# Two-photon absorption cross sections of pulsed entangled beams

Frank Schlawin  Max Planck Institute for the Structure and Dynamics of Matter,
Luruper Chaussee 149, 22761 Hamburg, Germany  University of Hamburg, Luruper
Chaussee 149, Hamburg, Germany  The Hamburg Centre for Ultrafast Imaging,
Hamburg, Germany<EMAIL_ADDRESS>
###### Abstract

Entangled two-photon absorption (ETPA) could form the basis of nonlinear
quantum spectroscopy at very low photon fluxes, since, at sufficiently low
photon fluxes, ETPA scales linearly with the photon flux. When different pairs
start to overlap temporally, accidental coincidences are thought to give rise
to a ’classical’ quadratic scaling which dominates the signal at large photon
fluxes and thus recovers a supposedly classical regime, where any quantum
advantage is thought to be lost. Here we scrutinize this assumption and
demonstrate that quantum-enhanced absorption cross sections can persist even
to very large photon numbers. To this end, we use a minimal model for quantum
light, which can interpolate continuously between the entangled pair and a
high-photon-flux limit, to derive analytically ETPA cross sections and the
intensity crossover regime. We investigate the interplay between spectral and
spatial degrees of freedom, how linewidth broadening of the sample impacts the
experimentally achievable enhancement.

††preprint: AIP/123-QED

##  I Introduction

It was recognized already in the early days of quantum optics that the quantum
statistics of a light field is intimately connected to the nonlinear optical
response it generates in a material [1, 2]. But it is only in recent years
with the advent of high-flux quantum light sources that this nonlinear regime
of quantum light-matter interactions has become experimentally accessible [3,
4, 5]. These experiments have since triggered a rapidly growing interest in
the use of quantum light in nonlinear spectroscopy [6, 7, 8, 9, 10]. Current
investigations range from the analysis of the photon statistics of light
emitted by photosynthetic complexes [11, 12, 13, 14, 15] to the improvement of
the signal-to-noise ratio in stimulated Raman scattering [16, 17, 18].

Arguably the most active field involves the use of entangled photons as
spectroscopic tools [19]. The archetypical process, which is widely believed
to encapsulate all the beneficial properties of entangled photons, is ETPA,
where an entangled pair is absorbed by a quantum system via far off-resonant
intermediate states. For such a process to occur, it is necessary for both
photons to be localised at the position of the quantum system at the same
time. This implies that the efficiency of this process may depend on the
spatial and the spectral correlations of the entangled pair - in addition to
the bunched nature of the entangled light field. In 1997 Fei et al. first
condensed the interplay of these effects into the elegant formula for the ETPA
cross section, [20]

$\displaystyle\sigma_{e}\simeq\frac{\delta_{r}}{A_{e}T_{e}}.$ (1)

Here, $\delta_{r}$ is the TPA cross section of randomly arriving photons. It
is divided by the so-called "entanglement area" $A_{e}$ and the "entanglement
time" $T_{e}$, which quantify the abovementioned momentum correlations and
spectral correlations, respectively. This result has since been generalized to
multiphoton cross sections [21]. The above cross section gives rise to the
detected ETPA rate, [20]

$\displaystyle R^{TPA}$ $\displaystyle=\sigma_{e}\phi+\delta_{r}\phi^{2}$ (2)

where $\phi$ is the photon flux density. Since entangled photons always arrive
in pairs, their ETPA rate scales linearly in the photon flux density [22, 23,
24]. When the flux increases, such that different pairs overlap in time,
random coincidences between uncorrelated photons are expected to give rise to
the classical quadratic scaling of the second term.

Despite the paramount importance of the two equations, they have never been
studied systematically in experiments [25]. Only one aspect - the linear
scaling of the ETPA rate, which is arguably by far their most important
prediction, has been investigated intensely. It was first observed in atomic
samples [26, 27, 3], and a tremendous amount of research has since focused on
extending these results to molecular samples. Some experiments report very
large ETPA signals in complex molecules [4, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37], with the help of coincidence detection [38], or with higher intensity
squeezed states [39]. Only one series of experiments explicitly demonstrated
the anticipated ETPA signal scaling after attenuation of the entangled photon
flux [40, 41]. Moreover, a growing number of experiments failed to detect any
clear ETPA signal at all [42, 43, 44, 45, 46, 47]. Beyond the scaling
behaviour, the dependence on the spatial and temporal correlations, as
predicted by Eq. (1), was not studied to date.

Theoretical arguments were advanced for why weak ETPA signals should be
expected in molecular samples [48, 49, 50, 51], highlighting the interplay
between molecular linewidth broadening and the achievable efficiency with the
use of spectrally entangled photons. Other theory works propose that entangled
photons may couple efficiently to very narrow transitions [52, 53]. Further
investigations compared the relative strength of one-photon transitions which
compete with the sought-after two-photon absorption [54, 48, 55, 56, 57, 58],
highlighted the role of interference between excitation pathways [59], the
optimization of measurements [60], or considered the possibility of spectrally
shaping the entangled two-photon wavefunction in order to optimise the ETPA
efficiency [61, 62, 63, 64]. Quantum metrology was used to analyze the
achievable enhancement compared to classical laser spectroscopy [65, 66, 67,
68, 69, 70].

In this paper, we aim to further this discussion by analyzing molecular ETPA
cross sections for pulsed, broadband entangled photons, taking into account
both spatial and spectral degrees of freedom. We use a minimal model for
entangled photons, which captures all the important physics, but allows us to
obtain analytical results, to first derive Eq. (1), and subsequently assess
the validity of Eq. (2). In the former goal, we show that Eq. (1) has to be
amended by a penalty factor which depends on the ratio between the broadening
of the molecular resonance and the bandwidth of the laser pulse generating the
entangled beam. In the latter case, we show that Eq. (2) is incorrect, as
coherent effects can survive up to very large photon numbers. In the limit of
a single spatial mode and a narrowband pump laser, our results largely agree
with the analysis by Raymer and Landes in Ref. 56. The paper is structured as
follows: In Section II, we discuss this model and highlight important
properties. Section III contains the perturbative calculations of the two
terms in Eq. (2). In Section IV, we then restrict our attention to the two-
photon limit, and derive Eq. (1) from our microscopic model. Going beyond the
two-photon limit, we discuss two limiting cases: in Section V, we consider the
situation where a single spatial mode is sufficient to describe the light
field, and in Section VI we consider ETPA with only a single active spectral
mode.

##  II Light-matter interactions & quantum light model

###  II.1 The quantized electromagnetic field

We want to describe the situation sketched in Fig. 1(a), where a nonlinear
crystal generates quantum light propagating along the $z$-direction with a
small transverse momentum ${\bf q}$. A thin slab containing sample molecules
is placed in the image plane of the optical system, where we set the origin of
our coordinate system $z=0$. In an ideal optical system, the positive
frequency component of the electric field operator in the image plane can then
be written as [71]

$\displaystyle{\bf E}^{(+)}({\bf\rho},z=0,t)$
$\displaystyle=\frac{i}{(2\pi)^{3/2}}\sum_{\sigma}\int_{0}^{\infty}\\!\\!d\omega\;\int_{\mathds{R}^{2}}\\!\\!\\!\\!d^{2}q\left(\frac{\hbar\omega}{2\epsilon_{0}}\right)^{1/2}$
$\displaystyle\times\left(\frac{\omega}{c^{2}k_{z}(\omega)}\right)^{1/2}\hat{a}_{\sigma}({\bf
q},\omega){\bf e}_{{\bf q},\sigma}e^{i({\bf q}\cdot{\bf\rho}-\omega t)},$ (3)

where $\hat{a}_{\sigma}({\bf q},\omega)$ is the photon annihilation operator
of a photon with transverse momentum ${\bf q}=(k_{x},k_{y})$, frequency
$\omega$, and polarization $\sigma$. The operators satisfy

$\displaystyle\left[\hat{a}_{\sigma}({\bf
q},\omega),\hat{a}^{\dagger}_{\sigma^{\prime}}({\bf
q}^{\prime},\omega^{\prime})\right]$
$\displaystyle=\delta_{\sigma,\sigma^{\prime}}\delta({\bf q}-{\bf
q}^{\prime})\delta(\omega-\omega^{\prime}).$ (4)

In addition, ${\bf e}_{{\bf q},\sigma}$ is the polarization vector, and the
transverse position in the image plane is ${\bf\rho}=(x,y)$. We employ the
small bandwidth approximation, where we evaluate the frequency-dependent
prefactor at the center frequency $\omega_{0}$ of the traveling light pulse.
We then have $k_{z}(\omega_{0})=n_{0}\omega_{0}/c$ with $n_{0}$ the refractive
index of the medium, and we further approximate ${\bf e}_{{\bf
q},\sigma}\simeq{\bf e}_{\sigma}$. The electric field operator becomes simply

$\displaystyle{\bf E}^{(+)}({\bf\rho},t)$
$\displaystyle=iE_{0}\sum_{\sigma}{\bf
e}_{\sigma}\int_{0}^{\infty}\\!\\!d\omega\;\int\\!\\!d^{2}q\;\hat{a}_{\sigma}({\bf
q},\omega)e^{i({\bf q}\cdot{\bf\rho}-\omega t)},$ (5)

where $E_{0}=(2\pi)^{-3/2}((\hbar\omega_{0})/(2\epsilon_{0}n_{0}c))^{1/2}$.

###  II.2 Light-matter interaction

The dipolar light-matter interaction Hamiltonian (in the interaction picture
with respect to molecular and field Hamiltonian) is given by

$\displaystyle H_{l-m}(t)$
$\displaystyle=\int_{\mathds{R}^{3}}\\!\\!d^{3}r\;{\bf\hat{d}}(t)\cdot\left({\bf\hat{E}}^{(+)}({\bf
r},t)+{\bf\hat{E}}^{(-)}({\bf r},t)\right),$ (6)

where ${\bf\hat{d}}({\bf r},t)={\bf\hat{d}}^{(+)}({\bf
r},t+{\bf\hat{d}}^{(-)}({\bf r},t$ are the molecular dipole operator and its
positive (negative) frequency components, respectively. We consider the
interaction of (quantum) light with a few-level sample system, as given by
Fig. 1(b). It is described by the Hamiltonian

$\displaystyle\hat{H}_{mol}$ $\displaystyle=\hbar\omega_{g}|g\rangle\langle
g|+\hbar\omega_{e}|e\rangle\langle e|+\hbar\omega_{f}|f\rangle\langle f|,$ (7)

where $|g\rangle$ is the ground state. In the following, we set $\omega_{g}=0$
without loss of generality. The Hamiltonian then consists of an intermediate
state $|e\rangle$ and final state $|f\rangle$. We can now expand the dipole
operator in these states, such that it reads in the Schrödinger picture

$\displaystyle\hat{{\bf d}}({\bf r})$ $\displaystyle=\delta({\bf r}-{\bf
r}_{mol})\left({\bf d}_{eg}|e\rangle\langle g|+{\bf d}_{ef}|f\rangle\langle
e|+H.c.\right).$ (8)

Here ${\bf r}_{mol}=({\bf\rho}_{mol},z_{mol})$ denotes the position of the
molecule. We assumed it to be point-like, which is an excellent approximation
for molecular light-matter interactions in the optical regime. We further
divide it into the positive and negative frequency components,

$\displaystyle\hat{{\bf d}}({\bf r})$ $\displaystyle=\delta({\bf r}-{\bf
r}_{mol})\left(\hat{{\bf d}}^{(+)}+\hat{{\bf d}}^{(-)}\right),$ (9)

where

$\displaystyle\hat{{\bf d}}^{(-)}$ $\displaystyle=\left({\bf
d}_{eg}|e\rangle\langle g|+{\bf d}_{ef}|f\rangle\langle e|\right)$ (10)

accounts for the excitation of the molecule (and ${\bf d}^{(+)}$ for the de-
excitation).

###  II.3  The PDC Model

In the sketch in Fig. 1(a), a laser pulse with frequency $\omega_{p}$,
bandwidth $\Omega_{p}$ and transverse beam area $\sim(2\pi/Q_{p})^{2}$ induces
spontaneous downconversion in a nonlinear crystal, generating two quantum-
correlated beams, traditionally called signal and idler. Since the purpose of
this paper is not to discuss the generation of PDC light in a multimode regime
in great detail (many excellent reviews can be found on this topic, e.g. Ref.
72), here we only introduce the bigaussian model, which allows us to carry out
many of the following calculations analytically. Crucially, this model
includes both spatial and spectral degrees of freedom, and is therefore
capable of describing spatiotemporal correlations [73, 74, 75, 76] \-
sometimes dubbed ”X-entanglement" [77] when the two degrees of freedom are not
separable.

####  II.3.1 Effective action

The effective action, which governs the downconversion process in the setup of
Fig. 1(a), can be written as [78]

$\displaystyle\Sigma_{eff}=\hbar\Gamma\int d\omega_{s}\int d\omega_{i}$
$\displaystyle\;\int d^{2}q_{s}\int d^{2}q_{i}\;F_{PDC}({\bf
q}_{s},\omega_{s};{\bf q}_{i},\omega_{i})$
$\displaystyle\times\hat{a}^{\dagger}_{s}({\bf
q}_{s},\omega_{s})\hat{a}^{\dagger}_{i}({\bf q}_{i},\omega_{i})+H.c,$ (11)

Here, the polarization index $\sigma$ of Eq. (5) is replaced by the signal and
idler fields, labelled "$s$" and "$i$", respectively. These two fields are
assumed to have a fixed polarization, which may be different (in type-II PDC)
or identical (in type-I PDC). In the latter case, we assume that the two
fields may still be distinguished, e.g. by their propagation direction.
$\Gamma$ is a dimensionless number, which quantifies the strength of the PDC
process. We have further introduced the joint spatiotemporal amplitude (JSA)
$F_{PDC}$, which we choose to be normalized, i.e.

$\displaystyle\int d\omega_{s}\int d\omega_{i}\;\int d^{2}q_{s}\int
d^{2}q_{i}\;\left|F_{PDC}({\bf q}_{s},\omega_{s};{\bf
q}_{i},\omega_{i})\right|^{2}$ $\displaystyle=1.$ (12)

In general, the JSA has a complicated structure which cannot be factorized
into spatial and spectral components [73], and which gives rise to high-
dimensional $X$-entanglement [77]. In this manuscript, we will use a
particularly convenient model, where the spatial and the frequency phase
matching functions are approximated by Gaussians. This is, e.g., appropriate
for non-collinear PDC [78]. The JSA then reads

$\displaystyle F_{PDC}({\bf q}_{s},\omega_{s};{\bf q}_{i},\omega_{i})$
$\displaystyle=F_{mom}({\bf q}_{s},{\bf
q}_{i})F_{spec}(\omega_{s},\omega_{i})$ (13)

with the momentum distribution

$\displaystyle F_{mom}({\bf q}_{s},{\bf q}_{i})$ $\displaystyle=\frac{1}{\pi
Q_{p}Q_{m}}\exp\left[-\frac{({\bf Q}_{s}+{\bf
Q}_{i})^{2}}{4Q_{p}^{2}}-\frac{({\bf Q}_{s}-{\bf
Q}_{i})^{2}}{4Q_{m}^{2}}\right]$ (14)

and the joint spectral amplitude

$\displaystyle F_{spec}(\omega_{s},\omega_{i})$
$\displaystyle=\frac{1}{\sqrt{\pi\Omega_{m}\Omega_{p}}}\exp\left[-\frac{(\Omega_{s}+\Omega_{i})^{2}}{4\Omega_{p}^{2}}-\frac{(\Omega_{s}-\Omega_{i})^{2}}{4\Omega_{m}^{2}}\right].$
(15)

Here we have introduced variables that are shifted with respect to their mean,
i.e. ${\bf Q}_{j}={\bf q}_{j}-{\bf q}_{j}^{(0)}$ and
$\Omega_{j}=\omega_{j}-\omega_{j}^{(0)}$. Due to energy and momentum
conservation, we have $\omega_{s}^{(0)}+\omega_{i}^{(0)}=\omega_{p}$ and ${\bf
q}_{s}^{(0)}+{\bf q}_{i}^{(0)}=0$. In the following, we will work with
degenerate downconversion with
$\omega_{s}^{(0)}=\omega_{i}^{(0)}=\omega_{p}/2$. This reduces some clutter in
our notation, but has no implications for the TPA cross section with off-
resonant intermediates states, which we will evaluate. In Eq. (14), the first
term in the exponential governs the distribution of the sum of the two
momenta, described by the width $Q_{p}$. This parameter is inversely
proportional to the transverse beam width $w_{0}$ of the laser pulse
triggering the downconversion process. In the following, we will characterize
the transverse width of the entangled beam by
$A_{p}=\lambda_{p}^{2}=(2\pi)^{2}/Q_{p}^{2}$. The second term in Eq. (14)
describing the distribution of the frequency differences is characterized by
the quantity $Q_{m}$, which is determined by the phase matching in the
nonlinear crystal [73, 79] and the emission angles of the PDC light which can
be collected by the optical system [78]. Eq. (15) is governed by similar
physics: the distribution of the sum of the two photon frequencies is governed
by $\Omega_{p}$, which is the bandwidth of the pump pulse generating the PDC
light. The distribution of the frequency differences $\Omega_{m}$ is again
related to the phase matching conditions in the nonlinear crystal.

The bigaussian form of Eqs. (14) and (15) is very convenient, since it allows
us to diagonalize the effective action (II.3.1) analytically. This can be done
with the help of Mehler’s formula for Hermite polynomials, which yields [78,
80]

$\displaystyle\frac{1}{\sqrt{\pi}}\exp\left[-\frac{1}{4}\frac{1+\zeta}{1-\zeta}(x+y)^{2}-\frac{1}{4}\frac{1-\zeta}{1+\zeta}(x-y)^{2}\right]$
$\displaystyle=\sqrt{1-\zeta^{2}}\sum_{n=0}^{\infty}(-1)^{n}\zeta^{n}h_{n}(x)h_{n}(y),$
(16)

where $0<\zeta<1$ and
$h_{n}(kx)=k^{1/2}(2^{n}n!\sqrt{\pi})^{-1/2}H_{n}(kx)e^{-(kx)^{2}/2}$111Note
that here the Hermite functions are normalized with respect to integrals over
space or frequency, i.e. $\int d\omega\;h_{n}^{2}(\omega)=1$.. Note that the
factor $(-1)^{n}$ is included for the case of Eq. (14) and (15), where the
first terms in the exponential have a smaller variance than the second terms,
i.e. where $Q_{p}>Q_{m}$ and $\Omega_{p}>\Omega_{m}$. In the case where this
condition is violated (e.g. when we evaluate the Fourier transform of these
functions), we use Eq. (II.3.1) without the factor $(-1)^{n}$.

We find for Eqs. (14) and (15)

$\displaystyle\zeta_{t}$
$\displaystyle=\frac{\Omega_{m}-\Omega_{p}}{\Omega_{m}+\Omega_{p}}$ (17)

and $\tau=1/\sqrt{\Omega_{m}\Omega_{p}}$. Likewise, for the momentum
correlation function we obtain, with $x_{j}=Q_{sj}q$, $q=1/\sqrt{Q_{m}Q_{p}}$,
and $\zeta_{q}=(Q_{m}-Q_{p})/(Q_{m}+Q_{p})$. Altogether, we can write the PDC
action (II.3.1) as

$\displaystyle\Sigma_{eff}$
$\displaystyle=\hbar\Gamma(1-\zeta^{2}_{q})\sqrt{1-\zeta_{t}^{2}}\sum_{{\bf
n}}\zeta_{t}^{n_{t}}\zeta_{q}^{n_{x}+n_{y}}\hat{A}^{\dagger}_{n_{t}n_{x}n_{y}}\hat{B}^{\dagger}_{n_{t}n_{x}n_{y}}+h.c.$
(18)

where we defined the Schmidt mode operators

$\displaystyle\hat{A}^{\dagger}_{{\bf n}}$ $\displaystyle=\int d\omega_{s}\int
d^{2}q_{s}\;h_{{\bf n}}({\bf q}_{s},\omega_{s})\hat{a}_{s}^{\dagger}({\bf
q}_{s},\omega_{s}),$ (19)

and

$\displaystyle\hat{B}^{\dagger}_{{\bf n}}$ $\displaystyle=(-1)^{|{\bf n}|}\int
d\omega_{i}\int d^{2}q_{i}\;h_{{\bf n}}({\bf
q}_{i},\omega_{i})\hat{a}_{i}^{\dagger}({\bf q}_{i},\omega_{i}).$ (20)

Here we use the short-hand notation ${\bf n}=({n_{t},n_{x},n_{y}})$, and

$\displaystyle h_{{\bf n}}({\bf
q},\omega)=h_{n_{t}}\left(\frac{\omega-\omega_{p}/2}{\sqrt{\Omega_{m}\Omega_{p}}}\right)h_{n_{x}}\left(\frac{q_{x}-q_{x}^{(0)}}{\sqrt{Q_{m}Q_{p}}}\right)h_{n_{y}}\left(\frac{q_{y}-q_{y}^{(0)}}{\sqrt{Q_{m}Q_{p}}}\right).$
(21)

The broadband operators inherit their orthonormality from the Hermite
functions, i.e. $[\hat{A}_{{\bf n}},\hat{A}^{\dagger}_{{\bf
n}^{\prime}}]=\delta_{{\bf n},{\bf n}^{\prime}}$, $[\hat{A}_{{\bf
n}},\hat{A}_{{\bf n}^{\prime}}]=0$ (likewise for idler mode $\hat{B}$), and
$[\hat{A}_{{\bf n}},\hat{B}_{{\bf n}^{\prime}}]=0$.

####  II.3.2  Output state

The quantum state of light generated by the PDC in a nonlinear crystal is
given by

$\displaystyle|\psi_{BSV}\rangle$
$\displaystyle=\exp\left[-\frac{i}{\hbar}\Sigma_{eff}\right]|0\rangle.$ (22)

Correlation functions of this state are most conveniently calculated in the
Heisenberg picture, where the input-output relations are readily derived as

$\displaystyle\hat{A}^{(out)}_{{\bf n}}$ $\displaystyle=\cosh(r_{{\bf
n}}\Gamma)\hat{A}_{{\bf n}}+\sinh(r_{{\bf n}}\Gamma)\hat{B}^{\dagger}_{{\bf
n}},$ (23) $\displaystyle\hat{B}^{(out)}_{{\bf n}}$
$\displaystyle=\cosh(r_{{\bf n}}\Gamma)\hat{B}_{{\bf n}}+\sinh(r_{{\bf
n}}\Gamma)\hat{A}^{\dagger}_{{\bf n}},$ (24)

where $r_{{\bf
n}}=(1-\zeta^{2}_{q})\sqrt{1-\zeta_{t}^{2}}\zeta_{t}^{n_{t}}\zeta_{q}^{n_{x}+n_{y}}$.
Using Eq. (5), the electric field operators in the image plane are then given
by

$\displaystyle{\bf E}_{s}^{(+)}({\bf r},t)$ $\displaystyle=E_{0}{\bf
e}_{s}\sum_{{\bf n}}h_{{\bf n},s}({\bf r},t)\hat{A}_{{\bf n}}^{(out)},$ (25)
$\displaystyle{\bf E}_{i}^{(+)}({\bf r},t)$ $\displaystyle=E_{0}{\bf
e}_{i}\sum_{{\bf n}}h_{{\bf n},i}({\bf r},t)\hat{B}_{{\bf n}}^{(out)}.$ (26)

Here, $E_{0}$ is defined in Eq. (5) with $\omega_{0}=\omega_{p}/2$, ${\bf
e}_{j}$ the polarization vector, and $h_{{\bf n},s}({\bf r},t)$ ($h_{{\bf
n},i}({\bf r},t)$) are the Fourier transforms of the Hermite functions (21),
i.e.

$\displaystyle h_{{\bf n},s}({\bf r},t)$ $\displaystyle\equiv\int
d\omega_{s}\int d^{2}q_{s}\;h_{{\bf n}}({\bf q}_{s},\omega_{s})e^{i({\bf
q}_{s}\cdot{\bf\rho}-\omega_{s}t)},$ (27)

and

$\displaystyle h_{{\bf n},i}({\bf r},t)$ $\displaystyle\equiv(-1)^{|{\bf
n}|}\int d\omega_{i}\int d^{2}q_{i}\;h_{{\bf n}}({\bf
q}_{i},\omega_{i})e^{i({\bf q}_{i}\cdot{\bf\rho}-\omega_{i}t)}.$ (28)

Finally, it is possible to calculate the Schmidt number explicitly in this
model [78]. This quantity measures the effective dimensionality of the state
of light. It is simply given by

$\displaystyle K$ $\displaystyle=K_{t}K_{x}K_{y},$ (29)

where we have $K_{t}=(\Omega_{m}/\Omega_{p}+\Omega_{p}/\Omega_{m})/2$, and
$K_{x}=K_{y}=(Q_{m}/Q_{p}+Q_{p}/Q_{m})/2$. A useful approximation for the the
Schmidt number, which we will use later, is obtained in the limit of strong
two-photon entanglement, where e.g. $\Omega_{m}\gg\Omega_{p}$ and $Q_{m}\gg
Q_{p}$, such that the number of spectral modes is given by the ratio of the
two numbers, respectively, i.e. $K_{t}\simeq\Omega_{m}/(2\Omega_{p})$ and
$K_{x/y}\simeq Q_{m}/(2Q_{p})$.

##  III Two-photon absorption cross section

###  III.1 From the excitation probability to the cross section

As discussed in detail below, we will use time-dependent perturbation theory
to calculate the two-photon excitation probability $P_{f}({\bf r}_{mol})$ of a
molecule at a position ${\bf r}_{mol}$ interacting with the pulsed quantum
light field, which we introduced in the previous section. The full TPA signal
(which could be measured e.g. by fluorescence) is then given by the sum over
all the molecules’ contributions, which are contained in a sample. Since this
will usually be a macroscopic number, we replace the summation over these
contributions by an integral over the sample volume and the average molecular
density $m_{0}$. Up to an unimportant constant which depends on the e.g. the
detection efficiency of the fluorescence photons, the measured signal is then
given by 222 Note that the factor $1/(2\pi)^{2}$ is introduced for
consistency, such that the photon number density
$\hat{a}^{\dagger}({\bf\rho},\omega)\hat{a}({\bf\rho},\omega)$ integrated over
space and frequency yields the mean photon number (III.1).

$\displaystyle S_{TPA}$
$\displaystyle=\frac{m_{0}}{(2\pi)^{2}}\int_{V_{sample}}\\!\\!\\!\\!d^{3}r_{mol}\;P_{f}({\bf
r}_{mol}).$ (30)

For the sample volume $V_{sample}$, we consider a thin slab of length $\Delta
z$ in the propagation direction of the light, such that we replace $\int
dz\simeq\Delta z$. The transverse extent $\Delta^{2}{\bf\rho}$ in contrast is
assumed much larger than the transverse width of the light field, such that we
can extend the integration boundaries to infinity in the transverse
directions.

The measured TPA rate is then given by the signal and the repetition rate
$f_{rep}$ of the laser which generates the entangled pulses,

$\displaystyle R^{TPA}$ $\displaystyle=S_{TPA}f_{rep}.$ (31)

To bring this rate into the form of a two-photon absorption rate (2), we write
it as

$\displaystyle R^{TPA}$
$\displaystyle=N_{mol}\sigma_{e}\phi+N_{mol}\delta_{r}\phi^{2}$ (32)

where the total number of molecules in the sample, which are irradiated by the
light, is given by $N_{mol}=m_{0}\Delta zA_{p}$. To establish the scaling with
the photon flux density, we write the photon flux density as

$\displaystyle\phi$
$\displaystyle=\langle\hat{N}\rangle\times\frac{f_{rep}}{A_{p}},$ (33)

where

$\displaystyle\langle\hat{N}\rangle$ $\displaystyle=\sum_{j=s,i}\int
d\omega\int d^{2}q\;\langle\psi_{BSV}|\hat{a}^{\dagger}_{j}({\bf
q},\omega)\hat{a}_{j}({\bf q},\omega)|\psi_{BSV}\rangle$
$\displaystyle=2\sum_{{\bf n}}\sinh^{2}(r_{{\bf n}}\Gamma)$ (34)

is the total photon number during one pulse, $f_{rep}$ accounts for the pulse
duration as above,

Figure 1:  (a) Schematic setup considered in this manuscript: A thin slab
containing the sample molecules is placed in the object plane of an imaging
system, such that near-field correlations of the light fields created by
parametric downconversion can be harnessed [73]. (b) Molecular level scheme:
the broadband photons generated in the PDC process trigger TPA events in the
molecule. The intermediate states $|e\rangle$, which are dipole-coupled to the
ground state, are far detuned from the centre frequency $\omega_{p}/2$ of each
photon. The final state $|f\rangle$ is near-resonant with the sum of the two
photon frequencies $\omega_{p}$. (c) Double-sided Feynman diagram representing
Eq. (37). For diagram rules, see e.g. Ref. 6.

###  III.2 Experimental setup and two-photon absorption probability

We next calculate the probability of entangled TPA by a point-like molecule at
position ${\bf r}_{mol}=({\bf\rho}_{mol},0)$ in the image plane of the optical
setup (see Fig. 1(a)). The two-photon absorption cross section can be
calculated from the probability to excite the final molecular state $f$ via
the absorption of two photons. Thus, we need to calculate

$\displaystyle P_{f}({\bf r}_{mol},t)$
$\displaystyle=\text{tr}\left\\{|f(t)\rangle\langle
f(t)|\mathcal{T}\exp\left[-\frac{i}{\hbar}\int^{t}_{t_{0}}\\!\\!d\tau
H_{l-m,-}(\tau)\right]\varrho_{0}\right\\},$ (35)

where the initial state of the light-matter system at time $t_{0}$ is given by
the state of light emitted by the PDC source, Eq. (22), and the molecular
electronic ground state,

$\displaystyle\varrho_{0}$ $\displaystyle=\varrho_{sys}\otimes\varrho_{f},$
(36)

with $\varrho_{sys}=|g(t_{0})\rangle\langle g(t_{0})|$ and
$\varrho_{f}=|\psi_{BSV}(t_{0})\rangle\langle\psi_{BSV}(t_{0})|$. The time
evolution is given by the Dyson series, and defined the time-ordering operator
$\mathcal{T}$. We also defined the light-matter interaction superoperator
$H_{l-m,-}$, which acts on the density matrix as
$H_{l-m,-}\varrho=H_{l-m}\varrho-\varrho H_{l-m}$. The interaction Hamiltonian
in defined in Eq. (6), and the electric field operator is given by the sum of
signal and idler fields, ${\bf E}={\bf E}_{s}+{\bf E}_{i}$. The final state
population can be calculated from Eq. (35) by expanding the exponential of the
Dyson series, and collecting the leading-order contributions.

Using the rotating wave approximation, $H_{l-m}\simeq{\bf d}^{(+)}\cdot{\bf
E}^{(-)}+{\bf d}^{(-)}\cdot{\bf E}^{(+)}$, and the limiting our analysis to
off-resonant intermediate states, we find the probability to excite the
f-state population is given by the Feynman diagram in Fig. 1(c) (for details,
see e.g. Ref. 51), which evaluates to

$\displaystyle P_{f}({\bf r}_{mol},t)$
$\displaystyle=\int^{t}_{t_{0}}\\!\\!d\tau_{4}\int^{\tau_{4}}_{t_{0}}\\!\\!d\tau_{3}\int^{\tau_{3}}_{t_{0}}\\!\\!d\tau_{2}\int^{\tau_{2}}_{t_{0}}\\!\\!d\tau_{1}\\!\\!\\!\\!\\!\\!\sum_{j_{1},j_{2},j_{3},j_{4}=s,i}\\!\\!\\!\\!\\!\\!2\Re\bigg{(}C^{mat}_{j_{1},j_{2},j_{3},j_{4}}(\tau_{1},\tau_{2},\tau_{3}.\tau_{4})C^{field}_{j_{1},j_{2},j_{3},j_{4}}({\bf
r}_{mol},\tau_{1},{\bf r}_{mol},\tau_{2};{\bf r}_{mol},\tau_{3},{\bf
r}_{mol},\tau_{4})\bigg{)},$ (37)

where

$\displaystyle
C^{mat}_{j_{1},j_{2},j_{3},j_{4}}(\tau_{1},\tau_{2},\tau_{3}.\tau_{4})$
$\displaystyle=$
$\displaystyle\frac{1}{\hbar^{4}}\text{tr}\left\\{\mu^{(+)}_{j_{1}}(\tau_{1})\mu^{(+)}_{j_{2}}(\tau_{2})\mu^{(-)}_{j_{4}}(\tau_{4})\mu^{(-)}_{j_{3}}(\tau_{3})\varrho_{sys}\right\\}$
(38)

is the matter correlation response function.We further defined the projections
of the molecular dipole operators on the polarization of the incoming light
field,

$\displaystyle\mu_{j}(t)$ $\displaystyle={\bf e}_{j}\cdot{\bf d}(t).$ (39)

Likewise, the correlation function of the field operators is given by

$\displaystyle C^{field}_{j_{1},j_{2},j_{3},j_{4}}({\bf r}_{mol},\tau_{1};{\bf
r}_{mol},\tau_{2};{\bf r}_{mol},\tau_{3};{\bf r}_{mol},\tau_{4})$
$\displaystyle=$ $\displaystyle\text{tr}\left\\{E^{(-)}_{j_{1}}({\bf
r}_{mol},\tau_{1})E^{(-)}_{j_{2}}({\bf r}_{mol},\tau_{2})E^{(+)}_{j_{4}}({\bf
r}_{mol},\tau_{4})E^{(+)}_{j_{3}}({\bf
r}_{mol},\tau_{3})\varrho_{f}\right\\}.$ (40)

The impact of the polarization degrees of freedom can be calculated with a
rotational average. For ETPA, this was done in Ref. [83], and we found that it
can account only for minor changes of the ETPA cross section. Thus, in the
following, we will neglect this procedure, and replace the dipole operators by
scalar quantities, i.e. $\mu_{j}\rightarrow\mu$.

####  III.2.1 The molecular correlation function

We will evaluate the molecular correlation function first. Using the Feynman
diagram in Fig. 1(c), we evaluate the correlation function straightforwardly
to find

$\displaystyle C^{mat}(\tau_{1},\tau_{2},\tau_{3}.\tau_{4})$
$\displaystyle=\sum_{e,e^{\prime}}\mu^{\ast}_{ge}\mu^{\ast}_{ef}\mu_{ge^{\prime}}\mu_{e^{\prime}f}e^{i(\omega_{eg}+i\gamma_{eg})(\tau_{2}-\tau_{1})}$
$\displaystyle\times
e^{i(\omega_{fg}+i\gamma_{fg})(\tau_{3}-\tau_{2})}e^{i(\omega_{fe^{\prime}}+i\gamma_{e^{\prime}f})(\tau_{4}-\tau_{3})}.$
(41)

This expression includes the summation over all the virtual states
$e,e^{\prime}$, through which the ETPA process can take place. We have further
added phenomenological dephasing rates $\gamma_{ge}$ and $\gamma_{ef}$, which
broaden the linewidths of the resonances. The correlation function is most
conveniently evaluated by changing to the time delay integration variables
$t_{1}=\tau_{2}-\tau_{1}$, $t_{2}=\tau_{3}-\tau_{2}$, and
$t_{3}=\tau_{4}-\tau_{3}$, and sending $t_{0}\rightarrow-\infty$. We are
further interested in the limit $t\rightarrow\infty$, i.e. in the final
population after the entangled pulse has interacted with the molecule. In this
limit, the $\tau_{4}$-integration simply evaluates to
$2\pi\delta(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{4})$. With the definition
of the electric field operator (5), we arrive at

$\displaystyle P_{f}({\bf r}_{mol},t\rightarrow\infty)$
$\displaystyle=2(2\pi)^{5}\Re\frac{E_{0}^{4}}{\hbar^{4}}\sum_{e,e^{\prime}}\mu^{\ast}_{ge}\mu^{\ast}_{ef}\mu_{ge^{\prime}}\mu_{e^{\prime}f}\int_{0}^{\infty}\\!\\!\\!d\omega_{1}\int_{0}^{\infty}\\!\\!\\!d\omega_{2}\int_{0}^{\infty}\\!\\!\\!d\omega_{3}\int_{0}^{\infty}\\!\\!\\!dt_{3}\int_{0}^{\infty}\\!\\!\\!dt_{2}\int_{0}^{\infty}\\!\\!\\!dt_{1}$
$\displaystyle\times
e^{i(\omega_{1}-\omega_{eg}+i\gamma_{eg})t_{1}+i(\omega_{1}+\omega_{2}-\omega_{fg}+i\gamma_{fg})t_{2}+i(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{fe^{\prime}}+i\gamma_{fe^{\prime}})t_{3}}$
$\displaystyle\times\sum_{j_{1},j_{2},j_{3},j_{4}}\text{tr}\left\\{\hat{a}^{\dagger}_{j_{3}}({\bf
r}_{mol},\omega_{3})\hat{a}^{\dagger}_{j_{4}}({\bf
r}_{mol},\omega_{1}+\omega_{2}-\omega_{3})\hat{a}_{j_{2}}(\omega_{2})\hat{a}_{{\bf
r}_{mol},j_{1}}({\bf r}_{mol},\omega_{1})\varrho_{f}\right\\}.$ (42)

Here, we use the mixed representation of the field operators,

$\displaystyle\hat{a}({\bf r},\omega)$
$\displaystyle=\frac{1}{2\pi}\int_{\mathds{R}^{2}}d^{2}q\;\hat{a}({\bf
q},\omega)\;e^{i({\bf q}\cdot{\bf\rho}+k_{z}(\omega)z)}.$ (43)

Evaluating the time integrals for the case of off-resonant intermediate
states, we replace

$\displaystyle\int_{0}^{\infty}\\!\\!dt_{1}\;e^{i(\omega_{1}-\omega_{eg}+i\gamma_{eg})t_{1}}$
$\displaystyle=\frac{i}{\omega_{1}-\omega_{eg}+i\gamma_{eg}}\simeq\frac{i}{\omega_{p}/2-\omega_{eg}},$
(44)

and likewise for the $t_{3}$-integration, where we find

$\displaystyle\int_{0}^{\infty}\\!\\!dt_{3}\;e^{i(\omega_{1}+\omega_{2}-\omega_{3}-\omega_{fe^{\prime}}+i\gamma_{fe^{\prime}})t_{3}}\simeq\frac{i}{\omega_{p}/2-\omega_{fe^{\prime}}}.$
(45)

This approximation assumes that the photon bandwidth $\sim\Omega_{m}$ is much
smaller than the detuning to the electronic resonances, as indicated in Fig.
1(b). We then obtain

$\displaystyle P_{f}({\bf r}_{mol},t\rightarrow\infty)$
$\displaystyle=\frac{1}{4}\left(\frac{\omega_{p}/2}{\hbar\epsilon_{0}n_{0}c}\right)^{2}\sum_{e,e^{\prime}}\frac{\mu^{\ast}_{ge}\mu^{\ast}_{ef}\mu_{ge^{\prime}}\mu_{e^{\prime}f}}{(\omega_{eg}-\omega_{p}/2)(\omega_{p}/2-\omega_{fe^{\prime}})}\int_{0}^{\infty}\\!\\!\\!d\omega_{1}\int_{0}^{\infty}\\!\\!\\!d\omega_{2}\int_{0}^{\infty}\\!\\!\\!d\omega_{3}\frac{1}{\pi}\frac{\gamma_{fg}}{\gamma_{fg}^{2}+(\omega_{fg}-\omega_{1}-\omega_{2})^{2}}$
$\displaystyle\times\sum_{j_{1},j_{2},j_{3},j_{4}}\text{tr}\left\\{\hat{a}^{\dagger}_{j_{3}}({\bf
r}_{mol},\omega_{3})\hat{a}^{\dagger}_{j_{4}}({\bf
r}_{mol},\omega_{1}+\omega_{2}-\omega_{3})\hat{a}_{j_{2}}(\omega_{2})\hat{a}_{{\bf
r}_{mol},j_{1}}({\bf r}_{mol},\omega_{1})\varrho_{f}\right\\}.$ (46)

Here we have taken the real part explicitly, using that the molecular response
encapsulated in the term $\sum_{e,e^{\prime}}\ldots$ is real.

####  III.2.2 The field correlation function

We next turn to the remaining field correlation function, where we can use the
input output relations (19) and (20) to obtain

$\displaystyle\sum_{j_{1},j_{2},j_{3},j_{4}}\text{tr}\left\\{\hat{a}^{\dagger}_{j_{3}}({\bf
r}_{mol},\omega_{3})\hat{a}^{\dagger}_{j_{4}}({\bf
r}_{mol},\omega_{4})\hat{a}_{j_{2}}(\omega_{2})\hat{a}_{{\bf
r}_{mol},j_{1}}({\bf r}_{mol},\omega_{1})\varrho_{f}\right\\}$
$\displaystyle=$ $\displaystyle f^{\ast}({\bf
r}_{mol},\omega_{1},\omega_{2})f({\bf r}_{mol},\omega_{3},\omega_{4})+g({\bf
r}_{mol},\omega_{1},\omega_{3})g({\bf r}_{mol},\omega_{2},\omega_{4})+g({\bf
r}_{mol},\omega_{1},\omega_{4})g({\bf r}_{mol},\omega_{2},\omega_{3}),$ (47)

where $\omega_{4}=\omega_{1}+\omega_{2}-\omega_{3}$ and we defined

$\displaystyle f({\bf r},\omega_{1},\omega_{2})$
$\displaystyle=e^{-i\omega_{p}t}\sum_{{\bf n}}\sinh(r_{{\bf
n}}\Gamma)\cosh(r_{{\bf n}}\Gamma)\left(h_{{\bf n},s}({\bf
r},\omega_{1})h_{{\bf n},i}({\bf r},\omega_{2})+h_{{\bf n},i}({\bf
r},\omega_{1})h_{{\bf n},s}({\bf r},\omega_{2})\right)$ (48)

and

$\displaystyle g({\bf r},\omega_{1},\omega_{2})$ $\displaystyle=\sum_{{\bf
n}}\sinh^{2}(r_{{\bf n}}\Gamma)\left(h_{{\bf n},s}({\bf r},\omega_{1})h_{{\bf
n},s}({\bf r},\omega_{2})+h_{{\bf n},i}({\bf r},\omega_{1})h_{{\bf n},i}({\bf
r},\omega_{2})\right).$ (49)

The Hermite functions of the signal mode (and similarly the idler modes) are
given by

$\displaystyle h_{{\bf n},s}({\bf r},\omega)$
$\displaystyle\equiv\frac{1}{2\pi}\int_{\mathds{R}^{2}}d^{2}q_{s}\;h_{{\bf
n}}({\bf q}_{s},\omega_{s})e^{i{\bf q}_{s}\cdot{\bf\rho}}$
$\displaystyle=i^{n_{x}+n_{y}}e^{i{\bf
q}_{s}^{(0)}{\bf\rho}}\;h_{n_{t}}\left(\frac{\omega-\omega_{p}/2}{\sqrt{\Omega_{m}\Omega_{p}}}\right)h_{n_{x}}\left(\sqrt{Q_{m}Q_{p}}x\right)h_{n_{y}}\left(\sqrt{Q_{m}Q_{p}}y\right).$
(50)

The function $f$ describes the absorption of a correlated photon pair, as it
contains only pairs of signal and idler functions, $h_{{\bf n},s}$ and
$h_{{\bf n},i}$, respectively, whereas the function $g$ corresponds to the sum
of the autocorrelation contributions of the two beams. We note that spatial
and spectral correlations, which are encoded in the Schmidt numbers $r_{{\bf
n}}$, are connected non-trivially in Eq. (III.2.2). The degrees of freedom can
be separated only in the weak downconversion limit $r_{{\bf n}}\Gamma\ll 1$,
when the light field is composed predominantly of temporally separate
entangled photon pairs. In the following, we will therefore consider three
limiting case: first, we consider the two-photon limit when the correlations
are in fact separable. Second, when the light field is confined to a single
spatial mode and only spectral correlations matter, and, third, when there is
only a single spectral mode, and spatial correlations affect the signal.

####  III.2.3 The full two-photon excitation probability

Inserting the field correlation function (III.2.2) into Eq. (III.2.1), we find
two contributions to the excitation probability,

$\displaystyle P_{f}({\bf r}_{mol},t\rightarrow\infty)$
$\displaystyle=P_{f}^{corr}+P_{f}^{unc},$ (51)

where $P_{f}^{corr}$ stems from the absorption of correlated pairs as
described by Eq. (48) and $P_{f}^{unc}$ from absorption according to Eq. (49).

##  IV The two-photon limit

We first analyze the two-photon limit of the output state (22). When $r_{{\bf
n}}\Gamma\ll 1$, we can approximate
$|\psi_{BSV}\rangle\simeq|0\rangle+\Gamma\ldots\hat{a}_{s}^{\dagger}\hat{a}_{i}^{\dagger}|0\rangle$.
The field correlation function (III.2.2) simplifies, using Eq. (II.3.1), to

$\displaystyle f_{low-gain}({\bf r}_{mol},\omega_{1},\omega_{2})$
$\displaystyle\simeq 2\times\Gamma
F_{mom}({\bf\rho}_{mol},\rho_{mol})F_{spec}(\omega_{1},\omega_{2}),$ (52)

where the Fourier transform of the momentum correlation function evaluated at
the position of the molecule ${\bf\rho}_{mol}$ reads

$\displaystyle F_{mom}({\bf\rho}_{mol},\rho_{mol})$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int_{\mathds{R}^{2}}d^{2}q_{s}\int_{\mathds{R}^{2}}d^{2}q_{i}\;F_{mom}({\bf
Q}_{s},{\bf Q}_{i})e^{i({\bf q}_{s}+{\bf q}_{i}){\bf\rho}_{mol}}$
$\displaystyle=\frac{1}{\pi}Q_{m}Q_{p}\exp\left[-Q_{p}^{2}{\bf\rho}_{mol}^{2}+i({\bf
q}_{s}^{(0)}+{\bf q}_{i}^{(0)})\cdot{\bf\rho}_{mol}\right].$ (53)

The factor 2 at the beginning of Eq. (52) stems from the fact that the
correlation functions (48) are symmetric with respect to the exchange
$\omega_{s}\leftrightarrow\omega_{i}$ and ${\bf Q}_{s}\leftrightarrow{\bf
Q}_{i}$ (the signal photon or the idler photon can be absorbed first,
respectively). These two excitation pathways contribute the same result and
yield the overall factor 2. Hence, the TPA rate (31) separates into a spatial
and frequency integrals, which can be treated separately,

$\displaystyle R^{TPA}_{low-gain}$ $\displaystyle=r_{0}p_{spat}p_{freq}.$ (54)

The prefactor $r_{0}$ can be read off from Eqs. (III.2.1) and (31),

$\displaystyle r_{0}$
$\displaystyle=m_{0}f_{rep}\Gamma^{2}\left(\frac{\omega_{p}/2}{\hbar\epsilon_{0}n_{0}c}\right)^{2}\sum_{e,e^{\prime}}\frac{\mu_{ge}^{\ast}\mu_{ef}^{\ast}\mu_{e^{\prime}f}\mu_{ge^{\prime}}}{(\omega_{eg}-\omega_{p}/2)(\omega_{p}/2-\omega_{fe^{\prime}})}.$
(55)

The spatial contribution $p_{spat}$, given by the modulus square of Eq. (IV)
integrated over the transverse position $\rho$, readily evaluates to

$\displaystyle r_{spat}$ $\displaystyle=\frac{\Delta
z}{(2\pi)^{2}}\int_{\mathds{R}^{2}}d^{2}\rho_{mol}\left(\frac{Q_{m}Q_{p}}{\pi}\right)^{2}e^{-2Q_{p}^{2}(x^{2}+y^{2})}$
$\displaystyle=\frac{\Delta zQ_{m}^{2}}{2\pi},$ (56)

and finally the frequency contribution is given by

$\displaystyle p_{freq}$
$\displaystyle=\int_{0}^{\infty}\\!\\!\\!d\omega_{1}\int_{0}^{\infty}\\!\\!\\!d\omega_{2}\int_{0}^{\infty}\\!\\!\\!d\omega_{3}\frac{1}{\pi}\frac{\gamma_{fg}}{\gamma_{fg}^{2}+(\omega_{fg}-\omega_{1}-\omega_{2})^{2}}F_{spec}(\omega_{3},\omega_{1}+\omega_{2}-\omega_{3})F_{spec}(\omega_{1},\omega_{2}).$
(57)

This last dimensionless quantity encodes the enhancement of the ETPA cross
section due to frequency entanglement. It was analyzed by Raymer et al. in
great detail [48, 43, 49, 50, 51], and depends crucially on the broadening
$\gamma_{fg}$ of the final molecular state. If the broadening is very small,
as proposed by Kang et al. [52] for ETPA in organic chromophores, one obtains
a very large enhancement. If it is as broad as suggested by Raymer and
coworkers for molecules in solution, the enhancement due to spectral
entanglement is almost completely eroded. With the bigaussian model for the
JSA (15), we can solve the above integrations analytically, when we extend the
integration boundaries to $-\infty$. In case of resonant excitation, where
$\omega_{p}=\omega_{fg}$, we obtain

$\displaystyle p_{freq}$
$\displaystyle=\frac{\Omega_{m}}{\Omega_{p}}\;\text{erfcx}\left(\frac{\gamma_{fg}}{\sqrt{2}\Omega_{p}}\right),$
(58)

where we find the scaled complementary error function erfcx. It quantifies how
efficiently the light field can activate the full two-photon response of the
molecule.

Figure 2:  The $\gamma_{fg}$-dependence of the ETPA cross section, Eq. (65),
is shown as a function of $\gamma_{fg}$ in units of the pump bandwidth
$\Omega_{p}$.

To obtain the photon flux scaling from Eq. (54), and hence extract the ETPA
cross section, we evaluate the photon flux density in the low-gain regime
$\phi_{low-gain}\simeq\frac{f_{rep}}{A_{p}}2\Gamma^{2}\sum_{{\bf
n}}r^{2}_{{\bf n}}=2\frac{f_{rep}}{A_{p}}\Gamma^{2}$. The second equality
follows from the geometric series (or, equivalently, from the normalization
(12) of the JSA). With Eq. (32), we then obtain the ETPA absorption rate

$\displaystyle\sigma_{e}$
$\displaystyle=\frac{\sigma^{(2)}}{A_{e}T_{e}}\text{eff}(\gamma_{fg}/\Omega_{p}).$
(59)

In the first term, we encounter the sought-after result (1) with the classical
TPA cross section [43]

$\displaystyle\sigma^{(2)}$
$\displaystyle=\left(\frac{\omega_{p}/2}{\hbar\epsilon_{0}nc}\right)^{2}\frac{1}{2\gamma_{fg}}\sum_{e,e^{\prime}}\frac{\mu_{ge}^{\ast}\mu_{ef}^{\ast}\mu_{e^{\prime}f}\mu_{ge^{\prime}}}{(\omega_{eg}-\omega_{p}/2)(\omega_{p}/2-\omega_{fe^{\prime}})},$
(60)

as well as the entanglement area

$\displaystyle A_{e}$ $\displaystyle\equiv\frac{(2\pi)^{2}}{Q_{m}^{2}}$ (61)

and the entanglement time

$\displaystyle T_{e}$ $\displaystyle\equiv\frac{2\pi}{\Omega_{m}}.$ (62)

In addition, the ’classical’ formula (1) is multiplied by the efficiency
function

$\displaystyle\text{eff}(x)$ $\displaystyle\equiv
x\;\text{erfcx}\left(\frac{x}{\sqrt{2}}\right).$ (63)

At large $x$, when $\gamma_{fg}\gg\Omega_{p}$, the function saturates to
eff$(x)\simeq 0.8$. This situation may be encountered in molecular TPA with cw
entangled photons, where you can simply use

$\displaystyle\sigma^{mol}_{e}$
$\displaystyle\simeq\frac{\sigma^{(2)}}{A_{e}T_{e}},$ (64)

in accordance with Fei et al. (1). At small $x$, in contrast, i.e. when
$\gamma_{fg}\ll\Omega_{p}$, we find eff$(x)\simeq x$. In this limit, the pump
bandwidth $\Omega_{p}$ is so large that it limits the efficient excitation of
the molecule. It is important to keep in mind, however, that the classical TPA
cross section is inversely proportional to the resonance broadening,
$\sigma^{(2)}\propto 1/\gamma_{fg}$. Therefore, the above discussion must be
understood at fixed $\gamma_{fg}$, and for varying pump bandwidth
$\Omega_{p}$. In particular, it does not imply that a large broadening that a
large broadening, which pushes eff$(x)$ towards saturation, would be
beneficial for ETPA. Instead, if we include the full dependence of
$\sigma_{e}$ on the broadening, we find that

$\displaystyle\sigma_{e}(\gamma_{fg})\propto\frac{1}{\gamma_{fg}}\text{eff}\left(\frac{\gamma_{fg}}{\sqrt{2}\Omega_{p}}\right).$
(65)

As shown in Fig. 2, this function only decreases with increasing broadening
$\gamma_{fg}$. When $\gamma_{fg}\ll\Omega_{p}$, we have eff$(x)\simeq x$, such
that the broadening dependence cancels and we obtain an ETPA cross section
with $\gamma_{fg}$ in Eq. (60) replaced by $\Omega_{p}$. When
$\gamma_{fg}\gg\Omega_{p}$, Eq. (63) becomes constant, such that
$\sigma_{e}\propto 1/\gamma_{fg}$.

##  V Single spatial mode limit

When we have $Q_{m}=Q_{p}$, signal and idler beams each propagate in one
respective spatial mode, but may still contain strong spectral quantum
correlations. In this case, the momentum correlation function factorizes
naturally into

$\displaystyle F_{mom}^{sep}({\bf q}_{s},{\bf q}_{i})$
$\displaystyle=\frac{1}{\pi Q_{p}^{2}}\exp\left[-\frac{({\bf Q}_{s}+{\bf
Q}_{i})^{2}}{4Q_{p}^{2}}-\frac{({\bf Q}_{s}-{\bf
Q}_{i})^{2}}{4Q_{p}^{2}}\right]$ (66) $\displaystyle=\frac{1}{\pi
Q_{p}^{2}}\exp\left[-\frac{{\bf Q}_{s}^{2}+{\bf
Q}_{i}^{2}}{2Q_{p}^{2}}\right].$ (67)

Consequently, we can drop the $n_{x}/n_{y}$-summations in the field
correlation functions (48) and (49), and replace the corresponding Hermite
functions by

$\displaystyle\tilde{h}({\bf r})$
$\displaystyle\equiv\frac{Q_{p}}{\sqrt{\pi}}e^{-i{\bf
q}^{(0)}{\bf\rho}-Q_{p}^{2}{\bf\rho}^{2}/2}.$ (68)

The field correlation functions are thus given by

$\displaystyle f_{1-mode}({\bf r},\omega_{1},\omega_{2})$
$\displaystyle=2\sum_{n_{t}}\sinh(r_{n_{t}}\Gamma)\cosh(r_{n_{t}}\Gamma)\tilde{h}^{2}({\bf
r})(-1)^{n_{t}}h_{n_{t}}\left(\frac{\omega_{p}/2+\omega_{1}}{\sqrt{\Omega_{m}\Omega_{p}}}\right)h_{n_{t}}\left(\frac{\omega_{p}/2+\omega_{2}}{\sqrt{\Omega_{m}\Omega_{p}}}\right),$
(69)

and

$\displaystyle g_{1-mode}({\bf r},\omega_{1},\omega_{2})$
$\displaystyle=2\sum_{n_{t}}\sinh^{2}(r_{n_{t}}\Gamma)|\tilde{h}({\bf
r})|^{2}h_{n_{t}}\left(\frac{\omega_{p}/2+\omega_{1}}{\sqrt{\Omega_{m}\Omega_{p}}}\right)h_{n_{t}}\left(\frac{\omega_{p}/2+\omega_{2}}{\sqrt{\Omega_{m}\Omega_{p}}}\right).$
(70)

The photon flux density becomes

$\displaystyle\phi$
$\displaystyle=2\frac{f_{rep}}{A_{p}}\sum_{n_{t}}\sinh^{2}(r_{n_{t}}\Gamma).$
(71)

###  V.1 Correlated contribution

We then find from Eq. (III.2.1) the correlated contribution of the final state
population

$\displaystyle P_{f}^{corr}({\bf r}_{mol})$
$\displaystyle=\left(\frac{\omega_{p}/2}{\hbar\epsilon_{0}n_{0}c}\right)^{2}\sum_{e,e^{\prime}}\frac{\mu^{\ast}_{ge}\mu^{\ast}_{ef}\mu_{ge^{\prime}}\mu_{e^{\prime}f}}{(\omega_{eg}-\omega_{0})(\omega_{0}-\omega_{fe^{\prime}})}\tilde{h}^{4}({\bf
r}_{mol})\sum_{n_{t},n_{t}^{\prime}}(-1)^{n_{t}+n^{\prime}_{t}}\sinh(r_{n_{t}}\Gamma)\cosh(r_{n_{t}}\Gamma)\sinh(r_{n_{t}^{\prime}}\Gamma)\cosh(r_{n_{t}^{\prime}}\Gamma)$
$\displaystyle\times\int_{0}^{\infty}\\!\\!\\!d\omega_{sum}\int_{0}^{\infty}\\!\\!\\!d\omega_{2}\int_{0}^{\infty}\\!\\!\\!d\omega_{3}\frac{1}{\pi}\frac{\gamma_{fg}}{\gamma_{fg}^{2}+(\omega_{fg}-\omega_{sum})^{2}}h_{n_{t}}(\omega_{sum}-\omega_{2})h_{n_{t}}(\omega_{2})h_{n^{\prime}_{t}}(\omega_{3})h_{n^{\prime}_{t}}(\omega_{sum}-\omega_{3}).$
(72)

When we extend the lower integration boundaries to $-\infty$, and use the
parity of the Hermite functions,
$h_{n}(\omega_{sum}-\omega)=(-1)^{n}h_{n}(\omega-\omega_{sum})$, we can use
the integral identity [84]

$\displaystyle\int_{-\infty}^{\infty}\\!\\!\\!\\!d\omega\;h_{m}(\omega)h_{n}(\omega_{sum}-\omega)$
$\displaystyle=\sqrt{\frac{n!}{m!}}\delta\omega^{m-n}L^{(m-n)}_{n}\left(\delta\omega^{2}\right)e^{-\delta\omega^{2}/2},\quad\text{for
}m\geq n,$ (73)

where $L^{(m-n)}_{n}$ is the n-th generalized Laguerre polynomial and
$\delta\omega=(\omega_{sum}-\omega_{p})/\sqrt{2\Omega_{m}\Omega_{p}}$, and
$w=(\omega_{sum}-\omega_{p})/\sqrt{2\Omega_{m}\Omega_{p}}$. Eq. (V.1)
evaluates to

$\displaystyle P_{f}^{corr}({\bf r}_{mol})$
$\displaystyle=\left(\frac{\omega_{p}/2}{\hbar\epsilon_{0}n_{0}c}\right)^{2}\sum_{e,e^{\prime}}\frac{\mu^{\ast}_{ge}\mu^{\ast}_{ef}\mu_{ge^{\prime}}\mu_{e^{\prime}f}}{(\omega_{eg}-\omega_{0})(\omega_{0}-\omega_{fe^{\prime}})}\tilde{h}^{4}({\bf
r}_{mol})\sum_{n_{t},n_{t}^{\prime}}\sinh(r_{n_{t}}\Gamma)\cosh(r_{n_{t}}\Gamma)\sinh(r_{n_{t}^{\prime}}\Gamma)\cosh(r_{n_{t}^{\prime}}\Gamma)$
$\displaystyle\times\int_{w_{0}}^{\infty}\\!\\!dw\frac{1}{\pi}\frac{w_{fg}}{w_{fg}^{2}+(\frac{\omega_{fg}-\omega_{p}}{\sqrt{2\Omega_{m}\Omega_{p}}}+w)^{2}}L_{n_{t}}\left(w^{2}\right)L_{n_{t}^{\prime}}\left(w^{2}\right)\exp\left[-w^{2}\right]$
(74)

where $w_{fg}=\gamma_{fg}/\sqrt{2\Omega_{m}\Omega_{p}}$ and
$w_{0}=-\omega_{p}/\sqrt{2\Omega_{p}\Omega_{m}}$. Sending
$w_{0}\rightarrow-\infty$, the remaining $w$-integration can be carried out
analytically or numerically by Mathematica with high efficiency (albeit the
integration becomes unstable for extremely highly entangled states). Still, it
is instructive to consider one limiting case first.

##### Narrow resonance

In the limit of a very narrow resonance, when $w_{fg}\ll 1$, we can replace

$\displaystyle\frac{1}{\pi}\frac{w_{fg}}{w_{fg}^{2}+(\frac{\omega_{fg}-\omega_{p}}{\sqrt{2\Omega_{m}\Omega_{p}}}+w)^{2}}$
$\displaystyle\simeq\delta\left(w+\frac{\omega_{fg}-\omega_{p}}{\sqrt{2\Omega_{m}\Omega_{p}}}\right).$
(75)

We then obtain on resonance ($\omega_{fg}=\omega_{p}$) from Eq. (V.1)

$\displaystyle P_{f}^{corr}({\bf r}_{mol})$
$\displaystyle=\left(\frac{\omega_{p}/2}{\hbar\epsilon_{0}n_{0}c}\right)^{2}\sum_{e,e^{\prime}}\frac{\mu^{\ast}_{ge}\mu^{\ast}_{ef}\mu_{ge^{\prime}}\mu_{e^{\prime}f}}{(\omega_{eg}-\omega_{0})(\omega_{0}-\omega_{fe^{\prime}})}\tilde{h}^{4}({\bf
r}_{mol})$
$\displaystyle\times\left[\sum_{n_{t}}\sinh(r_{n_{t}}\Gamma)\cosh(r_{n_{t}}\Gamma)\right]^{2}$
(76)

The expression can be interpreted in two limiting cases: At a low photon flux,
when $\Gamma\ll 1$, we can approximate, using the definition of the Schmidt
eigenvalues in Eq. (II.3.1),

$\displaystyle\sum_{n_{t}}\sinh(r_{n_{t}}\Gamma)\cosh(r_{n_{t}}\Gamma)\bigg{|}_{\Gamma\ll
1}$
$\displaystyle\simeq\sum_{n_{t}}r_{n_{t}}\Gamma=\Gamma\sqrt{\frac{\Omega_{m}}{\Omega_{p}}}\propto\langle\hat{n}\rangle^{1/2},$
(77)

The excitation probability thus becomes proportional to the photon flux. With
Eq. (71), we obtain the cross section

$\displaystyle R^{TPA,corr}_{low-gain,single-mode}$
$\displaystyle=N_{mol}\frac{\sigma^{(2)}}{A_{p}T_{e}}\frac{\gamma_{fg}}{\Omega_{p}}\phi,$
(78)

where $T_{e}=2\pi/\Omega_{m}$ is the entanglement time (62), and $A_{p}$ the
transverse beam area. This coincides with Eq. (59) of the previous section in
the limit of a narrow resonance, with the entanglement area $A_{e}$ (61)
replaced by the transverse beam area $A_{p}$. At high gain $\Gamma\gg 1$, we
approximate

$\displaystyle\sum_{n_{t}}\sinh(r_{n_{t}}\Gamma)\cosh(r_{n_{t}}\Gamma)\bigg{|}_{\Gamma\gg
1}$
$\displaystyle\simeq\frac{1}{4}\sum_{n_{t}}e^{2r_{n_{t}}\Gamma}\simeq\langle\hat{n}\rangle/2.$
(79)

This means, the excitation probability is proportional to the squared photon
flux, and we obtain the TPA rate at high-gain

$\displaystyle R^{TPA,corr}_{high-gain,single-mode}$
$\displaystyle=N_{mol}\sigma^{(2)}\frac{\gamma_{fg}}{2\Omega_{p}}\frac{1}{T_{pulse}f_{rep}}\times\phi^{2}.$
(80)

Here we encounter the product of the laser pulse duration
$T_{pulse}=2\pi/\Omega_{p}$ and the laser repetition rate $f_{rep}$. This
factor is also encountered in TPA with ultrafast lasers, and enhances the
pulsed TPA absorption cross section relative to the sample’s cw cross section
when $T_{pulse}f_{rep}\ll 1$.

###  V.2 Uncorrelated contribution

Just like in the above calculation of the correlated contribution, we find

$\displaystyle P_{f}^{unc}({\bf r}_{mol})$
$\displaystyle=2\left(\frac{\omega_{p}/2}{\hbar\epsilon_{0}n_{0}c}\right)^{2}\sum_{e,e^{\prime}}\frac{\mu^{\ast}_{ge}\mu^{\ast}_{ef}\mu_{ge^{\prime}}\mu_{e^{\prime}f}}{(\omega_{eg}-\omega_{0})(\omega_{0}-\omega_{fe^{\prime}})}\tilde{h}^{4}({\bf
r}_{mol})\sum_{n_{t},n_{t}^{\prime}}\sinh^{2}(r_{n_{t}}\Gamma)\sinh^{2}(r_{n_{t}^{\prime}}\Gamma)$
$\displaystyle\times\int
dw\frac{1}{\pi}\frac{w_{fg}}{w_{fg}^{2}+(\frac{\omega_{fg}-\omega_{p}}{\sqrt{2\Omega_{m}\Omega_{p}}}+w)^{2}}\left(\sqrt{\frac{n!}{m!}}w^{m-n}L^{(m-n)}_{n}\left(w^{2}\right)e^{-w^{2}/2}\right)^{2},$
(81)

where $m=\text{max}(n_{t},n_{t}^{\prime})$, and
$n=\text{min}(n_{t},n_{t}^{\prime})$.

Figure 3:  Single spatial mode  The ETPA resonance according to Eqs. (V.1) and
(81) is plotted (solid blue line) at different mean photon numbers, as
indicated, and (a) $\gamma_{fg}=0.1\Omega_{p}$ and $\Omega_{m}=1.5\Omega_{p}$,
(b) $\gamma_{fg}=\Omega_{p}$ and $\Omega_{m}=1.5\Omega_{p}$, (c)
$\gamma_{fg}=10\Omega_{p}$ and $\Omega_{m}=1.5\Omega_{p}$, (d)
$\gamma_{fg}=0.1\Omega_{p}$ and $\Omega_{m}=10\Omega_{p}$, (e)
$\gamma_{fg}=\Omega_{p}$ and $\Omega_{m}=10\Omega_{p}$, and (f)
$\gamma_{fg}=10\Omega_{p}$ and $\Omega_{m}=10\Omega_{p}$. The gray dashed line
indicates the uncorrelated contribution (81) to the full ETPA signal. Each
plot is normalized to its maximal value (i.e. on resonance).

##### Narrow resonance

We again replace the Lorentzian resonance by a delta-function (75). On
resonance, when $\omega_{fg}=\omega_{p}$, Eq. (81) then simplifies to

$\displaystyle P_{f}^{unc}({\bf r}_{mol})$
$\displaystyle=\left(\frac{\omega_{p}/2}{\hbar\epsilon_{0}n_{0}c}\right)^{2}\sum_{e,e^{\prime}}\frac{\mu^{\ast}_{ge}\mu^{\ast}_{ef}\mu_{ge^{\prime}}\mu_{e^{\prime}f}}{(\omega_{eg}-\omega_{0})(\omega_{0}-\omega_{fe^{\prime}})}\tilde{h}^{4}({\bf
r}_{mol})$ $\displaystyle\times\sum_{n_{t}}\sinh^{4}(r_{n_{t}}\Gamma).$ (82)

To extract the TPA cross section from this expression, we wish to express the
corresponding TPA signal as $\delta_{r}\phi^{2}$, where
$\phi^{2}\propto(\sum_{n}\sinh^{2}(r_{n}\Gamma))^{2}$, see Eq. (71). We thus
need to analyze the term $\sum_{n}\sinh^{4}(r_{n}\Gamma)$. Given the Schmidt
number $K$ of the PDC light, we expect that in the limit of many Schmidt
modes, we have $\sim K$ terms which contribute to this expression. In
contrast, the squared mean photon number should contain $\sim K^{2}$ terms.
Thus, in a highly entangled light field, when the parameter $\zeta_{t}$ in Eq.
(17) is close to one, we expect that

$\displaystyle\sum_{n_{t}}\sinh^{4}(r_{n_{t}}\Gamma)\sim\frac{1}{K}\left(\sum_{n_{t}}\sinh^{2}(r_{n_{t}}\Gamma)\right)^{2}.$
(83)

This should be true at least for sufficiently small photon numbers, when many
Schmidt modes contribute to the signal. In this regime, we obtain

$\displaystyle R^{TPA,unc}_{high-gain,single-mode}$
$\displaystyle=N_{mol}\sigma^{(2)}\frac{\gamma_{fg}}{\Omega_{m}}\frac{1}{T_{pulse}f_{rep}}\times\phi^{2}.$
(84)

Hence, in contrast to the correlated contribution (80), where the ratio
$\gamma_{fg}/\Omega_{p}$ appears, here it is the ratio
$\gamma_{fg}/\Omega_{m}$, i.e. the ratio between the molecular resonance and
the bandwidth $\Omega_{m}$ of the individual photons. As
$\Omega_{m}\gg\Omega_{p}$ in a highly entangled beam, this reduces the
uncorrelated contribution to the cross section considerably.

This behaviour again changes in a very high gain regime, when
$e^{r_{0}\Gamma}\gg e^{r_{1}\Gamma}$. In this regime, the largest Schmidt mode
becomes dominant, and the ETPA signal approaches that of a single-mode
squeezed state in this mode. Then the spectral correlations lose their
importance, and the uncorrelated TPA rate becomes exactly twice the correlated
contribution (80),

$\displaystyle R^{TPA,unc}_{veryhigh-gain,single-mode}$
$\displaystyle=2R^{TPA,corr}_{high-gain,single-mode}.$ (85)

###  V.3 Results

In the final part of this Section, we analyze the results obtained thus far,
and discuss physical implications.

Figure 4:  Single spatial mode  (a) ETPA Signal strength (V.1) and (V.2) vs.
the mean photon number with bandwidths $\Omega_{m}=50\Omega_{p}$ (blue, solid
line), $10\Omega_{p}$ (gray, dashed), and $1.5\Omega_{p}$ (black, dot-dashed).
Here, a narrow spectral resonance is assumed, see Eq. (75). (b) Ratio between
correlated and uncorrelated contributions, Eq. (86), for the same parameters
an in (a). (c) Ratio (86) as a function of the bandwidth at fixed mean photon
number $\langle\hat{N}\rangle=0.1$ (blue, solid), $1$ (gray dashed), $10$
(black, dot-dashed), and $100$ (red, dotted).

Spectral Resonance– In Fig. 3, we solve Eqs. (V.1) and (81) numerically to
show the change of the ETPA resonance as function of the broadening
$\gamma_{fg}$, the bandwidth $\Omega_{m}$ (or, equivalently, the amount of
entanglement), and the mean photon number $\langle\hat{N}\rangle$ per
entangled pulse. The simulations show the expected behaviour, where at small
photon numbers $\langle\hat{n}\rangle=0.1$, the signal is dominated by the
correlated contribution (V.1). As the mean photon number increases, the
incoherent part (81) becomes more dominant. However, the crossover photon
flux, where the uncorrelated events become more likely than the correlated
ones depends strongly on the broadening of the final state resonance and the
amount of entanglement (the Schmidt number) of the light field. For instance,
in Fig. 3(c), where the resonance is very broad and the light field is only
very weakly entangled, the uncorrelated events account for $\sim 50$ % of the
signal at $\langle\hat{n}\rangle=1$. Conversely, in Fig. 3(d), we simulate a
narrow resonance and strong entanglement of the incident light. Even at
$\langle\hat{n}\rangle=100$, the correlated contribution still accounts for
about $60$ % of the signal.

Intensity dependence of narrow molecular resonance– The transition of a
nonlinear optical signal from the separate photon pairs to a multi-photon
state was first analyzed in Ref. 85. In the following, we explore this
behaviour in the limit (75), were $\gamma_{fg}\ll\Omega_{p}$. The increase of
the full ETPA signal on resonance is shown in Fig. 4(a) for three different
bandwidths $\Omega_{m}$ ranging from $1.5\Omega_{p}$ (very weak entanglement)
to $50\Omega_{p}$ (very strong entanglement). At photon numbers below
$\langle\hat{N}\rangle\lesssim 100$, the strong quantum correlations in the
latter case afford a large advantage over the former one. Moreover, whereas
the linear signal scaling gives way to a quadratic scaling at
$\langle\hat{N}\rangle\simeq 1$ in the case of weak entanglement, it remains
linear up to $\langle\hat{N}\rangle\simeq 100$ in the strongly entangled case.
The crossover takes place when the mean photon number per Schmidt mode $n_{t}$
becomes of order one, i.e. when
$\langle\hat{A}^{\dagger}_{n_{t}}\hat{A}_{n_{t}}\rangle\sim 1$. Using the
Schmidt number $K$ as the effective number of modes, we expect this to be the
case when $\langle\hat{N}\rangle/K\sim 1$. In the case of large bandwidths,
$\Omega_{m}=50\Omega_{p}$, we have, using Eq. (29),
$K\simeq\Omega_{m}/(2\Omega_{p})$ and thus expect the crossover to take place
when $\langle\hat{N}\rangle\simeq 25$. This estimation is consistent with our
observations, given that the Schmidt number only gives an estimate for the
effective dimensionality of the entangled state.

However, this extended linear regime is not necessarily an advantage in terms
of absolute signal strength, as Fig. 4(a) also reveals: when we increase the
mean photon number even further, $\langle\hat{N}\rangle>100$, the weakly
entangled signal with $\Omega_{m}=1.5\Omega_{p}$ becomes stronger than the
other two. This is entirely due to the larger strength of the uncorrelated
contribution on resonance. Its bandwidth is narrower, and thus gives rise to a
narrower resonance, too [compare Fig. 3(a)]. On balance, this can be enough to
overpower the much larger correlated contributions of highly entangled states.
It appears that at these very large photon numbers, a few-mode or single-mode
entangled state is in fact preferable over a (highly entangled) multimode
state - at least insofar as the total signal strength is concerned. But in
many proposed applications of entangled photons for spectroscopy, one is
instead interested, e.g., in exploiting the quantum correlations of the
entangled pulses to control the excited states in a multilevel system [86, 87,
88, 89, 90, 91, 92, 93, 94, 95], and this control is enabled by the absorption
of pairs of correlated photons. The present results thus demonstrate that such
a control is feasible not only in a separate pair limit, where the overall
signal strength is very low. It can also be achieved with pulses containing
much larger photon numbers. However, these states may not be optimal for
inducing the largest possible ETPA signal, and one has to accept the
uncorrelated background signal.

We next analyze in more detail the relative size of correlated and
uncorrelated contributions,

$\displaystyle r_{rel}\equiv\frac{P_{f}^{corr}}{P_{f}^{unc}},$ (86)

where the two terms are calculated with Eqs. (V.1) and (V.2), respectively. It
is shown in Fig. 4(b) as a function of the mean photon number. Starting from
very large values when $\langle\hat{N}\rangle<1$, the ratio decreases steadily
with increasing photon number, until it saturates to $r_{rel}\rightarrow 1/2$,
which is the expected limiting behaviour according to Eq. (85). This limit can
be reached already at $\langle\hat{N}\rangle\simeq 10$ for a weakly entangled
state, and can be $\langle\hat{N}\rangle>10^{4}$ for strongly entangled ones.
Conversely, at fixed photon number, the ratio increases quadratically with the
bandwidth $\Omega_{m}$, as is shown in Fig. 4(c).

Crossover from narrow to broad molecular resonance–

Figure 5:  Single spatial mode  ETPA signal according to Eqs. (V.1) and (81)
is plotted vs. the broadening $\gamma_{fg}$ at fixed photon numbers (a)
$\langle\hat{N}\rangle=0.1$, (b) $\langle\hat{N}\rangle=1$, (c)
$\langle\hat{N}\rangle=10$, and (d) $\langle\hat{N}\rangle=100$, and with
bandwidths $\Omega_{m}=50\Omega_{p}$ (blue, solid line), $10\Omega_{p}$ (gray,
dashed), and $1.5\Omega_{p}$ (black, dot-dashed). The small insets show the
same plots, where each plot is normalized to its limiting value at
$\gamma_{fg}\rightarrow 0$.

So far, our discussion was limited to a very narrow molecular resonance. The
dependence of the ETPA signal on a finite resonance is investigated in Fig. 5,
where we solve Eqs. (V.1) and (81) numerically to obtain $R^{TPA}$. For a very
broad range of mean photon numbers $\langle\hat{N}\rangle$ and bandwidths
$\Omega_{m}$, we find that the signals saturate to a constant value when
$\gamma_{fg}\lesssim 0.1\Omega_{p}$. We have checked that this limit coincides
with the delta-function limits, Eqs. (V.1) and (V.2), which provide excellent
approximations in this parameter regime. When the broadening becomes similar
to or larger than the pump bandwidth $\Omega_{p}$, the signal starts to decay
with increasing $\gamma_{fg}$. If the mean photon number is small and the
signal is dominated by the correlated contribution to $P_{f}$, this decease is
independent of the photonic bandwidth $\Omega_{m}$. This can be seen in the
insets of Figs. 5(a) and (b), where the plots are normalized to their maximal
value, and the three plots corresponding to different bandwidths fall on top
of one another. In all cases, we have $R^{TPA}\propto 1/\gamma_{fg}$. At
larger photon numbers, when the incoherent contribution to $P_{f}$ becomes
important as well, this universal scaling behaviour is lost. As observed
already in Fig. 4(a), a state with narrower bandwidth (black, dot-dashed line)
generates a larger ETPA signal than a strongly entangled state (blue solid
line) when the resonance broadening is small. But when the broadening becomes
larger, this changes once more, and e.g. in Fig. 5(d), the strongly entangled
state generates the largest ETPA signal, when $\gamma_{fg}\gtrsim
100\Omega_{p}$ \- albeit a much smaller absolute signal than in the case of a
narrow resonance.

Figure 6:  Single spectral mode  (a) ETPA Signal strength according to Eqs.
(VI) and (VI) is shown vs. one spatial direction $x$, with $y=0$. The gray
shaded area indicates the uncorrelated contribution (VI) to the full ETPA
signal. (b) the same as (a), but with stronger spatial correlations with
$Q_{m}=10Q_{p}$.

##  VI Single spectral mode

We next turn to the analysis of the third limiting case, which we discuss in
this manuscript, where only a single spectral mode must be considered. Such a
state could be prepared, for instance, by spectral filtering of the PDC light
(at the cost of reducing the photon flux). This situation is easier to deal
with than that the previous Section V. It basically amounts to analysing the
probability of localising two photons simultaneously at the position of the
absorber. A molecular response function is not involved (or rather, it only
gives the same contribution to correlated and uncorrelated terms in the final
signal). In the following, we use the following single spectral mode function

$\displaystyle\tilde{h}(\omega)$
$\displaystyle=(\pi\Omega_{p}^{2})^{-1/4}\exp\left[-\frac{(\omega-\omega_{p}/2)}{2\Omega_{p}^{2}}\right]$
(87)

which we obtain from Eq. (15) by setting $\Omega_{m}=\Omega_{p}$. Adapting Eq.
(III.2.1) for this situation, we obtain

$\displaystyle
P_{f}^{corr}=\sigma^{(2)}2\gamma_{fg}\text{spec}(\omega_{p}-\omega_{fg},\Omega_{p},\gamma_{fg})$
$\displaystyle\times\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{n_{x},n_{y},n^{\prime}_{x},n^{\prime}_{y}}\\!\\!\\!\\!\\!\\!\sinh(r_{n_{x},n_{y}}\Gamma)\cosh(r_{n_{x},n_{y}}\Gamma)\sinh(r_{n^{\prime}_{x},n^{\prime}_{y}}\Gamma)\cosh(r_{n^{\prime}_{x},n^{\prime}_{y}}\Gamma)$
$\displaystyle\times
h_{n_{x}}^{2}(\sqrt{Q_{m}Q_{p}}x)h_{n^{\prime}_{x}}^{2}(\sqrt{Q_{m}Q_{p}}x)h_{n_{y}}^{2}(\sqrt{Q_{m}Q_{p}}y)h_{n^{\prime}_{y}}^{2}(\sqrt{Q_{m}Q_{p}}y).$
(88)

The uncorrelated contribution likewise simplifies to

$\displaystyle
P_{f}^{unc}=\sigma^{(2)}4\gamma_{fg}\text{spec}(\omega_{p}-\omega_{fg},\Omega_{p},\gamma_{fg})$
$\displaystyle\times\sum_{n_{x},n_{y}}\sum_{n^{\prime}_{x},n^{\prime}_{y}}(-1)^{n_{x}+n_{y}+n^{\prime}_{x}+n^{\prime}_{y}}\sinh^{2}(r_{n_{x},n_{y}}\Gamma)\sinh^{2}(r_{n^{\prime}_{x},n^{\prime}_{y}}\Gamma)$
$\displaystyle\times
h_{n_{x}}^{2}(\sqrt{Q_{m}Q_{p}}x)h_{n^{\prime}_{x}}^{2}(\sqrt{Q_{m}Q_{p}}x)h_{n_{y}}^{2}(\sqrt{Q_{m}Q_{p}}y)h_{n^{\prime}_{y}}^{2}(\sqrt{Q_{m}Q_{p}}y).$
(89)

In these expressions, we have combined the spectral integrals into the
function

$\displaystyle\text{spec}(\omega_{p}-\omega_{fg},\Omega_{p},\gamma_{fg})$
$\displaystyle=\int d\omega_{1}\int d\omega_{2}\int
d\omega_{sum}\frac{1}{\pi}\frac{\gamma_{fg}}{\gamma_{fg}^{2}+(\omega_{fg}-\omega_{sum})^{2}}$
$\displaystyle\quad\times\tilde{h}(\omega_{sum}-\omega_{1})\tilde{h}(\omega_{1})\tilde{h}(\omega_{sum}-\omega_{2})\tilde{h}(\omega_{2})$
$\displaystyle=\Re\tilde{w}(\omega_{p}-\omega_{fg}+i\gamma_{fg}),$ (90)

which describes the overlap between the spectral mode function and the
molecular response. In the present case, where we assumed Gaussian spectral
modes, it evaluate to the real part of the Faddeeva function $\tilde{w}$, as
the response is given by the convolution of a Gaussian and a Lorentzian, i.e.
by a Voigt profile. The only difference between the two terms in Eqs. (VI) and
(VI) - apart from a factor 2 - is the scaling behaviour
$\sim\sinh^{2}\cosh^{2}$ vs $\sim\sinh^{4}$ and the factor
$(-1)^{n_{x}+n_{y}+n^{\prime}_{x}+n^{\prime}_{y}}$, which reduces the
uncorrelated contribution, respectively. This contrasts with the previous
section, where correlated and uncorrelated contributions were convoluted
differently with the molecular response function, even in the limit of a a
very narrow resonance (75).

The spatial dependence of the of the signal according to Eqs. (VI) and (VI) is
shown in Fig. 6 for different mean photon numbers and a weakly entangled state
with $Q_{m}=1.5Q_{p}$, and a entangled state with $Q_{m}=10Q_{p}$. The
uncorrelated contribution is shown as a dashed, gray line. In the strongly
entangled case, the latter is much more narrowly peaked, signifying that ETPA
is possible only in a very narrow area with the largest photon density.

Figure 7:  Single spectral mode  (a) ETPA signal strength according to Eqs.
(31), (VI) and (VI) is shown vs. the mean photon number. (b) The ratio
$r_{rel}$, Eq. (86), extracted from the signals in (a).

In Fig. 7, we basically repeat the analysis of the previous section, and
analyze the scaling behaviour of the ETPA signal (31), which we obtain from
integrating Eqs. (VI) and (VI) over space. As before, we show both weakly
entangled states (with $Q_{m}=1.5Q_{p}$) and very strongly entangled ones
($Q_{m}=50Q_{p}$). Here we find that, in contrast to Fig. 4, the strongly
correlated signals always remain larger than weakly entangled ones. The reason
is straightforward: the spectral correlations are convoluted with the
molecular response, which gives rise to the intricate interplay we had
explored in the previous section. The spatial correlations, on the other hand,
manifest themselves a distinct scaling behaviour of Eqs. (VI) and (VI), which
are otherwise identical. Consequently, the advantage due to spatial quantum
correlations become smaller in relative terms as the mean photon number
increases, but it never goes away and may persist even to photonic states with
macroscopic photon numbers [5, 96].

##  VII Conclusions

In this manuscript, we have investigated entangled two-photon absorption cross
sections of pulsed entangled beams from the low- to the high-gain regime of
parametric downconversion. We have first presented a derivation of Fei’s
seminal formula for the ETPA cross section (1) starting from a microscopic
model of light. Eq. (1) relies on a factorization of the entangled photon
wavefunction into spatial and spectral components. When these degrees of
freedom are quantum correlated [73, 77], such that they cannot be factorized,
the above formula does not necessarily apply and new effects relying on the
joint spatio-temporal entanglement might be expected.

Our analysis shows that the formula (1) proposed by Fei et al. applies only in
the limit of a very narrow (compared to the pump laser bandwidth) molecular
resonance. A more realistic treatment, which accounts for a finite broadening,
gives rise to the result (59). The finite resonance broadening leads to a
reduction of the cross section when the experiment is conducted with
continuous-wave entangled photon pairs. This finding could explain some of the
unsuccessful attempts at detecting ETPA with continuous-wave entangled photon
pairs. Moreover, we have clarified the microscopic origin of the entanglement
time and the entanglement area, which appear in Eq. (1). These quantities are
directly connected to the amount of entanglement in the photonic state only in
certain limits (i.e. when $\Omega_{m}\gg\Omega_{p}$ and $Q_{m}\gg Q_{p}$).
Still, even in situations where this is not the case ( for instance, when both
$\Omega_{m}$ and $\Omega_{p}$ are large), small entanglement times and the
entanglement areas are always beneficial in enhancing the ETPA cross section.
In such a situation, we may still have quantum enhancement (due to the linear
scaling brought about by the two-photon nature of the state), but it is not
due to quantum entanglement. As both the entanglement time and the
entanglement area are controlled by properties of the nonlinear crystal where
the entangled photons are generated, these findings should be reflected in the
design of novel quantum light sources for spectroscopy [97].

We have further investigated the crossover from linear to quadratic regime of
ETPA. We have seen that ETPA can benefit from the same enhancement as TPA with
ultrafast lasers. We have further shown that anticipated crossover behaviour
according to Eq. (2) is incorrect, as the probability for the absorption of
correlated photon pairs also increases quadratically with the photon number at
sufficiently large photon fluxes. Thus, the cross section in the quadratic
regime is composed of two contributions, and quantum enhancement effects can
persist to very large photon numbers. They are lost only when the multimode
nature of the light field becomes negligible. We point out, however, that in
this very high-gain regime, corrections to the simple PDC model we employed in
this manuscript may become substantial and will require a numerically more
involved treatment of the light source [98]. In this discussion, we have
considered degenerate PDC. This is in fact the worst case scenario, where the
uncorrelated contributions can als drive TPA resonantly. In non-degenerate
PDC, this background would be suppressed.

In this manuscript, we have only treated the impact of spatial and spectral
correlations separately. As one can see from the basic equations we derived
here, their interplay may become highly nontrivial and give rise to
interesting new effects in the high gain regime, which could be very
appealing, e.g., for applications in nonlinear quantum imaging [99, 100, 101]
or spatially resolved spectroscopy. This will be an interesting direction for
future research. Overall, our work highlights the interesting effects one can
expect from nonlinear light-matter interactions of high-gain PDC light [102,
5, 96].

###### Acknowledgements.

I would like to thank Dr. Shahram Panahiyan for his helpful feedback on this
manuscript, and acknowledge support from the Cluster of Excellence ’Advanced
Imaging of Matter’ of the Deutsche Forschungsgemeinschaft (DFG) - EXC 2056 -
project ID 390715994.

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