Introduction to Atomic Physics

Today, Atomic Physics is a booming and growing field, with much of this due to major developments in the last 20 years. This includes technical advances, such as the diode and solid state lasers which have revolutionized table-top atomic physics. Advances in short, femptosecond optical pulses, and over the last few years, attosecond pulses, bring us to a period in which ultrafast physics is within our sights. Another important advance has been the development of supercavities, with a Q that is so high that photons can bounce back and forth many times before disappearing. Such "Cavity QED" methods enable engineering of the vacuum. More generally, these techniques allow engineering of atoms and light to access new physics, and allowing profound modification of basic properties of natural systems.

Access to ultracold temperatures, using laser cooling, truly opened the field of atomic physics in the mid 1980's. Laser cooling at that time was a research subject, giving insight into the mechanical effects of light: how light can transfer momentum to atoms. This was the state of the art of the field in the 1990's. But the story continued. From the initial mircoKelvin atoms, we've now reached nanoKelvin atoms, which access the physics of quantum degenerate gases and many body systems. Early atomic physics was about one and two body effects. Today, we study many body physics, with thousands to millions of particles which are strongly interacting.

There is another important direction: until the mid 1990's Atomic physics was about control - over single atoms and ions. If you can control a single ion, or cool a million atoms into a Bose-Einstein condenstate, you control a single particle, or put a system into a single state. Control of single photons is also sought. In this sense, Atomic Physics is reductionist, with a goal of attaining perfect control over single atoms and single particles, leaving perhaps the only thing left to control being zero atoms and zero photons (with humour).

A natural counterpart to perfect control is building up systems once again: engineering complex quantum systems with novel properties, using the ideas of quantum information, and entanglement, reaching into the vast unknown realm of Hilbert space where no man has been before. This is yet another important and recent direction in Atomic Physics.

For example, tecently, one of the most significant technical advances has been the frequency comb -- in many ways, as important to Atomic physics as the scanning tunneling microscope has been to condensend matter physics. The femptosecond frequency comb generator makes possible new extremes of precision time and distance measurement, in a simple way.

These developments have been recognized with an unsually large number of Nobel Prizes, in 1997, 2001, and 2005. These were very unsual, for being given for discoveries within 10 years of the award. For Atomic Physics to have been part of so many important developments was a key reason for this recognition.

The MIT 8.422 Course

MIT's 8.422 Atomic Physics II course is an advanced course in Atomic Physics which will capture some of these advances and give you the conceptual and fundamental background to understand and use these. In contrast, 8.421, Atomic Physics I, is perhaps more formal, and more systematic. That course starts with atoms, perturbation theory, Stark and Zeeman effects, two-level aoms, and so-forth: in a sense, the basic building blocks of atomic physics. In contrast, 8.422 is oriented at what the current frontiers of Atomic Physics are. As a result, this course has evolved considerably in the last few years, with a list of topics reflecting what we think are the interesting frontiers you might want to know about, across the field. 8.422 is in a sense taught from researchers to researchers.

Note that 8.422 does not require 8.421 as a pre-requisite. An undergraduate mid-level quantum mechanics class is usually adequate. The only downside of taking 8.422 first is that in a sense it can be anti-climatic, to see all the fun things first.

Paradigms of Atomic Physics

In the 50's and 60's, there was a great deal of emphasis on atomic structure, quantum electrodynamics, and what was an early precursor of things to come, optical pumping. Optical pumping was a revolution, because you did not just watch atoms, but rather, you manipulated them with lasers to put the atoms into many other levels. It meant that Atomic Physics took the step form observing Nature to engineering Nature.

In the 70's and 80's, laser spectroscopy brought a major revolution in resolution, with improvemens from $\sim$GHz scales to $\sim$MHz scales starting towards today's $\sim$Hz scales. This made accesible energy states that were previously uncontrollable.

These advances focused on accessing and controlling internal states of atoms. The motional state, on the other hand, was left to whatever the atoms had: atoms from molecular beams kept whatever velocities they had, and so forth. In the late 80's and early 90's, the showcase highlights of Atomic Physics was the development of light for controlling the external state. Laser cooling alloed the motion of atoms to be controlled, providing now full control of atoms. In the late 90's and early years of the new century, Bose-Einstein condenstates were realized, and new states of Fermions, and quantum states of entanled matter, were realized, providing access to strongly correlated many-body systems and quantum information science.

Topics gone by in Atomic Physics

There are some topics in Atomic Physics which are no longer covered in 8.422. In the 1990's there were numerous papers in laser cooling, for example, but this field is no longer such an active area of research. We therefore do not cover some of the more detailed aspects of topics such as sub-doppler cooling and VSCPT cooling.

We will also not have much to cover about atomic structure, although there are interesting frontiers with high-Z atoms and two-elecron atoms. And it may be an omission, but we will also not cover the physics of high intensity lasers; with femptosecond lasers, the electric field can exceed that of the natural atom, completely changing the dynamics. And we will only briefly touch cavity-QED in the strong coupling limit.

A basic dilema we face in presenting Atomic physics in this course is that due to the limited time we have, at some point the more we cover the less we truly uncover.

Topics covered in 8.422

We will cover:

• QED: quantum electrodynamics, from first principles, is essential. Typical quantum mechanics courses do not provide the understanding of open quantum systems which is necessary for atomic physics.
• Light-Atom interaction: a rigorous derivation of interactions is neded, for understanding concepts of laser cooling, and atom trapping
• Quantum properties of light
• Quantum gases
• Molecules: this was a part of the course perhaps 20 years ago, but went away for quite awhile. Very recently, though, techniques have been developed to create molecules, from atoms in a BEC, and to control single molecules.

An introductory topic in atomic physics

Let us consider a simple scenario, to provide perspective on this class and Atomic physics. In 8.421, in a sense, you could say that if you understood the Hydrogen atom, you understand the course material. After all, if you put a Hydrogen atom into an electric field, you get the Stark and Zeeman shifts, and much more. In 8.422 we need a different example, to epitomize what the class is about. We shall consider for this the physics of atoms in optical lattices. What happens when you have an atom in a standing wave of the electromagnetic field? To understand this, we need to know how atoms interact with light, through a Hamiltonian which is given as

$H=-{\vec {d}}\cdot {\vec {E}}_{\perp }\,,$

but there is a great deal of interesting physics behind this, particularly as we discover how a quantum description of the electric field can be needed. For example, the oscillating dipole has components aligned with, and 90$^{c}irc$ out of phase with the electromagnetic field. We will find that the in-phase component results in the stimulated light force, whereas the quadrature component gives rise to the spontaneous light force. It is the latter which gives the dominant physic needed to understand an atom in a standing wave.

The atom in the standing wave sees a potential due to the optical field,

$U=-\langle {\vec {d}}\cdot {\vec {E}}\rangle \,.$

This is due to the AC Stark shift, which shifts the phase of the atom by

$\Delta \omega \sim {\frac {\hbar \Omega _{1}^{2}}{4\delta }}\,,$

for an interaction strength parameterized by $\Omega$ and detuning of the atom from the standing wave field $\delta$. The potential is thus periodic.

Spontaneous emission gives some of the most important richness in the interactin of a standing wave with an atom. The pontaneous emission rate $\gamma =\Gamma P_{e}$ (where $P_{e}$ is the excited state population) is given by $\gamma =(\Gamma /\delta )(U/\hbar )$. As we shall see, this can be derived from the dynamics of the open quantum system defined by the light, the atom, and the vacuum into which the atom can spontaneously emit.

Conceptually, important dynamics occur because when the atom is in its ground state, and sits in the standing wave, it sees a different potential than when it is in its excited state. An atom in the ground state can thus climb up a potential hill, transition into its excited state, continue climbing up a hill, spontaneously emit back into the ground state, and repeat. This process, named Sysiphus cooling, is an example of how an atom can be cooled. We will learn about this through the formalism of dressed atom states.

An important natural limit exists in the picture of an atom in a periodic potential. If the potential is very deep, and the atom is not highly excited, the atoms cannot tunnel beween local wells, so we have to good approximation the system of a mechanical harmonic oscillator, with energy eigenstates that are equally spaced. We will find that this is how a single mode of the electromagnetic field is modeled, so in a sense this deeply trapped atom behaves like a single mode field. The physics of this are described in our coverage of the quantum properties of light. We will also return to these physics when describing the behavior of a single trapped ion, one of the most completely controlled systems, and one which is promising for implementation of quantum computers. Deeply trapped atoms can also be laser cooled using two laser beams, tuned to exchange single quanta of motional energy, much like optical pumping; this technique, called sideband cooling.

Optical lattices have recently become indespensible for realizing high accuraccy atomic clocks. Atoms are simple, pristine objects, but when they collide the physics become excessively complicated, giving rise to "collision shifts" in frequency. The ultimate concept for an atomic clock could therefore be called a "lattice clock," in which atoms are held separated by the periodic optical lattice potential, so that single atom physics can be accessed. On the other hand, the electric field of the optical lattice can give an undesured AC Stark shift; it turns out this can be avoided by using "magic wavelengths," at which the Stark shift of the excited and ground states are identical. This is not possible with two-level atoms, but with real multi-level atoms it can be realized.

Precision measurement of fundamental constants can also be achieved using particles in an optical lattice. Most models aside from the standard model predict that the electron should have a finite dipole moment, but if that exists, that will be very small. This can be tested very precisely with atoms trapped in an optical lattice, based on the idea that the electon's energy should change depending on the direction of the electric field. But it can perhaps be tested even better, with cold polar molecules in an optical lattice, because the electric field in a molecule can be much higher than one externally applied. Cold molecules thus sit at an interesting frontier in atomic physics.

Optical lattices can also be used transiently. If an optical lattice is pulsed on momentarily, an initially motionless atom will be left with no momentum, or $\pm 2\hbar k$ of momentum, and go flying off. The physics of this can be described in several intersting ways. In one picture, this has applied a stimulated transfer of momentum from photons. In another picture, though, by applying the periodic mechanical potential picture of the lattice, you can say that we have atomic diffraction applied by the lattice. In particular, you can consider the deBroglie wavelength of the atom, and realize that what you have is a phase grating, through which the matter wave of the atom diffracts. In this picture of "atom optics," many interesting scenarios can be envisioned. For example, by sending an atom through three standing waves, you can cause atoms to separate and recombine, realizing an atomic interferometer. These interferometers are now the most precise instruments known for measuring acceleration due to gravity, and rotation of the earth, providing the most advanced methods for modern inertial navigation.

Consider the case when the two laser beams used to make the optical lattice had slightly differernt frequencies. An atom can now take a photon from one beam, and re-emit it into the other beam,. An initial state $|p,E\rangle$ transforms into $|p+2\hbar k,E+\delta \omega \rangle$, where we can identify $2\hbar k$ as being a momentum transfer $\Delta p$. This indicates that by changing the energy of the lasers, one can probe relations such as $\omega =ck$, the dispersion relationship of the lattice, accomplishing what is known as "Bragg spectroscopy," measuring scattering as a function of momentum and energy. This is a well-known and important tool for spectroscopy of many-body systems.

Now consider a shallow optical lattice. In this case, the atoms interact with each other, forming Bloch waves, leading to band structure. When the atoms act as bosons, the resulting structure can have metallic properties, or act as a band insulator. Bosons in weak lattices are Bose-Einstein condensate superfluids, with the weak lattice changing the mass from the bare atomic masses $m$ to an effective mass $m^{*}$. When such a weak lattice is made deeper, a phase transition can occur, from the superfluid, to the Mott-Insulator state. This is a material which is an insulator, but without the traditional band-gap, which is believed to be important in high temperature superconductivity, and of considerable interest in condensed matter physics today.

Finally, recall that the atoms this a Mott-Insulator state have internal structure. Just considering the implications of each atom having two states leads to novel and interesting questions, because these states can act as an effective spin, giving rise to a magnetic material. For example, will the atoms in this material have ferromagnetic or anti-ferromagnetic ordering? What if the atoms are fermions, instead of bosons? Such questions are at the forefront of modern Atomic Physics; in the last 15 years, much of the focus in the field has been on the motion of atoms, but as these optical lattice scenarios and questions illustrate, in the future the next degree of freedom which will be of interest will be internal states, such as magnetic levels behaving like spin. Control of these for quantum simulations of magnetic materials will require even lower temperatures than what has been achieved so far.

Handouts

• Introduction (L1-3)
  • Introduction to Laser Cooling (2007) notes
    • Handouts: W.D. Phillips, Varenna notes S. Chu et al.
  • Introduction: the QED Hamiltonian (2007)
    • Handouts: Viewgraphs used in class; Appendix from Atom-Photon Interactions book; for further readong, a 500-page derivation and discussion of the basic equations of QED can be found in Cohen-Tannoudji, Claude, Dupont-Roc, Jaques, and Grynberg, Gilbert, Photons & Atoms, Wiley-Interscience, 1997.