# Intensity correlations in the forward four-wave mixing driven by a single
pump

A. A. C. de Almeida<EMAIL_ADDRESS>M. R. L. da Motta  S.
S. Vianna  Departamento de Física, Universidade Federal de Pernambuco,
50670-901, Recife, Pernambuco, Brazil

###### Abstract

We study the field intensity fluctuations of two independent four-wave mixing
signals generated in a cold rubidium sample as well as the transmission
signals. We employ an experimental setup using a single CW laser to induce the
nonlinear process in a forward geometry using either parallel and circular or
orthogonal and linear polarizations of the input fields. Even though the
spectra of each experimental configuration are significantly different due to
the distinct level structures of each scenario, both cases present
intensity–intensity cross-correlation of the four-wave mixing signals. We also
calculate the cross-correlation between the input fields and draft a
theoretical model that points that resonant phase-noise to amplitude-noise
conversion allows the observation of Rabi oscillations in the cross-
correlation curves.

††preprint: APS/123-QED

##  I Introduction

For the past three decades, there has been an interest in studying light
fluctuations when it interacts with matter. The pioneer experimental work of
Yabusaki et al [1] showed that one may obtain spectroscopic information using
the intensity fluctuations of a laser beam interacting with rubidium vapor. In
the sequence, R. Walser and P. Zoller [2] provided a theoretical framework to
explain this new type of spectroscopy and especially how the conversion of
phase-noise to amplitude-noise is at the root of this phenomenon. Ever since
these works, a large amount of research has been produced studying these
fluctuations in the light-matter interaction, with interesting results such as
the study of correlations and anti-correlations in electromagnetically induced
transparency [3, 4, 5, 6], controlling intensity noise correlations and
squeezing of four-wave mixing processes via polarization [7], and the
generation of correlated and anticorrelated fields via atomic spin coherence
[8].

There have been different approaches to the problems regarding these
fluctuations, concerning whether the analysis is in the frequency domain or
time domain. In this last approach, there is a set of works led by the group
of M. O. Scully [9, 10, 11], which are of interest to the problem we present
here. Our experiment uses a similar setup with a single cw laser, however,
along with the correlations between transmitted beams, we also investigate the
correlations between two non-linear signals generated by two independent FWM
processes.

In this sense, we present experimental observations of strong correlations
between intensity fluctuations of two FWM signals generated through the
interaction of laser light with a cold rubidium sample. The correlation
between the input laser fields is also detailed, with observations that agree
with the literature. We compare different polarization configurations, which
access distinct internal energy level structures. Since we use a cold atomic
sample, the system has a narrow Maxwell-Boltzmann distribution, so we can
study how the correlations behave as a function of the laser detuning. This is
an advantage of the cold system if compared to an atomic vapor, in which
several velocity groups can respond to the input laser even if one changes the
detuning, as long as it is inside the Doppler broadening range.

Furthermore, we observe an oscillatory behavior compatible with Rabi
oscillations [12] in the correlation functions. The intriguing feature is that
we can detect these oscillations long after the transient period, retrieving
the frequency information through the correlation function. This idea of
extracting an oscillation frequency using the correlation function has been
used in other contexts such as the observation of quantum beats in spontaneous
emission [13] or the observation of temporal beats in Raman Stokes fields
[14]. The theoretical model we build supports the idea that it is only because
of the fluctuations that we can detect this oscillatory behavior of the
system.

The paper is organized as follows: In Sec. II, we detail the experiment and
all the experimental results. In particular, we show the time series of all
four signals and the corresponding second-order correlation functions.
Moreover, we demonstrate how the correlation changes regarding variations in
the intensity and frequency of the input fields. Sec. III is devoted to
building a simple theoretical model that can provide insight into the physical
meaning of the results. We conclude by summarizing the relevant achievements
of this work in Sec. IV.

##  II Experimental setup and results

In the experiment, we use a single cw laser to generate two input laser beams
labeled by their wave-vectors $\vec{k}_{a}$ and $\vec{k}_{b}$, as presented in
Fig. 1. These two beams interact with a cold atomic sample of 87Rb atoms in a
magneto-optical trap (MOT), typically with $10^{8}-10^{9}$ atoms cooled to
temperatures of hundreds of $\mu$K. In the experimental configuration with
linear and orthogonal polarization, the input fields are aligned with the
atomic cloud using a polarizing beam-splitter (PBS). In the other case, with
circular and parallel polarization, we substitute the two PBS for beam-
splitters and add two quarter-wave plates before and after the atomic cloud.

Figure 1: (a) Simplified scheme of the experimental setup; (b) Wave-vectors of
the four signals (two FWM and two transmissions); (c) Hyperfine structure of
the $D_{2}$ line of 87Rb.

We are interested in the two FWM signals generated in directions
$2\vec{k}_{a}-\vec{k}_{b}$ and $2\vec{k}_{b}-\vec{k}_{a}$, as shown in Fig.
1(b). Therefore, we investigate processes in which two photons of one of the
beams are absorbed, and one photon of the other beam is emitted, generating
new coherent signals. The input beams are in an almost copropagating
configuration, with a small angle of 10 mrad between them to allow the spatial
separation of all four signals. This type of forward geometry is challenging
since scattered light from one beam might arrive at the detection position of
the other beams. We detect the two FWM signals and the transmissions of the
input beams $E_{a}$ and $E_{b}$ with avalanche photodiodes (APD) of the model
APD120A/M from Thorlabs.

The beams that induce the degenerate FWM processes are tuned near the closed
transition $\left|F=2\right\rangle$ $\rightarrow$
$\left|F^{\prime}=3\right\rangle$, of the $D_{2}$ line of 87Rb (see Fig.
1(c)). Since this is the same transition excited by the cooling laser of the
MOT, we use a temporal scheme to generate and acquire the signal of interest.
We shut down for 2 ms all trapping fields, that is, the cooling laser and the
anti-Helmholtz coils. This time interval is enough so that the atoms cannot
gain much speed and therefore move away from the center of the MOT. After
turning off the trapping fields, there is a 20 $\mu$s delay, allowing the
repump laser to properly prepare the atoms in the ground state
$\left|F=2\right\rangle$, where they can interact with the FWM inducing laser.
The repump laser is always active throughout the measurement to guarantee the
proper state preparation.

We lock the frequency of the input laser to a saturation absorption
spectroscopy peak and tune the laser frequency using an Acoustic-Optical
Modulator (AOM). In the time interval in which the MOT fields are off, we
acquire data from a time series with typically 100 $\mu$s intervals of all
four signals. In these measurements, one must be careful with the detuning
with respect to the resonance because for small detunings, or high laser
intensity, the radiation pressure can disperse the atomic cloud. Therefore,
there is a practical limitation in our experiment, since the FWM signal
increases when the input laser is closer to resonance and for higher laser
intensities, the same regime that increases the radiation pressure.

Given this limitation, the experimental configuration with linear and
orthogonal polarization has an advantage. The spectra, shown in Fig. 2(a), is
wider than the spectra for the circular and parallel case (see Fig 2(b)).
Furthermore, it has a dip around the resonance so that the maximum signal is
slightly off-resonance. The fundamental difference between each case is that
the linear and orthogonal polarization interacts with a sum of $\Lambda$
systems in the form of the Zeeman sublevels, while the circular case is
modeled by a pure two-level system. The presence of two degenerate ground
levels together with equally powerful input laser beams that scan their
frequency simultaneously induces a coherent population trapping that prevents
the signal from being generated on resonance [15, 16, 17].

Figure 2: FWM spectra with input laser intensity $I_{a}=I_{b}=10$ mW/cm2 for
(a) linear and orthogonal polarizations; (b) circular and parallel
polarizations.

The time series of the intensity fluctuations of all four signals and for the
two polarizations are presented in Fig. 3. The presented data is part of the
100 $\mu$s recorded time series and has been filtered with high-pass ideal FFT
filter with a cutoff frequency of 500 kHz, to eliminate any slow fluctuations
of the signals. In each figure, 3(a) for circular and parallel polarization
and 3(b) for orthogonal and linear polarization, we show the intensity
fluctuations versus time of the two FWM signals (red and blue lines) for an
input laser intensity of $I_{a}=I_{b}=3.3$ mW/cm2 and a detuning from the
excited state of $\delta/2\pi=70$ MHz. It is noticeable that fluctuations
behave similarly, even though they are not identical. Due to the experimental
difficulties regarding the radiation pressure, one can only achieve a signal
with a good signal-to-noise ratio far from resonance and with input lasers
with intensities close to or above the saturation intensity.

Figure 3: Time series of the intensity fluctuations for the FWM signal with
input laser intensity of $I_{a}=I_{b}=3.3$ mW/cm2, detuning from the excited
state of $\delta/2\pi=70$ MHz and (a) circular and parallel polarizations; (b)
linear and orthogonal polarizations. Time series of the intensity fluctuations
for the transmittance of the input lasers with $I_{a}=I_{b}=0.15$ mW/cm2,
$\delta/2\pi=15$ MHz and (c) circular and parallel polarizations; (d) linear
and orthogonal polarizations.

On the other hand, the intensity fluctuations of the input lasers can be
obtained a lot closer to resonance as long as the intensity is small. In Figs.
3(c) and (d) we show the time series of the input lasers (orange and green
lines) for a input laser intensity of $I_{a}=I_{b}=0.15$ mW/cm2 and a detuning
from the excited state of $\delta/2\pi=15$ MHz. It is clear that these results
are remarkably synchronized and should present near-perfect correlations, a
known result [11].

These correlations can be quantified with the second-order correlation
function $G_{ij}^{(2)}(\tau)$ [9, 10, 11, 8] for intensity fluctuations of two
optical beams with time delay $\tau$. It is given by

$G_{ij}^{(2)}(\tau)=\frac{\left\langle\delta I_{i}\left(t\right)\delta
I_{j}\left(t+\tau\right)\right\rangle}{\sqrt{\left\langle\delta
I_{i}\left(t\right)^{2}\right\rangle\left\langle\delta
I_{j}\left(t+\tau\right)^{2}\right\rangle}},$ (1)

where the $\delta I_{i,j}(t)=I_{i,j}(t)-\left\langle I_{i,j}(t)\right\rangle$
are the time-dependent intensity fluctuations with $\left\langle
I_{i,j}(t)\right\rangle$ being the average intensities of the laser fields and
$i,j=a,b,s1,s2$ the labels to designate the two input fields and the two FWM
signals, respectively.

Figure 4: Second-order correlation function $G_{ij}^{(2)}(\tau)$ between the
FWM signals with input laser intensity of $I_{a}=I_{b}=3.3$ mW/cm2, detuning
from the excited state of $\delta/2\pi=70$ MHz and (a) circular and parallel
polarizations; (b) linear and orthogonal polarizations. Second-order
correlation function $G_{ij}^{(2)}(\tau)$ for the transmittance of the input
lasers (dark brown line) and autocorrelation (orange line) with
$I_{a}=I_{b}=0.15$ mW/cm2, $\delta/2\pi=15$ MHz and (c) circular and parallel
polarizations; (d) linear and orthogonal polarizations.

We present the intensity fluctuations correlation functions
$G_{ij}^{(2)}(\tau)$ for the pairs of time series of Fig. 3 in Fig. 4. These
correlation functions have peaks at zero time delay with amplitudes (Pearson
coefficient) of $\approx 0.6$ for the FWM signals and over 0.95 for the
transmission signals. This confirms the expectation of Fig. 3 that there is a
strong temporal positive correlation in the intensity fluctuations of the
output signals. Moreover, in Figs. 4(c) and (d) we also present the
autocorrelation (orange curve) for the intensity fluctuations of the $a$ laser
beam. These curves are remarkably similar to the cross-correlation (dark brown
curves) of the two transmission signals, specially concerning the oscillations
near zero delay.

The cross-correlation we observe in the transmission beams, arises due to the
resonant phase-noise to amplitude-noise conversion [18, 2, 19, 20, 21]. The
resonant interaction with atoms plays a critical role in this result. If there
were no atoms, or if the input laser was not near resonance, there would be no
correlation. The point we raise here is that this conversion also happens to
the FWM signals, creating correlated fields, even though they come from
processes that cannot occur simultaneously for the same atom. Ultimately, the
fields are correlated because they all come from the same laser with the same
phase fluctuations. In the next section, we build a simple model that provides
insight on how the phase fluctuations of the input laser manifest themselves
in the detected signals.

Given this argument, it would be interesting to look at the correlations
between one of the input laser fields and one of the FWM signals. However, the
experiment limits this situation as there is not a suitable choice of
parameters to obtain both signals simultaneously. To achieve the maximum FWM
signal, one must increase the intensity of the input laser as much as the
radiation pressure on the atomic sample allows. On the other hand, to measure
the intensity fluctuations of the input laser, it cannot have a high
intensity, otherwise, the medium will saturate, and most of the detected
photons will not interact with the atomic cloud.

An intriguing feature of these correlation curves is that the width of the FWM
correlation peak seems different from the transmission correlation peak, which
is to be expected since they have different intensities and detunings. In Ref.
[9], a fairly similar experiment but using an atomic vapor and a magnetic
field to break the degeneracy of the Zeeman sublevels, the authors comment
that the widths of the correlation peaks are associated with a power
broadening of the single photon resonance in the Rb vapor. Therefore, we
repeated the time series for each pair in different intensities for a fixed
detuning. These results are presented in Fig. 5(a) for the FWM signals and
5(c) for the transmission beams, both with linear orthogonal polarization,
with normalized correlation functions to achieve a proper comparison of the
widths.

Figure 5: Normalized second-order correlation function $G_{ij}^{(2)}(\tau)$
between intensity fluctuations of FWM signals with linear and orthogonal
polarization (a) varying input laser intensity with $\delta/2\pi=85$ MHz and
(b) varying input laser detuning; (c) Transmission signals with linear and
orthogonal polarization varying input laser intensity with $\delta/2\pi=25$
MHz and (d) varying input laser detuning

In most cases, we increased the input laser intensity by a maximum factor of
3, and the width of the correlation peak did not change significantly. We
believe it is because the atomic medium is not truly saturated in any of the
measurements, either when we compare transmission or FWM signals. In the case
of the FWM signals, the input laser fields are above saturation intensity, but
the nonlinear signal itself is weak.

These results indicate that the second-order correlation function behaves
differently regarding changes in the detuning. To be able to investigate this
is an advantage of using a cold sample instead of an atomic vapor. In hot
systems, the Doppler broadening is significant, meaning that variations of the
detuning inside the Maxwell-Boltzmann curve will always find a resonant
velocity group.

In Figs. 5(b) and 5(d), we present the correlation between transmission
signals and between FWM signals, in the experimental configuration with linear
polarization, for three different detunings. The results are similar for
circular polarization. The most noticeable feature of these results is that
correlation curves do get wider as the frequency of the input laser approaches
the resonance. Moreover, far from resonance, an oscillation of the correlation
curves becomes clearer and with higher frequency. There are regions of
correlation, $G_{ij}^{(2)}(\tau)>0$, and, regions of anti-correlation,
$G_{ij}^{(2)}(\tau)<0$. This behavior was already apparent in the previous
correlation curves for the FWM signals (see Fig. 4(a) and 4(b)), as they were
all far from resonance. It is important to remark that the results of Fig.
5(d) are for a fixed input intensity of $0.15$ mW/cm2, while the results for
the FWM signals, in Fig. 5(b), are for different intensities in each case.
However, if one looks at the generalized Rabi frequency
$\tilde{\Omega}=\sqrt{\Omega^{2}+\delta^{2}}$, it is approximately equal to
the detuning since the Rabi frequency of the input beams is close to the
natural linewidth of the transition. We must vary the intensity in these
measurements to maximize the FWM signal in each measurement, otherwise, the
signal-to-noise ratio would not allow proper visualization of the correlation.

Since to obtain the correlation between intensity fluctuations of the
transmission signals, we can work with very low intensities, it is easy to
tune the laser frequency without pushing away the atoms, making it possible to
achieve a detailed map of the correlation as a function of detuning. We
present such a map in Fig. 6. In this graph the broadening of the central peak
near resonance becomes clearer. Furthermore, it is noticeable the presence of
oscillations near the central peak. It seems that the frequency of this
oscillation gets smaller near resonance. A Fourier analysis of the curves in
Fig. 6 shows (see Fig. 7) that they have a spectral component compatible with
$\tilde{\Omega}$.

Figure 6: Second-order correlation function $G_{ij}^{(2)}(\tau)$ as a function
of the detuning $\delta/2\pi$ between transmission signals with linear and
orthogonal polarization. The input laser intensity is $I_{a}=I_{b}=$0.15
mW/cm2.

Therefore, these oscillations in the second-order correlation function are
connected to the generalized Rabi frequency of the input laser. This indicates
that in the conversion process between phase fluctuations of the laser into
intensity fluctuations through the interaction with the atomic medium, the
intensity fluctuations oscillate with approximately the generalized Rabi
frequency [12].

One could expect to see this oscillation in the raw data, that is, in the time
series of Fig. 3. However, they are not noticeable in this case as these
measurements are taken long after the transient period when this oscillation
should be more noticeable. Furthermore, the signal we acquire is the average
signal of the light emitted by the atomic ensemble and not by a single atom.
In fact, a Fourier analysis of that data does not reveal any spectral
component in particular.

On the other hand, a higher-order measurement should be able to retrieve the
spectral information of the system [14, 13]. This is possible with the
intensity fluctuations correlation function $G_{ij}^{(2)}(\tau)$, which does
present a noticeable spectral component, as we already mentioned. The Fourier
analysis of these curves (Fig. 4, 5, and 6) shows that there is a spectral
component compatible with the generalized Rabi frequency, or since the laser
intensity is usually small, compatible with the detuning. It is expected that
this is an approximate result because even for a simple two-level system, the
presence of spontaneous emission decay modifies how the temporal solution of
the optical Bloch equations oscillates, but the values should be close to
$\tilde{\Omega}$.

To verify this claim, we plot in Fig. 7 the spectral component present in each
correlation curve (red dots) compared with the absolute value of the detuning
(solid line). There is a reasonable agreement between the two results,
supporting our argument. One can see in the transmission case (see Fig. 7(c)
and (d)) that near resonance, the spectral component of the correlation curves
moves away from the detuning as the Rabi frequency gets more relevant to the
generalized Rabi frequency.

Figure 7: Frequency of the oscillation in the correlation function for FWM
signals with (a) linear and orthogonal polarizations and (b) circular and
parallel polarizations; Transmission signals with (c) linear and orthogonal
polarizations and (d) circular and parallel polarizations. The solid lines are
the absolute value of the detuning.

One final comment on the result of the correlation between FWM signals is
related to why we observe a positive correlation and not a competition between
signals, that is, an anti-correlation. Yang et al [8] observed an anti-
correlation between FWM signals in an atomic vapor using a $\Lambda$-sytem.
This situation compares to our results using linear and perpendicular
configurations. Our case differs from theirs because the ground states are
degenerate, rendering a symmetrical system that forbids competition between
the fields. The results of Ref. [11] show how this degeneracy controls the
correlation by introducing an external magnetic field that can break the
degeneracy, and change from perfect correlation to anti-correlation.

##  III  Theoretical model

We employ the model of Ref. [11] to explain the main observed features in the
correlation results between the transmission signals. We extend the results of
this previous work by exploring the dependence of the correlation on the
detuning, with special attention to the presence of the Rabi oscillations. Our
system allows this analysis since the Doppler broadening can be neglected in
cold atomic systems. Therefore, we begin modeling the experimental results
with linear and perpendicular polarization configuration, which is connected
to a three-level system. To model the features of the circular and parallel
polarization scenario, we will use the same set of equations but eliminate one
of the ground states.

The treatment of the problem begins considering an electric dipole coupling as
the interaction Hamiltonian

$\hat{H}_{int}=-\hbar\sum^{3}_{j\neq
k}\left(\Omega_{l}+c.c.\right)\left|j\right\rangle\left\langle k\right|,$ (2)

where $\Omega_{l}=\frac{\mu_{jk}E_{l}}{2\hbar}$ ($l=a$ or $b$) is the Rabi
frequency with $\mu_{jk}$ being the transition dipole moment and $E_{l}$ the
electric fields. In the linear and perpendicular case these fields are
represented by

$\begin{split}\vec{E}_{a}&=\frac{1}{2}\left[\varepsilon_{a}(t)e^{-i\left(\omega_{a}t+\phi\left(t\right)-k_{a}z\right)}+c.c.\right]\frac{\left(\hat{\sigma}^{+}+\hat{\sigma}^{-}\right)}{\sqrt{2}};\\\
\vec{E}_{b}&=\frac{1}{2}\left[\varepsilon_{b}(t)e^{-i\left(\omega_{b}t+\phi\left(t\right)-k_{b}z\right)}+c.c.\right]\frac{\left(i\hat{\sigma}^{+}-i\hat{\sigma}^{-}\right)}{\sqrt{2}}.\end{split}$
(3)

where $\varepsilon_{l}$ is the amplitude of the electric field; $\omega_{l}$
is the optical frequency, $\phi(t)$ is the fluctuating phase, and
$\vec{k}_{l}$ is the associated wave-vector. The polarization vector is
represented in the circular base as it highlights how these fields interact
with the $\Lambda$-system.

We consider that the electric fields have a fluctuating phase $\phi(t)$,
described by a Wiener-Levy diffusion process [22]. For these processes, the
average of the stochastic variable is zero and the average of the two-time
correlation is given by
$\overline{\left\langle\dot{\phi}\left(t\right)\dot{\phi}\left(t^{\prime}\right)\right\rangle}=2D\delta\left(t-t^{\prime}\right)$;
where $D$ is the diffusion coefficient. Often in the literature, the
stochastic process chosen to represent this phase is the Ornstein–Uhlenbeck
[23, 24], which includes an extra term to the Wiener process with an
exponential function of the time delay.

Moreover, we introduce an extra simplification to the model: we consider that
the field $a$ is in one of the transitions while the field $b$ is in the
other, as Fig. 8 shows. That is, we only take one circular component of each
field. We do so to allow each one-photon coherence to oscillate with the
frequency of one of the input fields. Hence, the number of coupled
Stochastical Differential Equations (SDE) decreases significantly. A complete
treatment of this system, including even wave-mixing processes of superior
orders, is performed in Ref. [25]. In this model, they perform a Floquet
expansion of the density-matrix elements in the frequency of the input fields
and their combinations. One could include a stochastical phase in this last
model, but it would take the system from nine coupled SDEs as we write it in
this work to a few tens of equations. The numerical solution becomes unstable
and demanding in terms of computation time.

Figure 8: Simplified level scheme for the configuration with linear and
perpendicular polarization.

Using this Hamiltonian and considering the approximations we described, it is
possible to write Liouville’s equation

$\frac{\partial\rho_{jk}}{\partial
t}=-(i\omega_{jk}+\gamma_{jk})\rho_{jk}-\frac{i}{\hbar}\left\langle
j\right|[\hat{H}_{int},\hat{\rho}]\left|k\right\rangle,\\\ $ (4)

where $\gamma_{jk}$ is the decay rate of the density matrix element
$\rho_{jk}$, and $\omega_{jk}$ is the frequency of the $\left|j\right\rangle$
$\rightarrow$ $\left|k\right\rangle$ transition. The Bloch equations in the
rotating-wave approximation can be written as:

$\displaystyle\dot{\rho}_{11}$
$\displaystyle=-i\sigma_{12}\Omega_{a}+i\sigma_{21}\Omega_{a}^{*}+\Gamma_{21}\rho_{22};$
(5) $\displaystyle\dot{\rho}_{22}$
$\displaystyle=i\sigma_{12}\Omega_{a}-i\sigma_{21}\Omega_{a}^{*}-i\sigma_{23}\Omega_{b}^{*}+i\sigma_{32}\Omega_{b}-(\Gamma_{21}+\Gamma_{23})\rho_{22};$
$\displaystyle\dot{\rho}_{33}$
$\displaystyle=i\sigma_{23}\Omega_{b}^{*}-i\sigma_{32}\Omega_{b}+\Gamma_{23}\rho_{22};$
$\displaystyle\dot{\sigma}_{12}$
$\displaystyle=-\sigma_{12}\left(i\delta_{a}+\gamma_{12}-\dot{\phi}\left(t\right)\right)-i(\rho_{11}-\rho_{22})\Omega_{a}^{*}-i\sigma_{13}\Omega_{b}^{*};$
$\displaystyle\dot{\sigma}_{13}$
$\displaystyle=-\sigma_{13}\left(i\delta_{a}-i\delta_{b}+\gamma_{13}\right)-i\sigma_{12}\Omega_{b}+i\sigma_{23}\Omega_{a}^{*};$
$\displaystyle\dot{\sigma}_{32}$
$\displaystyle=-\sigma_{32}\left(i\delta_{b}+\gamma_{23}-\dot{\phi}\left(t\right)\right)-i(\rho_{33}-\rho_{22})\Omega_{b}^{*}-i\sigma_{31}\Omega_{a}^{*}.$

The $\sigma_{jk}$ terms are the coherence from equation (4) in the rotating
frame, whereas $\Gamma_{jk}$ are the decay rates of the populations. The
missing coherence equations are the complex conjugate of the ones presented.

Since the set of Eqs. (5) contains stochastical terms, we must solve them
numerically using Itô’s calculus. As previously mentioned, we take a typical
stationary stochastic process, the Ornstein-Uhlenbeck process, to describe the
phase fluctuations. This process satisfies the SDE:

$dX_{t}=\alpha\left(\gamma-X_{t}\right)dt+\beta dW_{t}$ (6)

where the Itô’s diffusive process $dX_{t}$ has a deterministic part and a
stochastic one. The deterministic term, the first one, has a magnitude of the
mean drift $\alpha$ while the asymptotic mean is $\gamma$. If $X_{t}>\gamma$
the drift will be negative and the process will go towards the mean. If
$X_{t}<\gamma$ then the opposite happens, the drift is positive and the
process moves away from the mean. As for the stochastic part, it is a Brownian
motion $W_{t}$ with a magnitude constant $\beta$.

We solve the system of SDEs using a stochastic Runge-Kutta for scalar noise
algorithm. This algorithm possesses a good accuracy for our problem, with a
thin distribution of residuals. We also use the same Brownian increment
$dW_{t}$ for both one-photon coherences, as the original fluctuation comes
from a single laser. Finally, we probed several choices of parameters of the
Ornstein-Uhlenbeck process, but the outcomes are not drastically different as
long as the variance of the process, given by $\beta^{2}/2\alpha$, is small.

Once the numerical simulation is complete, we have access to a theoretical
time series of all elements of the density-matrix. Therefore, for each of
these terms, we can calculate the second-order correlation function for the
intensity fluctuations. However, we must establish the link between the
density-matrix elements and the actual detected signal. To do so, we solve the
wave equation derived from Maxwell’s equations, neglecting the transverse
derivatives of the electric field. With a few algebraic manipulations and
using the adiabatic approximation we can obtain the simple differential
equation $\frac{\partial\Omega_{l}}{\partial z}=i\kappa_{2j}\sigma_{2j}$,
where $\kappa_{2j}=\frac{\omega_{l}N\mu_{2j}^{2}}{2\hbar\epsilon_{0}c}$, with
$N$ being the number of atoms.

Solving this equation leads to the fields we detect in the experiment after
they propagate in the sample. To do so, we use the fact that MOT diameter $L$
is much smaller than the Rayleigh length of the fields in play. Therefore, it
is adequate to consider the thin-medium regime, which implies that we can make
use of the equations in Ref. [11] and rewrite the second-order correlation
function of the intensity fluctuations of Eq. (1) as:

$G^{\left(2\right)}\left(\tau\right)=\frac{\left\langle\mathrm{Im}\left(\delta\sigma_{21}\left(t\right)\right)\mathrm{Im}\left(\delta\sigma_{23}\left(t+\tau\right)\right)\right\rangle}{\sqrt{\left\langle\left[\mathrm{Im}\left(\delta\sigma_{21}\left(t\right)\right)\right]^{2}\right\rangle\left\langle\left[\mathrm{Im}\left(\delta\sigma_{23}\left(t\right)\right)\right]^{2}\right\rangle}}.$
(7)

To obtain this last result, we neglected second-order terms when calculating
the field intensity.

Notably, in the model, as it is, only the transmission results can be directly
explored. However, as our numerical solution renders the elements of the
density-matrix in all orders, the nonlinear effects of wave-mixing are also
built in the one-photon coherence. Naturally, the stronger term should be the
lower-order one, which is indeed connected to the transmission.

##  IV Theoretical results

After solving the system of coupled SDEs, we have a numerical simulation of
the time series for the transmission signals. An example of a single
realization of such series is presented in Fig. 9 (a) for a detuning of
$\delta/2\pi=30$ MHz and a Rabi frequency $\Omega_{a}=\Omega_{b}=0.1\Gamma$.
We simulate the signal for 20 $\mu$s but neglect the first half of the series
to ensure that the transient period is not present. Employing Eq. (7), we
calculate the second-order correlation function for the series of Fig. 9(a)
and present the results in Fig. 9(b).

Figure 9: (a) Numerical simulation of a time series of the intensity
fluctuations for the transmission signals with input Rabi frequency
$\Omega_{a}=\Omega_{b}=0.1\Gamma$, detuning from the excited state of
$\delta/2\pi=30$ MHz and linear and orthogonal polarizations. (b) Second-order
correlation function $G_{ij}^{(2)}(\tau)$ between the time series of (a).

Since we use the same Brownian increment $dW_{t}$ for both signals, they must
be perfectly correlated as in Fig. 9(b). It is easy to introduce different
increments for each input field and control how large their correlation is by
stating that $dW_{t}^{(1)}=dW_{t}^{(2)}+\sqrt{1-\rho^{2}}dW_{t}^{(3)}$, where
$\rho$ ranges from zero to one; that is, one increment is equal to the other
with the addition of a third increment. However, we use a single cw laser, so
the stochastic phase each input field carries should be the same.

We presented the case of cross-correlation between input field intensity
fluctuations for the scenario with linear and perpendicular polarization. If
we eliminate one of the ground states and, therefore, all the equations and
terms connected to it in Eqs. (5), then we would have a two-level system. This
is the scenario with circular and parallel polarization. However, as the
results are remarkably similar to those in Fig. 9, we do not present them
here.

Figure 10: (a) Second-order correlation function $G_{ij}^{(2)}(\tau)$ as a
function of the detuning $\delta/2\pi$ between theoretical transmission
signals with linear and orthogonal polarization. The input laser Rabi
frequency is $\Omega_{a}=\Omega_{b}=0.1\Gamma$. (b) Frequency of the
oscillation in the correlation function of (a). The solid line is the absolute
value of the detuning

A theoretical map such as the one presented in Fig. 6 can be achieved and it
is presented in Fig. 10(a). It shows a broadening of the correlation peak
compatible with the behavior of the experimental result. Moreover, the
theoretical results highlight the oscillation of the correlation curve in the
generalized Rabi frequency.

A graph similar to the one shown in Fig. 7 is presented in Fig. 10(b) for the
transmission signals in the linear and perpendicular polarization case. This
graph agrees with the experimental results and therefore supports that the
frequency of the oscillation we see in the correlation curves is indeed well
described by the generalized Rabi frequency.

We must emphasize that our results were able to reveal this oscillating
behavior because the experiments were performed in a cold atom cloud. In a
vapor cell, for example, this signature would have been washed away due to the
atomic movement. The Doppler integration should change the observation of
these oscillations, as the correlation curve would contain the response of
several velocity groups.

##  V Conclusions

We have successfully demonstrated that there are temporal correlations between
intensity fluctuations of two distinct degenerate FWM signals in a cold
rubidium sample. It is noteworthy that this is the first demonstration of
these correlations in degenerate FWM processes, which due to this degeneracy,
do not present competitive signals and therefore have a positive correlation
between them. In a complementary scenario to ours are the results of Ref. [8]
that presents an anticorrelation between FWM signals due to the nondegeneracy
of the ground states.

Furthermore, since our cold atomic system allows a proper definition of
detuning, namely, there is not a significant Doppler broadening, we can study
how the correlation between FWM signals and between transmission signals
behave as a function of detuning. The results show that the system exhibit
Rabi oscillations that can be revealed by the second-order correlation
function long after the transient. The theoretical model from Ref. [11] was
used to provide numerical results for the transmission signals that support
the experimental findings. Even though the model only deals with the
transmission signals, it provides a good insight that the mechanism behind the
correlations, and the Rabi oscillations we see in them, is the conversion of
phase-noise to amplitude-noise due to the interaction of the laser with the
atoms. The FWM signals should follow a very similar behavior, so we believe
that the results regarding the cross-correlation between transmission signals
can be extrapolated to the nonlinear case.

###### Acknowledgements.

This work was supported by CAPES (PROEX 534/2018, No. 23038.003382/2018-39).
The authors would like to acknowledge the financial support from Brazilian
agencies CAPES and CNPq. A. A. C. de Almeida acknowledges financial support by
CNPq (141103/2019-1). M. R. L. da Motta acknowledges financial support from
CNPq (130306/2020-7) and CAPES (88887.623521/2021-00).

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