# Definitions for OPT

Absorption: Absorption is what happens when an electron absorbs the energy of an interacting photon and transitions into an excited state. An electron can only absorbs photons with an energy corresponding to the energy difference between the initial and final states (it is only allowed to be in discrete energy levels). The energy associated with the photon can be determined through the Planck Relationship (E=hv). If an incoming photon's energy does not correspond to an energy gap in the atom's allowed states, then the atom will not interact with the atom. What's more, certain transitions are not allowed because they would violate conservation of angular momentum.

In Relationship to this Lab: We will be exciting the different states of the Rubidium isotopes trapped inside the bulb by absorption. When the excited atoms decay into a state of lower energy, they emit a photon corresponding to the characteristic wavelength (i.e. the wavelength corresponding to a photon with the energy difference). An image of this process is shown here [1]. The amount of light which makes it through the gas tells us how much was absorbed. If no more light can be absorbed then it means that we have fully "pumped" the system into the highest M state (this is a subtle point which has to do with the hyperfine splitting of rubidium and the allowed transitions). Once it has reached this state the gas should be transparent to the incoming photons, and is in a so-called "dark state". The direction of the circularly polarized light (σ+ or σ− ) that pumps the rubidium atoms has its transitions dictated by the magnetic field orientation. When right-handed circularly-polarized light (σ+) is parallel to the magnetic field, it produces a $\Delta m_f = +1$ transition, but when its orientation is anti-parallel to the field it produces a $\Delta m_f = -1$ transition. Therefore, rather than changing the handedness of the incoming light it is easier to examine a switch in the orientation (i.e. the polarity) of the magnetic field.

Atomic Energy Levels:  An electron bound in orbit around a nucleus is in a potential wel. Once it is spatially confined to this space it can only occupy certain discrete energy values.  We call these values the energy levels.

In Relationship to this Lab: It is important to familiarize yourself with the energy level diagram of the two Rubidium isotopes. Since Rubidium is an Alkali-metal, it has one valence electron, and thus can be modeled as a one electron atom. The occupied orbitals of Rubidium are: $1s^2 2s^2 2p^6 3s^2 3p^6 3d^{10} 4s^2 4p^6 5s$. It is the 5s electron in the final orbital that acts like hydrogen’s electron. It is the transitions between the sub-levels of the 5s–5p transition that we will be concerned with. The difference between the two isotopes energy levels are determined by the different mass and charge of its nucleus.

Atomic Orientation: the atoms angular position with respect to an axis. The atom's orientation is affected by its energy, spins state, and other factors.

In Relationship to this Lab: When we apply a magnetic field to the system we are going to be creating two different splits in energy levels of the atomic structure of rubidium. One of those splits is called the Zeeman splitting and refers to the splitting that occurs in F = I + J. The splitting of the energy levels of F are determined by the orientation of the atom. This splitting only occurs in an applied external magnetic field. It would make sense that different orientations of the spin in alignment to the field would cause different energy levels to arise.

Buffer Gas: A buffer gas is an inert or nonflammable gas, meaning that it has an extremely low reactivity with other substances. We use these gases to add pressure to a system and control the speed of combustion with any oxygen present.

In Relationship to this Lab: Controlling the pressure of the bulb containing the rubidium isotopes allows us to control the rate of collisions of the atoms. The fewer collisions occur, the longer they remain in their excited states. The reason we want the buffer gas is to keep the Rb atoms localized within the volume of the bulb for long times is that we want to prolong the time it takes for them to diffuse onto the glass walls of the bulb. However adding this inert gas comes at a cost, due to the interaction of the isotopes with the gas the energy levels are smeared out (pressure broadened).  This can be countered through the heating of the bulb.

Breit-Rabi Equation: This equation yields the atomic energy shifts due to the Zeeman effect in an intermediate magnetic field. It may be applied to atomic systems with one valence electron in an S level.

In Relationship to this Lab: You will be using this equation to quantitatively examine the intermediate-field Zeeman effect. It allows you to calculate the spacing between Rubidium energy levels for a given magnetic field.

Degeneracy: In quantum mechanics, an energy level of a system is said to be degenerate if there exist multiple quantum states with that energy. The "degree" of the degeneracy for a given allowed energy is the number of different states with that energy.

In Relationship to this Lab: This lab examines the hyperfine structure of Rubidium. For example, the first excited state of Rubidium has degree 5 degeneracy in a typical potential (i.e. with no external magnetic field, etc.). The magnetic field generated by the Helmholtz coils adds a new term to the Hamiltonian of each atom. This new Hamiltonian has new solutions with new energies (Well, some of the new energies correspond to the original energies, since those states with zero angular momentum do not interact with the magnetic field. Thus, for them the Hamiltonian doesn't really change). This new system "lifts" the degeneracy because states which used to have the same energy now have different energies.

Discharge Lamp: This is a lamp of an artificial light source that generates its light by creating electrical discharges through plasma (essentially ionized gas). Once the gas is ionized the free electrons are accelerated by the electric field in the tube to collide with gas atoms. This collisions excite the atoms in the plasma. When they relax back into their lower energy states, they emit a photon of characteristic wavelength. These photons are then directed outwards. The emission spectra of these lamps vary amongst different atomic gas, an example of different spectra can be found here [2].

In Relationship to this Lab: The discharge lamp for this lab is located inside the smaller metallic square box located above the larger one. It is full of Rubidium gas. This means that the characteristic photons are matched to the rubidium inside the experiment, which is necessary for driving transitions.

Electric Dipole Transition: This is the effect generated by the interaction of an electron in orbit around a nucleus with the electromagnetic field. It is essentially a coupling of an electric particle to the electric component of an electromagnetic field. A transition can occur when an electron moves between orbitals of the atom, however these transitions are limited by selection rules. This transition is the most basic way that an atom can absorb light of a given angular momentum and energy.

In Relationship to this Lab: We will be dealing with the transitions between states within the two Rubidium isotopes, thus understanding the physical mechanism of these transitions is important. These transitions occur through stimulated or spontaneous emission as well as absorption.

Electron Configuration: The configuration of the electron in its orbit around the nucleus reveals a lot about the energy level of the system of the atom. The orientation of the orbit follows four values attached to the electron. The first is the principal quantum number (n) which is an integer dictating which shell the electron orbits and its distance from the nucleus. The second is the Azimuthal quantum number (l), commonly referred to as the orbital angular momentum, and describes the shape of the orbital. Next is the magnetic quantum number (ml) describing the quantum state of the electron and distinguishes the orientation of the orbit along the nucleus's axis. The final value is the spin quantum number (ms) which is fixed at 1/2 for electrons.

In Relationship to this Lab: This lab will need extensive knowledge of the electron configurations available to the single valence electron in the two isotopes as it will be the determining factor for how different energy states of the system split.

Equilibrium Distribution:  the distribution of energy state occupation levels (number of atoms in the ground state, number of atoms in the first excited state, etc.) in an isolated gas at a fixed temperature. The distribution of occupations changes when a system experiences radiation tuned to a specific energy gap (e.g. population inversion in laser systems). Optical pumping brings a system to a non-equilibrium distribution.

In Relationship to this Lab: We will optically pump a Rubidium gas to achieve a non-equilibrium distribution of atoms in the Zeeman sublevels of the atomic ground state. In particular, we will concentrate the occupations into one of the hyperfine levels of the atom.

Fine Structure: The corrections to the allowed atomic energy levels when relativistic effects and spin-orbit coupling are taken into account. The spin-orbit coupling is due to the interaction of the spin magnetic moment of the electron with the magnetic moment it produces during its orbit of the nucleus (moving charge creates magnetic field, and the moving charge interacts with this field because its also a little magnet). Due to this interaction, the orientation of the electron's magnetic moment with respect to its orbit affects its energy (the energy is proportional to $\mu \cdot B$).

In Relationship to this Lab: The splitting of these different energy levels will each react differently to an externally applied magnetic field and will allow us to see smaller effects such as Zeeman splitting.

Helmholtz Coil: The apparatus used to produce a uniform magnetic field in a region between the two circular coils (arranged like wheels on an invisible axle). By running an electric current around the loops, a magnetic field is generated in a direction along the invisible axle following the right hand rule. The separation, the radius, the number of turns and the current running through the coils all determine the strength of the magnetic field.

In Relationship to this Lab: We will be using Helmholtz coils in order to generate an external magnetic field that will produce the Zeeman splitting we wish to explore. We can also use these coils to counteract the effects of the earth's magnetic field. This is a key to understanding why the apparatus is tilted about an axis.

Hyperfine Structure:  Correction to the Hamiltonian by accounting for the finite nuclear spin.

In Relationship to this Lab: We will be examining the hyperfine structure of the isotopes just as we did the fine structure. It will further split the degenerate energy levels based on the spin orientation of the nucleus.

Interference Filter:  This filter is an optical filter designed to reflect one or more spectral bands or lines and transmit others. It has a nearly zero coefficient of absorption for all wavelengths of interest, allowing them to pass directly through.

In Relationship to this Lab: A filter at the opening of the discharge lamp allows photons of the desired resonant frequency to pass into the gas and excite rubidium isotopes to specific levels.

Larmor Frequency: When a magnetic dipole is angled with respect to a magnetic field, the field exerts a torque. This torque causes the magnetic dipole to precess in a plane perpendicular to the magnetic field's direction (like a spinning top in a gravitational field). The frequency of precession is called the Larmor frequency.

In Relationship to this Lab: It is the different orientations of the nuclear magnetic moments with respect to the field that will be splitting the energy levels of the Zeeman effect in different magnitudes. The Larmor precession frequency of the dipoles will be found in the strong DC field. If the circularly polarized light coming from the discharge lamp is in the same axis of rotation and direction as the Larmor precession we can see effects of nuclear magnetic resonance.

Line Width: The spectral lines associated with different atoms provide us with a lot of information about the state of the substance. The finite line width results from a number of different things. Some factors include: Doppler shifting, which results from the distribution of atomic speeds within the gas; energy-time uncertainty, which sets a lower limit on the possible width of the energy level as a function of the lifetime of the state; and pressure broadening, which results from a number of different effects due to the presence of collisions between atoms.

In Relationship to this Lab: The spectral line of the transitions between energy levels in the two isotopes broadened by the Doppler shift caused by the movement of the gas in a temperature range between 35$^{\circ}$ and 45$^{\circ}$. The pressure of the buffer gas also widens the spectral lines. If the line width of the different energy levels is too wide, the hyperfine structure will be indistinguishable. Fortunately, the system is prepared in such a way as to prevent that.

Linear and Circular Polarizers: A polarizer serves as an optical filter, allowing light of specific polarization to pass through. A linear polarizer blocks allows light polarized along a given axis to pass through. It converts unpolarized light into linearly polarized light. A circular polarizer converts unpolarized light into circularly polarized light. It also selectively absorb or pass clockwise and counter-clockwise circularly polarized light.

In Relationship to this Lab: Circularly polarized light induces atomic transitions that obey selection rules. In particular, circularly polarized light only allows transitions with $\Delta m = \pm 1$. Since we are pumping up, we want to radiate the atoms with right-handed polarized light (which only causes transitions with $\Delta m = +1$

LS Coupling: This refers to the angular momentum coupling and the interaction between the quantum numbers L and S (often referred to as Russell-Saunders coupling) and results in the fine structure of the atom. The S and L couple together to form a total angular momentum J = S + L. This approximation can be made in cases where the external magnetic fields are weak.

In Relationship to this Lab: The LS coupling determines the energy levels of rubidium. For example, for the ground state in 85Rb, L = 0 and S = 1/2, so J = 1/2; for the first excited state, L = 1, so J = 1/2 or J = 3/2, and so on and so forth.

Magnetic Dipole Moment: The vector that determines the torque experienced by a magnet when placed in an external magnetic field. Anything that generates a magnetic field (no matter how small) has a magnetic dipole moment. Typically written as  $\vec \mu$.

In Relationship to this Lab: All the spins of the electrons and nuclei of the rubidium isotopes have a magnetic dipole moment. These magnetic moments give rise to the fine structure of the atom. In particular, the level to which we want to pump the Rubidium corresponds to atoms with a specific orientation of their magnetic nuclear and electron dipole moments with respect to the magnetic field orientation.

Maxwell-Boltzmann distribution: the distribution of molecular speeds in an ideal gas model. It looks kind of like a squeezed over Gaussian; google it to see a picture.

In Relationship to this Lab: Since the atoms in the Rubidium are whizzing around with respect to the laboratory frame of reference, the photons they emit, as well as the photons they absorb from the discharge lamp, experience a Doppler shift. If the gas is too hot, then the average wavelength of the photons get Doppler shifted out of the range of absorption and we can no longer achieve a fully pumped state.

Modulation of Frequency: Modulating a frequency refers to encoding information in a carrier wave by varying the instantaneous frequency of the wave while keeping the intensity constant. Essentially the carrier wave's amplitude and phase remain constant while its frequency is altered.  An image of this process can be found here [3].

In Relationship to this Lab: We can use frequency modulations to look for the resonance frequency of the rubidium isotopes by pumping the coil currents with a modulated signal.

Nuclear Spin: This is a representation of the total angular momentum associated with the nucleus of an atom and is given by the symbol I. The nucleus of an atom does not distinguish between its orbital angular momentum and spin. The nuclear spin of individual protons are 1/2 similar to the electron, however the overall spin of the nucleus changes with varied atomic numbers.

In Relationship to this Lab: The different spins of the isotopes will determine the different energy levels and how they react to the external magnetic field. Since we are dealing with alkali-metal atoms that have a nuclear spin, the energy level structures become more complicated. When we add a spin of nucleus I to the Hamiltonian we find three new energy levels. This additional angular momentum term creates a new total angular momentum F = J + I. The energy levels vary due to the interaction via magnetic forces of the nucleus's spin with that of the electrons.

Paschen-Back Effect: If the external magnetic field passes the region of being a weak field then the splitting of the energy levels no longer follows the Zeeman splitting effect. The coupling between the orbital and spin angular momenta is disrupted and the pattern changes. This effect is known as the Paschen-Back effect and the S and L orbitals couple more strongly to the external magnetic field.

In Relationship to this Lab: We want to understand the limitations on the current we drive the coils with, past a certain intensity we will no longer notice the fine and hyperfine structure that we have come to expect. We will enter the strong-field regime where the splitting of energy levels behaves differently. More information about this regime can be found here [4] in section 3.

Quantum Numbers: These numbers describe the values of conserved quantities when dealing with the dynamics of a quantum system. They are usually determined by the acceptable solutions to Schrödinger wave equation.

In Relationship to this Lab: The numbers of a state give information about the orientation of the magnetic dipole of the atom, which we influence when we apply a magnetic field. Allowed transitions are labeled by the change in quantum number associated with the transition.

Radiative Lifetime: The time it takes an excited state to relax back into a lower energy state in the presence of radiation. If you invert the lifetime you get the spontaneous decay rate, which also corresponds to the natural line width of the emitted radiation.

In Relationship to this Lab: The fourth reference talks a little more about measuring these average times, for example the found  lifetimes of 27.70(4) ns for the 52P1/2 state and 26.24(4) ns for the 52P3/2 state.

Relaxation: The process through which an atom in an excited state emits a photon of characteristic frequency and returns to the ground state.

In Relationship to this Lab: We will want to know roughly the value of the relaxation time in order to understand how quickly we want to pump the system in order to keep it in its non-equilibrium population and how long we should give it to re-equilibrate. It will also be a good indicator when measuring the rate of change in the signal height when the RF is gated on or off.

Resonance: The response of a system with a natural oscillation frequency when driven by an external oscillation at its natural frequency.

In Relationship to this Lab: From Planck's relation, energy is proportional to oscillation frequency. As such, there is a natural wavelength associated with the energy difference between two atomic levels. An atom, when exposed to radiation, has a certain probability of absorbing a photon and transitioning to a new energy level. This probability has a peak when the photon to be absorbed has an energy equal to the difference between the initial and final energies. In other words, the photon is resonant with that atomic transition. We can measure the transitions between energy levels of Zeeman splitting in the two isotopes by finding the wavelength of resonant radiation.

Selection Rules: Selection rules constrain the possible transitions of a system from one quantum state to another. A detailed table with all the selection rules can be found here [5].

In Relationship to this Lab: The transitions in this lab come from absorption and emission of photons. These processes follow specific selection rules. In particular, the absorption of circularly polarized photons has special selection rules that is crucial to understanding optical pumping. This is part of the pre-lab.

Spontaneous Emission: This is the process by which a quantum system can undergo a transition to a lower energy state through a spontaneous decay thus emitting a quanta of energy. In order to explain how this is possible the argument must be extended into quantum field theory, which is beyond the scope of this class.

In Relationship to this Lab: The photons emitted through stimulated and spontaneous emission differ in their polarization, since the spontaneously emitted photons are not a result of an external excitation, they are unpolarized. They can further be absorbed by other surrounding atoms and lead to depolarization. This causes the selection rules to differ between the two forms of emission.

Stimulated Emission: This is similar to spontaneous emission, except the emission is caused due to the interaction of an incoming photon of a specific frequency that can interact with a previously excited atomic electron. The presence of an external, oscillating electric field is sufficient to prompt an atom to emit a photon and descend to a lower state.

In Relationship to this Lab: Stimulated emission is the mechanism behind the discharge lamp. More importantly, in order to verify that optical pumping has happened, we need to bring the system in and out of a pumped state. We can bring the system out of the pumped state through stimulated emission.

Spectroscopic Notation: a way of naming the different states of an atom or molecule. It includes information about the total spin, optical electron configuration, quantum number and angular momentum of the system. The general format for the notation follows the form N2S+1Lj, Where N is the principal quantum number, s is the total spin quantum number ((2S+1) is  the number of available spin states), L refers to the orbital angular momentum quantum number l however is written in the literal form S, P, D, F, ... for l = 0,1,2,3,... and j is the total angular momentum quantum number. It can be confusing, but it's a very useful notation to know. It's worth spending half an hour on some different examples to get the hang of it.

In Relationship to this Lab: The relevant states of the Rubidium isotopes will be referred to by their names, as per spectroscopic notation. It is the universal nomenclature for atomic physics and is important to know.

Zeeman Effect: The Zeeman effect refers to the splitting of atomic energy levels into several closely spaced lines due to an external magnetic field. This effect arises from the interaction between the magnetic field and the magnetic dipole moment of the atom. The magnetic field exerts a torque on the magnetic dipole. The change in potential energy of the system is proportional to both the magnitude of the applied magnetic field and the dipoles angle with respect to the orientation of the field. The interaction takes the general form: $\Delta E = \frac{e}{2m}(\vec{L}+2 \vec{S}) \cdot \vec{B} = g_{L} \mu_{B} m_j B$

In Relationship to this Lab: One objective of this lab is to measure the energy difference between Zeeman levels of the two isotopes, which will lead you to calculate the values of the nuclear spins. The external magnetic field will be generated by the two coils on either side of the metallic enclosing.

*This page was created by Segre Interns Daniel Naim and Hunter Akins.

References