Download Precision Spectroscopy in Alkali Vapor

Precision Spectroscopy in
Alkali Vapor
Undergraduate Project in Physics
Submitted by Orel Bechler
Supervised by Dr. David Groswasser
Department of Physics, Ben-Gurion University
Last revised: Sep 16, 2009
Table of Contents
1. Introduction
1
2. The Rubidium Electronic Structure
1
3. Atomic Clocks
5
4. Coherent Population Trapping (CPT)
7
5. Vertical Cavity Surface Emitting Lasers (VCSELs)
12
6. Rubidium Vapor Cells
16
7. Experimental Status in the Atom Chip Lab
18
8. Conclusion
20
9. Bibliography
21
1. Introduction
For many years, conventional atomic clocks were based on an alkali vapor cell contained
in a microwave cavity. The microwave radiation is applied to excite transitions between
the ground states of the atoms, which are extremely accurate, due to the long lifetime of
the ground states. This use of a microwave cavity is important, so that the microwave
field will be homogenous and the atoms will experience the same spatial interaction.
However, this sets a limit to the size of the atomic clock. In contrast, an atomic standard
based on the phenomenon of Coherence Population Trapping (CPT) has no need for a
cavity and therefore enables further miniaturization, now limited by the vapor cell
dimensions.
Furthermore, in recent years, the development of Vertical Cavity Surface Emitting diode
Lasers (VCSELs) that are directly current-modulated using a tunable microwave
oscillator facilitate the realization of the two phase coherent optical fields that are needed
for the observation of CPT. High precision CPT spectroscopy allows very accurate
measurement of the transition frequency between Zeeman sublevels belonging to the two
hyperfine manifolds. In Rubidium, the atomic clock operates between F=1, mf=0 state
and the F=2, mf=0 state (a “clock transition”). These mf states are not sensitive to
magnetic fields and can be used to construct a CPT-based atomic clock. In addition, CPT
between mf≠0 states can be used to probe magnetic fields.
This work summarizes the basic principles of CPT and describes the experimental
considerations and progress taking place at the atom chip group on the way for the
realization of a CPT-based clock/magnetometer.
2. The Rubidium Electronic Structure
Rubidium’s electronic structure includes a single valence electron in its outer shell, a
characteristic of all alkali metals, including Cesium, Sodium and Potassium. This unique
trait is what makes alkali atoms a relatively simple system to study and such popular
research system of choice in quantum optics.
1
The theoretical method required in determining Rubidium’s electronic energy levels
depends on the precision of the external electromagnetic field that is to be used
experimentally. Treating the problem as a simple hydrogen-like atom and solving
Schrödinger’s equation for a system containing a single negatively-charged electron
bound to the nucleus by an isotropic central Coulomb force, yields the most basic
electronic structure in which the atomic energy depends merely on the n (principal) and l
(angular momentum) quantum numbers. The energy splitting in this most elementary
treatment is in the hundreds of Terahertz region.
However, other interactions have to be taken into account in order to explain the detailed
level structure and observed spectrum. Those are described in [1] for example, include
the coupling between the orbital angular momentum, L, of the outer electronic shell and
its spin angular momentum, S. This “spin-orbit coupling” affects the energy levels of
atomic electrons and can be visualized as a magnetic field caused by the electron's orbital
motion interacting with the electron spin. The interaction energy is that of a magnetic
dipole in a magnetic field and takes the form:
 
 1
1 V  r    
 so     B eff   2 2 
LS
 2me c r r 
In this treatment, L and S are no longer good quantum numbers, and in order to describe
the system there is a need for a new quantum number that commutates with the perturbed
Hamiltonian. The total electron angular momentum is then given by:
  
J  LS
The corresponding quantum number J must lie in the range:
LS  J  LS
Resembling the formulation for the angular momentum and spin operators, the magnitude
of J is
J  J  1 and the eigenvalue of Jz is m j  . The energy of any particular level is
now determined by these values. For instance, the ground state of
87
Rb is characterized
by L=0 and S=1/2, so J=1/2  5 2 S1/2  , and the first excited state is described by L=1 and
S=1/2, so J could come out to be either 1/2  5 2 P1/2  or 3/2  5 2 P3/2  . The energy splitting
2
of the two excited states in this fine structure of the atom is around 7 THz, requiring a
more precise light source in order to distinguish between them.
Finally, a higher resolution of the electronic structure of an atom can be established by
coupling the total electronic angular momentum, J, with the total nuclear angular
momentum, I. The electron, in its orbital motion around the nucleus, feels a magnetic
field generated from the relative movement of the charged protons, which results in
further energetic shifts in the atomic spectra. Expressing the interaction Hamiltonian
using a similar treatment of a magnetic dipole in a magnetic field leads to the formulation
of the total atomic angular momentum, F, which is given by:
  
F IJ
As before, the magnitude of F can take the values:
J I F  J I
The value of I depends on the number of protons and neutrons that exist in the nucleus.
For
87
Rb and
85
Rb , for example, I=3/2 and I=5/2 respectively [2]. The two atoms are
distinguished from one another merely by the nucleus and the number of neutrons that it
contains, which consequently gives rise to fairly different electronic structures. For the
aforementioned
87
Rb ground state, J=1/2 and I=3/2, so F=1 or F=2. For the excited state
of the D2 line  5 2 P3/2  , F can come out to be either 0, 1, 2 or 3, and for the D1 excited
state  5 2 P1/2  , F can be either 1 or 2.
The energy splitting for this hyperfine structure is in the tens (in the excited state) to
thousands (in the ground state) of MHz region (figure 1).
3
Figure 1- Energy diagrams for both the 85Rb and 87Rb atoms, showing the D1 and D2 transitions,
along with the fine and hyperfine splittings in MHz [3]
According to figure 1 on the right,
87
Rb ’s ground state hyperfine splitting is about 6.8
GHz. These two levels typically have extremely long lifetimes that survive for years, in
contrast to the excited states having a lifetime of only several nanoseconds. Accordingly,
each transition experiences a lifetime broadening which is directly related to the
Heisenberg’s Uncertainty Principal:
E  t 

2
This relation is the fundamental motivation for using transitions between the 5 2 S1/ 2
ground states hyperfine levels for various applications in quantum technology such as
atomic clocks [4-6], magnetometers [7], slow and fast light [8], quantum memory for
photons [9], and more. In addition, the use of alkali atoms, such as Rubidium and
Cesium, each with a solitary valence electron in the outer shell, allows for a much
simpler system to manipulate. Constraining the single electron to just two or three of the
many available energy levels is much easier, which makes controlling an entire ensemble
of atoms possible, as well as considerably simplifying theoretical formulation.
4
3. Atomic Clocks
All mechanical clocks work by counting the vibrations of something which has a constant
frequency such as a pendulum. Unfortunately, the frequency of a pendulum is not
perfectly constant. It is affected by changes in temperature, air pressure and the strength
of gravity. This leads to a lack of precision in count, causing the clock to run too quickly
or too slowly.
As discussed in the previous section, the ground state hyperfine splitting of an alkali atom
can be used as an accurate reference for a periodic signal with a well-defined frequency
and an extreme spectral purity. In practice, the procedure is straight forward. The atomic
ensemble is exposed to microwave radiation while probing the absorption of a laser beam
tuned to resonance between the upper ground state and the excited states. If the
microwave frequency is not exactly tuned to the ground state splitting the absorption of
the probe beam is decreased. A conventional servo system that is able to detect the
variations from the atomic transition frequency is then applied to fix the microwave
frequency to the atomic reference. In practice, this is usually done by modulating the
laser light at low frequency and using a lock-in-amplifier as depicted in Figure 2.
This precise frequency can then be converted into a counter which can time the exact
duration of one second. In the Cesium atomic clock, for example, the frequency of a
microwave oscillator, thanks to an appropriate signal processing, is tuned to be in exact
coincidence with the frequency of a hyperfine transition which is known to a high
accuracy and is equal to 9,192,631,830 Hz [10, 11].
The frequencies measured by atomic clocks are much higher than those of a pendulum
but vary much less, so atomic clocks keep time much more accurately. Modern atom
clocks only gain or lose one second in thousands, or even millions, of years. This makes
the ground state hyperfine splitting of an alkali atom the best time frequency standard
mankind has ever known.
Figure 2 illustrates the relevant energy levels for understanding the conventional doubleresonance technique for interrogating the ground state hyperfine frequency for a gaseous
5
atomic ensemble [5]. In this conventional configuration, the atomic vapor is illuminated
by an optical source, resonant with the D1 or D2 transition between one of the hyperfine
ground states and an absorbing optical state, as well as exposed to a magnetic field,
oscillating at the hyperfine frequency (RF). The transmitted optical intensity is
monitored by a photo detector. In the absence of RF radiation, the absorption is limited
by the depletion of the resonant ground state due to optical pumping. When RF radiation
is applied, resonant with the ground state hyperfine splitting, the optically connected
ground state is repopulated, leading to enhanced absorption (reduced transmission) when
the RF frequency is tuned to the ground state hyperfine splitting. The optical
transmission as a function of RF detuning is indicated schematically in figure 3.
This “RF method” of interrogation is very precise and has been employed in laboratory
and commercial Rubidium frequency standards for over 30 years. The disadvantage of
this approach is in the hardware setup, which requires a resonant RF cavity, as depicted
in figure 4. This cavity is what limits further miniaturization of atomic clocks, its size
restricting clocks to the centimeter regime. Today, installment of atomic clocks on
satellites makes modern navigation systems very accurate, lowering their imprecision to
just a few meters in each direction. The desire to install atomic clocks on handheld
devices, such as GPS and cell-phones, has need for the ability to reduce their size even
further, requiring a different interrogation method for the atomic vapor.
Figure 2- Relevant energy levels for RF interrogation of a gaseous atomic ensemble [5]
6
Figure 3- Optical transmission as a function of RF detuning [5]
Figure 4- Hardware setup for the “RF method” of interrogation [5]
4. Coherent Population Trapping (CPT) [12-13]
In the CPT technique, an atomic sample is simultaneously interrogated by a pair of
collinear, circularly polarized laser beams, tuned so that each couples one of the ground
hyperfine states to the same optically excited state. The frequency difference of the two
external fields matches exactly the splitting between the two hyperfine sublevels of the
atomic ground state. In this specific setup, if both laser beams are phase-coherent,
coherence is generated in the atomic vapor, and atoms affected by the perturbation are
trapped in a non-interacting superposition of the two states, known as a “dark state”.
This method has no need for any sort of cavity, allowing for a much more compact
system in comparison with the “RF method”.
7
c
c
ac bc
b
b
a
a
Figure 5– The three-level
 system, illustrating the CPT transitions [13]
In this case of a three-level atom with two nearly-resonant laser fields, the state of the
atomic system is generally given by a superposition of all three states:
  t   ca  t  e iat a  cb  t  eibt b  cc  t  e ic t c
It is reasonable to assume that the states  a and  b are ground states with very long
lifetimes, and that the state  c is an excited state which decays to the ground states at
some average lifetime  c . In addition, each electromagnetic field interacts with only one
atomic transition. Since E1  1 cos  1t  couples the states  a and  c and
E2   2 cos 2t  couples the states  b and  c , the only non-zero matrix elements of
'
the interaction Hamiltonian are  'ac  1e i1t and  bc
  2 e i2t and their complex
conjugates, where
1 
ac1
2
2 
bc 2
2
are the Rabi frequencies for the two transitions and the parameter
8
ij   i  e  z  j
is the transition dipole moment, its value determined by intrinsic properties of the atom
and represents the strength of the interaction with the external electromagnetic field.
The interaction Hamiltonian (  ' ) is obtained by using the rotating wave approximation,
an approach in which the fast-oscillating part of the solution is neglected, conveniently
leading to Fermi’s Golden Rule [14].
Solving the time-dependant Schrödinger equation
i
 
   t  ;
t
 
0  
',
where 
yields the following equations for the coefficients of the wave function   t  :
.
i cc  1ei (1 ac )t ca  2 ei (2 bc )t cb
.
i cb   2 e  i (2 bc )t cc
.
i ca  1e  i (1 ac )t cc
Taking into consideration the excited state  c ’s finite lifetime, in order to fulfill the
requirement that the light field will propagate through atoms without any absorption, the
.
condition cc  t   0 has to be met. If this is true at all times, the atoms do not get excited
and no spontaneous emission takes place, therefore no energy dissipation occurs. In
other words no light is absorbed, even in the presence of the optical fields. From the first
equation it is clear that this is only possible when:
1ei (1 ac )t ca   2ei (2 bc )t cb  1ca   2cb ei[(2 bc ) (1 ac )]t
9
Namely, if 2  1  ab  b  a   hfs , the exponent is reduced to unity and a twophoton resonance is achieved. This means that the difference in the two external fields’
frequencies must equal the atomic hyperfine splitting. In this situation, the atomic system
can be prepared in the non-interacting quantum state:
 
 2 e  iat  a  1eibt  b
12  2 2
In order to demonstrate that this new state cannot be excited, it is necessary to calculate
the probability of the transition using the electric dipole operator:
Vdip  1e i1t  c  a   2 e i2t  c  b
 c Vdip  
1 2 e i (1 a )t
1   2
2
2
1  e
 i ( b a 2 1 ) t

e i (b a 2 1 )t  1
 c Vdip   0
Since 2  1  ab  b  a   hfs , as previously discussed, the transition dipole matrix
element for  becomes zero, and the transition is not allowed. It is possible to calculate
the transition probability of an additional quantum state  , given as:
 
1e  iat  a   2 e ibt  b
 c Vdip  
12   2 2
e  i (1 a )t
  2
2
1
2

1
2
  2 2  e i (b a 2 1 ) t
e i (b a 2 1 )t  1
 c Vdip   e i (1 a )t 12  2 2
10

 is called a coherent dark state. Atomic population that is pumped out of 
becomes trapped in  . This reduces the fluorescence intensity and makes the medium
appear “darker”. For this reason, CPT is often also called Electromagnetically Induced
Transparency (EIT). It is important to notice that  is a coherent superposition of two
atomic states, sensitive to their relative phase. This requires the two light fields to
maintain their relative phase at all times, so that the atoms can stay in their coherent trap
for as long as the perturbation persists. The above mathematical formulation can also be
carried out taking the phases of the electromagnetic fields into consideration. As long as
the difference in phase remains constant, the same result is obtained.
Figure 6 shows the CPT resonance peak, which is acquired by holding one of the light
fields constant at the proper transition frequency and scanning the other field around the
other transition frequency. When the aforementioned conditions are met, and the
frequency difference coincides with the splitting of the ground-state hyperfine sublevels,
a very sharp minimum absorption peak appears. This peak can be used as a frequency
standard using proper signal processing, very much resembling the technique used in the
conventional “RF method” mentioned in section 3.
Figure 6- Absorption and dispersion spectra measured in the vicinity of the CPT resonance as a
function of Raman detuning [13]
11
5. Vertical Cavity Surface Emitting Lasers (VCSELs) [5, 6, 12]
Using two physically separate lasers in order to generate the two optical fields required
for CPT can be problematic, since there is always a possibility for jumps in frequency
and phase that occur, among other reasons, as a result of changes in temperature. These
random jumps will make both sources lose correlation rather quickly, and coherence will
be lost.
In order to resolve this problem, it is possible to create two light fields with two different
frequencies out of one physical laser source by modulating its phase. The relative
frequency of the two fields can be set by an external generator, which creates a comb
made out of a carrier with sidebands (figure 7). Using this approach, random changes
that occur in the laser will make both fields jump in the same manner, maintaining
coherence between them.
VCSELs are excellent choices as lasers for use in CPT, since they can be phasemodulated by direct current modulation up to extraordinarily high frequencies (10 GHz).
This enables the use of VCSELs both in Cesium and Rubidium vapor, as modulation for
just half of the ground-state hyperfine splitting is generally needed. A picture of a
VCSEL diode is displayed in figure 8.
Figure 7- Modulation of a VCSEL at 6.8 GHz and 9 dBm of power [6]. The sideband/carrier
intensity ratio can be increased by increasing the modulation power.
12
Figure 8- Picture of a VCSEL diode [15]
Phase modulation can be written mathematically as [16]:
E  t   E0 cos c t  M sin M t 
where c is the carrier frequency, E0 is the amplitude, M is the modulation frequency
and M is the modulation amplitude, or modulation index. This equation can be rewritten
as a sum of Bessel functions:
E t  
E0
2

 J M e
n 
i ( c  nM ) t
n
Applying the Fourier transformation reveals a series of monochromatic electromagnetic
fields, each with amplitude equal to their respective Bessel function coefficient. The
carrier frequency is characterized by n=0, and the sidebands are spaced over integer
multiples of the modulation frequency. It is important to note that in order for the
sidebands to be of any appreciable amplitude, the modulation index must be large, which
is possible with a VCSEL.
Characterizing VCSELs and obtaining their output frequencies and intensities is possible
by using a Fabry-Perot cavity. A schematic sketch of the apparatus is shown in figure 9:
13
Figure 9- A schematic diagram of an optical setup for the measurement of phase modulation and
characterization of laser diodes [6]
The Fabry-Perot (FP) interferometer is, essentially, a set of two parallel mirrors, each
typically with 99 percent reflectivity. The mirrors are both aligned perpendicular to the
beam and the waves passing through them constructively interfere to create transmission
through the mirrors at frequencies separated by the free spectral range:
 
c
2d
where d is the distance between mirrors and c is the speed of light in vacuum. The free
spectral range parameter (FSR) determines, among other things, the resolution that can be
achieved by the FP. In a moving FP interferometer, one of the FP mirrors is fixed on to a
Piezo element, which allows variation of the distance between the mirrors by applying
electric voltage. This is depicted in more detail in figure 10:
Figure 10- A Fabry-Perot interferometer (resonator) apparatus,
showing the Piezo element (PZT) [17]
14
Inside the FP resonator, light waves are reflected back and forth between the mirrors. At
a given distance d the mirrors can only form waves which have the field intensity of zero
at both ends of the cavity. These waves are called oscillating modes and they must fulfill
the following condition in order to fit into the FB:
d  n

2
In order to obtain a characteristic spectrum for a specific diode laser, linear changes in the
mirror distance d take place periodically until the above condition is fulfilled for a
specific wavelength (or frequency). The light intensity which is transmitted through the
FP then reaches a photo detector to form a peak for that particular frequency. In this
manner, it is possible to isolate the different frequencies and their intensities and
spectrums such as the one displayed in figure 7 can be obtained.
Figure 11 shows a schematic diagram of a CPT apparatus; where the cavity from the “RF
method” was replaced by a quarter-wave phase retarder (λ/4), in order to circularly
polarize the laser source. As mentioned previously, an atomic clock based on this setup
can be miniaturized even further to the millimeter regime; the limit now is determined by
the size of the vapor cell.
Rb cell
Figure 11- A schematic diagram of a CPT apparatus [5]
15
The next section deals with atomic vapor cells and their contents. As will be explained,
buffer gases at different pressures influence the linewidth of the resonance peak in
different ways. Cell fabrication, therefore, is a very crucial part of this research and can
determine the quality of the obtained signal. Figure 12 is an example of a relatively
narrow CPT resonance peak, its width in the hundreds of Hz regime.
Figure 12- a relatively narrow CPT resonance peak, obtained from a
Cs vapor cell, containing 90 torr of N2/Ar buffer gas mixture.
The Lorenzian fit (in red) indicates a linewidth of γ=438 Hz [18]
6. Rubidium Vapor Cells [8, 19, 20, 21]
As mentioned in the previous sections, an atomic ensemble lies at the heart of many
quantum-optical technologies. Atomic clocks, magnetometers and other devices rely on
mechanisms such as CPT and the “RF method”, which require the manipulation of spin
states of atoms. Such processes necessitate high-quality state preparation and minimal
decoherence of the spin state of the atomic ensemble. In warm atomic vapor cells, spin
state lifetimes are often limited by wall collisions, which destroy spin coherence.
Coating the cell with paraffin (tetracontane) or other organic materials allows atoms to
undergo many wall collisions without losing their spin coherence and can prolong spin
16
lifetimes by up to four orders of magnitude. The smaller the vapor cell, the bigger the
influence the cell walls have, because of increasing surface-to-volume ratio.
Adding buffer gases such as He and N2 to the contents of the cell can also help reduce
decoherence by keeping Rb atoms away from the cell walls, as well as colliding with
each other. Rb-Rb collisions apparently differ from Rb-N2 collision, much resembling
the way collisions with an uncoated cell wall differ from collisions with coated cell walls.
Most atomic vapor cells therefore include both buffer gases and paraffin cell coatings,
resulting in narrow resonances (~1 Hz) that can be used as extremely accurate frequency
standards. Standard cell coating and filling procedures are theoretically simple but can be
technically complex, since the cell has to be filled with the desired materials while
impurities must be kept out. The technique usually includes a Pyrex manifold of some
sort (an example shown in figure 13) connected to a cell and a vacuum system. Coating
the interior of the cell is done by inserting a paraffin-coated rod and heating it to ~2000C
using an oven until the paraffin is vaporized. The cell is then cooled to room temperature
and tetracontane layers are formed on the cell walls. The process may be repeated to
improve uniformity of the coating, after which the rod can be taken out using a magnet.
Figure 13- A schematic diagram of a Pyrex manifold and vacuum system used to apply paraffin
coating and buffer gases to the inside of a cell [8]
Filling the cell with natural abundance Rb vapor is usually done by using an ampoule,
which can be broken by applying force or heating. If a specific isotope is required, an
RbCl salt can be used, in which case the process is more complex, since the Cl has to be
17
prevented from entering the cell during the process. This can be done by reacting RbCl
with BaN6, which will cause the release of the Rubidium isotope into the manifold and
the cell, along with the buffer gas N2. In addition, a solid BaCl residue is formed, which
can be removed rather easily by sealing and detaching that specific section of the
manifold. This process has been described briefly here, omitting a few intermittent
cleaning and vacuum pumping stages which can be added as required.
The procedure in the case of miniaturized vapor cells is quite different. The filling
technique is similar, but since small cell volumes of about 10 mm3 are desired, anodic
bonding is used in order to seal the cell after the coating and vapor are inserted. This
process is described more thoroughly in [22].
7. Experimental Status in the Atom Chip Lab
The Atom Chip group is trying to develop Quantum technology devices based on Rb
vapor cells. For this purpose, significant effort had been invested in learning the
traditional techniques for cell production, paraffin coating and buffer gas effects. This
work is part of this effort. Today, a new experimental setup is being built for making
such cells (figures 14-16). It is important to note that only several experimental groups
and manufacturers worldwide produce high quality cells. In addition, the fabrication
facility had purchased the equipment for anodic bonding of miniature vapor cells. Such
cells are produced only in NIST and making them in BGU will be a significant
technological achievement. Furthermore, we have investigated the modulation of
VCSELs at 795 nm. We were able to observe multiple spectroscopy signals from the
carrier and two (or more) side-bands. However, we were not able to observe the
sidebands using a FP interferometer. Further effort is required in order to optimize the
microwave current modulation efficiency and to purchase new diodes which today are
very expensive and are not always available in the market.
18
Figure 14- A diagram of the glass manifold used for cell production and
filling in the Atom Chip Lab
Figure 15- The glass manifold from figure 13 connected to the vacuum system
19
Figure 16- The entire cell-filling device including the glass manifold,
the vacuum system and the oven
8. Conclusion
Theoretical understanding of coherent population trapping is essential to the application
of the phenomenon in the field of frequency standards. The CPT method of interrogating
atomic vapor has opened avenues in the realization of a new type of miniaturized atomic
clocks and magnetometers. This sate-of-the art technique will be implemented in the
next generation of atomic clocks providing potentially simpler, smaller devices, enabling
their installation on handheld gadgets. The lack of a microwave cavity gives way to
further miniaturization, exposing new areas of research which are crucial to the field.
Now setting the limit to the size of atomic clocks, great efforts are put into development
of miniature atomic vapor cells. Production procedures must include feasible techniques
for coating the interior of the cell with paraffin and adding buffer-gases to its contents, as
well as the ability to insert only a specific isotope of the alkali metal while impurities are
isolated and kept out of the process. The better these conditions are fulfilled, the longer
spin coherence is preserved in the atomic ensemble, enabling the output of a narrower
resonance signal by the device.
20
9. Bibliography
[1] Daniel A. Steck, “Rubidium 87 D Line Data”, available online at
http://steck.us/alkalidata (revision 2.0.1), 2008.
[2] C. J. Pethick, H. Smith, Bose-Einstein condensation in dilute gases, Cambridge
University Press, Cambridge, p. 40, 2002.
[3] S. J. H. Petra, “Development of frequency stabilized laser diodes for building a
Magneto-Optical Trap”, a thesis, 1998.
[4] J. Vanier, “Atomic clocks based on coherent population trapping: a review”, Appl.
Phys. B., 81, 421-442 (2005).
[5] R. Lutwak, D. Emmons, W. Riley, R. M. Garvey, “The Chip-Scale Atomic ClockCoherent Population Trapping Vs. Conventional Interrogation”, 34th Annual Precise
Time and Time Interval (PTTI) Meeting.
[6] N. T. Belcher, “Development of a Prototype Atomic Clock Based on Coherent
Population Trapping”, a thesis, 2008.
[7] W. Farr, E. W. Otten, “A Rb-magnetometer for a wide range and high sensitivity”,
Appl. Phys. A., 3 (5), 367-378 (1974).
[8] M. Klein, I. Novikova, D. F. Phillips, R. L. Walsworth, “Slow light in paraffincoated Rb vapour cells”, J. Mod. Opt., 53 (16-17, 10-20), 2583-2591 (2006).
[9] C. H. van der Wal, M. D. Eisaman, A. Andre´, R. L. Walsworth, D. F. Phillips, A. S.
Zibrov, M. D. Lukin, “Atomic Memory for Correlated Photon States”, Science,
301, 196-200 (2003).
21
[10] http://www.sciencemuseum.org.uk/onlinestuff/stories/atomic_clocks.aspx
[11] G. Chartier, Introduction to Optics, Springer Science + Business Media, Inc., New
York, pp. 408-409, 2005.
[12] N. Belcher, E. E. Mikhailov, I. Novikova, “Atomic Clocks and Coherent Population
Trapping: Experiments for Undergraduate Laboratories”.
[13] R. Wynands, A. Nagel, “Precision spectroscopy with coherent dark states”, Appl.
Phys. B, 68, 1-25 (1999).
[14] I. N. Levine, Molecular Spectroscopy, John Wiley & Sons, Inc., New-York, pp.
117-121, 1975.
[15] http://www.ulm-photonics.de/
[16] C. L. Smallwood, “CPT and N-resonance phenomena in rubidium vapor for smallscale atomic clocks”, a thesis, 2005.
[17] repairfaq.ece.drexel.edu/sam/MEOS/EXP03.pdf
[18] R. Lutwak, D. Emmons, T. English, W. Riley, A. Duwel, M. Varghese, D. K.
Serkland, G. M. Peake, “The Chip-Scale Atomic Clock- Recent Development
Progress”, 35th Annual Precise Time and Time Interval (PTTI) Meeting.
[19] D.F. Phillips, A. Boca, R. L. Walsworth, “Evaporative Coating of Rb Maser Cells”,
notes, 1999.
[20] E. B. Alexandrov, M. V. Balabas, D. Budjer, D. English, D. F. Kimball, C. H. Li, V.
V. Yashchuk, “Light-induced desorption of alkali-metal atoms from paraffin
coating”, Phys. Rev. A, 66, 042903 (2002).
22
[21] J. Kitching, S. Knappe, L. Hollberg, “Miniature vapor-cell atomic-frequency
references”, Appl. Phys. Lett., 81 (3), 553-555 (2002).
[22] S. Knappe, V. Gerginov, P. D. D. Schwindt, V. Shah, H. G. Robinson, L. Hollberg,
J. Kitching, “Atomic vapor cells for chip-scale atomic clocks with improved longterm frequency stability”, Opt. Let., 30 (18), 2351-2353 (2005).
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