1 Introduction

Biological imaging and sensing is an important driving force behind the research of optics and chemistry. In particular, two-photon fluorescence (TPF) is widely used in visualizing biological subcutaneous tissues [1–4]. In the process of TPF, the target specimen absorbs two long-wavelength photons, which has low scattering, and emits a photon near their sum frequency. Compared with one-photon fluorescence, TPF provides an improved spatial resolution with a reduced phototoxicity (lower photon energy) and autofluorescence [1, 5]. However, similar to any multi-photon processes, the intensity of TPF is much lower than the single-photon fluorescence under the same excitation power, and thus requires high excitation intensity to observe the signal [6], which can result in damages of tissues and rupture of cells [7]. To solve this problem, efforts have been focused on finding more efficient fluorphores, such as quantum dot [8], rare earth doped nanoparticle (RENP) [9] and aggregation-induced emission luminogen (AIEgen) [10], to reduce the threshold of the excitation power. On the other hand, quantum light can offer a larger two-photon cross section than coherent light (laser) and thermal light [11]. In the past decades, quantum sources, including entangled photons and squeezed light, have been utilized in spectroscopy and imaging with an improved signal-to-noise ratio [12]. In contrast to the classical two-photon absorption (TPA), where photons arrive randomly and the TPA probability has a quadratic dependence with the excitation power, in the entangled TPA the exciting photons arrive in pairs, resulting in a linear power dependence [13–16]. Entangled TPA is particularly advantageous in the low-power excitation regime.

Recently, many experiments on entangled TPA or TPF have been reported [17–19]. A central quantity measured in these experiments is the entangled TPA cross section, \(\sigma _{e}\). In some of the experiments, the estimated \(\sigma _{e}\) is large, ranging from 10−17 to 10−21 cm2 [20–25]. To understand the meaning of these values, we consider the TPF generated by entangled photons,

$$ F \propto \sigma _{e} \phi _{e} + \delta _{c} \phi _{e}^{2}, $$
(1)

where \(\delta _{c}\) is the classical TPA cross-section with values generally from 10−47 to 10−50 cm4s [26], and \(\phi _{e}\) is the flux rate of entangled photon pairs per unit area. The first term \(\sigma _{e} \phi _{e}\) is contributed by the absorption of the entangled photon pairs, while the second term \(\delta _{c} \phi _{e}^{2}\) comes from the random absorption of photons from different pairs. When the TPF is dominated by the entangled photon absorption, i.e., \(\sigma _{e} \phi _{e} > \delta _{c} \phi _{e}^{2}\), we obtain \(\phi _{e}<\sigma _{e}/\delta _{c}<10^{26}\) cm−2s−1, according to the above data. If true, such quantum enhancement can be achieved in a wide range of power with ordinary entangled photon sources, which can be revolutionary for high-sensitivity spectroscopy and microscopy [27].

However, parallel experiments reported negative results, resulting in controversial arguments on the feasibility of such a technique in biological imaging [28–30]. In these experiments, after carefully considering the factors that can influence the data, no enhancement can be attributed to the entanglement. The estimated upper bound for the \(\sigma _{e}\) is about \(5\times 10^{-23}\) to 10−25 cm2, substantially smaller than the previously reported values. It is worth noting that the linear power dependence between the fluorescence signal and excitation power is not a sufficient condition to attribute the enhancement to the entanglement, because it may come from other processes, e.g., linear scattering [30], hot-band absorption (HBA) [31], etc. A comparison experiment with coherent illumination or by changing the time delay between entangled photons is required.

In order to clarify the debate on the quantum advantages in entangled TPA and TPF, we perform a series of experiments on novel materials including organic dyes, RENP, and AIEgen, which have been proved to have large cross sections in classical TPA. We first prepare a bright entangled photon source with a photon flux rate about \(1.2 \times 10^{12}\) pairs/s. The entanglement time, which is defined as the width of the two-photon correlation function, is measured to be \(T_{e}=50\) fs by a Mach-Zehnder (MZ) interferometer [16, 30, 32, 42]. We use a nonlinear crystal, where entanglement enhancement in the sum frequency generation (SFG) is verified, to test our experimental setup. In particular, we compensate the dispersion of entangled photon pairs to synchronize their arrival time in our samples. Using an ultrasensitive detector we observe classical TPF from 2,3-bis(4′-(diphenylamino)-[1,1′-biphenyl]-4-yl) fumaronitrile (TPATCN) [10, 33] with an excitation power as low as 7 μW, but see no signal with entangled photon pairs. We also exclude the effect of entanglement for observed signals when the samples are illuminated by quantum light sources. By illuminating RENP with nW entangled photons we observe TPF, which is proved to be the excitation by only the high frequency photons of the entangled photon pairs, indicated by a quadratic power dependence and verified by using a shortpass filter. In upconversion materials such as Indocyanine Green (ICG), we observe that TPF under the illumination of a laser can also have a linear power dependence, which is attributed to the HBA. These experiments provide guidance to find proper molecules for entanglement enhanced TPF.

2 Results

In the experiment, we generate high-flux entangled photon pairs from the spontaneous parametric down conversion (SPDC) in a periodically poled lithium niobate (PPLN) crystal pumped by a 532 nm continuous-wave laser with maxium power 1.6 W. The SPDC light has a large bandwidth (FWHM 173 nm, see Fig. 1f for joint spectral intensity), which allows us to have high-flux and temporally separable entangled photon pairs. However, the photons with different colors have considerable delays between each other due to the dispersion. Therefore, we use a pair of prisms to compensate the dispersion and also to filter out the residual pump. Compared with the four-prism compressor, such a two-prism scheme is more compact, symmetrical and easy to align [34]. Then the entangled photons are directed to a lightproof box, where a series of long-pass filters are mounted to block ambient light. In order to compare with the classical TPF, a 1064 nm laser beam is also guided in the lightproof box along the same path (not shown in Fig. 1).

Figure 1
figure 1

Experimental setup. (a) Preparation and dispersion compensation of the SPDC light. A periscope and two prisms are used to compensate the dispersion without introducing additional chromatic aberrations. The prisms are designed to work at Brewster angles. A hollow-roof retroreflector is used to reduce the height of the SPDC beam and make it pass through the two prisms for a second time. (b) Detection of the SFG in a PPLN crystal using a PMT. (c) Forward-fluorescence collection unit for liquid samples. (d) Backward-fluorescence collection unit for solid-state (opaque) samples. (e) Two-photon MZ interferometer for measuring the entanglement time and estimating the entangled photon pair rate. (f) Interferograms of coincidence counting and single-photon counting in Fig. 1(e). (g) Joint spectral intensity of entangled photon pairs measured from a time-of-flight (10 km-long optical fibre) spectrometer. The wavelengths of the photon pairs are obtained from their relative time delay in the fibre, under the condition that energy is conserved during the SPDC. HWP: half-wave plate, PBS: polarizing beamsplitter, BS: non-polarizing balanced beamsplitter, DM: dichroic mirror

Full size image

We use the SFG from the same PPLN crystal (20 mm-long, type-0) to characterize our entangled photon source. The set-up is shown in Fig. 1b [29, 34]. A photomultiplier tube (PMT) (Hamamatsu, H7421-40) with a quantum efficiency 45% at 532 nm is used to detect the SFG photons. The recorded photon counting rate is nearly 7000 counts/s. In order to test the effect of entanglement, we use two ways to attenuate the SPDC light. The first way is to pairwise reduce the entangled photons by rotating the polarization of the pump light, as shown in Fig. 2. This is based on the fact that only vertical polarization can satisfy type-0 phase matching condition in SPDC. We can effectively reduce the generation of the photon pairs. We observe that the SFG signal is linearly dependent on the power of the SPDC light, which is a signature of the entangled TPF. The second way is to randomly attenuate all SPDC photons by inserting neutral density (ND) filters in the SPDC light path. In such a way photons lose their entangled partners, and thus the TPF signal is quadratic to the power of the attenuated SPDC light. We also compare the results with those under coherent light illumination, showing an 80-fold quantum enhancement at 20 nW coherent light excitation.

Figure 2
figure 2

(a) The power dependence of the SFG rate of entangled photons in a PPLN crystal. By rotating the polarization of the pump light, the entangled photon pair rate can be reduced. The SFG has a linear dependence with the pumping power (gray-triangle). On the other hand, with ND filters attenuation, the entangled photon pairs are destroyed and the SFG signal is quadratically dependent on the pumping power (red-triangle). As a comparison, the SHG generated by a laser also has a quadratic dependence on the pumping power (purple-triangle). The solid lines are fitting curves. (b) The SFG rate as a function of the insertion length of the prisms. The shaded area for each curve obtained from six independent data sets represents three standard deviations

Full size image

To synchronize the arrival time of the entangled photons, we adjust the delay between signal and idler photons by changing the insertion length of the prisms. As shown in Fig. 2b, there is an optimized insertion length to achieve the maximum value of the SFG. We use a superconducting nanowire single photon detector (SNSPD) to characterize the photon statistics in the SPDC light. Two important quantities are measured, the coincidence counting rate and the entanglement time (Fig. 1e). Considering the overall detection efficiency, including coupling efficiency of optical fibres, quantum efficiency of detectors and transmission loss of optical elements, we estimate the entangled photon flux rate is \(1.2 \times 10^{12} \) pairs/s (see Fig. 1f.) The overall detection efficiency \(\eta _{A(B)}\) can be inferred from the average coincidence counting rate and the single counting rate of two detection channels according to \(\eta _{A(B)} = R_{CC} / R_{SC,B(A)}\), where \(R_{CC}\) denotes the coincidence counting rate and \(R_{SC,B(A)}\) is the single counting rate of channel B(A). The entanglement time can be obtained by fitting from either the coincidence counting (Fig. 1f, top) or the single photon counting (Fig. 1f, bottom)[42].

With this tunable and bright entangled photon source, we are ready to test the entangled TPF in several molecules that have large two-photon cross section. Rhodamine 6G (Rh6G) has been widely used in fluorescent imaging. We prepare a 3 mmol/L Rh6G dissolved in a methanol cuvette. Another sample is a dried TPATCN on a glass slide. Such TPATCN belongs to the family of AIEgen, which is free from concentration quenching and thus promising for biomedical imaging. We test the TPF in these two samples with laser and SPDC light (see Fig. 3). Setting 1 PMT count/s as the lower limit of the TPF signal, we observe the threshold power of classical TPF of Rh6G and TPATCN are 12 μW and 3 μW, respectively. The fluorescence is quadratic to the laser power.

Figure 3
figure 3

The power dependence of classical TPF of Rh6G and TPATCN. A 3 mmol/L Rh6G solution is prepared in methanol, while the TPATCN is dried powder. The logarithm slopes of Rh6G and TPATCN signal are 1.91 and 1.99, respectively. The integration time is 300 s. Error bars obtained from six independent data sets denote one standard deviation

Full size image

In general, the classical TPF rate can be estimated by [29]

$$ F_{c} = \eta YcV \delta _{c} \phi _{c}^{2}, $$
(2)

where \(F_{c}\) is the TPF rate with classical illumination, η and Y are the overall detection efficiency (including the fluorescence collection efficiency and the PMT efficiency) and the quantum yield of molecules (for Rh6G, \(Y = 0.8\) [35]), c is the concentration and \(\phi _{c}\) is classical light flux. Assuming that the effective light-matter interaction occurs within the Rayleigh length of the light beam, Eq. (2) becomes \(F_{c} = \pi \eta Yc \delta _{c} N_{c}^{2} / 2 \lambda \), where \(\lambda = 1064\) nm is the excitation wavelength and \(N_{c}\) is the total excitation photon flux rate. Considering the Rh6G two-photon cross section \(\delta _{c} = 1 \times 10^{-49}\) cm4s [36], we estimate the detection efficiency \(\eta =8.86\%\) from Fig. 3 (the theoretically calculated value is 7.3% with parameters of the optical elements, including the collection lens numerical aperture 0.78 and PMT quantum efficiency 0.4 around 600 nm).

We observe no entangled TPF by using SPDC light. We can set an upper bound on the \(\sigma _{e}\) of Rh6G via the entangled TPF formula,

$$ F_{e} = \eta YcV \sigma _{e} \phi _{e}, $$
(3)

where \(F_{e}\) is fluorescence rate using entangled photon pairs and \(\phi _{e}\) is the entangled photon pair flux. The absence of the entangled TPF signal indicates

$$ \sigma _{e} < \frac{F_{d}}{\eta Yc V \phi _{e}}, $$
(4)

where \(F_{d}\) is the lower limit we set in detecting the signal. Considering the beam waist 1.5 μm in the experiment, we obtain the interaction volume \(V = 1.3 \times 10^{-10}\) cm3 and the photon pair flux \(\phi _{e} = 1.3 \times 10^{19}\) pairs cm−2s−1. The upper bound of \(\sigma _{e}\) is \(4.9 \times 10^{-27}\) cm2, which is 5 to 6 orders smaller than previously reported values [19]. However, this estimation is in agreement with the Refs. [28–30]. More efforts are needed to improve the detection efficiency, entangled photon sources, and fluorescence yield of the molecules.

In all the materials we have attempted, only the \(\mathrm{Er}^{3+}\) and \(\mathrm{Yb}^{3+}\) doped core-shell \(\mathrm{NaYF}_{4}\) nanocrystals have TPF under the illumination of SPDC light. As a novel efficient luminescent material [37], it belongs to the family of RENP, which have narrow-band absorption and long fluorescence lifetime (microsecond). We can detect TPF from RENP with the SPDC light. However, as shown in Fig. 4a, the quadratic power dependence indicates that the observed TPF is resulted from the absorption of two photons from different entangled photon pairs. To verify this hypothesis, we use a shortpass filter that blocks the photons with wavelength longer than 1064 nm and find that the TPF has no change. This suggests that the TPF is solely due to the TPA of the photons with wavelengths shorter than 1064 nm. The entangled partners of these photons have no contribution. To investigate the optical absorption and TPF of RENP, we use a picosecond laser to scan the excitation wavelength and find that RENP has strong absorption and upconversion efficiency near 975 nm (see Fig. 4b)). We further measure the spectrum of the TPF when the sample is illuminated by 980 nm picosecond laser (Fig. 4c). Two dominate peaks at 540 nm and 660 nm are observed, which correspond to the two transitions in Fig. 4d. Therefore, these results evidence that such TPF, although generated with SPDC light, is due to the uncorrelated absorption of the short-wavelength photons.

Figure 4
figure 4

(a) The power dependence of the TPF of RENP under SPDC illumination. Error bars obtained from six independent data sets denote one standard deviation. (b) The TPF of RENP as a function of the excitation wavelength. A tunable picosecond laser (2 ps, 80 MHz repetition rate) is used to excite the sample. (c) The spectrum of TPF of RENP excited by 980 nm picosecond laser. (d) Energy levels and TPF processes of RENP. ET: energy transfer

Full size image

We shall be cautious on claiming the observation of entangled TPF in new materials. In particular, the linear dependence of the TPF on the pumping power can be attributed to other reasons. We use ICG, which can have upconversion of near infrared light assisted by thermal photons, to demonstrate such a possibility. In this case molecules excited to the vibrational excited states by ambient thermal light only need one photon to generate anti-Stokes fluorescence [38]. Therefore, the fluorescence is also linearly dependent on the power of the near infrared light, which can be mistaken for entangled TPF (see Fig. 5) [31]. Such HBA induced anti-Stokes fluorescence has been observed in various materials including dyes [39], AIEgen [40] and quantum dot [41].

Figure 5
figure 5

The power dependence of TPF of ICG illuminated by a 1064 nm CW laser. We use a scientific complementary metal oxide semiconductor (sCMOS) camera to record the fluorescence of ICG since the wavelength (850 nm) is out of the detection range of the PMT. The logarithm slope is fitted to be 1.05. Such a linear dependence is attributed to the HBA effect. Error bars obtained from six independent data sets denote one standard deviation

Full size image

3 Conclusion

We build a high-flux entangled photon source to investigate TPF in various materials. The entangled photon source is examined by the SFG in a nonlinear crystal. Entanglement enhancement in the nonlinear process is directly validated with photon pair flux rate more than \(1.2 \times 10^{12}\) pairs/s. However, in fluorescence molecules, we observe no TPF signal from the entangled photon pairs. The difference between the results in the PPLN crystal and the fluorescence molecules is majorly due to the following reason. The two-photon signal we observed from PPLN crystal is from a coherent SFG while the TPF in molecules is an incoherent process. The phase-matching effect greatly enhances the SFG. On the other hand, the state of matter of the fluorescent molecules can also affect the TPF efficiency, which has been considered to ensure these molecules are prepared in the most fluorescent states. We set an upper bound of entangled TPA cross section of Rh6G, which is 5-6 orders of magnitude lower than the positively reported values. Although we observe fluorescence in RENP when illuminated by the SPDC light, the signal is proved to be classical TPF. Special caution shall be paid to materials that have HBA, which can also have the linear power dependence of the fluorescence. In conclusion, although the entanglement can enhance SFG in nonlinear crystals, the molecules that can use this effect to improve the performance of two-photon imaging are yet to be found.