††thanks: These authors contributed equally to this work.
<EMAIL_ADDRESS>These authors contributed equally to this
work.
<EMAIL_ADDRESS>
# Hyperfine dependent atom-molecule loss analyzed by the analytic solution of
few-body loss equations

Kai K. Voges  Institut für Quantenoptik, Leibniz Universität Hannover, 30167
Hannover, Germany  Philipp Gersema  Institut für Quantenoptik, Leibniz
Universität Hannover, 30167 Hannover, Germany  Torsten Hartmann  Institut für
Quantenoptik, Leibniz Universität Hannover, 30167 Hannover, Germany  Silke
Ospelkaus<EMAIL_ADDRESS>Institut für Quantenoptik,
Leibniz Universität Hannover, 30167 Hannover, Germany  Alessandro Zenesini
Institut für Quantenoptik, Leibniz Universität Hannover, 30167 Hannover,
Germany  INO-CNR BEC Center, Dipartimento di Fisica, Università di Trento
and TIFPA-INFN, 38123 Povo, Italy

###### Abstract

We prepare mixtures of ultracold 39K atoms in various hyperfine spin states
with 23Na39K molecules in an optical dipole trap at a fixed magnetic field and
study inelastic two-body atom-molecule collisions. We observe atom-molecule
losses that are hyperfine dependent with a two-body loss rate far below the
universal limit. We analyze the two-body loss dynamics based on the derivation
of general and easy applicable analytic solutions for the differential
equations describing the loss of an arbitrary number $\gamma$ of particles in
a single collisional event.

##  I Introduction

Quantum gas experiments rely on the scattering properties of the neutral
particles [1]. Their collisions are crucial for example in rethermalization
during evaporative cooling [2] or Bose-Einstein condensation [3]. The control
of the collisions by means of Feshbach resonances [4] can lead to the
selective emergence or collapse of Bose-Einstein condensates[5, 6], formation
of liquid quantum droplets [7, 8] or creation of ultracold dimer molecules
[9]. However, short-range collisions are often accompanied by losses.
Especially resonant scattering conditions can enhance the losses from the
ultracold ensemble. The loss dynamics itself offer valuable clues to the
origin of these losses, for example the number of involved particles, the loss
mechanisms or the density of resonant states [10].
A prominent and timely example for few-body losses are ultracold heteronuclear
ground-state molecular gases. In all so far created ground-state molecular
many-particle gases two-body associated losses have been detected [11, 12, 13,
14, 15, 16, 17], where the particular loss rate coefficients are close to the
universal scattering limit [18, 19]. In the case of chemically unstable
molecules, such as the KRb molecule, those losses originate mainly from two-
body exothermic exchange reactions [19, 20, 21]. In contrast to that,
chemically stable ground-state molecules, such as NaK, NaRb and RbCs are not
affected by exothermic reactions [20], but still suffer from nearly universal
loss in collisions [14, 15, 12, 13, 17]. This is suspected to be due to sticky
collisions [22, 23]. Due to the high density of states in molecular collisions
long-lived tetramer complexes form and might get excited by light from the
optical dipole trap [24] such that both molecules get lost. This has been
detected in the relatively heavy KRb and RbCs molecules [25, 26], but remains
elusive in the light weight molecular systems of NaK and NaRb. Indeed,
experiments with these molecules suggest complex lifetime several orders of
magnitude larger than calculated [27] or even additional unexplained loss
mechanisms [28]. The same kind of investigation recently expands to ensembles
of molecule-atom systems, in which also sticky trimer complexes can form in
collisions of a single dimer molecule with a single atom. Near universal
losses due to light excitation have already been observed in KRb+Rb collisions
[29], likewise for an unexpected high complex lifetime. Again, in contrast to
the heavy systems, the light weight systems LiNa+Na and NaK+K have observed
deviations from that behaviour, where losses can be suppressed [30, 17] or
even be tuned by means of external magnetic fields [31, 32]. The complex
formation and the loss mechanisms are not fully understood and require further
fundamental investigations towards few- and many-body quantum physics and
chemistry. The data analysis of the loss dynamics involves the relatively
complicated use of sets of differential equations modeling the loss of the
particles and the associated temperature evolution in the trap. Numerical
solving and fitting routines make a fast analysis of the data inconvenient and
less intuitive.
In this paper, we report the observation of hyperfine dependent atom-molecule
loss in mixtures of 39K atoms prepared in various hyperfine spin states of the
electronic ground state and bosonic 23Na39K molecules in a single but fixed
hyperfine state of the rovibronic ground state. We observe the atom-molecule
two-body loss rate to be dependent on the hyperfine spin state of the 39K
atoms and far below the universal limit. We analyze the two-body loss dynamics
based on the derivation of analytic solutions to the differential equation
system for the time evolution of particle number and temperature in a
$\gamma$-body loss process.

In Sec. II we describe the general coupled differential equation system for
loss dynamics resulting from a $\gamma$-body loss process and the derivation
of the analytic solutions to this problem. Sec. III reports about our
experimental work. First, we summarize the preparation of atom-molecule
mixtures of 39K atoms in various hyperfine spin states and 23Na39K molecules
in our experimental apparatus. We then discuss the hyperfine spin dependent
loss measurements and their analysis using the analytic solution from Sec. II
for the two-body decay ($\gamma=2$). Finally, we discuss our results in Sec.
IV.

##  II Analytical model for the $\gamma$-body loss problem

The loss dynamics in pure molecular ensembles and atom-molecule mixtures
presented in this paper follow typical two-body losses. This is described by a
set of nonlinear differential equations where the loss rate coefficient
$k_{2}$ is used as a fit parameter. Nevertheless, such a set of equations can
be generally applied to the case of a $\gamma$-body problem, where $\gamma$ is
the number of particles interacting with each other during a single scattering
event. This is a universal problem in many fields of physics.

The typical approach to apply these systems to experimental data is to
laboriously fit numerically deduced solutions of the differential equation
system. The application of analytic solutions thus will be much more
convenient and facilitate the analysis of experimental data drastically. In
the following we derive these solutions for the loss problem in general with
$\gamma$ involved particles. This makes our solutions applicable for a broad
range of applications.

We consider a system of $N(t)$ particles of temperature $T(t)$ trapped in a
harmonic potential $U(x,y,z)$ with trapping frequency $\omega_{i}$ for the
direction $i$. Following the Boltzmann statistics, the size of the ensemble is
given by a Gaussian profile with width
$\sigma_{i}(t)=\sqrt{k_{B}T(t)/m\omega_{i}^{2})}$ 111Note, we define $2\sigma$
as the distance, where the density drops to $\frac{1}{e^{2}}\cdot n(0,0,0,0)$,
where $m$ is the mass of the particle and $k_{B}$ is the Boltzmann constant.
The density of the ensemble is given by

$n(t,x,y,z)=N(t)\prod_{i=x,y,z}\frac{e^{-i^{2}/(2\sigma_{i}(t)^{2})}}{\sqrt{2\pi}\sigma_{i}(t)}.$
(1)

If we limit the derivation to one- and $\gamma$-body particle loss processes
in the system, with loss rates $k_{1}$ and $k_{\gamma}$, the particle number
$N(t)$ and the temperature $T(t)$, assuming re-thermalization, result from a
system of coupled equations

$\begin{array}[]{*{20}{l}}{\dot{N}(t)\hfill}&=\hfill&{-k_{\gamma}\langle
n(t)^{\gamma}\rangle_{V}-k_{1}N(t)}\,,\\\
{\dot{T}(t)\hfill}&=\hfill&{+\frac{\Gamma_{\gamma}(t)}{3k_{B}N(t)}k_{\gamma}\langle
n(t)^{\gamma}\rangle_{V}}\,,\hfill\end{array}$ (2)

where the $\langle n(t)^{\gamma}\rangle_{V}$ is the volume integral of the
$\gamma$-body density $n(t,x,y,z)^{\gamma}$. The anti-evaporation is described
by the quantity $\Gamma_{\gamma}(t)$ and corresponds to the mean potential
energy

$\Gamma_{\gamma}(t)=\frac{3}{2}k_{B}T(t)-\langle
U(x,y,z)n(t,x,y,z)^{\gamma}\rangle_{V}\,.$ (3)

$\Gamma_{\gamma}(t)$ depends on time both through the temperature and the
density. Note that the one-body loss process does not contribute to the anti-
evaporation as it is not density-dependent.

By explicitly integrating the volume integrals over the Gaussian density
profiles, one reaches the following system of equations

$\begin{array}[]{*{20}{l}}{\dot{N}(t)\hfill}&=\hfill&{-\frac{k_{\gamma}C^{(\gamma-1)}}{\gamma^{3/2}}\
{\frac{{N(t)^{\gamma}}}{{T(t)^{(3/2)(\gamma-1)}}}}-k_{1}N(t)\,,\hfill}\\\
{\dot{T}(t)\hfill}&=\hfill&{\frac{k_{\gamma}C^{(\gamma-1)}}{\gamma^{3/2}}\left({\frac{{\gamma-1}}{{2\gamma}}}\right){\frac{{N(t)^{\gamma-1}}}{{T(t)^{(3\gamma-5)/2}}}}\,,\hfill}\end{array}$
(4)

where $C$ is $(m\omega^{2}/(2\pi k_{B}))^{(3/2)}$ and $\omega$ is the average
trap frequency of the trapping potential
$\omega=(\omega_{x}\omega_{y}\omega_{z})^{1/3}$. Note that we use the same
conventions as used in [23].

The first step to an analytic solution for $N(t)$ and $T(t)$ consists in
substituting $N(t)$ with $M(t)e^{-k_{1}t}$ to hide the one-body term

$\begin{array}[]{*{20}{l}}{\dot{M}(t)\hfill}&=\hfill&{-\frac{k_{\gamma}C^{(\gamma-1)}}{\gamma^{3/2}}{\frac{{M(t)^{\gamma}}}{{T(t)^{(3/2)(\gamma-1)}}}}e^{-(\gamma-1)k_{1}t},\hfill}\\\
{\dot{T}(t)\hfill}&=\hfill&{\frac{k_{\gamma}C^{(\gamma-1)}}{\gamma^{3/2}}{\frac{{\gamma-1}}{{2\gamma}}}\left({\frac{{M(t)^{\gamma-1}}}{{T(t)^{(3\gamma-5)/2}}}}\right)e^{-(\gamma-1)k_{1}t}.\hfill}\end{array}$
(5)

By dividing the first line by the second one in Eq. 5 we have:

$\frac{\dot{M}(t)}{\dot{T}(t)}=-\frac{2\gamma}{\gamma-1}\frac{M(t)}{T(t)},$
(6)

which leads to

$\frac{M(t)}{N_{0}}=\left(\frac{T(t)}{T_{0}}\right)^{-\frac{2\gamma}{\gamma-1}},$
(7)

where $N_{0}=N(0)=M(0)$ and $T_{0}=T(0)$ are the initial atom number and the
initial temperature. By introducing $M(t)$ and $N_{0}$ in the first equation
in Eq. 5, we get

$\frac{{\dot{M}(t)}}{N_{0}}\hfill=\hfill{-N_{0}^{\gamma-1}\frac{k_{\gamma}C^{(\gamma-1)}}{\gamma^{3/2}}{\frac{{(M(t)/N_{0})^{\gamma}}}{{T(t)^{(3/2)(\gamma-1)}}}}e^{-(\gamma-1)k_{1}t},}$
(8)

where $M(t)/N_{0}$ can be substituted using Eq. 7:

$\displaystyle\frac{d}{dt}\left[\left(\frac{T(t)}{T_{0}}\right)^{-\frac{2\gamma}{\gamma-1}}\right]\hfill=$
(9)
$\displaystyle\hfill{-\frac{k_{\gamma}(N_{0}C)^{\gamma-1}}{\gamma^{3/2}}{\frac{{\left(\left(\frac{T(t)}{T_{0}}\right)^{-\frac{2\gamma}{\gamma-1}}\right)^{\gamma}}}{{T(t)^{(3/2)(\gamma-1)}}}}e^{-(\gamma-1)k_{1}t}.}$

This differential equation for $T(t)$ has the solution

$\frac{T(t)}{T_{0}}=\left(1+k_{\gamma}\frac{\eta}{\alpha}\left(\frac{1-e^{-(\gamma-1)k_{1}t}}{(\gamma-1)k_{1}}\right)\right)^{\beta},$
(10)

where $\eta$ contains the starting conditions and trapping parameters

$\eta=\frac{1}{\gamma^{3/2}}\left(\frac{CN_{0}}{T_{0}^{3/2}}\right)^{\gamma-1}.$
(11)

The two constants $\alpha$ and $\beta$ depend only on $\gamma$ and can be
written as

$\alpha=\frac{4\gamma}{(\gamma-1)(7\gamma-3)}\,,$ (12)
$\beta=\frac{2}{7\gamma-3}\,.$ (13)

The solution for $N(t)$ is then obtained by combining Eq. 7 and Eq. 10.

$\frac{N(t)}{N_{0}}=\frac{e^{-k_{1}t}}{\left(1+k_{\gamma}\frac{\eta}{\alpha}\left(\frac{1-e^{-(\gamma-1)k_{1}t}}{(\gamma-1)k_{1}}\right)\right)^{\alpha}}\,.$
(14)

For $\gamma=1$, this solution collapses to the known exponential decay. The
solutions for $\gamma=2,3,4,5$ have the following $\alpha$ and $\beta$:

$\gamma$ | $\alpha$ | $\beta$
---|---|---
2 | 8/11 | 2/11
3 | 1/3 | 1/9
4 | 16/75 | 2/25
5 | 5/32 | 1/16

Note that the case $\gamma=3$ corresponds to the solution already presented by
Kraemer [34] although the solution of $T(t)$ in the thesis manuscript has a
typo on $\eta$. The limit for negligible one-body losses ($k_{1}\to 0$) can be
obtained just by noticing that $\lim_{k_{1}\to
0}\left(\frac{1-e^{-(\gamma-1)k_{1}t}}{(\gamma-1)k_{1}}\right)=t$, which leads
to:

$\begin{array}[]{*{20}{l}}T(t)/T_{0}&=\left(1+k_{\gamma}\frac{\eta}{\alpha}t\right)^{\beta},\\\
N(t)/N_{0}&=\left(1+k_{\gamma}\frac{\eta}{\alpha}t\right)^{-\alpha}.\end{array}$
(15)

In the following, we use our newly derived analytic solutions to analyze the
hyperfine dependent atom-molecule loss observed in our experiments.

##  III Hyperfine dependent atom-molecule loss

###  III.1 Preparation of atom-molecule mixtures

The ground-state molecule creation starts from mixtures of about $1.8\times
10^{5}$ ultracold 23Na and $0.6\times 10^{5}$ 39K atoms trapped in a 1064 nm
crossed beam optical dipole trap (cODT) with temperatures of 300 nK. The cODT
frequencies are measured by trap oscillations of the respective particles. For
39K atoms the trap frequencies are
$\omega_{\textrm{K},(x,y,z)}=2\pi\times(375.8(8.6),375.8(8.6),60.8(6.4))\,$Hz.
The trap frequencies for 23Na atoms are scaled by a factor of about $0.62$ due
to different dynamical polarizability and mass. The atoms are initially
prepared in the states
$\mathinner{|{f=1,m_{f}=-1}\rangle}_{\textrm{Na}}+\mathinner{|{f=1,m_{f}=-1}\rangle}_{\textrm{K}}$
for which the inter- and intra-species scattering properties are well known
[35]. Feshbach molecules are formed close to a Feshbach resonance at about 200
G in the atomic
$\mathinner{|{f=1,m_{f}=-1}\rangle}_{\textrm{Na}}+\mathinner{|{f=2,m_{f}=-2}\rangle}_{\textrm{K}}$
states [36]. The bound molecular state is populated by a short radio frequency
pulse of 400 $\mu$s. Immediately after molecule formation the transfer to the
ground state takes place. This is done within 12 $\mu$s via a stimulated Raman
adiabatic passage (STIRAP) to a single rovibrational hyperfine state
$\mathinner{|{m_{i,\text{Na}}=-3/2,m_{i,\text{K}}=-1/2,M_{J}=0,M_{i,\text{NaK}}=-2}\rangle}$
[17]. For detection of molecules the STIRAP is reversed and 39K atoms from the
Feshbach molecule state are imaged via a cycling transition. Typically, about
4000 ground-state molecules are generated per experimental cycle. The
determined trap frequencies of the 23Na39K molecules are
$\omega_{\textrm{NaK},(x,y,z)}=2\pi\times(195.6(8.3),195.6(8.3),33.1(1.3))\,$Hz.
The temperature of the 23Na39K molecules is measured by a time-of-flight
measurement on the 39K atoms after STIRAP reversal and Feshbach molecule
dissociation [17]. The temperature of the remaining 39K atoms is still 300 nK,
from which we conclude that neither molecule association nor dissociation lead
to heating of the molecules. After ground-state transfer the molecules are
still embedded in a bath of remaining atoms. 23Na atoms are usually removed as
fast as possible since the molecules undergo a chemical reaction with 23Na
atoms. This is done by a 500 $\mu$s resonant light pulse, which pushes the
atoms out of the cODT. To prevent optical pumping and thus incomplete removal
of the 23Na atoms, two resonant frequencies are used connecting the excited
state $\mathinner{|{f=2,m_{f}=-2}\rangle}_{\textrm{Na}}$ to the corresponding
low field ground states $\mathinner{|{1,-1}\rangle}_{\textrm{Na}}$ and
$\mathinner{|{2,-1}\rangle}_{\textrm{Na}}$. The notation is given in the basis
of the Zeeman regime with $\mathinner{|{f,m_{f}}\rangle}$, where $f$ is the
hyperfine number and $m_{f}$ its projection. After the removal of the 23Na
atoms, the molecules are left in a gas of 39K atoms in the
$\mathinner{|{1,-1}\rangle}$ hyperfine state. Fortunately, this atomic state
leads to very little losses of the molecules [17], which is favorable for all
further state manipulations of the mixture; see Fig. 1. For the preparation of
a pure molecular cloud 39K atoms are removed from the trap by a rapid
adiabatic passage (RAP) to the $\mathinner{|{2,-2}\rangle}$ state followed by
a $500\,\mu\textrm{s}$ resonant light pulse. For the study of atom-molecule
loss with 39K atoms in various spin states, the hyperfine state of the 39K
atoms is manipulated using RAPs and rf pulses. In our experiment, we prepare
six different 39K hyperfine states: $\mathinner{|{1,1}\rangle}$,
$\mathinner{|{1,0}\rangle}$, $\mathinner{|{1,-1}\rangle}$,
$\mathinner{|{2,-2}\rangle}$, $\mathinner{|{2,-1}\rangle}$ and
$\mathinner{|{2,0}\rangle}$, where the states are written in the basis
$\mathinner{|{f,m_{f}}\rangle}$.

###  III.2 Atom-molecule two-body loss coefficients

Atom-molecule collisions between 23Na39K molecules in the
$\mathinner{|{m_{i,\text{Na}}=-3/2,m_{i,\text{K}}=-1/2,M_{J}=0,M_{i,\text{NaK}}=-2}\rangle}$
state [17] and 39K atoms in one of the six different hyperfine states
mentioned above are studied in the optical dipole trap at a fixed magnetic
field of 200 G in the vicinity of the used Feshbach resonance. Apart from
losses due to two-body atom-molecule collisions, the decay dynamics of an
atom-molecule mixture is also affected by the two-body losses due to molecule-
molecule collisions, and it is therefore important to first understand in
detail the two-body decay of a pure molecular cloud.

Pure molecular collisions between 23Na39K molecules have already been studied
in our previous work [17]. One-body background losses for example from vacuum
background gas collisions [35] or photo-excitation of bialkali molecules are
negligible [37] and the dominating loss is observed to result from two-body
molecule-molecule collisions. We thus analyse the decay of a pure molecular
cloud using the analytic solution from Eq. 15 with $\gamma=2$. Accordingly,
$\alpha$ is $8/11$ and the constant $\eta$ contains the parameters of the
particles and the trap, which are the initial particle number and temperature
as well as trap frequencies; see Sec. III.1. We fit this model to the decay of
a pure molecular cloud and extract the two-body loss rate coefficient
$k_{2}=4.49(1.18)\times 10^{-10}\,\textrm{cm}^{3}\textrm{s}^{-1}$ at 200 G
close to the universal limit and consistent with the value reported in [17]
(see Fig. 1).

Knowing the background two-body loss resulting from collisions of two ground-
state molecules we can investigate ultracold collisions of molecules with 39K
atoms in various hyperfine states. We therefore prepare different atom-
molecule mixtures of 39K atoms in one of the six different hyperfine states
and 23Na39K molecules; see Sec. III.1. We measure the decay of the molecular
cloud as a function of time for each of the six different atom-molecule
mixtures and analyze loss dynamics using solutions from Sec. II, in particular
Eq. 14.

To apply Eq. 14 to this problem, one needs to include potassium losses into
the analytical model. Given that the density of the 39K gas is one to two
orders of magnitude larger than the density of the molecular gas, we can
safely assume that the atomic density does not change during the decay of the
molecular cloud. Therefore one can treat this like losses from vacuum
background collisions, although that does not account for possible re-
thermalization of the molecules with the potassium cloud. This results in a
molecular two-body decay with an additional exponential one-body decay on top,
originating from a two-body atom-molecule loss.

We thus analyze the loss dynamics of the molecular cloud considering molecule-
molecule collisions with the obtained $k_{2}$ and modeling atom-molecule loss
as a one-body (molecule) loss part with $k_{1}$ from Eq. 14 expressed as

$k_{1}=k_{2,\textrm{K}}n_{\textrm{K}}\,,$ (16)

where $n_{\textrm{K}}$ is the initial 39K density and $k_{2,\textrm{K}}$ the
actual two-body loss rate coefficient, including spatial density overlap of
the atomic and molecular cloud in the ODT, for the atom-molecule collision.
Note that, within the experimental and statistical uncertainties, the obtained
values of $k_{1}$, found with the solutions in Sec. II, are compatible with
the ones calculated for a system where the atoms and molecules thermalize and
therefore remain at a fixed temperature.
We extract the $k_{2,\textrm{K}}$ loss rate coefficients for the atom-molecule
collision for the six different hyperfine states of the 39K atoms. The results
are shown in Fig. 1 in the left panel. All state combinations show a loss rate
coefficient much smaller than the calculated universal limit [38], with one
state even showing a drastic suppression of the loss rate coefficient of
several orders of magnitude [17]. For comparison, we include the calculated
universal scattering limit as a solid black line in Fig. 1, the two-body loss
rate coefficient for 23Na39K+23Na (Fig. 1, middle panel) as well as the
molecule-molecule two-body loss rate coefficient (right panel). The lower part
of Fig. 1 shows the number of available energetically allowed loss channels
due to hyperfine changing collisions with and without preserved $M_{F}$ (left
and right axes, respectively). The loss rate coefficient for 23Na39K+23Na
collisions obtained by mean of our model is in agreement with the universal
limit and with the fact that for a chemical unstable reaction, all collisions
lead to loss of the involved atom and molecule. This is consistent with the
large losses observed during the sample preparation and the aforementioned
necessity to remove 23Na as quickly as possible from the trap. The numerical
agreement also allows us to verify that the applied model gives reasonable
output.

Figure 1: Loss in atom-molecule mixtures. The upper part shows the two-body
loss rate coefficients for the different molecular systems. The left panel
includes the loss rate coefficients for the 23Na39K+39K mixture with different
hyperfine states for 39K atoms (open circles). The values for 23Na39K+23Na
(gray diamond, middle panel) and 23Na39K (gray triangle, right panel) are
shown for comparison. Note, that the value for the
$\mathinner{|{1,-1}\rangle}$ state is so small, that it is not shown in the
graph. All measurements were performed at a temperature of 300 nK. The
horizontal lines represent the respective universal limit, calculated also for
300 nK. The solid gray bar diagram in the lower part shows the number of
available loss channels (left axes) due to hyperfine changing collisions for
the 23Na39K+39K mixture with the total
$M_{F}=M_{i,\text{NaK}}+m_{f,{\mathrm{K}}}$ preserved. The dashed bar diagram
indicates the number of all energetically possible decay channels (right
axes), if one assumes that $M_{F}$ is not preserved.

##  IV Discussion

Without including hyperfine interactions and possible coupling to trimer
states, the endoergic nature of the 23Na39K+39K system [20] would result in
stable and long-lived mixtures, as there are no two-body losses from chemical
exchange reactions.
However, near-universal decay which has been attributed to the coupling to a
large density of trimer states has been observed in various atom-molecule
systems with the exception of 40K87Rb+87Rb mixtures when both atoms and
molecules have been prepared in their lowest stretched states [19] and in
6Li23Na+23Na with both particles in their highest stretched states [30]. In
other state combinations of both examples losses consistent with the universal
limit have been reported [19, 16]. Meanwhile, for 23Na40K+40K collisions
numerous magnetic field tuneable Feshbach resonances in different hyperfine
states have been found and assigned to long-range states [32]. In fact, in
these experiments collisional resonances have been identified as an
enhancement of the loss rate coefficient in the vicinity of a resonance
starting from a near-universal background loss [31].

In contrast, for the bosonic 23Na39K+39K case we do report suppression of
losses far below the universal limit where neither the molecules nor the atoms
are in their stretched states at the same time, see Fig. 1. Although for the
bosonic case 23Na39K+39K no magnetic field dependent measurements have been
done so far, still the change of the loss rate coefficient for different spin
channels suggests a hyperfine dependent Feshbach structure, which is similarly
tuneable as in fermionic 23Na40K+40K collisions.
The effect of the changed loss rate coefficients and the loss of the particles
remains elusive. We rule out, that the losses of molecules originate from
hyperfine changing collisions between atoms and molecules with preserved total
$M_{F}$ of the atom-molecule system. For this purpose we determine the Zeeman
structure of the lowest hyperfine states of the 23Na39K+39K at the magnetic
field of 200 G, including hyperfine structure of the molecule in the Paschen-
Back and for the atom in the intermediate regime and compare the number of
possible $M_{F}=M_{i,\text{NaK}}+m_{f,{\mathrm{K}}}$ preserving inelastic loss
channels for atom-molecule mixture with the measured loss rate coefficients.
The results are shown in the lower panel in Fig. 1. The number of available
hyperfine changing loss channels does not correlate with the loss rate
coefficients. This indicates that the observed loss is not dominated by
hyperfine changing collisional processes.

In dipolar collisions, hyperfine changing processes can also lead to a non-
preserved total angular momentum projection $M_{F}$ [39, 40]. The dipolar
angular momentum of the 23Na39K molecule in absence of an externally applied
electric field is zero. Moreover, the only dipole moments in our 23Na39K
system are the nuclear ones, which are only relevant at short range. Since we
cannot exclude the possibility that some resonant processes amplify the
influence of the nuclear dipoles, we give in Fig. 1 the number of all
energetically possible decay channels even when $M_{F}$ is not preserved. We
find that this number does not correlate with the loss rate coefficients, thus
we can rule out a role of hyperfine changing processes just based on possible
decay channels.

One other possibility of losses from a collisional system are light-induced
losses of long-lived trimer complexes [24, 29], for example from the light of
the optical trap. Estimations of rovibrational as well as the hyperfine
spacing for the 23Na39K+39K complex give evidence for resolvable photo-
excitation resonances which might explain the increased loss rate coefficient
in certain hyperfine states of 39K atoms [41].

##  V Conclusion

In summary we investigated atom-molecule two-body loss in a mixture of
ultracold 23Na39K molecules and 39K atoms in six different hyperfine states
prepared in an optical dipole trap at a fixed magnetic field of around 200 G.
We observed atom-molecule loss that is hyperfine dependent with a two-body
loss rate far below the universal limit. We analyzed the two-body loss
dynamics based on the derivation of general and easily applicable analytic
solutions for the differential equations describing the loss of $\gamma$
particles in a single collisional event, which are easily implementable in
data analysis. In contrast to the numerous atom-molecule mixtures studied to
date, in which losses predominantly near or at the universal limit have been
observed, the light NaK+K system exhibits interesting properties that require
further investigation. Feshbach resonances and hyperfine-dependent collisions
have been observed and are far from being understood. New theory insight will
be required to understand the physical mechanisms underlying the observed
phenomena. Such insight will help to develop a complete understanding of atom-
molecule collisions and associated trimer formation and should ultimately be
applicable also to molecule-molecule collisions. Detailed understanding of
both molecule-molecule and atom-molecule collisions is crucial for further
advancement in the production of molecular gases close to quantum degeneracy
and associated research opportunities in quantum chemistry, quantum
simulation, quantum information and precision measurements.

## Acknowledgements

We thank Goulven Quéméner for enlightening comments and Jeremy Hutson and
Matthew Frye for insightful conceptions. We gratefully acknowledge financial
support from the European Research Council through ERC Starting Grant POLAR
and from the Deutsche Forschungsgemeinschaft (DFG) through CRC 1227 (DQ-mat),
project A03 and FOR2247, project E5. P.G. thanks the Deutsche
Forschungsgemeinschaft for financial support through Research Training Group
1991. A.Z. thanks Provincia Autonoma di Trento for financial support.

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# Fast generation of multi-component cat states under the Strong Rydberg
Dressing Regime

Mohammadsadegh Khazali  Department of Physics, University of Tehran, Tehran,
14395-547, Iran<EMAIL_ADDRESS>
###### Abstract

The study explores cat state formation in the strong Rydberg dressing regime,
uncovering the emergence of cat states despite the presence of all orders of
nonlinearities. This unexplored regime demonstrates potential for rapid cat
state formation, particularly beneficial for operation in 2D lattices in
Rydberg labs. Additionally, the paper discusses the potential for creating a
superposition of m coherent spin states (m-SCSS), where the maximum m is
determined by the number of atoms within the blockade radius $m=\sqrt{N}$.

This note discusses my previous studies on making cat states under a strong
dressing regime performed after a discussion with Nathan Schine 18 months ago.
Since a different preprint is now investigating the same regime of operation
Zha23  I present my notes as a complement to their interesting results.

## introduction

There is a significant effort in making cat states Omr19 ; Gri20 ; Kha16 ;
Kha18 . In the atomic system, it is equivalent to the superposition of two
coherent states pointing to opposite directions of the Bloch sphere e.g.
$({|{\theta=\pi/2,\phi=0}\rangle}+{|{\theta=\pi/2,\phi=\pi}\rangle})/\sqrt{2}$.
This state could improve the precision of atomic clocks and enhance the
sensitivity of quantum measurements and sensing. Here we propose a method to
make m-SCSS with $m=\sqrt{N}$ being the square root of the atom number. This
would help the entire atomic ensemble to simultaneously acquire $m$ different
quantum states at once and sense the corresponding physics or undergo the
corresponding operation.

While there are multiple measures for the quantumness, an interesting figure
of merit could be the number of different quantum states all the elements
could possess at the same time. This paper proposes an approach to apply non-
linear Rydberg interaction to form superposition of multiple CSS ideal for
simultaneous sensing at different quantum phases. Subsequent coherent
retrieval of the initial CSS under the non-linear interaction completes the
process for Ramsey interferometry.

In the previous Rydberg dressing studies, The operation in the weak dressing
regime was studied to isolate the second order of nonlinearity Kha18 ; Kha16
while the numeric results in Kha16  suggests that enhancing the dressing
strength within the weak dressing regime could lead to larger cat formation.
In making cat states in the typical 2D lattices, the weak dressing condition
put a limit on the achievable size of cat states. This note studies the cat
state generation under the strong dressing regime where the fast operation
allows significant enhancment the number of entangled atoms. Considering the
two dressing approaches in making cat states Kha18 ; Kha16 , the higher order
terms of non-linearity could destroy the superposition in the Lipkin-Meshkov-
Glip type model as the odd terms of non-linearity accompanied by the
transversal field destroys the symmetry as discussed in Kha18 . On the other
hand, this paper shows that in the Yurke-Stoller approach Yurk86 ; Kha16  the
effects of higher-order terms of non-linearity are less destructive. In fact,
the improved coherence and the new possibilities of entanglement engineering
justify investigating the strong dressing operation regime. Yurke-Stoller
discussed the cat formation of 2-SCSS under a single even term of nonlinearity
$U\propto N^{2k}$ and 4-SCSS under an isolated odd order $U\propto N^{2k+1}$.
This note shows that under a quadratic nonlinearity, the m-SCSS is realizable.
Surprisingly, under the strong Rydberg dressing regime where all the orders of
nonlinearities exist, a similar process is observed in a numerical study. The
superposition states of $m$-CSS are observed with lower fidelity but at a much
smaller time that allows operating over a larger number of atoms.

Despite the benefits of a strong dressing regime, the absence of an analytical
model describing the interaction profile at this regime Saf21 , has deprived
the mean-field studies on this unexplored range of parameters. This note
investigates an analytic form of the interaction profile encountering two and
three-body interactions, which gives an insight into the effects of lattice
geometry as well as controlling parameters on the orders of non-linearity.
Furthermore, it opens new opportunities to study the dynamics of Rydberg
dressed BEC Hen10 ; Kha21  under a stronger dressing regime applying the mean-
field theory.

##  I Dressing Interaction

When the entire ensemble is inside the blockade radius, the dressing laser
couples the state with no Rydberg excitation
$|\psi_{0}\rangle=\otimes_{i}\left|\phi_{i}\right\rangle$ (where
$\phi\in\\{e,g\\}$) to a state where only one of the atoms in the $|e\rangle$
level gets excited to the Rydberg level
$|\psi_{1}\rangle=\sum_{i}\left|\phi_{1}...r_{i}...\phi_{N}\right\rangle$ with
an enhanced Rabi frequency $\sqrt{N_{e}}\Omega_{r}$ and detuning $\Delta$,
where $N_{e}$ is the number of atoms in the excited state. To preserve the
blockade the laser detuning and interaction must have opposite signs
$\Delta.V<0$. Over the Rydberg dressing process, the Hamiltonian can be
diagonalized instantaneously

$D\equiv UHU^{\dagger}=\left(\begin{array}[]{cc}E_{-}&0\\\
0&E_{+}\end{array}\right),$ (1)

where
$E_{\pm}=\frac{\Delta}{2}(1\pm\sqrt{1+\frac{N_{e}\Omega_{r}^{2}}{\Delta^{2}}})$
and

$U=\left(\begin{array}[]{cc}\cos(\theta/2)&-\sin(\theta/2)\\\
\sin(\theta/2)&\cos(\theta/2)\end{array}\right)$ (2)

with $\theta=\tan^{-1}(\frac{\sqrt{N_{e}}\Omega_{r}}{\Delta})$. The
Schrodinger equation expressed in the dressed state basis $|\varphi>=U|\psi>$
is

$i\frac{\partial}{\partial t}\left(\begin{array}[]{cc}|\varphi_{-}\rangle\\\
|\varphi_{+}\rangle\end{array}\right)=\left(\begin{array}[]{cc}E_{-}&-i\dot{\theta}/2\\\
i\dot{\theta}/2&E_{+}\end{array}\right)\left(\begin{array}[]{cc}|\varphi_{-}\rangle\\\
|\varphi_{+}\rangle\end{array}\right).$ (3)

To avoid the scattering of population from the ground dressed state to the
excited dressed state, the coupling term
$\dot{\theta}=\frac{\sqrt{N_{e}}\Omega_{r}\dot{\Delta}-\sqrt{N_{e}}\Delta\dot{\Omega}_{r}}{N_{e}\Omega_{r}^{2}+\Delta^{2}}$
should be smaller than $E_{+}$.

Focusing on the ground dressed state, the exact light-shift the system would
experience is given by

$H_{\text{exact}}=\frac{\Delta}{2}(1-\sqrt{1+\frac{\hat{N}_{e}\Omega_{r}^{2}}{\Delta^{2}}}).$
(4)

Within the weak dressing regime $(\frac{N_{e}\Omega_{r}^{2}}{\Delta^{2}}\ll
1)$ one can Taylor expand the light-shift to find the Hamiltonian in the weak
dressing regime

$H_{\text{w}}=-\frac{1}{4}\frac{\hat{N}_{e}\Omega_{r}^{2}}{\Delta}+\frac{1}{16}\frac{\hat{N}_{e}^{2}\Omega_{r}^{4}}{\Delta^{3}}-\frac{1}{32}\frac{\hat{N}_{e}^{3}\Omega_{r}^{6}}{\Delta^{5}}+\frac{5}{128}\frac{\hat{N}_{e}^{4}\Omega_{r}^{8}}{\Delta^{7}}+O(\hat{N}_{e}^{4})$
(5)

Going to stronger dressing with larger
$N_{e}\frac{\Omega_{r}^{2}}{\Delta^{2}}$ activates the higher orders of non-
linearity and raises the many-body interaction terms in the Hamiltonian, see
Sec. IV.

##  II Making superposition of $m$-CSS

The two levels $|g_{i}\rangle$ and $|e_{i}\rangle$ for each atom are
equivalent to a spin $1/2$ system with Pauli matrices
$\sigma_{x}^{(i)}=(|g_{i}\rangle\langle e_{i}|+|e_{i}\rangle\langle
g_{i}|)/2$, $\sigma_{y}^{(i)}=i(|g_{i}\rangle\langle
e_{i}|-|e_{i}\rangle\langle g_{i}|)/2$ and
$\sigma_{z}^{(i)}=(|e_{i}\rangle\langle e_{i}|-|g_{i}\rangle\langle g_{i}|)/2$
acting on the atom at site $i$. We define collective spin operators
$S_{l}=\sum_{i=1}^{N}\sigma_{l}^{(i)}$. A coherent spin state (CSS) is defined
as a direct product of single spin states CSS

$|\theta,\phi\rangle=\otimes_{i=1}^{N}[\cos{\theta}|g\rangle_{i}+\sin{\theta}e^{i\phi}|e\rangle_{i}],$
(6)

where all the spins are pointing in the same direction, and $\phi$ and
$\theta$ are the angles on the (collective) Bloch sphere. The CSS can also be
represented as CSS

$|\eta\rangle=|\theta,\phi\rangle=(1+|\eta|^{2})^{-N/2}\sum_{N_{e}=0}^{N}\eta^{N_{e}}\sqrt{C(N,N_{e})}|N;N_{e}\rangle,$
(7)

where $\eta=\tan(\theta/2)e^{-i\phi}$,
$C(N,N_{e})\equiv\left(\begin{array}[]{c}N\\\ N_{e}\end{array}\right)$ and
$|N;N_{e}\rangle=\frac{1}{\sqrt{C(N,N_{e})}}\sum_{i1<i2<...<iN_{e}}^{N}|g_{1}...e_{i1}...e_{iN_{e}}...g_{N}\rangle$
is the Dicke state of $N_{e}$ excited atoms, where $|N;N_{e}\rangle$ is an
alternative representation of the $|J\,M\rangle$ basis with $N=2J$ and
$N_{e}=J+M$.

Considering the time evolution of the CSS of Eq. 7 under the dressing
Hamiltonian of Eq. 5, the linear term in $N_{e}$ preserves the CSS and only
generates a rotation around $J_{z}$ while the quadratic term causes spin-
squeezing over the Bloch sphere and when the CSS was squeezed over the equator
it starts to make a superposition of $m$-CSSs at $t_{m}=2\pi/m\chi$.

${|{\psi(t_{m})}\rangle}=\frac{1}{\sqrt{m}}\sum\limits_{k=1}^{m}{|{\theta=\pi/2;\phi=k\frac{2\pi}{m}}\rangle}$
(8)

Continuing the interaction at $t_{1}$, one can observe the revival of the
initial CSS. This revival can be used as proof for the successful creation of
a quantum superposition at $t_{m}$ since a statistical mixture of CSSs at
$t_{m}$ would evolve into another mixture of separate peaks.

For the weak dressing, where the third order of nonlinearity is negligible,
the operation time $t_{m}=2\pi/m\chi$ would match perfectly with the numerics
in Fig. 1. Going to strong dressing the time would be longer than $t_{m}$.
This is because the third order of nonlinearity has the opposite sign with the
second order, see Eq. 5, which makes the process a bit slower. However, the
trend remains positive in terms of enhancing coherence. Furthermore, there is
the possibility of observing m-SCSS multiple times. out of the above mentioned
order abnormalOrdering .

Here we discuss the physics that determines the maximum number $m$ of CSS that
could be formed in a superposition state. The population difference of the
$N_{e}$ in the initial CSS is given by $N_{e}^{max}-N_{e}^{min}=2\sqrt{N}$.
Hence the quadratic term of interaction would cause different rotation speeds
around $J_{z}$ for different ${|{N_{e}}\rangle}$ elements. This would stretch
the initial CSS over the equator. After the head and tail of the squeezed
state meet and go through each other they form constructive and destructive
interferences, which is shown as the superposition of CSSs, see Fig. 1. Hence,
the minimum required time to form the superposition is defined by the time
that the head and tail of the squeezed state are stretched over
$\Delta\phi=4\pi$ to spread the interference all over the Bloch sphere
equator. Considering the difference of speed under the quadratic term, the
minimum dressing time that is required for the superpositions to appear would
be
$t_{\text{min}}=\frac{4\pi}{(N_{e}^{max}-N_{e}^{min})\Omega^{4}/16\Delta^{3}}=\frac{4\pi}{2\sqrt{N}\Omega^{4}/16\Delta^{3}}$.
The m-SCSS would only be formed if its operation time $t_{m}=2\pi/m\chi$
occurs after the spread of interference all over the equator at
$t_{\text{min}}$, see Fig. 1. Hence the maximum number of $m$-CSS
superpositions that could be formed would be determined by the number of spins
in the operation $m=\sqrt{N}$. One should note that this argument is derived
at the weak dressing regime.

(a)  (b)  (c)  (d)  (e)  (f)

Figure 1: The Physics of Multi-Component Cat Creation: The quadratic
interaction term leads to varying rotation speeds around $J_{z}$ for different
$|N_{e}\rangle$ elements, spanning the range
$N_{e}^{max}-N_{e}^{min}=2\sqrt{N}$. This causes: (a) stretching of the
initial CSS over the equator, (b-e) constructive and destructive interferences
as the squeezed state’s head and tail meet and pass through each other, and
(f) formation of the first $m$-CSS superposition after a rotation difference
of $\Delta\phi=4\pi$, allowing for interference to spread all over the
equator.

##  III Advantage of strong dressing

For constant ensemble size $N$, larger $\Omega/\Delta$ is favorable for
enhancing the interaction-to-loss ratio. The dominant term of interaction in
the softcore is $\chi\propto N_{e}^{2}\Omega^{4}/8\Delta^{3}$ while the loss
rate from the Rydberg population $P_{r}=N_{e}(\frac{\Omega}{2\Delta})^{2}$ is
given by $P_{r}\gamma_{r}t_{m}$ with $\gamma_{r}$ being the loss rate from the
Rydberg levels Bet09 . Considering the symmetry of states around the Bloch
sphere’s equator the $N_{e}$ could be replaced by $N/2$ in this scaling
argument. Hence the interaction to loss ratio scales by
$\frac{N\Omega^{2}}{2\Gamma\Delta^{2}}\Delta$. This shows that approaching to
strong dressing regime, using large detuning, and a large principal number
enhances the interaction-to-loss ratio. Figure 2 plots the loss rate over the
m-SCSS generation as a function of dressing strength for a range that is
mostly in the weak dressing regime. The error bars are due to the presence of
a time interval over which the desired state is formed under the dressing
interaction. This would translate to an uncertainty of loss.

(a)  Figure 2: The scale of loss probability vs dressing strength in the weak
dressing regime. This simulation plots the loss over the cat formation under
the exact dressing interaction of Eq. 4 including all terms of non-
linearities. Here $N=18$ atoms are considered in a 2D arrangement with the
lattice constant of $a=532$nm Wei11 . The system is dressed by the Rydberg
state $|53P_{3/2},3/2\rangle$ via a laser with detuning $\Delta=0.1$GHz. The
operation is in the range where the second order of nonlinearity is dominant.
For the effects of higher orders, see Table I.

To investigate the validity of this statement in the strong driving regime,
the cat creation is simulated under the exact Hamiltonian of Eq. 4 at weak and
strong dressing regimes, see Table I. In this paper a 2D lattice with lattice
constant $a=532$nm Wei11  is considered. Dressing atoms with
$|53P_{3/2},3/2\rangle$ Rydberg state with laser detuning of
$\Delta/2\pi=0.02$GHz accommodates $N=48$ atoms well within the blockade
radius $R=R_{b}/2=(C_{6}/2\Delta)^{1/6}/2$. This would guarantee the
homogeneity of interaction experienced by all atoms, see Fig. 4.

In Table I, the first column corresponds to cases where the second order of
nonlinearity $\chi_{2}$ is isolated, in the second column partial involvement
of the third order $\chi_{3}/\chi_{2}=0.06$ is considered, and in the third
column the extreme case is applied where all orders of nonlinearity are
involved in ascending order. Here the fidelity of all the generated m-SCSS is
reported compared to the ideal case Eq. 8. Surprisingly, a similar process of
cat formation could be observed in the case that all orders of nonlinearity
exist. This deviates from the regime of Yurk86  that requires the isolation of
a certain order of nonlinearity.

The population loss is defined by $N_{e}(\Omega^{2}/4\Delta^{2})\Gamma t$ in
the weak dressing regime where $\Gamma$ is the decay rate of the Rydberg level
at the cryogenic environment of T=77K Bet09 . At strong dressing, the maximum
Rydberg population $P_{r}=(\Omega/2\Delta)^{2}N_{e}$ could not surpass
$P_{r}=1$ as long as the condition $V\gg\Omega$ persists. Since the operation
occurs on the equator of Bloch sphere $N_{e}\approx N/2$. While going to a
stronger dressing reduces the fidelity, the sharp diving of the operation time
and the loss rate far surpass the deviation from an exact m-SCSS. Hence,
depending on the applications, enhancing the coherence, or size of the
entangled state might justifies the reduction of the fidelity.

m-SCSS |  $\Omega/\Delta=.15$, $\chi_{3}/\chi_{2}=0.003$ |  $\Omega/\Delta=.36$, $\chi_{3}/\chi_{2}=0.07$ |  $\Omega/\Delta=10$, $\chi_{3}/\chi_{2}=50$
---|---|---|---
| Fid=61% loss=0.29 , t=790$\mu$s | Fid=57% loss=0.22, t=102$\mu$s | Fid=49% loss=0.005, t=2$\mu$s
m=2 |  |  |
| Fid=53% loss=.19, t=521$\mu$s | Fid=53% loss=0.15, t=70$\mu$s | Fid=51% loss=0.0038, t=1.4$\mu$s
m=3 |  |  |
| Fid=46% loss=.14, t=380$\mu$s | Fid=48% loss=0.1, t=49$\mu$s | Fid=39% loss=0.0027, t=1$\mu$s
m=4 |  |  |
| Fid=44% loss=0.12, t=316$\mu$s | Fid=44% loss=0.085, t=40$\mu$s | Fid=45% loss=0.0025, t=0.9$\mu$s
m=5 |  |  |
| Fid=46% loss=0.1, t=252$\mu$s | Fid=44% loss=0.068, t=32$\mu$s | Fid=46% loss=0.0019, t=0.7$\mu$s
m=6 |  |  |
| Fid=46% loss=0.08, t=221$\mu$s | Fid=42% loss=0.064, t=30$\mu$s | Fid=41% loss=0.0018, t=0.66$\mu$s
m=7 |  |  |
Table 1:  The transition from the weak to strong dressing regime.

##  IV The interaction profile at strong dressing regime

In Rydberg dressing an off-resonant laser weakly couples Rydberg levels to the
ground state atoms. This dressing results in a Plateau-type interaction that
has been formulated for weak dressing in a perturbative approach that covers
up to the two-body interaction Hen10 ; KhaThesis . Going to stronger dressing,
the effects of higher-order terms would magnify the three-body interaction.
This section looks into the analytical profile of the interaction up to the
third order of nonlinearity.

Considering $N$ atoms in an optical lattice, for each pair of atoms excited to
the Rydberg level $|r\rangle$ and separated by ${\bf x}_{ij}={\bf x}_{i}-{\bf
x}_{j}$, where ${\bf x}_{i}$ is the position of the $i$th atom, the binary
interaction is $V_{ij}=C_{6}/x_{ij}^{6}$. Here the quantization axis is
considered perpendicular to the lattice plane to preserve the isotropy of
interaction. The dressing potential $U({\bf x}_{1}..{\bf x}_{N_{e}})$ of the
state $|\psi({\bf x}_{1}..{\bf x}_{N_{e}})\rangle$ with $N_{e}$ atoms in
${|{e}\rangle}$ being dressed with Rydberg level ${|{r}\rangle}$, is
calculated under the condition of $(\frac{\Omega}{2\Delta})^{2}\ll 1$ by
applying perturbation theory

$\displaystyle\frac{U}{\Delta}|\psi\rangle=$ (9)
$\displaystyle({\sum\limits_{i=1}^{N}\hat{\sigma}_{ee}^{i}+\frac{1}{\Delta}\sum\limits_{i<j}V_{ij}\hat{\sigma}_{ee}^{i}\hat{\sigma}_{ee}^{j}})+{{\frac{\Omega}{2\Delta}}}({\sum\limits_{i=1}^{N}\hat{\sigma}_{ge}^{i}+\hat{\sigma}_{eg}^{i}})\,|G\rangle,$

where the first and second parentheses separate the unperturbed and perturbed
parts of Hamiltonians.

The contribution from the second and fourth-order perturbations are calculated
as Hen10

$U^{(2)}=-\frac{\Omega^{2}}{4\Delta}N;\quad
U^{(4)}=\frac{\Omega^{4}}{16\Delta^{3}}\sum\limits_{i<j}[\frac{1}{(1+\frac{V_{ij}}{2\Delta})}-1]$
(10)

where the former is the sum of the single atom light-shift and the latter
encounters the two-body interactions among atoms. In the limit of strong
interaction ($V\rightarrow\infty$) these terms reproduce the first two terms
of collective light-shift expansion of Eq. 5.

Going to a stronger dressing regime with a non-negligible third order of
nonlinearity $(N_{e}\frac{\Omega^{2}}{4\Delta^{2}})^{3}$ the sixth-order of
perturbation must be taken into account. Here different configurations of the
six photon transitions in the perturbative ladder are plotted in Fig. 3, which
includes the two-body as well as three-body interactions. The sixth correction
of the interaction profile will be given by

$\displaystyle
U^{(6)}=\frac{\Omega^{6}}{2^{6}\Delta^{5}}[-2N^{3}+(\sum_{k=1}^{N}(1))\sum_{i\neq
j}\frac{4}{2-V_{ij}/\Delta}$ (11) $\displaystyle+\sum_{i\neq
j}\frac{2}{2-V_{ij}/\Delta}(\sum_{k\neq
i}\frac{2}{2-V_{ik}/\Delta}+\sum_{k\neq j}\frac{2}{2-V_{jk}/\Delta})\times 2$
$\displaystyle-\sum_{i\neq
j}\frac{2}{2-V_{ij}/\Delta}(\sum_{i}\frac{1}{2-V_{ij}/\Delta}+\sum_{j}\frac{1}{2-V_{ij}/\Delta})$
$\displaystyle-\sum_{(ijk)\neq}(\frac{2}{2-V_{ij}/\Delta}+\frac{2}{2-V_{ik}/\Delta}+\frac{2}{2-V_{jk}/\Delta})^{2}\times$
$\displaystyle\frac{1}{3-V_{ij}/\Delta-V_{ik}/\Delta-V_{jk}/\Delta}]$

Comparing the terms in Eq. 10 and 11, one can see that the ratio of $\Delta/V$
in a lattice with a given geometry could be used as a knob to control the
relative strength of different nonlinear terms. For example, expanding this
trend to the higher order one can see that in a square/triangle lattice, the
fourth/sixth order of non-linearity would get enhanced around $\Omega=\Delta$.

Considering 3 atoms on an equilateral triangle, the interaction profile as a
function of interatomic distance is plotted in Fig. 4. The solid line shows
the interaction profile up to the two-body interaction $U^{4}$ while the
dashed line includes the effects of both two- and three-body interaction
$U^{(6)}$. In this figure, the light-shift of non-interacting atoms is
subtracted. To generate a homogeneous interaction all the atoms must be
confined inside a sphere with diameter $0.6R_{c}$ to keep the variations of
$U^{(6)}$ and $U^{(4)}$ below 7% and 10% respectively.

(a)  Figure 3: Perturbation paths used to calculate the $U^{(6)}$ in Eq. 11.
Indices $i,j,k$ labels distinguished excited atoms.  (a)  Figure 4: Spatial
profile of dressing interaction including two and three-body interactions.
Considering 3 atoms on an equilateral triangle, the interaction profile as a
function of interatomic distance is plotted. The solid (dashed) line contains
up to 4 (6) orders of perturbations. Here $\Omega/2\Delta=0.15$ $\Omega=1$MHz,
$R_{b}=3\mu$m.

Acknowledgement: The author acknowledges a fruitful discussion with Nathan
Schine.

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