Section III: Computational Analysis

Due to relative ease in design, ability to obtain quick results, and freedom in which one can adjust variables, computer modeling lends itself to many areas of study. Laser cooling and trapping is no different. In many recent publications, researchers have used computer models to show forces on single atoms in spontaneous force traps¹, forces on single atoms in one-dimensional magneto-optic traps², and motions of atoms in magneto-optic traps³. These simulations, however, are only useful for analyzing specific experimental conditions and often isolate only one variable for study.

Cool was developed to accurately simulate the physical phenomena that occur when laser cooling and trapping atoms in the laboratory. The program graphically depicts the position and velocity of a variable number of atoms in two dimensions as they are cooled and trapped. The algorithm accounts for absorption, spontaneous and stimulated emission, Doppler and Zeeman shifts, and radiation trapping. The user can adjust time step, atom velocity, laser intensity and detuning, and magnetic field gradient. The program can demonstrate optical molasses, a MOT, and a dark MOT. Simulations can be recorded and played back in the same graphical environment, along with .AVI movies from the actual experiment. All data from simulations can be stored in text files for further analysis. The simulation is a useful educational tool for explaining and investigating Doppler cooling and the magneto-optic trap. The program allows anyone to obtain computational results and compare them with published experimental results.

The development language for Cool is a combination of Microsoft Visual Basic and Visual C++. The reason for this combination is simple: Visual Basic offers a highly graphical, user-friendly framework and Visual C++ offers the speed of a compiled language. Therefore, the user-interface is written in Visual Basic and the laser cooling and trapping algorithm is written in Visual C++ in the form of a dynamic link library (DLL).

3.1 The Dynamic Link Library (DLL)

Cool calculates many of the physical phenomena encountered in laser cooling and trapping. This requires storage of numerous variables, calculation of several probabilities, and reliance on a random number generator. (The random number generator used is a lagged Fibonnacci random number generator based on code from an article by James⁴ and will not be discussed here.) The essential variables include atomic position, velocity, and state (ground or excited). The probabilities calculated are those of the events included in Cool. These events are absorption, spontaneous emission, stimulated emission, and radiation trapping. The DLL is designed so that it keeps track of changes in atomic position, velocity, and state due to specific events during a user defined time step. At the end of a time step, the new atomic position, velocity, and state are returned to the main program for graphical output. A complete flow chart of the algorithm can be seen in Figure 3.1.

3.1.1 Probability Calculation

Each atom has a probability per unit time of undergoing absorption, spontaneous emission, or stimulated emission. The atom's state, either ground or excited, determines what type of event is allowed to occur. For example, a ground state atom could never spontaneously emit a photon. Once this is determined the probability per unit time of an event is calculated.

The probability per unit time of a spontaneous emission, for this particular study, is simply the natural linewidth of the cesium 6s_3/2 to 6p_1/2 transition. This probability does not need to be calculated and remains a constant throughout the program. The probability per unit time of absorption or stimulated emission, however, is not constant and is given by the scatter rate equation mentioned in section 1.2.3. (It should be noted that by rearranging some of the terms in the scatter rate equation, a more computationally efficient form can be obtained.)

Since this program simulates a two-dimensional cooling and trapping apparatus, absorption or stimulated emission rates must be calculated for each one of four lasers. The scatter rate by each laser is determined by inserting the component of an atom's position and velocity that corresponds to the axis of the laser's propagation. Assume an atom has an x position of 1 mm and a y position of 2 mm. This atom also has an x and y velocity of 2 m/s and 4 m/s, respectively. In calculating the scatter rate by the laser propagating in the positive x direction the position and velocity used would be 1 mm and -2 m/s, respectively. Notice, that for this laser the x component of the velocity is negated. This is because in the scatter rate equation the velocity used is the velocity relative to propagation. A similar procedure is used to calculate the other three scatter rates for this same atom.

3.1.2 Event Direction

Now that the probability of an event are known, the direction of the event must be determined. For absorption and stimulated emission events, the direction of the event is calculated first by summing the scatter rates from each of the four lasers. A random number multiplied by that sum returns a number that is compared to certain sections of the sum. For example, the scatter rates of the four lasers are: L1 = 0.004, L2 = 0.008, L3 = 0.01, and L4 = 0.014. The sum of these probabilities = 0.036. Multiplying this value by a random number (for this example, 0.6), yields 0.0216 which is then compared with certain sections of the sum. For instance, the section due to laser 1 represents values between 0 and 0.004. The remainder of the sections are: 0.004 < L2 (0.004 + 0.008), 0.012 < L3 0.022, and 0.022 < L4 0.36. From this data the event occurred due to laser 3. This value is saved in a variable and passed to the event manager.

From the theory discussed in section 1.2.2, the direction of spontaneous emission is completely random. Therefore, in the case of a spontaneous emission a random angle is generated using the random number generated. This angle corresponds to the direction of the atom's recoil.

3.1.3 Event Time

At this point, one might be tempted to take the highest probability of an event, compare it with a random number, and decide at nanosecond intervals whether the event occurred. The atom's position, velocity, and state could then be updated given the event and subsequent events could be determined using this iterative method. This proves, however, to be a very intensive computation and tremendously reduces efficiency. For example, a typical probability per nanosecond of absorption is on the order of 10^-2. Therefore, with a nanosecond loop, each atom must cycle through the loop about 100 times before one absorption will occur!

Computational efficiency can be increased by a factor of 10 by calculating an event time for each atom. Consider a random coin toss. The rate of tossing a "heads" is p = 0.5. Likewise, the rate of tossing a tails is q = 1p = 0.5. After N tosses, the probability that a "heads" has been tossed is represented by the relation:

Plugging in some numbers we see that after one toss the probability, P, is 0.5. After two tosses we see that P = 0.75. After three tosses P = 0.875, and so on. As the number of tosses increases the probability of obtaining a "heads" approaches 1. It turns out that the probability relation can be written as:

We can now apply this logic to our system. Let "heads" equal an event (i.e. absorption, spontaneous emission, and stimulated emission) and let a "tails" correspond to no event. Furthermore, let a toss represent a nanosecond. The rate, p, now represents the scatter rate for absorption and spontaneous emission and the natural linewidth for spontaneous emission. If we choose our probability, P, to be a random number, z, between 0 and 1 we can solve for the time duration where:

We now have an equation which returns the event time given a known rate of an event with a random probability and have avoided the need to step through the algorithm at one nanosecond intervals. This technique greatly enhances computational efficiency.

To summarize, the DLL has a method for calculating and storing the direction and time an event occurs. The job now is to manage how these events affect the atoms' position, velocity, and state in a given time step.

3.1.4 Event Management

Each atom enters the event manager with an initial configuration. This configuration includes: x and y position, x and y velocity, state (which is "ground" initially), time when an event occurs, and direction of that event. In order to get a better understanding of the event manager, let's follow the progression of two different atoms through a time step.

First, assume we have a ground state atom with a certain position, velocity, event time, and event direction. The event time of the atom is compared with the user defined time step. If the time step is less than the event time, the time step is subtracted from the event time, the manager updates the new position of the atom (time step x velocity), and the next atom is analyzed. If the time step is greater than the event time the manager recognizes that an event (in this case absorption) will happen in this time step. The state of the atom is immediately converted from ground to excited. The velocity of the atom is adjusted by hk/m (3.5 mm/s) in the direction of the laser's propagation. Finally, a new event direction and event time are calculated for the atom. Since an excited atom may have either a stimulated or spontaneous emission, a determination of which type of emission will occur must be made. This determination is similar to the determination of event direction. The type of emission is stored in a Boolean unique to excited atoms. A new event time is calculated and compared, again, with the time step to check for the possibility of multiple events in a time step. (Since time steps are often on the order of 1000 ns, this measure is necessary because in such a time multiple events will occur.)

The excited atom is treated by a similar method as the ground state atom. The atom's event time is compared with the time step in order to determine if an event will occur. The event that pertains to an excited atom is some type of emission (stimulated or spontaneous). As described in the previous paragraph, the type of emission is stored in a Boolean. Given a stimulated emission the atom's velocity is changed by hk/m in the direction opposite to the laser's propagation. If spontaneous emission occurs the change in velocity is given by:

where is the random angle of emission. Once an emission event occurs, the atom is changed to the ground state, and a new absorption event time is calculated.

One event, which the user has the option to include, is radiation trapping. To review, radiation trapping is a net repulsive force experienced by atoms in a MOT.⁵ The force arises when an atom emits a photon (either by stimulated or spontaneous emission) and the photon is reabsorbed by a neighboring atom. Although physically significant, this phenomenon represents a serious computational dilemma. It means that every time an emission event occurs, the position of every other atom must be analyzed to see if it lies within a certain area of the photon's path. The algorithm has been optimized so that a quadrant check routine is performed on each atom which quickly eliminates atoms not in the same quadrant as the emission. After the quadrant elimination is performed, a capture window is calculated for the remaining atoms. This capture window is a constant cross sectional area of each atom which is user defined. If the emitted photon lies within the capture window of an atom, a re-absorption occurs such that the atom becomes excited and has a change in velocity given by:

where is the angle of emission. This computation, however, significantly decreases computational efficiency. Simulations without radiation trapping are similar to dark MOT experiments and those with the event correspond to light MOT experiments.

As mentioned earlier, the algorithm accounts for the four main atom/photon interactions in laser cooling and trapping. One other force not calculated in this program is beam attenuation through the cloud of atoms. ⁶ This force would add significantly to computation time and probably yield few results.

3.2 A Tour of the Screen

The main form of Cool is displayed in Figure 3.2. Across the top of the screen is the menu bar which contains many of the file, simulation, and help controls. Below the menu bar and on the left side of the screen are two boxes which correspond to the "Position" and "Velocity" charts. These charts show animated pictures of the atoms' position and velocity during an experiment. The box on the top right side of the form contains the graphs. These graphs include "Doppler Probability," "X Distribution," "Speed Distribution," and "Velocity vs. Time" and can be selected by clicking on the option buttons to the right of the graph. Below the graphs are two boxes labeled "Simulation Setup," and "Simulation Data." These two boxes contain information about computational setup and current data during a simulation. Underneath these boxes, to the left, is a section that is activated only during a light MOT simulation. This section gives information about radiation trapping. To the right, is a status bar which gives information about a recorded simulation. Underneath the status bar, is a slider which allows a user to quickly move to a certain location in a recorded simulation. Across the bottom of the screen on the left, are two status bars which indicate the percentage of atoms in the ground and excited states. To the right of these are two check boxes labeled "Optimize Calculation" and "Monitor" whose functions will be discussed later. Finally, to the right of the check boxes are three control buttons which begin, pause/resume, and stop a simulation.

3.3 The Menu Bar

The menu bar consists of a hierarchy of commands which control many of the functions of Cool. The user may access these commands by clicking on "File," "Simulation," or "Help." By pressing ALT-(underlined letter) of one of the three labels the drop down box of commands may be accessed as well.

3.3.1 File

Underneath the "File" label is a drop down box containing the file handling, data conversion, and printing routines available in the program. The first two options are Open Simulation and Record Simulation which allow the user to open and record a simulation. The simulation files have the extension .CSM. Only files with the .CSM extension are valid. These files contain information concerning computational setup and each atoms' data at a given time interval. If the Open Simulation is clicked, the user is prompted to choose from an existing .CSM file for play back. Clicking the Record Simulation option prompts the user to input a desired time step and total time, and then prompts the user to give the .CSM a name. The current simulation setup is written to the simulation file and a recording is now ready to begin. Creation of a library of simulations is often very efficient since a play back of a simulation is faster than running the actual simulation.

The Save Current Data function saves the current system setup, position, and velocity data for each atom at the particular simulation time the function was called. The information is stored in a text file (.TXT) named by the user. By clicking the Convert .CSM option, the user performs a similar operation as in the Save Current Data function except the simulation file often contains data for multiple time steps. The data from the

.CSM file is written to a text file of the same name (except with .TXT as the extension). Both of the above mentioned functions allow the user to further analyze saved or converted data in a spreadsheet program or other application. A sample of the text file created by one of the two operations can be seen below.

Cool Data at 76000 ns

Dark MOT

# of Atoms= 100 Max Init. X Vel.= 5

Max Init. Y Vel.= 5 Laser Intensity= 1

Laser Detuning= -1 Magnetic Field Grad.= 25

RMS Pos.= .1851247 RMS Vel.= .5829124 Temp.(K)= 2.717956E-03

Notice that the simulation setup data is listed at the top of the file and an atoms' position, velocity, and state are listed in columns below. If the text file were created using the Save Current Data function only one time step's worth of data would be stored. If the Convert .CSM function were used, another column indicating the simulation time would appear next to the data. It should be obvious that a simulation with numerous atoms and many time steps could potentially produce an extremely large file. When making a simulation, time steps and total time should be chosen wisely.

The next functions listed in the drop down box deal with printing and saving. Using the Print Graph function the user is able to print the current graph displayed on the form. The graphs include Doppler Probability, Speed Distribution, and X Distribution. The Save Graph As function saves the current graph as a bitmap (.BMP). Note that this is the only format in which the graph may be saved. The Print Chart and Save Chart As functions perform similar functions in similar formats as those above. Clicking on either of the functions reveals a sub-menu that allows the user to choose between the Position or the Velocity charts. Each of the above mentioned functions can be accessed by right clicking the mouse over the corresponding chart or graph selecting from a pop-up menu.

The final option in the File drop down box is the Exit function. The Exit function ends the program, and closes any open files.

3.3.2 Simulation

The drop down box under the section of the menu bar labeled "Simulation" contains functions essential to controlling and configuring a simulation. The Start function begins a laser cooling and trapping experiment. If a simulation is opened using the Open Simulation function the function is entitled Play. If a simulation is to be recorded the function is entitled Record. The Pause function pauses a simulation. Once a simulation is paused the function label is immediately changed to Resume. This allows a paused simulation to be restarted. The next function is the Stop function. Clicking this label stops a computation clearing all charts and graphs. If a Stop is performed during a play back the simulation file is closed and must be reopened using the Open Simulation function. During a recording, if a Stop is performed, the simulation is still valid but the file ends at the time when the Stop was performed. The three control buttons at the bottom of the screen labeled Start, Pause, and Stop have the same function as those above.

The next three items listed in the drop down box deal with the program's configuration. The item labeled Experimental Setup opens a new form seen in figure 3.3. This form allows the user to change many aspects of the experimental setup. Note that during a recording the System Setup is disabled and during a normal computation the simulation must be paused for changes to be made in the experimental setup. The experimental setup form contains sliders and input boxes contained in three sections. The section of the form labeled Atom Setup contains the information about the atoms in the experiment. The atoms in the experiment have a variable number, x velocity, and y velocity. When demonstrating Cool and a quick computation is desired, it is probably a good idea to use a small number of atoms (about 50). This allows the viewer to better see the atoms in the charts . If the user wants to analyze data, then a larger number (about 1000) will provide a better statistical sample. Note that the minimum and maximum number of atoms in the experiment is 1 and 5000. The initial velocity may have any value permitted by the sliders and is measured in meters per second. The atoms are given initial x and y velocities randomly within the ranges specified by the sliders.

The section labeled Laser Setup refers to information about the four counterpropagating trapping lasers. The user has the ability to control laser intensity, laser detuning, and photon momentum. The laser intensity is a factor of the saturation intensity discussed in section 1.2.2. The slider labeled Laser Detuning allows the user to vary the frequency by a factor of the natural linewidth with respect to the 6S_{1/2 6P3/2} transition in cesium. The detuning may either be to the "red" or the "blue" of the transition. The "red" detuning corresponds to a negative detuning and a "blue" detuning corresponds to a positive detuning. A detuning of zero means that the frequency of the laser is resonant with the transition. The photon momentum of the laser beams can be adjusted as well. The value of the photon momentum in the experiment is a factor of the recoil of a photon in resonance with the 6S_1/2-6P_3/2 of cesium.

The box labeled System Setup contains information on time step, magnetic field gradient, and radiation trapping arclength. The input box labeled time step allows the user to adjust the time interval in which the graphics are updated. The value of the time step has a minimum of 1 and a maximum of 25,000. The units of the time step are given in nanoseconds. The magnetic field gradient can be adjusted by changing the value of the slider and has the units of Gauss per centimeter. The radiation trapping arclength refers to the scale of an arbitrary capture window. Increasing the radiation trapping arclength increases the chance that an atom will absorb a photon emitted from a neighboring atom. The rate of radiation trapping can be monitored during a simulation and appropriately adjusted.

The control buttons labeled Ok, Cancel, and Apply allow the user to register the values and close the form. The Ok button registers the current values of the experimental setup with the program and closes the form. The Apply button simply registers the experimental setup but leaves the form open. This allows the user to observe how changes in the experimental setup alter aspects of the program without closing the form. The Cancel button closes the form and ignores any changes made to the experimental setup.

The next function in the Simulation drop down box is labeled Magneto-Optic Trap. This function allows the user to choose between a Light MOT or Dark MOT. Remember that the Light MOT includes radiation trapping and consequently is slower computationally. Moreover, when the Light MOT option is checked, a box appears on the screen which calculates the number of radiation trapping events per emission in a given time step. The default value, however, is the Dark MOT.

The Record Options function opens a new form on the screen as seen in figure 3.4. This form has two different input boxes. One is for a time step and the other is for the total time of the simulation. The form also displays the estimated size of the simulation in bytes which is a function of the time step, total time, and number of atoms. The user has the option of two types of recordings, either Record Simulation Data or Record RMS Data. The simulation data records, in a .CSM file, the data needed to play back a simulation. The RMS data records, in a text file, the RMS Position and Velocity at each time step during the simulation. The Record Options form is automatically presented to a user when the Record Simulation function is called. Note that when recording a simulation the time step in the Record Option form overrides the value of time step in the Experimental Setup form.

Figure 3.4. A screen capture of the Record Options form.

The final item in the simulation drop down box is entitled, "Cool Movies." Clicking this option opens a movie player which can be used to play .AVI files. The Movie Player is seen in Figure 3.5. This utility is particularly useful for playing movies from the actual experiment. The Movie Player has its own menu bar and command buttons across the bottom of the form to control its functions. To open a movie, click on File in the menu bar and select open. The user is prompted to open an .AVI. Once the movie is opened it can be played, paused, or stopped using the control buttons at the bottom of the form. The Movie Player can be exited by clicking the Exit control button or Exit in the drop down box of the File menu bar.

Figure 3.5. A screen capture of the Cool Movie Player. With this form, the user can play .AVI files of the real laser cooling and trapping experiment.



3.3.3 Help

The function labeled Help in the menu bar reveals the features added to help the user with program operation and theory. Clicking on Contents in the drop down box opens the Contents page of the Cool Help file. The Contents page contains an overview of the features of the help file. The next function in the drop down box is labeled Search For Help On. This function opens the help file search engine which allows the user to input key words or phrases in order to get help. The section labeled Atomic Theory when clicked lists the four atom/photon interactions calculated in Cool. Clicking on one of these events opens a form which shows a simple animation of the particular interaction. An example of the Atomic Theory form can be seen in Figure 3.6. The final item in the drop down box is labeled About. The About function displays a form containing the name, version, and designer of the program.

Figure 3.6. A screen capture of the Atomic Theory section of Cool. Forms like this one demonstrate the Atom/Photon interactions calculated in Cool. The one shown here demonstrates radiation trapping.

3.4 Charts

Cool offers the user two animated charts labeled Position and Velocity which present position and velocity data during an experiment. Position and Velocity charts are displayed in Figure 3.7. The Position chart by default shows the largest region (4 mm by 4 mm) that can be viewed in the program. This region represents the boundary for the atoms. At the start of a simulation atoms are randomly distributed in this region. As the simulation continues, the atoms' positions are updated at a user defined time step. Functions that pertain to the chart can be accessed by right clicking the mouse over the chart. When this is performed, a pop-up menu displaying Full View, Zoom, Copy, Save Position Chart, and Print Position Chart appears on the screen. The Full View function enlarges the Position Chart to fill the program form. The Position Chart is returned back to a normal view by right clicking on the chart and selecting Normal View. The Zoom function changes the scale of the chart and updates the axis labels accordingly. The user

Figure 3.7. The two charts pictured above are examples of Position and Velocity charts taken from Cool. These charts provide animated pictures of the atoms' position and velocity during a simulation.

has the option to zoom in or zoom out by clicking on the listed percentage. Clicking on the Copy function copies the chart to the Windows Clipboard so the chart can be pasted into another application. The Save and Print functions perform the same functions as the save and print functions in the menu bar. Note that when a chart is enlarged it is saved and printed with that particular size.

The Velocity chart is located underneath the Position chart. This chart displays an animation of the atoms' velocities during an experiment. The Velocity chart has similar functions as the Position chart with only one modification. This chart has very flexible scaling and may be zoomed in or out well beyond the default value. The default value is determined by the value of the initial x and y velocity (whichever is larger) in the Experimental Setup form. The atoms are randomly placed in the chart using the initial x and y velocity sliders before a simulation is started.

3.5 Graphs

Cool offers several graphs which are displayed in the box in the upper right portion of the screen. The box next to the graphs allows the user to select between graphs and recalculate a graph. The graph options include: Doppler Probability, Speed Distribution, X Distribution, Velocity vs. Time. The Doppler Probability graph displays the probability of absorption from a trapping beam versus the velocity of an atom. A positive velocity represents an atom moving toward the beam and a negative velocity denotes an atom moving away from the beam. The limits of the velocity axis are governed by the value of the initial x and y velocity sliders. The Speed Distribution graph counts the number of atoms within a certain velocity bin and plots the distribution. Similarly, the X Distribution graph calculates the number of atoms within a certain position bin in the X direction and plots the result. By clicking the Recalculate button both of these distributions can be updated since they often change throughout the course of a simulation. The final graph option is the Velocity vs. Time graph. This graph is a strip chart that plots the root mean square velocity of the system during a simulation. Note, that this option can only be activated before a simulation begins. Furthermore, the Recalculate button has no effect on this graph since it is updated continuously.

The graphs have many of the same functions available to them as do the charts. By right clicking on the mouse over the graphs, the user can save, print, or copy the graphs. The Velocity vs. Time graph can be scaled as well. The graphs, however, only provide a glimpse of the state of the system. Using the data conversion utilities the user can export data to a spreadsheet or other application for further analysis.

3.6 Miscellaneous Features

The boxes labeled Simulation Setup and Simulation Data contain all of the pertinent data of the program. The Simulation Setup box shows the number of atoms, the initial velocity settings, the laser intensity and detuning, and the magnetic field gradient. When the program is started each of these parameters has a default value. The Simulation Data box shows information concerning time, root mean square position and velocity, and temperature. All of these values are updated at a user defined time step. The time is given in units of nanoseconds. The RMS position and velocity are given in units of millimeters and meters per second, respectively. The temperature is determined using the RMS velocity and assuming the atoms are cesium. The temperature is in Kelvin.

At the bottom of the form, Cool offers several other items. The two status bars on the left side report the percentage of atoms in the ground and excited state. Notice, too, that the color green corresponds to ground state while red denotes the excited state. Next to these two status bars are two check boxes entitled Optimize Calculation and Monitor. The Optimize Calculation option when checked turns off the graphics during a simulation. This allows the algorithm to run much faster. During a recording of a simulation the Optimize Calculation check box is checked automatically. The Monitor function minimizes the main form and produces a smaller screen which shows the data listed in the Simulation Data box. The user can then work on another application and still monitor the progress of a simulation. Closing the Monitor form returns the main form to its normal size. Just above the check boxes, during a Light MOT simulation, is a box which reports the rate of radiation trapping. This percentage is determined by the ratio of radiation trapping events to the total number of emissions during a time step. The status bar and slider to the right show the progress of a recorded simulation or a recording. When a simulation is opened and the pause button is clicked the user can use the slider to move to different locations in the simulation file. During a recording the slider is disabled but the status bar shows the progress of the recording. Finally, at the bottom of the screen there are three control buttons. These buttons control the actions of a simulation.

3.7 Running, Recording, and Playing

Upon entering Cool a simulation can be run by simply clicking the Start button since all variables have a default value. The default values are such that the user should notice the velocity and the temperature decreasing over time. The default magnetic field gradient is zero, so no trapping will be evident. By changing the experimental setup, as described earlier, the user can test cooling and trapping versus specific variables.

In order to record a simulation, set the variables to the desired values and click on the Record Simulation function in the File menu bar. The user is then prompted to enter the desired time step and total time of the simulation. Notice that the time step is in units of nanoseconds and the total time is in units of seconds. As these values are being set a text box displays the estimated size of the simulation file in bytes. In order to play back the simulation, make sure the Record Simulation Data option is clicked. After these values are registered, the user is prompted give the simulation a name and may begin recording. The program will record the simulation and automatically stop the recording when the total time has elapsed.

The user can play back a simulation by clicking on the Open Simulation function in the File menu bar. The user is prompted to open the desired .CSM file. The .CSM is loaded into the program and the user may begin the simulation by clicking the button labeled Play.

3.8 Computational Results

The purpose in the design of Cool was to allow users to simulate laser cooling and trapping experiments. The next few pages contain detailed descriptions of computational experiments, an explanation of results given certain configurations, and how these results correlate to actual experimental results.

3.8.1 Experiment #1: Optical Molasses

As mentioned in section 2.2, optical molasses can be created by counterpropagating red shifted laser beams. The default values of Cool are configured such that optical molasses can be created by clicking Start once the program is opened. The value of variables for this simulation are:

The atoms in this experiment are randomly distributed in a 4 mm by 4 mm area. The x and y velocities for the atoms are randomly distributed between 5 m/s. The laser intensity of 1 means that the laser intensity is exactly the saturation intensity. A detuning of -1G corresponds to a frequency one natural linewidth to the red of the resonant transition. Figure 3.8 shows numerical and graphical results from Cool.

Figure 3.8a shows the numerical data taken from the simulation. Notice that the data is split into two columns. The one labeled Start shows the initial values of the system while the column labeled Equilibrium shows final values. The equilibrium time is defined to be the time where the position, velocity, and temperature values seem to oscillate around a number. Given the randomness introduced in the calculations, however, this value will not be exact. It is merely reported to give the user an idea of the computational time for a particular simulation and a place to make measurements. The root mean square position values at the start and equilibrium show that the atoms generally had no net change in their position. This is to be expected. Dramatic differences are noticed, however, in the initial and final velocity and temperature measurements. This demonstrates the radical cooling power of laser cooling and trapping. Figure 3.9 shows a plot of the RMS Velocity versus time and the equilibrium level.

Part b of figure 3.8 shows example charts from the program. The two charts on the left represent initial conditions while the two on the right show equilibrium conditions. (Remember: These charts are easily copied, saved, or printed by right clicking on the chart.) The position graphs for the most part show the same distribution of atoms. This is not surprising since the initial and final RMS positions were essentially the same. Comparing the velocity charts, one can see the dramatic change in the velocities. Notice that the velocity chart on the right has different axes than the one on the left. Also, notice that the final velocity distribution has circular symmetry denoting an equal cooling force from all directions.

Figure 3.9. Velocity versus time graph taken from Cool.

3.8.2 Experiment #2: The Dark Magneto-Optic Trap (dark MOT)

The next experiment will demonstrate the dark MOT. Remember that the dark MOT utilizes a magnetic field gradient to distort the energy levels of the cesium in such a way that the atoms experience a net inward force. The computational experiment is no different. In fact, the same experimental setup as in the previous experiment is used with the addition of a non-zero magnetic field gradient. The computational setup is as follows:

Clicking Start on the screen will begin the simulation. Notice that the atoms are randomly distributed in a 4 mm by 4 mm region with a random x and y velocity between 5 m/s. In both charts, the atoms should begin to converge to the center and quickly reach equilibrium values. The results for this experiment are displayed in Figure 3.10.

Dramatic changes are seen now in both the RMS velocity and position values. Notice that the final velocity and temperature values are similar to the optical molasses values denoting no real change in the cooling force. What is different, though, is the change in the RMS position values. It is easy to see the results of the net inward force caused by the magnetic field gradient.

Parts b, c, and d of figure 3.10 show the position charts from Cool at the initial and equilibrium time. (The velocity charts for this experiment were similar to those in figure 3.8.) The chart labeled b shows the initial distribution of the 100 atoms. Chart c shows the equilibrium chart. The atoms have clearly been compressed into a smaller region. Chart d shows a slightly different scale of chart c. In this chart, it is easy to see the new distribution. Again, the circular symmetry of the distribution denotes an even force from all sides. Notice also, that the atoms seem to have some distribution within the circle. This unique distribution will be the subject of the next experiment.

3.8.3 Experiment #3: Dark MOT versus Light MOT

The difference between a dark MOT and a light MOT is radiation trapping. Remember, radiation trapping is a net repulsive force due to the re-absorption of a photon by a neighboring atom. In the laboratory, a comparison of the dark and light MOT is rather difficult. Fortunately, using Cool, the comparison can be made with only a few mouse clicks. Let us now increase the number of atoms in the simulation from 100 to 5000 to obtain a better statistical sample and run a similar dark MOT simulation as described in Experiment #2. Therefore the dark MOT settings should be:

The computation should be run until equilibrium is reached. Pause the simulation and use the Save Current Data function in the menu bar to save the simulation data at equilibrium. The data will be in the form of a text file.

The program should then be prepared to compute the light MOT. This can be achieved by clicking on the Light MOT option in the menu bar. The radiation trapping arclength, which alters the rate of radiation trapping, can remain at the default value 2. The experimental values should resemble:

By clicking the Start button, the simulation will begin. Remember that the light MOT is much more computationally intensive than the dark MOT and will take longer to run. When performing this type of simulation the Optimize Calculation option should be utilized. The rate of radiation trapping can be monitored in the box labeled Light MOT. Once the simulation reaches an equilibrium value, save the equilibrium data into a text file.

Now that both simulations have been run and the data has been saved, export the text files to a spreadsheet program or similar application for analysis. Figure 3.11 shows a comparison of the equilibrium X position distributions of the dark and light MOT. Notice how the distribution of the light MOT is nearly flat across the top. This is due to the net repulsive force of the atoms in the MOT. The dark MOT, however, has a shape fit nicely with a Gaussian curve. The fit reveals a spring constant, k, of 3.8 x 10^-19 N/m. It should be noted that the spring constant is the slope of the graph in figure 1.7. This graph shows the position dependence of light force on atoms in a MOT. A theoretical calculation of the slope of this graph, inserting the proper detuning, laser intensity, and magnetic field gradient, yields a spring constant of approximately 4 x 10^-19 N/m which is strikingly similar to the computational result.

Figure 3.11. This is an X position distribution plot of a dark and light MOT at equilibrium.

It should be noted that there was a slight increase in the RMS velocity and temperature in the light MOT. This is because the radiation trapping provides another source of internal energy. The RMS velocity at equilibrium for the light MOT was about 0.17 m/s and the temperature was about 290 mK. Remember that the radiation trapping arclength value used by Cool is merely an approximation. In order to obtain values for the light MOT that match experimental values this parameter must be adjusted.

3.8.4 Experiment #4: Cooling and Trapping with Higher Velocity Atoms

This experiment was designed to simulate more realistic laboratory conditions. In the experiment described in Section II, the atoms are contained in a low pressure chamber at room temperature. The atoms in the sample have velocities governed by the Maxwell-Boltzmann distribution. Thus some of the atoms have velocities approaching zero while others have velocities much higher. The cooling and trapping force exerted by the lasers is only sufficient to stop atoms of certain velocities. The probability of absorption (net stopping force) of atoms with certain velocities can be ascertained from the Doppler Probability graph in Cool. A sample Doppler Probability graph with a detuning of -G is the subject of Figure 3.12.

Figure 3.12. This is a Doppler Probability graph from Cool corresponding to the configuration of this experiment. Notice that graph suggests a capture velocity of approximately 15 m/s.

The major change of variables in this experiment is the initial x and y velocities. Most of the other variables are the same as in other experiments. The experimental setup is as follows: (The choice of the dark MOT in this experiment is simply for computational speed.)

The simulation is started in a similar fashion to other experiments. One major difference that can be seen is in the velocity chart. The dimensions of this chart are now 25 m/s in the x and y direction. The results of this experiment are reported in figure 3.13.

Figures 3.13a and b show position charts from this experiment. The initial position chart a is the same as in other experiments. The equilibrium position chart b is different than other dark MOT position charts and warrants some explanation. In the center of the position chart one can see trapped atoms in the typical circular Gaussian distribution. Around the trapped atoms, though, are other atoms which in this picture seem to have had no net change from their initial position. Indeed, these particular atoms have not been trapped and are actually moving around the trapped atoms. The fact that they are not trapped is due to their high velocity and can better be demonstrated with the velocity charts.

Figures 3.13c, d, and e show the velocity charts from the experiment. The initial chart c shows 100 atoms randomly distributed between 25 m/s in the x and y direction. Charts d and e show equilibrium values. As in the position charts, we see that a group of atoms have converged around the origin. These are the atoms that have been cooled. Yet there are two more groups of atoms which provide an interesting insight into the meaning of the Doppler Probability. One group of atoms has clustered on the axes. These atoms had initial velocities in either the x or y direction that were within the capture velocity. (Refer to figure 3.12.) But, the velocity in the other direction was too high to be cooled and trapped. The second group of atoms had velocities in both directions that were too high to be cooled and are located off the axes and away from the origin. Chart e shows a zoomed in version of the equilibrium velocity chart showing that the cooled atoms in the experiment have similar values as those in experiments where all of the atoms were cooled and trapped. Figure 3.14 shows plots of the initial and equilibrium speed distributions for the experiment. Notice that there is a clear division between cooled and background atoms in the equilibrium plot.

Figure 3.14. Speed Distribution plots taken from Cool show a system with high velocity atoms in the background.

It should be noted, though, that given enough time all of these atoms would be cooled and trapped which is one point where the simulation diverges from experiment. This is due to the fact that atoms are confined to a region that is 4 mm by 4 mm and always in the laser beams. In real life, atoms of high velocity (background atoms) would pass straight though the beams and into the chamber experiencing little or no cooling. Furthermore, these background atoms collide with cooled atoms knocking them out of the trap. This phenomenon is not accounted for in the simulation.

3.8.5 Experiment #5: Effects of Detuning

As seen in the preceding experiment, the Doppler Probability dictates which atoms will be cooled and trapped. This graph is created from the scatter rate equation derived in section 1.3.3. In this experiment, we will look at how the detuning of the laser beams changes this Doppler Probability. We will examine what happens when the lasers have zero detuning and also when the lasers are "blue" shifted. Finally, we will examine how detuning affects the equilibrium temperature. Remember that "detuning" refers to the difference in laser frequency from the resonant transition in an atom. In Cool, this resonant frequency is the 6S_1/2 to 6P_3/2 transition in cesium which corresponds to about 3.52 x 10¹⁴ Hz. Using the program the lasers can be detuned as a positive or negative factor of the natural linewidth (G ).

Figure 3.15 shows several Doppler Probability graphs with different detunings. Notice that as the detuning changes so does the velocity where the maximum probability for absorption occurs. Note that a negative detuning such as in parts a and b have maximum probabilities when atoms are moving toward the laser. As in parts d and e, a positive detuning has a maximum probability when atoms are moving away from the beam. A zero detuning corresponds to a laser frequency equivalent to the resonant transition and favors positive and negative velocity atoms equally. It is easy to see, that the detuning of the laser determines what velocity atoms the photons will interact with and alters the computational results.

Using Cool, a simulation is created resembling previous experiments. Except, in this simulation, the detuning is set to zero. The experimental setup is as follows:

Once the simulation is begun, the RMS velocity and temperature slightly increase. Figure 3.16 shows a plot of the RMS velocity versus time.

Figure 3.16. This figure shows the RMS velocity versus time. Notice how the value slightly increases in time.

The initial RMS velocity of the system was 4.1 m/s, and after 1ms the value was about 4.3 m/s. Although the detuning is such that absorption cools as many atoms as it heats, there is a recoil heating due to spontaneous emission. This is an example of diffusion referred to in section 1.4.

Having seen that a zero detuning has a slight heating effect on the system, we will now give the lasers a positive detuning. A laser whose frequency is positively detuned is said to be "blue" shifted. A computational experiment is created in Cool using similar settings as in the previous simulation except the detuning is positive. This simulation's setup is as follows:

As expected, the RMS velocity and temperature start to increase. Figure 3.17 shows the results from this experiment. Parts a and b of figure 3.17 show the initial and equilibrium velocity charts. The initial velocity chart shows 100 atoms randomly distributed between 5 m/s in the x and y direction. The equilibrium chart shows the dramatic increase in velocity due to the positive detuning. Notice how the atoms are forming four symmetric groups about the origin. This denotes an equal heating force in all directions. Part c of the figure shows the rapid heating of the atoms during the first millisecond of the simulation.

We know from previous simulations that negative detunings cause the atoms to be cooled. But, does the equilibrium temperature vary with the negative detuning? This question is the subject of the next simulation. In this simulation, we are going to increase the number of atoms to 1000 to obtain a better statistical sample. The detuning is negative and variable and the rest of the settings are similar to those in other simulations. The setup is as follows:

For this simulation eleven different detunings were used and the equilibrium temperatures calculated. Figure 3.18 shows a plot of this data. Notice that the lowest temperature is between -0.7G and -0.5G . In calculating the Doppler limit, theorists predict that a detuning of -0.5G yields the minimum temperature.⁷ Remember that for a three-dimensional cooling and trapping apparatus the Doppler limit for cesium equals

125 mK. The Doppler limit calculated by the simulation is approximately 170 mK.

Figure 3.18. This plot shows equilibrium temperatures for various detunings. Notice that the minimum temperature is approximately 170 m K and occurs between -0.7G and -0.5G.

3.8.6 Experiment #6: Effects of the Magnetic Field Gradient

This experiment shows the effects of the magnetic field gradient on the system. In Experiment #2 we saw that with negatively detuned laser beams and a magnetic field gradient one can create a cold, trapped cloud of atoms. This same procedure will be used in this experiment except that each simulation will have a different magnetic field gradient. The computational setup is as follows:

Figure 3.19 shows the results from five simulations all with the same parameters except for the magnetic field gradient. Notice that the change in magnetic field gradient has little effect on the equilibrium RMS velocity or temperature. Yet as the magnetic field gradient increases the RMS position significantly decreases. Figure 3.20 shows a plot of the concentration of atoms versus the magnetic field gradient for the simulations. From the plot we see that the concentration increases linearly with an increasing magnetic field gradient.

Figure 3.19. This chart shows data obtained from the various simulations. Notice how the concentration changes with the magnetic field yet the temperature remains constant.

Figure 3.20. A plot of the concentration of trapped atoms at equilibrium given a specific magnetic field gradient. The upper and lower points at each gradient denote the range of concentrations at equilibrium.

3.8.7 Experiment #7: The Cooling Rate

In Section I we saw that counterpropagating beams that have the proper intensity and detuning exert a cooling force on the atoms. This force was given as:

where was the damping coefficient and v was the atom's velocity in the direction of the laser. This simulation is designed to analyze the rate at which atoms are cooled by the four counterpropagating beams. The simulation setup is as follows:

This simulation should be recorded using the Record RMS Data option in the Record Options form. Once this is done a text file is created that contains the RMS velocity, RMS position and the time. We are only interested in the change in RMS velocity to obtain a cooling rate graph. The cooling force should damp the atomic velocity such that:

Figure 3.21 shows a natural log plot of the data for this simulation. Notice that the data is nearly linear. The fit yields a slope of approximately -1.2 x 10⁴/s. Thus the cooling rate is given by:

Figure 3.21. This is a plot of the Ln(V_rms²/V_{o RMS} ²) versus time. The black line shows the data and the gray line is the fit. The cooling rate is approximately 1.2x10⁴/s.

For this particular setup, a theoretical calculation of the cooling rate yields approximately 1.1 x 10⁴/s. Again, the results from Cool have good agreement with theory.

3.8.8 Experiment #8: The Velocity Distribution of Cold Atoms

An important question to answer in laser cooling and trapping is: What is the shape of the speed distribution of the cold atoms at equilibrium? Ordinary atoms at room temperature distribute their speeds such that they can be described using Maxwell-Boltzmann statistics. Therefore, atoms in two dimensions have a speed distribution governed by:

where v is the speed in two dimensions, k_B is Boltzmann's constant, and T is the system's temperature in Kelvin. This simulation is designed to investigate whether the cold atoms at equilibrium obey Maxwell-Boltzmann statistics. The simulation setup is similar to the one used for the optical molasses except the number of atoms is increased to 5000 for a better statistical sample. This simulation is as follows:

The simulation was allowed to run until an equilibrium was reached and the data was saved using the Save Current Data function. The text file was imported into a spreadsheet program for analysis. Figure 3.22 shows the plot of the equilibrium speed distribution of the cold atoms. The data was fit with the two dimensional Maxwell-Boltzmann speed distribution function. The fit reveals a theoretical temperature of

190 m K which matched closely with the computational value (192 m K). It appears that laser cooled atoms distribute their speeds according to Maxwell-Boltzmann statistics.

Figure 3.22. A plot of the equilibrium speed distribution of optical molasses. The fit represents a Maxwell-Boltzmann speed distribution function for two dimensions. The temperature revealed from the fit is 190 m K.

3.8.9 Experiment #9: Random Walks: Watching a Single Atom being Cooled

We have seen that the atoms in a laser cooling and trapping experiment experience a strong cooling force. In Section I we also discussed briefly a heating or dispersion caused by spontaneous emission. One then might offer the question: What does this cooling and heating look like and how does it affect the atom? The question can be answered by recording a simulation of a single atom being cooled down to equilibrium. The simulation setup is as follows:

Figure 3.23. This is a plot in velocity space of a single atom being cooled. Notice the competing cooling and heating forces.

This simulation is recorded using the Record Simulation Data option in the Record Options form. The .CSM file is then converted to a text file and exported to a spreadsheet application for analysis. Figure 3.23 shows a plot of the atom in velocity space as it is cooled to equilibrium.

By starting a simulation with zero x and y velocities, we can observe the behavior of a single atom about the origin in velocity space. The simulation is configured the same as the previous simulation except the initial velocities are set to zero. Figure 3.24 shows a plot of the atom as it starts from rest and is heated by subsequent absorption and emissions.

Figure 3.24. A plot of a single atom in velocity space starting from rest. Notice how the atom is quickly heated.

The vertical and horizontal lines represent absorption or stimulated emissions and the diagonal lines represent spontaneous emissions. The atom mimics a random walk through velocity space until a certain point where an equilibrium is established. This point represents a balance between the cooling and heating forces (i.e. the Doppler Limit). Figure 3.25 shows the equilibrium the atom reaches if the previous simulation is allowed to continue.

Figure 3.25. This is a plot in velocity space of an atom at equilibrium in four counterpropagating beams. The atom initially had zero velocity and was heated until a balance between the cooling and heating force was achieved.

3.8.10 Experiment #10: Watching a Single Atom in a MOT

We have seen that the total force experienced by an atom in a MOT is a function of both velocity and position. The equation of motion for the atom in the apparatus has the form of a damped harmonic oscillator. This simulation is designed to investigate this equation of motion. As in the previous simulation we will observe a single atom. The difference in this simulation is that we will now employ a magnetic field gradient. The setup is as follows:

The simulation is recorded by choosing the Record Simulation Data option in the Record Options form. The .CSM file was converted to a text file and exported to a spreadsheet application for analysis. Figure 3.26 shows a phase space plot of the atom as it is cooled and trapped. Notice how the plot resembles that of a damped harmonic oscillator.

Figure 3.26. A phase space plot of a single atom as it is cooled and trapped in a MOT. Notice the plot resembles a phase space plot of a damped harmonic oscillator.