# Control with EIT: High energy charged particle detection

Aneesh Ramaswamy, Svetlana A.Malinovskaya
Department of Physics, Stevens Institute of Technology, NJ 07030, USA
<EMAIL_ADDRESS>
###### Abstract

The strong non-linear optical response of atomic systems in
electromagnetically induced transparency (EIT) states is considered as a means
to detect the presence of small perturbations to steady states. For the 3
level system, expressions for the group velocity and group velocity dispersion
(GVD) were derived and a quantum control protocol was established to account
for the change in the chirp spectrum of a probe pulse when the steady state
was perturbed. This was applied to the propagation of slow Cherenkov
polaritons in the medium due to the passage of a train of high-energy charged
particles (high energy particles). The choice of the initial steady state with
focus on the slow light condition and strong narrowly confined dispersion,
equated to the continuous trapping of Cherenkov polaritons in the medium along
a narrow group cone, allowing for non-trivial fields to accumulate.
Considering another medium prepared for the detection of the radiation,
sweeping of the control field and detuning parameters in the field-atom
parameter space showed the presence of optimal regions to maximize the first
order perturbation in the coherences creating changes in the optical responses
that modify the chirp spectra of probe pulses.

###### Contents

  1. 1 Introduction
    1. 1.1 A brief review on EIT
  2. 2 Developing a picture of EIT with three-level $\Lambda$ systems
    1. 2.1 Hamiltonian in the Field Interaction Picture
    2. 2.2 Lindblad Formalism: Finding EIT steady states
    3. 2.3 Optical response functions
  3. 3 Control scheme
    1. 3.1 The concept
      1. 3.1.1 Control using group velocity, group velocity dispersion and the chirp spectrum
    2. 3.2 Optical response functions in the control regime
  4. 4 Application: Detecting high-energy charged particles using EIT
    1. 4.1 A model using Fourier transformed Maxwell’s equations
    2. 4.2 Cherenkov radiation in the EIT regime
    3. 4.3 Perturbation to steady states: Developing a control method to detect high energy particles
      1. 4.3.1 Control method summary
  5. 5 Summary and vision for nanophotonics

##  1 Introduction

###  1.1 A brief review on EIT

The use of coherent interactions between light and matter yield phenomena in
which there is drastic change in the optical response function. EIT is one of
those phenomenon in which a narrow transparency window with strong non-linear
optical effects is achieved due to interference between quantum excitation
pathways. The result is an almost negligible amplitude and for an absorptive
transition that shows us Fano resonances in the optical response functions. Of
prime interest is the greatly reduced absorption and drastically increased
dispersion effects. Whilst its effects has analogues in CPT (Coherent
population trapping), which occurs in optically thin media, and ATS (Autler-
Townes Splitting); EIT is a phenomenon that involves modification of both the
optical and material states of the coupled light-matter system and hence
occurs in optically thick systems. Of particular interest is the study of
quantum fields in EIT systems as dark state polaritons, the pseudo-particle
arising from the entanglement of the propagating light with the dipole
transitions.

Theoretical studies have opened up several applications including generation
of non-classical atomic ensembles, high-resolution spectroscopy and reversible
quantum memories in optical systems. A proposed technique by Fleischhauer and
Mewes to greatly increase storage time for use in quantum memory is in
adiabatically reducing the control field’s strength to bring the light to a
halt, effectively mapping the light’s quantum state to the atomic spin
ensemble and then achieve reconstruction through restoring the control field’s
strength [DPS, StopDPS]. An experiment realization of this protocol in a
laser-cooled ensemble of a cloud of Rb-87 atoms in a magneto-optical trap
achieved a favourable result of 0.036 storage efficiency for a transport
distance of 1.26 mm (1/e size of the cloud) for a lifetime of 2.6 ms [transp].
The approach and results show a high promise for EIT in developing robust
mechanisms for quantum optical storage with increased transport distance,
though reducing decoherence due to natural lifetime and dephasing is still a
problem of study. EIT also has great promise in few photon systems and
creating schemes for resonant non-linear interactions that create large
single-photon phase shifts, which is of great interest in quantum information,
such as in the development of high fidelity quantum gates. There have been a
number of theoretical and experimental approaches to achieve high fidelity
whilst reducing the characteristic residual absorption of EIT [slpg, cqpg].

Further uses of EIT include transmission with negligible dissipation, non-
linear optics in the weak field regime, and slow light. Moreover EIT does not
require the highly controlled experimental setups (e.g ultra-low temperatures)
for these phenomena to occur. Whilst EIT was initially discovered in
atomic/molecular systems (chief of which is the three-level $\Lambda$ system),
it has been studied in optomechanical systems, plasmonics, coupled
microresonators, solid-state physics and photonic crystals[Whisp]. Especially
considering the rise of interest in quantum metrology using atomic systems,
EIT is a powerful candidate to investigate in the increased performance and
precision in atomic clocks, in high-resolution atomic interferometry and in
magnetoptic measurements. Traditional weaknesses of EIT including low signal-
to-noise ratio and fidelity, have been compensated for by the great
versatility in control protocols that can be developed to target phenomena of
interest. For example, an investigation into the frequency stability of atomic
clocks found higher stability when tuning EIT towards detecting magneto-
optically induced light polarization shifts rather than controlling the
intensity of EIT fields [guidry]. This has naturally led to a great interest
in EIT for quantum control theory for developing efficient practical protocols
for optimizing performance and precision.

Our study will be on the three-level $\Lambda$ system, a simple starting point
to gain a strong understanding in the features and challenges with EIT.

##  2 Developing a picture of EIT with three-level $\Lambda$ systems

We consider the 3-level $\Lambda$ system with stable energy levels $\ket{1}$,
$\ket{2}$ and metastable excited state $\ket{3}$. Only the 1-3,2-3 transitions
are allowed. We will be considering the dynamics with the 1st Born
approximation and the Markovian approximation. Most of our work will be based
on a Lindbladian master equation.

An archetypal example of such a system is for the Rubidium $D_{2}$ transition
fine splittings of the $5S_{1/2}$,$F=1$ and $5P_{1/2}$,$F=2$ energy levels. We
take our ground state, $\ket{1}$, as $[5S_{1/2},F=1,m_{f}=0]$, the excited
state, $\ket{3}$, as $[5P_{1/2},F=1,m_{f}=1]$ and the Raman ground
state,$\ket{2}$, as $[5S_{1/2},F=2,m_{f}=0]$. The 1-3 transition frequency is
377 THz (not including the contributions from the fine splitting differences)
and the Stokes shift is 6.384 GHz. The natural lifetime of the transition
$\tau=2\pi/\Gamma$=27.7 ns. The hyperfine transition dipole matrix elements
for our chosen transitions is 2.54 Debye[ldata].

We first develop our model’s interaction Hamiltonian.

Figure 1: Three level $\Lambda$ system with laser frequency $\omega_{L}$, Rabi
frequency $\Omega$, laser phase modulation $\phi$, one-photon detuning
$\Delta$, two-photon detuning $\delta=\Delta_{2}-\Delta_{1}$ and decay rate
$\Gamma$.

###  2.1 Hamiltonian in the Field Interaction Picture

We consider $\ket{3}$ to be the zero energy reference and we define the pump
$E_{1}$ and Stokes $E_{2}$ electric fields as below

$E_{j}(t)=E^{0}_{j}(t)\cos(\omega_{L_{j}}(t-t_{c})+\phi_{j}(t))$

The Rabi frequency is given by:

$\Omega_{j}(t)=\frac{|p_{j3}|E^{0}_{j}(t)}{\hbar}$

The dipole moment can be complex, but we introduce the phase function into the
oscillatory part of the field. We have:

$\displaystyle p_{j3}$ $\displaystyle=|p_{j3}|e^{i\theta_{j}}$
$\displaystyle\tilde{\phi}_{j}(t)=\phi_{j}(t)+\theta_{j}$

We now define the interaction Hamiltonian $H_{int}$ where:

$H_{int}=\hat{{U}}H_{int}\hat{{U}}^{\dagger}$

We use the unitary operator to go to the interaction picture and apply the
rotating wave approximation (RWA) where:

$H_{int}^{j3}=\dfrac{\hbar}{2}\Omega_{1}(t)(e^{i((2\omega_{L_{j}}+\Delta_{j})(t-t_{c})+\tilde{\phi}_{j}(t)}+e^{-i(\Delta_{j}(t-t_{c})+\tilde{\phi}_{j}(t)})$

$H_{int}^{3j}=(H_{int}^{j3})^{*}$

And all other terms are zero. We ignore the fast oscillating term and
transform back to the Schrodinger picture, and now consider the field
interaction picture using the unitary operator:

$\hat{{U}}(t)=e^{-i\omega_{L_{j}}t/\hbar}$ $\hat{{U}}(t)=e^{-iH_{0}t/\hbar}$

We arrive at the field interaction Hamiltonian

$H_{FE}=\dfrac{\hbar}{2}\begin{pmatrix}2\Delta_{1}&0&\Omega_{1}(t)e^{i\tilde{\phi}_{1}(t)}\\\
0&2\Delta_{2}&\Omega_{2}(t)e^{i\tilde{\phi}_{2}(t)}\\\
\Omega_{1}(t)e^{-i\tilde{\phi}_{1}(t)}&\Omega_{2}(t)e^{-i\tilde{\phi}_{2}(t)}&0\end{pmatrix}$

.

###  2.2 Lindblad Formalism: Finding EIT steady states

Now we consider the case for open dynamics. We use
$\Delta_{2}=\Delta_{1}-\delta$ and define the master equation with the
superoperator $\mathcal{L}$:

$\dot{\rho}=-\dfrac{i}{\hbar}\commutator{H_{FE}}{\rho}+\mathcal{L_{\mathcal{J}}}\rho$

$\displaystyle\mathcal{L_{\mathcal{J}}}\rho=\sum_{j\in\mathcal{J}}\lambda_{j}\mathcal{F}_{j}(\rho)$
$\displaystyle\mathcal{F}_{j}(\rho)=F_{j}\rho
F_{j}^{\dagger}-\dfrac{1}{2}\left\\{F_{j}^{\dagger}{F}_{j},\rho\right\\}$

$\displaystyle\\{F_{j}\\}=\\{\ket{1}\bra{3},\ket{2}\bra{3},\ket{1}\bra{1},\ket{2}\bra{2},\ket{3}\bra{3}\\}$
$\displaystyle\\{\lambda_{j}\\}=\left\\{\dfrac{\Gamma_{1}}{2},\dfrac{\Gamma_{2}}{2},\dfrac{\gamma_{12}+\gamma_{2}-\gamma_{3}}{2},\dfrac{\gamma_{12}-\gamma_{2}+\gamma_{3}}{2},\dfrac{-\gamma_{12}+\gamma_{2}+\gamma_{3}}{2}\right\\}$

$\mathcal{L}\rho=\begin{pmatrix}\Gamma_{1}\rho_{33}&-\gamma_{12}\rho_{12}&-(\dfrac{\Gamma_{1}+\Gamma_{2}}{2}+\gamma_{1})\rho_{13}\\\
-\gamma_{12}\rho_{21}&\Gamma_{2}\rho_{33}&-(\dfrac{\Gamma_{1}+\Gamma_{2}}{2}+\gamma_{2})\rho_{23}\\\
-(\dfrac{\Gamma_{1}+\Gamma_{2}}{2}+\gamma_{1})\rho_{31}&-(\dfrac{\Gamma_{1}+\Gamma_{2}}{2}+\gamma_{2})\rho_{32}&-(\Gamma_{1}+\Gamma_{2})\rho_{33}\end{pmatrix}$

.

We define $\Gamma_{j}$ to be the natural decay rate from the excited state to
the jth state and $\gamma$ are dephasing parameters to account for thermal
broadening and other losses of coherence. We then define the dephasing
parameters:

$\displaystyle\gamma_{31}$
$\displaystyle=\dfrac{\Gamma_{1}+\Gamma_{2}}{2}+\gamma_{1}$
$\displaystyle\gamma_{32}$
$\displaystyle=\dfrac{\Gamma_{1}+\Gamma_{2}}{2}+\gamma_{2}$
$\displaystyle\gamma$ $\displaystyle=\gamma_{12}.$

We find stationary solutions for this open system that corresponds to an EIT
state, we use the Gell-Mann matrices as our basis and solve for the Bloch
vector:

$\displaystyle\expectationvalue{\vec{\sigma}}=\Tr{\vec{\rho\sigma}}$
$\displaystyle\expectationvalue{\vec{\sigma}}=(\expectationvalue{\sigma_{1}},...,\expectationvalue{\sigma_{9}})$

We solve for the density matrix terms in the Bloch equations using the below
initial condition:

$\displaystyle\expectationvalue{\dot{\vec{\sigma}}(0)}=0$

We find the coherence terms, reducing equations to depend on one unknown
$\rho_{12}$, assuming the populations for determined parameters are known and
fixed:

$\displaystyle\rho_{13}$
$\displaystyle=\dfrac{i\gamma_{31}+\Delta_{1}}{2(\gamma_{31}^{2}+\Delta_{1}^{2})}\left(\Omega_{1}(t)e^{i\tilde{\phi}_{1}(t)}(\rho_{11}-\rho_{33})+\Omega_{2}(t)e^{i\tilde{\phi}_{2}(t)}\rho_{12})\right)$
$\displaystyle\rho_{23}$
$\displaystyle=\dfrac{i\gamma_{32}+\Delta_{2}}{2(\gamma_{32}^{2}+\Delta_{2}^{2})}\left(\Omega_{1}(t)e^{i\tilde{\phi}_{1}(t)}\rho_{21}+\Omega_{2}(t)e^{i\tilde{\phi}_{2}(t)}(\rho_{22}-\rho_{33})\right).$

The solution to the differential equation for $\rho_{12}$ are obtained by
substituting the above expressions for $\rho_{13}$, $\rho_{23}$:

$\displaystyle\dot{\rho}_{12}$
$\displaystyle=(-\Gamma^{\prime}+i\omega^{\prime})\rho_{12}+K.$
$\displaystyle\Gamma^{\prime}$
$\displaystyle=\gamma+\dfrac{\gamma_{31}}{4}\left(\dfrac{\gamma_{31}}{\gamma_{32}}\dfrac{\left(\dfrac{\Omega_{1}}{\gamma_{31}}\right)^{2}}{1+\left(\dfrac{\Delta_{2}}{\gamma_{31}}\right)^{2}\left(\dfrac{\gamma_{31}}{\gamma_{32}}\right)^{2}}+\dfrac{\left(\dfrac{\Omega_{2}}{\gamma_{31}}\right)^{2}}{1+\left(\dfrac{\Delta_{1}}{\gamma_{31}}\right)^{2}}\right)$
$\displaystyle\omega^{\prime}$
$\displaystyle=\delta+\dfrac{\gamma_{31}}{4}\left(-\dfrac{\Delta_{2}}{\gamma_{31}}\left(\dfrac{\gamma_{31}}{\gamma_{32}}\right)^{2}\dfrac{\left(\dfrac{\Omega_{1}}{\gamma_{31}}\right)^{2}}{1+\left(\dfrac{\Delta_{2}}{\gamma_{31}}\right)^{2}\left(\dfrac{\gamma_{31}}{\gamma_{32}}\right)^{2}}+\dfrac{\Delta_{1}}{\gamma_{31}}\dfrac{\left(\dfrac{\Omega_{2}}{\gamma_{31}}\right)^{2}}{1+\left(\dfrac{\Delta_{1}}{\gamma_{31}}\right)^{2}}\right).$
$K=\dfrac{\gamma_{31}}{4}\left(-\left(\dfrac{\gamma_{31}}{\gamma_{32}}\right)^{2}\dfrac{\Omega_{1}\Omega_{2}}{\gamma_{31}^{2}}\dfrac{(\rho_{11}-\rho_{33})e^{i(\tilde{\phi}_{1}(t)-\tilde{\phi}_{2}(t))}}{1+\left(\dfrac{\Delta_{2}}{\gamma_{31}}\right)^{2}\left(\dfrac{\gamma_{31}}{\gamma_{32}}\right)^{2}}\left(1+i\dfrac{\Delta_{2}}{\gamma_{31}}\dfrac{\gamma_{31}}{\gamma_{32}}\right)\right)\\\
+\dfrac{\gamma_{31}}{4}\left(\dfrac{\Omega_{1}\Omega_{2}}{\gamma_{31}^{2}}\dfrac{(\rho_{22}-\rho_{33})e^{i(\tilde{\phi}_{1}(t)-\tilde{\phi}_{2}(t))}}{1+\left(\dfrac{\Delta_{1}}{\gamma_{31}}\right)^{2}}\left(-1+i\dfrac{\Delta_{1}}{\gamma_{31}}\right)\right)$

In the weak field approximation, we generally see a significant oscillatory
behaviour only for the case of non-zero detuning. The scattering parameter
also depends mainly on dephasing between the two ground states when
$\gamma_{3j}$ is large. Now we solve for the initial value of $\rho_{12}$ at
the stationary point:

$\displaystyle\rho_{12}^{s}$
$\displaystyle=\dfrac{K}{\Gamma^{\prime}-i\omega^{\prime}}.$

Next we find the expressions for the steady-state coherences. The expression
of $\rho_{13}^{s}$ and $\rho_{23}^{s}$ is our main interest as it is a measure
of the dispersion and the absorption of radiation near the resonance of the
($\ket{1}\rightarrow\ket{3}$) transition and will be used to determine the
optical response functions

$\displaystyle\rho_{13}^{s}=\dfrac{\dfrac{\Delta_{1}}{\gamma_{31}}+i}{1+\left(\dfrac{\Delta_{1}}{\gamma_{31}}\right)^{2}}\left(\dfrac{\Omega_{1}}{\gamma_{31}}e^{i\tilde{\phi}_{1}(t)}(\rho_{11}-\rho_{33})+\dfrac{\dfrac{\Omega_{2}}{\gamma_{31}}\dfrac{K}{\gamma_{31}}\left(\dfrac{\Gamma^{\prime}}{\gamma_{31}}+i\dfrac{\omega^{\prime}}{\gamma_{31}}\right)e^{i\tilde{\phi}_{2}(t)}}{\left(\dfrac{\Gamma^{\prime}}{\gamma_{31}}\right)^{2}+\left(\dfrac{\omega^{\prime}}{\gamma_{31}}\right)^{2}}\right)$

$\displaystyle\rho_{32}^{s}=\dfrac{\dfrac{\Delta_{2}}{\gamma_{32}}+i}{1+\left(\dfrac{\Delta_{2}}{\gamma_{32}}\right)^{2}}\left(\dfrac{\Omega_{2}}{\gamma_{32}}e^{-i\tilde{\phi}_{2}(t)}(\rho_{22}-\rho_{33})-\dfrac{\dfrac{\Omega_{1}}{\gamma_{32}}\dfrac{K}{\gamma_{32}}\left(\dfrac{\Gamma^{\prime}}{\gamma_{32}}+i\dfrac{\omega^{\prime}}{\gamma_{32}}\right)e^{-i\tilde{\phi}_{1}(t)}}{\left(\dfrac{\Gamma^{\prime}}{\gamma_{32}}\right)^{2}+\left(\dfrac{\omega^{\prime}}{\gamma_{32}}\right)^{2}}\right)/$

We note that the system has a natural timescale dependent on the dephasing
parameters $\gamma_{3j}$. For the rest of this chapter, we will set
$\gamma_{31}=1$.

The above is the general results for the coherences for the steady-state
solution. In the weak-field approximation, the fields being smaller than the
natural decay rate implies that the steady state is unique as all the other
eigenvalues of the Lindbladian have negative real parts. With a model of the
microscopic dynamics understood, we now build the optical response functions.

###  2.3 Optical response functions

We find the expectation for the dipole density operator. First, we rotate from
the from the corotating field interaction picture back to the Schrödinger
picture using (9):

$\displaystyle\tilde{\rho}(t)=U(t)^{\dagger}\rho(t)U(t).$

When we rotate back to the ground reference frame, the coherence terms
oscillate with the frequencies of the fields driving the transitions:

$\displaystyle\tilde{\rho}_{3j}(t)=\rho_{3j}(t)e^{-i\omega_{L_{j}}t}.$

We compare the terms for the polarization field in terms of the electric field
and susceptibility tensor, and the density matrix description. Here
$\chi(\omega,E)$ is a generally non-linear function and depends on the
frequency and the amplitude of the electric field. Then the polarization
reads:

$\displaystyle\mathbf{P(\mathbf{r},\mathnormal{t})}$
$\displaystyle=\chi(t)*E(\mathbf{r},t)=\int_{-\infty}^{t}\chi(t-t^{\prime})E(\mathbf{r},t^{\prime})dt^{\prime}$

If we use a control protocol where we vary the field properties over time or
if the envelope is time dependent, it’s not obvious what the form of the
polarization will be.

In the case where we have a superposition of monochromatic plane waves, the
expression simplifies to:

$\displaystyle\mathbf{P(\mathbf{r},\mathnormal{t})}$
$\displaystyle=\sum_{k=1}^{n}(E_{0}/2(\chi_{k}(\omega,t)e^{-i\omega_{k}t}+\chi_{k}^{\dagger}(\omega,t)e^{i\omega_{k}t}))$

For the case where we have a general field of the form
$E(\mathbf{r},t)=E_{0}(\mathbf{r},t)cos(\omega_{l}t+\phi)$, where we take
$\phi$ to be constant, we get the following:

$\displaystyle\mathbf{P(\mathbf{r},\mathnormal{t})}$
$\displaystyle=(\chi(\omega,t)*E_{0}(\mathbf{r},\omega))e^{-i\omega_{l}t}+(\chi^{\dagger}(\omega,t)*E_{0}(\mathbf{r},\omega))e^{i\omega_{l}t}.$

For the rest of the chapter, we deal with CW light and monochromatic plane
waves.

$P=\epsilon_{0}\chi(\omega,E)E=\dfrac{\epsilon_{0}\hbar\Omega_{1}}{2|p_{13}|}(\chi^{*}_{1}(\omega,E)e^{i(\omega_{L_{1}}t+\phi_{1})}+\chi_{1}e^{-i(\omega_{L_{1}}t+\phi_{1})})\\\
+\dfrac{\epsilon_{0}\hbar\Omega_{2}}{2|p_{23}|}(\chi^{*}_{2}(\omega,E)(e^{i(\omega_{L_{2}}t+\phi_{2})}+\chi_{2}e^{-i(\omega_{L_{2}}t+\phi_{2})})$
(1)

$\expectationvalue{p}=\Tr{\rho
p}=\zeta_{1}(\omega,t)+\zeta_{2}(\omega,t)\equiv|p_{13}|/2(\rho_{13}e^{i(\omega_{L_{1}}t+\phi_{1}+\theta_{1})}+\rho_{31}e^{-i(\omega_{L_{1}}t+\phi_{1}+\theta_{1})})\\\
+|p_{23}|/2(\rho_{23}e^{i(\omega_{L_{2}}t+\phi_{2}+\theta_{2})}+\rho_{32}e^{-i(\omega_{L_{2}}t+\phi_{2}+\theta_{2})}).$
(2)

We have n as the number density of atoms/molecules:

$\displaystyle\expectationvalue{P}=n\expectationvalue{p}.$

We use a simple model of a homogeneous medium of a dilute weakly interacting
gas. Comparing (1) and (2), we see that we have to match oscillating factors
to get the electric susceptibility functions. Then it follows that:

$\displaystyle\chi_{j}(\omega,E)=\dfrac{2n}{\hbar\epsilon_{0}}\dfrac{|p_{3j}|^{2}}{\Omega_{j}}\rho_{3j}e^{-i\theta_{j}}.$

At this point, we have enough information to find other optical response
functions such as the complex refractive index, the phase and the group
velocities and the Group Velocity Dispersion (GVD). Now we focus our attention
in the region in the parameter space in which we develop our model for the EIT
optical response functions.

##  3 Control scheme

With the above results, now we can develop a scheme where we explore the slow
light phenomenon and the strong dispersive effects of EIT. Our goal is to
develop a control scheme for a given atomic system, which yields high
transmission of the probe pulses with the group profiles centered near
frequencies which is offset to $\omega_{31}$ by a chosen value, therefore
belonging to the resonance window. The parameters of the system+field are
chosen to satisfy this condition as well as to optimise changes in dispersion
properties, such as the group velocity, GVD, and the chirp spectrum, so that
small differences in the frequency spectrum of the probe pulses lead to
significant deviations in group profiles of pulses with differing spectra
propagating in the atomic medium.

###  3.1 The concept

We choose a control Rabi frequency relatively large compared to the pump Rabi
frequency which gives a peak near resonance with sharp dispersion and low
absorptivity, and we choose a value of the two-photon detuning to shift the
transparency window relative to the resonance. Choosing the non-zero detunings
does contribute to some population of the other states, though our conditions
are still valid for small detunings. For the rest of the parameters that give
us a spike in the susceptibility near resonance. We use units of $\gamma_{31}$
to make relevant parameters dimensionless (time is in
$(\gamma_{31}/2\pi)^{-1}$). We choose the following conditions:

$\displaystyle\begin{split}\gamma_{31}=1\\\ \Omega_{j},\Delta_{j}\ll 1\\\
\Omega_{1}\ll\Omega_{2}\\\ \rho_{11}\approx 1\\\
\omega^{\prime}\approx\Delta_{2}\nu_{2}-\Delta_{1}\nu_{1}\\\
\alpha=\gamma_{32}/\gamma_{31}.\end{split}$ (3)

Our next step is to refine the choice of parameters to optimise the optical
response of the system to a weak probe pulse of form
$E(t)=A(t)e^{i(\omega_{L_{1}}+x)t}$,
($\mu\absolutevalue{E(t)}\ll\hbar\Omega_{1}$). The polarization $P(t)$ gains a
chirp in the atomic medium and its chirp spectrum is given by
$P_{chirp}(\omega)=A_{\omega}(\omega-(\omega_{L_{1}}+x))\chi(\omega)$. We can
define the chirp more explicitly:

$\displaystyle
P_{chirp}(\omega)=\absolutevalue{A_{\omega}(\omega-(\omega_{L_{1}}+x))}\absolutevalue{\chi(\omega)}e^{i(\phi_{A}(\omega-(\omega_{L_{1}}+x))+\phi_{\chi}(\omega))}.$

Each frequency component gains a phase delay and a group delay,
$\tau_{p}(\omega)$, $\tau_{g}(\omega)$:

$\displaystyle\tau_{p}(\omega)=-\dfrac{\phi_{\chi}(\omega)}{\omega}$
$\displaystyle\tau_{g}(\omega)=-\dfrac{d\phi_{\chi}}{d\omega}.$

The above delays allow us to construct the chirp spectra for probe pulses that
can yield phenomena such as anomalous dispersion, group velocity direction
changes and pulse lengthening. We consider control of group velocity, GVD and
the chirp spectrum in the general case and for transform-limited pulses.

####  3.1.1 Control using group velocity, group velocity dispersion and the
chirp spectrum

If we wish to be in the neighbourhood of the minimum group velocities, we
choose parameters such that
$\left.\dfrac{d^{2}k^{\prime}}{d\omega^{2}}\right|_{\omega=\omega_{L_{1}}+x}=0$
(the local extremum of $\dfrac{dk^{\prime}}{d\omega}$ in the resonance
window). Here, the group velocity differences are small but we can focus on
controlling of the dispersion and extinction coefficients at low velocities.

To maximize the delay between the probe and pump pulses, we intend to maximize
$\Delta
T\propto\left(\dfrac{1}{v_{g}(\omega_{L1}+x)}-\dfrac{1}{v_{g}(\omega_{L1})}\right)$.
As a general approach, we choose parameters where the group velocities are
small in the interval $[\omega_{L1},\omega_{L1}+x]$ but the difference in
$v_{g}$ is still significant.

If the second-order Taylor series for $n^{\prime}(\omega)$ is a good
approximation in the interval, the GVD can be used to estimate the time
delay/chirp. We choose parameters such that $\dfrac{dn^{\prime}}{d\omega}$
resembles a linear function near our chosen frequency, say $\omega_{L1}$,
giving us local extrema for the GVD. We have $\Delta T\propto
GVD(\omega_{L1})x$

For a probe pulse with a Gaussian envelope, the duration of the pulse (related
to the standard deviation $\sigma$) is extended after traveling a distance $d$
in the medium. We define $\sigma_{ex}=\sqrt{\sigma^{2}+(0.5d\crossproduct
GVD(\omega)/\sigma)^{2}}$ as the extended pulse standard deviation.

In general, we can always design our optical response function to chirp the
spectrum of probe pulses to our purposes. Whilst not discussed in this work,
the use of chirped pulses for the driving and probe fields can subdue
undesired non-adiabatic effects during pulse propagation in a medium and add
more control to shaping of the optical response functions and the chirp
spectra of post-medium pulses[Klaas]. For sustaining EIT states in general
high dimensional systems, spectral modulation by methods such as linear and
sinuosidal spectral chirp can improve mitigation of decoherence effects and
adiabatically transfer populations[Rydmbs, Harmspec].

In the next section, we develop approximate forms for the optical response
functions and its derivatives in the control regime.

###  3.2 Optical response functions in the control regime

We consider the approximate form of the susceptibility
$\chi=\chi^{\prime}+i\chi^{\prime\prime}$ near resonance. We use
$\Delta\omega=\omega-\omega_{31}$ as the argument of the functions.

$\displaystyle\begin{split}\chi^{\prime}(\Delta\omega)\approx\dfrac{2n|p_{31}|^{2}}{\hbar\epsilon_{0}}&\left((\rho_{11}-\rho_{33})\Delta\omega+\dfrac{Q_{1}\Gamma^{\prime}(\Delta_{2}\nu_{2}-\Delta\omega\nu_{1})}{(\Gamma^{\prime
2}+(\Delta_{2}\nu_{2}-\Delta\omega\nu_{1})^{2})}\right.\\\ &\left.\text{
}+\dfrac{Q_{2}\Gamma^{\prime}\Delta_{2}}{(\Gamma^{\prime
2}+(\Delta_{2}\nu_{2}-\Delta\omega\nu_{1})^{2})}\right)\end{split}$ (4)
$\displaystyle\chi^{\prime\prime}(\Delta\omega)\approx\dfrac{2n|p_{3j}|^{2}}{\hbar\epsilon_{0}}$
$\displaystyle\left((\rho_{11}-\rho_{33})-\dfrac{Q_{3}\Gamma^{\prime}}{(\Gamma^{\prime
2}+(\Delta_{2}\nu_{2}-\Delta\omega\nu_{1})^{2})}\right)$ (5) $\displaystyle
Q_{1}=\dfrac{\Omega_{2}^{2}}{4\Gamma^{\prime}}(\alpha^{2}(\rho_{11}-\rho_{33})+(\rho_{22}-\rho_{33}))$
$\displaystyle\text{ }\text{ }\text{ }\text{
}+\dfrac{\alpha}{\nu_{1}}(\rho_{11}-\rho_{33})+(\rho_{22}-\rho_{33})$
$\displaystyle
Q_{2}=\alpha(\rho_{11}-\rho_{33})-\dfrac{\alpha}{\nu_{1}}(\rho_{11}-\rho_{33})+(\rho_{22}-\rho_{33})$
$\displaystyle
Q_{3}=\dfrac{\Omega_{2}^{2}}{4}(\alpha^{2}(\rho_{11}-\rho_{33})+(\rho_{22}-\rho_{33})).$

We see that the real susceptibility is the sum of a linear term that depends
on the population in the 1st ground state plus a distribution (similar to a
Fano resonance) with the maximum value $(Q_{1})/(2\Gamma^{\prime})$ and
$2\Gamma^{\prime}$ being the full width between the local extrema. The second
term is an overlapping Cauchy distribution with FWHM $\Gamma^{\prime}$ and
amplitude $Q_{2}\Delta_{2}(\Gamma^{\prime})^{-1}$. Similarly, the imaginary
part of the susceptibility has a constant term plus a Cauchy distribution with
FWHM $2\Gamma^{\prime}$.

Figure 2: Plot of the imaginary (left) and the real(right) part of the
susceptibility of the system for parameters $(\Omega_{2},\gamma_{12})$ for the
numbered configurations: i)$(1/35,4.3E-5)$, ii)$(2/35,4.3E-5)$
iii)$(1/70,4.3E-5)$ iv)$(1/30,1.7E-4)$ and $\Omega_{1}=0.1\Omega_{2}$.

Using our scheme, we assume the following approximations for the parameters:

$\displaystyle\Gamma^{\prime}\approx\gamma+\gamma_{31}\left(\dfrac{\Omega_{2}^{2}}{4}+\dfrac{\alpha\Omega_{1}^{2}}{4}\right)$
$\displaystyle\nu_{1}=(1-\gamma_{31}\Omega_{2}^{2}/4)>0$
$\displaystyle\nu_{2}=(1-\gamma_{31}(\alpha\Omega_{1})^{2}/4)>0.$

Below, we use the approximations for the dispersion and extinction
coefficients:

$\displaystyle n^{\prime}(\omega)\approx 1+0.5\chi^{\prime}(\omega)$
$\displaystyle
n^{\prime\prime}(\omega)\approx\dfrac{\chi^{\prime\prime}(\omega)}{2}.$

Figure 3: (Left) Plot of the group velocity (solid line) and the GVD (dashed
line) for the configuration in Sec 2 for two values of $\gamma_{12}$ 1.5 kHZ,
6.0 kHZ and $\Omega_{2}=1/35$, $\Omega_{1}=0.1\Omega_{2}$. (Right) Plot of
group velocity and the GVD showing anomalous dispersion and local minima.
Figure 4: (Left) Plot of the group velocity and the GVD showing anomalous
dispersion and local minima with $\Omega_{1}$ scaled by factor of 1.1x. Green
line is the asymptote where anomalous dispersion occurs and the group velocity
becomes infinite. We see how small perturbations in the pump Rabi frequency
cause non-trivial changes in the dispersion of a pulse propagating through the
medium. (Right) Plot showing the first derivative of the dispersion and the
zero and the local minima/maxima in the derivative correspond respectively to
anomalous dispersion asymptotes and the local maxima/minima of the dispersion.

We get the expression for the group velocity. For brevity, we define
,$f(\Delta\omega)=(\Gamma^{\prime})^{2}+(\Delta_{2}\nu_{2}-\Delta_{1}\nu_{1})^{2})$,
then:

$\displaystyle\begin{split}v_{g}(\Delta\omega)/c&\approx\dfrac{(f(\Delta\omega))^{2}}{\chi_{0}Q_{1}\Gamma^{\prime}\nu_{1}}\\\
&\crossproduct\left(\dfrac{(f(\Delta\omega))^{2}}{\chi_{0}Q_{1}\Gamma^{\prime}\nu_{1}}(1+0.5\chi_{0}(\rho_{11}-\rho_{33})\Delta\omega)\right.\\\
&\left.\text{ }\text{ }\text{
}+\dfrac{f(\Delta\omega)}{2Q_{1}\Gamma^{\prime}\nu_{1}}(Q_{1}\Gamma^{\prime}(\Delta_{2}\nu_{2}-\Delta\omega\nu_{1})+Q_{2}\Gamma^{\prime}\Delta_{2})\right.\\\
&\left.\text{ }\text{ }\text{
}+0.5(\omega_{31}+\Delta\omega)\left(\dfrac{(f(\Delta\omega))^{2}}{Q_{1}\Gamma^{\prime}\nu_{1}}(\rho_{11}-\rho_{33})-f(\Delta\omega)+\dfrac{2}{Q_{1}\Gamma^{\prime}}(\Delta_{2}\nu_{2}-\Delta\omega\nu_{1})\right)\right)^{-1}.\end{split}$
(6)

The above expression implies that we can get infinite and negative group
velocities if $\chi^{\prime}$ has a negative gradient region with a large
enough steepness. We note the group velocity about the saddle point near 0 of
the real susceptibility reads:

$\displaystyle\begin{split}\min_{\absolutevalue{v_{g}}}{v_{g,e}/c}&\approx\beta_{\text{min}}\left(1+\mathcal{O}\left(\dfrac{\Gamma^{\prime}}{\omega_{31}}\right)\right)^{-1}\\\
&=\dfrac{\Gamma^{\prime}}{2\chi_{0}Q_{1}\nu_{1}\omega_{31}}\left(1+\dfrac{\Gamma^{\prime}}{Q_{1}\nu_{1}\omega_{31}}\left(2\Gamma^{\prime}+\Delta_{2}\left(Q_{2}+(\rho_{11}-\rho_{33})\Gamma^{\prime}\dfrac{\nu_{2}}{\nu_{1}}\right)\right)\right)^{-1}\end{split}$

This is pretty close to the minimum (unsigned) group velocity, and we note
that the second term $\ll 1$. We see the strong dependence of the slow light
condition on $\Gamma^{\prime}$ and $\omega_{31}$. Whilst we can use weaker
fields to reduce the line broadening effect, the dephasing between the ground
states (and therefore $\gamma_{31}$) sets a hard limit on the lowest value of
$\Gamma^{\prime}$.

With this, we can calculate the group velocity dispersion. In our scheme, our
light fields are at least of THz frequencies, therefore the dominant terms in
the GVD are those which include $\omega$. We also use the fact that for a
dilute medium $\chi^{\prime},\chi^{\prime\prime}\ll 1$. Then:

$\displaystyle\begin{split}GVD(\Delta\omega)c&\approx-A\nu_{1}(\Delta_{2}\nu_{2}-\Delta\omega\nu_{1})f(\Delta\omega)^{-2}\\\
&\crossproduct(1+(\Delta_{2}\nu_{2}-\Delta\omega\nu_{1})f(\Delta\omega)^{-1})\end{split}$
(7) $\displaystyle A$
$\displaystyle=2\omega\chi_{0}\dfrac{\Omega_{2}^{2}}{4}(\nu_{1}(\alpha^{2}(\rho_{11}-\rho_{33})+(\rho_{22}-\rho_{33}))$
$\displaystyle+\Gamma^{\prime}(\alpha(\rho_{11}-\rho_{33})+(\rho_{22}-\rho_{33}))).$

For a light pulse, each frequency component $E(\omega)$ picks up a phase
$\Delta\phi(\omega)=k(\omega)d$ after it passes through a dispersive planar
medium of length $d$. The pulse exiting the medium will be the Inverse Fourier
Transform (IFT) of the post-medium $E(\omega)$ and the time difference between
peaks of two simultaneous pulses of frequencies $\omega_{1},\omega_{2}$ is
given by:

$T=d\left(\dfrac{1}{v_{g}(\omega_{1})}-\dfrac{1}{v_{g}(\omega_{2})}\right).$

Since the dispersion in the transparency window is strong in our scheme, a
transform-limited pulse will have its spectral components accumulate different
phases with the time-domain pulse broadening. The GVD is part of the estimate
($\text{GVD}(\omega_{0})(\text{bandwidth})d$) of the chirp introduced to the
pulse (valid for relatively small pulse bandwidths). Therefore, the group
velocity, GVD and the chirp spectrum are effective tools to monitor responses
to perturbations in the EIT medium by the way of determining how the
properties of the probe pulses change relative to the baseline case.

An example we will consider is the passage of high-energy charged particles
(high energy particle) that induce Cherenkov radiation production in the
medium. The probability of a photon being emitted by an individual high energy
particle event is rather small and unlikely to be detectable. Tuning the
medium to have Cherenkov radiation about frequencies which have slow group
velocities will effectively ensure the buildup of the radiation over time as
more events pass. In the dark polariton picture, the Cherenkov radiation will
couple to the atomic transitions and travel as polaritons. The effect will be
to introduce a growing electric potential that will affect the EIT state,
which will manifest in probe pulses having a different chirp spectrum upon
detection. We will explicitly consider the Cherenkov radiation produced in the
EIT medium and give an analytic form to the field in our parameter regime.

##  4 Application: Detecting high-energy charged particles using EIT

Traditional detection of high energy charged particles have been handled in
experiments using Cherenkov gas counters where ultra-sensitive detectors
measure the small yield of light (essentially 1 photon per 10,000 events). The
small yield of Cherenkov radiation, which can be calculated by the Frank-Tamm
formula, makes the use of such ultra-sensitive detectors a necessity. But
there has been a recent surge of interest in using the techniques in atomic
interferometry to measure low-energy physical phenomena; and using EIT with
atomic spectroscopic techniques to measure changes in the atomic medium has
been proposed in the 2000s [Slight].

We develop on these techniques and propose a semiclassical model which gives
an approximate analytic solution for the Cherenkov radiation using the optical
response functions calculated using the Lindbladian. The strong non-linear
dispersion and high transmission properties of EIT can be used to create
Cherenkov radiation with slow group velocities for a train of relativistic
charged particles. At a classical level, the slow light condition ensures that
the group profile of additive Cherenkov contributions will not smear and
constructively increase in intensity. This is owing to the the quantum dark-
polariton description which implies that the Cherenkov radiation develops
coherence with the transition dipoles after emission by the medium,
essentially travelling as slow-moving dark polaritons confined to the medium
for a duration proportional to the inverse of the group velocity. Here the
model is developed for the case of using a very dilute homogeneous gaseous
medium, but the results can easily be extended and optimised towards more
complex systems such as EIT diffraction gratings and crystals.

###  4.1 A model using Fourier transformed Maxwell’s equations

We consider a charged high energy particle traveling with relativistic
velocity $\mathbf{v}=\beta c\hat{z}$ and having charge q. We use the treatment
of a classical point particle, assuming that the width of its wavefunction is
much smaller than the wavelengths of emitted radiation. The below derivations
are in the far-field regime where we assume the particle is sufficiently far
away to neglect near-field radiation. The current density and its Fourier
transform in the space and time domain is given by:

$\displaystyle\mathbf{J}=\rho(x,y,z,t)\mathbf{v}=q\delta(z-vt)\delta(x)\delta(y)\mathbf{v}$
$\displaystyle\mathbf{J}_{\omega,k}=(2\pi)q\tilde{\delta}(\omega-\mathbf{k}\cdot\mathbf{v})\mathbf{v}.$

(where $\tilde{\delta}(z)$ is the generalized delta function and $q$ is the
charge measured in the lab frame $q=\gamma q_{0}$).

We solve for the fields produced as a result of this current when placed in a
homogeneous, anisotropic, non-magnetic medium driven to a EIT steady state. We
use the derived optical response function
($\epsilon(\omega)=1+\chi_{31}(\omega-\omega_{31})$) with the Fourier
transformed Maxwell equations to calculate the fields in the far-field limit.
The effects of retarded time are neglected in this derivation. We start with
the Fourier transformed Maxwell’s equations for an anisotropic, non-magnetic
medium:

$\displaystyle\omega\mathbf{k}\cdot(\epsilon_{0}\check{\epsilon}\cdot\mathbf{E_{\omega,k}})=$
$\displaystyle\rho_{\omega,\mathbf{k}}$ $\displaystyle
i\omega\mathbf{k}\cdot\mathbf{B_{\omega,k}}=$ $\displaystyle 0$
$\displaystyle\mathbf{k}\crossproduct\mathbf{E_{\omega,k}}=$
$\displaystyle-\omega\mathbf{B_{\omega,k}}$
$\displaystyle\mathbf{k}\crossproduct\mathbf{B_{\omega,k}}=$
$\displaystyle-i\mu\cdot\mathbf{J_{\omega,k}}+\omega\mu\epsilon_{0}\check{\epsilon}\cdot\mathbf{E_{\omega,k}}.$

And then introduce the scalar and vector potentials:

$\displaystyle\mathbf{E_{\omega,k}}=i\mathbf{k}V_{\omega,\mathbf{k}}+i\omega\mathbf{A_{\omega,k}}$
$\displaystyle\omega\mathbf{k}\cdot(\check{\epsilon}\cdot\mathbf{A_{\omega,k}})+(\mathbf{k}\cdot(\check{\epsilon}\cdot\mathbf{k}))V_{\omega,\mathbf{k}}=\dfrac{\rho_{\omega,\mathbf{k}}}{\epsilon_{0}}$
$\displaystyle\mathbf{k}\crossproduct(\check{\mu}^{-1}\cdot(\mathbf{k}\crossproduct\mathbf{A_{\omega,k}}))+\dfrac{\omega^{2}}{c^{2}}\check{\epsilon}\cdot(\mathbf{A_{\omega,k}}+\dfrac{c}{\omega}\mathbf{k}V_{\omega,\mathbf{k}})=-\mu_{0}\mathbf{J_{\omega,k}}.$

Here $\check{\epsilon}$ is the 3x3 rank 2 dielectric tensor and is a function
of frequency. We choose to take our coordinate system to be along the
dielectric axes $(e_{+},e_{-},e_{z})$ where the tensor is diagonal. We
consider the case where only one transverse mode is active and the medium is
nearly homogeneous. We use the Lorentz gauge condition

$\mathbf{k}\cdot(\check{\epsilon}\cdot\mathbf{A_{\omega,k}})+\dfrac{\omega\absolutevalue{\check{\epsilon}\cdot\check{\mu}\cdot\check{\epsilon}}}{c^{2}}V_{\omega,\mathbf{k}}=0$

to arrive at the Helmholtz equations for the potentials [Mxwell]:

$\displaystyle\mathbf{A_{\omega,k}}=(2\pi)\mu_{0}q\left(k^{2}-\dfrac{\omega^{2}}{c^{2}}\epsilon\right)^{-1}\tilde{\delta}(\omega-\mathbf{k}\cdot\mathbf{v})\mathbf{v}$
(8) $\displaystyle
V_{\omega,\mathbf{k}}=(2\pi)\dfrac{q}{\epsilon\epsilon_{0}}\left(k^{2}-\dfrac{\omega^{2}}{c^{2}}\epsilon\right)^{-1}\tilde{\delta}(\omega-\mathbf{k}\cdot\mathbf{v}).$
(9)

For lossy mediums, $\mathbf{k^{*}}\cdot\mathbf{E}\not=0$ in general. We will
see the presence of both tranverse and longitudinal modes.

We take the inverse spatial Fourier transform of $\mathbf{A}_{\omega,k}$:

$\displaystyle\mathbf{A}_{\omega}\cdot\hat{v}=\dfrac{1}{(2\pi)^{2}}\int_{\Gamma_{k}}\mu_{0}qv\left(k^{2}-\dfrac{\omega^{2}}{c^{2}}\check{\epsilon}\cdot\right)^{-1}\tilde{\delta}(\omega-\mathbf{k}\cdot\mathbf{v})e^{i\mathbf{k}\cdot\mathbf{r}}d^{2}k_{\perp}dk_{z}.$

Expressing it in cylindrical coordinates and carrying out integration over
$k_{z}$, we get:

$\displaystyle\mathbf{A}_{\omega}\cdot\hat{v}=\mu_{0}qe^{i\omega(z/v)}\dfrac{1}{(2\pi)^{2}}\int_{0}^{\infty}\left(k^{2}-\dfrac{\omega^{2}}{c^{2}}\check{\epsilon}\cdot\right)^{-1}\left(\int_{0}^{2\pi}e^{ik_{\perp}r_{\perp}\cos(\theta)}d\theta\right)k_{\perp}dk_{\perp}$

The angular integral is a representation of the zeroth Bessel function.

The above integral, for the $k_{\perp}$ integrand, has 2 poles $\pm
ik_{r}=\pm\omega c^{-1}\sqrt{\beta^{-2}-\epsilon(\omega)}$ in the complex
domain corresponding to forward and backward radially propagating modes. Since
the medium is dissipative, we consider only the physical forward propagating
pole and use a semi-circular contour with bumps excluding the poles.
$k_{z}=\omega v^{-1}$ fixes the possible values for the radial wavevector. For
the case of no loss, the Cherenkov condition is $\Re{\epsilon}>\beta^{-2}$. We
will use this later to determine the contributions for radiative modes and
evanescent modes:

$\displaystyle\mathbf{A}_{\omega}\cdot\hat{v}=\mu_{0}qe^{i\omega(z/v)}\dfrac{1}{2\pi}\int_{0}^{\infty}\dfrac{k_{\perp}J_{0}(k_{\perp}r_{\perp})}{k_{\perp}^{2}+\dfrac{\omega^{2}}{c^{2}}\left(\beta^{-2}-\epsilon(\omega)\right)}dk_{\perp}.$

The solution of the integral is the modified Bessel function $K_{0}$ evaluated
at $ik_{r}$. Considering the treatement in [Saff] in extending the lossless
case to the lossy case when we consider transforming to the time domain, the
frequency integration can be segmented into two regions
$\real(\epsilon(\omega)-\beta^{-2})\gtrless 0$, which correspond respectively
to the radiative and non-radiative regions. We define the radiative region:
$\Gamma_{1}=[\omega_{31}+a,\omega_{31}+b]$. We hence solve the final integral
for $\mathbf{A_{\omega}}$, and similarly define $V_{\omega}$:

$\displaystyle\mathbf{A}_{\omega}(\mathbf{r},\omega)\cdot\hat{v}=\begin{cases}\dfrac{\mu_{0}q}{4\pi}e^{i\omega(z/v)}K_{0}(ik_{r}r_{\perp})&\omega\in[\omega_{31}+a,\omega_{31}+b]^{c}\\\
\text{ }&\text{ }\\\
\dfrac{\mu_{0}q}{4\pi}e^{i\omega(z/v)}\dfrac{i\pi}{2}H_{0}^{(1)}(k_{r}r_{\perp})&\omega\in[\omega_{31}+a,\omega_{31}+b]\\\
\end{cases}$ (10)

$\displaystyle
V_{\omega}(\mathbf{r},\omega)=\begin{cases}\dfrac{q}{4\pi\epsilon_{0}\epsilon
v}e^{i\omega(z/v)}K_{0}(ik_{r}r_{\perp})&\omega\in[\omega_{31}+a,\omega_{31}+b]^{c}\\\
\text{ }&\text{ }\\\ \dfrac{q}{4\pi\epsilon_{0}\epsilon
v}e^{i\omega(z/v)}\dfrac{i\pi}{2}H_{0}^{(1)}(k_{r}r_{\perp})&\omega\in[\omega_{31}+a,\omega_{31}+b].\\\
\end{cases}$ (11)

In the far-field limit, $kr_{\perp}\gg 1$, asymptoptic expansion of the Hankel
function gives us $i\pi
H_{0}^{(1)}(k_{r}r_{\perp})=\sqrt{i\pi}e^{ik_{r}r}/\sqrt{k_{r}r}$. We can
define the steepest descent contours by using the phase term:

$\displaystyle\begin{split}&\dfrac{r_{\perp}}{c}\theta(\tilde{\omega},r,z,t)=\dfrac{r_{\perp}}{c}i\left(\dfrac{\tilde{\omega}c}{r}\left(\dfrac{z}{\tilde{v}}-t\right)+{\tilde{\omega}n^{\prime}_{\perp}}+i{\tilde{\omega}n^{\prime\prime}_{\perp}}\right)\\\
\end{split}$ (12)

The steepest descent contour will keep the imaginary part of the phase
constant whilst the real part will have an additional term $-p$. We define the
contour $\gamma_{\tilde{\omega}}(p)$, parameterized by real $p$ by:

$\displaystyle\theta(\gamma_{\tilde{\omega}}(p),r,z,t)=\theta(\tilde{\omega},r,z,t)-p$

About a point $\omega_{i}$, if we can define a local analytic expansion of
$\theta(\tilde{\omega},r,z,t)$ extended to the complex plane, we can define an
inverse function for $\theta$ and get
$\gamma_{\tilde{\omega}}(p)=\theta^{-1}(\theta(\omega_{i},r,z,t)+ip)$.

The method of steepest descent can be used to find any stationary points in
the case when $\chi^{\prime\prime}$ is very small. These group modes represent
points of constructive interference of wavefronts on a group cone with vertex
at $(0,vt)$. This involves finding the critical points of (12). Analyticity of
the refractive index ensures that the critical points will be saddle points
and there’ll be no local extrema in the complex plane. The critical
frequencies $\tilde{\omega_{i}}$ obey the below condition:

$\displaystyle t-\dfrac{z}{v}-r_{\perp}\dfrac{dk_{r}}{d\tilde{\omega}}=0.$
(13)

We deform the integration contour on the real interval to the complex plane
such that we pass through all critical points via their steepest descent
contours and all other contributions are on segments, which are very small.
For the time-domain fields in the radiative domain, we define the set of
critical points $\mathcal{S}_{c}$ with each critical point defined on contour
$\gamma_{\tilde{\omega}_{i}}(p)$. A larger upper limit $P$ will give a better
approximation of the integral

$\displaystyle
V(\mathbf{r},t)\approx\sum_{w_{i}\in\mathcal{S}_{c}}\dfrac{iq}{4\pi\epsilon_{0}v}e^{ir_{\perp}c^{-1}\theta(\tilde{\omega}_{i},r,z,t)}\dfrac{\sqrt{i\pi}}{\sqrt{2r_{\perp}}}\int_{0}^{P}\dfrac{e^{-r_{\perp}c^{-1}p}}{\epsilon\sqrt{k_{\perp}(\gamma_{\tilde{\omega}_{i}}(p))}}\dfrac{d\gamma_{\tilde{\omega}_{i}}}{dp}dp.$
(14)

In the case where the critical points are non-degenerate and represent local
maxima of the real part of the phase, we can use Morse’s Lemma for complex
functions to get a nicer form of the integral. Since the functions are
holomorphic, there exists a neighbourhood near non-degenerate saddle points
where phase $\theta(\tilde{\omega})$ is approximately quadratic. Accordingly,
we can define the asymptotic expansion of (14)

$\displaystyle\begin{split}V(\mathbf{r},t)&\approx\dfrac{-iq}{4\pi\epsilon_{0}r_{\perp}v}\sum_{\tilde{\omega}_{i}\in
S_{t,z,r_{\perp}}}\dfrac{e^{r_{\perp}c^{-1}i\theta(\tilde{\omega}_{i},t,z,r_{\perp})}}{(-\text{GVD}_{r}(\tilde{\omega}_{i}))^{1/2}}\left(\dfrac{1}{\sqrt{\text{k}_{\perp}(\tilde{\omega}_{i})}}+\mathcal{O}\left(\dfrac{c}{r}\right)\right).\end{split}$
(15)

The critical points will be complex for a dissipative system and will be
located near neighbourhoods of critical points for the non-dissipative system
if the dissipation is weak. In the case where no critical points exist or the
integration is difficult, we can calculate the Gauss-Laguerre quadrature of
the integral [GLQ]. However, when the dissipation is significant, the validity
of the Cherenkov condition and the simplicity of the fields is not generally
true.

Figure 5: Plots of real and imaginary parts of $k_{r}$, when the radiative
modes are in the anomalous dispersion region, with piecewise parabolic
approximations as functions of the frequency offset. For our particular
system+field choice, we divide the radiative region into 2 regions with common
endpoint at 7500 Hz. Resonant points where $\real\epsilon=\beta^{-2}$ are at
the intersections of real and imaginary parts, bounding the radiative region.

An explicit evaluation of the integral in terms of complex error functions can
be done in the far-field limit by approximating $k_{\perp}$ by a piecewise
function composed of parabolas. As shown in Figure 5, we consider the local
quadratic best fit in the region and expand it about a central frequency,
$\omega_{31}^{\prime}=\omega_{31}+x$, in region $[a_{i},b_{i}]$, $k_{r}\sim
k_{0}+\eta\Delta\omega+\sigma^{\prime}(\Delta\omega)^{2}$. We evaluate the
radial component of the E-field which has a Hankel function of first type with
order 1. We use the asymptotic far-field limit and we get the below integral:

$\displaystyle\begin{split}\mathbf{E}(\mathbf{r},t)&\approx\dfrac{iq}{8\pi\epsilon_{0}r_{\perp}v}\sqrt{ir_{\perp}\pi}\sum_{j\in
I}e^{i\omega^{\prime}_{31}(z/v-t)+ik_{0}r_{\perp}}\\\
&\crossproduct\int_{a_{j}}^{b_{j}}dw\text{ }e^{i(w(z/v-t)+r_{\perp}(\eta
w+\sigma^{\prime}w^{2}))}\dfrac{\sqrt{k_{r}}\hat{r}+\dfrac{\omega}{v\sqrt{k_{r}}}(1-\beta^{2}\epsilon)\hat{z}}{\epsilon(w)}\end{split}$
(16)

$\displaystyle\begin{split}\mathbf{B}(\mathbf{r},t)&\approx\dfrac{iq}{8\pi\epsilon_{0}r_{\perp}c^{2}}\sqrt{ir_{\perp}\pi}\sum_{j\in
I}e^{i\omega^{\prime}_{31}(z/v-t)+ik_{0}r_{\perp}}\\\
&\crossproduct\int_{a_{j}}^{b_{j}}dw\text{ }e^{i(w(z/v-t)+r_{\perp}(\eta
w+\sigma^{\prime}w^{2}))}\dfrac{\sqrt{k_{r}}}{\epsilon(w)}\hat{\phi}.\end{split}$
(17)

We approximate the part of the integrand outside the exponential by a Taylor
series in $w$ with coefficients $c_{j}$. The solution to (16) is a sum of
complex error functions. The integral of the exponential term is given below:

$\displaystyle\begin{split}G_{0}(a,b)=&i\tilde{\sigma}^{-1}\left[(e^{\tilde{\sigma}^{2}b^{2}r_{\perp}-it^{\prime}b}-e^{\tilde{\sigma}^{2}a^{2}r_{\perp}-it^{\prime}a})e^{\Phi^{2}t^{\prime
2}/(2\sigma^{2}r_{\perp})}\right.\\\
&\left.-e^{\tilde{\sigma}^{2}b^{2}r_{\perp}-it^{\prime}b}\text{erfcx}\left(\Phi\left(\dfrac{t^{\prime}}{\sqrt{2r_{\perp}}\sigma}+i\dfrac{b\tilde{\sigma}^{2}\sqrt{r_{\perp}}}{\sqrt{2}\sigma}\right)\right)\right.\\\
&\left.+e^{\tilde{\sigma}^{2}a^{2}r_{\perp}-it^{\prime}a}\text{erfcx}\left(\Phi\left(\dfrac{t^{\prime}}{\sqrt{2r_{\perp}}\sigma}+i\dfrac{a\tilde{\sigma}^{2}\sqrt{r_{\perp}}}{\sqrt{2}\sigma}\right)\right)\right]\end{split}$
(18)

$\displaystyle\begin{split}&\sigma=\absolutevalue{\tilde{\sigma}}\equiv\absolutevalue{\sqrt{2i\sigma^{\prime}}}\text{
}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{
}\Phi=e^{-i\arg(\tilde{\sigma})/2}\text{ }\text{ }\text{ }\text{ }\text{
}\text{ }\text{ }\text{ }\text{ }t^{\prime}=(t-z/v)-\eta
r_{\perp}.\end{split}$

The general expansion for any of the fields terms with expressions of form
$A\sum_{j\in I}c_{j}\int_{a_{j}}^{b_{j}}dw\text{ }e^{i(w(z/v-t)+r_{\perp}(\eta
w+\sigma^{\prime}w^{2}))}f(w)$ is given by:

$\displaystyle F(t,z,r)=A\sum_{j\in
I}c_{j}\sum_{k=0}^{N}i^{n}\left.\dfrac{d^{k}f}{dx^{k}}\right|_{x=\omega^{\prime}_{31}}\dfrac{d^{k}}{d(z/v-t)^{k}}G_{0}(a_{j},b_{j}).$
(19)

The expression above provides more information about the relations between the
parameters in the Cherenkov field and system parameters. It’s clear that
$\eta_{j}$ and $\sigma_{j}$ are both related to derivatives of the radial
wavenumber $k_{r}$ about the central frequency $x$ for each frequency interval
$[a_{j},b_{j}]$. We can estimate these values by looking at the susceptibility
and seeing how altering parameters like
$\Gamma^{\prime},\Delta_{2},\gamma_{12}$ affect the curvature and first
derivative of $k_{r}$. We see that decreasing the nonlinearity in the
susceptibility about region $I_{j}$ reduces the inhomogeneous broadening of
spectral components over propagation distance, suggesting that the linear
dispersion region of EIT is ideal for decreasing dissipation and for
controlling group velocities.

Figure 6: Formation of the group cone for coherent Cherenkov radiation,
showing the propogation of the group mode with angle $\theta$ and the
radiation propagation distances over transit time $\tau$ and time of Cherenkov
cone vortex $t$.

We finish this section by expanding on the geometry of Cherenkov radiation and
checking that the causality conditions are satisfied. For the case where we
have a single stationary point for a mode with group velocity $v_{g}$, we can
find the critical frequency $\tilde{\omega}$ that solves (13). The phase speed
at this frequency matches the charge velocity $v$ and results in the phase
matching condition where path difference $CF=2BF\sin(\theta)$. We can define
the transit time $\tau$ of radiation reaching $(r,z)$ at time $t$ (as in
Figure 6), where $\tilde{v}_{r}$ is the complex radial velocity or group
velocity for wavenumber $k_{\perp}$:

$\displaystyle\begin{split}\tilde{\tau}&=t-\dfrac{\sqrt{(z-v\tau)^{2}+r^{2}}}{\tilde{v}_{g}(\tilde{\omega})}\end{split}.$
(20)

The transit time $\tau$ connects the retarded time $t^{\prime}=t_{0}+\tau$ and
the charge position $vt^{\prime}$ that is the source for the radiation at
$(t,\mathbf{r})$. In the case of weak dissipation, the real group velocity
will be a relevant physical quantity and the below causality condition must be
satisfied:

$\displaystyle
t-\dfrac{z}{v}-r\sqrt{\dfrac{1}{v_{g}(\omega)^{2}}-\dfrac{1}{v^{2}}}\geq 0.$
(21)

Defining the group cone (13) and the retarded current density ensures that
this condition is automatically satisfied. To satisfy non-degeneracy, we
require that the complex GVD not to be identically zero.

Care must be taken here, as the real group velocity defined by
$v_{g}=\left(\real\dfrac{dk}{d\omega}\right)^{-1}$ is not always a reliable
measure of the signal velocity, especially in cases where we have significant
dissipation and superluminal group velocities. For our purposes, we assume
we’re working with timescales where the group shape is still is coherent. In
the next section, we’ll consider the fields in the case of very small
dissipation when we use the steepest descent method and the case of weak
dissipation where we numerically solve for the fields.

###  4.2 Cherenkov radiation in the EIT regime

In general, the introduction of a nonzero loss term breaks the Cherenkov cone-
like condition and dampens the field amplitude. For a large enough loss, the
highly anisotropic character of the radiation and angular sensitivity to the
charged particle’s speed is weakened. Ideally we would want to get
contributions close to the center of the transparency window, reduce the
gaseous medium’s particle density and introduce background refractive index
enhancement to reduce $\Im{\chi}$. For the case of EIT with Rubidium D1 line
spectra using fields with the linear polarization, we can select a transition
between levels with magnetic number $m=0$ for EIT and introduce a static
magnetic field to cause large enough energy level shifts for levels with the
same nonzero $m$. For large enough detuning, these transitions will introduce
a frequency-independent contribution to the susceptibility
$\epsilon_{\infty}$.

We use the control scheme outlined in Sec. 3. The frequency range where
Cherenkov radiation is transmitted is $[\omega_{31}+a,\omega_{31}+b]$, which
contains all frequencies that contribute non-trivially to the radiation. We
develop a scheme to evolve the fields corresponding to a certain mode
$\omega_{i}$ after some time $\Delta t$.

To define the time-evolving field, we first state the group velocity vector
for Cherenkov radiation:

$\displaystyle\mathbf{v}_{g,Ch}(\Delta\omega)=\dfrac{\beta^{-1}\hat{z}+n_{\perp}\hat{r}}{(1+\chi^{\prime})+0.5(\omega_{31}+\Delta\omega)\dfrac{d\chi^{\prime}}{d\Delta\omega}}$
(22)

Suppose the wave propagates for small time $\Delta t$, such that the group
profile remains coherent, from its initial point at $\\{t,z,r_{\perp}\\}$.
Since the peak propagates with velocity $\mathbf{v}_{g,Ch}$, the new positions
are given by
$r^{\prime}=r+(\mathbf{v}_{g,Ch}(\Delta\omega_{i})\cdot\hat{r})\Delta t$,
$z^{\prime}=z+(\mathbf{v}_{g,Ch}(\Delta\omega_{i})\cdot\hat{z})\Delta t$. We
can express $\Delta t+z^{\prime}-z=\alpha(vt-z)$ and $r^{\prime}-r=\beta r$
and they satsify the cone condition if $\alpha=\beta$. This condition gives us
the relation
$\dfrac{v-\mathbf{v}_{g,Ch}(\Delta\omega_{i})\cdot\hat{z}}{v\mathbf{v}_{g,Ch}(\Delta\omega_{i})\cdot\hat{r}}=v_{g,r_{\perp}}^{-1}$,
which can be obtained from (19). The cone condition, and the condition for
non-degeneracy, is then automatically satisfied and we can define the time-
evolving averaged Cherenkov Poynting vector, and the radiant power per unit
length at a fixed radius in the axial direction $\zeta_{z}=2\pi
r_{\perp}\expectationvalue{S}$, for modes
$\tilde{\omega}_{i}\in\mathcal{S}_{t,z,r_{\perp}}$:

$\displaystyle\begin{split}&\zeta_{z}(\mathbf{r}+\mathbf{v}_{g,Ch}(\Delta\tilde{\omega}_{i})\Delta
t,t+\Delta
t)=\sum_{\tilde{\omega}_{i}\in\mathcal{S}_{t,z,r_{\perp}}}\pi(r+(\mathbf{v}_{g,Ch}(\Delta\omega_{i})\cdot\hat{r})\Delta
t)\\\
&\crossproduct\mu^{-1}\Re{E_{r,\tilde{\omega}_{i}}(\mathbf{r},t)B^{*}_{\tilde{\omega}_{i}}(\mathbf{r},t)}e^{-2\Delta
t\Im{\mathbf{k}\cdot\mathbf{v}_{g,Ch}(\Delta\tilde{\omega}_{i})+\Delta\tilde{\omega}}}.\end{split}$
(23)

The peak intensity of the radiation is confined to a thin cylinder about the
z-axis because the dissipation and the spatial coherence depend only on the
bandwidth and radial wavevector. For small $r$, the spatial phase
$k_{r}^{\prime}(\omega)r$ for different frequencies is small enough that the
spectral components add coherently. The dissipation part
$k_{r}^{\prime\prime}(\omega)r$’s shaping for the spectral profile is
negligible for these distances. For large $r$, the phase differences are large
enough that the phase is rapidly changing with frequency and the contributions
to the field come from the critical points satisfying (13) or from nodes for
the Gauss-Laguerre quadrature. The coherent addition of spectral components
also depends on the retarded time through the relative spectral phase
$\Delta\omega t^{\prime}$, and hence on the radial components of group
velocity and GVD. It follows that minimizing bandwidth and the range of
$k_{r}$, for frequencies that contribute non-trivially to the radiant power,
increases the coherence distance and time. The background susceptibility
$\epsilon_{\infty}$ and the system+field parameters can be used to control the
radiation spectrum and the radial refractive index through manipulation of the
susceptibilities (4,5). EIT and the slow light condition allow for strong
correlations between spectral components such that the fields due to multiple
charges can stay in this coherence regime. With a significant enough
accumulation of Cherenkov polaritons in the medium, we can expect a change in
the optical response functions. In the next section, we consider the
perturbative effects to the EIT state due to small fields.

(a) $\real n_{r}$

(b) $\imaginary n_{r}$

Figure 7: Plots showing the radial refractive index $n_{r}$ for various values
of $\epsilon_{\infty}$. The decrease of $\imaginary n_{r}$ for large
$\epsilon_{\infty}$ proportionally decreases the dissipation and is
accompanied by increased and sharper dispersion.

(a) $\Omega_{2}=1/35\gamma_{31}$

(b) $\Omega_{2}=2/35\gamma_{31}$

Figure 8: Plots for $\zeta_{z}(\mathbf{r},t)$ with varying $r$ for different
times.

###  4.3 Perturbation to steady states: Developing a control method to detect
high energy particles

We can consider a couple of approaches in detecting the presence of Cherenkov
radiation. Using a sensor is a viable approach but we can also use the methods
of atomic spectroscopy and interferometry by tuning the optical properties of
the medium and its sensitivity to small perturbations.

Consider a setup of atoms in an EIT steady state exposed to Cherenkov
radiation described by the Lindbladian $\mathcal{L_{\mathcal{J}}}$. We assume
that $\mathcal{L_{\mathcal{J}}}$ has a unique steady state, with projection
operator on the steady state subspace given by
$\mathcal{P}_{\infty}=\sum_{\lambda_{k}}\ket{\psi_{\lambda_{k}}}\bra{\psi_{\lambda_{k}}}$,
and introduce the perturbation operator $g(t)\mathcal{O}$ (corresponding to
$-i\commutator{V}{\cdot}$ in the field interaction picture) to describe the
Cherenkov radiation-atom interaction. The time evolution from an initial state
is given by
$\mathcal{T}e^{\int_{-\infty}^{t}d\tau(\mathcal{L_{\mathcal{J}}}+g(t)\mathcal{O})}\ket{\rho_{s}}$.
We assume that the perturbation due to Cherenkov radiation is switched on at
$t=0$, $g(t)$ is slowly time-varying, and use the Kubo formula to solve for
the first-order perturbation [lindblad]:

$\displaystyle\mathcal{T}_{t}^{(1)}\ket{\rho_{s}}=\int_{-\infty}^{t}d\tau
g(\tau)e^{(t-\tau)\mathcal{L_{\mathcal{J}}}}\mathcal{O}\ket{\rho_{s}}$ (24)

In the case where $g(t)=G\Theta(t)$,
$\mathcal{T}_{t}^{(1)}\ket{\rho_{s}}=G\mathcal{L_{\mathcal{J}}}^{-1}\mathcal{O}\ket{\rho_{s}}$.
The above equation assumes we start from the steady state for a particular
choice of parameters. We then apply the perturbation to the system and
calculate the first order perturbation to the steady state using the Kubo
formula. The first order perturbation on a steady state describes a leakage
from the asymptotic state subspace [lindblad]. For the choice of parameters
$\Omega_{1}=0.1\text{ MHz}$, $\Omega_{2}=1.0\text{ MHz}$,
$\Delta_{1}=\Delta_{2}=0$, the Lindbladian is diagonalizable with 1 physical
state and 8 unphysical traceless states with negative real part of eigenvalues
$\lambda_{k}$,
$\left(\dfrac{\Omega_{1}^{2}+\Omega_{2}^{2}}{\gamma_{31}}\leq|\lambda_{k}|\leq\gamma_{31}\right)$.
This dissipative energy gap, $\Delta_{dg}$, is very similar to the curve of
$\dfrac{\Omega_{1}^{2}+\Omega_{2}^{2}}{\gamma_{31}}$. Increasing the
dissipation gap to large values $\Delta_{dg}$ will decrease the leakage from
the asymptotic subspace. We observe that small fields lead to almost no change
in the coherence for our chosen probe field as contributions due to dephasing
negates any population of the other states. For large Rabi frequencies, we
observe that the perturbation magnitude decreases, as the field strength
increases $\Delta_{dg}$ and $\Gamma^{\prime}$. However, as Figure 9 shows,
this effect is somewhat offset by choosing large $\absolutevalue{\Delta_{2}}$.
The perturbative effect seems to be strongest between the two sub-bands (where
the perturbation is minimized) and to the right of the second sub-band. There
are trade-offs in choosing smaller Rabi frequencies in having to deal with
larger absorptivity, while larger Rabi frequencies will broaden the optical
response lineshapes.

Figure 9 show regions in parameter spaces in which we can get significant
change to $\rho_{31}$, whilst retaining choice for the real and imaginary
parts, for small perturbations. In determining the full perturbation to the
steady state, we can configure the initial system state of the detection
system towards a desired change in the optical response functions
$n^{\prime},n^{\prime\prime}$. For example, we could be in the region with
maximized GVD and the perturbed state will modify the chirp spectrum of the
polarized field of the Cherenkov radiation or another probe field and cause a
delay in comparison to the pulses propagating in the unperturbed system. Small
perturbations can also be very useful if the peak frequencies of the Cherenkov
field or of another probe pulse are near the anomalous dispersion region, such
that the perturbative response will lead to phenomena such as superluminal
group velocities. Alternatively we could measure a different propagation angle
for Cherenkov radiation due to change in $\eta$. This method allow us to
observe the change in group profile/chirp spectrum of the Cherenkov radiation
or a sequence of probe pulses.

(a) $\real\mathcal{T}_{t}^{(1)}\rho_{13,s}$

(b) $\imaginary\mathcal{T}_{t}^{(1)}\rho_{13,s}$

Figure 9: Plots showing the real and imaginary parts of the perturbation
$\mathcal{T}_{t}^{(1)}\rho_{13,s}$, due to perturbation $G\mathcal{O}$ after 1
$\mu$s (color axis). $\Omega_{1}=0.1\Omega_{2}$, $\gamma_{12}=4.3E-5$ ,
$\Delta_{2}=-5.7E-4$, $\Omega_{pert}=2.7E-4$.

####  4.3.1 Control method summary

We explicitly state our control protocol for detection of high energy
particles. Our detection scheme is developed for two different systems of
interest with different objectives for control. The first is for the
accumulation of Cherenkov polaritons, it prioritises high transmission,
nonlinear modes and slow group velocity to increase the yield of radiation.
The second system is tailored towards the detection of the Cherenkov
radiation, involving choosing atomic and field parameters such that the
optical response functions in the medium will develop a desired change in
properties such as the chirp spectra of a probe pulse post-medium. We start
from the same base conditions as those listed in Sec 3.1.

For the first system, we note that the properties of the Cherenkov radiation
are determined by the parameters $\omega_{31}$, $\mathbf{r_{\perp}}$,
$\sigma$, $\eta$, $\phi$, $c_{0}$ and the bandwidth $[a,b]$ (from Sec 4.2). We
set the cutoff speed to be equal to $\beta$, choose a Cherenkov parameter set
to give us a desired form for the radiation, choose the system+field
parameters to shape the radial wavenumber function $k_{r}(\omega)$ and
determine the frequency interval $[\omega_{L1}+a,\omega_{L1}+b]$ (with $a,b$
being the solutions to $1-\beta^{-2}+\Re{\chi}=0$ in the window centered at
the peak frequency). The bandwidth for the coherent group mode is limited by
the EIT linewidth $\Gamma^{\prime}$ and adjusting the control Rabi frequency
$\Omega_{2}$ is one way to broaden the linewidth, decrease dissipation and
increase coherence times. The axial and radial group velocities are determined
using (22) and (6) which can then be used to find any critical points
satisfying (13) and then get the radiant power density (23). We can then
optimise the power density for a train of high energy particles associated
with some distribution of arrival times.

For the second system, we optimise the detection process by choosing the set
of system+field parameters such that the initial state is perturbed by the
Cherenkov field to a desired state, which has different optical responses. We
determine the regions in the plot of $\mathcal{T}^{(1)}(\rho_{s})$ in the
system+field parameter space which correspond to perturbed states we target.
For example, we can choose to have the initial state to cause high GVD for a
probe pulse and thus a significant enough perturbation that we get a
measurable chirp and pulse delay for propagation in the perturbed medium.

This concludes our treatment for detecting high energy particles using EIT in
atomic media. In the final section, we review our results and consider their
extension to more complex systems.

##  5 Summary and vision for nanophotonics

We derived the optical response functions for the three-level $\Lambda$ system
driven to a EIT state in an open weakly coupled system using the Lindblad
formalism to consider the effects of dephasing and decoherence after
determining the field interaction Hamiltonian. We used the Bloch vector
formalism to transition from the microscopic to the macroscopic picture where
we discussed the optical response functions (dispersion and absorptivity), and
showed how an EIT state with a choice of parameters would lead to a narrow
transparency window with strong non-linear dispersion effects that created
phenomena such as slow light and anomalous dispersion.

This led to our main focus of this chapter which is on developing quantum
control schemes to use EIT’s optical response and sensitivity as a
spectroscopic tool to detect phenomena such as the passage of high-energy
particles. Our model was developed in the regime of weak fields with
ultranarrow Fano resonances that capitalize on the properties of slow light
and dissipation-free transmission. The group velocity dispersion (GVD) and
features of the group velocity (such as in the presence of anomalous
dispersion) (Sec 3) were considered in this regard.

We discussed an application of our approach to the detection of high-energy
charged particles through the accumulative stimulation of Cherenkov radiation,
by proxy of the radiation behaving as slow moving Cherenkov polaritons in a
medium undergoing EIT (Sec 4), and derived a weak-field approximation
analytical model that allows us to calculate the group profile and its time-
dependence. We then considered how small perturbations would affect the steady
state configuration determining the leakage from the steady-state subspace,
the first order perturbation and the dissipation gap that is a measure of the
minimum decay rate of contributions from states outside the steady state
subspace. In tandem with the known optical response functions, we showed that
we can realize a control scheme that optimises the sensitivity of the optical
response functions to the perturbation.

We note that whilst we spend most of our attention on the atomic and field
properties, the high energy particles can also be controlled in a similar
manner to the atomic transitions. Work in the field of PINEM (Photon induced
near-field electron microscopy) [PINEM] shows that high energy charged
particle wavefunctions can be modulated over nanostructures with fringing
evanescent fields and enhance probing of a train of randomly arriving high
energy particles , introduce correlations between the particles, and increase
the perturbation to a bound electron state [Freel]. We can control the Rabi
phase for a random arrival process and increase coherent population build up
in comparison to a train of uncorrelated wavefunctions. Further approaches
include reverse engineering the high energy particle wavefunction from the
photon spectral density function as well as through optically probing the
atomic coherences [Cohlight].

Whilst this chapter focused on a particular atomic system, the results
presented here would have similar analogues in more complex systems, such as
photonic crystals and nanostructures, providing a strong relevance to quantum
information theory and optical storage. There is vast literature in research
of EIT in nanostructures [Polariton]. Coupling of the EM field to material
modes in the system leads to formations of quasiparticles similar to dark
state atomic EIT polaritons. Examples already researched include surface
plasmon polaritons, in the case of introducing a substrate below the
dielectric, or quasiparticles that form in bulk such as exciton-polaritons.

Surface plasmon polaritons (SPPs) show powerful applications in the production
of tunable power-enhanced Cherenkov radiation. In [GrapheneCH], a dielectric
substrate with a buffer layer deposited with multiple layers of graphene was
used to enhance the production of Cherenkov radiation due to an electron beam
by coupling the radiation to the SPP modes, resulted in increasing the
intensity of the radiation by more than 2 orders of magnitude. The properties
of the buffer, substrate and SPP modes can be used to control the intensity
increase and the radiation frequency. Of particular interest is the analogous
phenomenon of plasmon-induced transparency (PIT) realized in metamaterials
consisting of two plasmonic resonators coupled to a waveguide [EITPM]. One
scheme utilizes coupling between a dark and bright resonator to achieve a
transparency window and optical response functions similar to that of EIT in
atomic media. Graphene metamaterials have seen wide applications in this field
due to its low-loss, shiftable Fermi energies and electronic properties. A
number of different geometrical approaches have theoretically showed PIT
[EITtp, EITtp2] in the THz regime and this phenomena’s applicability to
quantum control. Due to the geometry of the resonators, spacings in the
metamaterial, and properties of the graphene layer give high controllability
for the electromagnetic response. The features of enhanced slow light and
trapping as well as the amplification of signals and the presence of PIT
suggests a possible extension for this chapter’s results.

Exciton interactions are another area where this research can be extended. The
strong interaction of excitons with matter and light, the capability to induce
EIT, and the formation of strongly non-linear exciton-polariton states
[GiantRyd] suggest a future direction for sensing high energy particles with
EIT in more complex media. One particular application is for systems with
Rydberg excitonic states, which yield significantly higher nonlinearities in
the optical response and few photon strongly correlated states that result
from the Rydberg blockade effect [GiantRyd]. With the introduction of the
blockade effect, EIT is disrupted and the nonlinearities are transferred to
the few photon states. Benefits of systems with Rydberg excitonic states
include greatly increased interaction strengths and applications of ultralow
field intensities causing reduced decoherence and line broadening effects.
With the traditional features of EIT including the transparency window and
slow light along with the strongly correlated few photon states and quantum
control aspect, there is good scope in detecting high energy particles through
the production and control of slow moving Cherenkov polariton-excitons.