# Accumulation phenomenon of dipole-dipole interaction induced by dark Rydberg
atoms

Ko-Tang Chen,1 Bongjune Kim,1,111Electronic address<EMAIL_ADDRESS>Chia-
Chen Su,1 Shih-Si Hsiao,1 Shou-Jou Huang,2 Wen-Te Liao,2,3,4 and Ite A.
Yu,1,4<EMAIL_ADDRESS>1Department of Physics, National Tsing Hua
University, Hsinchu 30013, Taiwan
2Department of Physics, National Central University, Taoyuan City 320317,
Taiwan
3Physics Division, National Center for Theoretical Sciences, Taipei 10617,
Taiwan
4Center for Quantum Technology, Hsinchu 30013, Taiwan

###### Abstract

We experimentally observed an accumulative type of nonlinear attenuation and
distortion of slow light in a laser-cooled electromagnetically-induced-
transparency (EIT) medium with the Rydberg state of $|32D_{5/2}\rangle$. The
present effect of attenuation and distortion was caused by the dipole-dipole
interaction between Rydberg atoms in a weak-interaction regime, i.e., the mean
distance between Rydberg atoms was far greater than the dipole blockade
radius. Our observation can be attributed to the atoms self-accumulated in
dark Rydberg states other than those in the bright Rydberg state
$|32D_{5/2}\rangle$ driven by the coupling field. We performed three different
experiments to verify the above hypothesis, to confirm the existence of the
dark Rydberg states, and to measure the decay rate from the bright to dark
Rydberg states. In the theoretical model, we included the decay process from
the bright to dark Rydberg states and the DDI effect induced by both the
bright and dark Rydberg atoms. All the experimental data of slow light taken
at various probe Rabi frequencies were in good agreement with the theoretical
predictions based on the model. The systematic study presented in this work
provides a better understanding of the Rydberg-EIT system.

##  I Introduction

Atoms in the Rydberg states possess a strong electric dipole-dipole
interaction (DDI) among themselves. The blockade effect arising from the DDI
is a versatile mechanism for quantum information processing
blockade_Zoller2000 ; Comparat_2010 ; SaffmanRMP ; Pfau_2013 . The combination
of the DDI and the effect of electromagnetically induced transparency (EIT)
can mediate strong photon-photon interactions Adams_2010 ; Fleischhauer_2011 ;
Lukin_Nature_2012 ; Gorshkov_PRL_2017 ; Hofferberth_2014 ; Rempe_swithch_2014
; Rempe_2014 ; Rempe_gate . Thus, Rydberg atoms have led to applications such
as quantum logic gates Lukin_2000 ; Saffman_2017 ; Lukin_PRL_2019 ; Rempe_gate
, single-photon generation Kuzmich_2012 ; Pfau_2018 ; ondemand_2020 , single-
photon transistors Hofferberth_2014 ; Rempe_2014 , and single-photon switches
Rempe_swithch_2014 . Furthermore, the system of Rydberg atoms in a vapor or an
array is also a platform for the study of many-body physics Bloch_2012 ;
dressing_2016 ; Gross_2016 ; Browaeys_2016 ; Lukin_2017 ; Lukin_2019 ;
dressing_2020 ; Lukin_2021 .

Most of the experimental studies on Rydberg atoms have been performed in a
strong interaction regime. In this work, we experimentally studied the DDI-
induced effect in a weak interaction regime, i.e., $(r_{B}/r_{a})^{3}<0.06$
where $r_{a}$ is the half mean distance between the Rydberg atoms, and $r_{B}$
is the blockade radius. The Rydberg state with a low principal quantum number
of $n=32$ and a low Rydberg-atom density equal to or less than 2.0$\times$109
cm-3 were employed in the experiment. Nevertheless, a high optical depth (OD)
equal to or larger than 70 and a low decoherence rate of 2$\pi\times$49 kHz in
the experimental system enabled us to observe the DDI-induced effect in the
weak interaction regime OurOE2020 ; OurCommunPhys2021 . Due to the EIT effect
EIT_review , the propagation delay time or photon-matter interaction time in
our high-OD medium was a couple of $\mu$s. The high-OD EIT medium of weakly
interacting Rydberg atoms can lead to realization of Bose-Einstein
condensation of Rydberg polaritons OurCommunPhys2021 .

In this work, we systematically studied the transmission of a probe field
propagating through a Rydberg-EIT system, in which the Rydberg state of
$|32D_{5/2}\rangle$ was employed. The probe field drove the transition from
the ground state of $|5S_{1/2}\rangle$ to the intermediate state of
$|5P_{3/2}\rangle$, while the coupling field drove the transition from the
intermediate state to the Rydberg states. Both the probe and coupling
frequencies were nearly resonant to the transitions, and the two-photon
resonance condition was accurately maintained. With a weak and short Gaussian
probe pulse, the propagation delay time of the pulse was about 1.8 $\mu$s
under the coupling Rabi frequency, $\Omega_{c}$, of 1.0$\Gamma$. The
experimental data of the output probe pulse was in agreement with the
theoretical prediction. However, when we employed a long Gaussian input probe
pulse, the output pulse was distorted and the experimental data were
inconsistent with the theoretical prediction, in which the DDI-induced
decoherence rate was considered. A larger input probe Rabi frequency,
$\Omega_{p0}$, resulted in a more severe distortion, while the condition of
$\Omega_{p0}^{2}\ll\Omega_{c}^{2}$ was always satisfied. The distortion
phenomenon of the long Gaussian pulse was unexpected.

To investigate the distortion phenomenon, we employed a quasi-continuous wave
(CW) probe field and measured its output transmission. It was expected from
the theoretical prediction that the transmission should increase to a steady-
state value. However, experimentally, the transmission reached the maximum and
then decayed to a steady-state value. A larger value of $\Omega_{p0}$ resulted
in a faster decay and a lower steady-state transmission. To explain the
unexpected decay behavior of the transmission, we suggest that the DDI-induced
decoherence rate, $\gamma_{\rm DDI}$, or the DDI strength increases over time.

After the quasi-CW probe field was switched off, we further applied a very
weak Gaussian probe pulse to measure $\gamma_{\rm DDI}$ of the system at a lag
time. The coupling field remained unchanged during the measurement, and
quickly removed the population in $|32D_{5/2}\rangle$. Since the weak probe
pulse produced little DDI, one could expect that $\gamma_{\rm DDI}$ would also
quickly decrease to nearly zero after the quasi-CW probe field was switched
off. On the contrary, $\gamma_{\rm DDI}$ slowly decreased to an insignificant
value with a decay time constant that was close to the lifetimes of the
Rydberg states, with $n$ around 32. This observation suggested that after the
quasi-CW probe field was switched off, the population still remained in some
Rydberg states and was not de-excited by the coupling field. As these Rydberg
states were not driven by the coupling field, we call them dark Rydberg
states. Similarly, $|32D_{5/2}\rangle$ is called the bright Rydberg state.

In the measurements of the long Gaussian probe pulse and quasi-CW probe field,
atoms in the bright Rydberg state were able to be transferred to those in the
dark Rydberg states. Possible transfer processes will be discussed in the
Discussion section. Since not interacting with the coupling field, the dark
Rydberg atoms accumulated over time. On the other hand, the DDI between the
dark and bright Rydberg atoms existed. A larger number or density of the dark
Rydberg atoms produced a larger DDI on the bright Rydberg atoms. Consequently,
the probe light suffered a larger DDI-induced decoherence rate in the EIT
system consisting of the bright Rydberg state. As the dark Rydberg atoms
accumulated over time, the DDI-induced decoherence rate increased with time.
The above description can explain the phenomena observed with the long
Gaussian probe pulse and the quasi-CW probe field.

To quantitatively study the accumulation phenomenon, we modified the optical
Bloch equations (OBEs) to include the process of the population transfer from
the bright to dark Rydberg states and took into account the DDI-induced
decoherence rate due to the population in the dark Rydberg states. We also
measured the transfer or decay rate of the atoms from the bright to dark
Rydberg states. The measured decay rate was linearly proportional to the
density of the bright Rydberg atoms. We modeled the decay process in the OBEs
and set the parameters of the decay rate according to the experimental result.
In addition, the DDI-induced decoherence rate due to the population in the
dark Rydberg states was determined by the experimental data of the quasi-CW
probe field. Finally, we compared the experimental data of the distorted
output probe pulse taken at different values of $\Omega_{p0}$ with the
theoretical predictions, which were calculated from the modified OBEs without
any adjustable parameters. The theoretical predictions were in good agreements
with the experimental data. Thus, the distortion of the output probe pulse in
the Rydberg-EIT system was well understood.

This article is organized as follows. In Sec. II, we present the distorted
output probe pulse shapes, which we cannot explain with the theoretical model
as it only considered the DDI between the bright Rydberg states under the EIT
condition. In addition, we show the necessity of the consideration of the dark
Rydberg states by measurements of the decay time constant, the steady-state
attenuation coefficient of the square input pulse, and the lifetime of the
dark Rydberg state. In Sec. III, we introduce a modified theoretical model to
describe the increasing attenuation coefficient over time depending on the
density of the Rydberg atom. We also report the method for estimating the
decay rate from the bright Rydberg state to the dark Rydberg state and compare
the experimental data with the theoretical prediction. In Sec. IV, we discuss
the possible transfer process from the bright to the dark Rydberg states.
Finally, we summarize the results in Sec. V.

##  II Observation of the Accumulation Effect

###  II.1 Experiment setup

The experiment was performed with a cigar-shaped cold 87Rb atom cloud produced
by a magneto-optical trap. The dimension of the cloud was $1.5\times 1.5\times
5.0$ mm3 cigar-shape MOT  and the temperature was about 350 $\mu$k OurPRA2019
; OurCommunPhys2021 . The transition scheme formed by the probe and coupling
fields is shown in Fig. 1(a). The probe field drove the transition between the
ground state $|1\rangle$ and the intermediate state $|3\rangle$, and the
coupling field drove the transition between state $|3\rangle$ and the Rydberg
state $|2\rangle$. States $|1\rangle$, $|2\rangle$, and $|3\rangle$
corresponded to the ground state $|5S_{1/2},F=2,m_{F}=2\rangle$, the Rydberg
state $|32D_{5/2},m_{J}=5/2\rangle$, and the excited state
$|5P_{3/2},F=3,m_{F}=3\rangle$ of 87Rb atoms, respectively. We prepared all
population in a single Zeeman state $|5S_{1/2},F=2,m_{F}=2\rangle$ by optical
pumping Natcomm_2014 . In the experiment, the $\sigma_{+}$-polarized probe and
coupling fields were used. Owing to the optical pumping and polarization of
the laser fields, the relevant energy levels $|1\rangle$, $|2\rangle$, and
$|3\rangle$ were considered as a single Zeeman state. The spontaneous decay
rate of $|3\rangle$ is $\Gamma=2\pi\times$6.07 MHz, and that of $|2\rangle$ is
$\Gamma_{2}=$ 2$\pi$$\times$7.9 kHz or 1.3$\times$$10^{-3}$$\Gamma$ Rb87Data ;
RydbergStateLifeTime . As the van der Waals interaction energy is denoted as
$\hbar C_{6}/r^{6}$, the Rydberg atoms in $|32D_{5/2},m_{J}=5/2\rangle$ have
$C_{6}=-2\pi\times$130 MHz$\cdot\mu$m6 Saffman_C6 .

Figure 1:  (a) Relevant energy levels and transitions of the laser fields.
$|1\rangle$ and $|3\rangle$ are the ground and intermediate excited states.
$|2\rangle$ is the Rydberg state driven by the coupling field, and $|4\rangle$
represents all the nearby Rydberg states of $|2\rangle$ that involve the DDI
effect. $|2\rangle$ and $|4\rangle$ are the bright and dark Rydberg states,
respectively. $\Gamma$, $\Gamma_{2}$, and $\Gamma_{4}$ represent the
spontaneous decay rates of $|3\rangle$, $|2\rangle$, and $|4\rangle$.
$\Omega_{p}$ and $\Omega_{c}$ denote the Rabi frequencies of the probe and
coupling fields. The two fields’ frequencies were stabilized to maintain the
two-photon resonance. As compared with $\Gamma$, the magnitude of the one-
photon detuning was negligible. (b) Sketch of the experiment setup. BS: beam
splitter; PBS: polarizing beam splitter; PD: photo detector; DM: dichroic
mirror; $L_{p}$, $L_{b}$: lenses; QWP: quarter-wave plate; CL: collimation
lens; MMF: multimode optical fiber; PMT: photomultiplier tube. Red and blue
arrowed lines indicate the optical paths of the probe and coupling beams.

The experimental scheme is shown in Fig. 1(b). Frequency stabilized laser
systems generated the probe and coupling fields. The details of the
stabilization method are described in Refs. OurPRA2019 ; OurCommunPhys2021 .
The probe and coupling fields were the first-order beams of the acousto-optic
modulators (AOMs, not drawn in the figure). The AOMs were used to control the
time sequence, shape the probe pulse, and adjust the frequency and the
amplitude of the fields precisely. Each field was sent to the atom cloud by
the polarization maintained optical fiber (PMF) after the AOM. To minimize the
Doppler effect, we used the counter-propagating scheme of the probe and
coupling fields. At the center of the cloud, the $e^{-1}$ full widths of the
probe and coupling beams were 130 and 250 $\mu$m, respectively. The pulse
shape and amplitude of the input probe field were monitored by a photo
detector (PD) before entering the atom cloud as shown in Fig. 1(b). The probe
field was detected by a photomultiplier tube (PMT) after passing through the
atom cloud. A digital oscilloscope (Agilent MSO6014A) acquired the signal from
the PMT and produced the raw data.

###  II.2 Theoretical model without the accumulation effect

The Rydberg atoms inside the cloud were considered randomly distributed
particles, similar to ideal gases, due to the assumption that the condition of
a weakly-interacting many-body system of Rydberg polaritons was satisfied.
Therefore, the mean-field model developed in Ref. OurOE2020  was adopted to
describe the DDI-induced behaviors in this study.

We initially considered only the states $|1\rangle$, $|2\rangle$, and
$|3\rangle$ shown by Fig. 1(a) in the theoretical model. The DDI-induced
decoherence rate ($\gamma_{\rm DDI}$) and frequency shift ($\delta_{\rm
DDI}$), which are caused by the population in the Rydberg state $|2\rangle$,
were taken into account. The optical Bloch equations (OBEs) of the density
matrix operator and the Maxwell-Schrödinger equation (MSE) of the probe field
are given below:

$\displaystyle\frac{\partial}{\partial t}\rho_{21}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\Omega_{c}\rho_{31}+i(\delta_{\rm
DDI}+\delta)\rho_{21}$ (1) $\displaystyle-$ $\displaystyle\left(\gamma_{\rm
DDI}+\gamma_{0}+\frac{\Gamma_{2}}{2}\right)\rho_{21},$
$\displaystyle\frac{\partial}{\partial t}\rho_{31}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\Omega_{p}(\rho_{11}-\rho_{33})+\frac{i}{2}\Omega_{c}\rho_{21}+i\Delta_{p}\rho_{31}$
(2) $\displaystyle-$ $\displaystyle\frac{\Gamma}{2}\rho_{31},$
$\displaystyle\frac{\partial}{\partial t}\rho_{32}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\Omega_{p}\rho_{21}^{*}+\frac{i}{2}\Omega_{c}(\rho_{22}-\rho_{33})$
(3) $\displaystyle-$ $\displaystyle
i\Delta_{c}\rho_{32}-\frac{\Gamma}{2}\rho_{32},$
$\displaystyle\frac{\partial}{\partial t}\rho_{22}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\Omega_{c}\rho_{32}-\frac{i}{2}\Omega_{c}\rho_{32}^{*}-\Gamma_{2}\rho_{22},$
(4) $\displaystyle\frac{\partial}{\partial t}\rho_{33}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\Omega_{p}^{*}\rho_{31}+\frac{i}{2}\Omega_{p}\rho_{31}^{*}-\frac{i}{2}\Omega_{c}\rho_{32}$
(5) $\displaystyle+$
$\displaystyle\frac{i}{2}\Omega_{c}\rho_{32}^{*}-\Gamma\rho_{33},$
$\displaystyle\frac{\partial}{\partial t}\rho_{11}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\Omega_{p}^{*}\rho_{31}-\frac{i}{2}\Omega_{p}\rho_{31}^{*}+\Gamma\rho_{33}$
(6) $\displaystyle\frac{1}{c}\frac{\partial}{\partial t}\Omega_{p}$
$\displaystyle+$ $\displaystyle\frac{\partial}{\partial
z}\Omega_{p}=i\frac{\alpha\Gamma}{2L}\rho_{31},$ (7)

where $\rho_{ij}$ ($i,j=1,2,3$) represents a matrix element of the density
matrix operator, $\Omega_{p}$ and $\Omega_{c}$ are the Rabi frequencies of the
probe and coupling fields, $\delta$ is the two-photon detuning, $\gamma_{0}$
is the intrinsic decoherence rate in the experimental system, $\Delta_{p}$ and
$\Delta_{c}$ are the one-photon detunings of the probe and coupling
transitions, and $\alpha$ and $L$ are the optical depth and length of the
medium.

The values of $\gamma_{\rm DDI}$ and $\delta_{\rm DDI}$ were obtained by the
analytic formulas of the attenuation coefficient ($\Delta\beta$) and the phase
shift ($\Delta\phi$), respectively, induced by the DDI in the steady-state
condition. The formulas were derived in Ref. OurOE2020 , utilizing a mean-
field model based on the nearest neighbor distribution, and they were
experimentally verified in Ref. OurCommunPhys2021 . At the condition of
$\delta=0$ and $\Delta_{c}\ll\Gamma$, the formulas of $\Delta\beta$ and
$\Delta\phi$ are given by:

$\Delta\beta\left(\equiv\frac{2\alpha\Gamma\gamma_{\rm
DDI}}{\Omega_{c}^{2}}\right)=\frac{2\pi^{2}\alpha\sqrt{|C_{6}|\Gamma}}{3\Omega_{c}}n_{R},$
(8)

$\Delta\phi\left(\equiv\frac{\alpha\Gamma\delta_{\rm
DDI}}{\Omega_{c}^{2}}\right)=\frac{\pi^{2}\alpha\sqrt{|C_{6}|\Gamma}}{3\Omega_{c}}n_{R},$
(9)

where $n_{R}$ is the density of the Rydberg-state atoms. Using the relation
$n_{R}=n_{a}\rho_{22}$ between $n_{R}$ and the density of all atoms, $n_{a}$,
we introduced the coefficient $A$, which is written as:

$A=\frac{\pi^{2}\Omega_{c}\sqrt{|C_{6}|}}{3\sqrt{\Gamma}}n_{a}.$ (10)

The coefficient $A$ represents the DDI-induced decoherence rate or frequency
shift per unit $\rho_{22}$. From Eqs. (8) and (10), we obtained:

$\gamma_{\rm DDI}=A\rho_{22}.$ (11)

Similarly, from Eqs. (9) and (10), we also obtained:

$\delta_{\rm DDI}=A\rho_{22}.$ (12)

Thus, $\gamma_{\rm DDI}$ and $\delta_{\rm DDI}$ were replaced by $A\rho_{22}$
in the OBEs. According to the values of $\Omega_{c}$, $C_{6}$, and $n_{a}$
used in the experiment, it was determined that $A=$ 0.76$\Gamma$. We set and
fixed $A$ to 0.76$\Gamma$ in all the theoretical predictions of this work and
the values of $\gamma_{\rm DDI}$ and $\delta_{\rm DDI}$ are linearly
proportional to the population of the Rydberg state, $\rho_{22}(z,t)$.

Furthermore, the effect of the DDI, i.e., $\gamma_{\rm DDI}$ and $\delta_{\rm
DDI}$, was ignored to obtain Eq. (3) due to the assumption of
$A\rho_{22},\Gamma_{2}\ll\Gamma$, which was the case in the experiment.
Throughout this study, the two-photon resonance condition of $\delta$
$(\equiv\Delta_{p}+\Delta_{c})=0$ was always maintained. Please note that the
term of $i(\delta_{\rm DDI}+\delta)\rho_{21}$ in Eq. (1), i.e.,
$i(A\rho_{22}+\delta)\rho_{21}$, played a negligible role in the output probe
transmission, since $\delta=0$ and
$4(A\rho_{22})^{2}\Gamma^{2}\ll\Omega_{c}^{4}$ in all the cases of this work.
To obtain Eqs. (5) and (6), we also considered that the population in
$|2\rangle$ rarely decayed to $|1\rangle$ and $|3\rangle$. When we made the
population in $|2\rangle$ all decay to $|1\rangle$ by adding the term of
$+\Gamma_{2}\rho_{22}$ to the right-hand-side of Eq. (6), the calculation
result changed very little because $\Omega_{p}^{2}\ll\Omega_{c}^{2}$ in this
work.

###  II.3 Distorted output pulses of slow light

Before the measurement, the values of the one-photon detuning and two-photon
detuning were set to zero and experimentally verified. We determined the
experimental parameters in the order of $\Omega_{c}$ $\rightarrow$
$\gamma_{0}$ $\rightarrow$ OD. The parameters were confirmed again after the
measurement in the reverse order of that before measurement. To measure the
values of $\Omega_{c}$ and $\gamma_{0}$, we intentionally reduced the OD to a
value between 15 and 20 by adjusting the dark MOT parameter to minimize the
effect of DDI. The measurement results enabled us to set and verify
$\Omega_{c}=$ 1.0$\Gamma$ and determine $\gamma_{0}=$ 8.0$\times
10^{-3}\Gamma$. Then, we employed a short and weak Gaussian probe pulse to
determine the OD by the delay time. The pulse was sufficiently weak such that
the DDI effect played a negligible role in the delay time. It was determined
that $\alpha$ (OD) = 70 in the experiment. The detailed methods are described
in Ref. OurPRA2019 . Following the definition of the half mean distance
between Rydberg polaritons, $r_{a}=(4\pi n_{R}/3)^{-1/3}$, the estimated
smallest $r_{a}$ in this experiment was 5.0 $\mu$m when $\Omega_{p0}$ was
$0.2$$\Gamma$ OurOE2020 . The blockade radius, $r_{B}$, was 1.9 $\mu$m
according to the formula of $r_{B}=(2|C_{6}|\Gamma/\Omega_{c}^{2})^{1/6}$
Lukin_Nature_2012 . $\Omega_{p0}=0.2\Gamma$ was the largest input probe Rabi
frequency and $\Omega_{c}=1.0\Gamma$ was kept throughout the EIT experiment.
Thus, it was considered a weakly-interacting many-body system of Rydberg
polaritons, which satisfied the condition of $(r_{B}/r_{a})^{3}\ll 1$.

Figure 2: Experimental observation that the slow light of a Gaussian pulse was
distorted and a larger value of $\Omega_{p0}$ made the distortion more severe.
(a) Experimental data of the output pulses versus time are shown by red, blue,
and green lines, and their input Rabi frequencies, $\Omega_{p0}$, of
0.05$\Gamma$, 0.1$\Gamma$, and 0.2$\Gamma$, respectively. Since the shape of
the three input pulses is very similar, we only plot the one with the
$\Omega_{p0}$ of 0.1$\Gamma$ as shown by the black line, which is scaled down
by a factor of 0.4. In the measurements, $\alpha$ (optical depth) = 70,
$\Omega_{c}=$ 1.0$\Gamma$, and $\gamma_{0}=$ 8.0$\times 10^{-3}$$\Gamma$,
which were determined experimentally OurPRA2019 . (b) Theoretical predictions
by solving Eqs. (1)-(7) with the above values of $\Omega_{p0}$, $\alpha$,
$\Omega_{c}$, and $\gamma_{0}$, and with the coefficient $A$ = 0.76$\Gamma$
given by Eq. (10).

Using the above experimental conditions, we measured the slow light data as
shown in Fig. 2(a). An input probe pulse with a Gaussian $e^{-1}$ full width
of 11.5 $\mu$s was used in this measurement. This input pulse was far longer
than that used in the determining the OD. Considering the values of OD and
$\Omega_{c}$, the delay time $\tau_{d}$ was expected to be $\sim$$1.8~{}\mu$s
as the result of a short input Gaussian pulse with the $e^{-1}$ full width of
0.66 $\mu$s. However, the peak position of the output pulse with
$\Omega_{p0}=0.05$$\Gamma$ of the long Gaussian input probe pulse was nearly
the same as the peak position of the input pulse (i.e., the delay time was
nearly zero). Furthermore, a stronger input pulse had an output peak position
that preceded the input peak position and showed a more distorted shape of the
output pulse.

The degree of distortion of the output pulse shape depends on $\Omega_{p0}$,
or equivalently the Rydberg atom density. One might guess that the DDI effect
could explain the distortion, and the experimental data of Fig. 2(a) could be
predicted by Eqs. (1)-(7) with the introduction of $\gamma_{\rm DDI}$ and
$\delta_{\rm DDI}$. To test this hypothesis, we calculated the theoretical
predictions as shown in Fig. 2(b) by solving Eqs. (1)-(7), which included the
effect of DDI. We used the experimentally-determined values of $\Omega_{p0}$,
$\alpha$, $\Omega_{c}$, and $\gamma_{0}$ in the calculation. Owing to the DDI
effect, i.e., the term of $\gamma_{\rm DDI}$ or $A\rho_{22}$ in Eq. (1), we
did observe a lower transmission with a higher value of $\Omega_{p0}$ in the
theoretical prediction. In each case of $\Omega_{p0}$, the theoretical
prediction gave a very similar delay time ($\sim$$1.8~{}\mu$s), which resulted
in a higher peak transmission than the experimental value. The theoretical
predictions did not agree with the experimental data.

###  II.4 Phenomenon of the DDI-induced attenuation increasing with time

Figure 3: Experimental observation of the accumulation effect, in which the
DDI-induced attenuation increased with time. In each panel, the black line
scaled down by a factor of 0.15 is the input pulse, the red line represents
the output pulse, and the blue line is the best fit of the tail part of the
red line. The best fit is an exponential-decay function. The values of
$\Omega_{p0}$ were (a) 0.08$\Gamma$, (b) 0.1$\Gamma$, (c) 0.15$\Gamma$, and
(d) 0.2$\Gamma$, respectively. In the measurements, $\alpha$ (optical depth) =
70, $\Omega_{c}=$ 1.0$\Gamma$, and $\gamma_{0}=$ 1.1$\times 10^{-2}$$\Gamma$,
which were determined experimentally OurPRA2019 .

To further study the phenomenon, we applied the long square probe pulse with
the input Rabi frequency $\Omega_{p0}$ of 0.08$\Gamma$, 0.10$\Gamma$,
0.15$\Gamma$, or 0.2$\Gamma$ under the constant presence of the coupling
field, as shown in Fig 3. The experimental parameters of $\alpha$ (OD) = 70,
$\Omega_{c}$ = $1.0$$\Gamma$, $\Delta$ = 0, and $\delta$ = 0 were kept the
same as in the measurements with the Gaussian input, but $\gamma_{0}$ was
changed to $1.1\times 10^{-2}$$\Gamma$ in the measurement.

Regarding the behavior of the output probe field in Fig. 3, we observed that
the transmission decreased (or the attenuation increases) with time, after it
reached the peak value in each of the measurements. The peak value of the
output probe transmission decreased with $\Omega_{p0}$. This was expected from
the DDI effect, i.e., the term of $\gamma_{\rm DDI}$ or $A\rho_{22}$ in Eq.
(1), where $\rho_{22}\approx\Omega_{p}^{2}/\Omega_{c}^{2}$. However, neither
the theoretical model nor Eqs. (1)-(7) described in Sec. II.2 were not able to
explain the phenomenon of the transmission decay or the increasing attenuation
over time. We fitted the decayed part of the experimental data with an
exponential-decay function, and determined the decay time constant and the
steady-state attenuation coefficient. Blue lines shown in Fig. 3 are the best
fits.

Figure 4: Experimental observation of a larger $\Omega_{p0}$ resulting in a
faster accumulation of DDI-induced attenuation. (a) Decay time constant,
$\tau_{sq}$, as a function of $\Omega_{p0}^{2}$. The values of $\tau_{sq}$
were determined by the best fits shown in Fig. 3. (b) The steady-state
attenuation coefficient, $\beta_{ss}$, as a function of $\Omega_{p0}^{2}$.
$\beta_{ss}=-\ln{(T_{ss})}$, where $T_{ss}$ is the steady-state transmission.
The values of $T_{ss}$ were determined by the vertical offsets of the blue
lines shown in Fig. 3.

Based on the best fits in Fig. 3, the decay time constant, $\tau_{sq}$, and
steady-state attenuation coefficient, $\beta_{ss}$, as functions of
$\Omega_{p0}^{2}$ are shown in Figs. 4(a) and 4(b), respectively. The value of
$\beta_{ss}$ was given by the absolute value of the logarithm of the
transmission at the steady state. Figures 4(a) and 4(b) clearly show that a
larger value of $\Omega_{p0}^{2}$ resulted in a faster decay and a larger
steady-state attenuation.

A higher Rydberg polariton density leads to a larger DDI-induced attenuation.
Thus, the data in Figs. 3 and 4 inferred that the Rydberg polariton density
increased with time. One can guess that some of the Rydberg population in
$|2\rangle$ was transferred to other states (denoted as $|4\rangle$), and that
the atoms in $|4\rangle$ did not interact with the coupling field but were
able to have a DDI with the atoms in $|2\rangle$. Due to the existence of the
DDI, $|4\rangle$ had to represent some nearby Rydberg states of $|2\rangle$.
The population in $|4\rangle$, which was not driven by the coupling field,
should have accumulated. As more population accumulated in $|4\rangle$, the
DDI-induced attenuation of $|2\rangle$ became large. Thus, the probe output
transmission decreased with time, exhibiting the accumulation phenomenon of
the DDI effect.

In Fig. 1(a), we introduce $|4\rangle$ to represent all possible nearby
Rydberg states of $|2\rangle$. The spontaneous or collisional decay process
could cause the transfer of the population from $|2\rangle$ to $|4\rangle$,
because no additional field was applied in the experiment. Since $|2\rangle$
was driven by the coupling field, it was called the bright Rydberg state. On
the other hand, $|4\rangle$ did not interact with any applied field and was
called the dark Rydberg state.

###  II.5 Evidence of dark Rydberg states

To verify the hypothesis based on the dark Rydberg state $|4\rangle$, we
measured the existing time of the DDI effect, after nearly all the atoms in
the bright Rydberg state $|2\rangle$ had been de-excited. The basic idea of
this measurement was as follows. Since the atoms in $|4\rangle$ did not
interact with any applied field, they should decay by themselves and have a
decay rate of $\Gamma_{4}$. Furthermore, $|4\rangle$ represented a number of
Rydberg states, so $\Gamma_{4}^{-1}$ must be close to the Rydberg-state
lifetime. Consequently, after the atoms in $|2\rangle$ had been removed, the
DDI effect should still exist in the system, and gradually decay with the time
constant of $\Gamma_{4}^{-1}$. Then, we explored the DDI-induced attenuation
as a function of time in the system by employing a very weak Gaussian probe
pulse and the coupling field. The probe pulse was weak enough to cause a
negligible DDI effect by itself.

The sequence of the input probe field is depicted in the inset of Fig. 5. A
70-$\mu$s square probe pulse of $\Omega_{p0}$ = 0.1$\Gamma$ was first
employed. The pulse duration was sufficiently long compared with the lifetime
of $|2\rangle$, which was approximately $20~{}\mu$s at room temperature. The
OD was 70 and the coupling field of $\Omega_{c}=$ 1.0$\Gamma$ was continuously
kept on during the measurement. After the square probe pulse was switched off,
the coupling field quickly de-excited the remaining population in $|2\rangle$.
The de-excitation time was estimated to be about 27 ns. We waited for a
certain time $\Delta t$ after the square pulse was turned off, and then
applied a weak Gaussian probe pulse to measure the DDI-induced attenuation,
$\beta_{G}$, as a function of $\Delta t$. The value of $\beta_{G}$ was equal
to the absolute value of the logarithm of the peak transmission of the output
Gaussian pulse. The input Gaussian pulse had the peak $\Omega_{p0}$ of
0.05$\Gamma$ and the $e^{-1}$ full width of 10 $\mu$s. Compared with the
70-$\mu$s square probe pulse, the weaker and much shorter Gaussian probe pulse
induced a negligible DDI effect.

Figure 5: Experimental observation of the lifetime of the dark Rydberg states
by the measurement of DDI-induced attenuation $\beta_{G}$ as a function of
$\Delta t$. Inset: the time sequence of the measurement. The coupling field
was continuously on during the measurement. Circles are the experimental data
and the black line is the best fit of an exponential-decay function. The decay
time constant of the best fit was 31 $\mu$s, which is similar to the lifetimes
of dark Rydberg states with the principal quantum number, $n$, of around 32.

We observed the exponential-decay behavior of $\beta_{G}$ versus $\Delta t$ as
shown in Fig. 5 in which the circles are the experimental data and the black
line is the best fit of an exponential-decay function. The decay time constant
of the best fit was 31 $\mu$s, which is about the lifetime of a Rydberg state
with a principal quantum number between 32 and 38. The atoms in $|2\rangle$
were quickly de-excited by the coupling field, after the square probe pulse
was switched off. If the atoms in the dark Rydberg state $|4\rangle$ could not
exist in the system, the value of $\beta_{G}$ would quickly drop to its
steady-state value due to the absence of the atoms in the bright Rydberg state
$|2\rangle$. The slow decay of $\beta_{G}$ demonstrated the existence of the
atoms in $|4\rangle$, which did not interact with the coupling field, but gave
the DDI effect and decayed slowly.

##  III Theory of the accumulation effect and experimental verification

###  III.1 Theoretical model with the accumulation effect

In the previous section, we discussed the necessity of the consideration of
dark Rydberg states in the system. To simulate the accumulative DDI effect
observed in the experiment, we improved the theoretical model described in
Subsec. II.2 by including the dark Rydberg state $|4\rangle$ and the decay
process from $|2\rangle$ to $|4\rangle$. The DDI effect between one atom in
$|2\rangle$ and one atom in $|4\rangle$ was also taken into account. Thus, the
OBEs can be given by:

$\displaystyle\frac{\partial}{\partial t}\rho_{21}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\Omega_{c}\rho_{31}+i(A\rho_{22}+B^{\prime}\rho_{44}+\delta)\rho_{21}$
(13) $\displaystyle-$
$\displaystyle\left(A\rho_{22}+B\rho_{44}+\gamma_{0}+\frac{\Gamma_{24}}{2}\right)\rho_{21},$
$\displaystyle\frac{\partial}{\partial t}\rho_{31}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\Omega_{p}(\rho_{11}-\rho_{33})+\frac{i}{2}\Omega_{c}\rho_{21}+i\Delta_{p}\rho_{31}$
(14) $\displaystyle-$ $\displaystyle\frac{\Gamma}{2}\rho_{31},$
$\displaystyle\frac{\partial}{\partial t}\rho_{32}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\Omega_{p}\rho_{21}^{*}+\frac{i}{2}\Omega_{c}(\rho_{22}-\rho_{33})$
(15) $\displaystyle-$ $\displaystyle
i\Delta_{c}\rho_{32}-\frac{\Gamma}{2}\rho_{32},$
$\displaystyle\frac{\partial}{\partial t}\rho_{22}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\Omega_{c}\rho_{32}-\frac{i}{2}\Omega_{c}\rho_{32}^{*}-\Gamma_{24}\rho_{22},$
(16) $\displaystyle\frac{\partial}{\partial t}\rho_{33}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\Omega_{p}^{*}\rho_{31}+\frac{i}{2}\Omega_{p}\rho_{31}^{*}-\frac{i}{2}\Omega_{c}\rho_{32}$
(17) $\displaystyle+$
$\displaystyle\frac{i}{2}\Omega_{c}\rho_{32}^{*}-\Gamma\rho_{33},$
$\displaystyle\frac{\partial}{\partial t}\rho_{44}$ $\displaystyle=$
$\displaystyle\Gamma_{24}\rho_{22}-\Gamma_{4}\rho_{44},$ (18)
$\displaystyle\frac{\partial}{\partial t}\rho_{11}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\Omega_{p}^{*}\rho_{31}-\frac{i}{2}\Omega_{p}\rho_{31}^{*}+\Gamma\rho_{33},$
(19)

where the coefficient $A$ represents the DDI-induced decoherence rate or
frequency shift per unit $\rho_{22}$ with a value given by Eq. (10), the
coefficient $B$ (or $B^{\prime}$) is similar to the coefficient $A$ and
represents the DDI-induced decoherence rate (or frequency shift) per unit
$\rho_{44}$, $\Gamma_{24}$ is the decay rate from $|2\rangle$ to $|4\rangle$,
and $\Gamma_{4}$ is the decay rate of $|4\rangle$.

The terms $A\rho_{22}+B^{\prime}\rho_{44}$ and $A\rho_{22}+B\rho_{44}$ in Eq.
(13) represent $\delta_{\rm DDI}$ and $\gamma_{\rm DDI}$, respectively. The
values of $\gamma_{\rm DDI}$ and $\delta_{\rm DDI}$ varied with space and time
due to the populations of $\rho_{22}(z,t)$ and $\rho_{44}(z,t)$ during the
propagation of the probe light. The decay rate $\Gamma_{24}$ was parametrized
as:

$\Gamma_{24}\equiv C(\rho_{22}+\rho_{44})+D,$ (20)

where the coefficient $C$ is the two-body decay rate per unit $\rho_{22}$ or
$\rho_{44}$, and the coefficient $D$ is the one-body decay rate. The atomic
density, $n_{a}$, of the system is a part of the coefficient $C$. In reality,
the values of $C(\rho_{22}+\rho_{44})$ and $D$ can be the collisional decay
rate between two Rydberg atoms and the spontaneous decay rate, respectively,
and the two decay processes transfer the population from $|2\rangle$ to
$|4\rangle$.

In Eq. (15), the DDI effect, i.e., $\gamma_{\rm DDI}$ and $\delta_{\rm DDI}$,
and the decay rates, i.e., $\Gamma_{24}$, and $\Gamma_{2}$, were ignored due
to the experimental condition of
$A\rho_{22},B\rho_{44},\Gamma_{24},\Gamma_{2}\ll\Gamma$. Please note that the
frequency shift induced by the population in each of the dark Rydberg states
might be positive or negative, and thus the net frequency shift resulted in
$|B^{\prime}|\leq B$. Furthermore, the term of
$i(A\rho_{22}+B^{\prime}\rho_{44}+\delta)\rho_{21}$ in Eq. (13) played a
negligible role in the output probe transmission, since $\delta=0$ and
$4|A\rho_{22}+B\rho_{44}|^{2}\Gamma^{2}\ll\Omega_{c}^{4}$ in all the cases of
this work.

The value of $A$ given by Eq. (10) was derived from Ref. OurOE2020  and
experimentally verified in Ref. OurCommunPhys2021 . We first measured the
coefficients $C$ and $D$ as described in Subsec. III.3. Using the known values
of $A$, $C$, and $D$, we then determined $B$ using the experimental data of
Fig. 3 as illustrated in Subsec. III.4. Finally, we compared the experimental
data of the slow light shown in Fig. 2 with the theoretical predictions
calculated using the experimentally determined values of $A$, $B$, $C$, and
$D$ in Subsec. III.4. The comparison was used to test the validity of the
theoretical model introduced here.

###  III.2 Experimental setup for measuring the decay rate of the bright
Rydberg state

Figure 6:  (a) Scheme of the two-photon transition, which moved the population
from $|1\rangle$ to $|2\rangle$. The energy levels here were the same as those
in Fig. 1. Please note that $|1\rangle\rightarrow|3\rangle$ is a cycling
transition and, thus, population of any undesired transition to $|3\rangle$
quickly decayed back only to $|1\rangle$. $\Omega_{a}$ and $\Omega_{b}$ denote
the Rabi frequencies of the two square pulses of the laser fields. We kept the
frequencies of $\Omega_{a}$ and $\Omega_{b}$ to the two-photon resonance, and
applied a one-photon detuning ($\Delta$) of 4$\Gamma$ to $\Omega_{b}$ or that
of $-4$$\Gamma$ to $\Omega_{a}$. $\Gamma$ and $\Gamma_{4}$ are the spontaneous
decay rates of $|3\rangle$ and $|4\rangle$, respectively. $\Gamma_{24}$ is the
decay rate from $|2\rangle$ to $|4\rangle$. (b) Optical paths of $\Omega_{a}$,
$\Omega_{b}$, and a clean pulse in the experiment. $\Omega_{b}$ was the same
as that of the coupling field, but had a different frequency. The clean pulse
was employed to wipe out the population in $|1\rangle$.

We designed an experiment to determine the coefficients $C$ and $D$ in Eq.
(20). The decay rate, $\Gamma_{24}$, of the atoms in $|2\rangle$ was measured
against the atomic density of $|2\rangle$, $n_{R}$. The procedure of the
measurement of $\Gamma_{24}$ was as follows. We first prepared a given number
of atoms in the ground state $|1\rangle$ and excited them to $|2\rangle$ with
a two-photon transition (TPT) pulse. The TPT had a large one-photon detuning
as shown in Fig. 6(a). After the TPT pulse, no light field was applied, and
the atoms in $|2\rangle$ decayed. Then, we waited for a certain time $\Delta
t$, and also depleted the residual atoms in $|1\rangle$. At the end of $\Delta
t$, the remaining atoms in $|2\rangle$ were brought back to $|1\rangle$ with
another TPT pulse. Finally, after the second TPT pulse we determined the
number of the returned atoms in $|1\rangle$ or, equivalently, measured the
absorption coefficient, $\Delta\beta$, of the weak probe field. The result of
$\Delta\beta$ was proportional to the remaining atoms in $|2\rangle$ after the
given $\Delta t$. By varying $\Delta t$ and measuring $\Delta\beta$, we
obtained the data points of $\Delta\beta$ as a function of $\Delta t$, and
fitted the data with an exponential function. The best fit gave the value of
$\Gamma_{24}$.

We employed the TPT scheme rather than the EIT scheme to place the population
in $|2\rangle$, because the TPT scheme is able to move a large portion of the
atoms in $|1\rangle$ to $|2\rangle$. In the measurement of $\Gamma_{24}$, we
added the pulses of $\Omega_{a}$ and $\Omega_{b}$, which formed the TPT pulse,
and the clean pulse as shown in Fig. 6(b). The coupling field ($\Omega_{c}$)
in the EIT experiment was utilized as $\Omega_{b}$, but we changed its
frequency to set the one-photon detuning, $\Delta$, to 4.0 $\Gamma$ for the
TPT as depicted in Fig. 6(a). The frequency of $\Omega_{a}$ was tuned such
that the TPT satisfied the two-photon resonance. A weak and resonant probe
pulse, denoted as $\Omega_{p^{\prime}}$, was used in the measurement. The
pulse of $\Omega_{p^{\prime}}$ had the same optical path as the field of
$\Omega_{p}$ shown in Fig. 1(b). The absorption coefficient of
$\Omega_{p^{\prime}}$ indicated the number of atoms in $|1\rangle$.

The polarization, propagation direction, and beam size of $\Omega_{b}$ (or
$\Omega_{p^{\prime}}$) were the same as those of $\Omega_{c}$ (or
$\Omega_{p}$) in the EIT experiment. The field of $\Omega_{a}$ was generated
by a diode laser, which was injection-locked by the same laser that generated
the field of $\Omega_{p}$ in the EIT experiment. Hence, the two-photon
frequency of $\Omega_{a}$ and $\Omega_{b}$ was stable. The pulse of
$\Omega_{a}$ was sent to the atom cloud by a PMF after passing through an AOM.
The clean pulse was used to remove the population in $|1\rangle$. After
passing through the AOM, the clean pulse had a frequency that was resonant to
the transition of $|5S_{1/2},F=2\rangle\rightarrow|5P_{3/2},F=2\rangle$. The
polarization of $\Omega_{a}$ and the clean pulse were $\sigma_{+}$ and
$\sigma_{-}$, respectively. There was a separation angle of $0.36^{\circ}$ (or
$0.40^{\circ}$) between the propagation directions of $\Omega_{a}$ (or the
clean pulse) and $\Omega_{b}$. Lens $L_{a}$ (or $L_{c}$) was used together
with $L_{p}$ to make $\Omega_{a}$ (or the clean pulse) a collimated beam with
the $e^{-1}$ full width of 3.6 mm (or 4.5 mm). The two beam sizes of
$\Omega_{a}$ and the clean pulse were sufficiently large to cover the entire
atom cloud.

Figure 7: Experimental demonstration of the two-photon transition (TPT) shown
in Fig. 6(a). In the main figure, we plot the remaining population in state
$|1\rangle$ after the TPT pulse as a function of the TPT pulse width,
$\tau_{p}$. In the inset, we show the timing sequence of the pulses of
$\Omega_{a}$ and $\Omega_{b}$ (to drive the TPT) and $\Omega_{p^{\prime}}$ (to
measure the population in $|1\rangle$), as well as their pulse shapes. We set
$\Delta=$ 4.0$\Gamma$, $\alpha$ (optical depth) = 0.5,
$\Omega_{a}=\Omega_{b}=$ 1.2$\Gamma$, and $\Omega_{p^{\prime}0}=$ 0.08$\Gamma$
in the measurement. Circles are the experimental data, which have similar
error bars. Only the last data point shows the error bar, which represents the
typical value. The black line is the theoretical prediction of a simple three-
level cascade system under the two-photon resonance. The prediction was
calculated with the decoherence rate of 6.5$\times$$10^{-2}$$\Gamma$, and the
above values of $\Delta$, $\Omega_{a}$, and $\Omega_{b}$.

To study the efficiency of the TPT pulse, i.e., the simultaneous pulses of
$\Omega_{a}$ and $\Omega_{b}$, we varied the TPT pulse duration, $\tau_{p}$,
and measured the population or atom number right after the TPT pulse as shown
in Fig. 7. The population in $|1\rangle$ was determined by the absorption
coefficient of $\Omega_{p^{\prime}}$. The timing sequence of $\Omega_{a}$,
$\Omega_{b}$, $\Omega_{p^{\prime}}$, and their pulse shapes are depicted in
the inset. In the main plot, the circles are the experimental data, and the
line is the theoretical prediction of a simple three-level cascade system
under the two-photon resonance, i.e., Eqs. (1)-(6) are utilized with
$\delta=0$, $\delta_{\rm DDI}=0$, $\gamma_{\rm DDI}=0$,
$\Omega_{p}\rightarrow\Omega_{a}$, and $\Omega_{c}\rightarrow\Omega_{b}$. To
calculate the prediction, we set $\Delta$, $\Omega_{a}$, and $\Omega_{b}$ to
the experimental values, and adjusted $\gamma_{0}$ to match the experimental
data. The fair agreement between the data and the prediction verified the
effect of the TPT pulse on the population. According to the study shown in
Fig. 7, we decided to use a 1.8 $\mu$s TPT pulse with $\Delta=$ 4.0$\Gamma$
and $\Omega_{a}=\Omega_{b}=$ 1.2$\Gamma$ in the measurement of $\Gamma_{24}$
because the transition probability after $\tau_{p}=$ 1.8 $\mu$s was
insensitive to the pulse width. Please note that the reduced population in
$|1\rangle$ was used to determine the population in $|2\rangle$, but during
the TPT pulse the excitation to the intermediate state $|3\rangle$ was not
negligible. Nevertheless, since the population in $|3\rangle$ decayed only to
$|1\rangle$ in a rather short time right after the TPT pulse, the excitation
of the population to $|3\rangle$ did not affect the determination of the
population in $|2\rangle$.

###  III.3 Determination of the decay rate from bright to dark Rydberg states

Figure 8: Determination of the decay rate of the bright Rydberg state
$|2\rangle$. The inset shows the time sequence of the measurement. The purple,
green, and red areas represent the two-photon transition (TPT) pulses, the
clean pulse, and the probe pulse, respectively. In the main plot, circles are
the representative data of the difference between the absorption coefficients
with and without the second TPT pulse, $\Delta\beta$, as a function of $\Delta
t$. The black line is the best fit of an exponential-decay function, which
determined the decay rate. In the measurement of these data points, $n_{R}=$
0.002 $\mu$m-3, and the values of $\Delta$, $\Omega_{a}$, $\Omega_{b}$, and
$\Omega_{p^{\prime}0}$ were the same as those in shown Fig. 7.

To obtain the decay rate, $\Gamma_{24}$, of the atoms in $|2\rangle$ as a
function of the density, $n_{R}$, of the atoms in $|2\rangle$, we performed
the measurement described in the first paragraph of Subsec. III.2. The time
sequence of the measurement is depicted in the inset of Fig. 8. We kept the
time difference between the peak of the clean pulse and the rising edge of the
second TPT pulse to about 400 ns, and that between the falling edge of the
second TPT pulse and the rising edge of the probe pulse to about 1 $\mu$s.
Each of the two TPT pulses had a duration of 1.8 $\mu$s. The TPT pulse was
able to move a fixed fraction (about half) of the atoms from the initial state
to the final state, i.e., $|1\rangle$ $\rightarrow$ $|2\rangle$ or $|2\rangle$
$\rightarrow$ $|1\rangle$. The residual atoms either were removed from the
system or did not affect the measurement of the decay rate. We varied $\Delta
t$ and measured the absorption coefficient, $\Delta\beta$, of the weak probe
pulse, $\Omega_{p^{\prime}}$. The representative data of the difference
between the values of $\Delta\beta$ with and without the second TPT pulse as a
function of $\Delta t$ are plotted in Fig. 8. Since the atoms in $|1\rangle$
right before the second TPT were all depleted by the clean pulse,
$\Delta\beta$ corresponded to the remaining atoms in $|2\rangle$ after a decay
time of $\Delta t$.

The data points in Fig. 8 were all taken at $\Delta\alpha=$ 2.9, where
$\Delta\alpha$ is defined as the difference of the values of OD before and
after the first TPT pulse. The two values of OD were about 5.2 and 2.3 in this
case. We were able to estimate the density of the atoms in $|2\rangle$ after
the first TPT pulse from $\Delta\alpha$ using the following relation:

$\Delta\alpha=\sigma_{0}n_{1}L=\sigma_{0}n_{R}L,$ (21)

where $\sigma_{0}$ is the absorption cross section of the resonant probe
transition from $|5S_{1/2},F=2,m_{F}=2\rangle$ to
$|5P_{3/2},F=3,m_{F}=3\rangle$, and $n_{1}$ is the density of the atoms in
$|1\rangle$ that were moved to $|2\rangle$ by the first TPT pulse.
Consequently, $n_{1}$ was equal to the initial Rydberg-atom density $n_{R}$,
which participated in the decay process of $|2\rangle$. As shown in Fig. 8, we
fitted the experimental data taken at a given $n_{R}$ with an exponential-
decay function. The decay time constant of the best fit gave the value of
$\Gamma_{24}$ of the given $n_{R}$.

Figure 9: The decay rate $\Gamma_{24}$ as a function of the density, $n_{R}$,
of the atoms in $|2\rangle$. Based on Eq. (21), the value of $n_{R}$ was
determined by $\Delta\alpha$, which was the difference of the values of OD
between before and after the first TPT. Circles are the experimental data.
Each data point was determined by the result of a series of measurements
similar to those in Fig. 8. The black line is the best fit of a straight line.

The circles in Fig. 9 are the experimental data of $\Gamma_{24}$ as a function
of $n_{R}$. Each circle represents the result of a series of measurements
similar to those shown in Fig. 8. We fitted the circles in Fig. 9 with the
fitting function of a straight line. The interception of the best fit
determined the coefficient $D$ defined in Eq. (20), which was
2.0$\times$$10^{-3}$$\Gamma$. The determined value of $D$ was comparable to
the spontaneous decay rate of $32D_{5/2}$ at room temperature, which is
$1.3$$\times$$10^{-3}$$\Gamma$ Rb87Data ; RydbergStateLifeTime . Furthermore,
the slope, $k$, of the best fit determined the coefficient $C$ defined in Eq.
(20) in the following way: Since $k\,n_{R}=C\rho_{22}$ and
$n_{R}=\rho_{22}n_{a}=\rho_{22}\alpha/(\sigma_{0}L)$, the coefficient $C$ is
given by:

$C=k\alpha/(\sigma_{0}L),$ (22)

where $\alpha$ is the OD used in the EIT experiment. During the measurements
of the data shown in Figs. 2(a) and 3(a)-3(d), the optical depth ($\alpha$)
was 70, we used the above equation to estimate that the value of $C$ was
$9.1$$\times$$~{}10^{-2}$$\Gamma$ at $\alpha=$ 70\. Possible processes of the
population transfer from the bright to dark Rydberg states are discussed in
Sec. IV.

###  III.4 Comparison between theoretical predictions and experimental results

Figure 10: Comparison between the experimental data of the long square probe
pulses and the theoretical predictions. Black lines represent the input probe
pulses and are scaled down by a factor of 0.15. Red lines represent the output
probe pulses under the constant presence of the coupling field. The
experimental data are the same as those shown in Fig. 3, where the input Rabi
frequencies $\Omega_{p0}$ are (a) 0.08$\Gamma$, (b) 0.1$\Gamma$, (c)
0.15$\Gamma$, and (d) 0.2$\Gamma$. Blue lines are the predictions calculated
using Eqs. (7) and (13)-(20). In the calculation, we set $\alpha$ (optical
depth) = 70, $\Omega_{c}=$ 1.0$\Gamma$, and $\gamma_{0}=$
1.1$\times$$10^{-2}$$\Gamma$, which were determined experimentally, the
coefficients $C=$ $9.1$$\times$$10^{-2}$$\Gamma$ and $D=$
$1.3$$\times$$10^{-3}$$\Gamma$, which were deduced from the result of Fig. 9,
and $A=$ 0.76$\Gamma$ given by Eq. (10). $B$ was adjusted to make the
predictions fit the data and its optimum value is 7.7$~{}\Gamma$.

After determining the values of $C$ and $D$ in the decay rate $\Gamma_{24}$
used in Eqs. (13), (16), and (18), we next utilized the experimental data of
Fig. 3 to determine the value of coefficient $B$ used in Eq. (13). Since the
measured value of $\Gamma_{24}$ was the total decay rate of $|2\rangle$ but
not merely the decay rate from $|2\rangle$ to $|4\rangle$, the determined
value of $B$ might account for the discrepancy. Figure 10 shows the comparison
between the experimental data of the long square probe pulses shown in Figs.
3(a)-3(d) and the theoretical predictions calculated with the modified model
described in Sec. III.1. In the calculation, we used the experimentally
determined values of the optical depth, the coupling Rabi frequency, and
intrinsic decoherence rate $\gamma_{0}$ in the system OurPRA2019 . We set $A$
(the DDI-induced decoherence rate per unit $\rho_{22}$) to the value given by
Eq. (10), and used the values of $C$ and $D$ derived from the result of Fig.
9. The only adjustable parameter in the calculation of the predictions was $B$
(the DDI-induced decoherence rate per unit $\rho_{44}$). Note that we set
$B^{\prime}$ = $B$ in the term of
$i(A\rho_{22}+B^{\prime}\rho_{44}+\delta)\rho_{21}$ in Eq. (13), but this term
played a negligible role in the calculation result of the output probe
transmission. A good agreement between the theoretical predictions and
experimental data was obtained at the coefficient $B=$ 7.7$~{}\Gamma$.

A single optimum value of $B$ was able to explain all the experimental data
taken at different input probe Rabi frequencies. This demonstrated the
theoretical model presented by Eqs. (13)-(20)was qualitatively valid.
Furthermore, we compared the experimental data of the Gaussian input probe
pulses as shown in Fig. 2 with the theoretical predictions. The values of the
parameters used in the calculation of the predictions were the same as those
in Fig. 10 except for the value of $\gamma_{0}$, which had a day-to-day
fluctuation of 1.5$\times$$10^{-3}$$\Gamma$. Figures 11(a)-11(c) show that the
experimental data were all in good agreement with the theoretical predictions,
manifesting that the theoretical model was also quantitatively successful.

Figure 11: Comparison between the experimental data of the Gaussian input
probe pulses and the theoretical predictions. Black lines represent the input
probe pulses and are scaled down by a factor of 0.4. Red, blue, and green
lines represent the output probe pulses under the constant presence of the
coupling field. The experimental data are the same as those shown in Fig. 2,
where the input Rabi frequencies $\Omega_{p0}$ are (a) 0.05$\Gamma$, (b)
0.1$\Gamma$, and (c) 0.2$\Gamma$. Gray lines are the theoretical predictions
calculated with Eqs. (7) and (13)-(20). All the parameters used in the
calculation here are the same as those used in Fig. 10, except for
$\gamma_{0}=$ 9.0$\times$$10^{-3}$$\Gamma$.

##  IV Discussion

Experimental observations of population transfer from one Rydberg state to
another, i.e., from a bright Rydberg state to the dark Rydberg state, have
been reported in several articles Martin_2004 ; Porto_2016 ; Porto_2017 ;
Hond_2020 ; Gorshkov_2020 . The underlying mechanisms of such transfers can be
transitions driven by a microwave field Martin_2004 , the spontaneous decay
induced by black-body radiation and vacuum fluctuations Porto_2016 ;
Porto_2017 ; Hond_2020 , and the DDI-induced antiblockade excitation and
state-exchange collision assisted by radiation trapping Gorshkov_2020 . The
mechanism investigated in this study was DDI-induced state-exchange collisions
without radiation trapping. We discuss these articles and compare their
results and experimental conditions with our observations and experimental
condition in the following paragraphs.

The authors in Ref. Martin_2004  measured the linewidth of the transition from
$|45D_{5/2}\rangle$ to $|46D_{5/2}\rangle$, while the entire Rydberg
population was initially prepared in $|45D_{5/2}\rangle$. As they moved half
of the Rydberg population to $|46P_{3/2}\rangle$ by applying a microwave
before the linewidth measurement, the measured linewidth was broadened. The
authors explained that the DDI between a $|45D_{5/2}\rangle$ atom and a
$|46P_{3/2}\rangle$ atom was stronger than that between two
$|45D_{5/2}\rangle$ atoms, resulting in the linewidth broadening. The
population transfer from the bright ($|45D_{5/2}\rangle$) to dark
($|46P_{3/2}\rangle$) Rydberg state was intentionally driven by a microwave
field, and no accumulative DDI effect was reported in Ref. Martin_2004 . Since
we did not apply any additional field to move the population from the bright
to dark Rydberg states, the physical mechanism in this reference is unable to
explain the accumulative DDI effect observed in our study.

On the other hand, the population in the dark Rydberg state could be produced
via the spontaneous decay from the bright Rydberg state induced by the black-
body radiation and vacuum fluctuation. In Ref. Porto_2016 , the authors
measured the spectrum of the two-photon transition from a ground state to the
Rydberg state $|18S_{1/2}\rangle$ with the 87Rb atoms trapped in a 3D optical
lattice. They observed that the measured linewidth was about two orders of
magnitude larger than the expected linewidth due to the DDI between two
$|18S_{1/2}\rangle$ atoms. Such a large linewidth was explained by the DDI
between an atom in $|18S_{1/2}\rangle$ and another in a nearby Rydberg state,
$|17P\rangle$ or $|18P\rangle$. The existence of the population in
$|17P\rangle$ and $|18P\rangle$ due to the spontaneous decay from
$|18S_{1/2}\rangle$ was further verified by Ref. Porto_2017 . In Ref.
Hond_2020 , the authors reported a similar phenomenon with the bright Rydberg
state of $|28D_{5/2}\rangle$ and the dark Rydberg states of
$|26F_{7/2}\rangle$, $|27F_{7/2}\rangle$, $|29P_{3/2}\rangle$, and
$|30P_{3/2}\rangle$. Nevertheless, the spontaneous decay was an one-body
process and did not depend on the Rydberg-atom density. As shown in Fig. 9,
the observed decay rate linearly depended on the atomic density of the bright
Rydberg state. Furthermore, it was also larger than the spontaneous decay rate
of $|32D_{5/2}\rangle$ used in our experiment, which was 2$\pi$$\times$7.9 kHz
or 1.3$\times 10^{-3}$$~{}\Gamma$. Thus, the decay from the bright to dark
Rydberg states observed in this study is unable to be explained by the
spontaneous decay.

Population transfer from the bright to dark Rydberg states can also be induced
by the direct antiblockade excitation antiblockade_PRL2007 ;
antiblockade_PRL2010  and state-changing Rydberg collisions
state_changing_collision_PRA2015 ; PRL_stateMixing2008 ; PRR_stateMixing2020 ;
PRL_stateMixing2004 . In Ref. Gorshkov_2020 , the authors drove the Rydberg-
EIT transition from a ground state to $|111S_{1/2}\rangle$ (the bright Rydberg
state) and detected the ions coming from a number of dark Rydberg states
(nearby states other than $|111S_{1/2}\rangle$) after an ionization pulse.
Compared with the experiment in our work, the experiment in the reference was
carried out under a high atomic density of 5$\times$1012 cm-3 and in the
strong-interaction regime of $(r_{B}/r_{a})^{3}>1$. Under such an atomic
density, the authors in Ref. Gorshkov_2020  explained that the effect of
radiation trapping was prominent, producing more atoms in the bright Rydberg
state. The atoms produced by the radiation trapping had all possible angular
momentum angles. Then, the strong-interaction regime enabled the DDI-induced
antiblockade excitation and state-changing collisions to generate dark Rydberg
atoms from the bright Rydberg atoms. These dark Rydberg atoms produced ions
after an ionization pulse, while the ions were clearly not able to come from
the bright Rydberg state $|111S_{1/2}\rangle$. The dark Rydberg atom also
resulted in the unexpected reduction of the output probe photon number. The
authors stated that the reduction is unable to be explained by the prediction
of the blockade effect with only the bright Rydberg state but no dark Rydberg
states.

In our experiment, the population transfer from the bright to dark Rydberg
states should be caused by the DDI-induced state-changing collisions. This
experiment was carried out under a much lower atomic density of 4$\times$1010
cm-3 and in the weak-interaction regime, i.e., $(r_{B}/r_{a})^{3}<$ 0.06, as
compared with the experiment in Ref. Gorshkov_2020 . Under such an atomic
density, the radiation trapping effect was not significant. Nevertheless,
according to the input power and propagation delay time of the probe pulse,
there were many bright Rydberg atoms in the system. In the weak-interaction
regime, it took these bright Rydberg atoms a considerable time to generate
dark Rydberg atoms via the state-changing collisions. Thus, the transmission
of the output probe light in our experiment required a longer time constant to
reach the steady state than that in Ref. Gorshkov_2020 . Furthermore, the
lifetime of the dark Rydberg atoms determined in Fig. 5 was about 31 $\mu$s,
which was very long compared with that determined in Ref. Gorshkov_2020 .
Consequently, a sufficient amount of the dark Rydberg atoms were accumulated
to strongly influence the transmission of the output probe light even in the
weak-interaction regime.

##  V Conclusion

In this work, we systematically studied the transmission of a probe field
propagating through a Rydberg EIT system, in which the DDI strength was in the
weak interaction regime. As shown in Fig. 2(a), we observed the distorted
output of a long Gaussian input probe pulse. The distortion was enhanced with
a stronger input probe Rabi frequency, $\Omega_{p0}$. Such a phenomenon was
unable to be predicted by the theoretical model, which only considered the DDI
between the bright Rydberg atoms as demonstrated by Fig. 2(b). According to
the further experimental data shown in Figs. 3-5, there existed atoms in the
dark Rydberg states which were not driven by the coupling field. The dark
Rydberg atoms came from the decay of the bright Rydberg atoms. Without the
interaction with the coupling field, the dark Rydberg atoms accumulated. The
dark and bright Rydberg atoms had the DDI effect with each other. As the
number of dark Rydberg atoms increased due to the accumulation, the DDI effect
on the bright Rydberg atoms became larger. Equivalently, in the EIT system the
DDI-induced attenuation of the probe field increased with time.

To simulate the accumulation phenomenon of the DDI effect, we introduced the
coefficients $A,B,C,$ and $D$ to the modified OBEs. $A$ and $B$ described the
DDI effect between the two bright Rydberg atoms and that between one bright
and one dark Rydberg atoms, respectively. $C$ and $D$ were associated with the
two-body (state-changing collisional) decay and one-body (spontaneous) decay
processes from the bright to dark Rydberg states, respectively. We measured
the decay rate of the bright Rydberg atoms, and the representative data of the
measurement are demonstrated in Fig. 8. The values of $C$ and $D$ were
determined by the decay rate of the bright Rydberg atoms as a function of
their density as shown in Fig. 9. Furthermore, the value of $A$ was
experimentally determined in Ref. OurCommunPhys2021  and that of $B$ by the
data shown in Fig. 10. Finally, the theoretical predictions, which were
calculated using the determined values of $A,B,C,$ and $D$, were in good
agreements with the experiment data of slow light as shown in Fig. 11. In
conclusion, this study provided a better understanding of the DDI effect under
a long interaction time SLP_Lukin ; SPL_2009 ; SLP_review , and pointed out a
possible issue in the realization of the Bose-Einstein condensation of Rydberg
polaritons DSP_BEC_Fleischhauer2008 ; OurCommunPhys2021 .

## Acknowledgements

This work was supported by Grant Nos. 109-2639-M-007-001-ASP and
110-2639-M-007-001-ASP of the Ministry of Science and Technology, Taiwan.

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