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The slow-light effect: An experimental study of light propagation through rubidium
vapour
Mark Zentile
Department of Physics, Durham University, South Road, Durham, DH1 3LE, UK
(Dated: June 28, 2012)
This report presents a study of the slow-light effect in a 2 mm long cell containing thermal
rubidium vapour. The model for the electric susceptibility previously developed in our group has
been used, along with a Fourier analysis, to model the pulse propagation and is then compared with
experiment. Two experimental methods were used where the results show agreement. Advantages
and disadvantages of the different techniques are discussed along with further improvements that
could be made. Measurements of absolute absorption have also been carried out to verify the electric
susceptibility theory, which was then used to verify that the presence of buffer gas contained within
one particular experimental cell cannot be considered negligible. A discussion of the future outlook
of our project is also presented.
CONTENTS
I. Introduction
A. Motivation for smaller cells
II. Theory
A. The phase shift and absorption from the
electric susceptibility
B. Equation for the electric susceptibility
C. Absolute absorption
D. Optical Pulse Propagation
1. Narrowband approximation
III. Experiments
A. Absolute Absorption
1. Experimental Method
2. Results
3. Discussion
4. Conclusion
B. Slow-light
1. Experimental Method
2. Results
3. Discussion
4. Conclusion
I.
1
2
3
3
3
4
4
4
5
5
6
7
7
9
9
9
10
11
11
IV. Future Outlook
12
A. Tunable laser lock using the Faraday effect in
high magnetic fields
12
B. Optical Switch
13
Acknowledgements
13
A. Fitting with simulated annealing
13
B. Laser locking by polarisation spectroscopy
14
C. A Monte-Carlo method for theoretical photon
counted pulses
14
References
15
INTRODUCTION
The interaction of light with atoms or molecules in
thermal vapours has become of great interest. The electric susceptibility of the vapour is essential in characterising this interaction, and once known can be used to
quantitatively predict a wide variety of physical phenomena. The aim of this report is to present an experimental
study of the slow-light effect in a 2 mm long rubidium
vapour cell. This report shows how the model for the
electric susceptibility, which has previously been developed in the slowlight project, has been applied for this
purpose.
There has been much interest in recent years on the
phenomenon of slow-light [1, 2]. It is characterised by
the situation where the light pulse velocity is much less
than that of continuous-wave light, and is realised in systems where there is a large derivative of the refractive
index with frequency of light. Slow-light increases the
amount of time a light pulse spends in the medium, and
so can greatly increase the atom-light interaction. This
means that processes, such as all optical switching [3],
image rotation [4] and an optical delay line [5] can be realised in experimental set-ups of modest length. A slow
light medium can also increase the sensitivity of interferometers [6, 7]. When combined with the Faraday effect
[8], a slow light pulse placed off resonance can act as a
sensitive atomic probe [9]. Electromagnetically induced
transparency (EIT) [10] is often used to create large pulse
delays [11, 12] due to the extremely (spectrally) narrow
transparent features that can be achieved. However, one
drawback of this is that it requires pulses with low bandwidth to avoid distortion, which corresponds to relatively
long (of the order of a microsecond) pulses. For communication purposes, shorter pulses means more information can be sent in any given time. However, slow light
using EIT gives the ability to store light [13], by switching the pump beam off while the pulse is in the medium,
giving the possibility of quantum memory [14]. For the
Slowlight project in AtMol we use < 800 ps pulse widths.
Knowledge of the electric susceptibility, χ, of the
2
atomic vapour is essential in understanding and performing experiments in pulse propagation. A spectroscopic
measurement is all that is required to determine quantities such as pulse delays (dispersion can be calculated
from absorption [15, 16]). However, a model of χ allows
experiments to be modelled and optimised before being
built. Also, it allows many phenomena other than pulse
propagation to be investigated, of which some have been
investigated in the slowlight project. One is quantitative
spectroscopy [17–20], which is essential in characterising
the properties of atomic or molecular transitions, and has
applications in measuring the Boltzmann constant [21–
23] and molecular fingerprinting [24], amongst many others. Another phenomenon investigated is that of Faraday
rotation [20, 25], which can be used to make an optical
isolator [26], dichroic beam splitter [27] or filter [28] and
a laser frequency locking signal [29], to name but a few.
The excellent agreement between experiment and theory already seen in references [17–20, 25] gives us confidence in our model. Due to equipment constraints, the
vapour cell that was used for most of the experimental
data in this report contained an unknown quantity of an
unknown species of buffer gas. Our model currently does
not include the effects of broadening and shifts due to
buffer gasses, and although it should be relatively simple
to incorporate empirically [30], it will still be very difficult
to identify the buffer gas species and pressure. However,
A study of the slow light effect should not be hindered
by the presence of the buffer gas, because as previously
mentioned, spectroscopic measurement can provide all
the information about pulse delay times. However, using
this cell for future experiments may be difficult since a
good theory prediction might be hard to achieve. This
is why measurements of absolute absorption have been
done to see if the buffer gas can be considered significant.
All our experiments are done with rubidium vapour,
rather than any other choice of atom or molecule, for
three reasons. The first is that our theoretical model for
χ is based on alkali metal atoms, due to relative simplicity of a hydrogen-like system with a quantum defect
[31]. The second is vapour pressure. Heating a 2 mm
vapour cell from room temperature to around 200◦ C allows us to go from an optically thin medium to an optically thick one, giving us a full range of control of this
experimental parameter. We have found that it is relatively simple to heat the vapour cell to this range of
temperatures using ceramic resistors. This contrasts to
sodium and potassium, where temperatures need to be
∼ 100◦ C and ∼ 300◦ C hotter respectively to achieve the
same vapour pressure [32], which creates more of an experimental challenge. The third reason is that Rubidium
D1 and D2 transitions are resonant at ∼ 795 nm and
∼ 780 nm respectively, where there exist relatively cheap
diode lasers that are convenient to work with.
A.
Motivation for smaller cells
The absorption of continuous-wave monochromatic
light by a medium of length L is given by the well known
Beer-Lambert law [33]:
I = I0 exp (−αL)
(1)
where I is intensity after the medium, I0 is the initial
intensity and α is the absorption coefficient. Interesting physical phenomena such as Rydberg blockade [34],
the cooperative Lamb shift [35, 36] and saturated opacity
[37], can be seen at large atom number densities. This
means that α will be large (see section II C) and so using
conventional cell lengths of a few centimetres the medium
becomes optically thick. This means that a probe beam
cannot be used to extract information about the sample. The solution to this is to make L much smaller
hence making the exponential of the product −αL nonnegligible.
Another reason to work with a small cell, and is of
particular interest for the future outlook of our project,
is the realisation of an optical switch [3, 38] using side
pumping (see section IV B). To access the same volume
of the dense atomic vapour for two crossed beams, a thin
cell is required (see Fig. 1).
FIG. 1. Schematic of an optical switch using a dense atomic
vapour with two crossed beams. The side beam is resonant
and so is fully attenuated at a short depth. For maximum
anisotropy of the ground state atoms, and hence maximum
efficiency, there should be as much overlap of the path of the
pump and probe beams as possible.
Another reason to work with smaller cells is to create
light and compact instruments [26, 39–41], that allow
mobile devices to be constructed.
3
II.
A.
THEORY
B.
The phase shift and absorption from the
electric susceptibility
For a particular transition, labelled j, we can write χj
as [17]
The electric susceptibility, χ, of an atomic vapour characterizes both its absorptive and dispersive properties.
We can see this by writing χ (a function of angular frequency ω) in terms of the relative permittivity [42],
r ≡
= 1 + χ,
(2)
0
where is the permittivity in the medium. In the electric
dipole approximation (magnetic response of the medium
is negligible at optical frequencies), the refractive index,
n is simply given by the square root of r [42]. The
definition of the refractive index is the ratio of the speed
of light in vacuum to the phase velocity in the medium.
With this it is easy to show
nω0
k = nk0 =
,
(3)
c
where k is the wave number in the medium, k0 is the wave
number of the incident light before entering the medium,
ω0 is the angular frequency of the incident light and c is
the speed of light in a vacuum. Now, using the equation
of a plane wave, at depth z in the medium and at time t
[42], we see that
~ = E~0 exp (i [kz − ω0 t]) ,
E
~ = E~0 exp (−nI k0 z) exp (i [nR k0 z − ω0 t]) ,
E
(4)
where n is allowed to be complex and is decomposed into
its real (nR ) and imaginary parts (nI ). Using the binomial expansion and noting that χ is small for atomic
vapours (at densities we can achieve experimentally) we
get
χ
(5)
n≈1+ ,
2
χR
nR ≈ 1 +
,
(6)
2
χR
nI ≈
.
(7)
2
This gives us the result
h
i
~ = E~0 exp − χI k0 z exp i k0 z − ω0 t + χR k0 z ,
E
2
2
(8)
where we can immediately identify a phase shift of
χR
k0 z.
(9)
∆φ =
2
Now using the relation [43]
n0 c ~ 2
I=
|E| ,
2
(10)
(valid for harmonic fields) and equation (1) we can also
identify the absorption coefficient,
α = k0 χI .
Equation for the electric susceptibility
(11)
χj (∆j ) = c2j
d2 N
sj (∆j ) ,
h̄0
(12)
where cj is the transition strength factor, d is the reduced dipole matrix element, N is the number density,
h̄ is the reduced Planck constant, 0 is the vacuum permittivity, sj (∆j ) is the line-shape of the transition. ∆j
is the angular detuning defined as ∆j ≡ ωL − ωj , where
ωL is the laser angular frequency and ωj is the natural
angular frequency of the transition. The total value of χ
is given by summing over all contributions
P from individual electric dipole transitions, χ (∆) = j χj (∆), and is
a function of the detuning around a defined global linecentre. For this report we have chosen, for convenience,
to use the weighted average of the transition frequencies
in the hyperfine manifold. In this report we use the D1
transition (52 S1/2 → 52 P1/2 ) and define the line-centre to
be (2π × 377 107 407.299 MHz). Transition frequencies
at a given magnetic field are calculated using methods
given in [20].
The line-shape factor sj (∆j ) is obtained by a convolution between the (complex) line-shape factor in the absence of Doppler broadening with a normalised Gaussian
distribution [17]. The line-shape factor in the absence of
Doppler broadening contains broadening due to natural
line width, Γ0 , and collisional broadening, Γcol . In a rubidium vapour cell, Γ0 is effectively equal for all hyperfine
transitions and magnetic sub-levels to the accuracy that
we can measure. For the D1 transition, the experimentally determined value of Γ0 = 2π × (5.746 ± 0.008) MHz
[44] is used. For a rubidium vapour cell without buffer
gas Γcol has only a contribution from self-broadening and
can be calculated using the treatment given in [19]. However, due to the presence of an unknown buffer gas at
an unknown pressure, the total Lorentzian line-width,
Γtot = Γ0 + Γcol , is allowed to float when fitting to
data. The collisional broadening parameter is again approximated to be equal for all hyperfine transitions and
their magnetic sub-levels, as well as isotopes. Also, there
should exist a frequency shift due to energy level changes
in collisions, but this has not been taken into account.
No deviation from the validity of these approximations
[45] has been seen in vicinity of experimental conditions
available to us [19], however this has not been tested in
the presence of buffer gas. All these considerations leave
a line-shape factor equal for all transitions, with the only
difference being the obvious shift in frequency due to the
hyperfine interaction.
The overall rubidium atomic vapour number density,
N , is given by an equation in [32]. It should be noted that
N is strongly temperature dependent. The fractions of
this number density that are either of the naturally abundant isotopes 85 Rb or 87 Rb area expected to be given by
4
the abundance in the metal reservoir. The atoms are assumed to be evenly distributed among the ground state
energy levels. This is justified as the Boltzmann factor
[46] is very close to one for the two ground states, while
being negligible when considering a thermal excitation to
either P-state. A weak probe beam must be used so as
to not pump the atoms into the dark ground state [47].
C.
Absolute absorption
Recalling the Beer-Lambert law given in equation (1),
we now define the transmission
T ≡
I
= exp (−α(∆, T )L)
I0
(13)
which can be seen as simply normalising the input intensity to one. This is convenient because it retains all
the relevant information and means measurements with
different initial intensity can be easily compared.
Equation (11) gives the absorption coefficient which is
a function of ∆ and temperature, T . Each individual χI,j
has the form of a Voigt profile [48], arising from a convolution between the Lorentzian line shape due to homogeneous broadening mechanisms (full width at half maximum given by Γtot ) and a Gaussian due to the MaxwellBoltzmann distribution of velocities at a given temperature.
When comparing theoretical and experimental spectra, several variables can be extracted from the data by
numerical fitting. If using a vapour cell containing rubidium that is not at its natural abundance, the ratio
between the isotopes 85 Rb and 87 Rb will need to be determined for the cell and can be extracted from fits. This
only needs to be done once and then the fraction can be
kept as a constant for future fits. The temperature of the
vapour often needs to be extracted from a fit because the
spectrum is a strong function of temperature and cannot
be measured precisely or accurately enough with our current equipment constraints. The Lorentzian width is fully
determined by the temperature of the cell, but as mentioned earlier, due to the presence of an unknown buffer
gas this also needs to be fitted. When a magnetic field B
is applied, its value can be extracted from a fit. No effort
is made to shield the vapour cells from the background
field, but it has been seen that assuming it to be zero
is a valid approximation for measurements of absolute
absorption [17].
Issues can arise when fitting with many parameters,
or the initial guesses to those parameters are poor. Appendix A describes these issues and how they are dealt
with.
D.
and absorptive properties. We have seen earlier how
to extract the phase shift and absorption of monochromatic continuous-wave light from the electric susceptibility. Pulses are characterised by a sharp change in
intensity with time and are clearly not monochromatic
continuous-wave light. The technique for dealing with
this is to write the pulse as a superposition of continuouswave light of different frequencies, we can then find the
phase shift and absorption of each component.
To do this, we first use equation (10) to relate the
intensity the initial pulse (as a function of time), with the
magnitude of its electric field. Given this initial electric
field (E(z = 0, t)), we perform a Fourier transform to
find the electric field in terms of angular frequency,
Optical Pulse Propagation
To determine how an optical pulse propagates through
a medium we need to know the medium’s dispersive
1
E (z = 0, ω) = √
2π
Z∞
E(z = 0, t) exp(−iωt)dt. (14)
−∞
To find the electric field at an a arbitrary depth z, we
then need to multiply by this result by exp(ik(ω)z) (see
equations (3) to (7)). Setting z = L gives us the pulse
angular frequency profile at the exit of the medium. We
can then use the reverse Fourier transform to construct
the electric field as a function of time at z = L,
1
E (z = L, t) = √
2π
Z∞
E(z = L, ω) exp(iωt)dω.
(15)
−∞
Now we can again use equation (10) to reconstruct the
intensity pulse.
The pulse propagation modelled in this report relies on
equation (12) to provide k(ω). This equation gives the
electric susceptibility as derived from the steady state solutions to the optical Bloch equations [43]. The pulses of
light we use for our experiments have a full width at half
maximum (FWHM) of < 800 ps, far less than the lifetime
of the excited states. However, equation (12) is still valid
for our pulses as long as we remain in the weak probe
regime. The pulse is considered weak if its effect on the
atomic ensemble is negligible, and so time derivatives on
the density matrix [49] should be correspondingly negligible.
1.
Narrowband approximation
The previous treatment is all that is required to model
pulse propagation but does not give a great deal of insight. In this section we make two approximations that
aid in understanding. We start by making the approximation that our pulses are Gaussian in shape, which is
not too bad an assumption when compared with measurements of the initial pulse profile. The range of frequencies that encompass ∼ 99% of the pulse is defined as
the ‘bandwidth’ and the peak frequency is known as the
carrier frequency. We then make the ‘narrowband approximation’, which essentially means that the change in
5
absorption and dispersion is small across the pulse bandwidth. With these approximations, the peak of the pulse
travels at the ‘group velocity’ [43], which we label vg . We
now define the group refractive index ng ≡ c/vg which is
given by the following equation [50]:
dnR (ω)
dω
(16)
Using this equation and our model for the electric susceptibility we can calculate ng for an alkali metal atomic
vapour. Fig. 2 shows the result of this for a 2 mm long
vapour cell containing rubidium in its natural abundance.
Notice that there are regions where ng is negative, which
corresponds to the pulse peak exiting the medium before
it has entered. These parts of the curve are regions where
the narrowband approximation breaks down or the pulse
is fully absorbed. However around these regions, ‘fast-
ng,( ×103 )
nR
Transmission
1.0
0.8
0.6
0.4
0.2
0.0
1.008
1.004
1.000
0.996
0.992
4
2
0
2
4
6
8
8
6
4
2
0
2
Detuning (GHz)
4
6
8
FIG. 2. Theoretical plots of transmission, the refractive index
for a continuous-wave beam (real part) and the group refractive index against detuning. The the colours correspond to
different cell temperatures as follows: red = 80◦ C, green =
100◦ C, blue = 120◦ C, cyan = 140◦ C, yellow = 160◦ C and purple = 180◦ C. The regions with the largest magnitude of the
group refractive index are regions of high absorption. The
grey highlighted region shows the bandwidth of the initial
pulse used for the experiment, with a carrier detuning of 300
MHz.
light’ pulses can be seen. Fig. 3 shows the result after
solving the discrete form of equation 15 numerically [51]
for a pulse with a carrier frequency that is resonant while
the medium is not optically thick. The peak of the pulse
after the medium arrives before a reference pulse travelling at the speed of light in vacuum. One thing to note is
that the pulse seen after traversing the medium is completely enveloped by the reference pulse on the leading
edge. This predicts that no more photons will arrive at
the detector at a given time then would have been the
case for the reference pulse. The topics of causality, energy transfer and signal velocities in a fast-light medium
are currently an area of much interest with much debate.
This is beyond the scope of this report.
0.8
Normalised Intensity
ng (ω) = nR (ω) + ω
1.0
0.6
0.4
0.2
0.0
2.0
1.5
1.0
0.5
0.0 0.5
Time (ns)
1.0
1.5
2.0
FIG. 3. Graph showing intensity against time for a reference
pulse travelling through vacuum (larger red curve) and the
pulse seen after traversing the medium (smaller blue curve).
The vertical dashed line crosses the blue curve at its peak. We
can see that the peak of the pulse traversing the medium arrives before a pulse travelling through vacuum. The medium
is modelled as atomic rubidium vapour in its natural abundance at 80◦ C in a 2 mm long vapour cell while the carrier frequency is set to -1.5 GHZ (resonant with the 85 Rb
Fg = 3 → Fe = 2, 3 Doppler broadened transition). Parameters have been chosen such that an experiment could be
performed in the future to verify this prediction.
In this report we present experiments where the slowlight effect is manifest. To clearly present the effect we
need to work in a region where the group refractive index
does not change vastly over the pulse bandwidth while
maintaining a relatively high value. Also the transmission should be high so that the output pulse is clearly
visible. Using Fig. 2 we have identified a region corresponding to the pulse bandwidth centred at 300 MHz
(shown as the grey shaded region), which best fits these
criteria.
III.
A.
EXPERIMENTS
Absolute Absorption
The aim of these experiments were to see if the presence of buffer gas in the 2 mm long 87 Rb vapour cell has
a considerable effect on transmission spectra. If experimental spectra agree with theoretical fits, we can confidently say that expected shifts due to the buffer gas are
negligible. Collisional broadening, however, can be accounted for by fitting the Lorentzian width (because collisions should be homogeneous), and so are not expected
to be to be an issue.
6
Experimental Method
Fig. 4 shows a schematic of the experimental apparatus used. A Toptica DL 100 external cavity diode laser
(ECDL) is used to produce the a coherent light beam.
The frequency of the laser beam is scanned around the
D1 line-centre (377 107 407.299 MHz). The beam passes
through a 50:50 beam splitter (BS) where a fraction of
the light passes through a Fabry-Pérot etalon (FPE) and
onto a photo-detector (PD). The remaining light is incident on a polarising beam splitter (PBS) where a fraction
of the light passes through and is then attenuated by a
neutral density filter (ND) before passing through the experimental cell. The light reflected at the PBS is used
to perform sub-Doppler hyperfine pumping spectroscopy
[52]. Linearly polarised light performs hyperfine pumping on the reference cell, before being attenuated by an
ND and passing though a quarter wave-plate (λ/4). The
beam is then reflected back to be counter propagating
to itself, passing again through the λ/4, ND and reference cell. The double pass through the λ/4 causes a π/2
rotation in the linear polarisation, ensuring the light is
now transmitted through the PBS and onto a PD. The
experimental cell is contained in an oven which allows a
laser beam to travel through. Outside the oven is an aluminium magnet holder which holds two permanent magnets on either side of the oven. The magnets allow the
laser beam to pass through the centre of them, and are
placed such that they create a magnetic field (B) parallel
to the laser beam. The magnets can be removed from
the holder when no magnetic field is required. The light
after the experimental cell is then incident onto another
PD. The signals from all the PDs are measured simultaneously on an oscilloscope. Using the method outlined in
[17], the FPE signal and sub-Doppler spectroscopy signal
are used to recalibrate the time axis of the oscilloscope
into a frequency axis.
nal gives a peak at fixed intervals in frequency, which
allows the scan to be made linear in frequency. The
frequency axis is then calibrated using the peaks of the
sub-Doppler signal as reference points, where the values
of the transitions are taken from [53] and the crossovers
are taken to be half way in between the transition values. If we apply this procedure to the sub-Doppler signal
2.5
Intesnity (Arbitrary Units)
1.
2.0
1.5
1.0
0.5
0.00
5
10
15
Time (Arbitrary units)
20
FIG. 5. Example of data raw data from the FPE signal (blue)
and sub-Doppler spectroscopy (red). The time axis is converted to a frequency axis by first linearising the data using
the FPE signal, and then using the sub-Doppler signal is to
calibrate the axis.
itself, we can extract a rough estimate of the accuracy of
the frequency calibration and linearisation on resonance.
Fig. 6 shows the result of doing this.
Freq. Diff. (MHz)
4.4
Intensity (Arbitrary units)
4.2
4.0
6
4
2
0
2
4
64 2 0 2 4 6
3.8
3.6
3.4
3.2
3.0
2.8 4
FIG. 4. Diagram of the experimental apparatus used for spectroscopy. The oven and magnet holder are not shown.
Fig. 5 shows an example of the both the FPE signal
and the sub-Doppler spectroscopy signal. The FPE sig-
2
0
2
Detuning (GHz)
4
6
FIG. 6. Graph of the sub-Doppler spectroscopy signal where
the intensity is plotted against the calibrated frequency. The
vertical lines correspond to known values for the transition
frequencies and crossovers. The inset shows the difference
between the peak positions and vertical lines. The difference
is less than 6 MHz for all peaks.
7
2.
Results
Fig. 7 shows the transmission through a 75 mm cell
containing rubidium in its natural abundance [55]. A
least squares fit was done to extract the temperature of
(20.70 ± 0.13) o C, which was the only free parameter.
The uncertainty on this value was taken as the difference
on the temperatures extracted from fits when the rubidium ratio was changed by ±0.01 (the uncertainty in the
rubidium ratio). At this temperature self broadening is
negligible [19] and so the Lorentzian width can be taken
to be just the natural line-width. Excellent agreement
is seen between theory and experiment, as shown by the
residuals.
Fig. 8 shows a plot of transmission against detuning for
a 2 mm long cell containing (98.20 ± 0.09%) 87 Rb. The
temperature, Lorentzian width and ratio of 87 Rb to 85 Rb
were extracted from fits to five separate spectra taken in
quick succession (less than 20 seconds). Notice that the
Lorentzian width is far higher than would be expected for
a cell containing just rubidium at this temperature. The
likely cause is buffer gas. This particular spectrum was
used to calculate the ratio of rubidium isotopes because
the 87 Rb and 85 Rb absorption peaks are both visible but
not optically thick. This meant that the fit to the ratio
should be most sensitive and hence most accurate. The
ratio extracted from this fit was used to fix the value for
all calculations using this cell. In Fig 9 a spectrum taken
at high temperature is shown. All the rubidium transition features have almost completely merged, giving a
broad region of high absorption.
1.00
Transmission
0.95
0.90
0.85
0.80
0.75
0.70
0.4
0.2
0.0
0.2
0.4
Residuals ( ×100)
The PD output voltage is linear with intensity, but
the background intensity varies with frequency due to
the laser scan and reflections in the set-up. The background can be quantified by finding known regions in
the absorption spectrum where the absorption should be
negligible and fitting a polynomial to those areas. This
works well for spectra which have relatively narrow features which correspond to low temperatures. However,
at high temperatures there may not be any region negligible absorption in the scan range. For a 2 mm cell this
problem can be solved by taking a ‘background’ spectrum when the cell is at room temperature, where the
absorption is negligible.
After the data was normalised and frequency calibrated, the data was fitted to theory using either a
Marquardt-Levenberg (ML) algorithm [54] or a simulated
annealing (SA) algorithm, or often both (see appendix
A). Five spectra were taken and the quoted fit parameters are the mean given with the standard error. A
thermocouple measurement was taken along with each
spectrum to ensure the temperature does not vary much
over the length of time taken to obtain five spectra (usually around 20 seconds). The thermocouple measurement
was not used to give the temperature of the vapour, this
was always extracted from the fit.
6
5
4
3
2
1 0 1 2
Detuning (GHz)
3
4
5
6
FIG. 7. Graph of the transmission against detuning in a 75
mm long cell containing rubidium in its natural abundance
(72.17% 85 Rb & 27.83% 87 Rb) [55]. The black curve shows the
experimental data while the red curve shows the theoretical
result after a least squares fit to the temperature using an ML
algorithm. Five spectra were taken (only one shown), each
of which were fitted to theory using the ML algorithm. The
temperature was found to be (20.70±0.13) o C. The statistical
variation in the five fits was very small and so the uncertainty
quoted was found using the functional approach [54], varying
the rubidium ratio by ±0.01. The bottom graph plots the
residuals of the data and theory curves in the top graph.
The result of fits to a spectrum taken at high magnetic
field are shown in Fig. 10. Two techniques were used to fit
the data. The first involved taking spectra with the magnets removed and fitting for temperature and Lorentzian
width only (magnetic field set to zero). The magnets
were then placed into the holder, and then the spectra
seen in Fig. 10 were taken and fitted to magnetic field
only. The other technique involved fitting the spectrum
for magnetic field, temperature and Lorentzian width using an SA algorithm followed by an ML algorithm. We
can see that the value of the Lorentzian width extracted
via the two methods strongly disagree.
3.
Discussion
Fig. 7 showed, as expected, that a long cell with no
buffer gas gives an excellent agreement between theory
and experiment. Also the value of the temperature extracted from the fit is in agreement with the thermocouple measurement. This agrees with the work in [17] and
gives us confidence in the theory for the electric susceptibility for rubidium vapour at low temperatures and no
buffer gas.
The measurement of the ratio of the rubidium isotopes
was taken from a fit to the spectra shown in Fig. 8. This
measurement should be insensitive to shifts and broadening due to buffer gas, because it is sensitive only to the
amplitude of the peaks (a strong function of number den-
1.0
0.8
0.8
Transmission
1.0
0.6
0.4
0.6
0.4
0.2
0.0
6
4
2
0
2
4
610
0.0
3
2
1
0
1
2
310
8
6
4
2 0 2 4
Detuning (GHz)
6
8
10
FIG. 8. 98.20% Rb87 cell in the presence of buffer gas. The
black curve shows the experimental data while the red curve
shows the theoretical result after fitting the temperature and
Lorentzian width with the ML algorithm. Five spectra were
taken (only one shown), each fitted to theory with the SA
algorithm to find the global minimum in parameter space.
Each of the five spectra were then fitted to theory using the
ML algorithm using the SA fit results as the input parameters. The temperature was found to be (90.17 ± 0.10) o C,
the Lorentzian width was Γtot = 2π × (165 ± 1) MHz and
the fraction of 87 Rb was found to be (98.20 ± 0.09)%. The
uncertainties quoted are the statistical fluctuation in the five
measurements. The bottom graph plots the residuals of the
data and theory curves in the top graph.
sity). Furthermore, the measurement should be insensitive to the overall value of the number density (and hence
temperature) because only the ratio of number densities
of the two rubidium isotopes is measured. For these reasons, the value of the ratio extracted from this measurement is trusted and was fixed for all subsequent fits.
The overall fit to theory seen in Fig. 8 and Fig. 9
is reasonably good, as seen by the residuals. However,
the values of the temperature extracted from the fits
disagreed with thermocouple measurements. The thermocouple measurement for the spectrum shown in Fig. 8
was ∼ 40◦ C higher than the fit, and it was ∼ 30◦ C higher
for the spectrum shown in Fig. 9. The accuracy of the
thermocouple was therefore brought into question. The
thermocouple calibration was checked using boiling water
and was found to agree with the defined value of 100◦ C.
The fit to the spectra in the long rubidium cell (Fig. 7)
was in agreement with the thermocouple measurement
when both were at ambient room temperature. Therefore, within the range of 20◦ C to 100◦ C, we expect the
thermocouple to be accurate. This rules out the thermocouple accuracy as the cause of the discrepancy, at
least for the lower temperature spectrum. One possible
explanation for the discrepancy may be that the thermocouple tip was placed in the cell heater at a place that
was hotter than that of the cell. Another explanation is
Residuals ( ×100)
0.2
Residuals ( ×100)
Transmission
8
8
6
4
2 0 2 4
Detuning (GHz)
6
8
10
FIG. 9. 98.2% 87 Rb cell in the presence of buffer gas. The
black curve shows the experimental data while the red curve
shows the theoretical result after fitting the temperature and
Lorentzian width with the ML algorithm. Five spectra were
taken (only one shown), each fitted to theory with the ML
algorithm. A fit using SA was used on each spectrum to
ensure that the fit parameters are in the global minimum.
The temperature was found to be (182.1 ± 0.4)o C, while the
Lorentzian width was Γtot = 2π × (170 ± 4) MHz. The uncertainties quoted are the statistical fluctuation in the five
measurements. The bottom graph plots the residuals of the
data and theory curves in the top graph.
that the formula, giving the number density at a given
temperature [32], is invalid for this particular cell. Buffer
gas pressures in the cell may be high, causing a smaller
rubidium number density for a given temperature. Also
if the cell is deficient in rubidium, that will also cause a
smaller number density at a given temperature. If there
is a smaller number density for a given temperature, fits
to theory will show a lower temperature than they should
(because number density is a stronger function of temperature than the Doppler width). This may then cause
the Lorentzian width to be fitted larger than it should,
to compensate for the small theoretical Doppler width.
Of the two methods used to fit the spectra taken
at high magnetic field, the one using a spectrum with
the magnets removed to measure the temperature and
Lorentzian width should be more reliable. However, we
expect the global minimum in parameter space to correspond to accurate values of the parameters if the theory
is valid. However, the Lorentzian widths disagree. One
explanation for the disagreement may be that the shifts
due to the buffer gas may be compensated by an increase
in Lorentzian width, for an optically thick medium when
the magnetic field is set to zero. However, when the
magnetic field is on and the spectrum no longer is optically thick, the fit to the magnetic field should be able to
account for shifts better than an increase in Lorentzian
width and so the Lorentzian width will be seen to be
smaller.
9
B.
1.0
Transmission
0.8
Slow-light
The following section shows the experiments that were
done to demonstrate the slow-light effect and to test the
theory developed to account for it. A variety of methods
were used, the results of which are discussed and will be
used to improve future experiments.
0.6
0.4
0.2
Residuals ( ×100)
0.0
4
2
0
2
4
10
1.
8
6
4
2 0 2 4
Detuning (GHz)
6
8
10
FIG. 10. 98.2% Rb87 cell in the presence of buffer gas. The
black curve shows the experimental data while the red curve
shows the theoretical result after fitting the magnetic field
with the ML algorithm. Five spectra were taken (only one
shown) and the magnetic field was found to be (895 ± 6)
Gauss. Fits to five spectra with the magnets removed gave the
values of the temperature (110.94 ± 0.02)o C and Lorentzian
width 2π × (127.7 ± 0.3) MHz. An SA fit of the temperature,
Lorentzian width and magnetic field showed that fit given by
the red curve is not in the global minimum. The blue curve is
a ML fit of the data after ensuring the fit is within the global
minimum, giving the magnetic field as (904.8 ± 0.6) Gauss,
temperature as (110.136 ± 0.014)o C and Lorentzian width as
2π×(91.8±0.4) MHz. The uncertainties quoted are the statistical fluctuation in the five measurements. The bottom graph
plots the residuals, with like colours corresponding the theoretical curves in the top graph. Normalised RMS deviation of
red (blue) curve to data = 1.5(1.2)%.
4.
Conclusion
We have seen that while our theoretical fits can account
for the vast majority of the structure of the spectrum in
our rubidium cell with buffer gas, an excellent agreement
is not obtained. Good fits are obtained, but the values of
the extracted parameters cannot be fully trusted. This
means the buffer gas cannot be considered negligible because our theory (in its current form) cannot account
fully for the spectrum due to shifts and the difference
in the number density at a given temperature. This cell
may not be suitable for future experiments because theoretical models that help design experiments, may not be
valid for this cell. However, this cell can still be used for
simple pulse propagation experiments, because we can
characterise the absorptive and dispersive properties of
the medium using a spectroscopic measurement. This
is done by simply allowing our model to fit to the data
such that a good agreement is obtained, while acknowledging that the parameters are more phenomenological.
A dispersion curve can then be generated from theory.
Experimental Method
Fig. 11 shows a schematic of the apparatus used for
the experiment. A Toptica DL 100 laser is used to generate continuous wave light. The beam passes through a
high extinction, Glan-Thompson polarisation beam splitter (PBS) linearly polarising the beam. The beam then
enters a Pockels cell [56], which rotates the plane of polarisation when a large voltage is applied to the Pockels cell.
The beam is then incident on another high extinction,
Glan-Thompson PBS which is crossed with respect to the
first PBS. The Pockels cell is attached to a voltage supply which can rapidly switch a large voltage. The crossed
PBSs with the Pockels cell create a pulse of light when
the plane of polarisation is rapidly rotated. The light
pulse is then attenuated by a neutral density filter (ND)
before entering the experimental cell. The experimental
cell is placed in an oven that is used to heat the cell and
change the vapour pressure. After traversing the cell the
pulse is then incident on either a fast photo-diode detector (FPD) or an avalanche photo-diode detector (APD).
The signal from the FPD or APD is then recorded on
a fast oscilloscope (∼ 160 ps rise time). The avalanche
photo-diode has an extremely fast response time (< 50
ps quoted from the manufacturer) and works as part of a
photon counting module. This means it needs many repetitions to accumulate a pulse profile. The FPD on the
other hand gives a profile over one shot, but has insufficient resolution to accurately characterise the pulse. The
FIG. 11. A schematic of the experimental apparatus. The
oven is not shown. Also not shown is a third PBS placed just
before the cell with a FPD measuring the output of the side
arm.
input pulse shape is important to characterise accurately
for an accurate theoretical prediction. The FPD can be
used to experimentally measure pulse delays, but are not
10
accurate enough for an excellent agreement with theory.
The APD can be used to gain a much higher resolution
but many repetitions need to be measured to accumulate
a pulse profile. The input pulse was characterised using
the APD because of the better time response. However,
it was noticed that the output from the FPD is not exactly the same with each repetition. The time when the
pulse arrives after the trigger of the oscilloscope changes,
and also the peak height of the pulse changes. Fig. 12
shows a plot of the peak voltage (proportional to peak
pulse intensity) against the arrival time of the peak. This
‘jitter’ will have the affect of broadening the pulse as measured by the photon counters. To quantify this effect a
Mont-Carlo algorithm [57] reproducing this distribution
was used to model the jitter and found that the broadening was negligible (< 20 ps increase in the FWHM, see
appendix C). Therefore, the input pulse profile obtained
by counting over many repetitions was taken as the initial
pulse on all occasions.
2.
Results
Fig.13 shows experimental plots of slow-light pulses.
The pulses were measured using a FPD, and show qualitatively that pulses of light can be significantly slowed
with respect to a reference pulse.
0.5075
0.5050
Peak voltage measured (V)
a phenomenological function in order to smooth it. To
measure a reference pulse, the laser was tuned far off
resonance and the pulse profile was formed after counting for the same amount of time. A third PBS placed
before the ND and the experimental cell (not shown in
Fig. 11) was used with a FPD to measure any change in
laser power when moving the frequency far off resonance.
This change in laser power was then factored out during
analysis.
A measurement of absolute absorption (like that in section II C) was taken soon after the pulse data was taken.
This transmission spectrum was then fitted to theory to
extract χ for the atomic vapour. This χ was then used
to create a theoretical output pulse profile using the off
resonance pulse as the initial pulse.
0.5025
0.5000
0.4975
0.4950
0.4925
0.4900
50.00 50.05 50.10 50.15 50.20 50.25 50.30 50.35 50.40
Time of peak pulse after trigger (ns)
FIG. 12. Graph of the peak voltage output of the FPD against
time after the trigger of the oscilloscope, when a pulse is incident on the FPD. There are 2011 points, each taken in quick
succession through the vapour cell when off resonance and
at room temperature. The distribution is consistent with a
Gaussian in voltage and time, with standard deviations measured as approximately 3 mV and 50 ps respectively.
When forming a pulse profile via photon counting over
many repetitions, a long time-scale of the order of an
hour was needed to gain a very good profile. However,
when measuring the output pulse on or near resonance
in a hot medium, this time-scale is enough to see the
laser shift frequency by a noticeable amount. To minimise laser drift over the counting period the laser should
be locked. When using the 98.2% 87 Rb cell the laser
was locked using polarisation spectroscopy to the 85 Rb
Fg = 3 → Fe = 2 transition (see appendix B). However,
when using the cell with rubidium at its natural abundance, this method of laser locking was unsuitable. To
compromise, a short counting time of around 10 minutes
was used to give a pulse profile, that was then fitted to
FIG. 13. Graph of intensity against time for slow-light pulses
in a 2 mm long cell containing rubidium vapour at its natural
abundance. The black curve is the pulse measured using a
FPD when the cell was at room temperature and optically
thin, and was assumed to have a negligible time delay. The
room temperature of 22◦ C was measured with a thermocouple. The other curves are pulse profiles measured using a FPD
at temperatures (shown next to the respective curve) that
were extracted from fits to transmission spectra. These temperature values were extracted from fits to absorption spectra.
The carrier detuning was approximately 300 MHz. The laser
was not locked.
Fig. 14 shows the result from measurements of pulses
after traversing the 98.2% 87 Rb cell. A FPD was used
for the measurements and the laser was frequency locked.
The theoretical predication in Fig. 14 was the result of
solving equation (15) numerically for a reference pulse
approximated to be a Gaussian FWHM of 900 ps. We
11
can see that the pulse is largely absorbed, with not much
delay.
FIG. 14. Graph of normalised intensity against time for pulses
after traversing a 2 mm long cell containing rubidium of which
98.2% is 87 Rb. The cell also contained buffer gas. The solid
blue curve is the pulse measured using a FPD when the cell
was at room temperature and was assumed to have negligible
delay or absorption. The solid pink curve was measured when
using a FPD when the cell was heated. The laser was locked
using polarisation spectroscopy to the 85 Rb Fg = 3 → Fe = 2
transition (≈ −1500 MHz detuning). The red dashed curve
is the theoretical prediction when a Gaussian with a FWHM
of 900 ps is taken as the reference pulse.
Fig.15 shows a delayed pulse as measured using an
APD as part of a photon counting module. The pulse
was measured by building up a profile over 11 minutes
and 30 seconds and then fitting a phenomenological curve
to the profiles. The carrier wave detuning of the pulse
was approximately 300 MHz and the reference pulse was
taken to be the pulse profile when the carrier wave was
far off resonance. This reference pulse, along with a measurement of absolute absorption to extract χ, allowed the
theoretical curve to be produced.
3.
Discussion
In Fig.13, it is shown that by increasing the cell temperature it is possible to get large time delays, where the
pulse can be delayed by more than the pulse width. A
rough calculation shows that group refractive indices of
∼ 400 and ∼ 1000 are seen for the two hottest temperatures. The FPD detectors do not have a very good time
resolution but they are sufficient to see the rough time
delay.
In regions where the absorption is large, the absorption
and dispersion often change rapidly with frequency. Frequency locking the laser is therefore advantageous when
performing experiments in these frequency regions. The
results in Fig.14 were taken when the laser was locked.
A FPD was used and so the laser would not have drifted
much over the course of the experiment, but the advan-
FIG. 15. Graph of intensity against time for a pulse of light on
and off resonance with the atomic vapour. The intensity axis
is normalised to the peak hight of the solid blue curve. The
blue markers show the count number (normalised) from the
APD, within a bin width of 20 ps, for a far off-resonance (∼15
GHz) pulse measured over many repetitions for 11 minutes
and 30 seconds. The red markers show the count number
(normalised) from the APD, within a bin width of 20 ps, for a
pulse carrier detuning of ≈ 300 MHz measured over the same
amount of time as the off-resonance pulse. The solid blue
and red curves are phenomenological fits to the blue and red
markers respectively. The time axis is zeroed at the peak of
the solid blue curve. The black dashed curve is a theoretical
prediction for the delayed pulse when taking the solid blue
curve to be the reference. The experimental cell was 2 mm
long containing rubidium in its natural abundance and was
at a temperature measured to be 137.6◦ C.
tage here for laser locking was that it also provides a
more accurate measure of the frequency. Fig.14 shows
reasonable agreement between theory and experiment,
but hindered by not being able to measure the pulses accurately enough with the FPD. An improvement can be
made to the data made in this graph by using the photon
counters
Fig.15 shows the result of using the photon counting
method and achieves a good agreement. The discrepancy
between theory and experiment may be due to the fact
that the profile of the measured pulses were lacking in
precision due to the relatively short counting time. Also
the laser was not locked which may have meant that the
value taken as the carrier frequency was not very accurate. A tunable laser lock would therefore improve this
experiment (see section IV A).
4.
Conclusion
Reasonable agreement between experiment and theory
has been observed. We have seen that APDs are preferred over FPDs because of their better time resolution.
We have also seen that locking the laser may improve
12
IV.
A.
FUTURE OUTLOOK
Tunable laser lock using the Faraday effect in
high magnetic fields
1
Ix − Iy
= cos (2φ) exp − α+ + α− L
I0
2
FIG. 16. Schematic of the proposed apparatus that can be
used for a laser lock. This arrangement is similar to that in
Fig.4 with the addition of a half-wave plate (λ/2) to rotate the
initial plane of polarisation, and a PBS after the cell with two
photo-diodes used to detect the beam on each arm of the PBS.
These two photo-diodes can be used as part of a differencing
detector to get a Faraday signal. The experimental cell will be
a 1 mm long micro-fabricated cell placed within a permanent
magnet (not shown) that produces a high magnetic field.
the beam is separated into horizontally polarised (Ix ) and
vertically polarised (Iy ) components. These components
are then measured on separate photo-detectors (PD).
By tuning the experimental parameters of magnetic
field, temperature and angle of the plane of polarisation
of the initial light (θ0 ), we can find a vast number of
(17)
where φ is the angle of the plane of polarised light with
respect to the x-axis after the cell, and α+ and α− are
the absorption coefficients for light driving σ + and σ −
transitions respectively. The rotation angle is given by
the phase shift of the two circularly polarized components
of the beam which can be found using our model for χ in
the circular basis. Also, α+ and α− can be given by our
model. Using our model we can now change experimental
parameters to find locking points. Fig.17 is one example
of this where a locking point at a detuning of 300 MHz
was found.
(Ix −Iy )/I0
In order to improve the photon counting method for
generating a pulse profile, it will be necessary to frequency lock the laser. Taking inspiration from reference
[29] we can use a Faraday signal to lock the frequency
of our laser. If we use a micro-fabricated 1 mm long
cell placed within a permanent magnet, magnetic fields
of up to around 0.6 T are achievable [26]. By changing the depth of the cell in the magnet, we can tune the
magnetic field. Fig.16 gives a schematic of the apparatus
that can be used to generate the laser locking signal. The
schematic is very similar to that in Fig.4. After the cell
locking points (zero crossings) over a range of about 50
GHz. To give an example of this, we first choose our
Faraday signal to be (Ix −Iy )/I0 (this is one of the Stoke’s
parameters [20]). This Faraday signal can then be given
by the following equation (taken from [20]),
Transmission
the accuracy of the experiment. Future experiments will
benefit from using a tunable laser lock along with the
APDs to get the most reliable experimental data.
We have also seen that using the room temperature cell
to characterise the reference pulse may be unreliable due
to reflection effects changing with temperature. Moving
the laser frequency off resonance, whilst accounting for
any changes in laser intensity, may therefore give better
results.
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6-10
-8
-6
-4
-2 0 2 4
Detuning (GHz)
6
8
10
FIG. 17. Theoretical transmission and (Ix − Iy )/I0 against
detuning. Shown with the vertical dashed line is a zero crossing which can be used as a locking point. The experimental
cell was modelled as 1 mm long, containing 99% 87 Rb. The
following experimental parameters were chosen: B = 2142
Gauss, θ0 = 2π/5 and T = 150◦ C
In general the signals obtained, like that in Fig.17, do
not have very sharp zero crossings and so may be less
precise than other locking schemes. For the type of pulse
propagation experiments shown in this report, less locking precision is needed when away from a resonant feature, for which this method may be suitable. However,
when on-resonance, we need a higher precision due to the
rapidly changing absorption and dispersion. For this we
can use a laser lock based on polarization spectroscopy
(see appendix B).
13
B.
Optical Switch
One possible future outlook for our project will be to
create an optical switch based on the controlled Faraday
rotation of a pulse [25]. To cause a Faraday rotation of
the polarisation of linear polarised light, we need to cause
a phase shift between the two circular components of the
light. This is can be done using a thermal atomic ensemble in two different ways (shown in Fig.18). One way is
to use a magnetic field to cause one of the σ transitions
to be more resonant than the other. The other way is
to use optical pumping to make the populations of the
magnetic sub-levels uneven. Using optical pumping we
FIG. 19. Diagram of the optical switch (adapted from a figure
made by Lee Weller). The initial polarisation vector of the
probe pulse (FARADAY ) is rotated by both the presence of
an applied magnetic field (B) and the pump beam. A change
in the angle of the polarisation of the pump beam (PUMP )
changes strength of the component of light driving either σ
or π transitions.
FIG. 18. Simplified energy level diagram showing 2 ground
state magnetic sub-levels and one excited state level. An
anisotropy in the strength of coupling to the probe beam is
achieved in two different ways. (a): A larger fraction of the
atoms are in one ground state rather than the other, causing a stronger coupling of the ensemble to the probe beam
for the transition with more atoms. (b): A magnetic field
lifts the degeneracy in the magnetic sub-levels causing one to
be shifted further from resonance while the other is shifted
towards resonance.
can change the amount of Faraday rotation of the probe
beam. If we can change the polarization by π/2, light
can be made to switch between orthogonal polarisation
channels by use of a PBS after the cell. This would create
an optical switch. Optical pumping alone is difficult to
create the required rotation, but it was shown in [25] that
combining this with a magnetic field makes this possible.
A counter-propagating pump and probe beam was used
as the arrangement in [25].
Our intention is to use these principles, and create an
optical switch for pulsed light that is pumped using a
beam that accesses the medium orthogonal to the direction of the probe beam. This way the medium can
be more uniformly pumped over the volume that the
pump and probe cross, and also the pump and probe
both address the same atoms across a wide range of velocity classes. This should increase the performance of
the optical switch. At present, the idea is to have the
probe pulse on the D1 line and the pump on the D2 .
ACKNOWLEDGEMENTS
The work in this report was carried out in collaboration
with Lee Weller, Charles Adams and Ifan Hughes.
Appendix A: Fitting with simulated annealing
Fitting many parameters at once can create issues with
finding the best fit parameters. When fitting theory to
experimental data some sort of cost function (a measure
of how much the fit deviates from the data) is employed,
and is minimised to find the best value of the parameters. The cost function lives in a space with the same
number of dimensions as parameters, and is often impractical (or even impossible) to completely explore it
within its allowed bounds. Therefore, a simple and fast
method is to make a guess of the parameters and then
vary the parameters, only allowing changes that decrease
the cost function and finish when each step is no longer
reducing the cost function very much. This method is
called ‘hill climbing’ (from the situation where the cost
function is defined with the different sign convention).
There are many hill climbing algorithms of which the
Marquardt-Levenberg method (ML) [54] is the one we
use. The problem with the hill climbing method is that
there is a real danger of falling into a local minimum and
not finding the best overall parameters. This problem is
sometimes caused by parameters compensating for each
other, and with more parameters, there can be many
more local minima.
Fig. 20 shows an example of how fitting using a
hill climbing algorithm can get stuck in a local minimum. Two fits to experimental transmission spectra were
done using the ML algorithm but with different starting
guesses to the three parameters. After calculating the
root-mean-square deviation of the fits to the data, we
can clearly see that one fit is better than the other. The
difference between some of the fit parameters is far larger
than the uncertainty in the fit parameters (as given by
the Hessian matrix used in a ML fit [54]). This means
14
that the worse fitting one must have been the result of
fitting in a local minimum.
1.0
Transmission
0.8
0.6
tial energy. An example of this is silicon dioxide which
forms glass when rapidly cooled. Annealing is analogous
to our problem, where the positions of the particles are
the many parameters and the internal energy of the system is the cost function.
In its simplest form the simulated annealing algorithm
takes the following steps:
0.4
1. Make initial guesses for the parameters and evaluate the cost function (F1 )
0.2
2. Change these parameters and evaluate the cost
function again (F2 )
Residuals ( ×100)
0.0
4
2
0
2
4
6
810
3. If F2 < F1 accept these new parameters.
8
6
4
2 0 2 4
Detuning (GHz)
6
8
10
FIG. 20. Graph of transmission against detuning for a 2
mm long vapour cell, containing 98.2% 87 Rb. The red and
blue curves are ML least-square fits to theory with different
starting guesses for the parameters. The starting parameters for the red (blue) curve were: B = 800(600) Gauss,
T = 131(120)◦ C, Γtot /2π = 120(100) MHz. The final values for the red (blue) curve were: B = 861(858) Gauss,
T = 93.0(92.8)◦ C, Γtot /2π = 262(314) MHz. The normalised RMS deviation of the red (blue) curve to data was
1.96(2.30)%.
The best method to get around this problem is by engineering the situation such that only one parameter need
be fitted at any one time. For example, if we want to take
data that we will later want to fit the temperature and
magnetic field to, we should first take a spectrum with no
applied field and then quickly apply the field and take the
data. This way the two spectra should have been taken
at the same temperature (assuming the vapour cell is
not rapidly changing in temperature). This way the first
spectrum can be used to extract the temperature when
the magnetic field is fixed at zero, and then we can fit
the magnetic field to the second spectrum while fixing
the temperature.
Fitting as few parameters as possible is always important, but due to the presence of buffer gas in the experimental cell, a fit to just one parameter is not possible.
The Lorentzian width is needs to be fitted along with the
temperature. This means a new algorithm needs to be
employed, one that will find the global minimum with
a high number of parameters and potentially bad initial guesses for the parameters. For this purpose a fitting
code using ‘simulated annealing’ [58] has been developed.
Simulated annealing uses the Metropolis algorithm [59]
to emulate the remarkable observation that when a substance is cooled to a crystalline solid slowly, it always
finds the state of minimum energy. The crystalline form
is one of low energy, but if the substance is rapidly cooled
it may form an amorphous solid of higher internal poten-
4. If F2 > F1 accept these new parameters with probability exp[(F1 −F2 )/kT ], where k is a scaling factor
(like Boltzmann’s constant) and T is ‘temperature’
parameter.
5. Reduce T and repeat the procedure until the system is ‘cold’, i.e T is very small.
The initial value of T and k need to be chosen appropriately along with a slow method of decreasing T . The
method that was used for decreasing T was setting the
new T value to T /(1+βT ), where β is a sufficiently small
number [60].
After applying the simulated annealing algorithm to
the data in Fig. 20, it was found that in fact both the ML
fits were not in the global minimum, with both having
vast overestimations of the Lorentzian width.
Appendix B: Laser locking by polarisation
spectroscopy
Polarization spectroscopy [61, 62] was used to frequency lock the laser when taking the data shown in
Fig. 14. Shown in Fig. 21 is the locking signal. We can
see that the difference between the locking signal, and
the accepted value of -1.497657 GHz [53] for the transition, is of the order of 10 MHz. Over time the height of
this signal may vary (due to batteries running low on the
detector). Therefore, the laser will be locked anywhere
between the peak and trough of the signal, without jumping to another zero crossing. The distance between the
peak and trough is approximately 20 MHz while the frequency calibration is accurate to approximately 5 MHz.
We can therefore say that when locking to this signal,
the laser detuning was −1.50+0.01
−0.02 GHz from the defined
global line-centre.
Appendix C: A Monte-Carlo method for theoretical
photon counted pulses
Pulses were measured using the photon counting
method have shown FWHM values of as low as 750 ps.
When using a FPD the pulse shape is not accurately
represented but it is assumed that it reacts the same for
each pulse. This assumption would mean that the measured peak arrival times should correspond directly to
15
malised to a peak hight of one. This result was then be
plotted on top of the initial pulse that would theoretically
be seen over just one shot (by a perfect FPD), shown in
Fig. 22. The graph shows four curves, two are for a theoretical reference pulse and two are for a delayed pulse
after traversing a medium with the same conditions as
for the experiment shown in Fig. 14. The two curves for
each pulse show the result for just one shot and the result
of the Monte-Carlo code respectively. The two curves lie
almost on top of each other, where the broadening is only
just visible. By inspection it is possible to see that the
broadening is less than 20 ps which is smaller than exper-
Intesnity (Arbitrary Units)
0.2
0.1
0.0
0.1
0.2
0.3
1.56
1.54
1.52
1.50
Frequency (GHz)
1.48
1.0
1.46
0.9
FIG. 21. Graph of the polarisation spectroscopy signal in the
region if the 85 Rb Fg = 3 → Fe = 2 transition. The vertical
dashed line shows the transition detuning of -1.497657 GHz.
the actual pulse peak arrival times. The ‘jitter’ observed
in Fig. 12 shows pulses arrived at different times over a
range of approximately 300 ps, which intuitively would
seem like the photon counted pulse may just be the result of a much thinner pulse broadened significantly by
photon counting over many repetitions. A Monte-Carlo
method to simulate this was therefore used to try to infer
the original width of the pulse.
The Monte-Carlo computer code works by first reproducing the sort of distribution seen in Fig. 12. The code
takes into account any possible linear trend in the data
by fitting a line of best fit to the distribution. Fig. 12
clearly does not have much of a linear trend, but a linear
trend has been seen in other measurements of the Jitter.
The 2-D distribution is then recreated by modelling the
1-D distributions in time and distance from the line of
best fit as Gaussian.
Once the distribution is recreated, pulses were modelled for the points in the distribution. This was done
by taking the point’s relative hight above the mean as
the relative normalised intensity of this pulse from the
mean. In a similar way, the pulse time was shifted by
the difference in time of the point from the mean. The
pulses were then all summed together and then renor-
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