# Supplemental Material

C. He and R.R. Jones  Department of Physics, University of Virginia,
Charlottesville, Virginia 22904-4714

##  .1 Rydberg Atom Excitation, Spatial Distribution, and Detection

In the experiments, the 300ns, 482nm Rydberg excitation laser pulse has a
bandwidth of approximately 3MHz and is produced by chopping a 482nm, $\sim$
100mW cw laser beam with an acousto-optic modulator. The excitation laser
drives atoms from the 85Rb 5$p_{3/2}$ upper MOT state to the nominal
32$p_{3/2},|m_{j}|$=3/2 level in the presence of a static electric field,
$F_{0}\sim$12V/cm. The 5$p\rightarrow$ 32$p_{3/2},|m_{j}|$=3/2 optical
transition is enabled by weak Stark mixing of the 32$p$ level with neighboring
$s$ and $d$ states. The excitation laser is focused to a diameter of 30$\mu$m
through the center of the 0.5-mm diameter MOT with sufficient intensity to
saturate the 5$p\rightarrow$ 32$p$ transition at $\simeq$ 50$\%$ excitation
probability, creating a cylindrical volume of 32$p_{3/2},|m_{j}|=3/2$ atoms.
The Rydberg excitation bandwidth is $\sim$ 6MHz, dominated by the natural
linewidth of the 5$p_{3/2}$ state. The 85Rb 32$p$ fine structure splitting is
greater than 3 GHz, so there is no laser excitation of 32$p_{1/2}$ atoms [1].
Similarly, for $F_{0}\sim$ 12 V/cm, the energy splitting between the nominal
$32p_{3/2},\absolutevalue{m_{j}}$=1/2 and
$32p_{3/2},\absolutevalue{m_{j}}$=3/2 is 150MHz. Thus, there is also
negligible excitation of atoms to the $32p_{3/2},\absolutevalue{m_{j}}$=1/2
states.

The atom density in the MOT is calibrated by measuring the fluorescence from
the 5$p$ atoms and is controlled by adjusting the background Rb vapor pressure
in the MOT vacuum chamber [2]. Prior to the 482nm laser pulse, the atoms in
the laser excitation volume are assumed to be randomly distributed with a
nearest neighbor distribution function

$G(R)=4\pi\rho R^{2}e^{-\frac{4}{3}\pi\rho R^{3}}~{},$ (1)

where $R$ is the separation between nearest neighbor atoms and $\rho$ is the
atom density. This distribution predicts the most probable and average
internuclear separations

$R_{0}=\sqrt[3]{\frac{1}{2\pi\rho}}$ (2)

and

$R_{avg}=\sqrt[3]{\frac{3\ln{2}}{4\pi\rho}}\simeq 1.01R_{0}~{},$ (3)

respectively.

Given the relatively low Rydberg principal quantum number, moderate atom
densities, and broad Rydberg excitation bandwidths in the experiments, the van
der Waals shifts associated with non-resonant Rydberg-Rydberg interactions are
negligible for all but the closest atoms in the ensemble. Specifically, the
Rydberg blockade radius [3] associated with a $\sim$ 6MHz frequency shift is
only $\sim$ 2.5$\mu$m. Even at the highest experimental Rydberg density,
$\rho$ = 3 $\times$ 109 atoms/cm3, blockade effects reduce the Rydberg
excitation probability in less than 10% of the atoms in the laser focal
volume. Accordingly, the Rydberg atoms can be assumed to be randomly
distributed with a saturated Rydberg atom density ($\rho$ = 0.5 - 3.0 $\times$
109 atoms/cm3) given by a fixed fraction ($\sim$0.3) of the measured MOT
density.

The parallel stainless steel field plates straddling the MOT suppress stray
electric fields and reduce inhomogeneities in the applied field to $\simeq$
0.03% of the nominal value [2]. Voltage pulses from an arbitrary waveform
generator (AWG) applied to the field plates facilitate Stark tuning of the
Rydberg atom energies relative to the dipole-dipole (DD) driven
32$p_{3/2},|m_{j}|$=3/2$\,+\,$32$p_{3/2},|m_{j}|$=3/2 $\leftrightarrow$
32$s\,+\,$33$s$ (i.e. $pp\leftrightarrow ss^{\prime}$) Förster-resonance. At
the end of each measurement, a high voltage pulse applied to the field plates
ionizes any Rydberg atoms in the interaction region, pushing the resulting
ions through a small slot in the field plates toward a microchannel plate
(MCP) detector. The time-dependent ion current from the MCP provides a measure
of the relative population in different Rydberg states via state-selective
field ionization [4].

##  .2 Stark Tuning Near the 32$\mathbf{p_{3/2},|m_{j}|=}$3/2 +
32$\mathbf{p_{3/2},|m_{j}|=}$3/2 $\leftrightarrow$
32$\mathbf{s+}$33$\mathbf{s}$ Förster-resonance

Figure 1: Stark map of 85Rb $32p_{3/2},\absolutevalue{m_{j}}=3/2$ (black
solid), $32p_{3/2},\absolutevalue{m_{j}}=1/2$ (black dash), $32s$ (blue), and
$33s$ (orange) single atom binding energies vs electric field.

Figure 1 shows the calculated binding energies of the nominal
32$p_{3/2},\absolutevalue{m_{j}}=$1/2, 32$p_{3/2},\absolutevalue{m_{j}}=$3/2,
32$s$, and 33$s$ Rydberg states in an isolated 85Rb atom as a function of
applied electric field, $F$. In the weak fields of interest in our experiments
(11-12V/cm), the eigenstates undergo small quadratic Stark shifts, but
principally retain the character of the zero field levels to which they are
adiabatically connected. Specifically, the percentages of the zero field
levels in these Stark eigenstates are: 32$p_{3/2},\absolutevalue{m_{j}}=1/2$
(98.3$\%$ at 11V/cm and 97.9$\%$ at 12V/cm);
32$p_{3/2},\absolutevalue{m_{j}}=3/2$ (99.0$\%$ at 11V/cm and 98.8$\%$ at
12V/cm); 32$s$ (99.4$\%$ at 11V/cm and 99.3$\%$ at 12V/cm); 33$s$ (99.1$\%$
11V/cm and 99.0$\%$ 12V/cm). Accordingly, we use zero field state-labels for
the Stark eigenstates and the corresponding radial and angular momentum
eigenfunctions when calculating matrix elements associated with laser
excitation of, and DD-interactions between, the Rydberg atoms.

The electric field $F_{0}$ at which the $pp\leftrightarrow ss^{\prime}$
Förster-resonance occurs is determined from the energies of “bare” atom pairs
with large internuclear separations $R\rightarrow\infty$, for which atom-atom
interactions are negligible. Figure 2 shows the relevant $pp$ and
$ss^{\prime}$ pair energies as a function of electric field. The
32$p_{3/2},\absolutevalue{m_{j}}$=3/2 + 32$p_{3/2},\absolutevalue{m_{j}}$=3/2
($pp$) and $32s$+$33s$ ($ss^{\prime}$) atom pair states are degenerate at
$F_{0}$ = 11.49V/cm, defining the resonance field, in good agreement with
experimental observations.

The energy splitting between the $pp$ and $ss^{\prime}$ states can be
precisely tuned, by varying the external electric field, $F$. Near resonance
($|F-F_{0}|\leq$ 0.5V/cm for our studies), the relative Stark shift of the
non-interacting $pp$ and $ss^{\prime}$ pair states can be approximated by a
linear function of $F$. This electric field to resonance “detuning” conversion
function is: $E(F)\simeq$ 170$(F-F_{0})\frac{\text{MHz}}{(\text{V/cm})}$.

Figure 2: Stark map of the energies of the four degenerate
32$p_{3/2},\absolutevalue{m_{j}}$=3/2 + 32$p_{3/2},\absolutevalue{m_{j}}$=3/2
($pp$ \- blue solid), eight degenerate 32$p_{3/2},\absolutevalue{m_{j}}$=3/2 +
32$p_{3/2},\absolutevalue{m_{j}}$=1/2 (blue dash), four degenerate
32$p_{3/2},\absolutevalue{m_{j}}$=1/2 + 32$p_{3/2},\absolutevalue{m_{j}}$=1/2
(blue dot) and eight degenerate $32s$ \+ $33s$ ($ss^{\prime}$ \- orange) non-
interacting atom pair states vs. electric field. The energies are plotted
relative to those of $pp$ and $ss^{\prime}$ at the $pp\leftrightarrow
ss^{\prime}$ resonance, which occurs in an electric field $F_{0}\simeq$ 11.5
V/cm.

##  .3 Dipole-Dipole Interaction Between Two Well-Separated Rydberg Atoms

Consider a pair of Rydberg atoms, each containing a positively charged ion
core and a highly excited electron. For ion separations $R$ much greater than
the spatial extent of the relevant Rydberg wavefunctions on the two atoms, the
lowest order term in the neutral atom-atom multipole coupling is dipole-dipole
(DD). The DD interaction between two atoms is [4]

$V_{DD}=\frac{\vec{r_{1}}\cdot\vec{r_{2}}}{R^{3}}-\frac{3(\vec{r_{1}}\cdot\vec{R})(\vec{r_{2}}\cdot\vec{R})}{R^{5}},$
(4)

where $\vec{r_{1}},\vec{r_{2}}$ are the positions of electrons 1 and 2
relative to their respective ions, and $\vec{R}$ is the position of ion 2
relative to 1 (Fig. 3).

We define $\hat{z}$ as the direction of an external electric field $\vec{F}$
in the lab frame, and express Eq. 4 using lab frame coordinates with
$\vec{r_{1}}=(x_{1},y_{1},z_{1}),\vec{r_{2}}=(x_{2},y_{2},z_{2})$ and
$\vec{R}=(X,Y,Z)$. The internuclear distance is $R=\sqrt{X^{2}+Y^{2}+Z^{2}}$,
and the direction of $\vec{R}$ can be defined using polar and azimuthal angles
$\Theta=\text{arctan}\frac{\sqrt{X^{2}+Y^{2}}}{Z}$ and
$\Phi=\text{arctan}\frac{Y}{X}$, respectively. This gives

$\begin{split}V_{DD}=\frac{1}{R^{3}}\big{[}&(x_{1}x_{2}+y_{1}y_{2}+z_{1}z_{2})\\\
&-3(x_{1}\text{sin}\Theta\text{cos}\Phi+y_{1}\text{sin}\Theta\text{sin}\Phi+z_{1}\text{cos}\Theta)\\\
&\cdot(x_{2}\text{sin}\Theta\text{cos}\Phi+y_{2}\text{sin}\Theta\text{sin}\Phi+z_{2}\text{cos}\Theta)\big{]},\end{split}$
(5)

in the lab frame. Although it is not immediately obvious from Eq. 5, for a
single pair of atoms, $V_{DD}$ is independent of the azimuthal angle $\Phi$
(describing rotations of $\vec{R}$ about the electric field axis, $\hat{z}$).
However, the independence no longer holds for ensembles with more than two
atoms where beyond nearest neighbor interactions play a role.

Figure 3: Coordinates relevant to the computation of the DD-interaction
between a pair of Rydberg atoms subject to an electric field,
$\vec{F}=F\hat{z}$.

Due to the substantial mass of the ions, and their relatively large
separation, $\vec{R}$ can be treated as a classical parameter. However, the
positions of the Rydberg electrons, $x_{1},y_{1},z_{1},x_{2},y_{2},z_{2}$ must
be treated using quantum mechanical operators. We exploit the symmetry of the
electronic angular momentum eigenstates to reduce the complexity of the
computation of $V_{DD}$, writing the Cartesian position operators for the two
Rydberg electrons in terms of spherical tensor operators [5, 6],
$x_{1,2}=\frac{1}{\sqrt{2}}r_{1,2}\left(C^{(1)}_{-1}-C^{(1)}_{1}\right),y_{1,2}=\frac{i}{\sqrt{2}}r_{1,2}\left(C^{(1)}_{-1}+C^{(1)}_{1}\right),z_{1,2}=r_{1,2}C^{(1)}_{0}$,
where $r_{1,2}$ are the radial coordinate operators for the two electrons.
This gives

$\begin{split}V_{DD}=\frac{r_{1}r_{2}}{R^{3}}\bigg{[}&-\frac{3}{2}C^{(1)}_{1}C^{(1)}_{1}\text{sin}^{2}\Theta(\text{cos}2\Phi-i\text{sin}2\Phi)\\\
&+\frac{3}{2\sqrt{2}}\left(C^{(1)}_{1}C^{(1)}_{0}+C^{(1)}_{0}C^{(1)}_{1}\right)\\\
&\cdot\text{sin}2\Theta(\text{cos}\Phi-i\text{sin}\Phi)\\\
&+C^{(1)}_{0}C^{(1)}_{0}(1-3\text{cos}^{2}\Theta)\\\
&+\left(C^{(1)}_{1}C^{(1)}_{-1}+C^{(1)}_{-1}C^{(1)}_{1}\right)\left(\frac{3}{2}\text{sin}^{2}\Theta-1\right)\\\
&-\frac{3}{2\sqrt{2}}\left(C^{(1)}_{-1}C^{(1)}_{0}+C^{(1)}_{0}C^{(1)}_{-1}\right)\\\
&\cdot\text{sin}2\Theta(\text{cos}\Phi+i\text{sin}\Phi)\\\
&-\frac{3}{2}C^{(1)}_{-1}C^{(1)}_{-1}\text{sin}^{2}\Theta(\text{cos}2\Phi+i\text{sin}2\Phi)\bigg{]},\end{split}$
(6)

where, in each product of two spherical tensor operators, the operator on the
left acts on the Rydberg electron in atom 1 and the operator on the right acts
on the Rydberg electron in atom 2. Notably, the projection of total electronic
angular momentum along the electric field axis, $M_{J}$, is not conserved in
the presence of both the static field and $V_{DD}$. $V_{DD}$ can couple states
for which: i) $M_{J}$ changes by 0, $\pm 1$, or $\pm 2$; and ii) the angular
momentum quantum numbers $\ell_{1}$ and $\ell_{2}$ for the individual
electrons each change by $\pm 1$.

##  .4 Pair Eigenstates and Energies Near the 32$\mathbf{p_{3/2},|m_{j}|=}$3/2
+ 32$\mathbf{p_{3/2},|m_{j}|=}$3/2 $\leftrightarrow$
32$\mathbf{s+}$33$\mathbf{s}$ Förster-resonance

The DD-coupling between atoms alters the bare Rydberg pair states and the
energies that are shown in Figs. 1 and 2. The effects are small when the
magnitude of the energy separation between the interacting states $|E|$ is
much greater than the magnitude of the relevant matrix element of $V_{DD}$
between them. The prefactor, $\frac{r_{1}r_{2}}{R^{3}}$, in $V_{DD}$ sets the
upper scale for those matrix elements, since the angular terms are of order
unity. In general, for a given Rydberg state with principal quantum number
$n$, the radial matrix elements connecting it to other levels are largest when
both states have very nearly the same effective principal quantum number
$\nu=n-\delta$, where $\delta$ is the quantum defect. In this case, the matrix
elements scale as $\nu^{2}$. Their magnitude decreases substantially for
differences in $\nu$ greater than 1 [4].

Our experiments focus on the effect of DD-interactions on initially excited
85Rb 32$p$ ($\nu\simeq$29.4) atoms. By far the largest relevant matrix
elements of $V_{DD}$ involve the 32$s$ or 33$s$ states ($\nu\simeq$ 28.9 or
29.9, respectively) or the 30$d$ or 31$d$ levels ($\nu\simeq$ 28.7 or 29.7,
respectively) [1]. For two Rb atoms separated by the most probable nearest
neighbor distance $R_{0}$ in a random ensemble at density $\rho$, those matrix
elements have maximum magnitudes on the order of $2\pi\rho\nu^{4}$, roughly
5MHz at a typical experimental Rydberg density, $\rho\sim 10^{9}$ atoms/cm3.
However, all Rydberg atom pair combinations, except 32$s$33$s$, are separated
in energy from 32$p$32$p$ by several cm-1 or more over the range of applied
fields we consider, greater than the largest DD matrix elements by at least
five orders of magnitude. Therefore, we can safely neglect all pair
combinations other than 32$p$32$p$ and 32$s$33$s$ when considering the DD-
driven dynamics of an ensemble of initially excited 32$p$ atoms. Similarly,
given that both the maximum DD matrix elements and the Rydberg excitation
bandwidth have magnitudes of several MHz, the 3.4GHz 32$p$ fine-structure
splitting ensures that 32$p_{1/2}$ states play no role in the dynamics of
isolated atom pairs [1]. Thus, the bare pair basis states that are potentially
relevant to the experiments are the 24 (sixteen 32$p_{3/2}$ \+ 32$p_{3/2}$ and
eight $ss^{\prime}$ levels) whose field-dependent energies are shown in Fig.
2.

Due to the differential Stark shift, in electric fields near the
32$p_{3/2},|m_{j}|$=3/2+32$p_{3/2},|m_{j}|$=3/2 $\leftrightarrow$ 32$s+$33$s$
(i.e. $pp\leftrightarrow ss^{\prime}$) Förster-resonance, the pair states
involving 32$p_{3/2},|m_{j}|$=3/2+32$p_{3/2},|m_{j}|$=1/2 and
32$p_{3/2},|m_{j}|$=1/2+32$p_{3/2},|m_{j}|$=1/2 are split from the
32$p_{3/2},|m_{j}|$=3/2+32$p_{3/2},|m_{j}|$=3/2 levels by $\sim$140MHz and
$\sim$280MHz, respectively (see Fig. 2). Accordingly, there is negligible
direct laser excitation of atom pairs involving 32$p_{3/2},|m_{j}|$=1/2, and
the DD couplings between the initially populated
32$p_{3/2},|m_{j}|$=3/2+32$p_{3/2},|m_{j}|$=3/2 pairs and the
32$p_{3/2},|m_{j}|$=3/2+32$p_{3/2},|m_{j}|$=1/2 or
32$p_{3/2},|m_{j}|$=1/2+32$p_{3/2},|m_{j}|$=1/2 levels are expected to be
weak.

Fig. 4 confirms that expectation, comparing two calculations of the pair
eigenenergies in the presence of the DD interaction at resonance ($F=F_{0}$),
as a function of the orientation angle $\Theta$. The solid black lines show
the energies computed using the 24 state basis (i.e. including
32$p_{3/2},|m_{j}|=$1/2 states). The orange and blue curves are the energies
computed using a reduced basis with only 32$p_{3/2},|m_{j}|=$3/2, 32$s$ and
33$s$ states. The energies are plotted relative to the bare
32$p_{3/2},|m_{j}|=$3/2 + 32$p_{3/2},|m_{j}|=$3/2 level. As expected, there
are only minor differences in the results of the two calculations. The
eigenenergies agree to better than 1MHz, with the largest differences at
angles where the DD-coupling is zero in the reduced basis. The
32$p_{3/2},|m_{j}|$=1/2 character in the eigenstates is also quite small.
Accordingly, starting from a pure 32$p_{3/2},|m_{j}|=$3/2 ensemble, for short
interaction times ($<$ 1$\mu$s) the DD-driven dynamics of atom pairs are
accurately described using the reduced basis. Over these brief timescales,
spontaneous emission, transitions between Rydberg levels due to black-body
radiation, and thermal and/or DD-interaction-induced atom motion can all be
reasonably neglected.

Figure 4: On-resonance pair eigenenergies vs alignment angle $\Theta$ in a
static field $F=F_{0}$, for an atom separation equal to the average nearest
neighbor separation for a random ensemble with a Rydberg density of $3\times
10^{9}$/cm3. The energies are plotted relative to the bare
32$p_{3/2},|m_{j}|$=3/2+32$p_{3/2},|m_{j}|$=3/2 level. The solid black curves
show the results from the full 24 state basis (i.e. including
32$p_{3/2},|m_{j}|$=1/2 states). The blue and orange curves are from the
reduced basis calculations (excluding 32$p_{3/2},|m_{j}|$=1/2 states). The
blue lines show eigenstates that are linear combinations of bare angular
momentum states with $\absolutevalue{M_{J}}$=1 (for $ss^{\prime}$) and
$\absolutevalue{M_{J}}$=3 (for $pp$). The orange lines show eigenstates that
are linear combinations of $pp$ and $ss^{\prime}$ with
$\absolutevalue{M_{J}}$=0. The lines at zero energy (only orange is visible)
are states that are antisymmetric linear combinations,
$\ket{ss^{\prime}}_{A}=(\ket{ss^{\prime}}-\ket{s^{\prime}s})/\sqrt{2}$, which
do not couple to $pp$.

##  .5 Approximate 2-Level DD-Coupled Systems

A distinct advantage of the reduced pair-state basis is that the level
structure and dynamics collapse to that of four independent, coupled two-level
systems, distinguished by the signs of the projections of total electronic
angular momentum for the individual 32$p_{3/2},|m_{j}|=$3/2 electrons in the
initially excited $pp$ state. Each of those distinct $pp$ basis states is
coupled to a single symmetric linear combination of $ss^{\prime}$ and
$s^{\prime}s$ states,
$\ket{ss^{\prime}}_{S}=(\ket{ss^{\prime}}+\ket{s^{\prime}s})/\sqrt{2}$, which
are also distinguished by the $m_{j}$ values of the two electrons. Four
additional basis states that are antisymmetric linear combinations
$\ket{ss^{\prime}}_{A}=(\ket{ss^{\prime}}-\ket{s^{\prime}s})/\sqrt{2}$ with
$(m_{j},m_{j^{\prime}})=$(1/2,1/2),(-1/2,-1/2),(1/2,-1/2) and (-1/2,1/2), do
not couple to $pp$ and, therefore, remain unpopulated and not relevant to the
dynamics. The basis vectors and corresponding DD matrix elements for the four
independent two-level systems are detailed below.

$\begin{pmatrix}\ket{pp}\\\\[2.0pt]
\ket{ss^{\prime}}_{S}\end{pmatrix}_{++}\;=\begin{pmatrix}pp(3/2,3/2)\\\\[2.0pt]
\frac{1}{\sqrt{2}}[ss^{\prime}(1/2,1/2)+s^{\prime}s(1/2,1/2)]\end{pmatrix}\;$
$V_{++}=-\frac{r_{ps}r_{ps^{\prime}}}{\sqrt{2}R^{3}}\text{sin}^{2}\Theta(\text{cos}2\Phi-i\text{sin}2\Phi)$
(7)

$\begin{pmatrix}\ket{pp}\\\\[2.0pt]
\ket{ss^{\prime}}_{S}\end{pmatrix}_{--}\;=\begin{pmatrix}pp(-3/2,-3/2)\\\\[2.0pt]
\frac{1}{\sqrt{2}}[ss^{\prime}(-1/2,-1/2)+s^{\prime}s(-1/2,-1/2)]\end{pmatrix}\;$
$V_{--}=V_{++}^{*}$ (8)

$\begin{pmatrix}\ket{pp}\\\\[2.0pt]
\ket{ss^{\prime}}_{S}\end{pmatrix}_{+-}\;=\begin{pmatrix}pp(3/2,-3/2)\\\\[2.0pt]
\frac{1}{\sqrt{2}}[ss^{\prime}(1/2,-1/2)+s^{\prime}s(1/2,-1/2)]\end{pmatrix}\;$
$V_{+-}=\frac{r_{ps}r_{ps^{\prime}}}{\sqrt{2}R^{3}}\left(\text{sin}^{2}\Theta-\frac{2}{3}\right)$
(9)

$\begin{pmatrix}\ket{pp}\\\\[2.0pt]
\ket{ss^{\prime}}_{S}\end{pmatrix}_{-+}\;=\begin{pmatrix}pp(-3/2,3/2)\\\\[2.0pt]
\frac{1}{\sqrt{2}}[ss^{\prime}(-1/2,1/2)+s^{\prime}s(-1/2,1/2)]\end{pmatrix}\;$
$V_{-+}=V_{+-}$ (10)

The radial matrix elements $r_{ps}$ =964 and $r_{ps^{\prime}}$=941 are
computed numerically.

For each 2-level system, with a detuning $E$ between the $pp$ and
$ss^{\prime}$ basis states, the Hamiltonian takes the simple form:

$H=\begin{pmatrix}0&V\\\ V^{*}&E\\\ \end{pmatrix},$ (11)

with eigenvalues

$E_{\pm}=\frac{1}{2}\Big{(}E\pm\sqrt{E^{2}+4\absolutevalue{V}^{2}}\Big{)}$
(12)

for $V$ = $V_{++}$, $V_{--}$, $V_{+-}$, or $V_{-+}$ depending on the initial
$pp$ state. By inspection, and as expected from the symmetry of the two-atom
geometry, the eigenvalues are independent of $\Phi$ for any of the four
possible values of $V$. Thus without loss of generality, we can set $\Phi$ =
0, ensuring that $V$ is real. The relevant eigenstates corresponding to the
respective eigenvalues $E_{+}$ and $E_{-}$ can then be written

$\ket{+}=\cos{\frac{\alpha}{2}}\ket{pp}+\sin{\frac{\alpha}{2}}\ket{ss^{\prime}}_{S}$
$\ket{-}=\cos{\frac{\alpha}{2}}\ket{ss^{\prime}}_{S}-\sin{\frac{\alpha}{2}}\ket{pp}$
(13)

where $\tan{\alpha}=\frac{2V}{E}$. The eigenvalues and eigenstates for
$V_{++}$ and $V_{--}$ ($pp,~{}\absolutevalue{M_{J}}=3$ and
$ss^{\prime},~{}\absolutevalue{M_{J}}=1$) are identical, as they are for
$V_{+-}$ and $V_{-+}$ ($pp,~{}\absolutevalue{M_{J}}=0$ and
$ss^{\prime},~{}\absolutevalue{M_{J}}=0$). For a random ensemble with a
density $\rho$ = 109 cm-3, the angle averaged DD-matrix elements for an atom
pair separated by the most probable nearest neighbor distance $R_{0}$ are:
$\langle V_{++,--}\rangle\simeq 2.0$MHz and $\langle V_{+-,-+}\rangle\simeq
0.66$MHz.

For a given pair of atoms with separation $R$ and orientation angle $\Theta$,
the eigenstates form an avoided level crossing as a function of the detuning
$E$ (or the applied field $F$) with a minimum curve separation of
2$V(R,\Theta)$ at $E$=0 (i.e. at $F=F_{0}\simeq$ 11.5 V/cm). Fig. 5 shows the
distribution of calculated avoided level crossings for a random ensemble at a
density $\rho=$ 2 $\times$ 109 cm-3. The gray scale reflects the number of
energy curves passing through each point. The dashed orange and blue lines
show the energy levels for an atom pair with the ensemble averaged value of
$V$.

Figure 5: Distribution of calculated energies for a random ensemble of DD-
coupled $pp$ and $ss^{\prime}$ atom pairs vs electric field. The distribution
of atom separations corresponds to that for nearest neighbor distances at a
density $\rho=$ 2 $\times$ 109 cm-3. The $pp$ atom pairs are assumed to have
equal probabilities for total electronic angular momentum projections,
$\absolutevalue{M}=3,0$. The gray scale reflects the number of energy curves
passing through each point. The dashed orange and blue lines show the
eigenenergies for a pair of atoms with the average $V$ for the random
ensemble. The vertical dotted lines show typical fields used for the quasi-
phase-matching sequences described in the main text.

##  .6 Dynamics of the 2-Level DD-Coupled Systems

In the experiments, all of the Rydberg atoms in the ensemble are initially
excited to 32$p_{3/2},|m_{j}|=$3/2 states in an electric field, $F$=12 V/cm,
such that they are well-detuned from resonance (see Fig. 5). As shown in Fig.
6, at $t=0$ the atoms are tuned on, or near, resonance using a voltage step or
a sequence of voltage steps with fast ($\simeq$ 2ns) rise (or fall) times,
allowing $pp$ atoms to interact via the DD-interaction. After a total
interaction time $T$ the atoms are decoupled by tuning them very far from
resonance and then field ionized to measure the population transfer
probability from $pp$ to $ss^{\prime}$. Within the nearest neighbor
approximation, the dynamics of each atom pair can be described using an
analytic expression.

Figure 6: Schematic of the experimental timing including laser excitation of
the 32$p$ Rydberg atoms (dark blue line), control of the DD-interaction
through tuning-field steps (orange line) relative to the resonance lineshape
(cyan line), and field ionization detection (magenta line).

If the detuning $E$ is constant throughout the interaction time, each initial
pair $\Psi(0)=\ket{pp}$ evolves to

$\Psi(T)=C_{pp}(T)\ket{pp}+C_{ss^{\prime}}(T)\ket{ss^{\prime}}_{S}$ (14)

where the amplitudes $C_{pp}(T)$ and $C_{ss^{\prime}}(T)$ are given by the
standard transformation [7]:

$\begin{split}\begin{pmatrix}C_{pp}(T)\\\\[4.0pt]
C_{ss^{\prime}}(T)\end{pmatrix}\;=U(E,T)\begin{pmatrix}C_{pp}(0)\\\\[4.0pt]
C_{ss^{\prime}}(0)\end{pmatrix}\end{split}$ (15)

with $C_{pp}(0)$= 1, $C_{ss^{\prime}}(0)$=0 and

$U(E,T)=\begin{pmatrix}\cos{\frac{\phi}{2}}-i\frac{E}{\Gamma}\sin{\frac{\phi}{2}}&i\frac{2V}{\Gamma}\sin{\frac{\phi}{2}}\\\\[7.0pt]
i\frac{2V}{\Gamma}\sin{\frac{\phi}{2}}&\cos{\frac{\phi}{2}}+i\frac{E}{\Gamma}\sin{\frac{\phi}{2}}\end{pmatrix}$
(16)

where $\phi=\Gamma T$ is the Rabi phase and $\Gamma=\sqrt{E^{2}+4V^{2}}$ is
the generalized Rabi frequency.

Accordingly, the population transfer probability to $ss^{\prime}$ is

$\absolutevalue{C_{ss^{\prime}}(T)}^{2}=\frac{4V^{2}}{\Gamma^{2}}\sin^{2}{\frac{\phi}{2}}~{},$
(17)

exhibiting Rabi oscillations as a function of $T$. The maximum population
transfer probability has a Lorentzian lineshape as a function of detuning,
with a full-width at half-maximum (FWHM) $\Delta E=4V$ (see Fig. 1 in the main
text).

##  .7 Dynamics During QPM Sequences

To implement the time-domain quasi-phase matching (QPM) sequences described in
the main text, the sign of the detuning is rapidly reversed at the start of
$N$ consecutive time zones, with the atoms spending a time $T/N$ at a constant
detuning $\pm E$ in each zone. In this case, the system evolution is described
by a product of constant detuning transformations

$\begin{split}\begin{pmatrix}C_{pp}(T)\\\\[4.0pt]
C_{ss^{\prime}}(T)\end{pmatrix}\;=U_{QPM,N}\begin{pmatrix}C_{pp}(0)\\\\[4.0pt]
C_{ss^{\prime}}(0)\end{pmatrix}\end{split}$ (18)

with

$U_{QPM,N}=[U(-E,T/N)\cdot U(E,T/N)]^{N/2}$ (19)

for an even number of zones.

For large detunings $E\gg V$, we have $\frac{E}{\Gamma}\simeq 1$ and
$\frac{2V}{\Gamma}\ll 1$. In this case, it is straightforward to show that the
$N$-zone QPM transfer matrix can be approximated as

$U_{QPM,N}\approx\begin{pmatrix}1&i\frac{2V}{\Gamma_{N}}\sin{\frac{\phi_{N}}{2}}e^{i\frac{\phi_{N}}{2}}\\\\[4.0pt]
i\frac{2V}{\Gamma_{N}}\sin{\frac{\phi_{N}}{2}}e^{-i\frac{\phi_{N}}{2}}&1\end{pmatrix}$
(20)

for $\Gamma_{N}=\Gamma/N$ and $\phi_{N}=\Gamma_{N}T=\phi/N$, provided
$2^{N/2}\left(\frac{2V}{\Gamma_{N}}\right)^{2}\sin^{2}{\frac{\phi_{N}}{2}}\ll
1$ so that the net population transfer to $ss^{\prime}$

$\absolutevalue{C_{ss^{\prime}}(T)}^{2}=\frac{4V^{2}}{\Gamma_{N}^{2}}\sin^{2}{\frac{\phi_{N}}{2}}~{},$
(21)

is small.

This expression is identical to that derived for constant detuning (Eq. 17),
except that the generalized Rabi frequency $\Gamma$ has been replaced by
$\Gamma_{N}$. In terms of population transfer, the net result of the QPM
sequence is an increase in the period of the off-resonant Rabi oscillations by
a factor of $N$ and an increase in the amplitude of those oscillations by a
factor of $N^{2}$. As discussed in the main text, the enhancement of the
maximum population transfer afforded by QPM enables the creation of more
uniform quantum states throughout an ensemble through the use of far off-
resonant interactions for which the variations in the generalized Rabi
frequency are substantially reduced. Notably, as seen in the experiments, for
large population transfers where the condition
$2^{N/2}\left(\frac{2V}{\Gamma_{N}}\right)^{2}\sin^{2}{\frac{\phi_{N}}{2}}\ll
1$ no longer holds, the QPM enhancement factor is less than $N^{2}$ and the
effective Rabi oscillations are no longer strictly sinusoidal.

Figure 7: Bloch sphere representation of the evolution of the state vectors
for 3 atom pairs subject to different DD-interaction strengths ($V$ = 0.09,
0.14 and 0.19 for magenta, red and blue, respectively) at the same detuning
from resonance, $E$ = 0.99, for a $N$ = 4 time-zone QPM sequence. Groups (a),
(b), and (c) show identical evolution, but from different viewing angles. The
time during the evolution is noted in the upper left corner of each of the
eight panes within the three groups. The subscripts on the time labels
indicate that the snapshot corresponds to conditions just before ($-$) or just
after ($+$) the detuning reversal at the interface between two time zones. The
time spent at constant detuning in each zone is approximately one-quarter of a
Rabi cycle, $T_{0}/4=\pi/2$. The thin dotted lines that form circles on the
surface of the sphere show the projected paths of the state vectors at the
current detuning. Thick solid lines show the paths actually traversed on the
Bloch sphere up to that point in time. Thin solid radial lines show the
locations of the Bloch vectors at that instant. Note that due to the reversal
of the direction of the Bloch vector rotation, the small differences in the
azimuthal phase angles due to the different coupling strengths do not grow in
time, as their magnitudes at then end of each time zone are limited to those
at the end of the first zone.

Figure 8: Bloch sphere representation of the evolution of the state vectors
for 3 atom pairs subject to a 4-zone QPM sequence, analogous to Fig. 7 except
that the time spent in each zone is approximately one-half of a Rabi cycle,
$T_{0}/2=\pi$.

Figure 9: Bloch sphere representation of the evolution of the state vectors
for 3 atom pairs subject to a 4-zone QPM sequence, analogous to Fig. 7 except
that the time spent in each zone is approximately one full Rabi cycle,
$T_{0}=2\pi$.

##  .8 Bloch Sphere Representation of State Dynamics During QPM Sequences

Examination of the Bloch sphere evolution of the state-vectors for individual
atom pairs provides additional insight into the origin of the $1/N$ scaling of
the generalized Rabi frequency as well as the mechanism underlying active
dephasing suppression with QPM (see Figs. 7-9). Consider several atom pairs,
initially in state $pp$ at a constant detuning $E$ from resonance. At $t$=0,
each experiences a different Förster interaction strength $V(R,\Theta)$ with
the same constant detuning $E\gg V(R,\Theta)$. The interaction causes the
state-vectors for each atom pair to move away from $\hat{z}$ (at a polar angle
$\theta$=0), precessing at similar Rabi frequencies $\Gamma(R,\Theta)\simeq
E$, around axes that are tipped at small polar angles $\chi(R,\Theta)\simeq
V/E$ relative to $\hat{z}$, with azimuthal angles $\beta=\pi$. As shown in
Figs. 7-9, the state-vectors for the individual atom pairs trace out circles
on the surface of the Bloch sphere, with the $ss^{\prime}$ transition
amplitudes $A_{ss^{\prime}}=e^{i\varphi}\sin{\theta/2}$ defined by their
respective angular coordinates $(\theta,\varphi)$. During each Rabi period,
the values of $\theta$ for the different state-vectors oscillate between 0 and
2$\chi$. Due to the differing values of $\chi$ and $\Gamma$ for the different
pairs, the state-vectors become angularly dispersed over a dephasing time,
$\tau$.

In a QPM sequence with $N=2$ time zones, $E\rightarrow-E$ at $t=T/2$, and the
precession axes for the state-vectors jump to a new azimuthal angle,
$\beta=0$, and the precession direction reverses. The paths traced by each of
the state-vectors now encircle $\hat{z}$, with radii that depend on their
polar angles $\theta$ at the instant of the detuning jump. As a specific
example, consider the situation where $T\simeq\pi/E$ (illustrated in Fig. 8
for $N$ = 4). In this case, all state-vectors complete approximately half of
one precession in the first zone, such that $|A_{ss^{\prime}}|$ is at (or very
near) maximum with $(\theta,\varphi)\simeq(2\chi,\pi)$ at the end of the first
zone ($t=T/2$). After the detuning reversal, the state-vectors rotate in the
opposite direction, with cone angles 3$\chi$ relative to their new precession
axes, arriving at new maximum tilt angles $(\theta,\varphi)\simeq(4\chi,0)$ at
(or very near) the end of the second time zone ($t=T$). Thus, for $N$=2 zones,
the maximum transition amplitude and the time required to reach that maximum
are twice as large as with constant detuning, consistent with
$\Gamma\rightarrow\Gamma/2$. Generalizing to $N$ zones (assuming $2N\chi<1$),
$\Gamma\rightarrow\Gamma/N$ in agreement with the analytical prediction.

Two principal factors are responsible for the suppression of dephasing through
QPM. First, the variation in the Rabi frequency is significantly smaller for
$E\gg E_{0}$, resulting in a substantial reduction in the ensemble phase
variation $\Delta\phi$ over any time interval. Second, since the phase
evolution is reversed in successive time zones (similar to a spin echo),
$\Delta\phi$ does not accrue over the total interaction time $T$. The effect
is distinct from an echo, however, because the coupling is present throughout
the system evolution and $\varphi$ advances at a non-constant rate within each
zone. Of particular importance, for each atom pair, $\varphi$ changes by $\pi$
during one Rabi cycle in zone 1 (where the path traced by the state-vector on
the Bloch sphere does not enclose $\hat{z}$), as compared to 2 $\pi$ in all
other zones. Therefore, the contribution to $\Delta\phi$ from zone 1 is over-
corrected (by a factor of 2) by the reversed phase evolution in zone 2. So, at
the end of zone 2, $\Delta\phi$ has the same magnitude, but the opposite sign
as compared to that after zone 1. The same overcompensation occurs in all
subsequent zones, so that at the end of the QPM sequence at $t=T$, the
magnitude of $\Delta\phi$ is equal to that at the completion of zone 1, equal
to that acquired during the initial interval $T/N$. Accordingly, if the
ensemble dephases in a time $T=\tau$ while interacting at constant detuning
then, with a QPM sequence, it will not dephase until the time spent in zone 1
is equal to $\tau$, i.e. until the total interaction time is $T=N\tau$. Stated
more directly, QPM extends the dephasing time by a factor of $N$. The greater
the detuning and number of zones, the greater the extension of the dephasing
time. This scaling holds even for large population transfers.

Inspection of the Bloch sphere evolution of the state vectors for atom pairs
with different coupling strength makes clear that the restoration of the
visibility of Rabi oscillations for an ensemble is truly due to a suppression
of dephasing. During any time zone, the magnitude of the probability
amplitudes $|A_{ss^{\prime}}|$ (as determined by $\theta$) for different atom
pairs can differ significantly, but the phase angles $\varphi$ for the quantum
states are nearly identical. The larger the number of zones, the smaller the
time $T/N$ spent in the first zone, leading to a proportional reduction in the
ensemble variation in $\varphi$.

##  .9 QPM Sequences for Suppressing Dephasing in Ordered Samples

Quasi-phase-matching sequences can also be very effective for suppressing
dephasing driven by resonant couplings in more ordered systems. One example
might involve DD-interactions between atoms held at different sites in an
optical lattice or tweezer array for which the internuclear separations are
nominally the same, but not perfectly identical [8, 9, 10].

Fig. 10 shows the time-dependent $pp$ population calculated for an ensemble of
DD-coupled Rb atom pairs, initially excited to the same $pp$ states relevant
to our experiments, but with a Gaussian distribution of internuclear
separations $R=3\pm$0.05$\mu$m and internuclear axes aligned to the external
electric field axis ($\Theta=0$). A configuration similar to this might be
experimentally realized using an array of optical tweezers.

The damped Rabi oscillations seen in the black curve in Fig. 10 reflect the
significant dephasing that results from on-resonance excitation, despite the
perfect pair alignment and narrow distribution of internuclear separations.
The magenta curve illustrates that the dephasing can be effectively eliminated
through the use of off-resonant interactions, albeit with a small maximum
population transfer, if the atoms are held at a constant detuning. The blue
and orange lines show that large amplitude Rabi oscillations can be restored,
despite the large detunings, using QPM sequences. Further, the frequency of
those oscillations can be chosen by proper selection of the detuning for a
given number of QPM zones. The 97$\%$ contrast of the oscillations in the
orange curve can be further improved by increasing the number of zones and/or
reducing the detuning. It is worth noting, however, that the higher contrast
oscillations typically deviate from perfect sinusoids (see orange curve in
Fig. 10).

Figure 10: Simulations of the time-dependent $p$ state population (normalized
to total population) for an ensemble of DD-coupled atom pairs oriented with
their internuclear axes aligned along the external electric field
($\Theta=0$), and an average internuclear distance $R$ = 3$\mu$m with a
standard deviation of 50nm (Gaussian distribution). The different curves show
the population transfer resulting from DD-interactions at constant detuning or
pulsed QPM control sequences. The black line shows the damped Rabi
oscillations that result when the atoms are tuned to the Förster-resonance.
The magenta line shows the small amplitude, high frequency Rabi oscillations
predicted for a large constant detuning $E=2\sqrt{N^{2}-1}V_{avg}$. The blue
line shows the evolution when the atoms are subjected to a QPM sequence with
$N$=4 time zones and a detuning $E=2\sqrt{N^{2}-1}V_{avg}$, chosen to given
the same Rabi frequency as in the resonant case. The orange curve shows the
Rabi oscillations obtained using a QPM sequence with a somewhat smaller
detuning, $E=1.4\sqrt{N^{2}-1}V_{avg}$, allowing for higher contrast
population transfer modulations ($>$97$\%$), but with a moderate reduction in
the effective Rabi frequency.

##  .10 Simulations Including Beyond Nearest Neighbor Interactions Using
4-Atom Groups

Although the resonant coupling between nearest neighbor atoms in an ensemble
can suppress interactions with more distant atoms, that suppression is not
complete and it weakens for larger detunings from resonance [11, 12, 13, 2].
In particular, non-tunable exchange or “excitation hopping” interactions of
the form $ps\leftrightarrow sp$, $ps^{\prime}\leftrightarrow s^{\prime}p$, as
described through Eqs. 5 and 6, can have a non-negligible effect on the
dynamics. We model these beyond nearest neighbor effects using groups of four
atoms, coupled pairwise through the matrix elements given by Eqs. 7-10 for the
resonant processes, and through analogous expressions (involving the square of
one of the two radial matrix elements $r_{ps}$ or $r_{ps^{\prime}}$ rather
than the product of the two) for the exchange processes. For the 4-atom
groups, the problem does not reduce to one that can be solved in closed form,
so the quantum dynamics within each 4-atom group are computed numerically.

To define each four atom group, 99 atoms, each described by Cartesian
coordinates $(x_{i},y_{i},z_{i})$ are first randomly placed in a cube, the
length of each edge measuring $\sqrt[3]{\frac{100}{\rho}}$, assuming a uniform
distribution along each of the three spatial coordinates. We then place an
atom at the center of the cube. This atom, along with its three nearest
neighbors define the group of four. The relative position vectors between each
pair $\vec{R_{ab}}=(R_{ab},\Theta_{ab},\Phi_{ab})$ are derived from their
respective Cartesian coordinates and used to calculate the relevant matrix
elements. At $t=0$, each of the four atoms are randomly assigned to one of the
two possible $32p_{3/2},\absolutevalue{m_{j}}$=3/2 states. The probability
amplitudes in all relevant 4-atom states are computed by integrating the time-
dependent Schrödinger equation, either for constant DD-interactions at a
specific detuning $E$, or for a given QPM sequence. After an interaction time
$T$, the total probabilities for finding atoms in the 32$p$, 32$s$ and 33$s$
states are recorded. Those probabilities are summed for 5000-10000 4-atom
groups and then normalized to unity for comparison with the data (see Figs. 2
and 3 in the text of the primary paper).

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