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ATOMIC AND MOLECULAR ACCELERATION VIA ONE DIMENSIONAL
OPTICAL LATTICE
By
Taylor Clark Lilly
Table of Contents
Tables .............................................................................................................................................................iv
Figures ............................................................................................................................................................. v
1
2
Motivation and Current Research ........................................................................................................... 1
1.1
Introduction.................................................................................................................................... 1
1.2
Related Research and Experiments ................................................................................................ 5
1.2.1
Optical Molasses ....................................................................................................................... 6
1.2.2
Magneto-Optical Traps (MOT) ................................................................................................. 7
1.2.3
Dipole Traps .............................................................................................................................. 8
1.2.4
Atom Optics ............................................................................................................................ 11
1.2.5
Optical Lattices – Low Density Gas ........................................................................................ 14
1.2.6
Optical Lattices – High Density Gas ....................................................................................... 16
Theory .................................................................................................................................................. 17
2.1
Laser Interference ........................................................................................................................ 17
2.2
Quantum Mechanics and the semi-Classical Approach ............................................................... 19
2.2.1
Schrödinger and Hamiltonian .................................................................................................. 19
2.2.2
Two Level Atom in an Oscillating Electric Field .................................................................... 22
2.2.3
Rabi Frequency........................................................................................................................ 22
2.2.4
State Probability – Coherent Evolution for a Single Laser ...................................................... 23
2.2.5
Spontaneous Emission and State Lifetime .............................................................................. 24
2.2.6
Natural Width .......................................................................................................................... 26
2.3
2.3.1
Density Matrix ......................................................................................................................... 26
2.3.2
OBE ......................................................................................................................................... 27
2.3.3
Saturation and Power Broadening ........................................................................................... 28
2.4
3
Optical Bloch Equations (OBE)................................................................................................... 26
Forces on an Atom in an Optical Lattice ..................................................................................... 29
2.4.1
Near-Resonant Forces on a Two Level Atom ......................................................................... 29
2.4.2
Moving Atom in a Standing Wave .......................................................................................... 30
2.4.3
Non-Resonant Optical Lattice ................................................................................................. 31
Proposed High Speed Investigation ...................................................................................................... 33
3.1
3.1.1
Experimental Setup ...................................................................................................................... 33
Apparatus................................................................................................................................. 33
ii
3.1.2
Atomic Beam Formation ......................................................................................................... 33
3.1.3
Hot Wire Detector and Channeltron ........................................................................................ 38
3.1.4
Accelerating Optical Lattice .................................................................................................... 39
3.2
4
Proposed High Temperature Investigation ........................................................................................... 43
4.1
5
Numerical Accompaniment ......................................................................................................... 41
Theoretical Framework ................................................................................................................ 43
4.1.1
Single Pulse ............................................................................................................................. 43
4.1.2
Multiple Pulses ........................................................................................................................ 44
4.2
Simulation Method – Preliminary Simulations ............................................................................ 45
4.3
Results and Discussion – Preliminary Simulations ...................................................................... 47
4.4
Possible Experimental Setup – Acoustic Measurement ............................................................... 53
Statement of Work................................................................................................................................ 56
5.1
High-Speed Experimental Development ..................................................................................... 56
5.2
High-Speed Numerical Development .......................................................................................... 56
5.3
High-Temperature Numerical Development................................................................................ 56
5.4
High-Temperature Experimental Development ........................................................................... 56
6
Bibliography ......................................................................................................................................... 57
7
Empirical Values and Their Sources .................................................................................................... 61
7.1
Cesium Properties ........................................................................................................................ 61
7.1.1
Ionization Potential .................................................................................................................. 61
7.1.2
Vapor Pressure ........................................................................................................................ 61
7.1.3
Collision Cross Section ........................................................................................................... 61
7.1.4
Transitions ............................................................................................................................... 62
iii
Tables
Table 2-1: Cesium D2-line information ......................................................................................................... 26
Table 3-1: Cesium atomic beam characteristics ............................................................................................ 38
Table 7-1: 1st Ionization potential of Cs ........................................................................................................ 61
Table 7-2: Vapor pressure of solid Cs ........................................................................................................... 61
Table 7-3: Vapor pressure of liquid Cs ......................................................................................................... 61
Table 7-4: Cs-Cs total collision cross section ............................................................................................... 61
Table 7-5: Cs 6p 2P3/2 ↔ 6s 2S1/2 transition frequency .................................................................................. 62
Table 7-6: Cs 6p 2P3/2 → 6s 2S1/2 transition lifetime ...................................................................................... 62
iv
Figures
Figure 1-1: Conceptual diagram of optical lattice-atom interaction ................................................................ 2
Figure 1-2: Proposed atomic acceleration experiment .................................................................................... 4
Figure 1-3: Notional diagram of a 3D optical molasses (Shaffer 2008) ..........................................................6
Figure 1-4: Conceptual diagram of the spatial constriction process in a MOT (Shaffer 2008) ....................... 7
Figure 1-5: Cs2 trapping in a radial dipole trap (Takekoshi, Patterson and Knize 1998) ................................ 9
Figure 1-6: Rb potential within a Laguerre-Gaussian beam profile (Kuga, et al. 1997) ................................. 9
Figure 1-7: Diagram of doughnut beam dipole trap experiment (Kuga, et al. 1997) .................................... 10
Figure 1-8: Notional 3D diagram of an axial dipole potential within an optical lattice (Alt 2004) ............... 11
Figure 1-9: Pulsed deflection of I2 and CS2 by the radial dipole force (Sakai, et al. 1998) ........................... 12
Figure 1-10: Deceleration of C6H6 by a non-resonant pulsed laser (Fulton, Bishop and Barker 2004) ........ 13
Figure 1-11: Optical lattice diffraction of an Na atomic beam (Moskowitz, et al. 1983) .............................. 14
Figure 1-12: Transposition of a single Cs atom in a standing wave dipole trap (optical lattice)
(Schrader, et al. 2001) .................................................................................................................... 15
Figure 1-13: Likelihood of an atom staying in an accelerated optical lattice (Schrader, et al. 2001) ............ 15
Figure 1-14: Velocity distribution for a pulse accelerated ensemble of CH4 (Barker and Shneider 2001) ... 16
Figure 2-1: Probability for an atom to be in the excited state versus 1/Γ, for various Ω
(Metcalf and van der Straten 1999) ................................................................................................ 24
Figure 2-2: Probability for an atom to be in the excited state versus 1/Γ, including spontaneous emission
(Metcalf and van der Straten 1999) ................................................................................................ 25
Figure 2-3: Cs D2 natural line shape.............................................................................................................. 26
Figure 2-4: Probability for an atom to be in the excited state versus τ, using the OBE
(Metcalf and van der Straten 1999) ................................................................................................ 28
Figure 2-5: Axial dipole force vs position, PLaser=100 mW, DLaser=1 mm, ωL= ω0-50Γ ............................... 31
Figure 3-1: Conceptual drawing of experimental apparatus .......................................................................... 33
Figure 3-2: Radial velocity distribution at the oven orifice (y or z axis) ....................................................... 36
Figure 3-3: Axial velocity distribution at the oven orifice (x axis) ............................................................... 36
Figure 3-4: Radial velocity distribution at the downstream orifice (y or z axis) ........................................... 37
Figure 3-5: Axial velocity distribution at the downstream orifice (x axis).................................................... 37
Figure 3-6: Spatial distribution at the downstream orifice (r=sqrt(y2+z2)) .................................................... 37
Figure 3-7: Expected detector signal vs transverse location (y axis) (MC) ................................................... 39
Figure 3-8: SIMIon calculation of 0.2 eV ion trajectories over 2π steradian ................................................ 39
Figure 3-9: Deflection vs start location, PLaser=100 mW, DLaser=1 mm, ωL= ω0-50Γ, a=1x106 m/s2 ............ 40
Figure 3-10: |Deflection vs initial velocity, PLaser=100 mW, DLaser=1 mm, ωL= ω0-50Γ, a=1x106 m/s2 ....... 41
v
Figure 4-1: Example of axial laser intensity [W/m2] for a 500 ps intervening time ...................................... 46
Figure 4-2: Translational (trn) and rotational (rot) temperature as a function of radial position ................... 47
Figure 4-3: Single pulse change in translational energy for varying initial temperature ............................... 48
Figure 4-4: Final temperature after 10 pulses as a function of intervening time for several gases................ 48
Figure 4-5: N2 final temperature after 10 pulses as a function of intervening time and pressure .................. 49
Figure 4-6: Final argon temperature for an intervening time of 0 ns............................................................. 50
Figure 4-7: Final argon temperature for an intervening time of 0.5 ns.......................................................... 51
Figure 4-8: Final argon temperature for an intervening time of 10 ns........................................................... 51
Figure 4-9: Argon pressure profile development for an intervening time of 10 ns ....................................... 52
Figure 4-10: N2 temperature as a function of pulse number for an intervening time of 1 ns......................... 52
Figure 4-11: Proposed acoustic measurement experimental laser setup ....................................................... 53
Figure 4-12: Ionization detection through microwave scattering experimental setup ................................... 54
Figure 4-13: Possible experimental acoustic wave detection gas cell ........................................................... 54
Figure 7-1: Comparison of published Cs vapor pressures ............................................................................. 61
vi
1
1.1
Motivation and Current Research
Introduction
The interaction of intense laser fields with atoms or molecules offers a strong, tunable method for
the remote control of atomic or molecular position, energy, and momentum. The application of lasers for
energy and momentum deposition can be used to address a number of high-speed and high-temperature
research questions associated with spacecraft interaction and space environment simulation. Lasers are
routinely used in low-temperature atomic physics.
However, their application towards energy and
momentum addition is contrary to their common use and has little experimental precedent (Shneider,
Barker and Gimelshein, Molecular transport in pulsed optical lattices 2007).
By building on the
fundamentals established by existing research, this study proposes to provide a proof of concept for using
high intensity laser fields for the creation of high enthalpy flows (high-speed / high-temperature). This
study will use experimental and numerical techniques to provide evidence for the applicability of the
concept to aerospace problems and offer a potential path for the realization of a production process for the
creation of such flows.
An atom or molecule subjected to an electric field will have a dipole moment induced by the
field’s distortion of its electronic structure. In turn, this moment has a potential energy associated with
immersion in the electric field. The specifics of this moment and potential are affected by resonances
within the atom or molecule and the frequency of the radiation field, but the principle is the same regardless
of radiation frequency. A spatial gradient in the field strength will yield a force pushing the atom or
molecule towards areas of least potential. The electric field of an interference pattern formed by two
counter-propagating laser fields takes the shape of a standing wave, or “optical lattice.” This lattice
oscillates over a period of half the constituent laser’s wavelength, yielding a strong spatial gradient which is
available to do work. By modulating the constituent laser frequencies, the lattice can be put in motion,
dragging trapped atoms or molecules along with it. Experimentally, a single atom has been subjected to
accelerations as high as 105 m/s2 using this method (Schrader, et al. 2001). Additionally, if the density of
the gas is high enough, the effect of intermolecular collisions offers the opportunity to continually deposit
1
kinetic energy into the gas. Numerical simulations suggest that this energy addition can increase the
average molecular temperature over 500 K for a single 1 ns laser pulse (Ngalande, Gimelshein and
Shneider 2007). A conceptual diagram of these processes can be seen in Figure 1-1.
Potential Depth
Stationary Potential
Accelerating Potential
Collision & Energy Transfer Oscillating Trapped Atom
Higher Density
Lower Density
Figure 1-1: Conceptual diagram of optical lattice-atom interaction
Aerospace research requires high fidelity terrestrial simulation of on-orbit and reentry flow
conditions. For on-orbit conditions, accurate reproduction of the incident flow on a spacecraft entails
neutral mono-energetic atomic beams at incident velocities on the order of 8 km/s (5 eV for O). Current
methods for creating such a flow include ionization, acceleration, and charge exchange (Livingston and
Blewett 1962) and laser heated nozzle expansions (Caledonia, et al. 1990). For ionization and charge
exchange, a gas is first dissociated and ionized using a discharge. The ions are accelerated and passed
through a charge exchange cell to neutralize the beam.
The drawbacks with this method include
complication with the ionization process due to space-charge limitations and the charge exchange process
due to scattering (Ketsdever 1995). For laser heated nozzle expansions, a high intensity laser breaks down
the gas to ionize and partially dissociate the molecular species. The ions readily absorb energy from the
laser field which in turn heats the gas. The expanding gas quickly recombines charged species to selfneutralize but still results in a mixed flow of atomic and molecular components (Minton, et al. 1991). The
proposed study will investigate the possibility of using laser fields below the breakdown threshold of the
gas to accomplish the ideal goal of neutral mono-energetic atomic and molecular beams with predictable
and easily changeable properties, thus avoiding the drawbacks of existing approaches.
The second area of interest, in addition to the creation of neutral beams, is energy deposition into
high-density gases. The interest here is aimed at large temperature increases in the target gas without
noticeable ionization.
For reentry conditions, the understanding of high-temperature, strongly non-
equilibrium gas flows surrounding the vehicle requires accurate data for the dominant energy transfer
2
processes and chemical reactions. This data is required in terms of both energy dependent cross sections
and temperature dependent rates (Park 1990) (Anderson 2006). Current methods for investigating these
flows involve the use of combustion, shock facilities (Jerig, Thielen and Roth 1990), and arc jets (Matsui,
et al. 2004). Combustion involves unwanted chemical species and reactions, shock facilities suffer from
temporal limitations in measurement (da Silva, Guerra and Loureiro 2006), and arc jets rely on ionization
and plasma heating which may affect the measurement of interest. The proposed laser energy deposition
technique only couples with translational energy modes of the gas allowing for well characterized initial
conditions. Such initial conditions are advantageous for the measurement of chemical reaction rates
without reliance on ionization or the formation of strong shocks.
The proposed study will begin by bridging the basis of research already utilizing near-resonant and
non-resonant atom-laser interactions with the application of energy and momentum deposition. This
application is contrary to the common application of removing energy and momentum from a gas in laser
cooling. The manipulation of atoms and molecules by light fields has been used extensively by the low
temperature atomic physics community (Karczmarek, et al. 1999), (Stapelfeldt, et al. 1997), and (Dall, et al.
1999). The near-resonant dissipative interaction of photon scattering and the non-dissipative forces related
to the dipole potential lend themselves to the requirements of energy reduction and tight spatial
confinement of single and ensembles of atoms. The non-resonant corollary has also been demonstrated for
atom trapping (Miller, Cline and Heinzen 1993). These well established techniques are routinely used to
provide cold atoms for use in spectroscopy, quantum computing (Steuernagel 2005), and quantum collision
experimentation (Stamper-Kurn, et al. 1998). The dipole force has also been demonstrated to affect atom
beam velocities through fiber guides (Renn, et al. 1995) and atom optics (Martin, et al. 1988). Research in
this field tends to use continuous wave lasers and alkali atoms to take advantage of well defined and
isolated electronic transitions in the atom which facilitate cooling through spontaneous emission and
confinement through resonance enhanced dipole forces.
To prove the concept of using lasers for the creation of high speed beams, the proposed study will
use counter-propagating, near-resonant laser fields to add transverse momentum to a thermal cesium beam.
The beam will originate in an oven, be collimated using an orifice, pass through a laser interaction region
(optical lattice) and be diagnosed by a hot wire detector and channeltron.
A single near-resonant
3
continuous wave laser will be split into two beams and passed through frequency modulators before being
interfered in the interaction region; these will form the optical lattice. As the atomic beam passes through
the lattice, atoms with relative energies less than the potential well depth will be restricted (in the laser axial
direction) within the standing wave potential pattern. By modulating the constituent beam frequencies the
motion of the lattice can be controlled. If the standing wave is accelerated, the atoms which are restricted
by the potential field will be deflected along the laser propagation axis. A diagram of the proposed
experiment can be seen in Figure 1-2.
Figure 1-2: Proposed atomic acceleration experiment
A companion numerical study will be conducted in order to improve understanding of the details
of the laser–gas interaction, as well as provide a tool for the optimization of the experimental setup and a
parametric study of the important influences. The nature of the proposed research, which includes the
modification of the distribution function of the gas molecules, implies that a kinetic, microscopic approach
will be used for any numerical investigation. Depending on the flow regime, a free molecule or collisional
kinetic solver will be used. For nearly-free molecule regime, a particle Monte-Carlo based numerical
simulation will be developed to predict the atomic trajectories of the cesium beam. Using the results of the
experiment to validate the numerical simulation, the developed code will be used to ascertain the setup
required to increase atomic velocities on the order of 8 km/sec, comparable to low earth orbit velocity. The
code will also be used to provide predictions of the flexibility and limitations of the process regarding its
production complexity, atomic fluxes, and energies.
In regards to using the dipole potential force for creating high temperature flows, the direct
simulation Monte Carlo (DSMC) based code SMILE will be adapted to include the non-resonant laser
interaction.
A modified version of the SMILE code will be used to simulate various gas and laser
4
parameters and experimental geometries to find an optimal experimental configuration. Two possible
experimental measurements will be investigated and optimized using numerical simulation. The first
measurement is a gas temperature increase in a cell due to repeated interaction with a pulsed optical lattice.
The temperature increase directly relates to the energy being deposited into the gas by the lattice. The
second is the measurement of a pressure pulse originating from the expansion of a heated gas volume
following its interaction with a single laser pulse. The measurement of the magnitude and dissipation of
the pulse can be used to ascertain the energy deposited to the gas. Based on the simulated efficiency of the
energy deposition of these two possible measurements, an experiment will be designed and performed.
1.2
Related Research and Experiments
The fine control of atoms and molecules by way of laser fields has its roots in low temperature
atomic physics. From this field grows application towards quantum computing and fundamental research
in Bose-Einstein condensate and quantum collisions. There is also a strong thread of research for the
control of atomic beams in a process analogously called “atom optics.” As the two areas are linked by the
laser interaction, they both will act as a basis from which the proposed study will set forth. There is a wide
range of phenomena, processes, and techniques which are associated with these two fields and must be
considered, if only to ascertain their applicability and the level of their effect. In order to recognize the
applicability of these processes, a look at past and present laser-atom and laser-molecule interaction
experiments is presented.
There are two ways of categorizing laser-particle interaction processes: by purpose and by laser
interaction regime. Most laser-particle experiments are performed with one of two purposes. The first is to
remotely control the particle’s momentum and energy. These techniques act as damping or dissipative
tools to “cool” the particle(s). The second purpose is that of spatial confinement or transposition. These
techniques are ideally conservative as to not add unintended energy to an already “cooled” system. In
regards to the interaction regime, the frequency of the laser in relation to resonances in the atom or
molecule has a strong impact on the form of the interaction. In this way processes may be described as
being in the resonant or non-resonant, i.e. far from resonance, regime.
5
1.2.1
Optical Molasses
The primary method for laser cooling is that of “optical molasses.” This is a near-resonant process
for momentum reduction.
In this technique atoms are cooled by velocity selective absorption and
spontaneous emission of Doppler shifted photons. The name of the technique was coined by the first
experimenters who likened the process to a particle in a viscous fluid (Chu, et al. 1985). The absorption
and emission process utilized for molasses cooling is present whenever operating in a near resonant regime.
The setup for 3D optical molasses entails six lasers arranged in three orthogonal counter-propagating pairs.
If all of the lasers are slightly detuned to the red (longer wavelength) from an electronic resonance the
Doppler shift resulting from their velocity towards a particular laser will cause the atom to absorb more
photons from that laser than its anti-parallel partner. The increase in absorbed photons increases the
momentum transfer from that laser’s beam to the atom, slowing the atom. It is important to note that
optical molasses does not spatially confine, it affects only the velocity of the atom. A notional diagram of
this setup can be seen in Figure 1-3.
Figure 1-3: Notional diagram of a 3D optical molasses
(Shaffer 2008)
This technique, using only one laser, can be applied to slow or accelerate an atomic beam instead
of a three dimensional cloud. An atomic beam of sodium was decelerated using a counter propagating laser
by frequency sweeping the laser to maintain an optimal Doppler shift throughout the process (Ertmer, et al.
1985). The beam was slowed from several hundred m/s to less than 90 m/s. Conversely, there is no reason
why an opposite frequency shift could not accelerate a beam to several hundred m/s. The shortcoming of
this technique is that the force imparted on the atoms is limited by the resonant photon absorption/emission
cycle which saturates as the population (ground/excited) of the atoms equalizes, thus there exists a
6
theoretical maximum force on the atom which is independent of the intensity of the laser. This limits the
time and distance required for acceleration and deceleration of an ensemble of atoms or molecules via the
scattering force (the force present in an optical molasses).
1.2.2
Magneto-Optical Traps (MOT)
The addition of a strong magnetic field to optical molasses allows for the utilization of Zeeman
shifting of the atom’s magnetic sublevels. Whereas Doppler shifting changes the likelihood of absorption
by shifting the photon frequency, Zeeman shifting changes the likelihood of absorption by shifting the
distribution of accepted photons by changing the energy of the transition. This technique augments an
optical molasses setup such that an atom not only preferentially absorbs photons due to velocity, but
position as well. It therefore offers a method to add spatial confinement to the near-resonant momentum
reduction. Such a setup is referred to as a Magneto-Optical Trap (MOT). The shape of the magnetic field
affects the probability field of the atom with regards to photon absorption. In a MOT, circularly polarized
light is required to couple to the appropriate magnetic sublevel to the appropriate laser and give the force
the desired directionality.
With a suitably shaped and sufficiently strong magnetic field, an optical
molasses laser configuration may act as both a cooling mechanism and a confinement force. The MOT is a
fairly standard apparatus to cool and constrict atoms from a background gas to central region for use in
other experiments. This may be useful to the proposed study if future experiments require a source for
dense cool atoms as a starting condition before acceleration. A conceptual diagram of the MOT spatial
confinement process in one axis can be seen in Figure 1-4.
Figure 1-4: Conceptual diagram of the spatial constriction process in a MOT
(Shaffer 2008)
7
1.2.3
Dipole Traps
As mentioned, in the presence of a strong electric field a dipole potential is induced on an atom.
This dipole potential can be exploited in a spatial varying electric field, such as a focused laser, to produce
a useful force on the atom or molecule. This dipole force is generally non-dissipative and can be used for
confinement and transposition of atoms without adding energy to the system. A complete review of dipole
traps can be found in (Grimm, Weidmüller and Ovchinnikov 1999).
1.2.3.1
Radial Dipole Traps
The radial intensity variance in a laser beam leads to a spatial gradient in the induced dipole
potential of an atom within that beam in the beam’s radial direction. This force, for red detuned lasers,
causes atoms to be forced towards areas of higher intensity (less potential), for a Gaussian beam this is the
core of the beam. There is no cooling of the atom due specifically to the dipole force, but other optical
techniques, such as molasses or Sisyphus cooling, can be used in conjunction with the dipole trap to
increase the phase space density of trapped atoms. This technique may be used with resonant lasers, but is
routinely used with far-from-resonance light to avoid complication due to photon scattering.
As an example, cesium dimmers (Cs2) have been trapped in this manner (Takekoshi, Patterson and
Knize 1998). A 17 W 10,600 nm (non-resonant) CO2 laser was focused to a waist diameter of 64 μm. The
pinching of the focus and the Gaussian shape of the beam effectively creates a three dimensional force
which pushes the molecules towards the focal point of the laser. Figure 1-5 shows a diagram of the
experimental setup. The CO2 beam is the radial dipole trap used to trap molecules which have been cooled
and isolated within a MOT. In this experiment other lasers were used to accomplish molecule ejection and
ionization for detection. This experiment utilizes the same dipole force that will be used in the proposed
experiment; however, there are few other similarities. This experiment is tuned far off-resonance to reduce
photon scattering and interference with the MOT which is used for the creation of the Cs2. By doing so the
magnitude of the dipole force is reduced relative to the laser intensity thus a powerful laser is required to
impart an appreciable force.
8
Figure 1-5: Cs2 trapping in a radial dipole trap
(Takekoshi, Patterson and Knize 1998)
If the frequency of the laser used was tuned to the blue side of an electronic transition instead of
the red, the sign of the force is reversed. The atom is then pushed away from regions of high intensity
towards those of low. If this were applied to a traditional Gaussian beam profile, the atoms would be
repelled from the center and the laser would act as an anti-trap. However, if the shape of the beam were to
look more like a doughnut, the minima in the center of the profile would act as a good dipole trap. In
addition to trapping, this shape has the advantage of placing the atoms in a region of low laser intensity
which reduces photon scattering, even near-resonance. An example of such a beam profile can be seen in
Figure 1-6. In this figure, the potential energy of a Rb atom in the beam is given as an equivalent
temperature, T ∝ E / k .
Figure 1-6: Rb potential within a Laguerre-Gaussian beam profile
(Kuga, et al. 1997)
Rb atoms have been trapped in such a “doughnut” trap using a Laguerre-Gaussian beam profile.
In this experiment, a 600 mW beam was focused to a radius of 600 μm. The Rb atoms were first cooled
and loaded into a MOT, then transferred to the trap. Because there is no axial confinement, as there is with
a pinched red-detuned dipole trap, the main trap beam was recovered, split, and brought through the ends of
the trap as plug beams. A diagram of the experimental trapping setup can be seen in Figure 1-7. This
experiment reinforces the versatility of using tailored light as an atom manipulator. Traditional cooling and
9
trapping setups use red-detuned Gaussian lasers because the combination yields an advantageous setup for
cooling and spatial confinement. This experiment shows that convention does not limit possibility. While
designing a new experiment, it is important to build on the experience of other experiments, including those
which choose opposing paths to the norm.
Figure 1-7: Diagram of doughnut beam dipole trap experiment
(Kuga, et al. 1997)
1.2.3.2
Axial Dipole Traps
For two counter-propagating lasers with the same frequency which are co-linearly polarized, the
electric field formed by their interference is a standing wave when averaged over the fast oscillating
(optical) terms. The same dipole force which pushes the atom to the center of a red-detuned radial dipole
trap now pushes the atoms towards the anti-nodes of the standing wave, where the light intensity is highest.
In comparison with radial gradients which vary over a distance of 10’s of μm. The standing wave formed
by visible lasers varies over fractions of that. These strong gradients allow for trapping of atoms with a
spatial confinement on the order of 200 nm. A diagram of the axial periodic dipole potential of an atom in
such an interference pattern can be seen in Figure 1-8 with the axial direction stretched 1500 times to show
individual wells. The shape of the wells comes from the interference pattern, the radial Gaussian shape of
the two beams, and if the beams are co-focal instead of collimated there is an additional shape due to the
axial variation in intensity. This is the cold atom basis for the proposed study. If the frequency of the two
lasers is slightly shifted, the entire interference pattern is put in motion. If the acceleration associated with
that motion is small enough (compared to the depth of the trap), the trapped atoms continue to be trapped
and are accelerated with the pattern. Again, this technique can be used with non-resonant or resonant light
and is primarily used for spatial confinement and not for cooling.
10
Figure 1-8: Notional 3D diagram of an axial dipole potential within an optical lattice
(Alt 2004)
1.2.4
Atom Optics
As opposed to creating long residence times within a dipole trap, the dipole force has also been
used for the purpose of controlling particle trajectories in a manner similar to controlling rays of light. In a
bit of role reversal, light is be used to diffract atomic beams much the same way that atoms are used to
diffract light. In this setup the atom is subjected to the laser for only a brief amount of time (the transit
time). In most cases there is little to no energy addition to the atom, it is simply redirected (diffracted). In
this manner the dipole force technically is being applied as a momentum adjuster instead of for spatial
confinement.
1.2.4.1
Radial Dipole Atom Optics
Iodine (I2) and carbon disulfide (CS2) molecules have been deflected by non-resonant pulsed
lasers using the radial dipole force (Sakai, et al. 1998). In these experiments two different lasers were used
independently to deflect an off axis jet from a pulsed micro-nozzle. The molecules where then ionized with
a third laser through multi-photon ionization and their time of flight detected by a set of multi-channel
plates. The first laser used was an Nd:YAG at 1064 nm with a pulse width of 14 ns and an energy of 10
mJ. This laser was focused to 7 μm which yields a peak intensity of ~9x1015 W/m2. The second laser was
a CO2 laser at 10,600 nm with a pulse width of 70 ns and an energy of 600 mJ. This laser was focused to
35 μm which yields a peak intensity of ~4.5x1015 W/m2. The velocity profile of the deflected molecules
can be seen in Figure 1-9. The entrance trajectory of the molecules and the location of the ionization beam
are also shown in the figure.
11
Figure 1-9: Pulsed deflection of I2 and CS2 by the radial dipole force
(Sakai, et al. 1998)
This work has the same intension as the proposed experimental study, to impart momentum to
particles. Unlike the proposed study, this work uses the radial dependence of one laser instead of the axial
interference of two lasers and is only capable of deflecting the atoms without adding additional kinetic
energy. This work also uses non-resonant lasers which have a weaker interaction proportional to the laser
intensity. Because of a weaker interaction, this work uses pulsed lasers instead of CW lasers to recuperate
the diminished force through increased intensity. This work acts as a pulsed central field deflector
analogous to a gravity turn of a spacecraft around a planet or the focusing of light rays through a lens.
A similar work slowed benzene molecules with a pulsed far off-resonance laser (Fulton, Bishop
and Barker 2004). The setup was similar to the above work with similar results; however, the goal in this
case was to cause a linear deceleration, not a redirection. The atoms were directed through the center of the
beam instead of around the core, slowing molecules as they traversed the beam diameter. An example of
their measured velocity profile can be seen in Figure 1-10. This work has direct connection to both the
acceleration of atoms for high speed flows and for energy deposition into a gas for high temperatures.
Although the pulse width was sort the reported deceleration was upwards of 108 g. This is an indication of
the potential of using the dipole force for accelerating atoms and molecules. This experiment also points
out the boundary for using this technique. It edges closer to the regime where the laser intensity will
breakdown the irradiated molecules, defeating the neutral goal of the process.
12
Figure 1-10: Deceleration of C6H6 by a non-resonant pulsed laser
(Fulton, Bishop and Barker 2004)
1.2.4.2
Axial Dipole Atom Optics
Instead of passing an atomic beam through a single laser, the interference pattern (optical lattice)
from two counter-propagating lasers can create a diffraction grating from which to scatter the beam over.
A single 0-20 mW on-resonance laser was retro-reflected and focused to 25 μm to diffract an atomic beam
of sodium (Moskowitz, et al. 1983). The laser standing wave added transverse momentum to the atoms
depending on the laser power (electric field strength). An example of the diffraction pattern can be seen in
Figure 1-11. This experiment holds particular relevance to the proposed cesium atomic beam experiment.
The proposed experiment will utilize a moving optical lattice to accelerate the atoms, however if the lattice
is left stationary, the same transverse momentum addition should still be present. The momentum addition
is not large, but may be measureable at the edge of the atomic beam transverse velocity distribution.
13
Figure 1-11: Optical lattice diffraction of an Na atomic beam
(Moskowitz, et al. 1983)
1.2.5
Optical Lattices – Low Density Gas
When two counter propagating lasers are crossed, the interference pattern created is a standing
wave. If three such standing waves are set up orthogonally to each other, the resulting three dimensional
periodic potential pattern is referred to as an optical lattice. This lattice can be used for atomic confinement
and sorting and is of particular interest to quantum computing. A one dimensional optical lattice, i.e. one
pair of lasers creating a standing wave, can be utilized to confine particles by combing the radial dipole
force and the axial standing wave. One such 1D optical lattice, tuned near atomic resonance, will be used
in the proposed study for the acceleration of particles.
1.2.5.1
Optical Conveyor
There is precedence for the use of optical lattices for particle transposition, though always with
lasers tuned far from resonance (Alt 2004),(Kuhr, et al. 2001),(Miroshnychenko, et al. 2006). There are
two reasons for staying away from resonance. The first is that the scattering of photons, thus random
heating, is inversely related to laser detuning, which makes large detuning advantageous for low
temperature experiments. Secondly, to make the interference pattern move the lasers must be detuned from
each other. By tuning far from resonance, the laser delta detuning from each other can be considered small
in relation to the detuning from resonance. This allows a simplifying assumption in the force derivation
which lets you treat the system as being constructed of only one laser frequency.
In (Schrader, et al. 2001) an atom is first loaded into a MOT before being transferred to a one
dimensional optical lattice. The lattice is then shifted from position A to position B. The lattice is then
shifted back from B to A and the atom is reloaded into the MOT. While in the MOT (A) and at position B
the atom can be irradiated with resonant light to ascertain its existence and position through fluorescence.
A breakdown of the particle transposition can be seen in Figure 1-12.
14
Figure 1-12: Transposition of a single Cs atom in a standing wave dipole trap (optical lattice)
(Schrader, et al. 2001)
The acceleration imparted on the atom was varied over four orders of magnitude. This gives reasonable
indication as to the usefulness of this technique for high velocity flows. The measured efficiency, given by
the likelihood that the atom stayed in the lattice, can be seen in Figure 1-13.
Figure 1-13: Likelihood of an atom staying in an accelerated optical lattice
(Schrader, et al. 2001)
The largest differences between this work and the proposed experimental effort are the frequency
detuning of the lasers and the number of atoms considered. By working closer to resonance the magnitude
of the force on the atoms can be increased. This comes at the cost of increased heating and increased
complexity for the derivation of the applied forces. Without the ability to simplify the problem to one laser
15
frequency, many of the approximations are inapplicable. Secondly, the proposed experimental effort will
work with a beam of atoms, instead of individual atoms.
The physical difficulties associated with using lasers for atomic or molecular acceleration does not
hinder its numerical investigation. The possibility of using frequency swept pulsed optical lattices to
accelerate an ensemble of methane has been numerical investigated (Barker and Shneider 2001). In this
study two far-off resonance pulsed lasers are interfered and their frequency swept over several GHz in a
few nanoseconds. The study does not address the problem of how to physically implement the frequency
sweep nor the effect of that implementation on the interferemetric stability of the system. This study does
address the optimal results which could be expected from future non-resonant systems.
Figure 1-14: Velocity distribution for a pulse accelerated ensemble of CH4
(Barker and Shneider 2001)
1.2.6
Optical Lattices – High Density Gas
The investigation of energy deposition to a continuum gas using a pulsed optical lattice has not
been experimentally investigated but has been numerically simulated for several configurations. The
closest experiment to utilize pulsed optical lattices for a comparable goal is that of (Pan, Shneider and
Miles 2004). In that experiment a pulsed optical lattice is used to create density gradients in an already
heated gas. The gas density is low enough that the trapped molecules do not collide with each other (thus
not continuum). This creates a density grating which another laser may be scattered off The only
molecules which are trapped by this effect are those with relative velocities close to that of the moving
optical lattice. By measuring the intensity of the scattered light as a function of the lattice velocity, the
relative number density of molecules at a particular velocity can be detected.
16
2
Theory
The proposed study will utilize the force imparted on a particle in a spatially varying electric field
originating from an induced dipole potential on the atom. For the high-speed flow experiments, the particle
will be a cesium atom and the electric field will be created by two counter-propagating continuous wave
lasers tuned near the D2 transition (6s 2S1/2 – 6p 2P3/2). Cesium will be used because its electronic structure,
like other alkali metals, is amenable to being treated as a two level system which is a great simplification
for the quantum mechanical derivation of near resonant forces. Compared with other alkali, cesium’s
transitions are readily excited by available lasers (852 nm) and there is an extensive experimental history
from which to draw lessons. For the high-temperature flow experiments, the particle will be methane,
molecular nitrogen, or argon and the electric field will be created by two counter-propagating pulsed lasers
tuned far from a resonance (1064 nm or 532 nm). These gases have been selected because of their static
polarization, relevance to aerospace problems, and to represent monatomic, diatomic, and polyatomic
species for theoretical extrapolation of the process to other gases. The theory associated with these two
forces differs due to the proximity to resonance. For non-resonant interaction a number of simplifying
approximations are valid. Therefore the two theories will be addressed separately to allow for a fuller
description of the more complex cesium near-resonant interaction.
The resonant interaction will be discussed based on the derivations of previous authors on atomic
and molecular manipulation by laser fields (Foot 2005), (Letokhov 2007), (Metcalf and van der Straten
1999). Following these authors, the atom-laser interaction will use the semi-classical approach. As such,
the laser fields will be considered classical electromagnetic waves, while the atom will be approached as a
quantum mechanical system. The quantum mechanical derivations used are in line with (Liboff 1991) and
(Cohen-Tannoudji, Diu and Laloë 1977).
2.1
Laser Interference
It is assumed that the path length of the experimental setup is much less than the coherence length
of the lasers used. The bandwidth of frequencies emitted by a laser, or the line width, is denoted by ΔωL.
17
Assuming that all photons of all frequencies start in phase, the length at which the lowest frequency and the
highest frequency are 2π out of phase is called the coherence length and can be calculated by (Hecht 2002)
Δlc =
c
2π c
=
Δν L ΔωL
(2.1)
For a narrow line laser with a line width of approximately 1 MHz, the coherence length is approximately
300 m. For a table top experiment with path lengths on the order of 3 m, the laser can be considered
coherent and therefore treated as the interference of two monochromatic plane waves.
The electric field of a propagating laser beam can be denoted in several ways where kL is the wave
number, ωL is the frequency, and E0 is the amplitude of the electric field oscillation of the laser.
e i ( k L x + ωL t ) − e − i ( k L x + ω L t )
2i
e i ( k L x + ω L t ) + e − i ( k L x + ωL t )
E ( x, t ) = E0 cos(k L x + ωL t ) = E0
2
E ( x, t ) = E0 sin(k L x + ωL t ) = E0
(2.2)
The first of two possible interference patterns for two counter propagating laser fields is a standing
wave. This is created by crossing two laser of the exact same frequency (usually by splitting one laser and
bringing it back on itself or retro-reflecting off of a mirror). The superposition of their electric fields means
that their field amplitudes are added, not their intensity, which is proportional to E02. The square of the
summation of their electric fields is given by
E 2 ( x, t ) = { E0 sin( k L x + ω L t ) + E0 sin( − k L x + ω L t )} = 4 E02 cos 2 ( k L x ) sin 2 (ω L t )
2
(2.3)
It can be seen that the electric field is separated into a spatial oscillation and a temporal oscillation. By
averaging over an oscillation period, the resulting electric field is given by
E 2 ( x, t ) = 2 E02 cos 2 (kL x) = E02 (1 + cos(2kL x) )
(2.4)
This represents a spatially oscillating electric field with a periodicity of half the constituent laser
wavelength.
The second possible interference pattern is a moving standing wave. In this case a frequency
difference between the two lasers causes the interference pattern to go into motion. The square of the sum
of their electric field amplitudes is given by
18
2
2
E 2 ( x, t ) = EL1
cos 2 ( k L1 x − ωL1t ) + EL2
cos 2 ( kL2 x − ωL2 t )
+ EL1 EL2 ⎡⎣ cos ( ( k L1 − k L2 ) x − (ωL1 − ωL2 ) t ) + cos ( ( k L1 + k L2 ) x − (ωL1 + ωL2 ) t ) ⎤⎦
(2.5)
When kL1≈-kL2 and ωL1≈ωL2, the interference term of the field has two components: one with a relatively
long spatial and short temporal period and the other with a short spatial and long temporal period. When
the gradient of eqn. (2.5) is taken, the portion with the long spatial period has a negligible impact. In
addition, the fast oscillating terms (cos2) can be time averaged to a constant value.
2.2
2.2.1
Quantum Mechanics and the semi-Classical Approach
Schrödinger and Hamiltonian
It is a postulate of quantum physics that for a well-defined observable, X, there exists an operator,
X̂ , such that a measurement of X produces values, x, which are eigenvalues of the relation
X̂ ϕ = xϕ
(2.6)
The quantum mechanical eigenvalue equation for the energy of a system is expressed by the timeindependent Schrödinger equation as
G
G
Hˆ ϕ (r ) = Eϕ (r )
(2.7)
where Ĥ is the Hamiltonian operator and E is the eigenvalue (i.e. energy) for the eigenfunction
(wavefunction), φ, which is expressed as a function of position and momentum in the atom-centric frame
G
such that r =< x, y, z, px , p y , pz > . It is this equation that is solved for the electron-nucleus system to attain
the electronic energy levels of an atom in the nth state, En. The expression of this relation stems from
Hamiltonian mechanics. Hamiltonian mechanics is an alternate formulation of classical (Newtonian)
mechanics where the Hamiltonian operator is an expression of the total energy, kinetic and potential, of the
system. By virtue of its formulation, the time rate of change of position can be found by taking the
derivative of the Hamiltonian with respect to momentum. Likewise the time rate of change of momentum
can be found by the negative derivative of the Hamiltonian with respect to position (this latter statement is
of particular importance as it represents the Hamiltonian equivalent of the force due to a spatially
inhomogeneous potential field).
19
As we will be concerned with a system changing in time, the evolution of the wavefunction of the
system must be considered. The evolution of the wavefunciton is given by the time-dependent Schrödinger
equation
i=
∂
G
G
ψ (r , t ) = Hˆ ψ (r , t )
∂t
(2.8)
This equation sets the relation between the time rate of change of the wavefunction and the Hamiltonian of
the system at that time. If one assumes that the Hamiltonian is independent of time (as it relates to the
energy of a closed system), the solutions to time-dependent Schrödinger equation (2.8) can be found by
separation of variables. The solution must include the spatial solution to the time-independent Schrödinger
equation (2.7) as well as the temporal solution to an ODE of the form f’(t)+f(t)=0. Thus eigenfunctions
(wavefunctions) of equation (2.8) will take the form
G
G
ψ n (r , t ) = ϕn (r )e− iE t =
(2.9)
n
where Ψn is the time-dependent wavefunction of the atomic system and En is the energy of the nth state. By
the principle of superposition, the total state of the atom is given by the sum of all the available states
(wavefunctions) scaled by a proportionality constant. This is expressed as
G
G
G
ψ (r , t ) = ∑ cn (t )ψ n (r , t ) = ∑ cn (t )ϕ n (r )e −iE t =
n
n
(2.10)
n
where |cn(t)|2 is the probability of finding the system in nth state at time t, which is described by the
wavefunciton Ψn.
At this point, consider a form of notation which operates on two states, giving the coupling
between them. In the “bra-ket” notation the integration of the complex congregate of the nth state times the
modification (by an operator) of the kth state over all of the atom-centric space is given by
G
G G
ϕ n Xˆ ϕ k = ∫ ϕ n* ( r ) Xˆ ϕ k ( r )d 3 r
(2.11)
Because the solutions to the Schrödinger equation are eigenfunctions of the problem and thus a basis for the
system, they are orthogonal. In this way <φn|φk>=0 for n≠k. In order for the normalization of the
wavefunctions to make sense, the sum of the expectation value of all available states must equal 1 (the
square of the wavefunciton amplitude is a probability which must to unity for all possible states).
20
∑ϕ
k
ϕk = 1
(2.12)
k
At this point, the stated equations are exact as no approximations have been made. However, the
case for an atom in a radiation field is unsolvable and must be addressed using some form of
approximation. Through the method of time-dependent perturbation theory, an atom is initially assumed to
be in an unperturbed state which satisfies the time-independent Schrödinger equation (2.7). A perturbation
is turned on at time t=0 and its effect on the energy of the system is described by a perturbation
Hamiltonian such that the total Hamiltonian for the system is given by
G
G
G
Hˆ (r , t ) = Hˆ 0 (r ) + ε Hˆ ′(r , t )
(2.13)
where ε is a parameter of “smallness” and will be addressed momentarily. Start with equation (2.8) and
substitute in equations (2.10) and (2.13). Finally start from the left and integrate over all space and all
states to consider the atom as a whole. The interim equation is then given by
i=
∂
G
G
ϕn ϕk cn (t )e −iEn t = = ∑∑ ϕn Hˆ 0 (r ) ϕk cn (t )e−iEnt = + ε ∑∑ ϕn Hˆ '(r , t ) ϕk cn (t )e−iEn t = (2.14)
∑∑
∂t n k
n
k
n
k
The n summation comes from equation (2.10) where n is the state for which we are interested while the k
summation comes from the integration over all possible states with which the nth state may couple. On the
left-hand side all terms <φn|φk>=0 for n≠k and through the normalization condition (2.12) the sum of all
n=k terms equals 1. On the right-hand side a similar argument is made after exchanging the unperturbed
Hamiltonian for a constant through the time-independent Schrödinger equation (2.7). The perturbation
term on the other hand does not satisfy the orthogonality of eigenfunctions because the wavefunctions are
modified by the perturbation operator. Using a final substitution to shorten the notation for the perturbation
G
G
G
G G
H 'nk (t ) = ϕ n Hˆ '( r , t ) ϕ k = ∫ ϕ n* ( r ) Hˆ '( r , t )ϕ k ( r )dr
(2.15)
the time rate of change of the coefficients cn(t) is given by
i=e − iEn t
=
d
cn (t ) + En cn (t )e− iEn t = = En cn (t )e −iEn t = + ε ∑ H 'nk (t )ck (t )e− iEk t
dt
k
=
(2.16)
It is straightforward to see that this equation can be simplified by cancelling the identical term and
multiplying both sides by the positive nth state exponential to simplify those terms to one. Replacing the
energy with the Bohr angular frequency ωnk = ( En − Ek ) = , equation (2.16) becomes
21
i=
2.2.2
d
cn (t ) = ε ∑ H 'nk (t )ck (t )eiωnk t
dt
k
(2.17)
Two Level Atom in an Oscillating Electric Field
At this point one of two further assumptions can be made. Either the perturbation is assumed to be
small, from either a short time evolution or weak field interaction, or the system can be reduced to a two
level system where the perturbation only couples between two energy states. Under the former assumption
a power expansion in ε is taken to converge the terms of cn(t) and express the probability of being in an
excited state as a first (or arbitrary) order approximation. This method is only valid in cases where the
probability of being in the excited state is small and is limited by complexity of taking the calculation to an
arbitrary power in the interest of accuracy. Under the second assumption, ε is set equal to 1 and the indices
are reduced to n,k=1,2. Because the forces under consideration for the proposed experiment will be acting
over long periods of time on an atom which readily behaves like a two level system, the second assumption
will be made. This resonance approximation is only appropriate when the perturbation frequency (laser
frequency) is close to the resonant frequency of the transition between states 1 and 2 (or ground and excited
1,2→g,e) such that all other states can be ignored. The pair of coupled equations thus derived from
equation (2.17) is then given by
d
cg (t ) = H 'ge (t )ce (t )e − iω0t
dt
d
i= ce (t ) = H 'eg (t )cg (t )eiω0 t
dt
i=
(2.18)
where ω0 = ω21, which is the center frequency for the transition.
2.2.3
Rabi Frequency
The next and most important step is to apply the appropriate perturbation function. At this point
the single laser derivation and the derivation appropriate for the interference pattern of two counterpropagating lasers diverge. However, regardless of the shape of the perturbing field, assume that it is made
up of an amplitude term (E0) and an oscillating term. Therefore the Hamiltonian for the perturbed atom can
be separated into similar parts, one for the energy of the perturbed state and the other for its oscillation in
time and space (lab frame). The perturbation Hamiltonian is calculated as an induced electric dipole
22
potential by using the electric dipole approximation. It is assumed that the electric field does not vary over
the space of the atomic wavefunction , i.e. λL>>rCs. This yields a Hamiltonian of
G
G
G
Hˆ '(r , t ) = −eE (t ) ⋅ r
(2.19)
In order to be substituted into equation (2.18), the Hamiltonian will need to be operated on by the
“bra ket” notation. It becomes convenient then to define a scalar value for the interaction strength between
the magnitude of the radiant field and the dipole matrix element (integral represented by the bra ket). The
Rabi frequency is defined as
Ω=
eE0
G
ϕ1 eεˆ ⋅ r ϕ2
=
(2.20)
where E0 is the magnitude of the electric field and ε is the polarization vector. It will be shown that this
definition yields a relevant characteristic of the atom in the radiant field, namely the frequency of
oscillation between the ground and excited states when taken independently of spontaneous emission.
Calculating the Rabi frequency is non-trivial, however there are several relations which will be introduced
later which can lead to a value. Until then, it is a convenient way to book-keep the interaction Hamiltonian.
2.2.4
State Probability – Coherent Evolution for a Single Laser
The probability coefficients given in equation (2.18) can be solved for a perturbing field of one
irradiating laser which takes the sine form of equation (2.2). Assume that the laser wavelength is different
than the transition wavelength, i.e. near resonance but not on-resonance, and let that detuning be denoted
by δ= (ωL - ω0). Given the initial condition that the atom starts with a probability of being in the ground
state equal to unity, the time development of the probability amplitudes (where |cg,e(t)|2 is the state
probability/population) is given by
δ
Ω′t
Ω′t ⎞ + iδ t 2
⎛
cg (t ) = ⎜ cos
− i sin
⎟e
Ω′
2
2 ⎠
⎝
Ω
Ω′t − iδ t 2
ce (t ) = −i sin
e
Ω′
2
(2.21)
Ω′ ≡ Ω 2 + δ 2
To reach this solution an approximation has been made. The Rotating Wave Approximation (RWA)
neglects terms of the order 1/ωL compared with terms of the order 1/δ. This is only valid for near resonant
situations where δ<<ωL. This derivation only addresses the resonant (stimulated) forcing of atomic energy
23
states by the irradiating laser. This does not address the quantum phenomenon of spontaneous emission,
which artificially resets the time development to the initial condition of the ground state probability equal
to unity. The evolution of the probability of finding an atom in the excited state is shown for Figure 2-1.
The solid line represents δ=0, the dotted line represents δ=Γ, and the dashed line represents δ=2.5Γ. Time
is given in units of 1/Γ and all lines assume Ω=Γ. Γ is the radiative width of the transition which will be
defined in the following section.
Figure 2-1: Probability for an atom to be in the excited state versus 1/Γ, for various Ω
(Metcalf and van der Straten 1999)
2.2.5
Spontaneous Emission and State Lifetime
The time an atom will spend in an excited state before spontaneously releasing a photon and
relaxing to the ground state is an important characteristic of the transition. First consider the atom as a
classical system with an excited electron as a classical harmonic oscillator. The excited electron then has
an electric dipole oscillating at an angular frequency. This oscillating dipole will radiate a power equal to
P=
e 2 x02ω 4
12πε 0 c 3
(2.22)
Consider the total energy of the harmonic motion as E = me x02ω 2 2 . The rate of the energy lost from the
system is equal to the power radiated such that
dE
e 2ω 2
E
E=−
=−
3
dt
6πε 0 me c
τ
(2.23)
For strong transitions, the lifetime, τ, from the classical approach is very close to the quantum mechanical
result.
24
Returning to the quantum mechanical view of the atom, the derivation of the spontaneous
emission of a photon from an atom becomes more complex. The same derivation for the stimulated
processes of an atom immersed in an oscillating electric field can be done for an atom in free space which
emits a single photon. This derivation requires the quantization of the emitted photon’s field and the
inclusion of vacuum fluctuations. The final result of that derivation gives the change in the probability of
being in the excited state as a function of time as
d
1 e2ω 3
G
c2 (t ) = −
ϕ1 r ϕ2
3
dt
2 3πε 0 =c
2
c2 (t )
(2.24)
This yields a quantum mechanical lifetime equal to
1
τ
=
e 2ω 3
G
ϕ1 r ϕ2
3
3πε 0 =c
2
(2.25)
Because the dipole transition matrix element (the bra ket notation part) is difficult to analytically calculate,
its value is often derived through the relation in equation (2.25) and experimental measurement of the
lifetime. This element can be related to the element in the Rabi frequency give a value to the “bra ket”
notation in equation (2.20). By adding this time to the evolution given in Figure 2-1, the average of 1, 10,
and 100 atoms is given in Figure 2-2 where Ω=Γ and δ=-Γ.
Figure 2-2: Probability for an atom to be in the excited state versus 1/Γ, including spontaneous emission
(Metcalf and van der Straten 1999)
25
2.2.6
Natural Width
Because of the finite lifetime of the excited state, the frequency of the emitted photon is not
completely determined as a discrete quantity. The shape of the probability, or spectrum, of frequencies
emitted by the atom during spontaneous emission is given by the Lorentzian shape
f nw (ω ) =
⎤
Γ
1 ⎡
⎢
⎥
2
2
2π ⎣⎢ (ω − ω0 ) + ( Γ 2 ) ⎦⎥
(2.26)
where Γ is the full width at half max (FWHM) of the distribution and ω0 is the center frequency of the
transition which correlates to ω21. Equation (2.26) is normalized such that the integral over all frequencies
is equal to 1. This width, Γ, is referred to as the natural line width and defines the radiative broadening of a
spectral line. The natural line width can be related to the lifetime of the transition by
Γ≡
1
(2.27)
τ
where τ is the mean lifetime of the transition. An example of the natural line shape of the cesium D2-line
can be seen in Figure 2-3 and a table of the relevant characteristics of the transition can be seen in Table
2-1.
x 10
-9
Arb. Units
15
10
5
-10
-5
0
(ω - ω ) / Γ
5
10
0
Figure 2-3: Cs D2 natural line shape
Transition Wavelength
Transition Lifetime
Natural Line Width
λ=2πc/ω0
τ=1/ Γ
Γ/(2π)
852.35 nm
30.5 ns
5.22 MHz
Table 2-1: Cesium D2-line information
2.3
2.3.1
Optical Bloch Equations (OBE)
Density Matrix
The density matrix is a way of describing the state of the atom at any particular time. The
diagonal terms of the density matrix give the probability (or population of an ensemble) of being in the two
26
states while the off-diagonal terms, or coherences, describe the coupling between the two states within the
field. The density matrix for a two level atom is given by
2
⎛ c1 ⎞ * * ⎛ c1
⎜
ρ = ⎜ ⎟ ( c1 c2 ) =
⎜ c c*
⎝ c2 ⎠
⎝ 2 1
c1c2* ⎞ ⎛ ρ gg
⎟=⎜
2
c2 ⎟⎠ ⎝ ρeg
ρ ge ⎞
ρee ⎟⎠
(2.28)
The time rate of change of the density matrix will be required to calculate the forces on the atom within the
field. The formulation of these rates (time derivative of the density matrix elements) is described by the
Optical Bloch Equations (OBE) which are optical equivalents to the Bloch equations for nuclear magnetic
resonance.
OBE
2.3.2
A reformulation of the density matrix elements and inclusion of the spontaneous emission of
photons as damping to the stimulated excitation of the atom ultimately results in the optical Bloch
equations:
d ρ gg
dt
d ρee
dt
d ρ ge
dt
d ρ eg
i *
( Ω ρeg − Ωρ ge )
2
i
= −Γρee + ( Ωρ ge − Ω* ρ eg )
2
i
⎛Γ
⎞
= − ⎜ + iδ ⎟ ρ ge + Ω* ( ρee − ρ gg )
2
⎝2
⎠
= +Γρee +
(2.29)
i
⎛Γ
⎞
= − ⎜ − iδ ⎟ ρ eg + Ω ( ρ gg − ρee )
dt
2
⎝2
⎠
*
where ρij ≡ ρij e−iδ t . For ρ eg = ρ ge
the steady state solution to the OBE yields
ρeg =
iΩ
2 ( Γ 2 − iδ )(1 + s )
s
1
ρ gg − ρee =
⇒ ρee =
1+ s
2(1 + s )
(2.30)
where the saturation parameter, s, is given by
s=
Ω
2
2
δ 2 + Γ2 4
=
s0
1 + ( 2δ Γ )
2
(2.31)
and the on resonance saturation parameter, s0, is given by
s0 = 2 Ω
2
Γ 2 = I I sat
(2.32)
27
It can be seen that as the strength of the external field increases (Ω→∞, s>>1), the population difference
asymptotes, ρgg-ρee→0. If these equations are used to calculate the probability of finding an atom in the
excited state, the resulting evolution can be seen in Figure 2-4. This evolution is the same as the evolution
shown in Figure 2-2 using the average of an infinite number of atoms.
Figure 2-4: Probability for an atom to be in the excited state versus τ, using the OBE
(Metcalf and van der Straten 1999)
2.3.3
Saturation and Power Broadening
The value for Isat is the intensity required for an on-resonance field to cause the population of the
excited state to be ¼ of the total population. The transition at this point is said to be “saturated” such that
further increases in laser intensity do not result in comparable increases in excited population. This
intensity is directly calculable and given by
Isat = π hc 3λ03τ
(2.33)
It has been mentioned that the rate at which the excited state decays is given by Γ times the population of
the excited state, equation (2.24). It is now possible to explicitly calculate the emission rate of spontaneous
photons as
γ p = Γρ ee =
s0 Γ 2
1 + s0 + ( 2δ Γ )
2
(2.34)
When s0→∞, the maximum photon emission rate is given by Γ/2. This is an important consequence for
processes that rely on the spontaneous emission of photons for momentum dispersion, e.g. optical
molasses. Note also that the rate of the photon scattering as a function of laser detuning has a shape with a
full width half max of Γ for s0<1. But, as the on resonance saturation parameter increases, the width of the
28
profile increases as well. This “power broadening” affects both the frequency range of emitted photons
from the transition as well as the photon frequencies which excite the atom.
2.4
2.4.1
Forces on an Atom in an Optical Lattice
Near-Resonant Forces on a Two Level Atom
Through the Ehrenfest theorem, the quantum mechanical equivalent to the classical force due to a
spatial gradient in potential energy is given by the expectation value of the gradient of the Hamiltonian
such that
F =−
∂Ĥ
∂x
(2.35)
The expectation value of the Hamiltonian is calculated by
A = ∑∑ cn*ck ϕn A ϕk
n
(2.36)
k
such that the force on an atom can be described by
⎛ ∂Ω * ∂Ω*
⎞
F = =⎜
ρeg +
ρ eg ⎟
∂x
⎝ ∂x
⎠
(2.37)
where x is the axial laser direction. It is important to note at this point that this describes all forces
(scattering and dipole) on the atom within the laser field as long as the matrix element, ρ, includes
relaxation due to spontaneous emission. In order to conveniently look at the variation of this forces with
space first redefine the gradient of the Rabi frequency as
∂Ω
= ( qr + iqi ) Ω
∂x
(2.38)
where (qr+iqi) is the logarithmic derivative of Ω separated into its real and imaginary components. This
allows the force to be more easily separated into a real derivative corresponding to the amplitude gradient
and the imaginary part corresponding to the phase gradient. Substituting equation (2.38) into equation
(2.37) yields
*
F = =qr ( Ωρ eg* + Ω * ρ eg ) + i =qi ( Ω ρ eg
− Ω * ρ eg )
(2.39)
29
This equation is general enough that it can be used for any situation where the OBE can be solved for the
density matrix elements. It is important to note that the notation of the complex Rabi frequency is for book
keeping purposes only and it should be made clear that the force is real.
2.4.2
Moving Atom in a Standing Wave
One last bit of complication is required before the force on an atom in a standing wave can be
considered and that is the atoms motion within the lattice. In the derivation so far spatial gradients in the
field have been ignored through the electric dipole approximation. If the atom is stationary, this is perfectly
reasonable. In order to now account for the atomic motion within the standing wave (and the spatial
gradients the moving atom will traverse), a first-order correction to the OBE is made such that the time
derivatives of the appropriate quantities are modified by the spatial gradient times the velocity. This
transforms the spatial gradient at a point into a time derivative. This first-order approximation is only
appropriate if the velocity of the atom relative to the standing wave is slow enough such that the state of the
atom adiabatically changes with the change in external parameters, e.g field strength/gradient. Under such
a condition the atom’s state at any time/position is considered steady state. These first order corrections are
given by
∂Ω ∂Ω
∂Ω
=
+ vx
∂t
∂t
∂x
∂ρeg ∂ρ eg
∂ρ eg
=
+ vx
∂t
∂t
∂x
(2.40)
In order to arrive at a general force for an atom in a standing wave, the following formulation is
used: the electric dipole for the Hamiltonian [eqn. (2.19)], the electric dipole approximation [the field does
not vary over the atomic volume], standing wave [eqn. (2.3)], the definitions for the Rabi frequency [eqn.
(2.20)] and OBE [eqn. (2.29)], first order correction to the OBE [eqn. (2.40)], and the total force on an
atom in a laser field [eqn. (2.39)]. The force on a moving atom within a one dimensional optical lattice is
given by
F = − = qr
sδ ⎛
(1 − s )Γ 2 − 2 s 2 (δ 2 + Γ 2 4) ⎞
⎜ 1 − v x qr
⎟
1+ s ⎝
(δ 2 + Γ 2 4)(1 + s ) 2 Γ ⎠
(2.41)
From the cos2 in electric field, the derivative terms are given by qi=0 and
qr = − k tan(kx)
(2.42)
30
In order for this equation to satisfy the slow velocity assumption made during the first-order correction to
the OBE, the following inequality must be true
1 kvx
1
2π Γ
(2.43)
For a cesium atom and a laser tuned near the D2 transition, this inequality is unity for a velocity of
approximately 28 m/s. This means that the above derivation is only appropriate for relative velocities
between the lattice and the atom of around 3 m/s or less.
Using equation (2.41) and substituting values consistent with the center of two 100 mW,
collimated, 1 mm, lasers with a detuning δ=-50Γ, the force acting on the atom at rest can be seen in Figure
2-5.
1
x 10
-18
Fstand [N]
0.5
0
-0.5
-1
-4
-2
0
Axial Position [m]
2
4
-7
x 10
Figure 2-5: Axial dipole force vs position, PLaser=100 mW, DLaser=1 mm, ωL= ω0-50Γ
The shape is defined first by the tangent in the derivation of the gradient and second by the oscillation in
the light intensity (variation in s) within the interference pattern given by
I=
2.4.3
cε 0 E ( x)
2
2
= 2cε 0 E02 cos 2 (k L x)
(2.44)
Non-Resonant Optical Lattice
For a radiating field which is tuned far below atomic or molecular resonances, the electric dipole
element is reduced to a static polarizability, α, as the field can be considered static for ωL<<ω0. This
removes much of the complication originating from the quantum resonance interaction. The interaction
between an atom or molecule and an optical lattice formed by far off-resonance laser pulses can be
modelled as the force on an atom in a static electric field given by (Boyd 1992)
1
F = −∇U = − α∇E 2
2
(2.45)
31
Using the equation for the square of the electric field given in equation (2.5), time averaging the fast
oscillating terms and neglecting the long spatial gradient, the resulting force acting on an atom or molecule
within the potential region is
1
F = −∇U = − α∇ ⎡⎣ EL1 EL2 (1 + cos(kΔ x − ωΔ t ) ) ⎤⎦
2
(2.46)
where kΔ= kL1-kL2 and ωΔ = ωL1-ωL2. Note that kΔ and ωΔ define the velocity of the moving standing wave,
ξ= kΔ/ωΔ, where the sign of ωΔ defines the direction of motion.
Because the static polarizability is much smaller than the resonant interaction, the magnitude of
the electric field needs to be greatly increased to attain the same force. Thus, with a non-resonant laser
interaction, pulsed lasers would be needed to form a sufficiently strong optical lattice. The intensity of two
laser pulses can be assumed to have a Gaussian shape in both space and time which can be described by
I (r , t ) = I max e
⎛ −4 ln(2)(t −t )2 −4 ln(2)r 2 ⎞⎟
0
⎟⎟
⎜⎜⎜
+
⎟
⎜⎝
τ2
D2
⎠⎟
fwhm
(2.47)
beam
where t0 is the time at which the peak intensity will pass a spatial location and is a function of the distance
from the laser, τfwhm is twice the time required for the intensity to decay to half its maximum value, and
Dbeam is the same for radial distance. By substituting the laser intensity for the electric field magnitude,
eqn. (2.46) in the axial direction becomes
Fx (x , r , t ) =
αkΔ
c ε0
I 1 (x , r , t )I 2 (x , r , t ) sin(kΔx − ωΔt )
(2.48)
while in the radial direction eqn. (2.46) becomes
Fr (x, r, t ) = α ( 1 + cos(kΔx − ωΔt ) )
8 ln(2)r
2
Dbeam
I 1(x, r, t )I 2 (x, r, t )
cε0
(2.49)
It should be noted that a gradient in the axial intensity profile caused by the temporal Gaussian shape was
neglected as (2π/kΔ)/(c τfwhm)<<1.
32
3
3.1
Proposed High Speed Investigation
Experimental Setup
In order to provide a proof of concept for the acceleration of atoms and molecules to high
velocities, an experiment which builds on existing research pertaining to resonant cesium manipulation will
be designed, built and tested. The purpose of the experiment will be to add transverse momentum to a
collimated cesium atomic beam using near resonant laser light interfered to form a 1-D optical lattice
perpendicular to the atomic motion. The lattice will be put in motion, and the atoms will be dragged along
with the potential, adding momentum in the axial laser direction (perpendicular to the atomic beam axis).
3.1.1
Apparatus
The experimental apparatus will consist of a vacuum chamber and accompanying equipment
(pumps, gauges, feed-through, windows, etc.), a cesium oven, an orifice to generate the atomic beam, a hot
wire ionizer and channeltron to diagnose the cesium beam. A conceptual sketch of the experimental
apparatus can be seen in Figure 3-1.
852 nm
Path (red)
Cesium
Path (black)
852 nm Laser
Deflected Cs
Modulators
Vacuum
Equip. (blue)
Hot Wire
Detection System
Cs Oven
Optical Lattice
Figure 3-1: Conceptual drawing of experimental apparatus
3.1.2
Atomic Beam Formation
The atomic beam is created by heating pure cesium in an oven. An orifice in the oven wall allows
atoms from the evaporated vapor to leave towards the test section. In order to determine the velocity
distribution and flux of the atoms leaving the oven, thus the initial conditions of the experiment, the flow
regime of the orifice must be identified. The Knudsen number, Kn, indicates what flow dynamics will
33
govern the regime, from continuum (Kn→0) where the flow would be treated as a sonic orifice expanding
into vacuum to free-molecular (Kn→∞) where the flow would be treated as a collisionless equilibrium flux
across a plane. The Knudsen number is defined by the ratio of the mean distance between an atom’s
intermolecular collisions and a length scale of interest and is given by
Kn ≡
λmfp
(3.1)
L
where λmfp is the mean free path and L is the characteristic length scale (the orifice diameter in this case).
One simplifying assumption for the calculation of kinetic properties of a gas is that the intermolecular
potentials can be modelled as hard spheres (Bird 1994). For the low energies (<1 eV) and pressures (<1
Torr) in the Cs oven, this approximation yields little difference from other, more complex, intermolecular
potentials. With this assumption, the expression for the mean free path is given by
1
λmfp =
(3.2)
2nσ T
where σT the collisional cross section of the cesium-cesium collision and n is the number density in the
cesium in the oven. The number density can be found through the ideal gas law, P=nkboltzT, and the vapor
pressure relation given in Table 7-3. A list of the experimental atomic beam properties are given in Table
3-1. As Kn>1 for the flow leaving the oven orifice, a free-molecular approach to the flow is appropriate,
though technically the flow is at the beginning of the transitional regime between free-molecular and
continuum. Treated as a free-molecular flow, the effect of intermolecular collisions is ignored.
The fluxal properties of the atoms leaving the orifice can be found by taking moments of the
velocity distribution within the gas. For a stationary gas in thermal equilibrium, the velocity distribution
can be considered Maxwellian and is given in polar coordinates by
⎛
⎞
m
f (v)dv = ⎜
⎟
⎝ 2π kboltzT ⎠
32
2
ve
⎛ m ⎞ 2
− ⎜⎜
⎟⎟ v
⎝ 2 kboltzT ⎠
sin(θ )dθ dφ dv
(3.3)
where v is the magnitude of the velocity vector or speed and m and T are the mass and temperature of the
atoms.
Note that the distribution is equal in all directions.
Since it is assumed that there are no
intermolecular collisions over a length scale equivalent to the orifice diameter, the equilibrium distribution
can be used at the entrance plane of the orifice to define the outflow conditions of the oven. If the orifice is
34
considered infinitely thin, thus all atoms entering the orifice exit the other side unperturbed, fluxal values
for the flow out of the oven (normal direction θ=0 and ϕ=π/2 ) are calculated per unit time per unit area as
β3
Q = n 3 2
π
∞ π 2 π
⎛
⎞
m
∫0 −π∫ 2 ∫0 Q ⎜⎝ 2π kboltzT ⎟⎠
32
v3e
⎛ m ⎞ 2
− ⎜⎜
⎟⎟ v
⎝ 2 kboltzT ⎠
sin 2 (θ ) cos(φ )dφ dθ dv
(3.4)
The number flux of atoms across that surface is given by setting Q=1 and multiplying by the area of the
orifice which simplifies to
nv '
nA 8kboltzT
N =
A=
πm
4
4
(3.5)
where A is the area of the orifice and n is the number density of the cesium in the oven.
To collimate the beam, another orifice is placed downstream of the oven orifice. To analytically
find the number of atoms which pass through the oven orifice and subsequently the downstream orifice, the
directional limits of integration on equation (3.4) would not be a hemisphere. Instead they would need to
be a function of radial position on the oven orifice (while integrating over the area) and the distance
between the two orifices, spanning the angles which cover the area of the downstream orifice as seen from
the differential area of the oven orifice. This quadruple integral is cumbersome to solve, even numerically.
Instead a Monte-Carlo (MC) simulation can be employed since the phase space distribution of atoms is
known at the oven orifice and their trajectories are unchanged by intermolecular collisions (the freemolecular assumption). The initial condition for the MC simulation along the radial axes of the orifice is
given by the Cartesian version of equation (3.4) in the form
12
⎛
m
⎞ G2
⎛
⎞ −⎜⎜⎝ 2 kboltzT ⎟⎟⎠vy,z G
m
f (vy,z )dv = ⎜
dvy,z
⎟ e
⎝ 2π kboltzT ⎠
(3.6)
In the normal direction, the distribution is modified by the normal velocity, as the distribution is defining
only atoms which pass through the orifice and exit the oven.
⎛
m
⎞ G2
G ⎛ m ⎞ −⎜⎜⎝ 2 kboltzT ⎟⎟⎠ vx G
f (vx )dv = 2vx ⎜
dvx
⎟e
⎝ 2kboltzT ⎠
(3.7)
The comparison between the MC start conditions used and the analytic distributions are given in Figure 3-2
and Figure 3-3. Note that the appropriate normalization has been applied such that the integral of the
35
analytic function over all possible velocities (-∞-∞ or 0-∞) or the sum of all histogram points times the bin
width equals 1.
Probibility [Arb. Units]
x 10
-3
2.5
2
1.5
1
0.5
0
-500
0
Velocity [m/s]
500
Figure 3-2: Radial velocity distribution at the oven orifice (y or z axis)
Probibility [Arb. Units]
x 10
-3
4
3
2
1
0
0
100
200
300
Velocity [m/s]
400
500
Figure 3-3: Axial velocity distribution at the oven orifice (x axis)
The MC simulation is then used to select only the atoms whose trajectories pass through the
downstream orifice. As the atoms leaving the oven orifice are distributed over 2π steradians, the small area
of the downstream orifice reduces the number flux into the test section considerably. However, by
trimming the atoms which leave the oven, the radial velocity distribution of the resulting beam is much
narrower than the thermal effluence of the oven. The radial velocity distribution at the downstream orifice
can be seen in Figure 3-4. The width of this distribution can be used to calculate an equivalent translational
temperature in the direction of the laser crossing. This temperature, when compared with the potential well
depth and acceleration of the lattice, gives an indication of the likelihood of an atom to be trapped by the
potential.
36
Probibility [Arb.Units]
1.5
1
0.5
0
-2
-1
0
Velocity [m/s]
1
2
Figure 3-4: Radial velocity distribution at the downstream orifice (y or z axis)
The axial velocity distribution can be seen in Figure 3-5. Note that the peak has shifted higher than the
oven effluence. This is due to faster atoms in the x-direction contributing more to the distribution since
they will have a smaller travel time between the oven and the downstream orifice. Less time travelled
allows for more lenient tolerance on the radial velocity which would eventually move them outside the
downstream orifice.
Probibility [Arb.Units]
2
x 10
-3
1.5
1
0.5
0
0
100
200
300
Velocity [m/s]
400
500
Figure 3-5: Axial velocity distribution at the downstream orifice (x axis)
The spatial distribution at the downstream orifice can be seen in Figure 3-6. This spatial distribution has
direct implications on what the resultant signal will be for the detector. Again, the atoms will move from
the downstream orifice unperturbed to the hot wire ionizer. Thus, from the result of the Monte Carlo
Probibility [Arb.Units]
simulation, the expected distribution of atoms at the detector can be calculated.
1500
1000
500
0
0
1
2
3
Radial Position [m]
4
5
x 10
-4
Figure 3-6: Spatial distribution at the downstream orifice (r=sqrt(y2+z2))
37
Oven Temperature
Oven Pressure
Oven Number Density
Orifice Diameter
Knudsen Number
Number Flux – Oven (eqn. (3.5))
Oven to 2nd Orifice
2nd Orifice Diameter
Number Flux – 2nd Orifice (MC)
Oven to Hot Wire
Toven
Poven
n
2Rorifice
Kn
N
dskim
2Rskim
N beam
ddet
350 K
0.017 Pa
3.5x1018 m-3
1 mm
8.6
1.6x1014 atoms/s
12” = 304.8 mm
1 mm
437.3 x106 atoms/s
24” = 609.6 mm
Table 3-1: Cesium atomic beam characteristics
3.1.3
Hot Wire Detector and Channeltron
To diagnose the atomic beam, a hot wire filament will be placed in the beam path. The likelihood
of a cesium atom to be ionized upon collision with the wire is a function of the wire material’s work
function, cesium ionization energy, and the temperature of the wire. If the work function of the wire
exceeds the ionization energy of the atom (>0.5 V) the atom will be ionized by contact with the wire
(Ramsey 1956). As the temperature of the wire increases, the likelihood of the cesium to adsorb to the
surface is reduced; however as the temperature increases the likelihood of the reflected atom to be ionized
is decreased as the residency of the atom on the surface is also reduced (Taylor and Langmuir 1937). There
exists then a temperature region where the absorption of cesium is sufficiently small and the likelihood for
ionization is sufficiently large that all atoms can be considered to reflect from the wire and be ionized.
The solid angle which the filament intersects is a function of the width of the filament, its radial
location from the centerline and its distance from the orifice. Assuming that the effect of the curvature of
the atomic beam shape on the ends of the interaction region can be neglected, i.e. the impingement region is
rectangular, the signal on the channeltron from ionized atoms can be seen in Figure 3-7 (from the MC
simulation). This figure assumes that all cesium atoms which strike the filament are ionized and detected
with a gain of 106 (advertised gain of the channeltron).
Sginal [pA]
8
x 10
5
6
4
2
0
-1.5
-1
-0.5
0
0.5
Transverse Position [m]
1
1.5
-3
x 10
38
Figure 3-7: Expected detector signal vs transverse location (y axis) (MC)
Assuming that the hot wire is at a 1 V potential relative to the chamber and the channeltron is at a
-2200 V potential, the trajectories of atoms distributed over 2π steradian can be seen in Figure 3-8. These
trajectories were calculated using the SIMIon simulation package. A 1 V potential is larger than the
potential required to drive the tungsten filament and heat it. The channeltron is modelled as a disc at the
bottom of the vacuum chamber. The ions are assumed to be singly ionized cesium with an initial velocity
equivalent to 1500 K (≈0.2 eV). The simulation is conducted in a full 3D environment with the figure
displaying the view perpendicular to the center of the ion initial trajectory distribution and the tungsten
filament.
Figure 3-8: SIMIon calculation of 0.2 eV ion trajectories over 2π steradian
3.1.4
Accelerating Optical Lattice
A full three dimensional MC simulation of the atomic motion and laser interaction is necessary to
predict the signal of the accelerated atomic beam at the channeltron. This simulation will be addressed in
parallel with the experimental execution. For now, an approximating simulation of the atomic motion in
just the laser propagation axis will be used to estimate the forces on a cesium atom in an accelerating
optical lattice and the magnitude of the expected signal. The time step of the simulation is chosen such that
an atom does not move a significant fraction of the lattice period in one time step. At each time step, the
39
velocity of the atom is updated by dividing the force exerted on the atom by the lattice, equation (2.41), by
its mass and multiplying by the time step. The position is then updated according to each atom’s new
velocity. This is essentially the direct integration of the equations of motion.
Using the approximating simulation, consider an ensemble of atoms starting with zero velocity.
Assume that they are evenly distributed, spatially, across one period of the optical lattice. In order to
simulate the acceleration of the optical lattice, the position of the atom within the lattice is given by x-1/2at2
and its velocity is given by vx-at. The lattice is accelerated constantly from an initial velocity of 0 m/s at
time t=0 to 4 m/s at time t=4 μs (acceleration equal to 106 m/s2). At this point the laser field is removed and
the atoms are allowed to coast at their final velocity for an additional 1.2 ms, roughly the time a thermal
cesium atom would take to travel 0.3 m, the length of the drift section of the proposed experiment. The
distribution of the final position of the atoms versus their initial location can be seen in Figure 3-9.
x 10
-3
Deflection [m]
5
4
3
2
-0.1
-0.05
0
Start Location [x/λ ]
0.05
L
Figure 3-9: Deflection vs start location, PLaser=100 mW, DLaser=1 mm, ωL= ω0-50Γ, a=1x106 m/s2
Note that Figure 3-9 does not include all of the points from –λ/4 to λ/4 (one lattice period). Atoms which
violated inequality (2.43) such that the left hand side was greater than 0.1 have been discarded as the force
equation used to calculate their motion is no longer valid. The points which remain represent 40.6% of the
initial even distribution across the lattice. This simulation indicates that for atoms which are accelerated by
the lattice for only 4 μs (the time a thermal atom would take to cross the lattice diameter), their
displacement from the center of the beam would be upwards of half a centimeter. Considering the
undisturbed beam would have a diameter approximately half that, the deflected atoms should be
distinguishable from the un-deflected atoms.
Now assume that an ensemble of atoms is evenly distributed, in velocity space, from -1.5 m/s to
1.5 m/s (the approximate transverse velocity spread of the proposed collimated thermal cesium beam) with
40
an initial position at the anti-node (Fx=0 @ t=0). The lattice is again accelerated constantly over the same
range as before. The distribution of the final position of the atoms versus their initial velocity can be seen
in Figure 3-9.
Deflection [m]
x 10
-3
5
4
3
2
-0.5
0
Velocity [m/s]
0.5
Figure 3-10: |Deflection vs initial velocity, PLaser=100 mW, DLaser=1 mm, ωL= ω0-50Γ, a=1x106 m/s2
Note that Figure 3-9 again does not cover the full range of the initial condition. Again the points which
violate inequality (2.43) are discarded. The points which remain represent 34.6% of the initial even
distribution in velocity space. Again, this indicates that the atomic beam initial conditions are conducive to
trapping and accelerating a reasonable fraction of the cesium atoms.
3.2
Numerical Accompaniment
The approximating simulation used to estimate the deflection of atoms from the accelerating
lattice is inadequate to make real predictions for the trajectory of cesium atoms in the experiment. A full
MC simulation of the experimental conditions and the atomic trajectories will be developed to optimize the
proposed experiment and to extend the results of the experiment to a configuration capable of accelerating
atoms to km/s. The numerical simulation will need to incorporate a more stringent interaction force as to
avoid the necessity of discarding atoms which violate inequality (2.43). Such a derivation will require the
interaction force to be considered from first principle and derived without the first order correction used in
section 2.4.2. Such a derivation, to the author’s knowledge, has not been done before for two near resonant
lasers of different frequencies.
From the rigorous derivation of the interaction force, the MC simulation code will proceed with
established numerical techniques for the direct simulation of gas flows. Using the initial conditions derived
above and verified through the un-accelerated experiment, the simulation will populate atoms in a
statistically accurate manner.
Their trajectories will be calculated through direct integration of the
41
equations of motion, which will include the spatially and time dependant force on the atom due to the
optical lattice. In order to produce statistically relevant results, the simulation must contain a large number
of atoms and will require a significant amount of time and computing resource to calculate the motion over
sufficiently small time steps.
42
4
Proposed High Temperature Investigation
Because there is even less experimental precedent for the proposed high temperature investigation
than the high speed experiment, it is prudent to begin the high-temperature investigation with numerical
simulations to identify the most appropriate experimental course. Numerical simulation lends itself to trade
studies that would be impossible or infeasible within the constraints of an experiment. The simulations will
parameterize not only possible configurations for a proposed experimental measurement, but look at
several of possible measurements to experimentally study. The proposed study will begin by numerically
investigating one of two possible experimental measurements. The first simulations will look at the heating
of a gas by repeated interaction with a pulsed optical lattice.
If these simulations suggest that an
experimental measurement would be infeasible, the second set of simulations will look at the pressure pulse
(acoustic signal) in a gas cell after a single interaction. From these numerical simulations an experimental
course will be chosen based on the magnitude of the simulated signal and the availability of experimental
apparatus (primarily the laser).
An initial run of the first set of simulations (temperature increase) has already been conducted to
test the integration of the lattice force within the simulation package. These simulations assume that many
sequential optical lattices are formed and look at the effect of the time between pulses as it relates to the
relaxation of the gas to equilibrium.
4.1
4.1.1
Theoretical Framework
Single Pulse
As the goal of this investigation is to add energy to a continuum gas (as opposed to a free-
molecular), the role of intermolecular collisions must be accounted for. It is important then to consider the
effect of a pulsed optical lattice from the perspective of the velocity distribution of the gas. As the fardetuned (non-resonant) dipole force is conservative, the difference in the maximum and minimum potential
energy in the field, or well depth ΔU, can be related to a maximum change in atomic or molecular velocity
that corresponds to a kinetic energy equal to ΔU. The change in velocity is given by Δv = 2ΔU m ,
43
where m is the atomic or molecular mass. Shneider and Barker describe the effect of an optical lattice on
the axial velocity distribution (velocity in the laser propagation axial direction) as limited to ±Δv of the
lattice velocity, ξ, resulting in a tendency towards a uniform distribution from ξ+Δv to ξ-Δv (Shneider and
Barker, Optical Landau damping 2005). For lattice potentials corresponding to a velocity range wider than
the bulk of the velocity distribution (high intensity), the influence of the optical lattice causes a general
broadening of the distribution.
This general broadening of the distribution constitutes energy addition to the gas which will be
redistributed as the gas returns to equilibrium through collisions. For laser pulse durations on the order of
nanoseconds or faster, there is insufficient time for atoms or molecules outside the interaction region to
travel into the interaction region or visa versa. With average collision frequencies of room temperature
gases on the order of 100 ps there will be collisions between the atoms or molecules within the interaction
region. However, as energy is added to the gas within that region, the mean kinetic temperature of those
atoms or molecules will increase, decreasing the efficiency of the energy deposition. As the interaction
region cannot absorb an infinite amount of energy and the short duration of the pulse limits thermal
diffusion, the amount of energy which may be deposited to a gas within one pulse width is limited. This
leads to the practical consideration of a train of pulses forming many sequential optical lattices.
4.1.2
Multiple Pulses
If multiple pulses are used (either through the use of a high speed pulsed laser or an optical
cavity), consideration must be given to the time between pulses and the relaxation of the gas to a local
equilibrium. This is only one facet of the problem, but is a beneficial first thrust for the testing of the
simulation and the familiarity with the simulation package.
Because of the decrease in deposition
efficiency with temperature and the time required to relax to equilibrium, there exists an optimum time
between the pulses in order to create the highest temperature at the center of the interaction region. In
order to create the optimum condition for successive pulses, adequate time must be given for the excited
axial translational mode to transfer energy to unexcited modes and relax to equilibrium, thus creating a
cooler gas which will more efficiently absorb energy from the lattice.
The preliminary simulations
consider three gases: argon, nitrogen, and methane. For monatomic species, the only consideration for
thermal equilibrium is translational relaxation. Assuming the variable hard sphere (VHS) intermolecular
44
potential for argon at 1 atm and 300 K, the mean collision time is 137 ps (Bird 1994). Since a VHS gas
needs 4 to 6 collisions to relax from a large deviation from equilibrium, an optimum intervening time is
expected to be about 700 ps. For the same conditions with nitrogen and methane, the time for translational
relaxation is approximately 500 ps and 320 ps respectively, though as molecules, internal modes must be
also considered. For nitrogen at 300 K, 900 ps is estimated to bring the rotational energy within 1/e of the
translational (Bird 1994). Thus the optimal intervening time for nitrogen should be some time after the
longest relaxation time (>900 ps). The vibrational relaxation is estimated at a timescale much longer than
translational or rotational and longer than thermal diffusion. Therefore at temperatures lower than the
characteristic vibrational temperature (for nitrogen ≈3000 K) vibrational relaxation does not influence the
optimal intervening time.
4.2
Simulation Method – Preliminary Simulations
The preliminary simulations follow the same procedure which will be applied to the proposed
numerical energy deposition investigation. One of the more prolific high energy pulsed laser is an
Nd:YAG which emits pulses at 1064 nm. The first harmonic for that laser is thus 532 nm in the green. The
preliminary simulations use 532 nm as the constituent laser frequency. An optical lattice formed by two
such laser pulses would have a lattice period of 266 nm. When interacting with a gas at atmospheric
pressures, this resulting Knudsen number is approximately 0.2 within the interaction region. Also, the
mean collision time is on the same order of magnitude as the laser pulse duration, approximately 100 ps.
To properly simulate this phenomenon, a kinetic approach must be used. Simulations (have and) will be
conducted using a modified version of the SMILE DSMC code (Ivanov, Kashkovsky, et al. 1997). The
SMILE code has been broadly applied and experimentally validated, see (Ivanov and Gimelshein 2003) and
the references therein. The code has been extended to include the gas-lattice interaction described in
section 2.4.3. The effect of the dipole force on the particles is modelled as a constant acceleration over the
time step. The majorant frequency scheme is employed for modeling molecular collisions (Ivanov and
Rogasinsky 1988).
The VHS model is used for modeling intermolecular interactions.
The Larsen-
Borgnakke model (Borgnakke and Larsen 1975) with temperature-dependent rotational and vibrational
relaxation numbers is utilized for rotation-translation and vibration-translation energy transfer.
45
For the preliminary simulations, the simulation domain was modelled axisymmetrically around the
optical axis of the two anti-parallel, counter propagating laser pulses. The pulses were assumed to come
from two 532 nm Nd:YAG lasers generating 393 mJ and having a full width half maximum of 100 ps. This
parameter is easily changed within the simulation allowing for a study of the effect of pulse width and
pulse energy on the energy deposition to the gas. The pulses are assumed to have identical or nearly
identical frequencies such that a one dimensional optical lattice with a constant velocity is formed at their
intersection. For the preliminary simulations, this velocity was 0 m/s. The pulses are assumed to be
focused and collimated to a diameter of 0.1 mm. In order to cover the temporal shape of the pulses, each
pulse simulation is run from -τfwhm to +τfwhm, for a total of 200 ps. In between pulses, or intervening times,
the gas is simulated without external forces using simulation parameters (time step and grid size) adequate
for the simulation of the relaxation of the gas. This portion of the simulation must be done separately from
the lattice interaction because the time step and grid size required to attain a believable result are quite
different for the gas-lattice interaction and the relaxation time periods. Therefore, the peak-to-peak time
between pulses would be 200 ps plus the intervening time. An example of this scheme can be seen in
Figure 4-1.
Pulse Simulation Width
Intervening Time
Figure 4-1: Example of axial laser intensity [W/m2] for a 500 ps intervening time
If the repetition of pulses was created by an optical cavity instead of a high frequency pulsed laser
(required for very short intervening times), the corresponding optical cavity size scales with the peak-topeak time, ranging from 60 mm for no intervening time (200 ps peak-to-peak) through 3060 mm for an
intervening time of 10 ns (10.2 ns peak-to-peak). The axial domain boundaries enclose 0.01064 mm of gas,
or 40 lattice periods.
With a laser pulse spatial length of approximately 30 mm, the axial domain
boundaries are simulated as periodic. The domain is assumed to be at the center of the crossing pulses.
The radial domain boundary includes the surrounding gas sufficiently to ensure temperature and pressure
perturbations did not propagate from the axis to the outer boundary over the simulated pulses, assuring no
46
discontinuity at the boundary. The radial boundary simulated an ambient equilibrium gas at the initial
conditions.
In order to cover a range of species, argon, nitrogen, and methane will be used as test gases. The
polarizability to mass ratio of these gases are, 2.715, 4.24 (Wolf, Briegleb and Stuart 1929), and 10.92
(Kang and Jhon 1982), [10-15 C m2 kg V-1] respectively. Sampling is conducted over the last 5% of the
simulation time. With between 5 and 15 million simulated particles in the computational domain, the
average statistical error for a given sampling cell is approximately 2%. Mentioned temperatures are
assumed to be at the centerline of the gas volume.
4.3
Results and Discussion – Preliminary Simulations
Since the pulses were modelled as Gaussian in space and the heating within the gas is related to
the intensity of the laser field, the final translational and rotational temperature profiles of the gas for a
single pulse should show a similar trend to the laser intensity. The result for different gases can be seen in
Figure 4-2.
650
Temperature [K]
600
550
500
450
CH4 - trn
N2 - trn
Ar - trn
CH4 - rot
N2 - rot
400
350
300
0.0E+00
5.0E-05
Radial Location [m]
1.0E-04
Figure 4-2: Translational (trn) and rotational (rot) temperature as a function of radial position
The short duration of the laser pulse does not allow for thermal diffusion to influence the profile shape over
a single pulse nor allow for the internal energy modes to equilibrate with the excited translational mode.
The vibrational modes are unexcited over a single lattice pulse. The relative magnitude of the temperature
profile for each gas is consistent with the polarizability to mass ratio given above.
47
Δ Translational Energy [n.d.]
Argon
1.00
Nitrogen
Methane
0.10
0.01
0
1000
2000
Initial Temperature [K]
3000
Figure 4-3: Single pulse change in translational energy for varying initial temperature
The deposited energy from the lattice decreases as initial gas temperature increases. This is because the
hotter, broader initial velocity distribution will be disturbed less significantly than a cooler, narrower
distribution. Figure 4-3 shows the change in translational energy for one optical lattice pulse for various
initial gas temperatures relative to the change at 300 K.
Overall Temperature [K]
1,900
1,700
1,500
1,300
1,100
900
Argon
Nitrogen
Methane
700
500
300
0.05
0.5
Intervening Time [ns]
5
Figure 4-4: Final temperature after 10 pulses as a function of intervening time for several gases
Figure 4-4 shows the final temperature of multiple gases initially at 300 K measured at the end of
10 sequential lattice pulses with zero lattice velocity. The highest temperature reached by the gas is
dependent on the polarizabilty to mass ratio and the available energy modes in the gas. After the first
pulse, nitrogen has a higher temperature than argon, due to its higher polarizabilty to mass ratio. However,
as deposited translational energy relaxes into internal modes, argon attains a higher overall temperature (as
energy is not being distributed over as many modes). The simulated optimum intervening time for argon
48
(denoted by the highest temperature) is around 700 ps, consistent with the time for translational relaxation.
The simulated optimum time for nitrogen is around 1 ns, consistent with the time for rotational relaxation.
Overall Temperature [K]
The simulated optimum time for methane is around 250 ps.
1,100
1,000
900
1.0 atm
800
0.5 atm
700
0.1 atm
600
0.1
1
Intervening Time [ns]
10
Figure 4-5: N2 final temperature after 10 pulses as a function of intervening time and pressure
Since the collision frequency is dependent on the number density of the gas, an inversely
proportional shift in the optimum intervening time should be observed with changing gas pressure. This
shift is limited for long intervening times by the effect of thermal diffusion removing energy from the
centerline of the volume. Figure 4-5 shows the shift in optimal intervening time for the centerline
temperature of nitrogen originally at 300 K. The intervening time shows a peak shift between 1.0 and 0.5
atm without a significant decrease in centerline temperature. At 0.1 atm, the peak should be approximately
10 ns according to the relaxation time; however, this is long enough for diffusion to become a significant
effect and effectively reduce the centerline temperature.
In addition to inadequate time for thermal relaxation, rapidly applied lattice pulses can actively
decelerate gas particles for intervening times at or near zero. Figure 4-6 shows the argon centerline
temperature evolution as a function of pulse number with no intervening time.
49
Final Temperature [K]
480
460
440
420
400
380
360
340
320
300
0
2
4
6
Pulse Number [n.d.]
8
10
Figure 4-6: Final argon temperature for an intervening time of 0 ns
As expected, there is a large increase in temperature for the first pulse, approximately 110 K. This is
followed by a considerably smaller energy addition for the second pulse, about 30 K, due to the lattice
interaction with a particle velocity distribution that has already been severely disturbed. However, the third
pulse shows a marked decrease in the temperature, about 30 K. A single 100ps pulse is not sufficiently
long enough to allow a large fraction of particles to cross a distance equal to half of a lattice spatial period,
133 nm. Since the axial dipole force is periodic, this means that over one pulse the particles are constantly
accelerated. However, by the third pulse, a large enough fraction of particles have travelled far enough that
the sign of the dipole force has changed. Thus the lattice during the third pulse acted to decelerate the
particles. With a peak-to-peak time of 200 ps, the particles can expect only one collision between pulses on
average. After the fourth pulse, the spatial distribution of particles is sufficient that there are a relatively
equal number of accelerated and decelerated particles. Temperature increase is then a function of the
Final Temperature [K]
partial thermal relaxation which the gas undergoes pulse to pulse.
1,300
1,200
1,100
1,000
900
800
700
600
500
400
300
0
5
Pulse Number [n.d.]
10
50
Figure 4-7: Final argon temperature for an intervening time of 0.5 ns
Figure 4-7 shows the temperature evolution of the centerline of the gas volume as a function of the
pulse number for a 0.5 ns intervening time. Temperature after the pulse is shown with (◊) and the
temperature after the intervening time (thus before the next pulse) is shown as (□). There is no significant
temperature loss due to thermal diffusion during the intervening time. The expected decrease in deposition
efficiency with temperature is observed as a 20% smaller temperature increase between pulse 9 and pulse
10 than between pulse 1 and pulse 2.
Final Temperature [K]
1,100
1,000
900
800
700
600
500
400
300
0
2
4
6
8
Pulse Number [n.d.]
10
Figure 4-8: Final argon temperature for an intervening time of 10 ns
If the intervening time is increased sufficiently, thermal diffusion mitigates the temperature
increase as it allows for the flow of thermal energy away from the centerline. Figure 4-8 shows the
temperature evolution at the centerline as a function of the pulse number for a 10 ns intervening time.
Intervening times resulting in excess of approximately 10 collisions will impart more total energy to the gas
volume, as the lattice is interacting with a cooler gas, but yield a smaller maximum temperature. Figure 4-9
shows the evolution of the pressure profile for the same test run. From pulse 4 to pulse 10, a linear
propagation of the pressure profile can be observed from a radial distance around 40 μm to 60 μm. The
propagation of the pressure profile is representative of diffusion and the cause of the temperature loss seen
in Figure 4-8 past pulse four.
51
2.6
Pulse 2
Pressure [atm]
2.4
Pulse 4
2.2
Pulse 6
2.0
Pulse 8
1.8
Pulse 10
1.6
1.4
1.2
1.0
0
20
40
60
80
Radial Distance [μm]
100
Figure 4-9: Argon pressure profile development for an intervening time of 10 ns
In order to assess the usefulness of the gas-lattice interaction as a method for creating high
temperature gases, a simulation was run over 50 pulses. Figure 4-10 shows the temperature evolution of
nitrogen initially at 300 K and 1 atm as a function of pulse number. The highest temperature reached was
2480 K. The reduction in deposited energy as a function of temperature can be seen by the concave down
temperature profile. In addition to reduced deposition efficiency, there is an increased effect of thermal
diffusion, ultimately limiting the centerline temperature. The increase in thermal diffusion at higher
temperatures may change the optimal intervening time as a function of the number of pulses used as the
benchmark. At higher temperatures shorter mean collision time and a higher effect of thermal diffusion
may require a shorter intervening time to reach higher temperatures.
Final Temperature [K]
2,800
2,300
1,800
1,300
800
300
0
10
20
30
40
Pulse Number [n.d.]
50
Figure 4-10: N2 temperature as a function of pulse number for an intervening time of 1 ns
52
4.4
Possible Experimental Setup – Acoustic Measurement
Assuming that the experimental measurement of heating a gas cell is not deemed viable, an
experiment will be designed and constructed for the measurement of acoustic signals induced by an optical
lattice in a gas cell. Initial tests will be conducted using an optical lattice with no velocity (ξ =0 m/s). This
is most efficient condition for energy transfer to the gas where the laser intensity creates a lattice with a
well depth deeper than the width of the velocity distribution. It is also the easiest to implement with pulsed
lasers. For the initial stage of the experiment, one 532nm, 800mJ (visible, 1650mJ IR), 5-7ns, seeded,
Nd:YAG laser will be used to set up the lattice. The laser pulse from the single laser will be split equally in
two beams to produce the two counter-propagating fields. This setup will require extensive alignment and
testing of the optics to assure the formation and placement of the lattice in time and space. Both spatial and
temporal beam profiling will be conducted to assure a full understanding of the optical inputs to the lattice
both before and during lattice forming experiments. A diagram of this setup can be seen in Figure 4-11.
Figure 4-11: Proposed acoustic measurement experimental laser setup
Before testing energy deposition to the gas by the optical lattice, experiments must first be
conducted to assure that neither of the constituent beams are above the ionization threshold for the test gas
conditions. Although no visible or audible indication may be detected to indicate breakdown, there may
still be a sufficient ionization to affect the experimental results. In order to verify sufficiently low
ionization in this experiment, a microwave diffraction test setup will be used to detect ionization below the
53
visible/audible threshold. Laser power will be reduced until no scattering is detected, at this point effects
from the ionization of the gas should be many orders of magnitude lower than the lattice interaction effect
for which the experiment is testing. A diagram of the ionization test setup can be seen in Figure 4-12.
Figure 4-12: Ionization detection through microwave scattering experimental setup
Once appropriate laser parameters have been determined for various test gas conditions, testing on
energy deposition to the gas as detected through acoustic waves will begin. In order to maximize the
measured effect, the initial test cell for this experiment will be a 200μm by 200μm, 4cm long capillary.
The sides of this capillary will be made of etched quartz for optical access. The top and bottom of the
capillary will be made of polished aluminum for ease of machining and mounting of the pressure sensors.
A diagram of this test cell can be seen in Figure 4-13.
Figure 4-13: Possible experimental acoustic wave detection gas cell
54
Ambient gas pressures will range from 0.1 to 1 bar using nitrogen, argon, and methane as test
gases. Pressure sensors will be located at each end of the capillary. These sensors will measure the
difference between the pressure of the gas within the capillary and that of the surroundings. In addition to
containing the pressure energy of the lattice pulse within a smaller area (as opposed to spherical dissipation
in free space) the capillary can offer the additional benefit of affecting the flow relaxation time through the
use of reservoirs. This relaxes the dynamic response requirements of the pressure sensors to attainable
experimental limits.
55
5
5.1
Statement of Work
High-Speed Experimental Development
•
Assemble and characterize un-perturbed cesium atomic beam apparatus
•
Assemble and characterize optical lattice laser and optical apparatus
•
Measure atomic beam distribution for stationary 1D optical lattice
•
Measure atomic beam distribution for constant velocity 1D optical lattice
•
Measure atomic beam distribution for accelerating 1D optical lattice
5.2
High-Speed Numerical Development
•
Derive resonant atom-lattice force equation without use of first order correction
•
Simulate un-perturbed atomic beam profile and compare with experiment
•
Integrate resonant atom-lattice force with atomic beam code and compare with experiment
•
Execute trade study over variable experimental parameters to validate fidelity of simulation
•
Use validated code to design an experiment capable of accelerating Cs to 8 km/s
5.3
High-Temperature Numerical Development
•
Use integrated non-resonant simulation package to replicate related experimental data (e.g. Pan)
•
Execute trade study to ascertain the steady state temperature increase in a gas cell
•
Notionally design experiment and run simulation to predict signal
•
Execute trade study to ascertain the magnitude of an acoustic wave in a gas cell
•
Notionally design experiment and run simulation to predict signal
•
Based on chosen experiment, simulate experimental conditions and compare
5.4
High-Temperature Experimental Development
•
Based on numerical investigation design and construct experiment
•
Vary experiment over input parameters (e.g. laser power, wavelength, interaction volume)
•
Compare with simulation
56
6
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7
7.1
Empirical Values and Their Sources
Cesium Properties
The molecular weight of cesium is 132.905 amu (Lide and Frederikse 1997).
7.1.1
Ionization Potential
Cs 1st Ionization Potential
EI=3.893905373±(9.6x10-7)
Units
EI =[eV]
Source
(Steck 2008)
Table 7-1: 1st Ionization potential of Cs
7.1.2
Vapor Pressure
The melting point of cesium is 28.44° C = 301.59 K (Lide and Frederikse 1997).
Equation
log10 PS = 10.5460 − 1.00 log10 T − 4150 T
Units
PS=[Torr], T=[K]
Source
(Taylor and Langmuir 1937)
log10 PS = 5.006 + 4.711 − 3999 T
PS =[Pa], T=[K]
(Lide and Frederikse 1997)
Table 7-2: Vapor pressure of solid Cs
Equation
log10 PL = 11.0531 − 1.35log10 T − 4041 T
Units
PL=[Torr], T=[K]
Source
(Taylor and Langmuir 1937)
log10 PL = 5.006 + 4.165 − 3830 T
P=[Pa], T=[K]
(Lide and Frederikse 1997)
Pressure [Pa]
Table 7-3: Vapor pressure of liquid Cs
280
300
320
340
360
Temperature [K]
380
400
Figure 7-1: Comparison of published Cs vapor pressures
7.1.3
Collision Cross Section
Cs-Cs Total Collision Cross Section
σ=2040±20%=2040±408
σ=2350±5%=2350±118
Units
σ=[Å2]
σ=[Å2]
Source
(Manista and Sheldon 1964)
(Estermann, Foner and Stern 1947)
Table 7-4: Cs-Cs total collision cross section
61
7.1.4
Transitions
Frequency
f0=351725718.509±0.055
f0=351725718.4744±0.0051
f0=351725718.50±0.11
Units
f0=[MHz]
f0=[MHz]
f0=[MHz]
Source
(Das, et al. 2006)
(Gerginov, et al. 2004)
(Steck 2008)
Table 7-5: Cs 6p 2P3/2 ↔ 6s 2S1/2 transition frequency
Lifetime
τ=30.55±0.27
τ=30.499±0.070
τ=30.57±0.07
τ=30.405±0.077
Units
τ=[ns]
τ=[ns]
τ=[ns]
τ=[ns]
Source
(Tanner, et al. 1992)
(R. J. Rafac, et al. 1994)
(R. J. Rafac, et al. 1999)
(Steck 2008)
Table 7-6: Cs 6p 2P3/2 → 6s 2S1/2 transition lifetime
62