# Incoherent control of two-photon induced optical measurements in open
quantum systems: quantum heat engine perspective

Md Qutubuddin and Konstantin E. Dorfman<EMAIL_ADDRESS>State Key
Laboratory of Precision Spectroscopy, East China Normal University, Shanghai
200062, China

###### Abstract

We present a consistent optimization procedure for the optical measurements in
open quantum systems using recently developed incoherent control protocol.
Assigning an effective hot bath for the two-entangled-photon pump we recast
the transmission of classical probe as a work in a quantum heat engine
framework. We demonstrate that maximum work in such a heat engine can exceed
that for the classical two-photon and one-photon pumps, while efficiency at
maximum power can be attributed to conventional boundaries obtained for three-
level maser heat engine. Our results pave the way for incoherent control and
optimization of optical measurements in open quantum systems that involve two-
photon processes with quantum light.

###### pacs:

Valid PACS appear here

##  I Introduction

Thermal engine [1] plays a pivotal role in many thermodynamical processes. In
the seminal work by Scovil and Schulz-DuBois [2], a maser heat engine was
formulated in the context of the detailed balance which results in the maximum
efficiency [3, 4] for a three level system operating between hot and cold
baths. The ascent of quantum heat engines (QHEs) attracted a significant
amount of attention in the last few decades and constitutes an important
research direction within the quantum thermodynamics, both theoretically [5,
6, 7, 8, 9, 10, 11, 12, 13] and experimentally [14, 15, 16, 17, 18, 19]. In
addition to the diverse development of QHEs [20, 21, 22], it has intrinsic
relationship with the real physical systems such as laser, solar cell[23, 24],
battery [25, 26], light harvesting [27], etc. While some of the promising
features of QHEs such as quantum coherence and entanglement [28, 29, 30, 31]
show a possibility for enhancing the maximum output power for resonantly
driven systems [32], the significance of entanglement in optical measurements
in open quantum systems from the QHE perspective has not been investigated so
far.

Recently, authors developed an incoherent control method of optical signals
[33] that views the pump-probe measurements as a QHE, which transfers energy
from the pump pulse to the probe pulse treating the dissipation to the
environment explicitly, while computing the work performed by the system via
the detected probe photons. In this method we have introduced an effective
thermal bath by combining a coherent pump pulse excitation of electronic
excited state of molecule with the thermal relaxation. The “incoherent”
control algorithm for the optical signals in open quantum systems is then
introduced based on the analogy with the QHE. It has been further shown that
the spectroscopic measurement for the probe pulse transmission can be improved
when the corresponding parameter regime is close to the limiting operation
such as the Curzon-Ahlborn limit etc.

In the course of the above developments we realized that the efficient
operation of QHE in the open quantum systems such as molecules is strongly
correlated with the ability to efficiently excite a particular electronic
state. In the same time the various molecular degrees of freedom such as
nuclear motion, the complex selection rules, and the associated dissipation
processes restrict the ability to control the excited states probabilities.
Moreover the latter are governed by the uncertainty relation between the
spectral bandwidth of the molecules and the temporal profile of the excitation
pulse. It has been further shown that the two photon excitation with the
entangled photons can efficiently control the multi exciton population
distribution in the complex molecules since these can violate the uncertainty
principle in the two-photon absorption measurements[34]. In addition to the
spectral selectivity, the entangled two-photon absorption probability scales
linearly with the pump intensity in contrast with the classical two-photon
absorption which is quadratic in the pump intensity [35]. This feature makes
it attractive for the low intensity applications in photo sensitive materials
such as biological molecules etc [36]. While a significant amount of
experimental [37] effort has been dedicated for the optimization of two-photon
absorption measurement using quantum light, it lacks a formalized theoretical
foundation due to the dissipative nature of the open quantum system. It is
therefore imperative to develop a consistent optimization procedure for such
optical measurements in connection with the fundamental frameworks such as
QHEs.

In this article we explore the QHEs analogy with the setup that involves two-
photon pump excitation with both classical and quantum states of light by
utilizing the effective thermal bath [33]. The two photons can be initially in
e.g., entangled twin photon state, which allows us to explore the effects of
entanglement in QHE operation. By presenting a consistent technique of
maximization of power and efficiency at maximum power for the two-photon
pumped QHE using incoherent control, one can manipulate the two-photon induced
fluorescence (TPIF) and pump-probe signals due to additional control parameter
(entanglement time) [38, 34] which does not exist classically.

##  II Effective Heat Bath

We consider a three-level molecular system with ground state $g$, single
excited electronic state $e$ and double excited electronic state $f$ (see in
Fig. 1). To keep the notations consistent with our previous work [33] we
denote vibrational states of electronic ground state as $0$ and $g$, while
vibrational states of double excited electronic state are $2$ and $1$, and $e$
and $e^{\prime}$ denote vibrational states of single excited electronic state.

Figure 1: (a) Schematic for the three-level molecule undergoing two-photon
pump - probe measurements. The pump field resonant with transition $g-2$
excites a vibrational wave packet in the higher vibrational state $2$ via the
intermediate levels $e$ and $e^{\prime}$, which relaxes to the lower energy
vibrational state $1$. The probe field then stimulates the emission from the
state $1$ to the excited vibrational level $0$ in the ground electronic state.
Finally, vibrational relaxation brings the system back to its ground state
$g$. (b) Equivalent three-level QHE with transitions between energy levels
$g-1$ and $g-0$ driven by hot (at temperature $T_{h}$) and cold (at $T_{c}$)
heat baths. The single-mode stimulated emission representing the work done by
the QHE occurs at $1-0$ transition with the Rabi frequency $\lambda$.

A two-photon pump field excites the molecule from $g$ to $2$ via virtual
states $e-e^{\prime}$ with the pump Rabi frequencies $\Omega_{i}$ and central
frequencies $\omega_{i}$, $i=1,2$. The vibrational state $2$ relaxes to $1$ by
emission of phonon. The stimulated emission $1-0$ due to interaction with
probe field of Rabi frequency $\lambda$ followed by the thermal relaxation via
interaction of $0-g$ transition with the cold bath which then brings the
system to its initial ground states. The total Hamiltonian of the system is
given by

$\displaystyle\hat{H}_{tot.}=\hat{H}_{0}+\hat{H}_{I}+\hat{H}_{I.V.},$ (1)

where subscript $I$ indicates the light-matter interaction of pump and probe
fields and $I.V.$ indicates the interaction with the vibrational modes.
$\hat{H}_{0}=\sum_{i}\omega_{i}|i\rangle\langle i|$ where
$i=g,0,e,e^{\prime},1,2$. The pump-molecule interaction Hamiltonian in the
rotating wave approximation reads

$\displaystyle\hat{H}_{I}(t)=-\hat{V}^{\dagger}(t)\sum_{i=1,2}(\hat{E}_{j}(t)+\hat{E}^{\dagger}_{j}(t)),$
(2)

where $\hat{V}(t)=\mu_{ge}\ e^{-i\omega_{e}t}\ |g\rangle\langle e|+\mu_{e2}\
e^{-i\omega_{2}t}\ |e\rangle\langle 2|$, and
$\hat{E}_{j}(t)=i\int_{0}^{\infty}d\omega_{j}\sqrt{\frac{\hbar{\omega}_{j}}{2\mathbf{V}\epsilon_{0}}}\hat{a}_{j}(\omega_{j})e^{-i\omega_{j}t}$,
where $j=1,2,$ indices denote the first and second photons, where
$\hat{a}_{j}$ and $\hat{a}^{\dagger}_{j}$ are the annihilation and creation
operators for the $j$-th photon that satisfy the commutation relation
$\left[\hat{a}_{j}(\omega),\hat{a}_{j^{\prime}}^{\dagger}(\omega^{\prime})\right]=\delta(\omega-\omega^{\prime})\delta_{jj^{\prime}}$,
$\mathbf{V}$ is the quantization volume and
$H_{I.V}=\sum_{m,i<j}b^{\dagger}_{m}|i\rangle\langle j|e^{-i\omega_{i}t}$. The
two-photon pump utilized in our work originates from the single photon sources
so the coupling to the system is naturally weak. In the case of multi photon
sources such as squeezed light the coupling can be moderate and yet, it is far
from being strong [39]. Hence, we consider that the couplings to pump fields
are weak so we can use the perturbation theory.

Assuming that all molecules are initially in the ground state, the density
matrix of interacting matter-field system at time $t$ is given in the
interaction picture by the time-ordered exponential superoperator

$\displaystyle{\hat{\rho}}(t)=\hat{\mathcal{T}}\text{exp}\left[-\frac{i}{\hbar}\int^{t}dt^{\prime}\hat{H}_{int,-}(t^{\prime})\right]\hat{\rho}_{mat}\otimes\hat{\rho}_{\text{field}},$
(3)

where $\mathcal{T}$ is a time ordering superoperator, the interaction
Hamiltonian superoperator $H_{int,\pm}$ is defined by its action on the
ordinary operator $X$ as:
$H_{int,-}X=H_{int}X-XH_{int}~{}~{}\text{and}~{}~{}H_{int,+}X=H_{int}X+XH_{int}$
[40],

Figure 2: (a) The set of double sided Feynman diagrams representing the
leading order contribution to the population of excited state due to
relaxation of $2-1$. There are a total of six pathways that contribute to the
matter field system: diagrams (a)–(c), and their complex conjugate.

$\hat{\rho}_{mat}$ denotes the ground state of the matter system, and
$\hat{\rho}_{\text{mat}}$ denotes the initial state of the light field. The
leading-order contribution to the population of level $1$ is given by the
convolution of four-point field and matter correlation functions using Feynman
diagram in Fig. 2 is obtained in Eqs. (A17), (A18) and (A19) in Appendix A.
The total population of state $1$ due to pumping transition $g-2$ followed by
the relaxation $2\rightarrow 1$ is given by
$\rho_{11}(t)=\rho^{a}_{11}(t)+\rho^{b}_{11}(t)+\rho^{c}_{11}(t)$ obtained in
Eq.(A20), and it recast as by introducing the detuning
$\delta=\omega_{2e}-\omega_{2e^{\prime}}$ and assuming the pump is tuned
midway between $e$ and $e^{\prime}$ states
$\omega_{0}=\frac{1}{2}(\omega_{2e^{\prime}}+\omega_{2e})$. The solution we
obtain the population of level $1$ from Eq. (A20) reads

$\displaystyle\rho_{11}(t)$
$\displaystyle=\frac{16\,\delta^{2}\,\tilde{\delta}^{2}\,\Omega^{4}_{p}\
(1-e^{-\Gamma_{2}(2n_{2}+1)})}{(2n_{2}+1)(\delta^{2}+4\sigma^{2}_{p})^{2}(\tilde{\delta}^{2}+4\sigma^{2}_{p}+)^{2}},$
$\displaystyle\rho_{gg}(t)$ $\displaystyle=1-\rho_{11}(t),\ \ $ (4)

where $\tilde{\delta}=\delta+2\omega_{2e^{\prime}}-2\omega_{e^{\prime}g}$.
Before proceeding to the QHE model, which is based on the perturbative
solution of the complete set of equations given by Eq. (A21) (Appendix A), we
first introduce an effective heat bath. To that end, we assume that the probe
field is much stronger than the coupling to the phonon bath that governs the
$2-1$ transition, which itself is stronger than that of the bath driving $0-g$
transition: $\lambda>>\Gamma_{2}n_{2}>>\Gamma_{c}n_{c}$. The latter condition
can be obtained in a variety of molecular systems [44]. Under these conditions
one can eliminate the state $0$ from the total system of Eq. (A21) and
consider only three states such that the coherent excitation $g-2$ is followed
by a relaxation $2-1$ in Fig 1b. The combined effect of the coherent
excitation $g\rightarrow 2$ followed by the phonon relaxation $2\rightarrow 1$
can be replaced by an effective thermal bath at temperature $T_{h}$ with the
average photon number
$n_{h}=\left[\text{exp}(\hbar\omega_{1g}/k_{B}T_{h})-1\right]^{-1}$ and
dephasing $\Gamma_{h}$. In this case state $2$ can be eliminated and the
corresponding equation of motion for the populations of $g$ and $1$ read

$\displaystyle\dot{\rho}_{11}=-\Gamma_{h}\left[(n_{h}+1)\rho_{11}-n_{h}\rho_{gg}\right],\
\dot{\rho}_{gg}+\dot{\rho}_{11}=0,\ \ $ (5)

which yields the time-dependent solution

$\displaystyle\rho^{th}_{11}(t)=\mathcal{N}_{th}n_{h}(1-e^{-\Gamma_{h}(2n_{h}+1)t}),\
\rho^{th}_{gg}+\rho^{th}_{11}=1,\ \ $ (6)

where superscript $"th"$ indicates the thermal bath and the normalization
$\mathcal{N}_{th}=\left[1+2n_{h}\right]^{-1}$. Following the approach outlined
in Ref. [33], the population of level $1$ excited by a coherent drive is given
in Eq. (4) and the $\rho_{gg}(t)$ is obtained using the population
conservation $\rho_{11}(t)+\rho_{gg}(t)=1$. This coherently excited
populations $1$ and $g$ match with the thermal bath driven populations $1$ and
$g$ given in Eq. (6). By matching the two we obtain $n_{h}$ and $\Gamma_{h}$,
that must satisfy

$\displaystyle n_{h}$ $\displaystyle=\frac{16\ \Omega^{4}_{p}\
(n_{2}+1)\delta^{4}}{(2n_{2}+1)(\delta^{2}+4\sigma^{2}_{p})^{4}-32\
\Omega^{4}_{p}\ (n_{2}+1)\ \delta^{4}},$ $\displaystyle\Gamma_{h}$
$\displaystyle=\Gamma_{2}\frac{(2n_{2}+1)(\delta^{2}+4\sigma^{2}_{p})-32\
\sigma^{4}_{p}(n_{2}+1)\delta^{4}}{(\delta^{2}+4\ \sigma^{2}_{p})^{4}},$ (7)

where we set $\omega_{2e^{\prime}}\simeq\omega_{e^{\prime}g}$. The effective
thermal bath parameters defined in Eq. (7) yield the quantitative population
match between coherent and thermal baths shown in Fig. 3a. The agreement
between the populations shown in Fig. 3b ensures the qualitative formation of
an effective bath.

Figure 3: (a)The population of ground $g$ and lowest excited state 1 obtained
using the coherent bath in Eq. (4) (solid lines) and the thermal bath in Eq.
(6) ( dot-dashed) using parameters in Eq. (7). (b) The difference between
populations of coherent and thermal baths $\rho_{c}-\rho_{th}$ for parameters
in Eq. (7). The parameters read
$n_{2}=n_{c}=100,\Gamma_{2}=\Gamma_{c}=0.002\text{ps}^{-1},\Omega_{p}=0.0078\text{ev}\
\text{and}\ \sigma_{p}=30.34\text{cm}^{-1}.$

##  III Quantum Heat Engine

We next obtain for the power and efficiency given by [42]

$\displaystyle P=i\hbar\lambda(\omega_{c}-\omega_{h})(\rho_{01}-\rho_{10}),$
$\displaystyle\dot{Q}_{h}=i\hbar\omega_{h}\lambda(\rho_{01}-\rho_{10}),$
$\displaystyle\eta=1-\frac{\omega_{c}}{\omega_{h}}.$ (8)

where $\omega_{h}=\omega_{1}-\omega_{g}$ and
$\omega_{c}=\omega_{0}-\omega_{g}$, $n_{h}$ and $n_{c}$, $\Gamma_{h}$ and
$\Gamma_{c}$ are the average occupation numbers and dephasing rates for the
hot and cold baths, respectively. To find the steady state solution for
$\varrho_{01}$ and $\varrho_{10}$ in Eq. (III), we follow the standard
procedure for the three level molecule given in the Supplementary Material of
Refs. [23] and [33]. The output power and efficiency for the three level QHE
then read

$\displaystyle P$
$\displaystyle=\frac{2}{3}\frac{\lambda^{2}\Gamma_{h}\Gamma_{c}(n_{c}-n_{h})(\omega_{c}-\omega_{h})}{(\Gamma_{h}n_{h}+\Gamma_{c}n_{c})(\lambda^{2}+\Gamma_{h}\Gamma_{c}n_{c}n_{h})},$
$\displaystyle\eta$ $\displaystyle=1-\frac{1}{c_{p}-c_{21}},$ (9)

where $c_{p}=\omega_{p}/\omega_{c}$ and $c_{21}=\omega_{21}/\omega_{c}$.

###  III.1 Classical-two-photon pump

We now recast the output power in Eq. (9) using Eq. (7) for in the high
temperature limit where $n_{c}=n_{1}\simeq T_{c}/\omega_{c}$,
$n_{2}=n_{h}\simeq T_{2}/\omega_{21}$, where $\omega_{c}=\omega_{0g}$. We then
introduce an effective temperature of the hot bath $T_{h}=(\Omega_{p}\
\Gamma^{2}_{2}\ /2\ \delta)^{1/2}$ and the dimensionless temperature scale:
$\tau=T_{c}/T_{h}$. The pump energy scale $c_{p}=\omega_{p}/\omega_{c}$, the
coupling scale: $\lambda^{\prime}=\lambda(\Gamma_{2}T_{c})^{-1/2}$ and the
pump pulse width scale: $\sigma^{\prime}_{p}=\sigma^{e}_{p}\ \Gamma_{2}/\delta
T_{c}$, where $\sigma^{e}_{p}=(\sigma^{2}_{p}-\delta^{2}/4)^{1/2}$. Eq. (9)
for dimensionless parameters given in Eq. (A22) can be finally maximized with
respect to dimensionless variable $c_{21}$ which yields

$\displaystyle
P^{max}_{C}=\frac{4uv\lambda^{\prime}\tilde{\tau}\big{(}2\,\mathcal{A}+2\,\alpha
u\,v+\tau^{8}\sigma^{\prime
8}_{p}c^{\prime}_{p}(mu+v)\big{)}}{3\tau^{8}\sigma^{\prime
8}_{p}\left(v-u\lambda^{\prime}\right)^{2}},\ \ $ (10)

where $\mathcal{A}=\sqrt{u\,v\,\left(\tau^{8}\,c^{\prime}_{p}\,\sigma^{\prime
8}_{p}\,+\alpha\,u\right)\left(\tau^{8}\,c^{\prime}_{p}\,\lambda^{\prime}\,\sigma^{\prime
8}_{p}+\alpha v\right)}$, $\tilde{\tau}=1-\tau^{8}\sigma^{\prime 8}_{p}$ and
$c^{\prime}_{p}=c_{p}-1$. The efficiency corresponding to the maximum output
power defined in Eq. (10) is given by

$\displaystyle\eta^{*}_{C}=1-\frac{1}{c_{p}+\frac{\tilde{\tau}\sqrt{uv\left(c^{\prime}_{p}\tau^{8}\sigma^{\prime
8}_{p}+\alpha u\right)(\tau^{8}\lambda^{\prime}c^{\prime}_{p}\sigma^{\prime
8}_{p}+\alpha
v)}+uv\,\tilde{\tau}\alpha^{2}}{\tilde{\tau}\,\tau^{8}\sigma^{\prime
8}_{p}\left(\alpha
v+\lambda^{\prime}\left(c^{\prime}_{p}\tau^{8}\sigma^{\prime 8}_{p}+\alpha
u\right)\right)}},$

where subscript $C$ specifies the efficiency of two-photons pump. We next
assume the weak dissipation regime i.e., $\omega_{c}>>\Gamma_{c}$ which yields

$\displaystyle\eta^{*}_{CW}=1-\frac{1}{c_{p}+\frac{\alpha^{2}uv}{\tau^{8}\sigma^{\prime
8}_{p}\left(\tau^{8}\left(c_{p}-1\right)\lambda^{\prime}\sigma^{\prime
8}_{p}+\alpha u\lambda^{\prime}\right)}},$ (12)

where subscript $CW$ indicates the classical efficiency in the weak coupling
regime.

Bound | $\eta^{*}_{CW}$ | $c_{p}$ | $\sigma^{\prime C}_{p}$
---|---|---|---
I | $0$ | $1$ | $\sqrt[8]{\xi\left(u-\sqrt{u^{2}-\frac{4uv}{\lambda^{\prime}}}\right)}$
I/II | $\frac{\eta_{C}}{2}$ | $\frac{2}{2-\eta_{C}}$ | $\sqrt[8]{\xi\left(u-\sqrt{u^{2}-\frac{2uv\left(2-\eta_{C}\right)}{\lambda^{\prime}}}\right)}$
II/III | $\eta_{CA}$ | $\frac{1}{\sqrt{1-\eta_{C}}}$ | $\sqrt[8]{\xi\left(u-\sqrt{u^{2}-\frac{4uv\sqrt{1-\eta_{C}}}{\lambda^{\prime}}}\right)}$
III/IV | $\frac{\eta_{C}}{2-\eta_{C}}$ | $\frac{2-\eta_{C}}{2(1-\eta_{C})}$ | $\sqrt[8]{\xi\left(u-\sqrt{u^{2}-\frac{8uv\left(1-\eta_{C}\right)}{\left(2-\eta_{C}\right)\lambda^{\prime}}}\right)}$
IV | $\eta_{C}$ | $\frac{2}{1-\eta_{C}}$ | $\sqrt[8]{\xi\left(u-\sqrt{u^{2}-\frac{4uv\left(1-\eta_{C}\right)}{\lambda^{\prime}}}\right)}$
Table 1: Parameters of the coherent bath corresponding to the QHE efficiency
bounds shown in Fig. 4, where
$\xi=\frac{\alpha}{2\left(1-\eta_{C}\right){}^{8}}$.  Figure 4: (a) $2D$
mapping of the efficiency at maximum power $\eta^{*}_{CW}$ in Eq. (12) vs
Carnot efficiency $\eta_{C}=1-\tau$ . (b) 2D mapping of the $c_{p}$ vs
$\eta_{C}$ corresponding to (a). (c) 2D mapping of the $\sigma^{\prime C}_{p}$
vs $\eta_{C}$ corresponding to (a)

Entire parameter space corresponding to the efficiency given by Eq. (12) can
be separated in four regions summarized in Table I represented by the colorful
2D shapes in Fig. 4a. Using the dimensionless pump frequency $c_{p}$ as a
control parameter which depends on the effective temperature ratio $\tau$, the
pump pulse bandwidth $\sigma^{\prime C}_{p}$ (the classical pump bandwidth of
$\sigma^{\prime}_{p}$) that depends on the dimensionless probe coupling field
$\lambda^{\prime}$, $\tau$ and $u,v$ and $\alpha$. We define the
characteristic efficiency values describing the boundaries between the four
regions correspond to $0$, $\eta_{C}/2$ (between I and II regions),
$\eta_{C}/(2-\eta_{C})$ (between III and IV), Carnot efficiency
$\eta_{C}=1-\tau$ upper bound of IV) and Curzon-Ahlborn (CA) limit [45]
$\eta_{CA}=1-\tau^{1/2}$ (between II and III). Note, that the two parameters
of the pump field: the frequency $\omega_{p}$ and the Rabi frequency
$\Omega_{p}$ which define an effective hot bath temperature $T_{h}$ and the
pump bandwidth $\sigma^{\prime C}_{p}$ can be controlled experimentally. Thus
the 2D parameter space $\\{\tau,c_{p}\\}$ and $\\{\tau,\sigma^{\prime
C}_{p}\\}$ shows a constrained relation between the two as seen in Figs. 4b
and 4c, respectively.

We now compare the two-classical-photon pump with our previous work [33] where
a single resonant pump has been taken to drive transition $g-2$. Let us
highlight some important points here. First, the range of the pump frequency,
in the single photon is $\omega_{p}\geq 2\omega_{c}$ while in the two photon
case $\omega_{p}\geq\omega_{c}$, which affirms that the size of the system can
be smaller. Second, the particular boundary is reached at different pump
parameters. For instance, a CA limit is obtained for the single photon pump at
$c_{p}\simeq 2/\sqrt{\tau}$ and for the two photons $c_{p}\simeq
1/\sqrt{\tau}$ and its corresponding Rabi frequency is
$\Omega^{CA}_{p}\simeq(4\omega_{p}/\omega_{c})^{4}T^{2}_{c}\delta/\Gamma^{2}_{2}$.
The factor of two, which appears in the other bounds as well, originates from
the quadratic scaling of the photon absorption probability with the input
intensity for the classical light.

###  III.2 Entangled-two-photon pump

We now consider the case when the two photons driving transitions
$g\rightarrow e$ and $e\rightarrow 2$ are entangled. A classical pump beam at
frequency $w_{p}$ directed into a crystal is down converted into entangled
photon pair - signal $(s)$ and idler $(i)$ with frequencies $\omega_{s}$ and
$\omega_{i}$ , respectively, as shown in Fig. 5a. We consider type-II down
conversion, which corresponds to orthogonally polarized signal and idler beams
which allows us to introduce the entanglement time and avoid complications
with the selection rules. The different group velocities along two
polarization axes create a time delay between the signal and the idler photons
represented by entanglement time $T$. Owing to energy conservation,
$\omega_{p}=\omega_{i}+\omega_{s}$. The photon pair is fully characterized by
the twin photon state amplitude
$\varphi(\omega_{i},\omega_{s})=\mathcal{A}(\omega_{i}+\omega_{s})\Phi(\omega_{i},\omega_{s})$,
where $\mathcal{A}(\omega)=\frac{A_{0}}{\omega-\omega_{p}+i\sigma}$ is a
Lorentzian envelop function of the pump photon with bandwidth $\sigma$
centered around $\omega_{p}$ and
$\Phi(\omega_{i},\omega_{s})=\text{sinc}\left[(\omega_{s}-\omega_{i})T/2\right]$,
originating from the phase matching inside the crystal (see Appendix B).

Figure 5: (a) The pump photon frequency $\omega_{p}$ is down converted into
two signal and idler photons with frequencies $\omega_{s}$ and $\omega_{i}$,
respectively. We consider spontaneous down conversion (SPDC) with energy
conservation $\omega_{p}=\omega_{i}+\omega_{s}$. The entangled photons drive
transitions $g\rightarrow e^{\prime}$ and $e^{\prime}\rightarrow 2$ and the
probe field then stimulates the emission from state $1$ to the excited
vibrational level $0$ of the ground electronic state.

The leading contribution to population of state $2$ calculated perturbatively
according to the ladder diagram in Fig. 2 and by following the same approach
given in Appendix A to obtained Eq. (4) and assuming that transition energy
$\omega_{21}$ is much larger than dephasing rate $\Gamma_{2}$, the final
population reads

$\displaystyle\varrho_{11}(t)$
$\displaystyle=\frac{\mathbb{N}^{2}\,{\Gamma_{2}}\,\tilde{n}_{2}\,\omega_{2e^{\prime}}\,\omega_{e^{\prime}g}\,\tilde{\omega}^{2}_{2g}\left(1-e^{-\Gamma_{2}\left(2n_{2}+1\right)t}\right)\theta}{\Gamma_{2}\left(2n_{2}+1\right)\left(\sigma^{2}_{p}+\tilde{\omega}^{2}_{2g}\right)^{2}},$
$\displaystyle\varrho_{gg}(t)$ $\displaystyle=1-\varrho_{11}(t)$ (13)

where
$\tilde{n}_{2}=n_{2}+1,\,\tilde{\omega_{2g}}=\omega_{2g}-\omega_{p},\theta=\text{sinc}^{2}[\frac{T(\omega_{2e^{\prime}}-\omega_{e^{\prime}g})}{2}]\
$, and
$\mathbb{N}^{2}=\frac{\mathcal{N}^{2}A_{0}^{2}\mu_{eg}\mu_{e^{\prime}e}\mu_{2e^{\prime}}\mu_{21}}{4\epsilon^{2}_{0}V^{2}}$
is a normalization. Assume for brevity, that the normalization $\mathbb{N}$ of
quantum state is same as that of a classical [46],
$\mathbb{N}=\Omega_{1^{\prime}}\Omega_{2^{\prime}}$. This ensures that all the
dimensionless parameters for the entangled case are the same as in the
classical case. Similarly to the classical case we now introduce an effective
hot bath characterized by the thermal photon occupation number $n_{h}$ and
dephasing rate $\Gamma_{h}$ which drives the transition $g\leftrightarrow 1$,
where the parameters of the bath are given by

$\displaystyle n_{h}$
$\displaystyle=\frac{\tilde{n}_{2}\,\theta\,\omega_{2e^{\prime}}\omega_{ge^{\prime}}\,\Omega^{2}_{1^{\prime}}\Omega^{2}_{2^{\prime}}\Delta^{2}}{\left(2n_{2}+1\right)\tilde{\Delta}^{4}-2\Omega^{2}_{1^{\prime}}\Omega^{2}_{2^{\prime}}\Delta^{2}\tilde{n}_{2}\omega_{2e^{\prime}}\omega_{ge^{\prime}}\theta}$
$\displaystyle\Gamma_{h}$
$\displaystyle=\frac{\Gamma_{2}\big{(}\left(2n_{2}+1\right)\tilde{\Delta}^{4}-2\,\tilde{n}_{2}\,\theta\,\omega_{2e^{\prime}}\omega_{ge^{\prime}}\Omega^{2}_{1^{\prime}}\Omega^{2}_{2^{\prime}}\Delta^{2}\big{)}}{\tilde{\Delta}^{4}},\
\ \ \ \ \ $ (14)

where $\tilde{\Delta}^{2}=\Delta^{2}+\sigma^{2}_{p}$, and
$\Delta=\omega_{2g}-\omega_{p}$.

Figure 6: (a)The population of ground $g$ and lowest excited state $1$
obtained using a coherent bath in Eq. (13) and a thermal bath $\rho_{th}$
using parameters in Eq. (14). (b) The difference between populations of
coherent and thermal baths $\varrho_{c}-\rho_{th}$ for parameters in Eq. (14).

Using the effective bath introduced for the system excited by the two
entangled photons in Eq. (14), we can perfectly match the populations of $g$
and $1$ driven by the thermal bath given in Eq. (6) as shown in Fig. 6. In the
two-photon entangled, pump bandwidth scale: $\sigma^{\prime
Q}_{p}=\sigma^{e}_{p}\Gamma_{2}/T_{c}\Delta$, where
$\sigma^{e}_{p}=(\sigma^{2}_{p}-\Delta^{2})^{1/2}$

Following the general approach outlined in Ref. [33] we apply the high
temperature limit for the phonon bath $i.e.,T_{2}>>\omega_{21}$, and the
maximum power w.r.t $c_{21}$ which is recast in terms of dimensionless
parameters given in Eq. (B16) yields:

$\displaystyle
P^{max}_{Q}=\frac{4uv\lambda^{\prime}\mathcal{W}\tau^{4}c^{\prime
2}_{p}(\sigma^{\prime Q}_{p})^{4}\left(\theta-\tau^{4}\sigma^{\prime
4}_{p}\right)}{3\theta\left(\mathcal{X}+\tau^{4}v\
c^{\prime}_{p}(\sigma^{\prime
Q}_{p})^{4}\right)\left(\mathcal{X}+\tau^{4}uc^{\prime}_{p}\lambda^{\prime}(\sigma^{\prime
Q}_{p})^{4}\right)},$ (15)

where
$\mathcal{X}=\mathcal{W}+\mathcal{E},\mathcal{E}=\alpha\,u\,v\,\text{sinc}^{2}[T\left(\omega_{2e^{\prime}}-\omega_{ge^{\prime}}\right)/2]$,
$\mathcal{W}=\sqrt{uv\left(\tau^{4}c^{\prime}_{p}\sigma^{\prime
4}_{p}+\mathcal{E}/v\right)\left(\tau^{4}c^{\prime}_{p}\lambda^{\prime}\sigma^{\prime
4}_{p}+\mathcal{E}/u\right)}$, and $c^{\prime}_{p}=c_{p}-1$. The corresponding
efficiency at maximum power is given by

$\displaystyle\eta^{*}_{Q}=1-\frac{1}{c_{p}-\frac{(c_{p}-1)\mathcal{E}}{\mathcal{W}+\mathcal{E}}},$
(16)

where subscript $Q$ denotes the two photon entangled pump. Similarly, in the
weak dissipating regime Eq. (16) can be recast as

$\displaystyle\eta^{*}_{QW}=1-\frac{1}{c_{p}+\frac{u\,v\,\alpha^{2}\,\theta^{2}}{\tau^{4}\lambda^{\prime}\sigma^{\prime
4}_{p}\left(\alpha\,u\,\theta+\left(c_{p}-1\right)\tau^{4}\sigma^{\prime
4}_{p}\right)}},$ (17)

where subscript $QW$ specifies the weak dissipating limit of two-photon
entangled (quantum) pump and
$\theta=\text{sinc}^{2}[T\left(\omega_{2e^{\prime}}-\omega_{ge^{\prime}}\right)/2]$.
Similar to the classical efficiency given in Eq. (12) the entire parameter
space of the respective quantum efficiency in Eq. (17) is also divided into
four regions summarized in Table 2. By comparing Tables 1 and 2, it is clear
that the four regions for $\eta^{*}_{CW}$ and $\eta^{*}_{QW}$ are identical
when $\sigma^{\prime C}_{p}=\sigma^{{}^{\prime}Q}_{p}$. The distinction
between $\sigma^{{}^{\prime}C}_{p}$ and $\sigma^{{}^{\prime}Q}_{p}$ originate
due to additional parameter $T$ and the different pump intensity scaling
(quadratic vs linear) [38] as mentioned in Tables 1 and 2. The effective
bandwidth for the two-classical and two-entangled photon pump vs $\eta_{C}$ is
depicted in Figs. 7a and 7b, respectively. Furthermore, the efficiency
corresponding to the maximum output power for the quantum light is more robust
than that for the classical light for the moderate range of $\tau$ as shown in
Fig. 7c and which will be discussed in the next subsection.

Bound | $\eta^{*}_{QW}$ | $c_{p}$ | $\sigma^{\prime Q}_{p}$
---|---|---|---
I | 0 | $1$ | $\sqrt[4]{\Xi\left(u-\sqrt{\frac{u\left(u\lambda^{\prime}-4v\right)}{\lambda^{\prime}}}\right)}$
I/II | $\frac{\eta_{C}}{2}$ | $\frac{2}{2-\eta_{C}}$ | $\sqrt[4]{\Xi\left(u-\sqrt{\frac{u\left(\lambda^{\prime}u-2v\left(2-\eta_{C}\right)\right)}{\lambda^{\prime}}}\right)}$
II/III | $\eta_{CA}$ | $\frac{1}{\sqrt{1-\eta_{C}}}$ | $\sqrt[4]{\Xi\left(u-\sqrt{\frac{u\left(u\lambda^{\prime}-4v\sqrt{1-\eta_{C}}\right)}{\lambda^{\prime}}}\right)}$
III/IV | $\frac{\eta_{C}}{2-\eta_{C}}$ | $\frac{2-\eta_{C}}{2(1-\eta_{C})}$ | $\sqrt[4]{\Xi\left(u-\sqrt{\frac{u\left(u\left(2-\eta_{C}\right)\lambda^{\prime}-8v\left(1-\eta_{C}\right)\right)}{\left(2-\eta_{C}\right)\lambda^{\prime}}}\right)}$
IV | $\eta_{C}$ | $\frac{2}{1-\eta_{C}}$ | $\sqrt[4]{\Xi\left(u-\sqrt{\frac{u\left(u\lambda^{\prime}-4v\left(1-\eta_{C}\right)\right)}{\lambda^{\prime}}}\right)}$
Table 2: The efficiency and pump scale are same as in Table I. The pump
bandwidth parameters of the quantum bath corresponding to the QHE efficiency
bounds in this table shown in Fig. 7, where
$\Xi=\frac{\alpha}{2\left(1-\eta_{C}\right){}^{4}}\text{sinc}^{2}[\frac{T(\omega_{2e^{\prime}}-\omega_{ge^{\prime}})}{2}]$.
Figure 7:  (a) The 2D mapping of $\sigma^{\prime Q}_{p}\text{vs}\ \eta_{C}$
for entangled photons given in Table. 2. (b) The 2D mapping of $\sigma^{\prime
C}_{p}\text{vs}\ \eta_{C}$ of two pump photons from Table 1. (c) The ratio of
efficiency at maximum power in Eqs. (17) and (12) vs $\tau$ for
$\text{sinc}^{2}\left[\frac{T(\omega_{2e^{\prime}}-\omega_{ge^{\prime}})}{2}\right]\sim
1$ and $\eta^{*}_{QW}>\eta^{*}_{CW}$ occurs for small $\tau$ also in inset
$\eta^{*}_{QW}<\eta^{*}_{CW}$ for considerably large value of $\tau$. The
parameters read:
$T_{2}=T_{c}=300\text{K},\omega_{p}=1.3\text{ev},\omega_{c}=0.012\text{ev},\Omega_{p}=0.023\text{ev},\lambda=0.1\text{ev},\delta=0.00003\text{ev},\sigma_{p}=200\text{cm}^{-1},\Gamma_{2}=0.71\text{ps}^{-1}$
and $\Gamma_{c}=0.025\text{ps}^{-1}$.

###  III.3 Maximum QHE power for the quantum and the classical two-photon pump

After maximization of QHE power with respect to the temperature and pump
bandwidth we obtain different scaling for the classical and the entangled two
photon pumps [41]. Fig. 8a shows numerically that in the specific temperature
range the maximum output power in entangled case can be larger than that in
the classical case. The quantum enhancement for the maximum output power
occurs for small $\tau$. In this case Eqs. (10) and (15) yield, respectively,

$\displaystyle P^{max}_{C}$ $\displaystyle=$
$\displaystyle\frac{\tau^{8}\left(c_{p}-1\right){}^{2}\lambda^{\prime}(\sigma^{\prime
c}_{p})^{8}}{3\,\alpha}$ (18) $\displaystyle P^{max}_{Q}$ $\displaystyle=$
$\displaystyle\frac{\tau^{4}\left(c_{p}-1\right){}^{2}\lambda^{\prime}(\sigma^{\prime
Q}_{p})^{4}}{3\,\alpha\,\text{sinc}^{2}\left[\frac{T(\omega_{2e^{\prime}}-\omega_{ge^{\prime}})}{2}\right]},$
(19)

where for brevity, $\sigma^{\prime C}_{p}=\sigma^{\prime
Q}_{p}=\sigma^{\prime}_{p}$, and from Eqs. (18) and (19) results, we have
$P^{max}_{Q}/P^{max}_{C}=\tau^{-4}\sigma^{\prime-4}_{p}\text{sinc}^{-2}[T(\omega_{2e^{\prime}}-\omega_{ge^{\prime}})/2]$
which gives $P^{max}_{Q}>P^{max}_{C}$ for
$\text{sinc}^{2}[T(\omega_{2e^{\prime}}-\omega_{ge^{\prime}})/2]\simeq 1$ and
$\sigma^{\prime}_{p}\tau<1$. The above analysis clearly indicates the relation
between the effective bath temperature, the entanglement time and the spectral
bandwidth of the optical fields as well as the system energy scale and its
effect on the optical measurements with the entangled light in the open
quantum systems. For instance in the limit of short entanglement time we can
achieve quantum enhancement even in highly anharmonic system as long as
$|\omega_{2e^{\prime}}-\omega_{e^{\prime}g}|<<1/T$. Similarly, for the long
entanglement time the quantum enhancement can be reached for nearly harmonic
system $(\omega_{2e^{\prime}}\simeq\omega_{e^{\prime}g})$. In the same time
inequality $\sigma^{\prime}_{p}\tau<1$ yields an additional requirement for
the pumping source such that
$\Omega_{p}>4\delta(\sigma^{e}_{p})^{2}/\Delta^{2}$.

##  IV Spectroscopic regime

So far we have discussed QHE regime in which the density matrix has been
solved nonperturbatively. We now focus on the pump-probe spectroscopic signal
derived by perturbative approach in light - matter interaction. Ref. [33]
shows the apparent connection between the thermodynamics of the QHE and the
spectroscopy which emerges as an incoherent control tool for the optimization
of optical measurements, which can enhance the yield of fluorescence, pump-
probe measurements, and improve the signal-to-noise ratio in a wide class of
the optical signals. Here we explore the class of two-photon pump - classical
probe signals using classical and entangled two-photon pumps. To that end the
coherence $\rho_{01}$ and $\rho_{10}$ which enters in the definition of Eq.
(III) can be calculated perturbatively.

Using Eq. (A21) (in Appendix A), while we keeping the leading order terms
following the Feynman diagram in Fig. 2, we substitute the solution for the
population $\rho_{11}(t)$ from Eq. (4) and solve for $\rho_{01}(t)$ for two-
photons pump, which yields

$\displaystyle\rho_{01}$
$\displaystyle=-\frac{32\,i\,\Gamma_{2}\,\lambda\,n_{2}\,\delta^{4}\,\Omega^{4}{}_{p}}{\left(\delta^{2}+4\sigma^{2}_{p}\right)^{4}\sigma_{\text{pr}}\left(\Gamma_{c}n_{c}+\Gamma_{2}n_{2}\right)\left(5\Gamma_{2}n_{2}+\Gamma_{c}n_{c}\right)},$
$\displaystyle\rho_{10}$ $\displaystyle=-\rho_{01}.$ (20)

Similarly, we can obtained the coherence $\varrho_{01}(t)$ while we keeping
the leading order terms following the Feynman diagram in Fig. 2, we substitute
the solution for the population $\varrho_{11}(t)$ from Eq. (13) and we get

$\displaystyle\varrho_{01}$
$\displaystyle=-\frac{2\,i\,\Gamma_{2}\,\lambda\,n_{2}\,\delta^{4}\,\Omega^{2}{}_{1^{\prime}}\Omega^{2}{}_{2^{\prime}}\text{Sinc}^{2}\left[\frac{T(\omega_{2e^{\prime}}-\omega_{ge^{\prime}})}{2}\right]}{\left(\Delta^{2}+\sigma^{2}_{p}\right)^{4}\sigma_{\text{pr}}\left(\Gamma_{c}n_{c}+\Gamma_{2}n_{2}\right)\left(5\Gamma_{2}n_{2}+\Gamma_{c}n_{c}\right)},$
$\displaystyle\varrho_{10}$ $\displaystyle=-\varrho_{01}.$ (21)

Utilizing the Eq. (20) we obtained the power for the classical two-photon pump
in Appendix C, Eq. (C1). After optimizing w.r.t $c_{21}$, the maximum power
yields

$\displaystyle\mathcal{P}^{max}_{C}=\frac{u\lambda^{\prime}\left(3c_{p}+5\alpha
u-3-\mathcal{C}\right)}{2\tau^{8}\sigma_{\text{pr}}\sigma^{\prime 8}_{p}},\ $
(22)

where $\mathcal{C}=\sqrt{5\left(c_{p}+\alpha u-1\right)\left(c_{p}+5\alpha
u-1\right)}$. Similarly, we optimize the power for the entangled two-photon
pump in Eq. (C2) and it maximum yields

$\displaystyle\mathcal{P}^{max}_{Q}=\frac{4\alpha
c_{21}u^{2}\left(c_{p}-c_{21}-1\right)\lambda^{\prime}\text{sinc}^{2}\big{[}\frac{T\left(\omega_{2e^{\prime}}-\omega_{ge^{\prime}}\right)}{2}\big{]}}{\tau^{4}\sigma_{\text{pr}}\left(c_{21}+\alpha
u\right)\left(c_{21}+5\alpha u\right)\sigma^{\prime 4}{}_{p}}.$ (23)

In appendix C, we demonstrated numerically that the maximum power for the
quantum light is much larger than that for the classical light within for the
moderate range of $\tau<1$ in Eqs. (23) and (22) (see Fig. 8b). The ratio of
maximum power for the corresponding equations reads

$\displaystyle\frac{\mathcal{P}^{max}_{Q}}{\mathcal{P}^{max}_{C}}=\tau^{4}\sigma^{\prime
4}_{p}\text{sinc}^{2}\big{[}\frac{T\left(\omega_{2e^{\prime}}-\omega_{ge^{\prime}}\right)}{2}\big{]}.$
(24)

For the short entanglement time
$\text{sinc}^{2}\big{[}\frac{T\left(\omega_{2e^{\prime}}-\omega_{ge^{\prime}}\right)}{2}\big{]}\simeq
1$ and in the limit of $\sigma^{\prime}_{p}\tau>1$ we obtain
$\mathcal{P}^{max}_{Q}>\mathcal{P}^{max}_{C}$. We therefore identified the
parameter regime where maximum power for the entangled two-photon pump is
enhanced compared to the classical case using perturbative regime. In
comparison to QHE (nonperturbative) regime the power increase due to the
entanglement in spectroscopic (perturbative) regime occurs when the bath
temperature ratio is $\tau>1/\sigma_{p}^{\prime}$, whereas in the former case
$\tau<1/\sigma_{p}^{\prime}$, which agrees with strong pumping
(nonperturbative) vs weak pump (perturbative) limit taken in these two cases.

The simulation of the maximum power for QHE regime given in Eqs. (18) and
(19), and for Spectroscopic regime given in Eqs. (22), and (23), respectively
shown in Fig. 8 (Appendix C). It shows that the quantum enhancement for the
power is achieved at different time scale: in the QHE regime at
$0<\tau<0.5\text{X}10^{-2}$, and in spectroscopic regime at $0<\tau<0.07$.

##  V Summary

It has been shown that the two photon absorption of entangled light may
enhance the Raman excitation [47] due to different intensity scaling at low
photon fluxes. In the present analysis the two photon absorption in open
quantum system regime benefits from additional control parameters using
incoherent control scheme by mimicking the QHEs. In this proposed model we
analytically explored the characteristics of two photon absorptions for the
classical and entangled pair of photons and their dependence on additional
degrees of freedom, due to which we get the maximum work, in both for weak and
strong intensity approximations. By using the approach of [33] we developed
connection between the thermodynamics of the QHE and the spectroscopy. The
transfer of entanglement to the system allows to optimize the detailed balance
in system-bath driven optical transitions in an open quantum system allowing
QHE to operate near thermodynamic cycle, which consequently provides an
enhanced yield of conversion between the pump and the probe fields. Our
results can be further extended to Raman, hyper-Raman and other techniques
that require additional control over illumination intensity and pump light
statistics.

##  VI ACKNOWLEDGMENTS

We gratefully acknowledge the support from the National Science Foundation of
China (No. 11934011), the Zijiang Endowed Young Scholar Fund, the East China
Normal University and the Overseas Expertise Introduction Project for
Discipline Innovation (111 Project, B12024). M.Q. acknowledges the support
from CSC Scholarship (CSC No. 2018 DFH 007778).

Appendix A: Effective heat bath

The master equation in Liouville space

$\frac{d\hat{\rho}}{dt}=-\frac{i}{\hbar}\left[\hat{H}_{int}(t),~{}\hat{\rho}\right],$
(A1)

We define the superoperator in Liouville space by acting on a arbitrary
operator $X$

$\hat{H}_{int,-}\hat{X}=\hat{H}_{int}\hat{X}-\hat{X}\hat{H}_{int}.$ (A2)

Now, the solution of Eq. (A1) can be written as a Dyson series in Liouville
space. Hence, we obtain the density matrix

$\hat{\rho}(t)=\mathcal{G}(t,t_{0})\hat{\rho}(t_{0}),$ (A3)

with the Liouville space Green’s function

$\mathcal{G}(t,t_{0})=\hat{\mathcal{T}}\left[-\frac{i}{\hbar}\text{exp}\int_{t_{0}}^{t}\hat{H}_{int,-}(\tau)\right]$
(A4)

where $\mathcal{T}$ is a time ordering superoperator which is defined by

$\displaystyle\mathcal{T}A(t_{1})B(t_{2})$
$\displaystyle\equiv\Theta(t_{1}-t_{2})A(t_{1})B(t_{2})+\Theta(t_{2}-t_{1})$
$\displaystyle B(t_{2})A(t_{1}),$ (A5)

where $A(t)$ and $B(t)$ are two arbitrary superoperators and $\Theta(t)$ is
the Heaviside function. A perturbative expansion of the Dyson series yields
the leading order contribution of Eq. (A3) reads off:

$\displaystyle\rho_{11}(t)$
$\displaystyle=\frac{1}{\hbar^{4}}\int_{t_{0}}^{t}d\tau_{1}\int_{t_{0}}^{\tau_{1}}d\tau_{2}\int_{t_{0}}^{t_{2}}d\tau_{3}\int_{t_{0}}^{\tau_{3}}d\tau_{4}<<H_{int,-}$
$\displaystyle(\tau_{1})H_{int,-}(\tau_{2})H_{int,-}(\tau_{3})H_{int,-}(\tau_{4})\rho({t_{0})}>>,$
(A6)

where $<<\cdot>>=\text{Tr}\left[\cdot,\rho(t)\right]$ represents the trace
with the density operator. The population of the excited state due to
relaxation of $2-1$ is represented by a diagrammatic Feynman ladder in Fig. 2
given by

$\displaystyle\rho^{a}_{11}(t)$
$\displaystyle=\frac{1}{\hbar^{4}}\int^{t}d\tau_{1}\int^{\tau_{1}}d\tau_{2}\int^{\tau_{2}}d\tau_{3}\int^{\tau_{3}}d\tau_{4}\langle\langle\mathcal{G}_{11,22}$
$\displaystyle(t-\tau_{1})\hat{\mathcal{V}}_{2e}\mathcal{G}_{2e,2e}(\tau_{1}-\tau_{2})\hat{\mathcal{V}}^{\dagger}_{2e}\mathcal{G}_{ee,e^{\prime}e^{\prime}}(\tau_{2}-\tau_{3})\hat{\mathcal{V}}_{e^{\prime}g}$
$\displaystyle\mathcal{G}_{e^{\prime}g,e^{\prime}g}(\tau_{3}-\tau_{4})\hat{\mathcal{V}}^{\dagger}_{e^{\prime}g}\rangle\rangle\langle\hat{E}^{\dagger}_{1}(\tau_{3})\hat{E}^{\dagger}_{2}(\tau_{1})\hat{E}_{2}(\tau_{2})\hat{E}_{1}(\tau_{4})\rangle,$
$\displaystyle=\frac{1}{\hbar^{4}}\int^{t}dt_{1}\int^{t_{1}}dt_{2}\int^{t_{2}}dt_{3}\int^{t_{3}}dt_{4}\langle\langle\mathcal{G}_{11,22}(t_{1})$
$\displaystyle\hat{\mathcal{V}}_{2e}\mathcal{G}_{2e,2e}(t_{2})\hat{\mathcal{V}}^{\dagger}_{2e}\mathcal{G}_{ee,e^{\prime}e^{\prime}}(t_{3})\hat{\mathcal{V}}_{e^{\prime}g}\mathcal{G}_{e^{\prime}g,e^{\prime}g}(t_{4})\hat{\mathcal{V}}^{\dagger}_{e^{\prime}g}\rangle\rangle$
$\displaystyle\langle\hat{E}^{\dagger}_{1}(t-t_{1}-t_{2}-t_{3})\hat{E}^{\dagger}_{2}(t-t_{1})\hat{E}_{2}(t-t_{1}-t_{2})$
$\displaystyle\hat{E}_{1}(t-t_{1}-t_{2}-t_{3}-t_{4})\rangle,$ (A7)

where
$t_{1}=t-\tau_{1},t_{2}=\tau_{1}-\tau_{2},t_{3}=\tau_{2}-\tau_{3},t_{4}=\tau_{3}-\tau_{4}$
and $\mathcal{V}$ is a time independent dipole operator in Liouville space.
Now, we transform the fields components in frequency domain and it can be
recast as

$\displaystyle\rho^{a}_{11}(t)$
$\displaystyle=\frac{1}{\hbar^{4}}\int\frac{d\omega_{1}\ d\omega_{1^{\prime}}\
d\omega_{2}\
d\omega_{2^{\prime}}}{(2\pi)^{4}}\langle\hat{\Omega}^{\dagger}_{1}(\omega_{1^{\prime}})\hat{\Omega}^{\dagger}_{2}(\omega_{2^{\prime}})\hat{\Omega}_{2}(\omega_{2})\hat{\Omega}_{1}(\omega_{1})\rangle\int_{-\infty}^{\infty}dt_{1}dt_{2}dt_{3}dt_{4}\mathcal{G}_{11,22}(t_{1})\mathcal{G}_{2e,2e}(t_{2})$
$\displaystyle\mathcal{G}_{ee,e^{\prime}e^{\prime}}(t_{3})\mathcal{G}_{e^{\prime}g,e^{\prime}g}(t_{4})e^{i\omega_{1^{\prime}}(t-t_{1}-t_{2}-t_{3})+i\omega_{2^{\prime}}(t-t_{1})-i\omega_{2}(t-t_{1}-t_{2})-i\omega_{1}(t-t_{1}-t_{2}-t_{3}-t_{4})},$
$\displaystyle=\frac{1}{\hbar^{4}}\int\frac{d\omega_{1}\ d\omega_{1^{\prime}}\
d\omega_{2}\
d\omega_{2^{\prime}}}{(2\pi)^{4}}\langle\hat{\Omega}^{\dagger}_{1}(\omega_{1^{\prime}})\hat{\Omega}^{\dagger}_{2^{\prime}}(\omega_{2^{\prime}})\hat{\Omega}_{2}(\omega_{2})\hat{\Omega}_{1}(\omega_{1})\rangle
e^{i(\omega_{1^{\prime}}+\omega_{2^{\prime}}-\omega_{1}-\omega_{2})t}$
$\displaystyle\mathcal{G}_{11,22}(\omega_{1}+\omega_{2}-\omega_{2^{\prime}}-\omega_{1^{\prime}})\mathcal{G}_{2e,2e}(\omega_{1}+\omega_{2}-\omega_{1^{\prime}})\mathcal{G}_{ee,e^{\prime}e^{\prime}}(\omega_{1}-\omega_{1^{\prime}})\mathcal{G}_{e^{\prime}g,e^{\prime}g}(\omega_{1}),\
\ \ \ $ (A8)

where
$\hat{\Omega}_{1}(\omega_{1})=\mu_{ge}\sqrt{\frac{\hbar{\omega}_{1}}{2\mathbf{V}\epsilon_{0}}}\hat{a}_{1}(\omega_{1})e^{-i\omega_{1}t}$,
similarly we define
$\hat{\Omega}_{2}(\omega_{2}),\hat{\Omega}^{\dagger}_{1}(\omega_{1}^{\prime})$
and $\hat{\Omega}^{\dagger}_{2}(\omega_{2}^{\prime})$.

The population Green’s functions
$\mathcal{G}_{11,22},\mathcal{G}_{ee,e^{\prime}e^{\prime}}$, and
$\mathcal{G}_{gg,00}$ originate from the solution of coupled transport
(relaxation) equations:

$\displaystyle\dot{\rho}_{22}$
$\displaystyle=-\Gamma_{2}(n_{2}+1)\rho_{22}+\Gamma_{2}n_{2}\rho_{11},$
$\displaystyle\dot{\rho}_{11}$
$\displaystyle=\Gamma_{2}(n_{2}+1)\rho_{22}-\Gamma_{2}n_{2}\rho_{11},$ (A9)
$\displaystyle\dot{\rho}_{e^{\prime}e^{\prime}}$
$\displaystyle=-\Gamma_{e}(n_{e}+1)\rho_{e^{\prime}e^{\prime}}+\Gamma_{e}n_{e}\rho_{ee},$
$\displaystyle\dot{\rho}_{ee}$
$\displaystyle=\Gamma_{e}(n_{e}+1)\rho_{e^{\prime}e^{\prime}}-\Gamma_{e}n_{e}\rho_{ee},$
(A10) $\displaystyle\dot{\rho}_{00}$
$\displaystyle=-\Gamma_{c}(n_{c}+1)\rho_{00}+\Gamma_{c}n_{c}\rho_{gg},$
$\displaystyle\dot{\rho}_{gg}$
$\displaystyle=\Gamma_{c}(n_{c}+1)\rho_{00}-\Gamma_{c}n_{c}\rho_{gg}.$ (A11)

Eqs. (A9) - (A11) can be recast as a Pauli master equation:

$\displaystyle\dot{\rho}_{ii}(t)$
$\displaystyle=-\sum_{ii,jj}\kappa_{ii,jj}\rho_{jj}(t),$ (A12)

where, $\kappa_{ii,jj}$ is the population transport matrix. In Eq. (A12), the
diagonal elements, $i=j$, $\kappa_{ii,ii}$ are positive, whereas the off-
diagonal elements, $i\neq j$, $\kappa_{ii,jj}$ are negative. The population
transport matrix satisfies the population conservation:
$\sum_{i}\kappa_{ii,jj}=0$. The evolution of the diagonal elements is defined
by the population Green function,
$\rho_{jj}(t)=\sum_{i}\mathcal{G}_{jj,ii}(t)\rho_{ii}(0)$. where
$\mathcal{G}_{jj,ii}(t)$ is given [43]

$\displaystyle\mathcal{G}_{jj,ii}(t)=\sum_{n}\xi_{jn}^{(R)}D_{nn}^{-1}\exp(-\lambda_{n}t)\xi_{ni}^{(L)},$
(A13)

where $\lambda_{n}$ is the $nth$ eigenvalue of left and right eigenvector
$(\xi_{n}^{(L)},\xi_{n}^{(R)})$ and $D=\xi^{L}\xi_{R}$ is a diagonal matrix.
Using Eq. (A13) we obtain for the population Green’s functions:

$\displaystyle\mathcal{G}_{00,gg}(t)$
$\displaystyle=\frac{n_{c}(1-e^{-t(1+2n_{c})\Gamma_{c}})}{(1+2n_{c})},$ (A14)
$\displaystyle\mathcal{G}_{ee,e^{\prime}e^{\prime}}(t)$
$\displaystyle=\frac{n_{e}(1-e^{-t(1+2n_{e})\Gamma_{e}})}{(1+2n_{e})},$ (A15)
$\displaystyle\mathcal{G}_{11,22}(t)$
$\displaystyle=\frac{(1+n_{2})(1-e^{-t(1+2n_{2})\Gamma_{2}})}{(1+2n_{2})}.$
(A16)

Utilizing Eqs. (A14 - A16) and Liouville space Green’s functions
$\mathcal{G}(\omega)=\frac{-(n_{i}+1)\Gamma_{i}}{(\omega+i\epsilon)\left[\omega+i(2n_{i}+1)\Gamma_{i}\right]}$,
where $n_{i}$ is the average phonon occupation number and $\Gamma_{i}$ is the
dephasing rate for the $i\leftrightarrow i-1$ transition. To examine field-
induced fourth order correlations of matter, we utilize the reduced density
matrix obtained by tracing out the field degrees of freedom, Eq. (A8) reads

$\displaystyle\rho^{a}_{11}(t)$
$\displaystyle=\frac{\Gamma_{2}\Gamma_{e}(n_{2}+1)(n_{e}+1)e^{-(\Gamma_{21}+\epsilon_{2})t}}{\hbar^{4}(\epsilon_{2}-\Gamma_{21})(\epsilon_{e}-\Gamma_{ee^{\prime}})}(e^{\Gamma_{21}t}\hat{\Omega}^{\dagger}_{2}[\
\omega_{2e}+i(\epsilon_{2}-\Gamma_{2e})]-e^{\epsilon_{2}t}\hat{\Omega}^{\dagger}_{2}[\
\omega_{2e}+i(\Gamma_{21}-\Gamma_{2e})])$
$\displaystyle\hat{\Omega}_{1}\left[\omega_{e^{\prime}g}-i\Gamma_{ee^{\prime}}\right]\big{(}\hat{\Omega}_{2}[\
\omega_{2e}+i(\epsilon_{e}-\Gamma_{2e})]\hat{\Omega}^{\dagger}_{1}[\
\omega_{e^{\prime}g}+i(\epsilon_{e}-\Gamma_{e^{\prime}g})]-\hat{\Omega}_{2}[\
\omega_{2e}-i(\Gamma_{2e}-\Gamma_{ee^{\prime}})]$
$\displaystyle\hat{\Omega}^{\dagger}_{1}[\
\omega_{e^{\prime}g}+i(\Gamma_{ee^{\prime}}-\Gamma_{e^{\prime}g})]\big{)},$
(A17)

where $\epsilon_{2}~{}\text{and}~{}\epsilon_{e}$ are the dephasing rate at
transition $2-1$ and $e-e^{\prime}$, respectively.

The population of vibrational state $1$ from Feynman diagram b and c:

$\displaystyle\rho^{b}_{11}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\hbar^{4}}\int^{t}dt_{1}\int^{t_{1}}dt_{2}\int^{t_{2}}dt_{3}\int^{t_{3}}dt_{4}\langle\langle\mathcal{G}_{11,22}(t-t_{1})\hat{\mathcal{V}}^{\dagger}_{2e}\mathcal{G}_{e2,e2}(t_{1}-t_{2})\hat{\mathcal{V}}_{2e}\mathcal{G}_{ee,e^{\prime}e^{\prime}}(t_{2}-t_{3})$
(A18)
$\displaystyle\hat{\mathcal{V}}_{e^{\prime}g}\mathcal{G}_{e^{\prime}g,e^{\prime}g}(t_{3}-t_{4})\hat{\mathcal{V}}^{\dagger}_{eg}\rangle\rangle\langle\hat{E}^{\dagger}_{1}(t_{3})\hat{E}^{\dagger}_{2}(t_{2})\hat{E}_{2}(t_{1})\hat{E}_{1}(t_{4})\rangle$
$\displaystyle=$
$\displaystyle\frac{\Gamma_{2}\Gamma_{e}(n_{2}+1)(n_{e}+1)e^{-(\Gamma_{21}+\epsilon_{2})t}}{\hbar^{4}(\epsilon_{2}-\Gamma_{21})(\epsilon_{e}-\Gamma_{ee^{\prime}})}(e^{\Gamma_{21}t}\hat{\Omega}_{2}\left[\omega_{2e}-i(\epsilon_{2}-\Gamma_{2e})\right]-e^{\epsilon_{2}t}\hat{\Omega}_{2}\left[\omega_{2e}-i(\Gamma_{21}-\Gamma_{2e})\right])\hat{\Omega}_{1}\left[\omega_{e^{\prime}g}-i\Gamma_{e^{\prime}g}\right]$
$\displaystyle\big{(}\hat{\Omega}^{\dagger}_{1}\left[\omega_{e^{\prime}g}+i(\epsilon_{e}-\Gamma_{e^{\prime}g})\right]\hat{\Omega}^{\dagger}_{2}\left[\omega_{2e}-i(\epsilon_{e}-\Gamma_{2e})\right]-\hat{\Omega}^{\dagger}_{1}\left[\omega_{e^{\prime}g}-i(\Gamma_{e^{\prime}g}-\Gamma_{ee^{\prime}})\right]\hat{\Omega}^{\dagger}_{2}\left[\omega_{2e}-i(\Gamma_{ee^{\prime}}-\Gamma_{2e})\right]\big{)}.$
$\displaystyle\rho^{c}_{11}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\hbar^{4}}\int^{t}dt_{1}\int^{t_{1}}dt_{2}\int^{t_{2}}dt_{3}\int^{t_{3}}dt_{4}\langle\langle\mathcal{G}_{11,22}(t-t_{1})\hat{\mathcal{V}}_{2e^{\prime}}\mathcal{G}_{2e^{\prime},2e^{\prime}}(t_{1}-t_{2})\hat{\mathcal{V}}_{e^{\prime}g}\mathcal{G}_{2g,2g}(t_{2}-t_{3})$
(A19)
$\displaystyle\hat{\mathcal{V}}^{\dagger}_{2e^{\prime}}\mathcal{G}_{e^{\prime}g,e^{\prime}g}(t_{3}-t_{4})\hat{\mathcal{V}}^{\dagger}_{eg}\rangle\rangle\langle\hat{E}^{\dagger}_{1}(t_{2})\hat{E}^{\dagger}_{2}(t_{1})\hat{E}_{2}(t_{3})\hat{E}_{1}(t_{4})\rangle$
$\displaystyle=$
$\displaystyle\frac{\Gamma_{e}(n_{e}+1)e^{-(\Gamma_{21}+\epsilon_{2})t}}{\hbar^{4}(\Gamma_{21}-\epsilon_{2})}(e^{\Gamma_{21}t}\hat{\Omega}^{\dagger}_{2}\left[\omega_{2e^{\prime}}+i(\epsilon_{2}-\Gamma_{2e^{\prime}})\right]-e^{\epsilon_{2}t}\hat{\Omega}^{\dagger}_{2}\left[\omega_{2e^{\prime}}+i(\Gamma_{21}-\Gamma_{2e^{\prime}})\right])\hat{\Omega}_{1}\left[\omega_{e^{\prime}g}-i\Gamma_{e^{\prime}g}\right]$
$\displaystyle\hat{\Omega}_{2}\left[\omega_{2g}-\omega_{e^{\prime}g}+i(\Gamma_{e^{\prime}g}-\Gamma_{2e})\right]\hat{\Omega}^{\dagger}_{1}\left[\omega_{2g}-\omega_{2e^{\prime}}+i(\Gamma_{2e^{\prime}}-\Gamma_{2g})\right].$

The The total population
$\rho_{11}(t)=\text{Re}(\rho^{a}_{11}+\rho^{b}_{11}+\rho^{c}_{11})$ is induced
by pulse with Lorentzian shape is defined $e.g.$
$\hat{\Omega}_{1}\left[\omega_{e^{\prime}g}+i\Gamma_{e^{\prime}g}\right]=\frac{\hat{\Omega}_{1^{\prime}}}{\omega_{e^{\prime}g}-\omega_{0}+i\sigma_{p}-i\Gamma_{e^{\prime}g}},$
where $\omega_{0}$ is the central frequency and $\omega_{e^{\prime}g}$ is the
$e^{\prime}-g$ transition frequency. Employing the approximation
$\Gamma_{21}=\Gamma_{2}(2n_{2}+1)>>\epsilon_{2}$, assuming that the transition
energy is much larger than the decay constant. The total population of level
$1$ by employing Eqs. (A17), (A18) and (A19) yields:

$\displaystyle\rho_{11}(t)=\frac{16(n_{2}+1)\,\Omega^{4}_{p}\tilde{\omega}^{2}_{2e^{\prime}}\,\tilde{\omega}^{2}_{e^{\prime}g}(1-e^{\Gamma_{2}(2n_{2}+1)t})}{(2n_{2}+1)(\sigma^{2}_{p}+\tilde{\omega}^{2}_{2e^{\prime}})^{2}(\sigma^{2}_{p}+\tilde{\omega}^{2}_{e^{\prime}g})^{2}},$
(A20)

where
$\tilde{\omega}_{2e^{\prime}}=\omega_{2e^{\prime}}-\omega_{0},\tilde{\omega}_{e^{\prime}g}=\omega_{e^{\prime}g}-\omega_{0}$
and we set
$|\Omega_{1^{\prime}}|=|\Omega_{2^{\prime}}|=|\Omega^{\dagger}_{1^{\prime}}|=|\Omega^{\dagger}_{2^{\prime}}|=|\Omega_{p}|$.

We consider a three-level molecular system with the ground state $g$, single
electronic excited state $e$ and double excited electronic state $f$ shown in
Fig. 1(a). we denote the viberational states of electronic ground states $0$
and $g$ and vibrational double excited states of electronic states $2$ and $1$
and $e$ and $e^{\prime}$ for single electronic states. The corresponding
equation of motion for the density matrix is given by

$\displaystyle\dot{\rho}_{gg}(t)$
$\displaystyle=i\Omega_{1}(\rho_{ge}(t)-\rho_{eg}(t))+\Gamma_{c}(n_{c}+1)\rho_{00}(t)-\Gamma_{c}n_{c}\rho_{gg}(t),$
$\displaystyle\dot{\rho}_{ee}(t)$
$\displaystyle=-i\Omega_{1}(\rho_{ge}(t)-\rho_{eg}(t))+i\Omega_{2}(\rho_{g2}(t)-\rho_{2e}(t)),$
$\displaystyle\dot{\rho}_{22}(t)$
$\displaystyle=-i\Omega_{2}(t)(\rho_{e2}(t)-\rho_{2e}(t))-\Gamma_{2}(n_{2}+1)\rho_{22}(t)+\Gamma_{2}n_{2}\rho_{11}(t),$
$\displaystyle\dot{\rho}_{11}(t)$
$\displaystyle=-i\lambda(\rho_{01}(t)-\rho_{10}(t))-\Gamma_{2}(n_{2}+1)\rho_{00}(t)+\Gamma_{c}n_{c}\rho_{gg}(t),$
$\displaystyle\dot{\rho}_{2e}(t)$
$\displaystyle=i\Omega_{2}(t)(\rho_{22}(t)-\rho_{ee}(t))+i\Omega_{1}\rho_{2g}(t)-\left\\{\frac{\Gamma_{2}(n_{2}+1)}{2}+i(\omega_{2e}-\nu_{2})\rho_{2e}\right\\},$
$\displaystyle\dot{\rho}_{eg}(t)$
$\displaystyle=i\Omega_{1}(\rho_{ee}(t)-\rho_{gg}(t))-i\Omega_{1}\rho_{2g}(t)-\left\\{\frac{\Gamma_{c}n_{c}}{2}+i(\omega_{eg}-\nu_{1})\right\\}\rho_{eg}(t),$
$\displaystyle\dot{\rho}_{2g}(t)$
$\displaystyle=i\Omega_{1}\rho_{2e}(t)-i\Omega_{2}\rho_{eg}(t)-\left\\{\frac{\Gamma_{2}(n_{2}+1)}{2}+\frac{\Gamma_{c}n_{c}}{2}+i(\omega_{2g}-\nu_{1}-\nu_{2})\right\\}\rho_{2g}(t),$
$\displaystyle\dot{\rho}_{10}(t)$
$\displaystyle=i\lambda(\rho_{11}(t)-\rho_{00}(t))-\left\\{\frac{\Gamma_{2}n_{2}}{2}+\frac{\Gamma_{c}(n_{c}+1)}{2}+i(\omega_{10}-\nu_{pr})\right\\}\rho_{10}(t),$
$\displaystyle\dot{\rho}_{e2}(t)$
$\displaystyle=\dot{\rho}^{\dagger}_{2e},~{}\dot{\rho}_{ge}(t)=\dot{\rho}^{\dagger}_{eg},~{}\dot{\rho}_{g2}(t)=\dot{\rho}^{\dagger}_{2g},~{}\dot{\rho}_{01}(t)=\dot{\rho}^{\dagger}_{10},$
(A21)

where $\Gamma_{2}/2$ is a dephasing rate and
$n_{2}=\left[\text{exp}(\hbar\omega_{21}/k_{B}T_{c})\right]$ is the average
phonon occupation number corresponding to $1\leftrightarrow 2$ at temperature
$T_{2}$.
The output power in Eq. (9) using Eq. (7) in the high temperature limit
becomes:

$\displaystyle
P_{C}\\!=\\!\frac{4\,T_{c}\,\omega_{c}\,c_{21}\,\tau^{8}u\,v\,\tilde{c}_{p}\lambda^{\prime}\sigma^{\prime
8}_{p}\left(\tilde{c}_{21}-\tau^{8}\left(\tilde{c}_{21}-c_{21}\right)\sigma^{\prime
8}_{p}\right)}{3\,(c_{21}\,\tau^{8}\,\sigma^{\prime
8}_{p}+u\,\tilde{c}_{21})\left(c_{21}\tau^{8}\lambda^{\prime}\sigma^{\prime
8}_{p}+v\tilde{c}_{21}\right)},\ \ \ \ $ (A22)

where subscript $C$ specifies the two photons pump and
$\tilde{c}_{p}=\left(c_{p}-c_{21}-1\right),\tilde{c}_{21}=\alpha-
c_{21},u=\Gamma_{2}\omega_{c}/\Gamma_{c}T_{c}$, $v=\Gamma_{c}/\omega_{c}$ and
$\alpha=T_{2}/\omega_{c}$, where $T_{2}$ is the phonon bath temperature of
level $1-2$.

Appendix B: Entangled states of two photon

The state of SPDC to first order in perturbation theory:

$\displaystyle|\psi\rangle=|0\rangle-\frac{i}{\hbar}\int_{t_{0}}^{t}\text{dt}\
\mathcal{H}_{I}(t)|0\rangle,$ (B1)

where $\mathcal{H}_{I}$ is the effective third-order interaction Hamiltonian
given by

$\displaystyle\mathcal{H}_{I}(t)=\epsilon_{0}\int_{V}\text{d}^{3}\bm{r}\chi^{(2)}E^{+}_{p}(\bm{r},t)\hat{E}^{-}_{s}(\bm{r},t)\hat{E}^{-}_{i}(\bm{r},t)+\text{h.c.},$
(B2)

where $\chi^{(2)}$ is the susceptibility tensor of rank $2$ which describe the
nonlinear crystal. $V$ is interaction volume covered by pump filed and the
pump field is simply chosen a classical plane wave along z direction.

$\displaystyle
E^{+}_{p}(z,t)=E_{p}\int\text{d}\omega_{p}\mathcal{A}(\omega_{p})e^{-i(k_{p}(\omega_{p})z-\omega_{p}t)},$
(B3)

where $\mathcal{A}(\omega_{p})$ is the pulse envelope function..

$\displaystyle\hat{E}^{-}_{j}(z,t)=\int\text{d}\omega_{j}\mathcal{E}(\omega_{j})\hat{a}^{\dagger}_{j}(\omega_{j})e^{-i(k_{j}(\omega_{j})z-\omega_{j}t)},$
(B4)

where $j=s,i$ and $\hat{a}^{\dagger}(\omega_{j})$ is a creation operator and
we have restricted the spatial integral to be over z coordinate only. We
assume that
$\mathcal{E}(\omega_{j})=\sqrt{\hbar\omega/\epsilon_{0}\textbf{V}}$, where V
is quantization volume is slowly varying over the frequencies of interest and
therefore we can bring it outside the integral. The interaction Hamiltonian
part of Eq. (B1) using Eqs. (B2), (B3) and (B4) is recast as

$\displaystyle\int_{t_{0}}^{t}\mathcal{H}_{I}(t^{\prime})\text{d}t^{\prime}=\mathcal{A}\int_{-\infty}^{+\infty}\text{d}t^{\prime}\int_{\frac{-L}{2}}^{\frac{+L}{2}}\text{d}z\int\text{d}\omega_{i}\text{d}\omega_{s}\text{d}\omega_{p}$
$\displaystyle
e^{-i(k_{i}(\omega_{i})+k_{s}(\omega_{s})-k_{p}(\omega_{p}))z}e^{i(\omega_{s}+\omega_{i}-\omega_{p})t}\mathcal{A}(\omega_{p})$
$\displaystyle\hat{a}^{\dagger}_{i}(\omega_{i})\hat{a}^{\dagger}_{s}(\omega_{s})+\text{h.c.},$
(B5)

where $L$ is the length of the crystal and
$\mathcal{A}=E_{p}\mathcal{E}(\omega_{i})\mathcal{E}(\omega_{s})$. For a
pulsed laser, we can assume that the pump field, and therefore the interaction
Hamiltonian, is zero before $t_{0}$ and after $t$. Therefore, we can extend
the limits of the integration over infinite time. Performing the time integral
yields $\delta(\omega_{i}+\omega_{s}-\omega_{p})$, which then allows the
$\omega_{p}$ integral to be evaluated, giving

$\displaystyle\int_{t_{0}}^{t}\mathcal{H}_{I}(t^{\prime})\text{d}t^{\prime}=-2\pi\mathcal{A}\int_{\frac{-L}{2}}^{\frac{+L}{2}}\text{d}z\int\text{d}\omega_{i}\text{d}\omega_{s}\mathcal{A}(\omega_{i}+\omega_{s})$
$\displaystyle
e^{-i(k_{i}(\omega_{i})+k_{s}(\omega_{s})-k_{p}\omega_{p})z}\hat{a}^{\dagger}_{i}(\omega_{i})\hat{a}^{\dagger}_{s}(\omega_{s})+\text{c.c.},\
\ $ (B6)

Evaluating the integral over z yields and substitute in Eq. (B1)

$\displaystyle|\psi\rangle$ $\displaystyle=|0\rangle+\frac{2i\pi
L\mathcal{A}}{\hbar}\int\text{d}\omega_{i}\text{d}\omega_{s}\mathcal{A}(\omega_{i}+\omega_{s})\Phi(\omega_{i},\omega_{s})$
$\displaystyle\hat{a}^{\dagger}_{i}(\omega_{i})\hat{a}^{\dagger}_{s}(\omega_{s})|0\rangle+\text{H.C.},$
(B7)

where $\Phi(\omega_{i},\omega_{s})=\text{sinc}(\frac{L\Delta{k}}{2})$ and
$\mathcal{A}(\omega_{s},\omega_{i})=\frac{A_{0}}{\omega_{i}+\omega_{s}-\omega_{p}+i\sigma}$
is the normalized band pump pulse of width $\sigma$. For narrow band pump
$\sigma\rightarrow 0$.

The essential character of the phase-matching function is better illustrated
when it is expressed in a simpler form obtained by making the Taylor
expansions

$\displaystyle k_{p}(\omega)$ $\displaystyle\approx$ $\displaystyle
k_{p0}+(\omega-\bar{\omega})\frac{\partial
k_{p}(\omega)}{\partial\omega}|_{\omega=2\bar{\omega}},$ $\displaystyle
k_{j}(\omega)$ $\displaystyle\approx$ $\displaystyle
k_{j0}+(\omega-\bar{\omega})\frac{\partial
k_{j}(\omega)}{\partial\omega}|_{\omega=\bar{\omega}}$ (B8)

Here, $2\bar{\omega}$ is the center pump frequency. Discarding all but the
first two terms yields by using Eq. (B8)

$\displaystyle\Delta k$
$\displaystyle=k_{s}(\omega_{s})+k_{i}(\omega_{i})-k_{p}(\omega_{s}+\omega_{i})$
$\displaystyle\approx(\omega_{s}-\omega_{i})(\frac{\partial
k_{s}(\omega_{s})}{\partial\omega_{s}}-\frac{\partial
k_{i}(\omega_{i})}{\partial\omega_{i}})$
$\displaystyle=(\omega_{s}-\omega_{i})(\frac{1}{v_{s}}-\frac{1}{v_{i}}).$ (B9)

Therefore, the phase matching factor reads:

$\displaystyle\Phi(\omega_{s},\omega_{i})=\text{sinc}(\frac{(\omega_{s}-\omega_{i})T}{2}),$
(B10)

where, $T=L(\frac{1}{v_{s}}-\frac{1}{v_{i}})$ is entanglement time,
characterizing the group velocity dispersion inside the SPDC crystal. The
output state of SPDC from Eq. (B1) is given by

$\displaystyle|\psi\rangle=\mathcal{N}A_{0}\int\int\frac{\text{d}\omega_{s}\text{d}\omega_{i}\Phi(\omega_{s},\omega_{i})}{\omega_{i}+\omega_{s}-\omega_{p}+i\sigma}\hat{a}^{\dagger}_{s}(\omega_{s})\hat{a}^{\dagger}_{i}(\omega_{i})|0\rangle,$
(B11)

where $\mathcal{N}$ is normalization constant.

For the classical light

$\displaystyle\langle\hat{E}^{\dagger}_{1}(\omega_{1})\hat{E}^{\dagger}_{2}(\omega_{2})\hat{E}_{2}(\omega_{2})\hat{E}_{1}(\omega_{1})\rangle$
$\displaystyle=|\langle
0|\hat{E}_{2}(\omega_{2})\hat{E}_{1}(\omega_{1})|\phi\rangle|^{2}$
$\displaystyle=|\hat{E}_{2}(\omega_{2})\hat{E}_{1}(\omega_{1})|^{2}$ (B12)

The Rabi frequency for the transition from $g-2$ via intermediate level $e$ in
a given classical field

$\displaystyle\Omega^{2}_{1}\ \Omega^{2}_{2}$ $\displaystyle=$
$\displaystyle|\mu_{eg}|^{2}|\mu_{2e}|^{2}|E_{2}(\omega_{2})E_{1}(\omega_{1})|^{2}$
$\displaystyle=$ $\displaystyle|\mu_{eg}|^{2}|\mu_{2e}|^{2}|\langle
0|E_{2}(\omega_{2})E_{1}(\omega_{1})\phi\rangle|^{2}.\ \ \ \ $ (B13)

For quantum light, the two-point field correlation function reads

$\displaystyle\langle
0|\hat{E}_{2}(\omega_{2})\hat{E}_{1}(\omega_{1})|\psi\rangle=\langle
0|\sqrt{\frac{\omega_{s}}{2\epsilon_{0}V}}\hat{a}_{2}(\omega_{s})\sqrt{\frac{\omega_{i}}{2\epsilon_{0}V}}\hat{a}_{1}(\omega_{i})|\psi\rangle$
$\displaystyle=\frac{\mathcal{N}}{2\epsilon_{0}V}\int\int\text{d}\omega_{s}\text{d}\omega_{i}\sqrt{\omega_{s}\omega_{i}}\Phi(\omega_{s},\omega_{i})\frac{A_{0}}{\omega_{i}+\omega_{s}-\omega_{p}+i\sigma}$
$\displaystyle\underline{\langle
0|\hat{a}_{2}(\omega_{s})\hat{a}_{1}(\omega_{i})\hat{a}^{\dagger}_{i}(\omega_{i})\hat{a}^{\dagger}_{s}(\omega_{s})|0\rangle}\rightarrow\delta_{2s}\delta_{1i}$
$\displaystyle=\frac{A_{0}\mathcal{N}}{2\epsilon_{0}V}\frac{\sqrt{\omega_{1}\omega_{2}}}{\omega_{1}+\omega_{2}-\omega_{p}+i\sigma}\Phi(\omega_{2},\omega_{1}).\\!$
(B14)

By combination of Eqs. (B10), (B11), (B13) and (B14) the quantum and
correlation functions recast as

$\displaystyle{\Omega}_{1}(\omega_{1}){\Omega}_{2}(\omega_{2})$
$\displaystyle=\frac{\mu_{eg}\mu_{2e}\mathcal{N}A_{0}}{2\epsilon_{0}V}\frac{\sqrt{\omega_{2}\omega_{1}}}{\omega_{1}+\omega_{2}-\omega_{p}+i\sigma}$
$\displaystyle\text{sinc}[\frac{(\omega_{2}-\omega_{1})T}{2}]$ (B15)

The output power in Eq. (9) using Eq. (14) in the high temperature limit
becomes:

$\displaystyle
P_{Q}=\frac{2\,u\,v\,c_{21}\tilde{c}_{p}\lambda^{\prime}\,\tau^{4}\sigma^{\prime
4}_{p}\left(2\,\theta\,\tilde{c}_{21}-\tau^{4}\left(2\tilde{c}_{21}-c_{21}\right)\sigma^{\prime
4}_{p}\right)}{3\left(u\ \tilde{c}_{21}\theta+c_{21}\tau^{4}\sigma^{\prime
4}_{p}\right)\left(v\
\tilde{c}_{21}\theta+c_{21}\lambda^{\prime}\tau^{4}\sigma^{\prime
4}_{p}\right)},\ \ $ (B16)

where subscript $Q$ denotes the two photon entangled pump and
$\tilde{c}_{21}=\alpha+c_{21},\tilde{c}_{p}=c_{p}-c_{21}-1,\theta=\text{sinc}^{2}[T\left(\omega_{2e^{\prime}}-\omega_{ge^{\prime}}\right)/2],\alpha=T2/\omega_{c},u=\Gamma_{2}\omega_{c}/\Gamma_{c}T_{c}$
and $v=\Gamma_{c}/\omega_{c}$.

Appendix C: Maximum power of QHEs and Spectroscopy

Using Eq. (20) in the definition of the power of Eq. (III) and applying
dimensionless parameters over the high temperature limit, the spectroscopic
power is given by

$\displaystyle\mathcal{P}_{C}=\frac{4\alpha
c_{21}u^{2}\left(c_{p}-c_{21}-1\right)\lambda^{\prime}}{\tau^{8}\sigma_{\text{pr}}\left(c_{21}+\alpha
u\right)\left(c_{21}+5\alpha u\right)\sigma^{\prime 8}_{p}},$ (C1)

where all dimensionless parameters defined in earlier sections. Similarly we
recast spectroscopic power for entangled two-photon source using Eq. (21)

$\displaystyle\mathcal{P}_{Q}=\frac{\alpha
c_{21}u^{2}\left(c_{p}-c_{21}-1\right)\lambda^{\prime}\text{sinc}^{2}\big{[}\frac{T\left(\omega_{2e^{\prime}}-\omega_{ge^{\prime}}\right)}{2}\big{]}}{2\tau^{2}\sigma_{\text{pr}}\left(c_{21}+\alpha
u\right)\left(c_{21}+5\alpha u\right)\sigma^{\prime 2}_{p}}.$ (C2)

The maximum output power given in Eq. (10) and Eq. (15) for two-photon
entangled and classical states respectively and their numerical simulation vs
$\tau$ shown in Fig. 8(a), we considered small interval of $\tau$ because only
in this regime it shows quantum advantages in the scale of $10^{-3}$ and
within the range of $\tau\in\left[0,0.0048\right]$ the maximum output power of
two-photon entangled pump larger than the two photon classical pump case. The
small value of $\tau$ correspond to the high intensity of the pump field,
because $\tau=T_{c}/T_{h}$, where $T_{h}\propto\sqrt{\Omega_{p}}$ and $T_{c}$
at room temperature.

Figure 8:  (a) The numerical simulation of maximum classical power vs $\tau$
and it shows the within the small range of $\tau$ shows quantum advantage. (b)
same as (a) for the spectroscopic power shows the quantum advantage for the
different regime of temperature scale. The vertical axis of (a) and (b)
correspond to the maximum classical and quantum output power of the non-
perturbed and perturbed regimes, respectively and in this simulation, the
quantum advantages differ in their magnitude as shown in Figs. The parameters
read:
$T_{2}=T_{c}=300\text{K},\omega_{p}=1.3\text{ev},\omega_{c}=0.012\text{ev},\Omega_{p}=0.023\text{ev},\lambda=0.1\text{ev},\delta=0.00003\text{ev},\sigma_{p}=200\text{cm}^{-1},\Gamma_{2}=0.71\text{ps}^{-1}$
and $\Gamma_{c}=0.025\text{ps}^{-1}$

Similarly, the maximum output power for spectroscopic regime in Eqs. (22) and
(23) shown in Fig. 8(b) and the quantum advantage shown in the regime of low
pump intensity. The analytical solution of maximum output power in the limit
of $\tau$ explained in main text.

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See pages - of tut5-zhangA.pdf