Direct observation of Rydberg-Rydberg transitions via CPmmW spectroscopy

Rydberg-Rydberg transitions of BaF molecules have been directly observed in our lab. The key to the experimental success is our ability to combine two powerful and new technologies, Chirped-Pulse millimeter-Wave spectroscopy (CPmmW) and a buffer gas cooled molecular beam source. CPmmW spectroscopy is a form of broadband (20 GHz bandwidth), high-resolution (50 kHz), free induction decay-detected (FID) spectroscopy with accurate relative intensities (10%), which is successfully applied to record Rydberg-Rydberg spectra of Calcium and Barium atoms. To extend CPmmW spectroscopy to a molecular system, I have constructed a new setup, a 20 K Neon buffer gas cooled molecular beam system, which generates of beam containing >1000 times more molecules and 10 times less translational velocity than a Smalleytype laser ablation supersonic beam source. Hundreds of molecular Rydberg-Rydberg transitions with high resolution and high dynamic range can now be recorded in a few hours. The success of this experiment suggests many applications in the near future, such as developing a user-friendly experimental method to study and manipulate Rydberg molecules, preparing a single molecular beam pulse that contains 10 state-selected core-nonpenetrating Rydberg molecules/molecular ions, and studying the electronic structure (dipole and quadrupole moments and polarizability) of the molecular ion core with unprecedented precision and completeness. In addition, strong collective effects (superradiance) have also been observed. Thesis Supervisor: Robert W. Field Title: Haslam and Dewey Professor of Chemistry


Acknowledgments
There are many people I wish to thank for their advice and collaborations in the past seven years. The first among them is my adviser, Bob Field. In one hand, Bob is not different from most PhD advisers I have heard. He has extreme enthusiasm on research, works hard and loves his students. He has a unique vision of small molecule spectroscopy and is good at conveying his ideas to us. However, in another hand, Bob is unlike any PhD advisers I have heard. He has seldom answered my questions directly, but always listens to me very carefully, and then give me his guidance through many instructive questions. He has never criticized my ignorance, but always encourages me to be an expert through carefully learning and thinking.
He has never gave me a job list to do, but always encourages me to think what is the most important. I have never felt any pressure from him, but always benefit and enjoy the light he shines in the front. In addition, he always set up a filter to shield out most of the bureaucratic things and leave me a freedom and relax research environment. At the same time, he always tries his best to leave opportunities to me of reporting our work and express our ideas at the center of the academic arena. You are not only a senior technician, but also my electronics adviser.
Thank you, Peter Giunta, for all that you have done for everybody in the group.
Last but certainly not least, I wish to thank my parents. Your constant support and encourage in the past thirty years keep me going forward. Step 1, we use lasers to impact on the area near the ion-core (blue area) to populate core-penetrating Rydberg states (ii). Corepenetrating states contain information about ion-electron interactions, which can reveal the information of chemical reactions. However, we

Contents
do not stop at this step because of the complexity of the spectrum and the fitting model.
Step 2, we use millimeter-wave pulses to impact on the area far from the ion-core (yellow area) to populate corenonpenetrating Rydberg states (iii). In the core-nonpenetrating states, the electron and ion-core are almost decoupled and can be controlled and measured separately. Even better, the Rydberg electron can be used as a sensitive probe to characterize the structures of the molecular ion-core. By fitting the core-nonpenetrating Rydberg spectrum by a perturbative model [1], we can extract the important parameters of the molecular ion-core, such as multipole moments and polarizabilities.
Step 3, theorists can implement the precisely measured information of the molecular ion-core into the collisional model of the core-penetrating as measured by millimeter wave spectroscopy. The frequency axis is plotted relative to a high precision measurement by Kleppner [7]. The frequency shift is due to the linear Stark effect, and the broadening comes from the inhomogeneous electric field in the interaction volume. 141 [3][4][5][6][7] The best Ramped-PFI spectrum of millimeter wave spectrum of Calcium 36p-36s transition with 5 MHz linewidth. The asymmetric lineshape might come from the asymmetric stray electric field. To achieve a acceptable signal to noise level, this spectrum is the average of the 10 spectra with the same bandwidth. The total data collection spends

Properties of Rydberg states
The Rydberg states of an atom or a molecule are highly electronically excited states [12].
The loosely bound Rydberg electron has an orbit radius far larger than the Bohr radius and spends most of its time far from the ion core. The electron/ion-core interaction can be described in zeroth order by the Coulomb interaction, where e is the electron charge, Z is the effective charge of the ion-core, ε 0 is the vacuum permittivity, and r is the distance between the electron and the ion-core.
where IP is the ionization potential, Ry is the Rydberg constant, n is the principal quantum number, and δ is the n-independent quantum defect, which is zero for all states of the Hydrogen atom. The energies and wavefunctions of the Rydberg states that belong to one Rydberg series can be calculated with well-known equations and have obvious periodic structures with n-dependent scaling rules.  [13]. Since many properties scale very non-linearly in n, Rydberg states with moderately high principal quantum numbers (n∼40) have much larger or smaller properties than low-lying excited states. For example, the gigantic electric dipole transition moment and polarizability provide a large handle for control of the internal and external degrees of freedom of the atoms by interactions with an external field or atom-atom collisions. The long radiative lifetime provides a large time-window for precise control and measurement [14,15,16,17,18].
In addition to atomic Rydberg states, molecules also have Rydberg states, which provide more valuable information [19]. Different from the closed-shell alkali atomic ion-core, the molecular ion-core has additional degrees of freedom, such as rotation and vibration, and a permanent electric dipole moment. The collisions between the Rydberg electron and the ion-core are not only elastic, but also can be inelastic with energy and angular momentum exchange, especially when the period of the Rydberg electron motion is resonant with the period of the ion-core rotation or vibration (Stroboscopic resonance) [20,21,22,23,24]. Stroboscopic resonances govern intramolecular chemical reactions and reveals the subtle mechanisms of the interactions between a heavy nuclei and a light electron.
Based on the strength of the interactions between the Rydberg electron and the ion-core, molecular Rydberg states can be classified into two categories, corepenetrating Rydberg states (CP) and core-nonpenetrating Rydberg states (CNP).
The Rydberg electron in core-penetrating Rydberg states with low angular momentum, , can penetrate into and collide with the ion-core, as (ii) in Figure 1-1. Due to the strong coupling between the Rydberg electron and the ion-core, the vibration and rotation quantum numbers of the ion-core and the angular quantum number of the Rydberg electron are not "good". The spectroscopy of core-penetrating states should be of special interest to chemists, because important information about chemical reactions can be extracted from the spectrum [25].
Molecular core-nonpenetrating Rydberg states are a unique and neglected class of matter [1,26,27,28,29]. They are defined as having negligible overlap between the Rydberg electron wavefunction and that of the ion-core. Their energies and wavefunctions can be described a priori by the hydrogenic Rydberg formula.
The Rydberg electron is essentially decoupled from its ion-core, leading to an atomlike electronic structure with "almost good" angular momentum quantum number of the Rydberg electron and vibration and rotation quantum numbers of the ioncore. Because molecular core-nonpenetrating states do not interact strongly with the non-spherically symmetric part of the ion-core, they are similar to atomic Rydberg states in the zeroth-order picture. Therefore, the Rydberg electron is extremely sensitive to external electric fields, which can induce gigantic electric dipole moments and polarizabilities. In addition, the molecular core-nonpenetrating Rydberg states are metastable with relatively long lifetimes [13,30,19]. Molecules in such long-lived, highly polarizable states are useful to researchers working on molecular slowing, cooling, and trapping [14,15,16,17,18]. Different from the complicated spectra of molecular core-penetrating Rydberg states, the spectra of molecular Step 1, we use lasers to impact on the area near the ion-core (blue area) to populate core-penetrating Rydberg states (ii). Core-penetrating states contain information about ion-electron interactions, which can reveal the information of chemical reactions. However, we do not stop at this step because of the complexity of the spectrum and the fitting model. Step 2, we use millimeter-wave pulses to impact on the area far from the ion-core (yellow area) to populate core-nonpenetrating Rydberg states (iii). In the core-nonpenetrating states, the electron and ion-core are almost decoupled and can be controlled and measured separately. Even better, the Rydberg electron can be used as a sensitive probe to characterize the structures of the molecular ion-core. By fitting the core-nonpenetrating Rydberg spectrum by a perturbative model [1], we can extract the important parameters of the molecular ion-core, such as multipole moments and polarizabilities.
Step 3, theorists can implement the precisely measured information of the molecular ion-core into the collisional model of the core-penetrating Rydberg states to reduce the number of fit parameters in the multichannel collision theory [2,3,4,5].
Molecular core-nonpenetrating Rydberg states are both a central object of scientific study as well as a tool for novel applications in spectroscopy and molecular dynamics [19,32,33,26,27], quantum chemistry [34,35,36], and fundamental physics [37,38,39]. to jump over the complicated molecular core-penetrating Rydberg states to easily controlled and measured molecular core-nonpenetrating Rydberg states, where the Rydberg electron is used as a sensitive probe to characterize the structure of the ioncore. Then, we feedback the precisely measured parameters of the ion-core into the molecular core-penetrating Rydberg states to to reduce the number of fit parameters in the multichannel collision theory. Our ultimate goal is to obtain a complete model that consists of fundamental molecular constants, but without any empirical or ab initio calculated parameters. We would hope to apply this model to calculate the "all-spectra and all-dynamics" in a broad range of energy [35,34,36,40,2,41].

Chirped-pulse millimeter-wave technique
Millimeter wave spectroscopy is a powerful tool for performing high precision measurements on atoms and molecules in the gas phase [42,43,44,7,45,46,47]. Compared to laser spectroscopy, millimeter wave spectroscopy can have >1000 times higher resolution and accompanying higher frequency accuracy. Millimeter wave spectroscopy has been widely used to determine molecular structures by the rotational spectrum [48,49,50], to detect astronomical and atmospheric molecules [51], and to induce pure electronic transitions in Rydberg states [52,53,7,54,45,55]. However, for a conventional frequency-stepping millimeter wave spectrometer, the dual goals of achieving high resolution and covering a broad bandwidth requires very long data collection times [56]. Fortunately, the recent invention of Chirped-Pulse Fourier-Transform Microwave (CP-FTMW) spectroscopy by Pate and co-workers breaks the limitation of the frequency-stepping spectrometer, and provides a method for obtaining broadband spectra with high resolution in a very short data collection time [57,58]. Barratt  With advanced phased-locked microwave electronics, all carrier frequency information is preserved and can be transformed into the frequency-domain. Due to the preservation of both amplitude and phase information, and the use of an advanced fast digitizer, the CPmmW technique has a much higher data collection rate (more than 10 5 improvement) than a frequency-stepping spectrometer. In addition, since all frequency components are collected simultaneously, the relative intensities in the frequency-domain are reliable and immunized from shot-to-shot fluctuations of the lasers and molecular beam source. CPmmW spectroscopy provides a combination of high sensitivity, broad single-shot spectral coverage (>20 GHz), high resolution (sub-100 kHz), and accurate (5%) relative intensities. This technique plays a central role in this thesis.

Buffer gas cooled molecular beam
The atoms or molecules of interest in my experiments are alkaline earth atoms (Calcium, Barium) and alkaline earth monohalide molecules (CaF, BaF). The conventional beam source for such species is a supersonic jet coupled with laser ablation, which has many advantages, such as low translational and rotational temperatures (<5 K), relatively small gas load (<10 ml/min at one atmosphere and room temperature), and production of a wide range of species [61,62]. However, fast beam velocities (1800 m/s for Helium, 600 m/s for Argon), large shot-to-shot fluctuations (>50%), and relatively low flux (<1×10 8 molecules/sr/pulse) significantly degrade the resolution, signal-to-noise ratio, and limit the applicability of CPmmW spectroscopy to our interested problems, such as superradiance of a dense atomic Rydberg gas, and pure electronic spectroscopy of molecular core-nonpenetrating Rydberg states.
The buffer gas cooling technique was pioneered by the De Lucia group in the 1980s [63,64] and has recently been extensively developed by the Doyle group [65,66,67,68,69,70,71] for application to ablated atoms/molecules. Compared to supersonic cooled molecular beam, the buffer gas cooling technique achieves comparable translational and rotational temperatures (<5 K), but obtains a much smaller beam velocity (reduced by a factor of 10) and larger number density (increased by a factor of 1000).
Compared to a supersonic cooled beam, the slow beam generated in the buffer gas cooling technique reduces Doppler dephasing, increasing our spectroscopic resolution by a factor of 10. In addition, the high number density not only linearly increases the detected signal in typical experiments, but also provides access to the optically thick regime for nonlinear experiments. Combining the number density and the spectroscopic resolution improvement, the buffer gas cooled molecular beam provides a factor of 10000 improvement.
• Create an extremely optically thick medium -Intermolecular interactions can induce or modify chemical reactions, especially via intermolecular radiative interactions, which is of considerable interest to chemists (e.g. J-aggregate) [85].
-Study the propagation of a coherent pulse in a highly nonlinear medium [86,87,88].

Thesis overview
In Chapter 2, I describe the experimental setups of two vacuum chambers, including:

Vacuum system for supersonic jet-cooled molecular beam
Smalley-type laser ablated supersonic jet-cooled molecular beam sources have been widely used to create cold metal clusters and diatomic radicals in the past fifty years [61,62]. This technique has many well-known advantages: (1)  Nitrogen cold traps, which simplifies the vacuum system dramatically. However, the price of the turbo pump is much higher than that of a diffusion pump with the same pumping speed. To avoid paying too much for unnecessary pumping capability of the turbo pump, I estimate required pumping speed carefully. Two parameters that limit where P ba is the average background pressure, P 0 is the backing pressure of the general valve, d is the diameter of the nozzle (cm), f is the experimental repetition rate, ∆t is the nozzle opening duration time, S is the pumping speed (L/s), and C is the nozzle conductance (L/cm 2 /s). 1 [94].
The transient background pressure can be calculated by: where P ∆ bt is the transient background pressure, τ is the background pressure relaxation time constant, and V is the volume of the vacuum chamber. This is due to the volume of our source chamber being large enough to dissipate the gas pulse very rapidly at early time, a rate of which is almost independent of the pumping speed. The 50 ms background pressure relaxation time with the turbo pump suggests that I cannot decrease the transient pressure significantly by increasing the pumping 1 In our experiment, P 0 =40 PSI, d=0.5 mm, f=20 Hz, ∆t=150 µs, S=1000 L/s for the turbo pump, 3000L/s for the diffusion pump, C He =45 L/cm 2 /s, C N e =20 L/cm 2 /s, C Ar =14 L/cm 2 /s, C He =16 L/cm 2 /s The required pumping speed for the turbo pump in the detection chamber is much slower due to the tiny gas flow through the 0.5 mm diameter skimmer. The pressure of the detection chamber with the turbo pump can be calculated by: where P ef f is the effective pressure before the skimmer, which is the local pressure of the molecular beam, and approximately 10 times higher than the transient background pressure (P bt ) of the source chamber. For typical experiments, the pressure in the detection chamber is <1 µtorr, which is acceptable for the TOF-MS and the MCP detector.

Cutaway source
The cutaway source was designed by the Duncan group to create metal-Noble gas clusters, as shown in Figure 2-3, Plot (a) [95,96]. Compared to the regular closed channel source, as shown in Figure 2-3, Plot (b), the molecular beam from the cutaway source is colder, more intense, and more stable. In addition, without the confinement properties of the cutaway source are helpful, except for the metal-rare gas clustering.  As short as possible (>10 µs, otherwise, the supersonic expansion cannot be well established). In my experiment, 150 µs, which is limited by the IOTA one pulse controller. Delay between the ablation laser pulse and gas pulse Usually, the sweet spot occurs when the ablation laser pulse meets the front edge of the gas pulse. This time delay is quite sensitive, and 30 µs off would decrease the molecular beam intensity by a factor of 5.

Miscellaneous
Several minor improvements to Gertrude are summarized as follows: • LIF experiment: To increase the detection efficiency, I built a cage system (LC6W, Thorlabs) with short focued legnth lens (f=5 cm) above and parabolic mirror below (f=5 cm). The detection solid angle is nearly π. In addition, to decrease the laser scattering, two 1' long baffle tubes with four quarter inch irises and two Brewster windows are implemented. The number of scattered photons per pulse detected by PMT with typical laser power is less than 3 without any filter.
• Skimmer position: To prevent molecular beam reflections from disturbing the beam propagation, a 0.5 mm diameter conical skimmer is placed 1.5" away from the wall between the source and detection chambers.
• Ion deflector: To prevent the ablated ions from propagating into the detection region of the TOF-MS, a DC ion deflector at 10 V is placed between two conical skimmers in the detection chamber. In addition, to shield the detector region from the stray electric field generated by the ion deflector, a gold shroud is mounted around the high voltage plate stacks.
• Fine tuning power supply: To minimize the stray electric field, two isolated metal plates in the detection region should have the same voltage. The D-603 TOF power supply has very good short-term and long-term voltage stability (V rms < 10mV ).
However, the tuning resolution of the output is 1 V. To achieve better tuning resolution, I connect a high precision, low voltage power supply (HP 6632A) to D-603 power supply in series. This combination power supply can provide up to 3 kV with <20 mV voltage fluctuation and <1 mV tuning resolution. • Faster General Valve controller: Ablated molecules in our cutaway source are typically distributed in a beam of 50 µs duration. The gas pulse, whose duration is longer than 50 µs, is unnecessary to enhance the signal, but degrades the performance of the molecular beam and requires a larger vacuum pump. It would be possible to modify our current IOTA-One General Valve controller to achieve shorter gas pulse duration (<50 µs). Thus, the background gas pressure in the source chamber could be lower, and the molecular beam properties could be improved. It would also give us some freedom to increase the backing pressure of the nozzle far beyond our current limiting pressure of 40 PSI.
• Faraday cage: There are two parallel plates in the interaction region with the same DC voltage (1.5 kV). A ∼2 µs ramped pulse is applied to the bottom plate, but not to the top plate. Our current setup adds the same voltages on both of these two metal plates with two independent 1.5 kV power supplies. The fluctuation of the relative voltages from the two power supplies is more than 20 mV, which is difficult to be decreased by fine tuning either power supply. To decrease such source of stray electric field, the best method is to connect the two plates to form a Faraday cage.
A uni-directional electric circuit with multiple diodes might be used to isolate the ramped pulse on the bottom plate.

Vacuum system for buffer gas cooled molecular beam
My first atomic Rydberg-Rydberg FID experiments are performed in the laser ablated supersonic beam apparatus, which is described in Section 3.2. The number density of Calcium Rydberg states created in the supersonic beam (∼1×10 5 cm −3 ) barely results in sufficient FID radiation signal to be detected by the CPmmW spectrometer.
However, due to a factor of ∼10 loss for molecule synthesis (eg. Ba + SF 6 → 10% BaF) and a factor of 10 loss for the rotational partition function (at 5 K), extension of the FID detected experiments to a molecular system is difficult in the supersonic beam apparatus. In addition, the supersonic beam has several other disadvantages:

Characterization of gas flow regimes
The Reynolds number is a dimensionless quantity that is used to characterize the gas flow. It is defined as the ratio of inertial forces to viscous forces: where n 0,b is the number density of the buffer gas in the cell, m is the mass of the buffer gas atoms, T 0 is the temperature of the cell (4 K for Helium buffer gas, 20 K for Neon buffer gas), v 0,b is the mean thermal velocity of the buffer gas in the cell (150 m/s), and A aperture is the aperture size of the cell output (d aperture =3 mm).
With a fixed temperature and aperture size, the buffer gas density is proportional to the flow rate, as shown in Figure 2 Effusive source (Re ≤ 1) In this region, the mean free path of the buffer gas (λ b ) is much larger than the aperture diameter, d aperture . There are no collisions as the molecules pass through the aperture. The behavior of the beam is completely determined by the thermodynamics in the cell and the geometry of the aperture. The density distribution of the gas in the beam can be separated into a spatial distribution and a velocity distribution: where n(R, θ) is the spatial distribution of the number density, f(v) is the normalized velocity distribution, R is the distance from the aperture, and θ is the angle away from the normal to the aperture. The beam temperature is the same as the equilibrium temperature in the cell. The normalized distribution of a classical gas follows the Maxwell-Boltzmann distribution: The velocity distribution in an effusive beam and mean forward velocity can be obtained from Eq.(2.8): Due to the complete thermal equilibrium in an effusive beam, the forward velocity spread and transverse velocity spread are identical and follow the thermal 1D Maxwell-Boltzmann distribution: The angular part of the number density distribution in an effusive beam can be assumed as a semi-isotropic distribution: where n 0 is the number density of the gas in the cell. The angular spread (FWHM) and divergence is: The above discussion applies not only for the buffer gas, but also for the target molecules, because there is complete thermal equilibrium at temperature T. However, Supersonic source (Re ≥ 100) In the supersonic region [61,94], the mean free path of the buffer gas (λ b ) is much smaller than the aperture diameter d aperture . Therefore, the buffer gas experiences thousands of collisions upon passing through the exit aperture. The beam properties are determined not only by the thermal equilibrium in the cell, but also determined by the dynamical flow of the gas, which is well described by the Navier-Stokes equation [94].
where γ is specific the heat ratio, which, for a monatomic buffer gas, is γ=5/3, D nozzle is the nozzle diameter, and A and x 0 are fitting parameters (A = 3.26 • The temperature of the supersonic beam decreases as the beam isentropically expands. During the expansion, thermal energy is converted into translational kinetic energy. Therefore, the internal energy (velocity spread) of the beam is reduced, but the average velocity of the beam is increased.
where T 0 is the temperature of the nozzle, m is the mass of the buffer gas molecules, and R is the gas constant.
• During the expansion, the local pressure and beam density decrease continuously: where P 0 and n 0 are the pressure and number density in the nozzle.
From Eq. (2.14) and (2.15), I obtain a direct relationship between cooling and expansion for a monatomic buffer gas: where R is the distance from the nozzle. The temperature and density of the beam decrease rapidly as the molecules fly away from the nozzle. However, this cooling process terminate where the gas density in the beam becomes sufficiently low that the mean free path is larger than the distance of the neighboring molecules. There, the gas in the beam ceases to behave as a fluid and the Navier-Stokes equation can no longer be used. Following, I estimate the effective cut-off boundary of the cooling process and evaluate the final thermal properties of the beam. This discussion not only can be applied for the supersonic expansion (Re ≥ 100), but also for the hydrodynamical expansion (1 ≤ Re ≤ 100).
In Figure where n(x) is the number density of the beam x cm away from the nozzle, σ is the collisional cross section for Neon-Neon or Neon-target molecule. A very rough but simple approximation of the average mean free path between positions 1 and 2 is: where v is the beam velocity. 3 The time intervals from the two frames should be the same, as ∆t 1 = ∆t 2 . Therefore: (2.20)  14) to calculate that the lowest temperature in the conventional supersonic beam is 3 K, which is consistent with the experimental measurements. In addition, I evaluate all other thermodynamic processes that occurs in the supersonic molecular beam at d boundary , such as the velocity and the beam divergence as following: where H(T 0 ) is the enthalpy of the gas in the nozzle at room temperature, is the enthalpy of the gas at the cut-off boundary in the beam at T=3 K.
• Forward velocity spread • Angular spread (FWHM) ∆θ ss and beam divergence ∆Ω ss The above discussions of the gas thermodynamics in a supersonic beam apply to the buffer gas, but not to the target molecules. Different from the effusive beam, the equilibrium condition of the buffer gas and target molecules in the supersonic beam is not having the same translational temperature, but having the same velocity and velocity spread. Therefore, the translational temperatures of the buffer gas and target molecules are different by a factor of m t /m b . However, due to the obscured lack of mass dependence of the rotational temperature, the equilibrium rotational temperature of the target molecules is still approximately the same as the translational temperature of the buffer gas. For example, when I seed BaF in a 4 K supersonic beam with Argon buffer gas, the translational temperature of the BaF molecules is ≈8 K, but the lowest rotational temperature of BaF molecules is still 4 K.  Figure   2-10 shows how the forward velocity changes ongoing from the effusive limit to the supersonic limit.
where v 0,t is the average velocity of the target molecules in the cold cell, and v 0,b is the average velocity of the buffer gas molecules in the cold cell. In an isentropic expansion, the forward velocity spread begin to decrease and the temperature of the target molecules in the buffer gas beam is cooled to a lower temperature than the cell temperature. The final temperature is determined by the Reynolds number and lies between the effusive and supersonic limits. The specific dependence is calculated by the method described in Section 2.2.1, and plotted in decreases linearly in the 1 ≤ Re ≤ 10 region, and then saturates to the supersonic limit. Similar empirical formulas can be constructed: In the intermediate region, the collisions between the buffer gas and target molecules not only increase the forward velocity, but also increase the transverse velocity spread.
In a zero-order picture, the model of transverse velocity spread should not be quite This diffraction effect modifies the Reynolds number (Re) to an effective transverse Reynolds number (Re tran = Re × d 2 cell /d 2 aperture ). Therefore, the transverse velocity spread is: However, the gas dynamics near the aperture is more complicated than this simple  increase of the transverse velocity spread relative to that of the forward velocity at low Reynolds number, the angular spread or the divergence of the beam decreases at low Re, and increases at high Re. By tuning the Reynolds number carefully, I can collimate the molecular beam and increase the beam density at a position that is far from the output aperture. The simulation predicts the behavior of the beam divergence, as shown in Figure 2-12. At Re ≈ 10, the minimum angular spread and divergence can be as small as: The divergence of the buffer gas beam (0.1π) is significantly smaller than that of the effusive (π) and the supersonic beams (0.4π). In addition, the beam collimation becomes better for heavier molecules.

Comparison of effusive beam, buffer gas cooled beam and supersonic beam
As a conclusion to this section, I plot the expected beam velocities and velocity spreads of three beam sources (room temperature effusive beam source, 20 K Neon

Introduction of species, thermalization, diffusion and extraction
In the previous section, I have described the gas dynamics through the output aperture and the formation of a beam. In this section, I describe the gas dynamics in the cell. The combination of the dynamics in the cell, through the aperture, and outside of the cell completely determines the properties of the molecular beam in the detection region. The gas thermodynamics in the cell are relatively simpler and effect the beam properties less sensitively. However, the cell design, which is not easily modified and optimized in situ, requires a complete understanding of the thermodynamics.

Introduction of species
There are three usual methods [70] to introduce the target molecules into the buffer gas cooled cell: (1) hot oven; (2) capillary filling; (3) laser ablation. The hot oven method has been widely used for alkali atoms and for some molecules that have high vapor pressure at ≈ 500-700 K. Capillary filling is used for molecules that are permanent gases at room temperature. The common advantages of these methods are: (1) formation a continuous beam; (2) a large flux. However, their limitations are: (1) requiring sophisticated thermal isolation between the hot and cold components; (2) not working with unstable molecules, such as radicals. Laser ablation is a method complementary to the hot oven and capillary filling methods, and has been widely used to create a long pulse (≈5 ms) for a variety of diatomic molecules or radicals.
The laser ablation technique requires application of a focused high intensity laser pulse to the surface of the solid precursor, which is located inside the cold cell, to create a dense plasma plume. Multiple chemical reactions occur simultaneously in the plasma. In a very short time, the plasma either recombines to neutral molecules/atoms, or creates free ions. For example, I apply a 25 mJ, 532 nm, 10 ns Nd:YAG laser pulse to ablate a BaF 2 precursor. The final products can be Ba, Ba + , BaF, BaF + , BaF 2 or F. Optimizing the yield of the target molecules while suppressing others is nontrivial.
I am unaware of a generic and reliable theory that would guide me into optimization of the laser ablation process. Therefore, for a specific molecule, such as BaF, I have spent several months to optimize the precursor preparation and parameters of the ablation laser. This optimization is discussed in detail in Section 2.4.

Thermalization
The initial temperature of ablated molecules is typically 1000 K to 10000 K. Before being pumped out of the cold cell, there should be enough collisions between the hot target molecules and cold buffer gas to decrease the target molecule temperature to the equilibrium temperature (20 K) of the cold cell. The number of collisions should be larger than: where T t (0) is the initial temperature of the laser ablated target molecules (5000 K), T b is the equilibrium temperature of the Neon buffer gas (20 K), N is the number of collisions, and m b and m t are the mass of the buffer gas (20 for Neon) and target molecules (157 for BaF). The minimum number of collisions for cooling BaF with Neon buffer gas is 50. The mean free path of a target molecule in the cold cell is: where σ b−t is the thermally averaged elastic collision cross section and is typically If we use f 0,b =5 SCCM flow rate, the mean free path is ≈0.1 mm. The total thermalization length is typically no more than 100 × 0.1 mm=1 cm. Therefore, the internal diameter of the cold cell should be larger than 1 cm.
The above discussion applies exclusively to translational thermalization. The typical rotational collision cross section is one order of magnitude smaller (σ rot = 10 −15 cm 2 ) than the translational collision cross section. However, due to the absence of a large mismatch mass factor ( m t /m b ) in the rotational cooling process, the rotational thermalization efficiency of each collision can be higher than that of translational thermalization. Combining these opposing considerations, the rotational cooling almost has the same efficiency as the translational cooling. However, the vibrational cooling is much less efficient, due to a tiny collisional cross section, σ vib /σ t−b <0.1%, where σ vib is the vibrational relaxation cross section and σ t−b is the translational relaxation cross section. Therefore, with typical experimental parameters, the vibrational temperature is similar to the ablation temperature. By measuring the vibrational temperature in the cell, I can approximately obtain the initial plasma temperature.

Diffusion and entrainment from the cold cell
After the target molecules are cooled to the cell temperature, they need to move toward the output aperture. There are two types of gas dynamics in the cell that transport the target molecules from the ablated spot to the aperture: (1) diffusion; (2) entrainment. Diffusion always occurs at equilibrium, and the rate is determined by the diffusion constant, as in Eq.(2.35). With a Brownian motion model, the estimated diffusion time constant is typically 1-10 ms. However, diffusion is not an efficient way to move the target molecules out of the cold cell, and the probability of a target molecule diffusing out of the cell is A aperture /A wall < 1%. Most target molecules impact and freeze on the wall of the cold cell, and are lost.
Entrainment of the target molecules out of the cell can be much more efficient than diffusion, and the rate is determined by the conductance of the output aperture: where V cell is the internal volume of the cold cell. The dynamics of the entrainment process is a single exponential decay with time scale τ pump in Eq.(2.36). The pumpout time is also typically 1-10 ms. Except for very high or very low buffer gas flow rate, these two processes occur simultaneously. The ratio of their contributions not only determines the extraction efficiency, but also determines the beam velocity and velocity distribution. I define a dimensionless parameter to characterize their contribution: To enhance the entrainment contribution, I can increase the buffer gas flow rate. To enhance the contribution of diffusion, I can design a larger cell. A simulation to predict the extraction fraction of the target molecules and the beam velocity vs. γ cell is plotted in Figure 2-14. The maximum extraction efficiency can be as high as 40%.
In typical experiments, this efficiency is usually around 10%.

Design of the buffer gas cooling chamber
Based on my understanding of gas dynamics, introduction of species, thermalization, diffusion, and extraction as discussed above, I have designed and constructed a Neon buffer gas cooling source to form a slow, high density, and cold molecular beam. In  However, transferring the liquid Helium to a heat isolated reservoir and recycling the evaporated Helium gas are nontrivial and require a lot of routine student time to manage. In addition, the price of liquid Helium is currently increasing rapidly. The The aluminum radiation shield is mounted on the first stage of the refrigerator, and the 4 K cold plate is mounted on the second stage. There are two stages of the gas cooling spiral units. One (i) is mounted on the radiation stage, and the other (ii) is mounted on the cold plate through a stainless steel adapter (iv). The cooling unit (ii) is heated by a resistive heater (vi), and its temperature is measured by a diode sensor (v). In typical experiments, the gas cooling unit (i) operates at 40 K and the gas cooling unit (ii) at 20 K. To minimize heat conduction, the gas tube between them is made of stainless steel which has poor heat conductance. The cold cell is mounted on the cold plate through a stainless steel adapter (iv). The heater and diode temperature sensor are mounted on the bottom surface of the cold cell, which are not visible in this photograph. (iii) is a flat cold plate, which has a 1 cm diameter hole at its center.  Helium gas filling needs to be performed only once every year.
There are two minor issues of the PT410RM refrigerator that demand attention: (1) The cooling capacity is relatively low. To achieve fast cooling, the cold unit should be designed to be as small as possible. To obtain a temperature below 4 K, I need to minimize all unnecessary heat transfer processes (contact heat transfer or blackbody radiation). (2) The power consumption of the compressor for 1 W cooling capacity is 8.9 kW. Most of the electric power is converted to heat dissipation in the power supply. I need low temperature (<20 • C) and large flow rate (2.3 Gallon per minute) of cooling water to carry away the excess heat. In addition, to make the compressor work stably and have a long lifetime, it is important for using clean and temperature stabilized cooling water. The lab chilled water is not qualified and the internal loop chiller usually does not have enough cooling capacity. My solution is to use a small chiller to directly cool the Cryomech compressor. However, before the hot water returns back to the chiller, it goes through a multi-plate heat exchanger that is cooled by the lab chilled water and carries away 95% of the heat. Thus, the cooling water in the compressor is always clean. The flow rate and temperature is controlled and stabilized by the chiller.

Vacuum pumps
Due to the high flow rate of continuous buffer gas input, a large vacuum pump is required to keep the pressure in the chamber low enough for the molecular beam experiment. The acceptable background gas pressure is no more than 30 µtorr at room temperature, in which the mean free path of the molecules is larger than 0.5 m.
Thus, the molecular beam is not significantly attenuated during its transportation to the detection chamber. If the input flow rate is 10 SCCM, the minimum required pumping speed is 4000 L/s, which is calculated by: where S is the pumping speed, Q is the input flow rate, and p is background gas pressure in the chamber. A conventional diffusion pump or turbo pump with 4000 L/s is very large and expensive. However, a cryogenic pump with similar pumping speed is relatively small and cheap. Even better, I do not need an external cryogenic pump. I can mount several 4 K cold plates around the molecular beam formation region. This is equivalent to locally implementing a large cryogenic pump. The pumping speed of the cold surface for Neon buffer gas is: where c is the capture coefficient, which describes the probability of capturing a gas molecule with a single collision with the cold surface. The capture coefficient of Neon buffer gas on a 4 K cold surface is ∼1. A is the total area of the 4 K cold surface, k B is the Boltzmann constant, m is the mass of the pumping gas, and T is the gas temperature, which is always referenced to room temperature to evaluate the pumping speed. In our apparatus, there is ∼500 cm 2 cold surface. Thus, the effective cryogenic pumping speed is ∼7000 L/s, which is sufficient to keep the pressure in the vacuum chamber below 30 µtorr. In addition to the cryogenic pump, I set up a 150 L/s turbo pump, which is used to pump the chamber, when the cryogenic pumping is not running. I always leave the small turbo pump on along with the cryogenic pump.
However, after the cold head is cooled to 4 K, turning off the small turbo pump does not cause an increase in the background pressure in the chamber any more.

Cold cell
The cold cell is the central part of the buffer gas cooled molecular beam source, as shown in Figure 2-19. It is made of copper for good heat conductance. The internal diameter of the cell is 1.2", which is large enough to cool the hot plasma particles to the buffer gas equilibrium temperature before they hit the cell wall. The distance between the ablation spot and the output aperture is ∼ 1.2", which is sufficiently short to extract more than 10% of the target molecules out of the cell. There is

Radiation shield
Based on the Stefan-Boltzmann law (Eq.(2.40)), the room temperature blackbody radiation is ∼500 mW/cm −2 , and ∼10% of the total power incident on the cell is absorbed by the 4 K cold surface (cold cell, cold plate, refrigerator head etc.) made of copper.
where J is the total power radiated per unit area, T is the absolute temperature, surface with ∼100 cm 2 area absorbs ∼5 W blackbody radiation power. Such high radiation power cannot be efficiently pumped by the second stage of the pulsed tube refrigerator. To decrease the blackbody radiation incident on the second stage of the refrigerator, it is necessary to surround the 4 K cold surface with a cool surface, the temperature of which is much lower than room temperature. I build an aluminum box, which is cooled by the first stage of the PT410 refrigerator, to surround the 4 K cold surface. The room temperature blackbody radiation is reflected and absorbed by the radiation shield. The heat dissipating power on the radiation shield is typically less than 15 W. The first stage of the refrigerator has enough capacity to pump such energy and maintain the radiation shield at ∼ 35 K. Due to the strong temperature dependence in Eq.(2.40), the blackbody radiation absorbed by the 4 K cold plate is decreased by a factor of 5000 and can be neglected. I choose aluminum, but not copper, to build the radiation shield, because it is light and relatively easily polished.
A seriously oxidized surface on the radiation shield would increase the absorption of the blackbody radiation significantly, which increases the temperature of the radiation shield. Therefore, I must polish the surface of the radiation shield every few months, or cover the outside of the radiation shield with several layers of thin aluminum-coated Mylar super-insulation.
In addition to a ∼40 K radiation shield, the Doyle group also suggests building a 4 K radiation shield to further block radiation heating of the cold cell. The temperature in their Helium buffer gas cooling experiment must be as low as possible. In addition, the 4 K radiation shield can also improve the performance of the cryogenic pump for Helium buffer gas. However, in our current Neon buffer gas cooling experiment, the required temperature is less than 8 K instead of less than 4 K. Therefore, we do not implement the second stage radiation shield, which should be added if we need to perform Helium buffer gas cooling experiments.

Temperature measurement and control
The cryogenic temperature measurement and control system are commercially available, but relatively expensive. I have decided to build our own system. In our system, I have two thermocouple sensors to measure the temperatures of the top and the bottom of the radiation shield, and four diode sensors to measure the temperatures of the components mounted on the second stage of the refrigerator (4 K cold head, 20 K cold cell, 20 K gas tube and 4 K cold skimmer). In principle, I can extend our current system up to 30 sensors without significantly increasing the cost. In addition to the low cost, using a Labview control program written by myself, I build a feedback loop to stabilize the temperature easily, and a more subtle automatic control program can be implemented in a few minutes.
There are two types of temperature sensors in our setup: (1) thermocouple sensor; (2) diode sensor. The Type K thermocouple sensor is very cheap and usually used for temperatures above 70 K. Below 70 K, the temperature response slope is very small, which decreases the measurement accuracy. With a high gain and low noise amplifier (AD 595), it is possible to measure the temperature to 1 K accuracy at T

Laser ablation
In my buffer gas cooling experiment, the target molecules are introduced by laser ablating a solid pellet in the cold cell. Preparation of pellets and optimization of the ablation laser to maximize the yield of the target molecules are much more complicated than my expectation. The performance of a well optimized experiment can be more than 1000 times better than an unoptimized experiment. Due to its importance, I discuss these details separately in Section 2.4.

Beam collimation
Generally speaking, the requirements for collimation of the buffer gas cooled beam are less strict than the requirements for the supersonic beam, because the expansion is gentle, and reflections from the wall of the vacuum chamber is less destructive than in the supersonic beam. Therefore, in typical experiments, instead of using a  The stability of the temperatures with PID control. With the temperature control system, the temperatures of the cold cell, gas tube and cold head can be stabilized with 0.03 K, 0.05 K, 0.08 K short-term fluctuations and no long-term fluctuations. When the ablation laser is fired, there is a temperature jump (*) of the cold cell. The PID control can damp this heat impulse by adjusting the heating power for a few seconds.
conical skimmer, a plate with a hole is adequate to extract the center of the beam without destroying the beam. However, due to the continuous buffer gas flow, the local pressure just in front of the output aperture can be very high, and the pressure distribution in the vacuum chamber can be very inhomogeneous. A gas dynamics simulation shows that without any local cryogenic pumping (the nearest cryogenic pump is placed 10 cm away), the pressure at 1 cm away from the output aperture with 10 SCCM flow rate is ∼1 mtorr, which is three orders of magnitude higher than the average pressure in the source chamber (∼ 1 × 10 −6 mtorr). Therefore, I need to add a local cryogenic pump in front of the output aperture to pump the large divergent buffer gas beam, while allow the center of the beam to pass through.
There are two usual ways to implement such a local cryogenic pump: (1) We can place a 4 K copper plate with a 1 cm diameter hole (called "cold skimmer") ∼5 cm away from the output aperture. In principle, most of the Neon gas is frozen on the surface of the cold skimmer and the effective pressure between the cold skimmer and output aperture should decrease significantly. However, the major disadvantage of this method is not all Neon gas will be captured with one collision on the cold skimmer (Helium gas would be worse). The pumping speed might not be as high as we expect. In addition, when a large amount of Neon gas is frozen on the surface (With 10 SCCM flow rate, there is ∼0.5 cm 3 /hour Neon ice generated on the cold skimmer), the cryogenic pumping of the cold skimmer might be very unstable. (2)   has a parabolic spatial distribution. In the interaction volume, the upper limit of the field inhomogeneity is along the long axis of the detection volume (10 cm length). 1% non-flatness compensated magnetic field only causes <14 kHz line broadening, which is smaller than our spectrometer resolution and can be ignored.
The magnetic field generated by the Helmholtz coils can be fine tuned by tweaking

Performance of the buffer gas cooling technique
Before transitting the CPmmW spectrometer from the supersonic beam system to the buffer gas cooling system, it is important to test the performance of the new apparatus. Most tests in this thesis employ LIF, which is well suited for measuring the number density and the rotational temperature of the target molecules. The diagram of the measurement system is shown in Figure 2

Rotational temperature
The rotational temperature is sampled by a low resolution LIF spectrum, as shown in Figure 2-24, Plot (a). I fit the measured spectrum to the simulated spectrum, generated by Pgopher, to obtain the rotational temperature [98].

Number density
The number density is measured by: where N is the total number of the target molecules in the excitation region, V is the volume of the excitation region, n is the number density of the target molecules, S is the voltage amplitude on the oscilloscope, τ is the fluorescence lifetime, e is the charge of the electron, G is the gain of the PMT, η P is the PMT detection efficiency, η S is the detection solid angle, η Y is the fluorescence quantum yield, and η P is the rotational partition function. A factor of 2 in the numerator comes from the ratio of the total number of molecules to the number of excited molecules at saturated excitation, and a value of 50 in the denominator is the input impedance(Ω) of the oscilloscope. For example, in a typical BaF LIF experiment at 20 cm downstream from the output aperture in the detection chamber, S=200 mV, τ BaF =20 ns, G=5×10 5 , .1 cm × 0.1 cm × 1 cm, and the measured number density is 3 × 10 8 cm −3 , which is three orders of magnitude larger than the number density in the supersonic beam.   Measuring the Doppler broadening is more difficult with CPmmW Rydberg spectroscopy, because the blackbody radiation always induces larger frequency broadening than the Doppler broadening. To completely characterize the dynamics of the molecular beam, I still need to perform high resolution absorption spectroscopy.

Stability
Compared to the supersonic beam, the buffer gas cooled beam not only has much higher number density and lower forward velocity, but also has superior shot-to-shot stability. Comparison of the shot-to-shot stability in the supersonic beam and in the buffer gas beam. We take 3000 shots with fixed laser power and frequency to make the histogram. The width of the LIF signal distribution in the supersonic beam is >50%, while that in the buffer gas cooled beam is <20%.
hopping induced frequency jitter. Therefore, the actual fluctuation of the buffer gas cooled beam source might be as small as ∼10%. A more reliable stability test should come from the high resolution absorption spectrum.

Comparisons of the effusive beam, the supersonic beam and the buffer gas cooled beam
I briefly summarize the most important parameters of the effusive beam, the supersonic beam, and the buffer gas cooled beam in Table 2.6. The number density is evaluated at a position 20 cm away from the beam source. Translational temperature, shot-to-shot stability, and Doppler broadening should be remeasured by high resolution absorption spectroscopy. However, the improvement in the buffer gas cooled beam over the supersonic beam is clear and satisfactory.

CPmmW spectrometer
Millimeter wave spectroscopy is a powerful tool for performing high precision measurements of atoms and molecules in the gas phase. Compared to laser spectroscopy, millimeter wave spectroscopy usually has much higher resolution and accompanying higher frequency accuracy [42,43,44,7,45,46,47]. However, for a conventional frequency-stepping millimeter-wave spectrometer, achieving high resolution and covering a broad bandwidth requires very long data collection times. For example, if a scanning spectrometer is used in conjunction with 10-20 Hz repetition rate Nd:YAG pumped lasers or pulsed molecular beams, to obtain a 20 GHz bandwidth spectrum with 20 kHz spectral resolution would require a month of data collection. In the early 1980s, the invention of cavity-enhanced Fourier-transform spectroscopy significantly increased the sensitivity and decreased the averaging time [56]. However, due to the mechanical tuning of the large cavity mirrors required for each frequency step, the data collection rate is still too slow for our experiments.
The recent invention of Chirped-Pulse Fourier-Transform Microwave (CP-FTMW) spectroscopy by Pate and co-workers shatters the limitation of the frequency stepping spectrometer, and obtains broadband spectra with high resolution in a very short data collection time [57,58]. Barratt

Spectrometer design
A schematic of the CPmmW spectrometer is shown in Figure 2

Optimizations
To improve the performance of the CPmmW spectrometer, such as decreasing the noise frequency amplitude, increasing the phase stability, and better collimating the output beam, I have performed several careful optimizations as follows: • The RF pulse generated by the AWG is cleaned by a DC to 2.2 GHz low pass filter (vi) to remove the Nyquist frequency (f samplerate − f ).  and the signal to noise ratio is unchanged. Therefore, the amplifier we use here is appropriate.
(2) If the input power is 0 dBm, the output power is still no more than 10 dBm, which saturates the amplifier. The effective gain of the signal is only 10 dB, but that of the noise is still 20 dB, which decreases the signal to noise ratio by 10 dB. Therefore, it would be better to choose an amplifier with smaller gain. (3) If the input power is -30 dBm, the output power should be -10 dBm. To obtain 10 dBm power, I have to use two amplifiers in series. Usually, the signal to noise ratio of one high gain power amplifier is better than that of two sequential low gain amplifiers. In addition, most high quality amplifiers are optimized at the typical output power level. Therefore, it would be better to choose an amplifier with higher gain. • Modulate the output amplitude. Different from the passive frequency multiplier, the output power of the active multiplier is independent of the input power. If the input power is above a threshold, the output amplitude is constant. For example, a Gaussian shape RF pulse generated by the AWG with FWHM = 100 ns is converted to a 100ns rectangular mmW pulse. To shape the output pulse, I use a voltagecontrolled fast attenuator (xix) with a user-defined pulse shape. In principle, I can create any pulse shape longer than 10 ns. • The relationship of the actual millimeter-wave frequency and the downconverted RF frequency read by the oscilloscope is: f IF is read by the oscilloscope, f syn is generated by the HP synthesizer (iv). I cannot determine the sign in Eq.(2.42) with one downconverted spectrum. The ambiguity

Performance of the CPmmW Spectrometer
To characterize our CPmmW spectrometer, I have performed several test experiments as follow: • Output power: The output power is measured by a DET-10 power detector (Millitech), and shown in Figure 2-29, Plot (a).
• Single frequency generation: Figure 2 • Phase noise: Phase noise is defined by: where φ(t) represents the short-term phase noise, which is a random function with similar time-scaled fluctuations as the carrier frequency. It is usually represented as a noise function: where S total is the signal total power, and S(f ) is the signal power of 1 Hz of bandwidth at a frequency f away from the carrier frequency. Short-term phase noise would ruin the phase sensitive measurements and coherent control. The phase noise of our spectrometer is directly measured by the oscilloscope, as shown in Figure 2-30, Plot (a). ϕ(t) represents long term phase noise (phase shift), which is slowly variating and causes unidirectional phase modulation. It would ruin the long term data average (π phase shift would completely cancel the signal). The phase shift of our spectrometer within an hour is no more than π/5, as shown in In addition, I also can correct the long-term phase shift automatically by data post-processing [59].
• Detection sensitivity: I define the detection sensitivity by S/N > 3 with 5000 averages. It is frequency dependent, and also depends on the coupling efficiency between the emitting and receiving horn. The typical detection sensitivity for our experiments is <10 nW.
• Broadband pulse generation: Figure 2 sequence with three segments. The first segment is a single frequency pulse with a triangle amplitude pulse envelope. The second segment is a single frequency rectangular pulse. And the last segment is a broadband chirped pulse.
• Intensity calibration: Since the data is recorded in the time-domain, the relative intensity in the frequency-domain of our spectrometer is not modulated by the and fixed (modulation period in frequency domain ∼GHz). They can be calibrated with a power meter detector, which itself is well-calibrated by the manufacturer.
The calibration of the propagation is more difficult, because the reflections from the horn, the Teflon windows, and the lens might form a low-Q millimeter-wave cavity, which would modulate the amplitude at different frequencies. The rate of this modulation depends on the cavity length and is usually very high (modulation frequency is ∼10 MHz). In addition, the envelope of the variation is very sensitive to alignment and can change day-to-day. Up to now, my best calibration has 10% uncertainty, which mainly is limited by the propagation modulation.

Implementation of the CPmmW Spectrometer in the supersonic cooled molecular beam apparatus (Gertrude)
Our first CPmmW spectrometer is setup on Gertrude, as shown in

Transmission mode
The schematic diagram of the transmission mode is in Figure 2

Reflection mode
The schematic diagram of the reflection mode is in Figure 2 (1), or form a pseudo-cavity to enhance some noise frequencies. In Section 2.3.3, I have employed several tricks to minimize such problems. However, some components cannot cover the full bandwidth of our spectrometer. Therefore, I find that near the lower and upper limits of our spectrometer, the noise frequencies increase significantly. Better filters, isolators, and amplifiers need to be explored and implemented to decrease such noise frequencies.
• Intensity calibration. The uncertainty of the relative intensities is 10%, which would be improved significantly if we could minimize the accidental reflections or calibrate the variation precisely. The intensity information might be very important for studying core-nonpenetrating Rydberg-Rydberg transitions.
• Waveguide feedthrough. The horns are placed outside of the chamber in the current setup. The millimeter wave radiation broadcast from the emission horn must transmit through the glass or Teflon window that is mounted on the vacuum chamber, and propagate a long distance to interact with the sample. The loss due to the transmission and propagation is significant. If we can build a low loss waveguide feedthrough, it will be possible to place the horns inside of the chamber. Mounting horns inside the chamber not only increases the coupling efficiency between the emitting and receiving horns, but also makes the alignment easier and more stable.
I have built a waveguide feedthrough by drilling a 0.1" × 0.05" rectangular hole with sharp corners on a copper plate with 1/4" thickness, and inserting and gluing an acrylic bar into the hole to hold the vacuum. The tested loss is 3 dB, which is still larger than our requirement (<1dB). Using a thin plastic film instead of a thick acrylic bar might decrease the loss significantly.
• Quadrature phase detection. In our CPmmW spectrometer, I use mixers to downconvert millimeter-wave to RF frequency. In signal processing, this mixing process is called quadrature-sampling. Our down-conversion mixing is not a completely coherent demodulation processes. The in-phase component still exists, but the quadrature component is lost. A direct ambiguity from this loss is that I cannot distinguish positive frequency and negative frequency after down-conversion.
I can separate them by changing the down-conversion frequencies, as in Section 2.3.3. However, if the spectrum is very dense, it is possible to mislabel the downconversion frequency shift. In addition, it requires recording every spectrum twice.
With quadrature phase detection, according to the direction of the quadrature signal, we can know the sign of the demodulation signal directly.
• Dual-channel CPmmW. One limitation of our current CPmmW spectrometer is that I cannot generate two different frequencies at the same time, because: (1) Our AWG has one output channel; (2) The active multipliers do not work well for more than one frequency component at the same time. A two-channel AWG, two independent frequency up-conversion arms and a W-band coupler can generate an overlapped pulse pair, which can be used for coherent population trapping in a three-level Rydberg system and other coherent control experiments.
• Extending to sub-millimeter wave region. The main obstacle in extending from the millimeter wave region to the sub-millimeter wave region is the power loss significantly from >10 mW to less than 1 mW. However, this is not a problem for Rydberg states, which have extraordinarily large electric dipole transition moments.

Laser ablation
In both apparatuses, Gertrude and Buffy, laser ablation is used to create the atoms or molecules entrained in the beams. Although this technique has been applied in many research fields for more than 30 years, there is still not a general theory or principle to guide experimentalists to systematically optimize the experimental conditions [99].
Each research group summarizes their own empirical rules for creating their target

Comparison between laser ablation of metals and laser ablation of salts
Laser ablation processes involve very complicated energy transfer, gas dynamics and relaxation processes. Different materials usually behave quite differently. Most metals are easily ablated or vaporized, because they readily absorb photons due to the absence of a band-gap. However, typical ionic salts have a large band-gap (∼10 eV) and require simultaneous absorption of two or three photons. This nonlinear process requires a much higher local laser electric field. Therefore, a more tightly focused laser beam with higher pulse energy is used to achieve better ablation efficiency (more plasma and less dust). Based on my experiment, I summarize the generic optimized parameters between laser ablation of metals and laser ablation of salts in Table   2.7. Laser ablation of metals is usually much easier than the laser ablation of salts.
Therefore, in this section, I focus on the optimization of the laser ablation of salts.

Laser focusing
From Table 2.7, I find that the laser ablation of salts requires a tightly focused laser.
In my experiment, the ablation laser beam is expanded and collimated to 1.

Pellet preparation
A well prepared ablation pellet should have the following properties: • The pellet is opaque at the ablation laser wavelength (532 nm) and can absorb laser energy efficiently.
• The pellet is non-fragile at relatively high ablation laser energy (∼20 mJ with 10 ns pulse length).
• The pellet has a high relative density (>90% of the crystal density).
Based on these criteria, I currently have a general procedure for pellet preparation as follows, which also might be improved in future.
(2) Grind the rough BaF 2 granules into a fine powder in a mortar.
(3) Barium fluoride is very hygroscopic. Therefore, keep the powder in a desiccator all the time. If the BaF 2 powder is already hydrated, I put it in a glass tube vacuum furnace and evaporate the water at ∼ 400 • C for two hours. If the water is not removed completely, chemical reactions might occur when the pellet is heated to high temperature in step 6: (4) Mix 50% BaF 2 powder (weight) with 50% CaF 2 powder. CaF 2 is used to improve the cohesiveness of the pellet. Then, the hydraulic press is released slowly over 3 minutes. The density of the target in this step should be more than 80% of that of BaF 2 crystal.
(6) Transfer the targets into a vacuum furnace with < 50 mtorr pressure. Slowly increase the furnace temperature to 800 • C for two hours and then decrease to room temperature over an additional two hours. The density of the target should be more than 90% of that of BaF 2 crystal. The choice of 800 • C was made after careful empirical optimization. If the heating temperature is too high, the pellet becomes too hard to be ablated, and most of the ablation products from such hard pellets are Ba atoms instead of BaF molecules. If the temperature is too low, there is a large amount of dust generated during the ablation.

Minimize the generation of dust
Ablation is a pretty brutal process, which might create much more dust than homogeneous plasma. The dust has no contribution to our molecular beam, but brings  In principle, the laser ablation process is very fast (<10 µs). Therefore, <100 Hz       require absolute calibration of the laser wavelength or an accurate ionization potential.

Experimental implementations
n * 1 and n * 2 are the principal quantum numbers of neighboring Rydberg states in a series (with the same orbital angular momentum quantum number l), and is the Rydberg constant.

Minimization of the stray electric field
To obtain a clean Rydberg spectrum, I must minimize the stray electric field in the interaction volume, which destroys the parity, induces significant Stark -mixing, and causes frequency shifts and broadening [13]. Due to their gigantic electric dipole transition moment for |∆n * | ∼ 1 transitions, Rydberg states are extremely sensitive to very weak stray electric fields. In Section 2.1.5, I describe the stable and finely adjustable high voltage power supply. Here, I describe the use of atomic Rydberg states as a probe to measure the stray electric field locally [55], and the use of a fine tunable power supply to minimize this field. The atomic Rydberg stray electric field "probe" has two modes: low sensitivity mode and high sensitivity mode.
The low sensitivity mode is used to minimize the forbidden transitions of Barium atoms(s-s or d-s) in the laser spectrum, as shown in Figure 3   sensitivity of this method is limited by the signal to noise ratio of the laser spectrum and can be estimated by: where ∆ is the energy difference of the bright states (p) and dark states (s or d), ∼3 GHz for n ∼60, which can be resolved by our pulsed laser. µ is the electric dipole transition moment of the neighboring bright and dark states, ∼ 5000 Debye.
Based on the current signal to noise ratio for a typical bright state (S/N ∼30 with 40 averages), I can detect transitions into dark states, the intensity of which is 5% that of the bright states (S/N∼2). Therefore, the expected sensitivity to the stray electric field E from Eq. The stray electric field not only mixes the bright state characters into the dark states, but also shifts the states by: 3) I substitute the parameters above into Eq. (3.3) and obtain the frequency shift ∼100 MHz with 1 V/cm stray electric field. It is difficult to precisely measure the 100 MHz frequency shift using 1 GHz linewidth pulsed laser. However, cw millimeter-wave radiation can be used to measure the frequency shift at ∼MHz resolution. Therefore, I can minimize the stray electric field by minimizing the millimeter wave frequency shift with a much higher sensitivity.
To measure the millimeter-wave frequency shift induced by the stray electric field, I perform an optical-optical-millimeter-wave triple resonance experiment on Calcium atoms. Millimeter-wave radiation is used to pump the low Rydberg states (n * ) to higher Rydberg states (n * +1 or n * +2), which are detected as described earlier by ramped PFI. The raw data obtained in the time-domain on the oscilloscope is shown In addition to the poor millimeter-wave resolution in the presence of the current stray electric field, to obtain a relatively smooth spectrum, as shown in Figure 3-7, I must record the same spectrum more than 10 times and average them to eliminate short-term and long-term fluctuations. Therefore, the data collection rate is very low. Thus, to obtain a broadband spectrum of molecular Rydberg states, as in  as measured by millimeter wave spectroscopy. The frequency axis is plotted relative to a high precision measurement by Kleppner [7]. The frequency shift is due to the linear Stark effect, and the broadening comes from the inhomogeneous electric field in the interaction volume. the optimal E 0 here is usually far below the spectrometer power limit). Therefore, the figure of merit of CPmmW Rydberg spectroscopy has an extra √ α improvement compared to rotational spectroscopy. (4) There are no high voltage electric circuits in this system. Therefore, the stray electric field is always negligible.
However, compared to the most CPmmW rotational spectroscopy experiments with permanent gas, the CPmmW Rydberg spectroscopy has its own challenges. The major one is the number density of the laser ablated alkali/alkali earth atoms, such as Calcium and Barium, or diatomic radicals, such as BaF and CaF, entrained in the beam is relatively low. In the supersonic beam, the total number of laser ablated particles is usually n<10 7 cm −3 . However, the 1% permanent gas seeded in the supersonic beam is usually n>10 11 cm −3 . In addition, the excitation efficiency of multi-color opti- However, these two dephasing mechanisms degrade the spectroscopic resolution and should be avoided in the regular high resolution spectroscopic experiments.
To avoid accelerating the cooperative and collisional dephasing, but still collecting enough emitting power from the sample, I must increase the interaction volume and decrease the number density. If the total number of Rydberg states is fixed, enlarging the interaction volume by a factor of M (decreasing the number density by a factor of M ) can keep the total emitting power, but decrease the cooperative dephasing by a factor of ∼ √ M and the collisional dephasing by a factor of M . In addition, the geometry of the sample also determines how much forward radiation power can be collected by the detection horn, as P ∝ √ SL, where S is the cross section of the sample perpendicular to the millimeter wave propagation direction, and L is the length of the sample along the millimeter wave propagation direction. Therefore, to maximize the forward emission, a cylindrical shape of the interaction volume is preferred to a disk shape. However, the diameter of the cross section must not be too small to avoid diffraction losses, as d ∼ λ, where d is the diameter of the millimeter wave beam, and λ is the millimeter wave wavelength. Near this limit, diffraction causes the forward propagating wave to expand significantly, which prevents efficient coupling into the detection horn. Based on these two considerations, I shape the typical interaction volume to a cylinder shape with S ∼10 cm 2 and L ∼10 cm by overlapping the expanded lasers, the collimated millimeter wave beam, and the nonskimmed molecular beam, as shown in Figure 2-33. In a typical laser ablation Calcium atomic supersonic beam, the maximum number density of Calcium Rydberg states in the interaction volume, which is ∼ 15 cm away from the nozzle, is ∼10 5 cm −3 .
Therefore, if fully polarized, the total number of the Rydberg emitters is ∼ 5 × 10 6 . In the Calcium experiment with d=3 cm, the saturation power of the pump laser is ∼100 µJ, and that of the probe laser is ∼1 mJ. It is better not to over saturate Ions not only destroy the coherence by collisions, but also create a stray electric field, which shifts the resonant frequency, as discussed previously. The power of the chirped pulse can be estimated using Eq.(3.5), with the calculated electric dipole transition moment.
where α is the bandwidth of the chirped pulse, T is the pulse duration, µ is the electric dipole transition moment, and S is the cross section area of the millimeter wave. I substitute typical values into this equation (α = 500 MHz, T = 10 ns, µ = 3500 Debye, S = 10 cm 2 ) and obtain the power required for full polarization to be 0.2 mW. Therefore, I need to set the variable attenuator to ∼20 dB attenuation (Full output power of CPmmW spectrometer is 30 mW).  pulse and FID radiation, in Section 3.2.5.

Rydberg state by transient absorption/emission
To evaluate the sensitivity of the CPmmW spectrometer, I need to measure the total number of the emitters in our interaction volume. One method is using Laser Induced Fluorescence (LIF) first to measure the total number of atoms in the ground state within a small interaction volume first, and then scale that value to a larger volume.
To convert to the total number of atomic Rydberg states, I must multiply by the laser pumping efficiency (At saturation power, each laser pumping step has 50% efficiency). The total number of millimeter wave photons can be obtained from the input millimeter-wave power, using the geometrical configuration parameters. I also can use the optimized π/2 pulse with precisely calculated electric dipole transition mo-     In Figures 3-13 and 3-14, the envelopes of the excitation pulses and the amplitudes of FIDs qualitatively agree with the predictions in Table 3.1. Non-zero FID at nominal π or 2π pulse associates with the inhomogeneous millimeter wave intensity, which deviates the pulse area. The phase measurement of the excitation pulse and FID are discussed in Section 3.2.5, which also agrees with Table 3

Interference of the chirped excitation pulse and FID radiation
Due to the gigantic electric dipole transition moment, for a moderately high number density of the atoms in a single Rydberg state, the amplitude of the induced macroscopic polarization is comparable to the amplitude of the excitation pulse. Therefore, I can not only detect the relatively intense FID radiation after the excitation pulse, but can also detect their interference during the excitation pulse. Because the interference amplitude is still < 20% of the excitation amplitude, I can still assume that the excitation and radiation processes are decoupled. The radiation during the

Extraction of Information from the radiation phase shifts
In Figure 3  Dr. Colombo [116,52] 10) The precise measurement of phase is determined by the precise measurement of resonance frequency, f 0 . The errors of φ F and φ P are determined by: where ∆f is the resonance linewidth and δf is the precision of the center frequency can be as small as 50 kHz. If the excitation pulse duration t P =500 ns, the error of 2πf 0 t P is only 0.05π. In addition, in the buffer gas cooled beam, the linewidth is one order of magnitude smaller (∼50 kHz), and the absolute value of δf also decreases by a factor of ten (∼5 kHz). Therefore, the phase error of 2πf 0 t P can usually be ignored relative to φ F and φ P . The basic procedure of the FFT phase extraction method is summarized in Figure 3  multiple transitions, I first decompose the FID into multiple single frequency signals: I evaluate the systematic error of the phase shift from the non-resonant excitation in a linearly chirped pulse as following: A linearly chirped pulse with chirp rate α can be represented by Eq. (3.13). The Fourier transformation of the linearly chirped pulse has a regular phase shift φ P , a constant phase shift π/4, and a quadratic frequency dependent phase shift term, which have contributions from non-resonant excitation. For a two-level system, it is easy to show that the non-resonant excitation contribution to the population transfer Therefore, if the detuning frequency is larger than 3 times the Rabi frequency (ω R ), non-resonant excitation can be ignored. I substitute ∆ = ω R −2πf 0 = 3ω R into the last phase term of Eq.(3.13) and find that the systematic phase error (δφ) should be proportional to E 2 /α (ω R = µE/ , E is the amplitude of the excitation pulse), as in Eq.(3.14). However, E and α are not independent, as in Eq.(3.15). P is the macroscopic polarization, µ is the electric dipole transition moment, and N is the total number of atoms. To keep a fixed excited macroscopic polarization, E must be proportional to the square root of the chirp rate α. Therefore, the systematic phase error is independent of the chirp rate, but depends on the Rabi flopping angle of the excitation, δφ ∝ P 2 . For the maximum Rabi flopping angle (π/2 excitation), I substitute typical experimental parameters into Eqs.(3.14), (3.15), and show that the systematic phase error is ∼ π/2. To decrease this error below π/20, I must reduce the excitation from π/2 to π/6, which significantly reduces the contribution of the non-resonant excitation.
In addition to reducing the Rabi flopping angle of the excitation, I can also apply two chirped pulses with opposite chirp directions. The systematic phase errors due to the non-resonant excitation coming from above and below the resonance cancels symmetrically. A simulation shows that based on the current signal to noise ratio and frequency-dependent E-modulations, this cancellation can reduce the phase error by a factor of 10.

millimeter wave photon-echo
The CPmmW technique has not only advantages in recording broadband spectra with high efficiency, but it also is convenient for control of the coherence and population of quantum states. In this section, I show a millimeter wave photon-echo experiment, which demonstrates that I have the ability to perform NMR type spinecho experiments [117,118] in the millimeter wave region [119, 120, 121] for Rydberg systems.
As usual, I apply a 10 ns, single frequency pulse (excitation pulse) with π/2 pulse area to polarize 36p-36s transition, and the resultant FID lifetime is dominated by inhomogeneous Doppler dephasing (T * 2 ). After a ∼2 µs waiting time (∆t > T * 2 ), another pulse with the same frequency but π pulse area (rephasing pulse) is applied.
The rephasing pulse reverses the phase evolution of all inhomogeneous components.
The FID revives (incompletely) after ∆t, and then dephases again, as shown in Figure   3-21.
The lifetimes of the initial dephasing FID and the subsequent rephasing echo are both determined by the inhomogeneous T * 2 . The difference is that the former one is T * 2 and the latter one is 2T * 2 . To bypass the inhomogeneous lifetime T * 2 and measure the homogeneous lifetime T 2 (T 2 > T * 2 here), I record a series of photon echo spectra with different waiting times, as shown in Figure 3

Populating high-states using a crafted pulse sequence
I show a proof of principle experiment in which a millimeter-wave pulse sequence is applied to transfer population from low-to high-Rydberg states. I apply two laser pulses to initially prepare 57d state, and then apply three 10 ns pulses with different frequencies to sequentially excite the 54f-57d, 56g-54f, and 54h-56g transitions.
All three pulses are not exact π pulses. Therefore, in addition to the population Figure 3-24: Populating high-states with a crafted pulse sequence. The inset plot shows a schematic diagram of the four energy levels and three transitions that contribute to the spectrum. The 57d state is initially pumped by the laser. Three 10 ns single frequency resonant millimeter-wave pulses sequentially move a part of population from =2 to =5. Since no pulse area is chosen to be exact π, each transition undergoes FID and therefore appears in the frequency spectrum.
transfer, strong FID radiation is observed, as shown in Figure 3-24. In principle, I can replace the single frequency pulses by chirped pulses to achieve nearly 100% population transfer efficiency for each step, and to climb up to higher-states beyond =5. This technique is applied in molecular Rydberg experiments to prepare core-nonpenetrating states, as discussed in Chapter 5 and Chapter 6.  After a 500 ns single frequency excitation pulse, there is an intense but short-lived early FID, followed by a weak but long-lived late FID. In the frequency domain (Inset plots in Figure 3-25), the early FID has a Lorentzian lineshape (a dominantly homogeneous lineshape is Lorentzian) with ∼ 800 kHz linewidth, and the late FID has a Gaussian lineshape (dominantly inhomogeneous dephasing lineshape is Gaussian)

Evidence for collective effects
with ∼ 300 kHz linewidth. To confirm that this effect arises from the optical density of the sample, I decrease the power of the two pumping lasers, which results in a decrease of the number density by a factor of 3. I observe that the amplitude of the early FID decreases by a factor of ∼10, and the linewidth is ∼ 500 kHz. However, the amplitude of the late FID only decreases by a factor of ∼ 3, and the linewidth is unchanged.

In addition to the deviation of lineshape and linewidth of the early FID from
McGurk's formalism [114], the curvature of the nutation is also inconsistent with the expectation summarized in Table 3

Experimental demonstration of the collective effects
In section 3.2.8, I list several pieces of evidence for collective effects. The limitations of the supersonic beam setup prevent observation of strong collective effects: (1) relatively low number density; (2) relatively large Doppler broadening. These two obstacles are overcome by the buffer gas cooled beam setup. Therefore, much stronger collective effects are expected to be observed. Figure 3-28 shows the envelopes of the radiation field of the Barium 40p-41s transition (the transition direction is downward).
At low number density, the coherence can only be established by the excitation pulse, and the amplitude is maximum at t=0, then decays as a single exponential. However, at high number density, the coherence initiated by the excitation pulse can be amplified in dense Rydberg gas. During this amplification process, the internal energy of a population inverted system is dumping. When the population difference is zero, the amplification terminates, and the coherence is maximal. Then the coherence decays exponentially. Different from the conventional FID picture, the coherence here is not only created by the external excitation pulse, but also contains a significant contri-  The early strong collective radiation comes from the dense part of the sample, but the late weak collective radiation comes from the dilute part. (2) According to Dicke's formalism [110], the early radiation comes from large Dicke states, and the late FID comes from small Dicke states. I believe both contributions exist and how they may be separated is discussed in Chapter 4. In addition, the explanation for the 1 MHz shift is also discussed in Chapter 4.

Chapter 4
Superradiance in a dense Rydberg gas Superradiance is one example of a collective effect, in which a coherence is created between several atoms contained in a small volume and propagates throughout the entire system by radiative coupling [122,111,110,109,123,112]. The details of the initial coherence (initial conditions) and its subsequent propagation (boundary conditions) represent a complete description of the system. This chapter discusses the possibility of using a dense Rydberg gas to observe and study detailed features of superradiance in the millimeter-wave regime. Compared to a millimeter-wave induced rotational transition, Rydberg-Rydberg transitions have enormous electric dipole transition moments and polarizabilities, which are sensitive to external and self-induced electromagnetic fields. In a dense Rydberg gas, a large group of atoms can share an electric field, and absorb and radiate collectively. With carefully designed initial and boundary conditions and the capability for angularly and temporally resolved radiation measurements, subtle light-matter interactions beyond the standard assumptions of isolated emitters and semi-classical approximations can be observed and microscopically characterized. Highly nonlinear coherent control mechanisms such as Self Induced Transparency (SIT) [124,125,126], Electromagnetically Induced Transparency (EIT) [88,87,86], and Autler-Townes effects [127,128] can also be studied.

What is superradiance?
In this section, I attempt to answer two questions: (1) What is the essence of superradiance? Is it a quantum phenomenon or a statistical phenomenon? (2) When is field quantization necessary and when is it not? For simplicity, I examine two abstract systems consisting of two particles. The classical system consists of two classical dipole oscillators, and the quantum system consists of two two-level dipole polarizable atoms. I calculate the differences in the radiative lifetime and frequency of a collective system and of an isolated system. The calculation methods in this section are not easily extended to a real system with many particles. However, it is easy to capture the fundamental mechanism of the light-matter interaction.

Classical picture
In the classical system, there are two dipole oscillators with the same radiative lifetime and resonant frequency, as shown in Figure 4-1 [8]. For simplicity, I assume the initial phases of the excited oscillators are the same, but this assumption is not generally required. If I assume that these two oscillators are oscillating and radiating Figure 4-1: Geometry for radiation by two dipole oscillators. Reprinted from [8] independently, the equation of motion can be described as damped oscillators: where γ is the radiative decay rate of an isolated oscillator, ω is the resonant frequency, and v i is the initial velocity of the oscillator. The initial total energy of these two . The envelope variation of the oscillator can be obtained by considering the excited state decay rate, γ, It is easy to obtain the radiative electromagnetic field in the far field and the total radiative energy by integrating over time and all angular coordinates, where S is the Poynting vector, R is defined graphically in Figure 4 and ξ is the dimensionless distance between the two dipole oscillators, which is defined as ξ = 2πd 0 λ (d 0 is the distance between the two oscillators, λ is the wavelength of the radiation, λ = 2πc/ω).
Typically, 3β sin ξ 2ξ = 0 and the radiative energy is thus not equal to the initial energy.
This violation of energy conservation is not acceptable. It might come from the failure of the initial assumption that the two oscillators are independent. Therefore, I include radiation coupling between two oscillators, where γ 12 (ξ) is the distance dependent radiative coupling term, which can be derived from classical electromagnetic theory. The solution of Eq.(4.4) is: (4.5b) The first term inside the bracket is the time-independent initial amplitude, which is the same as Eq.(4.2). The second term is time-dependent and modifies the decay rate (real part) and the radiation frequency (imaginary part). Figure 4-  power, The radiated power and initial power are now equal. And the radiation decay rate can be evaluated by, in which I assume that the two oscillators are placed at the same position. The effective radiation power decay rate is approximately 4γ instead of 2γ. For an Noscillator system, the rate is expected to be 2Nγ. Through the example of two classical dipole oscillators, we find that: (1) Superradiance can happen in a purely classical system; (2) The radiative lifetime is proportional to the number of dipole oscillators in a small volume.

Quantum picture
Although superradiant radiation does not require any quantum mechanism, we are often concerned with the emission of photons from microscopic particles, which must be described by quantum mechanics. In this section, I replace the two classical dipole oscillators with two two-level quantum dipole oscillators, as shown in Figure 4-3.
The two atoms are exactly the same, and all other interactions between them except for the radiative coupling can be ignored. Their populations, coherence and electric dipole transition moments can be described by a density matrix and a dipole matrix, where i=1,2 is the index of the two dipole oscillators, e represents the excited state, and g represents the ground state. The polarization of each dipole oscillator, which directly induces the radiation, is P i = T r (ρ i µ i ) = 2Re(ρ eg,i )µ eg,i (4.9a) P = P 1 + P 2 = 2Re(ρ eg,1 )µ eg,1 + 2Re(ρ eg,2 )µ eg,2 . (4.9b) The total polarization is the sum of the two individual polarizations. Coupling the two atoms with the radiation electromagnetic field, I calculate the dynamics of the atoms and radiation field similarly to the classical picture. However, the collective two-level model does not include dipole correlations in the atomic part, but in the radiation field part instead. This physical picture of the dipole-dipole coupling is hidden in the iterations of the atom-field coupling, which is opaque. Dicke proposed to use a stacked multi-level system to replace the two-level system, which will be shown to incorporate the dipole correlations in the atomic part [110].
The total polarization evaluated by the density matrix and dipole matrix of the Dicke states, Eq.(4.12), is the same as Eq.(4.9). P = T r (ρµ) = 2Re(ρ eg,1 )µ eg,1 + 2Re(ρ eg,2 )µ eg,2 (4.12) The collective two-level model includes the symmetry information in the Hamilto- where γ is the spontaneous decay rate of the excited state of an isolated atom, and The decay rate of the upper Dicke state is the same as the regular spontaneous decay rate, but that of the intermediate Dicke state is approximately twice as large.
The radiation decay rate can be obtained easily as in Eq. (4.15), which is the same as Eq.(4.7) derived from the classical theory,

Calculation methods
In the previous section, I used a two-particle model system to discuss the essence of superadiance. However, in most practical experiments, the number of atoms is much larger than two. In this section, I briefly review two conventional fully quantum mechanical calculation methods first, and then describe a simplified semi-classical calculation method, which is much easier to implement and fits the requirements of the Rydberg system. The next two sections follows the discussion of Gross and Haroche [130].
Extending the physical picture of the collective two-level system and Dicke states where ω 0 is the resonant frequency of the atom, D 3 i represents the population difference of each atom ( i D 3 i = M ), k and label the mode and the polarization of the radiation field, a † and a are the creation and annihilation operators of the radiation field, D α is the electric dipole transition moment, and V is an arbitrary quantization volume, which is much larger than the sample dimension.

Schrödinger picture
whereΦ andṼ are the wavefunctions and light-matter interaction term transformed into the interaction representation with the zeroth-order Hamiltonian H at + H rad .
With straightforward integration and iteration, I obtain an integro-differential equation to describe the state evolution of the atoms, To solve Eq.(4.18), I apply two approximations, the Born and Markov approximations, to cut off the past time integration over the atoms and all field modes.
The Born approximation neglects the build-up of correlation between the atoms and the field (Φ(t − τ ) →ρ(t − τ ) ⊗ |0 radrad 0|). The Markov approximation neglects the atom-field correlation time relative to the evolution time of the atomic system (ρ(t−τ ) →ρ(t)). These two approximations are always correct if there is no cavity to suppress the decay of the field coherence. The evolution of the atomic quantum state can be described by the superradiance master equation in the Schrödinger picture: To explicitly solve Eq. (4.19), it is necessary to know precise information about the atomic spatial distribution. When all atoms are confined in a small volume, it is pos-sible to make an approximation that all atoms are placed at the same position. This assumption automatically guarantees that the system is fixed in the completely symmetric Dicke state, which induces a perfect case of superradiance. The superradiant master equation can be simplified significantly. This approximation is questionable and discussed later. With this approximation, the real part of the master equation is ,

Heisenberg picture
The Schrödinger picture is good for describing problems in a small symmetrical subspace of the full Hilbert space. However, if the sample dimension is larger than the wavelength, not all atoms can participate in the collectivity and be described by Dicke states. Therefore, the Schrödinger picture is applicable, but inconvenient. The Heisenberg picture does not require Dicke states and is straightforward to extend it from a small volume to a large volume. In addition, the formalism of the Heisenberg picture is much closer to the semi-classical light-matter interaction.
In the Heisenberg picture, two new physical observables, which are equivalent to the wavefunction in the Schrödinger equation, are population difference and polarization, defined as The physical observables, such as population difference, N, electric field, E, and polarization, P, evolve according to the Heisenberg equation of motion, Equations.(4.28) separate the slow envelope variation of the electric field and polarization from the fast carrier frequency. More profoundly, this factorization also includes the Born-Markov approximation in the Schrödinger picture. is not easy to extract information about the emission rate and frequency shift, which are obvious in Eq. (4.19). However, the Heisenberg picture is convenient to obtain a numerical solution and does not require the assumption of a small volume. Therefore, to calculate the superradiant emission in a large volume with significant propagation effects, the Heisenberg picture is more advantageous. In addition, it is necessary to notice that, due to field quantization, N and P on one side and E + and E − on the other side are non-commuting. This non-commutation is necessary for initializing spontaneous radiation, but is unnecessary for typical propagation calculations. In addition, if the initial coherence is not established by spontaneous emission, as in our Rydberg experiments where initial coherence is created by an external field, field quantization is not necessary. The Heisenberg method can be simplified into the semi-classical theory directly, which is the subject of the next subsection. is treated as the local electric field, E input + E radiation , which must be evaluated iteratively until self-consistency is achieved. Therefore, the Hamiltonian of N atoms interacting with an electric field via an electric dipole transition moment is:

Simplified semi-classical calculation
where H 0 is the Hamiltonian of the N non-interacting atoms, ∆ω 1j = ω 1j − ω is the detuning of the millimeter wave frequency, µ 1j is the transition dipole moment between the initial state and the jth state. In this Hamiltonian, I assume that all atoms are placed at the same position. The effect of deviation from this assumption is included in the phenomenological longitudinal decays and transverse decays.
Numerical solution of the Bloch-Maxwell equations is achieved by transformation to a frame rotating at the electric field carrier frequency. Following the rotating wave approximation (RWA), the far off-resonant components are discarded. For a two-level system, the transformed equations are the conventional Bloch equations.
For a multi-level system, a group of coupled first-order partial differential equations is obtained, which has n-1 diagonal and 2(2n-3) density matrix elements. For the Maxwell equation, the rotating frame transformation also removes fast oscillations, which have no significant effect here. The transformation from the local time frame to a retarded time frame further simplifies the description to a first order ordinary differential equation: where ρ, V and µ are the density matrix, electric dipole interaction operator, and electric dipole transition moment in the non-rotating frame, σ and V r are the corresponding operators in the frame rotating at the electric field carrier frequency, and U is the unitary transformation matrix between these two frames. The polarization is described by: where P r and P i are the real and imaginary parts of the macroscopic polarization, which are induced by the real and imaginary parts of the electric field. N is the number density of molecules in a specific quantum state. The spatial dependence of the electric field is dε i,r dz = 2πω c P i,r . (4.33) The computer program described in Section 4.3 can numerically solve the coupled differential equations, (4.31), (4.32) and (4.33) to obtain the dynamics of superradiance [109].
The details of the calculation are discussed in the next section.

Semi-classical calculations of a dense Rydberg gas
In this section, I apply the semi-classical method to calculate the quantum state evolution of the Rydberg atoms and the dynamics of the superradiant millimeter-wave emission in a dense Rydberg gas [133]. The initial polarization can be induced by spontaneous photons, blackbody radiation photons, or external excitation photons.
However, the number of blackbody photons (the number of 100 GHz millimeterwave photons in one mode of room temperature blackbody radiation is about 80) and external excitation photons (the number of photons in a weak chirped pulse (2nW, 50ns), is more than 10 6 ) is typically much larger than the number of vacuum fluctuation photons. Therefore, only the initial polarization induced by the blackbody radiation and the external chirped pulse are considered in our typical experiments.
Due to the initial photon number being much larger than 1 in a mode, the semiclassical approximation is always valid. The millimeter-wave radiation can be treated as a classical electromagnetic field without a need for field quantization. I do not attempt to show all possible results of these calculations in thre present section. I only highlight the most interesting and unexpected results.

Two-level System
The coherence and population dynamics in an inverted two-level system are discussed here. The population in the upper state of the inverted two-level system can be prepared by optical or infrared pumping from an energetically remote non-Rydberg state. The lower state of the inverted two-level system is a neighboring Rydberg state with |∆n| = 1 or 2, and |∆l| = 1 without any initial population. The initial condition of the millimeter-wave transition is that the population starts in the upper state and both states are nondegenerate and energetically distant from any perturbing states. In a typical rotational spectrum with a small electric dipole transition moment, each molecule usually radiates independently without coupling to other molecules.
However, in a dense Rydberg gas with large electric dipole transition moments, many molecules are coupled together via a shared radiation field, and are therefore excited and radiate cooperatively.
I define the input pulse as a rapidly oscillating carrier frequency ω, and a slowly oscillating envelope E(t), where E r and E i are the real and imaginary parts of the complex electric field: For a single-frequency pulse: For a linearly-in-time chirped pulse: where α is the chirp rate.
The Hamiltonian of the semi-classical system is: where H 0 is the atomic Hamiltonian and µ is the electric dipole transition moment.
Considering the atomic two-level basis, the matrix form of the Hamiltonian can be written as Eq.(4.38). There are two ways to separate the zeroth-order Hamiltonian and interaction terms. The first form is in Eq.(4.38a), which treats the atomic Hamiltonian as zeroth-order. Therefore, the interaction term includes only the off-diagonal dipole interaction. The second form is in Eq.(4.38b), which treats the dressed atomic Hamiltonian as zeroth-order. Therefore, the interaction term not only includes the off-diagonal dipole interaction, but also a diagonal frequency detuning term. ω 0 is the frequency difference between the ground state and excited state of the atom. ω is the frequency of the external millimeter-wave field. The detuning is defined as ∆ω = ω 0 − ω. The second form is advantageous to transform into the rotating frame, which is shown in the following: The evolution of the density matrix is transformed into the interaction picture using a dressed-atom zeroth-order Hamiltonian S, as in Eq. (4.39).
where σ is the density matrix operator in the Schrödinger picture, V s and ρ are the interaction term and density matrix operator in the interaction picture, and tz/c is the retarded time. In the two-level atomic basis, the matrix elements of the density operator and converted operator Eq.(4.39) can be evaluated into three regular differential Eqns. (4.40).
For the conventional phase choice, the plus sign in Eq.(4.39) is for e| µ |g , and the minus sign is for g| µ |e . The diagonal matrix elements represent the populations of the two states and must be real. The off-diagonal matrix elements represent the coherences and are complex. I define: where ∆ℵ is the population difference between the ground and excited states, ℘ r and ℘ i are the real and imaginary parts of the macroscopic coherence, respectively. I use ∆ℵ, ℘ r , and ℘ i with clear physical meanings to replace the density matrix elements in Eq. (4.40), and obtain equations to describe the time-dependence of the real and imaginary parts of the macroscopic polarization and population difference.
where T 1 is the depopulation (longitudinal) decay lifetime, T 2 is the homogeneous decoherence (transverse) decay lifetime, κ = 2µ eg / , and ∆ℵ 0 is the population difference at equilibrium. A Mathematica program is written to numerically solve Eq.(4.43). With ℘ r and ℘ i , the macroscopic polarization can be calculated: where ω and k are the frequency and wavevector of the millimeter-wave field. The radiative electric field can be calculated by the Maxwell equation: where c is the speed of light, and z is the one-dimensional propagation distance. With a slow variation approximation of the electric field envelope as in Eq.(4.34), Eq.(4.45) can be simplified into: Thus, a complex second-order partial differential equation is simplified into two real first-order partial differential equations. In addition, the time-dependent term can be contracted into the spatial-dependent part in a retarded frame. Therefore, the fully simplified Maxwell equations are two real ordinary first-order differential equations, which can be integrated directly.
In the following, based on practical experimental parameters, I compare the dynamical behaviors and the information encoded in a millimeter-wave spectra for a 1 Debye rotational system vs. a 5 kDebye Rydberg system. Typical parameters used in this calculation are summarized in Table 4.1. The number densities of molecules in Rydberg states and non-Rydberg molecules in rotation-vibration levels listed in Table   4.1 come from typical experimental considerations. The pulse amplitude, 10 V/m, is the maximum millimeter-wave electric field created by our CPmmW spectrometer, which is used to polarize the rotation-vibration transitions with small electric transition dipole moments. A much weaker pulse, 5 mV/m, is chosen to polarize the  In a Rydberg gas with a large electric dipole transition moment, the initially prepared polarization does not directly produce FID radiation with a single exponential From Figure 4-10, at low number density, the radiation decay curve is a single exponential, the lineshape is Lorentzian, the lifetime is T 2 , and the population distribution is constant after excitation. At intermediate number density, the radiation decay curve has a broad maximum peak in the time-domain, and the linewidth is smaller than 1/T 2 . In addition, the population evolves as the radiation is emitted. At high number density, the radiation curve has a sharp maximum in the time-domain, the lineshape is distorted, and the linewidth is broadened. More than 50% of the population decays to the lower state in a short time as the system radiates. Such contrasting behavior is due to the difference between the strengths of the molecular radiation field and the input electric field. When the radiation field is weak compared with the input field, its contribution to the local electric field can be neglected. Thus, each molecule may be treated as interacting independently with a constant external electric field, and the final result is just the sum of each individual molecule's field.
But when the radiation is strong or the molecular response to the radiation is sensitive, I cannot ignore the effect that such radiation has on the emitters themselves.
This radiation field is shared by many molecules, therefore it couples them together and forces them to behave cooperatively.
The most unexpected feature in this calculation is, at intermediate number density, the radiation can be kept at a relatively low amplitude, but can last for a very long time (> T 2 ). This can be explained by the following physical picture. If there is only one atom in a system, and decoherence of the radiation occurs by some sort of dephasing mechanism (such as collisions, inhomogeneous field, etc.) in a time T 2 , the system would lose coherence completely. However, if there are several atoms, and a weak excitation pulse only polarizes one or a few of them, before the excited atoms undergo decoherence in time T 2 , they could transfer their coherence to neighboring atoms. Due to this transfer mechanism, a complete loss of coherence in the entire system takes a much longer time. However, coherence transfer needs to be driven by the internal energy of the system. Therefore, when the number density is very high, although coherence transfer can occur frequently (effective 1/T 2 is increasing), the internal energy of the system will be depleted quickly at the same time (effective 1/T 1 is decreasing). The net effect would decrease the lifetime of the coherence significantly at high number density. In the calculation shown in Figure 4-10, there is an optimized number density, at which the radiation persists for more than 20 × T 2 .

Extended sample source
In our typical millimeter-wave experiments, the sample is designed to have a cylindri- always been difficult to observe, except in a dense Rydberg gas.
In addition to calculating the propagation of the radiation in an extended sample, it is also easy to calculate the behavior of a long excitation pulse propagating in the sample. Different from previous calculations, a non-inverted two level system is used here with a π/2 pulse to maximally polarize the transition. Figure 4-12 shows how a 500 ns π/2 excitation pulse propagates in a long and dense Rydberg gas. Plot (a) shows a 3D plot of the pulse electric field, and Plot (b) shows a 3D plot of the polarization field. At the front of the sample, because the accumulated response radiation is small compared to the excitation pulse, the nutation exhibits π/2 curvature as I expect. However, at the back of the sample, because of the strong absorption and emission from the front, the radiation is comparable or even larger than the excitation pulse. The nutation deviates far from nominal π/2 curvature and has several nodes. I use such an effective local electric field to calculate the induced polarization, in Plot (b) and find that at the front of the sample, there is a

Three-level System
In principle, the semi-classical calculations of the three-level system are not different from those for the two-level system. The Maxwell equations are the same, and the number of Bloch equations increases quadratically with the number of states involved.
In this section, I derive the Bloch equation for an inverted Λ-type three-level system first, and demonstrate the existence of unique collective effects in this system by numerical calculation.
The matrix form of V S is (4.50) The matrix form of Eq.(4.49) is, In Eq.(4.51), I ignore the coherence between states |2 and |3 , because there is no direct dipole coupling between them and the coherence must be created by multiquantum excitation, which can be ignored in my typical experiments. This assumption might fail in a extremely dense Rydberg gas or with a strong external driving field, but such systems are beyond the scope of this thesis. As I discussed above, the diagonal terms of the density matrix represent the population and are real, and the off-diagonal terms represent the coherence and are complex. Therefore, The following matrix elements are relevent: The result is seven coupled real differential equations,  (4.54) is also based on the assumption that all atoms are placed at the same position. To include the spatial distribution, the multiplicative prefactor F is employed in the same way as in the two-level system. In principle, the same method can be used to extend this semi-classical calculation method to multi-level system.
However, the computation time increases quadratically with the number of levels.
Branching ratio of the radiation in an inverted Λ-type three-level system Consider a Λ-type, inverted three-level system, as shown in the inset of Figure 4  In addition, at early times, the amplification is less important, thus the intensity ratio is close to the oscillator strength ratio. However, at later times, the amplified polarization is completely dominated by the cooperatively radiated electric field from the coherently coupled molecules. This causes most of the radiated energy to be steered into the more polarizable channel. The intensity ratio is far from the independent-transition prediction based on the oscillator strength ratio.
In Figure 4

New experiments
The major challenges of experiments dealing with collective effects are [141]: (1) preparing a dense, homogeneous, and strongly interacting medium by optically thin optical pumping at short wavelength that can radiate at long wavelength in an optically thick manner; (2) creating an initial coherence in a precisely controllable manner; and (3) creating an arbitrarily shaped, sharply defined boundary of the initially interacting volume, the shape of which governs the propagation of the coherence. The combination of the recent upgrade of our molecular beam source (buffer gas cooled ablation source), optimization of our CPmmW spectrometer, and our achievement of an efficient scheme for preparing core-nonpenetrating Rydberg states, which is discussed in Section 6, enable a series of systematic experiments that observe the coherence generation, propagation, and response to external manipulations. Based on the preliminary experiments in Section 3.3 and the calculation results in this chapter, I believe that our dense Rydberg gas is unique for two reasons: (1) the initial Rydberg state is prepared by optical pumping from a valence state. Such a transition is optically thin, thus the sample is uniform; (2) the Rydberg-Rydberg transition is initialized and probed by millimeter-wave excitation and the induced FID radiation. This transition is optically thick and exhibits strong collective effects. For fundamental photonics research, the existence of uncoupled preparation and probing processes enables unprecedented control over the initial and boundary conditions of samples that exhibit cooperative effects. Qualitatively distinct from ultracold manybody collisional experiments performed at high density but small volume, our setup have a large, homogeneous, and geometrically well-defined interaction volume along with a precisely known total number of molecules. These conditions are essential for fundamental studies of radiation-induced collective inter-particle interactions.

Superradiance lineshape
The buffer gas cooled photoablation technique creates an intense molecular beam at a density of 1×10 9 cm −3 . Two or three crossed tunable laser beams, with carefully designed beam geometries, pump the molecules into a single Rydberg state in either a tube-like or disk-like volume. The maximum transition dipole moment of normally-used laser optical pumping schemes in the UV region is ∼1 Debye, which is in the optically thin limit (<5% laser beam attenuation occurs through L=2 cm and N density ∼1 ×10 9 cm −3 Rydberg gas). However, for |∆n| ≤ 1 Rydberg-Rydberg electronic transitions, the large electric dipole transition moments (∼5 kDebye) result in a system in the extreme optically thick limit (>99.9% millimeter-wave beam attenuation occurs through L=2 cm and N density ∼1 ×10 6 cm −3 Rydberg gas). A medium with such an enormous optical thickness will be an ideal test platform for the study of nonlinear phenomena, especially collective effects. In Section 3.

Superradiance anisotropy
In my experiments, the Rydberg system is prepared by incoherent laser excitations, and the initial coherence is established by a weak millimeter-wave pulse. Such coherence propagates along the pulse propagation direction. Therefore, I expect to detect forward radiation. In addition to the coherence initialized by the millimeterwave pulse, additional coherence can be established by the self-amplified superradiant emission. Such coherence prefers to propagate along the z axis of the sample (Here, I consider a cylindrically shaped sample in a cylinder shape, where z is the length of the cylinder). Therefore, I expect to detect radiation in the z direction. In my typical experiments, the excitation millimeter-wave pulse also propagates along the z axis. However, what happen if the direction preferences are different? More deeply, this question is equivalent to the discussion of the initial phase symmetry and evolution, as shown in Figure 4-15. Theorists have performed detailed calculations to predict the radiation dynamics of the timed atomic state and symmetric atomic state [142,9,143,144]. However, very few experiments successfully demonstrate the predictions. Here, I propose a new class of experiment in the Rydberg system. Figure   4-16 shows an experimental diagram for detecting the radiation anistropy. The initial coherence is generated by a millimeter-wave pulse from horn 1 or 3. Horn 2 is used to detect the radiation. Based on this geometry, a millimeter-wave excitation pulse  [9] from horn 1 prepares the timed atomic state, and that from horn 3 prepares the symmetric atomic state. To make the amplitudes of the initial coherences the same, it is possible to apply a slightly off-resonant excitation pulse, which is not attenuated significantly by the Rydberg atoms. The anisotropy of the radiation can be measured by the relative intensities I ⊥ and I . In addition to the amplitudes of the radiation, the radiation frequency might be different in different directions, which is discussed in the following.

Superradiance frequency shift
An atom interacting with the electromagnetic field by an electric dipole transition moment exhibits frequency shifts. For example, Weisskopf and Bethe predict that a radiation field can induce a small frequency shift in radiation from the 2s 1/2 state of Hydrogen atom [145,146]. The interaction between the electron and the virtual photon leads to the familiar Lamb shift. Scully has shown that this frequency shift can be amplified by a collective interaction in an ensemble, giving rise to a collective Lamb shift [142,9]. The collective Lamb shift has a similar mechanism to the ordinary Lamb Figure 4-16: A schematic diagram for detecting the radiation anistropy. cw laser absorption is used to calibrate the pulse-to-pulse number density fluctuations of the molecular beam source. The rectangular box represents the initially prepared Rydberg sample using two or three crossed tunable lasers (not shown). Three identical millimeter-wave horns are used to broadcast the excitation pulse or receive the radiation.
shift except that the virtual photon emitted by one atom can be absorbed by another.
The collective Lamb shift had not been observed until 2010 by Rohlsberger [147].
The usual difficulty of preparing a system with optically thin pumping and optically thick radiation is overcome in our system. However, due to the gigantic electric dipole transition moment of Rydberg-Rydberg transitions, other frequency shift mechanisms exist in our system, such as the Coulomb shift, the resonant collision shift, and the dipole-dipole shift, each of which must be characterized carefully. Fortunately, theorists find that the different frequency shift mechanisms have different dependence on the geometry of the sample [148,149,144]. We have the ability to systematically change the sample geometry by shaping the excitation laser beams, to isolate the collective Lamb shift. In addition to experimental difficulties, a precise measurement of the center frequency of the radiation is non-trivial. For example, in Figure 3

Rydberg mirror
In Figure 4 According to the Fresnel equation, it is easy to compute the transmitted electric field and the reflected electric field, where E 0 is the incident electric field, E 1 and E 2 are the reflected and transmitted electric fields respectively, and ε 1 and ε 2 are the permittivity of the vacuum and the Rydberg species respectively. The permittivity of the vacuum is real and of unit magnitude, and that of the Rydberg species is complex and can be divided into dispersion and absorption terms, √ ε 2 = n 2 + iκ 1 . I substitute this into Eq.(4.55), Therefore, the relationship between the transmitted wave and reflected wave is On resonance, n 2 =1, the ratio of the electric fields of the reflected wave to the transmitted wave is proportional to the absorption coefficient. For a dense Rydberg gas with a large absorption coefficient, the reflected wave is significant. In the quantum mechanical picture, the phase relationships of the three waves (incident, reflected and transmitted) are more complicated and discussed in detail by Yelin [150].
If a dense Rydberg gas can resonantly reflect the millimeter-wave pulse, this would enable use to detect the resonant signal at the back side of the sample, which is totally separate from the excitation pulse. It is similar to a transient absorption experiment, but the background is zero. With such a resonant Rydberg mirror, we would be able to separate excitation and radiation spatially. This spatial separation can not only increase the sensitivity to detect weak signals, but also can enable detection of fast decay process during the excitation. In addition, in this geometrical configuration, the Doppler shift instead of Doppler broadening limits the spectroscopic resolution.

Chapter 5
CPmmW spectroscopy of BaF

Rydberg-Rydberg transitions
I choose BaF as the first candidate on which to perform molecular Rydberg-Rydberg transition experiments, because: • The dissociation threshold of BaF lies higher in energy than the ionization limit (v + = 0 and 1) [151]. This unique feature eliminates nonradiative decay via predissociation for v + < 2. Therefore, all states belonging to v + = 0 Rydberg series are stable and have lifetimes similar to Calcium or Barium Rydberg states.
Thus, I have sufficient time to populate high-states by stepwise laser and millimeter-wave pulsed excitation.
• The states belonging to v + = 1 Rydberg series have similar spectroscopic structures to v + = 0 Rydberg series, but with much shorter lifetimes due to autoionization dynamics. Therefore, the detailed analysis of the v + = 0 Rydberg series can help to assign the spectra of v + = 1 Rydberg series. By comparing v + = 0 and v + = 1 Rydberg spectra, I can systematically study vibrational autoionization dynamics, which directly connects to the R-dependence quantum defects.
• v + = 1 Rydberg series with controllable and predictable non-radiative lifetimes provide us with a good test system for new quantum control techniques for lossless access to core non-penetrating states, see Section 6.
• Considering experimental convenience, our lab possesses a large amount of data from low-lying Rydberg states of BaF (4.4 ≤ n * ≤ 14.3) that simplifies our exploration and accelerate our assignment of the higher-n * Rydberg states, especially the core-nonpenetrating Rydberg states [152,153,154,155,151].

BaF experiment in the supersonic beam apparatus
Although the most important BaF FID spectra cannot be recorded on the supersonic beam apparatus, a high quality Optical-Optical Double Resonance (OODR) spectrum of BaF Rydberg states at n * ∼40 is still fairly necessary for the following FID experiments in the buffer gas cooled beam apparatus. Such OODR spectra were far more difficult to collect than I expected.  is calibrated by the Te 2 absorption spectroscopy, and the probe laser (539-542 nm) is calibrated by I 2 LIF spectroscopy.

Optimization of the BaF molecule generation
To obtain a stable and bright BaF molecular beam, I optimize several parameters of the beam source by detecting the LIF signal, which will be described in Section 5.1.3. The optimized parameters are listed in Table 5.1. The items with asterisk are the most sensitive and need to be optimized carefully.  The black curve shows that the rotational temperature of BaF increases with higher ablation laser power, and the red curve shows the decrease of the number density of BaF molecules in the vibrational ground state with higher ablation laser power. Plot (b) shows titration curves of Ba with different concentrations of SF 6 . The BaF yield is maximum at 0.1% SF 6 . With high concentration of SF 6 , most of the Barium appears to react with SF 6 to create BaF 2 , which cannot be optically detected here.

Laser Induced Fluorescence (LIF) and Resonance En-
hanced Muli-Photon Ionization(REMPI) spectra of BaF The transition C 2 Π 3/2 -X 2 Σ + has already been measured precisely. In this chapter, my purpose is not to refine the spectroscopic data. The LIF experiment is used to roughly measure the BaF number density per single rotational state, which is used for optimization of both the supersonic cooled beam source and the buffer gas cooled beam source. The REMPI experiment is used for optimizing the TOF-MS to achieve a better OODR spectra, as discussed in the next section.
In the LIF experiment, the supersonic molecular beam is skimmed by a 3 mm diameter conical skimmer, which is placed 3 cm downstream from the supersonic nozzle, and then crossed with the pump laser (6 cm downstream), which is collimated with 3 mm diameter. Because the fluorescence wavelength is almost the same as the excitation wavelength, which cannot be separated by a regular interference filter, I In addition to recording LIF spectra, I also record REMPI spectra by ion detection for several reasons: (1) Ion detection has much higher sensitivity (100% detection efficiency); (2) With a TOF-MS, I can separate the five different isotopologues of BaF by their different arrival times; (3) Optimizing REMPI spectra is helpful for the OODR with Ramped-PFI experiment, to be discussed in the next section. The details of the TOF-MS have been described in Jason Clevenger's thesis [93]. Here I only list several matters that require attention. The notation below can be found in Jason Clevenger's thesis.
• The amplitude of the extraction pulse on A1 should be below 250 V/cm. Otherwise, the ions with the same e/m cannot be focused in the time-domain. The detailed discussions and calculations are in Christopher Gittins' thesis [156].
• Although focusing the excitation laser can also increase the mass resolution of the TOF-MS, it reduces the total number of ions. With a 1 mm 2 loosely focused laser spot, I must tweak the ion optics carefully to focus the ions temporally and spatially. The resolution of our spectrometer is high enough to resolve all five isotopologues of BaF.

Pulsed Field Ionization detection (Ramped-PFI)
With the spectroscopic information on the intermediate C 2 Π 3/2 state of BaF, I use another pulsed dye laser (Probe laser) to populate Rydberg states with n * ∼40 and v=0. The ionization potential (IP) of BaF has been measured by Jakubek et al [151].
Using the Rydberg formula, I can approximately estimate the probe laser wavelength.
Because the Rydberg states with v=0 are bound states, I must apply a pulsed electric field to ionize them. As discussed in Section 3.1.2, I apply a ramped pulsed field, which provides additional information about n * . The threshold of the pulsed field ionization for a Rydberg state with n * is In Eq.(5.1), E is the threshold of PFI electric field in atomic units. The energy spacing between n * and n * +1 is ∼3 cm −1 for n * ∼40. However, the number of electronic-rotational transitions can be as large as 100. Therefore, a tiny stray electric field might couple multiple neighboring levels thereby washing out the periodic structure of the Rydberg series. Therefore, before I record the OODR data, I must minimize the electric field with the methods discussed in Section 3.1.3. Figure 5-6 shows the raw data of the OODR spectrum with Ramped-PFI detection. The pump laser is fixed and populates the N=2 C 2 Π 3/2 intermediate state of BaF. The probe laser is scanned across a ∼50 cm −1 range, which covers from n * =37 to the IP. In Although the spectroscopic transition density is very high and most of the structures The ramped pulse in this experiment is used to isolate the Rydberg states from the ions created by multi-photon ionization. This method decreases the background to zero and increases the signal to noise by a factor of 3 compared to conventional PFI with a rectangular pulse. A disadvantage is that I cannot separate the signals of different isotopologues. As mentioned, to obtain a spectrum as shown in Figure 5-7, I must minimize the stray electric field carefully, which is non-trivial. However, when the OODR experiment is used only for obtaining n * for the subsequent millimeterwave experiments, we use a much less demanding method. From Eq.(5.1) [13] and As in the REMPI experiment, I also measure the saturation power of the probe transitions. The typical saturation power is ∼10 mJ/cm 2 , which is much higher than that of the pump transitions.

Experimental implementations
There are no new experimental implementations for the BaF CPmmW experiment.
All necessary preparations have been discussed in previous sections. Here, I will outline the necessary steps as follows: • Molecular beam source: optimized buffer gas cooled molecular beam, as described in Section 2.2.
• Pump and probe lasers: two tunable pulsed dye lasers pumped by an injection seeded Nd:YAG laser. Their wavelength and powers can be set according to the LIF spectrum and OODR spectrum, which have been described in Section 5.1. Due to the high saturation power (100 mJ/pulse) of the probe transition, I cannot saturate this transition with a d=10 cm 2 expanded laser beam. A typical value of the pulse energy of the probe laser used in the BaF CPmmW experiment is 20 mJ, which is the maximum output of our pulsed dye laser.
• Implementation of the CPmmW spectrometer with the buffer gas cooling apparatus: reflection mode for maximizing the interaction volume, as described in Section 2.3.6.

Millimeter wave induced Rydberg-Rydberg transitions
with Free Induction Decay detection (FID) As far as I know, this is the first spectrum of FID-detected molecular Rydberg-Rydberg transitions in the world. However, I find two suspicious problems after carefully studying this spectrum: (1) One transition is much stronger than the others.
As we know, there should be several transitions, ∆N=0, ±1, ∆ =±1, from the same initial Rydberg state. Their electric dipole transition moments should not be different by a factor of 10.
(2) The linewidth of the strong transition is ∼400 kHz, which is larger than the resolution of the FT-window. However, the linewidth of the weak transitions are ∼150 kHz, which is limited by the FT-window. This is also unexpected, because I do not think that the lifetime of one Rydberg state is significantly shorter than that of the others. One possible explanation of these unexpected phenomena is superradiance. As I discuss in Section 4, in a system with multiple transitions, strong collective effects can focus intensity to one transition that has a relatively larger transition dipole moment.
The same as shown in Figure 4-14, although the strongest transition in Figure [157]. The essence of XCC is to record two spectra which contain a linear combination of several patterns, as shown in Figure ( re-descending robust estimator:

Recovering the millimeter-wave frequency
where i labels each data point, R and d are defined in Figure  What I need in the FID experiment is not pattern recognition, but noise reduction.
However, I can modify the XCC method to achieve our goal. Similar to the XCC method, I record the FID spectrum twice (spectrum 1 and spectrum 2) with the same experimental conditions. The standard method to reduce the noise is just to average them. My method is to apply a weighted average. The weight factor is in Eq.(5.2).
Different from XCC, the typical noise level in my spectrum is very large, especially for a weak signal. Therefore, it is not easy to obtain α. A wrong α could accumulate noise in a weighted average to create a fake signal. An easy solution is to replace the positive variable R in Eq.(5.2) by R-R baseline , which can be positive or negative, and R − R baseline = 0. Thus, the accumulation occurs only for "asymmetric" signal, but not for "symmetric" noise. A method to obtain R baseline has also been develpoed by Jacobson et al., in our group [158]. In addition, if I perform weighted averaging for many averages, the noise reduction rate can be much faster (determined by the setting of the weight parameter).

Laser-millimeter-wave 2D Spectrum
For spectroscopic purposes, I am interested in millimeter-wave transitions from not only a specific initial Rydberg state pumped by the probe laser. Therefore, to systematically obtain all millimeter-wave transitions from a wide range of initial Rydberg states, I must scan the probe laser. Figure 5-15 displays the raw data of a lasermillimeter-wave 2D spectrum. This 2D spectrum is produced by arranging several hundred FID spectra with different probe laser wavelengths together. Each FID spectrum is recorded for 10 seconds with 100 averages. The entire 2D spectrum with ∼200 millimeter-wave transitions is recorded in ∼1 hour.
It is easy to extract a millimeter-wave spectrum with a specific probe laser transition, as shown in Figure 5-16. Figure 5-17 shows two zoomed-in transitions from      After reducing noise, recovering the millimeter-wave frequency, and precisely measuring the laser transition wavelength, I can create a refined laser-millimeter-wave 2D spectrum, as shown in Figure 5-20. The sizes of the dots represent the intensity of the FID amplitude signal. For visual clarity, the relationship between the dot size and FID amplitude is chosen to be not linear. In addition, the resolution of the millimeter-wave spectrum is much higher than it appears in this 2D plot (vertical scale: ∼20 GHz, resolution: <150 kHz, >10 5 pixels along the vertical axis).

Connecting laser wavelengths to millimeter-wave transitions
There are many observed transitions in Figure 5 Plot (a) shows the first type of connection with one millimeter-wave photon: In the 2D spectrum, this means I can find two transitions with the same millimeterwave frequency y 0 , and the difference of the laser wavelength coordinates ∆x is equal to y 0 . In addition, the direction of the millimeter-wave transition with lower laser frequency should be up, and that with higher laser frequency should be down. Alternatively, the direction of the millimeter-wave transition can be extracted by the phase measurement discussed in Section 3.2.5. Strictly speaking, the parity selection rule is incompatible with such an assignment. However, if any state, such as a Π  shows the other type of connection via two millimeter-wave photons:

Stark demolition
As I mentioned in Section 1. demolition", is based on the super-sensitivity of core-nonpenetrating Rydberg states to an external DC electric field [13]. Figure 5-23 is a schematic diagram of the Stark demolition experiment. A <1 V/cm DC electric field is applied to couple one laser excited "bright" CNP state or a millimeter-wave excited "bright" CNP state, to other "dark" CNP states with high-(The "bright" states can be populated by probe laser only, or by one probe laser photon and one millimeter-wave photon. The "dark" states can be populated only when a DC field is involved). Due to small quantum defects, all CNP states (with up to n-1) are nearly degenerate. A small interaction induced by the Stark field can completely mix them. The coherence induced by the chirped millimeter-wave pulse is diluted and dephased into the large number of "dark" CNP states. Therefore, the FID radiation from all CNP-CP or CNP-CNP transitions will be undetectable. However, core-penetrating (CP) states, with their larger quantum defects, are far from the "dark" CNP states and cannot be mixed by a small Stark field. Therefore, the FID radiation from CP-CP transitions can be detected even in the presence of a small Stark field.
The experimental implementation for Stark demolition is simple. Two large mirror-polished stainless steel plates (4" × 6" × 1/16") with 3" separation are inserted into the detection chamber. The plates separated by this large distance (3") do not cause reflections with the millimeter-wave beam, laser beam, or molecular beam. A pulse generator creates a 0-16 V 10 µs rectangular pulse, which is temporally overlapped with the FT-window of the FID radiation. Figure 5-

millimeter-wave multiple resonance
In Section 3.2.7, I showed that a millimeter-wave pulse can not only induce FID radiation, but also is sufficiently strong to induce population transfer in an atomic Rydberg system. More than one millimeter-wave photon populates high-Rydberg states by multiple step resonance excitation. Different from the experiment in Section 3. To observe millimeter-wave multiple resonance spectra with more than two photons, I apply more than one chirped pulse, as shown in Figure  By the two examples above, I have demonstrated a millimeter-wave multiple resonance technique, which is an efficient method to rapidly detect a complex network of energy levels. However, the data analysis method I am using is preliminary and not elegant. Within a single multiple resonance spectrum, I cannot judge whether the transitions are sequential or parallel. Therefore, the laser-millimeter-wave 2D spectrum, as shown in Figure 5-20, might include many transitions of ν laser + ν mmW + ν mmW , which would increase the spectral complexity for analysis. Such information is likely encoded in the phase information and could be extracted directly with an appropriate algorithm.   Comparing the integrated FID signals in two spectra. Plot (a) shows the integrated FID signals vs. probe laser wavelength from n * =33 to n * =38. Plot (b) shows a much larger spectrum to spectrum vairations in a scheme from n * =39 to n * =50.

Problems and possible solutions
tion diagnostic. However, a major obstacle prevents me from applying such wonderful techniques to systematically collect data for detailed analysis. This is the stability of the system. Some FID signals seem to be "randomly" lost. For example, Figure 5-28 shows two sets of data taken in the same day with the same experimental conditions. Plot (a) shows the integrated FID signals vs. probe laser wavelength from n * =33 to n * =38. Most of the integrated FID signals from the two data sets are overlapped.
However, the intensity fluctuations of some transitions are very large, and a few integrated FID signals only show up in one data set. This problem seems to be even worse in the region from n * =39 to n * =50 shown in Plot (b). The difference between two spectra is quite large, and the number of signals in the two spectra is much smaller than I expect.
Through several diagnostic experiments, I find that the instability comes from the probe laser. The Sirah Precision Scan pulsed dye laser used as the probe laser, has ∼0.03 cm −1 short-term linewidth, which is in agreement with its specs. However, its output frequency has a long-term (∼20 s) fluctuation (∼0.06 cm −1 ). If the linewidth of the Rydberg←C transition is smaller than the frequency fluctuation, large intensity fluctuations are expected. To measure the linewidth, I scan the probe laser in a small step-size (0.01 cm −1 ), and plot the integrated FID amplitude, as shown in  shown evidence that the electric dipole transition moments between the C state and Rydberg states are much smaller than them from another state, such as the D state.
(3) I could apply a small electric field to Stark broaden the transition. I hope at least one of these possible solutions will fix the instability problem, especially near n * =43, in which n * region 76-98 GHz millimeter-waves induce ∆n * =1 transitions.

Molecular core-nonpenetrating Rydberg states
Core-nonpenetrating Rydberg states are defined as having negligible overlap between the wavefunction of the Rydberg electron and that of the ion-core, due to the centrifugal barrier [19]. The effective potential for an electron is: where Z ef f (r) is the r-dependent effective positive charge seen by the Rydberg electron, n and are the principal quantum number and angular momentum quantum number of the Rydberg electron. At a turning point, V l (r nl± ) = E nl = −R/n 2 , where R is the Rydberg constant (in cm −1 ). I set Z ef f (r) = 1, thus From equation 6.2, I know that when n >> l, the inner turning point position is almost independent of the principal quantum number n, but strongly dependent on the angular momentum quantum number . The inner turning points of the 3d, 4f, and 5g orbits can be calculated as 3.17, 6.35, and 10.59Å, respectively, compared to a typical ion-core diameter of 5Å. For most diatomic and small polyatomic molecules, states with Rydberg electron angular momentum ≥ 3, are core-nonpenetrating and can be described by the hydrogenic Rydberg formula, E n = 1/2n 2 [a.u.]. The corenonpenetrating electron is prevented by the centrifugal barrier from penetrating inside the extended electron distribution of the ion-core. All of the efficient mechanisms for exchange of energy between the Rydberg electron and the ion-core are turned off [19].
The Rydberg electron is essentially uncoupled from its ion-core, leading to an atomlike electronic structure with "almost good" quantum numbers and near-degenerate λ components [33,32,2,159,19].
The Rydberg electron of a molecular core-nonpenetrating state does not interact strongly with the non-spherically symmetric part of the ion-core. Therefore, it is extremely sensitive to an external electric field, which can induce kiloDebye  [14,15,16,17,18]. For spectroscopists, molecular core-nonpenetrating Rydberg states are also distinctive, because of easily recognizable and predictable patterns that arise from the highly restrictive "pure electronic" transition selection and propensity rules (∆ = ±1, ∆N + = 0, ∆v + = 0, ∆N = 0, ±1, + ↔ −) [19,30]. However, a simple spectrum does not imply the absence of information. With high-resolution spectroscopic techniques and an analytic perturbative long-range model, measured frequency shifts and intensity deviations from the hydrogenic Rydberg formula yield a complete picture of the electronic structure of the molecular ion-core [31,1].
The discussion above is based on the instantaneous Hamiltonian. However, because the Hamiltonian is explicitly time-dependent, nonadiabatic interactions can couple the adiabatic states causing some mixing of the short-lived "dark" state |2 into the "bright" states. To avoid such a loss process in STIRAP, I must reduce the diabatic interaction to a negligibly small value. The adiabatic condition can be derived from general considerations in time-dependent quantum mechanics. The Hamiltonian matrix element for the nonadiabatic coupling between our state of interest |a 0 and the perturbed state |a ± is a + |ȧ 0 . If the coupling matrix element is much smaller than the field-induced splitting, |ω ± − ω 0 |, such non-adiabatic interaction can be ignored, as shown in Eq.(6.7a). I substitute Eqs.(6.4) and (6.5) into Eq.(6.7a) and obtain the more practical adiabatic condition, as in Eq.(6.7). If I know the transition Figure 6-3: Adiabatic evolution in the STIRAP process. Plot (a) shows the counterintuitive pulse sequence. Plot(b) shows the evolution of the eigen-frequencies of the three eigen-states. Plot (c) shows the population evolution during the STIRAP process. Initial population in |1 is 100% transferred to state |3 . No population is moved into State |2 . dipole moments, I can apply Eq.(6.7c) to verify the adiabatic condition.
a + |ȧ 0 ω ± − ω 0 (6.7a) Θ ω ± − ω 0 (6.7b) If the laser pulses that I use have a smooth change in amplitude, such as Gaussian or even triangular, Eq.(6.7c) can be approximately simplified to: where Ω ef f = Ω 2 P + Ω 2 S , and ∆τ is the duration of the laser pulse. The left-hand side of this equation is proportional to the laser pulse energy. Eq.(6.7c) is satisfied for completely coherent lasers (FT-limited laser pulse). For most pulsed dye lasers, the output pulses are not FT-limited, and Eq.(6.7c) must be modified to: Ω 2 ef f ∆τ > 100 ∆τ 1 + ∆ω L ∆ω F T 2 Γ, (6.9) where ∆ω L is the actual bandwidth of the laser pulse, ∆ω F T is the FT-limited bandwidth of the laser pulse, and the parameter Γ depends on the spectral profile (Gaussian or Lorentzian) and is approximately unity.

Results of optical-millimeter-wave STIRAP calculation
The STIRAP technique has been widely used in many research communities. The duration and wavelength of the pulse-pairs range from fs-to-s and UV-to-rf [161,162,163,164,165]. However, in our proposed experiment, to induce transitions from n * =4, =2 via n * ∼40, =3 to n * ∼40, =4, the Pump pulse is in the visible (∼10 15 Hz) region and the Stokes pulse is in the millimeter-wave (∼10 11 Hz) region. In this section, the calculations of population transfer efficiency demonstrate quite conclusively the However, the STIRAP technique is not limited to CaF molecules, but is applicable to Rydberg states of other diatomic and polyatomic molecules. ns, τ f = 1 ps to 10 ns, τ g = 100 ns. Typically, the lifetime of the g state is still not long enough for high resolution millimeter-wave spectroscopy, but it is long enough to excite step-wise through it to a more stable h state.
In this section, I perform two calculations with different sets of parameters. In the first calculation, all parameters are based on the basic setup in our lab, such as long-pulse laser (100 ns), which is under construction in our lab, in combination with a Gaussian shaped millimeter-wave pulse. The detailed parameters are listed in Table   6.2.
Pulsed dye laser + rectangular millimeter-wave pulse In the calculation, the incoherent properties of the non FT-limited ns-pulsed dye laser are taken into account by the colored noise phase fluctuation model [163]. Different from a completely coherent radiation source, the electric field from an incoherent source has extra phase fluctuation terms: µ P E P (t) = Ω P (t) cos [ω 21 t + α(t)] (6.10a) I apply an algorithm developed by Vemuri et al. [166] to generate colored noise to simulate our dye laser pulses. And, as shown in Figure 2-30, the natural linewidth of the millimeter-wave source is 2 kHz in 1 ms. Its phase noise can be ignored in a 10-100 ns pulse. In Figure 6-5, Plot (d), I find that the population transfer efficiency without colored noise (ideal efficiency) is much higher than that with colored noise. I try to compensate the laser noise by introducing multiple frequency components in the millimeter-wave pulse. Figure 6-6, Plot (a) shows a similar STIRAP process as in Figure 6-5, Plot (c), but with a 2 GHz linearly chirped millimeter-wave pulse. In   toward the ideal efficiency significantly. I try to add two chirped millimeter-wave pulses to improve the compensation. With appropriate relative strengths and delay, I can achieve 60% average population transfer efficiency, as shown in Figure 6-6, Plot (d). The ideal efficiency is also increased, because the second chirp also moves the population from the g state to the more stable h state. I name these two methods CHIRAP (CHirped sTimulated Raman Adiabatic Passage). However, due to the complicated light-multi-level matter interactions in CHIRAP, the distribution of the population transfer efficiency is broader.
The calculated population transfer efficiency above is acceptable. However, the population transfer efficiency, even ideal efficiency, is still much lower than that in typical STIRAP experiments. From Eq.(6.7c), it is not difficult to discover that the sharp edge of the rectangular millimeter-wave pulse induces a significant nonadiabatic interaction. To minimize the population leakage due to such non-adiabatic interactions, it is necessary to change the shape of the millimeter-wave pulse from rectangular to a smoother one, such as Gaussian.
Long pulse FT-limited laser + Gaussian shaped millimeter-wave pulse The comparison shows that I must keep the total frequency of the two photons (sum or difference) in resonance with the energy difference of the initial and final state.       Figure 6-10 shows a comparison of the spatial transfer efficiency distribution for an unskimmed buffer gas cooled slow molecular beam and supersonically cooled fast molecular beam. Transfer efficiencies in the central diameter of the molecular beams are the same. However, due to the different Doppler widths, the slow molecular beam has a more uniform spatial distribution, which is more compatible with the large volume of the CPmmW experiment. To achieve an even better spatial distribution, I can simultaneously apply two or more millimeter-wave pulses with different frequencies, which are used to compensate the laser Doppler shifts.

Unwanted processes
The calculations above demonstrate the feasibility of high efficiency population transfer to high-core-nonpenetrating Rydberg states. However, two important processes, which might decrease the population transfer efficiency significantly, are not included in the calculations. In typical STIRAP experiments, the intensities of both the laser pulse and the millimeter-wave pulse are much higher than in the step-wise double resonance experiment. Intense laser pulses might induce multi-photon ionization, as shown in Figure 6-11, Plot (a). Intense millimeter-wave pulses might induce a nonresonant excitation, as shown in Figure 6-11, Plot (b). The former process causes population loss by creating ions, and the latter process causes population loss by creating unwanted states. In this section, I discuss these two unwanted processes and show that they would not interfere with my experiment with an appropriate setup.
• Laser-Multi-photon ionization To avoid multi-photon ionization, I need to derive the dependences of the efficiency of ionization and STIRAP on pulse duration, τ , for a fixed total pulse energy. To evaluate the effective pulse area of a bound-continuum transition, I need to integrate over the bandwidth of the pulse, E∆t d∆ω, (6.14) where M is the matrix element of the transition dipole moment, ∆ω is the bandwidth of the laser pulse, Γ is the coherent decay rate of the continuum states, and ∆t is the duration of the laser pulse. For continuum final states, Γ >> ∆ω. Therefore, Eq.(6.14) is approximately, where M/iΓ is independent of the laser pulse, and for a FT-limited pulse, ∆t∆ω is a constant. Therefore, the effective pulse area is only linearly dependent on the electric field, which is inversely proportional to √ ∆t for a fixed total pulse energy, A ef f ∝ 1/∆t. The effective pulse area of a bound-bound transition in STIRAP is straightforwardly calculated, Therefore, the figure of merit for avoiding ionization scales as τ . In addition to the pulse length, a FT-limited laser pulse is preferred to minimize multi-photon ionization due to its narrow bandwidth, which has smaller overlap with the continuum spectrum. By experiments, I have observed that the pulse energy threshold for two-photon (7 ns non-FT-limited dye laser pulse) ionization is ≈100 µJ/mm 2 . My proposed experiment use a 100 ns long, <10 µJ/mm 2 FTlimited laser pulse, far below the scaled ≈1 mJ/mm 2 multi-photon ionization threshold.
• Non-resonant millimeter-wave excitation The ratio of the resonant excitation and non-resonant excitation can be evalu-

ated by
A ef f ∝ M ∆ω + iΓ (6.17) where ∆ω is the off-resonance detunings, and Γ is the radiative lifetime. On resonance, ∆ω ∼0 and the matrix element of the transition is inversely proportional to Γ. Off resonance, ∆ω >> Γ and the matrix element of the transition is inversely proportional to ∆ω. Therefore, the ratio of the resonant excitation and non-resonant excitation is: 6.3 Adiabatically-focused STark-mixed Rydberg Orbitals (ASTRO) 6

.3.1 Non-Hermitian Hamiltonian
The non-Hermitian complex Hamiltonian is used to to describe bound-free interactions [19]. Different from the usual Hermitian Hamiltonian, the diagonal matrix elements of the non-Hermitian Hamiltonian can be complex. Therefore, its eigenvalues are also complex. The real part represents the energy of the eigenstate, and the imaginary part represents the lifetime. The Hamiltonian of a two-level system with finite lifetime can be written as, The parameters in the Hamiltonian are defined as: where ε 1 and ε 2 are the real energies, Γ 1 and Γ 2 are the decay rates, and V represents the interaction. The eigenvalues of the Hamiltonian are: (6.21) For example, in a two-level system, I assume that one state is a completely bound state, and the other one has a 1 GHz decay rate. Their energy spacing is 2 GHz.
When I apply a DC electric field to mix the two states, the real energy in Eq.(6.21) represents a Stark splitting, as shown in Figure (6-12), Plot (a). The imaginary energy shows a mixing of the decay rate, as shown in Figure (6-12), Plot (b). It is straightforward to extend this method from a two-level system to a multi-level system, which is used to analyze the ASTRO scheme in the following.

Description of ASTRO
The recipe for preparing core-nonpenetrating states through ASTRO is: • This is a two-laser 1+1 double resonance excitation scheme. Choose a bright, • Turn on a ∼10 V/cm electric field to mix the bright state (f state in CaF) into high-dark states. I assume Stark mixing only destroys the quantum numbers, but does not mix λ (projection of on the molecular axis). This assumption is valid for the Rydberg states that can be described by Hund case d. Detailed discussions of this assumption and a more general calculation method can be found in Petrovic's thesis [167]. With this assumption, all other electronic states with |λ| =|λ initial | or <|λ initial | are not involved. Due to the small applied electric field, all states with quantum defect δ >0.2 are excluded from this Stark-mixing process. As mentioned in Section 6.1, the strong propensity rules prevent N + , v + mixing also.
• Selectively populate one Stark state with chosen nominal " " character using a 10 ns pulsed tunable laser or >100 ns pulse amplified cw diode laser.
• Ramp down the electric field to zero at a maximum rate determined by a simple calculation. The selected " " state evolves adiabatically to a field-free state with pure character.
Several key features of this technique must be emphasized: • Although bright states with bright <4 usually have very short lifetimes (e.g., CaF fσ state, 1ns), the strong Stark mixing results in dilution of the fast decay rate into nearly all >4 dark states. This dilution effect increases the effective lifetime approximately by a factor of 100.
• The laser power used to populate Stark-mixed states must be higher than usual, because most Stark states have only a small amount of bright state character.
Usually, small | − bright | states have larger bright character. However, if the sign of the quantum defect of a target state is different from that of the bright state, the mixing angle will be negligibly small (e.g., the hφ, iφ, and kφ states of CaF have quantum defects opposite in sign to that of fφ, which is predicted by long-range model of CaF). To fill in this gap, I can use a mm/rf pulse to induce transitions into such states during/after the pulsed Stark field.
• As discussed above, the laser powers required for different Stark states compared to that for the field-free bright states are much larger. Based on the discussion in Section 6.2.3, a 10 ns 1 GHz bandwidth pulsed dye laser with sufficient power to populate Stark states might cause significant multi-photon ionization.
However, a >100 ns Fourier-Transform limited laser pulse populate the Stark states efficiently with minimal multi-photon ionization. The lifetimes of the Stark states at maximum electric field are sufficiently long to accommodate the long pulse laser excitations.
• The maximum rate at which the electric field must be decreased toward zero depends on nonadiabatic loss and dissociation decay loss. Minimizing both of these loss mechanisms make opposite demands on the ramp rate of the Stark field. The nonadiabatic loss rate is described by the Landau-Zener formula [168], as Eq.(6.22). The nonadiabatic transition probability is smaller if the ramp rate is smaller. The dissociation loss can be calculated by the effective decay rate, which is determined by the imaginary part of the diagonalized non-Hermitian Hamiltonian. The dissociation loss is smaller if the ramp rate is larger. The ramped rate needs to be a compromise of two opposite considerations, P = exp −2π |V | 2 |dE/dt| , (6.22) where V is the Stark interaction matrix element and dE/dt is the rate of change of the Stark energy.
• For most experiments with an external electric field in Rydberg states system, uncompensated stray electric fields limit the -purity of the final states [55].

Calculation results of ASTRO
In the above section, I describe the recipe for implementing the ASTRO scheme and discuss the general experimental considerations. In this section, I use CaF as a model system to demonstrate the feasibility of ASTRO by calculation [169,13]. The quantum defects and lifetimes of the |λ| = 1 Rydberg states of CaF with =4 to =7 are listed in Table 6.3. Because of their large quantum detects, states with smaller cannot be mixed by a 10 V/cm Stark field and are not included in this calculation.
States with larger are degenerate, because µ=0 and have infinite lifetime. The estimated electric dipole transition moments are calculated by a numerical method for non-hydrogenic Rydberg states. Thus, I construct a 32 × 32 non-Hermitian Hamiltonian for n=35, π states. As described in Section 6.3.1, the energies of the Stark states and their effective lifetimes are obtained from the real and imaginary parts of the diagonalized Hamiltonian, as shown in Figure 6-13. The mixing coefficient of f in each Stark state can be used to estimate the required excitation laser power, as in the fourth column in Table 6.3.  field, the dissociation decay rate is very large. In another range, it is almost zero.
This suggests designing a non-linearly ramped electric field. The ramped field has a relatively fast ramp rate in the fast dissociation decay regime to avoid dissociation loss, and has a relatively slow ramp rate in the slow dissociation decay regime to avoid nonadiabatic loss. Figure 6-15 shows the relative behaviors of both the nonadiabatic loss rate and the lifetime as a function of field. Compared to a linear ramp, it is more efficient to use a nonlinear ramp that decreases rapidly at large fields, and slowly at smaller fields, in order to obtain the transfer efficiencies shown in the sixth and seventh columns of Table 6.3.
The final discussion in Section 6.3.2 concerns the effects of a stray electric field.
The electric field efficiently couples near-degenerate states. If the ASTRO technique is applied to high-states, the requirement of minimizing the stray electric should be very strict. The last column in Table 6.3 lists the tolerance to stray electric fields for different, accessible states. For practical experiments, the chosen state must have a small, but non-zero quantum defect (0.005 < mod(µ) < 0.1). Thus ASTRO allows access only to those states in which I am most interested: the core nonpenetrating states with relatively small, non-zero quantum defects.
Including all of the considerations above, I ran a complete simulation to display the entire procedure of the ASTRO scheme, as shown in Figure 6-15. I find that: initially, the 10 V/cm DC electric field mixes a wide range of Stark states. With an Figure 6-14: Electric field-dependent dissociation lifetime and effective upper limit of the ramp rate for an adiabatic process. These two curves suggest the use of faster ramp at high electric field and a slower ramp rate at low electric field. appropriate non-linear ramped field, multiple Stark states focus to a single field-free core-nonpenetrating Rydberg state. With a moderately fast dissociation decay rate of the f state (τ ∼1 ns), the population transfer efficiency into g, h and i states can be more than 50%.
Appendix A

Rectangular Helmholtz coils
The magnetic field spatial distribution of the rectangular Helmholtz coils can be