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Observation of spatial ordering and blocked excitation in Rydberg gases
Ravensbergen, C.
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2013
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Download date: 17. Jun. 2017
University of Technology Eindhoven
Coherence and Quantum Technology
Observation of spatial ordering and
blocked excitation in Rydberg gases
C. Ravensbergen
Supervisors:
dr.ir. R.M.W. van Bijnen
dr.ir. S.J.J.M.F. Kokkelmans
dr.ir. E.J.D. Vredenbregt
C. Ravensbergen
2
Abstract
The ability to tune and control many properties of ultracold quantum gases has led to the
possibility of using these gases as a simulator for other more complex and inaccessible materials,
hence the name Quantum Simulator [1]. In this thesis we investigate the feasibility for using
Rydberg atoms as the donor-system on which to perform quantum simulations. Rydberg atoms
are neutral atoms of which a single electron is excited to a state with high principal quantum
number [2], and they exhibit a number of very useful and special properties. The van der Waals
forces between Rydberg atoms is multiple orders of magnitude larger than for ground state atoms
[3], and is the source for the long-range interactions and strong correlations [4]. The interaction
manifest itself in the excitation blockade, by which a Rydberg atom blocks, in its vicinity, the
excitation of all other atoms [5]. The mere excitation of Rydberg atoms causes the emergence of
strongly correlated many-body states, the most prominent of which is the formation of crystal
like structure of Rydberg atoms [6].
The numerical simulations developed in this thesis give a fast and accurate method to study
the dynamics of many thousands of interacting Rydberg atoms over long periods of time [7].
The simulations are based on solving rate equations with the Kinetic Monte-Carlo algorithm.
The simulations showed a blockade radius of 12.9 ± 0.1 µm, at the 87S Rydberg state and
under realistic experimental values, and strong suppression in the fluctuations of the number of
Rydberg atoms per cycle, quantified by a Mandel Q-factor below zero [8].
Control over Rydberg atoms is mainly exerted by the excitation laser. Here we implement a
method to shape the intensity of the laser in arbitrary patterns. These patterns can be smaller
than the blockade radius, which is the typical length scale of the blockade effect. The device
responsible for shaping the light is a Spatial Light Modulator, SLM. The SLM in combination
with a novel in-vacuo wavefront measurement enables us to implement Adaptive Optics in the
form of iterative aberration correction and image optimization. The Root mean square wavefront
error, after correction, was measured to be as low as λ/6.8. The smallest produced spots had a
diameter of 11 ± 1 µm (2σ) and were spaced by 14 ± 1 µm, thus future experiments could obtain
only a single Rydberg atom per spot.
Interactions between Rydberg atoms can be analyzed quantitatively by constructing a paircorrelation function from all the atom positions [9]. We measure the positions by field ionization
the Rydberg atoms and high resolution spatial imaging of the hereby obtained ions. The measured correlation functions are in excellent agreement with the simulations and literature, all
three exhibit, for example, the same blockade radius. The simulated and measured correlation
function also contain a strong indication, the presence of a maximum outside the blockade radius,
for the presence of an ordered, crystal like, structure in the Rydberg gas. The largest measured
blockade radius, of all performed experiments, was 16.3 ± 0.1 µm, found for the 103S state.
The Mandel Q-factor has proven to be unsuitable, in the current setup, to show reliable
blockade effects in an ultra-cold trapped Rydberg gas due to its large susceptibility to external
influences like the detection efficiency, laser frequency fluctuations and excitation beam shape.
3
C. Ravensbergen
Contents
1 Introduction
6
2 Theory of Rydberg atoms
8
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2
Rydberg states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3
Optical excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.4
Three-level atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.5
Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.6
Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.7
Open quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.8
Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.8.1
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.8.2
Simulations in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.8.3
Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.8.4
Mandel Q-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3 Experimental setup
3.1
Vacuum chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.2
Trapping and cooling lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.3
Excitation lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.4
Ionization and acceleration voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.5
Software for ion detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
4 Shaping the excitation light
4
29
36
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
4.2
Basic working principles of spatial light modulators . . . . . . . . . . . . . . . . . . .
36
4.3
Calculation and construction of arbitrary light patterns . . . . . . . . . . . . . . . .
38
4.4
Measuring aberrations of optical systems . . . . . . . . . . . . . . . . . . . . . . . . .
39
4.5
In vacuo aberration correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
Contents
5 Observation of Rydberg Atoms
51
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
5.2
Rydberg level scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
5.3
Detector Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
5.4
Measured correlation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5.4.1
S-states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5.4.2
D-states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
Mandel Q-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
5.5
6 Conclusion
63
5
C. Ravensbergen
1
Introduction
Research on ultra-cold gases was originally fueled by the desire to study quantum effects in a more
controlled environment than for example solid-state physics. But quite soon it became clear that
the applications where much more numerous and they range from incredibly stable clocks [10] and
interferometers [11] to more recently, quantum computer, teleportation and quantum simulators
[12].
A quantum computer does not use binary bits, with value 0 or 1, like a conventional computer but
instead uses so-called qubits which are a quantum superposition of two or more states meaning that
the value can be both 0 and 1. Quantum teleportation is the art of instantly transferring a state
from particle A to an arbitrarily distant particle B, for example between two Canary isles [13] using
photons. The last mentioned device, known by its general term quantum simulator, has attracted
a great deal of attention and generated many breakthroughs in the past few years. A quantum
simulator is in itself a quantum system with some special and very useful properties. For one it
behaves exactly like another quantum system but is much easier to study, manipulate or alter. This
gives the opportunity to simulate and optimize a complex system like for example a superconductor,
which is nearly impossible to do with a computer. Quantum mechanics is in itself a very powerful
and precise theory but its numerical application, to some quantum many-body problems, is very
limited today due to limited computational resources. Especially these many body computations
can be tedious and time consuming, while building a simulator to mimic all interactions and single
particle physics could potentially deliver much better results much faster.
A good quantum simulation has to fulfill certain requirements, it starts by mapping all the relevant
physics onto the particles, like the densities, correlations and interactions. Then one has to prepare
the desired initial state and perform the measurement [1]. For many unresolved problems, which
are impossible to simulate with conventional computers, could a quantum simulator lead to essential insights. These problems include: quantum phase transitions [14], high Tc superconductors,
superfluidity, complex spin systems and many more. Quantum simulators are also a useful tool to
reach and study regions which are unavailable in real systems or to verify theories like universal
physics [15] or gauge fields [16].
Given the need for these new computation methods, known by the general term “Quantum Information”, a suitable building block had to be found, analogous to the transistor. It is needed
to transform the conventional binary logic into the quantum logic of qubits, superpositions and
uncertainty. Suitable physical systems are numerous and vary from trapped ions [17], Josephson
junctions [18], diamond vacancies [19], quantum dots [20] to trapped neutral atoms [1]. They all
differ in size, from the size of a small chip to that of a full room, and in temperature, from room
temperature to temperatures close to absolute zero. Cooling and isolating qubits is the most important step to increase the qubits lifetime. These two properties can both be guaranteed in clouds of
trapped atoms, where the lasers, and sometimes a magnetic field, both cool and trap the atoms in
an ultra-high vacuum environment. Therefore ultra-cold neutral atoms are a promising candidate
for future quantum information systems [1], but gaseous neutral atoms lack the important property
that they do not exhibit strong interactions. Strong interactions are essential to correctly simulate
many of the interesting quantum many-body problems. Still, the advantages, of ultra-cold trapped
neutral gases, kindled a search for methods to create strong interactions. By exciting the atoms
into a Rydberg state one meets all the demands posed on a effective quantum information system.
6
1
Introduction
Another advantage or Rydberg atoms is the precise and relatively easy state preparation [5]. A
Rydberg atom is an atom of which a single electron is excited to a state with a high principal
quantum number. The choice of state defines the nature and magnitude of the interaction, leading
to the possibility to adjust interactions over an enormous range [6].
In this thesis, trapped ultra-cold neutral Rubidium Rydberg atoms will be studied through simulations and observations. We focus on highly excited Rydberg states to make the interactions as
strong as possible. The thesis starts with an explanation of the general experimental setup in Chapter 3. Control over Rydberg atoms mostly happens through the excitation laser, and in Chapter 4
we greatly enhance this control of the setup by actively shaping the light in arbitrary patterns. The
device responsible for shaping the light is a Spatial Light Modulator (SLM), and this is the first
time such a device has been added to Rydberg experiments. Its vast capabilities have enabled us to
develop a novel method to measure and correct aberrations in our optical setup, an essential step
towards optical lattices with only a single Rydberg atom per lattice site and eventually the creation
of logical quantum gates. Chapter 2 focuses on the characterization of Rydberg interactions and its
manifestations like the blockade effect, where one atom blocks the excitation of its surrounding cloud
of ground state atoms, and the self-assembly of Rydberg crystals. The interactions can be directly
made visible by measuring a correlation function. This special function contains much information
about the spatial dependence of the Rydberg interactions. It can, for example, be used to directly
measure the blockade radius which has recently also been performed by an other group [9], where
they used a much lower Rydberg state, as compared to this thesis, and without a random distribution of atoms. Most other experiments [21, 8, 22] measure the blockade effect indirectly without a
need for spatial resolution, which results in a single parameter called the Mandel Q-factor. We also
measure the Q-factor, also in Chapter 2, and compare our results with the other experiments and
explain the shortcomings of this parameter. A combination of all the results is given in Chapter 6
which also contains a short outlook and suggestions for future experiments.
7
C. Ravensbergen
2
2.1
Theory of Rydberg atoms
Introduction
Rydberg atoms are regular atoms with the distinction that one electron is excited to a high lying
state this means that the principal quantum number, n, is much higher than it is in the ground
state [2]. This introduces and enhances a number of characteristics. Their lifetime for one is much
longer than the lower excited states and scales as n3 . Another feature is a large orbital radius which
scales with n2 leading to enormous sizes up to a few microns [23]. This very large extension of the
wavefunction is the reason for most of the other extreme properties: they are easily polarized and
therefore very sensitive to external EM fields and even though they are neutral they have long range
interactions through the van der Waals interaction where the corresponding interaction strength coefficient, C6 , even scales with n11 [2].
Rydberg atoms have been studied for quite some time now. Original interest came from astronomy
because Rydberg atoms are the main source of interstellar radiation but research was mostly theoretical until the introduction of lasers. Nowadays Rydberg physics is used and studied in many
different areas like plasma physics, spectroscopy and, important in this thesis, ultra-cold gases. Here
Rydberg atoms are found useful for a number of reasons. Because the interactions depend extremely
on the electro-magnetic fields and electron state, interactions can be tuned over a wide range, while
their low temperature and density gives rise to long coherence times and low ionization losses.
The strong interaction manifests itself in a number of typical phenomena. One of which is the socalled blockade effect where Rydberg atoms block all excitations in their surrounding volume. This
blockade has been a topic of great interest because of its wide range of applications especially for
quantum information, simulation and computation [5]. The blockade effect can be used for storing
and processing quantum information which are essential building blocks for a quantum computer.
Other applications include: single particle sources for photons [24], ions [25] or electrons and many
nonlinear optics components [26].
Recent technological advances in the creation, manipulation and especially spatial observation of
Rydberg atoms have let to experimental confirmation of many exotic phenomena e.g. dipole blockade between two atoms [5], spatial ordering [9], Förster resonances [27], resonant energy transfer
and ultra-cold Rydberg molecules.
In this chapter first an overview of the physics relevant to Rydberg atoms is given, treating the
energy levels in Section 2.2, a general two level excitation in Section 2.3 and the extension of
this two level system to our three level system in Section 2.4. The essential Rydberg interactions
are being derived in Section 2.5 and a method to characterize them is presented in Section 2.6.
Spontaneous emission plays an important role in the formation of Rydberg gases and it is added to
the framework in Section 2.7. The addition of spontaneous emission changes the dynamics from a
coherent excitation to an incoherent one, and the populations of the states can now be described
by a set of rate equations also given in Section 2.7.
In Section 2.8 an algorithm is introduced which is able to exactly simulate our Rydberg system.
The excellent performance of the algorithm enables us to simulate thousands of atoms under typical
experimental conditions but for much longer timescales than would be possible in the lab. It is
therefore a perfect tool to simulate many-body Rydberg interactions like the blockade effect and
8
2
Theory of Rydberg atoms
crystallization. These simulations are performed in a 1-dimensional chain of equidistant atoms in
Section 2.8.2 and a randomly distributed 2-dimension gas in Section 2.8.3. The fluctuations in the
number of Rydberg atoms differs, due to the blockade, from a normal, Poissonian, excitation and
is discussed in Section 2.8.4.
2.2
Rydberg states
The outer shell of Rubidium contains only a single electron which means that it is an alkali metal.
This is the reason for the relatively simple excitation spectrum of figure 17. If this electron is in
a Rydberg state it is most of the time far away from the nucleus and surrounding inner electrons.
These then form a robust, hard to perturb, core leading to a near 1/r coulomb potential for the
electron. This means that the spectrum of the Rydberg states will be very similar to the Hydrogen
one. One can introduce a huge simplification of the quantum many-body problem by replacing all 36
inner electrons and the nucleus by this core and coulomb potential. The corresponding Hamiltonian
acting on the outer, or valence, electron is then given by:
1
α2
Ĥa = − ∇2 + Vcore (r) + 3 L̂ · Ŝ.
2
r
(1)
Here the first part is the kinetic energy while Vcore is the effective core potential that the electron
experiences. The final part in the spin-orbit coupling, where α is the fine-structure constant and
L̂ and Ŝ are the orbital and spin angular momentum operators. The units used throughout this
chapter are the Hartree atomic units where the length scale is the bohr radius, a0 , the energy unit
is the Hartree energy given by EH = mα2 c2 , with m the electron mass and c the speed of light. The
energy of the Rubidium levels has been measured with high precision yielding the following formula
which is very similar to the Hydrogen one [28]:
E(n, j, l) = −
Ry ∗
Ry ∗
=
−
,
n∗2
(n − δ(n, j, l))2
(2)
where Ry ∗ = 0.4999673250 is the Rydberg constant for Rubidium opposed to Ry ∗ = 1/2 for
Hydrogen, n∗ is the effective principal quantum number after correction with a small defect δ(n, j, l)
originating from the interaction of the valence electron with the other electrons when it penetrates
the core. The defect depends on the quantum numbers of the electron orbital defined by the principal
quantum number, n, and the total angular momentum number, j, which is the combination of the
orbital angular momentum, l, and the electron spin, s [28]. The defect is almost zero for Rydberg
states with l ≥ 3 due to the centrifugal barrier. The lowest order approximation for the quantum
defects are, ignoring the j dependence, for Rubidium given by δ(l = 0) = 3.13, δ(l = 1) = 2.64 and
δ(l = 2) = 1.35 [29].
9
C. Ravensbergen
2.3
Optical excitation
Optical excitation plays a key role in every cold atom experiment, especially for the initial conditions
like the density, temperature and amount of atoms. Our experiment involves exciting ground state
atoms via an intermediate level to a Rydberg state using two different lasers. The most appropriate
method of calculating excitation probabilities and rates, in this case, is a semi-classical treatment
with quantized atoms in a classical electro-magnetic field which remains unaffected by the atoms.
The atom light interaction Hamiltonian in this method takes the following form:
ĤAL = −µ̂ · Ê(t),
(3)
where µ̂ is the atomic dipole moment operator and Ê(t) the electric field operator of the laser light.
In this Hamiltonian the electric dipole approximation is used, this neglects the spatial dependence
of the electric field which is a valid approximation because the wavelength of the light is much longer
than the size of the initial electronic wavefunction. In equation (3) also the interaction of the atom
with the magnetic field has been neglected because its strength is several orders of magnitude lower
than that of the electric field. Calculations of the atomic dipole moments for Rubidium transitions
can be found in abundance in literature for example in [30], the definition of the matrix element is
given by:
µ̂ = da = qhn, l, j, mj |r̂|n0 , l0 , j 0 , m0j i.
(4)
Here r̂ is the displacement operator measuring the distance between the electron and the nucleus,
q is the charge of the electron, a is the unit vector, in the rotating frame, describing the atomic
transition polarization (σ ± -circular polarized and π-polarized) and the quantum numbers characterizing the electron states are given by: the principal quantum number, n, the orbital angular
momentum, l, the total angular momentum, j and the projection of the angular momentum along
the quantization axis, mj . Most matrix elements are zero except those obeying the selection rules
given by [28]:
l0 = l ± 1,
m0j = mj − µi ,
|j − j 0 | ≤ 1.
(5)
The Hamiltonian of equation (3) includes all possible electron transitions, but if the laser is only
resonant with a single transition, all but this transition can be neglected from the Hamiltonian
[31]. This simplifications means only two levels remain: the ground state, labeled from this point
on by |gi, and the excited state, |ei. After making the rotating wave approximation [31] the total
Hamiltonian, which is the combination of the atomic and interaction Hamiltonian, reads:
1
Ω
Ĥ = − ~δe σee + ~ (σge + σeg ),
2
2
(6)
where the detuning, δe = ωeg − ω, is the difference between the atomic and laser frequency, and
σij = |jihi| are the atomic raising, lowering and counting operators [32] and Ω is the classical Rabi
frequency given by:
dE
,
(7)
Ω=
~
where E is the electric field amplitude.
10
2
2.4
Theory of Rydberg atoms
Three-level atom
For Rydberg experiments s-states are extremely desirable due to the isotropic repulsive interactions.
But in order to excite an s-state one needs at least a two photon excitation in order to preserve
angular momentum. The excitation path would start with the Rubidium ground state depicted by
|gi, from which the electron is excited to the intermediate state |ei and finally the Rydberg state
which can either be a S or D state depicted by |ri in figure 1. In here the two Rabi frequencies and
detunings for the two excitations have also be depicted, where δr = ~1 |E(nS) − E(5P )| − ωlaser + δe .
Figure 1: Schematic representation of the three level Rydberg excitation scheme of
the experiment.
85 Rb
used in
Three level atoms exhibit many new effects which are not present for two levels like electromagnetically induced transparency where the atom becomes transparent for one of the two transitions
under certain conditions. Another phenomenon for three level atoms is the well known population
inversion [33], which is the essential technique on which lasers are based.
2.5
Interactions
With the ability to describe the coherent excitation of a single atom it is time to discuss the
interaction between multiple atoms. This Section will treat the interaction between two Rydberg
atoms both being in the same nS state. Since the atoms are neutral and have no permanent dipole
moment the lowest order of interaction at long distances will be of the induced dipole-dipole type.
The dipole-dipole interaction energy operator for two well separated Rydberg atoms is:
V̂dd (R) =
µˆ1 · µˆ2
(µˆ · R)(µˆ2 · R)
−3 1
,
3
R
R5
(8)
11
C. Ravensbergen
where R is the nuclear separation between the two Rydberg atoms and R is the length of this vector.
The second part of this equation gives an angular dependent interaction and can be omitted for S
states [34]. In the frozen gas limit the movement and kinetic energy of the atoms is negligible on
the timescales of the experiments. The strong interaction energy comes, like most cases in quantum
mechanics, from two nearly degenerate states. It happens to be the case that the energy of the
double excited Rydberg state consisting out of two S states, |S, Si, is degenerate with a double P
state, |P, P 0 i, as is shown in figure 2.
Single Excitation
Double Excitation
P'
S+P'
P+P'
Δ
S
P
S+S
S+P
Figure 2: The origin of the strong interactions in Rydberg gases comes from the near degeneracy
of the state with two S atoms with the state consisting out of one P and one P 0 atom.
These two states are coupled by the dipole-dipole interaction operator of equation (8). For large distances the interaction energy can be calculated perturbatively. First order perturbation interaction
energy is given by:
E1 = hφi , φi |V̂dd |φi , φi i = 0,
(9)
and it has to be zero because of the selection rules that apply for the dipole operator given in
equation (5). The second order perturbation energy gives the van der Waals interaction,
HvdW =
X hS, S|V̂dd |φ1 , φ2 ihφ1 , φ2 |V̂dd |S, Si
C6
= 6 D,
~∆(S, φ1 , φ2 )
R
(10)
φ1 ,φ2
where ~∆ is the energy difference depicted in figure 2, C6 is the van der Waals interaction coefficient
and D contains all the matrix elements with the angular momentum properties of the states [34].
This second order perturbation of the system by the interaction Hamiltonian gives the typical
van der Waals interaction with the 1/R6 behavior. A complete description including the actual
calculation of the C6 coefficients are performed in References [34], [3] and [4]. In this last paper by
Singer et. al. a table is listed for the C6 coefficients of Rubidium S, P and D Rydberg atoms. For
nS-nS interactions the coefficient can be estimated using the following formula:
C6 (nS − nS) = n11 (11.97 − 0.8486 n + 3.385 · 10−3 n2 ),
12
(11)
2
Theory of Rydberg atoms
which only depends on the principal quantum number, n, and is valid in the range between 30 to
95. The coefficients are always negative, meaning that the interactions between equal S states is
repulsive for all n. Typical the value for the van der Waals coefficient is −10−58 Jm6 (for n ' 60),
this gives a frequency shift of roughly 1 MHz at a distance of 7 µm. This frequency shift can be
regarded as an, atom and place specific, detuning. If this detuning becomes equal or larger than
the laser linewidth the excitation probability is considerably suppressed. This is the origin of the
blockade effect, it means that if atoms are too close to each other the interaction moves them out
of resonance and they can no longer be excited by the laser. An estimate of the blockade radius
can be made by equating the power broadened linewidth of the laser, given by the Rabi frequency,
and the van der Waals shift [9]:
|C6 | 1/6
Rb =
.
(12)
~Ω
In here the Rabi frequency, Ω, is the combined Rabi frequency of the two lasers given by:
s
Ω2ge Ω2er
+ δr2 .
Ω=
4δe2
(13)
In figure 3 an energy diagram is given which shows this interplay of distance, intensities and detuning.
Figure 3: Energy diagram of a double Rydberg excitation as a function of the inter-atomic distance.
Additionally the Rabi frequency of the excitation laser and the linewidth of the Rydberg level is shown.
At the blockade radius the energy shift of the double excited state |r, ri is equal to Rabi frequency of
the excitation laser. Courtesy to Löw et. al. [6].
Alongside S states, D states can also be accessible via a two photon excitation. These do not have
an isotropic interaction, which complicates the whole picture a bit, but generally have a much larger
dipole matrix element and are thus easier to excite. The reason for the anisotropic interaction can
be found in the non-spherical symmetric wavefunction of the Rydberg electron in a D state. This
results in a direction specific polarizability in respect to the quantization axis. Interactions between
13
C. Ravensbergen
D state Rydberg atoms can either have an attractive or repulsive nature depending on the principal
quantum number and orientation [35]. For the D5/2 state the interaction is attractive for n ≥ 43
and for the D3/2 state at n ≥ 59. The blockade effect for D states is no longer a simple blocked
sphere but can be regarded more or less like an ellipsoid, while the sign of the C6 coefficient does
make the interaction attractive or repulsive. The same line of reasoning as before leads again to a
distance specific detuning of the Rydberg level. For attractive interactions this is a red detuning
but the excitation probability is independent of the sign of the detuning. The anisotropy of the
interaction can be more than twice the value of the mean van der Waals coefficient in certain states
and directions and plays therefore an important role in the behavior of Rydberg D state gases.
2.6
Correlation functions
It is common in physics to characterize interactions by correlation functions, which are also called
coherence functions. The different orders of the correlation functions are used to study different
aspects of the interaction. Correlation functions give a quantitative measure of the correlation of a
property at two different points in time or space. For light shining on a detector the second order
correlation function is the correlation between the intensities on the detector [32]:
G(2) (r1 , r2 ) = hI(r1 )I(r2 )i.
(14)
This property is still dependent on the absolute value of the intensities and has to be normalized
to get the second order normalized correlation function:
g (2) (r1 , r2 ) =
hI(r1 )I(r2 )i
G(2) (r1 , r2 )
=p
,
hI(r1 )ihI(r2 )i
G(2) (r1 , r1 )G(2) (r2 , r2 )
(15)
where I(ri ) is the intensity on position ri on the detector. For particles the intensity has to be
replaced by the density. And by a summation over all particle positions, r1 , one arrives at the
radial density distribution function, also called the pair correlation function, schematically depicted
in figure 4.
Figure 4: Schematic representation of the pair correlation function g (2) (r).
This function gives the chance of finding another particle between the distance r and r + dr.
For completely structureless or uniform systems g (2) equals 1 (figure 5a). While for a crystal the
14
2
Theory of Rydberg atoms
correlation function is a series of delta peaks with height and spacing governed by the crystal
geometry (figure 5b). All other cases are somewhere in between these two extremes. The spatial
correlation function is a useful tool for Rydberg interactions in multiple ways. First of all is the
blockade directly visible in the correlation function because inside the blockade radius no other
Rydberg atom can exist and therefore the correlation function equals zero for r < Rb . And the
theoretical prediction of spatial structures and crystallization, due to the strong correlations in
Rydberg gases, would mean some oscillating behavior of g (2) for distances larger than the blockade
radius [36].
2
1.8
1.6
1.4
g(2)
g(2)
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
distance (a.u.)
0.8
(a) Non-interacting gas.
1
0
0.2
0.4
0.6
distance (a.u.)
0.8
1
(b) Perfect crystal.
Figure 5: Schematic correlation functions for two extreme cases.
15
C. Ravensbergen
2.7
Open quantum systems
Complete isolation of a quantum state means that it will remain in its state for ever while the
coupling to its surrounding induces decoherence and state mixing [37]. To be able to describe the
decoherence of quantum systems one can no longer use a pure state or wavefunction but has to use
statistical ensembles. A complete way to describe these systems is by use of the density matrix [37]:
X
ρ=
pi |ψi ihψi |,
(16)
i
with the normalization condition:
X
pi = 1.
(17)
i
The density matrix description can also be used for problems where the initial state preparation
is uncertain or to describe only one particle of an entangled pair. It is therefore a much more
experimental approach to quantum mechanics and we use it here to come to a description of a three
level atom with spontaneous emission. The expectation value of an operator Ô of a system with
density matrix ρ is given by:
hOi = T r(ρ O)
(18)
If all but one pi are zero, ρ describes a pure state. From the definition of the density matrix (16)
and the extension of the Schrödinger equation for the projector on pure states |φihφ|:
d|φihφ|
d|φi
dhφ|
=
hφ| + |φi
,
dt
dt
dt
(19)
with:
d|φi
= H|φi,
dt
one gets the Schrödinger equation for the density matrix:
i~
i~
dρ
= [H, ρ].
dt
(20)
(21)
The real positive diagonal terms of the density matrix are called ”populations”, they correspond
to the probability to find the system in the corresponding pure state. The off-diagonal terms are
called ”coherences” because they govern the phase differences between the states and are zero for
statistical mixtures. The total density matrix can be split in two reduced parts, one for the system,
ρA , and one for the bath, ρB . The evolution of the reduced density matrix of the system, ρA , from
time t to a time t + τ is described by a process:
X
ρA (t + τ ) = Lτ [ρA (t)] =
Mµ (τ )ρA (t)Mµ† (τ ),
(22)
µ
where L is the so-called ”quantum map” and Mµ are the Kraus operators depending on τ , the
interaction between the system and the environment and the density operator of the environment,
ρE . This expression is only valid under a few assumptions and restrictions. First the environment
or reservoir has to be large and the Markov approximation has to be valid, this means that the
system is a ”memoryless” stochastic process. This ”memoryless” property does not have to mean
that there can be no correlations in the time evolution but it has to mean that the system has
16
2
Theory of Rydberg atoms
a very short memory, τc , after which all correlations have vanished. We can now write down the
derivative of ρA :
dρA (t)
Lτ [ρA (t)] − ρA (t)
=
,
(23)
dt
τ
where τc τ Tprocess . After some manipulations we end up with the master equation written in
Lindblad form [37]:
X 1 †
dρA (t)
i
1 †
†
(24)
= − [HA , ρ] +
hi Li ρLi − Li Li ρ − ρLi Li ,
dt
~
2
2
i
where hi are the rates at which the corresponding process takes place. To construct the master
equation for a three-level Rydberg atom we follow the method used by Reference [38]. But in this
thesis both the intermediate, |ei, and the Rydberg level, |ri, can have a detuning. The lifetime
of the Rydberg state is considered much longer than the experimental time and therefore only the
excited state experiences spontaneous emission with rate Γ. The effective Hamiltonian for the three
level system is given by:
HA = ~δe |eihe| + ~δr |rihr| + ~
Ωge
Ωer
(|eihg| + |gihe|) + ~
(|rihe| + |eihr|) ,
2
2
(25)
where δe (δr ) is the detuning of the laser of the excited (Rydberg) state and Ωge is the Rabi frequency
of the first excitation step while Ωer the Rabi frequency of the second excitation step. In equation
(24) the Lindblad operator is given by:
N X
1 †
1 †
†
L[ρ̂] = Γ
Li ρ̂Li − Li Li ρ̂ − ρ̂Li Li ,
2
2
(26)
i
where Li are the quantum jump operators, they induce drastic changes to the system. In this case
this is the spontaneous emission of a photon leading to a loss of energy and a collapse of the electron
to the ground state:
Li = |gi ihei | and L†i = |ei ihgi |.
(27)
If we now write down all the separate elements of the master equation we get a set of coupled
differential equations, the optical Bloch equations (OBE):
Ωge
(ρge − ρeg ) + Γρee ,
2
Ωge
Ωer
= i
(ρeg − ρge ) + i
(ρer − ρre ) − Γρee ,
2
2
Ωer
= −i
(ρer − ρre ) ,
2
Ωge
Ωer
Γ
(ρee − ρgg ) + i
ρge − ρge + iδe ρge ,
= −i
2
2
2
Ωge
Ωer
Γ
= −i
(ρrr − ρee ) − i
ρge − ρer + iδr ρer ,
2
2
2
Ωge
Ωer
= −i
ρer + i
ρge + i(δr + δe )ρge ,
2
2
= (ραβ )∗ .
ρ̇gg =
ρ̇ee
ρ̇rr
ρ̇ge
ρ̇er
ρ̇gr
ρβα
i
(28)
17
C. Ravensbergen
1
ρ
1
−ρ
ee
exp(− Γ t)
0.8
ρ
gg
0.8
0.4
0.7
Probability
Probability
ρ
0.6
0.2
0
−0.2
0.5
0.4
0.3
−0.6
0.2
−0.8
0.1
0
0.5
1
1.5
2
time (s)
2.5
3
0
3.5
0
0.5
1
1.5
2
time (s)
−7
x 10
(a) Short time evolution of ρee − ρgg .
rr
0.6
−0.4
−1
gg
ρ ee
0.9
2.5
3
3.5
−5
x 10
(b) Long time evolution of the three level atom.
Figure 6: Time evolution of the populations of a three level atom with spontaneous emission,
obtained by direct numerical integration of the optical Bloch equations. The population difference,
ρee − ρgg (a), can be neglected for timescales larger than 1/Γ. The parameters are: Γ = 2π 6 MHz,
Ωge = 10Γ, Ωer = 0.3Γ, δe = 5 MHz and δr = −5 MHz.
Of course the proper normalization has to apply:
X
ραα = 1.
(29)
α
These equations are hard to solve, especially for many interacting particles. To simplify the problem
a few helpful features of Rydberg atoms will be included. First, the upper transition is much more
weakly driven due to the smaller dipole matrix element for Rydberg states, Ωer Ωge . End
secondly, to insure the stochastic behavior and fast dampening of the coherences the decay rate of
the excited state has to be much larger than the Rabi frequency of the Rydberg excitation, Γ Ωer .
Under these conditions the dynamics of the coherences can be adiabatically eliminated by setting:
ρ̇αβ = 0,
(30)
for α 6= β. With use of equations (30) and (29) one can rewrite the OBE’s and remove all dependence
on the coherences from the populations in equation (28). Because the dynamics of the two lower level
are much faster than those of the Rydberg level, and being still under the adiabatic approximation,
one can neglect the population difference ρee −ρgg for timescales t > 1/Γ ∼ 30 ns. This is illustrated
in figure 6a where the population difference is compared to exp(−Γt). The populations are obtained
by direct numerical integration of the optical Bloch equations with realistic values for the Rabi
frequency. Combining this result together with the time derivative of equation (29) one gets:
2ρ̇ee + ρ̇rr = 0.
(31)
This can now be used to get rid of the intermediate level and the final form for the population of
18
2
the Rydberg level is given by:
ρ̇rr = −
γ↑
ρrr + γ↑ ,
ρ∞
rr
Theory of Rydberg atoms
(32)
where ρ∞
rr (Ωge , Ωer , Γ, δe , δr) is the steady-state population of the Rydberg level and γ↑ (Ωge , Ωer , Γ, δe , δr )
is the excitation rate for the Rydberg level from the effective ground state population. This equation
is called the ”rate equation”. The next step will be a trick to include interactions in this method
without solving the many body Hamiltonian. If one assumes only van der Waals interactions between the particles one can model the interactions by applying an additional, atom and position
specific, detuning corresponding to the total interaction energy imposed by its surroundings:
δr (r) = δlaser +
1X
C6
−
~
|ri − rj |6
(33)
j6=i
This then leads directly to a system of many interacting three level atoms with the same differential
equation given by equation (32) but with different rates.
2.8
2.8.1
Simulation
Algorithm
In the previous section we found the uncoupled many body differential equations of the state
populations of interacting Rydberg atoms. In this section these equations will be solved numerically
for realistic systems. This is done via the so-called ”Kinetic Monte-Carlo” method. This method has
several advantages being that it gives exact numerical results, indistinguishable from real systems
evolving through a master equation, while still remaining orders of magnitude faster than molecular
dynamics simulations [7].
The term Monte-Carlo refers to a wide range of numerical algorithms that solve problems with
help of random numbers. They became commonly used with the introduction of computers and the
most famous one is the Metropolis algorithm. The Kinetic Monte Carlo, KMC, method is intended
to simulate the time evolution of stochastic Poissonian processes. With this method one is able to
solve the master equation, and the algorithms are able to simulate the evolution of a system over
typical much longer periods of time as compared to direct simulations such as molecular dynamics.
Because of the Markov approximation, made in the previous section, the system is completely
memoryless, it does not know how and when it entered the current state. In order to simulate such
a system one only has to know two properties: what will be the next evolution step and when will
it take place [39]. Master equations, like the OBE in equation (28), for systems with N states can
always be written in a general form:
N
N
l=1
l=1
X
X
dPk
=
Γkl Pl −
Γlk Pk ,
dt
(34)
where Pk is the probability of the system to occupy the state k and Γkl is the rate with which the
system moves from state l to k. The total rate, Γk , for the system to move away from state k is the
19
C. Ravensbergen
P
sum over all rates: Γk = l Γlk . From this rate one can calculate the probability that the system
still remains in state k at time t0 via:
!
Z t0
0
0
Γk (τ )dτ .
(35)
psurvival (t ) = Γk (t ) exp −
t
The KMC algorithm is now based on the ability to generate the reaction time t0 from equation (35)
and the corresponding new state l. This process of finding the reaction time is repeated at every
time step and it can be represented by the following steps:
ˆ Initialize the system to its given state, k, at actual time t.
ˆ Create a new rate list Γlk for the system where l = 1, ..., N .
ˆ Choose a random number r, 0 < r ≤ 1, and calculate the time, t0 , at which the next reaction
R t0 P
takes place by solving t
l Γlk (τ )dτ = −ln r.
ˆ Determine which particle undergoes this reaction by picking a P
random number, 0 < r0 ≤ 1,
and search for which particle, l, Rl−1 < r0 RN ≤ Rl where Rj = ji=1 Γik (t0 ) and R0 = 0.
ˆ Set the system to state l and modify the time to t0 . Return to the first step.
2.8.2
Simulations in One Dimension
We first apply this algorithm to a simple 1 dimensional chain of interacting atoms, either in state 0,
the effective ground state, or 1, the Rydberg state. This simple case helps to understand the complex
dynamics originating from the many different parameters and dimensions in play. These results can
then be used as a starting point for the actual experiment in the lab. The initial parameters for all
simulations are as follows: The one dimensional chain consists out of 7 equidistant (5 µm apart)
Table 1: The used values of the relevant parameters in the simulations.
Γ(MHz)
2π 6
Ωge (MHz)
3Γ
Ωer (MHz)
0.2 Γ
δe
0
δr
0
State
80S
Rubidium 85 atoms. The blue laser is resonant with the 80S Rubidium Rydberg level. The total
simulation time is 25 µs and we average over 300 cycles. The result is given in figure 7. Initially
all atoms are in the ground state, and have the same excitation chance. But the two outermost
atoms only have a single neighbor which in combination with the strong interactions means a much
higher excitation chance. These are the two atoms which are most of the time ’on’. This now is the
trigger for the spontaneous (quasi) crystallization of the whole chain. After 10 µs this has become
the dominant state of the system. One can clearly see the effect of the blockade radius: the atoms
next to an atom with a high excitation chance have an almost vanishing chance.
This time evolution of the chain of atoms can be studied in more detail by just plotting the probability as a function of the simulation time which has been done in figure 8. Initially the evolution
20
2
Theory of Rydberg atoms
Figure 7: Time evolution of the excitation probability of 7 interacting 80S Rydberg atoms. The
other relevant parameters are given in Table 1
of all the atoms is completely equal but when a certain amount of Rydberg atoms is present they
start to block each other and some atoms become more probable to be excited while they suppress
the excitations of others. The two red curves of figure 8 are the first and the last (nr. 7) atom of
the chain and they reach the highest excitation chance which automatically translates in the lowest
excitation chance for their neighbor atom nr. 2 and 6. This continues towards the most inner atoms
but slowly dampens out.
0.9
0.8
Rydberg Excitation Probability
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
time (µ s)
Figure 8: Plot of the time evolution of the Rydberg excitation probabilities, different, random,
colors represent the different atom positions.
21
C. Ravensbergen
The probabilities are the result of averaging 300 simulations. The maximum excitation probability
is 0.86 and this means that in 86% of the cases this atom was excited. If one wants to create clear
crystals in these simulations it is important to match the atom distances to the blockade radius
otherwise different options for the final state exists which translates in a much flatter distribution
see for example figure 9. The excitation probability is never 1, unfortunately this means that it is
impossible to create a deterministic final state. So even in this very small and controlled simulation
it is impossible to predefine the final number of Rydberg atoms.
0.7
Excitation Probability
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
Time (µ s)
Figure 9: Plot of the time evolution of the Rydberg probabilities without careful matching the
blockade radius to the atom distance (-10% as compared to figure 8). Again, the, random, color
coding represents different atomic positions.
An elegant way to visualize the match, and mismatch, of the blockade radius with the inter-atomic
distance is by varying the principal quantum number for an otherwise fixed system. This has been
shown in figure 10. For low n states the blockade radius is smaller than the average inter-particle
distance and no clear crystal is produced. However, starting from n = 35 the first match is found
giving a crystal with 7 excitations (5 visible and the two atoms at the edges). Increasing the principal
quantum number even further decreases the number more and more with in between regions of a
mismatch, see for example the profile at n = 55. The position of the first excitation peak as a
function of n is not linear and might be consistent with the approximated behavior of Section 2.5
where the blockade radius scales with n2.25 .
22
2
Position (a.u.)
0.8
Theory of Rydberg atoms
0.15
0.6
0.1
0.4
0.05
0.2
30
40
50
Principal Quantum Number
60
Figure 10: The excitation probability for a 1D chain of atoms for different principal quantum
numbers, n, ranging from 30S to 60S. The two outermost atoms have been left out for enhanced
contrast. Increasing n means a larger blockade radius, which results in fewer excitations.
The border of the simulation plays an important role in the formation of the crystal, and with the
current parameters it is impossible to create equal excitation probabilities for all lattice atoms. One
could solve this problem by introducing a spatial depended laser intensity with the obvious solution
of turning the laser off at all positions except right on top of the desired lattice spots. But so far
we’ve left out another important parameter, the detuning. The detuning can be used together with
the van der Waals interaction to enhance the formation of a crystal. By detuning the blue laser
one decreases the excitation probability but the interaction can shift it back into resonance. This
enhances Rydberg excitations exactly on the edge of the blockade radius, leading to the final state
distribution with an even more dominant position for the complete crystal, reflected by the higher
contrast in figure 11.
The crystal growth takes time, the final steady state distribution is only reached asymptotically and
this ’hitting’ time scales also with the number of atoms involved and the dimension of the problem.
Many excitations and de-excitations are needed before equilibrium is reached. This could lead to
experimental problems where the finite temperature of the atoms defines a limit for the timescale
on which the atoms can be regarded stationary. Hitting times can be reduced by increasing the
laser intensities, lowering the dimension or with help of a so-called chirp in the detuning [40].
23
C. Ravensbergen
1
0.9
Excitation Probability
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
position (a.u.)
Figure 11: Final excitation distribution similar to figure 7 but now with a slight detuning of the
blue laser, δr = −0.1 Γ, and 10 atoms. All other parameters are equal to those listed in Table 1
Notice the almost flat excitation distribution even though this is still a 1D chain with a uniform
laser intensity.
2.8.3
Correlation Function
The previous techniques show the behavior of multiple Rydberg atoms in 1D, these results depend for
a large part on the system geometry. By introducing a second dimension the influence of the border
becomes smaller but the geometry becomes much more complex. Here two different techniques are
shown to characterize and quantify the blockade and long-range interactions. Simulating is a very
easy and fast way to test these different techniques. The first technique is the already explained
(section 2.5) pair correlation function. The function is calculated by averaging many thousands of
identical simulation runs. Each starts with 1600 (202 ), randomly distributed, ground state atoms
with all parameters similar to the actual experiment (density=1.2 10−10 cm−3 which equals an
inter-atomic distance of 5 µm, laser intensities as listed in Table 1 and excitation times, ∼ 15 µs).
After a run all the distances between the Rydberg atoms are calculated and put in a histogram
with a fixed binning. Each binning of the histogram is divided by its corresponding area given by
2πrdr, where r is the mean distance of this binning and dr is the binning width. This histogram is
the second order correlation function G(2) (r) which includes all correlations in the cloud. We apply
a trick to remove the geometrical correlations introduced by the finite volume. By calculating two
correlation functions, once between atoms in the same run (figure 12a), which includes both the
Rydberg correlations and the geometrical correlations, and once between atoms in different runs
(figure 12b). The correlation function of atoms from different runs do not exhibit any Rydberg
correlations but do includes the geometrical correlations. It can therefore be used to remove the
geometrical correlation from figure 12a.
24
35
35
30
30
25
25
G(2) uncorrelated
G(2) correlated
2
20
15
20
15
10
10
5
5
0
0
20
40
60
distance (µm)
80
0
0
100
(a) With Rydberg correlations.
Theory of Rydberg atoms
20
40
60
distance (µm)
80
100
(b) Without Rydberg correlations.
Figure 12: Pair correlation functions for the 87S Rydberg atoms. One with pairs of atoms in
the same run and one with pairs of atoms from different runs. Both reflecting the geometry of the
simulated cloud and laser geometry.
By dividing these two correlation functions one obtains the normalized pair correlation function,
which is volume independent and only contains Rydberg correlations:
(2)
g (2) (r) =
Gcorrelated (r)
(2)
,
Guncorrelated (r)
(36)
and the result is shown in figure 13.
In figure 13 we clearly see the influence of the van der Waals interaction on the excitations. The
chance for finding two atoms at a distance smaller than 13 µm rapidly decreases until it completely
vanishes for distances smaller than 7 µm. Right outside the blockade radius a maximum at r =
16 µm is visible with height 1.37 which is the beginning of a few oscillations until it dampens out to
its final value of 1. The minimum has a depth of 0.92. The oscillation has a wavelength of 16 µm
which is also the distance to the first maximum. The blockade radius is defined as the distance
at which g (2) , for the first time, crosses unity [9]. The overall shape correlation function is very
much reminiscent of the pair correlation function of a liquid [41] and the experiment performed in
Reference [9]. The oscillations end at about 35 µm which is ∼ 50 times the radius of the electron
wavefunction, a clear indication of the long range interactions. The strength of interactions can be
characterized by na3 [6] with n being the density of the system and a the effective range of the
interactions, here given by the blockade radius. The density is in our case just the MOT density
which gives an interaction strength parameter of:
na3 ≈ 35,
(37)
which is much bigger than unity so we are indeed dealing with a strongly interacting system.
25
C. Ravensbergen
1.5
g(2)
1
0.5
0
0
20
40
60
distance (µm)
80
100
Figure 13: g (2) (r) for the 87S Rydberg level simulated in two dimensions and cut of at r > 100 µm.
Obtained by dividing figures 12a and 12b. The blockade radius, Rb , is 12.9 ± 0.1 µm.
2.8.4
Mandel Q-factor
Another, less direct, method to prove strong interactions and blockade effects is by looking at the
number of excited atoms. And more specifically at the fluctuations in the number of excited atoms.
The excitation of a single atom in an ensemble of non interacting atoms follows Poissonian statistics.
The interactions however correlate the excitations and one expects a transition to sub-Poissonian
statistics for strong enough interactions [42]. Intuitively this can be understood if one regards the
excitation of Rydberg atoms as marbles being stacked in a fixed volume. Each marble exists out of
a single Rydberg atom and its surrounding blocked cloud of ground state atoms, this is also called a
superatom. The stacking leads to a maximum number of ’marbles’ that fit into the volume heavily
lowering the fluctuations in the number of Rydberg atoms. In the ideal case this would lead to only
a single possible number of Rydberg atoms: a perfect number state. The Mandel Q factor can be
used as a quantitative parameter for the amount the statistics deviate from Poisson statistics. The
Q factor is defined by [43, 44]:
Q=
hN 2 i − hN i2
,
hN i2
(38)
and its value is 0 if the statistics are Poissonian. For Q factors higher than one the statistics are
called super-Poissonian, while statistics with a negative Q factor are called sub-Poissonian. SubPoissonian statistics mean that the width of the probability distribution is smaller than the square
root of the mean of the distribution. The minimum value of the Mandel Q factor is -1 and it is
reached in the case of a number state, a state with a fixed expectation value without fluctuations.
The 2D simulation does show strong sub-Poissonian statistics for the 87S Rydberg atoms as is
shown in figure 14. The obtained Q factor is: −0.86 ± 0.01. Experiments in this regime also find
Q-factors close to -0.9 [8]. The parameters used in the 2D, 87S, simulation are: 2000 runs, each
containing 292 ground state atoms. The laser parameters are equal to those of Table 1.
26
2
Probability
0.15
Theory of Rydberg atoms
Simulation
Poissonian
0.1
0.05
0
20
30
40
50
60
Number of Rydberg atoms
70
Figure 14: Comparison of the distribution of the number of excited Rydberg atoms in the simulation
and a Poissonian distribution with the same mean. Parameters are given in the text.
The Poisson distribution in figure 14 is given by:
hN iN −hN i
e
,
(39)
N!
with N , the number of Rydberg atoms. It is not possible to directly extract the blockade radius
from the Q factor or the measured distribution. But since it is only a single simple number, that
can be based on a much lower number of samples as opposed to the total correlation function, it is
well suited to quantify changes in the system as a function of for example: the principal quantum
number, the excitation time or the density. In figure 15 the value of the Mandel Q-factor is given as
a function of the excitation time, the time both the red and blue laser are turned on. Just like the
Rydberg crystal the Q factor needs time to reach its final value, but it happens much faster than
the crystallization of the chain. The Rydberg level is again 87S, but, to speed up the calculation,
the number of atoms is now 225.
P (N ) =
Mandel Q−factor
−0.2
−0.4
−0.6
−0.8
−1
0
2
4
6
Excitation time (µs)
8
10
Figure 15: The Mandel Q factor as a function of the excitation time, the time the Excitation
lasers are turned on. Each data point is calculated from a measurement like figure 14.
27
C. Ravensbergen
Both the correlation function and Q-factor are able to prove blockade effects. While the correlation
function provides much more information like the actual blockade radius, crystallization, Rydberg
lattice spacing, cloud geometry and much more, the Q-factor gives a simple quantity which can
already be extracted, with sufficient quality, out of just a few tens of excitation runs.
28
3
3
Experimental setup
Experimental setup
This part of the thesis will give an overview of the setup used to create and image Rydberg atoms.
Like the apparatus the chapter is divided into three parts, a vacuum chamber in which the experiment takes place, an optical system containing all the lasers and optical beam shaping elements
and an ion imaging and detection structure. The main part of the experiment has been depicted in
figure 16. The Rubidium-85 atoms are trapped and cooled in a Magneto-optical trap, MOT, with
the red laser wavelength at 780 nm. These atoms are then excited another time to a third level
with the blue beam at 480 nm to reach the desired Rydberg level. The Rydberg atoms can then be
ionized and accelerated towards the detector by an electric field created by an accelerator.
3.1
Vacuum chamber
The Rubidium is brought into the vacuum chamber by heating an oven to 30◦ C. The structure
of the chamber is such that the MOT coils are inside the vacuum as is the accelerator and two
mirrors for the MOT beams confining the atoms in the z direction, the direction in which the ions
are accelerated. The typical pressure inside the main chamber is below 3 10−9 mbar and only
pumped by an ion getter pump to decrease mechanical vibrations. The accelerator is designed to
Figure 16: Schematic side view of the vacuum chamber with the Rubidium atom cloud trapped in
a Magneto-optical trap (MOT), the first excitation beam in red and the second one in blue and the
accelerator used to ionize the atoms and transport them towards the detector.
allow for voltages up to 30kV while the direction of the field governs whether one receives ions or
electrons on the detector. The other part is dumped onto the MOT mirror depicted in figure 16.
All optical access is done through holes in the top and bottom of the accelerator. The holes are a
29
C. Ravensbergen
trade off between the best electric field geometry on one side and optical resolution / flexibility on
the other. The outer part of the accelerator is grounded while the high voltage is applied to the
inner shell. The distance between these two shells is 20 mm [45]. After ionization, the ions fly in
the z-direction through a 1.5 m long vacuum tube towards a Micro-Channel plate (MCP) shielded
from the vacuum by a grounded grid with a throughput of 50%. Inside the MCP the ions create
a shower of electrons from the material, similar to an electron multiplier, via secondary emission.
These electrons, accelerated by the voltage applied over the MCP, are converted into light by a
phosphor screen which is then imaged with a lens on a CCD camera (Apogee Alta U9000). The
MCP, manufactured by Photonis, is built out of two 40 mm sized glass discs containing tens of
thousand tubes with a radius of 5 µm each and a spacing of 8 µm. The camera is cooled to −15◦ C
to reduce the pixel noise. The specified detection efficiency of the MCP at the used voltages for
heavy ions is between 5 and 80% [46].
The resolution of the whole ion imaging system combined yields: the CCD pixels are 22.3 µm, the
magnification of the accelerator and camera lens combined is 46×, and therefore the size of a single
pixel on the camera corresponds to 0.48 µm in the MOT. Counterintuitively the magnification of
the beamline is independent of the accelerator voltage, the larger off-axis fields are canceled by the
higher velocity of the particles. This remarkable feature is determined via beamline simulations and
a calibration using different known apertures as is explained in Reference [47].
3.2
Trapping and cooling lasers
The Rubidium 85 atoms are trapped and cooled in a standard [31] 3D MOT with three orthogonal
pairs of circular polarized laser beams and a quadrupole magnetic field. The magnetic field is
created by a pair of coils in an anti-Helmholtz configuration meaning that the current in one coil
is reversed compared to the other. The MOT cloud, on average, contains about 2 × 108 Rubidium
atoms and has a diameter of 1 mm which gives a density up to 2 × 1010 cm−3 . In order to to
cool atoms effectively they have to scatter multiple thousands of photons and because 85 Rb does
not have a single closed transition two lasers are needed. The two used optical transitions are
illustrated in figure 17, the trapping laser or pump excites the ground state atoms (5S1/2 (F = 3))
to the 5P3/2 (F 0 = 4) state and is operated at 780 nm. The linewidth for this transition is 2π · 6
MHz and the saturation intensity is 1.64 mW cm−2 . Because the two 5S hyperfine states are only
separated by 2915.1 MHz, atoms will not only decay to the F = 3 state but also to the F = 2
metastable state where they would not interact with the light field anymore. To avoid the loss of
these atoms they are re-excited with the repump laser to the 5P3/2 (F 0 = 3) state and can then
decay back to either ground state. This is a closed transition and ensures the continues cooling of
all atoms which started in the ground state. The final temperature of the atoms is close to 200 µK,
which is slightly above the Doppler temperature (TD = 146 µK).
The trapping laser is a commercially available tapered amplifier diode laser system (Toptica DLX)
and produces up to 900 mW of optical output power. The laser is directly locked on a 85 Rb vapor
cell via modulation transfer spectroscopy which is extensively described along with the the rest of
the setup in [45]. Another commercial diode laser system (Toptica DL 100) is used to produce the
repump light. The output power is 100 mW and its frequency is shifted with respect to the pump
laser by 2915.1 MHz. This laser is locked by a scheme called frequency offset lock which utilizes
30
3
Experimental setup
F
4
121 MHz
5 P3/2
3
2
1
Trapping
Laser
5 S1/2
3
2
63 MHz
29 MHz
Repump
Laser
3036 MHz
Figure 17: Hyperfine splitting of the ground state, 5S1/2 , and excited state, 5P3/2 , of
optical transitions for the trapping and repump beam.
85 Rb
and the
not a vapor cell but a beat signal obtained by mixing the light with the already stabilized trapping
laser. The frequency of the beat signal is the same as the frequency difference between the two
lasers and can therefore be used to give the repump laser a constant frequency offset compared to
the pump laser.
3.3
Excitation lasers
Direct excitation of Rydberg atoms would introduce two limitations: one can only excite P states
and the corresponding laser, in the ultraviolet range, is not commercially available. One can avoid
these shortcomings by utilizing a two step excitation process, explained in Section 2.4. The first
transition, 5S1/2 → 5P3/2 , is exactly the same as used by the trapping laser in for the MOT but can
be detuned independently via a separate AOM (Acousto-Optic Modulator). This light travels in
the same direction as the ions and to prevent it from illuminating the detector it is angled slightly
upward by 5◦ . The second excitation step brings the atoms from the excited state 5P3/2 to the
desired Rydberg state which can be, due to conservation of angular momentum, a nS or nD state
where n can be any desired principal quantum number, like it is shown in figure 18. The wavelengths
resonant with these transitions lie between 484 and 478 nm. Transitions to these high lying states
have, because of the small overlapping wavefunction, a small dipole moment and, due to their long
lifetimes, very narrow natural linewidths, this will be treated in more detail in Section 2.2.
The red laser, exciting the first transition, has already been mentioned in Section 3.2. For the blue
transition a commercial high power frequency-doubled diode laser (Toptica TA-SHG 110) is used
and provides up to 250 mW of output power. The Rydberg excitation Cw laser (continuous-wave) is
switched on and off with a computer controlled acousto-optic deflector, AOD (Intraaction ADm-70),
which produces pulse lengths down to 150 ns and has a maximum repetition rate of 100 kHz. These
pulses are transported to the top of the experiment via a single mode fiber. Here they pass through
some beam shaping optics and are focused inside the MOT. A cylindrical lens can be added to
defocus the beam in one direction and create a laser sheet in the xy plane. The diffraction limited
spot size of the blue laser is ≈ 40 µm. Its wavelength is continuously measured by a wavemeter
(Lambdameter LM-007) with a precision of 10−7 .
31
C. Ravensbergen
Ionization Limit E=0
nS
E=0
nD
~480 nm
5P3/2
780 nm
5S1/2
Figure 18: Schematics of the two laser exciting electrons from the ground state, |5S1/2 i, to the
excited state, |5P3/2 i, and then to either S or D Rydberg state with variable principal quantum
number n. In the presence of an external field the ionization limit is lowered and the electron
becomes unbound in the continuum, as is explained in Section 3.4.
In the presence of a strong electric field the energy of the blue laser is well above the ionization limit
which means that one can ionize the atoms directly into the continuum with the same wavelength
as would be otherwise used to excite the Rydberg state in the absence of the external field. The
creation of these fields is explained in the next Section 3.4.
3.4
Ionization and acceleration voltage
The accelerator creates a strong electric field which serves two purposes. It is used to ionize the
atoms and transport them to the detector. It can be either operated at a DC voltage to immediately
ionize the excited atoms or an voltage pulse to ionize the Rydberg atoms after the excitation has
taken place. The range of the DC voltage is ±6 kV. The amplitude of the pulse is limited by the
switch, a homemade bipolar high voltage pulser based on two Belhke switches, at ± 2.5 kV. The
rise time of the pulses generated by this switch are below 100 ns. Outside of the accelerator the
electric field fans out which acts as a negative lens on the charged particles. This lens is used in
the setup to increase the resolution of the detector by a factor of 46x. The time of flight of the ions
is between 22 µs and 40 µs for resp. 5 and 2 kV. Ion-ion repulsion could be a considerable effect
at these timescales but due to the effective negative lens at the exit of the accelerator the mean
distances between the ions is already greatly increased after ∼ 1 µs. The complete geometry and
overlap between the field and laser has been clarified in figure 19. tor.
32
3
Experimental setup
MOT
Detector
Y
Z
X
Figure 19: Schematic representation of the geometry of the lasers with respect to the MOT and
detector. The red dots represent the shaped first excitation laser at 780 nm while the blue laser is
the second Rydberg excitation laser at 480 nm. The MOT size is in the actual experiment much
larger than the laser widths.
3.5
Software for ion detection
The physical part of detection scheme has been discussed in Section 3.1 governing the field shielding
grid, MCP, phosphor screen and CCD camera. In this Section the software, used to extract the ion
locations from the images, will be discussed. Images are recorded by the camera in the common
astronomical .fits (Flexible Image Transport System) file format. All further analysis is done in
Matlab exploiting its vast image processing toolbox. At the beginning of every measurement an
empty image, without ions, is taken. It is used to subtract the background, light not originating
from the phosphor screen. This background light illuminates the whole CCD with a near constant
intensity. Therefore it can be used to correct for the unequal response of the camera pixels and
MCP/phosphor screen combination. The phosphor screen has for example a dark spot right in the
middle of the field of view. In this spot the signal is still present but the overall intensity is lower.
A division by the background gives an equal sensitivity of the detection over the whole image. The
ion detection, and especially locating the ions, is a crucial part of this experiment. The MCP is
used in a single particle mode, this means that every ion which starts an electron avalanche gives a
detectable signal, such as that shown in figure 20b, and part of a cross-section in figure 20a. This
mode has not been used in the past on this setup. Every ion creates a two dimensional Gaussian
profile on the detector as can be seen in the example in figure 20a. Many different algorithms have
been developed to find the positions of certain structures in an image. Extensive research is available
on the detection of star positions in astronomy. These algorithms tend to be extremely accurate
but also quite slow. So here an algorithm from the field of real-time pattern recognition has been
chosen. These algorithms are mostly used in low computational power environments like embedded
systems in robotics or in big data applications like security cameras. The algorithm to find the
ion locations consists out of the following steps. First the images are cleaned up by: background
subtraction, detector calibration and removal of a 8 pixel border. It then continues with performing
a normalized 2D cross-correlation of the image with a two dimensional Gaussian, the result of which
is shown in figure 20c. The values of the cross-correlation range between -1, anti-correlation, and
+1, perfect correlation. Since the ion signal is an intensity signal, no anti-correlation exists. From
this cross-correlation image a binary image is constructed (figure 20d) based on a threshold. The
33
C. Ravensbergen
threshold is found by trial and error and is set for all experiments to 0.35. If the threshold is too
low a lot of false positives are found while a too high threshold means one misses a lot of ions.
The quality of the binary images is improved by filling out all the holes in the image which might
originate from single pixel defects like hot or cold pixels, the used Matlab command is: imfill. The
binary image consists out of two types of regions: ions are circular areas with value 1 while the rest
is zero. The middle of the circles can be found by calculating the center of mass of every connected
region. The connected regions are found with use of the Matlab command bwconncomp and all the
centers of mass are then obtained by calculating the ’Centroid’ property of all the connected regions
with regionprops. A crop of the final result is given in figure 20e, here the positions of the two
overlapping Gaussians from figure 20a have been interpreted as two separate ions even though one
of the two ions has no clear maximum.
34
Mean Intensity (a.u.)
3
Experimental setup
0.03
0.02
0.01
0
0
10
20
30
40
50
y (pixel)
10
10
20
20
30
30
40
40
y (pixel)
y (pixel)
(a) Part of the y cross-section of two very close ions (at x=50 in Fig. 20b).
50
50
60
60
70
70
80
80
90
90
100
100
20
40
60
80
x (pixel)
100
120
140
(b) The raw image.
20
40
60
80
x (pixel)
100
120
140
(c) Cross-Correlation.
10
20
30
y (pixel)
40
50
60
70
80
90
100
20
40
60
80
x (pixel)
100
120
(d) The binary image.
140
(e) Crop with the positions of two overlapping
ions.
Figure 20: Crops of a typical ion image and the modifications performed by the algorithm described
in the text.
35
C. Ravensbergen
4
4.1
Shaping the excitation light
Introduction
Today’s quantum physics research is more than ever trying not only to observe and describe phenomena but also trying to directly influence them, even down to the single particle range. The
field of ultra-cold quantum gases is no exception and to accomplish this job different ’handles’ on
the atoms have been developed. Due to the tiny scale and hard access most methods are based
on either optical, electric or magnetic interactions or a combination of these three. Especially for
trapping and cooling of atoms laser light has proven to be essential. To specifically influence only
certain cold atoms one has to structure the laser beam. This can be achieved in different ways.
One can extrapolate the idea of lenses to an array of micro-lenses [14], use different lasers for each
site like vertical-cavity surface-emitting lasers [48] or create interference patterns in standing wave
light fields [49]. Here a different method is chosen because of several advantages. The spatial light
modulator, or SLM, used in this experiment gives full control over the actual light pattern including
the intensity, phase distribution and the position of this pattern. SLM is a very generic term but is
used here to describe a device which alters the phase, and sometimes also the amplitude, of light.
The generated pattern is not fixed, like in many other methods, but can be changed freely and
very easy via a digital interface which gives much experimental flexibility. However, the interface
operates at a 60 Hz refresh rate, which is insufficient to change intensities at atomic timescales.
This chapter begins with a short summary of the working principles of spatial light modulators in
Section 4.2, and then continues with the theory on calculating the phase pattern corresponding to
the chosen arbitrary light pattern in Section 4.3. This is followed by a general explanation of the
limitations and imperfections of optical systems in Section 4.4. It concludes with the experimental
results and performance optimization in Section 4.5.
4.2
Basic working principles of spatial light modulators
A spatial light modulator can act as a reflective or transmitting element. Transmitting SLMs have
the advantage that all light has passed through the active layers as opposed to the reflective one
where for example some light will always reflect on the cover plate. The drawback of transmitting
SLMs is the inherent performance loss due to the restriction that all parts have to be light transmitting. Most parts of a reflective SLM do not have this limitation and they have another useful
feature: the light travels twice through the modulating layer. Transmitting SLMs use a LCD, short
for liquid-crystal display, while reflective SLMs utilize liquid-crystal on silicon, or LCOS, technology.
LCOS mirrors consist of a silicon substrate layer, with most of the electronic circuit, on which a
reflective layer is stacked. On top of this reflective layer the parallel aligned nematic liquid crystal
cells are placed together with the second electrodes layer, this time transparent, and a protective
glass substrate. The whole stack is shown in figure 21.
The placed of the SLM in the setup is in between the lasers and the vacuum chamber as is shown in
figure 23. The excitation light, shaped by the SLM, is mixed together with one of the MOT beams
at the polarizing beam splitter, after which they have identical paths.
36
4
Shaping the excitation light
Figure 21: Sketch of the various layers of a typical liquid-crystal on silicon SLM.
Figure 22: Implementation of the SLM in the setup. Source:[30].
37
C. Ravensbergen
4.3
Calculation and construction of arbitrary light patterns
This section will explain what effect modulating the phase of light in a certain plane has on the
intensity pattern in another plane. The section is based the extensive treatment of our Spatial Light
Modulator and the used algorithms of Reference [30]. To keep things simple it will all be based on
scalar diffractive optics. The light field in this approximation is given by a complex valued function
U (x, y, z) with both an amplitude and a phase, from which the time dependent phase exp(iωt) is
omitted. Assume the incident light is a plane wave passing through a rectangular aperture with the
same size as the SLM (Lx × Ly = 15 × 8 mm):
2x 2y
,
.
(40)
Ui (x, y, z = −∞) = U0 (x, y) rect
Lx Ly
The SLM, at z = −L, now superimposes a custom phase, eφ(x,y) , on this light field which becomes:
U (x, y, −L) = Ui eφ(x,y) .
(41)
This light is now imaged with a lens, and given that the Fresnel approximation holds [50] and the
SLM is placed exactly one focal length, f = 90mm, in front of the lens, then the light field in the
focal plane is given by:
i x0 y 0 h
0 0
φ(x,y)
0
Uf (x , y ) = F Ui (x, y) e
,
,
(42)
λf λf
where λ is the wavelength of the light (780 nm) and F is the two dimensional Fourier transform
y0
x0
evaluated at frequencies λf
and λf
. This whole process is visualized in figure 23a.
If the SLM is turned off, the intensity in the focal plane becomes the familiar sinc (sinc(a) ≡
sin(a)/a) function:
0 0 x
y
0 0
2
2
2
|U (x , y , f )| = sinc
sinc
,
(43)
∆x
∆y
here ∆x is the diffraction limited spotsize given by:
∆x =
λf
.
Lx
(44)
This length, also known as diffraction limit, is extremely important in optics, it defines the smallest
possible features one can image. For this reason we call it a ’focal unit’. The other important
length scale is special for SLMs and it originates from the limited amount of pixels on the SLM,
Nx × Ny . This results in a focal plane built up out of Nx focal units, of size ∆x, in the x direction
and Ny focal units, of size ∆y, in the y direction like it is depicted in figure 23b. These quick
calculations give an understanding of the limits of the SLM and the properties that define them.
The lens strength and size of the SLM define the resolution of the images while the number of pixels
gives the useful drawing area. The pixels in the SLM have another important effect: the Fourier
transform of equation (42) is essentially not continuous but discrete, with the maximum frequency
defined by Li and the number of frequencies by Ni .
The intensity in the focal plane, given in equation (42), is influenced by the phase imposed on the
light field by the SLM. It is a straightforward calculation to get this intensity pattern for a given
38
4
Shaping the excitation light
Figure 23: Schematic representation of the way imaging works with a SLM (a) and the influence
of the dimension and pixels of the SLM on the final image (b). Source: [30]
phase pattern but unfortunately, because we have no control over the intensity pattern in the SLM
plane, there is no one-to-one relation. For an arbitrary given final pattern |Us |2 there does not exist
a exact solution for the phase, φ, such that |Uf |2 = |Us |2 . Luckily many different methods have
been developed to calculate the phase for which |Uf |2 ≈ |Us |2 .
The Iterative Fourier Transform Algorithm (IFTA) is particularly well suited to create arrays of
diffraction limited spots [30]. The algorithm uses an iterative method to calculate the SLM phase
during which the light is propagated back and forth between the SLM plane and the focal plane.
At each plane the calculated light field is compared to the desired light field, being the laser shape
on the SLM plane and the desired intensity pattern in the focal plane, and the constraints are
reapplied. The separate steps of the algorithm have been depicted in figure 24.
4.4
Measuring aberrations of optical systems
Every optical system has imaging faults if one moves away from the optical axis, and even in most
systems no actual optical axis exists. This leads to a lower overall performance of the system. These
imperfections can be reduced by carefully designing all optical components and precise alignment.
These faults or aberrations can be quantified with help of a special set of functions called the Zernike
polynomials. These polynomials form a complete and orthonormal set inside the unit circle, and
39
C. Ravensbergen
Figure 24: The steps performed by the IFTA algorithm. An initial phase guess, φ0 , is propagated
back to the SLM plane and the Gaussian beam intensity constraints are applied. This is then
propagated to the Focal plane and the desired intensity constraints are applied after which the whole
process is repeated until the desired precision is achieved.
can be used to describe the wavefront of the light at any position. In the focal plane an ideal
wavefront is completely flat, so here any deviations are directly related to the aberrations. The
Zernike polynomials are given by the following equation [51]:
Znm (r, θ) = Rnm (r)cos(mθ)(m ≥ 0)
Znm (r, θ) = Rnm (r)sin(mθ)(m < 0).
(45)
Here the polynomials are split in an angular part, which only depends on the angular frequency, m,
and the polar angle, θ, and a radial part, R, also depending on the order, n, and the radial distance,
r. The radial part is given by:
n−m
2
Rnm (r) =
X
l=0
(−1)l (n − l)!
rn−2l .
l![ 12 (n + m) − l]![ 12 (n − m) − l]!
In figure 25 the first five orders of the total Zernike polynomials have been plotted.
40
(46)
4
Shaping the excitation light
Figure 25: Plot of the Zernike polynomials up to the 5th order.
The first two orders describe piston, tip and tilt, which are resp. an overall constant offset and
in-plane displacement of the wavefront. These have no meaning when describing a light field since
they do not influence the light pattern and are only related to the choice of origin. The next order
corresponds to a simple defocus (m=0) and astigmatism (m=±2). Astigmatism is the effect where
the two orthogonal axes have different focal lengths. These first three orders are called the low
aberration orders while the rest are the high orders of aberration. The next orders consist out of:
coma, trefoil, tetratoil, spherical and higher orders of the same effects. The polar nature of the
polynomials is indeed very neat in the almost exclusively circular world of optics, but for a square
SLM it definitely is not. Simply transforming the polar coordinates to Cartesian ones is not enough
because it breaks the orthonormality of the polynomials. The solution is to transform them and
use a method like Gram-Schmidt to make them orthonormal again, but this time over the area of
the SLM instead of the unit-circle [52].
Different methods exist to obtain the wavefronts from an optical system. One can for example
buy dedicated Shack–Hartmann wavefront sensors which are placed directly in the light field [53].
They exist out of an array of micro lenses each imagining a part of the wavefront onto a ccd chip
placed behind the lens array. A plane wave would create an equidistant array of spots on the ccd
sensor while a local deviation from this plane wave would mean that also this particular spot has a
displacement on the detector as depicted in figure 26.
So by collecting all deviations from the lenses the wavefront can be reconstructed, with a precision
limited by the resolution of the detector and the finite size of the lenses. But in our case this
41
C. Ravensbergen
perturbed wavefront
lenslet array
∆r
image plane
Figure 26: Schematic illustration of the working principles of a Shack-Hartmann wavefront sensor.
The local angle of the wavefront defines the spot deviation ∆x on the detector. Source: [54]
detection method cannot be used because we are operating inside a vacuum and the vacuum window
is probably the main source of aberrations. Therefore a different method had to be found to measure
the wavefront at the focal plane inside the MOT. The SLM enables a similar method like the ShackHartmann method with the difference that in this case not the lenses select the parts of the wavefront
of which the phase is detected. Instead the exact same setup is always used to image the light in
the MOT but only a small part of the initial Gaussian light beam follows this path while the rest
is directed somewhere else. The position of this spot inside the MOT can be indirectly measured
by letting the light ionize the MOT atoms and detect them with the ion imaging system. One
then scans over the whole SLM which area of the SLM is allowed to image inside the MOT and
measuring the deviation of this area. The combination of the area and its deviation gives a map of
the complete wavefront. The area selection by the SLM and the resulting optical path is shown in
figure 27. So nothing physical changes in the setup, shown in figure 22, but only part of the initial
beam is allowed, by the SLM, to travel to the MOT.
A spot position deviation in the MOT plane corresponds to a gradient of the phase in the SLM
plane given by:
2π
∇φ =
r,
(47)
λf
where φ is the phase of the light in the SLM plane, λ is the wavelength, f is the focal length and x
is the spot position deviation. Spot positions have been determined by fitting the ion images with
a Gaussian in both the x and y direction after removing the background intensity. From this fit the
following variables are obtained: the intensity, position and size. An example for such a fit is given
in figure 28 where both the original ion image and the two Gaussian fits have been depicted for two
different parts of the SLM.
42
4
Shaping the excitation light
SLM
Selected Wavefront
Lens
Image plane (MOT)
∆r
Figure 27: Schematic illustration of the working principles of the modified Shack-Hartmann wavefront sensor. The light is focused inside the MOT to create ions. By selecting only part of the
light-field, with the SLM, the local wavefront can be measured via its spot deviation, ∆r, and after
scanning the whole light-field the total wavefront can be reconstructed.
x (um)
400
x (um)
600
200
50
50
100
100
150
150
200
200
250
y (um)
y (um)
200
400
600
250
300
300
350
350
400
400
450
450
Figure 28: Typical distorted images of the Gaussian ion spots obtained after imaging only a small
circle of the SLM plane. Distances are given in µm and are in the MOT plane.
43
C. Ravensbergen
With the help of equation (47) and the measured spot deviations of figure 28 one has all the necessary
ingredients to measure the wavefront at the focal-plane inside the MOT. The SLM cannot only be
used to measure this wavefront but is able to correct it before the aberration occur. This form of
adaptive optics gives the possibility to try and improve aberrated systems, the process of which is
discussed in the next section.
4.5
In vacuo aberration correction
After the determination of the aberrations of the complete optical system it is in theory a straightforward job to correct aberrations with help of the SLM. The aberrations of the system can be
viewed as an additional phase modulation of the laser beam, performed by the system and not the
SLM. But by correcting the phase aberrations before they occur only the desired intensity pattern
in the focal plane remains. The total phase pattern on the SLM now consists out of two parts: the
phase that governs the intensity pattern and an additional phase to counter the aberrations of the
system, leading to smaller and less distorted features in the focal point.
In reality, however, changing the phase has some complications. It means that light will travel
along a different path leading to a different overall aberration. To compensate for this effect an
iterative method is chosen, where instead of trying to completely remove all aberrations in one step
the whole correction process is repeated multiple times. This of course relies on the assumption
that this iterative algorithm will converge to the aberration free solution or at least an acceptable
one. The speed of the convergence can be influenced by the choice of step size. Figure 29 shows the
aberrations of the uncorrected case. Here the in-plane deviation from the focal point is plotted as
a function of the area of the SLM.
60
30
spot position shift in x direction (µm)
spot position shift in y direction (µm)
3
3
20
1
10
0
0
−1
−10
−2
40
2
20
1
y (mm)
y (mm)
2
0
0
−1
−20
−2
−20
−3
−6
−4
−2
0
x (mm)
2
4
−3
6
−30
(a) Deviation in the x direction.
−40
−6
−4
−2
0
x (mm)
2
4
6
−60
(b) Deviation in the y direction.
Figure 29: Plot of the in-plane spot deviation from the focal point as a function of the original
position on the SLM.
As mentioned earlier in equation (47), spot position deviations are directly proportional to the
gradient of the phase in the SLM plane. In order to create a smooth aberration correction phase,
the gradient of the phase is fitted by the gradient of the rectangular Zernike polynomials mentioned
in Section 4.3. Another option would be to numerically integrate the measured gradient of the phase
but this is much more prone to noise than taking the gradient of the analytical Zernike polynomials.
44
4
Shaping the excitation light
Figures 30 and 31 show a comparison of the measured gradient of the phase and one based on the
fit with Zernike polynomials.
Measured Zernike gradient x
Measured Zernike gradient y
0
5
10
10
0
0
−10
0
−5
−10
−20
−10
5
4
2
0
−2
−4
y (mm)
−5
5
4
2
0
0
0
−2
x (mm)
−4
y (mm)
−5
x (mm)
Figure 30: Measured gradient of the phase in both the x and y direction. The z-axis units are given
in radians per meter multiplied by the dimension of the SLM to make it dimensionless.
Zernike gradient x
Zernike gradient y
5
10
10
0
5
0
0
0
−10
−10
−5
−5
−20
−20
−10
4
5
2
0
0
−2
y (mm)
−4
−5
−30
4
5
2
0
0
−2
x (mm)
y (mm)
−4
−5
x (mm)
Figure 31: Fitted gradient of the phase in both the x and y direction. The units and false color
axis are the same as in figure 30.
The fit does indeed converge perfectly on the data. In these figures only well-resolved, single
Gaussian, ion spots have been used for the fit, hence the data does not cover the complete size of
the SLM. Near the edges of the SLM the ion signal is almost completely lost. This loss of ions has
multiple reasons. Near the edges of the lenses the angle between the light and the glass becomes
bigger leading to a higher reflectivity and therefore lower intensity. Another possibility for losses is
when the spot deviations become bigger than the MOT/blue laser size and therefore the deviation
becomes undetectable. And finally: some light might simply be blocked by non-transparent items
like the accelerator.
The main feature of figure 30, in the y-direction, is the linear behavior which corresponds to a lens
strength error or astigmatism. A much higher order aberration is dominant in the x-direction. A
quantitative way to identify these aberrations is by looking at the value of the Zernike coefficients
given in figure 32. There is not a single dominating aberration visible apart from the coefficients of
the 3rd order (Zernike coefficient 3, 4 and 5) describing astigmatism and defocus.
45
C. Ravensbergen
2.5
Before Correction
After Correction
2
Phase (rad)
1.5
1
0.5
0
−0.5
−1
−5
0
5
10
15
Zernike coefficient
20
25
30
Figure 32: The Zernike coefficients up to the 28th order for corrected and uncorrected setup.
In figure 33a the phase pattern calculated from the Zernike coefficients is shown. The main feature in
this phase pattern is the circular cutoff at one side which we believe is due to the vacuum window.
The beam does not pass through the middle of this window but very much near one side, and
because the window is far from flat but becomes much thicker near the edges this would introduce
asymmetrical aberrations to our excitation beam. After four cycles of aberration measurement and
correction each time adding up the new Zernike coefficients to the old ones the phase pattern in
figure 33b was obtained.
Although these phase patterns are a very elegant way to depict the so-called flatness of the wavefront
they are in our case rather indirect because they are based on a fit. The actual measurement of
the corrected case is shown in figure 34. These are one to one identical in axes, viewing angle and
scaling as the plots in figure 30. Even though the spot deviations are almost non existent anymore
the useful area of the SLM has not increased by correcting the aberrations. The reason for this
might be the physical size of the apertures of the lenses and vacuum chamber.
Different methods exist to quantify the distortion of an optical system. These can be used to
monitor the progress of the aberration correction. Since we already found the Zernike coefficients
calculating the wavefront quality is a straightforward job and can be done by taking the RMS error
of the coefficients. The RMS wavefront error as a function of the number of correction iterations is
shown in figure 35. The final error corresponds to about λ/6.8 or 0.11 ± 0.1 µm. Even though this
is a huge improvement over the initial conditions it is not quite diffraction limited yet. A diffraction
limited system has a RMS wavefront error of λ/14 or lower [55]. But the aberration correction is
most likely limited by the detector resolution and mechanical vibrations in the setup. Considering
the enormous initial aberrations it is quite remarkable that our final system performs similar to
the famous Keck telescope, without their adaptive optics system, in Hawaii [56]. Figure 35 also
suggests that we have not reached the correction limit yet and it should be quite possible to correct
46
4
Fitted Phase Aberration
Shaping the excitation light
Fitted Phase Aberration
−6
6
−6
4
−4
1.5
−4
−2
2
0
0
0.5
−2
x (mm)
x (mm)
1
0
0
−0.5
2
−2
2
4
−4
4
−1
−1.5
−2
6
−4
−2
0
y (mm)
2
6
−6
4
(a) Aberrated phase pattern.
−2
0
y (mm)
2
(b) Aberration corrected phase pattern.
Figure 33: Phase patterns calculated from fitted Zernike coefficients. The non corrected case comes
directly from the values in figure 32 while the latter was constructed after four cycles of aberration
measurement and correction. Note that the false color axes of the images are unequal.
Measured Zernike gradient x
Measured Zernike gradient y
5
4
10
2
10
0
0
0
0
−2
−4
−10
−5
−10
5
2
0
0
−2
y (mm)
−5
x (mm)
5
2
0
0
−2
y (mm)
−5
x (mm)
Figure 34: Measured gradient of the phase in both the x and y direction after correction.
47
C. Ravensbergen
the system down to λ/10 or lower.
60
RMS wavefront error
50
40
30
20
10
0
−2
0
2
4
6
Iteration
Figure 35: The RMS wavefront error as a function of the number of correction cycles. The RMS
error after correction equals λ/6.8.
The following examples are meant to demonstrate the difference between the the corrected and
distorted system. In figure 36a the typical image of a 10x10 ion spot pattern in focus is given, the
distortions are clearly visible in the shape of the individual spots while the lattice remains mostly
intact. After the complete correction has been applied, to the system, the exact same image is
taken, 100 seconds camera exposure time at 10 Hz shot frequency, as is shown in figure 36b.
(a) Without aberration correction.
(b) After aberration correction.
Figure 36: Ion crystals of 10x10 spots with 30 µm spacing. The dark spot is a defect of the
phosphor screen and is independent of the excitation light pattern.
48
4
Shaping the excitation light
The correction makes the individual spots much more symmetric and smaller. More stray light is
focused back into the spot giving much brighter spots. The phosphor screen introduces some image
errors, most notably the dark spot in the middle of both images of figure 36. The camera is cooled
to −15 0 C but still some hot pixels are visible. The non-uniform response of camera and phosphor
screen can be removed with the use of a background image, but this does not work with the hot
and dead pixels since they contain no information. Some distortions and a constant rotation are
always present, unaffected by the correction. The rotation has been removed by rotating the images
by 5◦ . But the spots also deviate from a straight line and the angle between the horizontal and
vertical spots is not 90◦ . These aberrations are the same before and after correction which might
indicate that they are introduced by the ion imaging system and not the optical excitation. They
could for example originate from a not completely cylindrical symmetric electric field or an off-axis
ionization. Another possibility is the fact that the correction only works in the focal point and not
necessarily for the other points in this plane. No additional efforts have been made to correct these
aberrations because the performance is sufficient for the moment. The performance in our case is
quantified by two parameters, the spot size and the spot distance. The spot size gives information
about the optical performance of the whole system while the minimal spot distance is a combination
of the spot size and the pixelated nature of the SLM, as is explained in Section 4.3.
4
7000
5.2
x 10
data
fitted curve
6000
5
Intensity (a.u.)
Intensity (a.u.)
5000
4000
3000
2000
4.8
4.6
4.4
1000
−1000
0
4.2
data
0
fitted curve
10
20
30
40
50
x (µ m)
(a) Eight spots in x-direction with 14 ± 1 µm spacing and 11 ± 1 µm spot size (2σ).
4
0
50
100
150
200
y (µ m)
(b) Seven spots in y-direction with 25±3 µm spacing and 23 ± 5 µm spot size (2σ).
Figure 37: Intersections in the x (y) direction averaged over 7 y (x) pixels and fitted with the same
amount of Gaussians as there are spots.
The intersections from figure 37 have been taken in the limit of visibility. Our slightest decrease in
spot distance, one focal length of ∼ 5 µm, would result in no discernible pattern at all. A crop of the
original image is shown in figure 38. The camera binning is 2 compared to the normal binning of 5
in for example figure 36, this increases the camera resolution at the cost of noise. It is important
to notice the different spot sizes and separation in the x and y direction due to the shape of the
SLM. The diffraction limited spot size for the SLM is 4.7 µm in the x direction and 8.8 µm in y. The
measured spot size is: 11 ± 1 µm in the x-direction and 23 ± 5 µm for y. This means our measured
49
C. Ravensbergen
spots are ∼2.5 times larger than the theoretical limit in both directions, which almost equal to the
factor of 2 in the RMS wavefront error. The spotsize.
Figure 38: Cropped ion image with 14 µm spot spacing, 11 µm spot size and camera binning of 2.
A intersection with fit is given in figure 37.
In between spots the Intensity does not drop to zero. Naturally this can be due to the stray light
still present after correction but our imaging system exhibits a similar effect. This is shown in figure
39. Here the intersection of a single ions does not yield a single Gaussian. Instead it consist out of
two peaks (at the same position), the actual one and one which is about 3 times wider. If one now
images many thousands of ions one gets a large broad background in the vicinity of the signal, a
known artifact of MCP detectors [57].
1.2
Narrow Gaussian fit
Wide Gaussian fit
Image Data
Normalized Intensity (a.u.)
1
0.8
0.6
0.4
0.2
0
−0.2
0
10
20
30
distance (µm)
40
50
Figure 39: Intersection of a single ion fitted by a double Gaussian.
50
5
5
5.1
Observation of Rydberg Atoms
Observation of Rydberg Atoms
Introduction
In the previous section the process of shaping the excitation light has been discussed. This ability
opens up a fast way to change, in between experiments, the excitation volume. We are now able
to create, without any physical change to the setup, arbitrary shaped one and two dimensional
intensity patterns. And by one dimensional we mean that the intensity pattern is smaller, in width,
than the expected blockade radius. In this Chapter we do not use these advanced capabilities of the
SLM yet. We first want to understand the basic Rydberg excitation and interaction in our system,
and therefore use a simple Gaussian excitation profile for both the red and blue laser. In section 5.2
we perform a scans of the blue laser wavelength to detect the positions of the Rydberg levels. We
then perform a calibration of the detector by measuring the correlation function of non-interacting
particles in the form of directly extracted ions. And now everything is set to perform the correlation
function measurement of different Rydberg levels, the result of which is shown in Section 5.4. The
final part of this Chapter, Section 5.5, consists of the Q-factor measurement results.
5.2
Rydberg level scan
Every Rydberg experiment starts with a scan of the laser wavelength to find the Rydberg level.
Scanning the wavelength of an external cavity diode laser can be done in three separate ways. The
very coarse adjustment of the laser is done by adjusting the grating angle by hand with a micrometer
screw. This enables the tuning of the laser over a range of multiple nanometers. A finer method,
with a lower range, is adjusting this same angle but now with a Piezoelectric actuator which changes
the angle as a function of the applied voltage. And the last and most precise method is by adjusting
the current through the diode. All this time the wavelength is monitored by a wavemeter described
in Section 3.3. The precision of the wavemeter is about 10−7 which corresponds to a frequency
precision of 30 MHz, since we measure the wavelength of the not yet frequency doubled infrared
laser. This is still much higher than the specified linewidth of the blue laser given by: 2 MHz. Thus,
our only way to measure the bandwidth of the laser is by performing a Rydberg level scan. The
wavelength of the transitions can be calculated with equation (2). The detection itself is done by
measuring the intensity on the detector as a function of the detuning of the blue laser, while the
red laser is kept on resonance by the locking scheme discussed in Section 3.2. The result is shown
in figure 40. The state in theory nearest to the dialed in wavelength is the 91S level, its position
is chosen to have a detuning of 0 MHz. But we do not see a single resonance. There are three
distinctive resonances in figure 40.
The strong interactions between Rydberg atoms is mainly due to the large extension of the electron
wavefunction. This also means that the electron is extremely sensitive to external electric field. The
corresponding Stark map, a graph which shows the influence of an external magnetic field on the
energy of the levels, has been plotted in figure 41. It is clear that even for extremely small external
fields the states with l > 3 already cross, or come near, the S state. This makes identifying the
resonances in figure 40 extremely difficult if not impossible.
51
C. Ravensbergen
6
7
x 10
91S
6
Intensity (a.u.)
5
4
3
2
1
0
−1
−700
−600
−500
−400 −300 −200
Detuning (MHz)
−100
0
100
Figure 40: Rydberg population as a function of the excitation laser detuning. Three resonances
are visible. The theoretical position of the 91S level at zero external field is placed at the origin.
These are the things we know for certain: the wavemeter seems to have a constant error in the
measured wavelength of about -200 MHz. This value has been found by scanning lower Rydberg
levels and D states where only a single resonance is present. It is unclear however if we are able to
excite these higher angular momentum states and what their influence will be on the measurement.
For the remainder of this thesis the measured resonance will be called after the nearest theoretical
state at zero electric field.
−600
−500
← |91 0 0.5 0.5>
Energy (Mhz)
−400
−300
−200
−100
0
100
0
0.05
0.1
0.15
0.2
0.25
External field gradient (V/cm)
0.3
Figure 41: Stark map of the 91S Rydberg state. It shows the influence of the external electric field
gradient on the relative energies of the Rydberg states. A match with figure 40, based on the relative
distances of the three resonances, can be found at all the crossings between 0.08 and 0.25 V/cm.
52
5
Observation of Rydberg Atoms
The width of the peaks of figure 40 are governed by the linewidth of either the laser or the Rydberg
transition. The linewidth of the Rydberg transition is in the order of a few tens of kilohertz. So
the linewidth is dominated by the width of the excitation laser. The half-width at half-maximum
(HWHM) of the Lorentzian fit through the data is 9 MHz, much wider than the specified width, at
< 2 MHz, but it should be still good enough for a strong blockade effect to be present.
A scan of another level, 83S, started at a slightly lower wavelength, could clarify the level issue a
bit. The scan is shown in figure 42, and one recognizes the same double peak as was visible in figure
40 but it continues to much lower wavelengths (more negative energy in figure 41). This is a strong
indication that indeed the peak at -450 MHz in figure 40 is the S state since no other resonances
are visible at a higher detuning.
6
x 10
2
Intensity (a.u.)
83S
1.5
1
0.5
0
−100
0
100
200
Detuning (MHz)
300
400
500
Figure 42: Rydberg population as a function of the excitation laser detuning. Many resonances
are visible. The theoretical position of the 83S level at zero external field is placed at zero detuning.
5.3
Detector Calibration
With this method to find Rydberg levels we can start to do measurements. The simulations showed
that the pair correlation function gives the most insight in the Rydberg physics. So naturally this is
what one tries to measure first. The measurement suffers from a few complications the simulation
does not have. First of all, due to the size of the lasers, our environment isn’t 2D. The geometry
of the red laser has been extensively discussed in Chapter 4.5, it defines both the x and y spot
size. Much time has been spent to decrease the spot size of the blue laser which defines the depth
of our excitation volume. The Gaussian spot size is currently 50 µm (2σ). Because intensity is
at the moment more important than size the cylindrical lens, to create a laser sheet, was not in
place during all Rydberg experiments. Thus, the excitation volume has approximately the following
dimensions: 50 × 75 × 50 µm. The Rabi frequencies for the red laser can be up to 10 Γ, while the
Rabi frequency for the blue laser is much lower, it has a maximum of about 0.8 Γ. With Γ = 2π 6
53
C. Ravensbergen
MHz. Because our detection scheme is in two dimensions while the excitation is in three we expect
the correlation function not to go to zero for small distances but reach a finite value. Additionally
it is important to know the resolution of the detector, atoms closer spaced than the resolution will
be indistinguishable.
A calibration measurement can give answers to these detection problems. The idea is to do a
complete measurement and analysis without Rydberg interactions. Two options are available to
implement this: one could excite a Rydberg state where the blockade radius is much smaller than
the average particle distance or one could use a strong electric field to lower the ionization limit and
excite the atoms directly into the continuum. This last option is quite a lot easier to implement
because everything will remain untouched except a single trigger: the field has to be ramped up
before the laser excitation as opposed to after the excitation. The resulting correlation function is
given in figure 43. Normally, ions would have much stronger interactions than the Rydberg atoms.
But because the field, that also ionizes them, immediately accelerates them, the chance of creating
two ions, at the same time and place, is extremely small. The ions spent approximately 0.01 µs in
the atom cloud. For a perfect detector one would expect the correlation function from figure 5a, in
our case the detector is not perfect but starts to deviate from the theoretical curve at 8 µm.
1.6
Ideal gas
Fit
Measured ions
1.4
1.2
(2)
g (r)
1
0.8
1.1
1
0.9
0.8
0.7
0.6
0.4
0.2
4
0
0
20
6
40
60
Distance (µm)
8
10
80
100
Figure 43: Measurement of the normalized correlation function for non-interacting ions.
A fit of this curve is used in the remainder of this thesis to correct the data for the influence of the
detector. The fit is based on a very simple model: g (2) (r) = exp(−(a0 /r)a1 ), but describes the data
quite well. The fitted values for the coefficients are: a0 = 2.4 ± 0.2 and a1 = 2.6 ± 0.5. From the
insert in figure 43 we estimate a linear detection efficiency down to 8µm. For smaller distances the
detection efficiency goes gradually down and can still be compensated to values well below 4 µm.
This means that if a Rydberg blockade is present in our setup we should be able to directly observe
it.
54
5
5.4
5.4.1
Observation of Rydberg Atoms
Measured correlation function
S-states
Now to the choice of Rydberg level. Higher levels have a bigger blockade radius but are harder to
excite and the combination of these two means a much lower atom number. In order to measure a
smooth correlation function one needs a lot of measurements with many atoms. The used parameters
are given in Table 2.
Table 2: The used values of the relevant parameters in the simulations.
80S
87S
Γ(MHz)
2π 6
2π 6
Ωge (MHz)
4
4
Ωer (MHz)
7.5
6.6
δe
0
0
|δr |
< 10 MHz
< 10 MHz
τexcitation
15 µm
15 µm
Each measurement consists out of 1000 single shots each containing between 10 to 60 atoms depending on the laser intensities and Rydberg level. The repetition rate of the experiment is a very
low, without the readout time of the camera, the maximum is 10 Hz. Because of this long time the
laser will drift substantially and has to be, at the moment, corrected by changing the diode current
by hand. The laser is kept on resonance by maximizing the number of ions per shot. The number of
ions is detected via the current flowing through the MCP. Because the wavelength is only marginally
stable, reasonably high Rydberg states have been chosen to make sure that some blockade effect
remains present. In figure 44, the measured g (2) of 80S has been plotted. To compensate for the
dependence of the detection efficiency on the distance the whole data curve has been divided by
the fitted curve of figure 43. This has little influence except that it lifts up the data points for
small distances. The blockade radius is defined as the distance at which the pair-correlation function reaches unity for the first time [9]. The blockade radii and uncertainties, in all measurement,
are obtained as follows. The first part of the numerical correlation function, up to its maximum
value, is fitted with a third order polynomial. We use the numerical g (2) because: it has a much
smaller uncertainty, higher binning number and can be regarded as an independent fit of the data.
The excitation volume is not 2-dimensional, so two Rydberg atoms could be excited behind each
other, which leads to a non-zero value of the correlation function inside the blockade radius. The
measured blockade radius of figure 44 is: 13.5 ± 0.5 µm. The simulation shown in figure 44 is only
based on exactly the same parameters found in the experiments and given in table 2, as well as the
same density found in the MOT (ρ = 1.2 10−10 cm−3 ). The intensity of the red laser in the MOT
has been approximated due to the complex imagining method. The simulation shows an excellent
agreement with the data over the complete, experimental, distance range.
As mentioned earlier, the principal quantum number and the intensities of both lasers are the main
components that define the blockade radius. The geometry of the excitation volume largely gives
the deviation from the two dimensional correlation function from figure 13. The simulation and
data both show a maximum outside the blockade radius similar to the one measured in Reference
[9], this is direct proof of spatial ordering of the Rydberg excitations. And, outside the blockade
radius, the measured g (2) has the same value, and number of oscillations, as the simulations. And
55
C. Ravensbergen
1.6
Simulated 80S
Measured 80S
1.4
1
(2)
g (r)
1.2
0.8
0.6
0.4
0.2
0
10
20
30
Distance (µm)
40
50
Figure 44: Measurement of the normalized correlation function for the 80S Rydberg state and a
KMC simulation with the same parameters. The depicted error bars give the uncertainty of the mean
(SEM), the uncertainties of the simulation are 5 times smaller in magnitude. Rb = 13.7 ± 0.1 µm.
also inside the blockade radius are the measurement and simulation in perfect agreement. The
measured correlations functions equals, within measurement uncertainty, unity for large distances.
Data points at very small and large distances have been omitted, in all measurements, because of
the large uncertainties originating from the low number of atoms in these binnings. The error of
each data point is calculated by:
s(x)
S(hxi) = √ ,
(48)
N
where s(x) is the standard deviation of the sample and N the number of measurements. Because the
blue laser is not stabilized its detuning is unknown. The influence of the detuning on the correlation
function has been simulated and the results are shown in figure 45.
The value of the detuning could lie anywhere in this region and changes during the measurement.
Luckily, the behavior of g (2) is in a number of key areas almost independent of the detuning. The
three cases are near identical inside the blockade radius down to 9 µm. The only real changes happen
in the height and position, the lattice constant, of the maximum outside the blockade radius and
the number of oscillations [58]. A negative detuning is actually helpful for the emergence of a
crystal, it increases the Rydberg excitation probability at a certain distance interval. This interval
corresponds to the region where the Rydberg interaction shift cancels the laser detuning leading
only there to a resonant excitation. See for example figure 3 for an energy diagram.
The next Rydberg level measured with the same settings was 87S given in figure 46a. The effect of
the dipole blockade is again clearly present and equal to the simulated effect. But, in the height of
the peak, the main problem of measuring the correlation function becomes clear: the uncertainty of
each data point is substantial and overshadows the smaller peak in the simulation. Increasing the
56
5
Observation of Rydberg Atoms
Blue laser detuning δr
0 MHz
10 MHz
−10 Mhz
1.2
g(2)
1
0.8
0.6
0.4
10
20
30
distance (µm)
40
50
Figure 45: Simulated correlation functions of the 98S Rydberg level for three different values of
the detuning, δr , for the blue laser.
binning size would only solve part of the problem, while longer measurements are not achievable
due to the instability of the lasers.
1.6
1.6
Simulated 87S
Measured 87S
Simulated 103S
Measured 103S
1.4
1.2
1.2
1
1
(2)
g (r)
g(2)(r)
1.4
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0
10
20
30
40
Distance (µm)
50
(a) The 87S Rydberg state.
60
0.2
0
10
20
30
40
Distance (µm)
50
60
(b) The 103S Rydberg state.
Figure 46: Correlation functions, after correcting for the detection efficiency, and the corresponding KMC simulation. Even though n is in figure (b) much higher, the blockade radius,
Rb = 16.0 ± 0.1 µm (a) and Rb = 16.3 ± 0.1 µm (b), is comparable due to the increased laser
intensity.
With these settings it would be impossible to go to higher principal quantum numbers. So we
increased the laser intensity, the new values are given in table 3, which led to more atoms per shot.
This drastically reduces the uncertainty for each binning resulting in figure 47. The catch here is
that this also reduces the blockade radius. Even though the principal quantum number is much
higher than the previous figures the blockade radius is almost equal to the 80S Rydberg state. The
57
C. Ravensbergen
blockade radius in the measurement seems to be slightly smaller than the simulated one. The shape
of the peak in the measurement is a very good match of the simulation. Unfortunately does the
dimension of the excitations volume prevent us from seeing a local minimum behind the peak. Also
note the changing height of both the measurement and simulation at very small distances between
the different Rydberg states. This value is also directly influenced by the blockade radius. These
are the values of the parameters in these measurements and simulations:
Table 3: The used values of the relevant parameters in the 98S and 103S simulations.
98S
103S
Γ(MHz)
2π 6
2π 6
Ωge (MHz)
25
25
Ωer (MHz)
5.5
5.1
δe
0
0
|δr |
< 10 MHz
< 10 MHz
Simulated 98S
Measured 98S
1.1
(2)
g (r)
1
0.9
0.8
0.7
0.6
10
20
30
Distance (µm)
40
Figure 47: The 98S Rydberg state. Rb = 11.4 ± 0.1 µm.
The combination of all S state measurements, given in figure 48, shows the behavior of the blockade
radius as a function of the principal quantum number. The obtained blockade radius has to be
rescaled, to compensate for the changing laser intensities and Rabi frequencies, to obtain a setup
independent property before it can be compared to literature values. The
p scaled Rydberg blockade
radius is obtained by multiplying both sides of equation (12) with 6 Ω(n), where Ω(n)
p is the
combined, three level, Rabi frequency. The analytical curve in figure 48 is given by: 6 C6 (n)/~,
where the van der Waals coefficient has been determined with help of equation (11). Figure 48
shows the excellent correspondence, both in absolute value and scaling, between the literature value
of the van der Waals coefficient and the measured blockade radius, leading to the conclusion that
equation 12 is indeed an adequate approximation of the blockade radius. This also proves that the
correlation function is indeed a very comprehensive tool to characterize Rydberg interactions.
58
5
Observation of Rydberg Atoms
300
Measured
Scaled Rb (µm Ω1/6 )
280
Analytical
260
240
220
200
180
160
80
85
90
95
Principal quantum number, n
100
√
Figure 48: The blockade radius, scaled by 6 Ω, as a function
of the principal quantum number
p
6
C6 /~. Error bars are based on the
compared to the analytical value of this property given by
uncertainties of the fit.
5.4.2
D-states
Until now all experiments were performed on Rydberg S states, but the two photon excitation can
also excite D states, we call them D states because we are not able to distinguish the two finestructure levels: nD3/2 and nD5/2 . These D states have attractive anisotropic interactions. This
makes simulations a lot more complicated but experimentally these states are preferable because,
due to their relatively large dipole moment, they are much easier to excite. The measurement is
identical to the nS states. However, the simulation has to be altered to adjust for the changed
interactions. The quantum defect for nD states is given in Section 2.2, which is only a small
adjustment in the code. The van der Waals coefficient, C6 , also changes, both in magnitude and
size. A complete calculation of the energy shifts of interacting D and P state Rydberg atoms
is performed in Reference [35]. Due to the added computation complexity and lack of a clear
quantization axis the interactions in the simulations on D states have been kept isotropic by using
the mean of the C6 coefficient. The result of two D state measurements are given in figure 49.
Like for the S states is there a strong blockade present and in the case of figure 49a even a slight
maximum outside the blockade region. Even though the interactions are attractive the blockade
does not seem to differ much. Further studies might discover large differences between these two
types of Rydberg states but with the current setup they seem to be remarkably equal, except for a
simple scaling of the blockade radius.
59
C. Ravensbergen
1.6
1.6
Simulated 98D
Measured 98D
Simulated 99D
Measured 99D
1.4
1.2
1.2
1
1
(2)
g (r)
g(2)(r)
1.4
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0
10
20
30
Distance (µm)
40
(a) The 98D Rydberg state.
50
0.2
0
10
20
30
Distance (µm)
40
50
(b) The 99D Rydberg state.
Figure 49: Correlation functions after correcting for the detection efficiency and the corresponding
KMC simulation without anisotropic interactions.
The setup used in this thesis has a few experimental uncertainties, like the detuning of the blue
laser and the residual electric field. But still, the presented data is in excellent agreement with
the simulations, especially in the value and behavior of the correlation function inside the blockade
radius and the number of oscillations outside this region. The method presented here shows great
experimental promise. It could, for example, be used to visualize the time evolution of the selfassembly of Rydberg crystals. And combined with the optical lattice discussed in Section 4 or a
chirp in the detuning, discussed in Reference [40], it could be used to directly study many-body
effects in Rydberg gases. The results of the other method to prove blockade effects, the Mandel
Q-factor, will be discussed in the next section.
5.5
Mandel Q-factor
Originally, the Mandel Q-factor, explained in Section 2.8.3, seemed to be an excellent parameter to
prove the existence of the dipole blockade mechanism. Its calculation only need requires the number
of Rydberg atoms. This is a clear advantage for setups without the needed spatial resolution. But
this simplicity is also its main weakness, it is almost impossible to give the influence of a certain
parameter, like the blockade radius, the density or the excitation time, on the Q-factor. And the
detection efficiency plays a major role in the measured value of the Q-factor. But the detection
efficiency is uncertain in most cases. The influence of the detection efficiency on the Q-factor is
given by [43]:
Qmeasured = η Qactual (η < 1),
(49)
where η is the detection efficiency. The simulations show that for Rydberg levels near n = 80 the
fluctuations are already highly sub-Poissonian, with Q below −0.7. Although many experiments
have tried to measure sub-Poissonian Q-factors [22, 59, 8] it can be quite difficult [60]. These
experiments all use cavities, to stabilize their blue laser, and high ion detection efficiencies. These
properties are both absent in our setup. Since we already found all the Rydberg atom positions, is
60
5
Observation of Rydberg Atoms
it extremely easy to calculate the measured Q-factor. Which initially, only in the areas of maximum
intensity, gave the good results shown in figure 50. The Q-factor for the complete excitation volume
was above 0 for all measurements. The Q-factor for the low level, 72D, was close to 0 at while
the higher nD (n > 80) states all gave about the same result of Q = −0.3 ± 0.1. The Q-factor
and its uncertainty interval have been determined with the Matlab function normfit which not
only returns the mean and standard deviation, of a sample, but also their upper and lower bounds
(95%).
72D
87D
0.18
0.18
Data
Poissonian
0.16
0.14
0.14
Probability
0.12
Probability
Data
Poissonian
0.16
0.1
0.08
0.12
0.1
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
0
5
10
15
Rydberg Atom Number
(a) 72D Rydberg state.
0
0
5
10
15
20
Rydberg Atom Number
25
(b) 87D Rydberg state.
Figure 50: The measured probability of the number of Rydberg atoms in a single shot and a
Poissonian distribution with the same mean for different Rydberg levels. The average inter-particle
distance in figure 50a is higher than the theoretical blockade radius while in the latter figure 50b this
is not the case anymore.
During the second round of measurements no large negative Q-factors were found. Even though
almost every aspect of the experiment had been improved, ranging from the detector resolution
(factor of 5) to the intensity of the lasers (factor of 2), only much smaller Q-factors have been
measured (Q = −0.1 ± 0.2).
Where does this mismatch come from? The detector has stayed the same so equation (49) is not
able to explain the difference. One might start with the laser: fluctuations of the blue laser, both
in the detuning and the intensity, increase the total fluctuations of the Rydberg atom number. The
piezo which holds the external grating in place in the blue laser drifts in time. This drift is much
larger after a large displacement of its position. During the first measurement the piezo had much
more time to stabilize than during the second round. And there is the fact that the first round
had a much shorter measurement time due to the higher shutter frequency. This was only possible
thanks to the high camera binning which reduced the image size substantially. Unfortunately this
does limit the detector resolution which means it is impossible to calculate the correlation function
inside the blockade region. This drift in wavelength has not as big of an effect on the correlation
function, as is shown in the simulation of figure 45, as it is on the Q-factor. Which even ranges in
the simulation between -0.9 and -0.6 depending on the detuning.
61
C. Ravensbergen
99D
50
0.3
100
0.25
y (pixels)
Probability
150
0.2
0.15
0.1
200
250
300
350
400
0.05
450
0
0
2
4
6
Number of Rydberg atoms
8
(a) Comparison of the number of Rydberg atoms
in an image and the Poissonian distribution with
the same mean.
500
100
200
300
x (pixels)
400
(b) Figure of the sum of all images. The area used
for the determination of the Q-factor is depicted in
red.
Figure 51: Measurement of the fluctuations of the number of 99D Rydberg atoms in a certain
area. This area has been chosen to reduce the influence of laser intensity fluctuations.
This is enough reason to not continue, under the current experimental conditions, with the Q-factor
measurements. Because it seems that one either optimizes for the measurement of correlation
functions, or for Q-factors, while correlation functions need long measurements with many shots, a
significant Q-factor can already be obtained from only 50 shots [8].
The measurement of Q-factor could benefit from the following improvements to the setup. The
blue, Rydberg excitation, laser wavelength needs to be stabilized, this can either be achieved by
locking it to a stabilized reference cavity [61], create an error signal and lock the laser onto that.
The error signal can be obtained from electromagnetically induced transparency [62], EIT, or from
a novel type of vapor cell [63]. A reference cavity would be the best option because it does not only
stabilize the laser but also further reduces its bandwidth, essential for coherent excitations [64].
Another improvement to the setup would be to increase the density of the MOT, which is needed
if one wants to create a 2-dimensional environment with an acceptable number of atoms in it. And
finally, it is needed to determine, and remove, the residual electric field which not only affects the
level energy but also the lifetime of the Rydberg atoms.
62
6
6
Conclusion
Conclusion
Future quantum computers and simulators will rely on a number of key features. In this thesis it
is shown that ultra-cold trapped Rydberg atoms would indeed be a plausible candidate for such
systems. They possess strong interactions in the form of the blockade mechanism, the existence of
which has been shown in this thesis by direct measurement of the pair-correlation function. The
blockade has been found to be dependent on both the intensity of the lasers driving the transition and
the principal quantum number of the Rydberg state. The blockade radii, measured in this thesis,
show perfect agreement with literature values and numerical simulations. And even correlations
outside this blockade radius have been found by us in a number of measurements, which suggest
the formation of a liquid like [41] density distribution of Rydberg atoms in an otherwise completely
random gas. The combination of the shaped excitation light of the SLM and the strong blockade
of Rydberg atoms would also give the ingredients of a system where one could address a single
Rydberg atom, which is another important feature of a future quantum simulator.
Blockade effects are present in all measurements, both on S and D state Rydberg levels. And
some measurements clearly show a maximum, outside of the blockade radius, in the pair-correlation
function. This peak is a strong indication for for the rise of an ordered, crystal-like structure in the
Rydberg gas. The absence of this peak in some measurements could be due to the unknown, and
changing detuning, of the second excitation laser or the larger uncertainties in these measurements.
The highest measured blockade radius is 16.3 ± 0.1 µm (103S). This is larger than the smallest spot
size obtained with the SLM in the x-direction: 11 ± 1 µm. This means that, in future experiments,
it should not only be possible to create only a single Rydberg atom per spot, but also let a Rydberg
atom block the excitation in the neighboring spots.
The spots imaged with the SLM are almost diffraction limited, defined by an RMS wavefront error
of λ/14. This was made possible by iterative correction of the aberrations of the optical system. The
iteration consists of the following steps: measuring the wavefront at the focal plane inside the MOT,
fitting the wavefront with Zernike polynomials and adding the phase of the fitted polynomials to
the image phase on the SLM. The RMS wavefront error after correction was measured to be: λ/6.8.
The aberrations correction was limited by the number of performed iterations and the resolution of
the detector.
All Rydberg experiments have also been simulated with the developed Kinetic Monte-Carlo method.
This method is in excellent agreement with the present data, it reproduces the same, within the
measurement uncertainty, blockade radius and correlation functions as the experiment, giving a fast
an accurate way to investigate the feasibility of different types of experiments.
The Mandel Q-factor has proven to be unsuitable, in the current setup, to show reliable blockade
effects in an ultra-cold trapped Rydberg gas due to its large susceptibility to external influences like
the detection efficiency, laser frequency fluctuations and excitation beam shape. The determination
of the Q-factor would benefit from a number of setup improvements, of which the Rydberg excitation
laser frequency stabilization is the most important.
The methods performed in this thesis could be used to measure many more phenomena in future
experiments. The Q-factor can be used to study time-dependent properties of the Rydberg excitation. And the correlation function can be used to study the formation of Rydberg crystals by
the presence of long-range interactions. The incorporation of the SLM in the Rydberg experiment
63
C. Ravensbergen
can provide a tool to influence many properties of the Rydberg crystals, like the lattice spacing,
geometry and dimension. And eventually it could be used to implement various proposed quantum
gates and single site addressing for quantum computation purposes.
64
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