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 that 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 for 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 analytically derive ETPA cross sections and the intensity crossover regime. We investigate the interplay between spectral and spatial degrees of freedom and how linewidth broadening of the sample impacts the experimentally achievable enhancement.

## I. INTRODUCTION

It was already recognized in the early days of quantum optics that the quantum statistics of a light field are intimately connected to the nonlinear optical response it generates in a material.^{1,2} However, 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.^{3–5} These experiments have since triggered a rapidly growing interest in the use of quantum light in nonlinear spectroscopy.^{6–10} Current investigations range from the analysis of the photon statistics of light emitted by photosynthetic complexes^{11–15} to the improvement of the signal-to-noise ratio in stimulated Raman scattering.^{16–18}

^{19}The archetypical process, which is widely believed to encapsulate all the beneficial properties of entangled photons, is entangled two-photon absorption (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 localized 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 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}

^{,}

*δ*

_{r}is the two-photon absorption (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}

^{,}

*ϕ*is the photon flux density. Since entangled photons always arrive in pairs, their ETPA rate scales linearly with the photon flux density.

^{22–24}When the flux increase 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,^{3,26,27} 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,^{4,28–37} with the help of coincidence detection^{38} or with higher intensity squeezed states.^{39} Only one series of experiments on molecules explicitly demonstrated the anticipated ETPA signal scaling after attenuation of the entangled photon flux,^{40,41} verifying the two-photon nature of the observed signal. Moreover, a growing number of experiments failed to detect any clear ETPA signal at all.^{42–47} Beyond the scaling behavior, the dependence on the spatial and temporal correlations, as predicted by Eq. (1), has not been studied to date.

Theoretical arguments were advanced for why weak ETPA signals should be expected in molecular samples,^{48–52} highlighting the interplay between molecular linewidth broadening and the achievable efficiency with the use of spectrally entangled photons. Other theoretical works propose that entangled photons may couple efficiently to very narrow transitions.^{53,54} It was also pointed out how competing linear processes, such as hot band absorption, can give rise to the sought after linear scaling.^{55} Further investigations compared the relative strength of one-photon transitions, which compete with the sought-after two-photon absorption,^{48,56–60} highlighted the role of interference between excitation pathways,^{61} optimized measurements,^{62} or considered the possibility of spectrally shaping the entangled two-photon wavefunction in order to optimize the ETPA efficiency.^{63–66} Quantum metrology was used to analyze the achievable enhancement compared to classical laser spectroscopy.^{67–72} The role of spatial correlations in multiphoton processes was previously treated by Klyshko.^{73}

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 that 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. Our results collect spatial and spectral degrees of freedom in a single formalism, generalizing earlier analyses by Raymer and co-workers,^{48–52} Drago and Sipes,^{57} and Klyshko.^{73} 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. 52. The paper is structured as follows: In Sec. II, we discuss this model and highlight important properties. Section III contains the perturbative calculations of the two terms in Eq. (2). In Sec. 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 Sec. V, we consider the situation where a single spatial mode is sufficient to describe the light field, and in Sec. VI, we consider ETPA with only a single active spectral mode.

## II. LIGHT–MATTER INTERACTIONS AND QUANTUM LIGHT MODEL

### A. The quantized electromagnetic field

*z*-direction with a small transverse momentum

**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 to

*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

^{74}

**q**= (

*k*

_{x},

*k*

_{y}), frequency

*ω*, and polarization

*σ*. The operators satisfy

**e**

_{q,σ}is the polarization vector, and the transverse position in the image plane is

**= (**

*ρ**x*,

*y*). We employ the small bandwidth approximation, where we evaluate the frequency-dependent prefactor at the center frequency

*ω*

_{0}of the traveling light pulse. We then have

*k*

_{z}(

*ω*

_{0}) =

*n*

_{0}

*ω*

_{0}/

*c*, with

*n*

_{0}the refractive index of the medium, and we further approximate

**e**

_{q,σ}≃

**e**

_{σ}. The electric field operator becomes simply

### B. Light–matter interaction

**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 in Fig. 1(b). It is described by the Hamiltonian,

*g*⟩ is the ground state. In the following, we set

*ω*

_{g}= 0 without loss of generality. The Hamiltonian then consists of an intermediate state |

*e*⟩ and a final state |

*f*⟩. We can now expand the dipole operator in these states such that it reads in the Schrödinger picture,

**r**

_{mol}= (

*ρ*_{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,

**d**

^{(+)}for the de-excitation).

### C. The PDC model

In the sketch in Fig. 1(a), a laser pulse with frequency *ω*_{p}, bandwidth Ω_{p}, and transverse beam area $\u223c(2\pi /Qp)2$ induces spontaneous parametric downconversion (PDC) 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. 75), 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^{76–79}—sometimes dubbed “X-entanglement,”^{80} when the two degrees of freedom are not separable.

#### 1. Effective action

^{81}

^{,}

*σ*of Eq. (5) is replaced by the signal and idler fields, labeled “

*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. Γ is a dimensionless number that quantifies the strength of the PDC process. We have further introduced the joint spatiotemporal amplitude (JSA)

*F*

_{PDC}, which we choose to normalize, i.e.,

^{76}and that gives rise to high-dimensional

*X*-entanglement.

^{80}In this paper, we will use a particularly convenient model where the spatial and frequency phase matching functions are approximated by Gaussians. This is, e.g., appropriate for non-collinear PDC.

^{81}The JSA then reads

*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 $Ap=\lambda p2=(2\pi )2/Qp2$. 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

^{76,82}and the emission angles of the PDC light that can be collected by the optical system.

^{81}Equation (15) is governed by similar physics: the distribution of the sum of the two photon frequencies is governed by Ω

_{p}, which is the bandwidth of the pump pulse generating the PDC light. The distribution of the frequency differences Ω

_{m}is again related to the phase matching conditions in the nonlinear crystal.

^{81,83}

*ζ*< 1 and $hn(kx)=k1/2(2nn!\pi )\u22121/2Hn(kx)e\u2212(kx)2/2$.

^{84}Note that the factor (−1)

^{n}is included for the case of Eqs. (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 Ω

_{p}> Ω

_{m}. In the case where this condition is violated (e.g., when we evaluate the Fourier transform of these functions), we use Eq. (16) without the factor (−1)

^{n}.

*x*

_{j}=

*Q*

_{sj}

*q*, $q=1/QmQp$, and

*ζ*

_{q}= (

*Q*

_{m}−

*Q*

_{p})/(

*Q*

_{m}+

*Q*

_{p}). Altogether, we can write the PDC action (11) as

**n**= (

*n*

_{t},

*n*

_{x},

*n*

_{y}), and

#### 2. Output state

*E*

_{0}is defined in Eq. (5) with

*ω*

_{0}=

*ω*

_{p}/2,

**e**

_{j}the polarization vector, and

*h*

_{n,s}(

**r**,

*t*) (

*h*

_{n,i}(

**r**,

*t*)) are the Fourier transforms of the Hermite functions (21), i.e.,

^{81}This quantity measures the effective dimensionality of the state of light, and in the isolated pair regime (when Γ ≪ 1), it can be directly related to the entanglement entropy of a photon pair.

^{85}It is simply given by

*K*

_{t}= (Ω

_{m}/Ω

_{p}+ Ω

_{p}/Ω

_{m})/2, and

*K*

_{x}=

*K*

_{y}= (

*Q*

_{m}/

*Q*

_{p}+

*Q*

_{p}/

*Q*

_{m})/2. A useful approximation for the Schmidt number, which we will use later, is obtained in the limit of strong two-photon entanglement, where, e.g., Ω

_{m}≫ Ω

_{p}and

*Q*

_{m}≫

*Q*

_{p}, such that the number of spectral modes is given by the ratio of the two numbers, respectively, i.e.,

*K*

_{t}≃Ω

_{m}/(2Ω

_{p}) and

*K*

_{x/y}≃

*Q*

_{m}/(2

*Q*

_{p}).

## III. TWO-PHOTON ABSORPTION CROSS SECTION

### A. From the excitation probability to the cross section

*P*

_{f}(

**r**

_{mol}) of a molecule at a position

**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 of 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 with an integral over the sample volume and the average molecular density

*m*

_{0}. Up to an unimportant constant that depends on, e.g., the detection efficiency of the fluorescence photons, the measured signal is then given by

^{86}

*V*

_{sample}, we consider a thin slab of length Δ

*z*in the propagation direction of the light such that we replace

*∫dz*≃Δ

*z*. In contrast, the transverse extent Δ

^{2}

**is assumed to be much larger than the transverse width of the light field such that we can extend the integration boundaries to infinity in the transverse directions.**

*ρ**f*

_{rep}of the laser, which generates the entangled pulses,

*N*

_{mol}=

*m*

_{0}Δ

*zA*

_{p}. To establish the scaling with the photon flux density, we write the photon flux density as

*f*

_{rep}accounts for the pulse duration as above.

### B. Experimental setup and two-photon absorption probability

**r**

_{mol}= (

*ρ*_{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 of exciting the final molecular state

*f*via the absorption of two photons. Thus, we need to calculate

*t*

_{0}is given by the state of light emitted by the PDC source, Eq. (22), and the molecular electronic ground state,

*ρ*

_{sys}= |

*g*(

*t*

_{0})⟩⟨

*g*(

*t*

_{0})| and

*ρ*

_{f}= |

*ψ*

_{BSV}(

*t*

_{0})⟩⟨

*ψ*

_{BSV}(

*t*

_{0})|. The time evolution is given by the Dyson series, which defines the time-ordering operator $T$. We also defined the light–matter interaction superoperator

*H*

_{l−m,−}, which acts on the density matrix as

*H*

_{l−m,−}

*ρ*=

*H*

_{l−m}

*ρ*−

*ρH*

_{l−m}. The interaction Hamiltonian is defined in Eq. (6), and the electric field operator is given by the sum of signal and idler fields,

**E**=

**E**

_{s}+

**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.

*H*

_{l−m}≃

**d**

^{(+)}⋅

**E**

^{(−)}+

**d**

^{(−)}⋅

**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

The impact of the polarization degrees of freedom can be calculated with a rotational average. For ETPA, this was done in Ref. 87, and we found that it can account only for minor changes in the ETPA cross section. Thus, in the following, we will neglect this procedure and replace the dipole operators with scalar quantities, i.e., *μ*_{j} → *μ*.

#### 1. The molecular correlation function

*e*,

*e*′, through which the ETPA process can take place. We have further added phenomenological dephasing rates

*γ*

_{ge}and

*γ*

_{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}=

*τ*

_{2}−

*τ*

_{1},

*t*

_{2}=

*τ*

_{3}−

*τ*

_{2}, and

*t*

_{3}=

*τ*

_{4}−

*τ*

_{3}, and sending

*t*

_{0}→ −

*∞*. We are further interested in the limit

*t*→

*∞*, i.e., in the final population after the entangled pulse has interacted with the molecule. In this limit, the

*τ*

_{4}-integration simply evaluates to 2

*πδ*(

*ω*

_{1}+

*ω*

_{2}−

*ω*

_{3}−

*ω*

_{4}). With the definition of the electric field operator (5), we arrive at

*t*

_{3}-integration, where we find

_{m}is much smaller than the detuning to the electronic resonances, as indicated in Fig. 1(b). We then obtain

_{e,e′}… is real.

#### 2. The field correlation function

*ω*

_{4}=

*ω*

_{1}+

*ω*

_{2}−

*ω*

_{3}and we defined

The function *f* describes the absorption of a correlated photon pair, as it contains only pairs of signal and idler functions, *h*_{n,s}, and *h*_{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*_{n}, are connected non-trivially in Eq. (47). The degrees of freedom can be separated only in the weak downconversion limit *r*_{n}Γ ≪ 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.

#### 3. The full two-photon excitation probability

## IV. THE TWO-PHOTON LIMIT

*r*

_{n}Γ ≪ 1, we can approximate $|\psi BSV\u232a\u2243|0\u232a+\Gamma \u2026a\u0302s\u2020a\u0302i\u2020|0\u232a$. The field correlation function (47) simplifies, using Eq. (16), to

*ρ*_{mol}reads

*ω*

_{s}↔

*ω*

_{i}and

**Q**

_{s}↔

**Q**

_{i}(the signal photon or the idler photon can be absorbed first, respectively). These two excitation pathways contribute to the same result and yield the overall factor 2. Hence, the TPA rate (31) is separated into spatial and frequency integrals, which can be treated separately,

*r*

_{0}can be read off from Eqs. (46) and (31),

*p*

_{spat}, given by the modulus square of Eq. (53) integrated over the transverse position

*ρ*, readily evaluates to

*et al.*in great detail

^{43,48–51}and depends crucially on the broadening

*γ*

_{fg}of the final molecular state. If the broadening is very small, as proposed by Kang

*et al.*

^{53}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), when we extend the integration boundaries to −

*∞*, we obtain for resonant excitation, where

*ω*

_{p}=

*ω*

_{fg},

^{43},

*x*, when

*γ*

_{fg}≫ Ω

_{p}, the function saturates to eff(

*x*) ≃ 0.8. This situation may be encountered in molecular TPA with cw entangled photons, which you can simply use

*et al.*(1) as well as the very similar calculation in (51). At small

*x*, in contrast, i.e., when

*γ*

_{fg}≪ Ω

_{p}, we find eff(

*x*) ≃

*x*. In this limit, the pump bandwidth Ω

_{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,

*σ*

^{(2)}∝ 1/

*γ*

_{fg}. Therefore, the above discussion must be understood at fixed

*γ*

_{fg}and for varying pump bandwidth Ω

_{p}. In particular, it does not imply that a large broadening, which pushes eff(

*x*) toward saturation, would be beneficial for ETPA. Instead, if we include the full dependence of

*σ*

_{e}on the broadening, we find that

*γ*

_{fg}. When

*γ*

_{fg}≪ Ω

_{p}, we have eff(

*x*) ≃

*x*such that the broadening dependence cancels, and we obtain an ETPA cross section with

*γ*

_{fg}in Eq. (60) replaced by Ω

_{p}. When

*γ*

_{fg}≫ Ω

_{p}, Eq. (63) becomes constant such that

*σ*

_{e}∝ 1/

*γ*

_{fg}.

## V. SINGLE SPATIAL MODE LIMIT

*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

*n*

_{x}/

*n*

_{y}-summations in the field correlation functions (48) and (49), and replace the corresponding Hermite functions by

### A. Correlated contribution

*∞*and use the parity of the Hermite functions,

*h*

_{n}(

*ω*

_{sum}−

*ω*) = (−1)

^{n}

*h*

_{n}(

*ω*−

*ω*

_{sum}), we can use the integral identity,

^{88}

*w*

_{0}→ −

*∞*, the remaining

*w*-integration can be carried out analytically or numerically by Mathematica with high efficiency (although the integration becomes unstable for extremely highly entangled states). Still, it is instructive to consider one limiting case first.

#### 1. Narrow resonance

*w*

_{fg}≪ 1, we can replace

*ω*

_{fg}=

*ω*

_{p}) from Eq. (74),

*T*

_{e}= 2

*π*/Ω

_{m}is the entanglement time (62), and

*A*

_{p}is 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 Γ ≫ 1, we approximate

*T*

_{pulse}= 2

*π*/Ω

_{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}≪ 1.

### B. Uncorrelated contribution

#### 1. Narrow resonance

*ω*

_{fg}=

*ω*

_{p}, Eq. (81) then simplifies to

*δ*

_{r}

*ϕ*

^{2}, where $\varphi 2\u221d(\u2211n\u2061sinh2(rn\Gamma ))2$, see Eq. (71). We thus need to analyze the term

*∑*

_{n}sinh

^{4}(

*r*

_{n}Γ). Given the Schmidt number

*K*of the PDC light, we expect that in the limit of many Schmidt modes, we have ∼

*K*terms that contribute to this expression. In contrast, the squared mean photon number should contain ∼

*K*

^{2}terms. Thus, in a highly entangled light field, when the parameter

*ζ*

_{t}in Eq. (17) is close to one, we expect that

*γ*

_{fg}/Ω

_{p}appears, here it is the ratio

*γ*

_{fg}/Ω

_{m}, i.e., the ratio between the molecular resonance and the bandwidth Ω

_{m}of the individual photons. As Ω

_{m}≫ Ω

_{p}in a highly entangled beam, this reduces the uncorrelated contribution to the cross section considerably.

^{89,90}

### C. Results

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

*Spectral Resonance*: In Fig. 3, we solve Eqs. (72) and (81) numerically to show the change of the ETPA resonance as a function of the broadening *γ*_{fg}, the bandwidth Ω_{m} (or, equivalently, the amount of entanglement), and the mean photon number $\u27e8N\u0302\u27e9$ per entangled pulse. The simulations show the expected behavior, where at small photon numbers $\u27e8n\u0302\u27e9=0.1$, the signal is dominated by the correlated contribution (72). 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 contains only a few Schmidt modes, the uncorrelated events account for $\u223c50$% of the signal at $\u27e8n\u0302\u27e9=1$. Conversely, in Fig. 3(d), we simulate a narrow resonance and strong entanglement of the incident light. Even at $\u27e8n\u0302\u27e9=100$, 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. 91. In the following, we explore this behavior in the limit (75), where *γ*_{fg} ≪ Ω_{p}. The increase in the full ETPA signal on resonance is shown in Fig. 4(a) for three different bandwidths Ω_{m} ranging from 1.5Ω_{p} (very weak entanglement) to 50Ω_{p} (very strong entanglement). At photon numbers below $\u27e8N\u0302\u27e9\u2272100$, 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 $\u27e8N\u0302\u27e9\u22431$ in the case of weak entanglement, it remains linear up to $\u27e8N\u0302\u27e9\u2243100$ 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 $\u27e8A\u0302nt\u2020A\u0302nt\u27e9\u223c1$. Using the Schmidt number *K* as the effective number of modes, we expect this to be the case when $\u27e8N\u0302\u27e9/K\u223c1$. In the case of large bandwidths, Ω_{m} = 50Ω_{p}, we have, using Eq. (29), *K* ≃Ω_{m}/(2Ω_{p}) and, thus, expect the crossover to take place when $\u27e8N\u0302\u27e9\u224325$. This estimation is consistent with our observations, given that the Schmidt number only gives an estimate for the effective dimensionality of the entangled state. This was also observed in the analysis of cw PDC light in (52).

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, $\u27e8N\u0302\u27e9>100$, the weakly entangled signal with Ω_{m} = 1.5Ω_{p} becomes *stronger* than the other two. This is entirely due to the larger strength of the uncorrelated contribution to 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. However, 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,^{92–101} 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.

*r*

_{rel}→ 1/2, which is the expected limiting behavior according to Eq. (85). This limit can be reached already at $\u27e8N\u0302\u27e9\u224310$ for a weakly entangled state and can be $\u27e8N\u0302\u27e9>104$ for strongly entangled ones. Conversely, at a fixed photon number, the ratio increases quadratically with the bandwidth Ω

_{m}, as shown in Fig. 4(c).

*Crossover from narrow to broad molecular resonance:* So far, our discussion has been 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. (72) and (81) numerically to obtain *R*^{TPA}. For a very broad range of mean photon numbers $\u27e8N\u0302\u27e9$ and bandwidths Ω_{m}, we find that the signals saturate to a constant value when *γ*_{fg} ≲ 0.1Ω_{p}. We have checked that this limit coincides with the delta-function limits, Eqs. (76) and (82), which provide excellent approximations in this parameter regime. When the broadening becomes similar to or larger than the pump bandwidth Ω_{p}, the signal starts to decay with increasing *γ*_{fg}. If the mean photon number is small and the signal is dominated by the correlated contribution to *P*_{f}, this decrease is independent of the photonic bandwidth Ω_{m}. This can be seen in the insets of Figs. 5(a) and 5(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} ∝ 1/*γ*_{fg}. At larger photon numbers, when the incoherent contribution to *P*_{f} becomes important as well, this universal scaling behavior is lost. As observed already in Fig. 4(a), a state with a narrower bandwidth (black, dotted-dashed line) generates a larger ETPA signal than a strongly entangled state (blue solid line) when the resonance broadening is small. However, 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 *γ*_{fg} ≳ 100Ω_{p}—albeit a much smaller absolute signal than in the case of a narrow resonance.

## VI. SINGLE SPECTRAL MODE

_{m}= Ω

_{p}. Adapting Eq. (46) to this situation, we obtain

^{2}cosh

^{2}vs ∼ sinh

^{4}and the factor $(\u22121)nx+ny+nx\u2032+ny\u2032$, which reduces the uncorrelated contribution, respectively. This contrasts with the previous section, where correlated and uncorrelated contributions were convolved differently with the molecular response function, even in the limit of a very narrow resonance (75).

The spatial dependence of the signal according to Eqs. (88) and (89) is shown in Fig. 6 for different mean photon numbers and a weakly entangled state with *Q*_{m} = 1.5 *Q*_{p} and a entangled state with *Q*_{m} = 10 *Q*_{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.

In Fig. 7, we basically repeat the analysis of the previous section and analyze the scaling behavior of the ETPA signal (31), which we obtain from integrating Eqs. (88) and (89) over space. As before, we show both weakly entangled states (with *Q*_{m} = 1.5*Q*_{p}) and very strongly entangled ones (*Q*_{m} = 50*Q*_{p}). Here, we find that, in contrast to Fig. 4, strongly correlated signals always remain larger than weakly entangled ones. The reason is straightforward: the spectral correlations are convolved with the molecular response, which gives rise to the intricate interplay between pump bandwidth and the molecular broadening we explored in the previous section. The spatial correlations, on the other hand, are related to the probability of localizing two photons at the same position in the image plane.^{73} Since the molecules can be treated as point-like absorbers, this potential enhancement is always present. Consequently, the advantage due to spatial quantum correlations becomes smaller in relative terms as the mean photon number increases, but it never goes away and may persist even in photonic states with macroscopic photon numbers.^{5,102}

## VII. CONCLUSIONS

In this paper, we have derived entangled two-photon absorption cross sections of pulsed entangled beams from the low-to-high-gain regime of parametric downconversion. Starting from an established bigaussian model for the entangled beams, we have presented a derivation of Fei’s seminal formula for the ETPA cross section (1). It generalizes known results in the literature, which treat spatial or spectral aspects separately. Our analysis shows that Eq. (1) relies on a factorization of the entangled photon wavefunction into spatial and spectral components. It will be interesting to explore in future work what happens when this is not the case and these degrees of freedom are quantum correlated.^{76,80}

Moreover, we have related the entanglement time and the entanglement area, which appear in Eq. (1), to microscopic parameters in the quantum model of light. These names are rather unfortunate, as these quantities are directly connected to the amount of entanglement in the photonic state only in certain limits (i.e., when Ω_{m} ≫ Ω_{p} and *Q*_{m} ≫ *Q*_{p}). Still, even in situations where this is not the case (for instance, when both Ω_{m} and Ω_{p} are both 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. The enhancement provided by a short entanglement time in this case can be understood in the same way as an ultrafast laser pulse enhances the classical TPA rate. As both the entanglement time and the entanglement area are controlled by the 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.^{103}

We have further investigated the crossover from the linear to the 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 the anticipated crossover behavior 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, Eqs. (80) and (84), respectively, and quantum enhancement effects can persist for 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 paper may become substantial and will require a numerically more involved treatment of the light source.^{90} Finally, in this discussion, we have considered degenerate PDC. This is, in fact, the worst case scenario, where the uncorrelated contributions can also drive TPA significantly. In a non-degenerate PDC, this background would be suppressed.

In this paper, 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^{104–106} or spatially resolved spectroscopy. This will be an interesting direction for future research. Overall, our work highlights the interesting effects one can expect from the nonlinear light–matter interactions of high-gain PDC light.^{5,102,107}

## ACKNOWLEDGMENTS

I would like to thank Dr. Shahram Panahiyan for his helpful feedback on the manuscript and acknowledge the support from the Cluster of Excellence Advanced Imaging of Matter of the Deutsche Forschungsgemeinschaft (DFG)—EXC 2056—Project No. 390715994.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Frank Schlawin**: Formal analysis (lead); Funding acquisition (lead); Writing – original draft (lead).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

## REFERENCES

*n*-photon state

*x*entanglement: The nonfactorable spatiotemporal structure of biphoton correlation

Note that here the Hermite functions are normalized with respect to integrals over space or frequency, i.e., $\u222bd\omega hn2(\omega )=1$.

Note that the factor 1/(2π)^{2} is introduced for consistency such that the photon number density *â*^{†}(*ρ*, ω)*â*(*ρ*, ω) integrated over space and frequency yields the mean photon number (34).