Limitations in Fluorescence-Detected Entangled Two-Photon-Absorption Experiments: Exploring the Low- to High-Gain Squeezing Regimes

A key experimental feature of ETPA is the predicted linear scaling of the EPTA rate with excitation flux, not expected for most second-order nonlinear optical phenomena, which typically display quadratic scaling for classically driven processes. This linear flux scaling feature alone is cited as evidence of ETPA in many experiments that have observed signals attributed to ETPA[5, 6, 8].

In contrast to ETPA, TPA driven by coherent or thermal-like light lacks the linear-in-flux term because in those cases the probability for two photons to arrive within the molecular response time depends quadratically on the incident flux. For ETPA the $\sigma_{e}$ term arises due to the fact that entangled photons are produced in pairs, leading in the low-flux limit to a linear relationship with the photon flux.

In cases where the squeezed light is confined to a single transverse mode, such that $A_{e}$ equals the transverse mode area $A_{0}$, we have $n=\phi A_{0}T_{e}$, where the entanglement time $T_{e}$ equals the inverse of the full-width-at-half-maximum (FWHM) bandwidth B (in Hz) of the squeezed-light source $T_{e}=1/B$, assuming the SPDC pump laser bandwidth is much smaller than the SPDC bandwidth, $\Delta\nu_{p}<<B$. The case when the bandwidth of the squeezed light source becomes comparable to the molecular TPA linewidth is described by more complicated formulas, as explored in [22, 25].

If linear loss, e.g., an optical attenuator, is inserted between the SPDC source and the TPA sample, the flux scaling of the TPA rate in the low-flux regime is predicted to be quadratic with the amount of loss, because both photons in a pair are independently subject to the loss. This feature has been verified in sum-frequency generation (SFG) [27, 17], which is a separate nonlinear optical process with dynamics similar to that of TPA in the absence of intermediate resonances [12, 25]. This feature is also readily observable in photon coincidence counting experiments, which are insensitive to ultra-fast temporal or spectral effects. Observation of both the linear behavior (when varying the SPDC pump power) and quadratic behavior (when inserting loss) in the same experiment is a more robust validation than verifying linear scaling alone, since linear scaling in itself is a common phenomenon and can lead to mis-attribution of unrelated linear-optical effects to ETPA [18, 19, 20]. To date, only [21] has reported both effects simultaneously in molecular TPA.

It is worth noting that, as pointed out in [21], $\sigma_{e}$ is a quantity that depends on the molecular properties of the TPA sample as well as the detailed properties of the entangled light driving the interaction. This includes experimental conditions such as beam focusing, pump bandwidth, as well as the phase-matching bandwidth of the process generating the entangled photons. Because of this, it is ambiguous to report $\sigma_{e}$ without explicit consideration of these parameters. A modified quantity $\sigma_{e}\times T_{e}\times A_{e}$ is more physically meaningful for comparison of experiments with varying conditions.

The primary question we wish to address is what values can the parameter $f$ take on in realistic experiments? While theory indicates that $f$ should be on the order of 1, [14] the value needed to produce the ETPA efficiencies reported in other ETPA experiments[5, 21] is many orders of magnitude greater. The present paper reports further concrete experimental evidence that $f$ is not much greater than $1$ in solvated dye-like molecules.

The crossover between the linear and quadratic flux-scaling regimes of two-photon absorption and SFG rates for a narrowband two-photon absorbing sample occurs when the two terms in Eq. (1) are comparable, as described in [12, 25, 23]. This corresponds to $\phi=\sigma_{e}/\sigma^{(2)}$. Using $n=\phi A_{0}/B$, under the condition that $A_{e}=A_{0}$, this criterion is equivalent to having the number of photons per spectral-temporal mode equal $f$. This conclusion agrees with expectation when recognizing that $f$ is approximately equal to 1. The understanding is that when the number of photons per spectral-temporal mode exceeds 1, we leave the isolated-pair regime. Above the isolated-pair regime, photons from uncorrelated pairs overlap in time and generate excitation from TPA at rates comparable to ETPA, eliminating the benefit of temporal-spectral entanglement and reverting to quadratic flux scaling. At this flux, a transform-limited pulse of the same duration and intensity would have nearly the same ETPA efficiency [25].

Unfortunately, it is unfeasible to observe the incident flux at which crossover occurs directly, due to the extremely low efficiency of TPA. However, the crossover point in the absence of spatial-coherence effects can be understood as a feature of the SPDC process and the acceptance bandwidth of the nonlinear process rather than being a feature unique to the TPA process. In this regard, the response of SFG is analogous to that of TPA. Because of this similarity, we use SFG to constrain experimentally the flux at which the crossover flux occurs for our SPDC (squeezing) source. To do this we measure directly the crossover that occurs in SFG using broadband entangled photons of variable flux. The agreement with predictions ensures that our theoretical description of the interaction captures the relevant dynamics for TPA, giving us confidence in our reported results for TPA in molecules.

Here we make a few more clarifying remarks comparing the TPA rate with bright squeezed vacuum to that with classical light. As noted above, Eq. (3), the TPA rate of an instantaneous two-photon process is proportional to the degree of second-order coherence, $g^{(2)}(0)$, of the driving field [24, 28, 29, 30, 31, 25]. For classical coherent-state light arising from laser excitation we have $g^{(2)}(0)=1$ and for squeezed vacuum $g^{(2)}(0)=3$. For thermal-like light $g^{(2)}(0)=2$. Values of $g^{(2)}(0)$ greater than 1 can be thought of as an increase in TPA rate arising from the quadratic nature of the interaction and the increased fluctuations in the driving field leading to higher probability for two or more photons present in a single temporal-spectral mode. However, in the case of interest the two-photon transition has a finite bandwidth, and thus the bandwidth and spectral correlations of the incident field are important considerations. In the limiting case of exceedingly narrow TPA linewidth, $\Delta\nu_{f}$, in comparison to the bandwidth of the field $B$, the incoherent (spectrally uncorrelated) portion of the field has an exceedingly small overlap with the transition, and contributes negligibly to the overall TPA rate [12, 25]. On the other hand, the TPA rate arising from the coherent (correlated) portion of the field at high-gain is comparable to the TPA rate arising from classical light. In the case of a moderate TPA linewidth, the incoherent contribution is no longer negligible and the final TPA rate will lie somewhere between these two bounds. A more complete discussion of these effects can be found in [25].

In a previous publication we reported bounds on the ETPA efficiency of Rhodamine 6G (R6G) for entangled photons generated from continuous-wave (CW) excitation via SPDC centered around a wavelength of 1064 nm [17]. Subsequent reports observed signals attributed to ETPA in R6G along with the linear and quadratic dual-scaling behavior characteristic of nonlinear interactions with entangled photons, as discussed above [21]. This dual-scaling is a key feature of ETPA not previously observed in any other ETPA study. The reported non-zero ETPA fluorescence detection rates (on the order of 10 counts/s after background subtraction of approximately 215 counts/s) [21] did not contradict the bounds that had been set by our previous work[17]. That report motivated us to improve the threshold for detecting ETPA relative to our previous report and to attempt to detect ETPA signals under nearly identical conditions as in [21].

We made every attempt to replicate the exact experimental conditions of [21], for example the optical beam parameters, lens focal lengths, interlens distances, squeezed-light source, detection geometry, etc…, while making minor modifications to improve detection sensitivity, as described below and in the supplemental information [32]. As mentioned in the Introduction, we used both a CW laser as in [21] as well as a picosecond pulsed laser to generate squeezed vacuum in both low- and high-gain regimes, the latter allowing us to explore the regime of BSV-induced ETPA for comparison with the CW results and testing of theory predictions [25]. The pulsed version of the experiment enabled clear detection of TPA-induced fluorescence. In the case of BSV excitation, we verified the presence of the TPA process by confirming the fluorescence emission lineshape [32] and quadratic dependence of fluorescence intensity with excitation flux. In addition, we used the pulsed source to confirm the theory’s ability to predict properties of the SFG signal arising from squeezed vacuum, including observation of the crossover region at intermediate flux.

Fig. 1 shows a schematic of the replication experiment. The pump generating the SPDC was a CW laser with a wavelength of 532 nm and bandwidth of approximately $\Delta\nu_{CW}=5$ MHz (Coherent Verdi V5). A prism was used to separate any residual 1064 nm light generated by the laser. The beam was then focused into the down-conversion crystal with a 500 mm focal-length lens, resulting in an approximately 70 $\mu m$ beam waist. The down-conversion crystal was a 2 cm long magnesium-oxide doped periodically poled lithium-niobate (PPLN) crystal (Covesion MSHG1064-0.5-20). The crystal was phase-matched for collinear, non-degenerate type-0 down-conversion via temperature tuning in a home-built oven. The SPDC light generated in this process was collimated using a 100 mm focal-length achromatic lens with anti-reflection (AR) coating optimized for 532 nm and 1064 nm (Edmund Optics #33-211). The pump was separated from the SPDC beam via a harmonic separator mirror (Newport 10QM20HB.21). The SPDC beam then passed through a chopper wheel with 50% duty-cycle operating at 400 Hz. Any remaining 532 nm light was then blocked by two long-pass filters (Thorlabs FELH0900, FELH0750).

After preparation of the entangled photons, the beam was routed to the ‘sample-and-detection’ apparatus, which collected back-scattered light from the sample. The SPDC light was reflected off a dichroic mirror (Thorlabs DMSP0900), and focused onto the sample using a 3 mm focal-length aspheric lens AR-coated for near-infrared (NIR) light (Thorlabs C330TMD-B). The focal spot in the sample was near the diffraction limit so that the entanglement area was approximately equal to the spot size itself, that is $A_{e}=A_{0}$, as assumed in our theoretical discussions above. The fluorescent sample was mounted on a linear translation stage enabling optimized placement in the longitudinal direction of the beam. The results reported in this paper utilized a standard 1-cm quartz cuvette with approximately 1-mm thick quartz window thickness. A thin sample cell using a microscope coverslip as a window, similar to that used in [21], was also tested with no significant differences in observed signal rates in both pulsed and CW experiments. Fluorescence emitted in the backward direction was collected and collimated by the focusing lens and transmitted through the dichroic mirror onto a filter stack (Thorlabs FESH0850, FESH0650, Semrock NF-03-532). An 11 mm focal-length aspheric lens AR coated for visible wavelengths focused the fluorescence onto a free-space coupled avalanche photodiode (APD) operating in photon-counting mode (Laser Components Count-10B). The dark rate of the detector was measured to be approximately 3 Hz, which is approximately 2 orders of magnitude smaller than that reported in [21]. All elements of the experiment were light-shielded such that additional counts from stray light were not distinguishable from the intrinsic rate of dark counts of the detector alone.

Alignment was optimized by maximizing the measured TPA fluorescence count rate from the sample when driven by BSV generated from the pulsed laser pumping the same crystal used to generate the CW low-flux SPDC, as detailed in the next section. The pulse-pumped BSV source was also used to validate the differential measurements used to suppress any slow drifts in background noise. The differential measurements is defined as the difference between detected counts in time intervals in which the beam was either allowed to pass through or completely blocked by a rotating shutter, or ’chopper wheel’. There were approximately 400 open/closed cycles per second. Validation was carried out both at high signal rates ($10^{3}$ to $10^{5}$ cps) as well as at fluorescence signal rates below 1 cps, yielding consistent results at all tested measurement durations, validated up to a maximum duration of 60,000 s. Background measurements under four different experimental conditions:–with the laser off; SPDC beam blocked; phase-matching temperature detuned; and on blank (without dye) ethanol solutions–all yielded no detectable signal above the dark counts. For a thorough discussion of the differential measurements see supporting information [32].

The spectral characteristics of the CW SPDC at low gain were characterized via time-of-flight spectrometry. For this measurement the SPDC beam was coupled into a single-mode fiber via an 8-mm focal-length aspheric lens AR-coated for NIR (Newport KGA240-B-MT). The fiber-coupled SPDC light was then split by a 50:50 fiber beam splitter (Thorlabs TW1064R5F2B). One outgoing fiber was sent directly to a superconducting nanowire single-photon detector (SNSPD) (IDQuantique), and the second outgoing fiber was sent through 1 km of optical fiber (Nufern 780HP), which imparted the necessary optical dispersion for a time-of-flight measurement (converting frequency to time) before arrival at a second SNSPD. Due to the stochastic nature of the CW SPDC process, the photon pairs from the source arrive at the detector without reference to any external clock. However, the relative arrival time of each delayed photon contains spectral information due to the tight temporal correlation of the pairs [33, 34]. The measurement was calibrated using spectral filters with known sharp wavelength cutoffs. This measurement confirmed degenerate phase-matching around a central frequency of 1064 nm and a FWHM spectral bandwidth of 30 nm, as shown in Fig. 2.

To minimize the effects of drift and noise, we used ‘chopped’ measurements of ETPA rates. The ‘chopped’ measurement rates are defined as the difference between the detector counts measured when the SPDC beam is unblocked (open channel) by the chopper wheel and when it is blocked (closed channel). For a signal rate $R_{S}$ arising from fluorescence, the number of counts in a measurement time $T$ is expected to be $N_{S}=R_{S}T$ with uncertainty $\Delta N_{S}=\sqrt{R_{S}T}$. To determine whether a measured signal can be attributed to a process other than noise, we set the detection threshold by requiring the measured signal should be greater than 5 times the expected noise due to dark counts, that is we must have $N_{S}\geq 5\Delta N_{noise}=5\sqrt{N_{noise}}=5\sqrt{DT}$. Here $D$ is the dark count rate (assumed constant). This allows us to define the detection threshold rate $R_{det}=5\cdot 2\sqrt{D/T}$. The factor of 2 accounts for the 50% duty cycle of the chopped measurement scheme. For our experiments, $D=3$ Hz and $T=60,000$ s, giving $R_{det}=0.07$ Hz. Any signal with fluorescence rate above this detection threshold should be observable within our experiment. For comparison, note that the equivalent detection threshold rate in the previous experiment [21] would be approximately $R_{det}\approx 10\sqrt{200/2\times 10^{4}}=1$ Hz. All cited rates and powers are adjusted to account for the chopper duty cycle, and correspond to the average values while the beam is unblocked.

The PPLN crystal was pumped by a maximum CW power of 1 W, which generated approximately 200 nW of SPDC power measured after additional bandwidth filtering (Semrock 1055/70). In order to facillitate comparisons to BSV experiments, we cite the total SPDC power within this spectral window, without attempting to adjust for pair-production efficiency. Shorter-duration measurements at higher powers were also conducted without observing evidence of TPA, however a power of 1 W was chosen to minimize temperature-dependent and photorefractive effects in the SPDC crystal. Measurements of fluorescence generated via TPA of entangled photons in a cuvette containing a 5-mM solution of Rhodamine 6G (Sigma Aldrich 252174) in ethanol were made with a duration of 60,000 seconds, with no statistically significant difference between open and closed (chopper) channels. At this measurement duration, our detection threshold was 0.07 Hz (4.2 counts/minute). The measurement was repeated using a thin sample cell similar to the one used in [21]. It was also repeated using a sample of the R6G dye that was used in [21], graciously provided by the authors to rule out sample-based discrepancies. No significant differences between samples or sample cells were observed in the repeated measurements with the CW or the pulsed configuration.

Using the same PPLN crystal pumped by a pulsed laser, we investigated the behavior of TPA driven by the source at moderate gain. We then reduced the flux in an attempt to approach the low-gain regime with a source known to be capable of driving observable fluorescence from TPA of squeezed vacuum.

The pulsed SPDC source utilized pump pulses with 532-nm center wavelength, which were about 8 ps in duration (Lumera Laser Hyperrapid 50). The 8 ps pulse duration is far longer than the entanglement (correlation) time of the exciting light ($T_{e}=1/B=\text{110 fs}$), making the theoretical models discussed earlier valid for our experiments. The repetition rate was varied between 100 kHz and 5 MHz. Otherwise the experimental apparatus was unchanged from the CW case, with the beam alignment to the CW beam confirmed by coupling into single-mode fiber as well as overlap on a CCD camera (Hamamatsu ORCA-ER-1394) at the location of the crystal, and near the sample itself.

The BSV-induced signal can be unambiguously attributed to fluorescence from TPA, as verified by spectral measurements and quadratic scaling[32]. This is a major benefit of our current experiment, which no previous experiments in the ETPA literature have been able to achieve, due to the limited rates of detectable fluorescence.

To approach the low-gain regime, the average power of the BSV generated by the pulsed source was kept constant at 300 nW, which is approximately the maximum attainable power in the CW regime. At 100 kHz laser repetition-rate, this provided a clearly measurable TPA signal of 88 Hz. From here, the repetition rate of the laser was increased incrementally, with the pump intensity adjusted to ensure the average power of the generated BSV remained constant. Increasing the laser repetition rate at constant average BSV flux reduces the number of photons per pulse, and the measurements are reported in terms of photons per pulse.

Due to the quadratic scaling of TPA intensity with peak BSV power, the rate of TPA is reduced as the repetition rate is increased, despite the average power remaining constant. This remains true until the linear scaling regime is reached, after which the repetition rate does not impact the detected fluorescence rate.

The results of this experiment are shown in Fig. 3 in terms of per-pulse BSV intensities reported as average photon number per pulse. The quadratic dependence can be readily observed down to our minimum exposure flux. We were able to increase the repetition rate to 5 MHz while keeping the average BSV power constant at 300 nW. At higher repetition rates the pump power and thereby the BSV gain was insufficient to reach 300 nW of BSV. At 5 MHz repetition rate and 300 nW BSV power, TPA was still observable at a detection rate 0.9 Hz. Two further measurements were made at 5 MHz, with reduced average powers of 100 nW and 37 nW. These measurements took 4 hours, and 4 days respectively to reach the same statistical significance levels as previous measurements–indicative of the extremely unfavorable scaling behavior of measuring weak, shot-noise-limited, quadratic signals. The last two measurements also omitted the use of the chopper wheel, and instead used the laser pulse-train to distinguish ‘open’ and ‘closed’ channels in order to maximize the signal-to-noise ratio of our measurement.

The lowest-flux measurement was made with approximately 40,000 photon pairs per pulse, or approximately 400 pairs per spectral-temporal mode for our source. Notably, despite the benefit of being far above the isolated-pair regime—which is expected to be near 1 pair per spectral mode—at the same average powers as those attainable in the CW experiment, the fluorescence rate is well below that reported in [21]. This result is in agreement with the null result from our CW experiment.

A notable difference between our CW and pulsed experiments is the spectral bandwidth of the pump pulse, which determines the tightness of the spectral correlation between the entangled photon pairs [12, 23]. This constitutes a significant experimental difference between pulsed and CW versions of the experiment, since the bandwidth of the CW pump is orders of magnitude narrower than that of the pulsed pump. This difference is not expected to affect the TPA efficiency in R6G, since the TPA linewidth is much broader than the pulsed-laser bandwidth, and thus the correlations present in the pulsed case are sufficient to maximize resonant overlap with the two-photon transition [22].

To confirm another relevant point of our understanding of TPA arising from light generated via SPDC, we designed an experiment to compare the rate of TPA from a ‘classical’ (coherent-state) laser reference beam to the efficiency of TPA from BSV generated by our source. As discussed in the theory section, the relative efficiency for pulses of equal duration is expected to fall between 1 and 3, the former for narrow TPA linewidth and the latter for ultra-broad phase-matching[25, 24].

To ensure identical spatial characteristics of the beams, a modified configuration was used that included a spatial-mode filter, as shown in Fig. 4. This configuration used a 10-mm PPLN crystal (Covesion MSHG1064-1.0-10) rather than the 20 mm PPLN crystal which was used in the above experiments. In this second configuration, the chopper wheel was omitted and the BSV beam was coupled into a 5-cm single-mode optical fiber stub (Nurfern 780HP) using an 8-mm focal-length aspheric lens with NIR AR coating (Newport KGA240-B-MT ). The fiber stub was terminated in an FCPC connector. Its output could be coupled to a time-of-flight spectrometer, which verified the spectral and joint-spectral properties of the fiber-coupled BSV, as discussed in the supplemental information [32].

Coupling into the fiber stub was optimized on coincidence counts in the low-gain SPDC regime. The spectral characteristics of the coupled light were also verified via measurement of the joint spectral intensity (JSI) of the photons coupled into the fiber. These were compared with previous JSI measurements[32]. After verifying the SPDC coupled into the short fiber displayed the desired JSI, the fiber going to the spectrometer was decoupled and light was collimated via an 8-mm-focal-length aspheric lens and routed to the detection apparatus in a backward collection geometry as described above.

Fluorescence emitted in the backwards direction was collected and collimated by the focusing lens, and passed through the dichroic mirror through a filter stack (Thorlabs FESH0850, FESH0650, Semrock NF-03-532), before being coupled into a multi-mode fiber, via an 8-mm aspheric lens (Newport KGA240-B-MT). This multi-mode fiber was coupled to a spectrometer (Ocean Optics Flame) to verify that the spectral properties corresponded to known properties of fluorescence from Rhodamine 6G.

The classical laser reference beam was coupled into the same fiber stub as the BSV, with all of the subsequent optics remaining unchanged, guaranteeing identical spatial properties (up to chromatic variations due to the broad BSV spectrum).

To measure the temporal properties of the BSV, the beam after spatial mode filtering was routed to an autocorrelation experiment consisting of a Michelson interferometer prior to a 0.7-mm beta barium borate (BBO) crystal with broad phase-matching bandwidth. The pulse duration was used to calibrate the relative efficiency of the BSV compared to that of the classical reference laser. See further discussion in the supplemental information [32].

The results of the efficiency comparison are summarized in Fig 5. Both sources showed quadratic scaling as confirmed by power-law fits of the data. After normalizing by the duration of the pulses, the relative TPA efficiency of the BSV was measured to be a factor 1.8 times as efficient as the classical reference laser. This result is in good agreement with expectations for this factor to be in the range 1 to 3 for a TPA sample with moderately broad TPA lineshape (resulting from optical dispersion in the system and the finite TPA molecular linewidth.) Crucially, no evidence of enhanced efficiency is observable when using BSV compared to classical laser light to drive TPA.

The final experiment we describe uses sum frequency generation (SFG) as a model for TPA. The theories described in [12, 22] predict TPA scaling behavior with the squeezed light flux that changes from linear to quadratic at the crossover flux, as described earlier. The crossover is predicted to occur at at photon rate of approximately 1 photon per temporal mode. This crossover rate is an important parameter for estimating the low-gain efficiency if the low-gain efficiency cannot be measured directly. The moderate-gain BSV results shown in Fig. 3 confirmed that this crossover for ETPA does not occur until below 400 photon pairs per mode from our source.

To constrain this expected behavior, which cannot be observed in our ETPA data for lack of signal, we use SFG as a model as it has similar dynamics to TPA[12], with higher interaction strengths and thus higher and detectable efficiency.

This measurement was conducted using the 10-mm PPLN crystal used for the BSV experiment, as well as the pulsed laser source operating at 10 MHz repetition rate. The chopper wheel was omitted, and no mode filtering was employed. The squeezed vacuum was collimated using a 200-mm focal-length achromatic lens. After spectral filtering (Thorlabs FELH0900, FELH0750), the beam passed through a dispersion-compensating system (prism pair) to account for the second-order dispersion accrued in the optical system. The beam was then focused via a second achromatic, NIR AR-coated 200-mm focal-length lens into a nominally identical PPLN crystal under the same phase-matching conditions. The SFG acceptance bandwidth of the PPLN crystal was narrow compared to the bandwidth of the SPDC light driving the SFG, which ensures contributions primarily from the coherent contribution to the SFG signal.

After generation of the SFG beam, the remaining squeezed light was blocked by an interference filter stack (Thorlabs FESH0850, FESH0650, Semrock NF-03-532), and the resulting light was coupled into a multi-mode optical fiber before detection on an APD (Laser-components COUNT-Blue 10-B). The intensity of the BSV was measured via a multi-mode fiber-coupled power meter (Thorlabs S150C). At low BSV intensities, the sensitivity of the power meter was limiting. The intensity of the pump was also monitored, to ensure that no errors were introduced due to measurement limitations of the power meter.

Fig. 6 summarizes the result of this experiment. The crossover is measured to be near 250 photons per pulse, which is in good agreement with the predicted crossover for our source with  100 temporal modes. No evidence of a large discrepancy in the predicted efficiency of the low-gain process for SFG was observed.