# Photon echo in ring cavity: pulse area approach

Sergey A. Moiseev<EMAIL_ADDRESS>Ravil V. Urmancheev
<EMAIL_ADDRESS>Kazan Quantum Center, Kazan National Research Technical
University n.a. A.N.Tupolev-KAI, 10 K. Marx, Kazan 420111, Russia

###### Abstract

Pulse area approach has been established as a versatile analytical tool for
studying the resonant interaction between the light and the resonant atomic
ensemble. In recent years photon and spin echoes in cavity assisted schemes
become increasingly interesting. In this article we develop the photon echo
pulse area approach to describe primary and multi-pulse echo generation in the
atomic ensemble placed in the ring cavity. We show that the pulse area
approach predicts relative echo magnitudes and whether the system is operating
in a single- or a multi-pulse generation regime. We also analyze the
conditions needed for the realization of these generation regimes. This work
develops the pulse area theorem approach for analytical study of photon/spin
echoes in optical and microwave cavities and echo based protocols of quantum
memory.

##  I Introduction

Photon echo Kopvillem and Nagibarov (1963); Kurnit  _et al._ (1964) is an
optical realization of the Hahn spin echo Hahn (1950), which is a coherent
response of inhomogeneously broadened resonant atomic ensemble to the action
of two or more resonant light pulses. Since its discovery in the beginning of
the second half of the previous century it has been established as a reliable
and developing tool in nonlinear coherent spectroscopy, used to measure
transition relaxation times and quantum dynamics of different resonant media
Smallwood and Cundiff (2018); Cundiff and Mukamel (2013); Kosarev  _et al._
(2019); Beica  _et al._ (2020); Jin  _et al._ (2020); Welinski  _et al._
(2020); D’Alessandro and Sangalli (2022). It has also become the basis of the
number of photon echo quantum memory protocols Moiseev and Kröll (2001);
Tittel  _et al._ (2010); Ma  _et al._ (2021); Horvath  _et al._ (2021);
Heshami  _et al._ (2016); Chanelière  _et al._ (2018). Recent development of
modern optical and microwave integral technologies initiates study of
photon/spin echoes in optical and microwave cavities Zhong  _et al._ (2017);
Heller  _et al._ (2020); Moiseev  _et al._ (2021); Bustard  _et al._ (2022),
which are especially important for elaboration of quantum memory devices.

Description of the photon echo is based on a solution of complicated nonlinear
Maxwell-Bloch equations, which often compels to use only numerical methods
Weichselbaumer  _et al._ (2020). In optically dense media, the task is also
complicated by the presence of strong rephasing pulses also known as
$\pi$-pulses that control atomic coherence. The pulse area theorem McCall and
Hahn (1969) was proposed to partly bypass these difficulties and provide an
analytical tool to study general nonlinear properties of resonant pulse
propagation. The theorem was later developed to consider photon (spin) echoes
Hahn  _et al._ (1971); Moiseev (1987); Moiseev  _et al._ (2020), three-level
systems Eberly and Kozlov (2002) and atoms placed in a Fabry-Perot cavity
Moiseev and Urmancheev (2022).

In this work we develop the pulse area theorem to analytically study the echo
generation in a single-mode ring cavity, similar to our previous work on
Fabry-Perot cavityMoiseev and Urmancheev (2022). We obtain the general
equation describing the area of any pulse during a two-pulse echo generation.
We then solve this equation analytically for the incoming signal pulses and
numerically for the first three echo pulses. We also provide an approximate
analytical solution for the primary echo pulse. This allows us to study the
conditions for single- and multi-pulse echo generation that are in agreement
with experimental investigation Weichselbaumer  _et al._ (2020).

##  II Pulse area theorem

###  II.1 Basic equations

We consider an ensemble of N two-level atoms that is placed inside a single-
mode ring cavity with the mode central frequency $\omega_{0}$ being in
resonance with the atomic transition. The atoms occupy a length $L$ along the
optical axis $z$; $L$ is greater than the light wavelength $\lambda$ and
smaller that the cavity length $L_{c}$. We assume that the inhomogeneous
broadening of the atomic transition $\Delta_{inh}\gg\gamma,$ where
$\gamma=1/T_{2}$ is the homogeneous linewidth and $T_{2}$ is the coherence
time of a single atom, $T_{2}$ typically shorter than the lifetime of the
optical transition $T_{1}$. The electric field is described by a slowly
varying amplitude $\mathcal{E}(t)=\mathcal{E}_{0}a(t)$, where
$\mathcal{E}_{0}=\sqrt{\frac{\hbar\omega_{0}}{2\varepsilon_{0}(\varepsilon
L+L_{v})S}}$, $\varepsilon_{0}$ and $\varepsilon$ are the vacuum and the
atomic medium permittivities, $L_{v}=L_{c}-L$ and $V=SL_{c}$ denotes the mode
volume. We assume a uniform excitation of the sample, so the coupling constant
of the dipole interaction between the cavity mode and an atom $g$ is the same
for all the atoms: $g=\frac{d}{\hbar}\mathcal{E}_{0}$, where $d$ is the dipole
moment of the atomic transition.

We use the quantum Tavis-Cummings model for the interaction of $N$ two-level
atoms with the cavity mode and apply the input-output formalism of quantum
optics Walls and Milburn (2008) co couple the amplitudes of the cavity mode
$\mathcal{E}(t)$ to the amplitudes of the input $\mathcal{E}_{in}(t)$ and
output $\mathcal{E}_{out}(t)$ field modes (where
$\mathcal{E}_{in,out}(t)=\sqrt{\frac{\pi\hbar\omega_{0}}{\varepsilon_{0}S}}a_{in,out}(t)$,
$S$ is a cross-section of the light beam). In the limit of large number of
atoms Keeling (2009), these transfers to the system of semiclassical Maxwell-
Bloch equations McCall and Hahn (1969); Allen and Eberly (1975) for atoms and
the resonator field mode:

$\displaystyle\partial_{t}a$
$\displaystyle=-\frac{\kappa+\kappa_{in}}{2}a+Ng\langle
v\rangle_{\Delta}+\sqrt{\kappa}a_{in},$ (1) $\displaystyle\partial_{t}u$
$\displaystyle=-\Delta v-\gamma u,$ (2) $\displaystyle\partial_{t}v$
$\displaystyle=\Delta u-\gamma v+\Omega(t)w,$ (3) $\displaystyle\partial_{t}w$
$\displaystyle=-\Omega(t)v,$ (4) $\displaystyle a_{out}$
$\displaystyle=\sqrt{\kappa}a-a_{in},$ (5)

where $\Omega(t)=ga(t)$, $\kappa$ is a decay rate of cavity mode to the
external waveguide modes and $\kappa_{in}$ \- internal losses of the cavity;
$u,v$ and $w$ are the components of the Bloch vector, dependant on time $t$,
and detuning $\Delta$ of the atom; $\langle v\rangle_{\Delta}\equiv\int
G(\Delta)v(t,\Delta)d\Delta,$ where $G(\Delta)$ is the inhomogeneous line
shape. Equation (5) relates the cavity mode $a$ to the input and output modes
$a_{in}$, $a_{out}$ according to input-output approach Walls and Milburn
(2008).

We assume that a pulse of light comes at the moment $t_{c}=(t_{0}+t_{1})/2$,
where $t_{0},t_{1}$ are two distant time moments. There might have been
additional pulses before the time $t_{0}$ or after $t_{1}$, but there are no
other pulses in the time interval $(t_{0},t_{1})$, which is much longer that
the pulse duration $\delta t$. Now we multiply both parts of the field
equations (1) &(5) by $g$ and integrate over time $\int_{t_{0}}^{t_{1}}dt$ to
arrive to the general pulse area equation:

$\displaystyle\frac{\kappa_{S}}{2}\Theta$
$\displaystyle=\sqrt{\kappa}\Theta_{in}+Ng^{2}\int_{t_{0}}^{t_{1}}dt\int
d\Delta G(\Delta)v(t,\Delta),$ (6) $\displaystyle\Theta_{out}$
$\displaystyle=\Theta_{in}-\sqrt{\kappa}\Theta,$ (7)

where $\kappa_{S}=\kappa+\kappa_{in}$. We also considered that
$\int_{t_{0}}^{t_{1}}\partial_{t}a(t)dt=a(t_{1})-a(t_{0})=0$ when
$t_{1}-t_{0}\gg\delta t$.

The treatment of this equation is analogous to the case of photon echo area
theorem in free space Urmancheev  _et al._ (2019a); Moiseev  _et al._ (2020)
and Fabry-Perot cavity Moiseev  _et al._ (2020) obtained previously. This
allows us to obtain the general equation for the pulse area of an incoming
pulse or an arbitrary echo pulse (even in the presence of an external exciting
pulse with a pulse area $\Theta_{in}$ at the cavity entrance):

$\displaystyle\begin{split}\frac{\kappa_{S}}{2}\Theta&=\sqrt{\kappa}\Theta_{in}+\frac{\varkappa}{2}\Big{[}2v_{0}\cos^{2}\frac{\Theta}{2}+w_{0}\sin\Theta\Big{]},\\\
\Theta_{out}&=\Theta_{in}-\sqrt{\kappa}\Theta,\end{split}$ (8)

where $v_{0}$ and $w_{0}$ are resonant components of atomic polarization that
phase and lead to the emission of an echo signal at the time $t_{c}$;
$\varkappa=2Ng^{2}\pi G(0)=2Ng^{2}/\Delta_{inh}$ for the Lorentzian
inhomogeneous broadening $G(\Delta)$. Under classical treatment $\varkappa$
corresponds to atomic absorption per round trip inside the cavity:
$\varkappa=2\alpha L/t_{rt}$ ($\alpha L$ is the optical depth of the resonance
transition, $t_{rt}$ is the cavity round trip time).

###  II.2 Incoming pulses

The first incoming pulse arrives to the system in the ground state, so we
substitute $v_{0}=0,w_{0}=-1$ into Eq. (8) to get Chanelière (2014):

$\displaystyle\frac{\kappa_{S}}{2}\Theta_{1}=\sqrt{\kappa}\Theta_{in,1}-\frac{\varkappa}{2}\sin\Theta_{1},$
(9)

Figure 1: Transition pulse areas $\Theta_{1},\Theta_{out,1}$ versus the
incoming pulse area $\Theta_{in,1}$ in the impedance-matched ring cavity
($\varkappa=\kappa_{S}$) in normalized coordinates. Blue solid line shows the
pulse area inside the cavity $\Theta_{1}/\pi$ and the orange dashed line shows
the pulse area at the output of the cavity
$\Theta_{out,1}(2/\pi\sqrt{\kappa})$. The $(2/\sqrt{\kappa})$ factor conforms
the areas inside and outside of the cavity.

In the limit of weak signal pulse $\Theta_{1}\ll 1$ we have
$\Theta_{out,1}=\frac{\varkappa+\kappa_{in}-\kappa}{\kappa_{S}+\varkappa}\Theta_{in,1}$.
Analogous to electronics we can derive an impedance matching condition, when
$\Theta_{out,1}=0$ and the incoming pulse is fully absorbed:
$\xi_{im}=\frac{\kappa}{\varkappa+\kappa_{in}}=1$. Parameter $\xi_{im}$
defines the relative coupling strength of the interaction between the light
mode with atoms and free propagating modes, which plays an important role in
light-atoms dynamics. Behavior of pulse area in the impedance matching
condition was studied in Chanelière (2014) for $\kappa_{in}=0$.

Figure 1 shows the dependence of the transition pulse area inside the ring
cavity $\Theta_{1}$ and at the output of the cavity $\Theta_{out,1}$. Note
that pulse areas outside and inside the cavity have different units and we use
the additional factor $(2/\sqrt{\kappa})$ for pulse areas outside the cavity
to compensate for that. The most notable feature of Fig.1 is the sharp rise of
the output pulse area near the point $(2/\sqrt{\kappa})\Theta_{in,1}=\pi$. The
sharpness of the rise depends on the ratio between the coupling constants
$\xi=\varkappa/\kappa_{S}$ and is the sharpest at the impedance matching
condition. This feature is a more pronounced version of the self-induced
transparency effect in free space McCall and Hahn (1969): if the incoming
pulse area is smaller than the threshold value
$(2/\sqrt{\kappa})\Theta_{in,1}=\pi$ then the output pulse area gravitates to
0, otherwise it comes close to $2\pi(\sqrt{\kappa}/2)$. The introduction of
the cavity increases the field-atom interaction and thus the nonlinear effects
typical for the optically dense media appear.

For the second pulse we get an equation similar to (9)

$\frac{\kappa_{S}}{2}\Theta_{2}=\sqrt{\kappa}\Theta_{in,2}-\frac{\varkappa}{2}\cos\Theta_{1}\sin\Theta_{2},$
(10)

where initial atomic inversion is modified by the action of first pulse
$w_{0}=-\cos\Theta_{1}$. Solution of Eq. (10) is very similar to solving of
Eq. (9) when $\Theta_{1}<\pi/2$. Below we will immediately move on to the
study of the behavior of echo signals

##  III Photon echo pulse area

###  III.1 Total echo pulse area

Using Eqs. (9),(10) for the incoming pulse we can derive the pulse area of the
total pulse area of all echo pulses excited inside the two-level medium. To do
this we view both incoming pulses as a single composite pulse and write Eq.(9)
for this pulse:

$\displaystyle\frac{\kappa_{S}}{2}\Theta_{tot}=\sqrt{\kappa}(\Theta_{in,1}+\Theta_{in,2})-\frac{\varkappa}{2}\sin\Theta_{tot},$
(11)

here $\Theta_{tot}$ is the total pulse area of all pulses excited inside the
cavity. To calculate the total pulse area of all echoes inside and outside the
cavity we just need to subtract the areas of the incoming pulses:

$\displaystyle\Theta_{e,\Sigma}=$
$\displaystyle\Theta_{tot}-\Theta_{1}-\Theta_{2},$ (12)
$\displaystyle\Theta_{out,\Sigma}=$
$\displaystyle\Theta_{out,tot}-\Theta_{out,1}-\Theta_{out,2}.$ (13)

This can be used as an estimation tool for the echo pulse area. However Eq.
(8) allows to study each echo signal individually.

###  III.2 Linear solution

Before studying the particular echo signals let us analyse some general
properties of basic Eq. (8). An echo signal is irradiated without presence of
external light pulse ($\Theta_{in,e}=0$):

$\Theta_{e}=\xi\Big{[}2v_{0}\cos^{2}\frac{\Theta_{e}}{2}+w_{0}\sin\Theta_{e}\Big{]},$
(14)

where $\xi=\varkappa/\kappa_{S}$.

We can find from (14) that there is a particular solution: $\Theta_{e}=2\pi$
at $v_{0}\xi=\pi$, that is not longer available in its immediate vicinity and
is therefore practically unrealizable. The solutions $\Theta_{e}=\pi,3\pi$ are
also impossible, so realistically we have $0\leq\Theta_{e}<\pi$ and
$\pi<\Theta_{e}<3\pi$.

It is also convenient to rewrite Eq. (14) in the form

$\Theta_{e}=2\xi\sqrt{v_{0}^{2}+w_{0}^{2}}\cos\frac{\Theta_{e}}{2}\sin\frac{(\varphi-\Theta_{e})}{2},$
(15)

where $\varphi=2\arctan\\{-\frac{v_{0}}{w_{0}}\\}$ and from Eq. (15) we find
that $\Theta_{e}<\varphi$.

Let us first consider the case of a small signal pulse $\Theta_{1}$, which is
typical for optical quantum storage. It is particularly interesting since in
this regime the echo pulse shape can be identical to the incoming (first)
pulse shape, which means that the pulse area also characterises the energy and
efficiency of the outgoing pulse. Consider at the beginning the limit of a
weak signal pulse $\Theta_{1}$ and an echo pulse $v_{0}\ll 1$ and also that
$\Theta_{e}<1$. Then we can obtain the linear solution by keeping only the
terms of the first order $O(v_{0})$ in (14):

$\Theta_{e}=\frac{2\xi}{1-\xi w_{0}}v_{0},$ (16)

Let us consider two interesting cases. The first is the formation of a two-
pulse (primary) echo, and the second is the restoration of a suppressed echo
(so called ROSE-protocol). We assume that the control exciting pulses are
launched orthogonally to the resonator axis and have specified pulse areas
$\Theta_{2},\Theta_{3}$.

By taking into account the relaxation of the atomic coherence for the time of
the primary echo emission $t_{e1}=2\tau$, where $\tau$ is the time interval
between the pulses, we have
$v_{0,pe}=\Gamma_{\tau}\sin\Theta_{1}\sin^{2}\tfrac{\Theta_{2}}{2}$, where
$\Gamma_{\tau}=e^{-\gamma t_{e1}}$ is the decoherence term and
$w_{0,pe}=-\cos\Theta_{1}\cos\Theta_{2}$ Moiseev (1987); Moiseev  _et al._
(2020). By assuming weak pulse area in first signal pulse ($\Theta_{1}\ll 1$)
for the primary echo we have:
$v_{0,pe}\cong\Gamma_{\tau}\Theta_{1}\sin^{2}(\Theta_{2}/2)$,
$w_{0,pe}=-\cos\Theta_{1}\cos\Theta_{2}\cong-\cos\Theta_{2}$ and for ROSE-echo
Minnegaliev  _et al._ (2021):
$v_{0,rose}=\Gamma_{\tau}\sin\Theta_{1}\sin^{2}(\Theta_{2}/2)\sin^{2}(\Theta_{3}/2)\cong\Gamma_{2\tau}\Theta_{1}\sin^{2}(\Theta_{2}/2)\sin^{2}(\Theta_{3}/2)$,
$w_{0,rose}=-\cos\Theta_{1}\cos\Theta_{2}\cos\Theta_{3}\cong-\cos\Theta_{2}\cos\Theta_{3}$.
Using these formulas and assuming $\Theta_{2}=\Theta_{3}=\pi$ we get:
$v_{0,pe}=v_{0,rose}=\Theta_{1}$ and $w_{0,pe}=1$, $w_{0,rose}=-1$.
Substituting these values in Eq. (16) we obtain for the primary echo:

$\displaystyle\Theta_{pe}=\frac{2\xi\Gamma_{\tau}}{1-\xi}\Theta_{1}=\frac{2\frac{\varkappa}{k}\xi_{im}\Gamma_{\tau}}{(1+2\frac{k_{in}}{k})\xi_{im}-1}\Theta_{1}=$
$\displaystyle=\frac{\varkappa}{k_{in}}\Gamma_{\tau}\Theta_{1}=\frac{\varkappa}{\varkappa+k_{in}}\frac{k}{k_{in}}\Gamma_{\tau}\Theta_{1},$
(17)

and for the ROSE echo:

$\displaystyle\begin{split}\Theta_{rose}=&\frac{2\xi\Gamma_{2\tau}}{1+\xi}\Theta_{1}=\frac{\varkappa}{\varkappa+k_{in}}\frac{2\Gamma_{2\tau}}{1+\xi_{im}}\Theta_{1}\\\
=&\frac{\varkappa\Gamma_{2\tau}}{\varkappa+k_{in}}\Theta_{1},\end{split}$ (18)

where we have taken into account the impedance matching condition
$\xi_{im}=1$.

Thus, as follows from Eq.(17), the pulse area of the primary echo
$\Theta_{pe}$ increases with decreasing resonator losses $k_{in}$ and large
values may be larger than the pulse area of the signal pulse $\Theta_{1}$ when
$k\gg k_{in}$, $\varkappa\sim k_{in}$. This behavior indicates the
amplification of the primary echo signal in an inverted medium where it will
already be necessary to solve the nonlinear Eq.(14).

As it is seen in Eq.(18), unlike $\Theta_{pe}$, the pulse area of the ROSE
signal $\Theta_{rose}$ is always smaller than the pulse area of the signal
pulse $\Theta_{1}$ and only maximally approaches its value with a decrease of
resonator losses. This behavior corresponds to the fulfillment of the matching
condition in the signal field absorption describes the complete recover of the
signal pulse in the ROSE signal which is used in the impedance matched photon
echo QM schemes Afzelius and Simon (2010); Moiseev  _et al._ (2021);
Minnegaliev  _et al._ (2021). Thus, pulse area approach provides a simple tool
for studies of some basic properties of such QM schemes, especially, depended
on nonlinear properties of coherent light-atom interactions.

###  III.3 Nonlinear approximate solution

To obtain a more general analytical solution of Eq.(16) we decompose Eq.(14)
in series about the point $\Theta_{e}=0,v_{0}=0$. But this time we keep the
terms up to the third order of magnitude ($O(v_{0}^{3})$ or
$O(\Theta_{e}^{3})$). This way we arrive to the cubic equation:

$\Theta_{e}^{3}+\frac{3v_{0}}{2w_{0}}\Theta_{e}^{2}+\frac{6(1-w_{0}\xi)}{w_{0}\xi}\Theta_{e}-\frac{12v_{0}}{w_{0}}=0.$
(19)

Introducing $\zeta=(1-w_{0}\xi)/w_{0}\xi$, we find the discriminant

$\Delta=-108\Big{\\{}8\zeta^{3}+\frac{v^{2}_{0}}{w^{2}_{0}}\left[-\tfrac{3}{4}\zeta^{2}+18\zeta+36\right]+o(v_{0}^{3})\Big{\\}}.$
(20)

In the quantum memory case (small incoming pulse and strong rephasing pulse)
$\Delta<0$, which means that Eq.(19) has a single real root:

$\displaystyle\begin{split}\Theta_{e}&=-\frac{v_{0}}{2w_{0}}+\sqrt[3]{-\frac{\Delta_{1}}{2}+\sqrt{\Delta_{0}}}+\sqrt[3]{-\frac{\Delta_{1}}{2}-\sqrt{\Delta_{0}}},\\\
\Delta_{1}&=-\frac{v_{0}}{w_{0}}\left[12+3\zeta-\frac{1}{4}\left(\frac{v_{0}}{w_{0}}\right)^{2}\right].\end{split}$
(21)

where we introduced $\Delta_{0}=-\Delta/108$. This formula is valid for
$\Theta_{in,1}\ll 1$, increasing the signal area will eventually change the
sign of $\Delta$, at this point Eq. (19) starts having three real roots.

###  III.4 Primary echo

Now we take into account that all light pulses excite resonator mode. In our
previous paper Moiseev  _et al._ (2020) we described the algorithm of finding
resonant components of initial polarization and inversion $v_{0}$ and $w_{0}$
for each echo pulse. These results can be readily applied here. Substituting
the expression for phasing polarization $v_{0,pe}$ and atomic inversion
$w_{0,pe}$ for the primary echo signal (see above) in Eq. (14) we have the
nonlinear equation for the primary echo pulse area for the sample placed
inside the single mode ring cavity:

$\displaystyle\begin{split}\Theta_{e1}=\xi\Big{[}2\Gamma_{\tau}&\sin\Theta_{1}\sin^{2}\tfrac{\Theta_{2}}{2}\cos^{2}\tfrac{\Theta_{e1}}{2}\\\
&-\cos\Theta_{1}\cos\Theta_{2}\sin\Theta_{e1}\Big{]},\end{split}$ (22)
$\displaystyle\Theta_{out,e1}$ $\displaystyle=\sqrt{\kappa}\Theta_{e1}.$ (23)

Note that Eq. (22) contains the difference of two terms and when these two
terms become equal the echo area becomes zero.

Now we can compare the numerical solution of Eq.(22) with approximate
solutions obtained in the previous section. To test the accuracy of the
approximate solution we study the echo pulse area $\Theta_{out,e1}$ while
varying the first pulse’s incoming area $\Theta_{in,1}$. The second pulse area
we keep constant $(2/\sqrt{\kappa})\Theta_{in,2}=0.9\pi$. The term
$2/\sqrt{\kappa}$ accounts for the fact that pulse area inside the cavity
$\Theta$ differs from the incoming pulse area $\Theta_{in}$. To find the
values of $\Theta_{1},\Theta_{2}$ used to calculate $v_{0}$ and $w_{0}$ we
numerically solve Eqs. (9) and (10).

Figure 2 shows the comparison between the linear approximation Eq.(16), cubic
approximation Eq.(21) and numerical solution of Eq.(22). To plot the cubic
solution we solved Eq.(19), which has a single real root for small
$\Theta_{in,1}$. For larger values of $(2/\sqrt{\kappa})\Theta_{in,1}>0.75\pi$
Eq. (19) has three real roots from which we choose the root that maintains the
continuity of the solution. Figure 2 shows that the linear approximation works
very well in the region $\Theta_{e1}<0.2\pi$, but deviates rapidly after this
point. The cubic approximation works well for the primary echo across all
range of the incoming first pulse areas $\Theta_{1}\in[0,2\pi)$, the relative
error does not exceed 11%. Note that the introduction of decoherence in the
form of $\Gamma_{\tau}=0.5$ term in Eq.(22) brings the cubic solution and
numerical solution even closer. Decoherence makes the echo signal smaller and
consequently the difference between numerical and approximate solutions
becomes smaller too (see the caption for the Fig. 2).

The primary echo experiences a sharp drop and a change of sign near the point
$(2/\sqrt{\kappa})\Theta_{in,1}=\pi$. This is a consequence of the same
nonlinear nature that causes the sharp rise of the transmitted pulse area of a
single pulse in Fig. 1. We note that this drop is sharper in the case of the
impedance-matched ring cavity than in the case of Fabri-Perot cavity Moiseev
and Urmancheev (2022). The sharpness is higher in the case of the ring cavity
due to the uniform excitation of the sample. In the Fabry-Perot cavity the
field forms a standing wave, causing additional phase mismatch between
different regions of the sample Moiseev and Urmancheev (2022). The peak value
of the echo pulse area depends on the coupling constants ratio
$\xi=\varkappa/\kappa_{S}$. The echo pulse area is limited and does not exceed
$\pi$ for the impedance matched cavity.

Figure 3 shows the dependence of the primary echo area efficiency
$\Theta_{out,e1}/\Theta_{in,1}$ versus the incoming pulse area of the second
pulse $\Theta_{in,2}$ for the impedance matched cavity $\xi=1$ for negligibly
low resonator losses $k_{in}\ll k$. The incoming signal pulse area is constant
$(2/\sqrt{\kappa})\Theta_{in,1}=\pi/10$ . Figure 3 shows that the dependence
is periodic and peaks near characteristic points
$(2/\sqrt{\kappa})\Theta_{in,2}=\pi\cdot n$ where the efficiency exceeds $1$,
meaning that the echo pulse is amplified, similar to the free space echoes
Urmancheev  _et al._ (2019b); Moiseev  _et al._ (2020). The observed pulse
area amplification is also confirmed by the echo pulse energy investigations
Wang  _et al._ (1998); Cornish  _et al._ (2000); Ruggiero  _et al._ (2009).
The periodicity is broken in Fabry-Perot cavity, when the field forms a
standing wave inside the cavity, leading to a more complicated dependence
Moiseev and Urmancheev (2022).

Figure 2: Comparison between the numerical (blue solid curves) solution of
Eq.14, approximate linear solution Eq.(16) (green dot-dashed line) and
approximate cubic solution Eq.(21) (orange dashed lines);
$(2/\sqrt{\kappa})\Theta_{in,2}=0.9\pi,\xi=\varkappa/\kappa=1$. The difference
between the exact solution and the third order approximation does not exceed
11%. The error is even less if we introduce the decoherence term
$\Gamma_{\tau}=0.5$ (the second, closer to the $x-$axis set of solid blue and
dashed orange curves).

###  III.5 Secondary echoes

After the primary echo pulses there may be secondary echo pulses generated in
the medium under certain conditions. To calculate the pulse area of these
echoes we need to find the corresponding phasing resonant components of
$v_{0}$ and $w_{0}$ at the time of the pulse emission.

For the second echo at the time $3\tau$ we have Moiseev  _et al._ (2020):

$\displaystyle\begin{split}v_{0,e2}=&\Gamma^{2}_{\tau}\cos\Theta_{1}\sin\Theta_{2}\sin^{2}\tfrac{\Theta_{e1}}{2}\\\
&+\tfrac{1}{2}\Gamma^{2}_{\tau}\sin\Theta_{1}\sin\Theta_{2}\sin\Theta_{e1},\end{split}$
(24)
$\displaystyle\begin{split}w_{0,e2}=&-\cos\Theta_{1}\cos\Theta_{2}\cos\Theta_{e1}\\\
&-\Gamma^{2}_{\tau}\sin\Theta_{1}\sin^{2}\tfrac{\Theta_{2}}{2}\sin\Theta_{e1}.\end{split}$
(25)

These expressions we substitute into Eq.(8) and solve numerically for
$\Theta_{e3}$ and $\Theta_{out,e3}$. Similar but more complicated formulas are
derived for the third echo pulse Moiseev  _et al._ (2020). Those can be found
in the Appendix.

Figure 3: Relative primary photon echo pulse area
$\Theta_{out,e1}/\Theta_{in,1}$ at the output of the impedance-matched ring
cavity $\varkappa=\kappa_{S}$ versus the second incoming pulse area
$\Theta_{in,2}$ for $(2/\sqrt{\kappa})\Theta_{in,1}=\pi/5$ (blue solid curve)
and $(2/\sqrt{\kappa})\Theta_{in,1}=\pi/2$ (orange dashed curve). Figure 4:
Echo pulse area efficiency $\Theta_{out,e}/\Theta_{in,1}$ for the primary
(blue solid line), secondary (orange dashed line), and third (green dash-
dotted line) echoes versus the coupling strength $\xi=\varkappa/\kappa_{S}$ in
the Ring cavity;
$(2/\sqrt{\kappa})\Theta_{1}=\pi/2,(2/\sqrt{\kappa})\Theta_{2}=0.9\pi$. Red
dotted curve marked as $\Theta^{\prime}_{e1}$ corresponds to primary echo in
the case of lower first incoming pulse area
$(2/\sqrt{\kappa})\Theta_{in,1}=\pi/10$, the second pulse incoming area is the
same as for all curves. Blue vertical lines show the maximums of the primary
echo. The filled dashed blue line marked as ”$\Theta_{diff}$” plots the $0.1$
of the pulse area of the additional echo pulses excited after the third echo
in a classic $(\pi/2,0.9\pi)$ excitation, obtained as
$(\Theta_{out,\Sigma}-\Theta_{out,e1}-\Theta_{out,e2}-\Theta_{out,e3})/(10\Theta_{in,1})$
using Eq.(13).

Figure 4 shows the dependence of the first three echo’s pulse areas versus the
relative coupling strength $\varkappa/\kappa$. The incoming pulse areas are
$(2/\sqrt{\kappa})\Theta_{in,1}=\pi/2,\,(2/\sqrt{\kappa})\Theta_{in,2}=0.9\pi$,
close to the classic $(\pi/2,\pi)$ sequence of photon/spin echo Hahn (1950);
Kurnit  _et al._ (1964); Allen and Eberly (1975). The echo areas dependencies
have two characteristic regions. In the small coupling regime
($\varkappa/\kappa<0.5$) the primary photon echo is dominating the process and
the majority of the energy is emitted into the first echo pulse. In the strong
coupling regime ($\varkappa/\kappa>1$) all three of the considered echo
signals become comparable. The region ($0.5<\varkappa/\kappa<1$) is an
intermediate regime, in which the multi-pulse echoes are still small and each
consequent echo is smaller than the previous one:
$\Theta_{e,n}<\Theta_{e,n+1}$.

The coupling strength ”chooses” whether the system is in the single pulse and
multi-pulse echo generation regime. This feature has been confirmed
experimentally in the recent works Weichselbaumer  _et al._ (2020); Wang  _et
al._ (2022). In the strong coupling regime there are multiple echo pulses
generated inside the cavity. Although we calculate only the first three echo
signals, we can show that there are actually more secondary echo signals
generated by plotting the difference between the total echo pulse area of all
echo pulses generated inside the media. To do so we use Eq.(13), where we have
$\Theta_{diff}=0.1(\Theta_{out,\Sigma}-\Theta_{out,e1}-\Theta_{out,e2}-\Theta_{out,e3})/\Theta_{in,1}$.
The term $0.1$ is to agree the scales of different curves in Fig. 4.
$\Theta_{diff}$ is negligible in the single pulse regime meaning there are no
additional echo signals excited in the medium. In the strong coupling regime
it rises up to $0.5$ meaning that the total pulse area of the consequent
echoes is approximately 5 times larger than the signal pulse area
$\Theta_{in,1}$, which suggests the existence of multiple additional echo
pulses.

Each echo signal in Fig. 4 has a single maximum, corresponding to the single
impedance matching condition. Previously we derived the impedance matching
condition $\varkappa=\kappa$ for the signal pulse but each echo pulse also has
its own impedance matching condition depending on the incoming pulse areas. We
show this by comparing the primary echo curves for two different signal pulse
areas $(2/\sqrt{\kappa})\Theta_{in,1}=\pi/10$ (red dotted curve) and
$(2/\sqrt{\kappa})\Theta_{in,1}=\pi/2$ (blue solid curve). The curves peak at
different coupling strength, meaning that the primary echo pulse impedance
matching condition depends on the incoming signal pulse area. In turn the
second and third echo signals have their own impedance matching conditions and
peak at different coupling strengths (orange dashed and green dash-dotted
curves in Fig.4). This allows to optimize the impedance matching condition to
a specific generation regime.

##  IV Conclusion

We applied the photon echo area theorem to the two-level atomic ensemble
inside an optical ring cavity to find the pulse areas of the incoming pulses
and photon echo pulses up to the third echo pulse.

The approach successfully represents the nonlinear properties of the echo
generation process. The echo strongly depends on the incoming pulse areas as
well as on the coupling constant between the sample and the cavity mode. We
see that the pulse area theorem successfully captures the main nonlinear
features of the photon echo process: primary echo amplification in the inverse
medium and multi-pulse echo signal excitation. The strong point of the
approach is that it remains true for arbitrary input pulse areas. It can also
be useful in the analytical study of nonlinear patterns of photon echo in
multi-resonator systems Moiseev and Moiseev (2016); Perminov  _et al._ (2018),
which is the subject of subsequent research.

This work constitutes an important step towards establishing photon echo area
theorem approach as a general and reliable tool to analyze various photon echo
schemes inside optical cavities and an analytical alternative to the computer
simulations.

This research was supported by the Ministry of Science and Higher Education of
the Russian Federation (Reg. number NIOKTR 121020400113-1).

## References

  * Kopvillem and Nagibarov (1963) U. K. Kopvillem and V. R. Nagibarov, Fiz. Metal. i Metalloved. 15, 313 (1963).
  * Kurnit  _et al._ (1964) N. A. Kurnit, I. D. Abella, and S. R. Hartmann, Phys. Rev. Lett. 13, 567 (1964).
  * Hahn (1950) E. L. Hahn, Phys. Rev. 80, 580 (1950).
  * Smallwood and Cundiff (2018) C. L. Smallwood and S. T. Cundiff, Laser & Photonics Reviews 12, 1800171 (2018).
  * Cundiff and Mukamel (2013) S. T. Cundiff and S. Mukamel, Phys. Today 66, 44 (2013).
  * Kosarev  _et al._ (2019) A. N. Kosarev, S. V. Poltavtsev, L. E. Golub, M. M. Glazov, M. Salewski, N. V. Kozyrev, E. A. Zhukov, D. R. Yakovlev, G. Karczewski, S. Chusnutdinow, T. Wojtowicz, I. A. Akimov, and M. Bayer, Phys. Rev. B 100, 121401 (2019).
  * Beica  _et al._ (2020) H. C. Beica, A. Pouliot, P. Dowling, A. Carew, T. Vacheresse, G. Carlse, L. Marmet, and A. Kumarakrishnan, Phys. Rev. A 101, 033408 (2020).
  * Jin  _et al._ (2020) M. Jin, Z.-Q. Zhou, C.-F. Li, and G.-C. Guo, Physics Letters A 384, 126780 (2020).
  * Welinski  _et al._ (2020) S. Welinski, A. Tiranov, M. Businger, A. Ferrier, M. Afzelius, and P. Goldner, Phys. Rev. X 10, 031060 (2020).
  * D’Alessandro and Sangalli (2022) M. D’Alessandro and D. Sangalli, SciPost Phys. 12, 193 (2022).
  * Moiseev and Kröll (2001) S. A. Moiseev and S. Kröll, Phys. Rev. Lett. 87, 173601 (2001).
  * Tittel  _et al._ (2010) W. Tittel, M. Afzelius, T. Chaneliére, R. Cone, S. Kröll, S. Moiseev, and M. Sellars, Laser & Photonics Reviews 4, 244 (2010).
  * Ma  _et al._ (2021) Y.-Z. Ma, M. Jin, D.-L. Chen, Z.-Q. Zhou, C.-F. Li, and G.-C. Guo, Nat. Commun. 12, 4378 (2021).
  * Horvath  _et al._ (2021) S. P. Horvath, M. K. Alqedra, A. Kinos, A. Walther, J. M. Dahlström, S. Kröll, and L. Rippe, Physical Review Research 3 (2021), 10.1103/physrevresearch.3.023099.
  * Heshami  _et al._ (2016) K. Heshami, D. G. England, P. C. Humphreys, P. J. Bustard, V. M. Acosta, J. Nunn, and B. J. Sussman, J. Mod. Opt. 63, 2005 (2016).
  * Chanelière  _et al._ (2018) T. Chanelière, G. Hétet, and N. Sangouard, in  _Advances in Atomic, Molecular and Optical Physics_, Vol. 67 (Academic Press, 2018) pp. 77–150.
  * Zhong  _et al._ (2017) T. Zhong, J. M. Kindem, J. G. Bartholomew, J. Rochman, I. Craiciu, E. Miyazono, M. Bettinelli, E. Cavalli, V. Verma, S. W. Nam, F. Marsili, M. D. Shaw, A. D. Beyer, and A. Faraon, Science 357, 1392 (2017).
  * Heller  _et al._ (2020) L. Heller, P. Farrera, G. Heinze, and H. de Riedmatten, Phys. Rev. Lett. 124, 210504 (2020).
  * Moiseev  _et al._ (2021) E. S. Moiseev, A. Tashchilina, S. A. Moiseev, and B. C. Sanders, New Journal of Physics 23, 063071 (2021).
  * Bustard  _et al._ (2022) P. J. Bustard, K. Bonsma-Fisher, C. Hnatovsky, D. Grobnic, S. J. Mihailov, D. England, and B. J. Sussman, Phys. Rev. Lett. 128, 120501 (2022).
  * Weichselbaumer  _et al._ (2020) S. Weichselbaumer, M. Zens, C. W. Zollitsch, M. S. Brandt, S. Rotter, R. Gross, and H. Huebl, Phys. Rev. Lett. 125, 137701 (2020).
  * McCall and Hahn (1969) S. L. McCall and E. L. Hahn, Physical Review 183, 457 (1969).
  * Hahn  _et al._ (1971) E. L. Hahn, N. S. Shiren, and S. L. McCall, Physics Letters A 37, 265 (1971).
  * Moiseev (1987) S. A. Moiseev, Opt. Spectrosc. 62, 180 (1987).
  * Moiseev  _et al._ (2020) S. A. Moiseev, M. Sabooni, and R. V. Urmancheev, Phys. Rev. Research 2, 012026 (2020).
  * Eberly and Kozlov (2002) J. H. Eberly and V. V. Kozlov, Phys. Rev. Lett. 88, 243604 (2002).
  * Moiseev and Urmancheev (2022) S. A. Moiseev and R. V. Urmancheev, Opt. Lett. 47, 3812 (2022).
  * Walls and Milburn (2008) D. F. Walls and G. J. Milburn,  _Quantum Optics_ (Springer-Verlag Berlin Heidelberg, 2008) p. 425.
  * Keeling (2009) J. Keeling, Phys. Rev. A 79, 053825 (2009).
  * Allen and Eberly (1975) L. Allen and J. Eberly,  _Optical Resonance and Two-level Atoms_, Dover books on physics and chemistry (Dover, 1975) p. 256.
  * Urmancheev  _et al._ (2019a) R. Urmancheev, K. Gerasimov, M. Minnegaliev, T. Chanelière, A. Louchet-Chauvet, and S. Moiseev, Opt. Express 27, 28983 (2019a).
  * Chanelière (2014) T. Chanelière, Opt. Express 22, 4423 (2014).
  * Minnegaliev  _et al._ (2021) M. M. Minnegaliev, K. I. Gerasimov, R. V. Urmancheev, A. M. Zheltikov, and S. A. Moiseev, Phys. Rev. B 103, 174110 (2021).
  * Afzelius and Simon (2010) M. Afzelius and C. Simon, Phys. Rev. A 82, 022310 (2010).
  * Urmancheev  _et al._ (2019b) R. Urmancheev, K. Gerasimov, M. Minnegaliev, T. Chanelière, A. Louchet-Chauvet, and S. Moiseev, Opt. Express 27, 28983 (2019b).
  * Wang  _et al._ (1998) T. Wang, C. Greiner, and T. Mossberg, Opt. Commun. 153, 309 (1998).
  * Cornish  _et al._ (2000) C. Cornish, W. Babbitt, and L. Tsang, Opt. Lett. 25, 1276 (2000).
  * Ruggiero  _et al._ (2009) J. Ruggiero, J.-L. Le Gouët, C. Simon, and T. Chanelière, Phys. Rev. A 79, 053851 (2009).
  * Wang  _et al._ (2022) S. Wang, L. Yang, M. Shen, W. Fu, Y. Xu, R. L. Cone, C. W. Thiel, and H. X. Tang, Phys. Rev. Appl. 18, 014069 (2022).
  * Moiseev and Moiseev (2016) E. S. Moiseev and S. A. Moiseev, Journal of Modern Optics 63, 2081 (2016).
  * Perminov  _et al._ (2018) N. S. Perminov, D. Y. Tarankova, and S. A. Moiseev, Laser Physics Letters 15, 125203 (2018).

##  V Appendix

Signal components of the resonant polarization and inversion of the third echo
signal. We can readily use them in Eq.(14).

$\displaystyle\begin{split}v_{0}(7\tau/2,z)=&\tfrac{1}{2}\Gamma_{\tau}^{2}\times\\\
&\big{[}\sin\theta_{1}\sin\theta_{2}\cos\theta_{e1}\sin\theta_{e2}\\\
&+\cos\theta_{1}\sin\theta_{2}\sin\theta_{e1}\sin\theta_{e2}\\\
&+2\cos\theta_{1}\cos\theta_{2}\sin\theta_{e1}\sin^{2}\tfrac{\theta_{e2}}{2}\big{]}\\\
&+\Gamma_{\tau}^{4}\big{[}\sin\theta_{1}\cos^{2}\tfrac{\theta_{2}}{2}\sin^{2}\tfrac{\theta_{e1}}{2}\cos^{2}\tfrac{\theta_{e2}}{2}\\\
&-\sin\theta_{1}\sin^{2}\tfrac{\theta_{2}}{2}\cos^{2}\tfrac{\theta_{e1}}{2}\sin^{2}\tfrac{\theta_{e2}}{2}\big{]},\end{split}$
(A1)
$\displaystyle\begin{split}w_{0}(7\tau/2,z)=&-\cos\theta_{1}\cos\theta_{2}\cos\theta_{e1}\cos\theta_{e2}\\\
&-\Gamma_{\tau}^{2}\times[\sin\theta_{1}\sin^{2}\frac{\theta_{2}}{2}\sin\theta_{e1}\cos\theta_{e2}\\\
&+\cos\theta_{1}\sin\theta_{2}\sin^{2}\frac{\theta_{e1}}{2}\sin\theta_{e2}\\\
&+\frac{1}{2}\sin\theta_{1}\sin\theta_{2}\sin\theta_{e1}\sin\theta_{e2}]=\\\
&-v_{4}\sin\theta_{e2}+w_{4}\cos\theta_{e2}.\end{split}$ (A2)