Download Modulation Transfer Spectroscopy in Atomic Strontium

M ASTER S CIENCE DE LA MATIÈRE
École Normale Supérieure de Lyon
Université Claude Bernard Lyon I
Stage M2
Qing Fang
M2 Physique
Modulation Transfer Spectroscopy in Atomic
Strontium
Abstract :
We present a blue diode laser system for laser cooling and trapping of neutral strontium
atoms. The laser light is sent to a spectroscopy cell where a frequency discriminant is
generated by modulation transfer spectroscopy. The laser is locked to the 1 S0 → 1 P1
transition at 460.862 nm. After locking the laser to the transition frequency, the laser
light will be used to injection lock other diode lasers to generate the power required for
slowing, cooling, and trapping atoms at mK temperatures.
Key words : Strontium, modulation transfer spectroscopy, cold atoms
Supervisor : Sebastian Blatt
Max Planck Institut für Quantenoptik
Hans-Kopfermann-Straße 1, 85748 Garching bei München
Contents
1
Introduction
1
2
Theory of laser cooling
2.1 Semi-classical description . .
2.2 Zeeman slower . . . . . . . .
2.3 Magneto-optical trap . . . . .
2.4 Laser stabilization is necessary
.
.
.
.
3
3
5
5
5
3
Preparation of spectroscopy cell
3.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Preparation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
6
7
8
4
Modulation transfer spectroscopy
4.1 Absorption coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Basic calculation of error signal . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Experimental setup of saturation spectroscopy . . . . . . . . . . . . . . . . . . . . .
9
9
10
11
5
Laser frequency stabilization
5.1 Linear control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Frequency locking procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Characterization of locking stability . . . . . . . . . . . . . . . . . . . . . . . . . .
15
15
16
17
6
Conclusion and outlook
20
A Spectroscopy cell design
21
B Homebuilt Electro-optic modulator
22
C Injection-locked diode laser
26
Acknowledgments
28
Bibliography
29
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Chapter 1
Introduction
It has been shown a century ago by Einstein that light can be decomposed into photons. The momentum of a photon is transferred to the atoms which absorb it. As a laser is an optical device which
produces a stream of photons, a moving atom can be decelerated by scattering photons from a counter
propagating laser beam. By using a magneto-optical trap (MOT), which will be presented in the next
chapter, neutral atoms can be cooled to a temperature below 1 mK or even 1 µK (Nobel prize 1997 to
Cohen-Tannoudji, Phillips and Chu).
Cold atoms are widely used in quantum simulation [1], quantum metrology [2] and quantum
computation [3, 4]. These systems are a magnificent platform to study not only quantum degenerate
gases but also other fields. For instance, cooling molecules to ultracold (< 1 mK) temperature regimes
can study quantum chemistry [5] and atoms in optical lattices can simulate Hamiltonians of solid state
systems [6].
A variety of atoms can be laser cooled and trapped. Among them, alkaline-earth or alkaline-earthlike elements like strontium, calcium [7] and ytterbium [8] raise recent interest because their two
valence electrons lead to a distinctive level structure, such as narrow intercombination lines [9]. We
are using strontium atoms in our lab.
Strontium has four stable isotopes. Three of them are bosons (84 Sr, 86 Sr, 88 Sr), which have no
nuclear spin. The remaining isotope, 87 Sr, is a fermion with a nuclear spin of 9/2. Experiments with
strontium have been done in many fields. For instance, precision measurements using 87 Sr and 88 Sr
in optical lattices [10] and creating cold molecules such as Sr2 and SrF [11, 12].
The Sr energy levels used for laser cooling and trapping are shown in Fig. 1.1. Because of the
two valence electrons, strontium has singlet and triplet electronic states. Transitions between singlet
and triplet states are spin-forbidden therefore they are very narrow. The 1 S0 → 3 P0 transition has a
linewidth of about 1 mHz, and is used in optical clocks; 1 S0 → 3 P1 is used for the red MOT with a
Doppler temperature of 200 nK; the 3 P2 state is a metastable state which can be used to store atoms
in a magnetic trap [13].
During laser cooling on the 1 S0 → 1 P1 transition, Sr atoms in the 1 P1 state can decay into the
1
D2 state with a small branching ratio of 1:50 000. The 1 D2 state then decays into the 3 P2 and the
3
P1 states. The 3 P2 is a reservoir state where atoms cannot decay into the 1 S0 . The atoms can be
transferred back to the cooling cycle by adding repumping lasers on the 3 P2 → 3 S1 and 3 P0 → 3 S1
transitions. Atoms can decay into the 1 S0 state via the 3 P1 → 1 S0 intercombination line.
To investigate quantum many body physics, we need a reliable laser which can be tuned to the
desired atomic transition frequency. Nowadays, semiconductor lasers can produce many wavelengths
from ultraviolet to infrared [14] and their wavelengths can be tuned by changing the temperature and
the driving current. Commercial blue diode lasers at 461 nm have been invented in the market a few
years ago which allows us to get rid of the complicated second harmonics generation (SHG) laser.
We can increase power by using several blue laser diodes injection locked to the master laser.
In this report, we will discuss how to set up a stable blue laser system for addressing the 1 S0 → 1 P1
1
2
Coo
lin
g
Repumping
C
ling
o
o
Figure 1.1: Illustration of energy levels and transitions in neutral Sr which will be used in our experiment. The solid lines are transitions that are required for the MOTs. The 1 S0 → 1 P1 and the
1
S0 → 3 P1 transitions are used for cooling and the 3 S1 → 3 P2 and 3 S1 → 3 P0 transitions are used for
repumping. The figure is reproduced from Ref. [15].
transition frequency. A spectroscopy cell is used to generate an error signal for frequency locking. The
remainder of this report is structured as follows. In chapter 2, some basic theories of laser cooling are
given for interested readers. In chapter 3, we will discuss a few requirements for the spectroscopy cell.
In chapter 4, we present the generation of the frequency discriminant and our optical setup. In chapter
5, the locking stability is examined. Details of some components are listed in the appendix. This
report also serves as a reference for the next spectroscopy cell for the 1 P0 → 3 P1 intercombination
line.
Chapter 2
Theory of laser cooling
In this chapter, we present the basic theory used in our experiment for understanding what the stabilized laser will be used for. The basic principle of Zeeman slowers and MOTs is presented.
In our experiment, the spectral line will be broadened by different mechanisms. The main broadening mechanisms are Doppler broadening and power broadening. We will consider the power broadening effect when we measure the frequency spread of our error signal in a Doppler free spectroscopy
setup in chapter 4.
A single atom will experience photon recoil force when scattering photons. The strength of this
force depends on the laser power and detuning and we need to tune the laser power and frequency to
have a maximal scattering force.
In our experiment, we use this scattering force to set up our experiment to cool and trap strontium
atoms. The scattering force determines the length of the Zeeman slower which is used to decelerate
strontium atoms and the magnetic field setup of the MOT. The MOT allows trapping neutral atoms
and cools them to temperatures of about 1 mK [16].
After the Zeeman slower, the Sr atomic flux has a typical velocity around 30 m/s such that the
atoms can be captured by a MOT. The Doppler cooling limit can tell us what temperature we can cool
to and whether we need a second MOT or evaporative cooling to achieve µK temperatures. More
details of the semi-classical theory of light can be found in Ref. [17, 18]. The presentation and
notation in this chapter follows Ref. [19].
2.1
Semi-classical description
Power broadening We describe the interaction between an atom and light in a simplified two-level
atom consisting of a ground state and an excited state. Two-level systems are the simplest quantum
systems since the absorption and emission of light cannot exist in a one-level system. A general
two-level system is shown in Fig. 2.1.
In the density matrix description, a two-level system can be characterized by the optical Bloch
equations. In the steady state, the population of the excited state is
ρee =
s0 /2
1 + s0 + (2δ/Γ)2
where δ is the detuning (see Fig. 2.2, the frequency of the red-detuned laser is below the resonant
frequency and the frequency of the blue-detuned laser is above the resonant frequency), Γ is the decay
rate of an atom in the excited state, s0 = I/Isat is the saturation parameter and Isat = πhc/3λ3 τ is
the saturation intensity (h is the Planck constant, c is the speed of light, λ is the wavelength of light
and τ = Γ−1 is the lifetime of an atom in the excited state). The overall decay rate of the excited state
3
4
2.1. SEMI-CLASSICAL DESCRIPTION
Excited state
Excited state
Light
Light
0
Absorption
Red detuned
Blue detuned
Radiation
Ground state
Ground state
Figure 2.1: The two-level system with the absorption and radiation of light. This two-level
system is the simplest quantum systems since
absorption and emission of light cannot exist
in a one-level system.
Figure 2.2: Illustration of laser detuning δ.
The frequency ω of the red-detuned laser is
below the resonant frequency ω0 and the frequency of the blue-detuned laser is above the
resonant frequency.
should equal to the total scattering rate Γ0 of laser light. For decay rate Γ, we have
Γ0 = Γρee =
Γ/2
s0
√
1 + s0 1 + (2δ/(Γ 1 + s0 ))2
0
For s0 1, the scattering rate
√ saturates to Γ → Γ/2. The scattering rate has a Lorentzian shape
which is broadened to Γp = Γ 1 + s0 . This is called power broadening.
Scattering force Another property that can be deduced from the density matrix description is the
scattering force. The absorption of one photon gives a kick to the atom while the spontaneous emission has no preferred direction. The total scattering force is determined by the scattering rate and the
photon momentum ~~k as
Γ
s0
F~sc = ~~kΓ0 = ~~k
2 1 + s0 + (2δ/Γ)2
where k is the wave vector of light and ~ = h/2π is the reduced Planck constant. In the large intensity
limit, the above formula reduces to F~sc = ~~kΓ/2, which is the maximum force.
Cooling limit In the optical molasses where three pairs of counter propagating beams are used to
cool atoms, the total force from pair of counter propagating beams is
F~ =F~sc (ω − ω0 − ~k cot ~v ) − F~sc (ω − ω0 + ~k · ~v )
−2δ/Γ
2
=− 4~k s0
~v
(1 + (2δ/Γ)2 )2
= − α~v
where ω0 is the resonant frequency, ω is the laser frequency, and ~v is the velocity of the atom.
The spontaneous emission of photons causes a random walk of an atom which leads to a non-zero
value of the mean squared velocity. This heating rate must be balanced by the cooling rate given by
Fsc v = −αv 2 . As there are six beams, the total heat rate is 2Er × 2Γ0 . By using the equipartition
theorem 12 mhv 2 i = 12 kB T , the Doppler temperature is given by
~Γ 1 + (2δ/Γ)2
kB TD =
4
−2δ/Γ
where m is the mass of the atom and the minimum temperature kB TD = ~Γ/2 is obtained at δ =
−Γ/2 . Another cooling limit is the recoil limit kB Tr /2 = Er = (h/λ)2 /2m (the kinetic energy
of an atom which is initially at rest increases to Er after emitting a photon). For the blue MOT of
strontium, the Doppler temperature is 730 µK. For the red MOT, the Doppler temperature is about
180 nK, which would be larger than the recoil limit of 460 nK. These temperatures place fundamental
limits on how cold we can cool the atoms in the optical molasses technique.
CHAPTER 2. THEORY OF LASER COOLING
2.2
5
Zeeman slower
The blue MOT can only capture the atoms with velocities below ∼ 30 m/s. However, the average
velocity of the atoms moving out of an oven of 600 ◦ C (which is the temperature of our oven) is
about 500 m/s. According to the Maxwell-Boltzmann statistics, only a small amount of atoms can be
captured. Therefore, we need to build a Zeeman slower to decelerate the atoms.
A counter-propagating laser beam can decelerate the atomic beam as every photon reduces the
atom energy by ~k on average. However, the Doppler effect influences the laser cooling process such
that there is no interaction between light and atoms when the Doppler shift exceeds the linewidth of
the atomic resonance.
To keep decelerating atoms, we can either change the laser frequency which is called chirp cooling
[20] or detune the resonance frequency to compensate for the Doppler shift. The latter one is used in
Zeeman slowers by varying the magnetic field. The Zeeman effect should obey the condition:
gµB B(z)
= ω + kv
~
where µB is the Bohr magneton, B is the magnetic field and z is the spatial coordinate in the Zeeman
slower (z = 0 when v = v0 , and vp
0 is the initial velocity of atoms). Normally, a constant deceleration
a is considered such that v(z) = v02 − 2az . It gives the magnetic field as
1/2
z
B(z) = B0 1 −
+ B0
L0
ω0 +
where L0 is the length of the Zeeman slower and B 0 = ~(ω − ω0 )/µB is the bias magnetic field.
2.3
Magneto-optical trap
The MOT is used to trap and cool atoms. Three pairs of laser beams are added as in the optical
molasses technique. The laser beams have polarizations σ + and σ − as the σ + polarized light induces
only |mi → |m+1i transitions and σ − polarized light induces only |mi → |m−1i transitions. These
|mi indicates the magnetic sublevel of the ground and excited states.
A pair of counter-propagating laser beams with σ + and σ − polarizations gives a position dependent
force. This force is directed towards the zero crossing of the magnetic field when the laser frequency
is red detuned.
A quadrupole magnetic field is produced by two coils in anti-Helmholtz configuration (current in
opposite direction). The center of the coils has zero magnetic field and the magnetic field increases
linearly in every direction near the center. The force in the MOT is then
z
FM
OT =Fσ + (ω − kv − (ω0 +
gµB ∂B
gµB ∂B
z)) − Fσ− (ω + kv − (ω0 −
z))
~ ∂z
~ ∂z
= − βv − κz
This is a damped harmonic force to cool and trap the atoms. The spatial dependence of the magnetic field results in spatially dependent coefficients β and κ.
2.4
Laser stabilization is necessary
To achieve a stable system of laser-cooled atoms, laser frequency stabilization should be considered
in the first place. Noise in the laser output will perturb atoms in a MOT. Laser noise induced heating
will decrease the number of trapped atoms [21]. We thus need a stabilized laser system to cool and
manipulate atoms. To achieve this goal, we need to build a reliable spectroscopy cell to lock the laser.
Chapter 3
Preparation of spectroscopy cell
We use a spectroscopy cell to stabilize a blue diode laser to the 1 S0 → 1 P1 transition. The linewidth
of the 1 S0 → 1 P1 transition is 30.5 MHz. In the long term, the laser frequency will drift and move
out of this range. Therefore, frequency locking is required. We built a spectroscopy system where
the signal from a spectroscopy cell is used to feedback on the piezoelectric transducer or the driving
current of the laser. In this chapter, we will talk about how to build such a cell.
3.1
Requirements
A spectroscopy cell is a heated pipe where the metallic strontium is heated to generate strontium
vapour inside. We should consider the number density n(T ) of strontium vapour, the vapour pressure
P (T ), and the protection of windows from strontium coating.
When a low-intensity (s0 1) laser beam passes through the strontium vapour, it gets absorbed
according to an absorption coefficient
α = n(T )σ
(3.1)
where σ is the absorption cross section which is proportional to λ2 , and the transmitted intensity is
= −αI(z). Hence we obtain Beer’s law [22]:
calculated by dI(z)
dz
I(z) = I0 e−nσz
(3.2)
The pressure of strontium vapour in the cell is related to its number density. Changing the temperature can tune the vapour pressure. The pressure formula with experimentally determined coefficients
is given in Ref. [15] as
P (T ) = 10−9450/T +10.52−1.31log(T )
(3.3)
where the temperature is in units of K and the pressure is in mbar.
We can estimate the number density using the ideal gas law n(T ) = P (T )/kB T . We plot the
temperature dependence of the pressure (Eqn. 3.1) in Fig. 3.1 and the transmission (Eqn. 3.2) in Fig.
3.2.
The mean free path describes the average distance a particle travels between two successive collisions with other particles. We can estimate the mean free path of strontium atoms in a pure Sr vapour
by [23]
kB T
(3.4)
l=√
2πd2 P
where kB is the Boltzmann constant, d is the diameter of strontium atoms.
The pressure and temperature obtained from previous considerations give a mean free path which
is much longer than the length of a standard cell. This means that if a Sr atom starts to move towards
6
CHAPTER 3. PREPARATION OF SPECTROSCOPY CELL
Figure 3.1: Temperature dependence of Sr
vapour pressure. The spectroscopy cell operates at about 400 ◦ C. The melting point of
Sr is at 777 ◦ C.
7
Figure 3.2: Transmission of light through Sr
vapour. The cross section for scattering light
2
. The interaction region
is taken to be σ = 3λ
2π
of light and atoms is taken to be 57 mm (this
distance is given by the structure of baffle insert, see A.2.
the end of the cell, it will hit the window and stick to it. If many Sr atoms accumulate on the window, they form a reflective layer and the laser light cannot pass through. Then we cannot obtain a
spectroscopy signal.
One way to prevent the formation of such a layer is to heat the viewports to prevent Sr atoms from
staying on the windows, but this requires an expensive window which can resist high temperature. A
simpler solution is to add a buffer gas into the cell. Buffer gas atoms increase the collision probability
of Sr atoms such that the mean free path of Sr atoms is decreased drastically. After a few collisions,
the hot Sr atoms are deviated towards the wall. They will attach to the wall and cannot reach the
viewports. Ar gas is a suitable buffer gas for Sr atoms.
The buffer gas will not be the final solution when the atomic flux is large. A large atomic flux
happens when we heat the cell hot enough, especially when we test the spectroscopy cell at high
temperature. We should also optimize the length and the diameter of the cell to allow only a small
flux of atoms towards the viewports. This flux depends on the solid angle of viewport seen from the
center of the Sr vapour. There are not many choices for the length of the cell as we cannot increase
the breadboard infinitely. A clever way of decreasing the flux is to add a baffle insert, see Fig. A.2.
The strontium source is loaded by a long spoon into the cell and pushed to the barrel by the baffle
insert. Due to the length of spectroscopy cell, a better way of doing this is to weld an additional CF
16 above the barrel such that we can easily put Sr inside.
3.2
Design
The cell is 600 mm long and the inner and outer diameters are 37 mm and 40 mm respectively. We
have put a baffle insert inside the cell, the illustration is shown in the Fig. A.2. This stick has two
plates held where the separation is about the size of barrel. In the center of each plate, a hole of 6 mm
is made to allow light to go through the region of strontium vapour. The idea of this stick is to keep
strontium atoms confined inside the plates so that the viewports are hardly to be coated by strontium
atoms.
The viewports are placed at 10◦ with respect to the normal position to minimize the reflection of
laser light. The windows with a specific coating are chosen to get better light transmission. They are
from Pfeiffer which has an AR coating from 425 to 760 nm that suits both blue and red spectroscopy
cell. The flatness is below λ/4 and they can only be baked up to 200 ◦ C. The transmission and flatness
are tested to ensure their quality.
8
3.3. PREPARATION PROCEDURE
The cell is made from stainless steel and we have tested its quality by heating to 900 ◦ C. There
was no break or any slit on the cell, the vacuum quality is maintained.
The illustration of the spectroscopy cell is shown in appendix A.
3.3
Preparation procedure
The baking procedure is necessary for good vacuum conditions. The tube is wrapped with heating
wire and glass fiber in order to heat homogeneously. The stainless steel absorbs water vapour and the
baking procedure accelerates the outgassing. The gas is then pumped out during baking. More details
about vacuum technique can be found in Ref. [24]
The cell is pre-baked without viewports. We wrap a heating wire around the tube homogeneously,
then add glass fiber and aluminium foils to heat the tube to a temperature about 450 ◦ C for tens of
hours. The viewports should not be heated to a temperature higher than 200 ◦ C, so they are not used
in the pre-baking stage.
After the pre-baking, we put the cell into a plastic glove bag with Argon gas filled in. We change
two viewports and break the ampoule of strontium inside of the bag. Strontium is loaded into the cell
by a spoon and pushed to the barrel by the baffle insert. We need to make sure that there is no residual
air as strontium interacts strongly with oxygen and becomes a chalk like powder which can no longer
be used in further experiment.
About 5 g of Strontium is loaded into the barrel of the cell. It can be used for few years without
opening the cell. We can monitor the cell by using the pressure gauge and by adding buffer gas. The
pressure gauge is PCR 280 from Pfeiffer which is a combination of a diaphragm capacitive gauge and
a Pirani gauge. It can measure the pressure from 5 × 10−5 to 1500 hPa (hectopascal). The buffer gas
should be added during pumping for a reasonable time to eliminate any residual gas. A valve is used
to control the flow of Argon gas to attain the desired equilibrium pressure.
After adding two viewports and a pressure gauge, we re-bake our cell slowly to 200 ◦ C as these
new elements may cause a bad vacuum condition in the cell. This baking stage takes about two days.
We can also test the sealing quality of the cell with a helium leak detector. We put Helium gas at
every CF connection and measure the Helium pressure inside the cell.
Finally, we take off the heating wire and use a heating clamp only around the barrel. Two copper
blocks are added where water cooling is used. The heating region is covered again by glass fiber sheet
and aluminium foil.
Chapter 4
Modulation transfer spectroscopy
Modulation transfer spectroscopy is a pump-probe scheme where we send a saturated pump beam
(s0 > 1) through the strontium vapour and a weak probe beam in the opposite direction. It is a
saturated absorption spectroscopy and it has a signal resolution which is close to the natural linewidth.
It can achieve a steep signal for frequency discrimination to lock the laser frequency. The capture
range is typically a few times the natural linewidth hence it is usually below 100 MHz. Modulation
transfer spectroscopy has several advantages over other techniques: it has a zero background signal,
the zero crossing is centered on atomic transition and the closed atomic transitions dominate the
signals. Frequency modulation spectroscopy usually has a sloping background where an additional
demodulation is required and it also gives a open transition signal which is not suitable for laser
locking in closely spaced spectrum [25].
We will discuss the light absorption of atoms to explain why we need this spectroscopy.
4.1
Absorption coefficient
According to the Doppler effect, the moving atoms see different frequencies of the laser beam. In a
thermal gas, the atoms can be classified into different velocity classes. The frequency seen by each
k̂ (where λ is the laser
class is given by ω = ωl − ~k · ~v where ωl is the laser frequency, ~k = 2π
λ
wavelength and k̂ is the unit vector along the laser beam propagating direction) and ~v is the velocity
of atoms.
The laser light can be absorbed by the atoms in the velocity class satisfying ω0 = ωl − ~k · ~v .
The velocity distribution of atoms obeys Maxwell-Boltzmann statistics which has a Gaussian shape.
Hence, the absorption linewidth will not be Lorentzian but will be broadened to a Gaussian. This is
called the Doppler broadening and the FWHM is given by [26]
u
∆ωD ≈ 1.7 ω0
c
p
where u =
2kB T /m is the most probable speed for atoms. This can be about GHz which is
too large in comparison to the natural linewidth. We need a spectroscopy technique with linewidth
comparable to the natural linewidth.
Saturated absorption spectroscopy gives the absorption profile which has a Doppler broadening
background with a dip at the center of this profile. The dip is called Lamb dip which W.E. Lamb has
interpreted theoretically. The absorption coefficient is given by [27]
s0
(Γs /2)2
0
αs (ω) = α (ω) 1 −
1+
(4.1)
2
(ω − ω0 )2 + (Γs /2)2
√
where α0 is the unsaturated absorption coefficient, Γs = Γ 1 + s0 .
9
10
4.2. BASIC CALCULATION OF ERROR SIGNAL
The experimental signal of the Lamb dip will be presented later. We will calculate the error signal
in the next section.
4.2
Basic calculation of error signal
In the modulation transfer spectroscopy, the intense pump beam is propagating through an electrooptic modulator (EOM, details can be found in appendix B) to get two sidebands. Consider the single
frequency pump beam E = E0 sin(ω0 t) and the driven frequency of the EOM is ωm . An additional
time-dependent phase term is added to the pump beam
E = E0 sin(ω0 t + m sin ωm t)
where m is the modulation index given by the ratio of the driven voltage V and the half-wave voltage
Vπ , m = VVπ . ωm is called the modulation frequency. See appendix B.
By using Bessel functions {Jn (x)}, we expand the above expression to get a beam which has a
carrier term and two sidebands terms as
E = E0 sin(ω0 t + m sin ωm t)
∞
n
= E0 [Σ∞
n=0 Jn (m) sin(ω0 + nωm )t + Σn=1 (−1) Jn (m) sin(ω0 − nωm )t
m
m
≈ E0 [sin ω0 t + sin(ω0 + ωm )t − sin(ω0 − ωm )t]
2
2
o
1 n m
m
= E0 − exp[i(ω0 − ωm )t] + exp(iω0 t) + exp[i(ω0 + ωm )t] + c.c
2
2
2
Where the last two lines are valid for m 1 and we have two sidebands oscillating at ω ± ωm . This
is an ideal case as the absorption of light is not considered. In the real case, the intensity of light will
decrease after passing through the EOM. After the two sidebands are generated, the pump beam is
passing through the area of atomic vapour.
A probe field E = Ep sin(ω0 t) oscillates at the frequency ω0 . It propagates through the atomic
vapour in the opposite direction of the pump beam. These two beams interact in a nonlinear way
such that a four wave mixing process occurs between one sideband and the carrier of the pump beam
and the probe beam[25]. The new field beats with the probe beam creating a new signal at sideband
frequency ωm . According to [28, 29, 30], the lineshape of the beat signal for two sidebands is given
by
C
J0 (m)J1 (m)[(L−1 − L− 1 + L 1 − L−1 ) cos ωm t
S(ωm ) = p
2
2
2
2
Γ + ωm
+ (D1 − D 1 − D− 1 + D−1 ) sin ωm t]
2
(4.2)
2
where
Ln =
Γ2
Γ2 + (δ − nωm )2
is the Lorentzian resonance function and
Dn =
Γ(δ − nωm )
Γ2 + (δ − nωm )2
Γ is the natural linewidth and δ is the detuning. C is a constant number.
Using heterodyne detection, the absorption signal or the dispersion signal can be detected by
changing the phase of the local oscillator. We can plot the dispersion and absorption lineshapes by
CHAPTER 4. MODULATION TRANSFER SPECTROSCOPY
11
using the above formula. We use Γ = 2π × 30.5 MHz and ωm = 2π × 22 MHz, the lineshape is given
in Fig. 4.1.
(a)
(b)
(d)
(c)
(e)
Figure 4.1: Theoretical calculation of the error signals for different phases of the local oscillator using
Eqn. 4.2: phase difference is equal to (a) 0◦ ; (b) 45◦ ; (c) 90◦ ; (d) 135◦ ; (e) 180◦ .
4.3
Experimental setup of saturation spectroscopy
The experimental setting is shown in Fig. 4.2. Two cylindrical lenses are added to fix the elliptical
beam shape. The beam sampler reflects 1% of the beam intensity which is collected by the wavemeter.
We use 3.0 mW of input light from the optical fiber and split it into probe and pump beams on the
spectroscopy bread board.
The intensity difference of pump and probe beams is tuned using a half wave plate with a polarization beam splitter. As different light polarizations have different half wave voltages which give
different phase modulations, a Glan-Talor beam splitter is added to reduce phase noise. Two sidebands at 22 MHz are generated on the pump beam by the EOM. The size of the two beams can be
adjusted by using two telescopes. Currently, they are enlarged to 2.5 mm in diameter. As the polarization of pump and probe beams are perpendicular, two quarter wave plates are used to ensure their
polarizations are the same inside the cell. The probe beam is detected by a photodetector.
The overlap of pump and probe beams is adjusted by using four mirrors M1 to M4. A perfect
beam overlap and polarization adjustment is needed for maximum signal.
In the transmission profile in Fig. 4.3, the intensity of pump and probe beams are 2 mW and 0.9
mW, respectively. The related saturation parameters are 1.9 and 0.86 respectively. The data is fitted
by using Eqn. 4.1.
After setting up all optical elements, we can also adjust the heating power and buffer gas pressure
to optimize the error signal. The electronics used to obtain an error signal is shown in Fig. 4.4.
We find that when half of the probe beam is absorbed at the transition frequency, the error signal
is maximum. We adjust the temperature of the cell to 359 ◦ C and add minimum buffer gas where
the pressure gauge shows "under range". However, for the red spectroscopy cell the atomic transition
bandwidth is only 7.4 kHz. The cell should be heated to higher temperature to get enough Sr vapour
density for spectroscopy. For this reason, the buffer gas pressure should be increased.
12
4.3. EXPERIMENTAL SETUP OF SATURATION SPECTROSCOPY
Laser
Cylindrical
Lenses
(a)
M3
Collimation
2
Lenses
2
Glan-Taylor EOM Telescope
Polarizer
2
M1
PD
Telescope
M2
4
Spectroscopy Cell
M4
4
(b)
Figure 4.2: (a) Optical setup for the blue master laser. The spectroscopy setup shown in (b) is optimized to only use 3 mW light out of fiber. Components are taken from Inkscape ComponentLibrary.
Figure 4.3: The transmission signal and fit function given by Eqn. 4.1. The solid red curve is the data
which is fitted by the dashed blue curve. The temperature of the spectroscopy cell is 359 ◦ C (This
temperature is measured at the bottom of the barrel). The beam diameter is 2.5 mm and the intensity
of pump and probe beams are 2 mW and 0.9 mW, respectively.
CHAPTER 4. MODULATION TRANSFER SPECTROSCOPY
Function CH 1
Generator CH 2
Phase
Delay
l.o.
Line
13
Spectroscopy
Cell
EOM
r.f.
i.f.
PID
Low-Pass
Filter
SC 100
Offset V
PD
Voltage
Summer
PZT
+
Spectrum
Analyzer
Figure 4.4: The electronics for negative feedback frequency locking. CH1 and CH2 are two output
channels of a function generator. The phase of the local oscillator (l.o.) can be changed by adjusting
the different phases on the function generator or by changing the cable length. The photodiode (PD)
has two ports, one is the DC port, which is sent to oscilloscope for transmission signal, the other is the
AC port, which is sent to the radio frequency (r.f.) part of a frequency mixer. A proportional-integralderivative controller (PID) and a scan controller (SC) are used to control the piezoelectric transducer
(PZT) in the laser head.
(a)
(b)
(d)
(c)
(e)
Figure 4.5: Error signal for different phases. The scales of these signals in 20 mV per box in the
vertical axis and 10 ms per box in the horizontal axis. These data are taken at higher input beam intensity for better illustration. For different phases of the local oscillator, we observe: phase difference
is equal to (a) 0◦ ; (b) 45◦ ; (c) 90◦ ; (d) 135◦ ; (e) 180◦ . The error signals show good correspondence to
theoretical calculation in Fig. 4.1.
14
4.3. EXPERIMENTAL SETUP OF SATURATION SPECTROSCOPY
The error signal is also strongly influenced by the phase difference of the radio frequency and
local oscillator signals. We fix the phase of the radio frequency signal and tune the phase of the local
oscillator by directly adjusting the phase of the function generator or by changing the phase delay
line. Here we show the phase dependences of the error signal in Fig. 4.5.
We should choose a phase of local oscillator to make an error signal which has a zero crossing at
transition frequency. Besides, its magnitude should be as large as possible to avoid unlock by noise.
In Fig. 4.5, we can also observe a second error signal. This error signal is located at about 120
MHz below the first error signal. The isotope shift of 86 Sr is −124.8 MHz and its the second most
abundant isotope. Hence 86 Sr gives the second error signal.
Chapter 5
Laser frequency stabilization
In this section, we will briefly introduce a basic theory to study the dynamic properties of the laser
stabilization system and then talk about how to realize a reliable laser locking. For readers who are
interested in these topics, [31] is a good reference.
5.1
Linear control system
The general control system can be split into two parts. One is a control system which is called
controller and the other is a plant which represents the process being controlled. The control loop can
form a negative feedback such that when an error of output is detected, a response signal is generated
to have an opposite sign of initial out to correct it back to the set value.
Consider the gain function of the controller and plant as gc (t) and gp (t), respectively. The initial
input vi (t) is amplified to the convolution of gp and gc in the closed negative feedback loop by
Z
t
g(t − t0 ) (vi (t0 ) − vo (t0 )) dt0
vo (t) =
−∞
where g(t) is the total gain given by
Z
t
g(t) =
gp (t − t0 )gc (t0 )dt0
−∞
A useful way to consider the above equations is to use the Laplace transformation. For any ordinary function f (t), the Laplace transformation is
Z
F (s) = L[f ] =
∞
e−st f (t)dt
0
If we apply the Laplace transformation to g(t), the convolution transforms to a simple product
G(s) = Gp (s)Gc (s)
We can use the block diagram to show the relation of different blocks, where a block represents
the principle function of a system part. The relation between different blocks is indicated by arrows
and the summer with plus and minus sign at the inputs. The closed negative feedback loop has the
block diagram given in Fig. 5.1
15
16
5.2. FREQUENCY LOCKING PROCEDURE
Vi
+
p
s
c
s
Vo
-
Figure 5.1: The block diagram of a negative feedback loop, the overall transfer function is G(s) =
Gp (s)Gc (s). The + and − signs indicate negative feedback of the control loop.
From this diagram, the Laplace transformed input and output voltages are related by
Vo (s) = G(s) (Vi (s) − Vo (s))
The transfer function H(s) is
H(s) =
Vo (s)
G(s)
=
Vi (s)
1 + G(s)
For laser frequency stabilization, a frequency discriminant will generate a response signal when
laser frequency does not equal to the set frequency. This response signal is amplified by a loop filter
to control the piezoelectric transducer or driving current of the laser.
Assume the laser frequency f depends linearly on the amplified response signal f = f1 + κVc . Vc
is the voltage given by the control loop. It is always a reasonable assumption in the range of transition
linewidth. This dependence allows us to directly consider the transfer function between the input and
output frequencies.
In the control part, the output frequency f is compared to f0 which is the atomic transition frequency at zero crossing of the error signal. There is one difference between the block diagram 5.1
and our system. The feedback signal is proportional to the frequency difference ∆f = f − f0 .
Following the block diagram and the Ref. [31], the loop equation can be written by
F (s) = F1 (s) − G(s) × (F (s) − F0 (s))
then
F (s) = F1 (s)
G(s)
1
+ F0 (s)
1 + G(s)
1 + G(s)
where F (s), F1 (s) and F0 (s) are the Laplace transforms of f , f1 and f0 respectively.
We can characterize the frequency noise by considering the laser frequency jitter ∆F1 and the
reference frequency noise ∆F0 . Using above formula, we have
∆F = ∆F1
G(s)
1
+ ∆F0
1 + G(s)
1 + G(s)
In the regime where G(s) 1, ∆F1 is suppressed and only noise from the frequency discriminant
is important. In our case, this noise comes from the error signal zero crossing fluctuation.
5.2
Frequency locking procedure
The basic loop filters take the integral, differential or proportional of the error signal. Their combinations form the simplest controller. In our case, we use the Toptica PID 110 in combination with SC
100. The PID controller has a transfer function [31]
1
GP ID (s) = Kp + Ki + Kd s
s
CHAPTER 5. LASER FREQUENCY STABILIZATION
17
The laser head with all the electronic equipment form the experiment part are treated as a black box
because of no available analytic expression of the transfer function. We tune the PID to generate a
fast responding overall transfer function.
In order to get good laser stabilization, we need to adjust the temperature and current of the laser
to avoid mode hoping. After choosing an appropriate running condition, we can start to lock the laser
frequency in the following steps:
1. Before locking the laser frequency, we should set the laser frequency to the atomic transition
frequency. A simple way of doing that is to use an offset voltage and a voltage summer to adjust
the voltage on the PZT, see Fig. 4.4.
2. Turn up the scan amplitude to show the error signal on the oscilloscope, like Fig. 4.5 (a).
3. Choose the correct direction of PID action according to Fig. 5.2.
4. Adjust the set point of the PID controller to put error signal at zero crossing, then turn down
scan amplitude.
5. Keep the PID controller off and set the P, I and D trimpots farthest to the left where they have
minimum gain. Set the PID gain potentiometer to an intermediate value.
6. Turn on the PID controller and increase the I trimpot to get an initial lock. We can see an error
signal similar to Fig. 5.3 (a). Decrease I to get an error signal similar to Fig. 5.3 (c).
7. Increase the P and D trimpots until oscillation, similarly as in Fig. 5.3 (d). Decrease P and D to
get rid of the oscillation.
8. Check the locking stability by measuring the error signal noise with the spectrum analyzer.
5.3
Characterization of locking stability
To measure the stability of an oscillator, there are two basic ways. One is in the time domain, we
measure the mean of the output signal during fixed period to determine the signal variance. The other
is in the frequency domain, we use the power spectrum of output fluctuation to calculate the frequency
noise.
For a laser, it is usually hard to count the radiation circle precisely so usually people study the laser
stability in frequency domain. In our experiment, we can access the error signal spectrum density with
a Stanford sr 760 fft spectrum analyzer.
Assuming laser light has only phase noise φ(t) such that its amplitude E0 is constant. The laser
light can be described by [31]
E(t) = E0 sin(2πf0 t + φ(t))
where f0 is the laser’s frequency without noise. The instantaneous frequency is defined by
f (t) = f0 +
1 dφ
2π dt
The frequency fluctuation is given by
∆f (t) = f (t) − f0 =
1 dφ
2π dt
(5.1)
18
5.3. CHARACTERIZATION OF LOCKING STABILITY
Figure 5.2: The scales of these signals are 10 mV per box in the vertical axis and 50 ms per box in
the horizontal axis. The top signal is the error signal and the bottom signal is the PZT scan voltage.
The laser frequency decreases when the scan voltage increases, which can be confirmed as the isotope
error signal appears at a lower PZT voltage. The relation between scan voltage and laser frequency
determines the required gain sign of the servo loop.
(a) The laser frequency is locked. The P and
D trimpots are turned to zero and the error
signal is slightly above the red line.
(b) The laser frequency is locked. The P and
D trimpots are turned to zero and the error
signal is slightly below the red line.
(c) The laser frequency is locked. The P and
D trimpots are turned to zero and the error
signal is on the red line.
(d) The laser frequency is locked. Starting
from the settings in panel (c), the P and D
trimpots are increased to observe oscillation.
Figure 5.3: The scales of these signals are 10 mV per box in the vertical axis and 50 ms per box in
the horizontal axis. The panels show the error signal from the monitor output of the PID controller.
CHAPTER 5. LASER FREQUENCY STABILIZATION
19
By Wiener-Khintchine theorem, the power spectrum of phase noise Sφ (f ) is the Fourier transform
of its autocorrelation function Rφ (t)
Z
1 T /2
φ(t + t0 )φ(t)dt0
Rφ (t) = lim
T →∞ T −T /2
Z ∞
1
Rφ (t)eiωt dt
Sφ (ω) =
2π −∞
However, what we are interested in is not phase noise spectrum density but the frequency noise
spectrum density. According to Eqn. 5.1,
Sφ̇ (ω) = ω 2 Sφ (ω)
hence
hφ˙2 i = Rφ̇ (0) =
Z
∞
−∞
Sφ̇ (ω)dω
As the measurement from spectrum analyzer gives spectral density in positive frequencies, the
mean square frequency deviation h(∆f )2 i is given by the spectral density of frequency fluctuations
S∆f (f )
Z
∞
h(∆f )2 i =
S∆f (f )df
0
where the one-sided spectral density is used.
Typically, the free running diode laser has a linewidth of a few MHz. Frequency locking reduces
this linewidth smaller than the atomic transition linewidth. In our experiment, the linewidth of the
laser is measured by the density power spectrum of the error signal obtained from the spectroscopy
breadboard. The error signal amplitude is linearly dependent on the detuning frequency in the range
of the natural linewidth.
From Fig. 4.5 (a), we can measure the proportionality constant between the error signal amplitude
and frequency. We measure the density power spectrum of the error signal when laser is running
freely and when the laser is locked. The measurements are shown in Fig. 5.4.
√
Figure 5.4: (a) Power spectrum density (PSD in the figure, unit in µVrms / Hz) of the error signal,
(b) PSD is zoomed in. The solid red curve is for the free running laser close to the locking frequency
and the dashed blue curve is for the locked laser.
The noise is reduced after the locking procedure. From this in-loop measurement, we cannot
conclude that the laser linewidth is reduced. However, by integrating the error signal PSD, we can
estimate that the servo reduces the frequency excursions by a factor of 3 approximately. The main
contribution of noise after locking comes from low frequencies. They are harmonics of 50 Hz lab AC
supplies. The low frequency noise will be reduced to get a narrower linewidth.
Chapter 6
Conclusion and outlook
We built a system to stabilize the frequency of a Toptica diode laser system to a saturated absorption
spectroscopy. The laser light is split into a pump beam and a probe beam. The intense pump beam
is modulated by an EOM and then goes through Sr vapour to saturate atoms in excited state while
the weak probe beam propagates in the opposite direction. The transmission signal from probe beam
shows a Lamb dip in a Doppler broaden transmission profile. This is a basic saturation spectroscopy
scheme. The modulation frequency in pump beam is transferred to the probe beam via four wave
mixing process.
The transmitted probe signal is analyzed by electronic devices to give an error signal which has a
zero crossing at the atomic transition frequency. This error signal is used in a PID controller to lock
the diode laser frequency. The in-loop error signal is examined with a 100 kHz Fourier transformation
spectrum analyzer to ensure the good locking quality.
After locking the blue laser to the Sr transition, we will use it as the reference further blue diodes
lasers for power amplification. We designed a diode laser mount compatible with injection locking,
shown in appendix C.
In the blue laser system, acousto-tptical modulators (AOMs) will be added to shift frequency for
transverse cooling, the Zeeman slower and the MOT. A similar setup will be built for the red laser
system with an additional optical cavity to reduce the laser linewidth below 7.4 kHz. We are in the
process of designing the vacuum system for cooling and trapping of Sr based on these new laser
systems.
Many proposals will be possible to realize with strontium experiments. Recently, there was a
proposal about a toy model in QCD which can be realized in strontium optical lattice [32]. The
fermionic strontium atoms have nuclear spin I = 9/2 representing the SU(N ) spins model with
N ≤ 2I + 1. Among these SU(N ) models, certain effective low-energy (2+1) dimensional spin
ladder models of SU(N ) can produce the (1+1) dimensional CP(N − 1) model under the dimensional
reduction where the continuous limit is taken. Another proposal [33] discussed the possibility to study
lattice gauge theories in the strontium optical lattice. These proposals show a lot of exciting physics
in quantum simulation of particle physics via the strontium optical lattice.
20
Appendix A
Spectroscopy cell design
The spectroscopy cell we use was originally designed by A. Mayer. For the next version of the
cell, we would like to add a CF 16 flange on the top of the barrel such that it is convenient to load
strontium into the barrel. The cell which is used in the blue laser spectroscopy is shown in the Fig.
A.1. To prevent coating the viewports we added two baffles around the Sr reservoir (see Fig. A.2).
Additionally, the cell is water cooled.
10°
33,75
12,70
12,70
6,89
(a)
(b)
Figure A.1: (a) Blue laser spectroscopy cell and (b) windows. The window has a 10◦ angle with the
tube. This angle should be designed to minimize the reflection of laser light.
1,50
n
1,50
n6
4,0
0
Ø6,00
Ø6,00
Ø6,00
,00
Figure A.2: The baffle insert is put into the cell, the total length is 490mm and the distance between
two baffles are 57 mm. The hole in the center of baffle allows light go through Sr vapour while keep Sr
atoms stay in this region as the aperture of the hole is 6 mm in comparison with 40 mm cell diameter.
This small aperture reduce the flux of Sr towards viewport and prevent windows from Sr coating.
21
Appendix B
Homebuilt Electro-optic modulator
An electro-optic modulator (EOM) is an optical device to modulate the phase of laser beam. The main
part is a crystal whose refractive index depends linearly on the applied electric field. The electric field
is generated by two parallel plates attached to the crystal. The beam propagating through the crystal
acquires an additional phase proportional to the local electric field.
In order to make a good homebuilt EOM, we need to understand how the phase modulation can
be generated in the crystal and how we can maximize the modulation index.
Linear electro-optic effect
In certain crystals, the response of refractive indices are linear in the applied electric field [34]. In the
presence of an electric field, the index ellipsoid is changed from
y2
z2
x2
+
+
=1
n2x n2y n2z
to
1
n2
2
x +
1
1
n2
2
y +
2
1
n2
2
z +2
3
1
n2
yz + 2
4
1
n2
xz + 2
5
1
n2
xy = 1
(B.1)
6
where x, y, z are the principal axes of the crystal, see Fig. B.1. The changes in the
~ can be described by the elctro-optic tensor
presence of the electric field E
 


∆ n12 1
r11 r12 r13
∆ 12  r21 r22 r23  
n 2 



∆ 12  r31 r32 r33  Ex
n 3  = 

 
∆ 12  r41 r42 r43  Ey
n 4 



∆ 12  r51 r52 r53  Ez
n 5
r61 r62 r63
∆ n12 6
1
n2 k
in the
For our experiment, we have used lithium niobate where the non-zero components of electro-optic
tensor are [35]
r13
r22
r33
r42
=8.6 × 10−12 m/V
=3.4 × 10−12 m/V
=30.8 × 10−12 m/V
=28 × 10−12 m/V
22
APPENDIX B. HOMEBUILT ELECTRO-OPTIC MODULATOR
23
z axis
y axis W
AC V
voltage
x axis
L
Figure B.1: Illustration of an EOM with principal axes. The yellow surface indicates a copper strip
where AC voltage is applied to the crystal to generate the electric field. L is the length and W is the
width of the crystal.
We use the transverse geometry where the voltage V is applied along the z direction (transverse
to the electric field). The phase deviation is
φ=
πn3o r63 L
V
λ W
where no is the refractive index along the z axis and λ is the wavelength of the laser. When the phase
deviation is π2 , the voltage applied is called half-wave voltage
Vπ =
λ W
n3o r63 L
The minimum half-wave voltage is 135 V for the lithium niobate crystals in our experiments.
When the applied voltage is sinusoidal, sidebands can be generated as
sin(ω0 t + m sin ωm t) =E0 [Σ∞
n=0 Jn (m) sin(ω0 + nωm )t
∞
+ Σn=1 (−1)n Jn (m) sin(ω0 − nωm )t]
(B.2)
where Jn (m) are Bessel functions and m is the modulation index.
Impedance Matching
When we build an EOM from a bare one lithium niobate crystal, we need to consider impedance
matching in order to get maximum power from a given rf source. For a circuit with complex impedance,
the power theorem [36] shows that when the load impedance is equal to the complex conjugate of the
source impedance, the power transmitted to the load is maximal. In our case, as the bare crystal is not
impedance matched, we need to understand how to match the impedance to the rf source with typical
output impedance of 50 Ω.
There simplest impedance matching method is called L matching, see Fig. B.2. The idea is to
use series and shunt components to adjust load impedance. For every impedance matched circuit, we
define a quality factor (Q factor) to characterize it. This Q factor is equal to the ratio of the center
frequency of the circuit to its 3-dB bandwidth. For L matching, the Q factor is fully dependent on
load impedance RL and source impedance RS
r
RL
−1
Q=
RS
24
In order to get a high Q, we need to use three elements circuits, π matching or T matching. These
two circuits can be understood by a combination of two L matching circuits.
RS
RS
X1
X2
RL
RS
X2
X1
(a)
X3
RL
(b)
X1
X3
X2
RL
(c)
Figure B.2: (a) L matching. (b) π matching. (c) T matching. Figure reproduced from [36]
The Smith chart was created in the second world war to help radio frequency engineers to calculate
complex impedance and reflection coefficients Γ. For a load impedance ZL and a source impedance
ZS , the reflection coefficient is given by
Γ=
ZS − ZL
ZS + ZL
In the Smith chart, all the values are normalized by the source impedance.
A superposition of Smith chart and a 180◦ rotation of Smith chart is sometimes called YZ chart
as it shows both impedance Z and admittance Y. These charts are extremely useful and very easy to
learn for a layman to do impedance matching trials.
In our case, the capacitance of EOM crystal is specified 12.3 pF and we want to build a circuit
with resonance frequency at 22 MHz. We add two shunt capacitors and make an series inductor by
copper wire for a π matching. The complete matching circuit of our homebuilt EOM is shown in the
Fig. B.3.
Figure B.3: The matching circuit for the EOM. A indicates the source capacitors with a tunable
capacitance, B is the coil (inductance) and C is the load capacitor.
The approach of selecting capacitors and inductors are shown in the Fig. B.4
APPENDIX B. HOMEBUILT ELECTRO-OPTIC MODULATOR
0.
1. 4
1.
6
45
1.2
50
1.0
0.9
8
1.
0
2.
5 65
0.8
55
0. 7
60
6
0.
0
1.
50
20
10
5.0
4.0
3.0
2.0
1.8
1.6
0
1.
0
2.
8
1.
6
1.
1. 4
0
10
0.9 0.8
1
0.9
5
20
1
1
4
3
15
2
0.7
0.6
0.8
4
0.4
0.7
2.5
10
3
0.5
5
0.6
2
8
6
0.3
7
1.8 1.6
6
8
0.2
0.5
9
5
10
0.1
0.4
0.3
1.4
4
12
3
14
0.05
0.2
1.2 1.1 1
2
20
0.01
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1
15
TOWARD LOAD
10
7
1 1
30
—>
5
1.1
0
0.1
0 0
1.1
0.1
0 1
0.99
0.9
CENTER
1
1.1
0.2
1.2
1.3
0.95
1.2
4
1.3 1.4
0.6 0.8 1
1.4
1.6 1.8 2
1.5
3
2
3
1.5 1.6 1.7 1.81.9 2
0.9
1.3
<— TOWARD GENERATOR
2
1
3
1.2
0.4
0.8
1.4
0.7
1.5
2.5
0.6
1.6
0.5
1.7
4
5
4
10
5
3
0.4
0
4.
8
0.
18
0
-5
-2
5
-3
32
0.
0.
42
17
0
-6
0
-3
33
0.
0.
41
6
0. 4
5
-7 0
16
34
0.14
-80
-4
0
0. 15
0. 35
0. 4
RE
AC
0.2
0.2
CAP
A
C IT
0.12
0.36
0.37
I VE
CE
-1
-1 10
0.
0. 1
0.11
-100
-90
0.13
TA
N
CO
M
PO
N
EN
T
20
0.
jX
5. 0
0
0
0.
43
0.
0
-1 7
30
08
0.
09
0.
(-
3.
42
41
0. 4
0.39
0.38
Figure B.4: In our case, the load impedance is normalized to 2.1. We start at load impedance of 2.1 at
point L, the admittance is 0.48. We add a shunt capacitance of +j3.3 to point A where the admittance
is simply 0.48 + j3.3 so the impedance is 0.04 − j0.3. Then, a series inductor of j0.5 is added to
point B such that the impedance becomes 0.04 − j0.3 + j0.5 = 0.04 + j0.2. Finally a shunt capacitor
of +j4.7 is added to match the impedance of EOM at point S. The original Smith chart is taken from
[37].
Resonant frequency
We measure the frequency dependent of reflection index with a network analyzer. The resonant
frequency of the EOM is at 22 MHz, shown in the Fig. B.5.
Figure B.5: Resonant frequency of EOM, amplitude and phase plots.
20
6
10 15
4
0.3
1.8
ORI GI N
5
10
0.2 0.1
0
1.9
2
TR
TR S. RF S. AT
AN
AN W. L . W. TE
SM
SM P L
N
.
.
E O LO .
CO
CO AK SS SS [ dB
EF
EF ( C [ d CO ]
F, ON B] EF
F,
F
P ST.
E
or
P)
I
0.
45
1.4
1.2
1.0
0.9
1.0
5
-4
-5
1.2
0.8
-6
5
-5
0
0.9
0. 7
5
5 0.
46
0.8
0.7
0.6
0.5
0.4
0.3
6
0.
0
-6
50
20
10
0 —> WA V EL E
0.49
N GTH
S TOW
A RD
0.4 8
0
D <—
0.49
GEN
RD L OA
ERA
TOWA
0.4 8
±180
TOR
TH S
0. 47
170
-170
EN G
—>
V EL
0. 47
WA
0.
16 0
04
-90
90
-1 60
0.
85
-85
46
0.
15
05
0
I
N
)
80
DU C
-80
Yo
0.
05
TIV
(-jB
45
ER
0.
CE
EA
AN
T
C
P
TA
0.
E
44
0.1
75
NC
SC
-75
06
14
0.
40
SU
EC
0.
E
0
06 -1
OM
IV
44
T
0.
C
PO
DU
N
EN
IN
70
R
T
-7 0
,O
( j+
o)
X/
/Z
0.2
Zo
),
0
2.
1.
8
6
1.
1. 4
<—
2.
0
5.0
4.0
3.0
1.8
2.0
0
1.
1.2
1.0
04
0.
50
8
1.
0
1.
1.6
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
6
0.
5
5 0.
-6
0.
TR
TR S. RF S. AT
AN
AN W. L . W. TE
SM
SM P L
N
.
.
E O LO .
CO
CO AK SS SS [ dB
EF
EF ( C [ d CO ]
F, ON B] EF
F,
F
P ST.
E
or
P)
I
Y o)
(-jB
-75
05
0.
45
40
06
0.
-1
1. 4
1.
6
45
1.2
50
1.0
0.9
0.8
55
0. 7
60
6
0.
5 65
06
0.1
0.2
0
2.
0.
0.
44
0.
70
0
05
14
0.
45
0.
( j+
X/
Zo
),
T
EN
75
0
15
PO
N
CE
80
TA
N
RE
AC
0 —> WA V EL E
0.49
N GTH
S TOW
A RD
0.4 8
0
D <—
0.49
GEN
RD L OA
ERA
TOWA
0.4 8
±180
TOR
TH S
0. 47
170
-170
EN G
—>
V EL
0. 47
WA
0.
16 0
04
-90
90
-1 60
0.
85
-85
46
<—
46
0.
04
0.
50
-1
-80
20
10
5.0
4.0
3.0
2.0
1.6
1.8
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
CO
M
8
1.
0
RE
AC
1.
44
0.
1. 4
1.
6
45
1.2
50
1.0
0.9
0.8
55
0. 7
60
6
0.
5 65
0.
06
0.
44
0.
70
0
05
14
0.
45
0.
( j+
X/
Zo
),
EN
75
0
CO
M
TA
N
CE
15
PO
N
0.
6
0. 4
0.
0.39
0.38
80
T
-2 0
08
0.
CTI
VE
CTI
VE
0.
0.1
IN D
U
20
09
0.
IN D
U
0
N
TA
EP
SC
SU
0.
Y o)
1.
0.
43
0.
0
-1 7
30
20
10
(-jB
50
0.
0
-1
(-
-7 0
T
0.
8
0.
0
VE
TI
-1
-1 10
0.36
0.37
I VE
CO
M
50
CE
CE
0.12
CE
0.
3.
0
2.
IN
C IT
0. 1
0.11
-100
-90
0.13
TA
N
1.
0
0
8
R
O
),
Zo
CAP
A
0.14
-80
-4
0
0. 15
0. 35
5
5
-7 0
16
34
-4
-3
0.
0.
0
0.9
0
RE
AC
0
0
-6
0. 7
17
33
0.39
0.38
jX
/
0.
0.
EN
1.
-15
4.
1.
C
DU
-3
32
42
0.8
0.
0. 4
PO
N
5. 0
6
0.
08
41
0.2
0.2
5
0.
18
0
-5
-2
5
-6
20
09
0.
0. 4
0. 4
0.
-5
0.
-2 0
-5
-1
-1 10
0.36
0.37
I VE
0.
43
0.
0
-1 7
30
3.
TR
TR S. RF S. AT
AN
AN W. L . W. TE
SM
SM P L
N
.
.
E O LO .
CO
CO AK SS SS [ dB
EF
EF ( C [ d CO ]
F, ON B] EF
F,
F
P ST.
E
or
P)
I
0.12
0
6
1.
C IT
0. 1
0.11
-100
-90
0.13
jX
0.
6
-10
1. 4
CAP
A
0.14
-80
-4
0
0. 15
0. 35
(-
0.
3.
0
5. 0
0
1.2
5
-7 0
16
34
T
4.
1.0
-3
0.
0.
EN
0
8
5
0
PO
N
0.
8
-4
0
-6
CO
M
0.
0
0.9
17
33
CE
1.
0
0
20
0.1
0.2 0.2
-5
0.
0.
TA
N
1.
-15
4.
0. 2
0. 4
0. 4
50
0.8
-3
32
RE
AC
0.2
0.2
5. 0
3
6
0. 7
0.
18
0
-5
-2
5
-10
0
20
10
0.
6
5
0. 4
0. 4
0.
20
50
0.
0.1
-5
3.
0.2 0.2
0.
8
5
6
0.1
0. 4
0. 4
50
0.
8
5 0.
4.
6
0.
-6
0.
6
0.1
0. 2
6
-2 0
31
0.
0.
3.
0
0.
19
6
0
5. 0
0
8
0.
6
0. 2
N
TA
EP
SC
SU
0.
8
0.
0.
-6
0.
0.
8
4
VE
TI
1.
0
0
0.
8
0.
4
C
DU
1.
-15
4.
0.
0.
IN
5. 0
3
-75
-10
20
10
0.
05
20
50
4
R
,O
o)
/Z
0.2 0.2
0. 2
0.
0.
0.1
0. 4
0. 4
4
45
6
0.
40
3
0. 2
8
29
0.
0.
6
50
0.
0
-1
0.
8
0.
0. 28
0. 22
3
-4
0.
4
2
0.
0.
-7 0
0.
0.1
-2 0
0
21
-3
3
0.
06
20
0.1
0. 2
0.
RADI ALLY SCALED PARAMETERS
10040 20
40 30
0. 22
0. 28
0.25
0.26
0.24
0.2 7
0.2 3
0.25
0.24
0.26
0.2 3
0.2 7
REFL ECTION COEFF CIEN T IN DEG
REES
L E OF
ANG
ISS ON COEFFICIEN T IN
TRA N SM
DEGR
EES
L E OF
ANG
0.
21
29
30
0.
44
0.
0.
3
0 —> WA V EL E
0.49
N GTH
S TOW
A RD
0.4 8
0
D <—
0.49
GEN
RD L OA
ERA
TOWA
0.4 8
±180
TOR
TH S
0. 47
170
-170
EN G
—>
V EL
0. 47
WA
0.
16 0
04
-90
90
-1 60
0.
85
-85
46
0.
2
<—
0.
40
46
31
04
0.
50
-1
-80
0.
19
8
5
2
0.
0.
0.
1.9
0
0.2
10
0.4
10 15
5
0.2 0.1
0.3
1.8
20
6
4
0.3
5
10
5
3
0.
1.7
5
4
6
0.
0.
1.6
4
0.4
50
6
0.8
1.5
2.5
0.5
0.
0. 7
3
0.6
L
0.5
3
2
5. 0
0.6
1.4
0.7
6
1.0
1.5
0.8
0. 7
0.
0.9
1.6 1.8 2
1.5 1.6 1.7 1.81.9 2
0.8
4
0.7
0.
0.8
1.3
<— TOWARD GENERATOR
2
1
3
1.3 1.4
0.6 0.8 1
1.4
0.9
015
0. 4
8
1.2
1.2
4
1.2
0.4
0.9
3
A
0.9
1.1
1.3
0.95
1.0
0.8
1.2
0.9
CENTER
1
0.2
1.2
1.0
0.7
0.
1. 4
1.1
0.99
S
0
0.6
—>
5
1.1
0.1
0 0
0 1
4.
0
10
1.
0.5
TOWARD LOAD
10
7
0
1.4
0.4
15
1 1
30
1.6
0.3
0
0.2
1
0.1
20
0
3.
8
1.
20
0.01
1.2
1.2 1.1 1
2
18
32
6
0.
0. 4
0.
6
0.1
0.1
1.
0.2
0.
I
0
ORI GI N
1.8
3
14
0.05
6
1.
0
ZS=1.04
YS =0.96
8
1.4
4
12
8
RESI STANCE COMPONENT ( R/ Zo) , OR CONDUCTANCE COMPONENT ( G/ Yo)
0.2 0.2
1.
0.3
0.
0.
50
25
20
2.0
5
10
0.1
17
33
30
10
0
1.8 1.6
6
9
0.
0.
60
20
B] P or
R BS [ d F, E
SW d S EF ,
S O F
L O . C OEF
C
N. FL
RT R FL .
R
2
8
0.4
16
34
0. 4
50
2.
31
7
0.2
0.5
1. 4
0.
19
2.5
8
6
0.3
0.
35
10
3.0
0.
5
0.6
1.
B
4.0
0
3
10
4
0.4
0.7
6
3
-4
0.
2
4
3
0.5
1.
0.
0.6
0.8
0.
70
Shunt
capacitor
+j 4.7
0.
0.
5.0
29
5
15
2
0.7
0
0. 28
0. 22
0
0.
21
-3
0.9
40
0.2
0. 4
0. 2
0.
0.9 0.8
C
ITI
0
10
-2 0
10
20
1
1
1
0.2
0
20
0.1
0
R
O
AC
AP
0. 15
0. 35
80
)
/ Yo
(+ jB
5. 0
50
0.25
0.26
0.24
0.2 7
0.2 3
0.25
0.24
0.26
0.2 3
0.2 7
REFL ECTION COEFF CIEN T IN DEG
REES
L E OF
ANG
ISS ON COEFFICIEN T IN
TRA N SM
DEGR
EES
L E OF
ANG
RADI ALLY SCALED PARAMETERS
10040 20
40 30
2.
0. 22
0. 28
20
4
0
E
NC
0.36
90
8
29
30
21
2
ORI GI N
0.
1.9
0
0.
10
3
10 15
5
0.2 0.1
0.
1.8
20
6
4
0.3
2
1.7
10
5
3
0.
1.6
5
4
40
1.5
4
0.4
3
TA
EP
SC
SU
VE
1.
31
1.4
2.5
0.5
0.
1.3
3
0.6
6
1.2
3
2
0.
1.1
0.7
5
0.8
1.5
0.8
0.
0.
0.7
1.6 1.8 2
1.5 1.6 1.7 1.81.9 2
0.2
0.6
<— TOWARD GENERATOR
2
1
3
1.3 1.4
0.6 0.8 1
1.4
0.9
0.3
0.5
4
1.2
0.4
0. 2
0.8
0.4
1.3
0.95
8
0. 7
0.9
CENTER
1
0.2
1.2
50
0.4
1.1
0.99
20
0.5
0.3
—>
5
1.1
0.1
0 0
0.6
0.2
TOWARD LOAD
10
7
0
I
0.1
15
1 1
30
0 1
I
0
1
0.1
L
6
0.
1.0
20
0.01
0.
0.9
1.2 1.1 1
2
0.7
0.2
0.8
3
14
0.05
0.9
1.4
4
12
6
8
1.2
0.3
1.0
5
10
0.1
1.2
1.8 1.6
6
9
1. 4
2
8
0.4
5. 0
0. 4
0
7
0.2
0.5
4.
0
0.
6
2.5
8
6
0.3
0.
0.
1.
5
0.6
B] P or
R BS [ d F, E
SW d S EF ,
S O F
L O . C OEF
C
N. FL
RT R FL .
R
3
10
4
0.4
0.7
4
8
0.8
1.0
0.9
1.2
4
3
0.5
0.
1.
0. 7
1. 4
0.6
0.8
A
0
6
5
15
2
0.7
3
2.
31
6
1.
0.9
3.
015
10
1.
0.
19
0.
8
0.9 0.8
0.
0.
1.
10
20
1
1
1
0. 2
0
5
0
B] P or
R BS [ d F, E
SW d S EF ,
S O F
L O . C OEF
C
N. FL
RT R FL .
R
0
S
0. 4
ZA=0.04-j0.3
YA=0.48+j3.3
RADI ALLY SCALED PARAMETERS
10040 20
40 30
4.
1.
1.4
3
-4
0.
2
4
1.6
0.
0.
0.
2.
Shunt
capacitor
+j 3.3
1.8
29
4
2.0
0
0.
21
-3
0.
8
0
RESI STANCE COMPONENT ( R/ Zo) , OR CONDUCTANCE COMPONENT ( G/ Yo)
0.2 0.2
0.1
0.
3
0. 28
0. 22
0.
3.0
-2 0
0. 2
Series
inductor
+j 0.5
4.0
8
20
0
3.
6
0.
20
50
5.0
0.
0.
0.14
0.37
0.38
0.39
100
11 0
41
ZB=0.04+j0.2
YB=0.96 -j4.7
10
10
6
1.
6
8
10
20
0.
0.
0.
43
13
19
50
0.
25
12
0
07
32
0.
20
0.1
0.2
0.3
0.4
3
0.5
6
8
0
0.
0.6
0.
0.
Z L=2.1
YL=0.48
0. 4
1.
0. 2
0.7
0.1
0. 4
0.8
ZA=0.04-j0.3
YA =0.48+j3.3
A
0.9
1.0
1.2
1.4
1.6
1.8
2.0
3.0
4.0
5.0
10
20
50
RESI STANCE COMPONENT ( R/ Zo) , OR CONDUCTANCE COMPONENT ( G/ Yo)
0.2 0.2
L
0.
50
0.
18
42
0. 4
09
0.
08
0.
0.
30
5
S
0.
0. 4
ZB=0.04+j0.2
YB=0.96-j4.7
B
17
33
0.
20
5. 0
50
0.25
0.26
0.24
0.2 7
0.2 3
0.25
0.24
0.26
0.2 3
0.2 7
REFL ECTION COEFF CIEN T IN DEG
REES
L E OF
ANG
ISS ON COEFFICIEN T IN
TRA N SM
DEGR
EES
L E OF
ANG
10
0
0. 4
0
20
10
50
3.
6
1
10
0
0.
0.
0.
60
0.2
1.
5. 0
0. 22
0
4.
16
34
0.
015
0. 28
1.
0
29
30
4.
21
0.
8
ITI
0.
35
0. 7
0.
1.
0.
C
AC
AP
0. 1
0.
70
)
/ Yo
(+ jB
0.2
0.13
0.12
0.11
40
0.8
3
20
0
3.
6
8
TA
EP
SC
SU
VE
0.9
0.
2
0.
0.
6
40
0.
0.
R
O
0
31
0. 4
0
2.
0.
19
5
25
43
13
E
NC
8
0.
32
0.
0.
0. 4
07
11 0
41
1.
6
0.
50
12
0
6
0.
0.2
0.
18
42
1.2
0.
0.
30
09
0.
08
1. 4
0.
1.
0. 7
0.2
17
33
0. 2
0.
0.
0.
60
0. 35
80
0.
16
34
0. 15
0.36
90
1.0
0.
35
NORMALIZED IMPEDANCE AND ADMITTANCE COORDINATES
0.14
0.37
0.38
0.39
100
0. 4
4
0.
70
)
/ Yo
(+ jB
.0
5. 0
40
0.8
ITI
0.9
C
AC
AP
0. 1
0. 35
80
1.0
6
R
O
1.2
1.
0
TA
EP
SC
SU
VE
E
NC
8
0
0
3.
11 0
41
1.
43
13
2.
0
12
0
07
0.
4.
42
1. 4
0.
0.
09
0.
08
0.13
0.12
0.11
0. 15
0.36
90
0.
0.
0.
NORMALIZED IMPEDANCE AND ADMITTANCE COORDINATES
0.14
0.37
0.38
0.39
100
0. 4
4
0.13
0.12
0.11
0. 1
0.
NORMALIZED IMPEDANCE AND ADMITTANCE COORDINATES
25
Appendix C
Injection-locked diode laser
Injection locking works for demands of high power and spectral purity [38]. A relative weak single
frequency laser beam, which is also called the master beam, is sent into a slave laser which emit
an intense light at the exact master frequency. The presence of master beam signal kills all other
oscillation modes which are existing when the injected beam is removed. To achieve good injection
locking, the frequency difference between two beams should be small to amplify master beam by the
gain medium of the slave laser [39].
We use injection locking for blue laser amplification as it is the most direct way to amplify laser
intensity. Tapered amplifiers are good candidates for laser power amplification but they are only
available for red and infrared wavelengths. Other amplification methods such as an optical parametric
amplifier or second harmonic generation are much more complicated than simple injection locking.
We show a simple injection locking scheme in the Fig. C.1.
Slave Laser
Faraday
Isolator
To experiment
AOM
2
2
From Master Laser
Figure C.1: Simple injection locking scheme. The master beam is frequency shifted with an AOM
and then injected through the rejected part of a Faraday Isolator.
Blue diode laser mount design
We designed a blue diode laser mount for injection locking. The idea is that we want to use a new
electronic control in a larger PCB and we want to make the mount more stable and add potential water
cooling. Hence we need a larger housing and make laser mount heavier.
The previous laser mount has an external cavity for frequency selection and the material used
is aluminium. Injection locking doesn’t require an external frequency selection so we reduce the
total length of the mount. We change aluminium to copper as it is heavier and it has better thermal
conductivity. The part which holds laser diode is cooled by a Peltier cooler and is fixed to bottom
plate by plastic screws. Water cooling was not used in previous laser mount and it is added to see
whether it will help for better temperature stabilization.
26
APPENDIX C. INJECTION-LOCKED DIODE LASER
27
Figure C.2: Illustration of the blue diode laser mount. Four main parts are shown as A, B, C and D in
the figure. The copper block A is used to hold the laser diode collimation package and is attached by
plastic screws from the baseplate B. The baseplate is heavy and provides good mechanical stability.
Additionally, the baseplate can be water-cooled using the connectors C, which might help to achieve
better temperature stability. The diode laser is encased in a plastic housing (D) to prevent temperature
fluctuation from air flows.
Acknowledgements
It is an adventure for me to join the construction of the strontium experiment. I would like to thank
my supervisor Sebastian Blatt and the team leader Immanuel Bloch for giving me this opportunity to
finish the internship for my Master’s degree. It is a fantastic environment for research and study.
I would like to thank Sebastian for his supervision, advice, trust and patience. Sebastian has a
extremely profound understanding not only in physics but also how to construct a new lab. I acknowledge his kind guidance in my work.
Nejc Janša has shared with me his experience, knowledge besides our office. It was a pleasure to
work together and he helped me a lot. I thank him for every project we have worked on.
I would like to thank Rodrigo G. Escudera, Zhichao Guo, Yunpeng Ji and Stepan Snigirev for their
kind help. All the technicians, Anton Mayer, Oliver Mödl and Karsten Förster, thank you for your
excellent electronic and mechanic works. I would like to thank people in other labs for their support
of knowledge and every electronic and optic component. Thanks to Zhenkai Lu and Jae-yoon Choi
for their kind discussion and advice.
Finally, I would like to thank my parents for their encouragement and continuous support.
28
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