Download Creating Strontium Rydberg Atoms

RICE UNIVERSITY
Creating Strontium Rydberg Atoms
by
Xinyue Zhang
A Thesis Submitted
in Partial Fulfillment of the
Requirements for the Degree
Master of Science
Approved, Thesis Committee:
F.B. Dunning, Advisor
Professor of Physics and Astronomy
T.C. Killian, Vice Advisor
Professor and Chair of Physics and
Astronomy
Douglas Natelson
Professor of Physics and Astronomy and
Professor in Electrical and Computer
Engineering
Houston, Texas
April, 2013
ABSTRACT
Creating Strontium Rydberg Atoms
by
Xinyue Zhang
Dipole-dipole interactions, the strongest, longest-range interactions possible between two neutral atoms, cannot be better manifested anywhere else than in a Rydberg atomic system. Rydberg atoms, having high principal quantum numbers n 1
and dipole moments that scale as n2 , provide a powerful tool to examine dipoledipole interactions. Therefore, we have studied the production and production rates
of strontium Rydberg atoms created using two-photon excitation and have explored
their properties in two distinct experiments. In the first experiment, very-high-n
(n ∼ 300) Rydberg atoms are produced in a tightly collimated atomic beam allowing
spectroscopic studies of their energy levels and their Stark effects. Simulations using a
two-active-electron model, developed by our theoretical collaborators, allow detailed
analysis of the results and are in remarkable agreement with the experimental results.
The high density of Rydberg atoms achieved, ∼ 5 × 105 cm−3 , in this experiment will
allow studies of strongly interacting Rydberg-Rydberg systems. The second experiment, in which a cold strontium Rydberg gas is excited in a magneto-optic trap,
features an imaging technique offering both spatial and temporal resolution. We use
this technique to observe and study the evolution of an ultra-cold strontium Rydberg gas which reveals the importance of Rydberg-Rydberg interactions in the early
stages of this evolution. A strongly interacting Rydberg gas provides an opportunity
iii
to realize a very strongly-correlated ultra-cold plasma.
Contents
Abstract
ii
List of Illustrations
vii
List of Tables
ix
1 Acknowledgment
1
2 Introduction
4
2.1
2.2
2.3
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.1.1
Manybody Physics . . . . . . . . . . . . . . . . . . . . . . . .
7
2.1.2
Photonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.1.3
Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.1.4
Ultracold Neutral Plasma . . . . . . . . . . . . . . . . . . . .
11
2.1.5
Detecting and imaging ultracold Rydberg atoms . . . . . . . .
12
The Strontium Rydberg System . . . . . . . . . . . . . . . . . . . . .
13
2.2.1
Strontium . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3 Theoretical Results and Background
18
3.1
Traditional Treatment of Two-electron System . . . . . . . . . . . . .
18
3.2
Two Electron model . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.3
Rydberg Atoms in a electric field . . . . . . . . . . . . . . . . . . . .
26
3.3.1
Classical Picture . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.3.2
Sr Stark Map . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
v
4 Experiment Setups and Techniques
4.1
35
Frequency Locked Diode Laser System . . . . . . . . . . . . . . . . .
35
4.1.1
Frequency Double High Power Diode Laser System[57] . . . .
36
4.1.2
HeNe Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
4.1.3
Scanning Fabry-Perot Interferometer . . . . . . . . . . . . . .
39
4.1.4
Data Acquisition System . . . . . . . . . . . . . . . . . . . . .
40
4.1.5
Locking Scheme . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.1.6
Limitations and Alternatives . . . . . . . . . . . . . . . . . . .
41
4.2
Strontium Oven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.3
Other Experimental Apparatus . . . . . . . . . . . . . . . . . . . . .
47
4.3.1
Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.3.2
Interaction Region . . . . . . . . . . . . . . . . . . . . . . . .
48
Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.4.1
Selective Field Ionization . . . . . . . . . . . . . . . . . . . . .
49
4.4.2
Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . .
52
4.4
5 Sr Rydberg Atoms in a Collimated Atomic Beam
5.1
53
Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
5.1.1
Even Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
5.1.2
the odd isotope
Sr . . . . . . . . . . . . . . . . . . . . . . .
58
5.1.3
Stray Fields Impact On Spectra . . . . . . . . . . . . . . . . .
62
87
6 Ultracold Rydberg Gas Evolution
66
6.1
Experimental Setup Overview . . . . . . . . . . . . . . . . . . . . . .
66
6.2
Principal Processes in Probing Sr Cold Rydberg Gas . . . . . . . . .
69
6.2.1
Autoionization . . . . . . . . . . . . . . . . . . . . . . . . . .
69
6.2.2
Electron-Rydberg Collisions . . . . . . . . . . . . . . . . . . .
72
6.2.3
Penning Ionization . . . . . . . . . . . . . . . . . . . . . . . .
74
6.2.4
Blackbody Radiation induced Ionization . . . . . . . . . . . .
75
vi
6.3
Imaging Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
6.4
Results Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
7 Conclusion and Outlook
Bibliography
85
87
Illustrations
2.1
Crossover to Collective Many-body States . . . . . . . . . . . . . . .
8
2.2
Ultracold Neutral Plasma Creation Setup . . . . . . . . . . . . . . . .
12
2.3
Natural Linewidths and Transition Wavelengths of Principle Sr Levels
14
2.4
Sr+ Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.5
Cooling Transitions towards Quantum Degeneracy . . . . . . . . . . .
16
3.1
Measured and calculated quantum defects in the single-electron
excitation of strontium . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.2
Calculated Excitation Spectra . . . . . . . . . . . . . . . . . . . . . .
25
3.3
Hydrogen Rydberg Atoms in a Electric Field
. . . . . . . . . . . . .
27
3.4
NonHydrogen Rydberg Atoms in a Electric Field . . . . . . . . . . .
28
3.5
Stark Map of Strontium Rydberg Atoms . . . . . . . . . . . . . . . .
32
3.6
Parabolic States Distribution of Strontium Stark States . . . . . . . .
34
4.1
TOPTICA Diode Laser System Schematics . . . . . . . . . . . . . . .
36
4.2
Vapor Pressure Chart . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4.3
Schematic diagram of the oven assembly . . . . . . . . . . . . . . . .
45
4.4
Interaction Region . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4.5
Stark Map for Sodium . . . . . . . . . . . . . . . . . . . . . . . . . .
50
5.1
Excitation Spectra For Sr Isotopes . . . . . . . . . . . . . . . . . . .
55
5.2
N ∼ 312 Spectra for Overlapping
86
59
Sr and
88
Sr . . . . . . . . . . . .
viii
5.3
N ∼ 335 spectra
5.4
Anticrossing of
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
Sr . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
5.5
Stay Field Limited High N Spectra . . . . . . . . . . . . . . . . . . .
65
6.1
Experiment Schematic, Diagram and Timing . . . . . . . . . . . . . .
67
6.2
Sheet Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
6.3
Autoionization
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
6.4
l mixing schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
6.5
Laser-induced fluorescence imaging . . . . . . . . . . . . . . . . . . .
76
6.6
Dependence of the LIF signal on Rydberg excitation time . . . . . . .
77
6.7
LIF images of spontaneous evolution . . . . . . . . . . . . . . . . . .
79
6.8
Visible Ions in Spontaneous Evolution . . . . . . . . . . . . . . . . . .
80
6.9
Collision Time Vs Initial Inter-Rydberg atoms Distance . . . . . . . .
83
87
Tables
2.1
Scaling Laws for Rydberg Atoms . . . . . . . . . . . . . . . . . . . .
5
2.2
Principal Isotopes of Strontium . . . . . . . . . . . . . . . . . . . . .
14
5.1
Sr Quantum Defects [65] . . . . . . . . . . . . . . . . . . . . . . . . .
58
1
Chapter 1
Acknowledgment
When I asked Dr. Barry Dunning if I had the honor to join his group three years ago,
I thought I had small chance to be admitted. I had clear idea of what I had to offer:
my B.S. was in geophysics, my English was terrible, my experimental experience was
zero and there were very limited number of equipments that I could lift in the lab.
Yet, he decided to give me a chance. Because of my poor background, Barry had to
painstakingly teach me the basic experimental skills and to make me useful in the
lab. Over the past years, there was not even once has he lost his patience or said one
harsh word to me for the various mistakes I made (everyone who has worked with me
should understand how difficult that was since I can be very bullheaded and cheeky
sometimes). On the contrary, he is always encouraging me, helping me to improve
and never giving up on me. I never ever thought it is possible for a person to be
so nice and kind before I knew Barry and I am so grateful everyday for having him
as my advisor. In addition to his guidance on experiments, he has made so much
contribution to the writing of this thesis. He revised it from structure, contents to
grammar and even punctuations for more than ten times and he managed to upgrade
it from a mindless talking to a professional, well-written thesis. For that, I couldn’t
thank him enough!
I feel very lucky to have Dr. Tom Killian as my second advisor who is also a
remarkable teacher and a very kind person. He spent so much time in teaching me
techniques on laser, fiber, optics and electronics from scratch with great patience. At
2
early times, he would generously devote many hours of his day in just helping me
with optical setup and alignments and making sure that I am not doing anything
foolish. Moreover, he is always trying his best to help me understand concepts in
atomic physics even if that means he has to explain one thing for quite a few times in
different perspectives. Just like Barry, he tolerated my ignorance, weak background,
my sometimes very annoying personalities and has been trying to build something
out of me. His efforts greatly helped me make through my research and are very
much appreciated. Both Barry and Tom are such dedicated scientists who have been
great role models for me to look up to. Having the opportunities to work with both
of them makes me smile in my dream.
All the people in Barry’s and Tom’s groups have also played important roles
in helping me complete this work and lightening up my every day at school. My
senior students, Shuzhen Ye and Patrick McQuillen, who suffered the worst of me
and yet they are still my great friends and teachers. Mi Yan, Brian DeSalvo and
Trevor Strickler can always put aside their work and happily discuss all my whimsy
questions. Changhao Wang, Yu Pu, Ying Huang and Micheal Kelley never said no
to help me. Francisco Caremo is always a delight to work with. My buddies here at
Rice including Ernie Yang, Yang-Zhi Zhou, Zhentao Wang, Sidong Lei, Alicia Chang,
Ksenia Bets, Jason Ball and many more offered me their friendships that I have always
cherished. Every each of these people made my life here simply lovely and I’d love to
express my gratitude for all of them.
At last, I would also like to thank Dr. Han Pu, Dr. Stan Dodds, Dr. Huey W.
Huang, Dr. Anthony Chan and Dr. Randy Hulet for their helps, trusts and insights
and I want to thank Dr. Douglas Natelson for keeping setting aside his time to make
my thesis defense possible. I’d like to end this part with my greatest appreciation to
3
my grandmother whom I will always love unconditionally.
4
Chapter 2
Introduction
Atoms in which one electron is excited to a state of large principal quantum number n,
termed Rydberg atoms, have been studied extensively because they possess extreme
physical characteristics unlike those normally associated with atoms in ground or lowlying excited states. This is illustrated in Table 2.1 which lists a number of atomic
properties, their n dependencies and their values for selected n levels of interest in
this work. Since the classical Bohr radius of an atom scales as n2 , Rydberg atoms
are physically very large and many of their properties can be described in terms
of the classical Bohr model of the atom. Their binding energies, which scale as
n−2 are very small. Because of their large size and weak binding, Rydberg atoms
can be strongly perturbed and even ionized by modest external electric fields, the
threshold for ionization scales as n−4 . The classical electron orbital period which
increases as n3 , is large allowing, for example, application of electric field pulses
whose duration is much smaller than the electron’s orbital period. At high n the
spacing between adjacent levels which decreases as n−3 becomes very small. The
radiative lifetimes of Rydberg atoms are also very large resulting in very narrow
spectral features. Table 2.1 also includes other atomic properties pertinent to the
present study including their polarizability, dipole moment and hr−4 i,hr−6 i (which
determine the relative contributions to the polarization energy Wpol from dipole and
quadrupole core polarization).
The dipole moments listed in the Table 2.1 are strictly speaking transition dipole
5
Table 2.1 : Scaling Laws for Rydberg Atoms
Property
Scaling
Na(10d)
n = 50d
n = 312d
(a.u.)
orbital radius
n2
147a0
0.368µm
14.3µm
orbital period
n3
0.15ps
18.8 ps
4.56 ns
binding energy
1/n2
0.14eV
5.6meV
0.14meV
energy spacing
1/n3
0.023 eV
0.18meV
0.75µeV
ionization field
1/n4
30 kV/cm
48V/cm
32 mV/cm
radiative lifetime
n3
1µs
125µs
30.4ms
dipole momenthnd|er|nf i
n2
143 ea0
3.58×103 ea0
139 × 103 ea0
polarizability MHz cm2 /V2
n7
0.21
1.64 × 1010
6.04 × 1015
hr−4 i
2n5 (l+3/2)(l+1)(l+1/2)l(l−1/2)
hr−6 i
35n4 −5n2 [6l(l+1)−5]+3(l+2)(l+1)l(l−1)
8n7 (l+5/2)(l+2)(l+3/2)(l+1)(l+1/2)l(l−1/2)(l−1)(l−3/2)
3n2 −l(l+1)
6
moments, not a permanent electric dipole moment. Atoms in zero field don’t have
permanent electric dipole moments! In a classical picture, high lm states have near
circular orbits, and a near zero net dipole moment. For low l states, the highly
elliptical orbits will precess around the nucleus due to core scattering resulting again
a vanishing net permanent dipole moment. However, there are a number of ways
to create Rydberg atoms with a large permanent dipole moment. One novel way,
as suggested in [36] is to create trilobite molecules in a Bose-Einstein Condensate
which represent a class of ultra-long-range homonuclear diatomic Rydberg molecules
that possess a permanent electric dipole moment in the order of kilodebye and some
experimental success in this direction has been achieved [37].
Another approach to create quasi-one-dimensional states is to selectively excite
extreme red-shifted Stark states in the presence of a DC field. The selection rules
only allow creation of low-l Rydberg states through photo-excitation. For the alkali
and alkaline-earth metals, such states are difficult to polarize due to core scattering
and initially only display quadratic Stark effects in a DC field, indicating a small
induced dipole moment. However at higher fields these states can mix with the
extreme strongly-polarized components of the Stark manifold and can themselves
become very polarized.
Upon obtaining a good quality quasi-one-dimensional (quasi-1D) state, it is straightforward to convert it to a near-circular, two-dimensional, “Bohr-like” state by application of an appropriate electric field pulse perpendicular to the atomic axis.
2.1
Motivation
In the present work we are developing techniques to create strontium Rydberg atoms
as a first step towards new projects involving strongly-coupled manybody systems
7
and the creation of long-lived two-electron excited states. Many of the proposed
experiments will take advantage of dipole blockade.
Rydberg blockade [1] is a manifestation of the strong, long range interaction between Rydberg atoms. This interaction could be of the Van der Waals type which
varies with the inter-particle distance R as C6 /R6 , whereas for Rydberg states with
permanent dipole moments the interaction will be of dipole-dipole type with form
C3 /R3 . Because of these interactions, excitation of one Rydberg atom shifts the energy levels of neighboring Rydberg atoms and prevents their excitations using the
same narrow linewidth laser. The resulting “dipole blockade” radii can be large
(∼ 5µm at n ∼ 50 and ∼ 100µm at n ∼ 300). Although initially proposed as a
means to create fast quantum gates for neutral atoms [2], Rydberg blockade has been
shown to be an extremely versatile tool with many applications in areas such as condensed matter physics, plasma physics, nonlinear optics and quantum information. In
the past decade, exciting progress has been made in each field that could be paradigm
changing in the future.
2.1.1
Manybody Physics
Rydberg atoms, blessed with their large electric dipole moments, interact strongly
permitting the simulation of a wide variety of condensed matter systems. Furthermore, as pointed out in early theoretical work [3], quantum information processing
could be based on the collective states of mesoscopic atomic ensembles due to Rydberg
dipole blockade effects.
Rydberg blockade effects have been the subject of a lot of experimental interest.
One of the first demonstrations [6] involved tuning an np state to the middle of the
adjacent ns and (n + 1)s states through Stark shifts induced by application of a DC
8
field. A 30% suppression of Rydberg excitation was observed at a Förster resonance.
Direct van der Waals blockade was also observed [7]. In 2009, two independent groups
demonstrated [9, 8] the collective excitation of two blockaded Rydberg atoms [10].
It was shown theoretically that full control of the strength, shape and character of
the interaction potential is possible by weakly dressing Rydberg atoms contained in
a Bose-Einstein Condensate [11]. In addition, by adjusting experimental parameters
like the detuning, it is experimentally feasible to crossover from two body interactions
to many body interactions; see Figure 2.1.
Figure 2.1 : Crossover to Collective Many-body States. (a) ground states |gi dressed
with Rydberg states |ri which are excited by a two-photon transition via the intermediate state |pi. The total detuning for this two-photon transition is ∆, ∆ = ∆p + ∆r .
(b) Diagram for the crossover from two-body to many-body interaction, Ω = Ωr + Ωp .
Figure adapted from [11].
The quest for exotic quantum phases can also be realized in a blockaded ultracold
Rydberg ensemble. Inspection of the experimental data revealed the existence of a
dimensionless parameter and an algebraic scaling law (characteristics of a second order phase transition) for an ultracold Rydberg gas. In other words, a frozen Rydberg
system can be employed to study phase transitions in a precise, controllable manner.
9
Thus the capabilities that have been developed to coherently engineer the interactions
in a many-body system, and the abilities to address and manipulate these “superatoms” individually due to their huge size, demonstrate the potential of Rydberg
systems as quantum simulators [16]. Here “superatom” is a term used to symbolize the large spherical volume formed by a Rydberg atom and all the consequently
blockaded ground state atoms within its dipole blockade radius.
The crystalline phase can be explored with Rydberg atoms. Dipole blockade was
proposed as a means to obtain dynamical crystallization through the use of a chirped
laser pulse [12], the Rydberg excitation number being predicted to display a staircase
structure. The mechanism is easy to understand (assuming the Rydberg interaction
is repulsive, i.e. C6 > 0): with the excitation laser initially red detuned from the
Rydberg atom transition, the collective many-body ground state for the ensemble
will be one in which every atom is in its one-body ground state, as in the system’s
Fock number state |0i. As the laser chirps towards resonance with the Rydberg level,
a single atom in the ensemble will be excited to a Rydberg state whereupon further
excitation will be prevented by dipole blockade, so the system jumps to number
state |1i. When the laser is blue detuned enough to compensate the smallest dipoledipole energy shift, which is that between the first Rydberg atom and the furthest
ground state atom, that particular ground atom will be excited, the system jumping
to number state |2i, . . . As chirping continues, ground state atoms, in well-ordered
positions will be excited one by one resulting in a crystalline structure comprising an
array of dipole blockade superatoms.
Creation of a supersolid, a novel phase simultaneously displaying crystal rigidity
and dissipation-less flow, has been an experimental challenge for decades and so far
has not been achieved. This peculiar phase requires a repulsive two-body potential
10
that softens at short distances and a long system lifetime to allow formation and
observation of this phase. Theoretically, both requirements can be met in an ultracold
atomic ensemble in the Rydberg blockade regime [14, 15]. The artificial potential can
be mimicked and controlled by subjecting the Rydberg atoms to a homogeneous
electric field in which Rydberg atoms possess large permanent dipole moments. Use
of an off-resonant two-photon transition to properly “Rydberg dress” the ground state
atoms can significantly reduce the photon scattering, thereby increasing the lifetime
of the system.
2.1.2
Photonics
Photons don’t interact with each other. Thus entanglement between photons does
not come naturally. So far, experimentalists have resorted to spontaneous parametric down-conversion to make pairs of entangled photons [17]. There has been a lot
experimental and theoretical work suggesting an effective mapping of the strong Rydberg interactions in the collective ultracold ensemble to photons. For instance, the
electromagnetically induced transparency (EIT ) obtained by driving transition to a
Rydberg level [18] is non-linearly influenced by the character of the Rydberg-Rydberg
interactions [19]. Recent work has also shown [20] the potential of Rydberg system
to applications like a fast single photon source, quantum teleportation, and the fast
entangling of spin waves. The photon retrieved from a superatom has instant second
order auto-correlation g (2) (0) as small as 0.075 and a single photon generation efficiency of 10%, not far from that of its well-developed quantum dot counterpart [21].
11
2.1.3
Quantum Gates
One of the very first proposed applications of Rydberg blockade was the implementation of quantum gates in an ultracold neutral gas [2]. The strong, long-range
interaction between Rydberg atoms can be coherently turned on and off in a short
time which will result in very fast, high fidelity quantum gates. Two advantages
of quantum gates using neutral atoms are their straightforward scalability to create
multi-qubit registers and their weak coupling to external field noises. The controlledZ gate and CNOT gate which form a complete set of universal gates for quantum
computing have already been realized in the neutral system [22].
2.1.4
Ultracold Neutral Plasma
Ultracold neutral plasmas in which the Coulomb potential energy of interaction between its constituents ECoulomb , are greater than their thermal energies ET hermal ,
represent an exciting new frontier in plasma physics. Such plasmas can be created by
near-threshold photo-ionization of atoms contained in a cold cloud (see Figure 2.2).
The ionized photo-electrons, which have energies ∼ 1K, begin to escape the cloud and
leave their ion cores behind. This process quickly terminates as the cloud builds up
a net positive charge creating a Coulomb potential well from which further electrons
cannot escape. Photoionization, however, produces ions distributed throughout the
cold atom cloud, and the ions are therefore disordered. As the ions relax to a more
ordered state, they are heated on a timescale of 100 ns, resulting in disorder induced
heating (DIH) and ion temperatures of a few Kelvins rather than the mK temperatures characteristic of the parent laser-cooled neutral atoms. The plasma coupling
parameter τ = ECoulomb /EKinetic is thus dramatically reduced [34]. This can be mitigated by exploiting dipole blockade to create an ordered cloud of Rydberg atoms
12
Figure 2.2 : Ultracold Neutral Plasma Creation Setup Figure adapted from [33].
and then ionizing these through pulsed electric field ionization [32]. This will allow
creation of plasmas with much larger and controllable values of τ and the exploration
of a new plasma physics regime.
2.1.5
Detecting and imaging ultracold Rydberg atoms
Over the past few years, two major techniques have been employed to image ultracold
Rydberg atoms. The first exploits traditional electric field induced ionization. To
obtain an image, a position sensitive multi-channel plate detector (M CP ) is generally
required [23]. Higher resolution can be achieved by using Field Ion Microscopy as
in [24]. The magneto-optical trap (M OT ) is located at the center of a closed cage
made of 10 independent electrodes that are used to minimize stray fields. The imaging
electrode tip (125µm in radius) projects the field ionized Rydberg atoms onto the
MCP detector. By accumulating many images and studying their auto-correlation,
Rydberg blockade can be seen.
A second approach is optical in situ fluorescence imaging of the Rydberg atoms
by stimulated deexcitation and has produced the first observation of the spatially
13
ordered components of the Rydberg-blockade-induced many-body states that formed
inside of a mesoscopic system [25]. The images’ temporal and spatial resolution is
unprecedented.
2.2
The Strontium Rydberg System
Since strontium Rydberg atoms are the focus of the present work, their particular
properties are now discussed.
2.2.1
Strontium
Strontium, an alkaline earth metal with two valence electrons possessing both singlet and triplet levels, has been the subject of numerous studies in the literature.
The singlet 1 S0 ground state makes it immune from magnetic Zeeman splitting. In
addition, strontium has a wealth of bosonic (88 Sr,86 Sr,84 Sr) and fermionic (87 Sr)
isotopes. The relative natural abundances of these isotopes are listed in Table 2.2
together with their nuclear spins (only the
87
Sr isotope has a nuclear spin) and the
isotope shifts for the principal 5s21 S0 → 5s6p1 P1 transition. Transition wavelengths
and natural linewidths for transitions between its lowest lying levels are shown in
Figure 2.3. Excitation to a high-l Rydberg state leaves an optically active core ion
that behaves much as an independent ion. The energy level structure of the core ion is
shown in Figure 2.4. Absorption/Fluorescence on the 2 S1/2 → 2 P1/2 , 2 P3/2 transition
can then be used to image and manipulate strontium Rydberg atoms.
The properties of strontium highlighted above have enabled a number of interesting experiments that are outlined below.
14
Figure 2.3 : Natural Linewidths and Transition Wavelengths of Principle Sr Levels.
Intercombination lines are 1 S0 → 3 P . Figure adapted from [31].
Table 2.2 : Principal Isotopes of Strontium
Isotope
Atomic
Natural
I
Mass
Abundance(%)
F
1
S 0 → 1 P1
Scattering
Shift(MHz) Length(a0 )
84
Sr
83.913
0.56
0
-
-270.8
122.7
86
Sr
85.909
9.86
0
-
-124.5
823
7/2
-9.7
9/2
-68.9
11/2
-51.9
-
0
87
88
Sr
Sr
86.908
87.905
7.00
82.58
9/2
0
96.2
-1.4
15
Figure 2.4 : Sr+ Transitions
Frequency Standards Due to hyperfine mixing, the strongly forbidden transition
(∆s 6= 0, ∆j = 0), 5s2 1 S0 → 5s5p 3 P0 is weakly allowed for
87
Sr. The natural
linewidth of this transition is only 1mHz allowing its use as an atom frequency standard [40]. The atoms must be cooled to µK to reduce the Doppler shifts and to allow
trapping of a large number of atoms in an optical lattice. Trapping atoms in the
antinodes of the lattice results in an ac Stark shift on the clock transition. However,
a magic wavelength exists [39] for cancellation of the upper and lower Stark shifts and
results in a negligible light shift. Currently, the strontium optical lattice clock is the
best optical atomic frequency standard and has been used to measure fundamental
constants.
Quantum Degenerate Gases For the spinless strontium singlet ground state 1 S0 ,
the traditional technique of evaporative cooling in a magnetic trap is no longer applicable. Nonetheless, quantum degeneracy has already been achieved using all the
16
Figure 2.5 : Cooling Transitions towards Quantum Degeneracy. Solid lines are
driven by lasers, dashed lines are the spontaneous decay path. Figure adapted from
[30].
principle isotopes of strontium [27, 28, 29]. The procedures employed are similar and all-optical [27] and can be understood by reference to Figure 2.5. A blue
laser (461nm) red detuned from the 1 S0 → 1 P1 is used to Zeeman slow and twodimensionally collimate the atomic beam. Atoms are then further cooled in a 461nm
MOT. With repeated cycling, some atoms start to accumulate in 3 P2 level (through
path(5s5p)1 P1 → (5s4d)1 D2 → (5s5p)3 P2 transitions ). When sufficient atoms have
been trapped, a 3-micron laser pulse is used to transfer the 3 P2 atoms back to ground
state via the transitions (5s5p)3 P2 → (5s4d)3 D2 → (5s5p)3 P1 → (5s2 )1 S0 . The
461nm blue MOT is then extinguished, and a red MOT operating on the 1 S0 → 3 P1
transition is turned on to further cool the atoms prior to loading into an 1.06µm
optical dipole trap (ODT). The atoms are then further cooled by lowering the trap
depth, evaporative cooling resulting in degeneracy.
17
2.3
Thesis Outline
The main part of this thesis will focus on the results of two recent experiments [35, 50]
designed to study the excitation of very-high-n (n ∼ 300) Rydberg states and to
explore the evolution of cold Rydberg gases towards an UNP by imaging the core
ions.
18
Chapter 3
Theoretical Results and Background
3.1
Traditional Treatment of Two-electron System
Compared with the simple hydrogen atom, alkali Rydberg atoms are more complicated in the sense that the closed-shell-core can be penetrated and polarized. The
resulting effects can be well characterized by one l-dependent quantum defect δl .
However, for alkaline-earth elements, things are far more complicated due to the interaction of the two valence electrons. Even though one of the electrons is promoted
to a Rydberg level, the strong short-range scattering with the one-active-electron core
leads to strong configuration mixing. For each term S L, this interaction among configurations can be described in a set of parameters in Multichannel Quantum Defect
Theory (MQDT) [49, 48].
When two electrons are close r12 < r0 , they can exchange angular momentum, spin
and energy via their Coulomb interaction 1/r12 without violating their overall conservation. In this regime of free-energy-exchange, a proper set of basis wavefunctions
can only be obtained by diagonalizing a scattering matrix S. This yields a set of Φα
“eigenchannels” with eigenvalues µα . These eigenchannels are formed from a mixture
of different configurations. Energy exchange becomes negligible once r12 > r0 due to
the diminishing overlap of the wavefunctions of the two electrons and the falloff of
their interaction 1/r12 . The outer electron can then be described as a superposition
of collision channels. Each eigen-collision-channel is then a pure configuration labeled
19
by quantum number νi ,
νi =
p
R/(Ii − E),
(3.1)
where Ii is the ionization limit of the ion core in this collision channel, R is the mass
corrected Rydberg constant and E is the energy of the system(νi can be regarded as
the quantum defect for the ith channel).The eigenenergies of the system for any r12 are
found by connecting the two sets of wavefunctions in regimes r12 < r0 and r12 > r0
via a transformation matrix Uiα and applying appropriate boundary conditions at
infinity. The nontrivial solution requires
Det|Uiα sin π(νi + µα )| = 0.
(3.2)
The bound eigenenergies of the system are found by adjusting the µα and Uiα that
simultaneously satisfy Equation 3.1 and Equation 3.2 until agreement with experimental data is reached. This method can also determine the admixtures of the
different configurations. For example, for Sr J=2 bound states, it has been shown
that the most important channels are 5snd1 D2 , 5snd 3 D2 , 4dns 1 D2 , 4dns 3 D2 , and
5pnp 1 D2 . For the 5s15d 1 D2 state, there is almost a 40% admixture from 5snd 3 D2 .
The semi-empirical techniques of MQDT have been very successful and very widely
used since they encapsulate the complex spectra, and configuration interactions, into
a number of parameters. They have also motivated the search for ab initio methods
to calculate short range scattering. There has also been great success in combining
the eigenchannel R-matrix method with MQDT. The R-matrix method is a way to
variationally calculate the set of eigenchannels Φα inside of a volume r < r0 in a
given configuration space. Essentially, this ab initio method requires solving the
time-independent Shrödinger equation with trial wavefunctions.
To construct a proper set of trial wavefunctions, the foremost thing is to find the
20
appropriate Hamiltonian
H=−
∆21 ∆22
1
−
+ V (r1 ) + V (r2 ) +
.
2
2
r12
(3.3)
In the above Hamiltonian, V (r) is not known. Nevertheless it’s not hard to imagine
this potential should be some kind of l-dependent screening potential. Since a lot of
orbitals are extremely sensitive to this potential, there has been a lot of work trying
to optimize this model potential for different alkaline earth elements. It has been
shown that by using the optimized potentials, accurate spectra can be obtained. One
optimized model potential for strontium is the following,
1
V (r) = − {2 + (Z − 2)exp(−α1l r) + α2l rexp(−α3l r)},
r
α1 = 3.551, α2 = 6.037, α3 = 1.439.
In the Section 3.2, another original ab initio method to calculate strontium Rydberg spectra will be presented. It also employs an l-dependent model potential.
However, while it is not an R-matrix method as described above, it does give accurate spectra that match to our experimental results. This method was developed to
help analyze our experimental results by our collaborators in Vienna.
3.2
Two Electron model
To analyze the excitation spectra of strontium, we employ a two-active-electron
model. The Hamiltonian is written as
H=
p21 p22
1
+
+ Vl (r1 ) + Vl (r2 ) +
2
2
|~r1 − ~r2 |
(3.4)
As we are mainly interested in single-electron excitation, it is practical to reduce
the number of configurations so that the eigenenergies can be evaluated efficiently
21
by numerically diagonalizing the Hamiltonian. The basis states are constructed from
the excited states of the Sr+ ion
H=
p2
+ Vl (r)
2
(3.5)
ion
|φni ,li ,mi i
Hion |φni ,li ,mi i = Enlm
The eigenstates hφni ,li ,mi | and the eigenenergies Eni ,li ,mi can be obtained numerically using the generalized pseudo-spectral method. The generalized pesudospectral
method [60] is a numerical procedure for solving equations such as equation 3.5. It
used for optimal grid discretization of the radial coordinates and is especially well
suited for problems involving a Coulomb singularity. It requires a smaller number of
grid points yet provides higher accuracy. It also introduces a split-operator technique
in the energy representation that allows the wavefunctions to propagate efficiently
in time (It has been widely applied in Floquet studies of atomic processes in strong
fields.). The calculated energies agree quite well with those measured for the ion.
The matrix elements of the two-electron Hamiltonian 3.4 are evaluated using the
basis states defined by
|n1 l1 n2 l2 ; LM i =
X
m1 +m2 =M
[
C(l1 , m1 ; l2 , m2 ; L, M ) |φn1 ,l1 ,m1 i |φn2 ,l2 ,m2 i
p
2(1 + δn1 ,n2 δl1 ,l2 δm1 ,m2 )
±
C(l2 , m2 ; l1 , m1 ; L, M ) |φn2 ,l2 ,m2 i |φn1 ,l1 ,m1 i
p
] (3.6)
2(1 + δn1 ,n2 δl1 ,l2 δm1 ,m2 )
where L is the total angular momentum, M is a projection, and the ClebschGordan coefficients are given by
22


L 
√
 l1 l2
C(l1 , m1 ; l2 , m2 ; L, M ) = (−1)−l1 +l2 −M 2L + 1 

m1 m2 −M
(3.7)
The basis states symmetric (antisymmetric) with respect to the exchange of two
electrons are used to calculate the eigenenergies in the singlet (triplet) sector. For
singlet excitation spectra the quantum numbers (n1 , l1 ) of the outer electron may
vary over the range of the whole excitation spectrum but those (n2 , l2 ) of the inner
electron can be limited to near the ground state. Using such a truncated basis set, the
eigenvalues of the active-two-electron system can be evaluated. Since the principal
quantum numbers n1 , n2 of the basis describe the excited states of Sr+ ion and not
those of neutral strontium, the correct quantum number n of the Rydberg electron has
to be assigned to the calculated eigenstates of the two interacting electron according
to the known excitation series ( including perturber states ) in each L sector, i.e.
|nLM i =
XX
cn1 ,n2 ,l1 ,l2 hn1 l1 n2 l2 ; LM |
(3.8)
n1 ,l1 n2 ,l2
This is not straightforward as some states are hard to identify. For example, there
is a 4d5p state in the 1 P1 sector, yet, the calculation shows no state with dominant
4d5p character. For strontium only few perturbers affect the Rydberg series for single
electron excitation. Since they have relatively small energy, the highly excited states
are not directly affected. For the singlet sector the quantum defects of singly-excited
low-L states are plotted in Figure 3.1. The calculated results, which includes the
6 configurations (5s, 4d, 5p, 6s, 5d, and 6p ) for the inner electron, are compared
with the previous studies based on MQDT (Quantum defects for L > 3 are negligibly
small ). The calculations agree well with the measured results, which is expected
as the model potential employed is known to yield the correct quantum defect using
23
Figure 3.1 : Measured and calculated quantum defects in the single-electron excitation
of strontium (singlet). A two-active electron model is used with 6 configurations of
the inner electron. Measured results are marked by circles.
R-matrix theory. Only a small disagreement is seen for the P-and D-states where the
calculated values slightly underestimate the measured quantum defects. We also note
that, as seen in Figure 3.1, the quantum defects slowly increase with the principal
quantum number n especially for P- and D-states. The eigenstates for highly excited
states have contributions from the inner electron that are almost exclusively from the
5s state. Even a very small overlap with the other inner electron configurations shifts
the phase of the wave function near the origin greatly affecting the quantum defect.
The numerical method can be tested by comparing the zero-field excitation spec-
24
trum with the measured data (Figure 3.2). The measured spectrum is taken at
n ∼ 280 and the calculations at lower n, n ∼ 50 and n ∼ 30. To compare the
spectra of two different n the frequency axis is scaled so that the energy difference
between two adjacent levels (n and n − 1) becomes invariant for different values of n.
The calculated spectrum is derived from the dipole transition | h5s5p| z |5snli |2 and
convoluted with a Gaussian to match the measured linewidth. The positions of the
n 1 S0 states relative to the two adjacent degenerate n levels are observed to be invariant as the quantum defects of n 1 S0 states are nearly n-independent. On the other
hand, the peak positions of the n 1 D2 states vary with n mirroring the n-dependent
quantum defect. For example, the quantum defect is δd ≈ 2.31 for n = 50 and that
extrapolated for the limit of n → ∞ is δd = 2.38. Another interesting observation
is that the relative intensity of the n 1 D2 state to the (n + 1) 1 S0 state increases
with n. The excitation strength is sensitive to the quantum defect as it phase-shifts
the wave function near the origin and modifies the overlap with the 5s5p state. In
this case, the quantum defect around δd = 2.38 appears to maximize the relative
intensity and be suppressed away from it. This suggests that the relative intensity
can be used to confirm the size of the quantum defect. We note that the underestimate of the calculated quantum defect for 32d overemphasizes this effect slightly.
Calculations using a single-active-electron model with a model potential similar to
that described in [61] have also been performed. In this model, a single electron is
moving in a model potential that is numerically fit from known and extrapolated
quantum defects. The eigenenergies as well as quantum defects can be obtained quite
accurately while the oscillator strength fails to reproduce the measured spectra due
to an inaccurate description of 5s5p state.
25
Figure 3.2 : Comparison between measured (a) and calculated (b, c) excitation spectra
in zero field. (a) Measured excitation spectrum recorded at n ∼ 283. Results of twoelectron calculations at n ∼ 50(b) and n ∼ 30(c) employing six inner electron states
(4s, 4d, 5p, 6s, 5d, and 6p). The energy axis is scaled such that E0 = 1 corresponds
to the energy difference between neighboring n and n − 1 manifolds
26
3.3
3.3.1
Rydberg Atoms in a electric field
Classical Picture
As n becomes very large, the quantum mechanical behavior of the excited electron in
a Rydberg atom can be described by the classical Bohr theory. In a hydrogen Rydberg
atom, the electron follows an elliptical orbit that is given by r = L2 /(1 + ε cos θ) in
~ = ~r ×~p is the angular momentum.
polar coordinates, where ε is the eccentricity and L
The hydrogen atom is a special case because the energy levels are highly degenerate
in l and m which is a manifestation of the 1/r character of the Coulomb potential.
Correspondingly, in the classical picture, hydrogen Rydberg atoms have one more
~ = p~ × L
~ − r̂. In atomic
physical quantity that is conserved, the Runge-Lenz vector A
units, the magnitude of the Runge-Lenz vector is the eccentricity ε. On the other
hand, alkali or alkaline earth atoms, do not have this “accidental degeneracy” due to
core penetration and polarization. Their energies, characterized by E = −1/2(n−δl )2 ,
can be viewed as perturbed by the core. As a result, their Keplerian elliptical orbits
will precess about the nucleus just as Mercury’s perihelion precesses about the Sun.
For non-penetrating cases, the frequency of this precession is ∼
5
δ.
n3 l l
The differences between hydrogen and non-hydrogenic Rydberg atoms are magnified in a electric field. For hydrogen, quantum mechanically, application of degenerate
time-independent perturbation theory will lead to the linear Stark effect. The Stark
map of hydrogen is simply like a fan; see Figure 3.3. The Shrödinger equation can
be solved analytically in parabolic coordinates for a hydrogen atom in a electric field.
The parabolic eigenstates are labeled by n, n1 , n2 , m and these quantum numbers are
related by n = n1 + n2 + |m| + 1. As shown in Figures 3.3, the dipole moments
are the largest for the extreme Stark states which have parabolic quantum numbers
27
Figure 3.3 : Hydrogen Rydberg Atoms in a Electric Field
Left: the fanlike
Stark map of hydrogen showing linear Stark shifts for |m| = 0 states. Every cluster
of l states is often called√a hydrogenic manifold; adapted from [56]. The classical
ionization limit Wc = −2 E is shown by a heavy curve where the Stark states begin
to be broadened by field ionization. Quasi-discrete states with lifetime τ > 10−6 s
(solid line), field broadened states 5 × 10−10 s < τ < 5 × 10−6 s (bold line), and field
ionized states τ < 5 × 10−10 s (broken line). Right: The charge density distribution
of hydrogen atoms in a electric field |m| = 0. Each figure is a parabolic eigenstate
which is a superposition of many l states of hydrogen. Moving from the left to
right, top to bottom, these figures are designated by the parabolic quantum numbers
k = n1 − n2 = 7 to −7 which are the extreme blue components to the extreme red
components; figure adapted from [62].
28
Figure 3.4 : NonHydrogen Rydberg Atoms in a Electric Field
Left: Precession of a nearly Keplerian elliptical orbit of a Rydberg electron about the core
ion in an electric field. The precession is produce by adding an induced dipole term
−αd /2r4 to the Coulomb potential which corresponds to the effects of polarization
induced in the core. The top figure is with a negative α while the bottom one is
with a positive α; figure adapted from [63]. Right: The Stark map for potassium
|m| = 0 states, the anti-crossings that appear near level intersections are obvious.
Figure adapted from [56].
29
k = n1 − n2 ∼ n. Also because of the charge distribution, it is easier to ionize the
electron in the extreme red state (in the last subfigure).
For non-hydrogenic Rydberg atoms, the l-degeneracy is lifted by the interaction
with the core. For low-l states non-degenerate time-independent perturbation theory
results in a quadratic Stark shift. Though the time-average of the precession diminishes the existence of a permanent dipole moment in zero field, a small dipole moment
can be induced by, and interact with, the electric field applied. This behavior can
be visualized as a non-uniform precession of the Keplerian orbital; see Figure 3.4.
However, this effect is only apparent for the low-l states since the interaction with the
core falls off quickly with increasing l. The high l states are still essentially degenerate, so in the Stark map, display a near linear Stark shift. One subtle difference,
compared with hydrogen Rydberg atoms, is the anti-crossings that appear between
different Stark states as the electric field is increased. In the following subsection, a
calculated description of the behavior of Sr Rydberg atoms in an electric field will be
presented together with the experimental measurements.
3.3.2
Sr Stark Map
Figure 3.5 shows the calculated eigenenergies for singly-excited strontium (n ' 50)
states as a function of the strength, Fdc , of a dc field applied along the z axis. The
high-l states which are nearly degenerate at Fdc = 0 exhibit a linear Stark shift and
Stark states of two adjacent n manifolds first cross at a field strength of
Fcross '
1
.
3n5
(3.9)
As explained previously, for the low angular momentum (1 P1 , 1 D2 ) states only the
quadratic Stark shift can be observed. In Figure 3.5 the measured excitation spectra
30
of strontium around n = 310 are also plotted. In these measurements orthogonal
polarizations of the 461 nm and 413 nm were used to avoid excitation of the 1 S0 states
and simplify the excitation spectrum (The dc field is parallel to the polarization of
the 461 nm laser). This setup yields Rydberg states with the total magnetic quantum
number M = ±1. In order to compare the spectra for different values of n, the energy
axis is scaled by En − En−1 ' n3 and the field axis is by Fcross . Using two-photon
excitation, only n1 D2 states can be excited at Fdc = 0. With increasing strength
of the dc field, the nD states become coupled with other angular momentum states
and these l-mixed states have smaller oscillator strengths than the unperturbed Dstates. As the state merges with the linear Stark manifold, the l-mixing is so strong
that the effect of the core scattering becomes negligible. Thus the state can become
strongly polarized and almost indistinguishable from the extreme red-shifted strongly
polarized Stark states. The behavior of the “312D” level mirrors that observed in
earlier studies [61] at lower n, n ∼ 80 which data are also included in Figure 3.5.
Slight shifts of the energy levels seen in the calculated 52P and 52D states are due
to the underestimated quantum defects. This evolution of the n1 D2 states can be
visualized by plotting the distribution of the parabolic quantum number k
ρ(k) =
X
|H hn, k, m| |nStark iSr |2
(3.10)
n
or, equivalently, the distribution of Az (Az is the z-component of the Runge-Lenz
vector) as k corresponds to the quantized action of −nAz . Here, |n, k, miH are the
parabolic states of the hydrogen atom and |nStark i is the outer electron state for an
eigenstate of strontium in a dc field (The inner valence electron is almost exclusively
in the 5s state). Figure 3.6 displays the evolution of the k-distributions as a function
of Fdc for the state which is the 52D state (M = 1) at Fdc = 0. For weak fields,
the k-distribution spreads over a wide range between −n and n. This indicates that
31
the state is unpolarized. Since the Runge-Lenz vector indicates the orientation of
the Kepler ellipse in classical dynamics, a wide distribution of Az implies an ensemble of Kepler ellipses (with l ∼ 2) whose orientations are broadly distributed. For
non-hydrogenic atoms, such a distribution is formed by core scattering which changes
the orientation of the ellipse while keeping the eccentricity. A node near k = 0 is
m=1
also noticeable in the plot which mirrors a node of the spherical harmonic Yl=1
.
With increasing Fdc the node is shifting towards the negative k side and the biased kdistribution indicates that state is becoming increasingly polarized. Near the merging
with the neighboring Stark manifold the k-distribution becomes very narrow indicating a convergence towards a single parabolic state. The 52P state, on the other hand,
does not show any hints of polarization. This is because its dipole-coupled partners,
S- and D-states, are also hard to polarize. The 52D state is dipole coupled to the
50F state which merges with the Stark manifold at relatively weak Fdc and becomes
polarized. The polarization of the 52D state is, therefore, caused by the coupling
with this polarized “50F” state.
The evolution of the calculated dipole moment, hz1 + z2 i, of several eigenstates
around the “52D-state” is shown in Figure 3.6. The states nearly degenerate at Fdc
= 0 becomes polarized even for very weak fields and dipole moments are given by
h(z1 + z2 )i = (3/2)nk. For the isolated low-L states, the dipole moment grows almost
linearly in Fdc , i.e.
hz1 + z2 i = −αFdc
(3.11)
2
when Fdc ' 0. These linear shifts lead to the energy shift ∆E = −(1/2)αFdc
quadratic
in Fdc . The polarizability α can be approximated using second-order perturbation
32
Figure 3.5 : Stark Map of Strontium Rydberg Atoms Evolution of the excitation spectrum with increasing applied dc field in the vicinity of n ∼ 310( thick
red line). The thin solid lines indicate the calculated eigenenergies of singly-excited
strontium (n ' 50) in a dc field while the dashed blue lines denote the corresponding
excitation spectrum. The squares are the results of earlier measurements at lower n,
n ∼ 80 from [61]. Fdc is normalized to the crossing field strength Fcross ∼ 1/(3n5 )
and the energy is normalized by En − En−1 ' 1.
33
theory as
α=2
X X | hnLM | (z1 + z2 ) |n0 L0 M i |2
.
En0 L0 M − EnLM
n0 L0 =L±1
(3.12)
Numerical calculations show that the polarizability α is dominated by a single term
in the summation for the 50F state as the dipole-coupled state (50G) is almost degenerate in energy due to its almost vanishing quantum defect. The resulting large
polarizability leads to sizable energy shifts. For the 52P state, similarly, the coupling
to the 52D state dominates the summation in Equation 3.12. However, the large
energy difference (see Figure 3.5 ) due to the quantum defect suppresses the values
of α and, therefore, the state is hardly polarized. The 52D state is found between
two dipole coupled states, 52P and 50F, and is slightly closer to the 50F. Therefore,
the coupling with the 50F dominates over that with 52P leading to the polarization
towards the downhill side resulting in a larger polarizability than that for the 52P
state. We note that, judging from the quantum defect, the (n + 2)D state is found
slightly closer to the midpoint between the (n + 2)P state and nF state for higher
values of n. In fact, the polarizability of the D state appears to be smaller for the
312D state as the measured spectrum (Figure 3.5) shows a smaller energy shift than
that of the calculation for n = 50. With increasing field the growth of the dipole moment becomes non-linear in Fdc , implying the non-negligible role of the higher-order
perturbation terms, i.e., strong mixing with higher L states. Thus the states can become polarized through a superposition with those high-L states and, as seen for 50F
and 52D, their polarizations approach the maximum value of hz1 + z2 i = 1.5n2 a.u..
34
Figure 3.6 : Parabolic State Distribution of Strontium Stark States Left
hand panels show the probability distribution of the parabolic quantum number k(=
−nAz ) as a function of the dc field strength Fdc . The evolution of the states which
are, at Fdc = 0, the 52D state, 52P state and 50F state are plotted. The distribution
for the 52P state is truncated where it merges into a Stark manifold. On the right
hand side, the average dipole moment of selected states including 52D, 52P, 50F as
well as the downhill and uphill Stark states are plotted. Fdc is normalized to the
crossing field strength Fcross .
35
Chapter 4
Experiment Setups and Techniques
For very-high-n strontium Rydberg atom creation in a thermal beam, most of the
equipment is the same as that employed in previous, successful Rydberg experiments
on potassium which jump-started this experimental exploration of strontium. This
left us with only two major new construction projects, the laser system and the strontium oven. Therefore this chapter will concentrate on these new pieces of apparatus
and only make short comments on its other components. Finally, the experimental
techniques employed will be summarized.
4.1
Frequency Locked Diode Laser System
We use two Frequency Doubled High Power Diode Laser Systems from TOPTICA
PHOTONICS to drive the two-photon Rydberg atom excitation. In our applications,
both of them are required to be locked on a specific frequency with MHz accuracy
for 8 hours continuously. This is accomplished by locking them with respect to a
commercial frequency stabilized Helium-Neon laser via a scanning Fabry-Perot interferometer. Their absolute wavelengths are determined using a commercial high
resolution wavemeter which produces GHz accuracy. In the following, the major
components of the locking system will be described as an introduction to explaining the locking scheme later. In the last subsection, system limitations and a few
alternative locking schemes that might provide better performance will be discussed.
36
Figure 4.1 : TOPTICA Diode Laser System Schematics adapted from toptica.com
4.1.1
Frequency Double High Power Diode Laser System[57]
This tunable diode laser system is very compact and rugged. Its narrow linewidth
(MHz over millisecond timescales), high and stable output power (100mW after
doubling) and high tunability makes it perfectly suited for our applications. The
whole laser consists of an diode laser system coupled to an electronic control system.
The electronic control system contains plug-in modules for the DC and HV power, the
diode current, temperature control, the crystal temperature control, laser modulation
37
and regulation, and external interfaces in a 19” unit.
As shown in Figure 4.1, the laser source is a grating stabilized external cavity
diode laser system based on the Littrow-Hänsch scheme. With a cavity formed by
the front facet of the diode and a holographic optical grating next to the rear facet of
the diode, this scheme offers a much smaller linewidth (1M Hz) as compared with a
bare diode(100M Hz). Besides the wavelength can be tuned easily over a large range.
Coarse tuning is achieved by adjusting the angle of the grating via a micrometer
screw and fine tuning is obtained by scanning the cavity length via the piezo element
attached to the grating. Mode hop free tuning is achieved by feedforwarding a current
proportional to the scanning voltage to the diode. To maintain single mode operation,
both the temperature and current of the diode head need to be adjusted together with
the grating.
Upon leaving the master laser diode, the collimated infrared laser beam( 40mW
approximately) is focused, mode matched into another diode, the tapered amplifier
(T A) to achieve more power than is possible with a single-mode laser diode. The gain
bandwidth of the tapered amplifier is usually of order of some tens of nanometers.
Due to the diodes’ sensitivity to feedback, high-suppression-ratio optical isolators are
integrated in the optical path to avoid reflection. Finally the output from the TA
(300mW )is mode matched to couple it into the bow-tie-ring resonator for frequency
doubling. Phase matching of the crystal is sensitive to both the alignment and the
temperature. Second harmonic output powers of 100mW are obtained.
The stabilization of the doubling cavity is achieved mainly via two feedback
loops. Their common error signal is generated by applying the Pound-Drever-Hall
scheme [58]. An RF modulation fed into the current of the master laser head produces two sidebands which, together with the carrier are all sent to the doubling
38
cavity. Their reflection from the cavity is collected by a fast photodiode (shown in
Figure 4.1) whose output is then fed into the PDD 110 module (the Pound-Dever-Hall
Detector). PDD mixes this signal with the same RF local oscillator used before to
extract their DC phase information which is the output error signal. The error signal
is then fed into two loops. The slower feedback loop (a few kHz) is closed by the
PID regulator stabilizing the cavity length with the piezo element attached to one of
the mirrors in the doubling cavity. The fast loop (5MHz) is closed through adjusting
the master diode’s current to lock the laser frequency to the doubling cavity. In the
fast loop, the doubling cavity is the reference cavity. The combination of these two
loops gets rid of thermal and acoustic noise as well as fast frequency jitter and can
maintain a narrow linewidth.
The electronic plug-in modules can also communicate via the backplane of the
rack if the jumpers are set accordingly. The feedforward function, for instance, is
achieved by sending a portion of the scanning voltage to the current control module
of the diode head(DCC). Thus this voltage/current is added to the current setvalue on the DCC and output to the diode head. The external input of the SC 110
module, which controls scanning of the grating of the master laser diode, is connected
to the same line as the BNC input/output of the computer analog interface DCB.
Therefore, the input from the BNC connector to the DCB module will be added to
whatever voltage is set on the SC module and then output to the grating in the master
laser diode. This particular feature will be used in our locking scheme discussed later.
4.1.2
HeNe Laser
Frequency/Intensity stabilized HeNe reference laser at 632.816nm is a commercially
available unit. Frequency stabilization is attained by comparing the intensity balance
39
of two orthogonally polarized longitudinal modes and can offer a ±2MHz frequency
stability on an eight hour timescale. The laser is very sensitive to any retroreflections.
Given the lack of an optical isolator, we use a 20dB neutral density filter to block
reflections from the Fabry-Perot Interferometer (will be discussed later). In addition,
the interferometer must be intentionally misaligned slightly to prevent reflections back
to the HeNe laser.
4.1.3
Scanning Fabry-Perot Interferometer
The finesse of a Fabry-Perot Interferometer, a measure of the interferometer’s ability
to resolve spectral features, is essentially determined by the reflectivity of the two
mirrors on each end of the cavity F = nR/(1−R2 ). Thorlabs offers scanning confocal
Fabry-Perot Interferometers (etalons) but not with optical coatings suitable for our
needs. In order to achieve a finesse F = 200 for 633nm, 412nm and 461nm laser
light, we ordered an SA200 with customized coatings on both (confocal) mirrors.
The free spectral range(FSR) of this etalon (≈ c/(4L),where L is the mirror spacing)
is 1.5GHz. With better alignment, when the laser beam is on the optical axis, the
free spectral range becomes c/(2L).
This customized SA200 is driven by the matching SA201 Spectrum Analyzer Controller. This controller provides a saw-tooth waveform with adjustable ramp amplitude and rise time and can be externally triggered. In our experiment, we set the
amplitude of the ramp to just cover one free spectral range so that two peaks of
the HeNe laser can be observed on the output spectrum. We scan the etalon at
50Hz which given the FSR of the etalon will yield peaks having a temporal width of
∼ 100µs.
The optical length of the Fabry-Perot Cavity is also sensitive to air pressure,
40
airflow and temperature change. Since we are using this cavity to lock the laser
frequencies, it is important to make it as stable as possible. Therefore, we put the
whole scanning etalon into an aluminum housing specially made for this purpose.
This enclosure is air-sealed. Laser light enters and exists the two AR coated N-BK7
windows sealed by O-rings. A small hole running cables is sealed by vacuum epoxy
glue.
4.1.4
Data Acquisition System
Laser control is accomplished using Labview and an NI PCle-6341 Xseries DAQ board.
This particular DAQ board has eight analog inputs with a total sampling rate of
500kS/s and two analog output channels. There are also plenty of digital lines with
even higher reading and writing rates.
4.1.5
Locking Scheme
The essence of our laser locking is that by continuously scanning the Fabry-Perot
interferometer, the relative frequency difference between the target laser and the
reference HeNe laser can be monitored and can be locked to a fixed value by feedback.
Here is how this scheme implemented in our system: Three laser beams (the HeNe, the
461nm, and 413nm beams) are all sent into the etalon, and the resulting transmission
peaks from each laser are detected by three independent photo-diodes. The signals
from the photo-diodes are fed into three analog input channels of the DAQ board.
One more analog input channel is used to display the ramp signal from the SA201.
The Labview program first locks the Fabry-Perot cavity to the HeNe laser. This is
done by feedback; any change in HeNe peak position (with respect to the ramp) is
compensated by changing the offset of the ramp voltage from the SA201. This step
41
is necessary due to the fact that the Fabry Perot cavity drifts due to the thermal
expansion or electrical drifts even though it is somewhat thermally isolated in the
aluminum enclosure. The next step is to continuously compare the relative position
of the HeNe peak and the 461nm/412nm laser peaks and generate feedback signals to
compensate for any changes to the diode laser systems to make their relative position
stay at the set value. In this step, the feedback voltage is sent to the BNC input
of the DCB module of the corresponding laser system. So, ultimately, the feedback
is to the position of the grating. In addition, both lasers can be locked or scanned
anywhere over the whole ramp which is simply realized by modifying the set value of
their positions relative to the HeNe peak.
Our locking scheme as described is able to lock both diode laser system to a
range of ±2MHz stably over the course of a day. And it effectively compensates the
long-term thermal drift the diode laser system suffers and successfully satisfies our
experimental needs.
4.1.6
Limitations and Alternatives
There are a couple of factors limiting the bandwidth of our stabilization scheme like
the processing rate of the Labview program and the sampling rate of the DAQ board.
But the major limitation is from the necessity of scanning the Fabry-Perot Cavity
over a whole Free Spectral Range which requires 20ms. Faster feedback is possible
by increasing the scanning rate and, accordingly, the sampling rate.
Locking a laser, like the 412nm laser, that is not associated with direct atomic
transitions or very close to any reference laser (up to GHz frequency difference), is difficult. Otherwise, either saturation absorption could be performed and the linewidth
of the locking could be cut to 100kHz or, two lasers’ beat signal could be utilized
42
to perform a fast heterodyne locking scheme. There is one other approach as described in [59]. They first find a cavity length that coincides with the transmission
maximas of the target laser and the reference laser by generating a sideband from
current modulating the reference laser. Then they lock the cavity to the reference
laser and lock the target laser to the cavity by analog circuitry. The scan of the target
laser is obtained by scanning the sideband of the reference laser. This approach has
achieved sub-MHz linewidth. Another approach is an analog of saturation absorption spectroscopy. Using the EIT signal from the coherent two-photon transition, the
linewidth should also be greatly reduced.
4.2
Strontium Oven
While several strontium atom beam sources have been described, these are typically
designed to optimize beam fluxes rather than to achieve tight beam collimation. Although beam divergences can be reduced by use of multicapillary arrays, these alone
cannot provide the required collimation. Here beam divergence is controlled through
the use of a small oven aperture and tight beam collimation. Because the oscillator strengths for excitation of high-n Rydberg states are small, sizeable atom beam
densities are required to achieve reasonable Rydberg photoexcitation rates. Given
that the oven aperture is small, this requires the vapor pressure in the oven be high
necessitating operation at temperatures of ∼ 500 − 650◦ C . Such temperatures can
be reached using commercial resistive heating elements provided that thermal losses
are minimized. While the present source design builds on earlier practice, its use
of a commercial heater allows a particularly simple design that is straightforward to
fabricate. The source has proven reliable in operation and, with appropriate collimation, can provide an atomic beam with a divergence of ∼5mrad FWHM and densities
43
Figure 4.2 : common elements vapor pressure chart, downloaded online
44
approaching 109 cm−3 sufficient to enable a wide variety of experiments with high-n
strontium Rydberg atoms.
The present atom source is shown in Figure 4.3. Its central component is a
cylindrical stainless steel oven that is loaded with granular strontium metal from the
rear. Atoms emerge from the oven through an orifice in the form of a cylindrical canal
that is ∼ 0.5mm in diameter and ∼ 1.2mm long. The oven fits inside a commercial
spiral wound coil heater.This heater comprises a heating element that is electrically
isolated inside a stainless steel sheath using MgO. The heater includes an unheated
section at one end through which the heater leads enter and exit, and also contains
an internal thermocouple to monitor its internal temperature. In air, the heater is
rated for 350 W and operation at temperatures up to ∼ 820◦ C. The spacing of the
heater coils is adjusted such that they are closer together near the front of the oven
than at its rear. This ensures that the front of the oven is maintained at a higher
temperature than its rear to prevent the exit canal from becoming blocked through
condensation. To further limit such condensation, the exit orifice is offset towards the
side of the oven and the heating coils are extended well forward of the front of the oven
to provide it with a strong radiant heat bath. The heating coil fits inside a polished
cylindrical copper jacket. The low emissivity of copper minimizes the heater power
required to reach a given operating temperature. Although hot copper reacts with
strontium, the jacket is not exposed directly to strontium vapor and no problems with
such reactions have been encountered. The oven and heater are positioned within the
jacket using a single small screw;see Figure 4.3. The temperatures of the front and
back of the oven, and of the jacket, are monitored using thermocouples.
To ensure good thermal isolation, the copper jacket is held in place using a number
of small stainless steel mounting screws that pass through ceramic bushings. Two
45
Figure 4.3 : Schematic diagram of the oven assembly
46
pairs of horizontally-opposed screws position the jacket vertically within a “U”-shaped
aluminum support bracket. Horizontal positioning is achieved with the aid of a single
vertical screw that passes through a second mounting bracket that runs up and over
the front of the jacket. To allow for expansion upon heating a small clearance is
included between the mounting brackets and the heater jacket, and the mounting
screws are free to slide within their bushings. The support bracket is held in place
by an arm that is connected via a bellows to an x-y translation stage located outside
the vacuum region. This stage is used to position the oven orifice during initial
beam alignment. The whole oven assembly is mounted inside a water-cooled copper
enclosure. To limit the temperature rise of the support bracket and of the power
input end of the heater, these are each cooled by connecting them via copper braids
to the enclosure. A 4 mm-diameter aperture located ∼ 4cm from the oven orifice is
used for initial beam collimation. Final collimation is provided by a 0.5 mm-diameter
aperture ∼ 10cm from the oven. No significant changes in beam pointing have been
observed as a result of day-to-day cycling of the oven temperature.
The oven is typically operated at temperatures in the range ∼ 500 − 650◦ C. The
power input to the heater required to reach these temperatures is ∼ 25−50 W and the
internal temperature within the heater remains below 800◦ C. While the strontium
atom beam density cannot be simply measured directly, it can be inferred from earlier
measurements (using a hot wire ionizer) of potassium atom beam densities produced
by an oven having a similar orifice and operating at similar metal vapor pressures;
see Figure 4.2. These earlier measurements suggest that at operating temperatures
above ∼ 600◦ C strontium beam densities approaching 109 cm−3 should be obtained.
Measurements of the photoexcitation of strontium Rydberg states demonstrate that
sizeable beam densities are produced. While much of the increase in Rydberg produc-
47
tion can be attributed to differences in oscillator strengths, the data do indicate that
sizeable strontium atom beam densities are obtained and that large photoexcitation
rates can be achieved.
The present atom source has proven reliable in operation and even with day-to-day
cycling of the oven temperature no heater failure has occurred. While the lifetime
of an oven charge depends on the required beam density, its capacity has proven
more than sufficient to allow several months of operation before reloading becomes
necessary. However, if required, longer operational periods could be achieved by
simply scaling up the design (a wide range of suitable coil heaters are available).
4.3
4.3.1
Other Experimental Apparatus
Vacuum System
The whole vacuum system consists of two relatively separate sections, the oven section where the oven is mounted and the interaction region section, where ground
state strontium atoms are photo-excited to Rydberg levels and subsequently field
ionized and detected. These two sections are connected by a 500µm-diameter aperture through which the atomic beam passes. Therefore, for efficient pumping, the two
sections are differentially pumped using diffusion pumps, a Varian VHS-4 for the oven
section and VHS-6 for the interaction region section. Their pressures are measured
by two Bayard-Alpert ionization gauges and are typically ∼ 10−7 torr.
The laser beams enter the vacuum system through two Ø1” AR coated Thorlabs
N-BK7 broadband precision windows traveling in opposite directions. The windows
are mounted slightly off normal to the beams to avoid retroreflections. The vacuum
seal is accomplished by Parker O-rings. The 412nm laser beam is focused by a Ø1”
48
Figure 4.4 : Interaction Region
f =40cm Thorlabs N-BK7 Plano-Convex Lens to the beam waist, 170µm, in the
center of the interaction region.
4.3.2
Interaction Region
The interaction region is bounded by three pairs of 10cm×10cm electrodes (see figure 4.4). Except the bottom plate, 5 of them can be biased independently to cancel
the stray electric fields in three orthogonal directions so that local electric field can
be reduced to ≤ 50µV /cm. A circular electrode disk mounted from the top plate is
used for applying fast electric pulses (often called half-cycle pulses) in the ẑ direction
and shares its bias potential with the rest of the plate. Sometimes, half-cycle pulses
(HCP) are also applied to the side-electrode as shown in Figure 4.4. For detection,
the Rydberg atoms are field ionized by a voltage ramp which is applied to the bottom
49
plate. The resulting electrons exit through the 1” inch aperture on the bottom plate
covered with fine copper mesh to be detected by the funnel shaped aperture of a Dr.
Sjuts channel electron multiplier KBL 25RS mounted beneath the mesh.
4.4
Experimental Techniques
As mentioned earlier, we use a ramped electric field to field ionize the Rydberg atoms
and subsequently collect the resulting electrons. Field ionization enables state selective detection and is widely employed in Rydberg atom experiments.
4.4.1
Selective Field Ionization
Classical ionization occurs when the electron’s potential energy becomes less than its
~ the
total energy. For an electron of energy −1/(2n2 ) subjected to an electric field E,
~ · ~r which scales as −1/n2 + En2 . The classical
total potential energy is −1/r + E
ionization limit can be heuristically obtained by equating them, −1/n2 + En2 =
−1/(2n2 ), thus the ionization field scales as 1/n4 . Calculation of saddle point in the
potential yields 1/(16n4 ) as the classical ionization limit which is shown in the Stark
map for sodium in Figure 4.5. Rydberg atoms are indeed ionized around this limit.
However, before being completely ionized, the energy level of the electron is first
Stark shifted or/and Stark mixed according to its l and the rise time of the applied
field, and these effects determine the exact field at ionization. Consider the Stark map
for sodium in Figure 4.5. Whereas the high l manifolds display linear Stark shifts, the
low-l states display quadratic Stark shifts just as was discussed in Chapter 3. As the
field increases, different Stark states experience avoided crossings. If the electric field
strength increases slowly enough, these crossings are traversed adiabatically, the states
following the paths indicated by the solid lines and ionizing at fields ∼ 1/(16n4 ). On
50
Figure 4.5 : Stark map for Sodium showing evolution of the m = 0 states in an
increasing field. Color red represents d states (δd = 0.01), color green represents
p states (δp = 0.86) and color purple represents the s states (δs = 1.35). In this
particular case, d states are very close to the manifold which are the whole cluster of
high l states. In this graph, the manifold states are represented by the 5 states just
above d states. The dotted lines are the diabatic paths for s,p,d states to ionize while
the adiabatic paths are the solid black serpentine lines for each states. The colored
dots on the ionization curve are the ionization points for the adiabatic passages of
each state. This graph is adapted from [56].
51
the other hand, a rapid rise in the applied electric field will result in diabatic passage,
states will follow a linear energy shift after mixing with the extreme l states in the
manifold and they will be ionized in a order that is completely different from that
of their zero field energies; see the dotted lines in Figure 4.5. In our experiment,
ionization can be regarded as diabatic because the energy separation at the avoided
crossings are very small for n ∼ 300.
No matter if the ionization is diabatic or adiabatic, the electric field required, to a
good approximation, scales as 1/(n4 ). If a ramped field is applied, different states will
ionize at different field and thereby different times. The resulting ionization signal
vs. electric field will provide information about the Rydberg states initially present.
With careful analysis through “selective field ionization” (SFI), especially for n≤100
when the available resolution is high, it is possible to obtain considerable information
on the initial n distribution. However, when n ∼ 300, it is only possible to infer the
approximate n from SFI spectrum.
To generate a SFI spectrum, we measure two quantities using techniques adapted
from previous experiments on potassium for which the excitation rate of Rydberg
atoms was low. Following each laser pulse, there is only one or zero atoms excited
to the Rydberg level. So single particle detection can be employed using a channel
electron multiplier. This amplifies the incident electron and produces a large number
of secondary electrons that generate one large signal pulse for detection. Besides this
signal, we also measure the time duration from the start of the ramp to the time
when we get the signal from the channel electron multiplier. Since we know the time
evolution of the ramp, with these two quantities, we can obtain the electric fields
needed to ionize that Rydberg atom.
Although we don’t use the SFI spectrum directly in the beam experiments. It
52
produces a valuable detection scheme. In our strontium experiment, we found a ramp
that rises from 0 to 5 Volts in 5 µs is more than enough to ionize all the Rydberg
atoms. .
4.4.2
Data Acquisition
The very-high-n Rydberg experiment is triggered by a master pulser which immediately triggers a 461nm laser pulse for Rydberg excitation. About 10µs later, the
ramped field is applied to the bottom plate. Right before the ramp, an ORTEC 566
time-to-amplitude converter (TAC) is started by a trigger pulse and it is stopped by a
signal from the channeltron. The output pulse from the channeltron is then converted
into a TTL signal by a charge sensitive amplifier and is fed into a computer. This
computer also controls, scans the 412nm laser frequency, adjusts the bias voltages
on the plates of interaction region and controls the multiple half-cycle-pulses (HCPs)
that are used for engineering quasi-one-dimensional states in some other experiments.
53
Chapter 5
Sr Rydberg Atoms in a Collimated Atomic Beam
Initial experiments focused on the production of Sr Rydberg atoms, their spectroscopy, their photo excitation rates, and their excitation in a weak dc field.
Strontium atoms contained in a tightly-collimated beam are excited to the desired high-n (singlet) state using the crossed outputs of two frequency-doubled diode
laser systems. The two-photon excitation scheme employed utilizes the intermediate
5s5p1 P1 state and radiation at 461 nm and 413 nm. The laser beams, which typically have the same linear polarization, cross the atom beam traveling in opposite
directions. Since their wavelengths are comparable, the use of counter-propagating
light beams can largely cancel Doppler effects associated with atom beam divergence
resulting in very narrow effective experimental linewidths. As noted earlier, residual
stray fields in the experimental volume are reduced to ≤ 50µV cm−1 by application
of small offset potentials to the electrodes that surround it. Measurements are conducted in a pulsed mode. The output of the 461 nm laser is chopped into a series
of pulses of ∼200 ns to 1 µs duration and 20 kHz pulse repetition frequency using
an acousto-optic modulator (The 413 nm radiation remains on at all times). Following each laser pulse, the probability that a Rydberg atom is created is determined
by state-selective field ionization for which purpose a slowly-rising (risetime ∼1µs)
electric field is generated in the experimental volume by applying a positive voltage
ramp to the lower electrode. Product electrons are accelerated out of the interaction
region and are detected by a particle multiplier. The probability that a Rydberg atom
54
is created during any laser pulse is maintained below 0.4 to limit saturation effects.
This is accomplished by reducing the strontium atom beam density by operating the
oven at a lower temperature, and/or by reducing the laser powers.
5.1
Spectroscopy
Figure 5.1 shows excitation spectra recorded in the vicinity of n = 282 for various
detunings of the blue 461 nm laser selected to optimize the transitions 5s2 1 S0 →
5s5p 1 P1 in the different strontium isotopes. The relative frequency of these transitions together with other properties of naturally-occurring strontium are listed in
Table 2.2. The isotope shifts and hyperfine splittings in both Table 2.2 and Figure 5.1 are quoted relative to the dominant
88
Sr isotope. The blue 461 nm laser
beam was unfocused and had a diameter of ∼3 mm. Its intensity, ∼10 mW cm−2 ,
was selected to limit line shifts and broadening due to effects such as the ac Stark
shift and Autler-Townes splitting. Its pulse width, ∼ 0.5µs, was selected because
for shorter pulse durations the widths of the spectral features become increasingly
transform limited. The “purple” 413 nm laser beam was focused to a spot with a
full width at half maximum (FWHM) diameter of ∼ 170µm, resulting in an intensity
of ∼ 250W cm−2 . The frequencies of both lasers were stabilized and controlled with
the aid of an optical transfer cavity locked to a polarization-stabilized HeNe laser as
described in Section 4.1. This cavity allows uninterrupted tuning of the lasers over
frequency ranges of up to ∼800 MHz. The frequency axis in Figure 5.1 shows the
sum of the blue and purple photon energies. The spectrum obtained with the blue
laser tuned to optically excite the dominant
88
Sr isotope displays a series of sharp
peaks associated with the excitation of 1 D2 states with the n values indicated. The
spectrum also contains a series of smaller peaks associated with the production of 1 S0
55
Figure 5.1 : Excitation spectra recorded in the vicinity of n = 283 for different
detunings of the 461nm laser. These detunings, specified relative to the 88 Sr 5s21 S0 →
5s5p1 P1 transition, are indicated in the figures. The frequency axis shows the sum of
the 461 nm and 413 nm photon energies. The horizontal bars beneath the data
identify the features associated with excitation of 5snd1 D2 Rydberg states in the
88
Sr, 86 Sr and 84 Sr isotopes, together with the positions of features associated with
excitation of 87 Sr Rydberg states. Two red arrows in the top subplot point to the
3
D2 state of 88 Sr.
56
states. The widths of these features, ∼5 MHz FWHM, is attributed to a combination
of transit time broadening (the transit time of an atom through the purple laser spot
is ∼ 300 ns ) and fluctuations in the laser frequencies during the ∼ 1 s required to
accumulate data at each point in the spectrum. The probability that a Rydberg atom
was created at the peak of the 1 D2 features during each laser pulse was limited to
∼ 0.5 by substantially reducing the strontium atom beam density by operating the
oven at ∼ 500◦ C. Tests were undertaken at higher operating temperatures in which
the 413 nm laser beam was attenuated using neutral density filters to limit the excitation rate. These tests showed that, with the oven operating at 630◦ C and using
the full 413 nm laser power (∼ 70 mW), ∼10-15 Rydberg atoms can be produced per
laser pulse in the excitation volume of ∼ 5 × 10−5 cm3 . This corresponds to a typical
inter-Rydberg spacing of ∼ 130µm which is approaching those at which effects due
to Rydberg-Rydberg interactions such as blockade become important.
5.1.1
Even Isotopes
When the blue laser is tuned on resonance with the 1 S0 → 1 P1 transition for
as shown in the top subplot in Figure 5.1, the two 1 D2 of
88
88
Sr,
Sr peaks dominate the
spectra. If we normalize the frequency separation between singlet 1 D2 states with
consecutive n as 1, then the nearest 1 S0 state to a 1 D2 should be separated by
|(δ1 S0 mod1) − (δ1 D2 mod1)| = |0.269 − 0.381| = 0.112,
where δ is the quantum defect of the corresponding states; see Table 5.1. Inspection
of the figure shows that there is an S state lying 0.11 to the right of every 1 D2 state
in accordance with the quantum defect reported previously. The same calculation
can also be carried out for the 3 D2 state, which shows that its separation from the
57
nearest 1 D2 should be
|(δ3 D2 mod1) − (δ1 D2 mod1)| = |0.636 − 0.381| = 0.245.
As marked on the top subplot of Figure 5.1 by red arrows, a small signal from 3 D2
states can also be discerned at the expected relative positions. The small size is due to
the fact that ∆S 6= 0 transitions are forbidden in the Russell-Saunders LS-coupling
scheme. For heavy atoms like strontium, however, LS-coupling breaks down and
strong spin-orbit interactions mix the wavefunctions from singlet series and triplet
series. So intercombination transitions are weakly allowed for Sr. But the spin-orbit
interaction strength or equivalently, the fine structure (EF S ∼ (n∗)−3 ) diminishes
quickly as n increases and, this mixing gets substantially weaker for very high-n
Rydberg states. Therefore, the 3 D2 states are barely excited.
As the blue laser is red detuned from resonance with the 1 S0 → 1 P1 transition
in
88
Sr, the size of the
88
Sr features in the excitation spectra decreases steadily
and new features emerge associated with the excitation of Rydberg states of the
other isotopes. At a detuning of ∼ −122 MHz, which optimizes the 1 S0 → 1 P1
transition for 86 Sr, the excitation spectrum is dominated by the creation of 86 Sr 1 D2
states, although some residual excitation of
88
Sr →
86
88
Sr isotopes remains. The observed
Sr isotope shift in the series limit, +210 ± 5 MHz, is consistent with that
reported in earlier spectroscopic studies at lower n [45]. At a blue laser detuning of ∼
−273MHz which optimizes the 1 S0 → 1 P1 transition in the 84 Sr isotope, the excitation
of 84 Sr1 D2 Rydberg states becomes apparent. The observed 88 Sr → 84 Sr isotope shift
in the series limit, +440 ± 8M Hz, is again consistent with earlier measurements [45].
However, because the fractional abundance of the 86 Sr and 84 Sr isotopes in the beam
shown in Table 2.2, are much less than for the
atoms are created.
88
Sr isotope, many fewer Rydberg
58
Table 5.1 : Sr Quantum Defects [65]
Series
δ
5sns1 S0
3.269
5snp1 P1
2.730
5snd1 D2
2.381
5snf 1 F3
0.089
5sns3 S1
3.371
5snd3 D3
2.630
5snd3 D2
2.636
5snd3 D1
2.658
At blue laser detunings of ∼ −90 MHz, the sizes of the features associated with
the excitation of
88
Sr and
86
Sr Rydberg states are nearly equal, resulting in the
appearance of two separate interleaved Rydberg series. The
88
Sr to
86
Sr isotope
shift, ∼ 210M Hz, matches the energy spacing between adjacent Rydberg levels at
n ∼ 313, indicating that in the vicinity of n ∼ 313 the two interleaved Rydberg series
should overlap, the 312 1 D2 88 Sr Rydberg level, say, overlapping with the 313 1 D2 86 Sr
Rydberg level. Spectra recorded in the vicinity of n = 312 (see Figure 5.2) under
these conditions do indeed display only a single series of Rydberg features.
5.1.2
the odd isotope
87
Sr
Several new features are also observed in the excitation spectra at blue laser detunings
chosen to favor excitation of the
87
Sr isotope. These features do not conform to a
59
Figure 5.2 : Spectra recorded with the detunings ∆ shown in the vicinity of N ∼ 312.
In (a), 88 Sr is dominant. In (b), the detuning ∆ favors the 1 S0 → 1 P1 transition of
both 87 Sr and 86 Sr. The most discernible peaks for 87 Sr are highlighted by green
arrows and they will be explained in Subsection 5.1.2. The 1 D2 peaks for n86 Sr at
the first three ns are distinguishable from those resulting from residual excitation of
(n + 1)88 Sr. At higher n, they merge resulting in a single sharp spectral feature. In
(c), where the detuning mostly favors 86 Sr, small peaks from 88 Sr can be observed
on the left of the first two dominant n86 Sr peaks.
60
simple Rydberg series having the expected values of n. Similar behavior has been
observed previously in studies at lower n and assigned to a combination of strong state
mixing, hyperfine-induced singlet-triplet mixing, and interactions between states of
different n [47]. These complexities of
87
Sr Rydberg series are mainly caused by its
nuclear spin I = 9/2 which is absent for all the even isotopes. Major consequences of
the nuclear spin are now discussed.
Hamiltonian with Hyperfine Interaction
The total Hamiltonian, that includes the fine and hyperfine interactions, for a Rydberg
atom with two valence electrons n1 l1 s1 , n2 l2 s2 and a nuclear spin I, can be written as
H = H1 + H2 +
e2
+ β1 s~1 · ~l1 + β2 s~2 · ~l2 + A1 I~ · ~j1 + A2 I~ · ~j2 ,
r12
(5.1)
where Hi is the Coulomb interaction between the valence electron i and the Sr2+ core,
e2 /r12 is the Coulomb repulsion between the two valence electrons which leads to the
singlet triplet splitting, βi~s · ~li is the spin-orbit interaction of the ith electron and
Ai I~ · ~ji , j~i = s~i + ~li is its hyperfine interaction. For a strontium Rydberg atom of configuration 5snl, we can use the subscript 1(2) to denote the nl(5s) Rydberg (ground)
electron. The spin-orbit interaction and hyperfine interaction for the Rydberg electron are negligible when n is large since both the spin-orbit constant β and hyperfine
structure constant A scale as n−3 . Because the 5s electron has l2 = 0, j2 = s2 = 1/2,
the spin-orbit interaction is also zero and the hyperfine interaction is reduced to
A2 I~ · s~2 . Now we can rewrite the Hamiltonian as
H = H1 + H2 +
e2
+ A2 I~ · s~2 .
r12
(5.2)
Now, the quantum number describing the total angular momentum is F (for even
isotopes, only the first three terms remain and the Hamiltonian is characterized by J),
61
~ The last term in this Hamiltonian
the quantum number F~ being given by F~ = I~ + J.
will be referred as the Fermi contact interaction.
Rydberg Series Limit Shift
For the even isotopes of strontium, different Rydberg series like 3 S1 , 1 S0 , 3 D2 , 1 D2 will
converge to one common limit, the ion’s ground state 5s Sr+ 2 S 1 . This is described as
2
2
Enl = Iion −R/(n−δl ) where Iion is the ionization potential and δl is the corresponding
quantum defect. Therefore, the shifts between even isotopes are their normal mass
shifts as shown in last section. However, due to the Fermi contact interaction which
is essentially the hyperfine interaction of the ion, the ion state 5s Sr+ 2 S 1 splits into
2
F = 4 and F = 5 hyperfine states and they are about 2 − 3GHz [47] away from the
ion energy without a hyperfine structure i.e.,
87
Iion − 88 Iion ∼ ±2 − 3GHz. These
shifts, much greater than the energy spacing between the n and n + 1 levels when
n ∼ 300, contribute to the movements of
87
Sr peaks relative to the peaks of the even
isotopes in the measured spectra.
Hyperfine Induced Mixings
The Fermi contact interaction becomes increasingly important as n gets higher. When
60 < n < 100, this interaction (∼ 2 − 3 GHz) is comparable to the e2 /r12 term
in equation 5.2 but smaller than the energy spacing between two consecutive ns,
and can cause mixing between triplet and singlet levels of the same F within the
same n (hyperfine induced singlet-triplet mixing within the same n). Consequently,
their energies are shifted and triplets can be populated as strongly as singlets. As n
continues to increase, the Fermi contact interaction becomes larger than En − En−1
(the energy spacing between two adjacent ns), and states of the same F but different
62
Figure 5.3 : N ∼ 335 spectra The 3 D2 states of 88 Sr are circled by red ovals in
the top figure. The blue laser is tuned to favor the 1 S0 → 1 P1 transition for 87 Sr in
the second and the third subplots. Based on our experiences, the 87 Sr features are
most apparent at a detuning around −69M Hz. The last subplot is of a detuning
that favors excitation of 86 Sr.
ns can be mixed. The higher the n is, then states with ever increasing differences in
ns can be mixed. At n ∼ 300 , hyperfine induced n-mixing can lead to “anti-crossing”
in the spectra since the energy differences between states of different n can be tuned
semi-continuously by changing n as shown in Figure 5.3 and 5.4.
5.1.3
Stray Fields Impact On Spectra
Figure 5.5 shows Rydberg excitation spectra recorded, with the blue laser tuned
to the
88
Sr5s21 S0 → 5s5p1 P1 transition in the
88
Sr isotope, at successively higher
63
Figure 5.4 : Anticrossing of 87 Sr This graph is the detailed, break-down plots of
the third plot of Figure 5.3. Every subplot of this graph starts from a 1 D2 peak of
88
Sr and ends at next 1 D2 peak of 88 Sr. At the anticrossing point, where the energy
mismatch is very small, one of the state features disappears due to the cancellation
of the oscillator strength. The red line and green line follow the peak positions of two
merging states.
64
values of n. As expected, as n increases the peak number of Rydberg atoms created
decreases dramatically as a result of both the decrease in the oscillator strength and
the increasing width of the spectra features. For values of n ≤ 350, two well-resolved
Rydberg series are seen, corresponding to excitation of 1 D2 and 1 S0 states. With
further increases in n the spectra features begin to broaden significantly, their widths
having approximately doubled by n ∼ 400. For even larger values of n the background
level begins to increase significantly, but a strong, well-resolved Rydberg series is still
evident for values of n up to ∼ 460. For n > 500, however, it becomes increasingly
difficult to discern any Rydberg series. This degradation in the Rydberg spectrum
with increasing n can be attributed to the presence of stray background fields which
lead to Stark shifts and broadening. These effects become particularly important at
fields approaching those at which states in adjacent Stark manifolds first cross, given
by Fcross ∼ 1/(3n5 )a.u., i.e., ∼ 50µV cm−1 at n ∼ 500. This suggests that stray fields
of ∼ 50µV cm−1 remain in the excitation volume which is consistent with earlier
estimates of their size as described in Section 4.3.2.
65
Figure 5.5 : Excitation spectra recorded near the values of n indicated. The 461nm
laser is tuned to the 88 Sr5s21 S0 → 5s5p1 P1 transition. The frequency axis shows the
relative frequency of the 413nm laser during each scan
66
Chapter 6
Ultracold Rydberg Gas Evolution
Ultracold Neutral Plasmas (UNPs) and Ultracold Rydberg Gases are inter-related.
The free electrons and ions in a plasma can form Rydberg atoms via three body recombination (e− +e− +R+ = R∗∗ +e− , “R∗∗ ” is the abbreviation for a Rydberg atom).
Conversely, an ultracold Rydberg gas can spontaneously evolve (ionize) into plasma.
Both systems have been studied extensively over the last decade. Yet the mechanism
for some processes are still intriguing. Use of traditional selective field ionization or
simple electron detection prohibits direct investigation towards the “pre-ionization”
stage of the evolution of a Rydberg gas into a UNP. The unique imaging capability
offered by strontium’s optically-active core effectively mitigates this problem. This
chapter, based on paper [35], will describe the techniques we employed in probing cold
Sr Rydberg gas dynamics along with a short introduction to the operative processes.
Our results, in contrast to earlier studies, stress the role played by Rydberg-Rydberg
interactions in the initial phases of evolution of a Rydberg gas.
6.1
Experimental Setup Overview
As shown in Figure 6.1 and 2.2, about a billion strontium atoms are captured in a
magneto-optical trap after the Zeeman slower and cooled to ∼7mK. Both the cooling
lasers and the laser driving the 461nm 5s21 S0 → 5s5p1 P1 transition are derived from a
frequency doubled Ti-Sapphire laser stabilized by saturated absorption spectroscopy.
67
Figure 6.1 : a).Rydberg excitation beams and UNP ionization beams are shown with
respect to the imaging system. The fluorescence imaging beam not shown in the
figure is parallel to the Rydberg excitation beams. b). Pertinent energy levels for the
two photon transition to Rydberg level and the core transition used for imaging. c).
Timing for the experiment. Figure adapted from [35].
68
The atomic density has a Gaussian profile with a radius of ∼1mm. The Rydberg
atoms are excited by a two-photon transition. The 461nm laser is red detuned by
430MHz from the intermediate 5s5p1 P1 state and a 413nm laser is used for further
promotion (see detailed description in Chapter 4). The diameters of both the 461nm
and 413nm laser beams used for Rydberg excitation are much larger than the size of
the MOT so that the density of Rydberg atoms should follow the MOT profile. The
413 nm light remains on all the time (see the time sequence in Figure 6.1) while the
461nm laser light is pulsed on during the excitation.
In order to count the number of Rydberg atoms excited, we directly create a
UNP by photoionizing a small fraction of the strontium atoms after the Rydberg
excitation. The resulting electrons rapidly l change and ionize the Rydberg atoms
allowing fluorescence detection of the core ions. The photoionization is produced by
a 10ns dye laser pulse at 412nm. The 355nm light from a Nd:YAG laser pumps the
dye laser which can be tuned to set the velocity (energy) of the ionized electrons. For
our purposes, we set the frequency just above the ionization threshold so that only a
small but sustainable plasma is produced in the MOT.
For imaging the core ions, we utilize a 422nm laser beam to drive the 5s2 S1/2 →
5p2 P1/2 core transition. This laser is frequency doubled from a 844nm infrared diode
laser. It is locked to a scanning Fabry Perot cavity that is referenced to the 922nm
laser beam that is frequency doubled (461nm) and is stabilized by saturation absorption spectroscopy as mentioned previously. The conversion from scanning voltage to
the real frequency of the cavity is calibrated by using the 422nm laser for saturated
absorption spectroscopy in a 85 Rb cell. This is possible because the strongest peak in
the
85
Rb spectra (5s 2 S1/2 (F = 2) → 6s 2 P1/2 (F = 3)) is only 440MHz red-detuned
from the
88
Sr+ 5s 2 S1/2 → 5p 2 P1/2 transition. This 440MHz offset is easy to span
69
by an acousto optic modulator. This light is used for imaging the bare ions since it
directly drives an optical transition in every ion (we can saturate this transition ).
We can either image the light absorption or image the laser induced fluorescence from
the atoms. Frequently, interest centers on the local density of the ions. So instead of
imaging a whole cloud, we image a thin sheet of atoms that has well defined geometry;
see Figure 6.2. Of course, its finite size needs to be taken into consideration for the
final ion density calculation. We argue that this technique can also image the high-l
strontium Rydberg atoms because, a Rydberg electron in a high-l orbit has negligible
overlap with the core ion allowing the core ion to behave as an independent particle.
In contrast, as described in the next section, low-l(l < 6) Rydberg atoms do not yield
a fluorescence signal.
6.2
Principal Processes in Probing Sr Cold Rydberg Gas
Among a complex array of processes occurring in our experiment, the following are
the most influential ones despite the ongoing debates as to their relative importance.
6.2.1
Autoionization
Excitation of the |5si core electron in a Rydberg atom can lead to autoionization.
Autoionization is a special demonstration of multichannel coupling in the language of
MQDT. It is an inherent property of a two electron atom resulting from the interaction
between the two valence electrons. The term H12 = 1/r12 in the Hamiltonian not
only mixes the configurations of bound Rydberg states that are nearly degenerate (as
shown in red arrows in Figure 6.3 ) but also couples the doubly excited Rydberg series
|5pnli to the Rydberg continuum |5sεl0 i; see blue arrows in Figure 6.3. The rate of this
particular transition, i.e. autoionization rate, is determined by | h5pnl| H12 |5sεl0 i |2
70
Figure 6.2 : Optical probes for UNP using light resonant with the Sr+ transition.
(a) Absorption Imaging: the laser beam is absorbed by the ions in the UNP creating
a shadow that is recorded by a CCD camera. A complete absorption image can be
constructed by a weighted integration of images taken over many different frequencies
across the resonance to account for ions having different velocities. The ions’ absorption profile is a Voigt distribution. (b) Fluorescence Imaging of a sheet UNP: the
imaging laser beam propagates perpendicularly to the the camera and imaging axis.
A complete image construction of all the ions is the same as for absorption imaging.
Figure adapted from [69].
71
Figure 6.3 : Sr Rydberg atom autoionization The short solid lines are the strontium Rydberg levels converging to different series limits. The shaded regions are the
corresponding continua. The red arrows denote the coupling between bound, nearly
degenerate Rydberg levels in different configurations. The blue arrows denote the
coupling to the continuum of a Rydberg series that has lower series limit. Figure
Adapted from [56].
72
which obeys the Rydberg scaling law ∼ 1/n3 . Its l dependence is complicated and can
be understood in a scattering picture: the outer electron scatters the inner electron
to the ground state while itself gaining energy and being ionized. To facilitate this,
the outer electron must approach the inner electron and be moving rapidly. In low-l
states, the Rydberg electron moves in a elliptical orbital, the lower l is, the closer
the perigee is and the faster the electron moves around the perigee and of course the
stronger the autoionization is. As a result, the oscillator strength generally decreases
very rapidly as l increases. However, for very low l states, the overlapping of the outer
electron with the inner electron is also strongly affected by the coupling between the
core and the outer electron since it governs the phase of the wavefunction near the
core.
In fact, for very low l states, the transition rate is very high (∆t ≈ 0). So according
to the Heisenberg principle, ∆E∆t ∼ ~, the transition linewidth ∆E is very large. For
strontium in particular, the autoionization transition for 5s48s 1 S0 and 5s47s 1 D2 is as
broad as tens of GHz [35]. Since the imaging laser we are using is of narrow linewidth
∼ 1M Hz, the effective autoionization, having a rate about 1M Hz/10GHz ∼ 0.01%
of the total autoionization rate, is negligible for low-l states during the 500ns imaging
time. For high-l states, overlap with the core, i.e., scattering, becomes negligible.
6.2.2
Electron-Rydberg Collisions
Electrons with velocities comparable to the Rydberg electron can be produced easily
in many collision processes. Collisions between electrons and Rydberg atoms can lead
to l and m changing and even n changing if the incoming electron is energetic enough.
In comparison, collisions between the ions and Rydberg atoms are unlikely due to
their low velocities. Therefore the most important collision is the l changing collision
73
Figure 6.4 : l-mixing due to Rydberg-electron collision. Step A shows the mixing into
the manifold as the electron approaches. In step B, the population is randomized
among the manifold. Step C is the ionization from the manifold. Figure adapted
from [68].
between the Rydberg atom and the electron. Here is how it works (see figure 6.4): as
the electron approaches, the low-l Rydberg state follows the quadratic Stark shift and
gradually mixes with the high-l manifold. Depending on the magnitude of the electric
field, it can mix strongly into the nearby manifold states. Thus when the electron
has passed by, it has a good chance to stay in one of the manifold states as the field
decreases. Even after collision the atom is subject to a very small electric field which
can lead to a redistribution among neighboring states in the manifold. As a result,
the original low-l state is further l-mixed into l > n/2 states on average. Though
m is a good quantum number for Rydberg atoms in an electric field, every electron
is incident in a random direction, whereupon m has to be referenced to a new axis.
So m mixing is happening all the time. The n-changing collisions are possible if the
74
electron passes close to the Rydberg electron. In the electron’s strong field manifolds
belonging to different n will cross, and the original state could mix into nearby ns.
6.2.3
Penning Ionization
Collisions between two Rydberg atoms can lead to Penning ionization
R∗∗ (n) + R∗∗ (n) → R∗∗ (n0 ) + R+ + e− .
(6.1)
Penning ionization can be facilitated if the two Rydberg atoms are attracted to each
other by attractive dipole-dipole interactions or attractive Van der Waals interactions. Once the interparticle distance R is small enough that the interaction energy
is comparable to the binding energy of the Rydberg atom, one of the Rydberg electrons will be ionized and the other atom will be deexcited into a more deeply bound
state. Simulations in [70] show that in almost all likelihood, the ionized electron will
be energetic enough to escape on a nanosecond timescale. The binding energy of the
resulting Rydberg atom almost doubles its initial binding energy. The rest energies
will make the ion and Rydberg atom leave each other in high relative velocities but
small center of mass velocity since their total momentum is negligible initially (cold
atoms). There are also cases where, even for attractive interaction, Penning ionization doesn’t happen. As the initial attractive interaction draws two Rydberg atoms
together, their dipole moments or equivalently their Runge-Lenz vectors can precess
to directions such that the interaction changes to repulsive. Consequently, they begin
to move apart from each other. Indeed Penning ionization will almost never happen
if the Rydberg-Rydberg interactions are isotropically repulsive.
Low-l Rydberg atoms don’t possess a permanent dipole. Nevertheless, a dipoledipole interaction which scales as ±C3 /R3 , C3 ∝ n4 can be induced if there is a
75
resonance like a Förster Resonance nl + nl ↔ n1 l1 + n2 l2 . In the absence of both
a permanent dipole and a resonance, the interaction between two Rydberg atoms is
of the Van der Waals type and scales as ±C6 /R6 , C6 ∝ n11 which is the interaction
between instantaneous dipole moments. This force is termed London dispersion or the
instantaneous dipole-induced dipole force. For the initially excited low-l strontium
Rydberg atoms, the interactions are all Van der Waals interactions. The values of
C6 s for different n are calculated in [65].
6.2.4
Blackbody Radiation induced Ionization
Rydberg atoms, having small binding energies, are generally sensitive to the 300K
thermal environment since the blackbody radiation (BBR) can not only drive dipole
allowed transitions to nearby states but also photoionize them. Both effects will limit
the lifetime of Rydberg atoms with BBR decay rates that scale as ∼ kT /n2 . For
5s48s 1 S0 and 5s47s 1 D2 states, the BBR induced n,l changing rates are ∼ 1 × 104 s−1
and the BBR induced photoionization rates are ∼ 1 × 102 s−1 . Note, however, that
it takes many dipole allowed transitions to induce large changes in l and thus this
process may be unimportant in the present experiments.
6.3
Imaging Technique
A direct laser induced fluorescence imaging of all the Rydberg atoms can be made by
exciting a dilute plasma (based on its function, we call it a seed plasma) immediately
after Rydberg excitation as shown in Figure 6.5. The free electrons trapped in the
MOT by the plasma cause repetitive l-mixing collisions and the low-l Rydberg atoms
are rapidly converted into high-l states which are visible to the imaging light. This
process can happen on a nanosecond time scale enabling us to count the Rydberg
76
Figure 6.5 : Laser-induced fluorescence imaging of UNP and Rydberg atom clouds
using the 5s 2 S1/2 → 5p 2 P1/2 core-ion transition at 422 nm. (a) Image after Rydberg
excitation to the 5s48s 1 S0 state for texc ≈ 3 µs, which yields ∼ 8 × 105 Rydberg
atoms. Notice that the fluorescence signal is very small and the scale bar represents
1 mm. (b) Image after exciting the same Rydberg population as in panel (a) but
with superposition of a seed UNP containing ∼ 2 × 105 ions and electrons. Note the
increased fluorescence from the cigar-shaped region of Rydberg excitation. (c) Image
of a seed UNP identical to that in panel (b), but with no Rydberg excitation. (d)
Signal due to Rydberg excitation obtained by subtracting the signal due to ion cores
in the seed UNP. Figure adapted from [35].
atoms. If we vary the Rydberg excitation time, we should expect a linear increase
in the Rydberg excitation due to the fixed excitation rate determined by the Rabi
frequency. This behavior is observed in Figure 6.6.
6.4
Results Discussion
As we can see in Figure 6.6, without a seed plasma, there are no visible ion cores
below excitation times texc ∼ 2.5µs implying all the excited Rydberg atoms are still
in their low-l states. If we stop the excitation at the time, and let the Rydberg gas
77
Figure 6.6 : Dependence of the LIF signal from parent 5s48s 1 S0 Rydberg atoms on
the excitation time texc . Left: each image is shown as a function of time with (top)
and without (bottom) a seed UNP present. (Contributions to the LIF signals from the
UNP are subtracted and the scale bar represents 1 mm). Right:Number of visible ion
cores versus Rydberg excitation time for parent 5s48s 1 S0 states. Data recorded both
with and without a seed UNP present are included together with results obtained
when only the seed UNP was created. Qualitatively similar results are seen for the
5s47d 1 D2 state. Figure adapted from [35].
78
evolve by itself, what will happen? We examined this using both the 5s47d 1 D2 state
and 5s48s 1 S0 state with the same initial density and number of Rydberg atoms.
Figure 6.7 shows that in both cases the core ions gradually become more and more
visible. However, though the initial conditions are very similar, these two states
display rather different time evolutions. These differences can be seen in the total
number of visible ion cores as shown in Figure 6.8. These differences can be explained
by noting that both l-mixed Rydberg atoms and true ions are seen by the imaging
beam. Since a very weak seed plasma (see image (c) in Figure 6.5 and notice that it
has almost the same color bar as Figure 6.7) is able to provide a sufficient number
of trapped free electrons to l-mix all the Rydberg atoms and make them visible, the
majority of the visible ions in Figure 6.7 for 4.1µs evolution time cannot be true ions
otherwise the attendant electrons should have l-mixed almost all the Rydberg atoms
in the central region and made them visible in nanoseconds. Therefore, it is clear that
almost all the visible ions we see for 4.1µs evolution time should be l-mixed Rydberg
atoms resulting from collisions with some early electrons.
Now the question becomes where do the initial electrons (which collide with the
Rydberg atoms and further lead to l-mixing) come from? Whatever the mechanism is,
we can see the early electrons are produced first in the places where the initial Rydberg
atom density is highest. This is shown unambiguously in Figure 6.7. For each state,
the visibility is brighter in the higher Rydberg atom density region in each image and
the visibility propagates to the lower density region with increased evolution time.
Both BBR ionization and Penning ionization will lead to such density dependent
ionization. However, the BBR-induced ionization is slow and cannot explain the
difference seen between the behavior of S and D state. Penning ionization, on the
other hand, would.
79
Figure 6.7 : LIF images showing the spontaneous evolution of an ultracold gas of 1 S0
(top) and 1 D2 (middle) Rydberg atoms. The evolution time is indicated above each
image in µs and the scale bar represents 1 mm. The initial numbers and densities for
both states are identical, 8 × 105 and 2.2 × 108 cm−3 , respectively. Notice how the
S-state population evolves more quickly in both space and time. The bottom panels
show one-dimensional plots of the density integrated along the vertical direction.
The spatial development of the S state (red) leads that of the D state (blue). Figure
adapted from [35].
80
Figure 6.8 : Evolution of the number of visible core ions for 5s48s 1 S0 and 5s47d 1 D2
Rydberg atoms. The circles represent number calibrations performed by scanning the
imaging laser through resonance for a complete construction of the “ion” numbers.
In most of the experiments, the imaging laser is set on resonance and only one image
is recorded. Such single images are converted to the total “ion” number by using a
conversion factor since the “ion” number is varying in a rather fixed profile against
the imaging frequency. As we can see from the calibrated points, this simple trick
works very well. At late times the number of visible ion cores seen, ∼ 8 × 105 , agrees
reasonably well with the number of parent Rydberg atoms initially excited, ∼ 7×105 ,
as determined using a seed UNP. Figure adapted from [35].
81
In reference [65], it is theoretically predicted that S states should have isotropically
attractive Van der Waals interactions while the interactions between D states should
be mostly repulsive and only for some restricted range of orientations will they attract
weakly. So Penning ionization should be slow for D states but should happen a lot
more frequently for S states. The explains our LIF images very well, especially the
image at an evolution time of 4.1µs. For S states, collisions produce electrons, initially
at the center of the cloud where the Penning ionization is the strongest which will
collide and l-mix with Rydberg atoms before they escape from the cloud. However,
free electron production for D states mostly from BBR ionization is much slower so
the number of the early electrons and the l-changing rate, is much less resulting in
reduced visibility for the core ions. As we can see from Figure 6.7 and 6.8, S states
lead to a faster evolution of the overall visibility.
This contrast in visibility would be more easy to see for even shorter evolution
times tevol < 4µs since the D state would be completely dark. However, this is not
the case because it will take a long time for two nearest Rydberg atoms to Penning
ionize in the Van Der Waals interaction regime. Assuming that the potential energy
of a pair Rydberg atoms that experience an attractive force is completely converted
to kinetic energy as they accelerate towards each other, then the collision time can
be approximated using [70] as
Z
R0
T =
Rc
1
q
C6
(R
6 −
C6
)
R06
dR.
(6.2)
× 4/M
where R0 is the initial separation between the pair of Rydberg atoms and Rc is the
lower limit of their separation which can be approximated as the diameter of the
Rydberg atom ∼ 482 a.u. = 2304a.u.. Since Rc R0 , Rc can be treated as zero. For
82
strontium, we have [65]
C6 = 15 × n11 = 15 × 4811 = 4.675 × 1019 a.u.,
Z
0
1
√
x3
dx ≈ 0.43, M = 16.04 × 104 a.u..
1 − x6
(6.3)
(6.4)
Equation 6.2 can be reduced to
R4
T = 0×
2
r
16.04 × 104
× 0.43a.u.
4.675 × 1019
T ≈ 0.039 × (R0 in µm)4 µs.
(6.5)
(6.6)
Since collision time is very sensitive to the change in the R0 as shown in Figure 6.9,
it is important to correctly estimate R0 . If we use the nearest neighbor distribution
for the initial condition in Figure 6.7 as described in reference [35], it is possible
to have hundreds of Rydberg atom pairs separated by R0 < 3.5µm which yields
collision times that allow electron production on micro-second time scales. However,
if Rydberg blockade is important, the initial separation between two Rydberg atoms
can be no less than the blockade radius Rvdw which can be calculated assuming an
overall linewidth of 1M Hz
C6
= 1M Hz = 2.419 × 10−11 a.u.
6
Rvdw
(6.7)
Rvdw = 1.116 × 105 a.u. = 5.91µm.
(6.8)
This restriction will yield a minimum collision time of 47µs! Apparently, Rydberg
blockade effects don’t appear to be dominant. There are multiple possible explanations. For instance, equation 6.2 assumes the initial velocities of the Rydberg atoms
is zero. While this assumption holds in the sense that the initial 7mK thermal velocities have no fixed direction, it is possible, in some cases, to have two Rydberg atoms
83
Figure 6.9 : Collision Time Vs Initial Inter-Rydberg atoms Distance
84
moving towards each other with their thermal velocity ∼ 1m/s. This initial velocity, about 1µm per microsecond, can greatly reduce the total collision time. On the
other hand, the overall linewidth for the particular two-photon transition employed
can be a lot larger than the pure laser linewidth. Moreover since we don’t know the
exact detuning of the transition, it is possible that the interaction induced energy
shift between two very close Rydberg atoms matches the detuning which facilitate
closer Rydberg pair excitation. In fact, the strong AC Stark shifts due to 5s2 → 5s5p
transition could lead to an anti-blockade effect [71] that makes Rydberg blockade
inefficient. Also, stray fields may further complicate the Rydberg blockade.
After the very first phases of the dynamics, the initial differences in the behavior of
S states and D states due to their different interactions will be smeared out by other
effects. For example, l-mixed Rydberg atoms possess permanent dipole moments
and their Penning ionization rates will be larger than those expected for Penning
ionization due to the Van der Waals interaction. Following formation of the electron
trap, the l-mixed Rydberg atoms will dominate in the cloud and all memory of the
initial state will be lost.
In the previous studies [67], [66] and [68], it has been suggested that almost all early
electrons are generated by the 300K blackbody radiation and the Penning ionization
plays a negligible role in the dynamics until the l-mixed Rydberg atoms dominate
the Rydberg gas. However, according to our study, we believe Penning ionization in
the early stage of the evolution is at least as significant as BBR induced ionization in
producing the early electrons.
85
Chapter 7
Conclusion and Outlook
We have created strontium Rydberg atoms in two environments and demonstrated
several of the key experimental capabilities they offer which will directly enable a
variety of future studies.
We produced very-high-n (n ∼ 300) strontium Rydberg atoms in a collimated
atomic beam using two-photon excitation. Spectroscopically, we observed the normal
mass shifts of the even isotopes 86 Sr, 88 Sr, 84 Sr and the hyperfine induced mixings of
the odd isotope 87 Sr. By exciting Rydberg atoms in a DC field, we studied the Stark
shifts of various Rydberg states. A two-active-electron model was used to analyze the
data and provided results in excellent agreement with experiment. The derived “nD”
Stark state possesses a large permanent dipole moment in DC fields approaching
those at which states in neighboring manifolds first cross. This allows production of
a quasi-one-dimensional state and that can be engineered into circular or elliptical
states. In other words, the dipole moment of this Rydberg atom is not only very large
(when it is a quasi-one-dimensional state) but is also tunable (when it is manipulated
into states of varying ellipticity). Also, the ability to near simultaneously produce
many Rydberg atoms opens the opportunity to study Rydberg-Rydberg interactions.
The Van der Waals interaction between Rydberg atoms of such high n has not yet
been investigated either experimentally or theoretically. Additionally, we can create
planetary states by exciting the second valence electron and examine the interactions
between two excited electrons within one Rydberg atom.
86
We also studied ultracold strontium Rydberg gases by imaging light scattered
from the core ions of l-mixed Rydberg atoms, i.e. the laser induced fluorescence.
To induce rapid l-mixing and rapidly image all the Rydberg atoms, a weak ultracold
neutral plasma is utilized to introduce free electrons that collide with Rydberg atoms.
The temporal and spatial resolution of this imaging technique permits the study of
the evolution of an ultracold Rydberg gas. In the early stages of this evolution, Penning ionization as well as blackbody radiation induce photoionization and introduce
initial electrons that l-mix the low-l Rydberg atoms and make them visible to the
imaging light. We observed faster dynamics for Rydberg states that have attractive
interactions than for states that have mostly repulsive interactions which highlights
the importance of Penning ionization. In the future, we plan to exploit the strong
interactions between cold strontium Rydberg atoms to form a more highly correlated
ultracold neutral plasma.
87
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