# Effect of light-assisted tunable interaction on the position response
function of cold atoms

Anirban Misra1<EMAIL_ADDRESS>Urbashi Satpathi2
<EMAIL_ADDRESS>Supurna Sinha1<EMAIL_ADDRESS>Sanjukta Roy1
<EMAIL_ADDRESS>Saptarishi Chaudhuri1<EMAIL_ADDRESS>1 Raman Research
Institute, C. V. Raman Avenue, Sadashivanagar, Bangalore-560080, India.
2Department of Physics and Materials Science Engineering, Jaypee Institute of
Information Technology, A-10, Sector 62, Noida, UP-201309, India.

###### Abstract

The position response of a particle subjected to a perturbation is of general
interest in physics. We study the modification of the position response
function of an ensemble of cold atoms in a magneto-optical trap in the
presence of tunable light-assisted interactions. We subject the cold atoms to
an intense laser light tuned near the photoassociation resonance and observe
the position response of the atoms subjected to a sudden displacement.
Surprisingly, we observe that the entire cold atomic cloud undergoes
collective oscillations. We use a generalised quantum Langevin approach to
theoretically analyse the results of the experiments and find good agreement.

##  I Introduction

Cold atoms are excellent candidates for precision measurements Inguscio and
Fallani (2013) and for studying time dependent response Morawetz (2014); Bhar
_et al._ (2022) of atoms to external perturbations. Tracking the position
response function of cold atoms is expected to enhance the understanding of
many-body physics with cold atoms and quantum gases. One particular area of
interest has been precision molecular spectroscopy using ultra-cold atoms
which is known as photoassociation (PA) spectroscopy Thorsheim  _et al._
(1987); Miller  _et al._ (1993); Lett  _et al._ (1995); Stwalley and Wang
(1999); Weiner  _et al._ (1999); Lett  _et al._ (1993); Jones  _et al._
(2006); Rvachov  _et al._ (2018). The traditional detection scheme in such
experiments has always been to measure a trap loss where the photoassociated
molecules leave the trap Thorsheim  _et al._ (1987); Lett  _et al._ (1993);
Miller  _et al._ (1993); Côté and Dalgarno (1998); Stwalley and Wang (1999);
Weiner  _et al._ (1999); Hill  _et al._ (2021). It is well known that near
such PA transitions the inter-atomic interaction strength is significantly
modified due to dipole-induced dipole interactions even when the bound state
is not created Van Abeelen  _et al._ (1998); Fatemi  _et al._ (2000); Blatt
_et al._ (2011); Taie  _et al._ (2016); Nicholson  _et al._ (2015).

Here we report a technique for detecting changes in inter-atomic interactions
near a photoassociation resonance, without relying on existing trap-loss
measurements Thorsheim  _et al._ (1987); Lett  _et al._ (1993); Miller  _et
al._ (1993); Côté and Dalgarno (1998); Stwalley and Wang (1999); Weiner  _et
al._ (1999); Hill  _et al._ (2021). Our technique involves observing the
atomic position response to a sudden external light pulse in a Magneto-Optical
Trap (MOT), offering an alternative approach to study light-assisted
collisions between ultra-cold atoms. By adjusting the cooling laser parameters
and the magnetic field gradient, we can control the damping and spring
constants of the atomic motion. Such response measurements of cold atoms in
the MOT have been the subject of interest of many past studies Kim  _et al._
(2005); Beilin  _et al._ (2010); Hodapp  _et al._ (1995); Deng  _et al._
(2007) including some recent ones Satpathi  _et al._ (2017); Bhar  _et al._
(2022); Moon  _et al._ (2017); Baek  _et al._ (2023). Generally, motion of the
atoms in the MOT is well understood and the response function can be modelled
efficiently using a generalized Langevin formalism Ford  _et al._ (1988). The
corrections to such measurements taking into account the beyond two-level
approximation has also been the subject of a recent work Baek  _et al._
(2023). Here we present an experimental study and the corresponding
theoretical treatment of the motion of the atoms trapped in a MOT under the
influence of an additional laser beam which tunes the interactions between the
atoms. In the latter part of the paper, we present the details of our
experimental sequence, the observations and the theoretical analysis.

We notice a qualitatively interesting transition in the behaviour of position
response function of the trapped cloud. In a parameter space which is
overdamped, surprisingly the emergence of collective oscillations (which
occurs in underdamped regime) is observed in the position response function as
a result of tuning the interactions between the atoms. The cloud can be viewed
as being in contact with a bath of optical molasses where we tune the inter-
atomic interaction using the PA beam in the vicinity of a molecular resonance
Nicholson  _et al._ (2015).

##  II Measurement of position response function

###  II.1 Experimental set-up and method

Figure 1: Upper Panel: A schematic diagram of the experimental setup. In the
diagram, it can be observed that the driving beam and the PA-beam are merged
and sent through the same path towards the trapped 39K cloud in the vaccuum
chamber. Lower Panel: Pulse sequence generated from LabVIEW used in this
experiment. After the atoms are loaded in the MOT, the driving beam is
switched on for a duration of 0.5 s. $t_{1}$ ($t_{2}$) is the time-delay
between imparting the driving beam and measuring the response of the atoms
without (with) the PA-beam on. Both $t_{1}$ and $t_{2}$ are varied from 0.2 ms
to 30 ms (also called as $\tau_{postdrive}$) simultaneously.

A detailed description of our machine capable of trapping large numbers of 39K
and 23Na atoms in a Magneto-Optical Trap (MOT) can be found in Sutradhar  _et
al._ (2023). This setup enables fast loading of atoms into the 3DMOT with the
help of two independent 2D+MOTs, resulting in a high signal-to-noise ratio
fluorescence images for precise centroid positioning of the atomic clouds. In
our experiment, as schematically depicted in Fig. 1 upper panel, we introduce
two additional beams to the trapped cloud: a driving beam, red-detuned
$6.5\Gamma$ from the ${}^{39}K$ D2 line, and a Photoassociation (PA) beam from
a separate laser, tuned to a ${}^{39}K_{2}^{*}$ PA line at 390.976803 $THz$
Stwalley and Wang (1999), with an accuracy of $\pm$1 MHz. This PA line, chosen
for its relevance to single-photon excitation of ${}^{39}K_{2}^{*}$ just below
the $4P_{3/2}+4S_{1/2}$ level and being $\sim$ 1000 $\Gamma$ red-detuned,
minimizes photon scattering by cold 39K atoms while facilitating measurements
of strong molecular transitions.

The experiment is performed in two separate sequences and two different
settings. Initially, we prepare the 39K atoms, which have been laser-cooled
and trapped in a MOT. In our first setting, we allow only the driving beam
onto the MOT, keeping the PA-beam blocked. While performing the second part of
the experiment, we allow the driving beam and observe its effect on the cloud
keeping the PA-beam ON throughout the experiment. In either of these settings,
when we shine the driving beam with or without the PA-beam onto the trapped
39K atoms, we track the movement of the centroid of the cloud.

In order to give uniform optical force on the entire cross section of the 39K
atom cloud, the diameter of the beam has been kept at 10 mm which is
sufficient for a cloud of diameter $\sim 4$ mm. This beam shifts the center of
the trapping region to align with the zero point of the optical force. We give
a detailed timing sequence for measuring the response function (Fig. 1 lower
panel) of cold 39K atoms. Initially, cold atoms are loaded into the MOT for 10
s. Blocking the 2D+MOT path, a driving beam is applied for 0.5 s, allowing the
cold atomic cloud to reach a new equilibrium and to ensure that the dynamics
of the cloud is solely influenced by the driving beam. After switching off the
beam, we track the motion of the atoms via fluorescence imaging. A weak (0.05
$I_{sat}$) and off-resonant (5$\Gamma$ blue detuned) reference beam is
directed at the MOT, followed by capturing 3 frames with an Andor ICCD camera.
The reference beam prepared so as to not disturb the cold atoms, aids in
position tagging. The sequence involves: 1) capturing the driven atomic cloud
with the reference beam, 2) capturing without the atomic cloud but with the
reference beam, and 3) capturing with both the reference beam and 3DMOT beams
off to note background counts.

In the next phase of our experiment, we first observed the motion of the
atomic cloud under a driving beam (intensity 0.3 $I_{sat}$) alone. We then
added a PA-beam to this setup, following the same procedure as the previous
phase of our experiment. The motion of the cloud was recorded using the
previously described method, ensuring the post-drive duration
($\tau_{postdrive}$) matched with that of the driving beam-only observations.
After capturing the images, we turned off the MOT. Within our selected range
of parameters we observed the overall shift in the centroid position of the
cold atomic cloud to be around 3 mm. This shift is more or less the same
irrespective of the presence or absence of the PA-beam which has an intensity
($|E_{0}|^{2}$ per unit area) of 400 $mW/mm^{2}$.

###  II.2 Experimental determination of the response function from the average
displacement of the trapped cloud

In this section we briefly outline the experimental method of determination of
the response function from the measured average displacement discussed in ref.
Bhar  _et al._ (2022). The average displacement $\langle X(t)\rangle$ is
related to the position response function $R(t)$ by the equation,

$\displaystyle\langle X(t)\rangle$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{t}R(t-t^{\prime})f(t^{\prime})dt^{\prime}$ (1)

This gives the expectation value of the velocity on differentiation with
respect to time $t$, $\langle\dot{X}(t)\rangle$ :

$\displaystyle\langle\dot{X}(t)\rangle$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{t}\dot{R}(t-t^{\prime})f(t^{\prime})dt^{\prime}$
(2)

Here $f(t)$ is the external perturbing force, which in our experiment takes
the form of a “top-hat function”:

$\displaystyle f(t)$ $\displaystyle=$
$\displaystyle\begin{cases}f_{0},\text{for}\;-\infty<t<0\\\
0,\text{for}\;t\leq-\infty,t\geq 0\end{cases}$ (3)

Substituting $f(t)$ in Eq. (2), we get,

$\displaystyle R(t)=-\frac{\langle\dot{X}(t)\rangle}{f_{0}}$ (4)

Figure 2: $\langle X(t)\rangle$ for both the incidents when only the driving
beam is allowed to perturb the cold atomic cloud without and with the PA beam
being present. [Inset] (a) The average displacement of cold cloud without and
(b) in the presence of light-assisted interaction induced by the presence of
Photoassociation laser. The error bars indicate $\pm 1\sigma$ statistical
error estimated from several sets of data.

In the experiment we measure the average displacement $\langle X(t)\rangle$ of
the cold atomic cloud as shown in Fig. 2, and using the relation Eq. (4) we
extract the response function $R(t)$. We observe in the inset (b) of Fig. 2 a
collective oscillation of the cold cloud in the presence of the PA-beam while
all other parameters are kept the same as in the inset (a) of Fig. 2.
Therefore we conclude that the collective oscillation arises due to inter-
atomic interaction which is induced by the PA-beam. We record several sets of
data to confirm the response of the cold atoms. The measurement statistical
1$\sigma$ errors are reported in the insets of Fig. 2 along with the data
points.

###  II.3 The role of the photoassociation laser frequency

Figure 3: A schematic energy diagram showing an excited bound state formation
using specific angular frequency ($\omega_{L}$) excitation laser. The upper
(lower) potential energy curve shows the inter-atomic potential as a function
of the inter-atomic separation of the excited (ground) molecular manifold.

In Fig 3 a schematic diagram of the photoassociation (PA) process is described
Jones  _et al._ (2006). There are discrete angular frequencies ($\omega_{L}$)
at which the specific excited bound states can be addressed. Typically the
linewidths of such excited molecular bound-states are in the range of
$\sim$GHz. Near each of the molecular resonances the inter-atomic interaction
strength also diverges. In our experiment, we choose $\omega_{L}$ to be one
such molecular resonance angular frequency. Generally in a PA spectroscopy
experiment, the intensity of the PA laser is kept very high so as to induce
significant atom loss from the trap. In contrast, we deliberately keep the PA
laser intensity at a moderate level so that the atom loss is not significantly
high while the inter-atomic interaction is still strong at resonance.
Therefore, we can investigate the modification in the dynamics of the trapped
cloud. In the theoretical analysis, discussed above, $\omega_{L}$ is taken as
a parameter where the inter-atomic interaction is a maximum.

##  III Model of N-interacting particles in a thermal bath: Generalized
Langevin equation and position response function

This is a basic model for calculating the position response function for $N$
interacting particles, where $N$ is arbitrary. Using two interacting particles
(i.e. $N=2$) in contact with a thermal bath as a toy model, we have calculated
the response function of the centroid coordinate of the interacting particles
under the influence of a laser field.

###  III.1 Description of the two-body model

Starting from a Hamiltonian for a system of $N=2$ interacting particles
coupled to a heat bath of independent harmonic oscillators and integrating out
the bath degrees of freedom, we arrive at the quantum Langevin equation (QLE).
We extract the position response function of the system from this QLE. This
response function captures the collective behaviour observed in the
experiment.

Figure 4: A schematic representation of the two particle system centroid or
the centre of mass interacting with bath particles. The bath particles are
harmonic oscillators and are represented by small particles and they do not
interact with one another, hence the bath is called an IO bath. The system
particles are represented by the two big gray solid circles, the centroid of
which is shown by a dashed circle.

To start with we consider the coupled system bath model, where the system is
composed of two particles of the same mass ($M_{1}=M_{2}=1$) interacting with
one another under the influence of an external laser field $E(t)$. The bath is
composed of $K$ non interacting independent particles of mass $m_{\alpha}$
($\alpha=1,2...,K$) and frequency $\omega_{\alpha}$ ($\alpha=1,2...,K$). This
model is called the Independent-Oscillator (IO) model Ford  _et al._ (1988).
There is a coupling of strength $c_{\alpha}$ ($\alpha=1,2...,K$) between the
system coordinate (centroid coordinate $X_{c}=\sum_{i=1}^{2}X_{i}/2$) and bath
coordinates. The Hamiltonian of the coupled system-bath model is given by,

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{s}+H_{b}+H_{c}$ (5)
$\displaystyle H_{s}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}\left(\frac{P_{i}^{2}}{2}+\frac{\Omega_{0}^{2}X_{i}^{2}}{2}+d_{i}E(t)\right)+\sum_{i,j,i\neq
j}^{2}\frac{q^{2}}{2r_{a}^{3}}X_{i}X_{j}$ (6) $\displaystyle H_{b}$
$\displaystyle=$
$\displaystyle\sum_{\alpha}\left(\frac{p_{\alpha}^{2}}{2m_{\alpha}}+\frac{m_{\alpha}\omega_{\alpha}^{2}x_{\alpha}^{2}}{2}\right)$
(7) $\displaystyle H_{c}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\left(\frac{c_{\alpha}^{2}}{2m_{\alpha}\omega_{\alpha}^{2}}X_{c}^{2}-c_{\alpha}x_{\alpha}X_{c}\right)$
(8)

Here, $H_{s},H_{b},H_{c}$ are respectively the system Hamiltonian, bath
Hamiltonian and system-bath coupling Hamiltonian. $r_{a}=|X_{1}-X_{2}|$ is the
inter-atomic distance. A schematic diagram of the two particle centroid
interacting with independent bath particles is shown in Fig. 4. $\Omega_{0}$
is the oscillator frequency of each of the system particles, $d_{i},i=1,2$ are
the dipole moments of the two system particles, each of charge $q$. The recipe
for arriving at the QLE involves deriving the effective equations of motion of
the system coordinates $X_{i},P_{i}$ by integrating out the bath coordinates
$x_{\alpha},p_{\alpha}$. Using Heisenberg equations of motion we get the QLE
of the centre of mass coordinate of the interacting system :

$\displaystyle\ddot{X}_{c}+\Omega_{0}^{2}X_{c}+\frac{q^{2}}{r_{a}^{3}}X_{c}-2\int_{t_{0}}^{t}ds\gamma(t-s)\dot{X}_{c}=\zeta(t)-qE(t)$
(9)

where,

$\displaystyle\gamma(t)$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\frac{c_{\alpha}^{2}}{m_{\alpha}\omega_{\alpha}^{2}}\cos(\omega_{\alpha}t)$
(10) $\displaystyle\zeta(t)$ $\displaystyle=$
$\displaystyle\sum_{\alpha}c_{\alpha}\left[\left(x_{\alpha}(t_{0})-\frac{c_{\alpha}}{m_{\alpha}\omega_{\alpha}^{2}}X_{c}(t_{0})\right)\cos(\omega_{\alpha}t)+\frac{p_{\alpha}(t_{0})}{m_{\alpha}\omega_{\alpha}}\sin(\omega_{\alpha}t)\right]$
(11)

are the dissipation kernel and noise terms, and $E(t)=E_{0}\cos(\omega_{L}t)$,
$|E_{0}|^{2}$ per unit area of the beam is the intensity of the laser and
$\omega_{L}$ is the laser angular frequency. The pairwise interaction strength
is a maximum at $\omega_{L}$, not at the resonant absorption angular frequency
of photons by the atoms. The bath information is only retained in the
dissipation $\gamma(t)$ and noise $\zeta(t)$ terms. In the continuum limit,
$K\longrightarrow\infty$ the thermal noise satisfies the following spectral
properties,

$\displaystyle\langle\zeta(t)\rangle$ $\displaystyle=$ $\displaystyle 0$ (12)
$\displaystyle\langle\\{{\zeta(t),\zeta(t^{\prime})\\}}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega
Re[\gamma(\omega)]\hbar\omega\coth\left(\frac{\hbar\omega}{2k_{B}T}\right)e^{-i\omega
t}$ (13) $\displaystyle\langle[\zeta(t),\zeta(t^{\prime})]\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{\pi}\int_{-\infty}^{\infty}d\omega
Re[\gamma(\omega)]\hbar\omega e^{-i\omega t}$ (14)

The above relation is obtained considering that the bath is in thermal
equilibrium at the initial time $t_{0}$ and is given by distribution function,
$f(\omega)=1/(e^{(\hbar\omega/2k_{B}T)}-1)$. In the Fourier domain we can
write the solution to the QLE,

$\displaystyle X_{c}(\omega)$ $\displaystyle=$
$\displaystyle\frac{\zeta(\omega)-qE(\omega)}{-\omega^{2}+\omega_{s}^{2}-2i\omega\gamma(\omega)}=\frac{\zeta(\omega)-qE(\omega)}{(\omega-
s_{1})(\omega-s_{2})}$ (15)

Here, $\omega_{s}^{2}=\Omega_{0}^{2}+\frac{q^{2}}{r_{a}^{3}},s_{1},s_{2}$ are
the roots of the quadratic equation in the denominator of Eq. (15). In order
to extract the roots $s_{1},s_{2}$ we need to model the thermal bath
dissipation kernel $\gamma(t)$. Here we consider an Ohmic bath model Satpathi
_et al._ (2017) for which $\gamma(t)=2\gamma_{0}\delta(t)$ or
$\gamma(\omega)=\gamma_{0}$. For the Ohmic bath,
$s_{1}=-i\gamma_{0}-i\gamma_{s},s_{2}=-i\gamma_{0}+i\gamma_{s}$, where
$\gamma_{s}=\sqrt{\gamma_{0}^{2}-\omega_{s}^{2}}$. Using the fluctuation-
dissipation theorem Kubo (1966) which relates the Fourier transforms of the
response function $R(t)$ and the position correlation function
$C_{X}(t)=\langle\\{X_{c}(t),X_{c}(0)\\}\rangle/2$,

$\displaystyle ImR(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{\hbar}\tanh{\left(\frac{\hbar\omega}{2k_{B}T}\right)}C_{X}(\omega)$
(16)

we get $ImR(\omega)$ for two particles interacting with one another, coupled
to an Ohmic bath in the presence of a laser field,

$\displaystyle ImR(\omega)$ $\displaystyle=$
$\displaystyle\frac{\omega\gamma_{0}+q^{2}E_{0}^{2}\left[\delta(\omega-\omega_{L})+\delta(\omega+\omega_{L})\right]^{2}}{(\omega^{2}-s_{1}^{2})(\omega^{2}-s_{2}^{2})}$
(17)

Using Kramer’s Kronig relations we can calculate $ReR(\omega)$,

$ReR(\omega)=\frac{1}{\pi}P\int_{-\infty}^{\infty}\frac{\omega^{\prime}ImR(\omega^{\prime})}{(\omega^{\prime
2}-\omega^{2})}d\omega^{\prime}$

and hence combining the real and imaginary parts we get the frequency
dependent response function $R(\omega)=ReR(\omega)+iImR(\omega)$, which on
performing an inverse Fourier transform gives the time dependent response
function $R(t)$,

$\displaystyle R(t)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}d\omega R(\omega)e^{-i\omega t}$ (19)
$\displaystyle=$
$\displaystyle\frac{e^{-\gamma_{0}t}}{\gamma_{s}}\sinh{\gamma_{s}t}+Re\left(\frac{q^{2}E_{0}^{2}e^{-(\gamma_{0}-\gamma_{s})t}s_{2}\left[\delta(s_{1}-\omega_{L})+\delta(s_{1}+\omega_{L})\right]^{2}}{4\gamma_{0}\gamma_{s}(\gamma_{s}^{2}-\gamma_{0}^{2})}\right)$
$\displaystyle+Re\left(\frac{q^{2}E_{0}^{2}e^{-(\gamma_{0}+\gamma_{s})t}s_{1}\left[\delta(s_{2}-\omega_{L})+\delta(s_{2}+\omega_{L})\right]^{2}}{4\gamma_{0}\gamma_{s}(\gamma_{s}^{2}-\gamma_{0}^{2})}\right)$

In a similar way one can solve the case of three interacting particles and
find the response function. We have generalised the two particle system
analysis to the context of a three particle system and found the QLE,

$\displaystyle\ddot{X}_{c}+\Omega_{0}^{2}X_{c}+\frac{2q^{2}}{r_{a}^{3}}X_{c}-3\int_{t_{0}}^{t}ds\gamma(t-s)\dot{X}_{c}=\zeta(t)-qE(t)$
(20)

and the response function is given by,

$\displaystyle R(t)$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma_{0}^{\prime}t}}{\gamma_{s}^{\prime}}\sinh{\gamma_{s}^{\prime}t}+Re\left(\frac{q^{2}E_{0}^{2}e^{-(\gamma_{0}^{\prime}-\gamma_{s}^{\prime})t}s_{2}\left[\delta(s_{1}-\omega_{L})+\delta(s_{1}+\omega_{L})\right]^{2}}{4\gamma_{0}^{\prime}\gamma_{s}^{\prime}(\gamma_{s}^{\prime
2}-\gamma_{0}^{\prime 2})}\right)$ (21)
$\displaystyle+Re\left(\frac{q^{2}E_{0}^{2}e^{-(\gamma_{0}^{\prime}+\gamma_{s}^{\prime})t}s_{1}\left[\delta(s_{2}-\omega_{L})+\delta(s_{2}+\omega_{L})\right]^{2}}{4\gamma_{0}^{\prime}\gamma_{s}^{\prime}(\gamma_{s}^{\prime
2}-\gamma_{0}^{\prime 2})}\right)$

Here $X_{c}=\sum_{i=1}^{3}X_{i}/3$, $\gamma_{0}^{\prime}=3\gamma_{0}/2$,
$\gamma_{s}^{\prime 2}=\gamma_{0}^{\prime 2}-\omega_{s}^{\prime
2}$,$\omega_{s}^{\prime
2}=\Omega_{0}^{2}+2q^{2}/r_{a}^{3},s_{1}=-i\gamma_{0}^{\prime}-i\gamma_{s}^{\prime},s_{2}=-i\gamma_{0}^{\prime}+i\gamma_{s}^{\prime}$.

###  III.2 $N$-particles pairwise interacting in a thermal bath

We generalize the formalism further to a $N$ particle system. The QLE for a
$N$ particle system reduces to,

$\displaystyle\ddot{X}_{c}+\Omega_{0}^{2}X_{c}+\frac{Nq^{2}}{r_{a}^{3}}X_{c}-N\int_{t_{0}}^{t}ds\gamma(t-s)\dot{X}_{c}=\zeta(t)-qE(t)$
(22)

and the response function is given by,

$\displaystyle R(t)$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma_{0}^{\prime}t}}{\gamma_{s}^{\prime}}\sinh{\gamma_{s}^{\prime}t}+Re\left(\frac{q^{2}E_{0}^{2}e^{-(\gamma_{0}^{\prime}-\gamma_{s}^{\prime})t}s_{2}\left[\delta(s_{1}-\omega_{L})+\delta(s_{1}+\omega_{L})\right]^{2}}{4\gamma_{0}^{\prime}\gamma_{s}^{\prime}(\gamma_{s}^{\prime
2}-\gamma_{0}^{\prime 2})}\right)$ (23)
$\displaystyle+Re\left(\frac{q^{2}E_{0}^{2}e^{-(\gamma_{0}^{\prime}+\gamma_{s}^{\prime})t}s_{1}\left[\delta(s_{2}-\omega_{L})+\delta(s_{2}+\omega_{L})\right]^{2}}{4\gamma_{0}^{\prime}\gamma_{s}^{\prime}(\gamma_{s}^{\prime
2}-\gamma_{0}^{\prime 2})}\right)$ $\displaystyle=$
$\displaystyle\frac{e^{-\gamma_{0}^{\prime}t}}{\gamma_{s}^{\prime}}\sinh{\gamma_{s}^{\prime}t}+R_{2}(t)+R_{3}(t)$
(24)

where,

$\displaystyle
R_{2}(t)=\begin{cases}Re\left(\frac{q^{2}E_{0}^{2}e^{-(\gamma_{0}^{\prime}-\gamma_{s}^{\prime})t}s_{2}}{4\gamma_{0}^{\prime}\gamma_{s}^{\prime}(\gamma_{s}^{\prime
2}-\gamma_{0}^{\prime 2})}\right),\text{for}\;\omega_{L}=|s_{1}|\\\
0\end{cases}$ (25) $\displaystyle
R_{3}(t)=\begin{cases}Re\left(\frac{q^{2}E_{0}^{2}e^{-(\gamma_{0}^{\prime}+\gamma_{s}^{\prime})t}s_{1}}{4\gamma_{0}^{\prime}\gamma_{s}^{\prime}(\gamma_{s}^{\prime
2}-\gamma_{0}^{\prime 2})}\right),\text{for}\;\omega_{L}=|s_{2}|\\\
0\end{cases}$ (26)

Here $X_{c}=\sum_{i=1}^{N}X_{i}/N$, $\gamma_{0}^{\prime}=N\gamma_{0}/2$,
$\gamma_{s}^{\prime 2}=\gamma_{0}^{\prime 2}-\omega_{s}^{\prime
2}$,$\omega_{s}^{\prime
2}=\Omega_{0}^{2}+Nq^{2}/R^{3},s_{1}=-i\gamma_{0}^{\prime}-i\gamma_{s}^{\prime},s_{2}=-i\gamma_{0}^{\prime}+i\gamma_{s}^{\prime}$.
Note that the dissipation kernel and the noise terms remain the same as in the
two particle case, the only change appears in $X_{c}$. In the absence of
interactions when $\gamma_{0}^{\prime}>>\Omega_{0}$, we get the expected
overdamped behaviour in the position response function, however if we include
interactions keeping $\gamma_{0}^{\prime},\Omega_{0}$ unchanged we notice an
oscillatory behaviour in the position response function. This is rather
surprising but it can be understood from a comparison of the two time scales
$\gamma_{0}^{\prime-1}$ and $\gamma_{s}^{\prime-1}$. In the absence of
interactions, $\gamma_{0}^{\prime 2}>\gamma_{s}^{\prime 2}$ and both are real,
resulting in an exponentially damped behaviour of the position response
function. Inclusion of interactions leads to the condition,
$\gamma_{0}^{\prime 2}<\gamma_{s}^{\prime 2}$ where $\gamma_{s}^{\prime}$ is
imaginary, resulting in an oscillatory behaviour of the position response
function. The response function given by Eq. (28) captures the transition from
damped to oscillatory behaviour without and with interactions, respectively.
We have used Eq. (28) in the paper for comparing our theory with the
experimentally observed data.

##  IV Experimental Results and Comparison with theory

Figure 5: The normalized response function for both the incidents with only
the driving beam perturbing the cold atomic cloud in the overdamped regime
without and with the PA beam present.

Following the methods described in section II we measure the centroid position
of the cloud as a function of time and extract the position response function,
$R(t)$. We normalize R(t) w.r.t. its maximum value. We plot the normalized
position response function of the cold atomic cloud as a function of
$\tau_{postdrive}$. The resulting observation is summarized in Fig 5. The
transition from an overdamped motion to an underdamped oscillatory motion is
clearly visible in the same run of an experiment upon the addition of an
inter-atomic interaction induced by the PA-beam. The derivation of the
Generalized Quantum Langevin equation (QLE) and the extraction of the position
response function of the centroid of $N-$ interacting atoms under the
influence of an external laser are outlined in the section III. The QLE and
the position response functions are given respectively by the equations:

$\displaystyle\ddot{X}_{c}+\Omega_{0}^{2}X_{c}$
$\displaystyle+\frac{Nq^{2}}{r_{a}^{3}}X_{c}-N\int_{t_{0}}^{t}ds\gamma(t-s)\dot{X}_{c}=\zeta(t)-qE(t)$
(27) $\displaystyle R(t)$
$\displaystyle=\frac{e^{-\gamma_{0}^{\prime}t}}{\gamma_{s}^{\prime}}\sinh{\gamma_{s}^{\prime}t}+R_{2}(t)+R_{3}(t)$
(28) $\displaystyle R_{2}(t)$
$\displaystyle=\begin{cases}Re\left(\frac{q^{2}E_{0}^{2}e^{-(\gamma_{0}^{\prime}-\gamma_{s}^{\prime})t}s_{2}}{4\gamma_{0}^{\prime}\gamma_{s}^{\prime}(\gamma_{s}^{\prime
2}-\gamma_{0}^{\prime 2})}\right),\text{for}\;\omega_{L}=|s_{1}|\\\
0\end{cases}$ (29) $\displaystyle R_{3}(t)$
$\displaystyle=\begin{cases}Re\left(\frac{q^{2}E_{0}^{2}e^{-(\gamma_{0}^{\prime}+\gamma_{s}^{\prime})t}s_{1}}{4\gamma_{0}^{\prime}\gamma_{s}^{\prime}(\gamma_{s}^{\prime
2}-\gamma_{0}^{\prime 2})}\right),\text{for}\;\omega_{L}=|s_{2}|\\\
0\end{cases}$ (30)

Here $X_{c}=\sum_{i=1}^{N}X_{i}/N$, is the centroid displacement,
$\gamma_{0}^{\prime}=N\gamma_{0}/2$, $\gamma_{s}^{\prime 2}=\gamma_{0}^{\prime
2}-\omega_{s}^{\prime 2}$,$\omega_{s}^{\prime
2}=\Omega_{0}^{2}+Nq^{2}/r_{a}^{3},s_{1}=-i\gamma_{0}^{\prime}-i\gamma_{s}^{\prime},s_{2}=-i\gamma_{0}^{\prime}+i\gamma_{s}^{\prime}$
and $E(t)=E_{0}cos(\omega_{L}t)$ is the external laser field. $\omega_{L}$ is
the angular frequency of the photoassociation laser field which induces the
interaction between the atoms. In the experiment, we measure the average
displacement $\langle X_{c}(t)\rangle$ which is related to the position
response function $R(t)$ as, $R(t)=-\langle\dot{X_{c}}(t)\rangle/f_{0}$ II.
The strength of the interaction depends on $\omega_{L}$ following the details
of the two-body inter-atomic potential Jones  _et al._ (2006); Ridinger  _et
al._ (2011); however, in our description, we have considered $\omega_{L}$ to
be a parameter.

$\gamma_{0}^{\prime}$ pertains to the damping coefficient due to the optical
molasses and the frequency constant $\Omega_{0}$ stems from the magnetic field
gradient of the MOT. Three distinct cases appear for the motion of the atomic
cloud in the absence of inter-atomic interactions, depending on the values of
$\gamma_{0}^{\prime},\Omega_{0}$: an overdamped motion for
$\gamma_{0}^{\prime}>\Omega_{0}$, a critically damped motion for
$\gamma_{0}^{\prime}=\Omega_{0}$ and an underdamped motion for
$\gamma_{0}^{\prime}<\Omega_{0}$.Since we can choose a suitable parameter
space in our experiment to operate either in the overdamped or underdamped
regime, we consider the overdamped case. We have fitted the data using Eq.(28)
which is the position response function for $N$ interacting atoms under the
influence of an external laser, stemming from a generalised QLE (Eq.(27)).

Figure 6: The molasses parameters are chosen such that the normalized position
response function of the cold atomic cloud is in the underdamped regime. In
(a) The data has been recorded in the absence of the ’PA-beam’ and (b) in the
presence of the ’PA-beam’. The data has been fitted here with the equation of
the position response function which gives $\omega_{s}^{\prime}=2\pi\times 90$
Hz for (a) and $\omega_{s}^{\prime}=2\pi\times 102$ Hz for (b).

We extract a damping coefficient, $\gamma_{0}^{\prime}$, using the fit to the
data. This damping coefficient is $\sim 1.23\times 10^{-23}$ kg/sec. In the
absence of interactions when $\gamma_{0}^{\prime}>>\Omega_{0}$, we get the
expected overdamped behaviour in the position response function.

Figure 7: The normalized response function with the PA-beam kept ON at a fixed
intensity while its frequency is changed, once above and once below a
particular molecular transition frequency. Only when the laser is kept at the
molecular transition line ($\omega_{L}=2\pi\times$ 390.976803 $THz$), the
oscillatory behaviour can be observed whereas if the PA-laser frequency is
tuned to below or above the molecular transition line (which are 390.973435
$THz$ and 390.979431 $THz$ in the figure), the clear oscillatory nature is
absent.

However, an oscillatory behaviour emerges in the position response function
when we include atom-atom interactions keeping
$\gamma_{0}^{\prime},\Omega_{0}$ unchanged. This is rather surprising but it
can be understood from a comparison of the two time-scales
$\gamma_{0}^{\prime-1}$ and $\gamma_{s}^{\prime-1}$. In the absence of
interactions, $\gamma_{0}^{\prime 2}>\gamma_{s}^{\prime 2}$ and both are real,
resulting in an exponentially damped behaviour of the position response
function Eq.(28). Inclusion of interactions leads to the condition,
$\gamma_{0}^{\prime 2}<\gamma_{s}^{\prime 2}$ where $\gamma_{s}^{\prime}$ is
imaginary, resulting in an oscillatory behaviour of the position response
function Eq.(28).

We investigated the effect of inter-atomic interaction on the position
response function in the under-damped regime as well. In Fig 6, we summarize
our observation in this regime. In Fig 6(a) the normalised position response
function of the cold cloud (as a function of time) is shown without the light-
assisted atom-atom interaction, whereas in Fig 6(b) the atom-atom interaction
has led to a marked increase in the oscillation frequency by more than 10$\%$.
The response function given by Eq. (28) captures this enhancement of the
frequency of oscillations in the under-damped regime without and with
interactions, respectively.

Figs. 5 and 6 show a comparison between the experimentally measured response
function and the theoretical response function from Eq.(28) for the overdamped
and underdamped regimes respectively. The theoretical expression shows a very
good fit to the experimental data for both cases. The dipole moment of a 39K
atom in its ground state is much smaller than that of a Rydberg atom Ravets
_et al._ (2015), or an atom possessing high dipole moment such as Chromium
Stuhler  _et al._ (2005) at sub-micro Kelvin temperatures. Therefore, the
angular dependence of the dipole-dipole interaction has been neglected in our
theoretical modelling.

We conducted additional experiments where the PA-beam was kept ON at the same
intensity but its frequency was shifted away from the molecular transition
frequency. In Fig. 7, we show the response function for three different values
of $\omega_{L}$. Dramatically, the oscillatory behaviour is seen only when
$\omega_{L}$ is tuned to a molecular resonance whereas shifting $\omega_{L}$
slightly ($\pm 3$ GHz) takes it to an overdamped motion. This resonance
tracking is beyond the discussed theoretical model (Eq.(28)) because
$\omega_{L}$ is taken as a fixed parameter. The atom-atom interaction strength
as a function of $\omega_{L}$ is discussed in the section III.

In our magneto-optically trapped cold atomic cloud of ${}^{39}K$, the inter-
atomic interactions are described by the s-wave scattering length, $a$ (for
temperatures $\leq$ 2 mK). For large PA beam detuning from the atomic
transition ($|\delta|\gg\Gamma$, where $\Gamma$ is the decay rate of the
atomic transition), the scattering length $a(\omega_{L})$ is adjusted by the
PA laser frequency $\omega_{L}$ and the PA-beam intensity $I$ Nicholson  _et
al._ (2015); Blatt  _et al._ (2011) as
$a(\omega_{L})=a_{bg}+l_{opt}\frac{\Gamma_{m}}{\omega_{L}-(\omega_{n}+s_{n}I)}$.
Here, $a_{bg}$ is the background scattering length, $\Gamma_{m}$ is the
spontaneous emission rate from the excited molecular state, and
$\omega_{n}+s_{n}I$ is intensity-dependent shift of the resonance frequency.
The optical length $l_{opt}$, indicative of the PA line-strength, incorporates
the influence of the PA laser. With the background scattering length of
${}^{39}K$ atoms being negative ($-a_{bg}\simeq 33a_{0}$), the inherent inter-
atomic interaction is attractive and corresponds to an overdamped behavior of
the position response function without the PA beam. By applying the PA beam
near a molecular resonance the interaction alters, enabling a shift from an
overdamped to an underdamped behaviour of the position response function.

##  V Conclusion and outlook

In this study, we alter the inter-atomic interactions by exposing a cold cloud
of ${}^{39}K$ atoms to a beam tuned to one of the PA lines. Tracking the
position response function, we offer a novel approach to detect resonant
enhancement of interaction strength in cold atoms. Extending beyond existing
theoretical analysis Bhar  _et al._ (2022), our study incorporates light-
assisted inter-atomic interactions into a generalized Langevin formalism,
focusing on their impact on the position response function over a significant
range of time scales, while disregarding molecule formation due to its brevity
in this context. Our theoretical analysis aligns well with our experimental
observations, setting the stage for more intricate future models of inter-
atomic interactions as well as by including angle dependent dipole-dipole
interactions in quantum-degenerate clouds. Moreover, our experimental system
also allows us to study a mass-imbalanced cold atomic mixture of 23Na and 39K
which opens avenues to explore the position response function amidst varying
intra- and inter-species atomic interactions.

##  VI Acknowledgement

The authors acknowledge the funding from Department of science and Technology,
India. The authors also thank S. Majumder and G. pal for discussions as well
as helping with the instruments. The support from the mechanical and
electrical workshops in Raman Research Institute is appreciated for conducting
the experiments successfully.

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