Evaluation of Alternative Designs for a Magneto-Optical Trap

Yiyi Deng, Dr. John Noe, Professor Harold Metcalf
Laser Teaching Center, Stony Brook University

1. Abstract: This report examines several alternative designs for a rubidium magneto-optical trap (MOT) suitable for construction in an undergraduate laboratory. The analysis compares the characteristics of each design in terms of variables such as laser power, trapping capability, stability, ease of control, and monetary cost.

   A magneto-optical trap consists of three orthogonal pairs of counter-propagating circularly polarized laser beams in an anti-Helmholtz magnetic field. Within the trap, atoms are cooled down by exchange of momentum with the photons they absorb during their random motion. The Doppler effect, resulting from laser detuning and atomic motion, generates a velocity dependent force, while the Zeeman effect, resulting from the inhomogeneous magnetic field created by the anti-Helmholtz configuration, generates a position dependent force. The ability of the MOT to cool and physically confine atoms has made possible many novel projects not otherwise possible, such as Bose-Einstein condensation, atomic lasers, and studies of the atomic spectra of radioactive isotopes such as Francium (Fr).

   Despite the great complexity of a working MOT, there are a number of MOT setups which have been successfully developed in small undergraduate laboratories at, for example, Truman State University, Amherst College, and Bryn Mawr. This analysis presents a optimized hybrid design incorporating features of each setup that would be most suitable for construction in the Laser Teaching Center.

   This work was supported by NSF Grant PHY 0243935

2. Introduction: In the early part of this centry, the study of the "intrinsic properties of isolated atoms led to the dormulation of quantum mechanics' --- understanding matter at a more precise scale requires measurements made with greater precision. necessary precision is not easy achieve --> solids & liquids are difficult to isolate from neighbors and thermal motion of gas molecules makes precise measurements difficult; bottle the atoms in a bottle w/o physical walls -> walls are electromagnetic fields [W. D. Philips and H. J. Metcalf, Cooling and Trapping Atoms]
   • Purpose, Objectives & Motivation: Familiarity with lasers, ultrahigh vacuum, electronics, magnetic fields, light polarization, imaging systems, and photodetection are prerequisite for the successful implementation of laser cooling and trapping.
   • Applications/Relevance: ultra-high resolution spectroscopy, atom laser, intense atomic beams, nuclear physics (trapped radioactive atoms can be used in the fundament symmetry experiments, including those on Beta decay, atomic parity nonconservation, and the serach for parity and time-reveral violating electric dipole moment), interference fringes nonlinear optics with matter waves, ultrasensitive isotope trace analysis, ultracold atom collisions, cavity QED, facilitate spectroscopic measurements, improvements in atomic clicks, measurement of fundamental constants, detail formation of checmical bonding, laser cooling/em trapping --> manipulate atoms of antimatter, new states of matter (including neutral atomic BEC, ion liquids, crystals, collapsing and exploding condensates
   • Organization:
3. Theory: "salient feature of atom optics: small size of atomic deBroglie wavelengths relative to optical wavelengths

   Sat Spec Locking the frequency of a laser = very important; two many ways - saturated spectroscopy (through changes in intensity) and polarization spectroscopy (changes in polarization). Both operate on similar principles otherwise **** The eventual goal (a frequency locked laser) is the same; however, the latter presents more complications in alignment and requires more specialized equipment (to the point of setup specific construction). Thus, the saturated spectroscopy method is preferred in the undergraduate laboratory. Despite the term "lock," the frequency of a laser will drift with time. For this reason, a saturated (or polarization) spectroscopy "locked" laser can be further locked to a Fabry-Perot. This configuration extends the lifetime of the stability of the wavelength.

   When a laser beam passes through an atomic vapor cell, and frequency of the laser matches an allowed transition between a ground state and excited state of the atom, a photon can be absorbed by the atom. However, laser excitation causes random thermal excitation of atoms (because temperature α average kinetic energy) that results in a Doppler Shift of both the absorbed and emitted radiation. This occurs because the frequency of the absorbed and re-emitted radiation is dependent on atomic velocities. Thus, atoms are said to be Doppler Broadened when they absorb and emit radiation at different frequencies. Doppler Broadening conceals the details in the atomic hyperfine structure (which comes from the interaction of the nuclear moments with electric and magnetic fields and field gradients produced by the orbiting electrons) whose transition are closely paced. However, Doppler Broadening can be overcome by performing pump-probe saturation measurements.

   Hyperfine structure of an atom's absorption spectrum can be seen by using pump and probe beams. In pump-probe saturated absorption measurements, two counter propagating laser beams interact with the same atoms in the region where they intersect. The probe beam causes an atom to experience a set of transitions and the pump beam creates another set of transitions in the same atoms. Therefore, the field that reaches the detector is a function of both Doppler Broadened Peaks and Hyperfine structure.

   The pump and probe beams are created by splitting a single beam from a diode laser into two separate beams by using a 5% retarding plate. The weak 5% beam is called the probe beam and is directly sent through the beam. This beam causes transitions in the atoms that create Doppler-broadened peaks. The pump beam is the more intense beam, 95% of the original beam. Instead of going directly through the cell like the probe beam, the pump beam is redirected using two flat gold-coated mirrors to where it is precisely counter-propagating with respect to the probe beam. The mirrors deflecting the pump beam are aligned so that the pump and probe beams intersect through the entire length of the cell. The pump beam changes the density of atoms in the lower levels having particular velocity v _z, and then raise them to a higher energy level where the probe beam again interacts with those same atoms and cause Doppler broadened saturated absorption with hyperfine structure.

   In order for the atoms to undergo energy level transitions the frequency of the laser must be equivalent to the energy of the transition energy of the atoms.

   In past experiments, researchers have been successful with saturated absorption of 133-Cs and detection of cesium's Doppler-broadened peaks with hyperfine structure. In the atom cooling and trapping experiment, the saturated absorption tunes and locks the lasers to the proper transition of 87Rb. The laser beams from both the pumping and repumping lasers are split for saturated absorption, and the resultant saturated absorption spectra allows the lasers to be visually tuned to the right transitions in 87Rb.

   Trapping and Re-pumping: Trapping Rubidium is simply confining the Rubidium atoms to a small area. Before this can be accomplished, the Rubidium atoms must be cooled. In theory, cooling atoms is a simple matter - it implies a decrease in atomic velocity. The average velocities of atoms at room temperature are in the neighborhood of 102 to 103 m/s. To retard atomic motion, an external force opposite to atomic motion must be applied. Once the atoms are cooled, the force provided by the lasers ceases to be a strong opposing force. To keep the atoms from escaping the confined area, a non-homogeneous magnetic field and a position dependent force are introduced. When the atoms are cooled to a certain point (zero velocity), the atoms are no longer affected by the trapping laser (the atoms feel an equal force from each beam). During this time, however, the atoms are colliding with other atoms still present in the confinement area that causes the Rubidium to be kicked out of the confinement area. This dilemma can be solved by introducing a position dependent force. By making the force zero in the confinement area and allowing the magnitude of this restoring force to increase radially, the cooled atoms will be trapped in a small area.

   Classic MOT systems utilize six beams: three beams from the laser and the retro-reflection of these 3 beams after passing through the confinement area. These beams cannot be reflected perfectly back into the laser, as doing so would harm the diode. To continue trapping the atoms, the design incorporates another laser that allows for atomic transitions from another state that is out of the range of the trapping laser. The trapping laser is tuned to the 5S (F=2) to the 5P3/2 (F??=3) transition, while the re-pumping laser is tuned to the transition that occurs every 1 out of every 1000 transitions: 5S (F= 1) to 5P3/2 (F??=2). The re-pumping laser prevents the atoms from being stuck in this state.

   To keep the laser from wandering too far from the required wavelength, the trapping laser must be stable within a few megahertz. Additionally, because the number of atoms trapped is directly proportional to laser output, the laser output should be relatively high. Laser diodes are the best to use for this experiment because of its characteristic high output, low cost, and availability of wavelengths. Unfortunately, diodes have a single drawback - mode hopping. This occurs when one transition frequency overlaps another, i.e. the lasing mode of the frequency overlaps with another, and makes tuning very difficult, especially when attempting to lock the laser. The lack of a smooth transition between frequencies results in output jumps instead of an even background.

   Cooling Because atomic temperatures are directly related to the square of atomic velocities, cooling atoms involves physically decreasing atomic velocities. Thus, laser cooling involves a veolcity dependent force; that is, by using atomic transitions and the Doppler Effect, an 'atomic braking force' that increases with increasing atomic velocity is observed. More speficially, a laser's wavelength is detuned (shifted to the redder end of the EM spectrum) with respect to atomic resonance levels, so an atom will absorb photons that are traveling in the direction opposite to its own travel direction. This can happen because from the viewpoint of the atom, the START HERE moving towards a laser beam will absorb (and later, scatter) photons due to the Doppler

   is introtuduced: (etc etc)...However, there is a Doppler Limit a temperature at which the absorption/emission has been saturated by the availble quanta. More specifcally, the atom is too cold (or slow) to make use of the doppler shift in order to absorb photons. Consequently, the Zeeman effect is introduced (by the addition of a magnetic field) to split the atomic transitions, precluding the need for continuously blue-tuning the laser as the atoms cool.

   Bose Einstein Condensation is the name for the behavior of bosons as they reach very low temperatures and begin to occupy the lowest quantum states; such behavior is impossible for fermions (note the Pauli exclusion principle!). To observe the 'quantum clustering' of bosons (a term I coined and felt descriptive of the cooling process), they must be cooled. One method for achieving temperatures low enough is called laser cooling.

   Laser cooling utilizes laser beams to lower the temperature of a dilute atomic gas. Because the temperature of any material is a measure of the average kinetic energy of the atoms, we know from chemistry that

   KE _(average) = 1/2 m v ² = 3/2 K T

   To cool the gas, a velocity -dependent force must be applied to the atoms. Light can supply this force to the atoms in the form of photons for absorption. However, this can happen only if the photons carry a quanta of energy that corresponds exactly to an electron transition in the atoms; namely,

   E = h f

   When photons are absorbed, the energy of the atom is raised from the ground state to an excited state. When the atom decays back down to its ground state, it releases a photon. By the conservation of momentum, the atom will feel a force equal in magnitude and opposite in direction to that of the released photon (much like how the radiation pressure/refractile properties for optical tweezers!). According to literature, "the net force on a group of atoms will average to zero because the direction in which the photon is released in random." This seems like a highly Heisenburg-esque statement - how do you equate randomness with net position change? Heisenburg is the only explanation I can think of...but getting back to the velocity-dependent force - the candidate is the Doppler Effect, mathematically represented as:

   f' = f^o (1 + v/c)

   From this equation, it's clear that an atom moving towards a laser will see the light blue-shifted (higher frequency). Because the energy of a photon is related directly to frequency, the atom will -allegedly- feel a greater force from the laser and be decelerated. Thus, the laser must be 'red-detuned' to the atom's energy transition. This is perfect for the idea of a velocity dependent force: atoms with very little velocity will not 'feel' the Doppler force, while atoms with high velocities will 'see' the light closer to the energy transition, and have a higher probability of absorbing the photon (and later decaying and releasing a photon, thus effectively decreasing the average kinetic energy of the collection of atoms). A region of atoms that are confined and cooled by six laser beans in three dimensions is commonly known as optical molasses.

   Once the atoms are cooled, they must by trapped so that they will not wander out of the vicinity of the cell due to the random collisions (I hear a call for optical tweezers!). Trapping requires a position-dependent (or restoring) force, and Zeeman splitting with polarized light is the phenomenon that allows for trapping of cooled atoms. Zeeman splitting states that when an atom is under a magnetic field B, the quantum energy levels of its electrons split:

   Δ E = u m B,
   where
   u = Bohr Magneton
   m = 1, 0, -1 (magnetic quantum number)
   B = magnetic field

   Because a circularly polarized photon has angular momentum, LH or RH polarization dictate the specific transition that it can make (either m = 1 or m = -1). Thus, if a magnetic field varies lineraly as a function of distance from the center of the cell, x, then

   B = A x

   This tells us that an atom far from the center of the cell will have a large magnetic field imposed on it, and consequently, there will be a significant splitting of its energy transitions. One energy transition will be closer to the energy of the laser photons (keeping in mind that the laser is "red-detuned"). Next, the atom will be a calculated polarization of light in order to slected which of the split energy transitions to utilize and experience a force towards the center of the cell. Meanwhile, an atom near the center of the cell will experience a small magnetic field, causing little splitting of energy levels. Because the laser is "red-detuned," it will feel no force due to the laser. Thus, this is a successful plan for confinement of cooled atoms without warming them.

   Before trapping two isotopes of Rubidium, it is evaporated from a source into a trapping cell. The appratus is in a vaccuum (on the order of 10^-9 Torr) because higher pressures allow for background gas. Background gas is unwanted because it presents an opportunity for collision increase of the cooled vapor. Additionally, some sources note that the collision rate between the background gas and the cooled gas is actually higher than the trapping rate.

   see mot stuff

4. Experimental Methods: various schemes can be used to build a magneto-optical trap.
   • Mellish and Wilson: present a pyrimidal style laser cooling and trapping appratus that requires only a single beam (rather than the three pairs of orthogonal beams
     • More background fluorescence (due to mirror's reflections leads to more sample heating
     • Only one window to view/conduct experiments in
     • for example; cennot send a beam through
     • PRO: easier for ungraduate --- less alignment & more room for error
   • Wieman and Flowers:
5. Conclusions: Saturated Spectrscopy and Magneto-Optical Traps open doors to research in areas from optical lattices to Bose Einsten Condensation; for this reason, MOTs have been called "the workhorse of physics." The construction of such a setup involves principles from every many areas of physics, including electronics, quantum physics, and even thermodynamics. In short, MOTs serve as a starting point and a functioning MOT in an undergraduate laboratory has myriad research potential. Clearly, a setup accessible to undergraduates that has potential for future experiments is vital. This paper reviewed two major techniques for creating a magneto-optical trap: the classic six orthogonal counterpropagating beams, and the single beam in pyramidal mirrow. Through a close examination of experimental technique, cost and trap capability
6. Acknowledgements: see it on 2002 site THis research was supported by NSF Grant PHY 0243935
7. References: see journal --> reference again using program
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   20. and THIS
8. Appendix I [Components]:
9. Appendix II: [Glossary of Relevant Terms]:
   • Littrow Configuration: an angle/wavelength sensitive diffration grating can be used to narrow the wavelength spread of a laser.
   • Metcalf Configuration
   • Peltier Effect
   • Stark Effect
   • Zeeman Shift:
   • Doppler Effect