# CHARACTERIZATION OF DIM LIGHT RESPONSE IN DVS PIXEL: DISCONTINUITY OF EVENT
TRIGGERING TIME

###### Abstract

Dynamic Vision Sensors (DVS) have recently generated great interest because of
the advantages of wide dynamic range and low latency compared with
conventional frame-based cameras. However, the complicated behaviors in dim
light conditions are still not clear, restricting the applications of DVS. In
this paper, we analyze the typical DVS circuit, and find that there exists
discontinuity of event triggering time. In dim light conditions, the
discontinuity becomes prominent. We point out that the discontinuity depends
exclusively on the changing speed of light intensity. Experimental results on
real event data validate the analysis and the existence of discontinuity that
reveals the non-first-order behaviors of DVS in dim light conditions.

Index Terms—  Dynamic vision sensor, dim light conditions, event triggering
time, discontinuity, non-first-order behaviors

##  1 Introduction

Dynamic vision sensors (DVS) detect temporal contrast in light intensity,
different from conventional cameras sensing light intensity directly [1, 2, 3,
4, 5]. It imitates human retina, a logarithmic and first-order system
sensitive to a fixed light contrast but invariant to absolute light intensity
[6]. The humanoid behaviors facilitate the high dynamic range (more than 6
decades), low latency, and energy saving [7].

However, the first-order response does not suit for the DVS in dim light
conditions, due to the properties of DVS circuit[4]. The imperfect behaviors
in dim light conditions have been studied by researchers. To increase the
dynamic range, Delbruck and Mead[1] suppress the non-first-order response in
the conventional logarithmic light receptor by an adaptive element and cascode
feedback loop. This adaptive photoreceptor invented by Delbruck and Mead is
the prototype of recent DVS [8, 9, 10, 11]. Lichtsteiner, Posch, and Delbruck
[4] point out that the first-order response is contaminated by the photodiode
parasitic capacitor and resistance of the feedback metal-oxide-semiconductor
field-effect transistor (MOSFET) without further analysis. Hu, Liu and
Delbruck [12] design a widely-used DVS simulator, which forms the operation of
DVS in dim light conditions as an infinite impulse response (IIR) low pass
filter. Graca and Delbruck [13] research on the shot noise of DVS in dim light
conditions, showing the paradox between the first- and second-order systems.
Lin, Ma, et al. [14] use Brownian motion with drift for simulation, but the
simulated behaviors deviate partly from those of real DVS in dim light
conditions. The above researches contribute greatly to the developments of
DVS, but lack of further analysis of DVS in dim light conditions which are
quite common in practical applications.

In this paper, the typical DVS circuit is studied. We find that there exists
discontinuity of event triggering time, which is one of the main factors
deciding the DVS’s unsatisfied behaviors in dim light conditions. Based on our
analysis, the discontinuity extends with slow changing speed of light
intensity. Because the dim light conditions generally share small difference
of light intensity, the discontinuity will be prominent. By analyzing the
properties of DVS circuit, the discontinuity of event triggering time is
attributed to the charge and discharge of parasitic capacitor of photodiode.
Experimental results on real data of DVS validate the above analysis and will
be helpful for further improvements of DVS, such as design of event
simulators. For convenience, the meanings of symbols used in this paper are
illustrated in Table. 1.

Table 1: symbol meanings

symbols | meanings
---|---
$\rm M_{fb}$ | feedback MOS
$\rm M_{n}$ and $\rm M_{cas}$ | cascode MOS amplifier
$\rm M_{sf}$ | MOS source follower
$\rm M_{dp}$ | MOS providing difference of voltage
$\rm PD$ | photodiode
$\rm C_{1}$ and $\rm C_{2}$ | capacitors for differential amplifier
$\rm C_{J}$ | parasitic capacitor of PD
$L$ | light intensity
$I_{pd}$ | photocurrent across photodiode $\rm PD$
$V_{pd}$ | voltage of photodiode
$V_{p}$ | gate voltage of $\rm M_{fb}$ fed back by cascode
$V_{sf}$ | drain voltage of $\rm M_{sf}$ buffered from $V_{p}$
$V_{d}$ | source voltage of $\rm M_{dp}$
$\Delta Q_{e}$ | charge of difference in $\rm C_{J}$ during triggering of events
$\Delta t_{e}$ | time delay between consecutive triggering events
$\mu$ | changing speed of light intensity

##  2 Discontinuity of event triggering time

In this section, the typical DVS circuit is analyzed along with its operation
principle. The charge and discharge of parasitic capacitor of photodiode
causes a time delay between triggering events, leading to the discontinuity.
The non-first-order behavior is aroused by the duration of discontinuity that
depends exclusively on the changing speed of light intensity and photocurrent.

###  2.1 DVS circuit and operation principle

Fig. 1: DVS pixel circuit.

The typical DVS circuit of one pixel is illustrated in
${\rm{Fig.}\ref{fig:pixel_circuit}}$. When the light signal on the photodiode
$\rm PD$ is static without change, there will be few events other than some
noise induced by the junction leak current of diode DL. Assume that the last
event triggers at the time $t_{1}$. After time $\Delta t$, the incident light
on $\rm PD$ changes from $L(t_{1})$ to ${L(t_{1}+\Delta t)}$, and the source
voltage of the metal-oxide-semiconductor (MOS) transistor $\rm{M_{fb}}$ varies
from $V_{d}(t_{1})$ to ${V_{d}(t_{1}+\Delta t)}$. In case that the MOS
comparators $(\rm{M_{ONp}}\quad and\quad\rm{M_{OFFp}})$ detect that
${V_{d}(t_{1}+\Delta t)}$ overcomes predefined ON threshold -$\Theta_{ON}$ or
OFF threshold $\Theta_{OFF}$, an ON or OFF event is triggered, respectively.
The process is formulated as follows:

${V_{d}(t_{1}+\Delta t)}\quad\begin{cases}\geq\Theta_{OFF},&trigger\ OFF\
events\\\ \leq-\Theta_{ON},&trigger\ ON\ events\\\ others,&no\ events.\\\
\end{cases}$ (1)

The source voltage ${V_{d}}$ will be reset when the event triggers.

After the introduction of event triggering, we analyze the signal transduction
from light signal $L(t)$ to electrical signal $V_{d}(t)$ in the DVS circuit.
At time $t_{1}$, the photocurrent flowing across the photodiode $\rm PD$ is
$I_{pd}(t_{1})$. After time $\Delta t$, the change of photocurrent is induced
in the photodiode $\rm PD$ that satisfies:

$I_{pd}(t_{1}+\Delta t)-I_{pd}(t_{1})\propto L(t_{1}+\Delta t)-L(t_{1}).$ (2)

The differential amplifier with capacitors $\rm C_{1}$ and $\rm C_{2}$
amplifies $V_{sf}$ that is buffered from $V_{p}$ and then drives the gate of
MOS $\rm M_{dp}$ to control the source voltage $V_{d}$ [4]:

$\displaystyle\Delta V_{d}(t_{1})$ $\displaystyle=$
$\displaystyle-A\cdot\Delta V_{sf}$ $\displaystyle=$
$\displaystyle\frac{-A\cdot V_{T}\cdot\kappa_{sf}}{\kappa_{fb}}({\rm
ln}I_{pd}(t_{1}+\Delta t)-{\rm ln}I_{pd}(t_{1})),$

where $A=\frac{\rm C_{1}}{\rm C_{2}}$ is the gain of differential amplifier,
$\kappa_{sf}$ and $\kappa_{fb}$ are the slope factors of MOS $\rm M_{sf}$ and
$\rm M_{fb}$, respectively, and $V_{T}$ is the thermal voltage.

###  2.2 parasitic capacitor of photodiode

Actually, the photodiode voltage $V_{pd}$ changes with the light signal $L$,
despite the clamping by the feedback loop and cascode. The cascode of $\rm
M_{n}$ and $\rm M_{cas}$ senses the change of $V_{pd}$ with amplification. The
amplified result $V_{p}$ is fed back by $\rm M_{fb}$ to accommodate the change
of $I_{pd}$.

Fig. 2: distributed diagram of photodiode PD.

In the distributed diagram of photodiode $\rm PD$ given in
${\rm{Fig.}\ref{fig:distribute_photodiode}}$, the junction of $\rm PD$
consists of the parasitic capacitor $\rm C_{J}$. As the parallel resistance
$\rm R_{SH}$ is extremely high and the series resistance $\rm R_{S}$ tends to
be quite low [15], the charge and discharge of $\rm C_{J}$ is one of the main
reasons for the delay and discontinuity of event triggering time.

Ideally, when there is a light intensity change in time $\Delta t$, an
immediately stimulated current $\Delta I_{pd}=I_{pd}(t_{1}+\Delta
t)-I_{pd}(t_{1})$ will flow across $\rm M_{fb}$. However, in fact this is not
the case. In the distributed diagram of photodiode $\rm PD$ shown in
${\rm{Fig.}\ref{fig:distribute_photodiode}}$, $\Delta I_{pd}$ needs to charge
or discharge the parasitic capacitor $\rm C_{J}$ of $\rm PD$, firstly. The
process is not completed until the voltage of $\rm C_{J}$ reaches
$V_{pd}(t_{1}+\Delta t)$ to accommodate the stimulated current $\Delta
I_{pd}$.

###  2.3 time delay and discontinuity

The charge and discharge of parasitic capacitor $\rm C_{J}$ is time-consuming,
leading to a time delay between triggering events. If we evaluate the
triggering time of events, the distribution of triggering time is probably
discontinuous because there are few events in the time delay region.

As the photodiode $\rm PD$ is reverse-biased, the barrier capacitance is
dominant in $\rm C_{J}$. Compared with the built-in potential of photodiode
$\rm PD$ (usually hundreds of mV) [16], the voltage change of $V_{pd}$ is much
smaller (nearly a few mV) [17]. According to [16], the parasitic capacitance
of $\rm C_{J}$ is almost unchanged.

The change of electric charge $\Delta Q$ stored in the capacitor $\rm C_{J}$
is:

$\Delta Q=\Delta V_{pd}\cdot{\rm C_{J}}=\Delta I_{pd}\cdot\Delta t.$ (4)

Based on Eqs. 1 and 2.1, the change $\Delta V_{p}$ is fixed between two
consecutive events. In the small-signal analysis of cascode,

$\Delta V_{p}=-A_{cas}\cdot\Delta V_{pd},$ (5)

where $-A_{cas}$ is the small-signal gain of cascode.

Therefore, $\Delta V_{pd}$ and $\Delta Q$ are invariable between two
consecutive triggering events if we do not consider the influence by shot
noise, dark current, and leak noise. Assuming that $\Delta Q_{e}$ is the fixed
electric charge difference, there is a time delay between triggering events:

$\Delta t_{e}=\frac{\Delta Q_{e}}{\Delta I_{pd}}.$ (6)

Note that: 1. $\Delta t_{e}$ is not proportional to $\frac{\Delta L}{L}$
($\Delta\ln L$), which means that the time delay follows a non-first-order
system rule of DVS; 2. As $\Delta I_{pd}$ is only related to the threshold
($\Theta_{ON}$ and $\Theta_{OFF}$), $\Delta t_{e}$ is invariant to current
light intensity.

In fact, $\Delta I_{pd}$ needs some time to change because light intensity
varies with time. DVS voltmeter [14] supposes that $\Delta I_{pd}$ and $\Delta
L$ change linearly with time, which is a reasonable assumption in a short
period. In that case, $\Delta t_{e}$ follows:

$\Delta t_{e}\propto\frac{1}{\Delta I_{pd}}\propto\frac{1}{\mu\cdot T},$ (7)

where $\mu=\frac{L(t_{1}+\Delta t)-L(t_{1})}{\Delta t}$ is the changing speed
of light intensity and $T$ is the total time between the two triggering
events. As can be seen, the time delay $\Delta t_{e}$ is inversely proportion
to the changing speed of light intensity $\mu$.

The existence of $\Delta t_{e}$ delays the triggering of events, breaking the
inverse Gaussian distribution that is followed by the event triggering time
[14]. Because that there are few events triggering during the time delay
$\Delta t_{e}$, the distribution of event triggering time fluctuates
dramatically, appearing as the discontinuity in the histograms and probability
density functions (PDF). As a result, the length of discontinuity in the
histograms and PDFs of event triggering time is $\Delta t_{e}$.

In the dim light conditions, the difference of light intensity is generally
small, leading to a slow changing speed of light. Therefore, the time delay
$\Delta t_{e}$ is prominent.

##  3 EXPERIMENTS

In the section, we provide the existence of time delay $\Delta t_{e}$ and the
discontinuity of event triggering time based on real experimental data. The
details of time delay $\Delta t_{e}$ are further validated.

###  3.1 experimental datasets and settings

A real public DAVIS dataset [10], captured by DAVIS240 event camera, is used.
Inside the dataset, boxes 6dof scene including over 1.3 billion events is
applied. To evaluate the time delay and discontinuity of event triggering time
accurately, the frame-based videos are interpolated [18] by 10 times, as done
in [14, 12, 19]. We use the ITU-R recommendation $BT.\ 709$ for linear
conversion [20]. For the gray frame-based videos, the pixel value is treated
as the luma value, in the changing speed $\mu$ and light intensity $L$.

In the following experiments, the histograms of event triggering time are
collected under different light intensity $L$ and light changing speed $\mu$.
Here $L$ is specified as the average pixel value between the two consecutive
events.

###  3.2 results

(a) $\mu=50,\ L=10$

(b) $\mu=50,\ L=20$

(c) $\mu=50,\ L=30$

(d) $\mu=50,\ L=40$

(e) $\mu=50,\ L=50$

(f) $\mu=100,\ L=10$

(g) $\mu=100,\ L=20$

(h) $\mu=100,\ L=30$

(i) $\mu=100,\ L=40$

(j) $\mu=100,\ L=50$

(k) $\mu=200,\ L=10$

(l) $\mu=200,\ L=20$

(m) $\mu=200,\ L=30$

(n) $\mu=200,\ L=40$

(o) $\mu=200,\ L=50$

(p) $\mu=300,\ L=10$

(q) $\mu=300,\ L=20$

(r) $\mu=300,\ L=30$

(s) $\mu=300,\ L=40$

(t) $\mu=300,\ L=50$

(u) $\mu=400,\ L=10$

(v) $\mu=400,\ L=20$

(w) $\mu=400,\ L=30$

(x) $\mu=400,\ L=40$

(y) $\mu=400,\ L=50$

(z) $\mu=500,\ L=10$

(aa) $\mu=500,\ L=20$

(ab) $\mu=500,\ L=30$

(ac) $\mu=500,\ L=40$

(ad) $\mu=500,\ L=50$

Fig. 3: distributions of event triggering time. Blue lines indicate the
histograms of event triggering time and red dashed lines are the fitting
curves. For different $L$, the lengths of discontinuity remain stable,
demonstrating that the time delay $\Delta t_{e}$ is not related to current
light intensity. Different from that, the changing speed $\mu$ decreases the
length of discontinuity.

Fig. 3 shows the statistical distributions of event triggering time as the
histograms with blue lines in different conditions. The fitting curves above
the histograms, marked as red dashed lines, roughly follow the inverse
Gaussian distribution. As is exemplified in Fig. 4, there exists significant
discontinuity in the histograms, which is also shown in Fig. 3. It validates
the existence of the time delay $\Delta t_{e}$ between consecutive triggering
events caused by charge and discharge of parasitic capacitor of photodiode
$\rm PD$.

Fig. 4: exemplified discontinuity of event triggering time. (Here $\mu=50$,
$L=10$)

In Fig. 3, the lengths of discontinuity remain the same in each row but vary
in each column, which demonstrates that the time delay $\Delta t_{e}$ only
depends on $\mu$ rather than current light intensity $L$. Therefore, the time
delay $\Delta t_{e}$ follows a non-first-order behavior.

Table 2: time delay $\Delta t_{e}$

| current light intensity $L$
---|---
| 10 | 20 | 30 | 40 | 50
light changing speed $\mu$ | 50 | $9.0*e^{-3}s$ | $9.0*e^{-3}s$ | $9.0*e^{-3}s$ | $9.0*e^{-3}s$ | $9.0*e^{-3}s$
60 | $7.5*e^{-3}s$ | $7.5*e^{-3}s$ | $7.5*e^{-3}s$ | $7.5*e^{-3}s$ | $7.5*e^{-3}s$
70 | $6.5*e^{-3}s$ | $6.5*e^{-3}s$ | $6.5*e^{-3}s$ | $6.5*e^{-3}s$ | $6.5*e^{-3}s$
80 | $5.5*e^{-3}s$ | $5.5*e^{-3}s$ | $5.5*e^{-3}s$ | $5.5*e^{-3}s$ | $5.5*e^{-3}s$
90 | $4.5*e^{-3}s$ | $4.5*e^{-3}s$ | $4.5*e^{-3}s$ | $4.5*e^{-3}s$ | $4.5*e^{-3}s$
100 | $4.5*e^{-3}s$ | $4.5*e^{-3}s$ | $4.5*e^{-3}s$ | $4.5*e^{-3}s$ | $4.5*e^{-3}s$
150 | $3.0*e^{-3}s$ | $3.0*e^{-3}s$ | $3.0*e^{-3}s$ | $3.0*e^{-3}s$ | $3.0*e^{-3}s$
200 | $2.0*e^{-3}s$ | $2.0*e^{-3}s$ | $2.0*e^{-3}s$ | $2.0*e^{-3}s$ | $2.0*e^{-3}s$

11footnotemark: 1

The time delay $\Delta t_{e}$ is measured on the first discontinuous region,
as can be seen in Fig. 4. Because it is more accurate compared with others
that are influenced by the unstable changing speed of light.

We also evaluate the discontinuity length quantitatively in Table 2. The time
delay $\Delta t_{e}$ declines with the increasing of changing speed $\mu$,
which can also be seen in Fig. 3. Furthermore, the product of $\Delta t_{e}$
and $\mu$ is almost unchanged with the light intensity $L$. It proves our
analysis of the inversely proportional relation between them.

The time delay $\Delta t_{e}$ exists with a non-first-order behavior, unlike
the usual manner of first-order system of DVS. In the dim light conditions,
the changing speed of light is slow because the absolute difference of light
is small. As a result, the time delay $\Delta t_{e}$ is more significant.

##  4 CONCLUSIONS

In this paper, we study on the properties of DVS circuit, and find a new
behavior of DVS: the time delay $\Delta t_{e}$ and discontinuity of event
triggering time. It leads to a non-first-order behavior, different from the
usual manners of DVS. The time delay is inversely proportional to the changing
speed of light. In dim light conditions, the difference of light intensity is
small, slowing the light variation. As a result, the time delay and
discontinuity become prominent in dim light conditions. The experimental
results are also provided for validation.

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Nonclassical Properties of a State Generated by a Driven Dispersive Interaction
Naveen Kumar^1
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Arpita Chatterjee^2
Department of Mathematics,J. C. Bose University of Science and Technology, YMCA, Faridabad-121006, Haryana, India

We consider a cavity field state, which is created by the atom-cavity field's interaction in the presence of a driven field. The two-level atom passes through the cavity and is driven by a weak classical field. A photon number dependent Stark shift is induced by the atom's dispersive interaction with the cavity field. When the atom is in excited state $|a\rangle$, the output cavity field thus obtained is taken into consideration. With the help of the state evaluated, we investigate different statistical properties such as photon number distribution, Wigner function, Mandel's $Q$ parameter and squeezing.

§ INTRODUCTION

The production and manipulation of nonclassical light fields are of great interest in quantum optics and quantum information processing [1]. Nonclassical states have many real life applications, for example, squeezed states are used to reduce the noise level in one of the phase-space quadratures
below the quantum limit [2], entangled states produced
in down-conversion process are employed to test fundamental quantum features such as non-locality [3] and to
realize quantum information transmission schemes (cryptography [4, 5] or teleportation [6]). Quantum superpositions of fields with different classical parameters are used to explore the quantum or classical boundary and the decoherence phenomenon [7].

In this context, theoreticians and experimentalists have offered many different strategies to produce nonclassical states of the optical field. To produce a nonclassical state adding and/or subtracting photons to standard quantum states is the right approach. The coherent states after adding a number of photons exhibit the gradual shifting from spontaneous to stimulated regimes of light emission [11]. Also, subtraction of photons can be used to improve the Gaussian state's entanglement [12, 13] and quantum computing [14].

A single-photon Fock state is an essential resource in quantum information processing devices [15]. These one-photon Fock states can be prepared by using cavity QED experiments in which atoms interact one at a time with a high $Q$ resonator, a quantum $\pi$ Rabi pulse in a microwave cavity  [16] or by an adiabatic passage sequence of atoms in an optical cavity [17]. We report here how an interacting atom-cavity field system in presence of a weak classical field generates a nonclassical field.

In this article, we describe the nonclassical properties of the cavity field state. We consider the interaction between a two-level atom and a single-mode cavity field driven by a weak classical field. To find the desired state of interest we solve the time-dependent Schrodinger equation with the effective Hamiltonian of the atom-cavity field system. To check the nonclassical nature of the evaluated state, we discuss some properties like the Wigner function and Mandel's $Q$ parameter. The negativity of the Wigner function affirms the nonclassicality of the state. The study of these states provides an essential understanding of quantum communication and quantum teleportation technique.

§ STATE OF INTEREST

We consider a two-level atom interacting with a singlemode cavity field and driven by a weak classical field.
The Hamiltonian (assuming $\hbar = 1$) describing the atom-field interaction in the rotating-wave approximation is
[21, 22, 23]

\begin{eqnarray}\nonumber
H = \omega_0 S_z + \omega_a a^\dag a + g(a^\dag S^- + aS^+)
\epsilon (S^+ e^{-i\omega t} + S^- e^{i\omega t})\\
\end{eqnarray}

where all the notations have their usual meanings (cf. Table <ref>).
If the system's reference frame is rotated with respect to the driving field frequency $\omega$, the Hamiltonian is given by

\begin{equation}
H_f = \Delta S_z+\delta a^\dag a+g(a^\dag
\end{equation}

where $\Delta = \omega_0-\omega$, $\delta = \omega_a-\omega$.
For further simplification, considering  $\Delta=0$, the interaction Hamiltonian is

\begin{eqnarray*}
H_I & = & e^{iH_{f_0}t}H_{f_1} e^{-iH_{f_0}t}\\
& = & \frac{1}{2}g\left[|+\rangle\langle+|-|-\rangle\langle-|+e^{i2\epsilon t}|+\rangle\langle-|
- e^{-i2\epsilon t}|-\rangle\langle+|\right]\\
& & \times ae^{-i\delta t}+\mathrm{H.C.}
\end{eqnarray*}

\begin{eqnarray*}
H_{f_0} = \delta a^\dag a+\epsilon(S^++S^-)
\end{eqnarray*}

\begin{eqnarray*}
H_{f_1} = g(a^\dag S^-+aS^\dag)
\end{eqnarray*}

and the dressed states $|\pm\rangle = \frac{1}{\sqrt{2}}(|a\rangle\pm|b\rangle)$
are the eigenstates of $(S^++S^-)$ with the eigenvalues $\pm 1$.
In the strong driving regime, $\epsilon\gg\{\delta, g\}$. As a result, high-frequency oscillating terms can be discarded. Now our effective Hamiltonian $H_{\mathrm{eff}}$ is [24]

\begin{eqnarray}\nonumber
H_{\mathrm{eff}} =\frac{1}{2}g (aS^+e^{-i\delta t}+a^\dag S^- e^{i\delta t})
\end{eqnarray}

With help of this effective Hamiltonian $H_{\mathrm{eff}}$, the time-dependent Schrodinger equation  [2]

\begin{eqnarray}
\frac{i\partial | \psi(t) \rangle }{\partial t}= H_{\mathrm{eff}} |\psi (t)\rangle,
\end{eqnarray}

is now solved for an arbitrary state vector $|\psi(t)\rangle = \sum {c_{a, n}(t) |a, n\rangle +c_{b, n+1}(t) |b, n+1\rangle}(t)$ with probability amplitudes $c_{a,n}(t)$
and $c_{b,n+1}(t)$, $|a\rangle (|b\rangle)$ is the excited (ground) state. The equations of motion in terms of $\dot{c}_{a,n}(t)$ and $\dot{c}_{b,n+1}(t)$ are

\begin{eqnarray*}
\dot{c}_{a,n}(t) & = & -\frac{ig\sqrt{n+1}}{2}  e^{-i\delta t} c_{b,n+1}(t)\\
\dot{c}_{b,n+1}(t) & = & -\frac{ig\sqrt{n+1}}{2}  e^{i\delta t} c_{a,n}(t)
\end{eqnarray*}

Initially if the atom enters the cavity in the excited state $|a \rangle$ then $c_{a,n}(0) = c_n(0)$ and $c_{b,n+1}(0) = 0$, where $c_n(0)$ is the probability amplitude for the field
alone. A general solution with these conditions is

\begin{eqnarray*}
c_{a,n} & = & c_{n}(0)\left\lbrace \cos{\frac{\Omega_{n}t}{2}}-\frac{i\delta}{\Omega_{n}} \sin{\frac{\Omega_{n}t}{2}}\right\rbrace e^{\frac{i\delta t}{2}},\\
c_{b,n+1} & = & -c_{n}(0)  \frac{2ig\sqrt{n+1}}{\Omega_{n}} \sin{\frac{\Omega_{n}t}{2}} e^{\frac{i\delta t}{2}},
\end{eqnarray*}

where ${\Omega_n}^2=\delta^2 +4g^2(n+1)$. If the radiation field is initially in a coherent state then $c_{n}(0)=e^{-\frac{|\alpha|^2}{2}}\frac{\alpha^n}{\sqrt{n!}}$ [25]. The state vector $|\psi(t)\rangle$ describes the time evolution of the entire system of atom and cavity field but we now concentrate on some statistical aspects of the cavity field only. The field inside the cavity after departing the atom is
obtained by tracing out the atomic part of $\rho(t)$ as

\begin{equation}
\label{eq10}
{\rho(t)}_{f} = Tr_{a}[\rho(t)],
\end{equation}

where we have used the subscript $a (f)$ to denote the
atom (field).
This ${\rho(t)}_{f}$ will be of consideration in all of the further work to determine the statistical properties of the field left into the cavity.

Symbols used

$a^\dag$     creation operator

$a$       annihilation operator

$S_z$      inversion operator for the atom

$S^+$      raising operator for the atom

$S^-$      lowering  operator for the atom

$g$     coupling constant between the atom and the cavity field

$\epsilon$     coupling constant between the atom and the classical field

$\omega_0$    frequency for atomic transition

$\omega_a$    frequency for cavity mode

$\omega$       frequency for classical field.

$\rho$       density operator

$\ket{\psi}$       state vector

§ PHOTON NUMBER DISTRIBUTION

Photon number distribution (PND) is the probability distribution of finding $n$ photons in a given cavity field state and it can be obtained as the expectation of the density field in Fock state $|n\rangle$ as

\begin{eqnarray}
\label{eq4}
p(n) = \langle n|\rho|n\rangle
\end{eqnarray}

\begin{eqnarray}\nonumber
\label{eq5}
{\rho}_f & = & |\psi \rangle_f\,{}_f\langle \psi|\\
& = & \sum_{n} \Big[|c_{a,n}|^2|n\rangle \langle n|+ c_{a,n} c_{b,n+1}^*|n\rangle \langle n+1|\\\nonumber
& &  + c_{a,n}^* c_{b,n+1} |n+1\rangle \langle n| +
|c_{b,n+1}|^2 |n+1\rangle \langle n+1|\Big]
\end{eqnarray}

Substituting (<ref>) into (<ref>), the photon number distribution for the cavity field is obtained as

\begin{eqnarray}\nonumber
\label{eq6}
p(n) & = & |c_n(0)|^2\left[{\cos}^2\left(\frac{\Omega_n t}{2}\right)+\frac{\delta^2}{{\Omega_n}^2} {\sin}^2\left( \frac{\Omega_n t}{2}\right)+\frac{4g^2(n+1)}{{\Omega_n}^2}{\sin}^2\left( \frac{\Omega_n t}{2}\right)\right]\\
& = & |c_n(0)|^2
\end{eqnarray}

Here (<ref>) gives the final expression of PND, where $c_n(0)$ is the probability
amplitude of the field alone.

Photon number distribution of cavity field state is plotted against $n$ for $|\alpha|=0.5$.

In Fig. <ref>, we see the regular changes in PND as we are changing photon number $n$. It is observed that with increasing $n$, PND decreases exponentially.

§ WIGNER FUNCTION

The nonclassicality of a quantum state can be studied
in terms of its phase-space distribution characterized by
the Wigner distribution. For a quantum state $\rho$, the
Wigner function of the system is defined in terms of the
coherent state basis [2, 26, 27] as

\begin{eqnarray*}
W(\beta, \beta^*) = \frac{2}{\pi^2}e^{2|\beta|^2} \int d^2\gamma{\langle-\gamma|\rho|\gamma\rangle e^{-2(\beta^*\gamma-\beta\gamma^*)}},
\end{eqnarray*}

$|\gamma\rangle$ is a coherent state.
By using the relation [28]

\begin{eqnarray*}
\sum_{n=k}^\infty n_{C_k}\,y^{n-k} = (1-y)^{-k-1},
\end{eqnarray*}

the Wigner function can be expressed in series form as [29]

\begin{eqnarray}
\label{eqn8}
W(\beta, \beta^*) = \frac{2}{\pi} \sum_{k=0}^\infty \left[(-1)^k \langle \beta,k|\rho|\beta,k\rangle\right] ,
\end{eqnarray}

where $|\beta,k\rangle$ is defined as the displaced Fock state. The
partial negative value of the Wigner function is a one
sided condition for the nonclassicality of the related state
[30], in the sense that one cannot conclude that the state
is classical when the Wigner function is positive everywhere. For example, the Wigner function of the squeezed
state is Gaussian and positive everywhere but it is a well-known nonclassical state. For a classical state, the positivity of the
Wigner function is a necessary condition but not a sufficient one. Hence, a state with a negative region
in the phase-space distribution is essentially nonclassical.

Now the displaced number state $|\beta, k\rangle$ can be expressed
in number state basis as following

\begin{eqnarray}\nonumber
\label{eqn9}
|\beta, k\rangle & = &
D(\beta)| k \rangle\\\nonumber
& = & e^{-\frac{|\beta|^2}{2}}\sum_{m=0}^{k} \frac{ \beta^{*{m}}}{m!} \sqrt{\frac{k!}{(k-m)!}}\sum_{p=0}^{\infty} \frac{ \beta^p}{p!}\sqrt{\frac{(k-m+p)!}{(k-m)!}}|k-m+p \rangle
\end{eqnarray}

\begin{eqnarray}\nonumber
\langle \beta,k|n \rangle & = &
e^{-\frac{|\beta|^2}{2}}\sum_{m=0}^{k} \frac{(-\beta)^{m}}{m!} \sqrt{\frac{k!}{(k-m)!}} \sum_{p=0}^{\infty}\frac{ \beta^{*{p}}}{p!}  \sqrt{\frac{(k-m+p)!}{(k-m)!}}\langle k-m+p|n\rangle\\\nonumber
%	& = & e^{-\frac{|\beta|^2}{2}}\sum_{m=0}^{k} \frac{(-\beta)^{m}}{m!} \sqrt{\frac{k!}{n!}} \frac{(\beta^*)^{n-k+m} n!}{(k-m)!(n-k+m)!}\\\nonumber
%	& = & e^{-\frac{|\beta|^2}{2}} \sqrt{\frac{k!}{n!}} (\beta^*)^{n-k}\sum_{m=0}^{k}  \frac{(-|\beta|^2)^{m} n!}{(k-m)!m!(n-k+m)!}\\
& = & e^{-\frac{|\beta|^2}{2}} \sqrt{\frac{k!}{n!}} (\beta^*)^{n-k}L_{k}^{(n-k)}\left(|\beta|^2 \right),
\end{eqnarray}

where $L_{m}^{(k)}(x)=\sum_{n=0}^{m}  \frac{(-x)^{n} (m+k)!}{ n!(m-n)!(k+n)!}$ is the associated Laguerre polynomial [31]. Substituting (<ref>) into (<ref>) we get

\begin{eqnarray}\nonumber
W(\beta, \beta^*) & = & \frac{2}{\pi} \sum_{k=0}^{\infty} (-1)^k \bigg[c_{a,n}c_{a,n}^* \langle\beta,k|n\rangle\langle n|\beta,k \rangle+ c_{a,n}c_{b,n+1}^* \langle \beta,k|n \rangle \langle n+1|\beta,k \rangle\\\nonumber
& & + c_{a,n}^*c_{b,n+1} \langle\beta,k|n+1\rangle \langle n|\alpha,k\rangle +c_{b,n+1}c_{b,n+1}^* \langle \alpha,k|n+1 \rangle \langle n+1|\alpha,k\rangle\bigg]\\\nonumber
& = & \frac{2e^{-|\beta|^2}}{\pi} \sum_{k=0}^{\infty} (-1)^k k! \bigg[|c_{a,n}|^2 \frac{|\beta|^{2(n-k)}}{n!} \left\{L_{k}^{(n-k)} \left(|\beta|^2 \right) \right\}^2\\
& & +|c_{b,n+1}|^2  \frac{|\beta|^{2(n+1-k)}}{(n+1)!} \left\{L_{k}^{(n+1-k)} \left(|\beta|^2 \right) \right\}^2\bigg]
\end{eqnarray}

Wigner distribution of cavity field state as a function of Real part of ($\beta$) and Imaginary part of ($\beta$) for fixed $|\alpha|=0.02$ and (a) $n=4$, (b) $n=7$, (c) $n=10$.

In Fig. <ref>, the Wigner distributions for a fixed value of $|\alpha|$ and various distinct values of $n$ are shown. The cavity field exhibits nonclassical characteristics. When the number of photons is increased, keeping other parameters fixed, the Wigner function becomes less negative. As a result, a cavity field with more photons can prepare a classical Gaussian-like state.

§ MANDEL'S $Q$ PARAMETER

Next to determine the photon statistics of a single-mode radiation field we consider the Mandel’s $Q$ parameter defined by [32]

\begin{equation}
\label{eq14}
Q=\frac{\langle a^{{\dagger}^2} a^2 \rangle-(\langle a^{\dagger}a\rangle)^2} {\langle a^{\dagger}a\rangle}
\end{equation}

which measures the deviation of the variance of the photon number distribution of the considered state from the
Poissonian distribution of the coherent state. If $Q=0$, the distribution is Poissonian while for $-1\leq Q<0$ ($Q>0$), the field obeys super-Poissonian (sub-Poissonian) photon statistics. However, the lack of positivity in $Q$ is a sufficient condition for separating quantum states into nonclassical and classical regimes. For example, a state may be nonclassical even if $Q$ is positive [31]. The expectation values can be found as following

\begin{eqnarray*}
\langle a^{{\dagger}^2} a^2 \rangle & = & {}_f\langle \psi|a^{{\dagger}^2} a^2|\psi \rangle_f\\
& = & n(n-1)|c_{a,n}|^2+(n+1)n|c_{b,n+1}|^2 \\
\langle a^{\dagger}a\rangle & = & {}_f\langle \psi|a^{\dagger} a|\psi \rangle_f\\
& = & n|c_{a,n}|^2+(n+1)|c_{b,n+1}|^2
\end{eqnarray*}

Now substituting these values in (<ref>), we get the expression for  $Q$ for the cavity-driven field as

\begin{eqnarray}\nonumber
Q & = & \frac{n(n-1)|c_{a,n}|^2+(n+1)n|c_{b,n+1}|^2}{n|c_{a,n}|^2+(n+1)|c_{b,n+1}|^2}-\left(n|c_{a,n}|^2+(n+1)|c_{b,n+1}|^2\right)
\end{eqnarray}

Mandel’s $Q$ parameter for a cavity field
state as a function of $\alpha$ ($\alpha$ assumed to be real) with different
photon numbers.

In order to see the variation of the Mandel’s $Q$ parameter against the coherent state amplitude $\alpha$, it is plotted
in Fig. <ref>.
Here $Q$ curves go below zero for a certain limit of $\alpha$ which depicts that the cavity field enjoys sub-Poissonian character in a particular interval. It is observed that $Q$ usually exhibits super-Poissonian nature over the range of $\alpha$ and as $n$ increases, Mandel's $Q$ becomes more super-Poissonian.

§ SQUEEZING PROPERTIES

In order to analyze the quantum fluctuations of the
field quadratures, we consider two Hermitian operators which are combinations of photon creation and annihilation operators as

with $\left[\hat{x},\hat{p}\right] = i/2 $. They follow the Heisenberg uncertainty principle expressed as
$\langle(\Delta\hat{x})^2\rangle\langle(\Delta\hat{p})^2\rangle\geq\frac{1}{16}$, and thus the quadrature squeezing occurs whenever $\langle(\Delta\hat{x})^2\rangle<\frac{1}{4}$ or $\langle(\Delta\hat{p})^2\rangle<\frac{1}{4}$. The squeezing parameters $s_x$ and $s_p$ are introduced as [33]

\begin{eqnarray}\nonumber
s_x & = & 4\langle(\Delta\hat{x})^2\rangle-1\\\nonumber
& = & 2\langle{a}^\dag{a}\rangle+\langle{a}^2\rangle+\langle{a}^{\dag2}\rangle-
\langle{a}\rangle^2-\langle{a}^\dag\rangle^2-2\langle{a}\rangle\langle{a}^\dag\rangle,
\end{eqnarray}

\begin{eqnarray}\nonumber
s_p & = & 4\langle(\Delta\hat{p})^2\rangle-1\\\nonumber
& = & 2\langle{a}^\dag{a}\rangle-\langle{a}^2\rangle-\langle{a}^{\dag2}\rangle+
\langle{a}\rangle^2+\langle{a}^\dag\rangle^2-2\langle{a}\rangle\langle{a}^\dag\rangle,
\end{eqnarray}

Squeezing occur in $\hat{x}$ or $\hat{p}$ quadrature if $-1<s_x<0$ or $-1<s_p<0$, respectively. The expectations can be evaluated as

\begin{eqnarray*}
\langle{a}^\dag{a}\rangle & = & n |c_{a,n}|^2+(n+1)|c_{b,n+1}|^2\\
\langle {a}^2\rangle & = & 0\\
\langle {a}^{\dagger^2}\rangle & = & 0\\
\langle {a}\rangle & = & \sqrt{n+1} c_{a,n}^*c_{b,n+1} \\
\langle {a}^\dagger \rangle & = & \sqrt{n+1} c_{a,n} c_{b,n+1}^*
\end{eqnarray*}

Substituting these values, the expressions for the squeezing parameters can be found.

Plots of squeezing parameters $s_x$, $s_p$ against $\alpha$.

The negativity of squeezing is a sufficient criterion for identifying nonclassical quantum state regimes, rather than a necessary one. In fig. <ref>, both the parameters $s_x$, $s_p$ are plotted against $\alpha$ and for different $n=1, 2$ and $3$. There is no squeezing in any of the quadratures, showing that the parameter $s_x$, $s_p$ are not so useful for identifying nonclassicality in the resulting cavity field.

§ CONCLUSION

In the article, we have obtained a cavity field state. The atom-cavity field is interacting in presence of a driven weak classical field. Assuming  the field is initially in a coherent state and the atom enters the cavity in its excited state $|a\rangle$, the cavity field state is obtained. The statistical features, namely the photon number distribution, Wigner function, Mandel's $Q$, and squeezing parameters, are calculated. The nonclassicality of the state has been depicted by two of them; the Wigner function's negativity and the Poissonian statistics of Mandel's $Q$ parameter. We have observed that the Wigner function has a partially negative region in phase space, indicating the nonclassicality of the cavity field state. Also, the $Q$ is negative in a certain range of $\alpha$.
We have also seen that the $Q$ function mostly displays super-Poissonian features. Here the squeezing parameters fail to depict the nonclassical behavior of the cavity field.

We can conclude that the cavity field state is showing nonclassical nature. The study of nonclassical states gives us the basic knowledge of quantum fluctuations and leads to more efficient way for new communication strategies. These generated states are widely used for testing fundamental quantum features such as teleportation and cryptography. These states are also useful in quantum computing and have many applications in quantum information processing.

§ ACKNOWLEDGEMENTS

Naveen's work is supported by the Council of Scientific and Industrial Research, Govt. of India (Grant no. 09/1256(0004)/2019-EMR-I).
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