polarizations are used so that photons moving towards the center of the trap will be on resonance with the correct shifted atomic energy level, always driving the atom towards the center.
Atomic structure necessary for magneto-optical trapping
![Rubidium85 laser cooling](https://upload.wikimedia.org/wikipedia/commons/9/98/Rubidium85_laser_cooling.png)
As a thermal atom at room temperature has many thousands of times the momentum of a single photon, the cooling of an atom must involve many absorption-spontaneous emission cycles, with the atom losing up to ħk of momenta each cycle .
Because of this, if an atom is to be laser cooled, it must possess a specific energy level structure known as a closed optical loop, where following an excitation-spontaneous emission event, the atom is always returned to its original state.
85Rubidium, for example, has a closed optical loop between the
state and the
state. Once in the excited state, the atom is forbidden from decaying to any of the
states, which would not conserve
parity, and is also forbidden from decaying to the
state, which would require an angular momentum change of −2, which cannot be supplied by a single photon.
Many atoms that do not contain closed optical loops can still be laser cooled, however, by using repump lasers which re-excite the population back into the optical loop after it has decayed to a state outside of the cooling cycle. The magneto-optical trapping of rubidium 85, for example, involves cycling on the closed
transition. On excitation, however, the detuning necessary for cooling gives a small, but non-zero overlap with the
state. If an atom is excited to this state, which occurs roughly every thousand cycles, the atom is then free to decay either the
, light coupled upper hyperfine state, or the
"dark" lower hyperfine state. If it falls back to the dark state, the atom stops cycling between ground and excited state, and the cooling and trapping of this atom stops. A repump laser which is resonant with the
transition is used to recycle the population back into the optical loop so that cooling can continue.
Apparatus
Laser
All magneto-optical traps require at least one trapping laser plus any necessary repumper lasers (see above). These lasers need stability, rather than high power, requiring no more than the saturation intensity, but a
linewidth
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to iden ...
much less than the Doppler width, usually several megahertz. Because of their low cost, compact size and ease of use,
laser diodes
The laser diode chip removed and placed on the eye of a needle for scale
A laser diode (LD, also injection laser diode or ILD, or diode laser) is a semiconductor device similar to a light-emitting diode in which a diode pumped directly with e ...
are used for many of the standard MOT species while the linewidth and stability of these lasers is controlled using
servo
Servo may refer to:
Mechanisms
* Servomechanism, or servo, a device used to provide control of a desired operation through the use of feedback
** AI servo, an autofocus mode
** Electrohydraulic servo valve, an electrically operated valve that c ...
systems, which stabilises the lasers to an atomic frequency reference by using, for example,
saturated absorption spectroscopy In experimental atomic physics, saturated absorption spectroscopy or Doppler-free spectroscopy is a set-up that enables the precise determination of the transition frequency of an atom between its ground state and an optically excited state. The acc ...
and the
Pound-Drever-Hall technique to generate a locking signal.
By employing a 2-dimensional
diffraction grating
In optics, a diffraction grating is an optical component with a periodic structure that diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form of structura ...
it is possible to generate the configuration of laser beams required for a magneto-optical trap from a single laser beam and thus have a very compact magneto-optical trap.
Vacuum chamber
The MOT cloud is loaded from a background of thermal vapour, or from an atomic beam, usually slowed down to the capture velocity using a
Zeeman slower
A Zeeman slower is a scientific apparatus that is commonly used in atomic physics to slow and cool a beam of hot atoms to speeds of several meters per second and temperatures below a kelvin. The Gas-phase atoms used in atomic physics are often ...
. However, the trapping potential in a magneto-optical trap is small in comparison to thermal energies of atoms and most collisions between trapped atoms and the background gas supply enough energy to the trapped atom to kick it out of the trap. If the background pressure is too high, atoms are kicked out of the trap faster than they can be loaded, and the trap does not form. This means that the MOT cloud only forms in a vacuum chamber with a background pressure of less than 10
micropascal
The pascal (symbol: Pa) is the unit of pressure in the International System of Units (SI), and is also used to quantify internal pressure, stress, Young's modulus, and ultimate tensile strength. The unit, named after Blaise Pascal, is defined a ...
s (10
−10 bar).
The limits to the magneto-optical trap
![Rubidium 85 Magneto optical trap cloud 2](https://upload.wikimedia.org/wikipedia/commons/e/e1/Rubidium_85_Magneto_optical_trap_cloud_2.png)
The minimum temperature and maximum density of a cloud in a magneto-optical trap is limited by the spontaneously emitted photon in cooling each cycle. While the asymmetry in atom excitation gives cooling and trapping forces, the emission of the spontaneously emitted photon is in a random direction, and therefore contributes to a heating of the atom. Of the two ''ħk'' kicks the atom receives in each cooling cycle, the first cools, and the second heats: a simple description of laser cooling which enables us to calculate a point at which these two effects reach equilibrium, and therefore define a lower temperature limit, known as the
Doppler cooling limit
Doppler cooling is a mechanism that can be used to trap and slow the motion of atoms to cool a substance. The term is sometimes used synonymously with laser cooling, though laser cooling includes other techniques.
History
Doppler cooling was si ...
.
The density is also limited by the spontaneously emitted photon. As the density of the cloud increases, the chance that the spontaneously emitted photon will leave the cloud without interacting with any further atoms tends to zero. The absorption, by a neighboring atom, of a spontaneously emitted photon gives a ''2ħk'' momentum kick between the emitting and absorbing atom which can be seen as a repulsive force, similar to coulomb repulsion, which limits the maximum density of the cloud.
As of 2022 the method has been demonstrated to work up to triatomic molecules.
Application
As a result of low densities and speeds of atoms achieved by optical cooling, the
mean free path
In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...
in a ball of MOT cooled atoms is very long, and atoms may be treated as
ballistic
Ballistics may refer to:
Science
* Ballistics, the science that deals with the motion, behavior, and effects of projectiles
** Forensic ballistics, the science of analyzing firearm usage in crimes
** Internal ballistics, the study of the proc ...
. This is useful for quantum information experiments where it is necessary to have long
coherence time
For an electromagnetic wave, the coherence time is the time over which a propagating wave (especially a laser or maser beam) may be considered coherent, meaning that its phase is, on average, predictable.
In long-distance transmission systems, ...
s (the time an atom spends in a defined quantum state). Because of the continuous cycle of absorption and spontaneous emission, which causes
decoherence
Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wa ...
, any quantum manipulation experiments must be performed with the MOT beams turned off. In this case, it is common to stop the expansion of the gases during quantum information experiments by loading the cooled atoms into a
dipole trap.
A magneto-optical trap is usually the first step to achieving
Bose–Einstein condensation Bose–Einstein may refer to:
* Bose–Einstein condensate
** Bose–Einstein condensation (network theory)
* Bose–Einstein correlations
* Bose–Einstein statistics
In quantum statistics, Bose–Einstein statistics (B–E statistics) describe ...
. Atoms are cooled in a MOT down to a few times the recoil limit, and then
evaporatively cooled
An evaporative cooler (also known as evaporative air conditioner, swamp cooler, swamp box, desert cooler and wet air cooler) is a device that cools air through the evaporation of water. Evaporative cooling differs from other air conditioning s ...
which lowers the temperature and increases the density to the required phase space density.
A MOT of
133Cs was used to make some of the best measurements of
CP violation
In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge symmetry) and P-symmetry ( parity symmetry). CP-symmetry states that the laws of physics should be the ...
.
See also
*
Dipole trap
*
Zeeman slower
A Zeeman slower is a scientific apparatus that is commonly used in atomic physics to slow and cool a beam of hot atoms to speeds of several meters per second and temperatures below a kelvin. The Gas-phase atoms used in atomic physics are often ...
References
*
*
*
*
*
Liwag, John Waruel F. Cooling and trapping of 87Rb atoms in a magneto-optical trap using low-power diode lasers, Thesis 621.39767 L767c (1999)*
*
*
{{Lasers
Atomic, molecular, and optical physics
Particle traps