Many-body dipole interactions
Jes?us V. Hern?andez
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
August 9, 2008
Many-body dipole interactions
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee.
This dissertation does not include proprietary or classified information.
Jes?us V. Hern?andez
Certificate of Approval:
Michael S. Pindzola
Professor
Physics
Francis Robicheaux, Chair
Professor
Physics
Allen Landers
Associate Professor
Physics
Stuart Loch
Associate Professor
Physics
George T. Flowers
Interim Dean, Graduate School
Many-body dipole interactions
Jes?us V. Hern?andez
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Dissertation Abstract
Many-body dipole interactions
Jes?us V. Hern?andez
Doctor of Philosophy, August 9, 2008
(B.S., Kansas State University, 2003)
143 Typed Pages
Directed by Francis Robicheaux
This dissertation presents the study of controllable but strong long-range inter-
action between dipoles. In particular, we investigate the excitation and interaction
between atoms in a cold gas where the collisional time is much greater than the inter-
action time between neighboring Rydberg atoms. In addition to quantum systems, we
also examine the excitation properties of a collection of classical electric dipoles cre-
ated by optically driving metallic nanospheres. We use various theoretical techniques
to simulate these systems, including the direct numerical solutions to Schr?odinger?s
equation, a Monte Carlo method, and a simple coupled point-dipole model.
We first perform simulations involving the excitation of a collection of cold
atoms to Rydberg states. When the interaction energy between excited atoms is
large enough to shift multiply-excited states out of resonance with the tightly tuned
excitation laser, the number of atoms able to be excited is suppressed, creating a
dipole blockade effect. The blockade effect offers exciting possibilities in the control
of quantum bits, which is crucial for the development of quantum computing. We
iv
also examined the effects of density variation with respect to the the dipole blockade
with three different models.
We then simulate the coherent interactions between Rydberg atoms. If the atoms
are excited into states where the dipole-dipole interaction between them allows for
resonant energy transfer to occur, then one state can freely hop from one atom to the
next via the dipole-dipole interaction. We generated band structures for one, two,
and three dimensional lattices and characterized the nature of the coherent hopping.
This hopping is also studied in both a perfect and non-perfect lattice case which
should be possible to examine experimentally.
Next, we simulate the effect of special excitation pulses on a cold gas of atoms.
First a rotary echo sequence is used to examine the coherent nature of a frozen
Rydberg gas. If collective excitation and de-excitation is present with little or no
source of dephasing, after these pulses the system should be returned to a state
with few excitations, and a strong echo signal should occur. We investigate systems
that should display a perfect echo and systems where the interaction between atoms
reduces the echo signal. A spin echo sequence is also used on a system of coherent
hopping excitations, and we simulate how the strength of a spin echo signal is affected
by thermal motion.
Finally, we describe the dipole-dipole interactions between a linear array of opti-
cally driven metallic nanospheres. These classical model calculations incorporate the
v
full electric field generated by an oscillating electric dipole. The effects due to retarda-
tion of the generated electric field must be taken into account and several interesting
effects are explored such as the ability to preferentially excite specific nanospheres.
vi
Acknowledgments
First and foremost, I must thank my advisor Prof. Francis Robicheaux for his
patience, guidance, and support during all of my graduate work. I would also like to
thank him for it making possible for me study and learn physics at a higher level.
Without his sincere efforts and willingness to help me whenever I showed up at his
door, I doubt I would have this opportunity to thank him now.
I am obliged to the other members of my advisory committee, Prof. Michael S.
Pindzola, Prof. Allen Landers, and Prof. Stuart Loch for their time and insightful
suggestions towards the completion of this work. I knew during my time at Auburn
I could get help from any one of them at any time, and the welcoming atmosphere
helped foster a great learning experience.
I would be remiss if I didn?t thank my parents Jes?us and Debra Hern?andez for
instilling in me a sense of joy in discovery, and for always being there when I needed
them. I would also like to thank my sisters, Nica and Julia for putting up with me
for all of these years.
I must also thank Jessica Williams for her tireless editing efforts and always
believing in me.
Finally I would like to thank the National Science Foundation for their financial
support of the research presented in this dissertation.
vii
Style manual or journal used Journals of the American Physical Society (together
with the style known as?auphd?). Bibliograpy follows the style used by the American
Physical Society.
Computer software used The document preparation package TEX (specifically
LATEX) together with the departmental style-file auphd.sty.
viii
Table of Contents
List of Figures xi
1 Introduction 1
1.1 Dipole blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Coherent hopping of excitation . . . . . . . . . . . . . . . . . . . . . 4
1.3 The effects of rotary and spin echo sequences on a Rydberg gas . . . 6
1.4 Interactions between classical dipole moments: nanospheres . . . . . . 8
2 Dipole Blockade 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 The many-body wavefunction . . . . . . . . . . . . . . . . . . . . . . 13
2.3 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 First order interaction . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 Second order interaction (van Der Waals) . . . . . . . . . . . . 18
2.4 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 Pair correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.2 Number correlation . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Filling factor for excitations . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Phase gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.7 Dipole blockade at higher densities . . . . . . . . . . . . . . . . . . . 32
2.7.1 Blockade radius . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.7.2 Effects of density variation . . . . . . . . . . . . . . . . . . . . 39
2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Coherent Hopping of Excitation 47
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.1 Field free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.2 Static electric field . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2.3 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3 Simple band structures . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.1 Linear lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.2 Square lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3.3 Cubic lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
ix
3.4 Hopping in a small non-perfect lattice . . . . . . . . . . . . . . . . . . 63
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.6 Validity of the essential state model . . . . . . . . . . . . . . . . . . . 69
3.6.1 Field-free case . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6.2 Static electric field . . . . . . . . . . . . . . . . . . . . . . . . 71
4 The effects of rotary and spin echo sequences on a Rydberg gas 74
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 Rotary echo of a dense Rydberg gas . . . . . . . . . . . . . . . . . . . 75
4.2.1 Rotary echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 Spin echo for Rydberg hoppers . . . . . . . . . . . . . . . . . . . . . 86
4.3.1 Spin echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5 Interactions between classical dipole moments: nanospheres 96
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2 Coupled dipole method . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6 Summary 113
6.1 Dipole blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Coherent hopping of excitation . . . . . . . . . . . . . . . . . . . . . 116
6.3 The effects of rotary and spin echo sequences on a Rydberg gas . . . 117
6.4 Interactions between classical dipole moments: nanospheres . . . . . . 119
Bibliography 121
x
List of Figures
2.1 The two particle correlation function for a group of Rydberg atoms.
The top graph is for the second order (van Der Waals) interaction and
the bottom graph is for the first order interaction. The axes are the
distances between particles in the x and z direction. The grey-scale
indicates the probability of finding a second Rydberg atomata location
with respect to a Rydberg atom at the origin. Black is the smallest
probability. The correlation function drops to zero outside of 25 ?m
in the vdW case and outside of 50 ?m for the first order case. This is
due to the fact that there are no pairs that exist with distances greater
than 2R0, where R0 is the size of the uniformly distributed sphere. . 22
2.2 Q values as a function of the fraction excited (Pe) for different den-
sities. The solid line was calculated using a density ?0 ? 1.3 ? 1010
cm?3. Moving from left to right, the densities decrease as follows:
?0/8 (dashed), ?0/27 (dotted), ?0/64 (dash-dotted), and?0/512 (thick-
dashed). The inset is a blowup of the region where there are very few
excitations. Figure 3.2.a is for the vdW case and Figure 3.2.b is for the
dipole dipole interaction. The line Q = ?Pe results when the atoms
are uncorrelated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Figure 2.3.a is the maximum ?Ne? for atoms in a line of length L. The
solid line is for the perfect lattice case while the dashed line is when the
atoms are randomly distributed on the line. In Figure 2.3.b the solid
line is the probability of being in a state with exactly one atom excited
as a function of the length L for the perfect lattice case. The dashed
line is the probability of being in a state with only two atoms excited
and the dotted line is the probability of being in a state with three
atoms excited; all for the perfect lattice case. Figure 2.3.c is the same
plot as 2.3.b except for the randomly distributed geometry. Figure
2.3.d is also the same plot as 2.3.c and 2.3.b except using a Poissonian
distribution. The dash-dot line is the probability of zero atoms being
excited, the solid is for one excited, the dashed is for two excited, and
the dotted is for three excited. . . . . . . . . . . . . . . . . . . . . . 28
xi
2.4 Two regions of equal size and density are excited by a ? ? 2? ? ?
sequence of pulses in the following manner: group 1 is excited by a ?
pulse then group 2 is excited by a 2? pulse and finally group 1 is de-
excited by another? pulse. Figure 2.4.a shows the phase shift ??/? as
a function of the average maximum intergroup pair distance, ?Rmax?
,divided by the blockade distance Rb. The solid line is for the vdW
case (1/R6), while the dashed line is for the dipole-dipole case (1/R3).
Figure 2.4.b is ??/? as a function of ??min?/?0, where ?0 is the pair
energy of two excited atoms separated by Rb. The solid line is for the
many atom case. The dot indicates a phase shift of 0.9?. The dashed
line is for the perfect two particle case, where the two atoms are in the
center of each sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 (a) is the number of blocked atoms per excited atom, Nb, as a function
of the density at various ?0. (dashed, +) ?0 = 210 kHz, (dotted, ?)
?0 = 210/? kHz, and (dash-dot, *) ?0 = 210/(2?) kHz. The lines were
generated using the fact that Nb ??4/5 and the points were generated
by the many body wavefunction at ? = 20?s. (b) is the blockade
radius, Rb, as a function of the density at the same ?0?s as part (a). 37
2.6 (a) is the number of excited atomsNexc in a volume 4??2?? found at a
scaled distance ? =radicalbigr2c/?2 +z2/?z2 from the center of the MOT at
various ?0. (full) ?0 = 210 kHz, (dashed) ?0 = 210/? kHz, and (dot-
ted) ?0 = 210/(2?). ?0 = 210/(2?) kHz. (b) is Nb and Nexc/4??2??
as a function of ?. (full) ?0 = 210 kHz, (dashed) ?0 = 210/? kHz,
and (dotted) ?0 = 210/(2?). The vertical line is at ? = ?5, where
Nexc/4??2?? is maximum. . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7 The fraction excited versus excitation time for the many-body wave-
function calculation (dashed) and the simple sin2 model (full). (a)
is for a low density (5.6 ? 1010cm?3) and (b) is for a high density
(2.8?1012cm?3). Both calculations were done using ?0 = 210/? kHz 42
2.8 A comparison between the experimental data and the three models:
the number excited versus excitation time for the convolved many-
body wavefunction calculation (full), the simple sin2 model (dashed),
and the MC model (dotted). The (+) are experimental data points
(?0 = 210 kHz). (a) was calculated using a Rabi frequency ?0 = 210
kHz, (b) ?0 = 210/? kHz and (c) using ?0 = 210/(2?) kHz. . . . . . 44
xii
3.1 The scaled bandenergy (bandenergy divided byradicalbig8?/3(dna1,nb0)2/?x3)
as function of the wavenumber k for a linear array of atoms with one
p state and the rest s states. . . . . . . . . . . . . . . . . . . . . . . 57
3.2 (a) is the Brillouin zone for a square lattice with special points and
paths. (b) is the Brillouin zone for a simple cubic lattice with special
points and paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3 The scaled bandenergy (bandenergy divided byradicalbig8?/3(dna1,nb0)2/?x3)
as function of the wavenumber k for a square array of atoms with one
p state and the rest s states. . . . . . . . . . . . . . . . . . . . . . . . 60
3.4 The scaled bandenergy (bandenergy divided byradicalbig8?/3(dna1,nb0)2/?x3)
as function of the wavenumber k for a cubic array of atoms with one p
state and the rest s states. . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5 A schematic drawing of the setup for a system of (a) a slightly irregular
linear lattice (b) a slightly irregular lattice with the third site empty,
and (c) a slightly irregular 2?2 square lattice. . . . . . . . . . . . . . 64
3.6 The probability P for finding the n = 61 state on various atoms as a function of
time. The atoms are placed in small regions with a width of 3 ?m and separated by
a center-to-center distance of 20 ?m. In all cases the n = 61 state in initially in the
leftmost region. (a) For two regions, where the solid line is probability of finding
the n = 61 state in the leftmost region, and the dashed line is the probability of
finding it in the rightmost region. (b) For six regions (see Fig. 3.5.a), the solid line
is the probability of finding the n = 61 state in the leftmost region. The dashed
line is the probability of finding it in the adjacent region, and the dotted line is
the probability of finding it in the rightmost region. For six regions with an atom
missing in third region (see Fig. 3.5.b), the solid line is the probability of finding
the n = 61 state in the leftmost region. The dashed line is the probability of finding
it in the adjacent region, and the dotted line is the probability of finding it in the
rightmost region. Note the similarity between (a) and (c) which indicates that the
excitations cannot hop over the skipped region. . . . . . . . . . . . . . . . . 66
3.7 The probability of the n = 61 state being in a region as a function of
time for a slightly irregular 2?2 square lattice (See Fig. 3.5.c). The
solid line is the probability of finding the n = 61 state in the initial
region I, the dashed line is the probability of finding it in an adjacent
region A, and the dotted line is the probability of finding it in the
opposite corner region O. . . . . . . . . . . . . . . . . . . . . . . . . . 68
xiii
4.1 An illustration of a rotary echo sequence. The sign of the excitation
amplitude F is smoothly flipped after time ?p. In this case ?p = 350
ns, and is indicated by the dashed line. . . . . . . . . . . . . . . . . . 79
4.2 Number excited versus the timing of the sign change of the excitation
amplitude. This echo signal is for 5 isolated atoms. Note that there are
zero excitations when ?p = 250 ns, exactly half of the total excitation
time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3 Number excited versus the timing of the sign change of the excitation
amplitude. This echo signal is for 10 atoms in the perfect blockade
regime. Note the perfect echo signal at 250 ns. . . . . . . . . . . . . . 82
4.4 Number excited versus the timing of the sign change of the excitation
amplitude. The echo signal for (a) peak density of ? = 5.0?1012 cm?3
and (b) peak density ? = 1.5?1013 cm?3. . . . . . . . . . . . . . . . 84
4.5 Number excited versus the timing of the sign change of the entire
interaction Hamiltonian. This echo signal is for a high peak density of
? = 1.5?1013 cm?3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.6 Spin Echo signal in a 0K gas with exactly one hopper versus difference
in scaled relaxation times. When ??/thop = 0 a prefect signal is seen. 91
4.7 Spin Echo signal in a gas with exactly one hopper as a function of
temperature for various ? relaxation times. In each case ?1 = ?2 = ?.
The solid line is for a ? = thop, the dashed line is for ? = 2thop, the
dotted line is for ? = 3thop, the perforated line is for ? = 4thop, and the
chain line is for ? = 5thop. . . . . . . . . . . . . . . . . . . . . . . . . 92
5.1 A schematic drawing of a possible set up for a regularly spaced lin-
ear array of silver nanospheres. A wide (compared to the size of a
nanosphere of radius a) beam of light of frequency ? is propagated
along the array?s axis in the particular medium where the array is as-
sembled. The light absorbed and scattered by each MNS will in turn
excite neighboring MNS?s and a coherent wave of oscillating electric
dipoles will be produced. . . . . . . . . . . . . . . . . . . . . . . . . 97
xiv
5.2 The ohmic power as a function of the position of 10 MNS?s in a reg-
ularly spaced linear array. The center-to-center inter-particle distance
is d = 80 nm and the diameter, 2a, of each MNS is 50 nm. A plane
transversely polarized electromagnetic wave is propagated from left to
right along the axis of the array. All of the MNS?s absorb and scat-
ter the incident electromagnetic wave of magnitude 1 V/m. The solid
line is for a chosen frequency ? = 4.85?1015 rad/s (?diel = 259 nm)
when most of the power is in the first sphere. The dashed line is for
a frequency of ? = 4.62 ? 1015 rad/s (?diel = 272 nm) when most of
the power is in the last sphere. This asymmetry between first and last
MNS?s vanishes in the near field approximation at all wavelengths. . 103
5.3 The ohmic power as a function of the frequency, ?, of a plane elec-
tromagnetic wave propagating along the axis of a regular linear array
of MNS?s. The dimension of the array and MNS?s is the same as in
Figs. 5.1 and 5.2. All of the MNS?s absorb and scatter the incident
beam of light as it comes in from left to right. The solid line is the
ohmic power of the first (leftmost) MNS and the dashed line is for
the last (tenth) one. The dotted line is the power response for the
single MNS case. The inset is the same as the main figure, but using
only the near field approximation. Note that it is now impossible to
preferentially excite the first or last MNS by modifying the driving
frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.4 The same physical set up as Fig. 5.2, but this time only the first sphere
is externally excited. Plotted is the differential radiated power per solid
angle versus the frequency of the incident light. The solid line is the
power scattered in the forward direction and the dashed line is the
power scattered in the backward direction. . . . . . . . . . . . . . . 108
xv
5.5 Again the same physical set up as Fig. 5.4, but this time the first sphere
is externally excited into an L mode. Plotted is the differential radiated
power per solid angle versus the scattering angle ? for two frequencies,
where ? is the angle relative to the line of MNS?s. The solid line is the
power scattered at ? = ?SP = 5.0?1015 rad/s and the dashed line is
the power scattered when? = 5.5?1015 rad/s. In order to more cleary
show the asymmetry the dashed line is scaled by 2.0 i.e the amplitude of
the driving force is increased by about 40%. The intermediate electric
field of the oscillating dipoles gives the asymmetry. The inset also plots
differential radiated power per solid angle versus the scattering angle
for ? = ?SP, but here the MNS?s are forced to oscillate in phase. . . 110
xvi
Chapter 1
Introduction
A Rydberg atom is characterized by the high principal quantum number, n, of
the valence electron and its resultant exaggerated properties [1]. Since the size of the
electron?s orbit increases as n2, the size of a Rydberg atom can become quite large.
This large separation from the nucleus makes the electron highly sensitive to external
electric fields and enables Rydberg atoms to support large electric dipole moments.
The interplay between the dipole moments of Rydberg atoms has been an active topic
of study for the last few decades, from early experiments observing resonant Rydberg-
Rydberg collisions [2] and spectral line broadening [3] to its possible use in creating a
quantum computer [4?7]. Additional advancements in laser cooling and trapping have
also allowed for the investigation of Rydberg-Rydberg interactions without having to
compensate for the effects of thermal motion [8,9] creating a frozen Rydberg gas
which has offered unprecedented experimental control over large collections of atoms.
The ability to actively control strongly correlated quantum systems allows for the
exploration of the fundamental nature of many-body physics and opens the door for
various applications in quantum computing [10].
The large charge separation between the highly excited valence electron of a
Rydberg atom and positively charged core leads to a large dipole moment that scales
as n2 [1]. If the dipoles are directly induced by a static electric field, the interaction
between dipole moments is a first order one. In this case, the potential between dipole
1
moments falls off as 1/R3, faster than the 1/R nature of the Coulomb potential
between charges, however because the dipole moment grows as n2, the first order
interaction between Rydbergs can be increased or decreased by a factor of n4. In
the absence of an electric field it is possible for two excited atoms to experience a
second order force (van der Waals) by inducing dipole moments on each other. Using
standard second order perturbation theory, the van der Waals interaction will fall off
as 1/R6 and scale as n11 [11]. In either case, experimentalists are able to precisely
control the long range forces between cold atoms by changing the density of the
trapped gas and the principal quantum number of atoms [12].
1.1 Dipole blockade
Two advantages of the interaction between dipole moments of Rydberg atoms
are its controllable strength and long-range nature. Under certain conditions, the
finely tuned excitation of a single cold Rydberg atom can even prevent the excitation
of many other nearby atoms, which is known as the dipole blockade effect [13]. This is
a result of an energy shift due to the dipole-dipole interaction between excited states
pushing the multiply-excited state out of resonance [4].
One of the early motivations for developing the dipole blockade was for uses in
quantum computing. The strong interaction between Rydberg atoms allows for the
swift exchange of information [4], but there are significant technical challenges in
placing Rydberg atoms on a regular lattice. A dipole blockade can help ensure that
only one Rydberg atom is excited per lattice site. In essence, the dipole blockade offers
2
coherent manipulation over a large collection of atoms, enabling tight macroscopic
control over microscopic systems. This ability to precisely interact with quantum
systems is critical in the development of quantum computing.
The dipole blockade effect was first observed in 2004 with the excitation of a
dense ultracold gas of Rb atoms. By utilizing a second order dipole interaction
between Rydberg atoms, a suppression in the total number excited was observed
[14?16]. It has also been seen in other experimental setups which use first order
dipole interactions [17?19]. The dipole blockade effect has even been explored in
highly dense Bose-Einstein condensates [20].
As more and more experiments were able to create a dipole blockade effect,
it became important for the underlying physics of the dipole blockade to be well
understood. Because the dipole blockade can involve the interaction between many
atoms, many-body effects will play a large role. An early simulation of the dipole
blockade incorporated a simple mean field model which accounted for the many-
body effects in the system by adding an energy shift caused by the other excited
atoms to single-atom states [14]. While successful in describing the experimental
measurements, the simple mean field model worked well only for high laser intensities
and could not give information about the spatial relationships between excited atoms
[21]. Later simulations used Monte Carlo approaches [17,22?24] or a perturbative
technique [25]. A drawback of the Monte Carlo models is that they do not contain
information that can be obtained by using a wavefunction such as phase shifts or the
amplitudes of specific states within the full wavefunction.
3
In chapter 2 of this dissertation, we studied these many body effects by simulat-
ing the many-body wavefunction using a direct numerical solution of Schr?odinger?s
equation. As a result of using the many-body wavefunction we were able to cal-
culate many useful properties of the system such as the fraction excited, the 2D
two-particle correlation function between Rydberg atoms, number correlations, how
the excitations fill space, and the phase errors accumulated during certain quantum
information operations. We also compared our many-body wavefunction model to
two other different models against a recent experiment where very large numbers of
atoms are blockaded. This study lead to the realization that density variations in the
ultracold gas can have a large effect on the excitation dynamics of the system.
1.2 Coherent hopping of excitation
The energy shift created by the dipole-dipole interaction between atoms in a
frozen Rydberg gas can make it possible for resonant energy transfer to occur [8].
For example, when an atom in state A interacts via the dipole-dipole potential with
an atom in state B the atom in state A is converted to B? while simultaneously the
atom in state B is converted to A?. In this case, the energy cost for this transition
must be minimal: EA +EB ? EB? +EA?. As a result of this transition it appears
as though the characters of the two atoms have switched places or hopped [9]. This
coherent hopping of excitation states makes the gas behave more like an amorphous
solid than a gas, with excitations moving through the system like phonons, particles,
or holes [26].
4
A major impetus for investigations of resonant energy transfer in a frozen Ry-
dberg gas is the use of neutral atoms in quantum computing. In order to fully
understand and control a quantum computer composed of atoms in an ultracold Ry-
dberg gas, the spatial properties and dynamics of the excitations must be carefully
explained [27].
Resonant processes in a Rydberg gas were first experimentally observed in 1998
by Mourachko et al. [9] and Anderson et al. [8] where the energy-transfer resonances
in the excitation spectra were substantially broadened. The broadening of the line
widths indicated that many-body effects were just as important to the dynamics
of the system as two-body effects. Subsequent experiments have also verified the
importance of simultaneous multiple atom interactions [28?31]. These complicated
many-body interactions have also been studied in experiments investigating the effect
of the orientation between dipole moments on the hopping probability [32], and the
effects of a magnetic field on line broadening [33].
Since many-body phenomena in these systems are prevalent, the theoretical tools
used to study them must also be able to take into account this complicated nature.
Quantitative models have been performed to reproduce the major features observed in
experiments [8,9,33], but more precise calculations are required to completely explain
the measured linewidths. In fact, early models suggested that coherent hopping
could not occur unless the temperature of the gas was extraordinarily low [34], but
recent models have shown that hopping can occur at temperatures experimentally
available [26].
5
In chapter 3, we studied the coherent hopping of character between Rydberg
atoms by employing an essential states model and numerically solving Schr?odinger?s
equation. The essential states model only includes states in the simulation that are
degenerate or nearly degenerate. We investigated two separate cases: (1) where a
single s atom is placed in a regular array filled with p atoms, and the s state hops
from |spp???p? to |psp???p?, |pps???p?, etc., and (2) where atoms are excited in
a static electric field and the hopping occurs among the highest Stark states. Our
calculations were able to produce band structures, and the effect of slight irregularities
of the lattice on coherent hopping.
1.3 The effects of rotary and spin echo sequences on a Rydberg gas
In chapter 2, it will be shown that when the van der Waals interaction between
Rydberg atoms is large enough it can push the excited states out of resonance with
the laser. A Rydberg atom will block other excited states from occurring within a
radius rb. Within rb the excitation cannot be pinned to a single atom, but rather
it is de-localized over all Nb atoms inside the blockaded region. These so called
?superatoms? [35] then collectively oscillate at a rate dependent on the ?Nb. In
section 2.7.1 of this dissertation we investigated the impacts of density variation
on the collective oscillation rate and found that inhomogeneities in the density of
the frozen Rydberg gas lead to inhomogeneities in the collective oscillation rates.
The coherent nature of the system is therefore hidden to simple measurements that
integrate over the entire signal [36]. Luckily echo techniques from nuclear magnetic
6
resonance physics have been developed to overcome analogous inhomogeneities in
oscillations due to irregular magnetic fields [37,38]. A strong rotary spin echo signal
is seen when the sign of the excitation amplitude is reversed exactly half-way through
the total excitation time if no dephasing occurs aside from the excitation source [38].
In section 3.4, a coherent hopping of excitation is proposed between slightly
irregular lattice sites. An experiment with clearly separated excitation regions should
be able to detect the spatial nature of the coherent hopping. This type of spatially
resolved measurement would be impossible in regular gas, and the coherent nature
of the hopping would be hidden as the rate of hopping between atoms would be as
inhomogeneous due to the irregularity in spacing between atoms (the hopping time
between atoms is related to the inverse of the interaction energy between them: thop ?
R3/n4) [26]. This inhomogeneity due to spatial arrangement can also be overcome
by an echo technique; this time a regular spin echo signal can be observed when the
sample is excited via a special pulse sequence and outside sources of coherence such
as temperature are small [37].
There has been a recent experiment which excited ground state atoms to Rydberg
states in a very strongly interacting regime and proved the coherence of the system by
using the rotary echo technique [36]. They were able to flip the excitation amplitude
by using an RF field to shift the phase of the pulse by ?. There has also been an
experiment which used a regular spin echo technique to control the dephasing of a
single 87Rb atom [39].
7
In chapter 4, we investigated the effects of using echo sequences on collections
of Rydberg atoms by using the many-body wavefunction approach to numerically
solve Schr?odinger?s equation. Since our approach in chapter 2 took into account the
spatial correlations between Rydberg states and employed a pseudoparticle approach
similar to the superatom picture, our model was able to correctly reproduce the
echo signal seen experimentally that a simple mean field approach could not. Our
simulations demonstrated a perfect rotary echo when within the perfect blockade
limit, and illustrated the effects of increasing density on the strength of the rotary
echo signal. We also look at an ideal case where the sign of the entire Hamiltonian is
reversed, thus in effect perfectly reversing the time evolution of the system. We next
examined the effect of a spin echo sequence on the coherent hopping of excitation in a
gas of Rydberg atoms. When the temperature is 0?K, a system with only one hopper
will perfectly echo, regardless of the inhomogeneity of the gas. We also examined the
effect of nonzero temperatures on the coherence of the system.
1.4 Interactions between classical dipole moments: nanospheres
When excited by light, the electrons in a metallic nanosphere oscillate strongly
and coherently. This resonant behavior is a consequence of the finite size of the
nanoparticle and the restoring force of the nuclei [40]. The collective oscillation
of the electrons is known as the dipole plasmon resonance of the nanoparticle. In
fact, the optical excitation of plasmons is the most efficient process by which light
interacts with matter [41]. These driven nanoparticles behave like oscillating classical
8
dipole moments, creating their own time-dependent electric fields. In turn, these
generated electric fields interact with nearby nanoparticles and thus couple collections
of nanoparticles together producing coherent waves of oscillating dipole moments.
The study of metallic nanoparticles has been active for well over a century, dating
as far back as the research done by Michael Faraday [42]. There have since been sev-
eral models made to describe the optical properties of a single metallic nanosphere,
from the exact solution to Maxwell?s equations by Mie to the Drude model of di-
electrics [40]. More recent studies have focused on the optical response of collec-
tions of nanoparticle and their uses in nanolithography [43], as efficient light-guiding
nanolenses [44], spasers (surface plasmon amplification by stimulated emission of ra-
diation) that generate intense local nanoscale optical-frequency fields [45], and other
various possible uses as subwavelength optical devices.
Modern fabrication techniques using electron-beam lithography have allowed for
the precise control over the location and particle size of nanoparticles [46]. Because
a linear array is one of the easiest configurations to set up experimentally and theo-
retically, it has been extensively studied. Theoretical analyses of the optical response
of a linear array of nanoparticles have been performed in variety of ways including
the discrete dipole approximation [47], the multiple multipole model [48], the finite
difference time domain method [46], and the T-matrix method [49].
In chapter 5, we described the dipole-dipole interactions between a linear ar-
ray of optically driven silver nanospheres by using a coupled dipole approximation
model that used the full electric field of an oscillating dipole. Using this model, each
9
nanosphere is represented by a single point dipole that responds linearly to a driving
electric field. By using classical electrodynamics and the Drude model to model the
response of the nanospheres to light, we created self-consistent equations of motion
for the oscillating electric dipoles. By solving these coupled equations we could deter-
mine the size of the dipole moment (and the ohmic power absorbed and radiated) of
each nanosphere. Our simulations emphasized the importance of using the full field
in the calculations and the resulting asymmetric response observed as a result of the
retarded scattered light.
10
Chapter 2
Dipole Blockade
2.1 Introduction
Atoms excited into Rydberg states are large in size and thus able to support large
dipole moments. The long range interactions between these large dipoles have been
a popular topic of study over the last several years. [3,8,9,14,15,19,20,23,24] When
an ultracold gas of Rydberg atoms is suffiently cold and dense enough, the effect of
the motion of the atoms is small compared to the possibly large interactions between
them. These relatively strong and long range interactions between dipole moments
should dominate the physics in this ultracold regime, making it possible to study
interesting many-body effects in detail. One such effect is seen when the interaction
energy between Rydberg atoms is large enough to shift multiply-excited states out
of resonance with a tightly tuned excitation laser. With multiple excitation states
now blocked from occurring, the number of atoms able to be excited is suppressed.
This suppression in the number of excited atoms is known as the dipole blockade
effect [13].
The dipole blockade effect, was first described in Ref. [4] as a method for the
macroscopic control and operation of quantum logic gates. Jaksch et al. proposed
for the blockade to be generated by utilizing the first order interactions between
11
dipole moments created by exciting a gas of ultracold alkali atoms in a static electric-
field. With recent advancements in cooling and trapping, the first order static dipole-
dipole (SDD) interaction described in that proposal has been experimentally verified
to produce a dipole blockade in a gas of ultracold cesium atoms [17] . In Ref. [13]
the proposed interaction was between Rydberg atoms at F?orster resonance and a
successful dipole blockade was observed for this configuration as well [18]. A blockade
using the second order van der Waals (vdW) interaction has also been seen in various
experiments [14?16,19,50]. In this chapter we investigate both the first order SDD
and second order vdW interactions between large numbers of atoms. In both cases,
the initial state is coherent; in other words we do not allow the ground level to decay
or be repopulated. If a narrow enough bandwith laser is used to drive atoms from the
ground state to a high Rydberg state (n > 40) on resonance, the dominant process
will be the dipole allowed transition [51] and each atom can be treated as a strictly
two-level system.
The possibly large and long range interaction between pairs of Rydberg atoms
implies that any theoretical study of a large collection of N atoms must be able to
take into account the many-body nature of the system. The effect of many-bodies
has been modeled using a mean field [14], a Mont?e Carlo approach [17,22,23], and a
perturbative approach [25]. By a direct numerical solution of Schr?odinger?s equation
we were able to compute and retain various interesting properties of the many-body
wavefunction. Although the Monte Carlo procedures require less computational ef-
fort, they cannot be used to probe quantities such as phase shifts or the amplitudes
12
of certain pieces of the full wavefunction useful for quantum computing. The use of
a mean field in [14] works well for high laser powers, but is unable to give informa-
tion about the spatial correlations that can develop within the excited gas. Unless
otherwise noted, atomic units will be used throughout this paper.
2.2 The many-body wavefunction
In this section we will describe the techniques involved in the direct numerical
solution to the many-atom wavefunction. We will also discuss how we solved for
the wavefunction in a manner that allowed us to check for convergence. Once the
wavefunction has been propagated, it is possible to compute the number of excited
atoms and various correlation functions.
We begin by treating each atom as a purely two-level system with one level being
the initial tightly bound, non-decaying ground state |g?, and the other being a highly
excited Rydberg state |e?. For the purposes of this chapter the locations of the atoms
will be fixed in space. This is a reasonable approximation if the temperature of the
gas is low enough and the time duration of the exciting laser pulse is short enough.
The motion of the atoms must be small compared to distances where the interaction
strength between dipoles is dominant. For example, using conditions similar to the
experiment in Ref. [51], where the temperature of the gas is around 100 ?K and the
peak density is 1011 cm?3, the simulation time must belessorapproxeql200 ns in order to effectively
ignore the motion of the atoms. We expanded the wavefunction onto product states
13
of two-level systems (|a?1 ?|b?2 = |ab?, etc):
|?(t)? = agg...g(t)|gg...g?+aeg...g(t)|eg...g?+
??? +aee...g(t)|ee...g?+aee...e(t)|ee...e? (2.1)
=
summationdisplay
?
a?(t)|??.
Quite clearly the number of basis states needed to completely describe the wavefunc-
tion increases as 2N and becomes prohibitively large when N > 15. As it would
be impractical to do otherwise, we did not use all of the basis states in the expan-
sion but recursively eliminated them using the pseudoparticle approach described by
Ref. [21]. These pseudoparticles have an interaction strength with the laser ?W
times bigger than the single atom case, where the weight W is the number of atoms
in each pseudoparticle. In this method, real atoms (pseudoparticles with weight equal
to one) were randomly placed within a volume large enough to cover the region of
correlation; then these strongly blockaded atoms were recursively grouped together
to form pseudoparticles until the number of real atoms, N, was reduced down to the
desired number of pseudoparticles, Np. The recursion was as follows: (1) the nearest
neighbors j and k were found, (2) these two ?atoms? were joined and replaced by a
pseudoparticle ilocated at the center of mass positionvectorri = (Wjvectorrj+Wkvectorrk)/(Wj+Wk),
and (3) the weight of the created pseudoparticle is increased to Wi = Wj +Wk while
removing j and k from the simulation. The errors created by forcing correlations
between atoms can be controlled by increasing the number of pseudoparticles.
14
Now the interaction between excited pseudoparticles j and k, can be calculated
by averaging over the interactions V between all of the pairs of associated atoms:
Vjk = 1W
jWk
summationdisplay
n?j
summationdisplay
m?k
Vnm(Rnm), (2.2)
where Rnm is the distance between atoms n and m, which belong to pseudoparticles
j and k respectively. Another way to calculate the interaction would be to simply
use the positions of the pseudoparticles themselves. Now Vjk = V(Rjk). Clearly as
the number of pseudoparticles is increased the two different methods will give the
same interaction energies. We used the latter approach in our calculations because it
converged faster with respect to random geometries as long as enough pseudoparticles
were used.
2.3 The Hamiltonian
The Hamiltonian of this system is:
?H(t) = summationdisplay
j
?H(1)j (t) +summationdisplay
j<k
Vjk|ejek??ejek|
?H(1)j (t) = ?(??(t)+?(t))|ej??ej|+F(t)?0
2
radicalbigW
j(|gj??ej|+|ej??gj|) (2.3)
where Vjk is the interaction between two excited states for a pseudoparticle pair
(j,k). The detuning of the laser is ??(t), and ?(t) is a mean field energy shift due
to excited atoms outside of the simulated volume. Since our simulation box size
15
cannot be accurately approximated as infinite, we need to try to reduce erroneous
edge effects when attempting to model the entire gas. We accomplished this by
employing wrap boundary conditions, which also help to hasten convergence. The
single atom Rabi frequency between states |g? and |e? is ?0, and the number of atoms
in each pseudoparticle j, is Wj. F(t) describes the time dependence of the shape of
the laser amplitude. Equation 2.3 was also obtained by using the rotating wave
approximation. If we use parameters similar to [14], the laser frequency is in the UV
range and at a relatively low intensity, allowing for the rotating wave approximation
to be a good one. If, however, the parameters in [52] are used and a microwave
transfer pulse is introduced then careful consideration must be used when applying
this approximation. If the detuning is set to zero, the mean field is negligible, and the
system is perfectly blockaded (i.e. at most one excitation is possible), then a system
of N atoms evolves in an oscillatory manner between all the atoms in the ground
state and a symmetrical state with only one excited Rydberg atom. The frequency
of this oscillation is ?N?0. The dynamics of this two level system can be given as:
?
??ag(t)
ae(t)
?
??=
?
?? cos(?(t)/2) ?isin(?(t)/2)
?isin(?(t)/2) cos(?(t)/2)
?
??
?
??ag(0)
ae(0)
?
??, (2.4)
where ?(t) = 2?N?0integraltexttt0 F(t?)dt?. The probability of being in the excited state at the
end of the laser pulse is sin2(?(t)/2).
16
In order to solve Schr?odinger?s Equation numerically, we use the split operator
method:
|?(t+?t)? = e?i ?Hdiag(t)?t/2e?i ?Hoff?te?i ?Hdiag(t)?t/2|?(t)?, (2.5)
where ?Hdiag is the matrix of diagonal elements (Rydberg-Rydberg interaction) of ?H
and ?Hoff is the matrix of off-diagonal elements (laser interaction) of ?H, such that
?H = ?Hdiag + ?Hoff. At this point, the laser part of the interaction is approximated
via the second order Runge-Kutta method. The single time step error introduced
by using the split operator is of O(?t3), and by using second order Runge-Kutta the
norm ??(t)|?(t)? is no longer conserved. By using small enough time steps we are
able to minimize this error.
2.3.1 First order interaction
For a group of cold Rydberg atoms in a static electric-field E, the SDD interaction
between atoms j and k can be written as:
Vjk =
vectordj ? vectordk ?3(vectordj ? ?Rjk)(vectordk ? ?Rjk)
R3jk , (2.6)
where vectordj is the dipole matrix element for atom j. If the electric field is in the ?z
direction and the atoms are excited to the highest energy Stark state, then the dipole
17
matrix elements are: vectord = (3/2)n2?z. Finally, Eq. 2.6 can be rewritten as:
Vjk = 9n
4
4
1?3cos2(?jk)
R3jk , (2.7)
where ?jk is the angle formed between the vector pointing from atom j to k and the
direction of the electric field. Clearly the SDD interaction is proportional to 1/R3
and scales like n4. When using the SDD interaction It is also important to take
into account the relative orientation between atoms, at least in the spherical polar
direction.
2.3.2 Second order interaction (van Der Waals)
With no electric field, a second order interaction between two atoms in states ?0
and ?0 respectively, takes place:
Vjk = 1R6
jk
summationdisplay
?,?
|??|??|vectorrj ?vectorrk ?3(vectorrj ? ?Rjk)(vectorrk ? ?Rjk)|?0?|?0?|2
E? +E? ?(E?0 +E?0) . (2.8)
This can be more concisely written as Vjk = ?C6/R6jk and the C6 coefficient is
proportional to n11. There are several standard methods of the calculating the C6
[53,54]. We calculated the C6 coefficient while taking into account the spin-orbit
interaction by using second order perturbation theory and quantum defect theory.
The necessary wavefunctions were generated by direct numerical integration in the
radial direction using a square root mesh and the quantum defects given by Ref. [55],
and the angular depedence was found using angular momentum relationships outlined
18
in Ref. [56]. The resulting C6 was in good agreement with results that did not take
into account the spin-orbit interaction [53].
In this chapter we only looked at ns?ns interactions in order to eliminate any
directional dependence.
2.4 Correlation
After propagating the wave equation for a period of time, the fraction of excited
atoms can be easily determined by projecting the wavefunction onto the states |??:
Pe =
summationdisplay
?
N?
N |??|??|
2 =summationdisplay
?
N?
N |a?(?)|
2, (2.9)
where N? is the number of atoms excited in state |??. It is also possible to calculate
a spatial correlation function, which is the probability of finding two excited atoms
separated by a distance R divided by the probabilities of each individual atom:
C(R) = Pee(R)P(R)P2
e
. (2.10)
Pee(R) is the probability to find two atoms excited a distance R apart:
Pee(R) =
summationdisplay
j<k
|??jk|??|2, (2.11)
19
such that state |?jk? has excited atoms j and k a distance R apart. P(R) is the
probability of finding a pair of atoms separated by a distance R. By examining this
two particle correlation function, we can estimate Rb, the blockade radius [21].
2.4.1 Pair correlation
If we take advantage of the azimuthal symmetry of the Hamiltonian and take
a slice in the xz plane, we can look at the 2D pair correlation function. For this
simulation we used ten 85Rb atoms and randomly placed them using a uniform spatial
distribution in a sphere of radius R0 with no wrap boundary condition and no mean
field. For the vdW case R0 = 12.5 ?m and for the SDD case R0 = 25 ?m. In both
cases we excite via a ? pulse for 120 ns that is perfectly on resonance (?? = 0). A ?
pulse perfectly tansfers a two level system from the ground state to the excited state.
According to Eq. 2.4 when ?(t) = ?, the system is completely in the excited state.
This means that a ? pulse can be defined by 2?N?0integraltexttt0 F(t?)dt? = ?, or
integraldisplay t
t0
F(t?)dt? = ?2?N?
0
. (2.12)
The shape of the laser pulse F(t) is given by:
F(t) = S?
0?
e?(t/?)2, (2.13)
20
where S is related to the laser amplitude and? is the pulse duration. Using the above
shape of the laser pulse, a ? pulse occurs when the laser amplitude
S? = 12
radicalbigg?
N. (2.14)
The top plot in Fig. 2.1 is the pair correlation C(?x,?z) for the vdW interaction.
As expected, the vdW case is also symmetric in the polar direction. A reasonably
accurate blockade region can be defined by Rb. If an atom is excited then all of the
other atoms within a distance Rb are blockaded. In the dipole-dipole case (static
electric field in the ?z direction) the interaction potential depends on ?jk, the angle
between the electric field and the vector connecting atoms j and k. In fact when
cos(?jk) = ?radicalbig1/3, Vjk = 0; the interaction vanishes and the two atoms are indepen-
dent of each other. This can be seen in the bottom plot of Fig. 2.1; along the angle
cos(?jk) =radicalbig1/3 (?jk ? 55?), the blockade region is pierced. In order to have a well
defined Rb as in the vdW case, the geometry must be set up where cos(?jk) negationslash= ?radicalbig1/3
for as many j,k pairs as possible. This can be accomplished by placing atoms in a
thin plate on the xy plane; by decreasing the thickness we can adjust the number of
pairs that lie along the critical angle. If the atoms lie in the xy plane, then C is only
a function of R, C ?C(R).
The value of Rb can be estimated by finding the distance where Vjk ? planckover2pi1/?. Since
the correlation function is not a perfect step function [21], the value ofRb is not exact
unless a well-defined criteria is established. In this paper, the blockade radius for a
21
?x (?m)
?z (
?m)
 0  5  10  15  20  25
 0
 5
 10
 15
 20
 25
?x (?m)
?z (
?m)
 0  10  20  30  40  50
 0
 10
 20
 30
 40
 50
Figure 2.1: The two particle correlation function for a group of Rydberg atoms. The
top graph is for the second order (van Der Waals) interaction and the bottom graph
is for the first order interaction. The axes are the distances between particles in the x
and z direction. The grey-scale indicates the probability of finding a second Rydberg
atom at a location with respect to a Rydberg atom at the origin. Black is the smallest
probability. The correlation function drops to zero outside of 25 ?m in the vdW case
and outside of 50 ?m for the first order case. This is due to the fact that there are no
pairs that exist with distances greater than 2R0, where R0 is the size of the uniformly
distributed sphere.
22
given S/? is defined in the following manner: it was the diameter of the largest sphere
where the average maximum number of excited atoms within that sphere was 1.01.
As a test of our definition ofRb, we computed the probability of being in the blockade
state (|egg...g?+|geg...g?+???+|ggg...e?)/?N, and the phase of the amplitude of
the blockade state. When ?Ne? = 1.01, then the probability of being in the blockade
state is approximately 98%, and the phase shift is about 0.52?; for a perfect blockade
these should be 100% and ?/2.
2.4.2 Number correlation
When a collection of cold atoms is in a blockade configuration (i.e. the physical
parameters are such that the interaction energy between two Rydberg atoms is large
enough to shift the pair out of the two excitation resonance), the number of atoms
that can be excited (Ne) is suppressed. By taking repeated measurements of Ne, we
can find the relationship between the mean ?Ne? and the variance ?N2e???Ne?2. The
Mandel Q-parameter, Q ? ?N2e???Ne?2?Ne? ?1, is a useful quantity to compare the atom
counting statistics to a Poissonian distribution [57]. For a Poissonian distribution the
mean is equal to the variance, so Q = 0. In the case of blockaded atoms Q should be
less than zero; this corresponds to a sub-Poissonian distribution. The Q parameter
reflects the measure of how efficiently the system is blockaded. Recent experiments
have been able to measureQvalues [19], and in this section we will present the results
of our simulations of Q. By time propogating the many-body wave function, Q is
readily found. For a collection of N completely independent quantum atoms with a
23
probability Pe of being excited:
?Ne? =
Nsummationdisplay
j=0
?
?? N
j
?
??jPj
e(1?Pe)
N?j = NPe (2.15)
and
?N2e? =
Nsummationdisplay
j=0
?
?? N
j
?
??j2Pj
e(1?Pe)
N?j (2.16)
= (NPe)2 +NPe(1?Pe). (2.17)
So for the extreme uncorrelated case, Q = ?Pe (i.e. uncorrelated quantum systems
do not give Q = 0, but minus the fraction excited). The other extreme case is if Nb
atoms are located within a blockade region. Now only up to one atom can be excited,
thus:
?Ne? = NbPe (2.18)
?N2e? = NbPe. (2.19)
So Q = ?NbPe for the case of Nb atoms being within the blockade radius. In Fig. 2.2,
the Q value is plotted as a function of the fraction of atoms excited for different
densities: ?0 ? 1.3 ? 1010 cm?3, ?0/8, ?0/27, ?0/64, and ?0/512. In all cases, 10
atoms were used in the simulation. Figure 2.2.a was generated using the vdW (1/R6)
interaction, while Fig. 2.2.b used the dipole-dipole (1/R3). At the lowest density,
24
0 0.01 0.02 0.03 0.04 0.05
Pe
-0.05
-0.04
-0.03
-0.02
-0.01
0
Q
0 0.2 0.4 0.6 0.8 1
Pe
-1
-0.8
-0.6
-0.4
-0.2
0
Q
(a)
0 0.01 0.02 0.03 0.04 0.05
Pe
-0.05
-0.04
-0.03
-0.02
-0.01
0
Q
0 0.2 0.4 0.6 0.8 1
Pe
-1
-0.8
-0.6
-0.4
-0.2
0
Q
(b)
Figure 2.2: Q values as a function of the fraction excited (Pe) for different densities.
The solid line was calculated using a density ?0 ? 1.3 ? 1010 cm?3. Moving from
left to right, the densities decrease as follows: ?0/8 (dashed), ?0/27 (dotted), ?0/64
(dash-dotted), and ?0/512 (thick-dashed). The inset is a blowup of the region where
there are very few excitations. Figure 3.2.a is for the vdW case and Figure 3.2.b
is for the dipole dipole interaction. The line Q = ?Pe results when the atoms are
uncorrelated. 25
Q(Pe) ??1Pe, indicating a very low level of correlation. In essence, we have a group
of isolated two level systems. As the density is increased, the slope of Q(Pe) steepens
suggesting the system is becoming more and more highly correlated. Because the
dipole-dipole interaction has a longer range than the vdW case, the slope of Q(Pe)
is steeper for the former. The inset examines the system at low laser intensities, or
equivalently when few atoms are excited. It can been seen that the slope remains
relatively constant in both the vdW and dipole-dipole cases, showing the system is
still correlated even when few particles are excited. Although the excited atoms are
not in a blockade configuration with each other, the blockaded atoms near by certainly
are. These blockaded atoms are still conveying the information about the system to
the separated excited atoms. They are in the ground state because they are blocked
from being excited, not just because the fraction of excitiation is small. It is the
initial state of the system which holds information, even before the interactions are
turned on. By comparing the slope m of Q(Pe) to the perfectly blockaded case, we
can in effect measure how efficient the interaction is at turning the system from a
large collection of uncorrelated two level systems to a single two level system. In fact,
?N/m gives a rough indication of how many uncorrelated two level systems remain
in the collection.
2.5 Filling factor for excitations
If a group of atoms is all within Rb then at most only one excitation is allowed.
If every excited atom creates a bubble of radius Rb around itself, then in a given
26
volume the maximum number of excited atoms should be approximately the number
of bubbles that can fit into that volume. We examined how the maximum number of
allowed excitations increases as the volume containing the atoms is increased while
maintaining a fixed density. We focused on the case of the atoms placed in a line
because that situation should show the largest effect. We initially placed 5 atoms on
a linear lattice 1.7 ?m apart and excited the system to the maximum number excited
via a pulse 120 ns long. For these parameters Rb ? 8.2 ?m. We then increased the
number of atoms in the line and again excited to the maximum Ne. If the bubble
picture is correct, then the maximum number of excited atoms should remain about
1 until two bubbles of radius Rb can fit into the line; then the number of excitations
should jump to 2. We also looked at the non-lattice case where we randomly placed
the appropriate number of atoms in the lengths used in the perfect lattice case. In
Fig. 2.3a we plotted the maximum ?Ne? as a function of chain length L for both
the perfect lattice case and the random case. In either case, the average number of
excited atoms rises fairly linearly with chain length, and as soon as there is room for
more than one excitation, ?Ne? does not jump up to the next integer number excited.
The ?Ne? fills the region rather smoothly, so the bubble picture is not entirely correct.
Figs. 2.3.b and 2.3.c illustrate how states with excited atoms become available as L
is increased. The solid line is the probability of being in a state with only one atom
excited, the dashed line is for being in a state with only two excited, and the dotted
line is for three. The smoothness of Fig. 2.3.a can be explained for by looking at
the probabilities of being in various excited states. By the time the probability of
27
10 20 30
L (?m)
0
0.5
1
1.5
2
2.5
3
<N
e>
10 20 30
L (?m)
0
0.2
0.4
0.6
0.8
1
Prob
10 20 30
L (?m)
0
0.2
0.4
0.6
0.8
1
Prob.
10 20 30
L (?m)
0
0.2
0.4
0.6
0.8
1
Prob.
(a) (b)
(c) (d)
Figure 2.3: Figure 2.3.a is the maximum ?Ne? for atoms in a line of length L. The
solid line is for the perfect lattice case while the dashed line is when the atoms are
randomly distributed on the line. In Figure 2.3.b the solid line is the probability
of being in a state with exactly one atom excited as a function of the length L for
the perfect lattice case. The dashed line is the probability of being in a state with
only two atoms excited and the dotted line is the probability of being in a state with
three atoms excited; all for the perfect lattice case. Figure 2.3.c is the same plot as
2.3.b except for the randomly distributed geometry. Figure 2.3.d is also the same
plot as 2.3.c and 2.3.b except using a Poissonian distribution. The dash-dot line is
the probability of zero atoms being excited, the solid is for one excited, the dashed is
for two excited, and the dotted is for three excited.
28
being in a state with only one excited atom is down to around 50%, the probability
of being in a state with two excited has risen to about 50%. Figure 2.3.d is similar to
Figs. 2.3.b and 2.3.c except that what is plotted here is the probabilities of being in
certain states given a Poissonian distribution. As expected, it is quite different from
the sub-Poissonian distribution of our correlated system.
2.6 Phase gates
If the system is sufficiently sparse, the Rydberg atoms all act as isolated two
level systems or, in the parlance of quantum computing, each atom represents an
independent qubit. When a group of atoms is perfectly blockaded, it also forms a
single two level system. This group of atoms appears as a two level system, but it is
actually a collection of atoms so tightly correlated that they act as a single two level
system. If we had two isolated groups of atoms each of the same size and density such
that each group was perfectly blockaded, then we would have two qubits. If we made
both groups such that they both sat withinRb, then again we would have a single two
level system. As we move one group outside of the blockade region, pairs of intergroup
atoms will no longer strongly interact with each other allowing the possibility of a
third level: both groups containing an excited atom. When the largest intergroup
pair distance is greater than RB, both groups are now independent of each other and
we are back to two uncorrelated two level systems. If they are not too far apart,
however, Rpair,max lessorsimilar Rb, most of the intergroup pairs are still correlated, thus both
29
groups are as well. Being in such a state would be undesirable as it leaves us with
neither a single two level system or two uncorrelated two level systems.
A quantum gate transforms an initial state to another state. We created two
spheres of cold atoms, each of radius R0 = Rb/4, separated by a center to center
distance D. Within each sphere, we randomly placed 8 atoms. The two groups of
atoms are then subjected to the following sequence of pulses: group one is excited
by a ? pulse, group 2 is excited by a 2? pulse (S2? = 2S?), and finally group 1 is
de-excited by another? pulse. The excited atoms interact via the dipole-dipole or van
der Waals interaction, depending on the situation. We varied the mean interaction
energy between the two groups by increasing D. In the ideal case we can represent
each group as a two level system, so the initial state is the ground state |gg?. When
the first group is excited by a? pulse |gg????i|eg?. If both groups are independent
of each other then a 2? pulse will take ?i|eg? ?? i|eg?, but if the groups are both
within Rb then it is impossible to excite the second atom and this pulse leaves the
state unaffected: ?i|eg????i|eg?. The final ? pulse to the first atom will de-excite
it and mupliply the state by ?i: for the uncorrelated case i|eg? ?? |gg? and for
the blockaded case ?i|eg??? ?|gg?. When the groups are independent, there is no
accumulated phase shift; the sequence of pulses leaves the original state unchanged.
When the system is blockaded, a phase shift (??) of ? is acquired, making a phase
gate [13].
The top plot in Fig. 2.4 shows the phase shift ??/? as a function of the average
maximum intergroup pair distance, ?Rmax? ,divided by the blockade distance. As D
30
0.5 1 1.5 2 2.5 3
<Rmax>/Rb
0
0.2
0.4
0.6
0.8
1
??/pi
(a)
00.511.522.533.5
<?min>/?0
0
0.2
0.4
0.6
0.8
1
??/pi
(b)
Figure 2.4: Two regions of equal size and density are excited by a ??2??? sequence
of pulses in the following manner: group 1 is excited by a ? pulse then group 2 is
excited by a 2? pulse and finally group 1 is de-excited by another ? pulse. Figure
2.4.a shows the phase shift ??/? as a function of the average maximum intergroup
pair distance, ?Rmax? ,divided by the blockade distance Rb. The solid line is for the
vdW case (1/R6), while the dashed line is for the dipole-dipole case (1/R3). Figure
2.4.b is ??/? as a function of ??min?/?0, where ?0 is the pair energy of two excited
atoms separated by Rb. The solid line is for the many atom case. The dot indicates
a phase shift of 0.9?. The dashed line is for the perfect two particle case, where the
two atoms are in the center of each sphere.
31
is increased, the distance between the two furthest pairs will also increase beyond Rb.
This allows for the possibility of more than one atom to be excited, thus introducing
an error into phase shift. The solid line in Fig. 2.4.a is for the vdW case and the dashed
is for the dipole-dipole interaction. The rapid 1/R6 scaling of the vdW interaction
can be seen in the steep drop of the phase shift with increasing ?Rmax?. As expected,
when ?Rmax? is small ??/? approaches 1, and when ?Rmax? is large ??/? tends to
0. With every intergroup distance an intergroup pair energy can be calculated, so
with each average maximum intergroup pair distance there is an associated average
minimum intergroup energy, ??min?, where ? = V?/planckover2pi1. The solid line in Fig. 2.4.b is
??/? as a function of ??min?/?0, where ?0 is the pair energy of two excited atoms
separated by Rb. The dot indicates a phase shift of 0.9?. The dashed line is for the
perfect two particle case where the two atoms are in the center of each sphere. If a
phase error of less than 10% is desired, then the average minimum pair energy must be
greater than about 2.5. If phase error of less than 5%, is required then ??min?> 3.5.
The difficulty in reducing the error is evident in the flatness of the curve in Fig. 2.4.b
as ??/? goes to 1.
2.7 Dipole blockade at higher densities
This section focuses on calculations regarding Rydberg excitation of ultracold
atoms at higher densities than previously discussed. We simulated the physical setup
similar to Ref. [50]. In that experiment, a two-photon excitation scheme is employed
from the 5S1/2 to 5P3/2 and finally to 43S1/2. Due to a large detuning to the blue on
32
the 5S1/2 to 5P3/2 transition, the three levels can be reduced to an effective two level
system [50]. So for all intents and purposes, we will consider each atom as a strictly
two-level system: a tightly bound, non-decaying ground state, |g? (5S1/2), and an
excited Rydberg state, |e? (43S1/2). The atoms are excited by a narrow bandwidth
laser which is quickly and smoothly switched on for an excitation time ? < 20?s. In
previous sections the time scale for collective excitation was governed by the shape
of the laser pulse, but in this case the time scales are set by the energies involved
in the many-body interactions. To match the experiment, we will take the density
distribution of ground state atoms to be Gaussian:
?(r,x,y,z) = ?0e?(x2+y2)/?2?z2/?z2, (2.20)
where ?0 is the peak density, ? = 12?m is the width in radial direction, and ?z =
220?m is the width in the axial direction. Since the peak density of the gas is quite
high (?? 3?1012 cm?3) and the van der Waals interaction between excited Rydberg
atoms can be very large (C6 ?n11), including many-body effects is important.
We take into account three interactions: the interaction of the laser with the
atoms, the van der Waals interaction between two excited Rydberg atoms, and a
mean field energy shift between an excited Rydberg atom and excited Rydberg atoms
outside of the simulated box. The experimental setup in [50] was able to cool the gas
to 3.4 ?K. At such a low temperature, the motion of the atoms is still small compared
to common blockade radii (v? ? 0.6?m, Rb ? 5?m), so we fixed the atoms in space.
33
We again used the expansion of Eq. 2.1 to describe the many-body wavefunction and
used a psuedoparticle approach very similar to one used in Ref. [21] to reduce the
number of basis states. In Sec. 2.2 the atoms were randomly placed within a volume
large enough to cover the region of correlation and then recursively grouped together
to form pseudoparticles. The atoms were grouped according to distance, with the
nearest neighbors being joined until the appropriate number of pseudoparticles was
reached. The location of the pseudoparticle was the center of mass position of the
associated atoms. When the number of atoms, Na, is low, the N2a nature of this
recursion is not significant when it comes to computing time. However, when Na
reaches the thousands needed to simulate densities along the lines of Ref. [50], using
that recursion relation becomes computationally taxing. Here we took an alternative
approach by using a Sobol sequence to place the pseudoparticles first. The Sobol
sequence is a quasi-random sequence that fills space in a more uniform manner than
uncorrelated random points [58]. It avoids the clumpiness that occurs when filling
a space with a random sequence, thus leading to quicker convergence. Once the
pseudoparticles are placed, we generate a random position for an atom and, using
wrapped boundary conditions, find which pseudoparticle it is nearest. The weight W
of that pseudoparticle is then increased by one and the process is repeated until all
of the atoms have been accounted for. There will now be a Poissonian distrubution
of atoms per pseudoparticle.
The Hamiltonian of the system can be written using Equation 2.3 where Vjk =
?C6/R6jk is the two particle interaction between pseudoparticles j and k. The 43S1/2
34
state has a repulsive van der Waals interaction (C6 = ?1.67 ?1019). The detuning
of the laser ??(t) = 0, and for the van der Waals potential ?(t) ??Pe(t)20C6?L?3,
where Pe(t) is the fraction of atoms excited at time t and L3 is the volume size. The
time dependence of the shape of the laser is described by:
F(t) =
?
???
???
e?25(t?tr)4/t4r for t<tr
1 for tr ?t??
, (2.21)
where tr = 100 ns ?? is the ramp-on time and ? is the excitation time.
2.7.1 Blockade radius
An atom can be considered blockaded when the two particle van der Waals inter-
action shifts the doubly excited state out of resonance, C6/R6 > ?. The interaction
distance at which this occurs is called the blockade radius, Rb ? (C6/?)1/6. An en-
semble of Nb blockaded atoms oscillates between a ground state and a symmetrical
state with one excitation at the frequency ? = ?Nb?0. At high densities the num-
ber of atoms blocked per excited atom, Nb, is large and closely follows a Poissonian
distribution. When Nb is small, this increase in frequency is not significant; as ?Nb
grows large, this effect becomes more important. A simple estimation of Nb depends
on the local density and the volume that encloses the ensemble: Nb ??R3b. In order
to estimate Rb, the number of blockaded atoms must be found. For an ensemble
of Nb blockaded atoms, Rb = (C6/?Nb?0)1/6. In turn, an excited atom blocks all
35
other atoms within a spherical volume (4/3)?R3b, so for a uniform distribution ?,
Nb = (4/3)?R3b?. These two equations can be solved leading to Rb ? ??1/15 and
Nb ? ?4/5. The plots in Figure 2.5 are the results for Rb and Nb as a function of
density for ranges which are similar to those found in the MOT used by [50]. The
lines were generated by the following equation:
Nb = ?
parenleftBigg
4?
3
radicalbiggC
6
?0?
parenrightBigg4/5
, (2.22)
where ? is a fit parameter to match the data generated by the many body
wavefunction calculations. We used an ? = 1.075. As expected, as the density
increases, Rb decreases and the Nb increases. The size of the Rb is dependent on
the local density. The difference in Rb from the lowest density edges of the MOT
to the peak density in the center is substantial. The difference in Nb from peak to
edge densities is also quite large, which means that excited atoms on the edges of
the MOT will oscillate many times slower than ones near the center. We introduce
a scaled distance ? = radicalbigr2c/?2 +z2/?z2 from the center of the MOT to study the
spacial locations of the excitations within the MOT. The plots in Figure 2.6 are the
Nexc in a volume 4??2?? (where ?? ? ?) and the Nb at a given scaled distance ?.
Most of the excited atoms occur at about ? = ?5, which is about 6.7 ?10?3 times
the peak density. The plot in Figure 2.6.b is both Nb and Nexc as a function of ?. As
the atoms are found further from the center of the MOT the number of excited atoms
per volume generally increases and the number of atoms that are blocked greatly
36
100
101
102
103
104
 1e-04  0.001  0.01  0.1  1  10
N b
? (1012 cm-3)
(a)
100
101
 1e-04  0.001  0.01  0.1  1  10
r b (
?m)
? (1012 cm-3)
(b)
Figure 2.5: (a) is the number of blocked atoms per excited atom, Nb, as a function
of the density at various ?0. (dashed, +) ?0 = 210 kHz, (dotted, ?) ?0 = 210/?
kHz, and (dash-dot, *) ?0 = 210/(2?) kHz. The lines were generated using the fact
that Nb ? ?4/5 and the points were generated by the many body wavefunction at
? = 20?s. (b) is the blockade radius, Rb, as a function of the density at the same
?0?s as part (a).
37
 0
 20
 40
 60
 80
 100
 120
 140
 160
 0  0.5  1  1.5  2  2.5  3
N exc
/4pi?
2 ??
?
(a)
10-2
10-1
100
101
102
103
104
 0.5  1  1.5  2  2.5  3
N b
N exc
/4pi?
2 ??
?
  Nb
  Nexc
(b)
Figure 2.6: (a) is the number of excited atoms Nexc in a volume 4??2?? found at
a scaled distance ? = radicalbigr2c/?2 +z2/?z2 from the center of the MOT at various
?0. (full) ?0 = 210 kHz, (dashed) ?0 = 210/? kHz, and (dotted) ?0 = 210/(2?).
?0 = 210/(2?) kHz. (b) is Nb and Nexc/4??2?? as a function of ?. (full) ?0 = 210
kHz, (dashed) ?0 = 210/? kHz, and (dotted) ?0 = 210/(2?). The vertical line is at
? = ?5, where Nexc/4??2?? is maximum.
38
decreases from the Nb at the peak density. At ? = 0, Nb ? 103, while at ? = ?5,
Nb ? 10 which means the majority of the oscillations in the system will be about a
factor of ten times slower than oscillations at the peak.
2.7.2 Effects of density variation
Even within a volume contained by Rb, the density can vary enough to have an
effect. For example, if Rb ? 6?m, the diameter of a blockade is approximately ?.
This means that density (and correspondingly Nb) can vary by an order of magnitude
within a blockade region. This variation in time scales and Rb indicates that in order
to correctly model the entire gas, the non-uniform density distribution of the MOT
must be accounted for. In other words, the local fraction excited will depend on the
local density?and the excitation time?. Unfortunately, the many-body wavefunction
calculations utilize wrapped boundary conditions and a mean field in order to make
convergence possible, both of which depend on a constant density across the simulated
volume.
Given a density distribution, the total number of atoms excited to a Rydberg
state after an excitation time ? will be:
Nexc(?) =
integraldisplay
Pe(?,?)? dV, (2.23)
where Pe(?,?) is the fraction excited after excitation time ? for a density ?. We
calculated Pe(?,?) for various densities by solving the many-body wavefunction, but
39
in these simulations we assumed that the density does not vary strongly within a
blockade region. This condition does not hold up when using the parameters in
Ref. [50] and will lead to a loss of accuracy in the calculations, but we still hoped for
qualitative agreement with experiment. If, as in our case, the density distribution is
Gaussian, this can be rewritten as
Nexc(?) = 2??2?z
integraldisplay
Pe(?,?)
radicalbig
ln(?0/?) d?. (2.24)
In order to accurately integrate numerically over the density, we used a simple linear
interpolation to get Pe(?,?) for values between the calculated values. The accuracy
of this integration is determined by the number of calculated density points and the
grid size in density.
Simple Sinusoidal Model
As a check, we also developed a simple model based on the idea that a strongly
blockaded ensemble of Nb atoms oscillates at ?Nb?0. For the large densities looked
at in this section, the number of blockaded atoms for a certain density varies in a
Poissonian fashion from trial to trial. Using these criteria, an estimate of the fraction
excited as a function of ? and ? can be found:
Pe,est(?,?) =
angbracketleftbigg 1
Nb(?) sin
2radicalbigNb(?)?0
2 ?
angbracketrightbigg
Nb
, (2.25)
40
where the brackets, ?????Nb, indicate an average over a Poissonian distribution in
Nb. Figure 2.7 shows a comparison between the fraction excited versus time for
the many-body wavefunction calculation and the simple sinusoidal model. The left
figure is for a fixed relatively low density and the right is a fixed high density. As
expected, the time dependence of the two models is similar, but the simple model
tends to slightly overestimate the fraction excited. The oscillations in the many-body
wavefunction calculations are similar to the sinusoidal model, indicating the coherent
nature of the system. At a higher density these oscillations are noticeably faster,
but still coherent. If the density across the system does not drastically change, the
collective Rabi oscillation is evident. If the density does change, the resulting high
?N
b fluctuations will mask the collective excitations.
Monte Carlo model
Due to the small fraction of atoms excited to a Rydberg state, people have
applied a Monte Carlo (MC) approach toward studying this system. We used a very
simple MC model that only allowed for excitations, no de-excitations. We started
by randomly placing 15 000 000 atoms in a Gaussian distribution with the same
parameters as above. For every time step ?t, each atom j has a probability of being
excited given by:
Pj = ?2?0 (?0/2)
2
(?0/2)2 + ?V2j ?t, (2.26)
41
 0
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
 0.035
 0  5  10  15  20
f
excitation time (?s)
(a)
 0
 0.0002
 0.0004
 0.0006
 0.0008
 0.001
 0.0012
 0.0014
 0.0016
 0  5  10  15  20
f
excitation time (?s)
(b)
Figure 2.7: The fraction excited versus excitation time for the many-body wavefunc-
tion calculation (dashed) and the simple sin2 model (full). (a) is for a low density
(5.6 ? 1010cm?3) and (b) is for a high density (2.8 ? 1012cm?3). Both calculations
were done using ?0 = 210/? kHz
42
where ?Vj =summationtextkVjk =summationtextk ?C6/r6jk is the energy shift between atomj and every other
excited atom. ?Vj is updated every time a new atom has been excited. This MC model
essentially blockades an atom if |?Vj| ? ?0 which is consistent with the definition of
being blockaded as previously described in Secs. 2.4.1 and 2.7.1. The MC model has
the advantage of not needing to be convolved, so it can serve as a quantitative check
on the previous two methods. The overestimation of the simple sin2 model is seen
when we convolve the simple model over the density distribution as in Figure 2.7.2,
but the two calculations reach the saturation number of Rydberg atoms at about the
same excitation time.
We also compared the convolved data to recent experimental data [50]. The
experiment used a Rabi frequency of ?0 = 210 kHz. The simulated results, while
having the correct qualitative shape and within about a factor of 2 in the saturated
number of excited atoms, is off when it comes to the time scale for saturation. We
repeated these calculations using two slower Rabi frequencies in an attempt to match
the times cale of the experiment. Unfortunately, as ?0 is decreased so does the Nexc.
This trend was consistent across all three models. We could not perform a calculation
that would match both the time dependence and the Nexc of the experiment with
any of the available adjustable physical parameters. This suggests that only taking
into account excited pair interactions and laser interactions while not accounting for
a strong variance in density across a blockade region, is not adequate enough to
correctly understand this system.
43
 0
 1
 2
 3
 4
 5
 6
 7
 8
 9
 0  5  10  15  20
N exc
 (10
3 )
excitation time (?s)
(a)
 0
 1
 2
 3
 4
 5
 6
 7
 8
 9
 0  5  10  15  20
N exc
 (10
3 )
excitation time (?s)
(b)
 0
 1
 2
 3
 4
 5
 6
 7
 8
 9
 0  5  10  15  20
N exc
 (10
3 )
excitation time (?s)
(c)
Figure 2.8: A comparison between the experimental data and the three models: the
number excited versus excitation time for the convolved many-body wavefunction
calculation (full), the simple sin2 model (dashed), and the MC model (dotted). The
(+) are experimental data points (?0 = 210 kHz). (a) was calculated using a Rabi
frequency ?0 = 210 kHz, (b) ?0 = 210/? kHz and (c) using ?0 = 210/(2?) kHz.
44
2.8 Conclusions
By solving for the many body wavefunction we were able to calculate many useful
quantities such as the 2D two particle correlation function which shows the angular
dependence of the first order dipole-dipole interaction. When using the dipole-dipole
interaction to investigate theQparameter or anything else that requires a well defined
blockade region, special care is needed to make sure that the critical angle ?jk ? 55?
is unattainable to pairs of atoms. We also calculated the Mandel Q-parameter, a
useful quantity for measuring the degree to which a gas is blockaded. The non-
excited atoms within a blockade region still affect atoms outside Rb, thus even when
few atoms are excited and the gas is dense enough, the system is still correlated.
The Q-parameter can also be used to determine the reduction in the number of two-
level systems remaining in the gas. If atoms are placed on a one dimensional lattice,
and excited in a manner that maximizes the number excited, the average maximum
number excited grows smoothly as a function of the lattice length. The size of Rb
can be seen, however, if the probability of finding Ne is plotted as a function of L.
Since we solved for the wavefunction, we were able to examine the use of groups
of blockaded atoms as phase gates. We calculated the phase accumulated during a
sequence of pulses and generated the errors acquired by a non-perfect phase gate as
a function of the interaction energy. In order to operate a phase gate that returns
??> 0.9?, a rather large interaction energy is required.
45
We also performed three very different model calculations that are all in good
qualitative agreement with each other. For such a large system of atoms, being within
a factor of 2 in both time scale and Nexc is encouraging. However, the calculations do
not agree well enough with experimental data to suggest that the underlying physics
of the system is completely understood. The biggest concern is the different time
scales for saturation between the computational models and the experimental data.
Since we were not able to take into account the density variations over Rb, the first
step towards developing a more accurate model might be to develop a method that
can account for this density variance and also converge within feasible limits. This
would present a challenge to any mean field calculations, as the value of the mean field
energy shift would depend on the location of the pseudoparticle within the MOT. The
actual shape of the gas is also very important if accurate models are to be developed;
in all of the simulations the time scale is largely determined by the slower oscillations
found toward the edges of the MOT where the density is lower.
46
Chapter 3
Coherent Hopping of Excitation
3.1 Introduction
There have been many recent experimental and theoretical investigations of the
interactions between Rydberg atoms. These possibly strong interactions can lead to
interesting many-body effects if the energy separation between excited states is small
and outside sources of decoherence are minimized. Since atoms are neutral, the typical
interaction potential between them is the dipolar interaction which is proportional to
the size of atoms squared divided by the cube of distance between them. Since the
size of an atom scales asn2, wherenis the principal quantum number, the interaction
scales as (nn?)2/R3. The large size of Rydberg atoms and the density of the atoms
determines the relevant energy scales.
There are numerous experiments that have been successful in exciting atoms to
Rydberg states that allow for the resonant exchange of energy to occur [2,8,9,29,31,
59,60]. In other words, a pair of atoms in states A and B can interact in a manner
that converts A?A? and B ?B?. The energy cost for this transition must be very
small; in other words energy must be roughly conserved: EA+EB ?EA? +EB?. These
experiments were all done in environments cold enough for the motion of the atoms to
be minimal, and the transitions were coherent and in both directions. The spectrum
47
of Rydberg atoms that allow these resonant transfers is fundamentally different than
the single atom case; some many-body processes are occurring [9,52].
In this chapter we present the results of simulating a cold Rydberg gas that has
been spatially ordered. The atoms are placed in specific areas and are not allowed
to move during important time scales. While this is not experimentally realistic, the
underlying physics should still be of interest in probing the nature of the coherent
energy exchange. For example, we calculated band structures for a single p state
Rydberg atom in various arrays of s state Rydberg atoms.
We also investigated finite systems where the location of each atom is not exact,
but randomly placed within small regions. These regions are then arranged in regular
patterns. Within each region a single atom will start in the state A or B. This setup
is experimentally viable as long as the width of an excitation beam is smaller than
the distance between the next beam. Our simulations were able to study the effects
of defects in the optical lattice.
As in most Rydberg-Rydberg interactions, we employed an essential states pic-
ture that includes only the states that are degenerate or nearly degenerate. As long
as the separation between energy states is not too small this approximation is ac-
curate. Since the energy difference between two states scales as 1/n3, the essential
states approximation loses accuracy as n increases. If the transition rate between
states is longer than the timescales of the simulation we can effectively ignore these
transitions. The applicability of the essential states model is justified in section 3.6.
We will use atomic units throughout this chapter except where noted.
48
3.2 Theory
The Hamiltonian for two interacting Rydberg atoms is
?H = ?H1 + ?H2 +V12, (3.1)
where ?H1 and ?H2 are the Hamiltonians for the two Rydberg atoms and V is the
interaction potential between the two Rydbergs. In order to describe V we define the
following coordinates: vectorR12 (the vector between the nuclei of atoms 1 and 2), vectorr1 (the
vector between the nucleus and the electron of atom 1), and vectorr2 (the vector between
the nucleus and the electron of atom 2). We will assume that the Rydberg atoms are
not so close that the two electrons will overlap, so we do not have to worry about
symmetrization of the wavefunction The interaction potential is now:
V12 = 1R
12
? 1vextendsinglevextendsingle
vextendsinglevectorR12 ?vectorr1
vextendsinglevextendsingle
vextendsingle
? 1vextendsinglevextendsingle
vextendsinglevectorR12 +vectorr2
vextendsinglevextendsingle
vextendsingle
+ 1vextendsinglevextendsingle
vextendsinglevectorR12 +vectorr2 ?vectorr1
vextendsinglevextendsingle
vextendsingle
? vectorr1 ?vectorr2 ?3(vectorr1 ?
?R12)(vectorr1 ? ?R12)
R312 . (3.2)
In this chapter we will only look at the lowest order nonzero coupling, which is the
dipole-dipole interaction shown in Eq. 3.2.
49
3.2.1 Field free
If the atoms are excited in a field free environment then eigenstates will have a
well defined angular momentum, therefore the coupling should be calculated between
states of specific angular momentum on each atom. In order to evaluate these matrix
elements we rewrote Eq. 3.2 in terms of separate radial and angular pieces. Using
angular momentum relationships
V12 = ?8?
radicalbigg2?
15
r1r2
R312
2summationdisplay
?=?2
[Y1(?r1)Y1(?r2)]2?Y?2?(?R12), (3.3)
where Y?m is a spherical harmonic function, and [Y1(?r1)Y1(?r2)]2? means the two spher-
ical harmonics are coupled to a total angular momentum of 2 and the azimuthal ?
via the standard Clebsch-Gordon coefficients.
In the absence of an electric field, atoms in states A and B, respectively, are
eigenstates of angular momentum. These eigenstates have a degeneracy of 2?A + 1
and 2?B +1. When the two atoms are coupled together through V there are 2(2?A +
1)(2?B + 1) available states. The factor of 2 out front is a result of the fact that the
two atoms can either be arranged |AB? or |BA?. Since the dipolar interaction, V,
depends on the first order spherical harmonic Y1 for each atom, the diagonal terms in
the coupling matrix are all zero (?AB|V|AB? = 0). The Y1 dependence also means
that A couples to B only when |?A ??B| = 1, and if we set ?R12 = ?z, then mA +mB
is a conserved quantity.
50
The wavefunction for a Rydberg atom can be written in terms of a radial piece,
Rn?(r), and an angular piece, Y?m(?r):
?n?m = Rn?(r)r Y?m(?r). (3.4)
The matrix element between states ?AB| and |B?A?? is given by
VAB,B?A? = ?8?
radicalbigg2?
15
(dnA?A,nB?B)2
R312
2summationdisplay
?=?2
Y?2?(?R12)??AmA,?BmB|
? [Y1(?r1)Y1(?r2)]2?|?Bm?B,?Am?A?, (3.5)
where ?AB| means that atom 1 is ?nA?AmA| and atom 2 is ?nB?BmB|, and |B?A??
means that atom 1 is |nB?Bm?B? and atom 2 is |nA?Am?A?. The dipole matrix element
d is defined as
dnA?A,nB?B =
integraldisplay ?
0
rRnA?A(r)RnB?B(r)dr. (3.6)
In this chapter we will focus on the interaction between an s state and a p state.
This simplifies the math a great deal, and nonzero matrix elements of the coupling
potential now become
V1m,00;001m? =
radicalbigg8?
3
(dnA1,nB0)2
R312 (?1)
m?
?
?
?? 1 1 2
m ?m? m? ?m
?
??Y
2,m??m(?R12), (3.7)
51
where (???) is the usual 3?j coefficient.
3.2.2 Static electric field
Another simple case to look at is the Rydberg-Rydberg interactions in a strong
static electric-field pointed in the z direction. This electric field breaks the spherical
symmetry and creates states with substantial dipole moments. As opposed to the field
free case, there are now nonzero diagonal matrix elements of the coupling potential.
Another difference is the number of states that can be coupled together through V.
While this can possibily lead to interesting physics, it does complicate the study
between two atoms. In this case we will choose two states that couple strong enough
to allow us to ignore possible coupling to other states.
The Rydberg atoms in the static electric-field are in Stark states, and the dipole
interaction between them causes a mixing with other Stark states within in the same
n manifold [11]. This mixing can be suppressed by exciting both of the atoms to
the highest (or lowest) energy states of the Stark manifold, and by increasing the
separation between atoms. For the rest of the subsection we will only use the highest
Stark states so we can label atoms by only the principal quantum number n. As
in the previous subsection, the dipole matrix elements between states in different
52
n-manifolds is of utmost importance:
?n|z|n? ? 32n2
?n?1|z|n? ? 13n2
?n?1|z|n+ 1? ? 19n2
?n|x|n?? = ?n|y|n?? = 0. (3.8)
Notice that the x and y components of the dipole interaction are 0 in this approxima-
tion. In order to investigate the largest interactions, we will only use the case where
the |n?n?| = 1.
For two atoms the diagonal elements of the coupling matrix are given by:
Vn,n?;n,n? = Vn?,n;n?,n =
parenleftbigg3nn?
2
parenrightbigg2 1?3(?R
12 ? ?z)2
R312 , (3.9)
and the off-diagonal elements are given by
Vn,n?;n?,n = Vn?,n;n,n? =
parenleftbiggnn?
3
parenrightbigg2 1?3(?R
12 ? ?z)2
R312 . (3.10)
3.2.3 Coherence
In order for the coherent hopping of character from one atom to another to occur
the electron states of the Rydberg atoms cannot couple to outside degrees of freedom.
As long as the simulated system is isolated from free electrons or photons then the
53
only source of decoherence will be the relative motion of the atoms. This source of
decoherence can be ignored as long the atoms do not move very much during relevant
time periods [26].
3.3 Simple band structures
Coherent hopping can be observed when all of the atoms are on a regular lattice.
We started by studying one of the simplest examples: one p atom in a group of s
atoms. Since there are three types of p states (m = ?1,0,1), and there is only one p
atom in our simulations, the system can be fully described by two quantum numbers:
|?m?. Atom ? has p character with an azimuthal quantum number m while all other
atoms are in the s state. Since there are three types of p states, this also means that
three modes of hopping are possible, so we label each mode by ?. Since we have an
ordered lattice of atoms, a natural representation of an eigenstate of the system can
expressed as a superposition of Bloch-type waves:
?vectork,? = 1?N
summationdisplay
??m?
eivectork?vectorR??Um??(vectork)|??m??. (3.11)
In Eq. 3.11,vectork is the wavenumber, N is the total number of atoms, vectorR? is the position
of atom ?, and Um? is a unitary matrix for any fixed vectork. Using Eq. 3.1 the time-
independent Schr?odinger equation becomes:
?H?vectork,? = ??(vectork)?vectork,?, (3.12)
54
where ??(vectork) is the eigenvalue of ?vectork,?. Since the diagonal elements of the Hamiltonian
are the same for every state they can be removed while still maintaining the relevant
physics. By projecting the state ??m| onto Eq. 3.12 and using the matrix elements
from Eq 3.7 we get:
summationdisplay
m?
Hmm?(vectork)Um??(vectork) = Um?(vectork)??(vectork), (3.13)
where
Hmm?(vectork) = ? ??x3(?1)m?
?
?? 1 1 2
m ?m? m? ?m
?
??
?
summationdisplay
??negationslash=?
e?ivectork?vectorR??? Y2,m?m?(
?R???)
R3??? , (3.14)
with ?x being the spacing between atoms and vectorR??? = vectorR? ? vectorR??.
In order to examine the band structure of this system we will have to extend it
toward the N ??? limit. This is accomplished by increasing the number of atoms
in the above sum until convergence is achieved.
Since the trace of the Hmm? matrix is 0 for all vectork, the sum of the band energies
must be 0. So when vectork ?vector0 there must be bands with positive and negative effective
mass. A wavepacket centered aroundvectork0 has the group velocity for band ?:
vectorv?(vectork0) =
bracketleftBigvector
?vectork??(vectork)
bracketrightBig
vectork=vectork0 . (3.15)
55
3.3.1 Linear lattice
The simplest lattice is a line of equally spaced atoms. The band structure for
this system can be seen in Fig. 3.1. Since it is possible to propagate two different
ways in a transverse manner (where the lobes of the p orbital are perpendicular to
the line of atoms) two of the bands will be degenerate. When k is small the bands
have a quadratic dependence: ?? ? A? + B?k2. Using Eq. 3.15, the magnitude of
the group velocity is proportional to k at small k. Some bands will have a particle
character (positive group velocity proportional tovectork for smallvectork) while others will have
hole character (negative group velocity proportional tovectork for small vectork).
The two degenerate transverse bands have hole character while the longitudinal
band has particle character. This band structure exactly matches the band structure
of a linear array of optically driven plasmons in metallic nanoparticles [46]. Since
the transverse bands are degenerate and the sum of the band energies must be 0,
the two bands will cross each other only at ? = 0. If the lattice is perfect then the
transverse and longitudinal waves are not coupled to each other; the bands exactly
cross. Defects in the lattice could cause a breaking of the degeneracy in the transverse
case, and then the exact crossing would be replaced by an avoided crossing. If the
effect of imperfections is small then the coupling between transverse and longitudinal
bands would be localized to vectork where ?(vectork) ? 0.
56
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1
?
k (pi/?x)
Figure 3.1: The scaled band energy (band energy divided by radicalbig8?/3(dna1,nb0)2/?x3)
as function of the wavenumber k for a linear array of atoms with one p state and the
rest s states.
57
3.3.2 Square lattice
The next logical extension was to look at the bands for a square lattice??(kx,ky).
The Brillouin zone for a square lattice has three special points: the center (?)
(kx,ky) = (0,0), the center of the side (X) (?/?x,0), and the corner (M) (?/?x,?/?y).
A common way of presenting the bands is to plot the energy along the three lines
that connect the special points: ? connects ? and X, ? connects ? to M, and Z
connects X to M. These are shown in Fig. 3.2. The band energies are shown in
Fig. 3.3 for the special lines as a function of k = |vectork| =radicalbigk2x +k2y. The nature of the
eigenvectors allow for the labeling of the different bands. At the center and corner
of the Brillouin zone, ? and X respectively, two of the bands are degenerate and one
is nondegenerate. The nondegenerate band must correspond to the p state having
m = 0 character since this is the state whose wavefunction has a nodal plane in the
xy plane. This is analogous to a transverse wave, i.e., the lobes of the p state are
perpendicular to k. This band exactly crosses the others since the m = 0 character
does not couple to states with m = ?1 character. The two degenerate bands near the
center of the Brillouin zone have the lobes of the p state in the xy plane. The band
that rises linearly with k has a p state with lobes perpendicular tovectork (transverse-like)
while the other band has lobes parallel to vectork (longitudinal-like).
The bands near k = 0 with a transverse nature have band energies that change
linearly with respect to k. Using Eq. 3.15 it becomes evident that the group velocity
will be constant near k = 0, and does not depend on the direction of propagation.
Therefore these transverse bands behave like neither particles or holes, but rather
58
(a)
(b)
Figure 3.2: (a) is the Brillouin zone for a square lattice with special points and paths.
(b) is the Brillouin zone for a simple cubic lattice with special points and paths.
59
-0.6
-0.4
-0.2
 0
 0.2
 0.4
 0.6
 0.8
 1
 0  0.2  0.4  0.6  0.8  1  1.2  1.4
?
k (pi/?x)
Figure 3.3: The scaled band energy (band energy divided by radicalbig8?/3(dna1,nb0)2/?x3)
as function of the wavenumber k for a square array of atoms with one p state and the
rest s states.
60
like photons or phonons. It is interesting to note that the nondegenerate band has
a negative group velocity, which means that the a wavepacket moves in the opposite
direction to the wavenumber. The longitudinal wave acts like a hole or a particle
depending onvectork.
3.3.3 Cubic lattice
The Brillouin zone for a simple cubic lattice has four special points: the center
of the cube (?) (kx,ky,kz) = (0,0,0), the center of a face (X) (?/?x,0,0), the center
of an edge (M) (?/?x,?/?x,0), and a corner (R) (?/?x,?/?x,?/?x). We computed
energies along six paths that connect these points: ? which connects ? to X, S
which connects X to R, T which connects M to R, ? which connects ? to M, Z
which connects X to M, and ? which connects ? to R.
In Fig. 3.4 we plotted the band energies ??(kx,ky,kz) as a function of k =
radicalbigk2
x +k2y +k2z. Again the character of the eigenvalues let us label the different bands.
All four of the special points have degenerate states and the ?, T, and ? paths also
have degenerate bands.
While the cubic lattice is very simple, the resultant band structure it not. As
with the square lattice, the particle or hole nature of the bands is dependent on the
direction of vectork near k ? 0.
61
-0.8
-0.6
-0.4
-0.2
 0
 0.2
 0.4
 0.6
 0.8
 1
 1.2
 0  0.5  1  1.5  2
?
k (pi/?x)
Figure 3.4: The scaled band energy (band energy divided by radicalbig8?/3(dna1,nb0)2/?x3)
as function of the wavenumber k for a cubic array of atoms with one p state and the
rest s states.
62
3.4 Hopping in a small non-perfect lattice
The actual hopping of a Rydberg excitation can be seen experimentally by using a
CCD camera and state-selective field ionization. Since the states can be distinguished
by ramping on an electric field, we assumed the atoms are always in a strong electric
field that places the Rydberg atoms into an extreme Stark state. In this section we
will study the case where one atom is excited to the n = 61 state and the rest of the
atoms are in the n = 60 state. Each atom is randomly placed in a 27 ?m3 cube and
each cube is separated by a center-to-center distance of 20 ?m. In every case we also
had the electric field pointed perpendicular to the line or plane of atoms (?z? ?R = 0).
The first case we simulated was a line of atoms along the x-axis with the n = 61
atom starting in the leftmost region. In Fig. 3.6.a we plotted the result for the 2
atoms. The solid line is the probability of finding the n = 61 state in the left region,
while the dashed line is the probability of finding it in the right region. If the spacing
between atoms was fixed then the probability would oscillate between 0 and 1 at a
fixed frequency ? that depended on the distance. Since the atoms are not evenly
spaced, but randomly placed within regions, damping occurs. This damping is a
direct result of averaging over the range of allowed frequencies.
In Fig. 3.6.b we plotted the results for 6 atoms, with the leftmost atom being in
the n = 61 state. Again the solid line is the probability of finding the n = 61 in the
leftmost region, the dashed is the probability of finding it in the adjacent region, and
the dotted line is the probability of finding it in the rightmost region. The coherent
63
Figure 3.5: A schematic drawing of the setup for a system of (a) a slightly irregular
linear lattice (b) a slightly irregular lattice with the third site empty, and (c) a slightly
irregular 2?2 square lattice.
64
hopping of character from one region to next can clearly be seen as the probability
of finding the n = 61 state in the region next to the leftmost region peaks just after
the probability of finding it in the initial region drops off. The hopping is also seen
as the n = 61 state returns and the dashed line peaks just before the solid line. The
probability of finding the n = 61 character in the rightmost region peaks around 8
?s which is around the same time scale where the two atom case damps out to the
average value. This indicates that the coherence survives the spatial for long time
scales. In fact these probabilities continued to oscillate up to approximately 20 ?s.
In an experiment it might not be possible to perfectly fill every lattice site. In
Fig. 3.6.c we examined the effects of having an atom missing from a region. We
simulated the case of 6 regions with only 5 atoms. The configuration was as follows:
61,60, no Rydberg, 60,60,60. See Fig. 3.5.
We looked at the hopping from region to region in the same manner as the
previous figure. What immediately pops out is the similarity between Figs. 3.6.a
and 3.6.c. This implies the leftmost two regions behave like the two atom case. The
probability of jumping the gap in region 3 is very small. Since the dipole-dipole
interaction falls off like 1/R3, the strength of the interaction between regions 1 and
2 is a factor of 8 times larger than the interaction between regions 2 and 4. The
strongest coupling is between the nearest atoms, so much so that the pair of atoms
do not interact strongly with the rest of the atoms.
We moved from a linear array of regions to a 2? 2 square of regions with one
n = 61 atom and the rest n = 60. It is important to once again mention that the
65
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 1
 0  1  2  3  4  5  6  7  8  9  10
P
t (?s)
(a)
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 1
 0  1  2  3  4  5  6  7  8  9  10
P
t (?s)
(b)
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 1
 0  1  2  3  4  5  6  7  8  9  10
P
t (?s)
(c)
Figure 3.6: The probability P for finding the n = 61 state on various atoms as a function of time. The atoms are placed in
small regions with a width of 3 ?m and separated by a center-to-center distance of 20 ?m. In all cases the n = 61 state in initially
in the leftmost region. (a) For two regions, where the solid line is probability of finding the n = 61 state in the leftmost region,
and the dashed line is the probability of finding it in the rightmost region. (b) For six regions (see Fig. 3.5.a), the solid line is the
probability of finding the n = 61 state in the leftmost region. The dashed line is the probability of finding it in the adjacent region,
and the dotted line is the probability of finding it in the rightmost region. For six regions with an atom missing in third region (see
Fig. 3.5.b), the solid line is the probability of finding the n = 61 state in the leftmost region. The dashed line is the probability
of finding it in the adjacent region, and the dotted line is the probability of finding it in the rightmost region. Note the similarity
between (a) and (c) which indicates that the excitations cannot hop over the skipped region.
66
electric field is perpendicular to the plane of the array ?z? ?R = 0, which means that
there is no angular dependence to the dipole-dipole interaction between Rydbergs.
Only the distance between regions will effect the nature of the hopping. We labeled
the regions of the square in the following manner: the initial region (I), an adjacent
corner (A), and the opposite corner (O). The solid line in Fig. 3.7 is the probability
of finding the n = 61 state in region I, while the dashed line is the probability of
finding it in region A, and the dotted line is the probability of finding the n = 61
state in region O. The peaks in the figure suggest the n = 61 state seems to hop from
region I to A before hopping to region O.
3.5 Conclusions
By implementing an essential states model it is possible to explore the coher-
ent interactions between Rydberg atoms by solving the time-dependent Schr?odinger
equation. We examined two distinct situations where atoms are excited into Rydberg
states between which a resonant exchange of energy is allowed to occur. In the first
case there is no electric field, and the energy exchange is between an |sp? state and a
|ps? state. The situation where there is onepstate in a sea ofsstates was used to cal-
culate simple band structures. The band structure for a perfect linear array of atoms
was remarkably similar to what is seen in the bands for a line of driven nanoparticles.
The nature of the excitations near k ? 0 was also labeled as either particle-like or
hole-like depending on the curvature. The coherent hopping of excitation was also
investigated for a perfect square lattice and a cubic lattice. As the dimensionality
67
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 1
 0  1  2  3  4  5  6  7
P
t (?s)
Figure 3.7: The probability of the n = 61 state being in a region as a function of
time for a slightly irregular 2?2 square lattice (See Fig. 3.5.c). The solid line is the
probability of finding the n = 61 state in the initial region I, the dashed line is the
probability of finding it in an adjacent regionA, and the dotted line is the probability
of finding it in the opposite corner region O.
68
of the system increased the richness of the band structure also increased. Now in
addition to holes and particles, phonon-like hopping was found.
The second case consisted of Rydberg atoms excited to high Stark states in a
static electric-field. The coherent energy transfer of interest then became from |n1n2?
to |n2n1? and visa versa. In order to simulate a realistic experimental setup, the
atoms were not perfectly placed in a regular arrangement, but rather a nearly perfect
one. Even in this non-perfect case, the coherent hopping should last for substantial
time scales. If the nonregularity is so severe that there appears to be a hole in the
chain of atoms, the hole in the chain will block the excitation from hopping across.
3.6 Validity of the essential state model
The results in this section are based in the work done by F. Robicheaux and T.
Top?cu in Ref. [26]. In order to investigate the validity of the essential states model the
interaction between atoms must be carefully examined, in particular the transitions
to states outside of the ones used the essential states model. Since a Rydberg atom
can be reasonably modeled as hydrogen-like, the essential states model can be tested
by numerically solving the time-dependent Schr?odinger equation for two hydrogen
atoms.
69
3.6.1 Field-free case
The potential between two hydrogen atoms can be modeled by
V(vectorr1,vectorr2,R) = ? 1r
1
? 1r
2
+ r1r2R3 . (3.16)
As the distance, R, between the two atoms is increased to infinity, V effectively
becomes the potential for two isolated hydrogen atoms. The size of the isolated atoms
can be estimated from the classical turning radius (where V(r) = E), r = 2n2. At
minimum the distance between atoms should be large enough for no overlap between
the electron orbits. Naturally there are two cases, when both atoms start in the same
state n1 = n2 = n and when they start in different states n1 negationslash= n2. The second
case can be limited to looking only at the situation where |n1 ?n2| = 1 because of
the rapid fall of the dipole matrix elements with respect to |n1 ?n2|. The first case
is nondegenerate, while in the second case n1 = n, n2 = n + 1 is degenerate with
n1 = n+ 1, n2 = n. In either case, the nearest available states differ from the initial
state by energies of the order 1/n4. Since the relationship between nearby states is
the same for either initial state, the simpler nondegenerate case should be sufficient
to describe the limitations of the essential states model in the field-free case.
If the essential states model is to be used then the two atoms starting at n1 =
n2 = n should remain in the states n,n at the end of the simulation time. In other
words the probability of finding the atomsin anearbyn?,n?? stateshould be very small.
The simulation was performed by including anexpansion of basis states?n1(r1)?n2(r2)
70
until convergence occurred. The expansion was accomplished by scanning n1 and n2
through the states n??n to n+ ?n, and ?n was increased until the wavefunction
converged. The amount of mixing between states is an effect of the last term in the
potential V, r1r2/R3, so as R is decreased the transition between states should get
stronger.
In their paper the conclusion was that for less than 10% mixing between n-
manifolds, the distance between atoms should be greater than 10n2 as long asn< 80.
There also needs to be no accidental degeneracies. Using these findings the essential
states model should be accurate for n = 60 as long as Rgreaterorsimilar 2?m.
3.6.2 Static electric field
In the presence of a static electric field F pointed in the z direction, the interac-
tion potential can be written
V(vectorr1,vectorr2,R) = (z1 +z2)F + vectorr1 ?vectorr2 ?3(vectorr1 ?
?R)(vectorr1 ? ?R)
R3 . (3.17)
This potential gives rise to Stark splitting and the transitions within the n-manifold.
The Stark splitting can also lead to mixing of states between n-manifolds. This
mixing occurs when the field strength is F ? 1/(3n5), so the essential states model
will require F < 1/(3n5).
Since this model is based upon using two H atoms, the problem can be greatly
simplified by taking advantage of the scaled Runge-Lenz vectors and rewriting the
71
position operators vectorr1 and vectorr2 [61],
vectorr1 = 32n1vectorA1, vectorr2 = 32n2vectorA2. (3.18)
The scaled Runge-Lenz vectors and the orbital momentum operators can now be
replaced by a new set of commuting angular momentum operators, {vectorJ1, vectorJ2, vectorJ3, vectorJ4}
vectorL1 = vectorJ1 + vectorJ2, vectorA1 = vectorJ1 ? vectorJ2
vectorL2 = vectorJ3 + vectorJ4, vectorA2 = vectorJ3 ? vectorJ4. (3.19)
Equation 3.17 can now be rewritten
V = 32 [n1(J1z ?J2z) +n2(J3z ?J4z)]F + 9n1n24R3
?
braceleftBig
(vectorJ1 ? vectorJ2)?(vectorJ3 ? vectorJ4)?3
bracketleftBig
(vectorJ1 ? vectorJ2)? ?R
bracketrightBigbracketleftBig
(vectorJ3 ? vectorJ3)? ?R
bracketrightBigbracerightBig
. (3.20)
The magnitude of vectorJ1 and vectorJ2 are both j1 = j2 = (n1 ?1)/2 while the magnitude of vectorJ3
and vectorJ4 are both j3 = j4 = (n2 ?1)/2, and the azimuthal component of vectorJ, m, ranges
from ?j to j.
The mixing between Stark states within ann-manifold is suppressed as the spac-
ing between states is made large compared to the coupling matrix elements. The cou-
pling between states is determined by the separation while the spacing between Stark
states is proportional to the strength of the electric-field. When the simulations are
run and R is varied, the distance where the state-mixing rapidly changes from strong
72
to weak is the distance where the strength of the dipole electric-field from one atom
is comparable to the strength of the external field: n2/R3 ?F ?Rmin greaterorsimilarn2/3F?1/3.
The size of the minimum separation between atoms can be made smaller by increas-
ing the electric-field, but as mentioned above F cannot be too strong or mixing
between manifolds will begin to occur. When just under the largest allowed value
for F lessorsimilar 1/(3n5) is used Rmin greaterorsimilarn7/3. For the n = 60 case looked at in this chapter,
Rmin ? 2?m which is much smaller than the 20 ?m distances used.
For either the field-free or the static electric field case, the essential states model
is reasonable and quite accurate for the parameters used in this chapter.
73
Chapter 4
The effects of rotary and spin echo sequences on a Rydberg gas
4.1 Introduction
Advancements in cooling and trapping have opened up new opportunities for
investigating the properties of interacting many-body systems. In particular, the
creation of frozen Rydberg gases have made it possible to study correlated groups of
atoms [8,9]. At these low temperatures the motion of the atoms can be neglected
during the time scales of excitation, and the long range interactions between atoms
can be carefully studied.
One interesting consequence of the strong interaction between Rydberg atoms is
the suppression of excitation known as the dipole blockade effect [4,14]. While the
reduction in excitation has experimentally been seen by several groups [14?19,50],
it is of recent interest to measure the coherent collective behavior of the groups of
atoms that have been blocked from becoming excited [36,50].
The dipole-dipole interaction between Rydberg atoms can also lead to a situation
where two pairs of states (|AB?,|B?A??) are moved into resonance with each other
(EA +EB ?E?B +E?A) [8,9,31,52]. In this case, the system will oscillate between the
two states at a rate governed by the dipole-dipole interaction between them. With
more than two atoms involved in the system, the states appear to coherently hop
from atom to atom [26,30]. If the atoms are placed into regular lattice sites then a
74
direct observation of the coherent hopping can be detected [30], but in a random gas
the coherent nature of the hopping is hidden.
In this chapter we simulated the effect of echo sequences on coherent Rydberg
systems by using the many-body pseudoparticle wavefunction approach outlined in
chapter 2 and the essential states model used in chapter 3 to numerically solve the
Schr?odinger equation. In particular we investigated the rotary echo of a strongly
blocked Rydberg gas and the spin echo of a system of hopping excitations. The
approach taken in section 2.7 is particularly appropriate to use in the strong dipole
blockade regime because we explicitly correlate groups of nearby atoms and take into
account the spatial correlations between pseudoparticles that a simple mean field
model can not. The hopping dynamics are well described by the essential states
model.
4.2 Rotary echo of a dense Rydberg gas
When the system is especially dense, the correlations within a gas can become the
dominant factor in the dynamics of the system. An example of this is the coherent
Rydberg excitation of dense ultracold atoms [50]. In this case, the van der Walls
interaction (V(R) ? 1/R6) between excited states actively suppressed the number of
atoms able to be excited to Rydberg states, exhibiting a dipole blockade [4]. In section
2.4.1 we simulated a dipole blockade, and generated 2D correlation functions which
indicated that there was a minimum allowed distance between excitations called the
blockade radius Rb. Only one excited atom within the blockade region was allowed,
75
and the single excitation was de-localized across all Nb atoms contained within this
volume. The collective Rabi oscillation rate ? of this collection, or ?superatom? [35],
ofNb atoms was given by ? = ?Nb?0 where ?0 was the Rabi frequency of an isolated
atom. The results of section 2.7.1 showed that the size of the blockade region was
related to the density of the atoms within the region: Rb ? ??1/15, and ultimately
the collective oscillation rate of a superatom was dependent on the local density
by ? ? ?2/5. In a typical MOT, the density of the gas spans over several orders
of magnitude; therefore superatoms within the gas will oscillate over a wide range
of frequencies. In fact, by using the MOT parameters in Ref. [50], we discovered
in section 2.7.1 that most of the superatoms in the gas oscillate about ten times
slower than those located near the peak density. This inhomogeneity in density
(and therefore collective oscillation frequency) makes it very difficult for experimental
studies to directly measure the coherent nature of the system because the observable
is an integration over the entire sample [36].
Early studies in the field of nuclear magnetic resonance physics had to overcome
similar problems with inhomogeneities in magnetic fields which led to a wide range of
Larmor precession frequencies and obscured the resonant absorption of the driving RF
field [37,38]. In 1950, Hahn demonstrated the effectiveness of a ?spin echo? sequence
of pulses that was extremely effective in eliminating noise from the signal. In 1959
Solomon also demonstrated the successful use of a ?rotary echo? in doped water to
overcome the effects of inhomogeneities in magnetic fields.
76
More recently, there was an experiment which used a rotary echo technique to
prove the coherence of the excitation in a strongly blockaded ultracold gas [36]. The
experimental setup in Ref. [36] trapped and cooled atoms down to 3.8 ?K and ex-
cited them to 43S3/2 for up to 500 ns while keeping track of the total number of
excitations in the gas. At such a low temperature and short excitation time the
atoms are effectively motionless, so thermal motion can be disregarded as an outside
source of decoherence. In a system of ultracold Rydberg atoms, a substantial source
of inhomogeneity in the Hamiltonian is the variation in local density across the sam-
ple; the Gaussian shape of the density distribution in Ref. [36] certainly led to an
inhomogeneity in ?.
4.2.1 Rotary echo
In simplest terms a rotary echo sequence flips the sign of the excitation amplitude
in the Hamiltonian after a certain time ?p. The Hamiltonian describing the excitation
of a dense ultracold gas using the pseudoparticle approach was given in section 2.3:
?H(t) = summationdisplay
j
?H(1)j (t) +summationdisplay
j<k
Vjk|njnk??njnk|
?H(1)j (t) = ?(??(t) +?(t))|nj??nj|+F(t)?0
2
radicalbigW
j(|gj??nj|+|nj??gj|), (4.1)
where Vjk is the interaction between two pseudoparticles j and k. The number of
atoms associated with each pseudoparticle is given by W. The detuning of the laser
is ??(t), and ?(t) is a mean field energy shift due to excited atoms outside of the
77
simulated volume. The kets |gj? and |nj? correspond to atom j being in the ground
state and excited to the n manifold respectively, and |njnk? is the state where atoms
j and k are both excited. In Ref. [36], the rotary echo sequence was accomplished
by using an RF field to flip the sign of ?0 after the time ?p ??, where ? is the total
excitation time. In our simulations we used the following to model the excitation
profile:
F(t) =
?
???
???
???
???
???
???
???
???
???
???
?
e?5(t?tr)2/t2r for t?tr
1 for tr <t??p ?tr
e?2(t?(?p?tr))2/t2r ?e?2(t?(?p+tr))2/t2r for ?p ?tr <t??p +tr
?1 for ?p +tr <t?? ?tr
?e?5(t?(??tr))2/t2r for ? ?tr <t??.
(4.2)
In Eq. 4.2, tr is the ramping time for the laser. A Gaussian is used during ramp on
time, between sign changes, and ramp down time as a smooth transition to avoid an
instantaneous switch which could lead to unphysical, and therefore undesired high
frequency effects. Figure 4.1 illustrates an example of this type sequence.
When the system is sufficiently sparse, the energy shift due to the van der Waals
interaction (Vjk) between atoms becomes negligibly small compared to the width of
the excitation amplitude, and the ground state atoms are excited for a time ?p and
de-excited for a time ? ??p. This system of isolated atoms will be returned to zero
excitations if ?p = ?/2. A measurement of zero excitations is the perfect rotary echo
78
-1
-0.5
 0
 0.5
 1
 0  100  200  300  400  500  600
F(t)
time (ns)
Figure 4.1: An illustration of a rotary echo sequence. The sign of the excitation
amplitude F is smoothly flipped after time ?p. In this case ?p = 350 ns, and is
indicated by the dashed line.
79
case. Figure 4.2 is the result of simulating the excitation of a diffuse system of 5
atoms using a rotary echo sequence. The excitation amplitude ?0 was chosen so the
maximum number would be excited at 500 ns. These echo simulations compare the
number of excited atoms to the timing of the sign flip ?p. As expected, when ?p = ?/2
a perfect echo was recorded.
If the system is so dense that the vdW interactions are much greater than the
excitation amplitude then the system is in the strong blockade limit. In the extreme
case where all of the atoms are within a blockade radius the system has been reduced
down to a single superatom. This single isolated superatom will also be excited and
de-excited for the same amount of time, thus returning the system to a state of zero
excitations. In Figure 4.3, we plot the number excited versus?p for a perfectly blocked
system of 10 atoms. A perfect echo can be seen when ?p = ?/2 and the maximum
number excited is exactly 1.
A more interesting situation arises when the system is dense enough to create
blockades, yet large numbers of excitations are allowed to occur. Now the energy shift
due to the van der Waals interaction is comparable to the width of the excitation
amplitude. In this case the pseudoparticle approach used in section 2.7 is especially
useful in describing the spatial correlations between pseudoparticles which represent
groups of blockaded atoms. In this model, the van der Waals interaction is between
pseudoparticles and not between individual atoms themselves, so the inhomogeneity
in collective oscillations is explicitly included in the simulation. We simulated a rotary
echo sequence of excitation for two different peak densities. In each case we simulated
80
 0
 1
 2
 3
 4
 5
 6
 0  100  200  300  400  500
N exc
?p (ns)
Figure 4.2: Number excited versus the timing of the sign change of the excitation am-
plitude. This echo signal is for 5 isolated atoms. Note that there are zero excitations
when ?p = 250 ns, exactly half of the total excitation time.
81
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 1
 0  100  200  300  400  500
N exc
?p (ns)
Figure 4.3: Number excited versus the timing of the sign change of the excitation
amplitude. This echo signal is for 10 atoms in the perfect blockade regime. Note the
perfect echo signal at 250 ns.
82
the fraction excited at specific density points as we did in section 2.7.2 and convolved
the results over the density distribution given in [50]. The other parameters used in
the following three simulations were chosen to match those used in Ref. [36]. Plotted
in Figure 4.4.a is the echo signal as a function of ?p for a peak density of 5 ? 1012
cm?3. While the echo signal is not perfect at?p = ?/2, it is still prevalent because the
interaction between pseudoparticles is not as strong. When simulating at a higher
peak density of 1.5?1012 cm?3, the echo signal is weakened as in Figure 4.4.b by the
dephasing caused by pseudoparticle interactions.
Interactions between pseudoparticles are a significant source of dephasing which
prevents a perfect echo from occurring. This can be understood in the context of the
Hamiltonian for the system. While the off-diagonal elements (given by the excitation
amplitude) are reversed after ?p, the sign of the diagonal van der Waals interaction
elements remain unchanged, and the system does not perfectly evolve backwards in
time. If it were possible to switch the sign of the van der Waals interaction between
particles at the same time as the excitation amplitude then the sign of the entire
Hamiltonian would be flipped and all sources of dephasing would have to be external
such as thermal motion or ionization. Flipping the sign of the whole Hamiltonian
is effectively the same as reversing the sign of t and perfectly running the system
backwards in time. The lack of a perfect rotary echo signal would indicate outside
sources of decoherence on the gas and coupling to external degrees of freedom. In
Fig. 4.5 we simulated a system where the whole Hamiltonian gets switched in sign at
?p in the same manner as Eq. 4.2. The perfect echo indicated in our simulations is
83
 0
 500
 1000
 1500
 2000
 2500
 0  50  100  150  200  250  300  350  400  450  500
N exc
?p (ns)
(a)
 0
 500
 1000
 1500
 2000
 2500
 0  50  100  150  200  250  300  350  400  450  500
N exc
?p (ns)
(b)
Figure 4.4: Number excited versus the timing of the sign change of the excitation
amplitude. The echo signal for (a) peak density of ? = 5.0?1012 cm?3 and (b) peak
density ? = 1.5?1013 cm?3.
84
 0
 500
 1000
 1500
 2000
 2500
 0  50  100  150  200  250  300  350  400  450  500
N exc
?p (ns)
Figure 4.5: Number excited versus the timing of the sign change of the entire inter-
action Hamiltonian. This echo signal is for a high peak density of ? = 1.5 ? 1013
cm?3.
85
consistent with the notion of running the system backwards in time. We believe that
it might be possible to realize a time-reversable system of Rydberg atoms by using a
small electric field to diabatically push the system to states that experience a dipole
interaction of the same magnitude but opposite in sign compared to the field-free
states. A promising interaction would involve taking advantage of the resonance near
n = 43 for d?d Rb atoms.
4.3 Spin echo for Rydberg hoppers
In section 3.4, a possible experiment to measure the coherent hopping of exci-
tation between slightly irregular lattice sites was proposed. A key feature of this
proposed experiment was the clear separation of regions which would allow for the
spatial location of the hopper to be measured, but it would be impossible to measure
the coherent nature of the hopping in an inhomogeneous gas in this manner. The
amount of time it takes for an excitation to hop between atoms is inversely propor-
tional to the dipole-dipole interaction energy between them, thop ? R3 [26], so in an
irregular gas of Rydberg atoms the excitations would be hopping at different rates
and directions. Any measurement of hopping rates or locations would be hidden due
to the random placement of Rydberg atoms.
4.3.1 Spin echo
Like the rotary echo, the spin echo sequence was first used to overcome inhomo-
geneities in the magnetic field in NMR. Unlike the rotary echo, the spin echo isn?t
86
simply a consequence of running the system forwards and backwards in time for the
same duration. The spin echo works by making series of unitary transformations on
the generalized Bloch sphere describing the quantum system [37]. In particular we
used the following sequence of pulses to simulate a spin echo: first thesystem is excited
by a ?/2 pulse (which takes the ground state |g? to a mix of the ground and excited
state (|g??i|n?)/?2, and an excited state |n? to a mix of |n? ? (?i|g?+|n?)/?2),
then the system is allowed to relax for a time ?1. Next, the system is excited by a
? pulse (which takes the state |g? ? ?i|n? and |n? ? ?i|g?); again the system is
allowed to relax for ?2, and finally the system is excited via a ?/2 pulse. In summary
the sequence looks as follows:
?/2 ??1 ?? ??2 ??/2. (4.3)
If sources of dephasing and outside decoherence are negligible then the system will
exhibit a strong spin echo when the two relaxation times are equal ?1 = ?2.
We simulated the effect of the spin echo sequence of Eq. 4.3 on the following
system: a single ground state |g? atom in a sea of atoms excited to the Stark state
|n?? or |?0? = |gn?n????n??. This is, of course, a severe approximation to a real
experiment where the number of |g? state atoms will most certainly be greater than
one, but the effect of a spin echo sequence on this simplified system should be of
interest. The density of the gas was chosen so the average distance between particles
would be 13.5 ?m. The excitation laser coupled |g? to an excited Stark state |n?, but
87
not |g? to |n??. The two Rydberg states |n? and |n?? were chosen in a manner that
allowed for the resonant exchange of energy through the dipole-dipole interaction [26].
This means that the |n? state on atom j can coherently hop to any atom k with |n??
character. If the direction of the static electric field is in the ?z direction then the
likelihood of hopping will be determined by (nn?/3)2/R3jk, where |n?n?| = 1 are the
principal quantum numbers of the two Stark states, and Rjk is the distance between
the two atoms. If the atoms are not allowed to move during the simulation time
(i.e. the temperature is set to 0 K), then the final state of the system after a spin
echo sequence can be analytically derived for the single hopper case in a sea of N
|n?? atoms. For clarity, we first solved the two particle case for the final state of the
system:
|?f? = 12
braceleftbigg
? bracketleftbigcos(V?2)e?i(?g?1+?n?2) + cos(V?1)e?i(?g?2+?n?1)bracketrightbig|gn??+
+ibracketleftbigsin(V?2)e?i(?g?1+?n?2) + sin(V?1)e?i(?g?2+?n?1)bracketrightbig|n?g?+
?ibracketleftbigcos(V?2)e?i(?g?1+?n?2) ?cos(V?1)e?i(?g?2+?n?1)bracketrightbig|nn??+
+ bracketleftbigsin(V?2)e?i(?g?1+?n?2) ?sin(V?1)e?i(?g?2+?n?1)bracketrightbig|n?n?
bracerightbigg
?e?i?n?(?1+?2), (4.4)
where V = (nn?/3)2/R3jk was the off-diagonal matrix element for the dipole-dipole
interaction, ?g was the energy of the ground state, ?n, and ?n? were the energies of
88
the excited Stark states. When ?1 = ?2 = ?, the final wavefunction is simply
|?f? = [?cos(V?)|gn??+isin(V?)|n?g?]e?i(?g+?n+2?n?)?. (4.5)
Note that only |gn?? and |n?g? states remained, and the probability of finding an atom
in the Stark state |n? is zero. This was a perfect spin echo. For the case of a small
difference in relaxation times, ?2 = ?1 +? and ? is small, this case can be simplified
as well to
|?f? = ?
braceleftbigg
cos([?n ??g] ?2 )[cos(V?1)|gn???isin(V?1)|n?g?]
?sin([?n ??g] ?2 )[cos(V?1)|nn???isin(V?1)|n?n?]
bracerightbigg
?e?i(?g+?n+2?n?)(?1+?2 ). (4.6)
Now the probability of finding a |n? state atom is very small and proportional to
bracketleftbig(?
n ??g)?2
bracketrightbig2. For the general case of one |n? hopper in a sea of N |n??, the final
wavefunction is
|?f? = ?12
braceleftbigg
i
N+1summationdisplay
k=1
bracketleftbig?
ik(?2)e?i(?g?1+?n?2) ??ik(?2)e?i(?g?2+?n?1)
bracketrightbig|n
k?
N+1summationdisplay
k=1
bracketleftbig?
ik(?2)e?i(?g?1+?n?2) +?ik(?2)e?i(?g?2+?n?1)
bracketrightbig|g
k?
bracerightbigg
?e?iN?n?(?1+?2), (4.7)
89
where ?ik is the probability amplitude of finding the |n? state initially on atom i on
atom k, |gk? represents the state with |g? atom k, and |nk? represents the state with
|n? on atom k. By solving for the eigenvalues and eigenvectors of the Hamiltonian
given in Equations 3.9 and 3.10, we found the values for the hopping amplitudes ?.
When ?1 = ?2 = ?, the probability of finding a |n? state atom is zero, and the system
again displays a perfect echo:
|?f? = ?
N+1summationdisplay
k=1
?ik(?)e?i[?g+?n+2N?n?]?|gk?. (4.8)
In Figure 4.6 we plot the probability of finding no |n? atoms as function of the dif-
ference in scaled relaxation time. The difference in scaled relaxation time is ??/thop,
where ?? = ?1 ??2 and thop = 0.83 ?s is the time it takes for the excitation to hop
the average distance between atoms: 13.5 ?m. As expected, in a zero temperature
gas with one hopper the system exhibits a perfect echo at ??/thop = 0. When the dif-
ference in relaxation times becomes large, the system becomes more and more evenly
mixed.
While the previous discussion neglected the effects of temperature, by allowing
the particles to exhibit thermal motion and by solving for the hopping amplitudes
during every time step, we simulated a one hopper gas with an outside source of
decoherence. In order to account for the time dependence of the Hamiltonian we
used an exponential propagator and adjusted the position of each atom during each
time step. If the time steps are kept small and the change in position is also small then
90
 0
 0.2
 0.4
 0.6
 0.8
 1
-30 -20 -10  0  10  20  30
Prob. Ground
??/thop
Figure 4.6: Spin Echo signal in a 0K gas with exactly one hopper versus difference in
scaled relaxation times. When ??/thop = 0 a prefect signal is seen.
91
 0
 0.2
 0.4
 0.6
 0.8
 1
 0  100  200  300  400  500  600  700  800  900 1000
Prob. Ground
Temperature (?K)
Figure 4.7: Spin Echo signal in a gas with exactly one hopper as a function of tem-
perature for various ? relaxation times. In each case ?1 = ?2 = ?. The solid line is
for a ? = thop, the dashed line is for ? = 2thop, the dotted line is for ? = 3thop, the
perforated line is for ? = 4thop, and the chain line is for ? = 5thop.
the accumulated errors can be kept to a minimum. The results of our simulations are
shown in Fig. 4.7 as a plot of of finding zero |n? atoms as a function of temperature.
At every temperature point the two relaxation times were exactly equal, so if thermal
motioncaused no decoherence then a perfect echo should be seen. If the two relaxation
times between excitation pulses are just long enough to allow one hop to occur than
the effect of temperature is minimal, the |n? state has not diffused very much. In this
situation, the hopping of the |n? state can be limited to only the nearest neighbors;
92
the sequence of hops to get to the final state is very simple, directly from i to k.
When the relaxation time is increased to allow two hops, the effect of temperature is
again pretty minimal well into the 100s of microkelvins. Now the |n? state is limited
to nearest and next-nearest neighbors and the number of possible particles it can
hop to has increased by a factor of about 8, and for each one of these atoms their
motion contributes to the dephasing. By increasing the relaxation time, the number
of atoms involved in the hopping dynamics increases drastically for each interval of
thop. The effect of temperature is quite apparent when the number of allowed hops
is 4 or above, the echo signal has effectively vanished for temperatures greater than
300 ?K. As expected when the temperature gets closer to 0 K, a strong echo signal
is observed, no matter how many times the excitation is allowed to hop. It should be
noted that the even when the thermal motion is neglected, the dipolar force between
Rydberg atoms will cause acceleration. The large mass of Rb and the shortness of
thop, however, prevent this motion from being significant. For example, after 10 hops
(? 8.3 ?s) the attractive force between two atoms initially separated by 13.5 ?m will
have moved them only 0.01 ?m closer.
4.4 Conclusions
In summary, we have been able to investigate the effects of echo sequences on
systems of interacting Rydberg atoms. We simulated a rotary echo signal in a dense
ultracold Rydberg gas by using the pseudoparticle many-body wavefunction approach
used in chapter 2 to describe theblockade effect. Unlike simpler meanfield models, the
93
pseudoparticle approach directly takes into account high correlation between nearby
atoms, and thespatial correlations between pseudoparticles. While a mean field model
predicts the strong echo signal to be suppressed due to the strong interactions between
nearby pairs of individual atoms, the pseudoparticle method prevents such short
distances between Rydberg atoms. Our simulations showed a clear rotary echo signal
when the sign ofexcitation amplitude isswitched half-way throughthetotal excitation
pulse. When all of the atoms or pseudoparticles are perfectly correlated, a perfect
echo is seen. As the density between Rydberg atoms is increased the interactions
between pseudoparticles is increased and the echo signal is reduced.
We also simulated a system where the sign of the entire interaction Hamiltonian
is flipped, both the laser-atom interaction and the atom-atom interaction. Because
the coupling between Rydberg atoms is no longer a source of dephasing, any reduction
of the rotary echo signal would indicate an external source of decoherence. It might be
possible to experimentally realize such a system using a weak electric field to switch
the sign of van der Waals potential between two 44d5/2 Rb atoms at the same time
an RF field is used to switch the phase of the excitation amplitude.
We finally examined the spin echo signal of a single coherently hopping Rydberg
excitation in a gas. If thermal motion is neglected, we analytically showed that
the system will display a perfect echo when the relaxation time between excitation
pulses is exactly equal. Our simulations of temperature dependence on signal strength
indicated that as the number of allowed hops in increased, the effect of thermal motion
becomes drastically more significant. If more than 4 hops are allowed during each
94
relaxation time then the the temperature of the gas must be less than 300 ?K for any
discernable echo signal to be detected. Of interest of would be further studies into
the effects of multiple hoppers on the spin echo signal, and the how the coherence of
these systems respond to increasing temperatures.
95
Chapter 5
Interactions between classical dipole moments: nanospheres
5.1 Introduction
The interactions between many Rydberg atoms have many similarities with the
interactions between nanoparticles [62]. Using our knowledge of the dipole-dipole
interactions between Rydberg atoms gained in the previous chapters, we describe
the dipole-dipole interactions between driven metallic nanospheres. There have been
many theoretical and experimental studies probing the nature of a collection of opti-
cally driven metal nanospheres (MNS?s), their interactions, and various possible uses
as sub-wavelength optical devices [44,46?49,62?69]. Arrangements of MNS?s with
features smaller than the wavelength of light, ?, can be constructed and will interact
strongly with light tuned to the surface plasmon (SP) frequency (?SP) [40]. The
coupling of the MNS?s through the electromagnetic field produces a coherent wave of
oscillating dipole moments. This coupling allows information to pass through geome-
tries smaller than ? and causes the direction of the scattered light to strongly depend
on ?.
A simple and important geometry to study is a straight line or chain of equally
spaced MNS?s placed in a dielectric medium. This configuration is shown in Fig. 5.1
with a realistic particle size and spacing. In this chapter we will use an array of ten
nanospheres; it has been shown when using common parameters the infinite chain
96
Figure 5.1: A schematic drawing of a possible set up for a regularly spaced linear array of silver nanospheres.
A wide (compared to the size of a nanosphere of radius a) beam of light of frequency ? is propagated along the
array?s axis in the particular medium where the array is assembled. The light absorbed and scattered by each
MNS will in turn excite neighboring MNS?s and a coherent wave of oscillating electric dipoles will be produced.
97
limit is met at around ten particles. [67]. A technique for the fabrication of such
assemblies is electron-beam lithography, which allows for good control over particle
size and regular placement [62]. Since this is one of the simplest configurations to set
up experimentally and theoretically, it has been studied extensively. All of the MNS?s
in the linear array can be excited using a broad light beam at a specific frequency ?.
Theoretical analysis of a linear array of MNS?s has been done by several different
groups. The numerical methods developed to describe these arrays and their inter-
actions with an external electric field include the discrete dipole approximation [47],
the multiple multi-pole method [48], the finite difference time domain method [62],
and the T-matrix method [49]. The method used in this chapter can perhaps best be
described as the coupled dipole approximation. Each nanosphere will be described
by a single point dipole that linearly responds to electric fields. When the system is
driven at a specific frequency, ?, then every time dependent quantity can be written
as A(t) = A(0) exp(?i?t). We will solve a set of self-consistent linear equations de-
scribing the response of each electric dipole to the incident field and to the scattered
electric fields from the other particles. In doing so we will use the full electric field
from the oscillating dipoles and contrast these results with using only the near field
approximation.
We report several interesting effects apparent in this simple system that emerge
from treating the full field rather than the near field. We show that the ohmic
power deposited in a particular MNS can strongly depend on the position in the
line of particles even when all of the MNS are equally illuminated by a light beam.
98
Furthermore, we can reverse the ratio of power in the first sphere compared to the
last sphere with an extremely small variation in light frequency suggesting a device
for wavelength discrimination over this range of frequencies. Also, we will show that
the direction of the light emitted from the MNS array strongly depends on frequency
when only one of the spheres is driven. Thus, it should be possible to detect collective
properties of the MNS array, like inhomogeneous power distribution along the array,
using simple measurements of the far field radiation which could simplify experiments.
There are three uncoupled modes of propagation down this chain: two degenerate
transverse (T) waves which have the direction of the dipole moments, vectorp, perpendicular
to the line of MNS?s, and a longitudinal (L) mode in which vectorp are parallel to the chain
axis. All other forms of propagation can be described as a linear combination of T
and L modes. For the T modes, widely separated MNS?s can interact via the far
field of the scattered light; this is not possible for the L modes since there is no
scattered light parallel to the dipole moment. A very thorough study on the effect
of including the far field of scattered light on the dispersion relation of T modes was
done by Citrin [68]. For the geometry discussed in this chapter, the light beam will
be directed from left to right so only T modes are excited unless noted otherwise.
5.2 Coupled dipole method
In our calculations, we assume that the plasmon excitations caused by incident
light produce an oscillating dipole electric field. This assumption holds if the wave-
length of the incident light is much larger than the diameter, 2a, of the MNS and the
99
inter-particle distance dgreaterorsimilar 3a [48,70]. A more complete investigation of higher order
multi-pole effects in the near field has been done by Park and Stroud [48]; higher
order multi-pole effects become increasingly important as the inter-particle spacing
decreases but should give only minor, quantitative differences for the cases we present
here. The electric field produced by a single, periodically oscillating dipole with an
electric dipole moment vectorp is given by:
vectorE(vectorp,vectorR,k) = 1
4??
braceleftbigg
k2(?R?vectorp)? ?R e
ikR
R (5.1)
+ [3?R( ?R?vectorp)?vectorp ]
parenleftbigg 1
R3 ?
ik
R2
parenrightbigg
eikR
bracerightbigg
,
where k = ?/v is the wavenumber in the dielectric, v is the speed of light in the
dielectric medium [71], and ?R is the unit vector in the direction of vectorR (vectorR is the
displacement from the dipole vectorp) [70]. If we use the near-field approximation R??,
then we set k = 0 and the electric field is:
vectorEnear(vectorp,vectorR) = ? 1
4??
vectorp?3?R(vectorp? ?R)
R3 . (5.2)
In the near-field the electric field is dominant, but when using the full field this is
not immediately obvious. We can neglect the effects of the magnetic field however,
if the charge separation is small, i.e. the magnitude of the dipole is not very large
compared to the total charge times the radius a on the MNS.
100
Using the full electric dipole fields, we can construct self-consistent equations of
motion for a driven system of MNS?s. The equation of motion for a dipole driven by
an electric field can be parameterized as ?vectorp+??vectorp+?2SPvectorp??...vectorp = ?vectorE(t) where ?, ?SP,
?, and ? are constants; the physical values of these parameters are substituted in the
equation below. The terms of this equation have a familiar origin: the first term is
the acceleration, the second term is the damping from ohmic heating, the third term
is from the harmonic force, the fourth term is from radiation damping and the right
hand side is the driving term from the electric field. Substituting the oscillating form,
vectorpn(t) = vectorpn exp(?i?t), and substituting the physical values for the constants gives the
coupled dipole-dipole equations of motion:
bracketleftbigg
??2 ?i??+?2SP ?i29
parenleftBiga
v
parenrightBig3
?2p?3
bracketrightbigg
vectorpn =
1
3a
3?2
p4??
braceleftBigg
vectorE(ext)n + summationdisplay
n?negationslash=n
vectorE(vectorpn?,vectorRnn?,k)
bracerightBigg
(5.3)
where a is the radius of the MNS, vectorRnn? ? vectorRn? ? vectorRn is the center-to-center dis-
tance between two particles, vectorE(ext)n is external electric field at the nth nanosphere,
and vectorE(vectorpn?,vectorRnn?,k) has been defined as Eq.(5.2). In a previous study, the radiation
damping was erroneously taken to be negligibly small [66]. For the parameters of
this chapter, the radiation damping is the largest loss mechanism, but the radiation
damping does become less important as the radius of the MNS?s decreases. In the
ohmic damping term, ? is the inverse of the electronic relaxation time. In order to
match the bulk dielectric properties of silver, we use a value of? = 7.87?1013 s?1 [66].
101
The coupling strength is determined in large part by the bulk plasmon frequency ?p.
The value of wp has been defined such that: ?2p = (Ne2)/(?m?), where N is the total
number of conducting electrons per unit volume, e is the charge of an electron, and
m? is the optical effective electron mass [72]. For silver we used N = 5.85 ? 1028
electrons/m3 and m? = 8.7?10?31kg. We have calculated a value of ?p ? 9.3?1015
rad/s, and we will use ?SP ? 5 ? 1015 rad/s [66]. The solutions to these coupled
inhomogeneous linear algebra equations are the induced dipole moments, vectorp, on each
sphere. Because vectorp appears linearly, these coupled equations can be solved directly by
using standard linear algebra packages.
5.3 Results
The MNS?s act as both scatterers and detectors of the total electromagnetic field.
The induced dipole moment of an MNS is proportional to the local electric field,
while the ohmic power dumped into the MNS is proportional to the magnitude of the
dipole moment squared. In Fig. 5.2 the ohmic power of each of the ten individual
MNS is plotted for two specific frequencies: one placing most of the power on the first
MNS and the other directing most of the power down the chain. In both cases the
magnitude of the incident electric field is chosen to be 1 V/m to facilitate (through
linear scaling) the calculation of the power at other field strengths. The dramatic
difference in the distribution of power with relatively small changes in frequency is
evident. The near field approximation has been used in many studies of interacting
MNS?s, but does not correctly reproduce most effects in Fig. 5.2. The near-field
102
 0
 50
 100
 150
 200
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Power (eV/s)
position (?m)
Figure 5.2: The ohmic power as a function of the position of 10 MNS?s in a regularly
spaced linear array. The center-to-center inter-particle distance is d = 80 nm and the
diameter, 2a, of each MNS is 50 nm. A plane transversely polarized electromagnetic
wave is propagated from left to right along the axis of the array. All of the MNS?s
absorb and scatter the incident electromagnetic wave of magnitude 1 V/m. The
solid line is for a chosen frequency ? = 4.85 ? 1015 rad/s (?diel = 259 nm) when
most of the power is in the first sphere. The dashed line is for a frequency of ? =
4.62?1015 rad/s (?diel = 272 nm) when most of the power is in the last sphere. This
asymmetry between first and last MNS?s vanishes in the near field approximation at
all wavelengths.
103
approximation is most accurate when kd ? 1, d being the regular center-to-center
inter-particle distance. In the case of silver nanospheres, ?SP ? 5 ? 1015 rad/s
which gives a value of k ? 3.0 ? 107 m?1 in the dielectric. Typical inter-particle
spacings are in the range of 80 nm, so in these cases kd ? 2?/3. In the near-field
approximation, coupling terms that fall off as R?1 and R?2 are neglected. When
using the full electric field, certain phase dependent phenomena are now taken into
account that have been overlooked in earlier work [66]. It is important to recognize
that the dipole fields created by the now oscillating electric dipoles have individual
phases that vary with the distance from an MNS. The retarded electric field can either
add constructively or destructively down the chain to create localized regions of high
total electric field. Using the near field approximation, all of the MNS?s interact with
each other instantaneously and any phase is solely due to the phase of oscillation of
a single dipole. Using this approximation, the ohmic power is symmetric through the
midpoint of the array at all wavelengths. In contrast when using the full dipole field
there is a lag in inter-particle communication due to the finite speed of light. It is
this lag that allows the MNS?s to have differing phases that can coherently add or
subtract at specific locations in space.
It is also interesting to compare the ohmic power of the first MNS to the last
one while scanning over a range of frequencies. Remember that in our geometry the
light is directed down the line of MNS?s. Therefore, without the interaction between
MNS?s, each of the nanospheres would dissipate the same amount of ohmic power.
104
It can immediately be seen in Fig. 5.3 that within a certain band of frequencies the
ohmic power from sphere one is much greater than that of the last one.
Within another band of frequencies the opposite is true. A similar though lesser
effect is present even for the limit of two MNS?s. In order to clearly illustrate the
sensitivity of the array we also plotted the single particle response to an identical beam
of light in Fig. 5.3 as dotted lines. Note that the response of the first MNS follows
the response of a single sphere when the driving frequency is far off resonance, but
the response of the last sphere is strongly suppressed for all of the plotted frequencies
greater than resonance. This forward-backward asymmetry is not present when the
near-field approximation is used (inset of Fig. 5.3), and thus the asymmetry is due
solely to the retardation of the electric field. It is also interesting that the power in
the first sphere is greater than for a single sphere for almost all frequencies and is
roughly a factor of 3 times larger at the peak.
Naturally the next step is to investigate what is the cause of the forward-
backward frequency dependent asymmetry as seen in Fig. 5.3. Using the calculated
dipole moments and looking in the far field (r ? ?) limit we can determine the
differential radiated power per solid angle [70]:
dPrad
d? =
1
2
vk4
(4?)2?[ |
vectorP|2 ?|?r? vectorP|2], (5.4)
where
vectorP ?summationdisplay
n
vectorpn e?ik?r?vector?n (5.5)
105
 0
 20
 40
 60
 80
 100
 120
4.0 4.5 5.0 5.5 6.0
Power (eV/s)
? (1015 rad/s)
Power (eV/s)Power (eV/s)
 0
 200
 400
 600
 800
 1000
 1200
 1400
4.0 4.5 5.0 5.5 6.0
Figure 5.3: The ohmic power as a function of the frequency, ?, of a plane electro-
magnetic wave propagating along the axis of a regular linear array of MNS?s. The
dimension of the array and MNS?s is the same as in Figs. 5.1 and 5.2. All of the
MNS?s absorb and scatter the incident beam of light as it comes in from left to right.
The solid line is the ohmic power of the first (leftmost) MNS and the dashed line is
for the last (tenth) one. The dotted line is the power response for the single MNS
case. The inset is the same as the main figure, but using only the near field approxi-
mation. Note that it is now impossible to preferentially excite the first or last MNS
by modifying the driving frequency.
106
and k = ?/v = 2?/? is the wavenumber in the dielectric medium. In Eqs. (5.4)
and (5.5), ?r is the usual radial unit vector, vector?n is the displacement from the the first
nanosphere to the nth one, and |vector?n| = (n?1)d for a regularly spaced linear array.
We set up a regular linear array as in Fig. 5.1, but we changed the simulation
so that only the first (leftmost) MNS is excited to a frequency ?. Exciting a single
MNS can probably be realized by using an electron beam instead of optical radiation.
Optical spot sizes are of at least the order of ? in dielectric, while e-beams can have
spot sizes much smaller than the inter-particle separation d. We can see in Fig. 5.4
that there areindeed certain bands of frequency that cause the whole system to scatter
light in the backward direction and that these frequencies are the same frequencies
where the forward-backward asymmetry is realized. In fact the similarity between
Figs. 5.3 and 5.4 is quite pronounced showing that the asymmetric behavior is caused
by the coherent constructive or destructive interference of the radiated light from the
individual MNS. When driving the first MNS at certain frequencies, the total electric
field emitted by the MNS?s add constructively along the line which correlates with a
large amplitude at the last sphere. At other frequencies, the total electric field gives
destructive interference along the line which correlates with the small amplitudes at
the final sphere.
Another way to look at Fig. 5.4 is to perform a discrete Fourier transform of the
induced dipole moments: vectorp(k) =summationtextnvectorp(?r? vector?n)exp(?ik?r? vector?n). We examined p(k) at
various ??s by starting from about 4.0?1015 rad/s and increasing the frequency. At
first a clear peak could be seen in p(k) and this peak increased as we increased ?.
107
 0
 20
 40
 60
 80
 100
 120
 140
4.0 4.5 5.0 5.5 6.0
Power (arb. units)
? (rad/s)
Figure 5.4: The same physical set up as Fig. 5.2, but this time only the first sphere is
externally excited. Plotted is the differential radiated power per solid angle versus the
frequency of the incident light. The solid line is the power scattered in the forward
direction and the dashed line is the power scattered in the backward direction.
108
At around ? ? 4.8?1015 rad/s a clear peak could no longer be discerned, and p(k)
becomes very noisy. This implies that above about? ? 4.8?1015 rad/s we are trying
to drive the system outside the allowed photonic bands. This behavior matches closely
with Fig. 5.4, suggesting that scattering light in the forward direction is suppressed
due to driving within the photonic bandgap for this system. It also suggests that a
correct band structure calculation must take into account the full electric dipole field
and radiative damping.
When exciting the system into L modes, widely separated MNS?s do not interact
via the far field of the scattered light (?R?vectorp = 0). They do still communicate through
the near and intermediate fields. Unlike using the near field approximation, however,
the scattered field is still retarded. This retardation will once again cause MNS?s to
oscillate at various phases allowing for interference effects. In Fig. 5.5 we again use
the same physical set up as Fig. 5.1, and we only excite the leftmost MNS at a various
frequencies ?. This time we excite into an L mode (parallel to the chain). Plotted
in Fig. 5.5 is the differential power scattered per solid angle (dP/d?) versus the
scattering angle ? at two different frequencies. The inset is also dP/d?, but forcing
all of the MNS?s to oscillate in phase. This plot is reminiscent of the symmetric
diffraction pattern of light passing through slits spaced closely relative to the incident
wavelength. A clear asymmetry however, can be seen in the main plot. As seen
with the T mode case, the bulk of the scattered light can be preferentially aimed in
different directions by adjusting the frequency of the driving force.
109
 0
 10
 20
 30
 40
 50
 60
 70
 80
0.0 0.2 0.4 0.6 0.8 1.0
Power (arb. units)
? (pi rad)
Power (arb. units)
 0
 20
 40
 60
 80
Figure 5.5: Again the same physical set up as Fig. 5.4, but this time the first sphere is
externally excited into an L mode. Plotted is the differential radiated power per solid
angle versus the scattering angle ? for two frequencies, where ? is the angle relative
to the line of MNS?s. The solid line is the power scattered at ? = ?SP = 5.0?1015
rad/s and the dashed line is the power scattered when ? = 5.5?1015 rad/s. In order
to more cleary show the asymmetry the dashed line is scaled by 2.0 i.e the amplitude
of the driving force is increased by about 40%. The intermediate electric field of
the oscillating dipoles gives the asymmetry. The inset also plots differential radiated
power per solid angle versus the scattering angle for ? = ?SP, but here the MNS?s
are forced to oscillate in phase.
110
5.4 Conclusion
In summary, we have shown that when looking at a system of MNS?s, interesting
effects can be lost when using only a near-field approximation. Within certain closely
spaced bands of frequencies it is possible, using a spatially broad beam of light, to
excite specific MNS?s. In Fig. 5.3, the ratio of the power in the first sphere to that in
the last sphere is 1 near the crossing frequency ? 4.7?1015 rad/s; near this frequency,
the ratio varies by more than an order of magnitude in a small range of frequencies.
Various experimental techniques can detect when the surface plasmon of an MNS is
excited; thus, it seems possible to distinguish between two nearby wavelengths in a
small frequency range near ? 4.7?1015 rad/s using a device less than 1 ?m in size.
Exciting the surface plasmon in a particular MNS is closely related to the frequen-
cies where the whole system exhibits a large amount of forward or backward scattering
of light. This pronounced forward and backward scattering is caused within specific
ranges of frequency that allow the collection of MNS?s to constructively or destruc-
tively add their radiated light. This behavior has exciting experimental consequences
such as being able to infer the ohmic power dissipated in an MNS by looking at the
scattered light field. Rather than having to coat the MNS?s with specific dyes to
measure their output power, it should be possible to simply detect the amount of
forward and back scattered light.
The retardation of the incident and scattered light field removes the symmetry
through the middle of the line of nanospheres allowing a range of complex phenomena.
111
Only a few of these effects have been discussed in this chapter; many variations have
not yet been explored and will probably disclose more fascinating results.
112
Chapter 6
Summary
In this dissertation, we explored the effects of many body dipole interactions.
Primarily we studied the long range interactions between groups of Rydberg atoms
since the large size of the Rydberg atoms allowed for the creation of large dipole
moments. The dipole moments were either created via a second order interaction be-
tween excited atoms in a field free environment or induced via a first order interaction
such as being placed in a static electric field. When using the second order case, or
van Der Waals, the interaction between Rydberg atoms fell off as 1/R6, and when
using the the first order case the interaction fell off as 1/R3. In both cases, the excita-
tions in the system were tracked by solving the time dependent Schr?odinger equation
numerically. We also looked at a classical system of interacting dipole moments and
calculated the power absorbed and emitted by each classical dipole.
6.1 Dipole blockade
The first effect studied in this dissertation was the dipole blockade. A dipole
blockade is seen when the interactions between nearby Rydberg atoms push the sys-
tem out of resonance with a tightly tuned excitation laser, and the total number of
excitations is suppressed. We developed a many body wavefunction approach that
allowed us to study two particle correlations, number correlations via the Mandel
Q-parameter, how the excitations fill space, and the possible use of Rydberg atoms
113
in a phase gate. This many body wavefunction was used with the time-dependent
Schr?odinger equation and solved numerically by using the split operator method.
The 2D two particle correlation function clearly illustrated the existence of a
minimum distance between atoms at which multiple excitations were possible. In
the first order dipole-dipole case, the region defined by this minimum distance, Rb,
is dependent on the angle of orientation between atoms. There is a critical angle,
? ? 55?, where the dipole interaction between pairs of atoms vanishes, and it is
possible for two nearby atoms along this angle to both become excited. If a well
defined blockade region is desired then special care must be taken to exclude any
pairs of atoms from the excitation region that might be situated close to the critical
angle.
The Mandel Q-parameter was a useful quantity for measuring the degree to
which a group of Rydberg atoms were blockaded and how well correlated the system
is. Our calculations of the Q-parameter showed the level of correlation increased as
the density of the system was increased. Even with very few excitations, if the gas
of atoms was dense enough, the system was still well correlated. This was a result of
the non-excited atoms within a blockade region still affecting atoms outside of Rb.
We learned that if the atoms are placed on a regular linear lattice, the maximum
number of allowed excitations grows smoothly as the length of the lattice is increased.
We did not see a sudden jump from one excited atom to two excited atoms which
indicated that a bubble picture for excitations is not correct. The size of Rb was seen
when the probability of being in a state with a specific number excited was plotted
114
as a function of chain length. This plot also explained the smoothness seen in the
maximum number of possible excitations versus chain length. Our results also clearly
demonstrated the sub-Poissonian nature of the correlated system.
One of the bigger advantages to solving the many body wavefunction as opposed
to a mean field or Monte Carlo approach was that it allowed us to keep track of phase
changes. We examined the use of blockaded atoms as qubits in a phase gate. We
simulated a well defined collection of Rydberg atoms as qubits, and calculated the
phase accumulated after a series of pulses. We compared a perfectly functioning phase
gate to the non-perfect case and discovered that in order for the generated errors to
be small the interaction energy between the two qubits must be rather large.
We also performed three very different model calculations of a gas or Rydberg
atoms in the strongly blockaded regime: the many body wavefunction model using
pseudoparticles, a simple sinusoidal model, and a Monte Carlo model. In this regime
the density of atoms is much higher than in previous simulations. In this case we were
simulating around of 1.5?107 atoms. The three models were all in good qualitative
agreement with each other and within a factor of 2 in both Nexc and saturation
time. For such a large number of atoms this was encouraging, but the calculations
were not in agreement enough with experimental data to suggest that the physics
of the system was completely understood. We were unable to take into account the
variations in density overRb present in the experimental parameters of the MOT, and
we believe that in order for a theoretical model to accurately simulate this experiment
this density variation must be taken into account. This would present an imposing
115
challenge to any mean field model since the value of the mean field energy shift of an
atom would also have to vary with the location of the atom in the MOT. Nonetheless
our studies still provided a useful tool in studying some of the many body effects
present in a strongly blockaded regime.
6.2 Coherent hopping of excitation
In Chapter 3, we studied the coherent energy exchange between Rydberg atoms.
This exchange of character between two atoms is possible when the energy cost for
such a transition is very small. The coupling that regulates this hopping of states
is a dipole interaction that falls off as 1/R3. We examined two distinct cases where
atoms are excited into Rydberg states that allow a resonant exchange of energy to
occur via a dipole transition: the hopping from an |sp? state to a |ps? state in a field-
free environment and the hopping among the highest Stark states (|n1n2?? |n2n1?)
in a static electric field. In the second case we proposed a realistic experimental
setup that has been attempted. We simulated the coherent hopping of excitation by
using an essential states model that included only states that are degenerate or nearly
degenerate in our calculations.
In the field free situation we placed a single s Rydberg atom in a sea of regularly
placed p Rydberg atoms. We then solved the time-independent Schr?odinger equation
to generate band structures for various geometries. The band structure for a linear
array of of Rydberg atoms was remarkably similar to what is seen in the bands for
a line of driven metallic nanospheres. There were three modes of propagation: two
116
degenerate bands (thelobesof thehoppingpstateperpendicular to thearrayaxis) and
one nondegenerate band (the lobes of thepstate parallel to the array axis). The band
structures also allow us to label the nature of the p state hopping as either particle-
like or hole-like. When the geometry was expanded to two dimensions, phonon-like
hopping was also seen in the bands. In three dimensions the band structure becomes
feature-rich for such a simple system, with holes, particles, and many degenerate
modes of hopping.
We proposed an experimental setup consisting of Rydberg atoms placed into
high Stark states by a static electric field. In this case one atom is excited to an
n = 61 Stark state in an array of n = 60 Stark states. We simulated this system as it
evolved in time by numerically solving the time-dependent Schr?odinger equation via
anexponential propagator. Since the perfect placement of atomsis not experimentally
viable, we randomly placed each atom in a number of separated regions and averaged
our results over a large number of geometries. Our simulations indicated that the
coherent hopping of the n = 61 state should still last for substantial time scales. We
also simulated a case where the non-uniformity of the atoms was severe enough for a
region to be empty, and our results showed that while coherence of the system is not
destroyed, the n = 61 state is unable to hop across the hole in the chain of atoms.
6.3 The effects of rotary and spin echo sequences on a Rydberg gas
We next studied the effects of using echo sequences on a Rydberg gases. In the
first part of chapter 4 we simulated the rotary echo signal of an ultracold Rydberg
117
gas in the strongly blockaded regime. While in the second part we examined the
spin echo signal of a Rydberg gas that allowed for a coherent hopping of state to
occur. A rotary echo is accomplished in a frozen Rydberg gas by reversing the sign
of the excitation mechanism after a length of time ?p. When ?p is exactly half of
the total excitation time ? and there is no significant source of outside decoherence,
a strong signal should appear reflecting that the system has been returned to its
initial configuration. A spin echo is performed by exciting a gas using a ?/2 pulse,
letting it relax for set time, exciting it again with a ? pulse, letting it relax again for
second amount of time, and finally deexciting it with a final ?/2 pulse. When the
two relaxation times are equal and outside sources of decoherence are again small,
the gas should be returned to its initial state. We simulated the first case by using
the many-body pseudoparticle wavefunction approach and the split operator method
to numerically solve the time-dependent Schr?odingder equation. The second case
was solved by diagonalizing the interaction Hamiltonian and using an exponential
propagator to numerically simulate the evolution of the many-body wavefunction.
For the rotary echo case, a strong echo signal means the gas of Rydberg atoms
has been returned to a state with very few excitations. In fact if the system is
perfectly blockaded, our results show that the system should exhibit a perfect echo
where no excitations are present when ?p = ?. This is a result of the entire collection
of atoms oscillating in a coherent manner, and analagous to an isolated superatom.
A more interesting situation arose when the system was not completely blockaded,
and the interactions between pseudoparticles lead to dephasing and a supression of
118
the echo signal. As the density was increased the echo signal was weakened. If the
van der Waals interaction between pseudoparticles was also reversed in sign at the
same time the excitation amplitude reverses sign, a perfect echo is again seen. We
offered a system that might possibly allow for the force between Rydberg atoms to
be switched from attractive to repulsive.
Our simulations of the spin echos showed that if only one hopper is present, and
the thermal motion of the gas is ignored then a perfect spin echo signal will always be
seen when the relaxtion times are equal. The effect of thermal motion is very small
when the relaxation times are short enough to only allow a couple of hops to occur.
When four or more hops are allowed, however, the spin echo signal is destroyed for
temperatures greater than 300 ?K. The increase in the number of allowed hops leads
to the increase in the number of atoms that contribute to the dephasing.
6.4 Interactions between classical dipole moments: nanospheres
In chapter 5 we investigated the response of a system of metallic nanospheres
(MNSs) to a driving electric field. When excited by a tuned beam of light, the
surface plasmon on a MNS behaves as an oscillating dipole moment. In turn, these
oscillating dipoles generate their own time-dependent electric fields which interact
with other MNS. We reported several interesting effects that arise from this simple
system when the full electric field from the oscillating dipoles was used rather than
just using the near field approximation. In order to describe this system of driven
oscillating dipoles we developed a coupled point dipole model where each MNS is
119
treated as a point dipole that responds linearly to an electric field. For our results
we used up to ten Ag MNSs.
When all of the MNSs are placed in a linear array and are equally illuminated
by a beam of light, the power deposited in a particular MNS strongly depends on
its position in the system and the frequency of the driving electric field. In fact, the
ratio of the power dumped into the first MNS and the last MNS can be inverted
by making small changes in the frequency of the excitation beam. This effect is not
seen when the near field approximation is used because retardation of the scattered
full field introduces interference. The retardation of the scattered light removes the
symmetry through the middle of a line of MNS allowing a range of complex effects.
Our simulations showed the possibility of taking advantage of effects through ex-
citing experimental realizations such as finely tuned frequency discrimination and
nanolithography.
120
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