Adiabatons in a double tripod coherent atom-light coupling scheme

Viačeslav Kudriašov1$\star$, Hamid R. Hamedi1$\dagger$ and Julius
Ruseckas2$\ddagger$

1 Institute of Theoretical Physics and Astronomy, Vilnius University,
Saulėtekio 3, Vilnius LT-10257, Lithuania

2 Baltic Institute of Advanced Technology, Pilies St. 16-8, LT-01403 Vilnius,
Lithuania

$\star$<EMAIL_ADDRESS>, $\dagger$<EMAIL_ADDRESS>$\ddagger$<EMAIL_ADDRESS>
## Abstract

Optical adiabatons are specific shape-invariant pulse pairs propagating at the
reduced group velocity and without optical absorption in the medium. The
purpose of this study is to analyze and demonstrate adiabaton formation in
many level atomic systems. Here we focus on the analysis of five level M-type
and double tripod systems. It is found that M-type atomic systems are prone to
intensity dependent group velocity and pulse front steepening which prevents
the formation of long range optical adiabatons. In contrast, the double tripod
atomic system is quite favorable for the formation of optical adiabatons
leading to two different optical field configurations propagating with
invariant shape.

Copyright attribution to authors.
This work is a submission to SciPost Physics.
License information to appear upon publication.
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###### Contents

  1. 1 Introduction
  2. 2 Adiabatons in $\Lambda$-type atomic system
    1. 2.1 Equations of motion for atoms and fields
    2. 2.2 Coupled and uncoupled states
    3. 2.3 Adiabatic approximation
    4. 2.4 Short duration of propagation
    5. 2.5 Solution
  3. 3 Propagation of light pulses in M-type system
    1. 3.1 Equations of motion for atoms and fields
    2. 3.2 Coupled and uncoupled states
    3. 3.3 Adiabatic approximation
    4. 3.4 Matrix form of equations
    5. 3.5 Equal fields
  4. 4 Adiabatons in double tripod system
    1. 4.1 Initial equations
    2. 4.2 Coupled and uncoupled states
    3. 4.3 Adiabatic approximation
    4. 4.4 Matrix form of equations
    5. 4.5 Adiabatons
  5. 5 Discussion and conclusions

##  1 Introduction

Precise control of light-matter interactions is crucial in optics research.
Remarkable progress has been made using quantum interference in resonant media
[1, 2, 3], where materials can be manipulated into specific quantum states to
enhance or suppress light absorption. One notable technique that exemplifies
this approach is Electromagnetically Induced Transparency (EIT) [4, 2, 5, 6].
EIT allows a material to become transparent to a specific light frequency that
it would normally absorb. This technique underscores the power of quantum
interference in achieving precise control over the optical response of
materials.

Beyond transparency, EIT can lead to a wealth of fascinating effects,
including the formation of nonabsorbing dark states [7], slow group velocities
(slow light) [8], giant enhancement of nonlinearities [9, 10, 11, 12] and
other phenomena [13, 14, 15, 16, 17, 18, 19, 20, 21]. Most of these effects
can be easily observed in a simple three- or four-level atomic medium where
two or three laser fields (probe and pump) coherently interact with the atomic
system tuned to resonant optical transitions. More sophisticated interaction
effects with richer dynamics can be observed in light-atom configurations
operating with a larger number of atomic levels. This extends the
possibilities of coherent control significantly.

Typically, EIT-based research focuses on conditions where the probe field is
much weaker than the coupling field. However, in practical applications, such
as achieving extremely low group velocities, the intensity of the coupling
laser must be kept quite low. This requirement presents a challenge for ultra-
slow light propagation because the probe field needs to be significantly
weaker than the coupling field to maintain the validity of the weak probe
approximation. Consequently, the probe field also needs to be strong enough to
be detectable, complicating the situation as the weak probe approximation is
often not met.

This issue has led to an interest in studying the effects of stronger probe
fields in EIT scenarios. Grobe et al. [22] addressed this by finding an exact
solution to the coupled nonlinear Maxwell-Schrödinger equations for the
$\Lambda$ scheme under adiabatic conditions. This solution permits waves of
arbitrary shapes to propagate, retaining their shapes relatively unchanged
after some initial reshaping. These shape-preserving waves are known as
adiabatons. An optical adiabaton is a pair of shape-invariant pulses with
slowly changing envelopes that propagate at reduced group velocities without
optical absorption in the medium. Adiabatons represent a nonlinear pulse
propagation regime under EIT conditions, where the intensity of the probe
field is comparable to that of the control field, allowing the adiabatic
approximation to hold [23]. When the system deviates from adiabaticity, the
pulses experience absorption, attenuation, reshaping, and steepening effects.

The theoretical prediction of adiabatons in a $\Lambda$ system [22] was soon
experimentally confirmed [24]. Subsequent studies explored this phenomenon in
various systems and conditions [25, 26, 27, 28, 29, 30]. For instance, a four-
level double-$\Lambda$ scheme was used to demonstrate pulse matching and
adiabaton-type propagation [25]. Temporal reshaping and compression of a probe
pulse, due to its group velocity’s dependence on the control field, were also
studied [26, 27]. Two types of adiabatons with zero and non-zero energy in
$\Lambda$ and ladder schemes, respectively, were analyzed [28]. Additionally,
two pairs of adiabatically propagating pulses at different group speeds—fast
and slow adiabatons—were found in a tripod system [29]. Adiabatons for five-
level systems have only been analyzed in a simple five-level double-$\Lambda$
scheme [30]. This motivates us to explore the formation of adiabatons in more
complicated five-level schemes with more laser fields involved. The presence
of additional control fields will enhance the controllability in the formation
and propagation of shape-preserving adiabatons.

In this paper, we theoretically investigate the optical pulse propagation
under the adiabatic regime in other known multilevel atomic systems: M-type
[31, 32] and double-tripod [33, 34, 35, 36, 37]. Both M-type and double tripod
systems have five distinct energy levels, but different number of light fields
arranged in an M-shape or double tripod interaction configuration,
respectively. The latter scheme has some remarkable features compared to a
simple three level $\Lambda$ system. Particularly, in a double tripod system
there are two probe beams at different frequencies which couple together via
four coupling fields and atomic coherences. This property leads to the
formation of a two-component or spinor slow light (SSL) having some distinct
properties [33, 37]. In this kind of medium only the special combinations of
the probe fields (normal modes) propagate in the atomic cloud with the
definite (and different) velocities. Our study shows that the double tripod
configuration supports the adiabatic pulse propagation by exhibiting two
different configurations of the fields that propagate without changing their
shape. This regime, however, is not supported in all multi-level systems that
have an uncoupled (dark) state, for example, it is not possible in the M-type
coupling scheme.

The paper is structured as follows. To illustrate our approach, we first
derive some well-established analytical results and the adiabatic
approximation, as previously described in Refs. [22, 23], for a simple
$\Lambda$ system containing only two optical fields (Sec. 2). This serves as a
basis for our later analysis of more complex multilevel systems. Next, we
proceed to theoretical analysis of systems with more energy levels and
additional coupled fields. In Sec. 3 we examine light pulse propagation in a
multilevel M-type system, which notably does not exhibit adiabaton formation.
Then, in Sec. 4 we present calculations demonstrating adiabatonic propagation
in a double tripod system. Finally, we conclude with a discussion and
summarize our findings in Sec. 5.

##  2 Adiabatons in $\Lambda$-type atomic system

###  2.1 Equations of motion for atoms and fields

Figure 1: Three level $\Lambda$-type atomic system. Two laser beams with the
Rabi frequencies $\Omega_{0}$ and $\Omega_{1}$ act on atoms characterized by
two hyperfine ground levels $0$ and $1$ as well as an excited level $e$.
Parameter $\delta$ denotes two-photon detuning from resonance. Atoms are
initially in the ground level $0$.

We consider a $\Lambda$-type atomic system, shown in Fig. 1, involving two
metastable ground states $|0\rangle$ and $|1\rangle$, as well as an excited
state $|e\rangle$. Laser fields with the Rabi frequncies $\Omega_{0}$ and
$\Omega_{1}$ induce resonant transitions $|0\rangle\rightarrow|e\rangle$ and
$|1\rangle\rightarrow|e\rangle$, respectively. Applying the rotating wave
approximation (RWA), the atomic Hamiltonian in the rotating frame with respect
to the atomic levels reads

$H_{\Lambda}=-\frac{1}{2}\left(\Omega_{0}|e\rangle\langle
0|+\Omega_{1}|e\rangle\langle 1|+\mathrm{H.c.}\right)+\delta|1\rangle\langle
1|-\frac{i}{2}\Gamma|e\rangle\langle e|,$ (1)

where $\delta$ is a two-photon detuning. The losses in the Hamiltinian (1) are
taken into account in an effective way by introducing a rate $\Gamma$ of the
excited state decay. In order to simplify the mathematical description of the
system while keeping the relevant physical details, we characterize the state
of an atom using a state vector
$|\Psi\rangle=\psi_{0}|0\rangle+\psi_{1}|1\rangle+\psi_{e}|e\rangle$, as in
Ref. [22, 38], instead of more complete description employed in Refs. [23, 39]
that involves a density matrix.

The time-dependent Schrödinger equation
$i\hbar\partial_{t}|\Psi\rangle=H_{\Lambda}|\Psi\rangle$ for the atomic state-
vector $|\Psi\rangle$ yields the following equations for the atomic
probability amplitudes $\psi_{0}$, $\psi_{1}$ and $\psi_{e}$:

$\displaystyle i\partial_{t}\psi_{0}$
$\displaystyle=-\frac{1}{2}\Omega_{0}^{*}\psi_{e}\,,$ (2) $\displaystyle
i\partial_{t}\psi_{1}$
$\displaystyle=\delta\psi_{1}-\frac{1}{2}\Omega_{1}^{*}\psi_{e}\,,$ (3)
$\displaystyle i\partial_{t}\psi_{e}$
$\displaystyle=-\frac{i}{2}\Gamma\psi_{e}-\frac{1}{2}\Omega_{0}\psi_{0}-\frac{1}{2}\Omega_{1}\psi_{1}.$
(4)

On the other hand, the Rabi frequencies of the laser fields obey the
propagation equations

$\partial_{t}\Omega_{l}+c\partial_{z}\Omega_{l}=\frac{i}{2}g\psi_{e}\psi_{l}^{*}\,,\qquad
l=0,1\;$ (5)

where the parameter $g$ characterizes the strength of coupling of the light
fields with the atoms. It is related to the optical depth $\alpha$ as
$g=c\Gamma\alpha/L$, where $L$ is the length of the medium. For simplicity
here we have assumed that the coupling strength $g$ is the same for both laser
fields.

###  2.2 Coupled and uncoupled states

The Hamiltonian (1) can be rewitten as [23]:

$H_{\Lambda}=-\frac{\Omega}{2}\left(|e\rangle\langle\mathrm{C}|+|\mathrm{C}\rangle\langle
e|\right)+\delta|1\rangle\langle 1|-\frac{i}{2}\Gamma|e\rangle\langle e|\,,$
(6)

where

$|\mathrm{C}\rangle=\frac{1}{\Omega}(\Omega_{0}^{*}|0\rangle+\Omega_{1}^{*}|1\rangle),$
(7)

is a coupled state with

$\Omega=\sqrt{|\Omega_{0}|^{2}+|\Omega_{1}|^{2},}$ (8)

being a total Rabi frequency. Additionally we introduce an uncoupled state

$|\mathrm{U}\rangle=\frac{1}{\Omega}(\Omega_{1}|0\rangle-\Omega_{0}|1\rangle),$
(9)

which is orthogonal to the coupled state. The probability amplitudes to find
an atom in the coupled and uncoupled states read

$\displaystyle\psi_{\mathrm{C}}$
$\displaystyle=\langle\mathrm{C}|\Psi\rangle=\frac{1}{\Omega}(\Omega_{0}\psi_{0}+\Omega_{1}\psi_{1})\,,$
(10) $\displaystyle\psi_{\mathrm{U}}$
$\displaystyle=\langle\mathrm{U}|\Psi\rangle=\frac{1}{\Omega}(\Omega_{1}^{*}\psi_{0}-\Omega_{0}^{*}\psi_{1})\,.$
(11)

In terms of coupled and uncoupled states, the equations (2)-(4) for the atomic
amplitudes take the form

$\displaystyle i\partial_{t}\psi_{\mathrm{U}}$
$\displaystyle=\Delta\psi_{\mathrm{U}}+\Omega_{-}^{*}\psi_{\mathrm{C}}\,,$
(12) $\displaystyle i\partial_{t}\psi_{\mathrm{C}}$
$\displaystyle=-\Delta\psi_{\mathrm{C}}+\Omega_{-}\psi_{\mathrm{U}}-\frac{1}{2}\Omega\psi_{e}\,,$
(13) $\displaystyle i\partial_{t}\psi_{e}$
$\displaystyle=-\frac{i}{2}\Gamma\psi_{e}-\frac{1}{2}\Omega\psi_{\mathrm{C}}\,,$
(14)

with [23]

$\displaystyle\Delta$
$\displaystyle=i\frac{\Omega_{0}}{\Omega}\partial_{t}\frac{\Omega_{0}^{*}}{\Omega}+i\frac{\Omega_{1}}{\Omega}\partial_{t}\frac{\Omega_{1}^{*}}{\Omega}+\delta\frac{|\Omega_{0}|^{2}}{\Omega^{2}}\,,$
(15) $\displaystyle\Omega_{-}$
$\displaystyle=i\frac{\Omega_{1}}{\Omega}\partial_{t}\frac{\Omega_{0}}{\Omega}-i\frac{\Omega_{0}}{\Omega}\partial_{t}\frac{\Omega_{1}}{\Omega}-\delta\frac{\Omega_{0}}{\Omega}\frac{\Omega_{1}}{\Omega}\,.$
(16)

Here $\Omega_{-}$ describes non-adiabatic losses and $2\Delta$ represents the
separaton in energies between the uncoupled and coupled states.

###  2.3 Adiabatic approximation

Let us consider a situation where the total Rabi frequency $\Omega$ is
sufficiently large so that the conditions presented below by Eqs. (20) and
(24) hold. In this case the adiabatic approximation can be applied. To prepare
for the derivation of approximate equations we express $\psi_{e}$ from Eq.
(13):

$\psi_{e}=2\frac{\Omega_{-}}{\Omega}\psi_{\mathrm{U}}-\frac{2}{\Omega}(i\partial_{t}+\Delta)\psi_{\mathrm{C}}\,.$
(17)

On the other hand, Eq. (14) relates $\psi_{\mathrm{C}}$ to $\psi_{e}$ as:

$\psi_{\mathrm{C}}=-\frac{2i}{\Omega}\left(\partial_{t}+\frac{\Gamma}{2}\right)\psi_{e}\,.$
(18)

Since the excited state decay rate $\Gamma$ is considered to be large compared
to the rate of change of the fields, we neglect the temporal derivative
$\partial_{t}$ in the above equation. Substituting Eq. (17) into Eq. (18) one
gets

$\psi_{\mathrm{C}}=-2i\frac{\Gamma}{\Omega^{2}}(\Omega_{-}\psi_{\mathrm{U}}-(i\partial_{t}+\Delta)\psi_{\mathrm{C}})\,.$
(19)

We solve this equation iteratively with respect to $\psi_{\mathrm{C}}$,
assuming that $\Omega$ is large compared to the rate of non-adiabatic
transitions

$\Omega\gg|\Omega_{-}|\,.$ (20)

In the zeroth-order of the adiabatic approximation the coupled state is not
populated, $\psi_{\mathrm{C}}\approx 0$, so Eq. (10) yields

$\psi_{0}\approx\frac{\Omega_{1}}{\Omega}\,,\quad\psi_{1}\approx-\frac{\Omega_{0}}{\Omega}\,.$
(21)

Non-zero $\psi_{C}$ appears in the first-order approximation. Putting
$\psi_{\mathrm{C}}\approx 0$ on the right hand side (r.h.s.) of Eq. (19), one
arrives at the first order result for the amplitude of the coupled state

$\psi_{\mathrm{C}}\approx-2i\frac{\Gamma}{\Omega}\frac{\Omega_{-}}{\Omega}\psi_{\mathrm{U}}\,.$
(22)

Inserting this expression back in the r.h.s. of Eq. (19) we obtain the
expression containing the second-order correction

$\psi_{\mathrm{C}}=-2i\frac{\Gamma}{\Omega}\frac{\Omega_{-}}{\Omega}\psi_{\mathrm{U}}+4\frac{\Gamma^{2}}{\Omega^{2}}(i\partial_{t}+\Delta)\frac{\Omega_{-}}{\Omega^{2}}\psi_{\mathrm{U}}\,.$
(23)

The second order term should be much smaller than the first order one in the
r.h.s. of the above equation. Since the atomic population is concentrated in
the uncoupled state, $|\psi_{\mathrm{U}}|\approx 1$ and
$|\psi_{\mathrm{C}}|\ll 1$, one arrives at the following condition

$\frac{\Gamma|\Delta|}{\Omega^{2}}\ll
1\,,\qquad\frac{\Gamma|\Omega_{-}|}{\Omega^{2}}\ll 1\,.$ (24)

Now let us present the adiabatic expansion of the excited state amplitude
$\psi_{3}$. The zeroth-order approximation of Eq. (17) is $\psi_{e}\approx 0$.
In the first order, taking into account Eq. (22) and the condition (24), we
have

$\psi_{e}\approx 2\frac{\Omega_{-}}{\Omega}\psi_{\mathrm{U}}\,.$ (25)

Since the excited state should be weakly populated, the condition (20) is to
be imposed.

Finally, inserting the first-order adiabatic result Eq. (22) relating
$\psi_{\mathrm{C}}$ to $\psi_{\mathrm{U}}$ into Eq. (12) we obtain the
equation for the amplitude of the uncoupled state

$i\partial_{t}\psi_{\mathrm{U}}=\Delta\psi_{\mathrm{U}}-2i\Gamma\frac{|\Omega_{-}|^{2}}{\Omega^{2}}\psi_{\mathrm{U}}\,.$
(26)

The last term on the r.h.s. represents losses due to non-adiabatic
corrections. Similarly, inserting Eqs. (25) and (22) into Eq. (5) we get the
equations for the amplitudes of the radiation fields $\Omega_{0}$ and
$\Omega_{1}$:

$\displaystyle\partial_{t}\Omega_{0}+c\partial_{z}\Omega_{0}$
$\displaystyle=g\left(i\frac{\Omega_{-}}{\Omega}\frac{\Omega_{1}^{*}}{\Omega}-2\frac{\Gamma}{\Omega}\frac{|\Omega_{-}|^{2}}{\Omega^{2}}\frac{\Omega_{0}}{\Omega}\right)|\psi_{\mathrm{U}}|^{2}\,,\,$
(27) $\displaystyle\partial_{t}\Omega_{1}+c\partial_{z}\Omega_{1}$
$\displaystyle=g\left(-i\frac{\Omega_{-}}{\Omega}\frac{\Omega_{0}^{*}}{\Omega}-2\frac{\Gamma}{\Omega}\frac{|\Omega_{-}|^{2}}{\Omega^{2}}\frac{\Omega_{1}}{\Omega}\right)|\psi_{\mathrm{U}}|^{2}.$
(28)

Equations (26), (27) and (28) describe adiabatic propagation of the fields.
The second terms on the r.h.s. of Eqs. (27) and (28) describe non-adiabatic
losses.

###  2.4 Short duration of propagation

Let us now consider the case when the duration of the propagation
$\tau_{\mathrm{prop}}$ is much smaller than the life time of the adiabatons:
$\Gamma\frac{|\Omega_{-}|^{2}}{\Omega^{2}}\tau_{\mathrm{prop}}\ll 1$. The
propagation duration is of the order of $L/v_{g}$, where $v_{g}=c\Omega^{2}/g$
is the group velocity. Expressing the atom-light coupling strength via the
optical density $\alpha$, $g=c\Gamma\alpha/L$, we obtain the condition

$\left(\frac{\Gamma|\Omega_{-}|}{\Omega^{2}}\right)^{2}\alpha\ll 1,$ (29)

which should be satisfied. In this situation one can neglect the decay terms
in Eqs. (26), (27) and (28). Furthermore, since $|\psi_{\mathrm{U}}|\approx
1$, Eqs. (27), (28) reduce to

$\displaystyle\partial_{t}\Omega_{0}+c\partial_{z}\Omega_{0}$
$\displaystyle=ig\frac{\Omega_{-}}{\Omega}\frac{\Omega_{1}^{*}}{\Omega}\,,$
(30) $\displaystyle\partial_{t}\Omega_{1}+c\partial_{z}\Omega_{1}$
$\displaystyle=-ig\frac{\Omega_{-}}{\Omega}\frac{\Omega_{0}^{*}}{\Omega}\,.$
(31)

Combining Eqs. (30) and (31), the total Rabi frequency $\Omega$ obeys the
equation

$\partial_{t}\Omega+c\partial_{z}\Omega=0\,.$ (32)

On the other hand, Eqs. (30) and (31) provide the following equation for the
ratio $\chi=\Omega_{0}/\Omega_{1}$:

$\left(c^{-1}+\frac{g}{c\Omega^{2}}\right)\partial_{t}\chi+\partial_{z}\chi+i\delta\frac{g}{c\Omega^{2}}\chi=0\,.$
(33)

This equation has a similar form to the equation for the propagation of a weak
probe field affected by a stronger control field when
$|\Omega_{0}/\Omega_{1}|\ll 1$. Yet, in the present situation, such a
condition is not imposed.

###  2.5 Solution

Equation (32) has the following solution satisfying the boundary condition at
$z=0$:

$\Omega(z,t)\equiv\Omega(\tau)=\sqrt{|\Omega_{0}(0,\tau)|^{2}+|\Omega_{1}(0,\tau)|^{2}}\,,$
(34)

where $\tau=t-z/c$. To obtain a solution of Eq. (33) we change the variables
$t$ and $z$ to $\tau=t-z/c$ and $z$, respectively. Then

$\frac{g}{c\Omega^{2}(\tau)}\partial_{\tau}\chi+\partial_{z}\chi+i\delta\frac{g}{c\Omega^{2}(\tau)}\chi=0\,.$
(35)

Further we change the time variable $\tau$ to the stretched time

$\zeta(\tau)=\frac{c}{g}\int_{-\infty}^{\tau}\Omega^{2}(\tau^{\prime})d\tau^{\prime}\,,$
(36)

yielding

$\partial_{\zeta}\chi+\partial_{z}\chi+i\delta\frac{g}{c\Omega^{2}(\zeta)}\chi=0\,.$
(37)

A particular solution of this equation is

$\chi(z,\zeta)=\exp\left(ik(z-\zeta)-i\delta\frac{g}{c}\int_{-\infty}^{\zeta}\frac{d\zeta^{\prime}}{\Omega^{2}(\zeta^{\prime})}\right)\,.$
(38)

When $\delta=0$, a solution of Eq. (37) is an arbitrary function $f(\zeta-z)$,
fixed by the boundary condition at $z=0$:

$f(\zeta(t))=\frac{\Omega_{0}(0,t)}{\Omega_{1}(0,t)},$ (39)

or

$f(z)=\frac{\Omega_{0}(0,\zeta^{-1}(z))}{\Omega_{1}(0,\zeta^{-1}(z))}\,,$ (40)

where $\zeta^{-1}$ is a function inverse to the function $\zeta$. It follows
from Eq. (36) that if $\Omega(\tau)$ becomes constant after a certain time,
the stretched time $\zeta$ becomes a linear function of $\tau$. After that
time the obtained solution $f(\zeta-z)$ describes a shape-preserving
propagation [22].

We performed numerical investigation of the above analytical study by modeling
field equations with optical fields of particular temporal shape. Here, the
first optical field is described as a Gaussian pulse at the input:
$\Omega_{0}(0,t)=A\exp[-(t-t_{0})^{2}/\tau_{0}^{2}]$, where
$\tau_{0}=5\Gamma^{-1}$, $t_{0}=23\Gamma^{-1}$ and $A=\Gamma$. The second one
is simply a constant field: $\Omega_{1}(0,t)=1.5\Gamma$. Results of the
numerical solution are depicted in Fig. 2 which shows the adiabatonic
propagation regime for the optical fields after some propagation time.
Specifically, a shape-preserving combination of fields $\Omega_{0}(z,t)$ and
$\Omega_{1}(z,t)$ propagating with the group velocity $v_{g}$ can be seen on
the right part of Fig. 2. The upward pulse from $\Omega_{1}(z,t)$ field seen
in the left upper part of the figure is propagating with the speed of light.

Figure 2: Temporal dependence of pulse amplitudes in $\Lambda$ system. Dashed
lines correspond to pulse envelopes $\Omega_{0}$ and $\Omega_{1}$ at the input
($z=0$), solid lines show pulse envelopes at propagation distance
$z=70L_{\mathrm{abs}}$. Here $L_{\mathrm{abs}}=L/\alpha$ is resonant
absorption length. Amplitude shown on the vertical scale is measured in
$\Gamma$.

##  3 Propagation of light pulses in M-type system

###  3.1 Equations of motion for atoms and fields

Figure 3: Five level M-type atomic system. Four laser beams with the Rabi
frequencies $\Omega_{1,0}$, $\Omega_{1,1}$, $\Omega_{2,2}$ and $\Omega_{2,0}$
act on atoms characterized by three hyperfine ground levels $0$, $1$ and $2$
as well as two excited levels $e_{1}$ and $e_{2}$. Atoms are initially in the
ground level $0$.

Now we will consider a M-type atomic system (Fig. 3) involving three
metastable ground states $|0\rangle$, $|1\rangle$ and $|2\rangle$, as well as
two excited states $|e_{1}\rangle$ and $|e_{2}\rangle$. Four laser fields with
the Rabi frequencies $\Omega_{j,l}$ induce resonant transitions
$|l\rangle\rightarrow|e_{j}\rangle$; here the $j=1,2$ and $l=0,j$. Applying
the rotating wave approximation (RWA), the atomic Hamiltonian in the rotating
frame with respect to the atomic levels reads

$H_{\mathrm{M}}=\sum_{j=1}^{2}\left\\{-\frac{1}{2}\left(\Omega_{j,0}|e_{j}\rangle\langle
0|+\Omega_{j,j}|e_{j}\rangle\langle
j|+\mathrm{H.c.}\right)-\frac{i}{2}\Gamma|e_{j}\rangle\langle
e_{j}|+\delta_{j}|j\rangle\langle j|\right\\}\,,$ (41)

where $\delta_{j}$ are two-photon detunings. The losses in the Hamiltonian
(41) are taken into account in an effective way by introducing a rate $\Gamma$
of the excited state decay. The time-dependent Schrödinger equation
$i\hbar\partial_{t}|\Psi\rangle=H_{\mathrm{M}}|\Psi\rangle$ for the atomic
state-vector

$|\Psi\rangle=\sum_{l=0}^{2}\psi_{l}|l\rangle+\sum_{j=1}^{2}\psi_{e_{j}}|e_{j}\rangle$

yields the following equations for the atomic probability amplitudes
$\psi_{l}$, $\psi_{e_{j}}$:

$\displaystyle i\partial_{t}\psi_{0}$
$\displaystyle=-\frac{1}{2}\Omega_{1,0}^{*}\psi_{e_{1}}-\frac{1}{2}\Omega_{2,0}^{*}\psi_{e_{2}}\,,$
(42) $\displaystyle i\partial_{t}\psi_{1}$
$\displaystyle=\delta_{1}\psi_{1}-\frac{1}{2}\Omega_{1,1}^{*}\psi_{e_{1}}\,,$
(43) $\displaystyle i\partial_{t}\psi_{2}$
$\displaystyle=\delta_{2}\psi_{2}-\frac{1}{2}\Omega_{2,2}^{*}\psi_{e_{2}}\,,$
(44) $\displaystyle i\partial_{t}\psi_{e_{1}}$
$\displaystyle=-\frac{i}{2}\Gamma\psi_{e_{1}}-\frac{1}{2}\Omega_{1,0}\psi_{0}-\frac{1}{2}\Omega_{1,1}\psi_{1}\,,$
(45) $\displaystyle i\partial_{t}\psi_{e_{2}}$
$\displaystyle=-\frac{i}{2}\Gamma\psi_{e_{2}}-\frac{1}{2}\Omega_{2,0}\psi_{0}-\frac{1}{2}\Omega_{2,2}\psi_{2}\,.$
(46)

On the other hand, the Rabi frequencies of the laser fields obey the following
propagation equations

$\displaystyle(\partial_{t}+c\partial_{z})\Omega_{1,0}$
$\displaystyle=\frac{i}{2}g\psi_{e_{1}}\psi_{0}^{*}\,,$ (47)
$\displaystyle(\partial_{t}+c\partial_{z})\Omega_{2,0}$
$\displaystyle=\frac{i}{2}g\psi_{e_{2}}\psi_{0}^{*}\,,$ (48)
$\displaystyle(\partial_{t}+c\partial_{z})\Omega_{1,1}$
$\displaystyle=\frac{i}{2}g\psi_{e_{1}}\psi_{1}^{*}\,,$ (49)
$\displaystyle(\partial_{t}+c\partial_{z})\Omega_{2,2}$
$\displaystyle=\frac{i}{2}g\psi_{e_{2}}\psi_{2}^{*}\,,$ (50)

where the parameter $g$ characterizes the strength of coupling of the light
fields with the atoms.

###  3.2 Coupled and uncoupled states

The Hamiltonian (41) can be rewitten as

$H_{\mathrm{M}}=\sum_{j=1}^{2}\left\\{-\frac{1}{2}\left(\Omega_{j,j}|e_{j}\rangle\langle\mathrm{C}_{j}|+\Omega_{j,j}^{*}|\mathrm{C}_{j}\rangle\langle
e_{j}|\right)+\delta_{j}|j\rangle\langle
j|-\frac{i}{2}\Gamma|e_{j}\rangle\langle e_{j}|\right\\}\,,$ (51)

where $|\mathrm{C}_{j}\rangle$ are two coupled states which are unnormalized
and non-orthogonal to each other

$|\mathrm{C}_{j}\rangle=\chi_{j}^{*}|0\rangle+|j\rangle\,,\quad\mathrm{with\quad\chi_{j}=\frac{\Omega_{j,0}}{\Omega_{j,j}},}\quad
j=1,2\,.$ (52)

Additionally we introduce a normalized uncoupled state which is orthogonal to
both coupled states

$|\mathrm{U}\rangle=\frac{1}{N_{0}}(|0\rangle-\chi_{1}|1\rangle-\chi_{2}|2\rangle)\,,$
(53)

where

$N_{0}=\sqrt{1+|\chi_{1}|^{2}+|\chi_{2}|^{2}}\,,$ (54)

is a normalization factor. The amplitudes to find an atom in the coupled and
uncoupled states read

$\displaystyle\psi_{\mathrm{C}_{j}}$
$\displaystyle=\langle\mathrm{C}_{j}|\Psi\rangle=\chi_{j}\psi_{0}+\psi_{j}\,,$
(55) $\displaystyle\psi_{\mathrm{U}}$
$\displaystyle=\langle\mathrm{U}|\Psi\rangle=\frac{1}{N_{0}}(\psi_{0}-\chi_{1}^{*}\psi_{1}-\chi_{2}^{*}\psi_{2})\,.$
(56)

The initial atomic amplitudes then can be expressed as

$\displaystyle\psi_{0}$
$\displaystyle=\frac{1}{N_{0}}\left(\psi_{\mathrm{U}}+\frac{1}{N_{0}}\chi_{1}^{*}\psi_{\mathrm{C}_{1}}+\frac{1}{N_{0}}\chi_{2}^{*}\psi_{\mathrm{C}_{2}}\right)\,,$
(57) $\displaystyle\psi_{1}$
$\displaystyle=\psi_{\mathrm{C}_{1}}-\chi_{1}\psi_{0}=-\frac{\chi_{1}}{N_{0}}\psi_{\mathrm{U}}+\cdots\,,$
(58) $\displaystyle\psi_{2}$
$\displaystyle=\psi_{\mathrm{C}_{2}}-\chi_{2}\psi_{0}=-\frac{\chi_{2}}{N_{0}}\psi_{\mathrm{U}}+\cdots\,.$
(59)

The terms that are ommited in Eqs. (58), (59) contain amplitudes
$\psi_{\mathrm{C}_{1}}$, $\psi_{\mathrm{C}_{2}}$ and are not needed for the
first order of the adiabatic approximation.

The equations for the atomic amplitudes in terms of coupled and uncoupled
states take the form

$\displaystyle i\partial_{t}\psi_{\mathrm{U}}$
$\displaystyle=\Delta\psi_{\mathrm{U}}+\cdots\,,$ (60) $\displaystyle
i\partial_{t}\psi_{\mathrm{C}_{1}}$
$\displaystyle=V_{1}\psi_{\mathrm{U}}-\frac{1}{2}\Omega_{1,1}^{*}N_{1}^{2}\psi_{e_{1}}-\frac{1}{2}\Omega_{2,0}^{*}\chi_{1}\psi_{e_{2}}+\cdots\,,$
(61) $\displaystyle i\partial_{t}\psi_{\mathrm{C}_{2}}$
$\displaystyle=V_{2}\psi_{\mathrm{U}}-\frac{1}{2}\Omega_{1,0}^{*}\chi_{2}\psi_{e_{1}}-\frac{1}{2}\Omega_{2,2}^{*}N_{2}^{2}\psi_{e_{2}}+\cdots\,,$
(62) $\displaystyle i\partial_{t}\psi_{e_{1}}$
$\displaystyle=-\frac{i}{2}\Gamma\psi_{e_{1}}-\frac{1}{2}\Omega_{1,1}\psi_{\mathrm{C}_{1}}\,,$
(63) $\displaystyle i\partial_{t}\psi_{e_{2}}$
$\displaystyle=-\frac{i}{2}\Gamma\psi_{e_{2}}-\frac{1}{2}\Omega_{2,2}\psi_{\mathrm{C}_{2}}\,.$
(64)

The ommited terms contain amplitudes $\psi_{\mathrm{C}_{1}}$,
$\psi_{\mathrm{C}_{2}}$. Here

$\displaystyle V_{1}$
$\displaystyle=\frac{1}{N_{0}}(i\partial_{t}-\delta_{1})\chi_{1}\,,$ (65)
$\displaystyle V_{2}$
$\displaystyle=\frac{1}{N_{0}}(i\partial_{t}-\delta_{2})\chi_{2},$ (66)

are the coefficients describing the non-adiabatic coupling; the detuning
$\Delta$ is defined as

$\Delta=\frac{1}{N_{0}}i\partial_{t}\frac{1}{N_{0}}+\frac{\chi_{1}}{N_{0}}i\partial_{t}\frac{\chi_{1}^{*}}{N_{0}}+\frac{\chi_{2}}{N_{0}}i\partial_{t}\frac{\chi_{2}^{*}}{N_{0}}+\delta_{1}\frac{|\chi_{1}|^{2}}{N_{0}^{2}}+\delta_{2}\frac{|\chi_{2}|^{2}}{N_{0}^{2}}\,.$
(67)

Finally,

$\displaystyle N_{1}$ $\displaystyle=\sqrt{1+|\chi_{1}|^{2}}\,,$ (68)
$\displaystyle N_{2}$ $\displaystyle=\sqrt{1+|\chi_{2}|^{2}},$ (69)

are normalization factors for the unnormalized coupled states.

###  3.3 Adiabatic approximation

Let us consider a situation where the light fields are slowly changing and
have large intensities. In this situation we can apply adiabatic
approximation. Neglecting small amplitudes of the coupled states
$\psi_{\mathrm{C}_{1}}$ and $\psi_{\mathrm{C}_{2}}$ in Eqs. (61), (62) leads
to a pair of equations

$\displaystyle 2V_{1}\psi_{\mathrm{U}}$
$\displaystyle=\Omega_{1,1}^{*}N_{1}^{2}\psi_{e_{1}}+\Omega_{2,0}^{*}\chi_{1}\psi_{e_{2}}\,,$
(70) $\displaystyle 2V_{2}\psi_{\mathrm{U}}$
$\displaystyle=\Omega_{1,0}^{*}\chi_{2}\psi_{e_{1}}+\Omega_{2,2}^{*}N_{2}^{2}\psi_{e_{2}},$
(71)

with the solution

$\displaystyle\psi_{e_{1}}$ $\displaystyle=2W_{1}\psi_{\mathrm{U}}\,,$ (72)
$\displaystyle\psi_{e_{2}}$ $\displaystyle=2W_{2}\psi_{\mathrm{U}}\,,$ (73)

where

$\displaystyle W_{1}$
$\displaystyle=\frac{1}{\Omega_{1,1}^{*}N_{0}^{2}}(N_{2}^{2}V_{1}-\chi_{1}\chi_{2}^{*}V_{2})=\frac{1}{\Omega_{1,1}^{*}N_{0}^{3}}[N_{2}^{2}(i\partial_{t}-\delta_{1})\chi_{1}-\chi_{1}\chi_{2}^{*}(i\partial_{t}-\delta_{2})\chi_{2}]\,,$
(74) $\displaystyle W_{2}$
$\displaystyle=\frac{1}{\Omega_{2,2}^{*}N_{0}^{2}}(N_{1}^{2}V_{2}-\chi_{2}\chi_{1}^{*}V_{1})=\frac{1}{\Omega_{2,2}^{*}N_{0}^{3}}[N_{1}^{2}(i\partial_{t}-\delta_{1})\chi_{2}-\chi_{2}\chi_{1}^{*}(i\partial_{t}-\delta_{2})\chi_{1}]\,.$
(75)

On the other hand, Eqs. (63), (64) relate $\psi_{\mathrm{C}_{1}}$,
$\psi_{\mathrm{C}_{2}}$ to $\psi_{e_{1}}$, $\psi_{e_{2}}$ as

$\displaystyle\psi_{\mathrm{C}_{1}}$
$\displaystyle=-i\frac{\Gamma}{\Omega_{1,1}}\psi_{e_{1}}\,,$ (76)
$\displaystyle\psi_{\mathrm{C}_{2}}$
$\displaystyle=-i\frac{\Gamma}{\Omega_{2,2}}\psi_{e_{2}}\,.$ (77)

Neglecting the decay of adiabatons, Eq. (60) for the amplitude of the
uncoupled state becomes

$i\partial_{t}\psi_{\mathrm{U}}=\Delta\psi_{\mathrm{U}}\,.$ (78)

Inserting Eqs. (72), (73) into Eqs. (47)-(50) we get

$\displaystyle(\partial_{t}+c\partial_{z})\Omega_{1,0}$
$\displaystyle=i\frac{g}{N_{0}}W_{1}\,,$ (79)
$\displaystyle(\partial_{t}+c\partial_{z})\Omega_{2,0}$
$\displaystyle=i\frac{g}{N_{0}}W_{2}\,,$ (80)
$\displaystyle(\partial_{t}+c\partial_{z})\Omega_{1,1}$
$\displaystyle=-i\frac{g}{N_{0}}\chi_{1}^{*}W_{1}\,,$ (81)
$\displaystyle(\partial_{t}+c\partial_{z})\Omega_{2,2}$
$\displaystyle=-i\frac{g}{N_{0}}\chi_{2}^{*}W_{2}\,.$ (82)

Equations (78)-(82) describe adiabatic propagation of the fields.

###  3.4 Matrix form of equations

From Eqs. (79)-(82) we obtain that the total Rabi frequencies

$\displaystyle\Omega_{1}$
$\displaystyle=\sqrt{|\Omega_{1,1}|^{2}+|\Omega_{1,0}|^{2}}=|\Omega_{1,1}|N_{1}\,,$
(83) $\displaystyle\Omega_{2}$
$\displaystyle=\sqrt{|\Omega_{2,2}|^{2}+|\Omega_{2,0}|^{2}}=|\Omega_{2,2}|N_{2},$
(84)

obey the equations

$\displaystyle(\partial_{t}+c\partial_{z})\Omega_{1}$ $\displaystyle=0\,,$
(85) $\displaystyle(\partial_{t}+c\partial_{z})\Omega_{2}$ $\displaystyle=0.$
(86)

Combining Eqs. (79)-(82) we get the equations for the ratios $\chi_{1}$ and
$\chi_{2}$:

$\displaystyle(\partial_{t}+c\partial_{z})\chi_{1}$
$\displaystyle=ig\frac{N_{1}^{2}}{N_{0}}\frac{W_{1}}{\Omega_{1,1}}\,,$ (87)
$\displaystyle(\partial_{t}+c\partial_{z})\chi_{2}$
$\displaystyle=ig\frac{N_{2}^{2}}{N_{0}}\frac{W_{2}}{\Omega_{2,2}}.$ (88)

Introducing the two-component spinor

$\chi=\left(\begin{array}[]{c}\chi_{1}\\\ \chi_{2}\end{array}\right),$ (89)

and using Eqs. (74), (75), Eqs. (87), (88) can be written in the following
matrix form:

$(c^{-1}+\hat{v}^{-1})\partial_{t}\chi+\partial_{z}\chi+i\hat{v}^{-1}\hat{\delta}\chi=0\,,$
(90)

where

$\hat{v}^{-1}=\frac{g}{cN_{0}^{4}}\left(\begin{array}[]{cc}\frac{N_{1}^{2}N_{2}^{2}}{|\Omega_{1,1}|^{2}}&-\frac{N_{1}^{2}\chi_{1}\chi_{2}^{*}}{|\Omega_{1,1}|^{2}}\\\
-\frac{N_{2}^{2}\chi_{2}\chi_{1}^{*}}{|\Omega_{2,2}|^{2}}&\frac{N_{2}^{2}N_{1}^{2}}{|\Omega_{2,2}|^{2}}\end{array}\right)=\frac{g}{c}\left(\begin{array}[]{cc}\frac{1}{\Omega_{1}^{2}}\frac{N_{1}^{4}}{N_{0}^{4}}&0\\\
0&\frac{1}{\Omega_{2}^{2}}\frac{N_{2}^{4}}{N_{0}^{4}}\end{array}\right)(N_{0}^{2}I-\chi\chi^{\dagger}),$
(91)

is the matrix of inverse group velocity, while

$\hat{\delta}=\left(\begin{array}[]{cc}\delta_{1}&0\\\
0&\delta_{2}\end{array}\right),$ (92)

is the matrix of two-photon detunings.

###  3.5 Equal fields

Let us consider a particular case where $\Omega_{2,0}=\Omega_{1,0}$ and
$\Omega_{2,2}=\Omega_{1,1}$ at the beginning of the medium ($z=0$). We also
assume that $\delta_{2}=\delta_{1}$. In this situation $\chi_{2}=\chi_{1}$ at
all times and positions. Equation for the propagation becomes

$\left(c^{-1}+\frac{g}{c\Omega_{1}^{2}}\frac{N_{1}^{4}}{N_{0}^{4}}\right)\partial_{t}\chi_{1}+\partial_{z}\chi_{1}+i\frac{g}{c\Omega_{1}^{2}}\frac{N_{1}^{4}}{N_{0}^{4}}\delta_{1}\chi_{1}=0\,.$
(93)

This equation has a similar form to the equation for the propagation of the
probe field in the EIT configuration. Also, it has the same form as the
equation for $\Lambda$-type system. The group velocity is

$v_{g}=\frac{c\Omega_{1}^{2}}{g}\frac{N_{0}^{4}}{N_{1}^{4}}\,.$ (94)

From this equation we see that the presence of the second pair of beams
increases the group velocity of the adiabaton propagation because
$N_{0}>N_{1}$. It can also be seen that the parts of the pulse with larger
amplitude propagate faster. This leads to changing shape of the pulse and to
an instability of a wave front.

We conducted numerical study of the above field equations using the Gaussian
input fields similarly to the case of $\Lambda$ system: the first field at the
input of the atomic media is of the form
$\Omega_{1,0}(0,t)=A\exp[-(t-t_{0})^{2}/\tau_{0}^{2}]$, where
$\tau_{0}=5\Gamma^{-1}$, $t_{0}=23\Gamma^{-1}$, $A=\Gamma$; the second one is
constant $\Omega_{1,1}(0,t)=1.5\Gamma$. Results of the calculation are shown
in Fig. 4 which demonstrates significant pulse deformation and strong
steepening effect at propagation distances exceeding $25L_{\mathrm{abs}}$.

Deformation of pulse shape observed in M-type system has similar origin as a
self-steepening effect known in classical nonlinear optics [40, 41]. The
latter develops as a result of nonlinear dependence of the refractive index on
pulse intensity that leads to the intensity-dependent group velocity. Commonly
this effect is observed for the propagation of short laser pulses producing
sharp steepening of the trailing edge of the pulse (optical shock) and
accompanying spectral broadening. The effect of self-steepening and
accompanying pulse behavior has been studied extensively in numerous papers
[40, 41, 42, 43].

Figure 4: Dependence of pulse amplitude ratio $\chi_{1}$ on the propagation
distance in M-type system calculated at time instance of $t=26.5\Gamma^{-1}$.

##  4 Adiabatons in double tripod system

###  4.1 Initial equations

Figure 5: Five level double tripod atomic system. Six laser beams with the
Rabi frequencies $\Omega_{1,0}$, $\Omega_{1,1}$, $\Omega_{1,2}$ and
$\Omega_{2,0}$, $\Omega_{2,1}$, $\Omega_{2,2}$ act on atoms characterized by
three hyperfine ground levels $0$, $1$ and $2$ as well as two excited levels
$e_{1}$ and $e_{2}$. Atoms are initially in the ground level $0$.

Finally, we investigate a double tripod type atomic system (Fig. 5) involving
three metastable ground states $|0\rangle$, $|1\rangle$ and $|2\rangle$, as
well as excited states $|e_{1}\rangle$ and $|e_{2}\rangle$. Six laser fields
with the Rabi frequencies $\Omega_{j,l}$ induce resonant transitions
$|l\rangle\rightarrow|e_{j}\rangle$. Applying the rotating wave approximation
(RWA), the atomic Hamiltonian in the rotating frame with respect to the atomic
levels reads

$H_{\mathrm{2T}}=\sum_{j=1}^{2}\left\\{-\frac{1}{2}\left(\sum_{l=0}^{2}\Omega_{j,l}|e_{j}\rangle\langle
l|+\mathrm{H.c.}\right)+\delta_{j}|j\rangle\langle
j|-\frac{i}{2}\Gamma|e_{j}\rangle\langle e_{j}|\right\\}\,,$ (95)

where $\delta_{j}$ are two-photon detunings. The losses in the Hamiltonian
(95) are taken into account in an effective way by introducing a rate $\Gamma$
of the excited state decay. The time-dependent Schrödinger equation
$i\hbar\partial_{t}|\Psi\rangle=H_{\mathrm{2T}}|\Psi\rangle$ for the atomic
state-vector $|\Psi\rangle$ yields the following equations for the atomic
probability amplitudes $\psi_{l}$, $\psi_{e_{j}}$:

$\displaystyle i\partial_{t}\psi_{l}$
$\displaystyle=\delta_{l}\psi_{l}-\frac{1}{2}\sum_{j}\Omega_{j,l}^{*}\psi_{e_{j}}\,,$
(96) $\displaystyle i\partial_{t}\psi_{e_{j}}$
$\displaystyle=-\frac{i}{2}\Gamma\psi_{e_{j}}-\frac{1}{2}\sum_{l}\Omega_{j,l}\psi_{l}\,.$
(97)

Here we introduced $\delta_{0}=0$. On the other hand, the Rabi frequencies of
the laser fields obey the following propagation equations

$(\partial_{t}+c\partial_{z})\Omega_{j,l}=\frac{i}{2}g\psi_{e_{j}}\psi_{l}^{*}\,,$
(98)

where the parameter $g$ characterizes the strength of coupling of the light
fields with the atoms.

###  4.2 Coupled and uncoupled states

It is convenient to introduce the amplitudes of the two coupled states

$\psi_{\mathrm{C}_{j}}=\frac{1}{\Omega_{j}}\sum_{l=0}^{2}\Omega_{j,l}\psi_{l}\,,$
(99)

where

$\Omega_{j}=\sqrt{\sum_{l=0}^{2}|\Omega_{j,l}|^{2}}\,.$ (100)

In addition, there is one uncoupled state

$\psi_{U}=\frac{1}{N_{0}}\left|\begin{array}[]{ccc}\psi_{0}&\psi_{1}&\psi_{2}\\\
\Omega_{1,0}^{*}&\Omega_{1,1}^{*}&\Omega_{1,2}^{*}\\\
\Omega_{2,0}^{*}&\Omega_{2,1}^{*}&\Omega_{2,2}^{*}\end{array}\right|\equiv\sum_{l=0}^{2}A_{l}^{*}\psi_{l}\,,$
(101)

where $N_{0}$ ensures that the uncoupled state is normalized. Here

$\displaystyle A_{0}$
$\displaystyle=\frac{1}{N_{0}}(\Omega_{1,1}\Omega_{2,2}-\Omega_{1,2}\Omega_{2,1})\,,$
(102) $\displaystyle A_{1}$
$\displaystyle=\frac{1}{N_{0}}(\Omega_{1,2}\Omega_{2,0}-\Omega_{1,0}\Omega_{2,2})\,,$
(103) $\displaystyle A_{2}$
$\displaystyle=\frac{1}{N_{0}}(\Omega_{1,0}\Omega_{2,1}-\Omega_{1,1}\Omega_{2,0}),$
(104)

and

$N_{0}=\left[|\Omega_{1,1}\Omega_{2,2}-\Omega_{1,2}\Omega_{2,1}|^{2}+|\Omega_{1,2}\Omega_{2,0}-\Omega_{1,0}\Omega_{2,2}|^{2}+|\Omega_{1,0}\Omega_{2,1}-\Omega_{1,1}\Omega_{2,0}|^{2}\right]^{\frac{1}{2}}\,.$
(105)

Note, that

$N_{0}^{2}=\Omega_{1}^{2}\Omega_{2}^{2}-\left(\sum_{l=0}^{2}\Omega_{2,l}^{*}\Omega_{1,l}\right)\left(\sum_{l=0}^{2}\Omega_{1,l}^{*}\Omega_{2,l}\right)\,.$
(106)

The initial atomic amplitudes then can be expressed as

$\psi_{l}=A_{l}\psi_{\mathrm{U}}+\cdots\,$ (107)

The ommited terms in Eq. (107) contain amplitudes $\psi_{\mathrm{C}_{1}}$,
$\psi_{\mathrm{C}_{2}}$. Note, that the coefficients $A_{l}$ have the property

$\sum_{l=0}^{2}A_{l}\Omega_{j,l}=0\,.$ (108)

The equations (96), (97) for the atomic amplitudes in terms of coupled and
uncoupled states take the form

$\displaystyle i\partial_{t}\psi_{\mathrm{U}}$
$\displaystyle=\Delta\psi_{\mathrm{U}}+\cdots\,,$ (109) $\displaystyle
i\partial_{t}\psi_{\mathrm{C}_{j}}$
$\displaystyle=V_{j}\psi_{\mathrm{U}}-\frac{1}{2\Omega_{j}}\sum_{m}\psi_{e_{m}}\sum_{l=0}^{2}\Omega_{j,l}\Omega_{m,l}^{*}\,,$
(110) $\displaystyle i\partial_{t}\psi_{e_{j}}$
$\displaystyle=-\frac{i}{2}\Gamma\psi_{e_{j}}-\frac{1}{2}\Omega_{j}\psi_{\mathrm{C}_{j}}\,.$
(111)

Here,

$\displaystyle\Delta$
$\displaystyle=\sum_{l=0}^{2}A_{l}(i\partial_{t}+\delta_{l})A_{l}^{*}\,,$
(112) $\displaystyle V_{j}$
$\displaystyle=\frac{1}{\Omega_{j}}\sum_{l=0}^{2}A_{l}(i\partial_{t}+\delta_{l})\Omega_{j,l}=\sum_{l=0}^{2}A_{l}(i\partial_{t}+\delta_{l})\frac{\Omega_{j,l}}{\Omega_{j}}\,.$
(113)

The terms omitted in Eq. (109) contain amplitudes $\psi_{\mathrm{C}_{1}}$ and
$\psi_{\mathrm{C}_{2}}$.

###  4.3 Adiabatic approximation

Let us consider a situation where the light fields are slowly changing and
have large intensities. In this situation we can apply adiabatic
approximation. Neglecting small amplitudes of the coupled states
$\psi_{\mathrm{C}_{1}}$ and $\psi_{\mathrm{C}_{2}}$ in Eq. (110) leads to a
system of equations

$2\Omega_{j}V_{j}\psi_{\mathrm{U}}=\sum_{m}\psi_{e_{m}}\sum_{l=0}^{2}\Omega_{j,l}\Omega_{m,l}^{*},$
(114)

with the solution

$\psi_{e_{j}}=2W_{j}\psi_{\mathrm{U}}\,,$ (115)

where

$\displaystyle W_{1}$
$\displaystyle=\frac{1}{N_{0}^{2}}(\Omega_{2}^{2}\Omega_{1}V_{1}-\sum_{l=0}^{2}\Omega_{2,l}^{*}\Omega_{1,l}\Omega_{2}V_{2})$
$\displaystyle=\frac{1}{N_{0}^{2}}\sum_{l=0}^{2}A_{l}\left(\Omega_{2}^{2}(i\partial_{t}+\delta_{l})\Omega_{1,l}-\sum_{m=0}^{2}\Omega_{2,m}^{*}\Omega_{1,m}(i\partial_{t}+\delta_{l})\Omega_{2,l}\right)\,,$
(116) $\displaystyle W_{2}$
$\displaystyle=\frac{1}{N_{0}^{2}}(\Omega_{1}^{2}\Omega_{2}V_{2}-\sum_{l=0}^{2}\Omega_{1,l}^{*}\Omega_{2,l}\Omega_{1}V_{1})$
$\displaystyle=\frac{1}{N_{0}^{2}}\sum_{l=0}^{2}A_{l}\left(\Omega_{1}^{2}(i\partial_{t}+\delta_{l})\Omega_{2,l}-\sum_{m=0}^{2}\Omega_{1,m}^{*}\Omega_{2,m}(i\partial_{t}+\delta_{l})\Omega_{1,l}\right)\,.$
(117)

On the other hand, Eq. (111) relates $\psi_{\mathrm{C}_{j}}$ to $\psi_{e_{j}}$
as

$\psi_{\mathrm{C}_{j}}=-i\frac{\Gamma}{\Omega_{j}}\psi_{e_{j}}\,.$ (118)

Neglecting the decay of adiabatons, Eq. (109) for the amplitude of the
uncoupled state becomes

$i\partial_{t}\psi_{\mathrm{U}}=\Delta\psi_{\mathrm{U}}\,.$ (119)

Inserting Eq. (115) into Eq. (98) we get

$(\partial_{t}+c\partial_{z})\Omega_{j,l}=igW_{j}A_{l}^{*}\,.$ (120)

Equations (119) and (120) describe adiabatic propagation of the fields. For
the validity of the adiabatic apporximation one should require that the
population of the excited states be small compared to the population of the
uncoupled state. According to Eq. (109) this leads to the condition

$|W_{j}|\ll 1\,.$ (121)

From Eq. (120) we obtain that the total Rabi frequencies $\Omega_{j}$ obey the
equations

$(\partial_{t}+c\partial_{z})\Omega_{j}=0.$ (122)

In addition, we have

$(\partial_{t}+c\partial_{z})\sum_{l=0}^{2}\Omega_{2,l}^{*}\Omega_{1,l}=0.$
(123)

From Eqs. (122), (123) it follows

$(\partial_{t}+c\partial_{z})N_{0}=0\,.$ (124)

###  4.4 Matrix form of equations

Introducing the two-component spinors (columns)

$\hat{\Omega}_{l}=\left(\begin{array}[]{c}\Omega_{1,l}\\\
\Omega_{2,l}\end{array}\right).$ (125)

We can write Eqs. (120) as

$(c^{-1}\partial_{t}+\partial_{z})\hat{\Omega}_{l}+A_{l}^{*}\hat{v}^{-1}\sum_{n=0}^{2}A_{n}(\partial_{t}-i\delta_{n})\hat{\Omega}_{n}=0\,,$
(126)

where

$\hat{v}^{-1}=\frac{g}{c}\frac{1}{N_{0}^{2}}\left(\begin{array}[]{cc}\Omega_{2}^{2}&-\sum_{m=0}^{2}\Omega_{2,m}^{*}\Omega_{1,m}\\\
-\sum_{m=0}^{2}\Omega_{1,m}^{*}\Omega_{2,m}&\Omega_{1}^{2}\end{array}\right),$
(127)

is the matrix of inverse group velocity [37]. Note that

$\det\hat{v}^{-1}=\frac{g^{2}}{c^{2}N_{0}^{2}},$ (128)

and

$(\partial_{t}+c\partial_{z})\hat{v}^{-1}=0\,.$ (129)

Multiplying Eq. (126) by $A_{l}$ and summing we get

$\sum_{l=0}^{2}A_{l}(c^{-1}\partial_{t}+\partial_{z})\hat{\Omega}_{l}+\hat{v}^{-1}\sum_{l=0}^{2}A_{l}(\partial_{t}-i\delta_{l})\hat{\Omega}_{l}=0\,.$
(130)

Finally, using the matrix of inverse group velocity $\hat{v}^{-1}$ the
adiabatic condition (121) in double tripod system can be written as

$\frac{c}{g}\left|\hat{v}^{-1}\sum_{l=0}^{2}A_{l}(i\partial_{t}+\delta_{l})\hat{\Omega}_{l}\right|\ll
1\,.$ (131)

The condition should hold for each component of the column in vertical bars.

###  4.5 Adiabatons

Let us consider the situation where there are no two-photon detunings,
$\delta_{l}=0$. We will search for a propagating solution in the form
$\Omega_{j,l}=f_{j,l}\left(t-\frac{1}{v_{g}}z\right)$. Then
$\partial_{z}\Omega_{j,l}=-v_{g}^{-1}\partial_{t}\Omega_{j,l}$ and Eq. (126)
becomes

$v_{g}^{-1}\partial_{t}\hat{\Omega}_{l}=A_{l}^{*}\hat{v}^{-1}\sum_{n=0}^{2}A_{n}\partial_{t}\hat{\Omega}_{n}\,.$
(132)

Multiplying this equation by $A_{l}$ and summing over $l$ we obtain

$\hat{v}^{-1}\sum_{l=0}^{2}A_{l}\partial_{t}\hat{\Omega}_{l}=v_{g}^{-1}\sum_{l=0}^{2}A_{l}\partial_{t}\hat{\Omega}_{l}\,.$
(133)

We see that in this case $\sum_{l=0}^{2}A_{l}\partial_{t}\hat{\Omega}_{l}$
should be an eigenvector of a matrix $\hat{v}^{-1}$, with $v_{g}$ being the
eigenvalue. Note, that since the adiabatic condition (131) contains inverse
group velocity, the adiabaticity requirement for the solution with smaller
value of $v_{g}$ is stricter.

Let us write the eigenvector of the matrix $\hat{v}^{-1}$ corresponding to an
eigenvalue $v_{g}^{-1}$ as $(1,\xi)^{T}$. Since this eigenvector is equal to
$\sum_{l=0}^{2}A_{l}\partial_{t}\hat{\Omega}_{l}$, we have

$\sum_{l=0}^{2}A_{l}\partial_{t}\Omega_{2,l}=\xi\sum_{l=0}^{2}A_{l}\partial_{t}\Omega_{1,l}\,.$

Then Eq. (132) can be written as

$\displaystyle\partial_{t}\Omega_{1,l}$
$\displaystyle=A_{l}^{*}\sum_{n=0}^{2}A_{n}\partial_{t}\Omega_{1,n}\,,$ (134)
$\displaystyle\partial_{t}\Omega_{2,l}$ $\displaystyle=\xi
A_{l}^{*}\sum_{n=0}^{2}A_{n}\partial_{t}\Omega_{1,n},$ (135)

or

$\partial_{t}\Omega_{2,l}=\xi\partial_{t}\Omega_{1,l},$ (136)

with the solution

$\Omega_{2,l}(t)-\xi\Omega_{1,l}(t)=\Omega_{2,l}(0)-\xi\Omega_{1,l}(0)\,.$
(137)

Initially the pulses $\Omega_{1,0}$ and $\Omega_{2,0}$ are absent, thus
$\Omega_{2,l}(t)=\xi\Omega_{1,l}(t)$.

As a consequence of Eqs. (122), (123) the propagating solution obeys the
equations

$\Omega_{j}=\mathrm{const}\,,\qquad\sum_{l=0}^{2}\Omega_{2,l}^{*}\Omega_{1,l}=\mathrm{const}\,.$
(138)

These equations ensure that the velocity $v_{g}$ remains constant.

Let us consider the situation when $\Omega_{2,2}(0)=\Omega_{1,1}(0)$ and
$\Omega_{2,1}(0)=\Omega_{1,2}(0)$. Then the eigenvalues are
$v_{g}=\frac{c}{g}[\Omega_{1,1}(0)-\xi\Omega_{1,2}(0)]^{2}$ with $\xi=\pm 1$ .
From the equations it follows that $\Omega_{2,2}=\Omega_{1,1}$ and
$\Omega_{2,1}=\Omega_{1,2}$ for all times. The values of $\Omega_{1,1}$ and
$\Omega_{1,2}$ can be found by solving the equations
$\Omega_{1,2}-\xi\Omega_{1,1}=\Omega_{1,2}(0)-\xi\Omega_{1,1}(0)$,
$\Omega_{1,0}^{2}+\Omega_{1,1}^{2}+\Omega_{1,2}^{2}=\Omega_{1,1}^{2}(0)+\Omega_{1,2}^{2}(0)$.

Similarly to the previous cases, we used numerical calculations to demonstrate
the obtained effects. At the begining of the atomi media two optical fields
have Gaussian temporal shape:
$\Omega_{1,0}(0,t)=A_{1}\exp[-(t-t_{1})^{2}/\tau_{1}^{2}]$,
$\Omega_{2,0}(0,t)=A_{2}\exp[-(t-t_{2})^{2}/\tau_{2}^{2}]$, where
$\tau_{1}=\tau_{2}=5\Gamma^{-1}$, $t_{1}=t_{2}=23\Gamma^{-1}$, $A_{1}=1$,
$A_{2}=0$; other fields are constant with amplitudes
$\Omega_{1,1}(0,t)=\Omega_{2,2}(0,t)=1.5\Gamma$ and
$\Omega_{1,2}(0,t)=\Omega_{2,1}(0,t)=0.5\Gamma$. Corresponding numerical
results are depicted in Fig. 6. It can be seen that there is adiabatonic
regime for the fields $\Omega_{j,l}$ after some propagation distance. There
are two combinations of fields propagating with different group velocities.
The first combination seen on the right part of Fig. 6 has
$\Omega_{2,0}(z,t)=-\Omega_{1,0}(z,t)$ and is propagating with smaller group
velocity. The second combination having $\Omega_{2,0}(z,t)=\Omega_{1,0}(z,t)$
is propagating with larger group velocity and can be seen in the left part of
Fig. 6.

Figure 6: Temporal dependence of pulse amplitudes in double tripod system.
Dashed line corresponds to pulse envelope $\Omega_{1,0}$ at the input ($z=0$),
solid lines show pulse envelopes at propagation distance
$z=50L_{\mathrm{abs}}$. Amplitude shown on the vertical scale is measured in
$\Gamma$.

##  5 Discussion and conclusions

In summary, we have analyzed the propagation of optical pulses in multilevel
systems of M-type and double tripod under the adiabatic approximation. Both
systems contain five non-degenerate atomic levels and exhibit an uncoupled
(dark) state. We also compared our results to a three level $\Lambda$ system
which has been used as a well known reference system for our theoretical
approach. It was found that in the case of M-type atomic system the group
velocity depends on the amplitudes of the fields, thus different part of the
pulse propagate with different velocity. As a result, a region of the pulse
with large steepness appears and the adiabatic approximation becomes not valid
after some distance. In contrast, in the double-tripod system there are two
different configurations of the fields that propagate without changing shape
and the adiabatic propagation of light pulses is possible. The latter system
can be realized experimentally like the one used as a setup demonstrating SSL
formation in the atomic system [37]. In that study a double tripod scheme
utilizing laser-cooled Rb atoms loaded into magneto-optical trap was explored.
Using similar setup with the appropriate parameters the results of our
theoretical model presented in Fig. 6 can be verified experimentally. To
conclude, our study demonstrates that in the application to many level systems
the adiabatic propagation of light pulses is configuration-sensitive and is
not common to all multi-level systems having an uncoupled (dark) state.

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