AMO Physics notes

Alkali Atoms

Topic: atom light interaction and spectroscopy

Introduction

Alkali atoms are the primary systems in ultracold atomic physics. With a single valence electron outside a closed shell, they are simple enough to model theoretically yet rich enough to exhibit the essential physics of atomic structure and light-matter interaction. Their strong optical transitions enable both laser cooling and precision spectroscopy, making them ideal platforms for studying quantum phenomena from the single-atom to the quantum-degenerate regime.

1. Energy Level Structure of Alkali Atoms

1.1 The Central Field Approximation

An alkali atom consists of a positively charged nucleus (with atomic number $Z$) surrounded by $Z$ electrons. Of these, $Z-1$ electrons fill the closed shells of the noble-gas configuration (e.g., krypton for rubidium, xenon for cesium), leaving a single optically active valence electron. In the central field approximation, the valence electron moves in an effective spherically symmetric potential $V_{\text{eff}}(r)$ that includes the Coulomb attraction of the nucleus screened by the core electrons. The many-electron problem thus reduces to a single-particle Schrödinger equation: \begin{equation} \left[ -\frac{\hbar^2}{2m_e} \nabla^2 + V_{\text{eff}}(r) \right] \psi_{n\ell m_\ell}(\mathbf{r}) = E_{n\ell} \, \psi_{n\ell m_\ell}(\mathbf{r}). \end{equation} The eigenstates are labeled by the principal quantum number $n$ and the orbital angular momentum quantum number $\ell$. The energy $E_{n\ell}$ depends on $\ell$ as well as $n$ because the core electrons partially screen the nuclear charge; states with lower $\ell$ penetrate closer to the nucleus and experience a stronger effective attraction, lowering their energy. This $\ell$-dependence lifts the Coulomb degeneracy of hydrogen and is responsible for the ordering of levels: for a given $n$, the $s$-states ($\ell=0$) lie lowest, followed by $p$-states ($\ell=1$), $d$-states ($\ell=2$), and so on.

The ground state of all alkali atoms is an $n s \, ^2S_{1/2}$ state, where $n$ is the principal quantum number of the valence shell: $n=2$ for Li, $n=3$ for Na, $n=4$ for K, $n=5$ for Rb, $n=6$ for Cs, and $n=7$ for Fr. The first excited states are the $n p \, ^2P_{1/2, 3/2}$ levels. The strong electric-dipole transitions between the ground state and the lowest excited states are the celebrated D lines. The D$_1$ line connects $nS_{1/2}$ to $nP_{1/2}$, while the D$_2$ line connects $nS_{1/2}$ to $nP_{3/2}$. These are the transitions that make alkali atoms so useful for laser cooling and spectroscopy.

The wavelengths of the D lines for common alkali isotopes are given in Table 1.

[Table 1: D-line wavelengths (in vacuum) for common alkali atoms. $^{87}$Rb: D$_1$ = 794.979 nm, D$_2$ = 780.241 nm; $^{85}$Rb: D$_1$ = 794.979 nm, D$_2$ = 780.241 nm (small isotope shifts omitted); $^{133}$Cs: D$_1$ = 894.593 nm, D$_2$ = 852.347 nm; $^{23}$Na: D$_1$ = 589.756 nm, D$_2$ = 589.158 nm; $^{39}$K: D$_1$ = 770.108 nm, D$_2$ = 766.701 nm; $^{6}$Li: D$_1$ = 670.992 nm, D$_2$ = 670.977 nm.]

1.2 Fine Structure

The fine structure arises from the relativistic spin–orbit coupling of the valence electron. The spin–orbit Hamiltonian is \begin{equation} \hat{H}_{\text{SO}} = \xi(r) \, \hat{\mathbf{L}} \cdot \hat{\mathbf{S}}, \end{equation} where $\hat{\mathbf{L}}$ is the orbital angular momentum operator, $\hat{\mathbf{S}}$ is the spin operator ($s=1/2$), and $\xi(r)$ is proportional to $(1/r)\,\mathrm{d}V_{\text{eff}}/\mathrm{d}r$. In the coupled basis, the total electronic angular momentum is \begin{equation} \hat{\mathbf{J}} = \hat{\mathbf{L}} + \hat{\mathbf{S}}, \end{equation} with quantum number $J$ taking values $|\ell - 1/2|$ and $\ell + 1/2$ for $\ell > 0$. The spin–orbit interaction splits each $n\ell$ level into two fine-structure components: \begin{equation} \Delta E_{\text{FS}} = E_{J=\ell+1/2} - E_{J=\ell-1/2} = \frac{\hbar^2}{2} (2\ell+1) \langle \xi(r) \rangle. \end{equation} For the lowest $p$-states ($\ell=1$), this gives the $^2P_{1/2}$ and $^2P_{3/2}$ levels. The fine-structure splitting increases with $Z$: for $^{6}$Li it is only about 10 GHz, while for $^{133}$Cs it is approximately 16.6 THz (the D$_1$–D$_2$ separation of about 42 nm). The spectroscopic notation for an atomic term is \begin{equation} n^{2S+1}L_J, \end{equation} where $S$ is the total spin ($S=1/2$ for one valence electron, hence the doublet multiplicity $2S+1=2$), $L$ is the total orbital angular momentum (denoted by S, P, D, F, ... for $L=0,1,2,3,\dots$), and $J$ is the total electronic angular momentum.

1.3 Hyperfine Structure

The atomic nucleus possesses an intrinsic spin angular momentum $\hat{\mathbf{I}}$ with quantum number $I$. For the stable alkali isotopes, the nuclear spins are:

The nuclear spin gives rise to the magnetic dipole hyperfine interaction (and, for $I \ge 1$, an electric quadrupole interaction) between the nucleus and the valence electron. The hyperfine Hamiltonian is \begin{equation} \hat{H}_{\text{HFS}} = A_{\text{HFS}} \, \hat{\mathbf{I}} \cdot \hat{\mathbf{J}} + B_{\text{HFS}} \, \frac{3(\hat{\mathbf{I}}\cdot\hat{\mathbf{J}})^2 + \frac{3}{2}(\hat{\mathbf{I}}\cdot\hat{\mathbf{J}}) - I(I+1)J(J+1)}{2I(2I-1)J(2J-1)}, \end{equation} where $A_{\text{HFS}}$ is the magnetic dipole hyperfine constant and $B_{\text{HFS}}$ is the electric quadrupole hyperfine constant (nonzero only for $I \ge 1$ and $J \ge 1$).

The total atomic angular momentum is \begin{equation} \hat{\mathbf{F}} = \hat{\mathbf{I}} + \hat{\mathbf{J}}, \end{equation} with quantum number $F$ taking values $|I-J|, |I-J|+1, \dots, I+J$. The hyperfine interaction splits each fine-structure level into $2J+1$ (or $2I+1$, whichever is smaller) hyperfine levels labeled by $F$. The energy shift to first order (for $B_{\text{HFS}} \ll A_{\text{HFS}}$) is \begin{equation} \Delta E_{\text{HFS}}(F) = \frac{1}{2} A_{\text{HFS}} C + \frac{1}{2} B_{\text{HFS}} \, \frac{\frac{3}{2}C(C+1) - 2I(I+1)J(J+1)}{I(2I-1)J(2J-1)}, \end{equation} where $C = F(F+1) - I(I+1) - J(J+1)$.

For the ground state $^2S_{1/2}$ ($J=1/2$), there are two hyperfine levels: $F = I \pm 1/2$ (for $I \ge 1/2$). The hyperfine splitting of the ground state is of great practical importance. For $^{87}$Rb ($I=3/2$), $F=1$ and $F=2$, with a splitting of $\Delta E_{\text{HFS}} / h \approx 6.835$ GHz. For $^{133}$Cs ($I=7/2$), $F=3$ and $F=4$, with a splitting of about 9.193 GHz—this transition defines the SI second. The hyperfine structure of the D$_2$ line of $^{87}$Rb is illustrated in Figure 1.

[Figure 1: Full hyperfine energy-level diagram for the D$_2$ line of $^{87}$Rb ($5S_{1/2} \to 5P_{3/2}$). The ground state splits into $F=1$ and $F=2$ (6.835 GHz splitting). The excited state splits into $F'=0,1,2,3$. Allowed electric dipole transitions ($\Delta F = 0, \pm 1$, $F=0 \not\to F'=0$) are shown, including the cooling transition $F=2 \to F'=3$ and the repump transition $F=1 \to F'=2$.]

1.4 Zeeman Effect and Magnetic Sublevels

In the presence of a static magnetic field $\mathbf{B} = B\hat{z}$, the Zeeman Hamiltonian \begin{equation} \hat{H}_Z = \frac{\mu_B}{\hbar} (g_J \hat{J}_z + g_I \hat{I}_z) B \end{equation} lifts the degeneracy of the magnetic sublevels $m_F = -F, -F+1, \dots, F$. Here $\mu_B = e\hbar/(2m_e)$ is the Bohr magneton, $g_J$ is the electronic Landé $g$-factor, and $g_I \ll g_J$ is the nuclear $g$-factor. In the low-field regime ($\mu_B B \ll A_{\text{HFS}}$), the Zeeman shift is linear: \begin{equation} \Delta E_Z(F, m_F) = g_F \mu_B m_F B, \end{equation} where the hyperfine $g$-factor is \begin{equation} g_F = g_J \frac{F(F+1) + J(J+1) - I(I+1)}{2F(F+1)}. \end{equation} The multiplicity of magnetic sublevels ($2F+1$ states per hyperfine level) is essential for laser cooling, as it provides the "closed" and "open" transitions that enable optical pumping, repumping, and sub-Doppler cooling mechanisms.

2. Laser Cooling of Alkali Atoms

2.1 The Principle of Doppler Cooling

Laser cooling exploits the momentum exchange between photons and atoms to reduce the kinetic energy of an atomic vapor. The fundamental mechanism was proposed by Hänsch and Schawlow (1975) and independently by Wineland and Dehmelt (1975). Consider a two-level atom with transition frequency $\omega_0$ moving with velocity $\mathbf{v}$ in a laser field of frequency $\omega$ and wavevector $\mathbf{k}$. In the rest frame of the atom, the laser frequency is Doppler-shifted: \begin{equation} \omega' = \omega - \mathbf{k} \cdot \mathbf{v}. \end{equation} When the laser is tuned below resonance ($\omega < \omega_0$, "red detuning"), an atom moving toward the laser ($\mathbf{k}\cdot\mathbf{v} < 0$) sees the light shifted closer to resonance and preferentially absorbs photons, receiving a momentum kick $\hbar\mathbf{k}$ opposite to its velocity. The subsequent spontaneous emission is isotropic on average, so the net momentum transfer per scattering event is $\hbar\mathbf{k}$ along the laser direction. The result is a velocity-dependent radiation pressure force that, for two counterpropagating red-detuned beams, provides viscous damping.

The scattering rate for a two-level atom is given by the steady-state solution of the optical Bloch equations: \begin{equation} \Gamma_{\text{sc}}(v) = \frac{\Gamma}{2} \frac{s}{1 + s + (2\Delta(v)/\Gamma)^2}, \end{equation} where $\Gamma$ is the natural linewidth of the transition (FWHM), $s = I/I_{\text{sat}}$ is the saturation parameter ($I$ is the laser intensity, $I_{\text{sat}}$ the saturation intensity), and $\Delta(v) = \omega - \omega_0 - \mathbf{k}\cdot\mathbf{v}$ is the velocity-dependent detuning. The saturation intensity for a two-level atom is \begin{equation} I_{\text{sat}} = \frac{\pi h c \Gamma}{3\lambda^3}. \end{equation} For the D$_2$ line of $^{87}$Rb, $I_{\text{sat}} \approx 1.67$ mW/cm$^2$ (for isotropic light and cycling transition).

For two counterpropagating beams along the $\pm x$ directions, the net force on an atom with velocity $v_x$ is \begin{align} F(v_x) &= \hbar k \left[ \Gamma_{\text{sc}}(v_x) - \Gamma_{\text{sc}}(-v_x) \right] \nonumber \\ &\approx -4\hbar k^2 s \frac{-2\Delta/\Gamma}{[1 + s + (2\Delta/\Gamma)^2]^2} \, v_x \equiv -\alpha v_x, \end{align} where the last equality holds for small velocities ($k v_x \ll \Gamma$). The friction coefficient $\alpha$ is positive for red detuning ($\Delta < 0$). The minimum temperature achievable in Doppler cooling, the Doppler limit, is obtained by balancing the cooling rate with the heating rate due to the random nature of photon scattering (momentum diffusion): \begin{equation} k_B T_D = \frac{\hbar\Gamma}{2} \frac{1 + s + (2\Delta/\Gamma)^2}{2|\Delta|/\Gamma}. \end{equation} Minimizing with respect to $\Delta$ and $s$ yields \begin{equation} \boxed{k_B T_D = \frac{\hbar\Gamma}{2}} \qquad (\text{for } \Delta = -\Gamma/2, \; s \ll 1). \label{eq:Doppler_limit} \end{equation} For the $^{87}$Rb D$_2$ line ($\Gamma/2\pi = 6.07$ MHz), $T_D \approx 146$ $\mu$K. This temperature, while low, is typically above the recoil limit and well above the critical temperature for quantum degeneracy.

2.2 The Optical Molasses and Magneto-Optical Trap (MOT)

Three orthogonal pairs of counterpropagating, red-detuned laser beams intersecting at a common point create an optical molasses: a region of viscous damping in all three spatial directions. An atom entering the molasses experiences a velocity-dependent force that slows it down, effectively creating a "viscous medium" for atoms. However, optical molasses alone do not provide a spatially confining force; atoms can diffuse out of the beam intersection region over time.

To achieve both cooling and trapping, a magneto-optical trap (MOT) is employed. The MOT superimposes a quadrupole magnetic field on the optical molasses. The field is produced by two anti-Helmholtz coils: \begin{equation} \mathbf{B}(\mathbf{r}) \approx B_0 (x\hat{x} + y\hat{y} - 2z\hat{z}), \end{equation} with a field gradient $B_0$ typically of order 10 G/cm. The counterpropagating beams along each axis are given opposite circular polarizations ($\sigma^+$ and $\sigma^-$). Due to the Zeeman shift of the magnetic sublevels, an atom displaced from the trap center ($z>0$, say) is closer to resonance with the $\sigma^-$ beam (which pushes it back toward $z=0$) than with the $\sigma^+$ beam. This creates a position-dependent restoring force: \begin{equation} F_{\text{MOT}} \approx -\alpha \dot{\mathbf{r}} - \kappa \mathbf{r}, \end{equation} where $\kappa$ is the effective spring constant. The MOT combines Doppler cooling with spatial confinement, and it is the primary workhorse for producing cold atomic samples, routinely reaching temperatures of tens to hundreds of microkelvin and densities of $10^{10}$–$10^{11}$ cm$^{-3}$.

[Figure 2: (a) Schematic of a magneto-optical trap: six laser beams in $\sigma^+$/$\sigma^-$ configuration intersecting at the center of a quadrupole magnetic field. (b) Energy-level diagram showing the position-dependent Zeeman shift that provides the restoring force. (c) A photograph of a $^{87}$Rb MOT fluorescing in the center of a vacuum chamber.]

3. The Necessity of Repumping

3.1 Optical Pumping into Dark States

The simple Doppler cooling theory assumes an ideal two-level atom. Real alkali atoms have hyperfine structure, and this introduces a critical complication. Consider laser cooling on the $F \to F' = F+1$ cycling transition (e.g., $^{87}$Rb $F=2 \to F'=3$). The electric dipole selection rules permit only transitions with $\Delta F = 0, \pm 1$ (and $F=0 \not\to F'=0$). The excited state $F'=3$ can decay by spontaneous emission to the ground $F=2$ state (the "bright" state, which continues to interact with the cooling laser) or to the ground $F=1$ state (the "dark" state). The branching ratios are determined by the relevant Clebsch–Gordan coefficients.

Once an atom falls into the $F=1$ ground state, it is far off-resonance from the cooling laser (detuned by the ground-state hyperfine splitting of 6.8 GHz for $^{87}$Rb, which is vastly larger than the laser detuning $\Delta \sim \Gamma/2\pi \sim 6$ MHz). The atom ceases to scatter photons, experiences no cooling or trapping force, and is lost from the MOT. This process, called optical pumping into a dark state, occurs on a timescale of typically a few hundred microseconds. Without a remedy, the MOT would rapidly depopulate and extinguish.

3.2 The Repump Laser

The solution is a second laser frequency, the repump laser, tuned to the $F=1 \to F'=2$ (or $F'=1$) transition. This laser optically pumps atoms from the dark $F=1$ ground state back into the excited state, from which they can decay into the bright $F=2$ state and re-enter the cooling cycle. The repump laser is typically detuned close to resonance and has an intensity comparable to (or somewhat weaker than) the cooling laser. With both cooling and repump beams present, the atom undergoes a closed cycle of excitation and spontaneous emission, and the MOT operates continuously.

For $^{87}$Rb, the standard configuration is:

The energy-level diagram with the cooling and repump transitions is shown in Figure 1. Similar schemes apply to other alkalis:

3.3 Additional Repumping Needs: Off-Resonant Excitation

In some cases, even with the primary repump laser, atoms can be lost from the cooling cycle due to weak off-resonant excitation to other excited-state hyperfine levels. For example, in $^{87}$Rb, the cooling laser tuned to $F=2 \to F'=3$ can weakly excite $F=2 \to F'=2$ (detuned by $\sim 267$ MHz) and $F=2 \to F'=1$ (detuned by $\sim 424$ MHz). Atoms excited to $F'=2$ or $F'=1$ can decay to the dark $F=1$ ground state via allowed transitions. To counteract this, some experiments employ an additional "depumper" or a second repump frequency to clear out any residual population in $F=1$ more efficiently. For potassium and lithium, the smaller hyperfine splittings make such off-resonant processes more significant, and careful repumping strategies are required.

3.4 Dark SPOT MOT and Other Advanced Configurations

An ingenious variation is the dark SPOT MOT (Spontaneous-force Optical Trap), in which the repump beam is spatially blocked in a small central region. In the dark central spot, atoms are optically pumped into the dark state and no longer experience the radiation pressure forces. This reduces the radiation-trapping-induced density limitations, allowing the accumulation of much higher atomic densities (up to $10^{12}$ cm$^{-3}$) than in a standard MOT.

[Figure 3: Schematic of the optical pumping cycle in a MOT. The cooling laser drives the $F=2 \to F'=3$ cycling transition. Spontaneous decay to $F=1$ leads to loss unless the repump laser ($F=1 \to F'=2$) recycles the atom. The branching ratios and Clebsch–Gordan coefficients are indicated.]

4. Sub-Doppler Cooling Mechanisms

4.1 Polarization Gradient Cooling (Sisyphus Cooling)

The Doppler limit $k_B T_D = \hbar\Gamma/2$ was derived for a simple two-level atom. Early experiments with optical molasses, however, measured temperatures well below $T_D$—as low as a few microkelvin for cesium. This was explained by the presence of polarization gradient cooling, also known as Sisyphus cooling, a sub-Doppler mechanism that relies on the multi-level structure (Zeeman sublevels) and the spatial variation of the light polarization in a standing wave.

Consider two counterpropagating laser beams with orthogonal linear polarizations (lin $\perp$ lin configuration). The total electric field is \begin{equation} \mathbf{E}(z,t) = E_0 \left[ \hat{x} \cos(kz) e^{-i\omega t} + \hat{y} \sin(kz) e^{-i\omega t} \right] + \text{c.c.} \end{equation} The polarization of the resulting field varies periodically in space with period $\lambda/2$: it cycles from linear (at $z=0$) to circular (at $z=\lambda/8$) to linear orthogonal (at $z=\lambda/4$), and so on. An atom with multiple ground-state Zeeman sublevels experiences a spatially modulated light shift (AC Stark shift) due to the position-dependent coupling strengths to the various excited-state sublevels. As the atom moves through the polarization gradient, it repeatedly climbs a potential hill (losing kinetic energy) and is then optically pumped to a lower-energy sublevel (dissipating the potential energy via spontaneous emission). This "Sisyphus" process provides a friction force that can cool atoms to temperatures near the recoil limit: \begin{equation} k_B T_R = \frac{\hbar^2 k^2}{2m}. \end{equation} For rubidium, $T_R \approx 360$ nK.

4.2 The $\sigma^+$/$\sigma^-$ Configuration and Gray Molasses

Another common polarization gradient configuration is $\sigma^+$/$\sigma^-$, where the two counterpropagating beams have opposite circular polarizations. This produces a linearly polarized field whose direction rotates along the axis, forming a helix. The cooling mechanism is somewhat different and can give rise to gray molasses—so named because the atoms are cooled into a "gray" state (a velocity-selective dark state) with sub-recoil temperatures. Gray molasses on the D$_1$ line of $^{40}$K and $^{6}$Li has been particularly effective for all-optical production of quantum degenerate gases.

4.3 Raman Cooling and Velocity-Selective Coherent Population Trapping

Further cooling below the recoil limit can be achieved by Raman cooling and velocity-selective coherent population trapping (VSCPT). Raman cooling uses stimulated Raman transitions between ground hyperfine states to transfer atoms from high-velocity to low-velocity classes, combined with optical pumping to remove the entropy. VSCPT cools atoms into a velocity-selective dark state: atoms with near-zero velocity are coherently trapped in a superposition of ground states that does not interact with the light, while atoms with non-zero velocity absorb and re-emit photons, undergoing a random walk in momentum space until they fall into the dark velocity class. These techniques can achieve temperatures in the nanokelvin regime.

[Figure 4: (a) The lin $\perp$ lin polarization gradient: the electric field polarization as a function of position. (b) The Sisyphus cooling mechanism: an atom moving in the polarization gradient climbs a potential hill and is optically pumped to a lower energy state. (c) Sub-Doppler temperature measurements for cesium molasses, showing temperatures well below $T_D$.]

5. The Road to Quantum Degeneracy

5.1 Density and Temperature Limitations of the MOT

While the MOT is an extraordinarily successful tool, it has fundamental limitations. The maximum density in a MOT is limited to $\sim 10^{11}$–$10^{12}$ cm$^{-3}$ by two effects: (1) radiation trapping—spontaneously emitted photons are reabsorbed by other atoms, creating an effective repulsive force between atoms; and (2) inelastic light-assisted collisions between ground-state and excited-state atoms, which cause trap loss. The minimum temperature is typically a few tens of microkelvin due to residual heating mechanisms. The phase-space density $\varrho = n \lambda_{\text{dB}}^3$ (where $\lambda_{\text{dB}} = h/\sqrt{2\pi m k_B T}$ is the thermal de Broglie wavelength) in a MOT is typically $10^{-6}$ to $10^{-5}$, far below the value $\varrho \approx 2.612$ required for Bose–Einstein condensation (BEC). Further cooling and compression stages are therefore necessary.

5.2 Magnetic Trapping and Evaporative Cooling

After the MOT stage, atoms are transferred to a conservative trap, where the density can be increased without the limitations imposed by scattered light. The two most common traps are:

Once in a conservative trap, evaporative cooling is employed. The principle is simple: the highest-energy atoms (those in the tail of the Boltzmann distribution) are selectively removed from the trap. The remaining atoms rethermalize through elastic collisions to a lower temperature. By gradually lowering the trap depth (e.g., by reducing the RF frequency in a magnetic trap that induces spin-flips at a specific Zeeman energy), the sample cools while the density increases, dramatically boosting the phase-space density. Evaporative cooling is the critical final step that brought alkali atoms into the quantum degenerate regime, earning Cornell, Wieman, and Ketterle the 2001 Nobel Prize.

5.3 The Need for Repumping in Magnetic Traps

Magnetic traps confine only weak-field-seeking states. For $^{87}$Rb, the $F=2, m_F=2$ state is magnetically trappable, while $F=1, m_F=-1$ is also weak-field seeking but with a much smaller magnetic moment. To maximize the number of trapped atoms, it is essential to optically pump the atoms into the desired trappable state during the transfer from the MOT. This optical pumping step also requires careful repumping to ensure that all population is collected into the target state. Inelastic two-body collisions in the magnetic trap can change the hyperfine state, leading to trap loss; thus, the choice of the trappable state and the collision properties are critical considerations in BEC experiments.

5.4 All-Optical Routes to BEC

An alternative pathway to quantum degeneracy bypasses the magnetic trap entirely and uses only optical potentials. After the MOT, atoms are loaded into a far-detuned optical dipole trap or a crossed-beam optical trap, and evaporative cooling is performed by gradually lowering the optical trap depth (reducing the laser power). This "all-optical" approach has the advantage of simplicity and the ability to trap atoms in any hyperfine state, but it typically requires higher initial densities and suffers from photon-scattering-induced heating if the detuning is insufficient. For $^{87}$Rb, all-optical BEC was first demonstrated in 2001. More recently, all-optical cooling of $^{40}$K and $^{6}$Li to quantum degeneracy via gray molasses on the D$_1$ line has become a standard technique for producing ultracold fermionic gases.

[Figure 5: The experimental sequence for producing a Bose–Einstein condensate: (a) MOT loading phase, (b) compressed MOT and optical molasses, (c) optical pumping into the trappable state, (d) magnetic transfer and trapping, (e) RF-induced evaporative cooling, (f) absorption image of the BEC showing the characteristic bimodal distribution.]

References

  1. H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, New York, 1999). The classic textbook covering all aspects of laser cooling, the MOT, and sub-Doppler mechanisms, with extensive data on alkali atoms.
  2. C. J. Foot, Atomic Physics (Oxford University Press, Oxford, 2005). An excellent introduction to atomic structure, the central field approximation, fine and hyperfine structure, and the spectroscopy of alkali atoms.
  3. D. A. Steck, "Rubidium 87 D Line Data," available online at http://steck.us/alkalidata (revision 2.3.2, 2023). The definitive reference for $^{87}$Rb spectroscopic data, transition strengths, and saturation intensities.
  4. D. A. Steck, "Cesium D Line Data," available online at http://steck.us/alkalidata (revision 2.3.2, 2023). Analogous comprehensive data for $^{133}$Cs.
  5. W. D. Phillips, "Laser cooling and trapping of neutral atoms," Reviews of Modern Physics 70, 721–741 (1998). Nobel Lecture, providing a historical perspective and deep physical insight into laser cooling.
  6. C. Cohen-Tannoudji, "Manipulating atoms with photons," Reviews of Modern Physics 70, 707–719 (1998). Nobel Lecture covering polarization gradient cooling, Sisyphus cooling, and sub-recoil techniques.
  7. S. Chu, "The manipulation of neutral particles," Reviews of Modern Physics 70, 685–706 (1998). Nobel Lecture describing the development of optical molasses, the MOT, and atomic fountains.
  8. W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, "Making, probing and understanding Bose–Einstein condensates," in Bose–Einstein Condensation in Atomic Gases, Proceedings of the International School of Physics "Enrico Fermi" (IOS Press, Amsterdam, 1999). A comprehensive review of BEC production techniques, including evaporative cooling.
  9. R. Grimm, M. Weidemüller, and Yu. B. Ovchinnikov, "Optical dipole traps for neutral atoms," Advances in Atomic, Molecular, and Optical Physics 42, 95–170 (2000). The authoritative review of optical trapping and all-optical routes to BEC.
  10. D. Budker, D. F. Kimball, and D. P. DeMille, Atomic Physics: An Exploration through Problems and Solutions, 2nd ed. (Oxford University Press, Oxford, 2008). A problem-based approach covering alkali structure, optical pumping, and laser spectroscopy.