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
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
/Rb
0
0.2
0.4
0.6
0.8
1
??/pi
(a)
00.511.522.533.5
/?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 ?. 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