Ultracold photoassociation spectroscopy: Long-range molecules and atomic scattering

Kevin M. Jones, Eite Tiesinga, Paul D. Lett, and Paul S. Julienne

Rev. Mod. Phys. 78, 483 – Published 22 May 2006

Abstract

Photoassociation is the process in which two colliding atoms absorb a photon to form an excited molecule. The development of laser-cooling techniques for producing gases at ultracold $(< 1 mK)$ temperatures allows photoassociation spectroscopy to be performed with very high spectral resolution. Of particular interest is the investigation of molecular states whose properties can be related, with high precision, to the properties of their constituent atoms with the “complications” of chemical binding accounted for by a few parameters. These include bound long-range or purely long-range vibrational states in which two atoms spend most or all of their time at large internuclear separations. Low-energy atomic scattering states also share this characteristic. Photoassociation techniques have made important contributions to the study of all of these. This review describes what is special about photoassociation spectroscopy at ultracold temperatures, how it is performed, and a sampling of results including the determination of scattering lengths, their control via optical Feshbach resonances, precision determinations of atomic lifetimes from molecular spectra, limits on photoassociation rates in a Bose-Einstein condensate, and briefly, production of cold molecules. Discussions are illustrated with examples on alkali-metal atoms as well as other species. Progress in the field is already past the point where this review can be exhaustive, but an introduction is provided on the capabilities of photoassociation spectroscopy and the techniques presently in use.

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DOI:https://doi.org/10.1103/RevModPhys.78.483

Erratum

Publisher's Note: Ultracold photoassociation spectroscopy: Long-range molecules and atomic scattering [Rev. Mod. Phys.78, 483 (2006)]

Kevin M. Jones, Eite Tiesinga, Paul D. Lett, and Paul S. Julienne

Rev. Mod. Phys. 78, 1041 (2006)

Authors & Affiliations

Kevin M. Jones

Physics Department, Williams College, 33 Lab Campus Drive, Williamstown, Massachusetts 01267, USA

Eite Tiesinga, Paul D. Lett, and Paul S. Julienne

Atomic Physics Division, National Institute of Standards and Technology, 100 Bureau Drive, Stop 8424, Gaithersburg, Maryland 20899, USA

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Vol. 78, Iss. 2 — April - June 2006

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Figure 1
A schematic of the photoassociation (PA) process $A + B + γ \to (A B)^{*}$ . Two potential-energy curves are shown as a function of the internuclear separation of two atoms. The upward arrow labeled PA is the photoassociation transition, the downward dashed arrow labeled “decay” indicates radiative decay out of the excited molecular state. As drawn, the decay would produce two free atoms with higher energy than the initial collision. Transitions to lower or same energy atoms or to a bound-state molecule are also possible. For clarity the size of the incident thermal collision energy of the atom pair, $E_{th}$ , is greatly exaggerated. $E_{at}$ is the energy to excite one atom (here atom $A$ ) at infinite separation and $E_{b}$ is the binding energy of the molecular vibrational level. For many purposes one can consider the PA transition as occurring at the outer turning point, $R_{v +}$ , of the vibrational level of the excited molecular state.Reuse & Permissions
Figure 2
A sample PA spectrum of ${}^{23}{Na}$ at an initial temperature of $450 μ K$ . Electronically excited molecules, ${Na}_{2}^{*}$ , produced by PA are ionized by a second color laser and the resulting molecular ions, ${Na}_{2}^{+}$ , are detected and counted to produce the signal. The different peaks correspond to different rotational levels (labeled $J = 0$ to 4) within a single vibrational state $(v = 1)$ of a purely long range potential discussed in Sec. 2E. The natural linewidth of this molecular state is $20 MHz$ ; the observed linewidth is larger due to thermal broadening. The binding energy of this level is $\approx 47 GHz$ below the $3 {}^{2}S_{1 ∕ 2} + 3 {}^{2}P_{3 ∕ 2}$ atomic asymptote.Reuse & Permissions
Figure 3
Schematic representation of the initial low-energy scattering radial wave function $∣ E, ℓ ⟩$ and the final bound vibrational wave function $∣ b ⟩$ as a function of internuclear separation. The scattering wave function is calculated for the case $ℓ = 0$ and $R_{v +}$ is the classical outer turning point of the excited-state vibration. At very large separations the scattering wave function oscillates with a wavelength much larger than the scale of this figure.Reuse & Permissions
Figure 4
Modification of atomic scattering by an optical Feshbach resonance. (a) An intense laser tuned near a PA transition adds a new path to the quantum-mechanical interferometer, which produces the nodal pattern of the ground-state collisional state wave function. The arrows labeled “photoassociative absorption” and “stimulated emission” are driven by the same laser; the arrow labeled “spontaneous emission” [see Eq. (2)] represents an inelastic loss process for the collision. The multiple arrows labeled “reflected waves” indicate that at low collision energies reflection occurs not only at the inner turning point but over a wide range of internuclear separations. (b) Dressed-state picture of the same process. The excited bound state $∣ b ⟩$ with $n - 1$ photons has approximately the same energy as the initial collision state $∣ E, ℓ ⟩$ with $n$ photons. The shaded region represents the continuum of collision states, weighted according to the thermal energy distribution of an ultracold gas with maximum population of states near zero energy. Viewed this way the physics of PA is similar to that of magnetic Feshbach resonances, hence the name “optical Feshbach resonance.”Reuse & Permissions
Figure 5
Molecular spectroscopy using a sample of cold molecules produced by photoassociating cold atoms. PA of $450 μ K$ ${}^{23}{Na}$ atoms by a fixed-frequency laser produces a steady population of cold, electronically excited ${}^{23}{Na}_{2}$ molecules in a single known rovibrational level [here $1_{g} (3 ∕ 2)$ , $v = 52$ , $J = 1$ ]. A second, scanning laser promotes some of these molecules to doubly excited states. An ion detector counts the resulting ${Na}_{2}^{+}$ ions and this count rate is plotted as a function of the total energy of the two photons. Three rotational lines of the doubly excited state $J = 1, 2, 3$ are shown, each split into two “ $Λ$ -doubling” components. The rotational spacings for this Rydberg state are $\approx 100$ times larger than those for the state shown in Fig. 2, reflecting the fact that the vibrational wave function for the Rydberg state lies at much shorter internuclear separations.Reuse & Permissions
Figure 6
Schematic view of the lowest energy levels of the $Na$ atom. The splittings are greatly exaggerated: the actual $3 {}^{2}P_{3 ∕ 2} \leftrightarrow 3 {}^{2}P_{1 ∕ 2}$ fine-structure splitting is a factor of $\approx 1000$ smaller than the $3 {}^{2}S \leftrightarrow {}^{2}P$ splitting, while the ground-state $f = 2 \leftrightarrow f = 1$ hyperfine splitting is an additional factor of $\approx 300$ smaller still. The arrow indicates the optical transition typically used for laser cooling of $Na$ .Reuse & Permissions
Figure 7
(Color online) The ten energetically lowest nonrelativistic Born-Oppenheimer potentials of the sodium dimer as a function of internuclear separation $R$ . For the eight curves dissociating to the ${}^{2}S + {}^{2}P$ limit it is helpful to realize that for $R ≳ 15 a_{0}$ the $Σ$ potentials go as $\pm 2 C_{3} ∕ R^{3}$ , while the $Π$ potentials go as $\pm C_{3} ∕ R^{3}$ , where $C_{3}$ is the resonant dipole coefficient. The vertical arrow is meant to suggest typical minimum internuclear separations and typical largest energy detunings important for PA, in contrast to the more schematic Fig. 1. The curves are from 90). Potentials for other homonuclear alkali-metal dimers will be similar in shape.Reuse & Permissions
Figure 8
(Color online) The 16 relativistic adiabatic interaction potentials as a function of internuclear separation near the ${}^{2}S + {}^{2}P_{j}$ dissociation limit of ${Na}_{2}$ . Clearly visible is the splitting between the $P_{1 ∕ 2}$ and $P_{3 ∕ 2}$ states. Each potential is labeled by $Ω_{g ∕ u}^{\pm}$ . To distinguish between these states it is helpful to indicate the dissociation limit as well, e.g., $1_{g} ({}^{2}S + {}^{2}P_{3 ∕ 2})$ ; as a shorthand this is denoted $1_{g} (3 ∕ 2)$ . Since, usually, it is the attractive potentials that are of interest, that labeling is generally sufficient.Reuse & Permissions
Figure 9
The connection between the Born-Oppenheimer potentials of Fig. 7 (distance regions I and II) and the relativistic adiabatic potentials of Fig. 8 (distance regions III and IV) for homonuclear alkali-metal dimers. In each of the four distance regions the vertical order indicates the energy ordering of the states. Gerade (ungerade) states in one region connect only to gerade (ungerade) states in other regions. The dissociation limits are labeled for the specific case of $K_{2}$ . From 151, where the distance scales are defined.Reuse & Permissions
Figure 10
(Color online) A blowup of the relativistic adiabatic potential-energy curves shown in Fig. 8, showing the purely long-range $0_{g}^{-}$ and $1_{u}$ potentials. Some of the vibrational levels of the $0_{g}^{-}$ potential are indicated. Only attractive states are labeled. From 138.Reuse & Permissions
Figure 11
(Color online) The long-range hyperfine structure, at zero magnetic field, of a ground-state $3 {}^{2}S + 3 {}^{2}S$ Na dimer. The three curves are separated by the Na ground-state hyperfine splitting and behave as $- C_{6} ∕ R^{6}$ . The three curves are labeled by the separated atom hyperfine levels: $f_{a} + f_{b}$ . The dotted line indicates the location of the van der Waals length scale $R_{vd W}$ defined in Eq. (13).Reuse & Permissions
Figure 12
(Color online) Analysis of the hyperfine/rotation structure of the $v = 14$ vibrational level in the ${}^{23}{Na}_{2}$ lowest triplet (“ground”) state, $a {}^{3}Σ_{u}^{+}$ . From top to bottom, the panels show the vibrational position, the hyperfine structure calculated from the known atomic splitting (Fig. 6), the combined effects of rotation and hyperfine, and an experimental spectrum. Rotation simply adds a second copy of the hyperfine pattern, shifted up in energy. Solid lines are $ℓ = 0$ , dashed lines are $ℓ = 2$ . The hyperfine pattern is calculated assuming that this is a pure triplet state. The lowest $(f = 2)$ line in the calculated hyperfine pattern (marked with a small arrow) does not quite match the observed position (indicated by $*$ for each $ℓ$ ). This is the signature of singlet-triplet mixing. The data are from the experiment described in the article of 18, which also details the model.Reuse & Permissions
Figure 13
(Color online) Analysis of the complex hyperfine/rotational structure of a vibrational level in the ${}^{87}{Rb}_{2}$ $1_{g} (1 ∕ 2)$ state about $10 {cm}^{- 1}$ below the $5 {}^{2}S + 5 {}^{2}P_{1 ∕ 2}$ dissociation limit. Panel (a) shows the vibrational position (set to be zero frequency); panel (b) adds in hyperfine structure given by $A_{v} Ω ∙ ι$ , where $A_{v}$ is a vibrational-level-dependent hyperfine constant, and $ι$ is the projection of the total nuclear spin $I$ along the internuclear axis; panel (c) adds rotation where each line is now labeled by the total angular momentum of the molecule $F$ ; and panel (d) is an experimental trap-loss spectrum (53). The model is similar to that described by 161.Reuse & Permissions
Figure 14
(Color online) The long-range van der Waals potentials for several partial waves. The dots indicate the top of the barrier with height $E_{C} (ℓ)$ at $R_{C} (ℓ)$ defined in the text. As discussed in Sec. 3B, the last $s$ -wave bound state of the potential must lie between the boundaries indicated by the dashed lines, $E = 0$ and $E = - 39.5 \dots E_{vdW}$ . The outer classical turning points for all bound vibrational levels except the last must be less than $0.848 \dots R_{vdW}$ .Reuse & Permissions
Figure 15
(Color online) Characteristic long- and short-range forms of the $s$ -wave scattering wave function. Upper panel: Scattering length $A = 0$ , for $E ∕ E_{vdW} = 0.0001$ , $0.01$ , and 1 ( $k R_{vdW} = 0.01$ , $0.1$ , and 1). The effect of the $1 ∕ R^{6}$ potential is imperceptible at these distance scales. Lower panel: The threshold $s$ -wave scattering wave functions $C (E) f (R, E)$ as $E \to 0$ for $R \approx R_{vdW}$ , where the $1 ∕ R^{6}$ van der Waals potential dominates. The different curves are for different values of $A ∕ R_{vdW}$ . The wave function is truncated at very short range where the detailed behavior is determined by chemical bonding. The solid circles show the location of the last, $N$ th, node of the zero-energy wave function $f (R, 0)$ .Reuse & Permissions
Figure 16
The relationship between the nodal positions $z$ in the zero-energy $s$ -wave threshold scattering wave function and the scattering length $A$ . If the scattering length is zero, the position of the $N$ th node is approximately $0.998 R_{vdW}$ . This result is derived from the well-known analytical solutions (103) for the zero-energy scattering wave function in a van der Waals potential. Adapted from 140.Reuse & Permissions
Figure 17
The photoassociation line shape vs detuning $ν - ν_{0}$ (red is negative, blue positive) approximated by summing the contribution from 20 discrete collision energies in Eqs. (20, 22). For each collision energy the line shape is a Lorentzian, with width $Γ_{nat} = (2 π) \times 5 MHz$ and a height proportional to $\sqrt{E} \exp (- E ∕ k_{B} T)$ . The heavy line is the sum, and the lighter lines are, right to left, the contributions from $E ∕ h = 2, 6, 10, 14, \dots, 50 MHz$ with the height multiplied by 2 for clarity. Adapted from 64.Reuse & Permissions
Figure 18
(Color online) Comparison of observed line shapes for two rotational lines in the $0_{g}^{-} (3 ∕ 2) v = 1$ vibrational level shown in Fig. 2. For each line the zero frequency is the fitted resonance position $ν_{0}$ . In the fit the linewidth and temperature were held constant while the position and height were fit to the data in a limited region covering the blue side and peak of the line (approximately the region $- 50$ to $50 MHz$ ). The $J = 2$ line is assumed to arise from an initial, free atom, $s$ -wave collision and the $J = 4$ line from a $d$ -wave collision. Note that in comparison to the $J = 2$ line, the $J = 4$ line is broader and peaks further to the red of its threshold.Reuse & Permissions
Figure 19
The quantity $∣ ⟨ b ∣ E, ℓ ⟩ ∣^{2} (\partial E_{b} ∕ \partial b)^{- 1}$ vs $R_{C}$ for $s$ -, $p$ -, and $d$ -wave collisions at $E ∕ k_{B} = 200 μ K$ using a model potential with a scattering length of $690 a_{0}$ and using the reduced mass of ${}^{23}{Na}$ . The lines show the reflection approximation and points show the exact results evaluated for every tenth vibrational level in the excited potential. From 16.Reuse & Permissions
Figure 20
Calculated PA line at $1 mK$ for the $v = 64$ ${}^{3}Σ_{g}^{+}$ excited state of ${Li}_{2}$ . From 15.Reuse & Permissions
Figure 21
(Color online) Trap-loss spectrum in ultracold Ca. The fractional trap loss vs the detuning of the PA laser from the atomic dissociation limit. The loss is quite small, $≲ 1 %$ . The PA lines are vibrational levels in the $B {}^{1}Σ_{u}^{+}$ state of ${Ca}_{2}$ near the $4 s^{2} {}^{1}S + 4 s 4 p {}^{1}P$ dissociation limit. The nonlinear frequency scale is chosen to give equally spaced vibrational levels. Inset: An enlarged view of the $v^{'} = 55$ resonance. The label $v^{'}$ counts vibrational levels down from the dissociation limit rather than up from the bottom of the well. From 168.Reuse & Permissions
Figure 22
Bolometric detection of PA in ${He}^{*}$ held in a magnetic trap at $5 μ K$ . (a) The atom number, (b) the temperature in $μ K$ , and (c) the peak optical density, all plotted as a function of the PA laser detuning from the atomic $D_{0}$ line. Each point represents measurements on a new evaporated cloud after PA pulse illumination, thermalization, and ballistic expansion. Strong heating of the atomic cloud is observed when the PA laser is resonant with a molecular transition. The curve in (b) is a Lorentzian fit to the data with a width of $2.8 MHz$ . From 78.Reuse & Permissions
Figure 23
Excitation scheme for fragmentation detection of PA in K. One laser photoassociates atoms to an excited vibrational level above the curve crossing. This state predissociates to form a $4 {}^{2}P_{1 ∕ 2}$ atom, which is subsequently ionized and detected. From 152.Reuse & Permissions
Figure 24
Actual scheme for measuring the dissociation energy $D_{0}$ of the ground $X {}^{1}Σ_{g}^{+}$ potential of ${}^{23}{Na}_{2}$ . The line labeled PA is the $free \to bound$ photoassociation measurement, the other lines, 1, 2, and XA, represent $bound \to bound$ transitions. From 65.Reuse & Permissions
Figure 25
Trap-loss PA spectrum (Sec. 4B) showing the transition labeled PA in Fig. 24. The arrow indicates the fitted resonance position of the $J = 1$ rotational line of this $v^{'} = 165$ vibrational level of the ${}^{23}{Na}_{2}$ $A {}^{1}Σ_{u}^{+}$ state using the theory discussed in Sec. 3C. The fitted line shape is for a temperature of $k_{B} T ∕ h = 6 MHz$ and a natural linewidth $Γ_{nat} ∕ (2 π) = 24 MHz$ . The fit assumes $s$ -wave scattering and the validity of the Wigner threshold law. Zero frequency is an etalon fringe at $16 954.8749 {cm}^{- 1}$ . From 65.Reuse & Permissions
Figure 26
Fragmentation (lower) and trap-loss (upper) spectra of the predissociated long-range $1_{g}$ and $0_{u}^{+}$ states of ${}^{39}K_{2}$ . The fragmentation spectrum is produced using the scheme shown in Fig. 23. The inset in (c) details the line broadening of the eight vibrational levels in the dashed rectangle. From 152.Reuse & Permissions
Figure 27
(Color online) PA spectrum of ${}^{23}{Na}$ in a dark-spot MOT at $300 μ K$ . One vibrational level of each of the $1_{g} (1 ∕ 2)$ and $0_{g}^{-} (1 ∕ 2)$ potentials is shown. The zero frequency is at the $3 {}^{2}S (f = 2) + 3 {}^{2}P_{1 ∕ 2} (f = 2)$ atomic dissociation limit. The detection is via ionization by a second color laser, which modifies the relative heights of the $1_{g} (1 ∕ 2)$ and $0_{g}^{-} (1 ∕ 2)$ features. The overall height of each piece—the $1_{g} (1 ∕ 2)$ and the $0_{g}^{-} (1 ∕ 2)$ —has been separately scaled to match the theory. The lower trace is a theoretical calculation using a full many-channel treatment for the calculation of the transition dipole moments.Reuse & Permissions
Figure 28
(Color online) (a) PA spectrum of ${}^{23}{Na}$ in a dark-spot MOT at $450 μ K$ detected by trap loss. The zero frequency is at the $3 {}^{2}S (f = 2) + 3 {}^{2}P_{3 ∕ 2} (f = 3)$ atomic dissociation limit. (b) A theoretical calculation where the vibrational defects have been adjusted to match the experiment. (c) A calculation using the same model but with slightly different vibrational defects so that the $1_{g}$ and $0_{u}^{+}$ features switch positions. From 138.Reuse & Permissions
Figure 29
The $0_{g}^{-} (3 ∕ 2)$ potential curve for various alkali-metal dimers calculated using the Movre-Pichler model (Sec. 2E) describing the electrostatic interaction with $R^{- 3}$ , $R^{- 6}$ , and $R^{- 8}$ terms. For ${Rb}_{2}$ and ${Cs}_{2}$ , the curves obtained by including only the $R^{- 3}$ term are also shown (closed circles). No such differences are visible for ${Na}_{2}$ or $K_{2}$ on the scale of the figure. From 44.Reuse & Permissions
Figure 30
(Color online) PA spectrum of the lowest level, $v = 0$ , of the $0_{g}^{-} (3 ∕ 2)$ purely long-range potential of ${}^{23}{Na}_{2}$ detected by ionization. The solid curve through the experimental points applies the theory of Sec. 3C. The dashed line shows the fit shifted to simulate what would happen if retardation were not present. Adapted from 63.Reuse & Permissions
Figure 31
The 16 excited relativistic adiabatic potential curves of the $KRb$ molecule, showing that all eight of the potentials that dissociate to the lower two asymptotes are attractive and all eight of those dissociating to the upper two asymptotes are repulsive $(1 Å = 0.1 nm)$ . From 155.Reuse & Permissions
Figure 32
Photoassociation spectrum for ${}^{39}K {}^{85}{Rb}$ showing eight vibrational bands from seven electronic states. A preliminary state number and $Ω^{\pm}$ symmetry is given for each band. Also given above each symmetry label is $10^{3} B_{v}$ in ${cm}^{- 1}$ . From 150.Reuse & Permissions
Figure 33
Transmission and light-induced fluorescence PA spectra of H held in a 6-T magnetic trap. The feature at $Δ = - 134 GHz$ is a PA line. The large structure in the fluorescence spectrum around $Δ = - 80 GHz$ is the atomic transition. Inset: The PA line shape with a theoretical fit that accounts for the large magnetic-field gradient. The vertical line denotes the inferred position of the unbroadened line. From 102.Reuse & Permissions
Figure 34
Ungerade electronic potential curves for excited states reached in PA of metastable He. The curves result from an extended Movre-Pichler model taking account of the three fine-structure levels of the atomic $2 s 2 p {}^{3}P$ state. Three arrows indicate the purely long-range potential wells. From 78.Reuse & Permissions
Figure 35
Relative PA signal strengths (dots) for transitions to vibrational levels of the $A {}^{1}Σ_{u}^{+}$ state of ${}^{6}{Li}_{2}$ detected by trap loss. The binding energies vary from $220 GHz$ for $v = 79$ to $3.5 THz$ for $v = 62$ , and the PA strength is averaged over the three hyperfine features. Results of calculations of the signal strengths for three different values of the scattering length are shown. From 1.Reuse & Permissions
Figure 36
Square of $ℓ = 2$ partial-wave ground-state wave function for ${}^{87}{Rb}$ in the presence and absence of a shape resonance at a collision energy $E ∕ k_{B} = 0.3 mK$ . The classically forbidden region of the $d$ -wave barrier (see Fig. 14) is shown by the dashed vertical lines. Inset: A blowup of the shorter-range behavior relevant for PA. From 13.Reuse & Permissions
Figure 37
Comparison of experimental (closed circles) and theoretical (solid line) line shapes for the $J = 2$ $0_{g}^{-}$ (3/2) line of several vibrational levels of ${}^{39}K_{2}$ . The synthetic spectra were calculated assuming a cloud temperature of $400 μ K$ . The asymmetric line results from the separate contributions from $s$ and $d$ waves. The latter are enhanced because of the presence of a low-lying $d$ -wave shape resonance. From 17.Reuse & Permissions
Figure 38
Photoassociative spectrum of the $2 {}^{3}Π_{g}$ and $2 {}^{1}Π_{u}$ states of ${Na}_{2}$ below the $3 S + 3 D$ asymptote located at $29 173 {cm}^{- 1}$ . Inset shows the excitation scheme used in the experiment. From 125.Reuse & Permissions
Figure 39
Two-color ${Na}_{2}^{+}$ ion production in a MOT at $450 μ K$ . For each trace a PA laser is fixed to produce a steady population of molecules in a specific level ( $J = 2$ , $v = 0$ , 1, or 4) of the $0_{g}^{-} (3 ∕ 2)$ purely long-range state. A second laser is scanned and resulting ions counted and plotted as a function of the total energy of two photons. The zero is the $3 {}^{2}P_{3 ∕ 2} (f = 0) + 3 {}^{2}P_{3 ∕ 2} (f = 0)$ atomic dissociation limit. The traces are offset vertically for clarity. Below threshold the three traces show the same pattern of transitions to bound autoionizing levels; the extent of the above-threshold “Condon internal diffraction” spectrum depends on the available kinetic energy of the vibrational level. From 66.Reuse & Permissions
Figure 40
(Color online) Two-photon PA scheme for probing the structure of the electronic ground state(s), here labeled with quantum numbers appropriate for Cs. The two laser fields are denoted $L_{i}$ ; $Δ$ is the detuning of $L_{1}$ with respect to level $∣ 1 ⟩$ . The vibrational level at the ground-state asymptote with its rotational structure is called $∣ 2 ⟩$ . The gray shaded area above the lowest hyperfine asymptote 3+3 indicates the thermal distribution of atoms in the MOT $(∣ 0 ⟩)$ . Loss channels with rates $γ_{i}$ are shown as well. Typically, $γ_{1}$ , due to spontaneous decay or ionization by auxiliary lasers, is much larger than $γ_{2}$ . From 85.Reuse & Permissions
Figure 41
Line shapes observed in Autler-Townes PA spectroscopy of Cs as a function of the frequency of laser $L_{2}$ (see Fig. 40). The rotational series of a vibrational level $0.6 {cm}^{- 1}$ below the $(f = 3) + (f = 3)$ hyperfine asymptote is shown. In (a) laser $L_{1}$ is tuned to the PA transition to $∣ 1 ⟩$ , while in (b) laser $L_{1}$ is tuned about $20 MHz$ higher. The frequency scales are the same for (a) and (b). From 85.Reuse & Permissions
Figure 42
Trap-loss spectra of two-photon PA transitions to ground-state molecules in a Bose condensate of ${}^{87}{Rb}$ for four different peak condensate densities $n_{0}$ . The intermediate excited state was a vibrational level in the $0_{g}^{-} (1 ∕ 2)$ state with a binding energy of $23 {cm}^{- 1}$ . The peak densities are for (A) $n_{0} = 0.77 \times 10^{14}$ , (B) $1.22 \times 10^{14}$ , (C) $1.75 \times 10^{14}$ , and (D) $2.60 \times 10^{14} atoms ∕ {cm}^{3}$ . Each spectrum shows the fraction of atoms remaining in the condensate after illumination by two laser fields as a function of the laser-frequency difference. The resonant decrease in atom number arises from the formation of molecules by stimulated Raman transitions, followed by their subsequent loss from the trap. The increase in width and center frequency of the resonance with density arises from atom-atom and molecule-atom mean-field interactions. From 163.Reuse & Permissions
Figure 43
(Color online) Two-color PA of two atoms in a single lattice site of an optical lattice. (a) The transitions in terms of the internuclear separation. (b) The same process viewed in terms of the center-of-mass coordinate of the pair. The Raman transition shown can affect both the relative motion (i.e., the rovibrational state) and the center-of-mass motion. In (b) two possible transitions are shown; in the upper graph the center-of-mass motion is not affected while in the lower it is changed by one quantum. From 117.Reuse & Permissions
Figure 44
Two-photon PA of a weakly bound $ℓ = 0$ level just below the ground electronic $1 + 1$ hyperfine dissociation limit in ${}^{87}{Rb}_{2}$ . This state, located at $δ ∕ 2 π \approx 636 MHz$ , is of mixed singlet/triplet character. Transitions to this state were also shown in Fig. 42. The center-of-mass motion is resolved and leads to the observation of nearly equally spaced lines, consistent with quantized, nearly harmonic motion. The detuning is $Δ = - 2 π \times 900 MHz$ and the lattice depth is $27 E_{r}$ . From 117.Reuse & Permissions
Figure 45
Example of time-dependent PA for the investigation of a shape resosance. (a) The timing of laser pulses in a time-dependent PA experiment. All traces are spectra of the $0_{g}^{-} (1 ∕ 2)$ vibrational level at $12 573.04 {cm}^{- 1}$ . (b), (e) Single-pulse PA [timing case (a)]. (c), (f) Two-pulse PA [timing case (b)]. (d), (g) The differences between the single- and double-pulse spectra with delay time $T = 0$ and $0.9 μ s$ , respectively. The absence of a $J = 4$ signal in (d) shows that the population behind the $g$ -wave barrier has been depleted. This signal recovers with delay time as shown in (g). From 13.Reuse & Permissions
Figure 46
Creation of bound ground molecules by spontaneous emission from an excited bound state formed by PA. (a) The generic relation between a highly excited vibrational level of the excited state and bound states in the ground potential. The excited state has a good overlap with the initial collisional state but not with a bound ground state. Arrow (1) is the PA step, (2) is spontaneous decay back to hot atoms, and (3) is the desired molecule formation process. (b) The special situation which occurs for levels in the $0_{g}^{-} (3 ∕ 2)$ potential of ${Cs}_{2}$ . This potential has a “bump” at a range of internuclear separations where ground bound levels have outer turning points. Adapted from 35.Reuse & Permissions
Figure 47
Cold ${Cs}_{2}$ molecule production and trapping. (a) The same schematics of the PA process shown in Fig. 46. (b) The number of ${Cs}_{2}^{+}$ detected as a function of time after the PA laser is turned off. The initial drop is due to molecules leaving the small focal spot of the ionization laser. That some ions are detected long afterwards shows that molecules are trapped in the weak magnetic trap. Curves (1) and (2) are with and without MOT light present, showing that MOT light influences the lifetime of the molecules. (c) The same as (b) curve (2) but with a higher background gas pressure. (d) The molecular yield for various length PA pulses. The number is measured $60 ms$ after the end of the PA process [arrow in (b)]. From 143.Reuse & Permissions
Figure 48
(Color online) Ground-state molecule production from ultracold Na atoms in a dark-spot MOT with most atoms in the $f = 1$ hyperfine state. The upper trace (and upper-level diagram) shows an Autler-Townes “dip” spectrum as discussed in Sec. 7A. The dips indicate tunings of L2 which connect ground-state levels to the excited-state level $∣ e 1 ⟩$ . Laser L1 is fixed to PA atoms to this same $∣ e 1 ⟩$ level. Peaks occur when L2 acts as the PA laser. In the lower trace laser L2 is tuning over the same region as before, but now laser L1 is set to a different PA line, producing excited-state molecules in level $∣ e 2 ⟩$ , which decay by spontaneous emission producing ground-state molecules which laser L2 excites to $∣ e 1 ⟩$ . The peaks in the lower trace which line up with dips in the upper trace represent molecule formation peaks. From 41.Reuse & Permissions
Figure 49
(Color online) On-resonance PA rate coefficient as a function of laser intensity in a simplified model in which all collisions have a single energy $E = k_{B} T$ . All parameters are scaled by units defined in the text. The solid curves show the rate coefficient for three different finite temperatures. The $T = 0$ line is given by $K ∕ K_{0} = I ∕ I_{0}$ . The horizontal dashed line is the unitarity limit for collisions at $T = T_{0}$ . This upper limit is reached at $I = I_{0}$ for this temperature.Reuse & Permissions
Figure 50
(Color online) Light shift $S_{b}$ of a PA resonance in a Na Bose-Einstein condensate (BEC) as a function of laser intensity. The dotted line is a straight-line fit to the data as expected from the intensity dependence in Eq. (21). The solid line is a straight line with slope predicted by a theoretical calculation of the dipole matrix elements. Even though the strength of this PA line is almost entirely due to $s$ -wave collisions, a major portion of the shift comes from the presence of a nearby $d$ -wave shape resonance. Adapted from 95.Reuse & Permissions
Figure 51
(Color online) On-resonance photoassociation rate constant as a function of intensity in a Na BEC. The corrected data have been adjusted to account for inhomogeneous broadening. The theory line is again due to a coupled-channel calculation of the dipole matrix elements with no adjustable parameters. It includes the factor of 2 reduction in the rate as compared to a noncondensed gas, as described in the text. Adapted from 95.Reuse & Permissions
Figure 52
On-resonance photoassociation rate coefficient in a ${}^{7}{Li}$ gas near the transition to Bose condensation as a function of laser intensity. The open circles are experimental data, obtained from trap-loss analysis. The solid line is the theoretical prediction, with no adjustable parameters. The dashed line is an extrapolation of the theoretical result in the low-intensity limit. From 114.Reuse & Permissions
Figure 53
Photoassociation method for detecting a magnetic Feshbach resonance in collisions of ultracold ${}^{85}{Rb}$ atoms in the ( $f = 2$ , $m_{f} = - 2$ ) state. The three ground-state $5 {}^{2}S_{1 ∕ 2} + 5 {}^{2}S_{1 ∕ 2}$ potentials are similar to those shown for Na in Fig. 11. The initial scattering wave function $u_{init} (R)$ couples to a quasibound resonance state with wave function $u_{res} (R)$ . The energy of the bound state can be shifted into resonance by application of a static magnetic field. A laser field induces PA to an excited, rovibrational level $(v, J)$ in the $0_{g}^{-} (1 ∕ 2)$ potential, which then decays by spontaneous emission back to free atoms. The PA is detected by trap loss from a FORT. As the quasibound level tunes through zero energy the scattering wave function is modified, changing the PA rate. From 28.Reuse & Permissions
Figure 54
Photoassociation trap-loss spectra showing the $J = 0$ , $M_{F} = - 4$ peak in a vibrational level of the $0_{g}^{-} (1 ∕ 2)$ state $\approx 6 {cm}^{- 1}$ below the $5 {}^{2}S_{1 ∕ 2} + 5 {}^{2}P_{1 ∕ 2}$ dissociation limit for a succession of magnetic-field values, with a laser intensity of $0.1 W ∕ {cm}^{2}$ . This peak arises only from $s$ -wave scattering between two $5 {}^{2}S$ ( $f = 2$ , $m_{f} = - 2$ ) atoms. The loss of atoms is plotted as a positive signal. The relative Zeeman shift of successive peaks is removed so that they appear at the same laser tuning. From 28.Reuse & Permissions
Figure 55
Photoassociation line strength as a function of applied magnetic field for the $J = 2$ rotational line in two different vibrational levels of the purely long-range $0_{g}^{-} (3 ∕ 2)$ potential in ${}^{133}{Cs}_{2}$ . The excited vibrational level is detected by ionizing ground-state molecules formed by spontaneous emission from $0_{g}^{-} (3 ∕ 2)$ . The rapid variation in signal strength with magnetic field reflects variations in the ground-state scattering. From 75.Reuse & Permissions
Figure 56
Optical Feshbach resonance in a ${}^{87}{Rb}$ Bose condensate. (a) The atom number remaining after an intense PA pulse as a function of the detuning of this laser from line center. (b) The measured Bragg resonance frequencies, which are sensitive to the mean-field interaction in the condensate. (c) The values for the scattering length obtained from the data in (a) and (b) along with a fit by Eq. (27). The background scattering length of $\approx 100 a_{0}$ , shown by the horizontal line in (c), is consistent with other determinations. From 136.Reuse & Permissions

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