Section I: Theoretical Principles

In order to understand the theoretical principles behind laser cooling and trapping, one first must become familiar with the results of quantum mechanics which govern atomic structure. The next important idea is that photons can transfer momentum to atoms which allows light to exert a force on atoms. The velocity and position dependence of this force can be used to cool and trapped neutral atoms. This section, however, is not intended to provide an all-inclusive study or a rigorous derivation of these theories. Rather, it is devoted to highlighting the essential concepts for understanding how scientists have been able to create dense clouds of atoms with temperatures approaching absolute zero.

1.1 Atomic Structure

In 1853, Anders Jonas Angstrom discovered the most prominent spectral line of atomic hydrogen. As spectroscopic techniques improved, scientists such as A. A. Michelson began noticing finer structure within these spectral lines. At the same time, experimentalists working with blackbody radiation noticed a departure in measured results from classical theory. Einstein's publication, in 1905, on the photoelectric effect and experiments in Compton scattering confirmed the need for a more comprehensive treatment of matter. Over the next few decades, an enormous effort to redefine physics was undertaken. An excellent account of this effort, which resulted in quantum mechanics, can be found in Hänsch et al.¹ From quantum mechanics, we are able to accurately describe the structure of the atom which is essential to our understanding of laser cooling and trapping.

Neils Bohr's idea of stationary states produced a solar system like model of the atom. According to Bohr, the atom consists of a massive nucleus and electrons which orbit in quantized energy level. Each energy level is labeled with an integers, n, called the principal quantum number. Since the electron move about the nucleus, it also has angular momentum labeled by the letter l, an integer between 0 to n-1. The projection of the orbital angular momentum along a coordinate axis describes the magnetic quantum number m_l which is an integer between -l and l. The concept of electron spin, s = ½, was introduced by Uhlenbeck and Goudsmit in order to explain the wavelengths and intensities of spectral lines. P. A. M. Dirac confirmed this spin and theorized a proton and neutron spin in a theory which unified quantum mechanics and special relativity. The electron spin's projection along a coordinate axis is denoted by the letter m_s, either +½ or -½. The nuclear spin is labeled with the letter I and varies depending on the nucleus.

The notation for describing atomic structure was in place. The electrostatic attraction between the electron and the nucleus could be described by the principal quantum number, n. The combination of l and s gives an electron's total angular momentum, J. Magnetic coupling between the electron's orbit and spin causes an energy splitting between levels with different J called the fine structure. The fine structure is split again into the hyperfine structure denoted by the letter F. The hyperfine structure is due to a magnetic coupling between the electron's total angular momentum, J, and the nuclear spin, I.

Both Bohr and Planck theorized that an electron could change energy levels by absorbing or emitting a photon of energy h. But, only certain energy level changes are allowable due to selection rules which govern electron transitions. An electron in transition must have l = 1, m_l = -1, 0, +1, and m_s = 0. These ideas will be of great importance when we talk about atom/photon interactions.

Figure 1.1. An energy level diagram showing the various levels of the valence electron in atomic cesium. Note that the energy separations are not to scale.

Given this brief introduction to atomic structure, let us now turn our attention to the atomic structure of cesium which is the atom used in this project. Cesium is an alkali metal which has an atomic number of 55 and an atomic mass of 132.91 amu. The electron configuration is identical to xenon with the addition of a 6s valence electron. The pertinent energy levels of the valence electron in cesium can be seen in figure 1.1.

The natural linewidth is another important consequence of quantum mechanics that arises because of the finite lifetime of an excited state. Remember that the Heisenberg Uncertainty Principle, can be written as:

An energy level has a spread of energy given by E = hG due to its finite lifetime . The accepted value for the lifetime of cesium 6p_3/2 is t= 30.494 ns.² Each energy level, as in figure 1.1, can be thought of as a Lorentzian of centered about that level. The FWHM of the Lorentzian corresponds to the natural linewidth of that transition denoted by the Greek letter G. For cesium 6p_3/2, = 2p (5.2 MHz).³

1.2 Internal Variables of Atom/Photon Interactions

As mentioned earlier, atomic energy levels can be changed by electromagnetic radiation of a certain energy. Moreover, we saw that only certain transitions are allowed and that they have a natural linewidth. This section is intended to examine how photons alter the internal parameters of atoms.

1.2.1 Absorption

The process whereby a photon causes an atom to go from a ground state to an excited state is called absorption. Absorption occurs whenever light of energy h is incident upon atoms with transitions of similar energy. The effective area of an atom in the direction of an incident beam of light is called the atomic cross-section and for our experiment is given by:

where l is the wavelength of the light.⁴ This effective area of the atom will play an important role in determining the rate at which an atom absorbs a photon. Returning to the idea of selection rules, we must consider how the polarization of a photon affects absorption. The electron in transition must have l = 1, m_l = -1, 0, +1, and m_s = 0. This suggests that a photon has angular momentum but cannot change the spin of the electron. The change in the magnetic quantum number depends on the photon's polarization. Linearly polarized photons produce D m_l = 0. Circularly polarized photons produce Dm_l = -1, or +1 depending on the handedness; s⁺ photons correspond to a D m_l = +1, while s^- corresponds to a Dm_l = -1.

1.2.2 Emissions

Once an atom absorbs a photon, it can return to the ground state by one of two mechanisms. The first of these mechanisms is called spontaneous emission. Spontaneous emission is a direct result of the finite lifetime of an electron in a particular excited state. The rate of spontaneous emission is given by the natural linewidth unique to the transition. No external parameters affect the rate of this type of emission. Furthermore, the photon is emitted in a purely random direction.

The second type of emission is stimulated emission. Unlike spontaneous emission, stimulated emission occurs due to additional photons impinging upon the effective cross-section of a excited atom. The process begins with an excited atom and a photon and results in a ground state atom and two identical photons. The emitted photons are in phase and propagate in the direction of the incident radiation.

Consider a situation where the rate of spontaneous emission equals the rate of stimulated emission:

This intensity is called the saturation intensity, I_s, where: ⁵

For cesium, the saturation intensity equals 1.6 mW/cm².

1.2.3 Scatter Rate

Now that we have looked at the internal variables of atom/photon interactions, we can assign a probability to these interactions. The rate of spontaneous emission is given by the natural linewidth. The rate of absorption or stimulated emission is given by the equation:

where p is known as the saturation parameter.⁶ The saturation parameter has the shape of a Lorentzian and is of the form:

where I is the intensity of the incident beam and D = w_L - w_o is known as the detuning. The saturation parameter must be modified such that becomes D'= D + d_Doppler + d_Zeeman when including the effects of atomic motion and magnetic fields.

The first modification in the saturation parameter is due to the Doppler effect. The Doppler effect for electromagnetic waves is a purely relativistic effect due to the motion of atoms. It results in a frequency shift such that:

where v represents the component of the atom's velocity along the line of incident light.⁷ So, an atom moving toward a source observes a frequency higher than the real frequency and vice versa for an atom moving away from a source. The change in frequency due to atomic motion is:

For an atom moving toward the source (i.e. the positive change in frequency), we substitute k = 2p/l = w/c to obtain:

The second modification to the saturation parameter occurs when a magnetic field is applied to the system. Since atoms have a magnetic dipole moment, an applied B field causes a shift in the energy levels, due to the Zeeman effect, given by: ⁸

where B is a magnetic field and _B is the Bohr magneton (m_B/h = 1.4 MHz). Note that the B field can either be constant or a function of position, B(x). In our application, we will find it useful to use a position dependent magnetic field (i.e. a magnetic field gradient). The corresponding frequency shift due to the Zeeman effect is:

The equation can be simplified by using only or polarization for the 6s_1/2 -6p_3/2 transition in cesium so that m_l = 1.

1.3 External Variables of Atom/Photon Interactions

We have now seen several atom/photon interactions which change the internal state of the atom and their associated probability. Let us now turn our attention to how photons change the external parameters of the atoms.

1.3.1 Wave-Particle Duality

One of the most fundamental and revolutionary ideas of quantum mechanics is that matter and light have a wave and particle nature. This postulate can be verified by passing a beam of electrons through a diffraction grating. The electrons, often considered particles, will produce a wave-like interference pattern. The equation linking the wave and particle nature of matter was suggested by de Broglie who formulated:

where is often called the "de Broglie wavelength," h is Planck's constant, and p is linear momentum. Thus, matter having momentum, p, will have a specific de Broglie wavelength. Conversely, an electromagnetic wave having wavelength, l, should have a specific linear momentum given by:

1.3.2 Conservation of Momentum

As seen above, photons have linear momentum. When an absorption, spontaneous emission, or stimulated emission occurs, there is a transfer of momentum to the atom called a recoil. Figure 1.2 shows a photon with an initial momentum p₁ = h / l and a motionless atom. After the absorption, the atom has acquired the photon's momentum, h / l .

Figure 1.2. This chart shows the conservation of momentum during an absorption. Notice that after absorption the linear momentum of the photon has been transferred to the atom.

During a spontaneous emission, momentum is conserved as well. The photon, in spontaneous emission, transfers h / l worth of momentum to the atom while having h / l of momentum in the opposite direction. Figure 1.3 shows the conservation of momentum in spontaneous emission. Note that the total momentum before and after the emission equals zero. (Remember that spontaneous emission occurs randomly in all directions.)

Figure 1.3. This chart shows the conservation of momentum in spontaneous emission.

In stimulated emission an excited state atom emits a photon in the direction of an incident photon and returns to the ground state. The incident photon causes no recoil in the atom but the emitted photon causes a recoil opposite the incident photon. Figure 1.4 shows conservation of momentum in stimulated emission. We can see that although the total momentum of the system is zero the atom has a recoil opposite the incident photon.

Figure 1.4. This chart shows the conservation of momentum in stimulated emission. Note that the incident photon's momentum is not included in the total momentum of the system.

Consider a single atom in the beam of a laser. Let the beam propagate in the positive x direction and have an intensity such that the rate of spontaneous emission is much greater than the rate of stimulated emission. The total change in the x component of momentum of the atom given absorption, spontaneous emission, and stimulated emission can be obtained using the expression: ⁹

where n, j, and k represent integers such that n = j + k (j >> k) and represents a random angle between 0 and 2. The three sums represent changes in momentum due to absorption, spontaneous emission, and stimulated emission, respectively. After numerous absorptions, though:

This is because = 0 for j >> 1 and the stimulated emission term is negligible since j >> k.

1.3.3 Light Forces

In the example above we have seen that the change in p_x of an atom in a laser beam is primarily due to absorption. In section 1.2.3 we calculated a scatter rate absorption and saw that it could depend on atomic position and velocity. Combining these two ideas we can now determine the force that a beam of light exerts on an atom. It follows that:

From our notion of probability and change in momentum in the previous sections the total force exerted by the beam of light in the x direction is:

We now have an equation for the light force on an atom which is dependent on the frequency and intensity of the beam, and the velocity and position of the atom!!

Consider two laser beams counterpropagating along the x directions. Each beam has a frequency that is less than the resonant frequency for the transition. (The beam is said to be shifted "to the red" of the transition.) Figure 1.5 shows a plot of the force in the x direction versus the x component of velocity. The dotted line shows the force from each beam and the solid line is the net force on the atoms. Considering small velocities, we see that the force (often called the cooling force) is nearly linear such that:

where is the damping coefficient. This idea will be important in our discussion of the cooling limit.

Figure 1.5. This is a plot of the light force versus the atoms' velocities in the x direction. The detuning is -½ G and there is no applied magnetic field. The arrows represent the counterpropagating beams.

Now consider the case when one of the counterpropagating beams has s⁺ photons to excite m_l = +1 transitions, while the beam has s - photons to excite m_l = -1. Introducing an inhomogeneous magnetic field on the system such that B = Ax, we can shift the atomic energy levels such that we have a selective radiative force. First suggested by Dalibard and implemented by Raab et al.¹⁰ , figure 1.6 shows the setup of a magneto-optical trap (MOT).

Figure 1.6. This shows the configuration of a magneto-optical trap (MOT).

Atoms located in the x direction will be more likely to absorb photons from the laser on the left and atoms on the positive side of the axis will more likely absorb from the laser on the right. An atom in this system, then, experiences a net inward force. Figure 1.7 shows the light force on the atoms in a magnetic field gradient due to two beams. The dotted lines represent the force from an individual laser while the solid line is the net force. Again, the net force is nearly linear taking the form of the restoring force for a simple harmonic oscillator:

where k is the spring constant.

Figure 1.7. This plot shows the light force on the atoms due to a magnetic field gradient. Again, the detuning is -½ G and the magnetic field gradient equals 5 Gauss/cm.

Combining the damping and restoring force, we find F = - a v- kx. For small oscillations about the origin the equation of motion is:

where b =2a / m, w _o² =k / m (not the atom's resonant optical frequency), and m is the mass of the atom. This is the equation for a damped harmonic oscillator. In Section III, we will analyze this damped motion in greater detail.

1.3.4 Conservation of Energy

We have seen that two counterpropagating beams with a red detuning create both a cooling and restoring force on atoms. The idea of "cooling" with lasers, however, seems counterintuitive. This section will discuss how lasers use this cooling force to "cool" atoms. The net momentum change of an atom in a beam of light during a given time period can be thought of as a summation of absorptions. Although, the p_x for spontaneous emission equaled zero for numerous events we will see how this process leads to "heating."

Consider the momentum expression for an absorption between a photon moving in the positive x direction with momentum, p₁, and an atom moving in the negative x direction with momentum, mv₂. Some time later a spontaneous emission occurs sending the photon back in the negative x direction with momentum, p₁ leaving the atom with momentum,

mv₂. The momentum expression for the combination of events is:

Solving for the velocity of the atom after spontaneous emission we obtain:

The conservation of energy during absorption/spontaneous emission requires:

Substituting the equation for v₂' into the conservation of energy equation and solving the equation for p1c, the energy of the emitted photon, we find that:

This would indicate that the emitted photon has an energy greater than the absorbed photon if v₂ > (p₁² / m). Assuming that condition to be true then the atom had to lose energy. In other words, it was cooled! Notice, however, that as the atomic velocity approaches the recoil due to spontaneous emission (p₁² / m) the absorbed and emitted photons have about the same energy.

1.4 Temperature Limits

We have now applied the laws of conservation of momentum and energy to show that given a certain configuration of laser beams we are able to cool and trap atoms. Given this analysis we can see, however, that there is a limit to cooling.

In section 1.3.3, we calculated a damping coefficient, , which governed the rate of the cooling. Along with this cooling, though, there is always some heating or diffusion caused by spontaneous emission. The diffusion coefficient, D, can be determined by looking at this spontaneous diffusion. We can, therefore, compare the ratio of heating to cooling by comparing the two coefficients. Note, that this is very similar to the analysis of kinetic energy due to Brownian motion which states: ¹¹

where k_B is Boltzmann's constant and T is the temperature in units of Kelvin. Considering three dimensions and numerous events the minimum temperature, or Doppler limit, is given by: ¹²

For cesium in a three dimensional trap, we find that the Doppler limit equals 125 mK.

When experiments were performed, however, temperatures below the Doppler limit were observed. In 1988, groups at Stanford and Ecole Normale Superieure independently provided a theoretical explanation for this type of cooling.¹³ The theory explains a process called polarization gradient cooling¹⁴ which simply put states that an atom is more likely to absorb a photon at a lower energy and spontaneously emit it at a higher energy. This change in energy is due to changes in the atom's ground state in a radiation field. Thus, the ultimate temperature limit is given by the single recoil of a photon and is given as:¹⁵

where hk = h / l and m is the mass of the atom.

Sub-Doppler cooling processes, although applicable to our experiment, are not incorporated into the program described in Section III.