Is Heralded Two-Photon Excited Fluorescence with Single Absorbers Possible with Current Technology?

Abstract

The interaction between single or a fixed number of photons with a single absorber is of fundamental interest in quantum technology. The harnessing of light matter interactions at the single particle limit has several potential applications ranging from quantum communication and quantum metrology to quantum imaging. In this perspective, a setup for heralded two-photon excited fluorescence at the single absorber level is proposed. The setup is based on a heralded two-photon source utilizing spontaneous parametric down-conversion, entanglement swapping and sum frequency generation for joint detection. This perspective aimed at triggering a discussion about the study of TPA and TPEF with only very few photons. The feasibility of the scheme is assessed by estimating the performance based on state-of-the-art technologies and losses, with the conclusion that the realization appears to be very challenging, but not completely impossible.

1. Introduction

Maria Göppert-Mayer derived two-photon absorption (TPA) theoretically in her thesis [1] and the first experimental observation of TPA was achieved 30 years later, enabled by the invention of the laser [2]. Garrett and Kaiser observed fluorescence at 425 nm induced by light with a longer wavelength of 694 nm, marking a milestone in the field of “nonlinear optics”, which for many decades was attributed to be the study of light matter interactions at high light intensities involving many photons [3]. However, with the advent of both atomic physics and quantum optics experiments involving single particles of light and matter [4,5,6], the field of “quantum nonlinear optics” has recently also extended towards low intensities down to the single photon level [7,8].

Two-photon microscopy (TPM) is one of the state of the art high resolution imaging technologies used in the life sciences sector [9] exploiting TPA and subsequent fluorescence, two-photon excited fluorescence (TPEF). Advantages of TPM are the deeper penetration depth into the tissue and a drastically increased depth resolution (see for example review article by Zipfel et al. [10]). TPM enabled in vivo deep brain imaging, which is of great interest in neuroscience as it made the efficient detection of beta-Amyloid plaque, a protein linked to Alzheimer’s disease, possible [11]. Thus far, TPM is exclusively performed with classical light at relatively high light intensities, which is problematic because biological samples are very sensitive to optical damage [12]. Thus, reducing the optical load at the probe is paramount to prevent irreversible damage to living cells, which is done by using ultrashort pulses in TPM setups that still contain many photons.

Strategies to enhance TPA/TPEF are to utilize absorbers with an enhanced cross section, for example $\pi$-conjugated chromophores with dipolar, quadrupolar, octupolar and more extended branched structures [13] exploiting plasmonic effects on surfaces [14] or with nanoparticles [15], facilitate strong photon–absorber coupling or utilize light with tailored photon statistics [16,17,18,19,20].

The photon statistics of a light field is important for nonlinear processes [16] and enhanced TPA was experimentally achieved with “classical” light by utilizing thermal light instead of coherent light [19,20]. The utilization of entangled photons for TPA was proposed by Gea-Banacloche [17], as well as Javaninen and Gould [18], and later experimentally demonstrated with atoms [21] as well as molecules [22,23,24,25,26,27,28]. Teich and Saleh proposed to exploit entangled photons for TPM in 1997 [29] with an ideal two-photon or biphoton source. They later extended this idea towards virtual state spectroscopy [30] giving birth to the field of entangled photon or quantum spectroscopy, which has gained a lot of interest since a few years [31,32,33,34,35]. Entangled or correlated photons are still most commonly generated by spontaneous parametric down-conversion (SPDC) and highly efficient biphoton sources can be realized by using periodically poled (PP) materials with waveguide structures [36].

Thus far, most of the entagled TPA schemes involve rather many photons incident at the probe like bright squeezed vacuum [37], but in the context of quantum nonlinear optics the question arises: “To what extend can TPA or TPEF be observed with extremely weak fields down to the single or n-photon limit?”. Heralded single photon sources based on SPDC [38] have been demonstrated and Guerreiro et al. have done seminal work on detecting weak nonlinear interactions between single photons using a heralded scheme comprising multiple nonlinear crystals [39,40].

In this perspective, a scheme for heralded TPEF is proposed utilizing parts of the approach by Guerreiro et al. [40] and exploiting other enhancement strategies for TPEF, which has potential applications in quantum microscopy and quantum spectroscopy.

2. Fundamentals and Background

2.1. Classical TPA

TPA is referred to be the “simultaneous” (It should be noted, that this is of course an unphysical term. Here, simultaneous means that the two-photons have to arrive within the lifetime of the intermediate state.) absorption of two photons by an absorber. First a photon interacts with an absorber and brings an electron from a ground state $|{\psi }_g\rangle$ with an energy eigenvalue of ${\epsilon }_g$ to an intermediate (nonresonant) state $|{\psi }_i\rangle$. This intermediate state has a very short lifetime ${\tau }_i$ (on the order of femtoseconds) and a complex energy eigenvalue of ${\epsilon }_i+i{k}_i/2$. If a second photon arrives within that lifetime window, the electron can be excited to the final state $|{\psi }_f\rangle$ with an energy eigenvalue of ${\epsilon }_f$. In the case of TPEF, a fluorescence photon is emitted subsequently to the absorption process (see Figure 1).

The TPA rate, $R_{TPA}$, will depend quadratically on the intensity I or the photon flux density $\varphi$ of the light field, when using a classical light field:

$$R_{TPA}\propto I^2,$$

(1)

$$R_{TPA}={\delta }_{TPA}·{\varphi }^2,$$

(2)

with the classical TPA absorption cross section ${\delta }_{TPA}$:

$$\begin{array}{ccc}\hfill {\delta }_{TPA}& =& \frac{\pi }{2}{\omega }_1{\omega }_2\delta ({\epsilon }_f-{\epsilon }_g-{\omega }_1-{\omega }_2)\times |\sum _j\left[\frac{D_{21}^{\left(i\right)}}{{\Delta }_1^{\left(i\right)}-i{\kappa }_i/2}+\frac{D_{12}^{\left(i\right)}}{{\Delta }_2^{\left(i\right)}-i{\kappa }_i/2}\right]{|}^2,\hfill \end{array}$$

(3)

being on the order of:

$${\delta }_{TPA}\approx {\sigma }_{gi}{\sigma }_{if}·{\tau }_i,$$

(4)

Here, ${\sigma }_{gi}$ and ${\sigma }_{if}$ are the one-photon absorption cross sections, ${\omega }_1$ and ${\omega }_2$ are the frequencies of the photons (or fields) driving the TPA transition, ${\epsilon }_f$ and ${\epsilon }_g$ are the energy states of the final and the ground state, respectively, ${\Delta }_k^{\left(i\right)}={\epsilon }_i-{\epsilon }_g-{\omega }_k$ is an energy mismatch factor and $D_{21}$ and $D_{12}$ are the transition matrix elements $D_{kl}^{\left(i\right)}=\langle {\psi }_f|d_k|{\psi }_i\rangle \langle {\psi }_i|d_l|{\psi }_g\rangle$ with the electric dipole components $d_k,d_l$ with $k,l$ = 1, 2.

2.1.1. Enhanced Classical TPA

Mainly motivated to reduce photo-damage, a lot of effort has been made on the design of new absorbers with a large TPA cross section. Such chromophores can be tailored by designing donor acceptor groups, $\pi$-conjugated connectors and branching symmetry [41] including dendrimers [23]. A review on TPA chromophores with dipolar, quadrupolar, octupolar and more extended branched structures can be found in [13]. Furthermore, it was shown that by coupling fluorescent dyes to plasmonic nanostructures, such as spheres, it is possible to achieve a field enhancement resulting in an increased absorption, as well as increased fluorescence of dye molecules [42], which can be implemented on surfaces [14] or by utilizing nanoparticles [15]. Such nanoparticles can have tailored shapes and serve as nanoantennas [43,44]. Besides, enhanced absorption cross section can improve or even tailor the fluorescence by plasmonic effects [45]. The enhancement depends on the wavelength, the size of the nano-particle and also on the distance between the absorber and the fluorescent dye. In some cases, a fluorescence quenching was observed, therefore the spacer layer of the nano-particles (as sketched in Figure 2) has to be carefully adjusted depending on the dye and the excitation wavelength.

2.1.2. Strong Photon-Absorber Coupling

To achieve strong photon–absorber coupling, several schemes have been implemented. Most prominently, high-Q enhancement cavities have been used to ensure that the photon interaction with an absorber is enhanced [8]. However, while such cavities are excellent for fundamental tests in quantum optics, they are not suitable for many real world applications as they require extreme stability and lack scalability. For TPM applications, cavities are not well suited. Therefore, several optical interconnects for free space coupling of photons and single absorbers have been proposed and utilized, including the use of high NA lenses [46,47,48,49] or mirrors for trapped ions [50,51,52] or molecules [53,54,55]. For TPEF, it is important to not only consider the coupling between the excitation beam and the single absorber, but also to realize a high collection efficiency of fluorescence photons.

2.2. TPA and Photon Statistics

It is well known that higher order absorption processes will probe higher order correlation functions of the light fields [16,56,57,58,59]. In 1968, Mollow used pertubative theory to connect the degree of second order coherence with the TPA rate [16]. Mollow assumed, that the spectral width of the final excited energy state is large compared to the bandwidth of the field $\Delta {\omega }_f\gg \Delta \omega$, as typical for commonly used laser dye molecules. Mollow showed, that for weak stationary light fields the TPA rate is linearly proportional to the degree of second-order coherence function at zero time delay

$$R_{TPA}\left(\omega \right)=g^{\left(2\right)}\left(0\right){\left|D\left({\omega }_0\right)\right|}^{2}2\frac{\Delta {\omega }_f/2}{{(\Delta {\omega }_f/2)}^2+{(2\omega -{\omega }_f)}^2}{I}^2,$$

(5)

with I being the intensity of the light field, ${\omega }_f$ being the transition frequency, $\Delta {\omega }_f$ being the spectral full width at half maximum of the final state and $D\left(\omega \right)$ transition.

For this work, only the influence of the photon statistic on the TPA rate is of and interest. Thus, this can be simplified to:

$$R_{TPA}\left(\omega \right)=g^{\left(2\right)}\left(0\right)C{I}^2,$$

(6)

with C being a constant that incorporates the properties of the absorber. When comparing the TPA rates of coherent and thermal light, one finds

$$\frac{R_{TPA}^{thermal}\left(\omega \right)}{R_{TPA}^{coherent}\left(\omega \right)}=\frac{g_{thermal}^{\left(2\right)}\left(0\right)}{g_{coherent}^{\left(2\right)}\left(0\right)}=2.$$

(7)

In seminal work, Boitier et al. have recently demonstrated, that amplified spontaneous emission (ASE) can exhibit intensity correlations on an ultrashort timescale similar to thermal radiation [19]. They unraveled the photon bunching effect by TPA in a photo-multiplier tube utilizing an interferometric setup. This was the first experimental validation of Mollows and others predictions.

Jechow et al. [20] could later demonstrate that an enhanced signal for two-photon excited fluorescence (TPEF) can be achieved when utilizing the photon statistics of the excitation light field. They used ASE from a superluminescent diode [60] in comparison to coherent light [20] and measured the fluorescence from common fluorophores and quantum dots in solution. The ASE was characterized by a thermal like photon statistics and the TPEF rate was directly proportional to the measured degree of second order coherence of the light utilized light sources. This represents a principal step towards utilizing the photon statistics in multi-photon microscopy.

2.3. Entangled TPA

Two-photon absorption with entangled photons was first experimentally demonstrated with atoms [21] and later with molecules [22,23,24,25]. Theoretical work predicted that TPA with entangled photon pairs at low flux rates will show a linear intensity dependence (see Figure 3), contrary to classical light [17,18], which was proven experimentally by Dayan et al. using broadband biphotons [61].

Subsequent theoretical work by Dayan [62] generalized for two-photon processes showed that below a critical flux of ${\Phi }_x\approx {\Delta }_{SPDC}$, with ${\Delta }_{SPDC}$ being the bandwidth of the down-converted field, the linear part will dominate, while above ${\Phi }_x$ the quadratic, random term will dominate. The rate enhancement is [62]:

$$\frac{R_{TPA,correlated}}{R_{TPA,uncorrelated}}=\frac{{\Delta }_{SPDC}}{{\Delta }_{TPA}}·\frac{n^2-n}{n}·\frac{P_e({\tau }_i-{\tau }_s)}{g_{pump}^{\left(2\right)}\left(0\right)}.$$

(8)

With n being the photons per mode (n = 1 at ${\Phi }_x$), ${\Delta }_{TPA}$ being the bandwidth of the two-photon process, $g_{pump}^{\left(2\right)}\left(0\right)$ the degree of second order coherence of the light field pumping the SPDC and $P_e$ being a normalized term describing a possible delay between signal and idler. Thus, at low excitation intensities (or small values of n), a drastic enhancement with nonclassical light is expected. Using a single photon source will result in no TPA signal.

Teich and Saleh proposed to exploit entangled photons for two-photon microscopy in 1997 [29] and later extended this idea towards virtual state spectroscopy [30]. Since a few years, several novel theoretical work has spawned in the context of entangled photon spectroscopy [31,32,33]. The recent tutorial by Schlawin is highly recommended for further reading [34].

3. The Proposed Setup for Heralded TPEF

Here, a new scheme for heralded TPA/TPEF experiments, exploiting several of the aforementioned schemes altogether, is proposed. The proposed system is depicted in Figure 4 and should comprise:

• A heralded two-photon scheme utilizing PP waveguide crystals,
• A high coupling between absorber and excitation field and an efficient photon collector for the fluorescence,
• A single absorber with a plasmonic enhancement antenna, and
• State-of-the-art single-photon counting devices.

The light source of the setup is inspired by seminal work by Guerrero et al. that targeted applications in quantum communication [40]. SPDC in two separate nonlinear PP waveguide crystals (SPDC1 and SPDC2) is pumped by a common pump laser (e.g., a frequency doubled diode laser [63] or a diode pumped solid state laser). The pump laser radiation is separated into two beams at a beam splitter (BS) and subsequently coupled into the waveguide crystals using high NA lenses (see, e.g., [64] using PP lithium niobate(PPLN)). In each of the waveguides a certain amount of photon pairs are generated independently, depending on the material constant, the phase matching conditions, the waveguide architecture (length, width, etc.) as well as the pump power. The light exiting the waveguides is collimated, and the pump fields are filtered out using bandpass filters (F). The SPDC field will be centered around twice the pump wavelength (i.e., 1064 nm in our example).

In the next step, the signal and idler fields from the down-converted light fields created in each SPDC crystals are separated utilizing dichroic filters (DM1 and DM2). Subsequently, the idler field from SPDC crystal 1 (wavelengths larger than 1064 nm) and the idler field from SPDC crystal 2 (wavlengths shorter than 1064 nm) are directed towards a third periodically poled nonlinear waveguide crystal, where the two light fields are spatially overlapped and coupled into the waveguide.

Behind this waveguide crystal a single photon detector (SPC 1) is placed behind a bandpass filter. In this crystal up-conversion of the two independently generated broadband down-converted fields occurs, and a single up-converted photon is generated if two photons of the matching energy arrive at the waveguide crystal “simultaneously”. Therefore, only if two photon pairs are generated “simultaneously” in the SPDC1 and SPDC2 crystals, a joint detection of the two idler photons from different SPDC crystals is possible [40]. This heralds the presence of two entangled photons in the two signal beams.

In the setup by Guerreiro et al., the two remaining outputs of the SPDC sources were each directed towards independent single photon detectors and by correlation between the three SPCs, photon triplets were detected [40] to show the possibility to herald the two photon correlation by the SFG detection. In the setup proposed here, the signal fields from SPDC crystal 1 and SPDC crystal 2 are coupled into fiber optics with an optional integrated optical switch (OS) that is opened when a photon pair is heralded at the SFG crystal. Both fiber optics require an adequate length to compensate for the slow electronics that control the shutters. At least one of the two paths requires an additional tunable optical delay line to provide fine tuning ability of any optical path length difference between the two photons. Each optical field from the fiber optics is then directed towards a beam splitter (BS), where the two light fields are spatially overlapped. (Adjusting the delay line could be realized by placing another SFG crystal at this position.) The overlapped beams are then focused tightly onto an absorber inside a lightproof box using high NA optics (HNAL) [55]. The absorber is ideally a levitating single molecule with a large TPA cross section. One way to implement the levitating single particle is to use a Paul trap architecture as shown with fluorescent dyes [65], nanoparticles [66] and proposed for large biomolecules [67]. Fluorescence of the absorber is collected by another high NA optic (HNAL) [53] that could possibly be diffractive optics [68], or a (tapered) fiber [65,69]. On the detector side, a single photon detector or a sensitive camera [20] could be used.

4. Discussion

The proposed setup could in principle be applied to study TPA/TPEF with few photons in detail. As outlined above, one major application of entangled TPEF is TPM, where the use of entangled photons (biphotons) promise a drastic signal enhancement. Another promising recent finding in using entangled photons for TPM is that Lum et al. showed that nonclassical time and energy correlations survived past 200 $\mathsf{\mu }$m in milk and chicken tissue at room temperature, indicating that reductions in time-energy correlations are unlikely to be a limiting factor for deep tissue entangled TPA [70]. While some groups have shown large entangled TPA cross sections and enhancement factors from entanglement and even some indication for entangled TPEF with strong fields [24,25], some other groups have shown less optimistic performances and theoretical predictions [26,27,28]. Such enhancements have also been demonstrated recently in multiphoton processes up to the 4th order [37] using bright squeezed vacuum. While most of the other setups rely either on absorption architectures or on rather high fluxes of photon pairs, the setup presented here aims towards using very low fields.

The heralded TPEF setup outlined here is not the final leap towards TPM application, but could be utilized to test entangled two-photon absorbers at the low limit. To gain more insights at this side is important because as an enhanced signal with classical light could stem from a cascaded single-photon process and not from a real TPA process. Then, the entanglement would not provide a benefit [71]. By harnessing the absorption of only two photons in a heralded setup and by controlling the delay between the two entangled photons it appears also feasible to perform virtual state spectroscopy [30,34].

However, the question is if heralded TPEF with single absorbers is really feasible with the current technology at hand and what to expect from such a scheme in terms of performance. On the one hand, the setup relies on well known components and the heralding of photon triplets was recently demonstrated by Guerreiro et al. [40]. However, they had to integrate over an extremely large time window of 260 h to show a correlation peak of 80 counts over a background of about 10 counts. They were limited by the dark count rate of their SPC at the SFG detection side. Their standard single photon avalanche diodes had about 3 counts per second and a response time of several 100 ps [40]. Recent progress shows highly efficient superconducting nanowire single-photon detectors are possible with time resolution below 3 ps [72], at low dark count rates below 10${}^{-4}$ per second [73] and efficiencies of 80–90% at visible wavelengths [74]. However, it has to be noted that none of the devices showed this performance at the same time.

A rough estimation of the photon flux rates gives about ${\Phi }_{SPDC}\approx {10}^{11}$ biphotons/s assuming about 100 mW pump power at each SPDC crystal and an SPDC efficiency of 10${}^9$ biphotons/s · mW as achieved in [64] with a PPLN waveguide, which is probably two orders of magnitude higher than the flux achieved by Guerreiro et al. with a pulsed solid state laser and bulk PPLN crystals [40]. Assuming about the same upconversion efficiency as Guerreiro et al. of ca. 10${}^{-8}$, the (optimistic) SFG photon flux would be ${\Phi }_{SFG}\approx {10}^3$ photons/s, which is much higher than the low dark count rate below 10${}^{-4}$ per second that could be achieved with the superconducting nanowire single-photon detectors [73]. Optical losses are not included yet and could be near unity with fiber optics and waveguides. Typical losses are on the order of 50–90%, giving an overall loss of 10–70%. High NA optics have shown very strong atom photon coupling efficiencies of 90% when focusing tight light fields on isolated single atoms [55] or trapped ions [49], as well as high collection efficiencies of more than 50% for fluorescence photons with a special mirror [51]. Such a laser cooled ion would also give a scattering rate of about 30% at the low intensity limit, allowing fluorescence detection at the pW level [49]. However, this would be much lower for a molecule at room temperature. Even when assuming the extremely optimistic (and most likely wrong) prediction of Saleh and Teich [29] that the entangled TPA cross section could reach that of a single photon absorption cross section at the low photon limit, one would assume several orders of magnitude lowered fluorescence photon flux. Estimating the losses of the optics to be 10${}^{-1}$ and taking the SFG photon flux of ${\Phi }_{SFG}\approx {10}^3$ photons/s and the low dark count rate of 10${}^{-4}$, the efficiency of the entangled TPA would need to be better than 10${}^{-6}$. This may sound doable, but taking into account the demanding precision for the temporal “simultaneous” arrival of the entangled photons and the uncertainty for the TPA cross section enhancement, one might just end up in the noise floor.

5. Conclusions

In this perspective, the feasibility of heralded TPEF with single absorbers is discussed based on a proposed scheme extended from a setup that targets application in quantum communication [40]. Other building blocks include high NA optics and fluorescent collection schemes from single atom experiments [51,55] and enhancement by plasmonic nano-antennas [43]. Although the individual building blocks have not been tested altogether, there seems to be no obvious reason to rule out their combined functioning. However, despite utilizing current technologies, the implementation remains very challenging. One obstacle is the detection efficiency and dark counts of the detectors, which might be solvable with cryogenic superconducting nanowire single-photon detectors, which have very low dark count rates [73]. The other uncertainty lies in the efficiency of the TPA process itself. If enhancements are less than the optimistic predictions, the signal might be too weak. This perspective aimed at triggering a discussion about the study of TPA and TPEF with only very few photons. In conclusion, the complexity of the setup and the expected low signal prevent an immediate real-world application in TPM, but the setup might be a good tool for fundamental studies in the context of quantum microscopy and quantum spectroscopy [34].